Handbook of Materials Behavior Models VOLUME I
Deformations of Materials
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Handbook of Materials Behavior Models VOLUME I
Deformations of Materials
EDITOR JEAN LEMAITRE Universit# Paris 6 LMT-Cachan Cachan Cedex France
ACADEMIC PRESS A Harcourt Science and Technology Company
San Diego San Francisco New York London Sydney Tokyo
Boston
This book is printed on acid-flee paper. Copyright 9 2001 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. ACADEMIC PRESS A Division of Harcourt, Inc. 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http ://www.academicpress, com Academic Press Harcourt Place, 32 Jamestown Road, London, NW1 7BY, UK http ://www.aca demicpress, corn Library of Congress Catalog Number:
2001089698
Set International Standard Book Number: 0-12-443341-3 Volume 1 International Standard Book Number: 0-12-443342-1 Volume 2 International Standard Book Number: 0-12-443343-X Volume 3 International Standard Book Number: 0-12-443344-8
Printed in the United States of America 01 02 03 04 05 MB 9 8 7 6 5 4 3 2 1
CONTENTS
Foreword (E. van der Giessen) Introduction (J. Lemaitre) Contributors
VOLUME I
DEFORMATIONS OF MATERIALS
Chapter 1 Background on mechanics of materials Chapter 2 Elasticity, viscoelasticity
1
Chapter 3
Yield limit
69 125
Chapter 4
Plasticity
195
Chapter 5
Viscoplasticity
299
VOLUME II
FAILURES OF MATERIALS
Chapter 6
Continuous damage
409
Chapter 7
Cracking and fracture
537
Chapter 8
Friction and wear
673
VOLUME III
MULTIPHYSICS BEHAVIORS Multiphysics coupled behaviors
793
Chapter 10 Composite media, biomaterials
955
Chapter 9
Chapter 11 Geomaterials INDEX
1073 1179
vi
Contents
CHAPTER
1
Background on mechanics of materials 1.1 Background on modeling J. Lemaitre
1.2 Materials and process selection
15
Y. Brechet
1.3 Size effect on structural strength
30
Z. Bazant
CHAPTER
2
Elasticity, viscoelasticity 2.1 Introduction to elasticity and viscoelasticity
71
J. Lemaitre
2.2 Background on nonlinear elasticity
75
R. W. Ogden
2.3 Elasticity of porous materials
84
N. D. Cristescu
2.4 Elastomer models
91
R. W. Ogden
2.5 Background on viscoelasticity
95
K. Ikegami
2.6 A nonlinear viscoelastic model based on fluctuating modes
107
R. Rahouadj, C. Cunat
2.7 Linear viscoelasticity with damage
117
R. Schapery
CHAPTER
3
Yield limit 3.1 Introduction to yield limits J. Lemaitre
127
vii
Contents
129
3.2 Background on isotropic criteria D. Drucker
137
3.3 Yield loci based on crystallographic texture P. Van Houtte
155
3.4 Anisotropic yield conditions M. Zyczkowski
166
3.5 Distortional model of plastic hardening T. Kurtyka
3.6 A generalized limit criterion with application to strength, yielding, and damage of isotropic materials
175
H. Altenbach
187
3.7 Yield conditions in beams, plates, and shells D. Drucker
CHAPTER
4
Plasticity 4.1 Introduction to plasticity
197
J. Lemaitre
4.2 Elastoplasticity of metallic polycrystals by the self-consistent model
199
M. Berveiller
4.3 Anisotropic elastoplastic model based on crystallographic texture A. M. Habraken, L. Duchr
204
A. Godinas, S. Cescotto
4.4 Cyclic plasticity model with nonlinear isotropic and kinematic hardening: No LIKH model
213
D. Marquis
4.5 Muhisurface hardening model for monotonic and cyclic response of metals
223
Z. Mroz
4.6 Kinematic hardening rule with critical state of dynamic recovery N. Ohno
232
viii 4.7
Contents
Kinematic hardening rule for biaxial ratcheting
240
H. Ishikawa, K. Sasaki
4.8
Plasticity in large deformations
247
Y E Dafalias
4.9
Plasticity of polymers
255
J. M. Haudin, B. Monasse
4.10 Rational phenomenology in dynamic plasticity
265
J. R. Klepaczko
4.11 Conditions for localization in plasticity and rate-independent materials
274
A. Benallal
4.12 An introduction to gradient plasticity
281
E. C. Aifantis
CHAPTER 5
Viscoplasticity 5.1
Introduction to viscoplasticity
301
J. Lemaitre
5.2
A phenomenological anisotropic creep model for cubic single crystals
303
A. Bertram, J. Olschewski
5.3
Crystalline viscoplasticity applied to single crystals
308
G. Cailletaud
5.4
Averaging of viscoplastic polycrystalline materials with the tangent self-consistent model
318
A. Molinari
5.5
Fraction models for inelastic deformation
326
J. E Besseling
5.6
Inelastic compressible and incompressible, isotropic, small-strain viscoplasticity theory based on overstress (VBO) E. Krempl, K. Ho
336
ix
Contents
5.7
An outline of the Bodner-Partom (BP) unified constitutive equations for elastic-viscoplastic behavior
349
S. Bodner
5.8
Unified model of cyclic viscoplasticity based on the nonlinear kinematic hardening rule
358
J. L. Chaboche
5.9
A model of nonproportional cyclic viscoplasticity
368
E. Tanaka
5.10 Rate-dependent elastoplastic constitutive relations
377
E Ellyin
5.11 Physically based rate- and temperature-dependent constitutive models for metals
387
S. Nemat-Nasser
5.12 Elastic-viscoplastic deformation of polymers
398
E. M. Arruda, M. Boyce CHAPTER
6
Continuous damage 6.1
Introduction to continuous damage
411
J. Lemaitre
6.2
Damage-equivalent stress-fracture criterion
413
J. Lemaitre
6.3
Micromechanically inspired continuous models of brittle damage
417
D. Krajcinovic
6.4
Anisotropic damage
421
C. L. Chow, Y. Wei
6.5
Modified Gurson model
430
V. Tvergaard, A. Needleman
6.6
The Rousselier model for porous metal plasticity and ductile fracture
436
G. Rousselier
6.7
Model of anisotropic creep damage S. Murakami
446
Contents
6.8
453
Multiaxial fatigue damage criteria D. Sauci
6.9
Muhiaxial fatigue criteria based on a muhiscale approach
457
K. Dang Van
6.10 A probabilistic approach to fracture in high cycle fatigue
464
E Hild
472
6.11 Gigacycle fatigue regime C. Bathias
6.12 Damage mechanisms in amorphous glassy polymers: Crazing
488
R. Schirrer
500
6.13 Damage models for concrete G. Pijaudier-Cabot, J. Mazars
6.14 Isotropic and anisotropic damage law of evolution
513
J. Lemaitre, R. Desmorat
6.15 A two-scale damage model for quasi-brittle and fatigue damage
525
R. Desmorat, J. Lemaitre
7 Cracking and .fracture CHAPTER
7.1
Introduction to cracking and fracture
539
J. Lemaitre
7.2
Bridges between damage and fracture mechanics
542
J. Mazars, G. Pijaudier-Cabot
7.3
Background on fracture mechanics
549
H. D. Bui, J. B. Leblond, N. Stalin-Muller
7.4
Probabilistic approach to fracture: The Weibull model
558
E Hild
7.5
Brittle fracture D. Franc~ois
566
xi
Contents
7.6
577
Sliding crack model D. Gross
582
7.7
Delamination of coatings
7.8
Ductile rupture integrating inhomogeneities in materials
H. M. Jensen
587
J. Besson, A. Pineau
7.9
Creep crack growth behavior in creep ductile and brittle materials
597
T. Yokobori Jr.
611
7.10 Critical review of fatigue crack growth T. Yokobori
7.11 Assessment of fatigue damage on the basis of nonlinear compliance effects
622
H. Mughrabi
7.12 Damage mechanics modeling of fatigue crack growth
633
X. Zhang, J. Zhao
645
7.13 Dynamic fracture W. G. Knauss
7.14 Practical applications of fracture mechanics: Fracture control
661
D. Broek
CHAPTER 8
Friction a n d w e a r 8.1
Introduction to friction and wear
675
J. Lemaitre
8.2
Background on friction and wear
676
Y. Berthier
8.3
Models of friction A. Savkoor
700
xii
Contents
8.4 Friction in lubricated contacts
760
J. FrCne, T. Cicone
8.5 A thermodynamic analysis of the contact interface in wear mechanics
768
H. D. Bui, M. Dragon-louiset, C. Stolz
8.6 Constitutive models and numerical methods for frictional contact
777
M. Raous
8.7 Physical models of wear, prediction of wear modes
787
K. Kato
CHAPTER 9
Multiphysics coupled behavior 9.1 Introduction to coupled behaviors
795
J. Lemaitre
9.2 Elastoplasticity and viscoplasticity coupled with damage
797
A. Benallal
9.3 A fully coupled anisotropic elastoplastic damage model
802
S. Cescotto, M. Wauters, A. M. Habraken, Y. Zhu
9.4 Model of inelastic behavior coupled to damage
814
G. Z. Voyiadjis
9.5 Thermo-elasto-viscoplasticity and damage
821
P. Perzyna
9.6 High-temperature creep deformation and rupture models
835
D. R. Hayhurst
9.7 A coupled diffusion-viscoplastic formulation for oxidasing multiphase materials E. P. Busso
849
xiii
Contents
9.8
856
Hydrogen attack E. van der Giessen, S. Schl6gl
9.9
Hydrogen transport and interaction with material deformation: Implications for fracture
864
P Sofronis
875
9.10 Unified disturbed state constitutive models C. S. Desai
9.11 Coupling of stress-strain, thermal, and metallurgical behaviors
884
T. Inoue
9.12 Models for stress-phase transformation couplings in metallic alloys
896
S. Denis, P Archambault, E. Gautier
905
9.13 Elastoplasticity coupled with phase changes E D. Fischer
9.14 Mechanical behavior of steels during solid-solid phase transformations
915
J. B. Leblond
9.15 Constitutive equations of a shape memory alloy under complex loading conditions
921
M. Tokuda
928
9.16 Elasticity coupled with magnetism R. Billardon, L. Hirsinger, E Ossart
9.17 Physical aging and glass transition of polymers
944
R. Rahouadj, C. Cunat
CHAPTER 1 0
Composite media, biomaterials 10.1 Introduction to composite media
957
J. Lemaitre
10.2 Background on micromechanics E. van der Giessen
959
xiv 10.3
Contents
Nonlinear composites" Secant methods and variational bounds
968
P. Suquet
10.4
Nonlocal micromechanical models
984
J. Willis
10.5
Transformation field analysis of composite materials
996
G. Dvorak
10.6
A damage mesomodel of laminate composites
1004
P Ladev~ze
10.7
Behavior of ceramix-matrix composites under thermomechanical cyclic loading conditions
1015
E A. Leckie, A. Burr, E Hild
10.8
Limit and shakedown analysis of periodic heterogeneous media
1025
G. Maier, V. Carvelli, A. Taliercio
10.9
Flow-induced anisotropy in short-fiber composites
1037
A. Poitou, E Meslin
10.10 Elastic properties of bone tissue
1048
Stephen C. Cowin
10.11 Biomechanics of soft tissue
1057
S. C. Holzapfel
CHAPTER 1 1
Geomaterials 11.1
Introduction to geomaterials
1075
J. Lemaitre
11.2
Background of the behavior of geomaterials
1076
E Darve
11.3
Models for compressible and dilatant geomaterials N. D. Cristescu
1084
XV
Contents
11.4
Behavior of granular materials
1093
I. Vardoulakis
11.5
Micromechanically based constitutive model for frictional granular materials
1107
S. Nemat-Nasser
11.6
Linear poroelasticity
1118
J. W. Rudnicki
11.7
Nonlinear poroelasticity for liquid nonsaturated porous materials
1126
O. Coussy, P. Dangla
11.8
An elastoplastic constitutive model for partially saturated soils
1134
B. A. Schrefler, L. Simoni
11.9
Sinfonietta classica: A strain-hardening model for soils and soft rocks
1146
R. Nova
11.10 A generalized plasticity model for dynamic behavior of sand, including liquefaction
1155
M. Pastor, O. C. Zienkiewicz, A. H. C. Chan
11.11 A critical state bounding surface model for sands
1164
M. T. Manzari, Y. E Dafalias
11.12 Lattice model for fracture analysis of brittle disordered materials like concrete and rock
1171
J. G. M. van Mier
Index
1179
FOREWORD
We know that there is an abundance of models for particular materials and for specific types of mechanical responses. Indeed, both the developers of models and their users sometimes criticize this situation, for different reasons. The presence of different models that attempt to describe the same material and response is due not only to the personal style of their inventors, but also to a desirable element of competition that drives the progress in the field. Given this situation, the selection of the proper constitutive model from all the available ones can be difficult for users or even materials modelers when they are not experts in the field. This Handbook is the first attempt to organize a wide range of models and to provide assistance in model selection and actual application. End-users will find here either potential models relevent for their application and ready to be used for the problem at hand, or an entrance to the specific technical literature for more details. Recognizing the breadth of the field as well as the unavoidable personal touch of each approach, Jean Lemaitre has chosen to include in this Handbook the writings of as many as 130 authors. Drawing on his wide experience developing and using constitutive models for many materials, he has addressed his worldwide network of colleagues, all experts in their pertinent subject, to accomplish this difficult task. Yet, even though the Handbook covers an unprecedented range of materials and types of behavior, it is only a sample of currently available models, and other choices would have been possible. Indeed, more choices will become possible as the development of novel and improved material models continues.
Erik van der Giessen Koiter Institute Delft Delft University of Technology The Netherlands xvi
INTRODUCTION
Why a Handbook of models? Handbooks are often compilations of characteristic numbers related to well-established laws or formulae that are ready to apply. In this case of the behavior of materials, no unique law exists for any phenomenon, especially in the range of nonlinear phenomena. This is why we use the term model instead of law. During the past thirty years many models have been proposed, each of them having its own domain of validity. This proliferation is partly due to advances in computers. It is now possible to numerically simulate the "in-service life" of structures subjected to plasticity, fatigue, crack propagation, shock waves and aging for safety and economy purposes. The time has come to try to classify, compare, and validate these models to help users to select the most appropriate model for their applications. How is the Handbook organized? All solid materials are considered, including metals, alloys, ceramics, polymers, composites, concrete, wood, rubber, geomaterials such as rocks, soils, sand, clay, and biomaterials. But the Handbook is organized first by phenomena because most engineering mesomodels apply to different materials. 9 In the first volume: "Deformation of Materials," the first chapter is an attempt to give general methodologies in the "art" of modeling with special emphasis, on domains of validity in order to help in the choice of models, in the selection of the appropriate materials for each specific application, and in the consideration of the so-called "size effect" in engineering structures. Chapter 2 to 5 deal, respectively, with elasticity and viscoelasticity, yield limit, plasticity, and viscoplasticity. 9 The second volume is devoted to "Failure of Materials": continuous damage in Chapter 6, cracking and fracture in Chapter 7, friction and wear in Chapter 8. 9 In the third volume "Multiphysics Behaviors" are assembled. The different possible couplings are described in Chapter 9. Chapters 10 and 11 are devoted to special classes of materials: composites and xvii
xviii
Introduction
geomaterials, respectively, because they each corresponds to a particular modeling typed and moreover to a self-organized community of people. 9 In each chapter the different sections written by different authors describe one model with its domain of validity, its background, its formulation, the identification of material parameters for as many materials as possible, some advice on implementation or use of the model, and some references. The order of the sections follows as much as possible from physical and micromechanical oriented models to more phenomenological and engineering oriented ones. How to use the Handbook? 9 Search by phenomena: This is the normal order of the Handbook described in the "Contents". 9 Search by model name: Unfortunately, not all models have a name, and some of them have several. Look in the list of contributors, where the names of all authors are given. 9 Search by type of application: Each chapter begins with a chapter introduction in which a few words are written on each section. If you do not find exactly what you are looking for, please remember that the best model is the simplest which gives you what you need and nothing more! In case of any difficulty, get in touch with the author(s), whose address is given after the title of each section. Some personal comments. This Handbook has been initiated by the editor of "Academic Press" who gave me much freedom to organize the book. It took me two years to prepare the contents, to obtain the agreement of more than 100 authors, to ask for manuscripts, to ask again and again (and again for some of them!) to review and to obtain the final material. It was an exciting experience for which all actors must be thanked: the editors Z. Ruder, G. Franklin, and M. Filion, all the authors who are still my friends, my colleagues and friends from the LMT-Cachan who often advised me on subjects and authors and particularly Erik van der Giessen, who helped me in the selection of the subjects, who corrected the chapter introductions, and who agreed to write the foreword, Catherine Genin who was so kind and so efficient with letters, fax, e-mail, telephone, disks and manuscripts and answered so many questions in order to obtain the materials in due time. I must also mention Annie, my wife, who accepted 117 articles on the table at home!
Merci d tous, Jean Lemaitre Septembre 2000
CONTRIBUTORS
Numbers in parentheses indicate the section of authors' contributions. ELIAS C. AIFANTIS (4.12), Aristotle University of Thessaloniki, Thessaloniki, 54006 Greece, and Michigan Technological University, Houghton, Michigan HOLM ALTENBACH (3.6), Fachbereich Ingenieurwissenschaften, MartinLuther-Universitat Halle-Wittenberg, D-06099 Halle (Saale), Germany E ARCHAMBAULT (9.12), Laboratoire de Science et GSnie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France ELLEN M. ARRUDA (5.12), Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan C. BATHIAS (6.11), Laboratoire de M~canique de la Rupture, CNAM/ITMA, 2 rue Conte, 75003 Paris, France ZDENEK P. BAZANT (1.3), Northwestern University, Evanston, Illinois, USA AHMED BENALLAL (4.11, 9.2), Laboratoire de M~canique et Technologie, ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan, France ALBRECHT BERTRAM (5.2), Otto-von-Guericke-University Magdeburg, Universit/~tsplatz 2, 39106 Magdeburg, Germany YVES BERTHIER (8.2), Laboratoire de M~canique des Contacts, UMR CNRSINSA de Lyon 5514, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne Cedex, France xix
XX
Contributors
B. J. BESSON (7.8), Ecole des Mines de Paris, Centre des Mat~riaux, UMR CNRS 7533, BP 87, 91003 Evry Cedex, France J. E BESSELING (5.5),
[email protected] M. BERVEILLER (4.2), Laboratoire de Physique et M&anique des Mat~riaux, Ile du Saulcy, 57045 Metz Cedex, France RENt~ BILLARDON (9.16), ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France SOL R. BODNER (5.7), Technion Israel Institute of Technology, Haifa 32000, Israel MARY C. BOYCE (5.12), Department of Mechanical Engineering, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA YVES BRECHET (1.2), L.T.EC.M. BP75, Institut National Polytechnique de Grenoble, 38402 St Martin d'Heres Cedex, France DAVID BROEK (7.14), 263 Dogwood Lane, Westerville, Ohio, USA HUY DUONG BUI (7.3, 8.5), Laboratoire de M&anique des Solicles, Ecole Polytechnique, 91128 Palaiseau, France Electricit4 de France, R&D, Clamart, France ALAIN BURR (10.7), Laboratoire de Physico-Chimie Structurale et Macromol&ulaire, UMR 7615, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France ESTEBAN P. BUSSO (9.7), Department of Mechanical Engineering, Imperial College, University of London, London, SW7 2BX, United Kingdom GEORGES CAILLETAUD (5.3), Centre des Mat4riaux de l't~cole des Mines de Paris, UMR CNRS 7633, BP 87, F91003 Evry Cedex, France VALTER CARVELLI (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy SERGE CESCOTTO (4.3, 9.3), D4partement MSM, Universit4 de Liege, 1, chemin des Chevreuils bfit.B52/3, 4000 Liege, Belgique J. L. CHABOCHE (5.8), O.N.E.R.A., DMSE, BP 72, 92322 ChStillon Cedex, France and LASMIS, Troyes University of Technology, BP 2060, 10010 Troyes Cedex, France A. H. C. CHAN (11.10), School of Engineering, University of Birmingham, United Kingdom
Contributors
xxi
C.L. CHOW (6.4), Department of Mechanical Engineering, University of Michigan-Dearborn TRAIAN CICONE (8.4), Dept. of Machine Elements and Tribology, Polytechnic University of Bucharest, Romania N.D. CRISTESCU (2.3), 231 Aerospace Building, University of Florida, Gainesville, Florida OLIVIER COUSSY (11.7), Laboratoire Central des Ponts et Chaussees, Paris, France STEPHEN C. COWIN (10.10), New York Center for Biomedical Engineering, School of Engineering, The City College, New York CHRISTIAN CUNAT (2.6, 9.17), LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la Foret-de-Haye, 54500 Vandoeuvre-les-Nancy, France PATRICK DANGLA (11.7), Laboratoire Central des Ponts et Chaussees, Paris, France FI~LIX DARVE (11.2), EINP Grenoble, L3S-BP 53 38041 Grenoble, France YANNIS E DAFALIAS (4.8, 11.11), Civil and Environmental Engineering, The George Washington University, Washington, D.C. S. DENIS (9.12), Laboratoire de Science et G~nie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France CHANDRA S. DESAI (9.10), Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Arizona, USA RODRIGUE DESMORAT (6.14, 6.15), Universite Paris 6-LMS, 8, Rue du Capitaine Scott, F-75015 Paris, France MARTA DRAGON-LOUISET (8.5), Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France DANIEL C. DRUCKER (3.2, 3.7), Department of Aerospace Engineering, Mechanics Engineering Service, University of Florida, 231 Aerospace Building, Gainesville, Florida 32611 GEORGE J. DVORAK (10.5), Rensselaer Polytechnic Institute, Troy, New York L. DUCHENE (4.3), D6partement MSM, Universit8 de Liege, 1, chemin des chevreuils b~t.B52/3, 4000 Liege, Belgique FERNAND ELLYIN (5.10), Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada
xxii
Contributors
E D. FISCHER (9.13), Montanuniversit. at Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria DOMINIQUE FRAN~;OIS (7.5), 12cole Centrale de Paris, Chfitenay-Malabry, F92 295, France JEAN FRIS.NE (8.4), Laboratoire de M~canique des Solides, Universit~ de Poitiers, France E. GAUTIER (9.12), Laboratoire de Science et G~nie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, t~cole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France A. GODINAS (4.3), D~partement MSM, Universit~ de Liege, 1, chemin des Chevreuils bfit.B52/3, 4000 Liege, Belgium DIETMAR GROSS (7.6), Institute of Mechanics, TU Darmstadt, Hochschulstrasse 1, D 64289 Darmstadt ANNE MARIE HABRAKEN (4.3, 9.3), D~partement MSM, Universit~ de Liege, 1, chemin des Chevreuils b~t.B52/3, 4000 Liege, Belgique JEAN-MARC HAUDIN (4.9), CEMEF- BP 207, 06904 Sophia Antipolis, France D. R. HAYHURST (9.4), Department of Mechanical Engineering, UMIST, P 9 Box 88, Manchester M60 1QD, United Kingdom FRANCOIS HILD (7.4, 10.7), LMT-Cachan, 61 avenue du Pr4sident Wilson, F-94235 Cachan Cedex, France LAURENT HIRSINGER (9.16), ENS de Cachan/CNRS/Universit4 Paris 6, 61 avenue du Pr4sident Wilson, 94235 Cachan Cedex, France K. HO (5.6), Yeungnam University, Korea GERHARD A. HOLZAPFEL (10.11), Institute for Structural Analysis, Computational Biomechanics, Graz University of Technology, 8010 Graz, Austria KOZO IKEGAMI (2.5), Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan TATSUO INOUE (9.11), Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, Japan HIROMASA ISHIKAWA (4.7), Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan
Contributors
xxiii
HENRIK MYHRE JENSEN (7.7), Department of Solid Mechanics, 404, Technical University of Denmark, DK-2800 Lyngby, Denmark KOJI KATO (8.7), Tohoku University, Aramaki-Aza-Aoba 01, Sendal 980-8579, Japan JANUSZ R. KLEPACZKO (4.10), Metz University, Laboratory of Physics and Mechanics of Materials, lie du Saulcy, 57045 Met7, France W. G. KNAUSS (7.13), California Institute of Technology, Pasadena, California DUSAN KRAJCINOVIC (6.3), Arizona State University, Tempe, Arizona E. KREMPL (5.6), Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, New York European Organization for Nuclear Research, CH-1211 Geneve 23, Switzerland
TADEUSZ KURTYKA (3.5), C E R N -
PIERRE LADEVI~ZE (10.6), LMT-Cachan, ENS de Cachan/CNRSAJniversit6 Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France FREDERICK A. LECKIE (10.7), Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California J-B. LEBLOND (7.3, 9.14), Laboratoire de Mod~lisation en M~canique, Universit~ de Pierre et Marie Curie, Paris, France JEAN LEMAITRE (1.1, 2.1, 3.1, 4.1, 5.1, 6.1, 6.2, 6.14, 6.15, 7.1, 8.1, 9.1, 10.1, 11.1), Universit~ Paris 6, LMT-Cachan, 61, avenue du Pr6sident Wilson, F-94235 Cachan Cedex, France GIULIO MAIER (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy DIDIER MARQUIS (4.4), Laboratoire de M~canique et Technologie, Ecole Normale Sup~rieure de Cachan, 61 avenue du President Wilson, 94230 Cachan, France MAJID T. MANZARI (11.11), Department of Mechanics, National Technical University of Athens, 15773, Hellas, and Civil and Environmental Engineering, University of California, Davis, California JACKY MAZARS (6.13, 7.2), LMT-Cachan, Ecole Normale Superieure de Cachan, 61, avenue du President Wilson, 94235 Cachan, France and L35-Institut National Polytechniquede Grenoble, F38041 Grenoble Cedex 9, France
xxiv
Contributors
FREDERIC MESLIN (10.9), LMT-Cachan, ENS de Cachan, Universit6 Paris 6, 61 avenue du Pr6sident Wilson, 94235 Cachan Cedex, France ALAIN MOLINARI (5.4), Laboratoire de Physique et M&anique des Mat6riaux, l~cole Nationale d'Ing4nieurs, Universit~ de Metz, Ile du Saulcy, 57045 MetzCedex, France BERNARD MONASSE (4.9), CEMEF- BP 207, 06904 Sophia Antipolis, France HAEL MUGHRABI (7.11), Universit~it Erlangen-Nfirnberg, Institut f~lr Werkstoffwissenschaften, Martensstr. 5, D-91058 Erlangen, Germany N. STALIN-MULLER (7.3), Laboratoire de M4canique des Solides, 12cole Polytechnique, 91128 Palaiseau, France Z. MROZ (4.5), Institute of Fundamental Technological Research, Warsaw, Poland SUMIO MURAKAMI (6.7), Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603 Japan ROBERTO NOVA (11.9), Milan University of Technology (Politecnico), Department of Structural Engineering, Milan, Italy A. NEEDLEMAN (6.5), Brown University, Division of Engineering, Providence, Rhode Island and Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmark SIA NEMAT-NASSER (5.11, 11.5), Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, California R. W. OGDEN (2.2, 2.4), Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom NOBUTADA OHNO (4.6), Department of Mechanical Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan J. URGEN OLSCHEWSKI (5.2), BAM-V2, Unter den Eichen 87, 12200 Berlin, Germany FLORENCE OSSART (9.16), ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France M. PASTOR (11.10), Centro de Estudios y Experimentaci6n de Obras P~blicas and ETS de Ingenieros de Caminos, Madrid, Spain PIOTR PERZYNA (9.5), Institute of Fundamental Technological Research, Polish Academy of Sciences, Swir 21, 00-049 Warsaw, Poland
Contributors
XXV
GILLES PIJAUDIER-CABOT (6.13), Laboratoire de G~nie Civil de Nantes SaintNazaire, t~cole Centrale de Nantes, BP 92101, F-44321 Nantes Cedex 03, France A. PINEAU (7.8), ]~cole des Mines de Paris, Centre des Mat~riaux, UMR CNRS 7533, BP 87, 91003 Evry Cedex, France ARNAUD POITOU (10.9), LMT-Cachan, ENS de Cachan, Universit8 Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France RACHID RAHOUADJ (2.6, 9.17), LEMTA, UMR CNRS 7563, ENSEM INPL 2, Avenue de la For~t-de-Haye, 54500 Vandoeuvre-l~s-Nancy, France MICHEL RAOUS (8.6), Laboratoire de Mecanique et d'Acoustique, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France GILLES ROUSSELIER (6.6), EDF/R&D Division, Les Renardi~res, 77818 Moret-sur-Loing Cedex, France J. W. RUDNICKI (11.6), Department of Civil Engineering, Northwestern University, Evanston, Illinois, USA KATSUHIKO SASAKI (4.7), Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan A. R. SAVKOOR (8.3), Vehicle Research Laboratory, Delft University of Technology, Delft, The Netherlands R. A. SCHAPERY (2.7), Department of Aerospace Engineering and Engineering Mechanics, The University of Texas, Austin, Texas ROBERT SCHIRRER (6.12), Institut Charles Sadron, 6 rue Boussingault, F-67083 Strasbourg, France SABINE M. SCHLOGL (9.8), Koiter Institute Delft, Delft University of Technology, The Netherlands B. A. SCHREFLER (11.8), Department of Structural and Transportation Engineering, University of Padua, Italy L. SIMONI (11.8), Department of Structural and Transportation Engineering, University of Padua, Italy PETROS SOFRONIS (9.9), Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, Illinois DARRELL SOCIE (6.8), Department of Mechanical Engineering, University of Illinois, Urbana, Illinois CLAUDE STOLZ (8.5), Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France
XXVi
Contributors
PIERRE M. SUQUET (10.3), LMA/CNRS, 31 Chemin Joseph Aiguier, 13402, Marseille, Cedex 20, France ALBERTO TALIERCIO (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy EIICHI TANAKA (5.9), Department of Mechano-Informatics and Systems, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan MASATAKA TOKUDA (9.15), Department of Mechanical Engineering, Mie University, Kamihama 1515 Tsu 514-8507, Japan V. TVI~RGAARD (6.5), Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmark K. DANG VAN (6.9), Laboratoire de Mechanique des Solid, l~cole Polytechnique, 91128 Palaiseau, France ERIK VAN DER GIESSEN (9.8, 10.2), University of Groningen, Applied Physics, Micromechanics of Materials, Nyenborgh 4, 9747 AG Groningen, The Netherlands P. VAN HOUTTE (3.3), Department MTM, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium J. G. M. VAN MIER (11.12), Delft University of Technology, Faculty of Civil Engineering and Geo-Sciences, Delft, The Netherlands IOANNIS VARDOULAKIS (11.4), National Technical University of Athens, Greece GEORGE Z. VOYIADJIS (9.4), Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana MICHAEL WAUTERS (9.3), MSM-1, Chemin des Chevreuils B52/3 4000 Liege, Belgium YONG WEI (6.4), Department of Mechanical Engineering, University of Michigan-Dearborn, USA J. R. WILLIS (10.4), Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom A. TOSHIMITSU YOKOBORI, JR. (7.9), Fracture Research Institute, Graduate School of Engineering, Tohoku University, Aoba 01 Aramaki, Aoba-ku Sendaishi 980-8579, Japan
Contributors
xxvii
TAKEO YOKOBORI (7.10), School of Science and Engineering, Teikyo University, Utsunomiya, Toyosatodai 320-2551, Japan XING ZHANG (7.12), Division 508, Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China JUN ZHAO (7.12), Division 508, Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Y. ZHU (9.3), ANSYS Inc., Houston, Texas O. C. ZIENKIEWICZ (11.10), Department of Civil Engineering, University of Wales at Swansea, United Kingdom MICHA ZYCZKOWSKI (3.4), Cracow University of Technology, ul. Warszawska 24, PL-31155 Krak6w, Poland
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CHAPTER
1
Background on Mechanics of Materials
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SECTION
1.1
Background O I l Modeling JEAN LEMAITRE UniversitY. Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France
Contents 1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Observations and Choice of Variables . . . . . . . . . 1.1.2.1 Scale of observation . . . . . . . . . . . . . . . . . . . 1.1.2.2 Internal Variables . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 State Potential . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.2 Dissipative Potential . . . . . . . . . . . . . . . . . . . 1.1.4 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.1 Qualitative Identification . . . . . . . . . . . . . . . 1.1.4.2 Quantitative Identification . . . . . . . . . . . . 1.1.5 Validity Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Choice of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Numerical Implementation . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1
3 4 5 6 6 7 8 9 9 11 13 13 14
14
INTRODUCTION
M o d e l i n g , as has a l r e a d y b e e n said for m e c h a n i c s , m a y be c o n s i d e r e d "a science, a t e c h n i q u e , a n d an art." It is s c i e n c e b e c a u s e it is the p r o c e s s by w h i c h o b s e r v a t i o n s can be p u t in a logical m a t h e m a t i c a l f r a m e w o r k in o r d e r to r e p r o d u c e or s i m u l a t e r e l a t e d p h e n o m e n a . In m e c h a n i c s of m a t e r i a l s c o n s t i t u t i v e e q u a t i o n s relate l o a d i n g s as stresses, t e m p e r a t u r e , etc. to effects as strains, d a m a g e , fracture, wear, etc. Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
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It is a technique because it uses tools such as mathematics, thermodynamics, computers, and experiments to build close form models and to obtain numerical values for the parameters that are used in structure calculations to predict the behavior of structures in the service or forming process, etc., safety and optimal design being the main motivations. It is an art because the sensibility of the scientist plays an important role. Except for linear phenomena, there is not unique way to build a model from a set of observations and test results. Furthermore, the mathematical structure of the model may depend upon its use. This is interesting from the human point of view. But it is sometimes difficult to select the proper model for a given application. The simplest is often the more efficient event, even if it is not the most accurate.
1.1.2
OBSERVATIONS AND CHOICE
OF VARIABLES First of all, in mechanics of materials, a model does not exist for itself; it exists in connection with a purpose. If it is the macroscopic behavior of mechanical components of structures that is being considered, the basic tool is the mechanics of continuous media, which deals with the following: 1. Strain, a second-order tensor related to the displacement ff of two points: 9 Euler's tensor ~ for small perturbations. 1 ~ ij -
-~ ( u i j -3t- uj , i )
(1)
In practice, the hypothesis of "small" strain may be applied if it is below about 10%. 9 Green-Lagrange tensor A (among others) for large perturbations, if F is the tangent linear transformation which transforms under deformation a point M0 of the initial configuration into M of the actual configuration. 1
_v
(M0)
(_vT_v - 1)
(2)
_
With _F~ the transpose of F. 2. Stress, a second-order tensor dual of the strain tensor; its contracted product by the strain rate tensor is the power involved in the mechanical process.
1.1 Background on Modeling
9 Cauchy stress tensor _a for small perturbations, checking the equilibrium with the internal forces density f and the inertia forces pff, d2R
crijo + fi - oiii
with/ii - dt--T
(3)
9 Piola-Kirchoff tensor _S (among others) for large perturbations.
S _ - det(F_)~_F_- r
(4)
3. Temperature T. These three variables are functions of the time t.
1.1.2.1
SCALE OF OBSERVATION
From the mathematical point of view, strains and stresses are defined on a material point, but the real materials are not continuous. Physically, strain and stress represent averages on a fictitious volume element called the representative volume element (RVE) or mesoscale. To give a subjective order of magnitude of a characteristic length, it can be 0.1 mm for metallic materials; 1 mm for polymers; 1 0 m m for woods; 100 mm for concrete. It is below these scales that observations must be done to detect the micromechanisms involved in modeling: 9 slips in crystals for plasticity of metals; 9 decohesions of sand particles by breaking of atomic bonds of cement for damage in concrete; 9 rupture of microparticles in wear; 9 etc. These are observations at a microscale. It is more or less an "art" to decide at which microscale the main mechanism responsible for a mesoscopic phenomenon occurs. For example, theories of plasticity have been developed at a mesoscale by phenomenological considerations, at a microscale when dealing with irreversible slips, and now at an atomic scale when modeling the movements of dislocations. At any rate, one's first priority is to observe phenomena and to select the representative mechanism which can be put into a mathematical framework
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of homogenization to give variables at a mesoscale compatible with the mechanics of continuous media.
1.1.2.2
INTERNALVARIABLES
When the purpose is structural calculations with sets of constitutive equations, it is logical to consider that each main mechanism should have its own variable. For example, the total strain _8 is directly observable and defines the external state of the representative volume element (RVE), but for a better definition of the internal state of the RVE it is convenient to look at what happens during loading and unloading of the RVE to define an elastic strain ee and a plastic strain e_P such as ~j - ~je + ~;P
(5)
The elastic strain represents the reversible movements of atoms, and the plastic strain corresponds to an average of irreversible slips. All variables which define the internal state of the RVE are called internal variables. They should result from observations at a microscale and from a homogenization process: 9 isotropic hardening in metals related to the density of dislocations; 9 kinematic hardening related to the internal residual microstresses at the level of crystals; 9 damage related to the density of defects; 9 etc. How many do we need? As many as the number of phenomena taken into consideration, but the smallest is the best. Finally, the local state method postulates that the considered thermodynamic state is completely defined by the actual values of the corresponding state variables: observable and internal.
1.1.3 FORMULATION The thermodynamics of irreversible processes is a general framework that is easy to use to formulate constitutive equations. It is a logical guide for incorporating observations and experimental results and a set of rules for avoiding incompatibilities. The first principle is the energy balance: If e is the specific internal energy, p the density, co the volume density of internal heat produced by external
1.1 Background on Modeling
sources, and ~' The heat flux: fie. =
cr ij ~ ij -Jr- (.0 - - q i , i
(6)
where the sommation convention of Enstein applies. The second principle states that the entropy production i must be larger or equal to the heat received divided by the temperature Ps > -
(7) T
,i
If ~ = e - Ts is the Helmholtz specific free energy (this is the energy in the RVE which can eventually be recovered),
~,j~j - 0(~; + ~ir) -q~r~ > o
(8)
T This is the Clausius-Duhem inequality, which corresponds to the positiveness of the dissipated energy and which has to be fulfilled by any model for all possible evolutions.
1.1.3.1 STATEPOTENTIAL The state potential allows for the derivation of the state laws and the definition of the associate variables or driving forces associated with the state variables VK tO define the energy involved in each phenomenon. Choosing the Helmholtz free energy ~, it is a function of all state variables concave with respect to the temperature and convex with respect to all other VK, 0 = O(~, T , f , f , . . .
VK...)
(9)
or in classical elastoplasticity O = O ( F , ~_p, r , . . . v ~ . . .)
(10)
The state laws derive from this potential to ensure that the second principle is always fulfilled. a ij fiPij - ~_~ p
~ K "V K
T > - 0 q~r,~
(11)
They are the laws of thermoelasticity
oo ~J - P oe~ o~ S ~-
OT
(12)
(13)
Lemaitre
The associated variables are defined by
00 cr~j - p 0 ~ O0 AK -- p OVK
(14)
(15)
Each variable AK is the main cause of variation of the state variable VK. In other words, the constitutive equations of the phenomenon represented by VK will be primarily a function of its associated variable and eventually from others. VK -- g K ( . . . A K . . . )
(16)
They also allow us to take as the state potential the Gibbs energy dual of the Helmholtz energy by the Legendre-Fenchel transform ~* = ~* (_~,s,...AK...)
(17)
or any combination of state and associated variables by partial transform.
1.1.3.2
DISSIPATIVE POTENTIAL
To define the gK function of the kinetic equations, a second potential is postulated. It is a function of the associate variables, and convex to ensure that the second principle is fulfilled. It can also be a function of the state variables but taken only as parameters. q) -- q~(K,...AK..., gracl T;e_.e, T , . . . V K . . . )
(18)
The kinetic laws of evolution of the internal state variables derive from .p _ Oq)
eij - Oaij
% _
&o
(19)
(z0)
OAK
-
T
=
-
----------~
(21)
0grad T
Unfortunately, for phenomena which do not depend explicitly upon the time, this function (p is not differentiable. The flux variables are defined by the subdifferential of q~. If F is the criterion function whose the convex F - 0 is
1.1 B a c k g r o u n d o n M o d e l i n g
the indicatrice function of qo. (,o-- 0
if
F<0---+kP--0
(,o - - o c
if
F -- 0 ---~/~_P=/=0
-
(22)
Then, some mathematics prove that
.p_ OF/.
gij ~ vK = -
if
F--0
and
lb-0
(23)
OF ,{ OAK
"P-0
gij "r - 0
} if
F<0
or
F<0
(24)
This is the generalized normality rule of standard materials for which ~. is the multiplier calculated by the consistancy condition f - 0, f - 0.
1.1.4 IDENTIFICATION
The set of constitutive equations is fully defined if the two potentials ~ and ~0 take appropriate close forms: this is the qualitative identification. The numerical response of the constitutive equations to any input is obtained if the materials parameters take the appropriate values: this is the quantitative identification.
1.1.4.1
QUALITATIVE IDENTIFICATION
Assume an interest in several phenomena for which q internal variables have been identified. Which functions should one choose for J/(~e, e_P,T, V 1 . . . Vq) and q0(cr,A1... Aq, grad Y;~ ~, T, V1. . . Vq)? If a phenomenon is known as linear, the corresponding potentials are positive definite quadratic functions. For linear elasticity, for example, 1 -
(25)
where p is the density and E the Hooke tensor. If two phenomena I and J are known to be coupled, the corresponding potentials should verify m
9 a state coupling: 020/OVIc~Vj r 0 9 or an evolution coupling: 02qo/OViOVj ~ 0 If no coupling occurs 02d//c3V~OVj - 0 and 02qo/OVxOVj - O.
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Following is an example of elasticity coupled to damage represented by the variable D:
02~
ODO~ # 0 1 ~e -- ~pp EijkZg~jg~Z"HI (D)
+-- multiplication of functions
(26)
(27)
If such coupling would not have existed, we would have written 1
r
~e - -~pEijkz~ij~l + H2(D)
,--- addition of functions
(29)
that is, c926/c9Dc%~ - 0
0G
For nonlinear phenomena, often power functions are used, but for phenomena which asymptotically saturate, exponential functions are preferred. Often this choice is subjective. Nevertheless, micromechanics analysis may yield logical functions with regard to the micromechanisms introduced at microscale. It consists of the calculation of the energy involved in a RVE by a proper integration or an average of the elementary energies corresponding to the micromechanisms considered. Qualitative experiments are used to point out the tendencies of evolution, but they do not concern the potentials in themselves because simple direct measurements of energy is not possible. Measurements concern the evolution of variables: strain as a function of stress, crack length as a function of time, etc. This means that the potentials are identified from an integration of what is observed. For example, an observation of the secondary creep plastic strain rate as a nonlinear function of the applied stress in creep tests given by the phenomenological Norton law ~ p - (o/K) N is introduced in the dissipative potential as q~
1
if some multiaxial experiments show that the von Mises criterion is fulfilled (r~r is the yon Mises equivalent stress).
11
1.1 Background on Modeling
1.1.4.2 QUANTITATIVEIDENTIFICATION This is the weakest point of the mechanics of materials. All the parameters introduced in constitutive equations (Young's modulus E and Poisson's ratio v in elasticity, Norton's parameters K and N in creep, etc.) differ for each material and are functions of the temperature. Since there are thousands of different materials used in engineering and since they change with the technological progress of elaboration processes, there is no way to built definite, precise databases. Another point is that when a structural calculation is performed during a design, the definitive choice of materials is not achieved, and, even if it is, nobody knows what the precise properties of the materials elaborated some years after will be. The only solution is to perform the structural calculations with the models identified with all known information and to update the calculations each time a new piece of information appears, even during the service of the structure. This, of course, necessitates close cooperation between the designers and the users. 1.1.4.2.1
Sensibility to Parameters
When a model is being used, all material parameters do not have the same importance for the results: a small variation of some of them may change the results by a large amount, whereas a large variation of others has a small influence. For example, a numerical sensibility analysis on the parameters cry, K, and M on the shape of the stress-strain curve, graph of the simple model of uniaxial plasticity
-
+ K4/
(32)
shows that the more sensible parameter is Cry;by taking an approximate value of M ( M = 3 , 4 , 5 ) , it is always possible to adjust K in order to have a satisfactory agreement. But a good correlation with the set of available data does not prove that the model is able to give satisfactory results for cases far away from the tests used for the identification. Before any quantitative identification of a model is made, it is advisable to perform a sensibility analysis in order to classify by increasing order of sensibility the parameters, M, K, cry here for example, and to proceed as follows: 1.1.4.2.2 Rough Estimation of Parameters From all known data, make a first estimation of the parameters using all approximations in the model in order to have the same number of unknowns as the number of pieces of information. Eventually, take values of parameters corresponding to materials that are close in their chemical composition.
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Continue with the same example of the preceding plasticity model for a mild steel for which cry is known as 300 MPa. If the ultimate stress cru is known as 400 MPa for a plastic strain to rupture epu ~ 0.20, then taking M = 1 allows one to find K ~ 500 MPa. These approximate values of the parameters may be taken as a starting solution of an optimization process.
1.1.4.2.3 Optimization Procedure If now more experimental results are available, an optimization procedure may be performed to minimize the difference between the test data and the prediction of those tests by the full numerical resolution of the model. The least-square method is advantageously used. Unfortunately, in the range of nonlinear models, the minimization of the error function may have several solutions due to local minima or fiat variations for which the gradient methods converge extremely slowly. This is why the starting solution should be as close as possible to the optimized solution and why one should give different weight factors to the parameters in order "to help" the numerical procedure: small weight factors to less sensible parameters.
1.1.4.2.4 Validation The process is not finished until the model has been applied and compared to special tests which have not been used for the identification. Of course, the model should be applied to the identification cases, but this is only for checking the identification procedure. These validation tests must be as close as possible to the case considered for applications, and as far as possible from the identification tests m close or far in the sense of variables. For example: 9 biaxial tests if the tests of identification were uniaxial; 9 nonisothermal tests if the tests of identification were conducted at constant temperature; 9 tests with gradient of stress or of other variables; 9 different time scales; 9 etc.
The comparison between validation tests and prediction gives concrete ideas about the applicability of the models from the point of view of accuracy and robustness.
1.1 Background on Modeling
13
1.1.5 VALIDITY DOMAIN Sometimes people say that "a good model should only be used to interpolate between good tests." I do not agree with this pessimistic view because to interpolate between tests results a "good polynome" is sufficient. A model is something more. First, it includes ideas on the physical mechanisms involved; second, it is a logical formulation based on general concepts; and third, only after that, it is numbers. The domain of validity of a model is the closed domain in the space of variables inside which any resolution of the model gives an acceptable accuracy. For the preceding model of plasticity, this is 0 < a < 400 MPa, 0 < ~p <0.2 for a relative accuracy of about &plop ,~ 10% on plastic strain for a given stress. The bounds are difficult to determine; they are those investigated by the identification tests program, plus "motivated" extrapolations based on wellestablished concepts. Time extrapolation is the most crucial because the identification procedure deals within a time range of hours, days, or months, whereas the applications of models deal within a time range of years or decades. In such long periods of time phenomena of aging and changing properties can occur which may be not included in the models. Aging and change of properties by "in-service incidents" are certainly still open problems.
1.1.6 CHOICE OF MODELS The best model for a given application must be selected with much care and critical analysis. First of all, investigate all the phenomena which may occur and which have to be checked in the application: for example, monotonic or cyclic plasticity. Then determine the corresponding variables which should exist in the model: for example, cyclic plasticity needs a kinematic hardening variable. Check the domain of validity of the possible models in comparison to what is expected in the application and select the simplest that has a good ratio of quality to price, the quality being the accuracy and the price the number of materials parameters to identify. /t The choice of the model depends also on the available data to identify the material parameters for the material concerned. Fortunately, often the structural calculations are performed to compare different solutions in order to optimize a design. In that case, good qualitative results are easily obtained with rough estimations of the parameters.
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1.1.7 NUMERICAL
IMPLEMENTATION
The last activity in modeling is the numerical use of the models. Most of them, in mechanics of materials, are nonlinear and incremental procedures are used together with iterations. For example, in plasticity: 9 In a first step the incremental strain field is calculated by means of the kinetic equations from momentum equations. 9 The second step concerns the integration of the constitutive equations to obtain the increments of the state variables and their new values. 9 The third step consists in checking the momentum balance equation for the actual stresses; if violated the iteration process goes to step 1 until a given accuracy is obtained. The Newton-Raphson method is often used. Implicit scheme in quasi-static conditions or explicit scheme in dynamic conditions are used until the end of the loading history or if a divergence appears as a loss of ellipticity or a strain localization characteristic of softening behavior.
BIBLIOGRAPHY Ashby, M., and Jones, D. (1987). Engineering Materials, vols. 1 and 2. Oxford: Pergamon Press. Francois, D., Pineau, A., and Zaoui, A. (1998). Mechanical Behavior of Materials, vols. 1 and 2. Dordrecht: Kluwer Academic Publishers. Lemaitre, J., and Chaboche, J. L. (1995). Mechanics of Solid Materials, Cambridge: Cambridge University Press.
SECTION
1.2
Materials and Process Selection Methods YVES BRECHET L.T.P.C.M. BP75, Institut National Polytechnique de Grenoble. 38402 St Martin d'Heres Cedex, France
Contents 1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Databases: The Need for a Hierarchical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Comparing Materials: The Performance Index Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Design Procedure: Screening, Ranking, and Further Information, the Problem of Multiple Criteria Optimization . . . . . . . . . . . . . . . 1.2.5 Materials Selection and Materials Development: The Role of Modeling . . . . . . . . . . 1.2.6 Process Selection: Structuring the Expertise .. 1.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 16 19
22 24 26 26 28
1.2.1 I N T R O D U C T I O N Designing efficiently for structural applications requires both a proper dimensioning of the structure (involving as a basic tool finite element calculations) and an appropriate choice of the materials and the process used to give them the most suitable shape. The variety of materials available to the engineer (about 80,000), as well as the complex set of requirements which define the most appropriate material, lead to a multicriteria optimization problem which is in no way a trivial one. In recent years, systematic methods for materials and process selection have been developed [1-4] and Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
15
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implemented into selection softwares [5-7] which ideally aim at selecting the best materials early enough in the design procedure so that the best design can be adequately chosen. This selection guide is most crucial at the early stages of design: there would be no hope of efficiently implementing a polymer matrix composite solution on a design initially developed for a metallic solution. Selecting the most appropriate materials is a task which should be done at the very beginning of the design procedure and all along the various steps, from conceptual design to detail design through embodiment design. The coupling with other design tools should at the very least provide finite element codes with constitutive behavior for the materials which appear the most promising. A more ambitious program, yet to be implemented, is to interface these elements with expert systems which would guide the designer toward shapes or processes more suited to a given class of materials, and which ultimately would help to redesign the component in an iterative manner according to the materials selection. These methods require databases of materials and tools to objectively compare materials for a given set of requirements. The amount of modeling needed in these methods is still quite elementary. In the present paper, we will focus on the tools used to compare materials rather than on their implementation as computer software. The modeling involved in the performance index method (Section 1.2.2) is standard strength of materials. The search for an optimal solution sometimes requires more refined optimization techniques (Section 1.2.3). We will outline in Section 1.2.4 the possible use of micromechanics and optimization methods in the development of materials with the aim of meeting a giving set of requirements. In Section 1.2.5 we will illustrate the need to structure and store the expertise in process selection, and will outline the need for modeling in this area.
1.2.2 DATABASES: T H E N E E D F O R A HIERARCHICAL APPROACH Material selection methods are facing a dilemma: the structure of the databases and the selection tools have to be as general as possible to be easily adaptable to a variety of situation. But this general structure is bound to fail when the selection problem is very specific (such as, for instance, selecting cast alloys). The methodology for materials selection presented in this paper is a compromise in this dilemma. We will present first the generic approach, and then some specific applications. The idea is always to go from the
1.2 Materials and Process Selection Methods
17
most generic approach to the most specific one. In order to do so, the materials databases have to be organized in a hierarchical manner so that the selection at a given level orients the designer toward a more specific tool. Depending on the stages of design at which one considers the question of materials selection, the level of information required will be different [1]. In the very early stages, all possible materials should be considered, and therefore a database involving all the materials classes is needed. Accordingly, at this level of generality, the properties will be stored as ranges with relatively low precision. When the design procedure proceeds, more and more detailed information is needed on a smaller number of materials classes. Properties more specific to, say, polymers (such as the water intake or the flammability) might be referred to in the set of requirements. In the last stages of design, a very limited number of materials, and finally one material and a provider, have to be selected: at this level, very precise properties suitable for dimensioning the structure are needed. This progressive increase in specialization motivates a hierarchical approach to databases used in materials selection tools: instead of storing all the possible properties for a huge number of materials, which is bound to lead to a database loaded with missing information, the choice has been to develop a series of databases incorporating each a few hundreds of materials. The generic database comprises metals, polymers, ceramics, composites, and natural materials. Specialized databases have been developed for steels, light alloys, polymers, composites, and woods. More specialized databases coupling the materials and the processes (such as cast alloys, or polymer matrix composites) can then be developed, but their format is different from the previous databases. The set of requirements for structural applications is very versatile. Of course, mechanical properties are important (such as elastic moduli, yield stresses, fracture stresses, or toughness). These properties can be stored as numerical values. But very often, information such as the possibility of getting the materials as plates or tubes, the possibility of painting or joining it with other materials, or its resistance to the environment which can be potentially chemically aggressive, are also important. All the databases currently developed contain numerical information, qualitative estimates, and boolean evaluations. More recent tools [6] also allow one to store not only numbers, but also curves (such as creep curves for polymers, at a given temperature under a given strength). When a continuum set of data has to be stored, such as creep curves or corrosion rates, being able to rely on a model with a limited number of parameters (such as Norton's law for creep) considerably increases the efficiency of the storing procedure. For a database to be usable for selection purposes, it should be complete (sometimes needing
18
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some estimation procedure), it should not overemphasize one material with respect to the others, and it should contain data which are meaningful for all the materials in the database. The databases used in materials selection are of two types: either they list the materials which are possible candidates, or they store the elements from which the possible candidates are made. The first case is rather simple: provided a correct evaluation function is defined, the ranking of the candidates can be done by simple screening of the database. The second case, for instance, when the database lists the resins and the fibers involved in making a composite material, requires both micromechanical tools to evaluate the properties of the materials from the properties of its components, and also more subtle numerical methods that are able to deal with a much larger (virtually infinite) set of possible candidates. Steepest gradient methods, simulated annealing, and genetic algorithms are possible solutions for these complex optimization problems. In principle, one should try to select materials and processes simultaneously, since it is very often in terms of competition between various couples (materials/processes) that the selection problem finally appears: should one make an aiplane wing joining components obtained from medium-thickness plates of aluminum alloys, or should one machine the wing components inside a thick plate of a less quench sensitive alloy the wing together with the stiffeners? The coupling between processes and materials properties is still very poorly taken into account in the current selection procedures. Processes are also selected from databases of attributes for the different processes (such as the size of the components, the dimensional accuracy, or the materials accessible to a given process). The databases for process attributes have the same structure as the ones for materials, and the same hierarchical organization, and information can be numeric, qualitative, or boolean. Beside the variety of properties (for materials) and attributes (for processes) involved in a selection procedure, depending on the stage of selection, one is either confronted with a very open end set of requirements, or with always the same set of questions. In the first situation, one needs a very versatile tool, but because of combinatoric explosion, one cannot afford to deal with questions involving interactions that are too complex between various aspects, (such as "this shape, for this alloy, assuming this minimal dimension, is prone during casting to exhibit hot tearing"). On the other hand, when the selection becomes very focused (such as selection of joining methods), the set of requirements to be fulfilled has basically always the same format: it can be stored as a "predefined questionnaire" which allows more refined questions to be asked since they are in a limited number.
19
1.2 Materials and Process Selection Methods
1.2.3 COMPARING MATERIALS: THE PERFORMANCE INDEX METHOD The databases are the hard core of the selection procedure: up to a certain point they can be cast in a standard format, which has been used in CMS, CPS, and CES software. When selection reaches a high degree of specialization, more specific formats have to be implemented, and a questionnaire approach rather than an "open-end selection" might be more efficient. But a database would be of little use without an evaluation tool able to compare the different materials. Simple modeling allows one to build such a tool, but the price to be paid is that dimensioning of the structure using this method is very crude. One has to keep in mind that the aim is to identify the materials for which accurate structural mechanics calculations will have to be performed later on. Each set of requirements has to be structured in a systematic manner: What are the constraints? What are the free and the imposed variables? What is the objective? For instance, one might look for a tie for which the length L is prescribed and the section S is free (free and imposed dimensions), which shouldn't yield under a prescribed load P (constraints), and which should be of minimum weight (objective). The stress which should not exceed the yield stress is P - < ay (1) SThe mass of the component to be minimized is M = p.L.s
(2)
The constraint not to yield imposes a minimum value for the section S. The mass of the component is accordingly at least equal to
Therefore, the material which will minimize the mass of the component will be the one which maximizes the "performance index" I:
I =ay P
(4)
This very simple derivation illustrates the method for obtaining performance indices: write the constraint and the objectives, eliminate the free variable, and identify the combination of materials properties which measures the efficiency of materials for a couple (constraints/objectives). These performance indices have now been derived for many situations corresponding to simple geometry (bars, plates, shells, beams) loading in simple modes (tension, torsion, bending), for simple constraints (do not yield, prescribed
20
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stiffness, do n o t b u c k l e . . . ) , a n d for v a r i o u s objectives ( m i n i m u m w e i g h t , m i n i m u m v o l u m e , m i n i m u m cost). T h e y h a v e b e e n e x t e n d e d to t h e r m a l a p p l i c a t i o n s . T h e w a y to derive a p e r f o r m a n c e i n d e x for a real s i t u a t i o n is to: -
s i m p l i f y the g e o m e t r y a n d the l o a d i n g ; i d e n t i f y the free variables; m a k e e x p l i c i t the c o n s t r a i n t u s i n g s i m p l e m e c h a n i c s ; w r i t e d o w n the objective; a n d
- e l i m i n a t e the free v a r i a b l e s b e t w e e n the c o n s t r a i n t a n d the objectives.
TABLE 1.2.1 Classical performance indices for mechanical design for strength or stiffness at minimum weight. Objective
Shape
Loading
Constraint
Performance index
Stiffness design with a minimal mass E/p
Minimize the mass
Tie
Tension
Stiffness and length prescribed, section free
Minimize the mass
Beam
Bending
Stiffness, shape and length fixed, section free
E1/2/p
Minimize the mass
Beam
Bending
E1/3/p
Minimize the mass
Plate
Bending
Stiffness, width and length fixed, height free Stiffness length width fixed, thickness free
Minimize the mass
Plate
Compression
Buckling load fixed, length width fixed, thickness free
E1/3/p
Minimize the mass
Cylinder
Internal pressure
Imposed maximum elastic strain, thickness of the shell free
E/p
E1/3/p
Strength design with a minimal mass Minimize the mass
Tie
Traction
Strength, length fixed, section free
ae/p
Minimize the mass
Beam
Bending
Strength, length fixed, section free
a2/3/p
Minimize the mass
Plate
Bending
Strength, length and width fixed, thickness free
ale/2/p
Minimize the mass
Cylinder
Internal pressure
Imposed pressure, the materials shall not yield, thickness of the shell free
ae/p
21
1.2 Materials and Process Selection Methods
Table 1.2.1 gives some standard performance indices currently used in mechanical design. Many others have been derived, both for mechanical and thermo-mechanical loading [ 1,4]. A simple way to use the performance index is with the so-called selection maps shown in Figure 1.2.1: on a logarithmic scale the lines corresponding to equal performances are straight lines whose slopes depend on the exponents entering the performance index. Figure 1.2.1 shows one of these maps used for stiff components at minimum mass. Materials for stiff ties should maximize E/p, materials for stiff beams should maximize E1/2/p, and materials for stiff plates should maximize E1/3/p. These performance indices have a drawback, however: they are concerned with time-independent design, the component is made so that it
1000
1. Modulus-Density
"Youngs Modulus E (G = 3E/8 ; K,= E.)
MFA:INH}I
t, f
(p'~ 89
J
-'En~i,needn~
"/ Composites
tlO
/ f "0 0
m
/
J"
/ Engineedn,, Polymers
3x10 a 1.0
r
t HDPE
Guide Unes for Minimum
LDPE
0.1
~/
.3xlo =
0.01;. i
"/. 1
0.3
,I
t,,I
..s
~_..~ rV I ~
I
1.0
/r
Elastomers /
/
I
,,"=
/
,/
Cl
3
)
I
I
I
I I
Density, p (Mg/m3) FIGURE 1.2.1
I
10
Selection map for stiff light design [1].
80
22
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should fulfill its function when it starts being used, and it is assumed it will be so for the rest of its life. Of course, this is rarely the case, and one often has to design for a finite lifetime. As a consequence, for instance, in designing for creep resistance or corrosion resistance, a new set of performance indices involving rate equations (for creep or corrosion) has been developed [8,9]. The performance indices then depend not only on the materials properties, but also on operating conditions such as the load, or the dimensions, or the expected lifetime. For instance, large-scale boilers are generally made out of steel, whereas small-scale boilers are often made in copper. In principle, finite lifetime design is possible within the framework of performance indices, but the data available to effectively apply the method are much more difficult to gather systematically.
1.2.4 T H E D E S I G N P R O C E D U R E : S C R E E N I N G , RANKING, AND FURTHER INFORMATION, THE PROBLEM OF MULTIPLE" CRITERIA OPTIMIZATION The previous method allows one to compare very different materials for a given set of requirements formulated as a couple (constraint/objective). However, in realistic situations, a set of requirements comprises many of these "elementary requirements." Moreover, only part of the requirements can indeed be formated that way. A typical selection procedure will proceed in three steps: 1. At the screening stage, materials will be eliminated according to their properties: only those that could possibly do the job will remain. For instance, for a component in a turbine engine, the maximum operating temperature should be around 800C: many materials won't be able to fulfill this basic requirement, and can be eliminated even without looking for their other properties. 2. At the ranking stage, a systematic use of performance indices is made: the problem is then, among admissible materials, to find the ones which will do the job most efficiently, that is, at the lowest cost, with the lowest mass, or the smallest volume. The ranking will be made according to a "value function" which encompasses the various aspects of the set of requirements. The problem of defining such a value function for multiple criteria optimization will be dealt with in the next paragraph.
1.2 Materials and Process Selection Methods
23
3. For the remaining candidates that are able to fulfill the set of requirements efficiently, further information is often needed concerning corrosion rates, wear rates, or possible surface treatments. These pieces of information are scattered in the literature, and efficient wordsearching methods are required to help with this step. At the same step, the local conditions, or the availability of the different possible materials, will also be a concern. The three steps in the selection procedure are also a way to structure process selection. The screening stage will rely on attributes such as the size of the component and the materials from which it is made. The ranking step will need a rough comparative economic evaluation of the various processes, involving the batch size and the production rate. The last step will depend on the availability of the tooling and the will to invest. It appears from these various aspects of the selection procedure that a key issue is to build a "value function" that is able to provide one with a fair comparison of the different possible solutions. The performance index method is the first step in building this value function. The second step is to deal with the multicriteria nature of the selection process. This multicriteria aspect can be conveniently classified in two categories: it might be a multiconstraint problem, or a multiobjective problem (in any real situations, it is both!). In a multiconstraint problem (such as designing a component which should neither yield nor fail in fatigue), the problem is to identify the limiting constraint. In order to do so, further knowledge on the load and the dimensions is needed. A systematic method called "coupled equations" [10] allows one to deal with this problem. In a multiobjective problem (such as designing a component at minimum weight and minimum cost), one needs to identify an "exchange coefficient" [10] between the two objectives, for instance, how much the user is ready to pay for saving weight. These exchange coefficients can be either obtained from a value analysis of the product or from the analysis of existing solutions [4]. They allow one to compute a value function, which is the tool needed to rank the possible solutions. Both the value analysis and the coupled equation method provide one with an objective treatment of the multiple criteria optimization. However, they require extra information compared to the simple performance index method. When this information is not available, one needs to make use of methods involving judgments. The most popular one is the "weight coefficients method," which attributes to each criteria a percentage of importance. The materials are then compared to an existing solution. It must be stressed that the value function so constructed depends on the choice of both the weighting factors and the reference material. Weighting factors are difficult to evaluate; moreover, multiple criteria often lead to no solution at all
24
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due to an excessive severity. Multiple optimization also implies the idea of compromise between the various requirements. For this reason, algorithms involving fuzzy logic methods [3] have been developed to deal with the intrinsic fuzziness of the requirements (two values will be given, one above which the satisfaction is complete, one under which the material will be rejected). Proposed situations at the margin of full satisfaction will be proposed for evaluation, and the value function will be constructed so that it will give, for the same questions, the same evaluation as the user. This technique bypasses the difficulty in giving a priori value coefficients, since they are then estimated from the evaluation-proposed solutions. However, these methods still involve judgments (though in a controlled manner), and, when possible, the objective methods should be preferred. Once the value function is available, the selection problem becomes an optimization one. When the database is finite, the optimization can be performed by a simple screening of all the available solutions. The method has been extended to the optimal design of multimaterials components such as sandwich structures [11, 12]. The aim is then to simultaneously select the skin, the core, and the geometry for a set of requirements involving stiffness and strength, constraints on the thickness, objectives on the minimization of the weight, or the cost. For single criteria selection, an analytical method was derived [13]. For multiple criteria, such a method is no longer available, and the selection requires one to compute the properties of a sandwich from the properties of its components and its geometry, and to compare all the possible choices. In order to find the optimal solution, a genetic algorithm was used. The principle is to generate a population of sandwiches whose "genes" are the materials and the geometry. New sandwiches are generated, either by mutation or by crossover between existing individuals, and the population is kept constant in size by keeping the individuals alive with a greater probability when their efficiency (measured by the value function) is greater. In such a way, the algorithm converges very rapidly to a very good solution.
1.2.5 M A T E R I A L S S E L E C T I O N A N D M A T E R I A L S DEVELOPMENT: THE ROLE OF MODELING In the previous sections, we were interested in selecting materials and processes to fulfil a set of requirements. The only modeling needed at this stage is a simplified estimation of the mechanical behavior of the component, together with a clear identification of the constraints and the objectives. The value function allowing one to estimate the efficiency
1.2 Materials and Process Selection Methods
25
of the different solutions is itself a simple linear combination of the performance indices corresponding to the dominant constraints identified by a predimensioning. However, the same method has been applied to identify suitable materials whose development would fulfill the requirements. Composite materials are especially suitable for this exercise because their value relies partly on the possibility of tayloring them for application [14,15]. In order to design a composite material, one has to identify the best choice for the matrix, for the reinforcement, for the architecture of the reinforcement and its volume fraction, and for the process to realize the component (which might be limited by the shape to be realized). One needs relations, either empirical or based on micromechanics models, between the properties of the components of the composite and the properties of the material itself. Usually, the process itself influences the properties obtained, which are lower than the properties of the ideal composite that micromechanics models would provide. One could think of introducing this feature in the modeling through interface properties, but it is generally more convenient to store the information as "knock-down factors" on properties associated with a triplet matrix/reinforcement/process. Another application of materials selection methods using mechanical modeling is the optimal design of glass compositions for a given set of requirements: since the properties are, within a certain range, linearly related to the composition, optimization techniques such as a simplex algorithm are well adapted to this problem. When a continuous variable, such as the characteristics of a heat treatment for an alloy, is available and is provided, either through metallurgical modeling or through empirical correlation, the properties can be given as a function of this variable, and materials selection methods are efficient to design the best treatment to be applied to fit a set of requirements. However, the explicit models available for relations between processes and properties are relatively few. Recent developments using Neural net-works to identify hidden correlation in databases of materials can also be applied and coupled to selection methods in order to design the best transformation processes. Another recent development in selection methods aims at reverting the problem, that is, finding potential applications for new materials [4, 16, 17]. Several strategies have been identified: for instance, one can explore a database of applications (defined by a set of requirements and existing solutions) and find the applications for which the new material is better than the existing solutions. Another technique is to identify the performance indices for which the new material seems better than usual materials, and from there, to find out the applications for which these performance indices are relevant criteria.
26 1.2.6 PROCESS SELECTION: THE EXPERTISE
Brechet STRUCTURING
In addition to selection by attributes of the process, which is efficient in the first stages of selection, when one is confronted with a more specific problem, such as selection of a definite cast aluminium alloy or a definite extruded wrought alloy, or selection of a secondary process such as joining or surface treatments, one is faced with the need to store expertise. For instance, for selection of cast aluminium [3, 18] alloys, the key issue is not to define the performance index; the key issue is to select the alloy which will be possible to cast without defects. Mold filling and hot tearing are the central concerns in this problem. The ability to fill a mold or to cast a component without cracks depends on the alloy, on the geometry of the mold, and on the type of casting. Ideally, one would wish to have models to deal with this question. In real life, hot tearing criteria are not quantitatively reliable, mold-filling criteria are totally empirical, and moreover, the properties of the cast alloy are dependent on the solidification conditions, that is, on the thickness of the component. These dependences are part of what is known as expertise. The simplest way to store this expertise is build the set of requirements according to a predefined questionnaire corresponding to the expert behavior. The second option is to mimic the general tendency identified by the expert by a simple mathematical function (for instance, capturing the tendency to increased hot tearing with thinner parts of the component) and to tune the coefficients of these functions by comparing the results of selection by a software with the results known from the case studies available to the expert. Along these lines, selection methods for cast alloys [18], extruded alloys [19], joining methods [20,21], and surface treatments [4,22] have been developed to capture various expertises. Clearly, modeling is still needed to rationalize the empirical rules commonly used (such as the shapes which can be extruded or cast), or to evaluate the cost of a process (for instance, for joining by laser, or for a surface treatment one needs to find the best operating temperature, power, speed, etc.).
1.2.7 CONCLUSIONS The selection methods briefly presented in this chapter are recent developments. The use of modeling in these approaches is still in its infancy. In the last ten years, general methods and software have been developed to select materials, to select processes, and to deal with multidesign element conception and with multicriteria set of requirements.
27
1.2 Materials and Process Selection Methods TABLE 1.2.2 Name of the software
Selection softwares developed following the guidelines of the present paper. Objectives of the software
Comments/status
CMS
Materials selection, graphical selection using maps; many databases, generic or specialized
Commercially available
CPS
Process selection, graphical method
Commercially available
CES
Materials and process selection, databases for materials, for processes and links between databases
Commercially available; constructor facility for development of dedicated databases
Fuzzymat
Materials selection, multicriteria and fuzzy logic-based selection algorithm
Commercially available; development of specialized databases
CAMD
Materials and process selection for multidesign element conceptions; expert system to guide and analyze the elaboration of requirements
Fuzzycast
Selection of cast aluminium alloys; databases: alloys/processes/geometry
Fuzzycomposites
Design of polymer-based composites; databases: resin, reinforcements, processes, and compatibilities
Sandwich selector
Optimization of sandwich structures; genetic algorithm coupled with fuzzy logic
Fuzzyglass
Optimization of glass compositions; simplex coupled with fuzzy logic
Property of SaintGobain
Astek
Selection of joining methods; databases: processes and shapes
Property of CETIM
STS
Selection of surface treatments; database: processes/materials/objectives
VCE
Evaluation of exchange coefficients from existing solutions
MAPS
Investigation of possible applications for a new material
Property of Pechiney; expertise on casting processes, design rules
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Table 1.2.2 gives a list of selection tools developed along the philosophy described in this chapter. These generic methods have been specialized to various classes of materials and processes. In special situations, a coupling with modeling made possible the use of the present methods to develop new materials or new structures (composites, sandwich structures). For specific processes (casting, joining, extrusions, surface treatments), the selection procedure developed was closer to an expert system, following a predefined questionnaire. Various methods of finding applications for a new material have been put forward. Up to now, the choice has been to rely on empirical knowledge when available, and to keep the selection procedure as transparent and as objective as possible. The main reason for this paper to be included in a book on models in mechanics is to express the need now to couple more closely modeling to design so that one may go beyond empirical correlation and optimize both the choice of materials and their future development.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12.
13. 14. 15. 16. 17. 18. 19.
Ashby, M. (1999). Materials Selection in Mechanical Design, Butterworth Heinemann editor. Esawi, A. (1994) PhD thesis, Cambridge University. Bassetti, D. (1998) PhD thesis, Institut National Polytechnique de Grenoble. Landru, D. (2000) PhD thesis, Institut National Polytechnique de Grenoble. Granta Design, Cambridge Selection softwares: CMS (1995), CPS (1997), CES (1999). Bassetti, Grenoble, Fuzzymat v3.0 (1997). Software for materials selection. Landru, D., and Brechet, Y. Grenoble, (1999). CAMD. Ashby, M., and Brechet, Y. Time Dependant Design (to be published). Brechet, Y., Ashby, M., and Salvo, L. (2000). Methodes de choix des materiaux et des procedes, Presses Universitaires de Lausanne. Ashby, M., (1997). ASTM-STP 1311, 45, Computerization and Networking of Materials Databases, Nishijima, S., and Iwata, S., eds. Bassetti, D., Brechet, Y., Heiberg, G., Lingorski, I., Jantzen, A., Pechambert, P., and Salvo, L. (1998). Materiaux et Techniques 5:31. Deocon, J., Salvo, L., Lemoine, P., Landru, D., Brechet, Y., and Leriche, R. (1999). Metal Foams and Porous Metal Structures, Banhardt, J., Ashby, M., and Fleck, N., eds., MIT Verlag Publishing, p. 325. Gibson, L., and Ashby, M. (1999). Cellular solids, Cambridge University Press. Pechambert, P., Bassetti, D., Brechet, Y., and Salvo, L. (1996). ICCM7, London IOM, 283. Bassetti, D., Brechet, Y., Heiberg, G., Lingorski, I., Pechambert, P., and Salvo, L. (1998). Composite Design for Performance, p. 88, Nicholson, P., ed., Lake Louise. Landru, D., Brechet, Y. (1996). Colloque Franco espagnol, p. 41, Yavari, R., ed., Institut National Polytechnique de Grenoble. Landru, D., Ashby, M., and Brechet, Y. Finding New Applications for a Material (to be published). Lovatt, A., Bassetti, D., Shercliff, H., and Brechet Y. (1999). Int. Journal Cast Metals Research 12: 211. Heiberg, G., Brechet, Y., Roven, H., and Jensrud, O. Materials and Design (in press, 2000).
1.2 Materials and Process Selection Methods
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20. Lebacq, C., Jeggy, T., Brecht, Y., and Salvo, L. (1998). Materiaux et Techniques 5: 39. 21. Lebacq, C., Brechet, Y., Jeggy, T., Salvo, L., and Shercliff, H. (2000). Selection of joining methods. Submitted to Materials and Design. 22. Landru, D., Esawi, A., Brechet, Y., and Ashby, M. (2000). Selection of surface treatments (to be published).
SECTION
1.3
Size Effect on Structural Strength ZDENI~K P. BAZANT
Northwestern University, Evanston, Illinois
Contents 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
1.3.2 History of Size Effect up to Weibull . . . . . . . . . .
34
1.3.3 Power Scaling and the Case of No Size Effect. 36 1.3.4 Weibull Statistical Size Effect . . . . . . . . . . . . . . . .
38
1.3.5 Quasi-Brittle Size Effect Bridging Plasticity and LEFM, and its History . . . . . . . . . . . . . . . . . . .
40
1.3.6 Size Effect Mechanism: Stress Redistribution and Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.3.6.1 Scaling for Failure at Crack Initiation.. 43 1.3.6.2 Scaling for Failures with a Long Crack or Notch . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
1.3.6.3 Size Effect on Postpeak Softening and Ductility . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.4 Asymptotic Analysis of Size Effect by Equivalent LEFM . . . . . . . . . . . . . . . . . .
47 48
1.3.6.5 Size Effect Method for Measuring Material Constants and R-Curve . . . . . . .
49
1.3.6.6 Critical Crack-tip Opening Displacement, 6C~OD. . . . . . . . . . . . . . . . . .
50
1.3.7 Extensions, Ramifications, and Applications .. 50 1.3.7.1 Size Effects in Compression F r a c t u r e . . 50
*Thanks to the permission of Springer Verlag, Berlin, this article is reprinted from Archives of Applied Mechanics (Ingenieur-Archiv) 69, 703-725. A section on the reverse size effect in buckling of sea ice has been added, and some minor updates have been made. The figures are the same.
30
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
31
1.3 Size Effect on Structural Strength 1.3.7.2
Fracturing Truss Model for Concrete and Boreholes in Rock . . . . . . . . . . . . . . . 1.3.7.3 Kink Bands in Fiber Composites . . . . . 1.3.7.4 Size Effects in Sea Ice . . . . . . . . . . . . . . . 1.3.7.5 Reverse Size Effect in Buckling of Floating Ice . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.7.6 Influence of Crack Separation Rate, Creep, and Viscosity . . . . . . . . . . . . . . . . . 1.3.7.7 Size Effect in Fatigue Crack Growth.. 1.3.7.8 Size Effect for Cohesive Crack Model and Crack Band Model . . . . . . . . . . . . . . 1.3.7.9 Size Effect via Nonlocal, Gradient, or Discrete Element Models . . . . . . . . . . 1.3.7.10 Nonlocal Statistical Generalization of the Weibull Theory . . . . . . . . . . . . . . . 1.3.8 Other Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8.1 Hypothesis of Fractal Origin of Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8.2 Boundary Layer, Singularity, and Diffusion . . . . . . . . . . . . . . . . . . . . . . . 1.3.9 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Bibliography . . . . . . . . . . . . . . . . . . . . . . .
51 52 52 54 55 56 56 58 59 60 60 61 62 62 62
The article attempts a broad review of the problem of size effect or scaling of failure, which has recently come to the forefront of attention because of its importance for concrete and geotechnical engineering, geomechanics, and arctic ice engineering, as well as in designing large load-bearing parts made of advanced ceramics and composites, e.g., for aircraft or ships. First the main results of the Weibull statistical theory of r a n d o m strength are briefly s u m m a r i z e d and its applicability and limitations described. In this theory as well as plasticity, elasticity with a strength limit, and linear elastic fracture mechanics (LEFM), the size effect is a simple power law because no characteristic size or length is present. Attention is then focused on the deterministic size effect in quasi-brittle materials which, because of the existence of a non-negligible material length characterizing the size of the fracture process zone, represents the bridging between the simple power-law size effects of plasticity and of LEFM. The energetic theory of quasi-brittle size effect in the bridging region is explained, and then a host of
32
Ba~.ant
recent refinements, extensions, and ramifications are discussed. Comments on other types of size effect, including that which might be associated with the fractal geometry of fracture, are also made. The historical development of the size effect theories is outlined, and the recent trends of research are emphasized.
1.3.1
INTRODUCTION
The size effect is a problem of scaling, which is central to every physical theory. In fluid mechanics research, the problem of scaling continuously played a prominent role for over a hundred years. In solid mechanics research, though, the attention to scaling had many interruptions and became intense only during the last decade. Not surprisingly, the modern studies of nonclassical size effect, begun in the 1970s, were stimulated by the problems of concrete structures, for which there inevitably is a large gap between the scales of large structures (e.g., dams, reactor containments, bridges) and the scales of laboratory tests. This gap involves in such structures about one order of magnitude (even in the rare cases when a full-scale test is carried out, it is impossible to acquire a sufficient statistical basis on the full scale). The question of size effect recently became a crucial consideration in the efforts to use advanced fiber composites and sandwiches for large ship hulls, bulkheads, decks, stacks, and masts, as well as for large load-bearing fuselage panels. The scaling problems are even greater in geotechnical engineering, arctic engineering, and geomechanics. In analyzing the safety of an excavation wall or a tunnel, the risk of a mountain slide, the risk of slip of a fault in the earth crust, or the force exerted on an oil platform in the Arctic by a moving mile-size ice floe, the scale jump from the laboratory spans many orders of magnitude. In most mechanical and aerospace engineering, on the other hand, the problem of scaling has been less pressing because the structural components can usually be tested at full size. It must be recognized, however, that even in that case the scaling implied by the theory must be correct. Scaling is the most fundamental characteristics of any physical theory. If the scaling properties of a theory are incorrect, the theory itself is incorrect. The size effect in solid mechanics is understood as the effect of the characteristic structure size (dimension) D on the nominal strength c~N of structure when geometrically similar structures are compared. The nominal stress (or strength, in case of maximum load) is defined as crN = cNP/bD or cNP/D2 for two- or three-dimensional similarity, respectively; P = load
1.3 Size Effect on Structural Strength
33
(or load parameter), b structure thickness, and cN arbitrary coefficient chosen for convenience (normally cN---1). So crN is not a real stress but a load parameter having the dimension of stress. The definition of D can be arbitrary (e.g., the beam depth or half-depth, the beam span, the diagonal dimension, etc.) because it does not matter for comparing geometrically similar structures. The basic scaling laws in physics are power laws in terms of D, for which no characteristics size (or length) exists. The classical Weibull [113] theory of statistical size effect caused by randomness of material strength is of this type. During the 1970s it was found that a major deterministic size effect, overwhelming the statistical size effect, can be caused by stress redistributions caused by stable propagation of fracture or damage and the inherent energy release. The law of the deterministic stable effect provides a way of bridging two different power laws applicable in two adjacent size ranges. The structure size at which this bridging transition occurs represents a characteristic size. The material for which this new kind of size effect was identified first, and studied in the greatest depth and with the largest experimental effort by far, is concrete. In general, a size effect that bridges the small-scale power law for nonbrittle (plastic, ductile) behavior and the large-scale power law for brittle behavior signals the presence of a certain non-negligible characteristics length of the material. This length, which represents the quintessential property of quasi-brittle materials, characterizes the typical size of material inhomogeneities or the fracture process zone (FPZ). Aside from concrete, other quasibrittle materials include rocks, cement mortars, ice (especially sea ice), consolidated snow, tough fiber composites and particulate composites, toughened ceramics, fiber-reinforced concretes, dental cements, bone and cartilage, biological shells, stiff clays, cemented sands, grouted soils, coal, paper, wood, wood particle board, various refractories and filled elastomers, and some special tough metal alloys. Keen interest in the size effect and scaling is now emerging for various "high-tech" applications of these materials. Quasi-brittle behavior can be attained by creating or enhancing material inhomogeneities. Such behavior is desirable because it endows the structure made from a material incapable of plastic yielding with a significant energy absorption capability. Long ago, civil engineers subconsciously but correctly engineered concrete structures to achieve and enhance quasi-brittle characteristics. Most modern "high-tech" materials achieve quasi-brittle characteristics in much the same way by means of inclusions, embedded reinforcement, and intentional microcracking (as in transformation toughening of ceramics, analogous to shrinkage microcracking of concrete). In effect, they emulate concrete.
34
Ba~.ant
In materials science, an inverse size effect spanning several orders of magnitude must be tackled in passing from normal laboratory tests of material strength to microelectronic components and micromechanisms. A material that follows linear elastic fracture mechanics (LEFM) on the scale of laboratory specimens of sizes from 1 to 10cm may exhibit quasi-brittle or even ductile (plastic) failure on the micrometer scale. The purpose of this article is to present a brief review of the basic results and their history. For an in-depth review with several hundred literature references, the recent article by Ba~.ant and Chen [18] may be consulted. A full exposition of most of the material reviewed here is found in the recent book by Ba~.ant and Planas [32], henceforth simply referenced as [BP]. The problem of scale bridging in the micromechanics of materials, e.g., the relation of dislocation theory to continuum plasticity, is beyond the scope of this review (it is treated in this volume by Hutchinson).
1.3.2 HISTORY OF SIZE EFFECT UP TO WEIBULL Speculations about the size effect can be traced back to Leonardo da Vinci in the 16th century [118]. He observed that "among cords of equal thickness the longest is the least strong," and proposed that "a cord is so much stronger as it is shorter," implying inverse proportionality. A century later, Galileo Galilei [64], the inventor of the concept of stress, argued that Leonardo's size effect cannot be true. He further discussed the effect of the size of an animal on the shape of its bones, remarking that bulkiness of bones is the weakness of the giants. A major idea was spawned by Mariotte [82]. Based on his extensive experiments, he observed that "a long rope and a short one always support the same weight unless that in a long rope there may happen to be some faulty place in which it will break sooner than in a shorter," and proposed the principle of "the inequality of matter whose absolute resistance is less in one place than another." In other words, the larger the structure, the greater is the probability of encountering in it an element of low strength. This is the basic idea of the statistical theory of size effect. Despite no lack of attention, not much progress was achieved for two and half centuries, until the remarkable work of Griffith [66], the founder of fracture mechanics. He showed experimentally that the nominal strength of glass fibers was raised from 42,300 psi to 491,000 psi when the diameter decreased from 0.0042 in. to 0.00013 in., and concluded that "the weakness of isotropic solids ... is due to the presence of discontinuities or flaws . . . . The
1.3 Size Effect on Structural Strength
35
effective strength of technical materials could be increased 10 or 20 times at least if these flaws could be eliminated." In Griffith's view, however, the flaws or cracks at the moment of failure were still only microscopic; their random distribution controlled the macroscopic strength of the material but did not invalidate the concept of strength. Thus, Griffith discovered the physical basis of Mariotte's statistical idea but not a new kind of size effect. The statistical theory of size effect began to emerge after Peirce [92] formulated the weakest-link model for a chain and introduced the extreme value statistics which was originated by Tippett [107] and Fr~chet [57] and completely described by Fischer and Tippett [58], who derived the Weibull distribution and proved that it represents the distribution of the minimum of any set of very many random variables that have a threshold and approach the threshold asymptotically as a power function of any positive exponent. Refinements were made by von Mises [108] and others (see also [62, 63,103,56]. The capstone of the statistical theory of strength was laid by Weibull [113] (also [114-116]). On a heuristic and experimental basis, he concluded that the tail distribution of low strength values with an extremely small probability could not be adequately represented by any of the previously known distributions and assumed the cumulative probability distribution of the strength of a small material element to be a power function of the strength difference form a finite or zero threshold. The resulting distribution of minimum strength, which was the same as that derived by Fischer and Tippet [58] in a completely different context, came to be known as the Weibull distribution. Others [62,103] later offered a theoretical justification by means of a statistical distribution of microscopic flaws or microcracks. Refinements and applications to metals and ceramics (fatigue embrittlement, cleavage toughness of steels at a low and brittle-ductile transition temperatures, evaluation of scatter of fracture toughness data) have continued until today [37, 56, 77, 101]. Applications of Weibull's theory to fatigue embrittled metals and to ceramics have been researched thoroughly [75, 76]. Applications to concrete, where the size effect has been of the greatest concern, have been studied by Zaitsev and Wittmann [122], Mihashi and Izumi [88], Wittmann and Zaitsev [121], Zech and Wittmann [123], Mihashi [84], Mihashi and Izumi [85] Carpinteri [41,42], and others. Until about 1985, most mechanicians paid almost no attention to the possibility of a deterministic size effect. Whenever a size effect was detected in tests, it was automatically assumed to be statistical, and thus its study was supposed to belong to statisticians rather than mechanicians. The reason probably was that no size effect is exhibited by the classical continuum mechanics in which the failure criterion is written in terms of stresses and strains (elasticity with strength limit, plasticity and viscoplasticity, as well as fracture mechanics of bodies containing only microscopic cracks or
36
Ba~.ant
flaws) [8]. The subject was not even mentioned by S. P. Timoshenko in 1953 in his m o n u m e n t a l History of the Strength of Materials. The attitude, however, changed drastically in the 1980s. In consequence of the well-funded research in concrete structures for nuclear power plants, theories exhibiting a deterministic size effect have developed. We will discuss it later.
1.3.3 POWER SCALING AND THE CASE OF NO SIZE EFFECT It is proper to explain first the simple scaling applicable to all physical systems that involve no characteristic length. Let us consider geometrically similar systems, for example, the beams shown in Figure 1.3.1a, and seek to deduce the response Y (e.g., the m a x i m u m stress or the m a x i m u m deflection) as a function of the characteristic size (dimension) D of the structure; Y = Yof(D)
~P lf-i
,
log CN
Xk. Yield or strength -':'-.\ Criteria
I
4-I + , ....I D'
•
J
b
a
,
log D
Large-size
._. ","I" asymptotic .,-. expansion e(D
-~ -~,-"[: .-. cez 0
,,
, " " " . _- . . . .
Plasticity
.. . . . .
~
".X~
1~2
/'I
/ CN = B ft' (1 +~oo)-v2
"'-~~"q~ Small-size [',,~~ asymptotic I " , , , ~ Pexpansl~ Z '
o) Size-effectlaw I _o by asymptotic matching ;o ......
r
",.J \<'x,, \ X ~c/
]
.
log D (structure size)
FIGURE 1.3.1 a. Top left: Geometrically similar structures of different sizes, b. Top right: Power scaling laws. c. Bottom. Size effect law for quasi-brittle failures bridging the power law of plasticity (horizontal asymptote) and the power law of LEFM (inclined asymptote).
1.3 Size Effect on Structural Strength
37
where u is the chosen unit of measurement (e.g., 1 m, i mm). We imagine three structure sizes 1, D, and D' (Figure 1.3.1a). If we take size 1 as the reference size, the responses for sizes D and D' are Y = f ( D ) and Y' = f ( D ' ) . However, since there is no characteristic length, we can also take size D as the reference size. Consequently, the equation
f(D')/f(D) = f(D'/D)
(1)
must hold ([8, 18]; for fluid mechanics [2, 102]). This is a functional equation for the unknown scaling law f(D). It has one and only one solution, namely, the power law: f(D) = (D/c1) ~
(2)
where s = constant and c1 is a constant which is always implied as a unit of length measurement (e.g., i m, i mm). Note that Cl cancels out of Eq. 2 when the power function (Eq. 1) is substituted. On the other hand, when, for instance, f ( D ) = log(D/Cl), Eq. 1 is not satisfied and the unit of measurement, cl, does not cancel out. So, the logarithmic scaling could be possible only if the system possessed a characteristic length related to Cl. The power scaling must apply for every physical theory in which there is no characteristic length. In solid mechanics such failure theories include elasticity with a strength limit, elastoplasticity, and viscoplasticity, as well as LEFM (for which the FPZ is assumed shrunken into a point). To determine exponent s, the failure criterion of the material must be taken into account. For elasticity with a strength limit (strength theory), or plasticity (or elastoplasticity) with a yield surface expressed in terms of stresses or strains, or both, one finds that s = 0 when response Y represents the stress or strain (for example, the maximum stress, or the stress at certain homologous points, or the nominal stress at failure) [8]. Thus, if there is no characteristic dimension, all geometrically similar structures of different sizes must fail at the same nominal stress. By convention, this came to be known as the case of no size effect. In LEFM, on the other hand, s = - 1 / 2 (for nominal strength), provided that the geometrically similar structures with geometrically similar cracks or notches are considered. This may be generally demonstrated with the help of Rice's J-integral [8]. If log ~N is plotted versus log D, the power law is a straight line (Figure 1.3.1b). For plasticity, or elasticity with a strength limit, the exponent of the power law vanishes, i.e., the slope of this line is 0, while for LEFM the slope is -112 [8]. An emerging "hot" subject is the quasi-brittle materials and structures, for which the size effect bridges these two power laws.
38
Ba~.ant
It is interesting to note that the critical stress for elastic buckling of beams, frames, and plates exhibits also no size effect, i.e., is the same for geometrically similar structures of different sizes. However, this is not true for beams on elastic foundation [ 16].
1.3.4 WEIBULL STATISTICAL SIZE EFFECT The classical theory of size effect has been statistical. Three-dimensional continuous generalization of the weakest link model for the failure of a chain of links of random strength (Fig. 1.3.2a) leads to the distribution Pf(crs,,r)- 1 - exp [ - f v c[a(Z), oN)]dV(x)] which represents the probability that a structure fails as soon as macroscopic fracture initiates from a microcrack (or a some flaw) somewhere in the structure; t r - stress tensor field just before failure, Z - coordinate vector, V - volume of structure, and c ( t r ) - function giving the spatial concentration of failure probability of material ( - V r l X failure probability of material representative volume Vr) [62]; c(tr),~ ~-~iPl(tri)/Vo where oi -- principal
P1
) ) ) ) ) ) ) ) ) ) )
I
i
t
/
I ,,/
I
t
,1
~1
I
t I i 9 i !
I0
I \ I i , i I
-
I
'
ii
I
i l
I
", "
'
\ "
O
FIGURE 1.3.2 a. Left: Chain with many links of random strength, b. Right top: Failure probability of a small element, c. Right bottom: Structures with many microcracks of different probabilities to become critical.
39
1.3 Size Effect on Structural Strength
stresses (i = 1,2,3) and Pl(cr)= failure probability (cumulative) of the smallest possible test specimen of volume V0 (or representative volume of the material) subject to uniaxial tensile stress a;
Pl ( a )
--
a
--
au
SO
(4)
[113] where m, so, O"1 = material constants (m = Weibull modulus, usually between 5 and 50; so = scale parameter; a~ = strength threshold, which may usually be taken as 0) and V0--reference volume understood as the volume of specimens on which c(a) was measured. For specimens under uniform uniaxial stress (and a, = 0), Eqs. 3 and 4 lead to the following simple expressions for the mean and coefficient of variation of the nominal strength: 8"N -- s0r(1 + m-~)(Vo/V) 1/m
(Sa)
co -- [F(1 4- 2m-~)/F2(1 + m -~) - 1] 1/2
(sb)
where F is the gamma function. Since co depends only on m, it is often used for determining m from the observed statistical scatter of strength of identical test specimens. The expression for ~'N includes the effect of volume V which depends on size D. In general, for structures with nonuniform multidimensional stress, the size effect of Weibull theory (for err ~ 0) is of the type
aN ~ D -nalm
(6)
where nd= 1, 2, or 3 for uni-, two- or three-dimensional similarity. In view of Eq. 5, the value a w - aN(V/Vo) ~/~ for a uniformly stressed specimen can be adopted as a size-independent stress measure called the Weibull stress. Taking this viewpoint, Beremin [37] proposed taking into account the nonuniform stress in a large crack-tip plastic zone by the socalled Weibull stress:
O'W
_
Ii V0
(r)
where Vi (i - 1, 2 , . . . Nw) are elements of the plastic zone having maximum principal stress o'ii. Ruggieri and Dodds [101] replaced the sum in Eq. 5 by an integral; see also Lei et al. [77]. Equation 7, however, considers only the crack-tip plastic zone whose size which is almost independent of D. Consequently, Eq. 7 is applicable only if the crack at the moment of failure is not yet macroscopic, still being negligible compared to structural dimensions.
40
Ba~.ant
As far as quasi-brittle structures are concerned, applications of the classic Weibull theory face a number of serious objections: 1. The fact that in Eq. 6 the size effect is a power law implies the absence of any characteristic length. But this cannot be true if the size of material inhomogeneities is not negligible compared to the structure size D. 2. The energy release due to stress redistributions caused by macroscopic FPZ or stable crack growth before Pmax gives rise to a deterministic size effect which is ignored. Thus the Weibull theory is valid only if the structure fails as soon as a microscopic crack becomes macroscopic. 3. Every structure is mathematically equivalent to a uniaxially stressed bar (or chain, Fig. 1.3.2), which means that no information on the structural geometry and failure mechanism is taken into account [36]. 4. The size effect differences between two- and three-dimensional similarity (na = 2 or 3) are predicted much too large. 5. Many tests of quasi-brittle materials (e.g., diagonal shear failure of reinforced concrete beams) show a much stronger size effect than predicted by the Weibull theory ([BP]). (Please see the review [9]). 6. The classical theory neglects the spatial correlations of material failure probabilities of neighboring elements caused by nonlocal properties of damage evolution (while generalizations based on some phenomenological load-sharing hypotheses have been divorced from mechanics). 7. When Eq. 5 is fitted to the test data on statistical scatter for specimens of one size (V = const.) and when Eq. 6 is fitted to the mean test data on the effect of size or V (of unnotched plain concrete specimens), the optimum values of Weibull exponent m are very different, namely, m = 12 and m = 24, respectively [28]. If the theory were applicable, these values would have to coincide. In view of these limitations, among concrete structures Weibull theory appears applicable to some extremely thick plain (unreinforced) structures, e.g., the flexure of an arch dam acting as a horizontal beam (but not for vertical bending of arch dams or gravity dams because large vertical compressive stresses cause long cracks to grow stably before the maximum load). Most other plain concrete structures are not thick enough to prevent the deterministic size effect from dominating. Steel or fiber reinforcement prevents it as well. 1.3.5 QUASI-BRITTLE SIZE EFFECT BRIDGING P L A S T I C I T Y A N D LEFM, A N D ITS H I S T O R Y Qausi-brittle materials are those that obey on a small scale the theory of plasticity (or strength theory), characterized by material strength or yield
41
1.3 Size Effect on Structural Strength
limit ~r0, and on a large scale the LEFM, characterized by fracture energy Gf. While plasticity alone, as well as LEFM alone, possesses no characteristic length, the combination of both, which must be considered for the bridging of plasticity and LEFM, does. Combination of or0 and Gf yields Irwin's (1958) [69a] characteristic length (material length): g0 = EGy
~
(8)
which approximately characterizes the size of the FPZ (E = Young's elastic modulus). So the key to the deterministic quasi-brittle size effect is a combination of the concept of strength or yield with fracture mechanics. In dynamics, this further implies the existence of a characteristic time (material time): ~o = g o / v
(9)
representing the time a wave of velocity v takes to propagate the distance g0. After LEFM was first applied to concrete [72], it was found to disagree with test results [74, 78,111,112]. Leicester [78] tested geometrically similar notched beams of different sizes, fitted the results by a power law, aN o( D-", and observed that the optimum n was less than 1/2, the value required by LEFM. The power law with a reduced exponent of course fits the test data in the central part of the transitional size range well but does not provide the bridging of the ductile and LEFM size effects. An attempt was made to explain the reduced exponent value by notches of a finite angle, which, however, is objectionable for two reasons: (i) notches of a finite angle cannot propagate (rather, a crack must emanate from the notch tip), and (ii) the singular stress field of finite-angle notches gives a zero flux of energy into the notch tip. Like Weibull theory, Leicester's power law also implied the nonexistence of a characteristic length (see Ba~ant and Chen [18], Eqs. 1-3), which cannot be the case for concrete because of the large size of its inhomogeneities. More extensive tests of notched geometrically similar concrete beams of different sizes were carried out by Walsh [111, 112]. Although he did not attempt a mathematical formulation, he was first to make the doubly logarithmic plot of nominal strength versus size and observe that it is was transitional between plasticity and LEFM. An important advance was made by Hillerborg et al. [68] (also Peterson [93]). Inspired by the softening and plastic FPZ models of Barenblatt [2, 3] and Dugdale [55], they formulated the cohesive (or fictitious) crack model characterized by a softening stress-displacement law for the crack opening and showed by finite element calculations that the failures of unnotched plain concrete beams in bending exhibit a deterministic size effect, in agreement with tests of the modulus of rupture.
42
Ba~.ant
Analyzing distributed (smeared) cracking damage, Ba~.ant [4] demonstrated that its localization into a crack band engenders a deterministic size effect on the postpeak deflections and energy dissipation of structures. The effect of the crack band is approximately equivalent to that of a long fracture with a sizable FPZ at the tip. Subsequently, using an approximate energy release analysis, Ba~.ant [5] derived for the quasi-brittle size effect in structures failing after large stable crack growth the following approximate size effect law: aN -- Bao 1 4- Doo
or more generally"
-FaR
aN - Ba0 1 +
(10)
-FaR
(11)
in which r, B = positive dimensionless constants; Do = constant representing the transitional size (at which the power laws of plasticity and LEFM intersect); and Do and B characterize the structure geometry. Usually constant aR = 0, except when there is a residual crack-bridging stress ar outside the FPZ (as in fiber composites), or when at large sizes some plastic mechanism acting in parallel emerges and becomes dominant (as in the Brazilian splitcylinder test). Equation 10 was shown to be closely followed by the numerical results for the crack band model [4, 30] as well as for the nonlocal continuum damage models, which are capable of realistically simulating the localization of strainsoftening damage and avoiding spurious mesh sensitivity. Beginning in the mid-1980s, the interest in the quasi-brittle size effect of concrete structures surged enormously and many researchers made noteworthy contributions, including Planas and Elices [94-96], Petersson [93], and Carpinteri [41]. The size effect has recently become a major theme at conferences on concrete fracture [7, 35, 86, 87,120]. Measurements of the size effect on Pmax were shown to offer a simple way to determine the fracture characteristics of quasi-brittle materials, including the fracture energy, the effective FPZ length, and the (geometry dependent) R-curve.
1.3.6 SIZE EFFECT MECHANISM: STRESS REDISTRIBUTION AND ENERGY RELEASE Let us now describe the gist of the deterministic quasi-brittle size effect. LEFM applies when the FPZ is negligibly small compared to structural dimension D and can be considered as a point. Thus the LEFM solutions can be obtained by methods of elasticity. The salient characteristic of quasi-brittle
43
1.3 Size Effect on Structural Strength
materials is that there exists a sizable FPZ with distributed cracking or other softening damage that is not negligibly small compared to structural dimension D. This makes the problem nonlinear, although approximately equivalent LEFM solutions can be applied unless FPZ reaches near the structure boundaries. The existence of a large FPZ means that the distance between the tip of the actual (traction-free) crack and the tip of the equivalent LEFM crack at Pmax is equal to a certain characteristics length cf (roughly one half of the FPZ size) that is not negligible compared to D. This causes a non-negligible macroscopic stress redistribution with energy release from the structure. With respect to the fracture length a0 (distance from the mouth of notch or crack to the beginning of the FPZ), two basic cases may now be distinguished: (i) a0 = 0, which means that Pmax o c c u r s at the initiation of macroscopic fracture propagation, and (ii) a0 is finite and not negligible compared to D, which means that Pmax o c c u r s after large stable fracture growth.
1.3.6.1
SCALING FOR FAILURE AT CRACK INITIATION
An example of the first case is the modulus of rupture test, which consists in the bending of a simply supported beam of span L with a rectangular cross section of depth D and width b, subjected to concentrated load P; the maximum load is not decided by the stress al - 3PL/2bD 2 at the tensile face, but by the stress value 8 roughly at distance cf/2 from the tensile face / (which is at the middle of FPZ). Because 8 - al - a ] c f / 2 where a x - stress g r a d i e n t - 2 a l / D , and also because ~ - a - i n t r i n s i c tensile strength of the material, the failure condition F - a0 yields P/bD - aN -- a0/(1 - Db/D) where D b - (3L/2D)cf, which is a constant because for geometrically similar beams L / D - constant. This expression, however, is unacceptable for D _< Db. But since the derivation is valid only for small enough cf/D, one may replace it by the following asymptotically equivalent size effect formula:
aN -- a0
1+
(12)
which happens to be acceptable for the whole range of D (including D --~ 0); r is any positive constant. The values r = 1 or 2 have been used for concrete [12], while r ~ 1.45 is optimum according to Ba~-ant and Novfik's latest analysis of test data at Northwestern University [28, 29].
44
Ba~.ant
1 . 3 . 6 . 2 SCALING FOR FAILURES WITH A LONG CRACK OR NOTCH Let us now give a simple explanation of the second case of structures failing only after stable formation of large cracks, or notched fracture specimens. Failures of this type, exhibiting a strong size effect ([BP], [19, 21, 65, 69, 83, 104, 110]) are typical of reinforced concrete structures or fiber composites [119], and are also exhibited by some unreinforced structures (e.g., dams, due to the effect of vertical compression, or floating ice plates in the Arctic). Consider the rectangular panel in Figure 1.3.3, which is initially under a uniform stress equal to ~rN. Introduction of a crack of length a with a FPZ of a certain length and width h may be approximately imagined to relieve the stress, and thus release the strain energy, from the shaded triangles on the flanks of the crack band shown in Figure 1.3.3. The slope k of the effective boundary of the stress relief zone need not be determined; what is important is that k is independent of the size D. For the usual ranges of interest, the length of the crack at maximum load may normally be assumed approximately proportional to the structure size D, while the size h of the FPZ is essentially a constant, related to the inhomogeneity size in the material. This has been verified for many cases by experiments (showing similar failure modes for small and large specimens) and finite element solutions based on crack band, cohesive, or nonlocal models. The stress reduction in the triangular zones of areas ka2/2 (Fig. 1.3.3) causes (for the case b - 1) the energy release Ida- 2x(ka2/2)cr2/2E. The stress drop within the crack band of width h causes further energy release
._o2 L_o -~ h ~--
FIGURE 1.3.3 Approximatezones of stress relief due to fracture.
1.3 Size Effect on Structural Strength
45
Ub- ha~2/E. The total energy dissipated by the fracture is W - aGf, where Gf is the fracture energy, a material property representing the energy dissipated per unit area of the fracture surface. Energy balance during static failure requires that O(Ua+ Ub)/Oa- aw/aa. Setting a - D(a/D) where a/D is approximately a constant if the failures for different structure sizes are geometrically similar, the solution of the last equation for gma;N yields Ba~ant's [5] approximate size effect law in Eq. 10 with ~ R - 0 (Fig. 1.3.1 c). More rigorous derivations of this law, applicable to arbitrary structure geometry, have been given in terms of asymptotic analysis based on equivalent LEFM [10] or on Rice's path-independent J-integral [32]. This law has also been verified by nonlocal finite element analysis and by random particle (or discrete element) models. The experimental verifications, among which the earliest is found in the famous Walsh's [111, 112] tests of notched concrete beams, have by now become abundant (e.g., Fig. 1.3.4). For very large sizes (D >> Do), the size effect law in Eq. 10 reduces to the power law ~rN c~ D -1/2, which represents the size effect of LEFM (for geometrically similar large cracks) and corresponds to the inclined asymptote of slope - 1 / 2 in Figure 1.3.1c. For very small sizes (D << Do), this law reduces to aN - constant, which corresponds to the horizontal asymptote and means that there is no size effect, as in plastic limit analysis. The ratio f i - D/Do is called the brittleness number of a structure. For fl + oo the structure is perfectly brittle (i.e., follows LEFM), in which case the size effect is the strongest possible, while for fl---+ 0 the structure is nonbrittle (or ductile, plastic), in which case there is no size effect. Quasi-brittle structures are those for which 0.1 _< fi <_ 10, in which case the size effect represents a smooth transition (or interpolation) that bridges the power law size effects for the two asymptotic cases. The law (Eq. 10) has the character of asymptotic matching and serves to provide the bridging of scales. In the quasi-brittle range, the stress analysis is of course nonlinear, calling for the cohesive crack model or the crack band model (which are mutually almost equivalent), or some of the nonlocal damage models. The meaning of the term quasi-brittle is relative. If the size of a quasi-brittle structure becomes sufficiently large compared to material inhomogeneities, the structure becomes perfectly brittle (for concrete structures, only the global fracture of a large dam is describable by LEFM), and if the size becomes sufficiently small, the structure becomes nonbrittle (plastic, ductile) because the FPZ extends over the whole cross section of the structure (thus a micromachine or a miniature electronic device made of silicone or fine-grained ceramic may be quasi-brittle or nonbrittle).
46
Ba~.ant
Limestone
Carbon com )osite
Plasticity
',
o
.
~
.
.
.
.
-0.1
-; -o.2
-0.1
x
\~. BaJant, Gettn and Kozemi ,
0
(1991),,,
C
-0.5
X
\"
~'
E
0 Z Z
0
Daniel and Li
Si02 c e r a m i c
\
l
t
'~
-0.6
0
Sea ice
0 - o
o
2
o .
-1,0
-0.4
McKinney and Rice (1981)
a
B~t,
-0.2 11996),,. -1.2
t-
.~_
',2
o I
-0.5
D = 0.5 - 0.80 rn Dempsey et al.
-1.5 /
I
0.5 0 log D (Specimen size) P
-%
(1995)
40
,
I
80
0 ............. 9 0
""
I
W
-0.1
I i /" "-. ",/I
""'.
2
m
"Kink band
1
-0.2
'",O
-Slanted notch
-0.3 -
-0.25
0
0.25 log (D/Do)
0.50
0.7E
FIGURE 1.3.4 Top: Comparisons of size effect law with Mode 1 test data obtained by various investigators using notched specimens of different materials. Bottom: Size effect in compression kink-band failures of geometrically similar notched carbon-PEEK specimens [ ].
47
1.3 Size Effect on Structural Strength 1.3.6.3
SIZE EFFECT ON POSTPEAK SOFTENING
AND DUCTILITY The plots of nominal stress versus the relative structure deflection (normalized so as to make the initial slope in Figure 1.3.5 size-independent) have, for small and large structures, the shapes indicated in Figure 1.3.5. Apart from the size effect on Pmax, there is also a size effect on the shape of the postpeak descending load-deflection curve. For small structures the postpeak curves descend slowly, for larger structures steeper, and for sufficiently large structures they may exhibit a snapback, that is, a change of slope from negative to positive. If a structure is loaded under displacement control through an elastic device with spring constant Cs, it loses stability and fails at the point where the load-deflection diagram first attains the slope -Cs (if ever); Figure 1.3.5. The ratio of the deflection at these points to the elastic deflection characterizes the ductility of the structure. As is apparent from the figure,
t
t
ON
Small
y
~Large
"~Larger
~ .
,.
I
I
=
Rel.deft.
FIGURE 1.3.5 Load-deflectioncurves of quasi-brittle structures of different sizes, scaled to the same initial slope.
48
Ba~.ant
small quasi-brittle structures have a large ductility, whereas large quasi-brittle structures have small ductility. The areas under the load-deflection curves in Figure 1.3.5 characterize the energy absorption. The capability of a quasi-brittle structure to absorb energy decreases, in relative terms, as the structure size increases. The size effect on energy absorption capability is important for blast loads and impact. The progressive steepening of the postpeak curves in Figure 1.3.5 with increasing size and the development of a snapback can be most simply described by the series coupling model, which assumes that the response of a structure may be at least approximately modeled by the series coupling of the cohesive crack or damage zone with a spring characterizing the elastic unloading of the rest of the structure (Ba~.ant and Cedolin [17], Sec. 13.2).
1 . 3 . 6 . 4 ASYMPTOTIC ANALYSIS OF SIZE EFFECT BY EQUIVALENT LEFM To obtain simple approximate size effect formulae that give a complete prediction of the failure load, including the effect of geometrical shape of the structure, equivalent LEFM may be used. In this approach the tip of the equivalent LEFM (sharp) crack is assumed to lie approximately a distance cf ahead of the tip of the traction-free crack or notch, cf being a constant (representing roughly one half of the length of the FPZ ahead of the tip. Two cases are relatively simple: (i) If a large crack grows stably prior to Pmax o r if there is a long notch,
O'N = and (ii) if
Pmax o c c u r s
+ ~ + '(Czo)D V'g'( o)q + g( o)D
at fracture initiation from a smooth surface +
ON
(13)
+
v/g'(0)cf + g"(0) (c~/2D)
(14)
[10, 12] where the primes denote derivatives; g(~o) -- KII,/aND 2 2 and 7(~0) 2 2 Ki,,/cryD are dimensionless energy release functions of LEFM of e - ao/D where a0 = length of notch or crack up to the beginning of the FPZ; KII,, KI~ = stress intensity factors for load P and for loading by uniform residual crack-bridging stress at, respectively; cry > 0 for tensile fracture, but a v r 0 in the case of compression fracture in concrete, kink band propagation in fiber composites, and tensile fracture of composites reinforced by fibers short
49
1.3 Size Effect on Structural Strength
enough to undergo frictional pullout rather than breakage. The asymptotic behavior of Eq. 13 for D - + oo is of the LEFM type, a N - Cry o( D -1/2 Equation 14 approaches for D ---+ oo a finite asymptotic value. So does Eq. 13 if fir > 0.
1.3.6.5
SIZE EFFECT METHOD FOR MEASURING
MATERIAL CONSTANTS AND R-CURVE Comparison of Eq. 13 with Eq. 10 yields the relations: Do -- cfg'(a0)/g(0~0)
B~ro -- Crov/EGf/cfg'(cZo)
(15)
Therefore, by fitting Eq. 10 with aR = 0 to the values of aN measured on test specimens of different sizes with a sufficiently broad range of brittleness numbers fl = D/Do, the values of Gf and cf can be identified [20,31]. The fitting can be done best by using the Levenberg-Marquardt nonlinear optimization algorithm, but it can also be accomplished by a (properly weighted) linear regression of cr~2 versus D. The specimens do not have to be geometrically similar, although when they are the evaluation is simpler and the error smaller. The lower the scatter of test results, the narrower is the minimum necessary range of fl (for concrete and fiber composites, the size range 1:4 is the minimum). The size effect method of measuring fracture characteristics has been adopted for an international standard recommendation for concrete ([99], [BP] Sec. 6.3), and has also been verified and used for various rocks, ceramics, orthotropic fiber-polymer composites, sea ice, wood, tough metals, and other quasi-brittle materials. The advantage of the size effect method is that the tests, requiring only the maximum loads, are foolproof and easy to carry out. With regard to the cohesive crack model, note that the size effect method gives the energy value corresponding to the area under the initial tangent of the softening stress-displacement curve, rather than the total area under the curve. The size effect method also permits determining the R-curve (resistance curve) of the quasi-brittle material a curve that represents the apparent variation of fracture energy with crack extension for which LEFM becomes approximately equivalent to the actual material with a large FPZ. The R-curve, which (in contrast to the classical R-curve definition) depends on the specimen geometry, can be obtained as the envelope of the curves of the energy release rate at P = Pmax (for each size) versus the crack extension for specimens of various sizes. In general, this can easily be done numerically,
50
Ba~.ant
and if the size effect law has the form in Eq. 10 with err = 0, a parametric analytical expression for the R-curve exists ([20], [BP] Sec. 6.4). The fracture model implied by the size effect law in Eq. 10 or Eq. 13 has one independent characteristic length, cf, representing about one half of the FPZ length. Because of Eq. 15, the value of g0 is implied by cf if ~r0 is known. The value of cf controls the size Do at the center of the bridging region (intersection of the power-law asymptotes in Figure 1.3.1c, and cr0 or Gf controls a vertical shift of the size effect curve at constant Do. The location of the large-size asymptote depends only on Kc and geometry, and the location of the small-size asymptote depends only on or0 and geometry.
1.3.6.6
CRITICAL CRACK-TIP OPENING
DISPLACEMENT~
~CTOD
The quasi-brittle size effect, bridging plasticity and LEFM, can also be simulated by the fracture models characterized by the critical stress intensity factor Kc (fracture toughness) and 6crop; for metals see Wells [117] and Cottrell [50], and for concrete Jenq and Shah [70]. Jenq and Shah's model, called the two-parameter fracture model, has been shown to give essentially the same results as the R-curve derived from the size effect law in Eq. 10 with Cra = 0. The models are in practice equivalent because
Kc- V/-~
~CrOD--(llrc)v/8GfcflE
(16)
Using these formulae, the values of Kc and 6C~ODcan be easily identified by fitting the size effect law (Eq. 10) to the measured Pmaxvalues. Like the size effect law in Eq. 10 with ~rR = 0, the two-parameter model has only one independent characteristic length, g0 - Kc2/~ 9If cr0 is known, then 6crop is not an independent length because cf is implied by g0 and 6crop then follows from Eq. 16.
1.3.7 E X T E N S I O N S , R A M I F I C A T I O N S , AND APPLICATIONS 1.3.7.1
SIZE EFFECTS IN COMPRESSION FRACTURE
Loading by high compressive stress without sufficient lateral confining stresses leads to damage in the form of axial splitting microcracks engendered
1.3 Size Effect on Structural Strength
51
by pores, inclusions, or inclined slip planes. This damage localizes into a band that propagates either axially or laterally. For axial propagation, the energy release from the band drives the formation of the axial splitting fracture, and since this energy is proportional to the length of the band, there is no size effect. For lateral propagation, the stress in the zones on the sides of the damage band gets reduced, which causes an energy release that grows in proportion to D 2, while the energy consumed and dissipated in the band grows in proportion to D. The mismatch of energy release rates inevitably engenders a deterministic size effect of the quasi-brittle type, analogous to the size effect associated with tensile fracture. In consequence of the size effect, failure by lateral propagation must prevail over the failure by axial propagation if a certain critical size is exceeded. The size effect can again be approximately described by the equivalent LEFM. This leads to Eq. 13 in which err is determined by analysis of the microbuckling in the laterally propagating band of axial splitting cracks. The spacing s of these cracks is in Eq. 13 assumed to be dictated by material inhomogeneities. However, if the spacing is not dictated and is such that it minimizes erN, then the size effect gets modified as erN m C D - 2 / 5 q_ eroo
(17)
([BP] Sec. 10.5.11) where C, eroo = constants, the approximate values of which have been calculated for the breakout of boreholes in rock.
1 . 3 . 7 . 2 FRACTURING TRUSS MODEL FOR CONCRETE AND BOREHOLES IN ROCK Propagation of compression fracture is what appears to control the maximum load in diagonal shear failure of reinforced concrete beams with or without stirrups, for which a very strong size effect has been demonstrated experimentally [9,21, 69, 71, 91, 98,104, 109,110]. A long diagonal tension crack grows stably under shear loading until the concrete near its tip gets crushed as a result of parallel compression stresses. A simplified formula for the size effect can be obtained by energetic modification of the classical truss model (strut-and-tie model) [9]. The explosive breakout of boreholes (or mining stopes) in rock under very high pressures is known to also exhibit size effect, as revealed by the tests of Carter [47], Carter et al. [48], Haimson and Herrick [67], and Nesetova and Lajtai [90]. An approximate analytical solution can be obtained by exploiting Eschelby's theorem for eigenstresses in elliptical inclusions [27].
52 1.3.7.3
Ba~.ant
KINK BANDS IN FIBER COMPOSITES
A kink band, in which axial shear-splitting cracks develop between fibers which undergo microbuckling, is one typical mode of compression failure of composites or laminates with uniaxial fiber reinforcement. This failure mode, whose theory was begun by Rosen [100] and Argon [1], was until recently treated by the theory of plasticity, which implies no size effect. Recent experimental and theoretical studies [40], however, revealed that the kink band propagates sideway like a crack and the stress on the flanks of the band gets reduced to a certain residual value, which is here denoted as t~y and can be estimated by the classical plasticity approach of Budiansky [39]. The cracklike behavior implies a size effect, which is demonstrated (Ba~.ant et al. [22, 24]) by the latest laboratory tests of notched carbon-PEEK specimens (Fig. 1.3.4); these tests also demonstrated the possibility of a stable growth of a long kink band, which was achieved by rotational restraint at the ends). There are again two types of size effect, depending on whether Pmax is reached (i) when the FPZ of the kink band is attached to a smooth surface or (ii) or when there exists either a notch or a long segment of kink band in which the stress has been reduced to ay. Equations 13 and 14, respectively, approximately describe the size effects for these two basic cases; in this case Gf now plays the role of fracture energy of the kink band (area below the stress-contraction curve of the kink bank and above the ay value), and cf the role of the FPZ of the kink band, which is assumed to be approximately constant, governed by material properties. The aforementioned carbon-PEEK tests also confirm that case (ii), in which a long kink band grows stably prior to Pm~x, is possible (in those tests, this is by virtue of a lateral shift of the compression resultant in wide notched prismatic specimens with ends restrained against rotation).
1.3.7.4
SIZE EFFECTS IN SEA ICE
Normal laboratory specimens of sea ice exhibit no notch sensitivity. Therefore, the failure of sea ice has been thought to be well described by plastic limit analysis, which exhibits no size effect [73,106]. This perception, however, changed drastically after Dempsey carried out in 1993 on the Arctic Ocean size effect tests of floating notched square specimens with an unprecedented, record-breaking size range (with square sides ranging from 0.5 m to 80m!) [52, 53, 89]. It is now clear that floating sea ice plates are quasi-brittle and their size effect on the scale of 100 m approaches that of LEFM. Among other things,
53
1.3 Size Effect on Structural Strength
Dempsey's major experimental result explains why the measured forces exerted by moving ice on a fixed oil platform are one to two orders of magnitude smaller than the predictions of plastic limit analysis based on the laboratory strength of ice. The size effect law in Eq. 10 with aR = 0, or in Eq. 13 (with ar = 0), agree with these results well, permitting the values of Gf and cf of sea ice to be extracted by linear regression of the Pmax data. The value of cf is in the order of meters (which can be explained by inhomogeneities such as brine pockets and channels, as well as preexisting thermal cracks, bottom roughness of the plate, warm and cold spots due to alternating snow drifts, etc.). Information on the size effect in sea ice can also be extracted from acoustic measurements [80]. Rapid cooling in the Arctic can produce in the floating plate bending moments large enough to cause fracture. According to plasticity or elasticity with a strength limit, the critical temperature difference A Tcr across the plate would have to be independent of plate thickness D. Fracture analysis, however, indicated a quasi-brittle size effect. Curiously, its asymptotic form is not ATcr O( D-l/2 but ATcr o( D -3/8
(18)
The reason is that D is not a characteristic dimension in the plane of the boundary value problem of plate bending; rather, it is the flexural wavelength of a plate on elastic foundation, which is proportional t o D 4/3 rather than D. It seems that Eq. 18 may explain why long cracks of length 10 to 100 km, which suddenly form in the fall in the Arctic ice cover, often run through thick ice floes and do not follow the thinly refrozen water leads around the floes. In analyzing the vertical penetration of floating ice plate (load capacity for heavy objects on ice, or the maximum force P required for penetration from below), one must take into account that bending cracks are reached only through part of the thickness, their ligaments transmitting compressive forces, which produces a dome effect. Because, at maximum load, this part-through bending crack (of a variable depth profile) is growing vertically, the asymptotic size effect is n o t P / D 2 -- fiN OC. D -3/8 [105] but aN (x D -1/2. This was determined by a simplified analytical solution (with a uniform crack depth) by Dempsey et al. [54], and confirmed by a detailed numerical solution with a variable crack depth profile [23]. The latter also led to an approximate prediction formula for the entire practical range of D, which is of the type of Eq. 10 with aN = 0. This formula was shown to agree with the existing field tests [59, 60, 81].
54
Ba~.ant
1 . 3 . 7 . 5 REVERSE SIZE EFFECT IN BUCKLING OF FLOATING ICE An interesting anomalous case is the size effect on the critical stress for elastic buckling of floating ice, i.e., a beam or plate on Winkler foundation. Consider floating ice pushing against an obstacle of size d in the horizontal direction. Dimensional analysis [102] suffices to determine the form of the buckling formula and the scaling. There are five variables in the problem, h = ice plate thickness, P~, Et, p, h, d, and the solution must be have the form F(Pcr,Et, p,h,d,)=O, where Per = force applied on the obstacle, p = specific weight of sea water (or foundation modulus), and E ' = E/(1 - v2), v being the Poisson ratio. There are, however, only two independent physical dimensions in the problem, namely, the length and the force. Therefore, according to Buckingham's 1-I theorem of dimensional analysis [102], the solution must be expressible in terms of 5 - 2 , i.e., 3 dimensionless parameters. They may be taken as Po./E'hd, x/~/E'h, and d/h, where D = Eth3/12 = cylindrical stiffness of the ice plate. If the ice is treated as elastic, P~r/E~hd must be proportional to x/-~/E~h and d/h. Denoting aNcr = Pcr/hd which represents the nominal buckling strength (or the average critical stress applied on the obstacle by the moving ice plate), we conclude that the buckling solution must have the form
rrN~- K(d/h) X / ~ ~
(19)
where K is a dimensionless parameter depending on d/h as well as the boundary conditions. The interesting property of Eq. 19 is that aNc~ increases, rather than decreases, with ice thickness h. So there is a reverse size effect. Consequently, the buckling of the ice plate can control the force exerted on a stationary structure only when the plate is sufficiently thin. The reason for the reverse size effect is that the buckling wavelength (the distance between the inflexion points of the deflection profile), which is Lcr- g(D/P) 1/4 (as follows from dimensional analysis or nondimensionalization of the differential equation of plate buckling), is not proportional to h; rather, Lo./hoch -1/4, i.e., Lo. decreases with h. This contrasts with the structural buckling problems of columns, frames, and plates, in which Lcr is proportional to the structure size. Although the axisymmetric buckling of a cylindrical shell under axial compression is a problem analogous to the beam on elastic foundation, the reverse size effect does not exist because the equivalent foundation modulus is not a constant. Therefore, Eq. (refl-cr) must apply to it as well. Since the lowest critical stress for nonaxisymmetric buckling loads is nearly equal to
1.3 Size Effect on Structural Strength
55
that for the axisymmetric mode, the reverse size effect given by Eq. 19 must also apply.
1 . 3 . 7 . 6 INFLUENCE OF CRACK SEPARATION RATE, CREEP~ AND VISCOSITY There are two mechanisms in which the loading rate affects fracture growth: (i) creep of the material outside the FPZ, and (ii) rate dependence of the severance of material bonds in the FPZ. The latter may be modeled as a rate process controlled by activation energy, with Arrhenius-type temperature dependence. This leads to a dependence of the softening stress-separation relation of the cohesive crack model on the rate of opening displacement. In an equivalent LEFM approach, the latter is modeled by considering the crack extension rate to be a power function of the ratio of the stress intensity factor to its critical R-curve value. For quasi-brittle materials exhibiting creep (e.g., concretes and polymer composites, but not rocks or ceramics), the consequence of mechanism 1 (creep) is that a decrease of loading rate, or an increase of duration of a sustained load, causes a decrease of the effective length of the FPZ. This in turn means an increase of the brittleness number manifested by a leftward rigid-body shift of the size effect curve in the plot of log aN versus log D, i.e., a decrease of effective Do. For slow or long-time loading, quasi-brittle structures become more brittle and exhibit a stronger size effect [26]. Mechanism 2 (rate dependence of separation) causes it to happen that an increase of loading rate, or a decrease of sustained load duration, leads to an upward vertical shift of the size effect curve for log CrNbut has no effect on Do and thus on brittleness (this mechanism also explains an interesting recently discovered phenomenon - - a reversal of softening to hardening after a sudden increase of the loading rate, which cannot be explained by creep). So far all our discussions have dealt with statics. In dynamic problems, any type of viscosity r/of the material (present in models for creep, viscoelasticity, or viscoplasticity) implies a characteristic length. Indeed, since q = stress/strain rate ~ k g / m s, and the Young's modulus E and mass density p have dimensions E ,-~ k g / m s 2 and p ,-~ k g / m 3, the material length associated with viscosity is given by
_. G
vp
v -
(20)
where v = w a v e velocity. Consequently, any rate dependence in the constitutive law implies a size effect (and a nonlocal behavior as well). There
56
Ba~.ant
is, however, an important difference. Unlike the size effect associated with g0 or cf, the viscosity-induced size effect (as well as the width of damage localization zones) is not time-independent. It varies with the rates of loading and deformation of the structure and vanishes as the rates drop to zero. For this reason, an artificial viscosity or rate effect can approximate the nonviscous size effect and localization only within a narrow range of time delays and rates, but not generally.
1.3.7.7
SIZE EFFECT IN FATIGUE CRACK GROWTH
Cracks slowly grow under fatigue (repeated) loading. This is for metals and ceramics described by the Paris (or Paris-Erdogan) law, which states that plot of the logarithm of the crack length increment per cycle versus the amplitude of the stress intensity factor is a rising straight line. For quasi-brittle material it turns out that a size increase causes this straight line to shift to the right, the shift being derivable from the size effect law in Eq. 10 ([BP] Sec. 11.7).
1 . 3 . 7 . 8 SIZE EFFECT FOR COHESIVE CRACK MODEL AND CRACK BAND MODEL The cohesive (or fictitious) crack model (called by Hillerborg et al. [68] and Petersson [93] the fictitious crack model) is more accurate yet less simple than the equivalent LEFM. It is based on the hypothesis that there exists a unique decreasing function w - g(~) relating the crack opening displacement w (separation of crack faces) to the crack bridging stress cr in the FPZ. The obvious way to determine the size effect is to solve Pmax by numerical integration for step-by-step loading [93]. The size effect plot, however, can be solved directly if one inverts the problem, searching the size D for which a given relative crack length 0~- a/D corresponds to Pmax. This leads to the equations [25, 79] D
P max
--
the first represents an eigenvalue problem for a homogeneous Fredholm integral equation, with D as the eigenvalue and v({) as the eigenfunction; - x / D , x - coordinate along the crack (Fig. 1.3.6); ~ -- a/D, O~o-- ao/D;
where
57
1.3 Size Effect on Structural Strength
a, a0 = total crack length and traction-free crack length (or notch length); and C ~(~, ~'), C ~P(~) = compliance functions of structure for crack surface force and given load P. Choosing a sequence of a-values, for each one of them obtains from Eq. 21 the corresponding values of D and Pmax. These results have also been generalized to obtain directly the load and displacement corresponding, on the load-deflection curve, to a point with any given tangential stiffness, including the displacement at the snapback point which characterizes the ductility of the structure. The cohesive crack model possesses at least one, but for concrete typically two, independent characteristic lengths: go - EGf/~2 and *fl - EGF/a02 where GF -- area under the entire softening stress-displacement curve cr = f(w), and Gf = area under the initial tangent to this curve, which is equal to GF only if the curve is simplified as linear (typically GF~ 2Gf). The bilinear stress-displacement law used for concrete involves further parameters of the length dimension the opening displacement wf when the stress is reduced to zero, and the displacement at the change of slope. Their values are implied by Gf, G> Cro and the stress at the change of slope. The scatter of size effect measurements within a practicable size range (up to 1:30) normally does not permit identifying more than one characteristic length (measurements of postpeak behavior are used for that purpose). Vice versa, when only the maximum loads of structures in the bridging region between plasticity and LEFM are of interest, hardly more than one characteristic length (namely, cf) is needed.
~
P,u
a
. . . . ----
J r
I
,> w
FIGURE 1.3.6
D
Cohesive crack and distribution of crack-bridging cohesive stresses.
58
Ba~.ant
The crack band model, which is easier to implement and is used in commercial codes (e.g., DIANA, SBETA) [49], is, for localized cracking or fracture, nearly equivalent to the cohesive crack model ([BP], [97]), provided that the effective (average) transverse strain in the crack band is taken as ey = w / h where h is the width of the band. All that has been said about the cohesive crack model also applies to the crack band model. Width h, of course, represents an additional characteristic length, ~4 - - h. It matters only when the cracking is not localized but distributed (e.g., due to the effect of dense and strong enough reinforcement), and it governs the spacings of parallel cracks. Their minimum spacing cannot be unambiguously captured by the cohesive crack model.
1 . 3 . 7 . 9 SIZE EFFECT VIA NONLOCAL, GRADIENT~ OR DISCRETE ELEMENT MODELS The hypostatic feature of any model capable of bridging the power law size effects of plasticity and LEFM is the presence of some characteristic length, g. In the equivalent LEFM associated with the size effect law in Eq. 10, cf serves as a characteristic length of the material, although this length can equivalently be identified with 8CrOD in Wells-Cottrell or JenqShah models, or with the crack opening wf at which the stress in the cohesive crack model (or crack band model) is reduced to zero (for size effect analysis with the cohesive crack model, see [BP] and Ba~.ant and Li [251). In the integral-type nonlocal continuum damage models, ~ represents the effective size of the representative volume of the material, which in turn plays the role of the effective size of the averaging domain in nonlocal material models. In the second-gradient nonlocal damage models, which may be derived as an approximation of the nonlocal damage models, a material length is involved in the relation of the strain to its Laplacian. In damage simulation by the discrete element (or random particle) models, the material length is represented by the statistical average of particle size. The existence of g in these models engenders a quasi-brittle size effect that bridges the power-law size effects of plasticity and LEFM and follows closely Eq. 10 with ~rN = 0, as documented by numerous finite element simulations. It also poses a lower bound on the energy dissipation during failure, prevents spurious excessive localization of softening continuum damage, and eliminates spurious mesh sensitivity ([BP], ch. 13). These important subjects will not be discussed here any further because there exists a recent extensive review [18].
59
1.3 Size Effect on Structural Strength
1.3.7.10 NONLOCAL STATISTICALGENERALIZATION OF THE WEIBULL
THEORY
Two cases need to be distinguished: (a) The front of the fracture that causes failure can be at only one place in the structure, or (b) the front can lie, with different probabilities, at many different places. The former case occurs when a long crack whose path is dictated by fracture mechanics grows before the m a x i m u m load, or if a notch is cut in a test specimen. The latter case occurs when the maximum load is achieved at the initiation of fracture growth. In both cases, the existence of a large FPZ calls for a modification of the Weibull concept: The failure probability P1 at a given point of the continuous structure depends not on the local stress at that point, but on the nonlocal strain, which is calculated as the average of the local strains within the neighborhood of the point constituting the representative volume of the material. The nonlocal approach broadens the applicability of the Weibull concept to the case of notches or long cracks, for which the existence of crack-tip singularity causes the classical Weibull probability integral to diverge at realistic m-values (in cleavage fracture of metals, the problem of crack singularity has been circumvented differently m by dividing the cracktip plastic zone into small elements and superposing their Weibull contributions [77]). Using the nonlocal Weibull theory, one can show that the proper statistical generalizations of Eq. 10 (with aR = 0 ) and Eq. 12 having the correct asymptotic forms for D---+ oo, D - + 0, and m - + oo are (Fig. 1.3.7): Case (a)" Case (b)"
aN
--
Bo'o(fl2maIm +/~r)-l/2r
aN - o'0~"a/m(1 + r~l-'~a/m) 1/r
f l - D/Do ~ -- Db/D
(22) (23)
where it is assumed that rna< m, which is normally the case. The first formula, which was obtained for r = 1 by Ba~.ant and Xi [36] and refined for n ~ 1 by Planas, has the property that the statistical influence on the size effect disappears asymptotically for large D. The reason is that, for long cracks or notches with stress singularity, a significant contribution to the Weibull probability integral comes only from the FPZ, whose size does not vary much with D. The second formula has the property that the statistical influence asymptotically disappears for small sizes. The reason is that the FPZ occupies much of the structure volume. Numerical analyses of test data for concrete show that the size ranges in which the statistical influence on the size effect in case (a) as well as (b) would be significant do not lie within the range of practical interest. Thus the deterministic size effect dominates and its statistical correction in Eqs. 22 and
60
Ba~.ant z 0")
o
rh.
log D
a Z
D 0
~ . _
m
b
log D
FIGURE 1.3.7 Scaling laws according to nonlocal generalization of Weibull theory for failures after long stable crack growth (top) or a crack initiation (right).
23 may be ignored for concrete, except in the rare situations where the deterministic size effect vanishes, which occurs rarely (e.g., for centric tension of an unreinforced bar).
1.3.8 OTHER SIZE EFFECTS 1 . 3 . 8 . 1 HYPOTHESIS OF FRACTAL ORIGIN OF SIZE EFFECT The partly fractal nature of crack surfaces and of the distribution of microcracks in concrete has recently been advanced as the physical origin of the size effects observed on concrete structures. Bhat [38] discussed a possible role of fractality in size effects in sea ice. Carpinteri [43, 44], Carpinteri et al. [45], and Carpinteri and Chiaia [46] proposed the so-called multifractal scaling law (MFSL) for failures occurring at fracture initiation from a smooth
61
1.3 Size Effect on Structural Strength
surface, which reads GN =
v/A1 q-(A2/D)
(24)
where A1, A2= constants. There are, however, four objections to the fractal theory [11 ]: (i) A mechanical analysis (of either invasive or lacunar fractals) predicts a different size effect trend than Eq. 24, disagreeing with experimental observations. (ii) The fractality of the final fracture surface should not matter because typically about 99% of energy is dissipated by microcracks and frictional slips on the sides of this surface. (iii) The fractal theory does not predict how A1 and A2 should depend on the geometry of the structure, which makes the MFSL not too useful for design application. (iv) The MFSL is a special case of the second formula in Eq. 12 for r = 2, which logically follows from fracture mechanics;
A1 - EGf /cfg'(O)
A2 -- -EGfg"(O)/2cf[g'(O)] 3
(25)
[12]. Unlike fractality, the fracture explanation of Eq. 24 has the advantage that, by virtue of these formulae, the geometry dependence of the size effect coefficients can be determined.
1.3.8.2
BOUNDARY LAYER, SINGULARITY,
AND DIFFUSION Aside from the statistical and quasi-brittle size effects, there are three further types of size effect that influence the nominal strength: 1. The boundary layer effect, which is due to material heterogeneity (i.e., the fact that the surface layer of heterogeneous material such as concrete has a different composition because the aggregates cannot protrude through the surface), and to the Poisson effect (i.e., the fact that a plane strain state on planes parallel to the surface can exist in the core of the test specimen but not at its surface). 2. The existence of a three-dimensional stress singularity at the intersection of crack edge with a surface, which is also caused by the Poisson effect ([BP], Sec. 1.3). This causes the portion of the FPZ near the surface to behave differently from that in the interior. 3. The time-dependent size effects caused by diffusion phenomena such as the transport of heat or the transport of moisture and chemical agents in porous solids (this is manifested, e.g., in the effect of size on shrinkage and drying creep, due to size dependence of the drying half-time) and its effect on shrinkage cracking [96].
62
Ba~.ant
1.3.9 CLOSING REMARKS Substantial though the recent progress has been, the undersl[anding of the scaling problems of solid mechanics is nevertheless far from complete. Mastering the size effect that bridges different behaviors on adjacent scales in the microstructure of material will be contingent upon the development of realistic material models that possess a material length (or characteristic length). The theory of nonlocal continuum damage will have to move beyond the present phenomenological approach based on isotropic spatial averaging, and take into account the directional and tensorial interactions between the effects causing nonlocality. A statistical description of such interactions will have to be developed. Discrete element models of the microstructure of fracturing or damaging materials will be needed to shed more light on the mechanics of what is actually happening inside the material and separate the important processes from the unimportant ones.
ACKNOWLEDGMENT Preparation of the present review article was supported by the Office of Naval Research under Grant N00014-91-J-1109 to Northwestem University, monitored by Dr. Yapa D. S. Rajapakse.
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63
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90. 91.
92. 93.
94.
95. 96.
97. 98. 99. 100.
101.
102. 103. 104.
105. 106. 107. 108. 109.
67
and Materials Summer Conf., held at University of California, Los Angeles, June), AMD-MD '95, New York: Am. Soc. of Mech. Engrs. Nesetova, V., and Lajtai, E. Z. (1973). Fracture from compressive stress concentration around elastic flaws. Int. J. Rock Mech. Mining Sci. 10: 265-284. Okamura, H., and Maekawa, K. (1994). Experimental study of size effect in concrete structures, in Size Effect in Concrete Structures, pp. 3-24, H. Mihashi, H. Okamura, and Z. P. Ba~.ant, eds., London: E & FN Spon (Proc. of JCI Intern. Workshop held in Sendai, Japan, 1993). Peirce, E T. (1926). J. Textile Inst. 17: 355. Petersson, P.E. (1981). Crack growth and development of fracture zones in plain concrete and similar materials. Report TVBM-1006, Div. of Building Materials, Lund Inst. of Tech., Lund, Sweden. Planas, J., and Elices, M. (1988). Conceptual and experimental problems in the determination of the fracture energy of concrete. Proc. Int. Workshop on Fracture Toughness and Fracture Energy, Test Methods of Concrete and Rock. Tohoku Univ., Sendai, Japan, pp. 203-212. Planas, J., and Elices, M. (1989). In Cracking and Damage, pp. 462-476, J. Mazars and Z. P. Ba~.ant, eds., London: Elsevier. Planas, J., and Elices, M. (1993). Drying shrinkage effects on the modulus of rupture, in Creep and Shrinkage of Concrete (Proc., 5th Int. RILEM Symp., Barcelona), pp. 357-368, Z. P. Ba~.ant and I. Carol, eds., London: E & FN Spon. Planas, J., Elices, M., and Guinea, G. V. (1983). Cohesive cracks vs. nonlocal models: Closing the gap. Int. J. Fracture 63 (2): 173-187. Reinhardt, H. W. (1981). Massstabseinfluss bei Schubversuchen im Licht der Bruchmechanik. Beton and Stahlbetonbau (Berlin), No. 1, pp. 19-21. RILEM Recommendation (1990). Size effect method for determining fracture energy and process zone of concrete. Materials and Structures 23: 461-465. Rosen, B. W. (1965). Mechanics of composite strengthening, Fiber Composite Materials, Am. Soc. for Metals Seminar, Chapter 3, American Society for Metals, Metals Park, Ohio, pp. 37-75. Ruggieri, C., and Dodds, R. H. (1996). Transferability model for brittle fracture including constraint and ductile tearing effects - - in probabilistic approach, Int. J. Fracture 79: 309-340. Sedov, L. I. (1959). Similarity and Dimensional Methods in Mechanics, New York: Academic Press. Selected Papers by Alfred M. Freudenthal (1981). Am. Soc. of Civil Engrs., New York. Shioya, Y., and Akiyama, H. (1994). Application to design of size effect in reinforced concrete structures, in Size Effect in Concrete Structures (Proc. of Intern. Workshop in Sendai, 1993), pp. 409-416, H. Mihashi, H. Okamura, and Z. P. Ba~ant, eds., London: E & FN Spon. Slepyan, L.I. (1990). Modeling of fracture of sheet ice. Izvestia AN SSSR, Mekh. Tverd. Tela 25 (2): 151-157. Sodhi, D. S. (1995). Breakthrough loads of floating ice sheets. J. Cold Regions Engrg., ASCE 9 (1): 4-20. Tippett, L. H. C. (1925). On the extreme individuals and the range of samples. Biornetrika 17: 364. von Mises. R. (1936). La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalcanique 1: 1. Walraven, J., and Lehwalter (1994). Size effects in short beams loaded in shear. ACI Structural Journal 91 (5): 585-593.
68
Ba~.ant
110. Walraven, J. (1995). Size effects: their nature and their recognition in building codes. Studi e Ricerche (Politecnico di Milano) 16: 113-134. 111. Walsh, P. E (1972). Fracture of plain concrete. Indian Concrete Journal 46, No. 11. 112. Walsh, P. E (1976). Crack initiation in plain concrete. Magazine of Concrete Research 28: 37-41. 113. Weibull, W (1939). The phenomenon of rupture in solids. Proc., Royal Swedish Institute of Engineering Research (Ingenioersvetenskaps Akad. Handl.) 153, Stockholm, 1-55. 114. Weibull, W (1949). A statistical representation of fatigue failures in solids. Proc., Roy. Inst. of Techn. No. 27. 115. Weibull, W. (1951). A statistical distribution function of wide applicability. J Appl. Mech., ASME, 18. 116. Weibull, W. (1956). Basic aspects of fatigue, in Proc., Colloquium on Fatigue, Stockholm: Springer-Verlag. 117. Wells, A. A. (1961). Unstable crack propagation in metals-cleavage and fast fracture. Syrup. on Crack Propagation. Cranfield, 1: 210-230. 118. Williams, E. (1957). Some observations of Leonardo, Galileo, Mariotte and others relative to size effect, Annals of Science 13: 23-29. 119. Wisnom, M. R. (1992). The relationship between tensile and flexural strength of unidirectional composite. J. Composite Materials 26: 1173-1180. 120. Wittmann, E H., ed. (1995). Fracture Mechanics of Concrete Structures (Proc., 2nd Int. Conf. on Fracture Mech. of Concrete and Concrete Structures [FraMCoS-2]), held at ETH, Zurich), pp. 515-534, Freiburg: Aedificatio Publishers. 121. Wittmann, E H., and Zaitsev, Yu.V. (1981). Crack propagation and fracture of composite materials such as concrete, in Proc., 5th. Int. Conf. on Fracture (ICF5), Cannes. 122. Zaitsev, J. W., and Wittmann, E H. (1974). A statistical approach to the study of the mechanical behavior of porous materials under multiaxial state of stress, in Proc. of the 1973 Symp. on Mechanical Behavior on Materials, Kyoto, Japan. 123. Zech, B., and Wittmann, E H. (1977). A complex study on the reliability assessment of the containment of a PWR, Part II. Probabilistic approach to describe the behavior of materials. in Trans. 4th Int. Conf. on Structural Mechanics in Reactor Technology, pp. 1-14, vol. H, J1/11, T. A. Jaeger and B. A. Boley, eds., Brussels: European Communities.
CHAPTER
Elasticity and Viscoelasticity
2
This Page Intentionally Left Blank
SECTION
2.1
Introduction to Elasticity and Viscoelasticity JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France
For all solid materials there is a domain in stress space in which strains are reversible due to small relative movements of atoms. For many materials like metals, ceramics, concrete, wood and polymers, in a small range of strains, the hypotheses of isotropy and linearity are good enough for many engineering purposes. Then the classical Hooke's law of elasticity applies. It can be derived from a quadratic form of the state potential, depending on two parameters characteristics of each material: the Young's modulus E and the Poisson's ratio v. 1 ~k* -- 2---pAijkl(E'v)0"ij0"kl 0~t*
l+v E
e~j - p 0a ~
ao
(1)
v ~ crkka~j
(2)
E and v are identified from tensile tests either in statics or dynamics. A great deal of accuracy is needed in the measurement of the longitudinal and transverse strains (6e ~ -+-10-6 in absolute value). When structural calculations are performed under the approximation of plane stress (thin sheets) or plane strain (thick sheets), it is convenient to write these conditions in the constitutive equation. 9 Plane
stress
(033 --
0"13 -- 0"23 -- 0)" 1 E
Igll 822 1
Sym
v E
1
0
0
i lll ~22
(3)
0"12
g12 l+v E Handbook of Materials Behavior Models. ISBN0-12-443341-3.
Copyright 9 2001by AcademicPress.All rightsof reproductionin any formreserved.
71
72
Lemaitre 9 Plane
strain
(833 - - 813 - - 823 - - 0)"
0.22
-
0.12
i
Sym
01111 i
2 + 2/~
0
822
2#
812
(4)
vE 2 - - (1 + v ) ( 1 - 2v) with
E -- 2(1 + v)
For orthotropic materials having three planes of symmetry, nine independent parameters are needed: three tension moduli El, E2, E3 in the orthotropic directions, three shear moduli G12, G23, G31, and three contraction ratios v12, v23, v31. In the frame of orthotropy: 811 -
1
v12
v13
E1
E1 1
E1 •23
822
E2
E2 1
833
_
0
0
0
0
0
0
0
0
0
0
0
0.11
0"22
0"33
E3 --
1
] 823
2G23
(5)
1
Sym
0
2G31
831
0"23
O"31
1 i
2G12
812
0"12 _
Nonlinear elasticity in large deformations is described in Section 2.2, with applications for porous materials in Section 2.3 and for elastomers in Section 2.4. Thermoelasticity takes into account the stresses and strains induced by thermal expansion with dilatation coefficient ~. For small variations of temperature 0 for which the elasticity parameters may be considered as constant: sO =
l+v v E 0./j - ~ 0.khcS/j+ a06/j
(6)
For large variations of temperature, E, v, and a will vary. In rate formulations, such as are needed in elastoviscoplasticity, for example, the
73
2.1 Introduction to Elasticity and Viscoelasticity
derivative of E, v, and c~ must be considered.
~v =
E
ev - g e ~ a v + ~0av + b-0
E
~v - F0
~ a v + N 0av 0
(7) Viscoelasticity considers in addition a dissipative phenomenon due to "internal friction," such as between molecules in polymers or between cells in wood. Here again, isotropy, linearity, and small strains allow for simple models. Quadratric functions for the state potential and the dissipative potential lead to either Kelvin-Voigt or Maxwell's models, depending upon the partition of stress or strains in a reversible part and in an irreversible part. They are described in detail for the one-dimensional case in Section 2.5 and recalled here in three dimensions. 9 Kelvin-Voigt model:
ffij = i~(~,kk -+- 02~'kk)(~ij _Jr_2/2(gij _+_Op~,ij )
(8)
Here 2 and/.z are Lame's coefficients at steady state, and 0x and 0~ are two time parameters responsible for viscosity. These four coefficients may be identified from creep tests in tension and shear. 9 Maxwell model:
9 1 + v (rij + giJ = E ~
-
-E
(rkk +
r2 /
aij
(9)
Here E and v are Young's modulus and Poisson's ratio at steady state, and rl and r2 are two other time parameters. It is a fluidlike model: equilibrium at constant stress does not exist. In fact, a more general way to write linear viscoelastic constitutive models is through the functional formulation by the convolution product as any linear system. The hereditary integral is described in detail for the one-dimensional case, together with its use by the Laplace transform, in Section 2.5.
'?'ij(t)--
fo'
Jijkl(t- "C) ~dcrkl dr + ~-~Jijkl(t- "c)AO'Pk/
(10)
p=l
[J(t)] is the creep functions matrix, and Ao-~l are the eventual stress steps. The dual formulation introduces the relaxation functions matrix JR(t)]
O'ij(t) --
/o t Rijkl(t-
"C) dC,kl dr, +
RijklAgPkkl
(11)
p=l
When isotropy is considered the matrix, [J] and two functions:
[R] each reduce to
9 either J(t), the creep function in tension, is identified from a creep test at constant stress; J ( t ) = g ( t ) / r and K, the second function, from the
74
Lemaitre
creep function in shear. This leads to
~,ij
-
-
( J + K) |
Dcrij _ K| (~ij Dz wz
(12)
where | stands for the convolution product and D for the distribution derivative, taking into account the stress steps. 9 or M(t), the relaxation function in shear, and L(t), a function deduced from M and from a relaxation test in tension R(t) = ~r(t)/~; L(t) = M ( R - 2 M ) / ( 3 M - R) ~0 - L |
D( ekk ) 6~j Dr
+ 2M |
D~3ij
~
dr
(13)
All of this is for linear behavior. A nonlinear model is described in Section 2.6, and interaction with damage is described in Section 2.7.
SECTION
2.2
Background on Nonlinear Elasticity R. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Contents 2.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stress and Equilibrium . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Constrained Materials . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 75 76 77 78 80 81 82
2.2.1 VALIDITY The theory of nonlinear elasticity is applicable to materials, such as rubberlike solids and certain soft biological tissues, which are capable of u n d e r g o i n g large elastic deformations. More details of the theory and its applications can be found in Beatty [1] and Ogden [3].
2.2.2 DEFORMATION For a continuous body, a reference configuration, denoted by ~r, is identified and 0 ~ r denotes the b o u n d a r y of ~ r . Points in ~ r are labeled by their position vectors X relative to some origin. The body is deformed quasistatically from ~ r SO that it occupies a new configuration, denoted ~ , with Handbook of Materials Behavior Models. ISBN 0o12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
75
76
Ogden
boundary 0~. This is the current or deformed configuration of the body. The deformation is represented by the mapping Z::~r --~ ~ , SO that x = z(X)
XC~r
(1)
where x is the position vector of the point X in ~ . The mapping X is called the deformation from ~ r tO ~ , and Z is required to be one-to-one and to satisfy appropriate regularity conditions. For simplicity, we consider only Cartesian coordinate systems and let X and x, respectively, have coordinates X~ and x~, where ~, i C { 1, 2, 3}, so that xi--zi(X~). Greek and Roman indices refer, respectively, to ~ r and ~ , and the usual summation convention for repeated indices is used. The deformation gradient tensor, denoted E is given by F = Grad
Fia = OXi/OXo~
x
(2)
Grad being the gradient operator in Nr. Local invertibility of Z and its inverse requires that 0 < J = det F < oo
(3)
wherein the notation J is defined. The deformation gradient has the (unique) polar decompositions F = RU = VR (4) where R is a proper orthogonal tensor and U, V are positive definite and symmetric tensors. Respectively, U and V are called the right and left stretch tensors. They may be put in the spectral forms 3
u-
3
| ul,I
v-
i=1
Z
,vl,I | vl,I
i=1
where v (i) = Ru (i), i C {1,2,3}, 2i are the principal stretches, u (i) the unit eigenvectors of U (the Lagrangian principal axes), v (i) those of V (the Eulerian principal axes), and | denotes the tensor product. It follows from Eq. 3 that J =/~1,~2,~3 9 The right and left Cauchy-Green deformation tensors, denoted C and B, respectively, are defined by C = FTF = U 2
2.2.3
B = FF ~ = V 2
(6)
STRESS AND EQUILIBRIUM
Pr and p be the mass densities in Nr and N, respectively. The mass conservation equation has the form
Let
Pr = pJ
(7)
77
2.2 Background on Nonlinear Elasticity
The Cauchy stress tensor, denoted g, and the nominal stress tensor, denoted S, are related by S = jF-lo
(8)
The equation of equilibrium may be written in the equivalent forms div ~ + pb = 0
Div S + Prb = 0
(9)
where div and Div denote the divergence operators in ~ and ~r, respectively, and b denotes the body force per unit mass. In components, the second equation in Eq. 9 is
OS~i c3X~
-t- IOrbi = 0
(10)
Balance of the moments of the forces acting on the body yields simply a t = ~, equivalently S TFT= FS. The Lagrangian formulation based on the use of S and Eq. 10, with X as the independent variable, is used henceforth.
2.2.4
ELASTICITY
The constitutive equation of an elastic material is given in the equivalent forms
oqW
S - H(F) - - - ~
(F)
( ~ - G(F) --j-1FH(F)
(11)
where H is a tensor-valued function, defined on the space of deformation gradients E W is a scalar function of F and the symmetric tensor-valued function G is defined by the latter equation in Eq. 11. In general, the form of H depends on the choice of reference configuration and it is referred to as the response function of the material relative to Nr associated with S. For a given ~r, therefore, the stress in ~ at a (material) point X depends only on the deformation gradient at X. A material whose constitutive law has the form of Eq. 11 is generally referred to as a hyperelastic material and W is called a strain-energy function (or stored-energy function). In components, the first part of Eq. 11 has the form S~i = cgW/cgFi~, which provides the convention for ordering of the indices in the partial derivative with respect to E If W and the stress vanish in N'r, so that W(I) -- 0
OW OF (I) - O
(12)
where I is the identity and O the zero tensor, then Nr is called a natural
configuration.
78
Ogden
Suppose that a rigid-body deformation x* = Qx + c is superimposed on the deformation x = z(X), where Q and c are constants, Q being a rotation tensor and c a translation vector. The resulting deformation gradient, F* say, is given by F* -- QF. The elastic stored energy is required to be independent of superimposed rigid deformations, and it follows that W(QF) = W(F)
(13)
for all rotations Q. A strain-energy function satisfying this requirement is said to be objective. Use of the polar decomposition (Eq. 4) and the choice Q - - R ~ in Eq. 13 shows that W ( F ) = W(U). Thus, W depends on F only through the stretch tensor U and may therefore be defined on the class of positive definite symmetric tensors. We write
OW ou Biot stress tensor, which T =
for the (symmetric) X = (SR + RTST)/2.
(14) is
related
to
S by
2.2.5 MATERIAL SYMMETRY Let F and F' be the deformation gradients in M relative to two different reference configurations, Mr and ~tr respectively. In general, the response of the material relative to ~'r differs from that relative to ~r, and we denote by W and W' the strain-energy functions relative to Mr and ~'r. Now let P -- Grad X' be the deformation gradient of ~'r relative to Mr, where X' is the position vector of a point in M'r- Then F -- F'P. For specific P we may have W' - W, and then W ( F ' P ) - - W(F')
(15)
for all deformation gradients F'. The set of tensors P for which Eq. 15 holds forms a multiplicative group, called the symmetry group of the material relative to Mr. This group characterizes the physical symmetry properties of the material. For isotropic elastic materials, for which the symmetry group is the proper orthogonal group, we have W(FQ) = W(F)
(16)
for all rotations Q. Since the Q's appearing in Eqs. 13 and 16 are independent, the combination of these two equations yields W(QUQ 1) = W(U)
(17)
79
2.2 Background on Nonlinear Elasticity
for all rotations Q. Equation 17 states that W is an isotropic function of U. It follows from the spectral decomposition (Eq. 5) that W depends on U only through the principal stretches 21, 22, and 23 and is symmetric in these stretches. For an isotropic elastic material, ~ is coaxial with V, and we may write = aoI + or where a0,
Or
--
R1,
2i~/2
(18)
+ 0r B2
and a2 are scalar invariants of B (and hence of V) given by
(~W 013
0{1 - -
_1/2(0W
213
OW)
~ 1 -+- I1-~2
-1/20W
0{2 -- -213
c912
(19)
and W is now regarded as a function of I1, I2, and I3, the principal invariants of B defined by I1
-
tr(B) -- 212 + 22 + 2~,
-
(20)
2 2 + 2321 2 2 + 2122 2 2 12 -- 11112- tr(B2)] -- 2223
(21)
2j2j2 I3 -- det B -- j"~1"2"'3
(22)
Another consequence of isotropy is that S and ~ have the decompositions 3
S-
~
3
tiu (0 | v (i)
i=1
o-
~
i=l
o'iv(i) | v (i)
(23)
where r~i, i c { 1, 2, 3} are the principal Cauchy stresses and ti the principal Blot stresses, connected by
ti = ~02i -J27~ lcri
(24)
Let the unit vector M be a preferred direction in the reference configuration of the material, i.e., a direction for which the material response is indifferent to arbitrary rotations about the direction and to replacement of M by -M. Such a material can be characterized by a strain energy which depends on F and the tensor M | M [2,4,5] Thus, we write W(F, M | M). The required symmetry (transverse isotropy) reduces W to dependence on the five invariants I1, 12, 13, 14 = M. (CM)
15 = M - ( C 2 M )
(25)
where I1, I2, and/3 are defined in Eqs. 20-22. The resulting nominal stress tensor is given by S = 2WIF ~- + 2W2(Ili - C)F ~- + 213W3F-1 + 2W4M | FM + 2Ws(M | FCM + CM | FM) where Wi = cgW/cgli, i = 1 , . . . , 5.
(26)
80
Ogden
When there are two families of fibers corresponding to two preferred directions in the reference configuration, M and M ~ say, then, in addition to Eq. 25, the strain energy depends on the invariants 16 -- M ~" (CM ~)
17 = M t" (C2M I)
18 = M. (CM ~)
(27)
and also on M. M ~ (which does not depend on the deformation); see Spencer [4,5] for details. The nominal stress tensor can be calculated in a similar way to Eq. 26.
2.2.6
CONSTRAINED
MATERIALS
An internal constraint, given in the form C ( F ) = 0 , must be satisfied for all possible deformation gradients E where C is a scalar function. Two commonly used constraints are incompressibility and inextensibility, for which, respectively, C(F) = detF - 1
C(F) = M-(F~-FM) - 1
(28)
where the unit vector M is the direction of inextensibility in ~r. Since any constraint is unaffected by a superimposed rigid deformation, C must be an objective scalar function, so that C(QF) = C(F) for all rotations Q. Any stress normal to the hypersurface C(F) = 0 in the (nine-dimensional) space of deformation gradients does no work in any (virtual) incremental deformation compatible with the constraint. The stress is therefore determined by the constitutive law in Eq. 11 only to within an additive contribution parallel to the normal. Thus, for a constrained material, the stress-deformation relation in Eq. 11 is replaced by
OC
OW
OC
S -- H(F) + q c3---F= o--F- + q OF
(29)
where q is an arbitrary (Lagrange) multiplier. The term in q is referred to as the constraint stress since it arises from the constraint and is not otherwise derivable from the material properties. For incompressibility and inextensibility we have
OW
S - - ~ + qF -1
OW
S - - ~ + 2qM | FM
(30)
respectively. For an incompressible material the Blot and Cauchy stresses are given by T-
OW c3---U- pU-1
detU-
1
(31)
81
2.2 Background on Nonlinear Elasticity
and
OW
o
F--~-F- - pI
--
det F - 1
(32)
where q has been replaced by -p, which is called an arbitrary hydrostatic pressure. The term in ~0 in Eq. 18 is absorbed into p, and I3 = 1 in the remaining terms in Eq. 18. For an incompressible isotropic material the principal components of Eqs. 31 and 32 yield
ow ti = o02i
ow p,~?X
O'i -- '~i -~-~- -- P
(33)
respectively, subject to 212223 = 1. For an incompressible transversely isotropic material with preferred direction M, the dependence on 13 is omitted and the Cauchy stress tensor is given by o = - pI + 2WIB + 2 W 2 ( I 1 B - B 2) -+- 2W4FM | FM + 2Ws(FM | BFM + BFM | FM)
(34)
For a material with two preferred directions, M and M/, the Cauchy stress tensor for an incompressible material is o = - pI + 2WIB + 2W2 (11B - B2) + 2W4FM | FM + 2Ws(FM Q BFM + BFM | FM) + 2W6FM' | FM' + 2W7(FM' | BFM' + BFM' | FM') + W8(FM | FM' + FM' | FM)
(35)
where the notation Wi = OW/OIi now applies for i = 1, 2, 4 , . . . , 8.
2.2.7 BOUNDARY-VALUE PROBLEMS The equilibrium equation (second part of Eq. 9), the stress-deformation relation (Eq. 11), and the deformation gradient (Eq. 2) coupled with Eq. 1 are combined to give
/
Div - - ~
+ Prb - 0
F -- Grad x
x-
z(X)
X C Nr
(36)
Typical boundary conditions in nonlinear elasticity are x = ~(X) on c9~
(37)
S ~ N - z(F,X) on O~f
(38)
where r and z are specified functions, N is the unit outward normal to 0~r, and 0 ~ and 0 ~ f are complementary parts of 0 ~ . In general, z may depend
82
Ogden
on the deformation through E For a dead-load traction "c is independent of E For a hydrostatic pressure boundary condition, Eq. 38 has the form
--JPF-rN
on c9~'~
(39)
Equations 36-38 constitute the basic boundary-value problem in nonlinear elasticity. In components, the equilibrium equation in Eq. 36 is written
02xj ,;l~i~j cgX--------~B+ p,.bi - 0
(40)
for i c { 1, 2, 3}, where the coefficients d~i~j are defined by
02W d~i~j -- d~j~i -- OFi~OFj~
(41)
When coupled with suitable boundary conditions, Eq. 41 forms a system of quasi-linear partial differential equations for xi = zi(X~). The coefficients ~ffoci~j are, in general, nonlinear functions of the components of the deformation gradient. For incompressible materials the corresponding equations are obtained by substituting the first part of Eq. 30 into the second part of Eq. 9 to give
d~j
02xj Op + Prbi _ 0 Ox~Ox~ - Ox---~
det(OxiOX~) - 1
(42)
where the coefficients are again given by Eq. 41. In order to solve a boundary-value problem, a specific form of W needs to be given. The form of W chosen will depend on the particular material considered and on mathematical requirements relating to the properties of the equations, an example of which is the strong ellipticity condition. Equations 40 are said to be strongly elliptic if the inequality
d~i~jmimjN~N~ > 0
(43)
holds for all nonzero vectors m and N. Note that Eq. 43 is independent of any boundary conditions. For an incompressible material, the strong ellipticity condition associated with Eq. 42 again has the form of Eq. 43, but the incompressibility constraint now imposes the restriction m . (F-rN) = 0 on m and N.
REFERENCES 1. Beatty, M. E (1987). Topics in finite elasticity: Hyperelasticity of rubber, elastomers and biological tissues n with examples. Appl. Mech. Rev. 40; 1699-1734. 2. Holzapfel, G. A. (2000). Nonlinear Solid Mechanics. Chichester: Wiley.
2.2 Background on Nonlinear Elasticity
83
3. Ogden, R. W. (1997). Non-linear Elastic Deformations. New York: Dover Publications. 4. Spencer, A. J. M. (1972). Deformations of Fibre-Reinforced Materials. Oxford: Oxford University Press. 5. Spencer, A. J. M. (1984). Constitutive theory for strongly anisotropic solids. In Continuum Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures No. 282, pp. 1-32, Spencer, A. J. M., ed., Wien: Springer-Verlag.
SECTION
2.3
Elasticity of Porous Materials N. D. CRISTESCU 231 Aerospace Building, University
of Florida,
Gainesville, Florida
Contents 2.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Identification of the Parameters . . . . . . . . . . . . . . 2.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84 85 85 88 90
2.3.1 VALIDITY The methods used to determine the elasticity of porous materials a n d / o r particulate materials as geomaterials or powderlike materials are distinct from those used with, say, metals. The reason is that such materials possess pores and/or microcracks. For various stress states these may either open or close, thus influencing the values of the elastic parameters. Also, the stress-strain curves for such materials are strongly loading-rate-dependent, starting from the smallest applied stresses, and creep (generally any time-dependent phenomena) is exhibited from the smallest applied stresses (see Fig. 2.3.1 for schist, showing three uniaxial stress-strain curves for three loading rates and a creep curve [1]). Thus information concerning the magnitude of the elastic parameters cannot be obtained: 9 from the initial slope of the stress-strain curves, since these are loadingrate-dependent; 9 by the often used "chord" procedure, obviously; 9 from the unloading slopes, since significant hysteresis loops are generally present. 84
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
85
2 . 3 Elasticity of Porous Materials
i
60
ot
[MPa] 50 -
i
i
i
E ~I
/ / u /
//
~. 0
11 /i
I
0
/
," ,"
/
-
..n.- / .
0-=0.002 IV[Pa S "l
p
12 days
-
-~
-
SCmST
,..
,'."
"I
0.1
0.065 .-= '<
,'.
IV"
6
//
/
.,It"
_ _ . " ~ ' " "" "- "" ,/r"
-/
I
0.49
f
/
/
,=" /
/./-
~" ,,..
3~f
/
/
/
/
i
/
/ j
40
20
i
0.2
I
I
I
I
0.3
0.4
0.5
0.6
,
I
0.7
r~ [%]
2.3.1 Uniaxial stress-strain curves for schist for various loading rates, showing time influence on the entire stress-strain curves and failure (stars mark the failure points). FIGURE
2.3.2
FORMULATION
The elasticity of such response" by
materials
D - 2-~ +
can be expressed
(1
3~<
as "instantaneous
lll(tr:f)l
(1)
2-0
where G and K are the elastic parameters that are not constant, D is the strain rate tensor, T is the stress tensor, is the trace operator, and 1 is the unit tensor. Besides the elastic properties described by Eq. 1, some other mechanical properties can be described by additional terms to be added to Eq. 1. For isotropic geomaterials the elastic parameters are expected to depend on stress invariants and, perhaps, on some damage parameters, since during loading some pores and microcracks may close or open, thus influencing the elastic parameters.
tr( )
2.3.3 IDENTIFICATION OF THE PARAMETERS The elastic parameters can be determined experimentally by two procedures. With the one is determining the travel time of the two
dynamic procedure,
86
Cristescu
elastic (seismic) extended body/longitudinal and transverse waves, which are traveling in the body. If both these waves are recorded, then the instantaneous response is of the form of Eq. 1. The elastic parameters are obtained from K--p
Vp--~
where vs is the velocity of propagation of the shearing waves, vv the velocity of the longitudinal waves, and p the density. The static procedure takes into account that the constitutive equations for geomaterials are strongly time-dependent. Thus, in triaxial tests performed under constant confining pressure ~r, after loading up to a desired stress state 1: (octahedral shearing stress), one is keeping the stress constant for a certain time period tc [2, 3]. During this time period the rock is creeping. When the strain rates recorded during creep become small enough, one is performing an unloading-reloading cycle (see Fig. 2.3.2). From the slopes (1
1 ) - 1 ( 1
1 ) -1
(3)
of these unloading-reloading curves one can determine the elastic parameters. For each geomaterial, if the time tc is chosen so that the subsequent unloading is performed in a comparatively much shorter time interval, no significant interference between creep and unloading phenomena will take place. An example for schist is shown in Figure 2.3.3, obtained in a triaxial test with five unloading-reloading cycles.
y
st
t ~
9K
0
6G
Transvcrs~ strain ~2
~t
v/Z__
0
Axial Strain E1
FIGURE 2.3.2 Static procedures to determine the elastic parameters from partial unloading processes preceded by short-term creep.
87
2.3 Elasticity of Porous Materials 40 o I
-
i
I''
I
i
l
(3'3
[MPal
30
2o
Io
aB=22.5 M P a
y 0
#i --0.001
!
0
!
0.001
0.002
I
0.003
FIGURE 2.3.3 Stress-strain curves obtained in triaxial tests on shale; the unloadings follow a period of creep of several minutes.
If only a partial unloading is performed (one third or even one quarter of the total stress, and sometimes even less), the unloading and reloading follow quite closely straight lines that practically coincide. If a hysteresis loop is still recorded, it means that the time tc was chosen too short. The reason for performing only a partial unloading is that the specimen is quite "thick" and as such the stress state in the specimen is not really uniaxial. During complete unloading, additional phenomena due to the "thickness" of the specimen will be involved, including, e.g., kinematic hardening in the opposite direction, etc. Similar results can be obtained if, instead of keeping the stress constant, one is keeping the axial strain constant for some time period during which the axial stress is relaxing. Afterwards, when the stress rate becomes relatively small, an unloading-reloading is applied to determine of the elastic parameters. This procedure is easy to apply mainly for particulate materials (sand, soils, etc.) when standard (Karman) three-axial testing devices are used and the elastic parameters follow from 1
Az
K= 3 Ael + 2Ae2
1
Az
C = 2 Ael - Ae2
(4)
where A is the variation of stress and elastic strains during the unloadingreloading cycle. The same method is used to determine the bulk modulus K in hydrostatic tests when the formula to be used is K =
Aa At;v
(5)
with ~ the mean stress and ev the volumetric strain. Generally, K is increasing with ~ and reaching an asymptotic constant value when ~ is increasing very much and all pores and microcracks are closed
88
Cristescu
under this high pressure. The variation of the elastic parameters with z is more involved: when z increases but is still under the compressibility/ dilatancy boundary, the elastic parameters are increasing. For higher values, above this boundary, the elastic parameters are decreasing. Thus their variation is related to the variation of irreversible volumetric strain, which, in turn, is describing the evolution of the pores and microcracks existing in the geomaterial. That is why the compressibility/dilatancy boundary plays the role of reference configuration for the values of the elastic parameters; so long as the loading path (increasing a and/or z) is in the compressibility domain, the elastic parameters are increasing, whereas if the loading path is in the dilatancy domain (increasing z under constant a), the elastic parameters are decreasing. If stress is kept constant and strain is varying by creep, in the compressibility domain volumetric creep produces a closing of pores and microcracks and thus the elastic parameters increase, and vice versa in the dilatancy domain. Thus, for each value of a the maximum values of the elastic parameters are reached on the compressibility/dilatancy boundary.
2.3.4 EXAMPLES As an example, for rock salt in uniaxial stress tests, the variation of the elastic moduli G and K with the axial stress O"1 is s h o w n in Figure 2.3.4 [4]. The variation of G and K is very similar to that of the irreversible volumetric ~k,a, (MPa)
20
(OD
(K)
o"l(MPa)
20
18
18
16
16
14
14
12
12
10
I0
-
8
8
-
6
6
4
2
m
} 1
4 2 i
2
A
"31
3
4
5
6
7
$ (MPa)
2
4
6
$
I0
%
FIGURE 2.3.4 Variationof the elastic parameters K and G and of irreversible volumetric strain in monotonic uniaxial tests.
89
2.3 Elasticity of Porous Materials
~days
~d
24 22
(G')
(K) ~.68
20
2422 "
~.66
k,~
20
10"61 18
18
16
16
14 12
14 12
A = 0.27 - 0 . 5 ~
8
A=0~2_~
10
o
108
_-o.17
4 2
o
A =0.12
2
f
4
A=0.
1
f /
K,G x 10.3
3
4
I
I
I
5
6
7
]
I
r x 104
!
8(MPa)
2
4
6
!
~
8
10
t
~._
12%"-
FIGURE 2.3.5 Variation in time of the elastic parameters and of irreversible volumetric strain in uniaxial creep tests.
strain e~/. If stress is increased in steps, and if after each increase the stress in kept constant for two days, the elastic parameters are varying during volumetric creep, as shown in Figure 2.3.5. Here A is the ratio of the applied stress and the strength in uniaxial compression ac = 17.88 MPa. Again, a similarity with the variation of e~ is quite evident. Figure 2.3.6 shows for a different kind of rock salt the variation of the elastic velocities ve and vs in true triaxial tests under confining pressure pc = 5MPa (data by Popp, Schuhze, and Kern [5]). Again, these velocities increase in the compressibility domain, reach their maxima on the compressibility/dilatancy boundary, and then decrease in the dilatancy domain. For shale, and the conventional (Karman) triaxial tests shown in Figure 2.3.3, the values of E and G for the five unloading-reloading cycles shown are: E = 9.9, 24.7, 29.0, 26.3, and 22.3GPa, respectively, while G = 4.4, 10.7, 12.1, 10.4, and 8.5 GPa. For granite, the variation of K with a is given as [2]
/ (:o) Ko-KI
1-
,
if a < a0
(6)
Ko, if a _~ ao
with K0 - 59 GPa, K1 -- 48 GPa, and a0 - 0.344 GPa, the limit pressure when all pores are expected to be closed.
Cristescu
90 90
. . . . .
lol
30
100
~" 50] / ~" 30-.
I' ~Jn'~"'i~i oca '~ A u,,,ch,,lg -'1 a
'
i
............~ 4 1 ~ i ~ .........:..:.,4,.~~
2o
lo 998
~
,"
10.
0
5
10
15 Pc
20
25
(MPa)
30
35
40
o
o.s
1
Axial stratn 1%)
.97 1.s
FIGURE 2.3.6 The maximum of Vs takes place at the compressibility/dilatancy boundary (figures and hachured strip); changes of vp and Vs as a function of strain ( ~ - - 1 0 -5 s -I, Pc -- 5 Mpa, T = 30 ~ C), showing that the maxima are at the onset of dilatancy. (Reprinted with permission from Popp, T., Kern, H., and Schultze, O., in press, Permeation and development of dilatancy and permeability in rock salt, in N. D. Cristescu and H. R. Hardy, Jr., eds., Proceedings of the 5th Conference on Mechanical Behavior of Salt, Mecasalt V, Bucharest, 9-11, August 1999, Trans Tech. Publ., Clausthal.)
The same formula for a l u m i n a p o w d e r is /
\
(7) w i t h K c~ = 1 x 10 7 kPa the constant value toward w h i c h the b u l k m o d u l u s tends at high pressures, ~ = 10 7, b - - - 1 . 2 x 10 -4, a n d Pa - I kPa. Also for a l u m i n a p o w d e r we have E ( a ) := E ~ - Pail e x p ( - d a )
(8)
with E ~ = 7 x 105 kPa, fl = 6.95 x 105, a n d d = 0.002. For the shale s h o w n in Figure 2.3.3, the variation of K with a for 0 < a < 45 MPa is K(r~) "-- - 0 . 7 8 r 2 + 65.32rr - 369
(9)
REFERENCES 1. Cristescu, N. (1986). Damage and failure of viscoplastic rock-like materials. Int. J. Plasticity 2 (2): 189-204. 2. Cristescu, N. (1989). Rock Rheology, Kluver Academic Publishing. 3. Cristescu, N. D., and Hunsche, U. (1998). Time Effects in Rock Mechanics, Wiley. 4. Ani, M., and Cristescu N. D. (2000). The effect of volumetric strain on elastic parameters for rock salt. Mechanics of Cohesive-Frictional Materials 5 (2): 113-124. 5. Popp, T., Schultze, O., and Kern, H. (2001). Permeation and development of dilatancy and permeability in rock salt, in The Mechanical Behavior of Salt (5th Conference on Mechanical Behavior of Salt), Cristescu, N. D., and Hardy, Jr., H. Reginald, eds., Balkema.
SECTION
2.4
Elastomer Models R. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Contents 2.4.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Description of the Model . . . . . . . . . . . . . . . . . . . . . 2.4.4 Identification of Parameters . . . . . . . . . . . . . . . . . . 2.4.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91 91 93 93 94 94 94
2.4.1 VALIDITY Many rubberlike solids can be treated as isotropic and incompressible elastic materials to a high degree of approximation. Here the mechanical properties of such solids are described through the use of an isotropic elastic strainenergy function in the context of finite deformations. For general background on finite elasticity, we refer to Ogden [5].
2.4.2 BACKGROUND Locally, the finite deformation of a material can be described in terms of the three principal stretches, denoted by 21, 22, and 23. For an incompressible material these satisfy the constraint /~1/~2/~3-
1
o
(1)
The material is isotropic relative to an unstressed u n d e f o r m e d (natural) configuration, and its elastic properties are characterized in terms of a Handbook of Materials Behavior Models. ISBN 0-12-443341-31
Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
91
92
Ogden
strain-energy function W(,~1,~,2,,~3) per unit volume, where W depends symmetrically on the stretches subject to Eq. 1. The principal Cauchy stresses associated with this deformation are given by
cgW
0"i -- 2i o,r
p,
i C { 1, 2, 3}
(2)
where p is an arbitrary hydrostatic pressure (Lagrange multiplier). By regarding two of the stretches as independent and treating the strain energy as a function of these through the definition W ( 2 1 , 2 2 ) - W(21,22,211221), we obtain
oW
crl -cr3 - 21 0~1
oW
cr2- cr3 - ~20~--T
(3)
For consistency with the classical theory, we must have 02*
!/r
1) -- 0,
021022 (1, 1)
cgW(1 1) - 0 0 2 * - 2#, ~ , , 022 (1, 1) -- 4/,,
(4)
0~ E {1,2} where # is the shear modulus in the natural configuration. The equations in Eq. 3 are unaffected by superposition of an arbitrary hydrostatic stress. Thus, in determining the characteristics of lye, and hence those of W, it suffices to set 0"3 = 0 in Eq. 3, so that o~
o~
(5)
Biaxial experiments in which 21,22 and o1,0"2 are measured then provide data for the determination of ~fr Biaxial deformation of a thin sheet where the deformation corresponds effectively to a state of plane stress, or the combined extension and inflation of a thin-walled (membranelike) tube with closed ends provide suitable tests. In the latter case the governing equations are written P* -- 211221 c9~Yr 022
__cgW_ _1 oq* F* --- 0 2 1 2 22211 (~'~2
(6)
where P* = PR/H, P is the inflating pressure, H the undeformed membrane thickness, and R the corresponding radius of the tube, while F* = F/2rrRH, with F the axial force on the membrane (note that the pressure contributes to the total load on the ends of the tube). Here 21 is the axial stretch and 22 the azimuthal stretch in the membrane.
93
2.4 E l a s t o m e r M o d e l s
2.4.3
DESCRIPTION
OF THE MODEL
A specific model which fits very well the available data on various rubbers is that defined by N
W - Z
#"(21" + 22" + 2~" - 3)/~.
(7)
n=l
where #, and an are material constants and N is a positive integer, which for many practical purposes may be taken as 2 or 3 [3]. For consistency with Eq. 4 we must have N
Z
-
(8)
n=l
and in practice it is usual to take/.t,,0~, > 0 for each n In respect of Eq. 7, the equations in Eq. 3 become N
N
O-1 - - 0- 3 - - ~ fin(/~l" - - ~ " ) n=l
2.4.4
1 , . . . ,N.
IDENTIFICATION
0"2 -- 0- 3 - - ~ ~/n(~" n=l
-- ~")
(9)
OF PARAMETERS
Biaxial experiments with a3 = 0 indicate that the shapes of the curves of a l - or2 plotted against /~1 a r e essentially independent of 22 for many rubbers. Thus the shape may be determined by the pure shear test with 2 2 = 1, s o t h a t N
al-
a2-
Z
N
#n(21" - 1)
-a2-
n=l
~
~,(23 n - 1)
(10)
n=l
for 21 >_ 1, 23 _< 1. The shift factor to be added to the first equation in Eq. 10 when 22 differs from 1 is N
~--~ #,(1 - 22" )
(11)
n=l
Information on both the shape and shift obtained from experiments at fixed 22 then suffice to determine the material parameters, as described in detail in References [3] or [4]. Data from the extension and inflation of a tube can be studied on this basis by considering the combination of equations in Eq. 6 in the form cgW cgW 1 2 ,rl - a2 - ~1 ~ - & ~ -- ~1/~ - ~ ~2,ZlP ~
(12)
94
Ogden
2.4.5 HOW T O U S E T H E S T R A I N - E N E R G Y FUNCTION The strain-energy function is incorporated in m a n y commercial Finite E l e m e n t (FE) software packages, such as ABAQUS and MARC, and can be used in terms of principal stretches and principal stresses in the FE solution of boundary-value problems.
2.4.6
TABLE OF PARAMETERS
Values of the parameters corresponding to a three-term form of Eq. 7 are n o w given in respect of two different but representative vulcanized natural rubbers. The first is the material used by Jones and Treloar [2]: (Z1 - - 1 . 3 ,
~2 = 4.0, ~3 = - 2 . 0 ,
]A1 - - 0.69, #2 -- 0.01, #3 -- - 0 . 0 1 2 2 N m m -2 The second is the material used by James et al. [1], the material constants having been obtained by Treloar and Riding [6]: 0~1 - - 0 . 7 0 7 ,
]A1
-
-
~2 = 2.9, ~3 = - 2 . 6 2 ,
0.941, #2 -- 0.093, #3 -- - 0 . 0 0 2 9 N m m -2
For detailed descriptions of the rubbers concerned, reference should be m a d e to these papers.
REFERENCES 1. James, A. G., Green, A., and Simpson, G. M. (1975). Strain energy functions of rubber. I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19: 2033-2058. 2. Jones, D. E, and Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain. J. Phys. D: Appl. Phys. 8: 1285-1304. 3. Ogden, R. W. (1982). Elastic deformations of rubberlike solids, in Mechanics of Solids (Rodney Hill 60th Anniversary Volume) pp. 499-537, Hopkins, H. G., and Sevell, M. J., eds., Pergamon Press. 4. Ogden, R. W. (1986). Recent advances in the phenomenological theory of rubber elasticity. Rubber Chem. Technol. 59: 361-383. 5. Ogden, R. W. (1997). Non-Linear Elastic Deformations, Dover Publications. 6. Treloar, L. R. G., and Riding, G. (1979). A non-Gaussian theory for rubber in biaxial strain. I. Mechanical properties. Proc. R. Soc. Lond. A369: 261-280.
SECTION
2.5
Background on Viscoelasticity Kozo IKEGAMI Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan
Contents 2.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.5.2 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . 95 2.5.3 Static Viscoelastic Deformation . . . . . . . . . . . . . . . 98 2.5.4 Dynamic Viscoelastic Deformation . . . . . . . . . 100 2.5.5 Hereditary Integral . . . . . . . . . . . . . . . . . . . . . . . . 102 2.5.6 Viscoelastic Constitutive Equation by the Laplace Transformation . . . . . . . . . . . . . . . . . . . . 103 2.5.7 Correspondence Principle . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.1 VALIDITY F u n d a m e n t a l deformation of materials is classified into three types: elastic, plastic, and viscous deformations. Polymetric material shows t i m e - d e p e n d e n t properties even at r o o m temperature. Deformation of metallic materials is also t i m e - d e p e n d e n t at high temperature. The theory of viscoelasticity can be applied to represent elastic and viscous deformations exhibiting timed e p e n d e n t properties. This paper offers an outline of the linear theory of viscoelasticity.
2.5.2 MECHANICAL MODELS Spring and dashpot elements as s h o w n in Figure 2.5.1 are used to represent elastic and viscous deformation, respectively, within the framework of the Handbookof MaterialsBehaviorModels.ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
95
96
Ikegami
O"
1
k
a
a
(a) Spring element
(b) Dashpot element
FIGURE2.5.1 Mechanicalmodel of viscoelasticity.
linear theory of viscoelasticity. The constitutive equations between stress a and stress e of the spring and dashpot are, respectively, as follows: o" -- ke
de o" - r/~-~
(1)
where the notations k and r/are elastic and viscous constants, respectively. Stress of spring elements is linearly related with strain. Stress of dashpot elements is related with strain differentiated by time t, and the constitutive relation is time-dependent. Linear viscoelastic deformation is represented by the constitutive equations combining spring and dashpot elements. For example, the constitutive equations of the series model of spring and dashpot elements shown in Figure 2.5.2 is as follows: r/d~ de
(2)
This is called the Maxwell model. The constitutive equation of the parallel model of spring and dashpot elements shown in Figure 2.5.3 is as follows: de cr - ke + t / ~ This is called the Voigt or Kelvin model.
(3)
97
2.5 Backgroundon Viscoelasticity
k
C;
I
FIGURE2.5.2 Maxwellmodel.
There are many variations of constitutive equations giving linear viscoelastic deformation by using different numbers of spring and dashpot elements. Their constitutive equations are generally represented by the following ordinary differential equation: dr d2 cr dn cr poor + Pl ~ q- P2 - - ~ + . . . if- P,, dt n
&
d2~
d"~
(4)
= qo~ + q l - ~ + q 2 - ~ + . . . + q. dt"
The coefficients p and q of Eq. 4 give the characteristic properties of linear viscoelastic deformation and take different values according to the number of spring and dashpot elements of the viscoelastic mechanical model.
98
Ikegami
t E k
FIGURE 2.5.3
Voigt (Kelvin) model.
2.5.3 STATIC VISCOELASTIC DEFORMATION There are two functions representing static viscoelastic deformation; one is creep compliance, and another is the relaxation modulus. Creep compliance is defined by strain variations under constant unit stress. This is obtained by solving Eqs. 2 or 3 for step input of unit stress. For the Maxwell model and the Voigt model, their creep compliances are represented, respectively, by the following expressions. For the Maxwell model, the creep compliance is g- +
--
)
+ 1
(5)
where zM = JT/k, and this is denoted as relaxation time. For the Voigt model, the creep compliance is ~-kI1-exp(-~)1-
k[1-exp (- ~)
(6)
where ZK --q/k, and this is denoted as retardation time. Creep deformations of the Maxwell and Voigt models are illustrated in Figures 2.5.4 and 2.5.5, respectively. Creep strain of the Maxwell model
99
2.5 Background on Viscoelasticity
1/k
0 FIGURE 2.5.4
t
Creep compliance of the Maxwell model.
0
1/k
0 FIGURE 2.5.5
t
Creep compliance of the Voigt model.
100
Ikegami
increases linearly with respect to time duration. The Voigt model exhibits saturated creep strain for a long time. The relaxation modulus is defined by stress variations under constant unit strain. This is obtained by solving Eqs. 2 or 3 for step input of unit strain. For the Maxwell and Voigt models, their relaxation moduli are represented by the following expressions, respectively. For the Maxwell model,
) For the Voigt model, o ~ k
/8/
Relaxation behaviors of the Maxwell and Voigt models are illustrated in Figures 2.5.6 and 2.5.7, respectively. Applied stress is relaxed by Maxwell model, but stress relaxation dose not appear in Voigt model.
2.5.4 DYNAMIC VISCOELASTIC DEFORMATION The characteristic properties of dynamic viscoelastic deformation are represented by the dynamic response for cyclically changing stress or strain. p
I k
0 FIGURE 2.5.6
t
Relaxation modulus of the Maxwell model.
101
2.5 Background on Viscoelasticity
I
0
0 FIGURE 2.5.7
t
Relaxation modulus of the Voigt model.
The viscoelastic effect causes delayed phase phenomena between input and output responses. Viscoelastic responses for changing stress or strain are defined by complex compliance or modulus, respectively. The dynamic viscoelastic responses are represented by a complex function due to the phase difference between input and output. Complex compliance J, of the Maxwell model is obtained by calculating changing strain for cyclically changing stress with unit amplitude. Substituting changing complex stress ~ - exp(icot), where i is an imaginary unit and co is the frequency of changing stress, into Eq. 2, complex compliance J , is obtained as follows: 1
i
. J* . k . i cot/
1
.k
i
.ikcOl:M J' - iJ"
(9)
where the real part J ' = 1 / k is denoted as storage compliance, and the imaginary part J" = 1/kco'cM is denoted as loss compliance. The complex modulus Y, of the Maxwell model is similarly obtained by calculating the complex changing stress for the complex changing strain
102
Ikegami
e - exp(icot) as follows:
1 + (cozu) 2
-+ ik
1 + (cozu) 2
-- Y' -t-tY"
(10)
where Y'-k((cozM)2/(1 + (cozu)2))and Y " - k ( c o z u / ( 1 + (cozu)2)). The notations Y~ and Y" are denoted as dynamic modulus and dynamic loss, respectively. The phase difference ~ between input strain and output stress is given by Y" tan6 - - - - - y1
1
(11)
(_DTM
This is called mechanical loss. Similarly, the complex compliance and the modulus of the Voigt model are able to be obtained. The complex compliance is
J* - k
1[
I xE ]
1 + (COZK)2 - - i ~ 1 + (COZK)2
where,'1[ 1 I J" 11 -- k 1 + (COZK)2 and
(12)
]
-- k 1 + (coz/()2
The complex modulus is Y , -- k +
iCOZK =
Y~ + iY"
(13)
where Y ' - - k and Y " - - kcozi,:.
2.5.5 HEREDITARY INTEGRAL The hereditary integral offers a method of calculating strain or stress variation for arbitrary input of stress or strain. The method of calculating strain for stress history is explained by using creep compliance as illustrated in Figure 2.5.8. An arbitrary stress history is divided into incremental constant stress history dcr~ Strain variation induced by each incremental stress history is obtained by creep compliance with the constant stress values. In Figure 2.5.8 the strain induced by stress history for t~< t is represented by the following integral: e,(t) -- tToJ(t) +
~0tJ ( t -
dtTl
t' ) - - ~ dt'
(14)
103
2.5 Background on Viscoelasticity
or(t)
/
d cr '
o"0
0
t'
FIGURE 2.5.8
t'+dt' Hereditary integral.
This equation is transformed by partially integrating as follows:
dt' e(t) - ~r(t)J(O) -+-fo t ~(t' ) dJ(td ( t - t't') -~--~ Similarly, stress variation for arbitrary strain history becomes t d~ o'(t) -- g0Y(t) -Jr- Y(t - t' ) --d7 dt'
fO
(15)
(16)
Partial integration of Eq. 16 gives the following equation:
~(t) - g(t)Y(O) -Jr-foot ~(t' ) ddY((t t-- t')t') dt'
(17)
Integrals in Eqs. 14 to 17 are called hereditary integrals.
2.5.6 VISCOELASTIC CONSTITUTIVE E Q U A T I O N BY T H E L A P L A C E TRANSFORMATION The constitutive equation of viscoelastic deformation is the ordinary differential equation as given by Eq. 4. That is,
dktr Pk ~ -- ~
m
k=O
k=O
dke, qk dt k
(18)
104
Ikegami
This equation is written by using differential operators P and Q, Pa-
where P -
/=o
dk pk--d~ and Q -
Qe
(19)
m dk ~ qk 9 k=0 dtk
Equation (18) is represented by the Laplace transformation as follows.
~_~ pksk~ -- ~-~ qkskF. k=0
(20)
k=0
where 5 and ~ are transformed stress and strain, and s is the variable of the Laplace transformation. Equation 20 is written by using the Laplace transformed operators of time derivatives 15 and Q as follows: Q_ cr = =~ P _
n
(21)
m
where 15 - 2 Pk sk and Q - ~ qkS k. k=0
k=0
Comparing Eq. 21 with Hooke's law in one dimension, the coefficient ~//5 corresponds to Young's modulus of linear elastic deformation. This fact implies that linear viscoelastic deformation is transformed into elastic deformation in the Laplace transformed state.
2.5.7 CORRESPONDENCE
PRINCIPLE
In the previous section, viscoelastic deformation in the one-dimensional state was able to be represented by elastic deformation through the Laplace transformation. This can apply to three-dimensional viscoelastic deformation. The constitutive relations of linear viscoelastic deformation are divided into the relations between hydrostatic pressure and dilatation, and between deviatoric stress and strain. The relation between hydrostatic pressure and dilatation is represented by m k=0
k !
d %
n
dk
(22)
k=0
ptt(Tii -- Qtt c,ii m
" dhe//
(231
dk
where P " - k=0 ~ p ~ - ~ and Q " = k=0 ~ q~ ~-i-~. In Eq. 22 hydrostatic pressure is (1/3) crii and dilatation is eii.
105
2.5 Background on Viscoelasticity
The relation between deviatoric stress and strain is represented by
m ! dkrriJ ! s ~-'2Pk dt k k=o
dk~!. U
_1-"
k=0
qk
- Q' ,5 m
where P ' -
dk
~ p!~ ~ k-0
(24)
(25)
dk )_2 q~k-d~" In Eq. 24 deviatoric stress and strain
~
and Q ' -
k=0
! and r ~ respectively. are rrij The Laplace transformations of Eqs. 22 and 24 are written, respectively, as follows: /3,,ryii -- Q"~/,
(26)
where P " - f'"(s) and Q ! ! - Q"cr(s), and -I
-!
-
l-I
P (Yij -- QEij
(27)
where 1B,- 1B!(s) and Q ! - Q'(s). The linear elastic constitutive relations between hydrostatic pressure and dilatation and between deviatoric stress and strain are represented as follows: r r i i - 3K~ii !
!
cr/j -- 2G~/j
(28) (29)
Comparing Eq. 17 with Eq. 19, and Eq. 18 with Eq. 20, the transformed viscoelastic operators correspond to elastic constants as follows: @ 3K
-
_--
p!!
2G = - p!
(30)
(31)
where K and G are volumetric coefficient and shear modulus, respectively. For isotropic elastic deformation, volumetric coefficient K and shear modulus G are connected with Young's modulus E and Poisson's ratio v as follows: G-
2(1+v)
E K -- 3(1 - 2v)
(32)
(33)
106
lkegami
Using Eqs. 30-33, Young's modulus E and Poisson's ratio are connected with the Laplace transformed coefficient of linear viscoelastic deformation as follows: 3QtQ II E--
2/St~tl + / 5 I ~
(34)
p, Q,, _ p,, Q, v -
+
(35)
Linear viscoelastic deformation corresponds to linear elastic deformation through Eqs. 30-31 and Eqs. 34-35. This is called the correspondence principle between linear viscoelastic deformation and linear elastic deformation. The linear viscoelastic problem is the transformed linear elastic problem in the Laplace transformed state. Therefore, the linear viscoelastic problem is able to be solved as a linear elastic problem in the Laplace transformed state, and then the elastic constants of solved solutions are replaced with the Laplace transformed operator of Eqs. 30-31 and Eqs. 34-35 by using the correspondence principle. The solutions replaced the elastic constants become the solution of the linear viscoelastic problem by inversing the Laplace transformation.
REFERENCES 1. 2. 3. 4. 5. 6.
Bland, D. R. (1960). Theory of Linear Viscoelasticity, Pergamon Press. Ferry, J. D. (1960). Viscoelastic Properties of Polymers, John Wiley & Sons. Reiner, M. (1960). Deformation, Strain and Flow, H. K. Lewis & Co. Fll~ge, W. (1967). Viscoelasticity, Blaisdell Publishing Company. Christensen, R. M. (1971). Theory of Viscoelasticity: An Introduction, Academic Press. Drozdov, A. D. (1998). Mechanics of Viscoelastic Solids, John Wiley & Sons.
SECTION
2.6
A Nonlinear Viscoelastic Model Based on Fluctuating Modes RACHID RAHOUADJ AND CHRISTIAN CUNAT LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la For~t-de-Haye, 54500 Vandoeuvre-l~s-Nancy, France
Contents 2.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Background of the DNLR . . . . . . . . . . . . . . . 2.6.2.1 Thermodynamics of Irreversible Processes and Constitutive Laws... 2.6.2.2 Kinetics and Complementary Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2.3 Constitutive Equations of the DNLR . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Description of the Model in the Case of Mechanical Solicitations . . . . . . . . . . . . . . 2.6.4 Identification of the Parameters . . . . . . . . . 2.6.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107 108 108 110 112 113 113 115 115
116
2.6.1 VALIDITY We will f o r m u l a t e a viscoelastic m o d e l i n g for p o l y m e r s in the t e m p e r a t u r e r a n g e of glass transition. This physical m o d e l i n g m a y be a p p l i e d u s i n g integral or differential forms. Its f u n d a m e n t a l basis c o m e s from a g e n e r a l i z a t i o n of the Gibbs relation, a n d leads to a f o r m u l a t i o n of c o n s t i t u t i v e laws i n v o l v i n g c o n t r o l a n d i n t e r n a l t h e r m o d y n a m i c variables. T h e latter m u s t t r a d u c e Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
107
108
Rahouadj and Cunat
different microstructural rearrangements. In practice, both modal analysis and fluctuation theory are well adapted to the study of the irreversible transformations. Such a general formulation also permits us to consider various nonlinearities as functions of material specificities and applied perturbations. To clarify the present modeling, called "the distribution of nonlinear relaxations" (DNLR), we will consider the viscoelastic behavior in the simple case of small applied perturbations near the thermodynamic equilibrium. In addition, we will focus our attention upon the nonlinearities induced by temperature and frequency perturbations.
2.6.2 B A C K G R O U N D OF THE DNLR 2.6.2.1
THERMODYNAMICSOF IRREVERSIBLE
PROCESSES AND CONSTITUTIVE LAWS
As mentioned, the present irreversible thermodynamics is based on a generalization of the fundamental Gibbs equation to systems evolving outside equilibrium. Note that Coleman and Gurtin [1], have also applied this postulate in the framework of rational thermodynamics. At first, a set of internal variables (generalized vector denoted z) is introduced to describe the microstructural state. The generalized Gibbs relation combines the two laws of thermodynamics into a single one, i.e., the internal energy potential: e =
e(s,
~, n,... ;~)
(1)
which depends on overall state variables, including the specific entropy, s. Furthermore, with the positivity of the entropy production being always respected, one obtains for open systems:
dAis
7--a7-
T~
-
_js . V ~ - ~-~jk . V ~ h +
n.~
> 0
(2)
k=l
where the nonequilibrium thermodynamic forces may be separated into two groups: (i) the gradient ones, such as the gradient of temperature ~TT, and the gradient of generalized chemical potential V/~h; and (ii) The generalized forces A, or affinities as defined by De Donder [2] for chemical reactions, which characterize the nonequilibrium state of a uniform medium. The vectors js, jh, and ~ correspond to the dual, fluxes, or ratetype variables. To simplify the formulation of the constitutive laws, we will now consider the behavior of a uniform representative volume element (RVE without any
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
109
gradient), thus:
~-
A.~ > 0
(3)
The equilibrium or relaxed state (denoted by the index r) is currently described by a suitable thermodynamic potential ( ~ ) obtained via the Legendre transformation of Eq. 1 with respect to the control or state variable (7)- In this particular state, the set of internal variables is completely governed by (7):
Our first hypothesis [3] states that it is always possible to define a thermodynamic potential g, only as a function of 7 and z, even for systems outside equilibrium: g, - g,(~; z)
(5)
Then, we assume that the constitutive equations may be obtained as functions of the first partial derivatives of this potential with respect to the dual variables, and depend consequently on both control and internal variables; i.e., fl - fl(7; z) and A - A(7; z). In fact, this description is consistent with the principle of equipresence, as postulated in rational thermodynamics. Therefore, the thermodynamic potential becomes in a differential form: q m=l
j=l
Thus the time evolution of the global response, fl, obeys a nonlinear differential equation involving both the applied perturbation 7 and the internal variable z (generalized vector)" /} - aU. 9 + b.~:
(7a)
A - - t b . 9 - g.s
(7b)
This differential system resumes in a general and condensed form the announced constitutive relationships. The symmetrical matrix a u - c92g,/c97c37 is the matrix of Tisza, and the symmetrical matrix g - c92g,/&c3z traduces the interaction between the dissipation processes [3]. The rectangular matrix b- c32g,/cqzc97 expresses the coupling effect between the state variables and the dissipation variables. In other respects, the equilibrium state classically imposes the thermodynamic forces and their rate to be zero; i.e., A - 0 and A - 0. From Eq. 7b we find, for any equilibrium state, that the internal variables' evolution results directly from the variation of the control variables: :~r _ _g-1 .tb. ~ (8)
110
Rahouadj and
Cunat
According to Eqs. 7b and 8, the evolution of the generalized force becomes A = - g . ( ~ - ~')
(9)
and its time integration for transformation near equilibrium leads to the simple linear relationship A = - g ( z - z r)
(10)
where g is assumed to be constant. 2.6.2.2
KINETICS AND COMPLEMENTARY LAWS
To solve the preceding three equations (7a-b, 10), with the unknown variables being fl, z, z r, and A, one has to get further information about the kinetic relations between the nonequilibrium driving forces A and their fluxes ~. 2.6.2.2.1 First-Order Nonlinear Kinetics and Relaxation Times We know that the kinetic relations are not submitted to the same thermodynamic constraints as the constitutive ones. Thus we shall consider for simplicity an affine relation between fluxes and forces. Note that this wellknown modeling, early established by Onsager, Casimir, Meixner, de Donder, De Groot, and Mazur, is only valid in the vicinity of equilibrium: ~-L.A
(11)
and hence, with Eq. 10: = -L.g.(z-
z r) = -T -1 . ( z - Zr)
(12)
According to this nonlinear kinetics, Meixner [4] has judiciously suggested a base change in which the relaxation time operator z is diagonal. Here, we consider this base, which also represents a normal base for the dissipation modes. In what follows, the relaxation spectrum will be explicitly defined on this normal base. To extend this kinetic modeling to nonequilibrium transformations, which is the object of the nonlinear Thermodynamics of Irreversible Processes (TIP), we also suggest referring to Eq. 12 but with variable relaxation times. Indeed, each relaxation time is inversely proportional to the jump frequency, v, and to the probability pj = exp(-AF~'r/RT) of overcoming a free energy barrier, AF+'r. It follows that the relaxation time of the process j may be written:
z~ -- 1Iv exp(AF~'r/RT)
(13)
where the symbol (+) denotes the activated state, and the index (r) refers to the activation barrier of the Representative Elementary Volume (REV) near the equilibrium.
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
111
The reference jump frequency, v0 = kBT/h, has been estimated from Guggenheim's theory, which considers elementary movements of translation at the atomic level. The parameters h, ks, and r represent the constants of Plank, Boltzmann, and of the perfect gas, respectively, and T is the absolute temperature. It seems natural to assume that the frequency of the microscopic rearrangements is mainly governed by the applied perturbation rate, ~), through a shift function a(~):
v = uo/a('~)
(14)
Assuming now that the variation of the activation energy for each process is governed by the evolution of the overall set of internal variables leads us to the following approximation of first order: AF+
--
AF/'r + K z . ( Z - z r)
(15)
In the particular case of a viscoelastic behavior, this variation of the free energy becomes negligible. The temperature dependence obviously intervenes into the basic definition of the activation energy as
AF/'r -- AE +'r - T AS/'r
(16)
where the internal energy AE +'r is supposed to be the same for all processes. It follows that we may define another important shift function, noted a(T), which accounts for the effect of temperature. According to the Arrhenius approximation, AE +'r being quasi-constant, this shift function verifies the following relation: In(a;) a(T, Tref) - - AE+'r(1/T -
1/Tref)
(17)
where T r e f is a reference temperature. For many polymers near the glass transition, this last shift function obeys the WLF empiric law developed by William, Landel, and Ferry [5]: ln(a;) = C l ( T - Tref)/[c2 + ( T - Tref)]
(18)
In summary, the relaxation times can be generally expressed as Tj(T) = T;(Tref)a(T , Tref) a(~) a(z, z r)
(19)
and the shift function a(z, Zr) becomes negligible in viscoelasticity. 2.6.2.2.2 Form of the Relaxation Spectrum near the Equilibrium We now examine the distribution of the relaxation modes evolving during the solicitation. In fact, this applied solicitation, 7, induces a state of fluctuations which may be approximately compared to the corresponding equilibrium one. According to Prigogine [6], these fluctuations obey the equipartition of the entropy production. Therefore, we can deduce the expected distribution in
112
Rahouadj and Cunat
the vicinity of equilibrium as and
B-1/)jl/z[ / - ' -""j = l V
j=l
(20)
where z; is the relaxation time of the process j, pO its relative weight in the overall spectrum, and n the number of dissipation processes [3]. As a first approximation, the continuous spectrum defined by Eq. 20 may be described with only two parameters: the longest relaxation time corresponding to the fundamental mode, and the spectrum width. Note that a regular numerical discretization of the relaxation time scale using a sufficiently high number n of dissipation modes, e.g., 30, gives a sufficient accuracy.
2.6.2.3
CONSTITUTIVE EQUATIONS OF THE D N L R
Combining Eqs. 7a and 12 gives, whatever the chosen kinetics, ]~-aU.~ b.(z zr).z~ -1 a u .~ (]~ /t~). z~ 1 _
_
_
_
_
(21a)
To simplify the notation, z~ will be denoted z. In a similar form and after introducing each process contribution in the base defined above, one has
p--1
j--1
Tj
where the indices u and r denote the instantaneous and the relaxed values, respectively. Now we shall examine the dynamic response due to sinusoidally varying perturbations 7n = 7~ where co is the applied frequency, and i 2 = - 1 , i.e., ) ; , - icoT,. The response is obtained by integrating the above differential relationship. Evidently, the main problem encountered in the numerical integration consists in using a time step that must be consistent with the applied frequency and the shortest time of relaxation. Furthermore, a convenient possibility for very small perturbations is to assume that the corresponding response is periodic and out of phase:
fin -- flOexp(icot + ~)
and
fin
--
icofln
(22)
where ~0is the phase angle. In fact, such relations are representative of various physical properties as shown by Kramers [7] and Kronig [8]. The coefficients of the matrices of Tisza, a" and a r, and the relaxation times, l:j, may be dependent on temperature and/or frequency. In uniaxial
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
113
tests of mechanical damping, these Tisza's coefficients correspond to the storage and loss modulus E t (or G t) and E" (or G'), respectively.
2.6.3
DESCRIPTION
OF THE MODEL IN THE
CASE OF MECHANICAL
SOLICITATIONS
We consider a mechanical solicitation under an imposed strain e. Here, the perturbation 7 and the response fl are respectively denoted e and a. According to Eqs. 19 and 21b, the stress rate response, 66, may be finally written
~-
d ~ . ~_
~J-p~ a~'~ .= a(~) a(e, er) a(r,
j=l
rrer)zj(rref)
(23)
As an example, for a pure shear stress this becomes n 6612 -- ~
~ O'j 12 -- POGr 812 p0GU~;12 - j=l a@) a(8, 8r) a(T, Tref)'cG(Tref)
(24)
In the case of sinusoidally varying deformation, the complex modulus is given by
j-1
1 + icozo
(25)
It follows that its real and imaginary components are, respectively, o ' ( ~ ) - o u + ~,s" - o ~) ~ po 1 j=l 1 -t- co2(z'~) 2
j=l
2.6.4 IDENTIFICATION
1 -4- (-D2(TG)2
(26)
(27)
OF THE PARAMETERS
The crucial problem in vibration experiments concerns the accurate determination of the viscoelastic parameters over a broad range of frequency. Generally, to avoid this difficulty one has recourse to the appropriate principle of equivalence between temperature and frequency, assuming implicitly identical microstructural states. A detailed analysis of the literature has brought us to a narrow comparison of the empirical model of Havriliak and
114
Rahouadj and Cunat
Negami (HN) [9] with the DNLR. The HN approach appears to be successful for a wide variety of polymers; it combines the advantages of the previous modeling of Cole and Cole [10] and of Davidson and Cole [11]. For pure shear stress the response given by this HN approach is 1
G* -- G~N + (G~tN - G~N)[1 + (ic0"CHN)e]//
(28)
where G~IN, G~IN, ~, and /3 are empirical parameters. Thus the real and imaginary components are, respectively, Gt
_
r _ G~N) G~_tN-Jr-(GHN
Gt'
__
(GHN u r _ _ GHN)
COS(riO) [1 + 2CO"V~tNCOS(0OZ/2)+ (D2a152~
(29)
sin(riO) [1 + 2CO~r~tNCOS(0CTr/2)+ (D2~'~2~]/J/2
(30)
The function 0 is defined by 0 - tan-i (1 -c~~t------SN2sin(~ ~ )/ ~ + CO~I:~-iNCOS(0r / 2),]
(31)
Eqs. 28 to 30 are respectively compared to Eqs. 25 to 27 in order to establish a correspondence between the relaxation times of the two models: log('cOr) - 1og('rHN) q-jA/n -}- 0
(32)
where | A, and n are a scale parameter, the number of decades of the spectrum, and the number of processes, respectively. A precise empirical connection is obtained by identifying the shift function for the time scale with the relation v ~ - a(~)r~ r - a(co)r~ ~ - (tan(/30)~v~r \ (D'~HN ,]
(33)
This involves a progressive evolution of the difference of modulus as a function of the applied frequency: (G r -
G u) -
(OhN --
GhN)f G
(34)
The function f6 is given by (1 + tan2(/30))
fo -- cos(riO)[1 + 2co~rhNCOS(0CTr/2)+ co2~z2~1/3/2
(353
115
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
2.6.5 HOW TO USE IT In practice, knowledge of the only empirical parameters of HN's modeling (and/or Cole and Cole's and Davidson and Cole's) permits us, in the framework of the DNLR, to account for a large variety of loading histories.
2.6.6 TABLE OF PARAMETERS As a typical example given by Hartmann et al. [12], we consider the case of a polymer whose chemical composition is 1PTMG2000/3MIDI/2DMPD*. The master curve is plotted at 298 K in Figure 2.6.1. The spectrum is discretized 109
,..10l
i08
I0~
107
lift
m(nz)
106
I
I
I
0
t
I
10
FIGURE 2.6.1 10
1
'
" ....
,
10--2
20
Theoretical simulation of the moduli for PTMG (J).* ,
'i
~
....
~
,
......
|
j
......
$
Log 03(Hz) -40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 10 20 FIGURE 2.6.2
Theoretical simulations of the shift function a(co) and offo for PTMG.*
* PTMG: poly (tetramethylene ether) glycol; MIDI: 4,4'-diphenylmethane diisocyanate; DMPD: 2,2-dimethyl-1, 3-propanediol with a density of 1.074g/cm 3, and glass transition Tg = -40~
116
Rahouadj and Cunat
with A = 6, a scale parameter | equal to - 5 . 6 , and 50 relaxation times. The parameters G~N -- 2.14 MPa, G~N -- Gu - 1859 MPa, ~I~N -1.649 • 10 - 7 s , c z - 0.5709 and f l - 0.0363 allow us to calculate the shift function a(co) and the function fo which is necessary to estimate the difference between the relaxed and nonrelaxed modulus, taking into account the experimental conditions. Figure 2.6.1 illustrates the calculated viscoelastic response, which is superposed to HN's one. The function fo and the shift function a(co) illustrate the nonlinearities introduced in the DNLR modeling (Fig. 2.6.2).
REFERENCES 1. Coleman, B. D., and Gurtin, M. (1967). J. Chem. Phys. 47 (2): 597. 2. De Donder, T. (1920). Lecon de thermodynamique et de chimie physique, Paris: GauthiersVillars. 3. Cunat, C. (1996). Rev. Gcn. Therm. 35: 680-685. 4. Meixner, J. Z. (1949). Naturforsch., Vol. 4a, p. 504. 5. William, M. L., Landel, R. E, and Ferry, J. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Amer. Chem. Soc. 77: 3701. 6. Prigogine, I. (1968). Introduction h la thermodynamique des processus irr~versibles, Paris: Dunod. 7. Kramers, H. A. (1927). Atti. Congr. dei Fisici, Como, 545. 8. Kronig, R. (1926). J. Opt. Soc. Amer. 12: 547. 9. Havriliak, S., and Negami, S. (1966). J. Polym. Sci., Part C, No. 14, ed. R. E Boyer, 99. 10. Cole, K. S., and Cole, R. H. (1941). J. Chem. Phys. 9: 341. 11. Davidson, D. W., and Cole, R. H. (1950). J. Chem. Phys. 18: 1417. 12. Hartmann, B., Lee, G. E, and Lee, J. D. (1994). J. Acoust. Soc. Amer. 95 (1).
SECTION
2.7
Linear Viscoelasticity with Damage R. A. SCHAPERY Department of Aerospace Engineering and Engineering Mechanics, The University of Texas, Austin, Texas
Contents 2.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 2.7.4 Identification of the Material Functions and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 118
119 121
123 123
2.7.1 VALIDITY This paper describes a homogenized constitutive model for viscoelastic materials with constant or growing distributed damage. Included are threedimensional constitutive equations and equations of evolution for damage parameters (internal state variables, ISVs) which are measures of damage. Anisotropy may exist without damage or may develop as a result of damage. For time-independent damage, the specific model covered here is that for a linearly viscoelastic, thermorheologically simple material in which all hereditary effects are expressed through a convolution integral with one creep or relaxation function of reduced time; nonlinear effects of transient crack face contact and friction are excluded. More general cases that account for intrinsic nonlinear viscoelastic and viscoplastic effects as well as thermorheologically complex behavior and multiple relaxation functions are published elsewhere [ 10]. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
117
118
Schapery
2.7.2 BACKGROUND As background to the model with time-dependent damage, consider first the constitutive equation with constant damage, in which e and rr represent the strain and stress tensors, respectively, (1)
= {Sdcr} + ~r
where S is a fully symmetric, fourth order creep compliance tensor and el- is the strain tensor due to temperature and moisture (and other absorbed substances which affect the strains). The braces are abbreviated notation for a linear hereditary integral. Although the most general form could be used, allowing for general aging effects, for notational simplicity we shall use the familiar form for thermorheologically simple materials, { fdg} -
f (~ - ~') - ~ dt' -
/ol
f (~ - ~') --~ d~'
(2)
~'- ~(t')
(3)
where it is assumed f = g = o for t < o and
~-
/o dt"/ar[~r(t")]
Also, at(T) is the temperature-dependent shift factor. If the temperature is constant in time, then ~ - ~' = ( t - t')/ar. Physical aging [12] may be taken into account by introducing explicit time dependence in at; i.e., use ar = a t ( T , t") in Eq. 3. The effect of plasticizers, such as moisture, may also be included in at. When Eq. 2 is used with Eq. 1, f and g are components of the creep compliance and stress tensors, respectively. In certain important cases, the creep compliance components are proportional to one function of time, S = kD
(4)
where k is a constant, dimensionless tensor and D = D(~) is a creep compliance (taken here to be that obtained under a uniaxial stress state). Isotropic materials with a constant Poisson's ratio satisfy Eq. 4. If such a material has mechanically rigid reinforcements and/or holes (of any shape), it is easily shown by dimensional analysis that its homogenized constitutive equation satisfies Eq. 4; in this case the stress and strain tensors in Eq. 1 should be interpreted as volume-averaged quantities [2]. The Poisson's ratio for polymers at temperatures which are not close to their glass-transition temperature, Tg, is nearly constant; except at time or rate extremes, somewhat above Tg Poisson's ratio is essentially one half, while below Tg it is commonly in the range 0.35-0.40 [5].
119
2.7 Linear Viscoelasticity with Damage
Equations 1 and 4 give
e = {Dd(ka)} + er
(5)
= k,{Ed8} - ki{Ede, r}
(6)
The inverse is
where k1 = k -1 and E = E(~) is the uniaxial relaxation modulus in which, fort>o,
{DdE} = {EdD} = 1
(7)
In relating solutions of elastic and viscoelastic boundary value problems, and for later use with growing damage, it is helpful to introduce the dimensionless quantities
_ 1E{ Ea }
4 - ~-~{Eder }
u R - ~---~{Edu}
(8)
where ER is an arbitrary constant with dimensions of modulus, called the reference modulus; also, ee and e~ are so-called pseudo-strains and u R is the pseudo-displacement. Equation 6 becomes o -
Ce R -
Ce~-
(9)
where C =_ ERkI is like an elastic modulus tensor; its elements are called pseudo-moduli. Equation 9 reduces to the equation for an elastic material by taking E = ER; it reduces to the constitutive equation for a viscous material if E is proportional to a Dirac delta function of ~. The inverse of Eq. 9 gives the pseudo-strain eR in terms of stress, eR _ So- + erR
(10)
where S - C -1 - k/ER. The physical strain is given in Eq. 5.
2.7.3
DESCRIPTION
OF THE MODEL
The correspondence principle (CPII in Schapery [6, 8]), which relates elastic and viscoelastic solutions shows that Eqs. 1-10 remain valid, under assumption Eq. 4, with damage growth when the damage consists of cracks whose faces are either unloaded or have loading that is proportional to the external loads. With growing damage k, C, and S are time-dependent because they are functions of one or more damage-related ISVs; the strain ~r may also depend on damage. The fourth-order tensor k must remain inside the convolution integral in Eq. 5, just as shown. This position is required by the correspondence principle. The elastic-like Eqs. 9 and 10 come from Eq. 5, and thus have the appropriate form with growing damage. However, with healing
12 0
Schapery
of cracks, pseudo-stresses replace pseudo-strains because k must appear outside the convolution integral in Eq. 5 [8]. The simplification in Eq. 4 is not needed when damage and crack-face contact (if any) are independent of time. In this case, regardless of the number of different time-dependent components of the compliance tensor, the standard correspondence principle and relatively simple methods of analysis [5] may be used to predict effects of damage on local and global mechanical states. The damage evolution equations are based on viscoelastic crack growth equations or, in a more general context, on nonequilibrium thermodynamic equations. Specifically, let W R and W~ denote pseudo-strain energy density and pseudo-complementary strain energy density, respectively,
wR
1 - - -~C(F_, R -- eR)(e R -- e R) --
F
(11)
W~ - l~oo + ~ a + v
(12)
W~ = - W R + ae R
(13)
so that
and
cgWR OeR
eR = cgW~ Oo
(14)
The function F is a function of damage and physical variables that cause residual stresses such as temperature and moisture. For later use in Section 2.7.4, assume the damage is fully defined by a set of scalar ISVs, Sp (p =1, 2,... P) instead of tensor ISVs. Thermodynamic forces, which are like energy release rates, are introduced,
fP --
cgWa OSp
(15)
or
fp-
cgW~ cgSe
(16)
where the equality of these derivatives follows directly from the total differential of Eq. 13. Although more general forms could be used, the evolution equations for ,Sp - dSp/d~ are assumed in the form
Sp - Sp(Sq, fp)
(17)
121
2.7 Linear Viscoelasticity with Damage
in which Sp may depend on one or more Sq ( q - 1,... P), but on only one force fp. The entropy production rate due to damage is non-negative if
~_,fpSp ~ 0
(18)
p
thus satisfying the Second Law of Thermodynamics. It is assumed that when ]fp] is less than some threshold value, then Sp - O. Observe that even when the stress vanishes, there may be damage growth due to F. According to Eqs. 12 and 16, cgW~
1 c9S
c9~
0F
fP - c9S---~- 2 cgS--pp~a 4- -~p ~ 4- cgS--pp
(19)
which does not vanish when ~ - o, unless cgF/cgSp -- O. The use of tensor ISVs is discussed and compared with scalar ISVs by Schapery [10]. The equations in this section are equally valid for tensor and scalar ISVs.
2.7.4 IDENTIFICATION FUNCTIONS
OF THE MATERIAL
AND PARAMETERS
The model outlined above is based on thermorheologically simple behavior in that reduced time is used throughout, including damage evolution (Eq. 17). In studies of particle-reinforced rubber [4], this simplicity was found, implying that even the microcrack growth rate behavior was affected by temperature only through viscoelastic behavior of the rubber. If the damage growth is affected differently by temperature (and plasticizers), then explicit dependence may be introduced in the rate (Eq. 17). In the discussion that follows, complete thermorheological simplicity is assumed. The behavior of particle-reinforced rubber [1,4] and asphalt concrete [3] has been characterized using a power law when fp > o,
Sp - ( f p ) %
(20)
where 0~p is a positive constant. (For the rubber composite two ISVs, with C~1 ---- 4.5 and 0~2 = 6, were used for uniaxial and multiaxial behavior, whereas for asphalt one ISV, with 0~= 2.5, was used for uniaxial behavior.) A coefficient depending on Sp may be included in Eq. 20; but it does not really generalize the equation because a simple change of the variable Sp may be used to eliminate the coefficient.
122
Schapery
Only an outline of the identification process is given here, but details are provided by Park et al. [3] for uniaxial behavior and by Park and Schapery [4] and Ha and Schapery [1] for multiaxial behavior. Schapery and Sicking [11] and Schapery [9] discuss the model's use for fiber composites. The effects of ~r and F are neglected here. (a) The first step is to obtain the linear viscoelastic relaxation modulus E(~) and shift factor a~- for the undamaged state. This may be done using any standard method, such as uniaxial constant strain rate tests at a series of rates and temperatures. Alternatively, for example, uniaxial creep tests may be used to find D(~), after which E(~) is derived from Eq. 7. (b) Constant strain rate (or stress rate) tests of specimens taken to failure at a series of rates or temperatures may be conveniently used to obtain the additional data needed for identification of the model. (However, depending on the complexity of the material and intended use of the model, unloading and reloading tests may be needed [7].) Constant strain rate tests often are preferred over constant stress rate tests because meaningful post stress-peak behavior (prior to significant strain localization) may be found from the former tests. For isothermal, constant strain rate, R, tests, the input is R t - / ~ , where R--Ra~ and ~ - t / a ~ . Inasmuch as the model does not depend on temperature when reduced time is used, all stress vs. reduced time response curves depend on only one input parameter /~ regardless of temperature. Thus, one may obtain a complete identification of the model from a series of tests over a range of/~ using one temperature and different rates or one rate and different temperatures; both types of tests may be needed in practice for/~ to cover a sufficiently broad range. One should, however, conduct at least a small number of both types of tests to check the thermorheologically simple assumption. (c) Convert all experimental values of displacements and strains from step (b) tests to pseudo-quantities using Eq. 8. This removes intrinsic viscoelastic effects, thus enabling all subsequent identification steps to be those for a linear elastic material with rate-dependent damage. If controlled strain (stress) tests are used, then one would employ wR(w R) in the identification. However, mixed variables may be input test parameters, such as constant strain rate tests of specimens in a test chamber at a series of specified pressures [4]. In this case it is convenient to use mixed pseudoenergy functions in terms of strain and stress variables. Appropriate energy functions may be easily constructed using methods based on linear elasticity theory. (d) The procedure for finding the exponent a and pseudo Young's modulus in terms of one damage parameter is given by Park et al. [3]. After this, the
2.7 Linear Viscoelasticity with Damage
123
remaining pseudo-moduli or compliances may be found in terms of one or more ISVs, as described by Park and Schapery [4] using constant strain rate tests of bar specimens under several confining pressures. The material employed by them was initially isotropic, but it became transversely isotropic as a result of damage. Identification of the full set of five pseudo-moduli and the pseudo-strain energy function, as functions of two ISVs, is detailed by Ha and Schapery [1].
2.7.5 HOW TO USE THE MODEL Implementation of user-defined constitutive relations based on this model in a finite element analysis is described by Ha and Schapery [1]. Included are comparisons between theory and experiment for overall load-displacement behavior and for local strain distributions. The model employed assumes the material is locally transversely isotropic with the current axis of isotropy assumed parallel to the current local maximum principal stress direction. A procedure is proposed by Schapery [10] that enables use of the same model when transverse isotropy is lost due to rotation of the local maximum principal stress direction.
REFERENCES 1. Ha, K., and Schapery, R. A. (1998). A three-dimensional viscoelastic constitutive model for particulate composites with growing damage and its experimental validation. International Journal of Solids and Structures 35: 3497-3517. 2. Hashin, Z. (1983). Analysis of composite materials - - a survey. Journal of Applied Mechanics 105: 481-505. 3. Park, S. W., Kim, Y. R., and Schapery, R. A. (1996). A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete. Mechanics of Materials 24: 241-255. 4. Park, S. W., and Schapery, R. A. (1997). A viscoelastic constitutive model for particulate composites with growing damage. International Journal of Solids and Structures 34: 931-947. 5. Schapery, R. A. (1974). Viscoelastic behavior and analysis of composite materials, in Mechanics of Composite Materials, pp. 85-168, vol. 2, Sendeckyi, G. P., ed., New York: Academic. 6. Schapery, R. A. (1981). On viscoelastic deformation and failure behavior of composite materials with distributed flaws, in 1981 Advances in Aerospace Structures and Materials, pp. 5-20, Wang, S. S., and Renton, W. J., eds., ASME, AD-01. 7. Schapery, R. A. (1982). Models for damage growth and fracture in nonlinear viscoelastic particulate composites, in: Proc. Ninth U.S. National Congress of Applied Mechanics, Book No. H00228, pp. 237-245, Pao, Y. H., ed., New York: ASME.
124
Schapery
8. Schapery, R. A. (1984). Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media, in: International Journal of Fracture 25: 195-223. 9. Schapery, R. A. (1997). Constitutive equations for special linear viscoelastic composites with growing damage, in Advances in Fracture Research, pp. 3019-3027, Karihaloo, B. L., Mai, Y.-W., Ripley, M. I., and Ritchie, R. O., eds., Pergamon. 10. Schapery, R. A. (1999). Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage. International Journal of Fracture 97: 33-66. 11. Schapery, R. A., and Sicking, D. L. (1995). On nonlinear constitutive equations for elastic and viscoelastic composites with growing damage, in Mechanical Behavior of Materials, pp. 45-76, Bakker, A., ed., Delft: Delft University Press. 12. Struik, L. C. E. (1978). Physical Aging in Amorphous Polymers and Other Materials, Amsterdam: Elsevier.
SECTION
Yield Limit
3
This Page Intentionally Left Blank
SECTION
3.1
Introduction to Yield Limits J E A N LEMAITRE
Universit~ Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France
Almost all materials have a range of loading below which the only mechanism of strain is reversible movements of atoms giving rise to the mesoscopic phenomenon of elasticity. The limit above which permanent or irreversible strains exist is most often related to stress and is called yield stress. It is not easy to measure the yield stress of a material because it depends upon the value of the offset of permanent strain allowed to detect that this irreversible strain ~p begins to exist: 9 In engineering, a value of ep = 0.2 x 1 0 - 2 is usually taken to define the yield stress cry(0.2.). This means that at this level of stress the permanent strain is of the order of magnitude of the elastic strain. This is acceptable for comparing materials or for the perfect plasticity scheme. 9 But in modeling plasticity or viscoplasticity this definition is generally not acceptable. A much smaller value must be taken to predict the small values of plastic strains which may occur in structure analysis. A fraction of the elastic strain is a better definition, for example, Cry = ~r%=o.1~0. The point is that different models of plasticity may use different values of the yield stress for the same material! More details are given in Section 3.2. When "the" value of the yield stress from a tension test is known, the problem is often to compare it with a two or three-dimensional state of stress to know if a part of a structure is loaded in its elastic range or not. The corresponding scalar function of the stress components is called a yield criterion. Many yield criteria have been proposed, depending upon the properties of the materials and the purpose. Handbook of Materials BehaviorModels. ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
127
128
Lemaitre
Isotropic criteria are described in Section 3.2, with the von Mises criterion being widely used in structures calculation through its equivalent stress aeq.
O'eq _ _
aD ij a Dij
0 "D
is the stress deviator aDo --
aij
-
aHbij.
1
aHiS the hydrostatic stress aH -- -~ akk. Because of previous forming, many metals have a preferred orientation or texture, which can be characterized by pole figures (see Section 3.3). Because of this, the material is anisotropic, so that anisotropic yield criteria are needed (see Section 3.4), often involving distortion of the yield surface (Section 3.5). Since all criteria have a common mathematical structure, it is possible to write down a general framework for them (Section 3.6). For engineering purposes it is also possible to extend the local concept of yield criteria to structures like beams, plates or shells, as shown in Section 3.7.
SECTION
3.2
Background on Isotropic Criteria DANIEL C. DRUCKER Department of Aerospace Engineering, Mechanics & Engineering Service, University of Florida, 231 Aerospace Building, Gainesville, Florida 32611
An elastic- perfectly plastic stress-strain curve in simple tension or in simple shear is the simplest picture of material behavior from which to begin this discussion of yield limit criteria. This idealized response is linear elastic up to a yield limit at which large plastic deformation occurs at constant stress. All reasonable definitions of yield limit, such as the initial departure from linearity or a very small or a moderate or a fairly large plastic strain offset, give the same value. Much of early and present plasticity theory and its use in analysis and design springs from this picture, which appears to match the appreciable plastic deformation at the lower yield point observed in tests on standard tension specimens of mild steel. The actual abrupt jump in strain at the much higher upper yield point, which can be exhibited in a carefully performed test on a well-prepared annealed and homogeneous specimen, is hidden by inhomogeneity, appreciable residual stress, and not quite central loading of the usual specimen. A macroscopic stable quasi-static perfectly plastic response at the lower yield point may be appropriately assumed for mild steel in limit analysis and design and other aspects of structural engineering practice because the unstable dynamic response, the highly local jump from the upper yield point to appreciable plastic deformation at the lower yield point, is almost completely masked. In contrast, the initial stress-strain curve for most workhardening structural metals or alloys in simple tension or simple shear also plots as an elastic straight line in the beginning stage but then deviates only gradually from linear elastic, slowly getting flatter as the plastic response grows with increasing stress. Upon full or partial unloading from any state of stress and subsequent reloading, the behavior is essentially linear elastic until plastic deformation resumes, and the stress-strain curve then follows the curve that would have been followed in the absence of unloading. (Fig. 3.2.1). Handbookof MaterialsBehaviorModels.ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
129
130
Dmc~r Largeoffset% Moderateoffset% Small offset %
!
%
/
/
/
/
/
/
/
/
/
/
,
Strain FIGURE 3.2.1 Perfectplasticity and workhardening. Yield limit defined by small, moderate, or large offset or as by Taylor and Quinney. The concept of an initial or a subsequently established yield limit for the typical workhardening response involves both idealization and definition. A key element of the idealization is that the response of the material to stress levels below the current yield limit is purely elastic on loading, as it is to at least moderate unloading and subsequent reloading up to the current yield limit. However, we know that large numbers of mobile dislocations will be present, some of which will move under the smallest increase or decrease of stress in the nominally elastic range. The macroscopic strain produced by this dislocation motion on the atomic scale is small, but it is measurable with the advanced instrumentation now available. Idealizing the response by ignoring these and otherwise produced very small inelastic strains is almost always permissible, both in engineering practice for elastic or elastic-plastic structural analysis and design and in the usual description of the macroscopic behavior of the material in the laboratory. Idealization of materials of construction as time-independent in the vicinity of room temperature and below likewise ignores the small inelastic strain changes over time that do occur but that are not significant over the times of interest for the materials employed. The most convenient definition of an initial or subsequent yield limit normally follows from the choice of a value of plastic deformation that is meaningful for the applications in mind and that is consistently measurable. The use of the familiar 0.002 offset for the initial yield limit in tension certainly satisfies the need for material quality control, with its reproducibility and agreement among laboratories in the test community, but is likely to be considered much too large by a factor of 100 or more by those members of the experimental plasticity community interested in many of the interesting and important details of macroscopic elastic-plastic behavior. At room
3.2 Background on Isotropic Criteria
131
temperature and below, loading into the plastic range and then moderate unloading leaves most structural metals in a workhardened state with stressstrain plots for reloading usually showing a close to sharp departure from linearity when the new yield limit is reached. A small offset definition of yield then will give reproducible results, in contrast to the difficulty of its use for the initial curve in the unhardened state. Some materials do exhibit visible rounding of the stress-strain curve for reloading, the more so the fuller the prior unloading. A moderate offset definition of yield then avoids disagreement, but ignores possibly significant behavior. Broadening the picture from the simple tension or simple shear curve to general states of stress and strain is straightforward. The initial yield limit and subsequent or current yield limits are replaced by an initial yield limit function of stress and subsequent or current yield functions, each of which may be represented by a surface in stress space. The straight line elastic range generalizes to the region of stress space enclosed by the current yield surface. Subsequent changes of stress represented by motion of the stress point within the current yield surface cause purely elastic changes in strain and do not affect the current yield surface at all in the idealized material. Plastic strain in addition to elastic occurs only when the stress point moves out beyond the current yield surface. However, only one point on the yield surface is given by the current yield point in simple tension or in simple shear. The appropriate generalization from one dimension or two to the nine dimensions of stress (six are independent) is far from unique. Each material is different and will have different carryovers from one or two dimensions. Experiment alone can provide answers, and even experiments cannot cover all the possibilities. Assumptions are also needed whose validity can be tested only partially by comparing the actual behavior observed with the consequences predicted by the assumed generalization. In view of the essentially infinite complexity of the inelastic behavior of real materials and their permissible idealizations, isotropy (independence of material orientation) provides a simple and appealing hypothesis from which to start. The idealization of an isotropic material does serve as an excellent representation for the purely elastic macroscopic behavior of many polycrystalline metals and alloys, despite the often strongly anisotropic response of their individual crystals or grains. A sufficiently large assemblage of such anisotropic grains with sufficiently varied orientations of the crystalline planes will exhibit an isotropic elastic response on the macroscale of many grain diameters, because there will be no favored orientation. It was natural to carry over the simplicity of isotropy to yield limit criteria when the mathematical theory of plasticity was being developed. Tresca had shown experimentally that plastic deformation took place when a limiting shear stress was exceeded. A maximum shear stress limit criterion
13 2
Drucker a2
FIGURE 3.2.2
Tresca hexagon, Mises or J2 ellipse, and intermediate curve for plane stress.
independent of orientation is an isotropic criterion that plots as the familiar hexagon in the two-dimensional principal stress space of plane stress and as a hexagonal prism in three-dimensional principal stress space with its axis equally inclined to the three principal stress axes. It is a good first approximation to physical reality and became accepted as the best simple representation of the yield limit for metals and alloys. Mises, as an applied mathematician, did not take issue with this consensus but for analytic convenience preferred that a smooth surface be employed instead (Fig. 3.2.2). In the purely elastic range, stress determines strain and vice versa. At each stage of plastic deformation, it is the increment in strain, or rate of strain, that is determined by the increment in stress, or rate of stress, for a given state of stress at yield. In an analogy to potential functions that governed many other types of physical behavior, Mises proposed a normality rule, the yield function expressed in terms of stress serving as a potential function for the increment or rate of plastic strain. At each value of stress at yield (in pictorial terms, the stress point lies on the yield surface in stress space), the plastic strain rate or increment is given by a scalar multiple of the derivative of the yield function with respect to stress (the strain rate or increment vector in the strain space superposed on the stress space is normal to the yield surface at the stress point). The three principal stresses al, a2, a3 and their orientation in the material provide a complete description of the state of stress. For an isotropic material, the yield limit surface is determined by the values of the principal stresses without regard to their orientation. Each combination of them, such as the
133
3.2 Background on Isotropic Criteria
m a x i m u m shear stress, is also an invariant of the stress field. With the knowledge that shear stress is the primary driver of plastic deformation, it is reasonable to do what is always permissible, to consider any state of stress at a material point as a hydrostatic tension equal to the mean normal stress plus the stress deviator, which is a general representation of shear stress. This leads naturally to the choice of the three independent invariants of stress as the sum of the principal stresses J1 = o ' 1 - + - 0 " 2 + 0"3 or O'kk and the two additional invariants J2, J3 that appear in the cubic equation for the determination of the three principal stress deviators from the nine components of the stress deviation si) = ai) - ( 6 k k / 3 ) 6 i j . The principal stress deviators (Sl, s2, s3), just like the principal stresses themselves, are independent of the arbitrary choice of the rectangular Cartesian axes for the stress components. Therefore, when the coefficient of the cubic term in the equation is set as unity, the remaining coefficients and the remaining term are invariants, the same for all choices of axes. These invariants are the quadratic coefficient of the linear term, the second invariant of the stress deviation tensor, J2 -
[ ( a l - ~2) 2 + (~2 - ~3) 2 + a3 - a l ) 2 1 / 6 -
(s21 + s 2 + s32)/2
or
s~jsj~/2
and the third invariant of the stress deviation tensor, which is cubic in stress, J3
-
[(2o1
-
02
-
03)(202
-
03
-
o1)(203
or
-
o1
s~jsjkSk~/3
-
a2)]/27
-
(s 3 + s 3 + s 3 ) / 3
The coefficient of the quadratic term, the sum of the principal values, or the equivalent of J1 in the cubic for principal stress, is zero for the deviator. Any other invariant of stress can be shown to be a combination of J1, J2, and J3. J2 is the simplest isotropic smooth function of shear stress. It also has a number of physical interpretations that appeal strongly to many who are not enamored of purely mathematical arguments. Among these are the mean of the squares of the principal shears and the square of the octahedral shear stress, the shear stress on the planes whose normals make equal angles with the three principal directions of stress. Probably the most compelling of all is that postulating a limiting value of J2 for the purely elastic range is the same as postulating that plastic deformation will ensue when the elastic energy of distortion or shear strain energy exceeds a critical value. Appealing and useful as any such interpretation may be, it should be kept in mind that none are based firmly on physical laws. When J2 is set equal to a constant to match the yield point in a onedimensional test, it plots as an ellipse for plane stress and as a circular cylinder in three-dimensional principal stress space. This choice is known now as the Mises criterion of yield, although Mises himself recognized that far more
134
Drucker
general isotropic forms existed and did not restrict his normality rule to the second invariant form alone (Fig. 3.2.3). Although the assumption of isotropy of the initial and subsequent yield limits is mathematically convenient, it does not usually represent physical reality when any of the customary definitions of the onset of initial or of additional plastic strain are employed. It takes considerable effort to produce a structural metal or alloy that is initially isotropic in its inelastic response. Osgood obtained such a structural aluminum alloy from ALCOA. In a very interesting set of experiments on thin-walled tubes under proportional or radial loading obtained by increasing tension and interior pressure in constant ratio, he showed good correlation of the data with both the maximum shear stress criterion of Tresca and the second invariant of the stress deviation tensor of Mises. The experiments were performed with great care so that it was possible for me at a little later time to show that although each fit the experiments within 10%, neither provided the most appropriate match for the data. The actual isotropic shear stress criterion lay about halfway between these two basic forms. A simple combination of the second and the third invariant of the stress deviation tensor, J23 - 2.25J~, did correlate all the data remarkably well. Interesting and significant as this result is, the more important point is that initial isotropy is lost once significant plastic deformation occurs. The picture of isotropic hardening, subsequent yield surfaces, being simply enlarged versions of the initial yield surface, is unreal despite its ability to correlate the results of radial loading tests on initially isotropic material. Structural metals and alloys are quite anisotropic in their initial and subsequent plastic response a2
L i I | ~ I
lli
Ii
FIGURE 3.2.3
Radial loading and isotropic hardening.
3.2 Background on lsotropic Criteria
135
as defined by any of the usual definitions of yielding, including the moderately large 0.2% or 0.002 plastic strain offset. The mechanical treatment given prior to use often involves appreciable plastic deformation, both cold and hot. Such plastically deformed metal is highly likely to have directional properties to start as well as the related difference in the stress-strain curves in tension and compression exhibited by Bauschinger over 100 years ago. A workhardening metal subjected to shear in one sense will have a raised yield stress for that sense and a lowered yield stress for reverse shear, much as loading in tension will produce a raised yield stress in tension and a lowered one in compression. Furthermore, structural metals and alloys generally come with appreciable initial or residual stresses because of their prior thermal or mechanical treatment. The degree of anisotropy induced by plastic deformation is very large when very much smaller deviations from elastic response than 0.002 are taken to define the onset of plastic deformation. With 1000 times greater sensitivity, some reverse plastic deformation often will be picked up upon unloading a plastically deformed material to zero stress. The diameter of each current yield surface so defined is a small fraction of the current yield value in simple shear, and the shape of each current yield surface is far from the Tresca or Mises or any other isotropic criterion. At 100 times greater sensitivity still, the diameter of any yield surface so defined shrinks to zero. At the other extreme, when interest is focused on large plastic deformations of 1% (0.01) or more, initial isotropy and isotropic hardening obeying the Mises criterion provide a crude but not unreasonable approximation. Bauschinger and allied cross-effects, the changes of shape of successive yield surfaces and the motion of their centers, so prominent when small offset definitions are employed, almost disappear from view along with the plastic strains that occur as the stress point moves about inside the current Mises ellipse. Plastic deformation of single crystals and polycrystalline metals and alloys is caused primarily by shear stress that exceeds the purely elastic carrying capacity. Schmid's "law" for single crystals states that the shear stress on the plane of slip governs. The shear stress needed for the common ductile structural metals is in fact independent of the normal stress on the planes of slip under most circumstances with the normal stress of the order of the yield stress. However, at extremely high hydrostatic pressure, Bridgman did find a 10% increase in the shear stress needed, an increase he thought was expected from the compaction of the atomic structure, but not very significant for the theory of plasticity. On the other hand, hydrostatic pressure, or normal stress on shear planes, does have an appreciable effect on polymeric materials, as would be expected from their much more open molecular arrangements. Furthermore, the ratio of shear stress to normal stress on the plane of slip is well known to be the
136
Drucker
controlling variable for simple frictional materials and not surprisingly plays a strong role in soils and other granular media. When it is permissible and useful to carry over expressions for plastic deformation to such materials, an isotropic criterion of yield must include the hydrostatic pressure in addition to measures of shear stress. The most general isotropic criterion then will be a function of the first invariant of the stress tensor (the sum of the principal stresses) along with the second invariant of the stress deviation tensor (the Mises criterion) and its third invariant. The simplest of these forms, f - ~J1 q_j~/2, now termed the Drucker-Prager criterion with ~ constant, gives a linear increase of the required (mean) shear with added hydrostatic pressure. It is represented by a right circular cone in principal stress space. The apex of the cone is at the origin (zero stress) for a cohesionless material and lies on the triaxial tension line for a material with cohesion. The Tresca, or maximum shear stress criterion, may be modified similarly to give a linear variation of the maximum shear stress with hydrostatic pressure. It plots as a regular six-sided pyramid in principal stress space. A Mohr-Coulomb material, which postulates the frictional criterion of a limiting shear stress on the plane of slip proportional to the normal stress on that plane, is frequently used as a model for granular material. It also is represented by a six-sided pyramid in principal stress space, but its crosssection by a plane perpendicular to the axis is not a regular hexagon. Nonlinear instead of linear variation of shear stress with normal stress or pressure can be accommodated simply. Greater realism within the idealization of isotropy can be provided by a moving yield limit cap over the open end of the yield limit cone or pyramid in principal stress space. The cap permits matching of consolidation under hydrostatic pressure and under pressure plus moderate shear. It also can be employed to reduce the undesirably high volumetric expansion otherwise produced in each of the open-ended models of behavior by normality of the plastic strain increment to the current yield limit surface. These and other related topics are discussed clearly in "Soil Plasticity Theory and Implementation" by W. E Chen and G. Y. Baladi (Elsevier, 1985) and in the follow-up volume, "Nonlinear Analysis in Soil Mechanics" by W. E Chen and E. Mizuno (Elsevier, 1990). The simplicity of isotropic initial yield limits and subsequent yield limits (isotropic hardening) is a great help in computation and in obtaining a crude but very useful picture of the plastic response. However, good agreement in detail with what will actually happen when a structure or a specimen deforms in the plastic range requires much more realistic yield limits, as described in the other sections of this chapter and in subsequent chapters.
SECTION
3.3
Yield Loci Based on Crystallographic Texture P. VAN HOUTTE Department MTM, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium
Contents 3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Quantitative measurement of the texture. 3.3.2.1 Desired Form . . . . . . . . . . . . . . . . . . . . . 3.3.2.2 Calculation of ODF from Pole Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.3 Determination of ODF by Single Grain Orientation Measurements .. 3.3.2.4 Sample Homogeneity . . . . . . . . . . . . . . 3.3.2.5 Discretization of the ODF . . . . . . . . . 3.3.3 Cconstitutive model for polycrystals with texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.1 Plastic Potentials . . . . . . . . . . . . . . . . . . 3.3.3.2 Crystal Plasticity Model . . . . . . . . . . . 3.3.3.3 Exploitation of the Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137 139
139 140 141 142 143 144
144 145 150 153
3.3.1 BACKGROUND This s e c t i o n p r e s e n t s m e t h o d s that are suitable for d e r i v i n g the yield locus of a p o l y c r y s t a l l i n e m a t e r i a l from a physical m e a s u r e m e n t of its c r y s t a l l o g r a p h i c texture. In principle, these m e t h o d s are l i m i t e d to s i n g l e - p h a s e m a t e r i a l s in a n n e a l e d c o n d i t i o n , w i t h e q u i a x e d grains sufficiently small as to n e g l e c t m a t e r i a l h e t e r o g e n e i t y at the scale at w h i c h one w a n t s to use the yield locus. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
137
138
Van Houtte
Strain rate sensitivity is supposed to be positive or negative but small enough in absolute value so that the exponent kt--c3 log ~r/~ log ~ (with cr the flow stress and ~ the strain rate) can be sufficiently well approximated by the value zero. For materials like steel and aluminium and their alloys, this holds from room temperature up to about 200~ in which range experimental measurements of # yield values between -0.05 and +0.05, depending on alloy content, temperature, and strain rate. These experimental values are rarely published but can be found in the - - often confidential m files of industrial companies and research institutions. Methods as described in the present chapter have almost exclusively been applied to fcc, bcc, and hcp metals, although in principle they could be used with any type of ductile polycrystalline material. Typical applications are anisotropic finite element (FE)-based simulations of metal-forming processes, mainly sheet metal forming (car body parts, can stock, etc.) but also rolling and wire drawing. The purpose of such FE simulations is to predict the final shape and dimensions of the product, forming limits (failure), spring-back and/or residual stress, final texture and final grain size. Sometimes the yield locus models are applied to cases for which, strictly speaking, they have not been intended, such as materials which already have been deformed before. The results must then be judged with care but may still be valid, the induced error being smaller than errors resulting from other sources. One may wonder, why bother about deriving a yield locus from a crystallographic texture the hard way, instead of determining the coefficients of a Hill-type anisotropic yield locus (or another type of simple analytical yield locus expression) by fitting them to some tensile test data? The answer consists of two parts: 9 simple analytical yield locus expressions are never versatile enough to describe the true yield loci of engineering materials; 9 during routine FE simulations of a forming problem, the FE code may submit very unexpected multiaxial loading cases to the constitutive material model, which cannot possibly have been explored beforehand by mechanical tests. The topic of the present section is too complicated to be described in a few pages of equations and a few tables. Instead, I will explain which types of software or other tools are needed, and to which important aspects attention should be paid. Often references will not be made to the earliest original papers on the topic, but rather to comprehensive books such as Reference [1] or to review papers. This has intentionally been done to make it easier for students and practice-oriented readers to collect the necessary papers.
139
3.3 Yield Loci Based on Crystallographic Texture
For the industrial applications mentioned previously, the ultimate quality criterion is quantitative agreement between FE predictions and the actual behavior of the material studied. This can only be achieved if the greatest possible care is given to the quantitative measurement of the initial texture, which is the topic of the next section.
3.3.2 QUANTITATIVE MEASUREMENT OF THE TEXTURE 3.3.2.1
DESIRED FORM
Yield locus calculation is a special case of the calculation of physical properties from a material with crystallographic texture [1]. Let g be the crystallite orientation, and q(g) some property which depends on it. The simplest possible micro-macro transition is one for which the macroscopic value qm of the property is given by a weighted average: qm-/f(g)q(g)dg
(1)
in which the weighing factor f(g) is the "orientation distribution function" (ODF) which describes the texture of the polycrystal [2-3]. The crystallite orientation g may be conceived as a tensorial quantity [4], but it is most often expressed as a set of three Euler angles representing the rotation from an external reference system ("sample reference system") to a crystal reference system (the ( 1 0 0 ) axes in the case of fcc or bcc metals) [2-3]. The three Euler angles (991, ~, and (02 in Bunge's notation) are then denoted by the single symbol g. The averaging formula above is the simplest possible micro-macro transition; so at the very least, the texture must be known under the form of an ODF f(g) The ODF itself may be known as a discrete function in Euler space, or as a Fourier series [2-3]: L
M(O N(O
Z E E e=O
#=1
v
v=l
L is the maximum degree of the series expansion. The angular resolution that :.//V can be achieved by this series expansion is roughly 160~ Tr (g)is a harmonic function of degree ( of the three Euler angles. C~~ is a Fourier coefficient. In principle, the property q(g) can also be described as #v a Fourier series expansion with coefficients qe [2]. In that case, the average
140
Van Houtte
is given by 1 qm--
f=O
M(e) N(e)
2~,+1 ~
. pv
#=I
Zqe
,~v
C~v
(3)
v=l pv
in which qe is the complex conjugated of qe [2]. This expression is helpful for the study of the angular resolution required for the ODE Suppose that the Fourier coefficients of the property q(g) are all zero for f _> Lq. In that case, L in the above equation can be reduced to Lq, which sets the desired angular resolution for the ODF (see above) but also strongly limits the number of multiplications and additions required to evaluate the expression. For elastic problems, Lq is as low as 4. For problems related to yield loci of fcc or bcc pv metals, L need not be chosen higher than 12, the qe taking very low values for f > 12. As a result, Eq. 3 allows for an incredible gain in calculation time when evaluating integrals such as Eq. 1.
3.3.2.2
CALCULATION OF O D F
FROM POLE FIGURES
Companies like Bruker, Inel, Philips, Seifert, and others sell x-ray equipment for the measurement of "pole figures" which characterise the crystallographic texture of the material. (They can also be measured using neutron diffraction.) A pole figure is not an ODF; it is necessary to measure several pole figures for each material and then combine them by means of appropriate software in order to obtain the ODF [2-3]. Such software can also be purchased from these companies, or it can be obtained from university laboratories that have developed their own. In the latter case, the texture goniometer and its control unit must be capable of exporting pole figure files in a format that can be read by the user. Other important points are as follows: 1. A high angular resolution is NOT required (5 ~ is sufficient), as long as no intense pole figure peaks are completely "missed." If this is a problem, a sufficiently wide x-ray beam should be used. 2. Great care must be given to various "corrections" to be applied to pole figure data. These are: a. Background subtraction. This means that part of the intensity measured in each direction is ascribed to "background radiation" and is removed from the signal for further processing. All commercial pole figure measuring systems contain some procedure for background subtraction. However, it is sometimes not good enough. The effect of subtracting too much or too less background, combined with subsequent data processing to generate an ODE amounts to effectively redistributing
141
3.3 Yield Loci Based on Crystallographic Texture
intensity of the strong texture components into the random component or vice versa. All C-coefficients of the ODF will be erroneous, including those of low rank ~e, and the value of all texture-related properties will be wrongly estimated, irrespective of the angular resolution that is needed for the property considered. So one should thoroughly study the procedure for background correction contained in the standard software, and if it is not good enough, replace it with a sound procedure. Even better is to physically remove background radiation by, using, for example a set of Soller slits combined with a monochromator in the diffracted beam [5]. b. Defocusing correction. A good method is the one based on a pole figure measurement of a good powder sample with random texture. Here, also, the standard procedure provided with a commercial system should be evaluated with care. 3. Number of pole figures. For fcc metals, it is highly recommended that one measure four pole figures, namely, {1 1 1 }, {2 0 0 }, {2 2 0}, and {3 1 1 }. For bcc metals, these are {1 1 0}, {200}, {1 1 2}, and {103}. It is possible to work with three pole figures only if they are completely measured (inclination range from 0 ~ to 90~ which is rarely the case. Normally they are measured using the Schulz reflection method (0 ~ to 80 ~ range) [6-7]. In that case, four pole figures are recommended if quantitative applications are aimed at. The software used to convert the pole figures into an ODF should have a ghost correction procedure [3] if plastic properties are to be calculated. If that is not done, no information can be obtained about C-coefficients of odd rank 2. It has been said in the preceding that, for cubic materials, the series expansion degree L (see Eq. (2)) should not be lower than 12 if plastic properties are to be calculated. Now in cubic materials, the first nonzero odd /zv C-coefficients (or, for that matter, qe coefficients) which are encountered in Eq. 3 appear at ~e= 9, so they cannot be neglected. The pole figure to ODF conversion method may be based on the harmonic method or not [3], but at least, once the ODF is obtained, it is advantageous to convert it into a C-coefficient representation, so that Eq. 3 can be used.
3.3.2.3
DETERMINATION OF O D F
BY SINGLE GRAIN
ORIENTATION MEASUREMENTS An alternative method of texture determination consists of directly measuring the orientation of a large number of single grains of the polycrystalline material. This can be done by various methods. By far the most convenient of these makes use of an automated system of orientation determination using
142
Van Houtte
"electron back scattering diffraction" (EBSD) in a scanning electron microscope ([6] pp. 171-176, [8]). The electron beam of the microscope is automatically positioned on a series of points at the surface of a sample. At each point, an EBSD pattern is generated, recorded, and analyzed by a computer, leading to a set of Euler angles describing the local crystallite orientation. Several commercial companies offer systems like this. At the present date, these systems offer a spatial resolution of about 0.5 gm, an angular resolution in Euler space better than 1~ and the time to measure one orientation is about i sec. It is expected that in the near future this time will go down by a factor of 5 or more. The determination of an ODF is only one of the many possible applications of these instruments. It requires the acquisition of 10,000 or more single grain orientations. So with present technology, it takes 3 to 12 hours (depending on the number on measurements desired) to measure an ODE an automated job that can be done during the night. A list of 10,000 to 50,000 sets of Euler angles is obtained, which can be converted into a continuous ODF described by C-coefficients. The method has several advantages compared to pole figure measurements. Since the ODF is directly measured, pole figure to ODF conversion is not necessary. Also, both the even and the odd part of the ODF are directly obtained; as a result, no ghost correction is needed. The various pole figure corrections discussed in the previous section are not needed either. However, the method has some drawbacks: 9 The grain size should be at least 0.5 gm, although in the foreseeable future this may go down to 0.1 gm. 9 Material with a high dislocation density (after plastic straining) cannot be studied; however, in most cases, the problem can be solved by giving the material a prudent recovery treatment. 9 The purchase and maintenance of the equipment (including a dedicated scanning electron microscope) are more expensive than those of a texture goniometer using x-ray diffraction, though not that much.
