Nonlinear Solid Mechanics a Continuum Approach for Engineering

Nonlinear Solid Mechanics A Continuum Approach for Engineering

Gerhard A. Holzapfel Technical University Graz, Austria

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Contents

Preface

ix

Acknowledgements

xiii

1 Introduction to Vectors and Tensors

1

....

1.1

Algebra of Vectors

1.2

Algebra of Tensors

1.3

Higher-order Tensors .

1.4

Eigenvalues, Eigenvectors of Tensors

1.5

Transformation Laws for Basis Vectors and Components

1.6

General Bases

1.7

Scalar, Vector, Tensor Functions .

1.8

Gradients and Related Operators .

1.9

Integral Theorems .

1

...... ....

. ....

20

....

...

24 28

.........

...

9

32

40 . . . .

.

.... ....

44 52

55

2 Kinematics 2.1

Configurations, and Motions of Continuum Bodies

56

2.2

Displacement, Velocity, Acceleration Fields . .

61

2.3

Material, Spatial Derivatives

64

2.4

Deformation Gradient

70

2.5

Strain Tensors

76

2.6

Rotation, Stretch Tensors .

85

2.7

Rates of Deformation Tensors

95

2.8

Lie Time Derivatives . . . . .

106

. . . .

v

Contents

vi

109

3 The Concept of Stress 3.1

Traction Vectors, and Stress Tensors . .

3.2

Extremal Stress Values . . .

3.3

Examples of States of Stress

3.4

Alternative Stress Tensors .

109 119 123 127

131

4 Balance Principles 4.1 4.2 4.3 4.4 4.5 4.6

4.7

. ...... 131 138 Reynolds' Transport Theorem .... Momentum Balance Principles . . .... 141 Balance of Mechanical Energy . . ..... 152 Balance of Energy in Continuum Thermodynamics . . 161 Entropy Inequality Principle .... 166 . ... 174 Master Balance Principle . . ..... . ... Conservation of Mass

..... . . . . . .

179

5 Some Aspects of Objectivity 5.1

Change of Observer, and Objective Tensor Fields .

5.2

Superimposed Rigid-body Motions . . . . . . . .

5.3

Objective Rates . . . . . . . . . . . . . .

179 187 192

5.4

Invariance of Elastic Material Response . .

196

6 Hyperelastic Materials

205

.... . . . . . ....

6.1

General Remarks on Constitutive Equations . .

6.2

Isotropic Hyperelastic Materials . . . . .

6.3

Incompressible Hyperelastic Materials .

6.4

Compressible Hyperelastic Materials

6.5

Some Forms of Strain-energy Functions .

6.6

Elasticity Tensors . . . . . . . . .

6.7

Transversely Isotropic Materials .

..

........

206 212 222 227 235 252 265

vii

Contents

6.8

Composite Materials with Two Families of Fibers .

6. 9

Constitutive Models with Internal Variables

. . . .

6.10 Viscoelastic Materials at Large Strains . . . . 6.11

Hyperelastic Materials with Isotropic Damage

7 Thermodynamics of Materials

272 278 282 295

305

7.1

Physical Preliminaries . . . . .

7.2

Thermoelasticity of Macroscopic Networks

7.3

Thermodynamic Potentials . . . . . . . .

7.4

Calorimetry . . . . . . . . . . . . . . . . .

7.5

Isothermal, Isentropic Elasticity Tensors . .

7.6

Entropic Elastic Materials . . . . . . . . .

7.7

Thermodynamic Extension of Ogden's Material Model ..

7.8 7.9

Simple Tension of Entropic Elastic Materials Thermodynamics with Internal Variables . . .

306 311 321 325 328 333 337 343 357

371

8 Variational Principles 8.1

Virtual Displacements, Variations

8.2

Principle of Virtual Work . . . . .

8.3

Principle of Stationary Potential Energy .

8.4

Linearization of the Principle of Virtual Work .

372 377 386 392

8.5

Two-field Variational Principles .

402

8.6

Three-field Variational Principles

. . . . .

409

References

415

Index

435

Preface

My desire in writing this textbook was to show the fascination and beauty of nonlinear solid mechanics and thermodynamics from an engineering computational point of view. My primary goal was not only to offer a modern introductory textbook using the continuum approach to be read with interest, enjoyment and curiosity, but also to offer a reference book that incorporates some of the recent developments in the field. I wanted to stimulate and invite the reader to study this exciting science and take him on a pleasant journey in the wonderful world of nonlinear mechanics, which serves as a solid basis for a surprisingly large variety of problems arising in practical engineering. Linear theories of solid mechanics are highly developed and are in a satisfactory state of completion. Most processes in nature, however, are highly nonlinear. The approach taken has the aim of providing insight in the basic concepts of solid mechanics with particular reference to the nonlinear regime. Once familiar with the main ideas the reader will be able to specialize in different aspects of the subject matter. I felt the need for a self-contained textbook intended primarily for beginners who want to understand the correspondence between nonlinear continuum mechanics, nonlinear constitutive models and variational principles as essential prerequisites for finite element formulations. Of course, no single book can cover all aspects of the broad field of solid mechanics, so that many topics are not discussed here at all. The selection of the material for inclusion is influenced strongly by current curricula, trends in the literature and the author's particular interests in engineering and science. Here, a particular selection and sty le was chosen in order to highlight some of the more inspiring topics in solid mechanics. I hope that my choice, which is of course subjective, will be found to be acceptable. My ultimate intention was to present an introduction to the subject matter in a didactically sound manner and as clearly as possible. I hope that the text provides enough insights for understanding of the terminology used in scientific state-of-the art papers and to find the 'right and straightforward path' in the scientific world through the effective use of figures, which are very important learning tools. They are designed in order IX

x

Preface

to attract attention and to be instructive and helpful to the reader. Necessary mathematics and physics are explained in the text. The approach used in each of the eight chapters will enable the reader to work through the chapters in order of appearance, each topic being presented in a logical sequence and based on the preceding topics. A proper understanding of the subject matter requires knowledge of tensor algebra and tensor calculus. For most of the derivations throughout the text I have used symbolic notation with those clear bold-faced symbols which give the subject matter a distinguished beauty. However, for higher-order tensors and for final results in most of the derivations I have used index notation, which provides the reader with more insight. Terminology is printed in bold-face where it appears for the first time while the notation used in the text is defined at the appropriate point. For those who have not been exposed to the necessary mathematics I have included a chapter on tensor algebra and tensor calculus. It includes the essential ideas of linearization in the form of the concept of the directional derivative. Chapter 1 summarizes elementary properties which are needed for the vector and tensor manipulations performed in all subsequent chapters and which are necessary to many problems that arise frequently in engineering and physics. It is the prime consideration of Chapter 2 to use tensor analysis for the description of the motion and finite deformation of continua. The continuum approach is introduced along with the notion 'Lagrangian' (material) and 'Eulerian' (spatial) descriptions. In a systematic way the most important kinematic tensors are provided and their physical significance explained. The push-forward and pull-back operations for material and spatial quantities and the concept of the Lie time derivative are introduced. The concept of stress is the main topic of Chapter 3. Cauchy's stress theorem is introduced, along with the Cauchy and first Piola-Kirchhoff traction vectors, and the essential stress tensors are defined and their interrelationships discussed. In Chapter 4 attention is focused on the discussion of the balance principles. Both statics and dynamics are treated. Based upon continuum thermodynamics the entropy inequality principle is provided and the general structure of all principles is summarized as the master balance (inequality) principle. Chapter 5 deals with important aspects of objectivity, which plays a crucial part in nonlinear continuum mechanics. A discussion of change of observer and superimposed rigid-body motions is followed by a development of objective (stress) rates and invariance of elastic material response. Chapters 6 and 7 form the central part of the book and provide insight in the construction of nonlinear constitutive equations for the description of the mechanical and thermomechanical behavior of solids. These two chapters show the essential richness of the field. They are written for those who want to gain experience in handling material models and deriving stress relations and the associated elasticity tensors that are fundamental for finite element methods. Several examples and exercises are aimed at enabling the reader to think in terms of constitutive models and to formulate more com-

Preface

xi

plex material models. All of the types of constitutive equations presented are accessible for use within finite element procedures. The bulk of Chapter 6 is concerned with finite elasticity and finite viscoelasticity. It includes a discussion of isotropic, incompressible and compressible hyperelastic materials and provides constitutive models for transversely isotropic and composite materials which are suitable for a large number of applications in practical engineering. An approach to inelastic materials with internal variables is given along with instructive examples of hyperelastic materials that involve relaxation and/or creep effects and isotropic damage mechanisms at finite strains. The main purpose of Chapter 7 is to provide an introduction to the thermodynamics of materials. This chapter is devoted not only to the foundation of continuum thermodynamics but also to selected topics of statistical thermodynamics. It starts with a statistical approach by summarizing important physical aspects of the thermoelastic behavior of molecular networks (for example, amorphous solid polymers), based almost entirely on an entropy concept, and continues with a systematic phenomenological approach including finite thermoelasticity and finite thermoviscoelasticity. The stress-strain-temperature response of so-called entropic elastic materials is discussed in more detail and based on a representative example which is concerned with the adiabatic stretching of a rubber band. Typical thermomechanical coupling effects are studied. Chapter 8 is designed to cover the essential features of the most important variational principles that are very useful in formulating approximation techniques such as the finite element method. Although finite elements are not treated in this text, it is hoped that this chapter will be attractive to those who approach the subject from the computational side. It shows the relationship between the strong and weak forms of initial boundary-value problems, presents the classical principle of virtual work in both spatial and material descriptions and its linearized form. Two- and three-field variational principles are also discussed. The present text ends where conventionally a book on the finite element method would begin. There are numerous worked examples adjacent to the relevant text. These have the goal of clarifying and supplementing the subject matter. In many cases they are straightforward, but provide an essential part of the text. The symbol • is used to denote the end of an exercise or a proof. The end-of chapter exercises are for homework. The (almost) 200 exercises provided are designed to supplement the text and to consolidate concepts discussed in the text. Most of them serve the purpose of stimulating the reader to further study and to reinforce and develop practical skills in nonlinear continuum and solid mechanics, towards the direction of computational mechanics. In many cases the solutions of selected exercises are given and frequently used later in further developments. Therefore, it should be instructive for the reader to work through a reasonable number of exercises. Numerous references to supplementary material are suggested and discussed briefly

xii

Preface

throughout the book. However, for a book of this kind it is not possible to give a comprehensive bibliography of the field. Some of the references listed serve as a starting point for more advanced studies. The material in this book is based on a sequence of courses that I have taught at the University of Technology in Graz and Vienna. The mechanics and thermodynamics of solids are relevant to all branches of engineering, to applied mechanics, mathematics, physics and material science, and it is a central field in biomechanics. This book is primarily addressed to graduate students, researchers and practitioners, although it has also proven to be of interest to advanced undergraduates. Although I have tried to provide a textbook that is self-contained and appropriate for self-instruction, it is desirable that the reader has a reasonable background in elementary mechanics and thermodynamics. I feel that Chapters 2-5 and parts of the remaining chapters are well suited for a complete course on nonlinear continuum mechanics lasting two semesters or three quarters. A one semester or one quarter course in the nonlinear mechanics of solids would focus on Chapter 6, while a one quarter course in the thermodynamics of solids could be based on Chapter 7. Chapter I is the core for a course that provides the student with the necessary background in vector and tensor analysis. Chapter 8 is certainly not designed to train specialists in variational principles, but to form a basic one quarter course at the graduate level. I believe that the present textbook, in providing many applications to engineering science, is not too advanced mathematically. Of course, some of the results presented may be derived with the help of more advanced mathematics using theorems and proofs. I hope that this book will help pure engineers to teach nonlinear continuum mechanics and solid mechanics. Naturally, as the author, I take full responsibility for not doing it better. Comments and criticisms will be welcome and greatly appreciated. I have learnt that the spirit of modern continuum mechanics and the underlying mathematics are as important to the design of powerful finite element models as are insights in the theoretical foundation of constitutive models and variational principles. A successful transfer of that combination to the reader would indicate that my objective has been achieved.

Graz, Austria August 1999

Gerhard A. Ho/zap/el

Acknowledgements

When I was a postdoctoral student at Stanford University I worked with the late Juan C. Simo, Professor of Mechanical Engineering, to whom I owe my deepest thanks. He stimulated, influenced and focused my study and writing in recent years; his friendship, versatility and dedication to scientific excellence provided a unique learning experience for me.

I am particularly indebted to Ray W. Ogden, Professor of Mathematics at the University of Glasgow, who spent a lot of time in reading the entire manuscript and rectifying certain ineptnesses. His outstanding expertise in the field made working with him an inspiring pleasure. His detailed scientific criticism and suggestions for improvements of the text were of immeasurable help. Many others have contributed to the book. Here I mention my collaborators, whose gentle encouragement and support during the course of the preparation of the manuscript I gratefully acknowledge. In particular, the inspiring and detailed comments of D1: Christian A.J. Schulze-Bauer, from a background in physics and medicine, were extremely helpful. He let me filter this text through his sharp mind. Also, Christian T. Gasser, whose background is in mechanical engineering, has suggested a number of valuable improvements to the substance of the text. His profound remarks in class prevented me from getting away with anything. Also~ Elisabeth Pernkopf, a mathematician, deserves special thanks for her generous assistance. She gave much helpful advice on the preparation of this text and offered many suggestions for improving it. Special thanks are due to Michael Stadler for his productive discussions and to Mario 0. Pa/Ii for his patience in preparing the figures. I am grateful to each of these individuals without whose contributions the book would not have taken this shape. I want to thank the Department of Civil Engineering, Graz University of Technology, for providing an environment in which this project could be completed. My gratitude goes to Gernot Beer, Professor of Civil Engineering, for his outspoken support. I also wish to acknowledge the Austrian Science Foundation, which has influenced my scientific agenda through the financial support of several grants over the past eight years.

xm

xiv

Acknowledgements

The enjoyment I experienced in writing this textbook would not have been the same without the moral support of numerous friends. My thanks belong to all of them who tolerated my absence when I disappeared for many evenings and weekends in order to bring the ideas of this fascinating field to you, the reader.

1 Introduction to Vectors and Tensors The aim of this chapter is to present the fundamental rules and standard results of tensor algebra and tensor calculus permanently used in nonlinear continuum mechanics. Some readers may prefer to pass directly to Chapter 2 leaving the present part for reference as needed. Many of the statements are given without proofs. For a more detailed exposition see the standard texts by HALMOS [1958], TRUESDELL and NOLL [1992] or the textbooks by CHADWICK [1976], GURTIN [1981a], SIMMONDS [1994], DANIELSON [1997] and OGDEN [ 1997] among many other references on vectors and tensors. To recall the elements of linear algebra see, for example, the book by STRANG [ 1988a]. In this text we use lowercase Greek letters for scalars, lowercase bold-face Latin letters for vectors, uppercase bold-face Latin letters for second-order tensors, uppercase bold-face calligraphic letters for third-order tensors and uppercase blackboard Latin letters for fourth-order tensors; for example,

a, {3, (, . . . (scalars) A, B, C, . . . (2nd-order tensors)

a, b, c, . . . (vectors)

A

B, C, ... (3rd-order tensors)

A, Jm, C, . . . (4th-order tensors)

(1.1)

.

For equations such as u

= av =

{Ja = 1b ,

(1.2)

we agree that (1.2)2 refers to u = {Ja. Often the derivations of formulas need relations introduced previously. If this is the case we refer to these relations in a particular order, reflecting the consecutive steps necessary for deriving the formula in question.

1.1

Algebra of Vectors

A physical quantity, completely described by a single real number, such as temperature, density or mass, is called a scalar designated by a, /3, ')', ... A vector designated 1

2


by u, v, w, ... (or in other texts frequently designated by ''i', ~' 1!!' ... , or fi, 'V, 'lV, ... ), is a directed line element in space. It is a model for physical quantities having both direction and length, for example,force, velocity or acceleration. Two vectors that have the same direction and length are said to be equal. The sum of vectors yields a new vector, based on the parallelogram law of addition. The following properties

u+v=v+u,

(I .3)

(u + v) + w = u + (v + w) ,

( 1.4) (1.5)

u+o=u, u+ (-u) == o

(1.6)

hold, where o denotes the unique zero vector with unspecified direction and zero length. Let u be a vector and a be a real number (a scalar). Then the scalar multiplication au produces a new vector with the same direction as u if a > 0 or with the opposite direction to u if n < 0. Further properties are:

(n,B)u (er+ f3)u

Dot product.

= o{Bu) , = nu+ {Ju

(1.7)

,

(1.8)

a(u+v)=au+o:v.

( 1.9)

The dot (or scalar or inner) product of u and v, denoted by u · v (or

(u, v} ), is O
u · v = lul lvlcosO(u, v) ,

( 1.10)

where O(u, v) is the angle between two nonzero vectors u and v when their origins coincide. This product gives a scalar quantity with the properties

U·V

V·U

(I.II)

U·O

0

( 1.12)

a(u·v)+/J(u·w)

u · (nv + {3w) U· U > 0

and

u#o

' u. u = 0

(l.13)

U=O

(l.14)

The quantity lul (or !lull) is called the length (or norm or magnitude) of a vector u, which is a non-negative real number. It is defined by the square root of u · u, i.e.

Iu I =

(u · u)

1 2 '

>o

,

')

u-

= u. u

.

( 1.15)

1.1

Algebra of Vectors

3

A vector e is called a unit vector if lel = 1. A nonzero vector u is said to be orthogonal (or perpendicular) to a nonzero vector v if

O(u,v) =7r/2 .

with

( 1.16)

Thus, using ( l .10) we find the projection of a vector u along the direction of a vector e with unit length, i.e.

u · e == jujcosO(u, e) .

( 1.17)

For a geometrical interpretation of eq. ( 1. I 7) see Figure I. I .

u

\0

-~

-""------

l

_____~------·:

e Figure 1.1 Projection of u along a unit vector e.

Index notation. So far algebra has been presented in symbolic (or direct or absolute) notation exclusively employing bold-face letters. It represents a very convenient and concise tool to manipulate most of the relations used in continuum mechanics. It will be the preferred representation in this text. However, particularly in computational mechanics, it is essential to refer vector (and tensor) quantities to a basis. Additionally, to gain more insight in some quantities and to carry out mathematical operations among (higher-order) tensors more readily (see next section) it is often helpful to refer to components. In order to present coordinate (or component) expressions relative to a righthanded (or dextral) and orthonormal system we introduce a fixed set of three basis vectors e 1 , e2 , c:i (sometimes introduced as i,j, k), called a (Cartesian) basis, with properties

e 1 • e.,..,

= e 1 • e..3 = e?- · e..3 = 0

'

These vectors of unit length which are mutually orthogonal form a so-called orthonormal system.

4


Then any vector u in the three-dimensional Euclidean space is represented uniquely by a linear combination of the basis vectors e 1 , e2 , e3 , i.e. ( 1.19) where the three real numbers u 1 , 'U 2 , u 3 are the uniquely determined Cartesian (or rectangular) components of vector u along the given directions e1 , e2 , e:1, respectively (see Figure 1.2). The components of e 1 , e 2 , e3 are (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively. u

'lL]

Figure 1.2 Vector u with its Cartesian components u 1 , u 2 , u 3 •

Using index (or subscript or suffix) notation, relation (l.19) can be written as u = E~=t uiei or, in an abbreviated form by leaving out the summation symbol, simply as (sum over i

= 1, 2, 3)

,

( 1.20)

where we have adopt the summation convention, invented by Einstein. The summation convention says that whenever an index is repeated (only once) in the same term, then, a summation over the range of this index is implied unless otherwise indicated. We consider only the three-dimensional Euclidean space, which we characterize by means of Latin indices i, j, k ... running over 1, 2, 3. We denote the basis vectors by { ei he{ 1,2 ,3 } collectively. Subsequently, in this text, the braces { •} will stand for a fixed set of basis vectors and the symbol • for any tensor element. The index i that is summed over is said to be a dummy (or summation) index, since a replacement by any other symbol does not affect the value of the sum. An index that is not summed over in a given term is called a free (or live) index. Note that in the same equation an index is either dummy or free. Thus, relations ( 1.18) can be

1.1

5

Algebra of Vectors

written in a more convenient form as if

I

i

if

=J

,

#j

'

( 1.2 l)

which defines the Kronecker delta 8ii. The useful properties

8ii

=3

(1.22)

'

hold. Note that 8ii also serves as a replacement operator; for example, the index on becomes an j when the components 'Ui are multiplied by oii· The projection of a vector u = ·u.iei onto the basis vectors ei yields the j-th component of u. Thus, from eq. ( 1.20) and properties ( 1.13), ( 1.21) and ( l.22h we have

'Ui

( 1.23) Taking the basis {ei} and eqs. (l.13), (l.20), (l.21) and (l.22)2, the component expression for the dot product (I. I 0) gives

u ·v =

=

u;ei · viei

U.(Vjei · ei

= 'l1. I 'VJ + 'll2V2 + 'll:fV3

=

Ui'vi8ii

=

u.ivi

( 1.24)

,

which is commonly used as the definition of the dot product. Thus, we may derive the dot product of u and v without knowledge of the angle between u and v. In an analogous manner, the component expression for the square of the length of u, i.e. ( 1.15), is

lul 2 = U · U = •)

U;Ci •

')

•)

= 'llj + ll~ + 'll3

'lljej =

U(lljc)ij

=

'll(l/.;

.

( 1.25)

Note that in eqs. ( 1.24) and ( 1.25) one index is repeated, indicating summation over 1, 2, 3. In symbolic notation this is indicated by one dot.

Cross product.

The cross (or vector) product of u and v, denoted by u xv (in the

literature also u /\ v), produces a new vector. The cross product is not commutative. It is defined as

u xv= -(v x u) ,

( l.26)

UXV=O (O'.U) X V = U X (O'.V) =

u and v are linearly dependent ft (

U X V) ,

u · (v x w) = v · (w x u) = w · (u x v) , u x (v + w) = (u xv)+ (u x w) = u xv+ u x w .

( l.27) ( 1.28) (I .29)

( l.30)


6

If relation ( 1.27) holds with u and v assumed to be nonzero vectors, we say that vector u is parallel to vector v. From eq. ( 1.29) we learn that

=0

u · (u x v)

( l.31)

.

The magnitude of the cross product is defined to be

In x vi

= lnl lvlsinO(u, v)

,

0

< O(u, v) < " .

( 1.32)

It characterizes the area of a parallelogram spanned by the vectors u and v (see Figure 1.3). The right-handed cross product of u and v, i.e. the vector u x v, is perpendicular to the plane spanned by u and v (0 is the angle between u and v). uxv

v

.~

Area=

lox vi

I

I I

I I I

:

u

I I

f -u xv= v x u

Figure 1.3 Cross product of vectors u and v.

In order to express the cross product in terms of components we introduce the permutation (or alternating or Levi-Civita e-) symbol C.ijk' which is defined as

C.ijk

=

I ,

for even permutations of (i, j, k) (i.e. 123, 231, 312) ,

-I '

for odd permutations of (i, j, k) (i.e. 132, 213, 321) ,

0 ,

if there is a repeated index

( 1.33)

,

with the properties Eijk = C.jki == C.kij' C.ijk = -Eiki and Eijk = -c.J'ib respectively. Consider the right-handed and orthonormal basis {ei}, then e 1 x e 1 = e2 x e 2

==

C3 X e:1

=0

,

( 1.34)

1.1

7

Algebra of Vectors

or in a more convenient short-hand notation, with ( 1.33), ( 1.35) With relations ( 1.2 I), ( 1.34), ( 1.35) it is easy to verify that EiJk may be expressed as the determinant of a matrix ( 1.36)

where we have introduced the square brackets [•] for a matrix. The product of the permutation symbols Eijk Epqr is related to the Kronecker delta by the relation

( 1.37)

With (1.22) 1 and (l.22h we deduce from (l.37) the important relations Eijk Eijk

EXAMPLE 1.1

Solution.

=6

.

Obtain the coordinate expression for the cross product w

( 1.38)

=u

x v.

Taking advantage of eqs. ( 1.20) and ( 1.35) we find that (l .39)

with the three components

( 1.40)

( l.41) ( 1.42) Consequently, (u x v) · ek equals

Eijk'uivi.

•

Now, using expressions ( 1.36), ( 1.39).1, the vector product u x v relative to { ek} may be written as

u xv= det

e:!

e:1

u1

U2

'll:1

'Vt

'V2

'lJ3

[ e1

l

= c ijk 'lli'lJ.i ek

( 1.43)

8


The triple scalar (or box) product (u xv)· w represents the volume F of a parallelepiped 'spanned' by u, v, w forming a right-handed triad (see Figure 1.4).

uxv w

v

Volume= (u xv) ·w u

Figure 1.4 Triple scalar product.

By recalling definitions ( 1.29), ( 1.10) and ( 1.32), we have

l"

= (u x v) · w = (v x w) · u = (w x = ju x vi lwlcos8 = lullvlsinO lwlcosc5 .

u) · v ( 1.44)

~~

base area

height

Using index notation, then, from eqs. ( 1.20) and ( 1.35) we find with ( 1.2 I) the volume 1·'" to be

l"

= (u x v) · w = Eijk'lli'VfWk = (u2'V3 - 'U;fV2}'W1 + (u3'U1

-

'UJ V:1)W2

+ (u1 V2

-

'll2Vi)W:i

.

(I .45)

Hence, the triple scalar product ( 1.45):1 can be written in the convenient determinant form

(u x v) · w = det [

:~ :~~ ::~ ] 'U3

'V3

( 1.46)

'W3

Note that the vectors u, v, w are linearly dependent if and only if their triple scalar product vanishes (the parallelepiped has no volume). The product u x (v x w) is called the triple vector product and may be verified with ( 1.39).i, ( 1.38) 1, ( 1.22h and representations ( 1.20) and ( 1.24)4

9

1.2 Algebra of Tensors

U X

(v

X W}

= EijkUi(EmnjVmWn)ek = =

(8km8in -

= (u · w)v -

EkijEmn{UiVmWnek

c5kn8im)UiVmWnek

( 1.47)

(u · v)w ,

which is the so-called 'back-cab' rule well-known from vector algebra. Similarly,

(u x v) x w = (u · w)v - (v · w)u .

( 1.48)

The triple vector product is, in general, not associative, i.e. ( u x v) x w ¥= u x (v x w).

EXERCISES

1. Use the properties ( 1.5), ( 1.6), ( 1.8) and ( 1.9) to show that QO= 0

,

Ou= o ,

(-n)u = a(-u) .

2. By means of (l.30) and (1.28) derive the property

(au+ ,Bv) x w = a(u x w)

+ {3(v x

w) .

3. Prove the triple vector product ( 1.48) and show that the vector (u x v) x w lies in the plane spanned by the vectors u and v.

1.2 Algebra of Tensors A second-order tensor A, denoted by A, B, C ... (or in the literature sometimes written as A, B, C .. .), may be thought of as a linear operator that acts on a vector u generating a vector v. Thus, we may write

v=Au

( 1.49)

which defines a linear transformation that assigns a vector v to each vector u. Since A is linear we have

A(nu+v) = aAu+Av

( 1.50)

for all vectors u, v and all scalars a. Since most tensors used in this text are of order two, we shall often omit the adjective 'second-order'.

10


If A and B are two (second-order) tensors, we can define the sum A + B, the difference A - B and the scalar multiplication a A by the rules (A± B}u = Au± Bu ,

( 1.51)

(o:A)u = cr(Au) ,

( 1.52)

where u denotes an arbitrary vector. The important second-order unit (or identity) tensor I and the second-order zero tensor 0 are defined, respectively, by the relations Io = ul = u and Ou = uO = o for all vectors u. Note that tensor 0 maps every u to the zero vector o. Tensor product. The tensor (or direct or matrix) product or the dyad of the vectors u and v is denoted by u ® v (some authors use the notation uv). It is a secondorder tensor which linearly tran~jorms a vector w into a vector with the direction of u following the rule ( 1.53) The dyad, not to be confused with the dot or cross product, has the linearity property (u ® v)(crw + x)

= a:(u ® v)w + (u 0

( 1.54)

v)x .

In addition, note the relations (nu

+ t'v) 0

w

u(v ® w) (u 0 v)(w ® x) A(u ® v)

=

a ( u 0 w)

+ f3 (v ® w)

= (u · v)w = w(u · v) , = (v · w)u ® x = u 0 x(v · w) =

(Au)® v .

( 1.55)

,

( 1.56) ,

( 1.57)

(l.58)

Generally, the dyad is not commutative, i.e. u 0 v # v 0 u and (u ® v)(w ® x) # (w ® x)(u ® v). A dyadic is a linear combination of dyads with scalar coefficients, for example, a(u ® v) + (J(w 0 x). Note further that no tensor may be expressed as a single tensor product, in general, A = u ® v + w ® x # y ® z. Any second-order tensor may be expressed as a dyadic. As an example, the secondorder tensor A may be represented by a linear combination of dyads formed by the (Cartesian) basis {ei}, i.e. ( 1.59) We call A, which is resolved along basis vectors that are orthonormal, a Cartesian tensor of order two (more general tensors will be considered in Section 1.6). The nine

1.2

11

Algebra of Tensors

Cartesian components of A with respect to { ei}, represented by Ai1, form the entries of the matrix [A], i.e.

(A]= [

~~:

( 1.60)

A:11

where by analogy with ( 1.22h

( 1.61) holds. Relation ( 1.60) is known as the matrix notation of tensor A.

EXAMPLE 1.2 Let A be a Cartesian tensor of order two. Show that the projection of A onto the orthonormal basis vectors ei is according to ( 1.62) where Aii are the nine Cartesian components of tensor A.

Solution. find that

Using representation ( 1.59) and properties ( 1.53 h, ( 1.2 I) and ( 1.61 ), we

Aei == Atk(el ® ek)ej = Atk(ek · ei)e,

==

AtkOkjel

= Atiet

.

(1.63)

On taking the dot product of ( 1.63 ),1 with ei, the nine Cartesian components Aii are completely determined, namely

ei · Aei

= ei · A11e1 = Atj(ei · e,) == Ali8il = AiJ = (A )iJ .

( 1.64)

In ( 1.64) we have used the notation (A)ii for characterizing the components of A.

•

If the relation v · Av > 0 holds for all vectors v # o, then A is said to be a positive semi-definite tensor. If the stronger condition v · Av > 0 holds for all nonzero vectors v, then A is said to be a positive definite tensor. Tensor A is called negative semidefinite if v · Av < 0 and negative definite if v · Av < 0 for all vectors v # o, respectively. The Cartesian components of the unit tensor I form the Kronecker delta symbol introduced in eq. ( 1.21 ). Thus, ( 1.65)

12


We derive further the components of u ® v along an orthonormal basis { ei}. Using representation ( 1.62) and properties ( 1.53) and ( 1.23 )a, we find that

(u ® v) ii

= ei · (u ® v) ei = ei · u(v · ei) = (ei · u)vi ( 1.66)

where the coefficients uivi define the nine Cartesian components of u ® v. Writing eq. ( 1.66)4 in the convenient matrix notation we have U1V2

U1V3

U2V2

U2V:3

U3V2

U3V3

l

( 1.67)

Here, the 3 x 1 column matrix [ui] and the 1 x 3 row matrix [vi] represent the vectors u and v, while the 3 x 3 square matrix represents the second-order tensor u ® v. A scalar would be represented by a 1 x 1 matrix.

EXAMPLE 1.3 written as

Show that the linear transformation ( 1.49) that maps v to u may be

(l.68) where index .i is a dummy index.

Solution.

Using expressions ( 1.20), ( 1.59) and rules ( 1.53 ), ( 1.21) and ( 1.61 ), we find from ( 1.49) that viei

= Aiiuk(ei ®ej)ek = Aiiuk(eJ · ek)ei = Aiiuk8ikei = Aikukei .

(1.69)

A replacement of the dummy index k in eq. ( 1.69)4 by j gives the desired result (1.68). •

Dot product. The dot product of two second-order tensors A and B, denoted by AB, is again a second-order tensor. It follows the requirement {AB)u = A(Bu) for all vectors u.

( 1.70)

1.2

Algebra of Tensors

13

The summation, multiplication by scalars and dot products of tensors are governed mainly by properties known from ordinary arithmetic, for example,

A+B=B+A,

(1.71)

A+O=A,

(1.72)

A+ (-A)= 0

,

( I.73)

A+ (B + C) = (A+ B) + C , (a:A) = o:(A} = a:A , (AB}C = A{BC) = ABC ,

( 1.74) ( 1.75) ( 1.76)

A2 -AA '

( 1.77)

(A+B)C = AC+BC .

( 1.78)

Note that, in general, the dot product of second-order tensors is not commutative, i.e. AB # BA and also Au # uA. Moreover, relations AB = 0 and Au = o do not, in general, imply that A, B or u are zero. The components of the dot product AB along an orthonormal basis { ei }, as introduced in ( 1.18), read, by means of representation ( 1.62) and relations (I. 70), ( 1.63 ).i,

{AB)ii = ei · (AB}ei =

== ei · A(Bkiek) ==

ei ·

A{Bei)

= (ei · Aek)Bki

AikBki

( 1.79)

or equivalently ( 1.80)

For convenience, we adopt the convention that a repetition of only one index between a tensor and a vector (see, for example, eq. (1.68) with the dummy index j) or between two tensors (see, for example, eq. ( 1.80) 1 with the dummy index k) will not be indicated by a dot when symbolic notation is applied. Specifically that means for v = Au (vi = AifUj) and A == BC (Aii = BikCki) we do not write v = A· u and A = B · C, respectively.

Transpose of a tensor. The unique transpose of a second-order tensor A denoted by AT is governed by the identity v·ATu=u·Av=AV·U

( 1.81)

for all vectors u and v. The useful properties (l.82)

14

1

Introduction to Vectors and Tensors

(nA + ~Bfr {AB)'r

= aAT + (JBT = BTAT ,

( 1.83)

,

( 1.84)

(u®v)T = v®u

( 1.85)

hold. Hence, from identity ( 1.81) we obtain ei ·AT ei = ei · Aei, which gives, in regard to ( 1.62), the important index relation {AT)ii = Aii·

Trace and contraction. The trace of a tensor A is a scalar denoted by tr A. Taking, for example, the dyad u ® v and summing up the diagonal terms of the matrix form of that second-order tensor, we get the dot product u · v = 1Li'Vi, which is called the trace of u 0 v. We write tr( u ® v} = u · v = -urvi

( 1.86)

for all vectors u and v. Thus, with representation ( 1.59) and eqs. ( I.86)i, ( 1.21 ), ( 1.61) the trace of a tensor A with respect to the orthonormal basis {ei} is given by trA

= tr(Aiiei 0 ei) = Aiitr{ei ® ei) = Aii(ei · ei) = Aiioii ( 1.87)

= Aii ,

or equivalently

trA

= Aii = .A11 + A22 + A33

( 1.88)

.

We have the properties of the trace:

trAT

= trA

( 1.89)

,

tr{AB} = tr{BA) ,

( 1.90)

tr{A + B) = trA + trB ,

( 1.91)

tr{o:A)

= o:trA

(1.92)

.

In index notation a contraction means to identify two indices and to sum over them as dummy indices. In symbolic notation a contraction is characterized by a dot. A double contraction of two tensors A and B, characterized by two dots, yields a scalar. It is defined in terms of the trace by A :B

== tr(ATB) = tr{BTA)

= tr{ ABT) = tr{BAT) = B:A or AiiBii == BiiAii

.

( 1.93)

15


The useful properties of double contractions

I :A A : {BC)

= trA = A : I = {BTA) : C = {ACT)

( 1.94)

:B ,

A: (u ® v) = u ·Av= (u ® v) : A ,

( 1.95) ( 1.96)

( 1.97) = (u · w)(v · x) , ( 1.98) (ei ® ei) : (ek 0 el) = (ei · ek)(ei · e,) == 8ik8it Note that if we have the relation A : B = C : B, in general, we cannot conclude

(u ® v) : (w ® x)

hold. that A equals C. The norm of a tensor A is denoted by IAI (or llAll). It is a non-negative real number and is defined by the square root of A : A, i.e. ( l.99)

Determinant and inverse of a tensor. The determinant of a tensor A yields a scalar. It is defined by the determinant of the matrix [A] of components of the tensor, i.e. ( 1.100)

with properties

det(AB} detA T

= detA detB = detA .

,

(1.101) (I. I 02)

A tensor A is said to be singular if and only {f detA = 0. We assume that A is a nonsingular tensor, i.e. detA # 0. Then there exists a unique inverse A - t of A satisfying the reciprocal relation ( 1.103)

If tensors A and B are invertible, then the properties {AB)- 1

= B- 1A- 1

,

(A-1)-1 =A ' {a:A)- 1 = 1/o: A -I

( 1.105) ,

(A-l)T=(AT)-1, A-2 = A-1A-1

(1.104)

( 1.106) ( 1.107)

'

(I.] 08) ( 1.109)

16


hold. Subsequently, in this text, we use the abbreviation (A-l}T

= A-T

( 1.110)

for notational convenience.

Orthogonal tensor.

An orthogonal tensor Q is a linear transformation satisfying

the condition Qu·Qv

= u·v

(l.111)

for all vectors u and v (see Figure 1.5). As can be seen, the dot product u · v is invariant during that transformation, which means that both the angle () between u and v and the lengths of the vectors, lnl, lvl, are preserved. Qu

u

Q

lul

v

Qv

Figure 1.5 Orthogonal tensor.

Hence, an orthogonal tensor has the property QTQ = QQT = I, which means that Q- = QT. Another important property is that det(QTQ) = (detQ) 2 with detQ = ± 1. If detQ = +1 ( -1), Q is said to be proper (improper) orthogonal corresponding to a rotation (reflection), respectively. 1

Symmetric and skew tensors. Any tensor A can always be uniquely decomposed into a symmetric tensor, here denoted by S, and a skew (or antisymmetric) tensor, here denoted by W. Hence, A

= S + W, while (l.112)

In this text, we also use the notation symA for Sand skew A for W. Tensors Sand W are governed by properties such as or

or

1.2

17

Algebra of Tensors

which in matrix notation reads

(SJ =

[ S11

812

S12

S22

S13 823

813

823

Saa

l

(W] = [

-l~'12

ll'12

-TV1a

0

W13 12a H

-H123

0

l

(l.114)

In accord with the notation introduced, the useful properties

s : B = s: BT= s: ~(B +BT)

w: B =

-W: BT=

w: ~(B -

(l.115)

' BT) '

....,

S:W= 0

(1.116) ( 1.117)

hold, where B denotes any second-order tensor. A skew tensor with property W = - WT has zero diagonal elements and only three independent scalar quantities as seen from eq. (1.114)2. Hence, every skew tensor W behaves like a vector with three components. Indeed, the relation holds:

Wu= w x u ,

(l.118)

where u is any vector and w characterizes the axial (or dual) vector of skew tensor W, with property lwl = {1/V2)1WI (the proof is omitted). The components of W follow from (1.62), with the help of (1.118), (1.20), (1.35), ( 1.21) and the properties of the permutation symbol

lVii =

ei

·Wei= ei · (w x ej) =

ei · (wkek

= ei · (wk€kjtel) = Wk€kjl8il = CkjiWk = -€ijkWk •

x ei) (1.119)

Therefore, with definition ( 1.33) we have

= -w3 = -€13kWk = W2

H112 =

ltf/13

-c12kwk

,

(1.120)

,

(1.121) (1.122)

where the components H112 , lV13 , H123 form the entries of the matrix [W] as characterized in ( 1.114)2. The inversion of ( 1. l l 9h follows with relations ( 1.38)2 and ( 1.22)2 after multiplication with the permutation symbol €ijp ( 1.123)

18


which, after a change of the free index, gives finally Wk• --

-

~c. <.;.zJ·kl • lT·. lJ

and

1

2

( 1.124)

Projection, spherical and deviatoric tensors. Consider any vector o and a unit vector e. With reference to Figure 1.6, we write o = 011 +OJ_, with 011 and OJ_ characterizing the projection of o onto the line spanned by e and onto the plane normal to e, respectively.

u

Figure 1.6 Projection tensor. With (1.53) we deduce that

011=(o·e)e=(e0e)o=P~u,

( 1.125)

~

p~ OJ_

= o - 011 = o - (e ® e)o = (I - e ® e) o = P~o , ~

p;

where P~ and P~ are projection tensors of order two. The projection tensor P~ applied to any vector o maps o into the direction of e, while Pt applied to o gives the projection of o onto the plane normal to e (see Figure 1.6). A tensor Pis a projection if Pis symmetric and P 11 = P (n is a positive integer), with the properties p1c

+ pllc

=I

pllpll

= pllc

c

c

( 1.127)

' '

(1.128)

p;- p;- = p;- '

( 1.129)

p~ p;

( 1.130)

=0

.