3.3.2.4
SAMPLE HOMOGENEITY
It is frequently observed that the texture of the sample is not homogeneous. In the case of sheet material, the texture may vary with distance from the surface. It is wise to check for this phenomenon by doing texture measurements at various depths (after removal of part of the surface). If a texture gradient indeed exists, it is recommended to do several measurements and construct an average, or to perform texture measurements on crosssections or oblique sections, after which the measured texture must of course
3.3 Yield Loci Based on Crystallographic Texture
143
be rotated. University-developed texture software packages often contain routines for performing these operations (averaging several textures, rotating of textures).
3.3.2.5
DISCRETIZATION OF THE O D F
It is possible that in the applications that will follow (see next section), one intends to implement Eq. 1 by means of C-coefficients, using Eq. 3; in that case, it is not necessary to convert the measured ODF into a set of weighted discrete orientations. However, in the opposite case, such a procedure will be required. The set of discrete orientations can then be used by some software that models the plastic response of the material. The procedure used for the discretization must then satisfy a few important conditions: Sample symmetry should NOT be used to reduce the number of orientations in the set, unless one is sufficiently experienced to avoid mistakes when calculating the plastic properties. The method should be "unbiased." This requires some explaining. Suppose that one wants to compute an average value of a texture-related property by means of Eq. 1. Let qm be the "true value" obtained by using a nondiscretized ODE and ~lm the value obtained from the discretised ODE Normally ~lm will be different from qm. However, it is said that elm is "unbiased" (using terminology from statistics) when its "expected value" (i.e., its most probable a priori value) is equal to q=. Whether that will be the case or not depends on the discretization procedure and on the formula to calculate elm- When both are known, one can formulate a "statistical theorem" stating that ~lm is unbiased, and try to demonstrate it. To the knowledge of the author, for only one discretization method has such proof ever been published, namely, for the "statistical method," one of the two discretization methods explained in Reference [9]. In addition, the theorem also holds for a discretization method in which one would look at the intensity of the ODF f(g) for all points gi in Euler space on a 5 ~ grid, retain all these points in the discretization, and give them a weighting factor f(gi)sin ~i ((I)i is one of the Euler angles used by Bunge [2]). However, a commonly used discretization method consists in retaining only those grid orientations in the discrete set for which f(gi) > a, a being some threshold value between 0 and 1. This latter method is biased, and will lead to systematic errors on any ~lm value. Such ~lm values are unsuited for predictions of plastic properties.
144
Van Houtte
3.3.3 CONSTITUTIVE MODEL FOR POLYCRYSTALS WITH TEXTURE 3.3.3.1
PLASTIC POTENTIALS
The concept of a "plastic potential" seems most suitable to performing the micro-macro transition and to carrying the information about the plastic anisotropy of the material from the micromechanical model (based on crystal plasticity and on texture data) to the application, which could, for example, be a FE model for a forming process, or a model for predicting the forming limit of a sheet material. Plastic potentials have been developed both for stress space and for strain rate space (see, for example, Reference [10]). The discussion here will be limited to the latter case, because it makes a simple micro-macro transition possible. Let tIJm(D) be such plastic potential. Its numerical value is equal to ~rm, the macroscopic rate of plastic work per unit volume. D is the plastic strain rate. Elastic strains are neglected in this formulation. The Cauchy stress S which corrresponds to D is given by [101: 0qJm S/j -
v aDij
(4)
with lt-IJ m
v
=
01ffflm/c3Dij
(5)
D ij
v = 1 in the case of rate-independent plasticity, whereas for simple power lawbased viscoplastic material models: v =
1
1+#
(6)
in which # = 0 log a / 0 log ~ as in Section 3.3.1. A similar plastic potential can also be defined for individual crystallites. It would be denoted as W(d), d being the local strain rate. If a means is available for computing Wm(D) for a polycrystal with texture, then Eq. 4 can in principle be used to construct a constitutive model for the plastic behavior of the material. The Taylor theory ([ 11], [ 12] p. 371) provides a very convenient means computing tIJm(D) for a polycrystal with texture, since qJm is equal to the rate of plastic work per unit volume. This theory assumes that the local plastic strain rate d is uniform in each crystallite and equal to the macroscopic plastic strain rate D. It is known that in that case, the macroscopic rate of plastic work per unit volume Wm simply is the weighted average of the local rates of plastic work of all crystallites. The latter
3.3 Yield Loci Based on Crystallographic Texture
145
are denoted by W; in the Taylor theory, they depend on the macroscopic strain rate D (as is the case with Wm) and on the crystal orientation g. It is seen that W(g) takes the role of q(g) in Eq. 1 and X/Cmthat of q,,. Both are simple scalars, but they do depend on D. For each value of D, X/Cmcan be calculated from the texture using Eq. 1 or Eq. 3, provided that W(g) is known. In the next section, we will discuss how X/C(g) can be obtained from D using a crystal plasticity model. Note that the simple micro-macro transition that is presented here cannot be applied when the Taylor assumption is replaced by a more complex one, such as the set of assumptions used in self-consistent models [13]. The use of such models may lead to a somewhat more precise prediction of the yield locus, especially in the case of non-cubic materials. 9 However, this advantage will be completely lost if the recommendations in Section 3.3.1 are not scrupulously followed. 9 Certain techniques which in the case of the Taylor assumption can greatly reduce the computing time, such as the use of Fourier coefficients, can no longer be applied. This is particularly unpleasant in industrial applications involving large-scale FE modeling.
3.3.3.2
CRYSTAL PLASTICITY MODEL
In this text, it is assumed that one wants to obtain a yield locus, not that one wants to simulate deformation textures. The latter case deserves a long discussion of its own, not in the least because a truly general, reliable, and quantitative model for deformation texture prediction does not yet exist (see, for example, the discussion in Reference [14]).
3.3.3.2.1 The Bishop-Hill Theory The problem that must be solved is to calculate the local rate of plastic work for a crystallite with a given orientation g and for a given local strain rate d. Because of the Taylor assumption, d - D, normally one would now express D with respect to the reference system x c of the crystal lattice of the given crystallite ((1 0 0) axes in the case of a cubic metal), leading to a matrix D~. The following formula is used for the transformation from the sample frame to the crystal lattice frame (note: a superscript c indicates that a tensor is described in the crystal lattice frame):
D~ -tiktjlDkz
(7)
146
Van Houtte
in which [,,j]
-
COS ~1 COS ~02 -cos
-
-
q91 sin q92 -
sin (~1 sin ~o2 cos (I) sin tpl cos cp2 cos (I)
sin q91 sin (I)
sin (~1 cos ~2 -~- cos (ill s i n q~2 cos (I)
sin ~o2 sin(I) 1
-sin
cosq9 2 sin (/)
q~l sin~~
+
-cos
cos q91 costP2 cos (I) tpl s i n (I)
cos (I)
l
(8)
q)l, (I), and r are the Euler angles that describe the orientation of the crystal lattice [2]. In the stress-strain space associated with the x c frame, D~j in general represents a strain rate vector without special properties (i.e., it is not parallel with one of the stress axes, or perpendicular to it, etc.). In such case, the classical rate-independent Bishop-Hill theory for the plastic deformation of crystallites states that the deviatoric stress a~j needed to achieve this plastic strain rate will be one of a finite number of deviatoric stress states, called "vertices" [11,12]. These vertices depend on the nature of the slip systems available in the material, as well as on the ratios between the critical resolved shear stresses on these slip systems (CRSS ratios). In deviatoric stress space, the yield locus of the crystallite has the shape of a hyperpolyhedron defined by the previously mentioned vertices (Fig. 3.3.1). The strain rate vector that corresponds to a stress that is somewhere on a facet of this hyperpolyhedron (not on one of the edges of the facet) is normal to the facet. Since there is only a finite number of facets, there is only a finite number of such directions, whereas there are oo4directions possible in deviatoric stress-strain space. Hence the probability that D~j would have such direction is infinitesimal. For deviatoric stress states which are at the intersection of two facets, the strain rate vector must be normal to that line, in the sense that the unit vector which defines its direction must be a positive linear combination of the unit vectors normal to the two facets. So, for each such intersection, there a r e c o 1 such directions, whereas there are oo4directions possible for D~j. Again, the probability that D~j would have such direction is infinitesimal. This argument can be adapted for all higher-order intersections between facet planes, except when the intersection is a single point, i.e., a vertex stress. Only in that case, the probability that D~j would have such direction is not infinitesimal. All this can be summarized as follows. According to the Bishop-Hill theory, based on the generalized Schmid law, the deviatoric stress state which corresponds to a prescribed D~j normally is one of a finite number of vertex stresses, except in some special cases. The probability that such a special case occurs is infinitesimal. (This conclusion is true in the present context, i.e., the use of a single crystal yield locus to solve the Taylor theory. It would not
147
3.3 Yield Loci Based on Crystallographic Texture
o3
FIGURE 3.3.1 Closed yield locus of a "facet-eye" type as it would appear in a three-dimensional stress space. A Bishop-Hill yield locus of a crystallite is of this type, but in five-dimensional deviatoric stress space.
necessarily be true in other applications, such as the use of a single crystal yield locus to solve the interaction equation of a self-consistent model.) Bishop and Hill [15] and Kocks [16] have published the list of the 56 vertex stresses which exist for fcc metals that deform by {1 1 1} (1 1 0) slip with the same critical resolved shear stress on all slip systems. For bcc metals with {1 1 0 }{112 }( 1 1 0 ~ slip systems, there are 216 vertex stresses, which can be obtained from the present author, who has developed software that can automatically generate all vertices for any set of slip systems and any set of critical resolved shear stresses. cV represent such a vertex (for fcc metals, V is an index between 1 and Let ~7ij 56). Using Hill's Maximum Work Principle, it can now be stated that the deviatoric stress aijc which corresponds to a given plastic strain rate Dijc i s a vertex stress, and it is such that {7 ij D ij ~
o ij IJ ij
(9)
for all V. Finding the right value of V then comes down to selecting from the _cv,-,c cV which maximizes oij *-'ij It is an easy and an list of vertex stresses the crij extremely fast operation for a computer. The value found for aijcV is now called ~r~, and the rate of plastic work per unit volume in the crystallite is given by -
-
"D
(10)
in which the stress tensor ~ is a parameter which does not change when D undergoes infinitesimal variations. Its value is known in the crystal lattice
148
Van Houtte
frame, but it can be transformed to the sample frame: a'i)
--
thitlja~!
(11)
Since a is a constant with respect to D, W can indeed be regarded as a plastic potential for individual crystallite. As explained in Section 3.3.3.1, the macroscopic plastic potential kI'/m then is the weighted average of ~z for all crystallites, using the ODF f(g) as weighting function. The macroscopic stress is then in principle given by Eq. 4 with v-1 (rate-independent case), but it can easily be shown (by elaborating the equation) that the macroscopic stress is simply given by
Sij = (aij)m
(12)
in which (aij)m is the weighted average of aij for all crystallites. This finally makes it possible to calculate the macroscopic stress from the plastic strain rate, which in principle is sufficient to obtain the yield locus.
3.3.3.2.2 Rounding the Yield Locus The crystallite yield locus described in the previous section may be as correct as can be obtained from present theoretical knowledge combined with the usual lack of knowledge of material data such as current values of the critical resolved shear stresses on the individual slip systems. However, it has sharp corners and edges and fiat surfaces, and this is sometimes reflected in the macroscopic yield locus as well, which may feature regions with very strong curvature and other regions with very low curvature, especially for materials with strong texture. This may become a source of numerical instability in certain applications, such as FE codes which are notorious for becoming unstable when a somewhat exotic constitutive material law is used. The fundamental solution to this problem is of course to improve the stability of FE codes so that they can digest true material behavior, but in the meantime, we may have to try to substitute the sharp-edged single crystal yield locus by a more rounded one. This can be achieved by using the following yield locus for an individual crystallite [ 17]: /~+1
#'+ 1
as
-- 1
(13)
In this equation, #~ does not stand for the strain rate sensitivity, since the equation is still intended for a rate-independent material. ~s is the resolved shear stress acting on slip system s and as such a function of the macroscopic stress S ([11], [12] p. 333). The plastic strain rates are related to the S in the usual way through the normality rule. ~0 is a constant (with a value close to
149
3.3 Yield Loci Based on Crystallographic Texture
the critical resolved shear stress) and the as are the CRSS ratios. It can be shown that for 12' ~ 0 this yield locus tends toward the edgy Bishop-Hill yield locus; for low values of 12' such as 0.02, the yield locus is still a good approximation of the Bishop-Hill yield locus, but the corners and edges are rounded with a short radius of curvature. The larger 12' is, the more rounded the yield locus becomes, and the more it will differ from the Bishop-Hill yield locus. This method, however, has some serious drawbacks. For a given crystallite, it allows for a straightforward calculation of the plastic strain rate from the stress. However, for the application at hand, we must find the stress from the plastic strain rate. This can only be done by an iterative procedure. It is difficult to make software for this that is completely stable, especially for low values of 12'. The required calculation time is one or several orders of magnitude larger than for the Bishop-Hill method. For the rest, Eq. 12 can still be used to perform the micro-macro transition.
3.3.3.2.3 The Visco-plastic Model The viscoplastic model ([12], p. 358 and pp. 363-365) for crystallites is also frequently used to obtain a rounded yield locus. Strictly speaking, a yield locus does not exist in this case (see following). This model can be developed from a plastic potential (for an individual crystallite) in stress space: (~+i)
s
I
~s'~01
~b(cr) in Eq. 14 has the value of ~/, the rate of plastic work per unit volume in the crystallite; ?0 is a constant with the nature of a slip rate; 12is the strain rate sensitivity exponent as explained in Section 3.3.1 1:0 and as have the same meaning as in Eq. 13; and ~:s is the resolved shear stress acting on slip system s and as such a function of the local stress or. The local strain rates are obtained as follows [10]:
d~j -
~ 04, 12 + 1 c~r
(15)
This leads to the usual formulae of the viscoplastic model ([12] p. 358). It is seen that Eq. 14 is almost the same as Eq. 13; only the interpretation is different. Whereas Eq. 13 leads to a rounded yield locus that approaches the Bishop-Hill yield locus when 12'~ 0, the equation ~b(cr)=constant defines an equipotential surface in stress space which tends to the Bishop-Hill yield locus when/2 ~ 0. It then becomes clear that the value of/2 will control
150
Van H o u t t e
the radius of curvature of the vertices of the yield locus-equipotential surface. A user of FE software would want to have some control on this radius of curvature, in order to keep the FE algorithms stable. But one does not have the right to fine-tune # if one takes it seriously that it is the strain rate sensitivity exponent: in that case, it should be experimentally determined, and whichever value comes out should be used. Note that for numerical reasons, a negative value would make the use of the viscoplastic model totally impossible. This would be the case for certain temperature and strain rate ranges of several important aluminium alloys (with Mg atoms in solid solutions) and steel alloys (mainly IF steels). Therefore, when for practical reasons one decided to abandon the idea that # should be equal to an experimentally determined strain rate sensitivity exponent, it would be scientifically more honest to use Eq. 13 instead of Eq. 14, and, if needed, devise a different means incorporate strain rate sensitivity into the model. From a computational point of view, this method features the same problems as the previous one (Section 3.3.3.2.2), and it uses similar algorithms for solution. The local stresses cr are calculated from D by an iterative procedure (after identification of D with d in Eq. 15). Eq. 12 is then used for the micro-macro transition.
3.3.3.3
EXPLOITATION OF THE CONSTITUTIVE MODEL
3.3.3.3.1 Direct Exploitation Quite often, FE or other applications can be organized so that the constitutive model for the polycrystalline material with texture has only to provide answers to one type of question: "What is the stress S, when the plastic strain rate D is given ?" In that case, Eq. 12 can be used in direct combination with one of the three models for crystal plasticity, operating on a set of discrete weighted crystallite orientations which represent the texture of the material. However, this may be too costly in computing time, since one needs for cubic metals about 2000 crystallite orientations in the set to obtain an acceptable accuracy; for lower lattice symmetries, this number is much higher. Therefore, it may be advantageous to use a precalculated database, although this makes it more difficult to simulate texture evolution.
3.3.3.3.2 Using a Precalculated Database
This discussion will be limited to the rate-independent case. It will first be explained how the macroscopic plastic potential I'IJ m c a n be precalculated for
3.3 Yield Loci Based on Crystallographic Texture
151
a large number of directions in strain rate space. Then it will be explained how an analytical expression can be constructed that fits these data. This makes it possible to use Eq. 4 for frequent (and fast) calculations of S from D in a FE or other application.
3.3.3.3.2.1 Calculation of
~'1 m
for Directions in Strain Rate Space
In Section 3.3.3.1, it was said that kI/m is equal to the macroscopic rate of plastic work. It can always be written in the following form [10]: ~Pm(D) = [[D][ Hm(a)
(16)
in which
v/DijDi)
[IDII -
(17)
and D a
=
IIDII
(18)
a represents a direction in strain rate space. Hm(a) is a [unction which only depends on the direction of D, not on its length. In a similar way, a plastic potential W can be constructed for a crystallite with an orientation g: ~g(g, d) = [[d[[ H(g, a)
(19)
Since we use the Taylor assumption, d = D and the macroscopic plastic potential kl/m is equal to (qJ)m, the average over all crystallite orientations of qJ, using the ODF as weighting function. This then leads to
H~(a) = (H(g, a))~
(20)
This equation is of the same type as Eq. 1. A certain number of calculations should now be done once and for all, without using the texture of the polycrystal. For a given type of material, a crystal plasticity model should be used to produce a H(g, a) function for each direction a taken from a set which represents all directions in stress-strain space with a given angular resolution. To do so, it is in principle possible to use the model for d = a, and calculate X/r (Eq. 10) for every crystallite orientation on a grid in Euler space. H(g, a) then simply is equal to the values found for X/(. Convert each H(g, a) function into Fourier-coefficients HfV(a) [2] so that, in the future, Eq. 3 can be used instead of Eq. 1. In principle, these Fourier-coefficients establish the database. It is exploited as follows: Once the C-coefficients of the texture are known, Eq. 3 is used to calculate Hm for all directions a of the discrete set. So the function Hm(a) in Eq. 16 is known in a numerical form for a large number of directions a. In principle the
15 2
Van Houtte
m a c r o s c o p i c plastic potential kI'/m is n o w k n o w n ; it represents the textured e p e n d e n t m o d e l for the plastic a n i s o t r o p y of the material. The w o r k described in this section seems a formidable task, since there are a b o u t 50,000 directions a in deviatoric s t r e s s - s t r a i n space using an angular r e s o l u t i o n of 10 ~ However, the w o r k can be o p t i m i z e d to a very large extent, w h i c h leads to surprisingly s h o r t calculation times [4].
3.3.3.3.2.2 Analytical Model for Hm(a) As e x p l a i n e d in the p r e v i o u s section, Hm(a) is o b t a i n e d as n u m e r i c a l values on a discrete set of directions a. This is quite u n p l e a s a n t in view of Eq. 4: It will be necessary to m a k e partial derivatives of tIJ m (see Reference [10] for m o r e details), a n d it t h e n b e c o m e s desirable to have Hm(a) in an analytical form. This can be achieved as follows: 9 Let lap] be a vector r e p r e s e n t a t i o n of the m a t r i x indices ij are c o n t r a c t e d into a single one p [10].
[aij]; i.e.,
the two
4
I
-
I
3
I
4
(~'1/~cl
FIGURE 3.3.2 Yieldlocus of an A13004 alloy that has been cold-rolled. The stresses and strain rates are expressed with respect to coordinate axes which make angles of 45 ~ and 135~ to the rolling direction. 0-11--0"22section of the yield locus in nondeviatoric stress space. 0"33= 0 for this section, as well as the shear stresses. The yield locus is calculated from the texture, using a precalculated database describing the plastic anisotropy of fcc metals.
3.3 Yield Loci Based on Crystallographic Texture
153
9 Hm(a) can then be written as a polynomial of the nth order. It is recommended to set n=6. In practice, only the fifth- and the sixth-order terms are necessary, because all a-tensors are normalized (Eq. 18): Hm(a) - Fvqrsuapaqarasau -+-Gvqrsuvapaqarasauav
(21)
The coefficients Fvqr~, and Gpqrsuv have to be found by least-squares-fitting [18], or they can be found by more clever methods [19]. Note that the fifthorder coefficients Fpqr~, are zero for materials which have a central-symmetric yield locus. Equation 16 combined with Eq. 21 has been used to implement texturebased plastic anisotropy in FE simulations of sheet metal forming. The results are quite encouraging, although there are some drawbacks [18-20]. Figure 3.3.2 shows an example of a yield locus obtained by this method.
REFERENCES 1. Kocks, U. E, Tom~, C. N., and Wenk, H.-R. (1998). Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Material Properties, Cambridge: Cambridge University Press. 2. Bunge, H. J. (1982). Texture Analysis in Materials Science London: Butterworth. 3. Kallend, J. S. (1998). Determination of the orientation distribution from pole figure data, in [1], pp. 102-125. 4. Van Houtte, P. (2001). Fast calculation of average Taylor factors and Mandel spins for all possible strain modes. International Journal of Plasticity, 17; 807-818. 5. Van Acker, K., and Van Houtte, P. (1998). The effect of Soller slit and monochromator used for background reduction in texture measurements. Textures and Microstructures 30; 133-143. 6. Wenk, H.-R. (1998). Pole Figure measurements with diffraction techniques, in [1], pp. 126-177. 7. Hatherly, M., and Hutchinson, W. B. (1979). An Introduction to Textures in Materials, Monograph Nr. 5, London: The Institution of Metallurgists. 8. Wright, S. I. (1993). A review of automated orientation imaging microscopy (OIM). J. Computer-Assisted Microscopy 5; 207-221. 9. Toth, L. S., and Van Houtte, P. (1992). Discretization techniques for orientation distribution functions. Textures and Microstructures 19; 229-244. 10. Van Houtte, P. (1994). Application of plastic potentials to strain rate sensitive and insensitive anisotropic materials. Int. J. Plasticity 10; 719-748. 11. Aernoudt, E., Van Houtte, P. and Leffers, T. (1993). Deformation and textures of metals at large strains, in: Plastic Deformation and Fracture of Materials pp. 89-136, Mughrabi, H. ed., vol. 6 of Materials Science and Technology: A Comprehensive Treatment, (R. W. Cahn, P. Haasen, and E. J. Kramer, eds., Weinheim: VCH. 12. Kocks, U. E (1998). Kinematics and kinetics of plasticity, in [1], pp. 326-389. 13. Tom~, C. N. and Canova, G. R. (1998). Self-consistent modeling of heterogeneous plasticity, in [ 1], pp. 466-511. 14. Van Houtte, P., Delannay, L., and Samajdar, I. (1999). Quantitative prediction of cold rolling textures in low-carbon steel by means of the LAMEL model. Textures and Microstructures 31; 109-149.
154
Van Houtte
15. Bishop, J. E W., and Hill, R. (1951). A theoretical derivation of the plastic properties of a polycrystalline face-centred metal. Philos. Mag. 42; 1298-1307. 16. Kocks, U. E (1970). The relation between polycrystal deformation and single crystal deformation. Metall. Trans. 1; 1121-1143.
17. Van Houtte, P., and Rabet, L. (1997). Generalisation of the relaxed constraints models for the prediction of deformation textures. Revue de M~tallurgie-CIT/Science et G~ie des, Mat~riaux 94; 1483-1494. 18. Van Bael, A. (1994). Anisotropic Yield Loci Derived from Crystallographic Data and their Application in Finite-Element Simulations of Plastic Forming Processes, Ph.D. Thesis, Dept. MTM, Katholieke Universiteit Leuven, Belgium. 19. Arminjon, M., Bacroix, B., Imbault, D., and Raphanel, J. L. (1994). A fourth-order plastic potential for anisotropic metals and its analytical calculation from the texture function. Acta Mech. 107; 33-51. 20. Van Houtte, P., Van Bael, A., and Winters, J. (1995). The incorporation of texture-based yield loci into elasto-plastic finite element programs. Textures and Microstructures 24; 255-272.
SECTION
3.4
Anisotropic Yield Conditions MICHAL ZYCZKOWSKI
Cracow University of Technology, ul. Warszawska 24, PL-31155 Krakfw, Poland
Contents 3.4.1 Three Approaches to Formulation of Anisotropic Yield Conditions . . . . . . . . . . . . . . .
155
3.4.2 Generalizations of the Huber-Mises-Hencky Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
3.4.3 Generalizations of the Tresca Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
3.4.4 Generalizations of the Burzyfiski Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
3.4.5 Generalizations of the Hershey "Power Yield Condition". ............................... 3.4.6 Polynomial Yield Conditions for Planar Orthotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
160 162
3.4.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164
3.4.1 T H R E E A P P R O A C H E S T O F O R M U L A T I O N OF ANISOTROPIC YIELD CONDITIONS In uniaxial tension perfect plasticity is defined by the equation crx/cr0 = 1 during the process; ~r0 denotes here the yield-point stress in the direction x. Conditions of perfect plasticity, or briefly yield conditions, for isotropic materials in the general three-dimensional case can be written as a simple generalization of the preceding equation; namely, Crx is replaced by certain functions of stress invariants. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
155
156
ZYcz~ows~
At least three approaches are used to generalize isotropic yield conditions to the materials showing innate (for example, in monocrystals) or acquired (for example, strain-induced) anisotropy. First, the stress invariants are replaced by simultaneous (common) invariants of the stress tensor and of the material (or structural) tensors of plastic anisotropy, for example,
f (Hijcrij , HijklaijahZ, H@lmn ~rijcrklCrm,~, . . . ) -
1
(1)
where H~j, 1-I~jkl, 1-I@zm,, ... denote tensors of plastic anisotropy and Einstein's summation convention holds. It should be noted that the invariants shown in Eq. 1 are the simplest, but by no means general. For example, two tensors of the second order have 10 irreducible polynomial invariants; besides 3 invariants of aij and 3 of H~j, there exist 4 simultaneous invariants H~jcr~j, H~jajkcrhl, 1-I~jHjkcrkl, and H~jHjkahzal~, and only the first of them is shown in Eq. 1. The number of irreducible invariants of H@I and ~r~jamounts to 28 (plus the invariants of H@I itself, less important for construction of yield conditions) [1]. A detailed discussion on invariant formulation of anisotropic yield conditions is given by Rogers [21. Second, a certain transformed or modified stress tensor may be introduced. For example, Sobotka [31 and Boehler and Sawczuk [41 proposed independently to introduce the tensor ~ij
--
AijklCrhl
with
Ajihl -~ Ajikl -- Aijlk - Aklij
(2)
where Aijkl is also a certain tensor of anisotropy, and postulated the wellknown isotropic yield conditions to hold for anisotropic materials as well if aij are replaced by Fi). In many cases this approach proves useful, but it is less general than Eq. 1" for example, second-order material tensors may be obtained here from Ai)kl just by contraction, whereas in Eq. 1 they are independent. Mean transformed stress equals 1
1
4. = -~pp - -~AppkZ.hl
(3)
and deviatoric transformed stress
(4) Another approach resulting in a nonlinear transformation of various anisotropic yield surfaces into a hypershere in five-dimensional Ilyushin's space was proposed by Zyczkowski and Kurtyka [5]. The third way, with the strongest physical background, is to postulate a certain physical quantity in an anisotropic material to be constant when reaching the yield-point stress (for example, the elastic energy), and then to express strains in terms of stresses and assume this function to be constant
157
3.4 Anisotropic Yield Conditions
during a perfectly plastic process. This way, typical for isotropic materials, is rather seldom used in the case of anisotropy. Yield conditions for anisotropic materials are usually formulated as certain generalizations of the relevant conditions for isotropic materials. Therefore, they will be classified here according to the types of isotropic conditions that are subject to generalization.
3.4.2 GENERALIZATIONS OF THE HUBERMISES-HENCKY Y I E L D C O N D I T I O N The first anisotropic yield condition was proposed in 1928 by Mises [6]: 1-Iijkl~Tij~Tkl-- 1
with
IIijkl = IIjikl = 1-Iijlk = IIklij
(5)
These symmetry conditions reduce the number of moduli to 21. Further, the additional requirement of independence of Eq. 5 on the mean stress am reduces the number of moduli to 15. In this case, making use of the transformation Eq. 2, we can write Eq. 5 in the form SijSij
--
2k 2
(6)
Mises [6] also gives some energy considerations: Eq. 6 corresponds at the beginning of yielding to constant distortion energy, though in the general anisotropic case energy cannot be decomposed into pure distortion energy and pure volumetric energy. Change of signs of all stresses in Eqs. 5 or 6 does not change the expression as a whole; hence the materials described by these equations are isosensitive [7], or without strengthdifferential effect [8]. Equation 5 takes a much simpler form in the case of orthogonal anisotropy, called briefly orthotropy. Then there exist three mutually perpendicular principal directions of the tensor Flijhl. Choosing the reference frame x y z in such a way as to obtain its coincidence with the principal axes of orthotropy, we eliminate nine terms of the type axr.xy (in engineering notation) and three terms of the type "Cxy'Cyz, and in Eq. 5 only nine terms remain. Further, if we require the independence of the yield condition on the mean stress am, then three additional conditions must be satisfied, and Eq. 5 takes the form F(ay - @)2 + G(az _ ax)2 + H(ax - ay) 2 + 2LZ2z + 2MZ2zx+ 2Nz 2 -- 1 (7)
The notation in Eq. 7 is due to Hill [9], who considered this condition in detail. Equation 7 will be called here "the first Hill's yield condition." The moduli F, G, H, L, M, N can be expressed in terms of the yield-point stresses in uniaxial tension (or compression) in the directions x, y, z, denoted
158 TABLE 3.4.1
ZYCZKOWSKI
C o n s t a n t s in Hill's Yield C o n d i t i o n Eq. 7
Pure a l u m i n u m sheets
F (MPa -2)
G (MPa -2)
H (MPa -2)
N (MPa -2)
1.090 x 10 -4
1.457 • 10 -4
1.530 • 10 -4
6.876 x 10 -4
by 0"Ox, 0"Oy,0"oz, and the yield-point stresses in shear ZOxy,"COyz,Zozx: F -- -~
q 0"2Oz
r~ x
c - ~
~ ~ox
.-~
~o~y G
1
L - - 2"C2yz
1)
1
G
(8)
M - 2~--2
N-2%
In the particular case of plane stress, which is very important for engineering applications (e.g., in rolled sheets), 0"z- "Czx- "Cyz- 0 and we obtain a four-parameter equation 2
0"y
1
0-___.~x_~_ __ 0-2x 0"~y__
1
1 ~z
._} 0"27
l:xy 0-x0-y "}- -T,~xy --
1
(9)
Experimental data for rolled pure aluminum sheets obtained by Klepaczko [10] are gathered in Table 3.4.1 (x is rolling direction, y is transverse direction; in the planar case under consideration, L and M were not measured; units are converted from mm4/kG 2 to MPa-2).
3.4.3 GENERALIZATIONS YIELD CONDITION
OF THE TRESCA
The generalizations of the Tresca (principal shear) yield condition to anisotropic materials are very simple but only in a particular case: if general anisotropy is restricted to orthotropy and the principal directions of the stress state and of the orthotropy coincide. The corresponding system of six linear equations, taking into account the possible anisosensitivity of the material (strength-differential effect), was first derived in 1958 by Hu [11 ], and then independently by Ivlev and Capurso (see [7])" 0"1 - - 0"2
0"2 - - 0"3
~ + - - = 1 ooi+ rr030"1 - - 0"2
.
.
0"01-
.
(10)
0"2 - - 0"3
.
0"03+
~
1
159
3.4 A n i s o t r o p i c Yield C o n d i t i o n s
and the remaining four equations are obtained by cyclic permutation of the indices both for stresses and for plastic moduli. The symbols 0-01+, 0-01denote here positive yield-point stresses in tension and in compression along the axis "1", respectively, and so on. Another generalization of the Tresca yield condition, also allowing for anisosensitivity, was proposed by Berman and Hodge [12]" 0-1
0-3
m
=1,
a 0-3 -- 0-1
=1,
a
0"2
-
-
0-3
b 0-3 -- 0-2
e
=1 =1
0-2
-
-
0-1
~ = 1 c 0-1 -- 0-2 ~ = 1 f
(11)
The relation between the positive moduli a, b , . . . , f, and the moduli 0-01+, 0-01-, ... depends on algebraic ordering of the first set. For example, if a > b > c > d > e > f , we obtain 0-01+=f,
0-02+=c,
0-03+=e,
0-01-=d,
0-02-=f,
0-03-=b
(12)
It means that an additional constraint a 0 1 + - 0-02- is imposed, and the constant a cannot be determined from uniaxial tests in principal directions.
3.4.4 GENERALIZATIONS OF THE BURZYlqSKI YIELD CONDITION Burzyfiski [13] proposed in 1929 a three-parameter failure hypothesis for isotropic materials aa 2 + ba 2 + Cam - 1
(13)
which was later u s e d - in its general form or in confined f o r m s - by many authors more or less independently as a yield condition. Historical notes are given by Zyczkowski [7]. The symbol ae denotes here the Huber-MisesHencky effective stress. Equation 13 describes in the space of principal stresses an ellipsoid, a paraboloid, a cone, or a cylinder. Direct generalization of Eq. 13 to anisotropic materials may be written, for example, by using transformed stresses #ij (2, 3, 4) with just one anisotropy t e n s o r Aijkl 3
-~AppklC~ij Aijqr
Aokt
~a
-
+
ttqrt~ij aklaqr -
(14)
(AppkZ~kl) 2 + 5App~lGkl -- 1
A more general yield condition of this type was derived by Betten [1], who used two independent material tensors.
160
Z~cz~ows~
Another generalization, using several anisotropy tensors, was proposed by Goldenblat and Kopnov [14,15]: (l'-[ij(Tij) ~ "nt- (1-'Iijkl(Tij~kl) f3 AV (l-[ijklmn~ij(Tkl(Tmn) 7 -~- . . . - - 1
(15)
where ~, ]3, 7,-.. are arbitrary exponents, but the authors proposed to assume ~ = 1,/3 = 1/2, 7 = 1 / 3 , . . . (then the homogeneity of the function on the left-hand side is assured). They discussed in detail the case (16)
rli)~ri) + v/rlijhl~ri)akl = 1
which may be regarded as a certain generalization of the Prager-Drucker cone. The monograph [15] gives the method of experimental evaluation of the moduli ri O and 1-Iijkl and many working formulae for particular cases. Assuming in Eq. 15 ~ = ]~ = 1, we obtain a material characterized, in general, by 6 + 2 1 - - 2 7 material constants (in view of symmetry requirements imposed on II ). Additional restriction of independence from the mean stress Crm reduces the number of constants to 5+ 15--20; this case was considered in detail by Szczepifiski [16]. On the other hand, restriction to orthotropy reduces the number of constants to 3 + 9 -- 12, [ 17]. If we impose both restrictions simultaneously, the number of constants amounts 2 + 6 - 8. Making use of the extended Hill's notation Eq. 7, we obtain, in this case [ 18], F(O'y
- - O'z)2--[ -- G ( o " z - - 0"x) 2 q -
H(o" x -- Cry) 2
+ 2LZ,2z + 2Mz, 2 + 2N1:2
(17)
+ Pax + QCry - (P + Q)cr z = 1
This condition is particularly useful for describing the properties of composites, and its particular form restricted to planar a n i s o t r o p y - to rolled sheet metals. In the latter case the anisotropy is acquired because of previous deformation, and anisosensitivity is mainly due to the Bauschinger effect.
3.4.5
GENERALIZATIONS
OF THE HERSHEY
"POWER YIELD CONDITION" Hershey [19] proposed in 1954 the following "power yield condition" for isotropic materials, expressed in terms of principal stresses: ]0" 1 - - 0"2[m-~-[0"2
- - 0"3[m-nt-[0"3
- - 0"1[ m - -
2o'~" -- 2o'~"
(18)
161
3.4 Anisotropic Yield Conditions
where ab denotes the yield-point stress in equibiaxial tension, and m ~> 1 is not necessarily an integer. The cylindrical surface Eq. 18 for m = 2 and m = 4 turns into the Huber-Mises-Hencky cylinder, and for m -- 1 and m ~ oo into the Tresca hexagonal prism. Numerous papers generalize Eq. 18 to anisotropy, mostly to planar orthotropy, which is important, for example, for the description of yielding of rolled sheets. Hill [20] suggested the following generalization of Eq. 18 for planar orthotropy: gl0"11m_+_fl0-zlm_+_hl0-1
_
_
0-21m_+_C/120-1__ 0-2[m_+_b[20-2
_
_
0-1]m..+_CI0-1 _1_ 0-21m-- 0-~ (19)
where f + g + a + b + 2mc = 1 (second Hill's yield condition). Earlier, Hosford had discussed a particular case of this condition, namely, a = b = c = 0. Equation 19 has six free material constants, m, f, g, h, a, b. Another, more complicated generalization of Eq. 18, proposed by Hill [21] for planar orthotropy, is of the form 1(~1
-
-Jr- 0"2) q-hlO'l
0-2lmq-(0-12-t'- 0-2)(m-2)/2(0-1
-
-
0-2)(k0-1 lo2) -- 0-bin (20)
with four material constants m, h, k, l (third Hill's yield condition). When m - - 2 , Eq. 20 turns into the first Hill's yield condition Eq. 7 in different notation. Barlat and Lian [22] considered the case of different axes of orthotropy and of the stress state. They proposed the following yield condition: a[[K1 - K2lm+[K1 q- K2[m] q- (2 - a)]2K2[m= 20-~"
(21)
where K1 =
2
'
K2
0-x -2 h0-y. +p2,rxy2
(22)
with four dimensionless material constants m, a, h, p. Numerical values of these constants are given by Barlat et al. [23] for a rolled aluminum alloy sheet 2008-T4 with the thickness 1.24 mm and composition (in weight %) 0.60 Si, 0.13 Fe, 0.93 Cu, 0.06 Mn, and 0.40 Mg; they obtained m - 8, a - 1.24, h - - 1.15, p = 1.02, under the assumption that x, y, z denote, in turn, the rolling, transverse, and normal directions. In the previously mentioned paper [23], Barlat et al. also proposed a generalization of Hershey's yield condition Eq. 18 to the general case of triaxial stress state in an orthotropic material, if the principal directions of the stress state and of orthotropy do not coincide. In this case all six stress components appear, one has to solve a cubic equation to find the principal stresses, and the final yield condition takes the following compound form (via
162
ZYCZKOWSVa
TABLE 3.4.2
Coefficients of the functions in Eqs. 23-25
Material
m
a
b
c
f
g
h
2008-T4 2024-T3
11 8
1.222 1.378
1.013 1.044
0.985 0.955
1.0 1.0
1.0 1.0
1.0 1.210
Cardano's formulae): (312) m/2
2 cos-------~
+
2 cos
20
I
)m(
3re 6
+
-2 c o s ~ 6
2o-g' (23)
where 1 12 -- -~--4{[a(ay -- a z ) -- c(rrx -- ay)] 2 + [C(ax -- Cry) -- b(o" z -- ax)] 2 1 f 2 z 2 z + g 2Zzx 2 + h2z2) + [ b ( a z - ax) - a ( a y - az)] 2} + -~( 1 13 - -~-~[C(ax - Cry) - b ( a z - ax)][a(ay - a z ) - C(ax - ay)] x[b(az
- ax) - a(ay - az) ]
+fghzyzZvcz + [a(~y
-
v o~) -
_
1 -~{[C(ax C(~x -
_
~
rry)-b(az
_
ax)]
f
2 Z ;2, z
+ [b(o~ - ~x) - a(o~ - ,~)]h~ G }
(24) 0 - arccos
(25)
with seven dimensionless constants m, a, b, c, f, g, h. Table 3.4.2 gives their numerical values for rolled aluminum alloy sheets, namely, for 2008-T4 defined previously, and for 2024-T3 with the thickness 0.30mm and composition (in weight %) < 0.50 Si, < 0.50 Fe, 4.40 Cu, 0.60 Mn, 1.50 Mg [231.
3.4.6 POLYNOMIAL YIELD CONDITIONS PLANAR ORTHOTROPY
FOR
Some authors propose anisotropic yield conditions in the form of a polynomial, usually of the third or fourth degree.
163
3.4 Anisotropic Yield Conditions TABLE 3.4.3
Coefficients of the Function Eq. 26 Divided by A1
Material
A1
A2
A3
A4
A5
A6
A7
A8
A9
Al-killed steel Cu-(1/4)H
1.0 1.0
-2.60 -1.80
3.75 2.68
-2.79 -2.20
0.991 1.203
6.29 6.58
-7.72 -5.51
6.33 6.71
8.96 11.48
Gotoh [24] proposed for orthotropic materials the following fourth-degree yield condition: 3 if- A30"x0"y 2 2 + A4 o"x@ + A50-y4 A10"x++A20-x0"y (26) 2 2 A9r~y 1 + (A60"x2 + A70"x0"y + A80";)rxy + with nine material constants A1, A2, . . . , A9. Numerical values of these constants were found experimentally by the author for commercial Al-killed steel and Cu-(1/4)H rolled sheets, 0.8 mm in thickness, with the notation of axes as given previously. They are quoted in Table 3.4.3. Detailed comparisons with quadratic yield condition were also provided. Hill [25] proposed a planar yield condition for orthotropic sheets in the form of a third-degree polynomial expressed in terms of principal stresses acting along the in-plane axes of orthotropy. It was called by the author "a user-friendly theory of orthotropic plasticity" (fourth Hill's yield condition): F-~+
G
(p+q)-
~ =
1
(27)
where 0-Ol, ao2, and 0-b denote, in turn, yield-point stresses in uniaxial tension in directions "1" and "2" and in equibiaxial tension. Dimensionless constants p and q are free (to be determined from experiments), whereas c must satisfy the equation c 1 1 1 = --4 (28) 2 O-010"02
0"~1
0"22
0"b
3.4.7 FINAL REMARKS Averaged (homogenized) properties of structurally anisotropic materials (reinforced, ribbed, perforated) often show an even more significant dependence on the direction than naturally anisotropic ones. In the case of structural plastic anisotropy, two typical approaches may be distinguished: either a purely formal, phenomenological application of the general anisotropic yield conditions, discussed in the previous sections, or a more detailed analysis of the structure combined with a homogenization process.
164
Z~z~ow~,
S o m e e x a m p l e s of the latter a p p r o a c h will be g i v e n in the c h a p t e r d e v o t e d to heterogeneous media. L i t e r a t u r e d e v o t e d to a n i s o t r o p i c yield c o n d i t i o n s is v e r y a m p l e ; n u m e r o u s a d d i t i o n a l r e f e r e n c e s are g i v e n in R e f e r e n c e [7]. M o r e o v e r , m a n y e x p e r i m e n t a l data are p r e s e n t e d b y I k e g a m i [26].
REFERENCES 1. Betten, J. (1988). Applications of tensor functions to the formulation of yield criteria for anisotropic materials. Int. J. Plasticity 4: 29-46. 2. Rogers, T. G. (1990). Yield criteria, flow rules, and hardening in anisotropic plasticity, in Yielding, Damage, and Failure of Anisotropic Solids, pp. 53-79, Boehler, J. P., ed., London: Mechanical Engineering Publications. 3. Sobotka, Z. (1969). Theorie des plastischen Fliessens von anisotropen K6rpern. Z. Angew. Math. Mechanik 49: 25-32. 4. Boehler, J. P., and Sawczuk, A. (1970). Equilibre limite des sols anisotropes. J. de M&anique 9: 5-33. 5. 2;yczkowski, M., and Kurtyka, T. (1990). A description of distortional plastic hardening of anisotropic materials, in Yielding, Damage, and Failure of Anisotropic Solids, pp. 97-111, Boehler, J. P., ed., London: Mechanical Engineering Publications. 6. Mises, R. (1928). Mechanik der plastischen Form~inderung von Kristallen. Z. Angew. Math. Mechanik 8: 161-185. 7. Zyczkowski, M. (1981). Combined Loadings in the Theory of Plasticity, Alphen aan den Rijn: m Warszawa, Nijhoff m PWN. 8. Drucker, D. C. (1973). Plasticity theory, strength-differential (SD) phenomenon, and volume expansion in metals and plastics. Metall. Trans. 4: 667-673. 9. Hill, R. (1948). Theory of yielding and plastic flow of anisotropic metals. Proc. Royal Soc. A193: 281-297. 10. Klepaczko, J. (1969). O pewnym przypadku anizotropii. Mechanika Teoretyczna i Stosowana 7: 155-163. 11. Hu, L. W. (1958). Modified Tresca's yield condition and associated flow rules for anisotropic materials and its applications. J. Franklin Inst. 265: 187-204. 12. Berman, I., and Hodge, P. G., Jr. (1959). A general theory of piecewise linear plasticity for initially anisotropic materials. Archiwum Mechaniki Stosowanej 11: 513-540. 13. Burzyfiski, W. (1929). 0ber die Anstrengungshypothesen. Schweiz. Bauz. 94: 259-263. 14. Goldenblat, I. I., and Kopnov, V. A. (1966). A generalized theory of plastic flow of anisotropic media (in Russian), in Stroitelnaya Mekhanika (Rabinovich Anniversary Volume), pp. 307-319, Streletsky, N. S., et al., ed., Moskva Izdat: Lit. po Stroitelstvu. 15. Goldenblat, I. I., and Kopnov, V. A. (1968). Yield and Strength Criteria for Structural Materials (in Russian), Moskva: Mashinostroyenye. 16. Szczepifiski, W. (1992). On deformation-induced plastic anisotropy of sheet metals. Archiwum Mechaniki Stosowanej 44: 663-698. 17. Tsai, S. W., and Wu, E. M. (1971). A general theory of strength for anisotropic materials. J. Compos. Mater. 5: 58-80. 18. Ota, T., Shindo, A., and Fukuoka, H. (1959). A consideration on anisotropic yield criterion, in Proc. 9th Jap. Nat. Congr. Appl. Mech., pp. 117-120.
3.4 Anisotropic Yield Conditions
165
19. Hershey, A. V. (1954). The plasticity of an isotropic aggregate of anisotropic face-centered cubic crystals. J. Appl. Mech. 21: 241-249. 20. Hill, R. (1979). Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc. 85: 179. 21. Hill, R. (1990). Constitutive modelling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids 38: 405. 22. Barlat, E, and Lian, J. (1989). Plastic behavior and stretchability of sheet metals. Int. J. Plasticity 5: 51-66. 23. Barlat, E, Lege, D. J., and Brem, J. C. (1991). A six-component yield function for anisotropic materials. Int. J. Plasticity 7: 693-712. 24. Gotoh, M. (1977). A theory of plastic anisotropy based on a yield function of fourth order. Int. J. Mech. Sci. 19: 505-520. 25. Hill, R. (1993). A user-friendly theory of orthotropic plasticity in sheet metals. Int. J. Mech. Sci. 35: 19-25. 26. Ikegami, K. (1982). Experimental plasticity on the anisotropy of metals, in Mechanical Behaviour of Anisotropic Solids, pp. 201-242, Boehler, J. P. ed., Alphen aan den Rijn: Nijhoff.
SECTION
3.5
Distortional Model of Plastic Hardening TADEUSZ KURTYKA CERN- European Organizationfor Nuclear Research, CH-1211 Geneve 23, Switzerland
Contents 3.5.1 Background and Validity of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Formulation of the Model . . . . . . . . . . . . . . . . . . 3.5.3 Description of the Model . . . . . . . . . . . . . . . . . . . 3.5.4 How to use the Model . . . . . . . . . . . . . . . . . . . . . 3.5.4.1 Use of the model as a Yield Condition . . . . . . . . . . . . . . 3.5.4.2 Use of the Model as a Hardening Rule . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166 167 170 171 171 171 173
3.5.1 BACKGROUND AND VALIDITY OF THE MODEL Experiments in metal plasticity show that during plastic loading the development of acquired plastic anisotropy is reflected by complex transformations of the initial yield surface. These transformations usually consist of the following five elements: (1) translation (kinematic component), (2) proportional expansion (isotropic component), (3) affine (elliptic) deformation, (4) rotation, and (5) distortion, exceeding affine deformation. Most practically used models of plastic hardening are confined to the kinematic-isotropic components of these transformations, and neglect the 166
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
167
3.5 Distortional Model of Plastic Hardening
yield surface distortions. These effects are, however, systematically observed in experiments and may be essential for some important applications, such as for a correct modeling of the multiaxial ratcheting under nonproportional loadings. Several distortional models have therefore been proposed [1-6] to describe the yield surface distortions and to introduce the corresponding distortional hardening rules; however, there is as yet no widely accepted theory of this type applicable to engineering problems. Such models are thus addressed mainly to experimentalists and need further investigations. This is also the case of the distortional model presented in this paper, which has the advantage of providing a quite general description and a simple geometrical interpretation of the distortional effects. Here, the model is mainly presented as a distortional yield condition and is only outlined as a hardening rule, where it needs further experimental specification. The model is formulated for initially isotropic J2 materials [7] but may be generalized to a wider class of pressure-insensitive "deviatoric materials" [8, 9 ]. As a hardening rule [ 10, 11], the model follows classical relations of rate-independent elastoplasticity for small plastic strains.
3.5.2 FORMULATION
OF THE MODEL
The primary idea of the present model is to have the initial yield condition described by a hypersphere and to consider various transformations of such a hypersphere. The model is therefore conveniently formulated using vectorial representations of stress and strain-type deviators in the auxiliary fivedimensional Ilyushin's vector spaces [12]. In what follows, index notation will be used. For the sake of clarity, Latin indices will refer to physical coordinates (i, j , . . . , = 1, 2, 3), and Greek indices will be reserved for the coordinates of Ilyushin's vector spaces (o~, fi,..., = 1, 2 , . . . , 5). In these spaces the stress deviator S = {sij} will be represented as a stress vector a = {a~} with the components a~ defined here as: (1) '
\2
'
In such a space the Huber-von Mises yield condition sijsij - 2a2/3 (where a0 stands for the yield stress in uniaxial tension) is described by a hypersphere with the radius R0 = a0; -
,o
-
(2)
168
Kurtyka
The strain vector ~ = {8~} of the strain deviator E = {eij} is defined as 2 81 - - e l l ~ 8 2 - ~ \
2
2
g22 , 83 -- ~ e 1 2 ,
2
84 -- ~ g 2 3 ,
2
85 -- ~ { 3 3 1
(3)
and may be decomposed into its elastic and plastic part; ~ = ~e + ~p. The present distortional model may be treated as a generalization of the kinematic-anisotropic hardening models described by the quadratic yield condition v
-
-
=0
(4)
where C@1 is a fourth-order tensor of plastic anisotropy and aij is a deviatoric translation tensor (or back stress). In the vector stress space (Eq. 1) this condition is mapped (under usual symmetry conditions imposed on the anisotropy tensor) by a hyperellipsoidal surface, described by the quadratic form F = C~,8(a~ - ao~)(a~ - a~) - 1 = 0
(5)
with the symmetric matrix C -- { C~/~} corresponding to the tensor C = {Cijhz} (its components divided by the common factor 2a02/3) and the translation vector a = {a~} representing the translation deviator. The present model is based on a simple geometrical generalization of the hyperellipsoidal surface, illustrated for a two-dimensional case in Figure 3.5.1. The hyperellipsoid is treated here as a surface resulting from a projective mapping of five concentric hyperspheres with five (generally distinct) radii R~, and with five orthogonal projective directions coinciding with the directions of the eigenvectors of the matrix C, being also directions of the elliptic deformation of the yield surface, Figure 3.5.1a. A nonelliptic distortion of the surface is obtained by allowing the hyperspheres to be nonconcentric (Fig. 3.5.1b), with their centers translated with respect to the origin of the moving reference frame F~. These translations are described by five stress-type parameters m distortional vectors d(~). In the general case of such a surface, their directions may be arbitrary. However, as shown in Reference [13], experimental distortions of subsequent yield surfaces are sufficiently well described by a special, simplified case in which the distortional vectors are coaxial with the corresponding (i.e., having the same index a ) axes ~ . This reduces the distortional parameters to five scalar parameters denoted d~ - - the nonzero components of the five vectors d(~) in the moving reference frame F~.
FIGURE 3.5.1 Distortional model of subsequent yield surfaces (b) as a generalization of quadratic surfaces (a).
17 0
Kurtyka
3.5.3 DESCRIPTION
OF THE MODEL
For this simplified case of distortion the yield surface equation, expressed in terms of its geometrical parameters, is formally identical to that of the quadratic surface (Eq. 5); however, the matrix C is now a functional matrix, with the diagonal matrix of "eigenvalues" D = diag(Dy~) depending on stress components: C./~ - Q~.Q~/~D(-~), with
D(~) -- R~ + 2d(~,)Q(y),~(a,~ - a~) - d~,2
(6)
where indices in brackets are labels, ( 7 ) = 7, which are not subject to summation. In these formulae Q = {Qy~} is an orthogonal matrix describing rotation of the moving frame F~. In this coordinate system the equation of the distortional yield surface may be presented in the following "canonical" form: 5
"2
a~
-1--0,
with
c~-Q~(a~-a~)
(7)
0r
where F~ denote the components of the "active stress vector" ( a - a) in the moving frame. The preceeding yield surface model is characterized by the following set of geometrical parameters: Q, a, R~, d~, containing the rotation matrix Q (with 10 independent components), one vector parameter m the back stress a (with 5 components), and 10 scalar parameters R~, d~. The surface of this type is defined if R2 ___d2, and smooth if R2 > d2. Its convexity has been proven at least in a two-dimensional case [7]. Invariant aspects of the model and the equivalence between tensorial and vectorial formulations are treated in Reference [14]. In its most general form, with all R~ distinct and all d~ different from zero, the model accounts for quite general asymmetric distortions observed for nonproportional loading paths. In the case of proportional loads, the yield surface equation is largely simplified, since in this case the yield surface, at least for the initial Huber-von Mises material, should be rotationally symmetric with respect to the loading direction (Ilyushin's postulate of isotropy). This implies d~ = 0, R~ = R0 for 0~= 2, 3, 4, 5. In this case the rotation matrix Q is fully defined by any unit vector collinear with the loading direction, e.g., by e = a/]a] = ~P/]~P] . . . . , and the matrix C may be expressed as: -
+
+
-
-
(8)
where 6~ is a unit matrix. This expression may be treated as a generalization of the Baltov-Sawczuk tensor of plastic anisotropy for the case of the distortional model, which is defined here by only four parameters: ]a[, R1, R0, dl.
171
3.5 Distortional Model of Plastic Hardening
3 . 5 . 4 H O W T O USE T H E M O D E L 3.5.4.1
USE OF THE MODEL AS A YIELD CONDITION
The model may, first of all, be used to approximate experimental yield surfaces. Most frequently this will be done for the yield surfaces investigated in two-dimensional subspaces of the stress space (Eq. 1). In this case the distortional model corresponding to general, nonproportional loads is defined by seven parameters; e.g., for the al, a3 tension-torsion plane these are al, a3, R1, R3, dl, d3, q~, where q) is the angle of rotation of the moving reference frame, defining the rotation matrix. These parameters may be identified using the numerical method of leastsquares approximation described in Reference [13]. A sufficient number of experimental points of the yield surface, obtained in multiaxial tests, is necessary to perform the identification. As shown in [13], the model describes with good accuracy the experimental yield loci obtained for various materials and for different experimental definitions of yielding, both for proportional and nonproportional loading paths. Two examples are shown in Figure 3.5.2, corresponding to two experimental methods of yield surface determination, investigated either during the loading process (method of partial unloading, Figure 3.5.2a) or after complete unloading (Fig. 3.5.2b). The model may also be used as an anisotropic yield condition for the materials with previous plastic working (e.g., after rolling [9]).
3.5.4.2
USE OF THE MODEL AS A HARDENING RULE
Two approaches may be outlined here. First, we may choose as hardening variables the geometrical parameters of the yield surface, discussed previously, and specify separate evolution equations describing translation, rotation, size changes, and distortions of the yield surface. General and some specified forms of such evolution equations, as well as the corresponding stressstrain relations for this case, have been presented in Reference [10] and applied in Reference [11] to the simulation of Bui's experimental results shown in Figure 3.5.2b. The second approach, still to be explored, may be termed "tensorial" and consists in looking for a global evolution equation defining the parameters contained in the matrix C in the distortional yield condition. The matrix C itself is not convenient for this purpose, but may be presented as an inverse of a certain matrix N, C~/~ - N~-~1,
with
N ~ -- Q~Q~D(~)
(9)
FIGURE 3.5.2 The distortional model as best-fit to experimental yield surfaces of pure aluminum for nonproportional tension-torsion load paths. Experimental points according to (a) Phillips and Tang [15] - load path OAA'BC, (b) Bui [161 - torsion under constant compression, load path tangent to the initial yield surface.
3.5 Distortional Model of Plastic Hardening
173
representing a second-order tensor in the stress vector space (corresponding to a fourth-order tensor in physical coordinates) and linear in stress components. This tensor may be presented in the general form
N~fl = a~fl + ( o~, - a~,) B~fl~,
( 1 O)
where the second-order tensor A~fl (fourth-order tensor in physical coordinates) contains isotropic and affine anisotropic effects, and the thirdorder tensor B~fl~, (sixth-order tensor in physical coordinates) is responsible for the distortional effects. In this approach, the evolution equations must be specified for these two tensors. As a hint and "prototype" for these evolution equations, one may take the expression for the tensor N corresponding to the case of proportional loading, where it can be derived from Eq. 8 in the following finite form: N~fl = k16~fl + k2e~efl + k3(a~ - a~,)e~efle~,
(11)
where kl, k2, k3 ~ three scalar functions describing isotropic, affine, and distortional effects are related to the geometrical parameters; k l - R~, k 2 - R1z - R ~ - dlz, k 3 - 2dl, and should be specified from experiments. In this second approach to the formulation of the distortional hardening rule, further investigations should be directed toward a differential or integral generalization of this expression that is valid for general nonproportional loading paths and is combined with a relevant translation rule for the back stress. Some proposals of such generalizations, formulated for other distortional models (c.f. [5, 6]), could be reexamined in the context of the present model.
REFERENCES 1. Williams, J. E, and Svensson, N.L. (1971). A rationally based yield criterion for work hardening materials. Meccanica 6(2): 104-114. 2. Shiratori, E., Ikegami, K., and Yoshida, E (1979). Analysis of stress-strain relations by use of an anisotropic hardening plastic potential. J. Mech. Phys. Solids 27: 213-229. 3. Ortiz, M. and Popov, E. P. (1983). Distortional hardening rules for metal plasticity. Trans. ASCE, J. Eng. Mech. 109: 1042-1057. 4. Helling, D. E., and Miller A. K. (1987). The incorporation of yield surface distortion into a unified constitutive model. Acta Mechanica, Part I 69: 9-23, Part II 72:39-53 (1988). 5. Watanabe, O. (1987). Anisotropic hardening law of plasticity using an internal time concept (deformations of yield surfaces). Jap. Soc. Mech. Eng., Int. J. 30(264): 912-920. 6. Voyiadjis, G. Z., and Foroozesh, M. (1990). Anisotropic distortional yield model. J. Appl. Mech. 57: 537-547. 7. Kurtyka, T., and Zyczkowski, M. (1985). A geometric description of distortional plastic hardening of deviatoric materials. Arch. Mech. 37(4): 383-395. 8. Kurtyka, T., and Zyczkowski, M. (1984). Generalized Ilyushin's spaces for a more adequate description of plastic hardening. Acta Mechanica 52, 1-13.
174
Kurtyka
9. Zyczkowski, M., and Kurtyka, T. (1990). A description of distortional plastic hardening of anisotropic materials, in Yielding, Damage and Failure of Anisotropic Solids, pp. 97-111, Boehler, J. P., ed., London: MEP. 10. Kurtyka, T., and Zyczkowski, M. (1996). Evolution equations for distortional plastic hardening. Int. J. Plasticity 12(2): 191-213. 11. Zyczkowski, M., Kurtyka, T., and Wasik, K. (1995). Integration of evolution equations for distortional plastic hardening. Math. Modelling and Scientific Computing 5(2-4): 257. 12. Ilyushin, A. A. (1963). Plasticity (in Russian), Moskva: Izd. AN SSSR. 13. Kurtyka, T. (1988). Parameter identification of a distortional model of subsequent yield surfaces. Arch. Mech. 40(4): 433-454. 14. Kurtyka, T. (1990). Invariant formulation of a distortional model of plastic hardening. Mech. Teor. Stos. 28: 115-131. 15. Phillips, A., and Tang, J. L. (1972). The effect of loading path on the yield surface at elevated temperatures. Int. J. Solids Struct. 8: 463-474. 16. Bui, H. D. (1966). Ecrouissage des M&aux. C. R. Acad. Sc. Paris 262: 401-404.
SECTION
3.6
A Generalized Limit Criterion with Application to Strength, Yielding, and Damage of Isotropic Materials HOLM ALTENBACH Fachbereich Ingenieurwissenschaften, Martin-Luther-Universitdt Halle-Wittenberg, D-06099 Halle (Saale), Germany
Contents 3.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Description of the model . . . . . . . . . . . . . . . . . . . 3.6.4 Identification of the parameters . . . . . . . . . . . . 3.6.5 How to use the model . . . . . . . . . . . . . . . . . . . . . 3.6.6 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175 176
177 177 179 186 186
3.6.1 VALIDITY This is generalized equivalent stress criterion that is valid for materials in the brittle or ductile state and that can be used to describe the beginning of yielding, loss of strength or damage in structural elements, granular materials, and so forth, that a r e subjected to multiaxial stress states. The criterion is presented for isotropic materials and the case of m o n o t o n i c loading conditions at constant temperatures. In addition, the t i m e - i n d e p e n d e n t behavior is presumed. Finally, from the presented generalized criterion various special cases can be d e d u c e d and the limits of their applicability can be defined. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
175
17 6
Ahenbach
3.6.2 BACKGROUND The use of traditional engineering materials at elevated exploitation conditions as well as the use of new materials (e.g., composites or special alloys) requires the reformulation of the traditional limit criteria based on the equivalent stress formulations. With the help of the equivalent stress, one can compare critical multiaxial stress states in structural elements with the results of mostly uniaxial materials tests. On the other hand, a general expression based on some physical principles, etc., cannot be found. This is the reason for the great number of engineering proposals which are established in the literature. In uniaxial stress states, the material properties, such as the yield stress point cry or the ultimative strength cry, are characteristics which allow one to obtain, e.g., the beginning of yielding or the loss of strength. Both can be obtained from the stress-strain curve (mostly the engineering stresses and the engineering strains). In multiaxial (combined) stress states, we have to define an equivalent stress criterion. Considering that both the material properties characterizing the limit state cry, cry, etc., are scalars for the comparison certain criterion must be introduced mathematically. The criterion can be generally expressed as
f(crij;Ch)
=0
(1)
The function f is a scalar-valued, non-negative, homogeneous function. Note that the function f is a function of the stress tensor crij (i, j = 1, 2, 3) and several material constants Ck (k = 1, . . . , n) for fitting available data from tests. The main experimental information is based on data derived from simple tests under the condition of atmospheric pressure and loaded in tension, compression, and torsion (main tests in mechanical testing materials). On the basis of these tests, it is impossible to give recommendations about the equivalent stress value if the materials are, working, e.g., under hydrostatic pressure conditions or if the level and the kind of stress state have a significant influence on the material behavior. Experimental data demonstrating such behavior are reported, for instance, in Reference [1]. Equivalent stress expressions for isotropic materials are introduced by various authors. In general, they have the following form: f (r
r
r
G) = 0
(2)
where cri (i = I, II, III) are three independent invariants of the stress tensor. There is no unique answer to the question of what kind of invariants must be introduced: the principal stresses of the Haigh-Westergaard coordinates, etc. Examples of such invariants are presented, e.g., in References [1, 3, 6].
177
3.6 A Generalized Limit Criterion
3.6.3 DESCRIPTION OF THE MODEL Consider that the limit (failure) state occurs if plastic behavior, loss of stiffness, or damage starts. Then the following criterion can be introduced:
~q ___~
(3)
where aeq is the equivalent stress and a7 denotes a critical material properties (limit value) estimated in the tension test. With the help of the introduced inequality, we describe the equivalence between the uniaxial and the complex material behavior. If aeq < or7 the structural element works, e.g., in the elastic range (no plasticity) or we cannot obtain any failure. If r - - O"T we assume the limit state for the given material. Assuming isotropic behavior, the equivalent stress depends on the stress tensor invariants only. Taking into account the various possibilities for defining the stress tensor invariants below, we consider the following invariants (Haigh-Westergaard coordinates): 9 the first invariant of the stress tensor I1 ~ r 9 the von Mises stress arm = V/3/2sijsij with sij the stress deviator; and 9 the angle of the stress state ~ with sin 3~ - - ( 9 / 2 ) Sij Sjk Ski/CTv3M, ]~] ~ n/6. With respect to the introduced invariants, the equivalent stress expression can be suggested [2]:
r
following generalized
--- '~1 CYvM sin~ + 22CrvMCOS~ + 23 arm +/~411 ~ 2511 sin~ + 2611 COS~
(4)
The 2m (m = 1, . . . , 6) are scalar parameters which should be determined experimentally. Note that the generalized equivalent stress is sensitive, e.g., to the influence of the hydrostatic pressure and the stress state (for instance, tension or compression). The classical equivalent stress the von Mises stress can be deduced if ,~3 = 1 and all other ,~m are vanishing. Assuming the parameters to be constant, the proposed model is restricted by monotonic loading, in addition, hardening behavior cannot be modeled.
3.6.4 IDENTIFICATION OF THE PARAMETERS The six unknown parameters in the equivalent stress expression can be identified with the help of the following basic tests in which homogeneous stress states can be assumed. In this case we can estimate the parameters as functions of the material properties obtained in tests. Such material properties are, for example, the first yielding (in the case of plastic behavior of ductile materials) or the ultimative strength.
178
Ahenbach
Let us introduce three basic tests of the material testing: 9 uniaxial tension o11 - - 0 . T ; 9 uniaxial compression 0"11 - - - - 0 - C ; and 9 torsion O"12 D "~T. Here 0-c and zr are the limit values of the material in the case of compression or torsion. In addition, we can introduce tests reflecting the complex behavior. The introduction of such tests is not unique and depends on the experimental facilities. Here we consider the following tests: 9 thin-walled tubular specimen under inside pressure with the stresses 20"11 - - 0"22 ~ 0"R;
9 biaxial tension (thin-walled tubular specimen loaded by inside pressure and uniaxial tensile force) with the stresses 0-11 = F / A + 0.~/2, 0-22 = 0.~; and 9 uniaxial tension superposed by hydrostatic pressure with the stresses 0"11 = F / A - q, 0"22 = 0"33 = - q . 0-R is the limit of the circumferential stress which can be calculated from 0-R - p R / h , p is the inside limit pressure, and R, h are the radius of the middle surface and the thickness of the thin-walled tubular specimen. In the next test, 0-c is the limit of the circumferential stress, F is the tensile force, and A denotes the cross section area of the thin-walled tubular specimen. In this test a special combination of the uniaxial force and the pressure must be fulfilled: 0 - 1 1 - 0 " 2 2 - 0-. with 0-, - p R / h . For the third test the following condition must be valid: 0 . 1 1 - 2 / 3 0 - * * , 0 " 2 2 - 0 " 3 3 - - 1 / 3 0 . * * with 0 - * * - 3q (q denotes the hydrostatic pressure). The last condition guarantees that the first invariant vanishes. For comparison we calculate for each test the invariants I1, 0"vM, and ~. Finally, we put these values in the expression of the equivalent stress. For example, in the case of uniaxial tension we obtain I1 --0"T, 0 " v M - 0"T, ~ = -re/6, and
,~1 v/3
,~5 V5
- - ~ - + --~--22 + 23 + 24 - -~- + --~-26 -- 1
(5)
Providing the same calculations for all other tests, we get six equations (algebraic with respect to the unknown parameters 2m) containing six known material properties identified by independent tests. The solution of this
179
3.6 A Generalized Limit C r i t e r i o n
system of equations can be presented as / 26
__
1 [ 4 ar - 2 a r - 3 + 3 a-----S-r a r 6-3v/3\ aR rr a** a.
24
\ aR
(
25 - 2
-1+
rr -24+v/3
0"**
-2-
(6)
(7 y
21 - - 1 + m + 224 + v/326 O-C
1
/~3 =
2
--
( aT v ~ \ 2O'..
aT TT
Note that, in the case of other tests for complex stress states, we obtain values of the unknown 2m that are different from the presented solution.
3.6.5 H O W TO USE THE MODEL The generalized criterion can be used for the description of the limit state (beginning of plastic behavior, damage or loss of stiffness) for materials with a behavior depending on the stress state. Examples are different behavior in tension and in compression or if the hydrostatic stress state influence cannot be ignored. The proposed equivalent stress expression generalizes various well-known classical criteria. The expression contains six parameters, and we can classify the equivalent stress as a six-parameter criterion, which means that we need for identification purposes six independent tests. Let us discuss the special cases from the point of view of criteria with fewer parameters or based on fewer independent tests. The Huber-von Mises-Hencky equivalent stress mostly used in plasticity is based on one test that means aeq = arm <_ ay
(7)
At the same setting 23 = 1 and all other 2m = 0, we obtain the special criterion from the general form if we take into account that a ; = ay. From the solution of our system of equations we can now calculate with respect to the given
Ahenbach
180
values of ,~m in the case of H-M-H some relations for the material properties O ' T - - O'T - o"C
o-,
~
O'T ~ 1, o-, ,
v~ ,
=
o-R
2
~ - -~-
(8)
"~T
If these relations are fulfilled, we can conclude that the Huber-von MisesHencky equivalent stress is valid. In other words, if we obtain these combinations of limit values in tests, the use of the Huber-von Mises-Hencky equivalent stress can be recommended; otherwise we have to consider another expression for the equivalent stress. In Table 3.6.1 various criteria are
TABLE 3.6.1 Values of the Parameters ~,i, i = 1 , . . . , 6 in the Generalized Criterion in the Case of Particular Criteia Criterion
;tl
H-M-H
0 0
C-T-V
22
23
One-Parameter criteria 0 1 2v~ 0 3
24
,~5
26
0 0
0 0
0 0
21
v~ 2
0
0
0
0
G-L
- 51
v~ 3
0
1
0
0
Sd
-g
1
v/3
1
1
0
0
1+~ 3
4 (1 4- v)
0
1-2~ 3
0
0
Ma
V-B
6
2
Two-parameter criteria Mo
_l-x 3
0 l+z
0
0
0
v~(z+l) 3 0
1-z
B-M
2
~ 2-
o
o
C-P
0
o
v~
a~/
0
0
P-L
z-1 3
v~(1-Z)
3
g
1-z
0
0
Sa
0
x/~(I+z)
0
2
o
0
K
0
2f(1-
0
0
0
o
1 (al 4- a2 4- a3)
0
0
b3
0
0
0
a4-~
0
0
3
r
q
3
Three-parameter criteria P Ts
(2.~ - ~1 - ~ ) bl
H
~
(.1
- ~3)
b2 3
Four-parameter criteria B
Ta
(2c2 - Cl - c3)
o
4(C1--C3)
C4
1 (cl + c2 + c3)
0
0
dl
0
d2
d3
d4
3.6 A Generalized
181
Limit Criterion
presented as special cases of the six-parameter-equivalent stress given here. From the Table 3.6.2. one can get other information: the combinations of the material's limit values for which the special expressions of the equivalent stress can be recommended. In both tables the following abbreviations are used: 9 H-M-H for the H u b e r - v o n Mises-Hencky criterion 0-eq ~
0-vM
9 C-T-V for the C o u l o m b - T r e s c a - d e Saint Venant criterion Tmax - 1(0-1 -- 0-3)
or
0-eq - -
0-1 - - 0-3
9 Ma for the Mariotte criterion 1 0-eq -- 0-1 --~(0-2 4- 0-3)
9 G-L for the Galilei-Leibniz criterion 0-eq - - 0-1
9 Sd for the Sdobyrev criterion 0- eq __ 1 (0-vM 4- 0-1)
9 V-B for the de Saint Venant-Bach criterion 0-eq =
0-1 - - V(O'2 4- 0-3)
9 Mo for the Mohr criterion 0- eq - - O"1 - - Z0-3, Z
m
O'T 7cc
9 B-M for the Botkin-Mirolyubov criterion 0-eq -- 1[(1 4- Z)0-vM 4- (1 -- X)I1]
9 D-P for the Drucker-Prager criterion (r/ --~r for recalculation of the criterion)
Tmax - -
~/~
SijSij Jr-
0~I1 or 0-eq -- ~30-vM 4- ~/111 with r/
_
v5 v5~+1
TABLE 3.6.2 Restrictions for the Material Characteristics in the Case of Particular Criteia Criterion
9 OC
EI TT
EI UR
EL O.
LL a.s
1
1 1
One-parameter criteria 1
V'3
4
C-T-V
1
2
1
1
Ma
1 -
3
a
1 2
1
1
2 3
1
5
1-v
5 ( 1 + v)
H-M-H
2
2
G-L
0
1
1
Sd
1 -
&
&
2
2
V-B
v
1+v
4
2-v 2
Two-parameter criteria Mo
1
1
B-M
v%l+x)+3(1-x) 4
%
C-P
P-L Sa K
6
TABLE
-
Criterion
K
3.6.2 (continued)
>
--
vc
or -
Q
EL
LX
ZT
OR
0.
0 . .
Three-parameter criteria
Four-parameter criteria
6 C
-.
i.
184
Altenbach
9 P-L for the Pisarenko-Lebedev criterion Creq - - ZCrvM + (1 -
Z)0"I
9 Sa for the Sandel criterion O'eq --
O" I -4-1(1
- 2)0"2 - 20"3
9 K for the Koval'chuk criterion 2-r/ 2_v/~,
ff eq - - qo'vM + (1 - q ) ( o " 1 - 0"3) , q
t']
~
o'r
9 P for the Paul criterion with
ffeq - - a l O l -Jr-a202 -k- a303
al-
1
9 Ts for the Tsvelodub criterion ~eq -- b l ~ v M sin~ + b2~rvMCOS~+ b3~rvM
with
-- bl 4- v ~ b 2 + 2b3 = 2
9 H for the Hayhurst criterion (Teq -- 0~I1 q- ~ffvM -Jr- ~0"1
with
c~+fl+7-1
9 B for the Birger cirterion O'eq ~ C10-1 + C20"2 + C30"3 + C40"vM
TABLE
3.6.3
with
C1 A t - C 4 - 1
Relation between Limit Values in Tension and Compression [5]
Material High=strength cast iron Wrought iron Grey cast iron High=strength steel Metal ceramics (tungsten carbide based) Graphite Glass Note that in the case of H-M-H or the C-T-V criterion cr~/crc must be 1.
0.2-0.3 0.7-0.95 0.2-0.4 0.9-1.0 0.1-0.4 0.2-0.6 0.07-0.2
4.
TABLE
6
3.6.4 Ultimative Strength in Tension, Compression, and Torsion for Grey Cast Iron 141
Material
GG-15
GG-20
Ferritdperlite Perlite 150-250 200-300 600 720 170 230 9 0.25-0.42 0.28-0.42 UT ac 0.88-1.47 0.87-1.30 7T Note that in the case of the H-M-H or the C-T-V criterion g T / z T - & = 1.73 or 2. Structure Ultimative strength (tension), N/mm2 Ultimative strength (compression), Nlmm2 Ultimative strength (torsion), ~ / m m ~
3.
GG-25
GG-30
GG-35
250-350 840 290 0.30-0.42 0.86-1.21
300-400 960 345 0.31-0.42 0.87-1.16
350450 1080 400 0.32-0.42 0.88-0.013
186
Altenbach
9 Ta for the T a r a s e n k o c r i t e r i o n
0-eq -- dl0-vMCOS~ + d211 4- d 3 I l s i n ~ 4- d4IlCOS~
with
v ~ d l 4- 2d2 - 2d3 + x/3d4 - 2
Here 0-i, i -- 1, 2, 3 denote the principal stresses (0-1 ~ 0-2 ~ 0-3) which can be expressed by the introduced invariants as follows" 0-1 ~
20-vMsin(~ + ~ ) +
3
I1
~ 0-2 m
20-vMsin~+ I1 20-vMsin(~ + ~ ) + ~ 0-3 m 3 3
I1
(9) Note the restrictions of introduced parameters are included with respect to the experimental facilities (here with respect to the proposed tests).
3.6.6 PARAMETERS Tables 3.6.3 and 3.6.4 present experimental data for some materials for which the generalized limit state criterion can be used.
REFERENCES 1. Altenbach, H., Altenbach, J., and Zolochevsky, A. (1995). Erweiterte Deformationsmodelle und Versagenskriterien der Werkstoffmechanik, Stuttgart: Deutscher Verlag ffir Grundstoffindustrie. 2. Altenbach, H., and Zolochevsky, A. (1996). A generalized failure criterion for threedimensional behavior of isotropic materials. Engineering Fracture Mechanics 54(1): 75-90. 3. Chen, W., and Han, D. (1988). Plasticity for Structural Engineers. New York et al.: Springer. 4. Deutsches Institut fiir Normung, NormenausschuB GieBereiwesen: DIN 1691-GuBeisen mit Lamellengraphit (Graugu~) (Mai 1985). 5. Lebedev, A., Koval'chuk, B., Giginyak, E, and Lamashevsky, V. (1983). Mechanical Properties of Engineering Materials at Complex Stress States, Kiev: Naukova Dumka. 6. Zyczkowski, M. (1981). Combined Loadings in the Theory of Plasticity, Warszawa: PWNnPolish scientific publisher.
SECTION3
-
7
Yield Conditions in Beams, Plates, a n d S h e l l s DANIEL C. DRUCKER Department of Aerospace Engineering, Mechanics & Engineering Service, University of Florida, 231 Aerospace Building, Gainesville, Florida 32611
As discussed briefly in the section on isotropic yield criteria, the yield condition of a ductile metal or alloy is a matter of definition and then idealization as well. Inhomogeneity of test specimens at the grain level, stress concentrations produced by inclusions and other discontinuities on the microscale, and the presence of large numbers of mobile dislocations all combine to produce small but detectable plastic strain at an early stage of loading. The yield limit, expressed as a yield stress for uniaxial or shear loading or as a yield function in stress space for more general loading, is then determined by an offset of strain or deformation per unit length, or by an alternative definition that gives a similar result. When the offset chosen is of the size of the elastic strain, the yield limit is a stress level beyond which the macroscopic plastic deformation that occurs is widespread and at most weakly constrained by the elastic response of neighboring or distant regions. In this conventional and useful approach to stress-strain relations, what is labeled as pointwise behavior actually is the load-deformation behavior of a small unit block of material containing a sufficient number of grains to average out their quite different elastic and plastic responses along with the many disturbances on the microscale. The large initial and fabrication stresses present in most ductile structures similarly induce some highly contained but observable local plastic deformation on the macroscale well within what is nominally the purely elastic range of response. As at the specimen level, a useful yield load condition at the structural level is defined by a deflection or deformation offset for the plastically deforming regions of the size of the elastic response. It appropriately ignores not only the contained plastic action on the microscale but also the early stages of the macroscopic plastic response in one or more regions, a plastic response that initially is strongly constrained by the almost purely elastic response of neighboring material. The yield limit so defined is reached when enough regions deforming plastically are no longer so
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
187
188
Drucker
constrained by the remaining elastic regions nearby or far away that a plastic (really elastic-plastic) deformation mechanism can operate. A very useful picture for placing such a yield condition in proper perspective is provided by plastic limit analysis. When the material is idealized as rigid-perfectly plastic, the elastic strains go to zero (infinite modulus) and the material does not workharden in the plastic range. No deformation at all then takes place until the loads applied to the structure reach the limit condition and excessive deformation or collapse occurs. With the more realistic idealization of the material as elastic-perfectly plastic, and (as is customary and almost always permissible) the equations of equilibrium are satisfied in a fixed configuration, the same limit values and purely plastic collapse mechanism apply because the stress everywhere remains unchanged during collapse. An initially stress-free elastic-perfectly plastic homogeneous straight beam of rectangular cross section b x h under pure moment M serves as illustration. As the moment is increased from zero within the purely elastic range, the bending stress induced is linear with the distance from the neutral axis. The yield limit stress O-y(= a0)will be reached at the extreme fiber when the moment M reaches My = aobh2/6. Any appreciable initial stress present will produce yield much earlier at one extreme fiber or the other. More and more of the beam goes plastic as the moment is increased beyond the yield value. The early plastic response is strongly constrained by the elastic response of the remaining fibers. A further increase of the bending moment to 11aobh2/48 will still leave an elastic restraining core of half the depth of the beam. Full plasticity with only an asymptotic vestige of purely elastic constraint requires that M reach its limit of M0 = aobh2/4. This limit moment is 50% greater than My, the moment at first yield in the absence of initial stress, which in turn is greater than the moment producing yield in a beam with initial stress. The moment-curvature plot for the elastic-perfectly plastic material idealization only asymptotically flattens to zero slope at the limit moment as shown (Fig. 3.7.1). For an elastic-workhardening material, the moment-curvature plot in the plastic range continues upward and looks like the stress-strain curve but with a much earlier and more rounded transition from the initial high elastic slope below initial yield ay at My = aybh2/6 to the very much flatter elastic-plastic at larger M. Just as the stress-strain curve in tension may be idealized as elastic or perfectly plastic at a limit stress equal to the offset yield stress a0 > ay, the moment-curvature picture also can be so idealized at a limit moment M0 defined by an appropriate curvature offset of the size of the elastic curvature. Mo = aobh2/4 will be a fair approximation. Under a general transverse loading, the moment will vary along the beam. Like the plot of moment vs. curvature, a load-deflection curve for a statically
189
3.7 Yield Conditions in Beams, Plates, and Shells
Mo
Curvature FIGURE 3.7.1 Moment vs. curvature for an elastic or perfectly plastic material and a workhardening material.
indeterminate beam also looks like a stress-strain curve but with a still earlier and more gradual transition from the high slope elastic response to the relatively fiat plastic response. The first reaching of the yield stress cry at the extreme fiber at the most highly stressed cross section will be even less visible. The yield load, defined by a deflection offset of the size of the elastic response, will not be reached until enough cross sections are sufficiently close to fully plastic that the response of the beam would be almost that of a mechanism with plastic hinges. For example, under a uniformly distributed load q per unit length, a fixed-ended initially stress-free beam of length L and uniform cross section b • h will reach the initial yield stress Cry for the material at the extreme fibers at the fixed end when the bending moment there, qL2/12, reaches crybh2/6. As the load is increased further, the bending moments everywhere along the beam increase and more and more of the cross sections at and near the ends go plastic. When the end cross sections are fully plastic, at a bending moment of M0, the response away from the ends will still be primarily elastic. Not until the center section also reaches M0 and can operate as a (workhardening) plastic hinge is a fully plastic mechanism possible. In the perfectly plastic limit of no workhardening, when the limit moment M0 is reached at the center and both ends, the load-deflection curve for the beam goes fiat. The limit load q0, given exactly by qoL2/16 = M0 = crobh2/4 for the perfectly plastic idealization, is twice the load calculated to produce first yield. With workhardening, the limit moment M0 for plastic yielding across the full depth of the beam not only is well above crybh2/4, but somewhat exceeds crobh2/4. As the load is increased, the end moments
190
Drucker
increase above their limit value for full plasticity on the cross section so that the effective limit condition for the beam is not reached until this increase in moment resistance due to workhardening also is overcome. Despite all this complexity, the limit load calculated for perfectly plastic response with M0 = ~robh2/4 is a crude but not unreasonable measure of the yield load for the workhardening beam as defined by an offset in the load-deflection plot of the size of the elastic deflection. The effect of shear force V on the limit condition can become important when beams are very short or are loaded in some unusual way. When shear forces are significant, they do reduce the limit moment and have an effect on the moment-curvature relation in the plastic range. It is not strictly correct to replace the simple limit moment condition of M - M0 with a single limiting curve in combined moment and shear force space. The interaction of shear and moment for each beam is different for different loading and support conditions as well as for different yield limit criteria. For I-beams and WFbeams it is easiest to use the simplification that the flanges take the bending and the web carries the shear and so that they bypass the interaction. For a rectangular cross section, although the interaction curves do change somewhat with the loading and support conditions, the one relation M / M o -1 - ( V / V o ) 4 is not a bad approximation for all, where Vo - ~robh/2 when the Tresca criterion of yield is chosen. Most often the effect of the shear is small enough to be ignored. When detailed load-deflection calculations are required, they can be obtained by computation with moment-curvature relations determined directly by experiment or by full calculation from appropriate uniaxial stress-strain curves and knowledge of the initial state of stress. On rare occasions it may be desirable to obtain the history of stress and strain at some critical point in a structure rather than dealing with generalized stresses and strains such as moment and curvature. As computers and programs for their use become more and more powerful, it becomes easier and easier to calculate the stress produced at each point by the application of loads to a perfectly elastic three-dimensional body of complex geometry. With knowledge of the initial stress and the state of the material everywhere (or the assumption of zero initial stress everywhere at zero load), and completely specified geometry of structure and loading, the onset of yield can be determined for any yield criterion that is chosen. However, accurate computation of the subsequent history of the state of stress and strain under increasing load in the plastic range is difficult and time-consuming with even the simplest yield criterion and flow rule. Furthermore, our knowledge of the stress-strain behavior of any material rarely if ever is complete enough to place much credence in the details of the results of such computations with the most elaborate of criteria chosen to match the data available. Fortunately, approximations that employ
3.7 Yield Conditions in Beams, Plates, and Shells
191
such concepts as offset yield criteria and plastic limit loads are satisfactory for analysis and design under most circumstances. An alternative approach is to replace the elastic-plastic behavior with a nonlinear elastic behavior that matches the stress-strain curve for increasing loading in simple tension or shear. This pseudo-plastic assumption goes under the name of deformation or total theory of plasticity and can give useful results when the significant stresses everywhere continue to increase as the load on the structure or element increases. It is far less demanding computationally than the simplest of genuine plasticity approaches when calculations are made of local stresses, strains, and displacements. When the body is a structure composed of such elements as beams, plates, and shells, the use of beam theory, plate theory, and shell theory makes for great simplification over a true three-dimensional approach and provides a very good assessment of first yield and of yield limit conditions. For plates and shells, the analysis is in terms of generalized stresses such as bending moments, twisting moments, and membrane normal and shear forces per unit length of middle surface. Corresponding kinematic quantities or generalized strains are curvature, twist, and membrane extensional and shear strains. The usual linear elastic calculations have many valuable uses. However, the initial yield they predict is unrealistic unless initial stress, elastic stress concentrations at connections and changes of section, and imperfect geometry of structure and loading are taken into account. Also, as for the rectangular beam, little of importance happens when a point on the surface of a ductile plates or shell yields. There usually is so much constraint in the elastic-plastic regime that deviation from linear elastic response remains small at the considerably higher load when a yield moment is first reached at one or more cross sections located at some point or along some line in the middle surface. Again the replacement of workhardening by perfect plasticity at an appropriate offset yield level and of elastic-plastic calculations by plastic limit theory permits a reasonable estimate of yield limit loads to be obtained fairly easily, along with a clear picture of the essence of the response. An alternative analysis based upon the substitution of nonlinear elasticity (deformation theory) for plasticity (flow theory) also is computationally tractable and may provide some useful insight. The usual plate analysis considers bending moments Mx and My and twisting moments Mxy--Myx per unit length of middle surface, or their curvilinear coordinate counterparts, with in-plane or membrane forces taken as zero. Transverse shear does not appear in elementary plate analysis or in the traditional limit condition. The moments produce a state of plane stress parallel to the middle surface at each point of the plate. Initial, highly constrained, yield at the top and bottom of the plate and the much later condition of full plasticity throughout the depth of the plate therefore are
19 2
Drucker
governed by the same criteria as for plane stress with 6Mx/h 2 or 4Mx/h 2 replacing ax, and similar replacement of My for ay, and Mxy for axy. On the assumption of initially isotropic material with isotropic hardening, the picture of yield limit curves in the two-dimensional principal bending moment space M1 vs. M2 is just like the picture in two-dimensional principal stress space, concentric Mises ellipses or Tresca hexagons or something between them. In the three-dimensional space (Mx, My, Mxy) it becomes a set of concentric surfaces. Its section by the plane My = 0 or by Mx = 0 is a concentric set of ellipses with major to minor axes in a 2 to 1 ratio for the Tresca and a square root of 3 ratio for the Mises, just as for the plot of one normal stress ax or ay vs. shear stress axy. Adding membrane forces to the bending and twisting moment components adds greatly to the complexity of the yield condition. Membrane forces do occur in plates but can usually be ignored as secondary quantities, unless the plates are subject to special in-plane loading or undergo large deflections. However, one or more will be among the primary quantities for shells under load. In general, the plastic as well as the elastic analysis and design of shells requires consideration of six components of generalized stress, the three membrane force components along with the three moment components whose principal axes need not coincide with those of the membrane forces. Except for a few important simple geometries and loadings, precise step-bystep computation of the generalized strains (curvatures and in-plane strains) and displacements under changing load in the plastic range is very timeconsuming, even with the assumption of zero initial stress and initial isotropy along with isotropic workhardening. As discussed for the beam, a reasonable estimate of a yield load condition (corresponding to a deflection offset of the size of the elastic deflection) may be obtained from limit analysis with a perfectly plastic yield limit surface for the six components of generalized stress. In principle, such a yield limit surface may be determined by experiment with appropriate offset of elastic magnitude for the generalized strains, a far from trivial extension of the previously described determination of M0 from a moment-curvature experiment. The extension of the calculation of M0 = aobh2/4 for a beam is a far more attractive option. For a plate or shell element (the beam is a special case), the desired perfectly plastic yield limit surface in generalized stress space is determined by the perfectly plastic yield limit stress surface in conventional stress space that is selected. Each combination of generalized plastic strains gives a linear variation of plastic strains through the thickness of the plate or shell element. These local plastic strains determine the local states of limit stress from which the generalized stresses corresponding to each choice of generalized plastic strains can be computed directly. The resulting yield limit surface in six dimensions and its two- and three-
3.7 Yield Conditions in Beams, Plates, and Shells
193
dimensional sections as well are likely to be of quite complicated geometry and not well suited for computation. Upper and lower bounding surfaces (circumscribed or inscribed figures) or some reasonable approximation to simplify the computations often are employed instead. Unless the geometry and the loading are very simple, computation of the details of the load-deflection behavior of workhardening plates and shells is a very long and tedious incremental process with the most tractable of elasticplastic relations between generalized stress and strain. When the path of loading leads to continually increasing (generalized) stresses in the most important regions, replacement of the elastic-plastic response by a nonlinear elastic relation between generalized stresses and generalized strains may provide a useful alternative means of computation of load-deflection relations. A number of limit load solutions for rotationally symmetric plates and shells under axially symmetric loading can be obtained directly from substitution of the yield limit condition in the equations of equilibrium without explicit consideration of the geometry of the deformation. These "statically determinate" examples are of direct use and may be of help in estimating limit loads for more complicated problems. A plate of radius R and uniform thickness h supported by a concentric circular ring of radius a and subjected to a radially symmetric transverse load provides a simple instructive example. The differential equation of equilibrium relating the radial and the circumferential bending moments per unit length (the generalized stresses) to the loads is an ordinary differential equation in the radius r. When the region of the plate within the supporting ring is fully plastic, the yield condition that is chosen gives the radial moment in terms of the circumferential moment. The resulting onedimensional equilibrium equation in the circumferential moment may be integrated (numerically if necessary) outwards from the plastically isotropic center of the plate where the bending moment per unit length is M0 on all transverse planes. With the offset yield strength in simple tension a0 taken as the yield limit for the perfectly plastic idealization, M0 -- aoh2/4. When the transverse loading is a uniform pressure p over the radius a and the Tresca (maximum shear stress) criterion is chosen, the yield limit pressure P0 ranges from 6Mo/a2 for R/a = 1 (simply supported at r - a) to 1.88 times as much for a fixed support when R/a is equal to or greater than the natural log base e(=2.718). The overhang region deforms plastically in the intermediate range but remains rigid during collapse when R/a exceeds e. A long thin cylindrical shell of radius R and wall thickness h deforming plastically under axisymmetric outward radial loading was the first plastic shell problem studied. It is similarly "statically determinate." The yield limit curve in the two-dimensional generalized stress space of circumferential force per unit axial length (in general not acting at the middle surface) and axial
194
Drucker
bending moment per unit of circumference is obtained easily from any isotropic yield limit in two-dimensional stress space. Substitution for the circumferential force in the equation of equilibrium gives an ordinary differential equation for the axial moment which can be solved analytically or integrated numerically. I chose the Tresca yield condition for convenience and determined the plastic yield limit for an outward ring loading of P per unit length of circumference to be 1.82 Cro(h/R)(Rh)1/2. For a band of uniform outward pressure p per unit of area over a length 2c that is large compared with the characteristic length of the shell, (Rh)1/2, the plastic limit pressure is reached when the hoop stress pR/h over the central portion of the loaded region reaches cr0. For a narrow band of uniform pressure, 2c much less than (R/h) 1/2, the plastic yield limit P - p(2c) is increased over the plastic limit for a concentrated ring loading by only one fourth of 2c(croh/R). The next level of complexity involves the three-dimensional picture of yield limit surfaces and plastic limit loads for an axisymmetric pressureloaded shell with two components of membrane force and one active moment. A thin cylindrical vessel with a thin torispherical head subject to interior pressure was analyzed by Shield and Drucker, who employed a hexagonal prism approximation for the yield limit surface for simplicity, based on a Tresca yield condition. We explained the very worrisome catastrophic fractures that occurred in large storage tanks by showing that plastic limit loads were exceeded when thin vessels were designed according to rules that are entirely satisfactory for boilers and other thick pressure vessels. So-called brittle fracture is all too likely in structures of ordinary steel at or below near freezing temperatures when limit loads are exceeded. The smoother but more complex yield limit surface in three dimensions for a perfectly plastic material obeying the Mises yield condition was given by Onat and Prager. For still more complicated loadings and shell geometries, the limit surface exists in a higher dimensional space. With a very few exceptions, yield load limits for shells cannot be determined exactly even when highly simplified yield limit surfaces in generalized stress space are chosen and powerful techniques of computation are employed. Plates are far simpler than shells, but they too are generally too difficult. Limit loads based upon perfect plasticity do provide reasonable approximations to the desired yield loads. They in turn can be approximated using either or both the upper (kinematic) and lower bound (equilibrium) theorems.
CHAPTER
Plasticity
4
This Page Intentionally Left Blank
SECTION
q.1
I n t r o d u c t i o n to
Plasticity JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France Plasticity is the process by which solids change their shape in equilibrium with loading and permanently after unloading. This behavior is not explicitly related to time. In metals it is due to irreversible movements of dislocations inducing slips. The corresponding plastic or permanent strains may be very large in comparison to elastic strains: up to several hundred times in metal forming. However, they do not modify the phenomenon of elasticity related only to relative movements of atoms. The main features of plastic deformations are as follows: 9 The existence of a yield stress (~y in pure tension) below which a variation of stresses induces only elastic deformations (see Chapter 3). 9 Plastic incompressibility. The plastic strains eijP do not involve any volume change (which is not the case for the elastic strains). v --0
gkk
(1)
Furthermore, to a large extent, they are not sensitive to pressure or hydrostatic stress: a~ = akk/3. This allows one to write the constitutive equations of plasticity as a function of the stress deviator.
-
-
(2)
9 In three dimensions the yield function which represents the yield phenomenon plays a major role because it is used as a potential, the loading function f, to derive the plastic strain rate ~/~ through the normality rule
- Of(aD'J)2Oaijif{ ff ----O0
~3ij'v-- 0 iff < 0 orf < 0 Handbook of Materials Behavior Models. ISBN 0-12o443341o3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
(3)
197
198
Lemaitre
being the plastic multiplier determined by the consistency condition f=0. The existence of hardening, i.e., the actual yield stress as, is always associated with the highest level of stress which has been applied, Cry being the initial yield stress, f = 0 means that the state of stress has reached the largest previous value, f - 0 means that the state of stress increases and induces the same increase of the yield stress. Often, plasticity models differ by the choice of hardening variables. A simple hardening law for slip at the level of single crystals can describe realistic hardening of polycrystalline materials after appropriate homogenization over many differently oriented crystals. Examples of such homogenization procedures are the self-consistent model (Section 4.2) or the finite element method (Section 4.3). The phenomenological method based directly on the thermodynamics of irreversible processes introduces a scalar variable of isotropic hardening and a tensorial variable of kinematic hardening (Section 4.4) or more through multisurface theory (Section 4.5). Nevertheless, it is not enough when modeling additional phenomena such as recovery and racheting, which need more complex kinematic hardening rules (see Sections 4.6 and 4.7). Large deformations needed for metal forming are described in Section 4.8. All previous sections are more or less devoted to metals; the special case of the plasticity of polymers may be found in Section 4.9. In usual plasticity, the strain rate plays little role, but this is not the case in dynamic plasticity if the strain rate is above 10 or 100 S-I, as in car crashes or high-speed perforations. Then the stress is an increasing function of the strain rate in some way similar to viscoplasticity (see Section 4.10). When softening occurs, as in the large strain range or by coupling with damage, the phenomenon of localization may occur (Section 4.11). The width of the resulting shear bands and some effects of the microstructure may be captured by the introduction of strain gradients (see Section 4.12).
SECTION
4.2
Elastoplasticity of Metallic Polycrystals by the Self-Consistent Model M. BERVEILLER Laboratoire de Physique et M~canique des Mat~riaux, lie du Saulcy, 57045 Metz Cedex, France
Contents 4.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.1 Single-Crystal Behavior . . . . . . . . . . . . . . . 4.2.2.2 Integral Equation for ~(r) and v~(r) .... 4.2.2.3 Self-Consistent Approximation . . . . . . . . 4..2.3 How to Use a Self-Consistent Model . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199 199 199 201 201 202 202
4.2.1 VALIDITY The self-consistent model constitutes a simple way of predicting the effective behavior of inhomogeneous media, especially for metallic polycrystals, for which it may be considered a "good" approximation. In the case of elastoplasticity, self-consistent modeling allows one to take into account texture transformations, development of second-order internal stresses, intragranular hardening, and so forth. All these p h e n o m e n a are strongly path-dependent.
4.2.2 B A C K G R O U N D 4.2.2.1
SINGLE-CRYSTAL BEHAVIOR
The single crystal behavior is specified for mechanisms like crystallographic slip on slip systems (g) with n g and m g the unit slip plane and slip Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
199
200
Berveiller
direction vectors, respectively. If ~g denotes the slip rate on system g, the corresponding plastic strain rate in the case of multislip is given by gq/g with
g
l(mgin~4-m~nig )
with
s,jg
- l (m~n~"_ " - m _~ n i g)
g
and the plastic spin is w,j" -
(2)
g
From the definition of the intrinsic dissipation d = aij~,Pij, the resolved shear stress rg = aijR~ on a slip system appears as the driving force for the flux ~g. The dynamics of slip are assumed to follow the Schmid law, which defines a critical shear stress rg as depending on the amount of slip ~h on all the systems. One arrives at the following flow rules" ~g--o
ifzg
~g - 0
if l:g -- ~:c g and ig < ig
~g>o
if~:g-l: g a n d i g - i g
(3)
The change of z~g with 7h defines the hardening matrix:
if
-
Iqghgh
(4)
The elastoplastic tangent moduli 1 defined by 6 ( r ) - l ( r ) ' ~ . ( r ) is obtained from the following additional equations where C are the elastic moduli: -
(5)
~p -
~-~R9
One obtains lijhl - Cijhz - ~
c'.. "~Jevpg >q .~/~gh~h . . . ~n. CmnhZ
(6)
g,h
where M gh is the inverse of pgh _ Hgh 4- RpqCpq,.sR,. g hs. The apparently simple tensor 1 is, in fact, complicated since it depends on the active slip systems, on the stress state as well as the loading path. For polycrystals or polyphased materials, 1 depends on the orientation of the local lattices.
201
4.2 Elastoplasticity of Metallic Polycrystals by the Self-Consistent Model
4.2.2.2
INTEGRAL EQUATION FOR ~(F) AND 1~r
The polycrystal is submitted at its external boundary to a velocity fi(r) = (/~ 4 - ~ / ) : x so that the overall strain and spin rates /~ 4- W are uniform. A given/~ 4- ~ / i s responsible for local fields +(r) and 6-(r), which have to satisfy
(rijj = 0
equilibrium condtions
(7)
1
compatibility relations
gij -- -~ (bli 0 4- Rj,i)
(8)
constitutive equations
d(r) = l(r): ~(r)
(9)
A fictitious homogeneous medium with tangent moduli L~ is introduced, leading to fluctuations al satisfying
81(r) = l(r) - L~
(10)
Equilibrium and constitutive Eqs. 7-9 yield to a Navier-type equation: 0 9 j 4- ( cSlijkZkkl),j - - 0 LijkzUk,l
(11)
which is transformed into an integral equation thanks to the Green tensor G O of the homogeneous medium: ~(r) = E + ~ r ~ ( r - r') 9al(r') 9~(r')dV'
(12)
(13)
#(r) = w + s r (r- r'). az(r') 9
where F s and F A are the symmetrical and antisymmetrical parts of the modified Green functions, defined from the usual Green function G o by s
Fijk' =
1
2 (Gi~
4- Gj~
and
A
1
0
0
Fijkl - --~(Gikol - Gjh,i,).
The solution of the integral Eq. 12 allows one to calculate the strain rate concentration tensor A(r) and then the effective tangent moduli of the VER by Le= l: A. Eq. 13 allows one to calculate the local total and elastic spin (v~ and w~) needed for the evolution of the lattices orientation.
4.2.2.3
SELF-CONSISTENT APPROXIMATION
The exact solution of Eq. 12 is not easy to get in the general case, and various approximations are introduced.
202
BerveiUer
The Taylor-Lin model assumes that ~ =/~ and v~ -- W so that L e = l. In order to build the self-consistent model, the elementary problem of an inhomogeneous and ellipsoidal inclusion with volume Vi and tangent moduli li within a homogeneous matrix with moduli L~ has to be solved. From Eq. 12, one gets
- E+ f
.
-
l
-
I. ~
Jv i
(14)
It is well known from Eshelby [3] that ~ is uniform inside an ellipsoidal inclusion, so that the solution of Eq. 14 is given by ~i
E ~ Tii : (l i - L0): ~i
(15)
where the interaction tensor T ii depends only on the shape of the inclusion and the tangent moduli L~ The self-consistent approximation of Eq. 12 is obtained with a tangent moduli L~ of the reference medium that is chosen equal to the effective medium L e so that the consistency condition c5l : ~ -- 0 or L~ -- Le is satisfied.
4.2.3 HOW TO USE A SELF-CONSISTENT MODEL In the case of metallic polycrystals, the development or use of a self-consistent model requires the following: 9 the characterization of the microstructure: shape, size, and orientation of the grains (represented by ellipsoidal inclusions), orientation of the lattices (initial texture); 9 the definition of (isotropic or anisotropic) elastic moduli of the single crystal; 9 the definition of the mechanisms: number and orientation of the slip systems, associated Burgers vector, and the parameters describing the inelastic behavior: initial critical shear stress, elements of the hardening matrix. Some of these values are well established from metallographic experiments on single crystals. Other material parameters can be measured on single crystals or identified by inverse methods using experimental results on a polycrystal.
REFERENCES 1. Berveiller, M., and Zaoui, A. (1979).J. Mech. Phys. Solids 26: 325-344. 2. Berveiller, M., and Zaoui A. (1984).J. Engng. Mat. Technology 106: 295-298.
4.2 Elastoplasticity of Metallic Polycrystals by the Self-Consistent Model 3. 4. 5. 6.
203
Eshelby, J.D. (1957). Proc. R. Soc. (London) 241: 376-396. Hill, R. (1965). J. Mech. Phys. Solids, 13: 89-101. Lipinski, P., Krier, J., and Berveiller, M. (1990). Rev. Phys. Appl. 25: 361-388. Lipinski, P., Berveiller, M. Reubrez, E., and Morreale, J. (1995). Arch. Applied Mech. 65: 231311.
SECTION
4 .3
Anisotropic Elastoplastic Model Based on Texture ANNE MARIE HABRAKEN D~partement MSM, Universit~ de Liege, 1, chemin des chevreuils bfit.B52/3, 4000 Liege, Belgium
Contents 4.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Description of the model . . . . . . . . . . . . . . . . . . . 4.3.3.1 Stress-Strain Interpolation Model . . . . . 4.3.3.2 Stress Integration Scheme . . . . . . . . . . . . 4.3.3.3 Implementation of the Texture Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Identification of the Material Parameters . . . . 4.3.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204 205 206 206 208 208 209 210 212 212
4.3.1 VALIDITY This model predicts the anisotropic plastic behavior of metals on the basis of their crystallographic texture. It is dedicated to polycrystalline materials. Starting from the initial texture of the material, the model computes the texture updating due to plastic strain and takes it into account to define plastic behavior during the whole finite element simulation. At this stage, the model is available for face-centered (f.c.c.) and body-centred cubic (b.c.c.) metals but can be adapted to hexagonal compact lattices. Beneath this plastic model, a power-type isotropic hardening law and an isotropic elastic behavior are implemented.
204
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
205
4.3 Anisotropic Elastoplastic Model Based on Texture
4.3.2 FORMULATION In order to take the texture of the material into account during a finite element simulation, we use the full constraint (FC) Taylor model, which is summarized in text following. The velocity gradient L defined into a single crystal (which is assumed to be the same as the macroscopic one in an FC model) can be decomposed in a slip-induced part and in a rate of crystal lattice rotation coL" __k -- K s 9Ys + ~x.
(1)
where Ks are matrices constructed from the slip plane and the slip direction of each slip system, and 9s is the associated slip rate. Einstein summation is applied on s, representing one slip system. The velocity gradient may be split into a deviatoric strain rate and a rate of rigid body rotation:
{ ~ - sym(L) - As. fs
(2)
~_ - shw(I_) - ~ ~ +~_L
(3)
---(DP
with A~ - ~ym(~) and Z~ - ~kw(K~). The resolved shear stress acting on a slip system can be derived by projecting the microscopic stress on the corresponding As matrix: zs -- As" ~
(4)
The yield locus of a single crystal is then defined by the Schmid law: -'rLs <- ~s <-
~ +~s
~
-
~Ls -<
A=s ' ~
-< ~ + s
(5)
C where ~:___s are the so-called critical resolved shear stresses. Equal signs hold for plastic deformation, while the inequalities delimit the elastic domain. Several different combinations of slip rates may achieve the prescribed strain rate according to Eq. 2. Taylor's principle states that only the one which minimizes the frictional power dissipation in the slip systems is chosen:
Pz -- ~ " I f s l - min
(6)
In order to solve the minimization defined in Eq. 6 with the constraint of Eq. 2, we use a linear programming algorithm which computes the slip rates Ys and gives the microscopic stress [5].
206
Habraken
The macroscopic stress of the polycrystal is computed by averaging the microscopic stress on each crystal. In addition to providing the crystal lattice rotation for each crystal, the Taylor model also allows for the texture updating.
4.3.3 DESCRIPTION
OF THE MODEL
The method based on the Taylor assumptions described in the previous paragraph is able to compute the elastic limit in the deviatoric stress space for a particular prescribed strain rate; this is in fact one point of the yield locus. To use it in a finite element code, we have developed an approach where the yield locus is obtained by interpolation between points computed by the Taylor model. In order to reduce the amount of computation, our model does not know the whole yield locus; only the interesting part of it is computed. So, only this small part will be updated during texture evolution. In the first stage, the interesting part of the yield locus was represented by a set of hyperplanes, each one being a plane defined in the five-dimensional deviatoric stress space. As has been shown [1], the yield locus discontinuities bred by this very simple interpolation method result in convergence problems in the finite element code. That is the reason why a second method has been developed. For that second approach, no yield locus is defined and a direct stress-strain interpolation between Taylor points is achieved. In this case, the stress and the plastic strain rate continuity conditions are fulfilled, but, since there is no yield locus formulation, a particular stress integration scheme has to be used.
4.3.3.1
STRESS-STRAIN INTERPOLATION MODEL
This model is particular in the sense that it does not use a yield locus formulation, neither for the interpolation nor in the stress integration scheme. We use a linear stress-strain interpolation, described by Eq. 7. o" --1: . C , u
(7)
In this equation, _~ is a five-dimensional vector containing the deviatoric part of the stress; the hydrostatic part is elastically computed according to Hooke's law. The five-dimensional vector u is the deviatoric plastic strain rate direction; it is a unit vector. The ~ is a scalar describing the workhardening according to the exponential relationship of Eq. 8, Where the strength
207
4.3 A n i s o t r o p i c E l a s t o p l a s t i c M o d e l Based on T e x t u r e
coefficient K, the offset F ~ and the hardening exponent n are material parameters fitted to experimental data and F is the polycrystal induced slip. The z is used as the common critical resolved shear stress for all the slip systems in the Taylor model. - K. (r ~ + r)"
(8)
The interpolation is included in the matrix C of Eq. 7 and is based on the following method. We assume five directions: _ui (i=1,..., 5), advisedly chosen in the deviatoric strain rate space and the associated deviatoric stresses -r (i=1,..., 5) lying on the yield surface according to the Taylor model. These points will define the interpolation domain and will be called stress nodes. The size of the interpolation domain is characterized by the angle between the stress nodes and is a very important parameter for the model. Additionally, we compute the contravariant vectors ssi and u u i defined by Eqs. 9 and 10. ss~o c~j -
a~j
(9)
u u i e uj -- aij
(10)
With the use of those contravariant vectors we define intrinsic coordinates in the interpolation domain for any stress vector or* by projection according to Eq. 11 and for any plastic strain rate direction u* with Eq. 12. rli - - or* 9 ssi
(11)
rli -
(12)
u* 9 uui
The most important property of our stress-strain interpolation states that if the stress if* and the plastic strain rate direction u* physically correspond to the same point, then the intrinsic coordinates/']i computed for or* (Eq. 11) or for u* (Eq. 12) are equal. The interpolation is achieved with the use of those intrinsic coordinates to compute the stress or the strain rate with a common formulation: o- - ~
r/i 9/cr
(13)
~/i" u/
(14)
i
u -- ~ i
Putting together Eqs. 12 and 13 and the workhardening coefficient, we can compute the stress associated with a plastic strain rate direction and get the expression of the interpolation matrix C: --
T, " ~
~7i 9 U l l i 9 ~_
i
~ =C
(15)
208
Habraken
As long as the interpolation is achieved in the domain delimited by the five stress nodes, all the five t]i must remain between 0 and 1. When one of the five /]i becomes negative, it means that the current stress is out of the domain and an updating of the stress nodes must take place. The first applied updating method consists in finding five new stress nodes defining a new domain that contains the current stress direction. The improved updating method makes use of the adjacent domain. Therefore, only one new stress node is computed with the Taylor model, and four of the five old stress nodes are kept for the interpolation. The main advantages of this method are that it requires only one (instead of five) Taylor model call for an updating and it improves the continuity of the resulting yield locus and the continuity of its normal. A drawback is that it can only be used when the adjacent domain contains the current stress and then gives rise to positive intrinsic coordinates. This condition is satisfied when the stress evolves smoothly, which is often the case during a finite element simulation.
4.3.3.2
STRESS INTEGRATION SCHEME
As already mentioned, the stress integration scheme implemented with the stress-strain interpolation method is completely different from the classical radial return with elastic predictor; the main ideas are summarized in Figure 4.3.1, where, obviously, no yield locus formulation is used. As has been observed during several finite element simulations, this stress integration scheme is well adapted for a local yield locus description and induces a reasonable number updating processes of the interpolation domain.
4.3.3.3
IMPLEMENTATION OF THE TEXTURE UPDATING
In this model, not only is the texture used to predict the plastic behavior of the material, but the strain history of each integration point is taken into account in order to update the texture. The main ideas of the implementation are summarized in Figure 4.3.2. It should be noticed that the constitutive law is based on the interpolation method described earlier and on the Taylor model applied on the actual set of crystallographic orientations. These orientations are represented with the help of the Euler angles ranging from 0 to 360 ~ for (o1 and from 0 to 90 ~ for (o and (o2 so as to take crystal cubic symmetry into account but not the sample symmetry. A second loop on the elements and on the integration points has been added to achieve texture updating only on converged time steps.
209
4.3 Anisotropic Elastoplastic Model Based on Texture Initial stress state: gA Prescribed total strain rate"
_
~V
Compute a 1st approximation of the plastic strain rate direction: u..U_approx I-,, Compute the final stress state: (~B according to Uapprox and the input data ~v
Use the stress-strain interpolation method to find UB associated to ( ~ B
)._
UB is the new approximation
I
I
lEnd FIGURE
4.3.1 Stress-strain interpolation method: stress integration scheme.
During a large finite element simulation, it is not reasonable to achieve a texture updating for each integration point and at the end of each time step. That is why an updating criterion must be used to reduce computation time. This is still under investigation; at this stage, an updating occurs after a predefined number of time steps. A criterion based on a maximum cumulated plastic strain will also be examined. The lattice rotation of each crystal coL is computed with the Taylor model by subtracting the slip-induced spin coP from the rigid body rotation co included in the strain history. This can be directly deduced from the Taylor rotation composition equation (Eq. 3).
4.3.4 IDENTIFICATION OF THE MATERIAL PARAMETERS This model makes use of different aspects of the material properties: the elastic properties, the hardening behavior, the crystallographic texture, and the structure of the crystal lattice. Since the elastic and the hardening behavior are supposed to be isotropic, they can be obtained from classical tensile tests.
210
Habraken
Initial Texture Data
"13
?
"o
o
Orientation Distribution Function
o tD ta ta
o
Set of Crystallographic Orientations
Loop on the Steps Loop on the Iterations Loop on the Elements
"!1 m .
Loop on the Integration Points [~
Constitutive Law
2nd Loop on the Elements and Integration Points IE~
m m
tD
3 o o
Test if texture updating is necessary N
~> Updating process of the Set of Crystallographic Orientations according the strain history
F I G U R E 4.3.2
Flowchart
of the finite e l e m e n t code.
The initial texture of the material can be measured with a x-ray method. It is characterized by the C-coefficients of the orientation distribution function, and a representative set of crystallographic orientations must be computed from the measured texture [4]. Finally, the permitted slip systems of the crystal lattice must be introduced in the Taylor model data. For b.c.c, metals, we use the 24 {1 1 0} (1 1 1) and {1 1 2} (1 1 1) slip systems; for f.c.c, metals, the 12 {1 1 1} ~1 1 0) permitted slip systems are imposed.
4.3.5 H O W TO USE THE MODEL Since this method uses a refined representation of the plastic behavior, it requires some particularities in its implementation in a finite element code.
211
4.3 Anisotropic Elastoplastic Model Based on Texture 12 11
I
....~
10
,~::::~. ........%: ......... ,,
:~i~
.........~.............~ , ........
'~;:~.... i:~ ""
. . . . . .
I l
.t............... ............
~,..,::...:,.:~_
-
.............
~";:"~"....................
:~:., .............
~
.........~...'...-..........
~........
................................... ~'..~
6 different
.............. ~ ...............
cases
~
~..'~'~",~:~ ........... ................
........
"-'-~...........
""~"~ ........I....... gradients
~i~~.,
iiii;il................................................... :::':':=/~!!~:=~,i
~ ! , ..............
100
for 2 steels)
~ ~ " M "e a 0n error " = value
......................-2<: ................. ........................... .......................
.... ,--;:..;..:...:::::::::,..,..,o..,..,~= 9 "-.~.~,~,,,S-.~'~
10
/
~ .....'-.~ ................................
1000
10000
Number of crystallographic orientations FIGURE 4.3.3 Influence of the number of crystal orientations on accuracy (the reference result is taken as the one computed with 40,000 orientations).
First, it needs a large computation time. Indeed, when a new interpolation domain must be computed or when texture updating takes place, the Taylor model is called for each crystallographic orientation of the representative set and this can occur at each integration point of the finite element mesh. Figure 4.3.3 shows that the number of those orientations should not be smaller than 2000 in order to be accurate (mean error smaller than 1%), and the number of integration points can also be very high, depending on the structure to model. It should also be noted that this model needs a lot of disk space because one must store the three Euler angles of each crystallographic orientation; furthermore, since the texture evolution can be different from one point to another, one must store those orientations for each integration point. In order to achieve the texture updating, the strain history of each integration point must also be stored. For the interpolation method, the five points (7i -- _[/i (i=1,..., 5) computed by the Taylor model must also be stored in internal variable between the updating processes of the domain. The last particular point of this model is that a second loop on the elements and integration points (see Fig. 4.3.2) is required when the time step is converged to perform texture updating. So, a subroutine devoted to texture updating has been added in the finite element code.
212
Habraken
ACKNOWLEDGEMENTS T h e r e s e a r c h t e a m of P r o f e s s o r P a u l Van H o u t t e providing us with the texture analysis modules.
is a c k n o w l e d g e d
for
REFERENCES 1. Duch~ne, L., Godinas, A., and Habraken, A. M. (1999). Metal Plastic Behaviour Linked to Texture Analysis and FEM Method, Proc. 4 th Int. Conf.: NUMISHEET'99. 2. Duch~ne, L., Godinas, A., and Habraken, A. M. (1999). Texture Effects on Steel Sheet Behaviour Under Large Strain Simulations, Proc. 12 th Int. Conf.: ICOTOM'12. 3. Munhoven, S., Habraken, A. M., Van Bael, A., and Winters, J. (1996). Anisotropic Finite Element Analysis Based on Texture, Proc. 3 ra Int. Conf.: NUMISHEET'96, pp. 112-119. 4. Toth, L. S., and Van Houtte, P. (1992). Discretization techniques for orientation distribution function. Textures and Microstructures 19: 229-244. 5. Van Houtte, P. (1988). A comprehensive mathematical formulation of an extended TaylorBishop-Hill model featuring relaxed constraints, the Renouard-Wintenberger theory and a strain rate sensitivity model. Textures and Microstructures 8, 9: 313-350. 6. Winters, J. (1996). Ph. D. Thesis, Kathoulieke Universiteit Leuven, Leuven.
SECTION
4.4
Cyclic Plasticity Model with Nonlinear Isotropic and Kinematic Hardenings: NoLIKH Model DIDIER MARQUIS Laboratoire de M~canique et Technologie, Ecole Normale Sup~rieure de Cachan, 61 avenue du Pr~.sident Wilson, 94230 Cachan, France
Contents 4.4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
4.4.2 O n e - D i m e n s i o n a l Loading . . . . . . . . . . . . . . . . . .
214
4.4.2.1 Strain Partition . . . . . . . . . . . . . . . . . . . . . .
214
4.4.2.2 Elastic d o m a i n . . . . . . . . . . . . . . . . . . . . . .
215
4.4.2.3 Nonlinear Isotropic Hardening Rule (NoIH Rule) . . . . . . . . . . . . . . . . . . . . . . . .
215
4.4.2.4 nonlinear kinematic hardening Rule (NoKH Rule) . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.5 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 T h r e e - D i m e n s i o n a l Loading . . . . . . . . . . . . . . . . .
216 217 217
4.4.3.1 Strain Partition . . . . . . . . . . . . . . . . . . . . . .
218
4.4.3.2 Elastic D o m a i n . . . . . . . . . . . . . . . . . . . . . .
218
4.4.3.3 Nonlinear Isotropic Hardening Rule (NoIH Rule) . . . . . . . . . . . . . . . . . . . . . . . .
218
4.4.3.4 Nonlinear Kinematic Hardening Rule (NoKH Rule) . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.5 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . .
219 219
4.4.4 P r o c e d u r e of Material C o n s t a n t s Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
220
4.4.4.1 Identification from the Steady State Cycle . . . . . . . . . . . . . . . . . . . . . . . . . .
220
4.4.4.2 Identification of the Material Constant b . . . . . . . . . . . . . . . . . . . . . . . . . .
221
4.4.4.3 Identification of the Material Constant 221 Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
213
214
Marquis
4.4.1 INTRODUCTION This model was developed at the end of the 1970s to take into account the elasticity and cyclic plasticity of metals. A purely elastic behavior domain is assumed. The evolutions of this domain are driven by the plastic flow. Its size is governed by a so-called nonlinear isotropic hardening rule. Its position is governed by a so-called nonlinear kinematic hardening rule. This rule was introduced in a simple form by Armstrong and Frederick in 1966 and has been used extensively by Chaboche since 1976 to build different models to describe cyclic viscoplasticity and cyclic plasticity. The tension curve predicted by this model describes the nonlinear behavior in tension and the Bauschinger effect along a reverse loading (see Figure 4.4.1).
4.4.2 ONE-DIMENSIONAL LOADING In this section, the presentation of the model is restricted to a oneDimensional loading in tension.
4 . 4 . 2 . 1 STRAIN PARTITION The total strain is split in two parts: the elastic strain related to the stress through the elastic law and the plastic strain. ~ = ~e+ ~p, ~ being the total
%
%+R
ay+R
FIGURE 4.4.1 Stress-straincurve predicted by the NoLIKHmodel, Crybeing the yield stress; X being the kinematic hardening stress ; and R being the isotropic hardening stress.
4.4 NoLIKH model
215
strain, ee being the elastic strain, and eP being the plastic strain. The elastic strain is related to the stress through the elastic law: a - Ee, a being the stress and E being the Young's modulus.
4 . 4 . 2 . 2 ELASTIC DOMAIN The elastic domain is defined through a yield function f. This function depends on the stress, the kinematic hardening stress X, and the isotropic stress R.
f -la-x
-ay-R
The elastic domain is defined by the stress domain so that the yield function is negative (Fig. 4.4.2).
4 . 4 . 2 . 3 NONLINEAR ISOTROPIC HARDENING RULE ( N o I H RULE) The driving force for the evolution of the size of the elastic domain is the plastic strain. More specifically, it has been shown that the accumulated plastic strain p is the right variable to be used to describe the cyclic plastic behavior. This variable p is defined through its rate. The nonlinear isotropic hardening rule is given in a differential (or incremental) form:/~ = ]~P[; notation: ~ = dx/dt total derivative with respect to time; and/~ -- b(Q - R)p, b being a material constant and Q being a material constant; Q is the isotropic stress limit. The evolution rate of R depends on b. The asymptotic level of R for large values of p is given by Q.
f < 0 Elastic domain
T
1
0
X
FIGURE 4.4.2 Geometric interpretation of the elastic domain.
O
Marquis
216 The integrated form of the isotropic rule is R -- Q(1 - e x p ( - b . p ) ) with the initial condition R(0) - 0 .
4.4.2.4
NONLINEAR KINEMATIC HARDENING RULE
(NoKH
RULE)
The driving force for the evolution of the position of the elastic domain is the plastic strain. The nonlinear kinematic hardening rule of the NoLIKH model is given by a differential equation: 2
-
c~p
-
~(p)Xp
where C is a material constant and V(p) is a material function depending on p. ?(P) - )'o~ - (~'~ - 7o)exp(-coP) The constant co gives the rate of evolution of the function 7. Vo~ is a material constant: Vo~ = limp__,oo V(p); co is a material constant. The study of the NoKH rule shows that the value of the kinematic stress is b o u n d e d between two opposite limits. The NoKH rule leads to a hysteretic behavior between the kinematic stress X and the plastic strain (see Fig. 4.4.3). The initial value of X before any plastic flow is 0.
X / /
/
/
i
~'/
v
~P
1
/
.........................:~Tf// ........................ .......
FIGURE 4.4.3
X-eP
9-...r
curve. The plastic strain increases up to ~1 and then decreases.
217
4.4 NoLIKH m o d e l
4 . 4 . 2 . 5 FLOW RULE The flow rule gives the evolution of the plastic strain. The driving force is the stress. The plastic strain does not change if the stress is within the elastic domain or if the stress is at the yield limit (f = 0) and the loading is reversed (unloading condition): kP = 0 iff < 0 or iff = 0 and ]a - X].6 < 0. Note that the NoIH rule and the NoKH rule lead to ~P=O=~/5=Oandl~=Oand)(=O. Then, if f < O or if f = O and ] a - X].6-< O, the behavior is purely elastic. If the stress is at the yield limit and for a loading condition, then the plastic strain rate is given by ~pC - 7(p).X.sign(a
- X) + b(Q - R)
if f -
0and ] a - X . d
_> 0
Note that the preceding flow rule is such that, after a plastic strain increment, the stress remains at the yield limit. This flow rule is deduced from the condition called the consistency condition: f - 0. Table 4.4.1 summarizes the one-dimensional model.
4.4.3 THREE-DIMENSIONAL LOADING In this section, the NoLIKH model equations are given for a general loading.
TABLE 4.4.1 S u m m a r y of the O n e - D i m e n s i o n a l NoLIKH Model. Strain partition Yield function
F, =
f=
f e nt - F,p
a-X-ay-R
Flow rule ~P - -
C - ~,(p).X.sign(a - X) + b(Q - R)
N o l H rule NoKH rule
iff=0and a-X.6-_>0 ~P = 0 if f > 0 or if f = 0 and ]a - X .6" < 0 R = Q(1 - e x p ( - b . p ) ) w i t h p = ]~P]
2 = c~p -
~(p)Xp
with y(p) = Yoo - ()'oo - ? 0 ) e x p (-cop) Initial conditions
If gP = 0 then X = 0 and p = 0
218 4.4.3.1
Marquis STRAIN PARTITION
The total strain tensor is split in two parts: the elastic strain tensor and the plastic strain tensor.
4.4.3.2
ELASTIC DOMAIN
The elastic domain is given by a von Mises-type equation. A v o n Mises equivalent effective stress is defined as
The deviatoric stress tensor is defined by
-
,,,j - 3o k, ,j
with c~ij the Kronecker symbol: bij -- I if i = j and 5 0 = 0 if i --fi j. Xij is the kinematic stress tensor. The yield function is given b y f - (or - X)e q - Cry - R, and the elastic domain is defined by f < 0.
4.4.3.3 (NoIH
NONLINEAR ISOTROPIC HARDENING RULE RULE)
The driving force is the accumulated plastic strain p. This strain is defined through its rate:
-
./-- .p .p V 3 ~ij~iJ
With this new definition of p, the nonlinear isotropic rule (NoIH rule) is unchanged./~ = b(Q - R)/~, with b, Q being material constants.
219
4.4 NoLIKH model
4.4.3.4 NONLINEARKINEMATICHARDENINGRULE (NoKH RULE) The kinematic stress is a tensor Xij. The NoKH rule writes:
4.4.3.5 F L O W RULE The driving force of this rule is the effective stress tensor. The plastic strain does not change if the stress is within the elastic domain or if the stress is at the yield limit (f = 0) and the loading is reversed (unloading). - 0 iff < 0 or iff - 0 and laij - Xijl.dij < 0 A geometric interpretation of the loading-unloading condition is given in Figure 4.4.4. If the stress is at the yield limit and for a loading condition, the plastic rate is given by
.p _ 9 (rrk~ -- Xkl).dkl Bij -7 "r'C -2~/(p).Xmn 9(O'mn Xmn)/(O'y + R) + b(Q - R) -
rr~ -- Xij + R) 2
-
i f f -- 0 and (akDt - Xkl).~kl ~ O. The preceding flow rule is deduced from the consistency condition: f = 0. Table 4.4.2 summarizes the three-dimensional model.
ading
f
xi. ~ i ~
U~adin~~
FIGURE 4.4.4 Geometric interpretation of the loading-unloading condition in the deviatoric stress plane. The scalar product given by the condition is positive on one side of the tangent line to the yield limit and negative on the other side.
220
Marquis
TABLE 4.4.2 Summary of the Three-dimensional NoLIKH Model. Strain partition f -- (a - X)e q -
~y
-
R
Yield function With and Flow rule
~=~9
( ~ - xk,). ~
,~ - x,j
3 c - xv(p), x~.. (am. - X~.)/(.y + R) + b(Q - R) '--~~+ R)~' Z
i f f = 0 and (tTkt~-- Xkz).8 Z 0
= 0 if f < 0 or if f = 0 and laij - Xijl.dij < 0 NoIH rule
R = Q(1 - exp(-b.p))with/~ = V 3 " ~
NoKH rule with y(p) = Yor - (Yoo- Vo)exp (-top) Initial conditions
If ~P --- 0 then X = 0 and p -- 0
4.4.4 PROCEDURE OF MATERIAL CONSTANT IDENTIFICATION
4.4.4.1
IDENTIFICATION FROM THE STEADY STATE
CYCLE The first step of the material constant identification uses a steady state cycle after a strain cycling. The strain amplitude has to be chosen close to the strain amplitude encountered by the material under the actual loading (see Fig. 4.4.5). Figure 4.4.5 defines a procedure to determine some material constant. F r o m the stress strain loop given in Figure 4.4.5a, the value of the Young's m o d u l u s is defined and the stress-plastic strain loop is deduced in Figure 4.4.5b. From this loop, the size of the elastic domain is deduced. As the value of p is very high at the steady state, this size is 2 ( Q + Cry). If from a standard tension test the value of Cry is deduced, then the value of Q is given. The X-ep loop is deduced by a simple translation rule. From this loop, the material constant C is given by the tangent shape to the curve at X = 0. Through an estimation of the asymptotic value of X, it is possible to deduce 700.
221
4.4 NoLIKH model
i
li ~+ . . . . . . . . . . . . . . . . . . . . .
........................F i~, ................................... i.................~ [........................................... _~ +......................::.....................
(c)
FIGURE 4.4.5 Identification steps, a. Steady state cycle in the cr-~ plane, b. Steady state cycle in the cr-~p plane, c. Steady state cycle in the X-~p plane.
From this first step, assuming that C~yis known, it is possible to get E, Q, C and 7oo-
4.4.4.2
IDENTIFICATION OF THE MATERIAL CONSTANT b
The material constant b gives the evolution of the elastic domain along the cycling. Figure 4.4.6 gives the steps for deducing the material constant b. The value of b is deduced from the slope of the tangent at the R - p curve for p = 0. At this point, the value of the slope is bQ, and then b can be deduced.
4 . 4 . 4 . 3 IDENTIFICATION OF THE MATERIAL CONSTANT ~Y0 A first estimation of the constant 70 can be given through the examination of the successive loops along the cycling of the strain. Some typical configurations are given at Figure 4.4.7.
222
Marquis
R~ o"
2(R(p)~(~y)
1
,.
Ep
p
(a)
(b)
FIGURE 4.4.6 Identification steps of b. a. Determination of R (p) through the determination of the elastic domain of different cycles (possible choices N=5, 10, 20, and 50). b. R-p curve.
/....../N=50 ~-----N=20 > N=IO
o"
L y
Ep
J (a)
I,t~)
FIGURE 4.4.7 Typical cyclic curves, a. In this case, the different loops are deduced through a translation along the rr axis. It is typical Of a cyclic isotropic hardening. In this case the kinematic hardening does not change along the cycling. Then Y0 = Yo~. b. In this case, the different hardening curves are not parallel. This is an indication of a significant contribution of the kinematic hardening to the cyclic hardening. In this case the material constant Y0 has to be taken as greater than 7oo. A first try for the value of co can be around 10.
SECTION
4.5
Multisurface Hardening Model for Monotonic and Cyclic Response of Metals Z. MRoz Institute of Fundamental Technological Research, Warsaw, Poland
Contents 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.5.2 Multisurface Hardening Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.5.2.1 Discrete Set of Loading Surfaces... 224 4.5.2.2 Continuous Description of Field of Hardening Moduli . . . . . . . . . . . . . . . . 228 4.5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.5.1
INTRODUCTION
The quantitative description of inelastic material response under a variety of loading conditions poses a challenging task in formulating constitutive models. The extensive experimental data provide an input toward more accurate model formulations aimed at the quantitative simulation of deformational response for complex loading histories. In particular, such effects as strain amplitude-dependent hardening, ratcheting for nonsymmetric cycles, nonproportional hardening, etc., should be incorporated into the model. The multisurface hardening model was originally proposed by Mr6z [1,2] for metals and was next extended to soils [3]. A set of nested surfaces is assumed to represent the field of hardening moduli, and the evolution of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
223
224
Mr6z
surfaces occurring in the course of plastic straining represents the material stiffness variation. The memory of specific loading events is then naturally incorporated into the model through the position of centers of consecutive surfaces. The subsequent developments used either two or an infinite number of surfaces, thus providing continuous variation of hardening moduli [4-7]. Extensive review of models of this kind can be found in the articles by Mr6z [8] and Ohno [9].
4.5.2 MULTISURFACE HARDENING FORMULATION 4.5.2.1
MODEL
DISCRETE SET OF LOADING SURFACES
The multisurface formulation is based on the concept of nested surfaces corresponding to different hardening moduli. The active loading surface is assumed to translate toward the consecutive external surface, so when the stress point reaches this surface, both surfaces are tangential and the contacted surface becomes the active surface. In classical theory of plasticity, the concept of the yield surface is widely used. The yield surface specifies the interior elastic domain and incremental loading-unloading conditions. For elastoplastic deformation processes we can also introduce the response surfaces representing the measures of developed plastic deformation, such as accumulated plastic strain, plastic work, or hardening moduli. Since the hardening modulus occurs explicitly in the constitutive equation, its specification is most convenient. Assume the yield condition in the form F (~ = f(a" - o~(~ - a (~ = 0
(1)
and the associated flow rule
,_lOS(OSN.,i- )
-o
F(o _
. >o
or
/~p _ -~n~rn 1
F(O) - - 0
~rn - - 8 . n > O
(3)
where n denotes the unit normal vector to the yield surface at the stress point and Ctn = 8" n denotes the projection of the stress rate vector on n. The
225
4.5 Multisurface Hardening Model
hardening moduli K and H are interrelated; thus
K - n N~
I~Pl
(4)
(/~p. /~p) 89
so in the uniaxial case the modulus K is proportional to the tangent modulus of the stress-plastic strain curve. The back stress is denoted by ~(0) and the yield limit by a (~ Assume further that the state of hardening is specified not only by the yield condition but also by a set of nesting surfaces of constant hardening moduli K0, K1,K2, ..., K, defined by Eq. 4, where K0 denotes the hardening modulus on the yield surface and Ki a r e referred to particular surfaces F (i) - - f ( a
- or(i)) -
a (i)2 -- 0
(5)
where ~(i) denotes the back stress associated with the i - t h surface and a (i) specifies the size of the surface. The set of equations in Eq. 5, specify the internal irreversibility domains and together with the yield condition constitute a field of hardening moduli in the stress space. Once the stress state along the evolution path reaches the respective nesting surface, the hardening modulus changes discontinuously, and instead of Eq. 3, we have 1
/~P - ~ - ~ n6-., F (~
--
F (1)
=
. . . - - F (i) - - O,
~"
n (i) > 0
(6)
From proportional loading and unloading tests the hardening field can be identified. In fact, instead of plotting conventional yield surfaces defined by the assumed value of plastic strain, more meaningful information can be produced by specifying the field of hardening moduli, since this field indicates the stiffness anisotropy of the material and its variation. Consider first the case of proportional loading, for instance, tension and compression. Approximate the stress-strain curve by linear segments 0-1, 12, 2-3, 3-4, along which the tangent modulus is constant (Figure 4.5.1a). In the stress space this approximation can be visualized by introducing surfaces F (~ - - 0 , F (1) - - O , . . . , F (i) - - 0 defining the regions of constant hardening moduli in the stress space. When the stress point moving from 0 reaches the surface F (~ - 0 at 1, the yield surface starts to translate with the stress point toward a point 2, and from 2 the two surfaces F (~ - 0 and F (1) - - 0 move together until a point 3 is reached. Fig. 4.5.1b presents the field of hardening moduli after reaching the point 3. During unloading and reverse loading the surfaces translate backwards, and at 3 / the symmetric configuration with respect to 3 is obtained (Figure 4.5.1c). It is seen that when surfaces do not expand but only translate, the Masing hardening rule is obtained. For nonproportional loading it is assumed that the surfaces F (~ - 0, F (1) --0, F (2) - 0 . . . do not intersect but consecutively contact and push each
226
Mr6z
3
G
3'
(a)
1
1
(b)
(c)
FIGURE 4.5.1 Piecewise linear approximation of the stress-strain curve and the respective loading surfaces, a. Loading-unloading curve, b. Nested surfaces at the loading point 3. c. Nested surfaces at the unloading point 3'.
other. For convex, similar surfaces this is achieved by postulating that the instantaneous motion of the stress point P corresponding to F 0) = 0 is directed to the point R on F (1+1) ~ 0 with the same normal as that at P. Similarly, when F (~ = 0 and F (1) - - 0 are tangential to each other at P, the translation of P occurs along PR where R is an associated point on F (2) = 0 corresponding to the same normal. The position of the associated point R is determined from the proportionality relation for any two similar surfaces F 0) = 0 and F (l+1) - - 0; thus
0,(l+ 1) _ ~(l+ 1) __ 0"(l+ 1) R
(1)
aO) t a p
--
0~(l))
(7)
227
4.5 Muhisurface Hardening Model
(a)
(b)
FIGURE 4.5.2 Translation path K-L, b.
rule of loading surfaces, a. and the cyclic loading program along
and the vector fl, connecting P and R, is expressed as (cf. Figure 4.5.2a) PR--
~ - a 0 ) l [ (~
- O.(I))O'(R/+I) -- O.(I+I)0c(I) -- O.(I)0c(I+I)1
(8)
The relative motion of P with respect to R is assumed to occur along PR; that is, ) -
-
(9)
where fi is a scalar factor 9 Since 9(z) - dr(z) + (a (z) _ a(z))~O~ d(z) tre
(10)
and otO+1))d(a~l)l)
o.(1+1) _ (x(l+l) 1) R + (a~ +
(11)
then from Eq. (9) we obtain ~.(l+1)
~(z) _ ~(z+]) + ~p +
_
9
o.(l) o.(0(a~)- ~(z))
(12)
and the scalar fi can be determined from the consistency condition F 0 ) - 1 ~(Z)-0. When all surfaces F ( ~ F ( 1 ) - 0 , . . . F 0 - 1 ) - 0 are in contact with the surface F (l) - 0 and move with the stress point P, their
228
Mr6z
translation is governed by the motion of P and we have
a(pk) _ or(k) O'(z) p
_ 0~(z)
a(k) --
a(z),
k-
0, 1, 2 , . . . 1
(13)
Figure 2b presents the case when the stress component al oscillates between the values at K and L with fixed component a2. Now, the yield and subsequent nesting surfaces will translate with the stress point between K and L, so the steady cyclic state corresponds to progressing accumulation of the plastic strain component ~2 after each stress cycle.
4 . 5 . 2 . 2 CONTINUOUS DESCRIPTION OF FIELD OF HARDENING MODULI Let us now discuss a limit case of the muhisurface hardening rule by assuming that there is an infinite number of nesting surfaces specifying the field of hardening moduli. Referring to Figure 3, assume that in the loading process OA the stress point has moved into the elastic-plastic region and reached the exterior surface Fm ~-O. At A all surfaces F (k) = 0, k = 0, 1,... m - 1 are tangential to the surface Fm = 0 . If the loading process continues, the subsequent incremental response loading surface will be specified by the active surface Fm = 0 and the hardening modulus ascribed to this surface. Therefore all surfaces F (h) = 0, k -- 0, 1, . . . m - 1 will be passive and do not
0"2
2 Fm=0
EI.r
ol
ok/
pv
2'
(a)
(b)
FIGURE 4.5.3 Continuous description of field of hardening moduli, a. In the stress space, b. Loading-unloading curve.
229
4.5 Multisurface Hardening Model
enter the material description. Consider now the unloading process A - B . A t B the surface FI1 tangential to Fm = 0 at A is the active loading surface. Let us note that, knowing the position B, the center and the diameter of the active loading surface Fll = 0 are specified. During the subsequent loading event, such as A - B , for which the stress path is directed into the interior of the domain enclosed by Fm ~ 0, there is no need to trace the positions of all surfaces; only the active loading surface passing through B should be determined. When the loading path at B changes its orientation and follows B-C-D, a new loading event commences for which the active loading surface F12 = 0 is tangential at B to Ell -- 0 with its center on OIB. Thus at D, after the loading history O-A-B-D, only three surfaces characterize the previous history and are stored in the material memory: the maximal loading surface F m - - 0 , the first loading surface Fll = 0, and the second loading surface F12 -- 0. When the stress point on B - C - D reaches D, the surface Fl2 = 0 coincides with the surface Fll = 0 and the second loading event is erased from the material memory. Similarly, when the stress point reaches E, the loading path A-B-E is erased from the material memory and the state of material is represented by the maximal loading surface Fm -- O. Considering the first loading event and assuming the equations of F~ = 0 and Fll = 0 in the form Fm = f (a - oc(m)) - a (m)2 = 0
(14)
and FII --f(cl
-
o[ (I)) -
0 "(I)2 - -
0
for the loading path A-B, the following relations occur: a(z) aA _~(1) _ (a.A_ 0c(m))a ~
(15) (16)
from which 0cd) and a d) are determined once the position of the stress point B on Fll = 0 is specified. Similarly, for the second loading event, the active loading surface F12 = 0 is specified from similar relations. Assume that the hardening modulus on the active loading surface FI = 0 depends on the ratio of semidiameters of maximal loading and active surfaces with respect to the diameter of the yield surface; thus K -- K(r/),
a(z) -_ a(~ r/-----a(m) a(o)
(17)
It is seen that 1/= 0 when a d) = a (~ and 1/= 1 when ad) = a(m). The variation of K can be described by the relation K--Km+A
(18)
230
Mr6z
3
~1
0
F,=O
(a)
(b)
FIGURE 4.5.4 Two-surface description of field of hardening moduli, a. Configuration of surfaces, b. Stress-strain curve.
where A and p are the material parameters and Km denotes the value of K on the maximal loading surface. Consider now the other particular case when there are only two surfaces F 0 - 0 and Fz = 0 and the hardening modulus varies with the distance c~ between the surfaces F0 = 0 and F z - 0. Assume that this distance is measured by the length P R (Figure 4.5.4). The maximal value of ~ is ~m = a (z) - a (~ for the first loading and (~m - - 2(0"(l) -- o'(~ for subsequent unloading from the surface Fz -- 0, whereas the minimal value equals (~min - - 0. Assume that the hardening modulus K takes an infinite or very large value for c~- ~m and K - Kz for f i - 0 that is at the contact between two surfaces. The variation of K can be , for instance, expressed by the relation K--Kz+a
~m_~
(19)
where a and m are material parameters and ~m is the maximal value of c~ from the previous deformation history. Fig. 4 shows the loading, unloading, and reloading curves generated by this description. When the stress point reaches the surface Fz = 0, the subsequent response is described by the hardening modulus Kz. When Kz is constant, a linear hardening curve is obtained; when Kz = 0, the surface F z - 0 is a limit surface. The nonlinear variation of the hardening moduli for the two-surface model was discussed by Dafalias and Popow [4] and independently by Krieg [7].
4.5 Multisurface Hardening Model
231
4.5.3 C O N C L U D I N G REMARKS The present brief exposition provides the foundation for more refined formulations of multisurface constitutive models aimed at describing the cyclic response of metals, accounting for such effects as transient and steady state cyclic response, nonproportional hardening, memory of maximal prestress, ratcheting, etc. [8]. The fundamental model parameters are generated from uniaxial cyclic loading tests. However, additional parameters are to be introduced when transient states and nonproportional hardening effects are quantitatively simulated, cf. [10].
REFERENCES 1. Mr6z, Z. (1967). On the description of anisotropic workhardening. J. Mech. Phys. Solids 15: 163-175. 2. Mr6z, Z. (1969). An attempt to describe the behaviour of metals under cyclic loads using a more general work-hardening model. Acta Mechanica 7: 199-212. 3. Mrdz, Z., Norris, V. A., and Zienkiewicz, O. C. (1978). An anisotropic hardening model for soils and its application to cyclic loading. Int. J. Num. Arch. Meth. Geomech. 2: 202-221. 4. Dafalias, Y., and Popov, E. P. (1976). Plastic internal variable formalism of cyclic plasticity. J. Appl. Mech. 98: 645-651. 5. Mrdz, Z. (1981). On generalized kinematic hardening rule with memory of maximal prestress. J. Mech. Appl. 5: 241-259. 6. Trampczynski, W., and Mrdz, Z. (1992). Anisotropic hardening model and its application to cyclic loading. Int. J. Plasticity 8: 925-946. 7. Krieg, R. D. (1975). A practical two-surface plasticity theory. J. Appl. Mech. 42: 641-646. 8. Mrdz, Z. (1986). Phenomenological constitutive models for metals, in Modeling of Small Deformation of Metals, Zarka, J., and Gittus, J., eds., Elsevier Science Publishing. 9. Ohno, N. (1990). Recent topics in constitutive modeling of cyclic plasticity and viscoplasticity. ASME Appl. Mech. Rev. 43: 283-295. 10. Mrdz, Z., and Rodzik, P. (1996). On multisurface and integral description of anisotropic hardening evolution of metals. Eur. J. Mech., A/Solids 15: 1-28.
SECTION
4.6
Kinematic Hardening Rule with Critical State of Dynamic Recovery NOBUTADA OHNO Department of Mechanical Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
Contents 4.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Description of the Rule . . . . . . . . . . . . . . . . . . . . 4.6.4 Identification of the Material Parameters .... 4.6.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 4.6.5.1 Installation in Constitutive Models... 4.6.5.2 Implementation in FEM . . . . . . . . . . . . . 4.6.6 Table of Material Parameters . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
232 233 234 236 237 237 238 238 239
4.6.1 VALIDITY This is a kinematic hardening rule that enables us to represent multilinear, as well as nonlinear, stress-strain relations under cyclic loading. This rule, which is formulated by introducing a critical state in dynamic recovery, is capable of simulating well ratcheting and cyclic stress relaxation. Moreover, the rule has the advantage of allowing us to determine very easily the material parameters. A rate-independent constitutive model of plasticity based on this kinematic hardening rule is implemented in a commercially available FEM software by utilizing the implicit integration equation and consistent tangent modulus derived for the model, though the rule itself is valid for both rate-independent and rate-dependent materials.
232
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All fights of reproduction in any form reserved.
4.6 Kinetic Hardening Rule with Critical State of Dynamic Recovery 4.6.2
233
FORMULATION
We consider strain hardening and dynamic recovery for formulating the evolution equation of back stress ~ [1], and we suppose that ~ consists of several parts, i.e., ~ - ~ ~Zi [ 2 ] . Let ai be the deviatoric part of ~i. Let us assume that the dynamic recovery of ai has a critical state j~ - 0, in which the dynamic recovery of ai is activated so fully that ai cannot develop beyond the surface fi = 0 [3]. This surface, which is similar to the yield surface of perfectly plastic materials, is defined to be a hypersphere of radius ri in the space of ai as 3
fi - -~ ai " ai - r2i - O.
(1)
Then, we may consider an evolution equation of ai which has two kinds of dynamic recovery terms, the term becoming active in the critical state and the term operating irrespective of this state. By using of Heaviside's step function H and Macauley's bracket ( ), such an evolution equation of ai can be expressed as [4]
where (') indicates the differentiation with respect to time, ~i and #i are material parameters, p denotes accumulating plastic strain rate
and )~i is determined to have the following form using the conditions fi = 0 and f i - O:
2 i - ~P" ai_ #iP ri
(4)
The first term in the right-hand side in Eq. 2 expresses strain hardening, whereas the second and third terms deal with the dynamic recovery of ai. The second term H ( f i ) { J , i ) a i allows the dynamic recovery of ai to be activated so fully in the critical state j] - 0 that Eq. 2 can satisfy f i - 0 when j~ - 0 and 2i > 0. The third term #iaip, on the other hand, operates simply at all times in proportion to ai and p. Equation 2 has two special cases, #i = 0 and #i = 1. If #i = 0, Eq. 2 is reduced to the first version of the Ohno and Wang rule [3],
234
Ohno
~T / ," Z s
iLLi - ' 0
::,
O < ~ i <1 fli = 1
0 FIGURE 4.6.1
1/~i
~P
Evolution of ~i under uniaxial tensile loading.
On the other hand, if /2i = 1, ai never reaches the critical state j~ = 0 (Fig. 4.6.1), so that Eq. 2 becomes the Armstrong and Frederick rule [1], t i i = (i(3ri~,P-aip)
(6)
If ri is not constant, Eq. 2 can be extended to the following form, which satisfies fi - 0 when Ji - 0 and ,~i > 0 [4]" hi = (i
4.6.3
DESCRIPTION
rii~p - H(j~)()~i)ai-/~iaip
+--ai Fi
(7)
OF THE RULE
To describe some fundamental features of Eq. 2, we now consider uniaxial loading. A uniaxial form of Eq. 2 is
where j] - cti2 - r/2 and J~i- i~Po~i/ri - ~,I~Pl. For monotonic tensile loading, Eq. 8 is reduced to
~i = ~ (i(ri - lai~i)e, p,
I
O~
0 <_ ~i < ri,
(9)
O~i -~ ri ~
which has the solution
oq=ri[1-(1-
1 - exp(-lai(ie, P) \ lai
/
(10)
4.6 Kinetic Hardening Rule with Critical State of Dynamic Recovery
235
This tensile change of 0~i, which has a corner at o~i ---Fi, is illustrated in Figure 4.6.1. Two special cases of/-/i = 0 and /-/i--- 1 are dealt with in the figure as well. When/-/i = 0 , Eq. 9 represents the bilinear change of ~i, such as oq = r i [ 1 -
(11)
(1-~i~3P}]
On the other hand, when #i = 1, Eq. 9 gives 0q = ri[1 - exp(-~ieP)]
(12)
In this case, therefore, ~ reaches the critical state only asymptotically; in other words, the critical state does not become active at all. Figure 4.6.2 shows the monotonic tensile change of back stress 0~obtained by superposing ~1 to ~M. As illustrated in the figure, if 0 < #i < 1, the ~ versus eP relation has corners, since each ~ versus eP relation has a corner under tensile loading. Especially if #i = 0, the ~ versus eP relation is multilinear, as shown by the dotted line in Figure 4.6.2. Thus [i and ri are related with the coordinates of corners of the multilinear ~ versus eP relation as follows [5]: 1 ~(i) 0~(i) -- 0~(i- 1)
(13)
eli) 0~(i+1) -- 0~(i)
(i-l)
(i+1)
where ~(o) - 0 and 0qo) - 0. Figure 4.6.3 illustrates the effect of #~ on the shape of ~ versus ~P hysteresis loops as well as on ratcheting under cyclic loading. Let us remember that Eq. 2 has two kinds of dynamic recovery terms, i.e., the term becoming active
a = ~.~a i
a(3)
.....................
0~(2)
....: ..~...:;"
a(l)
___o:
~
"
,
, , I
.............. /4=0 O < P i <1
~176176176176 I
~
FIGURE 4.6.2
~..........
~ I.....
~3
Evolution of ~ and its parts under uniaxial tensile loading.
236
Ohno
O[max
amin (a) ~,li -- 0
~[~mflx
v
O[min (b) 0 < / 1 i < 1 F I G U R E 4.6.3 Hysteresis loops of back stress 0~ and plastic strain ep u n d e r uniaxial cycling b e t w e e n 0~max a n d 0~min. a. ]-/i -- 0. b. 0 < ]-/i ~ 1.
in the critical state j~ = 0 and that operating at all times in proportion to ai and p. The former causes the 0~ versus ep hysteresis loops to be closed multilinearly under uniaxial cyclic loading, as shown in Figure 4.6.3a. On the other hand, the latter makes the 0~versus eP hysteresis loops open to induce ratcheting. Thus the smaller ]-/i is set, the less opening Eq. 2 gives to the hysteresis loops, resulting in smaller ratcheting.
4.6.4 IDENTIFICATION PARAMETERS
OF THE MATERIAL
Equations 13 and 14 help us to determine the material parameters ~i and ri. Especially if both isotropic hardening and rate dependence are negligible, they
4.6 Kinetic Hardening Rule with Critical State of Dynamic Recovery
237
O'(1)~ 0"(o)
FIGURE 4.6.4 Uniaxial tensile stress-strain curve approximated multilinearly.
can be determined immediately by multilinearly approximating a monotonic tensile curve (Fig. 4.6.4) and using Eqs. 13 and 14 with g(i) replaced by cr(i), i.e., Eq. 13 and
where or(0) indicates initial yield stress. Even if rate dependence must be taken into account under negligible isotropic hardening, ~i and ri can be determined easily as follows: By assuming an appropriate viscoplastic equation kP = G ( a - ~), the ~ versus ~P relation is estimated from monotonic tensile experiments done at several strain rates, and then the ~ versus ~P relation identified is multilinearly approximated to use Eqs. 13 and 14, by which (i and ri are determined. If isotropic hardening is not negligible, the material parameter ri may change with plastic deformation. In this case, it is necessary to assume an appropriate function for each ri to simulate experiments. The material parameters #i (i = 1, 2 , . . . , M) are determined so as to simulate well uniaxial and/or multiaxial ratcheting experiments, though a constant can be assigned to all ~i. If it is not necessary to consider ratcheting and cyclic stress relaxation, we can take either #i = 0 or ~i = 1.
4.6.5 HOW TO USE THE MODEL 4.6.5.1
INSTALLATION IN CONSTITUTIVE MODELS
The present kinematic hardening model can be used as a translation rule of yield surfaces and viscoplastic potentials. For rate-independent materials, we
238
Ohno
may employ the following yield surface, which translates and expands: 3 (s - a ) ' ( s
(16)
- a) - y2 _ 0
where s denotes the deviatoric part of stress ~r, and Y indicates the size of the yield surface. Then, on the basis of the normality rule, the plastic strain rate can be expressed as /:p where n be E' --
--
3(~'n> n 2
(17)
EP
~ / - 3 / 2 ( s - a ) / Y , and EP is the plastic tangent modulus derived to
+ ~ i= 1
/
~,r, 1 -
tt, + H(~) Fi
#, ri
/1/
(18)
For rate-dependent materials, on the other hand, we may assume 2 where cr ef f - [ ( 3 / 2 ) ( s - a ) ' ( s indicates drag stress.
4.6.5.2
D
a)] 1/2, G is a viscoplastic function, and D
IMPLEMENTATIONIN FEM
For the rate-independent constitutive model mentioned in Section 4.6.5.1, an implicit integration equation and a consistent tangent modulus are available, so that the constitutive model is implemented using a user subroutine UMAT in ABAQUS [6]. The integration equation, which is based on return mapping, allows us to take large strain increments without sacrificing accuracy, especially if ]Ai is equal or close to zero. The consistent tangent modulus, which is derived for a general form of strain hardening and dynamic recovery of back stress, affords the parabolic convergence in solving the nonlinear equilibrium equation in FEM. These provide high efficiency and stability in computation in FEM, not only for isothermal but also for non isothermal problems.
4.6.6 TABLE OF MATERIAL PARAMETERS Table 4.6.1 shows the values of ~i obtained by applying the present kinematic hardening rule to ratcheting experiments of some materials. As seen from the table, ~ i tends to take small values close to zero. This means that ratcheting
4.6 Kinetic Hardening Rule with Critical State of Dynamic Recovery
239
TABLE 4.6.1 Values of Material Parameter ~i. Material 316 stainless steel Modified 9Cr-lMo steel IN738LC 1018 and 1026 carbon steels
Temperature R.T. 550~ 850~ R.T.
~i
0.02 0.03 0.12 0.35
and cyclic stress relaxation usually occur but much more slowly than the predictions based on the Armstrong and Frederick rule. For the material parameters ~i and ri, no example is given here, since they can be determined very easily from experiments (Section 4.6.4).
REFERENCES 1. Armstrong, P. J., and Frederick, C. O. (1966). A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N731, Berkeley Nuclear Laboratories, Berkeley, UK. 2. Chaboche, J. L., and Rousselier, G. (1983). On the plastic and viscoplastic constitutive equations. ASME J. Pressure Vessel Technol. 105: 153-164. 3. Ohno, N., and Wang, J.-D. (1993). Kinematic hardening rules with critical state of dynamic recovery. Int. J. Plasticity 9: 375-403. 4. Ohno, N., and Abdel-Karim, M. (2000). Uniaxial ratchetting of 316FR steel at room temperature. Part II: constitutive modeling and simulation. ASME J. Eng. Mater. Technol. 122: 35-41. 5. Jiang, Y., and Kurath, P. (1996). Characteristics of the Armstrong-Frederick type plasticity models. Int. J. Plasticity 12: 387-415. 6. Kobayashi, M., and Ohno, N. (1999). A return mapping algorithm for the strain hardening/ dynamic recovery type of kinematic hardening models in Trans. 15th Int. Conf. on Struct. Mech. in Reactor Technol., Vol. II, pp. 165-176, Korean Nuclear Society.
4.7
SECTION
Kinematic Hardening Rule for Biaxial Ratcheting HIROMASA ISHIKAWA, KATSUHIKO SASAKI Hokkaido University, N13, W8, Kita-ku, Sapporo, 060-8628 Japan
Contents 4.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 4.7.4 Identification of the Material Parameter . . . . 4.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 4.7.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
240 240 242 243 244 245 246
4.7.1 VALIDITY A kinematic hardening rule introduced here is valid for biaxial ratcheting deformation of metals. The hardening rule employs memory effects of back stress, and it controls the biaxial ratcheting strain.
4.7.2 FORMULATION Combining the von Mises stress hardening form with the kinematic hardening, the following yield function is chosen for cyclically stable material at temperature T:
f - ~1 c~j~ (~j - ~ j ) ( o ~ - ~ )
- ~1 R2 (~ , T)
0
(1)
-
where Cokl is the plastic-deformation-induced anisotropic coefficient tensor of fourth rank, aij and a/j are stress and the center of the yield surface, or back stress, R is the flow stress, and ~c is the hardening or softening 240
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
241
4.7 Kinematic Hardening Rule for Biaxial Ratcheting
parameter. Associated with Eq. 1, the modified Levy-Mises equation of cyclic plasticity is obtained from the normality of plastic strain increment to yield surface, a4 -
3d -gp 2 e C,jh~(okz - ~k,)
(2)
where d~ and d ~P are the plastic strain increment and the equivalent plastic increment, respectively. The Ziegler type of assumption, which is used as the evolution equation of the back stress, is modified by employing the memorization of the back stress. The modified Ziegler type of assumption is expressed by d~ij - (aij - ~,j)d/z +
n,jg(p~)
(3)
where nijg(pc) denotes the memorization of the back stress induced during preloading, and g(pc) is the function of the accumulated plastic strain Pc until the previous inversion of loading, g(pc) is assumed to be expressed by g(pc) - ~0 { 1 - ~exp ( - c ~ ) )
(4)
and ~0 in Eq. 4 is assumed to be expressed by ~0 - ~ / 2 ~hz~z
(5)
vJ In Eqs. 4 and 5, ~, r and ( are material constants, and nij in Eq. 3 denotes the unit normal vector to the yield surface expressed by C~jkl(~kz -- ~kz)
~'J - / 3
CmnopCmnqr(O'op -- (Xop)(O'qr - O~qr)
(6)
Using the consistency condition df -- 0 from Eq. 1, the evolution of the back stress increment is represented by 2R2 d~,j ----g--~ C~m~(am~ -- ~mn)(da~ - n~g(p~)) (a,j - ~,j)} dR
(7)
where dWP is the modified plastic work increment. The hardening and softening parameter is reasonably assumed to be a scalar proportional to the modified plastic work.
a ~ - a w , - Ra~ - (~,j -~,j)a 4
(8)
To represent the roundness of the stress-strain curve and to drive the nonlinear plastic modulus simply, the following power law is
242
lshikawa and Sasaki
employed: ~p --
(v)
O"
~
m(n)
K~
(9)
where E is Young's modulus and K is a material constant. ~ in Eq. 9 is the equivalent stress measured from the current center of the yield surface and expressed by ~ = ~/(2/3)Cijh, (aij - aij)(ak,- ak,). The quantities of ao(,,) and m(,,) in Eq. 9 are th4 reference stress at the proof strain ~0 - 500 I~m/m and the exponent of hardening in the uniaxial stress state. They are expressed by
Cro(,,)- Cro(oo){ 1 - ~ exp ( - c ~ ) }
(10)
mOO : m ( o o ) { 1 - f l e x p ( - P ~ ) }
(11)
and
In addition, R can be assumed to be expressed by R(,) = R(oo) 1 - 2exp
-77z
(12)
In Eqs. 11 and 12, Pc is the accumulated plastic strain until the previous inversion of loading; Pa in Eq. 12 is the accumulated plastic strain at the current stage. In Eqs. 10-12, ~ro(oo), ~, Co, m(oo), fl, Cl, R(oo), 2, and c2 are material constants prescribed by the history of loading or effected by a strain path memory. These material constants are determined from some basic experiments.
4.7.3 DESCRIPTION OF THE MODEL For combined cyclic tension-compression and shear straining, the following relation can be reduced.
d~t
I
B~ k
~
~m(n, 3 (Ctcrdcr+ 2Ctss&r + 2Cts~ds + 4Cssds)(Cto + 2Ctss)
2R2 .
_
+ 2c~{&
- n~g(p~)} + 4 G s { & - nsg(p~)}]~
243
4.7 Kinematic H a r d e n i n g Rule for Biaxial Ratcheting
_R(~176 R c2 x (Godo" + 2R 2 .
2Ct~sd~ +
F }m~") cro(oo) 2Ct~ods + 4Cssds)cr + ntg(pc)
_
do~ - - 5 - - [ C t o { d o t - ntg(p~)} + 2Ct~s{dot - ntg(pc)} + 2 C t ~ { & - n~g(p~)} + 4 G s { & - n~g(w)}]s
-R(~176 ~ }m~"' R c2 cro(oo) • (Ctcrdcr+ 2Ct~sdcr+ 2Ct~crds+ 4C~sds)s + nsg(pc)
(13)
where B = (3e,o/2cro(,,))(m(,,)+ 1), cr = crt- at a n d s = r - 0q with kinematic back stresses at and ~xs. de,Pt, d7t', do~t, and do~s are the axial plastic strain increment, the shear plastic strain increment, the axial back stress increment, and the shear back stress increment, respectively. Ct, Cts, and Cs are anisotropic coefficients.
4.7.4 IDENTIFICATION OF THE MATERIAL PARAMETERS To obtain the material parameters, a biaxial ratcheting test is required. The anisotropy coefficients are determined by the following equations referring to a yield surface during the biaxial ratcheting, which is shown in Fig. 4.7.1. ct
+
,
1( 3r2 2 '~ C s - - ~ ~cos20 + ~ s i n O|
-
- 7/
inOco
O,
(14)
/
where cry and l:y are axial and shear yield stresses determined as the stress at 50/.tm/m proof strain. O is the angle of rotation of the subsequent yield surface, and the value of the angle is 0 ~ The yield stresses can be determined from the yield surface shown in Figure 4.7.1 which shows, for example, the yield surface in the case of the biaxial ratcheting due to cyclic shear straining with a strain amplitude of AT/x/3 = 0.5% superimposed on an axial stress of 100 MPa. The deformation of the yield surface during the biaxial ratcheting can be assumed to be
244
lshikawa and Sasaki
.-""
/" 9 .///
-150
I
-~
O
I
"" ,~ 9
- -
"~""" /
150
"" ~"
I
I
100
\
",
~
-50
0
I
100
a,, .
.
150
I
50
"\
\
MPa
~,/
.so / j,," .....+I
~.. ....
-~..~. 9
_
.~.
-
-150 .~.
; From proportional
.../"
. . . . .
.
~/d
I /
/ i. /
; I n i t i a l y i e l d surface
loading
FIGURE4.7.1 Yield'surface during biaxial ratcheting. such that: 1. the shear yield stress decreases with an increase in the number of cycles, while the axial yield stress remains the initial yield stress; 2. the yield surface does not rotate; 3. the shear yield stress during the biaxial ratcheting can be calculated by the following equations:
ay = 165 MPa, x/~Zy -- 105 1 + 0.57exp (15) where A~ - I s ~ + A?---~23 and where et, is the biaxial ratcheting strain at the previous cycle during the biaxial ratcheting. Figure 4.7.2 shows the axial strain (biaxial ratcheting strain) due to cyclic shear straining with a strain amplitude of A~/v/3 - 0.5% superimposed on an axial stress of 100 MPa. The simulation agrees well with the experiment.
4.7.5 HOW TO USE THE MODEL Biaxial ratcheting deformation is ordinarily observed in real-life structures. The model can be incorporated into a general-purpose code such as FEM for structure analysis.
245
4.7 Kinematic Hardening Rule for Biaxial Ratcheting
......__,.--.J
oJ
1
"2 <
~
m
0.5
I
- 0.5
-
0.25 Shear
0
0.25
0.5
q,/d-3-, %
strain
(a) experiment
1.5
=,.___......._
2 < 0.5
- 0.5
- 0.25 Shear
0 strain
025 "l(/,1-3-3, %
0.5
(b) simulation FIGURE 4.7.2
Biaxial ratcheting strain vs. shear strain.
4.7.6 TABLE OF PARAMETERS The material parameters were identified for Type 304 stainless steel (see Table 4.7.1).
246
Ishikawa and Sasaki
TABLE 4.7.1
Material Parameters for Type 304 Stainless Steel, Temperature 20~
0"0(1) 0"0(2) O'0(oo) 0t (MPa) (MPa) (MPa) 206
142
147
CO R(oo) ~ (MPa)
0.07 0.1 245
C2 m(1) m(2) m(oo) 13
0.07 0.1 9
3.5
3.5
C1
~
~
C3
0.07 0 . 0 8 - 2 e -5 -50 1
REFERENCES 1. Ishikawa, H., Sasaki, K., and Nakagawa, T. (1994). JSME Int. J. Series A 37(4): 347-354. 2. Ishikawa, H., and Sasaki, K. (1998). Int. J. Plasticity 14(7): 627-646.
SECTION
4.8
Plasticity in Large Deformations YANNIS E DAFALIAS Department of Mechanics, National Technical University of Athens 15773, Greece, and Civil and Environmental Engineering, University of California, Davis, CA, 95616
Contents 4.8.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1.2 Constitutive Relations . . . . . . . . . . . . . . 4.8.2 Numerical Implementation . . . . . . . . . . . . . . . . . 4.8.30rthotropic, Kinematic, and Isotropic Hardening Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3.1 Analytical Description . . . . . . . . . . . . . 4.8.3.2 Model Constants . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
247 247 248 249 250 250 251 253
4.8.1 BASIC THEORY The following constitutive framework applies to large elastic and plastic deformations of materials which acquire initial and/or evolving anisotropic properties of different kinds. The internal material state is described macroscopically by scalar and tensorial internal variables ki and Ai, respectively. The particular model presented at the end addresses mainly orthotropic symmetries, which are common to sheet metals.
4 . 8 . 1 . 1 KINEMATICS The multiplicative decomposition of the deformation gradient F into elastic and plastic parts is expressed by Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
247
248
Dahlias
F=FeF p
(1)
The velocity gradient at the intermediate plastically deformed configuration is given by ~pFp- 1 _ (~pFp- 1)s + (~:pFp- 1)a -- D~ + (FPFp-1 )a (2) where subscripts s and a mean symmetric and antisymmetric, respectively. The DPo is the plastic rate of deformation. The plastic material spin (FPFp-1)a can be decomposed in two parts as (~:vFp-1)a -- s176-+- Wp (3) where (2) i and W/p are the pair of constitutive and plastic spins, respectively, associated with a tensorial internal variable Ai. The s i is the spin with respect to which the constitutive corotational rate of Ai will be expressed in the sequel. In general, there are as many pairs of constitutive and plastic spins as there are Ails. The intermediate configuration is determined within an arbitrary rigid body rotation without loss of generality. A particularly useful choice employed in the sequel is that of the spinless configuration [1], which is defined by (see Eq. 3) (FPFp-1)a = 0 =~ o i - - -W/p (4)
4.8.1.2
C ONSTITUTIVE RELATIONS
4.8.1.2.1 Elastic If ~ / - r e, Ai, ki) is the elastic strain energy per unit mass, which is function of the elastic Green strain tensor E e - (1/2)(FeTF e - I), ki and Ai, hyperelasticity is obtained by II-
[FeIFr
e-T -- Po - ~
(5)
where ~ is the Cauchy stress at the current configuration, rl the second PiolaKirchhoff stress, and Po the density at the spinless intermediate configuration. 4.8.1.2.2 Plastic A state H, ki, and Ai is plastic if it satisfies the analytical expression
f (H, Ai, ki) = 0
(6)
of a static (rate-independent) or dynamic (rate-dependent) yield surface. In
249
4.8 Plasticity in Large Deformations
the case of rate dependence, the dynamic yield surface expression of Eq. 6 depends also on a scalar-valued measure of the rate of deformation. The following rate equations are now postulated at the spinless configuration (recall Eqs. 2 and 4) DPo -- [:PF p-1 --
,INPo(H, A,,
k~)
xaf(n, A,, k,) A,-
2 ( A , - ~ A , + Ai~2f)
(7)
(8)
(no sum on i)
/~i = 2/7ei(11, Ai, ki)
(9) (10)
where Eq. 9 follows from the constitutive corotational rate Ai c -- A i - oiAi + Aioi = 2Ai(II, Ai, ki) (no sum on 0, the choice of the spinless configuration (toi = - W i p) and Eq. 8. The 2 is the plastic multiplier (loading index, for plasticity, and overstress function, for viscoplasticity). Invariance requirements under superposed rigid body rotation render all constitutive functions ~, f, NPo, ~/P, k/, and 2~ki isotropic in their tensorial arguments 11 and A i .
4.8.2 NUMERICAL IMPLEMENTATION By a straightforward adaptation of an incremental solution scheme developed by Aravas [2] for an isoclinic configuration to the present choice of the spinless configuration, Dafalias [1] proposed the following. At a material point the solution 11(n) AIn), k (n), FP, and F, at time t, is known. For a given F.+I at time t . + l - t. + At, one must determine II (n+l) , A i(n+l) , k(n+l) , and F{.+I ). With the "directions" NPo and ~/P of DPo and W/p, respectively, assumed constant over the time step At, and with (NPo). -- X. for simplicity of notation, the following algorithm can be stated based only on the five Eqs. (5,6,7,9,10): Fe+l -- Fn+IF p-1 I--a/~Xn + FP+l Ee+l
(~52) Xn
r Fn+lFn+ 1
e~ e 1 --(1/2)(Fn+lFn+
ir,+l
_
(11a) (11b)
I)
po(O~///OEz).+I
Ai(n+l)__ Ai(n)_~.~,~(--(n)Ai _ ~-~/p(n)Aln ) __[_Aln)~,~/p(n) )
(llc) (lld) (11e)
250
Dafalias
(llf)
f(II('+l), --iA("+l), ki('+1)) = 0
(11g)
where/~,I n), k/ -(") , and n7 9(') are functions of n(') , AI"), and kl") of the nth step, and the 62 is chosen as the primary unknown to be determined from Eq. 11g. Observe that throughout the development of theory and numerical implementation, no need for a specific corotational or convected stress rate arises. This is because the choice of a stress rate is not a fundamental constitutive ingredient in hyperelastoplasticity [ 1].
4.8.3
ORTHOTROPIC,
ISOTROPIC 4.8.3.1
KINEMATIC,
HARDENING
AND
MODEL
ANALYTICALDESCRIPTION
A specific constitutive model will be presented within the foregoing framework of theory and implementation. It will be most appropriate for the large inelastic deformation of materials which already exhibit a strong orthotropic texture because of previous manufacturing processes, such as sheet metals. The elastic response (Eq. 5), is left upon the user. A linear elastic orthotropic or even isotropic relation can be a good approximation if plastic strains are very large compared to elastic. With the notation S for the deviatoric part of If, A for a deviatoric backstress tensor, S - S - A for the effective deviatoric stress, and Ai -- Ni @ Ni ( i - 1, 2, no sum) with Ni unit vectors along two of the three orthotropic directions at the spinless configuration, Hill's quadratic yield criterion can be written in terms of invariants [3] as f-
(L + M - N)trS 2 + 2 ( N - L)tr(S2A1) + 2 ( N - M)tr(S2A2) + (4G + F + H - 2M)tr2(gA1) + (4F + G + H - 2L)tr2(SA2)
(12)
+ 2(2F + 2G - H + N - m - t)tr(gal)tr(gA2) - k 2 - 0 where F, G, H, L, M, N are the orthotropic parameters of Hill [4]. The scalarvalued internal variable k is a measure of the size of the yield surface. The tensor-valued internal variables are the A, A1, and A2. The NPo is defined by the normality rule, and according to Pereda et al. [5] takes the form (compared to [5] notice a small correction in Eq. 13d).
251
4.8 Hasticity in Large Deformations
N~ --
~-H
- (4)1
(1/3)((/)3 + 4)4))I + 4)2S +
--
q~3A1+ ~b4A2
+ (/)5(gA1 + AIS) q- ~6(SA2 -[- A2S) ~1
--
-(4/3)[(N-
L)tr(.SA1) + ( N - M)tr(SA2)];
(13a)
= 2(L + M - N)
(13b) 4)3 = 2 ( F + 4 G + H - 2 M ) t r ( S a l )
+ 2(2G + 2F - H - M - L +
N)tr(SA2) (13c)
~b4 = 2(G + 4F + H - 2L)tr(SA2) + 2(2G + 2F - H - M - L + N)tr(SA1) (134) 4)5=2(N-L);
4)6 = 2 ( N - M )
(13e)
For the evolution of the back-stress A the Armstrong-Frederick evanescent memory rule will be used, for which the constitutive corotational rate reads A c - hDPo -c[(2/3)(trDeo2)]l/2A- 2~t, in terms of two constants h and c. Based on Eq. 7, it follows that -- hNP~ - c[(2/3)(trNpo2)]t/2A
(14)
The Ai (i = 1, 2) are purely orientational internal variables; therefore, it follows that A c = 0, which yields A1 = ~t2 = 0
(15)
If the anisotropic parameters F - N vary independently, one must be able to define their evolution equations from experiments. If for simplicity one assumes that they maintain their initial value, it remains only to determine the variation of k, that is, the isotropic hardening part. It can be described by a scalar-valued evanescent memory rule such.as k _ ( H - C k ) [ ( 2 / 3 ) t r ( D P o 2 ) ] l / 2 , in terms of two constants H and C. From k = 2k and Eq. 7 it follows that -
( H - Ck)[(2/3)tr(NPoo2)] 1/2
(16)
It remains to specify the plastic spins W~ - 2 f ~ and We - 2f~P for the back-stress A and orientational tensors A1 and A2, respectively. Dafalias [6,7] and Loret [8] proposed the expression W~ = r/A(ADPo -DPoA ), which in conjunction with Eqs. 7 and 8 yields ~"~P
--
/']A(ANPo -- NPoA)
(17)
where NPo is obtained from Eq. 13. Combining the conclusions reached in the works of Dafalias [9-11], Zbib and Aifantis [12], Kuroda [13], and Levitas [14], one can propose to render Wp a function of the non coaxiality between
252
Dafalias
g -- S - A and DPo, i.e., WP = t/(gDPo - DPog). From Eqs. 7 and 13, it follows after some algebra that [ 11] ~"~P --M(SNPo - NoPS) - t/[(Z(F 4- 4G + H - 2M)tr(SA1)
+ 2(2F + 2G - H + N - L - M)tr(gAz))(SA1 - Alg) + (2(6 + 4F + H - 2L)tr(gA2) + 2(2F + 2G - H + N - C - M)tr(gA1)) (SA2 - A2S)
(18)
4- 2 ( N - L)(S2A1 - AIS 2) 4- 2 ( N - M)(S2A2 - A2S2)] If S is co axial with theorthotropicdirections Ni ( i - 1, 2,3), one has the spectral representation S - SIAl 4- $2A2 4- $3 ( I - (A1 4- A2)), where Si (1 = 1,2,3) are the principal values of S. Since A1A2 = A2A1 = 0 it follows from Eq. 17 that f~P - 0 in this case. In general, there may be other stress directions besides the orthotropic ones for which f~P = 0, [11].
4.8.3.2
MODEL CONSTANTS
Equations 12-18 provide all necessary information in closed form for the numerical implementation of Eqs. 11, with the only exception the specification of ~ for the hyperelastic relation, which is left up to the reader. The model constants associated with the use of the model are for the large part conventional. To be more specific, one has the following groups of such constants.
4.8.3.2.1
Elastic Constants
The choice of ~ will imply a set of elastic constants. For the simplifying assumption of linear isotropic elasticity, one needs only an elastic modulus and a Poisson's ratio.
4.8.3.2.2
Anisotropic
Coefficients
The values of F-N entering Eq. 12 depend on the degree of orthotropic intensity and its evolution for a given material. Their interpretation in regards to yield stresses in different directions can be found in Hill [4]. This can be used to determine their values experimentally.
4.8 Plasticity in Large Deformations
4.8.3.2.3
253
Kinematic Hardening
In reference to Eq. 14, one needs the values of h and c by a procedure which is standard in evanescent memory models. Notice that h/c represents a saturation measure for the back stress.
4.8.3.2.4
Isotropic Hardening
Similarly, in reference to Eq. 16, one needs the values of H and C. The ratio H/C is the saturation value of the isotropic hardening variable k.
4.8.3.2.5
Plastic Spin Coefficients
4.8.3.2.5.1 For Kinematic Hardening The value of r/a in Eq. 17 influences the saturation level of the shear stress in simple shear loading. Its determination is not straightforward, and the reader is referred to Dafalias [7] for further details.
4.8.3.2.5.2 For orthotropic re-orientation Finally, the value of 1/ in Eq. 18 will be of cardinal importance when orthotropic reorientation takes place. In a recent work [11], it was found that when setting r/-rl'/[(2/3)trS2] 1/2 at the absence of kinematic hardening, values of r / ' - - 1 0 0 , - 2 0 0 , - 3 0 0 can simulate successfully the orthotropic reorientation measured experimentally when orthotropic sheet metal samples are stretched in different directions, [15].
REFERENCES 1. Dafalias, Y. E (1998). The plastic spin: Necessity or redundancy? Int. J. Plasticity 14: 909-931. 2. Aravas, N. (1994). Finite-strain anisotropic plasticity and the plastic spin. Modelling Simul. Mater. Sci. Eng. 2: 483-504. 3. Dafalias, Y. E, and Rashid, M. M. (1989). The effect of plastic spin on anisotropic material behavior. Int. J. Plasticity 5: 227-246. 4. Hill, R. (1950). The Mathematical Theory of Plasticity, Oxford University Press. 5. Pereda, J. J., Aravas, N., and Bassani, J. L. (1993). Finite deformations of anisotropic polymers. Mechanics of Materials 15: 3-20.
254
Dahlias
6. Dafalias, Y. E (1983). Corotational rates for kinematic hardening at large plastic deformations. ASME J. Appl. Mech. 50: 561-565. 7. Dafalias, Y. E (1985). The plastic spin. ASME J. Appl. Mech. 52: 865-871. 8. Loret B. (1983). On the effect of plastic rotation on the finite deformation of anisotropic elastoplastic materials. Mechanics of Materials 2: 287-304. 9. Dafalias, Y. E (1993). On the evolution of structure variables in anisotropic yield criteria at large plastic transformations, in Proc. CNRS Intern. Colloquium No. 351, Villard-de-Lans, June 1983, in Failure Criteria of Structured Media, Boehler, J.P. ed., Rotterdam: A.A. Balkema. 10. Dafalias, Y. E (1984). A missing link in the formulation and numerical implementation of finite transformation elastoplasticity, in Constitutive Equations: Macro and Computational Aspects, in ASME Special Publication, Willam, K. J. ed., pp. 25-40. 11. Dafalias, Y. E (2000). Orientational evolution of plastic orthotropy in sheet metals. J. Mech. Phys. Solids, 48: 2231-2255.. 12. Zbib, H. M., and Aifantis, E. C. (1988). On the concept of relative and plastic spins and its implications to large deformation theories, I and II. Acta Mechanica 75: 15-33, 35-56. 13. Kuroda, M. (1997). Interpretation of the behavior of metals under large plastic shear deformations: A macroscopic approach. Int. J. Plasticity 13: 359-383. 14. Levitas, V. I. (1998). A new look at the problem of plastic spin based on stability analysis. J. Mech. Phys. Solids 46: 557-590. 15. Kim, K. H,. and Yin, J.J. (1997). Evolution of anisotropy under plane stress. J. Mech. Phys. Solids 45, 841-851.
SECTION
4.9
Plasticity of Polymers JEAN-MARC HAUDIN, BERNARDMONASSE CEMEF - BP 207, 06904 Sophia Antipolis, France
Contents 4.9.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4.9.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 4.9.3 Description of the Models . . . . . . . . . . . . . . . . 257 4.9.3.1 Yield Behavior . . . . . . . . . . . . . . . . . . . . 257 4.9.3.2 Overall Constitutive Equations . . . . 258 4.9.3.3 Discussion and Further Developments . . . . . . . . . . . . . . . . . . . . 260 4.9.4 Identification of Rheological Parameters .. 261 4.9.4.1 One-Dimensional Laws . . . . . . . . . . . . 261 4.9.4.2 Generalization to Three Dimensions . . . . . . . . . . . . . . . . . . . . . . 262 4.9.4.3 Extension to Complex Loading and High Strain Rates . . . . . . . . . . . . . 262 4.9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
4.9.1
VALIDITY
Plasticity of polymers refers to large deformations of a m o r p h o u s polymers in a temperature range below and near the glass transition temperature Tg, and of semicrystalline polymers below their melting temperature Tin. In such conditions, the mechanical behavior can be, in a first step, characterized by a n o m i n a l stress-strain curve, as obtained in a classical tensile test (Figure 4.9.1): ~N = F/So, eN = A L / L o , where E AL, So, and Lo are the load, the elongation, the initial section, and the initial length of the specimen, respectively. After an elastic and viscoelastic range I, in which the deformation is h o m o g e n e o u s and essentially reversible, the n o m i n a l stress reaches a m a x i m u m (yield point). Beyond this m a x i m u m , necking occurs and ON Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
255
256
Haudin and Monasse
III
IV
FIGURE.4.9.1. Nominalstress-strain curve of a ductile polymer. decreases (stage II). Afterwards, the neck stabilizes and its shoulders propagate until they reach the specimen heads (stage III). Then, the deformation is homogeneous again till rupture occurs (zone IV). In comparison with metals, the term plastic behavior concerns stages II to IV, in which a permanent deformation is observed after unloading and remains at the time scale of the experiments. Plasticity of polymers exhibits a number of specific features. First, deformation is generally heterogeneous because of the occurrence of necking or shear-bands. Therefore, the nominal curve of Figure 4.9.1 does not reflect the "true" mechanical behavior. Then, deformation is largely reversible. In the case of amorphous polymers, it can be completely recovered by annealing above Tg. Finally, the mechanical behavior is very sensitive to previous thermomechanical history (e.g., annealing prior to deformation), deformation path (induced anisotropy), and hydrostatic pressure.
4.9.2 BACKGROUND In crystalline materials, plasticity is interpreted in terms of the motion of linear defects or dislocations. Taking into account some similarities in the relationships between shear yield stress and modulus, many researchers have identified the elementary mechanism of plasticity in glassy polymers as the thermal activation of structural defects analogous to dislocations. Furthermore, polymers are viscoelastic materials, which generally exhibit several mechanical relaxations. Therefore, the existing models are based on two or three of the following ingredients: defects or sheared microdomains, thermal activation helped by applied stress, and viscoelastic transitions [6].
257
4.9 Plasticiy of Polymers
Semicrystalline polymers are very often deformed when the amorphous phase is in the rubberlike state. This mobility of the amorphous component allows, at the beginning of deformation, relative displacements of the crystalline lamellae (interlamellar slip, interlamella separation). Then, plastic deformation of polymer crystals takes place, essentially by crystallographic slip. At large strains, the lamellar crystals are broken into smaller crystalline blocks connected by amorphous fragments of molecules. This lamellar-tomicrofibrillar transformation occurs in the macroscopic neck. The ultimate step is the extension of the amorphous network. Consequently, the deformation of the crystalline phase can be modeled using micromacro approaches based on slip mechanisms, whereas strain-hardening at high strain can be treated with a rubber elasticity formalism. When the amorphous phase is in the glassy state, deformation combines the specific mechanisms of the amorphous and crystalline phases described previously.
4.9.3 DESCRIPTION OF THE MODELS 4.9.3.1
YIELD BEHAVIOR
From the physical ingredients listed in Section 4.9.2, many authors have focused their analysis on the yield behavior of glassy polymers. Only two examples will be considered here. The work of Bauwens-Crowet is not based on the concept of dislocations but on the thermal activation of viscoelastic transitions. In a temperature range where a single molecular process is involved, the yield stress O'y as a function of temperature T and strain rate + is represented by O'y k [ l n ( ~ ) ~ T g - V +
]
or k - ko exp
(
Q - o-yV) kW
(1)
which corresponds to an Eyring equation with two adjustable parameters: the activation energy Q and the activation volume V. k is the Boltzmann constant. The interpretation of yield stress in terms of elementary relaxations has been developed by Perez [8] in a sophisticated model involving the stressassisted activation of collective molecular movements related to fl transition. This analysis is consistent with the dislocation approach, since these molecular movements are confined in sheared microdomains. The model is able to predict the complex modulus in the whole temperature range of e and fl transitions, to describe the yield behavior, and to give access to the stressstrain curves as a function of temperature and strain rate.
258 4.9.3.2
Haudin and Monasse
OVERALL CONSTITUTIVE EQUATIONS
These equations are generally one-dimensional and intend to represent the whole true stress vs. true strain curve. Some of them consider only the plastic behavior, whereas in many cases all the strain components are integrated into the same formalism. They are either fully phenomenological, or more or less based on physical models presented in Section 4.9.2. They can be classified into three categories: 9 Multiplicative laws: or(e, 8) = K v F(e) G(8) 9 Additive laws: or(e, 8) = F(e) + G(8) 9 Differential laws: do/de = D(e, k) The o denotes the stress, 8 the strain rate, and e - fo 8(u) du the accumulated strain. Kp is a scale factor.
4.9.3.2.1 Multiplicative Laws The most popular multiplicative law, which is purely phenomenological, originates from the work of G'Sell and Jonas. In its most complete form [5], it is written as rr = K~ e x p ( a / T ) [ 1 - exp(-we)][1 + c e x p ( - b e ) ] e x p ( h e n ) 8 m
(2)
where the temperature dependence of Kp is described by an Arrhenius law: Kp = K e x p ( a / T ) . F(e) consists of three terms: 9 V(e)= [1-exp(-we)], which describes the initial elastic-viscoelastic part
of the curve; 9 C(e)= [l+c exp(-be)] to account for the stress peak observed at the yield when the amorphous phase is glassy; 9 H(e)= exp(he") for the hardening at high strain; n is generally close to 2, the value proposed in the initial version of the theory. The strain-rate sensitivity G(8) is represented by a power law. This equation is well adapted for tension. For other types of loading (simple shear, compression, multiaxial impact, etc.) the following expressions are preferred for H(e): simple shear
H(e) = exp(he)
(3)
simple shear
H(~) -- 1 + h~
(4)
multiaxial impact
H(e) = 1 + hlg+h2 ~2
(5)
259
4.9 Plasticiy of Polymers 4.9.3.2.2 Additive Laws These can be considered either as empirical or of the stress into an "effective" stress o* (8) mechanisms of plasticity) and an "internal" effects related to chain orientation). Following Jonas [4] have proposed this equation:
oi(~)
~r =
+
as based on the decomposition (activation of the elementary stress ai(g) (strain-hardening Andrews and Ward, G'Sell and
o* (/;)
= Kl[exp(2e) - e x p ( - e ) ] - m In ~o where the expression of the internal stress ai(e) derives from the rubber elasticity theory. Other additive laws can be found in the literature. For polycarbonate (PC), one can mention: 9 the Belcadi-Parisot law, modified to take into account the strain-rate sensitivity = as + a o ( e x p ( e / e c ) - 1) + fl In
(7)
9 the 0 law = erez+ Ks[1 + 0e exp(-we2)] + K e x p ( a / T ) exp(he2)~ m
(8)
where the elastic component Crel obeys Hooke's law and the second term describes the stress maximum at the yield. 4.9.3.2.3 Differential Laws The Vest-Amoedo-Lee law, proposed for amorphous polymers like PC, is based on a rheological model consisting of a Hookean spring (modulus E) in series with three parallel elements (yield element, Eyring dashpot, rubber elasticity element). This leads to the differential equation
(')
- K1 { exp(2~) - exp(-~) }
--~o
d__~ &=E
1
K -sinh
1+ V
~b~
e x p ( - c l e) 4kT
(93
260
Haudin and Monasse
where ~bo and Cl characterize the stress distribution in the parallel elements, K and V are related to the Eyring dashpot, and Kl[exp(2e)-exp(-e)] is the rubber elasticity term. In the differential G'Sell-Jonas model [4], the strain is decomposed into an elastic part related to the stress through the modulus E and into a plastic component. Plastic deformation is due to the creation and propagation of dislocation-type defects. The constitutive equation is written d-7- E 1 - - 7 exp ee
kT
(10)
where the internal stress o-i(8 ) is described by the complete equation derived from rubber elasticity models: cri(e) = K1 [exp(2e)-exp(-e)] + K2 [exp(e)-exp(-2~)]
(11)
and o-~ is the constant effective stress after the transient yield behavior. In its original version, the model is temperature- and strain rate-independent, but these dependences can be introduced.
4.9.3.3
DISCUSSION AND FURTHER DEVELOPMENTS
The principal merit of the analytical multiplicative and additive laws is their ability to describe, at given strain rate and temperature, the stress-strain curve in a simple mathematical formalism. They can be easily implemented in computer codes, especially in the multiplicative case. Nevertheless, they suffer from their empirical or semiempirical character. Therefore, there is an increasing demand for physically based models. For amorphous polymers, the G'Sell-Jonas differential model can be considered as a first attempt, but the stress determination requires a numerical integration and this law is not easily usable in computer simulations. In the same way, Perez's model [8], which correctly predicts the yield behavior and the beginning of plastic deformation, would be fully operational if it were completed by an entropic strain-hardening law. Once more, the stress-strain curve is calculated numerically. Concerning semicrystalline polymers, the plastic behavior of the crystalline phase can be predicted using models developed for metals (e.g., modified Sachs model or self-consistent model). They must be combined with approaches based on rubber elasticity to account for the extension of amorphous chains at high strain. Thanks to these models, the prediction of the macroscopic mechanical behavior from microscopic processes becomes possible in simple cases.
261
4.9 Plasticiy of Polymers
4.9.4
IDENTIFICATION
OF RHEOLOGICAL
PARAMETERS 4.9.4.1
ONE DIMENSIONAL lAWS
The identification of the rheological parameters of the overall constitutive equations presented in Section 4.9.3.2 is generally performed in mechanical tests corresponding in principle to a simple loading: tension, compression, and shear. Nevertheless, a number of problems have to be overcome. The first one is to prevent self-heating, which imposes one to work at a low strain rate (typically g _< 10 -3 s- 1). A major problem is the inhomogeneity of the deformation observed in many tests (see introduction). A solution is either to find homogeneous (e.g., plane-strain compression of thin films) or quasi-homogeneous (e.g., compression of cylinders with appropriate dimensions) conditions or to consider a mesoscopic scale where the deformation is assumed to be locally uniform (uniaxial tension, simple shear). For instance, in videocontrolled tensile tests, the deformation of hourglass-shaped specimens is followed by a video camera. Necking occurs in the minimum section, and the system measures in real time the minimum diameter D, the local radius of curvature Rc, and the load E The true stress, strain, and strain rate are then --
O"
4F
(
/1:D2 1 + -
1 In 1 +
e--
21n
(__~)
8--
2 dD D dt
(12)
The stress 4F/rcD 2 is corrected for triaxiality effects using the Bridgman factor. Finally, the tests should be performed at constant strain rate g. This is possible in video-controlled tensile tests by monitoring the cross-head velocity from diameter measurements. The numerical values of the rheological coefficients can be determined from true stress-strain curves either "by hand" from a careful analysis of the different regions of the curves, or as a whole using optimization programs based on the least-square method. More recently, inverse methods combining a direct model for the simulation of the test by the finite element method and an optimization module for minimizing the differences between predicted and measured physical data have been employed [10].
262 4.9.4.2
Haudin and Monasse GENERALIZATION TO THREE DIMENSIONS
The generalization of one-dimensional laws to three-dimensional situations requires the definition of equivalent quantities (stress, strain and strain rate) associated with yield criteria. Incompressible and isotropic criteria (Tresca, von Mises) have been used in the literature, but this raises two categories of problems: 9 The mechanical behavior of solid polymers is very pressure-dependent. Whatever the polymer, the yield stress strongly increases with hydrostatic pressure. It implies that symmetrical criteria in the stress space cannot predict the yield stress under different deformation paths. Conversely, nonsymmetrical criteria such as Mohr-Coulomb or pressure-modified von Mises criteria seem to be able to describe the stress at yield, which has been checked on a limited number of polymers. 9 An important anisotropy is developed at high strain, strongly depending on the deformation path: it is higher in tension than under compression or shear. Micro-macro approaches are able to interpret these phenomena in simple cases, but they are not yet available for engineering applications. Inverse methods provide an alternative solution.
4 . 9 . 4 . 3 EXTENSION TO COMPLEX LOADING AND HIGH STRAIN RATES In many practical cases, e.g., in impact tests or during the thermoforming process, the material experiences both complex loading and high strain rates. It has been shown previously that 9 Constitutive equations are determined in laboratory experiments in simple loading conditions and at low strain rates. 9 The induced anisotropy is strongly dependent on the deformation path. Micro-macro approaches are able to predict this anisotropic behavior, but only in limited cases and with important calculations. Therefore, the idea is to select a test in which matter is supposed to be strained in conditions close to those encountered in the actual application. From experiments performed using this test and for a given mathematical form of the constitutive equation, the rheological coefficients can be identified thanks to an inverse method [10].
263
4.9 Plasticiy of Polymers
4.9.5 RESULTS Table 4.9.1 gives the values of the coefficients of the simplified Eq. 1 o
--
Kp[1
-
(i')
e x p ( - w e ) ] exp(he2)/; m
for different types of loading (uniaxial tension, uniaxial compression) and various polymers: polyethylene (PE) [1], polypropylene (PP) [3], poly(oxymethylene) (POM) [7], polyamide 6-6 (PA 66). For PE, the values given for m take into account its variations with e, and for w its variations as a function of ~. For other polymers, these values have been averaged. Bisilliat and Billon [2] have s h o w n that for PC b o t h the phenomenological 0 law Eq. 8 and the physically-based G'Sell-Jonas differential law Eq. 10 were able to describe the stress-strain curve u n d e r uniaxial tension, including the stress peak at the yield transition. The numerical values of the coefficients of these laws at 20~ are given in Tables 4.9.2 and 4.9.3. For multiaxial impact, Tillier and Billon [9] have used the modified G'SellJonas multiplicative equation cr - -
Kv[1
-
exp(-we)](1 + hie -~ h2~2) exp(he 2) ~m
(13)
and identified the numerical values of the coefficients using an inverse m e t h o d (Table 4.9.4).
TABLE 4.9.1 Rheological Coefficients for Different Types of Loading and Various Polymers. G'Sell-Jonas Muhiplicative Law Eq. 1' Polymer PE
PP
Test
Temperature(~
Kp (MPa Sm)
-20
65
22
46
60
25
20 80 110 130 150 115 165 25 100 230
63.6 17.4 9.1 7.5 4.5 70.3 10.3 122 70 23
Tension
Tension
POM
Compression
PA 66
Compression
W
h
45 68 22 50 20 30 31 32 33 26 23 34 45 22 7 15
0.48 0.41 0.43 0.52 0.36 0.39 0.45 0.40 0.05 0.10 0.10 0.10 0.10
m 0.03 0.05 0.09 0.06 0.10 0.05 0.082 0.047 0.029 0.040 0.034 0.110 0.056 0.021 0.021 0.021
264
Haudin and Monasse
TABLE 4.9.2
0 Constitutive Equation for PC at 20~
K (MPas m)
a (K -1)
5.33
749
TABLE 4.9.3 Tension.
Uniaxial Tension.
0
h
m
Ks (MPa)
270
1.8
0.041
0.9
G'Sell-Jonas Differential Constitutive Equation for PC at 20~
E (MPa)
W/h
r (MPa)
a2
K1 (MPa)
2300
38.7
65
0.05
60
TABLE 4.9.4 20~ Ke (MPa
Uniaxial
K2 (MPa) - 80
Tillier-Billon Constitutive Equation for PP; Multiaxial Impact; Test Temperature
Sm)
17.4
W
hi
h2
m
106.8
1.04
0.325
0.045
REFERENCES 1. Aly-Helal, N. A. (1982). Etude et simulation du d4veloppement de la striction dans le poly4thyl~ne ~ haute densit4, th~se, Institut National Polytechnique de Lorraine, Nancy. 2. Bisilliat, M. L. (1997). Comportement m4canique d'un polycarbonate ~ grande vitesse de sollicitation. Etude exp4rimentale et simulation, th~se, Ecole des Mines de Paris. 3. Duffo, P., Monasse, B., Haudin, J. M., G'Sell, C. and Dahoun, A. (1995). Rheology of polypropylene in the solid state. J. Mater. Sci. 30: 701-711. 4. G'Sell C. and Jonas, J. J. (1981). Yield and transient effects during the plastic deformation of solid polymers. J. Mater. Sci. 16: 1956-1974. 5. G'Sell, C. (1988). Instabilit4s de d4formation pendant l'4tirage des polym~res solides. Rev. Phys. Appl. 23: 1085-1101. 6. G'Sell, C. and Haudin, J. M. (1995). Introduction h la m~canique des polym~res, Institut National Polytechnique de Lorraine, Vandoeuvre l~s Nancy. 7. Haudin, J. M., Monasse, B., Valla, T. and Glommeau, S. (1995). Numerical simulation of polymer forging: How to determine the rheological coefficients. Intern. Polym. Process. 10: 179-185. 8. Perez, J. (1992). Physique et m4canique des polym~res amorphes, Lavoisier Tec et Doc, Paris. 9. Tillier, Y. (1998). Identification par analyse inverse du comportement m4canique des polym~res solides. Applications aux sollicitations multiaxiales et rapides, th~se, Ecole des Mines de Paris 10. Tillier, Y., Massoni, E. and Billon, N. (1998). Inverse method for the characterization of mechanical behavior of polymers under biaxial high velocity loading, in Computational Mechanics. New Trends and Applications, Proc. 4 th World Congress on Computational Mechanics, Buenos-Aires.
SECTION
4.10
Rational Phenomenology in Dynamic Plasticity JANUSZR. KLEPACZKO Metz University, Laboratory of Physics and Mechanics of Materials, Ile du Saulcy, 57045 Metz, France
Contents 4.10.1 Range of Strain Rates a n d F u n d a m e n t a l s .. 265 4 . 1 0 . 2 I n s t a n t a n e o u s Rate Sensitivity a n d Rate Sensitivity of Strain H a r d e n i n g . . . . . . . . . . .
266
4 . 1 0 . 3 The F o r m a l i s m in M o d e r n C o n s t i t u t i v e Modeling .................................
268
4 . 1 0 . 4 Identification of M i c r o s t r u c t u r e . . . . . . . . . . .
270
4.10.5 O n e - p a r a m e t e r M o d e l of M i c r o s t r u c t u r a l Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 71
4 . 1 0 . 6 Recent A p p l i c a t i o n s . . . . . . . . . . . . . . . . . . . . . .
272
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
273
4.10.1 RANGE OF STRAIN RATES AND FUNDAMENTALS Dynamic plasticity in its general definition covers several decimal orders in strain rate, typically from 10E-4 1/S (quasi-static case) to 10E-6 1/S (highvelocity impact). Since plastic flow is a kinetic process, in general, the level of stress at constant microstructure depends on both strain rate and temperature. The low-temperature micromechanisms involved in dynamic plasticity operate typically from OK up to about 0.5 of the melting point. The microstrucuture can be defined in many ways; one possibility is to use the concept of internal state variables (one or more than one) [ 1], which take into account a microstructural evolution. Since the density of moving defects per Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
265
Klepaczko
266
unit volume that take part in plastic deformation is very large, it is convenient to operate by mean quantities; however, they are not precisely defined by statistical means. Such an approach is called rational phenomenology, represented in this paper. The architecture of physical laws used is called the formalism. The constitutive formalisms are a powerful tools for formulating physically based rate- and temperature-dependent constitutive models.
4.10.2
INSTANTANEOUS
AND RATE SENSITIVITY
RATE SENSITIVITY OF STRAIN
HARDENING Over the years numerous experimental data have been accumulated, too many to mention, which clearly demonstrate the existence of so-called strain rate history and temperature history ,effects [1-3]. Those effects have been observed for both metals and alloys as well as for single crystals [4]. The typical behavior in two common lattice structures, FCC and HCP, for temperatures less than about half of the melting temperature, is shown in Figure 4.10.1 [5]. According to the schematized responses to strain rate or temperature changes from 1-'i to l~r or from Ti to Tr, an instantaneous elastic response is observed as an increment of shear stress A~:s; this is the path ABCD in Figure 4.10.1a. The same happens when temperature is abruptly decreased, as is shown in the lower part of Figure 4.10.1a i.e., the path ABCD. Decrease of from l'i to l~r or increase of T from Ti to Tr produces responses shown in Figure 4.10.1b. Independently, whether a metal is deformed with a jump of strain rate or 1-"is changed with a temporary unloading, the result is similar, as is shown in Figure 4.10.1. In this paper notation of the shear strain will be as follows: F = tan 7, where 7 is the shear angle, and the time derivative: -- d F / d t . It is evident that the strain rate or temperature history effects are due to a somewhat different physics of strain hardening at different strain rates or temperatures. The significance of the incremental tests lies in the fact that they provide values of flow stress at two different strain rates with virtually the same microstructure. In conclusion, A~s (path BC in Fig. 4.10.1) is the measure of instantaneous rate sensitivity at constant microstructure. The entire stress difference Avs 4- Avh refers to the same initial strain 1-'i but to two slightly different microstructures, and consequently Al:h is the result of deformation history. The total increment of stress AI: = A~s 4- Al:h can be determined from the tests with two constant strain rates or temperatures. It is obvious that A~h can be associated with rate sensitivity of strain hardening.
267
4.10 Rational Phenomenology in Dynamic Plasticity a
b
?i<'i',; ri>T, T:T r
~>'Pr ; Ti~rr T:Ti
' ~
-= YY', iBt'~T:? ~Yp
~
);'i
I --
~L'- -
,Yr
Yp
Yp FIGURE 4.10.1
This rather loose term emphasizes the dependence of strain hardening on strain rate. The following three general definitions for strain rate sensitivity of flow stress can be introduced (Eq. 1). The definitions are referred to shear deformation; however, they are also valid in case of tension or compression. All rate sensitivities are defined as instantaneous quantities determined with jump tests [1]: fl+ - (c~-r/a log f')r,SfR;
rn~ - (a log "c/a log 1-')T,STR;
r/~ -- (a'C/af")f,SfR
(1) where fi is the rate sensitivity, m is the logarithmic rate sensitivity, and r/is the viscosity. The complete set of the rate sensitivities is fl, m, r/. The total rate sensitivities can be determined at two constant strain rates and the instantaneous values from jump tests [1]. For both cases, fl~, m+, r/+ and fi~, m~, r/r can be determined, respectively, for shear and tension/compression. All those quantities are interrelated, as is shown in Table 4.10.1.
268
Klepaczko TABLE 4.10.1
Rate sensitivity m~ ~
*m
*[3,
l/r,
1
rl,
T
F/~
T ~
~
T
T
1
T 1/[" Rate sensitivity ma =
*m 1
~
~
T
*fla 1/ rr
1 t/ty
T tic
a/~
T T
1 1/~
T
T 1
4.10.3 T H E F O R M A L I S M IN M O D E R N CONSTITUTIVE MODELING The so-called formalism in constitutive modeling is a set of assumptions formulated on the notions of materials science. In view of the complexity and diversity of dislocation behavior, the formalism, or the "architecture" of the constitutive relations, may differ depending how many particular microprocesses of plastic deformation are taken into consideration. Plastic flow of metals is determined on the atomic scale by the motion of defects (mainly dislocations). Since the defects are multiplied during plastic deformation, the strain hardening develops. On the other side the plastic flow is a kinetic process. Such an approach leads to a specific constitutive modeling. The formalism presented here had been introduced some time ago [5], and further developments have been reported later [1, 6, 7, 8]. The formalism differs from that proposed later by Mecking and Kocks [9] and by Estrin and Mecking [10], and applied recently to high strain rate plasticity [11]. The thermal activation strain rate analysis is employed for both the kinetics of glide and the kinetics of structural evolution. Three main factors contribute to the actual level of flow stress: strain, strain rate, and temperature. The evolution of the effective stress is associated with glide kinetics and short transients of flow stress (instantaneous rate sensitivity) [12], and the evolution of the internal stress is coupled with the formation history of the microstructure (rate-sensitive strain hardening). The model takes into account the evolution of the mean dislocation density, mobile dislocation density, formation of dislocation cells, the effect of grain diameter, and
269
4.10 Rational Phenomenology in Dynamic Plasticity
formation of twins. As a result, a quantitative description of strain rate history and temperature history effects in FCC, BCC, and HCP lattices can be accomplished. More recent numerical simulations have demonstrated the capabilities of the model. It has also been shown that at high strain rates (103S-1 ~ F < 106S-1) an excessive dislocation generation occurs in FCC metals which leads to a substantial increase of the mechanical threshold (the flow stress in absence of the thermal vibration of the lattice, i.e., at temperatures near the absolute zero). The notion is adopted that plastic deformation in shear is the fundamental mode of deformation. Another fundamental assumption is partition of the flow stress r into the effective stress ru and the internal stress r*; thus r -- z~ + z *
or in tension/compression
a -- a~ + a*
(2)
The assumption of additiveness according to Eq. 2 implies the existence of two sets of obstacles opposing the dislocation movements [13]. The first one, associated with the internal stress zu, is due to strong obstacles to dislocation motion like cell walls, grain walls, twins, etc. The secondary defects, while more numerous, like forest dislocations, Peierls barriers, and second phase particles, are supposed to be weak obstacles, and they can be overcome by moving dislocations with the assistance of thermal vibration of crystalline lattice. Those obstacles produce the effective stress z*. The kinetics of defect (dislocation) movements interrelates at constant microstructure, characterized by j state variables sj, the instantaneous value of effective stress z*, and the instantaneous plastic strain rate F. The expression for this process can be written as
["
--
V i
(T, sj)exp[-AGi(T, "c*, sj)/kT]
(3)
or after inversion r* - f * {sj[h(1~, T)], 1-', T}
(4)
where vi is the frequency factor, AGi is the free energy of activation, T is the absolute temperature, and k is Boltzmann constant. The subscript i indicates the i-th, so far unspecified, thermally activated micromechanism of plastic deformation. Generally AG depends on the effective stress r* in a nonlinear manner [1,13]. Note that only plastic strain and plastic strain rate enter into equations. The internal stress zu must be also rate and temperature-dependent via dynamic recovery processes, i.e., relaxation of long-range internal stresses due to dislocation annihilation and rearrangements of obstacles to dislocation motion. Thus the internal stress is
~ - f~ {sj[h(f',r)]}STR
(5)
270
Klepaczko
It is assumed that the recovery processes leading to a structural evolution may be thermally activated. It is the source of temperature and strain rate dependence of zu.. Both strain rate and temperature enter into Eqs. 4 and 5 in the functional form
"CsTa-f~{s),[h(F,
T)]}sTa + f * {sj, [h(F, T)],F, T}STR
(6)
where h(]~, T) indicates that the internal state variables.sj do depend on the history of F and T defined more exactly as F(F) and T(F). Consequently, the plastic response of a material is divided into two logic steps: the flow stress z depends on the current structure, defined by sj state variables and also on the current values of 1-" and T, and next the structure evolves with plastic strain. This is the fundamental assumption of the model. Both components of stress in Eq. 6 are written for a current state characterized by sj state variables. Since the microstructure undergoes an evolution, and the state of microstructure is defined by j variables, the microstructure evolution is assumed in the form of a set of j differential equations of the first order d s j / d F - fj[Sk, F(F), T(F)],
k - 1 . . . ,j.
(7)
A solution of the set defined by Eq. 7 provides current values of sj to be introduced into Eq. 6. Thus the flow stress can be calculated for any deformation history.
4.10.4
IDENTIFICATION
OF
MICROSTRUCTURE Flow stress or yield stress in polycrystalline metals and alloys can be related to characteristic spacing of obstacles to dislocation motion associated with a particular microstructure [13]. A range of characteristic spacing is possible. The following four are assumed as a satisfactory choice for the internal state variables: the microstructure is characterized by the mean distance L between forest dislocations, the mean value of a dislocation cell d, the mean value of a grain diameter D, and the mean distance between twins A Each of those obstacles to dislocation motion will contribute to the total value of the internal stress ~:u. An explicit expression for ~u can be written as [1,8] "r,p -- •llgbp 1/2 -Jr-o~2/t
+e3//
-+-e412
(8)
where # is the shear modulus, ~1 "~4 are constants which characterize dislocation/obstacle strength, b is the modulus of Burgers vector, the mean spacing between forest dislocations can be defined as L - 1/V/-fl, and 5 is an exponent which is equal to 1 for one-dimensional characteristic spacing L and "
"
271
4.10 Rational Phenomenology in Dynamic Plasticity
6 = 1/2 for two-dimensional spacings (d, D, A). A more detailed characterization of all variables p, d, D, and A can be found elsewhere [1, 6, 8]. The first three terms in Eq. 8 are related, respectively, to dislocation/ dislocation interaction, evolution of subgrain, and the effect of grain diameter the so called Hall-Petch term [1]. It is well known that at low temperatures and at very high strain rates some metals and alloys produce deformation twins. Thus the fourth term in Eq. 8 accounts for twin formation as a dislocation obstacle at the beginning of a certain strain level, usually F ~ 0.08, dislocations form cells with linear dimension d. Generally, the evolution of d(p) intensifies strain hardening with a larger effect at small strains.
4.10.5 ONE-PARAMETER MODEL OF MICROSTRUCTURAL
EVOLUTION
Common to all recent microstructure-related models is that the set sj of internal state variables is reduced to a single structure parameter which can be identified as the mean dislocation density p [1, 5 - 8 , 12]. A general form of the differential equation for structural evolution can be written as [5]
d p / d r - Mg(p, F ) - Ma(p, F, T)
(9)
where the difference M4f = Mg - Ma is the effective coefficient of dislocation multiplication [5, 14]. Once M4f is known in an explicit form, one can predict the current value of plastic strain F that will be accumulated in any process of deformation by integrating Eq. 9. A simple evolutionary relation has been proposed in [6]
dp/dF = MH- ha([", T)(p - Po) with
ha = k0(F/F0) -2m~
(10)
The initial condition for M, that is, M MII if F = 0, is satisfied automaticall.y when p = P0; P0 is the initial dislocation density. The solution for p with F and T as parameters yields the following relation: =
P = P0 + ka(F, Y){1 - e x p [ k a ( F , T)F]}
(11)
Introduction of the solution for p(Eq. 11) and expression for ka into Eq. 11 yields the final form for the ~u. The effects of d, D, and A in this contribution are neglected. The second component of the flow stress, that is, r*, can found in the explicit form using the generalized Arrhenius relation (Eq. 3). The frequency factor v, is defined as vi = Pmb2VD, where p,, is the mobile dislocation density and VD is the Debye frequency ( ~ 1013s-1). It is usually assumed for FCC and HCP structures that Pm = fP, where f is the fraction of the total dislocation density, f < 1, and typicallyf ~ 10 -3. For BCC metals the
Klepaczko
272
evolution of Pm is more complex; f is not a constant but varies, causing, for example, the upper and lower yield point and Luders bands. Inversion of the Arrhenius relation yields the explicit expression for ~*
I where "c~ is the threshold stress at 0 K. For FCC and HCP structures the threshold stress depends on the dislocation density, whereas for BCC structures it is approximately constant this is the Peierls stress barrier [1, 6, 13]. Application of this formalism leads to the explicit form of the constitutive relation with F and 7" as parameters, r(f', T ) - r~(F, 1")+ z * (F, T), thus
r. - cz~b{Po + ka(f" , T , [ 1 - exp(ka(f% T,F,]}I/2 + "Co [1-G0k--~TlnVi(Fm)] (13) where ka(r ~, T) is defined by Eq. 10 and vi is defined as discussed previously.
4.10.6
RECENT APPLICATIONS
The formalism in constitutive modeling with one state variable as the mean dislocation density p has been applied so far many times, and the results of numerical calculations have been reported in several papers [1, 3, 8, 12, 15] for both FCC and BCC structures. The two state variable model with evolution Pi and Pm, where Pi and Pm are, respectively, the immobile and mobile dislocation density, has recently been applied to calculate short and long transients for aluminum [12]. Those simulations for A1 in the form of ~(F) curves at different temperatures along with incremental changes of f~ do show relatively good agreement with experimental observations. Concerning FCC pure metals, numerical simulations of strain rate history effects were reported in Reference [6]. They more or less exactly depict experimental results for polycrystals of A1 deformed at different temperatures with jumps in strain rate [1, 6, 8]. Stress-strain curves for Armco iron and 1020 steel have been reported for a wide range of strain rates in References [6, 7, 8, 16]. An agreement with experimental data was achieved for shear strains, F <0.4, with a simple evolution equation without annihilation. A more realistic evolution equation, in the simplified form of Eq 14, was applied to model strain hardening and rate-sensitive plastic flow for XC18 steel (approximately AISI 1018) [15]. A complete set of material constants for this steel is given also in Reference [15]. The identification of material constants in the model for the range of dynamic strain aging temperatures, via experiments on XC18 steel, was reported in [7].
4.10 Rational Phenomenology in Dynamic Plasticity
273
In general, the formalism permits calculations of viscoplastic responses to different deformation histories for FCC, BCC, and HCP relatively pure metals and metallurgically stable alloys. It is flexible enough to include a variety of effects on flow stress at different strain rates and temperatures.
REFERENCES 1. Klepaczko, J. R. (1988). A general approach to rate sensitivity and constitutive modeling of FCC and BCC metals, in Impact: Effects of Fast Transient Loadings, p. 3, Guman, W. J., et al., eds., Rotterdam: Balkema. 2. Klepaczko, J. R., Frantz, R. A., and Duffy, J. (1977). History effects in polycrystalline FCC metals subjected to rapid changes in strain rate and temperature. Engng. Trans. 25: 3. 3. Duffy, J. (1983). Strain-rate history effects and dislocation structure, in Material Behavior Under High Stress and Ultrahigh Loading Rates, p. 21, Mescall, J., and Weiss, eds., New York: Plenum Press. 4. Morrone, A. A., and Duffy, J. (1986). Strain Rate and Temperature Effects During Dynamic
Deformation of Polycrystalline and Monocrystalline High Purity Aluminum Including TEM Studies, MRL Brown University Report, Providence. 5. Klepaczko, J. R. (1975). Thermally activated flow and strain-rate history effects for some polycrystalline FCC metals. Materials Sci. Engng. 18: 121. 6. Klepaczko, J. R. (1987). Modeling of structural evolution at medium and high strain rates, in Constitutive Relations and Their Physical Basis, Proc. 8-th Ris6 Symposium, p. 387, Ris6, Roskilde, Danmark. 7. Zeghib, N. E., and Klepaczko, J. R. (1966). Work hardening of mild steel within dynamic strain ageing temperatures. J. Materials Sci. 31: 6085. 8. Klepaczko,J. R. (1991). Physical-state variables: The key to constitutive modeling in dynamic plasticity. Nuclear Engng and Design 127: 103. 9. Mecking, H., and Kocks, U. E (1981). Kinetics of flow and strain hardening. Acta Metall. 29: 1865. 10. Estrin, Y., and Mecking, H. (1984). A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall. 32: 57. 11. Follansbee, P. S., and Kocks, U. E (1988). A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall. 36: 81. 12. Klepaczko, J. R. (1992). Short and long transients in dynamic plasticity of metals, modeling and experimental facts, in Shock-Wave and High-Strain-Rate Phenomena in Materials, p. 147, Meyers, M. A., Murr, L. A., and Staudhammer K. P., eds., New York: M. Dekker. 13. Kocks, U. E, Argon, A. S., and Ashby, M. E (1975). Thermodynamics and Kinetics of Slip, Oxford: Pergamon Press. 14. Gilman, J. J. (1969). Micromechanics of Flow of Solids, New York: McGraw-Hill. 15. Klepaczko, J. R., and Rezaig, B. (1996). A numerical study of adiabatic shear banding in mild steel by dislocation mechanics based constitutive relations. Mech. of Materials 24: 125. 16. Klepaczko, J. R. (1984). An engineering model for yielding and plastic flow of ferritic steels, in High Energy Rate Fabrication, p. 45, New York: ASME.
SECTION
q
.ll
Conditions for Localization in Plasticity and RateIndependent Materials AHMED
BENALLAL
Laboratoire de M~canique et Technologie, ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du PrEsident Wilson, 94235 Cachan, France
Contents 4.11.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . 4.11.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.3.1 N (or ~ ) Has Distinct Eigenvalues. 4.11.3.2 N and ~ Have Both a Double Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.1
274 275 276 276 278 280
SCOPE
We give in this paper closed-form expressions for the critical conditions at localization and the orientations of the corresponding plane for general rateindependent constitutive equations of the form Fif f < O o r f - 0 and f < O ~-0_'~
L--
H-F-~~
and f - 0
(1)
where c~ is the stress tensor, g the strain tensor, and l_ the tangent modulus. In Eq. 1, f is the yield function, ~ and [3 are second-order tensors assumed throughout to be coaxial, H = h + ~:I:-l: [3 is a scalar parameter, and F is the elastic tensor assumed, throughout to be isotropic (with Lam~ constants 2 and /~); h is the so-called hardening modulus. The dot is differentiation with respect to time. It is convenient to introduce the deviatoric and hydrostatic parts of 0~ and [3 such that cz = a + pl, [3 = b 4-ql with p = 1/3 tr(oO and q = 1/3 tr([3). Localization is understood in the classical sense, i.e.. as an
274
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
275
4.11 Conditions for Localization
instability of the inelastic deformation, and is described as a bifurcation into modes involving jumps of the velocity gradient across a planar band of normal n. The general localization condition reads det (n.H.n) <_ 0 [1-31.
4.11.2
METHOD
OF SOLUTION
We use here a geometrical method to solve in closed form the localization condition. This can also be done using Lagrange multipliers as proposed by Bigoni and Hueckel [4]. The localization condition for model (1) takes the form (see [5]) 1
H~(2 + 2/0 <_ (2 + 2~)4[((a + b).n)((a + b).n) - ((a - b ) n ) ( ( a - b)n)] - (2 +/.z)~[(n(a + b)n) 2 - (n(a - b)n) 2] +/_zp +2 qn(a + b)n -
(2)
~P + qn(a - b)n + #pq 2
Introducing the tensors N = a + b and M = a - b, then the respective magnitudes T+ and T_, and the respective normal components 2+, 2_ of the generalized stress vectors N.n and M.n in the n direction, defined by 22+ = n.N.n, T+ = (N.n).(N.n), s
= n.M.n, T_ = (M.n).(M.n)
(3)
the localiztion condition becomes 4/x(2 + 2#) _< H(2 + 2/x)(T+ - T_) - ( 2 + 2/x)(2 2 - 22_) 4- ~/[(Z+ 4- p 4- q)2 _ (s
4- P _ q)]2
(4)
In the (Z+, T+) plane ( in the (Z_, T_) plane), by elementary Mohr analysis, a given direction n is represented by a point lying in an admissible area T+ (in an area T_) and corresponding exactly to the classical admissible area in the Mohr diagrams associated with N and M. When denoting by Np (p = 1, 2, 3) and by Mp (p = 1, 2, 3) the eigenvalues of N and M, respectively, and by Xp their common eigenvectors, classical formulae in Mohr analysis show that the components ni of n in the principal frames of N and M satisfy, respectively, T+ 4- NiX+ + NjNk -- (ni)2(Ni - Nj)(Ni - Nk) T_ + Mis
+ MjMk -- (ni)2(Mi- Mj)(Mi- Mk)
(5) (6)
where (i, j, k) is a triplet with distinct numbers belonging to the set (1,2,3). We denote by Hc the critical value of H at localization.
276
Benallal
4.11.3 RESULTS Two situations should be analysed separately.
4.11.3.1
~ (OR ~)
HAS DISTINCT EIGENVALUES
When the eigenvalues of ~ (or those of ~ ) are distinct, the admissible area T+ (or T_) is a triangle (see Fig. 4.11.1.); moreover, one can compute from Eqs. 5 (or 6) the components of n, and then by Eq. 3 one can compute ~_ and T_ as functions of ~+ and T+ (or ~+ and T+ as functions of ~_ and T_) and upon substitution in the localization condition Eq. 4, end up with an expression involving only ~+ and T+ (or ~_ and T_ only). Clearly, localization will occur when the domain D+ (D_) represented by this inequality meets in the (~+, T+) plane ( in the (~_, T_) plane) the admissible triangular area T+ (T_) defined previously. Depending on the constitutive behavior at hand, localization can occur at the beginning of the inelastic process or only after a finite period of this process. In the
FIGURE 4.11.1 Geometrical interpretation of the localisation condition in the (~+, T+) plane. The admissible domain in this plane is the triangular area T+. The domain representing the localization condition Eq. 2 is delimited by the hyperbola C+.
277
4.11 C o n d i t i o n s for L o c a l i z a t i o n
first case, the critical conditions are trivially given by the initial inelastic state; from now on we consider only the second case, where they are obtained by the instant when the curve C+ delimiting D+ intersects for the first time the admissible area T+. This may happen when C+ (which is a hyperbola in our case) touches T+ at one of its vertices (in which case the normal to the localization plane is a principal direction of ~l and M), or when it becomes tangent to one of its sides ( in which case the normal to the localization plane lies in a principal plane of N and ~ ) . Therefore, it is concluded that the normal to a potential plane of localization lies always in a common principal plane of ~ and g4]. It remains then to consider the case where ~J and M have both a double eigenvalue, since because of their deviatoric nature, they cannot have a triple one. This case will be examined in Section 4.11.2. Now that the potential planes have been obtained, the critical conditions at localization must be obtained. Denote by (i, j, k) a triplet of distinct number belonging to the set (1, 2, 3). Let us assume that n lies in the principal plane spanned by xi and xj and call 0 the angle between xi and n (e c [0, re]). Computing Z + , T+, Z_ and T_, substituting the result in Eq. 4 with equality to get H, and maximizing H with respect to 0, one gets the stationnarity condition: (Ni - Nj)sin 20[(2 +/2)cos 2 0 ( N i - Nj) - l l { N i q- Nj if- 2(p + q)}]
=
(Mi - Mj)sin 20[(2 +/.t)cos 20(Mi - Mj) - #{Mi + Mj + 2(p - q)}]
(7)
This stationnarity condition has two solutions: either sin 20 = 0 and the corresponding normals to the localization plane are the principal directions xi or xj leading to the value hi or hj given by
hi-
4[L~ q-2g) - 232~ 4#+ ((
Ni(Nj - Nk) - Mi(Nj - Mk)
p + q)(Nj - Nk) - (p - q)(Mj - Mk)) ]z
412 [ 3,~ + 21a(Ni(Mj _ Mk) -- (Mi(Nj - Nk) + (2 + 2/2)(32 + 2/,) 4/,
(8)
2
+ (p + q)(Mj - Mk) - (p -- q)(Nj - Nk)
1
4/~[(Nj - Nk) 2 -- (Mj - Mk) 2]
}2 (Nj - Nk)2 - (Mj - Mk) 2
and a similar one for hi. cos 20 =
l.Z {Ni + Nj + 2(p + q)}(N, - Nj) - {M, + Mj + 2(p - q)}(M, - Mj) 2+/2
(9)
278
Benallal
and the corresponding value of h is given by hi-j - ~ 1{
4#(232+2P+ la)[Nk(Ni - Nj) - Mk(Mi - Mj) 2#
2
+ 32 + 2~ (p + q)(Nj - Nk) -- (p -- q)(Mj - M/)] 4p 32 + 2p q (2 + 2/.t)(3;t, + 2p) 8# {Nk(Mi- Mj) - M k ( N i -
Nj)}
2
+ (p + q)(Mi - Mj) - (p - q)(Ni - Nj)
§
9
[Nk(Mi
_ Mj) - M k ( N i -
Nj)] 2 }
(10)
1 ( N i - Nj) z - ( M i - Mj) z
This expression is of course valid only if - 1 < cos 20 < 1. The critical value hc of h at localization corresponds to the normal n leading to the greatest value of h. Using Eqs. 8 and 10, one can determine the domain of validity of each of the expressions hp and hp_q, p, q - 1, 2, 3. These results, too long to be reported here (see [5] for more details) are summarized in Table 4.11.1.
4.11.3.2 N AND ~/~ HAVE BOTH A DOUBLE EIGENVALUE Let (i+, j+, k+) and (i-, j - , k-) be two triplets of distinct numbers belonging to the set (1, 2, 3). When N and ~ have both a double eigenvalue associated with the principal directions xi+, xj+ and xi-, xj-, we have Ni+ = Nj+---1/2Nh+ and Mi+ = Mj+ = -1/2Mh+. The domains T+ and T_ degenerate to two segments, the equations of which are, respectively, 1_ _2 -- 0and T+ - vNk+E+ zl T_ ---~Mk-E-1 - ~(Mk-) - 1 (Nk+ )2 -- 0
(11)
and Eqs. 5 and 6 using also Eq. 11 allow are to compute only ,,2
)
2s 1 3Nk+ t-~(nk-
-2
2 E_ 3Mh_
1 3
(12)
When the directions Xk+, xk- coincide, it is still possible to compute via Eqs. 11,8 and 12 T_ and s as functions of T+ and s only (or T+ and s as functions of T_ and s only ) since one has s - -Mh-/211 - 3(nk+)2], T_ = (Mh-)2/411 + 3(nh+) 2] and similar relations for T+ and s Therefore, we can work, for instance, in the (T+, s plane and with the same arguments as previously stated; it is found that localization will occur when C+ touches the
4.11 1 Cntlcal condit~onsat localization for a + b
TABLE
+
# 0 and a - b # 0
w
Normal n to the localization planes Criucal value of h
M, distinct or N , d ~ s t ~ nor ct 22N, = *-a+ and MJ = f-a3 3 (1
A, 2 0, B Z 5 0. Bjt 5 0 A,@; 2 A,~BI, h,, L AJkB$ + 4A,k(Btk 1) 0 A, > 0, A,k > 0 B,, 2 0, B,k L 0 or A,, > 0, A,k i 0 B,, 2 0, A,k(B,, 1) L 0 or A , 10, A,k I 0 B , 5 - 1 , Btk 1 - 1 A@:, + 4Akt(Bkt 1) 5 0
#J )
2_ Nk = &-a+ 3 2M~ = &-a3
h,
n, n E n,,
+
* A,,
diMk
p{(Mh
-
2 =
(N, - N )
~f A ,
=0
n~ n,-k ~fA,k = 0
n, are the common principal d~recnonsof a
+
+ b and a - b, T,,
IS
the plane spanned by n, and nl. T,IS the cone with axls n, and angle
-
- Z ( P - 4))(Mz - M,l - 2(Nk - Z(P + q))(N, - NI)] 2 ( ~ P)(N, - N,)' - (M, - M,)']
+
Z(P - q)-)(MI - Mi) - (Nk - 2 ( p + q ) ) ( N , - NJ]} xi 2(2 + P)(N, - N,)' - (M, - MI)']
-
(M,
-
Mi)', B -
:-'
&
z
r
G n~ r ,
nt-j
+
2
3
fElr
?L
+
*
M, = &-a2-
2-
N, = f - g + 3 2MI = i-o3
-i
+ >
=
23
N, = f - o +
0 {(Mh - 2 ( p q I ( M , A,,
- 4 ) - (Nk - 2 ( p + q ) ) ( N , N I ) )
-
1
5 glven by:
nE
r,
280
Benallal
segment T+ at one of its extremities or becomes tangent to it. It is concluded that the normal to the localization plane is either the common principal direction associated with the simple eigenvalue (when the tangency point is (T+ = 0, ~+ = Nk+), or an arbitrary direction in the common principal plane associated to the double eigenvalue ( when T + - - 0 , 2E+ = Ni+ = Nj+), or a direction belonging to the cone the axis of which is the common principal direction ( tangency to the segment T+ but not at its extremities) with angle ~ such that (cos ~ ) 2 = (nk+)2 = (nk_)2, the right-hand sides being computed by Eq. 12 with E+ and E_ corresponding to the tangency point of the curve C+ with the segment T+. The corresponding values of h are still given by Eqs. 8 and 10, and the critical conditions are also summarized in Table 4.11.1. W h e n the directions x/+, Xk- do not coincide, it is not possible to work in the (T+, E+) plane or in the (T_, s plane. The alternative is to use Eq. Eq. 11, compute T+ and T_, and upon substitution in the localization condition, end up with an equation involving only E+ and E_. A similar geometrical analysis can then be performed in the plane (E+, s where the localization condition is seen to be represented again by a hyperbola H. The admissible area in this plane.is easily obtained by regiving that the components of the unit normal to the localization plane nk-, nk+ (given by Eq. 12) and the third one (nk-k+) - - 1 - - ( n k + ) 2 - - ( n k - ) 2 along the axis orthogonal to both xk+ and Xk-, be comprised between 0 and 1, i.e., 2]E+ 1 2 E_ 0<+ < 1 0<---+ --3Nk+ 3 - - ' --3Mk-
1 1 < 1 0< 3-- ' --3
2E+
3Nk.
2 E_ < 1 3Mk---
(13)
This is again a triangular area and localization occurs when the hyperbola H meets this domain at one of its vertices or becomes tangent to any of its sides. In this case, it is concluded that the situation is exactly the same as when N or M has distinct eigenvalues and all the results of Section 4.11.1 hold true.
REFERENCES 1. Rudnicki,J. W. and Rice,J. R. (1975). Conditions for localization of deformation in pressuresensitive dilatant materials. J. Mech. Phys. Solids 23: 371-394. 2. Rice,J. R. and Rudnicki, J. W. (1980). A note on some features of the theory of localization of deformation. Int. J. Solids Struct. Vol. 16: 597-605. 3. Borre G. and Maier G. (1989). Meccanica 24: 36-41. 4. Bigoni, D. and Hueckel, T. (1991). Uniqueness and localisation. I. Associative and nonassociative elastoplasticity. Int. J. Solids Struct. 28: 197-213. 5. Benallal, A. and Comi, C. (1996). Localization analysis via a geometrical method. Int. J. Solids Struct.
SECTION
4.12
Gradient Plasticity ELIAS C. AIFANTIS Aristotle University of Thessaloniki, Thessaloniki, 54006 Greece, and Michigan Technological University, Houghton, Michigan
Contents 4.12.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.3. Elementary Gradient Plasticity Models . . . . 4.12.3.1 Gradient Deformation Model 1 (Symmetric Stress [2,3,23]) . . . . . . . 4.12.3.2 Gradient Deformation Model 2 (Asymmetric Stress [17,18,25]) .... 4.12.3.3 Gradient Flow Model 3 (Symmetric Stress [2,3,23]) . . . . . . . 4.12.3.4 Gradient Flow Model 4 (Asymmetric Stress [17,18,25]) . . . . 4.12.3.5 Generalizations . . . . . . . . . . . . . . . . . . . 4.12.4 Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.5 Forming Limit Diagrams (FLDs) . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281 282 283 283 284 285 286 287 288 293 294 296
4.12.1 VALIDITY W h e n the scale of observation and the resolution of the experimental apparatus are comparable to the size of the evolving microstructure, conventional plasticity theory may not be capable of describing plastic flow and capturing the associated heterogeneity and scale effects. Typical examples of questions that remain u n a n s w e r e d or partially answered by standard models, including deformation, flow, and internal variable theories of plasticity, are (i) the determination of widths and spacings of shear bands; (ii) the determination of velocities of Liiders and PortevinChatelier bands; (iii) the d e p e n d e n c e of yield stress in torsion and b e n d i n g on Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
281
282
Aifantis
specimen size; (iv) the dependence of yield strength of composites on particle reinforcement size; and (v) the dependence of hardness on indenter dimensions, as well as the dependence of interracial failure of multilayer coatings on film thickness. These questions can be addressed by modifying the yield or flow stress of classical plasticity theory to include additional strain gradient terms to model phenomenologically the heterogeneous character of plastic flow. The newly introduced phenomenological coefficients may be interpreted on the basis of the underlying microscopic mechanisms of plastic deformation, and they can also be determined independently from mechanical tests especially designed to measure spatial characteristics of shear bands and size effects.
4.12.2 BACKGROUND The interest in higher-order gradient theories has been revived recently among researchers in solid mechanics and materials science because of the ability of the higher-order terms to model phenomena not previously captured by standard elasticity, plasticity, and dislocation theories that do not involve an internal length scale in their constitutive structure. An account of current work on gradient theory at various scales of observation ranging from nano and micro scales to meso and macro scales can be found in a recent review by Aifantis [1] and references quoted therein. As first discussed by Aifantis [2,3], the initial motivation for introducing gradient theory was the need for determining the width of shear bands and the wavelength of persistent slip bands, as well as the problem of mesh-size dependence in finite element solutions when the material enters into the strain softening regime. These issues were later considered in detail by Aifantis and coworkers [4-7], as well as by others [8-16] who produced variants of the initial model and elaborated upon numerical [8,9,11,15] and thermodynamic [12-16] aspects. Another set of problems pertaining to strain gradient hardening and size effects in torsion and indentation was identified by Fleck et al. [7,8], who introduced another gradient plasticity model to interpret the related experimental observations. This model or variations of it were adopted by a number of investigators [19-22] to consider plasticity problems at the micron scale and to interpret size effects observed in particle reinforced composites and indentation tests. More details on the aforementioned two basic gradient plasticity approaches can be found in the review articles by Aifantis [23,24] and Fleck/Hutchinson [25]. It is emphasized that even though both of these approaches are based on a gradient modification of a scalar equation (the yield
4.12 GradientPlasticity
283
condition) by incorporating gradient effects through the second invariant of the plastic strain tensor (effective or equivalent strain), the stress tensor remains symmetric in the first approach, and it becomes asymmetric in the second one. This is mainly because gradient effects are directly introduced through gradients of plastic strain in the first approach, and through gradients of plastic rotation in the second one. Physically, this was motivated by strain incompatibilities associated with necessary geometric dislocations in the second approach, while it can be attributed to inhomogeneous strain transport through mobile dislocations in the first approach. It is beyond the scope of this paper to elaborate on the advantages and disadvantages of the mathematical structure of the two theories, the associated boundary conditions, and the experimental determination of the gradient phenomenological coefficients. However, gradient plasticity theory is still at its beginnings, and one should not expect a general model to be invented for use in the vast number of applications at the micron and nano scales. In this connection, the point of view is advanced that a variety of gradient models may be used according to the particular situation at hand; in analogy, for example, to the variety of standard constitutive models that are commonly employed in conventional applications where heterogeneity and scale effects are not of interest.
4.12.3. ELEMENTARY GRADIENT PLASTICITY MODELS The deformation and flow counterparts of the simplest form of the previously mentioned two types of gradient plasticity theories involving symmetric or asymmetric stresses are briefly reviewed in the following text. The notation was purposely kept as close as possible to that of the original works; but sometimes different symbols were used to unify the presentation.
4.12.3.1
GRADIENT DEFORMATION MODEL 1
(SYMMETRICSTRESS [2,3,23]) This simple form of gradient modification of the deformation theory of plasticity involves the incorporation of the Laplacian of the effective or equivalent plastic strain in the expression for the effective or equivalent stress, i.e., = ~c(~) - cK72~
(1)
Aifantis
284
where the equivalent stress F and the equivalent plastic strain ~ are defined, as usual, by the relations F ~ .2S,jS,j, . . . . . e EijEij , with ~c(~)denoting the usual homogeneous part of tlow stress, c being a phenomenological gradient coefficient, Sij being the deviatoric stress tensor, and Eij denoting the deviatoric strain tensor which, for incompressible plastic deformation (ekk = 0), equals the strain tensor gij. By assuming the equivalent work condition Sij &ij = F6~, we obtain from the previous equations the following gradient-dependent stress-strain relations: -
2F
2
(2)
which should be supplemented by appropriate boundary conditions for the solution of boundary value problems.
4 . 1 2 . 3 . 2 GRADIENT DEFORMATION MODEL 2 (ASYMMETRIC STRESS [ 17,18,25]) In contrast to the previous "symmetric stress" gradient deformation theory of plasticity, Fleck et al. [17,18,25] have proposed a Cosserat-type "asymmetric stress" strain gradient plasticity theory. In this theory the stress tensor is aij 4-zij, with aij denoting the symmetric part and zij denoting the antisymmetric part ; the relevant equilibrium equations are (aji 4- ~ji),j = 0 and ~jk 4- ~eijklmpi,p --O, with mij denoting the deviatoric part of the couple stress tensor, whose hydrostatic part does not enter into the equilibrium equations and is therefore taken to vanish. In analogy to the previous symmetric stress model, the equivalent work condition is postulated in the 1 4- Uj,i), with ehh -- 0 form aw -- SijtSEij 4- mijtS)(.j i -- ~.,tS~, where Eij -- 8ij -- ~(ui,j denoting the usual strain tensor and Zij =
2K(~)
(3)
with s V/t~2 + L'-2rh2, # - v/3SijSij/2, and r h - v/3mijmij/2. With appropriate extra boundary conditions (e.g., for m/j), boundary value problems can be solved.
4.12 GradientPlasticity
285
4.12.3.3 GRADIENT FLOW MODEL 3 (SYMMETRIC STRESS [2,3,23]) This simple gradient modification of the flow theory of plasticity (the flow counterpart of Model 1) involves the incorporation of the Laplacian of equivalent shear plastic strain 7 (note that for convenience 7 is used instead of g in accordance with the notation previously adopted in related literature) in the yield condition, i.e., F -- ~ - (K(7)- cV2?) - 0
(4)
where the equivalent shear stress z and equivalent shear strain rate ? are defined as usual by z - v/xsijsij;? - fgdt; 3)- ~/2~k~. The corresponding associated flow rule reads
(5)
F_,ij-- ?~Tj~ij:=>8ij
The relevant elastic strain increment k~e -- ~.ke -- ~e is determined by Hooke's law, which may be written in the form dij=C~jke(~.ke--~r C~jke -- 23ij3ke + G(3ik@ + 3ie3jh),with (2, G) denoting the Lam~ constants. The plastic multiplier 9 in the flow rule given by Eq. 5 satisfies the following loading-unloading conditions: _
f 3)if F - 0 & a/j~ > 0 (loading) 0 if F < 0 (elasticity)
or a/jk/~ - 0 (neutral loading)
.p
or F -- 0 & a/jg/j < 0 (unloading)
(6)
and is determined by the consistency condition/~ = 0, which eventually gives
9 -- V29 -- -~ C ijkeF,k(
(7)
where H - - h + SijC~keSke/4"c 2 - h + G and h - c~c(?)/c~7. Equation 7 is a differential equation for 9 in contrast to the classical plasticity case (c = 0) where 9 is determined from an algebraic equation. Various procedures have been developed for evaluating Eq. 7 in relation to the solution of elastoplastic boundary value problems. For example, in the method of Muhlhaus and Aifantis [5], 9 is treated as an additional independent variable and Eq. 7 as an additional field equation, whereas necessary extra boundary conditions are deduced from an appropriate variational principle. To this end, the following generalized incremental potential functional ~'(1~i, 9)is defined
lfB (
.Sij"~ e (
.Skr
~,
286
Aifantis
1 {h72+ cVg. Vg} dV. The applied tractions ti act on with ~ [ 9 1 - 2fB surface ~tB and displacement-type boundary conditions are prescribed ~uB= ~B-~tB. By assuming that for 9 > 0, ~ - = 0 at equilibrium arbitrary infinitesimal variations of [fi, 9] and with 66i = 0 on GB, following relations are derived:
the on for the
f (rij'j~f'lidV-J~t[B[~ijllj--ti] (~(,lidA-0 JB fBfSij e 9 H9 q-cV29} ,~9d V ~-~-~Cijkfgk~,-
0
(9)
{cVT} 9n69 clA - 0 => ~nn - 0 or 69 - 0, on ~epB
f~epB
It is noted that Eq. 9 (first equation) leads back to the stress equilibrium relations and the standard traction-type boundary condition; the second equation leads back to the consistency condition; the third leads to a nonstandard boundary condition along the elastic-plastic boundary ~epB. The second of the conditions in the third equation is automatically satisfied on ~ B . These equations provide the basic framework for the corresponding finite element implementation as followed, for example, by de Borst and Muhlhaus [9].
4 . 1 2 . 3 . 4 GRADIENT FLOW MODEL 4 (ASYMMETRIC STRESS [ 1 7 , 1 8 , 2 5 ]) For completeness, we provide a condensed summary of this model (the flow counterpart of Model 2) for which the starting point is the replacement of the standard deviatoric Cauchy stress by a 13-dimensional stress vector s = (S O, ~e-lmji) comprising the five components of the deviatoric symmetric stress tensor Sij and the eight components of the deviatoric couple stress tensor g-lmji, as well as the introduction of a 13-dimensional plastic strain rate vector t~p - (~, t'~/~)comprising the five components of the plastic strain rate tensor eij"P and the eight components of the plastic curvature tensor t'~/~. The yield condition reads (10)
~)(~, r ) = x - r = o
where
Y
denotes
v/3SijSij +3vg-Zmijmij-
/..
the uniaxial flow stress and Z - X / ~ Z . Z V/G2 q--~'-2/'~12 is the overall effective stress. The
4.12
287
Gradient Plasticity
associated flow rule reads l~p_
1 a~. h(E) c~Ex
(11)
with the hardening rate h being chosen so that the uniaxial homogeneous tensile response is reproduced. The plastic work rate per unit volume is as before ivP - Siji?,p --[-mjixijp9 _ E . 1~p - Z/~v where the overall effective plastic strain rate is defined as E.P- E / h - v/}E v. EP ~/}~eij.p + ~ ez~p~p~j~j_ _
V/(~p)2 + ( ~ ) 2 , -
where ~p - V/2~iij "peij is the effective plastic strain rate and
~Xij ~j is the effective plastic curvature rate. It then turns out that the
flow rule Eq. 11 can be decomposed into the following equations: .p __ 3 Sij f~ eiJ -- 2h Z,
where
the
overall
effective
3 S0 3 ~g-lmji ...--~Sij +-~ ~ ~ - l m j i .
.
ef(p _
3 e-lmji f~
2h stress
rate
(12)
Z, ~; can
be
obtained
as
The stress rate is related to the elastic strain
rate through a higher-order elasticity theory, but further details on this issue are omitted for brevity.
4.12.3.5 GENERALIZATIONS Various generalizations or variations of these models have been proposed by the developers of these theories and their coworkers, as well as by other authors [26,27]. The purpose of the present introductory paper does not justify a review of them. Nevertheless, a few remarks are useful for the subsequent sections of the paper. We note that a straightforward generalization of the symmetric stress model is r - ~c(7) - c* (V?. VT)~-cV27
(13)
which was suggested at the same time as the original symmetric stress gradient theory. It is noted that two gradient coefficients are now introduced to incorporate the effects of both first and second strain gradients, and the coefficient m is usually taken equal to 1. Similarly, the asymmetric stress model has been generalized to include both stretch gradients and curvatures through, for example, the relationship 22 F 2 -- -~EoEij -4- f2.(1).(1) l'lijk 'lijk _Jr_tPcs.~)(,ij)(,ij
(14)
288
Aifantis
(3) refer to the orthogonal decomposition of where l]ijk - - llk,ij - - ~a(1) 'lijk -~- •(2) 'lijk + 1]ijk the displacement gradient [25] and (/~1,~cs) denote two independent internal lengths. The corresponding stress components are then obtained from the relations Sij - ~w(~)/~Eij, mijh -- ~w(~)/~]l) ). In this connection, it is pointed out that a recent elegant variant of the asymmetric stress model has been advanced by Gao et al. [21,22]. This is based on an expression for the effective stress of the form F = cryv/f2(~) + ~rT,where F - - ayf(~:) for the nongradient case, ~ = I/2~.ijF~+j, and r7 - 1 -~V/~lijh~lijh. The /
hardening length scale is obtained as r ~ 0.5(G/ay)2bVJ ~ 5 l-tm, with G denoting the shear modulus and b the magnitude of Burger's vector. The previously listed flow stress expression and the resulting model of mechanism-based gradient plasticity (MSG) is motivated by Taylor workhardening theory and recent indentation experiments [19], and rests upon certain assumptions to relate the deformation behavior between "microscale" and "mesoscale" material elements. It follows from the variety of strain gradient plasticity models discussed in this section that the search for a unified gradient plasticity theory may not be a reasonable task to undertake. Rather, different models may be used according to the particular application at hand. The form of these models should not only depend on the underlying physical mechanism of plastic deformation but also on the mathematical simplicity and robustness of numerical implementation. Accordingly, different internal lengths may be involved, and the associated gradient coefficients need to be determined from appropriate experiments. This view is further supported in the following two sections, where we focus on the symmetric stress gradient plasticity theory only and illustrate how it can be used to interpret size effects (Section 4.12.4) and to derive forming limit diagrams (Section 4.12.5).
4.12.4
SIZE EFFECTS
In this section we elaborate on the interpretation of size effects in torsion and bending of solid bars by using a gradient modification of the strength of materials approach. This modification amounts to employing a gradientdependent flow stress of the type of Eq. 13, i.e. "~ = /~(~) + C1(~7~ 9~7~) 1/2 -- C2~72~
(15)
The rest of the hypotheses (i.e, strain distribution, definitions of equivalent stress and strain, Hooke's law for the elastic strains, and moment vs. stress distribution relation) remain the same as in the standard mechanics of materials approach. Then, on the basis of Eq. 15 with ~c(?)= z0 ~ 0.5a0 to
4.12
289
Gradient Plasticity
denote a perfectly plastic behavior for the homogeneous response, it turns out that an explicit expression for the dependence of the initial yield stress Y (first occurring at the outer surface of the bar where the elastic-plastic boundary first appears; Y = GTswith G denoting shear modulus and 7s surface strain) on the size (radius ~ or height h) of the specimen's cross-section is obtained as follows:
(
r ( ~ ) - % ~2 + (c2/G)
)
(c]/G)e
_
(
)
Y f h ) - ao ( h / 2 ) h--/2
'
(16)
for torsion and bending, respectively. Figure 4.12.1a,b shows the fitting of experimental results obtained for the size dependence of the yield stress Y on the radius ~ of cylindrical bars subjected to torsion (J. L. M. Morrison, Proc. Inst. of Mech. Eng., vol. 142, pp. 193-223, 1939) and on the height h of rectangular beams subjected to bending (C. W. Richards, Proc. Am. Soc. Testing Mats., vol. 58, pp. 955-970, 1958). As already indicated, 0:o, ao) denote the yield stress in shear and tension for homogeneously deforming specimens. Also, in the case of bending, r is replaced by the axial stress a, ~c(7) is replaced by a0, and Cl is replaced by ~1, which measures the gradient effect of the axial strain e with E denoting Young's modulus; i.e., a superimposed bar is added to the gradient coefficient when the constitutive equation is expressed in terms .
0.62
,
9
,
.
,
.
,
-
,
9
,
.
Y(h) 360 (MPa)
c,/G =0.38 mm
Y(oQo.6o
x/c2/G =0.476 mm
Go
o~
cl/E=1.28mm
320
0.58
2801
0.54 240!
o 0.52
i
0
(a)
,
2
9
,
4
9
,
,
6
8
,
9
|
9
'
10
14
(x (mm)
(b) .
M/~)~3
' ,'0 ' ; ' ~0 h (mm)
-
12
,
.
,
.
,
.
,
.
,
.
,
.
,
2~
.
600
(MPa) 400
200 2e~ = 2 0 g m 2~t
[
3O g i n [
=
20~ = 1 7 0 g n ~
(c)
%'.0
i
'
0.2
i
'
0.4
,
i
0.6
.
.
.
0.8
.
.
,
1.0
1.2
9
,
1.4
9
9
1.6
7s
FIGURE 4.12.1 Quantitative comparison between theory and experiment: (a) Yielding behavior in torsion. (b) Yielding behavior in pure bending. (c) Hardening behavior in torsion.
Aifantis
290
of • instead of 7. The effect of the c2-term vanishes identically in the case of bending. Similarly, for the interpretation of the recent experimental results on increased torsional hardening with decreased wire diameter reported by Fleck et al. [17], Eq. 15 is replaced by z = ~0 + ky"+ ~yn-l~72]j; i.e., a powerhardening behavior is assumed for the homogeneous response and a similar behavior is assumed for the gradient coefficient, which is now straindependent. The corresponding expression for the size-dependent torque (M) vs. surface shear strain (7s) reads M 2re ---~=
+
7s +3
1 + n+
1~2
;1~---'e-
(17)
The fitting of the experimental data by this expression is shown graphically in Figure 4.12.1c. The values of the standard parameters n and k are obtained from tension data as n = 0.23 and k = 117 MPa. The value of the internal length parameter g is 5.1~tm for the (170, 30, 15)-~tm-diameter wires, and 4.1 l.tm for the (20, 12)-~tm-diameter wires. The value of the initial stress z0 is zero for all cases except for the 12-~tm-diameter wire for which z0 ~ 58 MPa. The corresponding internal length estimated by Fleck et al. through their asymmetric stress strain gradient theory is of the order of 2.6-5.1 ~tm. These findings, in themselves, suggest that a lot more work is required to determine the gradient coefficients, their possible dependence on strain, and the most convenient form of gradient dependence of the flow stress on plastic shear strain (and possibly its rate) to be used in particular applications. In connection with these findings, reference is made to a recently published article by Stolken and Evans [20] where the asymmetric stress strain gradient plasticity model of Fleck and Hutchinson is used to interpret the observed size effect in bending of thin foils. This problem can also be discussed by using symmetric stress strain gradient models of the type of Eq. 15. In fact, by employing a gradient-dependent strain energy density formulation which allows a direct comparison of the aforementioned two approaches, one may consider the form w - w(e)+ Cl]~7/3[-]-62[~7e] 2 where w ( e ) - e[3Epe + 4 X ~ o _ ] / 8 is the homogeneous part of the strain energy density, with g denoting the equivalent strain, ~c being the curvature, and (Cl,C2) being gradient coefficients. The assumed expression for the homogeneous part of the strain energy density is the same as in Stolken and Evans [20], with Ep denoting plastic modulus (hardening coefficient) and E0 effective yield strength. Finally, by using the relation M = dW/dtr ( W = fA wdA) for the applied moment, we can obtain a moment (M) vs. surface strain (es) relation depicted with solid lines in Figure 4.12.2a. The dotted lines correspond to the prediction of the Fleck-Hutchinson theory
(2/x/3) ly I
(a) Fitting of Stolken and Evans [20] thin foil bending experimental results with the symmetric stress and FIGURE 4.12.2 asymmetric stress strain gradient models. (b). Calculated yield stress (0.2% offset) vs. particle size for an Al-Si-Mg metal matrix composite, showing gradient plasticity solutions for different gradient coefficients c, and for a dislocation-based model. The values off indicate particle volume fraction. From Zhu, Zbib, and Aifantis 1281.