19


Every tensor A can be decomposed into its so-called spherical part and its deviatoric part, i.e. A= al+ devA rl•

u

(1.131)

or

1 1 1 = -trA =-(I: A)= -4-· 3 3 34 u

•

(l.132)

Any tensor of the fonn al, with a denoting a scalar, is known as a spherical tensor, while elev A is known as a deviator of A, or a deviatoric tensor. The deviatoric operator dev( •) is denoted by the short-hand notation, i.e. dev( •) = (•) - (1/3)tr( •)I, or dev( • )iJ = (• )ij - (1/3)( • )kk8iJ . Computing the trace of ( 1.131) we deduce with ( 1.92), ( 1.87) 5 , ( 1.22) 1 and ( l. I 32h that Aii = 3(Aii/3) + dev Aii. This relation yields the important property tr(devA)

=0

(l.133)

,

which means that the trace of the deviator of A is always zero.

EXERCISES

1. Let u be any vector and e any unit vector. Use eqs. ( 1.125), ( 1.126) and the triple vector product ( 1.47) to show that u can be resolved into components parallel, i.e. 011' and perpendicular, i.e. Uj_, to unit vector e, according to

u

= u 11 + u.l = (u · e)e +ex

(u x e) .

2. Prove eqs. ( 1.57) and ( 1.58) and show that

3. Let A be a tensor with matrix [A] and AT the transpose of A with matrix [A]T. Show that [AT) = [A]T. 4. Show by means of representation ( 1.59) and properties ( 1.98) and ( J .61) that the double contraction of the two tensors A and B yields the component form Aii Bii. 5. Starting from eq. (l.100), verify that detA (1.101).

= EijkAi 1Ai2 Ak3 and show property

6. Consider eq. (1.93) 1 and relations (l.103), (l.88) in order to obtain the property

A -·r :A=3 .

( 1.134)


20

7. Given that S is a symmetric tensor and W is antisymmetric, prove property (l.117). 8. Two tensors A and B are given in the form of their matrix representations

[A)

=

[

cos(}

sinO

0

-sinO

cosO

0

0

0

1

l

Verify that these matrices are orthogonal. Show that [B] a reflection. Give a geometrical interpretation.

[A] describes a rotation and

9. Find the axial vector of the skew tensor W = 1/2{u ® v - v 0 u). 10. Let A be a tensor whose matrix is

[A]= [

tt

tl

Find the spherical and the deviatoric parts of A.

1.3 Higher-order Tensors In the following we discuss higher-order tensors. Any tensor of order (or rank) n may be expressed in the form

Ai 1 i2 ••• in ei 1 © ei 2 0 · · · ® ein .

(1.135) (_

As ( 1.135) shows, a tensor of order n has 3n components Ai 1 i 2 ••• i,., provided with n indices i 1 , i 2 , ••• , in. In particular, a tensor of order zero has 3° = 1 component with 0 (no) index, which simply is a scalar a (i.e. a single real number). A tensor of order one has 3 1 = 3 components characterizing any vector v, i.e. 'Vi as indicated in ( 1.23 ). A tensor of order two, for example, A, has 32 = 9 components, i.e. Aij as indicated in ( 1.60).

Tensor of order three. A tensor of order three we distinguish by the notation A, B, C, ... According to (1.135), A may be expressed as (l.136) where Aiik characterizes the 33

= 27 components of A.

1.3

21

Higher-order Tensors

A particular example for a third-order tensor is the so-called triadic product of three vectors u, v, w, denoted by u ® v ® w, with the properties

(u®v)®w=u®v®w,

(1.137)

(u ® v 0 w)x = (w · x)u 0 v ,

(1.138)

(u ® v ® w) : (x 0 y)

= (v · x)(w · y)u

(l.139)

,

(u®v®w):l=(v·w)u.

( 1.140)

Hence, by analogy with the procedure which led to eq. ( 1.62) we may determine the components of A, i.e. ( 1.141)

The double contraction of a third-order tensor A with a second-order tensor B produces a vector. It is denoted by A : B and may be found with representations (l.136), (1.59) and properties (1.139), (1.21) and (l.61) as

A: B = AiikB1m(ei 0

ei ® ek) : (e1 0

em)

= AijkBtm(ej · eL)(ek · em)ei = AijkBtm8j18kmei = AijkBjkei .

EXAMPLE 1.4 expressed as

(1.142)

Let E denote the third-order permutation tensor, which may be

(1.143) Here, Cijk = (ei x ei) · ek are the 33 components of E which we have introduced as the permutation symbol (recall eqs. ( 1.33) and ( 1.35)). Show that the double contraction of E with the dyad of any vectors u and v, i.e. u ® v, gives the cross product of v and u.

Solution.

With expressions

(E)ijk

=

€ijk

and (u 0 v)1m =

'U(Vm

we may write (1.144)

Using properties (1.139), (1.21) and (1.22) 2 , we find that (l.145)

which is a vector with components

€ijk"UfVk.

Finally, by means of the permutation

22 property €ijk


= -€kji and expressions ( 1.43h, ( 1.26) we obtain

e : (u ® v) = v x u which is the desired result.

( 1.146)

'

•

Any tensor of order four, denoted by A, JR, C, ... , has 34 = 81 components, i.e. Aiikl, Biikl, Ciikl, .. ., respectively. We may express any tensor A in terms of the three Cartesian basis vectors, we write

Tensor of order four.

(l.147) in which Aijkl are the components of A. One example for a fourth-order tensor is the product of the four vectors u, v, w, x, denoted by u ® v 0 w ® x. We have the useful properties:

(u 0 v) 0 (w 0 x) (u 0 v ® w ® x) : (y 0 z)

= u ® v® w® x , = (w . y )( x . z)( u ® v)

(I. 148)

.

( 1.149)

With these properties we can show that the components Aiikl of A can be expressed as (1.150) Hence, the double contraction of A with a second-order tensor B is a second-order tensor, denoted by A : B. With representations (I. I 4 7), ( 1.59), and properties ( 1.149), ( 1.21) and ( 1.61) we find that

A: B

= AijklBmn(ei ® ei ® ek ® e,) : (em® en) = Aijk1Bmn8krn81nei ® ei

(_

(1.151)

= Aijk1Bk1ei ® ei .

Following the definitions above and by analogy with (1.53), (1.56) and (1.57), note the important rules

{A® B) : C A : (B ® C)

= A(B : C) = (B : C)A = {A : B)C = C(A : B)

,

( 1.152)

,

(1.153)

(A® B) : (C ® D) = (B: C){A ® D) = (A 0 D)(B: C) ,

(1.154)

and the property

A(B ® C)D

= (AB) 0

(CD) .

(1.155)

1.3

EXAMPLE 1.5

23

Higher-order Tensors

Show eq. (l.152) which is the analogue of eq. (l.53).

Solution. Express tensor A, B, C in terms of their basis vectors according to ( 1.59). With properties (1.148), (1.149), (1.21) and (1.61) we find that (A® B): C = AijBktCmn(ei ® ej) ® (ek ® e,): (em® en)

= AijBklCmn(ek · em)(e, · en)(ei ® ej) = AiiBktCmn8km81nei ® ei

= AiiBk1Ck,ei ® ei

( 1.156)

.

By using representation ( 1.59) and definition ( 1.93) we obtain AiiBk,Cklei ® ei A{B : C) = (B : C)A, which is the desired result. •

A further example for a fourth-order tensor is the tensor product D = A ® B, while A and Bare second-order tensors with components Aii and Bii, respectively. In index notation we may write Dijkl = AiiBkt· The unique transpose of a fourth-order tensor A denoted by AT is governed, by analogy with (I .81 ), by the identity B : AT : C

= C : A : B = {A : B) : C

( 1.157)

for all second-order tensors B and C. The properties of a fourth-order tensor

(A?)T =A ,

( 1.158)

{A®B)T = B®A

( 1.159)

hold. From the identity ( 1.157) we deduce the index relation, namely (AT) iJkl We now define fourth-order unit tensors Il and Il so that A=Il:A,

AT=Il:A'

= AkliJ. ( 1.160)

for any second-order tensor A. These transformations assign to any A the tensor A itself, and its transpose, respectively. The fourth-order unit tensors may be represented by

n= 8ik8jlei ® ej 0 Il = c5ilc5jkei 0 ej 0 where (Il)iikl = 8ik8i1 and (Il)ijkt respectively. Note that i= nT.

n

= ei ® ej ® ei ® ej ek ® el = ei 0 ej 0 Cj 0 ei

ek ® e,

'

(1.161)

,

(l.162)

= 8itc5jk define the Cartesian components of Il and Il,


24

The deviatoric part of a second-order tensor A may be described by means of a fourth-order tensor. From relations ( 1.13 I), ( 1.132 h and transformations ( 1. I 60) 1, ( 1.152) we conclude that

1

devA =A -

3

(1: A)I

1

= {Il - 31 ~I} : A

,

(1.163)

or with the definition of a fourth-order projection tensor P devA = P: A ,

(1.164)

The components of dev A and A are related through the expression elev Aii with Pijkl

= oik8il -

= Pijkl Akt,

(1/3)8ii8kl· EXERCISE

I. Two fourth-order tensors §, W are given in the form 1 w = -(Il 2

-

JI)

(observe the similar structure to eq. ( 1.1I2)). Show that

1

·r {A +A ) =§:A ,

2

~(A -

-

AT) =

w :A

,

which assign to any second-order tensor A its symmetric and skew parts, respectively.

1.4 Eigenvalues, Eigenvectors of Tensors The scalars Ai characterize eigenvalues (or principal values) of a tensor A if there exist corresponding nonzero normalized eigenvectors Di (or principal directions or principal axes) of A, so that (i

= 1, 2, 3; no summation)

(1.165)

.

To identify the eigenvectors of a tensor, we use subsequently a hat on the vector quantity concerned, for example, ft. Thus, a set of homogeneous algebraic equations for the unknown eigenvalues ,.\, i = 1, 2, 3, and the unknown eigenvectors ni, 'i = 1, 2, 3, is - o (A - ,.\·l)n· 1 l -

'

(i


.

( 1.166)

1.4

25

Eigenvalues, Eigenvectors of Tensors

Eigenvalues characterize the physical nature of a tensor. They do not depend on coordinates. For a positive definite symmetric tensor A, all eigenvalues ,.\i are (real and) positive since, using (1.165), we have,.\ =Di· Ani > 0, i = 1, 2, 3. Moreover, the set of eigenvectors of a symmetric tensor A form a mutually orthogonal (orthonormal) basis {ni} (the proof of this statement is omitted). For the system ( 1.166) to have solutions ni determinant of the system must vanish. Thus,

Principal scalar invariants.

#

o the

(1.167) where

( 1.168) This requires that we solve a cubic equation in ,.\, usually written as ( 1.169) called the characteristic polynomial (or equation) for A, the solutions of which are the eigenvalues Ai, i = 1, 2, 3. Here, Ii(A), i = 1, 2, 3, are the so-called principal scalar invariants of A. In terms of A and its principal values,.\, i = 1, 2, 3, these are given by

= .A1 + ,.\2 + ,.\3

11 (A)

= Aii =

! 2 (A )

1 = -(AiiAji - AiiAii) = -1 [ (trA) 2 - tr(A 2 )] = trA -1 detA 2 2 = ,.\ 1.A2 + .A1.A3 + A2A3 , = EijkA1iA2jAak = detA = A1A2A3 .

/3(A)

tr A

(I. I 70)

,

( 1.171) (l.172)

Sometimes we shall use the alternative notation

l3(A)

= IIIA

(1.173)

for the three principal invariants interchangeably. A repeated application of tensor A to eq. ( 1.165) yields A 0 ni = .Arni, i = 1, 2, 3, for any positive integer a. Using this relation and ( 1.169) multiplied by Di, we obtain the well-known Cayley-Hamilton equation

(1.174) It states that every (second-order) tensor A satisfies its own characteristic equation.

Spectral decomposition of a tensor. Any symmetric tensor A may be represented by its eigenvalues ,.\i, i = 1, 2, 3, and the corresponding eigenvectors of A forming an

26


orthonormal basis {Di}· Using the unittensor, by analogy with ( l.65h, i.e. I= Di 0Di, and relations (1.58), (1.165) and (l.55) we obtain an expression which is known as the spectral decomposition (or spectral representation) of A, i.e. 3

A =AI= (Aili) 0 Di =

L ,.\iDi ®Di

(1.175)

.

i=l

The components Aii of tensor A relative to a basis of principal directions follow with ( 1.59) by replacing ei with the three orthonormal basis vectors Di. With eqs. (l.165) and (1.21) we obtain (j


,

( 1.176)

which produces a diagonal matrix [A] in the form )q

[A)=

[

~

(l.177)

where the diagonal elements are the eigenvalues of A. This result may be obtained directly from the spectral decomposition ( 1.175) of A. For a pair of equal roots, i.e. ,.\ 1 = ,.\ 2 # ,.\3 ( ,.\ 1 = ,.\ 2 == ,.\), with one unique eigenvector D3 associated with ,.\ 3 , we deduce from (1.175) that

(l.178)

where P~3 , Pt3 denote projection tensors introduced in (1.125) and (1.126). In (1.178) the eigenvector D3 is perpendicular to a plane spanned by the mutually orthogonal . ~ d~ ~ eigenvectors n 1 an n2. The third case to distinguish, namely for equal roots ( ,.\ 1 = ,.\ 2 = ;\ 3 = ,.\), is

(l.179) where every direction is a principal direction and every set of mutually orthogonal basis denotes principal axes.

EXERCISES

1. Show that the quantities in eqs. ( 1.170)-( 1.172), i.e. / 1 , / 2 , / 3 , are indeed principal invariants of a tensor A under an orthogonal transformation of coordinates.

27

1.4 Eigenvalues, Eigenvectors of Tensors 2. Let a tensor A be given by A=cr(I-e1®e1)+f3(e1®e2+e2®e1) , where er, {j are scalars and e 17 e2 are orthogonal unit vectors. (a) Show that the eigenvalues Ai, i

= 1, 2, 3, of A are ,,\3

a (cr2 = 2 -

(b) Derive the associated normalized eigenvectors Di, i matrix representation read

4 + /3

=

2) 1/2.

1, 2, 3, which in

3. Let a non-symmetric tensor A be given by the tensor product e 1 ® e:h with the property e 1 • e 2 = 0. Determine the eigenvalues and eigenvectors of tensor A. 4. Let A be a given second-order tensor. Show that the invariants of its deviatoric part, according to ( 1.163 ), are

fdevA = 0 ,

1 2 /IdcvA=-?tr(devA) ,

(l.180)

~

IIIdcvA =

1 )3· . det.(devA) = 3tr(devA

5. Use the relations (I. I 70)-( I. I 72), ( 1. I 80) and (I. I 63) 1 to show that I Ic1evA' I I IdevA may be expressed in terms of IA, I IA and I I IA as

1 2 JldevA = JJA - 3/A ' 1 2 :J l/lc1cvA =///A - - /A//A +~IA · 21 3

(1.181)

Hint: To derive eq. ( 1.181) apply the Cayley-Hamilto11 equation in the form of (devA) 3 - lc1evA (devA) 2 + IldevA devA - Illc1evA I= 0.

28


1.5 Transformation Laws for Basis Vectors and Components Consider two sets of mutually orthogonal basis vectors which share a common origin. They correspond to a 'new' and an 'old' (original) Cartesian coordinate system which we assume to be right-handed characterized by the two sets {ei} and { ei} of basis vectors, respectively. Hence, the new coordinate system may be obtained from the original one by a rotation of the basis vectors ei about their origin (see Figure 1.7). We find from ( 1.10) that the cosine of the angle between ei and ei is represented by Qii = cosO(ei, ei) = ei · ei. Note that the first index on Qii indicates the 'old' components whereas the second index holds for the 'new' components.

''

O(c2, c:!) ' \ '~

''

'

~

' ~C2 -

Figure 1.7 Rotation of mutually orthogonal basis vectors ei into ei. (_

It is worthwhile to mention that vectors and tensors themselves remain invariant upon a change of basis - they are said to be independent of any coordinate system. However, their respective components do depend upon the coordinate system introduced, which is arbitrary. The components change their magnitudes by a rotation (and/or reflection) of the basis vectors, but are independent of any translation (for the characterization of a translation, see p. 62). We now set up the transformation laws for various components of vectors and tensors under a change of basis. Following the considerations above we have and

T-

ei= Q ei '

i '-1'J3 . -

,-,

'

(1.182)

where Q denotes an orthogonal tensor (seep. 16), with components QiJ which are the same in either basis. The components describe the orientation of the two sets of basis

1.5


29

vectors relative to each other. In particular, Q rotates the basis vectors ei into ei, while QT rotates ei back to ei. Using eqs. ( 1.59), ( 1.53 )2, ( 1.21) and rule ( 1.61 ), we find that

(1.183)

and

which is the analogue of eq. ( 1.63)4 • By comparing the equations above we may extract the orthogonality condition of the cosines, characterized by QTQ = QQT = I. Equivalently, expressed in index or in matrix notation

[Q]T(Q]

= [Q][Q]T = [I] ,

( 1.184)

where [Q] contains the collection of the components QiJ. It is an orthogonal matrix which is referred to as the transformation matrix. Note that [Q]T = [QT] (compare with Exercise 3 on p.19). In order to maintain the right-handedness of the basis vectors we have admitted only rotations of the basis vectors, consequently det[Q] = +1.

Vectorial transformation law. We consider any vector u resolved along the two sets {ei} and { ei} of basis vectors, i.e.

- = u . ei -

'lli

10

{ei}

( 1.185)

m

{ei}

(1.186)

We assume that the relation between the basis vectors ei and ej is known by the cosines QiJ. Combining ( 1.185) with ( 1.182) 1 and using ( 1.183) 1 and ( 1. 186), we obtain the vectorial transformation law for the Cartesian components of the vector u, 1.e.

or

(1.187)

(given here in both index and matrix notation). In an analogous manner, we find that or

[u] = [Q][ii] .

(I. I 88)

These equations determine the relationship between the components of a vector associated with the (old) basis { ei} and the components of the same vector associated with another (new) basis {ei}· It is important to emphasize that the equations [ii] = [Q]'I'[u] and [u] = [Q][ii] are not identical to ii = QTu and u = Qii, respectively. In ( 1.187)2 and ( 1. l 88h, [ii] and [u] are column matrices characterizing components of the same vector in two different coordinate systems, whereas ii and u are different vectors.

Tensorial transformation law.

To determine the transformation laws for the Cartesian components of any second-order tensor A, we describe its components along the


30

two sets { ei} and { ei} of basis vectors, i.e. Aij

==

ei . Aej

m

( 1.189)

Aii

= ei · Aei

m

(1.190)

Combining (l.189) with (1.182) 1 and using (1.183) 1 and (l.190), then the (rectangular Cartesian) components Aii and Aii are related via the so-called tensorial transformation law Aij

=

ei. Aej =

(Qkiek). A(Qmjem)

= QkiQmj(ek · Aem) or

[A]

= [Q]T[A] [Q] .

(l.191)

Transformation (A] = [Q]T[A](Q] relates the different matrices [A] and (A], which have the components of the same tensor A. In an analogous manner, we find that or

[A] == [Q][A][Q]T .

(l.192)

In addition to previous considerations, the equations (A] = [Q]T[A](Q] and [A] = [Q][A](Q]T differ from the tensor equations A = QTAQ and A = QAQT, relating two different tensors, namely A and A. Finally, the 3n components Aiih···in of a tensor of order n (with n indices j 1 , h, ... , in) transform as

(1.193) This tensorial transformation law relates the different components Ai 1 i~ ... in (along the directions e1 , e2 , e3 ) and Ai•h···jn (along the directions e 1 , e 2 , e:i) of the same tensor of order n.

Isotropic tensors. A tensor A is said to be isotropic if its components are the same under arbitrary rotations of the basis vectors. The requirement is deduced frdm eq. ( 1.1 91 ) as or

[A] = (Q]T(A] [Q] .

( 1.194)

Note that all scalars as well as zero tensors and unit tensors of all orders are isotropic tensors.

EXAMPLE 1.6 of order two.

Show that the unit tensor, as given in ( 1.65), is an isotropic tensor

Solution. In accord with manipulation ( 1.191 )4 we can show that the components of the unit tensor I are 8ii in any coordinate system. Namely, using ( 1.61) and ( 1.184)

1.5


31

we obtain

(1.195) which means that the components §ii do not change upon a rotation or reflection of axes.

•

By multiplying both sides of oii = 8ii with a scalar we can see that every scalar multiple of I is an isotropic tensor. Consequently, any spherical tensor in the form of A == a:I

(1.196)

or

is an isotropic tensor, where a denotes a scalar. The permutation tensor ( 1.143) is an isotropic tensor of order three with respect to rotation of coordinate axes (proper orthogonal transformations). With respect to reflection it emerges that the permutation tensor is not isotropic. It can be shown that the fourth-order unit tensors lI, Il and the tensor I ® I with components (I ® I)iikl = 8ii8k1 are also isotropic tensors. The most general isotropic tensor of order four is of the form

a(I ® I)

+ /3Il + 1Il ,

(1.197)

where a:, /3, 1 are scalars. The components are expressed according to definitions

( 1.161 ), ( 1.162) as (n{I ®I)

+ /3lI + 1Il)ijkl = a8ijOk1 + /38ik8j1 + 78il8jk· EXERCISES

1. Let a new right-handed Cartesian coordinate system be represented by the set {ei} of basis vectors with transformation law e2 = -sin0e 1 + cos/Je2 , e 3 = e 3 • The origin of the new coordinate system coincides with the old origin. (a) Find

e1 in terms of the old set {ei} of basis vectors.

(b) Find the orthogonal matrix [Q] and express the new coordinates in terms of the old one. (c) Express the vector u = -Ge 1 basis vectors.

-

3e2

+ e:i

in tem1s of the new set {ei} of

2. Considering eqs. ( 1.189) and ( 1.190), show that Aii

= Aii.

32


3. Assume that A is an isotropic tensor of order two, for example, nl. Show that the deviatoric part of A is always the zero tensor of order two, i.e.

dev(nl)

=0

.

(1.198)

(In general, equations within exercises are not numbered. However, particular results may be given equation numbers, for later reference.)

1.6 General Bases There are more general ways to introduce the notion of a vector and a tensor. A more general theory assumes the existence of a vector space and its so-called dual space. Thereby, a vector and a tensor of order n are introduced through a linear and a multilinear jimction, respectively. Within this broader framework we must distinguish between the scalar product (which is only defined between elements of a vector space and between elements of its dual) and the inner product. For a detailed explanation of the underlying theory the interested reader is referred to, for example, BOWEN and WANG [1976b] and OGDEN [1997, Section 1.4.3]. However, for the purpose we pursue in this text it is sufficiently general to work with a theory which assumes the existence of Euclidean vector spaces and which identifies one vector space with its dual so that the scalar product is identified with the inner product. Thus far we have worked with Cartesian tensors which are resolved along a set of orthonormal basis vectors. However, tensor quantities can be projected onto more general bases. The aim of this section is to present the fundamental rules and standard results for these projections onto the basis of the aforementioned identification.

General basis vectors. We now resolve vectors and tensors along a fixed ~et {gJie{i,2 ,3 } of basis vectors. The elements g 1 , g2 , g3 of this basis are called covariant basis vectors. They are assumed to be non-zero and non-parallel but they need not necessarily be orthogonal to each other or of unit length. The general basis vectors are linearly independent, requiring (g 1 x g 2 ) • g:1 =f 0 (the vectors do not lie in a plane). We now introduce the so-called reciprocal (or dual) basis of {g; }, i.e. the fixed set {gi}ie{i,2 ,a} (note that for a more general theory which assumes the existence of a vector space and its dual the reciprocal and dual bases are in different spaces). The elements g 1, g 2 , g:1 of the reciprocal basis are called contravariant basis vectors or reciprocal basis vectors and have superscripts rather than subscripts. The two bases {g;} and {gi} satisfy the condition . l

g . gj

=

. uj

.n

'

(1.199)

1.6 General Bases

33

where 8J is the (mixed) Kronecker delta. As in definition ( 1.21) 8J = 1 for i = j and o] = 0 for i # j. Basically, eq. ( 1.199) means that each vector of a basis is orthogonal to the two vectors of the other basis whose indices are different. For example, the contravariant base vector g 1 is orthogonal to the two covariant basis vectors g2 and g3 , as illustrated in Figure I .8. gl

Figure 1.8 General basis {gJ, g 1 is orthogonal to g2 and g3 . For subsequent use we introduce the useful definitions (1.200) of the quantities gii and 9ii. Since, from eq. ( 1.11 ), the dot product is commutative we may write gii = gii and 9ii = Yii· The quantities gii and .9ii with the same indices represent the squares of the lengths of the associated basis vectors; for example, g 11 = lg 1 12 • The quantities yiJ and YiJ with different indices represent the product of the lengths of the associated basis vectors and the cosine of the angle (} between the basis vectors; for example, g12

= lg 1 Ilg2 lcosO(g1 , g2 ). Thus,

Igi I -- (9ii )1/2

'

(l.201)

(i, j = 1, 2, 3; no summation) determine the geometrical characteristics (the so-called metric) of a given basis. Hence, gii and 9ii are customarily referred to as the metric coefficients.

EXAMPLE 1.7

Show that the properties

(1.202) hold.

34

Solution.

1 Introduction to Vectors and Tensors We tentatively write (l.203)

where aik are unknown quantities. Taking the dot product of (1.203) with gi and using relation ( 1.199), we obtain ( 1.204) This means that aik is only multiplied with the factor 1 if j = k otherwise it is multiplied with zero. Hence, we may write gi · gi = aii, which is the analogue of ( 1.200) 1 • The proof is similar for property ( 1.202)2. •

Taking the dot product of gi with gk and using ( 1.202) we obtain gi ·gk = gii 9klgi ·g1. Finally, using ( 1.199) and replacing the subscript l on 9kt by a j, we arrive at the reciprocal property

g

ij

9kj

= Ukri

'

(l.205)

which means that the matrix with components .
921

( 1.206)

9a1

The distinction between subscript and superscript is essential for arbitrary bases. It is important to note that afree (or live) index, which is not summed over, such as 1: and j in eq. ( 1.199), must be either a subscript or a superscript on both sides of an equati.Qn. In the following we will sometimes use the term 'index' to denote either a subscript or a superscript. Only if the basis is orthonormal may we write gi == gi and then no distinction between upper and lower indices is necessary. For this case we write all indices as subscripts, as in previous sections where we used a Cartesian basis.

Covariant and contravariant components of a vector.

Any vector u may be represented uniquely by a linear combination of the contravariant or covariant basis vectors. We write

(1.207) where 'lli are the covariant components of u (with respect to the general basis {gi} ), while 'Ui are the contravariant components of u (with respect to the general basis

1.6 General Bases

35

{gi} ). It is only in this theory, in which the reciprocal basis is identified with the dual basis, that the vector u has both covariant components ui and contravariant components ui. We shall call a vector with covariant components a covariant vector and a vector with contravariant components a contravariant vector. By analogy with eq. ( 1.22)2, the Kronecker delta o} has the useful (replacement) properties 8~ 'lli t

= ui

(1.208)

'

in which the index ion u is replaced by .i. Note that summation can only take place over a dummy (or summation) index if one index is a subscript and the other is a superscript, such as index i in eq. ( 1.207). Hence, the .i-th covariant and contravariant components of any vector u are the projections of u onto the bases {gi} and {gi}, respectively. Taking the dot product of (I .207) 1 with gi (and the dot product of ( l.207)2 with gi), we obtain, by means of properties (l.13), (1.199), (1.208) 1 (and (l.208h), the useful relations ll ·

u . gi

i

= 'UiUrij = 1lj = uigi . gi = ,ui8f = 'll,i

gj ==

Uig ·

gj

,

( 1.209)

'

(l.210)

which should be compared with eq. (I .23) for the case of a Cartesian basis. The contravariant components of any vector u may also be expressed in tenns of its covariant components, and vice versa. Thus, taking the dot product of u = u 1gi and u == uigi with gi and gi, respectively, on both sides gives with definitions ( 1.200) and eqs. ( 1.21 Oh and ( l .209)J, i.e. 'U· l

== •g·ZJ·Uj

( 1.211)

'

which are properties analogous to those given in ( 1.202). Note that the new vector components farmed depend on the original one. As indicated in eqs. ( 1.211 ), the set of six quantities gii (or .
lnl of the vector u is lnl == (_gii'lt(llj)1f2 == (giiuiu/)1/2 = (uiui)1/2 since lnl = (u · u) 1/ 2 • and the length

'

(1.213)

Covariant, contravariant and mixed components of a tensor. Similarly to the above, any second-order tensor A may be represented with respect to a basis consisting


36

of tensor products of the gi and the gi. Thus,

A -_ Aii gi

'°' gj -_ A ijgi

t0i VY

VY

gi -_ Ai·jgi

t0i VY

i g1 -_ A·i i g

t0i VY

(l.214)

gj '

where Aii and Aii are the contravariant and covariant components, whereas A~i and are the mixed components of the tensor A. The dot in front of each of the indices of the mixed components indicates the order of occurrence of these indices. Thus, in A~i the first index is 'contravariant' and the second 'covariant'. In the following we shall call a tensor with covariant (respectively contravariant, mixed) components a covariant (respectively contravariant, mixed) tensor (the connections between these terminologies are discussed in more detail by OGDEN [1997, Section 1.4.3]). Note that the mixed components A~i and Aji of a tensor A are equal if and only if the tensor A is symmetric (Aii = Aii and Aii = Aji). Instead of 32 components for a Cartesian second-order tensor we have now four sets of components of a general tensor of order two, i.e. 6 2 different contravariant, covariant and mixed components, in general. However, no more than 3 2 components are independent. Moreover, equations such as

A/

d

_A ik

,.4ijUk -

(1.215)

'

are analogous to (1.61). Now, with rule ( 1.53) and properties ( 1.199) and ( 1.215), the projection of A onto the general basis {gi} and its reciprocal {gi} gives

= gi. AgJ

( 1.216)

Ai.= gi. Ag·1 •J

( 1.217)

Aii

The four sets of components follow in an analogous manner to that given in Example 1.2 on p. 11. The tensor relations for Cartesian components introduced in previous sections may be generalized to arbitrary components by basically replacing the Kronecker delta with the metric coefficients and using the different types of indices in an appropriate way. ~f tensors A and B are referred to a general basis~ then the properties

(AB)ii

= Ail;B~

A ·· B --

IAI

r'l·t]·Bii

=A/~ Bki ,

(1.218)

-- AiiB.lJ· -- Ai·j B·i i

= (AijA ij) 1/2

(1.219)

'

( 1.220)

'

t r A -- Ai·i -- .,;-,1·ii -- .'J iiA ii -- 9iJ..4ii , 41

[ ..41· 1 .. ·2 detA = det ..4~1 A2·2 43 ..43·2 • ·I ')

hold.

A\ A2 ·=~

A:i.3

l

= clet

( 1.22 I)

[ •4·tI

4·I

.. 2

-

A?

.. 2

4·3

.. 2

.. I

4·')

4-:1

.,;4·1 :3

A.2 a A·3 3

l

(1.222)

1.6

37

General Bases

EXAMPLE 1.8 Using the fact that Io = ul = u holds for all vectors u, show that the four sets of quantities .'lii

= (I)ii

(1.223)

,

as defined in eqs. ( 1.200) and ( 1.199), are actually the components of the second-order unit tensor I. Verify the representation

( l.224) = .'liigi ® gi = gi 0 gi = gi ® gi , with the properties gii = gii and .9ii = 9Ji of the metric coefficients. Recall the form ( l.207h and eq. ( 1.2 I I ) 1 to obtain u = ·u.;gi = giiuigi. Solution. I= ,r/igi ® gi

Hence, use relation ( l .209h for the components ·u,1 and property ( 1.53) to obtain u = gii(u. g)gi = gii(gi 0 gi) u ' '

"I

( 1.225)

1'

where I = gi.igi 0 g1 is the unit tensor in the representation ( 1.224 ) 1 and .r/i == (I)ii are identified as the six cmztrava riant components of I with respect to the basis { gi}. In an analogous manner, starting from u = u.;gi, we may obtain representation ( 1.224 b and the six covariant components 9ii = (I)ij of I with respect to the reciprocal basis {gi}. Since the metric coefficients .'lii and gii are components of I, we also call I the Euclidean metric tensor. In order to show the mixed component fonn we apply ui = 8jui to u = 11.ig;. Then, with eq.( l .21 Oh and rule ( 1.53 ), we obtain

( 1.226)

where the Kronecker delta 8j (1}~1 represents the mixed components of the metric tensor I. Since I is a symmetric tensor we conclude that (1)~1 = (I)j. •

A generalization to higher-order tensors is a straightforward task. A general tensor of order n,, having n indices i 1 , i'2, ... , in, may be represented in the form

.Ai1i:?•••ing. 0 g. ® ... (g) g. t1

l:!

ln

'

'l i1 .,,-itb-. .ing

IOI

'IY

gi:!

IOI

tO\

VY•••VYg

in

,

( 1.227)

where only 3n contravariant components Ai 1 i:!···in or 311 covariant components Ai 1i:? ... iu are involved. Now, a general tensor of order n may have mixed components of various kinds according to, for example,

38


In a fixed set of orthonormal basis vectors the components given in ( 1.227) and ( 1.228) coincide (compare with the form ( 1.135) ).

EXERCISES

1. Let the covariant basis vectors g 1, g 2 and a vector u be given by

= 1, 2 is a set of orthonormal basis vectors. Compute the contravariant basis vectors and u = ng 1 + f3g 2 • Draw a picture and where {e 0

},

a

interpret the geometrical situation. 2. Consider the two sets of covariant basis vectors

[g1J

[giJ

= =

rn rn

Find the set of basis vectors which form a possible general basis. Compute the covariant and contravariant components of a vector joining the origin to the point (1, 2, 1). 3. Let g 1 and g 2 be covariant basis vectors which are given as

r Consider the two vectors u = g 1 + 3g2 and v = -g 1 + 2g 2 resolved along this basis and compute the scalar product u · v and the lengths lnl, lvl using (a) the contravariant components of u, v and the metric coefficients, and (b) the contravariant and covariant components only. 4. Let A be a second-order tensor and consider the linear transformation v = Au that assigns a vector v to each vector u. By using eqs. ( l .207) and ( 1.214}, show that the transformation may be represented by 'U·l --

4· · ·ttj .6

lJ

--

"4· i·i,UJ·

•

1.6

39

General Bases

5. The contravariant, covariant and mixed components of a second-order tensor A, as represented by ( 1.214), are related to each other. (a) Take ( 1.214) and multiply both sides by the basis vectors gk (and gk ). Using the rule (l.53) and properties (1.205) and (1.215), show that the relations between the various components of A are given by

A ij

= gik9jl A kl = gjk.tl4i-k = 9ik Ak•j I

A ii -_ A i·i A i·i

(1.229)

'

ik i'A _ ikA·i _ Jk ,ti g g kl - g k - g ..1-1.k '

= gik 4ik = 9ik Aki = g ikA kj == _qjk Aik .I

' •

Note that the effect of multiplying the tensor components A ik, Aib A~k' Ai'~ by the set of quantities gik (or 9ik) and summing on the index k is to raise or lower an index on the associated tensor components. The operations of raising and lowering indices have the same meaning as in ( 1.211 ). (b) Consider the second-order tensor A

= e 1 ® e1 + 2e 1 ® e 2 - e1 ® e 3 -e2 ® e1 + e2 ® e2 - 2e2 ® e3 +2e3 ® ei + eJ 0 e2 + 3e3 ® e3 ,

where ei, e 2 , e 3 form an orthonormal basis. Consider a (new) basis represented by the covariant basis vectors

From eqs.( 1.216) and ( 1.217) derive the contravariant, covariant and mixed components of A with respect to the general basis { gi} and its reciprocal {gi}. Check the results by applying relation ( 1.229). 6. Let an orthogonal tensor Q be represented by

Q == Q~jgi ® gj

=

Q/ gi ® gj

.

(a) Use the identity (l.81) to verify that QT = Q~jgi ® gi show that the orthogonality condition may be written as

Qj·kQi·k

-

ri uj

'

9kl Qi·kQ·' j

== 9ij

= Q/gi ® gi

and

.

(b) Show that under the transformation gi = Qgi, gi = QTgi, i = 1, 2, 3, the components of the Euclidean metric tensor I remain unchanged.

40


1.7 Scalar, Vector, Tensor Functions A tensor function is a function whose arguments are one or more tensor variables and whose values are scalars, vectors or tensors. The functions (B), u(B), and A{B) are examples of so-called scalar-valued, vector-valued and tensor-valued tensor functions of one tensor variable B, respectively. In an analogous manner, (u), v(u), and A{u) are vector functions of one vector variable u with the value of a scalar, vector and tensor, respectively. Next, we consider scalar functions of one scalar variable, such as time t, in more detail. Let =
( 1.230) The first derivative of the scalar function is simply ci>

= dcI>(t}/dt.

In general, the n-th derivative of u and A (for any desired n) denoted by d 11 u/dt 1i and dn A/dtn, is a vector-valued and tensor-valued function whose components are d 11 [ui(t)]/dtn and d 11 [..4ij(t)]/dtn, respectively. Then-th derivative of is simply d 11 cI>(t)/dtn. Note that if , u, A are constant quantities, then their derivatives are equal to zero. By applying the usual rules of differentiation, we obtain the identities:

u±v=ti±v'

Cilii = ci> u + cI> ti ' u0v = ti©v+u0v , A±B=A±B

T

·T

trA

= trA

( 1.232) (1.233) ( 1.234) (I .235)

A =A

.

c[231)

.

( 1.236)

Here and elsewhere the overbars cover quantities to which the dot operations are applied.

41

1.7 Scalar, Vector, Tensor Functions

EXAMPLE 1.9

Show the identity

( 1.237)

Solution. Starting from A - l A = I we find by means of the chain rule that A- 1A+ A- 1A = 0. Finally, by multiplying the relation with A - l from the right-hand side we obtain the desired result (I .237). •

Gradient of a scalar-valued (tensor) function. We consider a nonlinear and smooth scalar-valued (tensor) function {A) of one second-order tensor variable A. Thus, for each A, (A) results in a scalar. The aim is to approximate the nonlinear function at A by a linear function. Firstorder (Taylor's) expansion for at A yields, using property ( l.93)i,

cI>(A + dA) = {A} d =

+ d + o(dA)

a~~A) :dA =tr [ (a~~))

T

dA]

,

( 1.238) ( 1.239)

where d denotes the total differential. The remainder is characterized by o( dA), where o( •) denotes the Landau order symbol. It is a small error that tends to zero faster than dA ---+ 0, i.e.

. o(dA} hm jdAI =0 . dA-tO

( 1.240)

The term D(A)/DA (or gradcI>(A)) denotes the gradient (or derivative) of function cl> at A, characterizing a second-order tensor. In order to make the derivation clear, the tensor variable to be derived is sometimes indicated in the literature by a corresponding index, for example, gradA {A). Note that the first variation of cI> is developed in the same way as its total differential. In Chapter 8 the concept of linearization is treated in more detail.

EXAMPLE 1.10 relation

If the second-order tensor A is invertible, establish the important

BdetA _ d A A-T BA - et .

( 1.241)

42

Solution.

1 Introduction to Vectors and Tensors With ( 1.10 I) we find that det(A + dA) = det[A(I + A- 1dA)] = detAdet(I + A- 1dA) .

( 1.242)

By means of eq. ( 1.168), we conclude with A replaced by A - I dA and with ,.\ that

= -1

det(A- dA +I) = 1 + /A-1dA + IIA-1c1A + IIIA-1dA 1

= 1 + tr{A- 1dA) + o{dA)

( 1.243)

.

In eq. (1.243) we have used the fact that in regard to (l.171) and (l.172) the second invariant is quadratic in dA and the third invariant is cubic in dA. Therefore, both are suitably small and summarized by the remainder of order o( dA). Replacing cI> (A) in relations ( 1.23 8) and ( 1.239) by the scalar-valued function detA, we deduce that

det(A + dA)

= detA +tr [ (

DdetA f)A

'l'

)

l

dA + o(dA) .

(1.244)

On the other hand we find by means of (l.242h, (l.243h and property (1.92) that det(A + dA)

= detA(l + tr(A- 1dA)] + o{dA) = detA + tr(detAA- 1dA) + o{dA)

.

(1.245)

Using rule (l.93) 1 and property (1.83) we find by comparing (1.244) with (l.245h that EJdetA

aA : dA = {detAA - 1)T

: dA

= detA A - T

which proves finally eq. ( 1.24 I}, since this holds for all dA.

: dA ,

(1.246)

•

Gradient of a tensor-valued (tensor) function.

We consider a nonlinear hnd smooth tensor-valued (tensor) function A {B) of one second-order tensor variable B. Thus, for each B, A(B) gives the value of a second-order tensor. In order to compute the total differential dA we consider first-order (Taylor's) expansion for A at B. We obtain A{B +dB)= A(B) + dA + o(dB) ,

cIA

= 8A(B) . dB

8B

.

.

( 1.247) ( 1.248)

The term 8A(B)/ EJB (or grad 8 A(B)) denotes the gradient (or derivative) of function A at B, characterizing afourth-order tensor. Note that BA/ DA produces the fourthorder unit tensor Il, as given in eq. (I. I 6 I).

1.7

43

Scalar, Vector, Tensor Functions

EXAMPLE 1.11 Assume that A is an invertible smooth tensor-valued function with property A = AT. Show that

aA-

1

aA

(

)

1

.. = - ?(AU,1 Aijt +All' Ak}) . 11kl

( 1.249)

-

Solution. Starting from the identity that 8{A- 1A)/8A = ((])gives the fourth-order zero tensor 0, we continue in index notation and find by means of the product rule

8Ahr!

,, _ 8Ami

4mj + "..:.iim1 {) 4 = 0 ./ aA kl kl

(1.250)

.6

and finally, after multiplication with (A - l )jn from the right-hand side, and using property ( 1.61 ),

8Aj;,: A

a

,t -'~kl

4-1 -

mj-- jn -

aaAkl A~,!

r Umn

8Ai;ll a"4kl

-

A-1 8Ami A-1 im {)A

kl

jn

'

1 ( A-1 r r 4-1 4-1 r r 4- I ) =- 2 imUmkUjt/ jn +" imUm/Ujk-'I jn

A-I ,t-lA-1) = - -1 (A-1 ik In + _.""iil kn I

2

'

•

( 1.251)

A replacement of the free index n by .i gives ( 1.249). Looking at the indices in ( 1.251) it is important to note that aA- 1 /BA# -A- 1 ® A- 1• •

EXERCISES

1. Suppose that Q = Q(t) is a given orthogonal tensor-valued function of a scalar, such as time t. Show that QQT is a skew tensor with the property QQT = -(QQT)T. 2. If the tensor A(t) is invertible, prove the important relation

detA(t) = detA(t)tr(A - l (t)A{t)] = detA(t)A(t)-T : A(t) and check the result with the matrix representation of the tensor A(t) given by

where a denotes a scalar.

44

1 Introduction to Vectors and Tensors 3. Consider two smooth tensor-valued functions A and B of a tensor argument. (a) Show the properties

DtrA =I

DA

Dtr(AJ) 8A

'

= 3(AT)2

(1 252)

'

.

(n is a positive integer) .

(b) If A is an invertible smooth tensor-valued function, show that (l.253) ( 1.254) 4. Let be a smooth scalar-valued function and A, B smooth tensor-valued functions of a tensor variable C. Obtain the properties

= A . DB B . 8A ac · ac + · ac ' 8( A) = A D aA 0 ac ac + ac ·

8(A : B)

(l.255) (1.256)

5. Alternatively to ( 1.241), show that

DdetA 8A

= {AT)2 -

I AT i

+

I I 2

r .

(1.257)

Hint: By means of property ( 1.92) and scalar invariants ( 1.170)-( I. I 72) take the trace of the Cayley-Hamilton equation ( 1.174) and derive it with respect to tensor A, use result ( 1.252).

1.8 Gradients and Related Operators In this section we consider scalar, vector and tensor-valued functions that assign a scalar, vector and tensor to each material point x varying over some region of definition.

1.8

45

Gradients and Related Operators

We call a tensor A and a vector u whose components Aii and ui depend on three (Cartesian) coordinates Xi a tensor field A{x) and a vector field u(x) in space at a fixed time, respectively. A scalar field cI>(x) of a body is a function that assigns a scalar cl> to each material point x. Subsequently, we consider a smooth scalar field cI>(x) (for example, a temperature, density or mass field). First-order (Taylor's) expansion for at x yields by analogy with ( 1.238) and ( 1.239)

Gradient of a scalar field.

(x + dx) == (x) d

+ dcI> + o( dx)

a == - · dx

ax

,

( 1.258) ( 1.259)

'

denoting the total d(fferential of . The remainder o( dx) tends to zero faster than dx -t o. The total differential, characterized as the difference between the scalars at x + dx and x, may be written in index notation as ( 1.260) In the literature the partial derivatives of with respect to Xi is frequently denoted by the subscript comma, i.e. a,i· We introduce further the vector operator (or Nabla operator) V of vector calculus, i.e. (1.261) Then ( 1.259) may be written in a different form, by means of ( 1.261) d

= V · clx

(1.262)

(1.263)

The sum over the partial derivatives of cI> multiplied by the appropriate basis vectors is a vectorjield (8cI>/8:1:i)ei, called the gradient (or derivative) of the scalar field cI>(x). By analogy with ( 1.261 ), the dot product, cross product and tensor product of the vector operator V with a smooth vector or tensor field ( •) is governed by the rules

V®(•)

= ~:;/ ®ei

These relations turn out to be very useful for subsequent derivations.

. (1.264)


46

Concept of directional derivative.

We consider cI> (x) describing a scalar field that varies throughout a three-dimensional space. We recall that (x) = (xi, .1: 2 , x:1) = const denotes a so-called level surface characterized by all points x resulting in the same value cl>. All neighboring points x + clx on a level surface in space are therefore governed by the relation d = 0. A normal to the surface is defined by three values, namely 8cI> I I~ acI> I a:r2' 8 introduced in (1.263). We conclude that grad = 8. If we introduce a unit vector n at a point x normal to the surface (directed along grad), then, n = grad /jgradI (see Figure 1.9).

ax

8ID u

/

grad

= ax

n

x

Level surface (x)

= canst

Figure 1.9 Level surface with unit normal vector n and an arbitrary vector u at x.

We introduce further a vector u at the point x inclined at an angle IJ to the direction of gradcI>. We call the dot product gradcI> · u a directional derivative (or a Gateaux derivative) of at x in the direction of the vector u. Note that for the definition of a directional derivative the normalized vector u is often used. As we let u vary, the directional derivative gradcI> · u takes its maximum (fora fixed point x) when u approaches the direction of grad (compare with the expression for the dot product, i.e. ( 1.10) ). For this particular situation u points in the direction of the normal vector n and cos(} = 1. Similarly, gradcI> · u takes its minimum value (for a fixed point x) when u approaches the opposite direction of the gradient (i.e. when cos(}= -1).

The directional derivative of in the direction of n is called the normal derivative; we write

gradcI> · n

= jgradI

.

( 1.265)

For the case in which u is a unit vector the normal derivative represents the maximum of all the directional derivatives of at x.

1.8

47


For further consideration we briefly note the equivalent representation for the directional derivative of at x, namely

(l.266) characterizing the rate of change of along the (straight) line through x in the direction u (for a more detailed account see, for example, ABRAHAM et al. [1988] and MARSDEN and HUGHES [1994]). Usually D( •)is called the Gateaux operator and c is a scalar parameter. By means of the chain rule we immediately find from definition (l.266) that D 0 (x) = gradcI> · u. Note that the directional derivative satisfies the usual rules of differentiation, i.e. the chain rule, product rule and so forth. Later in the text we will see that the Lie time derivative, Section 2.8, and the variation and linearization of fields, Sections 8.1 and 8.4, are equivalent operations which are based on the powerful concept of the directional derivative.

EXAMPLE 1.12

Suppose that the scalar field

cI>(x)

= :z:i + 3x2xa

(1.267)

describes some physical quantity in space. Compute the directional derivative of in the direction of the vector u = 1/ J3( e 1 + e 2 + e3 ) at the point x with coordinates (2, -1, 0) and interpret the result. The parametric expression x +ell represents a line through the given point x with direction u. We find the components of x + en (for arbitrary, but fixed e) as

Solution.

{2 + c/V3, -1 + c:/J3, c/J3). Hence, according to definition ( 1.266) and given expression ( 1.267), D 0 (2, -1, 0) is the derivative of 2

(x +cu)= ( 2 +

c )

J3

+3(

-1

e ) e 4 ') c + J3 J3 = 3 c- + J3 + 4

(1.268)

with respect to the one variable e, evaluated at zero. Therefore,

Duib (2 -1 0) = -d '

'

de

(4-e + -J3e +4) 2

3

1

=- . e=O J3

(1.269)

As noted above the directional derivative may also be computed by the dot product grad · u. In matrix notation we equivalently find from the given expression ( 1.267) that 1 [ 3x3 2:i: 1 D0 cJ>(x) = [grad][u] = J3 3:r:2

l[ l 1 1 1

( 1.270)


48

which, evaluated at :c 1 = 2, x 2 = -1, x:l = 0, gives the desired result of 1/ J3. Hence, if we are located at point (2, -1, 0) and walk away in the given direction u, we recognize an increase in some physical quantity, at a rate of 1/ J3 per distance, which is here

lnl =

1.

•

Divergence of a vector field. The introduced vector operator V dotted into any smooth vectorfield u(x) (for example, aforce, velocity or acceleration.field) is called the divergence of u(x). It is a scalar.field denoted by divu (or V · u). With ( 1.264) 1 and eqs. ( 1.20), ( 1.21) and ( 1.22)2 we find the important rule

au.·

Ou ·

8-ui

divu = V · u = - 1 e·1 · ei = - 1 8·i = 8:ri 8:i-i J 8:i;i Du1 8u.2 8u3

=8x1 - +a.1;2 - +8:1:3 -.

If divu

( 1.271)

= 0, then the vector field u(x) is said to be solenoidal (or divergence-free).

Curl of a vector field. The cross product of the vector operator V with a smooth vector field u(x) is called the curl (or rotation) of u. It is a vector field denoted by curio (or V x u, or rotu). With (I .264h and eqs. ( l .20) and ( 1.35) we find that

curio= V

D-u.

X U

= D J ei

X ej

=

Xi

a'u.

Eijk EJ J ek

( 1.272)

Xi

and with definition ( 1.33)

( l.273)

If curio = o, then the vector field u(x) is said to be irrotational (or conservative, A or curl-free). We can show further that curl grad cI> = o , div curio

=0

.

( 1.274) ( 1.275)

We now consider any vector field u of the form u = grad cI>, where is some scalar field, called the potential of u. According to ( 1.274) it readily follows that the vector field u is automatically irrotationa/, since curio = o.

EXAMPLE 1.13 Show that curio is determined to be twice the axial vector w of the skew part of tensor gradu.

1.8

49


Solution. We denote the skew part of gradu as W, with components H'ii = 8ui/axi. With the index relation H'ii = -lVii for skew tensors and (l.124)2, we deduce from (l.272h that

•

(1.276)

Gradient of a vector field. The gradient (or derivative) of a smooth vector.field u(x) is defined to be a (second-order) tensorjield. It is denoted by gradu (or Vu, or au/ax or sometimes more explicitly by V ® u) and according to (1.264)3 is given as gradu

=V

®u

with Cartesian components (gradu)ij

[gradu]

(1.277)

,

= 8uif 8xi. In matrix notation this reads as 8u1

a'U1

8u1

Bx1

ax2 a'U2

8xa

8u2

=

8-ui 8 J

= - x· ei ® ei

8u2

-

8:1:1

8x2

8u3

GU3

au3

a:r1

8x2

8xa

( 1.278)

8:c 3

It is easily seen from eq. ( 1.278) that the sum tr(gradu) of the diagonal elements of the matrix is equivalent to the divergence of u characterized in eq. ( 1.271 ). Thus, tr{gradu)

= gradu : I = divu

(1.279)

.

In addition, the transposed gradient of a smooth vectorfield u(x) of position xis denoted by gradTu (or uV, or (8u/8x)T or sometimes more explicitly by u ® V). It 1s given as gradT U

=U 0

V

= ei 0

with Cartesian components (gradTu)ij

au = Bu· a:ti. ei ® ej 8 1

-

,

(1.280)

Xi

= 8ui/8xi.

Divergence and gradient of a (second-order) tensor field. The vector operator V dotted into any smooth (second-order) tensor.field A(x) is a vector.field denoted by div A (or V ·A), which is the divergence of A(x). With (l.264)i, representation (1.59) and properties ( 1.53 }, ( 1.21) and ( 1.61) we have

( 1.281)


50

Note that, especially in the mathematical community, the alternative definition div A = 8Aii/ 8xiei of the divergence of a second-order tensor field A is frequently used. The gradient (or derivative) of a smooth (second-order) tensor field A(x) is defined to be a (third-order) tensor.field. It is denoted by gradA (or VA, or DA/Dx, or more explicitly by V ®A). Using ( 1.264)3 and representation ( 1.59), we may write

gradA

=V

®A

=

DA·· 11

aXk. ei ® eJ ® eh

( 1.282)

.

Laplacian and Hessian. The vector operator V dotted into itself gives the Laplacian (or the Laplacian operator) \7 2 (or V · V, or in the literature sometimes denoted by

~).

With the rules ( 1.261 ) 1, ( 1.264) 1 and properties ( 1.2 I), ( l.22h we find that

" . v-(•) =

V · V(•) == V

2 a (.) a (.) · -ei =

D:ri

= a~(·~

oij

8xi83-j

Dxia:c:i

= a2(~)

ei · e·3

.

( 1.283)

8.'ti

The Laplacian V 2 operated upon a scalar field cI> yields another scalar field. One example is Laplace's equation \7 2 cI> = 0. Another example is the inhomogeneous Laplace's equation often referred to as Poisson's equation, i.e. \7 2 = w. The differential equations of Laplace and Poisson occur in many fields of physics and engineering, for example, in electrostatics, in elasticity theory, in heat conduction problems of solids and so on. The operator VV (or more explicitly denoted by V ® V) is called the Hessian. With the rules ( 1.261 )i and ( 1.264h we obtain

a(.)

VV(•)=V0V(•)=V®-a· ei= :Ci

a2 (.)

aXi a·Xj ei®ei

( 1.284)

for the Hessian. Note the properties

curlcurlu =grad (divu) - \7 2 u , 2

\7 (u · v)

? ? = v-u · v + 2gradu · gradv + u · v-v .

( 1.285) ( 1.286)

If a vector field u is both solenoidal (divu = O} and irrotational (curio = o ), then \7 u = o (see eq. ( 1.285)). For this case the vector field u(x) is said to be harmonic. If a scalar field cI> satisfies Laplace's equation \7 2 cI> = 0, then cI> is said to be harmonic. 2

EXERCISES

1. Consider a smooth scalar field = x 1:z:2 :r3 - :r: 1 • Find a vector n of unit length normal to the level surface cl>= canst passing through {2, 0, 3).

1.8


51

2. Assume a force with magnitude F acting in the direction radially away from the origin at point (2a, 3a, 2v'3c) on the surface of a hyperboloid cI>(x) == cI>(1: 1, 1: 2 , 'JI 2 ., I ., 2 I " . :1:3 ) = :r:1 a + 1:2 er - :z::1 c- = 1. Compute the components of the force located in the tangent plane to the surface at the point (2a, 3a, 2v'3c). 3. Consider a scalar field {:r 1 , :v2} = e:ra cos(3:i: 1 - 2:r 2 }. Compute the directional derivative of cI> in the direction of the line :1: 1 = :r2 /3 - 1 at the point with coordinates (0, 1), for increasing values of:1: 1• 4. If U = u(x) == :r1:1:2J:ae1 + :r1:r2C2 + :r1 e3, determine divu, curio, graclu, V7 2u, respectively. Verify that ( 1.285) is satisfied. 5. Apply the operator V to products of smooth scalar fields cl>, \JI, vector fields u, v and tensor fields A. Establish the important identities div( cI>u) = divu + u · grad ,

( 1.287)

div{cpA) = cI>divA + Agrad ,

( 1.288)

div(ATu) = div A· u +A : gradu ,

( 1.289)

div(u x v) = v ·curio - u · curlv , div(u®v) = (gradu)v+udivv , grad ( \JI)

( 1.290)

= (grad cp) W + grad \JI

,

gTad(u) = u ® grad + gradu , grad{u · v) = {gracfru) v + (gradTv) u , curl{ cI>u)

= grad x u + curlu

( 1.291)

,

curl(uxv) =udivv-vdivu+(gradu)v-(gradv)u.

(1.292)

6. Consider the inverse square law defined by the particular vector field x u =a lxFI '

with the positive constant a and x = :ciei. The region of u is the three-dimensional space, excluding the origin whose gradient is u. (

== -a/lxl>

52


1.9 Integral Theorems In the following we summarize some results of the integral theorems of Gauss and Stokes which are of essential importance in the field of continuum mechanics.

Divergence theorem. Suppose u(x) and A(x) are any smooth vector and tensor fields defined on some convex three-dimensional region in physical space with volume v, and on a closed surfaces bounding this volume (see Figure 1.10). n

Surface element ds "

Closed surface s

Volume v

Figure 1.10 Volume v and bounded closed surface s (with infinitesimal surface element ds and associated unit normal vector n).

Then, for u and A we have - without proof

/ u · nds = / divudv

or

v

s

J

Ands

=

s

{

.

'Uinids

=

divAdv

or

I

1

axi

v

s

J

J8u·-

Ai1·n1·ds

=

v

s

ti

I

dv

'

--dv aA;j

8x·J

(1.293)

(1.294)

(given here in both symbolic and index notation), where n is the outward uni1' normal field acting along the surface s, dv and ds are infinitesimal volume and surface elements at x, respectively. The important transformation of a surface integral into a volume integral (i.e. (1.293), (1.294)) is known as the divergence theorem (or Gauss' divergence theorem). Similar results hold for higher-order tensors. The surface integral J'l u · nI in theorem (l.294) and knowing that div(cI>I) = grad, recall identity ( 1.288), we consequently obtain from ( 1.294)

J

nds

s

=

J

graddv

v

or

J

nids

s

=

J;:: v

dv ,

( 1.295)

53

1.9 Integral Theorems

which is known as the Green-Gauss-Ostrogradskii theorem. For the sake of completeness the theorem of Stokes is briefly Stokes' theorem. summarized. It relates a surface integral, which is valid over any open surface s, to a line integral around the bounding closed curve c in three-dimensional space. We now introduce a tangent vector to c denoted by dx (with components chi) and an outward unit vector field n normal to s. Curve c has positive orientation relative to n in the sense shown in Figure 1. I I . The indicated circuit with the adjacent points 1, 2, 3 (1, 2 on curve c and 3 an interior point of the surface s) induced by the orientation of c is related to the direction of n (i.e. a unit vector normal to sat point 3) by the right-hand screw-rule (see Figure 1. I I). n

Surface element ds

dx Surfaces

Curve c

Figure 1.11 Open surface.

For a smooth vector field u defined on some region containing s, we have

f u · dx = Jcurio · ncls c

( 1.296)

or c

s

(the proof is omitted). Note that if the surface s is closed the integral on the left-hand side of eq. ( 1.296) reduces to zero, and then Is curio · nds = 0. The line integral fc u · dx represents the circulation of u around the closed space curve c.

EXERCISES

1. Let cl>, u, and A be smooth scalar, vector and tensor fields defined in v and on s


54

and let n be the outward unit normal field acting along the surface s. Show that

div(u)dv ,

( 1.297)

/ Ands = / div(cI>A)d-u ,

( 1.298)

/ cI>u · nds s

= / v

v

/ n x uds = / curludv , v

/ u 0 nds = / gradudv , ti

/ u ·Ands= / div(ATu)dv .

(1.299)

v

s

2. Let u and A be smooth vector and tensor fields defined in v and on .s and let n be the outward normal to s. Use index notation to show that

/ u x Ands=/ (u x div A+£ : AT)dv ,

(1.300)

v

where

e is the third-order permutation tensor as expressed in eq. ( 1.143 ).

3. Let cI> and u be smooth scalar and vector fields defined on s and c and let n act on s. Show that

.f

f c

c

dx = / n x gradcI>ds , s

u x dx = /[(divu)n - (gradTu)n]ds s

2

Kinematics

In the real world all physical objects are composed of molecules which are formed by atomic and subatomic particles. A microscopic system is studied by means of magnifying instruments such as a microscope. Microscopic studies are effective at the atomic level and very important in the exploration of a variety of physical phenomena. The atomistic point of view, however, is not a useful and adequate approach for common engineering applications. The microscopic approach is used in this text only briefly within Chapter 7. We use the method of continuum mechanics as a powerful and effective tool to explain various physical phenomena successfully without detailed knowledge of the complexity of their internal (micro)structures. For example, think about water which is made up of billions of molecules. A good approximation is to treat water as a continuous medium characterized by certain field quantities which are associated with the internal structure, such as density, temperature and velocity. From the physical point of view this is an approximation in which the very large numbers of particles are replaced by a few quantities; we consider a macroscopic system. Hence, our primary interface with nature is through these quantities which represent averages over dimensions that are small enough to capture high gradients and to reflect some microstructural effects. Of course the predictions based on macroscopic studies are not exact but good enough for the design of machine elements in engineering. The subject of continuum mechanics roughly comprises the following basic ingredients: (i) the study of motion and defonnation (kinematics), (ii) the study of stress in a continuum (the concept of stress), and

(iii) the mathematical description of the fundamental laws of physics governing the motion of a continuum (balance principles). The aim of the fallowing three Chapters 2-4 is to derive the essential equations within the basic fields (i)-(iii). Note that the provided results are applicable to all

55

2

Kinematics

classes of materials, regard.less of their internal physical structure. In -order to explain the macroscopic behavior of physical objects, :first of all we must understand the motion and deformation that cause stresses in a material (or are caused by stress.es) arising from forces and moments. Thus, to study the motion and (finite) d.eformation of a continuum, i.e. kinematics, is essential and mainly the aim -of the following chapter. The reader who requires additional information on the subject of this -chapte.r i-s referred to the monographs by, for example, MALVERN [ l 969], WANG and TRUESDELL .(1973], CHADWICK [1976], TRUESDELL [1977], GURTIN [1981.a], CIARLBT :[1988], TRUESDELL and NOLL [1992], MARSDEN and HUGHES [.1994) or OGDEN [1997].

2.1


Under an electron microscope we see the discontinuous atomic structure of matter. The molecules may be crystalline or .randomly oriented. Between the particles there are large gaps. Theories considering the discrete structure of matter are ·molecular and atomistic theories. They are based on a discrete particle approach. For macroscopic systems such theories tend .to become too complicated to yield the desired results and therefore do not.meet our needs. However, under certain circumstances the microscopic approach is indispensable for the study of physical phenomena. A review of atomistic models is :presented in the work of ORTIZ f1999, and references the.rein_]. Notation of a particle nod a continuum body. Macroscopic systems usually can be described successfully with a continuum a_pproach (macroscopic .approach). Such an approach leads to the continuum theory. The continuum theory has been developed .independently of the molecular and atomistic theory and is meeting our needs. A fundamental assu-mp.tion therein states that a body, denoted by B, may be viewed as having a continuous (or at least a piecewise continuous) distribution of matte.r .in space and time. The body is imagined as being a composition of a (continuous) set of parti .. cles (or continuum particles or material points), represented by P E B as indicated . p·1gure 'J-· I . m. lt is important to note that the notion 'particle' (or 'continuum .particle' or 'material point') refers to a part of a body and does not imply any assoc.iation with the point mass -of Newtonian mechanics or the discrete particle of the atomistic theory as .mentioned above. A typical continuum particle is an accumulation of a large number of molecules, yet is small enough to be considered as a particle. The behavior of a continuum partic.le is a consequence of the collective behavior of all the molecules constituting that particle. Hence, in a macroscopic study we are concerned with the mec.hanics of a body in which both mass and volume are continuous (o.r at least piecewise continuous) .func-

2.. 1 Configurations, and Motions of Continuum Bodies Current configuration

Reference configuration .

.·.

.

. .

,; :)ii:::::::::

.

57

:· .

x

:~i,: >: ; :

:. ·:... ::. :·' ·' ._: .·. <:. _: _: :·'. .:. .:.: .: ·.\ ~---··: :_· /: <:.·.:

\·/

....... _:;_:._:·.·.:..·...:.-.,.: ..

time t

=0 l

'• I

,,,

\ ',

x

,----~-------~~

B

•p

~,

./'

~,

'

',\

\

.

time t

\

'

,1

'~~------------~~---~'

r.--11~

e2

x

x 2' :1:2

Figure .2•.1 Configuration and motion of a continuum body.

tions of continuum particles. Such a body is caIIed a continuum body, or just a continuum. A continuum is determined by macroscopic quantities. It has macroscopic dimensions that are .much "larger than the intermolecular spacings. For the linkage of atomistic and continuum theories and for mixed atomistic/continuum computational schemes the reader is referred to ORTIZ :.[1999]. Configuration. Consider .a continuum body B with particle P E 8 which is embedded in the three-dimensional Euclidean space at a given .instant of time t, as indicated in Figure 2.1.

We introduce a reference frame of right-handed, rectangular coordinate axes at a fixed origin 0 with orthonormal basis vectors ea, a = 1, 2_, 3, according to properties introduced in ( l .18). As the continuum body B moves in space from one instant of

58

2

Kinematics

time to another it occupies a continuous sequence of geometrical regions denoted by f2 01 ••• , n. Hence, every particle P of B corresponds to a so-cal led geometrical point having a position in regions n01 ••. , n. The .regions which are occupied by the continuum body Bat a given time-tare known as the configurations of Bat that time t. The continuum body B may have .infinitely many configurations in space. The geometrical regions are determined uniquely at any instant of ti.me. Region n0 with the position .of a typical point X corresponds to a fixed reference time. The region is referred to as the fixed -reference (or undeformed) configuration of the body B. A region at ·inithd time t = 0 is referred to as the initial configuration. We agree subsequently that the .initial configuration coincides with the reference configuration; hence, the reference time is at t = 0. Note that in dynamics the initial configuration is often not chosen as the referenc-e configuration. Now, the po.int X has the position of a particle occupied by P E B at t = 0, and P may he identified by the position vector (or referential posUion) X of point X relative to the fixed origin 0. We now assume that the region 0 0 of space moves to a new region which is occupied by the continuum body B at a subsequent time t > 0. The configuration of Batt is its so-called current (or deformed) configuration. We relate a typical point X of the reference con.figuration to a point x of the current configuration occupied by a particle P EB at times t = 0 and t > 0, respectively. The :position vector (or current pos"ition) x serves as .a label for the .associated point x with .respect to the ·fixed origin 0. It is often convenient to call X po.int X, associated with the particle P E ·B at t = 0, and to call x point x (or place x)., associated _with P E B at :t.

n

The components of vectors X = )(AEA and x ·=:cu.ea are considered as being along the axes introduced. We .Jabel .1X";t, il = 1, .2, ·3, as the material (or referential) coordi~ nates -of point X and :r:a, a = 1, 2, 3, as the spatial (or current) coordinates of point x. ·we have assumed that the origins of the sets {EA} and {Ca} of basis vectors -coincide. ln addition, we have agreed to have the same reference frame for the reference and current configurations, which is why we set the basis {EA} identical to {ea}. We use just {ea} in the following. To denote scalar, vector and tensor-quantities we use uppercase letters when they are evaluated in the reference configuration, and lowercase letters for corresponding quantities in the current configuration. Sometimes we employ the index zero for quantities acting in the reference configuration (for example, n0 , p0 , a 0 ... ). This convention is often used in textbooks of continuum mechanics; however, it occasionally leads to con"fticts with the ·general aim of Chapter l where we agreed to use lowercase, bold-face letters for vectors and uppercase, bold-face letters for second-order tensors. For example. we use X for the position vector of a point corresponding to the reference configuration despite the convention of Chapter 1 that uppercase, bold-face Latin letters usually denote second-order tensors. A dear and unique notation .is difficult to establish, considering that certain quantities in continuum mechanics are characterized

2.1


59

(by unequivoca] symbols. · ··. Whenever index notation is employed, as for the components of vectors and tensors, ·:::c-~nventionally we use uppercase letters for the reference configuration (for example, -:_'.·_'."·t.b BA ... ) and lowercase letters for the current configuration (for example, ~1:a., ba .. .). :Motion. We .assume that the map X = ~ 0 ( P, ."t) . is a one-to-one correspondence :·between a particle P E B and the point X E Ho that 8 -occupies at the given instant of ·_:time t == 0 (see Figure 2.1). Furthermore, let the map"' act on B to produce the region at time t. The place x = ~(P_, t) that the particle P (evidently identified with X and !t). occupies at tis described by (in symbolic and index notation)

·.n

:r:<,

= Xa (..\'"1, .X2, . X:i, t)

(2J)

{?r. all X E f! 0 and for all times t . In (2.1) x is a vector field that specifies the place x of for all fixed t, .and -is called the motion of the body 8. The motion x carries points X 1·ocated at f2 0 to places x in the current configuration n. ·we assume subsequently that possesses continuous derivatives w..ith respect to space and time . ·. The parametric equation (2. I) determines successive positions x of.a typical particle g.:· _i~ space. All successive points together form a curve in the Euclidean space which call the path line (or trajectory) of the particle P. :>_· ... ·. The motion xis assumed to be uniquely invertible. Consider (x 1 t), the position of X, which is associated with the place x at time ·t, is specified uniquely by (2 ..I) as

:.x

x

\ve

·.point .......... ... ;

X

= x·~

1

(x,t)

,

(22)

\vith the inverse -motion, denoted by x-l., i.e. the .inverse of the mapping x~ as defined -~~. ~q. (2.1). For a given time t, the inverse motion (2.2) carries points located at .n to ·.p_o_i_nts in. the reference configuration n0 • .In (2.1) and (2 . 2) respectively the pairs (X, t) .... . :and (x, t) denote independent variables . _:· ... ~ A motion x of a body will generally change its shape~ position and orientation. ·Acontinuum body which is able to change its shape is said to be deformable. By a ~~formation x (or inverse deformation x- 1) of a body we mean a motion (or inverse :motion) of a body that is independent of time. .

·.'·::'-:=.==.:··~·~···:··."·."·'·:·····:·:·, ................... , .............,., ..... :~, ....... ,.,_... , ....................

··········· ······················:·:·:······."·······:·····:·.·.·.... : ....... ,...................................... ············ ················:·:·.··:'•." . ••••'··'·····.·:···'·."·:·.'."•."•"."•.'."·.··.··.·,.: ..·····."·.··,.·,··· . ·····,:•.•."•:.··

EXAMPLE 2.1 Let a deformed configuration of an initially .rectangular region with lengths L 1 and L 2 be given as shown in Figure 2 . 2. The time-dependent angle fJ is :given by O(t) = wl:, where w denotes the angular velocity. :_'.="·:·:.-:Determine the motion of a .particle given by the position vector XE .n 0 and time t In: particular, determine the -motion x = x(X, .t) at (L 1 /2, .£ 2 /2, 1) and at time t = "/4, ·with w = 1~

·60

2 Kinematics

:t =0 . f!o

Figure 2.2 Deformed configuration of an initially rectangular region.

Solution. The reference configuration of the region 0 0 of interest is characterized by (Xt, ""\2 , ..X3_) .at t = 0, with 0 < .X1 < L1 , 0 < .X2 < L2 and 0 < .1Ya < 1. The motion results in accord with (2.1)

= X2(..(Y1, .1Y2> ..t\3, t) = .X2 = L2/2 x3 = x3(.:Y1, . .Y2,..1Y3,t) = .x·3=1 . ,

:i:2

where c(t)

= tanO(t) is a positive scalar.

,

(2.3)

•

Material .and spatial descri.ptions.. The so-called material (or referential) description is a characterization o.f the motion (or any other quantity) with respect to the material coordinates {.X1 , ~~2 , ~Y3 ) and time t, given by eq. (2.1). In the material description attention is. paid to a particle, and we observe what happens to the particle as it moves. Traditionally the material description .is often referred to as the Lagrangian description (or :Lagrangian form). Note .that at t 0 we have the consistency condition X = x., .x·.:, = Xa. The so-called Eule-rian (or spatial) description (or Eulerian form)~ is .a charac ... terization of -the -motion (or any other quantity) with .respect to the spatial coordinates (:t.t, x 2 , x 3 ) and time t, given by eq. (2.2). In the spatial description attention is paid to a point in space, and we study what ·happens at the point as time changes. In :fluid mechanics we quite often work in the Eulerian description in which we refer all relevant quantities to the position in space at time t. It is not useful to refer the quantities to the material coordinates ..iY1i, A = 1, 2, 3, at t = 0, which are, in general, not known -in fluid mechanics. However, in solid mechanics we use both types

=

2.2

Displacement, Velocity, Acceleration Fields

61

of description. Due to the fact that the constitutive behavior of solids is often given in terms of material coordinates we often prefer the Lagrangian description.

EXERCISE

:I. Consider the two motions defined by the following sets of ·equations:

= o:t:i e + X

x

:1:2

= -fJ -

,

(8 +wt)

1e-111"cos:

.~,

where .n, /3, ·/, c5 and w are scalar constants and e denotes a fixed unit vector. Show that the path lines are straight lines and circles, respectively.

2.2 Displacement, Velocity, A-cceleration Fields Displacement field.

The vector field U(X, 't) = x(X, t) - X

(2.4)

represents the displacement fie=ld of a typical particle and relates its position X in the undeformed configuration to its position x in the de.formed configuration at time t (see Figure 2.3). Relation (2.4) holds for all particles. The displacement fie.Id U is a function of the referential position X and ti.me t, which characterizes the material description (Lagrangian form) of the displacement field. The displacement ·field in the spatial description {Eulerian form), denoted u, .is a function of the current position x and time t. We write

u(x, t)

= x - X(x, t)

.

(2.5)

Representation (2.5) specifies the current position x of a particle at time t which results from its referential position X(x, t) plus its displacement u(x, t) from that position. Representation (2.4) -expresses the displacement of a particle at time t in terms of its ·referential position X. The two descriptions are related by means of the motion x = x(X, t), name-Iy

U(XJ)

= U[x-t.{x, t), t] =

u(x, t) .

(2.6)

62

2

Kinematics

U(X, .t)

lime t = 0

= u(x, t)

x

x

lime t

Figure 2.3 Displacement fi.eld U of a typical particle.

Note that U and u have the same values. They represent functions of different arguments. The vectors U and u are referred to the material coordinates UA and the spatial coordinates ··11.a, respectively,. Since we agreed that the initial configuration coincides with the reference configuration, the displacements vanish in .the reference configuration. For a so-called rigid-body translation any particle moves an identical distance, with the same magnitude and the same direction at time t. For this case the displacement field is independent of X (it can be a function of time t).

Velocity and acceleration field.

.In solid mechanics the motion and the deformation of a continuum body are, in general, described in terms of the displacement ·field. However, the primary field ·quantities in fluid mechanics describing the :fundamental kinematic properties are the velocity fie.Id .and the acceleration field. The "first and second derivatives of the motion x with respect to time t are .perform.ed by holding X fixed. We obtain

V(X, t)

= Dx.~X, t) t

'

A(x ,t. )

= 8V(X, t) = D2 x(X, 't_) '" a·t a')t'

where V(X, t) and A(X, t) denote the material description of the veloc:ity field and .the acceleration field, respectively. They are functions of the material coordinates (.X 1 , ..Y2 , ~\:1 ) and time t representing the time rate of change of position and velocity .of a particle X at time t.

2.2

·63

Displacement, Velocity, Acceleration Fields

Quite often we need a formuJation in terms of the spatial coordinates (:1:J, :l:2 , :i::i) chm·acterizing a fixed location in spac-e at time t. As mentioned above, such a description is known as the Eulerian or spatial description in which now we like to define the velocity field and the acceleration field. :.":". . . The two -descriptions are transformed into -each other by using the motion X· In · · the spatial description, X may be expressed in tenns of x and t Hence, from (2. 7) we conclude by analogy with eq. (2.6) that by change of variables

V(X, f)

= V[x- 1 (x, t), tJ =

A(X, t)

= A[x- 1 (x, t), t;] = a(x, t) .,

v{x~ t)

(2.8) (2 ..9)

:·\< . where v(x, t) and a(x, t) denote the spatial description of the velocity ·field and the :<:·:":.--_ acceleration field_, respectively. The velocity components .and the accele·-ration components for V and A are de·:··;· noted by V4, AA, while for v and a we write Va,
:-: : EXAMPLE 2.2 A certain motion of a continuum body in the material description is : ·.:·_:· given in the form It

-

.I, l -

.,,t ,.-

,-t ,,..