292
Aifantis
employed by Stolken and Evans [201 for a value of their internal length lc = 6.2 gm. There is a departure of this prediction from the one reported in the aforementioned article of Stolken and Evans due to an error in their procedure when rescaling the moment and computing the corresponding theoretical values. The internal lengths ll and 12 appearing in Figure 4.12.2a are defined by ll = Cl/~o and 12 = v/c2/Ep. A more rigorous (as contrasted to the previously described mechanics of materials approach) boundary value problem approach implemented by finite element analysis has been used by Zhu, Zbib, and Aifantis [28] for interpreting size effects in metal matrix composites. It was found that the strength of metal matrix composites decreases with increasing particle size of reinforcement under constant volume fraction for all three materials studied: A1-Si-Mg, A1-A1202, and A1-TiB2. The results of the gradient theory were in agreement with both existing experimental data and available microscopic dislocation models for these materials, as shown in Figure 4.12.2b. The solutions are cumbersome and were evaluated numerically by employing a gradient plasticity model based on Eq. 1. In concluding this section on size effects, reference is made to an open problem of increasing current interest: namely, the determination of hardness and related mechanical properties from indentation tests where the size of the indenter varies from the micrometer to the nanometer regime. A deeper understanding of the mechanics and physics associated with the contact and penetration phenomena in such small volumes is important from both the scientific and technological points of view, with implications ranging from device miniaturization and computer disk drive manufacturing to magnetic recording and tribological effects on piston wear. Quite interesting phenomena, including discontinuous yielding and sizedependent hardness, have been reported recently in such small-volume regimes. Size effects, in particular, have been observed as the indenter diameter D is reduced from 20 gm to about 2 I.tm. The gradient dependent form of the flow stress given by Eq. 15 can be utilized, in principle, to provide an explanation for the dependence of hardness on the size of the indenter. By taking H to be proportional to the yield stress a(H ,~ 3a; Tabor's relation), we have [29] H = H0 +
c1]V~)]-s
(18)
where H0 is the hardness in the absence of gradient effects. By approximating y with an average value equal to 2h/D (where h is the indenter's depth and D is the equivalent indenter's diameter) and using the rough estimates Vy ,-, 2y/D, V27 ~ 4y/D 2, we can obtain from Eq. 18 a plot of H vs. D where the values of the gradient coefficients Cl and c2 are adjusted to fit the experimental data reported, for example, by Nix and Gao [19]. The results are
4.12
293
Gradient Plasticity
FIGURE 4.12.3 Fitting the experimental results of size-dependent hardness H vs. D according to the gradient-dependent constitutive equation Eq. 18 for (a) Cu (111) and (b) Ag (110) single crystals.
shown in Figure 4.12.3 with the parameters (H0, C1, C2) taking the values (0.595 GPa, 2500N/m, 0.414 10 -3N) for Figure 4.12.3a and (0.37GPa, 853 N/m, 0.164 10-3N) for Figure 4.12.3b.
4.12.5 FORMING LIMIT DIAGRAMS (FLDS) In this section the applicability of gradient plasticity theory to predicting spatial instability phenomena in manufacturing-related processes at the macroscale is presented. In particular, it is shown how gradient theory can help in improving the construction of forming limit diagrams (FLDs) for sheet metals. Some initial results on this topic were reported by Aifantis [29], and a detailed treatment can be found in a recent doctoral dissertation by Zhu [301. A physical picture of localized neck formation in sheet metal forming is depicted in Figure 4.12.4. A central goal is to avoid the occurrence of necking by operating in a safe regime of the applied strains during the forming process. This leads to defining corresponding FLDs. For sheet metal forming processes, a vertex-type theory of plasticity of the form [3] D ij =
7 r - 7~ , 2--~i
7__6.!.
(7iJ -k 2. c ,j
(19)
may be used to describe the deformation. The quantities (Do, aij) denote the strain rate and stress components, (1:,7) denote, as usual, the equivalent shear stress and shear strain, a dot denotes time differentiation, and a prime denotes deviatoric component. A gradient-dependent flow stress expression is
294
Aifantis
FIGURE 4.12.4 Localized neck formation in sheet metal forming (a-c). Schematics of the sheet/inclined neck and the forming limit diagram (d,e).
assumed, with its rate form given by = h~ - cV2~
(20)
with the hardening modulus h (h=~K(7)/~7, K ( 7 ) = h 7 ") given by h = hn7 n-1. For plane stress conditions (rT33 = 0) and proportional loading (fl = D22/D11 ~-const) a standard bifurcation analysis gives the following expressions for the shear band (or localized neck) angle Oc~ and the critical strain 7or: 2nf(fi) 1 + fl
(21)
2fl 2 + n(2 + fl)2 (2+fl)f([3)
(22)
fl < 0 =~ Oc,. - a r c t a n x / ~ ;
ilK0 =~ 0 o r - 0;
7c~ =
7~ -
where f ( f l ) - V/1 + fl + f12. These relations hold for the case c - 0, while more complex results are obtained for the case c 7~ 0, which, however, do not greatly affect the predictions of localized neck orientation. The obtained FLDs which correspond to the two cases c = 0 and c ~ 0, are given in Figure 4.12.5. More details on this topic, the use of Hill's anisotropic yield criterion, including gradient terms, and the effect of strain path can be found in the aforementioned doctoral dissertation [30].
4.12
295
Gradient Plasticity
140 C:
opa,'mized band angle ~'~Ex
0.40
.i~rimental
"-,
.~-~
(/~
results
t
so
~
40
-
20
-
Storen and Rice' result
~
!
......... I
,
I.
-0.21)
~0
.
.0.10
i
I
g.OO,
(UO
Minor
..
I
,.
!
0.31
,.
030
- ~=
|
"
I
'
~
-~
,20
-~t.
o
,oo
-ols
~
1
'
\
I
I 9
I"
I
9I
,
I
'
I
" I
"
I
: I --'r.~,r,,,~.,
d
=" o~b-_
~
;: c q 2 = 3 O M P a
-
.
A~A~
~ I , I , 0
10
20
| 30
minor
.$
~="re~
.
I~,00
9
O0 0 ~
,
i
,
0
"~
80,00
-~.~
40.II)
,
minor
,
60
40
strain
o~b-
9 I , 40
i
| 50
60
strain 9
I
"
Io 9 O
Acceptable Afftsct~d
9
o
- -
t I+. ~
0
~
20
_._1
Fractured Th eoreUeal
0oB
/
"
20.00 20.00
1
10.~
(d)
TheoreUcal
"
-!
140.00
Ao%
"
Fructured
120.00
'
I
Aff,~ct~d
[
i
"
'1"
AJcceptable AfY,~ctod
'
~
.~, 40.00 ~a~ 30.00
Fractured - -
m
Engineering
e ~ 50.00
"~
I
A~celPtable Aff,~ctod
9
~
-20
: I 9
Engineering '
.
I
:
-40-30-20-10
Io
60.00
"
-\
0
(c) '
I
-
140
~'~
, -40
:1o
60 --
l
'
.
(b) "
'6~
70.00
O
!o
o~.~~='"
O
0.~
Strain 180
80.00
I
o~=,i
|
-~
L
AIR
O C~" O
"~
'
l
--
fl.lO
0~
!
120
-
o~o
(a)
I Ak
zero ~xtension direction
30.00
,
I
-20.00
,
,
-10.00
,
0.00
,
I
I0.00
......
I
~.00
,,,
Minor engineering strain
I
30.00
,
0.00
40.(10
,
,
I
-20.00 ~
(e)
,
' II
0.~
,
I
20.00
,
I
10.00
,
I
60.00
,
1
80.(I
Minor engineering strain.
FIGURE 4.12.5 Predicted and experimental FLDs for (a) Aluminum 2036-T4, c = 0; (b) AKsteel, c = 0; and (c) AK-steel, c r 0. Comparison [301 of predicted and experimental results (Hecker 1975) of FLDs for (d) 2036-T4 Aluminum and (e) AK-steel using Hilrs anisotropic yield criterion.
296
Aifantis
ACKNOWLEDGEMENT The support of the General Secretariat for Research and Technology of Greece, the TMR Program of EU, and the U.S. National Science Foundation is gratefully acknowledged. The help of my graduate students X. Zhu and I. Tsagrakis with the fitting of some of the experimental data and of my undergraduate student D. Dodou with the typing and proofreading of the manuscript is also acknowledged.
REFERENCES 1. Aifantis, E. C. (1999). Gradient deformation models at nano, micro and macro scales. J. Engng. Mat. Tech. 121: 189-202. 2. Aifantis, E. C. (1984). On the microstructural origin of certain inelastic models. J. Mat. Engng. Tech. 106: 326-330. 3. Aifantis, E. C. (1987). The physics of plastic deformation. Int. J. Plasticity 3: 211-247. 4. Zbib, H.M. and Aifantis, E. C. (1989). A gradient-dependent flow theory of plasticity: Application to metal and soil instabilities. Appl. Mech. Rev. 42: 295-304. 5. Muhlhaus, H. B. and Aifantis, E. C. (1991). A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7): 845-857. 6. Vardoulakis, I. and Aifantis, E. C. (1991). A gradient flow theory of plasticity for granular materials. Acta Mech. 87: 197-217. 7. Zbib, H. M. and Aifantis, E. C. (1992). On the gradient-dependent theory of plasticity and shear banding. Acta Mech. 92: 209-225. 8. Belytschko, T. and Lasry, D. (1988). Localization limiters in transient problems. Int. J. Solids Struct. 24: 581-597. 9. de Borst, R. and Muhlhaus, H. B. (1992). Gradient-dependent plasticity: Formulation and algorithmic aspects. Int. J. Numer. Meth. Eng. 35: 521-539. 10. Vardoulakis, I. and Frantziskonis, G. (1992). Micro-structure in kinematic hardening plasticity. Eur. J. Mech. A/Solids 11: 467-486. 11. Tomita, Y. (1994). Simulations of plastic instabilities in solid mechanics. Appl. Mech. Rev. 47: 171-205. 12. Maugin, G. A. and Muschik, W. (1994). Thermodynamics with internal variables, Part I: General concepts. Part II: Applications. J. Non-Equil. Thermodynamics 19: 217-249, 250289. 13. Valanis, K. C. (1996). A gradient theory of internal variables. Acta Mech. 116: 1-14. 14. Polizzotto, C. and Borino, G. (1998). A thermodynamics-based formulation of gradientdependent plasticity. Eur. J. Mech. A/Solids 17: 741-761. 15. Menzel, A. and Steinmann, P. (2000). On the continuum formulation of higher gradient plasticity for single and polycrystals. J. Mech. Phys. Solids 48: 1777-1796. 16. Gurtin, M. E. (2000). On the plasticity of single crystals: Free energy, microforces, plastic strain gradients. J. Mech. Phys. Sol. 48: 989-1036. 17. Fleck, N. A., Muller, G. M., Ashby, M. E and Hutchinson, J. W. (1994). Strain gradient plasticity: Theory and experiment. Acta Metall. Mater. 42: 475-487. 18. Fleck, N. A. and Hutchinson, J. W. (1993). A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41: 1825-1857.
4.12
Gradient Plasticity
297
19. Nix, W. P. and Gao, H. (1998). Indentation size effects in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46: 411-425. 20. Stolken, J. S. and Evans, A. G. (1998). A microbend test method for measuring the plasticity length scale. Acta Mater. 46: 5109-5115. 21. Gao, H., Huang, Y., Nix, W. D. and Hutchinson, J. W. (1999). Mechanism-based strain gradient plasticity - - I. Theory. J. Mech. Phys. Solids 47: 1239-1263. 22. Huang, Y., Gao, H., Nix, W. D. and Hutchinson, J. W. (2000). Mechanism-based strain gradient plasticity - - II. Analysis. J. Mech. Phys. Solids 48: 99-128. 23. Aifantis, E. C. (1992), On the role of gradients on the localization of deformation and fracture. Int. J. Engng. Sci. 30: 1279-1299. 24. Aifantis, E. C. (1995). Pattern formation in plasticity. Int. J. Engng. Sci. 33: 2161-2178. 25. Fleck, N. A. and Hutchinson, J. W (1997). Strain gradient plasticity, in Advances in Applied Mechanics, pp. 295-361, Hutchinson, J.W., and Wu, T.W, eds. 26. Acharya, A. and Bassani, J. L. (1996). On non-local flow theories that preserve the classical structure of incremental boundary value problems, in IUYAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials, pp. 3-9, Pineau, A., and Zaoui, A. eds., Dordrecht: Kluwer academic Publishers. 27. Acharya, A. and Bassani, J. L. (2000). Lattice incompatibility and a gradient theory of crystal plasticity. Int. J. Mech. Sol. 48: 1565-1595. 28. Zhu, H. T., Zbib, H. M. and Aifantis, E. C. (1997). Strain gradients and continuum modeling of size effect in metal matrix composites. Acta Mech. 121: 165-176. 29. Aifantis, E. C. (1998). Recent progress on gradient theory and applications, in Material Instabilities in Solids, pp. 53-548, de Borst, R., and van der Giessen, E., eds., Chichester-New York: Wiley. 30. Zhu, X. H. (1999). Theoretical Analysis of Sheet Metal Formability. PhD. Dissertation, Michigan Technological University.
This Page Intentionally Left Blank
CHAPTER
Viscoplasticity
5
This Page Intentionally Left Blank
SECTION
5.1
Introduction to Viscoplasticity JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France
Viscoplasticity refers to the mechanical response of solids involving time-
dependent, irreversible (inelastic) strains. The deformation of essentially all metals is, to a certain extent, time-dependent. This dependence, however, becomes more pronounced at temperatures exceeding a third of the material's melting point. At the microscale, two different mechanisms may be responsible for the material's rate dependency. One mechanism pertains to vacancy formation and grain boundary sliding, which are essential processes in creep (under constant stress) and stress relaxation (under constant strain) phenomena. The second is the slip-induced plastic deformation due to the motion and multiplication of dislocations. At temperatures suitably below the melting point, dislocations overcome their short-range barriers through their thermal activation, which is a rate-controlling phenomenon. In addition, the interaction of moving dislocations with phonons and, at low temperatures, with electrons results in a viscous drag on the dislocations, affecting the overall viscoplasticity of crystalline solids. In polymers, viscoplasticity occurs in both crystallites and amorphous phases; the crystalline phases experience slip, whereas the amorphous phases experience chain segment rotations against the resistance of weak intermolecular interactions. In elasto-viscoplasticity, elasticity and strain hardening are phenomena identical to those observed in plasticity. Therefore, the models are of the same nature, and only a viscosity function is added. Often this is the Norton's power law, which relates the secondary constant plastic strain rate kp to the applied stress ~ through two temperature-dependent material parameters 2(r), N(~)
This 1929 isotropic model has been much improved and sometimes replaced by an exponential function to take into account the anisotropy in single Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
301
302
Lemaitre
crystals (see Sections 5.2 and 5.3) or to homogenize what happens in polycrystalline materials (see Sections 5.4 and 5.5). The difficulty in modeling is to distinguish the part of the stress which is responsible for strain hardening from the part contributing to the viscosity. There are several methods: the overstress is introduced in Section 5.6, no yield condition is required in Section 5.7, nonlinear isotropic and kinematic hardenings are applied in Sections 5.8 and 5.9, and a yield and memory surface is used in Section 5.10. Over suitable ranges of deformation, rate, and temperature, the viscoplastic response of many crystalline solids stems from the motion of dislocations. The resulting stress-strain relations are then dependent on the strain rate and strongly upon variations of temperature, which is another difficulty (see Section 5.11). Finally, the special case of polymers is described in Section 5.12. An essential feature of numerical calculations in elasto-viscoplastic structures is the time wise in time incremental procedure different from that is used in plasticity. It is always difficult to choose, automatically or not, the time increment that ensures the convergence of the calculation (see Sections 5.7, 5.8, and 5.10).
SECTION
5.2
A Phenomenological Anisotropic Creep Model for Cubic Single Crystals ALBRECHT BERTRAM1 and JURGEN OLSCHEWSKI2 ~Otto-von-Guericke-University Magdeburg, Universithtsplatz 2, 39106 Magdeburg, Germany 2BAM-E2, Unter den Eichen 87, 12200 Berlin, Germany
Contents 5.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Description of the Model . . . . . . . . . . . . . . . . . . . 5.2.4 Identification of the Material Parameters .... 5.2.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1
303 303 304 306 307 307
VALIDITY
The model describes the behavior of cubic single-crystalline superalloys in the high-temperature regime under monotonous creep conditions in their primary and secondary creep phase.
5.2.2
BACKGROUND
For many technical applications, especially in the high temperature regime, single-crystalline components show an improved damage and fatigue resistance in comparison to polycrystals. However, the description of the thermomechanical behavior becomes much more complicated for single crystals because of their genuine anisotropy. The creep compliance for some superalloys differs by one or two orders of magnitude with respect to their Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
303
304
Bertram and O l s c h e w s k i
D
C
!
cy
R
..d
o"
!
K FIGURE 5.2.1
crystallographic orientation. The designer has to take into account this effect, as well as the strong nonlinearity of the relation between force and creep rate. For the creep modeling of single crystals two conceptually different approaches exist. One is to use slip system kinematics and one-dimensional creep equations for each of them. The other is phenomenological and based on tensor function representations for the description of the anisotropy. The latter approach has the advantage of being simpler in numerical implementation and wider with respect to the physical creep mechanisms to be included in the description, since they are not limited to slip systemdominated creep. The creep behavior in its primary and secondary phase can be described by a three-dimensional and nonlinear generalization of a rheological model with two springs and two dampers (Fig. 5.2.1). The one-dimensional constitutive law is given by the ordinary differential equation e 4-
4- ~ +
e 4- D----~a -- --~ e, 4- ( C 4- K )'~
where ~ is the strain, a is the stress, C and K are elasticities, and D and R are viscosities. The nonlinearity is taken into account by the dependence of the viscosities on the stresses in the following form: D = Do exp ( - B a ) R -- Ro exp ( - B a )
with positive material constants Do, Ro, and B. Note that D and R are constant during monotonous creep tests.
5.2.3
DESCRIPTION
OF THE MODEL
In the three-dimensional case we generalize the preceding differential equation by means of the projection method described in References [1, 2].
5.2 A PhenomenologicalAnisotropic Creep Model
305
This results in three tensor-valued equations
T;'+
( ci Ki Ci) CiK i ci K ~ ~T+~T+~T T ; + ~ T i - - - D ~ - E ~ + ( c i +
Ki
) E ; ' , i - - 1, 2,
with the three cubic subspace projections of the strain tensor to a crystallographic vector base: 1
0
0
0
1
0
0
0
1
3
8ij with respect
I -1 i11ool i1ool El - - 1 / 3 tr(E)
0 0
E2
822 0 -- 1/3 tr(E) 0 1 0 0 833 0 0 1
E3 --
I 0 812 813 1 812 0 823 813 823 0
with tr(E) = 811 -Jr-822 + 833. The projections Ti of the stresses c~ij are obtained in an analogous form. These differential equations of second order can be reduced to a system of first order by introducing a stresslike tensor of internal variables S which is decomposed analogously into three parts S1, $2, $3, by 1 Ti (s)] as S i ' - - K i 0 t [E;(s)- ~T We obtain the following evolution equations:
E~
_
1
C i _t_ K i
T~ +
Ki
-~
+
-~
+
-~
T/-
-~ S i
Ci
S~ -- Ci + Ki [T~ +~-i-(T,- Si)] Note that for monotonous creep all terms including T~ are zero. This system can be integrated by standard algorithms such as explicit or implicit Euler schemes. The nonlinearity is again taken into account by the dependence of the viscosities on the stresses. For that purpose we also use the exponential form of the one-dimensional model but substitute the exponent by a linear form of
306
Bertram and O l s c h e w s k i
the nine cubic invariants Jj
Di = Doi exp ( - a i ) Ri = Roi exp ( - a i ) 9 j=l
with positive material constants Doi, Roi, aij, i = 1, 2 , 3 , j = 1 , . . . , 9. It turns out that the following invariants already give a sufficient accuracy:
.12 - - 0"110"12 -- 0"22 § 0"110"33 -- 0"23 § 0"220"33 -- 0"23
J5 = 0120"23013 -
+
+
+
+
+
all together taken with respect to a crystallographic basis.
5.2.4
IDENTIFICATION
OF THE
MATERIAL PARAMETERS In the present model, the following material constants appear: 9 Elasticities: Ci, Ki, i = 1, 2, 3 9 Viscosities: Doi, Roi, ~ij, i = 2, 3, j - 2 , 4, 5, 6 If we assume that all inelastic deformations are isochoric, we get D~-1 -- R~-1 -- 0 => Do1 -- Ro) -- 0. Therefore, the coefficients a lj, j = 1 , . . . , 9 are irrelevant and can be set equal to 1. When subjecting the material to resonance tests, the dynamic elastic constants Ci § Ki, i = 1, 2, 3, can be determined [5, 6]. The other constants can be identified by minimizing the distance between experimental tests and the response calculated by the model. It is desirable for this purpose to have creep data within the relevant load range from single crystals for a widely distributed set of orientations. This leads to an inverse problem of nonlinear parameter identification. The identification has been done for three superalloys: SRR99, CMSX4, and CMSX6.
307
5.2 A Phenomenological Anisotropic Creep Model TABLE 5.2.1
TABLE of Parameters for SRR99 at a Temperature of 760~
C1 - - - 0
K1 = 461.153 GPa D1 = 1020 h GPa R1 = 1 0 2 0 h G P a 0{22 = 12.4347 GPa -1 o~24 = 1.7874 10 -3 GPa -2 o~25 = 4.1673 10 -2 GPa -3 ~26 = 3.1164 10 -4 GPa -3
C2 = 42.559 GPa K2 = 28.071 GPa D2 = 2.5042 107 h GPa R2 = 1.4714 109 h GPa 0{32 -- 0.24928 GPa -1 0{34 = 41.783 GPa -2 o~35 = 4.0697 10 -3 GPa -3 o~36 = 5.0154 GPa -3
C3 = K3 = D3 = R3 =
[3]. 46.764 GPa 156.436 GPa 1.10735 109 h GPa 374842 h GPa
Applications to the superalloy CMSX6 can be found in Reference [41.
5.2.5
HOW TO USE THE MODEL
The entire model is given with respect to a crystallographic basis. It is capable of describing the creep behavior under monotonous loads in its primary and secondary phases. There exist extensions of the model to include the tertiary creep phase (damage) [7-9].
REFERENCES 1. Bertram, A., and Olschewski, J. (1991). Formulation of anisotropic linear viscoelastic constitutive laws by a projection method, in High Temperature Constitutive Modeling: Theory and Application, pp. 129-137, ASME, MD vol. 26, AMD vol. 121, Freed, A., and Walker, K. P., eds. 2. Bertram, A., and Olschewski, J. (1993). Zur Formulierung linearer anelastischer Stoffgleichungen mit Hilfe einer Projektionsmethode. Zeit. ang. Math. Mech. 73 (4-5): T401-T403. 3. Bertram, A., and Olschewski, J. (1996). Anisotropic creep modeling of the single crystal superalloy SRR99. Comp. Mat. Sci. 5: 12-16. 4. Bertram, A., and Olschewski, J. (1995). Computational modeling of anisotropic materials under creep conditions. Math. Modelling Sci. Comp. 5 (2-4): 100-109. 5. Han, J., Bertram, A., Olschewski, J., Hermann, J. W., and Sockel, H.-G. (1994). Identification of crystal orientation by resonance measurements. Zeit. ang. Math. Mech. 74 (4): T322-T325. 6. Bertram, A., Han, J., Olschewski, J., and Sockel, H.-G. (1994). Identification of elastic constants and orientation of single crystals by resonance measurements and FE-analysis, in Twenty Years of Finite Element Analysis: Review and Future Prospects. Special Issue of Int. J. Computer Appl. Techn. 7 (3/4): 284-291. 7. Qi, W, and Bertram, A. (1997). Anisotropic creep damage modeling of single crystal superalloys. Technische Mechanik 17 (4): 313-322. 8. Qi, W, and Bertram, A. (1998). Damage modeling of the single crystal superalloy SRR99 under monotonous creep. Comp. Mat. Sci. 13: 132-141. 9. Qi, w , and Bertram, A. (1999). Anisotropic continuum damage modeling for EC.C.-single crystals at high temperatures. Int. J. Plasticity 15 (11): 1197-1215.
5.3
SECTION
Crystalline Viscoplasticity Applied to Single Crystals GEORGES CAILLETAUD
Centre des Mat~riaux de l'Ecole des Mines de Paris, UMR CNRS 7633, BP 87, F91003 Evry Cedex, France
Contents 5.3.1 General Presentation and Validity Domain.. 5.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Equations for Practical Use . . . . . . . . . . . . . . . . 5.3.4 Identification of the Material Parameters . . . . 5.3.5 Numerical Implementation in Finite Element Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1
GENERAL
VALIDITY
PRESENTATION
308 310 311 312 314 314 317
AND
DOMAIN
In most of the single crystals, the p r e d o m i n a n t deformation m e c h a n i s m is plastic slip. F r o m a mechanical point of view, that means that the plastic strain rate of the material is the result of a sum of plastic strain rate coming from each active slip system. Depending on temperature range, plasticity can be either time-independent or time-dependent, but for each case the critical variable is the resolved shear stress. For small perturbations, the following definitions are valid: Strain rate decomposition: ~ - ~e +
Orientation tensor: m s _
308
-~(,~se 1
(1)
~p
r~s + r~s e~
~)
(2)
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
5.3 Crystalline Viscoplasticity Applied to Single Crystals
309
Inelastic strain rate: ~P -- ~
mS9 s
(3)
Resolved shear stress: ~:s _ gg:m s
(4)
S
The elastic part of the strain rate ~e is computed using the classical elasticity law respecting the symmetries of the single crystal. Some of the variables in the equations are indexed by s, which denotes a given slip system defined by the normal to the slip plane ffs and the slip direction n~s. These definitions are very general, since they only describe the "geometry" of the deformation mechanism. The constitutive equations have then to be written to correlate the resolved shear stress vs and the slip rate on each slip system 9s. During the deformation of the material, a rotation of the slip planes takes place (about one degree for 1% deformation), leading to a modification of the m s tensor for active systems. This can be taken into account in a "small rotation" formalism, the rotation being modeled by the antisymmetric part of the tensorial product r~s | ffs. On the other hand, more general expressions that are valid for large transformations can be read in classical papers [1, 2]. Twinning is another crystallographic deformation mode, in which a zone of the material is submitted to an apparent rotation and extension with respect to its initial position, which produces a given deformation tensor. The resulting strain can be modeled in the present formalism using a predefined value for m s and replacing 9s by a term depending on the twinned volume fraction [3]. On the other hand, the influence of climb can be represented by a term including the tensorial product ffs | ffs. According to the crystallographic phase of the material and to the temperature domain, one or several slip system families must be introduced. For FCC materials, the most common family is octahedral slip ({ 111} planes, (110) directions), but cubic slip ({001} planes, (110)directions)can also be found [4]. In the case of BCC materials, where no slip plane is present, a simple solution consists in considering {110} planes, and (111) directions. The case of HCP materials is much more complex. Several families are usually active according to the value of the ratio c/a (a being length of the prism side, c height of the prism) with respect to the theoretical value for perfect network, x/~/3: 9 for materials having a low c/a value (zirconium, magnesium, titanium) prismatic plane is predominant; 9 for materials having a high c/a value (zinc, cadmium), basal slip is the most active. In both cases, pyramidal slip can also be observed.
310
Cailletaud
FIGURE 5.3.1
Slip systems for (a) FCC and (b) HCP materials.
Examples of slip planes are shown in Figure 5.3.1 for FCC materials (Fig. 5.3.1a) and HCP materials (Fig. 5.3.1b).
5.3.2 FORMULATION The framework of thermodynamics can be used for a rigorous formulation of the hardening of the material, but several strategies can be used for choosing the potential [5,6]. In the following, the state variables are the elastic strain (tensor), a vector storing the isotropic variables on each slip system (components p~), and a vector for the kinematic variables (components c~). The size of these vectors is equal to N, the number of slip systems in the model. The treatment of the elastic part is classical and is not reported here. The "inelastic part" of free energy is chosen so that the corresponding hardening variables linearly depend on the state variables: r s -- roi-~- bi~i ~
hsrfl r
(5)
s x S = c I ~s
(6)
The terms hsr are the components of the interaction matrix which characterizes both self-hardening (diagonal terms) and cross-hardening between the different systems [7]. The model then uses a classical formalism, introducing for each system s a threshold fs and two evolution equations: fs = [~ - xS[- r ~
(7)
fis = b~(1 _ bipS)
(8)
: / s - bs ( sign ('cs -xS)-dIxs'~cI,,]
(9)
5.3 Crystalline Viscoplasticity Applied to Single Crystals
311
The full definition is achieved when the value of the accumulated slip rate bS= 19~1 is defined. For a viscoplastic model, a simple power function can be chosen: kl
s/.i
with (x} - max(x, O)
(10)
Most of the viscoplastic crystallographic models for single crystals use a "Norton law" for each slip system so that all the slip systems are active. The predominant systems are just found numerically because of the nonlinear character of the equations. In the present case, a true threshold is introduced, so that the numerical behavior of the model is better. On the other hand, the strain rate sensitivity can be adjusted using suitable values of the corresponding coefficients hi and ni (low values of ki). A time-independent version of the model can also be proposed: a rule has then to be specified to determine the active slip systems for a prescribed external load. In the preceding equations, it has been assumed that all the slip systems of a given family present the same behavior, so that the coefficients are indexed with I, and not s, I denoting the family of the system s. The initial resolved shear stress is roi, the hardening capability is defined by QI and bi, and the kinematic hardening by ci and dI. The viscosity is characterized by ki and hi. The next section describes the identification method.
5.3.3 EQUATIONS
FOR PRACTICAL USE
The model can be simplified and treated exactly like a classical macroscopic model. For an explicit integration, the starting point is the actual state of stress and internal variables. The values of vs can be used instead of p~ for the practical calculations. The calculations consist then of the following steps: 9 compute 9 knowing rate (Eq. 9 compute
the actual value of the hardening variables (Eq. 11); ~ on each slip system, compute the accumulated viscoplastic slip 12) and the evolution of the hardening (Eq. 13); the inelastic strain rate (Eq. 14).
In these equations, the index I has been dropped for the sake of brevity; the two types of isotropic hardening correspond to a part with the interaction matrix (index 1) and a part introducing self-hardening only (index 2). Note that a system will be active provided that its resolved shear stress z s is greater than x s + r ~ or less than x ~ - r ~. The present formulation gives a saturation of
312
Cailletaud
the hardening in both monotonic and cyclic loading. x ~ - c ~ ; ,-~ - eo + Q~ ~
h~(1 - e-~}
+ Q~{1 -
~-~}
(11)
r
(12) ~s = ~)s _ dcxSbS {P - ~
mSb~sign ("c~ - x~)
(13) (14)
S
5.3.4 IDENTIFICATION OF THE MATERIAL PARAMETERS The model has been mainly used for modeling cyclic loadings in Ni-base superalloys. Other materials have also been investigated, like copper, alloy 600, zircaloy, or zinc. Basically, it does not differ very much from a classical viscoplastic model with isotropic and kinematic hardening, so that the tests needed for the identification include: 9 cyclic tests for discriminating between isotropic and kinematic hardening and for evaluating the cyclic hardening or softening; 9 tests at various strain rates to characterize the viscous effect; 9 creep or relaxation test. In addition, specific tests must be made: (1) When several system families may be active, the loading directions must be chosen so that each family is excited. (2) In order to know better the value of the terms in the interaction matrix, nonradial loading paths should be realized to evaluate the so-called latent hardening. When these tests are absent, the two opposite solutions consist in choosing either a diagonal matrix hsr=6sr (Kronecker ~) or an "isotropic" hardening, hsr = 1. The most difficult challenge is to affect a value to the cross-hardening between two families. These terms have to be chosen after a discussion with the metallurgists. The identification process needs a numerical implementation of the model because, for a given orientation, the result in terms of strain rates is the result of a collection of individual slip contributions, and the real specimen orientation must be used (a small misorientation can change the initially active slip systems and may drastically influence the material response). Nevertheless, the model can be reduced to very simple expressions in the case of multiple slip, when several systems are equivalent. For example, if M slip
313
5.3 Crystalline Viscoplasticity Applied to Single Crystals
systems (among N) are equivalent for a pure tension loading under the stress cr, the Schmid factor being m, the inelastic strain rate in the tension direction can be written:
~P
_
_
Mm~ s -
mx~
MmlmO-
k
r)n
(15)
The model is then equivalent to a macroscopic model:
withK -
k
1
m (raM)l~ n'
X -
x
--' rtl
r R -- -m
(16)
For the case of FCC materials, it can easily be observed that a tension in the direction {001} will activate octahedral slip systems (m = 1 / v ~ , M = 8), and that a tension along {111} will activate preferentially cubic slip (m = v/2/3, M = 6) (depending on the respective value of the critical resolved shear stress for each f a m i l y . . . ) . This fact can be used to separate the action of the two families, since the model provides a different expression of the tensile curve for each tensile direction. The corresponding values for octahedral and cubic slip systems are reported in Table 5.3.1 (note that the exponent is the same for the crystallographic and the classical approach).
TABLE 5.3.1 Equivalencebetween the Crystallographic Model and a Classical Approach in Pure Tension. Coefficient
Value for multiple slip (m, M)
Value for 001 tension
Value for 111 tension
K
k
x/6k
3k
m(Mm)1/n
(8/V/-6)1/n
2(n+3)/2n
R0
ro --
v/-6r0
Q
m
Q
3ro v~
3Q
m
~Q
b
b mM
~b 8
b 2v~
C
c Mm 2
3c 4
3c 4
a Mm
v~a 8
a 2x/2
D
314
Cailletaud
5.3.5 NUMERICAL IMPLEMENTATION FINITE ELEMENT CODE
IN
The integration of these constitutive equations in a finite element code is classical. For each integration point, the method can be either an explicit method, like Runge-Kutta, or an implicit method, like a 0-method. For each case, the vector of the internal variables is built with the components of ~e, the N values of ~s, and the N values of vs. For the 0-method, the residual vector consists of three types of equations, the first coming from the strain decomposition (Eq. 17), the second from isotropic hardening (Eq. 18), and the third from kinematic hardening (Eq. 19). A v S m s sign ('cs - x s)
.Re -- Ae..- Ae..e - ~ 5
--
-
l- r
R~s -- A~xs - (sign(z s - x s) - d~)Av s
(i 7)
(18)
(19)
The unknowns are Aee, Avs, A~s. The system can be solved by a Newton method, so that the consistent tangent matrix is given numerically [8].
5.3.6 REMARK Many mistakes have been made in the recent literature concerning the behavior of single crystals. The most classical error consists in considering a cubic version of the Hill criterion for the description of cubic single crystals. Since this version introduces only one adjustable coefficient, it can only be fitted for describing the difference between a tension along 001 and a tension along 111, for instance. On the other hand, it has nothing to do with a general formulation, since the group of cubic symmetries contains a large number of invariants which are not represented by Hill's formulation [9]. For instance, such a model does not represent the large heterogeneities which can be observed during the torsion of thin tubes [9]. This effect is illustrated in Figure 5.3.2, which also shows the evolution of yield surface in tensionshear when two slip families are present: according to the ratio of tension to shear, the initial plasticity is not observed at the same location in the tube. A "cubic Hill" criterion predicts that plasticity is uniform along the circumference.
5.3 Crystalline Viscoplasticity Applied to Single Crystals
315
FIGURE 5.3.2 Location of the plastic zones at the onset of plastic flow for two shear-tension ratios, a. r/~r = v/3. b. ~/~r = 1. c. Yield surface in (110) and (110) regions.
316 TABLE 5.3.2
Cailletaud Table of Parameters.
K
octa
5
n
10
Q1
ro
bl
c
OFHC Copper, room temperature 0.8 3.6 0.6 3,400
d
Q2
b2
3600(a)
0.7
2200
- 140
6
(a) 0~s - ~s _ dOsbS, with ~s = 0.2 + 0.8 e x p - 2 3 v s
octa cube
700 1172
4.7 2.4
4.6 17.3
SC 16, 950 ~C 3.9 3.7 96,500 -4.4 2.7 77,000
1050 1056
octa cube
220 100
10 10
390 387
CMSX2, 650~ 400 3700 20,000 380 2400 50,000
1000 1000
octa cube
920 540
4.6 4.6
70 54
0 0
octa cube
88 96
15 15
245 377
octa cube
517 707
5 5
69 47
0 0
AM1,950~ -
21,600 90,500
2110 1130
octa cube
490 360
4.1 4.1
29 47
0 0
AM1, 1100~ -
100,000 4,000
1600 850
octa
20
10
80
octa
50
25
prism
81
8.8
Au4G, Room temperature* 25 30 5,000 Zr702, 200~ * 6.1 -5.6 10 1,330
178 100 94 184
6.5 3.2 2.6 2.1
53 197 144 292
prism basal p y r l (a) pyr21(c + a)
CMSX2, 950 ~ 120,000 40,000
1400 540
AM1, room temperature 0 730,000 0 186,000
3200 560
316SS, Room temperature* 65 8 2,200
70
Zy4, Room 17.6 20 62.2 169
temperature* 103 100,000 17 4,000 94 101
130
200 67
1600 850
Units: MPa, s. 1 The sign (*) indicates that the model has been used for modeling polycrystal plasticity, in a polycrystalline aggregate. The coefficients are then valid for one grain more than for a single crystal.
317
5.3 Crystalline Viscoplasticity Applied to Single Crystals TABLE 5.3.3
The Interaction Matrix Corner.
Copper, RT
hii = H1 = 1 h12 =
hi3 = h23 =
hi9 -- hl,ll
h 4 5 - - h 4 6 - - h 5 6 = h 7 8 --'- h 7 9 - - h 8 9 = h 1 0 , 1 1
=hlo,12=hll,12=H2=4.4 h38 = h 4 9 - - h4,11
-- h25 -- h2,12 -- h36 --
h7,12 -
-" h57 -- h6,10
h8,1o -- H 3 -- 4.75
h 1 4 = h 2 7 = h 3 , 1 o - - h 5 , 1 2 = h 6 8 - - h9,11 - - H 4 = 4 . 7 5 h 2 9 = h 2 , 1 0 = h 3 4 - - h 3 7 = h3,11 = h 3 , 1 2 = h 4 8 - - h 4 , 1 2 - ---:- h5,11 =
h67 =
h69 = h6,12 --
h7,11 =h8,11 =
h58 =
hs,lO
h 9 , 1 0 = h 9 , 1 2 - - H 5 --- 4 . 7 5
h i 8 = h i , 1 2 - - h2,11 - - h 3 5 = h 3 9 = h 4 7 - - h 4 , 1 0 - - h 5 9 = h6,11 =
Ni-base single crystals (CMSX2, AM1, SC16) 316SS, Au4G Zy4
h7,1o=
h8,12 -- H6 = 5
diagonal matrix hsr ~ 1
diagonal matrix
REFERENCES 1. Mandel, J. (1973). Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Struct. 9: 725-740. 2. Asaro, R. J. J. (1983). Crystal plasticity. J. Appl. Mech. 50: 921-934. 3. Staroselsky, A., and Anand, L. (1998). Inelastic deformation of f.c.c, materials by slip and twinning. J. Mech. Phys. Solids 46: 671-696. 4. Nouailhas, D., Pacou, D., Cailletaud, G., Hanriot, E, and R~my, L. (1994). Experimental study of the anisotropic behaviour of the CMSX2 single crystal superalloy under tension-torsion loadings, in "Advances in Multiaxial Fatigue", ASTM STP 1191, pp. 244-258, eds. Mc Dowell, D. L., and Ellis, R. 5. M&ic, L., Poubanne, P., and Cailletaud, G. (1991). Single crystal modeling for structural calculations. Part 1: Model presentation. J. Engng. Mat. Technol. 113: 162-170. 6. Forest, S., Cailletaud, G., and Sievert, R. (1997). A Cosserat theory for elastoviscoplastic single crystals at finite deformation. Archives of Mechanics 49 (4): 705-736. 7. Brown, T. J., and Kocks, U. E (1966). Latent hardening in aluminium. Acta Metall. 14: 87-98. 8. Besson, J., Le Riche, R., Foerch, R., and Cailletaud, G. (1998). Object-oriented programming applied to the finite element method. Part 2. Application to material behaviors. Revue Europ&nne des F.l~ments Finis 7 (5)" 567-588. 9. Cailletaud, G., and Nouailhas, D. (1995). Tension-torsion behavior of single-crystal superalloys: Experiment and finite element analysis. Int. J. Plasticity 11 (4): 451-470.
SECTION
0
4
Averaging of Viscoplastic Polycrystalline Materials with the Tangent Self-Consistent Model ALAIN MOLINARI Laboratoire de Physique et M~canique des Matk.riaux, Ecole Nationale d'Ing~nieurs, Universit~ de Metz, Ile du Saulcy, 57045 Metz Cedex, France
Contents 5.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 General Background . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Polycrystal Averaging . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Discussion of the Averaging Schemes . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1
318 319 321 324 325
NOTATIONS
Vectors and second-order tensors are underlined by a single bar: e.g., x_, _d are the position vector and the strain rate tensor. Fourth-order tensors are underlined by a double bar: A. Cartesian notations are used, and the convention of s u m m a t i o n on the repeated indices is adopted. The double contracted products between second-order tensors or between fourth- and second-order tensors are respectively defined as a_ " b__- aob j`
(A" a_)0 - A i j h l a l k
The volume average o f f on the whole aggregate is represented by (f).
318
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
5.4 Averaging of Viscoplastic Polycrystalline Materials
319
5.4.2 GENERAL BACKGROUND This paper describes polycrystal averaging for time-dependent behaviors. Large deformations are considered, and the model is aimed at describing the overall stress-strain response together with the evolution of the internal structure of the material (hardening of individual crystals, crystallographic texture, and grain morphology). Polycrystal averaging is based on several basic steps: 1. choice of a representative volume element (RVE) containing a large ensemble of grains statistically representative of the polycrystal; 2. characterization of the deformation mechanisms and of the constitutive law of individual grains; 3. description of the initial crystallographic texture (distribution of lattice orientations) and of the initial morphological texture (grain's shape and orientation of the principal geometrical axes); 4. grain interaction model; 5. evolution laws for the crystallographic and morphological textures. The averaging scheme presented in following text assumes that the deformation remains uniform within grains and ignores all deformation heterogeneities such as shear bands and dislocation cells. Despite these simplifications, it is worth noting that the proposed averaging scheme is quite successful in decribing the texture evolution and the overall plastic response. We consider a viscoplastic material constituted by a statistical homogeneous distribution of crystals. Elastic deformations are neglected. Each individual crystal (c), is governed by the following flow law: Of(c) dij = Osij (s)
(1)
where d O is the strain rate tensor, sOis the deviatoric Cauchy stress tensor, and f(c) is a strictly convex potential; see Reference [4] for details. The viscoplastic flow is assumed to be volume-preserving; therefore: dii - tr( d) - 0
(2)
Since the potential f(c) is strictly convex, the flow law (Eq. 1) can be inverted into sij =
Og(c) (d) Odij
(3)
320
Molinari
where g(C) is the strictly convex function conjugated to f(c) via the Legendre transform: g(_d) - sup [s_"_d-f(s_)]
(4)
s
The macroscopic strain rate potential F(S_) can be defined as the volume average of the local potentials [1]"
f-
The macroscopic flow law has the form OF D 0 -- ~-~0 (5)
(5) (6)
where S_ and _D are the macroscopic Cauchy stress and strain rate tensors, respectively. They are related to the microscopic stress and strain rate by volume averaging:
S_- (s)
(7)
p - (d>
(8)
Similarly, the macroscopic stress potential is defined by
(9) and we have OG Sij - ~jij (_D)
(10)
The problem addressed in this article is determining the macroscopic overall material response in terms of the local behavior of the phases. In other words, we would like to determine the macroscopic flow law (Eq. 6) from Eq. 1, or to determine the law (Eq. 10) from Eq. 3. The difficult step in the homogenization process is to calculate the local strain rates _d as a function of the macroscopic strain rate _D:
_d- _d(_D)
(11)
If the localization law (Eq. 1 i) is determined, the macroscopic stress potential is precisely defined from Eq. 9
O(D)-
(12)
Similarly, from the stress localization law
(13)
321
5.4 Averaging of Viscoplastic Polycrystalline Materials
we can derive the macroscopic strain rate potential" F(S_)- (f(s_(S_))
5.4.3
(14)
POLYCRYSTAL AVERAGING
In a polycrystalline material, grains with the same chemical composition, same shape, and orientation of the principal geometrical axes, and same crystallographic orientation can be considered as defining a particular phase. If the spatial distribution of the different phases presents a large degree of disorder, the classical self-consistent scheme is appropriate to get an approximate solution of the localization problems (Eq. 11 or 13). In the self-consistent scheme it is assumed that the phase (c) can be represented by an ellipsoid F having the properties of phase (c); F is embedded in an infinite uniform matrix having the overall properties of the aggregate (to be determined in a consistent way). The matrix is called the homogeneous equivalent medium (HEM). We shall address a localization problem a little more general than Eq. 11. The overall velocity gradient
OV _L- c3x i
(15)
is applied at the remote boundaries of the aggregate; _V is the macroscopic velocity field, and x_ is the position vector. The macroscopic strain rate _D and rotation rate El are the symmetric and antisymmetric part of _L. We are interested in the determination of the velocity gradient l_c in the ellipsoid Ic, in terms of L. The solution of the inclusion problem (ellipsoidal inclusion embedded in an infinite homogeneous matrix) has been obtained in a closed form by Eshelby in the case of linear elastic materials. For nonlinear material behavior, an approximate solution of the "nonlinear" inclusion problem can be obtained by using a linearization procedure. The method consists in approximating the material behavior by an affine response in a certain range of strain rates. Grain's interaction is described by the following interaction laws [3]" sf - S_- [A=~ coc - ~ - -B~
- (P~ 9(p~
(_d~ - _D)
(16)
(_dc - _D)
(17)
322
Molinari
where s_c is the deviatoric Cauchy stress in the crystal (c). The macroscopic deviatoric stress _S is the volume average of the local stresses
s_-
Zs< _ <
with fc being the volume fraction of grain (c). _dc and coc are the symmetric and antisymmetric part of the local velocity gradient lf. A ~ is a macroscopic stiffness viscoplastic modulus (fourth-order tensor) which depends on the macroscopic strain rate _D. Different choices of A~ are possible, depending on the type of linearization used. To each choice ofA ~ is associated a different model [7]. p0 and B_~_ are fourth-order tensors depe-nding on A ~ and on the shape of the i-nclusion (c). The interaction laws (Eqs. 16 and 17) are obtained by using a Green function technique [3, 4]. Green functions G~m and Hm associated with the modulus A ~ are defined by A~jklG~m,lj(X_
- x_I) -F Hm,i(x_ - x_I) -Jr- (~im(~(x_ - x_I) -- 0
G~m,k(x - x') - - 0
(18) (19)
with boundary conditions and
G~m -- 0
Hm - 0 at infinity
(20)
The Green function _/_/0 does not appear in the interaction laws (Eqs. 16 and 17); however it is a necessary ingredient for calculating the local hydrostatic pressure in terms of the macroscopic pressure. The tensors p0 and B~ are defined as m
E
pOhl _ o
--
1 1
0 o
_
o
0 o
_ Tjilk )
(21) (22)
with
r h,j - i
(23)
To obtain these results, a linearization procedure was used consisting in somehow replacing the nonlinear response of the HEM in the vicinity of the "working" strain rate _Dby an a[fine approximation (Fig. 5.4.1). The nonlinear response of the HEM 0G (_D*) D* --S* -0--6
(24)
323
5.4 Averaging of Viscoplastic Polycrystalline Materials ~
S
s_
IS* = ___a~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* + S o(__D) I
.
-
8D
80(8) _D_D D FIGURE 5.4.1 Linearization of the macroscopic constitutive law D_D_*--+ S* = OG/OD_* (D*) in the vicinity of the macroscopic strain rate D. Depending on the definition of the macroscopic viscoplastic stiffness modulus A_~ different averaging schemes are defined. The back-extrapolated stress is defined by the relationship in Eq. 26.
is approximated for values of _D*close to _D, by
_D* ~ S_* - a_40(p)" _D* + _S0(_D)
(25)
The back-extrapolated stress is defined as S_~(_D) -- S_- A=~(_D). _D
(26)
A possible choice for A__~ consists in taking the incremental tangent macroscopic modulus as defined by the incremental self-consistent scheme [1], see also [6, 7]"
d o - a_'g - (_a(~)~g- [_~ + _P~g. (a (~)~g - aug)] -~) 9([K + _P~g" (_a(~)'~ - a_'~)] -~) (27) where __a(c)tg is the tangent modulus at the crystal level defined as a(c)tg(d_C~-= 02g(C)/c3_dQd, g(C) being the stress potential of the crystal (c) ( s / - 0 g ~ / 0 4 ) ( s / - 0g~/0_a). _P~g is playing, with respect to A__tg, the role of p0 with respect to A ~ -K is the fourth-order tensor defined by K~jk~- 1 / 2 ( G a j z + a , a j k ) - 1/3a,j&z. The solution of the localization problem is obtained as follows. Assume that at a given stage of the calculations an approximation of A__tg is obtained. Then, using the interaction law (Eq. 16), where A 0_ __+ A__tg,p0 ___+ptg, and S_ --+ ~fcs_ c, together with the constitutive law s c = sf(_d~), a set of N nonlinear equations (N: number of grains) is obtained from which _d~ can be calculated in each crystal (c). From these values of d ~, new estimates of __a(c)tg a r e calculated. Then by solving the implicit equation (Eq. 27) a new estimate of A tg is obtained. Iterations are continued until m
m
Molinari
324
convergence is achieved. When _dc is calculated, the rate of rotation coc of the crystal (c) is given by Eq. 17. The localization problem has been solved now for a given material state (given grain's morphology, crystallographic texture, and resistance stresses of the slip systems). In the next time increment, the material state is updated, and a new localization problem has to be addressed [4]. Other choices are possible for A~ For instance, considering power-law materials with uniform strain rate sensitivity m, a simple relationship exists between the tangent and the secant modulus: As - limA tg (the notation ~=tg used for the macroscopic tangent modulus is changed with respect to Eq. 27 because the approach and the results are different). The interaction model associated with the choice A_~_ ~=tg_ mA=S' and the use of the relationship S_- Ag"_D in the interaction law (Eq. 16), define another type of tangent approach [2, 4]. A family of averaging schemes can be obtained by considering the choice A~ r/A_ -tg,_ where ~/ is a positive scalar. It can be shown that P~ therefore, the interaction law (Eq. 16) has the form: $7c __ ~_ __ ~(A.~tg _ ptg-,).
(dC _ D).
In the limiting cases where ~/--+ 0 and ~/--+ ce, the static model (stress uniformity: s f - _S) and the Taylor model (strain uniformity: _d~ - _D) are retrieved. The secant (resp. tangent) self-consistent approach corresponds to ~I- 1/m (resp. 1 / - 1).
5.4.4 DISCUSSION OF THE AVERAGING SCHEMES The tangent self-consistent schemes (A~ A tg_ or A~ A= -tg) predict softer grain interactions than the Taylor scheme, which assumes strain uniformity. As a consequence, the average number of active systems per grain is about two or three for the tangent self-consistent scheme, while it is closer to five for Taylor averaging. When a crystal has soft deformation mechanisms together with hard ones (as, for example, in HCP metals), the self-consistent model favors soft systems, while it is not so in general for the Taylor model. This has important consequences with respect to texture predictions. Sometimes less than five deformation mechanisms are available in each grain, the single crystal yield surface being then nonclosed. The standard Taylor averaging cannot be used in that case, while the self-consistent scheme is still operant. Grain shape evolution is modeled in a natural way in the self-consistent scheme via the tensor _p0 appearing in the interaction law. Changes in the
5.4 Averaging of Viscoplastic Polycrystalline Materials
325
grain's aspect ratio, which have important consequences for texture predictions, can be followed in a continuous way during the process. It is worth mentioning that the Taylor scheme can be modified into the Relaxed Constraint Taylor model to account for the existence of fiat parallel grains, but the continuous evolution of the grain's shape can hardly be described in that model. Finally, although the Taylor model satisfies compatibility but not equilibrium, it is of note that, in the self-consistent approach, equilibrium and compatibility are not accounted for between individual grains but are satisfied in the solution of the inclusion problem. Therefore, equilibrium and compatibility are somehow satisfied "in average" between a grain and the surroundings. Many applications of the foregoing averaging schemes, in particular of the tangent self-consistent approach, have been developed for FCC, BCC, and HCP metals, semicrystalline polymers, and geological materials such as rocks and ice; for a review see Molinari [4, 7]. These schemes have been successful in describing the overall stress-strain response together with the evolution of the internal structure of polycrystalline materials (texture evolution). Elasticity being neglected, applications are restricted to m o n o t o n o u s loading and large deformations. A discussion of averaging methods, including elastic effects, can be found in Molinari et al. [5].
REFERENCES 1. Hutchinson, J. W. (1976). Bounds and self-consistent estimate for creep of polycrystalline materials. Proc. Roy. Soc. A348: 101. 2. Lebensohn, R. A., and Tom~, C. N. (1993). A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys. Acta Metall. Mater. 41: 2611-2624. 3. Molinari, A., Canova, G. R., and Ahzi, S. (1987). A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall. 35: 2983-2994. 4. Molinari, A. (1997). Self consistent modelling of plastic and viscoplastic polycrystalline materials, in CISM Lecture Notes., pp. 173-246, Teodosiu, C., ed., Springer Verlag. 5. Molinari, A., Ahzi, S., and Kouddane, R. (1997). On the self-consistent modeling of elasticplastic behavior of polycrystals. Mech. of Materials 26: 43-62. 6. Masson, R., and Zaoui, A. (1999). Micromechanics-based modeling of plastic polycrystals: An affine formulation. J. Mech. Phys. Solids. 7. Molinari, A. (1999). Extensions of the self-consistent tangent model. Modelling Simul. Mater. Sci. Eng. 7: 683-697.
SECTION
5.5
Fraction Models for Inelastic Deformation j. E BESSELING [email protected]
Contents 5.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Description of the Model . . . . . . . . . . . . . . . . . . . 5.5.4 Determination of the Model Parameters .... 5.5.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1
326 327 327 331 334 335
VALIDITY
In the course of a fast breeder project in a period of approximately ten years a large number of experiments have been performed on tubular specimens, which were loaded in combinations of tension and compression, torsion, and internal pressure along carefully selected stress and strain paths. During this loading the strain rate was kept nearly constant, and the experiments were restricted to strains of a few percent. The materials tested were austenitic and ferritic stainless steels. As in earlier but far less numerous experiments on A1-Alloy specimens, it was concluded that the fraction model gives a quite accurate description for almost all loading histories considered. In particular, no evidence was found in any of the tests that the yield surface may develop corners or vertices in the course of loading [3]. Much less successful has been the application of the fraction model in the creep range. In modeling creep deformation, the deficiencies of the mathematical model do not constitute the problem as much as the large scatter that is found in experimental creep data. Tests under uniaxial stress with step loadings on Mg-Alloy specimens have shown less reproducibility than a sensible application of a complex model would require [4].
326
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright O 2001 by Academic Press. All rights of reproduction in any form reserved.
5.5 Fraction Models for Inelastic Deformation
327
5.5.2 BACKGROUND In elastic deformation the distribution of the internal energy on a microscopic scale is irrelevant to the macroscopic deformation problem. Thus the internal energy can be replaced by the internal energy of a fictitious homogeneous elastic continuum. In inelastic deformation, however, the dissipation of energy cannot in general be replaced by the dissipation in a homogeneous continuum. In the real material, inelastic deformation mechanisms are initiated at certain points and will spread in subsequent loading over the whole volume in the neighborhood of a material point of the continuum. In the fraction model the inhomogeneous energy dissipation is discretized in a phenomenological manner by conceiving the material to be composed of a limited number of portions, which can be represented by subelements of an element of volume dV, all subjected to the deformation of the continuum, but each with its own dissipative properties. Thus the history of an inelastic deformation process is recorded in terms of fraction stresses constituting the thermodynamic internal or hidden state variables and determining by their weighted sum the macroscopic stress tensor. From its initial formulation [1, 2] it has been stressed that the fractions are not to be identified as grains. As to linear rheological models, two or more Maxwell models in parallel can be interpreted as a fraction model [5]. Also, Melan's kinematic hardening model, advocated by Prager (who added the adjective kinematic to it), is contained in the fraction model. The combination of one ideally plastic or hardening fraction with one purely elastic fraction gives a kinematic hardening model with a consistent shift rule for the yield surface, as required by Ziegler [51.
5.5.3 DESCRIPTION
OF THE MODEL
The fraction model of a solid is based upon the concept of a so-called natural reference state, with respect to which changes in internal energy in thermodynamic state space can be defined by an invariant function of elastic strains and entropy. For a finite, possibly very small, neighborhood of a material point of the continuum model the geometrical configuration of the elementary particles of the real material in relation to each other is assumed to differ from the configuration in the natural reference state by a linear transformation of line elements in that point:
dr = Fedr
(1)
Here r is the position vector and d~ is the vector of the infinitesimal lineelement, which only in the case of purely elastic behavior is integrable to the
3 28
Besseling
position vector ~ in the undeformed state. In the case of purely elastic behavior, the vector ~ and the initial position vector ro are identical. In general, the line-element d~ will, however, change its length and orientation by inelastic deformation. The elastic d e f o r m a t i o n tensor, acting as objective (with respect to the material) state variable in the internal energy function, is defined by d r o d r - d~o d~ - 2d~o 07ed~), rle - 1 (FerFe _ I)
(2)
The rate at which locally d~ changes with time is characterized in the physical space by a tensor A v" d~ - Avd~ ~ d i ' - F v ~ - l & + V~Avv~-ldr - (L~+LP)dr
(3)
The velocity g r a d i e n t tensor L has been split into an elastic and an inelastic part. Also, the rate of deformation tensor D and the s p i n t e n s o r ~2 may be split into an elastic and an inelastic part: De
- ~1 (L ~+Le~ ), o~ - 89(L~ +L~ ~)
~,'~e __ 1 ( L e -
LET), np
__ 1 ( L P _ L P T )
(4)
We have ~e __ FeTDeFe D e _ D - D p
(5)
In the fraction model an element of volume is divided into N portions of size ~h, each with its own elastic deformation tensor r/eh and its own dissipation process, leading to a Dph. The rate of deformation tensor D remains the rate of deformation tensor of the fictitious continuum. Denoting the temperature by T, the mass density by p, the internal energy per unit mass by e, and the free energy per unit mass by f, we have for each fraction p f k = p e k - Ts, p f k = pfk(qek, T)
(6)
The fraction stresses crh are expressed in terms of the free energy by crk - p \0r/e kf
(7)
determinin~ the macroscopic stress tensor ~ by their sum with the volume fractions ~ " as weighting factors: N
N
- Z
1
1
(8)
1
For changes of the state variables T and a k only rate equations can be given because of the thermodynamic irreversibility of the inelastic deformation and heat conduction processes. In the fraction model the energy dissipation per
329
5.5 F r a c t i o n M o d e l s for I n e l a s t i c D e f o r m a t i o n
unit volume, q), is the weighted sum of the dissipation functions of the N fractions: N
- Z
r)
(9)
1
In inelastic deformation the energy dissipation is the amount of work per unit volume that is transformed into heat. For the rate of change of the temperature, we then have from Fourier's law of heat conduction with cv for the specific heat:
pc,,'if = div(Kgrad T)+cp
(10)
The scalar product of internal state variables and rates of inelastic deformation gives the rate of energy dissipation. Since the symmetric stress tensors crk contribute to the energy dissipation solely through the symmetric inelastic deformation rates Dph, it would be necessary to introduce additional state variables, producing dissipation with the skew symmetric tensor f~P in order that this tensor may differ from the zero tensor. In the fraction model the tensor f~P is taken to be the zero tensor and the constitutive equations for the tensors D pk are formulated in terms of energy dissipation functions q0k:
(,oh -
(Tk D pk} --(FekcrkFek~,D pk} --(crk, ilPk),D p k - FekW-lrjPkFek-1
(11)
Here the weighted sum of the tensors T k determines the stress tensor of Cauchy, T, that occurs in the equations of motion of the continuum. The tensors T k are state variables in state space, not subjected to the equations of motion that apply in the physical space. N
N
- Z
- Z
1
/12)
1
For a solid that permits a thermodynamic description, change of volume is according to experiments a purely elastic phenomenon. Hence tr D p k = tr l:lPk -- 0 and only the deviator of the stress tensors,
T a -- T -
(ltr T)I, cra -
contributes to the energy dissipation. of Tah and a ak. If the fraction has an function of the stresses through the thereby the required invariance with
(
c r - (tr cr)I
(13)
Then D pk and tjph must be proper duals isotropic behavior, the function r is a invariants of the stress tensor and has respect to rotation. Then we have in
DPk _ ~pk Talk Oq~Ok , cgTak
) -1
Oq~k
oqT
(14)
a tensor with the required duality properties. In case a fraction would have anisotropic behavior, the energy dissipation must be considered to be a
Besseling
330
function of the objective stresses crh and the derivatives of cpk with respect to crah provide the proper duality for the tensor/jph. The constitutive equations for the inelastic deformation of one fraction are presented for the general case. The superscript k, denoting the fraction, will be omitted. In terms of a yield function @ with a yield stress rrr, 9 -- @(or, crF)
(15)
the dissipation function is defined by
(16)
It determines the creep rate/jc by O~ _ ~
cra O~
' ~-d~a
O~
acra
(17)
Experiments with metal specimens show that at a certain stress level the dissipation rate starts to increase very rapidly, such that very high rates of strain have to be imposed for a small increase of stress. This implies a high nonlinearity of the dissipation function at these stress levels. A dissipation function with this type of high nonlinearity leads to great difficulties in numerical simulations of the inelastic deformation process. From the start of the theory of plasticity, a discontinuity in the constitutive description in the form of a yield surface has been introduced, by which these difficulties are overcome. The yield function defines a closed surface in stress deviator space, limiting the stress states that can be reached, 0z@z-1
(18)
The value - 1 for 9 corresponds to zero stress. The energy dissipation must be positive, and the dissipation function is defined such that it is equal to zero for + fl = 0. The parameter /3 introduces a threshold for the stresses, below which no energy dissipation takes place. For ~ = 1 this is the state of zero stress. We have 0
cp>0
for
~+fl>0,
else
cp--0
(19)
With the aid of the fourth-order tensors
~=a~f ~_(o~b-~d'~ ~a~/-'(~a~ ~ /'~ae'~ (20) o,7~o,~, \ FD-d~ U | t, F~d)
331
5.5 Fraction Models for Inelastic Deformation
the compression modulus C, and the coefficient of cubic thermal expansion e, the rates of stress are defined by the following: ifr
and
else if
+=0:
6-=(o~~
9 + fl > 0:
I-h~'il c-C~tI,
6- = ~g~ - / I c) - C e t I ,
else
(21)
d = ~~0 - C~2I.
Here h is a small positive or negative hardening coefficient that determines the change of the yield stress in plastic deformation. From the consistency condition r 0 follows h ~a-dad,~~0
+ ~
dF -- 0
(22)
When the material behavior is simulated by a number of fractions with different values for the yield stresses cry, with different values for the threshold values fl, and possibly with different values for the hardening coefficients h, the typical anisotropic, memory-sensitive characteristics of inelastic deformation are described by the model.
5.5.4
DETERMINATION
OF THE
MODEL PARAMETERS The validation experiments have mainly been restricted to materials which in their annealed state could be considered to have isotropic properties. Furthermore, these experiments did encompass strains not larger than a few percentage points in nearly time-independent plastic deformation. Here the determination of the model parameters will be discussed for this particular application. For small strains the stresses T h and ~rk are related by the orthogonal transformation R. It represents the rotation of the neighborhood of the material point of the continuum with respect to the initial configuration T k = RakR r. Because of the isotropic properties of the fractions, the fourthorder tensors ~ and ~ , as well as the yield function 9 and thereby the dissipation function ~0, are invariant under rotation. The von Mises yield condition is based upon the stress invariant Jz - 89 tr (od)Z. We have with the shear modulus G 3J2 -( I ) - - cr--~F
1,
o - G//~d + C( l t r t / - ~(T - To))I
(23)
Besseling
332
In view of its limited significance in the envisioned applications, a simple dissipation function suffices: qo --~,(T).sin h (0.88137 ~ + fl)
(24)
In this expression 7(T) contains the temperature dependence of the creep process. Since sin h 0.88137 = 1.0000, the dissipation rate of a fraction upon reaching the yield surface is equal to 7(T). Because of the linear term in the series expansion of this function, it proves to be much more suitable for the description of stress relaxation effects than the well-known power law of creep. For the higher power values the relaxation rate decreases too rapidly with the value of the excess stress 9 4-//. There remains the determination of the number N of fractions, their weighting factors 6k, the values of their yield stresses 0-~, the hardening coefficients hi, and the values of their threshold values ilk. Simple tests on tensile specimens may provide the necessary data. If more than four fractions are taken, a smoother representation of the stress-strain diagram is obtained, but the accuracy of the simulation of complex stress histories will not be greatly enhanced in terms of predicted stress values. Furthermore, the model is not very sensitive to the precise location of the breakpoints on the model stress-strain diagram that mark the onset of yielding of one more fraction. This is, of course, a property that is essential for the usefulness of the model. The determination of the model parameters for time-independent plasticity is best illustrated for an experimental stress-strain diagram approximated by the Ramberg-Osgood formula with material parameters E, 0-3, m. = 0"3
+
(25)
0"3
We specify the breakpoints by their values for the secant modulus Es, denoting the breakpoints with subscript p. For 0"p -- EsP~/pwe have
;:
1
1)1
The value of 0-3 corresponds to the secant modulus, which has a value of 0.7E. Usually a good choice for the breakpoints is given by the following values of the secant moduli:
Es/E = 0.98, 0.85, 0.7, 0.4, 0.1
(27)
Table 5.5.1 shows the successive determination of the breakpoints and of the values for ~k and a). The yield stresses a~ are the initial values of the flow stresses @ of the various fractions, which are subject to change because of
333
5.5 Fraction Models for Inelastic Deformation
TABLE 5.5.1 Determination of Fraction Parameters from Ramberg-Osgood Diagram with m = 10 a n d v = 0 . 3 .
p
Ef/E
~/~
E~,/~
1 2 3 4
0.98 0.85 0.7 0.4
0.713 0.906 1 1.140
(0.728) 1.066 1.429 2.873
A~/(EA~) 0.547 0.259 0.103 0.023 ~ h = 0.020
5
0.1
1.403
k
2~=~ 0 n
1 2 3 4
0.498 0.783 0.927
14.025
0~
~,~/~
0.498 0.285 0.144
0.713 1.091
0.073
1.495 3.139
1
isotropic hardening. This isotropic hardening may, in a first estimate, be described by the same value h for all fractions. The values of h k do not greatly affect the values of O k and ~), but they are essential because of the cumulative effect of the isotropic hardening. An estimate for h is needed for the determination of the values of Ok. A further refinement of the model with different values for h k is mainly of value for cyclic loading. With the aid of the well-known relations G
__
E E 2(1 + v)'C - 3(1 - 2v)'
we derive the following expressions: ~y _
~3
3
Erl k
2(1 + v) \ ~3
(1 - h) ~
0 ~-
1 -
2 v ~rk
3
(28)
1 - A~/(EzX~) 1 - ~(1 - > / A ~ / ( E A ~ )
These expressions show that in the case of incompressibility ( v - 1) the volume fractions follow directly from the slope of the stress-strain curve between breakpoints, while the initial yield stresses of the fractions are given by the strain at the breakpoints, multiplied by the elastic modulus. Note that for the slope of the curve from p = 1 to p = 2 the line is taken to start on the elastic part of the diagram. The value of Eql/a3 is not used. Even far below the true creep range of temperatures, inelastic material behavior may be accompanied by strong strain rate effects (differences in predicted stress values of the order of 10%). In the tensile test a considerable drop in stress can occur if the loading of a specimen is interrupted at constant strain for several minutes. In a first approximation the drop in stress in such a relaxation period turns out to be more or less constant for the whole range of stress over the inelastic part of the tensile curve. Based upon this observation,
Besseling
334 TABLE 5.5.2
Iterative Determination of Hardening Coefficients.
k
~kk(1 -- h k)
h k//h4
1 2 3 4
0.489 0.279 0.141 0.072
0 0.1557 0.322 1
E n =h l
I~nhn/h4
0 0.044 0.091 0.163
hk 0 0.019 0.039 0.120
~kk 0.489 0.285 0.147 0.081 1.002
k ft. hn/h4 ~--'~n=l 0 0.044 0.092 0.173
h~
~kk
0 0.018 0.037 0.115
0.489 0.284 0.147 0.080 1.000
the threshold stress value for each fraction may put equal to the yield stress, diminished by this drop of stress, Aa. The values of fl then follow from
Usually, for a good correspondence between the experimental curves for cyclic loading and the curves according to the fraction model, an adjustment of the hardening coefficients h k for the various fractions is needed. This adjustment is best carried out if the values are chosen such that the curve for the first reversed loading is well represented. Often good results are obtained with values for hk which are proportional to the difference between the yield limit ayk of the fraction k and the yield limit of the first fraction. Obviously, this implies h I - - 0, and as a result, the initial yield surface does not expand, leading to a pronounced Bauschinger effect and followed by a rapidly increasing hardening. For unequal values of h k the determination of their values is best solved by an iterative process. For the data from the Ramberg-Osgood formula considered in Table 5.5.1, this iterative process is illustrated in Table 5.5.2. The values of ~k (1 -- h k) remain constant in the iteration process, since they determine the slope between breakpoints. Their values are given by ~kh(1 - h) in accordance with the values of Table 5.5.1. After the values of h k have been calculated by running down the first column and up the second column of each iteration, the new values for @k follow from the known values of ~kk (1 - hi). Sufficient accuracy was obtained in two steps.
5.5.5 HOW TO USE THE MODEL Any finite element analysis of inelastic deformation problems will introduce the constitutive equations of the material in sampling (or integration) points
5.5 Fraction Models for Inelastic Deformation
335
of the elements. If these points are material points of the continuum, as they, in case of small strains, usually will be, the fraction model equations may be directly applied. Of course, for each sampling point in which inelastic deformation has occurred, the history of the internal state variables must be recorded. The history of the variables of all fractions that have contributed to inelastic deformation must be traced by storing the values of the fraction stresses with the model parameters. In the case of large deformations, the mesh of finite elements has to be redefined in a continuous or discontinuous process, but in either case the problem of keeping track of the material points of the c o n t i n u u m with their history arises.
REFERENCES 1. Besseling,J. E (1953). A theory of flow for anisotropic hardening in plastic deformation of an initially isotropic material. Nat. Aero. Res. Inst., Amsterdam, report $410. 2. Besseling, J. E (1958). A theory of elastic, plastic and creep deformations of an initially isotropic material. J. Appl. Mech. 25: 529-536. 3. Besseling,J. E (1985). Models of metal plasticity: Theory and experiment, in Plasticity Today, pp. 97-114., Sawczuk, A., and Bianchi, G., eds., Elsevier Applied Science Publishing. 4. Lambermont,J. H., and Besseling,J. E (1970). An experimental and theoretical investigation of creep, in Creep in Structures, pp. 38-63, Hult, Jan, ed., Springer-Verlag. 5. Besseling, J. E, and Giessen, E. van der. (1994). Mathematical Modelling of Inelastic Deformation, Chapman & Hall, 324 pages.
SECTION
5.6
Inelastic Compressible and Incompressible, Isotropic, Small Strain Viscoplasticity Theory Based on Overstress (VBO) E. KREMPL1 and K. HO 2 1Mechanics of Materials Laboratory,
Rensselaer Polytechnic Institute,
Troy, New York 2yeungnam University, Korea
Contents 5.6.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Isotropic VBO Model for Inelastic Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Reduction to a Simplified, Isothermal, Incompressible Form . . . . . . . . . . . . . . . . . . . . . . 5.6.4 The Determination of VBO Constants from Constant Strain Rate Data . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
336 337 343 343 346
5.6.1 GENERAL REMARKS P h e n o m e n o l o g i c a l e x p e r i m e n t s are o n e of the f o u n d a t i o n s of the v i s c o p l a s ticity t h e o r y b a s e d o n o v e r s t r e s s (VBO). A t e n e t of m a t e r i a l s s c i e n c e [11] is
336
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
5.6
Compressible and Incompressible, VBO
337
that "the current behavior of a material is determined by its current state and the current loading conditions" (italics added by the present authors). A properly designed specimen represents the material and serves as the integrator of all the micromechanisms. From these responses the continuum model has to be synthesized. It is clear that an experiment-based approach captures the physics of material behavior; after all, acting micromechanisms are deduced from similar, if not identical, tests. The aim is to create an experiment-based, physical, small strain model that can be exercised like a real, servo-controlled testing machine with strain measurement on the gage section. A time integration program for a system of stiff, ordinary differential equations [5] is the numerical testing machine. VBO has been used at low, high, and variable temperatures with one basic formulation. A static recovery term becomes negligible for low homologous temperature. Terms that are present for variable temperature equal zero when the temperature is constant. Creep and plasticity are not separately formulated. All material constants are allowed to vary with temperature. To efficiently utilize the available space, provide an overview, and have the equations collected in one place, Boxes are used. A nomenclature listing, Box 5.6.1, precedes the display of the coupled, nonlinear differential equations in Box 5.6.2 for inelastic compressibility. Inelastic incompressibility is introduced in Box 5.6.3, where a simplified tensorial and a uniaxial VBO are derived from the general theory.
5.6.2 I S O T R O P I C V B O M O D E L FOR INELASTIC COMPRESSIBILITY The VBO formulation was started by Cernocky and Krempl [2], who showed that the overstress dependence enabled the modeling of longtime asymptotic solutions with a slope that can be negative, zero, or positive. The asymptotic solution provides a means of modeling the so-called flow stress region and has been very useful in modeling real material behavior. This aspect had not been discussed by Malvern [26,27] and Perzyna [28]. Krempl [13,14] has shown the capability of the overstress model to reproduce normal and "paradoxical" inelastic deformation behaviors that were found in the literature and in specially performed experiments [ 1, 12, 18, 20].
338
Krempl and Ho
BOX 5.6.1
gl 1 ~g22
.in
~0~g 0
Nomenclature
Coefficient of thermal expansion
1/ temperature
Rate sensitivity coefficient
No dimension
Shape function, 1 > ~ > Et/E Decreasing or constant, Eqs. 3, 11
No dimension
Rate of deformation, true strain rate tensor
1/time
Axial and transverse components of true strain rate in a uniaxial test
1/ time
Constants for normalization
1 / time
Strain beyond which the asymptotic solution applies approximately
No dimension
Effective inelastic rate of deformation,
1 / time
/tr(~.in~.in) -- 1/
V i 7 ~--~,
Eqs. 2, 10
Overstress invariant, Eqs. 2, 9
Stress
Inelastic Poisson's ratio, 0 < t / < 0.5, constant
No dimension
Variable, total Poisson's ratio
No dimension
Elastic Poisson's ratio, constant
No dimension
True (Cauchy) stress tensor
Stress
Isotropic stress
Stress
Ac
Constant, controls the speed with which saturation is reached
No dimension
Ai,Af
Initial and final value of A
Stress
339
5.6 Compressible and Incompressible, VBO
E, Et
F-.t
Drag stress, constant, or a growth law; D > 0
Stress
Elastic modulus, tangent modulus at maximum strain of interest based on total strain rate
Stress
Tangent modulus based on inelastic strain rate
Stress
l~t--E/(1-~) Flow function, positive, increasing, F[0] = 0; see k. Dimensionless flow function F -
e9in0 F^
1/time
No dimension
Curve fit of data, P = F[F/D] - F[0]
No dimension
Kinematic stress tensor
Stress
Equilibrium stress tensor, Eqs. 3, 11
Stress
Effective equilibrium stress
Stress
Identity
No dimension
Viscosity function, positive decreasing, k[0] r 0, F = F / ( E k ) , Eqs. 2, 10
Time
Static recovery function, positive, increasing
1/ time
True (Cauchy) stress deviator tensor
Stress
Temperature
Degree Kelvin
Time
Time
Material time derivative of x
1 / time
( )a
Deviatoric quantity
[ ]
Denotes "function of"
340
Krempl and Ho
VBO is based on the standard linear solid [14, 17]. The spring in front of the Kelvin element remains linear but the Kelvin element itself becomes nonlinear. The viscosity of the dashpot is represented by a nonlinear, decreasing function of the overstress. The "spring" of the Kelvin element has nonlinear and hysteretic properties. Although the actual responses of the standard linear solid and of VBO can be vastly different, some general properties remain the same. The existence of long-time asymptotic solutions for constant loading rate can be offered as an example. Also, in both models infinite time is required to reach the equilibrium stress in a relaxation experiment. BOX 5.6.2
Inelastic-Compressible, Isotropic VBO
Flow Law d(.l+v
v _
m
(tr(~))l +
(
l+r/ Ek[F/D]
rl
(~-g)-
(tr(a
-
EkEr/D]
g))I)
Y viscosity function ~m
(1)
+ 0r ~th
Invariants
F 2 = ((1 + ~/)2tr((o"- g)(o" - g)) + 012 - 2~/)(tr(a - g))2)/(1 + 2~/2)
= ---L-F =
F[F]; ~ = ((1 + r/)2tr(gg)+ (r/2- 2t/)(trg)2)/(1 + 2t/2) flow
(2)
function
Growth law for the equilibrium stress = ~
+
~
equilibrium stress rate
(~;)t~ + ~
elastic hard.
~- g
r
k[r/D] A 7-/-fl--F ~ /
inelastic hardening
nrs "-
Y
9
dynamic recovery
+ (1 - ~ ) f -
(3)
R[~]g static recovery
Growth law for the kinematic stress
a-g
IF]a-g (4)
kinematic stress rate
Growth law for the isotropic stress
~
A
isotropic stress rate
= Ac (Af - A)d) rate-dependent, complex forms possible
341
5.6 Compressible and Incompressible, VBO BOX 5.6.3
Inelastic-incompressible, isotropic VBO
Deviatoric, simplified, formulation, isothermal, ~/- 0.5, without static recovery ,~ _ ,~e7 + ~
(6)
(~e,) a_ ((1 + v ) / E ) i , tr ,~et = ((1 - 2v)/E)tr d
- 2 Ek[ra/D] - 2 rd _ r
tr((s -- gd)(s -- gd));
?h - rd/Ek - F;
~ - r162
( ~-g~ r~
,~ ~
ra
~_ r
tr(gagd)
gd_fa.)A + ( 1-~)~'d
~ -- EtF [~l
Ek
~~
s FJ
(7)
(s) (9) (10)
(11)
(12)
Poisson's ratios based on strain rates The ratios v and ~/are constant and are limited by 0 _< v _< 0.5 and 0 _< ~/<_ 0.5. rl is variable q _ _ t2__ A = vir/E + (n(~r - g)/(Ek[F/D])) ~ olE + ((,~- g)l(Ek[r/D]))
(13)
Uniaxial equations e
[F~l'~-g;r-,~-g
gA -f)
(14)
+ (1 - ~)f
) - ~ ' r [~] ~-g=r t~'~'n
(15) (16)
/...~ X
wh~r~ ~ -
' - ~ and ~milar h~ b~n u~d. \3/
The flow law, Eq. 1, Box 5.6.2, consists of three parts: the elastic, the inelastic, and the thermal strain rates, which are added to obtain the total true strain rate. The elastic strain rate is given as a total time derivative to ensure a path-independent elastic strain for variable temperature and temperaturedependent material properties [21, 22]. Naturally, for isothermal conditions
342
Krempl and Ho
the terms containing temperature rate as a multiplier vanish. The inelastic rate of deformation can be formulated using the viscosity function k[F/D] or the flow function F[F/D], see Eqs. 2 and 10 for their interrelation. The overstress invariant 1-', the effective inelastic strain rate (rate of deformation) q~ and the effective equilibrium stress ~ are normalized to the uniaxial state of stress and are given in Eqs. 2, 9, and 10. In the absence of the static recovery term, the equilibrium stress evolves mostly in a rate-independent manner. The applied stress must be different from the equilibrium stress to affect inelastic deformation. The equilibrium stress can be thought of as a measure of the strength of the defect obstacles that have to be overcome. The growth law for the equilibrium stress g, Eq. 3, has elastic and inelastic hardening terms as well as a dynamic recovery term. The last term, the static recovery term, plays a significant role at high temperature and is zero at low temperature. The term containing f is introduced in Eq. 3 to ensure the existence of the asymptotic, long-time solution for any slope [16]. The growth law simulates the hardening/recovery format attributed to Bailey/Orowan [4]. Note that only this term contributes in a time-dependent manner. The growth law for the kinematic stress is given in Eq. 4. The kinematic stress is the repository for modeling the Bauschinger effect and for modeling the slope of the stress-strain diagram at the maximum strain of interest. The constant inelastic Poisson's ratio r/complements the elastic Poisson's ratio v. The actual, variable Poisson's ratio see Eq. 13, can be easily calculated. When modeling solid polymer deformation [1, 6, 17] inelastic compressibility has to be accounted for. Following Lee and Krempl [21], who treated the anisotropic case, the formulation of Eq. 1 was arrived at by Ho [6]. The similarity of the expressions for the elastic and the inelastic strain rates is obvious. For r / = 0.5 the usual forms are obtained and are given in Box 5.6.3 for a simplified version. The compressible solution may be applied to cavitating, but otherwise incompressible, materials. In Eq. 3 the equilibrium stress growth law is written in terms of stresslike quantities. The formulation can also be given in terms of the strain rates by using their respective definitions, [16]. The first term on the right causes a nearly elastic growth with an initial slope of ~kE and establishes the quasielastic region. The second term is needed for variable temperature [22]. The next positive term represents the inelastic hardening, which is followed by the dynamic recovery term. The factor/~ in the dynamic recovery term of Eq. 3 and marked by nrs is zero for normal VBO. It is the repository for modeling zero and negative rate sensitivity and other unusual but important rate effects [7-9]. The modeling of different rate effects is a rather unique capability of VBO. The isotropic or rate-independent stress A primarily models cyclic hardening/softening. The simple formulation of the growth law given in m
5.6 Compressible and Incompressible, VBO
343
Eq. 5 is sufficient to demonstrate constant strain amplitude cyclic hardening or softening. For nonproportional loading extra hardening can be observed with some metals and alloys. For the modeling of such behaviors special growth laws are needed, e.g., [3, 29, 31].
5.6.3 REDUCTION TO A SIMPLIFIED, ISOTHERMAL, INCOMPRESSIBLE FORM The model given in Box 5.6.2 can be simplified and reduced to the usual inelastic incompressible, deviatoric formulation. It is further assumed that no isotropic hardening takes place and that static recovery terms are not operating. Only the equilibrium stress and the kinematic stress change. With these simplifications VBO models viscoplasticity with kinematic type hardening. Starting with Box 5.6.2, the following simplifications are introduced: 9 Inelastic Poisson's ratio is set to 7 = 0 . 5 to model inelastic incompressibility. 9 The equations are separated into hydrostatic and deviatoric parts. 9 The shape function ~ is reduced to a constant with the same name and the same limits as stated in Box 5.6.1. 9 The flow function F and five other constants (A, D, E, Et, and ~) are needed to describe the inelastic behavior with the simplified model when fl is set equal to zero. Following Majors [25], the simplified model has been proposed by Maciucescu et al. [24] and has been used by (Tachibana and Krempl [30]) for high homologous temperature applications. The simplified model is capable of modeling nonlinear rate sensitivity in strain and stress-controlled loading, creep, relaxation, the Bauschinger effect, and cyclic neutral behavior.
5.6.4 THE DETERMINATION OF VBO CONSTANTS FROM CONSTANT STRAIN RATE DATA The simplified VBO is applied to test conditions where recovery effects are negligible. Therefore, Eqs. 13-16 apply. Recent room temperature data by Khan and Liang [10] on bcc alloys cover a wide range of strain rates. Figure 5.6.1 shows stress-strain diagrams for a Ta-2.5 W alloy with strain rates ranging from 10 -6 to 1 1/s. The high static strain rate data show typical
344
Krempl and Ho
bcc behavior. The stress-strain curves are nearly equidistant in the flow stress region. The elastic modulus E and the tangent modulus Et at the maximum strain of interest can be directly obtained from the graph. For the determination of the flow function the relationships derived from the asymptotic solution and their interpretation are important, see Eqs. 10-14 of Krempl [16]. While the model given in Box 5.6.3 is valid for every loading, the asymptotic solution applies when plastic flow is fully established, i.e., at strains ~ > 0.04 in Figure 5.6.1. The equilibrium stress g and the constant drag stress D have to be determined together with the flow function F. For nonlinear problems a normalized formulation is desirable. Accordingly, F - e0F where P is dimensionless and where e90i, is an arbitrary constant with dimension of reciprocal time. Substitution into Eq. 14 yields k / k 0 - / ~ [ F / D ] where k i , = ~ ( 1 - Et/E) has been used and where the arbitrary s0 is the total counterpart of %.'i" In determining the overstress, the unique evolution of g in the asymptotic region is factored in as well as the fact that the equilibrium curve is below the slowest stress-strain curve. The finite time endpoint of a relaxation test started at a strain ~ can be below the slowest stress-strain curve [19]. With these guides the equilibrium stress is estimated. Next the condition F I l l - 1 is imposed. From actual stress-strain curves for various strain rates such as shown in Figure 5.6.1 the overstress at a given 9i n
700
:
"
............................. ;
i ,~
400
i
300
A........... i ..................
;
............ i ..................
2OO 100 0.,,
| l
0
i I , ..... I .......................
I
0.05
....
0.1 True
:
]
l ;
l
"
0.15
o i vs OIE-02
l/s
0 IE-06 l/s
I '
0.2
Strain
Figure 5.6.1 The influence of strain rate on the stress-strain diagrams of a Ta-2.5W alloy at ambient temperature. The symbols are selected points taken off the original stress-strain diagrams in Figure 6 of Khan and Liang [10]. The thick lines are the numerical simulation of the influence of strain rate. The thin lines, top at a strain rate of 1000 1/s and bottom for 10 -8 l/s, are extrapolations to demonstrate that VBO can model a large range of strain rates.
345
5.6 Compressible and Incompressible, VBO
strain rate and a strain ~ in the flow stress region (~ > 0.04) is estimated. Here the overstress for the strain rate of 1 1/s was found to be equal to 246 MPa. These choices and the unity condition at the beginning of this paragraph yield D = 246 MPa and k0 = 1 1/s. Knowing D, Et, and the stress, the isotropic stress A can be easily calculated. Since the strain rates cover several decades but stresses do not, log(k/k0) = y is plotted as a function of the stress ratio F / D = x using an Excel trend line analysis or equivalent. It yields, in the present case, a linear polynomial approximation with R2=0.9969. The log-plot cannot be extended to the strain rate origin. To satisfy F[0] = 0 we set F = g0(1 - Et/E) (F [F/D] - F[0]) where P is the function obtained from the trend line analysis that has all the properties of F except that it is not exactly zero for zero argument. The constants are listed in Table 5.6.1. The value of ~ has to be determined next. Selecting a value and numerically integrating the constitutive equation is a good procedure. To avoid possible overshoots at the transition from quasi-elastic to inelastic behavior, ~ < . is recommended. It is also possible to tailor ~ so that "yield points" develop. Figure 5.6.1 shows such a yield point for the stress-strain curve of 1000 1/s. Examples of experimentally observed overshoots are given by Khan and Liang [10]. Numerical experiments using the data given in Table 5.6.1 were performed and the results are plotted in Figure 5.6.1 together with the experimental data. The correlation is very good. The same calculation was repeated with A = 170 MPa and D = 266 MPa with equally excellent results. There is no doubt that other values of A can be found that would match the stress-strain data. It appears that the stress-strain rate data are insufficient to determine the material constants of this "unified constitutive law" uniquely. Additional experiments of a different nature are needed to uniquely determine A (or D) and @. Creep or relaxation tests are potential candidates. Relaxation data are preferred since the strain rate range that can be obtained in a single run is exceptionally high [23, p. 264]. The task is then to determine both ~ and A so that the relaxation and the stress-strain data at
TABLE 5.6.1
VBO Material Data for Ta-2.5 W Alloy of Khan and Liang [10]
A MPa
D MPa
E GPa
/~t MPa
~0 1/s
190
246
178
660
1
F --
0.4
~o(i -
Et/E)(F[F/D]
log/~ = 7 . 2 7 8 ( F / D )
-
-
F[O]),
7.183
i/s
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Krempl and Ho
m
700500600 j !!!!!!!!!!!i iiiiii !iiiiiiiiiiiiiiiiiiii : 0
i400 ..................... o !L o i 300
:
!
o
......... =:i: o .................
..... ~........... r................. ?.................. ~....................
200 100 0
0
0.05
0.1
0.15
0.2
True Strain
Figure 5.6.2 Numerical simulations of a repeated strain rate change experiment using the data of Table 5.6.1. No strain rate history effect is modeled in the numerical and in the laboratory experiments. The numerical simulations exhibit over- and undershoots, which Figure 15 of Khan and Liang [10] does not. However, they have been observed in other laboratory experiments.
various strain rates are modeled. This can be done when such data are available. Numerical repeated strain rate change experiments are shown in Figure 5.6.2. At the transition points over- and undershoots are seen which are not found in the present data on Ta-2.5 W; see Figure 14 of Khan and Liang [10]. Examples of overshoots are displayed in Figure 4 of the same paper for Ta. Neither the experimental nor the numerical results show a strain rate history effect.
REFERENCES 1. Bordonaro, C. M., and Krempl, E. (1992). The effect of strain rate on the deformation and relaxation behavior of Nylon 6/6 at room temperature. Polymer Engineering and Science 32: 1066-1072. 2. Cernocky, E. P., and Krempl, E. (1979). A nonlinear uniaxial integral constitutive equation incorporating rate effects, creep and relaxation. International Journal of Non-Linear Mechanics 14: 183-203. 3. Choi, S. H., and Krempl, E. (1993). Viscoplasticity Theory Based on Overstress: The Modeling of Biaxial Hardening Using Irreversible Plastic Strain, Advances in Multiaxial Fatigue San Francisco, American Society for Testing and Materials. 4. Gittus, J. (1975). Creep, Viscoelasticty and Creep-Rupture in Solids, New York: Halsted Press, a Divison of John Wiley and Sons, Inc. 5. Hindmarsh, A. C. (1983). ODEpack, a Systematized Collection of ODE Solvers. Scientific Computing. Amsterdam, North Holland: R. S. e. a. Stepleman. 6. Ho, K. (1998). Application of the Viscoplasticity Theory Based on Overstress to the Modeling of Dynamic Strain Aging of Metals and to Solid Polymers, Specifically Nylon 66. PhD Thesis,
5.6 Compressible and Incompressible, VBO
7.
8.
9.
10.
11. 12.
13. 14. 15. 16.
17.
18. 19.
20.
21.
22.
23. 24.
347
Mechanical Engineering, Aeronautical Engineering & Mechanics, Rensselaer Polytechnic Institute, Troy, New York. Ho, K., and Krempl, E. (1999). Extension of the viscoplasticity theory based on overstress (VBO) to capture non-standard rate dependence in solids. International Journal of Plasticity (In press). Ho, K., and Krempl, E. (2000a). Modeling of positive, negative and zero rate sensitivity using the viscoplasticity theory based on overstress (VBO). Mechanics of Time-Dependent Materials 4: 21-42. Ho, K., and Krempl, E. (2000b). The modeling of unusual rate sensitivities inside and outside the dynamic strain aging regime. J. of Engineering Materials and Technology 123: 28-35. Khan, A. S., and Liang, R. (1999). Behaviors of three BCC metal over a wide range of strain rates and temperatures: Experiments and modeling. International Journal of Plasticity 15 (10): 1089-1109. Kocks, U. E, et al. (1986). Physical and Phenomenological Plasticity, p. 56, Los Alamos, NM: Los Alamos National Laboratory. Krempl, E. (1979). An experimental study of room-temperature rate sensitivity creep and relaxation of AISI Type 304 Stainless Steel. Journal of the Mechanics and Physics of Solids 27: 363-385. Krempl, E. (1987). Models of viscoplasticity: Some comments on equilibrium (back) stress and drag stress. Acta Mechanica 69: 25-42. Krempl, E. (1995). From the Standard Linear Solid to the Viscoplasticity Theory Based on Overstress, Computational Engineering Science, Hawaii: Springer. Krempl, E. (1995). The overstress dependence of the inelastic rate of deformation inferred from transient tests. Materials Science Research International 1 (1): 3-10. Krempl, E. (1996). A small strain viscoplasticity theory based on overstress, in Unified Constitutive Laws of Plastic Deformation, pp. 281-318, Krausz, A., and Krausz, K., eds., San Diego: Academic Press. Krempl, E., and Ho, K. (2000). An overstress model for solid polymer deformation behavior applied to Nylon 66, in Time Dependent and Nonlinear Effects in Polymers and Composites STP 1357: 118-137, Schapery, R. A., and Sun, C. T., eds., West Conshohocken, PA: American Society for Testing and Materials. Krempl, E., and Kallianpur, V. V. (1985). The uniaxial unloading behavior of two engineering alloys at room temperature. Journal of Applied Mechanics 52: 654-658. Krempl, E., and Nakamura, T. (1998). The influence of the equilibrium growth law formulation on the modeling of recently observed relaxation behaviors. JSME International Journal, Series A 41: 103-111. Kujawski, D., et al. (1980). An experimental study of uniaxial creep, cyclic creep and relaxation of AISI Type 304 stainless steel at room temperature. Journal of the Mechanics and Physics of Solids 28: 129-148. Lee, K. D., and Krempl, E. (1991a). An orthotropic theory of viscoplasticity based on overstress for thermomechanical deformations. Internationhl Journal of Solids and Structures 27: 1445-1459. Lee, K. D., and Krempl, E. (1991). Uniaxial thermomechanical loading: Numerical experiments using the thermal viscoplasticity theory based on overstress. European Journal of Mechanics, A/Solids 10: 173-192. Lemaitre, J., and Chaboche, J.-L. (1994). Mechanics of Solid Materials, Cambridge: Oxford University Press. Maciucescu, L., et al. (1999). Modeling the deformation behavior of a Pn-Pb solder alloy using the simplified viscoplasticity theory based on overstress (VBO). Journal of Electronic
Packaging 121: 92-98.
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25. Majors, P. S., and Krempl, E. (1994). The isotropic viscoplasticity theory based on overstress applied to the modeling of modified 9Cr-lMo steel at 538C. Materials Science and Engineering A186: 23-34. 26. Malvern, L. E. (1951). Quarterly of Applied Mathematics 8: 405. 27. Malvern, L. E. (1951). The propagation of longitudinal waves of plastic deformation in a bar exhibiting a strain rate effect. Journal of Applied Mechanics 18: 203-208. 28. Perzyna, P (1963). The constitutive equations for rate sensitive plastic materials. Quarterly of Applied Mathematics 20: 321-332. 29. Ruggles, M. B., and Krempl, E. (1990). The interaction of cyclic hardening and ratchetting for AISI Type 304 stainless steel at room temperature - - I Experiments and II Modeling with the viscoplasticity theory based on overstress. Journal of the Mechanics and Physics of Solids 38: 575-597. 30. Tachibana, Y., and Krempl, E. (1998). Modeling of high homologous temperature deformation behavior using the viscoplasticity theory based on overstress (VBO): Part III A simplified model. Journal of Engineering Materials and Technology 120: 193-196. 31. Tanaka, E. (1994). A non-proportionality parameter and a cyclic viscoplastic model taking into account amplitude dependency and memory effects of isotropic hardening. European Journal of Mechanics, A/Solids 13: 155-173.
SECTION
5.7
An Outline of the Bodner-Partom (B-P) Unified Constitutive Equations for Elastic-Viscoplastic Behavior SOL R. BODNER Technion-Israel Institute of Technology, Haifa 32000, Israel
Contents 5.7.1 5.7.2 5.7.3 5.7.4 5.7.5 5.7.6
Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the Basic B-P Model . . . . . . . . . Modifications of the Model . . . . . . . . . . . . . . . . . Identification of the Material Parameters . . . . Integration of Equations and Implementation into Computer Programs . . . . . . . . . . . . . . . . . . . 5.7.7 Table of Material Parameters . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
349 350 350 353 353 355 355 356
5.7.1 VALIDITY The B-P m o d e l of isotropic elastic-viscoplastic b e h a v i o r is i n t e n d e d to be r e p r e s e n t a t i v e of the m a c r o s c o p i c r e s p o n s e of metals a n d metallic alloys to l o a d i n g over a very wide range of strain rates, t e m p e r a t u r e s , a n d l o a d i n g circumstances. It is an i n c r e m e n t a l t h e o r y c o n s i s t e n t w i t h the r e q u i r e m e n t s a n d c o n s t r a i n t s of c o n t i n u u m m e c h a n i c s a n d t h e r m o d y n a m i c s a n d is Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
349
350
Bodner
considered "unified" in that plasticity and creep are particular response characteristics based on a single inelastic strain rate variable. The model is also "unified" in a more general sense by not separating a fully elastic region from the overall response (no defined yield stress). A small strain formulation is presented here; extensions to large strains have been developed.
5.7.2 F O R M U L A T I O N The B-P elastic-viscoplastic constitutive equations do not require a yield criterion or loading-unloading conditions. As a consequence, the elastic and inelastic strain rate variables are generally nonzero throughout the deformation history, and the same equations apply for all circumstances. The macroscopic response properties indicated by the model include strain rate sensitivity and temperature dependence of inelastic deformation, stress saturation under imposed straining, isotropic and directional hardening for both monotonic and reversed loadings (e.g., the Bauschinger effect), primary and secondary creep (by including thermal recovery of hardening), and stress relaxation. Hardening corresponding to resistance to inelastic straining is treated by introducing load history-dependent variables in the kinetic equation for the inelastic strain rate and by developing suitable evolution equations for their development with loading history. The basic set of equations described in Section 5.7.3 is applicable for many metals and loading conditions. For enhanced generality, a few modifications for special conditions have been proposed which are described in Section 5.7.4.
5.7.3 D E S C R I P T I O N O F T H E BASIC B-P M O D E L ( F R O M [ 1, 21) a. Additivity of elastic and inelastic strain rates:
b. Plastic flow law (direction of plastic straining) and incompressibility condition:
351
5.7 B-P Unified Constitutive Equations
where sij is the deviatoric stress, sij = erij- (1/3)akkaij, and 2 is a positive scalar function of stress, the overall hardening variable, and temperature, which can be treated implicitly. c. Expressions for the strain rate components:
/:~ -- (lIE)[(1 +
v)dij - VdkkSij]
(3)
which is Hooke's Law in rate form; and the proposed kinetic equation for inelastic deformation as a function of state quantities is
,~
- Do exp
- -~ it,o.2SS,] j o'df
(4)
Equation 4 is a general growth function that indicates that ~ is close to zero at low values of effective stress r~cff followed by a region of rapid growth leading to a saturation value. In Eq. 4, erdf is defined by ryeff- [(3/2)sidsij] 1/2, n controls strain rate sensitivity, and Z is the overall scalar hardening variable. The quantity Do is the presumed limiting value of inelastic strain rate in shear and is usually set to be 108 sec -1, although a lower value, 104 sec -1, has been used in exercises at low strain rates ( < 10sec -1) to avoid possible numerical problems. Particular cases of uniaxial stress ~11 and simple shear z12 are
/:P1 --
~
\
o1, f
Do exp - ~ \~-11/ J
(5)
(6) where ?12 and ~:12 are the engineering shear strain rate and stress. The hardening variable Z consists of isotropic and directional components, Z = ZI+ Z D, where Z ~ is due to resistance presumed to be uniformly distributed throughout the volume and Z D arises primarily on the developed slip planes of the materials and is therefore dependent on stress history and its current value. Evolution equations for their development are Z , ' ( t ) - m,[Z1-Z'(t)]~;Yp(t)-A1ZI[ Z'(t)Z,-Z2] r~
z
(7)
(8)
352
Bodner
which is the component of the directional hardening tensor flij in the direction of the current stress, uij, where
--
Z1
Vij(t)
(9)
Uij(t) -- aij(t)/[ahl(t)crkl(t)] 1/2
(10)
Vii(t) -- flij(t) /[flkl(t) flhl(t)] 1/2
(11)
and
which is the current direction of riO" For both components, the measure of hardening is the plastic work rate (,Vp - aije,~, and the second terms in Eqs. 7 and 9 represent thermal recovery of hardening with Z z - Z2 and flij - 0 as minimum (annealed) values. (Use of Z1 in Eq. 9 is for dimensional purposes.) Initial values are ZI(o) = Zo and, if annealed, rio(O) - O, and the saturation value for Z I is Z1 and that for Z D is Z 3. From its definition, Eq. 8, Z D changes sign with stress reversal, with Z staying positive, and thereby accounts for the Bauschinger effect. The equations indicate stress saturation under imposed straining and secondary creep when Z I and Z D reach steady values. Temperature effects are treated implicitly through its influence on the material parameters. Higher temperatures generally lead to decreased values of n, which result in increased rate sensitivity and lower stressstrain curves. The hardening terms, Z0,Z1, etc., and the hardening rates m ~ , m 2 can be temperature-dependent, e.g. [29], while the terms in the thermal recovery functions, A1, A2, rl, r2, and the elastic constants are generally temperature-dependent. Numerical listings or functional forms for the temperature dependence of the material constants are therefore part of the characterization. Except for particular thermal history effects, such as those due to dynamic strain aging, the use of material constants at the current temperature obtained from isothermal test data is usually a valid procedure even under nonisothermal loading [3]. For the condition of only isotropic hardening without thermal recovery, the material constants reduce to n, Z0,Zl,ml with Do fixed and the elastic constants provided. In the case of uniaxial stress and constant plastic strain rate, the relevant Eqs. 5 and 7 could be integrated analytically [4]. An investigation on the thermodynamic consistency of the B-P equations is described in Reference [26].
353
5.7 B-P Unified Constitutive Equations
5.7.4 MODIFICATIONS
OF THE MODEL
a. Generalization of hardening rates for enhanced correspondence of stressstrain curves of strongly workhardening materials with test data [5], e.g., pure copper and aluminum"
ml - mlb + (mla -- mlb)exp[--mlc(Z I - Z0)]
(12)
m2 -- m2b + (m2a -- m2b)exp(--m2c ZD)
(13)
b. Generalization of isotropic hardening rate to represent possible strain rate dependence:
[ mxa
-- M a
1--~-
r t~eff ) q]]
(14)
where mla is the initial value of ml according to Eq. 12, ~eff is the effective "O (deviatoric) total strain rate, and eeff and q are additional material constants" this generalization is applicable for copper (and probably aluminum and iron) at very high strain rates, > 104 see-1 [6]. c. Representation of possible increase of isotropic hardening due to nonproportional cyclic straining by the introduction of a measure of nonproportionality in Eq. 7 [2]. d. Representation of possible changes in the evolution equation for isotropic hardening due to repeated stress reversals, see Reference [7]. Note: repeated reversed loading conditions with nonzero mean stress could lead to ratcheting for some materials such as stainless steels; the method proposed by Ohno and Wang [27] could be adopted to Eq. 9 for representation of that effect. e. Introduction of continuum damage as a state variable [8] modification of kinetic Eqs. 4-6 by Z--~Z(1-co),
0_
(15)
where co is a scalar load history dependent damage variable with a proposed evolution equation that is a function of stress and damage.
5.7.5 IDENTIFICATION OF THE MATERIAL PARAMETERS Procedures have been developed by various investigators to obtain the material constants for the B-P equations from test data. A method that is fairly direct and simple is described in References [9, 10]. First, most of the parameters at a given temperature can be obtained from monotonic uniaxial stress-strain curves over a range of strain rates for which thermal recovery of
354
Bodner
hardening is not effective. The parameters include those associated with directional hardening based on the observation that the rate of directional hardening m2 is usually significantly larger than that for isotropic hardening ml. As a consequence, directional hardening will saturate more rapidly and have greater influence in the low strain inelastic range, i.e., the region slightly beyond the essentially elastic range. The method involves determining the stress-plastic strain relation from a uniaxial stress-strain curve at a constant applied strain rate and determining the quantity, /I - -
1 dO'll dslP1
0"11
(16)
as a function of stress. The resulting curve is generally bilinear, with an upper slope m2 and a lower slope ml. The intersection of the extended curve with = 0 would be the saturation stress at which the total strain rate is equivalent to the plastic strain rate. From this information obtained at various applied strain rates, the parameter n could be obtained from the dependence of the saturation stress on plastic strain rate using Eq. 5. The constants Z0, Z1, Z3 can also be determined by this procedure. Recourse is then made to creep tests or very slow straining tests to obtain the parameters for the thermal recovery of hardening. It is noted that cyclic test data (with reversed loadings) are not required by the preceding procedure but could be used for checking and possible adjustments. Reliance on stress relaxation tests and tests involving rapid increases or decreases of applied strain rate are not recommended as references for obtaining material parameters for the B-P equations. Relaxation tests cover only a limited range of strain rates, whereas strain rate jump tests involve complicated histories of elastic and inelastic strain rates. Those tests serve well for examining of the predictive capability of the equations. Some minor modifications of the preceding procedure are described in References [11, 22]. The parameter values obtained by these procedures are not necessarily unique unless further refined by the sophisticated methods of References [12, 13], but are usually suitable for practical purposes. In the early development of the B-P equations, the factor [(n + 1)/n] appeared as a coefficient to the (Z2/Cr2eff)term in Eq. 4. Adjustment of the hardening parameters obtained from that equation with the equation of the present article is obtained by setting
Zo -[(n + 1)/n](1/2")Zo, etc.
(17)
m
where the Z i values are those derived from the equation with the factor (indicated by * in Table 5.7.1). The other material parameters are not affected by the factor.
355
5.7 B-P Unified Constitutive Equations
5.7.6 TABLE OF MATERIAL PARAMETERS Material parameters obtained for the B-P formulation are available in the literature. A representative listing is given in Table 5.7.1.
5.7.7 INTEGRATION OF EQUATIONS AND IMPLEMENTATION INTO COMPUTER PROGRAMS Methods for integrating the mathematically "stiff" equations of the B-P formulation have been developed in References [14-16], and that of Reference [16] seems to have received most attention. With a suitable integration scheme, the constitutive equations could be readily included in most finite element and finite difference programs. Computer simulations based on the B-P equations have been performed with: MARC [10], EPIC-2 [16], ABAQUS [17, 18], STEALTH [28], and also special programs developed for particular investigations, e.g., [19-21].
TABLE 5.7.1
Material Parameters.
Low strain rates ( < 1 sec-1); Do = 104 sec -1 Ni base alloy, B1900 + Hf (RT to 1093~ Inconel 718 (650~ Hastelloy-X (RT to 538~ Astroloy (RT to 982~ Aluminum (high purity) (277~ * Aluminum alloy 8009 (RT to 275~ Aluminum alloy AMG-6 (Russian) (RT to 400~ Zirconium alloy (Zr-2.Swt% Nb) (250~ Eutectic solder, 63/37 Sn/Pb, ( - 4 0 to 100 ~C)
References [9, 10] [17] [11] [19] [12] [11] [22] [18] [23 ]
High to very high strain rates (>10 sec-1); D o-- 108sec -1 C1008 steel, HY100 steel, 1020 steel A16061-T6, A17039-T64
[24]
Nickel 200, W2-tungsten, Armco iron (all at RT)
also in [ 16] Copper (high purity) (RT) Copper (high purity) (RT to 800~ A533B steel ( - 6 0 to 175~
~ = 2000 S- I )
[61 [29] [25]
*Kinetic equation containing factor as discussed in Section 5.7.5. Additional information on material parameters appears in a recent review of the B-P equations [30].
356
Bodner
REFERENCES 1. Bodner, S. R., and Partom, Y. (1975). Constitutive equations for elastic-viscoplastic strainhardening materials. ASME J. Appl. Mech. 42: 385-389. 2. Bodner, S. R. (1987). Review of a unified elastic-viscoplastic theory, in Unified Constitutive Equations for Creep and Plasticity, pp. 273-301. Miller, A. K., ed., London: Elsevier Applied Science. 3. Chan, K. S., and Lindholm, U. S. (1990). Inelastic deformation under nonisothermal loading. ASME J. Engng. Materials Techn. 112: 15-25. 4. Merzer, A., and Bodner, S. R. (1979). Analytical formulation of a rate and temperature dependent stress-strain relation. ASME J. Engng. Materials Techn. 101: 254-257. 5. Khen, R., and Rubin, M. B. (1992). Analytical modelling of second order effects in large deformation plasticity. Int. J. Solid Structures 29: 2235-2258. 6. Bodner, S. R., and Rubin, M. B. (1994). Modeling of hardening at very high strain rates. J. Appl. Phys. 76(5): 2742-2747. 7. Bodner, S. R. (1991). Further development of a viscoplastic constitutive model for high temperature applications, in High Temperature Constitutive Modeling Theory and Application, pp. 175-184, Freed, A. D., and Walker, K. P., eds., New York: Amer. Soc. Mech. Engin. 8. Bodner, S. R., and Chan, K. S. (1986). Modeling of continuum damage for application in elastic-viscoplastic constitutive equations. Engng. Fract. Mechanics 25(5/6): 705-712. 9. Chan, K. S., Bodner, S. R., and Lindholm, U. S. (1988). Phenomenological modeling of hardening and thermal recovery in metals. ASME J. Engng. Materials Techn. 110: 1-8. 10. Chan, K. S., Lindholm, U. S., Bodner, S. R., and Walker, K. P. (1989). High temperature inelastic deformation under uniaxial loading: Theory and experiment. ASME J. Engng. Materials Techn. 111: 345-353. 11. Rowley, M. A., and Thornton, E. A. (1996). Constitutive modeling of the visco-plastic response of Hastelloy-X and aluminum alloy 8009. ASME J. Engng. Materials Tech. 118: 19-27. 12. Mahnken, R., and Stein, E. (1996). Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations. Int. J. Plasticity 12(4): 451-479. 13. Fossum, A. E (1997). Parameter estimation for an internal variable model using nonlinear optimization and analytical/numerical response sensitivities. ASME J. Engng. Materials Tech. 119: 337-345. 14. Tanaka, T. G., and Miller A. K. (1988). Development of a method for integrating timedependent constitutive equations with large, small or negative strain rate sensitivity. Int. J. Numer. Methods Eng. 26: 2457-2485. 15. Rubin, M. B. (1989). A time integration procedure for large plastic deformation in elasticviscoplastic metals. J. Math. and Physics (ZAMP), 40: 846-871. 16. Cook, W. H., Rajendran, A. M., and Grove, D. J. (1992). An efficient numerical implementation of the Bodner-Partom model in the EPIC-2 code. Eng. Fract. Mechs. 41(5): 607-623. 17. Li, K., and Sharpe Jr., W. N. (1996). Viscoplastic behavior of a notch root at 650~ ISDG measurement and finite element modeling. ASME J. Engng. Materials Tech. 118: 88-93. 18. Zeng, H., and Sharpe, W. N. (1997). Biaxial creep strains at notch roots Measurement and modeling. ASME J. Engng. Materials Techn. 119: 46-50. 19. Dexter, R.J., Chan, K. S., and Couts, W. H. (1991). Elastic-viscoplastic finite element analysis of a forging die. Int. J. Mech. Sci. 33: 659-674.
5.7 B-P Unified Constitutive Equations
357
20. Zhu, Y. u and Cescotto, S. (1991). The finite element prediction of ductile fracture initiation in dynamic metal-forming processes. J. Physique III, 1. 21. Nicholas, T., Kroupa, J. L. and Neu, R. W. (1993). Analysis of a [00/90 ~ metal matrix composite under thermomechanical fatigue loading. Composites Engineering, 3: 675-689. 22. Senchenkov, I. K., and Tabieva, G. A. (1996). Determination of the parameters of the BodnerPartom model for thermoviscoplastic deformation of materials. Int. Appl. Mechs. 32(2): 132-139. (Translated from Prikladnaya Mekhanika 32(2), 64-72, Feb. 1996.) 23. Skipor, A. E, Harren, S. V., and Botsis J. (1996). On the constitutive response of 63/37 Sn/Pb eutectic solder. ASME J. Engng. Materials Tech. 118: 1-11. 24. Rajendran, A. M., Bless, S. J., and Dawicke, D. S. (1986). Evaluation of Bodner-Partom model parameters at high strain rates. ASME J. Engng. Materials Techn. 108: 75-80. 25. Dexter, R. J., and Chan, K. S. (1990). Viscoplastic characterization of A533B steel at high strain rates. ASME J. Pressure Vessel Techn. 112: 218-224. 26. Bodner, S. R., and Lindenfeld, A. (1995). Constitutive modelling of the stored energy of cold work under cyclic loading. Eur. J. Mech. A/Solids 14(3): 333-348. 27. Ohno, N., and Wang, J. D. (1993). Kinematic hardening rules with critical state of dynamic recovery, Part I: Formulation and basic features for ratchetting behavior, Part II: Application to experiments of ratchetting behavior. Int. J. Plasticity 9(3): 375-403. 28. Rajendran, A. M., and Grove, D. J. (1987). Bodner-Partom viscoplastic model in STEALTH finite difference code. AFWAL-TR-86-4098, Materials Laboratory, WPAFB, OH, 45433. 29. Bodner, S. R., and Rajendran, A. M. (1996). On the strain rate and temperature dependence of hardening of copper, Proc. of the Conf. of the Amer. Physical Society, Topical Group on Shock Compression of Condensed Matter, Seattle, Washington (1995), AIP Press, pp. 499-502. 30. Bodner, S. R. (2000). Unified plasticity: An engineering approach, www.eJournal-ofmechanics.org
SECTION
5.8
Unified Model of Cyclic Viscoplasticity Based on the Nonlinear Kinematic Hardening Rule j. L. CHABOCHE O.N.E.R.A., DMSE, BP 72, 92322 Ch~tillon Cedex, France and LASMIS, Troyes University of Technology, BP 2060, 10010 Troyes Cedex, France
Contents 5.8.1 Domain of Application . . . . . . . . . . . . . . . . . . . . . 5.8.2 Formulation of the Model . . . . . . . . . . . . . . . . . . 5.8.3 Description of the Model . . . . . . . . . . . . . . . . . . . 5.8.3.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.4 Determination of the Parameters . . . . . . . . . . . 5.8.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1
DOMAIN
358 359 360 362 363 365 367
OF APPLICATION
The presented viscoplastic model pertains to the class of unified constitutive equations in which only one inelastic strain is considered as an i n d e p e n d e n t variable [1]. Such a theory is particularly useful for the inelastic analysis of c o m p o n e n t s or parts submitted to complex multiaxial and thermomechanical cyclic loadings. The use of only one plastic strain ep to describe rapid and slow inelastic deformation processes is an i m p o r t a n t simplification that is particularly useful u n d e r nonisothermal conditions. The viscoplastic formulation allows the stress state to overpass the purely elastic domain, introducing the viscous part of the stress (or the overstress). The ratei n d e p e n d e n t plasticity can still be m o d e l e d as a limiting case.
358
Handbookof MaterialsBehaviorModels.ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
5.8 Unified Model of Cyclic Viscoplasticity
359
These equations have been developed and applied essentially for metals and alloys, even though there are some capabilities to also model polymers through the limit case of a nonlinear viscoelastic model [2]. The domain of validity covers the whole range of temperatures between room temperature and very high temperatures, at least for situations where the material can be considered stable. Microstructural evolutions (phase changes, precipitations, aging) are not considered in the present article. The presented constitutive model can be exploited for monotonic or cyclic loading conditions or for more complex situations. Introducing static recovery effects at high temperature allows us to correctly describe pure relaxation and pure creep behaviors (including the secondary creep). The only aspects not covered here concern the special overhardening effects encountered under cyclic nonproportional loadings and the plastic strain range memorization effects that can take place in some metals and alloys that have a low stacking fault energy. These effects need additional variables and models [3, 4].
5.8.2 FORMULATION
OF THE MODEL
The model can be presented in a general thermodynamic framework using the notion of local state and a set of internal state variables for the description of the present state (reversible processes) and of the dissipative evolutions (inelastic processes and associated hardening effects). One defines a state potential (for example, the Helmholtz free energy) from which derives Hooke's law and the relationship between hardening variables and their conjugate forces. The existence of dissipative potentials allows us to derive in a consistent way all the rate equations for the internal variables associated with irreversible processes. This framework will not be recalled here (it can be found in Reference [5]). The principal state variables associated with viscoplasticity are the plastic strain itself and the hardening variables. The plastic strain ~ p is defined by the total strain partition in the case of small perturbations: ~ - E e 4- E p, where and ~ e are, respectively, total and elastic strain (we do not consider here the thermal expansion). The hardening description combines kinematic and isotropic hardening. In the first case we use a set of second-rank tensors E i (associated with short-range and long-range interactions at the level of the microstucture). The corresponding conjugate forces are the back stresses X i, whose sum X is considered as the current "center" of the elastic domain (at least when considering its intersection with the deviatoric plane). The isotropic hardening is considered an unique scalar r, to which is associated
360
Chaboche
the conjugate force R, the variation of the yield stress. The possibility for a drag stress evolution will also be considered for isotropic hardening. Therefore, kinematic hardening corresponds to a translation of the elastic domain in the stress space, though isotropic hardening corresponds to a change in its size. Each hardening variable obeys a similar evolution equation, within an hardening-dynamic recovery-static recovery format, represented formally for the variable a:
(I)
da - h(... ) dep - r D(... ) a dep - r s(... ) a dt
The first term is the direct hardening associated with the plastic strain increment (and the corresponding increase of "obstacles" with the dislocation density increase), and the second one is the dynamic recovery effect, which is proportional to the variable a itself and to the plastic strain increment. This second term globally represents the annihilations of dislocations by cross slip, that take place without the influence of thermal activation. The last term, the static recovery, needs time to take place. It is thermally activated (much more important at high temperature) and represents rearrangements of dislocation microstructures by climbing mechanisms (and also possible recrystallisations).
5.8.3
DESCRIPTION
OF THE MODEL
All equations are summarized in Table 5.8.1, on the left for their general anisotropic form, on the right in the particular case of uniaxial tensioncompression. They are written here for a constant temperature. The temperature dependency is discussed in Section 5.8.4. The different aspects and functions of the model are presented successively: 9 The part of the free energy corresponding to the stored energy is assumed to be quadratic in the hardening variables ~i and r. The same anisotropy is assumed for all hardening stiffnesses, with N i - - C i N. Back stresses X i and yield stress change R derive from this energy.-In the isotropic case, we have N - 3Ia, where I a is the fourth-rank identity tensor. 9 The yield surface is defined by Hill's criterion, with the generalized norm ][ a ]]n - (a" H" a) 1/2, which reduces to the von Mises yield criterion in the isotropic case, with II a I ] - ( 3 ~ . ~)1/2, where ~ - a gL Tr a ~I is the deviator of a. 9 The viscoplastic potential is normally taken as a power function off. In the viscoplasticity case f > 0 is the viscous stress or overstress. D is the drag stress, either a constant or depending on isotropic hardening (see ,-..,
_
,"4
361
5.8 Unified Model of Cyclic Viscoplasticity TABLE 5.8.1
General and Uniaxial Forms of Constitutive Equations. Uniaxial
General anisotropic Free energy: 8 9 Ci ~ i "N :~ i + 89bQ r2
~-,i Ci~
-+- 1 bQr 2
Yield surface:
f-I~-xll.-R-k_<
f=
O
a-X
-R-k<_O
plastic strain rate modulus:
Viscoplastic potential:
D (f) n+l ~ = 77-i
D (fln+l n----n - -+ i
fin + 1 \ 7 6 /
+
Plastic strain rate:
~. ( ~ - x) ~p =/5 S i g n ( a - X)
e2p-Pll~_xi
H
P - II ~p ,-,-~
p=
e.p
Back stresses:
x = ~2~ c~ ~'~i Rate independent kinematic hardening: ~ , = ~ p - ~ , ~ ~' F " Xip Back stress rate equations: 5(i = Ci N" i p - q57i N" F 9X ~/5
~i = ~p -- ~b~ 7
i = Ci i p - q57iX~ ~'p
Static recovery of kinematic hardening:
~, _- (~,)
-
,)r-
with: ~b~(Xi)= " c-7
(
x,
< = ( < ) ~ - r (x,)x,
Ix, I~ Mi
Static recovery on the back stress:
~, =
(x,)~
- c,<~(x,)
r: x,
Isotropic hardening: /" =/5 - ~ib - Tr
Q
Sign(R - Qr)
R - bQr
other functions for isotropic hardening: 1 4)(r) = 1 + axbr
D - K(1 + a~br)
362
Chaboche
following text). One finds the plastic strain rate modulus/~ by deriving f~ in terms of f ' p - 0~2/c9f. A viscoplastic potential with two additive terms is also mentioned, with two exponents (fl > 1), that could be useful to limit the rate dependency in the large rate regime. The normality rule then gives the three expressions for the viscoplastic strain rate ~p. It is also shown that i b - ]]~plIH-~. 9 The back stress, used to translate the elastic domain (and the equipotentials) in the stress space, is the sum of the individual ones: 9 The kinematic hardening rules write independently for each variable ~i. First we have indicated the rate-independent case, with the hardening and dynamic recovery terms. The fourth-rank tensor F indicates the anisotropy of the dynamic recovery. It is replaced by 3 1 a in the isotropic case. The scalar function ~b is used to couple the kinematic hardening and the isotropic one (Marquis's format [6]). On the subsequent line we indicate the corresponding equations for each individual back stress X i. 9 The static recovery of kinematic variables is then introduced: (~i)~ denotes the rate-independent two first terms (the solution for high rate loadings). In the static recovery term a power function of ]] X~]]r = (X
5.8. 3.1 REMARKS 9 The kinematic hardening takes place and evolves rapidly during monotonic as well as cyclic loading conditions. On the contrary, the isotropic hardening is used to describe a progressive cyclic hardening ( Q > 0) or softening (Q<0) taking place slowly in monotonic conditions and saturating progressively for the stabilized cyclic conditions (after 10, 100, or 1000 cycles), under which it becomes inactive. 9 The stress can be decomposed into additive terms. Limiting here to the uniaxial case, we have o"
Xi + [k + R ( r ) + D(r)l~pl 1/"] Sign(ip)
m
i
(2)
5.8 Unified Model of Cyclic Viscoplasticity
363
Such a decomposition clearly shows the respective contributions for isotropic hardening: R(r) for the size of the purely elastic domain and D(r) for the value of the viscous stress, which is depending on the plastic strain rate. The isotropic hardening also plays a role in the current tangent stiffness associated with kinematic hardening, by the function 4(r). 9 If necessary, it is possible to consider an elastic domain initially reduced to one point ( k - 0) and to minimize the influence of isotropic hardening on the yield stress, giving a very low value to Q (but not zero, in order to still have the evolution of D(r) and 4}(r), because r is independent of Q: neglecting static recovery, it integrates in closed form to give r - l / b [ 1 - exp(-bp)]). 9 The evolution equations for the kinematic hardening variables ~i are valid even for varying temperature conditions (for any kind of temperature dependency for the parameters). Contrarily, the back-stress rate equations must necessarily incorporate an additional term proportional to the temperature rate. This term is easily deduced from the thermodynamic potential by considering parameters Ci as depending on temperature when we express the time derivative of X i - C i N 9~i. 9 The nonlinear kinematic hardening rule generally induces a too important ratcheting effect under nonsymmetrical cyclic stress control (or mean-stress relaxation under cyclic strain control). Introduction of a threshold in the dynamic recovery term allows for better descriptions (Reference [5]).
5.8.4 DETERMINATION
OF THE PARAMETERS
The model is designed to be fairly general for both monotonic and cyclic conditions at low or high temperatures. There are three main aspects of the model to be determined successively: (1) the rate-independent response (an advantage of this unified model is that it is applicable when not considering the strain rate effect), (2) the viscoplastic response (but without the static recovery), and finally, (3) the static recovery effects. In each case one can separate between initial (monotonic) and stabilized cyclic conditions. The test data can be chosen with great flexibility, depending of the set of influence we try to determine. Clearly, if we want a model to be used in cyclic conditions, we need cyclic tests. Obviously, if we want the effect of strain rate (viscoplastic model), we need tests where the strain rate has varied by a large amount (they can be creep or relaxation for monotonic as well as cyclic conditions). Clearly also, if we want to identify static recovery effects, we need long-duration tests at high temperature (creep or relaxation or recovery tests).
Chaboche
364 Let us develop the three determination steps just mentioned:
1. Determination of hardening rules in a rate-independent scheme. We assume to have uniaxial monotonic and cyclic tests (with stress-strain responses at the stabilized cycle), all of them realized in approximately the same high strain rate regime (say, 10 -3 to 10 -2 s-i). From the cyclic curve, considering ip ~ Ct at the peak stresses, one can identify the following relation obtained by integrating the kinematic hardening models for uniaxial conditions: V x Ci
Ae v
Aa _ k* + Q + Zi_, / - - t a n h 7i ~ 2 " 7i
(3)
in which k* represents the sum k + (1 + aK)K[~1/n assumed to be constant approximately. Eventually 7i is replaced by 7i~boo = 7i/(1 + ax) if coupling between isotropic and kinematic hardening is considered. In practice, if we have a sufficient number of kinematic hardening models (three, for example), we can try to adjust k* + Q to be as small as possible. The third variable can be assumed to be linear (73 = 0), and the parameter C3 is adjusted as the tangent stiffness to the cyclic curve Ao - Aep for large strain ranges (2 or 3%. For the rapid kinematic hardening, we can take high values for 71 (typical values between 2000 and 10,000) to obtain a smooth transition between elastic regime and moderate plastic flow. C1, C2, ~'2 follow from the experimental data in the intermediate domain 0.02%
~Ci
o- - k* 4- Q[1 - e x p ( - b ep)] 4- . 7 (1 4- axb gv)[1 - exp(-viep)]
(4)
where 1 + axb ep is a linearization of q~ for low values of the accumulated plastic strain. Let us recall that we have already determined k* 4- Q and Ci/ 7i(1 4- ax) from the stabilized cyclic curve, b is more correctly identified by plotting the normalized maximum stress as a function of the quantity p = 2NAe v in one (or several) cyclic tests, where N is the current number of cycles. A few iterations allow us to finally obtain all the parameters k*, Ci, ?i, Q, b, and ax. 2. Determination of the viscosity function. We now use the available data in strain rate ranges between, say, about 10 .6 s -1 and 10 .2 s -1. The cyclic and monotonic uniaxial equations reduce to
Ao 2
A crint
= ~
2
1/n
4- k 4- K(1 4- aK)g,p
(s)
5.8 Unified Model of Cyclic Viscoplasticity
365
Functions cri,,t(ep) and Ar are completely known from the previously determined parameters. In the monotonic case we have assumed a negligible evolution of the drag stress (that can be easily corrected). From the difference in the log-log plots of r - r~i,t - k and Ar~/2 - &xi,,t/2 - k versus ~p (for test values of k) one easily determines aK. Then, exponent n and the corresponding values of k and K are obtained by trial and error (or by a least-square method). Their determination is greatly facilitated if pure relaxation tests are available, either in monotonic or in cyclic conditions. 3. Determination of static recovery. We use data available in the low rate regimes (~p < 10 -6 s -1) in the long-duration creep or relaxation tests. In those regimes the static recovery effect appears to change drastically the log-log plot of c r - r as a function of ~p (with significant stress drops). It indicates the lowering of O'int due to an increased contribution of time recovery. By successive approximations one can determine the parameters that concern these recovery effects: rni, "ci, Qr, mr, and 7r, taking all of the others as fixed. The determination is improved if special recovery tests are available, where after cyclic stabilization, a significant hold duration at high temperature (100 h, for instance) is applied after unloading (with a very low load, maintained to prevent any increase or decrease of plastic strain), followed by the application of the same cyclic loading as previously.
5.8.5 HOW TO USE THE MODEL Implementation in finite element (FE) codes of such unified constitutive equations can now be considered a standard operation. This is done as a local process, the constitutive equations being necessary only at each individual finite element integration point (independent of the others in classical continuum mechanics). This model is available in several general purpose FE codes (it is impossible to give a list here). Two integration methods can be used in the framework of the implicit Newton-Raphson algorithm (to solve simultaneously the overall equilibrium and the nonlinear behavior): The simplest way is through an explicit Runge-Kutta fourth-order method, with a special substepping technique and automatic time step control. Only the rate equations have to be explicitly written; i.e., 5~ = Y(y, t), on the state variables y = {s p, E j, r}, where t indicates the current total local strain control supposed to be known from a given global iteration. The only disadvantage of this way is the inability to deliver explicitly the consistent tangent stiffness matrix that should enforce the best convergence of the implicit scheme.
366
Chaboche
9 The fully implicit, or semi-implicit integration procedure (0-method), with the possibility to condense the determination to only one tensorial and one scalar equation [7], and eventually to only one scalar equation in the isotropic case. This method allows us to deliver the consistent tangent stiffness matrix, but it requires a number of lengthy explicit derivations (corresponding to the derivation of second-order rates ~). The determination of model parameters as indicated in Section 5.8.4 shows that the notion of one constitutive equation (one set of parameters) for a given
TABLE 5.8.2
Examples of Material Parameters for Rate-Independent Behavior.
Material
T
C1
71
Building steel 35NCD16 steel Nimonic 90 alloy Cobalt VO 795 TA6V alloy INCO 718 superalloy IN 100 superalloy " " " " 304 stainless steel 316 stainless steel " Z3CND17-12 SPH
20 20 20 20 350 550 700 800 900 1000 1100 20 20 600 600 ~
280,000. 224,000. 666,000. 142,000. 91,000. 210,000. 678,000. 615,000. 362,500. 139,500. 41,000. 120,000. 162,400. 24,800. 87,700. MPa
1300. 400. 1800. 400. 350. 420. 1750. 1530. 1200. 940. 700. 2000. 2800. 300. 1300.
TABLE 5.8.3
C2
72
22000.
10.
3000. 6750. 800. 3600. MPa
1. 25. 25. 25.
b
Q
1.6 2. 900. 800. 500. 200. 110. 6. 8. 10. 12.
-60. -180. -20. -30. -35. -15. -10. 60. 60. 60. 104. MPa
ax
-0.15 -0.18 -0.22 -0.22 -0.16
1.
k* 225. 980. 505. 85. 350. 610. 530. 405. 236. 162. 120. 120. 185. 70. 89. MPa
Examples of Material Parameters for Viscoplastic Behavior.
Material TA6V alloy INCO 718 superalloy IN 100 Superalloy " " " " 304 stainless steel 316 stainless steel " Z3CND17-12 SPH
T 350 550 700 800 900 1000 1100 20 20 600 600 oc
n 40. 40. 28. 14. 9. 7.5 6.5 11. 24. 12. 24. MPa
K 190. 140. 580. 630. 490. 450. 420. 118. 151. 150. 116. (MPa, s)
k 200. 500. 110. 80. 60. 30. 17. 70. 82. 6. 10. MPa
5.8 Unified Model of Cyclic Viscoplasticity
367
material is not necessarily representative of the true situation. Very often the material parameters should be depending on the considered application. The choice of the various functions and the parameter values should be appropriated with the range considered for the application: temperature domain, small or significant strains, monotonic or cyclic, strain rate domain, loadings with long holds or not, etc. Therefore, one should remember that the use without caution of a set of parameters given in the literature can lead to important misinterpretations. Tables 5.8.2 and 5.8.3 give some determined values, to be treated as examples.
REFERENCES 1. Chaboche, J. L. (1977). Sur l'utilisation des variables d'~tat interne pour la description de la viscoplasticit~ cyclique avec endommagement, in Probl~.mes Non Lin~aires de M~canique, pp. 137-159. Symposium Franco-Polonais de Rh~ologie et M~canique, Cracovie. 2. Chaboche, J. L. (1997). Thermodynamic formulation of constitutive equations and application to the viscoplasticity and viscoplasticity of metals and polymers. Int. J. Solids Structures 34(18): 2239-2254. 3. Chaboche, J. L., Dang-Van, K., and Cordier, G. (1979). Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. Division L, Berlin, SMIRT 5. 4. Lemaitre, J., and Chaboche, J. L. (1985). M~canique des Mat~riaux Solides, Paris: Dunod. 5. Chaboche, J. L. (1996). Unified cyclic viscoplastic constitutive equations: Development, capabilities and thermodynamic framework, in Unified Constitutive Laws of Plastic Deformation, pp. 1-68, Krauss, A. S., and Krauss, K., eds., Academic Press. 6. Marquis, D. (1979). Etude th~orique et verification exp~rimentale d'un mod~.le de plasticit~ cyclique. Th~se de 3~me Cycle, Universit~ Pierre et Marie Curie, Paris 6. 7. Chaboche, J. L., and Cailletaud, G. (1996). Integration methods for complex plastic constitutive equations. Comput. Methods Appl. Mech. Engrg. 133: 125-155.
SECTION
5.9
A Model of Nonproportional Cyclic Viscoplasticity EIICHI TANAKA
Department of Mechano-Informatics and Systems, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603 Japan
Contents 5.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
5.9.2 Formulation of a Model . . . . . . . . . . . . . . . . .
369
5.9.2.1 Fundamental Assumptions . . . . . . . .
369
5.9.2.2 Constitutive Equation of Inelastic Strain Rate . . . . . . . . . . . . . . . . . . . . . . . 370 5.9.2.3 Evolution Equations of Kinematic Hardening Variables . . . . . . . . . . . . . .
370
5.9.2.4 Structural Tensor and Nonproportionality Parameter . . . . . 371 5.9.2.5 Description of Amplitude Dependence of Cyclic Hardening.. 371 5.9.2.6 Evolution Equations of Isotropic Hardening Variable . . . . . . . . . . . . . . .
372
5.9.3 Identification of Material Constants and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
373
5.9.3.1 Identification by Use of Analytical Expressions of the Model . . . . . . . . .
373
5.9.3.2 Identification by Numerical Simulations of the Model . . . . . . . . .
375
5.9.3.3 Examples of Material Constants and Functions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
368
376 3 76
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
5.9 A Model of Nonproportional Cyclic Viscoplasticity
5.9.1
369
INTRODUCTION
This viscoplastic constitutive model describes the behavior of both cyclic hardening and softening under proportional and nonproportional loading. The model is formulated by modifying the viscoplastic model proposed by Chaboche et al. [1]. A macroscopic structural tensor to represent an internal dislocation structure is first formulated, and a rational nonproportionality parameter is defined. Then internal variables to express the amplitude dependence of cyclic hardening are introduced. These variables are incorporated into the evolution equations of the isotropic hardening variable in order to describe the history dependence of cyclic hardening and softening. The identification of material constants and material functions is also discussed in detail.
5.9.2
FORMULATION
OF A MODEL
5.9.2.1 FUNDAMENTALASSUMPTIONS The initial isotropy of a material is assumed. The rate of deformation tensor is separated into the sum of the elastic and the inelastic part. For brevity, a deviatoric tensor is expressed as a vector in a five-dimensional vector space of the von Mises type [2]. A set of orthonormal base vectors in the vector space is represented by n~ (~ = 1 , . . . , 5). In the following, a Greek index takes the value 1 , . . . , 5 and the summation convention on Greek indices is adopted. Then the component P~ of a vector P = P~n~ related to strain quantities or the component S~ of a vector S = S~n~ related to stress quantities is defined by the corresponding tensor component Pij or sij (i,j = 1 , . . . ,3) as follows:
(1)
P5 - 2p31/ x/3
S1 = (3/2)s11, $2 = x,/3(s11/2 + s22),
S~ = v/Ss31
$3 = v/-3s12~ 54 = x/~$23,
(2)
Tanaka
3 70 5 . 9 . 2 . 2 CONSTITUTIVE EQUATION OF INELASTIC STRAIN RATE
Based on the viscoplastic model [1], an inelastic strain rate t' and its magnitude/5 are expressed by
~, _ p (s - x) Is-xl
K _Q)n
p_ ( s- xl- k
(3)
(4)
where S and X are the vectors related to the deviatoric stress tensor and the kinematic hardening variable, and Q is the isotropic hardening variable. The k, K, and n are material constants. The superposed dot on a variable indicates the material time derivative, and the symbols ] ] and ( ) denote the magnitude of a vector and the Macauley bracket defined by
(x)-
x 0
if x > O if x < O
(5)
5 . 9 . 2 . 3 EVOLUTION EQUATIONS OF KINEMATIC HARDENING VARIABLES In this model, the nonlinear kinematic hardening rule [3] is adopted. The rate of X is represented by - X1 + X2 + X3
(6)
and their evolution equations are expressed as follows:
~(1 = C l ( a l P - Xl/~)
(7)
~ = c:(a:~- x~P)
(8)
~:~ = ~3(a3~- x~P)
(9)
where C1~ al, c2, a2, c3, and a3 are material constants.
371
5.9 A Model of Nonproportional Cyclic Viscoplasticity
5 . 9 . 2 . 4 STRUCTURAL TENSOR AND NONPROPORTIONALITY PARAMETER Material hardening and softening behavior under cyclic loading is influenced by the shape and the amplitude of a cyclic strain path. To describe the path shape dependence, a macroscopic internal state variable C representing an internal dislocation structure is introduced. Based on the discussion in Reference [4], the variable C is represented by a second-rank tensor; i.e., C = C~/3n~ | n/3 in the five-dimensional vector space (this tensor is a fourthrank tensor in the conventional three-dimensional space), and its evolution equation is formulated as
r=
G(u | u - C)I6
(10)
where cc is a material constant, the symbol | indicates the tensor product, and u = u~n~ is the normalized inelastic strain rate vector defined by u = ~'/P
(11)
In Eq. 10, u | u is the target value of C that represents the stabilized internal dislocation structure formed by the loading process u. By using C and u, a nonproportionality parameter A is defined by
/tr(CXC) - uoCXCu A-- V ~riC-X~ ,
(0<__A
(12)
where the symbol (tr) and the superscript (T) indicate the trace and the transpose of a tensor, and the symbol (o) expresses the inner product of vectors. The component expression of A is given by
_
/C~C~
A V
- u~C~C~u~
(0_~A_~ 1)
(13)
This variable always takes the value zero in the case of proportional cycles, and is approximately l / v / 2 in the case of the nonproportional cycles with a circular strain path.
5.9.2.5 DESCRIPTION OF AMPLITUDE DEPENDENCE OF CYCLIC HARDENING Next, to describe the amplitude dependence of cyclic hardening and softening, two internal variables Y and R are introduced. The variable Y
3 72
Tanaka
describes the center of the translation range of the inelastic strain P, and the relevant evolution equation is represented by ~I = r v ( P - r)l 6
(14)
where ry is a material constant. For example, in the case of uniaxial cycles of a constant strain amplitude with nonzero mean strain (the case of cyclic relaxation), the variable Y moves gradually from the initial location (usually the origin of the inelastic strain space) to the center of the inelastic strain range. The variable R to describe the mean radius of a cyclic strain path is defined by R = [ P - r[ In the case of circular cycles of amplitude A P / 2 , the AP/2.
5.9.2.6
(15) R is almost equal to
EVOLUTION EQUATIONS OF ISOTROPIC
HARDENING VARIABLE In this model the target value QH of the isotropic hardening variable is expressed by QI-I = A [ q N ( R ) - qI,(R)] + q e ( R )
(16)
where qN(R) and qI,(R) are the functions of R to describe the target values for the cases of nonproportional (A = 1) and proportional (A = 0 ) hardening, respectively. To take account of the memory effects of cyclic hardening and softening, a target value QL expressed by QL -- QH + CR(Qmax -- QH}
(17)
is also introduced, where cR is a material constant describing the ratio of permanent hardening, and Qmax is the maximum value of the isotropic hardening variable in the past. When cR equals unity, the memory effects are permanent, and when ce = 0, they are completely recoverable. The evolution equation of Qmax is described by Qmax - (Q),
Qmax = 0,
when Q - Qmax
(18a)
when Q < Qmax
(18b)
The evolution equation of the isotropic hardening variable Q is finally represented by = dI-I(QL - Q)ff - d s ( Q - Qz.)P
(19)
373
5.9 A Model of Nonproportional Cyclic Viscoplasticity
where d/-/and ds are the material constants describing the rates of hardening and softening, respectively.
5.9.3
IDENTIFICATION
CONSTANTS
OF MATERIAL
AND FUNCTIONS
This model has 14 material constants k, K, n, 6ll, 612, a3, Cl, C2, C3, di-i, ds, CR, rr, co, and two material functions qe(R) and qN(R). These constants and functions are identified by using analytical expressions and numerical simulations of the model. In the following, the vector component in uniaxial cases is represented by the corresponding lightface character.
5.9.3.1
IDENTIFICATION BY USE OF ANALYTICAL
EXPRESSIONS OF THE MODEL The analytical expression of stress for monotonic uniaxial tension is s - x + Q + k + Kf"~/"
(20)
In this equation, the rate effects are induced by the term KP 1/". Thus the values of K and n are evaluated by comparing the stress values at the equal inelastic strain value in the stress-inelastic strain curves at several constant strain rates. The value of k is evaluated as the stress at P = 0 in the (S - KP1/,) versus P curve. The material constants of the kinematic hardening variables are determined in the following. The linear hardening part in the stress-inelastic strain curve of monotonic tension is first assumed to be approximated by
X3 + Q = bP
(21)
The constant b is easily determined from the slope of the linear part. Then, we have X 1 + X 2 -- S - (bP + k + KP 1/n)
(22)
The terms on the right-hand side are evaluated by use of the stress in tension tests and the material constants identified previously, and the variables X1 and X2 are represented by Xi = a/{ 1 - exp(-cie)}
(i = 1, 2)
(23)
The constants al, a2, Cl, and c2 are estimated by fitting the value of the righthand side of Eq. 22 by Eq. 23.
3 74
Tanaka
In the saturated stage of cyclic hardening, the value of Q is almost constant, so 5(3 is approximated by 2K3 -- b['
(24)
In this case, the stress amplitude AS~2 of circular cycles of inelastic strain amplitude AP/2 with zero mean inelastic strain is represented by
aicike/2
(AS/2)2_
~=1 1 + (
+
+ k+
1 + (
(25)
where Qc is the saturated value of the isotropic hardening variable for the circular cycles. The material constants in Eq. 25 are already known, and thus the function value of Qc(AP/2) is evaluated by use of AS~2 obtained from the experiments. Furthermore, the shape of AS~2 versus P curve is mainly governed by the evolution of the isotropic hardening variable. Since Eq. 19 gives the analytical expression
Q(P) = Qc(AP){ 1 - exp(-dnP) }
(26)
the value dH is determined by fitting Eq. 26 to the experimental curve. Next, in the amplitude variation experiments of circular cycles, the target value QL in the subsequent cycles of smaller amplitude (AP2/2) after the saturated stage of cyclic hardening of larger amplitude (AP1/2) is specified by Eq. 17. Solving Eq. 17 for cR gives the relation ~L -- ~H
cR = Qm x -- QH
(27)
In this case, ~max = Qc(AP1/2) and QI-I= Qc(AP2/2), and the value of Qz.- QI-I is the difference between the two saturated stress amplitudes of the circular cycles of AP2/2 with and without the preceding cycles of AP1/2. The constant ds is determined from the condition that the analytical expression for Eq. 19 for the softening, i.e.,
Q = QI_ + [Qc(AP1/2) - Qz.]exp{-ds(V- Vo)}
(28)
describes the AS~2 versus P curve in the cyclic softening process of amplitude decrease experiments. In this equation, P0 and Qc(AP1/2) are the values of the accumulated inelastic strain and the isotropic hardening variable at the end of the preceding cycles.
5.9 A Model of Nonproportional Cyclic Viscoplasticity
375
5.9.3.2 IDENTIFICATION BY NUMERICAL SIMULATIONS OF THE MODEL The remaining constants rr, Cc, a3, c3 and the two functions qp(R) and qN(R) are determined by performing the numerical simulations of the model. First the value of ry is estimated by the condition that the transient behavior of Y obtained from the simulation of Eq. 14 for the uniaxial cycles with nonzero mean inelastic strain is similar to the resultant cyclic relaxation behavior. Next, the assumption of the function value qp(R) and the use of the linear hardening law (Eq. 24), together with the material constants identified previously, enable the simulation of uniaxial cycles by the constitutive model. Hence, by assuming the provisional relation qe(aP/2)=Q(AP/2) and by comparing the saturated stress amplitude versus inelastic strain amplitude relationship with the corresponding experimental results, we can obtain the better estimate of the function qp(R). The repetition of this procedure enables us to identify the material function qp(R). By the use of this function and the Qr for circular cycles, the other material function qN(R) is expressed by qN(AP/2) -- x/-2[Qc(AP/2) - qp(AP/2)] + qp(AP/2)
(29)
Next, for monotonic uniaxial loading, from Eq. 20 we have X3--S-
(Xl q- X 2 q- Q -ff k -ff K P 1/ n )
(30)
Then the value of the right-hand side can be evaluated by using the experimental value of stress S and the values of k + KP 1/n and X1 + X2 identified previously. By using the analytical expression of R, i.e., R = { 1 - e x p ( - r y P ) }/ry
(31)
and by the numerical integration of Eqs. 16-19, the value Q of the isotropic hardening variable in Eq. 30 can be evaluated, and the relation between X3 and inelastic strain P is determined from Eq. 30. The constants a3 and c3 are determined by the condition that the analytical expression of X3 that is given as i = 3 in Eq. 23 describes the preceding relation. Finally, the constant cc is determined by the simulation of nonproportional cycles such as the cruciform cycles that show intermediate hardening between the proportional and the circular cycles. Since all constants and functions except cc have already been identified, we can adjust the value of cc by comparing the simulation results of the nonproportional cycles with the saturated stress amplitude of the corresponding experimental results.
3 76 5.9.3.3
Tanaka
EXAMPLES OF MATERIAL CONSTANTS
AND FUNCTIONS For reference we s h o w a set of material constants and functions identified for 316 stainless steel at r o o m t e m p e r a t u r e . The e x p e r i m e n t s of p r o p o r t i o n a l a n d n o n p r o p o r t i o n a l cycles were p e r f o r m e d by the p r e s e n t a u t h o r a n d c o w o r k e r s [5, 6]. The value of 3 x 10-5sec -1 is c h o s e n for the inelastic strain rate. The material constants and the functions are K = 92 MPa,
n = 8, k = 75 MPa,
al
=
125 MPa,
a2 = 70 MPa, a3 = 310 MPa cl = 1 6 0 0 ,
CR=0.4,
c2=250,
c3=6,
d/-/=6,
ds=4,
(32)
ry=40, cc=200
qp(R) = 4000R + 28{1 - e x p ( - 2 5 0 0 R ) } qN(R) = 73100R § 209{1 - e x p ( - 1 7 0 0 R ) }
(MPa) (MPa)
The e x t e n s i o n of this m o d e l to the description of inelastic b e h a v i o r of 316 stainless steel in the range from r o o m t e m p e r a t u r e to 973 K is found in Reference [7]. It is found in References [4, 7] that this m o d e l can describe the cyclic h a r d e n i n g a n d softening b e h a v i o r of various c o m p l e x p r o p o r t i o n a l and n o n p r o p o r t i o n a l cycles with a m p l i t u d e variations a n d p a t h changes at r o o m and elevated t e m p e r a t u r e .
REFERENCES 1. Chaboche, J. L., and Rousselier, G. (1983). On the plastic and viscoplastic constitutive equations. Part 1: Rules developed with internal variables concept. Trans. ASME, J. Press. Vessel Technol. 105: 153. 2. Ilyushin, A. A. (1963). Plasticity, Moscow: Izd. Akad. Nauk (in Russian). 3. Armstrong, P. J., and Frederick, C. O. (1966). A mathematical representation of the multiaxial Bauschinger effects. CEGB. Report, RD/B/N731, Berkeley Nuclear Laboratories. 4. Tanaka, E. (1994). A nonproportionality parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and memory effects of isotropic hardening. European J. Mech., A/Solids 13: 155. 5. Tanaka, E., Murakami, S., and Ooka, M. (1985). Effects of plastic strain amplitudes on nonproportional cyclic plasticity. Acta Mechanica 57: 167. 6. Tanaka, E., Murakami, S., and Ooka, M. (1985). Effects of strain path shapes on nonproportional cyclic plasticity. J. Mech. Phys. Solids 33: 559. 7. Ohmi, Y., Tanaka, E., Murakami, S., Jincho, M., and Suzuki, T. (1995). Constitutive modeling of proportional/nonproportional cyclic plasticity for type 316 stainless steel applicable to a wide temperature range. Material Science Research International 1 (4): 247.
SECTION
5.10
Rate-Dependent Elastoplastic Constitutive Relations FERNAND ELLYIN Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada
Contents 5.10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2 Rate-Dependent Constitutive Model . . . . . . . 5.10.2.1 Stress-Strain Relation . . . . . . . . . . . . . . 5.10.2.2 Evolution of Centers of the Yield and Memory Surfaces . . . . . . . . . . . . . 5.10.3 Numerical Implementation . . . . . . . . . . . . . . . . 5.10.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10.1
377 378 379 380 382 383 386
BACKGROUND
A number of engineering alloys, such as type 304 and 316 stainless steels and high-strength titanium alloys, exhibit rate-dependent behavior even at ambient temperature; see Ellyin [2], who provides examples and references to experimental data. The main manifestations of rate dependency are loading rate sensitivity, creep, relaxation, and creep-plasticity interaction. A constitutive relation for rate-dependent materials, therefore, should be able to simulate the aforementioned behavior. Constitutive models in which all aspects of inelastic deformation such as plasticity, creep, and recovery are treated by one set of equations are generally referred to as "unified models." Examples of this type of constitutive relation are given in this chapter by a number of contributors, and an interested reader is encouraged to consult them for details. There is, however, experimental evidence that the effect of prior deformation on the subsequent creep or plastic deformation is quite different. For example, a plastic prestrain has a Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
377
378
Ellyin
greater contribution to the creep hardening than the same amount of creep prestrain [5]. This type of distinction cannot be accommodated by a unified creep-plasticity model, since in these models creep and plastic strains are not separable from each other. The constitutive model to be described here is of a coupled nature, in the sense that the effect of prior creep on the subsequent plastic deformation is incorporated in the rate-dependent elastic-plastic constitutive relation. Similarly, the influence of plastic prestrain on the subsequent creep deformation is taken into account. Thus the framework for this constitutive model is based on the concept that any loading sequence can be predicted by two separate but coupled models. The rate-dependent elastoplastic consti ~.,tive model for the creep deformation is given elsewhere [6, 7], and an interested reader is encouraged to consult these references.
5.10.2 RATE-DEPENDENT CONSTITUTIVE
MODEL
The uniaxial stress-strain response at different strain rates has a characteristic feature which indicates an increase of yield stress with the increasing applied strain rate. Another feature of these curves is related to the change of the tangent modulus at the elastic-plastic transition region with the loading rate change. To describe these observations in a multiaxial stress state, we introduce two types of hypersurfaces: a yield surface, ~by,which is the locus of the current elastic regime, and a stress memory surface, ~m, which defines the maximum equivalent stress, Creq,max, to which the material has been subjected in its previous history. The rate- and history-dependent yield surface is given by ~y
3 2
-- --S" S
_ q2
(~eq, lp) -- 0
(1)
in which g = 8 - 1/3 tr(8)l, and er = d - 0~. The 9 and q specify the center and radius of the yield surface. The equivalent strain rate,/;eq - - (2/34- 4) 1/2 with 4 - e - 1/3 tr(e)l, and le - f ( 2 / 3 deP:deP) 1/2, is the accumulated plastic strain length. Note that herein bold letters indicate tensorial quantity. The evolution rule of the size of the yield surface is determined from
dq
cgq
-
cgq
0~eq d~eq q- -~p dlp
(2)
An explicit relation of the radius of the yield surface can be obtained by
379
5.10 Rate-Dependent Elastoplastic Constitutive Relations
adopting the conventional logarithmic strain-rate dependency, i.e., q - q[1og(keq/~0q),
lp]
(3)
where geq'0iS a reference equivalent strain rate, usually taken to be the slowest one in the loading history. The accumulated plastic strain length, is used to describe cyclic strain hardening or softening phenomenon [2]. The stress memory surface, ~b~, is related to the maximum equivalent stress, ~ e q - ( 3 / 2 s " S) 1/2, experienced by the material during its past loading history. It is expressed by
lp,
3
~r~ -- ~ S "
~ -- R2(~
(4)
-- 0
in which g - ~ - 1~with 1/denoting the center and R the radius of the current memory surface. The evolution rules for the center of the yield surface, 0~, and that of the memory surface,//, will be described later on.
5.10.2.1 STRESS-STRAINRELATION The total strain rate is decomposed into its elastic and plastic parts, i.e.,
~=g+~p
(5)
where i~
1 -- ~ [(1 + v)6- - v tr(d)l]
Ofy(~gfy
/~P -- g O-~-a\ 0 ~ " d
(6)
)
(7)
with 1(1 g-~qTq2 ~ -
1)
and
fy-3-g'g2
(8)
In the preceding, Et is the current tangent modulus, E t - - Et(o'eq , tTeq, is both rate- and history-dependent, similar to q in Eq. 3. Inverting Eqs. 5-7, the stress rate is given by ~ -
E 1+ ~
lol+
lol1 - 2~
g-g
~
lp),
which
(9)
[(1 + ~)/E] + q
Depending on the relative position of the current stress point with respect to the yield and memory surfaces, three loading regimes are distinguished: (i) elastic loading, (ii) monotonic plastic loading, and (iii) plastic reloading.
380
Ellyin
Each of these loading regimes has a different evolution rule, which is described in text following. For a loading to be classified as the elastic loading, the stress point must be inside the current yield surface, or in the case of elastic unloading, s : d < 0. In either case, the stress-strain relation is specified by Eq. 6, or the first two terms on the right-hand side of Eq. 9. In the case of monotonic plastic loading, s : 6 " > 0, and the current stress point is on the yield surface. However, if the equivalent stress value increases, the stress memory surface will expand but will remain tangent to the yield surface at the current stress point. This condition is expressed by Ofu (a - ~) = 2 ~ ~ (a - fl)
&r fm(a-
&r ,8) - R 2 = 0
(i0)
where f y ( a - ~ ) is defined by the second equation of Eq. 8 and ~) = 3 / 2 ~ : ~. For the proportional loading fl = 0. The plastic loading in which the current stress point is inside the stress memory is termed the plastic reloading. For this loading case the memory surface does not change; however, when there is a transition from elastic to plastic loading, the size of the yield surface would be dependent on the current equivalent strain rate; see Eq. 2. For the latter two plastic loadings, the calculation of the tangent modulus are different, and they will be explained in text following.
fm(a-
5.10.2.2
EVOLUTION OF CENTERS OF THE YIELD AND
MEMORY SURFACES For the monotonic loading regime, the evolution rule for the center of the yield surface is given by O~ : / ~ ( a lim -- a y) (11) and o.li m ~ Rlim (o.li m _ ~g)
(12)
q where a y is the current stress point which is on the yield surface, and Rlim is a constant depicting the maximum limit of the stress memory surface radius. When Rlim>~q, then one approaches Ziegler's rule in which ~ll(~ry - ~ ) . The value of Et in Eq. 8 is calculated from a family of strain rate-dependent stress-strain curves, based on the current equivalent stress value and strain rate. This will be explained later on. For the plastic reloading, the evolution of
381
5.10 Rate-Dependent Elastoplastic Constitutive Relations
the center of the yield surface is determined from - ti(tr m - trY)
(13)
where trm is a point on the memory surface whose exterior normal is parallel to the normal at point try (see Fig. 5.10.1)" R_
trm _ --,,,(try _ ~)
(14)
q
The calculation of the tangent modulus, Et, requires attention for this loading case. It is related to the ratio of two distances between three stress points, namely, the stress at which plastic reloading took place (point C in Fig. 5.10.1), the current stress point tr y and tr", determined according to Eq. 13. For any point in the stress space, we define a ratio, r
-
-
-82
(15)
where 61 is the distance measured from the current stress point, try (point D in Fig. 5.10.1), to the corresponding point on the memory surface, trm (point E in the same figure), and 82 is the distance from the point of onset of reversed plastic flow (point C) to the current stress point, trY. The distance between 400
memorysurface
-
f
200 -
i
A
0 (2.
ZE
O ....
-2oo -! -400
initialyield ~: & memorysurfaces
-4O0
t
-200
0
I 200
i 400
(MPo) FIGURE 5.10.1 Schematic representation of yield and memory surfaces for monotonic and reversed plastic loading.
382
Ellyin
two points (1) and (2) in the stress space is calculated from d - [((s) - (1)).((2)_ (1))]1/2
(16)
where(~ ) and (s2) are the deviatoric stress values at the two points. Upon the determination of fi~ and c~2, the r value is calculated from Eq. 15. Then an equivalent stress corresponding to point D is determined from O'eq = (O'eq,max-~ rq)/(1
+ r)
(17)
This is the value at which the tangent modulus is calculated from the uniaxial stress-strain curve at the current value of the strain rate. This procedure for calculating the tangent modulus during reversed loading has several advantages in comparison to previously proposed methods [1]. An accurate determination of tangent modulus is of considerable importance in cyclic loading and strain energy calculation. For a monotonic loading, the center of the memory surface remains fixed; i.e., ~ - 0. However, in the case of cyclic loading with a mean stress, the evolution rule of the center of the memory surface is given by
0p
--=
cole
~(a mean - fl)
(18)
w h e r e o"mean is defined as the geometric center of the cyclic stress path in the
stress space, i.e., ~m~.n =
it+T /t+T ~
dl
dl
(19)
J t
where T is the period of cyclic loading and dl is the differential length of the curve. Note that Eq. 17 is applied only during unloading case where the radius of the memory surface does not change. In addition to the transient hardening during cyclic loading, there is an extra hardening due to nonproportional loading; see Ellyin [2], where procedure for its determination is explained.
5.10.3
NUMERICAL
IMPLEMENTATION
A computer subroutine has been prepared for the numerical implementation of the constitutive model. In its most general form the model contains seven material constants, in addition to a required set of rate-dependent uniaxial stress-strain curves. The following are determined from the latter set of stressstrain curves. Two elastic constants, E and v, are obtained from the elastic part of the stress-strain relations. The rate- and history-dependent constants, q q(~.eq, lp) and E t - Et(aeq,~.eq, lp), are determined from the family of =
383
5.10 Rate-Dependent Elastoplastic Constitutive Relations
rate-dependent stress-strain curves; see illustrative examples to follow. A twodimensional interpolation is used to obtain a continuous description of the dependency of the yield surface radius and tangent modulus on the equivalent strain rate, ~eq, and the accumulated plastic strain length, lp.
5.10.4 ILLUSTRATIVE EXAMPLES The rate-independent version of the previously described constitutive model was programed and inserted into the user-supplied subroutine of ADINA Finite Element Computer Code. Various examples ranging from uniaxial to multiaxial monotonic and cyclic loading, for both proportional and nonproportional loading paths, are given in References [2, 4]. This subroutine is now available for commercial use and may be obtained by writing to the author. The rate-dependent version has also been programed and can be implemented in any FEM Code which has an option for the user-supplied material model. Here a couple of representative examples of the model prediction and comparison with experimental results will be given. Figure 5.10.2 shows the experimental data and model predictions of uniaxial stress-strain curves for three different strain rates of a titanium alloy, TIMETAL 21S. For this
4ool
8.3X10-*
3OO 8 . 3 X I 0 -s 200 C/)
9
lw
"9
8 . 3 X I 0 -s
---,.
0.0
0.2
0.4
0.6
_Mode!Prediction
0.8
1.0
FIGURE 5.10.2 Experimental data and model prediction for titanium alloy TIMETAL 21S at 650~ temperature for three strain rates.
384 material,
Ellyin E = 8 6 6 0 0 MPa,
v = 0.34, Rlim -- 4 6 0 0 MPa, q(~eq) = 4 6 . 4 3 1 + 8.3 x 106 s - .1 Two e x a m p l e s of t a n g e n t
0 _ 0.3292[1og(~eq/~0q)] 2"99, and ~eq
m o d u l u s v a r i a t i o n w i t h the effective stress at r e f e r e n c e s t r a i n rates, are g i v e n in Table 5.10.1. A f u r t h e r e x a m p l e is d e p i c t e d in F i g u r e 5.10.3. In this e x a m p l e , a type 3 0 4 stainless steel t u b u l a r s p e c i m e n was s u b j e c t e d to cyclic axial a n d differential
a)
0,/.
/
I
o:,s
o . , 5
G C~-} . . . . . . . . . . o.'~S
~
--
'
-
G
(MPo)
. f, r o.~5
'
o"o C~Po)
FIGURE 5.10.3 Biaxial stress-strain response for the first 10 cycles at a fast strain rate 4 x 10 -3 s -1. a. Strain path. b. Test results: (1) axial stress vs. axial strain, (2) tangential stress vs. tangential strain, (3) axial stress-tangential stress trajectory, c. Model predictions [3].
I
O
\ • e~
ue~ eN
o
o
.~
F~
~
ce~ G
O ~t" 00
c5~
00
o O
O
e~
e~
e~
e~
0G
O O
o
o O ee~
O
v~
o
• ~
G
5.10 Rate-Dependent Elastoplastic Constitutive Relations
O
o
~"
e~
385
386
Ellyin
p r e s s u r e ( c o n s t a n t e x t e r n a l a n d cyclic i n t e r n a l p r e s s u r e ) . T h e axial a n d t a n g e n t i a l strains (ea, et) w e r e c o n t r o l l e d , a n d t h e y v a r i e d in a n o n p r o p o r t i o n a l m a n n e r in a s q u a r e d i a m o n d p a t t e r n as s h o w n in F i g u r e 5.10.3a. T h e biaxial s t r e s s - s t r a i n r e s p o n s e a n d the stress t r a j e c t o r y for the first ten cycles are s h o w n in the figure for a s t r a i n rate of keq - 4 x 10 -3 s -1. It is s e e n t h a t for this fairly c o m p l i c a t e d l o a d i n g , the m o d e l p r e d i c t i o n is in v e r y g o o d a g r e e m e n t w i t h the e x p e r i m e n t a l results. T h e m a t e r i a l p r o p e r t i e s for this type 3 0 4 stainless steel w e r e [3] E -- 196,000 MPa, v -- 0.25, q~nit _ 75.7 MPa, q~t _ 122 MPa, and qinit(~:eq) -- q0init[l+Cll~176 w i t h C1 - 0.153 a n d geq'0_ 6 • 10 -5 S-1.
The superscript "init" and "st" refer to the virgin and stable cyclic values of the cyclic stress-strain curves. The tangent modulus versus effective stress for the initial stress-strain curve at the reference strain rate is given in Table 5.10.1.
REFERENCES 1. Ellyin, E (1984). An anisotropic hardening rule for elastoplastic solids based on experimental observations. ASME J. Appl. Mech. 56: 499-507. 2. Ellyin, E (1997). Fatigue Damage, Crack Growth and Life Prediction, London: Chapman & Hall. 3. Ellyin, E, Xia, Z., and Sasaski, K. (1993). Effect of rate and rate history on plastic deformation: Experiments and constitutive modeling. Int. J. Plasticity 9: 951-959. 4. Ellyin, E, Xia, Z., and Wu, J. (1993). A new elastoplastic constitutive model inserted into usersupplied material model of ADINA. J. Computers Struct. 10: 283-294. 5. Kawai, M., and Ohashi, Y. (1986). Creep-plasticity interaction of austenitic stainless steels at elevated temperature. Proc. Int. Conf. On Creep, April 1986, Tokyo, Japan, pp. 454-464. 6. Xia, z., and Ellyin, E (1991). A rate-dependent inelastic constitutive model, Part II: Creep deformation including prior plastic strain effects. ASMEJ. Engng. Mater. Technol. 113: 324-328. 7. Xia, Z., and Ellyin, E (1993). An experimental study on the effect of prior plastic straining on creep behavior of 304 stainless steel. ASME J. Engng. Mater. Technol. 115: 200-203.
SECTION
5.11
Physically Based Rate-and Temperature-Dependent Constitutive Models for Metals SIA NEMAT-NASSER Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, California
Contents 5.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 5.11.3 Athermal Stress Component, Za . . . . . . . . . . . 5.11.4 Viscous-Drag Component, ra . . . . . . . . . . . . . . 5.11.5 Thermally Activated Component, ~* . . . . . . 5.11.6 Three-Dimensional Model . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.1
387 388 388 389 391 396 396
INTRODUCTION
Guided by the concept of dislocation kinetics [3] and accompanied by systematic experimental investigation, a physically based model has been developed [4-7] for several polycrystalline metals. The model addresses the plastic deformation of metals in the range of temperatures and strain rates where diffusion and creep are not dominant, and plastic deformation occurs basically by the motion of dislocations. Here, we summarize this model, provide a table of typical values of the constitutive parameters, which have been experimentally evaluated, present a few typical comparative results, and show how the model can be used for Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
387
388
Nemat-Nasser
three-dimensional computations. In what follows, z and ~ are the stress and the corresponding strain rate in a uniaxial-stress compression test. When the model is used in a three-dimensional setting, these quantities are the effective stress and the effective plastic strain rate, respectively; see Section 5.11.6.
5.11.2 MODEL DESCRIPTION It is assumed that plastic deformation occurs by the motion of dislocations and that the resistance to this motion corresponds to the flow stress. To simplify the interpretation of the experimental results, this stress is decomposed into three parts. One part, denoted by z*, is due to the short-range barriers that dislocations may overcome through their thermal activation. These barriers may include the lattice itself (Peierls stress), point defects such as vacancies and self-interstitials, other dislocations that intersect the slip planes, alloying elements, and solute atoms (interstitial and substitutional). The second part, za, is the athermal component, mainly due to long-range effects such as the elastic stress field of dislocation forests and grain boundaries. Finally, there is a remaining viscous-drag component, zd, which usually is important at high temperatures and high strain rates. Although the decomposition of the stress in this manner may be an oversimplification of a rather complex process [3], it has proved effective in interpreting the experimental results and in obtaining explicit constitutive relations of broad applicability. The flow stress is thus expressed as z = za + zd + ' c *
(1)
The stress, z, is a function of the strain rate, ~, temperature, T, and some internal parameters that describe the microstructure. The microstructure here refers to the grain sizes, the distribution of second-phase particles or precipitates, and the distribution and density of dislocations. In general, the most commonly used microstructural parameter is the average dislocation density, p. The microstructure evolves differently for different histories of and T, affecting the current value of p, accordingly. 5 . 1 1 . 3 A T H E R M A L S T R E S S C O M P O N E N T , Za The athermal part, Za, of the flow stress, z, is independent of the current value of the strain rate, ~. The temperature effect on Za is only through the temperature dependence of the elastic moduli, especially the shear modulus, #(T) [8]. Linear elasticity suggests that Za should be proportional
5.11 Physically Based Rate- and Temperature-Dependent Constitutive Models
389
to #(T). Hence,
= g(p,
(2)
where d6 is the average grain size, the dots stand for parameters associated with other impurities, and #0 is a reference value of the shear modulus. In a general loading, the strain 7 represents the effective plastic strain that is a monotonically increasing quantity in plastic deformation. In the present case, 7 defines the loading path in a uniaxial-stress deformation mode, and is also a monotonically increasing quantity, since 9 > 0 (by definition, the strain rate is viewed as positive even in uniaxial compression tests). Therefore, it may be used as a load parameter to define the variation of the dislocation density, the average grain size, and other parameters which affect "ca, i.e., ra = g[P(V), d6(7),.. "]P(T)/~0 = g,(v)~'(r)/*'o
(3)
Further, as a first approximation, we use a simple power-law representation of g(7) and choose an average value for #0 so that la(T)/la o ~ 1. Then, "ca may be written as (4)
"ca = ao + a17 n + . . . ,
where ao, al, and n are free parameters which must be fixed experimentally. At high strain rates, ao may be neglected; i.e., ao ~ 0.
5.11.4
VISCOUS-DRAG
COMPONENT,
"ca
It is experimentally observed that, even at very high temperatures where the thermal activation energy of the dislocations exceeds the short-range energy barriers, the stress increases with increasing strain rate, especially when the strain rate exceeds about 1000/s. Figure 5.11.1 illustrates this for AL-6XN stainless steel. It displays the flow stress for 0.001/s, 0 . l / s , and 3500/s strain rates, in terms of the temperature, for a common strain of 10%. From this figure, it is seen that the high-temperature flow stress at low strain rates (0.001-0.i/s) is insensitive to the strain rate, but not at the high strain rate of 3500/s. This increased strain-rate sensitivity is usually attributed to the electron- and phonon-drag effects on the mobile dislocations [9-12]. The viscous-drag stress, rd, is often related to the dislocation motion by "ca ~ MB~/b, where M is the Taylor factor, B is the drag coefficient, ~ is the average dislocation velocity, and b is the magnitude of the Burgers vector. Since ~ relates to the strain rate by 9 = PmbF;/M (where Pm is the mobile dislocation density), it follows that "ca=g[m2B/(Pmb2),9,T]. At high temperatures, and in the absence of creep, the flow stress is essentially
390
Nemat-Nasser
1400 AL-6XN, 7 =
10%
1200 .,,--,.
9
Q
E
1000
800
-
500Is
_
600
400
200
- :iiiiii i i i i i i i i i i i i i
|
i
|
!
200
i
|
|
|
i
|
400
|
i
i
I
i
i
i
600
i
I
i
800
i
i
i
I
1000
i
I
i
i
1200
Temperature (K) FIGURE 5.11.1
independent of the temperature, T, and we may assume
"Cd -- g[M2B/(Pmb2), ~] To examine the effect of the viscous drag, consider the experimental results [13] for AL-6XN stainless steel, plotted in Figure 5.11.2. From these data, it is seen that, when the strain rate exceeds about 1000/s, the flow stress increases rapidly. Experimental results of Reference [14] for tantalum show that the drag on dislocations is significant over a range of strain rates, from a few hundred to several thousand per second. Based on this [7], we set rd -- mo[1 -- exp(--e);)],
a -
M2B Pmb2zy
(6)
where m0 is a material constant which can be measured directly at a very high strain rate and high temperature, and B represents an effective damping coefficient affecting the dislocation motion. The value of ~ may be established empirically. It can be estimated that ~ = O(10 -4) when ~ is measured per second. For ~ 1, we have "~d,.~mo~O, and for e ~ 1, we have rd "-~ m0. These limiting cases may be used to simplify computations in threedimensional applications.
5.11 Physically Based Rate- and Temperature-Dependent Constitutive Models
391
600 AL-6XN, y = 10% To = 1,000K
550 A
9 experimental ...... calculated
500
v
450 ~ ." ,,o"/
ID
=
. . . . .
400
_-----.
~
,.
~
350 300 0.001
O. 1
10
1,000
100,000
Strain Rate (1 Is) FIGURE 5.11.2
5.11.5
THERMALLY ACTIVATED
COMPONENT,
z*
The thermally activated flow stress, z* in general, is a function of temperature, T, strain rate, ~, and the internal variables characterizing the microstructure of the short-range barriers. This microstructure may evolve with the temperature and strain rate histories. To obtain a relation between ~, T, and z *, let AG be the energy that a dislocation must overcome by its thermal activation. Reference [3] suggests the following relation between AG and z* representing a typical barrier encountered by a dislocation: AG-G0
1-
,
O0-e2fb
(7)
where 0 < p _< 1 and 1 _< q _< 2 define the profile of the short-range barrier to the dislocation, ~ is the stress above which the barrier is crossed by a dislocation without any assistance from thermal activation, and Go is the energy required for a dislocation to overcome the barrier solely by its thermal activation; 2 and g are the average effective barrier width and spacing,
392
Nemat-Nasser
respectively. We assume that ~ is related to AG by
where 7r = Pmbcoos here, k is the Bohzmann constant, and COo is the attempt frequency of a dislocation to overcome its short-range barrier. From Eqs. 7 and 8, obtain 9* --'~ 1 -
-~001n~r
(9)
References [15 and 3] suggest that p = 2//3 and q = 2 are suitable values for these parameters for many metals. Nemat-Nasser and coworkers have verified this suggestion for a number of metals, although other values may fit the experimental results better for certain cases; see Table 5.11.1 and Reference [ 14]. For some materials (e.g., fcc metals), the dislocations that intersect the slip planes are the most dominant short-range barriers. To account for the microstructural evolution which affects the average dislocation spacing, ~, it is assumed in Reference [5], that ~ = ~.o/f(?,T), where ~0 is a reference (e.g., initial) average dislocation spacing. In view of Eqs. 7-9, this gives, for this class of metals, --z~
T)
and
7r
_
?o
f(v,T)'
with
zo
Go --b2fo
and
7o-Pmbcoo~o (10)
It is reasonable to expect that the average dislocation density increases with straining (workhardening) and that it decreases with increasing temperature (annealing). Based on this observation, and guided by experimental results, we follow Reference [5] and assume that
f(?,T)-- --l+a i-
~mm
(ii)
TABLE 5.11.1 Constitutive Parameters for Indicated Commercially Pure Metal. Metal, bcc, f= 1
al(Mea)
n
P
q
Tantalum Molybdenum Niobium Vanadium
473 720 440 342
0.20 0.25 0.25 0.15
2/3 2/3 2/3 1/2
2 2 2 2
T~ 1100 2450 1680 1260
k (K_I) G---o 8.62 • 8.62 • 1.24 • 1.24 •
10-5 10-~ 10-4 10-4
~0(s-1 ) 5 x 108 1.45 • 107 3.5 • 106 3.5 • 106
5.11 Physically Based Rate- and Temperature-Dependent Constitutive Models
393
800 700 600
.
~
50O
398K" gSK
4OO
I00 0
0
0.1
16ooT . 1400|
0.2
.
0.3 0.4 '13"ueSm~,
.
.
.
0.5
0.6
0.7
.
M~
098K
g
o
0.00
0.05
0,10
0.15
0.25
0.20
0,30
0,35
True Strain
0,40
0,45
0,50
800 r [[ Nb, 8,000 S"t f.
600 ~
2
9
6
~-
~
K .
.
.
500K
.
400
200
0,00
700
F Im
6oo !-
O.10
0.20
0.30
0.40 0,50 Tree Strain
0,60
0,70
080
Vanadium, 8,000 s-1 . . . . . . . . . . .
T,,= ~ K
200 I00
0.00
0.05
0.10
0,15
0,20 0,25 Tn~ Strain
0.30
0.35
0.40
0.45
FIGURE 5.11.3 Comparison between model predictions and experimental results for indicated metal and constitutive parameters given in Table 5.11.1.
394
Nemat-Nasser
where Tm is the melting temperature and the index m is a free parameter which m u s t be evaluated from the experimental data. Combining Eqs. 9-11, we arrive at the following expression for z*:
z* - z ~ 1 -
-b2go' where
Tc is
- GooIn
. 7o
7o-bPmC~
f(2, T)
for
T
f(Y,T)--l+a
1-
7m
given by
To---
G~(lngf (?'- Tc)) -1 k
(13)
70
Note that z * - 0 for T > To. Note also that, at high strain rates, the temperature of the sample increases due to heating t h r o u g h plastic work. This temperature change can be c o m p u t e d by AT --
PoCv
zd7
(14)
where P0 is the mass density, Cv is the t e m p e r a t u r e - d e p e n d e n t heat capacity, and fl,,~l is the fraction of the plastic w o r k used to heat the sample. At finite strains and high strain rates, it is s h o w n in Reference [16] that fl is essentially 1. For most bcc metals, the lattice provides the main source of the short-range resistance to the dislocation motion. In this case, go = g = b, and hence a = 0 and f = 1 in Eq. (12). Table 5.11.1 gives typical values of the constitutive parameters for the indicated commercially pure bcc metals, and Figure 5.11.3 compares the model and experimental results. In these cases, the drag effect is neglected. Table 5.11.2 provides data
TABLE 5 . 1 1 . 2
Metal, p = 2/3, q= 2 OFHC Cu, annealed OFHC Cu, as received AL-6XN Ti-6A1-4V
C o n s t i t u t i v e Parameters for Indicated Metals.
k/Go
mo
al (MPa)
n
z~ (MPa)
(K-1)
220
0.3
46
4.9 • 10-5
2•
220
0.3
400
4.9 x 10-5
2•
900 685
0.35 0.05
630 1560
6.6 x 10-5 6.2 X 10-5
*NA = Not available.
9o(S-1)
(MPa)
~(s)
a
m
1010
NA*
NA*
20
0
101~
NA*
NA*
1.8
0
140 NA*
3 X 10 -4 NA*
2 X 1010 1.32 X 101~
5 1/2 2.4 1
5.11 Physically Based Rate- and Temperature-Dependent Constitutive Models
395
for annealed OFHC copper, AL-6XN stainless steel, and a titanium alloy. The corresponding comparison with the experimental results is shown in Figure 5.11.4 [17].
8O0 700
T /
I Solid Curves:ModelPredictions Dashed Curves:Experments
600
.-
I
I
.- ....
~7
---.
i
I
OFHC Copper
8,000/s
100 0 0
0.2
0.4
0.6
0.8
True Strain 2400
Dashed Curves: Experiments
~" 1600 m g
~-
Solid Curves: Model Predictions
Ti64
2000
,
l"
,
s"
. .~.,,,,~,.,..
77K
296K
1200
798K
800
.....
_
.....
400
3,700 s"
0.00
0.10
020
030
0.40
0.50
060
0.70
True Strain
2O0O
1600 /~,T a'77K
~
"~,
,= 0
"
0.00
•
"
J/! :
.
0.10
.
.
.
.
.
020
'
0.30
'
'
0.40
0.50
Tme Sb~in
FIGURE 5.11.4 Comparison between experimental and model results for indicated commercially pure metals and constitutive parameters given in Table 5.11.2.
396
5.11.6
Nemat-Nasser
THREE-DIMENSIONAL
MODEL
Let ~ denote the deviatoric true stress tensor and DP the plastic deformation rate tensor. We define the effective stress and strain rates by (3 "C - -
)1/2 "e " "e
( 3 ) 1 / 2 ,
f~ - -
De: D v
With f~(~,...) denoting the plastic potential [18, 19], we set 0f~ OP -- 7 c3~ For example, when f ~ - v ~ -
(15)
(16)
1:, we obtain P
OP=7~,
#=
(17)
Here, ~ is given by Eqs. 8, 7, and 1. Since ~* = r - (l:a + l:d), and l:d is a function of ~, the expression for ~ is not explicit. Hence, when the drag resistance is significant, special numerical algorithms are necessary in the applications. Let L be the elasticity tensor, and ~o be the Jaumann rate of the stress tensor. The final constitutive equation then is
~o = L" (D - O p)
(18)
where D is the deformation rate tensor; see Reference [19] for more details.
REFERENCES 1. Nemat-Nasser, S., Okinaka, T., and Ni, U (1998). A physically-based constitutive model for BCC crystals with application to polycrystalline tantalum. J. Mech. Phys. Solids 46 (6): 1009-1038. 2. Nemat-Nasser, S., Ni, L., and Okinaka, T. (1998). A constitutive model for FCC crystals with application to polycrystalline OFHC copper. Mech. Mater. 30 (4): 325-341. 3. Kocks, U. E, Argon, A. S., and Ashby, M. F. (1975). Thermodynamics and kinetics of slip, in Progress in Materials Science: Thermodynamics and Kinetics of Slip, p. 20, vol. 19, Oxford: Pergamon. 4. Nemat-Nasser, S., and Isaacs, J. B. (1997). Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloys. Acta Mater. 45:907-919. 5. Nemat-Nasser, S., and Li, Y. L. (1998). Flow stress of fcc polycrystals with application to OFHC CU. Acta Mater. 46: 565-577. 6. Nemat-Nasser, S., Guo, W. G., and Liu, M. Q. (1999). Experimentally-based micromechanical modeling of dynamic response of molybdenum. Scripta Mater. 40: 859-872. 7. Nemat-Nasser, S., and Guo, W.G. (2000). High-strain-rate response of commercially pure vanadium. Mech. Mater. 32: 243-260.
5.11 Physically Based Rate- and Temperature-Dependent Constitutive Models
397
8. Conrad, H. (1970). The athermal component of the flow stress in crystalline solids. Mater. Sci. Eng. 6: 260-264. 9. Follansbee, P. S., and Weertman, J. (1982). On the question of flow stress at high strain rates controlled by dislocation viscous flow. Mech. Mater. 1: 345-350. 10. Zerilli, E J., and Armstrong, R. W. (1992). The effect of dislocation drag on the stress-strain behavior of fcc metals. Acta Metall. Mater. 40: 1803-1808. 11. Chiem, C. Y. (1992). Material deformation at high strain rates, in Shock-wave and High-StrainRate Phenomena in Materials, pp. 69-85, Marca et al., eds., Marcel Dekker. 12. Regazzoni, G., Kocks, U. E, and Follansbee, P. S. (1987). Dislocation kinetics at high strain rates. Acta Metall. 35: 2865-2875. 13. Nemat-Nasser, S., Guo, W., and Kihl, D. (2001). Thermomechanical response of AL6-XN stainless steel over a wide range of strain rates and temperatures. J. Mech. Phys. Solids, in press. 14. Kapoor, R., and Nemat-Nasser, S. (1999). High-rate deformation of single crystal tantalum: Temperature dependence and latent hardening. Scripta Mater. 40 (2): 159-164. 15. Ono, K. (1968). Temperature dependence of dispersed barrier hardening. J. Appl. Phys. 39: 1803-1806. 16. Kapoor, R., and Nemat-Nasser, S. (1998). Determination of temperature rise during high strain rate deformation. Mech. Mater. 27 (1): 1-12. 17. Nemat-Nasser, S., Guo, W. G., and Cheng, J. Y. (1999). Mechanical properties and deformation mechanisms of a commercially pure titanium. Acta Mater. 47: 3705-3720. 18. Rice, J. R. (1970). On the structure of stress-strain relations for time-dependent plastic deformation in metals. J. Appl. Mech. 37: 728-737. 19. Nemat-Nasser, S. (1992). Phenomenological theories of elastoplasticity and strain localization at high strain rates. Appl. Mech. Rev. 45 (3): 519-545.
SECTION
5.12
Elastic-Viscoplastic Deformation of Polymers ELLEN M. ARRUDA1 and MARY C. BOYCE2 1Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 2Department of Mechanical Engineering, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
Contents 5.12.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12.2 Description of the Model . . . . . . . . . . . . . . . . . 5.12.3 Identification of Material Properties . . . . . . . 5.12.4 Applications of the Model . . . . . . . . . . . . . . . . 5.12.4.1 Model Results for Simple States of Deformation . . . . . . . . . . . . . . . . . . . 5.12.4.2 Modeling Complex Deformation States and Histories . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12.1
398 399 402 402 402 404 406
BACKGROUND
The strain-rate- and temperature-dependent inelastic deformation of glassy polymers has important applications in solid phase forming processes near the glass transition temperature, such as blow molding and film stretching; in impact absorption during large strain deformations; and in microscale deformation processes such as crazing. We use established and speculated physics of amorphous polymer deformation together with continuum mechanics formulations of finite strain kinematics to develop and implement a three-dimensional representation of the strain-rate-, temperature-, and pressure-dependent response of glassy polymers. The three-dimensional constitutive model for glassy polymer deformation is a finite strain elastic-viscoplastic adaptation of the standard linear solid
398
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
399
5.12 Elastic-Viscoplastic Deformation of Polymers
linear elastic spring viscoplastic element
FIGURE 5.12.1
rubbe r elasticity Langevin 'spring'
Schematic of the three-element elastic-viscoplastic model.
model from viscoelasticity. The elements of the glassy polymer model can be visualized in one dimension using the schematic in Figure5.12.1; this approach builds on the original one-dimensional model of Haward and Thackray [13]. The model elements include (1) a linear spring used to characterize the initial response as elastic, acting in series with a Kelvin element which is a parallel arrangement of (2) a thermally activated viscoplastic dashpot used to represent the rate- and temperature-dependent yield that monitors an isotropic resistance to chain segment rotation, and (3) a non-Gaussian molecular network spring that models the molecular orientation-induced anisotropic strain hardening. Constitutive descriptions for each of these elements are summarized within the context of a general finite strain deformation framework.
5.12.2
DESCRIPTION
OF THE MODEL
The representation of the kinematics begins with decomposition of the deformation gradient F -- Vxx: F = FeF p
the multiplicative (1)
where FP is the deformation gradient of the relaxed configuration obtained by elastically unloading to a stress-free state via F e-1. The elastic component of the deformation gradient, F e, is restricted to be a stretch only. F e = V e-- U e. Note that this results in no loss of generality, as shown in Boyce et al. [7].
Arruda and Boyce
400
The velocity gradient, L, can be expressed as the sum of the symmetric tensor D, the rate of deformation, and the skew-symmetric tensor W, the spin, and is given as follows: L = FF -1 = D + W = Le + FeLPF e-1
(2)
where the velocity gradient of the relaxed configuration, LP = FPF p-l, may be represented in terms of its symmetric and skew-symmetric components: Lp = D p + W p
(3)
where Wp is the plastic spin and DP is the rate of shape change in the relaxed configuration, which is constitutively prescribed through the characterization of the viscoplastic element. It may be shown, as in Boyce et al. [7], that Wp is algebraically prescribed without loss of generality as a result of the imposed symmetry on the elastic deformation gradient. The linear spring used to characterize the initial response of the material is constitutively characterized by the fourth-order tensor operator of elastic constants, ~oe: T -- l~-cpe[ln U e] (4) 3 where T is the Cauchy stress, (ln U e) is the Hencky strain, and J = det U e. The nonlinear rubber elasticity spring element introduces a convected network stress, T N, which captures the effect of orientation-induced strain hardening. The underlying macromolecular network orients with strain and has been found to be well modeled using the Arruda and Boyce [1] eightchain model of rubber elasticity. The network stress tensor (taken to be deviatoric) is given by
X/~ ~,~'-1 ( ' ~ ~ n ) [ B - ~,chainI] TN -- ~R '~chain
(5)
where B - FPFpr, and '~chain- [1 tr B] 1/2 is the stretch on each chain in the network. The material properties describing the strain-hardening characteristics are #R, the initial hardening modulus, and N, the number of rigid molecular units between entanglements. The Langevin function ~ is given by ~o(fl) _ coth(fl) - ~;
fl -
(6)
and its inverse provides the functionality that as the chain stretch, '~,chain, approaches its limiting extensibility ( v ~ ) , the stress increases dramatically. Temperature-dependent strain hardening modeled by this element assumes that the thermally equilibrated number of chains per unit
401
5.12 Elastic-Viscoplastic Deformation of Polymers
volume follows:
n(O) -- A - B exp
-~--~
(7)
where A represents the portion of the network that does not dissociate with strain and the term containing B represents the thermally dissociating network. Ea is the activation energy for thermal dissocation, R is the universal gas constant, and | is the absolute temperature. Mass is preserved, which provides n(|174 constant [4]. The viscoplastic element describes the rate- and temperature-dependent yield behavior. The effective shear stress on this element is found from the tensorial difference between the Cauchy stress, T, and the convected network stress, TN: 1 T* -- T - - FeTNFe~ (8)
J T* is the driving stress state, i.e., the portion of the total stress which continues to activate plastic flow. The deviatoric component of the driving stress state is denoted T *~, and it can be expressed in terms of its magnitude, r, and its tensorial direction, N: 1 "c--
2
1
N --~T*'
(10)
The rate of shape change DP in the viscoplastic element is assumed to be aligned with the deviatoric driving stress state: D p = ~PN.
(11)
where ~P is the plastic shear strain rate, which ensues once isotropic barriers to chain segment rotation are overcome. The magnitude ~P is taken to depend on the relative values of the effective shear stress, r, and the athermal shear strength, s, according to the relation:
[ -~xAS{
7P --7o exp -
- (~)
~}]
(12)
where 7o is the pre-exponential factor proportional to the attempt frequency, As is the zero stress level activation energy, and k is Boltzmann's constant. Strain softening is modeled after Boyce et al. [6] by taking the athermal shear strength, s, to evolve from its initial annealed value, So--0.077 p / ( 1 - v) (where # is the elastic shear modulus, and v is Poisson's ratio), to a
402
Arruda and Boyce
TABLE 5.12.1
Material Properties.
Elastic
Viscoplastic
Material
E (MPa)
Y
PC PMMA
2300 3200
0.33 0.33
Softening
Orientation hardening
~o (S-1)
As (S)
h (MPa)
so/Sss
#R (MPa)
2 (1015) 2.8 (107)
3.3 (10 -19) 1.9 (10 -19)
500 315
0.78 0.87
18 8
N 2.8 2.1
preferred state, Sss:
where h is the softening slope. Pressure dependence is taken into account by adding the pressure contribution, ~p, to the athermal shear resistance, where p is the pressure and ~ is the pressure-dependent coefficient of the material.
5.12.3 IDENTIFICATION OF MATERIAL PROPERTIES The constitutive description is thus summarized in Eqs. (4)-(13). Material properties are obtained by reduction of uniaxial compression data at different strain rates as discussed in References [2, 5]. Table 5.12.1 lists properties for polymethylmethacrylate (PMMA) and polycarbonate (PC).
5.12.4 APPLICATIONS OF THE MODEL 5 . 1 2 . 4 . 1 MODEL RESULTS FOR SIMPLE STATES OF DEFORMATION Figure 5.12.2 shows both experimental and constitutive model results for the cases of uniaxial compression and plane strain compression of PMMA. The material constants for the model are fit to the uniaxial compression data; the plane strain compression simulation is thus a true prediction. The model is found to capture the uniaxial compression data and then found to predict the plane strain compression data very well. This capability has also been shown on other amorphous polymers. The predictive capability demonstrates that the constitutive model contains the essential physics of the three-dimensional anisotropic nature of the network deformation behavior.
403
5.12 Elastic-Viscoplastic Deformation of Polymers
/
./
200
,____,,_
5q U~ l
5q
/
I .
100
~. $:'~<
.
.
.
"!" "'~" '
i .} il
0
I 0.0
. . . . . . . . . . . . uniaxial data . . . . . . plane strain data ~~-uniaxial eight chain --.------plane strain eight chain
0.5
1.0
1.5
TRUE STRAIN FIGURE 5.12.2 Deformation state-dependent response of PMMA at room temperature. (Reprinted from International Journal of Plasticity, Vol. 9, Arruda, E. M., and Boyce, M. C., Evolution of plastic anistropy in amorphous polymers during finite straining, pages 699-720, 9 1993, with permission from Elsevier Science.)
Figure 5.12.3 depicts both experimental and model results for the temperature dependence of strain hardening in PMMA; the results show uniaxial compression true stress vs. true strain curves at different temperatures. The simulations considered the temperature-dependent representation of the network structure given in Eq. (7). Note that data are needed at two temperatures to obtain the temperature-dependent material constants; the third curve is a prediction. The strain hardening is found to be lower with increasing temperature and is well predicted by the model. This same model was used to simulate the rate dependence of PMMA as shown in Figure 5.12.4, where both experimental and model results are shown. The expected increase in initial yield stress with increase in strain rate is observed and predicted; however, note the greater amount of softening observed at the higher strain rate. The highest rate ( - 0 . l / s ) does not provide enough time for heat transfer to occur, and there is a rise in the material temperature during straining which produces thermal softening in addition to strain softening.
404
Arruda and Boyce .
200
.
.
.
"
i
!
!
!
I
1
!
/ ./.
~176
-
I"'--I
...
25oc
CL,
~.
. o.~
&.,-.,,I ..
100
-
.
.
:':...---.~... .
....__
-----'~.'..,." ...... .
.
.
.
.
il 0.0
";
~
s
., "'~
.~
.
:
~
f
.
,'// ."
./
1.0
0.5 TRUE
FIGURE 5.12.3
.i'~:"
../
STRAIN
Isothermal response of PMMA at various temperatures. (Reprinted from
Mechanics of Materials, Vol. 19, Arruda, E. M., Boyce, M. C., andJayachandran, R., pages 193-212, 9 1995, with permission from Elsevier Science.)
The model considers the network orientation contributions to the total energy to be stored and not dissipated; therefore, only contributions from the driving stress act to generate heat. A full discussion is provided in Arruda et al. [4] as well as in Boyce, Montagut, and Argon [9]. The strong dependence of the stress-strain behavior on rate and temperature is particularly important in cases of processing polymers and designing polymer components for impact loading situations, where thermal softening can alter the expected deformation response of the material.
5 . 1 2 . 4 . 2 MODELING COMPLEX DEFORMATION STATES AND HISTORIES Inelastic deformation of polymeric materials, their products, and their processes typically involves inhomogeneous deformation the entire
5.12 Elastic-Viscoplastic Deformation of Polymers 200
,
....
'
il
.
.
.
- .
,
. --
,
'
'
405 '
' I
I
I
i
data data 0.1is theory -O.O01/s theory
0.1/s
-O.O01/s
. . . . . .,----
,'-q, u3 Dq
/ ....
>~...,... 100
.'
- ~
., ,:..:...-. " ~
...~,~.,.._ " ..~ . ~ . . . ~ 9. . . . . . . . : , ~ ' , ' - - - - - -
;
' "~
~
"
-----.~
: _ , ~
-"=
i.
E~
0
! .......
0.0
i
i
I
i
,,,I
0.5
,,,i
T R U E STRAIN
i
I
,
1.0
FIGURE 5.12.4 Strain rate-dependent response of PMMA at room temperature. (Reprinted from Mechanics of Materials, Vol. 19, Arruda, E. M., Boyce, M. C., and Jayachandran, R., pages 193-212, 9 1995, with permission from Elsevier Science.)
specimen or product is not deforming in a uniform, homogeneous manner. The classic example is the necking and neck propagation behavior of polymers during a tension test. Such inhomogeneous deformation can now be analyzed in detail using accurate constitutive models together with the finite element method. Neck propagation has been investigated using this constitutive model in several studies, including Boyce and Arruda [8] on waisted bars, Boyce, Montagut, and Argon [9] on thermomechanically coupled cold drawing, Hasan [12] on both cylindrical bars (see Boyce and Haward [11]) and thin sheets, and Wu and van der Giessen [20] on plane strain tension. Numerous other problems have also been examined using this constitutive model together with the finite element method. Problems include simple shear experiments [10], and hydrostatic extrusion of polymer cylinders [5], and thermomechanically coupled compression [4], as well as indentation of
406
Arruda and Boyce
single- and multilayered polymeric coatings [14-16], micromechanics of filled polymers [18,19], and stress fields at blunt notches [17]. Such detailed simulations provide better understanding of specimen, product, and process behavior and can be utilized in the design and development of multiphase polymer materials, polymeric products, and polymerprocessing operations.
REFERENCES 1. Arruda, E. M., and Boyce, M. C. (1993). A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41 (2): 389-412. 2. Arruda, E. M., and Boyce, M. C. (1993). Evolution of plastic anisotropy in amorphous polymers during finite straining. Int. J. Plasticity 9: 697-720. 3. Arruda, E. M., and Boyce, M. C. (1993). Effects of initial anisotropy on the finite strain deformation behavior of glassy polymers. Int. J. Plasticity 9: 783-811. 4. Arruda, E. M., Boyce, M. C., and Jayachandran, R. (1995). Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mech. Mater. 19:193-212. 5. Boyce, M. C., Parks, D. M. and Argon, A. S. (1988). Large inelastic deformation of glassy polymers, Part I: Rate dependent constitutive model. Mech. Mater. 7: 15-34. 6. Boyce, M. C., Parks, D. M., and Argon, A. S. (1988). Large inelastic deformation of glassy polymers, Part II: Numerical simulation of hydrostatic extrusion. Mech. Mater. 7: 35-47. 7. Boyce, M. C., Weber, G. G. and Parks, D. M. (1989). On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37: 647-665. 8. Boyce, M. C., and Arruda, E. M. (1990). An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers. J. Polymer Engin. Sci. 30: 1288-1298. 9. Boyce, M. C., Montegut, E., and Argon, A. S. (1992). The effects of thermomechanical coupling on the cold drawing process in glassy polymers. Polymer Engin. Sci. 32: 1073-1085. 10. Boyce, M. C., Arruda, E. M., and Jayachandran, R. (1994). The large strain compression, tension and simple shear of polycarbonate. Polymer Engin. Sci. 34(9): 716-725. 11. Boyce, M. C., and Haward, R. N. (1997). The post yield deformation of glassy polymers, in Physics of Glassy Polymers, Haward, R. N., and Young, R. J., eds., London: Chapman and Hall. 12. Hasan, O. A. (1994). Ph.D. Thesis, MIT Department of Mechanical Engineering, Cambridge: MA. 13. Haward, R. N., and Thackray, G. (1968). The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. Roy. Soc. Lond. Series A 302: 453-472. 14. Jayachandran, R., Boyce, M. C., and Argon, A. S. (1993). Mechanics of the indentation test and its use to assess adhesion of polymeric coatings. J. Adhesion Sci. Technol. 7: 813-836. 15. Jayachandran, R., Boyce, M. C., and Argon, A. S. (1994). Thermomechanical analysis of indentation behavior of thin PMMA coatings. Journal of Computer Aided Materials Design 2: 23-48. 16. Jayachandran, R., Boyce, M. C., and Argon, A. S. (1995). Design of multi-layer polymeric coatings for indentation resistance. Journal of Computer Aided Materials Design 2: 151-166.
5.12 Elastic-Viscoplastic Deformation of Polymers
407
17. Lai, J., and van der Giessen, E. (1997). A numerical study of crack-tip plasticity in glassy polymers. Mech. Mater. 25: 183-197. 18. Steenbrink, A. C., van der Giessen, E., and Wu, E D. (1997). Void growth in glassy polymers. J. Mech. Phys. Solids 45: 405-437. 19. Socrate, S., and Boyce, M. C. (2000). Micromechanics of toughened polycarbonate. J. Mech. Phys. Solids 48: 233-273. 20. Wu, P. D., and van der Giessen, E. (1995). On neck propagation in amorphous glassy polymers under plane strain tension. Int. J. Plasticity 11: 211-235.
This Page Intentionally Left Blank
CHAPTER
Continuous Damage
6
This Page Intentionally Left Blank
SECTION
6.1
Introduction to Continuous Damage JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du Pr~.sident Wilson, 94235 Cachan Cedex, France
Damage of materials means the progressive or sudden deterioration of their
mechanical strength because of loadings or thermal or chemical effects. It covers all related phenomena that occur from the virgin or reference state up to a mesocrack initiation. From a physical point of view, damage can originate from multiple causes: debonding of atoms, nucleation, or growth and coalescence of microcracks and microcavities. Despite the discontinuous nature of such processes at the microscale, continuous damage means a homogeneous modeling in which microcracks and microvoids are represented by a continuous variable in the sense of the mechanics of continuous media. Having said this, the scale at which this homogenization takes place needs to be established. Microscale is the size at which the elementary mechanisms occur, macroscale is the size of engineering structures which must not brake, and Mesoscale is the size for which the homogeneous constitutive equations are written: it is a material point in mathematics, a, "reasonable" representative volume element (RVE) in the mechanics of materials. Let us say, as an order of magnitude: 0.1 mm for metals = 1 mm for polymers and some composites 10 mm for wood and some composites -~ 100 mm for concrete Most of the models of continuous damage introduce a damage variable at the mesoscale related to the volume density p of microvoids (porosity or ductility) or more generally to the surface density D of defects (microcracks and Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
411
412
Lemaitre
microvoids) in any plane of the RVE.
(~V cavities P= 6Vo
D
O
O
~S defects ~So
6Vo and 6So being the volume size and the surface size of the RVE. These variables represent the weakness of the materials, which yields a decrease in the elastic rigidity, in the actual yield stress, and in many other properties. This coupling is often introduced in elastic and plastic constitutive equations through the thermodynamics of irreversible processes. The evolution of p or D represents the deterioration of the materials up to mesofracture. At the mesoscale, damage is observed as different phenomenological phenomena, including:
9 Brittle or quasi-brittle damage when fracture occurs without significant irreversible strain. See Sections 6.2, 6.3, and 6.15.
9 Ductile damage in metals subjected to large plastic strain at low temperature: T(K)< ~ 1Tm, the melting temperature. See Sections 6.4, 6.5, 6.6, and 6.14.
9 Creep damage due to large plastic strain in metals subjected to elevated temperature: T(K) > ~ 1Tm. See Sections 6.7 and 6.14. 9 Fatigue damage due to repetitions of loading either above the yield stress, known as low cycle fatigue (see Sections 6.8 and 6.14), or below the yield stress, known as high cycle fatigue (see Sections 6.9, 6.10, and 6.15), and even in the gigacycle range (see Section 6.11). 9 Some materials need special treatment even if the same formalism applies. This is the case for the formation of crazes in polymers (see Section 6.12), for the large difference of strength in tension and compression of concrete (see Section 6.13), or for the debonding of interfaces in composites (see Chapter 10).
SECTION
6.2
Damage-Equivalent Stress Fracture Criterion JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
Contents 6.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Description of the Model . . . . . . . . . . . . . . . . . . . 6.2.4 Identification of the Parameter cr~ . . . . . . . . . . 6.2.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1
413 413 414 415 415 416
VALIDITY
This is a fracture criterion that is valid for brittle and quasi-brittle materials but also for ductile materials as a first approximation for a quick estimation (at least much better than the von Mises equivalent stress criterion often used).
6.2.2
BACKGROUND
In the continuum theory of isotropic damage, a damage state variable D is introduced as the surface density of microcracks and microcavities. The thermodynamics of irreversible processes defines its associated variable Y, called the "energy density release rate" such that Y./~ is the power dissipated in the process of damage. With the concept of effective stress ai) r - D and the principle of strain equivalence, the Helmholtz state potential energy is written as -
1 -
r
-
r
2 p EijhleiJGl(1 -- D) + . . .
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
413
414
Lemaitre
where p is the density, E/jkl is the elastic tensor, and ~ is the elastic strain tensor. Then 0~t'
we
Y = - P oqD -- l _ D
We being the total elastic strain energy density dWe - 0-ij&~. In the case of linear isotropic elasticity coupled with damage, Off
0-ij
=
P ~j
e
1 -F v
0-0
E
1 -- D
~'ij =
or
v
0-k k ~ . .
E 1 - D ''j
where E is the Young's modulus, v is the Poisson's ratio, and 6ij is the Kronecker operator. 2 0-eq
We = 2E(1 - D)
Rv
where 0-eq is the von Mises equivalent stress;
where 0-~ is the stress deviator 0 - ~ - 0-~j- 0-n~j; an is the hydrostatic stress an - 89 0-kk; and R~ is the triaxiality function Rv =2(1 + v) + 3(1 - 2v)(an/rrCq) 2
6.2.3
DESCRIPTION
OF THE MODEL
The damage-equivalent stress is defined similarly to the plastic von Misesequivalent stress for plasticity as the one-dimensional stress 0-* which, for the same value of the damage, yields the same value of the elastic strain energy density as in the case of a three-dimensional state. For the one-dimensional case:
[a] -
0-*
0
O]
0
0
0
0
0
0
0-eq -- 0-*
1
0-H
1 =--
0-eq
3
Rv= 1
Writing the equality we(a*) -We(aij)" 0-*2
a e2qRv
2E(1 -**D)
2F (1 -- D)
415
6.2 Damage-Equivalent Stress Fracture Criterion defines the d a m a g e e q u i v a l e n t stress
~7" -- ~TeqP~v/2. Since the variable Y associated w i t h the d a m a g e D is r e l a t e d to the elastic strain energy, it is p o s t u l a t e d that the fracture of a r e p r e s e n t a t i v e v o l u m e e l e m e n t is g o v e r n e d by the total elastic strain e n e r g y density; that is, w h e n the d a m a g e - e q u i v a l e n t stress r e a c h e s the value of the u l t i m a t e stress at fracture in t e n s i o n (or c o m p r e s s i o n ) , ~ru is characteristic of each material. o'* ~-- o-u 0"*
6.2.4 IDENTIFICATION OF THE PARAMETER a~ T h e u l t i m a t e stress to r u p t u r e is o b t a i n e d from a tensile stress at m e s o s c a l e u p to fracture.
6.2.5 H O W TO USE THE MODEL T h e crack initiation in a s t r u c t u r e , or the fracture, will o c c u r in the z o n e w h e r e or* is m a x i m u m , w h e n o r * - ~ r u . T h e d a m a g e - e q u i v a l e n t stress is c a l c u l a t e d from the stress c o m p o n e n t s ~ij r e s u l t i n g from a s t r u c t u r e analysis.
TABLE 6.2.1
Table of Parameters
Material C 35 steel A 201 steel A 517 steel 316 st. steel 316 st. steel 304 st. steel 304 st. steel Inco 718 2024 A. alloy Concrete in tension Concrete in compression Ceramic alumina
Temperature, ~
o'u, MPa
20 20 20 20 600 20 593 550 20 20 20 20
458 412 790 760 650 950 700 1150 500 3 35 500
416
Lemaitre
This fracture criterion, proportional to the von Mises equivalent stress, takes into account the important effect of the triaxiality ratio a~i/aeq by the triaxility function Rv. See Table 6.2.1 for values of au corresponding to some materials.
REFERENCE 1. Lemaitre,J. (1994 and 1996). A Course on Damage Mechanics, SpringermVerlag.
SECTION
6.3
Micromechanically Inspired Continuous Models of Brittle Damage DUSAN KRAJCINOVIC Arizona State University, Tempe, Arizona
Contents
6.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Description of the Model . . . . . . . . . . . . . . . . . . . 6.3.4 Identification of Parameters . . . . . . . . . . . . . . . . 6.3.5 Model Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
417 418 418 419 420 420
6.3.1 VALIDITY The continuum models of damage evolution that are based on the micromechanics of damage are applicable to deformations during which the irreversible changes of microstructure are related primarily to the random site nucleation of microcracks and their stable growth. This class of deformation is typical of materials with inferior cohesive strength and (a) heterogeneous microstructure (such as composites) at any state of stress; or (b) all microstructures (including iron, carbon, many semiconductors, ceramics, epoxies, cement, rocks, bones, etc.) in the absence of longrange tensile stresses. These two materials are known as being damage-tolerant and damagesensitive, respectively. The damage in the form of microcracks may be attributed to manufacturing processes, environmental causes (such as corrosion, low temperature), applied loads, etc. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
417
418
Krajcinovic
6.3.2 BACKGROUND Micromechanical models are deduced by the process of homogenization, during which the effective parameters of the material points of a continuum are derived by considering the volume of material, defined by a large number of microcracks embedded in homogeneous matrix, that is centered by the material point. The smallest volume for which a considered effective parameter of the volume does not depend on the exact locations of microdefects is known as the representative volume element (RVE). When the RVE is small enough to consider the applied macrostresses and strains to be uniform, the model is local. Assuming this to be the case, the relation between volume-averaged stress and strains is
8ij -- Sijmn~mn
(1)
where the bar above the symbol stands for the average over RVE and Sijmn is the compliance tensor of the damaged volume. Assuming that the all tensors in Eq. 1 are differentiable, the rate form of the stress-strain relation is -
=
+
-e
-
+
-d
(2)
The dot above a symbol stands for the rates, and the superscripts e and d stand for elastic and damage. This assumption may be violated when normal stresses change the sign and a closed crack opens, or vice versa. Using positive and negative projection operators, the stresses and strain tensors are divided into positive and negative parts. Assuming that a scalar damage potential f~(F, S) exists and the normality rule is valid, the rate of the effective compliance may be derived from
Sijmn = 2 O0~"~(F, ~ S)
where
1_ _ F=~aijam.
(3)
is the thermodynamic force (that drives damage evolution) conjugate to the selected internal v a r i a b l e Sijmn (effective compliance).
6.3.3 DESCRIPTION OF THE MODEL The damage (microcracks) density D is concealed within the effective compliance of the RVE. Since the exact locations and geometry of all microcracks cannot be determined at each material point (onto which the RVE was mapped) and for all instances, the selection of a "best" choice of damage parameters is somewhat arbitrary. The choice of the effective compliance Sijmn(D) is hereby prescribed for these reasons:
419
6.3 Micromechanically Inspired Continuous Models of Brittle Damage
9 it can be relatively easily measured in the laboratory and in situ; and 9 it can be derived using methods of micromechanics.
Assume, for simplicity, that all microcracks are of penny-shaped geometry defined by their radius a and orientations (0, ~b) that are not correlated. Hence, the microcrack density function is w(a, O, dp) ~ O(a) + p(O, dp) (4) In this case the effective compliance of the RVE is
i,j== if=. -
+
a0
2/=
o,
-
+,j=.
In Eq. 5, S~m, is the compliance of the pristine material, Fom,, is the normalized compliance attributed to a single crack, and co = N(a3), the BudianskiO'Connell damage parameter for an isotropic distribution of penny-shaped microcracks. Tensor ~/jm,, defined by the double integral in Eq. 5, generalizes the Budiansky-O'Connell damage parameter to damages that are not isotropic. Analytical quadrature of Eq. 5 and close-form estimates of the effective compliance using one of the effective field models are available for dilute concentrations of penny-shaped cracks and linear slits in the reference provided below. Finally, the mean-field estimate (Eq. 5) of the effective compliance is based on many other assumptions. The most debatable assumption is that the released energy rate G is equal to the path-independent integral J. Equation 5 defines the thermodynamic state under described assumptions in all points if the microcrack density function (Eq. 4) is known. If the affinity (the difference between the force F driving the damage evolution and the thermodynamic force resisting the damage evolution) is small, the considered irreversible process can be approximated by a temporal sequence of states equilibrated by forces F. In this case the macropotential ~2(F, S) can be, at least in principle, deduced from the micropotentials that regulate individual microcrack growths.
6.3.4 IDENTIFICATION OF PARAMETERS In comparison to the determination of statistical distributions of microcracks, the measurement of effective compliance is rather simple. However, the weak aspect of the considered model, and many other continuum models of damage evolution, is related to rather arbitrary determinations of the potentials in Eq. 3. Hence, the evaluation of rates is often not very reliable. The thermodynamic force Fomn, conjugated to the damage parameter Sijmn(D), is related to the elastic energy release rate. These facts provide the
420
Krajcinovic
connection between the damage and fracture mechanics and may be used in the experimental measurements of damage needed to determine the model parameters. The task of identifying the macropotential from micropotentials is a nontrivial task that must reflect the statistical nature of the affinity as a microcrack grows through a material with a disordered microstructure.
6.3.5 MODEL RESTRICTIONS The considered continuum models, based on the mechanics of microcracking, belong to the class of mean-field models that relate average stresses and strains using effective material (or structural) parameters. The BudianskyO'Connell damage parameter, used in most, if not all, models, does not discriminate between a few large cracks and many smaller cracks as long as the product co = N(a 3) is the same. Since the onset and stability of propagation of large and short cracks are different, this damage parameter, along with reliance on averages, becomes questionable when the microcrack propagation and their clustering become a significant mode of damage evolution. As soon as the affinity of a single microcrack becomes larger than unity enough to render its growth unstable, the previous model and all other local continuum models lose validity. The affinity of a single crack depends on the local stress and local cohesive strength at its tip, which may be radically different from the averages. Hence, any model that does not consider the micro-scale disorder will be of limited utility for all but dilute concentrations of damage. A rather novel method using particle dynamic simulations may prove to be useful in determining constitutive equations in both quasi-static and dynamic deformations. In general, the described model is applicable to quasi-static deformations at relatively small microcrack densities, especially when a damage-sensitive material is subjected to large-range tensile stresses.
REFERENCES 1. Kachanov, M. (1993). Elastic solids with many cracks and related problems, in Advances in Applied Mechanics, pp. 259-443, vol. 29, Hutchinson, J., and Wu, T., eds., New York: Academic Press. 2. Krajcinovic, D. (1996). Damage Mechanics, North-Holland Series in Applied Mathematics and Mechanics, vol. 51, Amsterdam, The Netherlands: Elsevier. 3. Mastilovic, S., and Krajcinovic, D. (1999). High-velocityexpansion of a cavity within a brittle material. J. Mech. Phys. Solids 47: 577-610.
SECTION
6.4
Anisotropic Damage C. L. CHOW and YONG WEI Department of Mechanical Engineering, University of Michigan-Dearborn
Contents 6.4.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 6.4.2 Background on Damage Variables . . . . . . . . . . 422 6.4.3 Description of Damage-Coupled Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 6.4.4 Identification of Damage Effect Tensor M and Damage Characteristic Tensor J . . . . . . . . . . . . 425 6.4.5 How to Use It by Numerical Simulation .... 428 6.4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
6.4.1 VALIDITY The damage mechanics approach is based on the irreversible thermodynamics theory. In the last two decades, this theory has been widely applied to study the behavior of rocks, composite materials, and concrete, as well as ductile fracture, creep rupture, and fatigue failure of metals. One basic hypothesis in most damage models is that the damage can be viewed as a macroscopic state variable D, known as the internal state variable, by ignoring the details on the nucleation and growth of distributed microdefects in a material element. The success of a damage model to a large extent depends on the definition of the damage variable D that is related to the concept of effective stress tensor F introduced first by Kachanov [1] and developed thereafter by many others [2]. A scalar damage variable D was defined by Kachanov as the loss of an effective load-carrying cross-sectional area that is responsible for accelerating the tertiary creep strain rate. The concept has been used by researchers as the decrease of the elastic modulus in isotropic damage models. The Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
421
422
Chow and
Wei
orientation-independent scalar damage variable was, however, subsequently found to be inaccurate, since it implies that Poisson's ratio does not change. Both theoretical analysis [3] and experimental measurement [4,5] have revealed substantial damage-induced changes in Poisson's ratio that are associated with the decrease of elastic modulus. In effect, an initially isotropic material such as AL2024-T3 after gross plastic deformation has been experimentally observed to undergo anisotropic damage [8]. The deficiencies of scalar theories have therefore caused concern about their effectiveness in modeling material behaviors. In this section, several anisotropic damage models are presented. Damage variables are first introduced. Then the damage-coupled constitutive equations and the damage evolution equations are formulated within the framework of irreversible thermodynamics based on the Helmholtz free energy, the yield surface, and the damage surface. A distinct advantage of the damage models is their ability to be readily incorporated in the finite element analysis through a damage-coupled stiffness matrix. Finally, several applications of the damage models are presented to illustrate failure analysis in engineering structures subjected to either static and dynamic loading.
6.4.2 BACKGROUND ON DAMAGE VARIABLES According to the theory of damage mechanics, the effective stress tensor is applied to substitute for the Cauchy stress tensor in the development of constitutive equations of elasticity and plasticity and of damage evolution equations. The relationship between the effective stress tensor F and the Cauchy stress tensor cr is expressed as [2]
#-- M(D)'# = (I- D)-I"cT
(i)
where M(D) is the damage effect tensor and I is the unit tensor. Selection of the damage variable tensor D may be based on the phenomenological or micromechanics approach. Currently, most of the micromechanics models are less ineffective for practical applications. This is primarily due to the difficulties associated with the determination of the microcrack distribution and the development of the evolution equation. Therefore, phenomenological approaches are emphasized within the framework of irreversible thermodynamics. Accordingly, the Helmholtz free energy 9 is postulated based on the principle of equivalent elastic energy that "the elastic energy of the
423
6.4 Anisotropic Damage
damaged material is the same in form as that of an undamaged material except that the stress is replaced by the effective stress," i.e.,
pW -- WE +pWp = 8 9
1" e + pWp(q)
(2)
where p is the density, W E is the elastic energy, kI/p is the plastic part of free energy due to strain hardening, Co is the elastic tensor of the undamaged material, and q denotes a set of internal state variables for the strain hardening. With the definition of the effective stress in Eq. 1, the free energy can be derived in the Cauchy stress space as
p t p _ 89
M r ' C o 1" M ' a + ptpp(q) _= 8 9 C -1 --
c-l.a
+ ptpp(q)
M r :Co 1 :M
(3) (4)
where C is the elastic tensor for damaged materials. Accordingly, the damage energy release rate tensor Y, or the conjugate to the damage variable, is defined as
ov
(5)
Y = - P 0D
6.4.3 DESCRIPTION OF DAMAGE-COUPLED CONSTITUTIVE EQUATIONS The constitutive equation of elasticity can be derived in the Cauchy stress-true strain space by the free energy in Eq. 3 as
0tg = C - l . a
8e = p ~
(6)
8e is the true strain tensor. For elastoplastic damaged materials, following the von Mises theory, the yield surface is postulated in terms of the effective stress as where
Fp(e, R) = op -[Ro + R(e)] = 0
(7)
where ap is the equivalent stress 1
~, =
{~'-o
.~}~
(8)
Ho is the plastic characteristic tensor for undamaged material, Ro is the yield stress, p is the effective equivalent plastic strain, and R is the strain-hardening threshold. In the Cauchy stress-true strain space, Eq. 8 becomes
ap = 89a T. H" a
H = M ~ ' H o 9M
(9)
424
C h o w a n d Wei
For anisotropic material, the form of the tensor Ho is given as [6]
H0 =
g+ h
-h
-g
0
0
0
-h
h+f
-f
0
0
0
-g 0
-f 0
f+g 0
0 2r
0 0
0 0
0
0
0
0
2m
0
0
0
0
0
0
2n
(10)
where f, g, h, r, m, and n are parameters characterizing the current state of plastic anisotropy. For isotropic materials, f = g = h = 1 and r = m = n = 3. The constitutive equations of plasticity for damaged materials are derived in the Cauchy stress-true strain space as
~,P -- 2p cga (11)
oFp P - 2P O ( - R ) = 2p
where ep is the plastic strain tensor and 2p is the Lagrange multiplier. It is postulated that there exists a plastic damage surface Fd = 0 which separates the plastic damaging domain from the undamaging domain. The plastic damage surface is formulated with the thermodynamic conjugate forces of the plastic damage variables as
Fd(Y, B) = Yd --[Bo + B(w)] = 0
(12)
where the equivalent damage energy release rate Ypd is defined as
Ya - ( 89 "J" Y)1/2
(13)
Bo is the initial plastic damage threshold, B is the plastic damage hardening, w is the overall damage, and J is the damage characteristic tensor. The plastic damage evolution equations are derived in a thermodynamic framework as [7]
C3Fd _
--2d J : Y (14)
aFd
425
6.4 Anisotropic Damage
The overall damage rate v~ can be determined by the plastic damage surface in Eq. 12 as oqYa. ~
Cr
c3cr
~V -- c3ya OFa OD
(15)
dB
OY f dw
Then the damage evolution equation can be written as
15=-w
J:Y
(16)
2Yd
A failure criterion has been proposed based on the overall damage accumulation associated with the concept of the plastic damage surface. Based on this criterion, a material element is said to have ruptured when the total overall damage w in the element reaches a critical value Wc.
6.4.4 IDENTIFICATION OF DAMAGE EFFECT TENSOR M AND DAMAGE CHARACTERISTIC TENSOR J The generalized damage tensor D may be defined as a second-order, fourthorder, or higher-rank tensor. It is always true that a higher-rank tensor contains more information and provides a more accurate representation of the behavior of material degradation. The drawback in applying a higher-order tensor is often associated with the measurement difficulty of all the required material parameters and with inefficient numerical analysis. Some simplifications may consequently be necessary from a practical viewpoint. Several damage effect tensors M have been established with the introduction of a second-order symmetric damage tensor D [8-10]. Some of the proposed forms are summarized as
A.
Mz
0
0
0
0
0
1 1-D2
0
0
0
0
0
0
1 1-D3
0
0
0
0
0
0
1 V/(1-D2)(1-D3)
0
0
0
0
0
0
1 V/(1-D1)(1-D3)
0
0
0
0
0
0
1 V/(1-D1) (1-D2)-
(17)
426
B.
Chow and Wei
"I_-~ID~
0
0
0
0
0
0
1 l-D2
0
0
0
0
0
0
1 1-D3
0
0
0
0
0
0
2 2-(D2+D3)
0
0
0
0
0
0
0
0
0
0
M---
2 2-(Dl+D3)
0
(18)
0
2
2-(D1 +D2).
0
0
0
0
0
0
1 l-D2
0
0
0
0
0
0
0
0
0
0
0
0
o
o
0
0
0
0
o
o
o
o
C.M=
1 l-D3 .
1 1 + ~_~'~) ~(~-~
1 1 + ~---k) ~(~=~
o
"
o
-~--~ +_<~ 2 ~, (19)
D.
"eD~
0
0
0
0
0
0
eD~
0
0
0
0
0
0
eD3
0
0
0
0
0
0
0
0
0
0
0
0
e(DI+D3)/2
0
0
0
0
0
U--
(20)
r
0
r
where D1, D2, and D3 are the respective principal values of the damage variable D. The damage energy release rate Y for cases A, B, and C have been derived by means of Eq. 5 in Reference [10].
427
6.4 Anisotropic Damage
Models based on a fourth-order symmetric damage tensor D have been proposed [2, 11, 12]. One sample form of damage effective tensor is [2] -
1
0
0
0
0
0
v D1 1--v l-D1
1
0
0
0
0
v D1 1-v l-D1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
l--D1
(21)
M
_
1-Ds
0
Based on the assumption that original isotropic materials will remain isotropic under load, a damage effect tensor is expressed with two scalars D and # [13]"
M
~
1-D
"1
#
#
0
0
0
#
1
#
0
0
0
#
#
1
0
0
0
0
0
0
1-#
0
0
0
0
0
0
1-#
0
.0
0
0
0
0
-
(22)
1-#_
This model is for isotropic materials, but the changes in both Young's modulus and Poisson's ratio due to material degradation are taken into account. The damage characteristic tensor J in Eq. 13 may be defined as [13] -1
/1 /1
0
0
0
/1
1
/7
0
0
0
/1
/1
1
0
0
0
0
0
0
2(1-/1)
0
0
0
0
0
0
2(1-/1)
0
0
0
0
0
0
2(1-/1)
J ~..
where q is a material constant.
(23)
428
Chow and Wei
6.4.5 H O W TO USE IT BY NUMERICAL SIMULATION The procedure for formulating the finite element analysis with damage is similar in principle to the conventional finite element method analysis. However, the stiffness matrix C needs to be modified for the following equation when damage effects are taken into account. do" = C :dg
(24)
where dcr is the Cauchy stress increment and de is the true strain increment. The approach to deriving the damage-coupled stiffness matrix can be found in References [13,14].
6.4.6 APPLICATIONS Several CDM models have been applied successfully to characterize the failure analysis of different materials for ductile fracture, metal forming, and fatigue failure in the last decade. The failure analyses include the following: 9 9 9 9 9
Crack initiation and propagation in mixed-mode ductile fracture [14] Three-dimensional ductile failure analysis [15] Damage analysis in composite materials [ 16] Prediction of forming limit diagrams [ 17] Fatigue failure analysis [18,19].
REFERENCES 1. Kachanov, L. M. (1958). On the creep fracture time. Ivz. Acad. Nauk U.S.S.R. Otd. Tech. Nauk. 8:26-31. 2. Lemaitre, J., and Chaboche, J. L. (1990). Mechanics of Solid Mechanics, Cambridge University Press. 3. Case, E. D. (1984). The effect of microcracking upon the Poisson's ratio for brittle solids. J. Mater. Sci. 19: 3702-3712. 4. Cordebois, J. P., and Sidoroff, E (1982). Anisotropic damage in elasticity and plasticity. J. Mech. Theor. Appl. 1: 45-60. 5. Chow, C. L., and Wang, J. (1987). An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract. 33: 3-16. 6. Hill, R. (1950). The Mathematical Theory of Plasticity, Oxford: Clarendon Press. 7. Yu, L. G., Chow, C. L., and Duggan, B. J. (1993). An orthotropic damage model with damage field mobility (DFM) method for fatigue crack propagation, in Advances in Engineering Plasticity and Its Applications, pp. 363-370, Elsevier Science Publishers.
6.4 Anisotropic Damage
429
8. Chow, C. L., and Wang, J. (1987). An anisotropic theory of continuum damage mechanics for ductile fracture. Engineering Fracture Mechanics 27: 547-558. 9. Lu, T. J., and Chow, C. L. (1990). On constitutive equations of inelastic solids with anisotropic damage. Theoretical and Applied Fracture Mechanics 14: 187-218. 10. Chen, X. E, and Chow, C. L. (1995). On damage strain energy release rate Y. International Journal of Damage Mechanics 4: 251-263. 11. Hansen, N. R., and Schreyer, H. L. (1995). A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Structures 31: 359-389. 12. Cauvin, A., and Testa, R. B. (1999). Damage mechanics: Basic variables in continuum theories. Int. J. Solids Structures 36: 747-761. 13. Chow, C. L., and Wei, Y. (1999). Constitutive modeling of material damage for fatigue failure prediction. International Journal of Damage Mechanics 8: 355-375. 14. Chow, C. L., and Wang, J. (1989). Crack propagation in mixed-mode ductile fracture with continuum damage mechanics. Proc. Instn. Mech. Engrs. 203: 189-199. 15. Chow, C. L., and Chen, X. E (1998). An endochronic damage model for three-dimensional ductile failure analysis of double-edge notched thick-tension specimens. Proc. Instn. Mech. Engrs 212C: 25-34. 16. Chow, C. L., and Yang, E (1994). Elastic damage analysis of interlaminar stress distributions in sysmetrical composite laminates with edge delamination cracks. Proc. Instn. Mech. Engrs 208:1-11. 17. Chow, C. L., Yu, L. G., and Demeri, M. Y. (1997). A unified damage approach for predicting forming limit diagrams. ASME J. Eng. Mater. Tech. 119: 346-353. 18. Chow, C. L., and Yu, L. G. (1995). An anisotropic damage model for metal fatigue. Computational Mechanics '95 2: 1904-1919. 19. Chow, C. L., and Wei, Y. (1996). A fatigue damage model for crack propagation. Advances in Fatigue Lifetime Predictive Techniques, ASTM STP 1291, 86-99.
SECTION
6.5
The Modified Gurson Model V. TVERGAARD1 and A. NEEDLEMAN2 1Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmark 2Brown University, Division of Engineering, Providence, Rhode Island
Contents 6.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Description of the Model . . . . . . . . . . . . . . . . . . . 6.5.4 Identification of Parameters . . . . . . . . . . . . . . . . 6.5.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
430 430 431 434 434 435
6.5.1 VALIDITY The effect of microvoid nucleation, growth, and coalescence on the plastic response of structural metals is modeled. The formulation is valid for finite strains, and there are versions for both rate-independent and rate-dependent plasticity. Porosity is accounted for through a single parameter, the void volume fraction. Any effect of deformation-induced void shape changes is neglected. The accuracy of the model is highest for states of relatively high stress triaxiality of relatively low void volume concentration. In order to predict crack initiation and growth, a length scale needs to be incorporated into the problem formulation.
6.5.2
BACKGROUND
Ductile fracture in structural metals deformed at room temperature takes place by the nucleation, growth, and coalescence of microvoids. The voids generally nucleate by decohesion of second-phase particles or by particle fracture, and grow by plastic deformation of the surrounding matrix. Void
430
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All fights of reproduction in any form reserved.
431
6.5 The Modified Gurson Model
coalescence occurs either by necking down of the matrix material between adjacent voids or by localized shearing between well-separated voids. This process of progressive cavitation softens the material, eventually leading to plastic instability and/or fracture.
6.5.3
DESCRIPTION
OF THE MODEL
The basis for the material model is a modification of the flow potential due to Gurson for a progressively cavitating solid.
~e @ -- ~-~ 4- 2qd* cosh t 2 ~ /
- 1 - (ql
- 0
Here ~- is the matrix flow strength and
with o'ij being the Cauchy or true stress. The potential q~ reduces to that for a Mises solid (see Chapter 4) when f* - 0. The parameters ql and q2 were introduced to bring the predictions of the model into closer agreement with full numerical analyses of a periodic array of voids. The bilinear function f*(f) accounts for the effects of rapid void coalescence at failure and is given by
f*- {f f
where () denotes partial differentiation with respect to time. The rate of increase of the void volume fraction due to the growth of existing voids is determined from the condition that the matrix material is plastically incompressible,
fgrowth -- (1 - f) d~k Here, d/~ is the plastic part of the rate of deformation tensor (also called the velocity strain tensor).
432
Tvergaard a n d N e e d l e m a n
The contribution resulting from the nucleation of new voids is taken to be described by
~'b +-~ ~k~
Jnuc~at~on =
Nucleation for a given population of particles is generally regarded as either plastic strain-controlled, d > 0 and ~ = 0, or stress-controlled, d = ~ , and often taken to follow a normal distribution. For example, for strain-controlled nucleation, M = 0 and ~r
fN
exp
l(~-eN
when ~ - (B)max and ~ > O. Here, E and Et are, respectively, the Young's modulus and the tangent modulus of the matrix material, fN is the volume fraction of void-nucleating particles, eN is the mean nucleation strain, and SN is the standard deviation. There is a similar expression for stress-controlled nucleation. For a rate-independent matrix,
1
d~ - -~ pij qkz CrhZ where akz is the Jaumann rate of Cauchy stress and I
3 aij P iJ = 2 ~ 4- ~ ~ ij ,
l
3 aij q iJ -- 2 Fr 4- fl ~ ij
90~
f* (q2ahk~ c~ - ~ qlq2 sinh \ 2~" / '
H
=
-g
3a(1 -f)--~f-+
/3 - ~ +-g e-by
d +--~a E - Et 1 -f~,F 2
where c9~ - =
cosh
of
\2~/
-
qls*]
with 1 K--
f
(~_fc)/(ff
0~ fie _ f , = - 2 ~-~ 0~
_fc)
f >_f~
(Tkk qlq2 -~sinh \ 2~ /
gr
433
6.5 The Modified Gurson Model
We also note that 3alj +f*qlq2 sinh (q2akk'~aij 8.2 a \ 26- /
Oaij
The preceding expression for the plastic strain rate pertains to plastic loading, which is when 9 0 and qklakl/H > 0. Otherwise, d/~ - 0 . The rate of increase of the matrix flow strength is given by
EEt aijdPij E - Et ( 1 - f)a Assuming small elastic strains, so that the elastic response can be represented in terms of a hypoelastic relation, the stress-rate-strain-rate relation can be written as
(r0 -- Lijhzdkz where, for plastic loading,
Lijkl -- Lijkl
/
1 Lij mnpmn (Lklrsqrs) H + pmnLemnrsqrs
Here, L~jkzis the tensor of elastic moduli
Lijkl
-
-
E v 1+
(&~jz + &~jk) ~ 1 -- 2v
with v being Poisson's ratio. For a rate-dependent matrix material, the plastic part of the rate of deformation tensor is written as dp _ (1 - f)8"~ 0 r
ahz Oakz with ~ a function of the current state. For example, for power law strain hardening and strain rate hardening, an expression of the form
-
~0
o(1 -F~/eo)
has been used in applications, when m is the strain rate-hardening exponent, N is the strain-hardening exponent, and ko, ao, and ~o are a reference strain rate, strength, and strain, respectively.
434
Tvergaard and Needleman
The stress-rate-strain-rate relation takes the form e
!1 --f)o'._.~ O~ O0"mn
In the rate-dependent case, strain-controlled nucleation is written as fnucleation = ~ and stress-controlled nucleation is still characterized by the previous expression with d = ~. The evolution of ~r is determined from the consistency condition that q) - 0 during continued plastic deformation.
6.5.4 IDENTIFICATION
OF PARAMETERS
The elastic parameters E and v, as well as the matrix-hardening properties, such as an initial yield strength, ~ry, and a strain-hardening exponent, N, are determined from a uniaxial tension test for the matrix material, i.e., the material with f = 0. The parameters ql and q2 have been estimated by a number of micromechanical analyses and by experiments on sintered metals. Generally, the values are in the ranges 1.25 < ql < 2 and 0.9 < q2 < 1, and frequently used values are ql -- 1.5 and q2 = 1. For the coalescence model the parameters fc and ff have been estimated using both experiments and micromechanical analyses. The value ff = 0.25 often reasonable, andfc ranges from 0.03 to 0.15 when the initial void volume fraction ranges from 0.001 to 0.08. The initial void volume fraction and the nucleation parameters such as fN, eN, and sN differ a great deal from material to material. These parameters can be estimated by studying micrographs for test specimens at different stages of deformation.
6.5.5 HOW
TO USE THE MODEL
The material model is typically used in a numerical computation, either to predict flow localization or to predict final failure by void coalescence and crack growth. The onset of plastic flow localization is often predicted at small void volume fractions, so that coalescence has not yet played a role. Several studies of localization have used a simplified model, the M-K-model but localization is also predicted in full numerical computations.
6.5 The Modified Gurson Model
435
W h e n final failure b y v o i d c o a l e s c e n c e is p r e d i c t e d at a m a t e r i a l p o i n t , the m a t e r i a l loses its s t r e s s - c a r r y i n g c a p a c i t y at this p o i n t . In a finite e l e m e n t c o m p u t a t i o n it is p r a c t i c a l to r e p r e s e n t this in t e r m s of an e l e m e n t v a n i s h t e c h n i q u e , w h e r e the stiffnesses a n d stresses for the e l e m e n t are n e g l e c t e d s u b s e q u e n t l y . W i t h this p r o c e d u r e the m a t e r i a l m o d e l c a n p r e d i c t first d u c t i l e failure as w e l l as d u c t i l e c r a c k g r o w t h in a n u m e r i c a l analysis. D i s c r e t i z a t i o n i n d e p e n d e n t p r e d i c t i o n s of f r a c t u r e r e q u i r e a l e n g t h scale to be i n c o r p o r a t e d i n t o the p r o b l e m f o r m u l a t i o n .
REFERENCES 1. Becker, R., Needleman, A., Richmond, O., and Tvergaard, V. (1988). Void growth and failure in notched bars. J. Mech. Phys. Solids 36: 317-351. 2. Chu, C. C., and Needleman, A. (1980). Void nucleation effects in biaxially stretched sheets. J. Engin. Mat. Tech. 102: 249-256. 3. Gurson, A. L. (1977). Continuum theory of ductile rupture by void nucleation and growth, Part I: Yield criteria and flow rules for porous ductile materials. J. Engin. Mat. Tech. 99: 2-15. 4. Koplik, J., and Needleman, A. (1988). Void growth and coalescence in porous plastic solids. Int. J. Solids Struct. 24: 835-853. 5. Needleman, A., and Rice, J. R. (1978). Limits to ductility set by plastic flow localization, in Mechanics of Sheet Metal Forming, pp. 237-265, Koistinen, D. P., and Wang, N. M., eds., New York: Plenum. 6. Needleman, A., and Tvergaard, V. (1994). Mesh effects in the analysis of dynamic ductile crack growth. Eng. Frac. Mech. 47: 75-91. 7. Pan, J., Saje, M., and Needleman, A. (1983). Localization of deformation in rate sensitive porous plastic solids. Int. J. Fract. 21:261-278. 8. Tvergaard, V. (1981). Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17: 389-407. 9. Tvergaard, V. (1982). Influence of void nucleation on ductile shear fracture at a free surface. J. Mech. Phys. Solids 30: 399-425. 10. Tvergaard, V. (1990). Material failure by void growth to coalescence. Adv. Appl. Mech. 27: 83-151. 11. Tvergaard, V., and Needleman, A. (1984). Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32: 157-169.
SECTION
6.6
The Rousselier Model for Porous Metal Plasticity and Ductile Fracture GILLES ROUSSELIER
EDF/R&D Division, Les Renardidres, 77818 Moret-sur-Loing Cedex, France
Contents 6.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 6.6.2 Formulation [ 11-13] . . . . . . . . . . . . . . . . . . . . . . . 437 6.6.3 Description of the Model . . . . . . . . . . . . . . . . . . . 438 6.6.3.1 General Equations . . . . . . . . . . . . . . . . . . . 438 6.6.3.2 Extension to Viscoplasticity . . . . . . . . . . 438 6.6.3.3 Extension to Void Nucleation . . . . . . . . 439 6.6.3.4 Ductile Fracture . . . . . . . . . . . . . . . . . . . . . 439 6.6.4 Identification of the Material Parameters f0, 0"1, and lc ............................... 439 6.6.5 Implementation in Finite Element Codes . . . 441 6.6.6 Tables of Parameters . . . . . . . . . . . . . . . . . . . . . . . 442 6.6.6.1 "ESIS" Ferritic Steel at 0~ . . . . . . . . . . . 442 6.6.6.2 Other Steels . . . . . . . . . . . . . . . . . . . . . . . . . 444 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
445
6.6.1 VALIDITY In the f r a m e of c o n t i n u u m d a m a g e m e c h a n i c s a m o d e l for p o r o u s m e t a l pla st i c i t y is p r e s e n t e d . This m o d e l y i e l d s m a t e r i a l instability ( l o c a l i z a t i o n of d e f o r m a t i o n a n d d a m a g e in a p l a n e ) a n d c a n be u s e d to p r e d i c t d u c t i l e f r a c t u re of p l a i n a n d c r a c k e d s t r u c t u r e s in the f r a m e of a local a p p r o a c h to fracture.
436
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
437
6.6 T h e R o u s s e l i e r M o d e l
6.6.2 FORMULATION
[11-13]
In the continuum theory of damage, a damage state variable f is introduced in addition to the strain-hardening state variable p and the plastic strain tensor el. The variables p and f are scalars, and the model is theoretically limited to isotropic strain hardening and damage; in practice, the model can be used for anisotropic ductile damage under proportional loading. The thermodynamics of irreversible processes defines thermodynamical forces (P, F) such that the specific intrinsic dissipated power is O"
= p . i-f - e p - F f > O
(1)
where p is the density; p = Po in the undamaged reference state; and Po = 1 can be assumed if inertial effects are neglected. The variables E/P, P, F associated with the internal variables __eP,p, f derive from the specific free energy ~(~ - e__P,p,f) = ~(e_e, p,f) [8]: _~
p
The
W
is
0~t'
assumed
0~
0~t'
0~
e=Op' v-of
to
be
an
additive
(2)
function
W=
E .=~e + Uffp(p) + ~ f ( f ) . consequently, P - e(p), F - h - l ( f ) , and, inver~l~e . = sely, f - h(F). The existence of a plastic potential is postulated that depends on the first and second invariants of the stress tensor: Crm coupled with F and Ceq coupled with P:
~vM--Creq/P- P is the yon Mises plastic potential (and yield criterion q~M = 0), and P = R(p) is the strain-hardening curve of the material. The simplest form is assumed for the second term: ~m = g(crm/p)h(F). The normality rules gives 9
cg@~M -
/~ _ _ 2 0 ~
-- 1-2 Cg@m = 3pg'(r
(4)
3
0~
(5)
The variable f is defined as the void volume fraction, and the matrix material is postulated to be incompressible (the volumic elastic strain eem is neglected); therefore, p - ( 1 - f ) / ( 1 - f o ) and f - 3 ( 1 - f ) k P m. From the
438
Rousselier
latter equation it is g (9)
=
demonstrated D0-1
exp
[12, 13] that
0-m (-~1)
,
Const.-
F=
0-1 In
1 (ff)
(6)
where D and 0-1 are constant parameters. The third parameter of the model is the initial void volume fraction f0.
6.6.3 DESCRIPTION 6.6.3.1
OF THE MODEL
GENERAL EQUATIONS
The equations to be used in applications are resumed from the preceding section:
~b = 0-e-Aq- R(p) + Df0-1exp(0-~) =
(7)
0-
=p = _E. --
_
(8)
_
= P 20-e----~q' ~'Pm= -~pDf exp
~
f = 3(1-f)/~Pm
(9) (10)
where 0-eq= (3__~" _.~/2)1/2 is the von Mises equivalent stress, __~ = ~ - 0-ml is the stress deviator, 0-m = tr(_~)/3 is the hydrostatic stress, /~ = (2~" ~ / 3 ) 1/2, k~ = ~ + ~Pml, p is t-he cumulated plastic strain, f is the void volume fraction (initial value fo), P = (1 - f) / (1 - fo) is the relative density (initial value Po = 1; since fo is small, some authors simply take p = 1 - f ) , D, fo, 0-1 are constant parameters, and R(p) is the true stress-true plastic strain curve of the material.
6.6.3.2
EXTENSION TO VISCOPLASTICITY
R(p) is replaced by R(p,/~) and/~ is deduced from (I)= 0. For example, if R(p,/~) = R(p) + Mp 1/m, then P= <1[~
_
R(p) -+-Df0-1exp(-~l)] ) rn
where (x) = x if x > 0 and (x) = 0 if x < O.
0-m
(ii)
439
6.6 The Rousselier Model 6.6.3.3
EXTENSION TO VOID NUCLEATION
The rate of the void volume fraction is given by f - f g 3(1 - f g ) ~ and f,, - A p or other rate equations.
+fn with fg =
6 . 6 . 3 . 4 DUCTILE FRACTURE The model describes not only porous metal plasticity but also void coalescence and ductile fracture. Actually, void growth in the model results in strain softening, which promotes localization of deformation and damage in a plane. Crack initiation and propagation occur naturally in a theoretical or
finite element calculation, resulting from the constitutive relations only, without it being necessary to define a critical void volume fraction [11]. For practical reasons it may be useful to accelerate void growth beyond a given value f - fc, for example, with f = 3B(1 - f ) ~ if f > fc (B > 1). The value fc has to be chosen well beyond the void volume fraction at strain localization, so that it will not affect the local instability, fc is not a parameter of the model, in contrast with the Gurson-Tvergaard-Needleman model. It may be useful too, for large crack growth, to set the rigidity to 0 when f exceeds some given value ff [10]. Typical values are fc = 0.2, B = 3, and --0.9.
6.6.4 IDENTIFICATION OF THE MATERIAL PARAMETERS f0, o1, A N D lc From cavity growth measurements [12] and theoretical considerations, it was concluded that D can be considered as material-independent and that for most materials D--2
(12)
The identification is based on coupled mechanical testing and finite element analyses. To calibrate f0, quantitative metallographic examinations are recommended, but not mandatory. For steels, if these examinations are not available, f0 can be estimated from the chemical composition thanks to the Franklin formula (manganese sulfide and oxide inclusions):
[
f0 = 0.054 %S(wt) - %M--n(wt)] + O'055~176
(13)
The parameter al or the couple (Crl,f0) can be calibrated with smooth or notched round tensile specimens. A stiff testing machine is recommended,
440 Lol /
Rousselier
~d
o1(2)>o1(1 )
(1)
Load /
(2)
ecl
gc2> gcl
~ 0 = ~0- ~
FIGURE 6.6.1 Effectof parameters (O'1,f0) and lc on the load vs. diametral reduction curve of a tensile specimen.
and the measurement of the reduction of diameter A~b = 4)0 - ~b is mandatory. The load-versus-diameter reduction curve presents at ~b = ~bF a breakpoint of rapid decrease of load corresponding to the inflation of a macroscopic crack in the center of the specimen (Fig. 6.6.1). In the case of smooth specimens, the diameter reduction of the minimum section can also be measured with interrupted tests and the breakpoint determined from the residual diameters of broken specimens. The parameters are calibrated so that the experimental and numerical breakpoints coincide. The hardening curve R(p) can be calibrated with the same tensile experiments. A first try for al is the mean of 2R(p)/3 over the range of deformation p = (0, gv), gF = 21n(~b0/~br) experienced by smooth tensile specimens. If R(p) = Kp", then 2 Kg} 2 R(gF) 1 = 3 n+----~
(14)
O'1 = 3 n +
This equation and the Franklin formula give a first estimate of (O'l,f0) but cannot replace a complete calibration. The ductile crack propagation in the center of tensile specimens, or crack initiation and propagation in precracked specimens (compact tension [CT]), depend on the finite element model: element size lc, element type, symmetries, mesh geometry, etc. That is why (i) the additional parameter lc has to be calibrated for applications to cracked geometries, and (ii) the same finite element model has to be used for calibration with laboratory specimens and for applications to structural components. The parameter l~ can be calibrated with experiments on tensile specimens or fracture mechanics specimens (CT). With tensile specimens the slope of the load vs. diameter reduction curve after the breakpoint has to be well
441
6.6 The Rousselier Model
defined. It may be not the case if the testing machine is not stiff enough, lc is calibrated so that the numerical and experimental slopes coincide. With fracture mechanics specimens, the calibration is performed on the slope of the displacement vs. stable crack growth Aa curve, or on the slope of the JR curve (tearing resistance parameter vs. Aa). For many steels lc is in the range 0.4-0.6 mm. lc is related to the interparticle spacing.
6.6.5 IMPLEMENTATION IN FINITE ELEMENT CODES The following implicit algorithm is recommended. The symbols X - X- + AX are used, in particular f - f - + Af. For a known strain increment A_~, Af is given by the scalar implicit equation: [(~-)-+2#A~_d]
-31zAp-R(p-+Ap)+Dal(f-+
Af)exp
eq
(cry1)
-0
where
Ap
--
Bf(1 - f) D exp(am/pal)'
p
--
+3K A e m -
3B(1 - f ) (16)
With the implicit algorithm, _a can be computed from A~_,i.e., a = | e). The tangent operator derives consistently from this equation:
A a _ - dO(Ae__)/d(Ae) " Ae_.. The calculations have to be performed with large displacements and large deformations options activated. In the finite element model a reduced integration scheme has to be used: for two-dimensional models, 8-noded quadratic elements (2 • 2 integration order), or: linear quadratic elements (1 • 1 integration order) coupled with linear triangular elements. For crack propagation and precracked geometries a uniform mesh along the crack extension has to be used. Rectangular or square elements are preferred, with sides lx along the crack extension and ly in the perpendicular direction. The size lc of these elements is a parameter of the model, lc is related to the height ly of the elements. For symmetrical geometries (half model) and 8-noded quadratic elements l c - 2ly; rectangular elements with lx - 2ly are preferred. For complete geometries lc - ly; square elements with l x - ly are preferred (Fig. 6.6.2).
442
Rousselier
Cr~
Ick
e,y = gc
Or ~ck _ _ , ;. . . . . . .
9l . . . . . . ,!
I
! I
I
ly
,
---, ...... ,~ "'-
! !
, I ,I
.......
-, . . . . . . ! !
I I
I'
X___., . . . . . . . . . . . . . . .
Complete
FIGURE
II
model
6.6.2
I
]
I
t
l
4. . . . . . . | l | |
i
,~. . . . . . I i
.L . . . . . .
I
l,~ =~
-t . . . . I, I I
,
L. . . . . . . . . . . I
I
I
i ............ , I
Half model (symmetrical problem) Uniform
mesh
of precracked
geometries.
6 . 6 . 6 TABLES O F P A R A M E T E R S 6.6.6.1
" E S I S " FERRITIC STEEL AT 0 ~
A numerical round robin on the application of micromechanical models and the local approach to fracture for characterizing ductile tearing has been proposed by the European Structural Integrity Society (ESIS) [3]. It is based on the data of a 22 NiMoCr 3 7 steel (German designation). The chemical composition of this steel is given in Table 6.6.1. The Franklin formula 13 gives f0 = 0.00015. The stress-strain curve is R(p)--908p0 " 1 2 5 (MPa) for p > 0.1459. The experiments on smooth tensile specimens (~b0 = 6 mm) show breakpoints at eF = 1.038 and 1.160. Equation 14 gives (71 - - 541 and 548 MPa, mean value 545 MPa. A set of 14 finite element calculations has been performed by CEA (Saclay), MPA (Stuttgart), and EDF (Les Renardi~res); the numerical breakpoints are given in Table 6.6.2. A quadratic fit of s f0) is ev - 0 . 5 9 0 4- 0.0642 lnfo - [3.83 4- 0.922 lnfo 4- 0.0192(lnfo)2](~rl/1000) 4- (2.54 4- 0.468 lnfo)(crl/1000) 2
(17)
For fo = 0.00015 and O"1 = 545MPa, Eq. 17 gives e F - 1.081, in the interval of the two experimental values. This shows the good quality for this steel of the simple calibration method based on Franklin formula 13 and Eq. 14.
443
6.6 The Rousselier Model TABLE 6.6.1
Chemical Composition (Weight %)
Steel "ESIS" 304L Austenitic weld A48 Rotor steel A508 class 3 heat A A508 class 3 heat B A508 class 3 (NKS1) A508 class 3 (SC1) NKS 3 10MnMoNi 5 5
C
S
P
Mn
0.22 0.023 0.010 0.20 0.225 0.143 0.163 0.192 0.22 0.25 0.10
0.004 0.0013 0.0013 0.021 0.013 0.0054 0.008 0.002 0.012 0.011 0.004
0.007
0.88 1.64 1.45 0.97 0.71 1.265 1.375 1.395 1.32 0.68 1.29
Cr
Mo
V
Steel "ESIS" 304L Austenitic weld A48 Rotor steel A508 class 3 heat A A508 class 3 heat B A508 class 3 (NKS1) A508 class 3 (SC1) NKS 3 10 MnMoNi 5 5
TABLE 6.6.2
0.39 19.0 18.20 0.10 1.72 0.196 0.17 0.03 0.08 0.47 0.06
0.51 0.45 2.78 0.03 0.70 0.551 0.50 0.48 0.57 0.75 0.58
0.019 0.022 0.010 0.0063 0.007 0.007 0.012 0.009 0.007
0.035 0.0025 <0.01
Si 0.23 0.41 0.36 0.23 0.23 0.267 0.24 0.25 0.20 0.23 0.15
N
O
0.059 0.054 0.011
0.0061 0.068 0.008
0.007
0.0035
Ni 0.84 9.6 11.16 0.12 1.02 0.745 0.70 0.765 0.78 0.71 0.94
<0.01 0.01
"ESIS" Steel ( K = 908 MPa, n = 0.125); Numerical Mean Strain eF at BreakPoint
(Smooth Tensile Specimen) ~f0, al(MPai 0.00010 0.00020 0.00021 0.00032 0.00061 0.00200
312
445
0.514
0.989 0.929 0.924 0.872 0.796 0.629
519
600
655
700
800
1.011
1.078 0.882
1.148
0.872 0.694
0.945
For a different choice of parameters: f0 = 0.00010 and cr1 - - 4 4 5 M P a (~/: = 0 . 9 8 9 ) , the calculation of experiments on CT specimens gives lc- 0.4 mm (square elements). If the rigidity of the material is set to 0 for a very low value of the void volume fraction f = 3 ~ = 0.05, then lc has to be increased to 0.6 mm.
444 TABLE 6.6.3
Rousselier
Parameters
Steel
Orient.
"ESIS" [3]
Temp ~ 0
idem
idem
304L [9] Aust. weld [13] A48 [2] Rotor steel [ 13] A508 A[13] A508 B[13][1]
L (axial) T circum. circum. L (circ.) L (circ.)
idem idem [6] idem
T (axial)
NKSI [4]
L (axial)
idem idem
idem idem
SC1 [7] NKS3 [5]
T (circ.) L (axial)
idem idem idem idem
idem idem idem idem
10M55 [10]
L
idem idem
300 20 300 40 100 100 320 0 320 100
idem 220 290 100 220 260 290 320 20
K MPa
n
f0 10-5
O"1 M P a
lc mm
908 908 731
0.125 0.125 0.168
1010 974 795 795 795 1008 797 795 820 820 611 934 995 986 988 971 1030
0.366 0.060 0.1 0.1 0.1 0.14 0.1 0.1 0.13 0.13 0.184 0.113 0.121 0.113 0.111 0.104 0.118
15 10 69 10 200 50 16 18 18 50 50 7 7 7 60 50 50 50 50 50 1
545 445 500 565 240 490 445 445 410 445 365 445 400 370 350 430 415 435 445 460 445
0.4 a 1 0.4 0.4 0.4 0.55 0.55 0.55 0.55 b 0.55 0.55 0.55 0.55 0.55 0.4 0.4 0.4 0.4 0.4 0.55 b
D=2 except for 304L (D=1.9). Rectangular elements lx = 21y except for "ESIS", 304L, A508 class 3 heat B direction T(axial) at 0 Celsius, and 10 MnMoNi 5 5 (square elements lx = ly). (a) lc=0.6 mm with d~=0.05. (b) lc=0.8 mm with d~=0.05.
Equation 17 can give a first estimate of the parameters (0"1, fo) of a different steel with stress-strain curve R ( p ) - I
6.6.6.2 OTHERSTEELS Corresponding to the chemical compositions of Table 6.6.1, the parameters of various steels are given in Table 6.6.3. The parameter 0.1 depends on the stress-strain curve used in the calculation, if it is different from the real curve of the material (NKS1 steel), in a way more or less unpredictable by Eq. 14. The temperature dependence
6.6 The Rousselier Model
of
O"1
445
can be affected by dynamic strain aging (Portevin-Le Chatelier), for the ferritic steels of Table 6.6.3. The parameter lc
maximum at 220~
s e e m s to b e 0 . 4 m m for t h e ferritic s t e e l s w i t h M n ( % ) < 1.0 a n d 0 . 5 5 m m w i t h
Mn(%) = 1.2"1.4.
REFERENCES 1. Batisse, R. (1988). Contribution h la Mod~lisation de la Rupture Ductile des Aciers. PhD thesis, Universit~ de Technologie de Compi~gne. 2. Bauvineau, L. (1996). Approche Locale de la Rupture Ductile: application h u n Acier CarboneManganese. PhD thesis, Ecole Nationale Sup~rieure des Mines de Paris. 3. Bernauer, G., and Brocks, W. (2000). Numerical round robin on micro-mechanical models, Phase II: Results of task a. Technical report, ESIS-TC8 and GKSS/WMS/00/5. 4. Bethmont, M., Rousselier, G., Devesa, G., and Batisse, R. (1987). Ductile fracture analysis by means of a local approach, in Trans. of the 9th Int. Conf. on Structural Mechanics in Reactor Technology. 5. Bethmont, M., Rousselier, G., Kussmaul, K., Sauter, A., and Jovanovic, A. (1989). The method of local approach of fracture and its application to a thermal shock experiment. Nucl. Eng. Des. 119: 249-261. 6. Carassou, S. (1999). D~clenchement du Clivage dans un Acier Faiblement Alli~: R6le de l'Endommagement Ductile Localis~ autour des Inclusions. PhD thesis, Ecole Nationale Sup~rieure des Mines de Paris. 7. Eripret, C., and Rousselier, G. (1994). First spinning cylinder test analysis using a local approach to fracture. Nucl. Eng. Des. 152: 11-18. 8. Lemaitre, J., and Chaboche, J.-L. (1985). M~canique des Mat~riaux Solides, Paris: DunodBordas. 9. Marques Vieira, E. (1998). M~canique Non Lin~aire et Approche Locale de la Rupture. Application d l'Acier Inoxydable Aust~nitique Z2 CN 18-10 sous Chargements Complexes. PhD thesis, Universit~ Paris VI. 10. Pitard-Bouet, J.-M., Seidenfuss, M., Bethmont, M., and Kussmaul, K. (1999). Experimental investigations on the "shallow crack effect", on the 10 MnMoNi 5 5 steel, and computational analysis in the upper shelf by means of the global and local approaches. Nucl. Eng. Des. 190: 171-190. 11. Rousselier, G. (1981). Finite deformation constitutive relations including ductile fracture damage, in ThreeJDimensional Constitutive Relations and Ductile Fracture, pp. 331-355, NorthHolland. 12. Rousselier, G. (1987). Ductile fracture models and their potential in local approach of fracture. Nucl. Eng. Des. 105: 97-111. 13. Rousselier, G., Devaux, J.JC., Mottet, G., Devesa, G. (1989). A methodology for ductile fracture analysis based on damage mechanics: An illustration of a local approach of fracture. ASTM STP 995, pp. 332-354.
SECTION
6.7
Model of Anisotropic Creep Damage SUMIO MURAKAMI Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603 Japan
Contents 6.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2.1 Representation of Anisotropic Damage State . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2.2 Restriction and Interpretation of Damage Tensor . . . . . . . . . . . . . . . . . . . . . . 6.7.2.3 Definition of Effective Stresses . . . . . . . 6.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 6.7.3.1 Anisotropic Damage Variable and Effective Stress . . . . . . . . . . . . . . . . . . 6.7.3.2 Evolution Equation of Anisotropic Creep Damage . . . . . . . . . . . . . . . . . . . . . . 6.7.3.3 Constitutive Equation of Damaged Material . . . . . . . . . . . . . . . . . . . 6.7.4 Identification of the Parameters . . . . . . . . . . . . 6.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
446 447 447 448 450 450 450 451 451 451 452 452
6.7.1 VALIDITY This m o d e l was originally developed to describe the anisotropic state of creep damage and the related evolution equation of the damage. Since the m o d e l is based on the concept of effective area reduction caused by distributed microscopic cracks and cavities, it can also describe anisotropic states of damage observed in creep-fatigue, elastic-plastic, and elastic-brittle damage of
446
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
6.7 Model of Anisotropic Creep Damage
447
more general materials such as metals, composite materials, concretes, and rocks. The damage variable proposed here can describe the anisotropy with symmetry larger than or equal to orthotropy.
6.7.2 BACKGROUND The present model of three-dimensional anisotropic damage [1-3] has been developed by extending the isotropic damage theory of Kachanov [4].
6.7.2.1 REPRESENTATIONOF ANISOTROPIC DAMAGE STATE Let us take a representative surface element PQR of an arbitrary orientation in the current (actual) damage configuration Bt of a body, and represent the line elements PQ, PR, and the area of PQR by vectors dx, dy and vdA as shown in Figure 6.7.1a, where v is a unit normal vector of the element PQR. It is assumed that the principal effect of the material damage is the effective area decrease due to three-dimensional distribution of microcavities. Then the effective net area of vdA will be equivalent to a smaller undamaged area element v* dA* in a fictitious (equivalent) undamaged configuration Bf of
FIGURE 6.7.1 Definition of three-dimensional state of damage, a. Current damaged configuration, b. Fictitious undamaged configuration.
448
Murakami
Figure 6.7.1b which is mechanically equivalent to Bt. The symbols in Bf corresponding to those in Bt are denoted by an asterisk (*). Since the decrease in the effective area of vdA to v* dA* is related to the material damage, the damage state can be identified uniquely by specifying the transformation between vdA and v* dA* of the two configurations Bt and Bf properly. If we represent the fictitious deformation gradient from Bt to Bf by G and employ Nanson's theorem, the area vector v* dA* in Bf is related to the vector vdA in Bt as follows:
v*dA - (1/2)dx* • = K(G-1) r(vdA),
(1) K -- det G
By introducing a new tensor I - D to represent the tensor K(G-1) r in the form of K(c-x)
= i - D
(2)
Equation 1 can be written as follows: v* dA* = (I - D)(vdA)
(3)
where I denotes the unit tensor of rank two. Thus the second-rank tensor D of Eq. 3 is an internal state variable representing the anisotropic damage state of materials, and will be called a damage tensor with respect to the current damage configuration Bt.
6.7.2.2 RESTRICTION AND INTERPRETATION OF DAMAGE TENSOR In view of the requirement (v* dA* )(vdA) = [(I - D)(vdA)] 9(vdA) > 0
(4)
the asymmetric part of ( I - D) can be excluded without loss of generality. Thus we can assume symmetry of the damage tensor D, and D can be expressed in a canonical form 3
D -- ~
Dini | ni
(5)
i-1
where ni(i = 1, 2, 3) and Di are the principal directions and the corresponding principal values of the tensor D. Then we take two sets of principal coordinate systems OXlX2X3 and O*XlX2X3 of the tensor D, and construct two tetrahedra QPQR and
449
6.7 Model of Anisotropic Creep Damage
O*P*Q*R* as shown in Figure furnishes
6.7.2. Substitution of Eq. 5 into Eq. 3
3
v* dA* = ~
(1 - Di)dAini
i=l
(6a)
= n,dA; + n2dA~ + ,,3dA~ dAi - ( 1 - D i ) d A i
(no sum oni; i - 1 ,
2, 3)
(6b)
where dAi - vidA and dA/* - v/~dA* (i - 1, 2, 3) designate the facet areas of the tetrahedra perpendicular to the principal direction n,. According to Eq. 6, the principal values Di of the tensor D can be interpreted as the effective area reduction ratios or the effective void area densities on the three principal planes of D in Bt. Though the damage state corresponding to arbitrary distribution of microscopic cavities can be described by the second rank symmetric tensor D, the canonical expression (5) implies that the tensor D cannot describe the damage states which have more complicated symmetry than orthotropy.
FIGURE 6.7.2 Cauchy stress and effective stress in current damaged configuration and fictitious undamaged configuration, a. Current damaged configuration, b. Fictitious undamaged configuration.
450 6.7.2.3
Murakami
DEFINITION OF EFFECTIVE STRESSES
According to the notion of effective stress usually employed in damage mechanics [4-6], the effective stress with reference to the current damaged configuration Bt for the general states of damage can be defined as a stress a*, which will be induced in the tetrahedron O *P * Q *R * in Bf of Figure 6.7.2 if the surface element P* Q* R * is subject to the identical force vector t(V)dA as that on the element PQR in Bt (where t (v) denotes the stress vector on the surface element PQR). This stress a * corresponds exactly to the first PiolaKirchhoff states tensor concerning the reference configuration Bf, and is given as follows: a* = K-1Go" -- ( I - D)-Ia 9
(7)
where a is the Cauchy stress tensor in Bt. The effective stress a* of Eq. 7 is asymmetric. Since it is inexpedient to formulate constitutive and evolution equations of damaged materials by using the asymmetric tensor a*, we adopt the symmetric part of the Cartesian decomposition of a * S - [a ~is _ (1//2)[( I _ D)-lo. q_ a ( I - D)-ll
(8)
Thus it can be interpreted that the mechanical effect of the Cauchy stress a in Bt is magnified to the effective stress S of Eq. 8 as a result of the effective area reduction due to damage. Representation of a damage state with respect to the elastically unloaded damage configuration and the effective stress in reference to the initial undamaged configuration were also discussed in Reference [1].
6.7.3 DESCRIPTION
OF THE MODEL
Based on the theory just described the model of anisotropic creep damage can be developed as follows.
6 . 7 . 3 . 1 ANISOTROPIC DAMAGE VARIABLE AND EFFECTIVE STRESS A state of anisotropic damage and the increased effect of stress due to the cavity development are described by a second-rank symmetric damage tensor
451
6.7 Model of Anisotropic Creep Damage
D and a symmetrized effective stress tensor as follows [1, 2]" 3 D =
|
i=1
S -- (1/2)[(I- D)-l~r -k ~r(I- D) -1] where
(10)
Di and ni are the principal value and the principal direction of D.
6 . 7 . 3 . 2 EVOLUTION EQUATION OF ANISOTROPIC CREEP DAMAGE By employing the damage variable D of Eq. 9 together with the effective stress S of Eq. 10, and by taking account of the microscopic mechanisms of damage development, the evolution equation of isotropic creep damage [4-6] can be extended to the anisotropic case as follows [2, 3]"
/~ --B0~S(1) q- fltrS -k (1 - ~ - j~)SEQ] m • [tr{(I - D ) - I ( v (1) Q Y(1))}]g'-m[l]y(1) (~ y(1) .or_(1 -
SEQ --[(3/2)trS2D] 1/2, SD -- S - (1/3)(trS)I
(11a)
rl)I] (11b)
where (') denotes the material time derivative, and the symbols S (1), SEQ, SD, and v (1) designate the maximum principal value, the equivalent value of the effective stress tensor S, the deviatoric tensor of S, and the direction of the maximum principal stress. The symbols B, ~, m, ~(0 _< ~ <__1), /~(0 _
6 . 7 . 3 . 3 CONSTITUTIVE EQUATION OF DAMAGED MATERIAL A constitutive equation of the damaged material can be derived from the corresponding constitutive equation of undamaged material by employing the effective stress tensor S of Eq. 10 and by postulating the hypothesis of strain equivalence [5, 6] or that of energy equivalence [5, 6] in damage mechanics.
6 . 7 . 4 I D E N T I F I C A T I O N OF T H E P A R A M E T E R S The material parameters of Eq. 11 are identified by performing creep damage tests under combined tension and torsion, and then by fitting Eq. 11 together
452 TABLE 6.7.1 Materials
Murakami
Table of Parameters of Eq. 11 Temperature~
B
~
]3
m
~
OFHC copper
250
4.46 • 10 -13 (MPa)-5.55(h) -1
0.75
0.25
5.55
10.55
0.5
Nimonic 80A
750
9.48 x 10 -16 (MPa)-5-~ -1
0.46
0.54
5.0
5.0
1.0
with a proper damage-couple constitutive equation of creep to the corresponding results of the tests.
6.7.5 HOW TO USE THE MODEL The evolution equation of anisotropic creep damage (Eq. 11) can be effectively applied to problems where damage anisotropy has significant effects. These problems are usually found in the case of nonproportional loading. See Table 6.7.1.
REFERENCES 1. Murakami, S. (1998). Mechanical modeling of material damage. J. Appl. Mech., Trans. ASME 55: 280-286. 2. Murakami, S., and Ohno, N. (1981). A continuum theory of creep and creep damage, in Creep in Structures, Proc. 3rd IUTAM Symposium, pp. 422-444, Ponter, A. R. S., and Hayhurst, D. R., eds., Berlin: Springer-Verlag. 3. Murakami, S., Kawai, M., and Rong, H. (1988). Finite element analysis of creep crack growth by a local approach. Int. J. Mech. Sci. 30: 491-502. 4. Kachanov, L. M. (1986). Introduction to Continuum Damage Mechanics, Dordrecht: Martinus Nijhoff. 5. Lemaitre, J. (1996). A Course on Damage Mechanics, Berlin: Springer-Verlag. 6. Skrzypek, J., and Ganczarski, A. (1999). Modeling of Material Damage and Failure of Structures, Berlin: Springer-Verlag.
SECTION
6.8
Multiaxial Fatigue Damage Criteria DARRELL SOCIE Department of Mechanical Engineering, University of Illinois, Urbana, Illinois
Contents 6.8.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Description of the Models . . . . . . . . . . . . . . . . . . 6.8.4 Determination of Material Constants . . . . . . . 6.8.5 How to Use the Models . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
453 453 454 455 455 456
6.8.1 VALIDITY Two muhiaxial fatigue damage criteria are needed for metals. One criterion is used for ductile materials that fail by the nucleation and growth of shear microcracks. A second criterion is used for higher-strength materials where failure is governed by the nucleation and growth of mode I tensile cracks. The models provide an estimate of the fatigue life, 2Nf, for complex muhiaxial loading.
6.8.2 BACKGROUND Microcracks emanate from slip bands in materials subjected to cyclic plastic deformation. Once formed, these microcracks are driven by cyclic tensile or shear strains. A separate damage criterion is needed for each case (see Fig. 6.8.1). During shear loading, the irregularly shaped crack surface results in frictional forces that will reduce crack tip stresses, thus hindering crack growth and increasing the fatigue life. Tensile stresses and strains will separate the crack surfaces and reduce frictional forces. Fractographs from specimens that have failed in pure torsion loading show extensive rubbing and are Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
453
454
Socie
o
I I
As
l~ ,,
I I
" I
Damage due to cyclic tension strains (a) FIGURE 6.8.1
Damage due to cyclic shear stains (b)
a. Damage due to cyclic tension strains, b. Damage due to cyclic shear strains.
relatively featureless, in contrast to tension test fractographs where individual slip bands are observed on the fracture surface, thus providing experimental evidence for the influence of mean normal stresses.
6.8.3 DESCRIPTION
OF THE MODELS
The following damage model [1] may be interpreted as the cyclic shear strain modified by the normal stress to include the crack closure effects that were previously described.
+ k Crn'max~ = ~(2Nf) r' -AT(1 b' + 2 O'y f
7'f(2Nf) c'
where A7/2 --maximum shear strain amplitude; fin,max = normal stress on maximum shear strain amplitude plane;
try - yield strength of material; k - material constant. The left-hand side of the model describes the loading conditions, and the right-hand side describes the material resistance. Shear strain life properties are introduced and summarized in Section 6.8.4. The sensitivity of a material to normal stress is reflected in the value k/cry. This model not only explains the difference between tension and torsion loading but also can be used to describe mean stress and nonproportional hardening effects in complex multiaxial loading histories.
6.8 Multiaxial fatigue damage criteria
455
An alternate damage model is needed for materials that fail predominantly by crack growth on planes of maximum tensile strain or stress. In these materials, cracks nucleate in shear, but early fatigue life is controlled by crack growth on planes perpendicular to the maximum principal stress and strain. Smith et al. [2] proposed a suitable relationship that includes both the cyclic strain range and the maximum stress. This model, commonly referred to as the SWT parameter, was originally developed and has widespread use as a correction for mean stresses in uniaxial loading situations. The SWT parameter can also be used in the analysis of both proportionally and nonproportionally loaded components constructed from materials that fail primarily because of mode I tensile cracking. The SWT parameter for multiaxial loading is based on the principal strain range, Ael, and maximum stress on the principal strain range plane, O'n,max; Ag1
r~52 (2Nf)2~ + rrSe)(2Nf) b+c
T
The stress term in this model makes it suitable for describing mean stresses during multiaxial loading and nonproportional hardening effects.
6.8.4 DETERMINATION OF M A T E R I A L CONSTANTS Material constants are obtained from standard low-cycle fatigue tests of the material (see Table 6.8.1). As a first approximation or if test data from multiple stress states are not available, k -- 1 and ~y -- or).
6.8.5
HOW
TO USE THE MODELS
For a complex stress state the planes experiencing the largest amplitudes of shear and normal strains must be determined from the stress and strain
TABLE 6.8.1
Fatigue strength coefficient Fatigue strength exponent Fatigue ductility coefficient Fatigue ductility exponent Modulus
Axial
Shear
~r} b ~ c E
z~ ~ cr}/x/3 by ~ b y~ ~/v/3e~ cy ~ c G
456 TABLE 6.8.2
Socie Table of Material Constants 1045 steel
Fatigue strength coefficient, cry, MPa Fatigue strength exponent, b Fatigue ductility coefficient, s) Fatigue ductility exponent, c Modulus, E, GPa Shear fatigue strength coefficient, z~, MPa Shear fatigue strength exponent, by Shear fatigue ductility coefficient, ~) Shear fatigue ductility exponent, c~ Shear modulus, G Yield strength, ~zy, MPa Material parameter, k
948 -0.092 0.26 -0.445 210 505 -0.097 0.413 -0.445 80 358 0.3
304 stainless steel 1000 -0.114 0.171 -0.402 185 709 -0.121 0.413 -0.353 78 280 0.8
Inconel 718 1640 -0.060 2.67 -0.82 208 902 -0.055 4.31 -0.82 83 972 0.4
tensors. Once these planes are identified, the normal stresses are determined. Fatigue lives can be estimated from both models. Frequently, the expected mode of damage, tensile or shear, is not known. In this case, the lower of the two estimates is often used. See Table 6.8.2 for the material constants.
REFERENCES 1. Socie, D. E, and Marquis, G. B. (2000). Multiaxial Fatigue, Society of Automotive Engineers. 2. Smith, R. N., Watson, P., and Topper, T. H. (1970). A stress-strain parameter for the fatigue of metals. Journal of Materials 5 (4)" 767-778.
SECTION
6.9
Multiaxial Fatigue Criteria Based on a Multiscale Approach K. DANG VAN kaboratoire de Mechanique des Solid, Ecole Polytechnique, 91128 Palaiseau, France
Contents 6.9.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.4 Identification of the Parameters . . . . . . . . . . . . 6.9.5 How to use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.1
457 457 459 461 461 463
VALIDITY
Prediction of high-cycle fatigue resistance is of great importance for structural design. In spite of this clear industrial need, until now modeling of metal behavior in a high-cycle fatigue regime was often based on empirical approaches: the WShler curve and the Goodman-Haigh or Gerber diagrams are still the very popular tools for engineers. However, these concepts are not appropriate when studying the muhiaxial stress cycles that are frequently encountered on modem mechanical components. These muhiaxial stresses arise from factors such as external loadings, the geometry of the structure, which can induce muhiaxiality even if the loading is uniaxial, and finally residual stresses.
6.9.2
BACKGROUND
In order to derive a muhiaxial endurance fatigue criterion, an original method of computing based on a muhiscale approach was proposed by Dang Van. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
457
458
Dang Van
This method is quite different from existing fatigue approaches. It arises from the observation that generally the first fatigue damage processes begin in grains which have undergone plastic deformation, with the appearance of slip bands in some grains which broaden progressively with the applied cycles; this stage is then followed by localized damage corresponding to formation of intragranular microcracks; these microcracks can be arrested by grain boundaries, but they may also propagate. After a certain number of cycles, a main crack initiates, grows in size, and shields the other defects and consequently leads to the final rupture of mechanical structure. In a highcycle fatigue regime, even if it is necessary to have plastic deformation at the micro- or the mesoscale (corresponding to the grain size), most of the time no visible irreversible deformation at the macroscopic level can be detected. It is thus characterized by a large heterogeneity of plastic deformation from grain to grain: only certain misoriented crystals undergo plastic slips, and in this way a very heterogeneous distribution of microcracks can be observed. In this fatigue regime, the initiation of the first visible cracks takes a large part of the fatigue life of the structure. It demonstrates the importance of studying conditions governing the crack initiation process. In most of the existing fatigue models, these conditions are described with the help of macroscopicparameters which are evaluated according to different assumptions of homogeneity and isotropy. For example, engineers evaluate the stress 12 not at a point of a structure but over a finite volume V that defines the macroscopic scale which is used (see Fig. 6.9.1). Typically V is of the order of the dimension of the strain gauges. However, because the phenomena which cause fatigue initiation are microscopic, the local parameters (for example, local stress 6) differ from the macroscopic ones. Thus the use of classical macroscopic engineering parameters does not seem pertinent, since, i
l
x,t
Y'T tz' ]
_z_,g
xi
FIGURE 6.9.1 Differentscales of material description: the macro- and mesoscopicscales.
6.9 Multiaxial Fatigue Criteria Based on a Multiscale Approach
459
at that local scale, the material cannot be considered anymore as homogeneous. In particular, the local redistributions induced by the inhomogeneous incompatible strains ep, and as a consequence local residual stress p, are not accounted for.
6.9.3 FORMULATION The originality in Dang Van's proposal is precisely the use of local mesoscopic mechanical parameters a to derive fatigue resistance criteria. These parameters are evaluated from the macroscopic parameters thanks to a hypothesis of elastic shakedown. More precisely, it is postulated that, near the
fatigue limit threshold, the mechanical structure shakes down elastically at all scales of material description. Under this assumption, the precise knowledge of the local constitutive equations, which is not possible to evaluate, is not necessary. The physical interpretation of this hypothesis is that after a certain number of loading cycles the response is purely elastic (or at least the plastic dissipation rate becomes negligible). Then, using shakedown theorems (Melan's theorem and its generalization by different authors; (see, for instance, Reference [1]), it is possible to derive a method for estimating the apparent stabilized stress (tensorial) cycle at the macroscopic and mesoscopic level which intervenes in the proposed fatigue criteria, provided that the material is considered a structure made of grains of different crystallographic orientations. Theoretical developments of this theory are presented in Reference [2]. For practical applications, it is only necessary to remember how to derive local parameters from macroscopic stress cycles near the fatigue limit, which is presented in following text. The general relation between macroscopic and local stress tensor is
~(m, t) -- A.~(M, t) + p(m, t) This relation is well known in the theory of polycrystalline aggregates. In this equation A is an elastic localization tensor which depends on the microstructure; ~(M,t) is the macroscopic stress tensor at time t in the representative volume element V(M) surrounding M, and ~(m, t) and p(m, t) are, respectively, the local stress tensor and the residual stress tensor at any point m of V(M). For the sake of simplicity, let us assume that A - identity (elastic homogeneity); then p characterizes the local stress fluctuation in V(M). If elastic shakedown happens, then p must become independent of time after a certain number of cycles, so that the local plastic yield criterionf(m) is no longer violated. Assuming that an approximate elastic shakedown occurs if the loading cycles are near the fatigue limit, then there must exist a local fixed
460
Dang Van
C ~'--
m~ ~
D
FIGURE 6.9.2
Determination of the local residual stress p , at the shakedown state.
(independent of time) residual stress tensor p*(m) and a fixed set of local hardening parameters ~(m) such that:
t) = Z(M, t) + V (m) f(6(m, t), ~(m)) - fOg(M, t) 4- p* (m), ~(m)) _< 0 If the Mises criterion is chosen, then 6(m, t) belongs to the hypershere in five dimensional space, representing the limiting value of the Mises norm for which elastic shakedown is possible. Mandel et al. [1] showed that p* (m) can be approximately taken to be the center of the smallest hypersphere surrounding the loading path in the deviatoric macroscopic stress space represented in Figure 6.9.2. In this figure, S (resp. s) represents the deviatoric stress corresponding to Ig (resp. 6). Because of the shakedown hypothesis at all scales of material description, S at stabilization is also elastic. Finally, by that construction, the local stress state is known at any time t of the apparent stabilized state. One must still choose a fatigue criterion. Since the local stress is approximately known at any time t, it is natural to try to take account of the characteristic of the loading path (as in plasticity). Thus a reasonable fatigue criterion could be stated as follows. Crack initiation will occur in a critically oriented locus (usually corresponding to a grain) within V(M) that has undergone plastic deformation, if, for at least one time instant t of the stabilized cycle, one has
t)] ___ 0
for m c v(M)
6.9 Multiaxial Fatigue Criteria Based on a Multiscale Approach
461
In such a criterion, the current stress is considered, in contrary to most (or all) existing fatigue criteria. Damage arises over a precise portion of the loading path (or equivalently, over a precise time interval of the loading period). Since cracks usually occur in transgranular slip bands, the local shear acting on these planes is an important parameter. Moreover, the normal stress acting on these planes accelerates damage formation. However, this quantity is rather difficult to compute generally because it depends of the considered plane. For this reason, hydrostatic stress is preferred because it is much easier to use, being an invariant scalar. Furthermore, it can be interpreted as the mean value of the normal stresses acting on all the planes that pass through the considered point of the structure. Based on these remarks, F(a) is chosen to be a function of the local shear r and the local hydrostatic stress p.
6.9.4 IDENTIFICATION OF THE PARAMETERS The simplest criterion that can be conceived is a linear relation between these quantities, v(~) - ~ + a p - b
where a and b are material parameters that can be types of fatigue experiments: uniaxial tests of torsion on classical fatigue test machines. If f is in alternate tension-compression, and t is the twisting, then t -f/2 a - - ~ f/3 '
determined by two simple tension-compression and the fatigue limit strength fatigue limit in alternate
b-t
The safety domain (no fatigue crack initiation) is delimited by the two straight lines represented in Figure 6.9.3. The ordinate is the algebraic shear stress acting in an oriented direction, and the abscissa is the hydrostatic stress p. On the same figure the loading paths for fatigue limit in tension compression and in twisting are represented. Two more loading paths are shown. Path lP1 is nondamaging because it lies entirely within the straight lines that delimit the safe domain, whereas a small portion of path F2 induces damage.
6.9.5 HOW TO USE THE MODEL To check automatically the fatigue resistance of a structure is a rather difficult task, because at each point one has to consider the plane on which the loading path (z[t], p[t]) is a "maximum" relative to the criterion. This computation can
462
Dang Van "t
I
tension-compression
.
.
.
~
.
o
FIGURE 6.9.3 Fatiguelimit domain and loading paths.
be simplified as follows. The maximum shear stress according to Tresca's measure is calculated over the cycle period:
z(t) = Tresca[tT(t)] It is useful to notice that Tresca[tT(t)] = Tresca[s(t)] = MaxultT,(t ) -r The stresses r and trj(t) are principal local stresses at time t. The quantity d that quantifies the danger of fatigue failure defined by d - Max t
"c(t)
b-
p(t)
is calculated over the loading period. The maximum is to be taken over the cycle. If d > 1, the fatigue failure will occur.
Working this way, all couples (~, p) are situated in the positive part of ~. All facets which could be involved by the crack initiation are automatically reviewed. Couples (~, p) verifying the condition d > 1 are associated with specific facets. Therefore, the criterion also provides the direction of crack initiation. Another possibility is to use the octahedral shear J2[a(t)] instead of z(t). However, this method does not give the critical facets. Another interesting proposal derived from the multiscale approach was given by I.V.Papadopoulos: the fatigue limit for a given periodic loading 2Q(t) corresponds to the limit of the intensity 2 such that elastic shakedown is possible. Beyond this limit, plastic shakedown or ratchet phenomena will induce damage and fracture because of subsequent softening. The limit size k*
6.9 Multiaxial Fatigue Criteria Based on a Multiscale Approach
463
of the hypersphere surrounding the loading path (as explained previously) is one possible and natural way to characterize this state. If k* is greater than some limit value which depends on the local maximum hydrostatic stress in the cycle, fatigue will occur. This corresponding fatigue criterion is k* 4-~Pmax - fl > 0
As previously, the parameters ~ and fl can be identified by two different tests. By this method it is no longer necessary to describe the whole loading path once k, is determined. In many cases, the predictions are very similar to the current state methodology as presented previously.
REFERENCES 1. Mandel, J., Halphen, B., and Zarka, J. (1977). Adaptation d'une structure elastoplastique ecrouissage cinematique. Mech. Res. Comm. 4: 309-314. 2. Dang Van, K. (1999). Introduction to fatigue analysis in mechanical design by the multiscale approach, in C.I.S.M. Courses and Lectures N ~ 392 on High Cycle Metal Fatigue, from Theory to Applications, Dang Van, K., and Papadopoulos, I.V., eds., New York: Springer Wien.
SECTION
6.10
A Probabilistic Approach to Fracture in High Cycle Fatigue FRAN(~OIS HILD LMT-Cachan, 61 Avenue du PrEsident Wilson, F-94235 Cachan Cedex, France
Contents 6.1 o. 1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 General Framework . . . . . . . . . . . . . . . . . . . . . . 6.10.3 Description of a Probabilistic Model in High Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . 6.10.4 Identification of the Material Parameters .. 6.10.5 Numerical Implementation . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1
464 464 466 469 470 4 71
VALIDITY
Fracture data obtained in high cycle fatigue (HCF) are usually scattered for many different materials. In HCE most materials are loaded in their elastic domain. Their failure is often catastrophic, with no warning signs such as nonlinearities. Therefore, the probabilistic treatment of fracture in HCF closely follows the one used to model the failure of brittle materials [1]. The key distinction, however, is that stable propagation occurs at the microscopic level instead of unstable propagation at all scales. It follows that the framework of the weakest-link statistics [2] can be used under the assumption of a gradual change of the flaw distribution with the number of cycles.
6.10.2
GENERAL
FRAMEWORK
In many situations, initial flaws are randomly distributed within a structure. We assume that the flaw distribution is characterized by a probability density 464
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
6.10 A Probabilistic Approach to Fracture in High Cycle Fatigue
465
function f. The function f gives the flaw distribution at a given stage of the load history. It may depend upon several morphological parameters w (e.g., a flaw size denoted by a, a flaw direction characterized by a normal n [1]). The failure probability PF0 within a domain ~0 of volume V0 is given by the probability of finding a critical flaw. The latter distribution refers to the initial flaw distribution characterized by a probability density function f0. For a given load level, the set of flaws D splits into two subsets. The first subset, Dc, is related to the flaws that are critical (e.g., the energy release rate G[w; Q] _>Go, where Q is a loading parameter, and Gc a critical energy release rate). The second one, Dnc, is related to the flaws that are not critical (e.g., G[w; Q] < Go). The higher the load level, the larger Dc becomes with respect to Dnc. When propagation is unstable, the failure probability PFo(Q) of a domain f20 for a given loading parameter Q is expressed as [1]
PFo(Q)- I
JD
fo(w)dw
(1)
with, for instance, Dc(Q) = {wiG(w;Q) >G~}. In the case of stable propagation, the initial morphological parameters w evolve to become W after N cycles. In particular, bifurcation may take place (therefore _N ~ o, and other morphological parameters may be needed). The morphological parameters W are assumed to be uniquely related to their initial values w through deterministic functions of C 1 class that model the crack propagation law
w = r
Q, N)
(2)
For a given number of cycles N and a fixed Q, the failure probability Pr0(Q, N) is linked with the flaw density function fN:
Pro(Q, N) -- f fN(W; Q, N)dw JDc(Q)
(3)
e.g., Dc(Q) - {WIG(W; Q) _~Gc}. If no new cracks initiate during the loading, fn is related to fo by fn[~(W; Q, N)] - J - l ( w ; Q, N)fo(w) (4) where J denotes the Jacobian of the transformation defined in Eq. 2. Through Eq. 4, Eq. 3 can be written as Eq. 5, where Dc defines the initial flaws that become critical after N cycles:
Pro(Q,N)- f fo(w)dw JD/ (Q,N)
(5)
e.g., Dc (Q,N)= {wIG[~(w; Q,N); Q] >_Go}. Equation 5 constitutes a unified expression for the failure probability with or without stable propagation. In both cases, it relates the failure probability to the initial flaw distribution f0.
466
Hild
If we neglect the interaction between flaws, the cumulative failure probability Pv of a structure ~ of volume V can be derived in the framework of the weakest-link theory [2] and with the independent events assumption. The expression for Pv can be related to the failure probability Pro of a link by [1,21 P F - - 1 - e x p I~o fa l n ( 1 - P,o)dV 1
(6)
The failure probability PF0 as well as the failure probability Pv depend on the loading parameter Q and the number of cycles N. By means of Eqs. 5 and 6, a general relationship between the initial flaw distribution and the failure probability of a structure f~ can be derived:
exp[0/ln/
Equation 7 shows that the failure probability of a structure can be related to the initial flaw distribution even for materials exhibiting time-dependent behaviors. The time dependence is given in the definition of the set D~ (Q, N) of initial defects that become critical after N cycles when the applied load level is equal to Q. Equation 7 constitutes a generalization of the existing failure probabilities that are only valid under time-independent behavior [1]. It is worth noting that Eq. 7 allows for the competition of flaws of different sizes at different locations with different stress levels, and the weakest link is not necessarily a flaw located at the most loaded point(s) but the most critical flaw defined by G[~(F; Q, N); Q] _ Go. Even though this section addresses the case of HCF, the same formalism can be used to study subcritical crack growth of monolithic ceramics [3].
6.10.3 D E S C R I P T I O N OF A PROBABILISTIC M O D E L IN H I G H C Y C L E F A T I G U E In the following, we consider volume elements where the defects are described by penny-shaped cracks. These volume elements are subjected to cyclic and proportional loading conditions. The normal to the crack plane is assumed to be aligned along the maximum principal stress direction (i.e., mode I mechanism). This direction is constant during the whole stress history. If the stable crack extension is assumed to remain in a mode I configuration, a failure criterion can be rewritten in terms of stress intensity factors: Ycrv~ = Kc with
cr = max ai i=1,3
(8)
467
6.10 A Probabilistic Approach to Fracture in High Cycle Fatigue
where Y is a dimensionless shape factor dependent upon the modeled defect, and o"i are the principal stresses. The critical flaw size is expressed as
ac-
~
(9)
The threshold size ath, below which no propagation occurs, is related to a threshold stress intensity factor Kth by
[ Kth ]2
(10)
[Yg(R)aJ
ath-
where the function g models the effect of the load ratio R = closure condition [4, 5]. The function g is defined by [5]
Kmin/Kmax on the
1-R g(R) - 1 - mR
(11)
where m is a material parameter. The crack growth law is expressed in the framework of linear elastic fracture mechanics. A modified Paris' law models the crack growth rate da/dN [4, 5]
a < ath CAi~eeff when ath < a < ac
O
da _dN
when
+oo
(12)
when a _> a,
where C and n denote material parameters. The effective stress intensity factor range AKeff is expressed as Kmaxg(R) -- Kth
AKeff --
Kth
Kc
(13)
g(R)
For cyclic loadings, the flaw size distribution evolves with the number of cycles N (when a constant stress level a is applied): its value after N cycles is denoted by fN(a). By using the previous crack propagation law, the density fN can be linked to the initial density f0. By integrating the crack propagation law, one can determine the relationship between the critical defect size ac and the initial critical flaw size aco:
(a~M) q)
(~~0)_ - (P\ y aM
Kthg(R). (O'max~ n IK Kth I \Tt-t~thf Nv
with Sth -"
Kth
Yg(R)4
(14)
468
Hild
where the function r depends upon the value of the power n (X -- Xth)l-n[(1 -- n)x + Xth]
when n ~ l a n d n ~ 2
(2 -- n)(1 --n) q~(x) --
2[x +
Xthln(x -- Xth)]
2 ln(x
-- Xth)
when n - - 1
2X X
--
(15)
when n = 2
Xth
and aM is a characteristic defect size to be specified later on. It is useful to introduce a function ~ such that a ( 0 ) = ~[a(N)]. This function can be computed by using the previous results. Since the crack propagation is assumed to be deterministic, the probability of finding a defect of size a and range da after N cycles is equal to the probability of finding a defect of initial size ~(a) and range d~(a). If no new defects nucleate during the loading cycles, the density fN can be related to the initial density f0 by (16)
fN(a) -- fo(ff[a(N)]) da
where the coefficient dq/da comes from the change of measure (from da to d~[a]). The failure probability of a volume element is the probability of finding defect sizes greater than or equal to ac after N cycles: Pv0 --
fN(a)da
Under the previous assumptions, the failure probability as [6] Pv0 --
(17)
f~ct~176
Pro can be
fo(a)da
~ac0~~176
rewritten
(18)
where aco = ~k(ac)denotes the initial defect size that becomes critical after N cycles (i.e., equal to ac when N -- Nv). At the level of a structure f], the failure probability Pv is expressed as
Pv - l -exp[~o f ln(1- faco~~)fo(a)da)dV]
(19)
Equation 19 particularizes Eq. 7 when only one morphological parameter is used to characterize the defect distribution.
6.10 A Probabilistic Approach to Fracture in High Cycle Fatigue
6.10.4
IDENTIFICATION
469
OF THE MATERIAL
PARAMETERS Two different sets of parameters have to be determined. The first set is related to the crack propagation law: C, n, m, k = Kth/Kc, Y, and Sth. The second set is associated with the flaw size distribution. In this section, the flaw size distribution is modeled by a beta density function
a~(aM--a) ~ fo(a) -- , _~+B+I
(20)
DaflUM
where a, fl, and aM are material parameters. The constant B~ is equal to B(~ + 1, fl 4- 1), where B(., .) is the Euler function of the first kind. It can be noted that the size aM now corresponds to the maximum flaw size within the material. In addition to the parameters of the beta distribution, one has to consider the volume V0 containing on average one defect. Two different routes can be followed to identify the parameters of the previously mentioned model. The first one uses data obtained at the structural or sample level and deduces the behavior at the level of one volume element (i.e., link). The second one uses microscopic observations of the flaw distribution and determines the crack propagation law from global considerations. If possible, the latter can be compared to conventional fracture mechanics experiments. 9 If the only available data are W6hler plots, one can identify the material parameters by first considering the fatigue limits (usually defined for a conventional value of the number of cycles to failure, say 107). It must be remembered that there are as many fatigue limits as values of the failure probability. The fatigue limits correspond to the case where the initial critical flaw size ac0 used in Eq. 19 can be replaced by the threshold flaw size ath defined in Eq. 10. The following parameters can be identified: cz, fl, Sth, and V/Vo. Then, by considering one constant failure probability (say 50%), one can determine the parameters of the crack propagation law: C/aM, n. The constant k is usually on the order of 1/3 for metallic materials. If only one load ratio is considered, one may consider an approximated value for the constant m: m ~ 0.5. On the other hand, if at least two load ratios are available, the comparison of the fatigue limits for the same failure probability and the two load ratios allows one to identify the constant m. Table 6.10.1 shows the parameters obtained for an austempered spheroidal graphite (SG) cast iron [7]. 9 The second procedure is more time-consuming, but its predictive capability is usually higher than in the previous case. Systematic observations of fractured surfaces of fatigued samples using a scanning
470
Hild
TABLE 6.10.1 Material aM (~tm) Y Sth (MPa) 0~ ]~ V0 ( m m 3) k -- Kth/Kc
C/aM n m
Material P a r a m e t e r s for Two Grades of SG C a s t Iron Austempered SG cast iron [7] --175 1.75 17.5 V/112 1/3 3.4 • 10 -4
Ferritic SG cast iron [9] 400 2 105 2.3 18 340 1/3 5.9 • 10 -5
2.34 ~0.5
2.0 0.59
electron microscope (SEM) are performed to determine the flaw size distribution (here, ~,]~,aM, and V0). The value of V0 is usually representative of the gauge volume of the sample. The parameters of the crack propagation law are determined by considering one constant failure probability. It should be noted that a constant failure probability is equivalent to the same defect size in tension-compression [6]. If independent fracture mechanics experiments are available (e.g., propagation in a compact tension (CT) specimen), the identified values can be compared to them. This is particularly important when the defects distributed within the material behave like small cracks. For those cracks, the threshold stress intensity factor is different from that of large cracks [8, 9]. Table 6.10.1 shows the parameters obtained for a ferritic SG cast iron [9].
6.10.5 NUMERICAL IMPLEMENTATION This model is coupled with an elastic calculation. A commercial finite element code can be used when closed-form solutions are not available. The computation of the failure probability is then performed numerically in a postprocessor [10]. At each integration point i, the equivalent stress (e.g., the maximum principal stress; see Eq. 8) is computed. To this equivalent stress corresponds an initial flaw size ac0 that is computed by solving Eq. 14 through a Newton method. The failure probability Pvj of a finite element j depends upon the element volume Vj, the number ng of integration points for this element, and the weight wi of the integration point i Vj ng i--1
6.10 A Probabilistic Approach to Fracture in High Cycle Fatigue
471
w h e r e Pvi is the c u m u l a t i v e failure probability at a p o i n t i c o m p u t e d by u s i n g Eq. 18. T h e failure probability PF of the w h o l e s t r u c t u r e is t h e n c o m p u t e d as Pv-1-exp
Ij~__~ll n ( 1 -
w h e r e ne is the total n u m b e r of e l e m e n t s \( V -
Pvj)]
(22)
~jn~=l V i i . /
REFERENCES 1. Hild, E (2001). Probabilistic approach to fracture: The Weibull model, In Handbook of Materials Behavior, Nonlinear Models and Properties, this volume 558-565. 2. Freudenthal, A. M. (1968). Statistical approach to brittle fracture, In Fracture pp. 591-619, vol. 2, Liebowitz, H., ed., New York: Academic Press. 3. Hild, E, Marquis, D., Kadouch, O., and Lambelin, J.-P. (1996). Analysis of the failure due to subcritical crack growth. J. Eng. Mat. Techn. 118: 343-348. 4. Elber, W. (1970). Fatigue crack closure under cyclic tension. Eng. Fract. Mech. 2: 37-45. See also Elber, W. (1971). The significance of fatigue crack closure. ASTM STP 486: 230-242. 5. Pellas, J., Baudin, G., and Robert, M. (1977). Mesure et calcul du seuil de fissuration apr~s surcharge. Recherche a~rospatiale 3:191-201 (in French). 6. Hild, E, and Roux, S. (1991). Fatigue initiation in brittle heterogeneous materials. Mech. Res. Comm. 18: 409-414. 7. Hild, E, Billardon, R., and B4ranger, A.-S. (1996). Fatigue failure maps of heterogeneous materials. Mech. Mat. 22: 11-21. 8. C14ment, P., Angeli, J.-P., and Pineau, A. (1984). Short crack behavior in nodular cast iron. Fat. Fract. Eng. Mat. Struct. 7: 251-265. 9. Yaacoub-Agha, H., B4ranger, A.-S., Billardon, R., and Hild, E (1998). High cycle fatigue behavior of spheroidal cast iron. Fat. Fract. Eng. Mat. Struct. 21: 287-296. 10. Chantier, I., Bobet, V., Billardon, R., and Hild, E (2000). A probabilistic approach to predict the very high cycle fatigue behavior of spheroidal graphite cast iron. Fat. Fract. Eng. Mat. Struct. 23: 173-180.
SECTION
6.11
Gigacycle Fatigue Regime C. BATHIAS Laboratoire de M~canique de la Rupture, CNAM/ITMA, 2 rue Cont,, 75003 Paris, France
Contents 6.11.1 Experimental Device . . . . . . . . . . . . . . . . . . . . . . 6.11.1.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1.2 Specimen . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Gigacycle Fatigue of Alloys with Flaws . . . . 6.11.2.1 SN Curves up to 109 Cycles . . . . . . . 6.11.2.2 Initiation Zones at 109 Cycles . . . . . 6.11.2.3 Integration of the da/dn Curve . . . . . 6.11.2.4 Prediction of Gigacycle Fatigue Initiation from a Flaw . . . . . . . . . . . . . 6.11.3 Gigacycle Fatigue of Alloys without Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.3.1 Gigacycle Fatigue of Ti Alloys . . . . . 6.11.3.2 Fatigue Initiation of Ti Alloys . . . . . . 6.11.4 General Discussion of the Gigacycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.4.1 Initiation Zones at 109 Cycles . . . . . 6.11.4.2 Initiation Mechanisms at 109 Cycles . . . . . . . . . . . . . . . . . . . . . . . . 6.11.4.3 Role of the Inclusions . . . . . . . . . . . . . 6.11.4.4 Role of the Porosities . . . . . . . . . . . . . . 6.11.4.5 Role of the Grain Size . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......
474 474 475 476 477 479 479 480 481 482 482 486 486 486 486 487 487 487
W h e n the fatigue curve or SN curve is defined, it is u s u a l l y d o n e in reference to c a r b o n steels. T h e SN curve is g e n e r a l l y l i m i t e d to 107 cycles, a n d it is a d m i t t e d , a c c o r d i n g to the s t a n d a r d , that a h o r i z o n t a l a s y m p t o t e allows one to d e t e r m i n e a fatigue limit value for an a l t e r n a t i n g stress b e t w e e n 106 a n d 107 cycles. B e y o n d 107 cycles (Fig. 6.11.1), the s t a n d a r d c o n s i d e r s that the fatigue life is infinite. F o r o t h e r alloys, it is a s s u m e d that the a s y m p t o t e of the SN curve is n o t horizontal.
472
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
473
6.11 Gigacycle Fatigue Regime
100
Failure area (All samples are broken)
r~ 90
r/3
[..., 80 scatter
70 60 0 .<
50
9
9
9 n
m
n
~
40 30
IFatigue [ limit ?
20
0
0
I
10
1
FIGURE 6.11.1
I
10
2
I
3
I
4
unlimited fatiguelife I
5
10 10 10 CYCLES TO FAILURE, Nf
I
10
6
I
10
7
I
10
8
Typical SN curve is defined by international standardisation.
For fatigue limits based on 109 cycles, few results can be observed in References [1 to 6]. Until now, the shape of the SN curve beyond 107 cycles was predicted using the probabilistic method, and this is also true for the fatigue limit. In principle, the fatigue limit is given for a number of cycles to failure. Using, for example, the staircase method, the fatigue limit is given by the average alternating stress crD, and the probability of fracture is given by the standard deviation of the scatter (s). A classical way to determine the infinite fatigue life is to use a Gaussian function. Roughly speaking, it is said that aD minus 3 s gives a probability of fracture close to zero. Assuming that "s" is equal to 10Mpa, the true infinite fatigue limit should be aD--30MPa. However, experiments show that between aD for 106 and aD for 1 0 9 the difference is greater than 30MPa for many alloys. It is important to understand that if the staircase method is popular today for determining the fatigue limit, this is because of the convenience of this approximation. A fatigue limit determined by this method to 107 cycles requires 30 hours of tests to get only one sample with a machine working at 100 Hz. To reach 108 cycles, 300 hours of tests would be necessary, which is expensive. Thus a high-frequency test is needed to explore the gigacycle fatigue regime. The actual shape of the SN curve between 106 and 101~ cycles is a better way to predict risk in fatigue cracking. Since W6hler, the standard has been to represent the SN curve by a hyperbole more or less modified as follows.
474
Bathias
Hyperbole: Ln Nf=log a - L n era; other methods may be listed as: 9 W6hler: Ln Nf= a-b ~ra 9 Basquin: Ln Nf= a-b Ln aa 9 Stromeyer: Ln Nf-- a-b Ln (oa-c) Only the exploration of the life range between 106 and 101~ cycles will create a safer approach to modeling the gigacycle fatigue regime. When the crack nucleates from a defect, such as an inclusion or pore, it is said that a relation must exist between the fatigue limit and the crack growth threshold. However, the relation between crack growth and initiation is not obvious for many reasons. First, it is not certain that a fatigue crack grows immediately at the first cycle from a sharp defect. Second, when a defect is small, a short crack does not grow as a long crack. In particular, the effect of the R ratio or the closure effect depends of the crack length. Thus the relation between AKth and aD is still to be discussed. Another important aspect is the concept of infinite fatigue life. It is understood that below AKth and below CrD the fatigue life is infinite. In fact, the fatigue limit crD is usually determinated for Nf= 107 cycles. As has been shown [1-6], fatigue failure can appear up to 109 cycles and may be beyond 109 . The fatigue strength difference at 10 7 and 109 cycles could be more than 100 MPa. It means the relation aD versus AKth must be established in the gigacycle regime if any relation exists. The experiments show that there are several mechanisms, depending on the alloys and defects. It seems that there is no general relation between AKth and crD even at 109 cycles. But, when initiation depends of inclusions, a Murakami-type model appears to be efficient.
6.11.1 EXPERIMENTAL DEVICE 6.11.1.1 PRINCIPLE Since the first 20kHz machine was constructed in 1950 by Mason [7], ultrasonic fatigue testing has become less time-consuming. A schematic view of our USF system for this study is shown in Figure 6.11.2. The machine is essentially an ultrasonic machine constituted of a Branson power generator whose frequency is held at 20 kHz. The vibration of the specimen is induced with a piezo-ceramic transducer which generates acoustical waves to the specimen through a power concentrator (horn) in order to obtain more important stress and an amplification of the displacement. The resonant length of the specimen and concentrator is
475
6.11 Gigacycle Fatigue Regime
.. Axial ~ 1 \ alsplacem~nt I
stress I
I
I
I I
I
holln
~
~.I..~_L..~
FIGURE 6.11.2
~J
'I
~ . sample
Schema of a Piezoelectric Machine
calculated at a frequency of 20 kHz. The dynamic displacement amplitude of the specimen extremity U0, is measured by an optic fiber sensor, which permits one to measure the displacement from 1 ~tm to 199.9~tm with a resolution of 0.1 ~tm. A system of video-camera/television is used for the detection of crack initiation with a magnification of 140-200 times.
6.11.1.2 SPECIMEN The specimen design for the ultrasonic test is shown in Figure 6.11.3. The dimensions Ro, R1, R2, and L2 were fixed for all materials to facilitate the machining of specimens, and the specific length L1 w a s determined analytically to have a resonance frequency of the first longitudinal vibration (f = 20 kHz) and a maximum strain value in the middle section of the specimen (x = 0), which is capable of damaging materials. A detailed analysis has been given [1], and the obtained solutions can be summarized as follows: Specimen resonant length ( f = 20 kHz): L1 = ~ arc tan
tanh~fl L2)
atanh/oL2/]}
/1/
Bathias
476 Kt=1.03
R=31 2R1
.
.
.
.
]~1 ,.~I..d
L2 FIGURE 6.11.3
"1
Ultrasonic Fatigue Specimen
with 1
a = - - arc L2
cosh(R2/R1)
(2)
K-- 2rcf~
(3)
fl = v/a2 - K2
(4)
where f - v i b r a t i o n a l frequency (= 20,000 Hz), p - d e n s i t y , Ea= dynamic Young's modulus, MPa. Relation between Uo ( x = L) and e,.(x= 0) or
g/cm 3, and
am(x=O)
~ = Vo~(L1, L2)fl
(5)
~ = EdVo~(L~, L~)fl
(6)
with ~b(L1, L2) =
cos(KL1)ch(aL2)/sh(flL2)
(7)
where U o = m a x i m u m displacement amplitude at the extremity of the specimen, mm, em = maximum strain occurred in the middle section of the specimen, and am = maximum stress deduced from Hooke's law, MPa.
6.11.2
GIGACYCLE FATIGUE OF ALLOYS WITH
FLAWS How to model gigacycle fatigue initiation? First, a new SN curve must be determined up to 1010 cycles, which is, in fact, more than the fatigue life of all technological machines. Second, a new fatigue strength at 109 cycles has to be
477
6.11 Gigacycle Fatigue Regime
predicted using the regular statistical method. More specifically, prediction of gigacycle fatigue is based on two different mechanisms"
the
9 Initiation is related to flaws (inclusions, defects, pores), and prediction is derived from stress concentration, fracture mechanics, or the short crack approach. 9 Initiation is not related to defect; in this case, microstructure is a key parameter, such as grain size, interface, load transfer, and microplasticity. Thus the discussion of gigacycle fatigue prediction is split in two parts. The first part is devoted to alloys with flaws.
6.11.2.1
S N CURVES UP TO 1 0 9 CYCLES
In the specialized literature, few results were given on this topic until "Euromech 382" was held in Paris in June 1998 (3). In order to present an overview of gigacycle SN curves, four types of alloys have been selected: highstrength steels, spring steels, Ni base alloys, and cast iron (3-9). The experimental results (Figs. 6.11.4 to 6.11.7) show that specimens can fail up to 109 and beyond. It means that the SN curve is not an asymptotic curve. Thus the concept of infinite life fatigue is not correct, and the definition of a 10 -3 r
E
Z
!
!
!
y~8,i45~c
! ~ !i
i i ii
10-4 10-5
"~ 10 -6 10 -7 10 -8
10 -9
.......................... !...............i..........!........i ......i':~~
........;" .........!...............!...........i........i... iii'~" ..... ....
..........................i...............i..........i........~ " ~ ; ~ i .......................... i............... i.......... i-...~%~
...... ~
~ .........i...........i........!......i... i'i'i" ....... i............... i........... i........ i ...... i...... F.i-..i...
i.. .i. .i. .
..........................i...............i..........i.......i............... i...i...........................!...............i...........i........i......i...... i
i
i
i
ii
!
i. ~o,~i i i i i
.......................... .............................. ............ . . . . . . . . . . . .i............. .. .......... i i,, ii i i i........i......i......i....i...i... .......................................................... ;; ..........................................................
10 -10 10
FIGURE 6.11.4
A K (MPalm)
S-N curves for High Strength Steels
100
478
Bathias 950
900
850
9
x
9 9
E 800 t~
9
750
700
..................
1.E+04
1.E+05
,
, ,,,,,,l
1.E+06
,
, ,,,,||l
1.E+07
,
, ,,,,,,i
1.E+08
,
, ,,,,,,i
1.E+09
,
1.E+10
,
,,,,,, 1.E+11
Cycles to Failure
FIGURE 6.11.5
S-N curves for Spring Steels
400 350
iB II B
300
9
9 9
9
9
9R=0.8 seeded
9 ill
D,.
9R=0.8 standard
~ 25o
9R=0 standard
200
9R=0 seeded
150 0 0
,
,
, , 11111
,
,
, 111111
,
I
, , ,,l,l
l
l
I 111111
,
I
I,
11111
l
I
, , 11111
,
,
I 111,
1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 Cycles
FIGURE 6.11.6
to
failure
S-N Curves for Nickel Base Alloys
fatigue limit at 106-107 cycles is not conservative. Depending on the alloy, the difference between TDH at 106 and 109 can be 200 MPa. From a practical point of view, the gigacycle fatigue strength becomes the more realistic property for predicting very long life.
479
6.11 Gigacycle Fatigue Regime
.---. 320 ~300
Transition
v
E 280 ._E 260 X t~
E 24O ID
" 220
925 Hz
O
::3
|
920KHz
O 0~0
00
t~
t__..~
r- 200 O
O 180
1,0E+04 1,0E+05 1,0E+06 1,0E+07 1,0E+08 1,0E+09 1,0E+10
Nombre de cycles FIGURE 6.11.7
6.11.2.2
S-N Curves for Cast Iron
INITIATION ZONES AT 10 9 CYCLES
According to our own observations and those of Murakami [i0], the gigacycle fatigue crack initiation seems to occur essentially inside the sample and not at the surface, as is observed for some with shorter life. Therefore, we can create a model for three types of crack initiation in a cylindrical sample whose surface was polished, depending on whether it is low (104 cycles), megacyclic (106 cycles), or gigacyclic (109 cycles) fatigue. Let us say that for the smallest numbers of cycles to rupture, the crack initiation sites are multiple and on the surface; according to the standard, at 106 cycles, there is only one initiation site, but, for the higher number of cycles to rupture, the initiation is located at an internal zone.
6.11.2.3
INTEGRATION OF THE
da/dn
CURVE
Consider da/dn = CAK"H and A K = 0z/2) Act v/rca where Aa is the fatigue limit at 109 cycles for a given R ratio and a is the m a x i m u m size of the radius of the internal propagation. We compute the initial value of AK at the tip of the inclusion or at the pore. The average AK initial is given in Table 6.11.1 for a nickel-base alloy (N18).
480
Bathias TABLE 6.11.1 R ratio
AKth
-1 0 0.8
5.5 8 4.5
AKi particle MPav/m
AKi pore MPax/m
8.5 12 6.7
6 6.25 3.3
Concerning the particles, for a given AKi the crack growth rate is always higher than 10 -6 mm per cycle. It means that the fatigue life of the specimens should be less than 106 cycles instead of 109 cycles. Thus it is not realistic to say the crack grows as soon as the first cycle is applied. A nucleation time is operating. Concerning the pores, AKi is smaller than AKth for the same R ratio, but AKth is determined in air and for a long crack. In vacuum AKth is higher than in air. In this respect, it is reasonable to think that the crack initiates without nucleation from a pore.
6 . 1 1 . 2 . 4 PREDICTION OF GIGACYCLE FATIGUE INITIATION FROM A FLAW Few models are able to predict the effect of nonmetallic inclusions on fatigue strength. This may be because adequate reliable quantitative data on nonmetallic are hard to obtain. Murakami and coworkers [10] have investigated the effects of defects, inclusions, and inhomogeneities on fatigue strength of high-strength steels and have expressed the fatigue limit as functions of Vickers hardness HV (Kgf/mm 2) and the square foot of the projection area of an inclusion or small defect: x/area (btm). The fatigue limit prediction equation proposed by Murakami is as follows: O"w
+
(v/area) 1/6
[(1-
2
where C--1.45 for a surface inclusion or defect, C--1.56 for an interior inclusion or defect, and a = 0.226 4-HV x 10 - 4 . The model does not specify the number of cycles for which the stress aw is represented. According to experimental data, a modified empirical equation based on the Murakami model was proposed to estimate gigacycle fatigue initiation from inclusions and small defects. This model is especially accurate for high-strength steels. O"w
(v/area) 1/6
2
6.11 GigacycleFatigue Regime where / ~ - - 3 . 0 9 - 0.12 log Nf for an interior inclusion or defect, and 2 . 7 9 - 0.108 log Nf for a surface inclusion or defect. Figures 6.11.8 and 6.11.9 show examples of the small inclusions defects caused fatigue fracture of the specimen. The size x/area of inclusions at the fracture origin is about 20 btm. Table 6.11.2 compares the fatigue strength predicted by Eq. 2 and experimental results in the gigacycle regime.
481 /~-and the the
6.11.3 GIGACYCLE FATIGUE OF ALLOYS WITHOUT INCLUSION What does happen in alloys without inclusion in the gigacycle fatigue regime? To answer this question, titanium alloys were tested in crack propagation and in crack initiation. It is well known that in titanium alloys there is not any inclusion or porosity. In this condition, nucleation of fatigue crack cannot nucleate from defects.
FIGURE 6.11.8 GigacycleFatigue Initiation on Inclusion.
482
Bathias
6.11.3.1
GIGACYCLE FATIGUEOF TI ALLOYS
A Ti-6246 alloy was supplied by the RMI Company: the chemical composition is shown in Table 6.11.3. Four thermomechanical process (TP) routes - - a TPI(1), TPI(2), TP2, and T P 3 were used to produce the forging with different microstructures and attendant mechanical properties (Table 6.11.4). The fl-processed microstructures present a similar lamella a-phase morphology with a different primary ~ volume fraction and grain size in a transformed fl matrix. The ~ + fl process conduces to a bimodal structure with duplex lamellae and a globular primary alpha phase. Quantification of the morphological aspects has been performed to provide a complete description of various microstructures.
6.11.3.2
FATIGUE INITIATION OF TI
ALLOYS
With the same alloys it is found that crack initiation and fatigue can occur up to 109 cycles even if there is no inclusion or pore. Figure 6.11.9 presents SN curves depending on the thermal processing. At room temperature a significant difference can be observed in SN curves between the different TP conditions. Thus the TP3 material has comparatively
650
-
600 550
x
E
'
9
m
a. 500
9
mm
9 TP1 (1) 9 TP1 (2) r
450
....<..
400
TP2 9 TP3
"--.t.
350 300 250
.........
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E+10
Cycles to failure
FIGURE 6.11.9 GigacycleSN Curve for Ti 6246 at 300 K and R - - 1
TABLE 6.11.2 Comparison of Predicted Fatigue Strength and Experimental Results
HV Nf Jarea H (pm) Gxp ~ ~ ( 2 ) Err.% (2)
a,,,:
465 5.75=+8 20 900 760 724 -4.7%
465 8.76'+7 16 135 740 787 +6.4%
465 7.12e+8 13 25 750 775 +3.3%
345 4.92e+5 25 0 630 592 -6.0%
450 2.59=+5 20 0 760 763 +0.4%
445 4.5'+5 60.1 0 588 621 +5.6%
550 2.0e+7 14.1 0 862 862 0%
554 1.63e+6 28.9 240 883 902 +2.2%
445 1.45'+7 53 350 550 588 +6.9%
maximum stress applied to specimen (MPa), cr, (11, u, (2): fatigue strength estimated by Eqs. 1, 2 (MPa), error %: (ow-uexp)/aexp.
500 1.7e+8 25 650 780 762 -2.3%
484
Bathias
TABLE 6.11.3 TP Number 1 (1) 1(2);2&3
Chemical Composition of Ti-6246. Alloy Investigated (in wt%)
A1
Sn
Zr
Mo
C (ppm)
Cu (ppm)
Si (ppm)
Fe (ppm)
02 (ppm)
H2 (ppm)
N2 (ppm)
5.68 5.70
1.98 2.04
3.96 4.08
6.25 6.43
90 83
<50 <50
<50 <50
400 300
930 1100
44 28
80 70
TABLE 6.11.4
Mechanical Properties of Ti 6246 0"0.2 (MPa)
TPI(1) TPI(2) TP2 TP3
1003 985 924 1044
UTS (MPa)
A (%)
RA (%)
1135 1144 1043 1145
15 12 17 11.5
17 21 28 13
the best fatigue resistance (510 MPa); the TP 1 (1) and TP 1 (2) materials exhibit a lower response, with a fatigue limit estimated, respectively, at 490 and 400 MPa for 109 cycles, and the TP2 material has the worst fatigue limit resistance, with only 325 MPa. Note that the TP3 alloy gives the lowest AK threshold and the best fatigue limit. The SEM fractographic observations indicate that all the TPI(1) broken samples have systematic surface initiation (less than 40 ~tm of the external surface), whereas TPI(2), TP2, and TP3 have a systematic internal fatigue crack site. In TP2 conditions, the microstructure and, more particularly, colonies of the primary alpha phase (P) are showing through the fracture surface by backscattered electron observations and form a sort of facet (Fig. 6.11.10). It can be seen that the facets are oriented to the fracture plane: a feature common to all specimens. In conclusion, it is emphasized that the gigacycle fatigue regime is not always correlated with defects such as inclusions or pores. For Ti-6246, the gigacycle fatigue stength is associated with a transformed amount and a secondary alpha volume fraction. Internal fatigue initiation with quasicleavage facets in primary alpha phase has been shown. In those conditions, it is very difficult to get a general relation between AKt~ and A~D. A nucleation process must exist. In this case, a linear relation between yield stress and ~D in the gigacycle regime is found: crD -- - 1 1 8 4 4- 1.6 ~rv
485
6.11 Gigacycle Fatigue Regime
FIGURE 6.11.10
Fatigue Crack Initiation on ~ Primary Phase
486
Bathias
6.11.4 GENERAL DISCUSSION OF THE GIGACYCLIC FATIGUE 6.11.4.1
INITIATION ZONES AT 10 9 CYCLES
In the specialized literature, few results are given on this topic [1]. According to our own observations and those of Murakami [10], the gigacycle fatigue crack initiation seems to occur essentially inside the sample and not at the surface, as is observed for some with shorter life. Therefore, we can create a model for three types of crack initiation in a cylindrical sample with a polished surface, depending on whether it is low (10 4 cycles), megacyclic (10 6 cycles), or gigacyclic (10 9 cycles) fatigue. For the smallest numbers of cycles to rupture, the crack initiation sites are multiple and on the surface; according to the general opinion, at 10 6 cycles, there is only one initiation site, but, for the higher number of cycles to rupture, the initiation is located at an internal zone. What remains is to specify how and why some fatigue cracks can initiate inside the metal in gigacyclic fatigue.
6.11.4.2
INITIATION MECHANISMS AT 1 0 9 CYCLES
The explanation of the phenomenon is not obvious. It seems that the cycle plastic deformation in plane stress condition becomes very small in the gigacycle regime. In this case, internal defects or large grain size play a role, in competition with the surface damage. It means also that the effect of environment is quite small in the gigacyclic regime, since the initiation of short cracks is inside the specimen. Thus the surface plays a minor role if it is smooth. The effect of plane stress plasticity is evanescent compared to microplasticity due to defects or microstructure misfits. It means that internal initiation is correlated with stress concentration or load transfer.
6.11.4.3
ROLE OF THE INCLUSIONS
The inclusions can be some privileged crack initiation sites, especially if the R load ratio is high. Figure 6.11.6 presents an example of the N18 alloys. If the crack initiates from an inclusion or from a pore, it seems a relation between AKth and CrD at 10 9 cycles can exist. But it is not a general relation.
6.11 Gigacycle Fatigue Regime
6.11.4.4
487
ROLE OF THE POROSITIES
The porosities can initiate crack in competition with inclusions, so much that the load ratio is low, particularly in tension-compression.
6.1 1.4.5
ROLE OF THE G R A I N S I Z E
In titanium alloys, the inclusions and the porosities are not important. They are only grain anomalies that initiate cracks. In the T6A4 V and 6246 alloys, the internal initiation often occurs near long primary ~ platelets. Nevertheless, the secondary phase seems to also play an important role in the resistance at gigacycle fatigue, enhancing more or less the ~P/b platelet cracking [5]. The important influence of the microstructure on the fatigue resistance at a high number of cycles has already been mentioned. In this case, the relation between AKth and CrD is difficult to justify.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Stanzl-Tschegg, S. (1996). Ultrasonic fatigue. Fatigue 96: 1886-1887. Wells, J. M. (1982). Ultrasonic fatigue. AIME-ISBN member 0-89520-397-9. Bathias, C., Miller, K. J. and Stanzl-Tschegg, S. (1999). Gigacycle fatigue. FFEMS 22(7). Ni, J. (1991). Mechanical Behaviour of Alloys in Ultrasonic Fatigue. Th~se de Doctorat, CNAM. Jago, G., and Bechet, J. (1999). Influence of microstructure of (c~ +/3) Ti alloy on high-cycle fatigue and tensile behaviour test. FFEMS 22(8): 647-655. Tao, H. (1996). Ultrasonic Fatigue of Ti and Alloys at Cryogenic Temperature. Th~se de Doctorat CNAM. Mason, W. P. (1950). Piezoelectric crystals and their applications, Ultrasonics, pp. 161-164. Wu, T., and Bathias, C. (1994). Application of fracture mechanics concept in ultrasonic fatigue. Engng. Fract. Mech. 47(5): 683-690. Bonis, J. (1997). Th~se de Doctorat, Univ. Paris-Sud. Murakami, Y., and Endo, M. (1994). Effects of defects, inclusions and inhomogeneities on fatigue strength. Fatigue 16: 163-182.
SECTION
6.12
Damage Mechanisms in Amorphous Glassy Polymers: Crazing ROBERT SCHIRRER Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France
Contents 6.12.1 Basic microstructure of Glassy Polymers 6.12.1.1 Macromolecules . . . . . . . . . . . . . . . . . 6.12.1.2 Amorphous and Crystalline Structures . . . . . . . . . . . . . . . . . . . . . . . 6.12.1.3 Glassy and Rubbery States . . . . . . . 6.12.2 Crazing Versus Micro Shear Bands . . . . . 6.12.2.1 Local Deformation Mechanism ... 6.12.2.2 Craze Growth . . . . . . . . . . . . . . . . . . . 6.12.2.3 Craze Rupture and Running Crack Tip . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
488 489 489 490 491 491 496 498 499
6.12.1 BASIC MICROSTRUCTURE OF GLASSY POLYMERS Any mechanical damage mechanism is closely dependent on the microstructure of the material. In polymers these mechanisms are dominated by the long and flexible macromolecules. Since crazes arise mainly in amorphous glassy polymers, or at least in the glassy phase of multiphase polymers, the basic structural concepts of this type of material will first be summarized briefly.
488
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
489
6.12.1.1 MACROMOLECULES Macromolecules are long repetitive sequences of an elementary chemical structure called the monomer (Fig. 6.12.1). The backbone of this long chain is mainly composed of carbon atoms linked together, and since the angle between two carbon atoms has a fixed value, successive atoms are located at only partially random positions. The stiffness of the macromolecule results from the stiffness of the monomer itself and from the shape and size of the space it occupies. Figure 6.12.2 shows the physical environment of the macromolecule modeled as a more or less flexible tube in which the macromolecule creeps, the energy necessary for this creep being either thermal or mechanical. The principal basic property of a macromolecule is its weight: M, (the number weight) or Mw (the molecular weight).
6.12.1.2
AMORPHOUS AND CRYSTALLINE STRUCTURES
A polymer is a large condensed assembly of macromolecules, which in the solid state may have two quite different microstructures. The first is a glassy disordered structure in which the smallest elementary volume representing the material has nearly the size of a monomer (Fig. 6.12.3). The macromolecules are connected to one another through physical entanglements (reversible knots) or irreversible chemical bonds, and consistent mechanical properties are attained when each macromolecule is connected to at least two others. In this microstructure the basic parameter is the molecular weight between entanglements. The second solid state is a semicrystalline structure in which the macromolecules are arranged on more or less regular small rigid lamellae with flexible amorphous connecting macromolecules (Fig. 6.12.4). The lamellae adopt a radial disposition to form large rigid
/
~
"'4,
~ rn ( Dimension \-~--. / / Z--.-~C.~."m~ =-Angstroms ~ / / // ""~""
FIGURE 6.12.1
-~O^
A simple image of a macromolecule.
490
Schirrer
ee o
T
other molecules
~
FIGURE 6.12.2
"
The tube model.
Entanglement kn
Thermoplastic FIGURE 6.12.3
Crosslinked
Amorphous microstructure showing physical or chemical links.
spherulites. In these polymers, the smallest elementary volume representing the material is much larger than in amorphous polymers, ranging from several tens of micrometers to millimeters. The semicrystalline polymer may be regarded as a two-phase material.
6.12.1.3
GLASSY AND RUBBERY STATES
Amorphous polymers are in their glassy state below the glass transition temperature Tg and rubbery above this temperature. Below Tg, the short-range molecular interactions between nonlinked atoms are strong and local loads are carried from atom to atom. If a small elastic strain is applied to the polymer, almost all carbon-carbon bonds are stressed and the carbon-carbon cone angles are distorted. In the case of large plastic strains, the atoms (or larger molecular groups) may flip from one position on the cone to another,
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
491
Amorphous state
FIGURE 6.12.4
A typical semicrystalline microstructure.
leading to nonrecoverable deformations. Above Tg, the short-range interactions between nonlinked atoms vanish, entanglements remain the only connecting points in the material, and the atoms are free to twist on the carbon-carbon cone. The elastic properties are mainly due to the entropy of variation of the positions of entanglements, which is nearly proportional to the macroscopic strain. The linear reversible behavior of the rubbery polymer may reach an extension ratio of up to 10.
6.12.2 CRAZING VERSUS MICRO SHEAR BANDS 6.12.2.1
LOCAL DEFORMATIONMECHANISM
Figure 6.12.5 shows that a network of entanglements may follow the external strain, and this schema is close to true in either the glassy or the rubbery state. In the glassy state the interactions between atoms render the elongation irreversible. Apart from the new positions of the entanglements, the molecular structure of the polymer nevertheless remains basically unchanged, and the energy input into the material is converted into heat. Conversely, under certain circumstances some molecules may break and a cavity may occur (Fig. 6.12.6). Once cavities are created, the microstructure of the polymer alters dramatically: some macromolecules have broken and the material is now a "composite" of bulk and holes.
6.12.2.1.1 Ultimate Extension Ratio
Figure 6.12.5 also shows that the extension ratio ,~max of the fully extended macromolecule is roughly Lc/Lo, where Lc is the contour length and L0 the
Schirrer
492
Lc
Fullyelongated~ molecules
,4
Neck'~
FIGURE 6.12.5 Molecularelongation in the glassy state.
aro
S J
f
FIGURE 6.12.6 A cavity at the instant of creation.
geometrical distance between entanglements. In the case of a real polymer, 2max of the fully extended material must be calculated in the threedimensional situation. On a molecular basis:
Lc ~ LmMe/Mmo and
'~,max '~
Lc/k(Me) 1/2
(1)
where Me is the average molecular weight between entanglements, Lm the length of the monomer, Mmo the monomer molecular weight, and k a
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
493
constant. Lm and Mmo a r e known from the chemistry, and the unknown molecular value Me may be calculated from the rubbery plateau shear modulus GN, the density p, the temperature T, and the gas constant R:
M~ = pRT/G~
(2)
At extensions above /~max, the stress-strain constitutive equation may be simply modeled by an infinite stress-strain slope. In usual glassy polymers /~max lies between 2 and 10. 6.12.2.1.2 Cavitation under Hydrostatic Tension Unlike in metals, cavities do not preexist in glassy polymers. The cavity shown in Figure 6.12.6 may be created only if the closing forces due to surface tension in the cavity are smaller than the opening forces due to the hydrostatic tension. Moreover, the energy balance criterion between the elastic energy density available in the material and the surface energy consumed in the cavity must be fulfilled. Since the voids are quite small, close in size to the macromolecules (radius r0 of 1-10nm), a mechanical analysis of their equilibrium if fairly difficult. Another difference between voids in metals and polymers is that voids in polymers do not evolve into a porous microstructure but into crazes through a collective yield flow deformation process. Some models have nevertheless been developed on molecular bases. In the mechanical approach, a number of basis assumptions can be made. Thus the energy consumed is at least the thermodynamic surface energy Ws arising from the specific thermodynamic surface energy F, Ws = 4rrr2F. At the critical equilibrium point, the closing stress 2F/ro due to the surface tension acting at the boundary of the cavity is equal to the hydrostatic tension ah, and hence r0 = 2F/ah. In rubbers, for example, F is roughly 0.03Jm -2, and the critical value ah is approximately 10 MPa, which means that the void radius r0 is very small (6 nm). The elastic hydrostatic energy density stored in the body is a~/2kb where kb is the bulk modulus, and the elastic energy used to form the cavity is drawn from the bulk in its vicinity over a distance ~r0 proportional to the radius of the cavity. Prior to cavitation, the amount of energy stored in this vicinity is 4~ 3~ a h /26 k ~ , and it is only partly consumed to create the cavity. Finallly, there is the critical hydrostatic tension:
crh < 3kb/~ 3
(3)
It is not easy to determine what proportion ~ of the local elastic energy is consumed to form the cavity. This concerns the nonlinear mechanical behavior around the cavity, where nonlinear viscoelastic or even plastic flow takes place. In addition, the size of the volume from which the energy is drawn is so close to the molecular size that continuum mechanics is not
494
Schirrer
necessarily relevant. Experimental values of Crh ( ~ 30 MPa) and kb ( ~ 10 GPa) in glassy polymers indicate that only a very small part of the local energy is used to create the cavity surfaces in these polymers. Local yield consumes much more energy. 6.12.2.1.3 Craze Microstructure (Fig. 6.12.7) The local stress increase around an isolated new cavity may create other neighboring organized cavities to form a craze precursor. A craze inside a polymer has usually a penny shape and resembles a crack except that fibrils bridge the crack surfaces, carrying loads from one to the opposite surface. These fibrils are thin (10-100nm) and rather long (up to 10 ~tm), and they grow like micronecks. Therefore, the load Crc applied to the craze surface is a material property roughly constant along the craze boundary and similar to the yield stress Cry, while the craze shape is close to that predicted by the Dugdale or Rice model:
~-<,<~ and S(x/-~
'-~
-~
/'+,+(,'
~)
_ x i7-7
(4)
where K1 is the stress intensity factor and E* the plain strain modulus.
0.01 to O. 1/xm
Macromolecules
[, I,, 1 ]
~ Io. c
/
Fibril Vacuum
Craze tip
1 to 10 # m
>
FIGURE 6.12.7 The craze microsturcture.
495
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
6.12.2.1.4 Craze Micromechanics Figure 6.12.8 shows how the craze fibrils grow by extracting polymer from the bulk in the so-called fluid process zone. In the fibrils the polymer macromolecules are fully stretched. Some molecules, caught at their ends in two separate fibrils, must therefore break or create a connecting cross fibril. The mechanical parameters governing the extraction process are the local flow stress ac, the surface tension F, the fibril diameter D, the radius R connecting bulk and fibrils, and the process zone size h. Simple scaling laws of fluid mechanics show that the fibril diameter is of the order of
D
8r/
c
2R
h
Since the volume before and after fibrillation remains constant, the extension ratio in the fibrils is /]'fibril ~
(D 4- 2R)2/D 2
(6)
Values for/],fibril normally lie between 2 and 10.
6.12.2.1.5 Craze versus Micro Shear Band If a glassy polymer is subjected to a mechanical stress, local microscopic fluctuations may induce a cavity or a micro shear band. A first cavity is the necessary precursor of any subsequent craze. Several criteria must be satisfied to obtain crazes rather than micro shear bands, some of which have a
Macromolecule ~ , ~
h=D
Vacuum
Flow..
~
~--------Fibril
~
O'c
~
"
~
zone zone limi
Active "fluid"
Micro neck ~
Surface tension
FIGURE 6.12.8 The micromechanics of craze growth. The fibrils extract material from the bulk in the process zone h.
496
Schirrer
molecular basis whereas others are related to the mechanics of the material. In mechanical terms, cavitation occurs at a certain level of hydrostatic tension and cannot in fact occur in compression or pure shear. Figure 6.12.9 illustrates this crazing criterion: within the grey area there is no damage, whereas in the upper right zone (tension) crazes arise and in the lower left zone (compression) micro shear bands arise. Obviously, this mixed-mode criterion must be separately determined for each glassy amorphous polymer. A frequently used equation correlates the critical macroscopic crazing strain ~c and the principal stresses Crl, or2, or3 through constants A and B:
~c = A + B/(~I + ~2 + ~3)
(7)
In molecular terms, Eq. 1 defines the highest extension ratio 2max of the polymer. Equation 6 indicates the required micromechanical extension ratio ~,fibril in the craze fibrils, and if/~max < ~,fibril, it is clear that a craze cannot form. Finally, all these criteria depend on temperature, strain rate, and molecular weight and structure.
6.12.2.2
CRAZE GROWTH
As seen in Figure 6.12.10, the craze grows through a meniscus instability mechanism with a craze front resembling fingertips. Since the fibril growth is similar to a creep mechanism extracting material from the bulk, it seems
FIGURE 6.12.9 Von Mises criterion plotted together with the crazing criterion for a typical polymer (polymethylmethacrylate).
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
497
FIGURE 6.12.10 The meniscus mechanism governing craze growth.
reasonable that the craze front will move with a velocity Vc governed by a creeping law. In polymers, creep is generally a stress- and temperature-activated process. The stress acting on the craze surface is controlled by the external load and the craze shape by the stress intensity factor K 1 of Eq. 4. In this equation, to obtain the craze growth law in the basic mechanical situations like constant external load (creep mechanism) or linear increasing external load, the material properties crc and E have to be replaced by the equivalent timedependent properties ~c(t) and E(t). The local material property ere(t) is readily determined by recording the craze length as a function of time S(t), in an experiment at low constant K1 where the crack does not grow into the craze. (/~K 2 ~ 1/2
crc(t)- \8S(t),]
(8)
In most cases the growth of the craze length can be expressed as S(t) =
At n
where A is a constant depending on the loading conditions and n ~ 0.5.
(9)
498
Schirrer
Velocity Vc
Velocity Vc
>
Midrib
Q O
Propagation r-
FIGURE 6.12.11
6.12.2.3
The steady-state propagating craze at a crack tip.
CRAZE RUPTURE AND RUNNING CRACK TIP
Once a craze has formed and grown for a certain time, some fibrils may break. Although most models of fibril breakage are based on molecular theories and do not concern mechanics, the case of crazes breaking in their midrib region (Fig. 6.12.11) can be described in mechanical terms. This type of craze breakdown is quite common in brittle polymers. If the fibrils grow by extracting polymer from the bulk, their oldest part is the midrib. Thus, if the fibrils break in their midrib, the concept of lifetime z(~r) (the time ~ a structure survives a stress ~r) can be applied to the center of a fibril. Use of this concept is satisfactory from a molecular point of view, since the molecular theory holds that the fibrils break through a slippage mechanism (molecular creep) evolving to disentanglement. The model is particularly simple for a crack propagating with velocity Vc and having a craze S at its tip:
"r(r
= S/Vc
(10)
~c[8T(~Yc)Vc/7~]1/2
(11)
Using Eq. 4, KI
-
-
Equation 11 relates the macroscopic fracture toughness of a propagating crack to local material properties like the lifetime of fibrils and the Dugdale craze stress. Conversely, real measurements of craze length S and toughness K1 at a
6.12 Damage Mechanisms in Amorphous Glassy Polymers: Crazing
499
running crack tip may be used to obtain the local mechanical properties Z(ac) and ac.
BIBLIOGRAPHY 1. Crazing in Polymers, Vol. 2 in Advances in Polymer Science 91/92, Kausch, H. H., ed., Berlin: Springer Verlag 1990. 2. Narisawa, I., and Yee, A. E (1993). Crazing and fracture of polymers, in Structure and Properties of Polymers, Materials Science and Technology, A Comprehensive Treatment, pp. 698-765, vol. 12, Thomas, E. L., ed., Weinheim: VCH. 3. Ward, I. M., and Hadley, D. W. An introduction to the Mechanical Properties of Solid Polymers, New York: Wiley 1993. 4. Trassaert, P., and Schirrer, R. (1983). The disentanglement time of the craze fibrils in PMMA. Journal of Material Science 18: 3004-3010. 5. Schirrer, R., Lang, R., Le Masson, J., and Tomatis, B. (1984). The disentanglement time of the craze fibrils under cyclic loading. Polymer Eng. and Science 24: 820-824. 6. Fond, C., Lobbrecht, A., and Schirrer, R. (1996). Polymers toughened with rubber microspheres. Int. J. of Fracture 77: 141-159.
SECTION
6.13
Damage Models for Concrete GILLES PIJAUDIER--CABOT 1 and
JACKY MAZARS 2
1Laboratoire de G~nie Civil de Nantes Saint-Nazaire, Ecole Centrale de Nantes, BP 92101, 44321 Nantes Cedex 03, France 2 LMT-Cachan, ENS de Cachan, Universitd Paris 6, 61 avenue du Prdsident Wilson, 94235, Cachan Cedex, France
Contents 6 . 1 3 . 1 I s o t r o p i c D a m a g e M o d e l [4] . . . . . . . . . . . 6.13.1.1 Validity . . . . . . . . . . . . . . . . . . . . . . . .
501 501
6.13.1.2 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . .
501
6.13.1.3 E v o l u t i o n of D a m a g e . . . . . . . . . . .
502
6.13.1.4 I d e n t i f i c a t i o n of P a r a m e t e r s . . . . . 503 6.13.2 Nonlocal Damage ......................
503
6.13.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . .
504
6.13.2.2 P r i n c i p l e . . . . . . . . . . . . . . . . . . . . . . .
504
6.13.2.3 D e s c r i p t i o n of the M o d e l . . . . . . .
505
6.13.2.4 I d e n t i f i c a t i o n of the I n t e r n a l Length .............
............
6.13.2.5 H o w to Use the M o d e l . . . . . . . . . . 6.13.3 Anisotropic Damage Model ............
505 506 506
6.13.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . .
506
6.13.3.2 P r i n c i p l e . . . . . . . . . . . . . . . . . . . . . . .
507
6.13.3.3 D e s c r i p t i o n of the M o d e l . . . . . . .
508
6.13.3.4 I d e n t i f i c a t i o n of P a r a m e t e r s . . . . . 510 6.13.3.5 H o w to Use the M o d e l . . . . . . . . . . References ....................................
500
511 512
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
501
6.13 Damage Models for Concrete
6.13.1
ISOTROPIC DAMAGE MODEL
6.13.1.1
VALIDITY
This constitutive relation is valid for standard concrete with a compression strength of 30-40 MPa. Its aim is to capture the response of the material subjected to loading paths in which extension of the material exists (uniaxial tension, uniaxial compression, bending of structural members) [4]. It should not be employed (i) when the material is confined (triaxial compression) because the damage loading function relies on extension of the material only, (ii) when the loading path is severely nonradial (not yet tested), and (iii) when the material is subjected to alternated loading. In this last case, an enhancement of the relation which takes into account the effect of crack closure is possible. It will be considered in the anisotropic damage model presented in Section 6.13.3. Finally, the model provides a mathematically consistent prediction of the response of structures up to the inception of failure due to strain localization. After this point is reached, the nonlocal enhancement of the model presented in Section 6.13.2 is required.
6.13.1.2
BACKGROUND
The influence of microcracking due to external loads is introduced via a single scalar damage variable d ranging from 0 for the undamaged material to 1 for completely damaged material. The stress-strain relation reads: 8ij
1 + v0 - d) aij
Eo(1
v0 -
Eo(1
-
d) [akhcSij]
(1)
E0 and v0 are the Young's modulus and the Poisson's ratio of the undamaged material; eij and aij are the strain and stress components, and 5ij is the Kronecker symbol. The elastic (i.e., free) energy per unit mass of material is p~' = 1(1
-
d)sijC~
(2)
where C~ is the stiffness of the undamaged material. This energy is assumed to be the state potential. The damage energy release rate is ~/ 1 0 Y = - p - ~ - ~sijCijhlshl with the rate of dissipated energy: Opt,
502
Pijaudier-Cabot and Mazars
Since the dissipation of energy ought to be positive or zero, the damage rate is constrained to the same inequality because the damage energy release rate is always positive.
6.13.1.3
EVOLUTIONOF DAMAGE
The evolution of damage is based on the amount of extension that the material is experiencing during the mechanical loading. An equivalent strain is defined as i--1(<~i>4-
where (.)+ is the Macauley bracket and function of damage is
~i
(3)
are the principal strains. The loading
f(~, ~c) = ~ - ~
(4)
where tc is the threshold of damage growth. Initially, its value is ~c0,which can be related to the peak stress ft of the material in uniaxial tension:
(5)
f' ~c0 = E0
In the course of loading ~c assumes the maximum value of the equivalent strain ever reached during the loading history. If f ( ~ , t c ) - O
{d
~-h(~c) -~
and f(~,tc)=O,
then
with c~ ___ 0, e l s e / C~ ~- 0- 0
(6)
The function h(~c) is detailed as follows: in order to capture the differences of mechanical responses of the material in tension and in compression, the damage variable is split into two parts:
d = o~td t -+- o~cdc
(7)
where dt and dc are the damage variables in tension and compression, respectively. They are combined with the weighting coefficients at and 0~c, defined as functions of the principal values of the strains ~lj and ~ due to positive and negative stresses: t
e i j - (1
-1
-
-
t
d)CijklO'kl,
c
eij = (1 - d) -lcCijklcrkl
(8)
503
6.13 Damage Models for Concrete
~
-
52
,
9_ ~ _
~
-
(9) ..__
Note that in these expressions, strains labeled with a single indicia are principal strains. In uniaxial tension c~t = 1 and ~c = 0 . In uniaxial compression ~c = 1 and at = 0. Hence, dt and dc can be obtained separately from uniaxial tests. The evolution of damage is provided in an integrated form, as a function of the variable ~c: dt ~ 1 ~
~Co(1
-
At)
~c
~Co(1 - A c ) dc ~ 1 - ~:
6.13.1.4
At exp[Bt(~c- ~Co)] Ac
(10)
exp[B~(~c- ~Co)]
IDENTIFICATION OF PARAMETERS
There are eight model parameters. The Young's modulus and Poisson's ratio are measured from a uniaxial compression test. A direct tensile test or threepoint bend test can provide the parameters which are related to damage in tension (tc0, At, Bt). Note that Eq. 5 provides a first approximation of the initial threshold of damage, and the tensile strength of the material can be deduced from the compressive strength according to standard code formulas. The parameters (Ac, Bc) are fitted from the response of the material to uniaxial compression. Finally, fl should be fitted from the response of the material to shear. This type of test is difficult to implement. The usual value is f l = 1, which underestimates the shear strength of the material [7]. Table 6.13.1 presents the standard intervals for the model parameters in the case of concrete with a moderate strength.
TABLE 6.13.1 STANDARDModel Parameters E0 ~ 30,000-40,000 MPa v0 ~0.2 ~c0~ 1 x 1 0 - 4 0.7 <~At<~1.2 104~Bt<~5 • 104 1 ~
504
Pijaudier-Cabot and Mazars
5 |
!
|
|
|
|
!
-0,008
-0,007
-0,006
-0,005
-0,004
-0,003
-0,002
, ~~ -0,001 n
, 0,001
35J -40 Strain FIGURE 6.13.1
Uniaxial response of the model.
Figure 6.13.1 shows the uniaxial response of the model in tension and compression with the following parameters: E 0 - 30,000MPa, v 0 - 0.2, to0 = 0.0001, At = 1, Bt = 15,000, Ac = 1.2, Bc = 1500, ]~ = 1.
6.13.2
NONLOCAL
DAMAGE
The purpose of this section is to describe the nonlocal enhancement of the previously mentioned damage model. This modification of the model is necessary in order to achieve consistent computations in the presence of strain localization due to the softening response of the material [8].
6.13.2.1 VALIDITY As far as the type of loading is concerned, the range of validity of the nonlocal model is exactly the same as the one of the initial, local model. This model, however, enables a proper description of failure that includes damage initiation, damage growth, and its concentration into a completely damaged zone, which is equivalent to a macrocrack.
505
6.13 Damage Models for Concrete
6.13.2.2 PRINCIPLE Whenever strain softening is encountered, it may yield localization of strains and damage. This localization corresponds to the occurrence of bifurcation, and a surface (in three dimension) of discontinuity of the strain rate appears and develops. When such a solution is possible, strains and damage concentrate into a zone of zero volume, and the energy dissipation, which is finite for a finite volume of material, tends to zero. It follows that failure occurs without energy dissipation, which is physically incorrect [1]. Various remedies to this problem can be found (e.g., [5]). The basic idea is to incorporate a length, the so-called internal length, into the constitutive relation to avoid localization in a region of zero volume. The internal length controls the size of the region in which damage may localize. In the nonlocal (integral) damage model, this length is incorporated in a modification of the variable which controls damage growth (i.e., the source of strain softening): a spatial average of the local equivalent strain.
6.13.2.3
DESCRIPTION OF THE MODEL
The equivalent strain defined in Eq. 3 is replaced by its average ~:
~(x) -- V,.(x----) r
s)~(s)ds
with
V,.(x) -
r
s)ds
(11)
where ~ is the volume of the structure, Vr(X) is the representative volume at point x, and ~ ( x - s) is the weight function, for instance: r
4 [ ' xs['2)12 -
(12)
where l~ is the internal length of the nonlocal continuum. The loading function (Eq. 4) becomes f(~, Z) = e - Z. The rest of the model is similar to the description provided in Section 6.13.1.
6.13.2.4
IDENTIFICATION OF THE INTERNAL LENGTH
The internal length is an additional parameter which is difficult to obtain directly by experiments. In fact, whenever the strains in specimen are homogeneous, the local damage model and the nonlocal damage model are, by definition, strictly equivalent (~ = ~). This can be viewed also as a simplification, since all the model parameters (the internal length excepted)
506
Pijaudier-Cabot and Mazars
are not affected by the nonlocal enhancement of the model if they are obtained from experiments in which strains are homogeneous over the specimen. The most robust way of calibrating the internal length is by a semi-inverse technique which is based on computations of size effect tests. These tests are carried out on geometrically similar specimens of three different sizes. Since their failure involves the ratio of the size of the zone in which damage can localize versus the size of the structure, a size effect is expected because the former is constant while the later changes in size effect tests. It should be stressed that such an identification procedure requires many computations, and, as of today, no automatic optimization technique has been devised for it. It is still based on a manual trial-and-error technique and requires some experience. An approximation of the internal length was obtained by Bazant and Pijaudier-Cabot [2]. Comparisons of the energy dissipated in two tensile tests, one in which multiple cracking occurs and a second one in which failure is due to the propagation of a single crack, provided a reasonable approximation of the internal length that is compared to the maximum aggregate size da of concrete. For standard concrete, the internal length lies between 3da and 5da.
6.13.2.5
H o w To USE THE MODEL
The local and nonlocal damage models are easily implemented in finite element codes which uses the initial stiffness or secant stiffness algorithm. The reason is that the constitutive relations are provided in a total strain format. Compared to the local damage model, the nonlocal model requires some additional programming to compute spatial averages. These quantities are computed according to the same mesh discretization and quadrature as for solving the equilibrium equations. To speed the computation, a table in which, for each gauss point, its neighbors and their weight are stored can be constructed at the time of mesh generation. This table will be used for any subsequent computation, provided the mesh is not changed. Attention should also be paid to axes of symmetry: as opposed to structural boundaries where the averaging region lying outside the structure is chopped, a special averaging procedure is needed to account for material points that are not represented in the finite element model. The implementation of the nonlocal model in an incremental format is awkward. The local tangent stiffness operator relating incremental strains to incremental stresses becomes nonsymmetric, and, more importantly, its bandwidth can be very large because of nonlocal interactions.
507
6.13 D a m a g e M o d e l s for C o n c r e t e
6.13.3 ANISOTROPIC DAMAGE MODEL
6.13.3.1 VALIDITY Microcracking is usually geometrically oriented as a result of the loading history on the material. In tension, microcracks are perpendicular to the tensile stress direction; in compression microcracks open parallel to the compressive stress direction. Although a scalar damage model, which does not account for directionality of damage, might be a sufficient approximation in usual applications, i.e., when tensile failure is expected with a quasi-radial loading path, damage-induced anisotropy is required for more complex loading histories. The influence of crack closure is needed in the case of alternated loads: microcracks may close and the effect of damage on the material stiffness disappears. Finally, plastic strains are observed when the material unloads in compression. The following section describes a constitutive relation based on elastoplastic damage which addresses these issues. This anisotropic damage model has been compared to experimental data in tension, compression, compression-shear, and nonradial tensionshear. It provides a reasonable agreement with such experiments [3].
6.13.3.2 PRINCIPLE The model is based on the approximation of the relationship between the overall stress (simply denoted as stress) and the effective stress in the material defined by the equation t
0
e
or
cr0 -- CoktekZ
t
0
crij -- Cijkl
(cdamaged
-1
)klmnO'mn
(13)
e where alj is the effective stress component, ekZ is the elastic strain, and t--damag "~0kl ed is the stiffness of the damaged material. We definite the relationship between the stress and the effective stress along a finite set of directions of unit vectors n at each material point:
a
_
[1 - d(n)]nial)n )
,
v
_
[1 - d(n)]
i=1 [cri)nj - (nkCrnknt)nii2
(14)
where a and z are the normal and tangential components of the stress vector, respectively, and d(n) is a scalar valued quantity which introduces the effect of damage in each direction n. The basis of the model is the numerical interpolation of cl(n) (called damage surface) which is approximated by its definition over a finite set of
508
Pijaudier-Cabot
and Mazars
directions. The stress is the solution of the virtual work equation: ~,
find ~rij such that Vei)
47~ , ~ t * -3-aije'iJ = a~ ([(1 - d(n))nkat~tntni + (1 - d(n))(aijn ) - nkathlnlni)] . eij nj)d.Q (15) Depending on the interpolation of the damage variable d(n), several forms of damage-induced anisotropy can be obtained.
6.13.3.3
DESCRIPTION OF THE MODEL
The variable d(n) is now defined by three scalars in three mutually orthogonal directions. It is the simplest approximation which yields anisotropy of the damaged stiffness of the material. The material is orthotropic with a possibility of rotation of the principal axes of orthotropy. The stiffness degradation occurs mainly for tensile loads. Hence, the evolution of damage will be indexed on tensile strains. In compression or tension-shear problems, plastic strains are also of importance and will be added in the model. When the loading history is not monotonic, damage deactivation occurs because of microcrack closure. The model also incorporates this feature. 6.13.3.3.1 Evolution of Damage The evolution of damage is controlled by a loading surface f, which is similar to Eq. 4:
f(n)
=
nie_,eijnj
-
e,d --
z(n)
(16)
where Z is a hardening-softening variable which is interpolated in the same fashion as the damage surface. The initial threshold of damage is ea. The evolution of the damage surface is defined by an evolution equation inspired from that of an isotropic model: Iff(n*) -- 0 and n.~ de~nf > 0 rid(n)
then
=
gd[1 nt- c/(rl i
e
(n~*e~nj) 2
*
.
exp(-a(n~* ~o"J ~ * -~a)) n, de~.j
else dd(n*) = O,
,
(17)
dz(n) = 0
The model parameters are ea and a. Note that the vectors n* are the three principal directions of the incremental strains whenever damage grows. After an
509
6.13 Damage Models for Concrete
incremental growth of damage, the new damage surface is the sum of two ellipsoidal surfaces: the one corresponding to the initial damage surface, and the ellipsoid corresponding to the incremental growth of damage. 6.13.3.3.2 Coupling with Plasticity We decompose the strain increment in an elastic and a plastic increment: (18)
deij - de~j 4- d ~
The evolution of the plastic strain is controlled by a yield function which is expressed in terms of the effective stress in the undamaged material. We have implemented the yield function due to Nadai [6]. It is the combination of two Drucker-Prager functions F1 and F2 with the same hardening evolution: Fi --
~t
J24-
(19)
A.Itl , 3 - Bi w
where J~ and I~ are the second invariant of the deviatoric effective stress and the first invariant of the effective stress, respectively, w is the hardening variable, and (Ai, Bi) are four parameters (i = 1, 2) which were originally related to the ratios of the tensile strength to the compressive strength, denoted 7, and of the biaxial compressive strength to the uniaxial strength, denoted fl: A1 --
V/ 2 1 -- 7 1 +7'
A2 -- V ~ fl - 1 2 f l - 1'
B1 - -
2 x,/2
7 1 +7'
B2 -- X/~
2/3- 1
(20) These two ratios will be kept constant in the model: fl = 1.16 and 7 = 0.4. The evolution of the plastic strains is associated with these surfaces. The hardening rule is given by w = qpr + Wo
(21)
where q and r are model parameters, w0 defines the initial reversible domain in the stress space, and p is the effective plastic strain. 6.13.3.3.3 Crack Closure Effects Crack closure effects are of importance when the material is subjected to alternated loads. During load cycles, microcracks close progressively and the tangent stiffness of the material should increase while damage is kept constant. A decomposition of the stress tensor into a positive and negative part is introduced: a = (a)++(~r)_, where (a)+, and (a)_ are the positive and negative parts of the stress tensor. The relationship between the stress and the
510
Pijaudier-Cabot and Mazars
effective stress defined in Eq. 14 of the model is modified:
rrijnj --
[1 - d(n)]/rr) t+i;n; + [1 -dc(n)]lrr)2i;n;
(22)
where de(n) is a new damage surface which describes the influence of damage on the response of the material in compression. Since this new variable refers to the same physical state of degradation as in tension, de(n) is directly deduced from d(n). It is defined by the same interpolation as d(n), and along each principal direction i, we have the relation
dic-(
dj(1
2-(~iJ)/~z'
iC
[1, 3]
(23)
where ~ is a model parameter.
6.13.3.4
IDENTIFICATION OF PARAMETERS
The constitutive relations contain six parameters in addition to the Young's modulus of the material and the Poisson's ratio. The first series of three parameters (~d, a, ~) deals with the evolution of damage. Their determination benefits from the fact that, in tension, plasticity is negligible, and hence ~d is directly deduced from the fit of a uniaxial tension test. If we assume that in uniaxial tension damage starts once the peak stress is reached, ~d is the uniaxial tensile strain at the peak stress (Eq. 5). Parameter a is more difficult to obtain because the model exhibits strain softening. To circumvent the difficulties involved with softening in the computations without introducing any nonlocality (as in Section 6.13.2), the energy dissipation due to damage in uniaxial tension is kept constant whatever the finite element size. Therefore, a becomes an element-related parameter, and it is computed from the fracture energy. For a linear displacement interpolation, a is the solution of the following equality where the states of strain and stresses correspond to uniaxial tension: h4~ - Gf,
with 4~ -
[~t(~)nk~tklnzn~]njdf~de~j
(24)
where ~b is the energy dissipation per unit volume, Gf is the fracture energy, and h is related to the element size (square root of the element surface in a two-dimensional analysis with a linear interpolation of the displacements). The third model parameter e enters into the influence of damage created in tension on the compressive response of the material. Once the evolution of damage in tension has been fitted, this parameter is determined by plotting the decrease of the uniaxial unloading modulus in a compression test versus
511
6.13 Damage Models for Concrete
the growth of damage in tension according to the model. In a log-log coordinate system, a linear regression yields the parameter ~. The second series of three parameters involved in the plastic part of the constitutive relation is (q, r, w0). They are obtained from a fit of the uniaxial compression response of concrete once the parameters involved in the damage part of the constitutive relations have been obtained. Figure 6.13.2 shows a typical uniaxial compression-tension response of the model corresponding to concrete with a tensile strength of 3 MPa and a compressive strength of 40MPa. The set of model parameters is: E - 35,000 MPa, v - 0.15, f t - - 2.8 MPa (which yields e,a - 0 . 7 6 • 10-4); fracture energy: G f - - 0 . 0 7 N/mm; other model parameters: 0~ = 12, r = 0.5, q = 7000 MPa, COo -- 26.4 MPa.
6.13.3.5
H o w TO USE THE MODEL
The implementation of this constitutive relation in a finite element code follows the classical techniques used for plasticity. An initial stiffness algorithm should be preferred because it is quite difficult to derive a consistent material tangent stiffness from this model. Again, the evolution of
5.
|
-
Stress MPa
I
-_5, -20.
_
-15. -20. -25. -30.
I
-6.
-4,
I
-2,
I
I
O.
i
2.
4,
6.
Strain (xlO00) FIGURE 6.13.2 Uniaxial tension-compression response of the anisotropic model (longitudinal [1], transverse [2], and volumetric Iv] strains as functions of the compressive stress).
512
Pijaudier-Cabot and Mazars
damage is provided in a total strain format. It is computed after incremental plastic strains have been obtained. Since the plastic yield function depends on the effective stress, damage and plasticity can be considered separately (plastic strains are not affected by damage growth). The difficulty is the numerical integration involved in Eq. 15, which is carried out according to Simpson's rule or to some more sophisticated scheme.
REFERENCES 1. Bazant, Z.P. (1985). Mechanics of distributed cracking. Applied Mech. Review 39: 675-705. 2. Bazant, Z.P., and Pijaudier-Cabot, G. (1989). Measurement of the characteristic length of nonlocal continuum. J. Engrg. Mech. ASCE 115: 755-767. 3. Fichant, S., La Borderie, C., and Pijaudier-Cabot, G. (1999). Isotropic and anisotropic descriptions of damage in concrete structures. Int. J. Mechanics of Cohesive Frictional Materials 4: 339-359. 4. Mazars, J. (1984). Application de la m~canique de l'endommagement au comportement non lin~aire et h la rupture du b~ton de structure, Th~se de Doctorat ~s Sciences, Universit~ Paris 6, France. 5. Muhlhaus, H. B., ed. (1995). Continuum Models for Material with Microstructure, John Wiley. 6. Nadai, A. (1950). Theory of Flow and Fracture of Solids, p. 572, vol. 1, 2nd ed., New York: McGraw-Hill. 7. Pijaudier-Cabot, G., Mazars, J., and Pulikowski, J. (1991). Steel-concrete bond analysis with nonlocal continuous damage. J. Structural Engrg. ASCE 117: 862-882. 8. Pijaudier-Cabot, G., and Bazant, Z. P. (1987). Nonlocal damage theory. J. Engrg. Mech. ASCE 113: 1512-1533.
SECTION
6.14
Isotropic and Anisotropic Damage Law of Evolution JEANLEMAITRE 1 and
RODRIGUE DESMORAT 2
1Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France 2 Universit~ Paris 6-LM2S, 8, rue du Capitaine Scott, 75015 Paris, France
Contents 6 . 1 4 . 1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
513
6.14.2 Background ..............................
514
6.14.2.1 I s o t r o p i c Case . . . . . . . . . . . . . . . . . . . .
515
6.14.2.2 A n i s o t r o p i c Case . . . . . . . . . . . . . . . . . .
515
6 . 1 4 . 3 K i n e t i c L a w of D a m a g e E v o l u t i o n . . . . . . . . .
516
6.14.3.1 I s o t r o p i c Case . . . . . . . . . . . . . . . . . . . .
516
6.14.3.2 A n i s o t r o p i c Case . . . . . . . . . . . . . . . . . .
516
6.14.3.3 D a m a g e T h r e s h o l d . . . . . . . . . . . . . . . .
517
6.14.3.4 M e s o c r a c k I n i t i a t i o n C r i t e r i o n . . . . . 518 6.14.3.5 Different E v o l u t i o n of D a m a g e in Tension and Compression . . . . . . . . .
518
6 . 1 4 . 4 I d e n t i f i c a t i o n of T h e M a t e r i a l P a r a m e t e r s . .
519
6 . 1 4 . 5 H o w to U s e t h e M o d e l . . . . . . . . . . . . . . . . . . . .
522
6.14.5.1 Use it as a P o s t p r o c e s s o r . . . . . . . . . .
523
6.14.5.2 Use it in a F u l l y C o u p l e d S c h e m e . 524 Bibliography .....................................
524
6 . 1 4 . 1 VALIDITY Based on the thermodynamics of irreversible processes dealing with elastic energy and plastic strain at mesoscale, this law is able to model any kind of damage: ductile, creep, fatigue, and even quasi-brittle if it is used at microscale. According to the definition of the damage variable, the micromechanisms act only as the surface density of microcracks and microvoids in any plane of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
513
514
Lemaitre and D e s m o r a t
the Representative Volume Element. The materials act according to the numerical value of the parameters which quantify the law. The weakness of the damage law results from its generality; the accuracy of its prediction is much better when the identification of the parameters is performed from tests as close as possible to the case being considered for the application.
6.14.2 BACKGROUND The damage variable as previously defined is a scalar D if the hypothesis of isotropy is made. It is a second-order tensor _D if, as shown by microscopic observations, it is mainly driven by the plastic strain which makes it orthotropic. The specific Gibbs energy ~k* taken as the thermodynamic state potential is qualitatively written from experimental results of damage measurements in several directions: p6, =
l+v D D 3(1-2v) 2E HijO'jkHklrYli-I2E
rr2
+p~P
1 - riD n
(1)
where 9 E and v are the Young's modulus and the Poisson's ratio, p is the density; 9 a is the Cauchy stress, rr~ = a i j - an6ij is the stress deviator and a n - lahk is the hydrostatic stress; 9
/-/-
(1 - _D) -1/2"
9 D n = 89
9 r/is a material parameter needed to take into account the variation of the contraction coefficient with damage during a tension test ( r / ~ 3 for many materials); 9 ffP is the plastic part of the thermodynamic potential. If isotropy is considered, Dij - D~ij, H -- 1 / v / 1 - D, rI -- 1. The isothermal law of elasticity coupled with damage derives from e
0~*
- 'OG,;
(2)
from which the effective stress is ~
rrij
__
D
(HikCrklHl j
)D
o-H
-4- 1 - r l D n fiij
(3)
515
6.14 Isotropic and Anisotropic Damage Law of Evolution or ~
oiJ=l_
~ij
(4)
D
for the isotropic case (the superscript z) denotes the deviatoric part). The energy release rate density _Y, as the contracted product _Y: /5 is the power dissipated in the process of damage, derives from
(5)
Y~j= PODv
6.14.2.1 Y-
ISOTROPIC CASE
F2qRv/2E, where rYeqis the effective von Mises equivalent stress:
%~J
~
O'eq
---
(6)
1- D
R~ is the triaxiality function: e~--5(1+v)+3(1-2v)
crn
(7)
6 . 1 4 . 2 . 2 ANISOTROPIC CASE The expression of Yij will not be used. We will need instead the expression of the effective strain energy density, denoted Y and defined as for the isotropic case (scalar Y) as -
1
1
~2
aeqev
(8)
F/q 2
(9)
with the effective triaxiality function
Rv
5(1+v)+3(1-2v)
(7H
~ e q - (_H_~D_H)eq, F/q -- 1 -Y/DH
(10)
516
Lemaitre and Desmorat
6.14.3 KINETIC LAW OF DAMAGE EVOLUTION Here again, from qualitative experiments, it is possible to choose the dissipative potential from which derives the damage constitutive equation (11)
F = f + FD
where f is the von Mises plastic loading function written for kinematic hardening _X, isotropic hardening R, and the yield stress ay: (12)
f = (F_- X_)eq - R - ay
and where FD is the damage potential function.
6.14.3.1 ISOTROPIC CASE The damage evolution law is given by OF O~FD /)=,~-~=,~--~,
S FD--(s+I)(I_D)
(s/S+l (13)
where ,~ is the plastic multiplier and S and s are two material- and temperature-dependent parameters. The normality rule also defines the evolution of the state variable r associated with R as /" =
OF --;~--
p(1 - D),
/~ -- ( 2 ~ ) 1 / 2
(14)
Then 2 =/" is identified as the accumulated plastic strain rate ib multiplied by (1 - D ) . Finally, the istropic damage law of evolution is /)=
~ /~
(15)
6.14.3.2 ANISOTROPIC CASE An extension of the isotropic case is keeping the two main properties of the damage evolution driven by the elastic energy and the plastic strain. The dissipative potential is now chosen as F=f +
YO[dr[o
where [.[ stands for the absolute value of the principal values of the plastic
517
6.14 Isotropic and Anisotropic Damage Law of Evolution
strain tensor, so that the damage rate/)ij is always positive.
fgij--J~ ~
- (s)Slde--P J~
(17)
with i - d r / d t - J~ being always positive, the anisotropic damage law of evolution is /30 --
~Pl/j
(18)
I~1
(19)
or
if written in terms of principal values.
6 . 1 4 . 3 . 3 DAMAGE THRESHOLD Experiments show that the damage initiates when a certain value of the accumulated plastic strain pD is reached. This corresponds to the nucleation of micro-defects, and pD is related to the energy stored in the material ~bs. Unfortunately, standard thermodynamics gives a much too large value of Cs, and a correction must be added. Furthermore, the contribution of the kinematic hardening is small in the static case and negligible in the cyclic case:
/0
R(p)z(p)dp
A fo p~' (aeq --
fly)L'(X-m)/m
d/)~-
with
z(oc)-0
(20)
A /m with two material parameters A and m > 1 and Taking z(p)__~p(l-ml writing the equality of ~b~ for the general three-dimensional case and for the one-dimensional tensile case allows us to express PD as a function of the onedimensional threshold epD in pure tension (strain hardening saturated at the ultimate stress au), dp - A(au
mF
where (.} denotes the positive part. Then
PD - -
C'pD
(
~ _ ~ ~m _
- ffy)epD , 1/m
(21)
518
Lemaitre and Desmorat
for the monotonic case, and m
p D - - g'pD
Cru -- Cry o.mqax_}_ O.e~in --O'y" 2
for the periodic case; / S i j - 0 if ~s---~bD or in an equivalent manner if
PK_PD.
6.14.3.4
MESOCRACK INITIATION CRITERION
Because of the definition of the damage variable, a mesocrack initiates in a plane of normal ni when the corresponding damage reaches a critical value Dc. This corresponds to instability of atomic decohesion after localization of damage and strain in this plane. The damage in such a plane is given by the norm of the damage vector Dijnj or by the larger principal value DI. Max(D)1 = Dc ---+mesocrack initiation
6.14.3.5 DIFFERENT EVOLUTION OF DAMAGE IN TENSION AND COMPRESSION In compression there is partial closure of the microcracks or microvoids, and the evolution of the damage is slower than in tension. A way to model such a phenomenon is to introduce a crack closure parameter h, 0 < h < 1, in the effective stress ~. A thermodynamics analysis leads then to the same laws (Eqs. 15 and 18) of damage evolution with a change in the definition of the effective strain energy density Y (isotropy) or Y (anisotropy). 6.14.3.5.1 Isotropic Case We have ~ - a/(1 - D) in tension, and ~ - a/(1 - hD) in compression. The strain energy release rate, taking into account the different behaviors in tension and in compression, is
Y = 1~+ 2E
(or)+: (6)+ + h (rr} O - - D--)2(1
(o-) ~ hD) ~
(1 ~ -q - 2
D)
(1 - hD)2J
(22
)
519
6.14 lsotropic and Anisotropic Damage Law of Evolution
where (.)+ (resp. (.)_) denotes the positive part (resp. negative) of a tensor in terms of principal components and where (.) is the positive part of a scalar. ~2 /2E and to the initial damage law, The case h - 1 corresponds to g - CreqRv which is identical in tension and in compression. For most metals h ~ 0.2. 6.14.3.5.2 Anisotropic Case For the anisotropic damage model the stress deviator Cr~ and the hydrostatic stress Cr/-Iact as independent variables. This feature allows us to introduce the quasi-unilateral conditions in the model by simply taking the positive part of the stress deviator instead of the full stress tensor. The corresponding function to use in the damage law (Eq. 18) is 3(1 - 2v)
_ 1 + V(HCrDH)2 3E
---
+
(CrH)2
2E
(23)
where the "special" positive part cr~ is used, built with the eigenvalues/~i and the corresponding eigenvectors Ti of (/-/_aD), D
-
Eu - 1
Ti][_/--/-1Ti]
(,~i)
(24)
with the normalization T~H_-ITj -- 60. The h parameter is no longer needed to reproduce the same order of magnitude of differences between tensile and compression behaviors given by h ~ 0.2 in the isotropic case.
6.14.4 IDENTIFICATION PARAMETERS
OF THE MATERIAL
This model possesses six specific parameters which have to be identified for each material and which are, in fact, monotonic functions of the temperature. 9 S, s, h stand for the damage evolution; 9 epD, m stand for the damage threshold; the single parameter epD is sufficient to model the damage threshold in monotonic loading; 9 Dc stands for the mesocrack initiation. The Young's modulus E, the Poisson's ratio v, the ultimate stress cru, and the yield stress Cry are not specific to the damage law and are known from elsewhere (see Table 6.14.1) The proposed method is to first obtain rough values of the parameters from standard tests. This set of material parameters can then be used as a starting solution of an optimization procedure to minimize the difference between predictions by the model and available test results of any kind.
520
Lemaitre and Desmorat
TABLE 6.14.1
Table of material parameters
Materials
Temp. E v ~ (MPa)
~y ~ru (MPa) (MPa)
Bpu
s
Dc
h
2.4
1
0.2
0.2
1
7
1
0.15 0.2
0
1
0.2
BpD
m
0.44
1
0.10
S (MPa)
Steel AISI 1010
20
190,000 0.28 320
Stainless steel AISI 316
20
200,000 0.3
600
140,000 0.32
1
0.5
0.5
20
195,000 0.3
190
340
0.26
2
2.8
2
0.2
0.2
300 500 580
177,000 0.3 150,000 0.3 134,000 0.3
145 105 85
235 165 135
0.18 0.12 0.10
2 2 2
2.4 1.4 0.6
2 2 2
0.2 0.2 0.2
0.2 0.2 0.2
Steel SOLDUR 355
20
200,000 0.3
380
475
5.5 0.3
0.2
A1. alloy 2024
20
72,000 0.32 300
500
0.03
1
1.7
1
0.23 0.2
Ceramic A1
20
400,000 0.2
306
500
0.10
1
6
1
0.99 0.2
Concrete
20
30,000 0.2
2.5
3
0
1
2.5 10-7 1
Stainless steel 2 1/4 Cr Mo
700
260 6
0.25 0.025 1.3 0.3
0.2
0.2
(a) A s t a n d a r d t e n s i o n test gives 9 9 9 9
the u l t i m a t e stress ~ru; the yield stress (ry; the u l t i m a t e plastic strain ~pu from w h i c h ~p~) ~ ~pu; for the d a m a g e t h r e s h o l d still, m - i or 2 is a "good" value for a starting solution.
(b) A strain localization theoretical analysis s h o w s that the l o c a l i z a t i o n in t e n s i o n o c c u r s for ~/D~/~ 1. As in t e n s i o n (in d i r e c t i o n 1), we have D2 -- D3 ---- D 1 / 2 (from the d a m a g e e v o l u t i o n law [Eq. 18]) a n d ~ / ~ 3; t h e n
Dc -
3 Dlc
~ -
2r/
~
0.5
(25)
(c) T h e identification of the p a r a m e t e r s S a n d s n e e d s d a m a g e measurements;, it is best to l o o k for elasticity c h a n g e s i n d u c e d by d a m a g e d u r i n g very l o w cycle fatigue tests (this m e a n s a n u m b e r of cycles to failure of 10 to 100). F o r the o n e - d i m e n s i o n a l case of t e n s i o n in d i r e c t i o n 1, the law of elasticity c o u p l e d w i t h d a m a g e s h o w s that the d a m a g e in the d i r e c t i o n s 1, 2, a n d 3 is s i m p l y r e l a t e d to the Young's m o d u l u s E, the d a m a g e d m o d u l u s E1 in
521
6.14 Isotropic and Anisotropic Damage Law of Evolution
direction 1, the Poisson's ratio v, and the damaged contraction ratio ~12 by D1 ~ 1
E1
(26)
E D2 = D3 --~ 1
1+ v E 1 + 3v12 - 2v
(27)
E1
Then D1 and D2 may be evaluated from experiments if the strains gl and e2 are carefully measured during unloadings. This allows us to check the feature D1 = 2D2 in tension and to evaluate ~/~ 3 for most metals from
rlDH -- 1
E1 1 -- 2V E 1 - 2v12
DH -- 3D1
with
(28)
Referring again to the damage law written in uniaxial tension (UT) for which [~111]- / ~ and considering the strain hardening saturated at 8eq "~ tru, we have
d V x _ ~O'uRvU[ dp \ 2ES
or
dD1
o"u
og~--Log
~-~ +sLog Rvvz
[(
2 1-D1
with Rvuz
-
2 5(1 + v) + 3 ( 1 -
2v)
1-
D1
1 1--~
+
/1-'
(29)
Then the graph of Log (dD1/d~11) as a function of Log(Rvvz) allows for the identification of S and s (see Fig. 6.14.1).
I
i
i
i
SOLDUR 355
dD11
log dEp
-
-1 -0.1 FIGURE
-
oo I
I
0
Plane tension
log Rv
I
0.1
6.14.1. Determination of S and s.
I
0.2
522
Lemaitre and Desmorat
If some very low cycle fatigue tests are not available, take s = 1 or 2, consider Rvur ,-~ 1 (for the isotropic damage law we always have Rvur = 1), and obtain S from (7 u2
g,pR - - g,pD
S ~ ~---E
Dc
)1/~
(30)
Having the approximate values of au, ay, gpD, m, Dc, S, and s, it is possible to perform an optimization procedure for better values if additional test results are available. In fact, a sensibility analysis shows that the two main parameters that have to be adjusted are m and S. The optimization procedure consists of two steps. 1. First, we calculate each available experiment by performing the numerical time integration of the set of constitutive equations fully coupled with damage. Concerning the identification of the damage law, it is sufficient to calculate the Damage versus Accumulated plastic strain D(p) curves (damage and plastic strain are both directly measured) and to compare them with the experiments. A finite element computer code may be used, of course, but mathematics softwares are generally more efficient for uniaxial loading: such softwares allow us to define the material parameters as arguments of functions, and they allow us to draw curves in the programming pattern. 2. Next, we minimize the error between computations and experiments. This can be done on the full stress-strain curves up to failure, but again it is advised to do it on the D(p) curves. If complex loading needs to be considered for the identification process, a software such as SIDOLO is needed to drive the finite element computations and to optimize the full set of material parameters.
6.14.5
HOW
TO USE THE MODEL
For the anisotropic damage law, the set of equations to be solved for any case of loading is
D,j -
I pl,j,
? =
e qRv 2E
523
6.14 Isotropic and Anisotropic Damage Law of Evolution
with 2
Rv
~(1 + v ) + 3(1 - 2v)
Gq -- (_H_~D_H)~q, ~n
cr_______En
-1
1
ff H -- -~O'kk, _/-t -
D.
1
-- -~Dkk
(_1 - p)-~/2
The integration of the damage differential equation is performed until Max D1 = Dc, which corresponds to a mesocrack initiation. The work is fully similar for the case of isotropic damage.
6.14.5.1
USE IT AS A POSTPROCESSOR
The history of stress ~rij(t) and the history of plastic strain each point considered. They may come from -
p gij(t) are given at
an elastoplastic structure calculation for ductile failure; an elasto-viscoplastic calculation for creep or creep-fatigue failure; an elastic structure calculation with a local analysis by Neuber's method when small-scale yielding induces low cycle fatigue.
They lead then to the knowledge of the history of Y, at least if the dependency of the triaxiality function upon the damage is neglected. The time integration of the damage evolution law (Eq. 15 or 18) is performed "a posteriori" with an eventual initial damage D~ Dij -- D O -4-
/,0
I~_P(t) ijdt
if
p > PD
(31)
For better accuracy, the dependency of Rv upon damage has to be made explicit. For the uniaxial tensile case, we get [RvuT(D1)]-SdD1
--
- ~G,
depl
(32)
Ductile and creep damage as well as low cycle fatigue may be analyzed by this method. For brittle failure and high cycle fatigue, a two-scale damage model is needed, as explained in Section 6.15.
524 6.14.5.2
Lemaitre and Desmorat U S E IT IN A FULLY C O U P L E D S C H E M E
When the damage is not localized and if a high accuracy is needed, the full set of the elasto-(visco-) plastic constitutive equations coupled with damage needs to be solved as field variables. Such constitutive equations coupled to isotropic or to anisotropic damage need to be implemented in a finite element computer code (for example, as a subroutine UMAT for ABAQUS). For details, see Section 9.2. The price to pay is a very important time of calculation.
BIBLIOGRAPHY 1. Desmorat, R. (2000). Strain localization and unilateral conditions for anisotropic induced damage model, in Symposium on Continuous Damage and Fracture, pp. 71-79, Benallal, A., ed., Elsevier. 2. Desmorat, R. (2000). Mod~lisation et estimation rapide de la plasticit~ et de l'endommagement. Th~se d'Habilitation h Diriger des Recherches, Universit~ Paris 6. 3. Desmorat, R. (in press). Fast determination of localized plasticity and damage by energetic methods. Int. J. Solids Structures, submitted for publication. 4. Lemaitre, J. (1992). A Course on Damage Mechanics, Springer Verlag. 5. Lemaitre, J., Desmorat, R., and Sauzay, M. (2000). Anisotropic damage law of evolution. Eur. J. Mech. A/Solids 19: 187-208. 6. Sermage, J. R, Lemaitre, J., and Desmorat, R. (2000). Multiaxial creep fatigue under anisothermal conditions. Fatigue and Fracture of Engng Mater. & Struct., 23(3): 241-252.
SECTION
6.15
A Two-Scale Model for QuasiBrittle and Fatigue Damage RODRIGUE DESMORAT1 and JEAN LEMAITRE2 1Universit~ Paris 6-LM2S, 8, rue du Capitaine Scott, 75015 Paris, France 2 Universit~ Paris 6, LMT-Cachan, 61 avenue du Pr~.sident Wilson, 94235 Cachan Cedex, France
Contents 6.15.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
525
6.15.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
526
6.15.3 Description Of The Model . . . . . . . . . . . . .
527
6.15.3.1 Three-Dimensional Model for any Kind of Loading . . . . . . . . . .
528
6.15.3.2 Monotonic Failure of Brittle Materials . . . . . . . . . . . . . . . .
529
6.15.3.3 Formulae for Proportional Fatigue Loading . . . . . . . . . . . . . . .
530
6.15.3.4 Formulae for Two-Level Fatigue Loading . . . . . . . . . . . . . . .
532
6.15.3.5 Stored Energy-Based Damage Threshold for Multilevel Loading. 532 6.15.4 Identification of the Material Parameters 533 6.15.5 How to Use the Model . . . . . . . . . . . . . . . . .
534
6.15.6 Tables of Material Parameters . . . . . . . . . .
535
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
535
6.15.1 VALIDITY This model describes the progressive deterioration of solid materials up to a mesocrack initiation. It can be used to predict the state of damage and the conditions of crack initiation in mechanical components subjected to mechanical and thermal loadings. It applies to the quasi-brittle type of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
525
526
Desmorat and Lemaitre
damage as brittle failure and high cycle fatigue. In both cases the damage is always very localized at a microscale in the vicinity of a defect considered as a weak inclusion. Two scales are considered (in addition to the macroscopic scale of structures): the mesoscale or scale of the representative volume element (RVE) of continuum mechanics, and the microscale or scale of a microdefect (microvoids, microcracks) embedded in the RVE. The model is written for isotropy, but anisotropic damage (Section 6.14) may be easily incorporated. An initial elastic structure calculation (analytical or by finite element [FE] analysis) is needed to define the stress and strain fields at the mesoscale. The time histories of the stresses and strains at the most loaded point(s) of the structure are then the inputs of the model. The main output is the evolution of the damage as a function of loading, time, or of the number of cycles. The present analysis may also model failure with (visco-)plasticity at mesoscale. In that case the mesostresses and mesostrains may be the result of a mechanical analysis of the structure by means of an FE analysis or of any analytical or semianalytical procedure like the Neuber method following an elastic calculation.
6.15.2
BACKGROUND
The points that need to be recalled are as follows: 1. The description of elastoplasticity coupled or not to damage (see Sections 4.2 and 9.2). 2. The isotropic damage law of evolution described in Section 6.14. 3. The definition of two scales to model a weak inclusion embedded in the RVE. The fields in the inclusion (stress tensor a~, total strain and plastic strain tensors ~ , f, ~P ij , damage D ~ . .) have a #-superscript meaning microscale when the fields at the mesoscale of the RVE are simply denoted crij, ~ij, ~. Note that we consider damage at microscale only, and we will omit the #-superscript in that case by setting D ~ = D. The scale transition is made by using the Eshelby-Kr6ner localization law: the total and plastic strains at microscale are related to the strains at the mesolevel as F,~ -- F,ij -~ j~(F,~ p -- 4)
(1)
where fl is given by Eshelby's analysis of a spherical inclusion, =
2 4-5v 15 1 - v
(2)
6.15
527
A Two-Scale Model for Quasi-Brittle and Fatigue Damage
Equation 1 may also be written in term of stresses or effective stresses: ff~J -- 1 - D
:
O'ij
-
2G(1 - fl)(e~ p - ePij)
(3)
where v is the Poisson's ratio, G the shear modulus, and E the Young's modulus. 4. The definition of a proportional loading. A proportional loading at mesoscale corresponds to a stress field aij(M, t) of constant principal directions proportional at the considered point M to a given tensor Tij(M),
Teq -- V/3 TijDTijD -- 1
6i) -- a(t)Tij,
(4)
Tension-compression is of course a proportional loading.
6.15.3
DESCRIPTION
OF THE MODEL
Brittle and high cycle fatigue (HCF) damages are always very localized at a scale much smaller than the plastic strain. This is the reason to consider a two-scale model in which the damage occurs in a weak microinclusion embedded in a meso-RVE that is elastic (or elasto-[visco-]plastic) and free of damage. This hypothesis allows us to obtain the mesostress and strain by a classical structure calculation that is elastic (or elasto-[visco-]plastic) with no damage at mesoscale and to solve the constitutive equations of elasto-[visco-]plasticity coupled with damage as a postprocessor at microscale.
STRUCTURE CALCULATION
Elastic
t~i~(t) ei~(t) { E v
Plastic
~ ij (t) e ij (t) e ij (t)
Damage
p~
el.t
RiE (Yf
D
micro FIGURE 6.15.1
Elastic
Self-consistent scheme Microelement embedded in an elastic RVE.
MESO
E v
528
Desmorat and Lemaitre
The mechanical behavior of the material is modeled in two different manners related to the two different scales. At the mesoscale the material is considered elastic because brittle or HCF failures occur at states of stresses below or close to the yield stress. - At the microscale the behavior is modeled by elastoplasticity coupled with damage. The weakness of the inclusion is related to its yield stress a; taken equal to the true fatigue limit af of the material, below which we consider that no damage occurs. The elastoplastic properties of the inclusion are those of the material at mesoscale below and above the yield stress. To simplify, only linear kinematic hardening is assumed at microscale, but with the same plastic modulus C than at mesoscale. Since damage is considered at microscale, the isotropic damage evolution law of Section 6.14 will in fact be written in terms of microstress, the microenergy density release rate, and microaccumulated plastic strain, i.e., 13 = (Y~/S)Sp~. -
6.15.3.1 THREE-DIMENSIONALMODEL FOR ANY KIND OF LOADING The history of the plastic strains 4 ( 0 and of the stresses aij(t) at the mesolevel being inputs, the fields at microlevel are obtained from the time integration of the constitutive equations at microscale coupled together with the law of localization (Eq. 1) even for fatigue cyclic loading. The equations to be solved are those of elastoplasticity coupled with damage with linear kinematic hardening (yield function f ~ - (#9in which X~ is the back stress):
X_#)eq-af
~ = aj + fl(a~P - ~ ) r
=
1 + v.
(or Eq. 3)
v .~
c?~= 1 - D
2 (_~" - _X")e,/1 - D
X~" -- ~ 2 C ~ 7 ( 1 - D )
(5)
6.15
529
A Two-Scale Model for Quasi-Brittle and Fatigue Damage
/3--
p~
if
Cs>r
or
P~>pD
m
PD
O- u u
--
8pD
O'y
.ry~e~a x - + a~d, _
for cyclic loading
2 The strain energy release rate Y', taking into account the different behaviors in tension and in compression, is defined as Y~
l+v[(_a~)+:(_a~/+ =
2---Y
il--D)
vI(Trcr~)2
(rr~) .(rr~) ] +h
(1 -- hD)
(-Trcr/~)2]
-.
(6)
where (.) + (resp. (.)_) denotes the positive part (resp. negative) of a tensor in terms of principal components and where (.) is the positive part of a scalar. By the introduction of the crack closure parameter h (for most of metals h ~ 0.2), the consideration of the damage law 1 3 - (Y~/S)Sp ~' takes into account the partial closure of the microcracks or microvoids in the evolution of the damage. The case h - 1 corresponds to Y~ - 6~2R~'/2E (with the notations of eq - - v i Section 6.14) and to a damage law that is identical in tension and in compression. Finally, 2 = p ~ ( 1 - D) is the plastic multiplier calculated from the consistency condition f~ = 0 and f ~ - 0 in plasticity, and from Norton's law JL -- (f~/K) N in viscoplasticity. C is the kinematic hardening parameter, and K, N are Norton's parameters. When the damage D reaches the critical value Dc, there is initiation of a mesocrack.
6 . 1 5 . 3 . 2 MONOTONIC FAILURE OF BRITTLE MATERIALS For proportional monotonic loading at mesoscale aij = a(t)Tij, Teq = 1, it is possible to perform an analytical integration of the damage law up to D = De. The hypothesis of no plasticity at mesoscale is made, and the formulae obtained here apply to quasti-brittle materials. A damage threshold epD -- 0 is considered (take also m = 0).
530
Desmorat and Lemaitre
There is microplasticity for mesostresses larger than the fatigue limit af. A scalar e~ is defined to quantify it under the proportional loading assumption, when
Cr~f 8~P---~8~Tij 3 D
Ct) 6-" -- C/~ ~ 3G(1 -
and
fl)
(7)
In general, we have C/G ~ 1" this shows that for a _> r the von Mises stress at microscale remains quasi-equal to the limit fatigue (d"~ ~ 0 in the previous equation and then a~ ~ af). The time integration of the damage law (coupled with Eq. 7) allows us to calculate the damage D at a given mesostress a, 3G(1 - fl)(2ES) s
RS"*
fff.,I
by introducing 2 1 (1 - 2v)~ 2 Rv, ( ~ ) - - 7 ( 1 + v ) + 7
(9)
The function Rv, (~) varies significantly over the integration intervals, and an approximate formula for the stress to failure may be derived only for small values of s not far from 1"
err ,-~ ~f +
30(1 - fi)(2ES)SDc _2sDs (Tkk) f /XV*
(10)
For compression-like loading (a < 0), considering the parameter h within the comp damage law allows us to calculate the stress at failure in compression crR much larger than in tension and to show that 0"~ nsi~ -- tTf '~ comp
hS
(11)
oR I-os
6.15.3.3
FORMULAE FOR PROPORTIONAL
FATIGUE LOADING For proportional cyclic loading at mesoscale aij = a(t)Tij, Teq = 1, of constant stress amplitude Aaij = AaTi) between O'minand Crmax - - O'min+ A o ' , it is possible to obtain accurate formulae for the number of cycles to crack initiation. For simplicity we assume here that h = 1 and we limit the analysis to the case of elastic fatigue (Aa ~_ 2af).
6.15 A Two-Scale Model for Quasi-Brittle and Fatigue Damage
531
The model will predict failure only if the stress amplitude at mesoscale is larger than twice the fatigue limit (i.e., if Ao- > 2o-f). The plastic strain increment over one cycle fcyde p~dt is then
ap,
2(a~- 2~f)
6--N = 3G(1 - fl)
(12)
Damage will initiate after a number of cycles No when the damage threshold
pD is reached: No=PD @ , aN
[6G(1-fl) pD -- g,pD
Ou-Cry ]m
C
(13)
mo- - ~ f
The damage increment over one cycle will then be
aN =
3ES
3G(1 - fl)
if
Tkk -- TrT - 0
if
Tkk -- TFT yk 0
o-~s+lq)* (o-minTkk' o-maxTkk)
cSD _
\ o-f
aN
o-f
Tkk3G(1 -- fl)(2ES) s
(14)
(15)
where the adimensional function ~. is introduced, cp* (x, y) ~
RS~.({)d{ + 2
/x
R~. ({)d{
(16)
with Rv, defined by Eq. 9. The function qg* (x, y) can be easily calculated by using mathematics softwares. A simple conservative approximate formula may be derived only for small values of s, q)* (x, y) ~ (y - x - 2)[RSv.(x) + RSv.(y)]
(17)
The formulae for the number of cycles at crack initiation are finally
N
-N0+
3es
e
(1 -/3)D~
2(Ao-- 2o-f)
Tkk3G(1 -- fi)(2ES)~D~ -
NR
No -q- o-ffs+l(49* (o-min k,, o-f Tkk, o-max o-f Tkk)
if
Tkk -- 0
if
Tkk r 0
(18)
(19)
They exhibit 9 a mean stress effect in tension identical to the mean stress effect in compression (due to h = 1); a mean stress effect different in tension and in compression is obtained with h = 0.2; 9 no mean stress effect in shear as experimentally observed.
532
Desmorat and Lemaitre
FORMULAE FOR Two-LEVEL FATIGUE LOADING 6.15.3.4
We consider here a two-level periodic loading in the fatigue regime of stress amplitude At)"1 ~ O'maxl--O'minl and A~2 and of mean stress ~1 -- (O'maxl -Jr- O'minl)/2 and F2. N -- N1 cycles of level i = 1 are followed by N2 cycles of level i = 2 up to failure for a number of cycles N R - N1 q-N2. Each level (denoted i) applied alone leads to a number of cycles to failure NRi after a number of cycles up to damage initiation Noi. Two cases may be defined: (a) N01 < N1; damage initiation occurs during the first level. The number of cycles at failure NR -- N1 -Jr-N2 (N1 known) is determined by:
N,
+
NR1 - - No1
NR -- N1 _ NR2 --
No2
NR1
(20)
NR1 -- No1
(b) N01 ) N 1 , damage initiation occurs during the second level. The number of cycles to failure is given by
N1 } NR -- N1 = . ~NR2 No1 No2 No2
(21)
Because of the existence of a nonzero damage threshold (No/~ 0), the previous relationships lead to a rule of damage accumulation different from the classical Miner's rule N1/NR1 + N 2 / N R 2 1. In the diagram N1/NR1 versus N2/NR2, the rule obtained is bilinear with an angular point P of coordinates ([NR -- NOl]/NR2, Nol/NR1). It represents well the nonlinear accumulation of damage observed experimentally.
6.15.3.5
STORED ENERGY-BASED DAMAGE THRESHOLD
FOR MULTILEVEL LOADING Each level is periodic between Omi n-(i) and a~)ax at mesoscale, corresponding to von Mises stresses a ~'(i) eqmin and a l'(i) eqmax at microscale (stress amplitude A o . i _ O.(m/)ax_ ~ larger than 2crf). Damage will initiate after n levels when the stored energy density reaches a loading-independent threshold or in an equivalent manner when the accumulated plastic strain p~ reaches the
6.15
533
A Two-Scale Model for Quasi-Brittle and Fatigue Damage
loading-dependent threshold PD given by: / Cry(") Cqmin + ~(") cq. 2
. . . --
/
iIm
[eD
11 m ]
-(
P n"- 1 )
q-
(22) _qt_ k=l
eqmin -~2
eqmax
1/m -- O'f
[(p~)
1 -- ( P ~ - I )
x 1/m
/m] __ (0. u _ O'Y)g'P D
where p~ is the value of the accumulated plastic strain reached at the end of level (k).
6.15.4 I D E N T I F I C A T I O N OF THE MATERIAL PARAMETERS For each material at each temperature considered, the numerical values of two sets of parameters must be identified. 9 Parameters at mesoscale. The Young's modulus E, the Poisson's ratio v, the
yield stress try, the linear kinematic hardening parameter C, and the ultimate stress rru are identified from a classical tensile test at mesoscale. The damage threshold ~po may also be derived from such a test (see Section 6.14) or may be identified from two-level fatigue tests at the same time as the exponent m (next paragraph). The fatigue limit rrf needs some fatigue tests in order to obtain the quasi-asymptotic part of the W6hler curve. ~ Parameters at microscale. The damage strength S, the damage exponent s, the damage threshold ~pm, the exponent m, and the critical damage De can be determined only by a nondirect identification, since mechanical tests at microscale are not possible to perform. The W6hler curve and some two-level low cycle fatigue tests are the experimental data needed to identify the five parameters S, s, ~pm,m, and Dc by an optimization procedure, either by "manual groping" or numerically with a "good" starting solution deduced from the number of cycles to failure for periodic tension-compression tests (Eq. 19). The explicit formulae given for two-level fatigue tests may help in the identification of the damage threshold parameters (if such tests are available, of course). The angular point P already defined corresponds to the case N01 = N1 for which the damage initiates at the time of level change. This is a way to m e a s u r e N01 and to consequently adjust the values of ~po and m of Eq. 13.
534
Desmorat and Lemaitre
For brittle materials with no fatigue tests available, one can consider epD = 0, m = 0 (there is then no need of C), O'f of the order of cru/2, s = 1, and D~ = 0.2, and use Eqs. 8-10 and 11 to identify S and h. In general, the set of material parameters concerning the damage law (at microscale here) is identical to the set at mesoscale obtained for the singlescale damage model of Section 6.14, the parameter which needs to be carefully identified in fatigue being then the exponent m.
6.15.5
HOW TO USE THE MODEL
Failure of brittle materials and high cycle fatigue (HCF) of ductile materials are both calculated with the two-scale model (with different material parameters) with the same computer code, the postprocessor DAMAGE2000, which is the numerical implementation of the three-dimensional constitutive equations. The code DAMAGE2000 is able to handle any kind of loading, proportional or not, monotonic or fatigue, and fatigue periodic by block or random fatigue. To integrate the set of equations for a given history or(t), ~(t) of mesostress and strain, a strain-driven algorithm is used in the classical iterative Newton's method with the elastic predictor-plastic corrector procedure. A "jump in cycle" procedure based on stepwise linearized damage saves much computer time in the case of HCE The scheme for the numerical integration of the constitutive equations as well as the Fortran subroutine of the initial version of the two-scale model may be found in Reference [7]. For brittle materials, the simple formulae of Eqs. 8-11 may be used for failure under tensile, compression, or any other proportional loading. For HCF applications, the simple formulae of Eqs. 19 and 20-21 may be used for single- and two-level fatigue loading. For multilevel experiments the accumulation rule of the damage is (N1 - N01) N
+ ~
Nh ~-~
-- Dr
(23)
k=2
where for simplicity damage is assumed to initiate on the first level after a number of cycles N01 and where Nh cycles of periodic loading at level (k) are made. The total number of cycles NR is then given by Eq. 23 with NR - - N1 q- N2 q - . . . q- Nn, each number of cycles N 1 , . . . , N n - 1 being known. It has been confirmed that this model contains the following properties: effect of stress or strain amplitude, effect of mean stress in tension, no effect in shear, effect of compression (with h < 1), nonlinear accumulation (due to the existence of the damage threshold), effect of an initial hardening or initial
6.15
535
A Two-Scale Model for Quasi-Brittle and Fatigue Damage
damage, no effect of an overload, fatigue limit in two or three dimensions, bi- or three-axial fatigue, effect of nonproportional loading, and fatigue under any history or random loading.
6.15.6
TABLES OF M A T E R I A L P A R A M E T E R S
TABLE 6.15.1
Material
HCF of Steels (E = 200,000 Mpa, v = 0.3). ay (Mpa)
Cru (Mpa)
Stainless steel SOLDUR355
400 400
600 475
TABLE 6.15.2
Brittle Material.
Material
E(Mpa)
v
Concrete
30,000
0.2
~rf (Mpa)
C (Mpa)
epD
m
200 180
2000 500
0.005 0.025
1 1.3
cru(Mpa) 3
S(Mpa) 0.5 0.3
s
Dc
2 1 5.5 0.3
h 0.2 0.2
crf(Mpa)
epD
S(Mpa)
s
Dc
h
1.5
0
2.3 • 10 -8
1
0.2
0.05
REFERENCES 1 Desmorat, R. (2000). Mod~lisation et estimation rapide de la plasticit~ et de l'endommagement. Th~se d'Habilitation fi Diriger des Recherches, Universit~ Paris 6. 2 Desmorat, R. Fast determination of localized plasticity and damage by energetic methods. Int. J. Solids Structures (submitted for publication). 3 Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. London, A241: 376. 4 Kr6ner, E. (1961). On the plastic deformation of polycrystals. Acta Metall. 9: 155-161. 5 Lemaitre, J. (1992). A Course on Damage Mechanics, Springer Verlag. 6 Lemaitre, J., Desmorat, R., and Sauzay, M. (2000). Anisotropic damage law of evolution. Eur. J. Mech. A/Solids 19: 187-208. 7 Lemaitre, J., and Doghri, I. (1994). Damage 90: A post-processor for crack initiation. Comput. Methods Appl. Mech. Engrg. 115: 197-232. 8. Sauzay, M., and Lemaitre, J. (2000). Post-processeur de m~canique de l'endommagement. Notice 10, LMT-Cachan. 9. Sermage, J. P., Lemaitre, J., and Desmorat, R. (1999). A two scale damage concept applied to fatigue. Int. J. Fract. 97: 67-81.
This Page Intentionally Left Blank
CHAPTER
7
Cracking and Fracture
This Page Intentionally Left Blank
SECTION
7.1
Introduction to Cracking and Fracture JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
Fracture mechanics deals with existing cracks. But from a practical point of view, what is an initial crack? It can be a known defect created by a mesofracture event during a forming process, for example, which has the sharp shape of a crack. It can also be a known defect with a smooth shape which does not represent a crack in the sense of fracture mechanics. The fracture analysis of such a defect needs first a "crack" initiation procedure to be applied to initiate a sharp shape, and then the fracture mechanics procedure is followed. It can also be the final stage of damage giving rise to a mesocrack of the size of the representative volume element (RVE): an abstract concept. A way to match damage mechanics and fracture mechanics is to consider that the damage mesocrack of size a0 has consumed an energy which can be calculated in two ways: 3
Dc
9 a volume damage energy: a o fo YdD, Y being the associated variable to the damage variable D; 9 with a surface fracture mechanics energy: fjoa~ Gd(a2), G being the strain energy release rate, the associated variable to the crack surface. A development of this concept may be found in Section 7.2, but an approximation of microcracks or crack growth in mode I at constant stress allows us to write 2E Gc ao = cr~ Dc where Gc is the toughness of the material and Y the damage energy density release rate, (see Section 6.2) 6 e2qRv y
z
O"u2
2E(1 - D) Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
539
Lemaitre
540
with rru the ultimate rupture stress, E the Young's modulus, and Dc the critical damage at mesocrack initiation (De ~ 0.5 for many materials). Fracture mechanics relates the length or the surface and the shape of the cracks to the intensity of the loading related to the crack. 9 From the thermodynamic point of view, the load is expressed by the energy release rate G, the conjugated variable to the crack surface A which can be calculated from the overall potential energy W of a structure calculation: 0W G=~ OA 9 From the point of view of conservation of energy, loading is described in terms of contour integrals around the crack tip, such as the Rice integral:
J
=
Wen1
-
ffijnj OX1f ds
where we is the elastic energy density, ff is the normal to the contour C, nl being its component along the linear crack in a bidimensional medium, and ff is the displacement in the structure. 9 In a local analysis, crack loading is in terms of the stress intensity factors characterizing the singularity of the elastic stress field at the crack tip. For example, in a pure mode I of opening of the crack KI -- lim
r---,0
(0"222V/~)
r being the polar coordinate from the crack tip r = 0 and rr22 being the normal stress in the direction normal to the crack. This background is described in Section 7.3. Nevetheless, the probabilistic approach of Weibull directly relates the probability of fracture to the applied loads (see Section 7.4). Fracture mechanics considers different mechanisms of failure, depending upon the materials and the loadings.
9 Brittle failure, in which no or almost no plasticity avoids an instability phenomenon of sudden fracture (see Section 7.5, and Section 7.6, in which the friction between crack lips is taken into account). Delamination of coatings is described in Section 7.7. 9 Ductile or creep failures when plasticity or viscoplasticity induce stable crack growth (see Sections 7.8 and 7.9). 9 Fatigue failure for cyclic loading (see Section 7.10) where the crack grows at each cycle with the phenomena of crack opening and
7.1 Introduction to Cracking and Fracture
541
crack closure as explained in Section 7.11. It can also be related to damage concepts (Sections 7.11 and 7.12). It induces specific phenomena when dynamic effects occur (as discussed in Section 7.13). Fracture mechanics is finally an engineering tool for designing and controlling the structures in service for safety purposes (see Section 7.14).
SECTION
7.2
Bridges between Damage and Fracture Mechanics JACKYMAZARS1
and GILLES PIJAUDIER-CABOT2
1L3S-Institut National Polytechnique de Grenoble, 38041 Grenoble Cedex 9, France 2Laboratoire de G~nie Civil de Nantes Saint-Nazaire, Ecole Centrale de Nantes, BP 92101, 44321 Nantes Cedex 03, France
Contents 7.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 How to Use These Bridges . . . . . . . . . . . . . . . . . 7.2.4.1 Behavior of a Structure Using a Combined Approach of Damage and Fracture Mechanics . . . . . . . . . . . . . . . . . 7.2.4.2 Residual Strength of Initially Cracked Structures . . . . . . . . . . . . . . . . . . 7.2.4.3 Identification of the Internal Length.. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
542 543 544 545
545 547 548 548
7.2.1 VALIDITY The p u r p o s e of this section is to provide a view on the possible c o n n e c t i o n s b e t w e e n damage and f r a c t u r e m e c h a n i c s in the particular case of quasi-brittle materials (concrete, rocks, ceramics) for w h i c h linear approaches are realistic. The objective of this exercise is to offer the possibility to pass from one theory to the other d u r i n g a same calculation or to obtain, from one theory, i n f o r m a t i o n on h o w to use the other.
542
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
543
7.2 Bridges between Damage and Fracture Mechanics
7.2.2 BACKGROUND A unified way to present damage and fracture mechanics is through thermodynamics. It deals with energetic considerations, from which it is easy to relate local damage variables and global fracture variables. These considerations start with the assumption of a specific form of the free (reversible) energy stored in the material during straining. Let us emphasise that this section deals with the simplest possible forms of such energy. The state equations are deduced from the free energy defined as T = U-
TS
(1)
(U, internal energy; T, temperature; S, entropy). For the elementary volume at a given state of damage D, the free energy density is 1 D
u - ~ Aijkl~ijekt
(2)
For the overall body, damaged or partially cracked, the total free energy is written as 1
U - ~ Kq~
(3)
AijDkz is the local stiffness matrix at a given stage of damage, and eij is the local strain component. A load denoted as Q is applied to the structure, q is the corresponding displacement, and K is the global stiffness. Assuming linear elasticity and isotropic damage, the relationship between AijDkl and the initial stiffness of the undamaged material is AijDk,- Aijk,(1 -- D)
(4)
AijkZ is the stiffness matrix for the virgin material, with constant components depending upon the Young's modulus and the Poisson's ratio for an isotropic material which is linear elastic. At uniform and constant temperature the state laws provide the stress-strain relations and the definition of the energy release rates.
9 For the damaged material, Y is the damage energy release rate: OT ffij ~-- O~ij
--
Aijkl(1 -- D)~kl,
cOT 1 Y = OD -- - -2 Aijkl~'ij~kl
(5)
9 For the cracked structure (A is the actual area of the crack), G is the fracture energy release rate: cg~k cgT 1 2 OK Q - --~q - K q , G - - - ~ -- -~ q c3A
(6)
544
Mazars and Pijaudier-Cabot
The first and second principles of thermodynamics are completely satisfied if the Clausius-Duhem inequality is also verified. For the two considered cases, we obtain -Y/)_0, -GA_>0 (7) Since (-Y) is a quadratic function and K decreases when A increases (see Eq. 6), these equations imply that / 5 _ 0 and A___ 0, showing that irreversibilities correspond to microor macrocracking propagation.
7.2.3 EQUIVALENCES Considering the similarity of the two approaches, it seems natural to go from one concept to the other [5]. One possible method is to transform a given damage zone into an equivalent crack. This equivalence is thermodynamically acceptable if the consumption of energy is the same during the two processes. Considering the case of LEFM, the critical condition of crack propagation is - G = Gc (Gc is the critical energy release rate). Then, the equivalent progression dA of a crack equivalent to a state of damage in the same structure is the solution of a aA -
- r aD(x)
(8)
Conversely, it is possible to derive the fracture energy Gc from the distribution of damage around a macrocrack which propagated in the considered structure (Fig. 7.2.1). For this, one needs to know the distribution of damage around the macrocrack, which is approximated as follows. Consider an infinite body subjected to uniaxial tension in direction 1, cr~ with cr~ - 0 for i r 1 and j 7~ 1. We assume at this stage a distribution of damage denoted as D O and the corresponding strain field denoted as s ~ When small deviations from this equilibrium state are analyzed, the displacement field is the solution of the partial differential equations div((rij)- O. The evolution law of damage is nonlocal. It is generically denoted as D = f(~) where ? is the nonlocal strain defined in Chapter 6 of this volume [6]. The equilibrium equations have a nontrivial (e.g., nonhomogeneous) harmonic solution with a wavelength which is not indeterminate. In fact, the wavelength 2rt/co is proportional to the internal length of the nonlocal continuum (for more details, see Reference [2]). The wavelength is also a function of the evolution law of damage. The calculation of the approximated fracture energy performed with the smallest value of the wavelength calculated for a uniaxial tensile test and corresponds to mode I crack opening.
545
7.2 Bridges between Damage and Fracture Mechanics
FIGURE 7.2.1 fracture energy.
Geometry of the localization band assumed for the calculation of the
With the minimum wavelength, the distribution of damage perpendicularly to the crack direction is (the coordinate perpendicular to the crack path is denoted as y): D(y) - f'+~ o~(s)I/(y 4- s)ds r/(z) f7 with
--
COS((-OmaxZ)ifZ E
2 ~ m a ' 2(O--max
x r/(z) - 0 elsewhere
(9) The energy consumption due to crack propagation is the integral of the energy dissipation at each material point in the fracture process zone, which encountered damage:
r oo
JO
2
1~1-11
dy
(10)
1
7.2.4 HOW TO USE THESE BRIDGES 7.2.4.1
BEHAVIOROF A STRUCTURE USING A COMBINED APPROACHOF DAMAGE AND FRACTURE MECHANICS The structure considered is a compact tension specimen (Fig. 7.2.2a) tested by Mazars [3,4]. During the tests many observations and measures were made
546
Mazars and Pijaudier-Cabot
r)Ocm
thickness= 10
?s
(a)
KlkN/m| lOS !
.I
(b)
t \.---..c
e
"i j
o
1,.1
13 J
20
a
Q
~n
-
~
ffz
a~)
kN
20 l-e,
(c) 0 --
X
C
---test. --e-+-- caLc.
0
Opening I
I
o.~
o.z
t"1
t .~
BIT1
FIGURE 7.2.2 Compact tension specimen, a. Geometry. b. Evolution of the stiffness with the crack, (t) theoretical, (e) experimental, c. Global behavior. The calculation is performed using the damage-fracture combined approach.
7.2 Bridges between Damage and Fracture Mechanics
547
showing that the global behavior includes three stages: 9 OA, linear elastic; 9 AB, damage with microcracks, but without macrocrack; 9 BC, combination of both microcracking and macrocrack. To simulate this behavior, we propose two kinds of calculation: 9 from O to B with a nonlocal damage model; 9 from B to C with linear elastic fracture mechanic. The bridge from the first calculation to the other directly uses the equivalent crack concept previously presented; this necessitates predetermining the evolution K = K(A), A being the actual equivalent area of the crack equal to a.t (a and t, respectively, being length of the crack and thickness of the plate). See Figure 7.2.2b. The following parameters have been used: 9 nonlocal damage calculation: E = 34,500MPa, to0 = 1.23E - 04, At = 0.8, Bt = 20,000, lc = 3 0 m m (K0 being initial damage threshold, and At, Bt, and lc being non local damage parameters, see Reference [6]; 9 critical fracture energy at point B: QB = 18.9 kN, qB = 0 . 2 E - 03 m, KB -- 9.5E + 04 kN/m; (-dK/dA)B - 51E + 05 k N / m 3 Gc - 1/2q 2 (-dK/dA)B = 102 N / m 9 LEFM calculation: from Eq. 6 and as - G = G~ at propagation, one can deduce q = ( 2 G ~ ) / ( - d K / d A ) , from which comes Q = Kq. From Figure 7.2.2b it can be pointed out: 9 that the equivalent crack length at point B is a = 13 cm; 9 that the experimental curve which gives the evolution of the ratio Q/q (#K) versus the crack length measured directly on the surface of the specimen is very different from the theoretical one. It confirms that the real crack appears close to the maximum load and that the evolution on the surface is faster than inside the specimen. We may notice that the global behavior deduced from the G~ value is close to the experimental one and the value determined from the analytical calculation w is Gf = 115 N / m , which is close to the 102 N/m obtained for Go.
7.2.4.2
RESIDUAL STRENGTH OF INITIALLY
CRACKED STRUCTURES Given a crack observed in a structure, it is possible to transform the crack into an equivalent damage zone. Equation 9 provides the distribution of damage
548
Mazars and Pijaudier-Cabot
except at the crack tip. The radial distribution of damage at the crack tip is assumed to follow the same mathematical expression (in which the distance r to the crack tip replaces the coordinate y. This distribution of initial damage can be projected on a finite element mesh, and the response of the initially cracked structure can be computed. Bod~ et al. [2] tested such a procedure and found that it provided reasonable accuracy (10 to 30% error on blind predictions).
7.2.4.3
IDENTIFICATION OF THE INTERNAL LENGTH
Size effect tests directly provide the fracture energy of the material (see Reference [1]). Hence Eq. 10 yields a relationship between the evolution law of damage and the wavelength of the distribution of damage, or the internal length of the nonlocal model equivalently. Assuming that the evolution law of damage D = f(~) is completely known, the single unknown in Eq. 10 becomes the internal length. In practice, the evolution of damage is not necessarily entirely known. The model parameters in the nonlocal constitutive relations are obtained by fitting the computations of the size effect tests with the experiments. Equation 10, with the knowledge of the fracture energy Gc, is an additional piece of information which is helpful for reaching a good fit of the size effects tests more easily.
REFERENCES 1. Bazant, Z. P., and Planas, J. (1998). Fracture and Size Effect in Concrete and Other Quasi-Brittle Materials, CRC Press. 2. Bod~, L., Tailhan, J. L., PijaudieroCabot, G., La Borderie, C., and Clement, J. L. (1997). Failure analysis of initially cracked concrete structures. J. Engrg. Mechanics, ASCE 123: 1153-1160. 3. Mazars, J. (1984). Application de la m~canique de l'endommagement au comportement non lin~aire et h la rupture du b~ton de structure. Th~se de Doctorat ~s Sciences, Universit~ Paris 6, France. 4. Mazars, J. (1986). A description of micro- and macroscale damage of concrete structures, Engineering Fracture Mechanics 25: 729-737. 5. Mazars, J., and Pijaudier-Cabot, G. (1996). From damage to fracture mechanics and conversely: A combined approach. Int. J. Solids Struct. 33: 3327-3342. 6. Pijaudier-Cabot, G., and Mazars, J. (2000). Damage models for concrete. Chapter 6, this volume.
SECTION
7.3
Background on Fracture Mechanics Htyg DUONG BuI 1'2, J-B.
LEBLOND 3, N . STALIN-MULLER 1
1Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France 2 Electricit~ de France, R&D, Clamart, France 3 Laboratoire de Mod~lisation en M~canique, Universit~ de Pierre et Marie Curie, 8 rue du Capitaine Scott, 75015 Paris, France
Contents 7.3.1
Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
549
7.3.2
S t r e s s or Velocity F i e l d s Singularities? . . . .
550
7.3.3
Plane Strain, A n t i p l a n e Strain, a n d Plane S t r e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
551
Dynamic Stress Intensity Factors ..........
551
7.3.5
Small-Scale Yielding A s s u m p t i o n . . . . . . . . . .
552
7.3.6
Plasticity a n d D a m a g e . . . . . . . . . . . . . . . . . . . .
552
7.3.7
Viscoplastic Soliton .......................
553
7.3.8
Energy considerations ....................
553
7.3.9
Ductile Fracture ..........................
554
7.3.4
7.3.10 Fatigue Cracking .........................
555
7.3.11 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
556
References .......................................
556
7.3.1 V A L I D I T Y Linear elastic fracture mechanics (LEFM) is based on the analysis of cracks in linear elastic materials. It provides a tool for solving most practical problems in engineering mechanics, such as safety and life expectancy estimation of cracked structures and components. The main success of the theory is based precisely upon linearity, which makes it possible to combine very simply Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
549
550
Bui et al.
the theoretical, numerical, and experimental analyses of fracture. Today, stress analyses of the complex geometry of structures as well as of test specimens are provided by powerful computers using finite element methods. The computed stress-intensity factor KI in mode I, which governs the strength of the singular field near the crack tip, depends linearly on the applied load. Its critical value, related to the toughness Kic of the material, is obtained by measuring the critical load Fc at the onset of unstable crack propagation. Strictly speaking, this scheme is an ideal one and can be used only under restricted conditions involving the geometry of the specimen, the flow stress level, etc. For example, there exist empirical conditions on the crack length a which must be greater than b, the specimen thickness, and the flow stress a>>b>>2.5(Kic/aO) 2. These empirical conditions result from test data and correspond to the smallness of the process zone size compared to a length scale. It also expresses the condition of smallness of plastic deformation when compared to elastic strain. This is the small-scale yielding assumption.
7.3.2 STRESS OR VELOCITY FIELDS SINGULARITIES? Without experiments, it is not easy to know to what extent a theory may be "good" enough for practical use. Nor is it simple to do "rigorously" experimental work to validate the theory. The rigor consists of carefully checking the actual conditions of the experiments. For example, the toughness of material is determined by testing specimens subjected to monotonous increasing loads only when some onset of crack propagation is observed. In an elastic medium with a propagating crack with the velocity V, in plane strain or in antiplane strain loading, it is well known that the singular stress fields near the crack tip (r --. 0) are governed by the stress-intensity factors in modes j = I, II, III: Kj - l i m ( r ~ O)a2i(j)(2nr) 1/2
i(I) - 2, i(II) - 1, i(III) - 3
(1)
The velocity field dui/dt is also singular at the crack tip and yields an alternative mean for characterizing the strength of the singularity. As a matter of fact, near the crack tip, the velocity is related to the displacement by dui/dt ~ -VOui/Oxl, or to the crack opening displacement (COD) (I)i, because in the vicinity of the crack tip one has ~i = [ui(xl - Vt, x2)], where [. ] denotes the jump across the crack surface. Hence there exist three crack opening displacement intensity factors, in the form K}u) - l i m ( r --~ 0)?
~i(j)(27r,/r) 1/2,
i(I) -- 2, i(II) -- 1, i(III) -- 3
(2)
551
7.3 Background on Fracture Mechanics
7.3.3
PLANE STRAIN, ANTIPLANE
STRAIN,
AND PLANE STRESS Equation 2 is valid for plane strain modes I and II, with 32= #/(k + 1), and k = 3 - 4 v , /2 being the shear modulus, and v the Poissons ratio. The mode III fracture corresponds to the antiplane shear loading with the value 7 = #/4. We do not give here the usual value 7 for the plane stress case, because contrary to common belief, the formula for plane stress is questionable. To show that, remember that the plane stress assumption implies that the normal strain e33(X1,X2)-=--Y(0"11 -Jr-O'22)/E is a function of the first two coordinates Xl, x2 because aij does. The compatibility equation for g33(X1,X2) implies that all partial second derivatives with respect to Xl, x2 are equal to zero; hence it may be a linear function of the coordinates e33 (Xl, x2) = aXl + bx2 + c. Clearly, the compatibility condition is violated in the vicinity of the crack tip because precisely the strain component g33(r, 0) as given in the textbook is singular as r -1/2. This contradicts the linear function variation. Hence the common belief that for thin shells the formula for plane stress prevails near the crack tip is questionable.
7.3.4
DYNAMIC
STRESS INTENSITY
FACTORS
In quasi-static linear elasticity, both definitions, either by the stress (Eq. 1) or by the COD (Eq. 2), are strictly equivalent. However, in dynamic crack propagation, when inertial forces cannot be neglected, Eqs. 1 and 2 lead to two distinct crack tip parameters. It has been proved that there are universal relationships between these parameters:
K} - (v)Kj
(3)
with for mode j = I, II, III fx(V) -- {4fil (1 - flz2)}/(k + 1){4fllf12 fII(V)
-
-
{4fl2(1 - f122)}/(k + 1){4fllf12
(1
-
fl2)2}
(1 - fl2)2}
filI(V) = lift2 where flj - (1 - V2/c~) 1/2, with Cl the velocity of the compressive P-wave, c2 the velocity of the shear S-wave, and k = 3 - 4v [1]. The ratio fi is equal to 1 for V = 0, and greater than i for V ~ 0, tending to infinity when V approaches the Rayleigh velocity.
552
Bui et al.
The multiplicity of crack tip parameters raises the question, What is the true parameter measured in testing materials in dynamic fracture, stress- or velocity-intensity factors? If the onset crack propagation velocity V is not negligible, or undetermined, the critical state determining the toughness of the material from experimental results may be differently interpreted by inappropriate formulae for stress-intensity factors.
7.3.5 SMALL-SCALE YIELDING ASSUMPTION This assumption is essential in LEFM theory. It explains the apparent paradoxical statement that a linear elastic theory is capable of describing complex situations involving nonlinearity such as plasticity, viscoplasticity, damage, etc., which prevail in the process zone near the crack tip. Modern asymptotic analyses using some small parameter for example, the ratio of the process zone size to the crack length provide a mathematical justification of this well-known concept that everyone in engineering science is familiar with, without any mathematical tool. A good review paper on this topic is provided recently by Willis [17]. We shall consider some aforementioned nonlinearities.
7.3.6 PLASTICITY AND DAMAGE A plastic correction to the linear elastic solution was first given heuristically by Irwin. In the plastic zone, defined by the overstressed region, the normal stress cr22 is cut off to the constant yield stress or0. Later, an exact solution taking account of perfect plasticity was provided by Rice [15] for the antiplane shear loading of a semi-infinite crack. The plastic zone is a circle passing through the crack tip. According to modern terminology, the plastic field is the inner solution while the elastic field governed by the stress asymptotics 0"3i ~ Kiiir-1/2g(O) at infinity, r--+ oo (not for r--~ 0), is the outer solution. Rice [15] gives an exact solution matching inner and outer solutions. This solution has been generalized by Bui and Ehrlacher [3] to a notch with the slit width 2h ~ 0 and the notch geometry to be determined in such a way that cr3~ = cr0 along the notch. It is found that the notch is a cusped cycloid and that the elastic-plastic boundary is a curled cycloid. For the limiting case where ~r0--+ oo, one recovers Rice's solution for the crack h--+ 0 with the circular plastic zone. The notch solution can be interpreted as an elastic-brittle damage model where the material inside the slit has been totally damaged (zero stress state). The slit of thichness 2h is the wake damage zone, and the cusped cycloid is the damage front.
553
7.3 Background on Fracture Mechanics
7.3.7 VISCOPLASTIC
SOLUTION
The process zone is likely a high-stress region where viscous plastic flow may occur. An asymptotic inner solution for r--+ 0 has been provided by Hui and Riedel [10] in the case of steady state propagation of a crack, with the velocity V and without inertial force. The Norton and Hoff visco-elastoplastic law without threshold is considered in the form ~ - Cl0"r H- g I0"lm-ls
(m ~ 3)
(4)
(the superscript r means "rate", er is the strain rate, 0"r is the stress rate, s is the stress deviator, and 10"1is the von Mises deviator norm). In antiplane shear loading, the nonlinear equilibrium equation corresponding to this law has been given by Kachanov [11] in terms of the stress function ~(xl,x2):
-Vk(O~/OXl) + tl div{Igrad oIm-lgrad ~} - 0
(5)
(0"31 = --~//,2, 0"32 = ~/,1), with suitable boundary conditions on the crack tip. Readers interested in the numerical solution of Kachanov's equation can refer to Hui and Riedel [10]. We do not go into the details of their numerical solution, but consider qualitatively instead its essential behavior. The main feature of the solution is that there is an inner solution ~(r, 0, V) having the asymptotics ~ ~ C(V)r(m-2)/(m-1)g(O) for r--+ 0. Not only is the angular function g(0) uniquely determined, but the amplitude C(F) as well. There is no free parameter like the stress-intensity factor K or Rice's J-integral. We are faced with what is known in nonlinear physics as the soliton solution of a nonlinear wave, like the well-known Korteweg and de Vries soliton in fluid dynamics. The term soliton has the meaning of a solitary wave. The term viscoplastic soliton was coined for the first time in Bui [5]. Let us draw the consequence of an existing match between the inner soliton and the outer elastic solution 0"3i~ KiIIr-1/2 at infinity r--+ oo. A matching of solution, which can be achieved only by numerical methods, means that there exists a relationship between KII1 and the amplitude C(V). This theoretical analysis provides an explanation for the dependence of the toughness on crack velocity, likely due to a viscous phenomenon occurring in the process zone.
7.3.8 ENERGY CONSIDERATIONS Paralleling with the simple approach of LEFM, a more elaborate avenue based on energetic considerations was followed by various authors, and notably by Irwin in the late 1950s and 1960s. This other approach was initiated by Griffith in the 1920s. Griffith showed that an energetic analysis of a quasistatically growing crack led in a natural way to some propagation criterion
554
Bui et al.
involving a critical value of the so-called energy release rate G, which could be related to the derivative of the overall stiffness K (or compliance C) of the body considered with respect to the crack length a, G = -(1/2)u2dK/da (or G = (1/2)T2dC/da). This led to a simple interpretation of G in the overall strain-overall stress diagram. From there, Irwin proved a famous relation connecting G to the stress-intensity factors, for example, in plane strain mixed modes I and II G - (1 - v2)(KI2 + K~)/E
(6)
This established the equivalence of Irwin's propagation criterion involving the stress-intensity factor(s) and Grifftith's one involving the energy release rate. Also, Rice [15] and Cherepanov [6] simultaneously established the existence of a line integral noted J (or F in Russian), the contour of which must surround the crack tip and which possessed the following nice properties: first, it is independent of the integration contour; second, it is equal to G and thus, by Irwin's formula, connected to the stress-intensity factor(s). If the crack is not loaded in pure mode I, the expression of J involves all three stress-intensity factors, so that not all values of these three quantities can be deduced from that of J. However, at least in mixed mode I + II, the problem of separating the modes, that is, of deducing the values of both stressintensity factors from Rice-type integrals, was solved by Bui [4]. The main interest of Rice's J-integral is essentially numerical: indeed, it allows us to evaluate the stress-intensity factors through numerical integration on a path located far from the crack tip, thus circumventing the difficulty of such an evaluation from the unavoidably somewhat inaccurate values of the near-tip mechanical fields.
7.3.9 DUCTILE FRACTURE The first approach to ductile fracture, that is, fracture of materials undergoing considerable plastic deformation prior to failure, was based on Rice's J-integral. Indeed, the property of invariance of J with respect to the integration path remains true in nonlinear elasticity, which made it tempting to apply it to problems involving plasticity (unloading effects being disregarded). In this approach, propagation was assumed to occur when some critical value of J was reached, this critical value being allowed to depend upon the crack length. This theory was later named the global approach to ductile rupture, since it did not rely on any detailed micromechanical analysis of the mechanism of ductile rupture, that is, nucleation, growth, and finally coalescence of voids, through breaking of the
7.3 Background on Fracture Mechanics
555
inclusions-matrix interfaces or the inclusions themselves and subsequent plastic flow of the matrix. In contrast, the more modern theory of ductile rupture, named the local approach, is based on such an analysis. As years passed, the superiority of the latter approach has become clear, although the older one is still widely used in practical problems. A first, a seminal contribution was made by Rice and Tracey [16]; it consisted of an approximate analysis of the growth of a void in an inifinite plastic matrix loaded arbitrarily at infinity. Later, approximate criteria for porous plastic solids were proposed by Rousselier [14], who extended Rice and Tracey's void model, and Gurson [9], who proposed a model based on an approximate limit analysis of a typical elementary cell in a porous medium (hollow sphere). This model has become very popular. It was recently extended by Gologanu et al. [8] to incoporate void shape effects, which were neglected in Gurson's model of spherical voids. Void growth in plastic solids is thus now fairly understood and described by suitable models. Therefore, the theoretical analysis of coalescence has now become the major challenge in the local approach to ductile rupture. This phenomenon is very complex and is influenced by numerous factors, such as void shape, inhomogeneities in the distribution of cavities, the presence of a second population of secondary, smaller voids, etc. Significant contributions have already been made in this direction, but much remains to be done.
7.3.10 FATIGUE CRACKING Generally, cracks leading to failure of structures originate from fatigue phenomena. Classically, the fatigue life is divided into "initiation" and "propagation" phases; depending on observation scales, the relative part of each phase is variable. In structural analyses, the crack initiation size is a few millimeters. For metallurgists, it may be of the order of microns. Some authors neglect the initiation phase [13]. Paris's law gives the crack propagation rate per cycle d a / d N = C(AKI) m
(7)
for long cracks in terms of the amplitude of stress-intensity factor variations AKI. For these cracks, the defect will not grow if AKI 5f (the fatigue limit), and cracks initiate and propagate until final rupture. In Kitagawa's diagram [12], based on Paris's law and the fatigue limit concept, the size of crack initiation corresponds to the region of LEFM applicability, which is of the order of 100 g or more for steels. Below this size, which
556
Bui et al.
corresponds to the domain of short cracks, studied by metallurgists at the scale of some microns. Different empirical formulae are proposed to correlate the crack growth rate with the local shear stress acting on the gliding plane of microcracks and/or plastic sliding shear along these planes. Nevertherless, the way to evaluate these quantities is questionable; these local quantities differs from macroscopic stress fields because the medium is no longer homogeneous at the microscopic scale. The scale to be considered is the grain size. There are stresses induced by incompatible plastic strain in grains. An attempt to evaluate the incompatible stresses is provided by Dang Van [7] for high cycle fatigue. This model is based on an elastic shakedown hypothesis at all scales.
7.3.11
CONCLUSIONS
The assumption of smallmscale yielding allows us to apply LEFM to a variety of nonlinear situations, such as plasticity, damage, viscoplasticity, fatigue, etc. This is why LEFM is so popular in the engineering sciences. It explains the apparent paradoxical statement that a linear elastic theory can be used for describing complex situations involving nonlinearity. It also explains why a theory based on stress singularity does not contradict the small strain assumption of linear elasticity. LEFM is irrelevant in nonlinear phenomena in ductile rupture, with fully developed plasticity, with void growth, or in damage theory with localized phenomena. Today, such nonlinear phenomena can only be analyzed with some confidence by the so-called local approach to ductile rupture. However, there are still many engineering problems for which LEFM is still useful (for example, the fluidfilled crack problems, stress-corrosion cracking) or can be generalized to (cracks in electro-piezoelasticity, in coupled elasticity-electromagnetism, poroelasticity, etc.). Numerous unsolved problems in LEFM, particularly in the fields of inverse problems (crack detection, seismology, etc.) are also challenges.
REFERENCES 1. Achenbach,J. D., and Bazant, Z. (1972). Elastodynamicnear tip stress and displacementfields for rapidly propagating cracks in orthotropic materials.J. Appl. Mech. 97: 183. 2. Bui, H. D. (1977). Stress and crack displacement intensity factors in elastodynamics.4th Proc. Int. Conf. Fracture, vol. 3, Waterloo.
7.3 Background on Fracture Mechanics
557
3. Bui, H. D., Ehrlacher, A. (1981). Propagation of damage in elastic and plastic solids, in Advances in Fracture Mechanic, p. 533, vol. 3, Francois, D., et al., eds., Oxford-New York: Pergamon Press. 4. Bui, H. D. (1983). Associated path-independent J-integrals for separating mixed modes. J. Mech. Phys. Solids 31: 439. 5. Bui, H. D. (1993). Introduction aux probl~nes inverses en m~canique des mat~riaux, Paris: Eyrolles. English translation Inverse Problems in the Mechanics of Materials: An Introduction, CRC Press, Boca Raton, 1994. 6. Cherepanov, G. P. (1968). Cracks in solids. Int. J. Solids Struct. 4: 811. 7. Dang Van, K., and Papadopoulos, T. Y. (1997). High cycle metal fatigue: From theory to applications. 8. Gologanu, M., Leblond, J.-B., Perrin, G., and Devaux, J. (1997). Recent extensions of Gurson's model for porous ductile metals, in Continuum Micromechanics, chapter 2, pp. 61-130, Suquet, P., ed., Springer-Verlag. 9. Gurson, A. L. (1977). Continuum theory of ductile rupture by void nucleation and growth: Part I. Yield criteria and flow rules for porous ductile media. ASME J. Engng. Mat. Tech. 99: 2-15. 10. Hui, C. Y., and Riedel, H. (1981). The asymptotic stress and strain field near the tip of a growing crack under creep conditions. Int. J. Fract. 17: 409. 11. Kachanov, L. M. (1978). Crack under creep conditions. Izv. An SSR Mekhanika Tverdogo Tela 9(1): 57. 12. Kitagawa, H., and Takahashi, S. (1976). Applicability of fracture mechanics to very small cracks or cracks in the early stage. 2nd Int. Conf. on the Mechanics of Behavior of Materials. ICM2, Boston, Metal Parks, Ohio: American Metals Society. 13. Miller, K. (1997). The three thresholds for fatigue cracks propagation. Fatigue and Fracture Mechanics, pp. 267-286, vol. 27, Piascik, R. S., et al., eds., 14. Rousselier, G. (1981). Finite deformation constitutive relations including ductile fracture damage. Proc. IUTAM Symposium, on ductile fracture and 3D constitutive equations, Dourdan. Nemat-Nasser, S. N., ed., North-Holland. 15. Rice, J. R. (1968). Mathematical analysis in the mechanics of fracture, in Fracture, p. 2, Liebowitz, H., ed., New York: Academic Press. 16. Rice, J. R., and Tracey, D. M. (1969). On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17: 201-217. 17. Willis, J. (1997). Asymptotic analysis in fracture, Proceedings ICF9, vol. 4, 1849-1859, Karihaloo, B. K., et al., eds., Pergamon.
SECTION
7.4
Probabilistic Approach to Fracture: The Weibull Model FRAN(~OIS HILD Universitd Par/s 6, LMT-Cachan, 61 avenue du President Wilson, 94235 Cachan Cedex, France
Contents 7.4.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 General Framework . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Description of the Weibull Model . . . . . . . . . . 7.4.4 Identification of the Material Parameters . . . . 7.4.5 How to Use the Weibull Model: Numerical Implementation . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
558 559 559 563 564 565
7.4.1 VALIDITY In this section we will study the failure properties of brittle materials (e.g., monolithic ceramics, rocks, cement, concrete, some brittle-matrix composites). The overall behavior of these materials remains linear elastic. The strains to failure are assumed to be small c o m p a r e d to unity. Lastly, the toughness is at most on the order of a few M P a v ~ . All these materials are sensitive to initial heterogeneities (e.g., inclusions, porosities). These imperfections are created during the fabrication of the material and are usually r a n d o m l y distributed within the material or at the surfaces. They lead to a scatter of the failure load. Therefore, there is a need for a probabilistic treatment of failure.
558
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
559
7.4 Probabilistic Approach to Fracture: The Weibull Model
7.4.2
GENERAL FRAMEWORK
The (cumulative) failure probability PF will be determined for structures made of brittle materials. As soon as local crack propagation occurs at one location of the material, the whole structure fails. In other words, as soon as the weakest link is broken, the whole chain (i.e., the structure) fails. Within the framework of the weakest-link statistics [1], the local failure probability Pro within a domain ~0 of volume Vo is related to the global failure probability Pr of a structure f2: PF-1-
exp [~o s In (1-Pr;o)dV]
(i)
This relationship constitutes the basis for all existing models developed to describe the failure of brittle materials. The cumulative failure probability of an elementary volume Pro is the probability of finding a critical defect within a domain f2o. The latter refers to the defect distribution characterized by a probability density function f. The initial flaw distribution (f0) depends upon morphological parameters related to the modeling of the defect: first a size (a), then a defect orientation (described by a unit normal n), and then other parameters (denoted by w). For a given load level, the set of defects D splits into two subsets. The first one (De) is related to the defects that are critical (i.e., they lead to the failure of a link, and therefore of the whole structure). The second one (Dnc) contains the defects that are not critical. The higher the load level, the larger Dc becomes with respect to D,,c. When the local propagation is unstable, the cumulative failure probability Pro(Q) of a domain f20 for a given loading parameter Q is given by
P.(Q) -
fo(a. _.2. w) aa a.2 aw
(2)
c
This definition ensures that Pro lies between 0 (i.e., D c - ~" no defect is critical) and 1 (i.e., Dc = D: all defects are critical).
7.4.3
DESCRIPTION
OF THE WEIBULL MODEL
One considers a three-dimensional situation of penny-shaped cracks of radius a whose orientation is defined by a normal n. The failure criterion is defined as
Yll ( l,
Kc
(3)
560
Hild
with ,
0"2, 0"3, n)l I - - 0-1G(0-2, -0-3 -, n /
-
\0" 1
0" 1
(4)
-
where 0"1 > 0"2 > 0"3 are the principal stresses, Y a dimensionless geometrical parameter, and Kc the material toughness. Let ~2c(]l~l], a) denote the solid angle containing all the defect normals n for which the failure criterion is satisfied. By assuming an equiangular distribution and an unbounded maximum value of defect size, the failure probability Pro is expressed as
f Pro - - / . "-'~c
(ll_~ll) _
fo(a) da
4~
(5)
where a~([l~ll) denotes the smallest critical defect size. Similarly, the cumulative failure probability PFo can be written as
Pro fO 4~ -~1 1 - Fo
K2c
2
dn-
(6)
{ Y0-1G (0-~-~212 0-3, --,0.1 ~)) where Fo is the cumulative probability distribution of sizes. These two expressions are completely equivalent. This fact was only recognized in the last decade [21. When the initial distribution fo is equivalent to Ka -n for large values of the defect size a [31, the cumulative failure probability Pro is approximated by
Pro
k So /
(7)
I 2 ( n - 1) , 0-1, ~---11
with
K c ( n - 1 ) 1/2(n-l) K
So - u I 2(n - 1) 0-2 0"3 _ 2
0-1~
n/
~d0
~/2 G 2(n-l)
dO
(8)
0.2, --, /1 COS(.pd~9 d~// \0-1 0-1 -
(9)
where (.) denotes the Macauley brackets. The previous expression can be used to derive the cumulative failure probability described by a two-parameter Weibull law [4]" Pv = 1 - exp
1
(0-1(M)> So
I
m,
--(M) - - ( M ) 0-1
' 0-1
dV
(10)
where m - 2(n - 1) is the shape parameter and VoS~ the scale parameter. If a nonvanishing threshold stress Su below which no failure occurs can be defined
561
7.4 Probabilistic Approach to Fracture: The Weibull Model
(i.e., there exists a bounded maximum flaw size), a three-parameter Weibull law [4] can be defined: Ps-
1
1 - exp
'
(~1 (M)
S0
S.)
I m, - - (M) - - (M) ~
'0-1
dV
(11)
Initially, Weibull proposed to consider the normal stress along the direction n as a suitable equivalent stress [4] expressed in terms of the angles q~ and defined in Figure 7.4.1"
ow -
\0-1
0"1
0,
(12)
with Gw
, - - , ~J, r
-
cos 2 ~ j + - s i n 2~
0-1
cos 2 r
0-1
2r
(13)
Kc}
(14)
0-1
By using the angles r and ~, the solid angle is defined as
~c([l~ll,a)-
{~,
r
0-2, 0-3, ~, ~P)llV/-~-
and
I m,~l, (15) -- -2 fo=/2/'~/2 [ ( cos2~p + ~0-z sin2~p) cos 2(p + -o-3 - sin 2q~]m cos q~dq~ d~ 7/:
J0
0-1
(9-1
It is worth noting that the equivalent stress chosen by Weibull corresponds to an assumption of a mode I mechanism of failure (even though this notion was not yet discovered). Some other models enter the present framework. Of those, one may mention the model developed by Batdorf et al. [5-7]. The formalism of that model follows Eq. 5. Lastly, the model proposed by Evans and Lamon [8-10] uses yet another failure criterion and is closer to Eq. 7. All the models alluded to in this section are very similar. The key distinction between the three models is given by the failure criterion (described by the functions G) [2]. For all the models, the failure probability can be rewritten in the following form: PF-l-exp
- ~ \
So )
562
Hild
03
~o2
FIGURE 7.4.1
Definition of the orientation of a normal n by the angles ~k and ~o.
where a stress heterogeneity factor Hm can be expressed as 1 / ~m dV Hm = Vo'~ J~
(17)
with 0"2 0"3
= alI m,--,
, av -- Max (7l(m) > 0
0-1
(18)
f2
The stress heterogeneity factor characterizes the effect of the load pattern on the cumulative failure probability. An effective volume Veff can be defined as [ 11 ] Vr = VHm
(19)
When the latter is loaded in pure tension (i.e., Hm=l), it would lead to the same failure probability as the considered domain ~ subjected to any loading condition. Similarly, a Weibull stress can be defined as [12]
[-
(20)
(vHq ~/~
(211
with
-
563
7.4 Probabilistic Approach to Fracture: The Weibull Model
Lastly, the average failure stress dv and the corresponding standard deviation dv can be written as {TF
-
S0 ( g ~ m )
l/m F (
--
1+
1)
,
(22)
I'(] ._~_2)-1.2(1_~__1)
where F is the Euler function of the second kind. In some situations, the previous model is simplified by noting that I[m, 0-2/0-1, 0-3/0-1]--constant for a unidimensional stress field. There is no need to use the function I of Eq. 9. Instead, one may consider the maximum principal stress at each point [12]. The model in this form is commonly used in materials science.
7.4.4 IDENTIFICATION OF THE MATERIAL PARAMETERS For a two-parameter Weibull law, the identification procedure is straightforward. It consists in recasting Eq. 16 in the following form: In[In(l_
i
(23)
(VoS~'~ pF)] -- mln(o-F) - ln\vHm f
Therefore, in a Weibull plot [4], it is expected that the material data follow a straight line whose slope corresponds to the Weibull modulus. Once the Weibull modulus is known, the stress heterogeneity factor can be computed by using Eq. 17, and then by knowing the intercept, the shape parameter VoS'~
TABLE 7.4.1
Weibull Parameters for Various Materials
Material Sintered SiC SiC-B SiC-HIP SIC-150 SiC Nicalon fibers C fibers A1203 Si2N4 ZrO2
Weibull modulus m
Weibull stress (MPa)
9-10 27 8-9 15 3-4 4-5 8-10 8-9 10-15
400 550 600 250 1650-1750 1150 100-600 750-1350 200-500
Reference volume (mm 3) 1 1 1 1 3 10 -3 3 10 -3 1 1 1
564
nild
can be identified. One can note that a conventional least-squares method can be utilized to identify the Weibull parameters. For a three-parameter Weibull law, a least-squares method can be used as well. However, the identification procedure is not as simple as the previous one. It is worth mentioning that a maximum likelihood procedure is also well suited for the identification of statistical distributions such as the Weibull model. Table 7.4.1 summarizes data for different monolithic ceramics. It can be noted that the processing route influences the Weibull parameters of materials with the same name.
7.4.5 HOW NUMERICAL
TO USE THE WEIBULL MODEL: IMPLEMENTATION
This model is usually coupled with an elastic calculation even though some examples show that a nonlinear behavior may need a probabilistic description of fracture [12]. A commercial finite element code can be used when closedform solutions are not available. The computation of the failure probability is then performed numerically in a postprocessor. At each integration point i, the equivalent stress distribution Ii[m, 0-2/0-1, 0"3/0"1] is computed by using Eq. 9. The contribution Hmj of an element j is Hmj
= ~
~
0"}' i=1
m
0"2 0"3
0"li Ii m , ~ ,
- wi
0"1 0"1
(24)
where ng is the number of integration points of the considered element and 141i the weight of the integration point i. The stress heterogeneity factor H m is then computed 1
ne
-
/q jvj
(25)
j=l
where Vj denotes the volume of the element j and ne the total number of elements: /1 e
V- ~
Vj
(26)
j=l
Note that the computation of models described by Eq. 5 (e.g., the one proposed by Batdorf et al. [5]) is more difficult to carry out. Lastly, the cumulative failure probability Pv of the structure is computed by using Eq. 16 for different stress levels, since the stress intensity factor Hm is independent of the load level. This result is not true if a three-parameter Weibull model is used [ 14].
7.4 Probabilistic Approach to Fracture: The Weibull Model
565
REFERENCES 1. Freudenthal, A. M. (1968). Statistical approach to brittle fracture, In Fracture, pp. 591-619, vol. 2, Liebowitz, H., ed., New York: Academic Press. 2. Chao, L. Y., and Shetty, D. K. (1990). Equivalence of physically based statistical fracture theories for reliability analysis of ceramics in multiaxial loadings. J. Am. Ceram. Soc. 73: 1917-1921. 3. Jayatilaka, A. de S., and Trustrum, K. (1977). Statistical approach to brittle fracture. J. Mat. Sci. 12: 1426-1430. 4. Weibull, W (1939). A statistical theory of the strength of materials. Roy. Swed. Inst. Eng. Res. 151. 5. Batdorf, S. B., and Crose, J. G. (1974). A statistical theory for the fracture of brittle sructures subjected to polyaxial stress states. J. Appl. Mech. 41: 459-465. 6. Batdorf, S. B. (1977). Fundamentals of the statistical theory in fracture. In Fracture Mechanics of Ceramics, pp. 1-30, vol. 3, Bradt, R. C., Hasselman, D. P. H., and Lange, E E, eds., New York: Plenum. 7. Batdorf, S. B., and Heinish Jr., H. L. (1978). Weakest link theory reformulated for arbitrary fracture criterion. J. Am. Ceram. Soc. 61: 355-358. 8. Evans, A. G. (1978). A general approach for the statistical analysis of multiaxial fracture. J. Am. Ceram. Soc. 61: 302-308. 9. Lamon, J., and Evans, A. G. (1983). Statistical analysis of bending strengths for brittle solids: A multiaxial fracture problem. J. Am. Ceram. Soc. 66: 177-182. 10. Lamon, J. (1988). Statistical approaches to failure for ceramic reliability assessment. J. Am. Ceram. Soc. 71: 106-112. 11. Davies, D. G. S. (1973). The statistical approach to engineering design in ceramics. Proc. Brit. Ceram. Soc. 22: 429-452. 12. Beremin, E M. (1983). A local criterion for cleavage fracture of a nuclear pressure vessel steel. Met. Trans. A 14A: 2277-2287. 13. Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18: 293-297. 14. Chantier, I., Bobet, V., Billardon, R., and Hild, F. (2000). A probabilistic approach to predict the very high cycle fatigue behavior of spheroidal graphite cast iron. Fat. Fract. Eng. Mat. Struct. 123: 173-180.
SECTION
7.5
Brittle Fracture DOMINIQUEFRANCOIS F_cole Centrale de Paris, Ch~ttenay-Malabry, 92295, France
Contents 7.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 General Framework . . . . . . . . . . . . . . . . . . . . . 7.5.3 Plastic and Damaged Zones at the Crack Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3.1 Small-Scale Yielding . . . . . . . . . . . . . . . 7.5.3.2 Perfectly Plastic Zone . . . . . . . . . . . . . 7.5.3.3 Damaged Zone . . . . . . . . . . . . . . . . . . . 7.5.4 Brittle Fracture Mechanisms . . . . . . . . . . . . 7.5.5 Fracture Toughness Models . . . . . . . . . . . . . 7.5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 7.5.5.2 Deterministic Models for Metals... 7.5.5.3 Deterministic Models for Brittle Nonmetallic Materials . . . . . . . . . . . . . 7.5.5.4 Stochastic Models . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
566 567 568 568 569 570 572 573 573 573 574 575 576
7.5.1 VALIDITY In the case of brittle fracture, linear elastic fracture m e c h a n i c s applies, since no or little plastic d e f o r m a t i o n precedes fracture so that small-scale yielding conditions can be observed. The sizes of the plastic a n d of the d a m a g e d zones are studied. Triggering of brittle fracture requires stress c o n c e n t r a t i o n s w h i c h are p r o d u c e d either by small defects in the case of nonmetallic materials or by h e t e r o g e n e o u s plastic d e f o r m a t i o n in the case of metals. This c o r r e s p o n d s to a critical value of the overall m a x i m u m principal stress. Because of the r a n d o m distribution of the m i c r o s t r u c t u r a l fracture nuclei, the fracture stress is
566
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
7.5 Brittle Fracture
567
statistically distributed. The Weibull distribution is used. Early models to describe the fracture toughness in terms of the uniaxial stress-strain behavior equated the maximum principal stress at some distance ahead of the crack tip to the critical fracture stress. Using the Weibull distribution of the fracture stress obtained by testing small specimens, a fair estimate of the fracture toughness can be calculated and the influence of the crack front length, temperature, and irradiation can be estimated.
7.5.2
GENERAL FRAMEWORK
Usually a material is considered brittle when it breaks at low elongations, not exceeding a few percentage points. As a result, linear elastic fracture mechanics (LEFM) applies well, since the deformations near the crack tip remain very close to the elastic limit. The problem to be considered is to understand and determine the relations between the global mechanical behavior of the material, often determined by a uniaxial test, and the fracture toughness, that is to say, the resistance to crack propagation. This requires considerations about fracture mechanisms which take place near the crack tip, since a crude model of crack propagation consists in visualizing a tiny fictitious tensile specimen ahead of the crack tip whose fracture starts the propagation of the crack (Fig. 7.5.1). The notion of brittleness applies also to the fracture mechanisms. This excludes fracture that is the result of nucleation, growth and coalescence of cavities, since this phenomenon is essentially due to large local plastic deformation, even though it can happen for quite low overall elongations in some cases. This type of micromechanism will not be considered; it is covered in another chapter of the handbook. LEFM yields a stress and strain singularity at the crack tip which cannot exist for real materials. A local perturbation has to take place due to either microcracking or plastic deformation or both, which limits the stresses and strains at finite values. Thus, for LEFM to be valid, the plastic and the damaged zones at the crack tip must remain small enough with respect to the size of the specimen or of the structure. The conditions for this to be achieved will first be explained by analyzing these zones. A description will then be given of the main brittle fracture mechanisms in order to understand the fracture criterion and to evaluate the characteristic length needed in deriving the relations between microscopic and macroscopic behaviors. This will allow us to study the models which relate the brittle fracture criterion and the fracture toughness. The most recent ones take into account the random nature of the brittle fracture mechanisms. It will be shown how they are derived and
568
Fran~:ois
l
l
FIGURE 7.5.1 Principle of the local approach to fracture mechanics, allowing us to relate the mechanical properties of a representative volume element to the fracture toughness: the crack propagates when the stresses and the strains at a critical distance ahead of the crack tip reach the fracture criterion of a ficticious tiny specimen.
the important consequences which they have on the influences of various parameters on the fracture toughness.
7.5.3
PLASTIC AND DAMAGED
ZONES
AT THE CRACK TIP 7.5.3.1
SMALL-SCALE YIELDING
LEFM implies r -1/2 stress and strain singularities at the crack tip, whereas real materials are damaged or yield plastically so that the stresses cannot exceed a certain limit, t~c. Damaged and plastic zones perturb the LEFM solutions near the crack tip. Small-scale yielding is such that the LEFM singularity dominates at infinity. We keep the same designation for completely brittle materials even though they do not yield plastically. Under this condition of small-scale yielding, the sizes of the damaged or plastic zones are proportional to (K/crc) 2, K being the stress-intensity factor (Fig. 7.5.2). For LEFM to apply, they must remain small with respect to the size b of the ligament remaining ahead of the crack. We now need to describe in more detail these zones to obtain a good approximation of their sizes.
569
7.5 Brittle Fracture
c~0
(K/(~c)2 FIGURE 7.5.2 Accordingto the r -1/2 elastic stress singularity, in small-scale yielding the plastic (or damaged) zone is proportional to (K/aa)2. 7.5.3.2
PERFECTLY PLASTIC ZONE
A closed-form solution exists in mode III for the description of the plastic zone for a perfectly plastic material the flow shear stress of which is k [ 1 ]. The plastic zone is found to be a circle of radius ry such that 1(?)
ry --~-~
2 (1)
In mode I a fair approximation of the plastic zone size R is given by I(K~) 2 R -- prr
(2)
where Rp is the flow stress, and p is equal to 1 in plane stress and to 3 in plane strain. The model of Dugdale-Barenblatt is based on the representation of the plastic zone by a constant closure stress equal to the flow stress Re acting on the tip of a ficticious crack over a distance equal to the plastic zone size R (Fig. 7.5.3). It is valid for thin sheets in plane stress. In this model, in smallscale yielding, R is given by R - -8
(4)
570
Fran~:ois
Rp
It I FIGURE 7.5.3 In the Dugdale-Barenblat (or the Hillerborg) model, the plastic (or the damaged) zone is represented by the extremity of a fictitious crack which encloses this plastic (or damaged) zone, along which the closing stresses act.
This is not much different from the previous expression, Eq. 3. The smallscale yielding condition states that the ligament size b must be large compared to the plastic zone size. The condition for LEFM to be valid is conventionally written
The mode III solution for a perfectly plastic material allows us to calculate the strain within the plastic zone. This shows that there is a 1/r plastic strain singularity. This cannot exist for real materials, and large deformations and damage perturb this ideal solution.
7.5.3.3
DAMAGED ZONE
Bui and Ehrlacher [2, 4] derived a closed-form solution for the damaged zone around a propagating crack in a brittle material which is elastic and breaks when the maximum principal stress reaches a critical value aa. In their model the damaged zone supports no stress, so that the crack is similar to a notch filled with broken material. The thickness 2h of this damaged zone is given by the simple expression -
(4)
Similarly, Bui [3, 4] treated the problem of a propagating crack in a perfectly plastic material with a fracture strain 7g in mode III. He found that the
571
7.5 Brittle Fracture d a m a g e d zone w i d t h was given by 1
2h-
II
(5)
27R/7o - 1 ~
w h e r e 7o is the yield strain k/p, p being the shear m o d u l u s . As for the w i d t h of the plastic zone 2R, it is given by 2R--- 1 ~ i
E (
sin c o s - 1 - 2 h
h (
+-cosg
1 -2h
(6)
This e x p r e s s i o n r e d u c e s to Eq. 1 for a c o m p l e t e l y ductile material with no d a m a g e d zone (h = 0). The fictitious crack m o d e l of Hillerborg [5] is quite similar to the DugdaleBarenblatt one. The difference is that the closure stress d i s t r i b u t i o n at the crack tip is n o t c o n s t a n t b u t decreases from the b o u n d a r y b e t w e e n the d a m a g e d zone a n d the elastic singularity to zero at the crack tip (Fig. 7.5.4). D e p e n d i n g on the function describing this distribution, the size of the d a m a g e d zone is slightly different. This is m o r e realistic than the c o m p l e t e and s u d d e n d e s t r u c t i o n of the resistance of the material w h e n the fracture
FIGURE 7.5.4 The slip line field around a crack tip in plane strain yields, within the large square, the maximum principal stress which is indicated. Blunting of the crack tip assimilated to a circle of diameter c~, the crack tip opening displacement (CTOD), modifies locally the slip lines in the shape of logarithmic spirals. Ahead of the crack the maximum principal stress follows a logarithmic variation as a function of the distance to the tip.
572
FranCois
stress is reached, as in the model of Bui and Ehrlacher, since it represents better the postpeak behavior of materials like concrete. However, it suffers from the oversimplified representation of the damage zone by a thin strip ahead of the crack.
7.5.4 BRITTLE FRACTURE MECHANISMS
[6]
The stress needed to break atomic bonds, the theoretical fracture stress, is very large, of the order of E / I O , E being the Young's modulus. The reason that actual fracture stresses are orders of magnitude lower is the heterogeneous distribution of the stresses in the material. Various mechanisms can explain this fact. The first category of materials contains numerous microcracks. This is the case, for instance, of glass, the microcracks occurring at the surface, or of concrete, where they are scattered within the entire volume, particularly at the interfaces of aggregates. These microcracks create large stress concentrations such that the theoretical fracture stress can easily be reached. Each microcrack can be represented by a penny-shaped crack of radius a for which the stress intensity factor is given by KI-
2
-- c r V / - ~ 7~
(7)
where a is the uniform applied stress along the normal to the crack. Such a crack will propagate unstably when KI reaches a critical value related to the fracture energy ?c, that is to say, the energy needed to create a unit area of crack. The fracture stress aR is then given by
I[ ~Ey~],/2
aR -- -~
(i -
v2)aJ
(8)
v being the Poisson's ratio; aR is larger the smaller the microcrack size a. Since these microcracks all have different sizes, and furthermore, since they have different orientations, the fracture stress is statistically distributed. The second category of brittle materials, such as ceramics, can be microcrack-free but contain impurities in their grain boundaries which lower the fracture energy. It is then easy to trigger fracture by breaking the material which possesses the lowest resistance because of a combination of impurity level, grain size, and orientation. The third category includes the materials like metals which can deform plastically. The stress concentration needed to create a crack, which is either a cleavage along a particular crystallographic plane or an intergranular crack, originates from heterogeneous plastic deformation. This plastic deformation
7.5 Brittle Fracture
573
starts in the most favorably oriented grains for high shear stress on a crystallographic glide plane. Shear bands are formed, at the tip of which the stress can be high enough to cleave an adjacent grain or to break a grain boundary embrittled by impurities. Often cleavage takes place in nonmetallic inclusions because of the stress concentration, again created by heterogeneous plastic deformation. The stress concentration is the higher the longer the shear band, the length of which is related to the grain size. The fracture stress is inversely proportional to the square root of the grain size. It is important to note that in those materials fracture cannot occur before the elastic limit is reached. Thus in general it is found that the brittle fracture criterion is that of Rankine: a critical value of the maximum principal stress. In all cases this fracture stress, which is inversely proportional to the square root of the size of the characteristic microscopic triggering features, is statistically distributed because of the various sizes, orientations, and resistances of these fracture nuclei, whether they are microcracks, weak boundaries, or grains.
7.5.5 FRACTURE TOUGHNESS MODELS
7.5.5.1 INTRODUCTION The problem is now to relate the fracture toughness Kc, or, in terms of fracture energy, Gc, to the fracture stress aR. The solution is sketched in Figure 7.5.1. We need to equate the maximum stress level ahead of the crack tip to the fracture stress. The distribution of the maximum principal stress must be known. Furthermore, the maximum level must be reached over a critical distance which is related to the microstructure of the material.
7.5.5.2
DETERMINISTIC MODELS FOR METALS
7.5.5.2.1 Model of Wilshaw, Rau, and Tetelman [7] Wilshaw, Rau and Tetelman used the slip lines field around the crack tip to evaluate the stress distribution in the plastic zone in mode I. This slip lines field (Fig. 7.5.4) is valid in plane strain for a perfectly plastic material. It shows that, ahead of the crack, plastic constraint increases the maximum principal stress level up to (2 + r0k. However, the blunting of the crack tip produces a gradual logarithmic evolution of this stress from 2k at the very tip up to the preceding value at a distance equal to 1.9b, b being the crack tip opening displacement. It is assumed that the blunting gives a circular shape to
5 74
Fran~:ois
the crack tip. The maximum stress is then reached at the elastic-plastic boundary. Using the expression of the plastic zone size (Eq. 2), the fracture toughness Kzc is found by equating the maximum principal stress at a distance R from the crack tip to the fracture stress ag:
Kxc--3.16R v exp af
_1
X/~
(9)
7.5.5.2.2 Ritchie, Knott, and Rice Model [8] Ritchie, Knott, and Rice used the stress distribution ahead of the crack tip given by the Hutchinson, Rice and Rosenfield (HRR) field. The HRR field results from the J/r plastic strain energy density singularity deduced from the path-independent J integral. In small-scale yielding, J is equal to the strain energy release rate G. If the stress-strain law is written
(10)
--=0~ ~o
where ~0 and a0 are the yield strain and strength, the stress distribution is given by
G_
(
J
~ 1/(N+1)
ao \~aogor/
~(N, O)
(11)
where r and 0 are the cylindrical coordinates at the crack tip. By equating this stress at the characteristic microstructural distance p to the fracture stress, and using the relation between the strain energy release rate G and the stress-intensity factor, it is found that
KIc -- A (crR~ (N+1)/2a0X ~ -\Cro/
(12)
A is a coefficient close to 1, and p is equal to once or twice the grain size. 7 . 5 . 5 . 3 DETERMINISTIC MODELS FOR BRITTLE NONMETALLIC MATERIALS In the model of Bui and Erlacher [2, 4] the width of the damage zone h can be set equal to the characteristic microstructural size. Equation 4 then provides an evaluation of the fracture toughness: K k - aav/-p
(13)
where p would be the size of a volume element containing one microcrack.
7.5 BrittleFracture 7.5.5.4
5 75
STOCHASTIC MODELS
7.5.5.4.1 Fracture Toughness of Metals; Beremin Model [11] The stress distribution in the plastic zone near the crack tip being given by the HRR field (Eq. 11), it is possible to calculate the Weibull stress aw. A fair approximation is given by integrating the equation giving crw within a plastic sector of thickness B, the length of the crack front, of angle 0 and of radius R, the size of the plastic zone, ahead of the crack. This yields
E/
N+ 1 m= Cro Jk aw 2(N + 1) - m \~ao~o,/
-BO - R2-(m/(N+1)) Vo
(14)
JIc is the critical value of J which is equal to K2c(1- v2)/E in small-scale yielding. R being given by Eq. 2, the Weibull law (see Section 7.4) yields
IVo(~p)m KIc -- C - -
au
(1) log 1 - PR
1/4 RV
(15)
where C is a constant. Equation 15 includes a characteristic microscopic size as Vo, the volume corresponding to one cleavage nucleus, is related to the grain size. The average cleavage stress au and the Weibull exponent m are considered to be independent of temperature. The yield stress Rp decreases when the temperature increases, and since m is large, greater than 4, the fracture toughness increases with temperature in relation with the yield stress. The decrease of the fracture toughness after irradiation is also related to the increase of the yield stress under those conditions. Another important conclusion from this model is that the fracture toughness is size-dependent and varies as B -1/4, where B is the length of the crack front. This is well supported by experiments. 7.5.5.4.2 Fracture Toughness of Nonmetallic Materials A similar model can be considered for brittle nonmetallic materials replacing the plastic HRR field by the elastic singularity (Eq. 1). If the integration is done over a distance r equal to the damage zone size of the order of (Kic/au)2, an expression similar to Eq. 15 is found:
KIc--C[~-~
log(1 -
1pR) ] 1/4au
(16)
The size dependence of the fracture toughness is then the same as for the cleavage of metals.
5 76
Fran~:ois
REFERENCES 1. Rice, J. R. (1968). Mathematical Analysis In the Mechanics of Fracture, p. 191, vol. 2, Liebowitz, H., ed., Academic Press. 2. Bui, H. D., and Erlacher, A. (1980). Propagation dynamique d'une zone endommag~e dans un solide ~lastique fragile en mode III et en r~gime permanent. C. R. Acad. Sc. Paris, S~rie B, 290: 273. 3. Bui, H. D. (1980). Solution explicite d'un probl~me de frontKre libre en ~lasto-plasticit~ avec endommagement. C. R. Acad. Sc. Paris, S~rie B, 290: 345. 4. Bui, H. D., and Erlacher, A. (1981). Propagation of damage in elastic and plastic solids, in Advances in Fracture Research, pp. 533-551, vol. 2, Francois, D., ed., Oxford: Pergamon Press. 5. Hillerborg, A., Modeer, M., and Petersson, P. E. (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. In Cement and Concrete Res. 6: 773-782. 6. Francois, D., Pineau, A., and Zaoui, A. (1998). Mechanical Behaviour of Materials, vol. 2: Viscoplasticity, Damage, Fracture Mechanics and Contact Mechanics, Dordrecht: Kluwer Academic Publishers. 7. Wilshaw R. T., Rau C. A., and Tetelman, A. S. (1968). A general model to predict the elasticplastic stress distribution and fracture strength of notched bars in plane strain bending. In Eng. Frac. Mech. 1: 191. 8. Ritchie, R. O., Knott, J. E, and Rice, J. R. (1973). On the relationship between critical tensile stress and fracture toughness in mild steel. In J. Mech. Phys. Solids 21: 395-410. 9. Pineau, A. (1981). Review of fracture micromechanisms and local approach to predicting crack resistance in low strength steels, in Advances in Fracture Research, pp. 553-577, vol. 2, Francois, D., ed., Oxford: Pergamon Press.
SECTION
7.6
Sliding Crack Model DIETMAR G a o s s Institute of Mechanics, TU Darmstadt, Hochschulstrasse 1, D 64289 Darmstadt, Germany
Contents 7.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Simplified Models . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3.1 Displacement-Driven Model . . . . . . . . . . 7.6.3.2 Force-Driven Model . . . . . . . . . . . . . . . . . 7.6.4 Internal Variable Model for Brittle Inelastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . 7.6.6 How to Use the Model . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
577 577 578 579 579 579 580 581 581
7.6.1 VALIDITY This is a two-dimensional model describing crack growth u n d e r compressive loading in brittle materials and taking dry friction into account. Like others, it is considered a basic dissipative m i c r o m e c h a n i s m underlying macroscopic inelastic deformation, brittle damage, and splitting p h e n o m e n a at final failure of rocklike materials.
7.6.2 BASIC MODEL The model consists of a preexisting closed crack of length 2c with frictional crack faces u n d e r increasing remote compressive loading, given by the principal stresses cri with I~rll > I~r21 (Fig. 7.6.1a). Frictional crack face sliding may be activated if the M o h r - C o u l o m b - t y p e condition Tll2 / --
,Fc - - ~O'111~
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
577
578
Gross
FIGURE 7.6.1 driven model.
Sliding crack model, a. Basic model, b. Force-driven model, c. Displacement-
is fulfilled where Zl,2, , and 0"1,1, are the actual stresses along PP' and Zc and are the cohesion and friction coefficient, respectively. Once sliding with a local relative slip b(x'2) occurs, the crack is under pure mode II loading, giving rise to crack kinking with an angle 0 at a critical load and subsequently to the formation of wing cracks growing under mode I conditions. They align with the maximum compression direction after short initial curving. During crack growth the condition K I - - KIc
must be fulfilled. When unloading, the wings may partly close, and certain backsliding is possible. The change of complementary energy of the system because of crack sliding and open wings is given by Aqj - 2 c f0 Zl,2,(a,/~)d/~+2f0 ff(l) dl
and
if-
K~+K~E,
where/~ is the average slip over 2c, ff is the energy release rate, l is the wing length, KI, KH are the stress-intensity factors, and E' -- f E E/(1 - v2)
(
plane stress plane strain
7.6.3 S I M P L I F I E D M O D E L S The basic model may be replaced approximately by the models in Figure 7.6.1b,c, allowing a simplified calculation of the stress-intensity factors.
579
7.6 Sliding Crack Model
7.6.3.1 DISPLACEMENT-DRIVENMODEL (Fig. 7.6.1c) E'/~ sin 0 KI =
-}-
2V/2~(l + l**)
V ~ [0-11 COS2( 0
-4- (D) -4- 0-22
E'/~ cos 0 KII
-~
_
2 V/2rc(l + l** )
sin2(O q- (#) q- T12 sin 2(0 -4- qo)]
1
-
]
1:12 cos 2(0 + q~) - ~ (0"11 - 0"22) sin 2(0 + q~)
7.6.3.2 FORCE-DRIVENMODEL (Fig. 7.6.1b) F sin 0 KI - -
2v/2 (z + z*)
q-~[0"11
F cos 0 KII
~
-
-
2V/2lr(l + l*)
c o s 2 ( 0 - } - ~ ) + 0"22
E x / ~ ~12 cos 2(0 + r
sin2(0+ q~) + 1:12sin 2(0+ q~)] 1 - ~ (0"11 - 0"22) sin 2(0 + q~)
l** -- 0.27crcZ/32, wing length correction; l* - 0.27c, wing length correction; F - -2q:eff, splitting force; Te ff __ Tlr2 t - r l , Z ,r, effective shear stress; r __ 1 Tl~2~ - - - - ~ (O'11 --
0"22) sin 2rp
+ "/712 c o s 2(]9,
resolved
shear
stress.
7.6.4 INTERNAL VARIABLE MODEL FOR BRITTLE INELASTIC MATERIALS The macroscopic material behavior is described by a representative volume element (RVE) consisting of an elastic matrix material containing N distributed noninteracting sliding crack defects. The description is based on Rice's internal variable thermodynamic framework. Splitting the total strain increment as d~, = de~, + di ~,
the elastic strain increment is given by d e ~ - M" da
580
Gross
where
a2~ Mohz
--
OaijOakl ,
instantaneous complinance;
o = 89aijMijhlahl + A~t, Gibbs (complementary) energy density; M~
-- compliance of matrix material; 1
N
A - RVE area. The inelastic strain increment follows from 1 N
cgf~
where f~ is the set of thermodynamic forces in the actual state conjugated to the internal variables (microfluxes). Restricting the number of variables of each defect to /~, relative slip, l, length of straight wing cracks, 0, orientation of wings, the thermodynamic forces and conjugated microfluxes are identified as
fl-2CZl,2,,
d~l -db,
0~ f 3 = S0 l -ff~dl
f2-~, d~2 - dl,
d~3 - dO
The constitutive equations are completed by the three growth conditions
OK/ "FlP21 --- "/7c - - f l O ' l t l ~ ,
K I - - Kic,
(00 - - 0
7.6.5 MATERIAL PARAMETERS The following material parameters have to be known:
M~ , compliance of the matrix material; 2c, microcrack length; #x, friction coefficient; zc, cohesion; KIc, fracture toughness; COo - N c 2 / A , initial crack density.
581
7.6 Sliding Crack Model
As an example, approximate values for granite are E ~ -- 580002ptMPa; v~ --0.23; c - 5 . 1 0 -4 m; # = 0.65; 1:c = 12MPa; Kic = 0.7MPav/-m;
7.6.6
HOW
COo = 0.375;
TO USE THE MODEL
The model may be used: a) as a basic mechanism in nonsimplified form, implemented directly in numerical codes (BEM, FEM); b) with a reduced number of internal variables (simplified model) as material model, implemented in numerical codes (FEM, BEM).
BIBLIOGRAPHY 1. Basista, M., and Gross, D. (1998). The sliding crack model of brittle deformation: An internal variable approach. Int. J. Solids Structures 35: 487-509. 2. Horii, H., and Nemat-Nasser, S. (1986). Brittle failure in compression: Splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society of London 319: 337-374.
SECTION
7.7
Delamination of Coatings HENRIKMYHREJENSEN Department of Solid Mechanics, 404, Technical University of Denmark, DK-2800 Lyngby, Denmark
Contents 7.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 7.7.4 Identification of the Material Parameters .... 7.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 7.7.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
582 582 583 584 585 586 586
7.7.1 VALIDITY This is a method for evaluating the adhesive strength and the development of delamination of thin coatings under small-scale yielding conditions. The boundary between delaminated and bonded parts of the coating is treated as an interface crack front. The radius of curvature of the crack front has to be large compared to the thickness of the coating.
7.7.2 FORMULATION For a large class of problems including delamination of thin coatings, possibly with a thin adhesive interlayer, on a substrate, the analysis of the adhesive strength can be evaluated by a method which allows the boundary between the bonded and unbonded coating to be treated as an interface crack front. If the radius of curvature of the crack front is large compared with the thickness of the coating, the analysis further simplifies into a plate-type problem for the
582
Handbookof MaterialsBehaviorModels.ISBN 0-12-443341o3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
583
7.7 Delamination of Coatings
delaminated coating with boundary conditions along the crack front given by displacement continuity. For a coating which is thin compared to the substrate, displacement continuity is simply formulated by imposing the substrate deformations as boundary conditions along the crack front. For the plate problem, geometrical nonlinearities due to large deformations of the delaminated coating may be included. The interface fracture mechanics parameters are in turn computed by the effective normal (N) and shear (5) membrane stresses and the bending moment (M) in the delaminated region along the crack front as obtained by the plate problem. The generalized stresses N, 5, and M should be regarded as differences relative to the state in the bonded part of the coating just outside the crack front, which in general should be nonzero due to residual stresses and external loading of the composite system. A fracture criterion of the type G = G~V(~,, 4,)
is then formulated where G is the energy release rate, Gc is the (mode 1) fracture toughness and F is a toughness, function which describes the dependence of the adhesive strength on the phase angles of loading, ~ and ~b measuring the near tip mode mix.
7.7.3 DESCRIPTION OF THE MODEL The energy release rate and the phase angles of loading are given by 2) M 2 + h2N2~ + $2(1 + v) Eh 3 12 J Eh
G-6(1-v
tan ~ -
v ~ 2 M cos co + hN sin co
cos q~ = SV/1 + v
-V/12M sin co + hN cos co' where E and v are the Young's modulus and the Poisson's ratio for the coating, assuming isotropic elasticity, and h is the coating thickness. The angle co is a function of the elastic parameters for the materials on the two sides of the interface. The following empirically based toughness functions have been suggested on the basis of experimental observations: F - (1 + (22 - 1)sin2~ sin2~b + (23 - 1)cos2~) -1 F= l+(1-24)
tan21/r
F = 1 + tan2((1 - 25)~),
for~=rt/2 for ~b = rt/2
584
Jensen
where the parameters 2 lie in the range 0 to 1 and adjust the relative contributions of modes 2 and 3 to the fracture criterion. The following fracture criterion has been suggested on the basis of a simple micromechanical model where the variation of the fracture toughness with the phase angles of loading is assumed to be the result of rough crack faces getting in contact under shear-dominated loading conditions, including frictional sliding:
F --
1,
f o r O < 6 and 4) = n/2
(cos(O - a) - # sin(O - a)) -2,
for a < ~ < a + arccot # and 4) = n/2
Here,/z is the Coulomb friction coefficient between the crack faces and 6 is the angle between the normal to facets on the crack faces and the crack plane, assumed to be the same for all facets. For ff > ~ + arccot/t the crack faces are frictionally locked. Toughness functions formulated for combined mode 1 and 2 loadings (~b = re/2) can approximately be generalised to other loading conditions by substituting ff with 0 where
(Khi~) vq*c
cos 0 - Re
where e is the bimaterial index and
E
+
with subscript ( )~ denoting elastic constants of the substrate and fl the second Dundurs' parameter. Finally,
-
where 0~ denotes the first Dundurs' parameter.
7.7.4 IDENTIFICATION
OF
THE MATERIAL PARAMETERS The angle co is a function of the elastic mismatch in the bimaterial system; in most cases of interest 45 ~ ~ The elastic parameters for the coating and the substrate may be obtained by standard methods, including uniaxial testing.
585
7.7 Delamination of Coatings
The first Dundurs' parameter is under plane strain conditions given by E 1 -
Es v 2
1 -
v 2
1 -
v s2
O~ - - -
E 1 -
Es v s2
and the second Dundurs' parameter fi and the bimaterial index e for practical purposes may both be set to zero. The parameters Go, 2, 22, and 23, and, possibly, # and & if they are not directly measured, can be identified by fracture mechanics tests such as the cut test or the blister test. Residual stresses in the coating have a significant influence on the delamination mechanics and on triggering the active mode of delamination. The residual stresses may be determined by methods such as curvature measurement on coated wafers and x-ray diffraction lattice strain measurement.
7.7.5 HOW TO USE THE MODEL The model would typically be used to evaluate experimental results obtained by standardized tests for measuring adhesive strength and residual stresses, or it would be applied to design for safety against possible delamination modes. As high compressive stresses are often present in the thin coatings, interface crack propagation and buckling of the delaminated region are commonly observed as failure modes. Common delamination modes include edge delamination, buckling delamination, edge delamination buckling, and propagation of buckling delamination. For a straight-sided edge delamination parallel to a free edge (1 G--
-
v2)N 2 71; ' 4 ~ = 2 and
2Eh
6-c~
for N > 0
where N is the effective membrane stress component in the bonded coating perpendicular to the crack front, assuming that this stress is completely released in the delaminated region. For N < 0 the crack faces are closed. Instabilities of the crack front morphology frequently show up during &lamination propagation. When investigating for stability of the crack front shape, the mixed mode fracture criterion is compared on the original crack front and on a sligthtly perturbed front. The preferred shape of the front is that with the highest value of G/F(~,qS).
586
Jensen
7.7.6 TABLE OF PARAMETERS Data are available for the following model systems: Epoxy/glass Epoxy/glass Polyimide/glass Epoxy/plexiglass Mica/aluminium
22 = 24 -22 = 22 = 25 =
23 = 0.1-0.15 0-0.5 23 -- 0.1-0.15 0.3, 25 = 0.33 0.3
Gc GcGcG~ -
5-6 J / m 2 5-6J/m 2 2J/m 2 30J/m 2
Gc -
0.4-1.7J/m
2
REFERENCES 1. Jensen, H. M. (1991). The blister test for interface toughness measurement. Engineering Fracture Mechanics 40: 475-486. 2. Jensen, H. M. (1993). A numerical method for delamination in composites. Computational Materials Science 1: 319-324. 3. Jensen, H. M., Hutchinson, J. W., and Kim, K.-S. (1990). Decohesion of a cut prestressed film on a substrate. International Journal of Solids and Structures 26: 1099-1114. 4. Jensen, H. M., and Thouless, M. D. (1993). Effects of residual stresses in the blister test. International Journal of Solids and Structures 30: 779-795. 5. Jensen, H. M., and Thouless, M. D. (1995). Buckling instability of straight edge cracks. Journal of Applied Mechanics 62: 620-625. 6. Hutchinson, J. W., and Suo, Z. (1992). Mixed mode cracking in layered materials. Advances in Applied Mechanics 29: 63-191. 7. Suo, Z., and Hutchinson, J. W. (1990). Interface crack between two elastic layers. International Journal of Fracture 43: 1-18.
SECTION
7.8
Ductile Rupture Integrating Inhomogeneities in Materials (DRIIM) A. PINEAU and J. BESSON Ecole des Mines de Paris, Centre des Mat~riaux, UMR CNRS 7533, BP 87, 91003 Evry Cedex, France
Contents 7.8.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Formulation and Description . . . . . . . . . . . . . . . 7.8.2.1 Constitutive Equations . . . . . . . . . . . . . . . 7.8.2.2 Heterogeneities . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3.1 C-Mn Steel . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3.2 Hydrided Zircaloy . . . . . . . . . . . . . . . . . . . 7.8.3.3 Duplex Ferrite-Austenite Steel . . . . . . . 7.8.4 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 7.8.5 List of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
587 589 589 590 590 591 591 593 593 595 596
7.8.1 VALIDITY E n g i n e e r i n g m a t e r i a l s c o n t a i n i n c l u s i o n s or precipitates. T h e s e i n h o m o geneities either are u n d e s i r e d i m p u r i t i e s or have b e e n a d d e d to i m p r o v e the m e c h a n i c a l p r o p e r t i e s of the base material. T h e s e s e c o n d - p h a s e particles are p r e f e r e n t i a l sites for the initiation of voids (either by interface d e b o n d i n g or by particle fracture) w h i c h , in a ductile f r a c t u r e process, will g r o w a n d coalesce, l e a d i n g to final r u p t u r e . Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
587
588
Pineau and Besson
These second-phase particles cannot be perfectly homogeneously distributed in the matrix. Their distribution is spatially heterogeneous because particles tend to form clusters, whereas some regions of the material contain fewer particles. Their strength (or cracking resistance) may also be statistically distributed. The spatial distribution is characterized by a length scale parameter 2. Depending on the size of the sample or structure (L) compared to the characteristic length, two different situations can be envisaged: 2 <
589
7.8 Ductile Rupture Integrating lnhomogeneities in Materials
7.8.2
FORMULATION
7.8.2.1
AND DESCRIPTION
CONSTITUTIVE EQUATIONS
Inside clusters as well as in regions containing fewer defects, the material behavior is described using constitutive equations for porous materials. Two models are commonly used: Rousselier [3] and Gurson-TvergaardNeedleman (GTN) (see review in Reference [4]). They both use plastic yield surfaces (~b = 0) incorporating the effect of pressure which are given by Rousselier
GTN
f D exp ~b - - (1 -0-eq f)cry + ~00-y "1 2 0-eq
~b - --v + 2qlf, cosh
(3q20-m~
\ 2~y /
((1- f)0-1) -1 0"m
-1-
2 2
q~L
(1) (2)
where 0-eq is the von Mises invariant of the stress tensor q; 0-m is the mean stress 1/3 trace q; D, al, ql, and q2 are constant parameters; Cry is the flow stress of the undamaged material; and f is the porosity, jr, (GTN model) is some function of the porosity used to represent void coalescence. In most cases, the following simple form is used: f, = f if f
~p - fi ~
(3)
where ~ is the plastic multiplier obtained by the consistency condition q~ = 0. The evolution of the porosity by the mass conservation: f - (1 - f ) trace ~p
(4)
These models can be modified to account for the nucleation of new cavities. For strain-controlled nucleation (which is the most used nucleation law), Eq. 4 is then rewritten as f - (1 - f)trace ~p + fn
with fn - An/~
(5)
where fn is the amount of nucleated porosity. An is some material-dependent function of the state variables which relates the rate of nucleation fn to the cumulated plastic strain rate/~ defined by the model. In some cases, damage nucleation corresponds to the apparition of cracks in second-phase particles. This nucleation process does not correspond to the creation of cavities as in
590
Pineau and Besson
the case of debonding, so the volume fraction f is unchanged. In that case the evolution law o f f is given by Eq. 4. fn represents an effective porosity that is mechanically equivalent to the nucleated cracks. A total porosity ft is then defined as j~ = f +fn and used in lieu o f f in Eq. 1 or 2. fn can be computed from the crack density and the crack size, assuming that a crack has the same mechanical effects as a spherical pore having the same projected surface. This assumption is supported by recent extensions of Gurson's model to voided materials containing elliptical cavities [5].
7.8.2.2 HETEROGENEITIES To account for the heterogeneities in the material, it is necessary to define a characteristic length ~ together with a distribution function (~-) of the material random parameter corresponding to areas of size ~2 (resp. ~3) in the two-dimensional (resp. three-dimensional) case. The material is represented as a grid (although a better geometrical description such as Vorono'i cells could be used) in which each cell is randomly assigned a value of the parameter. An FE mesh is then superposed to this grid, as depicted in Figure 7.8.1. Each Gauss point corresponds to a grid cell and a given value. Note that the cell size 2 and the mesh size h are a priori not related. 7.8.3
IDENTIFICATION
Parameters of the constitutive equations, i.e., D, 0"1, ql, and qz, are usually chosen according to values commonly used for the Rousselier or Gurson
FIGURE 7.8.1 Schematic representation showing how a random value is attributed to each Gauss point of a mesh. The position of each Gauss point in the grid is used to determine the value of the random parameter.
7.8 Ductile Rupture Integrating lnhomogeneities in Materials
591
model. As long as damage remains limited (i.e., no macroscopic softening caused by damage), cry can be directly obtained from tensile experiments. According to the spirit of the "local approach to fracture," the characteristic length 2 and the distribution function ~- should be measured directly from the microstructure examination (e.g., using image analysis). Fitting of macroscopic data (displacement-load curve, crack advance) using ~- as the adjustment parameter is not advised. If some fitting appears necessary, realistic bounds for the parameters should be proposed based on microstructure examinations. According to the previous remarks, the identification procedure will strongly depend on the investigated material. In the following, it is illustrated using three very different materials: (i) a C-Mn steel containing MnS inclusions [6], (ii) Zircaloy containing various amount of zirconium hydride precipitates [7], and (iii) a ferrite-austenite duplex steel [8, 9].
7.8.3.1 C - M n STEEL In this type of material, ductile rupture is controlled by the growth and coalescence of voids initiated around large MnS inclusions which debond during the early stages of plastic deformation. These inclusions are not homogeneously distributed and tend to form clusters as depicted in Figure 7.8.2. The mean distance between clusters is equal to 250 ~tm, which is considered the characteristic length (2). Images of size 2 x 2 were then analyzed to determine the local inclusion content which corresponds to a local initial porosity which is considered as the random variable. The resulting distribution function is plotted in Figure 7.8.2.
7.8.3.2
HYDRIDED ZIRCALOY
In this example, ductile rupture is controlled by the nucleation of numerous cracks in brittle hydrides. This material is presented to illustrate how the nucleation kinetics can be measured based on microstructural observations. The nucleation kinetics can be experimentally determined using tensile necked specimen after rupture. Because of the presence of the neck, a strong plastic strain gradient exists. This allows, using a limited number of specimens, the establishment of a relation between plastic strain and crack density for different contents of hydrogen. The crack density is then converted into an effective nucleation kinetics as explained previously (Fig. 7.8.3).
592
Pineau and Besson
FIGURE 7.8.2 Distribution function of the local porosity measured using field of size 250 x 250~tm. The micrograph shows the heterogeneous spatial distribution of the MnS inclusions.
FIGURE 7.8.3 Experimental nucleation kinetics on hydrided Zircaloy for different contents of hydrogen. The micrograph illustrates the nucleation of cracks in the brittle hydrides and the subsequent void growth.
7.8 Ductile Rupture Integrating Inhomogeneities in Materials
593
FIGURE 7.8.4 a. Microstructure of a duplex ferrite-austenite steel showing a highly damage region surrounded by undamaged regions, b. Voronoi cells drawn around individual cleavage used to define damage zones.
7.8.3.3
DUPLEX FERRITE--AUSTENITE STEEL
In this duplex material the ferrite (30% vol.) is aged and embrittled so that cleavage cracks are nucleated during deformation. As the ferrite is surrounded by ductile austenite, cracks grow into voids. The observation of the material after deformation shows that damage is highly heterogeneous, as shown in Figure 7.8.4a. To quantify damage, tensile fiat samples were polished and subsequently deformed in order to reach different levels of plastic deformation. For each level, the surface of the specimen was observed using SEM, allowing us to determine the position of cleavage cracks. Vorono'i cells were then drawn around those cracks. A typical result is shown in Figure 7.8.4b. These observations were used to determine the surface crack nucleation rate. The effective nucleation porosity is then computed as in the previous case. As evidenced by the Vorono'i tesselation, cleavage cracks are not homogeneously distributed. The nucleation rate parameter An is therefore defined locally in the different regions. It was found that An can be considered as constant for a given area but that it varies spatially.
7.8.4 HOW TO USE THE MODEL Given a set of the different material parameters, the model is based on FE simulations of random structures as depicted in Figure 7.8.1. For each specimen type (smooth tensile bar, notched bar, cracked specimen), several simulations have to be performed in order to evaluate dispersions and mean values of mechanical properties. Figure 7.8.5 illustrates damage evolution in a notched bar. In particular, it was shown that, for a given average value of the
594
Pineau and Besson
FIGURE 7.8.5 Damage (white: no damage, black: broken) growth in a notched bar. Cracks propagate between highly damaged sites.
FIGURE 7.8.6 a. Comparison of experiment and simulated ductilities for different notch geometries (duplex steel). Averagevalues are represented by lines, b. Comparisonof experimental and simulated ductilities for different specimen sizes (C-Mn steel).
random parameter, heterogeneous materials have reduced mechanical properties compared to an ideally homogeneous material. This effect is mainly due to the interaction of highly damaged regions. Size effects are modeled using larger meshes for a given value of 2. In order to keep the same geometrical discretization of the microstructure, the element size should also be kept constant. Consequently, more freedom is needed to model large structures. Figure 7.8.6 illustrates the kind of results that can be obtained. Figure 7.8.6a compares experimental values of the ductility of smooth and notched bars to simulated results (TB: tensile bar, AEz: notched axisymmetric bars with ~ = 10 • R/dp, R: notch radius, ~b minimum diameter of the specimen). The ductility is defined as the reduction of area in the notch cross section, the initial diameter ~b is equal to l Omm. It can be noted that average ductilities for smooth tensile bars are equal to those observed
595
7.8 Ductile Rupture Integrating Inhomogeneities in Materials
on AElo notched bars. This is related to a size effect, since the deformed volume in a smooth bar is larger than in a notched bar that has the same initial diameter. Figure 7.8.6b compares experimental and simulated ductilities (Gurson and Rousselier models) for the C-Mn steel obtained for AElo notched bars with different initial diameters: 3.75, 6, 10, and 15 mm. It is observed that both dispersion and average values decrease with increasing size. This trend is correctly represented by the model.
7.8.5 LIST OF PARAMETERS TABLE 7.8.1
C-Mn Steel Random Parameter: Initial Void Volume Fraction
~,
d~/dx
250 gm
1 ~ l exp(~) fl~r(~)x-
Gurson ql
o~= 1.14, fl = 0 . 2 2 10 .2
TABLE 7.8.2
= 1.5 , q2 = 1.0,
8 = 3, fc =0.4%
Rousselier o-1
= 240MPa,
D = 1.5
Hydrided Zircaloy Nucleation Law (C~ hydride concentration [ppm])
An--bB exp (bp) with B = 5.10 -~ and b = 5.37 + 0.0006Cn
TABLE 7.8.3 2 i mm
Duplex Steel Random Parameter: Local Nucleation Rate An ~ Uniform An = 0 . 1 . . . 1 . 0
Gurson ql = 1.5, q2 = 1.0, no f,
Note: In the case of the duplex steel, it was not necessary to adjust an f, function. This is due to the fact that rupture is entirely controlled by a single mechanism and also to the integration of material heterogeneities in the model. On the other hand, an f, function had to be used in the case of the C-Mn steel. Examinations of fracture surfaces have shown that smaller particles (carbides) play a role during the coalescence of voids nucleated at MnS inclusions. This second population of defects is not accounted for by the model and has therefore to be represented phenomenologically.
596
Pineau and Besson
REFERENCES 1. Becker, R. (1987). Effect of porosity distribution on ductile failure. J. Mech. Phys. Solids 35 (5): 577-599. 2. Leblond, J. B., and Perrin, G. (1999). A self-consistent approach to coalescence of cavities in inhomogeneously voided ductile solids. J. Mech. Phys. Solids 47: 1823-1841. 3. Rousselier, G. (1987). Ductile fracture models and their potential in local approach of fracture. Nuclear Engineering and Design 105: 97-111. 4. Tvergaard, V. (1990). Material failure by void growth to coalescence. Advances in Applied Mechanics 27: 83-151. 5. Gologanu, M., Leblond, J. B., and Devaux, J. (1994). Approximate models for ductile metals containing nonspherical voids - - case of axisymmetric oblate ellipsoidal cavities. Trans. ASME, J. Eng. Mat. Technol. 116: 290-297. 6. Decamp, K., Bauvineau, L., Besson, J., and Pineau, A. (1998). Size and geometry effects on ductile rupture of notched bars in a C-Mn steel: Experiments and modelling. Int. J. Fracture 88 (1): 1-18. 7. Grange, M., Besson, J., and Andrieu, E. (2000). Anisotropic behavior and rupture of hydrided Zircaloy-4 sheets. Met. and Mat. Trans A. 31A: 679-690. 8. Joly, P., Meyzaud, Y., and Pineau, A. (1992). Micromechanisms of fracture of an aged duplex stainless steel containing a brittle and a ductile phase: Development of a local criterion of fracture, in Advances in Fracture/Damage Models for the Analysis of Engineering Problems, pp. 151-180, Giovahola, J., ed., ASME. 9. Devillers-Guerville, L., Besson, J., and Pineau, A. (1997). Notch fracture toughness of a cast duplex stainless steel: Modelling of experimental scatter and size effect. Nuclear Engineering and Design 168: 211-225.
SECTION
7.9
Creep Crack Growth Behavior in Creep-Ductile and Creep-Brittle Materials A. TOSI-IIMITSUYOKOBORI,JR. Fracture Research Institute, Graduate School of Engineering, Tohoku University, Aoba 01 Aramaki Aoba-ku Sendai-shi #980-8579, Japan
Contents 7.9.1 Creep Crack Growth Behavior in Creep-Ductile and Creep-Brittle Materials .. 597 7.9.2 Creep Damage Accumulation Behavior and Crack Growth Rate in Creep-Ductile Materials . . . . . . . . . . . . . . . . . . . . 600 7.9.3 Creep Crack Growth Behavior for Creep-Brittle Material . . . . . . . . . . . . . . . . . . . . . . 605 7.9.4 Prediction of Creep Fracture Life Throughout from Ductile to Brittle Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
7.9.1 CREEP CRACK G R O W T H BEHAVIOR IN CREEP-DUCTILE AND CREEP-BRITTLE MATERIALS [4] Experimental correlation between the creep crack growth rate and the C* parameter for Cr-Mo-V steel (a high-temperature creep-ductile material) and IN100 alloy (a high-temperature creep-brittle material) is shown in Figures 7.9.1 and 7.9.2, respectively [1, 2]. For the case of Cr-Mo-V steel, a non-unique correlation between d a / d t and C* in the lower creep crack growth region is clearly apparent. This region typically occupies 30-40% of the total creep fracture life [1]. In the case of IN 100 alloy, the behavior is different from that of the creep-ductile material. There is a tendency for crack growth rate to Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
597
598
Yokobori 100
|. . . . . .
"1
Cr- Mo-V
O
CT Specimen
!0-1 E E .
"0
.
"~ io-2 ._
9
Q
Ji
O
-
t, . 9 10-3 t
_
10-I
i
I
i0o
101
I,,
102
C* , kJ/m2/hr FIGURE 7.9.1
Creep crack growth rate plotted against C* [1] (Cr-Mo-V steel).
~ IN-I00(NO SG) B = 25.4mm T = 825 "C P 24500 N
Fracture
101 = 9: corresponds --= to C* decreasing Z O: corresponds --"
'
E -
-]
-
/
~j,O
././d" ./.~
tO
10o
10-I _ ~ e ~ -=- 90% of Fr~ture Life -EL/I I IIIIll 1 ! I!11111
lO0
101
I
102
C* , k J / m m 2 / h r
FIGURE 7.9.2
Creep crack growth rate plotted against C* [2] (IN100 alloy).
decrease initially, and the region where the data are concentrated near the lowest da/dt value occupies 80-90% of the total creep fracture life, as shown in Figure 7.9.2 [2]. Note that both creep crack growth rate and the load line displacement rates attain steady-state, constant values in this region, where the 80~90% of the total creep fracture life is concerned. The different characteristics between creep-ductile and creep-brittle materials which can be seen in Figures 7.9.1 and 7.9.2 are caused by the
599
7.9 Creep Crack Growth Behavior
differences in the behavior of creep crack growth and load line displacement. Creep crack growth curves of CT precracked specimens for Cr-Mo-V steel (a creep-ductile material) and IN100 (a creep-brittle material) were plotted against nondimensional time t/tf as shown in Figure 7.9.3 [3], where tf is the creep fracture life for each specimen. For the case of Cr-Mo-V steel, acceleration in the creep crack growth rate begins at a relatively early stage of creep fracture life, and it occupies approximately 60% of the total creep fracture life. On the other hand, in IN100 alloy, the creep crack growth curve is quite different from that for Cr-Mo-V steel. A linear relationship between creep crack extension and nondimensional time is observed, this region occupies the major portion of the creep fracture life, approximately 80%, and the accelerating portion occupies only 10% of the total life. As a result, the creep crack growth rate attains a constant value during the major portion of the total life. This region is called the steady-state region for which the growth rate is determined by the initial stress-intensity factor [2]. In the creep displacement curve for IN100 alloy, a linear relationship between load line displacement and nondimensional time is observed, as seen in Figure 7.9.4 [4], and it occupies a major portion of the creep fracture life, approximately 80%, similar to the creep crack growth curve. Similar characteristics were observed in structural strengthening materials such as a TiA1 intermetallic compound with full lamellar structure [4] and 12Cr-W-Co-B steel (TAF650) with lath martensite structure [4, 5].
l
'I
I
I
I
I
'
!
I
I I"
3
o ; IN 100 z~ ;Cr-Mo-V
',' /
-
/
E
1
0
0.2
0.4
0.6 t/tf
0.8
FIGURE 7.9.3 The relationship between creep crack length and IN 100, tf in creep fracture life [3].
1.0
tits
for Cr-Mo-V steel and
600
Yokobori
INIO0 2
0.6
0.4 "
~
'. . . . . . . . . . .
~
E
'~
0.2 ,q
0
0.2
0.4 0.6 t/tf
0.8
1
0
7.9.4 The experimental relationships between creep crack length and t/tf,, and load line displacement and t/tf. FIGURE
7.9.2 CREEP DAMAGE ACCUMULATION BEHAVIOR AND CRACK GROWTH RATE IN CREEP-DUCTILE MATERIALS [4] In situ observation on creep damage initiated from a notch tip in a double-edge notched specimen (DEN specimen) is shown in Figure 7.9.5 for Cr-Mo-V steel [4]. Tensile stress is applied in the horizontal direction. The dark region in Figure 7.9.5 shows the creep damage. In this region, voids are formed and their coalescence occurring at a grain boundary results in the formation of grain boundary microcracking as shown in Figure 7.9.6 [4]. Relative notch opening displacement time, A ~ / ~ 0, as the creep displacement and creep damage area D, are plotted against nondimensional time, as shown in Figures 7.9.7 and 7.9.8 [4]. Here, 9 is the notch opening value; ~0 is the initial value of the notch opening, i.e., the value before applying the load; and A~ = ~ - ~0 is the increment of the notch opening value 9 from the instant of load application [6]. All data of D and A ~ / ~ 0 are expressed by a unique master curve, respectively, as shown in Figures 7.9.7 and 7.9.8, which show an accelerating behavior almost throughout the curve that is independent of applied stress and temperature [4]. Some studies have performed the analyses of creep damage progression using the damage function, co, based on the Kachanov-Rabomov theory [7-11].
dco D* (Act 1 q- 3Bam + Cae) z d--7 = (1 + +)(I - co)`/'
(1)
601
7.9 Creep Crack Growth Behavior
FIGURE 7.9.5
Creep damage around the notch tip (Cr-Mo-V steel) [4].
FIGURE 7.9.6 Magnified photograph of creep damage. Microcracks are seen in grain boundaries (Cr-Mo-V steel) [4].
where D*,A, B, C, Z, and q5 are constants and A + B + C - - 1 , al is the maximum principal stress, am is the mean stress, ae is the effective stress, co is the damage function (0 _< co < 1), and O" 1 - - 3am - ae - a for the uniaxial tensile condition. Integrating Eq. 1, co is given by Eq. 2 for the uniaxial condition: -
(2)
where tf - (DaZ) -1, q 0 - 6. The characteristics of Eq. 2 are shown by the solid line in Figure 7.9.8, which is in good agreement with that of damage area, D, obtained by our
602
Yokobori
Cr-Mo-V DEN Specimen T~ 538 560
353 274 314
Xl & 0
353.
O
.o
Xll
(TOHOKU UNIV) --Catculation
SU.SSl~p,) Synr
io
~,
O
025
170 194
L=o
UI
J gkJf~agr
',, _ _ ' .
c,~'~
1
0.5
t/t,
FIGURE 7.9.7 The experimental relationship between notch opening displacement and t/tf. A~ = ~ - ~ s - ~ 0 : ~ o and ~s (the fracture time for each specimen) are notch opening displacement at time and the instantaneous one at the instant of load application, respectively. ~o is the initial notch opening value (notched specimen, Cr-Mo-V steel) [3].
.
,0
.
.
.
,
,
,
,,,
ij
DEN Specimen ('c) 600 6z~
E
'i94
"6r,o ss0
0.5
Crack Initiation
5|tess 5~'va)o~ (MP=)
remp.
.
.
4
17s
E!
ZZl ~9,
:I0
O
I
9 9
~i4
!
/
I = i
A 0
I
0.2
I
0.4
i
tlh
0.6
....
'
0.8
I
1.0
FIGURE 7.9.8 The experimental relationship between creep damage area size, D, and (notched specimen, Cr-Mo-V steel). Solid line is Eq. 2 [3].
t/tf
e x p e r i m e n t s . T h e r e f o r e , D is f o u n d to be c o r r e l a t e d to co. F r o m F i g u r e s 7.9.7 a n d 7.9.8, we can o b t a i n the r e l a t i o n s h i p b e t w e e n D a n d (A(I)/(I) 0) [4].
D-~-0.5 (A~00)~~
(3)
7.9 Creep Crack Growth Behavior
603
T h e s e voids in the creep d a m a g e r e g i o n c o n c e n t r a t e a h e a d of a m a i n crack a n d cause a front crack at this point. This links w i t h the m a i n crack, a n d creep crack g r o w t h occurs. T h e typical b e h a v i o r is s h o w n in F i g u r e 7.9.9 a n d 7.9.10 for SUS304 stainless steel u n d e r creep c o n d i t i o n s [4].
FIGURE 7.9.9 Microcrack initiation and coalescence with the main crack under creep condition, a. t/tf = 0.8. b. t/tf = 0.95 (SUS304 stainless steel, 650~ ag= 191MPa, tf= 68 h) [4].
FIGURE 7.9.10 Scanning electron micrograph of microcrack and creep damage around a main crack under creep condition (SUS304 stainless steel, 650~ Crg= 191MPa) [4].
604
Yokobori
From these results, it can be seen that the creep displacement curve is dominated by the damage accumulation behavior, which results in creep crack growth. For creep-ductile materials, the creep crack growth rate, da/dt, is related to the creep displacement rate, c$, in the range accelerated region except for the nose part (t/tf ~_ 0.4), as shown in Figure 7.9.11, as well as to the C* parameter (Fig. 7.9.1). From these results, mutual correlation among creep damage progression, creep displacement, and crack growth can be seen. The creep crack growth rate is obtained from Eq. 4 [12-16, 17-20,2,21,22]"
da dt where
0~1, 0~2,
and
C *al ~ C t 2 ~ (A(I)) ~ ~
exp(Q*)
(4)
are constants. From Eqs. 3 and 4 we have
0~3
da
a t = Bv(o , Co) where F is a function of o3 and co, and where B is constant. Therefore, the creep fracture life is given by Eq. 6:
tf --
/o"d t - L B(~:da
(f)) ~
--7 Ct ~
ioo
0~3
~ fa~a: Cda
*{X
(6)
exp(Q* )
I
!
Cr-Mo-Y
CT
I
%
Specimen
lO-i
E
~
"~ io-1
,t e
1o-3 I 10 -3
i
I,
,, ,,. L
10-2
10 -i
1oo
.
~. mm/hr FIGURE 7.9.11 The relationship between creep crack growth rate and load line displacement rate for Cr-Mo-V steel [3].
605
7.9 Creep Crack Growth Behavior
Therefore, co is found to be a dominant factor of creep fracture life for a precracked specimen and can be characterized by the dominant parameters of the creep crack growth rate, such as C* [12-15], Ct [16], ~s [17-201, and Q* [2,21,22], as is shown in Eq. 6. From these results, creep fracture lives of precracked and notched specimens for creep-ductile materials were found to be dominated by creep damage progression.
7.9.3 C R E E P C R A C K G R O W T H CREEP-BRITTLE MATERIAL
BEHAVIOR FOR
It has been shown by Yokobori and Ichikawa [23] that, for homogeneous structure materials, the macroscopic crack proceeds macroscopically perpendicular to the applied stress in a zig-zag crack growth manner that is independent of the orientation of the initial microcrack, under the assumption that the stress intensity at the tip of the nearest segment controls the further extension of the total (main) crack. On the other hand, for heterogeneous materials such as a full lamellar TiA1 intermetallic compound, many prospected zig-zag crack paths are distributed in terms of lowerstrength grain boundaries around the hard lamellae. In this case, the crack will proceed along the grain boundaries which are located as close as possible to the maximum stress intensity. In this way, for this case, the crack proceeds in a zig-zag crack growth manner that is macroscopically perpendicular to the applied stress. This behavior is in good agreement with the zig-zag crack growth observed by this experiment (Fig. 7.9.12) [4]. Now, in this case, if the length of the segment corresponding to zig-zag crack growth is assumed to be nearly equal, say, to the length of the grain size, then the macroscopic crack growth rate will be revealed as constant (equal velocity), because this segment size will be controlling the creep crack growth rate. That is, in such materials, with a microstructure which causes crack bridging or provides the stress shielding effect, the creep crack growth rate may not be controlled by the total length of the past zig-zag path, but by the nearest segment length of the main creep crack. Since the segment length may be nearly equal, a constant rate of subcritical crack growth and creep displacement occurs. This causes the characteristics of a constant rate of creep crack growth and creep displacement over a major portion of the creep crack growth and displacement curves, as shown in Figure 7.9.4. The same reasoning may be used with 12Cr-W-Co-B steel [4, 5] and IN100 alloys [2]. For these creep-brittle materials, the creep crack growth rate was shown to be dominated by the initial stress-intensity factor, Ki, and the temperature, T, as given by Eq. 7a [2]. Therefore, some modifications are necessary to obtain a dominant parameter which characterizes the creep crack
606
Yokobori
FIGURE 7.9.12 Creep crack path and fracture pattern for TiA1 intermetallic compound. T = 850~ crg = 87.75 MPa, tf= 178 h [4].
growth rates for these materials. The Q* parameter for creep-brittle materials was derived and is given by Eq. 7b [2]. Furthermore, since the Ct parameter includes a local stress component around a crack tip under a small-scale creep condition [16], it can be correlated to Q~rittle as given by Eqs. 7b and 7c. d~ dt
=fl (Ki, T)
(7a)
= A exp(Qbrittl~)
(7b)
= f2[Ct(k = Ki)]
(7c)
7.9.4 P R E D I C T I O N OF CREEP FRACTURE LIFE T H R O U G H O U T FROM D U C T I L E TO BRITTLE MATERIALS The Q* parameter is defined as a power coefficient of the exponential in the thermal activation process equation for da/dt for both creep-ductile and
607
7.9 Creep Crack Growth Behavior
creep-brittle materials [2, 21, 22].
da dt - A exp ( Q* )
(8)
where A is a constant independent of temperature. Q* can be written as
a~g -
4,g(o)
RT
(9)
where AHg is the activation free energy for crack growth and dpg(a) is a positive function of the local elastic-plastic stress field around a crack tip. Integrating Eq. 8, creep fracture life concerned with creep crack growth is given by
1
m, _ Aoag exp RT f
f~c da
(10)
Since ac ~ a0, Eq. 10 reduces to
ts=
1
(
AHg~
Aoa~ gexp - RT ,]
(11)
where A1 = Ao[(m1-2)/2] 6/0 "(m~-2)/2 On the other hand, with respect to the creep strain rate, many studies have been carried out with a thermal activation process, such as a diffusion mechanism [24], a viscoelastic creep mechanism, and a cavitational mechanism [24,25]. The creep strain rate, ~, is written by the following thermally activated equation: ~--Alexp (-AHg-dPc(a))RT
= Acagexp (- AHc~
RT/
(12a)
(12b)
where Ac and mc are constants. By multiplying both sides of Eqs. 11 and 12b, we get [26]
tf~ = Ma~exp - RT /
(13)
where m = m c - mg, AH*= AHc-AHg, and M--At~A1. In general, mc is not equal to mg and AHc is also not equal to AHg. Therefore, the right-hand side of Eq. 13 is in general a function of applied stress eg and temperature T. For instance, with respect to SiC as typical creep brittle materials, the relationship between the logarithmic value of the creep fracture life, log tf, versus that of the steady creep rate, log ~s, is experimentally expressed by a
608
Yokobori
10 2 I.. z:
I
i
101
10o E
I
IL,&O'~
I~20~'~'~
10-1
ID D
10-2 -
=
re" v
SiC
10-s
i0-4
Steady Creep Rate
i0-3 ~s
i0-2 , /hr
FIGURE 7.9.13 The relationship between creep fracture life and steady state creep rate for SiC as affected by temperature [26].
10 3 r
I
I
I
I"
10 2
101 ~=.
ID
~.
10 o Sht~
10-1 10-2 .
~,
9 -
ID L
10-3
FIGURE 7.9.14 Si3N4 [26].
I
!
10-5
10-~
I .......
10-3
|
I
10-2
10q
10 0
Steady Creed Rate ~;~ Ihr The relationship between creep fracture life and steady state creep rate for
linear line, and the line depends on the temperature and applied stress, as shown in Figure 7.9.13. This relationship is theoretically suggested by Eq. 13. On the other hand, for the case of m~ - mg and AH~= AHg, then the righthand side of Eq. 13 takes the constant value and the straight line becomes one and the same line independent of applied stress and temperature for Si3N4, as shown in Figure 7.9.14. [26]. Apparently, the equation is in accord with the formula of Monkmann and Grant [27], but it is to be noted that the
609
7.9 Creep Crack Growth Behavior 10 3 ,.-
'
"1
'
i
~
~.
I^-%
I
"-
L
.
I
10 2
101 ~D
10 -I
~
lo -2
oo
10 -3
10-6
' 10-5
10-~
Steady
, I
104
Creep Rate
I0 -2
E s
I
I0-I ,
100
/hr
FIGURE 7.9.15 The relationship between creep fracture life and steady state creep rate for various materials [26].
right-hand side of Eq. 13 differs with different m a t e r i a l s - it may be called creep ductility. In Figure 7.9.15, the relationship between log tf and log ks is plotted for various materials. From this figure, it can be seen that for the same fracture life, tf, the steady state creep rate, 4, will decrease, say, become brittle as the creep ductility given by the right-hand side of Eq. 13 will decrease. Detailed descriptions and correlation with the microstructure obtained by TEM (transmission electron microscope) may be found elsewhere [28]. These characteristics enable us to discriminate creep ductility and predict creep fracture life not only for creep-ductile materials but also for creepbrittle materials.
REFERENCES 1. Yokobori, A. T., Jr., and Yokobori, T. (1996). Engineering Fracture Mechanics 55: 493. 2. Yokobori, A. T., Jr., Uesugi, T., Yokobori, T., Fuji, A., Yamaya, I., Kitagawa, M., Yagi, K., and Tabuchi, M. (1998). J. Mater. Sci. 33: 1555. 3. Yokobori, A. T., Jr. (1997). Advances in fracture research, in Proc. of the Ninth Int. Conf. Fract. (ICF9th)., Karihaloo, B. L., et. al., eds., vol. 1, p. 39. 4. Yokobori, A. T., Jr. (1999). Engineering Fracture Mechanics 62: 61. 5. Yokobori, A. T., Jr., Takamori, S., Yokobori, T., Hasegawa, T., Kubota, K., and Hidaka, K. (2000). Key Engineering Materials 171-174: 131. 6. Yokobori, A. T., Jr., Yokobori, T., Kuriyama, T., and Kako, T. (1984). Proc. of the Sixth Int. Conf. Fract. (ICF6th), vol. 3, p. 2181.