(-,, ..-·\ l -

(.

."l. 2

:c•1-

'

= et . /\. J + e-t . l\. ·> "'\!"

'"I'

i,.i-

-

•

1

1

3

-

~

.,._. . ../\3 '

(2. l 0)

;_·for t > 0. Find the velocity and acceleration components in terms of the material ··_and spatial coordinates and time, i.e. '1~1 = V4( ..Y..:t, t), AA = AA( . .~A, t) and 'Va = ·.Va { Xa, t)" CLa = a,, (:i;,1 , t), respectively.

:_._ ~olution. For the given motion ~l;,1 = Xc1(.X 1, ..:Y2 , ..Y3 , t), the velocity and accelera:_.·:_.:Hon components in tenns of the material coordinates .XA •. A = l_, 2, 3, and time t foHow :::·._:._from eqs. (.2.7) and have the forms T /•

,.,. 1

4

== e t ...-·i."\... J + c -t. . .v· -·\ 2

\·... 4 1 = e·t . "'. it

·-t "\.·"' ... ·\.2

- e

'

'

T...

•/2

'\.·.. = e ·__..\·v· 1 ·- .e -·t ..:\.2

,

+ (::."-t ..-·\v·2

_,

{,

4 _ £,) ....·v·\ I

" 2 -

(2.11) A.:1

=0

.

(2.12)

... ·. ··:. . . In order to find the velocity and acceleration components in terms of the spatial \·:.:.:_::·coordinates ~r.," u = 1, 2, 3, .and Lime t, we substitute simply the given motion into '.:./.: :· .:·~e_rived relations (2.l l) and (2 ..12). Thus, according to eqs. (2~.8) and (2.9) we find that .

..

.

V1 a1

== :1:1

=

= V2

~r2

'

,

'l.13

(2.] 3)

·== 0 ' ll:J

= -0

.

Iii

(2 ..14)

64

2

Kinematics.

EXERCISES

I. Consider the motion of a continuum body given by the equations

:r1

= . .Y.1 (1 + nt:i)

,. ,. .{.··, - "A - ... '

,

(2.15)

~

')

where a is a constant. Determine the displacement, velocity and acceleration fields in each of the material and spatial descriptions.

2. A motion of a continuum body is defined by the velocity -components 3;1:1

'll1

=- ' l+t

V2

=l

:r:'>

~t

(2.16)

1

Assume that the reference configuration of the continuum body is at t the consistency -condition X

= O,
= x.

(a) Derive the particle path, Le. the :motion x == x(X, t). (b) Compute the velocity components in terms of the material coordinates ..YA and time ·t, i.e. V4 , il = 1, 2, 3, and the associated acceleration in the material and spatial descriptions .

2~3


We introduce the terminology of a material field in which the independent variables are (X, t), i.e. the referential position X with material coordinates .X.-i, A = 1, 2, 3, and the time ·t. In a spatial field the independent variables are (x, t), i.e. the current position (the place) x with spatial coord.inates :ca, -a = 1, 2, 3, and the time t. In the

fol.lowing we denote a smooth material field and a spatial field of some physical scalar, vector or tensor.quantity associated with the motion x as F = F(X, t) and f = .f (x, t), respectively. Material time der"ivative of a material field. Consider the material time derivative of a smooth material field :F(X;t) which is conventionally denoted by DF(X, 't)/Dt or by a superposed dot, Le. :F(X, t). Other names for it .in the literature are tota:J or substantial time derivative. lt is the derivative of :F with respect to time t (holding X ·fixed); thus, in symbolic

and index notation,

:t(X, t) = DF(X, t) = (OF(X, t)) Dt or

,r.( )-·.. .· t ) ·= (··D:F(.XA, {)'t

.r . . /\ .·l ..

1

ut

t))

x

(2 . 17) ~

)\ ,.\

2.3

65


The subscript X, (.-YA), indicates the variable which is held constant while taking the partial differentiation of :F. In the following the subscripts are dropped to reduce the cumbersome notation,. However, in cases where additional information is needed, they will be employed. The.material time derivative represents the rate at which the material field :F changes with time, seen by an observer following the path line of a particle.

EXAMPLE 2.3

Determine the material description of the velocity field V and the acce.leration field A of a particle at time .t as the time rate of change of the displacement of that particle. Solution.

V(X, t)

By virtue of eqs. (2.1) 2 and (2.4) we may derive from (2.7)

= (8U(X, t)) = (J

so that with V(X, t)

.at

,

A(X,t)

x

= (BV(X,t)) = V ,

at

(2.1.8)

x

= 82 U(X, t)/8t 2 = U, . .. A=V=U

(2.19)

results. With the terminology introduced above, V(X, t) and A(X, t) denote the material velocity field and the material acceleration.field, respectively. ·•

For some studies following in this text we introduce the useful relationship between the .material time derivative and the directional derivative. Considering F = F(X, 't), the material time derivative of :F is ·equal to the directional derivative of F in the direction of the velocity vector v, we write

DF(X, t)

Dt

= D..,,,.~~(x, t ) '

(2.20)

which ·.can be shown by applying the definition of the directional derivative, i.e. (.l .266). Material gradient .-of a material field. The material _gradient of a smooth material field F(X, t), denoted by Gra.dF(X, t) (with uppercase G), is obtained by deriving :F relative to the referential position X, (.XA), at a fixed time t, i.e.

GradF(X, t ) =

8:F(X, t)

ax . .

(2.21.)

Subsequently, the material dive·rgence and the material curl of :F are denoted by DivF and CurLr (with uppercase D and C), respectively. They are divergence and

66-

2

Kinematics ·.

curl differential operators characterizing quantities with respect to the reference con-

figuration . .

.

.

.

.

.

.

.

Spatfal ti.me derivative and spatial gradient of a spatial 6eld.. The spatial time derivative of a smooth spatial fie.Id /(x, t) _is the derivative off with respect to time t holding the current position x fixed. ]tis simply denoted by Df (x, t)/fJt representing the rate at which f changes with time1 seen by an observer currently stationed at x. In the literature the spat~al time derivative "is also referred to as_ the local time derivative. The s_patial gradient of f, denoted by grad.f (with lowercase g), is the derivative off with respect to the current position x, (~1:(1), at a fixed time t, i.e.

=

gracl/{x, t)

DJ(x {) ,' .Ox

.

(2.22)

Subsequently, the spatial divergence and the s:patial curl off are denoted by div l and curl/ (with lowercas-e d and c), respectively. They are differential operators characterizing quantities with respect to the current configuration.

Materia'I time derlvative of a spatial field. Finally, we consider a very :important operation, i.e. the material time derivative of a smooth spatia·1 field f (x, t), which is denoted by D.f (x, t)/Dt .or j(x, t). It is the time derivative off holding.X fixed. .In order to evaluate f we map .f to the material description as a first step, and then we take the material time derivative. The result we map back to the spatial description in order to obtain .

j(x, t)

= D.f(x, t) = ( D.f[x(~, t), t )] )

at

Dt

. X=x- J (x,t)

Let cJ> be a smooth spatial field which assigns a scalar (x, t) to each point x at t. By the chain rule we find from (2.23) that

<~(x,t;) = ( D(x, Dt

t))

. (Dx(X, t))

+ (a.(x, t;)) .Ux

x

Dt

t

,

(2.24)

X=x- •(x,t)

and by .means of eqs. (2.7) 1,. (248) and notation (2.22),

. cI>(x, t)

=

DcI~(x

~

or

' t

t)

=

.

8cI>(x, t) [) ·t

D-

.

+ grad(x, t) · v(x, t) , D
= -Dt = -Dt

-1-

D

(2.25)

-·fJb 8:1;b .

Herein, the first term on the right-hand side of eq. (2.25) denotes the spatial (or local) time derivative of the spatial scalar field , while the second term is called the convective rate of change of ·cJ>, which describes the changing position of a particle.

2.3

Material_, Spatial Der.ivativcs

67

By analogy with the above, the ·material tim-e derivative of a smooth spatial field v(x, J;), which is vector-valued, is given by

··( ) v x,t

= Dv(x,t) Dt = Dv(x,t) at . vo.

or

Dva.

(

) .

+gradv x,t v(x,t

Dva

) (2.26)

Dva

= -D -a Vti . t = -D ·t; + ··=1:b

where v = v{x, -tL (vu, a = 1, 2, 3), and its material ti.me .derivative v = v(x, :I:), (t}ll., a = 1, 2, 3), may also be viewed as the spatial description of a velocity and acceleration field, referred to brie-fly as the spatial velocity .field nnd the spatial acceleration .field, respectively. The spatial acceleration field we often denote by a = v, (c1u = 'Vu, n = l_, 2, 3). Hence, we may write eq. (2.26h as a

= -Dv + (gradv)v ' 0. t

(2.27)

or

(the arguments of the functions have been omitted for simplicity). The first term on the right-hand side, name.1y av I at, describes the ·local acceleration (local rate of change of the velocity field). The second term is quadratically nonlinear in the velocity field and describes the convective acceleration field (convective rate of change of the velocity field). Relation (2.27) turns out to be very useful, because the spatial acceleration field a(x, t) m~y be determined from v(x, t) without knowing the motion explicitly, which is an important question in applied :fluid mechanics, By virtue of (2.7) 1 and (2.8), the spatial velocity field v(x, t) may be expresseµ as the material time derivative of the motion x = x(X, t), i.e.

. ax

V=X=----

8t

or

.

Vu

iJ:c a

= :i;n = -at .,

(2.28)

and from (2.5) we find that v == ii. The spatial acceleration field re.ads by analogy with relation (2 .19) as

a=v=u .

(2.29)

A very useful short-hand notation of lhe ope.rator D( •) /Dt which acts on spatial fields reads, in symbolic and index notation,

D(•) Dt

= D(•) + grad(•)·v Dt

-

or

D(~)

D(•)

D(•)

- - = - - +-.-'Vo , .Dt D:r(L

ut

(2~30)

which should be compared with eqs. (.2.25) and (2.26). Relation (2.30) shows clearly the relationship between the ·material and spatial t\me derivatives.

68

--~---=--~---,~~""""":'''~-,,~,~-~,-~-.-~

2 Kinematics

..... ,,-.,:·---· - - · - - - ___ ,

..

-·--------------·-·-"-·'"·~,,.,.~~:,,, _,.,.,,:·~~~:~~-,.~

...

..,,,.~,,,,,, ~,,,,,,,,,~.,,.,,~

...

..,,,,~,,,,. ,_.,,,.,,,~~-,

....... ,.

,~,.,,_,~.,,.~

... -.,"':"""""-.-...... ,.,_,.,. ___ ,

~---,·-~·-.-- ~-·

.. -· -.-:·

EXAMPLE 2.4 In the current configuration of a continuum body a certain physical quantity is given in space and time according to t-1.

(x, t}

= - TXT ,

(2.31)

with :place x = :1:iei. The region of the spatial (scalar) field · =
as seen by the observer. Solution. In order to compute the material time derivative of the spatial fieid apply relation (2.25)2. Therefore, we need to compute the spatial time derivative and the spatial gradient of . The spatial time derivative of gives t- 2

o(x., t)

at

lxl

(2.33)

.

With definition (2.22) we .find directly from (2.31) the spatial gradient of cl>, i . e. gradqi(x, t) =

8 (x, t:)

ax

-1

x

= t lxl3 .

(2.34)

Note that the spatial vector field gTadcI> is harmonic and describes a point source at .a certain time t (compare with Exercise ·6 on p. 51 ). By virtue of the derived solutions (2.33), (2.34) and the given velocity field (2.32) we obtain finally (2.35)

which gives the desired solution {2.3 .l ), the material time derivative of at x and at • time t. , ·' .... ~··,, ... , •.... : ··'····· '··· ~, ... ~·· ,, '.·: ~'·'···." ... , .. : .. ,.,, ~ .. ,., •. , , ~···. ~·, .... ,, ~-, ~,, ... ~'·' ~'·'····: .,'·····~··.··'·-''' ~ .. , ........ ·,··: ,, .. ·.·~···: ,,.,.,,.,,._ .... ,·.·~·~''' ~- '·' ., .. : ... c ~··,: ...... : , ~···: '···~·······, ~, ... , '.·:: '·' ~'·'····.. ·......... : ~···· ,_., ... '···, ... '···." .... ·: '···."····: ·'···." '·~'··'·"~', ~, '."·"·'·"·"·'·"•"·'·~·,,: .······: ,. : ,.~···': ..... :: ,., : ... -~·:: ~.

at

Types of motions. .If the velocity field is independent of time~ i.e. av I = 0 and v = v(x), the associated motion is said to be steady and (2.27) reduces to a = (gradv)v. On the other hand, if the velocity field at each instant :is independent of position x, i.e. gradv = 0 and v = v{t), the associated motion is said to be uniform and (2.27) reduces to a = av I If the velocity .field has components of the form

at.

2.3

=

v-i ( :1;.1> :1;2 , t), v2

=

69

l\llaterial, Spatial Derivatives

v 2 ( x 1 , :r 2 , t ).,-.v:1

=

0 the associated mot.ion is caned plane. A motion is potential if there exists a spatial velocity field v = grad lJ>, where the spatial scalar fie]d is called the veloc-ity potential. A motion satisfying c;url v = o is irrotational. By recalling relation (1.274) we conclude that if a motion is potential then it is automatic.ally an .irrotational motion.

v1

EXERCISES

:1. For a one-dimens.ional problem the displacement field U the equation

= U (.X, t)

is given by

U =ct)( ,

with c denoting a constant. The relation between the spatial and the material coordinates is given by :r = (1 + ct) .X. Derive the first and second derivatives of U with respect to time t in both the material and spatial descriptions. 2. Recall Example 2.2 on p. 63. The spatial components a. 1 = ;r 1 , a 2 = J: 2 , a.~i = 0, i.e. eq. (2.14), are derived directly from Lhe acceleration in the material description, Le. eq. {2.12)~ by means of given motion (2.10).

Show that the spatial components a.a, a = 1., 2, 3, may also be derived from expression {2 ..27) using eq. (2.13 ), i.e~ ·v 1 = :1: 2 , v2 = :r 1, v:i = 0. 3. ln a certain region the spatial veloc.ity components of v = v(x, t) are given as

.·,l . l -_ _ (l;(··.a .I..

; .2·) (:,_l-/Jt + ,.J,.. t •l'2

·'

'l 1'•2

=

(\i ( -4;

'1~2.,r'' 1ol.I 2

• '

+ •.,.a) •2

Ll

L-

-{Jl

.'7

where n, /3 > 0 are given constants. Find the components of the spatial acceleration field a ;::: a(x.~ t) at point (1, 0, 0) and time t = 0.

4. Consider a plane motion defined by the velocity components V··1

...

=

{J (T) ( •-r•. 'l"• ·f··) ± ~· I. ~ ' ' 2 ' . I

[h:1

,

where cl> is a harmonic scalar field. Show that the plane motion is .irro.tational and find the spatial acceleration field.

70

2

Kinematics

2.4 Deformation Gradient One goal of th.is section is to study the deformation (i.e. the changes of size and shape) of a continuum body occurring when moved from the reference configuration Q0 to some current configuration n. Fo.r simplicity, we often omit the arguments of the tensor quantities in subsequent considerations. Deformation gradient. As we know from Section 2.1, a typical point X E 0 0 identified by the position vector X maps into the point x E n with position vector x . Now we want to know how curves and tangent vectors deform. Consider a material (or undeformed) curve x = r(~) c no, (. YA = rA(~)), where f, denotes a parametrization (see Figure 2.4_). The material curve is associated

with the reference -configuration n0 of the continuum body. Hence, the material curve is not .a function of time. During a certain motion x the material curve deforms into a spatial (or deformed) curve x = 1(~, t) c n, (:z;a = 1'n(~, t) ), at ti.me'/:.

r.~· ..

..:.·•. •..

..i'(~, t)

·.. .

.

. . . . . . ·.. ~ ..

...

.n

~'

i time. t" = 0 J.;

[

3

0

"Figure 2.4 Defonnation of a material curve r

... .... .

••• .

]

time t

··-.i

c no into a spatial curve I

C

n.

The spatial curve at a fixed time t is then defined by the parametric equation x = 1'(C t)

= x(r(€)J)

or

We denote the spatial tangent vector to the spatial curve as dx and the material tangent vector to the material curve as dX. They are de-fined by dx ==

,../(~ I ~'

·t)d ~c

'

dX

= I''(c)d.C ~ ~

'

(2.37)

with the abbreviation ( e )' = D( •) / D~. In the literature the tangent vectors dx and dX, vihich are infinitesimal vector elements in the current and reference configuration (see

2.4

71

Deformation Gradient

:. figure 2.4), are often referred to as the spatial (or deformed) line element and the :,·material (or undeformed) line element, respectively. ~.{~·:=.:·::·By using (2.36) and the chain rule we find that 1'(~, t;) = (Bx(X, t)/DX)r'(~) . . Hence, from eq. (2.37_) we deduce the fundamental relation .. ..

:

··:·: .=

.

dx

..

= F(X, t)dX

or

F (X, t_) =

·Bx (X, t) · DX

(2.38)

\vhere the definition ....

"f: . . ·· .

r:t

or

r aA

= Grad.x (X. t ) 1

G:rl"afI..\"~<\:fa = Doxa ~,. = ..--\A

-(2.39)

"_is to be used. The quantity Fis crucial in nonlinear continuum mechanics and is a pri·inary measure of deformation, called the deformation ·gradient. In general_, F has nine ·,~.o~ponents for aU t,, and it characterizes the behavior of motion in the neighborhood .of a. poi.n t. Expression (2.38) clearly defines a linear transfonnation which .generates .a vector Clx ·by the action of the second-order .tensor F on the vector dX. Hence, eq. (2.3.8) ·:~-e-rves as a transformation rule. Therefore, Fis said to be a two-point tensor involving points in two distinct configurations. One index describes spatial coordinates, :i:a, and _the other material coordinates., ..YA. In sum.ma.ry: material tangent vectors map (i.e. :~r~~sform) into spatial tangent vectors via the deformation gradient. .·.· ... _We suppose that the -derivative of the inverse motion x- 1 with respect to the current _·position x of a (material) point exists so that

_ ax-~ (x,·t) F (x, t) = 1

1

or

ax

-1 _ F 4a --

.

1 DxA _ 8 -

·xa

=-= .gradX(x, t;)

(2AO)

,,.. gt cl(1a~a . ..-\ A . .•.

the tensor r-L is the inverse of the deformation g-radicnt. It carries the spatial line element dx into the material line element dX according to the (linear) transformu.tion rule dX = .F- 1 (x, ·t)dx, or, in index notation, cLY;t = ~:;~-cb:a. Generally, the nonsingular (invertible, i.e,. detF ¥= 0) tensor F depends on X ·which d~notes a so-called inhomogeneous de-lorm.ation. A deformation of a body in ques.t~.()~ is said to be homogeneous if F does not depend on the space coordinates. The co-mponents f: A depend only on time. Every part of a specimen deforms as the whole ·does. The associated motion is called affine~ For a rigid-body translation for which the .diSplacement field .is independent of X we have F = I, f;1..i = c5aA· However, if there ~s no .motion we have F = I and x = X.

-~here

1

·72

2

Kinematics

For a .two-dimensional problem the deformation is given by the

EXAMPLE 2.5 explicit equations

:1: ·= 1 )

-

3v 1.v. 2 + ·-. . {\. J - - . i\.') 2 . 2 -

(2.41)

Determine the matrix representation ..of the deformation gradient and its inverse and study the deformation of a unit square with reg.ion n0 (see Figure 2.5). Consider a material line element a 0 and a spatial line element b with unit lengths. Show how a 0 deforms, and carry .out the :inverse ope.ration with b. For the components of a 0 and b take (1/ J2 1 1/ J2) and (1_, 0), respectively. Solution.. By recalling definitions (2.39) and (2.40) we find after some simple algebra with the given defonnation (2.4 I) that -?

[FJ =

.[

(2..42)

3/;

The given deformation x carries the unit vector :a0 into its new position character.ized by a (see Figure 2..5). By means of (2.42) 1 and the given components of a 0 we may specify the deformed .position of a 0 , i.e.

[a]= [F][ao]

= ·[

-2

-1

] [

1/v'2]

3/2 -1/2 • • 1//2

= .v~'2,c, :'( -31 ]

la!= 5.·1;•)- .

(2.43)

The inverse deformation x- 1 carries the unit vector b into its undeformed position characterized by b 0 • By analogy with the above, using (2.42.h and the given components of b, we may specify .the undeformed position of b, Le.

[ho]

= [F-) Jib] = [

-1/5

2/5 ] :[ 1 ] • 0 •

-:3/5 -4/5

1

= - -;-1 [ 1 ]: o '. 3 .

Iho I = .( -=-2D )

/'2

(2.44)

Observe that during deformation x the length of a 0 increases from l tip to 5 1/ 2 ., while during the inverse map x- 1 the length of b decreases from 1 up to (2/5) 112 (see Fig . . ure 2.5). Iii .................................. , .... ,,.. :.. , ...... ,, ........... ···:·:···:···,.. , .. :····'·''····:··' .................·,:.··:···=·····:···:'·". , ... ·.··:·':······.·:··.·:··:·:··,. ···':······'."•:'·."···'··'··'·:, .... , ·············=··:'''''·"·."'·':·'····::··:··'·······:··::·.. :·."·'····'."··':··:··'·"··'·"··'··:··..·., .. , ....... , .............................. .

2.4

73

Deformation Grad·ient

4 • ':"

."

,

•

~

•

•

~

•

•

, •,

,

...

,

,

:. •

(

••

,

:. ,

..

,

•

,

,

I

"•

"'.

,

"•

:

3 ,., ..,.., ........... ,., ... .

I

2 ··-- ---- ..

•

~

:~·::·::·:J

lbl = 1

r

1.-'- - - - . . . .

.··ao

laol =

1 ~

-1

-~

~

: 0

bar.·. ·· ·z • _:

C1 ...

J. · ' ...

... . .-!-:-.·~.

I

3

2

.•.•••

Figure 2.5 Deformation of a unit square showing the map x of a material line element a 0 into a and the inverse map x-1. of a spatial line element b into b 0 •

Displacement gradient tensor. To combine the deformation gradient with the displacement vector we deduce from (2.4) and definition (2.39) that

GradU = Gradx(X, t) - GradX

= F(X, t) -

I

or

8Ua

a-··\.~....4

=

FaA -

c5aA

·

(2.45)

The second-order tensor GraclU is called the d·isplacement gradient tensor in the mate.rial descr~ption. From (2.5) we find using definition (2.40) that gradu

= gradx -

gTadX(x, t)

=I - F- 1(x, t)

or

(2.46)

where the second-order tensor gradu is called the displacement ·gradient tensor in

the spatial description.,.

2

74

Kinematics ·

Note the following relationships between the material gradient and ·mate.rial divergence, and the spatial gradient and spatial divergence of the smooth scalar, vector and tensor fields, , u .and A~ respectively. The fields are defined on the current configuration of a continuum body. By the chain rule we have the useful properties

D

or

_1 D

-,l.= F...:\a --a}--~

a

gradu

= Gradu F- 1

.,div A::::: DivAF- T

Dn(l

or or

~,.l.,,, u.

~

. .c\. .."t

• 'a

=

Dua F_ 1

av· ..'\.A

Ab

'

8A11b = ~A~,b p-i .

a'l'

.

• ,. (t

a)··' B ..-·\.

EJn

(2.47)

(2.48)

-(2.49)

Nanson's -formula. We already know that points, curves,· tangent vectors, for -ex.ample, X, r, dX, map onto points, curves, tangent vectors x, I~ .dx~ respecti ve-ly. An arbitrary differential vector maps via the deformation gradient F (see the linear tra.ns.formation(2.3 8) ). However, .a unit vector N normal to an infinitesimal material (or undeformed) sur~ face e:Jcment dS does not map to a unit vector n normal to the associated infinitesimal spatial (or deformed) surt·ace element els via F, as shown in the following. We perform the change in volume between the reference and the current configuration at time t dv = .J(X, t)dlf _, (2.50)

J(X, t) == detF(Xl t) > 0 ,

(2.5 l)

in which J is the detenninant of the deformation gradient F, known as the volume ratio (or Jacobian determinant). In (2.50), dl/ and dv denote infinites·imal vo]ume ele.ments defined in the reference and current configurations called material (or un .. deformed) and spatial (or deformed) volume elements, respect.ive.ly. Further, we assume that the volume is a continuous (or at least .a p·iecewise continuous) function of continuum particles so that dV' = d . .Y 1d ..\"'2 d.Y:1 and dv = fh 1d:r: 2 d:r.:_1 (continuum idealization). Since :F is invertible we have J(X, t) = det.F(X,.t) # 0. Because of the impenetrabiJity of matter, i.e. volume elements cannot have· negative volumes, we reject J (X., t) < 0 which mathematically is possible. Consequently, the volume ratio J (X, t) > 0 must be greater than zero for all X E fl 0 and for all times t. The inverse of relation (2.5]) follows with identity (l.109) as J- 1- = detF- 1 (x, t) > 0, with F~1. introduced in eq. (2..40) . If there is no motion (F = land x = X), we obtain the consistency condition J = 1~ since det.F = detl = .L However, a motion (or a defonnation) with J = 1 (at every

2.4 Deformation Gradient

75

p.artide in every configuration and time t) is called isocho,ric or volume-preserving. It keeps the volume constant. In order to compute the relationship between the unit vectors n and N -consider an arbitrary material line element dX, which maps to dx during a certain motion X· We now ·express the infinitesimal volume element in the current con.figuration dv as a dot product. By means of (2.50) we have the following relation dv

with ds

= ds · dx =

(2.52)

,JdS · dX ,

= dsn and dS = dSN denoting vector elements of infinitesimally sma]) areas

~ie~ned

in the current and reference configurations~ respectively. . With transformation (2.38) and identity ( 1.81) we ·may rewrite eq. (2.52) as (FTds - JdS) ·dX

"

"'0

,,

=0

.

(2.53)

Since (2.53) holds for arbitrary material line e·lements dX_, we find that

ds .= JF

-~r

. dS ,

(2.54)

Wmch shows how the vector elements of the infinitesimally small areas ds and d~ on the current and .reference configurations are related. Relationship (2.54) is well-known as Nanson's formula.

EXERCISES .

..- 1. Consider a parametric curve in space of the form X

= r(~), with .

de-fining a ·helix. Find the length of the helix for 0 < ~ < II

Cf lr'(c;) Id~) .

.:. :· 2. In a deformation of a three-dimensional problem, the displac-ement components of u are found to be

1

'll·l..._·,l-4~ - r1· ·r·1 2

Compute the matrix repres.entations of F_,_ 1 and F and deduce that the defonnation is isochoric. 3. Let a -continuum body undergo a homogeneous de.fonnation whic.h is defined by x = x(X, t) = c(t) + A(t)X, where the components of the vector c and the tensor A a.re constants or time9'.dependent functions .

76

2

Kinematics

(a) Show that [F] ="{A] and interpret this result. (b) Consider the components of vector c .and tensor A given ·by

[c]

=[

n

(A]

0 = [ -12 0 2/3 0 .0

0

0

l .

v'3/2 /

where a, f3, / are constants. Show that a particle that lies on a spherical surface in the current configuration initially lies on the surface of an e] lipsoid. (c) Consider (b) and detennine how a unit nonnal N of an infinitesimal small area dS in the reference configuration deforms. Take N = 1/ v'3 (el - e 2 + eJ).

2.5 Strain Tensors We have l~arnt from the ·preceding section that the deformation _gradient is the fundamental kinematic (second-order) tensor in finite deformation kinematics that characterizes changes of .material elements during motion. The aim of this section is to determine these changes in the form ·of (second-order) strain tensors related to either the reference or the current configuration. Note that unlike displacements, which are measurable quantities, strains are based on a concept that is introduced to simplify analyses. Therefore, numerous definitions and names of strain tensors have been proposed in the literature. We discuss (and compare) the most com1non definitions of strain tensors established in nonlinear continuum mechanics. Material strain tensors.. We compute the change in length between two neighboring points X and Y, located in region 0 , occurring during a motion (see Figure 2.6). By neighboring we mean that X is 'close' to Y. The geometry in the referen.ce co11figuratio11 is given by

n

y = y + (X clX .=dean

and

X}

= x + IY de-=

Y-X

XI IY - XI

= x + dX Y-X

IY - XI ,

ao

'

= IY -·XI .

(2.55) (.2.56)

We denote the (material) length of the material line element dX = Y - X by de. It is and y E i.e. de = :f y - XI the distance between the neighboring points x E with ds /IX I << 1 ( JX I # 0). The unit vector a 0 , lao I = 1, at the referential position X

no

no,

·77

2.5 Strai·n Tensors

d~'~::.~tw;::::::::.':):r::,•

:<~ .,"!, !i: :r: : : :i~;:I: : ',

.~:."'..·. :.:.

'.., .,

:_.... ·... ·. :_:·.

time t

,.......

::·,-,:.:.:·

.....:_-.:_.:_:·.:··.'..

.... ~:\;:,'/-:·.:....... ·

Au 0 = ·Fao

fAl)ol

=A

time ·t = 0

laol = 1

Figure 2.6 Deformation of a material line element with Jeng.th de into a spatial line element with length Ade.

describes the direction of the material line element (which may be imagined as a fiber),

as illustrated .in Fi_gure 2.6. Hence., additional.ly, we find using (2.56).1 that dX · dX

= dc:a0 • dea0 = dc: 2

(2.57)

•

Note that the vector quantities dX and n0 are naturally associated with the reference ~on.figuration of the body. Certain motions transform the two neighboring points X and Y into their displaced positions x = x(X, t) and y = x(Y, t) of region .Q, respectively. We now ask how close is x to y. Using Taylor's expansion according to (1.238}, {.l.23.9), y may be expressed by means of (2.55), (2.56) and the deformation gradient (2.39), as

y

= x(Y, t) = x(X + dcao, t) = x(X_, t) + dcF(X, t_)ao + o.(Y -

X) ,

(2.58_)

78

2 Kinematks

where the Landau order symbol o(Y-X) refers to a small error that tends to zero faster than Y - X ~ o. With motion x = x(X, t) and (2.56)2, (2.56)3 it follows subsequently from (2.58) that y- x

= dcF(X, t).a0 + o(Y - X) = F(X, t)(Y - X) + o(Y -

X) ,

(2.59)

which clearly shows that the term F(Y - X) linearly approximates the relative motion .y - x. The ·more Y approaches X the better is the approximation.$ the smaller is de- =

1v-x1. Next, we define the stretch vector .Aa0 .in the direction of lhe unit vector a 0 at X E 0.o_, i.e.

An 0 (X, t)

= F(X, 't)ao

(2.60)

,

with length ,.\ = l.A110 I called stretch ratio or simply the stretch (see Figure 2.6). Then, the length of a spatial line element (originally in the direction of a 0 ), Le. the. distance between the two neighboring places x and y., is obtained from (2.59) 1. by neglecting terms of order dc: 2 • Using definitions (1.15) 1 and (2.60) we find with y - x:;::: dc-Fa 0 that (2.61)

.In summary: a material line element dX at .X with length de at time ·t = -0 becomes the length Ade at time "t. The stretch ,\ is a m·easure of how much the unit vector a 0 has stretched. We say that a line element is extended, unstretched or com-pressed according to A > 1, ,.\ :;:::; 1 or A < L. respectively. With definitions (l..15h, (2.·60) and property (1..8.1), the square of,.\ .is computed according to ,\

2

= Ano · A110 == Fao · Fao

rr = a 0 · F Fao = a 0 · Ca0

(2.62)

,

(2.63)

or

where we have introduced the right Cauchy-Green tensor C as an important strain measure in material coordinates (F is on the right). Frequently in the literature C is referred to as the Green deformation tensor. From (2.63) we learn that to determine the stretch of a fiber one only needs the direction a 0 at a point X E 0 0 and the se-condorder tensor C. Note that C is symmetric and positive definite at each X E f2 0 . Thus, and

u · Cu > 0

for all

u =!= o . (2.64)

2.5 Strain Tensors

79

·.: · Consequently, given the nine components Fa.4 , it is easy to compute the six components ·. CAB = Cn.-t via (2.63), but given CAH it is impossible to compute the nine components ·: ~1 .AA With definition (2.63) and eqs . (l. l.01.) and (2.51.) we find that

detC

= (detF) 2 .= J 2 > O

.

(2.65)

··:·The so-called Piola deformation tensor, denoted by B, is defined by the inverse of the ·..dghtCauchy~Oreen tensor, i.e. "B = c- 1, with c- 1 = (FTF)- 1 == F- 1 _F~T . As a further strain measure we define the change in the squared lengths, i.e.. (/\de )22 ·. ·clc • With (2.62)a, the use of the unit tensor I and eq. (2.57)2 we have (2.66)

E

1 T = -(F· F- I) 2

E~w = ~(Fa,i.Fi,11 -

or

(2.·67)

i5A11) ,

-

.·.· where the introduced normalization factor 1/2 wiH be evident within the linear theory. ·. · : · This expression describes a strain measure .in the direction of a 0 at point X E .n 0 . In _:_.· (2.67) we have introduced the commonly used strain tensor E, which is known as the · ~reen-Lagrange strain .tensor. Since I and C ~re sym.metric we deduce from (2.67) . . that E = ET also. So far the introduced strain tensors operate so]ely on )he material vectors a 0 , X. ·.· Thus, C, .E and their inverse are also referred to as material strain tensors. :·. · $pati.al strain tensors. In order to relate strain measures to quantities which are · ·:_·.:.associated with the current configuration we continue with arguments entirely similar ._:. · ~
=·

y .... ...

dx

= y + (x. -

= dfa

x)

y-x·

= x + IY - xi Iy-x I = x + dx

and

d£

= IY- xi _,

a=

y-x

ly-xl

(2.68)

.

(2.69)

::··>··The (spatial) .length of the spatial line -element dx == y - x is given by d€ = IY - xi, · wi_th d£/lxl ~ 1 (lxl # 0). The unit vector .a, lal = 1, acts at the current position x :_\. · .· ~nd points in the direction of the spatial line element, which is the direction of A110 (see :. > figure 2.7). Since I-al = 1., we find using (2.69) 1 that

dx · dx

= dea · dE.a = di~

(2.70)

·.The vector a may be viewed as a spatial element characterizing the direction .of a fiber .· in the current configuration.

80

2

Kinematics

time t

lal = 1 time l

=0

A-H 1 = F- 1a

IA;l·I =A-I

Figure 2.7 Deformation of a spatial line element with length d£ into n mate1ial line clement with length A- l d€.

In order to perform the relationship between ).no and a we recall eqs. (2.59)i, (2.61):.i and definition (2.69h to give Fa0 = ...\a. Hence, from (2.60) we deduce that

An 0

= A8

(2.71)

.

Note that the vector quantities a and dx are naturally associated with the current conjigu111tion of the body. Using Taylor's expansion, the associated position vector Y E S20 , which is described by the inverse motion x- 1 (y' ·t)' may be express-ed by -means of (2.68), (2.69) and the inverse of the deformation _gradient (2.40). By the chain rule we find, by anal·Ogy with eq. (2.58), that

Y

= x- 1 (y, t) = x- 1 (x + dfa, t) = x-l(x,t)+dEF- 1 (x,t)a+o(y-x)

.

(2.72)

81

2.5 Strain Tensors

In an analogous way to (2.-60), we now define the stretch vector .A 0 in the direction of the unit vector a at x E s-2, i.e. (2.73) The length of a material line element (originally in the direction of a) is obtained from (2. 72h ·by neglecting terms of order d£2 . With definitions ( 1.15) 1 and (2. 73 ), we obtain, by means of Y - X = deF- 1a that (2.74) w~ere the length of the inverse stretch vector -~imply the . r:·_.:·

.,x; 1 is the inverse stretch ratio ,.\ -1. (or

inverse stretch) (see Figure 2. 7) .

The square of A- 1 follows with definitions (1.15.h, (2.73) and property (1.81) as

(2.75) or

-1 l1ab

=

p-tp-1 A.a

Ab

(2.76)

·

T.he strain tensor b- 1 is the inverse of the left Cauchy-Green tensor b, which is defined by

(2.77)

or

_(F . is on the le.ft). In the literature the left Cauchy-Green tensor b is sometimes referred to· as the Finger d.eformation tensor. lt is an important .strain measure in terms of :spatial coordinates. The left Cauchy-Green tensor is symmetric .and positive definite at

each x E 0, ·...: . b

= FF'l'· = (F'1' F) rr = b'r

and

u · bu

>0

for all

u

#o

.

(2~ 78)

\VI"th definition (2. 77) and ( 1.. l 0 I), (2.5 I) we find consequently that detb

= {det.F) 2 = J 2 > O

.

(2.79)

:-. _·: \·.· As a last important strain measure we define the change in the squared lengths, i.e. de2 - (,.\- 1di) 2 . With (2.75h, the use of the unit tensor I and eq. (2.70h we find a relation expressed sole.Jy through quantities in fl. Thus,

~(dt2 2

(,\-

1

d£")2] = ~[dt2

e =!{I - rTF- 1 ) 2

2

-

or

(dfa). F-Tr 1(cl€a)]

= clx · edx

,

(2.80)

(2.81.)

82

2 Kinematics

Relation (2 . 81) describes a strain measure in the direction of .a at place x E 0. We have .introduced the commonly used symmetric strain tensor e, which is well-known as the EuJ.er·Almansi strain tensor. Since the strain tensors b, e and their inverse operate on the spatial vectors a, x, we call them spatial strain tensors. ·we now consider the case -C = FTF = I. From the relation (2.56h we know that the distance between any two neighboring points .X and Y located in the reference configuration .is de.- = IY - Xf. On the -other hand the distance between the associated neighboring points x and y located in the current configuration is given via (2.6 I )a, i.e. ,\de: = ly--xJ. For C = I we conclude from (2.62), 1 that the line element is unstretched, i.e. ;\ = 1, and consequently de = f y - xi = f Y - XI~ Hence, the distance between any two points is unchanged during such .a mot.ion. This means that there is no relative motion of points under x. Since C = I we conclude additionally from relation (2.67) that the strain tensor E vanishes identically, which means that the body does not change its size and shape {no changes in distances and angles). This particular motion, which preserves the distance between any pair of points of a continuum body, .is called a rigid-body motion and is dealt with in more detail in Section 5.2. Hence, a rigid-body motion induces no strains .and consequently no stresses. A body which is only able to undergo a rigid-body motion .is said to be a rigid body. The idealization that a body is rigid is often cons-idered in engineering dynamics. As already seen_, ve-ctor and tensor-valued Push . .forward, pull-ha cl\ operation. quantities may be resolved along triads of basis vectors belonging to either the reference or the current configuration. Additionally, there are two-paint tensors which are associated with both configurations, one ex.ample being the defonnation gradient (2.39). The transformations between material and spatial quantities are typically called a push-forward operation and a pulJ . .back operation (familiar .in differential geom . . etry) and are denoted by short-hand x* (•) and x;- 1( •), respectively. .In the literature the pull-back operation is often written as

x* (•).

ln particular, a push-forward is an operation which transforms a vector or tensorvalued quantity based on the reference configuration to the current configuration . Since the Euler-Almansi strain tensor e is defined with respect to spatial coordinates we can compute .it as a push-forward of the Green-Lagrange strain tensor E, which is given in terms of material coordinates. From eq.. (2.81) we conclude, using definition (2.67)~ that

e=

~(I - F-TF- 1) 2

=

F-T[~FT(I 2

F-TF- 1 )F]F- 1

= F-T[~{FTF - I)]F- 1 = F-TEF- 1 = x*(E) .

(2.82)

83

.2.5 Strain Tensors

A pull-back is an inverse operation, which trunsforms a vector or tensor-valued quantity based on the current configuration to the reference configuration. Similarly to the above, the pull-back of e is

E

= ~(FTF 2 =

.. :.=·.

I)

= FT[~F-T(FTF 2

1)F- 1 ]F

FT[~(I - F-TF- 1 )]:F . F'1'eF ? ··· ...

= x; 1(e)

.

(2.83)

As can._ be seen from eqs. (2.82) and (2.83) the transformations are based on multiplications by one description of the deformation gradient, i.e. F, F- 1, FT, F-T. Which

.·.

r~~m

of the deformation gradient we have to take depends on the tensor to be trans-

(::_.:· formed. Following ·MARSDEN and HUGHES [ 1994] we indicate covariant tensors by (• )·~ and contravaritmt tensors by ( • )d (for the notions 'covariant' and 'contravariant' the . ·. reader is referred to Section 1.6). The push-forward and pull-back operations on co" . . .:. .:. i1aria11t second-order tensors (such as Eb, C~, eb, (b- 1)~) are according to (2.84)

:.'.-.-:_.·An example was given previously in eqs. (2 ..82) and {2.83), Le. e = x*(E~) and E := : ·. :. : · :x; 1 ( eb), wh.ich provide the relations.hips between the -material and spatial quantities, _:_:·_:· .: . . ~~e. e

= F-TEF- 1 and E = FTeF, respectively.

· However, the push-forward and pull-back operations on con.travariant second-order tensors (suc11 as ( 1) tt, bu and most of the common stress tensors) are according to

c-

. . . ...

. ~: : ·.

(2.85) · : ·. ~-~ t~e following chapters we use covariant strain tensors in combination with con .. "travariant stress tensors. /·_:._._ . _. .: · For completeness we write down the push-forward and pull-back operations on . _covariant vectors~ i.e . (2.86) ..

. . and of contravariant vectors, i.e. (2.87) ·:_ . :·.- _.. Finally we provide the so-called P.iol.a transformation of a spatial vector field if~ u(x, t), with components u 0 , i..e.

(2.88)

2

Kinematics

where U - U(X, t) denotes a material vector field with the components UA. The transformation from u to U involves the pull-back of .u scaled by the volume ratio J,. The inverse of eq. (2.88), i.e. u = .1- 1x*{U), involves the push-forward operation on U here scaled by the inverse of the volume ratio, i.e . .1- 1• For a more complete source on the underlying concept the =mathematically oriented reader is referred to the book by MARSDEN and HUGHES (1994, .and references therein].

EXERCISES

1. For a given material point ex.press conditions on the right Cauchy-Green tensor C which ensure that (a) no stretch occurs in a specified directfon ao of a fiber,

(b) no change in the angle between a pair of specified directions (a 0 1 , a 0 2 ) takes place, and

(c) no change occurs in an infinitesimal surface element els placed in a plane perpendicular .to a given direction (set ds = dS).. 2. Let a body undergo a homogeneous deformation defined by r .c,.. 1 -- a..., Xt

·'

where a, /J, 'Y are constants. (a) Determine· the components of the material and spatial strain tensors C, E and b? e, respectively. (b) Take the values /J = -cos() and 'Y = --sinO and show that if a = 1 the strains are zero. Explain that for a = 1 the map corresponds to a rotation of magnitude fJ about the ){ 1-axis.

3. If the deformation of a body is defined by , .............

•1,3 -

..

...A3

'

the body is said to be in a state of plane strain. Defonnations and strains occur only .in planes :i:a = const .and do not depend on the :r3 -coordinate. Strains in the x 3 -direction are zero,. Determine the matrix representations of tensors F, E, e.

2.6

·.·.-·..

.Rotation, Stretch Tensors

85

4. Using relations (2.45h and (2.46:h show that the strain tensors E and =e may be expressed in terms of the displacement gradient tensor according to

E

= ~ (Gracfru + GradU) + ~GradTU GradU

(2.89)

·e

1 T = -{grad u + gradu) -

(2.90)

··.... ;:-·

i//::. -: .:·

.=

....... ·.··

·.··

.. :.

1 ( 8U11

·.

EA11 .... ·.. : : . . .

·.··.

]

T

.:..:.grad u gradu ., 2 2 or in index notation, in terms of material and spatial coordinates, as

= 9-

!1-:t·'"

DU.4)

u .."\.A

+a".\·""..JJ .J.. \

1 DUc 8Uc

+?fl)· . a~·r "-' u . .<'\. A .il. .11

1 ( Duh 8ua ) 1 Bue Due + -, - - -2 8:1:" [};1:,, 2 8:1:(J a~rb

.

Cab= -

:··(.>·.-::· :·.

with implied summations on C and c, respectively.

:···. _:·.=·:_:

,::/..:\.::::·:.·

{<2.6 R.otation, Stretch Tensors ......::.:-:.-·----------~------------~------~

:::_/·:.

.

:. _:._.:{in the following we decompose a local motion, characterized by the nonsingular (in'./ vertible) tensor F(X, t), into a pure stretch and a pure rotation. As above, the argu: . .\)iients of the tensors are omitted, for convenience. However, they will be employed ;:.::·;:-.:·:::\Vhen additional information is .needed. ·:.·

......

;/ P~tar decomposition. At each point X E 0 0 and each time t, we have the following i: (~nique polar decomposition of the deformation gradient F: ·. .

...... :......

F =RU= vR

or

(2.91)

RTR=I

V

= VT

(2~92)

///.: rhis is a fundamental theorem in continuum .mechanics,. :(::\\-.. : In (2.91.) U and v define unique, positive definite, symmetric tensors, which we call /_:·/._the right (or material) stretch tensor and the left (or -spatial) stretch tensor, respec:·_:.·:.:.:_:_·_:tively. They =measure local stretching (or contraction) along their mutually orthogonal ·.·. :eigenvectors, that .is a change of local shape. The right stretc·h tensor U (w:ith compo·:-. ' . ·.~ents U;u1) is defined with respect to the reference configuration while the left stretch ):):.~te_~s~r v (with components 'V.ab) acts on the current configuration. Note that in this .-//·\·.~ext the symbol U also stands for the material displacement field and v for the spatial /·'.(._"ye"locity field, respectively. :;_}:_· ·): }· ;·... The positive definite and symmetric tensors U and v are introduced~ so that .......

. :_.:_:.:_ :: ... :-:. __

...

u2 = uu = c

and

2

v = vv = .b '

(2.93)

.86

2

Kinema.tics

which is based on the square-root theorem; see, for example, GURTIN [198 la, p. 13]. Consider the relation detC = detb = J 2 we deduce with (2.93) and rule (1..101) the important property detU

= detv = J > 0

(2.94)

.

The unique R is a proper orthogonal tensor, with detR = 1_, called the rotation tensor. It measures the local rotation that is a change of local orientation. A rigidbody -rotation about a fixed origin is then characterized {f and only {f U = v = I, so that F = R. Hence, each material line element dX is rotated into a unique spatial 1.ine element dx (and vice versa) according to (2.38), i.e.

= R{X, -t)dX

dx

or

cb;a

= RaA cLY_.\

.

(2.95)

On the other hand if R = I, the .deformation is called pure stretch, for which ·.F = U = ·v. Using (2.95), the pointwise polar decomposition (2.91.) may be written as dx = .R(UdX)

dx

= v (RdX)

= RaA (U;1[Jd.X.r:d

or

d:1:,1

or

&r:a. =

'Vab

,

(Rb.•\ d..Y..:l) .

(2.96) (2.97)

.Relation (2.96) describes a pure stretch of dX by the material tensor U, (UAu), followed by a pure rotation performed by the rotation tensor R, (RaA)· The rotation tensor transforms the material vector UdX into the spatial vector dx. However, relation (2.97) describes a pure rotation of dX performed by the same rotation tensor R, (RaA), which transforms .the material vector dX into the spatial vector RdX. Th.is rotation is followed by .a pure stretch of RdX with the spatial tensor v, (vab), which gives the spatial vector dx. In both cases the rotation tensor R maps between the reference and the current configuration and, therefore, R is a two-point tensor like F. Relation F = RU is also known as the unique right polar decomposition, while F = vR is referred to as the unique left polar decomposition.

Show that R is proper orthogonal, i.e. RTR = I and dPtR = 1. Show further that the multiplicative decomposition F = RU is unique.

EXAMPLE 2.6

= Fu- 1 and eqs. (2.93)1. and (2.9?h we find that R'l'R = (Fu-lfr(Fu- 1) = u-TFTFu- 1= u~Tu u= (uu- 1fruu- 1 = I . (2.98) In addition, computing d<;~tR = det (Fu- 1 ) we deduce with rule ( 1.10 O and relations

Solution.

Using R

2

1

(2.51) and (2.94) that 1

detR = detF·u c1et-

=1

.

(2.99)

2.6

Rotation, .Stretch Tenso..S

87

. . . In order to show uniqueness of the polar decomposition we suppose that there exist .(i. ·pos.i tive definite, second-order tensors R and UT so that (2 . .100)

F =RU= RU RTR =I .· . ·Then, with ·C

-

-'r U=U

and

(2.101.)

= .FTF and eqs. (2.100)2 and (2.101) it follows that (2.102)

·: . · ·\vhich means using (2.93) 1 that U

.:.: R

= U, since C

= R. The proof is similar for F = vR.

has a unique square root, and hence

•

We discuss further a physical .interpretation of the right stretch tensor U. Look in :. · itie direction of n0 at point X E n0 , with lao I = 1. According to (2.60) we find using = RU that the stretch vector A00 may be expressed as ..

.( ¥.

Au 0

= RUao

(2.103)

.

;>·Then, the square of the stretch ratio ,.\ = (A00 • ~ao) 112 may be computed with (2.103 ), .-._:·.identities ( 1.8 .I), ( 1. .84) and relation RTR = I. Thus, we have

A2

= RU.ao · RUa 0 = n0 • U'I'(.RT R)Ua 0 = Uao · Uao = IUaof .

(2.104)

ao

. Note that the length of A.110 acting .at X E Ho along the direction of the unit vector depends only on a state of pure stretch, i.e. U; therefore we call U the stretch tensor. Another form of A2 may be obtained from (2..104:h using (2.92h, i.e.

.A 2

= a0 · U T Ua0 = a0 • U2a0

•

(2.105)

Consequently, the right Cauchy-Green tensor C (recall eq. (2.93)i) and the Gree.nLag.range strain tensor "E = (C - I) /2 do not include information about the rotation, that is experienced by a particle during motion. The deformation gradient F ·contains more information th.an strains do in general (rotation and strain-like information). Finaily we find the relation between v and U, and between b and C. By me.ans of the polar decomposition (2.91) we have v = FR'l'

= RURT

and

(2.106)

88

2

Kinematics

Consequently, with v2 = RU 2 RT and eq. (2.93) we find an important relation between the left and right Cauchy-Green tensors, namely

(2.107) A general formula for strain measures was introduced by SETH [1964] and HILL [.1970]. The generalized strain measures in the Lagrangian and Eulerian descriptions are defined as

!(un -

n. lnU ,

I) ,

.!.(vn - I) , n . Inv ,

if n-#0

if n

=0

:

}

(2.1.08)

where n is a rea] number (not necessarily an integer). For the special cases n = 0 and n = 1 we obtain strain tensors associated with the names .Henck)' and Blot, respectively, while for n = 2 and n = -2 we obtain the Green-Lagrange strain tensor (U 2 - 1)/2 (or also (v2 - 1)/2) and the Euler-Almansi .strain tensor (I - u- 2 )/2 (or also (I - v- 2 )/2), which we have introduced in (2.67) and (2.81 ), respectively. For more details see, for example, MAN and Guo [1993]. The Hencky strain tensor in the material and spatial form, Le. lnU and Inv, is of particular interest in .nonlinear constitutive theories. Hecause of the logarithmic functions, these tensors are decomposed additively into so-called volumetric and isochoric parts (for a more detailed exposition of the decomposition .procedure of strain measures see, for example, p. 228).

EXAMPLE 2.7

Assume a certain two-dimens.ional ·motion which is given in the form of the deformation gradient F(X, t) = Bx(X, 't)/DX. All tensor quantities are defined with respect to the orthonormal basis {ea}, c\! = l_, 2. Show that fo.r the case of two dimensions the polar decomposition F = RU may be given by the closed-form expression

U = [11 (C)

+ 2J(C)tt./2..[C + J(C)I]

(2.109)

for the right stretch tensor, and by the rotation tensor R which follows by means of the inverse of expression (2.109), i.e. R = F0- 1• In eq. (2.109), / 1 (C) = C11 + C22 and J( C) = (C 11 C22 )"112 characterize the first scalar invariant of the right Cauchy-Green tensor C and the volume ratio, respectively.

Solution.

In two dimensions, the characteristic polynomial of U gives a quadratic

equation in ,\, i.e .

.a= 1, 2 ,

(2.1.10_)

2.6

with the principal scalar invariants ·i 1 (U)

U11 lf22

-

89

Rotation, Stretch Tensors

= trU

= U11

+ U22

and i 2 (U) = detU =

Ur2 ·

Knowing from .the Cayley .. Hamilton equation (1.174) that any (second-order) tensor satisfies its own characteristic -equation, ·we may write instead of (2.110), U2 i 1 (U)U + i 2 (U)I = 0. By use of U2 = C and rearranging we ·may write the expli-cit expression for U, i.e. (2.l I 1)

which coincides with the relation obtained in, for -example, HOGER and CARLSON [1984] and TING [1985].

On taking the trace of the last equation (2.111) and knowing that trC = 11 ( C) and ~rl = 2 in two dimensions we are able to ex_press the first invariant of U in .terms of the (irst invariant of C . Thus, i1 (U)

= [!1 (C) + 2i2 (U)]11 2

fo addition, using the relation det·C

(2 . 112)

.

= det{UU) = (detU) 2 , where the property (l.101)

is applied, and recalling (2 . 65h we conclude that

i2(U) = detU = (detC) 1l 2 = J(C) .

(2.113)

Using eqs. (2 ..112), (2.l 13)a we find finally from (2.111) the desired result (2.109). -'•""'!'"···~-------

. . . -· . . ···-··'l.-.. •"1·:·-··"·-·"""1-1-••1-·····--"····''1 ... -~··

-··~--"·

•

... •11..,1,,1..,.1.1:-··:·1·-:"·"·''1"·:··.•1:• ..... ,~.... _ ............ ..,~...... ~........ , .... ~.• :•~, ............................. ~.,,,,..,,.,,,,,,_,,,,,,,,,,~ ... :1111 ... 1.;1111:•,,.,, • ..,,, ....... ~~· ... ~··.".·:··":'1"1•·,~··,,...,,.·,~··}'.·-1"1•~·~·11.-- .. --... - - ; .... ,,,, _____ , ... _.... ~....... :.:111.

Eigenvalues .and eigenvectors -of strain tensors" We introduce the mutually or~hogonal and normalized set of eigenvectors {Na} and their corresponding eigenvalues Aa, a = 1, 2, .3,, of the material tensor U as

UNa ·= Aa.Na

with

:jNal

= 1 ,

(L

= 1, 2, 3

.

(2.114)

Furthermore, after combining (2.93) 1 with (2.114) we obtain the eigenvalue problem for C, i.e. ·? 3 a' -- 1'.;.J'

.

(2.115)

Clearly U and C have the same orthonormal -eigenvectors, i.e. the set {Na}, called ~he princi-pal referential directions (or principal referential axes). However, the corresponding positive and real eigenvalues d.iffer. The eigenvalues of the symmetric tensor U are Aa, a = 1, 2, 3, called the principal s.tretches, while for the symmetric tensor C we find the squares of the principal stretches denoted by,,\~.

2 Kinematics

90 Hence, with RTR problem for v, i.e.

v(RNa)

I and v

=

RUR1\ we obtain from (2.1.14) the eigenvalue

= RURT(RNa) = RUNa = Au(RNa)

.,

a -- 1~ ').-, ,J')

•

(2.1.16)

Combining (2.93h with (2..1.16) we obtain, by analogy with (2.115), the eigenvalue problem fo.r b, i.e . (1.

== .1' ').. , 3 '

(2.117)

which means that the two tensors v and b have the same eigenvectors RN(a)' while their positive and real eigenvalues are"\, and .A~, respectively. The eigenvectors of v and b are those of U and C rotated with R. We conc.lude that the principal referential directions Na transfonn onto the principal spatial directions (or principal spatial axes) ·Du (which are mutually orthogonal and normalized eigenvectors of v and b) via

Da

= RNa

w"ith

.a= I, 2, 3 .

(2.118)

This means that the two-point tensor R rotates the material vectors Na into the spatial vectors Du. So the principa] directions of v and b may be obtained by rotating the corresponding principal directions of U and C by R. Relation (2. l.18) will prove useful in a moment. ·we summarize the four introduced (symmetric) strain tensors in the convenient. form of their spectral decomposition, for J\ 1 # ,\~ =I= Aa # ...\ 1, i.e. (2.119) .ci.=1

(2.120)

Note that the eigenvalues and the eigenvectors depend on the point and time t. In order to express the two-point tensors F and R in terms of principal strekhes and principal directions we employ the polar decomposition F = RU and eq. (2.119). Using (2.1 "18) 1 we find the deformation gradient in the form 3

F

= RL a=l

3

AaNa 0

Na = L

3

/\a{RNa)

®Na = I: /\1Da © Na ·

(2.12.1)

a=;= 1

Knowing that the unit tensor I may be expressed as

Nu 0 Na (see relation (L65)) we

2.6

Ro.tation, Stretch Tensors

.find by means of (2.118)-1 that 3

R

= RI = (RNa) 0 Nu = L Dn 0 N.a ·

(2.122)

.a=1

Note that both tensors involve principal directions in the reference and current configuration, which emphasizes the two-point character of these tensors . Since tensors If and R are, in _general, non-symmetri~, the representations (2.121) and (2. 122) may not be viewed as spectral decompositions in the sense introduced on p. 25. Hence, it is clear that the principal stretches ,\1' a = 1, 2, 3~ .in eq . (2.121 h may not be interpreted ~s the eigenvalues of the deformation gradient F.

EXAMPLE 2.·s

Suppose that the spectral decomposition of the right Cauchy-Green tensor C, Le. eq. (2. "119), is given. Since the principal stretches are functions of C we ·muy write /\, = ,.\a (C). For later use we introduce the following relations a~2

~=Na 0

DC

N,1

(a= 1, 2, 3)

for

(2.123)

for

(2.I 24)

for

(2.125)

(compare also with SIMO and TAYLOR [19.9la] or SALEEB et al. ..[.1992]). Prove this set of equations for these dis ti net cases.

Solution. ".Firstly we consider the case ..\a =F ..,\b .for
dC

= Lf2A,1dAaNa 0 N" + ,\~(dNa ® Nn + N" ® dNa)] .

(2. l.26)

n==I

Recall that the three vectors Nn., with INuJ = 1, form an orthonormal bas·is (Nn ·Nb = c~ab), so that ·r'f u · dNa = 0 (the change of a vector with constant length is always a:rthogonal to the vector itselt). lf we pre- and postmultiply eq. c2~ 126) with N" we find that (1.

= t, 2, 3

(2.127)

.

.Applying property ( 1.96) to the left-hand side of (2. I 27) we may write

N

11 •

dCNa =

2 Kinematics

92

= (8C/D..\a)d,.\a we have

dC: Na® Na, a= 1, 2, ·3, and by means of identity dC

ac 8

.

-

>.a d,.\a : Na 0 Na = 2>.ad>.a

a-c .

- =1

1 >.a >..a : Na ®Na 2 8

a

= 1, 2, 3

.

(2.128)

Knowing that the value 1 may be written as the contraction (BC/DA.a) (Ef>..a/8C), the last equation implies 8Aa/8C = (2..\a)- 1 Na 0 N0 • With he.Ip of the chain rule we obtain finally the basic relation (2.123). The other two cases are straightforward results, compare with the general considerations on the spectral decomposition of a tensor, in particular, relations ( 1.178) and (l J 79). • .... ~ ..................... ~·-·'"'·' ...... ~.~- ..... ~- ... -~ ........ _.,, ...... : .. -~ ...... - •

--~:~~~ .. ~ .. ,,. __ ,_,.,.,.,,~: .. ~ .. - - - ........ _

.. ~ .. J'; .... ~· .. ~--·-·•:·~: .......... ; ··--........ ..,, ...., .. :,.:-.. --·-,-~.•h

...., ............... - . ;

~·,- ............ ~ .... .,.;

· · - · - - - ......~-~- .. ~'--.''' , •••••• .,,··:·''~' _.,.:

,,.,_,~.,.

:, ... ,.'·'···:

,,.,~~·:,· , . , . , , . , _ . , ,~··:'•'''·"'•' · ' , , . , . , .. ~_, , , •• ~, . " ' ' · ' •• ,

Uniform and biaxial deformation, pure shear. Consider an extension or compression of a rod (with uniform cross-section) in the direction of the :1: 1-axis up to the stretch )q. The associated relation -x 1 = ,.\ 1.X1 defines a uniform deformation along the :1: 1-axis. For X1 > 1 we call the de.formation a uniform extension (or in the literature sometimes referred to as .the uniaxial extension), however, for )q < .1 we call it a uniform compression (or uniaxial compression). Uniform extensions or compressions in all three directions follow the relations (2.129)

If two stretches, for example,

)q, ,\ 2 , can be chosen arbitrarly and the third stretch

"is determined by the condition J = . \. 1\2 .,\ 3 = 1 (this type of deformation keeps the volume constant) the associated mode of deformation is often referred to as bi . . axial (although there are changes of lengths in all three directions). A typical biax.ial deformation is characterized in the following form ...\ 3

:t::J

1 v = \\··'"\.3 .AJi\2

.

(2.130)

An equibiaxial deformation is defined as the mode ·of deformation in which ,\ 1 = /\i =>..(and A3 = ,.\-·2 provided it .is isochoric). If ,\ 1 = ,\ 2 = A3 , then the continuum body undergoes a so-called uniform dilation, i.e. a uniform expansion or uniform contraction in all three directions. Next, we consider a thin sheet of material which is fixed along a paraJlel pair of edges normal to the e 2 direction. An extension with the principal stretch ,.\.1 in the plane normal to these edges results in a plane deformation (state of plane strain) of the form :I:·:1

1 X"'. 3 = '...-· AL

(2.131)

2.6

Rotation, Stretch Tensors

93

(provided that the plane deformation is isochoric), known as pure shear (or stripbiaxial extension). The associated principal stretches are ,.\ 1, ,.\2 = 1 and ,.\a = ~\} 1 , while R = I. The area remains constant in the plane spanned by the orthogonal eigenvectors n1 and n3 • Experimental results on a thin sheet of rubber under pure shear were "first published by RIVLIN and SAUNDERS . [.1951], see also the book by TRELOAR fl 975, Chapter 5]. The volume of a ·continuum body under biaxial or pure shear deformations according to kinematic relations (2.1.30) and (2.131) .remains constant, as we will see later in the text..

EXERCISES

1. Use (2..106) to show that vn = RUnRT., where n .is an integer.

2. The motion presented by eq. (.2.3) may be given in symbolic notation as x = x(X, t) = X + c( t ){ e2 • X}e1 , e 1 and e2 denoting -orthogonal unit vectors fixed in space, with the components (1, 0, 0) and (0, 1, 0), .respectively, and the parameter c(t) = tanlJ(t) > 0. The motion described causes a so-called simple shear deformation (also known as a uniform shear deformation), where the planes ;i; 2 = const. .are the shear planes and the direction along ~i· 1 is the shear direction. The .an_gle B{t_) is a measure of the amount of shear. (a) Compute the matrix representations of tensors F, ·C, b,

c- 1, b-1..

(b) Show that F, C and b may be expressed as

(c) Show that a simple shear defommtion is iso.choric. (d) Compute the principal stretches /\1 , a = 1, 2, 3, and the three principal referential directions N,1' i.e . the normalized eigenvectors of C. (e) Compute the angle between the principal referential directions e2~ ea = e1 x -e2 in terms of c(t).

N(

1

and el,

(f) Compute the three principal spatial directions Da, i.-e. the normalized eigen-

vectors of b.

Hint: The principal directions nu are simply obtained from changing .-e 1 and e2.

N(

1

by inter-

(g) By using results {d) and (t) determine the spectral decompositions of the Cauchy-Green tensors C = U2 , b = v2 , i.e. (2.11.9), (2.120), and the

94

2 Kinematics representations of the deformation gradient F and the rotation tensor R, as derived in eqs. (2.121 h and (2.122)3, respectively. (h) From (g) we have basically obtained the polar decomposition ·F = RU via the spectral decomposition of C. For the particu tar deformation of simple shear, com.pare (and check) your results for the right stretch tensor U and the rotation tensor R w.ith those obtained from the closed-form expression (2.109) and R = Fu- 1•

3. Rec.all the representation (2.l2lh, the polardecomposifion F =RU and (2.l 18)t and (2.119). Establish the following expressions relating .the two sets of principal directions {N.a} and { Dn}, a= 1, 2, 3, i.e .

(2.132)

4. A body undergoes uniform extension in all three directions .according to (2.129). Find the matrix representations of tensors F, R, U, E" c. 5. Consider the deformation gradient Fin the form of a 3 x 3 matrix with detF > 0. Write a computer program in -order to determine the right stretch tensor U and the rotation tensor R. Start the procedure by computing C = FTF and the eigenvalues and eigenvectors of C. Then use the spectral decomposition (2.119) for the tensor U, and by means of R = Fu- 1 finally find R.

na

6. Show that the second-order tensors N12 0 Na and 0=Da occurring in .eqs. (2. 119) and (2.120) may be obtained ·in the closed form, for ,.\ 1 =f:. ,.\2 i= ,,\ 3 =/= A1., as

N

N -

a®

,\2 c -

a- .a

(I, - ,\~)I+ /3,\;;-2c-1 Da

'

(2.133)

(2.134) (a

= 1, 2, 3; no summation) where the scalar D" must be nonzero, with

2.-\;! - / 1,,\~ + Ia>...;;

2

Da =

The first and third invariants of C, and also of b, are J.1 and 13 _, respectively. The closed form expressions for Na ®Na and fia@n,1 circumvent the explicit computation of the ei_genvectors. •

Hint: Relations (2.133) and (2.] 34) foUow from the Rivlin-Ericksen representation theorem (compare with p. 201 of this text), see BOWEN and WANG [1.976b], MORMAN -{1986] for an analytical treatment and inter alia SIMO and TAYLOR [ 199 .la] and MIEHE [ 1994] for a numerical treatment.

2.7

Rates of Dcforma.tion Tensors

95

2.7 Rates of Deformation Tensors Within this section we want to study how some of the tensor fields introduced above ~~a.nge with time, by knowing the motion x = x(X, .t:). ln particular, we study the rate which changes of shape, position .and orientation of a continuum body occur.

at

Spatia·J and material velocity grad"ient.. The derivative of a spatial velocity -field v(x, t) with respect to the spatial coordinates .is defined by I (x, t)

t) ( ) = 8v(x, Ox = gradv x, t

(2.135)

·Of

here given in symbolic and index notation. The spatial field I, in general a .nonsymmetric second-order tensor, is known as the spatial velodty gradient commonly used in both solid and -fluid mechanics. . The material time derivative of the de.formation gradient F gives, with definition . (2..3_9}1 and relation {2.7)i, :

..

.

F(X,t)

·

= !!_ (ax(X_,t)) at

:.·...:·_·.

=

ax

BV~~· t)

=

== ~

(ax(X,-.t)) ax at

GradV{X, t)

or

:::.... ..

.

.f;1A

av,l = .n v·

LJ..(\.4

'

(2.136)

.

where the time rate of change of the deformation gradient, i.e.

F = DF /Dt, is equal

·~o,.GradV,

cal.led the material velocity gradient. In eq. (2.136) we have used, ad_"~it.ionally, the property that the mate.riaJ time derivative commutes with the material

·gradient. Another valuable -.derivation of the time ·rate of change of the deformation gradient _.is'.by ·means of the directional derivative introduced in eq. (1.266). At .a given time t, :

-~~-·._compute the directional derivative of F in the direction of the velocity vector v at

·pqshion x. We obtain (2.137)

9ri comparing -eq. (2 . .1.36)4 with (2.137):1 .(by means of (2.8)2), we find finally that · .. ·. . . . .

DF(X, t) Dt

= DvF{X, t)

(2.138)

.:~:hi~h is merely an

application of relationship (2.20). :i·,.":... :..· .1he spatial velocity gradient I = gradv may be expressed through the material

_";v_~l~city gradient F = GradV. ·with definition (2. l 35)i, identities (2.8h, (2.7)t., the

96

2

Kinematics

chain rule and the -definitions of F and F- l, we find the usef~l relation I

= 8v = DX.(X, t:) DX ax ax .Ox = ~ (ax(X1 F-1

t)) DX

8t

= FF-l

F= lF .

or

(2 ..1.39)

This may be written in index notation as Dv(L

lall = -

8:i.;b

=

a.xA = a8~\u. . .xA a:1:b

-~--,-

8 ( Bxa )

-1

at axr1

F ..11.

·

-1

.

or

= F..AFAb

z;i

ruA --

l·a.brbA ~

·

(2.140)

EXAM.PLE 2.9 Obtain the relationship which expresses the rate .of change of a unit vector a (characterizing the direction of a fiber in the deformed configuration, as introduced in (2.69h, with reference to Figure 2. 7) .in terms of the spatial velocity gradient I in the form . I ,\ a=a-/\a.

(2.141)

Solution. The combination of eqs. (2.60) and (2.71) gives Aa 0 = Fa 0 = /\a. Taking the materia~ time derivative, we find, by means of the product rule, that

Fao = ;\a + ,.\a Hence, using F = IF and applying Fa 0 some straightforward algebra. 11

(2.142)

.

= Aa once more, we obtain eq. (2.14 l) after

.. , .............. ,,, ........ , ....... ·.··'''··=···'·:······· ...· ..... ·.·:··'······'··· ....,.,.:,············:····'·: .... ,., ............ ,., ................................................ , .....,_,.•.. ,.,.:, .......... ,,, ... :··:·····."·."·."· ............................... :···:·····:······:···.··········:··:··:····:······.·.·:·::::.•.• . ."•••."•'···:·::·:··"······:··:·:···:······."·:··'····.·

Next, we compute the material time derivative of F- 1 and F-T which will be useful later in the text. Starting from F- 1F = I we deduce, using the product ru.le and relation I = :FF- 1 (see (2.139)4 ), that

or

-1 F An

-

-

-

p-•z . Al' ·ba

(2 ..143)

(compare also with relation (1.237)). Consequently, from e.q . (2. l43h we obtain :1 -:FF- 1.. However, starting from F-TFT

.

=

= I we find that .

F-T-rT

.

= -F-TFT .

F-T ·= -F-TFT·F-'T = _fl'F-T

.(2.144)

2.7

97

Rat-cs of Deformation Tensors

which is the analogue of (2.143). The overbar covers the quantity to which the time differentiation is applied. We now additively decompose the spatial velocity gradient 1 according to

-l{x, t)

= d(x, t) + w{x, t_)

,

(2..145)

where we have defined

d

= !(1 + JT) = !{gradv + gradT v)

w

1 = -(I -

2

2

2

(2.:146)

= dT ,

1' 1 . T ·r I·)= -(gradv -- gTad ·v) = -w ,

2

(2.147)

or, in index notation, as (2.148)

'Wab

Dvb) = -Wba = -21 (ava -8:i;IJ - -a:i:a

(2.149)

,

.with IT= grad'l'v .and the associated components lba = avb/8xa. In eqs. (2 ..146) and (2..147) the symmetric part of the spatial ·velocity gradient I, i.e. the rate of deformation tensor (or in the literature sometim.es referred to as the rate of strain tensor), and the antisymmetric (skew) part of I, i.e. the spin tensor (or sometimes called the rate of rotation tensor or vorUcity tensor), .are denoted by d and respectively. Both d and-ware spatial fields involving only quantities acting on the current configuration,. We also observe that the -components dab and Wab are linear in ._the ·velocity components Va. Note that the spatial tensors I, d, ware viewed as cov~_iant second-order tensors.

w,

.

- ·- - · ' , · · - • , -

,_,., •• ,._. , •• ,_ •• ,,.,., ••• -.•• •••••••• -· , ••• ,, .,,.,., .. , ,, .. ,,.,,.,,:

''·'·"~'··: ·: ,.,.,.. ·c ._.,, .._,;, ~,,,....,,_,_., ....... , •• _,_,, ..... , •• ,.-•• - - . • , •.• ' ., ., .. - - - . ' ..• , , ..... , , . ,., .. ,, , ... , ... , ... ,,, ,.,.,,.,,,,,., , •• , , ... , , ....... , ... , •••• --······ • ., _, __ ...._.. , • •: '"': ,,., ... -.•·. · - -···-•: ,..,,-~,: ,,.. , :•"':

EXAMPLE 2 . 10 Express the spatial acceleration field a(x, t), as introduced in .·(2.27), in terms of the spin tensor w(x, t). s"olution. Expanding (2.27) with (I/2)grad(v 2 ) - {gradTv)v di~ectly fro.m the identity (l.291) by setting u = v), we obtain

a=

8v

1 •>· at + (gradv)v + 2gracl{v-) -

T

=0

(grad v)v .

(which follows

(2.150)

·With definition (2.14 7) 2 we end up with an ·essential relation

av + ?grad(v-) 1 ') + 2wv

a= -

8t

for the spatial acceleration.

(2.151)

-

•

.... :....... __ ,,,_..............:.·:···'·"·":''•"•"•"•'•"·'··.·.·:·'•'•''."•'•."••.,•."•·' .. ····'····· .............. , ..................... , ............................................................................., .................: .. •:: .. .-................... : ... ·.'·""'•"'·:= ..................................................... : ..... ,, ................ : .. •.•: .......................................................................................... : .. •:::.-.•···.·:: ......... : .......................... .

2 Kinematics

98

The antisymmetric spin tensor w may also be represented by its axial vector according to

(2.152)

WV= W XV

(compare with eq. (1.118)), for any vector v. By recalling eq. 'Wob

= -WtJa = -8v-,,/D:1;a, we have 2w = 2w

-Eabt.~wtJb.ec

= curlv .

w,

Cl .276)

and property

= Eabc8vb/8:I:, ec. Thus:t 1

(2.153)

l

where 2w is referred to as the vorticity vector (or s·p·in), which is of fundamental importance in fluid mechanics. The spatial vector field w = (cuilv) /2 is -called the angular velocity vector. Fo.r an frrotatio.nal motion, Le. curlv = o, the vorticity vector 2-w and (therefore) the sp.i n tensor w vanish .at each point. Note that by ·means of eq. (2.152) and the vorticity vector, as given in (2..15.3), the term 2wv in relation (2.1..5]) may be expressed as

2wv

= (curlv)

xv .

(2.154)

EXAMPLE 2.11 Ass~me a certain motion wherein the spatial velocity field v{xl t) is the gradient of a given.spatial scalar field (x, t), known us the potential of v (v = grad). Compute the spatial acc-deration field a(x., t) and show that a may also be derived from that potential .

Solution. Knowing that every potential motion is an irrotational motion, i.e. curlv = o (or equivalently w = o, s-ee eq. (2.154))~ we find that the third term in (2.151) vanishes. Consequently we have a== -Dv

Dt

1 ,,.. ) = -a (DcI>) 1 D (D~I> + -grad{v -- · + --- · -D)

Dt

2

Dx

2 ax

Dx

Dx ·

(2.155)

and with the property that .the spatial time derivative commutes with the spatial gradi .. ent_,

D a=-, .Ux

(acI> l D - +- · -(HP) Dt

2 8x

ax

=grad

-

(a)Dt 2

Relation (2.156) shows that if v is the gradient of a :potential from . II

~I>,

(2.156)

then a may be derived

2.7


99

Spatial velocity gradient in terms of i.J. We determine a relationship between the spatial velocity gradient I and the material time derivative -of the right stretch tensor, iJ, which is given in terms of material coordinates. By means of I = F.F- 1, the polar decomposition F = RU, the pro.duct rule of differentiation .and the property _RTR = I, we conclude that (2.157)

where R characterizes the time rate of change of the proper orthogonal rotation tensor R. From RRT = I we deduce directly upon differentiation that RRT + (RRT)'r = 0. Thus, (2.158.) which clearly s.hows that RRT is a skew tensor. In order to express the rate of defonnation tensor d and the spin tensor w in terms of U we adopt the additive decomposition of the spatial velocity gradient (2.1.45). Hence, using (2.l57h and property (2.158_), the definitions (2J46) 1 and (2.147) 1 g.ive 1 )RT d = Rs:vm(uu..

,

(2.159)

These relations show that d is not a pure rate of strain and w is not a pure rate of rotation . ...... , ..................................... :.····:········.·

·······················································."····:·········."····:····.···'·."························:········.

······:·.··:·:··:········

EXAMPLE 2,12 Show that for a rigid-body .rotation (.motion) the .rate of deformation tensor d vanishes and the identity w = ·R.RT holds. Solution. For a rigid-body rotation (motion) we know that F == R, which implies U = I and iJ = 0 for all X. Hence, we find from (2.159) 1 that d vanishes, meaning that I= w. On the-other hand, from (2.159h we conclude that (2.160)

which means lhat for the case of a rigid-body rotation the spin tensor w coincides with • the skew tensor RRT.

EXAMPLE 2.13 Suppose a rigid-body is rotating about an axis. The rotation is characterized by the angular velocity vector w = w(t_), as-depicted in Figure 2.8. An arbitrary point x of the body with current position x E n is moving around a circle with the spatial velocity v( x, t) relative to a fixed point 0. Show that the ·velocity

100

2 -Kinematics

9 =

w = w(t) v=wxx

n·.-._·.·._· . _·:_.

·Figure 2.8 Velocity v of a particle relative to a fixed po.int 0.

of the point x -may be expressed .as

v(x, t) ·= w(t)x = w(t) x x ,

(2~161)

where w is the time-dependent antisymmetric spin tensor. Solution. The rotation of the rigid-body may be described by the linear transformation x = R(t)X (compare with eq. (2.95)), with the proper orthogonal rotation tensor R and the referential position X of the point. The material time dedvative of x yields, by means of the product rule, eq. (2.28)i and X = RT(t)x7

v{x, t)

= R(t)X = R(t)RT{t)x

.