610
Yokobori
7. Kachanov, L. M. (1958). IZV. Akad. Nauk. SSSR, Otd.tekhn.nauk 8: 26. 8. Rabotnov, Yu. N. (1959). Vopr. Prochn. Mat. I Konstr. Izd. AN. SSSR, Moscow. 9. Lemaitre, J., and Plumtree, A. (1979). Journal of Engineering Materials and Technology, Trans. of the ASME 101: 284. 10. Bassani, J. L., and Hawk, E. (1990). Int. J. Fract. 42: 157. 11. Hayhurst, D. R. (1992). J. Mech. Phys. Solids 10: 381. 12. Landes, J. R., and Begley, J. A. (1976). ASTM STP 590: 128. 13. Ohji, K., Ogura, K., and Kubo, S. (1975). Trans. Jap. Soc. Mech. Engng. 44: 183. In Japanese. 14. Taira, S., Ohtani, R., and Kitamura, T. (1979). Trans. ASME, J. Engng. Mater. Technol. 101: 154. 15. Koterazawa, R., and Mori, T. (1977). Trans. ASME, J. Engng. Mater. Technol. 99: 298. 16. Saxena, A. (1986). ASTM STP 905: 185. 17. Yokobori, A., T., Jr., Yokobori, T., Kuriyama, T., Kako, T., and Kaji, T. (1986). Proc. of the Int. Conf. on Creep, JSME, IMechE, ASME 135. 18. Fuji, A., and Kitagawa, M. (1994). Advances in Fracture and Structural Integrity, p. 487, Panasyuk, V. V., et. al. eds., Pergamon. 19. Dogan, B., and Schwalbe, K-H. (1992). ASTM STP 1131, Erunst, H. A., Saxena, A., and McDowell, D. L., eds., p. 284. 20. Saxena, A., Dogan, B., and Schwalbe, K-H. (1994). ASTM STP 1207, Landes, J. D., McCabe, D. E., and Boulet, J. A. M. eds., p. 510. 21. Yokobori, A. T., Jr., and Yokobori, T. (1989). Advances in fracture research, in Proc. of the 7th Int. Conf. Fract. (ICF7th), p. 1723, vol. 2, Salama, K., et. al., eds. 22. Yokobori, A. T., Jr., and Yokobori, T. (1993). Creep and fracture of engineering materials and structures, in Proc. of Fifth Conf. London, p. 81, Wilshire, B., and Evans, R. W., eds., The Institute of Materials. 23. Yokobori, T., and Ichikawa, M. (1965). Proc. of the First Int. Conf. Fract., p. 1039, vol. 2, Yokobori, T., et. al., eds. 24. Lange, E E, Clarke, D. R., and Davis, B. I. (1980). J. Mater. Sci. 15: 611. 25. Evans, A. G., and Wiederhorn, S. W. (1974). J. Mater. Sci. 9: 270. 26. Yokobori, A. T., Jr., Yokobori, T., and Yamazaki, K. (1996). J. Mater. Sci. Lett. 15: 2002. 27. Monkmann, E C., and Grant, N. J. (1956). Proc. ASTM 56: 593. 28. Yokobori, A. T., Jr., Yano, T., and Mitchell, T. E. (1997). Materials Sci. Forum 233-234: 109.
SECTION
7.10
Critical Review of Fatigue Crack Growth TAKEO YOKOBORI School of Science and Engineering, Teikyo University, Utsunomiya, Toyosatodai, 320-2551, Japan
Contents 7.10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.2 Identification of the Power Exponent and Proportional Constant . . . . . . . . . . . . . 7.10.3 How to Use the Model . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
611 612 619 620
7.10.1 BACKGROUND There are many theories, models, criteria, and mathematical equations [1-19] for fatigue crack growth; however, these include discrepancies. Most predict some experimental characteristics but do not explain others. Thus it is necessary to examine what theories or models are more correct or to develop a new criterion. Regarding this complex problem, the following statements may be helpful. 1. The ln(da/dN) versus AK curve assumes an S-type or sigmoidal behavior (Fig. 7.10.1), that is, a first region, a second region, and a third region, including final catastrophic fracture. Different factors may be predominant for each region, the mechanism of which still remains unsolved. Thus if we attempt to describe behavior covering all three regions in terms of a single equation, it may be necessary to include many parameters which may not be correlated with the micromechanism. Also, it may even be awkward and thus may not be useful from the mechanism's point of view. It is unrealistic to assume that, in the first region, however small AK may be, the crack will still grow. Thus, as a matter of course, the effective Handbookof MaterialsBehaviorModels.ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
611
612
Yokobori
i,.,
,,J...
,
!I~
o "13 o .=.
J
,~//
',/,
z
T /
,,,J.
first I second )' regionl region /Ilhir( 'I ' /Ire;, reg;i g I I
,
I /' !/
I I
l i
I
I
II I!
!
,! i
L
I
KTH
FIGURE 7.10.1
I
A
~+
Kmln
=
Kfr
10g A K
Schematic illustration of fatigue crack growth rate versus AK.
stress-intensity factor AK~ff, smaller than AK, for instance, A K - A KTH, should be used instead of AK only, where AKTH is the threshold stressintensity factor. In the third region, it is also needless to say that da/dN will accelerate, and the factor controlling final catastrophic fracture should be included in the equations. Nevertheless, the maximum stress-intensity factor Km~x is, in general, different from KF~ (so-called static fracture toughness), and, instead, fatigue fracture toughness Kf~ is proposed [20, 21]. 2. If the mean stress is significant, then this effect should be included. As already described, however, the first step is to approach the case with a small or zero stress, and then to modify the equations to include it.
7.10.2 IDENTIFICATION OF THE POWER EXPONENT AND PROPORTIONAL CONSTANT This section concerns, from the foregoing point of view, the identification and the justification of the theories and the models on fatigue crack growth described in Section 7.10.1. Below or at room temperature, the plastic deformation near the fatigue crack is under small-scale yielding. Thus, if we assume only the energy balance criterion, the criterion determining the crack growth rate may obey
613
7.10 Critical Review of Fatigue Crack Growth
linear elasticity fracture mechanics. On the other hand, the fatigue crack growth rate is not completely predicted [19,22, 17] by linear fracture mechanics. This, in turn, means that in fatigue crack propagation the local plastic stress distribution may play a dominant part. This may also be inferred, since the plastic deformation near the fatigue crack tip may play an important role whatever the physical model may be. Thus for the case of fatigue, even under small-scale yielding it is necessary to formulate the local stress distribution near the crack tip. The result obtained is as follows: 2fl @ -- f(fl)Crcy
v/~@y
(1)
where AK is the stress-intensity factor; @y is the initial yield stress in cyclic straining (not static yield stress); fi is the cyclic strain-hardening exponent; f(fl) is some function of fl; and ~ is distant from the crack tip in the direction of crack. On the other hand, the experimental formula for the fatigue crack growth rate d a / d N for a simple case is expressed as [1] da - AAK n dN
(2)
where A and n are constants. The problem is how A and n are correlated with the material parameters and testing conditions. By using Eq. 1 as the local stress distribution near the crack tip, the mathematical formula for the fatigue crack growth can be derived for a nucleation model [6, 17], for a dislocation group dynamics model [18, 19], and for a vacancy diffusion model [22]. Thus d a / d N is given in the following form:
a--d =
B
(3)
where n is expressed by the equation shown in Table 7.10.1, according to the different models. That is, d a / d N of the type of Eq. 1 is in common with all three physical models as far as stress-intensity factor dependence is concerned. In Figures 7.10.2 and 7.10.3 the n experimentally obtained is plotted against d -1/2 (d = averaged ferrite grain diameter) and static yield strength asy. It should be noted that n decreases with an increase of d -1/2 or a decrease of d (or increase of static yield stress Crsy). It is interesting to note that n tends to be a slowly increasing function of ferrite grain diameter. In Figure 7.10.4, logx0A was plotted against n for plain low-carbon steels [23-24], 3Si-Fe [25], high-strength eutectoid steel [26,27], high-strength
614
Yokobori
TABLE 7.10.1 Power Coefficient n of AK in da/dN ( T = a b s o l u t e Boltzmann's constant) Dislocation group dynamics model
Nucleation model
m'
Vacancy diffusion model
l+~L(m+2) +
1+13 1
v
-2/s
(1 +p)(1 + [3)
m - H~/4kT,
= 2 . 2 6 ~ p ( 1 - v 2)
+
m" = constant p = measure of time increase of the energy vacancy concentration in the plastic region [22]
Hk = activation energy [18,191
[16,171
temperature, k--
.
.
n
4-
9
9
o
32|
---
0
I
I
I,
2 I
I0
,
,
I
I
3
4
1
I
15
5
......
m
6
I
I
20
7
I
I
8
9
....
I.
25
10 d" 89 " 89 I
,
(Y',~ ( kg/rnm 2 )
FIGURE 7.10.2 Value of n against ferrite grain diameter and static yield stress for low-carbon steels. C)" 0.05 weight percent carbon steel [23]" @" 0.08 weight percent carbon steel [24].
tempered martensitic steel [24, 28], and high hardened bearing steel [29, 30]. Figure 7.10.4 shows that lOgloA is expressed by an equation of the following type: log A -- log B - n l o g ( x / ~ a c y
)
(4)
which corresponds to the result obtained by equating the right-hand side of Eqs. 2 and 3. Figure 7.10.4 shows that lOgloB and logl0v/~acy in Eq. 3 are practically independent of static yield strength asy, but n is a slowly increasing function of the ferrite grain diameter, and thus indirectly a decreasing function of static yield strength for low-carbon steel and 3 weight percent silicon iron, respectively.
615
7.10 Critical Review of Fatigue Crack Growth A R- 0.27 A R,O.5
I0 %
% %%
",. /X
n 5
,,, i
o
;
z
,,
,q
.
.
.
.
.
.
.
a
5
t/,Fd- (mm't)
3'5 ............
4'O
"
45
O",y kg/mm z
FIGURE 7.10.3 Value of n against ferrite grain diameter and static yield strength for 3Si-Fe [25].
Equation 3 can be shown as in Figure 7.10.5. If n assumes various values instead of a fixed value, say 4 or 2, all the lines predicted by Eq. 3 cross one point corresponding to In AK = lnx/~acy or In AK/x/~acy = 1. The data for the case of low-carbon steel as affected by ferrite grain size show that n increases with the increase of d or the decrease of d -1/2, and this trend is manifested in Figure 7.10.5. It is interesting to note that in the region of the smaller da / dN (orA K), say, nearthefi rstregionincludingthethreshold, da/dNishi ghe
rforsmal lergrainsizesatthesamev alueof A K, asshownbyFi gure 7.10.5. Thetrendmay correspondtotheobserv ationsontitaniumal loy s [31]andontheeffectofgrainsi zeonK;H oflow - carbonsteel[32]. In Figure 7.10.6, n is plotted against static yield strength asy for both lowcarbon steels [23, 24] and 3 weight percent silicon iron [25]. It should be noted that n is not uniquely determined by ferrite grain size or indirectly by static yield strength, but is also affected by some other microstructural factor. This feature may also be reflected in the data for logloA and n for highstrength tempered-martensitic steel, eutectoid steel, and high hardened
616
Yokobori
-8 -9
l
-IO
o'11 m
0
-12 t
3
-I
-14 -15
+
-16 .
-17
0
.
.
I
.
.
2
.
.
3
.
.
4
.
.
5
.
6
-
7
8
9
rl--~
FIGURE 7.10.4 Plot of lOgl0A versus n for low-carbon steels and 3Si-Fe wRh various ferrite grain diameters, respectively. 9 0.05 weight percent plain carbon steel [23]; 0 : 0 . 0 8 weight percent plain carbon steel [24]; &: 3Si-Fe ( R - 0.27) [25]: • high-strength eutectoid steel [26,27]; Q high-strength tempered martensitic steel [24,28]; +: high hardened bearing steel [29, 30]. Data on high-strength and high hardened steels also are plotted.
bearing steel plotted as in Figure 7.10.4. This figure shows that, for the two former steels, the higher the static yield strength, the smaller the value of n, and that, on the other hand, for the more hardened bearing steel, the value of n is rather larger than those of the two former steels. All these features probably show that n may be a function of the strain-hardening exponent, especially of the cyclic strain-hardening exponent, and this measure may be related to ferrite grain diameter (indirectly related to the monotonic yield strength). If ~ is assumed to be the length over which the high value of AK is averaged, A K / x / ~ in Eq. 3 will mean a measure of high localized stress, and in this case local stress may be a criterion-determining factor. If crcy is assumed to be the same as crsy, then ~ in Eq. 3 should be proportional to 2 so that x/~crcy is independent of crsy as seen from the inverse of crsy,
617
7.10 Critical Review of Fatigue Crack Growth
da
Ind-R
---ns> na)n, i
ni
ds :>dl > d,
d~/' , ..nz
/
I
l
/
In[~'(:Tc
InZ~K - > FIGURE 7.10.5 Schematic illustration of the proposed relation of by ferrite grain size in the second region.
AK as affected
i
8.0 7.0 6.0 5.0
da/dN versus
.
4.03.0-
2.01.00
0
I
1
I
I
20
.I
I
30
static yield stress ,
,
I
.
I
40
,,I
I
50
(Tsy kg/mm t
FIGURE 7.10.6 Comparison of n for low-carbon steel and 3Si-Fe as a function of static yield strength changed by ferrite grain size. Arrow shows this value will be higher for R = 0.24-0.40. Legend is the same as in Figure 7.10.4.
618
Yokobori
t
Pororneter
nt
O'sy --~ FIGURE 7.10.7 Schematic illustration of monotonic strain-hardening exponent ~, and monotonic yield strength asy
decrease --- :: d
FIGURE 7.10.8 Schematic illustration of cyclic strain-hardening exponent ]~ and ferrite grain diameter d.
Figure 7.10.4. In this case, however, e should not necessarily be the same as the total length of the plastic zone, and therefore is not proportional to the 2 Also, it is concluded that v~asy is not proportional to AK, inverse of Crsy. because, if this were the case, then Eq. 3 would show no dependence on AK. Further, even if we assume that a criterion-determining factor is energy, it is still not established a priori that e should be proportional to the 2 inverse of asy. It is well known [33] that the static strain-hardening exponent 2 against static yield stress asy is as shown in Figure 7.10.7, which shows that 2 is also a function of some microstructural parameter such as the percentage of carbon in steels. It has been proposed [16,19,34] that strain-hardening characteristics, especially cyclic strain-hardening characteristics, play a role in fatigue crack growth. Furthermore, the cyclic strain-hardening exponent // has a similar dependence [35] on ferrite grain size or indirectly on static yield strength, as shown in Figure 7.10.8, which is similar to Figure 7.10.7. On the other hand, da/dN based on elastic-plastic treatment in the criteria [16, 19, 34] is expressed as proportional to AK", where n is an increasing
619
7.10 Critical Review of Fatigue Crack Growth
function of ft. From this and Figure 7.10.8 it may be inferred that the power coefficient n will show a decreasing function with a decrease of ferrite grain size (or indirectly with an increase of asy) with some other microstructural parameter, as shown in Figure 7.10.6. Concerning the other microstructural parameter, the difference in the configuration of dislocation cell structures between low-carbon steel and 3 weight percent silicon iron may be mentioned. Furthermore, acy is not necessarily equal to or proportional to static yield strength asy. Instead, acy is considered practically unaffected [35] by ferrite grain size, or directly by asy. For the foregoing reasons, Eq. 3, in which n is expressed respectively as three models in Table 7.10.1 will be plausible.
7.10.3 HOW TO USE THE MODEL From dimensional analysis and a comparison of mathematical equations for fatigue crack growth in the literature, the fatigue crack growth rate in the second region is expressed with sufficient generality for the case without significant mean stress as d--N= S
a
where ac and e are constants with dimensions of stress and length, respectively. By compiling the experimental data and using the foregoing equation, one reaches the following conclusions. 1. B and v~ac in the foregoing equation are practically independent of monotonic yield strength, but n tends to be a slowly increasing function of the ferrite grain diameter, and thus indirectly a decreasing function of the monotonic yield strength for low-carbon steel and 3 weight percent silicon iron, respectively. 2. The latter features mean that n is an increasing function of the cyclic strain-hardening exponent. 3. n may also be affected by some other microstructural factor. 4. From the experimental characteristics described in Section 7.10.2 and the theoretical models described in Section 7.10.1, it can be seen that crc in Eq. 1 corresponds to the initial yield stress in cyclic strain hardening, acy. 5. Furthermore, the characteristics of the power exponent n in Eq. 1 are in accord with those predicted by the theories described in Section 7.10.1.
620 6. From this point of view, most of the existing equations growth should be subject to some criticism, at least for and 3 weight percent silicon iron. 7. In the region of smaller da/dN (or AK), say, near including threshold, da/dN may be higher for smaller same value of AK.
Yokobori
for fatigue crack low-carbon steel the first region grain size at the
REFERENCES 1. Paris, P., and Erdogan, R. (1963). Transactions, American Society of Mechanical Engineers, Journal of Basic Engineering 85: 528. 2. Liu, H. W. (1961). Transactions, American Society of Mechanical Engineers, Journal of Basic Engineering 83: 23. 3. Paris, P. (1964). Fatigue: An Interdisciplinary Approach, Syracuse: Syracuse University Press, 107. 4. Raju, K. N., (1972). International Journal of Fracture Mechanics 8: 1. 5. Cherepanov, G. P., and Halmanov, H. (1972). Engineering Fracture Mechanics 4: 219. 6. Rice, J. R. (1967). Fatigue Crack Propagation ASTM STP 415: 247. 7. Weertman, J. (1965). Proceedings, First International Conference on Fracture, Sendai, Japan, 1: 153. 8. Weertman, J. (1973). International Journal of Fracture Mechanics 8: 125. 9. Mura, T., and Lin, C. T. (1974). International Journal of Fracture Mechanics 10: 284. 10. Lardner, R. W. (1971). Philosophical Magazine 17: 71. 11. Schwalbe, K. (1973). International Journal of Fracture 9: 381. 12. Pook, L. P., and Frost, N. E. International Journal of Fracture 53. 13. Tomkins, B. (1968). Philosophical Magazine 18: 1041. 14. McEvily, A. J. (1974). The Microstructure and Design of Alloys: The Metals, p. 204, London: 15. Donahue, R. J., Clark, H. M., Atanmo, P., Kumble, R., and McEvily, A. J. (1972). International Journal of Fracture Mechanics 8: 209. 16. Yokovori, T. (1969). Physics of Strength and Plasticity, pp. 327-338, Argon, A. S., ed., Cambridge: Massachusetts Institute of Technology Press. 17. Yokobori, T., and Ichikawa, M. (1968). Reports of Research Institute for Strength and Fracture of Materials 4: 45-53. 18. Yokobori, T., Yokobori, A. T., Jr., and Kamei, A. (1975). International Journal of Fracture 11: 781-788; also, Corrigenda (1976), 12: 519-520. 19. Yokobori, T., Konosu, S., and Yokobori, A. T., Jr. (1977). Proceedings, Fourth International Conference on Fracture, University of Waterloo, Waterloo, Ontario, Canada, vol. 1, pp. 665-682. 20. Yokobori, T., and Aizawa, T. (1970). Reports of Research Institute for Strength and Fracture of Materials 6 (1): 19-23. 21. Kawasaki, T., Nakanishi, S., Sawaki, Y., Hatanaka, K., and Yokobori, T. (1975). Engineering Fracture Mechanics 7: 465-472. 22. Yokobori, T., and Ichikawa, M. Reports of Research Institute for Strength and Fracture of Materials 6: 75. 23. Yokobori, T., Kawada, I., and Hata, H. (1973). Reports of Research Institute for Strength and Fracture of Materials 9 (2): 35-64.
7.10 Critical Review of Fatigue Crack Growth
621
24. Yokobori, T., Tanaka, M., Hayakawa, H., Yoshimura, T., and Sasahira, S. (1967). Reports of Research Institute for Strength and Fracture of Materials 3 (2): 39-71. 25. Ando, K., Ogura, N., and Nishioka, T. (1976). Preprint, Japan Society of Mechanical Engineers, No. 760-2, 153 (in Japanese). 26. Yokobori, T., Sawaki, Y., Shono, S., and Kumagai, A. (1976). Transactions, Japan Institute of Metals 17 (1): 1-10. 27. Yokobori, T., Sawaki, Y., Shono, S., and Kumagai, A. (1976). Reports of Research Institute for Strength and Fracture of Materials 12 (2): 29-54. 28. Yokobori, T., Kuribayashi, H., Kawagishi, M., and Takeuchi, N. (1971). Reports of Research Institute for Strength and Fracture of Materials 7 (1): 1-23. 29. Yokobori, T., and Nanbu, M. (1976). Reports of Research Institute for Strength and Fracture of Materials 2 (2): 29-44. 30. Yokobori, T., and Aizawa, T. (1977). Reports of Research Institute for Strength and Fracture of Materials 13 (2): 75-78. 31. Pineau, A. G., private communication. 32. Masounave, J., and Batlon, J. P. (1976). Scripta Metallurgica 10: 165. 33. Hollomon, J. H. (1945). American Institute of Mining Engineers Technical Publication No. 1879, 1. 34. Yokobori, T. (1969). Reports of Research Institute for Strength and Fracture of Materials 5: 19. 35. Yokobori, T., Ishii, H., and Koyama, S. (1979). Scirpta Metallugica 13: 515-517. 36. Yokobori, T. (1979). ASTM STP 675: 683-701.
SECTION
7.11
Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects HAi~L M U G H R A B I
Universitht Erlangen-Niirnberg, Institut fiir Werkstoffwissenschaften, Martensstr. 5, D-91058 Erlangen, Germany
Contents 7.11.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.3 Description of the Model: Intrinsic and Damage-Induced Nonlinear Compliance ... 7.11.4 Identification of the Material Parameters .. 7.11.5 Applications: Examples of Assessment of Fatigue Damage Based on Changes in the Nonlinear Elastic Compliance . . . . . . . . . 7.11.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
622 623 624 624
626 631 632
7.11.1 VALIDITY Fatigue damage in the form of cracks which reduce the load-bearing cross section gives rise to a nonlinear (elastic) compliance which superimposes on the intrinsic nonlinear elastic behavior. In this article, the theoretical principles for the assessment of fatigue damage (and crack propagation), based on the measurement of nonlinear compliance effects and on complementary data obtained from parallel replica studies of surface cracks, will be outlined. The applicability to fatigued plain (unnotched!) specimens of commercial alloys will be demonstrated. It will be shown that, under
622
Handbook of MaterialsBehaviorModels. ISBN 0-12-443341-3. Copyright O 2001 by Academic Press. All rights of reproduction in any form reserved.
7.11 Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects
623
favourable conditions, the novel technique proposed is suitable for obtaining in a nondestructive manner data on fatigue crack propagation, crack opening, and crack closure from fatigue experiments on plain specimens.
7.11.2
BACKGROUND
It has been shown earlier that the differential Young's modulus of an undamaged material do ED(o") -- deez
(1)
becomes noticeably stress-dependent at larger stresses in the sense that ED decreases (increases) at higher tensile (compressive) stresses [1]. This is a consequence of the anharmonicity of the atomic potential, which is probed increasingly when a material is strained to higher and higher elastic strains, giving rise to an increasingly nonlinear elastic behavior. The latter can be taken into account by adding a second-order term in Hooke's law (corresponding to a third-order term in the potential energy), so that O.
-
-
Eo " Sel + k" ~,el
(2)
The constant Eo corresponds to Young's modulus in the linear Hooke approximation in the limit of vanishing o., and k is a constant which describes the deviation from linear elastic behavior. The constant k is always negative. Wasserb/ich [2] has derived a relation which expresses the constant k in terms of the third-order elastic constants and has listed the computed values of the constant k (which he has called kM) and the elastic moduli Eo for a dozen different steels. From Eqs. 1 and 2 it follows that the differential Young's modulus is given by
ED-
~E2o+ 4ko.
(3)
The parameters Eo and k are real material constants which are closely related to the atomic potential curve of the material in question. In a pure elemental material, this relationship is well founded. In single-phase alloys or in more complex multiphase materials, the constants Eo and k should be viewed as effective material constants averaged over the constituents of the material. The material constants Eo and k can be determined in a cyclic deformation test on an undamaged specimen from the slopes of the hysteresis loop, i.e., from the ED-values, at the load reversal points in tension and compression and/or by intermittent elastic unloadings within a closed cycle, as described in Section 7.11.4; compare also Reference [1].
624
Mughrabi
Now, in the case of a damaged material, the load-bearing cross section decreases (increases) progressively with increasing tensile (compressive) stress. Therefore, in the stress-strain response curve, the tangent modulus decreases (increases) in a qualitatively similar fashion as the differential Young's modulus of the undamaged material. The difficulty is, of course, to separate the intrinsic nonlinear behavior from the additional nonlinear contribution which superimposes when the material is damaged and which can be considered as a direct measure of the damage, as will be described in the next section.
7.11.3 DESCRIPTION OF THE MODEL: INTRINSIC AND DAMAGE-INDUCED NONLINEAR COMPLIANCE In the absence of damage, the elastic stress-strain behavior will be well described by Eq. 2 with a stress-dependent differential (tangent) Young's modulus given by Eq. 3. It has been observed that, once damage develops, a relation such as Eq. 2 is still suitable for describing the elastic stress-strain behavior. Now, however, the tangent modulus ED and the parameters Eo and k lose their meaning as true material constants and begin to deviate, showing increasingly nonlinear behavior with reduced ED-values, as long as the cracks (cavities) are open. This behavior is also reflected in a decrease of Eo and an increase in magnitude of the negative parameter k [3]. According to Eisenmeier et al. [4,5], the difference AED between the intrinsic value of ED of an undamaged specimen and of a damaged specimen with opened cracks (cavities) can be expressed as
AEo - IE2o + 4kor - Eo(ar)
(4)
Here, Eo and k are the intrinsic material constants of the undamaged specimen, ~r~r is the stress at the tensile reversal point, and ED (~rir) is the stiffness of the damaged specimen at that point. Alternatively, AED can be obtained from a series of elastic unloadings within a closed cycle [1], as described in Section 7.11.4. The normalized value AED/Eo has been found to be a suitable damage parameter, as will be described in text following.
7.11.4 IDENTIFICATION OF THE MATERIAL PARAMETERS As a starting point, it is necessary to identify the intrinsic material parameters ED(~r), Eo, and k, which can all be obtained from a hysteresis loop of the
625
7.11 Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects
undamaged specimen. Here, it is important to plot the stress not vs. the total strain et but vs. the plastic strain epl = e , t - cr/Eo in the linear Hooke approximation. It should be noted that, in a plot of cr vs. et, one usually obtains (very narrow) loops that are inclined with a slope of approximately the Young's modulus. Frequently, these loops have unusual, sometimes sickleshaped forms [1]. Quite generally, they are not suitable for extracting the information which is of interest here. In the plot of o vs. gpl, the elastic slope becomes vertical in the limit of vanishing stress and corresponds to the intrinsic (linear) Young's modulus Eo. At finite positive and negative stresses, the elastic slopes will now deviate from the infinite (vertical) limit as a consequence of the intrinsic nonlinear behavior and will assume (high) positive and negative values, respectively. An example from the work of Biermann et al. [6] on the particulate reinforced metal-matrix composite (MMC) AA6061-A1203-15p-T6 is shown in Figure 7.11.1. There are two possibilities for obtaining ED(cr) from the hysteresis curve. First, it is sufficient to determine the extreme values of ED at the peak tensile and compressive stresses ~rT and rrc, respectively, from the slopes of the elastic lines immediately after stress reversal. With these two values, an optimal fit according to Eq. 3 with appropriate values of Eo and k can be obtained. A more accurate technique is to perform a number of elastic unloadings within
300 2OO 13_ ~ 100 :~
f <1/r
0
j r "'
s
r
-100
L"
-200 -300
-2
'
' 0
'
' 2
'
~ 4
'
6
~:p~/ 10 -4 FIGURE 7.1 l . l Hysteresis curve of aluminium-matrix composite AA6061-AIzO3-15p-T6 with elastic unloadings at different stresses in a plot of o vs. epl--e,t-cr/Eo. Ae,t/e= 3• -3, N=18189. Before the elastic unloadings, the anelastic relaxation was allowed to decay. From Reference [6].
626
Mughrabi
220
$===l
0
210
r o.
w
200
l.iJ
190
180 . . . . . . . . . . I -3OOO -2000
I ii -'K)O0
I 0
....
I,, 1000
I= 2000
~,,~
3000
o[MPa] FIGURE 7.11.2 Dependence of differential Young's modulus Eo of a bainitic variant of the roller bearing steel SAE 52100 on stress ~ at room temperature. Data points were obtained from elastic unloadings. The curve has been fitted to the data points according to Eq. 3. From Reference [1].
one closed cycle, as is also shown in Figure 7.11.1. Then, it is possible to determine ED(a) for each of the stresses from which the unloadings were performed. From an optimal fit of Eq. 3 to these values of ED, one can then obtain quite acurate values of Eo and k. For the example shown in Figure 7.11.1, Eo ----88.6 4- 0.3 GPa and k -- - 3 1 8 4- 20 GPa [6]. An example for the actual measured stress dependence of ED(cr) is shown for a bainitic variant of the roller bearing steel SAE 52100 in Figure 7.11.2, from Reference [1]. Once the behavior of the undamaged specimen has been well characterized, the onset and development of fatigue damage can be monitored continuously by repeating the procedures described already and by noting any changes in ED, Eo, and k, compared to the corresponding values of the undamaged specimen. Examples of the assessment of fatigue damage in different materials will be presented in the next section.
7.11.5 APPLICATIONS: EXAMPLES OF ASSESSMENT OF FATIGUE DAMAGE BASED ON CHANGES IN THE NONLINEAR ELASTIC COMPLIANCE As a first example of how to apply these concepts, the strain-controlled fatigue behavior of the fatigued aluminium-matrix composite AA6061-A1203-20p-T6 which is reinforced with A1203-particles with a volume fraction of fp - 20%
627
7.11 Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects
will be considered [3]. It is found that Eo and k (in tension, i.e., with open cracks) depend in an interesting manner on the total strain amplitude Ae,t/2, as shown in Figure 7.11.3 for N = Nf/2 cycles (Nf: number of cycles to failure). At low amplitudes, Eo and k (in tension) are constant (Eo = 97.9 GPa, k = - 5 0 2 GPa), as expected. Starting at about Ae,t/2 = 0.004, Eo begins to decrease and becomes constant again at Ae,t/2 = 0.007. The parameter k decreases continuously. From these results, it could be concluded that, at total strain amplitudes which exceed a value of about 0.004, particle fracture occurs. This conclusion is consistent with metallographic observations of broken particles. It should be pointed out that, in an earlier study of monotonic deformation, Lloyd [7] also related a decrease in Young's modulus to an increase of the fraction of broken particles. In the case of MMCs, there is a difficulty in relating the variations in the nonlinear compliance quantitatively to the loss of the load-bearing cross section, since the Young's moduli of the matrix ( ~ 68GPa) and of the particles ( ~ 380 GPa) differ very strongly. Moreover, since the damage is more or less homogeneously distributed internally and not in the form of discrete propagating cracks, the nonlinear variations of the compliance cannot be evaluated reliably in order to obtain information on crack propagation. As will be shown in the following, the approach described
100
-1000
98
A
0
O
- -800
9
96
--
E0
~3
i,
94
92
9- - o - - k
.@
6
N-Nf/2
0 " 0 . . . . "(5 " 0
f p - 20 vol. % g
i
90
I
2
,
I
4
- -600
,
RT
I
I
6
8
i
I
t
-400
10
A e t / 2 / 1 0 -3 FIGURE 7.11.3 Variation of Eo and k in the fatigued aluminium-matrix composite AA6061A1203-20p-T6 as a function of total strain amplitude A~t/2, indicating particle fracture at Aet/2 > 0.004. From Hartmann et al. [3].
Mughrabi
628
previously can be applied advantageously in the case of more homogeneous materials such as the cast magnesium alloy AZ91. In References [4,5], it has been demonstrated that the propagation of fatigue cracks can be studied advantageously by a combination of surface replica studies (compare Reference [8]) and measurements of compliance in order to obtain the differential stress-dependent Young's modulus ED, as defined in Eqs. 1 and 2; compare also Reference [9]. The modulus ED should be measured both as a function of the number of cycles and also intermittently within closed hysteresis loops. As shown in the following for the cast magnesium alloy AZ91, it is thus possible to obtain information on the propagation of cracks not only along the surface but also into the depth. In addition, from the in situ compliance measurements during a closed cycle, crack opening and crack closure stresses can be determined. Figure 7.11.4 shows a sequence of traces of cracks obtained from surface replicas taken intermittently in a fatigue test on the alloy AZ91 after different numbers of cycles, employing the technique first applied by Ebi [8]. This technique permits one to trace the evolution of the fatal crack to its origin shortly after its initiation. In the case of the cast magnesium alloy AZ91, many small cracks initiate almost simultaneously at contraction cavities [4,5,9]. Then, larger cracks form as a result of the coalescence of these microcracks, until one fatal crack propagates and failure occurs. _.~---._..s'-.-~ ~
I
N-
150
-"- -.,
N = 300
-'
cy
--
N = 705 0.3 m m
N = 906 FIGURE 7.11.4 Tracesof coalescing fatigue microcracks in the magnesium alloy AZ91 forming the main crack, as observed in a sequence of surface replicas after different numbers of cycles N. A~t/2 -- 5• -3, T -- 20~C. From Reference [4,5].
629
7.11 Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects
The loss of the load-bearing cross section is reflected directly in appropriate compliance measurements, making use of the Eqs. 1-3. Thus, Figure 7.11.5a shows an example of elastic unloadings during a cycle early in the fatigue test, when there still exists no fatigue damage. Here, the slopes of the unloading paths, which are vertical in the limit of vanishing stress, decrease a little at higher tensile stresses and become slightly negative at higher compressive stresses. This is typical of the intrinsic nonlinear elastic behavior. Figure 7.11.5b shows examples of intermittent elastic unloadings later in the test after fatigue damage has spread over a larger part of the cross section. Now the nonlinear effects described previously for the intrinsic nonlinear behavior become much more pronounced and reflect the superposition of an appreciable additional nonlinear component due to the fatigue damage. The variation during the cycle of the differential elastic modulus ED according to Eqs. 1 and 2 (with appropriate values of Eo and k) is plotted in Figure 7.11.6 as a function of the stress or. It should be noted that this figure shows two important features, namely, the intrinsic stress dependence of ED for the specimen with closed cracks (upper line and dashed extension) and the stress dependence with damage, after the (main) crack has opened at a tensile stress of about + 12.5 MPa (lower inclined line). Crack closure corresponds to the jump AED from the lower to the upper line at a compressive stress of about - 2 5 MPa, as the stress is decreased from tensile to compressive values. The behavior is well described by the damage parameter AED/Eo, which is related to the loss of the load-bearing cross section [4, 5].
lO0
lO0
50
50
0
0
a)
D
D
-50
- 1 0 0
-50 . -0.04
.
0.00
.
ct-cs/E0/%
. 0.04
. 0.08
100 0.00
,
0.02
,
0.04
,
0.06
8t-cs/Eo/%
FIGURE 7.11.5 Hysteresis loops in the form of cr vs. (~t-':r/Eo) of fatigued AZ91 with intermittent elastic unloadings, Aet/2 -- 2.25• -3, 20~ a. Early in fatigue life, N/Nf ,~ 10%. b. Late in fatigue life, N/Nf '~ 80%. From References [4, 5].
630
Mughrabi
42
41
4O
=39
38
~ m
n -
9 compression to tension 3 7 A , tension tq cprrtPres~ion . . . . -100-75-50-25 0 25 50 75 100 /MPa FIGURE 7.11.6 Differential elastic modulus ED (stiffness values), as determined by the intermittent elastic unloadings shown in Figure 7.11.5b. From References [4, 5].
14
16
12
9 zx --
10
crack length from replicas ZXED/E0 by unloading tests AED/E o from stiffnesses at
8
12
l
tensile peak stresses ~ n
=~~ 6
Art/2
=
5 x 10-3
/
T = 20 ~ o
/ J
4
/ -
8
/ /
~
4
[.~~ m <
2 M
0
~ 0
,
I 400
800 N
,
I 1200
,
0 1600
FIGURE 7.11.7 Evolution of crack length (main crack) at the surface (replicas) and in the bulk (see text), as derived from the damage parameter AEo/Eoin different ways. From References [4, 5]. Finally, in Figure 7.11.7, the development of the crack length is plotted, as deduced by different methods [4,5]. The damage parameter AED/Eo is approximately proportional to the length of the main crack, as observed on the replicas, up to about 80% of fatigue life and then deviates toward higher
7.11 Assessment of Fatigue Damage on the Basis of Nonlinear Compliance Effects
631
values. This behavior reflects the fact that the crack obviously first spreads mainly along the surface and then propagates into the bulk only at a later stage. The magnitude of the deviation reflects the mean depth of the crack into the bulk. In Figure 7.11.7, another set of values of AED/Eo, which were determined automatically solely from the stiffness values at the load reversal points in tension and compression, are also plotted; compare Eq. 4. The good agreement between these data and those derived from the elastic unloadings is emphasized. In summary, the experimental approach just described can be considered novel and has the merit of obtaining crack growth data (in a quasinondestructive manner) during a standard fatigue test on a plain (unnotched) specimen without the need to use a standardized fracture mechanics specimen. The work reported here has been complemented by a mesomechanical model of fatigue crack initiation and growth which describes the fatigue crack growth and fatigue life data very satisfactorily [10].
7.11.6 TABLE OF PARAMETERS The intrinsic nonlinear elastic compliance of a number of engineering materials has been studied in the past. The values reported for the material parameters Eo and k are listed in Table 7.11.1. The values of Eo are accurate within 1-2%, those of k within about 5%. The magnitudes of typical variations of Eo and k in fatigue-damaged specimens follow from inspection of
TABLE 7.11.1 List of Intrinsic Material Constants Eo and k, as Determined for Materials Temperature in ~
Eoin GPa
k in GPa
Steels: SAE 52100, bainitic SAE 52100, martensitic
20 20
205 203
-750 -720
[1] [11
Al-alloy, peak-aged: AA 7022
20
73
-285
[111
MMC: AA6061-A1203-15p-T6 AA6061-A1203-20p-T6 X2080-SiC-15p-T4
20 20 20
88.6 97.9 99
-318 -502 -420
[61 [31 [12]
400 20
99.5 111.3
-261 -185
[111 [11]
Material
Ti-alloys: IMI 834 ]%Ti Beta-C
Ref.
632
Mughrabi
Figures 7.11.3, 7.11.6, and 7.11.7. It is clear that such variations and the values of the corresponding damage parameter AED/Eomust be determined from case to case, employing the procedures previously outlined.
REFERENCES 1. Sommer, C., Christ, H.-J., and Mughrabi, H. (1991). Acta Metall. Mater. 39: 1177-1187. 2. Wasserbhch, W. (1991). Phys. Stat. Sol. (b) 164: 121-125. 3. Hartmann, O., Biermann, H., and Mughrabi, H. (1998). In Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, pp. 431-436, Rie, K. T., and Portella, P. D., eds., Elsevier Science Ltd. 4. Eisenmeier, G., Mughrabi, H., HSppel, H. W., and Ding, H. Z. (2000). In DFG-Kolloquium 2000 "Lebensdauervorhersage", pp. 153-164, Deutscher Verband ffir Materialforschung und-prffung e.V. 5. Eisenmeier, G., Holzwarth, B., HSppel, H. W., and Mughrabi, H. (2000). In Proceedings of ICSMA 12, Special Volume of Mater. Sci. Eng. A (in press). 6. Biermann, H., Beyer, G., and Mughrabi, H. (1996). In Verbundwerkstoffe und Werkstoffverbunde pp. 197-200, Ziegler, G., ed., DGM Informationsgesellschaft, Oberursel. 7. Lloyd, D. J. (1991). Acta Metall. Mater. 39: 59-71. 8. Ebi, G. (1987). Doctorate Thesis, Rheinisch Westf'hlische Technische Hochschule, Aachen. 9. Mughrabi, H. (2000). In Fracture Mechanics: Applications and Challenges, invited papers presented at 13th European Conference on Fracture (ECF 13), ESIS Publication 26, pp. 13-28, Fuentes, M., et al., eds., Elsevier Science Ltd. 10. Ding, H. Z., Eisenmeier, G., and Mughrabi, H. (2000). In Proceedings of Fourth International Conference FATIGUE 2000: Fatigue and Ductility Assessment of Materials, Components and Structures, pp. 209-217, Bache, M. R., et al., eds., EMAS Ltd. 11. Christ, H. J. (2000). Personal communication. 12. Maier, H. J., Rausch, K., and Christ, H.-J. (1996). In Proceedings of the Sixth International Fatigue Congress FATIGUE'96, pp. 1469-1474, vol. 3, Lfitjering, G., and Nowack, H., eds., Pergamon.
SECTION
7.12
Damage Mechanics Modeling of Fatigue Crack Growth XING ZHANG AND JUN ZHAO Division 508, Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China
Contents 7.12.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.2 Constitutive Relations . . . . . . . . . . . . . . . . . 7.12.3 Governing Equations of the Process Zone . . . . . . . . . . . . . . . . . . . . . . 7.12.4 Boundary Conditions of the Process Zone . . . . . . . . . . . . . . . . . . . . . . 7.12.5 Crack Growth Rate . . . . . . . . . . . . . . . . . . . . 7.12.6 Table of Parameters . . . . . . . . . . . . . . . . . . . 7.12.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
633 633 634 637 638 644 644 644
7.12.1 VALIDITY This approach is valid w h e n the crack length increment during one load cycle is very small in comparison with the total crack length.
7.12.2 CONSTITUTIVE RELATIONS The damage coupled constitutive relations of cyclic loading can be established by means of the strain equivalence principle [1] and the O s g o o d - R a m b e r g equation and Masing principle [2] in the vicinity of the fatigue crack Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
633
634
Zhang and Zhao
tip as follows: ~
-n
eij = ~(D
3b(1 ~ = 2(E(1)n
n-l~
tTe
sij
(1)
where !
gij -- Fij - c~ijFij/3,
cp -- 1 - D
Fe -- ~r
(2)
and F/j is the cyclic range of the stress component, D is the extent of damage, E is the Young's modulus, b is the coefficient of plastic term in the OsgoodRamberg equation, n is the exponent of previous term, and (1 is the magnification factor due to subsequent yielding. It must be mentioned that there are no crack surfaces in the current model; instead, there is a continuous, long, thin damage band (Fig 7.12.1). The increment rate of ~ per load cycle for a prescribed material point is given as
>
aN
l,
0,
(3)
_<
where N is the number of load cycles, C(R) is a known function of the cyclic stress ratio R, and Y is the cyclic range of the damage driving force and equal to 1
(4)
= -~ Cijklq)-2Fij~kl
where Cij~z is the compliance of the tensor component of the original sound material.
7 . 1 2 . 3 G O V E R N I N G E Q U A T I O N S OF THE PROCESS ZONE In the polar coordinate system depicted in Figure 7.12.1, the Airy stress function A corresponding to Fij can be expressed by
,4- ~rZ+2,4(0)
(5)
from which the stress components satisfying the equilibrium equations will be ~rr -- ~ r ' ~ r r ( O ) ,
~00 -- R r ~ o o ( O ) ,
~rO -- R r ~ r O
(6)
635
7.12 Damage Mechanics Modeling of Fatigue Crack Growth
traditiona~ t rack
(r,O)
~
x i-
ces zone crack length --------- da/dN FIGURE 7.12.1
A fatigue crack in the sense of damage mechanics.
Application of Eq. 2 gives
Srr ~-
St0 -- ~l'2"SrO,
SO0 -- ~r~goo(O),
~r2Srr(0),
~ = ~r~(0)
(7)
(8)
The extent of continuity (p is equal to (9)
q) = flr~'Fp(O)
It is known that the crack tip is a point of null continuity; then/2 > 0. According to the damage coupled constitutive equations, i.e, Eq. 1, we have
~,rr - -
rn(2-#)~,rr(O)
~
~oo - ~
r"(~-~)~oo(O)
~rO -- ~
rn(2-~)~,ro(O)
(10)
The equation of compatibility is
O~;rr
1 02~rr
r oqO2
02
2(O~,rO~
Or ~- ~ r 2(re'~176-
-
r\
r
O0 ,]
--
0
(11)
Substitution of Eq. 10 into Eq. 11 gives --H
--!
err -11-ClerO -t- C2~,rr + C3~,00 - - 0
(12)
where C1 =
- 2 ( S -~ 1),
C2 = -S,
C3 = S(S + 1),
S = n(2 - #)
(13)
Zhangand Zhao
636
Eq. 13 is a fourth-order nonlinear differential equation of A(0) and ~(0) containing the unknown eigenvalues 2 and/.t. Equations 4, 9 and 10 give us //~,~ 2
Y(0)
~F-- k-fiJ r2(~-~) E '
E
Y(O)--
CijkZFP-2~kZ#ij
(14)
Inserting Eq. 14 into Eq. 3 yields
d~o
d---N=
C (~/2p+2
E(p+ 1)
1)(2-#) ~p+l
r2(p+
(15)
Using Eq. 9 and Figure 7.12.2 and preserving the dominate terms, we have
d~ dq9 = flr~_l(~,si n 0 - #~cos 0) dN dN
(x6)
Comparison of Eq. 15 and Eq. 16 shows ~tsin 0 -/.t~cos 8 - _~p+l
(17)
2(p + 1 ) ( 2 - #) -- # - 1
(18)
1 dN
EP+1
(19
fl
Eq. 17 is another nonlinear ordinary differential equation of A(0) and ~(0).
I a fixed element J ~ ]
!
/,---
j I
/
/
I
!
i
X
da FIGURE 7.12.2
Geometric relations with respect to crack growth.
637
7.12 Damage Mechanics Modeling of Fatigue Crack Growth
7.12.4 BOUNDARY CONDITIONS THE PROCESS ZONE
OF
Because of the symmetry of a mode I crack, it can be confirmed that
(20)
A' (0) = A"' (0) -- ~' (0) -- 0 From Eq. 5, we may assume
A(0)- 1
(21)
Furthermore, ~(0) and ~"(0), needed in the integration, can be expressed using Eqs. 12, 17 and 18 as forms of -
-
(0)]
(22)
A reasonable scheme of boundary conditions of the process zone is q~(r, Od) = O,
aoo(r, Od) -- a~0(r, 0d) -- 0
(23)
where 0 < 0a < re. Physically, the domain of 0 E [Od, rc] can be recognized as a region full of totally damaged media. So far, there are three undetermined values, i.e., #, k ~(0), and 0d, to be solved by three conditions shown in Eq. 23. Equations 12 and 17 can be integrated under the boundary conditions of Eqs. 20, 21 and 23 with consideration of Eq. 22 by means of the fourth-order Runge-Kutta method with adjustable steps. Then, A(O) and ~(0) can be obtained, and 2 can be found by Eq. 18. The computational results are shown in the table of parameters (Section 7.12.6), from which it can be seen that the damage field order is a positive number less than 1, the stress field order 2 is positive, the strain field order n(2 - #) is negative, and Od is greater than re/2 and less than n. The boundary of the plastic process zone can be also confirmed. In accordance with SEP and Masing's principle, the subsequent yon Mises yielding criterion for damaged media is given by
=
as
(24)
where a~ is the initial yielding stress. Substituting Eqs. 2 and 9 into Eq. 24 results in 1
Figures 7.12.3 to 7.12.6 show the normalized angular distribution modes of stress, strain, continuity, and the profiles of the plastic process zones, in which COp- rp(0).
Zhang and Zhao
638
Lfan s ssI (a)
0.6
frO0
p=0.5, n=3.0
(b)
0.6
0.4
O.4
0.2
0.2
~
o
o'~
o 030
~.
60
90
120
0I ~
30 (d) 60p=1.590=9.0120 "'n ]
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
~
o
o
~
1 ~0 _0 40 60 80 100 120 140 ]
1 --~40
i
~
0.8
-
(plane stress)
(~0 L . I , 00n=~; i40
oa -
0.6
0.6
0.4
0.4
0
p=0.5, n-9.0
0.8
0
30
60
90 ANGLE
0
(plane strain)
ffoo
0
20
40
60
80
, I00
0 (degrees)
FIGURE 7.12.3 Normalizedangular distribution of stress. 7.12.5
CRACK GROWTH
RATE
It can be seen from Eq. 19 that 0~ and fl are required to determine the crack growth rate da/dN. For this purpose the demarcation of different zones needs to be performed first. Since the yield limit is higher than the fatigue damage threshold, the plastic process zone, elastic process zone, and nondamaged elastic influenced zone lying from the crack tip to the outer area are depicted
639
7.12 D a m a g e M e c h a n i c s M o d e l i n g of F a t i g u e C r a c k G r o w t h
a)
p~.5 :n=3.0
1.4
(plane s t r e s s ) ~
1.2 ~-
0.4
o.6
0
o.2
-0.4
-0.2
-0.8
~ 30
O2 ~-(c) ~r~
b)
p~i5, n=9.0
1
0.8
Z9
b
j 60
-0.6t
, 90
0
l 30
1.75 t
(d)
120
7 =2"0'~--3"0 --
~
i
I
60
90
120
p=l.5, n--9.0
1.5 1.25
Z
N
0.5
0.75
0
3.25
-0.5
0.25
-I 0
1.8 [
20
40
60
80
100
0.75 c O.
120 140
(e) p~.5, n=3.0
~
1.8 ~
, 20
~ I 40 ....60
I 80
I 100
I 120 140
(f) p-~.5, n=9.0
1.3 1.3 0.8
0.8
0.3
0.3
-0.2
~
-0.2
-0.7
-0.7
.1.2 t 0
~ 30
~ 60
1 90
1
-1.2 0
35
70
105
ANGLE 0 (degrees) F I G U R E 7.12.4
N o r m a l i z e d a n g u l a r d i s t r i b u t i o n of strain.
in Figure 7.12.7. ~ and fl can be determined by employing the connection conditions between neighboring zones. For a primary investigation, we merely consider the connection requirement along the radius of 0 - 0. In the plastic process zone, from Eqs. 10 and 17, we have
~0 - ar~80(O),
~,ij - ~
r"(~-')e, ij(O)
(26)
640
Zhang and Zhao
0.8
0.8
q~
0.6
0.6
0.4
0.4
0.2
0.2
(plane stress)
0 F-
30
Z
1
0
0.8
[.Z
60
90
120
9
I
30
60
I
\
90
12Q
0.8
q~
0.6
q~
0.6 0.4
0.4
(plane stress)
0.2
O 0
0
(d) p=~.
0.2
!
i
I
30
60
90
(plane stress) '0
120
I
I
'
40
60
80
9
.
.
.
.
.
~ i
,~
100 120 140 .
0.8
0.8
q~
0.6 -
q~
0.6
0.4 0.2
,
,.,,
~D
Z
0
I
0.4
9
(plane strain) I
30
,,
I
,,
60
0.2
\~ . 3
0
90
ANGLE FIGURE 7.12.5
70
35
105
0 (degrees)
Normalized angular distribution of continuity extent.
While in the elastic process zone, it can be known that [3] ~!f) = ~(e)rCd(e)
~!f) =
~(e) ~g(e)~ r r
~!f)(0)
(27)
It can be concluded that the nondamaged elastic influenced zone has the same asymptotic fields as those of a V-notch [3] (see Fig. 7.12.7 and note that the
641
7.12 Damage Mechanics Modeling of Fatigue Crack Growth
... y/o)
[_
Co)
(a) 1.5
1.8
1.2
x/m A
-1
,
.
L
* ~
-0.5
0
0.5
L
-0.6
l
I
9
'
-0.2
'
i
0.2
0.6
y/m
y/co
(d)
(c)
4.
15
L
x/co ,
-2.5
-1.5
-0.5
0.5
.
x/~o
1
1.5
2.5
y/o)
,
1
-0.8
-0.2
~
0.4
1
y/o~
(e)
(f) 1.6 1. 0.8
1
x/~ ,
-1
I
-0.5
.
l
0
'
0.5
t
'
1.5
FIGURE 7.12.6
'
x/~
]
2
9
-0.6
i
-0.2
0.2
0.6
I
1
[
1.4
Profiles of plastic process zones.
material of 0 > 0d has no stiffness), so the concerned stress and strain components are
e!v. ) -- I(~vr(-1/q)el; ) (0)
~,~v) _ ff~.vF(_l/q)~,l;)(0),
q > 2
(28)
To establish the requirements of connection, the radius at 0 - 0 of the boundary point of each zone is needed. From Eq. 25 the radius at 0 - 0 of the
642
Zhang and Zhao
J
Linear elastic process zone \
Damage zone
Non-damage linear elastic influenced zone
-
/ Plastic process zone
FIGURE 7.12.7
Demarcation of different zones around the crack tip.
boundary point of the plastic process zone can be determined as follows: 1
From Eqs. 27 and 3, the radius at 0 = 0 of the boundary point of the elastic process zone will be
/
1
% ~ % (0)% (o)
The requirements of connection between the plastic and elastic process zones are
~_
~:/,
~00 _ ~ /
at ~ _ ~ ,
6-
0
(31/
The connection conditions between the elastic process zone and the elastic influenced zone will be Fire ) - F [ [ ),
~--~)
at
r--toe,
0--0
(32)
Substituting Eqs. 26-28 into Eqs. 31 and 32, we have -~ = Q/<[q(~-~) /3
/ 3 - S/<~-q~
(33)
where
Q- (Q~Q~)-(~-.), s- (R~R~Q~-"Q~-")-~
(34)
643
7.12 Damage Mechanics Modeling of Fatigue Crack Growth 1
(35)
e, - (e,Q,) ~-~ e!:)(o) err(0) 1
Pp --
~laS ee(O)]
r
8!~) (0)
~
(36)
1 e~
-
--7-(~--2-~
-
Inserting Eq. 33 into Eq. 19 and taking note of Eq. 18 results in da - Ff(~ dN '
~2(p+1)
F.- c ~
(37)
EP +1
which express the crack growth rate. Particularly in the case of small-scale damage, the noninfluenced zone around the crack tip can be described by linear elastic fracture mechanics. Thus the conservative integral J can be applied to correlate/(~ with/(i, which is the cyclic range of the crack stress-intensity factor, as follows: 1 --2
~2 (.oel_(q/2)i(Od)E1 ~ j _ -~1 KI Kv
(38)
where I(0d) can be derived using Eq. 28. According to Eqs. 30 and 36, we can rewrite Eq. 38 as ~q ~2
Sv
= KI
(39)
(PeQe)l-(q/2)I(Od)
Solving/(v from Eq. 39 and substituting it into Eq. 37 leads to da dN = c* K 2,
c*
?. (PeQe)l_(q/2)i(Od)
(40)
This equation is verified by some experimental result mentioned in Reference [4].
644
Zhang and Zhao
7.12.6 TABLE OF PARAMETERS TABLE 7.12.1 Asymptotic Orders and Other Related Parameters for the Plastic Process Zone (v = 0.3): 1 = Plane Stress, 2--Plane Strain
State
p
n
#
2
n ( 2 - #)
Oa
Relevant Section of Figs. 7.12.3-7.12.6
1 1 1 1 2 2
0.5 0.5 2 1.5 0.5 0.5
3 9 3 9 3 9
0.5840 0.8026 0.3228 0.6599 0.9912 0.9092
0.4453 0.7368 0.2099 0.5918 0.9898 0.8789
-0.4161 -0.5922 -0.3387 -0.6120 -0.0084 -0.2727
119.3 116.5 125.6 127.6 90.4 98.8
a b c d e f
7.12.7
CONCLUSIONS
With the aid of a mode I fatigue crack model coupled with plastic damage, we find that: (1) the stress field has no singularity whereas the strain field still has a singularity in the vicinity of the crack tip; (2) the angular scope of the process zone (1 active zone) is confined to [0, Oa] with (0 < Od< re) in the upper semiplane. (3) the material of 0 E [0a, rc] is totally damaged and thus has no rigidity; (4) a Paris-type formula for crack growth rate is theoretically derived.
REFERENCES 1. Lemaitre, J. (1992). A Course on Damage Mechanics, Berlin: Springer, pp. 12-16. 2. Moskvigin, B. (1965). Plasticity for Variable Loading. Moscow: Publishing House of Moscow University, pp. 103-107. 3. Zhao, J. and Zhang, X. (1995). The asymptotic study of fatigue crack growth based on damage mechanics. Engng Fracture Mech. 55: 131-141. 4. Liu, H.W. (1991). A review of crack growth analysis. Theor. Appl. Fracture Mech. 16: 91-108.
SECTION
7.13
Dynamic Fracture W. G. KNAUSS California Institute of Technology, Pasadena, California
Contents 7.13.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13.2 Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.3 Mode I Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13.3.1 Initiation and Arrest . . . . . . . . . . . . . . 7.13.3.2 The Running Crack: Roughness and Branching . . . . . . . . . . . . . . . . . . . . 7.12.3.3 The Running Crack: Speed Limit.. 7.13.3.4 Crack Speed and the Stress-Intensity Factor . . . . . . . . . . . . 7.13.4 Mode II Fracture . . . . . . . . . . . . . . . . . . . . . . . . . 7.13.5 Crack Front Waves . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
645 647 651 651 653 654 655 656 657 658
7.13.1 VALIDITY Fracture mechanics is a mechanics discipline that addresses the failure of materials in terms of crack propagation. The issue of the generation of cracks from an apparent continuum is usually ignored or circumvented, so that only the propagation of preexisting cracks is considered. The objective of fracture mechanics is then to predict the conditions under which a crack will propagate as a function of the geometry, loading, and material properties. Of primary concern is the response of the material in the immediate vicinity of the crack tip, which essentially serves as a stress intensifier of the stresses or deformations applied far from the tip. With respect to material properties, one typically distinguishes between brittle and ductile solids, with the latter representing a large range of plastic responses. Ignoring, for the moment, viscoelastic and viscoplastic material behavior, quasi-static fracture mechanics is usually (minimally) concerned with the dependence of the failure process Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
645
646
Knauss
on loading conditions and domain configuration, with the r a t e of deformations playing a secondary role or none at all. The field of dynamic fracture mechanics covers a vast range of physical and analytical research endeavors which is too large to cover in a brief review. However, it appears useful to examine the latest understanding of fracture processes based primarily on experimental results, with purely analytical treatments taking on a decidedly secondary character. The specific field of dynamic fracture mechanics is typically characterized in terms of far-field loading and crack-tip local response characteristics that invoke (a) motions strongly influenced by material inertia and (b) high rates of deformation of the material in the crack tip vicinity. These latter conditions can generate a constitutive response that is significantly different from its quasi-static counterpart. Constitutive behavior that is independent of deformation magnitude and rate is the classical basis for all elastodynamic analyses. Historically, the most difficult problem obstructing improved understanding concerned the marriage between experiments and analyses: before the advent of extensive computational power, tractable analyses were limited to the infinite domain of linearly elastic solids containing mathematically sharp cracks. In contrast and by their very nature, experiments are restricted to finite domains of "real" materials. Much of the study of dynamic fracture phenomena was thus relegated to qualitative investigations. Nevertheless, several classical analytical studies honed our early understanding of crack motion under dynamic conditions. These early investigations drew heavily on dimensional analysis [5,33]. Yoffe's [42] description of the stresses at the tip of a finite crack translating across a plane (artificial healing at the trailing crack edge) established (1) the crack speed-dependent change in orientation of the principal stress circumferential to the moving crack tip, and (2) the inertia-driven decrease in the stress intensity with increasing crack speed. The similarly illuminating solutions by Broberg [3] for the crack expanding symmetrically at a constant rate, and by Baker [2] for the semi-infinite crack propagating in infinite domains established that cracks propagating along a single line (plane) possess speeds bounded by the Rayleigh wave speed. These fundamental contributions to the dynamic fracture literature pointed out that the stress intensity and thus the energy released by a crack propagating (in infinite domains) at arbitrary velocity decrease monotonically from its static value to zero as the crack speed approaches the Rayleigh wave speed. In a sequence of papers devoted to allow for loading on crack surfaces as an arbitrary function of time and space while the crack propagates in an infinite domain, Freund generalized and confirmed these results [9-12] and established a balance between the energy released by a moving crack tip and the flux of energy into the crack tip region [13]. As a result of all these
7.13 Dynamic Fracture
647
analytical developments, the obvious variable of interest was the crack speed, so that gradually virtually all dynamic fracture phenomena were examined in terms of the speed of crack propagation as the controlling parameter. This trend of viewing the crack speed as an independent descriptor was reinforced by numerous experimental observations that dynamically moving cracks tend to branch at a presumably critical speed. Hand-in-hand with these observations, it became abundantly clear that cracks in "real materials" never reached the theoretically possible Rayleigh wave speed during propagation. The most serious "excuse" voiced was that prior to reaching that speed a crack would always branch. For several decades the search was thus on to explain (a) the criterion for crack branching and (b) the reason for crack speeds at typically half or a third of the theoretically maximal Rayleigh speed. Sufficient information has surfaced during the last two decades [ 18, 27-31] which explains the significant differences between elastodynamic analyses and laboratory experiments: on the one hand, these discrepancies are the result of the idealization of the fracture process in the analyses, and, on the other, of the truly complicated details accompanying fracture in the small region around the crack tip when a crack propagates rapidly. We shall discuss first certain aspects of the relevant stress analysis.
7 . 1 3 . 2 S T R E S S ANALYSIS One of the perennial questions surrounding discrepancies between analytical (for linearly elastic materials) and experimental results concerned the accuracy with which analyses could explain the physical situation. Although numerous examples had been established for quasi-static situations over the last few decades, no comparative certainty existed with respect to the dynamic case. A sizable body of dynamic photoelastic stress analysis emerged during the 1970s and early 1980s. Qualitatively, the stress distribution around crack tips that were suddenly loaded or moving at high rates agreed with the asymptotic expansions, but their matching with initial and temporal boundary conditions was rather uncertain, since early solutions were prevalent only for the infinite domain. Besides the mainstay of photoelasticity for experimentally determining the stress and deformation state around dynamically affected crack tips, the late 1970s experienced a growing use of the caustic method, which depends on the out-of-plane surface deformations in the crack tip vicinity [17, 20, 21, 37]. Critical experiments have shown that this method does rather well when applied to brittle materials but is questionable in its accuracy where plastically deforming solids are involved. An example of the former case, as applied in dynamic situations [27,32], is shown in Figure 7.13.1. Here a comparison is
648
Knauss
0.3 Experiment Theory o Q.
0.2
0 o
--.[---~-----
2. .'2_ r c
,a,.,
_a
.t.
0.1
I
I
I00
200
Time (F sec)
(a)
C
0
0 (1. :s
9K, Experiment a, Experiment K,TheorY
1.0
8O
c
""
c
j 0.5
60 E
0
9
c
~
. 01
9
|
40
g)
~c x W
0
20
o o
0
-
0
(b)
'
I00 Ti m 9( F sec )
0
200
7.13 Dynamic Fracture
649
effected between the computational result of a time-dependent stress-intensity factor in a plate of Homalite 100 (a "brittle solid" by all engineering standards) and the computation of the stress-intensity factor following Freund's formulation [12] and using a well-calibrated load history applied to the crack surfaces. The experimentally determined stress-intensity factor corroborates the computed one very well, including the transient effects as the crack starts to move. For the case of plastically deforming solids, the situation is not so clear-cut. Although no analytical (closed form) solution for a crack moving dynamically through an infinite (or finite) domain exists in this case, results developed for static cracks shed light on this question without the complication of dynamic effects. In a study of crack tip deformations utilizing the highly precise methods of Twyman-Green interferometry (out-of-plane) and moir~-interferometry (in-plane) [34-36], displacement profiles were compared to finite element computations involving a thorough elastoplastic characterization of the material behavior (incremental J2-theory) and the maximal discretization allowed by a Cray Y2 computer. It was found that as long as plastic deformations do not play a significant role, linear elastostatics describes the physical displacement exceedingly well with a precision at the micron level. However, when the stresses exceed levels of those corresponding to about half the failure load, measurable deviations arose, which increased at the fracture point to differences on the order of 25%, as illustrated in Figure 7.13.2. Thus a sizable error can exist when out-of-plane deformations are used to deduce the in-plane crack tip stress field. The most troubling feature of this discrepancy is that the deformation field, as represented by optical fringes or by the shape of a caustic, looks just like it is supposed to, except that the numbers are incorrect, so that one has no a priori or practical criterion as to whether the caustic analysis is appropriate in the presence of plastic material behavior or not. One must deduce then that all methods for determining the in-plane stress-intensity field from out-of-plane deformations in non-linearly deforming solids are suspect, even if the best numerical methods available today are being utilized. This caution must thus also be applied to caustics as well as to gradient sensing methods [38] when significant plastic deformations arise at the crack tip. FIGURE 7.13.1 Comparison of analytically determined stress-intensity histories (solid lines) with experimentallydetermined values (solid dots) for two loading conditions on the crack faces. a. "Point force" applied to crack faces, b. Uniform pressure, including the effects of initial crack propagation. It is essential in the experimental determination of the stress-intensity factor that the domain from which the optical (caustic) information is gathered is of the proper size to all the transient information that has radiated out from the crack tip to the "initial curve." (Reproduced with kind permission by Kluwer Academic Publishers from Ravi-Chandar, K., and Knauss, W. G., Dynamic crack-tip stresses under stress wave loading: A comparison of theory and experiment, International Journal of Fracture, 20, pp. 209-222, 1982.)
650
Knauss e :
20
o~
xa/t
= 0.50
-20 -40 m e" O e
o
_E .,4
-60 -80
", - 1 0 0 /
-120
-140
/
//
~-~,Expt, ............ FEN,
/
/
/
: : : Expt, 52.3 .... FEN, 52.3 , 9, E x p t , 7 3 . 5 -. FEN, ~ 7 3 . 5
/
-160 -180
3 5 . 0 kN 3 5 . 0 kN
" .0
' .5
.
. . 1.0
.
.
.
kN kN kN kN
1.5
rtt
(a) e = •
150
~
x3/t
= 0 50 , : : Expt, ............ FEM,
125
3 5 . 0 kN 3 5 . 0 kN
: :
= E x p t , 5 2 . 3 kN FEN, 5 2 . 3 kN ' - : E x p t , 7 3 . 5 kN FEN, * 7 3 . 5 kN
100 0 f_ ~ E v
75
50
25 L8~ _- . -. o ..... ..............................."........................................................ b ...... ~ .....
'O0 9
(b)
.25
.50
.75
1.00
r/t
1.25
1.50
1.75
2.00
2.25
FIGURE 7.13.2 Comparison of computationally and experimentally determined displacements at the tip of a crack in 4340 steel, a. Displacement normal to the specimen surface along crack extension line. b. In-plane surface displacement parallel to the crack along line at 60 ~ with respect to crack extension. (Reproduced with kind permission by Kluwer Academic Publishers from Schuhheisz, C., Pfaff, R. D., and Knauss, W. G., An experimental/analytical comparison of threedimensional deformations at the tip of a crack in a plastically deforming plate, III. Comparison of numerical and experimental results, International Journal of Fracture, 90, pp. 47-81, 1998.)
7.13 Dynamic Fracture
7.13.3
651
MODE I FRACTURE
Let us turn next to considerations of dynamic brittle fracture in mode I deformations, and examine first the question of initial crack propagation under impulsive loading, followed by observations on running cracks.
7.13.3.1
INITIATION AND ARREST
One refers to initiation when a crack starts to propagate from rest, and to when the reverse occurs. Criteria for initiation have not been established uniformly, other than achieving a critical stress intensity, which does not address the situations under very rapidly rising stress pulses or such pulses of very short duration. Curran et al. [6] observed in spall experiments that the stress to generate fracture from a continuum and to propagate it increases rapidly as the initiating stress pulse becomes shorter (times are measured in fractions of microseconds). They offered a reaction rate type of criterion as an explanation, insinuating that to fail (chemical) bonds in the solid requires a certain time frame which should be governed by the molecular structure. In experiments employing preexisting cracks in Homalite 100 plates subjected to ramp pressure histories with increasing initial loading rates applied to the crack flanks, Ravi-Chandar and Knauss showed that a similar phenomenon prevailed, though the time scale was somewhat longer, namely, in the microsecond range. In an attempt to explain this phenomenon on a purely mechanical basis, Liu, Knauss, and Rosakis [19] drew on experience gained from Ravi-Chandar's and Knauss's experiments on dynamically propagating cracks and the importance of a dynamically governed process zone (discussed in text following). They postulated that in order for fracture to start, a sufficiently large domain of material at the crack tip (process zone) must be developed to a critical stress level before further crack advance can take place. To achieve this condition in shorter and shorter times with a squareroot singular stress field that emanates from and grows at the crack tip requires that the stress level of the boundary loading must increase ever more rapidly andfor to larger and larger values. This simple consideration is examined analytically with the experimental data as illustrated in Figure 7.13.3. For "slow" loading histories such that the stressqntensity field is well established over a sizable domain around the crack tip at all times, crack propagation starts at a critical stress intensity factor, say Kic. Note that the basic idea underlying this phenomenon is devoid of any dynamic arrest
652
Knauss 9 ExperimentalResult 1.2 "~ 0
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~
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~ I!
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~q
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.,--4
0.0 0
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40
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80
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120
Time to Fracture, tcr (#sec) FIGURE 7.13.3. Relationbetween the instantaneous stress-intensity factor Kdc and the time to initiate crack propagation tcr. The quantity r denotes the size of the zone over which the stressintensity field at the crack tip has to be established before fracture propagation can begin (plane strain and example). (Reproduced with kind permission by Kluwer Academic Publishers from Liu, C., Knauss, W. G., and Rosakis, A. J., Loading rates and the dynamic initiation toughness in brittle solids, InternationalJournal of Fracture, 90, pp. 103-118, 1998.)
argument, other than the transient inertial evolution of the stress field at the tip of a crack. The inference of this result for Currans et al.'s [6] spall crack generation is that, while an existing crack already possesses a stress-intensity field due to the finiteness of the crack, the initiation from a flawed "continuum" m u s t occur from defects smaller than the tip zone of a macroscopic crack. This requires still higher stress levels to cause fracture in their vicinity than the macroscopic crack, if the times to achieve this are to be very short. Both observations fit the Curran et al. experiments. The process of crack arrest is not a simple time reversal of the initiation process. As will be evident, the history of crack propagation is important, even in brittle elastic solids. Thus crack arrest does not occur at the same value of the stress-intensity factor as initiation of crack motion, but the value of the arrest stress-intensity factor Kia is typically smaller (Kza < Kzc) by about 10% [28]. Reinitiation after arrest occurs a g a i n within experimental u n c e r t a i n t i e s - at the same stress-intensity level as the original initiation value Kzc.
7.13 Dynamic Fracture
653
7 . 1 3 . 3 . 2 THE RUNNING CRACK" ROUGHNESS AND BRANCHING Once the crack propagates, it does so often by generating increasing surface roughness along the crack path, which has been historically divided into "mirror, mist, and hackle." This division is arbitrary inasmuch as the roughness increases continuously along the crack path as measured in terms of an rms-value [30]. Although the roughness is somewhat related to the crack speed, it is more clearly associated with the magnitude of the stress-intensity factor, since the roughness increases with the latter. The "mirror, mist, and hackle" characteristic is thus primarily a consequence of the fact that in earlier test configurations the stress-intensity factor increased with the growing crack, which occasions an increase in the process zone [29]. Branching is thus connected to the nature and evolution of the process zone. To demonstrate this feature more clearly, Ravi-Chandar and Knauss [29] photographed a crack running at high speed close to the (empirical) crack branching conditions, viewing the crack so that the domain ahead of the crack through the plate thickness was visible in real time. This observation revealed that the different zones of "mirror, mist, and hackle" are associated with varying numbers of microfractures ahead of the main crack. The mirror zone produces (at the scale of the optical resolution) a virtually defect-free zone ahead of the main crack, while the hackle zone was associated with numerous microfractures ahead of the main crack front. Thus the propagation of a (main) crack in a brittle material really occurs through the generation of a process zone in the form of a cloud of microcracks ahead of the main crack which then coalesce to form the new macrocrack. These microcracks occur in a band accompanying the main crack; the wider this main crack is, the higher is the instantaneous stressintensity factor. The process of crack branching is then a consequence of these microcracks generated "off-axis" ahead of the main crack and parallel to it. Their stress fields interact with that of the main crack, which determines their further propagation direction as well as their linkup with the main crack. Drawing on the analytical (quasi-static) analysis by Pucik [25] for interacting cracks, one can see clearly that cracks, in relative positions to each other as would occur in the micro-/macrocrack problem, lead to growth of the leading end of the microcrack away from the macrocrack, thus forming the initiation of the branch. This behavior is corroborated by real-time photographs of crack branches that consistently start at the small scale parallel to the main crack and then turn continuously.
654 7.13.3.3
Knauss
THE RUNNING CRACK" SPEED LIMIT
A long-standing puzzle in dynamic crack propagation was the observation that experiments produced crack propagation speeds that are significantly below the maximum value (Rayleigh wave speed) allowed or predicted by linearized elastodynamics. These values are typically one third to one half of this "theoretical" value. The observation that the fracture process occurs through a "cloud of microcracks," the stress fields of which interact with each other, establishes quickly that the time for this microcrack interaction is not considered in typical elastodynamic models. Clearly, establishing an interaction between microcracks requires more time than the idealized continuum model (a single, sharp macrocrack) allows. At the same time, one observes that the multiple microfractures result from high stresses at the crack tip; the higher these stresses, the more microfractures one would develop. Thus, if the stress level is reduced so that fewer or no microcracks develop, then the crack might possibly propagate at its maximal allowed speed, the Rayleigh surface wave speed. Reduction of the crack tip stress levels is achieved if the material possesses vanishing cohesive strength. To simulate that condition, one needs to generate a continuum with all the proper physical properties (wave speeds) but with (nearly) zero strength. This condition may be achieved by manufacturing plates with an increasingly weak plane, such as obtained in imperfect bonding or welding of Homalite 100 or Plexiglass half-planes. This proposition was explored experimentally by Washabaugh and Knauss [40, 41], who rebonded previously broken (or machined) plates of Plexiglass under controlled temperatures, pressure, and time to produce plates containing a weak plane (the future "crack" propagation plane) of controllable strength. The weakest plate bond was determined by the need to handle the plates in the laboratory. Plates generated in this manner possessed virtually homogeneous properties - - no adhesive layer - - with a variably weak line or plane along the path normally taken by a crack propagating through a plate possessing the material's full strength. By performing tests on plates possessing decreasing cohesive strength, it was found that the propagation increases asymptotically toward the Rayleigh wave speed, the highest speeds achieved being 90 4-5% of that velocity. These results illustrate that, in principle, elastodynamic theory does indeed predict the correct propagation behavior for an idealized material. The idealization consists in viewing fracture to occur along a vanishingly thin strip and under sufficiently small stresses at the crack tip so that the stresses and strains follow the linearity assumption underlying the material description for stress analysis purposes. When this linearity assumption is violated, a different crack propagation response results.
7.13 Dynamic Fracture
655
Related situations of crack propagation along weak planes arise in the fracture of crystals as the crack propagates along a (weak) cleavage plane so that "off-axis" microcracking or damage is minimized or eliminated. Field [8] has recorded crack propagation in crystals and has demonstrated that cracks do indeed propagate at 80 to 90% of the respective Rayleigh wave speeds. This result, together with that described previously for isotropic plates, suggests very strongly that it is the generation of multiple fractures at the crack tip rather than the intrinsic elastic properties (crack or wave speed) of the solid that limit the rate of crack growth.
7.13.3.4
CRACK SPEED AND THE STRESS-INTENSITY FACTOR
In the beginning of dynamic fracture investigations (1960s), analyses were based on the assumption that the energy required to form new crack surfaces was a material constant. Experiments did not support this idea in the context of dynamically moving cracks, though quasi-static crack initiation corroborated this idea rather well. Thus investigators moved to treating this quantity as a function of crack speed; see, e.g., Freund [9, 13]. This suggestion went hand-in-hand with the earlier notion that crack tip velocity was an important and independently controlling parameter. In fact, the crack speed should be viewed as a direct consequence of the (history of the) stress-intensity factor, as its dependence on the roughness of the fracture surface indicates. At any rate, this emphasis on crack speed as a controlling parameter precipitated the search for, or determination of, the velocity-dependent toughness, or the supposedly one-to-one relation (K-v curve) between the stress-intensity factor and the crack speed. Experiments on large plates simulating the infinite domain no wave reflections and large crack extensions - - do not support this idea. Instead, as Figure 7.13.4 indicates, the crack speed can be totally independent of the stress-intensity factor (within the resolution of the experimental setup), and seems to be dictated more by the history by which the stress-intensity factor was initially established to make the crack propagate. Thus there appears to be no one-to-one relation between the speed of a dynamically moving crack and the stress-intensity factor, at least not for brittle materials. It has been suggested that this observation does not carry over to plastically deforming materials; however, the data supporting this notion have been typically obtained under conditions similar to those which produced the continuous K-v curve in Figure 7.13.4 rather than large plates with far running cracks and in the absence of interaction with stress waves reflected from the specimen boundary. It is also noteworthy that crack speeds can undergo finite jumps as
656
Knauss 500
-
400
f 300
E >-
i-m (,.)
o
_.1
2_00
>
Y
o ,,,::( nr (_) I00
L 0-5
I I -0
STRESS INTENSITY FACTOR, (MPa.qm)
FIGURE 7.13.4 Crack propagation velocities in (brittle) Homalite 100. The "horizontal" lines represent measurements of the tip motion of cracks in large plates as they grow with no or minimal influence of stress waves reflected from the plate boundaries. The solid curve represents the postulated "unique" relation between the instantaneous stress-intensity factor and the resulting crack speed derived from relatively small fracture specimens as delineated in Dally [7]. (Reproduced with kind permission by Kluwer Academic Publishers from Ravi-Chandar and Knauss [28-31].)
a result of an interaction of the crack tip with a stress wave of suitable magnitude [30, 31]
7.13.4 MODE II FRACTURE Following extended considerations of quasi-static fracture along interfaces during the last decade, and specifically of the role of shear stresses in such situations, the influence of mode II deformations in dynamic conditions has been addressed more recently. These investigations have been coupled with examinations of the development and role of shear-banding in dynamic fracture scenarios.
7.13 Dynamic Fracture
657
The first experimental study explicitly devoted to dynamic crack propagation under substantial shear loading in PMMA, Araldite, and steel was offered by Kalthoff [16]. The interesting result emerged that, as the deformation rates increased, the crack propagation changed from nearly quasi-static propagation governed by, essentially, a maximum stress criterion, to a mode that is heavily influenced by the evolution of a shear-band structure, from which a crack then emanates. A more recent and broader effort to elucidate the role of shear stresses on the dynamic propagation of cracks was undertaken by Rosakis and collaborators. Starting from the behavior of interfacial separation under dynamic loading conditions, this team examined the propagation of a separation along a weak plane in sheet material, much as Washabaugh and Knauss [40] did for mode I loading. In contrast to the results established by the latter, these shear-dominated separations produced tip speeds that exceeded the shear wave velocity of the material (Homalite 100, an optically clear polyester). Associated with these "crack growths" were sizable contact zones. Relevant linearly elastodynamic stress analyses, drawing on energy flow into the tip regions of mode I and of mode II separations [4, 14,39] confirmed that this different behavior is admissible for shear-governed separations, but not for mode I-driven failures. Although these analytical results provide necessary fracture conditions in agreement with the observations in both mode I and II conditions, they shed no direct light on the physical reason for this propagation response. In view of the fact that the mode II-dominated propagation behavior occurs, apparently, only with solids containing a weak plane, it is thus possible that the inhomogeneous dilatational stress field ahead of the "crack tip" causes (multiple) separations, thus "dragging" a composite shear zone along. If the dilatational or normal stress components ahead of the crack tip are insufficient to cause failure across the weak plane, presumably damage propagation will follow a different path (or zero path) as governed by different local constitutive behavior. Crack propagation phenomena that occur with rates bounded by the shear speed of the (isotropic) solid and the dilatational speed are associated with weak planes or fissures under pressure that possess no or weak intrinsic cohesion, such as is the case potentially in the slippage of earth faults. Indeed, suprashear-speed slip motion has been reported in such situations [1, 24].
7 . 1 3 . 5 C R A C K F R O N T WAVES A phenomenon suggested on the basis of analytical work, but not yet demonstrated explicitly in the laboratory, is the spreading of a wave attached
658
Knauss
to the front of a propagating crack and propagating along the crack front. Starting from the three-dimensional numerical spectral analysis of Geubelle and Rice [ 15], Morrissey and Rice [22] examined the response of a crack front to a small perturbation of fracture energy. They found a wave propagating along the future crack front persistent for as long as the computations could be carried out. These computations were performed for a crack growing with a law connecting the cohesive stress to the separation, which translates, in the limit of the singular crack model, to growth at a constant fracture energy. This result motivated Ramanathan, Ertas, and Fisher [26] to use the Movchan and Willis [23] small-perturbation elastodynamic analysis of the front of a three-dimensional planar crack to examine whether such a wave was actually implied by the equations of elastodynamics. After extending the formulation by Willis and Movchan to describe growth under constant fracture energy, they showed that such a propagating mode, localized along the crack front, did indeed exist. Whether this mode of crack front deformation is observable and significant or important in the dynamic crack propagation process needs to be seen. Although there are many features of dynamically generated fracture surfaces, some of which may speak to this phenomenon, the most prevalent type, which has been discussed and analyzed extensively, is the parabolic surface feature. These features result from the interaction of a straight crack front with failure at a point ahead of the crack and off the crack propagation plane. By contrast, if crack front waves generate features at all, they should lead to straight line characteristics (constant crack speed) emanating from a point; perhaps more careful attention to fracture surface interpretations can begin to resolve this question. On the other hand, because such a perturbation would have to occur at the crack front and in the plane of propagation, and because flaws away from that plane have a much higher probability of occurrence, it is possible that the crack front waves and associated features become apparent relatively seldomly.
REFERENCES 1. Archuleta, R. J., and Day, S. M. (1980). Dynamic rupture in a layered medium: The 1966 Parkfield earthquake. Bulletin of the Seismological Society of America 70 (3): 671-689. 2. Baker, B. R. (1962). Dynamic stresses created by a moving crack. J. Appl. Mech. 29: 449-458. 3. Broberg, K. B. (1960). The propagation of a brittle crack. Arkiv f6r Fysik 18 (10): 159-192. See also Broberg, K. B. (1999). Cracks and Fracture, Academic Press. 4. Broberg, K. B. (1989). The near-tip field at high crack velocities. Int. J. Fract. 39 (1-3): 1-13.
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659
5. Craggs, J. W. (1963). Fracture criteria for use in continuum mechanics. Fracture of Solids, Metallurgical Society Conferences 20: 51-63. 6. Curran, D. R., Seaman, L., and Shockey, D. A. (1987). Dynamic failure of solids. Physics Reports 147: 253-338. 7. Dally, J. w. (1979). Dynamic photoelastic studies of fracture. Exp. Mech. 19: 349-361. 8. Field, J. E. (1971). Brittle fracture: Its study and application. Contemp. Phys. 12: 1-31. 9. Freund, L. B. (1972). Crack propagation in an elastic solid subjected to general loading: I. Constant rate of extension. J. Mech. Phys. Solids 20: 129-140. 10. Freund, L. B. (1972). Crack propagation in an elastic solid subjected to general loading: II. Nonuniform rate of extension. J. Mech. Phys. Solids 20: 141-152. 11. Freund, L. B. (1973). Crack propagation in an elastic solid subjected to general loading: III. Stress wave loading. J. Mech. Phys. Solids 21: 47-61. 12. Freund, L. B. (1974). Crack propagation in an elastic solid subjected to general loading: IV. Obliquely incident stress pulse. J. Mech. Phys. Solids 22: 137-146. 13. Freund, L. B. (1976). Dynamic crack propagation. ASME, AMD 19: 105-134. 14. Freund, L. B. (1979). The mechanics of dynamic shear crack propagation. J. Geophys. Res. 84: 2199-2209. 15. Geubelle, P. H. and Rice, J. R. (1994). A spectral method for 3-dimensional elastodynamic fracture problems. J. Mech. Phys. Solids 43 (11): 1791-1824. 16. Kalthoff, J. E (1988). Shadow optical analysis of dynamics shear fracture. Opt. Eng. 27 (10): 835-840. 17. Kalthoff, J. E (1993). Shadow optical method of caustics, in Handbook on Experimental Mechanics, pp. 407-476, Kobayashi, A.S. ed., VCH Publishers. 18. Knauss, W. G. and Ravi Chandar, K. (1985). Some basic problems in stress wave dominated fracture. Int. J. Fract. 27: 127-143. 19. Liu, C., Knauss, W. G. and Rosakis, A. J. (1998). Loading rates and the dynamic initiation toughness in brittle solids. Int. J. Fract. 90: 103-118. 20. Manogg, E (1964). Anwendung der Schattenoptik zur Untersuchung des Zerreissvorgangs von Platten, Dissertation, Freiburg, Germany. 21. Manogg, P. (1964). Schattenoptische Messung der spezifischen Bruchenergie w~hrend des Bruchvorgangs bei Plexiglas, Proc Int Conf Phys Non-Crystalline Solids, Delft, The Netherlands, pp. 481-490. 22. Morrissey, J. W., and Rice, J. R. (1998). Crack front waves. J. Mech. Phys. Solids 46 (3): 467-487. 23. Movchan, A. B., and Willis, J. R. (1995). Dynamic weight-functions for a moving crack: 2. Shear loading. J. Mech. Phys. Solids 43 (9): 1369-1383. 24. Olsen, K. B., Madariaga, R., and Archeluta, R. J. (1997). Three-dimensional dynamic simulation of the 1992 Landers earthquake. Science 278 (5339): 834-838. 25. Pucik, T. A. (1972). Ph.D. Thesis, California Institute of Technology. 26. Ramanathan, S., Ertas, D., and Fisher, D. S. (1997). Quasistatic crack propagation in heterogeneous media. Phys. Rev. Lett. 79 (5): 873-876. 27. Ravi-Chandar, K., and Knauss, W. G. (1982). Dynamic crack-tip stresses under stress wave loading: A comparison of theory and experiment. Int. J. Fract. 20: 209-222. 28. Ravi-Chandar, K., and Knauss, W G. (1984). An experimental investigation into dynamic fracture: I. Crack initiation and arrest. Int. J. Fract. 25: 247-262. 29. Ravi-Chandar, K., and Knauss, W. G. (1984). An experimental investigation into dynamic fracture: II: Microstructural aspects. Int. J. Fract. 26: 65-80. 30. Ravi-Chandar, K., and Knauss, W. G. (1984). An experimental investigation into dynamic fracture: III: On steady-state crack propagation and crack branching. Int. J. Fract. 26: 141-154.
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31. Ravi-Chandar, K., and Knauss, W. G. (1984). An experimental investigation into dynamic fracture: IV: On the interaction of stress waves with propagating cracks. Int. J. Fract. 26: 189-200. 32. Ravi-Chandar, K., and Knauss, W. G. (1987). On the characterization of the transient stress field near the tip of a crack. JAM 109: 72-78. 33. Roberts, D. K., and Wells, A.A. (1954). The velocity of brittle fracture. Engineering 178: 820-821. 34. Schultheisz, C., Pfaff, R. D., and Knauss, W. G. (1998). An experimental/analytical comparison of three-dimensional deformations at the tip of a crack in a plastically deforming plate, I. Optical interferometry and experimental preliminaries, Int. J. Fract. 90: 1-25. 35. Schultheisz, C., Pfaff, R. D., and Knauss, W G. (1998). An experimental/analytical comparison of three-dimensional deformations at the tip of a crack in a plastically deforming plate, II. Material characterization and finite element analysis, Int. J. Fract. 90: 27-46. 36. Schultheisz, C., Pfaff, R. D., and Knauss, W G. (1998). An experimental/analytical comparison of three-dimensional deformations at the tip of a crack in a plastically deforming plate, III. Comparison of numerical and experimental results, Int. J. Fract. 90: 47-81. 37. Theocaris, P. S. and Joakimides, N. (1971). Some properties of generalized epicycloids applied to fracture mechanics. J. App. Mech. 22: 876-890. 38. Tippur, H. V., Krishnaswamy, S., and Rosakis, A.J. (1991). A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results. Int. J. Fract. 48: 193-204. 39. Wang, W., Huang, Y., Rosakis, A. J., and Liu, C. (1998). Effect of elastic mismatch in intersonic crack propagation along a bimaterial interface. Eng. Fract. Mech. 61: 471-485. 40. Washabaugh, P. D., and Knauss, W G. (1994). A reconciliation of dynamic crack velocity and Rayleigh wave speed in isotropic brittle solids. Int. J. Fract. 65: 97-114. 41. Washabaugh, P. D., and Knauss, W. G. (1995). The effect of aligned defects on the propagation speed of a dynamic crack in PMMA, Special Issue of the Int. J. Solids. Struct. 32 (17/18): 2481-2496. 42. Yoffe, E.H. (1951). The moving Griffith crack. Phil. Mag. 42: 739-750.
SECTION
Z14
Practical Application of Fracture Mechanics Fracture Control DAVID BROEK 263 Dogwood Lane, Westerville, Ohio
Contents 7.14.1 General Background . . . . . . . . . . . . . . . . . . . . . . 7.14.2 Residual Strength (Permissible Crack Size) 7.14.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . 7.14.2.3 Calculation of Residual Strength ... 7.14.2.4 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.2.5 Material Properties . . . . . . . . . . . . . . . . 7.14.3 Fatigue Crack Growth Analysis . . . . . . . . . . . 7.14.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.3.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.3.3 Limitations and Accuracy . . . . . . . . . . 7.14.3.4 Material Data . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
661 664 664 664 665 667 667 667 667 668 670 671 671
7.14.1 GENERAL BACKGROUND Fracture control of structures is the concerted effort by designers, metallurgists, production and maintenance engineers, and inspectors to ensure the safe operation of large structures without catastrophic fracture failures. Very seldom does a fracture occur because of an unforeseen overload on the u n d a m a g e d structure. Usually, it is caused by a structural flaw or a crack: because of repeated service loads a crack may develop and grow slowly in size. Cracks impair strength. Thus, during the continuing d e v e l o p m e n t of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
661
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Broek
the crack, the structural strength decreases until it becomes so low that the service loads cannot be carried anymore, and fracture ensues. If fracture is to be prevented, the strength should not drop below a certain safe value. This means that cracks must be prevented from growing to a size at which the strength would drop below the acceptable limit. In order to determine which size of crack is admissible, one must be able to calculate how the structural strength is affected by cracks; and in order to determine the safe operational life, one must be able to calculate the time in which a crack grows to the permissible size. Analysis then must provide information on crack growth times and on structural strength as a function of crack size. This type of analysis is called damage tolerance analysis. Damage tolerance analysis has two objectives, namely to calculate: 1. the effect of cracks on strength (margin against fracture); 2. the crack growth as a function of time. Figure 7.14.1 shows diagrammatically the effect of crack size, a, on strength. The strength is expressed in terms of the load, P, the structure can carry before fracture occurs (fracture load). This implies that a limit should be set, namely, that the residual strength should never be less than Pp, a limit set by rules and regulations issued by engineering societies or government authorities.
Pres
I
(LOAO)_ |
Pu = JP~ pp =gp~
I i I !
FIGURE 7.14.1 Residualstrength in the presence of cracks; strength of new structure (a -- 0) is Pu --jPs.
663
7.14 Practical A p p l i c a t i o n of Fracture Mechanics
I
ao ,,
tI FIGURE 7.14.2
,,
,j
H
time
Crack growth curve (schematically).
If the shape of the residual strength diagram is known, and Pe is prescribed, the largest permissible crack size can be obtained. The first objective is to calculate the residual strength diagram of Figure 7.14.1. Knowing that the crack may not exceed a e is of little help unless it is known when the crack might reach ap. The second objective of the damage tolerance analysis is then to calculate the crack growth curve, shown diagrammatically in Figure 7.14.2. Under the action of normal service loading, the cracks grow by fatigue, stress corrosion, or creep. The permissible crack, ap, following from Figure 7.14.1 can be plotted on the curve in Figure 7.14.2. Provided one can calculate the crack growth curve in Figure 7.14.2 one obtains the time H of safe operation (until at, is reached). If ao, is for example, an (assumed or real) initial defect, then the component or structure must be replaced after a time /-/. Alternatively, ao may be the smallest detectable crack. Then, the time between inspections must be less than/-/. At an inspection at time tl, the crack will be missed, because ao is the detection limit. If the next inspection were to take place/-/hours later, the crack would have reached ap already, which is not permitted; i.e., the inspection interval must be less than H; it is often taken as H/2. Better methods for determining the inspection interval are based on the probability of detection. In any case, the time of safe operation by whatever means of fracture control follows from/-/. In turn,/-/emerges from the damage tolerance analysis, provided that both the residual strength diagram (ap) and the crack growth curve can be calculated to obtain H. Hence, the practical application of
664
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fracture mechanics consists of two parts: (a) calculation of the residual strength diagram, and (b) calculation of the crack growth curve.
7.14.2 RESIDUAL STRENGTH (PERMISSIBLE CRACK SIZE) 7.14.2.1 VALIDITY Validity is limited to larger crack sizes, as shown in text following. Also, in the case of extensive plastic deformation, validity is limited to cases where the load is constant or increasing.
7.14.2.2 BACKGROUND The basic criterion for fracture is
dU dW d--a= da
(1)
The left-hand term is the available energy when the crack size a increases by da. The right-hand term is the fracture resistance, i.e., the work required to make fracture over da possible. Further, these quantities are for a unit thickness, so that the dimension is LF/L2= F/L (e.g., N/m). When the equation is just satisfied, fracture proceeds in a stable manner. Fracture instability (uncontrollable fracture) occurs when
d2U d2W da2 ~- da2 In the case of constant load, the strain energy increases, so that dU/da is the surplus work done by the load. In the case of constant displacement, the load does no work, and dU/da is the release of strain energy. In both cases dU/da is equal to the absolute value of the change in strain energy. Since during unloading from an advanced plastic state there is no release of strain energy, the equations in which U is the strain energy are valid only for constant or increasing load. Shorthand denotations for dUIda are G (elastic only) Jd, Jpz; those for dWIda are R (quasi-elastic), Jc, JIc, Ja. Expressions such as G = R, J = JR, .]el 4-Jpl = Ja, Jzc are all equivalent to Eq. 1. Suppose the material's stress-strain curve can be described by = o/E § ( - / - o ) " = - / E §
665
7.14 Practical Application of Fracture Mechanics
Then the general expression for Eq. 1 becomes
Ca2a/E + H(n) an+la/F =JR
or
O +Jpz --JR
or J~z +Jpi =JR
which can be confirmed easily by checking the dimension (FL/L 2) of (strain) energy per unit area of crack extensions. In these equations, a is the measure of the remotely applied stress, and C and H(n) are dimensionless multipliers depending upon a/L where L is a generalized length parameter. The factor H also depends upon the strainhardening exponent n. The factor C equals fl2v/n , where fl is the geometry factor in the expression for the stress intensity K = flav/na. Hence
nfl2(a/L)a 2 a/E + H(a/L,n)an+la/F =JR The material's fracture resistance, JR, can be measured by subjecting a sample with a crack, a, to a fracture test in which the fracture stress (remote), a, is measured. With fl and H known for the geometry, and E and F from the stressstrain curve, JR, can be inferred by substitution of the known values in the left-hand side of the equation. Stress, crack size, fl, and H increase while fracture is in progress. This permits measurement of the (rising) fracture resistance, JR, up to the point of fracture instability (the point of instability depends upon the geometry and the loading system).
7.14.2.3
CALCULATION OF RESIDUAL STRENGTH
When the fracture resistance, JR, is known from measurements, the same equation can be used to calculate the fracture stress of a structure (geometry with certain fl and H) containing a crack of size a. There are two extreme cases. When the structure's deformation is primarily elastic, the value of the second term in the equation is negligible with respect to the second. If plastic deformation is extensive, the value of the second term becomes so large that the first term is negligible. First consider the quasi-elastic case. Ignoring the second term, the equation becomes
~]~20"2a/E = JR
or
G= R
Using K = flav/na, where K is called the stress intensity factor, this can be reduced to
K---- v/EJR
or
K =- v/ER or
K-- Kc
When the fracture resistance JR or R is (assumed to be) rising only moderately, the fracture can be assumed to become instable immediately. Then the
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fracture strength follows from fl~47~a-- K c
or
r
~- K c / f l 4 ~ a
(2)
which is the simplest form of linear elastic fracture analysis. If the first term is negligible (mostly plastic) and again the rise of resistance is ignored, the equation reduces to
Han+la/F=JR
or
J--JR
or
J=JIc
(3)
so that the fracture stress follows from fffr -
(FJR/HCl) l/n+1
(4)
For (rare) intermediate cases in which both terms must be included, the algebraic equation to be solved is of a degree without solution. (Computer) iteration is then required to obtain the unknown fracture stress, a, from the equation. 2a 2 + ~a "+1 = JR
(5)
In all cases the equation must be solved for a range of "a" values; when the results are plotted, one obtains the residual strength diagram from which the permissible crack size, ap, can be obtained, as explained previously. It can be seen immediately from Eqs. 2, 4, and 5 that the fracture stress tends to infinity when "a" tends to zero. Despite all claims, elastic-plastic fracture mechanics does not solve this basic problem. Artificialities that introduce the limit load in the equation are just that; they solve nothing and are turgid language to cover up turbid thinking. Using simple limit load analysis to calculate the fracture stress for cases with small cracks is the best solution. For most practical problems, the preceding calculation of the residual strength diagram and permissible crack size is more than adequate, provided KC, KIC, Jic, o r JR values are used that properly reflect the state of stress (usually this means properly accounting for thickness). Also, if the fracture resistance is rising considerably, these properties should be defined at the fracture instability point in the test, despite the fact that the instability point is not unique (if the test specimen is at least somewhat representative of a structure: a center-cracked specimen is, but a compact specimen is not). The calculation is easily done with a spreadsheet program, but the use of software with libraries for fl and H is advisable. More sophisticated analysis recognizes that the fracture resistance is rising. In that case, R, KR, or JR curves must be available from tests. The analysis procedure then proceeds as follows: 1. Compute afr from Eqs. 2, 4, or 5, whichever is applicable. 2. Compute dU/da by numerical differentiation of the left-hand terms of Eqs. 2, 4, and 5 for constant stress at the given a.
667
7.14 Practical Application of Fracture Mechanics
3. Compute dJR/da or dR/da from the measured resistance curve at given a (numerical differentiation). 4. If dU/da < dJR/da, go to step 1 with a = a + Aa, else instability is reached. This iteration can be accomplished in many ways. Dedicated software usually contains a very fast iteration scheme.
7.14.2.4 ACCURACY Scatter in the material property (fracture resistance) is often so large that it makes errors in computational procedures negligible. Consider Eq. 3. If n = 10, the measurement of JR in two tests, one yielding a 10% higher fracture stress than the other (normal scatter), gives rise to a "scatter" of 1.111= 2.85. Hence, tests on the same material may give values of JR from 10,000 to 28,500 (units). This seems to be an extreme (and unacceptable) scatter in material properties. However, in the calculation of the residual strength this scatter is immaterial, because in using Eq. 4 we obtain or-- (10,000) 1/11 -- Crl
or
cr -- (28, 500) 1/11 -- 1.1~1
with only a difference of 10% in the calculated fracture stress. This is as good as any engineering analysis gets. In elastic fracture mechanics the effect of a difference in fracture resistance is obviously larger (i.e., proportional).
7 . 1 4 . 2 . 5 MATERIALPROPERTIES Values for Kc, K~c, Jic, JR, etc. for different alloys have not been systematically compiled. No numbers can be q u o t e d - even as e x a m p l e s - because doing so would be deceitful.
7.14.3 FATIGUE CRACK GROWTH ANALYSIS
7.14.3.1 VALIDITY During most fatigue crack growth (the greater part of the crack growth life), the stress intensity is low and, therefore, plastic zone sizes are small. This means that the use of an elastic parameter such as the stress-intensity factor,
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K, does not put serious limitations on the validity. Besides, the fatigue crack rate data, da/dN, i.e., the increase of a resulting from the increase in the number of cycles, N used for the analysis were interpreted on the basis of K. If any intrinsic errors occur in the test data interpretation, they are for the most part eliminated when the interpretation process is reversed during integration. Limitations of validity are mostly due to engineering judgments that are necessary to decide (postulated) crack shape, interpretation of future random or semirandom load sequences, the decision whether or not to account for retardation, and the accounting for the effect of a changing environment on da/dN. For cases with complicated load sequence, it is hardly possible to predict future crack growth curves for a structure within a factor of 2. Of course, repredicting a test result is no measure for prediction made for the "real world."
7.14.3.2 PROCEDURE The analysis consists of the numerical integration of
da/dN = f (AK, R) where AK is the range of K from minimum to maximum in a cycle, R the ratio of minimum and maximum stress in a cycle, a the crack size, and N the number of cycles. The integration is
N=
/ ap a0
da
f ap
f(AK, R) =a0
da = / aPFt (a ) da f(flAav/rca, O'minKmax) a0
The geometry factor, fl =f(a/L), must be obtained first (L is a generalized length parameter). It can be found for many geometries in handbooks [1], but compounding and superposition may be necessary [2]. If the loading is of constant amplitude, Aa and R are constants. Then integration is straightforward, and can be performed in small steps, Aa, during which F~(a) is presumed constant. Steps of A a - - 0 . 0 1 a are sufficiently small to keep integration errors below 1%. Steps of Aa = 0.1 a will keep integration errors generally below 3%. While f(AK, R) may be represented by a simple curvefitting equation such as the Paris or Walker equations,
da/dN = CAKm (Paris)
or
da/dN = CAKn/(1 - R)p (Walker)
it is just as easy to let the computer interpolate in the actual rate diagram presented as a tabular file in the form of AK vs. da/dN for a range of R-values. For simple cases the integration can be performed easily enough with a spreadsheet program. However, a dedicated computer program is preferable
7.14 Practical Application of Fracture Mechanics
669
because it can include databases for da/dN, /3, and stress spectra and can accommodate random and semirandom loading. Assuming a simple Paris curve fit, the integration then proceeds as follows: 1. 6/i
2. AK = fl(ai) Acrx/rcal 3. d a / d N = CAK m (Paris curve fit, or interpolate in rate file) 4. A a = 0.01ai 5. A N = A a / ( d a / d N ) 6. ai+ 1 ~ a i ~ Aa; N i + 1 - - N i q- A N
7. ai = ai+ l , N i - - Ni+ l
8. If ai < ap then go to 1 With Aa = 0.01ai the crack size doubles in 70 steps, so that growth from 1 to 64 mm takes 420 steps. The computer provides the answer in a split second. In general, the loading is not of constant amplitude, but A~r and R are different in every cycle. This may give cause to retardation: a high load causing less growth during subsequent cycles. The cyclic sequence may be random, semirandom, or follow a preset pattern which must be known by (submitted to) the software. Some software permits the use of an exceedance diagram from which realistic load sequences are generated automatically. Most software permits only a tabular input file of stress cycles and number of repeats, and possibly the option of a random pick from the file. The integration then proceeds as follows for every load cycle: o
2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
c/i
fl(ai) from library of geometry factors Acri, Ri from load sequence generated A K i = ~ A c r v/gCl i
d a / d N =f(AK, R) interpolation in rates from library
Sequence accounting for retardation; slow down 0~ da / dN -- o~ da / dN N i + l - - N i q- 1 ai+l ~
ai + l * d a / d N
ai ~ ai+l~ N i - - N i + l
If ai < ap then go to 1
Despite the great speed of present-day computers, crack growth covering many millions of cycles may take considerable computing time (several minutes up to several hours). Dedicated software is available. Development costs of software may run very high (several millions of dollars), depending upon the conveniences offered.
670 7.14.3.3
Broek
LIMITATIONSAND ACCURACY
Despite claims to the contrary, retardation models (most software provides options for several models) are simplistic and inaccurate: a mere change of yield strength by 10% (normal scatter) may give rise to a factor of 2 in crack growth life. Besides, elimination of one or a few of the highest loads in the sequence may easily change the crack growth life by another factor of 2 to 3. Consequently, for nonmilitary applications, retardation is usually not applied in the analysis (conservative). Considering the other uncertainties, this is most probably the best thing to do in view of the following: a. Most analysis is done for postulated (not real) cracks. A postulated initial crack of 0.5 mm easily gives a life twice as long as a postulated 1-mm crack. For part-through cracks the calculated life may vary by a factor of 10, depending upon the assumed aspect ratio of crack depth to length. b. The purpose of the analysis is to predict future crack growth, but the future loads and load sequences are unknown. Extrapolation of past experience is the best we can do. This is acceptable for loads occurring frequently, but the load statistically occurring once in the structure's life may or may not occur. If retardation is ignored, the effect is negligible, but with retardation the difference may be a factor of 2. c. Most structures experience changing environments: if the chemical environment doesn't change, the temperature certainly does. Besides, cyclic frequencies are usually lower than those used in the laboratory, and chemically affected processes, such as crack growth, are time (frequency) dependent. The differences in da/dN for different frequencies, environments, and temperatures may easily be a factor of 2, but the usual is a factor of 5 to 10. Using laboratory rate data would then introduce uncertainties as large as those factors. The typical engineering approach is to use a weighted average of da/dN data (i.e., weighted with regard to time spent at low and high temperature, moisture level, etc.). Such engineering pragmatism is certainly defendable, but certainly not accurate. In view of these remarks, no prediction of future crack growth can be very reliable. One may hope to be within a factor of 2. Any academic refinements in stress analysis, geometry factors, and retardation models pale in the light of reality. In the end, the resulting crack growth analysis depends solely upon engineering judgments and ever so slightly upon academic niceties of fracture mechanics.
7.14 Practical Application of Fracture Mechanics
7.14.3.4
671
MATERIAL DATA
Fatigue crack growth rates are a function of AK and R. Apart from this dependence there is a considerable effect (often factors of 2 to 3, sometimes more) of (1) environment, (2) cyclic frequency, (3) direction in material, (4) temperature, and (5) heat treatment. Consequently, the rate data for a single alloy (if reasonably complete) cover a dozen or so pages. The only systematic data compilation is the USAF Damage Tolerance Handbook covering aerospace materials. Otherwise data must be obtained from scattered publications.
BIBLIOGRAPHY 1. Rooke, D. P., and Cartwright, D.J. (1980). Compendium of Stress Intensity Factors, Her Majesty's Stationery Office. 2. Broek, D. (1988). The Practical Use of Fracture Mechanics, Kluwer Academic Publishing.
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CHAPTER
Friction, Wear Coating, Corrosion
8
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SECTION
8.1
Introduction to Friction and Wear JEAN LEMAITRE
UniversitF. Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
The surface phenomena in mechanics of materials are difficult subjects because the "representative volume element" does not exist anymore. 9 It involves friction when two solid bodies glide one on the other on their surface boundaries and a force develops against their relative movement. This force is due to a "film" in which microfractures or microdiscontinuities occur. It is called a "third body" in Section 8.2, and it exists even in lubricated contacts with interaction or not on the friction resistance for hydrostatic low-pressure lubrification or for hydrodynamic high-pressure lubrification (see Sections 8.3 and 8.4). Phenomenological models such as the Coulomb friction are widely used, but the phenomenon can be analyzed in the framework of thermodynamics (see Section 8.5). Its use in structural calculations together with the corresponding algorithms is described in Section 8.6. 9 It includes wear due to the fracture of microroughness of the two surfaces in contact, a phenomenon that engineers have tried to avoid forever, but its prediction is difficult (see Sections 8.2, 8.5, and 8.7).
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
675
SECTION
8.2
Background on Friction and Wear YVES BERTHIER Laboratoire de M~canique des Contacts, UMR CNRS-INSA de Lyon 5514, Batiment 113, 20, avenue Albert Einstein, 69621 Villeurbanne Cedex, France
Contents 8.2.1
Localization of the Problem . . . . . . . . . . . . . . .
677
8.2.2
Scale of A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . .
677
8.2.3
Approach and Procedure ..................
678
8.2.4
History and Co n s eq u en ces of a Divergence
679
8.2.5
Fluid Third Bodies .......................
680
8.2.6
Solid T h i r d B o d i e s . . . . . . . . . . . . . . . . . . . . . . .
680
8.2.6.1 F r i c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .
681
8.2.6.2 W e a r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
682
8.2.6.3 E v a l u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . .
682
8.2.7
Contact Mechanics .......................
683
8.2.8
Industrial Selection Criteria ...............
684
8.2.9
Evaluation ................................
685
8 . 2 . 1 0 Life o f a C o n s t a n t w i t h a Solid Third Body ...................................... 8.2.10.1 D e s c r i p t i o n . . . . . . . . . . . . . . . . . . . . . . . 8.2.10.2 E v a l u a t i o n . . . . . . . . . . . . . . . . . . . . . . . .
686 686 688
8.2.11 Nomenclature and Concepts ..............
689
8.2.11.1 Tribological Triplet . . . . . . . . . . . . . . . .
689
8.2.11.2 T h e Tribological Circuit . . . . . . . . . . .
691
8.2.11.3 E v a l u a t i o n . . . . . . . . . . . . . . . . . . . . . . . .
692
8.2.12 Solution of a Problem . . . . . . . . . . . . . . . . . . . .
693
8.2.12.1 C o n t e x t . . . . . . . . . . . . . . . . . . . . . . . . . .
693
8.2.12.2 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .
693
8.2.12.3 T h e M e c h a n i c a l Viability of a Solution ........................ 8.2.12.4 I n t e r p r e t a t i o n s . . . . . . . . . . . . . . . . . . . .
676
695 696
Handbook of Materials Behavior Madels. ISBN 0-12-443341o3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
677
8.2 Background on Friction and Wear 8.2.13 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
696 697 697
8.2.1 LOCALIZATION OF THE PROBLEM Although friction and wear are common parameters, they nonetheless raise scientifically and technically difficult questions [1]. Indeed, the friction coefficient, the ratio of the components of a force, is a dimensionless scalar with practically no physical content and therefore is "independent from everything, though nonetheless dependent on nearly everything." As for wear, its definitions, units, and sign vary as a function of application: positive variation of mass related, for example, to the oxidation of detached particles, or negative variation related to a real loss of material, or surface degradation without loss of mass, or volume worn per unit of distance traveled over normal load, or wear power linking wear volume to the energy dissipated, etc. Therefore, friction and wear are not magnitudes in the normal meaning of the word, nor are they intrinsic properties of materials in contact, but properties of use depending in particular on the mechanism that encompasses the bodies in contact, which is why they do not appear in any legal system of units. Publications on these subjects range from atomic to mechanical scales [2]. Because of lack of an indisputable arbiter, which could come from in vivo instrumentation of the contact, the hypotheses announced in publications have become certitudes which when taken out of context, spread false ideas, such as "the harder it is the less it will wear." The multitudinous bibliography available means that it is always possible to find the interpretation on the shelves of a tribology department that is necessary to justify in good faith anything and its contrary, though rarely to find a solution to a problem. The main problem is not the lack of information but the lack of structuring tools to identify the parameters that effectively govern the life of a contact and link the interpretations and correlations ranging from the infinitely small to the relatively large. Thus it is above all necessary to define the scale of the analysis.
8.2.2 SCALE OF ANALYSIS When the apparent surface area of the contact is less than about 10 ~tm, the flatness of the body in contact can allow the action of a monolayer and
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Berthier
interactions at the atomic level if still lower. In such cases, it is possible to calculate friction on the basis of physical laws [3-5]. On the contrary, for contacts of larger size, i.e., for most industrial contacts, it is necessary to take a global and thus mechanical approach. This is the scale dealt with by this chapter and on which friction, wear, and lubrication constitute the components of tribology, a discipline still in the process of forging its tools and in which knowledge and know-how are equivalent. Consequently, and at the risk of disappointing the reader of this handbook on materials, it is not possible to compile "material and coating couple solutions"; all that can be proposed is an approach to investigating and solving a problem.
8.2.3 APPROACH
AND PROCEDURE
The solution of friction and wear problems runs counter to the expectations of engineering offices that are accustomed to dealing with highly formal disciplines such as fluid and structural mechanics and design software requiring intrinsic values. This situation must first we understood in order to: 9 analyze the relative role of materials and the mechanism contained by the contact; 9 explain why friction and wear are not functions that can be bought and why their solution demands skill, which in turn demands feedback from experience; 9 acquire basic concepts and a nomenclature that structure information, permit understanding, set in context, and solve a problem by avoiding false ideas and the obvious and hasty generalizations brought about to a great extent by the requirement for intrinsic properties. This last point may come as surprise to some, but "materials with hexagonal structures have low friction coefficients" can still be heard, whereas it is well known that graphite and molybdenum bisulfate, both of which have hexagonal structures, have high friction in a vacuum and in an ambient environment. Likewise, there are erroneous images related, for example, to the representation of roughness profiles (Fig. 8.2.1), with scales that are so different for the abscissa and ordinates that they bring to mind the contact between two mountains, whereas simply restoring the scales would give the correct image of a contact between two surfaces. This false image of interpenetrating roughnesses has led to models removed from reality. It is obvious that this interpenetration of roughness can
8.2 Background on Friction and Wear
679
FIGURE 8.2.1 Steel spectrum, a. Classical representation, b. Harmonization of abscissa and ordinate scales.
exist, but it stems from either design error or deliberate will. The main source of error stems from a static perception of the dynamic phenomena brought into play in a contact. This state perception is supplied by the observation of the bodies brought into contact after their disassembly. It is precisely this that could be qualified as the "tribological Pompeii complex," since it requires the reconstruction of a dynamic view on the basis of postmortem examinations. This is the context in which the chapter attempts to provide a method of interpretation to find a way through the jungle of friction and wear in order to analyze a problem, explain a fault, and transpose test results to a real contact. The difficulties of these tasks are often exacerbated by the fact that the first two bodies of a mechanism do not belong to the same society or to the same division of the same society. The problem of physical interface therefore becomes a problem of human interface. Solutions to the problems, often badly represented, are postponed. Living with them can be a profitable compromise. Indeed, although wear has a cost, it is also source of profit. The solutions can be obtained more easily than is thought, since the interest of friction and wear lies in their complexity. Since a tribologist cannot be trained on the strength of just a few pages, we shall not deal with classical phenomena such as adhesion, abrasion, erosion, etc., already well described elsewhere; rather, emphasis will be given to ways of offsetting conceptual weaknesses by structuring feedback form experience, and to the use of contact mechanics and a phenomenological approach.
8.2.4 HISTORY AND CONSEQUENCES OF A DIVERGENCE A simple way of understanding the situation is to compare the changes of the two main components of tribology, which are, on the one hand, lubrication,
680
Berthier
meaning "in thick film," and, on the other, friction, wear, and so-called solid lubrication. If mechanics is the science of movement, then tribology is the science that controls this movement (M. Godet 1990). This is why our history and that of tribology are related [6]. Our joints use tribology, as do bowel movements and the friction of the eyelid on the eye. This friction occurs on a film of a thickness in the region of 16 ~m and whose pressure can reach 0.5 MPa [7]. This situation illustrates the basic configuration of the mechanical approach of tribology in which the two bodies, or the first bodies, are not in contact but always separated by a third that can be fluid or solid. Although the notion of the fluid third body used in hydrodynamics and hydrostatics is banal [8], this is not the case for solid third bodies, which are constituted by particles detached from the first bodies or introduced artificially between them. The histories of fluid and solid third bodies have caused them to attain different levels of formalism, which now influence their use by engineering offices.
8.2.5 FLUID THIRD BODIES The Egyptians already mastered the technology of fluid third bodies as they lubricated their transport sled runners with silt. The relation between viscosity and friction then became apparent by observing the differences of friction in presses used to produce olive oil in comparison with grape juice. The mechanical measurement of viscosity permitted mechanics to go beyond the chemical nature of the third body and lay the foundations of lubrication, which accompanied industrial evolution. This was largely thanks to the large quantities of mineral oil made available by the discovery of petroleum and to the products derived from it, such as additives, synthetic oils, etc, and, obviously, to the models formulated by O. Reynolds in 1886, which made viscosity predicable [8] by calculating the friction, loads, and speed gradients in film. It is only recently that new questions have been raised [9, 10] because of the increasing stresses applied to mechanisms.
8.2.6 SOLID THIRD BODIES This notion, introduced by Maurice Godet in 1974, provides a degree of freedom which permits harmonization of points of view from materials science, physical chemistry, and mechanics, the objective being to achieve convergence toward a single approach to friction, wear, and, at a later stage, fluid and solid third bodies.
8.2 Background on Friction and Wear
681
8.2.6.1 FRICTION The analysis of friction was first done via the tests carried out by L. de Vinci, who formulated the law # - TIN where T is the tangential force, N the normal force, and # the friction coefficient. Not long after came the notions of rolling friction and static and dynamic sliding introduced by R. Hooke (1660), who based on his works on the deformation of solids. This led him to study the formation of an "agglomeration" at the entry of the contact. Then, Amontons restored the law # - T/N, which often, and erroneously, bears the name Coulomb's law, despite his being its main critic [11], since he had observed that it was only true when the bodies in contact were coated with old "cart grease" which could thus be assimilated with a fluid third body. Coulomb, and above all B~lidor (1737), modeled or, more exactly, attempted to conceptualize friction on the basis of displacing macroscopic spherical asperites (Fig. 8.2.2), used later at the atomic scale with Bohr's model (1920), and then at the scale of tectonic plates. This geometric conceptualization has resulted in reliable models for these two extreme scales: i.e., atomic level and tectonic plates. On the contrary, work on the mechanical scale suffered a decisive setback in 1882 when Reynolds abandoned his study of friction and wear because they were "too complex." In fact, he abandoned friction and wear to materials specialists whose priority was the mass production of fault-free materials. These specialists only carried out friction and wear tests when necessary, creating islands of competency rather than a generalized corpus of knowledge such as that provided by viscosity for fluid third bodies. Bowden and Tabor [12] raised the bar by using a relatively mechanical approach, though this was interpreted physically, which limited its mechanical repercussions. The ensuing increase in the number of surface characterization tools made possible by vacuum technology led to the predominance of physical-chemistry approaches. This predominance has increased over the last ten years with characterization techniques in ambient
FIGURE 8.2.2 Conceptualizationof friction, a. Coulomb'sview. b. B~lidor'sview. c. Bohr'sview.
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Berthier
environments. In addition to chemical analysis, this permits one to apply stress to examine surfaces by carrying out, for example, atomic force microscopy and therefore "enact friction" at the nanometric scale.
8.2.6.2 WEAR Although wear and friction are often coupled, science seems to have taken an interest in wear much later than friction. Apart from a Scottish study on the loss of weight of coins in the eighteenth century, there were no developments until 1940 [13], when Achard attempted to model wear, leaving a law that bears his name and which is practically the basis for all the other laws concerning wear. It should be remembered that the loss of weight in coins meant their loss of function. Consequently, "the measurement of wear by loss of weight" was in direct relation with the loss of function, which is not the case of every application. Despite this, the loss of weight remains the most usual unit of wear. At present, the literature offers over a hundred laws on wear [2], each of which is no falser than the other. After a so-called wear test, it is always possible to find a function that correlates several parameters. The development of data processing software encourages this explosion of correlations. This is not a problem in itself insofar as modeling does not amount to describing; all that is needed is to establish a univocal relationship between the input data and output, i.e., wear, by using more or less reasonable adjustment parameters. This explains why, for example, Young's modulus of one of the materials in contact in certain wear laws can have a positive or negative sign assigned to it.
8.2.6.3 EVALUATION The situation has become such that friction and wear are not modeled but rather correlated with parameters that belong more to the cultures and means of investigation of researchers than to reality, the effect of which has amplified the isolation of skills and eliminated the concern for generality. In order words, when disciplines such as lubrication, structural mechanics, etc., expressed the relations of mechanical balance and then saw how materials were a part of this generality of friction and wear, they complied results from experimental and theoretical tests but failed to go further in search of the logical response, i.e., without attempting to analyze the reality of the contact.
8.2 Background on Friction and Wear
8.2.7
CONTACT
683
MECHANICS
Contact mechanics is not often taken into account by friction and wear studies that are highly experimental. This is a shame, since, like lubrication, it has maintained the same direction through time, thereby making it efficient in spite of its hypotheses, which are nonetheless clear on closer inspection. It started with the works of Newton, who pressed glass spheres on metal discs in order to evaluate the deformations. Though not providing any tribological results, these works were to have a future under the name Newton's rings. The decisive barrier was broken down by Hertz in 1881. He found an analytical solution to the calculation of the stress field in the contact of a cylinder on a semi-infinite mass. Progress was then continual: the taking into account of any normal load by Boussinesque (1885); then Carter (1926), Cattan~o (1938), and Midlin (1949) accounted for rolling and sliding conditions. A new step forward was made in 1958 by K. L. Johnson, who more or less gave the field of contact mechanics its current analytical framework. Using these works as a basis, Kalker progressed from 1963 to 1973 in the area of rolling contacts and introduced numeric methods. Their works have become References [14, 15]. From 1985, Kalker's works were extended to coatings and more generally to multilayers, including cracking under tribological stresses [16, 17]. These stresses owe their specificity to the joint presence of normal and tangential forces which lead to modeling and material characterization problems which will be dealt with later. Since the beginning of the nineties, the finite element method has been used increasingly in contact mechanics [18-20]. Developed for structural calculations, and therefore for volumes, it should, however, be used with caution for contact and therefore surface problems. Indeed, although it has become commonplace in structural mechanics, much remains to be done at the level of so-called contact elements, i.e., the management of contact conditions. The restrictions of contact mechanics dictated by the use of conditions with debatable limits such as "Coulomb's friction," and the still painstaking taking into account of solid third bodies and major deformations, can be overcome easily by varying the different parameters. It then becomes an excellent experimental analysis aid that permits: 9 varying a single parameter at a time, which is practically impossible under experimental conditions; 9 targeting the expertise, for example, of a metallographic section by defining the most deformed zone. Since considerable enlargements are required for observation, it is often unrealistic to carry out an expertise of an entire contact.
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The contribution of contact mechanics is decisive in calculating stress and deformation fields in order to predefine solutions, i.e., to define their mechanical viability. Indeed, it is unrealistic to take a tribological view of an undersized contact configuration. Tribology is not a miracle remedy, and it is difficult to divine stress fields in three dimensions. The effect of a coating is even more difficult to identify since it can act as a material by its thermoelastic characteristics or act geometrically via its thickness. This is one of the first steps toward solving a friction and wear problem (see Section 8.2.12).
8.2.8 INDUSTRIAL SELECTION CRITERIA As early as 1915, due to lack of scientific data and faced by the emergence of new materials, coatings, and surface treatments, technicians in industry attempted to formulate selection criteria for materials and their operating limitations in order to solve their problems. Before this, the conquest of the West tentatively opened the way for solid lubrication with the discovery of molybdenum bisulfate in the natural state and graphite, known to the Indians, who spoke of "fields of stones that slide." Despite this, solid lubrication remained restricted to specific applications, such as glass molding, satellite mechanism joints, electric motor contacts, etc. The first criterion was contact pressure limit (1915), which was followed by output limit (pressure • speed) (1920), the chemical compatibility of the materials in contact (1940-1965), the role of the chemical environment (1955-1970), wear coefficients (1958), and interface temperature (1964), which appeared at the same time as the arrival of polymers in mechanical construction. Initial advances were made from 1970 to 1974 in drawing up utilization properties with operating condition sheets. Drawing away rather too quickly from the standpoint of intrinsic properties and because of a lack of scientific structure, this approach was succeeded by the more classical approach of wear cards (1986), then by damage cards (1989) using dimension parameters in an attempt to privilege extrapolations of geometry from one card to another. The failure of these cards will be explained during the presentation of the "life of a contact" with a solid third body. If confirmation was still thought necessary that friction and wear are not intrinsic properties of materials, it came with the widely varying results (30%) from the series of friction tests carried out by many laboratories of different countries [21]. Since 1992, emphasis has been given to dissipated energy models by attempting to retrace wear on the basis of damage criteria formulated for the
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volumes, using the most accurate parameters in order to draw close to tribological stresses. It should be noted that since 1970-74, when operating condition cards were developed, approaches linked to advances in computer science have been applied, with the use of expert systems, experiment plans, and neuronal networks. None of these has given complete satisfaction. An experiment plan permits the study of the parameters it contains, without making it possible to find those that have not been predicted. However, in tribology, the problem of identifying parameters is frequent. Since the period mentioned, the only tangible result is a succession of fashions lasting from 5 to 8 years, during which the "hard" alternates with the "soft," as well as massive materials, coatings, and surface treatments.
8.2.9 EVALUATION An area of imprecision has been maintained with respect to the role of materials, since tribology did not carry sufficient scientific weight to channel the development of new materials for which it offered one field of application among others, bringing to mind the tribological infatuation with ceramic materials. With hindsight, tribological applications without at least a film of water, graphite particles, or, more generally, a third body inserted in the contact to save the situation are unknown. Only one material has been designed for a tribological application, namely, braking. It is produced by the Englishman Frood under the brand name Ferodo and, up to the recent prohibition of asbestos, used one of the components of asbestos. One product has evolved to fulfil its tribological function; this is the tire whose optimization of volume has permitted the control of surface properties and thus of adherence. This control, combined with mechanics, materials science, and physical chemistry, as well as with the couple volume/surface, is exceptional in tribology, though it demonstrates the path to be taken. On the periphery of contact mechanics, friction and wear have been combined in models of all types, since they provide a field of application to model developers and reassure tribologists who are "lacking science." This is not too serious, except when a company wishing to develop its product attempts to use the model in reverse to ascertain the modifications and parameters to be applied. Since the model is not generally adjusted to physical parameters, the results are surprising. What then, given this technological and scientific reality, is the reality of the life of a contact with a solid third body?
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8.2.10 LIFE OF A CONTACT A SOLID THIRD BODY
WITH
This "life," observed from the mechanical standpoint, is described schematically using the operation of a disc brake. It could just as well have been the contact of a magnetic tape over a reading head, a wheel on a rail, a bow on the strings of a violin, steel sheet on a stamping press, a stud on the disc of a tribometer, and even that of the head of an atomic force microscope on a surface. Consequently, the reader can transpose the vocabulary of this description to that of his own application.
8.2.10.1 DESCRIPTION This brake mechanism, placed vertically (Fig. 8.2.3), it composed of a disc and two linings, themselves composed of circular blocks which are fixed to the support. During braking, the linings are clamped on the disc by an articulated system controlled by a pneumatic cylinder while the application of braking torque is ensured by a connecting and fixed to the upper part of each lining. This architecture ensures that the balance of forces causes the contact's geometry to become dynamic, implying that the braking torque depends on the direction of rotation of the disc. This is the initial mechanical effect. It should be borne in mind that the friction is measured by way of a tangential force or torque, via a mechanism and not directly in the "contact." Likewise, during the measurement of the tangential forces required to move the point of an atomic force microscope. This measurement is carried out via the point holder blade whose bending also to the point-surface contact taking on a
-~--
I
]Pad _ I
- 7 -
i
FIGURE 8.2.3 Brakearchitecture.
Air jack
i
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FIGURE 8.2.4 Thirdbody. a. On a pad. b. Third body "in movement".
dynamic geometry. Consequently, the mechanism containing the contact always plays a major role. Let us continue the analysis on the level of the blocks. Because of the machining tolerances, previous operating conditions, dynamic geometry, etc., the blocks do not come under identical stress. Some of them heat and expand more than others. These differential expansions are more or less absorbed by the stiffness of the support plate. Here again, this is a mechanical effect that controls the distribution of the normal force on each of the blocks and the dynamic geometry. Careful examination of a block (Fig. 8.2.4) when relaxing the brake reveals the presence of "entities of the third body" of a length of about 10 gm and a thickness of several micrometers, mostly stemming from the blocks. The "entities of the third body" correspond to "particles in movement" in the mechanical meaning of the term, without prejudging the size of the elementary particles, in the physical meaning" which compose them. The morphology of these entities leads one to believe that they can slide against each other, which is confirmed by visualization tests using a high-speed camera, and which is made possible by a transparent first body. These visualizations show flows at the scale of the instantaneous contact zones which group around twenty entities of the third body. These zones relay each other. Consequently, after opening the contact, a layer of the third body appears to adhere strongly to one of the first bodies but which "flowed" during movement. Furthermore, the relays between the instantaneous contact zones mean that the instantaneous pressures of the contact can be from five to ten times greater than the apparent pressure. The flows lead to variations of thickness of the third body and thus of the behavior of the brake. These flows depend on the stiffness of the mechanism and that of the clamping system: the
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pneumatic or oil-actuated cylinder and its control law. This is an interaction between the mechanism and the third body. The particles can be drawn along at the scale of one or more blocks and even be reintroduced in the contact during a revolution of the disc or, on the contrary, fall onto the road. In the latter case, they are definitively lost from the contact and constitute wear particles. Thus wear can be treated as flows of third body that are permanently lost from the contact. This explains, for example, that for the same materials and apparent pressures of equivalent contacts, the change of a lining-disc geometry to a disc-disc geometry leads to different wear values. When, under mechanical action, the particles transit in the contact, they "sheer," creating "fresh" surfaces on which the physical chemistry of the environment will or will not have time to react. Depending on the case, this permits the particles to reconstitute, oxidize, adhere to the first body, etc. For a mechanical engineer, this competition between mechanical kinetics and physical chemistry can be expressed in a law of behavior: the rheology of the third body. This rheology is controlled by brake lining manufacturers empirically using loads placed in the blocks from which most of the third body comes. It should be noted that imposing the source of the third body from only one of the first bodies is a means of simplifying the problem and thus of better controlling the friction and wear that depend on the interaction between the mechanism, the first and third bodies as a function of the materials, the physical chemistry and the mechanics. To be complete, what is required is the explanation of the steps of conception and birth of the third body, since what has been described is its actual life. This description remains valid for a contact in alternating motion, even when of low magnitude, as in the case of fretting-corrosion. This problem requires the distinction between the detachment of the particles and their oxidation when they circulate in the contact. An amalgam is often made of detachment and oxidation, which amounts to considering oxidation as the cause of the phenomenon, whereas it is only a consequence. In tribology, inversions between causes and consequences that bias the solution to the problem are common.
8.2.10.2 EVALUATION This dynamic view of solid third body flows: 9 gives credit to the idea that wear can be approached in terms of flows and rheology of the third body by taking into account the role of the mechanism, the first and solid third bodies making up the tribological
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triplet and three disciplines, namely, mechanics, physical chemistry, and materials science; 9 explains the inadequacy of the vocabulary in the bibliography, which gives an overly static view of contacts and shows that materials are put under stress at pressures reaching several GPa, well over the traditional range of their characterizations. The interaction between the elements of the tribological triplet explains that friction and wear test results include the signature of the tribometer (mechanism) used and that friction and wear are not intrinsic properties of materials but of a tribological triplet. This explains the problems of extrapolating one contact to another. Solving problems requires the management of the tribological triplet, though this management is of an essentially phenomenological nature, since tools for characterizing solid third body flows are still in the process of construction. This is why a nomenclature is proposed to facilitate the application of the phenomenological work.
8.2.11 NOMENCLATURE AND CONCEPTS
8.2.11.1 TRIBOLOGICALTRIPLET 8.2.11.1.1 Mechanism This is the entire system which contains the contact and which imposes mechanical and physical-chemical stresses. Observations are made too frequently only at the scale of the contact, i.e., by forgetting that the balance of the forces is exerted by the mechanism, whence the dynamic geometry of the contact which can be considerably different from its static geometry. 8.2.11.1.2 First Bodies and the Role of Materials The first bodies are materials that enclose the contact and respond to tribological stresses. This response can be qualified as both indirect and direct. It is indirect when the properties and thermoelastic properties of the first bodies respond to the stresses transmitted by the mechanism by deforming. These deformations contribute to defining, with the third body, the elementary contact zones where they then respond directly to local tribological stresses (Fig. 8.2.5), schematically, by cracking and/or by
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FIGURE 8.2.5 Metallographicsection of the top of the rail.
changes of phase and structure. Abrasion, adhesion, etc., which have been omitted voluntarily here, can also be considered specific responses that are either desired or that can be avoided. Stresses here are qualified as tribological, since a contact is the only case where a normal force and an imposed movement coexist, implying a sustained tangential force that affects volumes of material from a few nanometers to a hundred cubic micrometers. The notion of response has been chosen, since the start of crack formation and structural changes are under study. In fact, the conditions of the formation of incipient cracks are still to be understood, as are the changes of phase and structure which lead by fragmentation to the detachment of particles of elementary size ranging from a nanometer to several micrometers. To draw attention to the fact that these changes of phase and structure are induced more by pressure and major deformations than by temperature, they are called superficial tribological transformations (STTs). This remark may cause surprise, since their chemical composition makes it tempting to use a diagram of equilibrium to trace back to a formation temperature and therefore a contact temperature. This logic omits the fact that pressure can have the same effects as temperature [22, 23]. Consequently, care must be taken with the "temperature mperature reflex." STTs have been observed with metallic and nonmetallic materials and appear to be a fairly generic case for the source flow of the third body. The materials differ in their incubation times for producing STTs, the depths of the latter and then the rheology of the third body that they have contributed to produce. In this nomenclature, a coating is an integral part of the first body, although layers of oxides, pollution, etc., already present on the first bodies belong to the third body. To get around the difficulties of analyzing these layers, they are qualified in this mechanical approach on the basis of one of their functions, which is to "screen" (hence their name) the adhesion forces between the first bodies.
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8.2.11.1.3 The Third Body This is a volume of material that includes the screens present between the first bodies. It is an "operator" that separates the first bodies and transmits the load (bearing) between the first bodies while accommodating the greater part of their difference in speed. Its three functions can be highlighted by following its deformations and its position in the contact. Models of it are being developed at present [24-26]. In a mechanical approach, it is sometimes of interest to geometrically distinguish the screens whose thickness is usually less than 5 0 n m and the solid particles circulating in the contact, which constitutes a "more or less continuous film" of a thickness of about 10 pm and which is the volume part of the third body. The third body is called natural when its volume stems from one or both of the first bodies. It is termed artificial when it is introduced into the contact, as in the case of coating one of the fist bodies with graphite, molybdenum bisulfate, etc.
8.2.11.2
THE TRIBOLOGICAL CIRCUIT
8.2.11.2.1 Flows The flows of the third body composing the tribological circuit are defined for an elementary contact in two dimensions (Fig. 8.2.6). Internal source flow QiS corresponds to the detachment of particles due to STTs, cracking, bonding, etc. It leads to the formation of the natural third body. External source flow Q~ stems from the introduction of that artificial third body in the contact. Internal flow Qi is the flow of the third body that circulates between the first
i
natural
i
"~
FIGURE 8.2.6
Qr
dx
"
Tribological circuit.
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bodies. External flow Qe is the flow of the third body that escapes from the contact. It is divided into a recirculation flow Qr and a wear flow Qw. The recirculation flow is composed of the third body reintroduced into the contact, drawn, for example, by one of the first bodies and which will contribute once again to accommodating the speed. On the contrary, wear flow Qw is composed of the third body which is definitively ejected from the contact and which can no longer participate in speed accommodation.
8.2.11.2.2 Rheology The rheology of the solid third bodies is evaluated on the basis of their "cohesion" and "ductility." The term cohesion has been chosen by analogy with the cohesion of powder environments [27], although in the case of solid third bodies this is rather more akin to compacting. The term ductility has been chosen by analogy with the plastic flow characteristics of metals. Cohesion and ductility are determined on the basis of observation (optical, electronic) coupled with the analysis of images of the morphology and texture of solid third bodies. This determination may appear surprising, but at present observation is often the method that disturbs the measurement the least. Even characterization by nano-indentation disturbs the layer of the third body too much; measurement is already handicapped by its surface condition. Furthermore, the range of stresses applied with nano-indentation is quite different from that brought into play in the contact. The cohesion and ductility values evaluated at a given scale are correlated with friction values. Obviously, this phenomenological procedure is unsatisfactory, though it is preferable to static characterizations, which are much further from reality.
8.2.11.3 EVALUATION When carried out well, characterization, even at the phenomenological level, of the rheology of solid third bodies is a means of tracing local conditions in the contact and is thus a tool for extrapolation between different tribological triplets. It is also a means of coding feedback form experience and a way of getting round the complex of Pompeii by delayed in vivo instrumentation.
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8.2.12 SOLUTION OF A PROBLEM 8.2.12.1 CONTEXT As with fluid third bodies, a solution to the problem of friction and wear that would satisfy an engineering office would take many years. It is therefore necessary to offset the lack of continuity of interpretations and available models by a forthright approach that integrates mechanics, materials science, and physical chemistry. This means that in order to solve a problem, and therefore manage the interaction between the elements of the tribological triplet, engineering offices must combine rigorous design with a scientific approach. They must at least have acquired a tribological culture, which they lack, since although the curriculum followed by technicians includes courses on materials, very few include one on tribology. This is perhaps why the "natural" solutions are materials, i.e., the first bodies. However, the cost and loss of reliability of solutions increase when they concern first the mechanism, then the first bodies, and, lastly, the third body. It is obvious that this section cannot give an exhaustive description of the approach taken to solve a problem; it prefers to focus on only a few often neglected points.
8.2.12.2 STRATEGY The first step is to determine whether it is possible to get round the problem by accommodating movements with elastic deformations such as spring rod joints and joints composed of multilayer materials, or by using thick film lubrication which can be modeled and whose third body has known properties. What is more, they can be readily purchased; however, solid third bodies are produced in situ, thus subjecting the mechanisms to their properties. It should not be forgotten that one of the three functions of the third body is to separate the first bodies. This separation occurs via the bearing, whose friction is in some way the price to be paid, and hence an energy dissipation that a fluid third body evacuates better than a solid third body. This thermal aspect remains the attribute of fluid third bodies, though for applications over 220~ only solid third bodies are suitable. Although the solid third body is the only way out, one strategy can be the following. Technologically, it entails obtaining a third body composed of materials similar to those present in the analysis of the life of a contact and
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whose flows will be as low as possible [28]. This can be done by ensuring that the source flow comes from one of the first bodies. The internal flow can be limited by ensuring the reconstitution of the entities of the third body. This reconstitution depends on the rheology of the third body, and thus its source. The scale of activation of the flows depends on roughness, among other things. If it is too smooth, there is a risk of observing the internal flow activated throughout the contact: the entire third body will constitute the external flow. However, greater roughness will limit the activation zones of the internal flow and can reduce the latter considerably. This explains why a progressive improvement of a solution of the surface condition is no longer possible beyond a roughness threshold value. Scientifically, the problem is to control this situation rather than impose it on the contact. This requires iteration between modeling, tribometry, and expertise. By expertise is meant structured feedback from experience. The objective of these iterations is to: identify the parameters that have a significant impact on the contact's functions and define the real conditions of the contact to be simulated in tribometry; establish for an application the criteria for transportation with tribometry, in order to solve the problem while taking care for future applications to reduce testing times and develop simplified tests that permit engineering offices to design products more quickly. "Accelerated or truncated" tests should be avoided; they have often shown themselves to be dangerous, since they do not truncate the right parameters. Wear flows are not always activated when stresses are greatest. In tribology, care must be taken not to concentrate overly much on the obvious at the expense of neglecting the real contact conditions that activate the tribological circuit. Iterations between modeling, tribometry, and expertise are possible because models are analytical tools that allow varying and .exacerbating parameters more easily than in experiments; this is so for both industry and tribometry. A tribometer is merely a specific mechanism that does no more than bring the materials of the real mechanism into contact. This is followed by an expertise of the prototype, carried out with both models and the first tribometry results, which allows for identification of the flows of the third body and the stresses actually brought into play. Thus it reconstructs the dynamics of the contact by distinguishing the mechanical effects from the material effects. It is only at this stage that tribometry can be efficiently used to sort from a given result the indirect and direct contributions of the
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materials. Moreover, it is only at this stage that an experimental approach has significance. This strategy provides solutions for both the mechanism and the materials in contact. Generally, the solutions for the mechanism formulated by structural mechanics studies are easier to control than the solutions for materials sensitive to physical chemistry. Obviously, the mechanical solutions imply taking into account all the elements of the mechanism involved in the sequence of forces and moments and not the contact alone. It is worthless to attempt to control interactions between the elements of the tribological triplet and the three disciplines if the potential solution is not mechanically viable, i.e., if the indirect response implies excessive direct stresses.
8.2.12.3
THE MECHANICAL VIABILITY OF A SOLUTION
A serious mechanical analysis assisted by contact mechanics and including thermomechanics permits one to minimize direct stresses in at least 80% of industrial cases and privileges the STT response rather than the cracking response. This mechanical analysis must be carried out for the entire mechanism in order to avoid everything that can from being in the contact. Care must also be taken that the engineering office complies with the initial conditions recommended, since they very first moments of the actual life of the contact depend on the design phase. Therefore, cleaning and machining conditions are decisive. For example, the same roughness value expressed in Ra, but obtained by a genuine section or by folding, will have significant consequences on the rheology of the third body. Likewise, two coatings with the same chemical compositions but of different morphologies will have different tribological behaviours. The source flow will be different, as will be the trapping of the third body and thus the internal flow, etc. The property gradients of injection molded polymer first bodies often depend on the thickness of the latter; thus their direct response will change during the activation of their source flow. The loads required to improve the volume properties are not necessarily those required to optimize the rheology of the third body. For example, certain loads permit one to reduce the cohesion of the third body and limit the extent of activation of the internal flow. Consequently, since the surface is borne by the volume, optimization of the surface and the volume require knowledge of the main parameters of a given application; hence the usefulness of the feedback from experience.
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8.2.12.4 INTERPRETATIONS During extrapolations of configuration from one contact to another, it is important to be wary of solutions that give satisfaction without the reasons being understood. This is often the case for coatings and surface treatments that can intervene in different ways by: 9 substituting their response for that of another first body and thus imposing the source flow (direct role); 9 modifying the stress field because of their thickness and/or their thermoelastic properties relative to those of the substrate (indirect role); 9 adding residual constraints (direct role); 9 because of their morphology, trapping the third body and thus acting on the internal flow. The effects can be difficult to disassociate. For example, shot-blasting leads to residual stresses and cupules on the surface that trap the third body. Still on the subject of coatings, one should use multilayer models rather than attempt to guess "hard layer on soft substrate" effects with and without residual stresses. It is also illusory and even dangerous to implement a solution concerning one of the elements of the triplet without evaluating the consequences on the others. Feedback at the ultimate stage of a contact's life provides too much information for it to be used easily. It is preferable to get feedback at the beginning of the contact's life on that which is at the source of the problem and evaluate the correct sequence between cause and effect. For example, a high wear rate means a high wear flow, which can stem from: 9 a high source flow that "pushes" the third body toward the exterior of the contact; 9 opening of the contact caused by a dynamic geometry, which then leads to reactivating the source flow. The solution to these problems will differ according to each case.
8.2.13
COMMENTS
Though it does not provide "material couple" solutions, the objective of this approach is to have: 9 added the "mechanism" dimension to that of "materials"; 9 substituted a dynamic perception in place of a static one;
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9 drawn attention to the interaction of the elements composing the tribological triplet and highlighted that it is pointless to experiment with a solution if its "mechanical" viability has not been ascertained. The interaction between the elements composing the tribological triplet of mechanics, materials science, and physical chemistry makes friction and wear a crossroads discipline whose limited formalism could transform it into a dead-end science if the identification of the sequence of phenomena leading to the activation of the tribological circuit is incorrect. Although quality departments possess criteria for controlling material volume properties, they are unable to apply the evaluation criteria required to ensure the reproductibility of the properties of the "skin of the first bodies" necessary to determine the tribological reproducibility of machining and surface cleaning. This is a subject on which much work remains to be done to define efficient measures.
ACKNOWLEDGMENTS The author acknowledges Maurice Godet, who, with the third body, opened the way for the work carried out by the Solid Third Bodies Team of the Laboratory of Contact Mechanics of INSA-Lyon, France, and whose views form the basis of this article.
REFERENCES 1. Berthier, Y. (1996). Maurice Godet's third body approach, in Proceedings of the 22nd LeedsLyon Symposium: The Third Body Concept: Interpretation of Tribological Phenomena, Tribology Series 31, Elsevier, pp. 21-30. 2. Ludema, K. (1996). Third Bodies in close fitting contacts, in Proceedings of the 22nd LeedsLyon Symposium: The Third Body Concept: Interpretation of Tribological Phenomena, Tribology Series 31, Elsevier. 3. Israelachvili, J. (1992). Intermolecular and Surface Forces, 2nd ed., Academic Press. 4. Singer, I. L., and Pollock, H. M., eds. (1991). Fundamentals of friction: Macroscopic and microscopie processes, NATO ASI Service, Series E: Applied Sciences, vol. 220. 5. Rigney, D. A., and Hammerberg, J. E. (1998). Unlubricated sliding behavior of metals. Reprint from Materials Research Society, MRS Bulletin 23 (6). 6. Dowson, D. (1979). History of Tribology, London: Longman. 7. Bou-Said, B. (1999). Contact ceil-paupibre: Aspects tribologiques. Journ~e de la Science, INSAUCB Lyon. 8. Fr~ne, J., Nicolas, N., Degueurce, B., Berthe, D., and Godet, M. (1997). Hydrodynamic Lubrication, Tribology Series 33, ed. D. Dowson Amsterdam. 9. Jullien, A., Meurisse, M.-H., and Berthier, Y. (1993). Fractionized thin film lubrication: A new regime of lubrication in a carbon based material-oil-glass contact. Lubrication Engineering, Journal of the STLE, 49 (6): 479-486.
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10. Mazuyer, D., Boure, Ph., Georges, J. M., Lubrecht, A. A., and Lorentz, G. Formation and adherence of boundary lubricating layers with water-based lubricant in a concentrated elastohydrodynamic contact. ASME-STLE International Tribology Conference (in press). 11. Coulomb, C. A. (1773). Essai sur une application des r~gles de maximis et minimis quelques probl~mes de statique, relatifs h l'architecture. Mbnoires de mathbnatiques et de Physique de l'Academie Royale des Sciences, Paris, pp. 343-382. 12. Bowden, E P., and Tabor, D. (1964). The Friction and Lubrication of Solids, parts I and II, Oxford: Clarendon Press. 13. Achard, J. E (1953). Contact and rubbing of fiat surfaces. J. Appl. Phys. 24: 981-988. 14. Kalker, J. J. (1990). Three Dimensional Elastic bodies in Rolling Contact, Kluwer Academic Publishers. 15. Johnson, K. L. (1985). Contact Mechanics, Cambridge University Press. 16. Lamacq, V., and Dubourg, M. C. (1999). Modelling of initial crack growth and crack branching conditions under fretting conditions. Fatigue and Fracture of Engineering Materials and Structures 22: 535-542. 17. Lamacq, V., and Dubourg, M. C. (2000). Stage II crack propagation direction determination under fretting fatigue loading: A new approach in accordance with experimental observations, Symposium Fretting Fatigue Current Technology and Practices, Salt Lake City, ASTM STP 1367, ed. D. W. Hoeppner, V. Chandrasekaran, and C. B. Elliott III, pp. 463-450. 18. Zienkiewicz, O. C., and Taylor, R. L. (1991). The Finite Element Method, 4th ed., McGraw-Hill. 19. Raous, M., Jean, M., and Moreau, J. J. (1995). Contact Mechanics, Plenum Press. 20. Baillet, L., Desrayaud, C., Brunet, M., and Berthier, Y. (1996). Numerical and experimental analysis of ironing of thin sheel metal. Numisheet 96, edited by J. K. Lee, G. L. Kinzel, and R. H. Wagoner, pp. 72-79. 21. Czichos, H., Becker, S., and Bexow, J. (1989). International multilaboratory sliding wear tests with ceramics and steel. Wear, 135, pp. 171-191; and Czichos, H. Becker, S., and Bexow, J. (1987). Multilaboratory tribotesting: Results from the Versailles Advanced Materials and Standards Programme on Wear Test Methods, Wear, no. 114, pp. 109-130. 22. Rigney, D. (1997). Comments on the sliding wear of metals. Tribology International 30 (5): 361-367. 23. Cai, B. C., Kuhlmann-Wilsdord, D., and Nelson, R. B. (1991). A study of shear deformation and work hardening of metals under pressures to 2100 MPa. Material Science and Engineering A 138: 33-47. 24. Heshmat, H. (1995). The quasi-hydrodynamic mechanism of power lubrication. Part II: Theory and rheology of triboparticulates. Tribology Transactions 38 (2): 269-276. 25. Busquet, M., Descartes, S., Berthier, Y. and Torrance, A. A. (1999). Wear: Influence of detached particles trapped in the contact when a hard cylindrical asperity slides over a soft smooth surface, in Proceedings of the 26th Leeds-Lyons Symposium: Thinning Films and Tribolo~cal Interfaces, Tribology Series, Elsevier. 26. Iordanoff, I., and Berthier, Y. (1998). First steps for a rheological model for the solid third Body, in Proceedings of the 25th Leeds-Lyon Symposium: Lubrication at the Frontier: The Role of the Interface and Surface Layers in the Thin Film and Boundary Regime, Tribology Series 36, Elsevier, pp. 551-560. 27. Brown, R. L., and Richards, J. C. (1970). Principles of Power Mechanics: Essays on the Packing the Flow of Powders and Bulk Solids, Pergamon Press. 28. Mahe, L., and Berthier, Y. (2000). The third body flow approach: Wear of filled PTFE. International Tribology Conference, Nagasaki, Oct-Nov. 2000. 29. Dowson, D., and Wright, V., eds. (1981). Introduction to the Biomechanics of Joints and Joint Replacement, London: MEP.
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30. Souchon, E, Berthier, Y., and Renaux, P. (1998). Helical scan head and tape contact behaviour: Optimization of tribological and magnetic aspect. 31 (8): pp. 479-484. 31. Jacquemard, P., Meurisse, M.-H. and Berthier, Y. (1996). From the phenomenology to the concepts which flow from the third body, in Proceedings of the 22nd Leeds-Lyon Symposium The Third Body Concept: Interpretation of Tribological Phenomema, Tribology Series 31, Elsevier, pp. 91-102.
SECTION
8.3
Models of Friction A. R. SAVKOOR
Vehicle Research Laboratory, Delft University of Technology, Delft, Netherlands
Contents 8.3.1 Introductory Background . . . . . . . . . . . . . . . . . 8.3.2 Global Model of Amontons-Coulomb Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Apparent Deviations from the Basic Coulomb Friction . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3.1 Influence of Normal Pressure (Load) on Friction . . . . . . . . . . . . . . . . . 8.3.3.2 Influence of Sliding Velocity on Friction (Static and Stationary Sliding) . . . . . . . . . . . . . . . . . 8.3.4 Friction of Viscoelastic Materials (Polymers and Elastomers) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4.1 Friction of Elastomers in Contact with Hard Solids in Stationary Sliding . . . . . . . . . . . . . . . . . . 8.3.4.2 The Falling Friction-Velocity Curve and Stick-Slip . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Nonstationary Characteristics of Friction.. 8.3.5.1 Nonstationary Friction Model . . . . . . 8.3.6 Dynamics of Friction Models for Nonstationary Sliding: Frictional Vibration and Stick-Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6.1 Dynamic Behavior Assuming a Purely Velocity-Dependent Friction. 8.3.6.2 Identification of Model for Friction of Elastomers Based on Bifurcation Characteristics . . . . . . . . . . 8.3.6.3 Comparison of Friction Models (Case A and Case B) . . . . . . . . . . . . . . . 8.3.7 Influence of Normal Vibration on Friction.
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Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
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8.3 Models of Friction 8.3.8
C o n t a c t and Friction of D e f o r m a b l e Solids: The Local C o u l o m b Friction Model . . . . . . .
725
8.3.8.1 The Local Model of C o u l o m b Friction . . . . . . . . . . . . . . . . . . . . . . . . . . .
726
8.3.8.2 Contact of Solids Subjected to Purely N o r m a l Load . . . . . . . . . . . . . . .
727
8.3.8.3 Hertz Theory of Contact for Purely N o r m a l Loading of Solids . . . . . . . . . . 8.3.9
728
The T r a n s m i s s i o n of Tangential Forces b y Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
730
8.3.9.1 Transition from Static to Kinetic Friction by Partial Slip (0 < T _
730
8.3.9.2 Irreversible Effects, the Dependence on Loading History and Contact Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.10 Friction in Rolling C o n t a c t of Solids . . . . . .
732 734
8.3.10.1 Rolling Friction of a Free-Rolling Wheel . . . . . . . . . . . . . . .
734
8.3.10.2 Friction Model for Traction in Rolling Contact . . . . . . . . . . . . . . . .
735
8.3.11 M a t h e m a t i c a l Models of Friction . . . . . . . . . .
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8.3.11.1 Model Based on Plasticity T h e o r y . . 740 8.3.11.2 Dahl Model Based on Hysteresis Behavior . . . . . . . . . . . . . . . 8.3.11.3 Extension of Dahl Model . . . . . . . . . .
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8.3.11.4 Mathematical F r a m e w o r k for F o r m u l a t i o n of Contact and Friction Problems . . . . . . . . . . . . . . . . .
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8.3.12 Physical Modeling of Friction . . . . . . . . . . . . .
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8.3.12.1 Characteristics of Surfaces and the Adhesion Model of Friction . . . . 742 8.3.13 E l e m e n t a r y Physical T h e o r y of Friction . . .
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8.3.13.1 Plastic Deformation of Asperities under C o m b i n e d N o r m a l and Tangential Loading . . . . . . . . . . . . . . .
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8.3.14 Models Based on Contact, Adhesion, and Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . .
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8.3.14.1 The JKRS Adhesion Model . . . . . . . .
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8.3.14.2 Tangential Loading of Adhesive Contact, Modeling of Static Friction . . . . . . . . . . . . . . . . . . . .
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8.3.15 Models Based on Friction on the Atomic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.16 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1
INTRODUCTORY
756 756 757
BACKGROUND
Friction is a common phenomenon that arises whenever solid bodies are presented together and certain parts of their surfaces come into contact. The contact enables solids to transmit forces reciprocally across their contact surface (also referred to as the interface). The resultant contact force acting on a solid may be resolved into two components in the directions of the normal and the tangent to the interface. The two components are the normal reaction force and the force of friction. Both the normal reaction and the friction forces depend on the applied force, or the load that is pressing the solids into contact. The essential characteristic of friction is that it is a force that resists relative motion of material points within the contact interface even though there is no "obvious" impediment to sliding. Friction is generally characterized by a coefficient defined as the ratio of the tangential and the normal components of the contact force. The coefficient of friction (#) depends on the normal load, the relative motion, the material properties of solids, and even more critically on surface and interface conditions (any foreign materials in the interface). Although one is aware that friction is a surface-sensitive phenomenon, it is a common practice to associate the coefficient of friction with the materials of the pair of solids. This is because of the difficulty of specifying precise surface and interface conditions except in ver general terms such as by saying "dry or wet," or by providing a detailed description of preparing or "cleaning" surfaces prior to performing friction experiments. It should be emphasized that friction is not purely a property of the solid materials, or solely that of the surfaces or the interface. Friction involves a complex interaction between the properties of solid materials, their surfaces, and any contaminating material introduced or existing in their contact interface. Therefore, the coefficient of friction is not a fundamental physical quantity, and it could be regarded merely as a convenient nondimensional representation of the force of friction. The dominant properties of interest concerning friction are as follows: 1. solids: the mechanical properties related to deformation (e.g., elastic, viscoelastic, plastic) of bulk material; 2. surfaces: the geometry on the macroscopic (overall shape) as well as on the microscopic (surface roughness, including texture) scales and the
8.3 Models of Friction
703
physicochemical properties (surface energy, adhesion, and fracture strength) of adsorbed surface films formed in a gaseous (e.g., air) environment; 3. interface contaminants: physical and chemical properties of any foreign material introduced intentionally) e.g., oil, solid lubricants) or inadvertently (e.g., water or dust) in the contact interface. Clearly, it is common knowledge that lubrication of the surface by fluids has a strong influence on friction, generally that of reducing friction substantially. The present chapter excludes discussion of lubrication resulting from interfacial liquids and restricts the scope to nonlubricated conditions or "dry friction." In practice, even in the dry state the physical and chemical properties of the surface and the interface are significantly different from those of the bulk material. One of the most important causes of this difference is the presence of a gaseous environment and/or traces of ill-defined foreign contamination. The difficulty of determining and controlling the inevitable films and traces of contamination on solid surfaces is to a large extent responsible for the relatively poor repeatability and large scatter of experimental results of friction. Consequently, only the strong and consistent trends are sufficiently robust to warrant the development of quantitatively reliable and useful models of friction. Furthermore, the ill-defined surface chemistry leads to a lack of definitiveness in the absolute magnitude of p that is tabulated for various materials in the open literature. Because friction arises in most mechanical contrivances, some understanding of its behavior was needed even in age-old times. The historical evolution of friction and lubrication is documented in the work of Dowson [1], which describes the pioneering scientific studies of da Vinci, Amontons, and Coulomb. It is interesting to note that despite the large differences in physical and chemical properties of solid materials such as ferrous and nonferrous metals, polymers, ceramics, concrete, wood, etc., the range of # quoted in the open literature in dry conditions is (0.2 < # < 1). For a number of pure metals with clean surfaces (outgassed in high vacuum), values of # from 5 to 200 have been recorded [2, 5]. Paradoxically, soft solid materials such as elastomers (rubber) that exhibit significant adhesion and support tensile loads yield large and even quite absurd values of # (p < 0). The force of friction is the outcome of mechanical and physicochemical processes involving the entire system that consists of the solid materials, including their surfaces and the interface. The phenomenon of friction may be classified in many different ways. A customary way is to look at the overall pattern of relative motion between macroscopic solids and classify friction according to the three distinct patterns: (1) static friction, (2) sliding friction, and (3) rolling friction.
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However, because friction is concerned with relative motion between contacting surfaces rather than that between bulk solids, it is more appropriate to classify the distinct manifestations of friction into only two basic types: "static" and "kinetic" friction. A large part of this chapter concentrates mainly on the modeling of static and kinetic friction; the technologically important case of friction in rolling contact will be described in a separate section. Modeling friction generally implies construction of theoretical models that describe the nature and behavior of frictional force of a mechanical system that consists of solids making contact in an inert, well-defined environment (such as ambient air at constant temperature, pressure, and relative humidity). A tacit assumption is that thermodynamic equilibrium prevails so that the properties of the solids, their surfaces, and the environment are practically invariant in the time span of interest. The modeling of friction is broadly pursued along two distinct but complementary paths that accentuate the differences in the requirements and interests of the engineering and the scientific community. In technological applications the emphasis is on the development of "functional" models or the so-called laws of friction primarily as tools that suit the purpose of engineering design. The modeling approach focuses directly on friction at the mesoscopic scale because the main concern is not how or why friction arises but the ability to characterize friction in terms of its effects on the dynamic behavior of mechanical components or systems involving friction. The development of such models rests on the examination of the salient and repeatable features of the frictional behavior of a certain class of solid materials in well-defined basic experiments on specially designed tribometer devices. The observed features are then structured in some compact mathematical form. It is then necessary to carry out a large-scale experimental program and establish from that the domain of validity of models for a particular class of materials. Considering the inevitable variability and anomaly of experimental data on friction, the time and effort of the investigator often limit the validation phase of the development. It is for this reason that, for quantitative estimation of friction, practicing engineers are advised to measure and identity friction of complex devices in the actual operating situation. In contrast to engineering modeling, scientific interest in modeling friction arises primarily from its enigmatic physical mechanism. The mechanism is enigmatic because it is hard to find evidence of any impediment to sliding of solids that can explain friction in unequivocal terms. The theoretical analysis of the mechanism of friction is motivated primarily by a desire to understand and establish the root cause of friction as reflected by the choice of microscopic or even atomic scale used to develop the models. The starting
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point for development of microscopic physical models uses two notions: (1) a conjecture of the physical mechanism of friction and (2) the choice of a relevant scale for describing the process of friction. The modern approach to modeling the physical mechanism of friction on the microscopic scale of surface asperity has become established since the profound work of Bowden and Tabor [3]. Their adhesion and ploughing model, to be described in a later section, is the most widely used physical model of friction. More recently physicists have taken a lively interest in developing models and conducting experiments on friction at the atomic or nanoscale. Specific experiments and models are developed to shed light on the fundamental aspects of frictional behavior on the scale of atoms, but the generality of the experimental results and theoretical models and their relevance to friction on the mesoscopic scale of solids is not yet clear. Hence the essential findings of friction on the nanoscale will be reported briefly in a separate section. Interestingly, early research on friction by Coulomb proceeded simultaneously along both the aforementioned paths but met with different degrees of success. The principal mechanism of friction as conjectured by Coulomb is based on a model of contact between extensive and wavy surfaces of perfectly rigid solids. The surface asperities interlock upon contact, therefore sliding of surfaces necessitates the application of a sufficiently large external force that pulls the solids up the slopes and across the tiny summits of the asperities. From a rudimentary analysis this mechanism results in a value of/, ~ tan(0), where 0 is the average slope of an asperity, independent of the load and the size of the contact. The model ignores the deformation of materials, and it does not account for the order of magnitude of either friction or the energy dissipation by friction observed in sliding experiments. On the other hand, the functional relations defined by the "laws of friction" of Amontons and Coulomb along the second path, even today, yield the most viable, albeit approximate, model of friction. The functional models of friction for describing the dependence of friction on operational variables such as the normal contact force and the relative motion between contact surfaces of solids will be considered first.
8.3.2
GLOBAL MODEL OF
AMONTONS-COULOMB
FRICTION
The most commonly used model of friction is that based on the AmontonsCoulomb law of friction. It is referred to as the Coulomb friction model. In it original form the model is essentially global because it deals with relations between resultant contact forces and overall motion of solids considered as
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S~vkoo~
"rigid bodies" subjected to a self-equilibrating system of forces. A tacit hypothesis in the Coulomb friction model is the unilateral nature of contact. Most elementary courses in mechanics acknowledge as an obvious inference from everyday experience that contact can support large normal compressive forces, limited only by the strength of the solid materials, but hardly any normal tensile forces at all. However, significant tangential forces can be transmitted readily by friction provided the solids are pressed together by a normal compressive force. The static friction that prevents relative motion of solid bodies in a state of rest has a finite limit, whereas the kinetic friction that opposes sliding is practically constant over a wide range of speed. The laws of friction describe the influence of a constant normal load, the contact area, and the relative motion (static and sliding) on friction. 9 Friction is proportional to the normal reaction force acting in the contact. 9 Friction is independent of the macroscopic area of contact between the solids. 9 Friction opposes relative motion between solids, both in the state of rest (zero relative velocity) and in the state of sliding (nonzero relative motion). The proportionality implied by the first law applies both to the limiting value of static friction and the kinetic friction. Referring to Figure 8.3.1a, for solids in equilibrium the components of applied forces normal and tangential to the contact area, respectively, P (such that P > 0) and T, are balanced by the corresponding components, namely, Fn (the normal reaction) and Ft, the force of friction in the contact interface. If Fn < 0 A P + Fn = 0 ' * T + Ft = 0
and
if Fn = 0
~
Ft = 0
(1)
According to the global version of the Coulomb friction model, the state of relative motion between the solids when both P and T are applied depends on
FIGURE 8.3.1 a. Friction in uniform relative translation of a rigid block on a large plane and rigid solid, b. The global model of Coulomb friction, c. friction in relative rotation of rigid solids in contact.
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8.3 Models of Friction
the ratio of the tangential component (Ft) and the normal component (Fn) of the reaction force. The two possible states of tangential relative motion are as follows. 1. If the relative surface velocity Vs z 0, friction opposes the applied force and prevents the relative motion between solids. if Vs -- 0 --~ Ft -- - T sgn(T) ~ Fts where Fts is given by F t s - -#o Fnlsgn(T)
(2)
Ft is the force of static friction, and Fts is the m a x i m u m value or the limiting value of the force of static friction; it is customary to express Fts in terms of the coefficient of (limiting) static friction
#o = IFt~l/IFnl
(3)
2. When sliding takes place, i.e., for V~ ~ O, the force of kinetic friction Ftk opposes the sliding speed. For Vs ~: 0 -~ Ft = Ftk where Ftk = -/~klFnlsgn(Vs)
(4)
The force Ftk is the force of kinetic or sliding friction, and #k is the coefficient of kinetic friction. In the Coulomb friction model #s and #k are assumed constant and equal. The determination of the frictional moment for relative rotation of rigid bodies (Fig. 8.3.1c) requires additional assumptions concerning the distribution of normal load and the introduction of the "local version" of the basic Coulomb friction model. The essential behavior of static and kinetic friction in the Coulomb model can be demonstrated with simple experiments where the tangential force is prescribed and the relative motion is observed (Fig. 8.3.2). Remarkably, the technological importance and robustness of the first of the so-called Coulomb's laws that enunciates direct proportionality between the frictional and the normal forces, albeit as a first approximation, are acknowledged universally. This law has inspired the development of physically based models of friction. The other two laws are much less robust.
FIGURE 8.3.2 Simpleand direct experimental setups to establish the nature and the limiting force of static friction and the transition to kinetic friction.
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Application and validity: The elementary so-called Coulomb friction model in Figure 8.3.1b is used extensively in several textbooks (see, e.g., [4]) and in several engineering applications. The solids are modeled by rigid bodies, and the main interest is only in the absence or presence of gross relative motion between solids. The model is valid when the relative motion of sliding is much larger than the preliminary displacement in static friction (see the section on deformable solids). Furthermore, because the dependence of friction on relative motion is highly simplified, it is not adequate for describing friction under dynamic sliding conditions.
8.3.3 APPARENT DEVIATIONS FROM THE BASIC COULOMB FRICTION Friction experiments with various materials indicate marked deviations from the basic Coulomb model concerning the dependence of friction both on the normal contact force and the relative motion between the solids. The nature of deviations reported in the open literature is generally observed in experiments performed under conditions of stationary sliding, that is, for constant normal load (P or F,) and uniform relative motion with sliding speed (Vs > 0). Relatively few studies report data concerning measurements of static friction (Vs = 0). The experimental determination of the limit of static friction by observing incipient relative motion between material points in contact proves to be a difficult task, especially because of the roughness of the surfaces. It should be noted that the dominant trends of friction behavior commonly "observed" in a limited range of variables, assuming that several other parameters which may influence friction, such as the temperature and the geometrical and physicochemical properties of the solids, their surfaces, and the contact interface, are maintained constant.
8 . 3 . 3 . 1 INFLUENCE OF NORMAL PRESSURE (LOAD) ON FRICTION The influence of normal load P on friction is expressed alternatively as that of the nominal normal pressure p, with p = P/A, (where A, is the area of nominal contact).Friction of some pairs of materials, such as steel, aluminum, and wood, is almost independent of the normal load, in accordance with Coulomb's first law over a wide range of loads, as illustrated by Rabinowicz [5]. If the surface and interfacial properties are not altered by normal loading and friction, the trends for the pressure dependence of the limiting static
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gk
P FIGURE 8.3.3 Influence of normal pressure (normal load) on coefficients of static and kinetic friction. The dotted lines indicate the range of pressures (not very low or high) used in experiments.
coefficient #0(P) and the kinetic coefficient/.zk(p) at some nonzero, constant sliding velocity V~ are shown in Figure 8.3.3. Over a fairly broad range of pressures, both /t0(p) and #k(P) decrease with increasing pressure p (or normal load P),
Its oc. (p)k
and
#k
(5)
The physical explanation of the influence of the nominal pressure or normal load on friction through the index n is based on the Bowden and Tabor theory of friction, which postulates direct proportionality between friction and the area of true contact of solids. This aspect will be described later in the section on physical models of friction. As a rough approximation, the influence of increasing normal pressure is generally to scale down the limiting static coefficient /t 0 and the kinetic friction-speed curve 12k(Vs). However, it is worth noting that with some materials, notably metals, surface conditions may be affected by loading. Surface films of oxides, especially on certain metal surfaces formed in air, are relatively brittle. The films may rupture when the normal load (or pressure) exceeds some critical value, resulting in bare metal-to-metal contact that leads to strong adhesive bonding. The critical normal load depends upon the nature of the roughness of the surface. For instance, for unlubricated copper on copper, the coefficient of friction of 0.4 obtained in the range of low loads increases beyond some critical load rapidly to a high value of nearly 1.6 ([5] Fig. 4.9).
8.3.3.2
INFLUENCE OF SLIDING VELOCITY ON
FRICTION (STATIC AND STATIONARY SLIDING) From the experimental results of friction for different materials and surface conditions (all dry), it is found that kinetic friction (#k) varies with the sliding
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I
....
.
I
.
.
.
.
.
.
.
.
.
.......
Vs
vs
(a)
(b)
Vs (c)
vs (d
FIGURE 8.3.4 Typical shapes of friction-sliding speed curves for stationary sliding (constant speed) with constant normal load for different solid materials having nominally smooth and dry surfaces.
velocity. The different forms of the coefficient of friction #, all as an odd function of the relative (sliding) velocity Vs, are shown for Vs in Figure 8.3.4, assuming a constant normal load pressing the solids together. All these forms have in common that there is a finite limit #0 for the static friction. For Vs - 0 ,
# < #0
(6)
where #0 is the limiting coefficient of static friction. For Vs -r O, # _
(7)
where #h is the coefficient of kinetic friction. In some cases, as depicted in Figure 8.3.4a, friction behaves according to the Coulomb model. In the "stiction" variant of Figure 8.3.4a, the static friction is relatively high and the transition from static (#0) to kinetic (#h) friction may be discontinuous, as in Figure 8.3.4b. More generally, one finds that friction varies with sliding velocity; it decreases initially to some minimum value and then shows an increasing trend at higher velocities, as in Figure 8.3.4c. This curve has a form similar to the so-called Stribeck curve [6] for lubricated contact, but the rising part of the curve occurs very rarely with dry contact. The variation of kinetic friction with sliding velocity may be caused by different physical mechanisms. It may be due to the influence of surface strains, strain rate, and the rise in temperature during sliding on either the surface properties or on the bulk properties of the solids. Friction of metals generally does not vary with speed over the modest range of sliding speeds encountered in most engineering applications. However, over a much broader speed range the friction versus speed curves for some metallic pairs such as steel on indium and steel on lead are qualitatively of the form shown in Figure 8.3.4d [5, Fig. 4.13]. In the range of very low speeds friction for some metals is found to increase with sliding speed, while at high speeds friction shows a decreasing trend with increasing speed.
8.3 Models of Friction
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It is highly unlikely that the behavior at very low speed can be attributed to any variation of the bulk properties of metals, suggesting that the trend reflects the influence of speed on surface properties. On the other hand, the decrease in friction at high velocities may be explained by frictional heating leading to high surface temperatures, which influence not only the surface properties but also the material properties of solids near the subsurface region. In extreme conditions the heating may cause severe wear and surface melting of material. Other more specific but generally weaker trends reported elsewhere, such as the anisotropy of friction (directional behavior) that depends on bias in surface texture or material anisotropy, are not sufficiently general to be considered in modeling friction. In contrast with friction of metals, the trend shown in Figure 8.3.4d is intrinsic to many polymers where strain rate and temperature exert a dominant influence on bulk material properties. The coefficient of friction increases in the range of low velocities until a maximum value and then decreases with further increase in velocity. In particular, viscoelastic materials such as elastomers exhibit a pronounced variation of friction with speed and also with temperature. This will be considered separately in the following section. It is necessary to emphasize that there is a certain amount of ambiguity in the interpretation of experimental data in regions where friction decreases with increasing sliding speed because of the nonstationary nature of sliding. The nonstationary aspect and its consequence for the measurement of friction in a dynamic situation will be discussed in Sections 8.3.4.2 and 8.3.6.
8.3.4 FRICTION OF VISCOELASTIC MATERIALS (POLYMERS AND ELASTOMERS) Polymers and elastomers are increasingly important because of the rapid growth in their industrial usage. In particular, much work on friction has been devoted to the sliding of PTFE (polytetrafluoroethlene) and elastomers mainly because of their extraordinary tribological properties. PTFE and related polymers are well known for their properties of low adhesion (nonstick pans) and low friction (bearings) but unfortunately these materials display poor wear resistance unless additional modification of the material is engineerd. Industrial elastomers consisting of long chain molecules with sufficient chemical cross-linking can sustain large deformation and display high friction with relatively low wear (automobile tires). The friction of most polymers is closely related to their viscoelastic behavior. The coefficient of friction increases with the sliding velocity until it attains a maximum value at a certain speed,
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followed by a decrease of the coefficient when the speed is increased further. The main factor that governs the variation in mechanical properties and friction is the flexibility of the polymer chains. A broad classification in the order of decreasing structural flexibility is linear crystalline polymers, linear amorphous polymers at temperatures below their glass transition, and highly cross-linked thermosetting polymers. At low speed of sliding the coefficient of friction of PTFE sliding on PTFE is very low (#k of 0.07 to 01). With increasing speed the coefficient increases to its maximum value (/.tk = 0.3 at Vs = 0.5 m/s), decreasing thereafter as speed increases [7]. The study of friction of both amorphous and crystalline polymers clearly brings out the important role of the bulk material of solids. Similar experimental studies of Bahadur and Ludema [8], Steijn [9], and Tanaka [10] on PTFE and other linear crystalline polymers such as PE (polyethylene), Polypropylene, and Nylon show strong influences of sliding speed on friction at room temperature (22~ With increasing (constant) temperatures the friction-speed curve shifts correspondingly toward a higher velocity. For polymers below the glass transition, an increase in temperature from T1 to T2 causes the friction curve to shift from a speed of V1 to V2 on a logarithmic scale according to the Arrhenieus equation:
log(V2/V1) = ( Q / R ) ( 1 / T 2 - l/T1)
(8)
where Q = activation energy and R = universal gas constant.
8 . 3 . 4 . 1 FRICTIONOF ELASTOMERSIN CONTACT WITH HARD SOLIDS IN STATIONARY SLIDING The kinetic friction of elastomers on relatively hard solids with nominally smooth surfaces exhibits a marked dependence on sliding speed and temperatures. Interestingly, Grosch [ 11] observed a strong correlation between the influence of speed and temperature on friction and the welldocumented dependence of viscoelastic deformation on strain rate and temperature. The principle of time-temperature equivalence that holds for the deformation behavior of thermo-rheologically simple and linear viscoelastic materials such as elastomers also applies to friction of such materials. This principle can be used fruitfully to design experiments to study the influences of speed and temperature on friction separately. Friction of a test specimen of elastomer is measured at a constant temperature in a limited range of low sliding velocities driven by a tribometer device. The device is mounted inside a chamber with a controlled temperature environment (air), and the highest driving speed of the tribometer is restricted to the order of 10 -2 m/s to avoid
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8.3 Models of Friction
FIGURE 8.3.5 a. Construction of a full master curve (bell-shaped) for the isothermal frictionspeed curve by shifting partial curves measured at different temperatures using the WLF transform, b. The shifting of the isothermal friction master curve at different temperatures along the logarithmic speed axis.
any significant rise in temperature due to frictional heating. The results of measurements in the limited range of sliding velocity yield a partial frictionspeed curve at a constant temperature because friction may be regarded as a quasi-isothermal process. A similar set of friction-speed curves for the limited range of sliding speeds is obtained at different ambient temperatures, typically from - 6 0 ~ to +70 ~ The curves in Figure 8.3.5a illustrate the strong influence of temperature on the isothermal friction-speed curve of Styrene-butadiene rubber compounds sliding on nominally smooth and hard solids. On the right of the same figure is shown how such partial curves, appropriately shifted, appear as segments that merge into a single master curve at a desired value of temperature. The shift corresponding to a test temperature T along the logarithmic speed axis defined by ar is given by an empirical relation such as the well-known WLF transform (instead of the Arrhenieus equation): log ar --
- 8 . 8 6 ( T - T~) 101.5 + T - Ts
(9)
where Ts is a characteristic temperature with Ts ~ Tg + 50~ and Tg is the glass transition temperature of the elastometer. The construction of the master curve enables one to examine the influence of sliding speed on isothermal friction at the reference temperature over a very wide range of speed. At higher temperature the master curve of isothermal friction shifts toward higher speeds (Fig. 8.3.5b). The shape of the isothermal master curve of friction versus speed determined from experiments (e.g., [11]) can be represented by an empirical relation [12]: ltk(Vs) = lao + (/am - / . t 0 ) e x p ( - ( h 2 / 2 )
~na(]Vs]/Vm))
(10)
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Savkoor
In this expression, #0 is the coefficient of static friction, /Am is the maximal coefficient for [Vs[ = Vm, and h is a dimensionless parameter reflecting the width of the speed range in which friction varies significantly. The width corresponds approximately to that of the rate-dependent viscoelastic properties such as the creep or relaxation spectra of elastomers. Friction increases with increased speed if [Vs[~,Vm, and it decreases if [Vs[ ) Vm.
8 . 3 . 4 . 2 THE FALLING FRICTION--VELOCITY CURVE AND STICK-SLIP One of the major difficulties in the measurement and interpretation of the stationary friction-speed relationship is the persistence of frictional vibration (stick-slip) that arises in the range of sliding velocities where [Vs[ ~ Vm. Despite keeping a constant driving speed of the tribometer, it becomes generally impossible to impose the velocity of sliding because of the persistence of friction-induced vibration known commonly as "stick-slip." The sliding velocity and the applied tangential force driving the tribometer exhibit strong fluctuation with large peak-to-peak values. In isothermal sliding of most rubber compounds, this phenomenon occurs at low temperatures where friction decreases with an increase in velocity. The magnitude of the fluctuation depends on the driving velocity, and, interestingly, this magnitude becomes smaller with increasing velocity, as seen in Figure 8.3.6a. For modeling friction the stick-slip phenomenon poses two problems: first, whether and how to determine the stationary friction-velocity curve in this range, and second, how to model the behavior of friction under conditions of nonstationary sliding. The determination of stationary friction in the failing part of the frictionvelocity curve at low ambient temperature was realized by providing damping in the tribometer system that is sufficiently large (positive) to overcompensate the negative damping associated with the decreasing friction-veloicity function. In the range of very low velocity, a convenient and effective way to produce sufficient positive damping is to use the rising part of the frictionvelocity curve of the elastomer. The practical implementation based on this solution led to the construction of the tandem tribometer device shown in Figure 8.3.6c [13]. The test specimen that slides isothermally at low temperatures inside a climate-controlled chamber is mechanically coupled to the stabilizing specimen sliding at room temperature. The tandem device has proved successful in the suppression of stick-slip. Figure 8.3.6b displays the traces of the tangential force measured with a force transducer for different driving speeds of the tribometer. The force
8.3 Models of Friction
715
FIGURE 8.3.6 a. Isothermal friction-velocity curve showing the stick-slip region in the falling part of the curve, b. Tangential force plots with and without stick-slip for different driving velocities (specimen: SBR compound on stainless steel at -25~ normal load 1.5 N, nominal area 30 mm2). c. Schematics of the tandem tribometer arrangement for suppressing the stick-slip phenomenon, d. Stationary friction-velocity curve in the falling part of the curve determined with the aid of the tandem tribometer (specimen: SBR on stainless steel at -45~ normal load 1.5 N, nominal area 30 mm2) [13]. oscillation (marked "P;') resulting from the stick-slip that arises in the base configuration of the tribometer is successfully suppressed with the aid of the stabilising pair of the tandem arrangement, as evidenced by the steady trace of force (marked "B"). The tandem tribometer device is depicted in Figure 8.3.6c. The stationary friction-velocity curve (free of stick-slip) determined with the tandem tribometer is seen in Figure 8.3.6d. It is interesting to note that although the success achieved in suppressing n o n u n i f o r m sliding (stick-slip vibration) enabled the determination of the entire stationary friction-velocity curve, two crucial questions remained unanswered. First, does the model based purely on the stationary friction curve hold under conditions of stick-slip found using the base arrangement of the tribometer? Second, can the stationary friction curve by itself explain the successful suppression of stick-slip achieved using the tandem arrangement? The answers to both these questions depend upon the outcome of an analysis of the nonlinear dynamic behaviour of the base and the tandem versions of the
716
Savkoor
tribometer, where the friction model is defined solely by the stationary friction-velocity curve. The analysis reported in Section 8.3.6 shows that the answer to the first question negative. The answer to the second question requires additional work on the identification of an accurate dynamic model of the modified version of the tribometer used in the experiments. However, it is not fruitful to dwell here on the issue of stationary friction because recent experimental findings discussed in the following section suggest that modeling the nonstationary aspects of friction may be important.
8.3.5
NONSTATIONARY
CHARACTERISTICS
OF FRICTION The global Coulomb model that assumed a constant #s with no explicit dependence on time of contact and a constant Ph where /zk = / z s cannot explain the observed dynamic p h e n o m e n o n (frictional vibrations and stickslip) of mechanical systems involving friction. It is frequently noted in experimental studies that friction of various materials displays significant nonstationary behavior that is not taken into account in the formulation of either the Coulomb model or a kinetic friction model that defines the friction as a function of only the instantaneous sliding velocity. The d o m i n a n t trends of nonstationary behavior discovered in experimental work using different tribometer devices are illustrated in Figure 8.3.7. These trends have been discussed by various investigators and are summarized in a comprehensive report by Oden and Martins [14]. Creep of bulk material resulting in the growth of the contact area and the time-dependent microscopic process of t10
lg)l - - .~V1
gk
~k(V2)~___._~
!.................q
__" .............. v . _ w , v,
, ....................
t
t
t
Time of static contact
V = drivingvelocity (Tangential load rate) b
Transientresponseof friction to step changein sliding velocity c
FIGURE 8.3.7 Typicaltrends for friction at normal load observed in experiments, a. Influence of static contact time on Ps. b. Influence of the rate of tangential loading on #s (V3 > V2 > V1). c. Transient response of friction to a step in sliding velocity Vs from V1 to V2 > V1 depends on the history of motion (state) and may include an instantaneous response. The dashed line applies to the rising part [#k(V2) > ~Ak(V1)], and the dotted line applies to the falling part [#h(V2)<#k(V1)] of the stationary friction curve.
8.3 Models of Friction
717
adhesive bonding are offered as plausible physical explanations for the behavior of static friction (Figure 8.3.7a). The effect of loading rate (driving velocity) on static friction observed by some workers is sketched in Figure 8.3.7b. It is claimed that this trend is distinct from that depicted by Figure 8.3.7a. However, it is difficult to establish from such experiments when the limit of static friction is reached. The effect of tangential loading rate (driving velocity) requires additional investigation, and it is generally not considered in most models of friction. The history-dependent transient nature of friction was noted in the early work of Rabinowicz [5] on metals and by Schallamach [15] on rubber on glass. Both these investigators suggested that the instantaneous value of friction depends on past sliding motion over a certain characteristic distance. The effect is attributed to the surface roughness or more generally to "conditioning" of sliding surfaces. The-transient frictional sliding of rocks has been investigated thoroughly by Dieterich [16], whose experiments were motivated by the phenomenon of earthquakes caused by geological faults. The response of/~k to a step change in sliding speed from Vl to V2 (V2 > V1), the sliding speeds assumed to be imposed) is shown in Figure 8.3.7c. The distinctive features are: 1. an instantaneous response of friction (positive jump for V2 > V1 and vice versa); 2. a subsequent asymptotic decay to the stationary value of friction corresponding to the new speed. The new stationary value may be larger or smaller than the initial one, depending upon whether V1 is in the rising or falling part of stationary friction curve/.Zk(Vs). This work has led to the development of "rate and state dependent friction" models developed by Ruina [17] and by Rice and Ruina [18]. The history dependence is represented by a state variable description that takes into account the effect of the past relative motion of solid surfaces. Since the work on rocks, the trend has been found in experiments using various other materials such as PMMA, PTFE, and wood. The more wellestablished trend of nonstationary behavior of friction is the noninstantaneous asymptotic decay of friction, whereas the noninstantaneous response of friction that acts as a linear viscous contribution is a relatively new feature that merits more detailed investigation in the case of elastomers.
8.3.5.1
NONSTATIONARY FRICTION MODEL
A model based on work in References [15] and [18] was recently proposed by Meijaard and Savkoor [19]. It applies to the friction-speed curves such as
718
Savkoor
those in Figure 8.3.7c. For any sliding velocity Vs there are two terms that contribute to the instantaneous coefficient of friction. The first term is that due to an instantaneous response #i(Vs) that depends only on the current sliding velocity, and the second is the internal state defined by a variable 0 that reflects the past history of sliding motion. In nonuniform sliding, the state variable 0 satisfies a first-order differential equation
b = (1/'Cs +
IV l/dh)( k(V )
-
o)
(11)
For constant Vs, the state variable 0 approaches the stationary value #h(Vs) asymptotically. The time constant in the sliding phase is: 1/(1/1:s + [Vsl/dk). The model (Eq. 11) describes the rise in static friction (for Vs = 0) with the time of static contact as portrayed in Figure 8.3.7a with a time constant of rs. For large sliding velocities, the equation approximately describes the behavior discovered by Rabinowicz with one state variable, that the instantaneous value of friction is the mean of the friction-velocity function over a past slid distance dh. In nonstationary sliding, the instantaneous value of the friction force is given by #(t) = #(Vs, O) = O sgn(Vs) + ~i(Vs)
(12)
The model contains two functions, #h and #i, and in addition two fixed parameters, rs and dh, that have to be identified from experiments.
8.3.6 DYNAMICS OF FRICTION MODELS FOR NONSTATIONARY SLIDING: FRICTIONAL VIBRATION AND STICK-SLIP The modeling of the functional relation between friction, the nominal normal pressure (contact force), and the relative motion relies essentially on experimental identification of the relation of the dynamic behavior of simple and well-defined mechanical systems such as a tribometer. The influence of friction curves on the dynamics of sliding solids can be studied using a simple tribometer that is represented by a one-degree-of-freedom system consisting of a block with mass m restrained by a spring with rate k and a viscous damper of rate c. As shown in Figure 8.3.8, a block rests on a rigid surface that is pulled by some electromechanical device at a constant velocity V. Although the driving velocity V of the tribometer is imposed, the device cannot prescribe the relative sliding velocity V~ of the slider. It is given by V~ = V - Yr
719
8.3 Models of Friction
FIGURE 8.3.8 A basic one-degree-of-freedom model of a tribometer to study frictioninduced vibrations.
The equation of motion for this system for the horizontal displacement x of the body from the undeformed state of the horizontal spring is m~ + cs + kx = #N
(13)
In the nondimensional form the equation reads + 2~2 + x = N#
(14)
Here the time scale is chosen as v/(m/K) and the length scale as the deflection of the horizontal spring caused by the maximal static friction force, while = (c/2)/v/(mK) is the relative damping. The stability of the sliding motion for the one-degree-of-freedom tribometer may be analyzed using Eq. 14 together with the friction model. In the case of Coulomb friction (Fig. 8.3.4a), the stationary solution is globally stable. Therefore, at constant normal load the model does not explain the occurrence of stick-slip. The discontinuous static-to-kinetic friction model of Figure 8.3.4b was treated Blok [20] in a brief study on the frictional oscillation. He stated that stick-slip instability occurs if the tangential force decreases with increasing speed of sliding and found that the stability of the stationary sliding is governed by two dimensionless parameters: vv~ D -- (#o - # k ) N
c and the relative damping ~ - 2x/-k-~
(15)
For a sufficiently stiff systems and at higher velocities a minimal amount of damping is sufficient to suppress the stick-slip motion and guarantee that the stationary solution is stable Considering that most materials have some intrinsic damping, the model of Figure 8.3.4b cannot explain the stick-slip phenomenon observed with nonmetallic materials at high speeds.
8 . 3 . 6 . 1 DYNAMIC BEHAVIOR ASSUMING A PURELY VELOCITY-DEPENDENT FRICTION This section considers the influence of the shape of the friction-speed curve on the stability of motion of a single degree of freedom sliding system shown
720
Savkoor
in Figure 8.3.8. Rewriting the equation of motion (Eq. 14) for this case: = N I a ( V - 5c). For every prescribed (positive) driving velocity of the surface, there is a unique stationary solution given by x = ~h(V). For convenience, taking the value N = 1 upon linearization of the equation of motion (Eq. 14) in this solution yields the equation for small perturbations, 5c + 2~5c + x = N # ( V s )
~+
~#k 2~+~-V-Ts(V) z ~ + k a c - 0
(16)
The prefix A denotes a perturbation of a variable with respect to its nominal value. The term in the square bracket will be denoted by ~c, which may be regarded as an effective damping. K-
2~ +~-V-Ts(v)
(17)
The stationary solution is stable if the effective damping ~c > 0, and it is unstable when ~c < 0. This latter case can arise provided the friction force decreases with increasing sliding velocity, such as with the shapes in Figure 8.3.4c and d. For the case when ~c = 0, a Hopf bifurcation occurs and the stability is determined by the third derivative of the function #(V~). The solution is unstable if the third derivative is negative and a subcritical, or hard, bifurcation occurs. Conversely, if it is positive, the solution is stable [see 21], and a supercritical, or soft, bifurcation occurs. Since the third derivative is negative for most of its decreasing part, this bifurcation is often of the hard type and a large-amplitude limit cycle, mostly of the stick-slip kind, is found. Apparently the stationary solution can be stabilized by adding a sufficiently large damping, whereas other measures, such as modifying the slider mass or the stiffness, give no solace.
8 . 3 . 6 . 2 IDENTIFICATION OF MODEL FOR FRICTION OF ELASTOMERS BASED ON BIFURCATION CHARACTERISTICS The dynamic stability and bifurcation behavior of the one-degree-of-freedom tribometer model will be considered now in the context of experiments on friction of an elastomer sliding on a nominally smooth surface of a relatively rigid solid (steel). The dynamics of sliding friction on the tribometer system shown in Figure 8.3.8 is considered, and the predicted dynamic behavior of the system of the two friction models, one without the state variable (case A), and the other including a state variable (case B), is compared. In both cases the influence of driving speed V of the tribometer on the dynamic behavior will be examined (V is treated as the bifurcation parameter). The
8.3 Models of Friction
721
speed d e p e n d e n c e of/~k on Vs in stationary sliding is represented by Eq. 10 with the following parameter values" /t o - 1, #m - 4, Vm -- 0.1, h -- 0.35, and ( - 0.01.
8.3.6.2.1 Case A: A Purely Velocity-Dependent Coefficient of Kinetic Friction, #k -- Pk (V s) Inserting Eq. 10 that defines the friction model for an elastomer and using the parameters given in Eq. 17 it turns out that the stationary solution is stable (uniform sliding motion) for V < 0 . 1 0 0 5 and for V > 3 7 . 8 4 . The stationary solution is unstable for driving speeds in the interval 0 . 1 0 0 5 < V < 1 8 . 2 0 3 . The results of the numerical analysis of dynamic behavior of the model assuming a purely s p e e d - d e p e n d e n t coefficient are s u m m a r i z e d in Figure 8.3.9 (the velocity extremes and the period of the limit cycles are shown). It is interesting to note that because the third derivative of the friction function is positive at the low-velocity boundary, the corresponding Hopf bifurcation is stable and a stable limit cycle is found. The limit cycle grows rapidly in amplitude with increasing driving velocities. At the u p p e r b o u n d a r y the third derivative is negative and an unstable limit cycle originates. At V = 37.84, the two limit cycles annihilate each other in a saddle-node bifurcation. The maximal velocity in the limit cycle motion (almost horizontal line) is approximately but not exactly equal to the drive velocity. The apparent
Normalised sliding velocity extremes z ~** ~.5i ..................... ? ~ - , 1.0
Normalised period ............................................................................... m 2 **,, Stable limit cycle
0.5 0 t
- 0.5
t
1
- 1.5
0.01
O.1
r
--0 ["
Drivin~ velocity V 1
10
1O0
FIGURE 8.3.9 Velocityextremes, normalized with respect to driving speed V, and the period of the limit cycles (divided by 2zc) for ~ = 0.01 [22]. Solid lines show stable branches, dashed lines the unstable branches (the horizontal dash line is the unstable stationary solution). Period is shown by dotted line.
722
Savkoor
"sticking" phase is not ideal because the actual sliding speed is either slightly smaller than, or exactly equal to, or a little larger than the velocity of the driven surface as found from the numerical results (not presented here). More importantly, the force variations are large, and such variations do not decrease sufficiently with increasing driving velocity.
8.3.6.2.2 Case B: Friction Model Including a Hidden State Variable 0 Such that #h = #~(0, Vs) The dynamics of the same tribometer system are considered now with the nonstationary friction law (including a state variable 0, according to Eqs. 11 and 12). In the present analysis the instantaneous response term #i will be taken to be zero because there is no known experimental evidence hitherto concerning the instantaneous response of the friction of elastomers. The additional parameters in the nonstationary friction law are # i ~ 0 , Ts - - 25, dk = 2.5. Linearization of Eqs. 11, 12 and 14 around the stationary solution for the driving speed V yields a s + (2r +
+ ax + a0 = 0
A0 + (1/zs + V/dh)(#gAYr + A0) = 0
(18)
Here #~ is the derivative of the friction function #k (Vs). The stability condition expressed by the Routh-Hurwitz criteria is: 2~[1 + (1/'rs + V/dh)(#~ + 2~)] + (1/'rs + V/dk)2(#~ + 2~) > 0 Therefore, the following condition stationary solution:
is sufficient
+ 20 > 0
(19)
for stability of the
(20)
It is the same as that which holds for the model in case A without a state variable. Hence the stationary motion is always stable in the rising part of the stationary #k(Vs) curve. However, an examination of the necessary condition for stability, given by Eq. 19, reveals that stable stationary solutions are possible for driving velocities even in the falling part of the stationary friction curve. The characteristic dynamic behavior represented by bifurcation and limit cycles is presented in Figure 8.3.10. The Hopf bifurcation occurs at V = 0.1973 and V = 17.966. There is a saddle-node bifurcation for the limit cycle for V = 24.037. It is interesting to note that the stationary sliding motion is stable in the velocity range (0.1005< V <0.1973) where friction decreases with increasing velocity and where (#~ + 2~)< 0.
8.3 Models of Friction
723
FIGURE 8.3.10 Velocityextremes and period of the limit cycle as a function of the sliding surface velocity for the friction model with a state variable [22]. Drawn lines denote extremes in stable branches, the horizontal dashed line is for the unstable stationary solution, the curved dashed line shows the extremes in the unstable branch, and the dotted line denotes the period.
8 . 3 . 6 . 3 COMPARISON OF FRICTION MODELS (CASE A AND CASE B) Comparison of the results shown in Figures 8.3.9 and 8.3.10 shows that stable stationary sliding for case B is possible over a significantly wider range of velocity than that in case A. The nonstationary behavior of friction has a stabilizing influence on the dynamics of the tribometer system. The variation of sliding velocity in case B is considerably smaller, especially with respect to the negative velocity. Figure 8.3.11 shows the variation of friction force during one period of the limit cycle at the relatively high driving speed V -- 15. The most striking difference between the predicted behavior with the two models becomes apparent from the magnitude of variation of the friction force for the larger driving velocities. In particular, as the driving speed increases, the variation of the friction force displayed by the model including the state variable (case B) becomes significantly smaller, which is in agreement with the experimentally observed trend shown in Figure 8.2.6a. This clear trend is not predicted by the model, assuming a purely velocity-dependent friction (case A). These results lead to the conclusion that the stationary friction-velocity relation by itself is insufficient for describing nonstationary sliding motion such as stick-slip and that the friction model must include the state dependence of friction such as
724
Savkoor
FIGURE 8.3.11 Comparisonof the sliding velocity and friction force predicted by the two friction models for the driving velocity V - 15. The results for the purely velocity-dependent friction (case A) are shown by dashed lines and those for the model including an internal state variable (case B) by drawn lines [22]. that in Eqs. 11 and 12. The present analysis illustrates the application of bifurcation and stick-slip behavior for identification of the friction model.
8.3.7 INFLUENCE ON FRICTION
OF NORMAL VIBRATION
It is worth noting that the frictional vibration and stick-slip that occur in sliding may be influenced by the occurrence of normal vibration. Most engineers are aware of the fact that in many practical situations friction between contacting components is generally reduced by the normal vibration of such components. As pointed out by Blok, the decay of frictional force may be due to the falling # as discussed before or to a decrease in the normal force with increasing velocity, but # remains constant as in the Coulomb model. Among others, Tolstoi [23] and Oden and Martins [14] have stressed the importance of normal vibrations in friction. Tolstoi suggested a physical mechanism based on the adhesion theory of friction that leads to a coupling of normal motion and tangential motion. However, while such a coupling mechanism is certainly active in the contact of viscoelastic solid materials, it may not necessarily be the principal source of frictional vibration in general, especially at a very low speed of sliding. Clearly a sliding surface is commonly subjected to some significant excitation in the normal direction because of the dynamics intrinsic to the mechanical system of machine components and because of the inevitable roughness of surface. Additionally, there is usually
725
8.3 Models of Friction
some significant coupling between the normal and tangential elastic fields; normal pressure affects the tangential displacement and vice versa. Oden and Martins characterize the interface response of a multiple-asperity contact for normal deformation (separation) by a nonlinear spring-damper system. The normal stress an is related to the normal deformation Wn as given by an - - - - ( C n W m" q-
bnwl"w)
(21)
and the tangential response is modeled according to a slightly modified Coulomb model connecting shear tractor ~ and the sliding velocity Vs, where the normal pressure term is replaced by a nonlinear spinglike term in w. r = -Gwm~sgn(Vs)
(22)
The constants in the equations, cn, b,,, mn, and ln, for the normal and G, me for the tangential deformation are determined experimentally. The authors pointed out that these representations do not always agree with the experimental data for static contacts. The equations are used in conjunction with a two-degree-of-freedom mechanical system (tribometer) to study the stick-slip motion induced by normal vibration of a sliding block on a plane. It is suggested that the high sensitivity of the frictional force to normal separation couples the normal and the tangential vibration and may actually cause the intrinsically Coulomb friction to manifest itself as a decreasing friction-velociy relation at low velocities. Furthermore, these authors studied elastodynamic problems of sliding friction for the Coulomb model and addressed the issue of computational difficulties arising from the Coulomb model while crossing zero sliding velocity.
8.3.8 CONTACT AND FRICTION OF DEFORMABLE SOLIDS: THE LOCAL COULOMB FRICTION MODEL When deformable solids are pressed together, a finite area of contact C is formed and the contact force is generally distributed nonuniformly inside the contact area. The local loading on a surface element in the contact can be defined in terms of the normal and shear components of traction. Generally, the contact mechanics analysis ignores normal adhesion between surfaces whereby the action of normal traction in the contact is assumed unilateral. The shear traction at any point in the contact arising from friction depends on the normal traction and the relative motion between the surfaces at this point. A simple and effective way of defining this relationship is based on the
726
Savkoor
pointwise application of the global Coulomb model. The relationship is the local Coulomb friction model that applies inside the contact of deformable solids. Whether or not relative motion can occur at any point inside the contact depends on the ratio of the tangential and the normal traction at that point. Therefore, the macroscopic frictional behavior of the solids, in particular the transition from static to kinetic friction of solids, essentially depends on the distribution of normal and shear traction in the contact. The contact mechanics of deformation and traction of solids is treated extensively by Johnson [24] and Kalker [25]. The treatment in the following sections briefly presents the typical features of contact and friction of deformable solids based on the local Coulomb model, which is widely used to study the friction of deformable solids. The model successfully explains the transition from static to sliding friction of solids through partial slipping of points in contact for both sliding and rolling contact situations. It is also useful for predicting the overall contact behavior for monotonic loads and other more complex patterns of loading.
8.3.8.1
THE LOCAL MODEL OF COULOMB FRICTION
The local model of Coulomb friction applies at any point inside the contact (C) (see Fig. 8.3.12). It determines whether or not the local shear traction (~x, l:y) due to friction is sufficiently large to initiate sliding, depending upon the local normal pressure p and the coefficient of friction #. At a point in the contact interface where z ___ #p, static friction (frequently referred to as adhesion) prevents relative motion of surface points in contact (local slip). If slip occurs at any contact point, then the local shear traction equals 9 - #p. The model rules imply that the distribution of normal pressure has a strong
FIGURE 8.3.12 Left: normal pressure and shear traction inside the contact (C). Right: components of traction and slip in a point inside the contact (C).
727
8.3 Models of Friction
influence on the relative motion of points in contact. The model also requires that at any contact point the traction and slip vectors are collinear and oppositely directed. Generally, in the case of rolling and sliding contact, the local sliding motion is expressed in terms of relative velocity (Vsx, Vsy). If (Vsx, Vsy) -- 0 ~
(T,2 + z2) 1/2
for
(Vsx, V~y) ~= 0 -0 rx - -lapV~x / v/V2~ + V2y /
and
(23)
-
/
§
In static contact problems, the sliding motion is defined instead, by the relative surface displacement of the solids at the point in question. The local sliding motion at a contact point depends upon the relative rigid body motion (shift) of the solids at points remote from the contact and the differential surface displacement due to deformation of the solids.
8 . 3 . 8 . 2 CONTACTOF SOLIDS SUBJECTEDTO PURELY NORMAL LOAD The size and shape of the contact area (C), the normal approach, and the normal pressure p due to normal load P can be determined from the knowledge of the geometry of the undeformed solids and their bulk material properties. The solid materials respond to loading by elastic, plastic, or viscoelastic deformation, depending upon the load and their properties. In the following, only elastic response of solids will be considered. The analysis of contact under purely normal load involves the solution of a mixed boundary value problem of continua using well-established methods of contact mechanics. Following Dundurs [26], the nature of contact between elastic solids depending upon the undeformed shape of the solids may be classified as advancing or receding, depending upon how the contact evolves with increasing load. The contact is of the "advancing type" if an increase in normal load leads an enlarged contact such that it includes new surface points of the solids in addition to those already existing inside the initial contact. The Hertzian contact is a typical example of this type of contact, which is both common and technically important. Conversely, if increasing load does not lead to a growing contact area with additional surface points, the contact is classified as the "receding type." The receding contact occurs, for example, when a rigid punch with a fiat-face and constant cross section indents a
728
Savkoor
semi-infinite elastic solid. The size and shape of the contact area are prescribed by the cross section of the punch, irrespective of the imposed normal load. When solids of dissimilar materials are subjected to a purely normal load, the presence of friction in the contact will produce radially symmetric shear traction, possibly causing slipping in some part of the contact.
8.3.8.3
HERTZ THEORY OF CONTACT FOR PURELY
NORMAL LOADING OF SOLIDS In many technical applications of interest, the material behavior is linear elastic and the size of the contact is small in relation to the dimensions of the solids (approximately a semi-infinite body). The analysis of the global contact deformation is usually performed by ignoring adhesion, that is, by assuming a unilateral contact. Additionally, the presence of surface asperity and the influence of friction in the contact are neglected. The latter condition implies that there is no or only a small interaction between tangential and normal fields (low ~). If the interaction is strong, the normal and tangential fields are couple and the analysis becomes more complex. The Hertz theory analyzes advancing contact of two deformable solids with smooth, contraformal shapes, ignoring friction. As the solids approach closer, initial contact in their undeformed state occurs in a point (point contact) or line (line contact). Under the action of a normal load, the solids deform, resulting in conformity of parts of their surfaces over a small (with respect to the dimensions of the solids) but finite area of contact. The contact deformation can be calculated by treating the solids as semi-infinite bodies. The contact area is generally an ellipse defined by its semima]or axis, semiminor axis, and an eccentricity. The Hertz theory of contact can be used to calculate the normal approach (compliance) of bodies, the size and shape of the contact, the contact pressure, and the internal stresses within the bodies. For illustrative purposes, the main results of the Hertz theory under the aforementioned assumptions are presented here for the case of contact between two spherical solids. The contact between two elastic solids of revolution with radii R1 and R2 and with material constants El, Y1 and E2, v2 under a normal load P results in a circular area of radius a and a central deflection ~. The central deflection ~ is the normal approach between material points remote from the contact area.
a ( pR, ,J3 and __~_f
~
( 16~g~,2/ 9p2 1j3
(24)
729
8.3 Models of Friction
The geometric and elastic properties of the two solids are combined into two constants, an equivalent radius R* and an equivalent elastic modulus E*. 1 R*
=
1 R1
}
1
and
R2
1 E*
=
l-v~ E1
}
l-v~ E2
(25)
The distribution of normal pressure in the contact area has an elliptical form
p(r)
_
3P
(
2~a2
1-
a2j
(26)
The normal pressure distribution in this Hertzian contact is depicted in Figure 8.3.13 (left). Domain of validity: The theory does not consider the following aspects arising from contact between real-world solids: (i) contact friction, (ii) surface roughness, and (iii) adhesion between solids. Finite friction in the contact produces shear traction in the contact between solids made from materials with generally dissimilar elastic constants. Excepting few special material combinations (identical materials or materials with Poisson ratio of 0.5, which holds approximately for elastomers), the dissimilarity leads to interaction between elastic fields produced by normal and tangential traction. For small values of # this interaction introduces minor modifications in the Hertz formula but at the expense of a considerable increase in complexity of the analysis [24, 25]. The surface roughness results in disperse true contact with high local pressure, but its effect on the global (smoothed-out) contact behavior becomes significant
FIGURE 8.3.13 Left: pure normal load (Hertz). Centre: tangential force applied to a Hertzian contact without slip. Right: partial slip caused by a tangential force applied to a Hertz contact with finite friction.
730
Savkoor
only for very small normal loads. Similarly, the effect of surface adhesion on the normal contact, which is discussed in the section on fracture mechanics models, is significant under very light loads only.
8.3.9
THE TRANSMISSION
FORCES
OF TANGENTIAL
BY F R I C T I O N
The effects of friction in the contact of deformable solids are significant when an additional tangential force is applied to the normally loaded bodies at points remote from the contact area. Of special interest is the role of friction in the transition from a static to a sliding contact. When solids are subjected to both normal and tangential forces, the distribution of normal pressure and shear traction in the contact area of solids is nonuniform and the two profiles are generally dissimilar. In the case of deformable bodies, the local normal pressure and shear traction are conceived as the normal and shear loads at each point in the area of contact. The aforementioned interpretation of traction as a local loading prompts the formulation of the local model of Coulomb friction for the contact between deformable solids. In a manner analogous to the global model, the ratio of the shear traction z and the normal traction p at any point inside the contact area determines whether or not sliding can occur in that point.
8.3.9.1
TRANSITIONFROM STATIC TO KINETIC FRICTION BY
PARTIAL SLIP (0 ~ T ~ # P ) The analysis requiring a local interpretation of the classical Coulomb model has been presented independently by Cattaneo [27] and Mindlin [28]. The tangential mixed boundary problem is formulated by assuming the half-space model of solids. In the case of a circular Hertzian contact, the simpler tangential problem considered first is that where slipping is prevented (infinite friction) across the entire contact area. The tangential boundary conditions applied to the surface of each half-space take the following form: 1. A uniform tangential displacement (shift) u = ~ is prescribed inside the entire contact circle r < a. 2. There is no traction outside the contact circle r > a, hence 9 = 0. n
The result of the linear elastic analysis for the shear traction ~ and the shift c~ using the above boundary conditions is given by Eq. 27 below and is
731
8.3 Models of Friction
presented in Figure 8.3.13 (centre).
T2
& - 8---a where 2 = [(2 - Vl)/G1 + (2 - v2)/G2] and
r(r) - ~ ( aT
2 . r 2) -1/2," r < a
(27)
It is interesting to note that the shear traction z is axially symmetric in magnitude only; it is directed everywhere parallel to the applied force T. Clearly, for any finite value of T, the traction is unbounded at the boundary: r--. a, z(r)--+ oo, whereas p(r)--. O. Based on the local version of the Coulomb law, one may conclude, as Cattaneo and Mindlin did, that slip must commence spontaneously at point close to the boundary of the contact area as soon as any tangential force is applied. Because the magnitude of z is axially symmetric, one would expect slipping to start at the boundary and with increasing T spread radially inwards toward the center. For tangential forces 0 _< T _ a, hence r ---0. The results found by Cattaneo and Mindlin for the circular locked region of radius c, and the shear traction both in the locked (r < c) and the annular slip region (C < r < a), read 1/3
a-
7P (28)
and -
-
-
-
3/~P /2 2rca2(1 - r2/a2) 1 - ( c / a ) ( 1 - r2/c2) 1/2 in 0 _< r _< c; 3#N 2tea2(1 - r2/a 2) 1/2 in c K r K a
732
Savkoor
The expressions for tangential shift c~ and the tangential compliance are c~ --
32#N{1-(1-T) 8a ~
2/3} and
d~ 2( Ty dT=--4a 1 - ~ - ~ /
1/3 (29)
These results may be viewed as close approximations because traction and slip vectors are not precisely collinear in the region of slip. The results according to Eqs. 28 and 29 for the annular slip region, the variation of shear traction and the tangential displacement (shift) for the Hertzian circular contact subjected to a tangential load (0< T
8.3.9.2
IRREVERSIBLE EFFECTS, THE DEPENDENCE ON
LOADING HISTORY AND CONTACT HYSTERESIS Kinetic friction that arises in the region of slip is essentially an irreversible process. It is not surprising, therefore, that shear traction in the contact during tangential unloading differs from that during loading. We start with the situation described in the previous section where solids under a constant normal load are subjected to a tangential load T = T1, such that T1
]
(30)
The resulting traction, shown in Figure 8.3.14, is obtained similarly by superposition. It is of interest to note that the compliance at the instant of unloading at T = T1 is the same as the initial compliance during the first loading, i.e., at T - 0. If the unloading proceeds until the contact is load-free, i.e., T2 = 0, there remains a displacement shift or permanent set. In order to remove this permanent set, unloading must continue in the reverse direction until ~ - 0. The situation of cyclic loading and unloading (load reversal) by tangential forces applied to solids in contact under a constant normal load occurs in
733
8.3 Models of Friction
FIGURE 8.3.14 to T2 ( < T1).
Shear traction zl (for the initial tangential load T1) and z2 following unloading
FIGURE 8.3.15 normal load.
Hysteresis loop of friction-displacement in cyclic tangential loading at constant
many mechanical joints. The force-displacement curves generated during a cycle of loading and unloading form a closed loop as sketched in Figure 8.3.15. This loop is the well-known "hysteresis loop" that encloses an area which represents the energy dissipation during the cyclic process. The energy dissipation results in heat generation within an annular slip region inside the contact. The rise in the contact temperature depends on the dissipated power and the thermal properties (heat conductivity and diffusivity) of the materials. In the long run such cyclic loading causes fretting wear of solid surfaces and under more severe loading conditions it may lead to catastrophic wear or seizure as a result of thermoelastic instability. Initially a tangential force T1 is applied to solids pressed under a normal load P. The tangential force is then cyclically varied between T1 and - T 1 . The energy dissipated per cycle is given by [301
lOa
~-P
- -
(31)
734
Savkoor
The foregoing analysis is based on the assumption of a constant coefficient of friction. In reality, slipping in the annular region generates heat and causes surface damage or conditioning that may continuously modify the value of # at points inside the region of slip.
8.3.10 FRICTION OF SOLIDS
IN ROLLING CONTACT
An important class of industrial applications successfully makes use of components or elements involving rolling contact of solids such as the wheels and tires of transportation vehicles. The rolling wheel or tire is capable of generating resultant frictional forces of desired magnitude and direction simply by controlling its relative motion m the "slip" m and these forces can be used in turn to control the motion of vehicles. The force generated can cover a wide range, from a very small value, barely sufficient to overcome resistance encountered by a cruising vehicle, to sufficiently large values required for acceleration, braking, or turning of a vehicle.
8.3.10.1
ROLLING FRICTION OF A FREE-ROLLING WHEEL
If solids are assumed to be rigid, one obtains a "point" or a "line" contact that results in the ideal situation of "pure rolling" of solids. However, pure rolling does not occur in the rolling of deformable solids where the size of the contact is finite. "Free rolling" along a straight path of wheels (deformable solids) is the closest approximation to the ideal situation of pure rolling. When rolling is with a constant speed along a straight path, the resistance to rolling is the "roiling friction." Energy lost in the cyclic inelastic deformation of materials ("hysteresis") is the dominant mechanism of roiling friction of free-rolling solids. Free rolling is accompanied by a minute amount of sliding in some small pockets inside the contact, but the resultant tangential force contributed by the almost (antisymmetric) self-equilibrating shear traction within the contact to free rolling is relatively insignificant. The viscoelastic normal deformation of tire carcass, tread, and sidewalls results in a distinctly asymmetric normal pressure distribution within the contact (Fig. 8.3.16). The resultant normal reaction shifts forward in the direction of rolling motion, which is equivalent to a moment resisting rolling. The free-rolling motion is sustained by a minimal tangential force (driven wheel shown in Fig. 8.3.16) or, in the case of a driving wheel, by a minimal torque applied to the driving axle.
735
8.3 Models of Friction
FIGURE 8.3.16 a fiat road.
Rolling resistance of a wheel (tire) in stationary, free straight rolling on
The principal mechanism of rolling resistance is the pronounced asymmetry in the distribution of normal pressure during roiling caused by the inelastic (viscoelastic) nature of deformation of solids. As illustrated in Figure 8.3.16 the asymmetry is mainly in the pressure p, and it is insignificant in the shear traction z. The rolling resistance is represented by the moment Mr. For a free-rolling wheel under a normal load, the linear velocity V and the angular velocity co are related through the effective roiling radius re of the wheel, where ro > r~ > rh
(32)
Here r0 is the radius of the undeformed wheel and rh the height of the deformed wheel above the ground. The rolling resistance of automotive and aircraft tires is of the order of 1% of the normal load. The structural materials, mainly rubber, undergo large, cyclic, and viscoelastic deformation during roiling. Typically, with steel wheels of railway vehicles roiling on steel rails, the much smaller contact deformation re ~ ro. The small deformation coupled with relatively low hysteresis of steel results in a very low rolling resistance of only 0.1%.
8.3.10.2 FRICTION MODEL FOR TRACTION IN ROLLING CONTACT The relative rigid body motion between tWO solids making contact may involve both translation and rotation, each with three components resolved
736
Savkoor
along the normal and tangent to the plane of contact. In most cases of interest, the normal component of the relative translation is taken to be zero (constant normal load and deformation). In addition, one of the two components of rotation in the tangent plane that gives rise to "rolling" is taken as zero (rolling in one direction). This leaves the following components of relative motion: two components of relative translation (longitudinal and lateral) and the two rotational components. One of the rotation components is the rolling in the plane of the wheel, and the other rotation normal to the contact plane is termed the "spin." The components of relative motion of interest are shown in Figure 8.3.16, where rolling takes place about the y-axis. The relative rigid body motion is generally expressed as deviation from the state of free rolling. A convenient measure of this deviation normalized with respect to the rolling velocity Vr is the "wheel slip" or "overall rigid body slip," usually referred to simply as "slip" ("creep ratio" in railway vehicle terminology). In the case of railway wheels or automobile tires, the countersolid (rail or road) in contact is at rest. In applications for traction, the emphasis is usually on describing the functional relation between components of resultant contact force (and moments) and the corresponding slip or creep ratio. The relation is established at a constant speed of rolling (the rolling velocity is taken as the linear velocity of the wheel). The transmission of tangential forces (or a moment) through the rolling contact of deformable solids is inevitably accompanied by partial slipping. The contact area is divided into regions of adhesion (no slip or stick) and the remaining regions of slip. The division is based on the local Coulomb friction model of Eq. 23, with #k = #0- The no-slip region generally begins at the leading edge (boundary) of contact and ends where shear traction builds up to a level that cannot be sustained by the available static friction. The shear traction 0:x, l:y) satisfies Eq. 22, connecting it with the local sliding velocity (Vsx, Vsy) according to the local Coulomb friction model. With increasing tangential force, the region of slip expands at the expense of the region of adhesion, and this growth continues with increasing force until the entire contact begins to slide. The determination of the extent and location of the slip region and the shear traction in the contact is again a mixed boundary value problem in the mechanics of rolling contact. The complexity of the solution directly depends upon that of the friction model used (# as a function of the slip). Most rolling contact problems in the open literature are solved by assuming the local Coulomb friction model. The kinematic input variables for the rolling and slipping motion of the wheel and the contact forces and moments of interest are presented in Figure 8.3.17.
8.3 Models of Friction
FIGURE 8.3.17 with slip.
737
The wheel-slip input and output of contact forces and moments for rolling
The longitudinal slip ex, the lateral slip ey, and the rotational slip or "spin" q5 are defined by
e,lx)" -
-- (Ore.
Vr
'
e( y)
_ __VY
Vr
and
4~-
(Dza
Vr
(33)
with a = semi-contact length and Vr ~ Vx. The longitudinal and lateral components of local sliding velocity or microslip at a point (x, y) inside the contact area are then defined by
Ou)
Vsx(X, y) - Vr ~, -- 4)y + ~ ( W sy ( X , y ) - - Mr
O~) 8 y Jr- ~) x -[- -~x
Ou
-- -Ot Ov
(34)
c~t
Here u and v are, respectively, the x and y components of the surface displacement of a point (x, y) inside the contact, and the second term in each equation (time derivatives) is the nonstationary part of the motion. The surface displacements (u, v) are brought about mainly by shear traction (rx, ry) and to small extent by the normal pressure p. Conversely, the shear traction influences normal displacements and thereby influences both the normal pressure p and the size and shape of the contact area. This set of complex elastic interactions increases the complexity of solution techniques used to solve contact problems. In many cases where the interaction of normal and tangential elastic fields may be ignored, the local tangential surface
738
Savkoor
displacement (u, v) can be expressed in terms of shear traction (Zx, Zy) with the aid of the surface influence functions
u(x, y) = f /
[~x(~,tl)G,~,(x, y; ~,~?) + ~y(~,tl)Gxy(X, y;
n)Jd an
c
v(x, y) =
Fl)Gyx(X, Y; ~, rl) + "Cy(~,Fl)Gyy(X, y; ~, rl)Jd~drl
(35)
c
For the steel-on-steel contact (railway wheel on rail), the classical solutions for a semi-infinite elastic solid subjected to point load on its surface is taken as the influence functions for the contact problem [25]. No such functions are available in the case of pneumatic tires; therefore, and in view of the complexity of its construction, the material properties, and the geometrical nonlinearity, approximate structural models that define the influence function are developed. Even with the relative simplicity of the Coulomb model, the effort towards a solution of the aforementioned equations is not trivial. It involves a numerically intensive process to determine the regions of adhesion and slip according to the local Coulomb model. Efficient numerical techniques developed recently are based on variation principles; these methods enable direct determination of the adhesion and slip regions. The solution for tractions can be integrated to yield the resultant forces and moments. The three more important ones for the longitudinal and lateral forces and the aligning torque (or spin moment) are
Fx = //~cxdxdy;
Fy = / / ' c y dxdy; Mz - / /
c
c
(X'Cy- y'cx)dxdy
(36a)
c
The other two components are the overturning moment and the rolling moment:
Mx -- f f y'Cz dxdy c
and
My=/fX'Cx dxdy
(36b)
c
The technically important results for the overall behavior of rolling and slipping wheels for constant normal load and constant rolling velocity may be expressed formally in terms of a function (q):
{Fx, Fy, Mx} = ~(~x, ~y, c~;V)
(37)
The typical trends of the longitudinal force-slip relation predicted by the Coulomb friction model and those from measurements under practical conditions are compared in Figure 8.3.18 (right). The differences become pronounced in the region of large slip where the frictional behavior deviates
8.3 Models of Friction
739
FIGURE 8.3.18 Left: transmission of tangential forces in stationary rolling (V, co, #, and s are constants) with increasing longitudinal slip. Right: Comparison between typical traction curve from measurement and that according to Coulomb friction. The regions of adhesion and slip, the latter, starting from the trailing edge of the contact, spreads with increasing Fx toward the leading edge.
significantly from the simple Coulomb model. The slope of the traction-slip curve at the origin where the slip region tends to zero is known as the slip stiffness or creepage coefficient. For steel-on-steel contact of the Hertzian type (railway wheel on rail), the slope can be determined exactly for any slip mode. The slope is known after Kalker's work [31] as Kalker creep coefficients. For tire-on-road contact, such results can be obtained either through an elaborate finite element analysis or empirically from experiments on full-scale tires. Application and validity: The essential simplicity of the local version of the Coulomb model has facilitated analysis of technically important contact problems of complex mechanical systems. The local Coulomb friction model is extensively used in the analysis of contact forces, slip, energy dissipation (damping), and wear of solids in contact in static, oscillatory (contact damping in joints), and roiling (wheels) situations. The local Coulomb model proves highly effective in applications where mainly the overall (or average) behavior of solids in contact (resultant forces and moments versus slip) is of interest and experimental data of the actual behavior are available. The predicted behavior can be made to fit the actual experimental data by selecting an appropriate value for the coefficient of friction #0 (#h = #0). However, the detailed behavior, such as the size and location of the slip region and the local sliding velocity, which affect wear and the temperature distribution in the contact, is not reflected correctly
740
Savkoor
because the friction model does not take into account the actual pointwise variation of p.
8.3.11 MATHEMATICAL MODELS OF FRICTION 8.3.11.1
MODEL BASED ON PLASTICITY THEORY
The similarity between a number of trends of frictional behavior and those related to elastoplastic deformation of materials or to the hysteretic behavior of materials has inspired some researchers to develop models of friction. A Coulomb friction model based on the framework of the theory of plasticity has been developed by Curnier [32]. In an analogy with the elastic and plastic deformation of solids, the distance between points on surfaces is decomposed into a no-slip (adherence) component and a slip component both in normal and tangential directions. It enables a unified treatment of the nonpenetration condition (unilateral) for normal deformation and the tangential slip between surfaces. The slip criterion of the Coulomb model is similar to the von Mises criterion for plastic yielding of bulk metals. The slip rules are similarly based on the flow rules of the plasticity theory and defined in terms of a convex potential. However, as pointed out by Klarbring [33] and Oden and Martins [14], the local friction law does not follow the normality rule of classical plasticity. The analogy is useful if the normal pressure in the contact is known or can be prescribed a priori. The contact problem of deformable solids is posed in the framework of variational inequalities. The reader is referred to the original work containing the author's model of "standard generalized friction." De Souza Neto et al. [34] developed models that assume friction to be a function of the density of frictional work along lines similar to the theory of workhardening elastoplasticity.
8.3.11.2
DAHI. MODEL BASED ON
HYSTERESIS BEHAVIOR The modeling of damping of joints and bearings in mechanical systems motivated Dahl [35] to extend the global Coulomb friction model by including the hysteretic behavior of frictional contact. As discussed earlier, the transition from static to sliding friction of deformable solids takes place by partial slipping that gives rise to contact hysteresis during a tangential loading cycle. This displacement-dependent effect may be incorporated into a global model of Coulomb by suitable modification of the discontinuity at
741
8.3 Models of Friction
V s - 0 The Dahl model is described in terms of the slope of the frictiondisplacement function dl-
~ 1--#0
sgn 1 - ~---sgn(2) #0
(38)
The displacement derivative does not depend explicitly on t or on x, and the value of the coefficient of friction # is bounded within the limits (-#0, #0). The Dahl model may be considered as a hysteresis operator, and the index i is thought to describe ductile behavior for i _> 1 and brittle behavior for i < 1. The hysteresis cycles with the nonsmooth global Coulomb model, the stiction model, and their regularized smooth behavior are shown in Figure 8.3.19. The friction model may be expressed in terms of sliding velocity by putting d# d# dt = dx 2 and writing fi--~
(
1-
Putting i -- 1 and writing ~ -
sgn 1-Y-~-sgn(~) #o
)
Vs, the equation may be cast into
fi - ~ ( Vs - # lVS In this form it is similar to the nonstationary relation described previously by Eqs 11 and 12 in the section describing the evolution of friction depending on a state variable
............
~........ ~176 ...... ~. . . . . . . . . . ~. . . . .
........
I
a) Global Coulomb
I
9
b) Dahl model
FT
9
i . . . . . . ~176 ...... ~176176176176 ..... i| .... ~176176176176176176 ~176176
...........................
i ..........................
8
I c) Stiction
d) Smoothed stiction
FIGURE 8.3.19 Hysteresis behavior of Coulomb and stiction models (a and c) and their smoothed versions (b and d).
742
8.3.11.3
Savkoor
EXTENSIONOF DAHL MODEL
The Dahl model has been developed as a hysteresis operator and has been generalized by Bliman and Sorine [36], especially for dealing with control of mechanical systems. The Coulomb model without and with stiction and the corresponding smoothed versions are depicted in Figure 8.3.19. Canudas de Wit [37] has also studied the Dahl model in the context of large-scale mechanical systems. Bliman and Sorine proposed an extension of the Dahl model to a second-order differential equation to consider the (Stribeck form) velocity dependence of friction.
8.3.11.4 MATHEMATICAL FRAMEWORK FOR FORMULATION OF CONTACT AND FRICTION PROBLEMS Klarbring [33] has presented a general mathematical framework for formulating unilateral problems of contact and friction and for solving such problems numerically based on the methods of mathematical programming. The contact problems of solids, including a thin layer separating the bodies, are considered by Ganghoffer et al. [38]. A thin layer of surface film called "interphase" is modeled in a thermodynamic formulation using viscoplastic and nonlinear incompressible elastic constitutive laws for the material. The model permits the physically significant description of damage and wear caused to the interphase by adhesion friction. The discussion of these and other mathematically oriented models is beyond the scope of this chapter.
8.3.12 PHYSICAL MODELING OF FRICTION 8 . 3 . 1 2 . 1 CHARACTERISTICS OF SURFACES AND THE ADHESION MODEL OF FRICTION The development of physical models of friction is based on two important observations concerning the nature of real-world surfaces and especially those employed in engineering applications (see Fig. 8.3.19). First, practically all solid surfaces are geometrically rough and wavy with the heights of peaks and valleys small in relation to the dimensions of the macroscopic solid. The second equally important observation is that surfaces are covered with films that inevitably form under the influence of the ambient gaseous environment. The terms smooth and rough, used sometimes for qualifying, respectively, low
8.3 Models of Friction
743
and high friction of surfaces, can be misleading because under dry conditions friction generally decreases with increasing surface roughness. Generally, the geometrically smooth and chemically clean surfaces are those that exhibit very high friction under dry operating conditions. In lubricated contacts, where smooth surfaces tend to separate normally due to the intrusion of thin hydrodynamic films, the surface roughness can prevent the formation of such films and thereby enhances friction by promoting direct contact between surfaces. Surface roughness plays an important role in dry friction, but its role is much more subtle and indirect than merely that of obstructing relative motion.
8.3.12.1.1 Contact Geometry and the Nominal and True Contact Under a given normal load, the overall region of contact of solids is determined by considering the macroscopic geometry of solids (ignoring surface roughness) and the material properties of the solids. The region is termed the nominal area of contact and the pressure is known as the nominal pressure. Because the surfaces are rough, the regions of "true" or "real" contact between solids are dispersed within the nominal area. Under light and moderate load, the true area of contact between solids occurs near the asperity summits, and, as a rule, it is only some small fraction of the nominal contact area (Fig. 8.3.20). Consequently, the normal pressure in the true contact is considerably higher than the nominal pressure based on macroscopic geometry of solids with perfectly smooth surfaces. In the contact between solids like metals and alloys, such high pressure may cause plastic deformation of surface asperity. The first contact between initially undeformed and oxide-free metallic solids often leads to plastic deformation of asperities. After some repeated loading and unloading cycles (running in), plastic shakedown occurs, and therefore even metallic surfaces subsequently resist loads by elastic deformation of asperities. The surface roughness can be represented conveniently by
FIGURE 8.3.20 Contact between solid surfaces covered with asperity and films of oxides and contaminants.
744
Savkoor
considering asperity heights (and tip radii) to be distributed randomly. Greenwood and Williamson [39] presented a simple and useful model of surface roughness for engineering surfaces (Fig. 8.3.21). The asperities on engineering surfaces display a random Gaussian height distribution with standard deviation a. For simplicity, the shape of the asperities in the neighborhood of the summits is assumed to be spherical with a uniform tip radius ]~ that represent the sharpness of the surface texture (see Fig. 8.3.21). The tip radius and the standard deviation are generally correlated with the average slope of an asperity. From the viewpoint of material reversibility or damage, the onset of plastic deformation is an important consideration. A simple criterion for delineating the elastic and plastic nature of contact deformation is given by the parameter known as the plasticity index ~. An estimate of ff may be given based on the Greenwood-Williamson model for describing the surface topography of surfaces employed in engineering practice. Denoting the plastic yield pressure of the softer metal in terms of its hardness H and the equivalent elastic modulus for the two solids by E*, the plasticity index ff is given by O- F
~-z
9
(39)
The superscript star indicates the combined quantity for the two solids and their surfaces. The two constants E*,/~* are the equivalent modulus and radius of asperity tips, and a* denotes the equivalent standard deviation of the peak height of asperities. Since a* and/~* are related to the average slope of an asperity, it implies that the average asperity slope governs the transition from elastic to plastic contact. 8.3.12.1.2 Elastic Contact In many engineering applications, surfaces of newly manufactured products are "run in" during some initial period by repeated loading and sliding to polish or wear out their initially high and relatively sharp asperities. The plastic deformation of asperities of metals is practically inevitable during the
Ti ra ioso po it
Son oviatioo
. . . . . . .
asperityheights FIGURE 8.3.21 Asperitymodel of surface topography (Greenwood-Williamson).
745
8.3 Models of Friction
running-in process, whereas with some other materials running in may result in brittle fracture of some of the tall asperities. Generally, the repetitive contact involving loading and unloading cycles results in modification of surface profiles and, in the case of metals, in workhardening of surfaces. Both these effects imply that, after running in, the surface asperities of most solid materials (including metals) respond to any subsequent loading mainly by elastic deformation (and viscoelastic deformation in the case of polymers). The true contact at or near the rounded tips of surface asperities may be visualised as a Hertzian microcontact (see Fig. 8.3.20). Assuming linear elastic deformation and the Hertzian formula to apply to an asperity contact, the size of any microcontact is proportional to the two-thirds power of the normal load carried by an asperity tip. The latter quantity is represented by the "smoothed-out" nominal pressure p that would result if the surfaces of solids were assumed to be ideally smooth. Because of the statistical dispersion of asperity heights, an increase in normal load (nominal pressure) brings an increasing number (roughly in proportion to the load) of asperity tips into contact. Although a single asperity contact grows in a Hertzian manner, the total area of true contact, At, is Ar oc pn where 2/3 < n G 1
(40)
Under light load (the tail of the Gaussian distribution approximately tending to an exponential distribution), the mean value of the area of true contact of an asperity remains approximately constant. The result is that the true area is nearly proportional to the normal load (n ~ 1). The same relation is obtained if the loading of asperities should cause plastic instead of elastic deformation of asperities, assuming that the plastic yielding is characterized by a constant parameter value such as the hardness of the softer metal. The model explains why the true area of contact is nearly proportional to the normal load, irrespective of the mode of deformation of asperities. Since both normal and tangential forces are transmitted across the true contact, the true (real) area plays an important role in modeling friction. The distinction between the true and the nominal areas of contact is essential for understanding the physical basis of how the properties of solids affect contact and friction.
8.3.13
ELEMENTARY PHYSICAL THEORY
OF FRICTION The modern approach to modeling the physical mechanism that explains friction as a resistance to motion arising from adhesion and irreversible
746
Sa~koo~
deformation of near surface material is credited to Bowden and Tabor [2, 3]. They argued that the proximity of atoms and molecules of solids that is responsible for cohesion of any single solid should act similarly to bring about adhesion between atoms and molecules of different solids. The interaction is mainly concentrated in the region where the contact is intimate between two distinct solids. However, they considered the interaction of surface asperity on the microscopic scale of true contact as the scale relevant for modeling both adhesion and the process of friction between real-world solids. Their so-called adhesion theory of friction has been generally accepted as a sound qualitative explanation of the major mechanism of friction. The Bowden and Tabor theory envisages dry friction to result from two mechanisms. The primary one is brought about by adhesion between surfaces in the true area of contact of solids, and the secondary source is the inelastic deformation while the asperities plough across the material in regions surrounding the true contact. However, the contribution of "ploughing" to friction strongly depends on that due to adhesion. Friction by pure ploughing can occur when adhesion is absent. Ploughing without adhesion involves deformation of a relatively small volume of material. Even if the deformation should occur in a largely irreversible manner, it would account for p of utmost 0.15 corresponding to the average slope of asperity of engineering surfaces. For most solid materials, the contribution due to ploughing would be still smaller. Adhesion significantly enhances the contribution of the ploughing component of friction. The importance of the true contact is linked to the adhesion mechanism because forces between atoms and molecules of the solid surfaces have a relatively short range of action. This interaction between surfaces over the rest of the nominal contact region is insignificant because of their larger separation. Adhesion between surfaces of solids in the true contact area acts as a joint or weld that restrains relative motion between the surfaces of solids. The restraining force is the friction, and hence the limiting force of friction depends ultimately on the strength of the adhesive joint or "junction." In essence, the theory considers the force of friction to depend on the true area of contact as well as on the strength of the individual junctions. The presence of surface films generally tends to reduce adhesion, leading to the formation of weak junctions. The low shear resistance of junctions formed between surface films explains why friction coefficients observed in ordinary atmospheric conditions rarely exceed a value of unity. Assuming that the joint separates or fractures along or in the vicinity of the area of true contact, the force of friction, Ft may be expressed in the form
Ft -- Arzf
(41)
747
8.3 Models of Friction
The area of true contact Ar and the ultimate fracture strength in shear zf are the two essential variables that determine friction. Irrespective of the mode of deformation of asperities, if n ~ 1: FT Ar c< P yields 12- P cx: "cf
(42)
Assuming that l:f remains constant (a property of either one of the materials or the interface), this rudimentary analysis explains the physical reason behind the Coulomb friction model, which postulates # being independent of P. More generally, for values of n between 2/3 and 1: #-~
FT
oc T,s
pn-1
(43)
Assuming again a constant ZS, this gives # oc pk where p is the nominal contact pressure and k = n - 1 varies between - 1 / 3 and zero, as found generally from friction experiments. If the joint fractures along a path that cuts through the subsurface layer within the bulk material, resulting in wear, the value of zf will be close to the shear strength of a thin subsurface layer of the solid material. The basic structure of this model has been refined and extended, amongst others, by Bowden and Tabor. First, the extended model takes into account the criterion for plastic deformation under the combined influence of the normal pressure and the shear stress. Second, the model considers the effect of contaminating surface films in reducing the shear strength of the adhesive joint. The contact between freshly prepared metallic specimens generally causes plastic deformation of asperities because of the intense pressure in the area of true contact. The plastic deformation determines in turn the size of the true contact Ar :Ar = P/H, where H is the hardness of the softer of the two metallic solids. Hence, the area of true contact Ar increases almost in proportion with the normal load P. Assuming Ar = P/H yields 12 = FT/P = z.f/H, and this gives for many metals an estimate of 12 ~ 0.2 independent of the normal load as in the Coulomb model, but this estimated value is much lower than that observed in experiments. Bowden and Tabor later extended this rudimentary calculation by including the effect of shear traction on the plastic deformation of asperity.
8.3.13.1
PLASTIC DEFORMATION OF ASPERITIES
UNDER COMBINED NORMAL AND TANGENTIAL LOADING Under the combined action of pressure and shear traction, plastic flow is predominantly in the direction of the normal. It leads to an increase in the
748
Savkoor
area of true contact. This process, known as "junction growth," may be studied on the basis of a detailed analysis with a suitable yield criterion (von Mises or Tresca) and flow rules of the plasticity theory for specific geometrical shapes and material properties of asperities. Friction experiments on relatively soft metals suggest an elementary empirical relation for describing the combined action of average pressure p and shear stress 1: on yielding. p2 if_ or
_ p2
(44)
with Ar = P/p;
T = Arz, s = e ( ' c s / p ) =
P'c s
(192 -- ~'s2) 1/2
-+ ~ ~ (p2 _ ~gs2) 1/2
(45)
Large growth of junction occurs with clean and soft metals in vacuumproducing values of # > 10. In certain experiments of friction of metals such as indium and platinum, the results suggest a value of a = 12, but the value -- 3 is found appropriate with gold, silver, copper, and platinum. However, the value found now is too large compared to the coefficient of friction of the order of unity found in air. In the ordinary environment, shearing of relatively weak surface films of oxides or contaminants may be responsible for early termination of the junction growth. Consequently, the values of # for metals rarely exceed unity in the ordinary atmospheric environment. The new extension of the Bowden and Tabor theory offers a reasonable explanation based on the relative weakness of the adhesive joint in comparison with the inherent shear strength of both the solid materials. The shear strength r,f, in this case is the ultimate strength of a thin layer of interfacial material in adhesive failure. Amongst others, Briscoe and Tabor carried out experiments on thin films of polymers (from a few nanometers to a micrometer thick) to determine the influence of normal pressure p on the shear strength r,f. These investigations suggest that the shear strength zf increases linearly with increasing normal pressure, resembling the bulk behavior of polymers. It is given by
'~S = "~SO+ ~P
(46)
where ~:SOand ~ are empirically determined constants, ~SObeing considerably smaller than the corresponding value for the bulk polymer and the values of being in the range of 0.08 to 0.57. The lowest values apply to PTFE films, and the highest values are obtained with PVC (polyvinylchloride) films. Considering the pressure dependence of shear strength Ft = A , ~ f = Ar('Cfo +" ap) and assuming P ,.~ A r p , tx - F t l P ~
"cS--9-~+ o~ P
(47)
8.3 Models of Friction
749
For very large pressure ~---+ ~ and the pressure dependence vanishes resulting, in Coulomb-like behavior. Validity of the model: The basis of the theory is the adhesion between surface atoms and molecules of the contacting solids that bonds the solids and resists external forces applied to separate the bodies. It explains friction as the tangential force resisting the rupture of the adhesive bonds between surfaces. The theory has had a testing time because under ordinary conditions there is no obvious evidence of any tangible resistance offered by the postulated adhesive bonding to normal separation of solids in contact. The paradox has been resolved more recently by analyzing the fracture mechanics of an adhesive joint by normal separation. The other major problem arises from the rudimentary nature of deformation analysis, especially the fracture criterion used to model the strength of the adhesive joint. Some progress made in addressing these two problems is described briefly in the following section. Notwithstanding the oversimplified nature of analysis, the basic qualitative conclusions of this theory have been widely accepted in the field of tribology.
8.3.14 MODELS BASED ON CONTACT, ADHESION, AND FRACTURE MECHANICS If the basic mechanism of friction is based on adhesion in the true area of contact of asperities, it should follow that the force of friction may be determined by the resistance of the adhesive junction to fracture in shearing. Similarly, such a junction should be expected to resist fracture when subjected to a purely normal force, although the strength of the junction in tension may be quite different from that in shear. A miniaturized version of Hertzian contact between spheres is a suitable geometrically simple model of contact between asperities near their smooth and rounded peak. As mentioned earlier, the "adhesion paradox" remained the most vulnerable element of the hypothesis underlying the Bowden and Tabor model of adhesive friction. The explanation of why the contact adhesion is generally intangible becomes clear from the analysis of Johnson et al. [40] and that carried out independently by Sperling [41]. The JKRS (Johnson-KendalRoberts and Sperling) (or JKR) analysis extends the basic framework of contact mechanics by introducing two new elements. The first one is that adhesion is modeled as a constraint on relative normal displacements of surface points in contact that can support tensile tractions. The second element is to represent the action of interatomic forces by an additional source of energy the surface energy (the work of adhesion). The latter energy is the product of the specific surface energy 7 and the area of contact.
750
Savkoor
The global energy balance equation, including a surface energy term in the spirit of the Griffith theory of brittle fracture, has been discussed thoroughly by Rice [42]. Applying the laws of thermodynamics to the process of propagation of a Griffith crack in a solid, Rice [43] stated the basic restrictions that must hold irrespective of the specific structural details on an atomic scale. The entropy production rate A is 1
A - T /(G
- ~)~ ds
(48)
where G is the elastic energy release rate or the generalized force due to the stress field acting on the crack, ~ the specific work of adhesion (in an inert environment or vacuum), and g the length of the crack (the dot to denote time derivative). The second law requirement of non-negative entropy production is Az0
=~ ( G - 7 ) ~ > 0
(49)
The equality applies only when the process is energetically reversible. This assumption is implicit in the use of the reversible work ~ in the JKRS equations. In a reactive environment, the expression of surface energy has to be modified [43]. The JKRS energy balance approach was utilized to investigate the influence of adhesion on the Hertzian contact between spherical surfaces of macroscopic solids with ideally smooth surfaces. The approach has also been extended to investigate the influence of adhesion between rough surfaces using a Greenwood-Williamson rough surface model of micro-Hertzian contact of individual asperities.
8.3.14.1 THE JKRS ADHESIONMODEL Assuming thermodynamic reversiblity, the JKRS adhesion model minimizes the total potential energy (free energy) of the system with respect to the size of the contact. The model predicts that adhesion between two counterformal solids pressed into contact by a normal load produces contact of a larger size than that according to Hertz theory for the same load. For the point contact configuration of a rigid sphere (radius R) pressed against an elastic semiinfinite solid (half-space), the contact area formed under a purely normal load P0 has a circular shape with a radius ao. Under the influence of adhesion (specific adhesion energy ~), the radius of contact increases to a radius al. =
(50)
751
8.3 Models of Friction
The most tangible evidence of adhesion is the significant resistance against separation of solids when a tensile force is applied. The contact can be sustained under the action of a small tensile force. The equilibrium remains stable until the tensile "pull-off" force P attains a critical value Pp0. The contact cannot sustain tensile loads larger than Ppo given by
Ppo -- - 1.5 ~nR
(51)
An annular region adjoining the boundary of the enlarged contact area supports tensile traction. The normal traction in the adhesive contact is 1/2 r2/a2)1/2 1 P(r) -- 2rca 2 (3P1 (1 - r2/a 2) - 6 x T K a 3 (1 }
(52)
Here P1 is the value of Hertzian (ignoring adhesion) normal load that corresponds to the enlarged contact of radius al. The second term involving the surface energy term ~ gives rise to a local field of singular tensile traction. Figure 8.3.22 shows the normal traction in the adhesive contact which varies from a compressive pressure in the central region to a singular tensile traction towards the contact boundary. It is also seen in this figure that the deformed shape of the surface at the contact boundary has a vertical tangent characteristic of the square-root singularity of the traction. Considering the geometry of the gap surrounding the contact as an external crack, the JKRS model falls neatly into the standard framework of linear elastic fracture mechanics (LEFM). It admits alternative formulations in terms of the streesintensity factor, the energy release rate G (or the J integral). In terms of the intensity factor KI corresponding to the crack-opening mode in the linear elastic fracture mechanics: tTij]r___~ a = KIp-1/2~ at equilibrium, KI-
-
(53)
FIGURE 8.3.22 Influenceof adhesion on the surface contour of a solid and the normal pressure in the "larger than Hertzian" contact area with radius al. A region near the contact boundary bears tensile traction.
752
Savkoor
The size of the contact corresponds to the stress-intensity factor at equilibrium (crack) of radius al. The constant KIE expresses the fracture toughness of ideal solid-to-solid interface in resisting the crack movement by normal separation. The interfacial strength KIF in mode I (normal separation) is related to the surface energy (the reversible work of adhesion) ~, as given by Eq. 52 if the process of adhesion is reversible, as is assumed in the JKRS analysis. Domain of validity: The resolution of this paradox by Johnson et al. [40] is undoubtedly an important milestone in the tribology of adhesion between solid materials. The JKRS model applies to Hertzian contact of solids (considered semi-infinite with respect to the extent of the contact) under a purely normal load and it extends the analysis load by including normal adhesion. Just as in the Hertz analysis, the model ignores interfacial friction. It holds for elastically similar materials. In the contact between dissimilar materials, shear traction arises at the interface due to friction even under purely normal loads. However, the influence of contact friction leads only to a minor modification of the JKRS model. The more serious limitation is the assumption that adhesion is a reversible process. There is insufficient experimental evidence concerning any damage to surface films that might occur during the process of normal separation.
8.3.14.2 TANGENTIAL LOADING OF ADHESIVE CONTACT, MODELING OF STATIC FRICTION As discussed earlier, adhesion in the form of static friction that constrains relative tangential motion (slip) in the contact of solid surfaces is a wellestablished concept in the field of contact mechanics of solids (both static and rolling). From a basic physical viewpoint, it would be more consistent to interpret the role of adhesion more explicitly, as a "joint" that imposes a general constraint on all components of relative displacements. It includes the JKRS case, where adhesion resists only the normal separation of surfaces. In this sense slip may be envisaged as a local failure or fracture by tangential separation of the adhesive joint, and the Coulomb friction model has no fundamental significance other than as one of the possible criteria of interfacial failure. More generally, the prediction of slip in the friction model is based on an appropriate criterion of fracture mechanics rather than the friction "law" of Coulomb. The model based on fracture of an adhesive joint implies that the role of normal pressure in the process of friction, if any, can be regarded as minor and indirect. It is conceivable that normal pressure exerts
753
8.3 Models of Friction
some influence on the strength of the interface through some mechanism such as compaction of the boundary layer. 8.3.14.2.1 Effect of a Tangential Force on Contact in Adhesion (Initial Peeling of Surfaces) The effect of a tangential force on the contact of solids with Hertzian configuration has been analyzed by Savkoor and Briggs (S-B) [44]. The starting point of their analysis is the contact of spherical bodies under a purely normal load in the presence of adhesion such that the contact area formed is larger than the Hertzian area as described by the JKRS theory. The application of a (constant) tangential force T (say, along the x-axis) makes an additional energetic contribution which depletes the total energy of the loaded system. The system seeks a new state of equilibrium where elastic energy is released at the expense of the adhesion energy of the surface in contact. Starting from the initial situation (JKRS) of purely normal loading of solids in the presence of adhesion, the application of a tangential force T results in a new equilibrium solution for the contact area of radius al, given by a31 _ 3R(14E- v2) P0 4- 3~,rcR 4-
6~,rrRP04-
9)'2/l;2R 2 -
16 ZK r
(54)
Clearly this result of the S-B analysis shows that the initially "greater-thanHertzian contact" under a purely normal force (JKRS model) decreases monotonically with increasing tangential force. The tangential force, as it were "peels" the adhesive joint by a process of normal separation of surfaces of the solids. The problem can also be formulated in terms of the stress-intensity factors of the linear fracture mechanics theory. These intensity factors which represent the strength of the stress-singularities (square-root type) in points close to the boundary of the contact area may be derived easily from the solutions of the contact problem for normal and tangential traction. The singularity of the normal tensile traction at the contact boundary yields the stress-intensity factor KI. The singular shear traction directed along T may be decomposed into in-plane KII and antiplane KIII components. The joint contribution of the stress-intensity factors of modes I II, and III to the total energy release rate G is 1 2 G - ~--~{(1 - v)(K 2 4- K2II) 4- KIII}
(55)
The equilibrium size of a Griffith crack (contact) is determined by the condition that G equals the reversible work of adhesion 3) of the interface. While all three modes (stress-intensity factors, KI, KII, and Kin) occur
754
Savkoor
simultaneously and contribute jointly to the energy release rate, the actual crack growth is supposed to take place by the opening mode only. The necessity of invoking some assumption regarding the prospective path of crack growth arises because the specific mode of fracture cannot be determined from the energy balance criterion alone. The result predicted by this model in the form of Eq. 54 is in qualitative agreement with experiments with glass on rubber described in Reference [44] and the experimental observations of Barquins [45] for similar materials. In friction experiments of Homola et al. [46] on mica, there was no observable effect of tangential force on the size of the contact area. Surprisingly, the JKRS adhesive contact appears to persist even under negative (tensile) normal load. According to Eq. 53, a real solution requires that T is smaller than a limit force To given by To -- 4-
4
(27~RPo4- 372~2R2)1/2
(56)
At the point T - To, the system attains a state of neutral equilibrium. The equilibrium becomes unstable for higher values of T because the energy release rate exceeds the rate of energy absorption by the surface. The force To is the peeling limit of static friction. It is of interest to note that the peeling limit force To can be significant because it does depend upon the normal load P0. Consequently, the tangential force, even in this early phase of static friction, is already much larger than the pull-force in normal adhesion. For T ) T o , there can be no equilibrium and the authors came to the conclusion that the instability will cause the size of the contact to decrease rapidly by peeling unto its Hertzian size corresponding to the normal load. The last conclusion has not been verified experimentally for elastomers.
8.3.14.2.2 Interfacial Strength (Toughness) in Mixed Mode Loading (Transition from Peeling to Sliding) The criterion of fracture in the previous work is based on the assumption that interfacial strength in the opening mode is the work of adhesion 7 of creating two free surfaces. The parameter 7 is constant regardless of the presence or absence of the shear intensity factor Kn. Indeed, if one assumes an ideal Griffith-like brittle fracture, it should be possible to achieve perfectly reversible processes of normal adhesion (JKRS) and incipient static friction. Recent experimental work on interfacial strength under mixed mode loading indicates that strength or toughness increases with the contribution of mode II. Following Hutchinson [47], the typical behavior of interfacial toughness Gc operating under a mixed mode loading
755
8.3 Models of Friction
(KI and Ku) is expressed by Gc(~t) - Go (1 4-tan2~p)
(57)
where Go(if) = tan -I(Ku/K~). The factor ~ denotes the so-called phase of the stress-intensity factor of the combined mode. This would mean that Ku contributes not only to the applied energy release rate for propagating the crack but also to the resistance of the interface to fracture in the opening mode. Recently, in the context of the S-B model, Johnson [48] took into account the increase in fracture strength (toughness) due to the influence of mode II. He was able to explain why the stable peeling observed by Briggs may continue with increasing T well beyond the critical force To. If tan ff = 1, the effect of increasing Ku is compensated fully by the increased strength, and the tangential force has no effect on the size of the JKRS. It is suggested that this situation may apply to the experiments [46] on thin sheets of mica, which seem to show the JKRS contact is sustained during frictional sliding. However, no specific details are reported concerning their observations of the transition from the static to the sliding friction. Moreover, as discussed earlier, the inerpretation of these results is not yet clear because the JKRS analysis does not apply to such experimental configurations. The physical explanation of the effect of Ku on the interracial toughness may be sought in the microprocesses of blunting the crack tip or shielding by small-scale plastic deformation. It would seem that these effects are inherently irreversible from the energetic viewpoint and perhaps also from considerations of material (damage) in the interracial layer. This implies that the interfacial toughness Gc may not be related to the reversible work of adhesion 7. Different equilibrium states (solutions) may be reached depending upon how the contact approaches the particular state. More detailed experimental work is needed to check the aspect of reversibility of both JKRS and S-B analyses in order to understand these basic aspects. Qualitatively, the important aspect that comes to light is the interaction between adhesion and friction which results in peeling of the adhesive joint in a manner similar to that caused by a purely normal tensile force. The transition from static to kinetic friction after the peeling phase is modeled in a manner that is qualitatively similar to that in the Cattaneo-Mindlin problem described in Section 8.3.9. However, unlike in the Cattaneo-Mindlin model, where slip is initiated according to the local Coulomb model, the analysis based on the fracture mechanics approach [49] considers slip as a mode II fracture and includes a constant (relatively low) postfracture residual strength independent of the normal pressure. Application and validity: the JKRS model of adhesion and the S-B model of peeling caused by a tangential force have been used to study the group behavior of small particles (powders). A tacit assumption in the theory is that
756
Savkoo~
the contact fails or grows in a self-similar manner. This is only approximately true, because the energy contributions of modes II and III are not equal. Consequently, the shape will not remain truly circular. Ignoring the small deviation from axial symmetry, the main physical assumption in the theory is that the energy released goes towards the initiation of the "opening mode" of the crack. The local slip is considered as a mode II fracture and the Coulomb friction law is not invoked. It is important to emphasize that the models described here are considered speculative, and hence they are of interest to research scientists rather than to practicing engineers.
8.3.15 M O D E L S B A S E D ON FRICTION ON THE ATOMIC SCALE More recently, mainly physicists have taken lively interest in modeling friction on the nanoscale of atoms, supported by experimental work on clean and smooth surfaces using specialized apparatus and nearly ideally defined geometry and materials of the test specimen. The foremost aim is to study the nature of contact and friction under ideal and well-defined conditions by removing much of the complexity arising from roughness and contamination of practical surfaces. Various hypotheses have been put forward based on atomic interactions where the primary dissipation of mechanism is explained by the production of phonons. Computational models to simulate atomic interactions are made, and the simulation of interactions is based on the equations of molecular dynamics described by Landman et al. [50]. The models have successfully predicted the influence and structural behavior of thin and well-defined liquid films squeezed between ideal solid substrates. However, such simulations are highly computation intensive, and it remains to be seen how long it takes before issues of adhesion and friction of practical surfaces can be tackled with this approach.
8.3.16 CONCLUDING
REMARK
There is yet no unified quantitative theory that can reliably predict friction from the knowledge of all measurable physical variables that are deemed to govern friction in any specific situation. The presence of several kinds and variable amounts of ill-defined foreign materials (contaminants) in the interfacial region makes it virtually impossible to derive quantitative estimates of the absolute magnitude of friction from basic physical properties of solids. For engineering calculations the interfacial frictional properties are generally
8.3 Models of Friction
75 7
characterized by the local Coulomb model of friction based on a nondimensional number known as the coefficient of friction. The coefficient of friction depends on the normal load, the relative motion, the material properties of the solids, and even more critically on surface conditions (any foreign materials in the interface). Although the precise modeling and controlling of friction, even today, remains in its infancy, considerable progress has been made in understanding the nature of friction that led to the formulation of models expressing the dependence of friction on a number of important operating variables. The models describe functional relations for extrapolating the magnitude of friction that is empirically determined at some fixed values of the variables. The modeling of the constitutive relation of friction takes into account the following elements: 1. the magnitude of the static friction and its dependency on the time of static contact; 2. the variation of the stationary value of kinetic friction with sliding velocity in uniform sliding (without stick-slip); 3. the influence of interfacial temperature on the magnitudes of static and kinetic friction; 4. the dependence of the magnitudes of static and kinetic friction on the normal pressure; 5. the transient nature of friction in nonstationary sliding motion that includes at least one hidden state variable. However, a proper application of such relations requires understanding of the physical background and the domain of validity of each model. When interpreted correctly, the models offer adequate guidance for avoiding pitfalls and saving labor involved in extensive empirical testing under real-life, practical situations.
REFERENCES 1. Dowson, D. (1984). History of Tribology, New York: Longman. 2. Bowden, E P., and Tabor, D. (1954). The Friction and Lubrication of Solids, Part I, Oxford: Clarendon Press. 3. Bowden, F. P., and Tabor, D. (1964). The Friction and Lubrication of Solids, Part II, Oxford: Clarendon Press. 4. McGill, D. J., and King, W E. (1965). Engineering Mechanics: Statics, 3rd ed., PWS Publishing Company, Chapter 6. 5. Rabinowicz, E. (1995). Friction and Wear of Materials, 2nd ed., John Wiley. 6. Williams, J. A. (1994). Engineering Tribology, Oxford: Oxford University Press.
758
Sa~koo~
7. McLaren, K. G., and Tabor, D. (1963). Friction of polymers at engineering speeds: Influence of speed, temperature and lubricants. Paper 18, Lubrication and Wear Convention, I. Mech. E. London, May, pp. 52-57. 8. Bahadur, S., and Ludema, K. C. (1971). The viscoelastic nature of the sliding friction of polyethylene, polypropylene and copolymers. Wear 18: 109-128. 9. Steijn, R. P. (1967). Friction and wear of plastics. Metals Engineering Quarterly, American Society of Metals, pp. 9-21. 10. Tanaka, K. (1984). Kinetic fiction and dynamic elastic contact behaviour of polymers. Wear 100: 243-262. 11. Grosch, K. A. (1963). The relation between the friction and visco-elastic properties of rubber. Proc. Roy Soc. Lond., A274: 21-39. 12. Savkoor, A. R. (1966). Some aspects of friction and wear of tyres arising from deformation, slip and stresses at the ground contact. Wear 9: 66-78. 13. Savkoor, A. R. (1974). Adhesion and deformation friction of polymers on hard solids, in Advances in Polymer Friction and Wear of Polymers, pp. 69-119, vol. 5b, Lee, L. H., ed., New York and London: Plenum Press. 14. Oden J. T., and Martins, J. A. C. (1985). Models and computational methods for dynamic friction phenomena. Comp. Mech. Appl. Mech. Engng. 52: 527-634. 15. Schallamach, A. (1963). Abrasion and tyre wear, in The Chemistry and Physics of Rubberlike Substances, Bateman, ed., London: McClaren. 16. Dieterich, J. H. (1979). Modeling of rock friction, 1: Experimental results and constitutive equations. J. Geophys Res. 88: 2161-2168. 17. Ruina, A. L. (1983). Slip instability and state variable friction laws. J. Geophys. Res. 88 (12): 10359-10370. 18. Rice, J. R., and Ruina, A. L. (1983). Stability of steady frictional slipping. J. Appl. Mech. 50: 343-349. 19. Meijaard, J. P., and Savkoor, A. R. (1997). Modeling and suppression of stick-slip in dry friction, Asia-Pacific Vibration Conference in Kyongju (Korea), Korean Soc. Mech. Eng., Kangnam-ku, Seoul, pp. 27-32. 20. Blok, H. (1910). Fundamental mechanical aspects of boundary lubrication. SAE Journal 46 (2): 54-68. 21. Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H. (1981). Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press. 22. Savkoor, A. R., and Meijaard, J. P. (2001). Application of bifurcation theory to the identification of the constitutive relations for sliding friction, in IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics, pp. 233-248, Narayan, S., and Iyengar, R. N., eds., Jan. 4-8, Chennai (India), Dordrecht: Kluwer Academic Publishers. 23. Tolstoi, D. M. (1967). Significance of the normal degree of freedom and natural normal vibrations in contact friction. Wear 10: 199-213. 24. Johnson, K. L. (1985). Contact Mechanics, Cambridge: Cambridge University Press. 25. Kalker, J. J. (1990). Three-dimensional elastic bodies in rolling contact, in Solid Mechanics and Its Applications, Dordrecht: Kluwer Academic Press. 26. Dundurs, J. (1975). Properties of elastic bodies in contact, in IUTAM Symposium on the Mechanics of the Contact between Deformable Bodies, DePater, A. D., and Kalker, J. J., eds., Enschede (the Netherlands). Delft University Press, pp. 54-66. 27. Cattaneo, C. (1938). Sur Contatto di due corpi elastici. Rend. Accad. Naz. dei Lincei 27 (6): 342-348, 344-443, 474-478. 28. Mindlin, R. D. (1949). Compliance of elastic bodies in contact. Trans. ASME Ser. E. J. Appl. Mech. 16: 259-268.
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8.3 Models of Friction
29. Mindlin, R. D., and Deresiewicz, H. (1953). Elastic spheres in contact under varying oblique forces. J. Appl. Mech., Trans ASME 75: 237-245. 30. Deresiewicz, H. (1974). Bodies in contact with applications to granular media, in R. D. Mindlin and Applied Mechanics, pp. 105-147. Herrmann, G., ed., Elmsford, NY: Pergamon Press. 31. Kalker, J. J. (1967). On the Rolling Contact of Two Elastic Bodies in the Presence of Dry Friction. Ph.D. dissertation, Delft University of Technology, Delft (the Netherlands). 32. Curnier, A. (1984). A theory of friction. Intern. J. Solids Structure 20 (7): 637-647. 33. Klarbring, A. (1985). Contact Problems in Linear Elasticity. Doctoral dissertation, Linkoping Studies in Science and Technology, No. 133, pp. 1-211, Linkoping. 34. De Souza Neto, E. A., Hashimoto, K., and Owen, D. R.J. (1996). A phenomenological model for frictional contact accounting for wear effects. Phil. Trans. R. Soc. London A354: 819-843. 35. Dahl, P. R. (1976). Solid friction damping of mechanical vibrations. AIAA Journal 14 (12): 1675-1682. 36. Bliman, P. A., and Sorine, M. (1993). Friction modeling by hysteresis operators, application to Dahl, stiction and Stribeck effects, in Models of Hysteresis, pp. 10-19, Harlow: Longman Science and Technology. 37. Canudas De Wit, C., Astr6m, K. J., and Braun, K. (1987). Adaptive friction compensation in DC motor drives. IEEE Journal of Robotics and Automation RA-3 (6), 681-685. 38. Ganghoffer, J. E, and Schultz, J. (1995). A deductive theory of friction. Wear 188: 88-96. 39. Greenwood, J. A., and Williamson, J. B. P. (1966). The contact of nominally fiat surfaces. Proc. Roy. Soc. London A195: 300-314. 40. Johnson, K. L., Kendall, K., and Roberts, A. D. (1971). Surface energy and contact of elastic solids. Proc. Roy. Soc. London A324: 301. 41. Sperling, G. (1964). Eine Theorie der Haftung yon Feststofteilschen an festen Koerpern. Doctoral dissertation, Fakultaet der Machinenwesen, T. H. Karslsruhe (Germany). 42. Rice, J. R. (1978). Mathematical analysis in the mechanics of fracture, in Fracture: An Advanced Treatise, pp. 191-307, chap. 3, Liebowitz, H., ed., New York: Academic Press. 43. Rice, J. R. (1978). Thermodynamics of the quasi-static growth of Griffith cracks.J. Mech. Phys. Solids 26: 61-78. 44. Savkoor, A. R., and Briggs, G. A. D. (1977). The effect of tangential force on the contact of elastic solids in adhesion. Proc. Roy. Soc. London A356: 103. 45. Barquins, M. (1987). Adherence, frottement et usure des elastomeres. Kautschuk/Gummi Kunstoffe 40 (5): 410-438. 46. Homola, A. M., Israelachvili, J. N., McGuiggan, P. M., and Gee, M. L. (1990). Fundamental experimental studies in tribology: The transition from "interfacial" friction of undamaged molecularly smooth surfaces to "normal" friction with wear. Wear 126: 65-83. 47. Hutchinson, J. W. (1990). Mixed mode fracture mechanics of interfaces, in Metal-Ceramic Interfaces, Acta Scripta Metallurgica, pp. 295-306, Pergamon Press. 48. Johnson, K. L. (1996). Continuum mechanics modeling of adhesion and friction. Am Chem. Soc., Langmuir 12: 4510-4513. 49. Savkoor, A. R. (1992). Models of friction based on contact and fracture mechanics, in Fundamentals of Friction Macroscopic and Microscopic Processes, pp. 111-133, Singer, I. U, and Pollock, H. M., eds., NATO ASI Series E. Dordrecht: Kluwer Academic Press. 50. Landman, U., Luedtke, W. D., and Ringer, E.M. (1992). Molecular dynamics simulations of adhesive contact formation and friction, in Fundamentals of Friction: Macroscopic and Microscopic Processes, pp. 463-505, Singer, I. L., and Pollock, H. M., eds., NATO ASI Series E. Dordrecht: Kluwer Academic Press. o
SECTION
8.4
Friction in Lubricated Contacts JEAN FRI~NE1 and TRAIAN CICONE2 1Laboratoire de M~canique des Solides, Universit~ de Poitiers, France 2Dept. of Machine Elements and Tribology, Polytechnic University of Bucharest, Romania
Friction is the resistance to motion that occurs when one body slides over another. The definition can be extended to the motion of a solid body in a fluid. The resistive force acting in the opposite direction of the relative motion is known as the friction force. The friction force has two forms: the force required to initiate sliding, known as the static friction force, and the force necessary to maintain the sliding, called the kinetic friction force. In 1902, R. Stribeck [1] presented experimental results on friction in plain and rolling-element bearings. His results were summarized on curves, which show the variation of friction versus relative speed. These curves are very general and can be used to present the different phenomena that occur in lubricated contacts. Figure 8.4.1 shows the variation of friction versus surface velocity for a plain journal bearing. Similar results can be presented for any other type of conformal contact like bearings (i.e., plain or tilting pad thrust bearings and journal bearings). The Stribeck's curve can be also plotted for a lubricated nonconformal contact like that in gears, cams, and rolling-element bearings (i.e., ball and roller bearings). The original Stribeck's curve was only a small part of the curve given in Figure 8.4.1. This figure shows that different regimes of lubrication can occur in a lubricated sliding contact, depending on the relative speed. These regimes also depend on the pressures occurring in the lubricating film. For conformal contacts (distributed contacts with low contact pressure), four different regimes of lubrication can be put in evidence. In the case of nonconformal contacts (concentrated contacts with high contact pressure), regime IV is impossible to be found. Table 8.4.1 shows the different lubrication regimes that can be found in lubricated contacts. Generally, the transition from one regime to another cannot be precisely defined. These regimes will be successively described in following text. 760
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
761
8.4 Friction in Lubricated Contacts
FIGURE 8.4.1 The Stribeck's curve. TABLE 8.4.1 Regimesof Lubrication
Regime I Regime II
Conformal contacts: low contact pressure
Nonconformal contacts: high contact pressure
Boundary lubrication
Extreme pressure (EP) lubrication
Mixed lubrication
Regime III
Hydrodynamic ( H D ) lubrication
Regime IV
Superlaminar flow
Elastohydrodynamic (EHD) lubrication
Regime I is called boundary lubrication for conformal contacts, whereas for nonconformal contacts, this regime is known as extreme pressure
(EP) lubrication. Boundary lubrication is defined [2] as a condition of lubrication in which friction between the two surfaces in relative motion is determined by the properties of the surfaces and by the properties of the lubricant, other than viscosity. A very thin lubricated film (of order of 100 A) is formed on metallic surfaces, either by physical adsorption of polar molecules or by chemical adsorption. This film separates the opposite surfaces in relative motion, reducing friction and preventing surface damage (Fig. 8.4.2). Polar oils such as esters or fatty acids are very efficient in forming lubricant layers. The coefficient of friction is in the range of 0.05 to 0.15, depending on the metal sliding combination and on the nature of the lubricant. The main feature of boundary lubrication in comparison with solid-tosolid contact is that although the reduction in the friction coefficient is less
76 2
Fr~ne and Cicone
FIGURE 8.4.2 Boundary lubrication model.
important (2 to 10 times), the reduction in wear is quite impressive (by a factor of 102 to 104). For very high contact pressures and for high shear stresses the adsorbed layers do not remain on the metal surfaces, so that extreme pressure (EP) films must be generated in the contact to protect the surfaces and to prevent seizure. An extreme pressure film is formed by nonreversible chemical reactions between special extreme pressure additives, which are included in the lubricant, and the metallic surface. These additives can be lubricant molecules with sulphur, chlorine, and phosphorus or inorganic films formed in the contact on metal surfaces (due to local pressure, shear stress, and temperature). The friction coefficient is also between 0.05 and 0.15, depending on the metal surfaces and the nature of the lubricant. Regime III corresponds to hydrodynamic (HD) lubrication for low-pressure films, or to elastohydrodynamic (EHD) lubrication for high-pressure films. For low-pressure films (i.e., mean pressure between 10 s and 10 7 Pa) and negligible elastic displacements of the bearing surfaces, the hydrodynamic regime is observed. The film thickness is quasi-proportional to lubricant viscosity and to surface relative speed. In this case, the Reynolds equation can be used and the bearing characteristics can be calculated [3]. For a sliding contact in isothermal conditions, the friction force is given by
F - k #VS
(1)
have
where # is the fluid viscosity, V is the relative speed, S is the area of the contact, have is the average film thickness, and k = 0.4 to i is a correction coefficient depending on the bearing configuration. The hydrostatic (HS) lubrication is similar to hydrodynamic lubrication except that the pressure field does not influence the friction. Equation 1 can also be used. For an unloaded circular journal bearing, the friction torque, Ca, is given by Ca -- 2rc#c~
C
(2)
763
8.4 Friction in Lubricated Contacts
where # is the dynamic viscosity, co is the angular velocity, R is the bearing diameter, L is the bearing length, and C is the bearing clearance. With the increase in load and the consequent increase in eccentricity, the value calculated with Eq. 2 gives a good approximation up to an eccentricity ratio of = 0.6, since the friction torque remains practically constant. For greater eccentricities, the friction torque sharply increases so that it is 4 times greater at e - 0.9 (i.e., for highly loaded bearings). Figure 8.4.3 shows the variation of the nondimensional friction torque, Ca, versus the eccentricity ratio, e, for various L/D ratios, in the case of a plain journal bearing. It is worth mentioning that, in the case of hydrostatic and hydrodynamic bearings, the use of the friction coefficient must be avoided. Its values can vary between zero or a very low value (10 -6 or 10 -7) to infinite, depending on the speed and on the load. The following two examples are very suggestive: 1. The Hale telescope on Mount Palomar [4] is supported by three groups of two hydrostatic thrust bearings. The telescope mass is approximately 450 000 kg, and it is driven by only a 60 W electric motor; this power is considerably in excess. Correspondingly, the friction coefficient is less than 10 -6 . The very low friction is due to the fact that the rotational speed of the telescope is only i rev./24 h.
Notations
q=
i|
Id
-
60 _
I
............. L I D = 0. 25 ......
L/D=I
/!'
LID = O. 5
]|
L/D= 2 ..____..
L / D = = <
bearing diameter, D.
-
e eccentricity ratio, e - - - ; C
where, e is the eccentricity of the
e
cr
-
shaft in respect with the bush center.
Ii 'e
40
/
-
I ," I, ~
2O
0
._r "
~
.
.~
_
0
I 0,5
,
[ 1
w
eccentricity ratio, e FIGURE 8.4.3
Friction torque in an HD journal bearing [3].
764
Fr6ne and Cicone
2. The hydrodynamic journal bearings of a centrifugal pump working in a vertical position do not carry any load except the very small unbalance. The friction torque is given by Eq. 2, and correspondingly, the friction coefficient can be written as
co f - RW
(3)
Since the applied load, W, is almost zero and the friction torque, Ca, is not equal to zero, the friction coefficient, f, tends to infinite. These two examples illustrate that the friction coefficient should not be used for hydrostatic and for hydrodynamic lubrication. For high-pressure films (i.e., for mean pressure from 2.108 up to 3.109 Pa), both the surface elastic displacements and the lubricant viscosity variation with the pressure should be taken into account. The viscosity increase with pressure (called the piezoviscous effect) can be as high as 50 times the viscosity at atmospheric pressure. On the other hand, the elastic deformations of the surfaces, even small, can be two or three times the minimum film thickness, so that cannot be neglected. Similar effects appear in high compliant conformal contacts, even at medium contact pressures (for example, in connecting-rod bearings). This regime is called elastohydrodynamic (EHD) lubrication. Because of the complexity of the phenomena, there is no simple equation to calculate friction. Usually frictional effects are defined in terms of traction coefficients and are given in graphical forms. Some typical traction curves for the roller-bearing case are presented in Figure 8.4.4. Regime II corresponds to mixed lubrication in which several mechanisms act simultaneously. This regime is a transition regime between boundary lubrication and hydrodynamic lubrication for low-pressure films or between extreme pressure lubrication and elastohydrodynamic lubrication for highpressure films. In these cases, by viscous effect, surface motion carries on the lubricant into the contact film to generate hydrodynamic or elastohydrodynamic pressure, which tends to separate the contact surfaces. Because of waviness and roughness effects, part of the contact is fluid film lubrication and part of the contact is still in the boundary lubrication regime or the extreme pressure lubrication regime (Fig. 8.4.5). As the friction for full fluid film lubrication is smaller than that corresponding to boundary or extreme pressure lubrication, the overall friction decreases as the hydrodynamic or elastohydrodynamic effects increase. Regime IV is characterized by superlaminar flow in hydrodynamic lubrication. This regime occurs for high-relative-speeds, low-viscosity lubricants or very large dimensions.
765
8.4 F r i c t i o n in L u b r i c a t e d C o n t a c t s 1"5'i
l.o
I I
1
,'- . . . . . . . . . . '
..........................i-...................... !. . . . . . . .
/
i
="1 .................. "
i i
~
_.I
<
....... - - . ~ ~
i.~:/:.~
-10.0
.,,@"*-
i/Li"
; -S.O
.......
!.................
i
-o.,-I ......... :,..,L-i-,,,:--:.tTJ4-
-I.S
~ ~ ' ' : " - - " ' ~ , ~
t ..............
o.o-t ................... - . . . . . .
i !
.................... i ..................
i 0.0
. . . . . . . . . S.O
SLIDIN@ RflTC (UI-b'2)/(UI+U2)
lY.)
0
G"
I.o+7
a
G -
mlli~(lm
m i r a
+
G -
l i B e l
i W l
l.e+6 e l l loH N I N I
i
OOl
S,e+5 I
,ISlSll
I
-- --I
X
G -
2.0+5
o
Q -
1.i+5
iO.O
G - transversal m o d u l u s of elasticity F I G U R E 8.4.4
Typical traction curves for a cylindrical contact [5].
F I G U R E 8.4.5
Mixed lubrication model.
In journal bearings, Taylor vortices first appear, and then turbulent flow is gradually developed. The transition between laminar flow and Taylor vortices is defined when the
Taylornumber, T=pcoRC v~~- 9t V/~~ /x
reaches the critical value Tc = 41.2 - 38~ + 63.33~ 2 [6].
(4)
766
Fr6ne and Cicone
L/D = 1
q~ .
,.
" J
C/R = 0.003
,
,,
Id 20o
=
C~-,~':237o | ~:,;74o
g
| |
,
7~o
(~) ~ ' :
17776
100
o
FIGURE 8.4.6
o.s
eccentricity ratio e
1
Nondimensional friction torque in superlaminar regime [6].
It is generally assumed that the turbulent flow is fully developed when the Reynolds number, ~R = pVh/#, is twice the critical value, which marks the appearance of first vortices. Nonlaminar flow in the film drastically increases the friction dissipation in the bearing. The turbulent shear stresses, Zxy, on the bearing surface can be approximated by: Zxy(Y- 0, h ) -
h c3p ~_V ~0.94 _ ~ ~x + (1 + 0.0012. )
(5)
where p is the fluid pressure and p is the fluid density. For the plain journal bearing case, the nondimensional friction torque, Ca, largely increases with the average Reynolds number, as shown in Figure 8.4.6. For thrust bearings, some kind of G6ertler vortices appear before turbulent flow. The transition, which marks the end of laminar flow, is obtained for the critical Reynolds number: 580 < ~Rc < 800 (6)
8.4 Friction in Lubricated Contacts
767
REFERENCES 1. Stribeck, R. (1902). Die Wesentlichen Eigenschaften der Gleit und Rollenlager. Zeitschrift Vereines deutche Ingenieure. 46 (38): 1341-1348, 1432-1438; 46 (39): 1463-1470. 2. O'Connor, J. J., Boyd, J., and Avallone, E. A. (1968). Standard Handbook of Lubrication Engineering, McGraw-Hill Book Company. 3. Fr~ne, J., Nicolas, D., Degueurce, B., Berthe, D., and Godet, M. (1997). Hydrodynamic Lubrication. Bearings and Thrust Bearings, Tribology Series No. 33, Amsterdam: Elsevier. 4. Fuller, D. D. (1956). Theory and Practice of Lubrication for Engineers, New York: John Wiley and Sons. 5. N~lias, D. (1999). Contribution h l'~tude des roulements, Habilitation h Diriger des Recherches, INSA de Lyon, Universit~ "Claude Bernard" Lyon I. 6. Fr~ne, J. (1974). Regimes d'~coulement non-laminaire en films minces. Application aux paliers lisses. Th~se de Doctorat es Sciences Physiques- Universit~ "Claude Bernard" Lyon.
SECTION
8.5
A Thermodynamic Analysis of Wear Hug DUONG BuI, MARTA DRAGON-LouISET, and CLAUDE STOLZ Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France
Contents 8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 General Thermodynamical Considerations on Wear Phenomena . . . . . . . . . . . . . . . . . . . . 8.5.2.1 Analysis of Entropy Production ... 8.5.3 Global Approach of the Interface Behavior Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3.1 Interface Study . . . . . . . . . . . . . . . . . . . 8.5.3.2 Wear Criteria . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
768 769 770 771 771 775 775
8.5.1 INTRODUCTION Wear phenomena due to contact and relative motion between two solids depend on the loading conditions and material mechanical behaviors and are characterized by a loss of materials. Particles are detached from sound solids ~1, ~'~2when some criteria are satisfied at the boundaries Fi (i - 1, 2) between ~'~i and the complex medium ~23, called hereafter the "interface", with some wear rates ~i (normal component of the velocity). The interface is a complex medium made of detached particles, eventually a lubricant fluid, and damaged zones. Macroscopic descriptions of the interface are known in the literature [1-4]. Recently, Dragon-Louiset [5,6] and Stupkiewicz and
768
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
8.5 A Thermodynamic Analysis of Wear
769
Mr6z [7] introduced microscopic scale models in order to derive macroscopic laws. A thermodynamical analysis of the evolution of the moving surfaces Fi is provided by Zmitrowicz [8], Dragon-Louiset and Stolz [9], and Stolz and Pradeilles-Duval [12]. The evolutions of the damaged surfaces are complex, particularly in the transient phase of the interface life. However, for particular geometry of the interface, and for the steady state case, with constant fluxes of materials, continuous wear states can be studied experimentally. There are useful wear criteria, such as Archard's law [10], but most of them depend on experimental conditions and cannot be used for other geometry or for different loading conditions. In contrast with the eulerian description used in References [5, 6], we reconsider the model of the interface given in References [5, 6, 11], using rather the lagrangian approach. In this paper, the interface description is based on given macroscopic laws and differs from the one derived from microscopic considerations in Dragon-Louiset [5,6]. The local approach allows us to make a distinction between mechanical quantities evaluated on a given geometry and for specific loading conditions and intrinsic ones associated with any moving wear surfaces Fi. Our approach to wear criteria is based on the energy release rate like quantities, similar to Griffith's theory in fracture mechanics. All these quantifies, some of them accessible to experiments, make it possible to better define wear criteria and w e a r rates.
Throughout this article, we make the small perturbations assumption.
8.5.2 GENERAL THERMODYNAMICAL CONSIDERATIONS ON WEAR PHENOMENA We consider a system consisting of two sliding contacting solids (~'~1 and ~22) and the contact interface zone f)3, having some mechanical properties which are assumed to be known and described by usual laws of continuous media (Fig. 8.5.1). Such interface laws may be understood in a macroscopic sense, as average or homogenized through the thickness of ~23 (for a model of macroscopic laws issued from microscopic considerations, see DragonLouiset [5, 6]). The interface thickness is small compared to the size of the tribological system; it results from damage mechanisms occurring in sound solids at Fi. Outside the damaged process zones in both solids the behavior laws are described by the free energies ~/i and the dissipation potentials di. As a result of wear mechanisms, the boundaries Fi m o v e with the velocities t~in i in the reference configuration. Along the boundary, we define If ]ri - f / + - f / with the (+) sign in ~2i, ( - ) sign in F13.
770
Bui et al.
1
~3
r2 FIGURE 8.5.1
The contact interface.
By expressing the conservation laws in ~"~i and across Fi as in References [6,9,12], we get the following equations for the quasistatic case: 9 Mass conservation (v: velocity; [2i" mass flux; p: density; a: stress): /5 + div(pv) - 0
in f2i,
Pi = Pir
on Fi
(1)
9 Momentum conservation: div2pta - 0 in f2i,
ni" [a]r,- 0
on Fi,
(2)
9 Energy balance (q: heat flux; ~: strain; Z: internal energy; s: entropy; T: temperature): pj~=a : ~-divq
in ~2i,
Pi[6 + sT]r,-ni.a.[V]r,+[q]r, .ni = 0
(3) on El.
On each moving surface, using the second conservation law (Eq. 3), one can show that there exists a heat source along Fi, with the density
#iGth -- Pi[6 + sT]r,-ni.a.[v]ri
8.5.2.1
(4)
ANALYSIS OF ENTROPY PRODUCTION
The internal entropy production is positive and consists of different kinds of separately positive contributions: 9 volumic thermal conduction: Dth --
- q . V T / T 2 >_ 0
(5)
9 volumic intrinsic mechanical irreversibility: Dm -- -~1(~" g r a d v - p ( ~ + s T ) ) _>0
(6)
771
8.5 A Thermodynamic Analysis of Wear
9 and surface (or line in two-dimensional problems) irreversibility terms: 1 Dr -- -~ (#i[0]r,-ni.~.[v]vi) >_ 0 If at any point of Fi the mass flux vanishes ~i = 0, the displacement u is continuous, the velocity jump is [v]r i = 0, and then Dr = 0. This means that the last dissipation Dr is a characteristic of material loss, i.e, the part of dissipation due to wear.
8.5.3 GLOBAL APPROACH OF THE INTERFACE BEHAVIOR LAWS At each front Fi, the normal ni is directed toward the sound solid. The interface of thickness e = 2h(x, t) is defined by its mean surface F. The total dissipation per unit area of F is given by
D --
D3dz +
.
~
F,
where D3 is the volumic dissipation of the interface medium:
D3--{-q.VT/T
2 +-1T(a" g r a d v - p(q~ + s~))}f~3
(9)
The total global dissipation associated with each surface contact element contains two distinct terms, D3 due to friction associated with relative movements of solids, and terms on Fi due to the fluxes of material losses (ill and #2)For a given mechanical behavior of the interface ff~3, issued either by experiments or by micromechanical considerations as was done in References [5,6], the resolution of the evolution equations of a strip of height e(x), subjected to uniform loadings, with the surfaces z = +h sliding with the relative velocity V, may give separately the dissipation rates by friction and by wear.
8.5.3.1
INTERFACESTUDY
The displacement is assumed to be continuous across the sound solids and the interface. Let us describe the tribological system f~l, f~2, f~3 by
772
B u i et al.
considering the interface f~3 as an abstract medium with some surface energy ~s and dissipation potential Ds to be defined hereafter. The mean surface 1-" is defined by S(x, t) -- 0. At each point x of 1", with the normal n(x), we define the surfaces r'l and r'2, respectively, by x 4. h(x,t)n(x) and x - h(x, t)n(x). We introduce the convected differential Drf of functions f (scalar, vector) with respect to F, Drf(x, t) := limav-,o
f (x 4. endt, t 4- at) - f (x, t) dt
In particular, we have for the evolution of the middle surface DrS(x, t) -- 0,
Dr(x 4. h(x, t)n(x)) - r
Dr(x + h(x, t)n(x)) -- r n, Dr(x) -- en,
Dr(n(x)) -- divr(r
At each Fi, the displacement u is continuous, and the velocity fields v satisfy the Hadamard condition D r [ u ] / = 0, or equivalently
= 0
[v]r ,4.r
(10)
The total dissipation takes the form
D=
D3dz+
.
(11)
where Gi and G3i are the same expressions as given in References [5, 6]:
Gi = PiOi - ni.~i.Vui.ni
(12)
G3i -- P3il//3i - ni.~3i.Vui.ni
(13)
In Eq. 11, the two dissipation terms are of different natures. The first one, volumic at the origin (or at a microscopic scale in Reference [5]), becomes a surface dissipation by the thickness average process, per unit contact area (the macroscopic scale in Reference [5]). It corresponds to friction, which depends on the characteristics of f~3. The complementary terms in Eq. 11 are due to wear rates r which depend on the characteristics of both media ~'~i and ~"~3 9
773
8.5 A Thermodynamic Analysis of Wear
Let us consider the isothermal evolution of the interface under uniform temperature. The global dissipation is written in the following form:
D-l{f
(a3"gradv3-P3~3)dz+~~i(Gi-G3i)}>_O/
(14)
It follows that a global approach of wear can be done by means of surfacefree energy defined a priori on F. The interface of thickness 2h is characterized by the free energy per unit volume ~3 and the local strain ~. We define the surface energy by
- (2ps(x)h(x)) -1 f p(x,z)~3(e(x 4- zn))j(x,z)dz
(15)
J2 h
where j(x, z) is a scalar which takes account of the curvature of F (case of wear in rotors, for example). The associated mass Ps is defined as Ps - ( 2 h ( x ) ) - l f
p(x,z)j(x,z)dz
(16)
J2 h
The section of the interface of thickness 2h is in equilibrium with external loads. Since the displacement fields are continuous on Fi, the surface energy ~s is a function of the displacement fields and internal parameters a, discussed later. The global potential energy of the tribological system can be written as d.tl i
all
where ul = u(x 4- hn, t) and u2 = u ( x - h n , t). At equilibrium, cr -- picg~i/c% and divcr -- 0 on F~/. In the case of the nonviscous interface, one has g.ni = 2p~hc3~s/OUion Fi. In the case of the viscous interface, for a given dissipation potential D3 --d(k; 0~) a similar average through the thickness can be done giving the surface dissipation Ds(Vl, v2; 0~):
2PshDs(vx,v2) -- f
pd(k(x 4- zn);
~)j(z)dz
(17)
d2 h
Equation 17 yields = 2psh(a
,s/OU, + aDs~Ova)
on F/
The macroscopic description given here ignores the details on the microscopic level, so that the thickness h(x) is the relevant parameter geometrically defining the interface, while a describes the given physical properties of the interface. In a microscopic analysis of the interface, as given in References [5, 6, 11], the parameter ~ also contains a description of the
774
Bui et al.
constituents, i.e., the set of volume fractions of debris {991,992} or the mass fractions {ml, m2}, together with their physical properties, etc. It can be shown that in the steady state case when only one solid, for example, f~2, is subjected to wear and when damage occurs in an elastic-brittle manner, both q92 and m2 are equivalent to the interface thickness 2h(x). In a microscopic approach to wear, the set of internal variables for general constitutive laws of the constituents is {991,992, ~1, ~2,-..}. The variation of the potential energy can be written as
dP _ --d fn P,~i[e(u)]dco + ~d [ ~r 2psh~/s(U 1 u2,~)dA] dt dt i which becomes after simple rearrangements d P = ~ pil~/i[~3(u)]do) - j ; pil//i~idA _F ~F {Dr(2pshOs)+2pshg/sdpT}dA
dt
,
where ? is the curvature. Similarly, the average process through the thickness of the mass conservation law yields
/r{
Dr(2psh) + 2pshdpT}dA - - P i l l + P2~2
(18)
Equation 18 makes it possible to relate the internal variables c~, h, or other relevant variables defining the interface properties appearing in its left-hand side to the wear rates q51, ~b2. Equation 18 is another form of the mass conservation law, more suitable for the interface description. Its right-hand side may be considered as the source terms feeding the contact interface. The variation of the potential energy P can be written in the form --
dt
=
,/r, n . o - . v +
dA
-
,
PiOi4idA+
0s[p4~] 2
idA -
, ni.~i.vidA
2Psh-~ui . (v, + Vu~.~b,n,)+ 2 p s h - ~ . Dr(~) dA
(19)
We obtain the dissipation
dt
~lr, +
n.o-.v dA
-
-
{p,r
-
ni.~i.Vui}.~b,nidA
2 p s h - ~ . Dr(~/- Cs(pCk)~-2psh---~. Dr(u~) dA
We recover the dissipation because of the interface movement. The first term corresponds exactly to Gi, and the complementary ones correspond to the interface contribution. In particular, if 2h is the internal parameter c~, since Dr(2h)n+ 2hDr(n) = (q51 -~b2)n, the previous expression takes a form similar to the
8.5 A Thermodynamic Analysis of Wear
775
global dissipation in w h i c h G3i is given by G3i - Gsi - ~ksPi - 2psh c9o~ In this approach, the interface is in equilibrium at any time. This expression shows that Gsi are p a r a m e t e r s directly associated with the interface model.
8.5.3.2
W E A R CRITERIA
8.5.3.2.1 Local M o d e l A wear criterion can be defined as a relationship b e t w e e n the n o r m a l velocity ~)i and the associated t h e r m o d y n a m i c force ~i ~ - - G i - G3i. O n e can define a dissipation potential d, a convex function of 7i, such that ~bi = Od/Oyi. The simplest criterion is similar to Griffith's criterion in fracture mechanics, for w h i c h d(7i) is the s u p p o r t i n g function of the convex 7i-< Tic. The wear criterion of m e d i u m (i) d e p e n d s on ~:c, Tic, G3i, i.e., on five parameters.
8.5.3.2.2 Interface Model There are simplifications w h e n the free energy and the potential dissipation have simple forms ~/s(ul, u2, cz) "-- f ( u l - u2, ~), Ds(Vl, v2, ~) := g(vl - v2, c~) We can s h o w that, at first order, [o].n = 0 and the viscous dissipation term is r e d u c e d to 2psh(cgDs/c% ) " [Di-(ui)] 21" Since the velocities q~i are small c o m p a r e d to the relative velocity V, we have 2psh(cgDs/c% ). DF(ui)l 1 "~ 2Psh(cgDs/Ov).V, w h i c h means that the dissipation is essentially due to friction. Such an a p p r o x i m a t i o n is a s s u m e d in References [5, 6, 11].
REFERENCES 1. Georges, J. M., Millot, S., Loubet, J. L., and Tonck, A. (1993). Drainage of thin liquid film between relatively smooth surfaces. J. Chem. Physics 98 (9): 7345-7359. 2. Godet, M. (1990). Third bodies in tribology. Wear 136 (1): 29-45. 3. Singer, I. L., and Wahl, K. J. (1999). Role of Third Bodies in Friction and Wear, 1999 MRS Workshop Series: Tribology on the 300th Anniversary of Amonton's Law, San Jose, California. 4. Meng, H. C., and Ludema, K. C. (1995). Wear models and predictive equations: Their form and content. Wear 181-183: 443-457. 5. Dragon-Louiset, M. (2000). Modfilisation microm~canique de l'interface d'un syst~me tribologique dans une approche de l'usure continue. M~ca. Ind. 1: 1-6.
776
Bui et al.
6. Dragon-Louiset, M. (2001). On a predictive macroscopic contact-sliding wear model based on micromechanical considerations. Int. J. of Solids and Structure 9 (38): 1625-1639. 7. Stupkiewicz, S., and Mr6z, Z. (1999). A model of third-body abrasive friction and wear in hot metal forming. Wear 231: 124-138. 8. Zmitrowicz, A. (1987). A thermodynamical model of contact, friction and wear: I. Governing equations. Wear 114: 135-168. 9. Dragon-Louiset, M., and Stolz, C. (1999). Approche thermodynamique des ph~nom~nes d'usure de contact. C.R.A.S. Paris 327: 1275-1280. 10. Archard, J. E (1953). Contact and rubbing of fiat surfaces. J. Appl. Physics 24 (8): 981-988. 11. Dragon-Louiset, M., Bui, H. D., and Stolz, C. (2000). On Prandtl's lifting equation arising in wear mechanics. Archives of Mechanics 52: 547-567. 12. Stolz, C., and Pradeilles-Duval, R. M. (1996). Approches ~nerg~tiques de la propagation dynamique de discontinuit~ m~canique. C.R.A.S. Paris 322 (IIb): 525-532.
SECTION
8.6
Constitutive Models and Numerical Methods for Frictional Contact MICHEL RAOUS Laboratoire de MF,canique et d'Acoustique, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France
Contents 8.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Models and Identification of the Parameters 8.6.2.1 Unilateral Contact . . . . . . . . . . . . . . . . . . . 8.6.2.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3.1 Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3.2 Algorithms and Numerical Parameters . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
777 778 778 779 782 782 784 786
8.6.1 VALIDITY The following models are relevant for dealing with unilateral contact, which means that no penetration into the obstacle can occur and the separation is free (the effective contact area depends on the loading), and friction, which means that a frictional force occurs on the parts of the b o u n d a r y where contact has been established. The formulations and algorithms are given with the following assumptions: 9 The contact occurs between a deformable body and a rigid obstacle (an extension to the case where contact occurs between two deformable bodies is given as a remark). Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
777
778 9 9 9 9 9
Raous The deformations are small. The solid is elastic. The problem is quasi-static. No adhesion is involved. No physico-chemical forces are involved in the contact.
Extensions of these models to finite deformation, nonlinear behavior, dynamic problems, and coupling with adhesion can be found in Reference [1] and in the references given therein.
8.6.2 MODELS A N D IDENTIFICATION OF THE PARAMETERS
8.6.2.1 UNILATERALCONTACT 8.6.2.1.1 Model CI: Strict Unilateral Contact (Signorini Conditions)
UN _<0 / FN
_ 0
(i)
MNFN -- 0
where F, the unknown contact force, and u, the displacement defined on the part on the boundary where the contact is liable to occur, are decomposed into their normal and tangential components (n is the outward unit vector to the boundary): u = UN n + u r , F = FN n + F t . The complementarity relations (Eq. 1) satisfactorily describe unilateral contact with the two possible conditions: either there is no contact, when UN < 0 and FN 0, or contact occurs, when UN = 0 and FN ~_ O. This is a nonsmooth problem, and the complementarity problem (Eq. 1) implies that FN is a multivalued application of UN (see the graph in Fig. 8.6.1). No parameters are required here. Comment 1: In the case where an initial gap between the solid and the obstacle has to be taken into account, UN has to be replaced by ( u n - d ) where the function d, defined on the boundary, is the initial distance between the solid and the obstacle. When the obstacle moves, the function d is also used to define its movement. Comment 2: For dealing with the contact between two deformable bodies, in the case of small deformations, a point-to-point relationship can be used, making a simple change of variables: if A (solid 1) and B (solid 2) are two opposite points, let uA - UA and u s - UA--UB. =
779
8.6 Constitutive Models and Numerical Methods for Frictional Contact
FN
FU
v
UN uN
FIGURE 8.6.1
Graph of the Signorini law.
FIGURE 8.6.2
Graph of the compliance law.
8.6.2.1.2 Model CII" Compliance This model allows a penetration which is controlled by a strong reacting force as follows: FN - - - - C n ( U N ) + "
where (UN)+ is zero when UN < 0 (no contact), and (Uu)+ = Uu when UN >_ 0. This problem is a much more regular one, because FN is now a function of UN. Its graph is given in Figure 8.6.2. This model is simpler for both computational and mathematical reasons. However, some penetration, depending on the choice of parameters C, and m,, occurs. Identification of Cn and m.: Strict identification of C. and m~ is not possible. The choice of these parameters is often based on computational reasons. Mechanical estimates can be obtained by considering the penetration tolerance relative to the current normal pressure. Analytical or computational considerations on the behavior of surface asperities can be helpful.
8.6.2.2 FRICTION The general form of a friction law can be written
LIv ll _
]
(3)
if []Fr][ -- f then fir is colinear and opposite to Fr where various choices of the sliding threshold f can be made. The corresponding laws are presented on Figure 8.6.4.
780
gao~s
8.6.2.2.1 Model FI: Coulomb Law, f =/~[FN[ This is the most classical friction law (# is the friction coefficient). The graph is presented in Figure 8.6.3. Identification of the friction coefficient #: This is a very difficult problem, first, because this coefficient characterizes a pair of materials, and second, because it depends strongly on various conditions (the surface machining condition, the temperature, the hygrometry, etc). Tables of friction coefficients are therefore rarely given. As a qualitative indication, it can be said, for example, that the range of magnitude is about 0.1 to 0.3 for a metal-metal contact and may be larger than 1 when rubber is involved. Specific measurements can be performed on a sample of the pair of materials using a tribometer. Whenever possible, it is recommended to identify the friction coefficient on the structure itself by adjusting it by comparing a finite element calculation with the experimental results obtained under elementary loading (see Reference [ 1]).
8.6.2.2.2 Model FII: Tresca Law, f = g The function g is given and defined on the contact boundary. In this case, the sliding limit does not depend on the normal force. This model cannot be directly coupled with unilateral contact. It can be convenient when the normal force is known (in the case of polymer injection, for example). It is also used as an intermediate problem in some iterative algorithms dealing with Coulomb friction. Identification of the function g: When the normal pressure on the contact boundary is known, g can be obtained by multiplying it by/z; otherwise, some
Fr
UT
FIGURE 8.6.3 Graph of the Coulomb law (in two dimensions).
8.6 Constitutive Models and Numerical Methods for Frictional Contact
781
iterative procedure has to be used to determine an appropriate value of the function g on each contact point.
8.6.2.2.3 Model FIII: Variants
8.6.2.2.3.1 Coulomb-Orowan Law, f -
min(/~lFN], k)
The parameter k is usually the elastic limit of the material. It can be used when very high pressures are involved.
8.6.2.2.3.2 Shaw Law, f = ak The function e is defined by e = FR/Fc where FR is the nominal (effective) contact surface depending on the flattening of the asperities, and Fc the whole surface where contact is liable to occur. This function, depending on the normal compression, has to be determined either analytically or numerically.
8.6.2.2.4 Model FIV: Models with a Variable Friction Coefficient A dependence of the friction coefficient on the sliding velocity (the most classical), the normal pressure, the temperature, or other quantities can be introduced. The Stribeck law (see "Friction in Lubricated Contacts" by J. Frene and T. Cicone in this volume) is an example of this dependence, which can be used to account for the transition between the dry friction and hydrodynamic regime. Note: However attractive this kind of model may be, one must not forget that it involves a loss of uniqueness of the solution (even for small friction coefficients) and a lack of existence results. This can lead to serious problems as far as the reliability of the numerical results is concerned.
8.6.2.2.5 Model FV: Regularized Coulomb Law A regularized formulation of the Coulomb law is obtained when F; is given as a function of ft; (instead of the multivalued application), FT = --pcp~(ft;)[FN]. For two-dimensional problems, square root, hyperbolic tangent, or piecewise polynomial functions are often used. In these cases, a regularization parameter has to be chosen, and it has to be noted that sliding always occurs (large or very small sliding when it approximates the stick condition). The strict stick situation occurs only when FT -- 0.
782
gaous FT
FIGURE 8.6.4
Coulomb
FN Various friction laws (in two dimensions).
8.6.2.2.6 Model FVI: Compliance Model As in the case of unilateral contact, a compliance law is introduced for the friction law:
IIF~II < C,(u~)+t and if [[F~-[I< Ct(UN) mt + then ur - 0 if
IIFTII
-
'l /
(4)
Ct(RN)~ t then f~T is colinear and opposite to F~-
where Ct and mt are two parameters. The case where Ct =/.zCn and m t = m n corresponds to the classical Coulomb law associated with the compliance model for unilateral contact. The same difficulties arise here, as for Cn and ran, for identifying Ct and rot.
8.6.3 NUMERICAL METHODS 8.6.3.1
FORMULATIONS
Both the displacements (unilateral conditions) and the velocities (friction law) are involved. When a displacement formulation is used, an incremental formulation has to be introduced. At each step, a "static" problem will be solved, involving extra terms computed at the previous step and characterizing the memory of the previous loading history. This section focuses on the resolution of this associated "static" problem. The first three types of formulation involve models based on the strict contact relationships (models CI, FI, FII, FIII, FIV), and the fourth type involves models based on the regularized contact relationships (models CII, FV, FVI).
8.6 Constitutive Models and Numerical Methods for Frictional Contact
783
8.6.3.1.1 An Implicit Variational Inequality Eliminating the contact forces leads to an implicit variational inequality. For example, a Signorini problem with Coulomb friction can be written as problem P1.
Problem PI: For each time tk+l, find R k+l C K such that: a(ll k+l, W-ll k+l ) + j ( t l k+l, w - - R k ) - - j ( l l k+l, llk+l--ll k) ~ L k + l ( w - ll k+l )
Vw
K
(5)
where a(., .) is a bilinear form associated with the elasticity mapping, L(.) is a linear form associated with the current loading, the nondifferentiable functional j(v,w) (associated with the friction) is j ( u , v ) = fr'c la FN(U)[ [[vr[[ds, and the convex K characterizes the unilateral conditions K = {v C U/VN <_ O o n F c } where Fc is the part of the boundary where contact is liable to occur.
8.6.3.1.2 A Complementarity Problem Another approach consists in writing the problem in the form of a linear complementarity problem, introducing two new variables by separating the tangential displacement into left and right sliding parts. It is then written, after finite element method (FEM) discretization and condensation in reducing the problem to the contact variables (partial inversion of the linear parts), as problem P2. P r o b l e m P2: Find F C Re, u E RP such that
Mu=F* +F
i= 1...p i -- 1 . . . p
Fi ~ 0, ui ~ 0 Filai --
0
I
(6)
where M and F* are, respectively, a nonsymmetric matrix and a loading vector deduced from the FEM problem by condensation (taking into account the change of variables associated with the choice of new variables), F and u are the contact forces and the contact displacements, and p is the number of contact degrees of freedom. This two-dimensional formulation can be extended to three-dimensional problems using a polygonalization of the Coulomb cone.
784
gaols
8.6.3.1.3 Lagrange Multipliers Another alternative is based, as in the previous formulation, on keeping the contact forces F as variables (Lagrange multipliers). This time, the unilateral conditions are kept in the form of Eq. i but the friction conditions are written under the Kuhn-Tucker conditions (analogous to plasticity). This is written as problem P3 where K* is the condensed regular FEM matrix. Problem P3: Find F E RP, A~ E RP such that K*u =
F* + F
llN < 0, FN ~ 0, UNFN - - 0
fs < O, AUT- -A~ cgfs/OFT, A~ >_ O, A~ fs = 0 -
I IF II -
(7)
,IF I
8.6.3.1.4 A Smooth Nonlinear Formulation When compliance, regularized models, or penalty techniques are used, nonlinear variational equalities are obtained. Penalty can be regarded as a numerical form of the compliance laws previously introduced. The problem is then written in the generic following form. Problem P4: Find u E RP such that K * . = F* +
(8)
+
where ~)N(1,IN) is given either by the compliance law (Eq. 2) or by penalty (for example, e[UN]+), and ~bT(UT)will be adjusted during the iterations in order to satisfy the Coulomb law.
8.6.3.2
ALGORITHMS AND NUMERICAL PARAMETERS
To solve problem P1, a fixed point procedure can be introduced to determine the tangential friction force FT: then, at each step n + 1, we have to solve a Tresca problem with a given sliding threshold gn which can be written as the following minimization problem Plbis. The solution of P1 is obtained when the process g,+l = #[FN(u")] has converged. Problem Plbis: Find un such that J(u n) < J(v)Vv E K with
J(v) = 1/2a(v, v) - L(v) + f r g"llv:rilds C
(9)
8.6 Constitutive Models and Numerical Methods for Frictional Contact
785
This minimization problem under constraints (Vv E K) of a nondifferentiable functional can be solved using various algorithms. 8.6.3.2.10verrelaxation Method (SSOR) with Projection (Problem Plbis) An optimum relaxation parameter has to be determined by carrying out a trial procedure. This method is very robust, but it can be costly when extended to nonlinear problems. 8.6.3.2.2 A Gauss-Seidel Method Accelerated with an Aitken Procedure (Problem Plbis) No numerical parameters are required here. The same comments as for Section 8.6.3.2.1 can be made. 8.6.3.2.3 Conjugate Gradient Method with Projection and Regularization (Problem Plbis) A special procedure is used in this case to preserve the conjugation relationship between the descent directions. To compute the gradients, it is necessary to regularize the friction law, which introduces numerical parameters (see model FV). A specific preconditioning procedure has to be used. A regularization parameter is required here. 8.6.3.2.4 Lemke Method (Problem P2) The complementarity problem P2 is solved using a mathematical programming method such as the Lemke method. This is a direct mathematical programming method based on pivoting techniques similar to the Simplex method. It is a very powerful method. No special parameters are needed here. 8.6.3.2.5 Augmented Lagrangian Method with Radial Return (Problem P3) Problem P3 is solved with an augmented Lagrangian procedure which introduces a penalty by defining /:N =
-
E
uN)
786
aaous
A Newton-Raphson algorithm is associated with a radial return procedure. This method is similar to a procedure used in plasticity in finite deformations. It is a very powerful method, the accuracy (as well the computational cost) depends on the choice of ~1- and ~N and on the number of augmentations performed. 8.6.3.2.6 Newton-Raphson Method for the Penalized Problem (Problem P4) Problem P4 is a classical nonlinear problem in which the normal contact force is directly computed during the resolution and the friction force is adjusted during the iterations in order to verify the Coulomb law. The choice of the penalty parameters (or of the parameters of the compliance law) is delicate because it strongly affects the accuracy of the solution (the penetration can be large) and the computational time. These parameters are usually made to gradually decrease during the iterations. The accuracy of the solution always has to be checked. 8.6.3.2.7 Iterative Procedure on Boundary Conditions A simple procedure which is sometimes used is that based on iterations on adjusting boundary conditions in order to satisfy the unilateral and the friction conditions: at each iterate, UN is set to zero when penetration occurs, fi~- is first assumed to be zero, and when [[F~-[[ > #[FN[ the previous condition is released but a friction force is applied, which is colinear and opposite to ~and such that [[F~[[ = #[FN[. Despite the lack of convergence results, this procedure can be an efficient way of dealing with simple linear problems. In other cases, the procedure may not converge.
REFERENCES 1. Raous, M. (1999). Quasistatic Signorini problem with Coulomb friction and coupling to adhesion, chapter 3 in New Developments in Contact Problems, pp. 101-178, Wriggers, P., and Panagiotopoulos, P., eds., CISMCourses and Lectures, 384, New York-Wien: SpringerVerlag.
SECTION
8.7
Physical Models of Wear Prediction of Wear Modes Koj~ I~TO Tohoku University, Aramaki-Aza-Aoba 01, Sendai 980-8579, Japan
Contents 8.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 8.7.4 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 8.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7.1
787 787 788 791 791 791
VALIDITY
This article describes a ceramic wear mode map in order to predict two wear modes in unlubricated self-mated sliding of ceramic.
8.7.2
BACKGROUND
Specific wear a m o u n t ws ( m m 3 / N m ) and friction coefficient/~ are observed in unlubricated self-mated sliding of A1203, ZrO2, and SiC. Two types of wear surfaces are recognized: In type I, wear generates a smooth surface, as shown in Figures 8.7.1 and 8.7.2b. In type II, wear generates a rough wear surface, as shown in Figures 8.7.1 and 8.7.2c. The wear surface of type I is observed when Ws is smaller than about 10 . 6 m m - 3 / N m , and the wear surface of type II is observed when ws is larger than about 10 .6 m m 3 / N m , as shown in Figure 8.7.3. So called mild wear corresponds to the wear of type I, and severe wear of type II is generated by Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
787
788
Kato
E =
E
6 5
I
rr
(/) W
2
~o ~. E
.E X m :S FIGURE 8.7.1
AI208 Pin/AI203 Dis i Temperature T=20oC Unlubricated condiUon
W = 3 N, v = 0.1 m/s 9 W = 3 N, v = 2.7 nYs
O
4 2
0
Type II Type I
~
0
, I , , , I . . . . .
2 4 6 e 10 Numberof friction cycles N, 1104 cycles
i
Representative surface roughness changes of alumina with number of friction cycles.
(a)
2gm~ 100gm
Z
k,~ (c)
FIGURE 8.7.2 Wearsurface profiles, a. Initial surface, b. Smooth wear surface, c. Rough wear surface. A1203/A1203.
having surface crack propagation in a brittle manner. In general, mild wear is considered to be a practically useful wear mode without lubrication.
8.7.3 DESCRIPTION OF THE MODEL It is supposed that a surface crack propagates when the following equation is satisfied: ~O'maxX/~ ~ KIc
(1)
789
8.7 Physical Models o f Wear
E
1 0 "1
" 10 ~
Type I
AI203 ZrO2
O
A
mc
1O's
.
[]
~"
Ceramics
9
Unlubricated sliding condition ~mAm~. A
9
_ ~ ~ ~
o h~
"~ 1 0 . 9 - , , , Or) 0
Type II
0~
o
I , , ,
0.2
9
9
Aj~P'Ou O0_e
O-
|
o
I ~ , , I ~,, I,,,I,,, 0.4 0.6 0.8 1
1.2
Friction coefficient I~ FIGURE 8.7.3 Distribution of specific wear amount unlubricated siding of ceramics against themselves.
ws and friction coefficient # in
where d is the preexisting crack length, O'max the maximum tensile stress at the tip of the crack, KIC the fracture toughness of the material, and fl a constant. O'max is induced mechanically by contact load and friction, and thermally by frictional heating. Mechanically induced O'ma x given by 1-2v O'max -- Pmax
3
4-by Av
8
/ ~
(2)
where Pmax is the maximum Hertzian contact pressure, # the friction coefficient, and v the Poisson's ratio. Equations 1 and 2 give the following equation for a crack propagation:
Scm = (X-[- XOlA)PmaxX/rd KIc _>
(3)
where v is supposed as 0.25. The parameter So,m, shows the mechanical severity of contact. Thermally induced O'ma x is given by E~ eft=l_ v
(4)
where E is Young's modulus, ~ a coefficient of thermal expansion, v the Poisson's ration, and AT the temperature difference due to the heat pulse.
V~to
790
By considering the friction heating at asperity contacts, AT is expressed as follows:
/VWHv
V
(5)
where V is the sliding velocity, W the load, Hv the hardness, 7 the heat partition ratio, /2 the friction coefficient, k the thermal conductivity, p the density, c the specific heat, and C1 a proportional constant. On the other hand, thermal shock resistance A Ts is given experimentally as follows: AT, =
1 (1-v)Kic
(6)
By introducing Eqs. 4-6 into Eq. 1, one obtains the following expression for a crack propagation:
Sc,t = - ~
kpc
- ~1
(7)
The parameters Sc,t shows the thermal severity of contact. Experimental results observed in Figure 8.7.3 give Figure 8.7.4, where mild wear is at Sc,m<6, Sc,t
FIGURE 8.7.4 Wearmap of ceramics for mild wear and severe wear regimes.
791
8.7 Physical Models of Wear
TABLE 8.7.1
Table of Parameters
Material (test no.)
A1203 (1-3)
A1203 (4)
ZrO2 (5)
SiC (6)
Mean grain size Dg, /~m Vickers hardness Hv, GPa Fracture toughness KIc, MPam 1/2 Young's modulus E, GPa Poisson's ratio v Density, 103 k g / m 3 Thermal expansion ~, x 10 -6 K -1 Thermal conductivity 2, W(mK) Specific heat c, J/kgK Thermal shock resistance ATs, K
5 16.5 3.5 390 0.24 3.9 7.8 29.3 795 200
5 16.6 3.5 345 0.23 3.9 8.0 29.0 586 200
1 14.5 7.4 220 0.3 6.1 9.2 2.9 460 280
6 31.4 3.5 400 0.24 3.1 3.9 80.0 795 500
8.7.4
TABLE O F P A R A M E T E R S
See Table 8.7.1.
8.7.5
HOW TO USE THE MODEL
For the case of unlubricated self-mated sliding of ceramics, the possible wear mode of mild wear or severe wear can be predicted by determining the values of Sc,m and Sc,t.
BIBLIOGRAPHY 1. Adachi, K., Kato, K., and Chen, N. (1997). Wear map of ceramic. Wear 203-204: 291-301.
This Page Intentionally Left Blank
CHAPTER
Multiphysics Coupled Behavior
9
This Page Intentionally Left Blank
SECTION
9.1
Introduction to Coupled Behaviors JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
All previous chapters, more or less, have dealt with one physical phenomenon even if several micromechanisms are involved, but in practice sometimes two or more phenomena may occur simultaneously with interactions between them. It is this coupling which is the subject of Chapter 9. For example, the cooling of a steel from high temperature may induce a phase transformation, which induces heterogeneous volume change, which induces internal stresses, which induce plastic strains, which induce damage, which may induce cracks, etc. Unfortunately, this is what can happen in welded structures: the damage changes the elastic strains, which change the stresses, which change the phase transformation. The thermodynamics of irreversible processes is intensively used to build models of coupled phenomena because if internal variables for each phenomenon may be qualitatively identified, the coupling is contained directly in the state and dissipative potentials, provided their quantitative identification can be performed. The state potential ~ is a function of all the states variables vi, ~t = I p ( y l , Y 2 , Y 3 , . . . , y i )
from which the associated variables (or thermodynamical forces Ai) derived as
are
A i ~OYi
There is state coupling between the phenomena i and j if 02~p
OviOvj
r
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
795
796
Lemaitre
and uncoupling if
02 q~ ~ = 0 Ov~Ovj The dissipative potential is a function of the associate variables
q) - q)(A1,A2,A3, . . . ,Ai) from which the kinetic evolution laws are derived according to Dq0 vi =
DAi
There is kinetic coupling between phenomena i and j if DZq)
DAiDAj
#0
The difficulty is, of course, to choose the proper variables Vi to choose the form of the function ~, and to choose the form of the function q0. The various models differ essentially in the choices made for vi, ~, and q~. Damage induces an elastic and plastic softening effect which must be taken into account for precise calculation of metal forming or the limit state of ductility and fatigue. See Section 9.2 for isotropic damage, Sections 9.3 and 9.4 for anisotropic damage, and Sections 9.5 and 9.6 for the additional effect of temperature. Coupling oxidation with viscoplasticity is described in Section 9.7, and interaction with hydrogen may be found in Sections 9.8 and 9.9. Section 9.10 is a general description of disturbed state in a hierarchical framework. Metallurgical couplings such as phase transformations inducing plasticity in metallic materials are important in many industrial processes such as quenching, welding, casting, surface treatments. Basic aspects suitable for numerical simulations may be found in Sections 9.11, 9.12, 9.13, and 9.14. Shape memory properties of some alloys are the result of another coupling between martensite and austenite exchanges (see Section 9.15). Modeling the coupling between elasticity and magnetism needs a good representation of hysteresis behavior (see Section 9.16). Aging is also an important phenomenon to take into consideration for long time range structures because it modifies the strength of materials, in particular polymers (see Section 9.17).
SECTION
9.2
Elastoplasticity and Viscoplasticity Coupled with Damage AHMED BENALLAL Laboratoire de M~canique et Technologie, ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du PrEsident Wilson, 94235 Cachan, France
Contents 9.2.1
Application ................................
9.2.2 Background
...............................
797 797
9.2.3 Description of the Model ...................
799
9.2.4 Identification ..............................
800
9 . 2 . 5 H o w to U s e t h e M o d e l . . . . . . . . . . . . . . . . . . . . .
800
References .......................................
801
9.2.1
APPLICATION
The model to be described herein was developed mainly for metals and metallic alloys at room or high temperatures. It can model their behaviors, progressive degradation, and final rupture along monotonic or cyclic loadings. It can also be used to analyze ductile rupture, creep rupture, and low-cycle fatigue.
9.2:2
BACKGROUND
Within continuum damage mechanics and in the case of isotropic damage, a scalar variable D is introduced which measures the average effects of the Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
797
798
BenaUal
degradation properties of the material on its mechanical response. In the context of continuum thermodynamics, this is an internal variable which completes the set of variables describing the behavior of the material. In the case of elastic-plastic or elastic-viscoplastic materials, the other main variables are the plastic strain, the accumulated plastic strain p describing isotropic hardening, and the internal strain a dealing with kinematic hardening effects. The coupling between the deformation behavior and the degradation of the material is usually undertaken by the effective stress concept rY = a / 1 - D associated with the strain equivalence principle. The total strain ~ is partitioned into an elastic part and an inelastic part. The free energy potential is of the form 1
tp _ ~p(1 -
1
D)Eijhl(ekl- ~ l ) ( e k ! - e~l) +-~Caijaij + g(p)
with E/jkl being the matrix of elastic constants. In the case of isotropy we have
E/jhl -- (K-2G)5/jc~hl
q-G(bihbjl "]-t~il(~kj)
where K and G are the bulk and shear moduli, respectively. This leads to the state laws .
.
.
.
e
-
+
cO~ up
-
4)
-d(p)
cgtP x~j - - p Oa~j -Ca~j
Y---p-ff-~-- We R, X, and Y are the forces associated with the internal variables p, a, and D, respectively. R is the size of the elastic domain, X is the back stress, and Y is the elastic strain energy. A very common form for g(p) is
g(p) = Q[1 - exp(-bp)] where Q and b are material-dependent parameters.
799
9.2 Elastoplasticity and Viscoplasticity Coupled with Damage
9 . 2 . 3 D E S C R I P T I O N OF T H E M O D E L Beside these ingredients, the model is based on a yield domain defined by a yield function: f(F, X, R ) = J 2 ( F - X ) - R - a y where J2(e - x) - x / ~ ( g i j - xij)(gij - x/j) is the second invariant of the active stress ~ - X. S is h ~ t stress deviator, and Cry the initial yield stress in uniaxial tension. The model is also based on an inelastic potential XijXij ~(~, x, R) - J ~ ( ~
- x) +
4~
y2 + 2-~ - R - Oy
involving the material-dependent parameters S and y and leading to the evolution laws for the internal variables .p _ ~
OF
e ~j
O a ~j = 2 ( 1 -
3
.
Sij--Xij
D ) P J - ~ d - - TX)
OF p-i-~-i OF
oqF
Y
The inelastic multiplier 2 is obtained by the consistency condition f - 0 for rate-independent elastic-plastic material. For rate-dependent elastic-viscoplastic materials, it is given by the viscosity law i - l(,(f)) // where ~/ is a material parameter and 9 a positive increasing function of f. Currently used forms for 9 are the power law and the exponential formulae, given respectively by ~(f)-(f)",
*(f)-exp(
f-
1)
Here K and n are also material-dependent parameters. The model is completed by 9 a local initiation criterion often defined by a damage threshold in terms of a critical accumulated plastic strain PD, i.e., such that D = 0 if p _
800
Benallal
9 a rupture criterion often defined by critical value Dc for the damage parameter.
9.2.4 IDENTIFICATION The identification procedure is dependent on the rate-dependent or rateindependent character of the material. For both types of materials, elastic and hardening properties are first obtained during the first stages of tests, when no damage is present. These are generally obtained by cyclic tests under strain control. For proportional loadings, tension and compression tests are usually sufficient. For rate-independent materials, the unloading branches of the stress-strain loops define the elastic domain. Its center represents exactly the back stress X (in the loading direction), and its size gives the amount of istropic hardening through 2(R + Oy). Experimental measures of the position of the center and its size and their plots versus the plastic strain or the cumulated plastic strain allow us to obtain the hardening parameters C, y, and the function g(p). For rate-dependent materials, the unloading branches of the stress-strain response also include the viscous stress. Therefore, before carrying out the previous procedure, one must first identify viscous properties. This is usually done by carrying out tests at different strain rates or more simply by ending the cyclic tests after saturation of the hardening by a relaxation period. Finally, obtention of the degradation properties is obtained by measuring the evolution of Young's modulus after damage has started. The decrease of this modulus gives the damage parameter and its plot versus the plastic or cumulated plastic strain and allows us to obtain the parameter S. For a general procedure identification, see Reference [3].
9 . 2 . 5 H O W T O USE T H E M O D E L The model has been implemented both in its rate-independent and ratedependent forms in ABAQUS through the UMAT subroutine. Generalized integration schemes have been used together with Newton-Raphson procedures. The corresponding consistent tangent operators were defined. For the rate-independent case, the onset of localization into planar bands (or loss of ellipticity) is signaled in order to avoid mesh dependency. The loss of the ellipticity criterion is implemented. Also, gradient and nonlocal procedures are being studied.
9.2 Elastoplasticity and Viscoplasticity Coupled with Damage
REFERENCES 1. Lemaitre, J. (1994). A Course on Damage Mechanics, Springer Verlag. 2. Lemaitre, J., and Chaboche, J. L. (1989). Mechanics of Solid Materials, Cambridge. 3. Benallal, A. Thermoviscoplasticit~ et endommagement des structures.
801
SECTION
9.3
A Fully Coupled Anisotropic Elastoplastic Damage Model SERGE C E S C O T T O 1
WAUTERS MICHAEL 2
ANNE-MARIE HABRAKENI~ and Y. ZHU3 1 University of Liege, Liege, Belgium 2 MSM_I, chemin des Chevreuils B52/3 4000 Liege, Belgium 3ANSYS Inc., Houston, Texas
Contents 9.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . 9.3.2.2 Thermodynamic Frame . . . . . . . . . . . . . . 9.3.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3.1 Anisotropic Elasticity and Damage . . . . 9.3.3.2 Anisotropic Plastic Yield Surface . . . . . . 9.3.3.3 Damage Evolution Law and Damage Surface . . . . . . . . . . . . . . . . . . . . . 9.3.4 Calibration of the Model . . . . . . . . . . . . . . . . . . . 9.3.4.1 Elastic Parameters . . . . . . . . . . . . . . . . . . . 9.3.4.2 Plastic Parameters . . . . . . . . . . . . . . . . . . . 9.3.4.3 Damage Parameters . . . . . . . . . . . . . . . . . . 9.3.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
802 803 803 804 805 805 805 807 808 808 809 809 810 813
9.3.1 VALIDITY This m o d e l p r e d i c t s the d a m a g e g r o w t h a n d fracture a p p e a r a n c e in ductile materials. Initially p r o p o s e d by Z h u a n d Cescotto [3], it has b e e n d e v e l o p e d in the case of sheets, especially for d e e p d r a w i n g p r o c e s s e s [2].
802
Handbookof MaterialsBehaviorModels. ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
803
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model
Important characteristics of this macroscopic model meter identification and the anisotropic evolution of plastic surfaces computed from the energy equivalence drawing simulations, described in Zhu and Cescotto validity.
are its easy parathe damage and assumption. Deep [3], confirm its
9.3.2 BACKGROUND
9.3.2.1 BAsic CONCEPTS This damage model is included in the continuum theory of damage. The damage in the material is represented by a variable D corresponding to an average material degradation affecting stiffness, strength, and anisotropy. It reflects various types of damage at the microscale level, such as nucleation, growth and coalescence of voids, and microcracks. In the present model, D is a vector of three components, the damage in each orthotropic direction of the sheet: _D--(D1
D2
D3) 7
The well-known concept of effective stress is used:
with the "damage effect" tensor M(D) of fourth order defined by its diagonal: . 1 1 1 1 m_M(D)--diag 1 - D, 1 - D2 1 - D3 V/(1 - D2)(1 -- D3)
1
]
•
V/(1-D1)(1-D2)
The principle of energy equivalence is taken into account. It states that the complementary elastic energy stored in the damaged material has the same form as the one for a fictitious undamaged material except that the true stress tensor is replaced by the effective stress tensor. This principle is able to take into account not only the apparent Young moduli decrease but also the Poisson's coefficients decrease.
P)=
p)
804 9.3.2.2
Cescotto et al THERMODYNAMIC FRAME
The Helmholtz free energy takes the following form:
p~(_~, ~, _D, ~, fl)- w~(~_~, ~, _D)+ ~p(~, ~)+ ~d(;, fl) where We(ee, T, D_) is the elastic strain energy, T is the temperature, ~p(T, ~) is the free energy due to the plastic hardening, e_ is the total strain tensor, @d(T, fl) is the free energy due to the damage "hardening," e~ is the elastic part of the Strain tensor, a is an internal variable representing the cumulated plastic strain, and fl is an internal variable representing the cumulated damage. Since the thermodynamic force _a (Cauchy or true stress tensor) is associated with the elastic strain ee, a thermodynamic force y can be associated with the damage tensor _D:
Y-
a~,
aw~(~_~,~, _D)
POD =
OD
aW~(~_,~, _D) = -
OD
y is called the damage energy release rate; the negative sign of y corresponds to the energy restitution due to damage growth. The forces associated with the cumulated plastic strain 0~and cumulated damage fl are, respectively, R and B, called the plastic hardening threshold and the damage strengthening threshold. With the hypothesis of uncoupling between mechanical plastic and damage dissipations, the second law of thermodynamics yields for an isothermal process: a_. " ~.p - Rd~ ~ O
-
Y [)-
B fi > O
This induces the existence of a plastic dissipative potential and a damage dissipative potential, chosen in this associated theory frame as the plastic yield criterion and the damage evolution criterion. Fp(a_, D_, R ) -
0
Fd(Y, B) -- 0
By introducing the Lagrange multipliers 2p and ~d, we define the 9 function: -- cr " ~,p - R ~ -
Y I ) - B f i - ,~pFp - ),dFd
If the criteria Fp and Fd are satisfied, the current values of _~, R, _Y, B, and _D will make the 9 function have a stationary value. We can build the following
805
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model
evolution laws" OOo = o Oa OOo OR
-0o
~
O@ OY - o
~.p - j.pOFp Oa
-
-
, o~p
o,
9
OFd
~ O__--xdgi
=o
a
/~--~d~
OFd
9.3.3 MODEL DESCRIPTION 9.3.3.1
ANISOTROPIC ELASTICITY AND DAMAGE
When the material is damaged, the constitutive elastic law is given hereafter as m
_a - __C~_e e with __Ce the elastic stiffness matrix of the damaged material. Using the principle of energy equivalence, the following relation can be written:
-_c~ - M -~ ( p ) _ c ~ -~ (p) In the case of orthotropic materials, the damaged elastic tensor ~e -1 is recalled in Reference [2].
9.3.3.2
ANISOTROPIC PLASTIC YIELD SURFACE
In this model, the plastic yield surface is chosen as the Hill's one: rp(_~, D, R) -- ~p(_e, R) -- ~ q - Ro - R ( ~ ) -- 0
with R0 the initial elastic stress threshold and ae----~the effective anisotropic equivalent stress:
806
Cescotto et al
where 7 is the back-stress tensor and H the plastic characteristic Hill tensor for the fictitious undamaged material: m
H
G+ H
-H
-G
0
0
0
-H
H+ F
-F
0
0
0
-G
-F
F+ G
0
0
0
0
0
0
N
0
0
0
0
0
0
L
0
0
0
0
0
0
M
F, G, H, L, M, and N are parameters characterizing the current state of plastic anisotropy. For a strain-hardening material, the uniaxial stress in one direction varies with an increase of plastic strain, and therefore the anisotropic parameters should also vary, since they are a function of the current yield stress. To determine them for the current state, we consider that the plastic work should be the same in each direction. For instance, in the case of a linear workhardening material, we have (Fig. 9.3.1):
with i - 1 to 6 (three tensile curves and three shear curves) and Eti is the slope of the stress-plastic strain curve i. We build the following ratios in terms of the new equivalent stress aF" 2 ai
=
H -- al q- a2 - a3 --
G-
(\ LZ /- ~ } (a2 - a2) - - q-- -a2i -
al -a2
~yi
~
~Pli
--F a3
F---al-Fa2-4-a3
~Pleq
FIGURE 9.3.1.
Etref
F,-
N -- a4 L-
as
M--a6
807
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model
In the present model, the rolling direction (direction 1) is taken as the reference one, so a l - 1. Using the evolution laws, we can develop the following expressions:
- OFp M H___M(_r- ~) ~p = 2p-~_ r -- 2p 2-~eq
/~ _ (1
_
m)",~vb-~OR
isotropic hardening rule
f~ _ m2p( 0"- 7_) OR r162 O--d + ~-Z' - ~_'~--
Fp < 0
2p > 0
plastic flow rule
kinematic hardening rule
2pFp- 0 plastic loading-unloading rule
where m is the ratio of kinematic to isotropic hardening and 7~ is the deviatoric tensor of Z"
9 . 3 . 3 . 3 DAMAGE EVOLUTION LAW AND DAMAGE SURFACE By analogy to the plasticity, a damage criterion, chosen as a quadratic homogeneous function of the damage energy release rate Y, is proposed [1]" Fd - Y~q - Bo - B(fl) -- 0
with the equivalent damage energy release rate Yeq defined thanks to the damage charateristic tensor J: m
A suitable tensor J, simple enough to be applied and able to describe the damage growth, hs been proposed by Zhu and Cescotto [3]"
s-2
/ Jv~J3
J3 J
In the case of damage hardening materials, the equivalent damage energy release rate increases with an increase of the total damage growth. As for the H matrix components, the anisotropic parameters should also vary. Again, we suppose that for a current state of damage, the damage work done in each direction should be the same. In the case of a linear damage hardening
808
Cescotto et al
characterized by its slope Dt, we have:
_ (Yeq) 2_
Yeq Gq) -}- W02i
with i = 1 to 3 (the three principal directions of an orthotropic material). In this model, the reference direction is the rolling direction and J1 = 1. Similarly to the plastic flow, the anisotropic damage evolution laws are characterized as follows:
COFd_--J~d J-Y D_ -
- i d OY_ -
COFd
2Y~q
OB
l} - 1~~
Fd
2d >_ 0
~ -
-- ;[
- 1d ~
-
id
damage evolution rules
damage hardening rule
2dFd -- 0
damage loading-unloading rule
9.3.4 CALIBRATIONOF THE MODEL All the parameters of this model can be identified only with tensile tests. These tests are characterized by ~, the angle between the rolling direction of the sheet and the axial direction of the sample (Fig. 9.3.2).
9.3.4.1 ELASTICPARAMETERS Tensile tests are done in the domain of small displacements for the directions 0~= 0 ~ 45 ~ and 90 ~ They allow us to compute the Young's moduli and the Poisson's coefficients. For sheets, tensile tests in the thickness direction are
rolling l direction
#
axialdirection
(~
1 ~ 1/ FIGURE
transversal direction 9.3.2.
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model
809
not possible; we assume the following relationships or use values deduced from texture measures: E3=
9.3.4.2
(El + E2) 2
and
G13-G23-G12
PLASTIC PARAMETERS
We need to know the stress-strain curves in the three orthotropic directions and in the three shear planes. Uniaxial tensile tests in the domain of large displacements are done. We use a statistical method which consists of tests in the directions ~ - 0 ~ 15 ~ 30 ~ 45 ~ 60 ~ 75 ~ and 90 ~ For a given plastic work level, we minimize a functional of the corresponding stresses in these seven directions, leading to the Hill's parameters E G, H, and N ( N - M - L for sheets) [4]. It is then possible to deduce the stress-strain curves in the thickness direction and in the shear plane 1-2. The shear curves in the planes 1-3 and 2-3 are supposed to be equal to the one in the plane 1-2. The model is able to use muhilinear stress-strain curves, with the number of points chosen by the user. Obviously, the model needs to be furnished with effective stress-strain curves. Once we know the damage parameters of the material, these six effective stress-strain curves can be computed analytically from the true stress-strain curves. _ g - M(_D)._~ and
9.3.4.3
~-
M -1 (_D).g e
DAMAGE PARAMETERS
From the theory, we have the following relationship for a uniaxial test in direction i: Y,=
Ei(1 - D i ) 3
We need the three damage curves, characterized by the initial value of Yi and the slope Dti of the damage curve (hypothesis of a linear behavior). If we suppose that the damaging phenomenon begins with the entry in plasticity, we have:
El(1 - Di) 3
Ei
810
Cescotto
e t al
To find the effective curves, we compute the damage values (D1 D2 D3) associated with a given stress value. It corresponds to the resolution of the system hereafter: yi =
cr2 El(1 - Di) 3 = DtiDi + Yio
This yields a function cri(Di) presenting a maximum for a precise value of D i. If we write the expression of Crmax according to Di, we obtain the following relation: 27Ei
0"imax -- V 256Dt 3 (Dti + Yi0)2
Physically, O"i max should have the same value as the maximum stress on the real stress-strain curve. Therefore, if we know this value, knowing Yio and Ei, it is possible to determine Dti. This leads to the conclusion that, for this model, no particular damage test is necessary to find the damage parameters. This fact is simply linked to the strong hypothesis of a linear damage curve.
9.3.5 H O W TO USE THE M O D E L The model has been implemented in the nonlinear finite element code LAGAMINE developed by the MSM team; the coupled integration scheme is described in Zhu and Cescotto [3]. We only introduce the six effective stressstrain curves, the Poisson's coefficients, and the damage parameters. The predictions of the model can be illustrated by Figures 9.3.3 and 9.3.4, which describe the effect on a uniaxial tensile loading. Figure 9.3.3 shows the negative influence of the damage on the plasticity. It compares the plastic surface obtained by using the true stresses (with damage) and the one which is defined in the fictitious case of no damaging phenomenon (using effective stresses). As it can be observed, the elastic zone is reduced by the damage. Figure 9.3.4 illustrates the shape evolution of the yield locus during the deformation process. It can be seen that the anisotropic behavior of the material varies with the deformation. This anisotropy variation is intensified by the damaging phenomenon. See Table 9.3.1.
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model section of yield surfaces in the sigl-sig2 domain (SPX1250 steel) Influence of the damage (epseq = 10%)
FIGURE 9.3.3.
section of yield surfaces in the sigl-sig2 domain (SPXI 250 steel) Shape evolution with the equivalent plastic strain growth
FIGURE 9.3.4.
811
812
o~
r eq"
.B t~
.2~
t~ t~
O4
0~
~D
~-4
00 O0
tt~
eq
~D
eqeq
eq
r
~
V:
~ cr
r'~
o~
oo eq
C e s c o t t o et al
9.3 A Fully Coupled Anisotropic Elastoplastic Damage Model
813
REFERENCES 1. Cordebois, J.P., and Sidoroff, E (1979). Damage induced elastic anisotropy, EUROMECH 115, Villard de Lans. 2. Habraken, A.M., et al. (1997). Calibration and validation of an anisotropic elasto-plastic damage model for sheet metal forming, Mc NU97, Voyiadjis, ed., "Damage in Engineering Materials", Studies in Applied Mechanics, Elsevier. 3. Zhu, Y., and Cescotto, S. (1995). A fully coupled elasto-visco-plastic damage theory for anisotropic materials, Int. J. Solids Structures 32 (11): 1607-1641. 4. Noat, P., et al. (1995) Anisotropic 3-D modelling of hot rolling and plane strain compression of A1 Alloys, in Simulation of Material Processing: Theory, Methods and Applications, Shen and Dawson, eds.
SECTION
9.4
Model of Inelastic Behavior Coupled to Damage G E O R G E Z . VOYIADJIS
Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana
Contents 9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . 9.4.3 Discussion of the Results for the Elasto-Viscoplastic Damage Analysis . . . . . . . . 9.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
814 814 817 818 820
9.4.1 INTRODUCTION The coupling of damage and inelastic deformation in materials have been studied only recently by Ju [1], Johansson and Runesson [2], and Voyiadjis and Park [3]. Both Ju [1] and Johansson and Runesson [2] made use of the effective stress utilizing a scalar measure of isotropic damage. Voyiadjis and Park [3] made use of the effective configuration by invoking the kinematics of damage through the use of a second-order damage tensor. The damage mechanism of materials has been studied extensively by Lemaitre [4], Krajcinovic [5], Murakami [6], and Voyiadjis and Park [7].
9.4.2 THEORETICAL FORMULATION Damage variables can be presented through the internal state variables of thermodynamics for irreversible processes in order to describe the effects 814
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
9.4 Model of Inelastic Behavior Coupled to Damage
815
of damage. One of the most crucial aspects of continuum damage mechanics is the appropriate choice of the damage variable, since the accuracy and reliability of the developed damage model are mostly dependent on the suitable selection of the type and numbers of these variables. Damage is expressed as follows [8]: 3 k=l
Since the elasto-viscoplastic response of the damaged material is considered here, both hardening effects due to viscoplasticity and damage together with previous above description of microcrack distribution can be introduced as hidden internal state variables in the thermodynamic state potential. The form of this potential in terms of observable and internal state variables can be given as follows: ~, = ~,(~, ~r, ~, p, ~,, ,~, F)
(2)
where the p and ~ variables characterize the isotropic and kinematic hardening in plasticity and viscoplasticity, respectively, and the tr and F variables characterize, respectively, the isotropic and kinematic hardening in damage. In Eq. 2, T characterizes the temperature, and ~e is the elastic component of the strain tensor. In order to account for loading rate dependency and to regularize the localization problems, a viscous anisotropic damage mechanism needs to be implemented. Such a model accounts for retardation of the microcrack growth at higher strain rates. The rate-dependent plastic strain rate [9-11] and the damage rate are given as follows:
~p = II~pll ,ep + I1~11.~pd
(3a)
and
-II~Vpll ndVp+ II~fl nd
(3b)
Superscripts do not indicate a tensorial character but only a particular state of the variable, such as elastic (e), plastic (p), viscoplastic (vp), damage (d), viscoplastic-damage (vpd), etc. Superscripts in this work do not imply tensorial indices but only describe the type of material inelasticity. In Eq. 3, ]l~vPll and I1~11 are the magnitudes of the plastic strain rate and damage rate which can be decomposed into a product of two functions [12] using the Zener parameters such that: ]l~,vplt - ~vp(T)ZVP ~ 0
(4a)
I~11- ~d(T)Zd ~ 0
(4b)
816
Voyiadjis
The unit tensors n vp, n vpd, n avp and n a are used to identify the direction of flow of the plastic strain and damage, respectively. The dynamic potentials F vp and G d for viscoplasticity and damage given by Voyiadjis and Deliktas [2000] are used in this work. As pointed out earlier, the internal state variables are introduced in the material model to represent the true response of the material due to the variation of the microstructure when subjected to external forces. The anisotropic structure of the material is usually defined in two forms, either as material-inherited or deformation-induced. The anisotropic nature of the composite material is material-inherited anisotropy. However, at the local level its constituents are isotropic materials. Therefore, the use of a micromechanical model to analyze the composite material deals with deformation-induced anisotropy. This phenomenon is characterized in the theory by using internal variables for the hardening terms and by using the second-order tensorial form of the damage variable. The general form of the internal variables can be defined as follows [12]: Ah -- hardening-dynamic recovery - static recovery
(5)
The hardening terms represent the strengthening mechanism, and the recovery terms represent the softening mechanism. The hardening and dynamic recovery terms evolve with the deformation due to either plasticity or damage or both. The static recovery term evolves with time. The evolution equations of the internal variables for the rate-dependent behavior are described as follows: f i - 3_ HV p (kvp 2 \
~, _ Qvp
3 IlXll 2 L vp
l ~vpll dvp" ~ /
II~Pll
1 - ~-~
_ ~vpx " by p
- ovpT(r )
(6a)
(6b)
X b vp -iX[---~
(6c) n vp 9X .
d vp -
(1 - pvP) bvP + pVp
n vp
IIX I
(6d)
Similarly, the evolution equations for the hardening variables of damage can be written analogously to that of plasticity as follows: - ~
_ Qa
2 L d I1~11 d d
(K) 1 - ~-a
I1~11 - ~d~(~)
- ~dr.
(7a)
(7b)
817
9.4 Model of Inelastic Behavior Coupled to Damage
bd =
F
F
(7c) F
r - (; -
+
rl d
ilrli
(74)
where p~P defines the nonproportionality condition. In these equations, H ~p, L~p, Q~P,H a, La, and Qa are the model parameters. In a general state of deformation and damage, the effective stress tensor # is related to the Cauchy stress tensor ~ by the following linear transformation [131:
(8) where M is a fourth-order linear transformation operator called the damage effect tensor. Depending on the form used for M, it is very clear from Eq. 8 that the effective stress tensor ~ is generally nonsymmetric. One of the symmetrization methods is given by [14]. The elastoplastic stiffness for the damaged material can be obtained by using the incremental relation of Hooke's law in the effective stress space as follows: a - / ~ 9~e (9) The effective stress rate, ~-, and the elastic component of the effective strain rate, ~, in Eq. 9 can be transformed into the damage configuration by using Eq. 8 such that 6 - _ m - 1 . /~ . m - 1 . ~e (10) The final constitutive equations are given as follows [11] 6---E a" ( k - Z p'6-),
6--- ( I + E a'Zp) -1 .E a . k _ D .
(11)
where Ea represents the elastic damaged stiffness and is defined by E d -- m-1
. ~.
m -1
(12)
D in Eq. 11 represents the elastoplastic damaged stiffness [11].
9.4.3 DISCUSSION OF THE RESULTS FOR THE E L A S T O - V I S C O P L A S T I C D A M A G E ANALYSIS The special-purpose computer program DVP-CALSET, is used for the numerical simulation of uniaxial loading of laminated systems. The numerical results using the proposed model show good correlations with the experimental results on titanium metal matrix composites reinforced with
818
Voyiadjis
silicon carbide fibers for the laminate layups of (90)8s, which are tested under uniaxial tension. The proposed model is also validated by showing good
agreement of the numerical results with the experimental observations on the characteristic behavior of the metal matrix composite at different strain rates and temperatures. The computational analysis of the viscoplastic damage model is performed for the laminate systems of (90)8s at elevated temperatures of 538~ and 649~ in Figure 9.4.1 and for different strain rates in Figure 9.4.2. The viscoplastic and damage model parameters are given in Tables 9.4.1 and 9.4.2
9.4.4 C O N C L U S I O N A coupled incremental damage and plasticity theory for rate-independent and rate-dependent composite materials is presented in this work. Damage is characterized kinematically here through a second-order damage tensor for each material constituent, and its physical interpretation is also presented. This is related to the microcrack porosity and crack density within the unit E l e v a t e d T e m p e r a t u r e s (538 C & 6 4 9 C) 90_(8s) Ti-15-3/SCS6 Strain Rate=0.0001
300.0
........ ~.
.
.
.
.
.
.
.
~x~,..~t= n.u., (s;sc)
: ........
!
~ Proposed Model (649)
At- - -A Proposed Model (538C) -, Experimental Resulta(649C) a ~ .A.
JIf
9 .-,a- ~ a -
9
~v -
~"
200.0
e~
f /
= e
|
/
1
/ 100.0
0.0
~.
0.000
,
....
I ..........
0.002
,
, I ...........
,
0.004
I
0.006
,
,
0.008
Longitudinal Strain
FIGURE 9.4.1 Comparisonof viscoplastic damage model with experimental results [15] of the (90)s, layup at different elevated temperatures of 538~ and 649~
819
9.4 Model of Inelastic Behavior Coupled to Damage (0/90) T i - 1 4 A I - 2 1 N
....
0.015
i
'
I.
.
.
.
.
". . . . . . . . . .
!
' ! f i 0.010
IH strainrate:0.01(fiber)....... 9
I
9
i - - - strain rate--0.01(matrix) [ strain rate=0.001(matrix) [A .....A strain rate=0.001(fiber) t a--It strain rate=0.0001(fiber) [ -- 7:::strain rate,0:0001(matrix)
i i
A
~.,
b/SCS-6
.f
g 0.005
0.000
0.0
FIGURE 9.4.2
.
.
.
.
.
1000.0
.
.... i
.
I
2000.0 Stress(MPa)
..........
3000.0
Viscoelastic Model Parameters
Properties
T= 21~
T-- 21~
T-- 21~
(GPa) (GPa) ~ry (MPa) /_/vp (MPa) L vp (MPa) D vp (MPa) nl
400 92.4 689 5000 100 840 5.4
393 72.2 45 50,000 85 450 1.55
370 55 15.5 50,000 75 85 1.3
Ef
Em
TABLE 9.4.2
Properties
~m vm
~f vf
4000.0 |
Strain rate effect on the damage variable ~b of the 90~8s) layup.
TABLE 9.4.1
/~m
t ......
Rate-Dependent Damage Parameters
T= 538~ 0.07 0.78 0.0003 0.04 0.78 0.00018
T= 649~ 0.0047 1.2 0.00003 0.0045 0.82 0.00005
820
Voyiadjis
cell and characterizes separately the damage in each material constituent. Damage is characterized here in terms of two fourth-order tensors, the damage in the matrix and the damage in the fiber. The damage in the fiber characterizes both the internal damage in the fiber due to the cracks and voids as well as the damage due to debonding. The present formulation seems to be robust and efficient for the type of problems presented in this work.
REFERENCES 1. Ju, J. W. (1989). Energy based coupled elastoplastic damage theories: Constitutive modeling and computational aspect. Int. J. Solids Struct. 25: 803-833. 2. Johansson, M., and Runesson, K. (1977). Viscoplastic with dynamic yield surface coupled to damage. Comput. Mech. 20: 53-59. 3. Voyiadjis, G. Z., and Park, T. (1999). Kinematics description of damage for finite strain plasticity. J. Eng. Sci. 37: 803-830. 4. Lemaitre, J. (1984). How to use damage mechanics. Nucl. Eng. Design 20: 233-245. 5. Krajcinovic, D. (1983). Constitutive equations for damaging materials. J. Appl. Mech. 50: 355360. 6. Murakami, S. (1988). Mechanical modeling of material damage. J. Appl. Mech. 55: 281-286. 7. Voyiadjis, G. Z., and Park, T. (1995). Local and interfacial damage analysis of metal-matrix composites. Int. J. Appl. Mech. 33(11): 1595-1621. 8. Voyiadjis, G. Z., and Park, T. (1997). Anisotropic damage effect tensors for the symmetrization of the effective stress tensor. J. Appl. Mech. ASME 64: 106-110. 9. Perzyna, P. (1963). The constitutive equations for rate sensitive plastic materials. Appl. Math. 20: 321-332. 10. Perzyna, P. (1971). Thermodynamic theory of viscoplasticity. Adv. Appl. Mech. 11: 313-354. 11. Voyiadjis, G. Z., and Deliktas, B. (2000). A coupled anisotropic damage model for the inelastic response of composite materials. Comput. Methods Appl. Mech. Engr. 183: 159-199. 12. Freed, A. D., Chaboche, J. L., and Walker, K. P . (1991). A viscoplastic theory with thermodynamic considerations. Acta Mechanica 90:155-174. 13. Murakami, S., and Ohno, N. (1981). A continuum theory of creep and creep damage, in Creep in Structures, pp. 422-444, Ponter, A. R. S., and Hayhurst, D. R., eds., Berlin: Springer. 14. Cordebois, J. P., and Sidoroff, E (1979). Damage-induced elastic anisotropy, in Mechanics of Behavior of Anisotropy Solids~N~ Comportement Mechanique Des Solides Anisotropes, pp. 1922, Boehler, J. P., ed., MartinuesNijhoff. 15. Majumdar, B. S., and Newaz, G. M. (1992). Inelastic deformation of metal-matrix composite: Part I. Plasticity and damage mechanism. CR-189095, NASA.
SECTION
9.5
Thermo-ElastoViscoplasticity and Damage PIOTR PERZYNA
Institute of Fundamental Technological Research, Polish Academy of Sciences, Swictokrzyska 21, 00-049 Warsaw, Poland
Contents 9.5.1 Basic a s s u m p t i o n s a n d definitions . . . . . . . . . .
821
9.5.2 C o n s t i t u t i v e P o s t u l a t e s . . . . . . . . . . . . . . . . . . . .
823
9.5.3 F u n d a m e n t a l A s s u m p t i o n s . . . . . . . . . . . . . . . . .
824
9.5.4 I n t r i n s i c M i c r o d a m a g e Process . . . . . . . . . . . . .
825
9.5.5 K i n e m a t i c H a r d e n i n g . . . . . . . . . . . . . . . . . . . . . .
827
9.5.6 T h e r m o d y n a m i c R e s t r i c t i o n s a n d Rate-Type C o n s t i t u t i v e Relations . . . . . . . . . . . . . . . . . . . . .
829
9.5.7 I d e n t i f i c a t i o n P r o c e d u r e . . . . . . . . . . . . . . . . . . . .
830
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
832
9.5.1 BASIC ASSUMPTIONS A N D DEFINITIONS Let us assume that a c o n t i n u u m body is an open b o u n d e d set ~ c [~3, and let ~b : ~ ~ ~ be a C 1 configuration of ~ in ~ . The tangent of ~b is denoted F = T~b and is called the deformation gradient of ~b. Let {XA} and {x a} denote coordinate systems on ~ and ~ , respectively. Then we refer to ~ C ~3 as the reference configuration of a c o n t i n u u m body with particles X E ~ and to ~ = ~b(~) as the current configuration with points x c ~ . The matrix F(X,t)=cgc~(X,t)/cgX with respect to the
coordinate bases EA(X) and %(x)is given by F ~ ( X , t) - - ~ - x ( X ,
where a mapping x -
t)
(1)
~b(X, t) represents a motion of a body N.
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
821
Perzyna
822 We consider the local multiplicative decomposition F - F e. F p
(2)
where (Fe) -1 is the deformation gradient that releases elastically the stress on the neighborhood #5[JV'(X)] in the current configuration. Let us define the total and elastic Finger deformation tensors b--F.F
r,
b e - - F e . F d-
(3)
respectively, and the Eulerian strain tensors as follows: e - l(g-
b-l),
ee - l ( g - be-l),
(4)
where g denotes the metric tensor in the current configuration. By the definition 1 eP
e - e e -- l(be-~ - b -1)
(5)
we introduce the plastic Eulerian strain tensor. To define objective rates for vectors and tensors, we use the Lie derivative. 2 Let us define the Lie derivative of a spatial tensor field t with respect to the velocity field t) as 0 L~t -- ~b. ~-~(~b*t)
(6)
where ~b* and ~b. denote the pull-back and push-forward respectively. The rate of deformation tensor is defined as follows: d ~ - L v e ~ = ~ 1L o g - ~ ( g1a c v c l b + g c b uc la) ea | eb
operations,
(7)
where the symbol b denotes the index lowering operator and | the tensor product,
Ov a and 7~c denotes the Christoffel symbol for the general coordinate systems {xa}. The components of the spin m are given by 1.
vc
l(cgva
Ov~'~
(9)
1For a precise definition of the finite elastoplastic deformation, see Perzyna [1]. 2The algebraic and dynamic interpretations of the Lie derivative have been presented by Abraham
et al. [2]; cf. also Marsden and Hughes [3].
9.5 Thermo-Elasto-Viscoplasticityand Damage Similarly, de=L~e and
823
e~ ,
dV = L ~ e p
d=dr
(10)
p
(11)
Let ~ denote the Kirchhoff stress tensor related to the Cauchy stress tensor (r by "e -- J(r
PRef ~ P
--
(12)
where the Jacobian J is the determinant of the linear transformation F(X, t ) - (O/OX)qb(X, t), and pR4(X) and p(x, t) denote the mass density in the reference and current configuration, respectively. The Lie derivative of the Kirchhoff stress tensor T E T2(~ ') (elements of T2[~~'] are called tensors on 50, contravariant of order 2) gives 0 Lot -- #5. ~-~(~b*"r) --
{ F.~-~[F-0
1 9 (,C. ~.) F ._ I . r] Fr }
~b
-1
(13)
= + - (d + o~).~ - v. (d + r where o denotes the composition of mappings. In the coordinate system Eq. 13 reads a 0 c 1A cCdFdlS )F~ (Lo*) ab -- F~-~(F
&cab &cab c = ' Ot
+ 07
v
cbova -
.c -~x c -
ZacOrb
(14)
Ox ~
Equation (14) defines the Oldroyd rate of the Kirchhoff stress tensor (cf. Oldroyd [4]).
9.5.2 CONSTITUTIVE POSTULATES Let us assume that (i) conservation of mass, (ii) balance of momentum, (iii) balance of moment of momentum, (iv) balance of energy, and (v) entropy production inequality hold. We introduce the four fundamental postulates: 1. Existence of the free energy function. It is assumed that the free energy function is given by -- ~(e, F, O; p)
(15)
where e denotes the Eulerian strain tensor, F is the deformation gradient, 0 is the temperature, and # denotes a set of the internal state variables.
824
Perzyna
To extend the domain of the description of the material properties and, in particular to take into consideration different dissipation effects, we have to introduce the internal state variables represented by the vector/~. (2) Axiom of objectivity (spatial covariance). The constitutive structure should be invariant with respect to any diffeomorphism (any motion) ~: 5p ~ 5" (cf. Marsden and Hughes [3]). Assuming that ~ : 5p ~ 5e is a regular, orientation-preserving map transforming x into x t and T~ is an isometry from TxSP to Tx, SP, we obtain the axiom of material frame indifference (cf. Truesdell and Noll [5]). (3) The axiom of the entropy production. For any regular motion of a body ~ the constitutive functions are assumed to satisfy the reduced dissipation inequality 1
1 9 " d - (r/0 + ~) - ~--~q. grad 0 _> 0
(16)
PRef where PRef and p denote the mass density in the reference and actual configuration, respectively, 9 is the Kirchhoff stress tensor, d is the rate of deformation, q is the specific (per unit mass) entropy, and q denotes the heat flow vector field. Marsden and Hughes [3] proved that the reduced dissipation inequality (Eq. 16) is equivalent to the entropy production inequality first introduced by Coleman and Noll [6] in the form of the Clausius-Duhem inequality. In fact, the Clausius-Duhem inequality gives a statement of the second law of thermodynamics within the framework of mechanics of continuous media; cf. Duszek and Perzyna [7]. (4) The evolution equation for the internal state variable vector /l is assumed in the following form: LvB = rh(e, F, 0, ~)
(17)
where the evolution function lh has to be determined on the basis of a careful physical interpretation of a set of the internal state variables and analysis of available experimental observations. The determination of the evolution function lh (in practice a finite set of the evolution functions) appears to be the main problem of the modern constitutive modeling.
9.5.3 FUNDAMENTAL
ASSUMPTIONS
The main objective is to develop the rate-type constitutive structure for an elastic-viscoplastic material in which the effects of the plastic non-normality,
825
9.5 Thermo-Elasto-Viscoplasticity and Damage
plastic strain induced anisotropy (kinematic hardening), micro-damaged mechanism, and thermomechanical coupling are taken into consideration. To do this it is sufficient to assume a finite set of the internal state variables. For our practical purposes it is sufficient to assume that the internal state vector p has the form p = (gP, ~, ~)
(18)
where eP is the equivalent viscoplastic deformation, i.e., gP --
d p" d p I
/
dt
(19)
The symbol ~ is volume fraction porosity and takes account of microdamaged effects and 9 denotes the residual stress (the back stress) and aims at the description of the kinematic hardening effects. Let us introduce the plastic potential function f - f ( J l , J 2 , 8,p), where J1, J2 denote the first two invariants of the stress tensor ~ - 1: - 0~. Let us postulate the evolution equations as follows: dp = A P ,
4=E,
Loe=A
(20)
where for the elasto-viscoplastic model of a material we assume (cf. Perzyna [1, 8-101) A-T-~I/(I)( f-
1))
(21)
Here Tm denotes the relaxation time for mechanical disturbances, the isotropic workhardening-softening function K is K = K(gP, ~), ~)
(22)
(I) is the empirical overstress function, the bracket (.) defines the ramp function,
Of
Of
-I
and E and A denote the evolution functions which have to be determined.
9.5.4 INTRINSIC
MICRODAMAGE
PROCESS
An analysis of the experimental observations for cycle fatigue damage mechanisms at high temperature of metals performed by Sidey and Coffin [11] suggests that the intrinsic microdamage process very much depends on the strain rate effects as well as on the wave shape effects. In the tests in which
826
Perzyna
duration of extension stress was larger than duration of compression stress (in single cycle), decreasing of the fatigue lifetime was observed and the fracture mode changed from a transgranular fracture for the fast-slow wave shape, to an intergranular single-crack fracture for equal ramp rates, to interior cavitation for the slow-fast test. To take into consideration these observed time-dependent effects, it is advantageous to use the description of the intrinsic microdamage process presented by Perzyna [12, 13] and Duszek-Perzyna and Perzyna [14]. Let us assume that the intrinsic microdamage process consists of the nucleation and growth mechanism. 3physical considerations (cf. Curran et al. [16] and Perzyna [12, 13]) have shown that the nucleation of microvoids in dynamic loading processes which are characterized by a very short time duration is governed by the thermally activated mechanism. Based on this heuristic suggestion and taking into account the influence of the stress triaxiality on the nucleation mechanism, we postulate for rate-dependent plastic flow --lh
[
m* (~q)'In- l:n(~' ~q' E P ) ' - 1]
(24)
where k denotes the Boltzmann constant, h~(~,~) represents a void nucleation material function which is introduced to take account of the effect of microvoid interaction, m ~ (~) is a temperature-dependent coefficient, rn(~,~,~ p) is the porosity, temperature, and equivalent plastic strain dependent threshold stress for microvoid nucleation,
defines the stress-intensity invariant for nucleation, ai ( i - 1,2,3) are the material constants, J1 denotes the first invariant of the stress tensor ~ = ~ - 0~, and j~ and, J~ are the second and third invariants of the stress deviator -
-
For the growth mechanism we postulate (cf. Johnson [17]); Perzyna [12,13]; Perzyna and Drabik [18, 19]) 1 g* (r ~)[~g _ ~eq(r ~, EP)]
(26)
x0
where Tm xo denotes the dynamic viscosity of a material, g* (r ~) represents a void growth material function and takes account for void interaction, 3Recent experimental observation results (cf. Shockey et al. [15]) have shown that the coalescence mechanism can be treated as a nucleation and growth process on a smaller scale. This conjecture very much simplifies the description of the intrinsic microdamage process by only taking account of the nucleation and growth mechanisms.
827
9.5 Thermo-Elasto-Viscoplasticity and Damage
Zeq(~,O, eP) is the porosity, temperature, and equivalent plastic strain dependent void growth threshold stress, Ig -- blJ1 -Jr-b2
(27)
~t qt_ b3 (j~)3
defines the stress-intensity invariant for growth, and bi (i = 1,2,3) are the material constants. Finally, the evolution equation for the porosity ~ has the form 4= h
exp Tm
kO
-
~
-
,
Tm Xo
(28) This determines the evolution function ~. 9.5.5
KINEMATIC
HARDENING
For a constitutive model describing the behavior of a material under cyclic loading processes, the evolution equation plays the crucial role for the back stress ~, which is responsible for the description of the induced plastic strain anisotropy effects. We shall follow some fundamental results obtained by Duszek and Perzyna [20]. Let us postulate Lo~ -----a ( d p, ~, 0, ~)
(29)
Making use of the tensorial representation of the function A and taking into account that there is no change of 9 when ~ = 0 and d p = 0, the evolution law (Eq. 29) can be written in the form (cf. Truesdell and Noll [5]) Lo~ -- r/ldP + r/2~ + r/3dp2 + /]4~2 + r/5 (dP 9~ + ~ - d p)
(30) where q l , . . - , r/s are functions of the basic invariants of d p and ~, the porosity parameter ~, and temperature 0. A linear approximation of the general evolution law (Eq. 30) leads to the result L ~ = r/ldP + ~2~
(31)
This kinetic law represents the linear combination of the Prager and Ziegler kinematic hardening rules (cf. Prager [21] and Ziegler [22]). To determine the connection between the material functions ql and r/2, we take advantage of the geometrical relation (cf. Duszek and Perzyna [20]) (Lv~- rdP): Q = 0
(32)
828
Perzyna
where -1
+
(33)
and r denotes the new material function. Equation (32) leads to the result
(34) Finally, the kinematic hardening evolution law takes the form
(35) where FI(~, ~) --- /]I,
F2(~, ~) - - F -- /71
(36)
It is noteworthy to add that the developed procedure can be used as general approach for obtaining various particular kinematic hardening laws. As an example, let us assume that the evolution function A in Eq. 29 instead of d p and ~ depends on d p and 0~only (cf. Agah-Tehrani et al. [23]). Then, instead of Eq. 35 we obtain (37) where
~ = ~,
P:Q ~ = -u(~, ~)~ ~:Q
(38)
When the infinitesimal deformations and rate-independent response of a material are assumed and the intrinsic microdamage effects are neglected then the kinematic hardening law (Eq. 37) reduces to that proposed by Armstrong and Frederick [24]. The kinematic hardening law (Eq. 37) leads to the nonlinear stressstrain relation with the characteristic saturation effect. The material function ~1(~, ~) for ~ = ~0 and & = ~0 can be interpreted as an initial value of the kinematic hardening modulus, and the material function ~2(~, ~9) determines the character of the nonlinearity of kinematic hardening. The particular forms of the functions ~1 and ~2 have to take into account the degradation nature of the influence of the intrinsic microdamage process on the evolution of anisotropic hardening.
9.5 Thermo-Elasto-Viscoplasticityand Damage
829
9.5.6 T H E R M O D Y N A M I C R E S T R I C T I O N S AND RATE-TYPE C O N S T I T U T I V E RELATIONS Suppose the axiom of the entropy production holds. Then the constitutive assumption (Eq. 15) and the evolution equations (Eq. 20) lead to the following results - OR4 -~e' rl -
- 0---~
08'
- ~-~q. grad 8 >_ 0.
(39)
The rate of internal dissipation is determined by
~=
c9~
[0~ /2
-0--~'L~
cql~ (
~ y 3 +-~-a" riP+
r 2 P: Q'~]A - 0 ~
~:Q /
c0--~-"
(40)
Operating on the stress relation (first part of Eq. 39) with the Lie derivative and keeping the internal state vector constant, we obtain (cf. Duszek-Perzyna and Perzyna [14]) P]~-~(@(-~- 1 ) )
Lo'r,-ff'r
(41)
where
o
2 ' e - PR4 ~
eth
o
-- --PR4 0 e & 9 "
(42)
Substituting ~ into the energy balance equation and taking into account the results (third part of Eq. 39 and Eq. 40) gives p~il -
-div q +
pS"i
(43)
Operating on the entropy relation (second part of Eq. 39) with the Lie derivative and substituting the result into Eq. 43, we obtain pCp~
-
-div q + ,9 p cg'r. d
+ PZ* "r, 9d e + PZ * *
PRef O~
(44)
where the specific heat
cp -
- 8 08------~
(45)
830
Perzyna
and the irreversibility coefficients Z* and Z** are determined by Z* - - -
[ ( c 3 ~ _ 02ff)V ~ (0~) ~-'~c9,9oq@ 4- ~ - , 9
X. . . .
c32~)) ( r2P:Q ) ] 1 cgOOot " rlP4Q ~" P
- - ~ - ,9 0,9c9r
(46)
so, a set of the constitutive equations of the rate type has the form as follows 1< (f)> Lvl: -- ~ e . d - ~6vth~ _ [(~(,e + gl: + l:g)" P]T--mm ~ - 1 ,
pcp8 = - d i v q + O
h.(r Tm
exp
p 0~. d+pX*~ml < ~ ( f PRef c9~
m* (,9) II'n - "on(C,,9, eP)[ k,9
1< ( f ) > Lvot = ~ 9 - 1
-1
)>
"'P+PX
..
4,
g. -
1 + ~r---v~ [Is
-
[~1(~, 0)P - (2(~, 0)0t].
"Ceq(~,,9, sP)], (47)
All the material functions and the material constants should be identified based on available experimental data. 9.5.7 IDENTIFICATION PROCEDURE To do the proper identification procedure, we first make an assumption of the material functions (cf. Dornowski and Perzyna [25]). The plastic potential function f is assumed in the form (cf. Perzyna [26] and Shima and Oyane [27])
f--- {]2 -']- [111(0) q- 112(~)~]J-~21}1/2
(48)
where
111(~) --" 0,
112(~) = const
(49)
The isotropic workhardening-softening function ~c is postulated as (cf. Perzyna [12] and Nemes and Eftis [28]) tc = ~(eP, ~q,r = {Ks(O)- [Ks(0)- Ko(~q)]exp[-0(0)eP]} 1 -
(50)
831
9.5 Thermo-Elasto-Viscoplasticity and Damage
where ~o* ~, ~9- ~Qo a(~) - a * - a** ~, /3(~) -/3*-/3** ~, ~ = Oo The overstress function O ( f - 1] is assumed in the form ~o(~) = ~o
-
(51)
/
x
The evolution equation for the kinematic hardening parameter 0c is assumed in the form of Eq. 37 with
r (r ~) - r - el* g, r162 ~) - r - r g
(53)
The evolution equation for the porosity ~ is postulated as
where (cf. Dornowski [29]) '
~o(0)
1
-
Ig - b l J l + b 2 ~ 2 ~(r
~, ~p) - ~ ( ~ ) ( 1 - r -[~(o) Cl (8) - const,
F(~o,~,~q) =
1
~ {2~(~)
- ~o(O)]F(~o, ~, o)} c2(8) = const,
r
1_~0
As in the infinitesimal theory of elasticity, we assume linear properties of the material, i.e., o~q~ -- 2#1 + 2(g | g)
(56)
where # and ~. denote the Lam~ constants, and the thermal expansion matrix is postulated as
~th
=
(2# + 32)0g
(57)
where 0 is the thermal expansion constant. To determine the material constants assumed, we take advantage of the experimental observations presented by Chakrabarti and Spretnak [30]. They
832
Perzyna
TABLE 9.5.1 Material Constants for AISI 4340 Steel Ks* = 809 MPa 6" = 14.00, ~9o= 293 K 2 = 115.38GPa ~1 = 15.00 GPa C1 - - 0 . 2 0 2
4o = 6
x 10 - 4
Ks** = 228 MPa 6** = 3.94 IF = 0.20 0 = 12 x 1 0 - 6 K -1 ~1" = 4.22 GPa c2 = 6.7 x 10-2 ~ = 0.85
Ko = 598 MPa ~* = 9.00 PRef= 7850 kg/m 3 Tm = 2.5ms ~ = 69.60 bl = 1.00
;~ = 0
Ko* = 168 MPa ~** = 2.53 ~ = 76.92 GPa m= 1 ~'2" = 19.60 b2 = 1.30 Cp= 455J/kg K
investigated the localized fracture m o d e for tensile steel sheet s p e c i m e n s s i m u l a t i n g b o t h plane stress and plane strain processes. The material used in their s t u d y was AISI 4340 steel. The principal variable in this fiat s p e c i m e n test was the w i d t h - t o - t h i c k n e s s ratio. Variation in s p e c i m e n g e o m e t r y p r o d u c e s significant changes in stress state, directions of shear bands, a n d ductility. T h e y found that fracture p r o p a g a t e d consistently along the shear b a n d localized region. Let us n o w consider the adiabatic d y n a m i c process for a thin steel plate u n d e r the c o n d i t i o n of plane stress state. In fact, we idealize the initial b o u n d a r y value p r o b l e m investigated by C h a k r a b a r t i a n d Spretnak [30] by a s s u m i n g the velocity-driven adiabatic process for a thin steel plate. The p r o b l e m has b e e n solved by using the finite difference m e t h o d . In n u m e r i c a l calculations it is a s s u m e d that V0-
1.5m/s,
t0-501ts,
tf-800its
The material of a plate is AISI 4340 steel. Based on the best curve fitting of the e x p e r i m e n t a l results o b t a i n e d by C h a k r a b a r t i and Spretnak [30] for the stress-strain relation, the identification of the material constants has b e e n done, (Table 9.5.1). The application of the constitutive equations (Eq. 47) of a t h e r m o - e l a s t o viscoplastic m o d e l of materials for the solution of various initial b o u n d a r y value p r o b l e m s (evolution p r o b l e m s ) has b e e n recently presented; cf. L o d y g o w s k i and Perzyna [31] and D o r n o w s k i and Perzyna [25, 32].
REFERENCES 1 Perzyna, P. (1995). Interactions of elastic-viscoplastic waves and localization phenomena in solids, in Nonlinear Waves in Solids, Proc. IUTAM Symposium, August 15-20, 1993, Victoria, Canada, Wegner, L. J., and Norwood, E R., eds., ASME Book No AMR 137, pp. 114-121.
9.5 Thermo-Elasto-Viscoplasticity and Damage
833
2 Abraham, R., Marsden, J. E., and Ratiu, T. (1988). Manifolds, Tensor Analysis and Applications, Berlin: Springer. 3 Marsden, J. E., and Hughes, T. J. R. (1983). Mathematical Foundations of Elasticity Englewood Cliffs: Prentice-Hall. 4 0 l d r o y d , J. (1950). On the formulation of rheological equations of state. Proc. Roy. Soc. (London) A 200: 523-541. 5 Truesdell C., and Noll, W. (1965). The nonlinear field theories. Handbuch der Physik, Band III/ 3, pp. 1-579, Berlin: Springer. 6 Coleman, B. D., and Noll, W. (1963). The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13: 167-178. 7 Duszek, M. K., and Perzyna, P. (1991). The localization of plastic deformation in thermoplastic solids. Int. J. Solids Structures 27: 1419-1443. 8 Perzyna, P. (1963). The constitutive equations for rate sensitive plastic materials. Quart. Appl. Math. 20: 321-332. 9 Perzyna, P. (1966). Fundamental problems in viscoplasticity. Advances in Applied Mechanics 9: 343-377. 10 Perzyna, P. (1971). Thermodynamic theory of viscoplasticity. Advances in Applied Mechanics 11: 313-354. 11 Sidey, D., and Coffin, L. E (1979). Low-cycle fatigue damage mechemism at high temperature, in Fatigue Mechanism, Proc. ASTM STP 675 Symposium, Kansas City, Mo., May 1978, Fong, J. T. ed., Baltimore, pp. 528-568. 12 Perzyna, P. (1986). Internal state variable description of dynamic fracture of ductile solids. Int. J. Solids Structures 22: 797-818. 13 Perzyna, P. (1986). Constitutive modelling for brittle dynamic fracture in dissipative solids. Arch. Mechanics 38: 725-738. 14 Duszek-Perzyna, M. K., and Perzyna, P. (1994). Analysis of the influence of different effects on criteria for adiabatic shear band localization in inelastic solids, in Material Instabilities: Theory and Applications, ASME Congress, Chicago, 9-11, November 1994, Batra, R. C. and Zbib, H. M. eds., AMD-Vol. 183/MD-Vol.50, pp. 59-85, New York: ASME. 15 Shockey, D. A., Seaman, L., and Curran, D. R. (1985). The microstatistical fracture mechanics approach to dynamic fracture problem. Int. J. Fracture 27: 145-157. 16 Curran, D. R., Seaman, L., and Shockey, D. A. (1987). Dynamic failure of solids. Physics Reports 147: 253-388. 17 Johnson, J. N. (1981). Dynamic fracture and spallation in ductile solids. J. Appl. Phys. 52: 2812-2825. 18 Perzyna, P., and Drabik, A. (1989). Description of micro-damage process by porosity parameter for nonlinear viscoplasticity. Arch. Mechanics 41 895-908. 19 Perzyna, P., and Drabik, A. (1999). Micro-damage mechanism in adiabatic processes. Int. J. Plasticity (submitted for publication). 20 Duszek, M. K., and Perzyna, P. (1991). On combined isotropic and kinematic hardening effects in plastic flow processes. Int. J. Plasticity 7: 351-363. 21 Prager, W. (1955). The theory of plasticity: A survey of recent achievements (J. Clayton Lecture). Proc. Inst. Mech. Eng. 169: 41-57. 22 Ziegler, H. (1959). A modification of Prager's hardening rule. Quart. Appl. Math. 17: 55-65. 23 Agah-Tehrani, A., Lee, E. H., Malett, R. L., and Onat, E. T. (1987). The theory of elastic-plastic deformation at finite strain with induced anisotropy modelled isotropic-kinematic hardening. J. Mech. Phys. Solids 35: 43-60. 24 Armstrong, P. J., and Frederick, C. O. (1966). A mathematical representation of the multiaxial Baushinger effect. CEGB Report RD/B/N731, Central Electricity Generating Board.
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Perzyna
25 Dornowski, W., and Perzyna, P. (2000). Localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings. CAMES 7: 117-160. 26 Perzyna, P. (1984). Constitutive modelling of dissipative solids for postcritical behaviour and fracture. ASME J. Eng. Materials and Technology 106: 410-419. 27 Shima, S., and Oyane, M. (1976). Plasticity for porous solids. Int. J. Mech. Sci. 18: 285-291. 28 Nemes, J. A., and Eftis, J. (1993). Constitutive modelling of the dynamic fracture of smooth tensile bars. Int. J. Plasticity 9: 243-270. 29 Dornowski, W. (1999). Influence of finite deformation on the growth mechanism of microvoids contained in structural metals. Arch. Mechanics 51: 71-86. 30 Chakrabarti, A. K., and Spretnak, J. W. (1975). Instability of plastic flow in the direction of pure shear. Metallurgical Transactions 6A: 733-747. 31 Lodygowski, T., and Perzyna, P. (1997). Localized fracture of inelastic polycrystalline solids under dynamic loading processes. Int. J. Damage Mechanics 6: 364-407. 32 Dornowski, W., and Perzyna, P. (1999). Constitutive modelling of inelastic solids for plastic flow processes under cyclic dynamic loadings. Transaction of the ASME, J. Eng. Materials and Technology 121: 210-220.
SECTION
9.6
High-Temperature Creep Deformation and Rupture Models D. R. HAYHURST Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, United Kingdom
Contents 9.6.1 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . . . . .
835
9.6.2 Single D a m a g e S t a t e Variable T h e o r i e s .. 8 3 6 9.6.2.1 A l u m i n i u m Alloy at 150~ 210~
and
...........................
9.6.2.2 C o p p e r at 2 5 0 ~
.................
836 838
9.6.2.3 A u s t e n i t i c Stainless Steel at 5 5 0 ~
.........................
9 . 6 . 3 M u l t i d a m a g e S t a t e Variable T h e o r i e s . . . .
839 839
9.6.3.1 N i c k e l - B a s e d Superalloy, N i m o n i c 80A Tested at 7 5 0 ~
..............
841
9.6.3.2 A l u m i n i u m Alloy Tested at 150~
.........................
842
9.6.3.3 Ferritic Steel a n d A s s o c i a t e d W e l d Materials at 6 4 0 ~
..........
845
9 . 6 . 4 I d e n t i f i c a t i o n of M a t e r i a l P a r a m e t e r s . . . .
847
9.6.5 H o w to U s e t h e M o d e l . . . . . . . . . . . . . . . . . .
847
References ....................................
847
9.6.1 B A C K G R O U N D The mathematical modeling of primary, secondary, and tertiary creep is addressed in this paper, with emphasis being placed on both uniaxial and Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
835
836
Hayhurst
multiaxial behaviour. The single state variable damage models due to Kachanov [1] and to Robotnov [2] are first presented and used to model aluminium alloys, copper, and austenitic stainless steels. Copper and the aluminium alloys have been selected since they represent materials with extreme types of multiaxial rupture behaviour, and austenitic stainless steel represents intermediate behaviour. The single state variable theory gives a good representation for these materials, since there is one dominant damage mechanism. Multistate damage variable theories [3] have been used to model those materials where either a more accurate representation is required than can be achieved by a single damage state theory, or the synergy between the mechanisms is so distinct that a multidamage state variable model is necessary. In this paper an aluminium alloy tested at 150~ is used to illustrate the former category and a Nimonic 80A superalloy and ferritic steels are used to represent the latter category. In the next sections, the single damage state variable theory is presented first, and is followed by the multidamage state variable models. The equations for each material are presented in turn, followed by the values of the constitutive parameters.
9 . 6 . 2 S I N G L E D A M A G E STATE V A R I A B L E THEORIES These theories were calibrated by Hayhurst [4,5] against experimental uniaxial data, and, in addition, they were developed for multiaxial stress conditions by Hayhurst [4, 6]. Hayhurst [6] used isochronous rupture loci, as shown in Figure 9.6.1, to characterise multiaxial stress rupture behaviour. In this figure the plane stress conditions are normalised with respect to the uniaxial stress or0 required to give a specified lifetime. In the next section the behaviour of aluminium alloys is presented.
9.6.2.1
ALUMINIUM ALLOY AT
150~
AND
210~
The behaviour of the alloy tested at 210~ has been investigated by Hayhurst [4, 5] for uniaxial stresses, and by Hayhurst [6] for multiaxial stresses. The alloy is specified as B.S.1472; it is a precipitation-hardened alloy tested after a 7-hour temperature soak period. The behaviour of a second alloy, tested at 150~ has been investigated by Kowalewski, Hayhurst, and Dyson [7] for uniaxial stresses. The alloy is again specified as B.S.1472 and was tested following a 12-hour temperature soak.
9.6 High-Temperature Creep Deformation and Rupture Models
' m cz = 1/ maximu tension stress// law_
837
~ -L
l
,
2 effective stress law FIGURE 9.6.1 Schematicplot of plane stress isochronous rupture loci. 9.6.2.1.1 Uniaxial Behaviour The uniaxial deformation, damage, and rupture behaviour for the materials tested at 150~ and 210~ is given by Eqs. 1 and 2, and the material constants, given in Table 9.6.1, may be traced to the references given previously. =
(1)
o,)"
CO-- Mtrxtm/(1 + ~b)(1 - o9) ~
(2)
The stress ranges, or, over which the parameters hold are: 9 270MPa > cr > 220MPa at 150~ c.f. Kowalewski, Hayhurst, and Dyson [ 7] 9 70 MPa > cr > 30 MPa at 210 ~ c.f. Hayhurst, Dimmer, and Chernuka [8].
9.6.2.1.2 Multiaxial Behaviour The multiaxial behaviour for materials tested at 150~ Eqs. 3 and 4.
~,ij - G{oq[~"+l(crke)/(n
+ 1)]/cgcrij}tm/(1
and 210~
- co)"
CO-- M{~t71 -Jr-(1 - o~)tye}Ztm/(1 + ~b)(1 - co) r
is given by (3)
(4)
838
Hayhurst TABLE 9.6.1 Material Constants in Units of MPa, % Creep Strain, and Hours, for Aluminium Alloys at 150~ and 210~ Parameter E/MPa n m Z ~b :r G M (.of
150 ~
210 ~
71.10 • 103 11.034 -0.3099 8.220 12.107 0 3.511 x 10 -27 2.569 x 10 -22 0.99
60.03 • 103 6.900 -0.200 6.480 9.500 0 1.15 x 10 -15 1.79 • 10 - 1 4 0.99
where ~(00he) is a homogeneous potential function of degree one in stress, O~ is the maximum principal tension stress, rye(= 3SijSij/2) 1/2) is the effective stress, sij(= 00ij - sij00~h/3) is the stress deviator, and the material constants are given in Table 9.6.1. The multiaxial stress rupture criterion for both aluminium alloys has been determined as close to g - - 0 , i.e., a maximum effective stress rupture criterion. This has been supported for other alloys by Hayhurst, Brown, and Morrison [9], Hayhurst, Dimmer, and Morrison [10], and Hayhurst and Storakers [ 11].
9 . 6 . 2 . 2 COPPERAT 2 5 0 ~ The material for which data are presented here is commercially pure copper which has been given a 12-hour temperature soak prior to testing at 250~ The uniaxial behaviour is given by Eqs. 1 and 2, and the material data are provided in Table 9.6.2. Data are provided for two bars from different batches of manufacture which may be traced to Hayhurst, Dimmer, and Morrison [10] for bar 1, and to Hayhurst, Brown, and Morrison [9] for bar 2. The value of the multiaxial stress rupture parameter ~ -- 0.70 has been verified using notch bar tests by Hayhurst, Dimmer, and Morrison [10]. The stress range, a, over which the parameters hold are: 9 45 MPa > 00 > 25MPa at 250~ Chumuka [8] 9 65MPa > 00 > 30MPa at 250~ Morrison [9].
bar 1; c.f. Hayhurst, Dimmer, and bar 2; c.f. Hayhurst, Brown, and
839
9.6 High-Temperature Creep Deformation and Rupture Models TABLE 9.6.2 Material Constants in Units of MPa, % Creep Strain, and Hours, for Two Different Copper Bars of the Same Composition Tested at 250~ Parameter
Copper bar 1
Copper bar 2
E/MPa n m Z ~b
66.24 x 103 5.00 -0.43 3.19 6.00 0.70 3.21 x 10 -10 1.89 x 10 -7 0.99
66.24 x 103 2.97 -0.79 1.21 3.83 0.70 1.28 x 10 -6 6.02 x 10 -4 0.99
G M cof
9.6.2.3
AUSTENITIC STAINLESS STEEL AT 5 5 0 ~
The material for which data are presented here is AISI 316 Stainless Steel which has been given a 12-hour temperature soak prior to testing at 550~ The uniaxial behaviour is given by Eqs. 1 and 2, and the material data are provided in Table 9.6.3. The value of the muhiaxial stress rupture parameter ~ - - 0 . 7 5 has been verified using notch bar tests by Hayhurst, Dimmer, and Morrison [10]. It is shown in Figure 9.6.2 as an isochronous locus, where it is compared with the experimental results of Chubb and Bolton [12] for a similar material at 600 ~C. The stress range, or, over which the parameters hold are: 9 350 MPa > o-> 160 MPa at 250~ son [10].
c.f. Hayhurst, Dimmer, and Morri-
9 . 6 . 3 M U L T I D A M A G E STATE VARIABLE THEORIES The need for synergistic multidamage state variable theories has been addressed by Dyson, Verma, and Szkopiak [13] and by Othman, Hayhurst, and Dyson [3]. The approach recognises the presence of competing softening mechanisms which interact through global deformation processes. Each physical mechanism is quantified by a single parameter, and its evolution is described by a single rate equation. This results in a set of coupled differential
840
Hayhurst TABLE 9.6.3 Material Constants in Units of MPa, % Creep Strain, and Hours, for an Austenitic Stainless Steel Tested at 550~ Parameter
316 Stainless Steel
E/MPa v n m Z r
169.617 x 103 0.300 1.737 -0.940 0.478 1.914 0.750 1.383 x 10 -5 2.774 x 10 -3
G M
(.of
0.99
!
d" dV
-0.8
~ = 0.75
r
-018
I
1
I
I
, -0.4
-0.4
9. . . .
I
0p
0;8
0.4 0
experiment
o
---0.• ~n/r FIGURE 9.6.2 Comparison of isochronous loci for ~ -- 0.75 with experimental results obtained for 316 Stainless Steel at 600~ [12].
equations which, when solved together with strain rate equations, leads to the description of the creep curve. Presented in the next sections are the equations for the superalloy Nimonic 80A tested at 750~ the aluminium alloy at 150~ and ferritic steel tested at 640~
9.6 High-Temperature Creep Deformation and Rupture Models
841
9 . 6 . 3 . 1 NICKEL-BASED SUPERALLOY~ NIMONIC 8 0 A TESTED AT 750~
9.6.3.1.1 Uniaxial Behaviour The uniaxial equations used by Othman, Hayhurst, and Dyson [3] to describe the damage evolution, creep strain, and failure of this material are as follows: k=A(sinhBa)/{(1 CA(1
051 ~-
-
0.) 1 ) ( s i n
-o91)(1 -002)"}
(5)
h B a ) / ( 1 - o92)"
(6)
o52 = DAN(sin hBa) / (1 - 091 )(1 -- 092)n n = Bae coth(Bae)
whereN-- ~1 0
for for
(
(7) (8)
al > 0 al < 0
The two damage state variables represent physical mechanisms which operate simultaneously, i.e., softening which takes place due to grain boundary cavity nucleation and growth, (.02, and to the multiplication of mobile dislocations, o91. The calibration of the equations is carried out as described by Othman, Hayhurst and Dyson [3], and the resulting material constants are given in Table 9.6.4. 9.6.3.1.2 Multiaxial Behaviour The set of constitutive multiaxial equations for this material is given by the following: (9)
T = ( A / B ) ( c o s h Bae)
de0
dT
1
dt = ~ij - ds 0 (1 - COl)(1 - oo2)~
(sin hBae)
3Asij =
(1
-
Ol)(1
-
(11)
n
051 = CA(1 - col)(sin h B a ) / ( 1 - oo2)n h B a e ) / ( 1 - o)1)(1 - co2)n
052 = D A N ( a l / a ) ~ ( s i n
whereN- ~1 0
t
for for
(10)
al > 0 al _< 0
(12) (13)
842
Hayhurst TABLE 9.6.4 Material Constants in Units of MPa, % Creep Strain, and Hours, for the Nickel-Based Superalloy Nimonic 90, Tested at 750~ Nickel-based superalloy
Parameter E/MPa A B
200 x 103 2 x 10 -4 16 • 10 -3
C D v
300 2 2
of
1/3
Note that neither uni- nor multiaxial equation sets include primary creep. The multiaxial rupture stress criterion v has been calibrated using torsion test data by Dyson and Loveday [14]. The failure criterion in this model is cof = 1/3, as defined by the creep constrained cavitation model, in contrast to cof = 0.99 for the single damage state variable model. The equations and constitutive parameters are valid in the stress, a, range: 9 600 MPa > a > 100 MPa at 750~
9.6.3.2
c.f. Dyson and Loveday [14].
ALUMINIUM ALLOY TESTED AT
150~
9.6.3.2.1 Uniaxial Behaviour The uniaxial constitutive equations are given by:
- A sin h{ Ba(11 - e H) } /(i -co2 )n /I = (h~/a)(1
-
(15)
H/H*)
q) -- (Kc/3)(l- ~)4
(16)
cb2 -- DAN sin h { Ba(1 1 -e- H ) } /(i - co2)" n
_ fBa_~(1 -_H) coth{Bae(1 ((i-~)
where N -
}
0
H)
(i-~) }
f 1 for
/
(14)
for
O"1 >
0
o1 < 0
(17)
(18)
9.6 High-Temperature Creep Deformation and Rupture Models
843
The creep constrained cavitation damage parameter 092 is included in Eq. 17 together with two new state variables. The first, which is included in Eq. 15, describes primary creep using the variable H, which monotonically increases to its saturation value H* when primary creep has been concluded. The second variable 9 describes the physics of aging to lie within the range of 0 to 1, for mathematical convenience. The corresponding material constants are given in Table 9.6.5. 9.6.3.2.2 Multiaxial Behaviour
The multiaxial generalisation takes place through a potential function based on the effective stress, and both the aging variable q) and the dislocation density variable H are treated as scalars. The evolution of the damage variable 092 is dependent on the magnitude of the maximum principal tension stress, and the muhiaxial stress state sensitivity is governed by the term (Crl/Cre) v in Eq. 24. This equation can be integrated to yield the isochronous loci given in Figure 9.6.3.
W -- (a/B)cos h i de/j dt
=
9 gij
-
3A
--
/_:/__
2r
Ba-!(1 -- -/-/)} (1 - q~)
dW 1 so (1 - 092)
(2o)
~Bae(1-H)}
sin h [
(~ - ~ i
_ / (1 - o92)n
hA ),(1 - H / H * ) s i n h{ B~ - H)} o'e(1 - o92 (1 - ~)
@P = (Kc/3)(1 - ~)4
whereN-
~1 0
[
(21)
(22) (23)
fBa~(1 - H) i~--~i }
(o2--DAN{~}sinh[
(19)
for for
(24)
~rl/Cre>0 al/a~_<0
It may be observed that the shapes of the loci due to the term (al/~r~) v in Eq. 24 are essentially the same as those given in Figures 9.6.1 and 9.6.2 for the function [~rl + (1 -~)~r~] z given in Eq. 4. It is worth noting that Kowalewski, Hayhurst, and Dyson [7] contrasted the quality of the creep curve representation provided by the single and multiple state variable theories. They observed that the latter gave a very
844
Hayhurst
TABLE 9.6.5 Material Constants in Units of MPa, % Creep Strain, and Hours, for an Aluminium Alloy at 150 ~ Parameter
Aluminium alloy
E/MPa A B D h H* Kc
71.1 • 103 2.960 • 10 -9 7.167 • 10 -2 6.630 1.370 • 105 0.2032 19.310 x 10 -5
col
0.3
1.5
,0
[,,.I
",_~
d
~o 0.0
-0.5
-1.0 - 1.0
--0.5
0.0
0.5
1.0
1.5
N o r m a l i z e d stress Y~2
FIGURE 9.6.3 Isochronous rupture loci for biaxial plane stress determined by damage evolution due to creep constrained cavitation with aging using Eqs. 20-24 for ao= 262 MPa. The loci are given for the range of values on the stress state sensitivity index v marked on the figure v= 12, 10, 7 , . . . , 0.
accurate prediction of the tertiary part of the creep curve. The equations and constitutive parameters are valid in the stress range, a, given by: 9 270 MPa > a > 220 MPa at 150~ Dyson [7 ].
c.f. Kowalewski, Hayhurst, and
9.6 High-Temperature Creep Deformation and Rupture Models
845
9 . 6 . 3 . 3 FERRITIC STEEL AND ASSOCIATED WELD MATERIALS AT 640~ Data are presented here for materials encountered in ferritic steel steam pipe butt welds. The parent pipe material considered is a 0.5Cr 0.5Mo 0.25V ferritic steel which has been characterised by Perrin and Hayhurst [15]; the weld material considered is derived from a 0.5Cr 0.5Mo 0.25V weld filler. The behaviour of the intermediary phases--heat affect zone (HAZ) and Type IV; see Figure 9.6.4, has been characterised using a technique used by Wang and Hayhurst [16]. The technique involves property ratios and a knowledge of the constitutive equations for the parent material. The constitutive equations are now considered. 9.6.3.3.1 Uniaxial Behaviour The equations developed by Perrin and Hayhurst [15] are as follows: ~-Asinh /2/--
(1-~)(1-co2)
(25)
H/H*)
(26)
OP)4
(27)
(hk/cr)(1
-
OP -- ( K c / 3 ) ( 1 -
B~(1-/q) } oh2 -
sin h
DAN
(1 - (I))(1 - o)2) f
where N -- J 1 0
t
w',~_r) ~ - T A L
for for
O-1 > 0 Crl < 0
[
/ FIGURE 9.6.4
Schematic diagram showing material phases in across-weld.
(28)
846
Hayhurst
Since all material phases of the weld are closely similar, the same equations and the associated physical processes are likely to be the same, and the uniaxial behaviour is described by Eqs. 25-28. C o m p a r i s o n of this equation set with Eqs. 14-17 for the a l u m i n i u m alloy reveals that the only difference is in the term for 1/(1 - co2), which relates to creep deformation; they therefore have very similar modeling capabilities. The material constants are presented for these materials in Table 9.6.6. It is w o r t h noting that the HAZ properties are the same as those of the parent materials.
9.6.3.3.2 Multiaxial Behaviour As in the cases of the superalloy and the a l u m i n i u m alloy, the generalisation to multiaxial stress takes place using a potential function based on the effective stress and the multidamage state variables. The multiaxial stress rupture criterion is being expressed t h r o u g h the term (al/ae) v in Eq. 33.
tp _ (A/B)cos h de,ij
dip
3sij.A
dt = ds--o~j= 2ae __( /:/-- h A 1 -
o'e
(1 - (I))(1 - co2)
{
(29)
Bae(1- H)
(30)
}
sin h (1 - (I))(1 - co2)
H/H*
)sin h
{
Bae(l-H)
}
(i - (I))(i -r
q) = (Kc/3)(l - q))4
cb2 -- DAN ~
sinh
(31)
(32)
i
(l_(i))(l_co2)
1
(33)
TABLE 9.6.6 Material Constants in Units of MPa, % Creep Strain, and Hours, for Ferritic Steel Parent, Weld Metal, HAZ, and Type IV Materials at 640~ Parameter
E/MPa A B D h H* Kc v cof
Parent
160 x 103 1.6783 x i0-s 0.2434 1.2845 2.4326 x 105 0.5929 4.7378 X 1 0 - 4 2.8 1/3
Weld Metal
160 x 103 3.3731 x i0-s 0.2508 1.2845 2.5128 x 105 0.6052 9.4689 X 1 0 - 4 2.8 1/3
HAZ
160 x 103 1.6783 x i0-8 0.2434 1.28451 2.4326 x 105 0.5929 4.7378 • 1 0 - 4 2.8 1/3
Type IV
160 x 103 9.4250 X 10 -7 0.1308 3.125 1.4750 • 1 0 4 0.6091 2.5720 x 10-3 2.8 1/2
9.6 High-Temperature Creep Deformation and Rupture Models
whereN=
~1
{0
for for
847
crl/Cre>O 0"1/(7e ~ 0
The ranges of stress, or, over which the equations are valid at 640~
are:
9 40 MPa > o- > 20 MPa for parent 9 115 MPa > a > 20 MPa for weld 9 140 MPa > o- > 20 MPa for HAZ
640~
c.f.Perrin and Hayhurst [15]
9 160 MPa > a > 55 MPa for Type IV
9.6.4 IDENTIFICATION OF MATERIAL PARAMETERS First, material data are required u n d e r a uniaxial state of stress, and care must be taken to achieve constancy of temperature control and to ensure axiality of loading. Second, data are required u n d e r at least one state of multiaxial stress, for example: biaxial tension cruciform test pieces [6, 17], Andrade shear discs [ 11], torsion tests [ 18], and axisymmetrically n o t c h e d bars [ 10]. In this way, the multiaxial stress rupture criterion of the material can be d e t e r m i n e d and calibrated.
9.6.5 H O W TO USE THE MODEL Any of the constitutive equations reported here may be used in nonlinear finite element codes for elasto-viscoplasticity. In order to solve the coupled multidamage state variable equation, it will usually be necessary to write specific user subroutines to enable the equations to be interfaced with the main finite element c o m p u t e r programme. In all cases it is r e c o m m e n d e d that the numerical routines be checked against published data to verify accuracy, stability, and convergence.
REFERENCES 1. Kachanov, L. M. (1960). The Theory of Creep (English translation ed. A. J. Kennedy), chs IX, X, Boston Spa, Wetherby: British Library. 2. Rabotnov, Yu. M. (1969). Creep Problems in Structural Members (English translation ed. E A. Leckie), ch. 6, Amsterdam: North Holland.
848
Hayhurst
3. Othman, A. M., Hayhurst, D. R., and Dyson, B. E (1993). Skeletal point stresses in circumferentially notched tension bars undergoing tertiary creep modelled with physicallybased constitutive equations, Proc. R. Soc. (London) 441: 343-358. 4. Hayhurst, D. R. (1970). Isothermal Creep Deformation and Rupture of Structures. Ph.D. thesis, Cambridge University. 5. Hayhurst, D. R. (1973). Stress redistribution and rupture due to creep in a uniformly stretched thin plate containing a circular hole. J. Appl. Mech. 40: 244-250. 6. Hayhurst, D. R. (1972). Creep rupture under multi-axial states of stress. J. Mech. Phys. Solids 20:381-390. 7. Kowalewski, Z. L., Hayhurst, D. R., and Dyson, B. E (1994). Mechanisms-based creep constitutive equations for an aluminium alloy. J. Strain Analysis 29(4): 309-316. 8. Hayhurst, D. R., Dimmer, P. R., and Chernuka, M. W. (1975). Estimates of the creep rupture lifetime of structures using the finite element method. J. Mech. Phys. Solids 23: 335-355. 9. Hayhurst, D. R., Brown, P. R., and Morrison, C.J. (1984). The role of continuum damage in creep crack growth. Phil. Trans. R. Soc. (London) A311: 130-158. 10. Hayhurst, D. R., Dimmer, P. R., and Morrison (1984). Development of continuum damage in the creep rupture of notched bars. Phil. Trans. R. Soc. (London) A311: 103-129. 11. Hayhurst, D. R., and Storakers, B. (1976). Creep rupture of the Andrade Shear Disc. Proc. R. Soc. (London) A349: 369-382. 12. Chubb, E. J., and Bolton, C. J. (1980). Stress state dependence of creep deformation and fracture in AISI type 316 stainless steel, in Proc. Int. Conf. on Engng Aspects of Creep 15-19 Sept. 1980, Sheffield 1, paper C201/80, p. 48, London: Institute of Mechanical Engineers. 13. Dyson, B. E, Verma, A. K., and Szkopiak, Z. C. (1981). The influence of stress state on creep resistance: Experiments and modelling. Acta Metall. 29: 1573-1580. 14. Dyson, B. E, and Loveday, M. S. (1981). Creep facture in Nimonic 80A under tri-axial tensile stressing, in Creep in Structures, 1980 (IUTAM Symposium, Leicester, U.K.) (ed. A. R. S. Ponter and D. R. Hayhurst), 406-420. Berlin: Springer-Verlag. 15. Perrin, I. J., and Hayhurst, D. R. (1999). Continuum damage mechanics analyses of Type IV creep failure in ferritic steel crossweld specimens. Int. J. Press. Vess and Piping 76: 599-617. 16. Wang, Z. P., and Hayhurst, D. R. (1994). The use of supercomputer modelling of hightemperature failure in pipe weldments to optimise weld and heat affected zone materials property selection. Proc. R. Soc. (London) A446: 127-148. 17. Hayhurst, D. R. (1973). A bi-axial-tension creep-rupture testing machine. J. Strain Analy. 8(2): 119-123. 18. Trampczynski, W. A., Hayhurst, D. R., and Leckie, E A. (1981). Creep rupture of copper and aluminium under non-proportional loading. J. Mech. Phys. Solids 29(5/6): 353-374.
SECTION
9.7
A Coupled DiffusionViscoplastic Formulation for Oxidasing Multiphase Materials ESTEBAN P. Busso
Department of Mechanical Engineering, Imperial College, University of London, London, SW7 2BX, United Kingdom
Contents 9.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Description of the Model . . . . . . . . . . . . . . . . . . . 9.7.4 Identification of Material Parameters . . . . . . . 9.7.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 9.7.6 List of Material Parameters . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
849 849 850 852 852 853 855
9.7.1 VALIDITY The coupled diffusion-constitutive approach is used to describe the effect of the phase transformations caused by local internal and external oxidation processes on the constitutive behaviour of multiphase metallic materials exposed to an oxidising environment.
9.7.2 BACKGROUND The oxidation of multiphase metallic alloys is an important factor leading to local microcrack formation at or near bimaterial interfaces. A proper Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
849
850
Busso
description of the local deformation processes requires that the constitutive behaviour of the materials be coupled with the dominant diffusion and oxidation mechanisms. Without a proper description of such phenomena, the crucial evolving stress conditions, as the material undergoes the transition from an unoxidised to a fully oxidised state, cannot be predicted [1]. Examples of material systems which undergo selective oxidation are those present in thermal barrier coatings, widely used in both aerospace and landbased gas turbines, and single crystal silicon wafers used in the manufacture of LSI devices by the microelectronics industry. The coupled oxidation-constitutive material framework recently proposed by Busso [2] (see also [3] and [4]) accounts for the diffusion of oxidant species through a multiphase material and the subsequent oxidation of one of its phases (e.g., Al-rich or Si-based), and incorporates the effect that the local volumetric expansion of the newly formed oxide has in the generation of inelastic volumetric strains and residual stresses. It relies on the level of oxygen concentration at each material point to identify whether oxide precipitates have formed, in which case the material at that location is treated, through a self-consistent approach, as an homogeneous equivalent material consisting of multimetallic phases and oxide products.
9.7.3 DESCRIPTION
OF THE MODEL
A hypoelastic formulation for the oxidising metal which accounts for nonisothermal effects can be expressed in terms of the Jaumann derivative of the overall aggregate stress, T, as V = ~a [D T
Din]
- 3~cc~0 1
(1)
where T - t - WT + TW, and D is the stretching tensor, 0 is the time rate of change of the absolute temperature, ~ is the thermal expansion coefficient, and W is the total material spin. The inelastic stretching tensor associated with the deformation, D in , is expressed in terms of the average inelastic stretching rate due to slip, D or, and a transformation term, D tr, which represents the nonrecoverable deformation rate induced by the oxidation of one of the metallic phases. Then,
The average inelastic stretching rate tensor D cr is defined as the volume fraction weighted sum of the inelastic strain rate tensor of each individual phase, ei,'crDcr --- E i fi ei,'cr where ~ is the current volume fraction of phase i.
851
9.7 A C o u p l e d Diffusion-Viscoplastic Formulation
Moreover, D tr is assumed to be controlled by the rate of change of an internal state variable associated with the oxidation kinetics, f. Here, f is related to the oxide volume fraction and varies from 0, before the oxidation begins, to 1, when the internal oxidation is complete. Thus
Dtr
__finif
p
(3)
+ e~ 1
where f~ni is the initial volume fraction of the oxidation-prone phase, 1 is the second-order unit tensor, P is a coefficient which depends on the shape of the oxide particles, and S and T r are the norm and deviatoric component of the aggregate stress tensor T. The dilatational term in Eq. 3 represents the volume change due to oxidation, and the deviatoric one accounts for any deviations from isotropy associated with the oxide formation. The formulation considers that oxidation begins at each material point once the local oxygen concentration, Co >_ Cocr, where Coc,. is a temperaturedependent critical concentration. The evolution of the internal variable f is based on nucleation kinetics, f-
( 1 - f ) S/p Vp for
Co >_ Cocr,
else f -
0
(4)
where Np is the rate of increase of oxide precipitates per unit volume, and Vp is the average volume of each oxide particle. The local behaviour of the coexisting metallic phases and oxide products is described through a homogenization procedure based on self-consistent relations which depend on an interphase accommodation tensorial variable, A i , for each phase i. The evolution of the stress rate tensor of each individual oxidising and non-oxidising phase, O'i, is given by O'i-
"F -3t-
2#(1
-- f l ) ( D cr
_ ei.Cr _ Ai }
(5)
Here fl is Eshelby's elastic accommodation factor and ~t is the shear modulus. In this equation, A~ is the time rate of change of Ai which is, in turn, expressed in terms of a corresponding phase accommodation tensorial variable Si,
$, where/2/(f, 0) is a dimensionless homogenization function, and the index m denotes the total number of coexisting phases. The phase accommodation variable evolves according to the following relation:
r
e,cy m 9
o)s, le,~,cFI
(7)
852
Busso
9.7.4 IDENTIFICATION OF MATERIAL PARAMETERS The true mean dilatational strain caused by the internal oxidation of one of the metallic phases, e~ in Eq. 3, can be determined from the corresponding chemical reaction. Let V0 be the material volume before the chemical reaction takes place, and V that of the oxidized material. Then, e~
-
1 -
V
3 Ln -Vo
(8)
The critical local oxygen concentration level, Co~r (Eq. 4), can be numerically calibrated from oxidation data, such as oxide thickness vs. oxidation time, at different temperatures. The time constant defined by z = 1/(/qpVp) can be determined from the knowledge of the kinetics of the oxidation reaction, and its value is expected to be temperature-dependent. The thermoelastic properties of the homogenised material are obtained from the thermoelastic behaviour, of each individual phase using existing homogenisation (e.g., self-consistent) relations. The inelastic strain rate tensor for each individual phase is defined by a power law relation of the form
e i -- ~ Ai exp -
ff~' ff~
where cri~ is the deviatoric stress tensor for the phase i, and ffi is the corresponding equivalent stress, 6 i - (3 &i: di) 1/z" The material constants Ai, Qi, and ni in Eq. 9 are determined from bulk material creep data for each individual phase. The dimensionless homogenization function, /:/(f, 0), needs to be calibrated from experimental data and detailed unit cell finite element calculations of the oxide-metal system at each stage of the oxidation process, that is, for a range of values of f and temperatures. Finally, the oxide shape parameter P can be deduced from the transformation strain ratios extracted from oxidation experiments, e.g., thin film curvature vs. oxide thickness measurements.
9.7.5 H O W TO USE THE M O D E L Use of the coupled formulation in complex boundary value problems representative of service generally requires its numerical implementation into the finite element method. This can be done using either fully explicit or implicit integration schemes [2, 3, 4].
853
9.7 A Coupled Diffusion-Viscoplastic Formulation
9.7.6 LIST OF MATERIAL PARAMETERS The formulation was calibrated in Reference [3] for an oxidising plasma sprayed thermal barrier coating (TBC) system consisting of a typical NiCoCrA1Y metallic coating and a 8% Y-stabilised zirconia top layer. It was then used to investigate the effect of the thermally grown oxide on the stresses at or near the m e t a l - t h e r m a l l y grown oxide interface, which are known to be responsible for microcrack nucleation. The microstructure of a typical NiCoCrA1Y alloy in the 950-1000~ temperature range consists of a 52.6 vol.% of fl-NiA1 and a 47.4 vol.% 7-Ni solid solution phase. Thus f~ni = 0 . 4 7 4 , and fi2ni = 0 . 5 2 6 . The average thermoelastic properties of the metallic coating and the alumina oxide at the oxidising temperature of 950~ are:
Eox = 340 GPa
Vox = 0.18
0Cox= 9.15x10 -6 1/~
Ecoat = 133 GPa
Vcoat = 0.30
~Zcoat = 17.0X10 -6 1/~
and the creep constants for the Ni solid solution phase (1), and the polycrystalline NiA1 phase (2) are: nl = 4 . 6
Q1 = 2 2 7 K J / m o l
A1 = 5 . 2 8 3 1 / ( s M P a nl)
n2 = 5.5
Q2 = 245 KJ/mol
A2 = 3 . 7 3 x 1 0 -3 1/(s MPa "2)
From the primary oxidation reaction, 52 NiA1 + ~1 0 2
~
51 A1203 + 52 Ni
(10)
one finds that e ~ - 0.122. Also, typically, z - 0.5 to 1 hour at 950~ and P = 0.24 for the oxidation of zirconium (value not available for NiCoCrA1Y). The average critical concentration relative to the value specified at the coating surface found to accurately describe the oxidation data at 950~ was Cocr= 1.45 X 10 -3. The diffusivity of oxygen in both the unoxidised and oxidised coating was taken as Do = 2.64 x 10 -20 m2/s. Finally, typical values for the homogenisation function H at 950~ calibrated from unit cell finite element calculations are given in Table 9.7.1. Figure 9.7.1 shows a comparison between the metallic coating creep data and the corresponding behaviour predicted by the self-consistent model. Also included are the individual phases' creep data. Figure 9.7.2, on the other
TABLE 9.7.1 Values of the Dimensionless f Homogenisation Function H at 950~ Material with a Maximum Oxide Volume Fraction of 100%
f
0.00
0.20
0.35
0.57
0.61
H
4961
2834
1707
434
0
for a
854
Busso ....
2.2
I ....
I ....
I ....
I ....
I ....
--o--Ni 9 n.. -
2-
n
NiAI
-41- NiCoCrAIY (Data) Self-Consistent Model
.,.,.j- ; :.: :'~"
1.8-
CD L
1.6-
0
1.4-
. . ' =
..i
-," "
.p
,,;,
o
I21"
v ~
.--I
1.2-
, 1 -4.5
,
,
,
I I
,
,
,
,
I I
-4
,
,
,
,
I I
-3.5
,
,
,
-3
Log(Creep
,
I I
,
,
,
,
-2.5
rate,
I I
,
,
,
-2
,
-1.5
l/s)
FIGURE 9.7.1. Comparison between the two-phase NiCoCrA1Y alloy creep data and the corresponding behaviour predicted by the self-consistent model. Also included are the individual phases' creep data.
3.0 z~ [] o F E unit cells
Data SC Model
9 .~
2.5
[]
ti
f = 57 %
[]
n
35%
j
-.--- 2 0 % L_
2.0
"'--0.0%
or) v
o J
O
1.5
1.0 -5.0
,
i
i
i
i
-4.0
i
i
!
i
I
I
I
I
-3.0 Log(Creep
rate,
i
I
-2.0
i
I
I
i
-1.0
l/s)
FIGURE 9.7.2. Comparison between accurate reference creep solutions for the two-phase NiCoCrA1Y alloy containing different oxide volume fractions (f) with the corresponding predictions obtained from the coupled diffusion-viscoplastic model.
9.7 A Coupled Diffusion-Viscoplastic Formulation
855
h a n d , p r e s e n t s a c o m p a r i s o n b e t w e e n a c c u r a t e r e f e r e n c e c r e e p s o l u t i o n s for the t w o - p h a s e NiCoCrA1Y alloy c o n t a i n i n g different o x i d e v o l u m e fractions w i t h the c o r r e s p o n d i n g p r e d i c t i o n s o b t a i n e d f r o m the c o u p l e d diffusionviscoplastic model.
REFERENCES 1. Busso, E. P. (1997). Coupled phenomena in bimaterial interface regions at high temperatures, in Mecanismes et Mecanique des Materiaux Multifunctionnels, pp. 49-52, Perreux, D., and L'Excellent, C., eds., MECAMAT, Paris, France, Proc. Colloque National MECAMAT 97, Aussois, France. 2. Busso, E. P. (1999). Oxidation induced stresses in ceramic-metal interfaces. Journal de Physique IV, 9: 287-296. 3. Busso, E. P., Lin, J, Sakurai, S., and Nakayama, M. (2000). A Mechanistic Study of OxidationInduced Degradation in a Plasma-Sprayed Thermal Barrier Coating System. Part I: Model Formulation. Acta Materialia, 49(9): 1515-1528. 4. Busso, E. P., Lin, J., and Sakurai, S. (2000). A Mechanistic Study of Oxidation-Induced Degradation in a Plasma-Sprayed Thermal Barrier Coating System. Part II: Life Prediction Model. Acta Materialia, 49(9): 1529-1536.
SECTION
9.8
Hydrogen Attack E R I K VAN DER GIESSEN a n d
SABINE M .
SCHLOCL
University of Groningen, Applied Physics, Micromechanics of Materials, Nyenborgh 4, 9747 AG Groningen, The Netherlands
Contents 9.8.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.3 Description of the Model . . . . . . . . . . . . . . . . . . . 9.8.3.1 Thermodynamic Model . . . . . . . . . . . . . . 9.8.3.2 Damage Mechanics Model . . . . . . . . . . . 9.8.4 Parameter Identification . . . . . . . . . . . . . . . . . . . . 9.8.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . . 9.8.6 List of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8.1
856 856 859 859 860 861 862 863 863
VALIDITY
The model described in this article is intended to describe the secondary and tertiary stages of creep at elevated temperatures u n d e r hydrogen-rich conditions, as typically e n c o u n t e r e d in petrochemical installations. The associated damage is grain b o u n d a r y cavitation leading to intergranular fracture. This p h e n o m e n o n , k n o w n as hydrogen attack (HA), occurs p r e d o m i n a n t l y in low-carbon, low-alloy ferritic steels (typically Cr-Mo steels).
9.8.2
BACKGROUND
At temperatures exceeding approximately 0.3 times the melting temperature, steels creep at a steady strain rate which is often conveniently described by the
856
Handbookof MaterialsBehaviorModels.ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
857
9.8 Hydrogen Attack
Norton power law (1)
"C -- BG'~, ge
with the creep exponent n and the temperature-dependent parameter B ( T ) . For uniaxial tension, ere and e.Ce are the applied stress and the creep strain rate, respectively. Under multiaxial stress, they denote the Mises equivalent quantities: ~ -
~3
I i %%,
, % - ~j - ~ma~j,
ge
1 Crm - ~ Crkk
,/r D,jD,j
(2)
(3)
V3
with aij denoting Cauchy stress and Dij the stretching or strain rate. The termination of the secondary creep regime in low-carbon ferritic steels in hydrogen-rich environments is caused by the initiation and growth of grain boundary cavities, filled with a methane-hydrogen gas mixture [6]. The socalled hydrogen attack initiates by hydrogen molecules in the gas atmosphere dissociating and the hydrogen atoms diffusing into the steel. Some get trapped at discontinuities occurring mainly between grain boundary carbides and matrix, where the hydrogen reacts with the carbon in the steel to generate methane. The methane molecules are too large to diffuse away and generate an internal pressure inside the cavity which drives subsequent growth. The deformation mechanisms involved during cavity growth are grain boundary diffusion and dislocation creep. Their relative contributions are determined by the cavity density, the internal gas pressure (4)
p m = PCH, -Jr-p.~
and the stress state in the material, O'm, t9"e and the stress a, normal to the grain boundary. For cavities with a spherical-caps shape (see Fig. 9.8.1), radius a, and mean spacing 2b, the rate of change of the cavity volume, -- - n a 3 h ( O )
3
h(O) -
(1 + cos O) -1
2 cos 0
in 0
(5)
can be expressed as [71
9~ - max [.L Vaiff
. L ., , V a f f + V. ,o.] + Vo.
(6)
with the maximum criterion in this expression originating from two modes in which the cavity can grow, depending on its size and the stress state. In Eq. 6,
858
van
FIGURE 9.8.1
der Giessen and Schl6gl
Spherical-cap-shaped cavities on a grain boundary.
the diffusive (diff) and creep (cr) contributions are given by 9L
Pm Jr-
Vdiff
--
4~
f--max
an
In (i/f) - 1 (3 - f ) ( 1 - f ) '
(b) 2, a + l . 5 L
'
L-
(7) (~O'e,~eC)1/3
(8) ft, = ( n - 1)(n + 0.4319)/n 2
~. = 3 / ( 2 n ) , 9H
Pm q -
Vdiff -- 4 ~
cl 2
an
In (i/f) -- 1 (3 --f)(1 --f)' 1 1 - (0.87a/b) 3/n
Vcnr -- 2Tce'Cea3h(r
X
(Zn
ae
(10)
-4---
n sign (am + Pm)
provided that Pm + am > ae (which is typically the case in HA). The enhanced straining in the tertiary regime is primarily due to cavity growth. The contribution of growth to the overall strain is controlled by the average rate of separation between the grains adjacent to the grain boundary, -- Vcav/Trb2 -- ~(pm, am, ae, a,;a,b)
(11)
The cavities grow until coalescence, when microcracks are formed along the grain boundaries, and this finally leads to intergranular failure. A working definition for cavity coalescence is when a/b = 0.7.
859
9.8 Hydrogen Attack
9.8.3 DESCRIPTION OF THE MODEL Hydrogen attack implies a coupling between the transport of hydrogen and carbon, the chemistry of methane formation and evolution of the microstructure, as well as the mechanics of cavity growth and creep. Full details are not yet understood, but the common assumption is that transport and chemistry occur on time scales that are much shorter than the expected lifetime of the components that are typically subject to hydrogen attack. The nucleation time of cavities is also neglected. An upper bound of the attack is then obtained by assuming that the methane gas inside the cavities is in thermodynamic equilibrium at all instants. This allows a simpler model which consists of a thermodynamic part (describing the methane pressure as a function of temperature and the composition of the material) and a damage mechanics part (describing the damage evolution under a given methane pressure and applied stress state).
9.8.3.1
THERMODYNAMICMODEL
The methane pressure stems from the reaction of carbides with the hydrogen and depends on the various phases in the steel. In case of an alloy carbide MxCy containing Cr, Mo, V, and Fe, the chemical reaction is of the type
(CrycrMOyMoVyvFeyFe)xCy + 2yH2 ~ yCH4 + XycrCr + XyMoMO + xyvV + XyFeFe (12) where YCr, YMo, YV, and YFe are the concentration parameters of Cr, Mo, V, and Fe, respectively, in the carbide MxCy (Ycr + YMo + YV + YFe = 1). An upper limit to the methane pressure is obtained by assuming this reaction to be in equilibrium, so that YflCH4 nt- XycrflCr -+- XyMoflMo -+- Xyv#v + XyFe//Fe -- 2yflH 2 -- /'/MxCy=0
(13)
where #x denotes the chemical potential of the component x. Knowing the chemical potentials of the metallic components at a given temperature (see later) as well as the pressure-dependent chemical potential of H2,
IIH2 -- [202 -+- R T In PH2
(14)
(/~~2 is the chemical potential of one mol H2 in the standard state where 1 arm), the methane chemical potential #cm is solved from Eq. 13. The methane pressure can then be solved from the expression
PH2--p0=
].lOll 4 --
0
]../OH 4
+ i T lnfcm,
where C(T) can be found in [2, 5].
fCH4 -- Pcm exp{C(T)pcm}
(15)
860
van der Giessen and Schl6gl
The chemical potentials of Cr, Mo, V, and Fe in Eq. 13 are those of these elements dissolved in the ferritic (bcc) matrix. The thermodynamic treatment of such a system is complex, but the solution is available in the literature [5], which gives PCr, PMo, #V, and ~AFe a s a function of the composition and temperature. The chemical potential PMxCy of the reacting carbide also depends on its crystal structure, composition, and temperature. Neglecting changes in the carbide composition during hydrogen attack, this chemical potential is also now known [5] for all most likely carbides in Cr-Mo steels.
9.8.3.2
DAMAGE MECHANICS MODEL
At the macroscopic continuum level, the constitutive equation in terms of Cartesian coordinates reads ~7ij - - ~ i j k l ( D h l
-
(16)
DkClc - ~F~kl)
where ~rij denotes the Jaumann derivative of Cauchy stress and Dcc the strainrate contribution due to creep and grain boundary cavitation. ~'ijkz is the tensor of elastic moduli, ~cPiJkl- 1 -4-v in terms of Young's modulus E and Poisson's ratio v, and ~ is the cubic thermal expansion coefficient. If there were no cavitation, Dcc would simply be equal to the power-law creep rate 9c 3 o'lj
(18)
D~ -- s e 2 ae
with s9ec determined by the macroscopic effective stress ae through Eq. 1. If there is cavitation, D~ c also accounts for the macroscopic rate of deformation caused by cavitation through the average separation rate ~ through Eq. 11. The way in which this is done depends on the mode of cavitation, as mentioned in the previous section. When creep deformations of the grains are significant, the effect of grain boundary cavitation can be described in terms of a penny-shaped crack model for which Dijcc reads _CCp
v,j
-
.C
[3 6iJt ( 1 -4- p ~ 3 ( : _ 1 1 ) +
(K n k Crkln K l --
\
-
G nK)2)
K--1
d-p
K--1
n~n~ n d- 1
n kahln I - a~ ae
"
(19)
9.8 Hydrogen Attack
861
This expression accounts for the contribution of three orthogonal families of cavitating facets with unit normal vector n//< (K -- 1 , . . . , 3). The parameter p is related to the density of cavitating facets and is given by p = 0.168 x (n + 1)/V/1 + 3 / n when all are potentially cavitated. The term n K k Crkln~ is the component of the macroscopic stress normal to grain facets K. The ~,/< in Eq. 19 is the normal stress transmitted by the facets K. This stress is determined from the condition that the cavitation rate c~(%/<) caused by cavitation under a facet normal stress of ~,/< according to Eq. 11 be equal to the .K opening rate CSp of a facet-size, penny-shaped "crack" (radius R) in a creeping solid bridged by the same stress, 9K _ _4 CSp rc
1+
nk r
-- % ~,Ce2R
(20)
O-e
In the other extreme mode of deformation, where creep of the grains is negligible, the macroscopic strain rate is ~rg
D CijCrg -- ge"C 2 (Tij 'Jr- K --~ cSij
(21)
and consists of a pure (yet small) creep part and a dilatational part due to isotropic cavitation on all facets. The cavitation rate C~rg is equal to the separation rate from Eq. 11 with the facet normal stress a, being taken equal to the macroscopic hydrostatic stress am. The coefficient /c depends on the grain shape and is typically equal to 0.3. The two extreme modes are combined simply by selecting one (Eq. 19) or the other (Eq. 21). A possible criterion for selecting the mode is the maximum value of the cavitation rates predicted in each of the two modes. If C~rgdelivers the largest value, Dcc to be substituted into Eq. 16 is taken to be equal to that in Eq. 21; otherwise the expression in Eq. 19 is used. Other criteria are conceivable but have not been used so far.
9.8.4 PARAMETER IDENTIFICATION The model requires various types of parameters related to the following 9 Composition of the steel, including its carbides. The composition of the carbides can be determined by EDS chemical analysis and their crystal structure by diffraction analysis. 9 Microstructure, including the grain size and the average half-spacing 2b between carbides on grain facets. The parameter R is obtained from the average grain volume Vgr as R ~ (0.042 Vgr)1/3
862
van der Giessen and Schl6gl
9 Material properties, including creep and diffusion parameters. At a given temperature, the creep parameters B and n in Eq. 1 are obtained by fitting to the minimum creep rate in a standard creep curve. According to Frost and Ashby [ 1], the temperature dependence of B is conveniently incorporated by writing 90
B(T) -- B(To) -~- exp
\l-~oJJ
(22)
in terms of the value of B at another temperature To and the activation energy Qv (J/atom) (k is Boltzmann's constant). Values of the latter for various steels are listed in the literature [1]. The diffusion parameter in Eqs. 7 and 9 is defined as ~ _ DB~Bf~ k---~- exp --k--T
(23)
in terms of the grain boundary diffusion coefficient DB and the corresponding activation energy QB (J/atom). Furthermore, C~S is the boundary thickness and f~ the atomic volume. The values of DB~B, QB, and f~ can be found in the literature [3].
9.8.5
HOW
TO USE THE MODEL
The damage model (Eq. 16) has the standard form of a viscoplastic or creeplike constitutive equation. It can therefore be incorporated into existing codes through a modification of the creeplike term Dcc. This modification will need to include a solution procedure for the nonlinear matching condition (Eq. 20); a straightforward secant method has been found to be effective [8]. Because of the high nonlinearity of creep, numerical stability requires attention just as in any creep analysis. A relatively simple rate tangent operator derived in [8] has proved to be efficient in the integration of Eq. 16 [4]. The calculation of the methane pressure is straightforward in principle but requires knowledge of the temperature (either constant or an outcome of a transient thermal analysis) and the composition at each point in the body. The necessary equations as summarized in Section 9.8.3.1 can be found in full detail in [5]. The Fortran code of the expressions is available from the authors upon request. A simplified version of the model can be used when the component or material is under autoclave conditions, i.e., at a constant temperature without external loading. In that case, the voids are subjected only to the gas pressure pm and the material does not creep at a macroscopic scale. Then, the void
9.8 Hydrogen Attack
863
growth equations (Eqs. 6-10) can be simply used to integrate the cavity size a in time until cavity coalescence takes place.
9.8.6
LIST OF PARAMETERS
Most of the parameters in the model are strongly dependent on the material and therefore cannot be listed. Following are physical constants and parameters that are applicable for a wide range of materials: 9 k--- 1.38 x 1 0 - 2 3 j / K 9 ~--- 1.18 x 10 -29 m 3 9 tc ~ 0.3 9 ~ ~ 70-75 ~
REFERENCES 1. Frost, H. J., and Ashby, M. E (1982). Deformation-Mechanism Maps, Oxford: Pergamon Press. 2. Odette, G. R., and Vagarali, S. S. (1982). An equation-of-state for methane for modeling hydrogen attack in ferritic steels. Metall. Trans. A, 13A: 299-303. 3. Parthasarathy, T. A. (1985). Mechanisms of hydrogen attack of carbon and 2.25Cro1Mo steels. Acta Metall., 33: 1673-1681. 4. Schl/Sgl, S. M., and Van der Giessen, E. (1999). Hydrogen attack in a welded reactor. J. de Phys. IV 9: 137-146. 5. Schl6gl, S. M., Van Leeuwen, Y., and Van der Giessen, E. (2000). On methane generation and decarburization in low alloy Cr-Mo steels during hydrogen attack. Metall. Mater. Trans. A31: 125-137. 6. Shewmon, P. (1987). Synergism between creep ductility and grain boundary bubbles. Acta Metall. 35: 1317-1324. 7. Van der Burg, M. W. D., Van der Giessen, E., Needleman, A., and Tvergaard, V. (1995). Void growth due to creep and grain boundary diffusion at high triaxialities. J. Mech. Phys. Solids 43: 123-165. 8. Van der Burg, M. W. D., Van der Giessen, E., and Tvergaard, V. (1998). A continuum damage analysis of hydrogen attack in in a 2.25Cr-1Mo vessel. Mat. Sci. and Eng. A241: 1-13.
SECTION
\
9.9
Hydrogen Transport and Interaction with Material Deformation: Implications. for Fracture PETROS SOFRONIS Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, Illinois
Contents 9.9.1 N o n - h y d r i d e - f o r m i n g s y s t e m s . . . . . . . . . . . . . .
865
9.9.1.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
865
9 .9 .1 .2 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . .
865
9.9.1.3 D e s c r i p t i o n of the M o d e l . . . . . . . . . . . .
865
9.9.1.4 Identification of P a r a m e t e r s . . . . . . . . . .
86 6
9.9.1.5 H o w to Use the M o d e l . . . . . . . . . . . . . . .
86 7
9.9.1.6 P a r a m e t e r s . . . . . . . . . . . . . . . . . . . . . . . . . .
86 8
9.9.2 H y d r i d e - F o r m i n g Systems . . . . . . . . . . . . . . . . . . 9.9 2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
868 868
9.9 2.2 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . .
869
9.9 2.3 D e s c r i p t i o n of the M o d e l . . . . . . . . . . . .
869
9.9 2.4 Id entification of P a r a m e t e r s . . . . . . . . . .
870
9.9 2.5 H o w to Use the M o d e l . . . . . . . . . . . . . . .
871
9.9 2.6 P a r a m e t e r s . . . . . . . . . . . . . . . . . . . . . . . . . .
872
Acknowledgements ..............................
872
References .......................................
872
864
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
865
9.9 Hydrogen Transport and Interaction with Material Deformation
9.9.1 N O N - H Y D R I D E - F O R M I N G SYSTEMS
9.9.1.1 VALIDITY The model applies to a variety of systems, such as fcc, bcc, hcp, pure metals (e.g., Ni, Fe, Mo, W), solid solutions (e.g., steels), precipitation-strengthened systems (e.g., aluminum alloys), and intermetallics (e.g., Ni3A1 and FeA1). It can be used to calculate the local hydrogen concentration qualified by hydrostatic stress and trapping induced by plastic straining. Then by using information on whether the fracture is strain- or stress-controlled, one can predict the location of the first microcracking event in a specimen.
9.9.1.2 BACKGROUND The interaction between solute hydrogen atoms and an applied stress field results from the hydrogen-induced volume and local moduli changes that accompany the introduction of the solute hydrogen in the lattice [11, 23]. In regions of tensile hydrostatic stress and softened elastic moduli, interstitial hydrogen has a lower chemical potential. As a consequence, diffusion through normal interstitial lattice sites (NILS) is generated toward these regions, tending to eliminate the gradients of the chemical potential. Regions with compressive hydrostatic stress or hardened elastic moduli are depleted. Transported hydrogen through NILS diffusion can interact with and accumulate at various microstructural heterogeneities, such as dislocations, grain boundaries, inclusions, voids, surfaces, and impurity atoms described as traps, and from there initiate fracture [7]. Trapping characterizes the fact that interstitial solute atoms often find interstices associated with lattice imperfections to be energetically preferable to NILS. It is emphasized that trap characteristics are evolutionary in nature. Deformation affects dislocation density and structure, changes the void population and size, and influences the behavior of inclusions and grain boundaries in their activity as traps.
9.9.1.3
DESCRIPTION OF THE MODEL
The governing equation of transient hydrogen diffusion that accounts for trapping and hydrostatic stress drift is [9, 22]
D dCL
(DVIqCL
)
oqNTdgP ,i
866
Sofronis
where ( ),, = 0( )/Oxi, d/dt is the time derivative, Cc = OLflNL is the hydrogen concentration in NILS measured in atoms per unit volume, 0L is the occupancy of NILS, fl is the number of NILS per solvent atom, Nc = Na/VM, NA = 6.0232 x 1023 atoms per mole is Avogadro's number, VM is the molar volume of the host lattice, Cr = 0raNt is hydrogen concentration in number of atoms per unit volume in trapping sites, 0r is the occupancy of the trapping sites, a is the number of trapping sites per trap, Nr, which is a function of the local effective plastic strain, i.e., Nr = NT(eP), denotes the trap density measured in number of traps per unit volume, D is the hydrogen diffusion constant through NILS, Vn is the partial molar volume of hydrogen in solid solution, R is the gas constant equal to 8.31J mole -1 K-1, T is the absolute temperature, Deff is an effective diffusion constant given by
Deft = D/(1 + OCr/oqCL),
(2)
aij is the Cauchy stress, and the standard summation convention over the range is implied for a repeated index. The relationship between CL and Cr in Eq. 2 is established through Oriani's equilibrium theory, which requires Or
0L
1 - Or -- 1
-
OLK'
(3)
where K = exp(WB/RT) is the equilibrium constant, and WB is the trap binding energy. As a result,
OCT
O~NT KIlNL
OC-----~=[1+ K-lflNr]2
9.9.1.4
(4)
IDENTIFICATION OF PARAMETERS
Information on binding energy values and the experimental techniques used to determine the binding energy of hydrogen to vacancies, solutes, solutedefect complexes, dislocations, internal interfaces, and isolated metal clusters can be found in the work of Meyers et al. [16]. By way of example, the trap density and binding energy for pure iron can be determined by correlating experimental results on measured permeation transients and time lags [8] with full numerical solutions to the diffusion equation (Eq. 1). The analysis should be repeated at several levels of macroscopic uniaxial strain to obtain the functional relationship Nr = Nr(~P). When trapping is associated with dislocations, a model estimation for NT can be devised by correlating Nrwith the dislocation density and the lattice structure. Thus, in a bcc lattice
9.9 Hydrogen Transport and Interaction with Material Deformation
867
Nr = vr2p/a, where p(eP) is the dislocation density, a is the lattice parameter, and the assumption of one trap per atomic plane threaded by a dislocation was made. The flow stress in unixial tension is required as a function of hydrogen concentration.
9.9.1.5
H o w To USE TUE MODEL
Numerical implementation in a finite element formalism is given in the work by Sofronis and McMeeking [22], Lufrano et al. [14], and Krom et al. [9]. Solution to Eq. 1 requires knowledge of the local values of he hydrostatic stress and effective plastic strain. This is obtained by coupling the solution process of Eq. 1 to that of the elastoplastic deformation of the material. Solution to the latter is carried out by considering the hydrogen dilatational effect [17] through a contribution to the deformation rate tensor (symmetric part of the velocity gradient) Dij - - ~
In 1 +
3f]
where c is the total hydrogen concentration (in NILS and trapping sites) measured in hydrogen atoms per solvent atom, Co is the corresponding initial hydrogen concentration in the absence of any straining, Av is the volume change per atom of hydrogen introduced into solution that is directly related to the partial molar volume of hydrogen V , = A V N A in solution, and k2 is the mean atomic volume of the host metal atom. In the case of small strain plasticity, Eq. 5 reduces to eij So far, research in non-hydride-forming systems [2, 3] has established two viable mechanisms of embrittlement, namely: (i) hydrogen-enhanced localized plasticity and (ii) hydrogen-induced decohesion. However, a fracture criterion which would allow a relevant model to have predictive capabilities has not yet been devised. Hence, the following discussion should be viewed only as suggestive and by no means as reflecting all of the fundamental physics underlying the embrittlement effect. Once the solution to the coupled initial boundary value problem is obtained for a given initial concentration of hydrogen, the following cases may be assessed [24]: (i) in the case of brittle intergranular fracture, the location of the first microcracking event should be expected to occur at locations where both stresses and hydrogen accumulation are high; (ii) in the case of ductile fracture by void nucleation at inclusion, the event will likely take place again in regions where stresses and hydrogen concentrations are high; (iii) in the case of fracture occurring predominantly by plastic flow
868
Sofronis
localization, microcracking can be expected to initiate at locations where the effective plastic strain and the local hydrogen concentrations are high.
9 . 9 . 1 . 6 PARAMETERS 9.9.1.6.1 Iron and steels [7, 22] W b - 6 0 k J/mole; N r - NT(gp) is calculated from experimental data by Kumnick and Johnson [10] and a relevant plot can be found in the work of Sofronis and McMeeking [22]; parameter 0~can vary from 1 to several powers of 10 [6]; f l - 1 (maximum NILS concentration of 1 hydrogen atom per solvent lattice atom); VM -- 7.116 cm3/mole; V/~ -- 2.0 cm3/mole; and for T _> 325K the diffusion constant is given by D - - 2 . 1 • 10-7(m2/s) exp(-6.88kJmole-1/RT) [26]. For a stress-free solid, the initial NILS concentration Co in equilibrium with hydrogen gas at pressure p and temperatures well above 300 K is given by Sievert's law as Co - 1.989 • 1026 x/~(atoms/m3) exp ( - 28.6kJmole- 1/RT). 9.9.1.6.2 Niobium [12, 13] W b - 29.2kJ/mole; a dislocation-based model [24] for the trap density is formulated (cf. Section 9.9.1.4) by assuming that the dislocation density measured in length per cubic meter varies linearly with effective plastic strain [5], such that ev
P-
{P0+~ 1016
for for
eP<0.5 ev >_0.5'
(6)
where P0 - 101~ line length per cubic meter denotes the dislocation density for the annealed material, and 7 - 2.0 • 1016 line length per cubic meter is a proportionality constant; parameter c~ can vary from 1 to several powers of 10; f l - 1 (maximum NILS concentration of 1 hydrogen atom per solvent lattice atom); VM - 10.852 cm3/mole; VH - 1.88 cm3/mole; for T _> 273 K the diffusion constant is given by D - 5.0 • 10-8(m2/s)exp(-10.215kJmole-X/RT); and since niobium is a high-solubility system, NILS initial concentrations for the stress-free lattice can vary from 0 to 1 H/Nb.
9.9.2 HYDRIDE-FORMING SYSTEMS
9.9.2.1 VALIDITY The model applies to systems that fail predominantly by hydride formation and cleavage. These are systems in which hydrides are either stable or can be
9.9 Hydrogen Transport and Interaction with Material Deformation
869
stabilized by the application of a stress field. Examples are the IVb and Vb metals and their alloys (Ti, Zr, V, Nb, Ta), as well as a number of other metals such as Mg and A1. This hydride mechanism is supported by microscopic observations [21, 25] and thermodynamic calculations [4].
9.9.2.2 BACKGROUND In hydride-forming systems, embrittlement occurs by hydride formation in severe stress raisers such as crack tips and is followed by cleavage of the brittle hydride [3, 12, 13]. The phenomenon is intermittent, with the crack propagating through the hydride and stopping when it reaches the matrix. Subsequently, new hydride forms either autocatalytically [21] or because of external loading, and then cleavage reinitiates [27]. Stress-induced hydride formation is a consequence of the volume dilatation, of the order of ,-o15%, which accompanies hydride precipitation [18]. It has been experimentally shown that hydrides form in regions of hydrostatic tensile stress even at temperatures which are above the solvus temperature in the absence of stress. The hydride formation is a result of the enhanced hydrogen concentration in the area of tensile stress [11] and the decreased chemical potential of the hydride relative to the solid solution in the same stress field [4, 18].
9.9.2.3
DESCRIPTION OF THE MODEL
The terminal solid solubility (also termed the solvus concentration), measured in hydrogen atoms per solute, of hydrogen in solution in a material under external stress a 0 is given by Cs
chem+ AG~_~ sur + AG~_~ mech) /RT ] , -- B exp[( AG~_~
(7)
where B is an experimentally determined constant, AGche~n is the chemical Gibbs energy change in the hydride formation, AGSU~ is the free energy needed for the creation of the interface, and AG~_~ mech is the total mechanical free energy of hydride formation. The mechanical energy AGme~h involves the elastic work done on the system (hydride + matrix) and is stored as elastic energy, the plastic work of accommodation, and the work done by the applied stress against the matrix displacement upon hydride formation [1, 13]. The dominant mode for diffusion is through the solid solution, while trapping is ignored. Diffusion through the hydride phase is extremely slow. Let CL denote the hydrogen concentration in the lattice expressed in hydrogen atoms per unit volume of solid solution phase when the hydride volume
870
Sofronis
fraction f is less than 1. At a given time t with the composition of the composite (solid solution 4- hydride phase) defined pointwise by its hydride volume fraction f such that 0 ~ f ___ 1, the hydrogen diffusion equation [12, 22] is written as follows:
C~CL( DcCL'i-DcCLVHoqkk,i), Q---~-3RT ,i
(8)
where
Q-
1--f 1
if 0 ~ f < l if f - - 1
(9) '
Dc is an effective diffusion constant for the hydrogen diffusing through the composite material given by
Dc
_ f (1 -f)Ds
Dh
if
0 ___f< 1
if f -
1
'
(10)
and Ds and Dh represent the diffusion constants of hydrogen in solution with metal and in the hydride, respectively.
9.9.2.4
IDENTIFICATION OF PARAMETERS
Standard solubility techniques are used to determine the solvus concentration as a function of temperature in the absence of externally applied stress. Those may be either direct by measuring the hydrogen concentration in solid solution [20] or indirect, e.g., through measurements of electrical resistivity or dilatometry or internal friction [1, 18]. The mechanical energy of hydride formation AGme~h is calculated by solving a separate elastoplastic boundary value problem (no hydrogen diffusion involved) in which the formation of a hydride precipitate is effected by a transformation strain (volume dilatation) in a plastically deforming matrix under externally applied loads that span the range of possible values from purely hydrostatic to purely deviatoric. The calculated values of the mechanical energy are then tabulated in order to be used as a subroutine module for the numerical solution of Eq. 8 [13]. Correlating the experimentally measured solvus concentration as a function of temperature with the calculated value of the accommodation energy mech in the absence of external applied stress [13], one determines the AG~_fl chem 4- AW'sur value of the sum of the chemical and free energy, AG~_fl ,-,,_,~_fl, for hydride formation at a specific temperature. It should be noted that this new approach to the energetics of hydride formation is an improvement over previous models, which, based on purely elastic considerations, separated the
9.9 Hydrogen Transport and Interaction with Material Deformation
871
mechanical energy into an accommodation component and an interaction component.
9.9.2.5
H o w xo USE THE MODEL
A finite element solution of Eq. 8 provides the local hydrogen concentration and the hydride volume fraction in a specimen under load as a function of time. At each time step, the local hydrogen concentration in the solid solution phase and the associated change in the hydride volume fraction are calculated by the lever rule in a continuum sense pointwise. The details of the numerical implementation are given in the work by Lufrano et al. [12]. Part of the solution procedure is the pointwise calculation of the terminal solid solubility of hydrogen as a function of stress. This is done through Eq. 7 o n c e AGchefl m_ qAG~_flsuris known and the module for the calculation of AG~_flmechis established. Solution to the transient diffusion problem needs to be coupled to a code for the solution of the elastoplastic boundary value problem that accounts for the hydrogen and hydride-induced volume dilatation. As in the case of nonhydride-forming systems, the part of the deformation rate tensor denoting the hydrogen effect is given by Eq. 5 with the term (c - Co)AV/~2 replaced by
eT _ ~ (C -- co)Oh if f -- 0 ( (1 - f ) ( c - co)Oh +f(0hyd -- CoOh) if f ~ 0"
(11)
In Eq. 11, c and Co are the local and initial hydrogen concentrations in hydrogen atoms per metal atom, respectively, Oh is the lattice local dilatation when a hydrogen atom dissolves in solution with the metal [17], and 0hya is the volume dilatation of a material element that is 100% hydride. An averaging approach is used to estimate the hydride size directly ahead of a crack tip during the numerical simulation. Suppose that at a given time t the region directly ahead of the crack tip along the axis to symmetry has a hydride volume fraction distribution f(r), where r is the distance from the tip and f - 0 for r > r0. In view of the large values of hydrogen diffusivity, an individual hydride particle in this region could be as large as a - f r o in length, where f is the average hydride volume fraction in the region 0 _< r _< r0. The fracture toughness in the presence of hydrogen K/c is defined [13] as the level of the applied load measured in terms of the applied stressintensity factor at the moment when the hydride particle size a at some location r will reach a critical size at which the fracture of the hydride is energetically favored in a Griffith sense, namely a-
acrit(r) --
27sE g(1 - v2)a2(r)"
(12)
872
Sofronis
The parameter Ys is the surface energy of the hydride phase, E is Young's modulus, v is Poisson's ratio, and a is the local stress ahead of the crack in the direction normal to the axis of symmetry. Of course, such a prediction for the fracture toughness is a conservative one, since the presence of a void ahead of a blunting crack tip due to cracking of a hydride particle does not necessarily lead to fracture.
9.9.2.6 PARAMETERS 9.9.2.6.1 Niobium [1, 12, 13] Oh =0.174; 0hyd = 0 . 1 2 ; the activation energy for diffusion through the hydride phase can be assumed equal to three times the activation energy for diffusion through the solid solution phase; ?s -- 5.04 Joules/m2; and terminal solid solubility is given by mech
c~-3.74exp
6k;/mo, R--T fexp\
RT
f
(13)
9.9.2.6.2 Zirconium [15, 19, 20] Oh -- 0.12 based on V/-/= 1.67 cm3/mole; 0hyd -- 0171; VM -- 13.85 cm3/mole; Ys can be calculated by considering that the fracture toughness of the pure hydride phase is ~-, 5.0 M P a v ~ ; D - - 2.17 • 10-7(m2/s)exp(-35.1kJmole-1/ RT); the activation energy for diffusion through the hydride phase can be assumed equal to three times the activation energy for diffusion through the solution solution phase; and terminal solid solubility is given by 10
0
R--T
/ exp \
RT / "
(14)
ACKNOWLEDGEMENTS This work was supported by the Department of Energy under grant DEFGO296ER45439. The author would like to thank Prof. H. K. Birnbaum for many helpful discussions on the subject.
REFERENCES 1. Birnbaum, H. K., Grossbeck, M. L., and Amano, M. (1976). Hydride precipitation in Nb and some properties of NbH.J. Less Comm. Met. 49: 357-370.
9.9 Hydrogen Transport and Interaction with Material Deformation
873
2. Birnbaum, H. K., and Sofronis, P. (1994). Hydrogen-enhanced localized plasticity: A mechanism for hydrogen related fracture. Mater. Sci. Eng. A176: 191-202. 3. Birnbaum, H. K., Robertson, I. M., Sofronis, E, and Teter, D. (1997). Mechanisms of hydrogen related fracture: A review, in Corrosion Deformation Interactions CDF96 (Second International Conference, Nice, France, 1996), pp. 172-195, Magnin, T., ed., The Institute of Materials, Great Britain. 4. Flannagan, T. B., Mason, N. B., and Birnbaum, H. K. (1981). The effect of stress on hydride precipitation. Scr. Met. 15: 109-112. 5. Gilman, J. J. (1969). Micromechanics of Flow in Solids, New York: McGraw-Hill Book Company, pp. 185-199. 6. Hirth, J. E, and Carnahan, B. (1978). Hydrogen adsorption at dislocations and cracks in Fe. Acta Metall. 26: 1795-1803. 7. Hirth, J. E (1980). Effects of hydrogen on the properties of iron and steel. Met. Trans. llA: 861-890. 8. Johnson, H. H., and Lin, R. W. (1981). Hydrogen and deuterium trapping in iron, in Hydrogen Effects in Metals, pp. 3-23, Bernstein, I. M., and Thompson A. W., eds., Metallurgical Society of AIME. 9. Krom, A. H. M., Koers, R. W. J., and Bakker, A. (1999). Hydrogen transport near a blunting crack. J. Mech. Phys. Solids 47: 971-992. 10. Kumnick, A. J., and Johnson, H. H. (1980). Deep trapping states for hydrogen in deformed iron. Acta Metall. 28: 33-39. 11. Li, J. C. M., Oriani, R. A., and Darken, L. S. (1966). The thermodynamics of stressed solids. Z. Physik Chem. Neue Folge 49: 271-291. 12. Lufrano, J., Sofronis E, and Birnbaum, H. K. (1996). Modeling of hydrogen transport and elastically accommodated hydride formation near a crack tip. J. Mech. Phys. Solids 44: 179-205. 13. Lufrano, J., Sofronis, E, and Birnbaum, H. K. (1998). Elastoplastically accommodated hydride formation and embrittlement. J. Mech. Phys. Solids 46: 1497-1520. 14. Lufrano, J., Symons, D., and Sofronis, E (1998). Hydrogen transport and large strain elastoplasticity near a notch in alloy X-750. Eng. Fracture Mech. 59: 827-845. 15. Lufrano, J., and Sofronis, P. (2000). Micromechanics of hydride formation and cracking in zirconium alloys. Computer Modeling in Engineering Science 1: 119-131. 16. Meyers, S. M., et al. (1992). Hydrogen interaction with defects in crystalline solids. Rev. Mod. Phys. 64: 559-617. 17. Peisl, H. (1978). Lattice strains due to hydrogen in metals, in Hydrogen in Metals I, Topics in Applied Physics, pp. 53-74, vol. 28, Alefeld, G., and Volkl, J., eds., New York: Springer-Verlag. 18. Puls, M. P. (1984). Elastic and plastic accommodation effects on metal-hydride solubility. Acta Metall. 32: 1259-1269. 19. Puls, M. E (1990). Effects of crack tip stress states and hydride-matrix interaction stresses on delayed hydride cracking. Metall. Trans. 21A: 2905-2917. 20. Shi, S.-Q., Shek, G. K., and Puls, M. P (1995). Hydrogen concentration limit and critical temperatures for delayed hydride cracking in zirconium alloys. J. Nucl. Mater. 218: 189-201. 21. Shih, D. S., Robertson, I. M., and Birnbaum, H. K. (1988). Hydrogen embrittlement of titanium: in situ TEM studies. Acta Metall. 36: 111-124. 22. Sofronis, P., and McMeeking, R. M. (1989). Numerical analysis of hydrogen transport near a blunting crack tip. J. Mech. Phys. Solids 37: 317-350. 23. Sofronis, P., and Birnbaum, H. K. (1995). Mechanics of the hydrogen-dislocation-impurity interactions--I. Increasing shear modulus. J. Mech. Phys. Solids 43: 49-90. 24. Taha, A., and Sofronis, P. (2001). A micromechanics approach to the study of hydrogen transport and embrittlement. Eng. Fracture Mech. 68: 803-837.
874
Sofronis
25. Takano, S., and Suzuki, T. (1974). An electron-optical study of//-hydride and hydrogen embrittlement of vanadium. Acta Metall. 22: 265-274. 26. V61kl, J., and Alefeld, G. (1978). Diffusion of hydrogen in metals, in Hydrogen in Metals I, Topics in Applied Physics, pp. 53-74, vol. 28, Alefeld, G., and Volkl, J., eds., New York: Springer-Verlag. 27. Westlake, D. G. (1969). A generalized model for hydrogen embrittlement. Trans. ASM 62: 1000-1006.
SECTION
9.10
Unified Disturbed State Constitutive Models CHANDRA S. DESAI Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Arizona
Contents 9.10.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.3 Identification of Material Parameters . . . . . . 9.10.3.1 Tests for Finding Parameters . . . . . . . 9.10.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.5 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX 1 DSC/HISS Parameters for Typical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
875 876 877 878 878 879 879 883
9.10.1 VALIDITY The disturbed state concept (DSC) is a unified approach for constitutive modeling of materials and interfaces and joints under thermomechanical and environmental loading. It allows one to consider, in a hierarchical framework, various behavioral features of materials, such as elastic, plastic, and creep strains, microcracking leading to degradation or damage, and stiffening or healing. Degradation and stiffening are incorporated by using the idea of disturbance (D). The DSC and its specialized versions have been found to provide a satisfactory characterization of a wide range of materials, such as geologic, concrete, asphalt concrete, ceramics, metals, allows (solders), and silicon. Details of the theoretical development of the models, their use and validation for various materials and interfaces and joints, and their Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
875
876
Desai
implementation in a computer (finite element) procedure and validation for a wide range of field and laboratory-simulated practical boundary value problems are given in various publications; only typical studies are listed under References.
9.10.2
FORMULATION
The basic incremental DSC equations are derived as
(i) where a, i, and c denote observed or actual, relative intact (RI) and fully adjusted (FA) responses, respectively, ~ and e are the stress and strain vectors, C is the constitutive matrix, D is the disturbance, dD is the increment or rate of disturbance, and d denotes increment. Although D is often treated as scalar, its tensorial form can be introduced in the DSC equations. In the DSC, it is considered that at any stage during deformation, a material element is composed of a mixture of two or more reference materials. For the dry material, one of the reference materials is in the RI state, whose behavior is expressed by using such continuum theories as elasticity, plasticity, or elastoviscoplasticity. The RI material continuously transforms to the material in the FA state because of the internal self-adjustment of the material's microstructure. It is the consequence of relative particle motions and/or microcracking; in the limit, the FA material is assumed to approach an invariant state. Behavior of the FA can be characterized by using various assumptions, e.g., (1) it can carry hydrostatic or isotropic stress and no shear stress and act like a constrained liquid, and (2) it can continue to carry a limiting shear stress under a given hydrostatic stress and continue to deform in shear without any change in volume, as in the critical state concept [15] and act like a constrained liquid-solid. If the material in the FA state is treated as a "void" as in the continuum damage concept [16], it can carry no stress at all. Since the material in the FA state is surrounded by the RI material, the foregoing two idealizations are considered to be more realistic compared to the "void" assumption in the damage concept. Furthermore, the DSC model allows for the coupling and interaction between the material parts in the RI and FA states. Such coupling is not allowed in the damage concept. As a result, the DSC model implicitly allows for the neighborhood or nonlocal effects, and external enrichments such as gradient and Cosserat theories and microcrack interaction are not required. The disturbance, D, acts as the coupling and interpolation mechanism and leads to the observed behavior in terms of the behavior of the RI and FA material parts.
9.10 Unified Disturbed State Constitutive Models
877
Specializations: If D = 0, Eq. 1 leads to the continuum models in which the RI constitutive matrix, C i, can be based on linear (nonlinear) elasticity, plasticity, or elasto-viscoplasticity. If D # 0, Eq. 1 provides for microcracking and degradation or stiffening (healing). If the terms related to the FA state (c) are ignored, Eq. 1 leads to the classical damage model.
9.10.3 IDENTIFICATION PARAMETERS
OF MATERIAL
The DSC model allows for the flexibility to choose specialized versions, such as the elastic, elastoplastic, viscoplastic, and disturbance (degradation). Hence, the user needs to specify parameters related to the version chosen for a given material and application need. The basic parameters are listed in following text, along with the explanation for various quantities: 9 Linear elastic: Young's modulus, E, and Poisson's ratio, v; or shear modulus, G, and bulk modulus, K; 9 Plasticity Classical: yon Mises: yield stress, Cry, or cohesion, c. Drucker-Prager: cohesion, c, angle of friction, ~b; Continuous yielding: Hierarchical Single Surface (HISS) (see "Function"): Ultimate (failure): slope of ultimate envelope, 7; yield surface shape, fl; bonding stress, R; Transition (compaction to dilation): n; Hardening or yielding: parameters al and r/I; 9 Creep: fluidity, F; flow function, N; 9 Disturbance: A, Z, Du; 9 Thermal effect: Coefficient of thermal expansion, ~7-; Parameter dependence: pr and 2; 9 Yield function: F -J-2D - (-0~]1 + 7]12)( 1 - fiSt) -~ 0; 9 Growth function: ~ -- al/~"1; 9 Creep (viscoplastic [17]): drip - F($)(OF/cg~) and ~ b - (F/Fo)N; multicomponent DSC or overlay models allow for viscoelastic and viscoplastic creep; 9 Disturbance" D - Du (1 - e -A~,); 9 Temperature dependence: p(T) - p(Tr)(T/Tr)C; r,J
where J1 and J2D are the first invariant of the stress tensor, Crij, and the second invariant of the deviatoric stress tensor, Sij, the overbar denotes nondimensional value with respect to the atmospheric pressure, Pa, ]1 - J 1 + 3R, R is
878
Desai
l-3/2 , J3D is the third invariant of Sij , the bonding stress, Sr = (X/~/2)J3D "d2D and ~D are the trajectories of total and deviatoric plastic strains, respectively, Pa is any parameter (see preceding), Tr is the reference temperature (e.g., 300 K), and T is any temperature. The yield surface, F, plots as continuous in various stress spaces, and the associated plasticity models are referred to as HISS (hierarchical single surface).
9.10.3.1
TESTS FOR FINDING PARAMETERS
Details of the procedures for finding the preceding parameters from laboratory tests are given in Desai [1]. The parameters have physical meanings in that they are related to specific states during deformations. The parameters are found from uniaxial tension and compression, shear, biaxial, triaxial (01,02 = 0"3), multiaxial (01 r 02 r 03), creep and relaxation, and/or cyclic thermomechanical tests. The disturbance parameters are found on the basis of the degradation response under static or cyclic tests in terms of stress, volumetric (void ratio), effective stress (pore water pressure) for saturated materials, and nondestructive properties such as ultrasonic P- and S-wave velocities or attenuation. The parameters are expressed in terms of temperature and strain rates.
9.10.4 IMPLEMENTATION The DSC model is implemented in linear and nonlinear finite element procedures for the solution of problems under static, dynamic, and repetitive loading, and involving dry and porous saturated materials. The finite element procedures have been used to predict the behavior of a number of practical problems in civil and mechanical engineering, electronic packaging, and semiconductor systems. The incremental constitutive equations, Eq. 1, are expressed as d a a = C Dsc d 8i, where C v s c is the constitutive matrix given by r
,-,,a
e',,a
C Dsc
-
(1 - D ) C
i +
D(1 + or C c + R; a"
(2)
where ~ is the relative motion parameter and ~r is the relative stress vector based on go_ gi. The incremental finite element equations with C Dsc at step
879
9.10 Unified Disturbed State Constitutive Models
n are derived as
I"
~V (1 - Dn) Brdan+l i dV + i Dn B ~dcrn+ldg c + f
(3)
B; (~ n - ai)dDn - dV --
which in terms of displacement increment
dqi are expressed as ,-,a
f BrcDSCBdV dq i
-d Q
l'l Or -
(4)
1
where B is the transformation matrix and d Q is the vector of applied loads. Various techniques for the incremental-iterative solutions for Eq. 4 are given by Desai [1]. The finite element procedures have been used to successfully predict the observed behavior of a wide range of simulated and field problems in civil and mechanical engineering and electronic packaging. ,-,,a
e,,a
9.10.5 PARAMETERS Parameters for typical engineering materials such as a clay, sands, rocks, concrete, rock salt, ceramic composite, and solder alloy (40/60, Pb/Sn) are given in Appendix 1. Parameters for many other materials and interfaces and joints are given elsewhere, e.g., Desai [1].
A P P E N D I X 1 DSC/HISS PARAMETERS FOR TYPICAL MATERIALS In view of space limitations, details of the physical properties of the materials and associated tests are not presented; they can be obtained from the references cited. Parameters for Saturated Marine Clay-DSC Model Parameter
Value
E, MPa (psi)
10 (1500) 0.35 0.047 0.00 2.80
v fl n
Parameter
Value
~
0.0694
2
0.169 0.903 1.73 0.309
eoc A Z
880
Desai
Parameters
for Saturated
Marine
Clay-DSC
Model
(Continued)
Parameter
Value
Parameter
Value
hi h2*
0.0001 0.78
Du
0.75
* ~ = hl/~ h2, a is the hardening in growth function, eoc is the initial void ratio, and ~v is the trajectory of volumetric strains.
Parameters
for Sands:
HISS Models Dry sand
Parameter
Ottawa
Leighton Buzzard
Munich
Saturated sand
E, MPa (psi) v 7 fl n
262 (38,000) 0.37 0.124 0.494 3.0 2.5 • 10 -3 0.370
R
0.00 0.265
63 (9200) 0.21 0.105 0.747 3.20 0.1258 1355 0.001 1.11 0.00 0.35
140 (20,420) 0.15 0.636 0.60 3.0
al* ~/1
79 (11,500) 0.29 0.102 0.362 2.5 0.135 450 0.0047 1.02 0.00 0.290
~c * O~ =
hi** h2 h3 h4
al - 0.16 x 10 -4 t/1 - 1.17 0.00 --
all~ ill
** a = hlexp [-h2~(1 -
r
+ ;,,r
Note: a is the hardening or growth function; tc is the nonassociative parameter.
Parameters
for Rocks
and Concrete
Parameter
E, MPa (psi) v 7 fl0* ill* n t/1
Rocks Soapstone
Sandstone
Concrete
9150 (1328 • 103) 0.0792 0.0470 0.750 0.0465 7.0 0.177 x 10 -2
25500 (3700 • 103) 0.11 0.0774 0.767 0.0020 7.20 0.467 x 10 -2
7000 (106) 0.14 0.113 0.8437 0.027 7.00 9 x 10 -3
881
9.10 Unified Disturbed State Constitutive Models
(Continued)
Parameters f o r R o c k s a n d C o n c r e t e Parameter
/71 R, MPa (psi)
Rocks Soapstone
Sandstone
Concrete
0.747 1.067 (155)
0.345 2.90 (420)
0.44 2.72 (395)
K A Z Du
668 1.50 0.875
* fl = floe-e~J~ ** Fs = [ e x p ( f l l / f l o J 1
) -
flSr] -0"5
Parameters for Rock Salt at Different Temperatures Temperature parameter
296 K
336 K
350 K
473 K
573 K
673 K
E (GPa) v aT (l/K) ( • 10 -5)
34.13 0.279 3.8
32.19 0.287 4.2
31.59 0.290 4.3
27.49 0.310 5.5
25.15 0.324 6.4
23.35 0.336 7.2
7 fl n (average)
0.0516 0.690
0.0384 0.620 3.92
0.0349 0.590
al ( • 10-9) /71 (average)
1.80
0.97
F( • 10-3/day) N (average)
4.95
6.11 3.0
0.95 0.474 6.54
Temperature dependence:
( T ) -0"462 E(T) =
33.92 3 - ~
( T ) 0224
;v(T) = 0.28 3--0-6
~(r) = 3.85 • ~0-~ (3-~0) ~
-0.91 ~(r) = 0.05 3-0-6
;fl(r) = 0.68 3--0-6
0{(T)= /a~-~/ /3~0/-0334 ;F(T) 300
- 5.0
(3~0 /
1.70
0.0173 0.450
10.77
0.0111 0.380
14.8
0.0076 0.33
19.35
882
Desai
Elasticity and Plasticity Parameters for Solder (Pb40/Sn60) at ~=0.002/sec" /~0-model Temperature (K)
208
E (GPa) v =r (l/K) X 10 -6
273
26.1 0.380 2.75
6/1 ( X 10 -6)
I/1 ~/1 (average) ~ry, yield stress (MPa) Bonding stress, R (MPa)
24.1 0.395 2.93
0.00083 0.0 2.1 8.3 0.431
n
348 22.45 0.408 3.11
0.00082 0.0 2.1 2.93 0.553 0.615 31.724 288.20
37.241 395.80
373 22.00 0.412 3.16
0.00082 0.0 2.1 1.25 0.626
0.00081 0.0 2.1 0.195 0.849
20.690 175.20
15.172 122.10
Creep Parameters for Pb40/Sn60 Solder at Different Temperatures Temperature
298 K
313 K
333 K
373 K
393 K
Fluidity parameter ln(F) Parameter N Average
0.578
2.058
3.475
4.61
6.96
2.665 2.67
2.645 2.67
2.667 2.67
2.448 2.67
2.74 2.67
Disturbance
for Pb40/Sn60
Parameters
Solder
at Different
Temperatures
(Du = 1 . 0 0 ) Temperature
223~
308~
398~
423~
Z Average A
0.7329/0.8697 0.676 0.056/0.072
0.5214/0.6031 0.676 0.188/0.1298
0.6973/0.5914 0.676 0.0496/0.146
0.6612/0.7224 0.676 0.197/0.169
Temperature Dependence of Parameters for Pb40/Sn60 Solder Parameter
P3oo
c
E v ~r y
23.45 (GPa) 0.40 3 x 10 - 6 ( l / K ) 0.00082 0.05 • 10 -4 240.67 (MPa)
-0.292 0.14 0.24 -0.034 -5.5 -1.91
R
883
9.10 Unified Disturbed State Constitutive Models Temperature
Dependence
Parameter F A
of Parameters P30o 1.80/sec 0.102
for Pb40/Sn60
Solder c 6.185 1.55
Note: Other parameters are not affected significantly by temperature; hence, their average values are used.
REFERENCES 1. Desai, C. S. (1999). Mechanics of Materials and Interfaces: The Disturbed State Concept, Boca Raton, Florida: CRC Press (in press). 2. Desai, C. S. (1995). Constitutive modelling using the disturbed state as microstructure selfadjustment concept, Chapter 8 in Continuum Models for Materials with Microstructure, Miihlhaus, H. B., ed., John Wiley. 3. Desai, C. S., Basaran, C., and Zhang, W. (1997). Numerical algorithms and mesh dependence in the disturbed state concept. Int. J. Num. Meth. Eng. 40: 3059-3083. 4. Desai, C. S., Chia, J., Kundu, T., and Prince, J. L. (1997). Thermomechanical response of materials and interfaces in electronic packaging: Parts I and II. J. Elect. Packaging, ASME 119: 294-309. 5. Desai, C. S., Dishongh, T. J., and Deneke, P. (1998). Disturbed state constitutive model for thermomechanical behavior of dislocated silicon with impurities. J. Appl. Physics 84: 11. 6. Desai, C. S., and Ma, Y. (1992). Modelling of joints and interfaces using the disturbed state concept. Int. J. Num. Analyt. Meth. Geomech. 16: 623-653. 7. Desai, C. S., and Salami, M. R. (1987). A constitutive model and associated testing for soft rock. Int. J. Rock Mech. Min. Sc. 24(5): 299-307. 8. Desai, C. S., Samtani, N. C., and Vulliet, L. (1995). Constitutive modeling and analysis of creeping slopes. J. Geotech. Eng., ASCE, 121(1): 43-56. 9. Desai, C. S., Somasundaram, S., and Frantziskonis, G. (1986). A hierarchical approach for constitutive modelling of geologic materials. Int. J. Num. Analyt. Meth. Geomech. 10(3): 225252. 10. Desai, C. S., and Toth, J. (1996). Disturbed state constitutive modeling based on stress-strain and nondestructive behavior. Int. J. Solids Struct. 33(11): 1619-1650. 11. Desai, C. S., and Varadarajan, S. (1987). A constitutive model for quasistatic behavior of rock salt. J. Geophys. Res. 92(B 11): 11445-11456. 12. Desai, C. S., and Whitenack, R. (2000). Review of models and the disturbed state concept for thermomechanical analysis in electronic packaging. J. Electronic Packaging, ASME (in press). 13. Desai, C. S., Zaman, M. M., Lightner, J. G., and Siriwardane, H.J. (1984). Thin-layer element for interfaces and joints. Int. J. Num. Analyt. Meth. Geomech. 8(1): 19-43. 14. Katti, D. R., and Desai, C. S. (1994). Modeling and testing of cohesive soil using the disturbed state concept. J. Eng. Mech., ASCE, 121: 648-658. 15. Roscoe, K. H., Schofield, A. N., and Wroth, C. P. (1958). On yielding of soils. Geotechnique 8: 22-53. 16. Kachanov, L. M. (1986). Introduction to Continuum Damage Mechanics, Dordrecht: Martinus Nijhoff Publishers. 17. Perzyna, P. (1966). Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9: 247-277.
SECTION
9.11
Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors TATSUO INOUE Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, Japan
Contents 9.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.2 Continuum Thermodynamics and Fundamental Framework . . . . . . . . . . . . . . . . . . 9.11.3 Stress-Strain Constitutive Equation . . . . . . . 9.11.3.1 Plastic Strain Rate . . . . . . . . . . . . . . . . . 9.11.3.2 Viscoplastic Strain Rate . . . . . . . . . . . . 9.11.4 Heat Conduction Equation . . . . . . . . . . . . . . . . 9.11.5 Kinetics of Phase Transformation . . . . . . . . . 9.11.6 Some Simulated Results of Engineering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
884 885 888 888 889 891 891 893 895
9.11.1 INTRODUCTION Coupling among metallic structures, including the molten state, temperature, and stress and/or strain occurring in processes accompanied by phase transformation, sometimes is one of the predominant effects of such industrial processes as quenching, welding, casting, and so on. Figure 9.11.1 shows the schematic representation of the effect of metallo-thermomechanical coupling with the induced phenomena [1-5]. When the temperature distribution in a material varies, thermal stress (@) is caused in the body, and the induced phase transformation (@) affects the structural distribution, which is known as melting or solidification in solid-liquid transition and pearlite or martensite transformation in the solid
884
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
885
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
(DThermalstress
IX X .ea, I TEMPERATURE
_•
~
X X heat
Temperature
XX
dependenceXX
.e,o "1 "
STRESS/STRAIN
I
& Transformation// (~ plasticity//Stress/strain
ofstructureXX
//dependence
//ofstructure
FIGURE 9.11.1 Metallo-thermo-mechanics: coupling between temperature, stress/strain, and metallic structure.
phase. Local dilatation due to structural changes in the body bring out the transformation stress (@) and interrupt the stress or strain field in the body. In contrast to these phenomena, which are well known in ordinal analysis, arrows in the opposite direction indicate coupling in the following manner. Part of the mechanical work done by the existing stress in the material is converted into heat (@) which may be predominant in the case of inelastic deformation, thus disturbing the temperature distribution. The acceleration of phase transformation by stress or strain, which is called stress- or straininduced transformation (@) has been treated by metallurgists as one of leading parameters of transformation kinetics. The arrow numbered by ((~)) corresponds to the latent heat due to phase transformation, which is essential in determining the temperature. The purpose of this section is to present the governing equations relevant to simulating such processes involving phase transformation when considering the effect of the coupling mentioned. Formulation of the fundamental equations for stress-strain relationships, heat conduction, and transformation kinetics based on continuum thermodynamics will be done in the first part, and a list of some examples of the numerical simulation of temperature, stress-strain, and metallic structures in the processes of quenching, welding, and casting will be presented.
9.11.2
CONTINUUM
FUNDAMENTAL
THERMODYNAMICS
AND
FRAMEWORK
Consider a material undergoing structural change due to phase transformation as a mixture of N kinds of constituents [6]. Denoting the volume fraction
886
Inoue
of the Ith constituent as ~I (see Fig. 9.11.2), the physical and mechanical properties x of the material are assumed to be a linear combination of the properties xi of the constituent as N
I--1
with N
I--I
EN_I is the summation for suffix I from 1 to N. All material parameters appearing in following text are defined in the manner of Eq. 1. The Gibbs free-energy density function G is defined as
where
1 p
~
where U, T, ~/, and p are the internal energy density, temperature, entropy density, and mass density, respectively. Elastic strain rate ~ in Eq. 3 is defined .i as the subtraction of inelastic strain rate ~ij from total strain rate ~ij, that is, 9e
9
.i
~ij ----~ij - ~ij
(4)
The thermodynamic state of a material is assumed to be determined by stress aij, temperature T, temperature gradient gi(--grad T), and a set of internal variables of inelastic strain ~iij, back stress ~ij, and hardening parameter ~c related to inelastic deformation, together with the volume fraction of the constituents ~I. Then, the general form of the constitutive equation can be
FIGURE 9.11.2
C o n c e p t of m i x t u r e .
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
887
expressed as
(
i
)
(~)
(
i
)
(6)
(
i
)
(7)
G = G aij, T, gi, ~'ij, oqj, 1r ~I
rl -- rI aij, T, gi, ei3, ~ij, ~c, ~
q i - q i crij , T, g i , e ij , czi3 , ~c, ~ I
(8)
~ij -- ~ij aij, T, gi, ~ij, ~ij, ~c, ~I
Here, qi is the heat flux. The evolution equations for the internal variables are defined in the same form as Eqs. 5-8, i.e.,
~
i(
~
/
(9)
i
)
(10)
~ij -- ~ij aij, T, gi, ~ij, ~zij, ~c, ~I
(
OCij -- O~ij Gij ,
T, gi,
F,ij, (xij, K,, ~I
(
i
(
i
~. -- ~. aij, T, gi , ~ij, ~
~c, ~x
)
(11)
)
(12)
When the strong form of the Clausius-Duhem inequality 9
e
.i
- p ( o + ~t) - ~,j~j + ~,j~,j >_ o
(13)
giqi <_ 0
(14)
with
is applied, the constitutive relationships in Eqs. 5-8 are reduced to G = G Gij , T, ~ij' CXij' K,
(15)
~j - p ~ o
(16)
~j, r,~j,~j,~,~,
rI = - p --~ G aij, T, sij , o~ij, K, ~i
qi = - k
( aij, T, ~ij,i ~zij, ~c, ~I ) gi
(17) (18)
Here, Fourier's law has been used in Eq. 18 with thermal conductivity k.
888
Inoue
9.11.3 STRESS-STRAIN CONSTITUTIVE EQUATION To obtain an explicit expression for the elastic strain in Eq. 16, the Gibbs freeenergy G is assumed to be determined by that of constituent GI in the form of Eq. 1 as N
G a0, T, e0, ~0, K, ~I
~IG~ a 0, T, e0, ~0, tc I=l
When GI is divided into the elastic and inelastic parts as
(
i
GI (Tij, T, F.ij, ~ij, K,, ~I
) -- GIe(cYij, T)
,(i
)
..qt_GI T, g.ij, cxij , K,
(20)
we can derive the elastic strain from Eq. 16 by expanding elastic part G~ around the natural state, ai) = 0 and T = To, in terms of the representation theorem for an isotropic function;
N
N
o I=l
I=1
(21)
Here, El, vI, ~I, and/31 correspond to Young's modulus, Poisson's ratio, thermal expansion coefficient, and dilatation of the Ith constituent, respectively.
9.11.3.1 PLASTIC STRAIN RATE Assume that the evolution of back stress ~0 of the yield surface and hardening parameter tc can be determined by
(22)
~j - C(T, ~, ~)~ i
where ~i represents the equivalent inelastic strain rate. To take into account the effect of changing structural fraction ~i, we take the form of the yield function as
(
i
F -- F ai), T, ei) , ~i), to, ~I
)_[
]
3(si) _ ~i))2 I/2_K(T ' to, ~z)
(24)
where si)(= cri) - 1 6i)ahh) represents the deviatoric stress component. Employing the normality rule and the consistency relationship, the final form of the
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
889
time-independent inelastic strain rate, or plastic strain rate, reads:
with
_1 ___( oqF oqF + ~ OF {-- (2 ~ ~ ) 1/20qF} 6 C~{7mnOF.in OGmnOO~mn O{7mnO-~mn
(26)
Equation 25 means that plastic strain is induced not only by stress, but also by the temperature and phase change.
9.11.3.2
VISCOPLASTIC STRAIN RATE
The elastic-plastic constitutive relationship is suitable for describing the material behavior at relatively low temperature. However, time dependency or viscosity might be predominate at a higher temperature level, particularly when the material behaves like a viscous liquid beyond its melting point. In order to analyze such processes as welding and casting, in which melting and solidification of the metal are essential phenomena, adequate formulation of the viscoplastic constitutive model is needed. Malvern [7] and Perzyna [8] proposed a viscoplastic constitutive equation for time-dependent inelastic "i in the form strain rate F,ij 1 OF
~iij -- -~-~(l~l(F) ) O~Tij
with the static yield function
(
F=
f r
r,
i) -1
(27)
(28)
K where # and K denote the stress, and the coefficient of viscosity and the static flow, and (~(F))
_ fo,
if ~(F) _< 0 if > 0
(29)
Equation 27 indicates that the inelastic strain rate is induced in an outer direction normal to static yield surface F, and that the magnitude of the strain rate depends on the ratio of excess stress (f - K) to flow stress K. If we adopt the flow rule (Eq. 27) to the liquid state, the flow stress tends to vanish (K--+ 0) and the yield surface F expands infinitely (F--+oo), which implies that the strain rate is infinite at low stress. To compensate for such an
890
Inoue
inconsistency occurring in a liquid, a modification to Eq. 28 is made such that F -- f ( CYij , T, F,iji , ~ij ) - - K ( T , ~c {I)
(30)
When we take the simple forms of functions ~ and F, ~(F) = F
(31)
(32) Equation 27 can be reduced to g/j
~
1 - [3(shl- 0~kl)(Skl- ~hl)]1/2 (sij - a/j)
This constitutive relationship may be relevant to region with high viscosity, as well as to a normal body [9]. For instance, when flow stress K equals strain rate ~,ij (= k~j + kb)is given by l+v v 1 ~.ij = E (rij - -~ (rkkfiij + ~
(33)
a liquid-solid transition time-independent plastic zero in Eq. 33, the total
sij
(34)
when the effect of temperature and phase change is neglected for simplicity. This equation is equivalent to the Maxwell constitutive model for a viscoelastic body. When the elastic component of shear deformation is small enough compared with the viscoplastic component, as is usual for a viscous fluid, the Newtonian fluid model
(35) is obtainable from Eq. 34 by neglecting the elastic shear strain rate. Furthermore, when the elastic volume dilatation e~h is removed from Eq. 35, we have aij = 2#~,ij - pcSij
(36)
which represents the model for an incompressible Newtonian fluid. In the limiting case for an inviscid material (# = 0), ~(F) in Eq. 27 tends to infinity and
3---~I/r A 1
(37)
should hold to give the form 9i = A ~ij
OF D-~/j
(38)
891
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
Parameter A can be easily determined by applying the consistency relationship, and thus we get back again to the previous discussion for timeindependent plastic strain. From the considerations just mentioned, the constitutive relationship developed in Eq. 33 seems to be useful for a wide range of metals, from inelastic solids to viscous fluids.
9.11.4 HEAT CONDUCTION
EQUATION
When we adopt Eqs. 15-18 to the energy conservation law 9
pO-
Oqi
-
(39)
o
the equation of heat conduction
02T
N
0~
f OH .i
OH
OH
= pr
)
(40)
holds with enthalpy density H (= G 4-T~/) and latent heat li due to the increase of the Ith phase OH li=o~ I (41) The fifth term on the left-hand side of Eq. 40 denotes the heat generation by inelastic dissipation, which is significant when compared with the elastic work represented by the fourth term, and the third term arises from the latent heat through phase changes. Hence, it can be seen that Eq. 40 corresponds to the ordinal equation of heat conduction, provided that these terms are neglected.
9.11.5 KINETICS OF PHASE TRANSFORMATION During phase transformation, a given volume of material is assumed to be composed of several kinds of constituent ~I as expressed in Eq. 1. We choose four kinds of volume fraction: liquid ~L, austenite ~A, pearlite ~p, and martensite ~M, and other structures induced by precipitation by recovery effect, say, during the annealing process. When austenite is cooled in equilibrium, bainite, ferrite, and carbide are produced in addition to pearlite, but for brevity all these structures resulting from a diffusion type of transformation are called pearlite. The nucleation and growth of pearlite in an austenitic structure are phenomenologically governed by the mechanism for a
892
Inoue
diffusion process, and Johnson and Mehl [10] proposed a formula for volume fraction ~p as ~p = 1 - exp(-Ve) where
Ve means
(42)
the extended volume of the pearlitic structure given by Ve --
/o
4nR(t-
(43)
~)3nd~
Here, R(t) is the radius of the pearlite particle at time t, and n denotes the number of nucleating particles per unit time. Bearing in mind that the value of R is generally a function of stress as well as temperature, Eq. 43 may be reduced to Ve -
f (T, aij) (t - T.)3d~
(44)
Function f(T, 0) can be determined by fitting the temperature-timetransformation (TTT) diagram or continuous-cooling transformation (CCT) diagram without stress, and f(T, aij) may be given by the start-time or finishtime data for pearlite transformation with an applied stress, an example of which is shown in Figure 9.11.3 [12]. The empirical relationship for the austenite-martensite transformation is also obtainable by modifying the kinetic theory of Magee [11]. Assume that the growth of a martensite structure is a linear function of the increase of the difference AG in free energy between austenite and martensite as d~M -- -~(1 - ~M)~d(AG)
/
~o.~ I
./
/ ii
///
~o~ / / / I0
r162
X' /"
~if
/// /j/
10 ~
Time t , s
(45)
oo:
:
0.P,:,
~ ~,~ 10 3
FIGURE 9.11.3 Dependenceof stress on pearlite reaction (AISI4340 steel, 343~
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
893
Regarding the free-energy G as a function of temperature and stress, we can obtain the form of {M by integrating Eq. 45 as {M - 1 - exp [~bI ( T - M~) + q~2(a,j)]
(46)
The function ~b2(crij) is identified by the such data as shown in Figure 9.11.4 [14]. For the case of solidification, we employ the well-known lever rule (see Fig. 9.11.5) as an example, and the volume fraction of austenite is
~
(TL- T)/mL - (~-
(47)
r,,)/,,,,, + (~, - r)/,n,~
where TL and TA denote the liquidus and solidus temperatures, respectively, and mL and mA are gradients of the liquidus and solidus temperatures with respect to the carbon content in the phase diagram.
9.11.6 SOME SIMULATED RESULTS OF ENGINEERING PROCESSES Simulation of some typical engineering processes involving phase transformation has been made by the authors. Following is the list of topics
1.0-
o
0.8-
--o-. Tens i o n --e-- C o m p r
o
gl r m
0s
cl
g
o 0.4 r
.9 r
o1,. 0.2
LL
0s )
FIGURE 9.11.4
/ /
tltll,
2," .......,/:~
_
500 1000 Stress O, MPe
_
_
1500
Dependence of stress on martensitic reaction (29%C steel, 247~
894
Inoue
I--
o
Carbon content C FIGURE 9.11.5 Schematic phase diagram.
with references. The results of heat treatment are those simulated by the finite element CAE system "HEARTS" (HEAt tReaTment Simulation program), which is available for the analysis of the heat treatment process in this stage [14-17]. Quenching Quenching of infinite cylinder [14] Carburized quenching of ring and gear wheel [14, 18]. Induction hardening of ring and gear wheel [19, 20] Quenching of Japanese sword [21, 22, 23] Welding Butt-welding of plates [24] Bead-on-plate [25] Casting Continuous casting of a slab [26] Thin slab casting [27] Centrifugal casting [28]
9.11 Coupling of Stress-Strain, Thermal, and Metallurgical Behaviors
895
REFERENCES 1. Inoue, T., Nagaki, S., Kishino, T., and Monkawa, M. (1981). Ingenieur-Archives 50(5): 315-327. 2. Inoue, T. (1988). Thermal Stresses, pp. 192-278, Hetnarski, Richard B., ed., North-Holland. 3. Inoue, T. (1990). Computational Plasticity: Current Japanese Materials Research, pp. 73-96, vol. 7, Inoue, T., et al., eds., Elsevier Applied Science. 4. Inoue, T., and Raniecki, B. (1978). J. Mech. Phys. Solids 3: 187-212. 5. Inoue, T. (1987). Berg- und Huttenmannische Monatshefte 3: 63-71. 6. Bowen, R. M. (1976). Continuum Physics, Vol. 3, New York: Academic Press. 7. Malvern, L. E. (1969). Introduction to the Mechanics of Continuous Medium, Prentice-Hall. 8. Perzyna, P. (1979). Advances in Applied Mechanics, pp. 315-354, vol. 9. 9. Wang, Z.-G., and Inoue, T. (1985). Material Science and Technology 1: 899-903. 10. Johnson, A. W., and Mehl, R. E (1939). AIME 135: 416-458. 11. Magee, C. L. (1979). Phase Transformations, p. 115, London: Chapman Hall. 12. Bhattacharyya, S., and Kel, G. L. (1955). ASM 47: 351-379. 13. Onodera, H., Gotoh, H., and Tamura, I. (1976). Proc. First JIM Int. Syrup. New Aspects of Martesitic Transformation, pp. 327-332. 14. Inoue, T., Arimoto, K., and Ju, D.-Y. (1992). Proc. First Int. Conf. Quenching and Control of Distortion, Chicago, pp. 205-212. 15. Inoue, T., Arimoto, K., and Ju, D.-Y. (1992). Proc. 8th Int. Congr. on Heat Treatment of Materials, Kyoto, pp. 569-572. 16. Inoue, T. (1998). Mathematical Modelling of Weld Phenomena, pp. 547-575, vol. 4, Cerjak, H., ed. 17. Inoue, T., and Arimoto, K. (1997).J. Materials Engineering and Performance, ASM 6(1): 51-60. 18. Yamanaka, S., Sakanoue, T., Yosii, S., Kozuka, T., and Inoue, T. (1998). Proc. 18th Conf. Heat Treating, ASM International, pp. 657-664. 19. Inoue, T., Inoue, H., Uehara, T., Ikuta, E, Arimoto, K., and Igari, T. (1996). Proc. 2nd Int. Conf. Quenching and Control of Distortion, Cleveland, pp. 55-62. 20. Inoue, T., Inoue, H., Ikuta, E, and Horino, T. (1999). Proc. 3rd Int. Conf. Quenching and Control of Distortion, Prague, pp. 243-250. 21. Inoue, T., and Uehara, T. (1995). Proc. Int. Symp. Phase Transformations during the Thermal/ Mechanical Processing of Steel, The Metallurgical Society of the Canadian Institute of Mining, Metallurgy and Petroleum, pp. 521-524. 22. Inoue, T. (1997). Materials Science Research International 3(4): 193-203. 23. Inoue, T. (1999). Proc. 8th Int. Conf. on Mechanical Behaviour of Materials, Victoria, vol. 2, 458-468. 24. Wang, Z.-G., and Inoue, T. (1985). Material Science and Technology 1(1): 899-903. 25. Sakuma, A., and Inoue, T. (1995). Proc. 5th Int. Symp. on Plasticity and Its Current Applications, Sakai-Osaka, pp. 721-724. 26. Inoue, T., and Wang, Z.-G. (1988). Ingenieur-Archives 58(4): 265-275. 27. Inoue, T., and Ju, D. Y. (1991). Advances in Continuum Mechanics, pp. 389-406. Bruller, O., Mannl., V., and Najar, J., eds., Springer-Verlag. 28. Du, D. Y., and Inoue, T. (1996). Materials Science Research International 2(1): 18-25.
SECTION
9.12
Models for Stress-Phase Transformation Couplings in Metallic Alloys S. DENIS, P. ARCHAMBAULT and E. GAUTIER Laboratoire de Science et G~nie des Mat~riaux et de M&allurgie, UMR 7584 CNRS/INPL, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France
Contents 9.12.1 Domain of Application . . . . . . . . . . . . . . . . . . . . 9.12.2 Theory Used to Derive the Models . . . . . . . . 9.12.3 Description of the Models . . . . . . . . . . . . . . . . 9.12.3.1 Kinetics Models . . . . . . . . . . . . . . . . . . . 9.12.3.2 Thermomechanical Model . . . . . . . . . 9.12.4 Identification of the Material Parameters .. 9.12.4.1 Metallurgical Characterization . . . . . . . . . . . . . . . . . . . 9.12.4.2 Thermomechanical Characterization . . . . . . . . . . . . . . . . . . . 9.12.5 Implementation in Codes . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
896 897 897 897 899 901 902 902 903 903
9.12.1 DOMAIN OF APPLICATION The models described in this article, which concern phase transformation kinetics and thermomechanical behavior, can be used for many steels and high-strength aluminium alloys submitted to various thermal histories (heating, cooling, isothermal holding) and to stress states that lead to small plastic strains (a few percent). Typical applications are the modeling of heat treatment processes (quenching, surface hardening [induction, laser, electron beam hardening], case hardening) and also welding processes (for the heataffected zone behavior).
896
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
9.12 Stress-Phase Transformation Couplings in Metallic Alloys
897
9.12.2 THEORY USED TO DERIVE THE MODELS The kinetics models rely on the general theory of solid-solid phase transformations [1]. In order to predict kinetics for various thermal histories, we have developed either a global approach based on the Johnson Mehl Avrami formalism that assumes an additivity principle (for diffusiondependent transformations in steels or titanium alloys), or an approach that explicitly includes nucleation and growth laws (for precipitation in aluminium alloys). The thermomechanical behavior of the material during the phase change is described by phenomenological behavior laws that include the thermoelastic and plastic-viscoplastic behavior of the stable multiphase material (mixture rules) and the effect of the phase transformations (volume changes, transformation plasticity) as proposed by several authors (for example, [2]). For precipitation at very small volume fractions, the approach through mixture laws is no longer valid, and a model describing both mechanical softening due to heterogeneous precipitation (according to the literature about solute hardening [3]) and hardening due to homogeneous precipitation (by including dislocation-precipitate interactions [4]) has been proposed.
9.12.3 DESCRIPTION OF THE MODELS 9.12.3.1
KINETICS MODELS
9.12.3.1.1 Global Description The model uses isothermal transformation kinetics (IT diagrams) to calculate the anisothermal kinetics on heating and cooling by applying an additivity principle [5]. Thus, for steels, the austenitization kinetics on heating as well as the kinetics of decomposition of austenite (into ferrite, pearlite, bainite, martensite) during cooling can be described. We recall here only the model on cooling. For diffusion-dependent transformations, incubation and growth periods are treated separately. The incubation period is determined according to Scheil's method: the transformation during continuous cooling begins when the sum S becomes equal to unity:
S =fZi Ati/z(Ti) where f is a heredity factor that has been introduced to take into account nonadditivity of incubation at the transition from the pearlitic to the bainitic domain, z(Ti) is the incubation time at temperature Ti, and Ati is the time step.
898
D e n i s et al.
The diffusion-dependent transformations (in steels, the formation of proeutecto'id ferrite or cementite, pearlite, or bainite) are modeled according to the law developed by Johnson-Mehl Avrami and Kolmogorov: Yh = y~ax[1 - exp (-bk(T)t hi(r)] where Yk is the volume fraction of constituent k, y~nax is the maximum that can be formed at a given temperature, and nk and bk are coefficients that are determined for each temperature from the IT diagram (t = 0 corresponds to the end of the incubation period). The transition from isothermal to nonisothermal kinetics is done through the fictitious time method [6]. For martensitic transformation, the progress of the transformation is calculated using the relation established by Koistinen and Marburger: YM -- y~[1 - exp (-aM(Ms -- T)] where yM is the volume fraction of martensite, yy is the remaining austenite fraction at Ms, and aM is a constant. This model takes into account the effect of austenitic grain size on the transformation kinetics on cooling as the effect of a carbon enrichment of austenite due to a previous ferritic transformation on the subsequent bainitic and martensitic transformations. Moreover, the influence of a stress state has been included: for diffusion-dependent transformations, the acceleration of the transformations due to a stress state is taken into account through phenomenological laws relating ~c(Ti), nk, and bk to the von Mises stress; for martensitic transformation, Ms is related to the hydrostatic stress and to the von Mises stress. The effects of carbon content heterogeneities (such as those introduced by carburizing [5] or inheritated from solidification segregations [7]) are included as well. A hardness calculation is associated with this phase transformation model, which implicitly takes into account the effect of the morphology of the different constituents. Recently, the description of tempering kinetics on heating and selftempering kinetics on cooling has also been added for low alloyed steels [8]. A similar modeling approach has been used for the prediction of the precipitation of the 0~ phase from the 1~ phase in a titanium alloy during continuous cooling [9]. This global approach is efficient in predicting the volume fractions of the different constituents during heat treatments of industrial alloys. But, since it does not take into account the nucleation and growth rates explicitly, it does not allow one to properly take into account the effects of hot forming on the subsequent transformations on cooling (enhancement of nucleation).
899
9.12 Stress-Phase Transformation Couplings in Metallic Alloys
9.12.3.1.2 Nucleation and Growth Modeling Considering the precipitation, as in aluminium alloys, the nucleation and growth of the precipitates have been modeled in order to predict the volume fractions of precipitates during a cooling process and also their mean radius and their distribution as well as the chemical composition of the remaining solid solution. The model developed [ 10, 11] includes both heterogeneous and homogeneous precipitation, which occur, respectively, at higher and lower temperatures during cooling. The equilibrium conditions are described by the solubility product. The heterogeneous nucleation rate is thus expressed as: lq = Iq0)~(Ntot- N~t)exp [-(AG* + Qa)/RT] where lq0 is a constant, 2 is a vacancy acceleration ratio, Ntot and Nact are the total and active density of heterogeneous nucleation sites, and Qd is the diffusion activation energy. AG* is the critical free enthalpy variation that depends on the precipitate surface energy 7 and on the variation of the nucleation free enthalpy variation AGv. For cylindrical precipitates, we have: A G * - 224/81rco~Nf73/aG 2 where N is the Avogadro number, c~ is a geometrical parameter, and f is a wetting function representing easier nucleation on heterogeneous sites. Homogeneous nucleation differs through the morphology of the precipitates (spheres), the interfacial energy, the wetting function, which disappears, and the elastic energy, which has to be taken into account. Postulating quasi-stationarity, the growth rate can be expressed as a function of the precipitate-matrix interface concentration XInt of element i" at~at
-
ss
-
xl
-
int
)
where X~~ and X~ are, respectively, the concentration of element i in the solid solution and in the precipitate, and, Di is the diffusion coefficient of element i. Local equilibrium is considered at the interface. Because of the relative large size of precipitates, the Gibbs-Thomson effect has been neglected. This model has been applied to an A1-Zn-Mg-Cu alloy (7000 series) considering the precipitation of the r/phase.
9.12.3.2
THERMOMECHANICAL MODEL
The modeling of the thermomechanical behavior of a material undergoing a phase transformation must include the thermoelastic and plastic-viscoplastic behavior of the stable multiphase material and the effect of the phase transformations.
900
D e n i s et al.
In order to be able to describe the great complexity of real materials like steels, mainly macroscopic phenomenological behavior laws have been developed [2] (see also review paper [12] for other references). It is generally assumed that the total strain rate is an addition of different contributions: 9t
9e
9th
9tr
9tp
9i n
eij = eij + eij + ~i) + eij + eij
(1)
9 9 e/j is the elastic strain rate, which is related to the stress rate by Hooke's law. Young's modulus and Poisson's ratio have to be temperature and microstructure dependent. (Here, microstructure means "volume fractions of the different phases"). 9 eij is the thermal strain rate, which takes into account the thermal expansion coefficients % of the different phases and their dependence on temperature: e~h --ZkYk f = ~ ( r ) d r . 9i n 9 eij is the inelastic strain rate: either the plastic strain rate when no viscous effects are considered or the viscoplastic strain rate. It is calculated using the classical theory of plasticity or viscoplasticity with the associated hardening rules (isotropic and/or kinematic) or is obtained from a micro-macro approach. All material parameters (yield stress, hardening parameters, strain rate sensitivity) are to be considered as temperature and microstructure dependent. Mixture rules are generally assumed. In the case of chemical composition heterogeneities, material properties depend also on local composition. Presently, a dependency with carbon content can be taken into account. In addition, it should be mentioned that taking hardening into account is quite complex when a phase transformation occurs. Models have been proposed to account for some possible recovery of strain hardening during a phase transformation; i.e., the new phase "remembers" or does not remember part of the previous hardening.k~jr is the strain rate due to the volume change associated with the different phase transformations: 9th
~,tijr- ~.,kYk~,tkr. tp 9 e/j is the transformation plasticity strain rate. It is generally assumed that 9
it is proportional to the stress deviator (discussion about this assumption can be found in [ 12]): - 3/2
f'
where Kk and f are experimental parameter and function, and s o are the components of the stress deviator. When considering aluminium alloys, the mechanical behavior of the supersaturated solid solution during cooling can be described through Eq. 1. But, because of the very low precipitate volume fractions, the deformation
901
9.12 Stress-Phase Transformation Couplings in Metallic Alloys
associated with the volume change is negligible as transformation plasticity. A thermo-elasto-viscoplastic behavior has been considered, and the flow stress is written: O'ss = 0"0 -I- H(gvP) n -[- K(~,vP) m
where or0 represents a threshold stress, H(eVP)" is the hardening due to the deformation, K(~,vP) m is the viscous stress, evp and kvp are, respectively, the viscoplastic strain and strain rate, H and n are hardening parameters, and K and m are the consistence and the strain rate sensitivity. All parameters depend on temperature. For considering the effects of precipitation on the mechanical behavior, we consider that the flow stress of the alloy can be written: ~r = Crss + A~r. This assumes that the precipitation process only influences the yield stress and has no influence on the hardening behavior. Act accounts for solute and precipitation hardening mechanisms in the following way. For heterogeneous precipitation, precipitates are non coherent and do not lead to hardening. Thus the mechanical behavior is mainly sensitive to the depletion of the solid solution (decrease of the solute content). According to literature about solute hardening, the mechanical softening can be expressed as [11, 13, 14] t) =
n
n
where X~s~ is the concentration of element i in the supersaturated solid solution, ai are temperature-dependent coefficients, and n is a temperatureindependent coefficient. Homogeneous precipitation (which is coherent or semicoherent) increases the yield stress of the alloy. The modeling of this hardening accounts for the dislocation-precipitate interaction which leads to [11]: Acrho(T, t) = g(f~, r), where f~ and r are, respectively, the volume fraction and the radius of the precipitates. The resulting precipitation-induced mechanical effect is then: Aa(T, t) = A~rhe -+- A~rho, where Aahe is negative and ACrho is positive.
9.12.4
IDENTIFICATION
OF THE MATERIAL
PARAMETERS In general, we perform material characterizations (metallurgical and thermomechanical) at constant temperatures after rapid cooling from the austenitization temperature for steels or from the solutionizing temperature for aluminium or titanium alloys. Anisothermal tests are mainly used to validate the models. (In some particular cases, they are used to get parameters that cannot be obtained through isothermal tests.)
902 9.12.4.1
D e n i s et al.
METALLURGICAL CHARACTERIZATION
The global isothermal transformation kinetics are measured at different temperatures by dilatometry and/or in situ resistivity measurements (in titanium and aluminium alloys resistivity is much more sensitive to phase transformation than dilatometry). In addition, microstructural analysis and quantitative image analysis of the isothermally transformed specimen allow one to obtain the equilibrium volume fractions at the different temperatures. (They can also be obtained through thermodynamic calculations of multicomponent systems.) The hardness of the different phases depending on the transformation temperature are also obtained. For the nucleation and growth model, identification of nucleation sites and precipitated phases (chemical composition) and determination of mean precipitate size evolutions (by transmission electron microscopy) are done after isothermal interrupted quenching tests. Different parameters of the model can be taken from the literature (equilibrium diagram, diffusion coefficient), but some other parameters (like nucleation density, wetting function, or surface energy) are unknown. Thus they are determined through comparisons between calculated and experimental results. Kinetics measurements during continuous cooling tests (at constant cooling rates) and microstructural analysis of the specimen are also performed in order to validate the metallurgical calculations (comparison between measured and calculated kinetics, final microstructures and hardnesses). These tests also give access to the thermal expansion coefficients of the different phases and to the volume changes associated with the phase transformations. Of course, additional experiments are necessary to quantify the effect of the parent phase grain size on the transformation kinetics or the incidence of chemical composition variations of the alloys.
9.12.4.2
THERMOMECHANICAL CHARACTERIZATION
On one hand, the mechanical behavior of the individual phases (austenite, ferrite, pearlite, bainite in steels or supersaturated solid solution in aluminium alloys) is obtained through tensile tests (at constant deformation rates) followed by relaxation tests at constant temperatures after formation of the required phases. These tests are performed on our thermomechanical testing apparatus (DITHEM) and allow us to identify the evolutions of the yield stress, the hardening parameters, and the parameters describing the viscous effects versus temperature for the different phases (i.e., by using an optimization software like SIDOLO).
9.12 Stress-Phase Transformation Couplings in Metallic Alloys
903
On the other hand, the effects of the phase transformations on the material behavior must be quantified. The changes in mechanical properties due to precipitation in aluminium alloys have been quantified by performing tensile tests for different holding times at a constant temperature. Thus, for example, the decrease of the yield stress related to heterogeneous precipitation and consequently to the depletion of the solid solution has been obtained [13]. In the case of alloys that, in addition to the change in mechanical properties induced by the new phase, exhibit transformation plasticity (like steels, titanium alloys), experimental studies aim to quantify this additional deformation. Typical tests consist in cooling down rapidly a specimen at the transformation temperature and in applying a constant stress before transformation starts and maintaining it all along the transformation. From the dilatometry and/or resistivity measurements, we get both the transformation plasticity amplitudes versus the applied stress and the effect of the applied stress on the transformation kinetics (for ferritic, pearlitic, bainitic transformations in steels, for example). Moreover, experiments in which the material is submitted to a tensile test either during its isothermal transformation or during continuous cooling transformations allow us to validate Eq. 1 [ 12]. Let us mention that numerous steel characterizations have been performed, but for confidentiality reasons only few of them can be found in publications (see, for instance, [12]). 9.12.5
IMPLEMENTATION
IN CODES
Our global approach for describing phase transformation kinetics during heating and cooling in steels has been implemented in different finite element codes (in house software [5] and different industrial softwares like LAGAMINE, REFPROE SYSWELD, FORGE) in order to predict the coupled thermal-metallurgical-mechanical processes during heat treatments of steels (quenching, surface hardening, case hardening) .The kinetics model for precipitation and the associated thermomechanical model have been introduced in the finite element code MARC for calculating temperature fields, metallurgical fields, and residual stresses and deformations of quenched high-strength aluminium work pieces [14]. REFERENCES 1. Christian, J. W. (1975). The Theory of Transformations in Metals and Alloys. Pergamon Press. 2. Sj6str6m, S. (1994). Physical, mathematical and numerical modelling for calculation of residual stress: Fundamentals and applications, Proc. 4th Int. Conf. on Residual Stresses ICRS4 pp. 484-497, James, M. R., ed., Bethel: Society of Experimental Mechanics.
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3. Sigli, C., Vichery, H., and Grange, B. (1996). Proc. ICAA-5, 5th Int. Conf. on AI Alloys, Vol. 1, pp. 391-396, Transtech Publications, Switzerland. 4. Ardell, J. A. (1985). Precipitation hardening. Metall. Trans. A, 16A: 2131-2165. 5. Denis, S., Archambault, P., Aubry, C., Mey, A., Louin, J. Ch., and Simon, A. (1999). Modelling of phase transformation kinetics in steels and coupling with heat treatment residual stress predictions. Journal de Physique IV France 9: 323-332. 6. Pumphrey, W. I., and Jones, E W. (1948). Inter-relation of hardenability and isothermal transformation data. JISI 159: 137-144. 7. Louin, J. Ch., Denis, S., Combeau, H., Lesoult, G., Simon, A., Aliaga, Ch., and Massoni, E. (2000). Effect of solidification segregations on phase transformation kinetics and on the development of internal stresses and deformations during cooling of steels, in Proc. 5th European Conference on Residual Stresses, pp. 205-210, B6ttger, A. J., Delhez, R., and Mittemeijer, E. J., eds., ECRS5, Trans Tech Publications Ltd, Switzerland. 8. Aubry, C., Denis, S., Archambault, P., Simon, A., and Ruckstuhl, E (1998). Modelling of tempering kinetics for the calculation of heat treatment residual stresses, Proc. ICRS5 (Int. Conf. on Residual Stresses), pp. 412-417, Ericsson, T., Oden, M., and Andersson, A., eds., Link6ping University, Sweden. 9. Laude, E., Gautier, E., Archambault, P., and Denis, S. (1996). Cin~tique de transformation des alliages de titane en fonction du traitement thermom~canique. Etude exp~rimentale et calcul. La Revue de M~tallurgie-CIT/Science et G~nie des Mat~riaux, 1067-1078. 10. Godard, D., Gautier, E., and Archambault, P. (1999). Modelling heterogeneous precipitation kinetics in a A1-Zn-Mg-Cu aluminum alloy, in Proc. PTM'99, International Conference on SolidSolid Phase Transformations, pp. 145-148, Koiwa, M., Otsuka, K., and Miyasaki, T., eds., The Japan Institute of Metals, Kyoto. 11. Godard, D. (1999). Influences de la precipitation sur le comportement thermom~canique lors de la trempe d'un alliage A1- Zn-Mg-Cu. Th~se de Doctorat de I'INPL, Nancy, France. 12. Denis, S., Archambault, P., Gautier, E., Simon, A., and Beck, G. (1999). Prediction of residual stress and distortion of ferrous and nonferrous metals: Current status and future developments, Proc. 3rd Int. Conf. on Quenching and Control of Distortion, pp. 263-276, Totten, G. E., Liscie, B., and Tensi, H. M., eds., ASM International; Journal of Materials Engineering and Performance (accepted October 2000). 13. Godard, D., Archambault, P., Houin, J. P., Gautier, E., and Heymes, E (1998). Mechanical softening kinetics at high temperatures in an A1MgZnCu alloy: Experimental characterization and microstructural interpretation, Proc. ICAA-6 pp. 1033-1038, Vol. 2, Sato, T., et al., eds., The Japan Institute of Light Metals. 14. Archambault, P., Godard, D., Denis, S., Gautier, E., and Heymes, E (1999). Prediction of heat treatment residual stresses: Application to quenching of high strength aluminium alloys including precipitation effects, in Proc. 7th Int. Seminar on Heat Treatment and Surface Engineering of Light Alloys, pp. 249-258, Lendva, J., and Reti, T., eds., Hungarian Scientific Society of Mechanical Engineering.
SECTION
9.13
Elastoplasticity Coupled with Phase Changes E D. FISCHER Montanuniversitht Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
Contents 9.13.1
Validity ..................................
905
9.13.2
Background
..............................
905
Description
of the Model .................
907
9.13.3
9.13.3.1
Transformation Kinetics ...........
907
9.13.3.2
Deformation Process ..............
909
9.13.4
Identification .............................
9.13.5
H o w to U s e the m o d e l . . . . . . . . . . . . . . . . . . . .
913
9.13.6
Table of Parameters
913
......................
References .......................................
9.13.1
911
913
VALIDITY
The transformation-induced plasticity (TRIP) term as an additional strain rate term to the "classical" plastic strain rate term is valid for ductile metals like steel under diffusional or displacive (e.g., martensitic) phase transformations subjected to a monotonic loading path. The transformation kinetics described herein is valid for many technically relevant metals subjected to nearly any paths of heating or cooling. The selection of mechanical material data is typical of heat treatment simulations, welding simulations, or surface treatment simulations by induction hardening, carburizing, nitriding, etc.
9.13.2
BACKGROUND
In a recent review paper, Fischer et al. [1] defined TRIP as "significantly increased plasticity during a phase change. For an externally applied load for Handbook of Materials BehaviorModels. ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
905
906
Fischer
which the corresponding equivalent stress is even small compared to the normal yield stress of the material, plastic deformation occurs . . . . " This softening has its origin in the fact that during a phase transformation a certain part of the material (say, a microregion) may change its volume and, occasionally, its shape, too. An internal stress state is necessary to achieve compatibility between the neighboring material and the transforming microregion. This internal stress state is superimposed by the load stress. The total stress state, e.g., in the case of a volume change of several percentage points, leads to plastification of the neighboring material and often of the microregion itself. One may speak about a plastic accommodation of the transformation eigenstrain. Finally, a macroscopic plastic deformation of the specimen can be observed. Among several authors, Leblond et al. [2] suggested a splitting of the plastic strain rate ~ (the dot represents a time derivative) into a classical plastic strain rate ~c representing plastification without any phase change "TP and an extra term ~ij corresponding to plastification without any changes of external loading and only effected by the phase change process. Assuming only two phases, the parent phase and the product phase, the volume fraction of the product phase is denoted as ~. Based on some elementary research by Greenwood and Johnson as well as Mitter (see Fischer et al. [1]), the following relation for eij9~P has eventually been suggested:
~,~e
3 d(.p(r / ~- -~ K d~ ij~
(1)
where K is a coefficient depending on the transformation volume change 5, "TP (p(~) is a function describing the evolution of e~j with ~ for a constant loadstress state, and S~j is the global (mesoscopic) stress deviator and corresponds to the stress deviator in the standard analysis of a work piece. The original derivation of the TRIP strain is based on S~j being constant in time. However, in the current formulation of Eq. 1, S~j is allowed to vary with time. Furthermore, ~c is usually denominated as ~. Equation 1 is considered to be valid for both a diffusional (time-controlled) and a martensitic (temperatureand stress-controlled) transformation. However, this cannot be held for the martensitic transformation, especially under nonmonotonic loading and temperature paths, since the martensitic transformation is characterized by a volume change 5 and a shape change (transformation shear) y of a single transforming microregion. Therefore, in addition to the previously mentioned elastic-plastic accommodation process, often referred to as the "GreenwoodJohnson effect," an orientation process takes place due to the formation of preferred variants in the martensitic microregions that may arrange themselves in some (partially)"self-accommodating" groups. This orientation
9.13 Elastoplasticity Coupled with Phase Changes
907
process, often referred to as the "Magee effect," may contribute in the same magnitude to the irreversible length change of elastic plastic material as the "Greenwood-Johnson effect." Thus shape memory alloys "thrive" exclusively on the orientation process! However, if monotonic loading paths are considered, a relation like Eq. 1 can be applied, if the coefficient K is taken from a "Greenwood-Johnson" diagram. Concerning an improved TRIP term for martensitic transformation, the reader is referred to a recent paper by Fischer et al. [3]. However, it must be said that this research is not yet finished. Finally, the book edited by Berveiller and Fischer [4] is recommended; it presents an extensive treatment of both types of transformations in various kinds of metals.
9.13.3 DESCRIPTION OF THE MODEL The whole model must consist of two more or less connected parts: 9 the transformation process itself, i.e., the metal-physical part described in following text as "transformation kinetics"; and 9 the deformation process described as the "deformation process." External variables to the model are both the load-stress state and the temperature field T(x, y, z, t). The evolution of the temperature field is obviously coupled with the phase change process by the production or extraction of heat (the "latent" heat or "transformation" heat or "recalescence" heat). This coupling must not be omitted. Here, the reader is referred to the chapter on heat treatment and corresponding data, e.g, the specific heat (heat capacity). Usually the coupling of the temperature field with plastic dissipation is weak and consequently need not be taken into account.
9.13.3.1
TRANSFORMATION KINETICS
In practice, a picture on the transformation kinetics is reflected by the time (t)-temperature (T)-transformation (T-T-T) diagram (continuous cooling diagram [CC], cooling transformation diagram [CT], continuous cooling transformation diagram [CCT]). This diagram is based on t-T curves for standard specimens. A typical T-T-T diagram for a certain steel grade is depicted in Figure 9.13.1. The cooling velocity ~ = dT/dt may vary along the curve. Several points are marked describing the type of the phase and the corresponding volume fraction of the phase under consideration. Typically,
908
Fischer
"i . . . I....I IIIit111 ~111
l l|ll
I IIII I I Ill tllLI
III L Illl I I|il
I I Ill
.li......
I
'~:! 0.1
! II t IIiH,,. I Il[I
111150o
lil I t Ill,**
I1 Ix~IIXN:[I~_klII
I xll
I I r~,,,,\l, llh,,! "~1 I ,~,, 1
10
10 ~
~0~
10 ~
Ilil i lH -~
11:oL,,~,H,o 105
10~
Time, seconds
FIGURE 9.13.1 Typical CCT diagram for a certain steel grade with AC3=830~ Abbreviations: A austenite, B bainite, F ferrite, M martensite, P pearlite. The numbers in the individual areas (as 1, 2, 10, 30, etc.) stand for the volume fraction of the corresponding formed phases. The numbers
at the termination of the curves stand for a certain hardness, e.g., DPH. several phases can be formed. The reader is referred to textbooks such as References [5, 6]. The T-T-T diagram must be transferred into the calculational procedure. Leblond et al. [7] made a proposal for how to perform this. For sake of simplicity this proposal is briefly explained by assuming only one phase transformation with the transformed volume fraction ~. Details can be taken from Leblond et al. [7] as well as from the SYSWELD-Manuals [8]. The transformed volume fraction ~ is described by the following differential equation: d~
dt =
Ceq(T)-~
z(r)
f(]-)
(2)
where ~eq(T) is the equilibrium volume fraction of the phase corresponding to a given temperature T and can only be estimated from the T-T-T diagram. ~eq(T) could be found in the "isothermal" T-T-T diagram (isothermal transformation diagram [IT]), which, however, is often not available. The functions z(T) and f(~') must be selected such that the curves in the T-T-T diagram are approximated in a satisfying way. For example, SYSWELD [8] provides a specific routine for constructing ~(T) and f(~F). Equation 2 can also be applied to the martensitic transformation, whose kinetics are usually described by the Koistinen-Marburger relation (see
9.13 Elastoplasticity Coupled with Phase Changes
909
Reference [9]) in total quantities and not in rate form as {(T)
-- l - e x p [ - b ( M s
- T)]
for T < M s
(3)
It follows that ~eq - 1, z - 1 / b , f ( S f ' ) - - ' F . It should be mentioned that ~(T) in Eq. 3 should also include a stress-dependent term. A constant load stress shifts the martensite start temperature Ms to a higher temperature Mo; (M~-Ms) is of the order of magnitude of some degrees centigrade. The influence of the stress state on ~(T) plays a significant role in the case of shape memory alloys (SMAs) and a minor role in the case of steels. The kinetics explained previously deal with a cooling path. In the case of steel and many other alloys, the situation is totally different with respect to a heating path from room temperature to a high temperature level, say, ca. 1000~ The transformation is totally "unsymmetric" with respect to a t - T cycle. All the phases mentioned remain more or less unchanged until the AC1 temperature, the austenite start temperature, is reached. At this stage a diffusional transformation to austenite starts, which stops at AC3. Of course, AC1 and AC3 depend on the heating rate. Very high heating rates shift AC1 and AC3 to remarkably higher temperatures. The T-T-T diagram offers mostly AC3 and sometimes AC1 for slow heating and no further information on the heating transformation kinetics. It is interesting to note that some recent experiments give clear evidence that also a martensite-to-austenite displacive "back transformation" may occur in steels at higher temperatures, say, _> 600 ~ see Reference [3]. Finally, it is trivial to state that in the case of several phases the volume fraction {i of each individual phase, i - 1,... , n, must add up to 1! The reader is also referred to some further papers [10-12] dealing with the mathematical modeling of phase transformations and temperature fields. These papers also deliver some relevant data for the different phases.