(2.162)

For a rigid-body .rotation we know from the last example that the skew tensor RRT coincides with the spin tensor w (see eq. (2.160)), which gives the desired expression (2J61) 1 • Relation v{x, t) = w{t) x x .is well-known from rigid-body dynamics. In conclusion, relation (2.161) shows that the spin tensor w is associated with the angular velocity of a .rotating rigid-body -characterized by the vector w. In fact, w is simply the axial vector of the skew tensor w. Eq. (2.161) represents a nice physical interpretation of relations (l.118) and (2.152). •

·Material time derivatives -of some strain -tensors. Our present starting point involves material strain tensors and the.ir derivatives with respect to time t. In particular, we compute the material time derivative of the Green-Lagrange strain tensor, E, the

2. 7


101

·_. right Cauchy-Green tensor, -C, and introduce the rotated rate of deformation tensor DR. From the definition of the Green-Lagrange strain tensor (2.67), and with the pro.duct ..:":._rule, eq. (2.139),1 and the rate of deformation -tensor d introduced in (2.146)i, we find

-..-_.that

E = ~(FTF + FTf) = ~(FTITF + FTIF)

-

...

(.2 . .1.63)

The .material time derivative -of tensor E is also known as the material str.ain rate tensor E. As can be seen from (2.163) . i, it is simply the pull-pack of the covariant rate of deformation tensor d, which we may write as E = x;- 1 ( d~) = FTdF (see rule (2.84h). Now we may show that the directional derivative of the Green-Lagrange strain tensor E in the direction of v equals lhe material strain rate tensor E, as given in (2.163). Using eq . (2.138) and the common properties of the product rule, we find using defini:tion (2.67) that (2.:164) and with (2.163) 1,

DE(X, t) = DvE(X. t) . Dt ·. '

(2.1.65)

The resulting relation (2.16.5) may also be found immediately from (2 . 20). The ti.me rate of change nf the right Cauchy-Green tensor C follows from defini. . tions (2.63), (2.67) as C = 2E. With eq. (2.163);"1 we obtain .

C

= '2E. = 2F'l'dF

.

(2.166)

By replacing F with the rotation tensor R in (2.163).-i we obtain Du. = RTdR, which is known as the rotated rate of deformation tensor D1t. Using the polar decomposition F = RU and the sym.metry UT = U we find from (2."166)2 that C = 2U(RTdR)U = 2UDnU. FinaUy, we have an alternative expression for Dn_.,

i.e.

oil = RTdR = !c-112cc-112 2

(2.167)

where c- 1 / 2 = u- 1 is in accordance with (2.93J1. In the following we consider spatial strain tensors and compute a relationship between the material time derivative of the le.ft Cauchy-Green tensor, b, and the spatial tensors I and b. Recall the definition (2.78) of b and use the product rule :in order to

102

2 Kinematics

obtain

. b = "FFT = FFT + FFT ·•

= (FF- 1)FFT + (FFT)F-TFT

.

(2.168)

Then, we find the important relationship

b =lb+ bfr

,

(2.169)

where the definition (2.139),1 of the spatial velocity gradient I is to be used. ············:·····························.•:,,.............................. , ... :·.······················."···· .. ··.·.·:·:·.·:,,... :·····:································'''•:'•,: ....., .. ,,....: .... : .................................. ,, .............,.,, .......... , ................................................ , .. :.": ........, .. ·:······························."·.···-;.....:... , ....... .

EXAMPLE 2.14

Show the useful relation

e= d -

fl'e - el

(2.170)

for the tnaterial time derivative of the Euler-Almansi strain tensor e.

Solution. Recall the definition of the spatial strain tensor ·e, i.e. eq. (2.81 ). Using the .product rule we obtain 1 . 1 . . e= - -F-TF- 1 = - ~(F-TF-.t + F-TF-.1) . (2.l Tl)

2

2

Hence, with the derived relations (2.144h and (2.143.h and the definition of the EulerAlmansi st.rain tensor (2.81), we obtain

e = - ~(-JTF-TF-1 = 9.-1 [lT (I - 2e) +(I = d - IT e - el ,_

F-TF-11)

2e)l]

where the definition (2.146) 1 of the rate of deformation tensor d is to be used.

(2.:172) II

l\tlaterial time derivatives of spatial line, surface and volume elements.. We consider "first spatial and material line elements dx E Hand dX E Q0 , as introduced in (2.37), and compute the material time derivative of dx. ·we know from (2.38) that line elements map via the deformation gradient according to dx = F(X_, t)dX. By means of .the product rule .and relation (2 ..l 39),1 we find that dx

= "FdX = FF- 1dx = Idx

or

As can be s-een, I = .grad v is a spatial tensor field transforming dx in.to dx.

(2.173)

103

2. 7 Rates of Deformation Tensors

. ·:· _·: ·.··.· . Before proceedin_g it is necessary to provide the relation for the material time :._. ·.derivative of the volume ratio J = detF > 0. Using the chain rule we obtain sim_;: : pJy. j = 8J/DF : F. We just .need to specify the term fJ.J/fJF which results from ._. . relation (1.241) by taking F .instead of A. Hence, we may write

DJ = JF-T

DF

'

(2.174)

.

/ fonsequently, with relation (2.174), F = IF and properties (l.95), (l.94), (l.279) we .-:. -:. . find expressions for the material time derivative of the scalar field J, namely

} = JF-T : .F = JF-T : IF = JF-T.FT : I = JI : gradv = Jtr(gradv) = Jdivv

.

or

(}vu

J=J-D . . :i:a

(2..175)

.·_._. ··using the additive split 1 = gTadv = d +wand knowing that the trace of a skew tensor . . ·.·.is zero, we deduce from (2.l 75)n an important alternative expression for .i, namely

J = Jtrd

or

J

= Jdaa

·

(2.176)

By recalling relationship (2.20), the material time derivative of J may also be eval-

as the directional derivative of J in the direction of .the velocity vector v. Thus, ··_::_-:·:< ~e also may write DJ(X, t)/.Dt = DvJ(X 1 t). {:::_<· . . ·_·_·.... A motion with J = 1 we called isochoric (keeping the volume constant, dv = <_ . . ·_. . coust). From ·eqs. {2 . .175) and (2. I 76) we may deduce alternative expressions for J = 1 ·:(: · . . · o:r.dv = ·Const, namely, } = 0, F-T : F = 0, clivv = 0 or trd = 0. In summary, ·_the following six statements characterize necessary and sufficient conditions for an :i._:_.·. isochoric .motion and are equivalent to one another: ·\>i_.· uated

J

=1

F-T:

F=

'

0

dv

= const , divv = 0 '

. J=O' . } trd

=0

(2.177)

A continuum body is said to be i.ncompressi:ble if every motion it undergoes is : :·: :.·_· . isochoric. Consequently, for every motion of an incompressible body each of the con. . . . ditions in (2 . .1.77) holds. The condition F-T : F = 0 is essential in the treatment of . . -:.:" motions of incompressible solids, while divv = 0, trd = 0 is of fundamental .impor:·:. fance in fluid mechanics. If the deformation behavior of a continuum body .is restricted, :;/·:· . . we say that the body is subjected to so-called internal ·constraints. In particular, the :_.-·._.· conditions formulated in (2.177) characterize the most important .internal constraint ·. _. known as the internal kinematic constraint, or more precisely the incompressibility . ·:···constraint.

104

2

Kinematics

Now we perform the material ti-me derivative of a··vector element ds of infinitesimally sma11 area -defined in the current configuration. .Applying Nanson's formula (2.54), i.e. ds = JF-TdS, the product rule and eqs. (2.l75)n and (2.144h, we obtain

cis =

(JF-T + .JF-T)dS = (divvl - IT)JF-TdS

= divvds -

(2.178)

fr ds ,

with the second-order unit tensor I (note that the spatial velocity gradient I has a si-milar symbol as I),. In .index notation eq. (2. l 78h reads as

_._ av,, dsa

= ~dsu c.Xl:b

avb

- · ds1, . 8.'Z:a

(2.179)

Expression (2.178.h (and (2.179)) represent the relation between the rate of change of the infinitesimal spatial vector area in terms of ds, the divergence of the spatial velocity field v and the transpose of the spatial ·velocity gradient I. Finally, the material ti.me derivative of the spatial volume element dv = J d 1/ gives by means of the product rule

dv where the relation J

..

= Jd1l" = divvdv

or

(2.180)

= J divv is to be used. EXERCisgs

1. Show that in a moving continuum the spatial velocity field v and the spatial vorticity fie"ld .2w are related to .their material time derivatives by the identity

cudv = 2w + 2wdivv -

{gradv)2w .

(2.181)

.Hint: Take the curl of relation (2.1.51) by considering eqs. {2.153) and (2.154), and then us-e property ( 1.292) with ( 1.27 5). 2. Consider a deformed fiber characterized by the vector a, with fal = 1. Show that the material time derivative of the logarithmic stretch ratio ,.\ at a particle along that direction a is given by

ln,\ =a· da .

(.2.182)

Project the tensor d onto the orthonormal basis vectors e.u and give a physical interpretation of the rate of deformation tensor d. In particular, discuss the diagonal components daa and the off-diagonal components dab (.a # b), a, lJ 1, 2, 3,

=

of the matrix [d] . Compute the dot product of vector a and the vectors -of equation (2.141) and use the fact that a = 0 (since a· a = 1), and a · wa = a · -(w x a) = -0 (since u · (u x v) = O_, whic.h should be compared with -eq. ( l.31 )).

.Hint.:

a·

2.7

105


3. By combining eq. (2 ..141) with eq. {2.182) show that the material time derivative of a unit vector a may be expressed as

a=la - (a. da)a . (2.183) = 1, 2, 3; denote the three eigenvalues of d and fia

We now assume that ~a' a its three associated normalized eigenvectors. For the particular case in which a is an eigenvector of the rate of deformation tensor d, show by means of (2.183) and decomposition l = d + w, that ·

Do. with Ga

= WDa =

W X

th~

eigenvalue problem don = lnAa.

Ila

a= 1, 2, 3 ,

1

= aafia

(a

=

(2,_,_l·.84)

1, 2, 3; no summation), where

Obviously, the spin tensor w is a measure for the rate of change of the eigenvectors of d, which gives a physical interpretation of the spin tensor w.

4. Recall Exercise 2, p. 93, with the motion x = X + c( t) (e2 • X)e 1, the -orthogonal unit vectors -ei, e2 , e3 = e 1 x e2 and the parameter c(t) = tanO(t) > 0. (a) Show that the spat!al velocity field v(x, t) .may be given by v = i~(t)(e 2 ·x)e·.z, where c(t)

= tanO(t)

is called the shear rate (for typical ranges of shear rates of-some specific materials see, for example, BARNES et al. [1989]). (b) Based on result (a) compute the rate of deformation tensor d(x, t), the spin tensor w(x, t) and the angular velocity vect:or w{x, t), i.e. the axial vector of the spin tensor, in terms of ei, e 2 and e3 . (c) Show that the rate of -de.formation tensor .may be expressed in its spectral form :1

d

=L

ltaDa

0 D.a ,

a=l

and hence compute the three eigenvalues of d~ .i.e. malized eigenvectors of dt i.e. the set { Da}.

O:a,

.and the three nor-

5. Consider the representation of the .deformation gradient (compare with (2.1.21) 2): :1

F(X, t)

=L

Aa(t){R(t)Nn) ®Na ,

(2..185)

o=l

where the principal referential directions Na (orthonormal eigenvectors) are assumed not to change in time. (a) Compute the rate of deformation tensor d(x, t), the spin tensor w(x, t) and the .angular velocity vector w(x, t)~

106

2· Kinematics

(b) Bas-e.d on representation (2.185), find ric tensors w and RRT coincide.

wd + dw

uu-

1

and show that the antisymmet-

= (divv)w

.

2.8 Lie Time Derivatives Consider a spatial field f f (x, "/;) characterizing some physical scalarT vector or tensor quantity in space and time (for the relevant notation recall Section 2.3 ). .I.n the following we compute the change of f relative to a vector field v which is commonly known as the Lie time derivative off, denoted by 1 v(f). The Lie time derivative of a spatial field f is obtained using the following concept: 1

(i) compute the pull-back ope.ration of f to the reference configuration; as a result

we obtain the associated material fie:ld :F(X, t;) ~ x; 1 (/ (x, t) ); (ii) take the .material time derivative of :F, i.e. :F, and

(iii) carry out the push:fonvard operation of the result to the current configuration. This technique is simply summarized as· (2.186)

Since the material time derivative can be obtained from the directional derivative according to relation (2.20) we may apply the concept of directional derivative to eq. (2.186). Consequently, we express the Lie time derivative of .f = f (x, t) as the directional derivative (se-e also WRIGGERS [1988]). Hence, eq. (2.186) reads equivalently as (.2.187) In summary: the Lie time derivative of the spatial field .f is the push-forward of the directional derivative of the associated material field :F = x.:;- 1(.f) in the direction of the vector v, identified as the velocity vector. By recalli.ng definition (I. .266), we may specify the directional derivative of :F at the reference configuration in the direction of the velocity vector v as d

DvF = - F(X +EV) l~=O .

1

(€

If

= (x_, t)

is a .function that assigns a scalar

~

(2.188)

to each point x at time t, the Lie

2..8 Lie Time Derivatives

107

···time derivative of coincides with the ·material time derivative of cJ>. Thus, £ v () -<:· === (x, t). ·..The concept of Lie time derivatives occurs throughout consfitutive theories in com'.· put~n.g stress rates (see Section 5.3). In addition_, it is a powerful concept used in the .·:-·_. . process of linearization and variation. In fact, it emerges that the variation or the lin.: ~.a.rization of a spatial field is formally equivalent to the Lie time derivative .according ·: to the rule (2. I 87) (see Chapter 8 for ·more details) . . :_ -·:--··.....,,, ............. , .............. , .......·.·:·:·.···················:········.······:·:·:·'·"·· ..·:·.-· ........................................................ :··:··:·,·:···'""•"···················:·.·············."·."··:·:·'''".......................................................... :····· ..................... , .......................... :••."••."·····

. EXAMPLE 2.15 Show that the Lie time derivative of the Euler-Almansi strain ten.-: . ··sore is the rate of deformation tensor d, i.e. ..

(2.189)

.·.· . _·Derive this result by using first the rule (2. l 86) and then the concept of directional ·derivative, in particular eqs. (2. 1·87), (2.188) . ._:_.· ...=.·.

·: Solution. Recall the pull-back operation of.the covariant Euler-Almansi strain tensor ·.· ·.e which gives the Green-Lagrange strain tensor E = F~reF, .i.e. eq. (2.83) 4 • Then . . ·.·recall that the push-forward operation on the material strain rate tensor E is the rate of
(2. l90)

. . . Another way to obtain eq. (2.189) .is through relations (2.187) and (2..188). Eq. (2.187) : . gives (2.1.9.l)

With reference to eq. (2.188) we must determine the directional derivative of the ·._._Green-Lagrange strain tensor E in the direction of v, which is, by reference to (2. l. 65), ·., . . . the material strain rate tensor E. Finally, from eq. (2.l9lh, we find that £v(e) = . . :. . ~-rr(D\.E)F- 1 = F-TtF- 1 = d. .• · . . _ _ ,,.,,:.,.-: .. , ........ , .. , ...... ,, .... , .. , .. , ...... :•.••."•.··············, ... , .......... , ................... , ... :··,···=····:·•.••."•.•,·····················:···:·.·:···:·:··':·:·'··.":••,,·:,.•,., .. :·····:····.······."•.".".•••:.·.··.····:···'·."·'·."·'•."••.":'•."••.",····'·::··,,·:···:··,.• .. •., ..........·.·····:·=···..·..·............,.,............ :, ... ,, ....... :.·:·•."•".•••."·········

EXERCISES

l. Establish the properties

108

2

Kinematics

2. Consider a contravariant spatial vector field u with its material time derivative .u = au/at+ (gradu)v (see eq. (2.30)). Recall relations (2.87) and (2. l43h and show that the Lie time derivative of u .is, according to the rule (2 . .186),

.l'v(u~) = ~; + (gradu)v -

lu .

(2.192)

The Concept of Stress

./ . I~. the previous chapter smne kinematical aspects of the motion and deformation of ::·i:\:· i1 continuum body were discussed. Motion and deformation give rise to interactions _(-.-_:.·between the material and neighboring material in the inte.rior part of the body. One /}consequence of these interactions is stress, which has physical dimension force per unit //:._-:_·o.f area. The notion of stress, which .is responsible for the deformation of materials, .is }._:\::~·rucial in continuum mechanics.

~:-: :/._:: ._.·:.· . In particular, we consider a deformable body during a finite motion. For that body \/(::·_~e introduce the concept of stress and discuss the properties of traction vectors and ::_\,:::>stress tensors in different descriptions. We continue with a brief review of the de~ :~>i;:ter;inination of extremal stress values, noting that this topic has already been studied :·i:-.:::-:\!.xtensively in numerous other boo~ks on mechanics.

F/ , Since the current configuration of many problems, especially those involving solids, ·:. :/i~ ..not known_, it is therefore not convenient to work with stress tensors

which are ex( p'tessed in terms of spatial coordinates. For some cases it is more convenient to work \,r.ith stress tensors that are referred to the reference configuration or an intermediate .... · .... ·.

i"_::_::·._.:

:

(:::·::.-:~.~-n_figuration. Some important forms of alternative stress measures, which are

associ-

)} .·ated with the names of Pio/a, Kirchhoff, Biol~ Green_, Nagluli or Mandel, are provided . ·.·.·. \:._:._:"·._":-__.·: ...

:f{3.l. Traction Vectors, and Stress Tensors [H~re we introduce surface tractions which are related to either the current or the refer/ _:._-· -~-~·c·e configuration. They may be expressed in

terms of unique stress fields acting .on a /:.:{. _ribimal vector to a plane surface. Stress components such as normal and shear stress-es {://·~[~:·introduced. ):::_:·:_·..;:_::::-: ..

({:.Surface tractions.

We focus .attention on a deformable continuum body B -occu"i,/pying an arbitrary region of physical space with boundary surface an at time t, as '(shown in Figure 3.1. .:=..

n

::·.··:.·

109

1.10

3 The Concept of Stress

. . . . . . . . . ·... : .. : . ·. ·.··: : .. · ·.··: :: _:· :. ·.·: · ...

Reference configuration

. . ... -' .. "·: -.~.· : ...·...· ':.

.····.

....'

./

:

//

.:

.'

:

. . .. :. . : .. . .. . . . . . . . . . . : .

:

_:

.

.: . -~ .

.

:_·:

Current -configuration

:.

..... :. ·.. . .·.. :·::· ....

:

-:

·. ·..

.

.

:_:.·:·: .· ..

. .....

.

·. ·.:

.

........ . .

...·

.·

. ...

.·: ............·..

:~, ,.,,··

··•

.... ... =··:··· :

.... :········..........-···

..

...

.4·

___.........__ t

_,:·

:··

time t

=0

ano

time ·t

an

X2~ :t2

-Figure 3.1 Traction vectors a·cting on infinitesimal surface -elements with outward unit nonnals.

We postulate that arbitrary forces act on parts or the whole of the .boundary surface (called external forces), and on an {imaginary) surface within the interior of that body

(called internalforce,rj) in some distributed manner. Let the body now be cut by a plane surface which passes any given point ·x E fl with spatial coordinates :1;a at time t. As illustrated in Figure 3.1, the plane surface s.eparates the deformable body into two portions. We focus attention on that part of the.free (continuum) body lying on the tail of a unit vector n .at x, directed along the outward normal to an infinitesimal spatial surface element -els E an. Since we consider interaction of the two port.ions, forces are transmitted across the (internal) plane surface. We denote an infinitesimal resultant .-(actual) force acting on a surface element as df. For .purposes which will be made dear in Section 4.3 we om:it distributed so-called resultant couples not occurring in the classical formulation of continuum mechanics. For a couple we can think of a pure torque. Initially, before motion occurred, the continuum body B was in the reference (undeformed) configuration at the reference time t = 0 and has occupied the .region H0 of

3.1

Traction Vectors, and Stress Tensors

111

:1::1:::i~bysica1 space with boundary surface 8110 • The quantities x (with spatial coordinates i{'.-::~
::r::\'.·(;:i}~'ference configuration. :?:-:;.:·;:).\i(:~: _: / ..: According to Figure 3. l we claim that for ·every surfac.e element

:~~i}ln . :_:_._·.. : : · : : : . : . : . - . · .

=..

df = td,q = TdS ,

(3.1.)

·.

t = t (x., t, n) ,

T ·= T(X_, t, N)

(3.2)

:::;:::':ere, t represents the Cauchy (ortrue) traction vector (force measured per unit sur./}face area defined in the current configuration), exerted on d.<1 with outward normal n. :-_({·'.:_".:·.:·The vector T rep.resents the 6.rst Piola-Kirchhoff (or nominal) traction vector (fo.rce r:;>\.·1neas.ured per unit surface area defined in the reference configuration), and points in the ::.\\\:::i?s~me direction as the Cauchy traction vector t. The (pseudo) traction vector T does not '.(/.\1-~scr.ibe the .actual intensity. lt acts on the region 0 and is, in -contrast to the Cauchy i(tC'action vector t, a function of the referential position X and the outward normal N to ·.>·:::\the boundary surface 80 0 • This circumstance .is indicated in Figure 3.1 in the form of :::)\:):··_a dashed line for T. Relationship (3.2) is Cauchy's postulate. The vectors t and T that act across the surface elements ds and dS' with respective :.:'.:.:..:.::-·normals n and N are .referred to as surface tractions or in some texts as contact forces, \(stress vectors or just 'loads. Typical surface tractions are contact and friction forces or /\are ca.used by liquids or gases, for example, water or.wind.

i( ~auchy's stress theorem.

There exist unique second-order tensor fields u and P so

: ·:.:.:··.that

= u(x, t)n T(.X, t;, N) = P(X, .t).N t(x_, t, n)

or

or

.·J;1

= ~1 A J\T.~t

}

(3.3)

). (the proof is omitted), where u denotes a symmetric spatial tensor field c'1Hed the / . (:auchy (or true) stress tensor (or simply the Cauchy stress); while P characterizes a _:_._ . tensor field called the first Piola~Kirchhoff (or nominal) stress .tensor (or si.mp1y the ::.· Piola stress). The index notation in relation (3.3) reveals that P, like F, is a two-point i._>. tensor in which one index describes spatial coordinates :i~a, and the other material co_··:·:· ordinates .XA. In Section 4.3, p . 1.47, we establish that the Cauchy stress tensor u is ·/· symmetric, under the assumption that .resultant couples are neglected . .Relation (3.3), which combines the surface traction with the stress tensor, is one of the most important ax.ioms in continuum mechanics and is known as Cauchy's stress :. theorem (or in the literature sometimes referred to as Cauchy's law). Basically it ·states that if traction vectors such as tor T depend on the outward unit normals nor N, ·then they must be linear inn or N, respectively.

.112


An immediate consequence of (3.3) is the following .relationship betwe-en t, T and the correspond.i ng normal vectors, i.e.

t(x, t, n) = -t(x, t, -n)

(3.4)

T(X, t, N) = -T(X, t, -N)

or

for all unit vectors n and N.. This is known as Newton's (third) law of action and reaction (see Figure 3.2). .....

•..,

°'··,: .

•' _,··~:.

N

'·

.. '· :,

.........:·'' ..

.... ... ..... . ...... · . . dS

t

, .....

.

·."'

•'\.........

. ·

-T

n

..·.'=-.. ,::""•

·\_ ___...,,.. T '·.,

., ~....

~-··"'···-. ..

·. ds

)·

-t ....·: , ..·~ ,·: ·:· .....·

·=;·'='·:··"

-N

,.. ,. ............ :=···

....

"'"•.) ..... , tJ ··l-.1,

·.... :·

time t.

time t = 0

F·igure 3.2 Newton ts (third) law of action and reaction..

To write Cauchy's stress theorem~ for example (3.3)1., in the more convenient matrix notation which is useful for computational purposes, we have

(t] [t)

= :[

ti t2 ] f:1

.

(u] =

= [u](n] '

[~~: 0":11

0"12

a1:1

a22

rr2a

0:12

a:m

(3.5)

]

[n] =

[ n2 n1 ]

(3.6)

n:i

where [.u] is usually calJed the (Cauchy) stress matrix. Finally, we ·find the relation between the Cauchy stress tensor u and the first PiolaKirchho:ff stress tensor P. From eq. (3.1) we obtain with eqs. (3.3) and (3.2) the important transformation

t(x, t, n)ds

= T(X

1

t,N)dS ,.

3.1

113

Traction Vectors, and Stress Tensors

= P(X, t)NdS

.u(x, t)nds

.

(3.7)

.;·.:using Nanson's formula, i.e. -eq. {2.54), P may be written in the form or

PaA

= JaabF..ib- l

(3.8)

·

The passage from u to P and back is known as the Pio la trans/ormation (compare with _:·the definition on p. 83). Strictly speaking, in order to obtain the two-point tensor P, :.../ . with components PaA, we have performed a Piola transformation on the second index : : :. of tensor .u, with components <1ab· .. For convenience, we omit subsequently the arguments of .the tensor quantities. The ::-.:explicit expression for the symmetric Cauchy stress tensor results as the inverse of .·:relation (3.8), i.e. i·

·

·

or

(3.9)

>. \v_hich necessarily implies

PFT

= FP~r

.

(3.10)

...

Conse-quently, the second-order tensor Pis, in general, not symmetric and has nine :::::Independent com_ponents PaA·

>.·::_:·:_-.:.: .

;.·:·:·.····· '.· • - ._ . _ ............................ -

.... _ . ______ .. , .......... :-•.-: ........ : .................... 11..;o--c ........ :: ........... - : ... ·-•-'':•';''•'••'''"'"·'"":"'···""':"'"-:''"'-~ ....................- .......... :"' .....- ... - - - - - . - - - - - - - - . - - · - - .... -

=:_
•••• - , .................. - · ...··:'·'''~'·"''·"'·'·''''''"'''''''~''•'•'."''·"'· ............ : ... •......................................................................................... : ...

A deformation of a body is described by

·:;-:::)·:_:·:.·:. 1•. .t3 -

·...

ly 3 . .- 3

-

~

.

(3.11)

:::::}'he Cauchy stress tensor for a certain point of a body is given by its matrix represen·/fation as

[u] .= .·.· ..·.:.

[

0 0

0 50

0 0

0

0

0

l

kN/c1n2

•

(3.12)

:_:}Jetermine the Cauchy traction vector t and the first Piola-Kirchhoff traction vector T ::.··:·_:·~ding on a plane, which is characterized by the outward unit normal n = e2 .in the :._:·_,c·~·i.rrent configuration. ,.··.·:·

·.::

:}-.Solution.

From the given defonnation (3.11) we find the components of the de.for(:·:.fu.ation gradient and .its inverse as ~: \ .:·:. .. 0 -6 0 ·.·.·.. ... (3.13) [F] - 1/2 0 0 :. ... ·:·

: :- .

:;..:::.::·::·.:·:.:

[

0

0

1/3

l

ll4


while detF = J = 1. The components of the .first Piola-Kirchhoff stress tensor read, according to (3.8), as

[P] = .J[o-][F-T]

=[

+~ ~ l

2

kN/cm

(3.14)

.

In order to find the outward unit normal N in the reference configuration we recall Nanson's formula NdS = J- 1FTnds. Hence, with the transpose of matrix [F], J = 1 and knowing that n = e2 we find that

NdS thus, N

1

= 2e1ds

(3.15)

,

= e 1• Finally, using Cauchy's stress theorem,

[t) = [u](n)

= [ 5}

] kN/cm

2

1~0

[TJ = [Pl[NJ = [

,

l

kN/cm

2

(3.16)

i.e. t = 50e 2 and T = lOOe~h respectively. As can be seen, t and T have the same direction. The magnitude of Tis twice that o.f t, because_, in view of (3.15), the deformed area is half the undeformed area. II ...... ,............... ,...... ,.. :··· ................ :.. :·'''"'"··· ...................... •:··•.""· ......................... ······.···········t·:·'·:·····.····."·t."•t·:·····:····.-. ..·.:····:·:··.·······t.··t·····:·········:··.··:··············.·-.··,··:···········.···:·.···················.···················:··:·········.······:···:··········

We project the unique Cauchy stress tensor u along an orthonormal set {ea} of basis vectors; then, according ta { 1..62), we find that Stress components.

with

(3.17)

In view of Cauchy's stress theorem (3.3) 1, te,, = ueb., b = 1, 2, 3, characterize the three Cauchy traction vectors acting on surface elements whose outward norma.ls point in

the directions e 1, e2, e~h respectively. We use the notation tc = t(x, t, ea) to .indicate explicitly the dependence of the traction vector t on the basis vectors e11 • In matrix notation, the columns of [uJ can be .identified as the components of traction vectors acting on planes perpendicular to e 1, ·e2 , e3 , respectively. We may write. 0

(3.18)

(3.19) t . _,

-~,,

= ue~· = a 1··e 1 + rJ·riC·> + a~i·1e•.} "

,,

......

-

' ..,

•l

(3.20)

(see the three faces of a cube illustrated in Figure 3.3), characterizing the .state of stress at a certain point.

3.1

115

Traction Vectors, and S:tress Tensors

The traction ve.c.tors on any surface element are detennined uniquely by the set of _given quantities all 11 , called the stress components of the Cauchy stress tensor u. Since the Cauchy stress tensor .is sy.mmetri.e we have six independent stress components acting at a certain point of a body, with a·12 ·= a 21 , a 1a = -a~u, a23 ·= a 32 . For each stress component .a0 b we adopt the mathematically logical convention that :the first i~dex characterizes the component of the vector .t at a point x in the direction of the associated base vector Ca_, and the second index characterizes the plane that t is acting on. The plane is described by its unit normal, i.e. in our case the direction of the base ::(vectoreb (see TRUESDELL and NOLL [1992]).

·~>::: . .

!'. Figure 3.3 Positive stress components or the traclion vectors te acting on the faces of a cube. 0

. . : ..... :_-. _

'!i)\. It is important to note that some authors (in particular, those in the engineering ·'.//c:ommunity) reverse this convention by identifying the first index with the plane and iXfth~ second index with the vector component (see ·MALVERN [ 1969]).

/i}/>· ;w··EXAMPLE 3.2

i:;>Sri1ution. f(i{hrive..

Project the Cauchy traction vector te 1 onto e2 and interpret the result.

Recall relation (3.1.8) 1 and use the analogue of representation

(.l.59)~

we

(3.21)

(\fWith rule (1.53) and repeated applications of eqs. (1.21) and (l.61) to eq. (3.21b we

116


obtain

(3.22) Note that the quantity a 21 is the component of te 1 in the direction of e 2 (compare with Figure 3.3). • ·-··~·-

.....

.

.

~------=----" -<(•--.-~ ·~. -.:Y-••""C":"---... - . .• J,~_,,_••'~'-•

,. .... ,__;-.. - - , · - . - - . -·•-=---(V . ~•_..:..,•"'~--'""'':-·---:"-"··-,·,--~-~~

In .order to introduce a sign convention for the stress components look at Figure 3.3. We denote those faces of a cube seen by the observer as positive. The so-called negative faces of the cube are not visible to the observer. When the stress components aab are positive scalars, then the components of the traction vectors tc11 are defined to be .positive, as illustrated in Figure 3.3. A negative vector component on positive faces of the cube points .in the ·opposite direction. Vector components on a negative face are just the opposites of those ·on the associated positive face.

Normal and shear stresses. Let the Cauchy traction vector tn ·= t{x, t, n) for a given current position x E fl at time t act on an arbitrary oriented surface element The surface is characterized by an outward unit norm.al vector n and a unit vector m embedded in the surface satisfying the property m · n = 0..

.,.... ...

.......

_ ,..,. ... \_tn

= tU + tk

,,

/tU=crn n

Figure 3 . 4 Normal and shear stresses at current position x..

According to Figure .3.4, t 0 may be resolved into the sum of .a vector along the_ normal n to the plane, denoted by and a vector perpendicular to n, denoted by ~:·. With the rule ( 1.53) we find that

t!L

t;

(323)

3.1

117

Tracthm Vectors, and Stress Tenso.rs

t; = (m · t )m = P;t 0

(3.24)

0

(compare with the equivalent equations (1J25) and (.1.126)), where P.U and P* are projection tensors of order two defined by

r; ·= m ·0 m = I -

P~=n0n,

(3.25)

n®n ,

with properties according to ( 1.127)-(1.130). The lengths of :tH and are cal led the normal stress .a and shear stress (or tan gen .. . . ·tial stress) T acting on the surface element cons-idered, respectively. Using eq. (3.3).i the explicit expressions for a and rare

t;

..

T

a = n · tn ·= n · un

or

= m · tn = m · un

or

= n.ll a,111·n,, ·r = nl,a a al1 '11b a

(3.26)

'

(3.27)

\. If a > O_, normal stresses are said to be tensile stresses while negative normal stresses _·:_:./ (cr < 0) are known .as -compressive stresses. Tr.ad.itionally, in soil .mechanics the \_::(compressive stresses are characterized as positive and the tensile stresses as negative. \:/.!.t is important to note that a tensile stress is a fundamental different type of loading -~://than a cmnpression stress. However, the sign of a shear stress has no intrinsic physical ).::'.relevance, the type of loading is the same. }/·•.· From Figure 3.4 we find the useful geometrical relation

tn = tll + t;

~(

tll = an

with

,

t; = rm

(3.28)

(3 .2:9)

·Of

mx> Consider the schematic diagram of Figure 3.3. Each component D"a (a = 1, 2, 3; i.{\#o summation), i.e. cr 1L, a22 , cr:1:.h is a norma] stress acting in the direction of the unit 11

;:\{{;._Vector .normal to a surface element. These stresses are the diagonal elements of the !/((~.t.~ess matrix [u] gjven in eq. (3.6h. Each remaining c·ompone.nt a 0 b (a # b)~ i..e. ·/):·:_d·1~, a 1a, ... , is a shear stress acting tangentially to a surface ele:ment. Shear stresses .i:ff~re the off-diagonal elements of the stress matrix [u]. :._:.:: ~-::

::- ": .·.

'{::::(:·:}) ..·_::· ..

t&i/u:l.

vecto::t:~::::

Consider Cauchy traction {!f;\;}i·r.:=:·: ·.··.:.at x with normals n and ii. Show that

i"f

on two infinitesimal surface elements

~t:t

> if and only if the associated stre:s·:nso:

3 xis symmetric.

( .JO)

ll8

3

The Conc~pt ·of Stress

2. At a certain point of a deformable body the Cauchy stress components a.re given with respect to a ·:i: 1 , :c 2 _, :1:a-coordi.nate .system as

1 4 [ -2

[u] =

l

_')

4 0 0

~

. kN/m2

3

(a) Find the components of the Cauchy traction vector tn and the length of the vector along the normal to the plane that passes throu:gh this point and that is parallel to the p.lane 2-:r 1 + 3:i: 2 + X:J = 5. (b) Find the length of tn and the angle that t 11 makes with the normal to the.

p.lane.

.a

( c) Find the stress components a.b along a new orthonormal basis {ea}, a = 1, 2, 3, given by the following transformation

(compare with Section 1.5), where basis.

{Ca} characterizes the old orthononl)a~

3. Consider the Cauchy stress matrix in the form

[.er]=

[

,. ..2

3.112

0

3'l''···2

0

-:1:1

0

-:1;1

0

b:{:2:i:a ">

l

kN/c.m2

(3.31):

and find the components of the Cauchy traction vector t 0 at the· point with coor- ·. dinates (1/2,

V3/2, -1) on the surface :1:i + :z:§ + ~1::~ = 0.

4. At a -certain point of a body the components of the Cauchy stress tensor are given by

2

5 3

l

[u] = . 5 1 4 . kN/cm2 [

3

4

•

3

{a) Find the components -of the Cauchy traction vector tn at the point on the_/ plane whose normal has direction ratios .3 : 1 : - 2. (b) Find the normal and shear components of t on that plane.

3.2 Extremal Stress Values

119

:- .:· 3..2 Extremal Stress Values . \·:Normal and shear stresses vary in magnitude, direction and location. The designer of :\·:-.a structure must prove that a certain material will not fail as a result of these stresses. \Magnitudes, directions and Jocations of extremal stress values must be identified. We ·_(_:~:p~~sent briefly the standard results on extremal stress values used in practical engineer-

::/_J;?.g~ .

?. -,M~ximum .and minimum normal stresses.

Consider the Cauchy traction vector .:f:·t(~; t_, n) = u{x, t)n on an arbitrary oriented surface element throu_gh any po.int x E \\Aiftime t. First of all, we wisl1 to find the unit vector n at x, indicating the direction of .<."·:;11a.ximwn and minimum values of normal stresses a. In order to obtain these maximum and minimum (or so-called extremal) values of ::~}-.//.~~-may app1y the Lag-range-multiplier method and claim the stationary position of f/a . f~nctional .C to be

n

/:!

~>t.:/·:~·:: ·,_ . .: :

.C (n, ,\ *)

= n · un -

,,\ * (In I2

-

1) ,

(3.32)

··:.·. . ).:=:":.:·:··· ~ :· ..

..:_,:-(or, .in :::.-.::::.-.=:·.·.: : .

index notation, £(na' ;\ *)

./·):}:.,-'.:::<·:··:

;):'~e~~. lnl

= naacJbnb -

,..\ * (n,,n(l - 1) .

(3.33)

1 = 0 (n,.na - 1 = 0) characterizes the constraint (auxiliary) condition ':i(:~rid ,,\ * the La.grange multi.plier. :iit}·.::::;:,_ ~or the stationary position of£, the derivatives 8£/8n 0 and 8£/8.,\* ·must vanish, \=/T~~~'~·,"w.ith resp.ect .to (3.33), 2 -

":··:·=:.. :··· ..... ·

ai)t:,nc = O"a11(c~bcnn + 1Sacnb) = 2(aca·nci -

8£

a,..\*

= nllna -

,,\ *nc)

=0

A*(2na811c) (3.34)

.,

1=0

(3.35)

i\W~'haVe used 8na/an,, = Oat.. property (l .22h and the fact that the stress tensor

(T

is

(:fs·§frifuetric. In symbolic notation (3.34h and (3.35) read as

::,:t;:~{

o-n - ,, ·n = o ,

<

lnl

=1

.

(3.36)

fi!:)')\O~ince this is an eigenvalue problem the general results and rules of Section 1.4

;/(hold.

The Lagrange multiplier ,,\ * .may be identified as .an eigenvalue. Adopting the \:ff~tevant notation of Section 1.4, we must solve homogeneous algebraic equations for {:(J~~".tinknown eigenvalues,..\:, a = 1, 2, 3, and the unknown eigenvectors ila, a, = 1, 2, 3, (:)fo'· :the form ;;:~:.:}~:v:.·r./~-;,_-:·_:·:... :·

: ..

:!:l:/:':'.l>:_\·:_::(:':...) ..... : : ..

(a

= 1_, 2, 3; no summation)

(3..37)

120

3 The Conce.pt of Stress

(compare with eq. (l.166)). In order to obtain the eigenvalues we must solve the characteristic polynomial-of u, which is cubic in 1\:., i.e.