9.13.3.2
DEFORMATION PROCESS
The deformation process is modeled according to the standard concept of elastic- (visco-) plastic materials. Therefore, only some specific details on the material data with respect to phase change phenomena are given. The following data are needed for a stress analysis: 9 Young's modulus E ( T ) . For steel E ( T ) is mostly the same for all phases. In the case of shape memory alloys, Young's modulus may differ significantly for the austenitic (7-) and martensitic (~'-) phase. 9 Poisson's ratio v, which is only weakly dependent on the temperature.
910
Fischer
9 Integral coefficient of thermal expansion (CTE), which differs significantly between the austenitic (7-) phase and the ferriticpearlitic-bainitic-martensitic (~- or ~-) phases. Usually the CTE depends weakly on the temperature and increases slightly with increasing temperature. The corresponding volume change is usually assumed to be isotropic and is 3 CTE. (T-Tref), Tref being the thermal strain-free reference temperature. 9 Volume change due to phase transformation, c~. As mentioned in Section 9.13.2, a phase transformation is accompanied by a volume change. The corresponding standard test is the dilatometer test. A typical value for for the transformation of Fe-based alloys from the 7- to ~- (or ~-) phase is _< 0.04. In the case of shape memory alloys, the value c~is nearly 0! If several phases are present, a "mixed" value for c~ is applied as c~~inl ~i~Si,i being number of phase, c~ibeing volume change of phase i, ~i see previous text. 9 Yield stress Criy(T, ~), i being number of phase; for details see Section 9.13.4. 9 In the case of several phases present, a mixture rule for the yield stress must be installed. For two phases, e.g., ~ and 7, nowadays a nonlinear mixture rule is applied, cryaX(T) -- [1 -f(~)]aVy(T)+f(~)ay(T),
~ + ~ = 1
(4)
f(~) is given from numerical studies. A representative data set can be found in Table 9.13.1. 9 Type of hardening; for details, see Section 9.13.4. 9 Transformation-induced plasticity (TRIP) term ~/'. For the sake of completeness, a current formulation of the TRIP strain rate is presented which is implemented in user-supplied material law formulations in several programs or is obligatory in the program SYSWELD [8]. The theoretical background has been explained in Section 9.13.2. The term k~.e adds up to the other strain rate terms as 9
.e
9t h
9TP
where ~ is the elastic strain rate, k~ is the conventional plastic strain rate or ~ is the conventional viscoplastic strain rate, e0 is the thermal strain rate, and k0 is the total strain rate. 9
9th
TABLE 9.13.1 WeightingFunction f(~) for Nonlinear Mixture Rule ~
0
12.5%
25%
50%
75%
100%
f(~)
0
0.044
0.124
0.391
0.668
1
911
9.13 Elastoplasticity Coupled with Phase Changes
In accordance with the notation used in Eq. 1, the following data and function are proposed:
K - 2c~/a~yh(oeqCr~y)
(6a)
q~({) = {(1 - In {)
(6b)
where { corresponds to the fraction of the 0~- or 0(-phase, h(oeq/a~y) is a correction factor, which is 1.0 for Oeq/O~ < 1/2, and Oeq is the equivalent stress to S/j.
9.13.4
IDENTIFICATION
The kinetics data can partially be identified from a T-T-T diagram. However, ~eq to a certain phase is often difficult to establish. A respresentative data set for various steel phases is given in Table 9.13.2. Usually the steel specimen is austenitized at 850 to 1000~ and held for some time (e.g., half an hour). Then the cooling process is started and plotted in a T-T-T diagram. A typical data set for martensitic transformation is Ms = 420~ b = 0.03. A typical austenitization interval ranks between A C 1 = 7 0 4 ~ and AC3 = 830~ A typical data set for E, v, CTE is E ( 2 0 ~ 210Gpa, E ( 1 0 0 0 ~ 105 Gpa, v=0.3, CTE~ ~ 12.10-6C ~ for 20~ C T E ~ 18.10-6C ~ for 20~ The transformation volume change c~ is obtained from a standard dilatometer test. Please check for the presence of a texture (orthotropic plastic anisotropy) in the specimen. A sure indication will be that c~will not be distributed isotropically. With respect to the yield stress aiy(T, ~) and the type of hardening, usually uniaxial stress-strain curves for monotonic loading are used as input data to the programs. Of course, data for all phases in the relevant temperature range are needed. It should be mentioned that very often the material data for a specific phase cannot easily be found, since only a given mixture of the phases is thermodynamically stable (or metastable). This forces us to extrapolate a
TABLE 9.13.2 Phase Transformation Intervals for a High-Strength Steel Phase Ferrite Pearlite Bainite Martensite
Start T(~ 704 675 537 279
End T(~ 675 537 279 20
dT/dt
~ (total) %
- 2.1 -2.1 - 1.9 - 2.6
5 25 55 100
912
Fischer
large amount of necessary data. It is highly recommended to consult a metallurgist for this problem. Usually cry(T,e) decreases with increasing temperature and fixed strain. However, there exist metals like Intermetallics, where cry(T,e) may increase with increasing temperature. Furthermore, the yield stress for the austenite cr~(20~ e) is usually smaller than that of ferritic, a a t pearlitic, bainitic, or martensitic (~- or ~'-) phase cry(20~ e) or Cry at room temperature. At higher temperatures, say >600~ the situation may change, and the ~- or ~'-phase becomes the softer phase. If a temperature cycle is applied for a temperature interval including the "crossing" point of the yield stresses of both phases, ratcheting will occur even in free specimens without any load stress; see details in Siegmund et al. [13] or Silberschmidt et al. [14]. That means that the specimen will change its size monotonically during a temperature cycle. Obviously, this could be extremely dangerous for devices. However, modern alloying techniques allow us to control the temperature a a t dependence of cr~ or cry and cry, for example, by a certain amount of nitrogen. Both cases are now possible, a 7-phase with a yield stress higher than that of the ~- or ~-phase and vice versa for a temperature range between 20~ and 1000~ Here the reader is referred to modern duplex steels; see also Silberschmidt et al. [ 14]. Since, for example, a heat treatment may lead to several changes of the sign and orientation of the principal stress components in a material point, one has to consider, at least for some of the possibly developing phases, how they behave under cyclic stressing. Of course, standard hardening mechanisms like isotropic hardening or kinematic hardening are available in most of the programs. However, one must keep in mind that especially during heating the dislocation density may be reset by recrystallization to nearly its original amount. That means that very careful control over the amount and the kind of hardening must be exercised in the simulation of thermal cycles. Specifically, almost no hardening occurs for martensite if the alloy contains some carbon. a t In addition, the existence of a very high yield strength (cr~ > 1500 MPa) is usually accompanied by a small fracture strain (only a few percentage points). The only additional mechanical quantity, the coefficient K, in the TRIP term (see Eqs. 1), 6a, needs some comments. Classical TRIP tests (sometimes also called "creep" tests) must be performed to find K by applying a constant load ]~ on a specimen (e.g., weight on a wire) and then performing the transformation by cooling. The irreversible length change J-P (e~-P= e - 6 / 3 - e e) must now be represented in relation to the load stress. The data points should lie on a straight line for []El
9.13 Elastoplasticity Coupled with Phase Changes
913
9.13.5 H O W TO USE THE MODEL T h e TRIP term ~/~.P,Eq. 1, m u s t be a d d e d to the o t h e r c o n t r i b u t i o n s of the total .TP 9F u r t h e r m o r e , a c o n t r o l over the t r a n s f o r m a t i o n k i n e t i c s m u s t s t r a i n rate eij be i m p l e m e n t e d . O n l y few g e n e r a l p u r p o s e p r o g r a m s a l l o w for b o t h ; o n e is the p r o g r a m SYSWELD [8]. T h e p r o p e r i m p l e m e n t a t i o n is by n o m e a n s an easy p i e c e of w o r k . At least an o p t i o n for a u s e r - s u p p l i e d m a t e r i a l d e s c r i p t i o n , e.g., a UMAT in ABAQUS [15], m u s t be available. T h e c o m p u t a t i o n a l a l g o r i t h m s are the s a m e as t h o s e for e l a s t o - ( v i s c o ) p l a s t i c b e h a v i o r , b u t w i t h an a d d i t i o n a l s t r a i n r a t e c,9ijrP , w h i c h is n o t c a l c u l a t e d via a c o n s i s t e n c y r e l a t i o n b u t p r o v i d e d e x t e r n a l l y like a c r e e p s t r a i n rate.
9 . 1 3 . 6 TABLE OF PARAMETERS Several data are given in Section 9.13.4, where only two tables are mentioned.
REFERENCES 1. Fischer, E D., Sun, Q.-P, and Tanaka, K. (1996). Transformation-induced plasticity (TRIP). Appl. Mech. Rev. 46:317-364. 2. Leblond, J. B., Mottet, G., and Devaux, J. C. (1986). A theoretical and numerical approach to the plastic behaviour of steels during phase transformations. I. Derivation of general relations; II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34:395-410, 411432. 3. Fischer, E D., Reisner, G., Werner, E., Tanaka, K., Cailletaud, G., and Antretter, T. (2000). A new view on transformation induced plasticity (TRIP). Int. J. Plasticity 16: 723-748. 4. Berveiller, M., and Fischer, E D. (1997). Mechanics of Solids with Phase Changes, CISM Courses and Lectures No 368, New York, Springer. 5. Krauss, G. (1980). Principles of Heat Treatment of Steel, Metals Park, Ohio: American Society for Metals. 6. Transformation and Hardenability in Steels (1967). No editors, Symp. Climax Molybdenum Comp. and Univ. Michigan, The Univ. Michigan Extension Service. 7. Leblond, J. B., and Devaux, J. (1984). A new kinetic model for anisothermal metallurgical transformations in steels including effect of austenite grain size. Acta Metall. 32: 137-146. 8. SYSWELD + 2.0: Reference Manual (1997), SYSTUS International, ESI Group: Simulation of Welding and Heat Treatment Processes, Framasoft + CSI. 9. Koistinen, D. P., and Marburger, R. E. (1959). A general equation describing the extent of the austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels. Acta Metall. 7: 59-60. 10. Denis, S., Farias, D., and Simon, A. (1992). Mathematical model coupling phase transformations and temperature evolutions in steels. ISIJ International 32: 316-325. 11. Hunkel, M., Lfibben, T., Hoffmann, E, and Mayr, P. (1999). Modellierung bei bainitischen und perlitischen Umwandlung bei Stfihlen. HTM (Hhrterei-Technische Mitteilungen--Zeitschrift ffir Whrmebehandlung und Werkstofftechnik) 54: 365-372.
914
Fischer
12. Centinel, H., Toparli, M., and Ozsoyeller, L. (2000). A finite element based prediction of the microstructural evolution of steels subjected to the Tempcore process. Mechanics of Materials 32: 339-347. 13. Siegmund, T., Werner, E., and Fischer, E D. (1995). On the thermomechanical deformation behavior of duplex-type materials. J. Mech. Phys. Solids 43: 495-532. 14. Silberschmidt, V. V., Rammerstorfer, E G., Werner, E. A., Fischer, E D., and Uggowitzer, P. J. (1999). On material immanent ratchetting of two-phase materials under cyclic purely thermal loading. Arch. Appl. Mech. 69: 727-750. 15. ABAQUS, www.hks.com
SECTION
9.14
Mechanical Behavior of Steels during Solid-Solid Phase Transformations JEAN-BAPTISTE LEBLOND Laboratoire de Mod~lisation en M~.canique, Universit~ Pierre et Marie Curie, 8 rue du Capitaine Scott, 75015 Paris, France
Contents 9.14.1 Validity of the Models Proposed . . . . . . . . . . 9.14.2 Theory Used to Derive the Models . . . . . . . . 9.14.3 Description of the Models . . . . . . . . . . . . . . . . 9.14.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.3.2 Case of Ideal Plastic Phases . . . . . . . . 9.14.3.3 Case of Isotropically Hardenable Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.3.4 Case of (Linearly) Kinematically Hardenable Phases . . . . . . . . . . . . . . . . 9.14.4 Identification of Material Parameters . . . . . . 9.14.5 Numerical Implementation . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
915 916 916 916 917 917 918 919 919 920
9.14.1 VALIDITY OF THE MODELS PROPOSED The models p r o p o s e d herein for the plastic behavior of solids during solidsolid transformations are valid for m a n y steels; see examples provided in Reference [1]. They are also applicable to other, nonferrous metals, as s h o w n by G r e e n w o o d and J o h n s o n [2] in their pioneering w o r k involving a first, simple version of the m o r e refined m o d e l described in following text for ideal plasticity. They are not applicable to shape m e m o r y alloys, for which the transformation is a quasi-elastic p h e n o m e n o n . Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
915
916
Leblond
9.14.2 THEORY USED TO DERIVE THE MODELS The models presented here do not involve any ad hoc, adjustable parameter, but simply the usual thermomechanical characteristics of the material. They are not derived from heuristic considerations but from (approximate) homogenization [3-7]. It is supposed that the anomalous plastic behavior during phase transformations, and especially the so-called transformation plasticity phenomenon, arise from the sole microscopic mechanism proposed by Greenwood and Johnson [2], with that suggested by Magee [8] being disregarded. In spite of this limiting assumption, the models proposed are often experimentally found to be sufficient, as already mentioned.
9.14.3 DESCRIPTION OF THE MODELS
9.14.3.1 NOTATIONS 9 s: stress deviator 0"7r . yield stress of the parent (y) phase 9 0"~. r yield stress of the product (g) phase 9 O"Y:"global" yield stress 9 z: volume fraction of the product (g) phase 9 T: temperature 9 f(z), 2ptg(z): functions given in Table 9.14.1 below 9
9 9 9 9 9 9 9
h(X)" function given by h(X) - ( 1 if 0 < X < 0.7 1+5(X-0.7) if 0 . 7 < X _ ~ 1" 9c p . ~p _ ~tp + ~,~p+ e,T total plastic strain rate ~tp. transformation plastic strain rate /~cp. classical plastic strain rate arising from variations of the stresses cp e~ : classical plastic strain rate arising from variations of the temperature AV/V: difference of specific volume between the two phases (function of temperature) E: Young's modulus (assumed to be the same for both phases) 2i: thermal expansion coefficient of phase (i) 9
TABLE 9.14.1 The functionsf(z)
and
g(z)
z
0
0.125
0.25
0.50
0.75
1
f(z) g(z)
0 0
0.0186 2
0.101 3
0.392 1.75
0.672 1.75
1 1
9.14 Mechanical Behavior of Steels during Solid-Solid Phase Transformations
917
9 ~ff: hardening parameter of phase (i) (isotropic hardening) 9 ai: center of yield locus of phase (i) (kinematic hardening) 9 0: transformation "memory coefficient" for hardening ( - 0 for ferritic, pearlitic, and bainitic transformations; - 1 for martensitic transformations)
9.14.3.2
CASE OF IDEAL PLASTIC PHASES
With 0-eq -_
If
( 7 ) 1S / 2"
and 0-r -- [1 - f (z)]0- 7Y + f (z)0-~Y "
S
0-eq < 0-Y :
~,tp = _1__1_A V h (0-eq'~ s(ln Z)Z; 0-7
3 (1 - z)g(z) scFq;
~7If
0-eq ~
3 (27_2~)z(lnz)s ~. 0"7
0-Y:
~P = ,~ s
9.14.3.3
where
,~ > 0 is indeterminate. m
CASE OF ISOTROPICALLY HARDENABLE
PHASES
With 0-eq_ and -
s" s
,
F q=
~P 9~P
918
Leblond
If ffeq < aY. 1
~tp =
kcp =
3
/~P = ~
~ff
_
_
2
3(1-
AV
z) V
3
AV h
(a~q~s o n
z)~;
(1 - z ) g ( z ) s (req;
(2~, - 2~)z(ln z) s 7";
(~eq'~(ln Z)Z + g (Ez )
,~q + 2 ( ~
2~) z In z ~_;
-
1-z
If ffeq _ crY. ~,p _
3 ~eq
2 ffeq
~ff = keq ;
e4s + o_~4.
# s _ e~_ ~
9.14.3.4 CASE OF (LINEARLY) KINEMATICALLY HARDENABLE PHASES With ay -- h r ( T ) b ~ ,
~q-
a~ - h~(T)b~, a -
E7
(s - a) " (s - a)
11'~ ,
~,e~ E~/s a~)/s a~)]1j2 and
a Y-
[1 - f ( z ) ] c r ~
(1 - z)a~, + za~,
- 2cffe q (s - a~)
~eq_
(~
~ p . ~p
+f(z)a~ Y 9
9 g,
)1j2,
9.14 Mechanical Behavior of Steels during Solid-Solid Phase Transformations
If
919
~7eq < flY"
_ ____1 AV h\~v]f'~q'~(s - aT) (ln z)~:; Y V
0"7
3 (1 - z)g(z) ( s - aT)dseq;
k~p _ 3 (27
2~)z(lnz)(s
aT)T;
0"7
I~ : 1----~zl (gP +e'cP +e/rP); If
Io~ - -zZb~ + 0ZbT"z
ffeq __ flY:
~p _- _ ~Fq 3 (s - a)"
2 tTeq
Z
9.14.4 IDENTIFICATION PARAMETERS
Z
OF MATERIAL
Purely mechanical parameters (Young's modulus, yield limit, and hardening slope of each phase) can be obtained through standard tests at temperatures sufficiently high or low for a single phase to be present. The thermal expansion coefficients of the phases and the difference of specific volume between them can be obtained through (stress-free) dilatometry tests. If one wishes to assess the accuracy of the formulae given previously for the transformation plastic strain rate, one must perform in addition dilatometry tests with some external stress applied, which requires more sophisticated equipment.
9.14.5
NUMERICAL
IMPLEMENTATION
The numerical implementation of the models described in preceding text is quite easy. For each time step, one should first correct the "elastically computed" (that is, assuming the total minus thermal strain increment to be purely elastic) stresses to account for the values of the transformation plastic strain increment and the classical plastic one given previously (implicit or
920
Leblond
explicit algorithm). If the equivalent von Mises stress calculated in that way is smaller than the global yield stress, the treatment is finished. If it is greater, then the previous corrections should be abandoned and a classical projection method employed to ensure satisfaction of the criterion at the final instant of the time interval considered and of the flow rule during this interval (implicit algorithm).
REFERENCES 1. Leblond, J.-B. (1990). Qualification exp~rimentale du module de plasticit~ de transformation, FRAMASOFT+CSI Internal Report no. CSS/UNT.90/4022. 2. Greenwood, G. W., and Johnson, R. H. (1965). The deformation of metals under small stresses during phase transformations. Proc. Roy. Soc. A 283: 403-422. 3. Leblond, J.-B., Mottet, G., and Devaux, J.-C. (1986). A theoretical and numerical approach to the plastic behaviour of steels during phase transformations. I: Derivation of general relations. J. Mech. Phys. Solids 34: 395-409. ~ 4. Leblond, J.-B., Mottet, G., and Devaux, J.-C. (1986). A theoretical and numerical approach to the plastic behaviour of steels during phase transformations. II: Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34: 411-432. 5. Leblond, J.-B., Devaux, J., and Devaux, J.-C. (1989). Mathematical modelling of transformation plasticity in steels. I: Case of ideal-plastic phases. Int. J. Plast. 5: 551-572. 6. Leblond, J.-B. (1989). Mathematical modelling of transformation plasticity in steels. II: Coupling with strain hardening phenomena. Int. J. Plast. 5:573-591. 7. Devaux, J., Leblond, J.-B., and Bergheau, J.-M. (2000). Numerical study of the plastic behaviour of a low alloy steel during phase transformation, Proceedings of the "First International Conference on Thermal Process Modelling and Computer Simulation", Shanghai, China. 8. Magee, C. L. (1966). Transformation Kinetics, Microplasticity and Ageing of Martensite in Fe-Ni. Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh.
SECTION , ,
9.15
Constitutive Equations of a Shape Memory Alloy under Complex Loading Conditions MASATAKA TOKUDA Department of Mechanical Engineering, Mie University, Kamihama 1515 Tsu 514-8507, Japan
Contents 9.15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 921 9.15.2 Validity (Domain of Applications) . . . . . . 922 9.15.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 9.15.3.1 Modeling of Phase Transformation System . . . . . . . . . . . . . . . . . . . . . . . . 923 9.15.3.2 Modeling of Crystal Component 924 9.15.3.3 Model of SMA Polycrystal . . . . . . 925 9.15.4 Constitutive Equations . . . . . . . . . . . . . . . . 925 9.15.5 Identification of Material Parameters... 926 9.15.6 How to Use the Model . . . . . . . . . . . . . . . . . 927 9.15.7 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7
9.15.1 I N T R O D U C T I O N The deformation mechanism of shape m e m o r y is the martensitic phase transformation induced by applied stress and temperature. Quite unique, complicated, and interesting mechanical properties can be obtained by applying the simultaneous change of temperature and stress, which has six independent components (i.e., by applying the complex loading conditions). Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
921
922
Tokuda
The proposed constitutive equations can predict the complicated nonlinear deformation behavior of a shape memory alloy.
9.15.2 VALIDITY (DOMAIN OF APPLICATIONS) Roughly speaking, the quite complicated deformation behavior observed in the shape memory alloy subjected to the complex loading path can be predicted (at least qualitatively) by the proposed constitutive equations. The reliability was confirmed experimentally by applying the combined loads of axial force and torque to the thin-walled tubular specimen of a Cu-based shape memory alloy. Figure 9.15.1 shows one example of comparisons between the experimental and predicted results. Figure 9.15.1a shows the given thermomechanical (complex) loading path to the thin-walled tubular specimen: 1-2: proportional loading of torsion stress z, axial stress or; 2-3: increase of temperature T while keeping the combined stress state constant; 3-4: decrease of temperature T while keeping the stress state constant; 4-5: decrease of shear stress z while keeping the axial stress a and
373K,, 6
T
:::..".. c,z"
(a) 295k .2 o" 0.015
,
,
,,
o
,
CalculatedResult
0.015
9
24
~. 0.01
ExperimentalResult
4
~. 0.01 3
3
0.005 5
0 0.01 c
(b)
~
~o.oo5
0 0
2
0.02
.
.
0
.
.
.
.
.
.
0.01 C
.
0.02
(e)
FIGURE 9.15.1 An example of comparisons between experimental and predicted results. (a) Thermo-mechanicalloading path. (b) Calculatedresult of strain response. (c) Experimental result of strain response.
9.15 Constitutive Equations of a Shape Memory Alloy
923
temperature T; 5-6 and 6-7: increase and decrease, respectively, of temperature while keeping the stress state constant: and 7-8: proportional unloading to zero stress state. Figure 9.15.1b,c show the computational (predicted) result and experimental result. As the figure shows, the computational result coincides with the experimental results qualitatively very well.
9.15.3 MODELING The formulation of constitutive equations is based on the semimicroscopic mesomechanical approach by considering the multilayered structure of a polycrystalline shape memory alloy (SMA). The formulation procedure is composed of the following stages.
9.15.3.1 SYSTEM
MODELING OF PHASE TRANSFORMATION
The basic phase transformation mechanisms of SMA are the temperatureinduced and stress-induced martensite phase transformations. The deformation induced by these martensite phase transformations is considered to be the twin-type (shear) deformation that has a negligibly small change of volume and that is controlled by the temperature and the stress tensor. This twin-type (shear) deformation occurs only in the crystallographically determined (easy twin) direction on the crystallographically determined (easy twin) planes. This "twin (shear) deformation system," with a specific direction and a specific plane, is named the phase transformation system, and it may correspond to the "slip system" in crystal plasticity, whose mechanism is dislocation slip. When SMA is subjected to thermal loading (the change of temperature) only, the martensite phase transformation occurs and produces the (twin-type) shear deformation. However, in this case, no macroscopically significant deformation of crystal grain can be observed. This phenomenon can be explained as follows: in the case of temperature change without any external (applied) or internal stress, the phase transformation can occur simultaneously in every transformation system with different (randomly distributed) orientations, and thus the induced shear (twin) strains cancel each other on average. On the other hand, in the case of the stress-induced phase transformation, the phase transformation occurs in the phase transformation systems with the preferable orientation to the applied stress state. Thus only the shear strains with specific direction are summed up, and
924
Tokuda
consequently, the significant macroscopic inelastic deformation can be observed in the crystal component (as well as in the polycrystal). Our engineering interests are in the macroscopic inelastic deformations, the force produced when the deformation is constrained, and their combination. Therefore, only the case when the material is subjected to some stresses is considered in the proposed constitutive equations. The effect of temperature is incorporated as the temperature effect on the shear stress which is necessary for the phase transformation, i.e., on the critical (resolved) shear stress. Details of the (critical) shear stress 1: and shear strain ~; relation formulated in the previously mentioned scheme are given in References [1, 2].
9.15.3.2
MODELING OF CRYSTAL COMPONENT
The deformation property (stress-strain-temperature relation) of the crystal grain component with the phase transformation systems described in Section 9.15.3.1 is formulated as follows, according to the crystal plasticity manner. The number of phase transformation systems in the crystal grain is assumed to be M (for example, M = 24 for the martensite phase transformation of Cu-based SMA). When the crystal grain is subjected to the uniform stress crij (l, j = 1,2,3), the shear stress 1: resolved on the m-th (m = 1,2,...,M) phase transformation system can be obtained by the following equation: 3
"C(m) = ~ O~(m)ijO'ij i,k=l
(1)
The coefficient ~x(m)ij in Eq. 1 is the so-called generalized Schmid factor, defined as follows: O~(m)ij = [S(m)in(m)j q- S(m)jn(m)i]
(2)
S(m)i and n(m)i (i = 1, 2.3) are the unit vector along the shear direction and the unit vector normal to the transformation plane of the m-th phase transformation system, respectively. By using the obtained resolved shear stress ~:(m), the shear strain increment dT(Pm ~) of the m-th phase transformation system can be estimated by using the model of phase transformation system explained in Section 9.15.3.1, when only the m-th system becomes active. If some systems become active simultaneously in the crystal grain component, some corrections are necessary in order to incorporate the interactions among the phase transformation systems (see details in Reference [1]). where
925
9.15 Constitutive Equations of a Shape Memory Alloy
9.15.3.3
MODEL OF S M A POLYCRYSTAL
When the stress-strain-temperature relation is derived on the basis of the stress-strain-temperature relation of its crystal grain component explained in Section 9.15.3.2, the interactions among grain components have to be incorporated. That is, each grain component has its own stress and strain, depending on its own orientation in the polycrystal, because each grain has the anisotropy related to the phase transformation system. Thus the complicated interaction among grains appears to satisfy the compatibility condition by strain and the equilibrium condition of stress in the polycrystal. Thus the nonuniform stress and strain distributions appear even if the applied stress is quite simple, for example, the uniaxial tension of a solid bar. The effects of the nonuniform distribution of stress and strain on the mechanical properties of polycrystalline materials are very important from the viewpoint of path dependency, especially when the strain or stress path is complex. This kind of interaction among grains can be taken into account by using one of the well-developed mechanical models of inhomogeneous solids: the selfconsistent model. According to the self-consistent model, the following equation can be obtained: -
s,j -
o(4
-
(3)
where s{h) /j is the deviatoric stress component of the k-th grain component embedded in the polycrystal, SO is the averaged (macroscopic) deviatoric ^PT(k) stress, qj is the (deviatoric) phase transformation strain components of the Pr is the averaged (macroscopic) phase transformation strain, and k-th grain, E~j G is the averaged shear modulus of the polycrystal. The coefficient e is still under discussion, and several values of c~ are proposed, for example, 0~= 0 (for the stress constant model: the so-called Maxwell model) = 0.2 (for the modified Kr6ner-Budiansky-Wu model by BerveillerZaoui) = 1.0 (for the KBW original model) = 2.0 (for the strain constant model: the so-called Voigt model) = oo (for the inelastic strain constant model: the Taylor model)
9.15.4
CONSTITUTIVE
EQUATIONS
The constitutive equations based on the modeling mentioned in Section 9.15.3 not generally be expressed in a compact closed form but in the flow chart for computing because some iterative (convergence) computation is necessary. In a special case, for example, the constitutive equations based on
926
Tokua.
the stress constant model can be expressed in a closed form when the strain path can be given. It should be noted that the proposed constitutive equations can be described symbolically in the form of an internal variable theory whose .PT(k) internal variables are Y(m) , in the following manner: 3
.PT(n)x J c Fijkl(Spq ' 2 , T " y(m ) )u,.~kl
dEij -- Z k,l=l
.PT(n)~,jx.,
+ Fij (Sij, 2, T 9Y(m) )UL, .Pr(k)~n T + Fij (Spq, Z, T ' y ( m ) ju
.Pr(k)
dy(m )
3
-
(k)
(4)
.Pr(k)
~ H(m)Pq(Sij' 2 ' T" Y(m) )dSpq p,q=l
, (k) (Spq ' T" Y(m) .Pr(k)~.~v )Uz-~
-Jr- /-/(m)
T .PT(k)~j T + I-I(m)(k ) (S~j,2, T" y(m) j.
(5)
where Y(m)'PT(k)( m - 1, 2,..., M, k - 1, 2,..., N) is a shear strain of the m-th transformation system in the k-th grain component, N is the number of grain components of the polycrystal model, and 2 is the volumetric part of the macroscopic (averaged) stress of the polycrystal model.
9 . 1 5 . 5 I D E N T I F I C A T I O N OF MATERIAL PARAMETERS The following material parameters have to be determined experimentally. All of them can be determined by two simple uniaxial tension (or torsion) tests with different temperatures. G: shear modulus K: bulk modulus H: shear strain hardening parameter of critical shear stress fl: temperature hardening parameter of critical shear stress ZoMs: shear stress for the martensite phase transformation starting at a reference temperature To ZOAF: shear stress of reverse (austenite) phase transformation finishing at a reference temperature To y ~ : maximum shear strain of a phase transformation system. The number M of phase transformation systems in a single crystal grain component and the generalized Schmid factor % are determined by the
9.15 Constitutive Equations of a Shape Memory Alloy
927
crystallographic structure of selected materials (can be found in the handbook of metals). The number N of crystal grain components in the polycrystal model is determined from a compromise of accuracy and computational time. From the author's experience, N--100 is recommended.
9.15.6
HOW
TO USE THE MODEL
The proposed constitutive equations were first formulated in order to predict the unique behavior of a shape memory alloy under complex loading conditions which are difficult to conduct experimentally. To this purpose, the proposed constitutive equations are quite successful and give us quite interesting information which cannot be obtained experimentally. On the other hand, the proposed constitutive equations may not yet be convenient for the stress-strain analysis of a shape memory structure by the finite element method or other numerical analyses. In this analysis, we need a much faster computer or some more simplification of constitutive equation which does not need the iterative computation.
9.1 5.7 P A R A M E T E R S The proposed constitutive equations have been used only for the Cu-based shape memory alloy, whose chemical composition is Cu-10 wt%A15 wt%Mn5 wt%Zn. The values of the material parameters for this material are as follows: G = 16.62ptGPa H = 5.0 x 1 0 2 MPa f l = 5 . 0 x 10MPa/K toms = 90.0 MPa(To = 25~ "COAV = 70.0 MPa, (To = 25~ PT
M, /VC5 -- 2.SO/o
The sets of parameters for other materials are at present unknown.
REFERENCES 1. Tokuda, M, Ye, M, Bundara, B, and Sittner, P (1999) 3D constitutive equations of polycrystalline shape memory alloy. Archive of Mechanics 51(6): 847-864. 2. Tokuda, M, Ye, M, Takakjura, M, and Sittner, P. (1998). Thermo-mechanicalbehavior of shape memory alloy under complex loading conditions. International Journal of Plasticity 15(2): 223-239.
SECTION
9.16
Elasticity Coupled with Magnetism RENI~
BILLARDON,LAURENTHIRSINGERand FLORENCEOSSART
ENS de Cachan/CNRS/Universitd Paris 6, 61 avenue, du PNsident Wilson, 94235 Cachan Cedex, France
Contents 9.16.1 General Validity . . . . . . . . . . . . . . . . . . . . . . . . . .
929
9.16.2 Hereditary Model: Preisach Model . . . . . . . .
929
9.16.2.1 Background . . . . . . . . . . . . . . . . . . . . . .
929
9.16.2.2 Description of the Model . . . . . . . . . .
929
9.16.2.3 Description of the Coupling with Elasticity . . . . . . . . . . . . . . . . . . . . 9.16.2.4 9.16.3 Internal between 9.16.3.1
931
Identification of the Parameters . . . . 931 Variable Model: State Coupling Elasticity and Magnetism . . . . . . . . 932 Background . . . . . . . . . . . . . . . . . . . . . . 932
9.16.3.2 Validity of the State Coupling . . . . . 933 9.16.3.3 Possible Choices for Internal Variable X . . . . . . . . . . . . . . . . . . . . . . . . 9.16.4 Magnetoelastic Scalar Reversible Behavior: Anhysteretic Behavior X = 0 . . . . . . . . . . . . . .
933 934
9.16.4.1 Description of Models for the Anhysteretic Magnetization . . . . . . . .
934
9.16.4.2 Description of Models of Anhysteretic Magnetostriction Strain . . . . . . . . . . .
935
9.16.4.3 Description of a Scalar Model . . . . . 936 9.16.4.4 Identification of the Parameters . . . . 937 9.16.5 Magnetoelastic Hysteretic Behavior: Model Proposed by Jiles and Atherton (X = Mhys) 937 9.16.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . 937 9.16.5.2 Description of the Evolution L a w . . 938 9.16.5.3 Identification of the Parameters . . . . 939
928
Handbook of Materials Behavior Madels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
929
9.16 Elasticity Coupled with Magnetism
9.16.6 Magnetoelastic Hysteretic Behavior: Model Proposed by LMT-Cachan (X -- Hhys) . . . . . . 9.16.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . 9.16.6.2 Description of the Model . . . . . . . . . . 9.16.6.3 Identification of the Parameters .... 9.16.6.4 How to use the Model . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.16.1
940 940 940 941 942 942
GENERAL VALIDITY
This chapter is devoted to different scalar magnetic hysteresis models that are valid for isotropic soft ferri- and ferromagnetic materials subjected to uniaxial magnetic excitations and uniaxial elastic stresses.
9.16.2 HEREDITARY MODEL: PREISACH MODEL 9.16.2.1
BACKGROUND
The Preisach model [1] consists of a collection of bistable elementary hysteresis operators y~/~(H), whose switching fields are denoted 0c and (Fig. 9.16.1). This collection is described by the so-called Preisach distribution #(c~,/~), which gives the weight of the operator 7~/~ and is usually represented by its isovalues in the Preisach plane (g,]~) (Fig. 9.16.2). The magnetic behavior of the material is modeled by the following formula:
M(H/-J
J~ >~
(1)
The model has two important properties [2]: 9 The extrema of the applied field H are detected and stored until they are erased by an extremum with a larger magnitude. 9 Minor loops are closed and congruent (no reptation).
9.16.2.2
DESCRIPTION OF THE MODEL
The Preisach model is currently recognized as a static scalar hysteresis model with well-defined properties and the more accurate prediction of minor loops. Numerous ideas have been proposed in order to extend the initial scalar model, either by assuming that the distribution #(~, ]~) depends on some state
930
Billardon et al
7~p
+1
13
a
H
-1 FIGURE 9.16.1 Elementary hysteresis operator definition, a>_fl so that hysteresis always causes dissipation. The magnetization of the material M(H) saturates for H>_Hsat. Hence, there is no operator such that a > Hsat or fl < Hsat.
FIGURE 9.16.2 Isovalues of the Preisach distribution. Hc being the coercive field, the distribution has a sharp maximum at (Hc,-He) and vanishes for a > Hsat or fl < - Hsat.
variables of the system, or by changing the behavior of the elementary operator 7~. Some examples are as follows: 9 Moving model [ 5 ] : the distribution #(a, fl) depends on the current magnetization M, which improves the description of minor loops. 9 Dynamic model [6]: 7~/~ switchings are damped, just as the actual motions of magnetic domain walls are "damped" by eddy currents. 9 Mayergoyz vector model [3]: the classical Preisach model is distributed over the space. 9 Charap vector model [7]: 7~/~is an anisotropic vector operator, including rotation of the magnetization M.
9.16 Elasticity Coupled with Magnetism
9.16.2.3
931
DESCRIPTION OF THE COUPLING WITH
ELASTICITY The mechanical state and in particular the stress state of the material have a great influence on its magnetic properties. Two different approaches have been proposed to build a scalar Preisach-type hysteresis model that accounts for the influence of a uniaxial stress: 9 Identification of the Preisach distribution as a function of stress cr so that #(~, fl, or) [8]: results obtained for standard Fe-3%Si laminations do not exhibit a simple evolution law. 9 Definition of the elementary operator as a function of stress so that 7~fl(H, if) [9]: experimental identification has to be done from a set of M(H, or) plots. The second approach can be used to build an anisotropic vector model based on the definition of vector operators 7~fl(H,~r) [10]. Since the magnetomechanical couplings are fundamentally anisotropic and different in tension and compression, using a vector elementary operator improves the scalar description of hysteresis even along a fixed direction. The classical Preisach model is easy to implement and use. All the needed details are found in the literature. However, none of its generalizations preserves the simplicity and well-defined properties of the initial model. Using them requires some rules of thumb and know-how.
9.16.2.4
IDENTIFICATION OF THE PARAMETERS
The identification of the model consists of linking the distribution #(~, fl) to experimental data. Two approaches may be used: 9 The distribution #(~,fl) is assumed to be described by an analytical function whose parameters are derived from the minimization of some global criterion (surface or global shape of the major loop) [4]. 9 The Preisach plane is discretized and the distribution is calculated at each point by using a set of minor loops [3]. The higher the number of loops, the higher the accuracy of the model.
932
Billardon et al
9.16.3 INTERNAL VARIABLE MODEL: STATE COUPLING BETWEEN ELASTICITY AND MAGNETISM 9.16.3.1
BACKGROUND
Isothermal situations are considered. The specific enthalpy ~P is partitioned into four terms [14]. rid m and ~P~m, respectively, correspond to the mechanical state of the material and to the magnetoelastic couplings. W~a, and ~Yffl~hys, respectively, correspond to the magnetic state of a comparison ideal material (with the same behavior as the anhysteretic response of the real material) and to the hysteretic part of the magnetic behavior of the real material. Hence, p~P(H, X, o-) = ptPm(o") + pT~m(O', H; X) + pqA~an(H) + pq2~hys(H;X)
1
= --a" 2
IF-1" a -
/o e~(a, H; X)" da - /o
#oMano(h) 9dh
+ ptr~hys(H; X)
(2)
where n:, ~,, Man0, respectively, correspond to the elastic moduli tensor, the magnetostriction strain tensor, the anhysteretic magnetic response. H, a and X, respectively, correspond to the magnetic field, stress tensor and the internal variable(s) representative of magnetic hysteresis. Subscript 0 refers to zero stress state. State laws are derived from the specific enthalpy ~P used as state potential: --
/~0M1 ---
OptP OH
#oY =
0o" =
9o" + e~(a, H, X)
-
OPtP,hY~
Opge
Op~2~,hys
0x
0x
0-----g--+
~0 `~Oe~'.
(3)
do
+ fo g -b--~(a, Oq~3#. H; X)" da
(4)
(5)
and Clausius-Duhem inequality becomes: Di -- - p q J - e" 5"-/toM. I:I = #o(M1 - M). I:I + / t o Y - X >_ 0
(6)
This condition, which must be satisfied for any physically admissible evolution, gives an important limitation to the possible expressions of the evolution laws for internal variables X.
933
9.16 Elasticity Coupled with Magnetism 9.16.3.2
VALIDITY OF THE STATE COUPLING
This state coupling has been experimentaly validated for pure nickel [13] and silicon iron alloy [14]. In the case of anhysteretic magnetoelastic evolutions ( X - 0), the Clausius-Duhem inequality (Eq. 6) simplifies and variable M1 appears as magnetization M - Ma,(a,H). Hence, the state law (Eq. 4) gives the expression of the anhysteretic magnetic response of the material when it is subjected to elastic loadings: c9~a~'" H) 9da #oMan(a, H) -- #0Man0(U) 4- f0 --~-(a;
(7)
where ea~,(a,U ) = e~(a,U,X-- 0) denotes the anhysteretic magnetostriction strain of the material when it is subjected to elastic loadings. Derivation of this relation with respect to stress cr leads to the so-called Maxwell expression:
OMan
tt~ cOa (a; H)
9.16.3.3
cg~n .
-
-
- ~ ( a ; H)
V(a, H)
(8)
POSSIBLE CHOICES FOR INTERNALVARIABLE X
Further assumptions must be made to complete the magnetoelastic model: 9 Internal variable X must be defined. In the case of a purely phenomenological approach, this definition may be directly deduced from the modeling of the magnetization response M(a, H, X). Different assumptions can be made. For instance, one may introduce a partition of magnetization M so that: -
M=Man(a,H)+Mhys
-
and or a partition of magnetic field H so that:
M=Man(a,U-Hhys)
and
X=Mhys
X=Hhys
(9)
(10)
These variables are depicted in Figure 9.16.3 in the uniaxial case. 9 A kinetic law for internal variable X that satisfies the Clausius-Duhem inequality (Eq. 6) for any magnetomechanical loading must be introduced: X = X(a, H, I:I, X)
(11)
9 A kinetic law for magnetostriction strains e~(a,H,X) must also be introduced.
Billardon et al
934
M~
~
Actual magnetic state
= ea~ r162
Hhvs
Magnetic Field H FIGURE 9.16.3 Definition of variables Mhys and Hhys in the uniaxial case.
9.16.4 MAGNETOELASTIC SCALAR R E V E R S I B L E BEHAVIOR: A N H Y S T E R E T I C BEHAVIOR X = 0 9 . 1 6 . 4 . 1 DESCRIPTION OF MODELS FOR THE ANHYSTERETIC MAGNETIZATION The anhysteretic magnetic response of a material free of any mechanical loading has been modeled by Frohlich [19] as Mo.0(H) -
Za~
+ Z.0H
(12)
where Manoand H, respectively, denote Mano and H moduli and where material parameters Zao and Ms are, respectively, related to the initial slope of the anhysteretic curve and to the saturation magnetization. By analogy with the behavior of paramagnetic materials, Jiles and Atherton [25] used the Langevin function to model the magnetic anhysteretic curve, such that:
Mano(H) -- M s ( c o t h ( ~ ) --~e)
with He -- H + ~M
(13)
where He denotes the effective magnetic field as introduced by Weiss to account for ferromagnetism (~ is the mean molecular field constant representing interactions of magnetic moments), whereas a denotes a material
935
9.16 Elasticity Coupled with Magnetism
parameter that depends on temperature and that controls the anhysteretic curve shape. From phenomenological considerations [18], the anhysteretic curve has also been modeled by the following relation:
H)
Mano(H)-- 2MS Arctan ( 2nZ a ~ n
(14)
This latter expression gives a better fit of the experimental response of Fe-3%Si alloys than Eq. 13 within the range of + 3000 A/m.
9 . 1 6 . 4 . 2 DESCRIPTION OF MODELS OF ANHYSTERETIC MAGNETOSTRICTION STRAIN For isotropic materials, it is reasonable to assume that the anhysteretic magnetostriction strain tensor takes the following form [22]:
o
o /
~an•
0
sym.
eanZ(H)
(15) (u,,,v•
where ull, v• and w• respectively, denote unit vectors that define a direct frame with direction ull parallel to magnetization vector M. Besides, e~' ,, and anll ea,,• respectively, denote the magnetostriction strains in the direction of magnetization and in any direction transverse to the direction of magnetization. Since the magnetization process is at quasi-constant volume, the transverse magnetostriction is such that [19]: 1 ~
3
(MaM(H)/
(16)
where 2s denotes the saturated magnetostriction strain along the saturation direction. For a randomly oriented polycrystalline cubic material, an estimation of 2s is obtained by averaging the magnetostriction of grains and neglecting the strain incompatibility between neighboring grains [21,22], such that: '~'s -- 52~100 -4- 3 ~111
(17)
where 2100 and 2111, respectively, denote the magnetostriction constant in [100] and [111] crystallographic directions of the cubic single crystal.
936
9.16.4.3
Billardon et
al
DESCRIPTION OF A SCALAR MODEL
When a uniaxial stress cr is applied in the direction of the magnetic field, Jiles and Atherton [25] have proposed to modify the effective magnetic field in the anhysteretic curve (Eq. 13) with
He -- H + aM 4
de ~ anllr #0 dM
(18)
where ea~llr denotes the magnetostriction strain in the direction of the applied uniaxial stress. According to Eqs. 15 and 16, its expression is ea~,ll~ = ~ 2s cos 2 0 -
Mss
(19)
where 0 denotes the angle between the direction of the applied stress rr and the direction of the applied magnetic field H. Eventually, the anhysteretic behavior of a material subjected to a uniaxial stress is predicted by the following expression of the anhysteretic curve Man:
M=Man(H,M, rr)-Ms(coth(-H~)--~e )
(20)
with
He, M, rr -- H + aM + --4-~
c~
-
M~
On the other hand, it appears that the inverse of the initial slope of the anhysteretic curve depends linearly on the applied uniaxial stress [18,20]. For sake of simplicity, the magnetic behavior of the material is assumed to remain isotropic. Finally, anhysteretic c u r v e Man is modeled by the following expression: M = M~,(H, rr) = 2M~ Arctan
7~
Za(rr
(22)
with
~al(O ") =
Z a1 + A . rr
(23)
where rr denotes the value of the uniaxial stress applied in the direction of the magnetic field. In other words, the magnetoelastic state coupling (derivative of Eq. 8 with respect to H) is approximately a constant, denoted by - A z 2, in the vicinity of zero magnetization:
02Man
OH cgrr(rr'H = O) = -
1 02~
anl-------~l(~r,H = 0) = -Az2(rr)
110 oq2H
(24)
937
9.16 Elasticity Coupled with Magnetism
9.16.4.4
IDENTIFICATIONOF THE PARAMETERS
The model parameters, four for the former (Ms, a, 0~, 2s) and three for the latter (Ms, Xa0, A), are identified from the anhysteretic curve measured on samples subjected to constant uniaxial stress. See Table 9.16.1.
9.16.5 MAGNETOELASTIC HYSTERETIC BEHAVIOR: MODEL PROPOSED BY JILES AND ATHERTON ( X - Mhys) 9.16.5.1 VALIDITY The model proposed by Jiles and Atherton [25] is a scalar magnetic hysteresis model. It is coupled with elasticity; i.e., it accounts for the effect of uniaxial stresses on magnetisation of bulk isotropic materials. Whereas good predictions can be obtained for major loops, minor loops are approximated: the monotonic condition dM/dH, always positive, is not always fulfilled just after reversal of the applied magnetic field. Table 9.16.1 Materials
Table of Parameters Ms
XaO
(103 Mm)
a
o~
21oo
)[111
2s
(Mm)
(10 -3 )
(10 -6)
(10 -6 )
(10 -6 )
N.O. Silicon Iron Alloy (M330-50A)
1190
39300
N.O. Silicon Iron Alloy (M450-50E)
11151700
3100041000
43426
0.0910.72
N.O. Silicon Iron Alloy (M600-50A)
12301700
3400070000
80538
0.1780.92
Low Carbon Steel
13971600
216010600
3421100
Carbon Steel (AISI 4130)
1350
375
Nickel
480
Ferrite Core
380
420
Terfenol D Tbo.3Dyo.TFel.9
765790
11
23
-4.5
6.5
0.641.6
21
-21
-4.2
2100
2
21
-21
-4.2
-24
-32.8
30
10 -3
7000
3235
1600
10001200
-46 90
A (10 -6 MPa -1) 0.925
-5.73
6100
938 9.16.5.2
Billardon et al DESCRIPTION OF THE EVOLUTION LAW
This model is based on the following principal hypotheses: 9 Even for real materials with defects (which are the source of hysteresis), the anhysteretic curve exists and corresponds to the magnetic behavior of a comparison ideal material free of any defect: the relationship between anhysteretic magnetization Man and magnetic field H is reversible nonlinear, such that:
where He and a, respectively, denote the effective magnetic field and a material parameter. 9 From the molecular field theory proposed by Weiss, the expression of the effective magnetic field He is the following (Eq. 18): d~~
He -- H + o~M + #o
anl_______2~. dM
(26)
9 To account for reversible magnetization due to reversible magnetic domain wall bowing and reversible rotation, magnetization M is partitioned into reversible, Mrs, and irreversible, Mirr, parts such that: (27)
M - - Mrev nu Mirr
9 The reversible magnetization component Mr~ is supposed to take the form: Mrev = c ( M a n - M i r r ) (28) where c denotes a material parameter. 9 Magnetic behavior irreversibility due to pinning-unpinning of magnetic domain walls during the magnetization process is associated with supplementary energy, noted Ezoss, needed to overcome these pinning sites:
Ezo~ --
(29)
k dgirr
where k denotes a material parameter characterizing the average energy required to break pinning sites. The energy supplied to material dE is either stored in the material in magnetostatic energy form or dissipated by hysteresis. In the case of anhysteretic behavior of the material, this supplied energy is completely stored in the material, such that: aE = M ( H ) a H -
= Ma
(H)aH
(30)
and consequently dMir-------~ r - - M a n - Mirr
dH
(31)
k
According to gqs. 27 and 28 and to the effective magnetic field equations (Eqs. 19 and 26), the evolution law (Eq. 31) of the internal variable M~rr,
9.16 Elasticity Coupled with Magnetism
939
which describes magnetic irreversibility, becomes: dMirr Man - Mirr d----H = 6k - o~* (Man - - Mirr)
(32)
with ~ * - - 0 ~ + 4 p0 M2
(cos0
where 6 denotes the sign function of the rate of magnetic field I:I - dH/dt, that is to say: 6=1
if / : / > 0
and
6=-1
if /-)<0
(34)
It can be noticed that, in this model, Eq. 9 takes the following form Mhys = (1 -- c)(Mirr(H,o- )
9.16.5.3
--
gan(H,~) )
IDENTIFICATION OF THE PARAMETERS
This magnetic hysteresis model coupled with elasticity proposed by Jiles and Atherton requires one to identify six parameters: saturated magnetization Ms and magnetostriction 2s, magnetic moments interaction parameter c~, shape parameter for anhysteretic curve a, average lost energy parameter k, and reversibility coefficient c (see Table 9.16.2). Parameters Ms, 2s, and c~are identified by accurate physical measurements, and their values are given in the literature. From the viewpoint of applications, some parameters can be directly identified from the hysteresis loop. By derivation of Eq. 31 with respect to H, parameter c appears as the ratio between initial susceptibilities (at origin) measured on the first magnetization curve )~in and on the anhysteretic curve )~an0 [26]: =
Zin
(35)
)(,anO
The parameter k can also be defined as: +
aH where Hc and )~Hc,respectively, denote the coercivity force and the slope of the hysteresis loop at Hc. Lastly, the parameter a can be identified as: a=
c~Ms
k
3 k-He
(37)
940
Billardon et al
Table 9.16.2
Table of Parameters
Materials
Ms (103 A/m)
a
(A/m)
k (A/m)
(10 -3 )
s
(10 - 6 )
N.O. Silicon Iron Alloy (M330-50A)
1190
N.O. Silicon Iron Alloy (M450-50E)
11151700
43426
0.0910.72
0.0850.1
40-36
N.O. Silicon Iron Alloy (M600-50A)
12301700
80538
0.1780.92
1.210 - 3 0.04
114104
Low Carbon Steel
13971600
3421100
0.641.6
1.1 10 -30.2
247400
-4.2
Carbon Steel (AISI 4130)
1350
2100
2
0.25
1800
35
Nickel
480
Ferrite Core
380
30
10 -3
0.95
10
Terfenol D Tbo.3Dyo.7Fel.9
765790
7000
32-35
90
32803950
9.16.6
6.5
-32.8
MAGNETOELASTIC
10001200
HYSTERETIC
BEHAVIOR: MODEL PROPOSED BY LMTC A C H A N ( X - Hhys) 9 . 1 6 . 6 . 1 VALIDITY T h e m o d e l p r o p o s e d by L M T - C a c h a n is a scalar m a g n e t i c hysteresis m o d e l . It is c o u p l e d w i t h elasticity; i.e., it a c c o u n t s for the effect of u n i a x i a l stresses ( a p p l i e d in the s a m e d i r e c t i o n as the a p p l i e d m a g n e t i c field) o n m a g n e t i z a t i o n of b u l k isotropic materials.
9.16.6.2
D E S C R I P T I O N OF THE M O D E L
This model is based on the following principal hypotheses: 9 Magnetic field H is partitioned into reversible anhysteretic and irreversible parts denoted by Ha. and Hhys, respectively, so that: H = H~ + H~y,
VM
(3S)
941
9.16 Elasticity Coupled with Magnetism
9 The anhysteretic curve exists for a real material and the relation between magnetization M and anhysteretic magnetic field Hat, is reversible nonlinear, as introduced in Section 9.16.4 by Eqs. 23 and 24:
M--Man(Han'a)
-- 2MSArctan(2za(a)~s)7~
(39)
~(a 1 (0") -- ZaO1 + A . a
(40)
with
where a denotes the value of the uniaxial stress applied in the direction of the magnetic field. 9 For sake of simplicity and as a first approximation, it is assumed that a uniaxial stress mainly affects the anhysteretic curve and has no significant influence on the hysteretic behavior. The evolution law of internal variable Hhys is chosen such that, on one hand, Eq. 6 is satisfied. From phenomenological considerations it has been proposed that:
I-Ihys -- ZO~Iif/:/> 0
and
Hhys ~ Hy
or
if/:/< 0
and
Hhys ~ -Fly (41)
I-Ihys D
H~ - sign(H)Hhy~ ZoI:t if I:t > 0 and Hhys > Hy Hc - Hy or if I:t < 0 and Hhys < - Hy
(42)
where constants Z0, He, and Hy denote material parameters, respectively, related to the initial slope of the first magnetization curve, the coercive magnetic field, and the so-called yield magnetic field.
9.16.6.3
IDENTIFICATION OF THE PARAMETERS
This magnetic hysteresis model coupled with elasticity proposed by LMTCachan requires the identification of seven parameters: saturated magnetization Ms and magnetostriction 2s, initial slopes of the anhysteretic curve (M vs. Ha,,), Za0 and of the first magnetisation curve (M vs. H), Z0, coercive force Hc, yield magnetic field Hy, and stress dependence coefficient of the anhysteretic curve initial slope A (see Table 9.16.3). To identify these parameters, major hysteresis loops with first magnetization curve and anhysteretic response of stressed material are needed.
942 Table 9.16.3
Billardon et al Table of Parameters
Materials
Ms (103Mm)
%aO
N.O. Silicon Iron Alloy (M330-50HA)
1190
39300
N.O. Silicon Iron Alloy (M450-50E)
I i 151700
3100041000
N.O. Silicon Iron Alloy (M600-50A)
12301700
3400070000
Low Carbon Steel
1397
216010600
10-100
-4.2
Carbon Steel (AISI 4130)
1350
375
10-100
-4.2
-5.73
Nickel
480
Ferrite Core
380
420
10
Terfenol D Tbo.3Dyo.FFel.9
765790
11
3000
10001200
6100
9.16.6.4
Hc
Hy
(Mm)
(Mm)
2s (10 -6 )
A (10-6MPa) -1
10
6.5
0.925
%0 0.85
60
HOW TO USE THE MODEL
The constitutive equations (Eqs. 41 and 42) could be solved iteratively by applying a 0-method and a pure Newton algorithm [29]. REFERENCES 1. 2. 3. 4.
5. 6. 7. 8.
Preisach, E (1935). Uber die magnetische nachwirking. Z. Phys. 94: 277-302. Bertotti, G. (1998). Hysteresis in Magnetism. Academic Press. Mayergoyz, I. D. (1991) Mathematical Models of Hysteresis. New York, Springer-Verlag. Bertotti, G., Fiorillo, E, and Soardo, G. P. (1987). Dependence of power losses on peak magnetization and magnetization frequency in grain-oriented and non-oriented 3% SiFe. IEEE Trans. Magn. 23(5): 3520. Torre, E. Della (1991). Existence of magnetization dependent Preisach models. IEEE Trans. Magn. 27(4): 3697-3699. Bertotti, G. (1992). Dynamic generalization of the scalar Preisach model of hysteresis. IEEE Trans. Magn. 28(5): 2599. Davidson, R., and Charap, S. H. (1996). Combined vector hysteresis models and applications. IEEE Trans. Magn. 32(5): 4198-4203. LoBue, M., Basso, V., Fiorillo, E, and Bertotti, G. (1999). Effect of tensile and compressive stress on dynamic loop shapes and power losses of Fe-Si electrical steels. J. Magn. Magn. Mat. 196-197: 372-374.
9.16 Elasticity Coupled with Magnetism
943
9. Berqvist, A., and Engdhal, G. (1991). A stress-dependent magnetic Preisach hysteresis model. IEEE Trans. Magn. 27(6): 4796-4798. 10. Sasso, C. P., Basso, V., LoBue, M., and Bertotti, G. (2000). Vector model for the study of hysteresis under stress. J. Appl. Phys. 87: 4774-4776. 11. Barbier, G. (1995). Proposition d'un module de couplage magn~to-m~canique pour les mat~riaux ferromagn~tiques doux, Rapport de DEA de M~canique, Universit~ Paris 6, LMT-Cachan. 12. Bassiouny, E., et al. (1998). Thermodynamical formulation for coupled electromechanical hysteresis effects. International Journal of Engineering Science 26 (12): 1279-1295. 13. Gourdin, C., et al. (1998). Experimental identification of the coupling between the anhysteretic magnetic and magnetostrictive behaviours. Journal of Magnetism and Magnetic Materials 177-181: 201-202. 14. Hirsinger, L., et al. (2000). Application of the internal variable formalism to the modeling of magnetoelasticity, in Studies Applied Electromagnetics and Mechanics, Vol. 29: Mechanics of Electromagnetic Materials and Structures, pp. 54--67, Yang, J. S., and Maugin, G. A., eds., IOS Press. 15. Maugin, G. A. (1998). Continuum Mechanics of Electromagnetic Solids, North-Holland. 16. Maugin, G. A. (1991). Compatibility of magnetic hysteresis with thermodynamics. Int. J. Appl. Electromag. Mat. 2: 7-19. 17. Maugin, G. A., and Muschik, W. (1994). Thermodynamics with internal variables. J. NonEquilib. Thermodyn. 19: 217-289. 18. Gourdin, C., et al. (1998). Finite element implementation of an isotropic internal variable magneto-elastic hysteresis model, in Non-Linear Electromagnetic Systems, pp. 625-628, Kose, V., and Sievert, J., eds., Amsterdam: IOS Press. 19. Jiles, D. C. (1991). Introduction to Magnetism and Magnetic Materials, Chapman & Hall. 20. Sablik, M. J. (1989). Modeling stress dependence of magnetic properties for NDE of steels. Nondestr. Test. Eval. 5: 49-65. 21. du Tremolet de Lacheisserie, E. (1990). Magnetostriction: Theory and Applications of Magnetoelasticity, CRC Press. 22. du Tremolet de Lacheisserie, E. (1999). Magnetisme, 1: Fondements, Presses Universitaires de Grenoble. 23. Calkins, E T., Smith, R. C., and Flateau, A. B. (2000). Energy-based hysteresis model for magnetostrictive transducers. IEEE Trans. Magn. 36(2): 429-439. 24. Hernandez, E. Del Moral, Muranaka, C. S., and Cardoso, J. R. (2000). Identification of the JilesAltherton model parameters using random and deterministic searches. Physica B 275:212-215. 25. Jiles, D. C., and Atherton, D. L. (1986). Theory of ferromagnetic hysteresis. Journal of Magnetism and Magnetic Materials 61: 48-60. 26. Jiles, D. C., Thoelke, J. B., and Devine, M. K. (1992). Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis. IEEE Trans. Magn. 28: 27-35. 27. Jiles, D. C. (1995). Theory of the magnetomechanical effect. Journal of Physics D: Applied Physics 28: 1537-1546. 28. Sablik, M. J., and Jiles, D. C. Coupled magnetoelastic theory of magnetic and magnetostrictive hysteresis. IEEE Trans. Magn. 29: 2113-2123. 29. Gourdin, C., Hirsinger, L., and Billardon, R. (1998). Finite element implementation of an internal variable magneto-elastic hysteresis model, in Studies in Applied Electromagnetics and Mechanics, Vol. 13: Non-linear Electromagnetic Systems, pp. 625-628, Kose, V., and Sievert, J., eds., IOS Press.
SECTION
9.17
Physical Aging and Glass Transition of Polymers RACHID RAHOUADJ a n d CHRISTIAN CUNAT LEMTA, UMR CNRS 7563,ENSEM INPL 2, avenue de la For~t-de-Haye, 54500 Vandoeuvre-l~s-Nancy, France1
Contents 9.17.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
945
9.17.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
945
9.17.3 Description of the Model . . . . . . . . . . . . . . . . .
947
9.17.3.1 Model for Simulation of the Volume Recovery . . . . . . . . . . . . . . . . 9.17.4 Identification of the Parameters . . . . . . . . . . .
947 950
9.17.4.1 Identification of the Parameters of Version 1 for the Volume Recovery . . . . . . . . . . . . . . . . . . . . . . . . 950 9.17.4.2 Identification of the Parameters of Version 2 for the Volume Recovery and Apparent Specific Heat . . . . . . . 950 9.17.5 How to Use the Model . . . . . . . . . . . . . . . . . . . . 9.17.5.1 Isothermal Behavior: The Case of the Volume Recovery . . . . . . . . . .
951 951
9.17.5.2 Nonisothermal Behavior: The Case of the Specific Heat . . . . . . . . . 9.17.6 List of Parameters . . . . . . . . . . . . . . . . . . . . . . . .
953 953
9.17.6.1 Isothermal Volume Recovery of PVAc . . . . . . . . . . . . . . . . . . . . . . . . . . . .
953
9.17.6.2 Anisothermal Aging and Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.17.6.3 Comments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
944
954 954 954
Handbook of Materials Behaviar Madels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
9.17
Physical Aging and Glass Transition of Polymers
945
9.17.1 VALIDITY In the case of glassy materials near the glass transition, the aging transformation is usually associated with the recovery of a thermodynamic equilibrium state. It concerns various physical properties, such as the mechanical compliance, the refractive index, the apparent volume expansion, and the enthalpy related to the apparent specific heat. The last-mentioned is measured during continuous heat treatment. Examples of experimental results of isothermal volume recovery and apparent specific heat of polymers are analyzed near the glass transition as functions of heat treatment and temperature, respectively. The present theoretical modeling is based on an irreversible thermodynamics approach called "the distribution of nonlinear relaxations" (DNLR).
9.17.2 BACKGROUND The theoretical background [1, 2], has been briefly described in the article entitled "A Nonlinear Viscoelastic Model Based on Fluctuating Modes", Section 2.6. This modeling is in accordance with the fundamental works on internal variables by Coleman and Gurtin [3]. In the frame of a modal analysis of the dissipation, we have shown that the constitutive relationship can be written as -- E
--
o
.
p)j amq7 q -j_,
-P~
(1)
(tim - -
where 7q and tim refer to the perturbation and to the corresponding response components, respectively, and j to a normal dissipation mode. The indexes r and u denote the relaxed (or equilibrium) and the unrelaxed states, respectively. The parameter pO represents the relative weight of the process j, j = 1,... ,n, in the overall relaxation spectrum. The parameter a u is the symmetrical matrix of Tisza. Referring to the theorem of equipartition of the entropy production, [2], the initial relaxation spectrum near equilibrium can be defined as
k=l
Note that this distribution of relaxation times, ~, is also assumed to be operational for completely frozen states as for the beginning of the aging transformation.
946
Rahouadj and
Cunat
The data for volume recovery are shown in Figure 9.17.1a-c. The representation in Figure 9.17.1c is preferred to others for comparing theoretical simulations and experiments by means of the effective relaxation time " c e f f - - 6 / 6 and the volume variation c~ due to nonequilibrium, as defined by Kovacs [4, 5]. Some nonlinearities come obviously from the statistical nature of the relaxation times distribution, whereas other nonlinearities may find their origin in the dependence of relaxation times with the departure from equilibrium states. This major effect can be taken into account by introducing a shift function, aa, into the relaxation spectrum. Then, the relaxation times may be defined as inversely proportional to the jump frequency and to the probability of occurrence of an activated state (denoted by the symbol + ). Thus one has -
aF;'r(r/+
-
( a E +,r -
(3/
assuming that AE+ # - AE +'~, Vj = 1 , . . . , n . From this relation, the relaxation times become h exp(AF[/Rr)-
r a(t)
(4)
raS['~)/Rr]
~ x p [(AE+, r and a ( t ) - aa - e x p ( K ~ f i / R T ) . To simplify the present analysis, we will consider the simplest case where 6 = t r a c e ( ] / - / ~ ) . Equation 1 can be rewritten more explicitly as
w h ~ r ~ *i -
~,
,Liquid Glass~,~
Recovery]/~ Tg2 ~ s ; I T o
Tgl T
(a)
"ln('t;eff) T / 8=V (To,t) -Vr(T0,oo) ~ _ L o g
~
t
Glass 1
Glass 2 (b)
(c)
FIGURE 9.17.1 Asymmetryof the recovery response with respect to the sign of deviation from equilibrium, a: Thermal sequences; b. and c Two different representations of the same responses.
9.17
947
Physical Aging and Glass Transition of Polymers
follows:
{ ,
- [ -(S1-S~)'''-(Sj-S;)'''-(S"-S~")] (~1 - < )
~;) .
(~j -
1/1:1 i
1/'cj
(5)
(.. - e )
i/rn when the system is submitted only to a restrictive thermal or thermomechanical solicitation, T and 6- being independent. The parameters C, ~, and J represent the specific heat, the tensor of expansion coefficients, and the tensor of compliance, respectively.
9.17.3
DESCRIPTION
OF THE MODEL
In this section, two specific applications of the DNLR are briefly presented, the volume recovery and the apparent specific heat. Further details are given elsewhere by Marceron and Cunat [6].
9.17.3.1 MODEL FOR SIMULATION OF THE VOLUME RECOVERY In the experiments of Kovacs [4, 5], used perturbation 7, (or solicitation) corresponds and the response tim tO each component of conditions, the thermomechanical coupling relation (instead of Eq. 5):
to illustrate our purpose, the to the absolute temperature T, the strain tensor. Under these leads to a specific tensorial
.,.;
j=l
"cj(T, ~5)'
s with
g-
j=l
gj
(6)
For an isotropic medium, the tensor of one-dimensional expansion verifies %u _ %u 1, the trace of the strain tensor being coupled to the temperature variations. In other words, the volume strain may also be introduced by
_~.f v--; -
1 ._a, v j - v f Vr 2_, j=l
rj
(7)
948
Rahouadj and Cunat
where V - - E j n l Vj, and the volume V r corresponds to the overall representative volume element at the relaxed state. Indeed, the experimental data by Kovacs for volume recovery of PVAc suggest the approximation V V~ [4, 5]. The variable ~u represents the coefficient of isotropic expansion of the frozen amorphous phase. The previous relation (Eq. 7) has been actually used in our simulations of volume variations during recovery tests. The time integration of this equation has been performed in a numerical way using our spectral distribution ( (~,j = 1 , . . . , n) and the specific spectral shift function a ( t ) . The actual relaxation spectrum is supposed to be continuous. Its modeling near the equilibrium has been approximated using a discrete distribution of 50 modes equidistributed over six decades of the time scale [2]. Two versions (called versions 1 and 2) corresponding to different levels of approximation of the DNLR formalism have been developed to describe the volume recovery under various thermal histories.
9.17.3.1.1 Version 1" A First Approximation without Entropic Coupling Version 1 is the simplest application of the DNLR approach. The coupling between the thermodynamic variables has been neglected. This version has led to the expression of the volume recovery as a function of the thermal history. Thus, referring to Eq. 7, the equilibrium condition may be written:
--=~rf W
(8)
Combining Eqs. 7 and 8 and introducing the volume variation (AV - V - Vr) and the relaxation times [~:j -zya(AV/Vr)] gives"
V----7--Thus, by considering the definition: AV/V
r -
trace (~) - 5
(i0)
Equation 9 becomes in its final form
~= a~f- ~
(ii)
where the factor of nonlinearity due to the deviation from equilibrium, K,~ -- K,~(T), can be temperature-dependent.
949
9.17 PhysicalAgingand Glass Transition of Polymers
9.17.3.1.2 Version 2: A Second Approximation with Entropic Coupling To establish a more complete modeling, we need to consider the entropic coupling effects explicitly contained in the general relation (Eq. 5). Indeed, this relation shows that both volume and entropy simultaneously relax during aging transformations. Consequently, the activation entropy AS~ - Sd+ - Sj involves another type of nonlinearity by means of the evolving relaxation times, Tf (T, S). Thus we will present a more accurate version of the DNLR, called Version 2, which consists in solving a new complete differential system involving simultaneously the entropy and the volume variations. As a first example, each isothermal evolutions (T = 0), during volume recovery, will be characterized by
~__~.~
Sj-Sf
;=~ ff (~, s) a(aS)
with
S- ~Sj,
and S~ - ~ S ;
;=~
(12a)
;=~
The adequate shift function and relaxation time are given by a ( A S ) - exp( Ks(S -
.
h
= exp (KsAS~\ RT /I
- 7(as+. + f,'0 g at'/l
"rj (T,S) - k--~ exp
RT
(12b)
(12c)
The unknown initial value A S 0 - S u - S r becomes a new adjustable parameter calculated for each isothermal experiment in the integration of Eq. 12a. In the case of PVAc, this parameter is not temperature-dependent, in the explored range. Thus Eq. 7 becomes 5d S) a(AV/V r) d
(12d)
~
where the shift function, a(aV/W)= exp(K,sS/RT), corresponds to the definition (Eq. 11) of version 1. Integrating numerically the set of equations (Eqs. 12a-12d) provides the fitting of the isothermal curves of volume recovery. The second example on PVAc is concerned with the variation of the apparent specific heat during heating or quenching processes. From Eq. 5, it appears that the apparent specific heat may be expressed as a function of the entropy:
c yP
-
:rs
r ( sj - s; ) ~ - C~ - -~ ;=1 ? a--(-A-~)
(13a)
An alternative formulation, with C~pp-/://~', is based on the enthapy H obtained by the Legendre transformation of the Helmholtz free energy instead
950
Rahouadj and Cunat
of the entropy. This is the choice of Aharoune [7], who analyzed the DSC response (differential scanning calorimetry) of various glasses with the following relation: 1~
(Hi -- H~)
ACp pp" - ACp - -T fl-~j-1~-~fa~A-H))
(13b)
where ACp - Cp - CpF and a ( A H ) - exp (Kh(H 7 H~))
9.17.4 IDENTIFICATION
(13c)
OF T H E P A R A M E T E R S
For all examples presented here, we have verified that a discrete distribution of 50 relaxation modes, equidistributed over six decades of time respecting Eqs. 1 and 2, is operational.
9.17.4.1
IDENTIFICATION OF THE PARAMETERS OF VERSION 1 FOR THE VOLUME RECOVERY From Eq. 11, in the case of volume recovery, four parameters must be adjusted, i.e., A~-0~"-0~ r, AE+'r, ASj+__ 'r corresponding to the longest relaxation time, and the factor K6. The identification of A~ is obtained directly from the Kovacs experimental data by estimating the initial deviation from equilibrium, 60 = V(To, t = O) - Vr(To, t = oo), for a given temperature jump and with the approximate relation 60 - A0~(Tg- To); see Figure 9.17.1b and 9.17.2. The three other parameters, AE+'r, AS)+__'~, and K6(T), are estimated for each isothermal by minimizing the deviations between theoretical simulations and experimental results, according to the classical algorithm of Gauss-Newton (Fig. 9.17.2).
9 . 1 7 . 4 . 2 IDENTIFICATION OF THE PARAMETERS OF VERSION 2 FOR THE VOLUME RECOVERY AND APPARENT SPECIFIC HEAT Version 2 involves six parameters in Eqs. 12a-12d, i.e., the four previous ones, A0~(T), ASj=., +,r AE +'~, and K,5(T), completed by two others which may depend on temperature:
9.17
951
Physical Aging and Glass Transition of Polymers ...................... o
30 C
32.50
10 3 8 -5 -4
~3
o- L o g (~eff) \
~...~.,,
-2
k,,'o I " ~"
21
0
FIGURE 9.17.2 Experimental results (symbols, Kovacs) and theoretical predictions of the model, version 1.
9 ASo(T)= S " - S r, representing the difference of entropy between the unrelaxed and relaxed states; 9 Ks(T), giving the nonlinearity due to the entropic recovery. To analyze the observations by Kovacs giving 6 = 6(T, t), we assume that As is not temperature-dependent. The other five parameters, i.e., AS+'r, AE +'r, K~(T), ASo(T), and Ks(T), are optimized for all isothermal evolutions using the numerical method mentioned previously. In addition, in order to simulate the apparent specific heat on the base of Eqs. 13b and 13c, one needs to identify four parameters using the same method, i.e., ACp, AS, AE, Kh, and ASo. In spite of the case of the volume recovery, the correction due to the entropic coupling has no significant effect on the specific heat.
9.17.5
HOW TO USE THE MODEL
9 . 1 7 . 5 . 1 ISOTHERMAL BEHAVIOR: THE CASE OF THE VOLUME RECOVERY Version 1 provides a simple and fast calculation of the theoretical response by numerical integration of the set of equations without any entropic coupling. These predictions are similar to those from the KAHR model developed by Kovacs and coworkers [8] and from the model suggested by Moynihan et al. [9]. But, as shown in Figure 9.17.2, such a simplified version failed in the cases of expansion due to the aging process, especially near the equilibrium, when 6 -~ 0. Version 2, which accounts for the entropic coupling and which is formally contained in the DNLR framework, leads to more accurate results of volume recovery (Fig. 9.17.3).
952
Rahouadj and Cunat
The most interesting aspect of such a model is, of course, its ability to give some precise predictions for various thermal histories. Figure 9.17.4 gives us a comparison between theory and experiments for complex thermal loading, i.e., sequences of quenching, preannealing, and heating treatments. One can
FIGURE 9.17.3 Theoretical curves (version 2) and experimental results for the effective relaxation time at different temperatures T and To, PVAc.
FIGURE 9.17.4 Memory effects induced by preannealing treatments: prediction of version 2 and experimental results (symbols).
9.17 PhysicalAging and Glass Transition of Polymers
953
see the good compatibility between version 2 and the isothermal volume recovery data from Kovacs.
9.17.5.2
NONISOTHEaMAL BEHAVIOR: THE CASE OF
THE SPECIFIC HEAT We have chosen here to describe the behavior of PVAc, whose specific heat is given by Volkenstein and Sharonov [10]. Figure 9.17.5 gives an example of simulations of heating combined with preannealing. The parameters are completely consistent with those found for the volume recovery experiments.
9.17.6 LIST OF PARAMETERS 9.17.6.1
ISOTHERMALVOLUME RECOVERY OF PVAc
9.17.6.1.1 Version 1: Study at 40~ A~--4.65 • 10 -4 K -1 AE +'r -- 712,700J/mol.at; ASj+__'~- 1991J/mol. K; K~-- - 2850 • 103 J/mol.at. 9.17.6.1.2 Version 2: Study at Various Temperatures A0r • 10 -4 K -1 AE + ' r - 712,000J/mol; ASj+='r,- 1980J/mol.K; A S o - llJ/mol.K; K~ -- 43,797 4- 2141 T - 2 7 . 7 T2; Ks -- 82 4- 22 T.
FIGURE 9.17.5 Experimentalresults (symbols) and theoretical simulation (model version 2) for the apparent specific heat of PVAc.
954 9.17.6.2
Rahouadj and Cunat
ANISOTHERMAL AGING AND SPECIFIC HEAT
AE+'r-715.2kJ/mol
(compare with 712.7kJ/mol for version 1 and 712kJ/mol for version 2); and Kh----11870, referring to the enthalpydissipation coupling. AS0 is identical to the previous values and has no significant effect on the responses.
9.17.6.3 COMMENTS It can be concluded from Figure 9.17.3 that the coupling between the entropy variation and the volume recovery leads to a better agreement between experiments and theory. Furthermore, we have examined the predictive ability of this version to traduce the complex thermal histories of volume relaxation as proposed by Kovacs. The obtained agreement, illustrated in Figure 9.17.4, shows a promising ability in spite of the observed small gaps, which are certainly due to the ignorance of the actual thermal rates. Figure 9.17.5 compares DSC experimental curves from Volkenstein and Sharonov [10] with theoretical simulations. Note also that the activation parameters governing the kinetics of isothermal recovery near the equilibrium are very close to those of the apparent specific heat.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Cunat, C. (1985). Th~se de Doctorat d'Etat, INPL, Nancy, France. Cunat, C. (1996). Rev. G~n. Therm. 35: 680-685, Elsevier, Paris. Coleman, B. D., and Gurtin, M. (1967). J. Chem. Phys. 47(2): 597. Kovacs, A. J. (1954). Ph.D. thesis, Paris. Kovacs, A. J. (1963). Fortschr. Hochpol. Forsch. 3: 394. Marceron, P., and Cunat, C. (1999). Submitted to J. Mech. Tim. Dep. Mat. Aharoune, A. (1991). Ph.D. thesis, INPL, Nancy, France. Kovacs, A. J., Aklonis, J. J., Hutchinson, J. M., and Ramos, A. R. (1979). J. Polym. Sci., Polym. Phys. Ed. 17: 1097. 9. Moynihan, C. T., et al. (1976). Ann. N.Y Acad. Sci. 279: 15. 10. Volkenstein, M. V., and Sharonov, Y. (1961). Vysokomol. Soed. 3: 1739.
CHAPTER
Composite Media, Biomaterials
10
This Page Intentionally Left Blank
SECTION
iO.1
Introduction to Compo.site Media JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du President Wilson, 94235 Cachan Cedex, France
All previous chapters have been concerned with phenomenological models which apply irrespective of the precise material. Here a special chapter is devoted to formalisms by which the overall behavior is obtained by homogenization of the properties of different constituents. The class of materials includes metals considered as mixtures of different phases, fibers reinforced composites, laminates, concrete considered as an assembly of rocks, sand, and cement, reinforced polymers, etc. Also included are biomaterials such as bones or soft tissues. Geomaterials are treated in another special chapter because of the particular formalism imposed by porosity, dilatancy, etc. A second reason is that, as for geomaterials, it is a composite community active in its own field. The objective of the approaches in this chapter is to express the parameters in a constitutive law for the overall deformation behavior in terms of the properties of the constituents, as well as their distribution (random, organised in layers, etc.) and shapes (spherical particles, fibers, etc.), i.e., the microstructure. Assuming these are known, there is a rich toolbox of techniques in the literature for linear elastic constituents, which are summarized in Section 10.2. For composite materials where one or more of the phases is nonlinear, the methods are less well developed and more difficult. Section 10.3 summarizes some recent developments for nonlinear materials. Almost all available approaches assume that the size of the component made out of the composite is much larger than the typical length scale of the microstructure. As miniaturization continues, one may approach the limits of this assumption and it may be necessary to use nonlocal models, as explained in Section 10.4. In several applications, not only is the material subjected to mechanical loading, but also thermal strains, phase transitions, etc., may take place simultaneously. One of the methods discussed in Section 10.2, based on eigenstrains or transformations, is ideally suited to incorporate such effects; this is discussed in Section 10.5. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
957
958
Lemaitre
The elementary mechanisms may be observed by microscopes of different kinds, from which it is possible to deduce simple laws, such as the Schmitt slip law in plasticity, the friction law, or the decohesion law. There quantitative identification is difficult because direct measurements at microscale are often impossible to perform. Microhardness and nano-indentation tests on metallic crystals or pullout tests of fibers in composites or microtension tests are used but accuracy is poor. Often the only way is an indirect identification from tests at mesoscale. It is inherent to homogenization that it applies to deformation behavior but not to fracture since this is a localization phenomenon. However, homogenization techniques have been developed to describe the influence of damage inside the material, treated as an evolving additional constituent. For laminate composites, this is demonstrated in Section 10.6, while Section 10.7 is concerned with the behavior of ceramic matrix composites. Section 10.8 is an example of an actual failure analysis, rather than a damage approach, to materials with a particular microstructure. The special case of reinforced polymers at their viscous state during extrusion is treated in Section 10.9. Finally, biomechanics using similar tools is described in Section 10.10 for bones and in Section 10.11 for soft tissues.
SECTION
10.2
Background on Micromechanics ERIK VAN DER GIESSEN
University of Groningen, Applied Physics, Micromechanics of Materials, Nyenborgh 4, 9747 AG Groningen, The Netherlands
Contents 10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959 10.2.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 10.2.3 Homogenization Techniques . . . . . . . . . . . . . . 962 10.2.3.1 Mean-Field Approaches . . . . . . . . . . . . 963 10.2.3.2 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 10.2.3.3 Cell Methods . . . . . . . . . . . . . . . . . . . . . . 965 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967
10.2.1 INTRODUCTION Although the precise origin is not clear, the term micromechanics is usually associated with the description of the overall behavior of heterogeneous materials such as composites. This section summarizes some basic notions and results taken from reference books such as Nemat-Nasser and Horii [9] and Suquet [10], which will serve as background for the more detailed contributions in the rest of this chapter. Many of the articles in this chapter use the direct notation of tensors, denoted by boldface letters. For example, a second-order tensor is denoted by a, with components aij on an appropriate Cartesion basis {ei}. The inverse is denoted by a superscript - 1 , while a superscript T or t denotes the transpose. The dot product is usually used for operations like a . n = ~ijnjei, but for the product of a second-order and a fourth-order tensor we simply write L e Lijklc,kleiej (although some authors write the same as L:e). The dyadic or tensorial product of two vectors, ab, is a second-order tensor with components aib). Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
959
960
van der Giessen
10.2.2 BASIC CONCEPTS Almost all engineering materials are inhomogeneous at some length scale; i.e., they consist of different phases or constituents. Depending on the material and on the scale of observation, the material's microstructure can be granular, such as for a polycrystalline metal, or can be viewed as a composite with one phase serving as a matrix in which the other phases are embedded as inclusions: particles, fibres, voids, etc. Composite modeling, as discussed in this chapter, becomes relevant when the typical dimension d of the phases is much smaller than L, the characteristic length of the body under consideration or the "wavelength" of its loading (Fig. 10.2.1). This then leads to a decoupling of the two scales so that: 9 spatial variations in the stress and strain fields at the microstructural scale, denoted by tr(x) and a(x), respectively, influence the macroscopic response only through their averages; 9 gradients of macroscopic fields, such as E(X) and E(X), and composition are not significant for the response at the microstructural scale. The coupling between the continuum descriptions at the two scales is provided by the appropriate averaging, denoted by brackets: E=(tr},
E=(~}
(1)
The averaging is performed over a representative volume element (RVE) (Fig. 10.2.1). This RVE has to be (i) large enough so that statistically meaningful averaging over the microstructure is possible, i.e., l >> d, but (ii) small enough that macroscopic gradients can be neglected, i.e., l << L. Once the RVE has been identified (region ~), averaging can be carried out in terms of volume averages; for example,
1 f~ q(x)dV
(2)
Since the stress and strain fields a(x) and a(x) satisfy equilibrium and compatibility, respectively, div a = O,
~ = l(Vu + Vur)
(3)
the macroscopic stress and strain can be expressed in terms of the traction fields t(x) and the displacement fields u(x) over the boundary c3~2of the RVE (with unit outer normal n):
Z = (~1 = -~1 L (xt + tx)dA,
E = <.> =
1 / ~ (un + nu)dA
(4)
961
10.2 Background on Micromechanics
FIGURE 10.2.1 A body made of an inhomogeneous material can be represented as its macroscopic scale X as a continuum when its characteristic length scale L is much larger than the characteristic dimension of the microstructure d. The macroscopic continuum then describes the average behavior of representative volume elements (RVEs) of the inhomogeneous material at the microscopic scale x. It is sometimes convenient to decompose the total volume of the RVE into the volumes ~(r) (sometimes denoted ~ ) of each phase r. Averages can then be c o m p u t e d also over each phase separately; for example,
_ 1~
(q)r - n(r)
(rl q ( x ) d V
With the aid of such phase averages, the total average (q) according to Eq. 2 can be rewritten as
(q) - ~ c(r)(q)~ Y
where c (r) is the volume fraction of phase r.
(6)
962
van der Giessen
10.2.3 HOMOGENIZATION
TECHNIQUES
Once the averaging is defined, the major challenge is to compute the microscopic fields for a given microstructure. We summarize a few of the most well-known approaches for linear materials; Section 10.3 will discuss some approaches for nonlinear materials. For linear elastic materials, the microscopic fields depend linearly on the corresponding macroscopic fields. This is expressed as
~,(x) = A(x)E,
or(x) = B(x)E
(7)
in terms of the fourth-order localization or concentration tensors A and B for strain and stress. From Eq. 4, it follows that
(A(x)) = (B(x)) = II
(8)
with I! the fourth-order unit tensor with Cartesian components c~ijc~kz.The problem thus is to find these localization tensors for the given microstructure. When these are known, the overall constitutive law can be readily obtained. When the microscopic elasticity (Hooke's law) is written as
~(x) = L(x)e,(x)
(9)
with the moduli L(x) being piecewise equal to one of the phase moduli L (r), the overall constitutive law is
Z,-L*E
(10)
where the overall or effective stiffness tensor L* is given by L* -- (LA) -- ~
c(r)L(r)A (r)
(11)
r
with A (r) -- (A)r. Equivalently, when the microscopic constitutive equation is given in terms of the compliance tensor M, ~(x) = M(x)tr(x)
(12)
the macroscopic equations become
E=M*Z,
(13)
with the overall, effective compliance tensor M* given by
M* - - ( M B ) - E c(r)M(r)B(r)
(14)
l"
Thus the remaining challenge is to find the localization tensors A or B. Since realistic microstructures tend to be so complex that closed-form solutions are beyond the present capabilities, numerous approximate techniques have been proposed. Some of these, usually referred to as meanfield methods, treat only the phase-average localization tensors A (r) or B (r),
963
10.2 Background on Micromechanics
since these are sufficient to compute the effective moduli; cf. Eqs. 11-14. Methods which actually compute A ( x ) or B(x) evidently provide more details about the microscopic fields, but are practically always numerical. We mention a few of the most popular ones.
10.2.3.1 MEAN-FIELDAPPROACHES There are many practical examples of composite materials in which the volume fraction of inclusions in the matrix is so small that the interaction between the inclusions can be neglected. Hence, in such so-called dilute cases, the inclusion can be considered to live in an infinite matrix. This allows us to make use of some central results by Eshelby [3] for inclusions in a matrix with the same elastic properties but with an eigenstrain in the inclusions through the concept of equivalent homogeneous inclusions. This involves replacing the inhomogeneous inclusion of the composite with a homogeneous inclusion having the proper eigenstrain or transformation strain (see further in Section 10.5). When the inclusions are modeled as ellipsoids (including spheres as well as discs and needles as limiting cases), Eshelby's [3] key result is that the strains inside such inclusions are uniform and related to the unconstrained eigenstrain through a fourth-order tensor S. Hence, the localization tensors A (i) and B (i) in the inclusions (r = i; r = m for the matrix) are uniform. It can be shown [6] that they are given by A (i) -- [I + S M ( m ) ( L (i) - L(m))] -1,
B (i) -- L(i)[I q- S M ( m ) ( L (i) - L(m))]-lM(m)
(15) This is enough for two-phase systems to determine the overall moduli as L * = L(m) + c i (L(i) _ L(m))A(i)
(16)
since the phase-average localization tensors in the matrix are correlated because of Eq. 8. The Eshelby tensor S here depends only on the properties of the matrix and on the shape of the inclusions. Explicit expressions can be found in, e.g., Reference [8]. For instance, the Cartesian components Sijkl for spherical inclusions in an isotropic matrix with bulk modulus k (m) and shear modulus ~(m) are (17)
964
van der Giessen
with 3k(m) = 3k (m) + 4#(m) '
6(k(m) + 2/*(m)) fl = 5 - ( 3 ~ ) + 4/z(m))
(18)
With these, the overall bulk and shear moduli, k* and #*, are obtained from Eq. 16 as c(i) (k(i) - k(m)) k* = k (m) + 1 + o~(k(i)/k(m) - 1)'
#(m) #* =
c(i) (#(i) _ lz(m)) + 1 + fl(#(i)/la(m) _ 1)
(19)
The dilute approximation loses accuracy when the inclusion volume fraction is larger than several percentages (depending on the contrast, i.e., the difference between L (i) and L(m)). Then, the interaction between inclusions must be accounted for. The Mori-Tanaka method [7] uses an approximation for this by using the Eshelby equivalent inclusion concept but replacing the actual stress on the inclusion by the matrix average stress. The most straightforward way of proceeding [1] is to replace Eq. 7 by
~(i) 9
a(i)o(m)
-- ~ ,~dil,~
~
~r(i) _ n(i),.(m) ~dil v
(20)
.
with A'ai l and B~dil the dilute localization tensors according to Eq. 15. Straightforward algebra to eliminate the matrix averages from these expressions leads to the Mori-Tanaka localization tensors A (i, = [ ( 1 - c(i')I +c(i'A(~)]-lA~iil,
B (i, = [ ( 1 -
c(i))I--~--c(i)B(dilJ-1B~ i)
(21)
Estimates of the overall properties of porous elastic materials, i.e., those with a relatively large contrast in properties of the "inclusions" compared to the matrix, are useful up to void volume fractions of c (i) = 0.25. So-called self-consistent methods were initially devised for composites, such as polycrystals, in which it is not obvious which phase is the matrix and which is the inclusion [6]. However, it was shown later that they deliver useful estimates for a much wider range of materials. The key idea is to account for inclusion interaction in an approximate manner by embedding the inclusion not in the matrix but in a medium with the overall elastic moduli. Thus one uses, for example, the dilute estimate (Eq. 16) rephrased as Lsc = L (m) + ci(L (i) - L(m))A *
(22)
with A* now being a function of the elastic moduli Lsc of the composite. Hence, Eq. 22 is implicit and has to be solved iteratively. Alternative more elaborate versions have been derived subsequently, see for example [9, 2].
965
10.2 Background on Micromechanics
10.2.3.2 BOUNDS According to Eq. 11, the overall modulus L* is not simply the (Voigt or Taylor) average (L(x)) of the microscopic modulus tensor. A similar conclusion holds for the (Reuss or Sachs) average ( M ( x ) ) in relation to M*--(L*) -1. However, it follows from minimum potential energy and minimum complementary energy considerations that the two averages do provide bounds on the actual overall modulus: (M) -1 --
(
~
c(r)M (r)
)_1
~ L* ~ (L) -- ~
y
c(r)L (r)
(23)
Y
Much tighter bounds on the actual overall modulus can be obtained from a variational principle due to Hashin and Shtrikman [4]. The details are beyond the scope of this section, but the key idea is (i) to estimate the average strain in each phase r by treating it as an ellipsoidal inclusion in a reference matrix with elastic modulus L~ as discussed previously; (ii) to use these as trial solutions in variational theorems. The classical Hashin-Shtrikman bounds apply to composites with a statistically isotropic distribution of particles, and Willis [11] has generalized the approach to more general cases.
10.2.3.3 CELLMETHODS The actual calculation of the microfields is frequently carried out by cell methods. The starting point is the idea that the fields at the microscale can be computed if the RVE with the distribution of phases is modeled as a body subjected to boundary conditions that are consistent with the macroscopic state. There are two distinct approaches: either one prescribes the boundary displacements u(x) in accordance with the macroscopic strain E as u(x) = E . x
oncg~
(24)
so that Is(u)) = E, or one prescribes uniform boundary tractions, t(x) = E . n(x)
so that ( t r ( t ) ) lemma holds:
on0fl
(25)
E. In either one of these boundary conditions, Hill's [5] (trg) = EE
(26)
But Eqs. 24 and 25 are not equivalent (except in the limit d / l ~ 0) and will only give the same fields away from a boundary layer (more on this in Section 10.4).
966
van der Giessen
FIGURE 10.2.2 a. A unit cell in a material with a periodic distribution of phases, b. A unit cell (dashed square) that has two reflection symmetries. Only a quarter (solid square) needs to be analyzed, by imposing periodic and symmetric boundary conditions on the periodic (~) and symmetric (~r) boundaries.
By definition, the RVE contains many phases, which makes the computation usually impractical. If the distribution of phases exhibits symmetries, these can be used to define a much smaller u n i t cell. Numerical solution of the microscopic fields in such cells may be quite feasible with current computing facilities to very high degrees of accuracy. In case of periodic microstructures, the microfields are periodic, and the proper boundary conditions are u(x)
- - E . x 4- u *
onc~f~
(27)
with the fluctuations u* prescribed periodic (see Fig. 10.2.2a). For these boundary conditions, there are no boundary layer artifacts, and Hill's lemma (Eq. 26) continues to hold. In particular cases, even simpler boundary conditions can be applied. The most well-known ones are the symmetry boundary conditions. These apply to (i) unit cells that have reflection symmetries (ii) under macroscopic states that respect the same symmetry. A typical example is shown in Figure 10.2.2b for a composite with a square packing of particles (in two dimensions; fibers in three dimensions) subject to a macroscopic strain with principal directions coinciding with the directions of packing. Periodic boundary conditions in
967
10.2 Background on Micromechanics s u c h a case i m p l y t h a t t h e cell b o u n d a r i e s
remain straight; the symmetry
b o u n d a r i e s r e m a i n s t r a i g h t b e c a u s e of s y m m e t r y .
REFERENCES 1. Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater. 6: 147-157. 2. Christensen, R. M. (1979). Mechanics of Composite Materials, New York: John Wiley & Sons. 3. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. Lond. A241: 376-396. 4. Hashin, Z., and Shtrikman, S. (1963). A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11: 127-140. 5. Hill, R. (1963). Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11: 357-372. 6. Hill, R. (1965). A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13: 213-222. 7. Mori, T., and Tanaka, K. (1973). Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21: 571-574. 8. Mura, T. (1987). Micromechanics of Defects in Solids. Dordrecht: Martinus Nijhoff. 9. Nemat-Nasser, S., and Hori, M. (1993). Micromechanics: Overall Properties of Heterogeneous Materials, Amsterdam: North-Holland. 10. Suquet, P (1997). Continuum Micromechanics. CISM Lecture Notes 377. 11. Willis, J. R. (1977). Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25: 185-202.
SECTION
10.3
Nonlinear Composites: Secant Methods and Variational Bounds PIERRE M. S UQUET LMA/CNRS 31 chemin Joseph Aiguier, 13402, Marseille, Cedex 20, France
Contents 10.3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . .
969
10.3.2 Secant Methods . . . . . . . . . . . . . . . . . . . . . . .
970
10.3.2.1 N o n l i n e a r Local P r o b l e m . . . . . . .
970
10.3.2.2 L i n e a r i z a t i o n . . . . . . . . . . . . . . . . . . .
971
10.3.2.3 Secant M e t h o d s in G e n e r a l . . . . . 972 10.3.2.4 A Secant M e t h o d Based o n F i r s t - O r d e r M o m e n t s (Classical Secant M e t h o d ) . . . . . . . . . . . . . . . .
973
10.3.2.5 A Secant M e t h o d Based o n Second-Order Moments (Modified Secant M e t h o d ) . . . . . .
974
10.3.2.6 E x a m p l e : D e f o r m a t i o n T h e o r y of Plasticity . . . . . . . . . . . . . . . . . . . . 10.3.3 V a r i a t i o n a l B o u n d s . . . . . . . . . . . . . . . . . . . .
975 978
10.3.3.1 Effective Potentials . . . . . . . . . . . . .
978
10.3.3.2 B o u n d s . . . . . . . . . . . . . . . . . . . . . . . .
978
10.3.3.3 C o n n e c t i o n w i t h the Secant M e t h o d Based o n S e c o n d - O r d e r Moments ......................
981
Acknowledgements ...........................
982
References ....................................
982
968
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
10.3.1
969
INTRODUCTION
The problem addressed here is that of the effective behavior of nonlinear composites. By composites we understand not only man-made materials but also all types of inhomogeneous materials which are ubiquitous in nature (polycrystals, wood, rocks, bone, porous materials). Consider a finite volume element V of such a composite material, large enough to be representative of the composite microstructure and nevertheless small enough for the different phases (grains or different mechanical phases) to be clearly distinguished. This representative volume element (RVE) V is composed of N distinct homogeneous phases Vr, r = 1 , . . . , N, the behavior of which is characterized by a nonlinear relation between the (infinitesimal) strain and stress fields, ~ and a, ~(x) -- ~(r)(a(x))
when x is in phase r
(1)
This constitutive relation corresponds either to nonlinear elastic behavior within the context of small strains, or to finite viscous deformations when ~ and a are interpreted as the Eulerian strain rate and Cauchy stress, respectively. When the RVE V is subjected to an average strain ~, it reacts to this strain by an average stress 8. The relation between 8 and ~ is the effective constitutive relation of the composite. The question addressed here is: Can predictions be made regarding
this effective constitutive relation, given the constitutive relations of the phases and some (often limited) information about the composite microstructure? When the phases are linear, there exists a large body of literature partially answering this question. However, when the phases are nonlinear, there are very few really nonlinear schemes to analyze the global, as well as the local, response of nonlinear composites. Most methods are heuristic and are extensions or modifications of the secant method (which we will discuss in Section 10.3.2) and of the incremental method (which we will not discuss here). Most of these schemes proceed in three successive steps: 1. First, the constitutive relations of each individual phase are linearized in an appropriate manner. This is done pointwisely and serves to define a linear comparison solid with local elastic moduli which, in general, vary from point to point. 2. Then, the problem is reduced to that of estimating the effective properties of a linear comparison solid with a finite number of phases. To this aim, an approximation is introduced by assuming that the local moduli are piecewise uniform. In most cases (but not all), the regions
970
Suquet
where the moduli are uniform are precisely the domains occupied by the material phases. 3. Finally, the effective linear properties of the linear comparison solid are estimated or bounded by a scheme which is relevant for the type of microstructure exhibited by the linear comparison solid. These linear effective properties are used to estimate the nonlinear effective properties of the actual nonlinear composite. Following the seminal work of J. Willis [1], more rigorous results, namely, bounds for the nonlinear effective properties of composites, have been developed in the past ten years by Ponte Castafieda [2], Willis [3], and Suquet [4], among others. Ponte Castafieda's variational procedure [5], which will be briefly recalled in Section 10.3.3, is probably the most rigorous bounding theory available to date. Other methods, less rigorous but sometimes more accurate - - for instance, the second-order procedure of Ponte Castafieda [6] or the affine procedure of Masson and Zaoui [7] - - w i l l not be discussed here. Some connections do exist between the heuristic secant methods and the more elaborate bounding techniques, and we will briefly outline them in Section 10.3.3.3, following Suquet [8, 9]. More details can be found in the review papers by Ponte Castafieda and Suquet [10] and Willis [11] (see also several contributions in Reference [12]).
10.3.2 SECANT METHODS 10.3.2.1 NONLINEARLOCAL PROBLEM The local stress and strain fields within V are solutions of the local problem consisting of the constitutive equations (Eq. 1), the compatibility conditions satisfied by ~, and the equilibrium equations satisfied by a: ~(x) -- f#(~)(a(x)) in phase r,
~ -- ~I(Vll -~- V I I t)
div(a) - 0
in V
(2)
Several classes of boundary conditions can be considered on ~V (all conditions being equivalent in the limit of a large RVE under appropriate growth conditions on the functions f#(r)). For definiteness, we will assume uniform tractions on the boundary er(x).n(x)= ~.n(x)on ~V. Once Eq. 2 is solved (at least theoretically), the spatial average ~ of the strain field g can be taken: -(g),
where
(.)-~
.dx
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
971
Then the relation between this average strain and the imposed average stress defines the effective constitutive relation of the composite: -- ff(~)
(3)
10.3.2.2 LINEARIZATION It is in general impossible to solve the nonlinear local problem (Eq. 2) exactly. Therefore, a first step in most nonlinear schemes is to write the constitutive relations (Eq. 1) in the form (the reference to the phase is implicit here): ~(x) = M(~r(x)) : ~r(x) + t/(x)
(4)
where M and t/have to be specified. Two particular choices, giving rise to two broad classes of models referred to as secant and tangent models and schematically depicted in Figure 10.3.1 (several different choices are reviewed in Gilormini [13]), correspond respectively to the following linearlizations: secant : M ( ~ ( x ) ) = Msct(~(x)),
t/= 0
(5)
tangent : M ( ~ ( x ) ) ~- Mtgt(cr(x)),
I1 = eo
(6)
and
Note
that
there are several choices for Msc t in (Eq. 5) for which Consequently, the secant moduli are not uniquely defined
Msct(O') : o" - - ~ ( o ' ) .
FIGURE 10.3.1 Secantand tangent moduli in a tensile uniaxial test. The secant and the tangent compliance are the inverse of the secant and tangent stiffness Lsct and Ltgt, respectively.
972
Suquet
(this observation was made by Gilormini [13]). The tangent moduli are defined (uniquely in general) a s Mtgt(~r) = d~(a)/da. We will not discuss the second choice (Eq. 6), namely, the class of tangent methods (the interested reader is referred to References [6, 13, 14] for additional details).
10.3.2.3
SECANT METHODS IN GENERAL
Using the equivalent writing (Eq. 5) of the constitutive relations (Eq. 1), the local problem (Eq. 2) can be reformulated as
8(x) - M~(a(x)) 9a(x) in phase r,
~ - 89
+ Vut),
div(a) - 0 in V (7)
This problem can be considered a linear elasticity problem for a composite with infinitely many phases, since the moduli M~rc~(~(x)) may vary from point to point. Equation 7 is therefore not simpler than the original (Eq. 2). A simplifying assumption is introduced by assuming that the secant moduli take piecewise uniform values M (r) on subdomains Wr, r= 1 , . . . , M, which coincide with, or are contained in, the physical domains Vr. For simplicity we will assume in the following that the domains Wr and Vr coincide. In addition, the uniform moduli are assumed to be evaluated at some "effective stress" 8 (r) for the phase r: M
-
(8 )
The simplified secant problem now consists of the following: ~(x) = M (r) 9a(x)in phase r,
~
l ( V u + Vut),
div(a) - 0
in V (9)
Assuming for a moment that the M(r)'s are given, Eq. 9 is a problem for a linear N-phase composite called the linear comparison composite. The determination of the M(r)'s is made possible through Eq. 8. These relations involve the effective stress 8 (r) (which has not been specified yet, but this will be done in the next two paragraphs), which itself depends on the average stress 8. Therefore, Eq. 8 is a closure condition which renders the problem nonlinear. In summary, any secant method involves three steps: 1. A linear theory providing an expression for M as a function of the moduli M (r) of the individual phases in the linear comparison solid.
973
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
2. The resolution of N nonlinear tensorial problems for the N unknown tensors M (r)"
M(r) _ ~l(r) 9,Lsct (~(r)) ~
tT(r) - -
function of {M (r) }r-1
,...,
N and
(10)
Note that these equations depend on the particular expression ("function of") chosen for the effective stress F(r). Different choices give rise to different secant methods. 3. Once the N nonlinear tensorial problems (Eq. 10) are solved, the overall stress-strain relation is given by
-- M
9 t~
(11)
where /~/ is the effective compliance derived by means of the linear theory of step 1 from the individual moduli M (r) as determined in step 2.
10.3.2.4 A SECANT METHOD BASED ON FIRST-ORDER MOMENTS (CLASSICAL SECANT METHOD)
It remains to define the effective stress F(r). In the classical secant method, this effective stress is set equal to the average stress over phase r (it should be emphasized that the stress field under consideration is now that in the linear comparison solid): ~(r) __ t~(r) __ (O.)r '
where
(')r--
~-~
. dx
(12)
r
This "first order moment" of the stress over phase r can be expressed in terms of the overall stress 8 by means of the "stress-localization" tensor B(r)" ~(r)_
(B)r
. ~
(13)
Most linear theories provide (more or less) explicit expressions for the "stress-localization" tensors B(r) as functions of the individual compliances M (r). A typical example will be given in following text. Equation 13 completes Eq. 10.
974
Suquet
1 0 . 3 . 2 . 5 A SECANT METHOD BASED' ON SECONDORDER MOMENTS (MODIFIED SECANT METHOD) The classical secant method has several serious limitations. One of them is illustrated by its unphysical prediction for the response of nonlinear porous materials under hydrostatic loadings. Consider an RVE composed of an incompressible matrix with voids and subject to an hydrostatic stress. The average stress in the matrix is hydrostatic (the average stress in the voids is 0). Since the matrix is incompressible, it is insensitive to hydrostatic stresses. Therefore, the secant compliance associated with a purely hydrostatic stress by Eq. 12 always coincides with the initial compliance (under zero stress) of the material. The secant method applied with Eq. 12 predicts a linear overall response of the porous material. However, the actual response of the porous material is nonlinear, since the local stress state in the matrix is not hydrostatic (analytic calculations can be carried out on the hollow sphere model to prove this point explicitly), even if the average stress is hydrostatic. The occurrence of shear stresses in some regions of the RVE introduces nonlinearities both in the local and overall responses of the RVE which are not taken into account by Eq. 12. This observation has motivated the introduction of theories based on the second moment of the stress field, in particular form by Buryachenko [15], in approximate form by Qiu and Weng [16], or in general and rigorous form by Suquet [8] and Hu [17]. It is indeed observed that in many cases of interest the secant compliance Msct depends on the stress through the "quadratic stress" Msct(tT) -- Msct(~cP),
where
5P - 89t7 |
tT
(14)
Therefore, rather than expressing M (r) in terms of an "effective stress" t~(r), one can express M (r) in terms of an effective "quadratic stress" ~(r). A very natural choice for this effective quadratic stress is 5t' ( r ) - (ft')r= 89 |
~)r
(15)
This effective "second-order moment" of the stress over phase r has definite advantages over the first-order moment used in the classical secant method. For instance, it better accounts for local fluctuation of the stress. To see this, ~ (r) note that ~ i j i j - (~Oc~ij)r 9 Therefore, as soon as ,7 is nonzero in a (nonnegligible) region of phase r, the second-moment ~(r) of the stress does not vanish. In particular, the overall response of porous materials under hydrostatic loading, as predicted by the secant method based on the "second-order moment," is nonlinear (as it should be) and close to the exact solution [16]. The modified secant theory consists in solving Eq. 10 together with the definition (Eq. 15) of the effective stress of phase r.
9 75
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
In practice, one has to compute ~(r) for the linear comparison solid. This can be done analytically by means of a result previously used in different contexts by several authors (see, for instance, Kreher [18]). Consider a linear composite composed of N homogeneous phases with elastic compliance M (r). Let ]~'I(M(1),..., M(r),..., M (N)) be the overall compliance tensor of this composite, and let tr denote the stress field in this linear composite. Then:
1 ~4 (tr | tX)r= c-~ t~ 9c~M(r) " ~
(16)
A detailed proof of this result can be found in References [9, 18] (among others). In conclusion, the nonlinear systems of equations to be solved to complete step 2 of the secant method read as: ,.~a(r) _
1 2c(r)
c~]Vl c~M(r) " G,
8"
M (r) -- '/l(r) 9,LSCt ( ~ ( r ) ~/ k
(17)
1 0 . 3 . 2 . 6 EXAMPLE: DEFORMATION THEORY OF PLASTICITY A rather general form of stress-strain relations for isotropic elastic-plastic materials in the context of a deformation theory is given by the following: -~+~P,
~r
trm
s 2#0'
- - f f i + ~
~P=
3 p (r 2 G~q
s
(18)
where Crm-(1/3)tr(~r) is the hydrostatic stress, ereq-[(3/2)s" s] 1/2 is the von Mises equivalent stress (s being the stress deviator), and p(Geq) is the inverse of the plastic hardening curve Cr~q(p) of the material. An alternate writing of Eq. 18 is
~,
-
M~a(o) " ~r,
1'!1
Osct(Geq) K
with Msct(~r) -- -~ J - - F ~
(19)
where 1
1
--
= ~
1 =
3p(Creq) + ~
(20)
and where J and K are the fourth-order tensors which project any secondorder tensor on its hydrostatic and deviatoric parts:
j-li|
K-I-J
Suquet
976
i and I being the identity for symmetric second-order and fourth-order tensors, respectively. Since we will be manipulating isotropic tensors throughout this section, simplified (and classical) notations will be helpful. Any isotropic fourth-order tensor B with minor and major symmetries can be decomposed as B "- bm J -4- baevK, where bm = B :: J and ba~ = (1/5)B :: K. We use the compact notation B - {bm, ba~}. The algebra over isotropic (and symmetric) tensors is then very simple: for two such tensors B and C, one has B :: C {bmcm, ba~vCa~} and (B) - 1 - {(1/bm), (1/ba~)}. The response of the phases as described by Eq. 19 is linear for purely hydrostatic loadings (characterized by a constant bulk modulus k), and nonlinear in shear (characterized by a stress-dependent secant shear modulus #set). Note that Msct, which depends on tr through ~req only, is indeed a function of the quadratic stress 5p defined by Eq. 14. Indeed, ~req is itself a function of 5P: treq - (3 s " s) 1/2 - (3K :: 5~) 1/2. Consider now a two-phase material, where both phases are elastic-plastic and obey the constitutive relations (Eq. 19) with different material constants. We further consider the case where the phases in the composite are arranged isotropically in a particle-matrix configuration, corresponding, for instance, to the case of spherical particles of phase 1 randomly distributed in phase 2 (which is the matrix). The linear comparison composite is a two-phase linear composite with the same microstructure as the initial nonlinear composite and with isotropic phases characterized by a compliance tensor M (r) = {m(r)/3, o(r)/2}. Regarding the first step of the three-step procedure outlined previously, namely, a theory for the effective properties of the linear comparison solid, the Hashin-Shtrikman formalism is known to provide, in most cases, an accurate estimate of the effective properties of isotropic linear composites with particle-matrix microstructure. The corresponding estimate reads as
/~ : M (2) + c (1)(M ( 1 ) - M (2)) " B (1), B (1) : (I--I- c(2)~ " (M ( 1 ) - M (2)))-1
(21) where c(1) and c(2) are the phase volume fractions and where Q - {3qm, 2qaev} with
qm
1-~ m(2),
qa~
1-fl 0(2) ,
~
3k (2) 3k(2)+4#(2 ),
fl
2 3k (2)+6# (2) ~3k(2 ) + (4/3)#(2)
Then, the effective compliance/~1 = {rh/3, 0/2} and the stress-localization tensors B(r) {b(mr), h(r)} read, respectively: W,d g v
10.3 Nonlinear Composites: Secant Methods and Variational Bounds b~ ) _
1+
1
C(2)qm(m (1)
1(1-
-
-
/11(2)) '
c(1)b(ml)),
977
b(ml) --
1 + c(2)qae~(0 ( 1 ) -
aev
c-~ 1 - c (1)h(1) L,,de V
1(
0(2)) '
) 9
Regarding step 2 of the procedure, namely, the non-linear equation which stems from the choice of the "effect stress," the classical method makes use of the average stresses ~(r) = B(0 : 8. However, since Msct(a) depends on ~r only through the von Mises stress, the only useful information in the average stress is the von Mises equivalent stress F~q)- ~,ae~Creq.h(r)Therefore, the nonlinear equations to be solved in the classical secant method read
1 0(r) __ /~(r) " sct (t~;)) -- ~0r) .~t-
3p(r) (t~;) ) tT~;) ,
(~;) __ .(r) Ddev O'eq
(22)
where the coefficients h(r) ~'aev depend on the 0 (r) (this is where the nonlinearity comes into play). As for the secant method based on the "second-order moment," using again the fact that the secant compliance depends on the stress only through the von Mises stress, it is sufficient to compute the following quantities ~r- -- (~Te2q]r .'31/2 These quantities can be calculated by means of Eq. 16. The resulting nonlinear systems of equations finally read (see Suquet [9] for more details):
oCr)
~](r)
/ =(r)'~
1
"scttCYe q ) -- - ~ + _-(2)
a -- r b
__
r
)
3
3P (r) f =(r)'~ "(r) r eq
'
~Teq
Ddev(Yeq'
bt~ev )1/2
c(X)m(1)b m(1)2 - c(2)m(2)b(2m)2 ) "dev -- 5
dev
30(2)
(23)
_~_~m(2))
Note that the effective stress in phase 2 (matrix) is now sensitive to the overall hydrostatic pressure (which is not the case in the classical secant method).
978
Suquet
10.3.3 VARIATIONAL BOUNDS 10.3.3.1
EFFECTIVE POTENTIALS
In this section it is further assumed that the constitutive behavior of the individual phases derives from a potential, or strain-energy function w(e), or equivalently a stress-energy function w*(~r), in such a way that the (infinitesimal) strain and stress fields, e and a, are related by (~),
~-
or equivalently
with w(~) + w* (a) = tT :
c~w* ~ = --~-(~),
(24)
In the composite, the potentials w and w* de~pend on the phase under consideration (and are denoted by w (r) and (w (r)) , respectively). The solutions u and ~ of the local problem (Eq. 2) have variational properties which are essential to deriving bounds on the effective energy of nonlinear composites. These properties, called minimum energy principles, also permit us to define the effective potentials of the composite. For simplicity, we shall only consider here the "minimum complementary-energy principle": v~*(~)-
inf , ~(~)
~ f ( ~ ) - {~, d i v ( ~ ) - O i n V , T.n-~.nonc~V}
(25) It can be checked [10] that the effective constitutive relations obtained by averaging the strain field solution of Eq. 2 or equivalently Eq. 25 derives from the potential v~*, which is therefore the effective complementary-energy of the composite: -
--~(e).
10.3.3.2 BOUNDS 10.3.3.2.1 Ponte Castafieda's General Theory To obtain bounds on the effective potential v~* which are sharper than the Voigt and Reuss bounds, we introduce an inhomogeneous linear comparison composite with compliance M(x) and complementary-energy
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
w o (x, v ) -
979
89v " Mo(x) " T. Writing w* as w* - w o 4- w o , one obtains
~*(8)--
inf
(w*(~)=
inf
( ( ( w * - w o ) ( ~ ) ) 4 - ( w o(~))) (26)
> ( inf (w o ('t))'~ - (v(x, Mo(x))) \~x(~)" "j
=
(Mo,
V(Mo)
where Wo is the effective complementary energy of the linear comparison solid and v(x, Mo(X)) -- sup [Wo (x, r ) - w* (x, z)], ,r
V(Mo) - (v(x, Mo(x)))
(27)
Since Eq. 26 is valid for any choice of Mo(x), one has ;,*
sup M0(x)>0
(Mo,
- V(Mo)]
(28)
The inequality (Eq. 28), due to Ponte Castafieda, gives a rigorous bound I on the nonlinear effective properties of the composite (through the potential ~ * ) in terms of two functions: 9 Wo (M0, i f ) _ 1 8 " /~/0 " F is the elastic energy of a linear comparison composite made up of phases with compliance Mo(x) at point x; the linear comparison solid is chosen among all possible comparison composites by solving the optimization problem (Eq. 28). The difficulty lies in the precise determination of the energy Wo for a linear comparison solid consisting of infinitely many different phases. 9 The role of v(x, .) is to measure the difference between the nonquadratic potential w*(x, .) and the quadratic energy Wo(X, .) of the linear comparison solid. This function is difficult to compute in general, but a bound can be easily computed for the class of materials considered in following text. The problem of bounding the effective properties of a linear composite with infinitely many phases being too difficult, we reduce it by minimizing over a smaller set of compliance fields Mo(x), namely, those fields which are uniform on each subdomain Vr. With this smaller set, the supremum
1For a broad class of material behavior (discussed in Section 10.3.3.3), the inequality (Eq. 28) is in fact an equality and is strictly equivalent to the variational characterization of ~* given in Eq. 25.
980
Suquet
in Eq. 28 even smaller: sup M(or)>O
r--l, ..., N
r=l ..... N /
(29) The linear comparison composite is now an N-phase composite, with compliance M(0r) uniform throughout phase r. There is a similarity (and even more, as will be discussed in the next paragraph) with the secant methods in that a linear comparison solid is introduced in both approaches. Note, however, that here the elastic moduli in the linear composite are determined by means of an optimization procedure, whereas they were deduced from the (somehow arbitrary) choice of an effective stress in the case of the secant method. This optimization procedure leads to a rigorous bound for the potential ~*. Remark: Most of the nonlinear bounds available to date are bounds on the energy of the composite. Only in specific situations do these bounds give bounds on the stress-strain relations of the composite. This is the case for power-law materials for which, due to the Euler theorem for homogeneous functions of degree n + 1, one has 8 : ~ = (n + 1)~* (8). Therefore, any bound on ~* gives a bound on the overall strain ~ in the direction of the applied stress 8. However, no information on the other components of the strain is provided. Interestingly, a method for bounding directly the stress-strain relation has been recently proposed by Milton and Serkov [19].
10.3.3.2.2 Complementary Energies Depending Only on the Quadratic Stress We consider here a broad class of behaviors corresponding to potentials w* which depend on the stress tensor ~ only through the quadratic stress ~ , w* ( ~ ) - G(~), for some appropriately chosen function G and where ~ = ~1 r | ~. G is further assumed to be a convex function of ~ . This class of materials contains in particular all materials with a complementary energy in the form w* (~r) - ~ ~m2 -[- ~(Creq) 9 For this class of materials, the function v(0 (corresponding to v in phase r) can be bounded from above by
where
~= 89174
(30)
10.3 N o n l i n e a r C o m p o s i t e s : S e c a n t M e t h o d s a n d Variational B o u n d s
981
(G(r)) * denotes the Legendre transform of the convex function G(r): (G(r)) * (M) - sup [M "" 3-- - G(r)(J-)]
(31)
Y
Note that in gq. 31 the supremum is taken over all symmetric fourth-order ~-, whereas the supremum in Eq. 30 is restricted to rank-one symmetric ~-. The lower bound (Eq. 29) reduces to g,*(8) >_ sup
(1
8 " /VI0({M(0r)})
M~,)>0
"8r = l .... , N
~--~c(r)(G(r)) * (M(0r))
)
r=l
(32)
1 0 . 3 . 3 . 3 CONNECTION WITH THE SECANT METHOD BASED ON SECOND-ORDER MOMENTS When the complementary energy of the constitutive phases depends only on the "quadratic stress" ~ , the constitutive relation (Eq. 24) can alternatively be written as c3G -Msa(Sf) " t~, where Msa(Sf) -- ~-~ (Sf), (33) 0G* or equivalently 5f - 0M (Msct) We are now going to inspect in more detail the optimality conditions for the moduli M(0r) derived from the optimization problem (Eq. 32). Assuming stationarity with respect to these moduli, the optimality conditions read as -18 "
2
-0/~o -"0"-0M(oo
_
c(r)0(G(r') 0M */M~r)\ ~ )
But according to Eq. 16, the first term in this equality is nothing other than the average second-order moment of the stress in the linear comparison solid. Therefore: ~f(r)
1 - ~{o" | O')r=
0M
Making use of Eq. 33, the optimally conditions finally amount to solving the following systems of nonlinear equations: M(or) -- M~:~(ff(r)),
with
This systems coincides with Eq. 17.
ft,(r) _ (r-------51 2c o'- " 0M~r )0/~~ . 8
982
Suquet
In conclusion, it has been shown that the optimal moduli M~0r) in the variational procedure coincide with the secant moduli M (r) determined by the (more heuristic) secant method based on the second-order moment described in Section 10.3.2.5. In other words, the variational procedure can be interpreted as a secant method. It has, however, the definite advantage of delivering a clear rigorous bound on the effective properties of the nonlinear composite.
ACKNOWLEDGEMENTS This paper was written while the author was a Visiting Associate at the California Institute of Technology for the year 2000-2001. The financial support of Caltech is gratefully acknowledged. Stimulating discussions with P. Ponte Castafieda are gratefully acknowledged.
REFERENCES 1. Willis, J. R. (1989). The structure of overall constitutive relations for a class of nonlinear composites. IMA J. Appl. Math. 43: 231-242. 2. Ponte Castafieda, P. (1991). The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39: 45-71. 3. Willis, J. (1991). On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids 39: 73-86. 4. Suquet, P. (1993). Overall potentials and extremal surfaces of power law or ideally plastic materials. J. Mech. Phys. Solids 41:981-1002. 5. Ponte Castafieda, P. (1992). New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids 40: 1757-1788. 6. Ponte Castafieda, P. (1996). Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 44: 827-862. 7. Masson, R., and Zaoui, A. (1999). Self-consistent estimates for the rate-dependent elastoplastic behavior of polycrystalline materials. J. Mech. Phys. Solids 47: 1543-1568. 8. Suquet, P. (1995). Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castafieda's nonlinear variational procedure. C.R. Acad. Sc. Paris, 320, S~rie IIb: 563-571. 9. Suquet, P. (1997). Effective properties of nonlinear composites, in Continuum Micromechanics, vol. 337 of CISM Lecture Notes. pp. 197-264, Suquet, P. ed., New York: Springer-Verlag. 10. Ponte Castafieda, P. Ponte, and Suquet, P. (1998). Nonlinear composites, in Advances in Applied Mechanics, pp. 171-302, vol. 34, van der Giessen, E., and Wu, T.Y. eds., New York: Academic Press. 11. Willis, J. R. (2000). The overall response of nonlinear composite media. Eur. J. Mech. A/Solids 19: $165-$184. 12. Suquet, P. (1997). Continuum Micromechanics, vol. 337 of CISM Lecture Notes, Wien: SpringerVerlag. 13. Gilormini, P. (1996). A critical evaluation of various nonlinear extensions of the selfconsistent model, in Micromechanics of Plasticity and Damage of Multiphase Materials, pp. 6774, Pineau, A., and Zaoui, A., eds., Dordrecht: Kluwer Acad. Pub.
10.3 Nonlinear Composites: Secant Methods and Variational Bounds
983
14. Masson, R., Bornert, M., Suquet, P., and Zaoui, A. (2000). An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids 48: 1203-1227. 15. Buryachenko, V. (1993). Effective strength properties of elastic physically nonlinear composites, in Micromechanics of Materials, pp. 567-578, Marigo, J. J., and Rousselier, G., eds., Paris: Eyrolles. 16. Qiu, Y. P., and Weng, G. J. (1992). A theory of plasticity for porous materials and particlereinforced composites. J. Appl. Mech. 59: 261-268. 17. Hu, G. (1996). A method of plasticity for general aligned spheroidal void of fibre-reinforced composites. Int. J. Plasticity 12: 439-449. 18. Kreher, W. (1990). Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Solids 38: 115-128. 19. Milton, G. W., and Serkov, S. K. (2000). Bounding the current in nonlinear conducting composites. J. Mech. Phys. Solids 48: 1295-1324.
SECTION
10.4
Nonlocal Micromechanical Models J. R. WILLIS Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Contents 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 The Effective Relation . . . . . . . . . . . . . . . . . . 10.4.4 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Statistically Uniform Media . . . . . . . . . . . . 10.4.6 Gradient Approximations . . . . . . . . . . . . . . 10.4.7 Variational Formulation for the Effective Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.8 Concluding remarks . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1
984 985 987 988 990 990 991 992 995
INTRODUCTION
Virtually any composite material has a microstructure that needs to be treated as random. Many composites are obviously disordered: polycrystalline metal or glass-fibre reinforced material is an example. Even composites with highly correlated microstructures can be viewed as random. Consider, for instance, the idealised case of a specimen comprising a matrix containing a perfectly periodic array of identical inclusions. The spacing of the inclusions is a very small fraction of a typical dimension of the specimen, and it is most unlikely that the exact location of any one inclusion (in terms of which the positions of all the others are fixed) will be known. Thus it is appropriate to regard the specimen as one chosen at r a n d o m from a set of macroscopically identical specimens, distinguished from one another by the exact location of one chosen inclusion within one particular periodic cell. In practice, observation
984
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
985
10.4 Nonlocal Micromechanical Models
of stress or strain at some position x within the specimen will result in the measurement of some average in the vicinity of x - for instance, by use of a strain guage or some optical device. It is appropriate, therefore, to seek an effective constitutive relation between such local averages of stress and strain. Unfortunately, this is difficult to accomplish, and it is usual instead to seek a relation between the ensemble means of the stress and the strain. This expedient is adopted throughout virtually the whole of physics, and it will be adopted without further comment here, although for composites at least the distinction between local spatial average and ensemble average can be confronted explicitly; it could also in principle be evaluated quantitatively in some particular case, but no such study is known to the author.
10.4.2
FORMULATION
The composite, in general, will be taken to occupy a domain f~ and to be made of n constituents, or phases, perfectly bonded across interfaces. The stress and strain in material of type r (r = 1, 2,..., n) are related so that
a = Lre, ~ij
=
(Lr)ijklekl,
(1)
the second form (in which the usual summation convention on repeated suffixes is employed) giving the explicit meaning of the first more concise form which will be used in the sequel. The elastic constant tensor L(x) of the composite then takes the form
L(x)- ~ trZ~(~)
(2)
r=l
where Zr(X) is the characteristic function of the region occupied by phase r, taking the value 1 when x lies in phase r and zero otherwise. The elastic constant tensors Lr are known and correspond to particular materials, but the functions Zr that specify the microgeometry are random. The probability pr(X) for finding material of type r at x, and the probability prs(X,X I) for finding simultaneously material of type r at x and of type s at x' are pr(X) = (Zr(X)), prs(X,x') = (Zr(X)Zs(X')),
(3)
the angled brackets signifying the ensemble average. If the composite is statistically uniform, Pr becomes a constant, equal to the volume fraction of phase r, and P,s becomes a function of ( x - x ' ) only. (Actually, these conditions by themselves only require the medium to be "statistically secondorder stationary.")
986
Willis
The (vector) equation of equilibrium of the actual medium is diva+f=0,
xe~2
(4)
which is to be solved in conjunction with the constitutive relation
and suitable boundary conditions applied over the boundary ~f2 of ~2. Prescribing the displacement, u -- u ~ say, over ~2 is one possibility; another is to prescribe the traction a . n = a ~ where n is the (outward) normal to 0f2. Under such conditions, the stress, strain, and displacement of the composite are defined. They could, in principle, be determined for each realisation of the composite and then their ensemble means calculated. The purpose of developing an effective constitutive relation is to allow the direct determination of the ensemble means. First, averaging Eq. 4 gives d i v ( a ) + f = O, x ~ ~
(6)
It is assumed here that the body force f is sure, for simplicity. The more general case of configuration-dependent body force has been treated recently by Luciano and Willis [1]. The exact constitutive relation (Eq. 5) is replaced by the effective relation <~) = Left
(7)
where L eg is a nonlocal operator. If it were known, it is a reasonable assumption that Eqs. 6 and 7 could be solved in conjunction with the ensemble averaged boundary condition, either ( u ) = u ~ or < a ) - n = a0. n over ~2. There is, however, a serious limitation. It will be seen that the effective nonlocal operator can be constructed in terms of a certain Green's function for the domain ~2, and this Green's function depends on the form of the boundary conditions. Thus, implicitly, at least the type of boundary condition has an influence on the form of L eff. This influence will only manifest itself, however, in some "boundary layer" in the vicinity of ~f2. Outside of this layer, L eg will take the same form as it would in an infinite body. This is the only case for which explicit results are known. Any attempt to use the infinite-body form right up to the boundary 0~2 would require the replacement of the exact (mean) boundary condition by some "effective" boundary condition, if such a condition could be proved to have a meaning. This question has not even been addressed to date. In the sequel, the problem will be avoided by considering only an infinite domain, loaded by a body force f which has compact support, coupled with the requirement that all fields should tend to
987
10.4 Nonlocal Micromechanical Models
zero as Ixl ---* oo. The question of boundary conditions will be briefly returned to at the end of the article.
10.4.3
THE EFFECTIVE
RELATION
It is useful to introduce a comparison medium, with elastic constant tensor L0, and to replace Eq. 5 by a = Log + "c
(8)
The "stress polarisation" 1: should satisfy 9( x ) = ( I . ( x ) - I . o ) e ( x )
(9)
Substituting Eq. 9 into Eq. 4 gives div(L0e) + div 9+ f = 0, x e f]
(10)
which, together with the relevant boundary condition, implies that u ~ - E~oI:
u-
(11)
Here, u ~ = G0f is the solution of the problem defined by Eq. 10 together with the boundary condition, Go representing the corresponding Green's function operator; Er is the operator obtained from it by integration by parts to move the operation corresponding to the divergence on 9 onto the Green's function. Differentiating Eq. 11 with respect to x and symmetrising then gives the expression e = e ~ - Foz
(12)
for the strain e. Explicitly, the operator Fo has kernel
(Co)~jkz(X, x')
c~2(G0)ik (x,
--
,Vx;&~ x') I'~j),~kZ)
(13)
where the bracketing of suffixes implies symmetrisation. Combining Eqs. 9 and 12 now gives an equation that must be satisfied by 1: or, equivalently, an equation that must be satisfied by e. In the case that the medium is weakly heterogeneous, so that L(x) differs only slightly from L0, the equation for e was solved by iteration, many years ago, by Beran and McCoy [2]. Such analysis demonstrates that the operator L qf is unavoidably nonlocal. Here, however, concern will be for the general case of large fluctuations. For this purpose, it is useful to consider the equation for 9 in the form (L - Lo)-1"r + F o ~ = e ~
(14)
988
Willis
Averaging Eq. 12 allows e ~ in Eq. 14 to be expressed in terms of (e), so that (L - Lo)-l'r + F o ( z - (z)) = ( e )
(15)
Formally, the solution of this equation is 9 = l"(e)
(16)
and it follows by averaging Eq. 8 that t d = to + ( r )
(17)
The perturbation theory contained in Reference [2] provides a series representation for T, yielding L eff -- ( L ) - ( ( L - L0)F0(L - ( L ) ) ) + . - .
(18)
The operator F0 may be expressed in the form ro -
+
(19)
where F~~ is the corresponding operator for an infinite body and F~m is an "image" correction, which comes from the boundary conditions. The kernel of F~~ is a homogeneous function of degree - 3 (in three-dimensional space). Except when the point x of interest is close to the boundary 0f2 (relative to the scale l of the microstructure), the kernel of the "image" term remains bounded, even when x' is close to x. In this case, F0 may be replaced, asymptotically, when I/L ~ 1, where L is a length characteristic of the domain f2, by F~~ and then the effective operator Ldf reduces to the form appropriate to an infinite body. As remarked previously, it is only in this case that any explicit calculations have been performed.
10.4.4
BOUNDS
Even in the infinite-body limit, L eff cannot be found exactly. It is, however, possible to obtain useful information by bounding the energy in the composite. Bounds which involve no more information than the one- and two-point probabilities pr and P,.s can be found by using the HashinShtrikman formalism, as developed in Reference [3] and exploited further in Reference [1]. Considering the case of zero prescribed displacement over c9f2 (including, as a limiting case, the problem for an infinite body), the energy in one realisation of the composite (elastic stored energy plus the energy associated with the body-force loading) is o~ _ _1 f
d~
frGf dx
(20)
where G represents the Green's function operator of the actual composite.
989
10.4 Nonlocal M i c r o m e c h a n i c a l Models
Ensemble averaging gives
(~)
__~ if
-
f r G q f f dx
(21)
dn Here, G qf = ( G ) is precisely the Green's function operator associated with the effective medium, which is defined by the effective modulus operator Lqf. Application of the Hashin-Shtrikman formalism permits the development of bounds for (G), implying corresponding restrictions on the operator L eff, as detailed in References [1] and [3], for example. The result is that a HashinShtrikman approximation GHs to G qf follows by solving, in place of the exact equation (Eq. 15) for ~, the Hashin-Shtrikman equations /1
pr(Lr - L0)-l~r -t- ~
{F0(prs - prps)}'Cs -- pr(ens)
(22)
s--1
where 9 is approximated as 11
'~(X)- ~ "~r(X))(,r(X)
(23)
r=l
the functions Vr being nonrandom, and eiqs is the corresponding approximation to e. The notation { K ( p r S - prps)} is used for an integral operator whose kernel is K(x, x')[ prs(X, x') - pr(X)ps(X')]. If the solution of Eq. 22 is expressed as
"Or-s Trs(Ps(eIqs))
(24)
the corresponding Hashin-Shtrikman approximation to Lelf is
LHs -- Lo + ~ r=l
prTrsPs
(25)
s=l
The associated Hashin-Shtrikman approximation GIqs to G eff provides the approximation
- --~l s FHS --
(26)
for ( g ) . It is an upper bound to ( g ) if L0 is chosen so that L r - Lo defines a positive (semi-)definite quadratic form for each r, and a lower bound if L r - L0 defines a negative (semi-)definite form for each r.
990 10.4.5
Willis
STATISTICALLY UNIFORM
MEDIA
For a statistically uniform medium, Pr is constant, and Pr~ is a function of ( x - x ~) only. Also, taking the medium as infinite in extent, F0 is replaced by F~~ whose kernel is also a function of ( x - x'). All operators become convolutions, and Eq. 22 can be reduced to algebraic form by taking Fourier transforms. Also, Plancherel's theorem allows Eq. 21 to be expressed
(,f ) = - 89f ]* (k)Geff(k)](k) dk d
(27)
where the tilde denotes the Fourier transform and * denotes complex conjugate transpose. It follows that (~eff_ Gns) defines a Hermitian form that is negative or positive semidefinite when L0 is such that (Lr - L0) defines a quadratic form that is positive or negative semidefinite for each r.
10.4.6
GRADIENT APPROXIMATIONS
We write the overall constitutive relation
(a)(x) = (LeYf(e))(x) -- f~ K(x,x')(e)(x') dx'
(28)
The kernel function K decays to zero rapidly as I x - x~[/l --* oo, where l is a characteristic length associated with the microstructure. Therefore, when the loading is such that the resulting mean strain ( e ) varies slowly relative to the scale /, the result of applying the operator Lqf to ( e ) can be evaluated, asymptotically, by approximating (r ~) by the first few terms in its Taylor expansion about x ~ = x: in symbolic notation, (e(x')) ~ (e)(x) + ( x ' - x). V(e)(x) + 89 - x) | (x' - x)] 9[ v |
(29) V](eS(x) + . . .
It follows that
(tr)(X) ~, Lh~
q- M(x) . V(e)(x) 4- N(x) " (V | V)(e)(x) + . . . (30)
where
I.h~
-- f K(x,x') dx'
M(x)
-- f K(x,x')(x' - x) dx'
991
10.4 Nonlocal Micromechanical Models
N(x) = 89J K(x,x')[(x' - x) | (x' -- x)] dx'
(31)
The tensor L n~ is the tensor of effective moduli in the "homogenisation limit" l/L ~ O. If the medium is statistically uniform, then K depends on ( x - x~). The tensors L h~ M, and N become constants; it will usually be the case that K is an even function of its argument, so that M = 0. An alternative representation is also possible: L h~
= g:(o)
N ---l(Vh |
Vk)K(0)
(32)
Thus the gradient approximation represented by Eq. 30 follows from the small-k expansion of the Fourier transform of L eft. The corresponding Hashin-Shtrikman approximations follow from parallel treatment of the Fourier transform version of Eq. 25. The constants were developed from this standpoint by Drugan and Willis [4]. 10.4.7 VARIATIONAL FORMULATION FOR THE EFFECTIVE MEDIUM For the infinite medium or, more generally, for the finite medium with displacements prescribed over the boundary, the solution of Eqs. 4, 5, and the boundary conditions minimises the energy functional ~(u)
- f ~ { 89 " L " e - f . u } dx
(33)
over displacement fields u that satisfy the boundary conditions. It follows that the family of solutions, defined over the entire set of realisations, minimises
over all random fields u, defined over f~ and the sample space, and satisfying the given boundary conditions on c3f~. The minimum value of o~ is N, and the minimum-value of ( ~ ) is (N). It follows from simple mathematics (not physics) that the solution of Eqs. 6 and 7 minimises the functional
~-eff ( ( u ) ) - ~
{ 89
" Leff ( e ) - f . ( u ) } clx
(35)
over fields ( u ) that satisfy the given boundary conditions ((u) = u ~ on 0f~).
992
Willis
The minimum value of ~eff is (zeSf)m/n -
89 f .
(u)dx
(36)
that is, ( ~ ) , precisely. There is, however, no relation between Z efS and ( Z ) more generally. Consider now the gradient approximation,
( a ) -- Lh~
+ N :(V | V ) ( e )
(37)
which applies under the assumptions specified in deriving Eqs. 32. If this constitutive relation is assumed to hold, right up to the boundary Off, then the problem specified by Eqs. 6 and 37 requires another boundary condition in addition to the given displacement condition. It is natural to associate with it the functional
cS((u)) - f~ {l(e)Lh~
-- l ( V ( e ) ) . N. (V(e)) - f . ( , ) } dx
(38)
which is stationary at the solution if the additional boundary condition on c~2 is taken as the "natural" one, n. N. (V(e)) = 0. There is, however, no direct physical reason why this condition should be applied: its need is a consequence of having adopted the constitutive relation (Eq. 37) up to the boundary, where its derivation has lost validity. There is another complication, perhaps still more severe. The functional may or may not be convex, depending on the constant tensor N. Thus the stationary value need not be a minimum, and, when it is not, it is possible that the boundary value problem just proposed, consisting of Eqs. 6, 37, the given boundary condition, and the "natural" boundary condition, may not have a unique solution; it is also possible that the desired solution could be unstable. This is not in conflict with the basic physics or mathematics: the gradient approximation is only good when the gradients are small. This problem was highlighted before, in the context of "weakly heterogeneous" analysis, by Beran and McCoy [2].
10.4.8 CONCLUDING
REMARKS
This article has shown how nonlocal effective constitutive response can be deduced from explicit consideration of the micromechanics of deformation of a composite. The relation in general involves an integral operator. In the case of a statistically uniform medium, the integral takes convolution form, and the effective response can equally well be described in terms of the Fourier transform of the integral operator. It cannot be determined exactly, but
10.4 Nonlocal Micromechanical Models
993
bounds on the energy can be developed and these, in turn, provide restrictions on the integral operator, which can be expressed purely algebraically in Fourier space, in the case of statistical uniformity. Figure 10.4.1 (reproduced from Reference [51) presents a sample result. The example is two-dimensional (representing, for example, a cross section of a fibre-reinforced material under conditions of plane strain). The composite is a matrix (r = 2) containing square inclusions (r = 1), at volume fraction 0.16, arranged periodically so that each period is a square. The length of each side of the square period is normalised and taken equal to 2re. Both matrix and inclusions have Poisson's ratio 0.25. Young's modulus is 16 for the inclusions and 8 for the matrix. The two-point probability pll (from which the other Prs can be derived, for a two-phase composite) is exactly periodic, its period being the square. The composite is statistically uniform because, as discussed in Section 10.4.1, the location of any one inclusion is treated as random, with uniform distribution over one period. Thus the effective properties can be developed in Fourier space. The figure gives three pairs of curves. One of each pair was obtained by taking L0 = L1 (to give an upper bound for the energy), and the other was obtained by taking L0 = L2 (to give a lower bound for the energy). The component ([Hs)llxl(k) is displayed, plotted in the three cases k - (k, 0), k - (k, k ) / x / 2 , k - (0, k). The associated gradient approximations follow from the quadratic approximations to the curves, in the vicinity of k = 0. It can be seen that the signs of the quadratic terms are not always the same; it can also be seen that the gradient approximation is accurate only when the magnitude of k is 0.1 or smaller. The results shown in Figure 10.4.1 are considered to be of some interest because the model is completely unambiguous and (in principle) realisable. In particular, the statistics of the medium are defined without any approximation, and the meaning of the ensemble average is completely explicit. Some results that were presented as bounds for nonlocal properties were produced several years ago by Diener, Hiirrich, and Weissbarth [6]. However, their assumed two-point probability was at best an approximation, and their results did not show the complexity displayed in Figure 10.4.1. The results apply, of course, only to the types of elastic composite for which they have been derived. They may, nevertheless, throw some light more generally on nonlocal models. Most nonlocal models rely on some underlying averaging. It is common practice (see, for instance, Reference [7]) to postulate some relation involving an integral which may, in turn, be developed under an assumption of small gradients to give a gradient approximation. Gradient approximations may also be postulated directly (for instance, Reference [8]). The micromechanical considerations presented here show that the form of the nonlocal kernel may be quite complicated, and the associated gradient approximation may result in a problem that is globally unstable, or ill-posed:
FIGURE 10.4.1 Plots of "upper-bound and "lower-bound estimates of (k), for a particular two-dimensional, two-phase periodic composite with square periodic cell, for the cases k = (k, 0), k = (k, k ) / f i , k = (0,k).
10.4 Nonlocal Micromechanical Models
995
it depends on the signs, and these do not always come out as might be desired! In all such theories, there remains the serious question of assessing the influence of the boundary, which itself interacts nonlocally with adjacent inhomogeneities. Although only linear elasticity has been considered here, it is, in fact, possible to develop a similar formulation for certain classes of nonlinear problems (starting from a Hashin-Shtrikman formulation as initiated by Talbot and Willis [9]; see also Reference [ 10]). However, for such problems, the resulting equations are nonlinear, and nonlocality can only be addressed by iteration, leading directly to a gradient approximation. The more exact form of the effective relation in the nonlinear case is at present entirely unknown.
REFERENCES 1. Luciano, R., and Willis, J. R. (2000). Bounds on non-local effective relations for random composites loaded by configuration-dependent body force. J. Mech. Phys. Solids 48: 18271849. 2. Beran, M. J., and McCoy, J. J. (1970). The use of strain gradient theory for analysis of random media. Int. J. Solids Struct. 6: 1267-1275. 3. Willis, J. R. (1983). The overall response of composite materials. J. Appl. Mech. 50: 12021209. 4. Drugan, W J., and Willis, J. R. (1996). A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44: 497-524. 5. Luciano, R., and Willis, J. R. (2001). Non-local effective relations for fibre reinforced composites loaded by configuration-dependent body forces (in preparation). 6. Diener, G., Hfirrich, A., and Weissbarth, J. (1984). Bounds for the non-local effective elastic properties of composites. J. Mech. Phys. Solids 32: 21-39. 7. Barenblatt, G. I., and Prostokrishin, V. M. (1993). A model of damage taking into account microstructural effects. Eur. J. Appl. Math. 4: 225-240. 8. Leroy, Y., and Molinari, A. (1993). Spatial patterns and size effects in shear zones: A hyperelastic model with higher-order gradients. J. Mech. Phys. Solids 41: 631-663. 9. Talbot, D. R. S., and Willis, J. R. (1985). Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35: 39-54. 10. Talbot, D. R. S., and Willis, J. R. (1997). Bounds of third order for the overall response of nonlinear composites. J. Mech. Phys. Solids 45: 87-111.
SECTION
10.5
Transformation Field Analysis of Composite Materials GEORGE J. DVORAK Rensselaer Polytechnic Institute, Troy, New York
Contents 10.5.1 Overview of the Method . . . . . . . . . . . . . . . . . . 10.5.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . 10.5.3 Evolution of Inelastic Deformation . . . . . . . . 10.5.4 Modeling of Damage Evolution . . . . . . . . . . . 10.5.5 Concentration Factor Estimates . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
996 997 999 1000 1001 1002 1002
10.5.1 OVERVIEW OF THE METHOD The transformation field analysis (TFA) method evaluates averages of local and overall strain and stress fields caused in elastic heterogeneous solids, such as composite materials, laminates, and polycrystals, by piecewise uniform distributions of transformation strains or eigenstrains; these may be caused by thermal changes, phase transformations, moisture absorption, or inelastic deformation. Also, equivalent eigenstrains applied in modeling of local cracks, voids, and other damage modes can be included in the analysis. Moreover, the method provides a system of differential equations describing evolution of the local transformation fields under varying overall loads, according to local constitutive relations prescribed in the constituents. Here we discuss applications to inelastic deformation and damage evolution in composites. A more complete description of the method can be found in References [ 1-4]. 996
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
997
10.5 T r a n s f o r m a t i o n Field Analysis of C o m p o s i t e Materials
10.5.2 GOVERNING EQUATIONS Consider a sufficiently large representative volume V of a composite material consisting of two or more perfectly bonded phases. The phases may have any physically admissible elastic symmetry and microstructural geometry. Each phase r in volume Vr is a homogeneous elastic solid, of constant stiffness Lr and compliance Mr=L~ -1 defined in a fixed overall Cartesian coordinate system x in V. The volume V may be subdivided into several local volumes Vr, r=1,2, ..., N, g V~=V, such that each contains only one phase material, although, in a more refined evaluation of the local fields, any given phase may reside in more than one volume V~. The volume V is subjected either to surface displacements on S of V that cause uniform overall strain s ~ in V, or to tractions derived from a uniform overall uniform stress ~r~ on S. Moreover, a uniform temperature change A0 and a piecewise uniform field of transformation strains pr E V~ may be prescribed in V The local constitutive relations in the phases are then written as
~r(X) -- Lrgr(X) -[- [rAO -'[- ~r
(1)
gr(X) = Mr~rr(X) --k-mrAO + Pr where mr is a thermal strain vector of linear coefficients of thermal expansion, |r = - - L r m r is a thermal stress vector, and ~r = - L r ~ r is a local transformation stress or eigenstress. Overall constitutive relations are written in the analogous form a = L~ ~ + lAO + ~
~ = Ma ~ + mAO + p
(2)
where L and M are overall stiffness and compliance satisfying LM = I, 1 = - L m is the overall thermal stress vector, and /, = - L p is the overall transformation stress caused by the /*r distribution in a fully constrained volume V. The thermal strains and transformation strains in Eq. 1 are assumed to be distributed such that both the overall stress r at s ~ = 0 and strain ~ at ~r~ = 0 are uniform in V. All strains are assumed to be small. Local averages of the stress and strain fields in Eq. 1 can be related to prescribed sets of the independent load contributions s ~ A0, Pr, or ~0, A0, ~r, in the form [1] N
~,s -- As~,~ + ~ r= 1
N
Dsr(mrAO + Pr)
~rs -- Bs~r~ + ~
Fsr(lraO + ~,r)
(3)
r= 1
where r, s=1,2 ,..., N. The As and Bs are the mechanical concentration factor tensors [5] and Dsr, Fsr are certain eigenstrain and eigenstress concentration factor tensors [1-4] which evaluate the contribution of a uniform transformation in Vr C V to the field average in Vs c V. Both are averages of the respective influence functions that define the local fields; they depend
Dvorak
998
on the local and overall elastic moduli and on the shape and volume fractions of the phases, and are therefore constant within certain temperature intervals. Like the Eshelby tensor S, the self-induced factors Dss and Fss contribute both the residual field caused in Vs by the eigenstrains and the eigenstrain themselves [6]. In the heterogeneous aggregates considered herein, the transformation concentration factors can be related to the mechanical factors. In two-phase systems r, s=0~, fl, where each phase contains a single, uniform transformation strain or stress, there are exact connections between the respective influence functions; c.f. Reference [1], Eqs. 123-126: Dr~(X) - [ I - Ar(x)] ( L ~ -
L~)-IL~
Dr~(X) -- - [ I - Ar(x)] (L~ - L f l ) - I L 3
(4)
Fra(X ) -- [ I - lr(X)] (Mo~ -- Mfl)-lMor Frfl(x ) --
-[I-
Br(x)] (Mo~ -
M~)-IM~
In multiphase systems, and in two-phase systems loaded by several piecewise uniform local transformations, such connections can be found in terms of estimates of the mechanical concentration factors and overall stiffness or compliance that are provided by certain averaging methods [2, 3, 7], which are briefly reviewed in Section 10.5.4. Dsr .
(I . As). (Ls .
Fsr -- ( I -
L) -l(~srI
erA Tr )Lr
(5)
Bs) (Ms - M) -1 (C~srI - crBTr )Lr
where I is a (6 • 6) identity matrix, Cr=Vr/V,and 6sris the Kronecker symbol, but no summation is indicated by repeated subscripts. The averaging methods are valid for either two-phase systems of arbitrary phase geometry or for multiphase systems where all phases have the same shape and orientation [8]. In more detailed evaluations of the local fields in unit cells and other material models, the transformation concentration factors are best evaluated with the finite element method, as described in References [4, 9]. Volume averages of the local fields (Eq. 3), obtained with substitutions from Eq. 4, or with any other admissible evaluation of the respective concentration factors, are related to the overall averages by
N ~, -- ~ Crgr mAO + Ill -r--1 N tT -- ~ CrGr |AO d- ,~ -r- 1
N ~ CrlTr (mrAO + ~lr) r--1 N ~ CrAT (|rA0 -~- ~,r) r= 1
(6)
These are distinct connections that must be satisfied separately by the averages of total fields and the thermal and other transformation fields [3, 5].
10.5 Transformation Field Analysis of Composite Materials
999
Finally, by invoking the well-known expressions for overall stiffness and compliance [5], L-
N ~_~ crLrAr r= 1
N
M-
~ crMrBr r= 1
(7)
evaluation of the local fields in Eq. 3 using Eq. 4 or 5 is reduced to finding the mechanical concentration factors Ar Br; see Section 10.5.5. Note that Eq. 5 is valid for As, At, L and Bs, Br, M derived with an averaging method; however, relations Eqs. 4, 6 and 7 must be satisfied by all admissible evaluations of concentration factors or influence functions.
10.5.3
EVOLUTION
OF INELASTIC
DEFORMATION Inelastic, thermal, and any other transformation strains are deformations remaining at a material point after an instantaneous elastic unloading from the current to zero local stress. This agrees with the additive strain and stress decompositions (Eq. 1) and implies that elastic-plastic, viscoelastic, viscoplastic, and any other inelastic strains ~rin (X) can be regarded as eigenstrains acting on an elastic solid in superposition with mechanical and thermal loads. The local constitutive equations (Eq. 1) can thus be replaced by derr(X) - Lrd~,r(X) +
lrdO + dante(x)
(8)
d"r(X) - Mrd~r(X) + mrdO -t- d"~n(x)
where the relaxation stress da re(x) - - L r d , ~ (x). As long as Lr, lr, Mr, and mr remain constant, Eq. 8 holds for both instantaneous and accumulated increments. Inelastic constitutive laws typically specify when a material starts to deviate from the linear elastic response, and then relate either the local relaxation stress to the past history and current increment of the local strain, or the inelastic strain to the history and current increment of the local stress. If these fields are uniform in Vr, this can be written as dtTr--~.~r(gr -- flr)dgr nt- ~rdO d g r - JHr(~rr - ~r)d6r nt- ~rd0
(9)
indicating that ~~ or d//r are, respectively, functions of the current back strain ]/r or back stress 0~r, which together w i t h / ' r and mr depend on past deformation history. Separating the inelastic from the total components and
1000
Dvorak
introducing instantaneous inelastic stiffness and compliance tensors yields da~ e - ~'Prde, r + ~PdO
de.~ - ,.///l,Pdar + ~Prd0
(10)
When substituted into the incremental forms of Eq. 3, this provides the following equations for evaluation of the local strain and stress increments: N
N
de,~ + E DsrMr~'~q;'Prdgr -- a~ d~~ + E Dsr(mr - M~g~)dO
r=l
r:l
N
N
da~ 4- E FsrLr't/dPrd~rr
=
(11)
B~ da~ + }~ Fsr(er
r=l
- -
Lr~Pr)d0
r=I
In multiphase systems, and in all systems which undergo a significant departure from elastic response, these equations are best written for refined subdivisions of V, with all concentration factors evaluated by the finite element method and solved numerically. Of course, closed-form solutions can be easily obtained for two- or three-phase systems; these appear useful in an analysis of contained inelastic deformation [9]. In any case, it is desirable to write the result in the form d~r = a:/rd~ + a~rdO d~rr - ~ r d a + drdO
(12)
where a/r, a~r, and Nr, dr are instantaneous mechanical an thermal strain and stress concentration factor tensors for the local volumes Vr. The overall response of the inelastic composite medium can be derived as described in [2, ~ l .
10.5.4
MODELING
OF DAMAGE EVOLUTION
As a simple application of the TFA method to damage analysis, consider a two-phase aggregate of volume V consisting of a matrix ( r = 1), the still bonded reinforcements ( r = b ) , and debonded reinforcements ( r = d ) . Incremental loading is applied to the undamaged aggregate by an overall uniform stress a ~ and by an equivalent eigenstrain Pa that simulates the effect of debonding in r = d. The average stress aa is usually zero, but any other value may be supported in r=d if obtained from an independent analysis, preferably as a function of a ~ or 5. Also,/,~ - / , ~ - 0. Adjusting Eq. 3 (second equation) to these requirements yields three equations for the local stresses at any given loading step k: a~ _ B l a ~ _ FlaL2p k ak aka -- B2a~ - F~aL2p~ k _
~rb
B2a~
k
k
-- FbdL2p d
(13)
1001
10.5 Transformation Field Analysis of Composite Materials
This respects the fact that both r = b and r = d volumes have the same shape and stiffness L2 and thus also the same concentration factor B2. However, since two different eigenstrains, /,~ and pdk, reside in phase L2, all Frs are distinct and evaluated using Eq. 5 (second equation). Partial debonding in r = d is simulated by the equivalent eigenstrain obtained from Eq. 13 (second equation): llkd -- M2(Fd) )k(B2a ~ -- akd)
(14)
Averages of the local stresses are then to be found as (15) ~bk -- B2trk0 q- c k B 2 ( i _ ckB2)-i (B2trk0 _ 6k)
(16)
where r is the known local stress in the debonded phase, possibly equal to zero. The concentration factors B1 and B2 are evaluated in the undamaged two-phase composite. The first right-hand terms are the stresses in the undamaged system, and the second terms are contributed by the eigenstrain (Eq. 14) simulating partial debonding in r =d. It can be verified that Eq. 6 (third equation), written here as Cla~ +cka k + Gka bk- a 0k, is satisfied by Eqs. 15 and 16. The local strain averages are obtained using the stresses in Eq. 1 (second equation as) k The overall strain is found from ~ relation
=
M2 k
ECrgr, or from the overall constitutive
ak -- Ma~ + Pk
(18)
where M is the overall elastic compliance of the undamaged system, and the overall eigenstrain Pk simulates the contribution to the overall strain by the current state of debonding in r =d; it can be evaluated from the generalized k--,TPd" k Together with a criterion Levin formula (second part of Eq. 6) as Pk -- cdu2 that specifies the probability of debonding as a function of stress in the bonded phase, Eqs. 15 and 16 can be converted to an incremental form and solved for damage increments dcak under overall stress change from
a'~ to a~ + da'~.
10.5.5
CONCENTRATION
FACTOR ESTIMATES
Heterogeneous solids are often modeled as aggregates of phases of ellipsoidal shape, possibly embedded in a common matrix. Under the homogeneous
1002
Dvorak
b o u n d a r y conditions described in Section 10.5.1, the local strain or stress averages in the phases are the first right-hand terms in Eq. 3, with estimates of the concentration factors provided by the self-consistent [5] or Mori-Tanaka [10] averaging methods. These belong to a m u c h wider, recently identified family of m e t h o d s [7] which consider each phase as a solitary ellipsoidal i n h o m o g e n e i t y e m b e d d e d in a large volume of a certain comparison m e d i u m L0 that is remotely loaded by a uniform image strain or stress field. Estimates of the mechanical strain concentration factors follow from
Ar = Air
crAIr
(Air) -1 - I -
S Lol(L0 - Lr)
(19)
where S denotes the Eshelby tensor evaluated in Lo. The Lo must be chosen such that the matrices L o - L r and L o - L o are b o t h positive or negative semidefinite. The overall stiffness estimate then follows from Eq. 7 (first equation) and 19; if Lo is used in Eq. 19, then Eq. 7 (first equation) provides the Walpole [11] form of the H a s h i n - S h t r i k m a n bounds. A convenient form that approximates the self-consistent estimate of the overall stiffness in t w o phase systems where L2-L1 is positive semi definite is L0 = L1 + c2(L2 - L1) D
ACKNOWLEDGEMENTS Work leading to the present results was sponsored by grants from the Army Research Office and the Office of Naval Research. Drs. Yapa D.S. Rajapakse and Mohammed Zikry served as program monitors.
REFERENCES 1. Dvorak, G. J. (1990). On uniform fields in heterogeneous media. Proc. Roy. Soc. (London) A 431: 89-110. 2. Dvorak, G.J. (1992). Transformation field analysis of inelastic composite materials. Proc. Roy. Soc. (London) A 437: 311-327. 3. Dvorak, G. J., and Benveniste, Y. (1992). On transformation strains and uniform fields in muhiphase elastic media. Proc. Roy. Soc. (London) A 437: 291-310. 4. Dvorak, G. J., Bahei-E1-Din, Y. A., and Wafa, A. (1994). Implementation of the transformation field analysis for inelastic composite materials. Computational Mechanics 14: 201-228. 5. Hill, R. (1965). A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13: 213-222. 6. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. (London) A 241: 376-396. 7. Dvorak, G. J., and Srinivas, M.V. (1999). New estimates of overall properties of heterogeneous solids. J. Mech. Phys. Solids 47: 899-920, 2207-2208.
10.5 Transformation Field Analysis of Composite Materials
1003
8. Benveniste, Y., Dvorak, G.J., and Chen, T. (1989). On the diagonal and elastic symmetry of the approximate stiffness tensor of heterogeneous media. J. Mech. Phys. Solids 39: 927-946. 9. Fish, J., Pandheeradi, M., and Shephard, M. S. (1997). Computational plasticity for composite structures based on mathematical homogenization: Theory and practice. Comp. Meth. Appl. Mech. Engng. 148: 53-73. 10. Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater. 6: 147-157. 11. Walpole, L. J. (1966). On bounds for overall elastic moduli of inhomogeneous systems: I.J. Mech. Phys. Solids 14: 151-162; II. ibid., 289-301.
SECTION
10.6
A Damage Mesomodel of Laminate Composites PIERRE LADEVl~ZE LMT-Cachan, ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France
Contents 10.6.1 Domain of application . . . . . . . . . . . . . . . . . . . 10.6.2 Basic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.3 The Single-Layer Model . . . . . . . . . . . . . . . . . . 10.6.3.1 Damage Kinematics . . . . . . . . . . . . . 10.6.3.2 Damage Evolution Law . . . . . . . . . . 10.6.3.3 Coupling between Damage and Plasticity (or Viscoplasticity) . . . . 10.6.4 Identification of the Material Parameters . 10.6.4.1 Tensile Test on [0 ~ 90~ Laminate . . . . . . . . . . . . . . . . . . . . . . . 10.6.4.2 Tensile Test on [+45 ~ -45~ Laminate . . . . . . . . . . . . . . . . . . . . . . . 10.6.4.3 Tensile Test on a [+67.5~176 Laminate . . . . . . . . . . . . . . . . . . . . . . . 10.6.5 How Does One Use the Model? . . . . . . . . . . 10.6.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1004 1005 1006 1006 1008 1010 1011 1012 1012 1012 1013 1013 1013
10.6.1 DOMAIN OF APPLICATION A m a j o r c h a l l e n g e in c o m p o s i t e d e s i g n is to calculate the d a m a g e state of a c o m p o s i t e s t r u c t u r e s u b j e c t e d to c o m p l e x l o a d i n g at a n y p o i n t a n d at a n y time u n t i l final fracture. D a m a g e refers to the m o r e or less g r a d u a l d e v e l o p m e n t s of m i c r o c r a c k s w h i c h lead to m a c r o c r a c k s a n d t h e n to r u p t u r e ; m a c r o c r a c k s are s i m u l a t e d as c o m p l e t e l y d a m a g e d zones. A s o l u t i o n for c o m p o s i t e s , especially l a m i n a t e c o m p o s i t e s , is b a s e d o n w h a t we call a
1004
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
10.6 A Damage Mesomodel of Laminate Composites
1005
damage mesomodel. It is a semidiscrete model for which the damage state is locally uniform within the mesoconstituents. For laminates, it is uniform throughout the thickness of each single layer; as a complement, continuum damage models with delay effects are introduced. The mesomodel has been used for various continuous fiber laminates, essentially for aeronautical and spatial applications.
10.6.2
BASIC ASPECTS
An initial step is to define what we call a laminate mesomodel [13, 14]. At the mesoscale, characterized by the thickness of the ply, the laminate structure is described as a stacking sequence of homogeneous layers through the thickness and of interlaminar interfaces (see Figure 10.6.1). The main damage mechanisms are described as fiber breaking, matrix microcracking, and debonding of adjacent layers (see Figure 10.6.2). The single-layer model includes both damage and inelasticity. The interlaminar interface is defined as a two-dimensional mechanical model which ensures traction and displacement transfer from one ply to the next. Its mechanical behavior depends on the angle between the fibers of two adjacent layers. A priori, 0~ ~ interfaces are not introduced. Herakovich, in his book [11], calls this theory the "mesoscale composite damage theory." The damage mechanisms are taken into account by means of internal damage variables. A mesomodel is then defined by adding another property: a uniform damage state is prescribed throughout the thickness of the elementary ply. This point plays a major role when trying to simulate a crack with a damage model. As a complement, delayed damage models are introduced. One limitation of the proposed mesomodel is that material fracture is described by means of only two types of macrocracks: 9 delamination cracks within the interfaces; 9 cracks orthogonal to the laminate's midplane, each cracked layer being completely cracked through its thickness.
FIGURE 10.6.1 Laminatemodeling
1006
Ladev/~ze Fibre-matrix interface debonding 9 ,'md Matrix micro-cracking
Brittle fibre rupture ,-...........~,
Adjacent layer debonding (Delamination) FIGURE 10.6.2
Damage and failure mechanisms.
The layers - - in our sense m are assumed to be not too thick. Another limitation is that very severe dynamic loadings cannot be studied; the dynamic wavelength must be larger than the thickness of the plies. Two models have to be identified: the single-layer model [16] and the interface model [1,6]. The appropriate tests used consist of tension, bending, and delamination. Each composite specimen, which contains several layers and interfaces, is analyzed in order to derive the material quantities intrinsic to the single layer or to the interlaminar interface. Various comparisons with experimental results have been performed to show the possibilities and limits of our proposed computational damage mechanics approach for laminates [2, 8, 19, 20]. The single-layer model is presented here. A similar model is used for the interface.
10.6.3 THE SINGLE-LAYER MODEL The single-layer model and its identification procedure are detailed in Herakovich [11]. Complements concerning the compression behavior and the temperature-dependent behavior can be found in Allix et al. [3, 4]. Here we present the latest version, which takes into account the layer size effects [23].
10.6.3.1 DAMAGE KINEMATICS The composite materials under investigation in this study (e.g., carbon-fiber/ epoxy-resin) have only one reinforced direction. In the following, subscripts 1, 2, and 3 designate the fiber direction, the transverse direction inside the layer, and the normal direction, respectively. The energy of the
1007
10.6 A Damage Mesomodel of Laminate Composites
damaged material defines the damage kinematics. Using common notations, this energy is
r
ED =2(1-- &) L i71
(~0
~03,~
(,,7, ,,o,~
Eo
-
(~0
~03,~
+
- t, Eo + -k-7-,) <''' "~ - t, Eo + ~ ) +
~
<'=<'~
[ (
(--0"22) 2 (--0"33) 2 217~ + 217--~- +
+ (1 - d)\ ~ +
t , ~ + -k-7-,) <'''"=
1
(0"22
89( 1 - a ' )
G~ "
E~
(O'33> 2
+
E~
) (1)
G~
is a material function which takes into account the nonlinear response in compression [3], and dF, d, and d' are three scalar internal variables which remain constant throughout the thickness of each single layer and which serve to describe the damage mechanisms inside. The unilateral aspect of microcracking is taken into account by splitting the energy into a "tension" energy and a "compression" energy; (.) denotes the positive part. The thermodynamic forces associated with the mechanical dissipation are o
)
-
2(1
-
+
o7-7 + c7,/
1 ra, - D-dT<
, :<~, -
2(1 - dF) 2
E1
(vo 2 vo,~
- \ Fo + ~)
4-
;~(<-o-11>) Eo
(vo 3 vo,
~=~
;2/
Here, unlike in previous papers, ((.}) denotes the integral value through the thickness, not the mean value.
1008
Ladev/:ze
10.6.3.2 DAMAGE EVOLUTION LAW From experimental results, it follows that the governing forces of damage evolution are
r - [Yd + bYd,], Y ' -
[Yd, + b' Yd],
(3)
where b and b' are material constants which balance the influence of the transverse energy and the shear energy. For small damage rates and quasistatic loading, we get
d--fd(Y_ 1/2)
for d _< 1
d t --fd,(Y_ tl/2)
ford'_<
dv - fv(Ylv/2)
fordv_
1
(4)
where V_[t - - SlAp Fly , T ~ t
and fd, fd', and fv are material "functions"; progressive damage evolution (generally defined by a linear function) and brittle damage evolution (defined by a threshold) are both present, fF is generally associated with a brittle damage mechanism. The model stays valid for a rather large temperature range [4]. At room temperature, a typical material function fa is given in Figure 10.6.3. For large damage rates, we have introduced a damage model with delay effects: c~-- 1 [1 - e x p ( - a ( f a ( Y 1/z) - d>)]
if d < 1, d - 1 otherwise
Tc
c~'-- --1 [1 - e x p ( - a ( f d , ( Y 'l/z) - d'>)]
if d' <1, d ' -
1 otherwise
(5)
"Fc
dF -- 1 [1 - exp(-a(fr(Y'~/2)
gc
- dr>)]
if de < 1, dr - 1 otherwise
The same material constants, ~ and a, are taken for the three damage evolution laws. For this damage model with delay effects, the variations of the forces Y, Y' and Yr do not lead to instantaneous variations of the damage variables d, d' and dF. There is a certain delay, defined by the characteristic time z~. Moreover, a maximum damage rate, which is 1/%, does exist. A first identification consists of taking half the Rayleigh wave speed combined with the critical value of the energy release rate. Let us also point out here that a clear distinction can be made between this damage model with delay effects and viscoelastic or viscoplastic models: the characteristic time introduced in the damage model with delay effects is several orders of magnitude less than
1009
10.6 A Damage Mesomodel of Laminate Composites 0.5
9 ~,", . . . . .
-,
9
0.4
0.3
0.2
0.1
O(
0
0.2
.
.
. 0.4
. yd~
.
. 0.6
.
. . 0.8
~''' 1
1.2
14
(MPa ~
FIGURE 10.6.3 Shear damage material function vl/2 *'a ~fa( yj/2) of the single-layer for the M55J/M18 material at room temperature. in the viscous case. This characteristic time is, in fact, related to the fracture process.
10.6.3.2.1 Remarks
9 Two damage variables are used to describe the damage associated with matrix microcracking and fiber-matrix debonding. They seem to account for all the proposed damage kinematics, including that starting from an analysis of the microcracks. Many works have established, experimentally or theoretically, a relation between the microcrack density and our damage variable d, which can be very useful for the identification of a damage fatigue model. 9 What we call the single layer is the assemblage of adjacent, usual elementary, plies of the same direction. The damage forces, being integral values through the thickness of the single layer, can be interpreted as energy release rates. It follows that the damage evolution law of the single layer is thickness-dependent. For single layers which are not too thick, such damage evolution laws include results coming from shear lag analyses. Consequently, the size effects observed, for example, in t e n s i o n - are produced by both the single-layer model and the interface model through a structure problem. This theory, which is very simple, works very well for most engineering laminates; however, it cannot be satisfactory for rather thick layers. A first solution is to modify the damage evolution law, using the thickness as a parameter. 9 The damage variables are active for [0 ~ 90~ laminates even if the apparent modulus does not change. The model predicts this hidden damage [15].
10 10
Ladev~ze
9 For fatigue loadings, we introduce: d = ds + dF
d ' - d~ + d~
(6)
where ds and dts are the quasi-static part of the damage defined by Eq. 4 or 5. dF and dtF denote the fatigue part characterized by the following fatigue evolution laws:
o& ON = a(a, [Ya + bYd,]) oa'~ = d(a', [yd,])
(7)
ON
where a, a ~ are two material functions and [.] denotes the maximum value over the cycle.
1 0 . 6 . 3 . 3 COUPLING BETWEEN DAMAGE AND PLASTICITY (OR VISCOPLASTICITY) The rnicrocracks, i.e., the damage, lead to sliding with friction, and thus to inelastic strains. The effective stress and inelastic strain are defined by r~11 = 0"11 F22 -- -(-0"22) + (1 - d'--------~rY33- -(-0"33) + (1 - d'--------~ rY12--1_d ~llp -- ~llp
~23=1_d
r~31 1 - d ~
~22p -- (~22p)(1 - - d t) - ( - ~ 2 2 p )
9
,
~12p ~ /;12p(1 -- d)
~23p -- ~23p(1 - d)
(8) ~33p -- (/;33p)( 1 - d r ) -
(-~33p)
.
831p = 831p(1 - d)
ei)p for/j C {1,2,3} denotes the usual inelastic strain. The idea is to apply classical plasticity or viscoplasticity models to effective quantities. A very simple plasticity model is defined by the following elastic domain:
f(",R) - [ G + ~23 + ~]~ + ~ ( ~
+ ~3)] ~/~ - e - eo
(9)
Hardening is assumed to be isotropic, which means that the threshold R is a function of the cumulated strain p; p--, R(p) is a material function, p being defined by p --
.~0t dt[e,12 2~2 -2 2.2 1 2,2 2.2 1/2 p 4- ~;23p+ ~31p 4--~(e,22p + e33p)]
(10)
1011
10.6 A Damage Mesomodel of Laminate C o m p o s i t e s
140
120
100 /
80 _
Yo
60-
9
e o --"
4020-
Test A Test
B
Test C
Identified
curve
_
0 0
I
'
0.2
FIGURE 10.6.4
I
0.4
'
!
0.6
'
I
0.8
'
I
1
'
I
1.2
'
"i
1.4
"~
Hardening curve at room temperature for IM6/914.
a is a material coupling constant. The yield conditions are .
/~f--0,
pZ0, fS0
1 ~ij 9 ~JP--2PR+R0 z a2Fij 9 ~ijp--P R + R o
for i:~ji,j E {1,2,3)
(11)
for/ E {2,3)
An example of such a hardening curve is given for the T300-914 material in Figure 10.6.4.
10.6.4 I D E N T I F I C A T I O N OF THE MATERIAL PARAMETERS The single-layer model and the interface model have been identified for various materials. Aside from the elastic constants, the model depends on: 9 three coupling coefficients b, b', a2; 9 the damage "functions" fd, fd', fF describing progressive and brittle evolutions; 9 the hardening function p---+R(p); 9 the function 25 defined practically by one parameter characterizing the compressive stiffness loss in the fiber direction (see Reference [1]). The identification is developed here for low-stiffness matrixes. It is based on three canonic tests: [0 ~ 90~ [+45~176 , [+67.5~176 The measured experimental quantities are:
1012
Ladev/~ze
F/S, where
9 the tension F, which is related to the macrostress 0"L by 0"L S is a specimen's section; 9 the laminate's axial strain e[ and the transverse one e r.
Consequently, the inelastic strains and moduli variations are determined.
10.6.4.1
TENSILE TEST ON [ 0 ~
90~
LAMINATE
This test defines fF, i.e., in many cases, the fiber's limit tensile strain.
10.6.4.2
TENSILE TEST ON
[+45~ -45~
LAMINATE
The following relations allow us to reconstitute the ply's shear behavior: qr
0"L 0"12 ~-
2
el2 -- (e L - e ~ ) / 2 -- el2 e + el2p
(12)
qr
0"11 - - 0"L
For m a n y materials, one has ~
> (~11, ~22)
0"L ~
0"22
Consequently, the transverse stress and strain do not affect the behavior. The damage function fd is defined using Y - v/2G~ el2e (1 - d) - 0"~/2G~ el2e The hardening function
p ~ R(p) is identified from R 4- R0
0 * -- G12el2e
p -- fo ~*2e2(1 -
10.6.4.3
(13)
TENSILE TEST ON A
(14)
d)dep
[+67.5~176
LAMINATE
b ~ can be taken to zero for m a n y materials. Then, the stresses and strains in the upper layer are 0"11 - -
~,
~
S0"L o ~
0"22 -- S 0"L -
0"12 -- S'0"L
~o~O~; + ~.~o~ ~
-
0 -- 67.5 ~
~o~O~.o
(~;
-
qr
~ )
(15)
10.6 A Damage Mesomodel of Laminate Composites
1013
where s, s t, s" are coefficients depending on 0 and the single layer's characteristics. This test allows one to identify the two coupling coefficients b and a 2 and the material function fa'. A complete numerical simulation of the model is needed here.
10.6.5
HOW
DOES ONE USE THE MODEL?
The complete damage mesomodel of laminate composites can be introduced in a finite element code. Some commercial codes propose it. However, the mesomodel of the single layer is sufficient when delamination does not occur. A simplified approach which has been used extensively is to introduce it in a finite element postprocessor in order to predict the intensities of the different damage mechanisms inside the different layers. The data is a finite element solution calculated under the assumption of elastic behavior.
10.6.6
EXTENSIONS
A first extension, in which microcracks can occur orthogonally to fiber directions, was introduced for ceramics composites [10]. Models for threedimensional and four-dimensional carbon-carbon composite materials are given in Reference [9, 18, 21]. More refined damage models are necessary for most ceramics composites [19,20]. They follow the anisotropic damage theory, which includes microcracks opening and closure effects introduced in Reference [ 12]. The extension to impact problems has been studied in Allix et al. [5, 6].
REFERENCES 1. Allix, O., and kadev~ze, P. (1992). Interlaminar interface modelling for the prediction of laminate delamination. Composite Structures 22; 235-242. 2. Allix, O. (1992). Damage analysis of delamination around a hole, in New advances in Computational Structural Mechanics, pp. 411--421, Ladev~ze, P., and Zienkiewicz, O.C., eds, Elsevier Science Publishers B.V. 3. Allix, O., Ladev~ze P., and Vitecoq, E. (1994). Modelling and identification of the mechanical behaviour of composite laminates in compression. Composite Science and Technology 51; 35--42. 4. Allix, O., Bahlouli, N., Cluzel, C., and Perret, L. (1996). Modelling and identification of temperature-dependent mechanical behaviour of the elementary ply in carbon/epoxy laminates. Composite Science and Technology 56; 883-888.
1014
Ladev~ze
5. Allix, O., and Deft, J. E (1997). Delay-damage modeling for fracture prediction of laminated composites under dynamic loading. Engineering Transactions 45; 29-46. 6. Allix, O., Guedra-Degeorges, D., Guinard, S., and Vinet, A. (1999). 3D analysis applied to low-energy impacts on composite laminates. Proceedings ICCM12, pp. 282-283, Masard, T., and Vautrin, A., eds. 7. Allix, O., Leveque, D., and Perret, L. (1998). Interlaminar interface model identification and forecast of delamination in composite laminates. Composite Science and Technology 56; 671-678. 8. Daudeville, L., and Ladev~ze, P. (1993). A damage mechanics tool for laminate delamination. Journal of Composite Structures 25; 547-555. 9. Dumont, J. P., Ladev~ze, P., Poss, M., and Remond, Y. (1987). Damage mechanics for 3D composites. Int. J. Composite Structures, 119-141. 10. Gasser, A., Ladev~ze, P., and Peres, P. (1998). Damage modelling for a laminated ceramic composite. Materials Science and Engineering A 250(2); 249-255. 11. Herakovich, C. T. (1998). Mechanics of Fibrous Composites, J. Wiley. 12. Ladev~ze, P. (1983). On an anisotropic damage theory; Report no. 34-LMT-Cachan (in French); Failure Criteria of Structured Media, Boehler, J. P ed., Balkema (1996), 355-364. 13. Ladev~ze, P. (1986). Sur la m~canique de l'endommagement des composites, in ComptesRendus des JNC5, pp. 667-683, Bathias, C., and Menk~s, D. eds., Paris: Pluralis Publications. 14. Ladev~ze, P. (1989). About a damage mechanics approach, in Mechanics and Mechanisms of Damage in Composite and Multimaterials. pp. 119-142, Baptiste, D.ed., MEP. 15. Ladev~ze, P. (1992). A damage computational method for composite structures. J. Computer and Structure 44(1/2); 79-87. 16. Ladev~ze, P., and Le Dantec, E. (1992). Damage modeling of the elementary ply for laminated composites. Composite Science and Technology 43(3); 257-267. 17. Ladev~ze, P. (1992). Towards a fracture theory, in Proceedings of the Third International Conference on Computational Plasticity, pp. 1369-1400, Owen, D. R. J., and Hinton, E. eds., Cambridge: Pineridge Press. 18. Ladev~ze, P, Allix, O., and Cluzel, C. (1993). Damage modelling at the macro- and mesoscales for 3D composites, in Damage Composite Materials, pp. 195-215, Voyiadjis G. ed., Elsevier. 19. Ladev~ze, P (1995). A damage computational approach for composites: Basic aspects and micromechanical relations. Computational Mechanics 8; 142-150. 20. Ladev~ze, P. (1995). Modeling and simulation of the mechanical behavior of CMCs, in HighTemperature Ceramic-Matrix Composites. pp. 53-63, Evans, A. G., and Naslain, R. eds. (Cereamic Transaction). 21. Ladev~ze, P., Allix, O., Gornet, L., Leveque, D., and Perret, L. (1998). A computational damage mechanics approach for laminates: Identification and comparison with experimental results, in Damage Mechanics in Engineering Materials, pp. 481-500, Voyiadjis, G. Z., Wu, J. W., and Chaboche, J. L. eds., Amsterdam: Elsevier. 22. Ladev~ze, P., Aubard, X., Cluzel, C., and Guitard, L. (1998). Damage and fracture modeling of 4D CC composites, in Damage Mechanics in Engineering Materials, pp. 351-367, Voyiadjis, G., Wu, J., Chaboche, J. L., eds., Elsevier. 23. Ladev~ze, P. (2000). Modelling and computation until final fracture of laminate composites, in Recent Developments in Durability Analysis of Composite Systems, pp. 39-47, Cardon, A. H., et al. eds., Balkema.
SECTION ,
,
10.7
Behavior of Ceramic-Matrix Composites under Thermomechanical Cyclic Loading Conditions FREDERICK
A. LECKIE1, ALAIN BuRa2 and FRANCOIS HILD 3
1Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 2 Laboratoire de Physico-Chimie Structurale et Macromol~culaire, UMR 7615, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France 3 Universitr Paris 6, LMT-Cachan, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
Contents 10.7.1 O v e r v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1015
10.7.2 B a c k g r o u n d : State P o t e n t i a l of a [0,90] Fiber-Reinforced Composite .............. 10.7.3 D e s c r i p t i o n of t h e M o d e l . . . . . . . . . . . . . . . .
1017 1019
1 0 . 7 . 4 I d e n t i f i c a t i o n o f t h e M a t e r i a l P a r a m e t e r s . 1021 10.7.5 H o w to U s e t h e M o d e l . . . . . . . . . . . . . . . . . . .
1022
10.7.6 Table of Parameters . . . . . . . . . . . . . . . . . . . . .
1022
References ......................................
1023
10.7.1 O V E R V I E W A constitutive law is proposed for ceramic-matrix composites (CMCs) which models matrix cracking, interface sliding and wear, fiber breakage, and fiber pullout and creep (of the matrix). These different mechanisms induce loss of stiffness, inelastic strains, creep strains, hysteresis loops, and crack closure. The features are analyzed within the framework of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1015
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L~c~e et al.
continuum damage mechanics (CDM) by the introduction of physical internal variables identified previously in material science investigations. The intention, then, of the present study is to develop a continuum description of the damage processes which is mechanism-based and which may be used to describe the behavior of CMCs under the conditions of multiaxial stress occurring in practice. Since crack spacing at saturation is small in most CMCs, CDM is an appropriate means of describing degradation, since changes in elastic moduli measured on a macroscopic level provide a simpler and more robust means of measuring damage than does microscopic measurement of crack density, which requires the average of many readings before reliable values are established. By combining CDM with micromechanical studies which are mechanismbased, constitutive equations are developed which lend themselves to the finite element procedures commonly used in practice. The CDM formulation applied to reinforced composites is written within the framework of the continuum thermodynamics. The first step in establishing such a model is to identify the internal variables which define the state of the material. The second is to determine the expression of the state potential in terms of the state variables, and the third to define the evolution laws of the internal variables. Composites consisting of a ceramic matrix reinforced by continuous ceramic fibers are candidates for application in components which operate at temperatures in excess of those which are normal for metallic structures. In spite of the fact that the constituents of the CMC are both brittle, it has been demonstrated that following matrix cracking, sliding occurs at the fiber-matrix interface which causes inelastic deformations. The presence of matrix cracks and inelastic deformations may impart to the material the ability to redistribute stresses. The ability to redistribute stress is an important property since design studies indicate that working stresses for CMC components are sufficiently high for matrix cracking to be unavoidable in regions of stress concentration occurring at the junctions and penetrations which are a feature of engineering components. The model is integrated into a finite element system (ABAQUS) and is used to estimate the behavior of a representative structures under monotonic [1], cyclic [2], and creep [3] loading conditions. Design and lifting procedures are able to deal with thermal loading, creep, and cyclic loading just by extending the original formulation to isochronous analysis [4]. It is observed that some CMCs have the ability to redistribute stress so that, as for plasticity, the presence of initial stress concentrations does not compromise the performance of the component, because it is reduced during the loading.
1017
10.7 Behavior of Ceramic-Matrix Composites
10.7.2 BACKGROUND: STATE POTENTIAL [0, 9 0 ] F I B E R - R E I N F O R C E D C O M P O S I T E
OF A
This part deals with the derivation of a constitutive law for a [0,90] (laminated or woven) composite submitted to multiaxial loads in plane stress conditions. The method is based upon the construction of the properties of the composite from the properties of the constituents and the stacking sequence of the layers. The initial behavior of the matrix is assumed to be isotropic. The presence of cracks leads the behavior to become anisotropic. The assumption is made that cracking occurs normal to the y-direction (e.g., maximum principal strain direction) in the matrix. Under the hypothesis of a monotonic loading condition, only one damage variable is needed to model matrix cracking, and that is denoted by Dmy. The study of a cracked system normal to one direction shows that the Young's modulus along that direction and the shear modulus are altered and that the expression of the elastic energy density of the matrix is
IE .[~,~ + 2Vm(1 - Dms)e,mx~e,myy + (1 - Dmy)e,2yy] Om = ~
1 - v~(1 - Vmy)
(1)
Jr- 2 G m ( D m y ) f"rnx 2 y
with
Gm
( Dmy ~
-
1+
1
1 - Dmyf 2(1 +
(2) Ym)
where Era, Vm, Gm are the initial elastic properties of the matrix. The components of the strain tensor of the matrix ~m expressed in the x - y frame are denoted by emx~, emyy, and emxy. The fibers are aligned along the i-direction. The fiber breaks are assumed to be perpendicular to the fiber direction and are described by a damage parameter Dfl. Therefore, the elastic energy density of the fibers is given by 1F
let-(1 0s - ZL-'-~
- Dfl)411
+ 3~f8r .,.s_
-4-
2Gf(Dfl)412
(3)
with Gf(Dfl ) -
1+
Dr1) 1 1 - Dfl 2(1 q-- Vf)
(4)
1018
Leckie et al.
where Ef, vf, Gf are the initial elastic properties of the fiber. The components of the strain tensor of the fiber ~f expressed in the x-y frame are denoted by 8./11, 8./22, and ~f12. A layer consists of fibers aligned along the 1-direction embedded in the matrix. To determine the behavior of this layer, microinterface compatibility conditions are written in terms of strains _d- and stresses ~r_t on the layer level. It is more convenient to write the conditions in the 1-2 frame. The following equations are derived by using Voigt's approximation in the fiber direction and Reuss's approximation for the transverse properties: m
emil - eS11 = e~l
fm
O'mll
-~-ff
fm em22 +j~
O'fll = 0"~1
eS22 - ~2L2
fm~m12+fs
8./12 -- 8L2
(5)
0"m22 = 0"f22 = o'L20"m12 -- O'f12 = o'L2
(6)
where j~, fm denote the volume fraction of the fiber and the matrix, respectively, and cr~ and s~ are the components of the stress and strain tensors __aL and _s._E_ in the 1-2 frame. The solution of the previous system yields
~L _. EL(Dmy, Dfx )'8_L
(7)
where EL(Dmy,Dfl) is the stiffness tensor of a layer which is dependent upon all danTage variables defined at the constituent level. The elastic energy density associated with matrix cracking and fiber breakage at the layer level is expressed as ~r
1 eL:E _ L(Dmy, Dfl):e _ L
(8)
The elastic behavior of a [0,90] composite system is determined by applying classical laminate theory
__ ~00__ ~90 and __a- foo __aoo+f9o ~90
(9)
where foo and f9o denote the volume fraction of the 0 ~ and 90 ~ layers, and ~oo, _aoo and _~9o,{r9o the strain and stress tensors in the 0 ~ and 90 ~ layers. The overall behavior of the composite is defined as 00
90
00
00
--" E.-E-(Dmy'Dmy,Dfx,Dfl ) " ~
(10)
with
E(DCmO D9Oy,D?O,D~O) __ fO0 E(DOO,D~O ) -t- f90 E(V9Oy,D?O ) where
E(D~17690
00
(11)
90 ,D)I ) is the fourth order elastic tensor of the composite,
w
which is a function of all damage variables on the constituent level for all
1019
10.7 Behavior of Ceramic-Matrix Composites
layers. The elastic energy density associated with matrix cracking and fiber breakage can be written on the composite level as ~r = f00 r o0 + f9o ~ 90 (12) where r a n d ~90 are the elastic energy densities of the 0 ~ and 90 ~ layers. Inelastic strains are essentially due the interface sliding between the fiber and the matrix. From a micromechanical point of view, sliding can take place as soon as a crack is bridged by fibers. The analysis of these sliding systems leads to the following expression of the stored energy density [5]:
--2E\~i1 + d-~22//+2 G t,,d-~12)
(13)
with E---
4 fOOEOOf9OE9O 3 fOOEOO_4_f9OE90
and
G-
foo oOOf90690 foo GOOjr_f90 G9o
(14)
where E~176 is the Young's modulus of the 0 ~ layer in the fiber direction 00, E90 is the Young's modulus of the 90 ~ layer in the fiber direction 90, G ~176 is the shear modulus of the 0 ~ layer, G 9~ is the shear modulus of the 90 ~ layer, d11, d22, and d12 are damage quantities related to sliding, and ~i11, ~i22, and s are the inelastic strains.
10.7.3 DESCRIPTION
OF THE MODEL
The following expression of the free energy density for [0,901 CMCs can be obtained: 1
O - - -~(~= - ~i - ~ c ) E ( D ~ 1 7 6
90
00
90
_
,Dr1 )'(~= - ~i - ~c) +
~S
(15)
where ~_cis the creep strain tensor. From this expression, the associated forces to each internal variable are given by partial differentiation:
0r a_ - - c3---~'
0r Y=
c3O '
0r y-
c3d '
0r X=
c3~ i '
0r S-
c3~ c
(16)
D00 .N90 .N00.N90 00 90 where D -__ ft,-.mx,,-.mx,~.-fl ,~flJ% and Y - {Y~; Y~; y~O; yfO}, d - { d l l , 9 d22, 9 d12) and y = {y11; y22; y12]:. A priori, it is necessary to know thirteen evolution laws associated with the thirteen internal variables. But only the six following evolution laws are necessary:
9 One law related to matrix cracking Dm = Dm(Ym), because the variables D ~176 and Dgm~ correspond to only one single mechanism, and therefore have
1020
Leckie et al.
the same evolution law
9 One law related to fiber breakage Df = Df(Yf), since D~~ and D 9~ have the same evolution law, because fiber breaks are perpendicular to t~e fiber direction:
Df--l-exp
-
~f0
;
(18)
9 Two laws related to the inelastic strains, since debonding and sliding mechanisms in the 0 ~ and 90 ~ layers arc identical, and thus ~i~l and ei22 have the same evolution law. The second one concerns ei12. In a unidimensional analysis [5] it has been observed that the back stress X is only a function of the applied stress or. If the relation is extended to a multiaxial state, then the inelastic strain tensor __eiis a [unction of the stress tensor or: 8io~oc - - 8i t \
and
tTth
8i12
--
8is
sign(a12)
Tth
(19)
with an equivalent shear stress dependent upon the hydrostatic pressure 9
(20)
where ee - 11, 22 and {.} are the Macauley brackets. When numerous cycles are performed, the evolution of the inelastic strains as a function of the number of cycles is written as FIN]-
ei~(N)
1
--
(1 - exp [ - ( l ~
-
1])"])
(21)
where gi~(N) is the inelastic strain after N cycles and ~i~ the inelastic strain for the first loading ( N - 0). 9 Two laws related to in-plane debonding d - d(y), since dll and d12 are assumed to have the same evolution law. These last evolution laws are not needed to derive the behavior of the composite. Neither the state laws nor the evolution laws are explicit functions of the damage variables d, which are needed to express the stored energy. 9 Similarly, two laws related to creep are needed. For the normal strains, the evolution law is assumed to be independent of the temperature and is written as ~E~
exp
1021
10.7 Behavior of Ceramic-Matrix Composites
and for shear strains, the evolution law is explicitly dependent on the temperature: ~c12[t]- ~12o exp b T o - 1
10.7.4
IDENTIFICATION
kZo/
t-~2o
.
(23)
OF THE MATERIAL
PARAMETERS The identification procedure is carried out by using the results of pure tension tests along two orientations: one in the direction of a set of fibers (0 ~ and one at + 45 ~ Each test involves a series of loading and unloading sequences. The first step is to determine the elastic properties, which are usually given by the manufacturer. The ones of the matrix may be degraded by processing (e.g., porosity or microcavities). The initial elastic properties of the matrix are determined from measurement of the initial Young's moduli E ~176 and E 45, respectively, for a 0 ~ and + 45 ~ tension test. The next step is to identify the parameters of the relevant evolution laws. The fitting procedure can be summarized in the following way: 9 The variation of Dm with Ym obtained from the experimental data of the unloading slopes for different maximum stress levels of a tension test at +45 ~ 9 The evolution law of the inelastic tensile strain, gill, is obtained from the value of the unloading strain at the maximum stress of each unloading loop in the tension test performed in the 0 ~ direction (i.e., direction of the fibers). 9 The evolution law of Df with Yf, modeling fiber breakage, results in the fitting of the unloading slope for the maximum stress levels of a tension test at 0 ~ 9 The evolution law of the inelastic shear strain, gi12, is obtained from the value of the unloading strain at the maximum stress of each unloading loop in a tension test at 4-45 ~ 9 The cyclic properties can be identified with either a tensile or a 4-point bend test. By measuring the evolution of the inelastic strain gi~fl with the number of cycles, the parameters of the function F can be identified. 9 To identify creep properties, one needs more tests because creep is sensitive to stress levels and temperature range. Four different tests are used: one tensile test at 0 ~ and three tensile tests at + 45 ~ The different steps are summarized in Table 10.7.1.
1022 TABLE 10.7.1.
Leckie et al. Experimental Tests Needed for the Identification of the Model
Number of experiments
Type of test
Tension tests with unloading
Fatigue
Tensile 4-point bend Creep test
test
Evolution law
@ 0~
Gioc~
Df
@ 4- 45 ~
Dm
~i12
60 o
FIN]
@ 0~
~caa (O'1, T1)
Creep test
@ + 4 5 ~ for 3 sets
(0"2,T1) (0"3, ]"3)
Bc12
10.7.5 HOW TO USE THE MODEL This model is fully implemented and maintained in the commercial finite element code ABAQUS via a UMAT routine. Integrated laws are easy to implement and reduce the computation cost, especially for fatigue and creep loading conditions, by using the isochronous analysis [4]. Several applications of the model can be found for SiC/SiC composites [14]. All these examples have a full description, in terms of the distribution of tensile and shear stresses, inelastic strains, and matrix damage, of tension tests on plates with a hole and bend test on a plate. Some of the computations are compared to experimental measurements. To simplify the identification procedure, an assistant was written by using the same routine developed for the finite element code. This quasi-automatic identification closely follows the different steps summarized in Table 10.7.1.
1 0 . 7 . 6 TABLE O F P A R A M E T E R S The identification procedure is applied to various [0,90] laminated or woven CMCs. Some examples are given in Table 10.7.2. It can be noted that the material parameters modeling elasticity, matrix cracking, interface sliding, fiber-breakage, and fiber pullout of SiC/SiC composites are valid for the whole temperature range (i.e., from the room temperature to 1200~ Furthermore, the chemical degradation of the fiber
1023
10.7 Behavior of Ceramic-Matrix Composites TABLE 10.7.2
Material Parameters for Various CMCs SiC/SiC
Material parameter
Value
Matrix elastic modulus Matrix Poisson's ratio Matrix volume fraction Weibull modulus Normalizing energy Saturation parameter
Em Ym fm mm Ymo Dsat
157 0.25 0.55 1.6 0.60 0.99
Fiber elastic modulus Fiber Poisson's ratio Fiber volume fraction Weibull modulus Normalizing energy
Ef vf ff mf Yfo
200 0.25 0.45 4.5 42.5
Tensile inelastic strain constant Threshold tensile stress Exponent Shear inelastic strain constant Threshold shear stress Exponent
8int ~rth nii 8ins zth n12
4.99 10 -5 135 1.0
Fatigue amplitude Normalizing value Exponent
7 a n
0.984 6.96 1.56
Time power law exponent Normalizing time Shear creep strain constant Temperature coefficient Stress power law exponent Time power law exponent Normalizing temperature Normalizing stress Normalizing time
nt ti0 s120 b ncr ns To ~0 tl20
0.451 70.2 1.35 10 -3 28.1 4.2 0.4 1204 34.5 100
Unit GPa
MPa
GPa
MPa
MPa
5.0 10 - 5
59.1 2.0
MPa
C/C Value 9.5 0.1 0.56 1.15 1.21 0.99 280 0.1 0.44 NA large NA NA NA 4.52 10 -2 41.9 3.45
A1203/A1203 Unit GPa
MPa
GPa
MPa
MPa
MPa
NA NA NA
h
~ MPa h
NA NA NA NA NA NA NA NA NA
Value 40 0.25 0.60 NA NA 0.01 380 0.2 0.40 NA large 2.68 10 -5 66.4 1.74 2.61 10 -6 22.5 25.3
Unit GPa
MPa
GPa
MPa
MPa
MPa
NA NA NA
h
~ MPa h
NA NA NA NA NA NA NA NA NA
h
~ MPa h
coating is not accounted for by the present model. For the two other materials, only room temperature data are reported in Table 10.7.2.
REFERENCES 1. Burr, A., Hild, E, and Leckie, E A. (1997). Continuum description of damage in ceramic-matrix composites. Eur. J. Mech. A/Solids 16: 53-78.
1024
Leckie et al.
2. Burr, A., Hild, E, and Leckie, E A. (1998). The mechanical behaviour under cyclic loading of ceramic-matrix composites. Mater. Sci. Eng. A250: 256-263. 3. Burr, A., Hild, E, and Leckie, E A. (1998). Behaviour of ceramic-matrix composites under thermomechanical cyclic loading conditions. Comp. Sci. Tech. 58: 779-783. 4. Burr, A., Hild, E, and Leckie, E A. (1998). Isochronous analysis applied to the behavior of ceramic-matrix composites, in Proceedings ofJNCll, pp. 1343-1349, Vol. 3, Baptiste, D., and Lamon, J., eds., Paris: AMAC. 5. Burr, A., Hild, E, and Leckie, E A. (1995). Micro-mechanics and continuum damage mechanics. Arch. Appl. Mech. 65: 437-456.
SECTION
10.8
Limit and Shakedown Analysis of Periodic Heterogeneous Media GIULIO MAIER, VALTER CARVELLI,
and
ALBERTO TALIERCIO
Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
Contents 10.8.1 Introduction: Basic Concepts and Applicability Domain . . . . . . . . . . . . . . . . .
1025
10.8.2 Limit Analysis by the Static Approach. 1027 10.8.3 Limit Analysis by the Kinematic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1029
10.8.4 A General Finite Element Linear Programming Procedure . . . . . . . . . . . . . .
1031
10.8.5 Shakedown analysis . . . . . . . . . . . . . . . . . . 10.8.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
1032 1034 1034
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1035
10.8.1 I N T R O D U C T I O N : BASIC C O N C E P T S AND APPLICABILITY DOMAIN The engineering motivations for what follows may be clarified first by referring to two typical examples of ductile heterogeneous media" (i) perforated steel plates frequently employed in power plants" (ii) structural components made of metal-matrix fiber-reinforced composites (MMCs). In both these representative cases, overall inelastic analyses must be carried out in terms of average (or "macroscopic") stresses __Z- (~=(x)) and strains E - (~(x)). Therefore, at first a homogenization procedure is required, i.e., the Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1025
1026
M a i e r et al.
characterization of a homogeneous "equivalent material" through a microstructural analysis based on the local constitutive models for each constituent (or "phase"), say, ~=(x_,t) = f(e(x, z), 0 < z < t), and on the geometry of the texture at the "microscale"; see, e.g., Reference [1]. From here onwards, (.) denotes averaging over a suitable representative volume; underlined symbols denote vectors, doubly underlined ones second-order tensors, and dots time derivatives. Consider engineering situations in which deformations up to collapse can be reasonably expected to be "small" (this implies linear kinematic compatibility and rules out the influence of deformations on equilibrium) and the material behavior can be realistically interpreted as elastic-plastic and stable in Drucker's sense (hence: convex yield surfaces; associated flow rules; no softening), either perfectly plastic or hardening with saturation. In such situations, if it is experimentally, or otherwise, ascertained that the dominant dissipative phenomenon is plastic yielding (no debonding at interfaces between phases; no damage or microfractures), then, in view of overall structural analyses, the main feature to be assessed for the equivalent homogeneous material is its "strength," i.e., the plastic failure locus (or "yield surface") in the space of the average stresses =E and, hence, the "macroscopic strength domain," G h~ defined by that locus. If the structure is subjected to thermal and/or mechanical external actions ("loads") fluctuating in time ("variable repeated," in particular cyclic), it is of interest to assess in the E space also "shakedown domains" "c-.ho~ with respect SD to sets of assigned "loading domains" ~2 of variable repeated Z ( t ) E D. Specifically, the question to answer is as follows: given one ~2 ("basic" loading domain), what is the amplification factor, or "shakedown limit," sso, such that the cumulative (integral over space and time) dissipated energy D is unbounded (i.e., limt~oo D ( t ) = oo) for factors /~ > SsD, and it is not so if /~ <_ ssD, namely, if yielding processes eventually vanish (i.e., the system "shakes down"). Unboundedness of D characterizes three distinct "ultimate limit states": (a) incremental collapse (otherwise called "ratcheting"); (b) alternating plasticity (or "plastic shakedown"), with no divergence of deformed configuration in time; and (c) "plastic collapse" in the special case where the load domain shrinks to a point, say, Z0 (in this case, sso coincides with the yield limit So, which amplifies E0 ~ on the yield = leading to a point E_-0 surface of the homogenized material). The following circumstances are worth noting: (i) G h~ contains every ~hom "" SD and represents an essential feature of the homogenized material model, whereas the shakedown domains c-horn depend also on the set of basic loading ~" S D domains considered; (ii) shakedown (SD) theories and methodologies apt to compute SsD (and hence any ~hom~ "-'So ~ cover as special cases those meant to provide So and, hence, Gh~ (iii) the present conception of strength is in
10.8 Limit and Shakedown Analysis of Periodic Heterogeneous Media
1027
principle nonconservative (though practically quite realistic for prevailing ductile behaviors), since it does not allow for undesirable mechanical events (such as excessive deformations or cracks) that might occur for # < SSD; (iv) "direct" methods for shakedown analysis (SDA) and limit analysis (LA), with respect to evolutive (step-by-step) inelastic analysis techniques are consistent with frequent lack of information on the loading history (only fluctuations intervals are often known); also, they are generally advantageous in terms of computational effort and therefore more suitable to multiple parametric studies in design processes. The fundamentals of direct (nonevolutive) methods for LA and SDA of heterogeneous media will be presented in the subsequent sections according to the following criteria and further restrictions: (i) periodicity assumptions and consequent reference to a representative volume (RV) with periodicity boundary conditions, typical of homogenization theory; (ii) von Mises perfectly elastic-plastic local material model, defined by a single parameter (e.g., tensile strength), in addition to the elastic moduli. The present subject is rooted in classical plasticity and homogenization theories and, hence, the relevant literature is enormous. The few items concisely presented herein are selected by application-oriented criteria, and only a few closely related references are cited for details. On direct methods, a state-of-the-art and a fairly comprehensive up-to-date pertinent bibliography can be found in Reference [2].
1 0 . 8 . 2 L I M I T ANALYSIS BY T H E STATIC APPROACH The classical static (or "lower bound," or "safe") limit theorem can be stated as follows: a solid will not collapse if and only if there is a ("statically admissible") stress field which everywhere fulfills equilibrium and local strength limitations (or "yield conditions"). Its validity is guaranteed, as mentioned in Section 10.8.1, by linear kinematics and perfect plasticity, stable in the sense of Drucker's postulate. By applying this statement, the strength domain Gh~ in the space of average stresses Z for a periodic heterogeneous medium whose RV occupies the volume V with boundary ~V can be defined as follows (see, e.g., Reference [1]):
o
-
{zlz
=
div a(x) - O_ in V;
v; a(x). n(x) antiperiodic on OV}
(I)
1028
Maier et al.
where G(x) denotes the (convex) strength domain of the heterogeneous material at any point x of the RV, and n is the outward unit normal to OV. A possible discontinuity surface for the microscopic stress field can be also accounted for in Eq. 1 by interpreting div a__in the sense of the theory of distributions. A domain Go contained in G h~ (and, hence, conservative, i.e., representing lower bounds to the strength of the homogenized material) can be computed by selecting over the RV special statically admissible microscopic stress fields that comply with the periodicity boundary conditions on OV. For unidirectional fiber-reinforced composites, a simple domain inside G h~ is [3, 4]" Gr = { E
= ~m + O'e_x (~) C_x;
0"__m C Gm;
-d-
(2)
where 6+ - vf (dr: - if:) having set Gm as the strength domain of the matrix material; g?, O" -+ m a s the uniaxial strength of fibers and matrix in tension (+) and compression (-); vf as the fiber volume fraction; o as a scalar stress variable; and | as the dyadic product. Under several (also multiaxial) stress conditions, Eq. 2 gives fairly accurate estimates of the macroscopic strength of the material [4, 5]. In particular, if both phases comply with von Mises strength criterion (which is the case, e.g., for MMCs), Eq. 2 analytically provides a lower bound, say EL, to the macroscopic uniaxial strength of the composite, ~.hom, for any orientation 0 of the applied stress E to the fibers (the superscripts 4- are omitted, the material behavior being symmetric in tension and compression). These bounds, visualized by the dashed line in Figure 10.8.1b for the MMC case of Figure 10.8.1a, reads [3]:
(__ 1
>
3 sin20 2
"q-
(_2 am
(3)
3~ 2sin 20 1 - ~ s i n 20
ifO
sin
0
1 -
if O* < 0 < 3 sin2 0 (3)
The orientation O* can be computed simply by equating the two expressions for E/.
10.8 Limit and Shakedown Analysis of Periodic Heterogeneous Media
1029
FIGURE 10.8.1 Limit and shakedown analysis of a metal-matrix composite, a. Representative volume, b. Bounds on the macroscopic uniaxial strength by the kinematic approach, Eq. 6 (solid line) and static approach, Eq. 3 (dashed line) [4]. c. Finite element mesh and incremental collapse mechanism for 0 = 30 ~ d. Uniaxial plastic collapse (dashed line) [ 11] and shakedown limit (solid line) vs. direction 0.
1 0 . 8 . 3 L I M I T ANALYSIS BY T H E K I N E M A T I C APPROACH The kinematic (or "upper bound," or "unsafe") limit theorem, subject to the same validity restrictions as the static one, can be expressed in the following terms: a solid will collapse if and only if there exists a mechanism (i.e., strain and displacement rates obeying geometric compatibility) such that the work done by the external forces exceeds the total dissipated energy (the local density of which is uniquely defined by the plastic strains). On the basis of this statement, the strength domain Gh~ can be formally defined as follows:
1030
Maier et al.
(4)
rch~ being the dissipated power over the RV associated to ~, namely:
Tch~
1 fVzr(~(x_))dV, subject to:
,_(x)-~1
r
- g
w_ +
;
(5)
h_(x) -- ~ . x + ~_(x_); ~_periodic over V
where IV[ is a volumetric measure of the RV; fi denotes the microscopic velocity field; ~ is the associated strain rate field; and zt is the specific dissipated (plastic) power (i.e., mathematically, the support function of the local strength domain G at point x E V). Here again, discontinuity surfaces for the microscopic velocity field can be accounted for in Eq. 5 by interpreting ~7fi in the sense of the theory of distributions. Domains that bound G h~ from outside can be obtained by considering simple "failure mechanisms" for the RV, either characterized by uniform strain rates or by "failure planes" that split the RV element into blocks rigidly moving with respect to each other [4]. For instance, upper bounds Ev to the macroscopic uniaxial strength of MMCs can be computed by means of mechanisms characterized by slip planes cutting the matrix and intercepting the midpoints of a couple of opposite or adjacent sides of the RV (see Reference [4] for further details). These bounds depend on the fiber volume fraction vf and the spatial arrangement of the reinforcement. For fiber evenly spaced according to a regular hexagonal pattern, provided that the percentage of reinforcement is not too high (Yf < ~V/3/8), an upper bound to s is
Eh~
_< E U ( 0 ) - min
2Fm
2Fm
6 + Fm; V~ sin 20' sin0
3 1-~sin20 (6)
It is worth noting that the two bounds, s Eq. 3, and Ev, Eq. 6, coincide at 0 - O; that is, the uniaxial strength of the composite along the fibers can be exactly predicted and turns out to be
10.8 Limit and Shakedown Analysis of Periodic Heterogeneous Media
1031
= ~ + F m -- (1 - vf )Fm + vfFf
(7)
~,h~
This is a well-known formula which is widely employed in applications and is usually called the "rule of mixtures." The bounds presented in this and the preceding section can be extended to materials with different strengths in tension and compression [4]. Also, debonding at the fiber-matrix interface can be accounted for, provided it does not imply a softening behavior [5].
10.8.4 A GENERAL FINITE ELEMENT LINEAR PROGRAMMING PROCEDURE The applicability of the convenient closed-form bounds was seen to be limited (they are particularly accurate for uniaxial stresses and moderate vf). A numerical method for LA of composites can be developed by a static approach through the following sequence of operation phases (for basic concepts of the method, see, e.g., References [6, 7]; for specific details, see References [8]). (a) Perform a piecewise linear approximation (PWL) of the local material model. (b) Generate a finite element model of the RV. (c) Consider as loads an average stress state ~0 amplified by a load factor #. (d) Maximize # under statical admissibility constraints on the discretized stresses, namely, subject to: linear equations expressing the average link between #~0 and the stress field; equilibrium and periodicity on OV; and linear inequality which enforces the PWL yield conditions. (e) Interpret the optimal (max) value as collapse multiplier (or safety factor) s s with respect to ~0 (i.e., s~0 as a point belonging to the boundary of G h~ and the optimal vector as one of the possible stress states at collapse. If a kinematic approach is adopted, the last two phases become: (dt) Minimize the total dissipated power (a linear function of plastic multiplier rates) under kinematic admissibility constraints on the modeled velocity fields and non-negative plastic multiplier rates, namely: linear equation which expresses splitting of velocity fi_ into an addend linear in space, related to ~, and a periodic addend r periodicity on OV and geometric compatibility of r normalization of the "external power" ~0 " ~. (et) Interpret the optimal (min) value as safety factor sk and the optimal vector as one of the possible collapse mechanisms. This sketchy outline can be supplemented by the following remarks. (1) At the finite element interfaces discontinuities can be modeled for stresses under equilibrium constraints on tractions, for strains under neither detachment nor compenetration conditions on displacements (see Reference [8]).
1032
M a i e r et al.
(2) If the local PWL strength loci in (a) are inscribed in the original ones, and if the field modeling (b) and the constraint in (d) are such that static admissibility is complied with everywhere, then ss is a lower bound on the exact, "safety factor" s. Similarly, if phase (a) generates a circumscribed PWL approximation and (d~) satisfies kinematic admissibility exactly, then s h = sv >__s. However, often in practice a tight bracketing of s is computationally cumbersome to achieve; then secant approximation (a) and only approximate enforcement in (d), or (d~), are preferable, leading to sS~ s, or s k ~ s, with discrepancy unknown in sign but reducible by refinements in phases (a) and (b) at the price of increased computing cost (which for a linear programming (LP) problem in "normal form" is roughly proportional to m n 3 if m and n are the numbers of variables and constraints, respectively). (3) The previously outlined LA procedures are centered on LP with the following peculiar features: (i) LP software (based on the classical Simplex method and its variants) is widely available and often adopted to large-scale computing; (ii) any LP problem uniquely defines its "dual" LP problem, which can be interpreted as a kinematic formulation if the primal is a static one, and vice versa. Modern LP solvers provide the solution to both, but process the more economical one. (4) The PWL phase (a) leading to LP implies a drastic increase in the number of variables (an increase which can be reduced by a suitable trial-anderror procedure). If (a) is omitted, the phases (d) and (d~) lead to (convex) nonlinear programming (NLP) problems, by far more laborious to solve than the LP problem. The trade-off between LP and NLP formulations of LA is not resolved in general, much depending in practice on the availability of ad hoc software. 10.8.5
SHAKEDOWN
ANALYSIS
For periodic heterogeneous elastic-plastic solids subjected to an assigned loading domain ~2 of (uniform) variable-repeated average stresses _Z(t) c ~2, the following theorems can be proved (see, e.g., References [7, 9, 10]). (a) Shakedown will occur if, and only if, there is a field of self-stress a s such that, at any time and everywhere in the RV, the sum of it and of the (local) elastic stresses a__e due to ~ complies with the (local) yield conditions and the periodicity on the boundary of the RV. (b) Shakedown will occur if, and only if, for all "admissible plastic strain cycles" (APSC) the cumulative dissipated energy is larger than, or equal to, the ("external") work done by the elastic stresses a__e due to the concomitant external actions. By APSC is meant a history over a
10.8 Limit and Shakedown Analysis of Periodic Heterogeneous Media
1033
time interval T of a fictitious imposed plastic strain field, the time integral of which over T is compatible with displacements complying with the periodicity conditions as in Eq. 5. (c) If ~2 is a polyhedrical region (possibly as a PWL approximation of the input data), for SD analysis purposes it can be replaced by the set of its, say m, vertices" this reduces q~(t) to a sequence ~ and any APSC to a corresponding sequence of finite increments e_if,j = 1 , . . . , m. The kinematic method outlined in following text turns out to be computationally rather effective, in the writers' experience. The formulation based on (b) and (c), on traditional finite element discretization (displacement modeling), and on Mises elastoplasticity reads:
~-- rnin ~ i
j
E
ciJin:(~i~)' subject to"
i
(8)
j
Here indices i and j run over the sets of Gauss points (C i and Ji being Gauss weights and Jacobians) and vertices of f~, respectively; Eq. 9 enforces plastic incompressibility, compatibility at the APSC end (_V being the nodal displacement vector allowing for periodicity, B i the compatibility matrix for point i), and normalization of the total "external work" along the APSC, respectively. Eqs. 8 and 9 represent a convex NLP problem with nonsmooth objective function (since the Mises dissipated power rc is nonsmooth). The solution algorithm, described and satisfactorily employed in References [11, 12], is centered on the enforcement of Eq. 9a (with empirical suitable choice of the relevant penalty factor 0~) and on an iterative procedure for solving the (nonlinear) Kuhn-Tucker optimality conditions. Remarks: (1) The optimal value ~ is an approximation of sst), not a bound. (2) If f~ shrinks to a point, SDA reduces to LA and elasticity could be proved to become immaterial. (3) If a PWL approximation is adopted for the local material models, SDA becomes an LP problem, as a generalization of the LA formulations of Section 10.8.4. (4) History-dependent post-SD quantities (e.g., relative residual displacement of two points of the RV) can be bounded from above by bounding techniques, which, in PWL formulations (possibly accounting for hardening), can be reduced to a quadratic programming followed by an LP problem (see, e.g., Reference [7]).
1034 10.8.6
Maier et al. EXAMPLES
The first illustrative example concerns a ductile MMC characterized by the RV shown in Figure 10.8.1a and by fiber vs. matrix ratios 1.86 for volumes and 8.7 for yield stresses (both are Mises materials). The homogenized uniaxial plastic collapse limits and SD limits (for fluctuations between 0 and E) versus the angle 0 from the fiber axis have been computed by the bounding formulae of Sections 10.8.2 and 10.8.3 and by the finite element procedure summarized in Section 10.8.5 (432 finite element with bilinear interpolation, c~ = 106). These results are comparatively visualized in Figure 10.8.1b, d. The second and third example concern LA and SDA of perforated metal plates, regarded as two-dimensional plane-stress systems and depicted in Figure 10.8.2a and Figure 10.8.3a as for RV and finite element mesh. Representative results are visualized in Figure 10.8.2b and Figure 10.8.3b, and compared in the latter to experimental data with excellent agreement.
10.8.7
CONCLUSIONS
What precedes is intended to provide introductory and orientative information on the title subject. In closing, the reader's attention is drawn to the following remarks.
.6
I
~
~
~
x2A 0.4
_?,2
I
0.2
xI
/ 0.0
. . . . . . . .
0.0 (a)
(b)
0.2
I I
I
' I
0.6
0.4 El /
FIGURE 10.8.2 Perforated plate with circular holes (volumetric ratio 0.2). a. Representative volume, mesh of 392 finite element and biaxial average stress state, b. For 0 = 45~ loci of plastic collapse (dashed line) and of shakedown limit (solid line) for rectangular load domains t2 (m -- 4).
1035
10.8 Limit and Shakedown Analysis of Periodic Heterogeneous Media
,, I,, I,, I,, I,, I,,
1.0
Z/~
0.8
-
m
0.6 0.4
di
- -~Xl
L 0.2 i 0.0
h
''1''1''1'' 0
(a)
(b)
15 30
I''l''t 45
60
75
90
0~
FIGURE 10.8.3 Perforated plate with rectangular holes (h = 0.1f; d - 0.7f). a. Representative volume, mesh of 224 finite element and uniaxial average stress state, b. Plastic collapse limit vs. direction 0, present method (solid line) and experimental results by Litewka et al. [13] (dots).
(a) For understanding and implementation of the computer-oriented methods outlined in Sections 10.8.4 and 10.8.5, details can be found in References [8, 11, 12]. Moreover, a variety of alternative techniques is available nowadays in the literature: most of them are referenced in the review paper [2]; specifically on composites, i m p o r t a n t contributions (not cited here for space limitations) are due to G. Dvorak, E Leckie, A. Ponter, E Rammerstorfer, D. Weichert, and others. (b) Besides plastic collapse and lack of s h a k e d o w n , susceptible to be analyzed by the direct (nonevolutive) m e t h o d s dealt with herein, there are other ultimate or unserviceability limit states to be studied by generally m o r e laborious time-stepping (evolutive) m e t h o d s (exp o u n d e d elsewhere in this book) for the integrity assessments, based on their texture, of h e t e r o g e n e o u s materials.
REFERENCES
1. Suquet, P. (1985). Elements of homogenization for inelastic solid mechanics, in Homogenization Techniques for Composite Media, CISM Lecture Notes in Physics pp. 193-278, vol. 272, Sancheg-Palencia, E., and Zarei, A., eds., Wien: Springer. 2. Maier, G., Carvelli, V., and Cocchetti, G. (2000). On direct methods for shakedown and limit analysis. Eur. J. Mech. A/Solids 19: $79-$100. 3. de Buhan, P., and Taliercio, A. (1991). A homogenization approach to the yield strength of composite materials. Eur. J. Mech. A/Solids 10: 129-154.
1036
Maier et al.
4. Taliercio, A. (1992). Lower and upper bounds to the macroscopic strength domain of a fiberreinforced composite material. Int. J. Plasticity 8: 741-762. 5. Taliercio, A., and Sagramoso, R (1995). Uniaxial strength of polymeric-matrix fibrous composites predicted through a homogenization approach. Int. J. Solids Struct. 32: 2095-2123. 6. Maier, G. (1970). A matrix structural theory of piecewise-linear plasticity with interacting yield planes. Meccanica 5: 55-66. 7. Cohn, M. Z., and Maier, G. (1979). Engineering Plasticity by Mathematical Programming, New York: Pergamon Press. 8. Francescato, R, and Pastor, J. (1997). Lower and upper numerical bounds to the off-axis strength of unidirectional fiber-reinforced composites by limit analysis methods. Eur. J. Mech. A/Solids 16: 213-234. 9. Maier, G. (1969). Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: A finite element, linear programming approach. Meccanica 4: 250260. 10. K6nig, J. A. (1987). Shakedown of Elastic-Plastic Structures, Elsevier. 11. Carvelli, V., Maier, G., and Taliercio, A. (2000). Kinematic limit analysis of periodic heterogeneous media. Computer Modelling in Engineering and Science 1(2): 15-26. 12. Carvelli, V., Maier, G., and Taliercio, A. (1999). Shakedown analysis of periodic heterogeneous materials by a kinematic approach. Mechanical Engineering (Strojnicky Casopis) 50: 229-240. 13. Litewka, A., Sawczuk, and A., Stanislawka, J. (1984). Simulation of oriented continuous damaged evolution. Jour. de MEch. Th~or. et Appl. 3: 675-688.
SECTION
10.9
Flow-Induced Anisotropy in Short-Fiber Composites ARNAUD POITOU and FRI~DI~RIC MESLIN LMT-Cachan, ENS de Cachan, Universit~ Paris 6, 61 avenue du PrEsident Wilson, 94235 Cachan Cedex, France
Contents 10.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.2 Constitutive Relation . . . . . . . . . . . . . . . . . 10.9.2.1 Volume Average . . . . . . . . . . . . . 10.9.2.2 Statistical Average . . . . . . . . . . . . 10.9.3 Explicit Calculations for --p ]E . . . . . . . . . . . 10.9.3.1 Suspension of Spheres . . . . . . . 10.9.3.2 Flow around an Ellipsoidal Particle Immersed in an Anisotropic Suspension . . . . . . 10.9.3.3 Results and Discussions . . . . . . 10.9.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.9.1
1037 1038 1038 1039 1040 1041
1042 1042 1046 1046 1046
INTRODUCTION
Short-fiber composites are made of a thermoplastic matrix into which shopped fibers are immersed (fiber length a r o u n d 0.1 mm, diameter a r o u n d 0.01 mm, volumetric concentration a r o u n d 15 to 30%). Composite parts are manufactured with molding processes similar to those used in the polymer industry (injection molding, extrusion). During the process, the filled polymer flows at a molten state and becomes anisotropic because of its deformation. The problem is coupled because the flow induces a rotation Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1037
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Poitou and Meslin
of the particles and thus an anisotropy, but the orientation of the particles modifies the material's behavior and thus indirectly affects the anisotropy. The aim of this article is to give a brief overview of this problem and to outline contributions from different scientific communities (solid mechanics modeling and low Reynolds number hydrodynamics). The application of a differential model to this problem is original in this context.
10.9.2 CONSTITUTIVE RELATION Two main steps permit one to derive macroscopic constitutive equations for anisotropic suspensions: (i) a now standard homogenization procedure which leads to volume average quantities if the orientation of the particles is perfectly defined, and (ii) a statistic description of the orientation.
10.9.2.1 VOLUME AVERAGE Basic equations are obtained with a volume average procedure [1]. Let f~ be a representative volume of our macroscopic scale, which contains many particles located in f~i. Let z, u, and d denote, respectively, the microscopic stress tensor, velocity vector, and strain rate tensor, and let n be a normal vector to the particle boundary (Figure 10.9.1). m
Pa_~cle (~2i)
Volume ~2
FIGURE 10.9.1 Representativevolume at a macroscopic scale.
1039
10.9 F l o w - I n d u c e d A n i s o t r o p y in S h o r t - F i b e r C o m p o s i t e s
For a given fiber orientation p in the suspension, for a Newtonian ambient fluid (viscosity r/), and if inertia terms can be neglected, the corresponding macroscopic variables T, v, and D are related by m
=T - - - e I d--- +
--P
2r/D(v) + =Ep _
lzJ ~n | xas ~
(i)
(2)
i 0f~,
In this expression, P is the volume averaged pressure and =Ep describes the contribution of the particle to the averaged extra stress tensor.
10.9.2.2 STATISTICALAVERAGE The fiber orientation p is given in an average sense by its probability distribution ~(p) so that the macroscopic statistical averaged stress tensor (i.e., the stress tensor used in engineering computations) writes: m
z_z_- f 2(p_)~,(p__)ap
(3)
For most engineering applications, the exact calculation at each point of O(P) is neither necessary nor possible. This orientation information is correctly approximated with use of the second-order and fourth-order orientation tensors:
-
f e | e ,(eldp; 8 = f e | e | e | e ,(el#
(41
The flow around a single particle in a Newtonian fluid gives a first approximation for the evolution of one particle orientation. From Jeffery's calculations [15], for example, if 2 denotes the shape factor of the particle, and if f~ and D are respectively the volume averaged vorticity and strain rate tensors" z~ - i dP_dt= _~(V) "pm+ r{D(V) p_ -- Tr[D(V) p . p]p} with r - 22 +-----~
(5)
The evolution equation of the probability distribution is given by
dO(e)
~
O(p)~_
- ~
D op~
(6)
D is a rotation diffusion coefficient, vanishing at zero shear rates, which intends to take into account the hydrodynamic interactions between fibers.
1040
Poitou and Meslin
The evolution equation for the second-order orientation writes:
(~---~ =
dt
-
. a -
a.
f]
) ( -
r
D . a +
a . D
) ---2ra4"
tensor then
D
(7)
This equation does not permit one to determine the orientation tensor if the velocity field is given, because it involves the fourth-order orientation tensor. The evolution equation for the fourth-order tensor would similarly involve the sixth-order tensor, etc. So, in order to close this equation, it is necessary to introduce a phenomenological relation between the secondand the fourth-order orientations (closure approximation), which can take different forms [2, 3]. For example, the simplest one is quadratic and writes:
a4-__a|
10.9.3
(8)
EXPLICIT CALCULATIONS
FOR
--p
The calculation of ---p Z as a function of the macroscopic velocity field v requires one to solve a microscopic problem which can take different forms according to the level of approximation. Two main kinds of situations have been considered. Within the first one (slender bodies theory), the stress distribution at the particle boundary is reduced to a multipolar distribution. The Batchelor's model [4] is deduced, which has been modified by Shaqfeh and Fredrickson [13]. All of these models can be written in the following form: Z
- ~ p
- 2#Tr[D(v)(p ~
~
~
| p)]p | p
(9)
In this expression, p is a unit vector aligned along the principal axis of the fiber, and/2 is calculated as a function of the particle concentration and of its shape factor, which is assumed to be very large. The exact expression depends on the level of approximations. The interaction between fibers is accounted for in a certain sense but for long or very long fibers only. A second class of models deals with ellipsoidal or spheroidal particles. This case has been extensively studied for linear elastic materials [5, 6]. However, it has not been much studied for suspensions, probably because Eshelby's works [7] are not popular in fluid mechanics.
10.9 Flow-Induced Anisotropy in Short-Fiber Composites
10.9.3.1
1041
SUSPENSION OF SPHERES
The case of a suspension of non-Brownian spheres is meaningful because the suspension remains Newtonian (viscosity rl~q). In this case, if q~ denotes the volumetric rate of spheres, and if there is no hydrodynamic interaction, the dilute approximation leads to Einstein's formula: E ---p
5r/q0D (v_) => r/e q - - r / ( 1
+ 2.5q0) (dilute model)
(10)
To account for hydrodynamic interactions, a very natural way consists in calculating -E- p in assuming that, instead of being immersed in the ambient fluid of viscosity r/, the consequences of hydrodynamic interactions are summarized by placing the sphere in an ambient fluid of viscosity r/eq (selfconsistent model). Z
=p
-- 5r/eqq0D(v_) => r/cq -- ~//(1 - 2.5q~) (self-consistent model)
(11)
For small but noninfinitesimal volumetric rates, the self-consistent model allows one to derive (with a very different approach than the original one) the Batchelor's approximation: ~eq -- ~1(1 + 2.5q~ + 6.25q~2)
(12)
However, the self-consistent model suffers from strong limitations because the equivalent viscosity increases to infinity when q ) = 40%. For this reason, another approach is needed to ameliorate Eq. 11. Following Christensen [8], we assume that the spheres are added progressively, step by step. At each step, the suspension is macroscopically a viscous fluid of viscosity rleq(q)). We then add a quantity d(p of spheres. This quantity is small enough that Einstein's formula for dilute suspension remains valid. The only difficulty consists in noting that the reference ambient fluid is the suspension of volumetric rate q~, so that the volumetric rate induced by d(p is d(p/(1-q~): dE=p - 5~/eq i dq~_~o--D(v_) =~ JTeq-- ~//(1 - q~)2"5(differential model)
(13)
This differential model leads to the same kind of expression as the Krieger and Dougherty [9] one. It does not hold anymore when direct contact between particles is to be considered because the global behavior of the suspension is then neither linear nor homogeneous. In the following, we extend Eqs. 11 and 13 to ellipsoids. Two difficulties must then be overcome. The first is that a suspension of nonspherical particles is non-Newtonian. The second is that the calculation is more technical.
1042
Poitou and Meslin
1 0 . 9 . 3 . 2 FLOW AROUND AN ELLIPSOIDAL PARTICLE IMMERSED IN AN ANISOTROPIC SUSPENSION 10.9.3.2.1 Algebraic Preliminary Result
Differential approach:Particles
are added progressively, so that each time one has to deal with a dilute suspension of spheroids. However, the problem is here specific, because the suspending fluid is a suspension, which is nonNewtonian. The system to solve is similar to Eq. 13: ~/dc# d(21 =
d(22 --
(1 -- ~)S1212
2~/dc# -
(18)
-
~/dr
d(23 =
(1 - r
with initial conditions: (2i = 217 for ~# = O.
10.9.3.3
RESULTS AND DISCUSSIONS
The numerical results [10] have been obtained by computations. In order to compare these results with other existing theories, the stress tensor will be written in a similar form as the one introduced by Tucker [11]: T = -PI=d + 2r/{d + Ns [(p | p)d + d(p | p)J +
Tr[(p |
p)dJ (p | p)}
(19)
with these notations:
Ns
--
(23
~
(21
(21 =
(20)
Np 3((22(21) 2((23(21) 2(21
(21
/]1 contains all the isotropic contributions to the viscosity (from both the solvent and the particles), while anisotropic contributions of the particles are represented by Np and Ns. However, as in Batchelor's dilute model, we find for both differential and self-consistent schemes that Np is always greater than Ns, even for particles of small aspect ratio. So, the anisotropic contributions by the particles are essentially described by Np. The product r/1 Np is equal to the parameter # of Shaqfeh's model, and we compare spheroidal model to slender bodies model. Figure 10.9.2 evidence the differences between the dilute
1043
10.9 Flow-Induced Anisotropy in Short-Fiber Composites 35
m
30
-
-
,
Din~.ti.1
/
.... Sl~qfeh & Fte4kickson ..... t,~.,t, ,~ A.-,,~t.o,,g . . . . . . . . . Batchelor
25
/
/
,m
I
[
//
!
m
/
!
/ -
m
20 rl, Np
15 / 10
/
/
/
i
.,/, "
5
p
0
--77L
I
] ii 0. l
0
,2~,kcd.~.,,-.--.-~',""~. . . . . . . . I
I
0.2 0.3 Fiber volume fraction
0.4
0.5
400
g
1 Self oonsistant --- - - Differential .... Shaqfeh & Fredrickson ..... Dinh & Armstrong .......... B a ~ h e l ~
350 300 250
l l [
t / ,t
[ /
/
/
~1 Np 200
/
L - 20
150
/
100
,,~
~
"
-...
/
/
/
s~ 'S
~'
50
/
,.,,,t
..
::."
. .....
0
0.1
0
1 lO s
.
.
.
0.2 0.3 Fiber volume fraction .
.
.
.
.
.
.
.
.
.
0.4
0.5
.
/ ......
,,
8104 ........
6104
/
I
~
ql Np 4 104
9
2104
9
o
~=350 /
O. 1
FIGURE 10.9.2
~
/
............ "~. . . . . . . . . . . .
/
/
/
,~j SS
0.2 0.3 Fiber volume fraction fix N p
i
,,"
/
~,-.a-~.".-.'r'-.~..'f-.'.r::-.-'
0
/
t
j
.., 9
,
0.4
0.5
as a function of fiber volume fraction.
1044
Poitou and Meslin 80 7O ....
60
Self eortsistant - . Differential Shaqfeh & Fredrickson modified
50
/ /
rim Np 40
,r
~=5
30
/
20 10 0 0.1
0.2 0.3 Fiber volume fraction
0.4
0.5
700 84 Self eonsis "rant . - --, Differential .... Shaqfeh & Fredrickson
600
/ / /
I
500
~iNp
I
400 ~,=20
300
9 ,S/
/
/
200
::'
:I
100 0
~
~
~
0.1
0
~ ~i
~ " ~' ~'"~~ r I 0.2 0.3 Fiber volume fraction
I
0.4
0.5
5
llO
/ _
8 10 4
Self consistant
_ r~ifr~.~.~l
....
~ /
/
Shaqfeh & Freddeksan [
/
6104
/
z
rh 410
4
~, = 350
1 J J
2 10 4
i ]
e
t
/
/ /
sss J Ss
,~ , - , ' ' "
0 0 FIGURE 10.9.3
0.1
i 0.2 0.3 Fiber volume fraction
Modified r/1
Np as a function
i 0.4
0.5
of fiber volume fraction.
10.9 Flow-Induced Anisotropy in Short-Fiber Composites
FIGURE 10.9.4
1045
Injection molding simulation.
spheroidal approximation (Batchelor spheroid), the Dinh and Armstrong's model [14], and Shaqfeh and Fredrickson's slender bodies model and our selfconsistent and differential model, for three shape factors of the particles. It can be seen that even for a shape factor ). of 350, neither Shaqfeh and Fredrickson's nor Dinh and Armstrong's model is correct because, for very low values of cp, the curves are not tangent to Batchelor's dilute approximation. This point has been already mentioned by Ranganathan and Advani [12], who proposed a correction which accounts for the finite aspect ratio of the particles. Figure 10.9.3 compares the same models (modified for slender
1046
Poitou and Meslin
bodies models) for various fiber shape factors. It is to be noted that, as for hard sphere models [8], the self-consistent approach overestimates the rheological parameters (this self-consistent model evidences also a vertical asymptote for a volumic rate less than 1). The differential model lies very near the modified Shaqfeh and Fredrickson's model for the large aspect ratio (in the semidilute range). Thus, in this sense the differential model generalizes Shaqfeh and Fredrickson's model in accounting for an averaged interaction between the finite aspect ratio's fibers. In particular, we can apply the differential model for prolate spheroids.
10.9.4
EXAMPLE
Figure 10.9.4 illustrates a three-dimensional modeling for which an injection molding process has been simulated. Details of the computation algorithm can be found in Reference [16]. In this representation, the arrow depicts the orientation tensor a. The orientation of the arrow shows the direction of the eigenvector associated with the largest eigenvalue, and its color shows the magnitude of this eigenvalue. In other words, the arrow shows the preferential direction for the fiber orientation and the color shows the intensity of the alignment. The composite is assumed to enter the mold in an isotropic state (each eigenvalue is then 1/3). The shearing alignment is seen at the entrance, particularly near the walls. On the other hand, near the flow front a disalignment is clearly evidenced.
10.9.5
CONCLUSION
The flow of short-fiber composites induces an anisotropy which can be controlled by simulations. The scientific field concerned with this question lies at the interface between fluid and solids mechanics. The control of anisotropy is to be correlated with prediction of rupture for the part at a solid state.
REFERENCES 1. Batchelor, G. K. (1970). The stress system in suspension of force free particules. J. Fluid. Mech. 41: 545. 2. Advani, S. G. and Tucker, C. L. (1990). Closure approximations for three-dimensional structure tensors. J. Rheol. 34: 367. 3. Dupret
10.9 Flow-Induced Anisotropy in Short-Fiber Composites
1047
4. Batchelor (1970). Slender bodies theory for particles of arbitrary cross section in Stokes flows. J. Fluid. Mech. 44: 419. 5. Gilormini, P. and Vernusse, P. (1992). Tenseur d'Eshelby et problme d'inclusion dans le cas isotrope transverse incompressible. C. R. Acad. Sci. Paris 314: 257. 6. Mura, T. (1982). Micromechanics of Defects in Solids. Martinus Nijhoff. 7. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London A241: 376. 8. Christensen, R. M. (1990). A critical evaluation for a class of micro-mecanics models. J. Mech. Phys. Solids 38: 379. 9. Krieger, I. M. and Dougherty, T. J. (1959). A mechanism for non-Newtonian flow in suspension of rigid spheres. Trans. Soc. Rheol. 3: 137-152. 10. Meslin, E (1997). Propridtes rheologiques des composites fibres courtes l'tat fondu. PhD thesis LMT-ENS de Cachan, France. 11. Tucker, C. L. (1991). Flow regime for fiber suspensions in narrow gaps. J. Non-Newt. Fluid. Mech. 39: 239. 12. Ranganathan, S. and Advani, S. G. (1993). A simultaneous solution for flow and fiber orientation in axisymetric diverging flow. J. Non-Newt. Fluid. Mech. 47: 107. 13. Shaqfeh and Fredrickson, G. (1990). The hydrodynamic stress in a suspension of rods. Phys. Fluids A2: 7. 14. Dinh, S. M. and Armstrong, R. C. (1984). A rheological equation of state for semiconcentrated fiber suspensions. J. Rheol. 28: 207. 15. Jeffery, G. B. (1922). The motion of ellipsoidal particles immersed in viscous fluid. Proc. R. Soc. A102: 161. 16. Chinesta, E (1999). Mod~lisation num~rique en mise en forme des polym~res et c~ramiques: diff~rents problems de transport. PhD (HDR), LMT ENS de Cachan.
SECTION
i0. I 0
Elastic Properties of Bone Tissue STEPHEN C. COWIN New York Center for Biomedical Engineering, School of Engineering, The City College, New York
Contents 10.10.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10.2 Cortical and Cancellous Bone . . . . . . . . . . . 10.10.3 The Elastic Model for Cortical Bone . . . . 10.10.4 The Source of the Elastic Constant Data for Cancellous Bone . . . . . . . . . . . . . . 10.10.5 The Analysis of the Elastic Constant for Cancellous Bone . . . . . . . . . . . . . . . . . . . 10.10.6 The Elastic Constant Data for Cancellous Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1048 1049 1050 1052 1053 1054 1055
10.10.1 VALIDITY Bone is a natural composite material. It may be modeled as a linearly anisotropic elastic material for all but the rarest cases within the physiological range of loading. Large strains occur only with trauma to the bone and irreversible damage to the microstructure. There are measurable viscoelastic effects in bone, but they are generally not significant in a stress analysis of a whole bone. A m e t h o d of data analysis for a set of anisotropic elastic constant m e a s u r e m e n t s of a material with variable composition has been applied to a database for cancellous bone. For these materials the identification of the type of elastic symmetry is complicated by the variable composition 1048
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright O 2001 by Academic Press. All rights of reproduction in any form reserved.
1049
10.10 Elastic Properties of Bone Tissue
of the material. A method of data analysis, described in following text, permits the identification of the type of elastic symmetry to be accomplished independent of the examination of the variable composition. This method of analysis may be applied to any set of elastic constant measurements, but it is illustrated here by application to an extraordinary database of cancellous bone elastic constants. The solid volume fraction or bulk density is the compositional variable for the elastic constants in this case. The final result is the solid-volume-fraction-dependent orthotropic Hooke's law for cancellous bone.
10.10.2
CORTICAL AND CANCELLOUS
BONE
Bone is a remarkable and extremely complex tissue. It has at least two major functions. First, it is the material from which the skeleton is made, which provides mechanical support and protection for the organism. The bone tissue of the skeleton is continually changing to adapt its form and structure to this task. From the birth to the death of the organism it modifies its structure by laying down new bone tissue and resorbing old. A second function of bone is to store the minerals, particularly Ca §247needed to maintain a mineral homeostasis in the body by regulating the concentrations of the key blood electrolytes, including Ca §247H § and HP04. Bone differs from the other tissues because of its greater stiffness and strength. These properties stem from its being a composite material formed by the deposition of a mineral, apatite or hydroxyapatite, in a frame of collagen. The stiffness of bone is the key to understanding why it differs so greatly from other tissues. At the macroscopic level there are two major forms of bone tissue, called compact or cortical bone and cancellous or trabecular bone. Cortical or compact bone is a dense material with a specific gravity of almost two in humans and a little over two in cattle. It forms most of the outer shell of a whole bone, a shell of variable thickness. Cancellous bone generally exists only within the confines of the cortical bone coverings. Cancellous bone is also called trabecular bone because it is composed of short struts of bone material called trabeculae (from the Latin for "little beam"). The connected trabeculae give cancellous bone a spongy appearance, and it is often called spongy bone. The mechanical properties of bone, other than the elasticity, have been documented. These include water content, viscoelastic properties, mechanical
1050
Cowin
s t r e n g t h , p e r m e a b i l i t y , m e c h a n o - e l e c t r o k i n e t i c effects, d a m a g e etc. O n e w i l l find d a t a o f this t y p e c o l l e c t e d i n C o w i n [4,6].
mechanics,
10.10.3 THE ELASTIC MODEL FOR CORTICAL BONE The elastic model for cortical bone tissue is anisotropic linear elasticity. Tables 10.10.1 and 10.10.2 below list the components of the orthotropic elastic compliance coefficients for cortical bone. The first table is for human bone, and second is for bovine and canine bone. The elastic coefficients in the anisotropic Hooke's law [8,13,16] are the three Young's moduli E l , E2, and E3 in the three perpendicular coordinate directions, the six Poisson's ratios denoted by subscripted v's (only three of the six Poisson's ratios are independent since v12//E1 z v21/E2, Vl3/E1 - v 3 1 / / E 3 , and v23//E2 = v32/E3) , and the three shear moduli G23, G13, and G12. In this notation the 3 direction is coincident with the long axis of the bone; the 1 and 2 directions are radial and circumferential, respectively. In the row indicating material symmetry, ORTH stands for orthotropy and TI stands for transverse isotropy. In the row indicating the testing method, U stands for ultrasound and M stands for standard machine testing. The entry NM stands for not measured.
TABLE 10.10.1
Group Bone Symmetry Method EI(GPa) E2(GPa) E3(GPa) G12(GPa) G13(GPa) Gz3(GPa) v12 v13 v23 v21 v31 v32
Technical Constants for Cortical Human Bone Reilly and Burstein [ 15 ] Femur TI M 11.5 11.5 17.0 3.6 3.3 3.3 0.58 0.31 0.31 0.58 0.46 0.46
Yoon and Katz [21] Femur TI U 18.8 18.8 27.4 7.17 8.71 8.71 0.312 0.193 0.193 0.312 0.281 0.281
Knets and Malmeisters [11 ] Tibia ORTH M 6.91 8.51 18.4 2.41 3.56 4.91 0.49 0.12 0.14 0.62 0.32 0.31
Ashman
et al. [2] Femur ORTH U 12.0 13.4 20.0 4.53 5.61 6.23 0.376 0.222 0.235 0.422 0.371 0.350
7 TABLE 10.10.2 Technical Constants for Cortical Bovine and Canine Bone
9
c.
Group
Lang I121
Reilly and Burstein [I51
Ambardar and Ferris [ I ]
Van Buskirk et al. [17]
Bone Symmetry Method El GPa) Ez(GPa) EdGPa) GdGPa) GdGPa) GdGPa) v12
Bovine phalanx TI
Bovine femur TI M 10.2 10.2 22.6 NM* 3.6 3.6 0.51 NM NM 0.51 0.36 0.36
Bovine femur TI U 6.97 6.97 20.9 2.2 6.9 6.9 0.55 0.15 0.15 0.55 0.44 0.44
Bovine femur ORTH
v13 "23
v21 V31
v32 *Not measured
u 11.3 11.3 22.0 3.8 5.4 5.4 0.484 0.203 0.203 0.484 0.396 0.396
u 11.6 14.6 21.9 5.29 6.29 6.99 0.302 0.109 0.205 0.380 0.206 0.307
Burris [3] Bovine femur TI M 13.3 13.3 18.1 3.5 5.0 5.0 NM NM NM NM 0.22 0.22
Burris 131 Bovine femur TI +
u 10.79 12.24 18.90 3.38 4.47 5.96 0.45 0.24 0.22 0.51 0.42 0.33
Ashman et al. 121 Canine femur
~i
m
1052
Cowi~
10.10.4 THE SOURCE OF THE ELASTIC C O N S T A N T DATA F O R C A N C E L L O U S B O N E The elastic constant results for cancellous bone presented here are based upon an analysis of a database consisting of 141 human cancellous bone specimens. This database, reported by van Rietbergen et al. [ 18, 19] and Kabel et al. [9], is superior to previous databases because the authors provide the entire set of anisotropic elastic constants without an a priori assumption of a particular material symmetry and without an assumption of the direction in which the maximum Young's modulus occurs. This database is unique in many different ways, the most important of which is the large number of specimens and its method of construction, but particularly because it is not based entirely on measurements of real specimens. The database of elastic constants of 141 human cancellous bone specimens employed here was constructed by imaging real specimens and then computationally determining their elastic constants. This cyberspace method of construction is thought to be more accurate than the conventional mechanical testing procedures for evaluating the elastic constants of human cancellous bone. The determination of the elastic constants of cancellous bone by conventional mechanical test procedures is very difficult. The basic problem is that, because of the size of the human body, it is difficult to obtain specimens of cancellous bone that are more than 5-mm cubes. The logical way to test small cubes such as these is by compression testing. However, compression testing is highly inaccurate for cancellous bone because of (1) the frictional end effects of the platens, (2) the near impossibility of identifying, a priori, the grain directions in a bone specimen and thus of cutting a specimen in the grain directions, (3) the stiffening effect of the platens on the bone near the platens, and (4) the unpredictable inhomogeneity of the specimen. The construction of the database of elastic constants of 141 human cancellous bone specimens employed here is a relatively inexpensive method of determining the full set of anisotropic elastic constants for a small specimen of cancellous bone by a combination of imaging the specimen [9,10,14] and subsequent evaluation of the effective elastic constants using computational techniques based on the finite element by van Rietbergen et al. [18,19]. Once the image of the specimen was in the computer and a finite element mesh was generated, a sequence of loadings [18,19] was applied to the specimen and the responses were determined. The sequence of loadings was sufficient in number to determine all 21 elastic constants. Thus no material symmetry assumptions were made in the determination of the constants. Quantitative stereological programs were used to determine the solid volume fraction q~ of each specimen. These are the data employed in the
10.10 Elastic Properties of Bone Tissue
1053
determination of the elastic constants for cancellous bone recorded in following text. In this method, the actual matrix material of the trabeculae comprising the bone specimen is assumed to have an axial Young's modulus Et. The value of Et may be fixed from a knowledge of the axial Young's modulus for the tissue, or from the shear modulus about some axis, or by measuring the tissue modulus Et itself. For purposes of numerical calculation, Et was taken to be 1Gpa [9,10,14,18,19]. However, since these are linear finite element (FE) models, the FE results can be scaled for any other modulus by multiplying the results with the new value of Et (in GPa). The tissue modulus Et thus is a scale factor that magnifies or reduces all the elastic constants. The cancellous bone elastic constant results are presented here as multiples of Et.
10.10.5 THE ANALYSIS OF THE ELASTIC CONSTANT FOR CANCELLOUS BONE The elasticity of cancellous bone is complicated to analyze, report, and record because of the highly inhomogeneous porous nature of this bone type. The elastic constants depend upon the volume fraction of solid matrix material present (one minus the porosity). In the case of porous isotropic materials, for example, it is customary to regress the Young's modulus against the solid volume fraction and obtain expressions for the Young's modulus E as a function of the solid volume fraction ~; for example, E - ( c o n s t a n t ) ~n. For cancellous bone a method must be available for construction similar to representations for an inhomogeneous anisotropic porous solid. Such a method is described in the next paragraph. Many materials are anisotropic and inhomogeneous because of the varying composition of their constituents. The identification of the type of elastic symmetry is complicated by the variable composition of the material, which makes the analysis of the elastic constant measurement data difficult. A solution to this problem in which identification of the type of elastic symmetry and analysis of the variable composition are separated, and then analyzed independently, was described in Cowin and Yang [5] and applied in Yang et al. [20] to bone softwood, and hardwood. The method consists of averaging eigenbases, that is to say, the bases composed of the orthogonal sets of eigenvectors of different measurements of the elasticity tensor, in order to construct an average eigenbasis for the entire data set. This is possible because the eigenbases, composed of eigenvectors, are independent of composition whereas the eigenvalues are not. The eigenvalues of all the anisotropic elastic coefficient matrices can then be transformed to the average eigenvector basis
1054
Cowi,
and regressed against their compositional parameters. This method treats the individual measurement as a measurement of a tensor instead of as a collection of individual elastic constant or matrix element measurements, recognizing that the measurements by different investigators will reflect the systematic invariant tensorial properties of a material, like eigenvectors and eigenvalues. This method for averaging different measurements of the anisotropic elastic constants for a specific material has advantages over the traditional method of averaging the individual matrix components of the elasticity or compliance matrices. Averaging invariants removes the effect of the reference coordinate system in the measurements, while the traditional method of averaging the components may induce errors because of the various reference coordinate systems and may distort the nature of the symmetry. This averaging process explicitly retains the orthonormality of the eigenvector basis. The results of Cowin and Yang [5] provided a means of extending the empirical method of representing the Young's modulus E as a function of the solid volume fraction ~b (for example, E = (constant) ~bn) to all the elastic constants of an inhomogeneous anisotropic material. In Cowin and Yang [5] this method was applied to feldspar, and it was discovered that the eigenvectors, but not the eigenvalues, were relatively independent of material composition. That result was extended by Yang et al. [20] to three natural, porous materials: cancellous bone, hardwood, and softwood. These works have established this method of analysis as a valid approach to the construction of anisotropic stress-strain relations for other compositionally dependent materials. This new method of analysis also identifies the type of elastic symmetry possessed by the material. No a priori assumption as to the type of elastic symmetry is made. The type of symmetry is identified from the character of the eigenvectors that are calculated. For the cancellous bone considered here, the analysis shows that human cancellous bone has orthotropic elastic symmetry at the 95% confidence level.
10.10.6 THE ELASTIC CONSTANT CANCELLOUS BONE
DATA FOR
Applying the method of analysis just described to the 14 I-specimen database discussed in the section before last, explicit representations of the solid volume fraction ~b dependent orthotropic elastic constants of human cancellous bone were obtained. These explicit representations are E1=1240
Et~ 1"80, E2 - 885 Etq~1"89, E3- 529 Et~ 1"92, 2G23- 533 Et~ 2"04, 2613----
1055
10.10 Elastic Properties of Bone Tissue 633
Etq~197,
2Glz=973
Et~bl98;
v23=0.256
~b- ~ 1 7 6 v 3 2 = 0 . 1 5 3
q5- ~ 1 7 6
v 1 3 - 0 . 3 1 6 q5 -o.19, v 3 1 = 0 . 1 3 5 ~b- ~ 1 7 6 v 1 2 = 0 . 1 7 6 05- ~ a n d v21 = 0 . 1 2 5 q5- ~ Et has the d i m e n s i o n of stress, a n d the o t h e r n u m b e r s m u l t i p l y i n g all ~b's r a i s e d to a p o w e r are d i m e n s i o n l e s s . T h e s q u a r e d c o r r e l a t i o n coefficients (R 2) for the o r t h o t r o p i c elastic coefficients are as follows: for 1/E1, R 2 = 0.934; for 1/E2, R 2 = 0 . 9 1 7 ; for 1/E3, R 2 = 0 . 8 7 9 ; for 1/(2G23), R 2 = 0 . 8 7 0 ; for 1/(2G13), R 2 = 0.887; for 1/(2G12), R 2 = 0.876; for -v12/E1, R 2 = 0.740; for -v13/E1, R 2 = 0 . 8 4 1 ; and for 1~21/E2, R 2 = 0 . 6 6 6 .
REFERENCES 1. Ambardar, J. D., and Ferris, C. D. (1976). A simple technique for measuring certain elastic moduli in bone. Biomed. Sci. Instrum. 12: 23. 2. Ashman, R. B., Cowin, S. C., Van Buskirk, W. C., and Rice, J. C. (1984). A continuous wave technique for the measurement of the elastic properties of bone. J. Biomechanics 17: 349. 3. Burris, C. L. (1983). A Correlation of Quasistatic and Ultrasonic Measurements of the Elastic Properties of Cortical Bone. Ph.D. dissertation, Tulane University, New Orleans, Louisiana. 4. Cowin, S. C., ed. (1989). Bone Mechanics, Boca Raton, FL: CRC Press. 5. Cowin, S. C., and Yang, G. (1997). Averaging anisotropic elastic constants data.J. Elasticity 46: 151-180. 6. Cowin, S. C., ed. (2001). Bone Mechanics Handbook, Boca Raton, FL: CRC Press. 7. Cowin, S. C., Van Buskirk, W. C., and Ashman, R. B. (1987). The properties of bone, in Handbook of Bioengineering, Skalak, R., and Chien, S., editors-in-chief, New York: McGrawHill. 8. Hearmon, R. E S. (1961). An Introduction to Applied Anisotropic Elasticity, Oxford: Oxford University Press. 9. Kabel, J., van Rietbergen, B., Odgaard, A., and Huiskes, R. (1999). Constitutive relationships of fabric, density and elastic properties in cancellous bone architecture. Bone 25: 481-486. 10. Kabel, J., van Rietbergen, B., Dalstra, M., Odgaard, A., and Huiskes, R. (1999). The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone. J. Biomechanics 32: 673-680. 11. Knets, I., and Malmeisters, A. (1977). Deformability and strength of human compact bone tissue, in Mechanics of Biological Solids: Proc. Euromech Colloquium 68, p. 133, Brankov, G., ed., Sofia: Bulgarian Academy of Sciences. 12. Lang, S. B. (1970). Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bovine bones. IEEE Trans. Biomed. Eng. 17: 101. 13. Lekhnitskii, S. G. (1963). Theory of Elasticity of an Anisotropic Elastic Body, San Francisco: Holden Day. 14. Odgaard, A., Kabel, J., van Rietbergen, B., and Huiskes, R. (1997). Fabric and elastic principal directions of cancellous bone are closely related. J. Biomechanics 30: 487-495. 15. Reilly, D. T., and Burstein, A. H. (1975). The elastic and ultimate properties of compact bone tissue. J. Biomechanics, 8: 393. 16. Saada, A. S. (1974). Elasticity Theory and Applications, Pergamon. 17. van Buskirk, W. C., Cowin, S. C., and Ward, R. N. (1981). Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J. Biomechanical Eng. 103: 67.
1056
Cowi.
18. van Rietbergen, B., Odgaard, A., Kabel, J., and Huiskes, R. (1996). Direct mechanical assessment of elastic symmetries and properties of trabecular bone architecture. J. Biomechanics 29: 1653-1657. 19. van Rietbergen, B., Odgaard, A., Kabel, J., and Huiskes, R. (1998). Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions. J. Orthop. Res. 16: 23-28. 20. Yang, G., Kabel, J., van Rietbergen, B., Odgaard, A., Huiskes, R., and Cowin, S. C. (1999). The anisotropic Hooke's law for cancellous bone and wood. J. Elasticity, 53: 125-146. 21. Yoon, H. S., and Katz, J. L. (1976). Ultrasonic wave propagation in human cortical bone. 11. Measurements of elastic properties and micro-hardness. J. Biomechanics, 9: 459.
10.11
SECTION
Biomechanics of Soft Tissue GERHARD A. HOLZAPFEL Institute for Structural Analysis, Computational Biomechanics, Graz University of Technology, 8010 Graz, Austria
Contents 10.11.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.2 Background on the Structure of Soft Tissues: Collagen and Elastin . . . . . . . . . . . 10.11.2.1 Collagen . . . . . . . . . . . . . . . . . . . . . . 10.11.2.2 Elastin . . . . . . . . . . . . . . . . . . . . . . . . 10.11.3 General Mechanical Characteristics of Soft Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.4 Description of the Model . . . . . . . . . . . . . . . 10.11.5 Representative Example: A Model for the Artery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.5.1 Residual stresses . . . . . . . . . . . . . . 10.11.6 Identification of the Material Parameters 10.11.7 How to Use the Model . . . . . . . . . . . . . . . . . 10.11.8 Table of Parameters . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1057 1058 1059 1060 1061 1062 1065 1066 1068 1069 1069 1070
10.11.1 VALIDITY A n efficient c o n s t i t u t i v e f o r m u l a t i o n a p p r o x i m a t e s all types of soft tissues w i t h a r e a s o n a b l e a c c u r a c y over a large s t r a i n range. W e r e q u e s t a s i m p l e c o n s t i t u t i v e e q u a t i o n w i t h o n l y a few m a t e r i a l p a r a m e t e r s i n v o l v e d that a l l o w for an " e x p l a n a t i o n " of the m a t e r i a l r e s p o n s e of tissues in t e r m s of their s t r u c t u r e . In a d d i t i o n , w e r e q u e s t t h a t the c o n s t i t u t i v e f o r m u l a t i o n is fully three-dimensional and consistent with both mechanical and mathematical Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1057
1058
Holzapfel
requirements, applicable for arbitrary geometries, and suitable for use within the context of finite element methods in order to solve complex initial boundary-value problems. The presented general model is a fully three-dimensional material description of soft tissues for which nonlinear continuum mechanics is used as the fundamental basis [10, 18]. It has the special feature that it is based partly on histological information (i.e., the microscopic structure of organs and tissues). The general model describes the highly nonlinear and anisotropic behavior of soft tissues as composites reinforced by two families of collagen fibers. The constitutive framework is based on the theory of the mechanics of fiber-reinforced composites [26] and is suitable to describe a wide variety of physical phenomena of soft tissues. The performance and the physical mechanism of the model are presented in Reference [11]. As a representative example, the general model for soft tissues is specified to predict the mechanical response of healthy and young arteries under physiological loading conditions [12]. The model neglects active components, i.e., contracting elements with biochemical energy supply which are controlled by biological mechanisms, and is concerned with the description of the passive state of arteries. The models are suitable for predicting the anisotropic elastic response of soft tissues in the large strain domain. A suitable constitutive and numerical model that is general enough to describe the finite viscoelastic domain is documented in Reference [11]. The presented models do not consider acute and long-term changes in geometry and/or the mechanical response of tissues due to, for example, drugs, aging, and disease. When soft tissues are subjected to loads that are beyond their physiological range, the load-carrying fibers of the tissue slip relative to each other. In clinical procedures tissues may undergo irreversible (plastic) deformations [12] which are of medical importance. Constitutive equations for describing plastic deformations of, for example, arteries are proposed in References [8, 27].
10.11.2 BACKGROUND O N T H E STRUCTURE OF SOFT TISSUES: COLLAGEN AND ELASTIN What do we mean by soft tissues? A primary group of tissue which binds, supports, and protects our human body and structures such as organs is soft connective tissue. In contrary to other tissues, it is a wide-ranging biological material in which the cells are separated by extracellular material. Connective tissues may be distinguished from hard (mineralized) tissues such as bones for their high flexibility and their soft mechanical properties. This article
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discusses connective tissue from the point of view of material science, biomechanics, and structural engineering (for more details, see, for example, Reference [6], Chapter 7). Examples of soft tissues are tendons, ligaments, blood vessels, skins, or articular cartilages, among many others. Tendons are muscle-to-bone linkages that stabilize the bony skeleton (or produce motion), and ligaments are bone-to-bone linkages that restrict relative motion. Blood vessels are prominent organs composed of soft tissues which have to distend in response to pulse waves. The skin is the largest single organ (16% of the human adult weight). It supports internal organs and protects our body. Articular cartilages form the surface of body joints (which is a layer of connective tissue with a thickness of 1-5 mm), distribute loads across joints, and minimize contact stresses and friction. Soft connective tissues of our body are complex, fiber-reinforced, composite structures. Their mechanical behavior is strongly influenced by the concentration and structural arrangement of constituents such as collagen and elastin, the hydrated matrix of proteoglycans, and the topographical site and respective function in the organism.
I0.11.2.1 COLLAGEN Collagen is a protein which is a major constituent of the extracellular matrix of connective tissue. It is the main load-carrying element in a wide variety of soft tissues and is very important to human physiology (for example, the collagen content of the [human] Achilles tendon is about 20 times that of elastin). Collagen is a macromolecule with a length of about 280nm. Collagen molecules are linked to each other by covalent bonds that build collagen fibrils. Depending on the primary function and the requirement of strength of the tissue, the diameter of collagen fibrils varies (the order of magnitude is 1.5 nm [17]). In the structure of tendons and ligaments, for example, collagen appears as parallel-oriented fibers [1], whereas many other tissues have an intricate disordered network of collagen fibers embedded in a gelatinous matrix of proteoglycans. More than 12 types of collagen have been identified [17]. The most common collagen is type I, which can be isolated from any tissue. It is the major constituent in blood vessels. The rodlike shape of the collagen molecule comes from three polypeptide chains which are composed in a right-handed triple-helical conformation. Most of the collagen molecule consists of three amino acids: glycine (33%), which enhances the stability of the molecule, proline (15%), and hydroxyproline (15%) [23].
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The intramolecular crosslinks of collagen gives the connective tissues the strength which varies with age, pathology, etc. (for a correlation between the collagen content in the tissue, % dry weight, and its ultimate tensile strength, see Table 10.11.1. The function and integrity of organs are maintained by the tension in collagen fibers. They shrink upon heating breakdown of the crystalline structure (at 65~ for example, mammalian collagen shrinks to about one-third of its initial length, [6], p. 263).
10.11.2.2 ELASTIN Elastin, like collagen, is a protein which is a major constituent of the extracellular matrix of connective tissue. It is present as thin strands in soft tissues such as skin, lung, ligamenta flava of the spine, and ligamentum nuchae (the elastin content of the latter is about five times that of collagen). The long flexible elastin molecules build up a three-dimensional (rubberlike) network, which may be stretched to about 2.5 times the initial length of the unloaded configuration. In contrast to collagen fibers, this network does not exhibit a pronounced hierarchical organization. As for collagen, 33% of the total amino acids of elastin consists of glycine. However, the proline and hydroxyproline contents are much lower than in collagen molecules. The mechanical behavior of elastin may be explained within the concept of entropic elasticity. As for rubber, the random molecular conformations, and hence the entropy, change with deformation. Elasticity arises through entropic straightening of the chains, i.e., a decrease of entropy, or an increase of internal energy (see, for example, References [9]). Elastin is essentially a linearly elastic material (tested for the ligamentum nuchae of cattle). It displays very small relaxation effects (they are larger for collagen).
TABLE 10.11.1 MechanicalProperties [6, 15, 25] and Associated Biochemical Data [30] of Some Representative Organs Mainly Consisting of Soft Connective Tissues Material
Ultimate tensile strength (MPa)
Tendon Ligament Aorta Skin Articular cartilage
50-100 50-100 0.3--0.8 1-20 9-40
Ultimatetensile Collagen strain (%) (% dry weight) 10-15 10-15 50-100 30-70 60-120
75-85 70-80 25-35 60-80 40-70
Elastin (% dry weight) <3 10-15 40-50 5-10
10.11 Biomechanics of Soft Tissue
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10.11.3 GENERAL MECHANICAL CHARACTERISTICS OF SOFT TISSUES Before describing a model for soft tissues, it is beneficial and instructive to gain some insight into their general mechanical characteristics. Soft tissues behave anisotropically because of their fibers, which tend to have preferred directions. In a microscopic sense they are nonhomogeneous materials because of their composition. The tensile response of soft tissue is nonlinear stiffening, and tensile strength depends on the strain rate. In contrast to hard tissues, soft tissues may undergo large deformations. Some soft tissues show viscoelastic behavior (relaxation and/or creep), which has been associated with the shear interaction of collagen with the matrix of proteoglycans [16] (the matrix provides a viscous lubrication between collagen fibrils). In a simplified way we explain here the tensile stress-strain behavior for skin, an organ consisting mainly of connective tissues, which is representative of the mechanical behavior of many (collagenous) soft connective tissues. For the connective tissue parts of the skin the three-dimensional network of fibers appears to have preferred directions parallel to the surface. However, in order to prevent out-of-plane shearing, some fiber orientations also have components out of plane. Figure 10.11.1 shows a schematic diagram of a typical J-shaped (tensile) stress-strain curve for skin. This form, representative for many soft tissues,
FIGURE 10.11.1 Schematicdiagram of a typical (tensile) stress-strain curve for skin showing the associated collagen fiber morphology.
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differs significantly from stress-strain curves of hard tissues or from other types of (engineering) materials. In addition, Figure 10.11.1 shows how the collagen fibers straighten with increasing stress. The deformation behavior for skin may be studied in three phases: I, II, and III. Phase I. In the absence of load the collagen fibers, which are woven into a rhombic-shaped pattern, are in relaxed conditions and appear wavy and crimped. Unstretched skin behaves approximately isotropically. Initially low stress is required to achieve large deformations of the individual collagen fibers without requiring stretch of the fibers. In phase I the tissue behaves like a very soft (isotropic) rubber sheet, and the elastin fibers (which keep the skin smooth) are mainly responsible for the stretching mechanism. The stressstrain relation is approximately linear, and the elastic modulus of skin in phase I is low (0.1-2.0 MPa). Phase II. In phase II, as the load is increased, the collagen fibers tend to line up with the load direction and bear loads. The crimped collagen fibers gradually elongate and interact with the hydrated matrix. With deformation the crimp angle in collagen fibrils leads to a sequential uncrimping of fibrils. Note that the skin is normally under tension in vivo. Phase III. In phase III, at high tensile stresses, the crimp patterns disappear and the collagen fibers become straighter. They are primarily aligned with one another in the direction in which the load is applied. The straightened collagen fibers resist the load strongly, and the tissue becomes stiff at higher stresses. The stress-strain relation becomes linear again. Beyond the third phase the ultimate tensile strength is reached and fibers begin to break. The mechanical properties of soft tissues depend strongly on the topography, risk factors, age, species, physical and chemical environmental factors such as temperature, osmotic pressure, and pH, and the strain rate. The material properties are strongly related to the quality and completeness of experimental data, which come from in vivo or in vitro tests having the aim of mimicking real loading conditions. Therefore, to present specific values for the ultimate tensile strength and strain of a specific tissue is a difficult task. Nevertheless, Table 10.11.1 attempts to present ranges of values of mechanical properties and collagen and elastin contents (% dry weight) in some representative organs mainly consisting of soft connective tissues.
1 0 . 1 1 . 4 DESCRIPTION OF THE MODEL At any referential position X of the tissue we postulate the existence of a Helmholtz free-energy function ~. We assume the decoupled form tp _ U(X;J) + tP(X; (~, A1, A2)
(1)
10.11 Biomechanics of Soft Tissue
1063
where U is a purely volumetric (dilatational) contribution and ~ is a purely isochoric (volume-preserving) contribution to the free energy q~. Here _ ~T~ denotes the modified right Cauchy-Green tensor and F - J - 1 / 3 F is the unimodular (distortional) part of the deformation gradient F, with J - det F > 0 denoting the local volume ratio. In addition, in Eq. 1, {A1, A2} is a set of two (second-order) tensors which characterize the anisotropic properties of the tissue at any X. The structure tensors A1 and A2 are defined as the tensor products a0~ | a0~, where a0~, ~ - 1,2, are two unit vectors characterizing the orientations of the families of collagen fibers in the (undeformed) reference configuration of the tissue (see Figure 10.11.2). Since most types of soft tissues are regarded as incompressible (for example, arteries do not change their volume within the physiological range of deformation [2]), we now focus attention on the description of their isochoric deformation behavior characterized by the energy function q~. We suggest the simple additive split
-- ~iso (X; I1 ) -~- ~aniso (X; I1, I2 )
(2)
of_ ~ into a part ~iso associated with isotropic deformations and a part tPani~o associated with anisotropic deformations. This is sufficiently general to capture the salient mechanical feature of soft tissue elasticity as described in Section 10.11.3 (a more general constitutive framework is presented in References [8, 11, 12]). In Eq. 2 we used I1 - ( ~ ' I for the first invariant of tensor C(I is the second-order unit tensor), and the definitions If (C, a01) - C ' A 1 ,
I2 (C, a02) - (~'A2
(3)
of the invariants, which are stretch measures for the two families of collagen --4r --~ fibers (see, for example, References [10, 26]). The invariants 11 and 12 are squares of the stretches in the directions of a01 and a02, respectively. Isotropy is described through the invariant I1 and anisotropy through I~ and Is
FIGURE 10.11.2 Arrangement of collagen fibers in the reference configuration characterized by two unit vectors a01, a02 at position X.
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Since the (wavy) collagenous structure of tissues is not active at low stresses (it does not store strain energy), we associate tPiso with the mechanical response of the noncollagenous matrix of the material (which is less stiff than its elastin fiber constituent). To determine the noncollagenous matrix response, we propose to use the isotropic neo-Hookean model according to (: (I1 - -
3),
(4)
where c > 0 is a stresslike material parameter. However, to model the (isotropic) noncollagenous matrix material any Ogden-type elastic material may be applied [ 18]. According to morphological findings at highly loaded tissues, the families of collagen fibers become straighter and the resistance to stretch is almost entirely due to collagen fibers (the tissue becomes stiff). Hence, the strain energy stored in the collagen fibers is taken to be governed by the polyconvex (anisotropic) function
--
-- 1/ 1 --1}
where kl > 0, k3 > 0 are stresslike material parameters and k2 > 0, k4 > 0 are dimensionless parameters. According to Eqs. 2, 4, and 5, the collagen fibers do not influence the mechanical response of the tissue in the low stress domain. Because of the crimp structure of collagen fibers, we assume that they do not support compressive stresses, which implies that they are inactive in compression. Hence the relevant part of the anisotropic function (Eq. 5) is omitted for this case. If, for example, I 1 > 1 and 12 > 1, then the soft tissue responds similarly to a rubberlike (purely isotropic) material described by the energy function (Eq. 4). However, in extension, that is when I1 ) 1 or I2 > 1, and the collagen fibers are active and energy is stored in the fibers. Equation 1 enables the Cauchy stress tensor, denoted ~, to be derived in the decoupled form O' =
G'vo 1 -+- ~
with
G'vo 1 - -
pI,
(0,)
8 _ 2j_ldev ~ O-~ ~T
(6)
with the volumetric contribution avol and the isochoric contribution ~ to the Cauchy stresses. In the stress relation (Eq. 6), p=dU/dJ denotes the hydrostatic pressure and dev(o) furnishes the deviatoric operator in the Eulerian description. The operator is defined as dev(o) - (o) - ] [ ( o ) " I]I, so that dev( 9 : I = 0. Using the additive split (Eq. 2) and particularizations (Eqs. 4 and 5), we get with the third part of Eq. 6 an explicit constitutive expression for the
10.11 Biomechanics of Soft Tissue
1065
isochoric behavior of soft connective tissues in the Eulerian description, i.e.,
2 ff = c dev 1~+ Z
2 ~ d e v (a~ | a~)
(7)
0~--1 where 13 = FFr denotes the modified left Cauchy-Green tensor,
1= 0~aniso 0I; = k l ( I i - 1 ) e x p
=
0T2
=k3(I2-1)ex p
[~2 (I; - 1) 2]
(8)
-
(9)
are (scalar) response functions, and a~ = Fao~, ~ = 1, 2, are the Eulerian counterparts of the unit vectors ao~. The specific form of the proposed constitutive equation (Eq. 7) requires the five material parameters c, kl, k2, k3, k4 whose interpretations can be partly based on the underlying histological structure, i.e., matrix and collagen of the tissue. Note that in Eq. 7, orthotropic (kl = k3, k2 = k4), transversely isotropic (kl = 0 or k3 = 0 ) , and isotropic hyperelastic descriptions (kl = k3 = 0) at finite strains are included as special cases.
10.11.5 REPRESENTATIVE EXAMPLE: A MODEL FOR THE ARTERY In this section we describe a model for the passive state of the healthy and young artery (no pathological changes in the intima, which is the innermost arterial layer frequently affected by atherosclerosis) suitable for predicting three-dimensional distributions of stresses and strains under physiological loading conditions with reasonable accuracy. It is a specification of the constitutive framework for soft tissues stated in the previous section. For an adequate model of arteries incorporating the active state (contraction of smooth muscles), see Reference [22]. For a detailed study of the mechanics of arterial walls, see the extensive review in Reference [13]. Experimental tests show that the elastic properties of the media (middle layer of the artery) and adventitia (outermost layer of the artery) are significantly different [31]. The media is much stiffer than the adventitia. In particular, in the unloaded configuration the mean value of Young's modulus for the media, for several pig thoracic aortas, is about an order of magnitude higher than that of the adventitia [32]. In addition, the arterial layers have different physiological tasks, and hence the artery is modeled as a thickwalled elastic circular tube consisting of two layers corresponding to the
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media and adventitia. In a young nondiseased artery the intima (innermost layer of the artery) exhibits negligible wall thickness and mechanical strength. Each tissue layer is treated as a composite reinforced by two families of collagen fibers which are symmetrically disposed with respect to the cylinder axis. Hence, each tissue layer is considered as cylindrically orthotropic (already postulated in the early work [20]) so that a tissue layer behaves like a socalled balanced angle-ply laminate. We use the same forms of strain energy functions (Eqs. 4 and 5) for each tissue layer (each layer responds with similar mechanical characteristics) but use a different set of material parameters. Hence, Eq. 2 takes on the specified form 2
- ----T(II cM/. -- 3) + 2k2Mklw~l{exp k2M(I:~-1)2]-1
(10)
~1 M
%
-
3)
+ 2--~
1/
~1
We end up with a two-layer model incorporating s/x material parameters, three for the media M, i.e., cM, klM, k2M, and three for the adventitia A, i.e., CA,
klA, k2A. The invariants associated with the anisotropic parts of the two tissue layers are defined by I1) - ~; " A1) and I2j - C 9A2j, j - M, A. The structure tensors A U, A2) are given by Alj=aol)|
A2j=ao2j|
j=M,A.
(12)
Employing a cylindrical coordinate system, the components of the unit (direction) vectors aolj and ao2) read in matrix notation
[aolj] --
I~
cos]~j
,
[ao2)]--
sin ]~j
I ~1 cos~j
, j-- M,A,
(13)
--sin ]~j
and ~j, j - M, A, are the angles between the collagen fibers and the circumferential direction in the media and adventitia (see Figure 10.11.3). Small components of the (collagen) fiber orientation in the radial direction, as, for example, reported for human brain arteries [5], are neglected.
10.11.5.1
RESIDUAL STRESSES
It has been known for some years that arteries which are excised from the body and not subjected to any loads are not stress-free (or strain-free) [28]. If, for example, the media and adventitia are separated and cut in a radial
10.11 Biomechanics of Soft Tissue
1067
direction, the two arterial layers will spring open to form open (stress-free) sectors, which, in general, have different opening angles (see, for example, the experimental studies [29] for bovine specimens). In general, the residual stress state is very complex, and residual stresses (strains) in the axial direction may also occur. Note that residual stresses result from growth and remodeling mechanisms [21, 24]. By considering the arterial layers as circular cylindrical tubes, we may characterize the reference (stress-free) configuration of one arterial layer as a circular sector, as shown in Figure 10.11.4. For each arterial layer of the blood vessel a certain opening angle c~ can be found by experimental methods. The importance of incorporating residual stresses associated with the load-free
FIGURE 10.11.3 Load-free configuration of an idealized artery modeled as a thick-walled circular tube consisting of two layers, the media and adventitia.
FIGURE 10.11.4 Cross-sectional representation of one arterial layer at the reference (stressfree) and load-free (stressed) configurations.
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(but stressed) configuration into the computation has been emphasized in, for example, References [4, 12]. Consideration of residual strains has a strong influence on the global pressure and radius response of arteries and also on the stress and strain distributions across the deformed arterial wall. For analytical studies of residual stresses see, for example, the works [14, 22], which contain further references. Therefore, it is essential to incorporate the residual stresses inherent in many biologic tissues. One possible approach to consideration of the influence of residual stresses on the overall three-dimensional stress behavior is to measure the strain energy from the load-free (stressed) configuration and to include the residual stresses [19]. Another approach is to start with the energy function relative to the stress-free (and fixed) configuration, as assumed in the presented models, and determine the deformation required to reach the load-free (stressed) configuration. Figure 10.11.4 shows the crosssectional respresentation of one arterial layer at the load-free configuration obtained from the reference configuration by pure bending. With the condition of incompressibility, the radius r of an arterial layer in the load-free configuration may be computed from the radius R of the associated reference configuration as [12]
r= g
/R 2 - R/2
2zt
(14)
where ri, Ri a r e the internal radii associated with the two configurations. The (constant) axial stretch is denoted by 2z, and the parameter k is a convenient measure of the tube opening angle in the stress-free configuration.
10.11.6 IDENTIFICATION OF THE MATERIAL PARAMETERS Preferred directions in soft tissues are well specified by the orientation of prolate cell nuclei. They can be identified in microphotographs of appropriately stained histological sections. By visual inspection there exists a high directional correlation between smooth muscle cells and collagen fibers. Hence, the bell-shaped distribution of collagen fiber orientations may be obtained from an image-processing analysis of stained histological sections. The angle fl (and thus the unit vectors a01, a02) may be identified as the mean value of the corresponding statistical distribution. Values of the material parameters associated with the model for soft tissues are then obtained by fitting the equations to the experimental data of the soft tissue of interest by using standard nonlinear fitting algorithms, such as the
1069
10.11 Biomechanics of Soft Tissue
Levenberg-Marquardt algorithm. If the mean values of the orientation of cell nuclei (collagen fiber) may not be identified experimentally, it is suggested to treat the collagen fiber orientations as additional (phenomenological) "material" parameters.
10.11.7
HOW
TO USE THE MODEL
The energy functions are well suited for use in nonlinear finite element software, which enables complex boundary-value problems to be solved. Aspects of finite element implementation and numerical analysis of the model are presented in Reference [11]. Furthermore, computations may be carried out with some of the commercially available mathematical software packages such as Mathematica or Maple, which allow symbolic computation. Based on Mathematica, in Reference [12] a numerical technique for solving the bending, axial extension, inflation, and torsion problem of an artery is described.
10.11.8
TABLE OF PARAMETERS
Values of the parameters correspond to the Eqs. 10 and 11 and are given for a representative carotid artery from a rabbit (experiment no. 71 in Reference [7]). The material parameters c, kl, k2 and the (mean) values of collagen fiber angles ~ are summarized in Table 10.11.2. In the adventitia many collagen fibers run closer to the axial direction of the artery, whereas in the media the collagen fibers tend to run around the circumference. The fiber angles ~ are meant to be associated with the reference (stress-free) configuration, and we assume that they are the same in the load-free (stressed) configuration. The difference in fiber angle which occur due to bending to the load-free configuration (see Figure 10.11.4) is small, so that it has a negligible influence on the stress-strain analysis of arteries.
TABLE 10.11.2 Parameters for a Carotid Artery from a Rabbit (Experiment no. 71 in Reference [7]) in reference to Eqs. 10 and 11 Media
Adventitia
cM= 3.0000 (kPa) klM = 2.3632 (kPa) k2M-- 0.8393 (-) ]~M--29.0~
CA= 0.30000 (kPa) 0.5620 (kPa) k2A -- 0.7112 (--) ]~A--62.0~ klA
=
10 7 0
Holzap fel
By using a wall thickness of 0.39mm (adopted from Reference [3]) and assuming that the media occupies two-thirds of the arterial wall thickness, the parameters in Table 10.11.2 predict the characteristic orthotropic behavior of a carotid artery under combined bending, inflation, axial extension, and torsion, as documented in Reference [12].
REFERENCES 1. Betsch D. E, and Baer, E. (1980). Structure and mechanical properties of rat tail tendon. Biorheology 17: 83-94. 2. Carew, T. E., Vaishnav, R. N., and Patel, D.J. (1968). Compressibility of the arterial wall. Circ. Res. 23: 61-68. 3. Chuong, C. J., and Fung, Y. C. (1983). Three-dimensional stress distribution in arteries. ASME J. Biomech. Engr. 105: 268-274. 4. Chuong, C.J., and Fung, Y. C. (1986). Residual stress in arteries, in Frontiers in Biomechanics, pp. 117-129, Schmid-Sch6nbein, G. W., Woo, S. L.-Y., and Zweifach, B. W., eds., New York: Springer-Verlag. 5. Finlay, H. M., McCullough, L., and Canham, P. B. (1995). Three-dimensional collagen organization of human brain arteries at different transmural pressures. J. Vasc. Res. 32: 301312. 6. Fung, Y. C. (1993). Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., New York: Springer-Verlag. 7. Fung, Y. C., Fronek, K., and Patitucci, P. (1979). Pseudoelasticity of arteries and the choice of its mathematical expression. Am. Physiological Soc. 237: H620-H631. 8. Gasser, T. C., and Holzapfel, G. A. (2001). Rate-independent elastoplastic constitutive modeling of biological soft tissues: Part I. Continuum basis, algorithmic formulation and finite element implementation. Int. J. Solids and Structures (in press). 9. Hoeve, C. A. J., and Flory, P. J. (1958). The elastic properties of elastin. J. Am. Chem. Soc. 80: 6523-6526. 10. Holzapfel, G. A. (2000). Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Chichester: John Wiley & Sons. 11. Holzapfel, G. A., and Gasser, T. C. (2001). A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications. Comput. Methods Appl. Mech. Engr. 90: 4379-4403. 12. Holzapfel, G. A., Gasser, T. C., and Ogden, R. W. (2001). A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity (in press). 13. Humphrey, J. D. (1995). Mechanics of the arterial wall: Review and directions. Crit. Rev. Biomed. Engr. 23: 1-162. 14. Johnson, B. E., and Hoger, A. (1998). The use of strain energy to quantify the effect of residual stress on mechanical behaviour. Math. Mech. of Solids 3: 447-470. 15. Martin, R. B., Burr, D. B., and Sharkey, N. A. (1998). Skeletal Tissue Mechanics, New York: Springer-Verlag. 16. Minns, R. J., Soden, P. D., and Jackson, D. S. (1973). The role of the fibrous components and ground substance in the mechanical properties of biological tissues: A preliminary investigation. J. Biomech. 6: 153-165. 17. Nimni, M. E., and Harkness, R. D. (1988). Molecular structure and functions of collagen, Collagen, pp. 3-35, Nimni, M. E., ed., Boca Raton, FL: CRC Press.
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18. Ogden, R. W. (1997). Non-linear Elastic Deformations, New York: Dover. 19. Ogden, R. W., and Schulze-Bauer, C. A.J. (2000). Phenomenological and structural aspects of the mechanical response of arteries. In Mechanics in Biology, Casey, J., and Bao, G. eds, New York, The American Society of Mechanical Engineers (ASME). AMD-Vol. 242/BED-Vol. 46, 125-140. 20. Patel, D. J., and Fry, D. L. (1969). The elastic symmetry of arterial segments in dogs. Circ. Res. 24: 1-8. 21. Rachev, A. (1997). Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions. J. Biomech. 30: 819-827. 22. Rachev, A., and Hayashi, K. (1999). Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries. Ann. Biomed. Engr. 27: 459-468. 23. Ramachandran, G. N. (1967). Chemistry of collagen, Treatise on Collagen, pp. 103-183, Ramachandran, G.N., ed., New York: Academic Press. 24. Rodriguez, E. K., Hoger, A., and McCulloh, A. D. (1994). Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27: 455-467. 25. Silver, E H., Christiansen, D. L., and Buntin, C. M. (1989). Mechanical properties of the aorta: A review. Crit. Rev. Biomed. Engr. 17: 323-358. 26. Spencer, A. J. M. (1984). Constitutive theory for strongly anisotropic solids, Continuum Theory of the Mechanics of Fibre-Reinforced Composites, pp. 1-32, Spencer, A.J.M., ed., Wien: Springer-Verlag, CISM Courses and Lectures No. 282, International Centre for Mechanical Sciences. 27. Tanaka, E., Yamada, H., and Murakami, S. (1996). Inelastic constitutive modeling of arterial and ventricular walls, in Computational Biomechanics, pp. 137-163, Hayasi, K., and Ishikawa, H. eds., Tokyo: Springer-Verlag. 28. Vaishnav, R. N., and Vossoughi, J. Estimation of residual strains in aortic segments, in Biomedical Engineering II: Recent Developments, pp. 330-333, Hall, C.W., ed., New York: Pergamon Press. 29. Vossoughi, J., Hedjazi, Z., and Borris, E S. (1993). Intimal residual stress and strain in large arteries, in Bed-Vol. 24, 1993 Bioengineering Conference ASME, pp. 434-437. 30. Woo, S. L. Y., Gomez., M. A., and Akeson, W. H. (1985). Mechanical behaviors of soft tissues: Measurements, modifications, injuries and treatment, in The Biomechanics of Trauma, pp. 107-133, Nahum, A. M., and Melvin, J., eds., Norwalk: Appleton Crofts. 31. Xie, J., Zhou,J., and Fung, Y. C. (1995). Bending of blood vessel wall: Stress-strain laws of the intima-media and adventitia layers. J. Biomech. Engr. 117: 136-145. 32. Yu, Q., Zhou, J., and Fung, Y. C. (1993). Neutral axis location in bending and Young's modulus of different layers of arterial wall. Am. J. Physiol. 265: H52-H60.
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CHAPTER
Geomaterials
11
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Introduction Geomaterials
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to
JEAN LEMAITRE Universit~ Paris 6, LMT-Cachan, 61 avenue du PrEsidentWilson, 94235 Cachan Cedex, France
Geomaterials are "natural" materials such as soils, rocks, sand, clay, salt, etc., which are used without any technological transformation. A specific chapter is devoted to them because their behavior simultaneously involves many characteristics which do not follow the elementary classification: pressure dependency, dilatancy, time effects, internal friction, water saturation or not, fabric dependency, etc., and also because there is a specific community devoted to geomaterials. The method of modeling does not differ from that for other materials, but specific variables are introduced to represent specific effects and the identification of material parameters often needs "in situ" tests and three axial tests. A general background is given in Section 11.2 with special emphasis on the use of two potentials for two plastic mechanisms and an incrementally nonlinear model. Section 11.3 shows how to identify nonassociated elastic viscoplastic constitutive equations taking into account closing and opening of microcracks. Micro- and mesomechanical considerations are described in Section 11.4 for granular materials in which elastoplastic behavior induces shear banding and failure. Micromechanics allows for modeling the effects of friction and induced anisotropy which are beyond the classical normality rule (Section 11.5). The general framework of linear poroelasticity may be found in Section 11.6, and nonlinear poroelasticity is described in Section 11.7 for liquid nonsaturated porous materials. Elastoplasticity for partially saturated soils is discussed in Section 11.8, taking into account the capillary effect of suction. Strain-hardening models for sand, clay, and rocks are developed in Section 11.9. Extension to dynamic behavior, including liquefaction, is given in Section 11.10. Bounding surface of plasticity may be established for sandy soils which are either drained or not, in monotonic and cyclic loading conditions (Section 11.11). Finally, Section 11.12 discusses a lattice model; i.e., a network of linear elements such as springs, trusses, or beams captures the main properties of fracture of concrete with a relatively small number of parameters.
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1075
SECTION
11.2
Background of the Behavior of Geomaterials FI~LIX DARVE I'INP Grenoble, L3S-BP 53 38041 Grenoble, France
Contents 11.2.1 General Background . . . . . . . . . . . . . . . . . . . . . 11.2.2 Incrementally Nonlinear Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Identification Of These Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1
1076 1079 1082 1083 1083
GENERAL BACKGROUND
Specific features of the behavior of geomaterials are as follows: 9 pressure dependency, because of a failure criterion of the MohrCoulomb type, which is, roughly speaking, a generalization of the solid friction law for continuous media; 9 nonassociated plastic strains, because the normality condition to a Mohr-Coulomb yield surface would imply too large a dilatancy under shear stress; 9 incrementally nonlinear properties, because the experiments have not allowed us to exhibit finite domains of linearity for the relation between d (or do) and ~ (or de). Here lie the basic reasons why nonlinear elasticity, associated elastoplasticity, and nonassociated elastoplasticity with a single plastic potential are today generally considered restricted frameworks for developing realistic constitutive models for geomaterials. Indeed, many constitutive relations for soils and rocks are incorporating two plastic potentials (usually one is linked to the 1076
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1077
11.2 Background of the Behavior of Geomaterials
compressibility properties and the second to those of shearing), and few are thoroughly incrementally nonlinear. Thus we propose first a global overview of the widely used constitutive relations for geomaterials and then a schematic presentation of one incrementally nonlinear model. The general formalism of incremental constitutive relations is given by Fh (de,, do', dt) = 0
where F is a tensorial function which depends on the material and on its stress-strain history by means of state variables and memory parameters h. For rate-independent materials, we can consider: de = Gh(da) or da = G~-I(de)
where G and G -1 are homogeneous functions of degree 1, because of the rateindependency condition. According to Euler's identity, de, - cg,dacda -- Mh
do"
or
d o --- O(de) de, - Ph
de,
where the elastoplastic tensors M and P are homogeneous functions of degree 0 with respect to dcr and de, respectively, and where
Let us note: u = a, /la l and v = d /Id l
We call now "tensorial zone Z" any domain of incremental linearity in dcr space (or, equivalently, in de space) [ 1]: Vu E Z : M ( u ) =_ M z
While any elastic constitutive relation is characterized by a unique tensorial zone (these relations are so-called incrementally linear), conventional elastoplasticity with one unique plastic potential has two tensorial zones (the so-called loading domain and unloading domain). The behavior of geomaterials is often described through four tensorial zones, which correspond to two plastic potentials and two loading-unloading criteria: 9 plastic mechanism 1 and related loading-unloading rule 1: cgf l 9dcr(O or)0
0or
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Darve
9 plastic mechanism 2 and related loading-unloading rule 2: cga
: da(O or)O
The four tensorial zones are defined by the intersections of both hyperplanes (in da space) whose equations are given by cgfl"da-O cga
and c9f2" -~a
da - 0
On these hyperplanes a continuity condition must be fulfilled by the elastoplastic tensors linked to the adjacent tensorial zones, as, for example, Of 1. Vdasuch t h a t ' - ~ a- d a - O
Of2. a n d - ~ a- da > 0
(M ++ - M -+)da =_ 0 where M § is associated with the tensorial zone defined by c9f1" da > 0 and c9f2" cga -~a d a > 0
and M -+ to the one defined by
fl
cgcr :
~f2. da
da < O a n d ~
> O
The introduction of such two plastic mechanisms is an attempt to describe the directional variation of M (or equivalently P) with respect to dff (respectively, d~), which is only roughly taken into account in conventional elastoplasticity with one elastic tensor Me for "unloading conditions" and one elastoplastic tensor M ep for "loading conditions." If now we continue to increase the number of tensorial zones from one in elasticity, two in conventional elastoplasticity, four with two plastic potentials, etc., we obtain for an infinite number of tensorial zones the case of the incrementally nonlinear constitutive relations where M (or P) is varying in a continuous manner with da (or, respectively, &). This class of incrementally nonlinear models includes as particular cases endochronic relations [2] and hypoplasticity [3-5] We present in the next paragraph an incrementally nonlinear relation of second order [6]. Another point of interest in modeling geomaterial behavior is the question of hardening parameters. Here also the specific complexity of the irreversible behavior of geomaterials has led us to try to generalize the usual isotropic and kinematic hardening variables by taking into account a rotational hardening (for example, Lade [7]). This rotational hardening allows us to describe, as an
1079
11.2 Background of the Behavior of Geomaterials
example, the rotation of the yield surface in (q, p) plane when the consolidation process is not isotropic but of the oedometric type. Finally the third point which must be briefly evoked here is the modeling of the viscous behavior, which is quite an important phenomenon for clays and which cannot be negligible for rocks in certain situations. There are two possible ways to take into account viscous strains by considering two different decompositions of the incremental strains: (1) : de = de ep 4- de v, where there is a distinction between the elastoplastic, instantaneous strains (de ep) and the viscous, delayed strains (deV); or (2) : de = de e 4- de vp, where the distinction is made between the elastic, reversible strains (de e) and the viscoplastic, irreversible strains (devp). For this second class of models the viscoplastic strains are usually treated by a viscoplastic potential as proposed by Perzyna
[8].
11.2.2 INCREMENTALLY NONLINEAR CONSTITUTIVE RELATIONS Returning to the relations: de = Mh (u)da or da = Ph (v)de
and considering the six-dimensional associated spaces, we develop the components of M or P in polynomial series expansions as follows: 1 2 3 M~ -- M~ 4- M~yu~ 4- M~6uyu6 4-... 2 3 p ~ _ p 1 4- P~,,v~,4-P~,,,~v~,v~ 4-...
Finally, 1
1
2
de~ - M ~ d a ~ + -[ydiM~da~da? 4- . . . 1 1 2 da~ - P~de,~ +-(~lP~?de,~de,~ +...
The first terms of the right-hand parts of these equations correspond to hypoelastic relations, and both the first terms are the general expression of incrementally nonlinear models of second order. In order to determine the expression of M 1, M 2, p1, and P2, one assumes three supplementary hypotheses: 9 The incremental relations are orthotropic. 9 There is not any crossed terms: 2 - 0 a n d P 2~ - 0 . Vfl ~ 7 9M~/~
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Darve
9 The shear behavior is incrementally linear:
2 = 0andP~& 2 = 0 V~ _> 4orV7 _> 4 9M~& In orthotropy axes the first model (so-called direct) and the second model (so- called dual) are thus taking the following expressions. For the direct model:
1
do'11 1 dt~111 = A &22 + i ~B dg22 dg33 d033
I I
(d~11)2
(d~22) 2
(d0-33)2
&12 = &r12/2G3
dg23 = do'23/2G1 d~;31 = dcr31/2G2 and for the dual model: t'
d~22
1
(&11)2
dt~11 1 -- C de22 -[- ]-~[D (d~22)2
I
do'33
dg33
i~.,-,i
(dg33) 2
d0"12 = 2G3 de12 d0"23 = 2G1 dg23 d0"31 : 2G2 de31
A, B, C, D are 3 x 3 matrices which depend on state variables and memory parameters, and G1, 62, G3 are the shear moduli. In order to determine A and B (direct model), and C and D (dual model), we proceed to an identification, respectively, for the direct model with "generalized triaxial paths" and for the dual model with "generalized oedometric paths." "Generalized triaxial paths" refer to triaxial compressions (index"+ ", d~i ~ 0) or extensions ( i n d e x " - " , do'i < 0)with two constant and distinct lateral stresses (denoted by crj and ~rk). Let us notice for the direct model:
(cgai'~
andV~ = - \c3eiJ
aj,a~
"Generalized eodometric paths" refer to oedometric compressions (index" + ", dei > O) or extensions ( i n d e x " - " , & i <0) with two constant and distinct lateral strains (denoted by ~j and ek).
11.2 Backgroundof the Behaviorof Geomaterials
1081
Let us notice for the dual model:
(0 ,3 a n d K ~ - \00"i) ~j ~gk
Finally, let us consider
V~+ <+
1 N + --
V12+
v~ +
_ e~-
V~+ e~
1
E~ v3+ E~
and N - i n the same way 1
and for the dual model:
-[ 01+ K~+02 + K3~+0; 2+ - [,,-2+n+ ..~ ..~ o~ K~+o + K~+O; K{+O~ O;
a n d S - i n the same way
It is now possible to prove that A - I(N+ + N-)and B -- I(N+ - N-) C=~
~++~-
andD-~
-~-
Besides these thoroughly incrementally nonlinear models, it is also possible to consider incrementally piecewise linear relations which are based on eight tensorial zones in do" space (or, respectively, in de space) and which have the same tensorial structure as an elastoplastic model with three plastic potentials. These so-called "octolinear" models are given by
Id111 Id111 id111 de22 -- A do.22 de33 do33
q- B Ido.221
[do'33[
and for the dual model by
I
do.111 do.22 - C do.33
111 d
de111 de22 + D 1&221 de33
]&331
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Darve
11.2.3 IDENTIFICATION OF THESE CONSTITUTIVE RELATIONS As we have just seen, the identification of the direct model is performed on triaxial paths and for the dual model on oedometric paths. We have developed sophisticated expressions of Ei and Vi (and, respectively, Oi and Ki) [9], and but also very simple expressions that are particularly useful for inverse analyses in geomechanics. We are only presenting here these last simple formulations [10] for the direct model: 9 The plastic limit surface is given by a Mohr-Coulomb criterion which is characterized by cohesion C and friction angle 9 Inside the limit surface, the tangent Young's moduli are equal to
Ei - -
EoV~oo
(p, mean pressure and Po reference pressure) for sands and to clays. The tangent Poisson's ratios ~ are equal to: Vo.
Ei -Eo p
for
9 On the limit surface, the moduli are taken as arbitrarily small and the Poisson's ratios are deduced from the dilatancy angle: 9 After a stress reversal the tangent moduli are mutiplied by k. For the dual model: 9 The plastic limit surface is of the Mohr-Coulomb type (C, cp) as for the direct model.- Inside the limit surface, the tangent oedometric moduli are equal to Oi - rToeao ~/-Pfor sands "-'0
Vv0
and to Eoedo
0
pforclays
The tangent lateral pressure coefficients K/j are equal to ko for oedometric compression and to ka for extension. 9 On the limit surface, the tangent oedometric moduli are divided by 0~ and the tangent lateral pressure coefficients are equal to the plastic passive pressure coefficient. 9 After a stress reversal, the tangent moduli are multiplied by k.
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11.2 Background of the Behavior of Geomaterials
11.2.4
PARAMETERS
Table 11.2.1 and Table 11.2.2. present typical values of the parameters which have been defined in 11.2.3 TABLE 11.2.1
Table of parameters for the Direct Model
Materials
C(MPa)
~o(~
Eo(MPa)
Vo
~(~
k
Sand Clay
0 0 --~ 0.3
30 ~ 40 20 ~ 35
50 --~ 120 10 --~ 80
0.25 0.30
-3
~ 10 - 5 --~ 15
3 ---~6 3 ---~6
TABLE 11.2.2
Table of parameters for the Dual Model
Materials
C(MPa)
q,(o)
E~ed(MPa)
ko
kd
~
k
Sand Clay
0 0 --~ 0.3
30 --~ 40 20 --~ 35
60 --~ 150 20 --~ 100
0.3 ---, 0.6 0.4 --~ 0.7
0.1 --~ 0.4 0.2 ~ 0.5
3 --~ 12 2~ 7
3 --~ 6 3~ 6
REFERENCES 1. Darve, E (1990). The expression of rheological laws in incremental form and the main classes of constitutive equations, in Geomaterials: Constitutive Equations and Modelling, pp. 123-148. Darve, E, ed., Elsevier applied Science. 2. Valanis, K.C. (1971). A theory of viscoplasticity without a yield surface. Arch. Mech. 23: 517-551. 3. Dafalias, Y. E (1986). Bounding surface plasticity: I. Mathematical foundation and hypoplasticity. J. Eng. Mech. 112(9): 966-987. 4. Kolymbas, D. (1984). A constitutive law of the rate-type for soils. Position calibration and prediction, in Constitutive Relations for Soils, pp. 419-437, Gudehus G., Darve, E, and Vardoulakis, I., eds., Balkema. 5. Chambon, R., Desrues, J., Hammad, W., and Charlier, R. (1994). Cloe a new rate-type constitutive model for geomaterials. Int. J. Num. Anal. Geom. 18(4): 253-278. 6. Darve, F (1990). Incrementally non-linear constitutive relationships, in Geomaterials; Constitutive Equations and Modelling, pp. 2t3-238,. Darve, E, ed., Elsevier Applied Science 7. Lade, E, and Inel, S. (1997). Rotational kinematic hardening model for sand. Part I. Concept of rotating yield and plastic potential surfaces. Comp. Geotech. 21(4): 183-216. 8. Perzyna, P. (1963). The constitutive equations for work-hardening and rate-sensitive plastic materials. Proc. Vibrational Problems 4(3): 281-290. 9. Darve, E, and Dendani, H. (1988). An incrementally non-linear constitutive relation and its predictions, in Constitutive Equations for Granular Soils, pp. 237-254, Saada, A.S., and Bianchini, G., eds., Balkema. 10. Darve, E, and Pal, O. (1998). A new incrementally non-linear constitutive relation with 5 material constants, in Computer Methods and Advances in Geomechanics, pp. 2445-2454, Yuan, ed., Balkema.
SECTION
11.3
Models for Compressible and/or Dilatant Geomaterials N.D. CRISTESCU 231 Aerospace Building, University of Florida, Gainesville, Florida
Contents 11.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Description of the model . . . . . . . . . . . . . . . . 11.3.4 Identification of Material Parameters ..... 11.3.5 How to Use the model . . . . . . . . . . . . . . . . . . . 11.3.6 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1
1084 1085 1087 1088 1089 1090 1092
VALIDITY
The model to be used for geomaterials must describe the main mechanical properties exhibited by such materials, such as instantaneous response, time effects (creep, relaxation, rate effects), and irreversible volumetric deformation (compressibility and/or dilatancy). The main mechanisms governing the deformability of Geomaterials are closing and opening and microcracks. This article shows what kind of testing is needed to formulate a nonassociated elastic-viscoplastic constitutive equation for such materials. One starts by defining the compressibility/dilatancy boundary and that of instantaneous failure, both to be incorporated into the constitutive equation. Then triaxial tests are used to determine the energy of deformation, the yield function, and finally the viscoplastic potential. The elastic parameters are determined from small unloading/reloading cycles which follow a short period of creep. The models can describe transient and steady-state creep, irreversible damage developed in time or irreversible compaction in time, convergence of the walls in underground excavations, landslides, etc.
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Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1085
11.3 Models for Compressible and/or Dilatant Geomaterials
The models can be used for any kind of rock or geomaterial to describe the deformation in time (creep) and irreversible volumetric changes, i.e., compressibility and/or dilatancy. Also, the m o d e l can be used to describe c o m p a c t i o n and/or creep flow of p o w d e r like materials in bulk, such as ceramic powders, pharmaceuticals, food, etc.
11.3.2 FORMULATION The model can be formulated using the data obtained in triaxial testing of geomaterials. For instance Figure 11.3.1 shows the volumetric behavior of rock salt in true triaxial tests, where cr is the m e a n stress and 9 is the octahedral shear stress. The locus in the cry-plane where one is passing from compressibility to dilatancy is the compressibility/dilatancy boundary. For rock salt it is
x(~ , ~) . . . . ~
O'.
+fl
~
+f2--
(1)
G.
where fl and f2 are material constants and or, is the unit stress. In reality this b o u n d a r y is a strip of incompressibility which, for convenience, is a p p r o x i m a t e d by a curve (Fig. 11.3.2). The u p p e r curve s h o w n in Figure
,r
25
.......
i ..........................
I ..............
i .................
i ........................
ROCK SALT
[MPa]
20 § _
~ /
"~ " -
"I"-
~ =40 MPa
-..,.....~..-.~:~, "~,
_
;." # , ,"~,§ r /§ I _.4 .§ § § I ..~-.... ~ t .... 0,05 0.I 0.15 02
~v [%1
FIGURE 11.3.1 Dependency of volumetric strain on octahedral shear stress for rock salt in true triaxial tests.
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Cristescu
FIGURE 11.3.2 incompressibility.
Domains of compressibility and dilatancy, separated by a strip of
11.3.2 is the short-term failure surface, which for rock salt is Y(cr, r) 9-- - r ~ -1:- s ft.
"c
q - ~cr
(2)
G.
where r and s are material constants. The model is based on the following assumptions: the material is homogeneous and isotropic; the material displacements and rotations are small; the geomaterial exhibits instantaneous response (i.e., the two extended body elastic (seismic) waves can propagate in that material); the yield stress can be close to zero; the material exhibits time effects (transient and steadystate creep as well as loading rate effect) for any stress state; and the constitutive domain is bounded by the short-term failure surface. With these assumptions the constitutive equation is of the form
D - 2----G-+- 3K
2G r
+ k~ 1
H (T ) ~
+ ks - ~
(3)
where D is the strain rate tensor, T is the stress tensor, 1 is the unit stress tensor, H(T) is the yield function, F(T) is the viscoplastic potential for transient creep, and S(T) is the potential for the steady-state creep. All these functions depend on stress invariants. Further, k~- and ks are two viscosity coefficients, and K and G are the elastic parameters (generally variable) which
1087
11.3 Models for Compressible and/or Dilatant Geomaterials
are defining the "instantaneous response" of the geomaterial.
/o t ~r~,Ivdt + /o t T "
W(t) =
D'Idt
(4)
is the irreversible stress work per unit volume, with "prime" standing for deviator and
H(r) = W(t)
(5)
is the stabilization boundary for transient creep. The term OF/OT can be replaced by an irreversible strain rate orientation tensor N(T), which is easier to formulate from tests.
11.3.3
DESCRIPTION
OF THE MODEL
The yield function is determined by performing triaxial tests from which W(t) is obtained following various loading paths, and thus the boundary (Eq. 5) is determined. The functions F (or N ) and S are determined starting from the expression for the irreversible volumetric strain rate deformation
9
;
,@ -- kr 1
)
w(t) or
u(r)
N +
os ks-~
( kr{ or
1
w(t)
H(I')
)
t,-(N)+ k~N
(6)
where the right-hand-side derivatives are with respect to the mean stress ~. These functions must satisfy the inequalities
OF OS Oa > 0, ~ - 0, in the compressibility domain OF
OS
OF
OS
0--a = 0, ~
- 0, in the C/D boundary strip
(7)
0---~< 0, ~--~a< 0, in the dilatancy domain This is obtained by incorporating both expressions X and Y (see (Eqs. 1 and 2) into the functions F and S. For instance, for F one can use
k Or
o~ =
x(o-, ,~),-I-'(o-) ~-(g7 ~5
[e~(e)+ ~]
(8)
where tp(a) is determined from the hydrostatic tests and G(F) from the deviatoric part of the triaxial tests (in many cases G can be disregarded). Here is the equivalent stress r - ( 2 / 3 ) # - [(2/3)Iir,] 1/2 and IIr, is the second invariant of the stress deviator. With this procedure dilatancy and/or compressibility, as well as short-time failure, is described exactly by the constitutive equation.
1088
Cristescu
11.3.4 IDENTIFICATION PARAMETERS
OF MATERIAL
First to be determined are the elastic parameters K and G. The procedure is described in this volume in "Elasticity of Porous Materials," Section 2.3. The yield function H is determined in the two stages of triaxial tests: hydrostatic and deviatoric; thus H(a, F) := HH(a) 4- HD(a, F), where generally
:=
ho sin c o ~ +
+ hl
i f a ~ ao
ho + hi
(9)
if a _> ao
where h0, hi, co are material constants, a , -- 1 (unit stress), and a0 is the pressure closing all pores and microcracks. Further, HD is a polynomial in with coefficients depending on mean stress. For rock salt: HD(a,z)'=-A(o')
z
+B(a)
z
+C(o')
z
(10)
The viscoplastic potential F is determined by starting from Eq. 8, integrating with respect to a, and then differentiating with respect to F (or z), and using the formula /tO-a-- 3
1
(11)
~(~)/
and the triaxial data for the irreversible strain rate components. The potential S is determined starting from volumetric deformation in long-term creep tests performed for various stress states. From these tests follow
i~Xvls __ k s O_~ . _
b
if X < 0
"c
(12)
if X > 0
0
from where follows for rock salt in the dilatancy domain OS bm ks--~z = n-+ 1
z
a
Here b, m, n, and p are material constants.
p + ~0",
z
5
1 0".
(13)
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11.3 Models for Compressible and/or Dilatant Geomaterials
11.3.5 H O W TO USE THE MODEL Two boundary value problems can be described: (a) creep under constant stresses, and (b) a general variation of both stresses and strains in long-term intervals. In the first case, an "instantaneous" loading results in a stress and strain distribution at time to (the elastic solution), and afterwards the stresses are held constant (or nearly constant) and the variation of the strains are obtained from
1 Wx(to)) OF H cgT~j( 1
-
exp
O r , n r,n.[kT(to --
(14)
Tkl
H cgThz c9S + --6-~-~ks(t -- to) olij
where e~ are the elastic strains. This formula can be used for creep tests, gravitational compaction, and any deformation by creep when stresses are nearly constant. For instance, for a radial convergence u of a vertical borehole of initial radius a at time to, we have
u
p--6h
a
a
2G
r
ar
_21-_
I1 W~-(to)) OF 1 oFH ~~ {1 exp I~ cgF w
1}
~1~c9 S (15)
ah
where p is the internal pressure, is the far-field stress and r is the cylindrical coordinate. For the second kind of problem, when, because of the change of geometry, or for other reasons, the stresses are also varying during creep, one has to introduce the constitutive equation in a code (see examples in Cristescu [1] and Cristescu and Hunsche [4]). Another example is compaction of a geomaterial (generally particulate material) under its own weight. The volumetric deformation describing
1090
Cristescu
compaction in time, as obtained from
I1 eZv(t) _
11.3.6
W(to)\OF H(T) I 0 o
(16)
PARAMETERS
For rock salt (tests performed by Hunsche), a2
C1
A(cr) "-- al -]- ------~, B(cr) "-- bl cr q- b2, C(~r) "~r al - - 7 •
-21MPa, a 2 -
+ c2 +c3
6 . 7 3 x 1 0 -12 MPa, bl - 1 . 5 7 x 1 0 - 6 MPa, b 2 -
1.7•
10 -5 MPa, Cl - 26.12 MPa, c2 -- - 0 . 0 0 1 5 9 MPa, c3 -- 3134, and ft. = 1 MPa. The viscoplastic potential for rock salt is krF(~, z) -= or.
[Y(~' z)]4+ - 5 flplZ(Z) +
+ 3f~p~[Z(r)] -~(f2p~ +f~p2)Z(~) +-~
-
3
p2 +f~p3 ---~p~
p3 - ~-~-p2 z(z) - ~-~fp3 In Y(o, z)+
p3 - ~fp2
[Y(cr, ~)]2
~
(G(z) + 1) + g(z)
with "C
~
G(z) " - U l - - + O'.
.-
"C
+u3
"C
+u4
-
0".
with Ul - 0 . 0 3 6 ,
u2
Y(o,
iT.
u2 - - 0 . 0 0 2 6 5 , u3 - 5.256x 10 -5, and u4 - 1.57x 10 -12.
11.3 Models for Compressible and/or Dilatant Geomaterials
1091
FIGURE 11.3.3 Yield surfaces (dotted lines) and viscoplastic potential surfaces (interrupted lines) for saturated sand. In the domain cgF/cga > 0, c9H/c9~< 0 the sand is instable.
The potential S for rock salt is given by Eqs. 12 and 13 with - l x l 0 -14 s -1 , m - 5, n - - 0 . 1 , p - 3 x 10 -13 MPa S-1. For other geomaterials, such as dry or saturated sand, granite, andesite, coal or cement concrete, see Cristescu [1] and Cristescu and Hunsche [4], and for ceramic powder see O. Cazacu et al. [2], and Cristescu et al. [3]. As another example, in Figure 11.3.3 is shown the yield surface (dotted lines) for saturated sand, with ~7 H(~, e ) ' a , _#/3) ~+be+c~ b
-
and a - 4.8• 10 -7 (kPa) -1, b - 0.0013. The viscoplastic potential surfaces are shown as dotted lines with OF ( - 8 + 2fcr) v/F 07 = hi (2f + 0~)cr- (1 + 0~/3)e
1092
Cristescu
where the expression at the denominator is just that involved in the shortterm failure and - F + 2 f a - 0 is the equation of compressibility/dilatancy boundary (~ = 1.34, f = 0.56). Sand is dilatant when c3F/c3a< 0 , compressible when c3F/c3a > 0, and unstable if c3F/c%r > 0, c3H/c3a< O.
REFERENCES 1. Cristescu, N. (1989). Rock Rheology, Kluwer Academic. 2. Cazacu, O., Jin, J., and Cristescu, N.D. (1997). A new constitutive model for alumina powder compaction, KONA. Powder and Particle, 15: 103-112. 3. Cristescu, N.D., Cazacu, O., and Jin, J. (1997). Constitutive equation for compaction of ceramic powders, in IUTAM Symposium on Mechanics of Granular and Porous Materials, pp. 117-128, Fleck, Norman, ed., Kluwer Academic. 4. Cristescu, N.D., and Hunsche, U. (1998). Time Effects in Rock Mechanics, Chichester-New YorkWeinheim-Brisbane-Singapore-Toronto: John Wiley and Sons.
SECTION
11.4
Behavior of Granular Materials IOANNIS VARDOULAKIS
National Technical University of Athens, Greece
Contents 11.4.1 Micro- and Macromechanical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 11.4.2 Elastoplastic Behavior of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 11.4.3 Shear-Banding and Postfailure Behavior .. 1102 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
11.4.1
MICRO- AND MACROMECHANICAL
CONSIDERATIONS Granular materials are random assemblies of small grains in very large numbers. Here situations are considered where the grains are always in contact with their neighbors. In a first approximation the grains are assumed to be spherical and granular materials are classified on the basis of their grain size distribution curve and its statistical moments, like the mean grain diameter. The geometric fabric of a granular medium at a given configuration in space is the three-dimensional truss with its nodes located at grain centers [ 1 ] (Fig. 11.4.1). The property which differentiates granular materials from other solids, like, e.g., metals, is their pronounced pressure sensitivity, which is attributed to the existence of internal friction and which is developing at grain contacts. The corresponding static fabric is a three-dimensional truss with its bars along intergranular grain contact forces. Intergranular forces must always lie inside Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1093
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Vardoulakis
FIGURE 11.4.1 Force and centroid's fabric.
or be tangential to the local friction cone of the corresponding grain contact. Thus, because of friction at grain contacts, the centroid's and force's lattices do not coincide. Accordingly, granular kinematics and granular statics are not linked to each other in a fashion that is amenable to simple analysis. This inherent difficulty of the micromechanical description of granular media with internal friction explains why until now phenomenological as opposed to micromechanical models seem to be more useful for engineering applications. However, in the last 20 years considerable progress has been made in computer-aided micromechanical simulations of granular assemblies [2-4]. Granular materials are porous media. From a continuum mechanics point of view, their porous or granular structure is described by macroscopic properties of the granular assembly such as the porosity n and bulk density p = (1 - n ) p s, where Ps is the density of the grain. Grain displacements are averaged over a representative elementary volume (REV), and the macroscopic velocity vi and the rate of deformation Dij fields can be computed [5];
Dij -- -~ \cgxj +
c~xi}
(1)
Since grains are practically incompressible (Ps = const.) and mass convection is in most cases negligible, mass balance yields that volume changes are
1095
11.4 Behavior of Granular Materials
simply due to changes in porosity o Dkk -- -~-~ln (1 - n)
(2)
The intergranular forces, ti, acting at grain contacts are linked to a local Cauchy stress tensor ai), which is assumed to fulfill equilibrium with negligible body forces. Through this local stress tensor a macroscopic aij stress can be computed as a volume average assigned to the center of a considered REV: = aq
1L
V~v
#ij d V = v)
xitj dS VRV.V
(3)
V~v)
In the case of a granular medium which is fully saturated with a pore fluid of low viscosity (e.g., water), an 'effective' stress is defined as a measure of the intergranular forces. We notice that in granular media the mixtures theory decomposition of the total stress in partial stresses (as is the case in gases) is not justified experimentally. Experimental evidence [6] and micromechanical considerations [7] support the idea of the total stress decomposition according to Terzaghi's [8] (empirical) effective stress principle into an effective stress and into a pore-water pressure (Tij -- (Ttij - pwC~ij
(Pw > 0) 1
(4)
The macroscopic (effective) stress tensor is decomposed in deviatoric and isotropic part, 1 a ~j -
s ~j + -~ a k k ~3~j
(5)
For a spherical REV and statistically uniform contact-normal distribution, it can be shown [9] that the mean normal traction p over all intergranular contacts is given by the first invariant of the stress tensor, I l a = akk, and the m e a n shear traction rm or the second invariant of the deviator, 3" =
=
p-
I ~ / 3 ; z~ -
T-
J2~
(6)
A simple micromechanical interpretation of the third stress invariant is not known [9]. However, the most lucid selection of a third stress invariant follows if one considers the ratio of m a x i m u m to mean shear stress. Let, for example, a2 _< a3 < al be the principal stresses for a given state of stress and "/73,max ~ [0"1 -- 0"2]/2; then one can show that this ratio is a simple function of 1Compression negative.
Vardoulakis
1096 the so-called stress invariant angle of similarity 0q:
(7)
"~3,max- ~_~Isin (n/3 4- ~s)l where cos 3oq =
3x/3 J3~ 2 ]3/2
(0 < O~s <
n/3)
(8)
d2s
11.4.2 ELASTOPLASTIC BEHAVIOR OF GRANULAR MATERIALS Typical examples of granular geomaterials are sands, silts, and psammitic rocks (sandstones). For slow deformation processes, granular materials exhibit predominantly nonlinear stress-strain behavior upon loading as well as upon unloading and reloading. Deformations are mostly irreversible and practically strain-rate-insensitive. Granular geomaterials are good examples of deformation-rate-insensitive plastic solids. The behavior of granular materials is strongly history-dependent. Thus finite constitutive laws are excluded, and constitutive equations can only be written in the form of evolution-type equations that relate an objective (effective) stress rate dij to the rate of deformation tensor Dij. The response function depends strongly on the direction of the strain rate, resulting generally in nonlinear rate-type equations. The assumption of rate independence restricts dij to being a first-degree homogeneous function of Dij. As an example, we refer here to Chambon et al. [10], who suggested recently the following simple incrementally nonlinear model: #ij ~- Tij(o'ij, Dij, . . .) -~- AijI[DI~I[I4- BijklDkl 4-""
(9)
In the frame of classical elastoplasticity theory, we distinguish roughly among two such "directions," namely, between loading and unloading. Accordingly, the rate of deformation tensor is decomposed into an elastic (reversible) and a plastic (irreversible) part,
Dij = D~ + ~
(10)
Upon virgin loading the response is relatively soft, since both elastic and plastic strains are generated. Elastic strains are relatively small, and they reflect reversible, nonlinear deformations of grain contacts. The elastic stiffness that relates the stress rate to the elastic deformation rate,
1097
11.4 Behavior of Granular Materials
is stress-dependent, and accordingly it is derived from a suitably constructed elastic strain energy function [ 11, 12]. In most cases plastic deformations dominate, since they reflect irreversible grain rearrangements, which in turn include grain contact detachment and intergranular slip. The state of stress of a given granular material point (REV) during plastic yielding is restricted by the so-called plastic yield condition, which has the form
F(aij, ~F, . . .) - 0
(12)
The parameter ~P in the argument list of E and possibly other scalar- and tensor-valued variables, is used to describe, to some degree of approximation, the history of plastic deformation, as this is reflected in the evolving force fabric. The yield condition of granular materials is expressing the constraint that is imposed by internal friction. Macroscopically, this results in pressuresensitive behavior, which is expressed by linking the shear stress intensity T to the mean (effective) normal stress though a constraint of the form
F(crij, ~ )
-
-
T - fc(q - P) -- 0
(13)
where f c O F , . . . ) is the macroscopic or Coulomb friction coefficient. The yield condition is visualized in stress space by the corresponding yield surface Flw_const. = 0 (Fig. 11.4.2). With the concept of yield surface one can generalize the one-dimensional concepts of loading and unloading to more complicated stress paths. This is done by assuming a) an elastic domain "inside" the yield surface and b) that plastic strains are generated only if the stress increment points in the direction of growth of the yield surface.
FIGURE 11.4.2 Coulomb-typeyield surface and dilatancy flow rule in (T, p) stress subspace.
1098
Vardoulakis triaxial extension
st
~crio,xial
ompresslon
t O~
s3
o 2 ~
(a)
,
fin
(b)
FIGURE 11.4.3 a. Deviatoric trace of the Mohr-Coulomb yield surface in principal stress space. b. Mohr circle of stresses with linear yield envelope and the definition of ~0m and c. In granular materials the response under continued loading is practically independent of the intermediate principal stress, as is shown in Figure 11.4.3. Mathematically this is expressed by the so-called Mohr-Coulomb yield condition [9, 13], with sin (tim fc
sin (a~ + n/3) + ~
1 cos (a~ + n/3) sinq~m
(14)
In Eq. 14 ~Pm is called the mobilised Mohr-Coulomb angle of internal friction, and reflects an average value of the internal friction, which at any instant of the deformation process is mobilised at grain contacts inside the REV. The parameter q is related to the cohesion c of the geomaterial, c = q tan 4~m. Both mobilised friction and cohesion are assumed to be functions of the plastic history parameter, sin ~bm = #(RJ), c = c(W). Besides internal friction, granular media are distinguished from other plastic solids by the phenomenon of dilatancy. Dilatancy means plastic volume increase under shear and is observed in relatively dense packings. Loose packings undergo instead plastic volume decrease during shear (contractancy). Dilatancy or contractancy are measured by the plastic volumetric strain-rate 9P = I~hh and are attributed to grain rearrangements due to grain slip, which in turn is measured by the second invariant of the deviator of the rate of deformation tensor, gP =,/2D{P.D'Y. (D~i, = iePaii/3 + D'i~). Since elastic deformations are subordinate to plastic ones, Eq. 2 tells us that dilatancy or
1099
11.4 Behavior of Granular Materials
contractancy will be directly evident in the experiment through visible (detectable) changes in porosity [14]. Dilatancy is an internal constraint for the plastic deformation of the form
dvP
vP -- dRg p =~ -@gp - d R ( V )
(15)
The parameter dR in this relation is called the Reynolds [15] dilatancy coefficient. The dilatancy constraint is depicted in Figure 11.4.2 as the corresponding vector of the plastic deformation rate, attached to the current stress state. By selecting an appropriate plastic potential function Q = Q(O.ij ' trff), the dilatancy constraint, Eq. 15, follows from the so-called
plastic flow rule - cv ~
Oa ~
(16)
Introducing the mobilized dilatancy angle ~m (Fig. 11.4.4), and in analogy to Eqs. 13 and 14, we have
Q(crij, W) - T + dRp + const. dR
(17)
sin ~m 1 sin (~s 4- rt/3) + ~ c o s (~s + rt/3)sin ffm
P 12
FIGURE 11.4.4 Mohr-circle of plastic strain rates and the definition of ~m-
(18)
1100
Vardoulakis
The phenomenological parameters, r and ~Jm, in the definitions of the yield function F and plastic potential Q, are assumed to describe the essential future of the structure of the force- and centroid's fabrics, respectively. As already mentioned in Section 11.4.1, these two fabrics do not coincide, and the corresponding plasticity theory must require distinct functions that specify the yield condition and the flow rule (F ~ Q). The way to relate plastic yielding to plastic flow in granular materials is to consider the fact that plastic work, r p = r D~ij
(19)
should after all express the work done by intergranular forces during grain slip. Let fc be the "true" interparticle friction coefficient; then the previous requirement is fulfilled if dR = fc + fc
(20)
This is Taylor's friction-dilatancy rule [16-18] and is interpreted as follows: From the point of view of energy dissipation, granular materials behave like incompressible frictional materials, with constant friction coefficient fc (Fig. 11.4.5).
FIGURE 11.4.5 Taylor's friction-dilatancy rule in simple shear conditions [16].
Equation 20 is the simplest generalization of the so-called normality condition2 of the classical (associated) flow theory of plasticity. Indeed, for ideally smooth particles f ~ - 0 , dR =fc form Eq. 20, and yield and plastic potential functions coincide. This is consistent with the micromechanics of granular media, since the condition fc = 0 also means that the granular medium is nondissipative and that the centroid's- and force fabrics coincide (cf. Fig. 11.4.1). The flow rule, (Eq. 16) and the corresponding flow theory of plasticity are called nonassociated, because no further restriction is imposed on the 2The normality condition of the flow theory of plasticity is stating that (a) stress and plastic strain rate are coaxial, and (b) in stress space representations the plastic strain rate vector is normal to the yield surface at the considered state of stress.
1101
11.4 Behavior of Granular Materials
dilatancy coefficient, except that energy dissipation is never negative:
~vP >_ 0 ~ dR < fc
(21)
Taylor's rule (Eq. 20) satisfies the previous constitutive inequality and provides a micromechanically motivated constitutive law for the dilatancy coefficient. Dense sands are dilatant, since fc > fc and dR > 0; loose sands are contractant, since fc
aij = CijhlDki
Cijhl -
Cijhl
Dkt
with:
9 Plastic stiffness tensor:
Cght=(1) cgQce cgF Cpqkl H Cgamn mnijCgap-----~q e
9 Plastic modulus: H = Ho + Ht > 0
9 Hardening (softening) modulus: OF{ > 0 9 hardening Ht - - - ~ - ~
<
0 9 softening
9 Snap-back threshold value for the softening modulus:
Ho -
> o
9 ( 9 Foeppl-Macauley brackets" 1 if 9 F -- 0 andBkl~,kl > 0 OF e 0 if " F < 0 or F = 0 and B~le,ki < 0 ;Bij = ~ Cklij
1102
Vardoulakis
ep
C0kl is a quasi-linear operator and in the case of associative plasticity is satisfying major symmetry conditions. We notice that for granular materials nonassociativity of the flow rule usually holds only for the volumetric component of the plastic strain rate. At the same time, the deviatoric normality is assumed to hold [21]; i.e.,
OF &rij
OQ Oaij
11.4.3 SHEAR-BANDING AND POSTFAILURE BEHAVIOR Otto Mohr [22] published in the year 1900 the original strength theory of cohesive-frictional or Mohr-Coulomb materials. These materials fail under shear by forming a set of conjugate slip lines. Granular materials are good examples of Mohr-Coulomb materials, since failure is manifested in these materials in the form of conjugate shear-bands (Fig. 11.4.6). A shear-band is a narrow zone of intense shear with a thickness that is a small multiple of the mean grain diameter [23]. In the past 25 years extensive work on shear-banding in granular media has been initiated by the works of
FIGURE 11.4.6 Conjugateshear-bands in perlite (volcanic soft rock in Melos island, Greece).
11.4 Behavior of Granular Materials
1103
Vardoulakis [24] and Desrues [25]; see Reference [9] for an extensive literature review. Shear localization induces intense intergranular slip and dilatancy of the material inside the localized zone [14], which is due to grain rearrangement, grain slip, and rotation [26]. Increasing porosity naturally reduces the coordination number of the granular assembly (i.e., the number of contacts per grain), yielding progressively to macroscopic material softening inside the localized zone. For equilibrium reasons the material outside the localized zone is unloading (Fig. 11.4.7). From the micromachanical point of view, an important structure that appears to dominate localized deformation is the formation and collapse (buckling) of grain columns. We notice that in order to account for these effects, higher moments concerning the grain geometry must be accounted for, such as their ellipticity, angularity, etc., which in turn lead to a basic asymmetry of shear stress and to micropolar effects [27,28]. At any rate, localization of deformation leads to a change of scale of the problem so that phenomena occurring at the scale of the grain cannot be ignored anymore in the modeling process of the macroscopic behavior of the material. Under these circumstances, it appeared necessary to resort to continuum models with a microstructure, which allow us to some degree to describe localization phenomena. These generalized continua contain additional kinematical degrees of freedom and/or consider higher deformation gradients. These observations have prompted the extension of classical continuum mechanical descriptions for granular media past the softening regime by resorting to the so-called Cosserat [9,29] or gradient models [9,30,311. In a recent paper, Zervos et al. [32] presented a new unified gradient elastoplasticity theory for cohesive/frictional, dilatant materials, where
FIGURE 11.4.7 Shear-bandmodel as a thin layer of strain-softening material embedded between elastically unloading half-spaces.
1104
Vardoulakis
gradient terms were considered in the elastic regime as well, and the stress rate equation reads as follows:
As a result, the order of the governing equations remains the same everywhere in the deforming solid throughout the loading history. The consistency condition of the flow theory of plasticity, which in this case is a differential equation, is solved analytically in an approximate fashion [9]. Therefore, only displacements need to be discretized in a finite element formulation, where a C 1 three-noded triangle with quintic interpolation for the displacement field was implemented. The ability of such a theory to model progressive
FIGURE 11.4.8 Finite-elementpostlocalization analysis of the biaxial test, using second-gradient elastoplasticity theory [31].
11.4 Behavior of Granular Materials
1105
localization of deformation is demonstrated by simulating the biaxial test on a weak sandstone. After the critical load level is reached, a shear-band is formed (Fig. 11.4.8). The material inside the band is in the softening regime and continues to deform plastically. The material outside unloads elastically. The inclination and thickness of the shear-band and the load displacement response of the sample are not affected by mesh refinement.
ACKNOWLEDGEMENTS This article is a partial result of research supported by funds of GSRT of Greece through the program PENED 99 ED 642.
REFERENCES 1. Satake, M. (1982). Fabric tensor in granular materials, in IUTAM Conf. on Deformation and Failure of Granular Materials, pp. 63-68, Delft, Balkema. 2. Cundall, P. A., and Strack, O. D. L. (1979). A discrete numerical model for granular assemblies. G~otechnique 29: 47-65. 3. Jean, M. (1995). Frictional contact in collections of rigid or deformable bodies: A numerical simulation of geomaterial motion, in Mechanics of Geomaterial Interfaces, Selvadurai, A. P. S., ed., Elsevier. 4. Emeriault, E, and Cambou, B. (1996). Micromechanical modelling of anisotropic non-linear elasticity of granular medium. Int. J. Solids Structures 33: 2591-2607. 5. Bagi, K. (1996). Stress and strain in granular assemblies. Mech. Materials 22: 165-177. 6. Bishop, A. W., and Skinner, A. E. (1977). The influence of high pore-pressure on the strength of cohesionless soils. Phil. Trans. Roy. Soc. London 284: 91-130. 7. Oka, E (1996). Validity and limits of the effective stress concept in geomechanics. Mech. Coh. Frict. Mat. 1: 219-234. 8. Terzaghi, K. V. (1936). The shearing resistance of saturated soils, in Proc. 1st ICSMFE Cambridge, pp. 54-56. vol. 1. 9. Vardoulakis, I., and Sulem, J. (1995). Bifurcation Analysis in Geomechanics, Blackie Academic and Professional. 10. Chambon, R., Desrues, J., Hammad, W., and Charlier, R. (1994). A new rate-type constitutive model for geomaterials. Int. J. Num. Anal. Meth. Geomechanics 18: 253-278. 11. Loret, B. (1985). On the choice of elastic parameters for sand. Int. J. Num. Anal. Meth. Geomechanics 9: 285-287. 12. Lade, P. V., and Nelson, B. (1987). Modelling the elastic behavior of granular materials. Int. J. Num. Anal. Meth. Geomechanics 11: 521-554. 13. Chen, W. E, and Han, D.J. (1988). Plasticity for Structural Engineers, Springer. 14. Desrues, J., Chambon, R., Mokni, M., and Mazerolle, E (1996). Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. G~otechnique 46: 529-546. 15. Reynolds, O. (1885). On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. Phil. Mag. 2(20): 469--481. Also: Truesdell, C., and Noll, W.
1106
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
28. 29. 30. 31. 32.
Yardoulakis
(1948). The Non-Linear Field Theories of Mechanics, Handbuch der Physik Band III/3, section 119, Springer. Taylor, D. W. (1948). Fundamentals of Soil Mechanics, John Wiley. Rowe, P. W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Roy. Soc. 269: 500-527. De Josselin de Jong (1976). Rowe's stress-dilatancy relation based on friction. Gr 26: 527-534. Mroz, Z. (1963). Non-associate flow laws in plasticity. Journal de M~canique 2: 21-42. Nguyen, Q. S., and Bui, H. D. (1974). Sur les mat~riaux ~lastoplastiques h ~crouissage positif ou n~gatif. Journal de M~chanique 3: 322-432. Gudehus, G. (1972). Elastic-plastic constitutive equations for dry sand. Arch. Mech. Stosowanej 24: 395-402. Mohr, O. (1900). Welche Umsthnde bedingen die Elastizithtsgrenze und den Bruch eines Materials? Zeitschrift des Vereines deutscher Ingenieure 44: 1-12. Roscoe, K. H. (1970). The influence of strains in soil mechanics. G~otechnique 20: 129-170. Vardoulakis, I. (1977). Scherfugenbildung in Sandkoerpern als Verzweigungsproblem, Doktorarbeit, Universitaet Karlsruhe. Desrues, J. (1984). La Localization de la DEformation dans les Mat~riaux Granulaires. Th&e de Doctorat et Science, USMG & INPG, Grenoble. Zervos, A., Vardoulakis, I., Jean, M., and Lerat, P. (2000). Numerical investigation of granular kinematics, in Mechanics of Cohesive-Frictional Materials (in press). Oda M. (1997). A micro-deformation model for dilatancy of granular materials, in ASCE/ ASME Symposium on Mechanics of Deformation and Flow of Particulate Materials, pp. 24-37, Chang, C.S., Misra, A., Liang, Ry., and Babic, M. eds. Bardet, J.-P., and Vardoulakis, I. (2000). The asymmetry of stress in granular media. Int. J. Solids Struct. 38: 353-367. Mfhlhaus, H.-B., and Vardoulakis, I. (1987). The thickness of shear bands in granular materials. Gr 37: 271-283. Vardoulakis, I., and Aifantis, E. C. (1991). A gradient flow theory of plasticity for granular materials. Acta Mechanica 87: 197-217. Vardoulakis, I., and Frantziskonis, G. (1992). Micro-structure in kinematic-hardening plasticity. Eur. J. Mech.~Solids 11: 467-486. Zervos, A., Papanastasiou, P., and Vardoulakis, I. (2001). A finite element displacement formulation for gradient elastoplasticity, Int. J. Numer. Meth. Engng. 50 (6): 1369-1388.
SECTION
11.5
Micromechanically Based Constitutive Model for Frictional Granular Materials SIA NEMAT-NASSER
Center of Excellencefor Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, California
Contents 11.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Back Stress and Fabric . . . . . . . . . . . . . . . . . . 11.5.3 Yield Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.4 Inelastic deformation Rate Tensor . . . . . . . . 11.5.5 Dilatancy, Friction, and Fabric . . . . . . . . . . . 11.5.6 Elasticity Relations . . . . . . . . . . . . . . . . . . . . . . 11.5.7 Rate Constitutive Relations . . . . . . . . . . . . . . 11.5.8 Material Functions . . . . . . . . . . . . . . . . . . . . . . 11.5.9 Illustrative Example . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1107 1109 1110 1110 1111 1112 1112 1114 1114 1116 1116
11.5.1 INTRODUCTION The response of frictional granules (i.e., granular masses w h i c h s u p p o r t the applied loads t h r o u g h contact friction) is strongly dominated by anisotropy or fabric w h i c h is invariably i n d u c e d u p o n shearing u n d e r confinement. C o n c o m i t a n t features of this characteristic of frictional granules are their dilatancy in m o n o t o n i c shearing and their densification u p o n s u b s e q u e n t unloading. A further remarkable p r o p e r t y of Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1107
1108
Nemat-Nasser
materials of this kind is that they can actually undergo reverse inelastic sheafing against an applied shear stress when a monotonic sheafing under confining pressure is followed by unloading [1]. The energy required for
plastic deformation against the applied shear stress is supplied by the work done by the confining pressure going through the accompanying shear-induced volumetric contraction. Experiments show that, under relatively small confining pressures, the following features are among the essential characteristics which must be captured in modeling the inelastic deformation of granular masses: 9 An initial densification (decrease in void volume), the magnitude of which decreases as the void ratio (the ratio of void volume to the volume of the solid) approaches a minimum value. 9 If the sample is dense (i.e., its void ratio is close to the corresponding minimum value), then the initial (small) densification is followed by dilatancy (increase in void volume), which continues until a critical void ratio is attained asymptotically. 9 If at a certain stage during the course of dilatancy, the shearing is reversed and the shear strain is gradually decreased to its initial zero value (completing half of a strain cycle), then there is always a net amount of densification, this amount decreasing as the void ratio approaches its minimum value. 9 If the sample is loose (i.e., its void ratio is larger than a critical value), then the sample may densify continuously until the critical void ratio is reached asymptotically. Based on micromechanics, a plasticity model which includes both isotropic and kinematic hardening has been developed [2] to describe the response of frictional granules under small confining pressures. The model includes pressure sensitivity and dilatancy, as well as the effect of induced anisotropy and the noncoaxiality of the inelastic strain rate and the stress tensors, and does capture the basic features listed previously. The basic hypothesis is that the deformation of frictional granular masses occurs through simple shearing accompanied by dilatation or densification (mesoscale), depending on the microstructure (microscale) and the loading conditions (continuum scale). The microstructure and its evolution are defined in terms of the fabric a n d its evolution, and this fabric is characterized in terms of a back stress (kinematic hardening). In this theory, the fabric affects both the inelastic and the elastic deformation, in a most profound way. An expression for the overall elasticity tensor is developed in terms of a fabric tensor.
11.5 Micromechanically Based Constitutive Model for Frictional Granular Materials
11.5.2
1109
BACK STRESS AND FABRIC
The term fabric is used to define the overall anisotropic distribution of the granules, their contact forces, the associated voids, and other microstructural parameters which are responsible for the anisotropic behavior of the granular mass. Assuming a uniform sample, here we measure the granular fabric by the distribution of the contact unit normals. Denote by n a typical contact unit normal, and let E(n) be its distribution-density function. Hence, E(n)dD represents the fractional number of contact unit normals whose directions fall within the solid angle dFl. Various aspects of this representation of the fabric of a granular mass are discussed by Mehrabadi et al. [3], Kanatani [4], and Subhash et al. [5]. Here, we use the second-order approximation of E(n), given by 1
1
E(n) - 2 n ( r - 1) (1 + E) " (n | n ) -
2 n ( r - 1) (1 + E/)ninj)
(la)
where r = 2 in two dimensions and r = 3 in three dimensions; 1 with components 5/) is the identity tensor; E is a second-order symmetric and 1 deviatoric tensor with components E/) -- 4( < ninj > --~5/)) in two dimensions, and E/)= 1 5 / 2 ( < n i n j > - 1 / 3 6 / ) ) in three dimensions, where < ... > denotes volume average; and the repeated indices are summed. The back stress is defined by
fl = x/2 pJ , J - -fiE
(lb, c)
where p is the pressure, a n d / i is an effective friction coefficient which is used as a normalizing factor. The minus sign in Eq. lc ensures that the major principal direction of fl coincides with the direction of the least density of the contact unit normals. The fabric tensor J is used as an internal variable to represent the effects of the microstructure. Tensors E, fl, and J are all deviatoric. The internal resistance to deformation in frictional granules can now be represented by a stress tensor v which consists of a hydrostatic pressure, - p l , the (deviatoric) back stress fl corresponding to the fabric tensor, and a
deviatoric tensor S representing the internal isotropic Coulomb friction, -- - p l
+ fl + S
The deviatoric tensor S is called the stress difference.
(2)
1110
Nemat-Nasser
11.5.3 YIELD CRITERION The yield condition is defined in terms of the isotropic Coulomb friction, using the stress difference tensor S, as follows: f-
1
9 - F(p, A, ~) <_ O, 9 - (1S'S)-2
(3a, b)
where A is the accumulated volumetric strain and ~ is the effective inelastic strain, defined in terms of the deviatoric part, DP, of the plastic deformation rate tensor DP, as follows: -
/o
~ dt, ~ = x/2p" DP
(3c, d)
where p is defined in the following text by Eq. 4c. As has been discussed in Nemat-Nasser and Shokooh [6], A can be related directly to the void ratio, e, and M - O F / O p is the isotropic friction coefficient. Indeed, in two dimensions, M - s i n r where qS~ is the internal friction angle [2]. With this in mind, the yield condition is written as f-'c-
pM <_ 0
(3e)
in what follows.
11.5.4 INELASTIC DEFORMATION TENSOR
RATE
Generalizing the double sliding models proposed by Spencer [7, 8], de Josselin de Jong [9], Mehrabadi and Cowin [10], and Balendran and NematNasser [11, 12], it is shown by Nemat-Nasser [2] that the inelastic deformation rate tensor, DP, can be expressed as D p -- y9~P + I/a(1 (4s) - p | p)" DI + ~yB1 1.
(4a)
or, in component form, as (45) where l(4S) is the fourth-order symmetric identity tensor; D I is the deviatoric part of the deformation rate tensor D; and S -x/~ (4c) is a unit tensor coaxial with S. In Eq. 4a,b, r / = +1 with the sign chosen such that the associated rate of work is positive; B is the dilatancy parameter which
11.5 Micromechanically Based Constitutive Model for Frictional Granular Materials
1111
relates the shear-induced volumetric strain rate to the shear strain rate; and > 0 is the noncoaxiality coefficient. In the literature, the part of the plastic deformation rate tensor which is noncoaxial with the stress difference is expressed in terms of the Jauman rate /~, which is orthogonal to p i.e., 0 /, 9 -- 0; see References [13-15]. S i n c e [(1 (4s) - / t l (~/tl) " D'] 9y = 0, a direct relation exists between the two representations; this is discussed in a forthcoming book by the author. In general, the response of frictional granular masses is dominated by their fabric. Therefore, the noncoaxiality term in the plastic strain rate can be neglected in many cases, especially at small pressures; i.e., c~- 0 may be assumed.
11.5.5 DILATANCY, FRICTION,
AND FABRIC
An expression for the dilatancy parameter B is obtained by setting the rate of frictional loss equal to the rate of distortional plastic work. To arrive at an explicit result, assume that the rate of frictional loss (per unit volume) is given by
ivy = ~pMy
(5a)
where Mf is an effective friction factor, a function of the void ratio and the fabric. Then, neglecting the work associated with the noncoaxiality term in Eq. 4a, b, from DP 9v -- ivf, obtain - B - Mj - (M +
-J"
(Sb, c)
In this equation, #f represents the effect of the fabric anisotropy on the dilatancy. Since Mf > 0 and M = r/p is generally very small, the dilatancy parameter B is initially negative (densification) for an initially isotropic virgin sample, until #f becomes suitably large, as/~ and J tend to become coaxial (in general, 0 < M << 1, for small pressures; i.e., most geomaterials have a very small elastic range). When the direction of shearing is reversed, #f changes its sign and attains a negative value. Hence, in reversing the direction of shearing, the dilatancy parameter B becomes negative, predicting that the sample would densify. This is in agreement with essentially all experimental observations. As the reversed shearing continues, the fabric anisotropy evolves and J changes with the applied stress. For the model to predict positive dilatancy in continued monotonic, say, shearing, the evolution of J must be such that J tends to become coaxial with/l, renderingJ : /~ positive again. The evolution equations for the fabric tensor J, as well as for the overall effective friction coefficient, Mf, and the radius of the yield surface defined by pM~ must thus reflect these basic physical requirements.
1112
Nemat-Nasser
Since, from Eq. 4a, b,/)~kk -- 9B, it follows that
(5d)
l+e so that the void ratio e is given by e -- ( 1 + eo)exp(fo ~B dy) - 1
(5e)
where e0 is the initial void ratio. As is seen, the exponential dependency of the void ratio e on the dilatancy parameter B makes the result very sensitive to the variation of the parameter B with the fabric parameter /if in Eq. 5b. Comparison with experiments has shown that Eq. 5b may have to be modified to reduce the effect of the fabric parameter on B, by replacing/if in Eq. 5b with ~/if, where 0 < ~ ___ 1.
11.5.6 ELASTICITY RELATIONS The elasticity of an assembly of frictional granules depends on the fabric and the applied pressure. Let L be the instantaneous elasticity tensor. Based on two-dimensional deformation of granular materials, it can be shown that the elasticity tensor is given by L -- 2G(1 (4s) _ 1/31 | 1) + 2G/tfl |
+ v/-2/((/,fl | 1 + 1 |
+ K1 | 1 (6a)
where G, G, K, and,/( are pressure-dependent moduli (all functions of the void ratio), and 1~ 1 1 '% = x/213 ' fl - (111"1J ) ~ = pJ ' J - (J "J)~ (6b-d) In the present formulation, the fabric tensor, 1~, affects both the elastic and inelastic response of the material.
1 1 . 5 . 7 RATE C O N S T I T U T I V E
RELATIONS
The final constitutive relation is now given by o
"e -- L " ( D -
D p)
(7a)
11.5 Micromechanically Based Constitutive Model for Frictional Granular Materials
1113
or, in component form, by 0 "r,ij -- Lijkl (Dkl - Dl~kz)
(7b)
For the evolution of the fabric, assume 0
0
(8a, b) where A is a material function, depending on the fabric measure, say, J, and possibly its history (e.g., the last extreme state of the fabric, in cyclic loading). In monotonic loading, the induced fabric (anisotropy) will eventually saturate. Hence, A must then tend to zero. Upon reverse loading, a jump in the value of A is expected; see the example in Section 11.5.9. The effective strain rate ~) is obtained from the consistency condition; i.e., f - 0 for continued plastic deformation. In the present case, we obtain (9a)
A1 -4- A2
-- H + O + Gla} + lafl(B + M)[( + MKB where A 1 - v/2(eVlz + GlAflVfl) -~- (~(]Afl -4- MK)Vkk -nt- x/~M~(Dfl
In these equations, # f l = p : pfl=/-tf/J, D~,--D : p, D f l - - D : p f l , and H is the workhardening parameter. To obtain an explicit expression for H, consider Eqs. 3e and 5c, and assume that M in Eq. 3e is a function of the void ratio e and the fabric parameter J, i.e., M = M (e, J); for small pressures, the assumption F(p, A, 7) = pM(e, J) is appropriate, but not for large pressures, where the dependence on the pressure may be nonlinear. From the consistency relation f = 0 at constant pressure, we thus obtain 0
dF
,8
p_
(1+
+
+1
A
(9d)
For a small elastic range, M ~ 0 and hence H ~ pA. In this case, the elastic volumetric strain rate may be neglected; i.e., Dhh ~ INChk = 9B. When, in addition, we set ~ = 0, then Eq. 9a reduces to
"); - -
pA + G + G/.t~
(9e)
1114
Nemat-Nasser
11.5.8 MATERIAL FUNCTIONS In this theory, there are three basic material functions which characterize the inelastic response of the granular mass. These are: (1) M, which defines the elastic range of the material. At small pressures, M~0. (2) Mf, which defines the overall frictional resistance of the material and, in general, is a function of the void ratio e and the fabric measure/~f. The following form for this function has been suggested by NematNasser and Zhang [16], based on experimental results of Okada [17]" Mf = (a 4- b ( e - em)nl)(/AO --/Af) n2
(X0a)
where em is the minimum value of the void ratio e, and #0 is a reference value of the fabric parameter #f. (3) A, which defines the evolution of the fabric tensor through the back stress. In monotonic loading, the induced anisotropy (fabric) is eventually saturated. To simulate this, A must tend to zero in such loadings. Then, upon unloading, A must jump to a finite positive value [16]. In addition, there are four elastic moduli, G, G, K, and,/~, all of which depend on the void ratio and the pressure. They may be approximated by the following general form: A - A0(1 4- c(eM - e) n3)
(10b)
where A stands for any of the four elastic moduli. In expressions Eq. 10a, b, the free parameters must be established based on experimental results, as has been illustrated by Nemat-Nasser and Zhang [16]. An example, given by these authors, is used in following text for illustration.
11.5.9 ILLUSTRATIVE EXAMPLE Okada [17] has performed experiments on large hollow cylindrical samples of Silica 60, for which e m - - 0 . 6 3 1 and eM = 1.095. Some of the results are examined and modeled by Nemat-Nasser and Zhang [16], where details of the modeling are given. The samples have 25-cm outside and 20-cm inside diameters, and 25-cm height. They are cyclically sheared under a constant confining pressure in a drained condition. A pure shearing with Zrr = zoo = Zzz = - p and ~rO = ~rz = 0 may be assumed, using a polar coordinate system. Setting 0 -- 1 and z - 2, and assuming a negligible elastic
'
1115
11.5 M i c r o m e c h a n i c a l l y Based C o n s t i t u t i v e Model for F r i c t i o n a l G r a n u l a r Materials
,0
Q.
v ~
'''
0.8 0.6 0.4
0.2
0
or"
c"
O3
I''''
I''''
I''''
I''''
I''
''
I''
' '
~
/ / / 1
7
0.0 -0.2
o3
I''''
-0.4 -0.6 -0.8 -1.0 -4.0
,,,, .... , .... -3.0 -2.0
, ....
. .. -'-"~" ~'
, ....
i ....
-1.0 0.0 1.0 S h e a r Strain( % )
. .... , .... 2.0 3.0
4.0
F I G U R E 11.5.1.
range, we obtain eli ~/}12 - +pAg, 9 - +
2D12
1 + A/K
> 0, ~c -- (G +
C_,)/p
(lla-c)
The combined elastic modulus K has the functional form given by Eq. 10b, and for A one may use K g(t) A(t) = 1 + d g(t) - gel"4'
D12(t) dt
--
( l l d , e)
where ~e is the value of ~(t) attained just before unloading. From Eqs. 11a-e, the shear stress q512 --g.Oz is obtained as a function of the strain ~. A typical example is given in Figure 11.5.1 , where the solid curve is the experimental result [17] and the dashed curve is the theoretical result. The material constants used are: K = 130, c = 1.42, d = 1.40 x 103, and n4 = 1.1. The pressure is 195 Pa. The void ratio is obtained from Eq. 5e with B -- - M r + ~'#f, #f = 4 -312
p
(12a, b)
The results for the same example are shown in Figure 11.5.2, where the experimental data are presented by open circles. The material constants are: a = 1/4, b = 0.1, n l - - 0.40, n2 -- n3 -- 1, and ~ = 3/4.
1116
Nemat-Nasser
0.72
' ' ' '
I ' ' ' '
I ' '
''
I ' ' ' '
I ' ' ' '
I ' ' ' '
i ....
, ....
, ....
I ' ' ' '
I
'
~
~
~
0.70 0 .,I.-a
rr "0 0 >
0.68
0.66
0.64
0.62
.... -4.0
' .... -3.0
' .... -2.0
-1.0 0.0 1.0 Shear Strain( %
, .... 2.0
, .... 3.0
4.0
)
FIGURE 11.5.2.
ACKNOWLEDGEMENTS T h e w o r k r e p o r t e d h e r e h a s b e e n s u p p o r t e d by the U.S. N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t C M S - 9 7 2 9 0 5 3 , w i t h the U n i v e r s i t y of California, San Diego.
REFERENCES 1. Okada, N., and Nemat-Nasser, S. (1994). Energy dissipation in inelastic flow of saturated cohesionless granular media. G~otechnique 44(1): 1-19. 2. Nemat-Nasser, S. (2000). A micromechanically-based constitutive model for frictional deformation of granular materials. J. Mech. Phys. Solids 48(6-7): 1541-1563. 3. Mehrabadi, M. M., Nemat-Nasser, S., and Oda, M. (1982). On Statistical description of stress and fabric in granular materials. Int. J. Num. Anal. Methods in Geomechanics 6: 95-108. 4. Kanatani, K. (1984). Distribution of directional data and fabric tensors. Int. J. Eng. Sci. 22(2): 149-164. 5. Subhash, G., Nemat-Nasser, S., Mehrabadi, M. M., and Shodja, H. M. (1991). Experimental investigation of fabric-stress relations in granular materials. Mech. Mat. 11(2): 87-106. 6. Nemat-Nasser, S., and Shokooh, A. (1980). On finite plastic flow of compressible materials with internal friction. Int. J. Solids Struct. 16(6): 495-514. 7. Spencer, A. J. M. (1964). A theory of the kinematics of ideal soils under plane strain conditions. J. Mech. Phys. Solids 12: 337-351.
11.5 Micromechanically Based Constitutive Model for Frictional Granular Materials
1117
8. Spencer, A. J. M. (1982). Deformation of ideal granular materials, in Mechanics of Solids, pp. 607-652, Hopkins, H. G., and Sewell, M. J. eds., Oxford: Pergamon. 9. de Josselin de Jong, G. (1959). Statics and Kinematics in the Failable Zone of Granular Material. Ph.D. thesis, University of Delft. 10. Mehrabadi, M. M., and Cowin, S. C. (1978). Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26: 269-284. 11. Balendran, B., and Nemat-Nasser, S. (1993). Double sliding model for cyclic deformation of granular materials, including dilatancy effects. J. Mech. Phys. Solids 41(3): 573-612. 12. Balendran, B., and Nemat-Nasser, S. (1993). Viscoplastic flow of planar granular materials. Mech. Mat. 16: 1-12. 13. Rudnicki, J. W., and Rice, J. R. (1975). Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23: 371-394. /4. Nemat-Nasser, S. (1983). On finite plastic flow of crystalline solids and geomaterials. Journal of Applied Mechanics (50th Anniversary Issue) 50(4b): 1114-1126. 15. Nemat-Nasser, S. (1992). Phenomenological theories of elastoplasticity and strain localization at high strain rates. The Journal of Applied Mechanics Reviews 45(3, Part 2): 519-545. 16. Nemat-Nasser, S. and Zhang, J. Constitutive relations for cohensionless frictional granular materials. J. of Plasticity (in press). 17. Okada, N. (1992). Energy Dissipation in Inelastic Flow of Cohensionless Granular Media. Ph.D. thesis, University of California, San Diego.
SECTION
11.6
Linear Poroelasticity J. W. RUDNICKI Department of Civil Engineering,
Northwestern University, Evanston, Illinois
Contents 11.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2.1 Isotropic Response . . . . . . . . . . . . . . 11.6.3 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.3.1 Plane Strain . . . . . . . . . . . . . . . . . . . . . 11.6.4 Material Parameters . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.1
1118 1118 1121 1122 1123 1124 1125
VALIDITY
Linear poroelasticity is a theory that includes the coupling between linear diffusion of a mobile species and the stress and deformation of a linear elastic porous solid. This theory has been widely applied not only to soils and rock masses infiltrated by groundwater but also to coupling of fluid flow and deformation in biological materials and diffusion of hydrogen in metals. Although departures from linear behavior are common in actual materials, linear poroelasticity is an improvement over idealizations that neglect either diffusion or deformation entirely. Because parameters describing nonlinear material behavior are difficult to obtain with accuracy, poroelasticity often offers a more practical approach than complex, nonlinear models. Furthermore, the insights gained from linear poroelasticity can be valuable for interpreting more complex models.
11.6.2
FORMULATION
The general development of linear poroelasticity was first given by Biot [1], but a revealing presentation of the constitutive relations by Rice and Cleary
1118
Handbook of Materials BehaviorModels. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1119
11.6 Linear Poroelasticity
[2] is followed here. Detournay and Cheng [3] have recently given a detailed discussion of the theory with a variety of solutions, especially for borehole problems, and an introduction to numerical formulations. In addition to the total stress aij and the (infinitesimal) strain of the solid matrix z/j, two additional variables are needed to describe the fluid phase (or other diffusing species). One choice for these is the fluid mass per unit reference volume of porous solid m and the pore fluid pressure p. The latter is defined as that pressure in an imaginary reservoir of fluid connected to the material point that equilibrates any mass flow to the porous solid. If a "point" of the porous solid is regarded as a representative volume element containing differently oriented fissures, then the assignment of a single scalar pore pressure to the point assumes that the time scale of deformation is slow enough to allow pressure equilibration within this element. Linear relations for the stress and alteration of fluid mass content from a reference value m0 in terms of the strain and pore pressure take the form (la)
aij -- Lijkle, kl -- AijP
m-
mo = Rij~ij + Qp
(lb)
where Lijkl, Aij, and Rij are arrays of factors reflecting the symmetries of aij and eij. The first of these has the usual form of the linear elastic relation with the stress replaced by the effective stress, aij + AijP. The fluid mass content can be written as the product of p, the density of homogeneous fluid, and v, the apparent volume fraction of pore space, m = pv. Linearizing this relation and substituting into Eq. lb yields v -
vo . . . . . .
1
P OR'J'F''J
+ -
p ( PO\
Q-
P0V0~ --~f fl
(2)
where the last term results from the linearized density change and Kf = PoP/(P - Po) is the bulk modulus of the pore fluid. An additional constraint on constitutive parameters arises from the relation for changes in the Helmholtz function per unit mass ~b(~ij,m) for isothermal deformation :
ddp = ~ijdgij -[- ladm
(3)
where the chemical potential # is given by -
p(p/
(4)
(McTigue [4] has given a formulation for linearized porothermoelasticity.) A Legendre transformation to the potential ~ = ~b - #m yields
1120
audnicki
Because Lijkz- oq26/O~ijO~kl, these moduli satisfy the usual elastic symmetry with respect to interchange of the first and last pair of indices. In addition, the Maxwell relation that results from computing second derivatives of ~ with respect to ~ij and p in either order is Orrij
Op
O~
=
&~j
(6)
Substituting Eqs. la and 2 yields
Rij -- PoAij
(7)
The remaining constitutive relation is Darcy's law:
(8) where qi is the fluid mass flow rate (relative to the solid phase) per unit area, gf/is the body force per unit mass of fluid, and ~cij is a permeability tensor, required to be positive definite by the second law of thermodynamics. The permeability ~cij is more typically expressed as kij/rl, where r/is the viscosity of the fluid and kij is a permeability tensor with dimensions of (length) 2, frequently measured in units of darcies (1 darcy=10-12m2). Alternatively, the permeability is expressed as an equivalent velocity k = yo~Cwhere Yo is the weight density of the fluid. If deformation occurs very slowly, fluid mass diffusion will equilibrate any alterations of pore fluid pressure. In this drained limit, the pore pressure change vanishes. Thus the moduli Lijkl are those appropriate for drained deformation. In the contrasting undrained limit, rapid deformation allows no time for fluid mass alteration. Setting m = mo in Eq. lb and substituting into Eq. la yields /A ~j - L~jk~kz
(9)
where
L~jkl - Lijhl -4- Q- l poAijAhl
(10)
are the moduli appropriate for undrained deformation. The pore pressure induced by undrained deformation can be expressed as 1
p -- --~Bij~ij
(11)
where
3 P oAmn Cmnij Bij = Q _ PoAmnCm,pqAp q and Cpqrs is the inverse of Lijkz.
(12)
1121
11.6 Linear Poroelasticity
11.6.2.1
ISOTROPIC RESPONSE
Although Cheng [5] has recently given an advantageous formulation of anisotropic poroelasticity, applications assuming isotropy are much more common because of the difficulties of obtaining a complete set of material parameters. For isotropy, the tensors Aij, Bij, and tcij reduce to e, B, and tc times the Kronecker delta, 6ij, respectively. The coefficients ~ and B are commonly referred to as Biot's parameter and Skempton's coefficient, respectively. The modulus tensor Lijkl assumes the usual form
Lijkl -- O(~ik~jl + c~iT~jk) 4- ( K - 2G/3)~ij~kz
(13)
with inverse
1{ l(~prc~qs-}-c~pst~qr)- 1 +v vc~eq~s}
Cvq~s -- 2G
(14)
where G and K are the drained shear and bulk moduli, respectively; and v is the Poisson's ratio for drained response, related to G and K by v - ( 3 K - 2 G ) / 2 ( G + 3K). The undrained modulus tensor has the same form as Eq. 13. Evaluation of Eq. 10 indicates that the shear modulus is the same for drained and undrained deformation and that the bulk modulus for undrained deformation/~ is given by Ku = K + ~2p0Q-1
(15)
The expression for B, obtained from Eq. 12, is B = (Ku - K)/o~Ku
(16)
Thus, in the isotropic case, Eqs. la and lb reduce to O'ij =
2G~ij + ( K - 2G/3)bije, kk -- o~Pbij
(17a)
m - m o - - - ~ 5akk +
(17b)
and the expression for the apparent volume fraction (Eq. 2) reduces to t) - v0 -- ~ ~O'kk+
Kf
Insight into the meaning of ~, B, and K, can be obtained by considering the special loading a 0 = - a S 0, p = a. For idealized circumstances delineated by Rice and Cleary [2] (following an observation of Reference [6]), the resulting volume strain and fractional change of pore space are equal to - a divided by the bulk modulus of the solid constituent K~. More generally, the moduli entering these expressions, denoted here (and in Reference [2]) by K~ and K~, will differ from Ks. Equating these expressions to those obtained from
1122
audnicki
Eqs. 17a, 17b and 18 yields the following relations: = 1 - K/KIs
(19a)
l/K- 1/K~s B = l/K-
l/K: + v0(1/Kf- 1/K')
1 / K - 1/Kts + Vo(1/Kf - 1/K") Ku = ( 1 / K - 1/K~s)K/Ks' ' + Vo(1/Kf - 1~Ks)"
(19b)
(19c)
The undrained response can also be expressed in terms of the undrained Poisson's ratio, v., related to K. by v. = (3/(. - 2 G ) / 2 ( G + 3/(.) or, in terms of v, B, and ~ as vu--v+
(1 + v)(1 - 2v)~zB 3-(1-2v)~B
(20)
11.6.3 FIELD EQUATIONS The relevant field equations are the usual ones of solid mechanics, equilibrium and strain displacement, and, in addition, conservation of fluid mass. Equilibrium is expressed as {Yij, i ~ Fj (X, t) -- 0
(21)
where Fj is the body force per unit volume and (..-),i denotes c9(...)/cgxi. The (small) strain displacement relation is ~ij = (ui,j + uj, i)/2
(22)
Conservation of fluid mass is qk,k + cgm/c)t = H(x, t)
(23)
where H is a fluid mass source. Substitution of Eq. 22 into Eq. 17a and the result into Eq. 21 yields (K + G/3)e,j + Gu;,/i + Fj - ~p,; = 0
(24)
where e = ~hk = Ili,i is t h e dilatation. Thus gradients in pore pressure act as a distribution of body forces. Consequently, if the distribution of pore pressure is known, the displacements and stresses can be determined by superposition of the solution of Eq. 24 for a Dirac singular distribution of body force (point force). In general, however, the pore pressure field is coupled to the stress field and cannot be determined independently. Alternatively, Eq. 17 can be
1123
11.6 Linear Poroelasticity
used to eliminate p in favor of m: (K. + O/3)ej + Ouj, ,, + Vj - ~(K.m - j K) CZPo
= 0
(25)
Substituting Darcy's law (Eq. 8), specialized to isotropy and a uniform permeability, into Eq. 23 gives -
Po~Cp,kk + oqm/Ot- H(x, t)
2 - po~:~,k
(26)
The divergence of Eqs. 24 and 17b can be used to eliminate p from Eq. 26. The result is a diffusion equation for the fluid mass content
Ot
cm,kk -- H(x, t) + P~
~(Ku + 4G/3)Fkk' - PO~,k
(27)
where Eqs. 16 and 17a have also been used. The diffusivity c is C---If,
(Ku - K)(K + 4 0 / 3 ) r + 4G/3)
(28)
or, as given by Rice and Cleary [2] in terms of B, v, and vu, (29) where, as they note, the first bracket is the drained elastic modulus for onedimensional strain and the second is unity for incompressible constituents.
11.6.3.1 PLANE STRAIN Two-dimensional solutions can provide a reasonable idealization of a variety of geomechanical problems. An important simplifying feature of the plane strain formulation, noted by Rice and Cleary [2], is that the governing equations can be expressed entirely in terms of the stress and pore pressure. For plane strain deformation in the xy plane, the compatibility equation, in the absence of body forces, can be expressed as
v (O x + Oyy + 2 p) - 0
(30)
and the fluid mass diffusion equation, without source terms, becomes (cV 2 - O/Ot)[Crxx+ ~yy + (2~//#)p] = 0
(31)
where V2(...) is the two-dimensional Laplace operator and the material constants enter only in the combinations r / - 3(vu - v)/2B(1 + vu)(1 - v)
(32)
# = (vu - v)/(1 - v)
(33)
1124
Rudnicki
If the boundary conditions can be expressed in terms of the stress and pore pressure, then Eqs. 30 and 31 and two of the three equilibrium equations (Eq. 21) suffice to determine these quantities.
11.6.4 MATERIAL PARAMETERS The drained elastic constants and two additional parameters are needed to characterize a linear poroelastic solid. Ideally, the drained elastic constants would be measured on a saturated sample deformed very slowly so that no pore pressure changes are induced by fluid flow. In practice, however, the drained elastic constants are often assumed to be equal to the values obtained on dry samples. An undrained test, in which fluid exchange between the sample and the surroundings is prevented, would, in principal, suffice to determine the remaining two porous media parameters, for example, K, or vu and/3. Unfortunately, these and other tests to determine the porous media parameters are difficult and not yet standard. Consequently, it is frequently necessary to resort to the assumption that both Kts and K~ are equal to the bulk modulus of the solid constituents Ks and then to calculate c~,/3, etc., from Ks, the porosity v0, and the pore fluid bulk modulus Ky using expressions such as Eqs. 19a and 19b. Values of v, and/3 determined in this way by Rice and Cleary [2] in their Table 1 for six rock types range from 0.29 to 0.34 and from 0.51 to 0.88, respectively. Values of c~are not given but can be calculated from the values in the table and range from 0.2 to 0.7. Values for three additional rock types listed by Detournay and Cheng [3] also fall within this range. Limiting ranges of c~,/3, and vu are easily obtained from Eqs. 19a, 19b and 20. From the first, it is evident that ~ approaches unity if the drained bulk modulus is much less than the bulk modulus of the solid constituents. If, in addition, the value of K is much less than the bulk modulus of the pore fluid, so that K << Kf/Vo, then /3 also approaches unity and v, = 1/2. These approximations are appropriate for most soils. If, on the other hand, the pore fluid is very compressible so that vo/Kf >> 1/K > 1/Kts, 1/K~s, then B ~_ Kf/voK and approaches zero. In this limit v, = v and, thus, v
11.6 Linear Poroelasticity
1125
but also involves a combination of moduli. Again, because large fractures and joints serve as conduits for fluid flow in situ, field values of the diffusivity are typically much larger than those measured in the laboratory. Roeloffs [7] schematically summarizes values from both laboratory and field data ranging from nearly 105 to 10 -11 m2/s. Laboratory values for the rocks tabulated by Rice & Cleary [2] are about 10 -2 to 10 -4 mZ/s except for the Berea sandstone, for which c = 1.6m2/s. A variety of observations related to earthquakes suggest diffusivities in the range 0.1 to 1.0m2/s, a range that may be representative of conditions near many faults, but values an order of magnitude smaller or larger are not unusual.
REFERENCES 1. Biot, M. A. (1941). General theory of three dimensional consolidation. J. Appl. Phys. 12: 155-164. 2. Rice, J. R., and Cleary, M. P. (1976). Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophy. Space Phys. 14: 227-241. 3. Detournay, E., and Cheng, A. H-D. (1993). Fundamentals of poroelasticity, in Comprehensive Rock Engineering: Principles, Practice and Projects, pp. 113-171, Vol. 2, Fairhurst, C., ed., New York: Pergamon. 4. McTigue, D. E (1986). Thermoelastic response of fluid-saturated porous rock. J. Geophys. Res. 91: 9533-9542. 5. Cheng, A. H.-D. (1997). Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci. 34: 199-205. 6. Nur, A., and Byerlee, J. D. (1971). An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res. 76: 6414-6419. 7. Roeloffs, E. A. (1988). Hydrologic precursors to earthquakes: A review. Pure and Applied Geophysics (PAGEOPH) 126: 177-209.
SECTION
11.7
Nonlinear Poroelasticity for Liquid Nonsaturated Porous Materials O L I V I E R COUSSY~ PATRICK D A N G L A Laboratoire Central des Ponts et Chauss~es, Par/s, France
Contents 11.7.1 An Energy Approach of Poroelasticity: Thermoynamic Potentials . . . . . . . . . . . . . . . . 11.7.2 Incremental Equations of Nonlinear Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.3 Assumption of Negligible Volume Change for the Solid Constituent . . . . . . . . . . . . . . . . 11.7.4 The Equivalent Internal Fluid Pressure zc. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7.1
AN ENERGY APPROACH
POROELASTICITY:
1126 1128 1129 1130 1132
OF
THERMODYNAMIC
POTENTIALS The approach of poroelasticity presented here is principally an extension to nonsaturated situations of the pioneering work of M.A. Biot on the modeling of deformable liquid saturated porous media (see in the references some of his classical papers). The extension to partially saturated porous media is mainly based upon thermodynamics of deformable open continua [7, 8, 12]. The porous material considered here is formed of a porous solid skeleton saturated by two fluids. These fluids are a wetting liquid, index ~ = l, and a nonwetting gas, index ~ = g. Submitted to external actions, an element d~ of
1126
Handbook of Materials Behavior Models. ISBN0o12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
11.7 Nonlinear Poroelasticity for Liquid Nonsaturated Porous Materials
1127
this porous material simultaneously deforms and undergoes variations of its fluid mass contents m~. Referring the deformation to that of the porous solid skeleton, let ~ij be the strain tensor components. Let also be the components of the (total) stress tensor supported by both the solid skeleton and the saturating fluids. For reversible (here also isothermal for sake of clarity) processes, the first two laws of thermodynamics applied between times t and t + dt to the open deformable system df~ yield the balance of its free energy in the form:
ffijdsij 4- gldmz 4- ggdmg - d t P = 0
(1)
In this equation ~P is the free energy per unit of volume d~ of porous material, and g~ is the Gibbs potential of fluid 0~per mass unit. Recall that: g~ - r
+ P~ P~
(2)
where ~ is the free energy per mass unit, p~ is the fluid pressure, and p~ is the volumetric mass. The term g~dm~ = ~k~dm~ + p~/p~ dm~ accounts both for the free energy convectively supplied by fluid a (term 6~dm~) and for the work required to make the infinitesimal mass dm~ actually enter the open porous material element d.Q(termp~/padm~). The term rr/jde/j accounts for the additional work required to produce the deformation of volume dO. Equation 1 is similar to the Gibbs-Duhem equation encountered in physical chemistry [1]. Poroelasticity corresponds to the constitutive equations of a saturated porous material deforming reversibly due to the elasticity of its solid constituent. For nonsaturated materials, poroelasticity accounts in addition for reversible capillary effects, not considering the hysteresis associated with such effects. In these conditions Eq. 1 holds for any process and, consequently, dqJ has to be the exact differential of some function W(e/j, m~) of the independent state variables (eij, m~). As a result, the state equations of poroelasticity derive from the potential function qJ according to: 0qJ cg~P cr/J-cge/j ga-cgm~ a - 1, v, a (3) Let now q~ be the potential defined by [10] ~Ps = V - m . ~ .
(4)
where ~,~ is the free energy of fluid ~ per mass unit. Since ~P represents the free energy of all matter currently contained in d~, ~P~ represents the free energy of the skeleton, including the fluid-fluid and solid-fluid interfaces, the latter possessing their own surface free energy. Let us also introduce the (Lagrangian) partial porosity 4~= related to fluid ~ and referring to the initial volume d~ such that ~b~d~ represents the current volume of d.Q occupied by
1128
Coussy and Dangla
fluid a. Therefore, we write ms -- p~b~
(5)
Recall also that
c~g~
1 =
--
(6)
Combining Eqs. 1 and 4 to 6, one derives
aijdgij -k pld~ 1 + pgd~g - d t P s - 0
(7)
Equation 7 turns out to be the free energy balance of the skeleton as just defined (i.e., including the interfaces). In fact, pzdc~z + pgdq~g represents the external work supplied by the liquid and the gas through the pressure they exert on the interface with the skeleton and reversibly stored by the latter in the form of free energy d~s. As a result of Eq. 7 the state equations of poroelasticity can be alternatively written in the form
By contrast to Eq. 3, Eq. 8 addresses separately the constitutive equations of the skeleton, while submitted to the internal pressures Pz and pg of the liquid and the gas. In turn, Eq. 5 can then be viewed as a consistency continuity equation ensuring the fullness of the porous space. The set of state variables (eij , PZ, Pg) often proves to be more convenient as primary variables than the set of state variables (eij, ~z, 4~g). Let us then introduce the Legendre-Fenchel transform H~(eij, Pz, Pg) of ~Ps(eij, q~z, q~g), reading
Hs -- ~tffs - Pill - Pg~g
(9)
A combination of Eqs. 8 and 9 yields [10]
OHs ,~,j = &~j
OHs Op~
C~g--
oqHs Oqpg
(10)
11.7.2 INCREMENTAL EQUATIONS OF NONLINEAR POROELASTICITY For many porous materials, potential qJs cannot be approximated by quadratic functions of their arguments, since experimental evidence shows that constitutive equations (Eq. 10) do not rely on linearly the set of variables (r ~)l, ~g) tO the conjugate s e t (gij, Pz, Pg). That means that the secondorder derivatives of potential function qJs involved in the differentiation of
11.7 Nonlinear Poroelasticity for Liquid Nonsaturated Porous Materials
Eq. 8 are However, behavior behavior. the form
1129
not constant and do depend on the state variables (e/j, pl, Pg). a usual assumption consists of considering that the deviatoric is linear and is not coupled with the nonlinear volumetric For an isotropic material, the potential function reduces then to
Hs = Hs~(e,, Pl, Pg) + 2G eoeji
(11)
1 where e is the volumetric strain e - g,ii and e0 - e o -~e6 v is the deviatoric strain tensor. A differentiation of state equations Eq. 10 yields
da = K(e, Pz, p g ) d e - bl(e, pz, p g ) d p z - bg(e, pt, pg)dpg
(12)
d~p l = Nil(g, Pl, pg)dpl-Jr-N/g(g, Pl, pg)dpg-+-bl(e,, Pl, pg)de
(13)
dc~g- Ngg(e,, Pl, pg)dpg + N/g(e, Pl, pg)dpl + bg(e, Pl, pg)de,
(14)
dsij - 2G deij
(15)
1 where a--5aii is the overall mean stress, and sij - aij - a a i j is the deviatoric stress tensor. K is the (drained) tangent bulk modulus, as coefficients bl and bg can be considered as tangent Blot's coefficients. Finally, G is the standard shear modulus.
11.7.3 ASSUMPTION OF NEGLIGIBLE VOLUME CHANGE FOR THE SOLID CONSTITUENT In many evolutions of usual porous materials, the volume of their solid part does not change significantly when compared to the change of the volume of the porous space. The volumetric strain e is then only due to the variation of the total porosity 4 ) - ~bz + 4)g, reading e - ~b - 4)0 - q~l + ~bg - ~b0
(16)
where ~0 stands for the initial overall porosity. Equation 16 implies that only two variables among the set (e, qSl,~bg) are independent. We choose (e, q5l) so that Eqs. 16 and 7 yield ( a i j q t-
Pg6ij)dgij - pcd~ z - d~s - 0
(17)
where Pc - P l - Pg is the capillary pressure. Under the assumption of small volume change for the solid constituent, Eq. 17 shows that ~s depends only on (eij, ~bl). 0% 0~F~ 6ij -Jr-Pgbij - - Og, ij Pc -- Oqbt (18)
1130
Coussy and Dangla
Let W(eij, Pc) be the Legendre-Fenchel transform of ~Ps(e0, ~bl): (19)
W = ~Ps 4- pcqbl The state equations (Eq. 10) reduce then to
OW (Yij 4- Pg~ij - - Oeij
OW ~)1- Opc
(20)
Equation 11 becomes
W-
Ws~(e, Pc)4- 2G eijeji
(21)
Differentiating Eqs. 20 and 16, one finally obtains d(~ + pg) - K(e, p~)& + b(e, p~)ctpc
d~--N(~,
e~)@c +
(22)
b(~, p~)d~
(23)
de = d(/) - dctbl + dckg
(24)
while Eq. 15 remains unchanged. A comparison between the set of Eqs. 12-14 with the set of Eqs. 22-24 shows that the assumption of negligible volume change for the solid constituent entails b-bl
bz4-bg-1
N-Nu---Nlg-Ngg
(25)
In addition, note that Maxwell's relations associated with Eq. 20 are =
e
pc
pc
e
11.7.4 THE EQUIVALENT INTERSTITIAL FLUID PRESSURE It remains to identify K(e, Pc), b(e, Pc), and N(e, Pc). A combination of the definition of the liquid water saturation, Sl = ~bz/~b,with Eqs. 23 and 24 yields
d?dSl - -Ndpc 4- ( b - Sz)&
(27)
In reversible (i.e., hysteresis associated with capillary effects is neglected) and nondeformable case (i.e., e - 0, ~ - ~b0), the previous equation reduces to
~0dSz = -N@~
(28)
p~ = p~(S~)
(29)
yielding A very common macroscopic assumption which can receive some support and also some limitations from microscopic considerations [6] consists in
11.7 Nonlinear Poroelasticity for Liquid Nonsaturated Porous Materials
1131
assuming that Eq. 29 still holds in the deformable case (i.e., ~: 0, q5 = ~b0 + e). In other words, provided that the current ratio $l of the porous space occupied by the wetting liquid does not change, the capillary pressure is not affected by the strain. Then, a comparison of Eq. 29 with Eq. 27 leads us to identify Biot's coefficient b as the liquid water saturation Sz:
(30)
b =
Equation 30 and Maxwell's equation (Eq. 26) shows then that the bulk modulus K depends only on e: K = K(e)
(31)
A substitution of Eqs. 30 and 31 into Eq. 22 entails (32) where rc is the pressure defined by - - p g --
fo
S l ( Z:v ) d z:v
(33)
For Sz = 1, the capillary pressure being always non-negative, Eq. 33 reduces to rc = pz. Thus, provided that later on the porous space remains saturated, i.e., dSz = 0, we identify drc = dpl and relation (32) reduces to d(o + Pl) = K(e)d~
(34)
Referring to the saturated case, i.e. Eq. 34, pressure rt involved in relation 32 can be interpreted in the nonsaturated case as an equivalent interstitial fluid pressure. In fact, relation 32 altogether with definition 33 ensures the continuity with the relation holding in the saturated case. Indeed, for negligible volume change for the solid constituent, Eq. 25 implies bz + bg = 1 and Eq. 34 can be directly and exactly recovered by letting dpz = dpg = dp in Eq. 25. When the volume changes for the solid constituent are not negligible, noting bl + bg = r, the latter procedure yields +
=
(35)
In the linear isotropic poroelastic saturated case, provided that the solid constituent is homogeneous and isotropic, Biot's relation (See, for instance, Reference [8]) provides fl as a function K = 1 (36) Ks of the drained bulk modulus K and the bulk modulus Ks of the solid constituent. Based upon Biot's relation, Eq. 32 is extended in the form d(a + fin) = K(e)de
(37)
1132
Coussy and Dangla
According to Eq. 37, ~r +/3~z governs the deformation of the solid skeleton (now not including the interfaces) over the whole range of saturation. Indeed, in the case of negligible volume change for the solid part of the skeleton (K/Ks << 1, fl ~ 1) and saturated situations ( S z - 1), Eqs. 33 and 37 indicate that the so defined effective stress cr 4-/3~z reduces to the celebrated Terzaghi effective stress a 4- p [13]. Hence, the stress ~r 4- fl~z can be viewed as a generalized effective stress for liquid nonsaturated porous materials. However, by contrast to general constitutive Eqs. 12-14, the effective stress concept is based upon assumptions and has to be experimentally checked. Indeed, for negligible volume changes of the solid part of the skeleton, a saturated experiment (i.e., Sl = 1, rr - pl, d[cr + pt] - K[e]&) must lead to the same identification of function K(g) as the one given by a free swelling experiment (i.e. ~ - O, drc - K p [ e ] d e ) [111. The previous constitutive relations concern only reversible evolutions. This approach can be extended to account for irreversible processes, in order to include both the hysteresis associated with capillary effects and the plastic deformation of the solid skeleton [9].
REFERENCES 1. Atkins, P. W. (1990). Physical Chemistry, 4 th ed., Oxford University Press. 2. Biot, M. A. (1941). General theory of three dimensional consolidation. Journal of Applied Physics 12: 155-164. 3. Biot, M. A. (1962). Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics 12: 155-164. 4. Biot, M. A. (1972). Theory of finite deformation of porous solids. Indiana University Mathematical Journal 33: 1482-1498. 5. Biot, M. A. (1977). Variational Lagrangian-thermodynamics of non-isothermal finite strain: Mechanics of porous solid and thermomolecular diffusion. International Journal of Solids and Structures 12: 155-164. 6. Chateau, X., and Dormieux, L. (1998). A micromechanical approach to the behaviour of unsaturated porous media, in Poromechanics, A Tribute to M.A. Biot, Thimus, J.-E et al., eds., Proceedings of the Biot conference on poromechanics, Balkema. 7. Coussy, O. (1989). Thermodynamics of saturated porous solids in finite deformation. European Journal of Mechanics, A/Solids 8: 1-14. 8. Coussy, O. (1995). Mechanics of Porous Continua, John Wiley & Sons. 9. Dangla, P., Malinsky, L., and Coussy, O. (1997). Plasticity and imbibition-drainage curves for unsaturated soils: A unified approach, in Numerical Models in Geomechanics, NUMOG VI, Pietruszczak and Pande, eds., Balkema. 10. Dangla, P., and Coussy, O. (1998). Non-linear poroelasticity for unsaturated porous materials: An energy approach, in Poromechanics, A Tribute to M. A. Biot, Thimus, J.-E et al., eds., Proceedings of the Biot conference on poromechanics, Balkema.
11.7 Nonlinear Poroelasticity for Liquid Nonsaturated Porous Materials
1133
11. Dangla, P., Coussy, O., Olchitzky, E., and Imbert, C. (1999). Non linear thermo-mechanical couplings in unsaturated clay barriers, in Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials, Ehlers, W., ed., Proceedings of IUTAM Symposium, Kluwer Academic Publishers. 12. Sih, G. C., Michopoulos, J. G., and Chou, S. C. (1986). Hygrothermoelasticity, Martinus Nijhoff Publishers. 13. Terzaghi, K. (1925). Principles of Soil Mechanics, A summary of Experimental Results of Clay and Sand, Eng. News Rec., 3-98.
SECTION
11.8
An Elastoplastic Constitutive Model for Partially Saturated Soils B.A. SCHREFLERand L. SIMONI Department of Structural and Transportation Engineering, University of Padua, Italy
Contents 11.8.1 Validity of the Model . . . . . . . . . . . . . . . . . . . . 11.8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . 11.8.4 Applications and Parameters . . . . . . . . . . . . . 11.8.4.1 Tests on Compacted Kaolin . . . . . . 11.8.4.2 Tests on Clays . . . . . . . . . . . . . . . . . . 11.8.4.3 Material Parameters Identification in an Oedometric Test for Hydrocarbon Reservoir Sandstone . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1134 1135 1136 1140 1140 1142
1143 1144 1144
11.8.1 VALIDITY OF THE MODEL This m o d e l has been developed by Bolzon et al. [1] for partially saturated geomaterials where capillary effects are of importance. It is an extension of the generalized plasticity m o d e l for fully saturated soils dealt with in previous sections. In the presented version a yield surface is used to take into account matric suction (also called capillary pressure), which has to be treated as an i n d e p e n d e n t variable.
1134
Handbook of Materials Behaviar Models. ISBN 0w12o443341o3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1135
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils
The model accounts for the following typical situations: 9 Dependence of soil stiffness on suction; i.e., unsaturated specimens exhibit a lower volume change than saturated ones when, e.g., subjected to the same increment of vertical stress. However, when the unsaturated samples are soaked and hence saturated, the soil exhibits a significant compression strain under constant stress. 9 When suction decreases (wetting), the geomaterial may exhibit first swelling (in the elastic range) and then collapse (irreversible compression). The model has been applied successfully to partially saturated clays and sands. It has the most important features of the well-known Barcelona model [2]. Moreover, because of the choice of the stress variables (see next section), it can easily be applied to the dynamic analysis of problems where both fully and partially saturated zones are present, as, for instance, in the seismic analysis of earth dams, dikes, and slopes.
11.8.2
BACKGROUND
Vectorial notation is used in this chapter. When dealing with the mechanics of partially saturated porous media, we have to define a suitable stress measure: it is commonly accepted that two independent stress dimension parameters (a combination of total stress tensor and water and gas pressure) are needed to describe the deformational behavior [3]. In the following, the assumed stress variables are soil suction s, say, the difference between gas and water pressure ! (pg and Pw, respectively), and Bishop's effective stress ~rij. The latter is a combination of total stress crij and pressure acting on the solid, represented by a weighted average of water and gas pressure. As a weighting function, water saturation Sr is assumed [3]; hence the stress measure takes the form CYlij ~
tTij --
ISrpw + (1
- Sr)pglc~ij
-- tTij --
IPg-
Srsl(~iJ
(1)
with soil suction s defined as -
pg -
/2/
Soil suction and water saturation may be mutually related by means of the following relationship [2]: Sr -- 1 -- m tanh(ls)
(3)
where m and l are material parameters ( S r - 1 - m represents irreducible saturation, i.e., the limiting value of Sr as suction approaches infinity). In Eq. 1, parameter Sr represents a phenomenological measure of the capillary
Schrefler and Simoni
1136
effects, through its experimental relationship with suction. Bishop's stress definition recovers the Terzaghi effective stress definition, usually assumed in fully saturated soil mechanics when saturation equals one; hence the consistency condition between stress measures is guaranteed. W h e n one is observing the mechanical behavior of a partially saturated geomaterial, different deformation histories are obtained depending on the water content (or saturation or relative humidity) and also the so-called h y d r i c p a t h (or suction path); i.e., a forced change in saturation results in a sample deformation. Figure 11.8.1 shows these effects in terms of volumetric strain for a chalk.
11.8.3 MODEL DESCRIPTION The elastoplastic model is now defined in the space (p~, q, s) of the mean stress p~, the deviatoric stress q, invariants of Bishop's stress, and suction s: pt _ O'li 3 = p - Srpw - (1 - Sr)pg - p - pg + Srs _
q -- X ~ 2
~
--
180 160 -140120-
ZijZij,
(5)
Zij -- aij - p~6ij
P l l 0 Rh =0.1 ~S ,,"
~, 1 0 0
O~
.."
,.o~""
. . . . . P122 Rh=0.2
o""........
.............. P123 Rh=O. 6
......
80 60 40
(4)
....... Pl31 R h = l
.....' .-"
.............
2o; 0
I
I
I
I
-0.03
-0.08
-0.13
-0.18
Volumetric strain FIGURE 11.8.1 Volumetric strain of a chalk sample under isotropic compression at different levels of suction (Rh= relativehumidity). (Redrawn with permission from Rock mechanics: Proceedings of the 35th US symposium, Lake Taboe, 4-7 June 1995, Daeman, Jaak J.K. and Richard A. Schhz (eds.), 1995, Balkema.)
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils
1137
From Eq. 4 it results that changes in the mean stress p' may be induced both by changes in gas pressure and changes in suction and saturation. As shown by Pastor et al., Section 11.10, this volume, generalized plasticity does not explicitly define the yield and potential surfaces but directly assumes the loading-unloading direction vector n and the direction vector defining the plastic flow ngL. When we refer to fully saturated conditions, i.e., when we use Terzaghi's definition of effective stress and invariants of Eqs. 4-5 must be evaluated for Sr - 1, these direction vectors can be given as functions of the stress ratio r / = q/p~, e.g., following Nova [6], as 1
{dr; 1 }T, df - (1 + 00(Mf - r/),
~ { d g ;
1 } T dg -- (1 + o~)(Mg - ~),
(6)
(7)
where 0~, Mg, and Mf are material parameters; see Section 11.7. Whereas c~can be assumed as independent of suction, parameters Mg and Mf must depend on suction in partially saturated problems (Laloui et al. [7], for Mf). Further assumptions for n and ngL are, however, possible, for instance, n--ngL, which results in associative plasticity. For the purpose of introducing the effects of suction, we define now a yield and potential surface, which are obtained by integration of Eqs. 6 and 7 as -q
1
g--q-Mgp'(1
-
+1)[1-(~-~)~]
-o
(8)
-- 0
(9)
where pf and pg are integration constants which determine the size of the surface but have no influence in defining the respective normals. Even though Eqs. 8 and 9 are formally the same as for fully saturated materials (Pastor et al., this volume), they must be assumed in (p~,q,s) space when dealing with partially saturated problems (see Fig. 11.8.2). In addition to the aforementioned dependence of Mg and Mf on suction, parameters pg and pf also depend on suction, as suggested by experimental evidence [8]. This dependence will be discussed later. Once the direction vectors have been defined, irreversible (or plastic) strains deV are related to stress increments da ~ by the relationship
de,p _ 1 n Tdal ngL H
( 1O)
1138
Schrefler a n d S i m o n i
f(p' s)
Elasti~zo._
___
,
,,~.
p
r
,p"
s SS s sis ~,
.'.'.'d /
J
s
s 9 s s sS ~,
M/
p,~
_
F I G U R E 11.8.2
Yield surfaces in (pt, s) a n d (pt, q) planes.
where H represents the plastic (or hardening) modulus. As usual, the strain increment d8 can be decomposed into an elastic part, d8e, and an elastoplastic one: dn = dg e + d8p
(11)
Recalling that specific volume v is the ratio between the sample volume and that of the solid, i.e., v = 1 4-e, the total volumetric strain is defined as ev .
.
-- V0
v0
.
.
e-
gO
. . 1 +e0
/l -- /l 0
1-n
(12)
where e is the void ratio, n is the porosity, and subscript 0 refers to initial values. Following the critical state models [9], the total volumetric strain (recoverable and not) associated with the change in the hydrostatic component of the Bishop's stress tensor pt can be linearized as dev = 2(s) dp___~ ~ v0 p'
(13)
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils
1139
In this equation, 2(s) is the soil compressibility, i.e., the slope of the virgin loading line in the (e, log p') plane during a triaxial test. Its dependence on suction is assumed to be of the type 2(0)
2(s) - 1 + as
(14)
where 2(0) is the compressibility of the saturated sample and a a material parameter. Other representations of the dependence of the compressibility parameter on suction can be found in the literature which have the advantage of giving a more correct shape of the yield surfaces, e.g., Reference [2]. They present, however, computational disadvantages because of their complexity and require at least two material parameters, which is more demanding for identification purposes. In a similar way, the elastic part of the volumetric strain can be written as ~c(s) dp'
dee = v0 p'
(15)
to(0) being the slope of the elastic unloading line in the (e, In p') plane during a triaxial test. A dependence on suction of to(s) similar to Eq. 14 is supposed in the following. Some authors (e.g., Laloui et al. [7]) assume the elastic compressibility modulus tc independent of suction. This is, however, in contrast with the results of Figure 11.8.1 and probably depends on the range of applied forces, larger for rock-type materials. For volumetric deformation during an isotropic virgin compression loading (0 = 0), by substituting Eqs. 6-7 into Eq. 10, the plastic flow can be cast in the form devP =
1
dp'
HoHw p'
(16)
with the assumption that H = HoHwp': Hw is related to partially saturated behavior, whereas term H0 depends on the material characteristics in fully saturated conditions. Comparison of Eq. 16 with Eqs. 13 and 15 and accounting for Eqs. 11 and 14 gives the hardening modulus during isotropic compression and the hardening effects due to suction, respectively, as 1 +e0 H0 -- 2(0) - tc(0)' Hw -- 1 + as
(17)
For generic (not isotropic) stress paths and for deviatoric effects, the reader is referred to the chapter by Pastor et al., this volume. Using the same symbols, the plastic modulus results in H = Hop'HwHI(Hv + I-Is)
(18)
1140
Schrefler and Simoni
and can be further enhanced to account for memory effects and plastic unloading. Once the dependence on suction of the plastic modulus has been defined, we have to introduce the same effects in the yield and potential surface equations (Eqs. 8 and 9). Experimental observations show that parameter pf is increasing with suction [3]. Given the initial yield stress P~0i for saturated conditions, the dependence of pf on suction is assumed as
(19)
pf _ ply~(S) -- PlyO~ + is
Parameter i has to be determined by interpolation of experimental data to obtain an increasing function of suction when water saturation is less than one. Volumetric hardening is controlled by irreversible volumetric strain; hence, the evolution of the yield surfaces in the (pt s) plane is given as 1+as '
'
._7-
(20)
This equation has been obtained by requiring that, when moving from one yield surface (e.g., the initial one) to another one, the related plastic volumetric strain must be the same independently of the starting point and the followed path. The same variation with suction can be assumed for the parameter pg, which means that yield and potential surface expand with the same law when suction is increasing. This is suggested by the possibility of assuming associative plasticity, in which case f _= g.
11.8.4 APPLICATIONS
AND PARAMETERS
We show here two back calculations for kaolin and clay and one determination of parameters for a hydrocarbon reservoir sandstone by means of an inverse identification procedure. These examples present typical features of unsaturated soil behavior.
11.8.4.1
TESTS ON COMPACTED KAOLIN
Data for a compacted kaolin have been obtained from the extensive experimental investigation conducted by Josa [10]. Some of these data are given in Table 11.8.1. Data of the present model not defined in Reference [10] are calculated assuming an intermediate value of 0.3 MPa for the mean net
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils
1141
stress ~ (mean total stress in excess of air pressure) in the range of the applied pressures. Volume changes resulting from the present model, due to isotropic compression at different but constant suctions, s - - 0 . 0 6 M P a and s - - 0 . 0 9 MPa, are c o m p a r e d with experimental data in Figure 11.8.3. The initial m e a n stress P'y0 is equal to 0.045 MPa, and bulk m o d u l u s tc is equal to 0.015. According to the experimental results, the specific volumes at the beginning of each loading path considered here have been assumed equal to 1.888 for a sample tested at s - 0 . 0 6 M P a , and equal to 1.893 for s - - 0 . 0 9 MPa. Good agreement is achieved between model predictions and experimental data.
TABLE 11.8.1
Material Data for a Partially Saturated Kaolin [10]
s (MPa)
Sr
p'y (MPa)
,;t,(s)
0.04 0.06 0.09
0.838 0.826 0.812
0.099 0.120 0.158
0.100 0.085 0.070
1.9
1-86 ""
s = 0-09 MPa
1.82 "\
s = 0-06 MPa
1-78
1-74
.
i
008
.....
o'16
o 2.
o132
j
o.
n
p: M P a
FIGURE 11.8.3 Specific volume plotted against mean net stress (p-Pa) under isotropic compression at constant suction: comparison with experimental data (dots, reproduced from G~otechnique,vol. 40, 1990, Alonso, E.E., Gens, A., and Josa, A., A constitutive model for partially saturated soils, pages 405-430).
1142 11.8.4.2
Schrefler and Simoni TESTS ON CLAYS
A characteristic behavior of partially saturated soils was indicated by the experiments performed by Escario and S~iez [11] on clay samples. As shown in Figure 11.8.4 (dots), along the loading path corresponding to suction decrease from 3.5MPa to zero at constant vertical external pressure (0.2 MPa), the material exhibits first swelling and then collapse. The main features of this behavior can be reproduced as follows. Data listed in the quoted paper and given in Table 11.8.3 are not enough to characterize the material completely according to the present model. Back calculation has yielded the following missing parameters: Vo = 2.3, m = 0.8, l = 2 MPa, a - 0.1 MPa-~, Py0 = 0.07 MPa. Only the contribution to volume changes due to the isotropic state of stress p = 0.2 MPa is considered, neglecting volume changes induced by soil dilatancy in the nonassociative plasticity model, depending on the values of the confining pressure. With these assumptions, the predicted volume changes plotted in Figure 11.8.4 are in good qualitative agreement with the experimental data. s: MPa 3.5q
3
9
1.5
1
0.5
--0.02
0
0.02
0.04
~V
FIGURE 11.8.4 Specificvolume plotted against suction under suction changes at constant net stress: comparison with experimental data (dots, reproduced from Escario, V., and S~iez,J., Measurement of the properties of swelling and collapsing soils under controlled suction, in Proceedings 3rd I.C.E.S., 1973, Haifa, pages 195-200).
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils 11.8.4.3
1143
MATERIAL PARAMETERS IDENTIFICATION IN
AN OEDOMETRIC TEST FOR HYDROCARBON RESERVOIR SANDSTONE The last example deals with a laboratory experiment performed at IKU, SINTEE Trondheim, on behalf of AGIP (Italian National Petroleum Company) [12] on a silty consolidated sandstone sample extracted from a gas bearing formation in the Northern Adriatic basin at a depth of 3400 m. Effective porosity, in situ water saturation, and irreducible saturation of the material were previously obtained, and then the specimen underwent an oedometric test. The loading was scheduled as follows: the sample at in situ saturation (0.38-0.45) was first stressed with an initial hydrostatic phase presenting ~r-rate equal to 0.01 MPa/s until err ----0.5 MPa. This was followed by an uniaxial phase with ~rz-rate of 0.004 MPa/s until crz reached 35 MPa; the sample was then held at constant stress level and water was injected for 25 hours until full saturation was attained. During this period of time, volumetric changes of the specimen were recorded, as during the phases of stress changes. Once the full saturation was reached, a second uniaxial phase, at constant water content, with stress rate of 0.004 MPa/s till about 110 MPa was performed. The test also included unloading cycles to determine the elastic behavior and recoverable deformation. The water injection test (hydric path) simulated the behavior of the gas reservoir rock during artificial water injection or during the flooding associated with gas extraction. The model response at the end of the identification procedure is shown in Figure 11.8.5. The agreement is very satisfactory, in particular for the increase of volumetric strain caused by water injection. The values of the identified parameters are shown in Table 11.8.2.
FIGURE 11.8.5 Resultsof the identification.
1144
Schrefler and Simoni
TABLE 11.8.2
Identified Material Parameters for a Reservoir Sandstone
l
m
a
2(0)
i
pf
K(O)
flo
~1
~/f
0.9898
0.9867
0.9797
0.0122
1.0792
9.9490
0.0051
0.9675
1.0648
1.0016
TABLE 11.8.3
Material Parameters for Partially Saturated Materials
Reference
tc
2(0)
py (MPa)
i
Escario and Saez [11] Alonso et al.[2] Cui and Delage [13]
0.035 0.015 0.015
0.105 0.14 0.20
0.07 0.055 0.35
0.55 1.18 0.7
Finally, a set of parameters obtained from tests in the literature is listed in Table 11.8.3.
ACKNOWLEDGMENTS We thank AGIP for allowing us to use unpublished data concerning saturation experiments carried out for AGIP by IKU, SINTEF (Trondheim).
REFERENCES 1. Bolzon, G., Schrefler, B.A., and Zienkiewicz, O.C. (1996). Elastoplastic soil constitutive laws generalised to partially saturated states. Gkotechnique 46: 279-289. 2. Alonso, E.E., Gens, A., and Josa, A. (1990). A constitutive model for partially saturated soils. Gkotechnique 40: 405-430. 3. Fredlund, D.G., and Morgenstern, N.R. (1977). Stress state variables for unsaturated soils. J. Geotech. Engng. Div., ASCE, 103: No. GT5, 447-466. 4. Schrefler, B.A., Simoni, L., Li, Xikui, and Zienkiewicz, O.C. (1990). Mechanics of partially saturated porous media, in Numerical Methods and Constitutive Modelling in Geomechanics, (CISM lecture notes), pp. 169-209, Desai, C.S., and Sioda, G., eds., Wein: Springer-Verlag. 5. Brignoli, M., Santarelli, EJ., and Papamichos, E. (1995). Capillary effects in sedimentary rocks: Application to reservoir water-flooding, in Rock Mechanics, Proc. 35th U.S. Symposium, pp. 619-625, Daemen, J.J.K., and Schltz, R.A., eds., Rotterdam: Balkema. 6. Nova, R. (1982). A constitutive model for soil under monotonic and cyclic loading, in Soil Mechanics: Transient and Cycling loads, Pande, G.N., and Zienkiewicz, O.C., eds., Chichester: Wiley.
11.8 An Elastoplastic Constitutive Model for Partially Saturated Soils
1145
7. Laloui, L., Geiser, E, Vulliet, L., Li, X.L., Bolle, A., and Charlier, R. (1997). Characterization of the mechanical behaviour of an unsaturated sandy silt, in Proc XIVth Int. Conf. on Soil Mechanics and Foundation Engineering, Hambourg, pp. 347-350. 8. Fredlund, D.G., Morgenstern, N.R., and Widger, R.A. (1978). The shear strength of unsaturated soils. Canadian Geotech. J. 15: 313-321. 9. Wood, D.M. (1990). Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press. 10. Josa, A. (1988). An Elastoplastic Model for Partially Saturated Soils. PhD thesis, ETSICCP, Barcelona. 11. Escario, V., and Sfiez, J. (1973). Measurement of the properties of swelling and collapsing soils under controlled suction, Proc. 3 ra I.C.E.S., Haifa, pp. 195-200. 12. Papamichos, E., and Schei, G. (1998). Characterization of Adriatic soft weak sediments for subsidence studies, IKU, SINTEE Tronfheim, Norway, Report Nr. 33.0693.00/01/01/98. 13. Cui, Y.J., and Delage, P. (1996). Yielding and plastic behaviour of unsaturated compacted silt. G~otechnique 46" 291-311.
SECTION
11.9
"Sin fonietta Classica"'. A Strain-Hardening Model for Soils and Soft Rocks ROBERTONOVA Milan University of Technology (Politecnico), Department of Structural Engineering, Milan, Italy
Contents 11.9.1 A Short History of the Model . . . . . . . . . . . . 11.9.2 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9.3 Formulation for Uncemented Materials... 11.9.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9.5 Parameter Determination . . . . . . . . . . . . . . . . 11.9.6 Formulation and Parameter Determination for Cemented Materials . . . . . . . . . . . . . . . . . . 11.9.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1146 1147 1147 1148 1150 1152 1153 1153
11.9.1 A SHORT HISTORY OF THE MODEL The model that will be presented here is the result of successive modifications of a constitutive law originally conceived to model virgin sand [1]. That law was characterized by a nonassociated flow rule and isotropic strain-hardening, depending on both deviatoric and volumetric plastic strains [2]. Two important features were the possibility of modeling dilatant behavior even for normally consolidated dense sand and the occurrence of static liquefaction for loose sand. The model was successively applied to clays [3]. A threedimensional generalization with different mathematical expressions for loading function and plastic potential (with similar shape to the original version) was given in Reference [4]. The model was then extended to cover soft rock behavior [5]. The last is the version that will be presented here. More
1146
Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
11.9 A Strain-Hardening Model for Soils and Soft Rocks
1147
recent developments concern mainly two aspects: the effect of induced anisotropy, taken into account by means of a mixed kinematic-isotropic hardening [6,7], and the degradation of the mechanical properties of soft rocks due either to bond crushing [8] or to chemical and temperature effects [9,10]. The unusual name given to the model is a homage to Prokofiev's Sinfonia Classica, op. 25, which the author composed while conjugating contemporary music and a classical framework.
11.9.2 VALIDITY The model is applicable in geotechnical problems concerning a wide spectrum of soils (gravels, sands, silts, clays) and isotropic soft rocks (tufts, marls, calcarenites, chalks) either in dry or fully saturated conditions. It can also be applied to describe the behavior of particulate materials of nongeological origin (e.g., rice, wheat grains, powders). Either monotonically varying or oligocyclic loadings are considered. The model is appropriate, therefore, for describing neither ratcheting nor cyclic mobility of sand. It is also not suitable for describing highly anisotropic materials such as shales.
11.9.3 FORMULATION MATERIALS
FOR UNCEMENTED
The model is elastic-plastic strain hardening. Hardening (or softening) is assumed to be isotropic. Stresses are intended as effective stresses and are taken as positive in compression as well as strains. The plastic potential g and the loading function f are defined as follows: g - 9(2 - 3)ln -p' Pg f-
~/J3r/§9(2 - 1)J2r/= 0
(1)
?J3~§ Q-~(y- 1)J2~ ~_ 0
(2)
"-t
3]~(?- 3)ln -p' pc
where p,
-
1 ,
(3)
J2~ ~ rlijrlij
(4)
J3~ - ~j~jk~k~
(5)
1148
Nova
and the stress ratio tensor t/ij is defined as _ sij
r/ij - p,
(6)
Pc is the hidden variable which determines the size of the elastic domain and controls hardening or softening in the following way: Pc
~
gW+ { ( d h d k ) 2 + *
(7)
where
e{ - ga,j _
-
3
(8)
v'hk
(9)
When f < 0 the behavior is assumed to be hypoelastic: 9e
1 /~/
(10)
Hypoelastic strain rates are added to the plastic ones to give total strain rates when f - 0 and f - 0. It is assumed further that no principal stress can become negative (tension cut-off) and that there exists a small initial elastic nucleus delimited by a surface given by f = 0 with pc = pco. It is also assumed that the elastic volumetric compliance cannot be larger than Be/Pco.
11.9.4
PARAMETERS
In total, the constitutive law is characterized by seven nondimensional parameters and a reference pressure Pco. Only two constitutive parameters fully characterize plastic potential and loading function: fl and 2. The parameter pg is unessential since only the derivatives of g matter. 2 is linked to the so-called characteristic state [11]. In axisymmetric (socalled triaxial) compression
7
9 - M2 2M3 + 3 - M 2
(11)
where M is in turn linked to the mobilized friction angle ~b'CV at the characteristic state (associated with zero plastic volumetric strain rate) by the
11.9 A Strain-Hardening Model for Soils and Soft Rocks
1149
relation 6 sin 4b'c~ 3 - sin 4b'~
M --
(12)
4b'~ typically ranges from 22 ~ to 36 ~ the lower value being typical of plastic clays and the higher one of angular sands. Accordingly, 7 can vary between 3.44 and 4.41. fi controls the deviation from normality, which is only due to the spherical part. If fi = 3, normality holds. In general,/7 is smaller than 3. Typical values are fl = 2 for clays and/7 = 1.2 for sands. The parameters Bp and B< are the logarithmic volumetric compliances under isotropic loading, relative to plastic and elastic strains, respectively. The value of Bp depends on the type and density of the material tested. For a dense silica sand the order of magnitude of Bp is 0.01, for a loose sand the order of magnitude can be two or four times larger, and for a virgin clay it ranges from 0.02 to 0.1. The ratio BdBp is typically of the order of 0.2. However, the higher Bp is, the lower the ratio between the two parameters tends to be. The other elastic parameter, L, is linked to the shear modulus G. It can be derived, in fact, that in a test at constant isotropic pressure G-
p'
(13)
2L"
The apparent elastic shear modulus is therefore assumed to vary linearly with the isotropic pressure. The two parameters { and ~ are linked to the dilatancy at failure, df, in axisymmetric conditions
k~
-3
+
(14)
where 2 e - ~(el - e3)
(15)
depends on the type and density of the soil. It is zero for normally consolidated kaolinitic clay and for loose sand (and can be negative for carbonate sands when particle crushing takes place). The denser the sand, the higher the { value. A typical value for dense sand is 0.3, while a silty clay can have { = 0.1. When { is positive, dilatancy at failure is negative (expansion) in drained tests, while in undrained tests the stress path has a characteristic hook across the phase transformation line [12]. Indeed, the model predicts that the phase transformation line and the characteristic state coincide. ~ is generally ten times smaller than { and can be put equal to zero for most practical purposes. In this case the absolute value of the dilatancy at failure is
1150
Nova
the same in compression and extension. Experimental data show, however, that there is actually a small difference that can be accounted for by ~. The value of P,o is important only for small stress levels. It can be assumed to be of the order of a few dozen of kPa. In a boundary value problem, e.g., a shallow foundation, this means that the behavior of the elements close to the free surface is assumed to be elastic, for the first loading steps, at least.
11.9.5 PARAMETER DETERMINATION All the constitutive parameters, with the exception of Pco, have a clear physical meaning. Their determination from experimental data is straightforward, and only few triaxial tests are necessary. From an isotropic test with a cycle of unloading-reloading it is possible to determine Be and Bp (Fig. 11.9.1). In fact, their sum gives the slope of the straight line (in a semilog plot) connecting volumetric strains to isotropic effective pressure. Be is instead the slope of the unloading-reloading branch. These parameters can be conceptually linked to the traditional compressibility indices Cc and Cs, respectively. The other elastic parameter, L, can be obtained by the initial slope of the stress-strain curve in a pl constant test (Fig. 11.9.2). As already mentioned, ~ and ~ govern the plastic dilatancy at failure via Eq. 14. They can be determined from the results of a drained compression and a drained extension test, taking due care of the fact that sgn ep is different in the two cases. If for the sake of simplicity ~ is assumed to be zero, only the triaxial compression test is necessary to determine ~, as shown in Figure 11.9.2.
p#
F
p'
F CE tan'l(
p,o
D
m. v
8
B
D
8
FIGURE 11.9.1 Schematic results in isotropic loading in arithmetic and semilogarithmic plot.
1151
11.9 A Strain-Hardening Model for Soils and Soft Rocks
0"1 - 0"3 ~
3 p' tan- ' -~-~ rlf p'
FIGURE 11.9.2 Schematic results of a p' constant test on dense sand. ~k o"1 -
0- 3
O" 1 - -
z/f r/.
/,/
/
E
0" 3
r/M
C
...... -
....
A
"P'
v
FIGURE 11.9.3 Schematic stress path and stress-strain behavior in an undrained test on a medium loose sand.
The stress level at failure ~/f depends on df and on 7. Once df has been determined, 7 can be derived from the measured value of t/f, which is directly linked to the friction angle 4)' via a relationship formally identical to Eq. 12. An alternative way to determine y is to perform an undrained test (Fig. 11.9.3). The inclination M corresponding to y via Eq. 11 can be obtained by connecting the origin with point D on the stress path curve, characterized by a vertical tangent. Note that if 7 is determined this way and df as in Figure 11.9.2, the limit state is uniquely determined. The traditional limit value appears then to be determined by the inclination of the characteristic state and by the dilatancy, as in traditional soil mechanics. Note further that the asymptotic level q, reached by the stress path in the undrained test is slightly lower than qf. Finally,/~ can be determined by fitting the calculated curve to the data of the drained test, since the hardening modulus is proportional to //, which therefore controls the specimen deformability. Because of such a proportion-
115 2
Nova
ality, the value of fl can be calibrated also on the value of the coefficient of earth pressure at rest K0 or the inclination of the "instability line" /111[13] (Fig. 11.9.3).
11.9.6
FORMULATION
DETERMINATION
AND PARAMETER
FOR CEMENTED
MATERIALS
The effect of bonding between grains is taken into account by the introduction of two parameters, Pt and Pm, defined as in Figure 11.9.4. Formally, the equations of the plastic potential and yield function are the same as Eqs. 1 and 2, but the arguments of such functions are p* and q~ instead of p~ and r/ij, respectively, where
P* = P' + Pt *
__
nij
(16)
Sij
F
(17)
Hardening is controlled by three parameters: Pt, Pm, and Ps, where the latter varies with plastic strains as does p, in Eq. 7, and P, - - p p ,
(18)
and Pm is assumed to be proportional to Pt. In total, three additional constitutive parameters are introduced: the initial value of Pt, Pto, the proportionality constant ~, and a decay parameter of the bond strength, p. Since in the tensile range failure occurs at the tension cut-off, Pro is nothing else than the absolute value of the tensile strength in uniaxial extension (which can be evaluated by performing a direct tension or a brasilian test). A typical value of Pro for soft rocks is between 100 and 200 kPa.
vp
'~OT"
p,
"'"
Pm
-'
FIGURE 11.9.4 Yield locus for cemented and uncemented soil.
11.9 A Strain-Hardening Model for Soils and Soft Rocks
1153
An isotropic compression test allows the yield value Pco to be determined. Since Ps0 is small, ~ can be evaluated as a first approximation as the ratio between Pco and Pto. Such a value is usually between 10 and 20. Finally, the parameter p depends on the bond fragility, since it measures the rate at which the bonds are broken and the soft rock is transformed into a soil. It can be determined only by fitting theoretical results to experimental data in an isotropic compression test. The higher the p value, the faster the compression curve will join the corresponding curve for an uncemented material. Such a parameter can usually vary between 0.5 and 5.
11.9.7 EXAMPLES Examples of model performance can be found in Reference [4] (two types of sands in three-dimensional loading conditions), Reference [5] (sand, clay, grouted sand, tuff, and limestone in triaxial compression tests), and Reference [14] (natural and artificial calcarenite, marl in drained and undrained conditions).
REFERENCES 1. Nova, R., and Wood, D. M. (1979). A constitutive model for sand in triaxial compression. Int. J. Num. Anal. Meth. Geomech. 3: 255-278. 2. Nova, R. (1977). On the hardening of soils. Archi. Mech. Stos. 29 (3): 445-458. 3. Nova, R. (1982). A constitutive model for soil under monotonic and cyclic loading, in Soil Mechanics: Transient and Cyclic Loading, pp. 343-373, Pande, G .N., and Zienkiewicz, O. C., eds; Wiley. 4. Nova, R. (1988). Sinfonietta classica: An exercise on classical soil modelling, in Constitutive Equations for Granular Non-cohesive Materials, pp. 501-520, Saada and Bianchini, eds., Balkema. 5. Nova, R. (1992). Mathematical modelling of natural and engineered geomaterials. General lecture 1st E. C. S. M. Munchen, Eur. J. Mech. A/Solids 11, (Special issue): 135-154. 6. di Prisco, C., Nova, R., and Lanier, J. (1993). A mixed isotropic-kinematic hardening constitutive law for sand, in Modern Approaches to Plasticity, pp. 83-124. Kolymbas, ed. 7. di Prisco, C., Matiotti, R., and Nova, R. (1995). Theoretical investigation of the undrained stability of shallow submerged slopes. GF.otechnique 45 (3): 479-496. 8. Lagioia, R., and Nova, R. (1995). An experimental and theoretical study of the behaviour of a calcarenite in triaxial compression. Gr 45 (4): 633-648. 9. Nova, R. (1986). Soil models as a basis for modelling the behaviour of geophysical materials. Acta Mechanica 64: 31-44. 10. Nova, R. (2000). Modelling weathering effects on the mechanical behaviour of granite, in Perspective and Developments of Hypoplasticity, Kolymbas, D., ed. 11. Luong, M. P. (1980). Ph~nom~nes cycliques dans les sols pulv~rulents. Revue Franfaise de G~otechnique 10: 39-53.
1154
Nova
12. Ishihara, K., Tatsuoka, E, and Yasuda, S. (1975). Undrained deformation and liquefaction of sand under cyclic stresses. Soils and Foundations 15 (1): 29-44. 13. Lade, P. V. (1992). Static instability and liquefaction of loose fine sandy slopes. Jour. of Geot. Engrg., ASCE 118, (1): 51-71. 14. Lagioia, R., and Nova, R. (1993). A constitutive model for soft rocks, Int. Symp. Hard soils-soft rocks, Athens, pp. 625-632.
SECTION
11.10
A Generalized Plasticity Model for the Dynamic Behavior of Sand, Including Liquefaction M. PASTOR1, O.C. ZIENKIEWICZ2 and A.H.C. CHAN3 1Centro de Estudios y Experimentaci6n de Obras Pfablicas and ETS de Ingenieros de Caminos, Madrid, Spain 2Department of Civil Engineering, University of Wales at Swansea, United Kingdom 3 School of Engineering, University of Birmingham, United Kingdom
Contents 11.10.1 Range of Application of Model . . . . . . . . . . 11.10.2 Framework: Generalized Plasticity Theory 11.10.3 Model Description . . . . . . . . . . . . . . . . . . . . . 11.10.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10.5 Material Properties . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.10.1
1155 1156 1157 1160 1162 1162
R A N G E OF A P P L I C A T I O N OF M O D E L
Classical plasticity models are not able to predict basic phenomena encountered in dynamic loading of saturated soils, such as the accumulation of plastic strain and the cycles, compaction, and pore pressure buildup under undrained conditions. To overcome such difficulties, a simple model was proposed by Pastor, Zienkiewicz, and Chan [1] in the framework of the generalized plasticity theory of Zienkiewicz and Mroz [3]. The model is able to reproduce the behavior of dense and loose sands under quasi-static and dynamic loading, and it can be applied to the following situations:
9 static liquefaction of very loose sands and the instability phenomena found during undrained loading of medium-dense specimens; Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1155
1156
P a s t o r et al.
9 material softening of dense sands under drained conditions; 9 zero volume change at the characteristic state line; 9 densification of granular materials during drained cyclic loading; 9 liquefaction and cyclic mobility of loose sands under cyclic undrained loading. 11.10.2
FRAMEWORK:
GENERALIZED
PLASTICITY THEORY The relation between the increments of stress and strain for a material whose response does not depends on time can be written as de = C : da
where the fourth-order constitutive tensor C depends on (i) the state of stress and strain, (ii) the past history, and (iii) the direction of the stress increment da. In the case of soils, where coupling between skeleton and pore water exists, constitutive laws are formulated for the solid skeleton via the effective stresses a' = a + pwI, where a is the total stress tensor in the mixture, p~ is the pore pressure, and I is the second-order identity tensor. In what follows, and for the sake of simplicity, we will denote by a the effective stress unless otherwise specified. Concerning sign conventions, compression is negative and pore pressure compression is negative [2]. As far as the last requirement is concerned, the dependency on the direction of da can be introduced in several ways, and the approaches of Darve and the Grenoble group [4], the hypoplastic law of Dafalias [5], and the hypoplastic model of Kolymbas et al. [6] are worth mentioning here. The simplest approach consists of introducing a normalized direction n for any given state of stress such that all possible increments of stress are separated into classes, referred to as "loading" and "unloading": de,z . = C L
: da
de, u = C v : da
for
n : da>0(loading)
for
n : da<0(unloading)
The limit case n : da = 0 separating both classes of behavior will be called "neutral loading." To ensure continuity between loading and unloading, it can be shown that tensors CL and Cv have to be of the form 1
CL/U - C e + HL/UngL/U | where ngL and ngu are arbitrary tensors of unit norm and HL/u two scalar functions defined as loading and unloading plastic moduli. It can be very
11.10 A Generalized Plasticity Model for the Dynamic Behavior of Sand
1157
easily verified that both laws predict the same strain increment under neutral loading where both expressions are valid and hence nonuniqueness is avoided. As for such loading, the increments of strain using the expressions for loading and unloading are de,L=Ct. : d ~ = C e : d~ de,v = C v
: d ~ = C ~ : d~r
This suggests that the strain increment can be decomposed into two parts: de, = de, e + de,p
where the elastic and plastic components are de, ~ = C e : d a
1
de,p =
(ng/_/v |
n) " d a
Hc/v
To account for softeningbehavior (H/_/u
: da
for for
n : d~e>O(loading) n : da e
where d a e is given by d~e =
C e- 1 :
de,
All these relations can be cast in an equivalent vectorial form, substituting the fourth-order tensor by matrices and the second-order tensors by vectors. This simpler notation will be employed in the following sections.
11.10.3 MODEL DESCRIPTION The proposed general law includes scalars Hi_lu and directions n and ng/_lv, which have to be determined in order to fully characterize the material behavior. First of all, the direction of plastic strain increment ngL is obtained from triaxial tests, assuming that the ratio between total volumetric and deviatoric components is a good estimate of the ratio of the plastic components. The dilatancy law is approximated as
dd desP ~ dg -- (1 + 0~)(Mg - ~/)
1158
P a s t o r et al.
where a and Mg are material parameters and r/ is the stress ratio r / = q/p, where q = 3 V ~ and p = - t r ( a ) / 3 . Both parameters can be obtained from drained triaxial tests. It is interesting to note that Mg characterizes the states at which no volume change occurs. This is the characteristic state line of Habib and Luong [7], which coincides with the projection of the critical state line on the plane (p-q). From this point, direction ngL is obtained as
ngTL = (rigo, rigs) with
dg
and
ngs ---- - - T - - - ~ _
v/X+
Next, direction n will be assumed to be given by the law
(no, with =
no
1 df and ns = V/1 + df V/1 + d~
where df = dg = (1 + oOMf - tI. In these equations, a new material parameter Mf has been introduced. It can be seen that an associated plasticity model can be produced by choosing Mf = Mg. In the case of granular materials, it has been suggested to use the approximate relation Mf/Mg = Dr to estimate Mf [1]. The plastic modulus during virgin loading is postulated to be of the form [ 1]
HL - Hop'Hf {Ho + Hs}
( )4
where
limit the possible states, and where
=
being the accumulated plastic deviatoric strain ~ = f des.
11.10 A Generalized Plasticity Model for the Dynamic Behavior of Sand
1159
In these equations, the three new material parameters H0, 13o, and ~1 have been introduced. Concerning H0, it can be obtained from isotropic stress paths, but 130 and ~1 have to be obtained by finding the values which better fit experiments. The model so far developed needs to be completed by including plasticity during unloading and a memory function to describe past events. The latter aspect can be introduced by assuming that directions n and ngL do not differ in loading and reloading, and by taking the plastic modulus HL as
HL = Ho.p'.HS(no + n~)HDM where HoM is a discrete memory factor given by
( 7x) where ( was defined previously as
and ? is a new material constant which has to be chosen to provide the best fit to loading-reloading experiments. Finally, we characterize plastic behavior during unloading by introducing the plastic modulus during unloading H. as for M-~ug[> 1 for ]M~-ug_<1
-- Huo and assuming that direction ngv is
n g v - (nguo, ngus) T where
,,~.o - -ab~(n~o) and
ngus
=
+ngs
In this way volumetric deformation during nature is of a contractive nature, as observed in experiments. Elastic behavior is characterized by constants Keo0 and Go, the bulk and shear moduli of the material.
1160
P a s t o r et al.
11.10.4 APPLICATIONS We will show here some simple applications to the undrained behavior of sands of different relative densities under monotonic undrained loading and cyclic loading. Figure 11.10.1 shows stress paths in the p-q space followed by samples of different densities during undrained triaxial loading. The predictions agree reasonably well with the experiments of Castro [8]. In the case of very loose specimens, it can be seen how liquefaction develops. It is interesting to note that even if there is a decrease in the deviatoric stress, the mobilized stress ratio, which in frictional materials is a measure of the mobilized strength, is always increasing. The material keeps during the whole process a tendency to densify, as the pore pressure is always increasing. The stress path approaches the origin, with an effective hydrostatic stress approaching zero. In this limit, the soil behaves no longer as a solid, but rather as a viscous fluid. Under cyclic loading, liquefaction is the consequence of two factors, the accumulation of pore pressure during cyclic loading, and finally, liquefaction in a last cycle (in the same way that happens under monotonic loading). Figure 11.10.2 shows a comparison between experimental data and model predictions for a very loose sand under undrained cyclic loading. Liquefaction develops in the last cycle. p, = 392 KPa T~:st.
800 ~
Dr
a
b
c :~ r/a
29% .44%
O" A
47%
:.o
66%
600.
a
r
=,0
400
9>
--
(c)
200
.Z, "13,, 200
40:0
,,
600:.
iMean effective confiniagpressure p' (KPa) F I G U R E 11.10.1 U n d r a i n e d behavior of BANDING at different densities (data from Castro
[8]).
1161
11.10 A Generalized Plasticity Model for the Dynamic Behavior of Sand
. ~ . ; ~ . ....
~.0
L4
U
,I.II
,.o,. ~.m
i O
-. . . . . . .
..
p l l it.~ll o (~r
__
,. 9, ~
.,' ~.S~tU I
m ~ ~,,, .~,,
pntlr~-tl
(d)
'*
Lo:
9
,,m
I.D
FIGURE 11.10.2 Liquefaction of a very loose sand under cyclic loading (experimental data from Castro [8]).
Pn.,,evvo
i.~r,=
Pie
|j.
~
-. -go
"~2.
I~I~)
~,
..~
9~ m l .
9;,',' 91,.aa|,
.'",., u~.
~
,o ;~.-
~_
.
-
.
.
.
.
?;j/ .
.
.
.
.
.
.....
UI
LO
9II
~q"
V"V.V;/ 9
~;li
,
:,
L5
:6
,
m" flail
t 'r
:
,.
.
.. ,l D
.,4.0
,4.~
-ILO
r
IIA
FIGURE 11.10.3 Cyclic mobility of loose Niigata sand (experimental results from Tatsuoka []).
1162
Pastor et al.
TABLE 11.10.1 Material Properties
Keo0 Go
Mf Mg H0 ~0
~1 7 Huo
Yu
Fig. 11.10.1 and 11.10.2 (Dr =0.29)
Fig. 11.10.1 (Dr -- 0.44)
Fig. 11.10.1 (Dr --- 0.47)
Fig. 11.10.1 (Dr -- 0.66)
Fig. 11.10.3
35,000 52,500 0.4 1.5 350 4.2 0.2 4
35,000 52,500 0.545 1.32 350 4.2 0.2 1.0
35,000 52,500 0.570 1.12 350 4.2 0.2
35,000 52,500 0.72 1.03 350 4.2 0.2
65,000 30,000 0.71 1.5 800 3.8 0.16 1
600
200
2
2
--
--
--
--
--
--
250
5
If density increases, the sand will exhibit cyclic mobility rather than liquefaction. Figure 11.10.3 shows cyclic mobility of loose Niigata sand. As density is higher, the stress path crosses the characteristic state line and, because of the t e n d e n c y to dilate, turns towards the right. However, the a c c u m u l a t i o n of pore pressure shifts the cycles to the left, w h e r e the m e a n effective confining pressure is small. There, a m o m e n t arrives at w h i c h large d e f o r m a t i o n s occur, as s h o w n in the figure.
11.10.5 MATERIAL PROPERTIES T h e material properties are s h o w n in Table 11.10.1.
REFERENCES 1. Pastor, M., Zienkiewicz, O. C., and Chan, A. H. C. (1990). Generalized plasticity and the modelling of soil behaviour. Int. J. Numer. Anal. Methods Geomech. 14: 151-190. 2. Zienkiewicz, O. C., Chan, A. H. C., Pastor, M., Schrefler, B., and Shiomi, T. (1999). Computational Geomechanics, John Wiley and Sons. 3. Zienkiewicz O. C., and Mroz, Z. (1984). Generalized plasticity formulation and application to geomechanics, in Mechanics of En~neering Materials, Desia, C. S., and Gallaher R. H., eds., John Wiley and Sons. 4. Darve, E, ed. (1990). Geomaterials: Constitutive Equations and Modelling, Elsevier Applied Science. 5. Dafalias, Y. E (1986). Bounding surface plasticity, h Mathematical foundation and hypoplasticity. Journal of Engineering Mechanics ASCE 112: 966-987.
11.10 A Generalized Plasticity Model for the Dynamic Behavior of Sand
1163
6. Kolymbas, D. (1991). An outline of hypoplasticity. Archive of Applied Mechanics 61: 143-151. 7. Habib, P., and Luong, M. P. (1978). Sols pulvurulents sous chargement cyclique, in Materiaux and Structures Sous Chargement Cyclique, Ass. Amicale des Ingenieurs Anciens El~ves de l'Ecole Nationale des Ponts et Chauss~es (Palaiseau, 28-29), pp. 49-79. 8. Castro, G. (1969) Liquefaction of Sands. Ph.D. thesis, Harvard University, Harvard Soil Mech. Series no. 81.
SECTION
ii.ii
A Critical State Bounding Surface Model for Sands M A J I D T. MANZARI 1 a n d YANNIS E DAFALIAS 2
1Civil and Environmental Engineering, The George Washington University, Washington, D.C., USA 2Department of Mechanics, National Technical University of Athens, 15773, Hellas, and Civil and Environmental Engineering, University of California, Davis, California, USA
Contents 11.11.1 Triaxial Space Formulation . . . . . . . . . . 1164 11.11.1.1 Basic Equations . . . . . . . . . . . . 1165 11.11.1.2 Critical State . . . . . . . . . . . . . . 1166 11.11.2 Multiaxial Stress Space Generalization 1167 11.11.3 Implementation and Model Constants 1169 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170
11.11.1 TRIAXIAL SPACE FORMULATION The following constitutive model applies to sandy soils at different densities and pressures which do not cause crushing of the grains. It is a general purpose model for multiaxial, drained, undrained, monotonic, and cyclic loading conditions, within the general framework of critical state soil mechanics and bounding surface plasticity. The presentation is a direct derivative of the work by Manzari and Dafalias [1] and includes some additional expressions. The basic concepts and related equations of the sand plasticity model will first be presented in the classical triaxial space where q = or1 - ~r3, p : ( 1 / 3 ) ( ~ r l + 2cr3), ~q = ( 2 / 3 ) ( e l - ~3), and ~v = ~1 + 2~3, w i t h ~i and ~i (i = 1,3) being the p r i n c i p a l stress and strains (or2 = or3,
82
=
1164
83). Handbookof MaterialsBehaviorModels.ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
11.11 A Critical State Bounding Surface Model for Sands
1165
11.11.1.1 BASICEQUATIONS The elastic relation will be assumed hypoelastic for simplicity, given in terms of the r a t e s / / = dq/dt and ib = dp/dt as 9e _ // eq 3 6 O - Go
.e _ /5 ~
(1)
/ < - Ko
(2)
where G and K are the elastic shear and bulk moduli, respectively, Pat is the atmospheric pressure, and the exponent a is usually given a default value of a =0.5. The elastic range is represented by the shaded wedge shown in Figure 11.11.1 in the q, p space, whose straight line boundaries Oc and Oe constitute the yield surface described analytically by f=~/-~:Fm=O
(3)
where the stress ratio r / = q/p and the dimensionless (stress ratio type) quantities c~ and m are shown in Figure 11.11.1. The c~-line is the bisector of the wedge angle, and 2 mp measures the wedge "opening." Equation 3 implies that upon constant r/ loading no plastic loading occurs, which is approximately correct if the p is not high enough to cause crushing of the sand grains and/or the sand sample is not very loose. Based on Eq. 3, plastic deformation occurs only when r/is on f = 0 and there is a change dq = Odt pointing outwards f = O. In this case the plastic rate equations are given by
O
(4/
FIGURE 11.11.1 Schematicrepresentation of the yield, critical, dilatancy, and bounding lines in
q, p space.
1166
Manzari
and Dafalias
where Kp and D are the plastic modulus and dilatancy, respectively. Since .e + ~v 9p, it is a trivial exercise to combine Eqs. 1 and 3 in ~q = ' e~q+ ~ a n d ~ v = ~v order to express ~q and ~v in terms o f / / ( o r ~) and/~, and vice versa. There remains the very important task of specifying Kp and D. In reference to Figure 11.11.1, assume that the line shown with a slope Mcb represents a peak stress ratio for a given state. Henceforth, subscripts c and e imply association of a quantity with triaxial compression and extension, respectively. Such value of Mcb is a bound for ~/; hence, within the framework of bounding surface plasticity one can write a stress ratio "distance"-dependent plastic modulus expression such as K~ -- h ( M ~ - ~)
(5)
in terms of a model parameter h. Similarly, the line shown with a slope M~ in Figure 11.11.1, represents the phase transformation line, or for better naming, the dilation line. According to standard dilatancy theory, one can write
D -- A ( M ~ - ~1)
(6)
with A another model parameter. Hence, Eqs. 5 and 6 determine Kp and D via M~, Mca, h and A, and the model is complete.
11.11.1.2 CRITICALSTATE The critical state in soil mechanics is defined as a triplet of qc, Pc, and void ratio ec values, at which unlimited plastic deviatoric strain occurs at zero volumetric strain rate. Such a critical state is defined simultaneously in the q,p space by t/c--qc/pc=Mc, and in the e - p space by ec = (ec)ref- 2ln(pc/pref), where Me, 2, and (ec)ref (for a chosen pref) are standard soil constants. The Mc is related to friction angle, and its corresponding visualization in q, p space is shown as a line of slope Mc in Figure 11.11.1. If left as is, the previous formulation will not meet the critical state requirements. For example, it follows from Eqs. 4, 5, and 6 that as ~/ approaches M~, which may be assumed to be equal to Me, Kp~ 0 and ~ ~ 0 while ~ > 0 and ~v p < 0 since M~ > M~. This implies unlimited dilation, contrary to physical expectation (negative volumetric strain rate means dilation). Furthermore, a fixed M~ does not allow for the softening response in drained loading observed in many dense sand samples. The remedy is to consider variable M~ and Mca, such that at critical state Mcb = Mca = Me. If the so-called state parameter ~ = e - ec [2], is used as a measure of "distance" from critical state in the e, p space (e and ec refer to the same p), the idea put forth by Wood et al. [3] and supplemented by Manzari
11.11 A Critical State Bounding Surface Model for Sands
1167
and Dafalias [1] in regards to the concept of 0~ in Eq. 3 can be expressed by M~-- (zc + m = Mc - kbc~
(7)
where kcb is a material constant and ~cb is the "bound" for c~ in the same sense that M~ is the bound for r/. The second important modification refers to the variation of Mca and was proposed in Reference [1] as = o~c 4- m -- Mc 4-
(8)
where k~ is a material constant and a~a is a back-stress dilatancy ratio corresponding to the stress dilatancy ratio Mca. Observe that for ~ < 0 (denser than critical), M a c < M c < M ~ , while for ~ > 0 (looser than critical), Mc~
(9)
which defines the critical back-stress ratio ~ in terms of m and the critical stress ratio Mc. Equations 1-8 provide a complete constitutive model in triaxial space. It only requires the specification of Me, k~, and kea in triaxial extension as well to describe reverse and cyclic loading. In such case, Eqs. 5 and 6 utilize the ensuing values of M~ and M~ in extension, following from Eqs. 7 and 8.
11.11.2 MULTIAXIAL GENERALIZATION
STRESS SPACE
The multiaxial stress generalization of the model follows standard procedures [1]. Equation 1 becomes ~=
s 2--G
P ~ - K
(10)
where bold-face characters imply tensor quantities and e e and s are the deviatoric elastic strain and stress tensors, respectively. Equation 2 remains as is. Equation 3 generalizes to f -- [(s - pat):(s - pot)] 1/2 - X//-i/3 m p -- 0
(11)
where ~ of Eq. 3 generalizes to the back-stress-ratio tensor at and : implies the trace of the product of two tensors. The plastic strain rate equations are expressed in the following in terms of i', where the deviatoric stress ratio tensor r - s/p is the generalization of r/. After some algebra which involves the O f / O ~ = n - (1/3)(n : r)I with I the identity tensor and n given in following text, one has for the deviatoric and
1168
Manzari
and Dafalias
volumetric plastic strain rates, respectively, ~P -- < L > n - - < L > ( 2 / 3 ) - 1 / 2 ( r - a t ) L
1
~P -- < L > D
2Gn" ~ - K(n'r)~v -
+
2O
-
(12a)
(12b)
D(n
Equations 12 are the generalizationof Eqs. 4 involving the plasticmodulus Kp
and dilatancy D, with the added explicit feature of the loading index L expressed either in terms of/" or in terms of ~ and ev. A combination of Eqs. 10 and 12 together with ~ - he + ~P, ev - ~e + ~v p yields 6 -- E ep" /;-- [Ee -
(2Gn + K D I ) | ( 2 G n - K(n 9r)I)] Kp 4- 2 G - KD(n i~) _ "~
(13)
for the total effective stress rate 8 in terms of the total strain rate ~ and the elastoplastic tangent stiffness moduli E~. The latter is given explicitly by the bracketed quantity of the last member of Eq. 13, where | means tensor product and E ~ is the well known isotropic elastic moduli tensor which in component form is given by E~jkz -- 2 G ( 6 i k 6 j l - (1/3)cSijcSkz) + K6ij6kz. The Kp and D in the muhiaxial space will be obtained by generalization of corresponding triaxial concepts and equations. First, the bounding, dilatancy, b ~c, d and ~cc in Eqs. 7 , 8, and 9, and critical triaxial back-stress ratios c~c, correspondingly, are generalized as the bounding, dilatancy, and critical surfaces whose traces on the principal stress ratio ri = s i / p 7z-plane are shown in Figure 11.11.2. The To-plane trace of the yield surface with its center a, Eq. 1 l, is also shown in Figure 11.11.2 as a circle. For a stress ratio r and
/~.~
~ t~,~_. ~
./ // ,t~'~~ ,,t ,, / / ' n'~J "
r2 = ---~
Yield Surface
i~'.. ) BoundingSurface r - ' . ~ , . - ~ CriticalSurface ~ ancy Surface
...................
-
tl~+~
_
s3
r3----~
FIGURE 11.11.2 Schematic representation of the yield, critical, dilatancy, and bounding surfaces. (Reproduced with permission from Manzari, M. T., and Dafalias Y. E (1997). A critical state two=surface plasticity model for stress. Geotechnique47: 255-272.)
11.11 A Critical State Bounding Surface Model for Sands
1169
associated n, the "image" back-stress ratio tensors ~ , ~d, and ~ are defined as the intersection of the n direction emanating from the origin with the foregoing three surfaces. Their scalar-valued norms are analytically given by ~bo - - g ( O ) M c
-
gb(0)kcbff -- m
(14a) (14b)
m
~co - - g ( O ) M c -
(14c)
in terms of a third stress invariant, the Lode angle 0 as shown in Figure 11.11.2, entering the interpolation functions g, gb, and gd in order to account for the variation from triaxial compression to triaxial extension. Observe that for ~ - 0, ~ - ~ - ~, and the three surfaces collapse into the critical one. The corresponding image tensor quantities follow according to ~ - x/~~n, with a = b, d, or c (one must distinguish between the tensor a~ and its norm ~). The kinematic hardening is given by & - < L > h • (~b0 - ~ ) in terms of a model parameter h, which, together with the consistency condition f = 0 applied to Eq. 11, yields for t/l = 0 the value of the plastic modulus as a)'n
(15)
D -- A ( ~ - ~): n
(16)
K p - - ph(ctbo --
Similarly, the dilatancy D is given by
Observe that a combination of Eqs. 3, 5-8 yields for the triaxial case that K p - h ( ~ - ~) and D - a ( ~ - ~ ) , hence, Eqs. 15 and 16 are their direct generalization. In applications, h and A may be constant or functions of the corresponding distances b - ( a ~ - ~ ) ' n and d - ( ~ - ~)" n, respectively, Figure 11.11.2. In Reference [1] the expression h = h o ] b ] / ( b r e f - ]b[) was used in terms of a model constant h0. The dependence of Kp, and by extension of D, on a distance between a stress-type quantity ~ and its "image" ~b0 on a surface is the classical constitutive feature of bounding surface plasticity.
11.11.3
IMPLEMENTATION
AND
MODEL
CONSTANTS The model is a usual bounding surface plasticity model, and its implementation follows standard procedures. The reader is referred to Manzari and Prachathananukit [4] for details of a fully implicit implementation. The model constants are summarized and divided in categories in Table 11.11.1, together with a set of typical values in parentheses employed in Reference [1].
1170
Manzari and Dahlias
TABLE 11.11.1 Model Constants Elastic
Critical state
State parameter
Hardening
Dilatancy
Go (3.14 x 104) K0 (3.14 x 104) a (0.6)
Mc (1.14)
kcb (3.97) k~a (4.20)
h0 (1200)
A (0.79)
2 (0.025) (ec)ref (0.80)
The most peculiar to the model among the foregoing constants are the kcb and kca (and their corresponding value keb, kea in extension). The kcb can be obtained from Eq. 7 and the experimentally observed values of the peak stress ratio Mcb and state parameter ft. Similarly, the kca can be obtained from the observed value of Mca when consolidation changes to dilation together with the corresponding value of ft. These presuppose knowledge of the critical state line in e-p space. For different Mcb, Mca, and ~ts, different k~ and kca may be determined. It is hoped that these values do not differ a lot, and then an average value is the overall best choice. The h0 and A are obtained by trial and error (there are some direct methods also). All constants can be determined by standard triaxial experiments.
ACKNOWLEDGEMENTS M.T. Manzari would like to acknowledge partial support by the NSF grant CMS-9802287, and Y.E Dafalias by the NSF grant CMS-9800330.
REFERENCES 1. Manzari, M.T., and Dafalias, Y.E (1997). A critical state two-surface plasticity model for sands. Geotechnique 47: 255-272. 2. Been, K., and Jefferies, M.G. (1985). A state parameter for sands. Geotechnique 35: 99-112. 3. Wood, D.M., Belkheir, K., and Liu, D.E (1994). Strain softening and state parameter for sand modelling. Geotechnique 44: 335-339. 4. Manzari, M.T., and Prachathananukit, R. (2001). On integration of a cyclic soil plasticity model. Int. Journal for Numerical and Analytical Methods in Geomechanics, 25: 525-549.
SECTION
11.12
Lattice Model for Fracture Analysis of Brittle Disordered Materials like Concrete and Rock J.G.M. VAN MIER Delft University of Technology, Faculty of Civil Engineering and Geo-Sciences, Delft, The Netherlands
Contents 11.12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.12.2 Background of the Model . . . . . . . . . . . . 11.12.3 Elasticity Parameters . . . . . . . . . . . . . . . . 11.12.4 Fracture Analysis . . . . . . . . . . . . . . . . . . . 11.12.5 Fracture Parameters . . . . . . . . . . . . . . . . . 11.12.6 Computational Procedure . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1171 1172 1172 1174 1175 1175 1176
11.12.1 I N T R O D U C T I O N Lattice-type models have a long history, but the development of useful models dates back to the end of the 1980s, when they were reinvented in statistical physics for simulating pattern growth in random media. Examples are fracture, dielectric breakdown of fuse networks, diffusion-limited aggregation, etc. The overviews written by Herrmann and Roux [2], Charmet et al. [1], Van Mier [8], and Krajcinovic and Van Mier [3] provide excellent introductions and examples of applications to this type of modeling. In this contribution we will deal with a lattice model for fracture analysis of concrete only. Handbook of Materials Behavior Models. ISBN 0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
1171
1172 11.12.2
van Mier BACKGROUND
OF THE MODEL
In a lattice model the material is schematized as a network of linear elements, i.e., springs, truss, or beam elements that are used to describe dependency relations between the nodes in a lattice. The lattice is a regular or irregular construction in two or three dimensions. Examples are regular triangular lattices or triangular lattices with randomly varying element lengths. Triangular lattices are preferred since they yield more realistic estimates of Poisson's ratio [5, 7]. Earlier lattice models, for example, Herrmann and Roux, were based on square lattices, which result in a zero Poisson's ratio and when fracture is considered not very realistic crack patterns. A lattice construction of linear elements does not resemble the real microstructure or mesostructure of, for example, concrete or rock. The effects from material structure can be included in the model by superimposing an image of the relevant material structure on top of a lattice. Properties are then assigned to the lattice elements, depending on the specific location in the material structure. The image of the material structure can either be a computergenerated idealized structure [5] or a digital image from the real material [6]. Another method of introducing effects from material structure, i.e., randomness associated with variations of local material properties, is to assign lattice element properties according to a statistical distribution. For materials like concrete, the method of assigning properties by superimposing a "real" material structure seems the best method. An example of an overlay of a computer-generated structure on top of a regular triangular lattice is shown in Figure 11.12.1 The figure suggests that the lattice elements should be small enough to capture sufficiently small detail from the material structure. Indeed, the ratio between the length of a lattice element and the smallest material structural feature should be selected correctly. The advantage of a lattice as "computational backbone" for numerical materials science is the fact that simple laws can be introduced to simulate fracture. These aspects will be treated in Section 11.12.4.
11.12.3
ELASTICITY PARAMETERS
The first step in an analysis is the determination of the lattice element size and the elastic properties of the global lattice. From this point on, only triangular lattices made with beam elements and a particle overlay will be considered. Three phases are generally distinguished in a lattice with particle overlay, namely, the aggregate and the matrix phases, and the interfacial transition zone. In Figure 11.12.1 b, c the lattice elements falling in these three phases
11.12 Lattice Model for Fracture Analysis of Brittle Disordered Materials
1173
FIGURE 11.12.1 Example of an overlay of a computer-generated particle structure (a) on top of a lattice of different fineness: beam length l = 0 . 5 mm (b) and 2.0 mm (c), after Reference [9].
are given different line thicknesses. Each of these phases has own elastic properties. For convenience, all lattice elements are selected with the same cross-sectional properties, which makes the tuning of the elastic properties like the Young's modulus and the Poisson's ratio more straightforward. The first step is to set the length l of the beam elements as a fraction of the smallest aggregate particle drain. Experience learns that a ratio of dmin/l between 3 and 5 gives a good balance between computational costs and obtained accuracy. Next the beam height is set. The value for hll depends on the overall Poisson's ratio of a lattice in relation to the Poisson's ratio of the relevant material. Schlangen and van Mier [7] derived a relation between v and h/l for twodimensional regular triangular lattices as follows: v--
4
--1
3 + (h/l) After l has been decided on the basis of the particle size, h is defined depending on the required Poisson's ratio, w h i c h - for c o n c r e t e - is usually
1174
van Mier
in the order of 0.15 to 0.20. The global Young's modulus can now be set: ratios of EJEm and Eb/Em have to be defined. The subscripts a, m, and b stand for aggregate, matrix, and bond zones, respectively. The thickness of the beam elements is set to the thickness of the structure that is analyzed. For a lattice with random beam length, Vervuurt [10] derived the relation between average h// and the Poisson's ratio and Young's modulus from numerical analyses. In principle the procedure is identical to the one sketched previously. A closed-form solution does not exist for random triangular lattices.
11.12.4
FRACTURE
ANALYSIS
Fracture can be modeled in a very simple and straightforward manner by removing in each load step the beam element with the highest (effective) stress-over-strength ratio. According to the beam distribution over the three material phases, different strength values are assigned to beams falling in the respective phases. A strength criterion is used because this simplifies the computational procedure and limits the duration of the computations. Other criteria might just as well be used. Generally, however, when iterative procedures must be adopted because nonlinear fracture laws are introduced, the computational effort will certainly grow out of bounds. The important part is the definition of effective stress in the beams. The simplest approach would be to use an effective stress based on the normal force N in each beam only, i.e., N
-X
where A - - b x h is the cross-sectional area of the beam. At this stage one might wonder why truss elements are not used. Removing lattice elements simulates fracture, and upon a certain amount of removals unstable regions might develop in the finite element mesh. This might impair the solution of the problem, i.e., specifically the inversion of the stiffness matrix. A second possible effective stress is a combination of normal load and bending moment as proposed by Herrmann and Roux [2] following
N o~([Mi[]Mj[)max
o- ss- i
9
'w
Here Mi and Mj are the bending moments in nodes i and j of the beam element, and W-bh2/6 is the section modulus. ~ is a coefficient which regulates the amount of flexure that is taken into account. In fracture
11.12 Lattice Model for Fracture Analysis of Brittle Disordered Materials
1175
simulations of concrete this effective stress has proven to be quite effective, especially in cases when the external loading is tensile or a combination of tension and (global) shear. In case of global compression, this effective stress seems not very successful, and currently the search for a better candidate is under way. A Mohr-Coulomb-type effective stress might be a better option in that case. Recent analyses of a Brazilian splitting test revealed that the development of a shear cone in the parts of the specimen adjacent to the loading platens could be captured by means of a Mohr-Coulomb criterion. In that case, however, the vertical splitting crack did not appear, see Lilliu et al. [4].
11.12.5
FRACTURE
PARAMETERS
The model requires a limited number of fracture parameters. They are the respective strengths of the beams falling in either of the material phases aggregate, matrix and bond. Next to that, when the second effective stress "law" is used (see previous section), the coefficient ~ must be specified. The determination of the coefficient ~ is not straightforward. For example, in order to simulate crack face bridging in concrete subjected to tension realistically, the value of c~has to be selected close to zero (i.e., in the order of 0.005-0.010 [6]. For compressive failure, different values of c~yield different results, and the ratio between the uniaxial compressive and uniaxial tensile strength is significantly affected. In that case, the highest ratio between the uniaxial compressive and uniaxial tensile strength is also obtained for a relatively low value of c~, but at the cost of an increased brittleness [7]. In van Mier [8] additional information on the development of inclined crack planes under uniaxial compression is shown. Table 11.12.1 contains examples of parameter settings used in many of the analyses carried out to date; see van Mier [8] for an overview.
11.12.6
COMPUTATIONAL
PROCEDURE
The fracture simulation is carried out as follows. After the elastic properties have been determined, a unit test load is applied to the structure to be analyzed. For each beam element the effective stress is computed and divided by the strength of the phase in which it is situated. The beam with the highest stress-over-strength ratio is then removed from the mesh and a new linear elastic analysis (under the application of a unit test load) is carried out in order to decide which beam has to be removed next. After the first beam has been removed, the first point in the load-deformation
1176
van Mier
TABLE 11.12.1 Parameters for the Fracture Lattice Model (Values for Normal Strength Concrete Considered as a Three-Phase Model) Beam length 1 Beam height h Beam thickness b
l
Young's moduli
Ea/Em = 75.000/25.000 (MPa/MPa) Eb/Em = 25.000/25.000 (MPa/MPa)
Fracture strength
ft,a/ft,m=lO/5 (MPa/MPa) ft, b/ft,m = 1.25/5 (MPa/MPa)
Flexural parameter a
a =0.005
diagram is known, namely, by elongating the P-b diagram until P reflects the real fracture load of the considered lattice element. Thus ordinary scaling is applied. The computation is stopped after the last beam element has been removed and a complete fracture separates the structure into one or more pieces.
REFERENCES 1. Charmet, J. C., Roux, S., and Guyon, E. (1990). Disorder and Fracture, New York: Plenum Press. 2. Herrmann, H. J., and Roux, S. (1990). Patterns and Scaling for the Fracture of Disordered Media, Elsevier Applied Science Publishers B.V. (North Holland). 3. Krajcinovic, D., and Van Mier, J. G. M. (2000). Damage and Fracture of Disordered Materials, CISM Courses and Lecture Notes # 410, Wien/New York: Springer. 4. Lilliu, G., Van Mier, J. G. M., and Van Vliet, M. R. A. (1999). Analysis of crack growth of the Brazilian test: Experiments and lattice analysis, in Progress in Mechanical Behaviour of Materials, Proceedings ICM-8, Ellyin, J., and Provan, J. W., eds., Victoria, Canada, May 16-21, Vol. I "Fatigue and Fracture," pp. 273-278. 5. Schlangen, E., and Van Mier, J. G. M. (1992). Experimental and numerical analysis of the micro-mechanisms of fracture of cement-based composites. Cem. Conc. Comp. 14(2): 105118. 6. Schlangen, E. (1993). Experimental and Numerical Analysis of Fracture Processes in Concrete. Ph.D. thesis, Delft University of Technology, Delft, The Netherlands. 7. Schlangen, E., and Van Mier, J. G. M. (1994). Fracture simulations in concrete and rock using a random lattice, in Computer Methods and Advances in Geomechanics, pp. 1641-1646, Siriwardane, H. J., and Zaman, H.H., eds., Rotterdam: Balkema. 8. Van Mier, J. G. M. (1997). Fracture Processes of Concrete: Assessment of Material Parameters for Fracture Models, Boca Raton, FL: CRC Press.
11.12 Lattice Model for Fracture Analysis of Brittle Disordered Materials
1177
9. Van Mier, J. G. M., and Van Vliet, M. R. A. (1999). Experimentation, numerical simulation and the role of engineering judgement in the fracture mechanics of concrete and concrete structures. Constr. Build. Mater. 13: 3-14. 10. Vervuurt, A. (1997). Interface Fracture in Concrete. Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
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INDEX
2 1/4 Cr Mo stainless steel, material parameters for, 520 10 MnMoNi 5 5 steel chemical composition of, 443 material parameters for, 444 35NCD16 steel, material parameters for, 366 304 stainless steel material parameters for, 246, 366, 386, 456 stress-strain curves for, 384, 385 304L steel chemical composition of, 443 material parameters for, 444 316 stainless steel, material parameters for, 239, 366, 376 316SS, material parameters for, 316 1018 carbon steel, material parameters for, 239 1020 steel, material parameters of, 355 1026 carbon steel, material parameters for, 239 1045 steel, material constants for, 456 4340 steel, stress analysis of, 650
A A48 steel chemical composition of, 443 material parameters for, 444 A508 class 3 (NKS1) steel chemical composition of, 443 material parameters for, 444 A508 class 3 (SC1) steel chemical composition of, 443 material parameters for, 444
A508 class 3 heat A steel chemical composition of, 443 material parameters for, 444 A508 class 3 heat B steel chemical composition of, 443 material parameters for, 444 A533B steel, material parameters of, 355 AA6061-A1203-15p-T6 elasticity of, 625 material parameters of, 631 strain-controlled fatigue behavior of, 626-627 AA6061-A1203-20p-T6, material parameters of, 631 AA 7022 alloy material parameters of, 631 Additive laws, related to polymers, 259 Adhesion models of friction Bowden-Tabor, 742-743, 749-750 JKRS, 750-752 Adhesive bonding, and static friction, 716-717, 752-756 Aging transformation of glassy polymers, 945 equations of, 945-947 isothermal behavior, 951-953 nonisothermal behavior, 953 parameters for, 950-951,953 simulation of volume recovery, 947-950 and specific heat, 953, 954 Airy stress function, 634 AISI 316 stainless steel high-temperature creep behavior of, 839, 840 material parameters for, 520 1179
1180 AISI 1010 steel, material parameters for, 520 AISI 4340 steel, material parameters for, 832 A1-6XN, material parameters for, 394 A16061-T6, material parameters of, 355 A17039-T64, material parameters of, 355 Alloys deformation of, 135 gigacycle fatigue of, 476-477, 481-486 microstructure of 270-272 rate-dependent behavior of, 388 shape memory, 921-927 stress-phase transformation couplings in, 896-904 Alumina, elasticity of, 90 Aluminum material parameters of, 355 plasticity of, 272 Aluminum alloy 2024, material parameters for, 52O Aluminum alloy 8009, material parameters of, 355 Aluminum alloy AMG-6, material parameters of, 355 Aluminum alloys FLDs for, 295 high-temperature creep behavior of, 836-838, 842-845 A1203 material parameters of, 791 Weibull parameters for, 563 AI203/A1203 CMC, material parameters of, 1023 AM1, material parameters for, 316 Amontons-Coulomb friction, 705-708 Amorphous polymers in glassy state, 490-491 in rubbery state, 491 Anisotropic creep damage, 446-447 constitutive equation of, 451 effective stresses in, 450 evolution equation of, 451 interpretation of damage tensor, 448-450 model of, 450-451 parameters of, 451-452 representation of, 447-448 Anisotropic damage, 421-422 damage rate in, 425 equations for, 423-425, 514-515, 516-517, 522-523 kinetic law of evolution, 516-517 models of, 428
Index numeric simulation of, 428 in tension and compression, 518, 519 tensors in, 425-427 variables related to, 422-423 Anisotropy Burzynski yield condition, 159-160 formulation of yield conditions under, 155-157 Hershey "power yield condition," 160-162 Hill's yield condition, 157, 158 Huber-Mises-Hencky yield condition, 157-158 Tresca yield condition, 158-159 Antiplane strain, 551 Arbitrary hydrostatic pressure, 81 Armstrong-Frederick evanescent memory rule, 251 Arrhenius approximation, 111 Arteries cross-section of, 1067-1068 elasticity of, 1065-1066 material parameters of, 1069-1070 residual stresses on, 1066-1068 Asperities, 745 geometry of, 743, 744 plastic deformation of, 747-749 Astroloy, material parameters of, 355 Au4G, material parameters for, 316 Augmented Lagrangian method for analyzing friction, 785-786 Austempered SG cast iron, material parameters for, 470 Austenitic weld steel chemical composition of, 443 material parameters for, 444 AZ91 alloy elasticity of, 629, 630 fatigue crack growth in, 628
B Beams, 191 yield values for, 188-190 Belcadi-Parisot law, 259 Beremin model, 575 Biomaterials bone tissue, 1048--1055 soft tissue, 1057-1070 Birger criterion, 184
1181
Index Bishop-Hill theory, 145-146 Bodner-Partom model applicability of, 349-350 equations for, 350-352 material parameters for, 353-354, 355 modifications of, 353 uses of, 355 Bone described, 1048-1049 cancellous, 1049, 1052-1055 cortical, 1049, 1050-1051 elasticity of, 1050, 1053-1055 mechanical properties of, 1049-1050 types of, 1049 Boreholes, breakout of, 51 Botkin-Mirolyubov criterion, 181 Boundary layer effect, 61 Boundary lubrication, 761-762 Bounding surface model for sands constants for, 1170 critical state and, 1166-1167 equations for, 1165-1166 implementation of, 1169 in multiaxial stress space, 1167-1169 triaxial space formulation of, 1164-1167 Bowden-Tabor model of friction, 742-743, 749-750 Braking, 685 architecture of, 686-687 block-level analysis of, 687-688 Branching, of cracks, 653 Brittle damage, 412 continuous models of, 417-420 localization of, 527 monotonic failure, 529-530 Brittle fracture, 540 damaged zone around, 570-572 LEFM analysis of, 567-568 mechanisms of, 572-573 plasticity and, 569-570 probability of, 558 sliding crack models of, 577-581 small-scale yielding in, 568-569 triggering of, 566-567 Weibull model of, 559-563 Brittleness number, 45 Budianski-O'Connell damage parameter, 419 Bui-Erlacher model, 574 Building steel, material parameters for, 366 Burzynski yield condition, 159-160
c C fiber, Weibull parameters for, 563 C/C CMC, material parameters of, 1023 C-Mn steel ductile rupture of, 591,592 ductility of, 594 material parameters of, 595 C1008 steel, material parameters of, 355 Cancellous bone, 1049 elastic constant analysis for, 1053-1054 elastic constant data for, 1054-1055 specimens of, 1052-1053 Capillary pressure, 1134 Cast iron SN curves for, 479 strength figures for, 185 tension/compression ratios in, 184 Casting, phase transformation aspect of, 894 Cauchy stress tensor, 5, 77 Cauchy-Green deformation tensors, 76 Cavities, in glassy polymers, 493-494 Ceramic A1, material parameters for, 520 Ceramic-matrix composites (CMCs), 1015 damage to, 1016 material parameters of, 1021-1023 model of, 1019-1021, 1022 state potential of, 1017-1019 Ceramics brittle fracture of, 572 wear map of, 790 Clausius-Duhem inequality, 7 Clay elastoplasticity of, 1142 marine, saturated, disturbed state parameters for, 879-880 material parameters for, 1083 CMSX2, material parameters for, 316 Cobalt VO 795, material parameters for, 366 Cohesion, of third bodies, 692 Cohesive crack model, 56-57 Collagen, 1059 chemistry of, 1059 in connective tissue, 1060 mechanical properties of, 1060 types of, 1059 Compact bone, 1049 Complementarity, 783 Compliance law, 779
1182 Composites behaviors of, 958 biomaterials, 1048-1070 ceramic-matrix, 1015-1023 flow-induced anisotropy in, 1037-1046 laminate, 1004-1013 limit analysis of, 1025-1032 micromechanics of, 959-967 nonlinear, 969-982 nonlocal models of, 984-995 shakedown analysis of, 1032-1034 short-fiber, 1037-1046 transformational field analysis of, 996-1002 types of, 957 Compressible/ dilatant geomaterials, 1084-1085 boundary between compressibility and dilatancy, 1085, 1086 material parameters of, 1088, 1090-1092 model for, 1087, 1089-1090 short-term failure surface for, 1086 Compression, damage evolution in, 518-519 Compression fracture propagation of, 51 size effects in, 50-51 Concrete anisotropic damage model for, 507-512 brittle fracture of, 572 crack closure effects in, 509-510 disturbed state parameters for, 880, 881 evolution of damage in, 502-503, 508-509 fracture analysis of, 1171-1176 internal length of, 505-506 isotropic damage model for, 501-504 material parameters for, 520, 535 microcracking in, 501,507 nonlocal damage model enhancement for, 504-506 scale of observation for, 5 shear failure of, 51 strain softening in, 505 Conjugate gradient method for analyzing friction, 785 Constants, size effect method for measuring, 49-50 Constraint stress, 80 Contact elastic, 744-745 nominal area of, 743 pressure and, 743-744 Contact hysteresis, 732-733
Index Contact mechanics, 683-684, 685 nomenclature of, 689-692 Contact pressure limit, 684 Copper high-temperature creep behavior of, 838, 839 material parameters of, 355 Cortical bone, 1049 bovine, 1051 canine, 1051 elastic model for, 1050 technical constants for, 1050 Coulomb friction coefficient, 1097 Coulomb friction model, 705-708, 780 apparent deviations from, 708-711 local, 725-730 regularized, 781 Coulomb-Orowan law of friction, 781 Coulomb-Tresca-de Saint Venant criterion, 181 Coupled behaviors, 795 elasticity with damage, 802-808 elasticity with magnetism, 929-941 elastoplasticity with damage, 797-801, 802-813 elastoplasticity with phase changes, 905-913 elasto-viscoplasticity with damage, 817-820 inelastic behavior with damage, 814-820 plasticity with damage, 1010-1011 stress with phase transformation, 896-904 thermo-elasto-viscoplasticity with damage, 821-832 viscoplasticity with damage, 797-801, 1010-1011 viscoplasticity with diffusion, 849-855 Coupled equations, 23 Cr-Mo-V steel creep crack growth in, 597, 598, 599, 604 creep damage in, 600, 601,602 Crack band model, 58 Crack front waves, 658-659 Crack growth analysis of, 667-671 creep, 597-599, 605-609 curve of, 663 fatigue and, 611-620 geometry of, 635-636 prediction by Gurson model, 434 prediction by Rousselier model, 439, 441 rate of, 638-644
Index Crack opening displacement, 550 Crack tip, 498 damaged zones around, 568-572, 634-636, 642 elastic process zone at, 638, 640, 642 LEFM analysis of, 567-568 opening displacement at, 50 plastic process zone at, 568-572, 637, 638, 639 running, 498 stress analysis at, 647-650 velocity field at, 550 Cracks arrest of, 652 branching of, 653 effect of closure of, 509-510 failure at, 43-46 initial, 539 initiation of, 518, 651-652 permissible size of, 664-667 prevention of, 662 propagation of, 555-556, 788-789 reinitiation of, 652 residual strength after, 547, 662 running, 653-655 velocity of, 550, 655-656 Crazing growth of, 496-497 mechanism of, 491-496 vs. micro shear bands, 495-496 micromechanics of, 495 microstructure of, 494 rupture, 498 Creep and hydrogen attack, 858 influence on fracture growth, 55 multidamage state variable theories of, 839-847 single damage state variable theories of, 836-839 and static friction, 716 Creep crack growth in creep-brittle materials, 597-599,605-606 in creep-ductile materials, 597-599 prediction of, 606-609 Creep damage, 412 analysis of, 523 anisotropic, 446-452 in creep-ductile materials, 600-605 Creep failure, 540
1183 Creep functions, in viscoelasticity, 73, 74 Creep ratio, 736 Critical state, defined, 1166 CRSS ratios, 146 Crystallites crystal plasticity model of, 145-149 visco-plastic model of, 149-150 yield loci of, 146-147 Crystals anisotropic creep model for, 303-307 deformation of, 135 elastoplasticity of, 199-201 viscoplasticity of, 308-317 Cu-10AI-15Mn-5Zn shape memory alloy, 927 Cubic single crystals creep modeling of, 304-306 material parameters for, 306-307 Current configuration, defined, 76 Cyclic hardening, amplitude dependence of, 371-372 Cyclic plasticity, 214 flow rule, 217, 219 and material constant identification, 220-222 nonlinear isotropic hardening rule, 215216, 218-219 nonlinear kinematic hardening rule, 216, 219 one-dimensional model of, 214-217 three-dimensional model of, 217-220 Cyclic strain, damage caused by, 453-454 Cyclic viscoplasticity, 358-359 model of, 359-363, 365-367 nonproportional, 369-376
D Dahl model of friction, 740-741 extensions of, 742 Damage anisotropic, 421-428, 805 brittle, 417-420 creep, 446-452 coupled with elasticity, 802-808 coupled with elasto-viscoplasticity, 817-820 coupled with inelastic behavior, 814-820 coupled with plasticity in laminates, 1010-1011 coupled with thermo-elasto-viscoplasticity, 821-832
1184 Damage (continued) coupled with viscoplasticity, 797-801, 1010-1011 defined, 411 in ductile materials, 802-813 evolution of, 513-524, 808, 1000-1001 and fracture mechanics, 542-548 modeling of, 513-514 multiaxial fatigue, 453-456 and nonlinear compliance, 625 plasticity and, 552 threshold of, 517-518 types of, 412 Damage cards, 684 Damage evolution, 513-524 laws of, 808 in laminates, 1008-1010 modeling of, 1000-1001 Damage mechanics modeling, 633-634 and crack growth rate, 638-644 equations of, 634-637 table of parameters for, 644 Damage mesomodel of laminates applicability of, 1004-1005 concepts of, 1005-1006 extensions of, 1013 material parameters for, 1011-1013 single-layer, 1006-1011 use of, 1013 Damage surface, 807 Damage tolerance analysis, 662 Damage-equivalent stress fracture criterion, 413-414 model of, 414-4 16 Damage-sensitive, defined, 417 Damage-tolerant, defined, 417 Darcy's law, 1124-1125 Databases formats for, 19 for materials selection, 16-18 types of, 18 de Saint Venant-Bach criterion, 181 Debye frequency, 271 Deep drawing steel, material parameters for, 812 Deformable solids contact point of, 725 Coulomb friction model of, 726-727 friction of, 726 under normal load, 727-730
Index Deformation analysis of, 91-92 defined, 76 inelastic, 326-335, 404, 999-1000 inhomogeneous, 404-406 and plasticity, 247-253 of polymers, 255-257, 261-262, 398-406 and shape memory, 921-927 types of, 95 and viscoplasticity, 273 Deformation gradient tensor, 76 Deformation theory of plasticity, 191 equations for, 975-977 Delamination, 582-583 model of, 583-586 parameters for, 584-585 Differential laws, related to polymers, 259-269 Diffusion coupled with viscoplasticity, 849-855 size effects caused by, 61 Diffusion-viscoplastic model of oxidizing materials equations for, 850-851 material parameters for, 852, 853-855 use of, 852 Dilatancy, 1098 angle of, 1099 and compressibility, 1084-1086 of frictional granular materials, 1111-1112 Dilatation, and hydrostatic pressure, 104-105 Dissipative potential, 8-9 Distortional model, of plastic hardening, 167-170 uses of, 171-174 Disturbed state constitutive models applicability of, 875-876 equations for, 876-877, 878-879 implementation of, 878-879 material parameters for, 877-878 DNLR (distribution of nonlinear relaxations), 945 equations of, 112-113 kinetics of, 110-112 and mechanical solicitations, 113 parameters for, 113-116 thermodynamics of, 108-110 Drucker-Prager criterion, 136, 184 Ductile damage, 412 analysis of, 523 microvoids and, 430-431
1185
Index Ductile fracture, 540 anisotropic elastoplastic damage, 802-813 equations of, 589-590 fracture mechanics of, 554-555 inclusions and, 590 DRIIM model of, 587-588, 593-595 modeling of, 439 parameters for, 590-593 Ductile rupture integrating inhomogeneities in materials (DRIIM), 587-588 use of, 593-595 Ductility size effect on, 47-48 of third bodies, 692 Dynamic fracture mechanics applicability of, 645-647 crack front waves in, 658-659 of mode I fracture, 651-656 of mode II fracture, 656-657 stress analysis, 647-650 Dynamic plasticity, 198 constitutive modeling of, 268-270 instantaneous rate sensitivity in, 266 microstructures and, 270-272 range of strain rates in, 265-266 strain hardening in, 266-267
E Elastic contact, 744-745 Elastic strain, 6 Elasticity anisotropic, 805 of bone, 1048-1055 boundary conditions in, 81-82 of constrained materials, 80-81 coupled with damage, 802-808 coupled with magnetism, 929-941 of elastomers, 91-94 equations of, 77-78 nonlinear, 75-82 in orthotropy, 72, 78-80 of porous materials, 84-90 Elastin, 1059 characteristics of, 1060 mechanical properties of, 1060 Elastohydrodynamic lubrication, 762, 764 Elastomers, 91-92, 711 elastic parameters of, 93, 94 elasticity of, 93
friction model based on bifurcation characteristics, 720-724 stationary sliding on hard solids, 712-714 strain-energy function and, 94 Elastoplasticity anisotropic model of, 204-212 coupled with damage, 797-801,802-813 coupled with phase changes, 905-913 of crystals, 199-201 of granular materials, 1096-1102 of partially saturated soils, 1134-1144 of polycrystals, 201-202 rate-dependent constitutive model of, 377-386 Elasto-viscoplastic damage analysis equations for, 816-617 results for, 817-818 variables in, 818-819 Elasto-viscoplasticity, 301,302 Bodner-Partom equations for, 349-367 coupled with damage, 817-820 model of, 399-406 of polymers, 398-406 Electron back scattering diffraction (EBSD), 142 Energy balance, 5-6 Entropy production, and wear, 770-771 Epoxy/glass, delamination parameters of, 586 Epoxy/plexiglas, delamination parameters of, 586 Equivalent interstitial fluid pressure, 11301132 Eshelby tensor, 963 Eshelby's elastic accommodation factor, 851 Eshelby-Kr6ner localization law, 526 ESIS steel chemical composition of, 443 material parameters for, 444 strain parameters for, 443 Euler's tensor, 4 Eulerian principal axes, 76 Eutectic solder (Sn/Pb), material parameters of, 355 Extreme pressure lubrication, 761,762
F Fabric, 1109 Failure at crack initiation, 43 at long cracks or notches, 44-46
1186 Fatigue crack growth, 611-612 analysis of, 667-671 assessment of, 628 damage mechanics modeling of, 633-644 equations of, 612-620 grain size and, 613-617 size effect in, 56 yield strength and, 613-616 Fatigue crack growth analysis accuracy of, 670 applicability of, 667-668 limitations of, 670 mathematics of, 667-669 variables affecting, 671 Fatigue damage, 412 assessment of, 626-631 calculation of, 534-535 equations for, 623-624 under loading, 528-529 at mesoscale, 533 at microscale, 533-534 modeling of, 525-535 under multilevel loading, 532-533 nonlinear compliance effects and, 622-632 under proportional cyclic loading, 530-531 under two-level loading, 532 types of, 412 Fatigue failure, 540-541 fracture mechanics of, 555-556 FE (finite element)-based simulations, 138 implementation of kinematic hardening rules in, 238 under outlying conditions, 148, 150 Ferodo, 685 Ferrite-austenite steel ductile rupture of, 593 material parameters of, 595 Ferrite core, material parameters of, 937, 939, 941 Ferritic SG cast iron, material parameters for, 470 Ferritic steel, high-temperature creep behavior of, 845-847 Fiber composites, kink bands in, 52 Fictitious crack model, 56 Films high-pressure, 764 low-pressure, 762 First bodies, 689
Index Fluid pressure, equivalent interstitial, 1130-1132 Fluid third bodies, 680 thermal aspect of, 693 Formalism, defined, 266 Forming limit diagrams (FLDs), 293-294, 295 Fractality, and size effects, 60-61 Fraction model of inelastic deformation, 326-327 equations for, 327-331 parameters for, 331-334 use of, 335 Fracture brittle, 540, 558, 559-563, 566-581 ductile, 439, 540, 554-555, 587-595, 802-813 hydrogen transport and, 865-872 influences on growth of, 55-56 mathematics of, 664-665 mode I, 651-656 mode II, 656-657 modeling of, 1174-1176 Weibull model of, 558-565 Fracture mechanics, 34-35 applications of, 661-671 aspects of, 540-541 and damage, 542-548 defined, 539 dynamic, 645-658 energy issues in, 553-554 fatigue crack growth analysis, 667-671 geometric approach to, 544-547 linear elastic, 31, 37, 41-42, 48-49 precision of, 667 and residual strength, 665-667 size effect analysis of, 49-50 stress and, 550-552 thermodynamic approach to, 543-544 Fracture resistance, 665 dynamics of, 665, 666 Fracture toughness, 573 models for metals, 573-574, 575 models for nonmetals, 574, 575 Friction, 675, 677 and adhesion, 742-745, 749-754 analysis of, 678-679 augmented Lagrangian method for, 785-786 classification of, 703 coefficient of, 702 complementarity issues, 783
1187
Index compliance law for, 781-782 conjugate gradient method for, 785 contact mechanics and, 683-684, 742 Coulomb, 705-709, 780 described, 693-697, 702 Gauss-Seidel method with Aitken procedure for, 785 historical analyses of, 681-682, 703, 760 industrial criteria involving, 684 iterative analysis of, 786 kinetic, 730-734 Lagrange multipliers in, 784 laws of, 706, 779-782 Lemke method for, 785 load and, 708-709 local Coulomb model of, 725-729 lubrication and, 703, 760-766 mathematical models of, 738-741 mechanical analysis of, 695 modeling issues, 704-705 in nanoscale, 705, 756 Newton-Raphson method for, 786 nonstationary characteristics of, 716-724 numerical formulations of, 782-786 overrelaxation method for, 785 partial slip and, 730-732 physical modeling of, 741-744 physical theory of, 744-748 and plasticity, 738, 746-748 properties of, 702-703 and rolling, 734-738 scale of analysis for, 677-678 sliding and, 709-711 smooth nonlinear formulation of, 784 static, 752-756 stationary, 714-716 surface roughness and, 717 transmission of tangential forces by, 729731 and unilateral contact, 777-786 variable, 781 velocity-dependent, 719-720, 721-722 vibration and, 724-725 viscoelasticity and, 711-716 viscosity and, 680 Friction force, 760 Friction torque, 762, 763 nondimensional, 766 Friction-velocity curve, 714-716 Frictional granular materials
anisotropy of, 1107 back stress in, 1109 case study of, 1114-1116 characteristics of, 1108-1109 dilatancy of, 1111-1112 elasticity of, 1112 fabric of, 1109 inelastic deformation rate of, 1110-1111, 1114 material functions of, 1114 rate constitutive relations for, 1112-1113 yield criterion for, 1110 Frictional vibration, 714-716 in nonstationary sliding, 718-724 Fully coupled anisotropic elastoplastic damage model applicability of, 802-803 calibration of, 808-810 damage parameters for, 809-810 elastic parameters for, 808-809 equations for, 805-808 plastic parameters for, 809 use of, 810-811
G G'Sell-Jonas law, 263 Galilei-Leibniz criterion, 181 Gauss-Seidel method with Aitken procedure for analyzing friction, 785 Generalized limit criterion, 175-176 equations for, 179-186 model of, 177 parameters for, 177-179 Geomaterials behavior of, 1076 compressible and dilatant, 1084-1092 defined, 1075 equations for, 1077-1079 frictional granular, 1107-1116 granular, 1093-1105 nonlinear constitutive relations of, 10791083 partially saturated soils, 1134-1144 poroelasticity of, 1118-1132 sands, 1155-1170 soft rocks, 1146-1153 soils, 1146-1153 Gerber diagrams, 457
1188 Gibbs equation, applied to DNLR, 108-110 Gigacycle fatigue, 412 of alloys without inclusion, 481-486 crack initiation in, 479 of flawed alloys, 476-477, 486-487 initiation mechanisms of, 486 initiation zones of, 479, 486 prediction of, 480-481 SN curve shape, 472-474 ultrasonic testing of, 474-476 Glass brittle fracture of, 572 tension/compression ratios in, 184 Glassy polymers cavitation in, 493-494 crazing in, 491-495,496-499 macromolecules in, 489 micro shear bands in, 495-496 microstructure of, 488-491 structures in, 489-490 Goodman-Haigh diagrams, 457 Gradient plasticity approaches to, 282-283 asymmetric stress models of, 284, 286-287, 288 elementary models of, 283-287 and forming limit diagrams, 293-295 and size effects, 288-293 symmetric stress models of, 283-284, 285-286, 287 Granite elasticity of, 89 material parameters for, 581 Granular materials described, 1093-1095 elastoplasticity of, 1096-1102 forces in, 1095-1096 frictional, 1107-1116 postfailure behavior of, 1103 shear-banding of, 1102-1105 Graphite, 684 tension/compression ratios in, 1.84 Green-Lagrange tensor, 4 Greenwood-Williamson model, 744 Guggenheim's theory, 111 Gurson model, modified, 430 described, 431-434 parameters for, 434 use of, 434-435
Index
H Hadamard condition, 772 Haigh-Westergaard coordinates, 176, 177 Hardening, 198 under cyclic loading, 371-372 described, 166-167 distortional, 167 equations for describing, 167-169 model of, 170-174 rate-independent, 364 types of, 198 Hardening coefficients, determination of, 334 Hashin-Shtrikman equations, 989 Hastelloy-X, material parameters of, 355 Hayhurst criterion, 184 Helmholtz free energy, 804 Helmholtz specific free energy, 7 Helmholtz state potential energy, 413 Hereditary integral, 102-103 Hershey "power yield condition," 160-162 Hertz theory of contact, 728-730, 739 High-cycle fatigue, 412 flaw distribution and, 464-466 localization of, 527 material parameters in, 469-470 prediction of, 457-463 probabilistic model of, 466-471 High-strength steel, material parameters for, 812 Hill's yield condition, 157, 158 Homalite 100 crack propagation in, 656 stress analysis of, 648, 649 Homogenization bounds of, 965 cell methods of, 965-967 for linear elastic materials, 962 mean-field approaches to, 963 self-consistent methods of, 964 Hooke's law of elasticity, 71 Huber-Mises-Hencky yield condition, 157158, 179-180, 181 HY100 steel, material parameters of, 355 Hydrocarbon reservoir sandstone, material parameters of, 1143-1144 Hydrodynamic lubrication, 762, 763 example of, 764 Hydrogen attack conditions for, 856-858
1189
Index damage mechanics model of, 860-861, 862-863 equations for, 859-861 model parameters for, 861-862 thermodynamic model of, 859-860 Hydrogen transport and cracking, 865 and embrittlement, 869 equations for, 865-866, 869-870 in hydride-forming systems, 868-872 model parameters for, 866-868, 870-872 in nonhydride-forming systems, 865-868 Hydrostatic lubrication, 762, 763 example of, 763 Hydrostatic pressure and dilatation, 104-105 effect on polymers, 135 Hyperelastic material, defined, 77 Hyperelasticity, 248 Hysteresis, and friction, 740-741
I Identification, 9-10 quantitative, 11-12 IMI 834, material parameters of, 631 IN 100 superalloy creep crack growth in, 597, 598, 599, 605 material parameters for, 366 IN738LC, material parameters for, 239 Inclusions, and crack initiation, 476-477, 486 INCO 718 superalloy material constants for, 456 material parameters for, 355, 366 Incompressibility, 80 Inelastic behavior, coupled with damage, 814-820 Inelastic compressibility, VBO model for, 337, 340-343 Inelastic deformation, 327 evolution of, 999-1000 model equations for, 327-331 model parameters for, 331-334 Inelastic incompressibility, VBO model for, 341,343 Inelastic strain rate, 370 Inextensibility, 80 Initial crack, 539 residual strength after, 547 Injection molding, 1045
Interface temperature, 684 Internal length, determination of, 505-506, 548 Internal variable magnetoelastic model, 932 assumptions of, 933 state coupling validity, 932-933 Iron hydrogen transport parameters for, 868 stress-strain curves for, 272 Irreversible processes, thermodynamics of, 5-6 Isotropic damage equations for, 514-516 kinetic law of evolution, 516 in tension and compression, 518-519 Isotropic hardening nonlinear, 198, 215-216, 218-219 variable, 372-373 Isotropy generalized limit criterion in, 175-186 yield limits in, 129-136
J JKRS model of friction, 750-752 Junction growth, 748
K Kalker creep, 739 Kaolin elastoplasticity of, 1140-1141 partially saturated, material parameters for, 1141 Kelvin-Voigt model of viscoelasticity, 73, 96, 98 creep deformation of, 98-100 dynamic deformation and, 102 relaxation behaviors of, 100-101 Kinematic hardening in biaxial ratcheting, 240-246 material parameters under, 238-239 multilinear, 232-239 nonlinear, 198, 216, 219, 358-367, 370 and thermo-elasto-viscoplasticity, 827-828 Kinetic coupling, 796 Kinetic friction irreversibility of, 732-734 transition to, 730-732 Kinetic friction force, 760
1190 Kinetic law of damage evolution in anisotropy, 516-517 applications of, 523-524 in isotropy, 516 parameters of, 519-522 Kink bands, size effects in, 52 Koval'chuk criterion, 184
L Lagrangian principal axes, 76 Laminates damage coupled with plasticity, 1010-1011 damage evolution in, 1008-1010 damage kinematics in, 1006-1008 described, 1005-1006 material parameters for, 1011-1013 single-layer model of, 1006-1011 Large deformations hardening model for, 250-253 hyperelasticity in, 248 kinematics of, 247-248 plasticity equations for, 248-250 Lattice model of brittle disordered materials background of, 1172 elasticity parameters in, 1172-1174 fracture analysis in, 1174-1175 fracture parameters for, 1175, 1176 history of, 1171 procedure of, 1175-1176 Lemke method for analyzing friction, 785 Levy-Mises equation, 241 Limit analysis applicability of, 1025-1027 examples of, 1034, 1035 kinematic approach to, 1029-1031 linear programming procedure for, 1031-1032 static approach to, 1027-1029 uses o f, 1035 Linear elastic fracture mechanics (LEFM), 31, 37, 41-42, 544, 549-550 and brittle fracture, 567 equivalent, 48-49 small-scale yielding assumption in, 552 stress in, 550-552 uses of, 556 Linear poroelasticity, 1118 equations of, 1118-1120 field equations for, 1122-1123
Index in isotropy, 1121-1122 material parameters of, 1124-1125 plane strain in, 1123-1124 Linear viscoelasticity with damage, 117 equations for, 118-119 functions and parameters for, 121-123 model of, 119-121 Linearity, 9 Liquefaction of sand, 1155-1156 under loading, 1160, 1161 LMT-Cachan model, 940-941 Loading effect on friction, 708-709 plastic deformation under, 747-748 Loading function, 197 Localization, 198 conditions for, 276-280 equations for, 275 Low-carbon steel, material parameters of, 937, 939, 941 Low-cycle fatigue, 412 Lubrication boundary, 761-762 elastohydrodynamic, 762 extreme pressure, 761,762 hydrodynamic, 762, 763, 764 hydrostatic, 762, 763 mixed, 764
M M330-50HA, material parameters of, 937, 941 Macromolecules, in glassy polymers, 489 Macroscale, defined, 411 Magnetism coupled with elasticity, 929-941 internal variable model of, 932-933 Preisach model of, 929-931 Magnetoelastic hysteretic behavior, 937-939 LMT-Cachan model of, 940-941 Magnetoelastic scalar reversible behavior models of anhysteretic magnetization, 934-935 models of anhysteretic magnetostriction strain, 935 parameters for, 937 scalar model of, 936 Mariotte criterion, 181 Mass conservation equation, 76-77 Material constants, identification of, 220-223
Index Material heterogeneity, and size effects, 61 Materials selection, 15-16 databases for, 16-18 design procedure for, 22-24 modeling in, 24-25 multiple optimization, 23-24 performance indices and, 19-22 procedure for, 22-23 software for, 27-28 Matric suction, 1134 Maximum shear stress criterion, 136 Maxwell model of viscoelasticity, 73, 96, 97 creep deformation of, 98-100 dynamic deformation and, 101-102 relaxation behaviors of, 100 Mechanical loss, 102 Mesocracks, initiation of, 518 Mesoscale, 5 defined, 411 Metal ceramics, tension/compression ratios in, 184 Metallo-thermomechanical coupling, 884-885 heat conductive equation for, 891 phase transformation kinetics, 891-894 plastic strain rate and, 888-889 stress-strain constitutive equation for, 888-891 thermodynamics of, 885-887 viscoplastic strain rate and, 889-891 Metals anisotropic elastoplastic model of, 204-212 brittle fracture of, 572-573 coupled behaviors in, 884-894 deformation of, 135 elastoplasticity of, 199-202 fracture toughness models of, 573-574, 575 microstructure of 270-272 multiaxial fatigue in, 453-456 multisurface hardening model for, 223-231 plastic hardening of, 166-173 scale of observation for, 5 Mica/aluminum, delamination parameters of, 586 Microcracks, 411,418-419 propagation of, 453-454 Micromechanical models of brittle damage, 417-420 of composites, 984-995 Micromechanics equations of, 960-962
1191 history of, 959 homogenization in, 962-967 nonlocal models of, 984-995 Microscale, defined, 411 Microvoids, 411, 412 coalescence of, 430-431 modified Gurson model of, 430-435 prediction of, 434 Miner's rule, 532 Mises criterion of yield, 132-134, 135 Mixed lubrication, 764, 765 Mobilized dilatancy angle, 1099 Mobilized Mohr-Coulomb angle of internal friction, 1098 Mode I fracture arrest of, 652 initiation of, 651-652 Mode II fracture, 656-657 Modeling described, 3-4 domain of validity, 13 formulation of, 6-9 identification of variables, 9-12 implementation of, 14 in materials selection, 24-25 model choice, 13 of nonlinear phenomena, 10 scales of observation in, 5-6 variable choice in, 4-6 Modified 9Cr-lMo steel, material parameters for, 239 Modified Gurson model, 430 described, 431-434 parameters for, 434 use of, 434-435 Modulus of rupture test, 43 Mohr criterion, 181 Molybdenum, material parameters for, 392 Molybdenum bisulfate, 684 Mori-Tanaka method, 964 Multiaxial fatigue criteria, 457-459 formulation of, 459-461 model of, 461-463 parameters for, 461 Multiaxial fatigue damage criteria, 453 models of, 454-456 multiscale-based models of, 457-463 Multifractal scaling law (MFSL), 60-61 Multiple optimization, 23-24 Multiplicative laws, related to polymers, 258
1192 Multisurface hardening model, 198 applicability of, 223-224 formulation of, 224-230 hardening moduli, 228-230 loading surfaces in, 224-228 N Natural configuration, defined, 77 Natural reference state, 327 Necking, 255-256 of polymers, 405 in sheet-metal forming, 293, 294 and void coalescence, 431 Newton-Raphson method for analyzing friction, 786 Nickel, material parameters of, 937, 941 Nickel alloys, SN curves for, 478 Nickel base alloy B1900+Hf, material parameters of, 355 NiCoCrA1Y, material parameters for, 853-855 Nimonic 80 alloy, high-temperature creep behavior of, 841-842 Nimonic 90 alloy, material parameters for, 366 Niobium hydrogen transport parameters for, 868, 872 material parameters for, 392 NKS 3 steel chemical composition of, 443 material parameters for, 444 NoLIKH model, 214 one-dimensional, 215-217 three-dimensional, 217-220 Nominal stress tensor, 77 Nonlinear compliance, 622-623 damage-induced, 624 intrinsic, 624 material parameters for, 624-626 Nonlinear composites, 969-970 secant methods for, 970-977 variational bounds of, 978-982 Nonlinear elasticity, 73 deformation, 75-76 stress and equilibrium, 76-77 Nonlinear kinematic hardening-based model, 360, 362 equations for, 361 formulation of, 359-360 material parameters for, 363-365,366 use of, 365-367 Nonlinear kinetics, 110-111
Index
Nonlinear poroelasticity described, 1126 equivalent interstitial fluid pressure in, 1130-1132 incremental equations of, 1128-1129 strain in, 1129-1130 thermodynamics of, 1126-1128 Nonlocal micromechanical models applicability of, 984-985 bounds for, 988-989 equations of, 987-988 formulation of, 985-987 gradient approximations in, 990-991 for statistically uniform media, 990 utility of, 992-993,995 variational formulation for, 991-992 Nonproportional cyclic viscoplasticity, 369 analytical expressions of, 373-374 assumptions regarding, 369 equations for, 370-373 numerical simulations of, 375 Normality rule, 197 Norton's parameters, 11 Norton's power law, 301 Nylon, friction of, 712
O Objective strain-energy functions, defined, 78 OFHC copper, material parameters for, 316, 394 Orientation distribution function (ODF), 139-140 calculation from pole figures, 140-141 determination by single grain orientation, 141-142 discretization of, 143 sample homogeneity and, 142-143 Orthotropy, elasticity in, 72 Output limit, 684 Overrelaxation method for analyzing friction, 785 Oxidizing multiphase materials, 849-850 diffusion-viscoplastic model of, 850-855 material parameters of, 853-855
P Partial slip, and transition from static to kinetic friction, 730-732
Index Partially saturated soils capillary effects in, 1134 equations for, 1136-1140 material parameters for, 1140-1144 suction of, 1135-1136 Paul criterion, 184 Pb40/Sn60 solder, material parameters for, 882-883 Peeling, 753-754 transition to sliding, 754-756 Peierls stress barrier, 272 Performance indices, 19-20 classical, 20 drawbacks of, 21 use of, 21 Phase transformations coupled with elastoplasticity, 905-913 coupled with stress, 896-904 engineering processes involving, 893-894 kinetics of, 891-894 shape memory and, 923 solid-solid, 915-920 Piezoviscous effect, 764 Piola-Kirchoff tensor, 5 Pisarenko-Lebedev criterion, 184 Planar orthotropy, yield conditions in, 162-163 Plane strain, 72, 551 in linear poroelasticity, 1123-1124 Plane stress, 71,551 Plastic flow rule, 1099 Plastic hardening described, 166-167 distortional, 167 equations for describing, 167-169 model of, 170-174 Plastic limit analysis, 188 Plastic multiplier, 198 Plastic potential, 144-145 Plastic slip, 308, 315 Plastic strain, 6, 269 Plastic yield surface, anisotropic, 805-807 Plasticity concepts in, 197-198 coupled with damage in laminates, 10101011 cyclic, 214-221 and damage, 552 of damaged materials, 424 defined, 197
1193 deformation theory of, 191,975-977 dynamic, 265-272 equations describing, 248-249 and friction, 740 gradient, 281-296 in large deformations, 247-253 localization in, 274-280 of polymers, 255-264 of porous metal, 436-445 theory of, 40 transformation-induced, 905-913 yield limits and, 127-128 Plasticity model for sand behavior applicability of, 1155-1156 equations for, 1157-1159 generalized plasticity theory, 1156-1157 and liquefaction, 1155-1156 uses of, 1160-1162 Plate theory, 191-193 Poisson effect, 61 Poisson's ratio, 11, 71 Polar decompositions of deformation gradient, 76 Pole figures, calculation of ODF from, 140-141 Polycarbonate behavior of, 259 material parameters of, 402 Polycrystals averaging of, 321-325 crystal plasticity model of, 145-149 deformation of, 135 elastoplasticity of, 199-202 homogenization of, 964 microstructure of 270-272, 984-995 plastic potentials of, 144-145 precalculated databases of potentials of, 150-153 rate-dependent model for, 388-390 static model of, 324 Taylor model of, 324, 325 temperature-dependent model for, 391-396 viscoplasticity of, 149-150, 318-325 yield loci of, 137-153 Polyethylene, friction of, 712 Polyimide/glass, delamination parameters of, 586 Polymers, 107-108 additive laws relating to, 259 in complex loading conditions, 262
1194 Polymers ( c o n t i n u e d ) "~ damaged, viscoelasticity of, 117-123 deformation of, 398-399 differential laws relating to, 259-269 DNLR examination of, 108-115 effect of hydrostatic pressure on, 135 elasto-viscoplastic deformation of, 398-406 friction of, 711-712 G'Sell-Jonas model of, 260 glass transition of, 945-954 glassy, 488-499,945-954 in high strain conditions, 262 multiplicative laws relating to, 258 one-dimensional deformation of, 261 plasticity of, 255-264 scale of observation for, 5 semicrystalline, 257 three-dimensional deformation of, 262 yield behavior of, 257 Polymethylmethacrylate behavior under compression, 402-404 material parameters of, 402 Polypropylene, friction of, 712 Poroelasticity linear, 1118-1125 nonlinear, 1126-1132 Porosity, and crack initiation, 487 Porous materials characteristics of, 84-85 dynamic procedure for analyzing, 85-86 elastic parameters for, 85-88 elasticity of, 85 example analyses of, 88-90 static procedure for analyzing, 86-87 Porous metal plasticity, modeling of, 436-445 Postpeak softening, size effect on, 47 Precracked geometries, analysis of, 441, 442 Preisach model, 929-930 implementation of, 931 parameters for, 931 Pressure, and contact, 743-744 Principal stretches, 76 Process selection, 15-16, 26 software for, 27-28 Proteoglycans, 1059 PTFE (polytetrafluoroethylene) characteristics of, 711 coefficient of friction of, 712
Index PVAc specific heat of, 949 volume recovery of, 948, 953
Q Quantitative identification estimation of parameters, 11-12 optimization, 12 sensibility to parameters, 11 validation, 12 Quasi-brittle damage, 412 calculation of, 534-535 under loading, 528-529 at mesoscale, 533 at microscale, 533-534 modeling of, 525-535 under multilevel loading, 532-533 under proportional cyclic loading, 530-531 under two-level loading, 532 Quasi-brittle materials, 33 creep in, 55 defined, 45 size effect in, 40-43 Quenching, phase transformation aspect of, 894
R R-curve, size effect method for measuring, 49-50 Radial loading, 134 Ramberg-Osgood formula, 332, 333 Ratcheting, 198 biaxial, 240-246 incidence of, 244 Rate-dependent constitutive model for metals, 387-388 description of, 388 stress component in, 388-389 viscous drag component in, 389-391 Rate-dependent elastoplastic model, 377 equations of, 378-379 examples of use of, 383-386 numerical implementation of, 382-383 stress-strain relation in, 379-380 yield and memory surface centers, 380-382 Rational phenomenology, defined, 266 Recovery, 198
Index Reference configuration, defined, 75 Regularized Coulomb law, 781 Relaxation near equilibrium, 111-112 nonlinear kinetics and, 110-111 Relaxation functions, in viscoelasticity, 73 Representative volume element (RVE), 5 Residual strength, 547, 662 calculation of, 665-667 Response function, defined, 77 Reynolds dilatancy coefficient, 1099 Reynolds number, 766 Rheology of solid third bodies, 692 Ritchie-Knott-Rice model, 574 Rock salt compressibility/dilatancy of 1083, 1084, 1090 disturbed state parameters for, 881 elasticity of, 88-89 viscoplastic potential of, 1090-1091 Rocks fracture analysis of, 1171-1176 disturbed state parameters for, 880, 881 soft, strain-hardening model for, 1146-1153 Rolling friction, 703 of free-rolling wheel, 734-734 and traction, 735-740 Rotor steel chemical composition of, 443 material parameters for, 444 Roughness, 742 promoting friction, 743 and wear, 787-788 Rousselier model, 436-438 equations for, 438-439 implementation of, 441 parameters for, 439-445 Routh-Hurwitz criteria, 722 Rubber, 91-92 elastic parameters of, 93, 94 elasticity of, 93 strain-energy function and, 94 Running cracks, 653 velocity of, 654-656
s SAE 52100 steel elasticity of, 626 material parameters of, 631
1195 Sand critical state bounding surface model for, 1164-1170 cyclic mobility of, 1162 dry, disturbed state parameters for, 880 dynamic behavior of, 1155-1162 liquefaction of, 1155-1156, 1160, 1161 material parameters of, 1083, 1162 saturated, disturbed state parameters for, 880 Sandel criterion, 184 Sandstone disturbed state parameters for, 880, 881 hydrocarbon reservoir, material parameters of, 1143-1144 Saturated marine clay, disturbed state parameters for, 879-880 Saturated sand, disturbed state parameters for, 880 SC16, material parameters for, 316 Scaling, 32 physical laws of, 33 Schmid's law, 35, 205 Sdobyrev criterion, 181 Sea ice buckling in, 54-55 size effects in, 52-53 Secant methods of analyzing nonlinear composites based on first-order moments, 973 based on second-order moments, 974-975, 981-982 classical, 973 and deformation theory of plasticity, 975-977 equations for, 972-973 linearization, 971-972 modified, 974-975,981-982 for nonlinear local problem, 970-971 Selection mapping, 21 Semicrystalline polymers, deformation of, 257 Shakedown analysis, 1029 applicability of, 1025-1027 described, 1032-1033 examples of, 1034, 1035 kinematic approach to, 1033 Shale, elasticity of, 89 Shape memory deformation mechanism of, 921-922
1196 Shape memory (continued) equations describing, 925-926 material parameters for, 926-927, 927 modeling of crystal component, 924 modeling of phase transformation system, 923 modeling of polycrystal, 925 relevancy of, 921-922 Shaw law of friction, 780 Shear effects on limit condition, 190 stress and strain of, 131-133, 135 Shear bands in glassy polymers, 495-496 in granular materials, 1102-1105 Shell theory, 191, 193-194 Short-fiber composites anisotropy of, 1038-1040 Batchelor approximation of, 1045 described, 1037-1038 Dinh and Armstrong model of, 1045 equations describing, 1039-1042 flow around ellipsoidal particles in, 1042 models of, 1042-1046 Shaqfeh and Frederickson model of, 1045 statistical analysis of, 1039 suspension of non-Brownian spheres in, 1041 volume average analysis of, 1038-1039 SiC material parameters of, 791 Weibull parameters for, 563 SIC-150, Weibull parameters for, 563 SiC-B, Weibull parameters for, 563 SiC-HIE Weibull parameters for, 563 SiC Nicalon fiber, Weibull parameters for, 563 SiC/SiC CMC, material parameters of, 1023 Signorini conditions, 778-779, 783 Silica 60, inelastic response of, 1114-1116 Si2N4, Weibull parameters for, 563 Single crystals common errors regarding, 314 creep modeling of, 304-306 equations applicable to, 311-312 material parameters for, 306-307, 312-313, 316 plasticity of, 314, 315 slip in, 308, 309, 310, 312-313 under strain, 309 under stress, 309
Index twinning of, 309 viscoplastic model of, 310-311,314-315 Size effects, 31-32 analysis by equivalent LEFM, 48-49 boundary layer effect, 61 caused by diffusion phenomena, 61 caused by Poisson effect, 61 for cohesive crack model, 56-57 in compression fracture, 50-51 for crack band model, 58 described, 32-34 deterministic, 35-36 in fatigue crack growth, 45 fractality and, 60-61 gradient plasticity approach to, 288-293 history of, 34-36 to measure material constants and R-curve, 49-50 mechanism of, 42-43 via nonlocal models, 58 power scaling and, 36-37 in sea ice, 52-55 statistical theories of, 35 in torsion of solid bars, 288-293 Weibull theory of, 33, 35, 38-40 Skin deformation behavior of, 1062 stress-strain curve for, 1061 Sliding and friction, 703, 709-711 transition from peeling to, 754-756 Sliding crack model, 577-578 for brittle inelastic materials, 579-580 displacement-driven, 578, 579 force-driven, 578, 579 uses of, 581 Slip, 736 and adhesive contact, 752-756 components of, 737 of crystal, 309, 310, 312-313 partial, 730-732 plastic, 308, 315 region of, 736 stick-slip, 714-716, 718-724 Smoothness, 742 Soapstone, disturbed state parameters for, 880 Soft rocks, strain-hardening model for, 1146-1153 Soft tissue, 1057-1058 components of, 1059-1060
Index defined, 1058 example of, 1065-1066 material parameters of, 1068-1070 mechanical characteristics of, 1061-1062 model of, 1062-1065, 1069 structure of, 1058 types of, 1059 Softening, 198 under cyclic loading, 371 Software, for materials and process selection, 27-28 Soils partially saturated, elastoplasticity of, 11341144 strain-hardening model for, 1146-1153 Solder, material parameters for, 882-883 SOLDUR 355 steel, material parameters for, 520, 535 Solid third bodies, 680 cohesion and ductility of, 692 contact with, 686-689 dynamic view of flows, 688-689 rheology of, 692 thermal issues with, 693-694 Solid-solid phase transformation models, 915 equations for, 916-919 ideal plastic phases, 917 implementation of, 919-920 isotropically hardenable phases, 917-918 kinematically hardenable phases, 918-919 material parameters for, 919 theory behind, 916 Soliton, defined, 553 Spring steel, SN curves for, 478 SRR99, material parameters for, 307 Stainless steel, material parameters for, 535 State potential, 7-8, 795 Static friction, 703, 760 Static recovery, determination of, 365 Steel fatigue crack growth in types of, 612-619 FLDs for, 295 hydrogen attack of, 856-863 hydrogen transport parameters for, 868 material parameters for, 245-246 SN curves for, 477 solid-solid phase transformations of, 915-920 stress-strain curves for, 272
1197 table of compositions, 443 table of parameters, 444 tension/compression ratios in, 184 Stick-slip, 714-716 in nonstationary sliding, 718-724 Stored-energy function, defined, 77 Strain angular distribution of, 639 antiplane, 551 cyclic, 453-454 elastic, 6 inelastic, 370 plastic, 6 plane, 72, 551, 1123-1124 relation to stress, 105 variables for, 4, 6 Strain gradients, 198 Strain-energy function, 94 defined, 77 Strain-hardening model for soils applicability of, 1147 for cemented materials, 1152-1153 history of, 1146-1147 parameters for uncemented materials, 1148-1152 for uncemented materials, 1147-1148 Strength theory, 40 Stress angular distribution of, 638 constraint, 80 integration into elastoplasticity model, 208 plane, 71,551 relation to strain, 105 tribological, 690 variables for, 4-5 Stress analysis, 647-650 interferometric, 649 photoelastic, 647 Stress deviator, 197 Stress difference, 1109 Stress tensors Cauchy, 5, 77 nominal, 77 Stress-intensity factor, and crack velocity, 655-656 Stress-phase transformation couplings, 896 importance of, 903 kinetics models of, 897-899 material parameters for, 901-903
1198 Stress-phase
Index transformation
couplings
(continued) nucleation and, 899 thermomechanical model of, 899-901 theory behind, 897 Stress-strain interpolation model of elastoplasticity, 206-208, 209 Stretch tensors, 76 Stribeck law of friction, 780 Stribeck's curve, 760, 761 Styrene-butadiene rubber, friction of, 713 Suction, 1134 soil stiffness and, 1135 Summation of Eustein, 7 Superficial tribological transformations (STTs), 690 Superlaminar flow, 764 Surfaces characteristics of, 742-743 contact between, 743-744 SUS304 stainless steel, creep damage in, 603, 605 Symmetry group, defined, 77
T Ta-2.5 W alloy material parameters for, 345 stress-strain curves of, 344 TA6V alloy, material parameters for, 366 TAF650 steel, creep crack growth in, 599 Tangent self-consistent model, 318 averaging in, 321-324 equations for, 319-321 steps in, 319 Tangential forces and adhesive contact, 752-756 components of, 737 partial, 730-732 region of, 736 Tantalum, material parameters for, 392 Tarasenko criterion, 186 Taylor number, 765 Taylor's friction-dilatancy rule, 1100 Taylor-Lin model, 202 Temperature and flow stress, 391-392 and plasticity, 268-270
Temperature-dependent constitutive model for metals, 387-388 description of, 388 stress component in, 388-389 thermal component in, 391-395 three-dimensional, 396 viscous drag component in, 389-391 Tension, damage evolution in, 518-519 Terfenol D, material parameters of, 937, 939, 941 Texture anisotropic elastoplastic model based on, 204-212 determination of, 209-210 updating of, 208-209 Thermal activation strain rate analysis, 268269 Thermoelasticity, 72-73 laws of, 7-8 Thermo-elasto-viscoplasticity conditions for, 824-825 coupled with damage, 821-832 equations for, 821-823 identification of, 830 and kinematic hardening, 827-828 microdamage in, 825-827 postulates of, 823-824 thermodynamic issues, 829-830 0 law, 259 Third bodies defined, 691 flows of, 691-692 fluid, 680 rheology of, 688 solid, 680, 686-689, 692 Ti 6246 chemical composition of, 484 gigacycle SN curve for, 482 mechanical properties of, 484 Ti alloys gigacycle fatigue of, 481--486 grain size and fatigue in, 487 Ti-6A1-4V, material parameters for, 394 TiA1 intermetal, creep crack growth in, 599, 605, 606 TiC, material parameters of, 631 TIMETAL 21S, stress-strain curves for, 383, 384 Trabecular bone, 1049 Traction, friction model for, 735-740
1199
Index Transformation field analysis (TFA) applicability of, 996 concentration factor estimates, 1001-1002 equations for, 997-999 and evolution of inelastic deformation, 999-1000 to model damage evolution, 1000-1001 Transformation plasticity phenomenon, 916 Transformation-induced plasticity (TRIP), 905 described, 905-907 deformation in, 909-911 identification of, 911-912 model of, 907-911, 913 transformation kinetics in, 907-909 Tresca criterion, 136 Tresca law of friction, 780 Tresca yield condition, 158-159 Tribology, 679-680 components of, 697 flows in, 691-692 nomenclature of, 689-692 views of, 685 Tribometry, 694 Tsvelodub criterion, 184 Twinning, of single crystals, 309 Two-parameter fracture model, 50 U Unilateral contact, 777 compliance and, 779 strict, 778-779 V Validity, 12 domain of, 13 Value functions, 23, 24 Vanadium, material parameters for, 392 Variational bounds minimum energy principles and, 978 and modified secant method, 981-982 Ponte Castafieda's theory of, 978-980 quadratic stress and, 980-981 VBO model applicability of, 336-337 constants for, 343-346 for inelastic compressibility, 337, 340-343 for inelastic incompressibility, 341,343 nomenclature of, 338-339 Vest-Amoedo-Lee law, 259
Vibration effects on friction, 724-725 frictional, 714-716 Viscoelasticity, 73, 95 correspondence principle of, 104-106 crack growth and, 120 creep functions in, 73, 74 dashpot model of, 95, 96 dynamic deformation, 100-102 equations of, 103-104 hereditary integral, 102-103 Kelvin-Voigt model of, 73, 96, 98 linear, with damage, 117-124 Maxwell model of, 73, 96, 97 nonlinear model based on fluctuating modes, 107-116 relaxation functions in, 73 spring model of, 95, 96 static deformation, 98-100 Viscoplastic soliton, 553 Viscoplasticity coupled with damage, 797-801, 1010-1011 coupled with diffusion, 849-855 in crystals, 308-317 cyclic, 358-367 defined, 301 isotropic model of, 301-302 modeling issues in, 302 nonproportional cyclic, 368-376 overstress and, 336-346 of polycrystals, 318-325 as response to deformation, 273 Rousselier model of, 438 Viscosity function, determination of, 364-365 Void nucleation, Rousselier model of, 439 von Mises criterion, 10
W Wear, 675, 677 analysis of, 678-679 chemical environment and, 684 coefficients of, 684 contact mechanics and, 683-684 criteria for, 775 described, 693-697 dissipated energy models of, 684 entropy production and, 770-771 equations for, 769-775 historical analyses of, 682
1200 Wear (continued) industrial criteria involving, 684 interfaces and, 771-775 mechanical analysis of, 695 and nonviscous interface, 773 physical models of, 787-791 scale of analysis for, 677-678 thermodynamic analysis of, 768-775 and viscous interface, 773 Wear cards, 684 Weibull model of brittle failure, 559-563 parameters for, 563-564 uses of, 564 Weibull statistical theory of size effect, 33, 35, 38-40 nonlocal statistical generalization of, 59-60 Weight coefficients method, 23 Welding, phase transformation aspect of, 894 Wilshaw-Rau-Tetelman model, 573-574 WLF empiric law, 111 W6hler curve, 457, 469 Wood, scale of observation for, 5 Workhardening, 129-130 yield stress and, 135, 188 Workhardening/softening function, 830 Wrought iron, tension/compression ratios in, 184
X X2080-SiC-15p-T4, material parameters of, 631
Index Y Yield limits, 127-128 in anisotropy, 155-164 in beams, plates, and shells, 187-194 distortional model of, 167-174 generalized criterion for, 175-186 in isotropy, 129-136 in planar orthotropy, 162-163 Yield loci, 137-138 calculation of, 139-143 rounding of, 148-150 Yield stress, defined, 197 Yield surface, 224 Young's modulus, 11, 71
Z Z3CNG17-12 SPH, material parameters for, 366 Ziegler type assumption, 241 Ziegler's rule, 380 Zircaloy, hydrided ductile rupture of, 591,592 material parameters of, 595 Zirconium, hydrogen transport parameters for, 872 Zirconium-Nb alloy, material parameters of, 355 Zr702, material parameters for, 316 Zr02 material parameters of, 791 Weibull parameters for, 563 Zy4, material parameters for, 316
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