(3.38) with the three principal stress invariants lu of tensor u (com.pare with eqs. ( 1.170)( 1. l72)), Le.

f..1 (u)

= trcr = .,\~ + ,,\~ + /\; ,

J2 {u)

= ~ [{tru) 2 -

-

tr( cr2 )] = /\7 A;+ A7 /\; + ,\;,\;

(3.39)

,

(3.40) (3.4.1)

The three eigenvalues ,,\~ are here called principal normal stresses and typically are denoted by a,1' a = 1, 2, 3. The principal values ·'1a include both the maximum and minimum normal stresses among all planes passing through a given x. The corresponding three orthonormal e.igenvectors iia, which are then characterized through refation (3.37), are the principal directions of u. The normal stress is stationary along these principal directions. Their related norma] planes are known as principal ,planes. Since the stress tensor u is sym·metric the set of eigenvectors form a mutually orthogonal basis. Shear stress components vanish at normal (principal) planes, since obviously aH eigenvectors are normal to their respective principal planes. We -may rep.resent u in the spectral form 3

u(x_, t)

=L

O"utln ® fiu

(3.42)

(see Section 1.4 ), satisfying the eigenvalue problem uita = alliia, a = 1, 2, 3. Note that only for isotropic materials do the introduced three principal directions fia of u coincide with the principal directions, as defined in Section 2.6 ..For a brief explanation the reader is referred to Section 5.4, in particular to ·eq. (5.88), and also to Section 6.2.

Maximum .and minimum shear stresses. The next goal is to find the direction of the unit vector n at x that gives the maximum and minimum values of shear stresses·T. In the following we choose the eigenvectors fia, a = 1, 2 1 3, of u as a possible set of basis vectors. Then, according to the spectral decomposition (3..42), all non-diagonal· components of the matrix [u] vanish. The components cr 1 ~ a 2 , a 3 form the only entries of the stress matrixT and [u] is 'diagonalized'. The Cauchy traction vector ·tp then appears in the simple form (3.43).:

n

where n = n 1n1 + n 2 2 + n:-ifi 3 denotes the .unit vector normal to an arbitrary oriented_·:_ surface element. The normal stress .and shear stress on that surface element follow from:·.·,

121

3.2 Extremal Stress Values

eqs. {3.26)i, (3.29) and have the forms -

CJ -

n •

t0

= -a1 n 21 + a2n22 + a3.n32

= lt1112 -

T2

(3.44)

a2

(3.45)

nu

:·./ where the relations n = naila and (3AJh with the property ·ob = 6ab (according to :). eq. (1.21)) are to be used. · With the constraint (auxiliary) condition lnl 2 - 1 = 0 (~ana - 1 = 0) we are now .·able to eliminate n 3 from eq. (3.45h. Since the principal stresses a 1 , o-2 , a:i are known, ..:._.r2 is a function of n 1 and n2 only. In order to obtain the extremal values of r 2 we >differentiate T 2 with respect to n 1 and n.2 and identify with zero:

:-:·.:·. a 2

;c\

{}:I

= 2n1 (a1 -

a3) { a1 -

2((a1 - a3)ni + (a2 - l13)nm

O";i -

, (3.4.6)

..

:·.·

=0

..

Yet we must find the solutions for the stationary shear stress directions, i.e. tor the {:\:complete set of principal directions. The first (obvious) solution of (3.46) is obtained \'./::,.by taking n 1 = n 2 = 0 and n 3 = ±1. If we consider further the two cases n1. = n 3 = 0 ({(~Jlcl n2 = na = 0 as the associated component pairs, we find that n 2 = ±1 and r/)1·} = ±1, respectively. In summary, with n = nafia we obtain the three respective unit :_::.:.

~.\\·.<

.':· ·,-. :..

:·::.· .

. :.::-. ·.

"ifi~§fresponding

D

= ±03

·D

= ±D·>

D

= ±01 ,

(3..47)

to the set { 0 0 } of principal dir:ct:ons of u. Obviously, this first solution !/\):d~scribes the principal stress planes upon which T is minimized (in fact r = 0), as r//~g:rt:M1y concluded from <3 .45). /)\<._~,::.\\._The second solution of (3.46) may be found by assuming n 1 = O_, which gives, from :_/;.)(~3_" •.46h, the component n 2 = ±l/v'2. The condition lnl = I leads to n:1 = ±I/J2~ ::\:~N:_):i.Oing so for n 2 0 we find n 1 = n 3 = ±1/./2, while .for ·na = 0 we obtain the '.:~{};B_Q"inponents n 1 = n 2 = ± 1/./2. These results substituted back into (3 .45) give the &\{Wssociated extremal values r 2 • In summary,

=

:·... ·....

::_:(:.:·:·.·:· .... ·.· ·.. ·:·:=.:::.

:.~:/("= . <····. ..

·.

·: :····.·. .... :: .. :·.=.-·:.-·:·.·:· . ...........·.: .. ·:·

1

Vi-· v'2

\W.V\\'.::. _:: . : : ..": .=.: -~ =. : : .

.. :.:·::.":::·.·. ·.-·::·.".":::: . . ··.·.::.::.·

1

n = ±-n-, ± - 0 3 n

1

= ± v2 rnn1 ± ·

n=

1 ±-01

/2

1 1n·na v2

1 ±-n2

v'2

2

1 ') = 4(£12 - 0"3)~

(3.48)

')

1 =4 (l1t -

(3.49)

T

T-

')

£13)-

(3.50)

122


Consequently, the maximum magnitude of the shear stress denoted by Tmax is given by the largest of the three values of (3.48)2-(3..50) 2 . We obtain 1

Tmax

= 2lamax -

t.Tmin·j

(3.5 l)

'

where Gmax and D"min denote the maximum and the minimum magnitudes of principal stresses_, respectively. It is important to note that the maximum shear stress acts on a plane that is shifted about an angle of ±45° to the principal plane in which the maximum and .minimum principal stresses act (compare (3.47) with (3.48}1-(3.50h),. In addition, it can be shown that the normal stress -a which is associated with Tmax has.the value.a= lamax + O"minl/2. EXERClS.ES

l. A Cauchy stress tensor u, whose components a(j1, depend on the coordinates ·.rz: 1 :r: 2 , :1: 3 , is given in the form of its matrix representation l

0 0

-/3:1:a where .a and {3 are constants. For a certain point x, given by its coordinates (0, /3 2 , n), find (a) the three principal stress invariants of tensor u ..

(b) Compute the principal stress components and the associated principal directions of stress. Verify that the principal directions of stress are mutually orthogonal. (c) Compute the maximum magnitude of shear stress and the plane on which it acts . 2. The Cauchy stress matrix is given as

7 0

[u]

=

0 [: 14

8 0

·i

140 · kN/cm2

•

-4

(a) Compute the principal stress components .and the associated principal di-

rections. (b) Compute the maximum shear stress and the plane on which this maximum

shear stress acts.

3.3

Examples or States of Stress

123

3. At a given current position x E Q consider a plane whose unit normal n makes equal angles with each of the three principal directions .Da of stress. Such a plane is typically called an octahedra·I plane characterized by the unit normal vector n = naila,

= 1, 2, 3, with components n 1 = n 2 =

n:-1 = .1/

.../3.

(a) With reference to eqs. (3.26) 1 and (3.29) verify that the normal stress and the shear stress on the octahedral plane are O'oct

Toct

=

I

3{ O"t +
1 [(a11 - a22 )2 = 3 2

+6(a12

= 31 [(01 -

+ (an -

,

•) + (aa:1 - u1 i)~')

aa:1)~

+ a2a2 + a:n) 2 J1./.2 0'2) 2

+ ( 02 -

o;i} 2

+ (a:i - 0"1) 211/2 ,

where a 1, a 2 , u;1 are the principal stresses. The normal and shear stress on the octahedral plane are typically denoted by O"oct and T0 ct,, respectively. (b) Find the traction vector, the normal and shear components on the octahedral

plane for .a certain point at which the principal stresses are a.1 = 2 kN/cm 2 _., a2 = a, a 3 = 11 kN/cm2, where n is a constant. Detennine a so that Toct is the maximum shear·stress .

/ 3.3 Examples of States of Stress <· . Since ~1 Cauchy traction vector at given (x, t) .is defined for each surface element repre:/·sented by the outward unit .normal n, we may find infinitely many traction vectors. In

!\ .this section we show some important cases.

: : ·. ·State of stress.

The set of pairs { (t, n)} at a given po.int is ca11ed the s.tate of stress,

·:-·· ~ompletely .determined by the stress tensor u at this given po.int and time t according <;-- ~? Cauchy's stress theorem. Since t .is a linear combination of n in three dimensions, :/. . t_~ree linear independent pairs (t, n) form a complete basis. The state of stress is said ·:.· . ·_to be homoge:neous if the stress tensor does not depend on space coordinates at each

· time t. · We now _give some important examples for different states of stress.

/·_· . (i) A pure normal stress state at a certain point is ·given by the stress tensor

u ·(see Figure 3.5(a)).

= a(n~n)

.or

(3.52)

124


Post-multiplying (3.52) with the unit vector n we find using rule (1.53) and (3.28)2 that

un = a(n ® n)n

=a (n · n) n =an= tll . ~

n

(3.53)

1

tll

Evidently, is along (or opposite) n (compare with Figure 3.4). This stress state is characterized by a normal stress a, the shear stress is zero. The stress a characterizes either pure tension (if a > 0) or pure compression (if a < 0). Consider a rectangular Cartesian coordinate system. If a 11 = er = const and all other stress components are identically zero, then we have either uniform tension or uniform compression in the x 1-direction. In the literature these stress states are often referred to as uniaxial tension and uniaxial compression, respectively. This may be imagined as the stress in a rod (with uniform cross-section) generated by forces applied to its plane ends in the x 1 -direction, which is one principal direction of stress. A uniform tension or compression of a uniform cylindrical rod leads to a deformation, which we had already called uniform extension or compression (see p. 92). A three-dimensional stress state in which all shear stress components vanish, i.e. a ab = 0, a i= b, is said to be triaxial, while in a biaxial stress state (which is associated with biaxial deformations) we have a pair of non-vanishing normal stresses, i.e. a 1 , a 2 ; a 2 , a 3 ; a 1 , a 3 • However, in an equibiaxial stress state (which is associated with equibiaxial deformations) the non-vanishing normal stresses have the same value. (ii) A pure shear stress state at a certain point is given by the stress tensor

u = r(n ® m + m ® n)

(3.54)

or

(see Figure 3.5(b)). Here, the stress r is related to the directions of the unit vectors n and m, with property m · n = 0. Post-multiplying (3.54) with the unit vector n we obtain with rule (1.53) and (3.28h

un

= r(n®m+m®n)n = r((m · n) n + (n · n) m) =rm= t;. .

(3.55)

~~

0

I

Evidently, t~ is tangential to the surface and along (or opposite) m, i.e. perpendicular to n (compare with Figure 3 .4 ). This stress state is characterized by a pure shear stress T (or'" pure tangential stress) and the normal stress is zero. Consider a rectangular Cartesian coordinate system. If a 12 = a~n = T = const and all other stress components are identically zero, then this stress state is characterized by a uniform shear stress. A state of uniform shear stress leads to a uniform (or simple) shear deformation, as illustrated in Figure 2.2 (see also Exercise 2 on p. 93 ).

3.3

125

Examples of States of Stress

(a)

(b)

t~ =Tm

lt~I

n

=T

n

tll =an

(c) n

tU

=-po

Figure 3.5 Examples of stress states.

(iii) A hydrostatic stress state at a certain point is given by the stress tensor

u

=-vi

(3.56)

or

(see Figure 3.5(c)). Within a rectangular Cartesian coordinate system that means that we have normal stresses a 11 = a 22 = a 33 = -p and no shear stresses on any plane containing this point, i.e. a 12 = a~a = a:11 = 0. Post-multiplying (3.56) with the unit vector n, we obtain, using (3.28)2,

un

= -(vl)n =

-vn

= t~! .

(3.57)

Evidently, tU is opposite to the normal vector n. The relationship un = -pn holds only if n is a principal direction of u. Note that any three mutually orthogonal directions may be regarded as principal directions (compare also with eq. ( 1.179)). This stress state is characterized by a scalar p, known as the hydrostatic pressure. In ge~eral, the hydrostatic pressure is a scalar function of time t, and in the literature is often introduced with the opposite sign. One example of stress state (3.56) is in an (elastic) fluid (without motion) that is not able to sustain shear stresses. Taking the trace of (3.56), we obtain

p

=-

1 3

-tru

or

1 p = - 3a"u '

(3.58)

126


.indicating that the pressure of the (elastic) ·fluid is a mean pressure. The three stress fields described for a certain point x at .t (with u constant) correspond to the states, as illustrated in Fi_gure 3.5. (iv) A plane stress state at a certain point is given by the relation (359) and a 11 , a 22 _, a 12 are functions of the coordinates :z; 1 .and :r.2 only. Consequently, the ~r~ 1 -direction, as represented by it3 , is a principal direction of stress with a zero car.. responding principal stress a~i:1 • The other two principal directions acting in a pfane normal .to ila are inclined at an angle fJ with the :r; 1 and x 2 direction, where 2a12

tan2B = - - - a1.1 -

(3.60)

0"22

The corresponding maximum and minimum -stresses are given by 1

2(0"11

1 + 0"22) ± [4 (0"11 - 0"22)- +
') ]

J j'> -

1

(3.61)

defining a biax.ial stress state. The maximum shear stress Tmax for a plane stress state will be the largest of the three values of (3.48}~-(3.50) 2 s-ince cr:1 = 0. The planes of extremal shear stresses form angles of ±45° with the .planes of the principal stresses. A plane -stress state occurs at any unloaded surface in a continuum body and is of practical interest. EXERCISES

I. For the case of plane stress, show that -e-q. (3.38) reduces to eq. (3.61 ). 2~

Assume a plane stress state in a rectangular cube bounded by the planes :r: 1 ±a, :I: 2 = ±b, :1;3 = ±c. The state of stress at a po.int with coordinates ;1~ 1 , ~r: 2 , :1::1 in the cube is given by

where n, /3 are constants. For a point with-coordinates (a/2, -b/2,-0), determ:ine. (a) the principal normal .stresses and the associated principal directions,

(b) the planes_, characterized by the unit normal n_, that give the maximum and minimum shear stresses and the magnitude of the extremal shear stress.

(c) Find the total Cauchy traction vector on each face of this rectangular cube.

127

AHernative Stress Tensors

3.4

3.4 Alternative Stress Tensors Numerous definitions and names of stress tensors have been proposed in the literature. Each definition has advantages .and disadvantages. In the following we discuss stress tensors used for practical nonlinear analyses ..Most of their components do not have a direct physical interpretation. Often it is convenient to work with the so-called Kirchhoff stress tensor r, which differs from the Cauchy stress tensor by the volume ratio J. It is a contravariant spatial tensor field parameterized by spatial coordinates, and is defined by T

or

=Ju

Tut>

= ,Ja ab

(3.62)

•

We introduce further the second Piola-Kirchhoff stress tensor S which does not admit a physical .interpretation in terms of surface tractions. The contravariant material tensor field is symmetric .and parameterized by material coordinates. Therefore, .it often represents a very useful stress measure in computational mechanics and in the formulation of constitutive equations, in particular, for solids., as we will see in Chapter 6. . The second Piola-Kirchhoff stress tensor is obtained by the pull-back operation on the contravariant spatial tensor field T~ by the motion x, which is, according to (2.85h, SA H

or

-lp-l = F Au 1J b Tub 1

Hence, the Kirchhoff stress tensor is the push-forward of S,

i.e.~

•

(3.63)

using (2.85}1, (3.64)

or

Using eqs. (3.6.3)2, (3.62) and (3.8) we obtain the Piola transformation relating the two stress fields S and u, i.e . S

or

SA 0

== .IF-tuF-T = F- 1p =ST

= p-lp · Aa n IJ

= ~JF.-t An-tF-l · ~I.lb fTub

=

(3.65)

c

•.J II A

with its inverse, or

rr

1-'

-

ub -

'

7- l F.uA .t'.bJJ' .'fi' S A.IJ.

•

(3.66)

From eq. (3.65) we find a fundamental relationship between the first Piola-Kirchhoff stress tensor P introduced in (3.8) and the symmetric second Piola-Kirchhoff stress tensor S, i.e.

P= FS

or

(3.67)

128

3 ·The Concept of Stress

In addition to the four stress tensors -given above, we .introduce another important -quantity. .It ·is a material stress tensor formally defined as Tn

= .RTP

(3.68)

or

The non-symmetric tensor Tu, which -is not in general positive definite, is known as the Biot .stress tensor. With (3.67) and the polar decomposition F = RU we deduce from (3.68) that Tn = RT (FS) = US. Herein, R and U denote the rotation tensor (with defR = 1) and the (positive definite) symmetric right stretch tensor, respectively. They are according to the polar decomposition of the deformation gradient F (see Section 2~6). Multiplying eq. (3.68) by R from the left-hand side, we obtain the polar decompos.ition P=RTn

or

(3.6.9)

for the first Piola-Kirchhoff stress tensor, which is in analogy with that for F. Since the .Biot stress tensor Tn is not positive definite this decomposition .is not unique, in general. Other examples of stress tensors are the symmetric so .. called corotated Cauchy stress tensor uu, as introduced by Green and Naghdi, and the Mande·J stress tensor ~ which is in general not symmetric. These tensors are defined with respect to an intermediate configuration. In .order to obtain the corotated Cauchy stress tensor take relation (3.66) .and use the symmetric right stretch tensor U instead of F. Then apply relation (3.65}1 and the polar decomposition -F = RU to find (3.70)

The .Mande] stress tensor is defined to be E =CS

(3.71)

which is often used to describe .inelastic (plastic) materials. EXERCISES

l. Consider an infinitesimal resultant (pseudo) force dfn = (TnN)dS, where TnN denotes the Biot traction vec.tor. Using (3.68) and the linear transformations (3.3) 2 and (3 . .1)2, verify that dfn

= .R'rdf

,

which shows that d.fn -only differs from elf (.introduced in (3 . .l )) by the rotation tensor RT.

3.4 Alternative Stress Tensors

129

2. The Cauchy stress components at a certain point are given with respe-ct to a x1., :1;rcoordinate system with the associated set { e 0 } , a = 1, 2, ·of orthonormal basis vectors as

[u] = .·[ 21

1 ] kN/m2 .

-3

Assume a .rotation of {e0 } into the new set {e0 } of orthonormal bas.is vectors according to

Compute the components of the corotated Cauchy stress tensor u 0 •

4

Balance Principles

In this ch~pter we provide the classical balance principles and -discuss some of their important consequences. The fundamentaJ balance principles, i.e. conservation of mass, the momentum balance principles and balance of energy, are valid .in all branches of continuum mechanics. They are applicable to any particular material and must be satisfied for all times. We also discuss another fundamental set of laws that are expressed as inequalities, such as the second law of thermodynamics. The section devoted to continuum thermodynamics specifically addresses balance of energy and the entropy .inequality principle. Finally, the structure -of principles is summarized as the ·master balance (inequality) principle.

4.1

Conservation of Mass

Every continuum body B possesses mass, denoted by tn. It .is a fundamental physical .property com.manly de·fined to be a measure of the amount of a material contained in the body B. In order to perform a macroscopic study we assume .that mass is continuously (or at .least piecewise continuously) distributed over an arbitrary region (of physical space) with boundary surface an at time t. The mass .is a scalar measure (a positive number) which is invariant during a motion. We exclude concentrated masses such as those used in class"ical Newtonian mechanics.

n

~losed

and ope.a systems. We define a system as a quantity -of mass or .a particular c~llection of matter in space. The complement of a system, i.e. the mass or region outside the system, we call the surroundings, while the surface that separates the system from its surroundings we can the boundary or wall of the system (se.e Figure 4. l ). A closed system (or control mass) consists of a fixed amount of mass in a properly selected region .in space with boundary surface which depends on ti.me t (see Figure 4.1). No mass can -cross (enter or leave) its boundary, but energy, in the form of work or heat, can cross the boundary. The volume of a closed system does not have

.an

n

13.1

132

4

Balance Principles

SURROUNDINGS

Boundary or wall of the system

Control surface

Fixed amount of mass; .Energy can cross the boundary

Figure 4.1 Closed and open

Fixed amount of volume; Mass and energy can cross the control boundary systems~

to be fixed. If even energy does not interact between the system and its surroundings, then we say that the boundary is insulated. Such a system is called mechanically and thermally isolated, which is an idealization for a physical system. There always exist electromagnetic and other types of forces which permeate the space. Note that no physical system .is truly isolated. An open syste.m (or control volume) consists of a fixed amount of volume of a properly selected reg.ion flc which .is independent of time t (see .Figure 4.1). The enclosing boundary of a control volume, over which both mass and energy can cross (enter or leave), is called a control surface, which we denote by

anc.

Conservation of mass.

In non-relativistic physics mass cannot be produced or

destroyed. It is assumed that during a motion there are neither mass sources (reservoirs that supply mass) nor mass sinks (reservoirs that abs-orb mass), so that the mass m of a body is a ·conserved quant-ity. Hence, if a particle has a certain mass in the reference

4.1


133

configuration it must stay the same during a motion. Considering a closed system, obviously that holds for the total mass too. We write

rn..(no)

= rn(n) > O

,

(4.1)

which holds for all times :t. Relation (4.1) is a statement of a fundamental mechanical law known as the conservation of mass. The ·boundary surfaces in the reference and configurations, with volume ll and v, are denoted by n 0 and !1, respectively. Note that the mass -rn is independent of the mot.ion and of the region occupied by the body. Hence, the material time derivative of the mass 1n gives D -n1..(no) Dt ·

D

= --rn(n) =o . Dt

(4.2)

The differential form of eq. (4.1) reads

drn(X)

= drn{x, t) > 0

(4.3)

1

·with the infinitesimal mass clement drn. The mass of n 0 and !1 is characterized by continuous (or at least piecewise continuous) scalar fields, i.e. p0 = p0 (X) > 0 and p = p(x, t) > 0, respectively. They denote physic.a·1 properties of the same particle. Property p 0 is called the .reference mass density .(or just density) and pis -called the spatial ·mass dens.ity during a motion j( = x(X, -t) . The spatial muss density, a1so known as the density in the motion, depends on place x E n and ti.met throughout the body. Note that {Jo is time-independent and intrinsically associated with the reference configuration of the body. He.nee,. p0 . depends only on the position X chosen in configuration 0 0 • If the density does not depend ·on X ·E n0 , i.e. if Gradp0 === o, the configuration is said to =be homogeneous. The mass densities at the points X and x are defined by the limit

.. ~7n(no) Pn(X) = .:l F{f?o)-ro hm ~F(n / ·"' · ·o.) , \Vhere

~1n denotes

p(x, t) =

..

~rn(.s-2)

hm ~ v (n) ~u(n)-+O .

(4.4)

a continuous function of incremental mass of an :incremental vol:· ume element in the reference and current configurations, which we have denoted by :·Liv· and ~'IJ, respectively. Note that ~ l . ~ (.n0 ), il·v (n), actually must not tend :to zero since then the limit of p0 , p, would show a discrete distribution according to the atomistic stmcture of matter. Therefore, to obtain representative averages, .6. ll(n 0 ) and ~v(n) must be large in terms of an atomistic scale and s.mall in lenns of a length scale of a certain physical ·problem. Usually the ratio of the length scale of a physical problem and the length sea-le of an incremental volume ele~nent -il ll(!1 0 ) and ilv(n) is of the order 1oa -or

more.

134

4 Balance Prlndples

In the differential form eq. (4.4) reads drn(X)

= p0 (X)dlt

drn(x., t)

,

= p(x, t)dv

,

(4.5)

with the standard infinitesimal volume elements d l·=" and clv defined in the reference and current configurations, respectively. Substituting eq. (4.5) into (4.3) we o·btain Po(X)dl/

= p(x, t)dv > 0

(4.6)

which means that volume increases when density decreases. By integrating the infinitesimal mass over the entire region, we .find the total mass rn of that region. He.nee, an alternative expression for (4.6) reads

rn

=/

fJo{X)dV = / p(x, t)dv no n

= coust > 0

(4.7)

for all times t, which impHes the rate form

·1h = -Dni Dt

1· p(x,. t)dv = 0 .

D = -. Dt,

(4.8)

n

Hence, conservation of mass requires that the material time derivative of m. is zero for all regions of a body which change with time (mass remains unchanged during the motion of .n). An equation which holds at .every point of a continuum and for all times, for ex.ample, eq. {4.6), is referred to us the local (or differential) form of that equation (local means pointwise). An equation in which physical quantities over a certain region of space are integrated is referred to as the global (or integral) form of that equation; see, for example., eq .(4.. 7). Consequently we may say that (4.6) is the local form and (4.7) is the global form of conservation of mass . .In general, local forms are .ideally suited for approximation techniques such as the finite difference method while global forms are the best to start with when the finite ·element method is employed.

n

Continuity mass equation. We want to find a relationsh.ip between the reference mass density Po (X) E n0 and the spatial mass density p( x, t) E n. By recalling eqs. (2.50) .and (2.51), i.e. dv = J(X;t)d l/, J = detF(X, f) > 0, we may change the variable of integration in eq. (4.7) from x = x(X, t) to X and we obtain the identity

/ [po(X} - p(x(X, t), f ).J(X, t)]dY C')

.l .. 0

=0

.

(4.9)

4.1

135

Conservation of M·ass

By assuming that Vis an arbitrc11J' volume of region in (4.9) must vanish everywhere. Hence, Po(X)

n0 , we condude that the integrand

= p(x(X, t_L t).J(X, t)

,

(4.l 0)

holds for .all X E n0 • It represents the continuity mass equation (continuity stands for constancy of mass) in the material (or Lagrangian) description which is the appropriate descri pt.ion in solid mechanics. Since the reference mass dens.ity p0 is independent of time we find simply from (4.10) that

Dpo(X) _ . (X) _ O

at -

{Jo

-

(4. l l)

~

which is the rate form of (4.10) in the material description.

EXAMPLE 4.1 Show how the spatial mass density p = p(x~ t) changes with time. In particular, derive the rate form of continuity mass equation in the spatial (or Eulerian) description, which .is (expressed in terms of the velocity components)

rJ(x, t)

+ p(x, t)divv{x, t)

= 0

j1 + p ~;~: = 0 ,

or

(4.12)

or in the two equivalent forms

Dp~~, t) + gradp(x, t) · v(x, t) + p(x, t)divv(x, t) = 0 8p(x, . (p (x, t )v (x, t )). = Dt t) +div

o

,

(4.13) (4.14)

for all x E H and for all times t. Solution.

Since 110

= 0 we obtain from (4.10) that D (

Dt pJ

)

. = -· pJ = O

(4.15)

(for simplicity written without arguments of the scalar quantities). In order to express eq. (4.15) .in terms of the spatial velocity components we find using the product rule ~_nd J = Jdivv, i.e. eq. (2.175){;, that

pJ = i)J + p} = J(p + pdivv)

=0

,

(4.16)

where the material time derivative of the spatial density function p is, having regard to (225), g.iven by the explicit expression

.

p=

Dp Dt

Dp

= Dt

+gradp·v

(4.17)

136"

4 .Balance Principles

Since J > 0 we deduce from (4.16)2 the desired result (4.12), which is the corresponding local forn1 of (4.8) 2. With the material time de.rivati ve of the spatial density function (4.17) and by means of identity (I ~287) we may obtain from (4.12) the two equivalent forms (4.13) and (4 ..14 ).

·• :.-=···· . ·····:,•... ,.................. , ....,..,,............,.... :,.:,:·:,::.,,;o,,,.......• __ ,,,,.,-; .. ,,.,,

.• ,_..,,,,,,,,,, ......... ~, ... , .. , .•. ,, ... :····:·:-· .. ····:,, .. ,,,, ... : .. ,,, ...................................... ,
lf the density of a -continuum body is constant at any particle, then from relation (4.12) we find with p = 0 a kine:matical restriction which characterizes un isochoric (volume-preserving) motion, i.e. divv = 0 (compare also with eqs. (2.177)5). A continuum body is inco:mpressible if every motion it undergoes satisfies p = 0. The rate forms of continuity mass equations (4.12)-( 4. "I 4) show how the spatial mass dens.ity p changes as time changes. They represent the continuity mass equation in the spatial description, which is the appropriate description in.fluid dynamics.

Conservation of ·mass for an open system. Snmetimes we work with an open system given by a region nr. and boundary control. surface an(.. At a certain time ta control volume contains the mass ·nz.(t) = fnc p(x~ t)dv. Since the re.gion of integration f2c does not depend on t., integration and differentiation commute and we may write . . -rn(t)

= -D /& p(x, t.) dv = Dt.

He

/~ 8 p( x, t) . dv . . t n,:

a.

(4.1.8)

App.lying the divergence theorem for a fixed amount of volume f2c we have using (l .297)

/ div(p(x, t)v(x, t) )dv

=/

p(x, t)v(x, t) · nc!H ,

(4.19)

an~

Or

where n denotes the outward unit vector field perpendicular to the boundary control surface The tenn .l~nc pv. nds determines the flux of {JV out of across Drlc. Integrating the continuity mass equation in the form of ( 4.1.4) over a certain region and using eqs. (4.18) and (4.19), we obtain the conservation of mass for a control volume in the global form, .i.e.

anc.

nc

nc

D .;· p(x, t)dv Dt flc

= - .;· p(x, t)v(x, t) . nds

.

(420)

one

Relation (4.20) .asserts that the material ti.me derivative of the .mass inside a control volume is equal to the flux -of pv ente.ring f1e across Df1c:. The global form (4.20) -is widely used in fluid dynamics.

nc

4.1


137

EXERCISES

1. A velocity field of a plane motion has components of the form V3

=Q

,

where a and /3 are positive constants. Assume that the spatial mass density p is independent of the curren_t position x so that gradp = o. (a) Express p so that the continuity mass equation :is satisfied.

(b) Find a condition for which the given motion is isochoric. 2. Tuo motions of a continuum body are given in the form

x = (1 + a:(t)t)X , with the scalar function a:(t) and the set {ea}, a = 1, 2_, 3, of orthogonal unit vectors. Find expressions for the spatial mass density pin terms of p0 so that the continu-

ity mass equation is satisfied. 3. Consider a spatial scalar field = (x, t) and a spatial vector field u = u(x, t). Use the rate form of the continuity -mass -equation and obtain the identities D Dt

p-

=

D(p) · 8t

+ div{pv)

'

p Du

Dt

= a~u) + div(pv ® u) .t

, (4.21)

where v denotes the spatial velocity field. 4. An irrotational motion of an incompressible continuum is given by the s.patial velocity field v = -grad. Show that for this case the scalar field is harmonic. 5. By means of continuity mass equation (4..12) and identity (2.181.) show that the vorticity vector 2w is .related to the spatial mass density p and to the spatial velocity vector v by

Dt (2w) = curl Dv D + (gradv)2w p . t

p -.

0

which is known as the Beltrami vorticity ·equation.

,

.138

4 Balance Principles

4.2 Reynolds' Transport Theorem Suppose we have a spatial scalar field = (.x, t) describing some physical quantity (for example, mass, internal energy, entropy, heat or entropy sources) of a particle in space per unit volume at time t:. Assume to be smooth, so that it is continuously differentiable. Hence, the present status of a continuum body in some three-dimensional region n with volume v at given time t may be characterized by the scalar-valued function

l(t) = / (x, t)dv . n

(422)

The aim is now to compute the material time derivative of the volume integral I(t). Since the region of integration depends on time t, integration and time differentiation do not commute. Therefore, as a first step I(t) ·must be transformed to the reference configuration. By .changing variables using the motion x = x(X, -t) and the relation dv = J(X_, t)dll we find the time rate of change of I(t) to be

n

. o;·

l(t) = Dt. (x, t)d'IJ n

Dr (x(X, t),. t)J(X, t)dl'~

= Dt.

.

(4.23)

no

Since the region of integration is now time-independent, integration and differentiation commute. Hence, as a second step, from (4.23h we obtain, using the product rule of differentiation,

gt / (x, t)dv n

= /

[ (x(X, t), t)J(X, t)

+ (x(X, t), t).i(X, t)] dV ,

(4.24)

no

where 4~ denotes the material time derivative of the spatial scalar fie.Id .according to relation (2..25). In a last step we undo the change of variables and convert the volume integral back to the current configuration. By means of eq. (2.l 75)fi, dv = J(X, t)d\/ and motion x = x(X, t), we find finally that D ;· (x, t)clv = ;·((x(X, t), t)

D.t,

n

'

+ (x(X, t), t) ·~~~· t~]J(X, t)dF f

nu

.[. . (x, t) /

,t

.

}(X, t) J

+ cl>(x, t) J(X, t)

dv

n

/ ( (x, t) + cl>(x, t}divv(x, t) ]clv n

(4.25)

4.2

:_\~;(;:·: .. ·

139

Reynolds' Transport Theorem

:::/ID the following the arguments of the tensor quantities are dropped in order to simplify i::i(/tbe notation. However, in cases where additional information is needed, they will be ::'{iemployed. Hence, relation (4.25):1 reads as

~t ./ dv = / (divv)clv ·.:._:·.·_.·::

n

·..

(4.26)

,

n

;:':!}~here we have assumed smoothness of the spatial velocity field v. Other forms of the time rate of change of the integral (4.22) result from (4.26)

.:' 'by means of the material time derivative of the spatial scalar field
_eq. {2.25), and the product rule (l .287), i.e.

E_ Dt.

1· dv = .f (O + grad
divv)dv . 8t:

n

n

. = . (div( v) /

+

D {)t

(4.27)

)dv ,

n

/~nd finally, using the divergence theorem according to (l.297), -D /" clv

.Dt. n

=

I

v · nds +

. an

;· D
' n

-.-dv .

(4.28)

{)t

.·.· ..

:'./.·.: · TI.ie first term on the right-hand side of eq. (4.28) characterizes the rate of transport \(Or the outward normal flux) of v across the surface out of region n, which is (:(_::_assumed to be fixed. This contribution arises from the moving region. The second }):.·_term denotes the local time rate of change of the spatial scalar field cJ> within region n. :/:··:·In (4.28) n denotes the outward unit normal .field acting along DH. Relation {4.28) is :\:·_.·referred to as Reynolds' transport theorem. Another very useful relationship is obtained by considering the scalar-valued func .. .· tion

on

l(t) = ./ p(x, t)W(x, t)dv ,

(4.29)

n

:;_ . which is deduced from eq. (4.22) by changing into fJ°W, where \JI denotes a smooth . . · spatial scalar field describing some physical quantity of a particle in space per unit :_.:··mass at time t. W·ith reference to eq. (4.26) the time rate of c·hange off(t) is .then given as

-D ;· p'lldv Dt

.n

= ;·-· (p\JI + p\Jldivv)dv .

n

.

(4.30)

140

4 ·Balance Princip-Jes

We now apply the product rule of differentiation and the rate form (~.12) of continuity mass equation to the first term of the integrand in (4.30). We find p\J! = pW + p\JJ = p\J! - p\J! divv. As a consequence of the continuity mass equation, we obtain

gt/ p(x, t)W(x, t)dv =/ p{x, t)~(x, t)dv n

.

(4.31)

n

Assume \JI = 1 so that the volume integral i(t) .in (4.29) represents mass within . region n (see eq. (4.7)). For that cas-e, eq. (4..31) reduces to (4.B"h~ since \lf = 0.

EXAMPLE 4.2 Show that the important relation (4.31) may also be obtained.by changing the scalar-valued integral (4.29) over space into an integral over mass.

Solution. that

Using the infinitesimal mass element drn = pdv (see eq. (4.5h) we find

=.J~ Dw{x, Dt t) dm = .(

D .( \JI (x, t )( p (x, t)dv). Dt n

n

(

).(

p x, t \JI x, t

)dv ,

(4.32)

n

since the .infinitesimal mass element is not affected by the material time derivative. .,.,,,.,_ .•• ,,.,,,.,,_,..,.,,,,_,,, .• ,,.., ....

,.,_,_,,,.,.,.~.-·-·-'':'·'."'''"'':'·,,,

___ .,_,,,.,,.,,.,,..., ______ ,.,,,,:.... _, ____ ,,_,,,,,._,, ___

·-:''~··,.,-::·-,-

II

..,--.•,,,.,,.,:,.,:-··-·... --·-''''•'"':''" .. "'"--.·---,·'''''·'''".,. _____ _,,.,, .• -::•·'·'"."''-''·'-·-···'·"'''·''''··.·:,1, •.• ,.,,,,,,,,.,,, •. ,,

EXERCISES

1. Derive the rate form of continuity ·mass equation (4. 12) in the spatia·1 description, i.e. /j + pdivv = 0, by applying relation (4.26) and using eq. (4.8)2.

2. Essentially the same statements as derived in eqs. (4.26), (4.27.h, (4.28) and (4.31) result for the material time derivative of a vector-valued volume integral (or tensor-valued volume -integral) by regarding either or \JI as a smooth (continuously differentiable) spatial vector field u = u(x_, t). Show that

gt/ud·v = /(u+udivv)d'V= n

n

=

.!

/[~~ +div(u®v)]clv n

~; dv

u(v · n)ds + /

,

(4.33)

D ./' pudv = . plid-v , Dt

(4.34)

an

n

1·

n

n

where the material time derivative of u is ti denotes the spatial velocity.

=au/at+ {gradu}v. The vector v

4.3

141

Momentum Balance PrinCiples

4.3 Momentum Balance Principles In this section we -describe balance of .linear and angular momentum for a closed and an open system, essential in .continuum mechanics. These principles are valid for the whole or arbitrary parts of a continuum body B. In addition, we de.rive Cauchy's first equation of motion and show the symmetry of the Cauchy stress tensor. Balance of linear and .angular momentum in spatial and material description. Consider a continuum body B with a set of particles occupying an arbitrary region n with boundary surface an at time t. We consider a closed system with a given motion x = x{X, t), spatial mass density p = p(x, t) and spatial velocity field v ·= v(x, t). We define the total linear ·momentum L (or in the .literature .sometimes called translational momentum) by the vector-valued function

L(t)

= ./ p(x, t)v(x, t)dv = ./ Pn(X)V(X, t)dV fl

(4.35)

,

Ou

and the total angular momentum J relative to a fixed point (characterized by the position vector x0 ) as

J(t)

= ./ r x p(x, t)v(x, t)dv = /

r x p0 (X)V{X, t)dF .

(4.36)

no

n

We used the identity (2.8), conservation of mass in the form of pdv definition of the position vector r, i.e.

r(x)

=X-

Xo

= x(X_,:t) -

Xo .

= p0cfl/ and the (4.37)

Jn the literature the angular ·momentum J is often .referred to as the moment of momentum or the rotational momentum. Momentun1 equations (4.35) and (4.36) are formulated with respect to the current and reference configurations with associated quantities p, v, dv and p0 , V, dl/, respectively. Linear momentum and angular momentum per unit current and reference volume are defined as the products pv, {Jo V and r x pv_, r x p 0 V, respectively. To avoid congestion we often omit the arguments of the tensors for much of the remainder of this section. The material time derivatives of linear and angular momentum (4.35h and (4.36h -of the particles which fill an arbitrary region n result in fundamental axioms called momentum balance principles for a continuum body. We postulate the balance of linear momentum as

.

L(t)

Dr pvdv = Dt. D ;· PnVcW

= Dt.

n

no

= F(t) ,

(4.38)

142

4

Balance Principles

and the balance of angular momentum (or balance of moment of momentum or balance of rotational momentum) as

. J(t)

D = Dt.

f r x pvdv = Dt. D f r x p VdV = M(t) 0

,

(4.39)

no

n

which are given in both the spatial and material descriptions. In relations (4.38) and (4.39), F(t) and M(t) are vector-valued functions. They characterize respectively the resultant force and the resultant moment (or resultant torque), i.e. the moment of F about x0 • The momentum balance principles are generalizations of Newton's first and second principle of motion to the context of continuum mechanics, as introduced by Cauchy and Euler. The contributions to linear momentum L and angular momentum J of a body are due to external sources, i.e. F and M, respectively. If the external sources vanish linear and angular momentum of the body are said to be conserved. By virtue of relation (4.34) we may rewrite the balance principles as

L(t) = ./ pVdv n

=

./PoVdll

(4.40)

= F(t)

flo

j(t) = / r x pVdv = ./ r x µ0 Vdlf = M(t)

n

(4.41)

flu

v,

r x

Here, we have used the relation r x v = r x since = = v according to (4.37) 1 and (2.28) 1 , and consequently r x v = v x v = o. The spatial and material acceleration fields are characterized by and V(compare with Section 2.3 ). The so-called inertia forces per unit current and reference volume are denoted by and p0 V, respectively. In the following we define the structure of forces acting on a continuum body. Consider a boundary surface an of an arbitrary region n which is subjected to the Cauchy traction vector t = t( x, t, n) (force measured per unit current surface area of BO), as introduced in Section 3.1. The unit vector n is the outward normal to an infinitesimal surface element ds of DO. Furthermore, let b == b{x, t) denote a spatial vector field called the body force. It is defined per unit current volume of region 0 acting on a particle, as illustrated in Figure 4.2. Note that the symbol b should not be confused with the strain tensor introduced in eq. (2.77). A body force is, for example, self-weight or gravity loading per unit volume, i.e. b = pg with the spatial mass density p and the (constant) gravitational acceleration g. Hence, the resultant force F and the resultant moment M (about a point x 0 ) on the body in the current configuration have the additive fonns

v

pv

F(t) = / tds + ./ bdv ,

an

n

(4.42)

4.3

143

Momentum Balance Principles

b t

.1,

..1\•1 X•11)

time t

Figure 4.2 Structure of forces acting on the current configuration.

M(t) =

.!

r x tds +

{}fl

.!

(4.43)

r x bdv .

fl

Finally, by virtue of eqs. (4.38) and (4.39) the global forms of balance of linear momentum and balance of angular momentum may be given in the spatial description as

D .;· pvdv = .;· tds Dt

it .!

n

;m

r x pv
.!

an

n

+ ;· bclv fl

r x tels +

.!

(4.44)

,

(4.45)

r x bdv .

n

These equations are fundamental in continuum mechanics. For the balance of angular momentum (4.45) we have assumed the restriction that distributed resultant couples are neglected. If we consider resultant couples throughout a body in motion, then the balance of angular momentum (4.45) reads as D ;· (r x pv + p)dv D,

t.

n

= ;· (r x t + m)ds + ;· (r x b + c)du .

an

.

.

(4.46)

n

Here, m is the distributed assigned coupled traction vector per unit current area acting on the boundary surface an while c is the distributed assigned body couple (or also called body torque) per unit volume acting within the volume of region n. The spin angular momentum (or intrinsic angular momentum) per unit current volume is

144

Balance Principles

4

denoted by p. A continuum without distributed resultant couples is called .non-polar. If any. couple acts on parts of the continuum we say that the continuum is polar. Polar continua are not considered in this text. For a detailed study of .polar continuum mechanics se·e, for example, TRUESDELL and TOUPIN [1960] or MALVERN [1969]. In order to express the ·momentum balance principles in terms of materia·1 coordinates we introduce the {pseudo) ·body force -called the reference body force B = B (X, t). It acts on the region n and is, in contrast to the body force h~ referred to the reference .position X and measures force per unit reference volume. With volume change dv = JdF and motion x = x(X, ·t) we find the transformation of the body force terms of eqs. (4.44) and (4.45) in the form

/ b(x, t)dv = /b(x_(X, t}, t)J(X, t)dl' n no or in the loca] form as B(X~ t) =

J(X? t)b(x, t)

=/

B(X, t)dF

(4.47)

no

or

(4.48)

Using the first P.iola-Kirchhoff traction vector T = T(X, t, N) introduced in (3.1), relations (4.38), (4.39), (4.48) and d-v .JdV.~ we conclude from (4.44) and (4.45) that

gt.! g;·

=

poVd'V =

nu

t .

.!

ano

r x p0 VdV

TdS +

.!

BdV ,

(4.49)

no

= .f r x TdS + .;· r x BdV

,

(4.5~_)

no ano no which are the global forms of balance of linear momentum and balance of angular momentum~ respective·ly, in the material description. Equation of motion in s.patia:J and material descript.ion. A necessary and suffi.:.. cient condition that the momentum balance principles (4.44) and (4.45) are satisfied is the existence of a spatial tensor field .u so that t(.x, t, n) = u(x, t)n (see eq. (3.3)i) . .By computing the integral fonn of Cauchy's stress theore.m (3.3)i and by using_-:·; divergence theorem ( 1.294), which converts the surfac~ integral into a volume integral,-.. -.} we find that ·· ·

/t(x, t, n)ds = / u(x, t)nds = / divu(x, t)dv , (4.51) an an n .. ·. where u :is the sym-metric Cauchy stress tensor. By substituting this result into the·;:) balance of linear momentum (4.44), and using (4 . 38}, (4.40), we may show that.~/.:;: satisfies the Cauchy's .first equation of motion .··'/) : . ·.:·/{ -.:_

/ (divu + b - pV )dv = o n

(4.52)/ ... -::-:-

4.3

145

:Momentum Balance Principles

here presented in the global form. This relation is supposed to hold for any volume v. Hence, we may deduce Cauchy"'s first equation of motion in the local fonn, i.e. divu

+ b = pv

or

(4.53)

for each point x of v and for all times t. The differential equation of motion is here presented with respect to the current configuration. Note that the material time derivative of the spatial velocity field v is, according to (2.26), given as v = av/ at+ (gra
+b = o

or

(4.54)

?->N.hich is referred to as Cauchy's equation of equilibrium in elastostatics. A spatial ·/·s.tress field satisfying divu = o is called self-equilibrated. :·.·...

rlt:XAMPLE 4.3

Show that Cauchy's first equation of motion may also be written in /:?rile important equivalent form

8(pv)

.

- - · = div(u - pv ® v) .~ ·; ·: ·;. ":· ::..:··. :·: ... ·.:

at

:

+b

.

(4.55)

ii;~6Jution.

In order to reformulate the term pV on the right-hand side of eq. (4.53) we :::::_;;W6piy (2.26), the product rule and the continuity mass equation in the form of (4.1.4) to

:Jr:hbtain .

pv

Dv

+ p(gradv)v =

==

p8t

=

a~v) + vdiv(pv) + t

EJ(pv) Dp {) · - ~v + (graclv)(pv) ·t

ut

(gradv)(pv) .

-

· (4.56)

:::with (l .290) we find finally the identity pV =

O~v) + div(pv ® v) ·t

(4.57)

.::::'(Which also comes from eq. (4.21)2 with u rep1aced by v). Substituting (4.57) into !J::::~=~,_:.·~·3) gives the desired result. •

146

4

Balance Principfos

For solid bodies, it is sometimes more convenient to work with the material description. Hence, we rearrange the equations of motion (4.52) and {4.53) in .terms of quantities wh.ich are refeITed to the .reference configuration. To begin with, we introduce the P.iola ·identity, which .is ............ .........

~

__ .. ,

...,,,.,,,.

__ ,,,., ...,.,,.,

. . . . . .. . . .. . ·:" ..

_piy(!~~-~~)·· . · o~

D(.!~4~)

or

= 0 .

(4.58)

D~\:A

.~-~·· ... . . . . .

To prove the important ide.ntity (4.58) we pick any region n0 of a continuum body with boundary surface Bnn and apply the divergence theorem twice. With (1.298), Nanson's formula (2.54) and ( 1.294), we obt~in simply that

Pmol

'/ no

. Div(JF-'1')dV

= .;· JF-'1'NdS imo

= /1nds

\. 1·

. ; nd.s

an

.!

= ~dv = o n

im

.

(4.59)

II

o

Now with this identity, Piola transformation (3.8), the product rule and the symmetry of u we may take the -divergence of the first Piola-Kirchhoff stress tensor P with respect .to the material coordinates, i.e.

, _ Div?. · . Div( J o-F~_'r) = .Div['!{,l_F.'.~~r)] .,, -·,··-···· = .l(Piv«:T)F-T +CT bfv(JFjT) = .J(D•vO')F-T . .

(";<-·,,,,... ... .

'

.

'V"

...··

_,,

---· .. _,,__ ..'~------

(4.60)

•. ··: '•·. ,,.,. ·:·· -C'''·'"'~·- ... --~.-~-,, ..... -

0

By recalling relation

(~.49) ·.........

we obt&;l.ill the·lransformation ... :·'"

.: ...

~

·.:·=

;:

(4.6.1)

or

Combining this resu]t and e-qs. (4.47)2 and (4.40) with Cauchy's first equation-nf motion (4.52) we obtain, after a change of variables and use of dv = Jell/,

/(Di\•P + B - p0 'V)dF no

=o

(4.62)

.

It is the global form of the equation of motion in the reference configuration. Since the volume v (and therefore l/') is arbitrary, we obtain the associated local form, i.e. DivP + B == pnV

or

o~i.4

•..

a:v . + Ba = Po l . .n

(4.63)

..··\A

in which the independent variables are (X.~ t) . The equilibrium counterpart of (4.63) is given simply by setting V~ zero.

{ f. ·~), equal to

4.3

147


Symmetry of the Cauchy stress tensor. The Cauchy stress tensor u is symmetric, as can be seen from the global form of balance of angular momentum (4.45) as follows. Knowing Cauchy's stress theorem (3.3) 1 and the divergence theorem, as given in.( 1.300), we are able to convert the first term on the right-hand .side of eq. (4.45) to a volume .integral according to

Ir xtels= Ir xo-nds = /(r x

diva-+ e : uT)dv ,

iJn

an

.(4.64)

n

where E denotes the third-order permutation tensor introduced in eq . ( 1.143). With (4.:64) and relations (4..39), (4.41) we are now ab.le to rewrite (4.45) as

/ r

X

(pV - b - divo-)d'V = / £: uTclu ,

(4.65)

n

n

Using the equation of motion (4.53) and the fact that the current volume v is arbitrary~ we conclude that

e: U ..

'(\

=

or

0

(4~66)

which holds at each point x of the region and for all times 't. The double contraction UT gives a vector with components CalJr~ac1" which must be zero. We see that

.e:

.

.a3·>- -

·
= ·o

a~n

'

-

0"1.2

=0

.

(4.67)

;· This relation is satisfied, {f and 011/y if the Cauchy stress tensor u is symmetric, i.e. or

a alJ

= a,J.(l

•

(4.68)

.............

The crucial result (4.68) is a local consequence of the balance of angular momen·.· tum (4.45), often referred to as Cauchy's second equation of.motion. From eqs. (3.62) and (3.65) 1 we deduce that the Kirchhoff stress tensor T and the second Piola-Kirchhoff stress tensors are also symmetric. However, from eq. {3.67) we find that the .first Piola. Kirchhoff stress tensor P is, in general, not symmetric as indicated in (3.10). . . . Note that for a polar continuum (resultant couples are not zero) the symmetry prop.;> ~rty does not hold any longer {u ¥- u'I) and therefore eq. (4.68) may also be viewed ·:. · ?s'a constitutive equation (compare with Exercise 5 on p.. 152). .....

:

._:_.:·_·.

an

n

148

4

Balance Principles

In other words, given the Cauchy stress tensor u = u(x, t) (that is defined to be a smooth function of x) and a motion x = x(X, t), then the system of forces (t, b) may be uniquely determined. The Cauchy traction vector t follows from Cauchy's stress theorem (3.3)i, while the body force bis obtained from the equation of motion (4.53). The spatial mass -density p therein results from the continuity mass equation. The pair (u, x) defines a so-called dynamical process, or just a process, for convenience.

EXAMPLE 4.4 A dynamical process ( u, x) is given by the Cauchy stress tensor u in the form of the matrix (4.69)

where a and f3 are scalar constants., and by a motion according to Example 2.2 (see · :z; = e·t . v· .:"" x = ef. ..v· eq. ( ?-··10)) . , 1.e. -·\. 1 - e-t . .,_\.:h -\. 1 + e-t . .,.-.-\ 2 , .rz:a = . i\.i/·3 .i:1.or t > ·0.. 1 2 Find the system of forces (in matrix notation) so that Cauchy's first equation of motion (4.53) and continuity mass equation in the form of eq. (4.10) are satisfied. The Cauchy traction vector t is assumed to act at a point x of a plane tan_gential to a sphere, given by equation

= ~z:i + :z;~ + :r:5.

In order to determine the components of the traction vector t from Cauchy's stress theorem we must find the unit vector .n normal to the surface 4>, which is n = grad /lgradI = x/lxl (see Figure 1.9). Hence, the com_ponents oft take on the values

Solution.

(4.70_) In order to determine the components of the body force vector ht. we rewrite Cauchy's equation of motion so that (b] = p(v] - [divu], which .requires the computation of p, [v] and [divu]. The spatial mass density p follows from the inverse of relation (4 ..10), i.e. p = ,J- 1p 0 , where the volume ratio J is for the given motion (2.10) equal to 2. The components of the spatial acceleration .field are a.·1 = ·u-1 = :1: 11 a 2 = 'lJ2 ::;::: :r:~h a:-J = 'lJ3 = 0 (compare with Example 2.2, eq. (2.14)) and the divergence of the Cauchy stress tensor gives a vector with components (2:1: 1 + o:x~, 2x 2 , 0).

4.3

149

Momentum Balance .Principles

Finally, the components of b are given unique]y as

(4.71)

_.,

which together with (4. 70)a determine the system of forces. ..

, ,,., .. , ... , .. ,,,.,., ............ ,.,,.,,,.,,, .. , ... , .. ,.,,,

II

....... ,.,,, ..... ,., ... , ........ ... , ... , .... ,,,,_ ... ,,,,.,, ... ,.,,, .... , ..... ,,,.,,,.,, .. ,, ..... , ... , ... ,.,, .. ,,.,,,,. ....... , .. ... , .. , .. ,, .. ,.,,,., .. ·... ,,,,,.,.,, .. ,.,,,,, .... ,.,.,, .... , ...,.,,,,,:··''"•'·····"'''•''''•'•''•'·:''''''''···'·'''''''•''':'•''·:···'"''''.. '"' .. ''''"':'·... ,,, ... , .. :•'•"' ,

,

..

·::··''':, , .. ,

.... ; .... , ..

We call a flow a set of quantities, such as the velocity field v, the spatial .mass density p and the sy1nmetric tensor field u, .i.e.. (v~ p, u), which is associated with the system of forces (t, b). To each system of forces consistent with the ·momentum balance principles (4.44) and (4.45) there corresponds exactly one flow and vice versa. We call a flow (v_t p, u) steady if the associated spatial quantities are independent of time, i.e. av I = 0, 8p/ 8.t = 0, Du /8t = 0 and consequently v = v(x), {J = p(x), ( j ·= u{x). Since, for a steady flow, av I Dt: = 0, we have with reference to eq. (2.15

at

n

. V•V

= V .. ry1 g.rad(. .V'""'))

(4.72)

}

iW

where the relation v · w v = 0 is to be used (since w is motion (4.53) for a steady flow has the form v · divu

+v·b =

skew)~

pv · *grad(v2 ) ....,

•

Hence, the equation of

(4.73)

By analogy with the types of motion introduced on p. 68, a flow is said to be a potential flow :if the ve1ocity field is v = gradq>, with a spatial sca]ar field . Then the spatial acceleration ·fi.e.ld a = v has the form of (2. l 56h a.nd the equation of mot.ion (4.53) takes on the form divu + b

1 ·>) = pgrad ( -D + -(grad
(4.74)

A flow satisfying curlv = o is irrotational. By recalling relation (l.274) we conclude that potential .flows are irrotational. If the flow .is steady and irrotational we find that the spatial acceleration field v on the right-hand s.ide of the equation of motion (4.53) is simply (4.75) v. = 1 grac1( v-')) ~

-

9

since av I Dt =

0

and curlv

= o.

Balance of linear .and -angular ·momentum for .an open system. We set up finally the momentum balance principles for an open system, i.e. a selected region in .space with control volume and enclosing control surface 8f2c, independent of time. By

nc

150

4

Balance Prin:ciples

integrating the equation of motion (453) over

nc: we find that

/ divudv + / bch1 nc ne

=

.!

(4.76)

p\rthl .

nc

With divergence theorem (:1.294_) and Cauchy's stress theorem (3.3) 1, the first term in eq. (4. 76) yields, by analogy with (4.5 .I), / divudv

=

nc

.!

und.<; = /

(4.77)

tds .

Dnc

One

The term on the right-hand side of eq. (4.76) follows by integration over the control volume !le

.I

. p\rdv

1· (pv 0 v)ndH . + .1·

=.

n(~

~t

iJ( JV)

dv

He

~ D ;· - • (v · n)pvds + Dt pvclv one nc

I

(4.78)

We have used eq. (4.57)., the analogue of the divergence theorem O.298), ·the fact that the volume is time-independent and rule (L53) 1• Consequently, by substituting eqs. (4.77h and (4.78) 2 into (4.76) we arrive finally at the ha.lance of linear momentum for a control volume in the form

1·

Dt D . pvdv

1·

1· bdv

= . [t - (v · n)pv]ds +.

n(;

Oflc

.

(4..79)

n(!

It states that the rate at which pv .(linear momentum) changes i-~_...the· control volume nc equals the traction vector acting on the boundary control surfac¢ anc of.nc and the flux of {JV entering across o.f2c plus the body force b acting on fie. In an analogous manner we can show the balance of angular .momentum for a control volume, i.e.

~t / flc

r x pvdv = / r x (t - (v · n)pv)d8 + / r x bell! , ant~

(4.80).

n~?

which has a similar interpretation.

EXERCISES

I. Consider a certain point g in the current con.figuration which we refer to as the· mass center of an arbitrary region of a body B with mass ·m.. It is defined·.:

n

4.3

151


(independently of the choice of the point Xo) as

g(t)

= m~n) /

p(x, t)xdv .

n

Considering conservation of mass, ni(f2 0 ) = 11i(H), this relation may be expressed by change of variables as l/1n(!10 ) •/~'0 Po(X)x(X, t:)dl/ . ~

(a) Differentiate g with respect to time and show that the average spatial velocity g and the average spatial acceleration .g of n are given ~y

g(t)

= mtn) /

p(x, t)v(x, t)dv ,

g(t)

n

= ,.1n tn) { p(x, t)v(x, t)dv . .•

.

n

(b) With reference to -eqs. (4.35) l and (4.40) -show that these relations imply L (t) = ,,.n (n) g( t) ,

L(t)

= ni(fi)g(t) =

F(t) ,

which characterize the motion of the mass center and which we already know from Newtonian mechanics (they simple govern the Newtonian par.. tide). Note the following .important property of -the mass center: linear momentum of .a given arbitrary region n of a body B at t is the same as that of a particle with mass nt(fl) concentrated at the mass center of that region. Additionally, the resultant force F of a given region n is equal to the mass of that region multiplied by the acceleration of .its .mass center. 2. A material vector field U is given by its Piola trnnsformation U = JF- 1u. Using the Piola identity (4.58) shows that DivU = ,Jdivu, i.e. in index notation 8UA/8.XA = J8ua/8:ta. 3. Consider the Cauchy .stress distribution for a .continuum in equilibrium given with reference to a rectangular x 1_, ~c 2 , :i:3 -coordinate .system. The components of the Cauchy stress tensor .u are given in .the form

Find the body force ·b that acts on this continuum. 4. The Cauchy stress distribution of a continuum is expressed with respect to a rectangular ·:r. 1 , :1:2 , :i: 3 -coordinate system. The matrix representation of the Cauchy

4

152

Balance Principles

stress tensor u is given in the form

where a and f3 are scalar -constants. (a) Find ·cJ>{i:2 , ~J; 3 ) so that the given spatial stress field satisfies divu

= o.

(b) Consider (i: 2 , :1:a) from (a) and find the C~uchy traction vector t on the p··~_ane

w = ·:r 1 + x· 2 + :1; :i •

5 . ·A polar continuum in motion is .acted on by .resultant couples in addition to the linear momentum pv, the surface tract.ion t and the body force b. Consider a spin angular .momentum p and a body couple c acting over an arbitrary region n and a coup.le traction vector m on its boundary surface

.an.

(a) Show that the presence of couples does not affect the linear momentum principle, and hence the equations of motion ( 4.53) are still valid. (b) Show that balance of angular momentum (4.46) requires that

divM

+ c + E: uT = ·P

.or

oAiad

a:"Cd

+Ca

. + Eabcar.b ·=Va.

': (4.81)

where M is called the couple stress tensor. It is a linear operator that acts on the outward unit ve·ctor field n (perpendicular to an) generating the vector m according to

.m(x, t, n)

= M(x, t)n

or

which is analogous to Cauchy's stress theorem (3.3):i. (c) Show that the three partial differential equations (4.81.) imply that the stress tensor . u is not symmetric.

4.4 Balance of Mechanical Energy In this section we consider only mechanical energy. Other forms of energy, such as thennal, electric, magnetic, chemical or nuclear, .are neglected. For this consideration the balance of energy is not an additional statement to be satisfied, it is a consequence of Cauchy's fi.rst equation of motion (balance of linear momentum).

4.4

Balance of 1"1echanical Energy

153

For subsequent studies we assume a dynamical process given by a symmetric Cauchy stress tensor u = u(x, t) (i.e. a smooth function of x) and a motion x = x{X, t) which de.forms an arbitrary region ton.

no

External mechanical power, ·stress power, kinetic energy.

·we consider a .set of

particles occupying a region n in space with boundary surface an and define quantities first in terms of spatial coordinates. The external mechanical power or the rate of external ·mechanical work 'Pext is defined to be the power input on a region n at time t done by the system of forces (t,b),i.e.

'Pcxt(t)

=It· +I vcls

an

Pe:ct(t)

or

b · vdv

n

= /

(4.82)

tav11 ds + / bavadv .

an

n

The dimension of 'Pext is work per time that is equal to power. As usual, the spatial velocity field is denoted by v = The scalar quantities t · v and b · v give :the exte nzal mechanical power per unit current surfaces and current volume·v, respective.1y. The kinetic energy Kofa continuum body occupying a region n at time tis basically a generalization of Newtonian mechanics to continuum mechanics. We have the definition

x.

JC(t)

!

= . 2I pv2dv = n

1· 2

1 pv · vdv

(4.83)

or

n

The stress power or the rate of internal mechanical work Pint which describes the response of a .region n at time t, done by the stress field, is defined by the scalar

'Pint(t) =

f n

u:

d&u = / tr{uTd)dv n

Pint(t) =

or

f

aabdabdv . (4.84)

n

For a rigid-body motion the stress power Pint is zero since the rate of deformation tensor d, as characterized in eq,. (2.146), vanishes (recall Example 2.12 on p. 99).

Balance of mechanical energy in spatial description. ff only mechanical energy is considered, balance of -mechankal -energy (in the literature sometimes -called the theore.m of power-expended) follows on using eqs. (4 ..82)-(4.84). Thus,

D

Dt /C(t} or

Dn/'l Dt . ..

-

?.pv 2dv +

+ 'Pi11t(t) = 'Pext(t) ,

!

~

n

u : ddv =

1·

, an

t · vds

(4,.85)

+

r

. n

b · vdv .

(4.86)

154


It states that the rate of change of kinetic energy K. of a mechanical system plus the rate of internal mechanical work (stress-power) Pint done by internal stresses equals the rate of external mechanical work (external mechanical power) Pext done on that system by surface .tractions t and body forces b. Hence, the rate of change of kinetic energy JC contains contributions from internal as well as external sources. Note that the kinetic energy K. is not a conserved quantity, since the first term in eq. (4.86) does not vanish, in genera.I. If 'Pext is zero, then we have a problem of free vibration, while if D/C/Dt is zero the problem is called quasi .. static (the associated quantities can still depend .on time).

In order to prove (4.85_) we .look first at the tenn J~m t · vds in eq. (4.86). With Cauchy's stress theorem t = un, divergence theorem ( 1.299) and the product Proo.f

rule (.l .289) we find that

/t ·

vds

nn

= /{un) · vds = an = / (divu · v + u

./ div(uTv)chJ n

: gradv)dv .

(4.87)

n After expansion with the scalar pv · v (where v denotes the spatial acceleration field) we may rewrite the external mechanical power 'P<~xt" i.e. eq. (4.82), as

Pcxt(t) = ./(pV · v + u : grndv)dv + ./ S
n

(4.88)

o

W.ith (2. l35) and decomposition (2.145) we find that u : gradv = u ; I = u : (d + w). Since the Cauchy stress tensor u is symmetric and the spin tensor w is .antisymmetric we find, using property (l .117), the result u : .gradv = u : d. Consequently, with the equation of motion (4.53) we deduce from (4.8.8) that

Pcxl(t) = .l(pV · V + O' : d)dv

1

(4.89)

n

which means that the skew part of the spatial velocity gradient I, ·i.e. the spin tensor w, does not contribute to the rate of work. By v · v = (v: v)/2 and relation (4.34), which allows D(•)/Dt to be written in front of the integral, we conclude from (4.89) that .(4.90)

4.4

"Balance of Mechanical Energy

155

which, by means of eqs. (4.83) and (4.84), is the left-hand side of the ·balance of mechanical energy (4.85), i.e. the rate of change of kinetic energy JC plus the stress power Pint· Ill Next, we introduce a fund.a.mental quamity .at current position x E n and time t which is the sum of all the microscopic forms of energy called the interna·I energy. It is a (thermodynamic) state variable denoted by ec = ec(x, t) (or in other -texts frequently designated by u) and is defined per unit current volume. The internal energy possessed by a continuum body occupying a certain region n, denoted by e' is .expressible as

£(t)

=/

ec(x, t)dv .

.(4.91)

n The term 'internal energy' was introduced by Clausius and Rankine in the second half of the nineteenth century. Since only mechanical energy is considered, we state that the rate of work done on the continuum body by internal stresses, i.e. the stress power Pint' ·equals the rate of internal energy E. We write

Pint(t)

D

= -0 t t'(t) .

(4.92)

.

Now, eq. (4.85) may be ·expressed in terms of the internal energy, i.e. (4.93) or, when written in the explicit form,

1·

1·

2

!

D ( I pv +ec)dv=. t·vds+. h·vdv. Dt. 2 n an n

(4.94)

The term on the Ieft""'.hand side in the form

•

f

1

')

( -? pv.A.I

. + Cc ). dv

(4.95)

n

characterizes the tota:J energy. It is the sum of the kinetic and internal energies . Note that the contributions to the total energy are only due to external sources, i.e. the external mechanical power Pext· Jn order to express the terms of the balance of mechanical energy (4.85) with respect to material ·coordinates at time t, we ·must establish the ·external mechanical power Pext' the kinetic energy K and the stress power 'Pint .in the material description.

Balance of mechanical energy in materhil descri.ption.

156

4 ·nalance P-rinciples

By means of relation (3.l) .and identity (2.8), the ·first term on the right--hand side

of eq. (4 . 82) transforms into

ft·

f

vds =

T · VdS ,

(4.96)

an

ano and with the local form (4.48), dv == JdF and (2.8) the remaining term in eq. (4.8.2) gives

f

f

b · vdv =

(4.97)

B · VdV .

·n no Hence, the external mechanical power Pext can be obtained as

Pext {t)

=

I

. ano or

P~xt(t) =

f

T · V dS +

f T..i

no

v.1dS +

B · V dV

f

(4.98)

BA V.idV

ano no With identity (2.8) and .conservation of :mass (4.6) the kinetic energy JC is, in view of {4.83 ), given as . 1

r 2

K(t) = . 2Po V dV / no K(t)

or

=

I

=.

1

,.

2Po v. VdV

no

J~Po l~1

(4.99)

l-':.1cJll .

no Next, we write the rate of internal mechanical work (stress-power) Pint in terms of the first Piola-Kirchhoff stress tensor P. For that, recall eq. (4.84)h the additive decomposition of the spatial velocity gradient, Le. I = d + w = :FF- 1 and conclude that the (skew) spin tensor w acting on the symmetric Cauchy stress tensor -u yields zero (.u: w ~ 0) . Hence, by means of the property (1.95) and eq. (2.50) we Jind that

fer: ddv = .fu: (FF- )dv 1

Pint(t) =

n

n

.!

uF-T: Fdv = ./ JuF-T: FdV (4.100) n no With the Pio]a transformation (3.8) and manipulations according to (1.93) 1 we obtain from (4 . .I 00) 4 the equivalence for the rate -of internal mechanical work, i.e. 1'int(t)

=

f

P : Felli

no or

Pint(t) =

=

.! no

.!

tr{PTF)dlf

no PaAP1v1dV

(4.101)

4.4

157

Balance of Mechanical Energy

Substituting (4.98), (4.99)i, (4.1.01) 1 into (4.85), we have -finally, equivalently to (4.86), the balance of mechanical energy in the material description, .i.e.

~ ;· ~fJo V 2d\! + Dt.

-

.

f P : FdV = ;· T · VdS + .;· B · VclF ~

.

(4.102)

no no
=

.!

ec(x, t)dv =

n

.!

ec(x(X, t), t)J(X, t)d'V

no (4. .103)

./ e(X, t)dV no with the important transformation

e{X,t) = J(X,t:)ec(x;t) .

-(4.104)

Adopting the expression (4.99)t :for the kinetic energy IC and (4.103)a for the internal energy E.., we may write the total energy J~(pv 2 /2 + ec)dv in the mate.rial description as

(4.105). Using the equivalence of eqs. (4.82) and {4.9.8) and the total energy in the form of (4.105) we obtain finally from (4. 94) the desired result

gt /(~poV2 + e)dV == / ~

T · VdS

+

.!

B · Vdl/ ,

(4.106)

~

OOo

which may -directly be -derived from the balance of mechanical energy ( 4~ I 02). This requires eq. (4.92) with (4.101) 1 and (4 . 103)a, leading to the transformation /~ P : '

.ll{)

Fd1/ = D/Dt J~20 ed1/.

Alternative expressions for ·the -stress power.

.In the following we write the stress

power 'Pint in different equivalent versions. We start from eq. (4.84) 1 • By using (2.50)., the pull-back of the rate of deformation tensor d, .i.e. the inverse of relation (2.163) 4 , the rule (1.95) and the stress relation (3.65)t we find Pint for a region r2 0 to be

Pint(t} =/Ju: ddF

=I

no

no

Ju: F-TEF- 1dF

.!

/ .JF- 1uF-T: EdV = S: EdF no no

(4.107)

158

4

Balance Principles

with the second Pio.la-Kirchhoff stress tensor S and the material strain rate tensor E. . Another alternative form for the stress power Pint may be obtained by use of C = 2E (see eq. (2.l66)i) .and the non-symmetric .Mandel stress tensor :E = CS wh.ich is de.fined in an intermediate configuration. Starting from eq. (4 . .107h we have, by means of Cl .95), (4.108)

where the symmetry of the right Cauchy-Green tensor C is to be used. Next, we recaU the rotated Cauc.hy stress tensor u 11 = RTuR (see eq. (3.70h), the rotated rate of deformation tensor DR = RTd.R (see eq. (2.167) 1) and the relation .R- 1 =RT. Then, from (4.84}1 we find, using eq. (2.50), that

'Pint(t) =/JU': ddV = / JRTo-R: R- 1dR-Tc]l/

no

no

=/

lU'u : DadV ,

(4.109)

no where the integrand has been manipulated .according to

EXAMPLE 4.5 and show that

CI .95).

Express the stress power Pint in terms of the Biot stress tensor Tn

Pini(t)

=/

P: FdV

nu

=/

symTu: t'.klV ,

(4. ·110)

no

where the notation sym( •) is used to .indicate the symmetric part of T 13 ., and U is the material time derivative of the right stretch tensor U, which is symmetric (compare with Section 2 ..6). Solution. Starting from (4.101) 1 and using the polar decomposition F product rule of differentiation and the identity RTR =I, we have

.! no

p : FdV

=/ no

= RU, the

p: [R(RTR)U + Rt'.J]dlf

.!

/ p: RRTFdV + p: Rt'.JdV no no

(4.1 H)

With the definition of the symmetric Kirchhoff stress tensor r = PFT (see Sect.ion 3.4) and the Biot stress tensor Tu = RTP (see eq. (3.68)), we Jind from eq. (4.11 ·1 h using

4.4

Balance of Mechanical Energy

159

manipulations .according to (1.95) that

/ P : Fc1v = / PFT : RRTdv + / aTp : Uc1v no no flo

/ r : RRTdV no

+/Tu :

iicIV

(4..112)

no

Knowing that r is a symmetric tensor and RR'T' is a skew tensor accordin_g to (2.158), we conclude by analogy with property ( 1.1.17) that r : RRT = 0, consequently the first term in (4.112}~ vanishes. Since U is a symmetric tensor the second term gives the desired result (4.11 Oh. II

In summary, the alternative expressions for the .rate of internal mechanical work Pint a.re the relations (4.1.07_)i, (4.IOl)i, (4.107) 4., (4.108h, (4.l09h and (4.1 lOh, and we have finally the important identities w;n 1 (t)

= P: F. = S: E. = 'E: 2.1 c -1 C. : Dll = sy1nTn : U .

= Ju : d

= Juu

(4."I 13)

The double contraction of a stress tensor and the associated rate of deformation tensor (:for example, P : F) describes the re'll physical power during a dynamical process, i.e. the .rate of internal mechanical work (or stress power) per un.it reference volume, denoted by w 1nt. In this sense the stress fields Ju, P, S, ~, Ju u, syn1 Tu are said to be work conjugate to the strain "fields d, F, E, c~ 1 c/2, D1t, U, respective·ly. Hence, . for example, P and F is said to be a work COJ\jugatc pair (for a more comprehensive survey of work conjugate pairs see, for -example, ATLURI [1984.J). Conservative system. Let the scalar-valued functions Ilext .and flint be called the potential energy of the external loading (or the external potential energy) and the total strain energy (or the internal .potentia~ -energy) of the body, respectively. The sum of IIl~xt and TI int we cal I the total potential energy (or frequently referred to as the energy functional) I1 of the mechanical system_, i.e. (4..11.4)

A mechanical system. is known as conservative if the external mechanical power Pffxt expended on a certain reg.ion of the body and the internal mechanical -power P1nt

·are expressible as Pc::.t(t) = -

DTI(~Xt ("t)

Dt

.

(.

= -Ilc::.t t) ,

(4.U5)

160

4 Halance Principles

with

Ilim(t)

=I

WdV , (4.116)

no where \JI represents the s·tra·in-energy function (referred to briefly as the strain -energy or stored energy) defined subsequently per unit reference volum.e rather than per unit mass .and dealt with in Section 6J in more detail. In the literature the strain-energy function is sometimes denoted by nr . By means of the assumptions (4.115) and (4.116), the balance of mechanical energy (4.85) implies that the sum of the total potential energy IT and the kinetic energy K is conserved (constant) during a dynamical process (u, x). We write Ilexi (t)

+ Ilint ( t) + IC( t) = canst

~

(4.117)

.It is important to note, that, for example, most surface tractions and al.I problems associated with dissipation of energy (due to external or intenrnl friction, viscous or pfastic effects) lead to non-mechanical energies which are non-conservative in Lhe sense that they cannot be derived from a potential. EXERCISES

1. lf only mechanical energy is considered, show that the energy equation is merely Cauchy's first equation of motion. Thus, take the dot product of each term of divu + b = pv with the spatial velocity field ·v and integrate the result over the current region .n with volume v. 2. A hydrostatic stress state at a certain point is given by the Cauchy stress tensor in the form of u = -pl. (a) Show that the stress ._power Wint per unit referential volume is given by

·. \ \Vinl

.

= JCT : d =

JDp ]J-

µ

Dt

·

(b) Formulate the balance!~of mechanical energy for this particular stress state and conclude that if the considered motion is isochoric and if there exists no external mechanical power, the kinetic energy· is conserved (K = const). 3. A rigid-body is rotating about a fixed point 0 with angular velocity w. Show that the kinetic energy may be expressed as

JC(t) =

~w · Dw

,

D

=I

p((x · x)I - x ® xjd'IJ ,

n where the current position of an arbitrary point in the rig-id-body is characterized by x. Note that.Dis called the lnertia tensor of the body relative to 0 occupying a region n.

4.5

4..5

Balance of Energy in Continuum Thermodynamics

161


In this section we consider both mechanical and thermal energy which are essential in many problems of physics and engineering. The effect of other energies (such as electric, chemical ... ) on the behavior of a continuum is a topic beyond the scope of this text. In a thermodynamic context the conservation of mass and the momentum balance principles are supplemented by the balance of energy and the entropy inequality law. The development of the concepts of energy and entropy has been one of the most important achievements in the evolution of physics. For an extens·ive account of these conce.pts see, for example, TRUESDELL and TOUPIN [ 1960, Chapter E], the two-volume work of l<.ESTIN [1979], or SILHAVY [.1997] and W:rLMANSKI [1998]. In this section we discuss briefly the balance of energy equation in a thermodynamic context. To begin with, we :introduce some important terminology common in thermodynamics.

General notation. A continuum which possesses both mechanical and thermal energy is called a thermodynamic continuum. We say that the thermodynamic state or the condition of a sys.tern is known if all quantities throughout the entire system are known. All quantities characterizing a system .at a certain state are cal.led thermodynamk state variables. They are macroscopic quantities and, in general, they depend on position and time. For example, the thermodynamic state of a thermoelastic solid can be represented in a seven-dimensional state space with six variables corresponding to the strains and one to temperature. The function that describes a certain state yariable is called a thermodynamic state function. Any equation that interrelates state variables is called equation of state or constitutive equation. Any change that a system undergoes from one thermodynamic state to another is called a thermodynamic process. The path that connects the two states is parameterized by time t. If a system returns to its initial state at the end of a thermodynamic process this system is said to have undergone a so-called cycle (initial and final states are identical). A non .. equili=brium state is a state of imbalance (there exists a gradient of temperature and velocity), while an equilibrium state is a state of balance (of uniform temperature and zero velocity). To study non-equilibrium or equilibrium states is a central goal of continuum thermodynamics. A system within equilibrium has no tendency to change when it .is isolated from its surroundings. If there is no change in the values of the state variables at any :particle of the system with time we say that this system is in thermodynamic equilibrium or ~hermal equilibrium. The transition from one state of thermodynamic equilibrium to another is studied in thermostatics. A process in a system that remains close to a state

162


of thermodynamic equilibrium at each time is referred to as a quasi-static process or quasi-equilibrium process. A quasi-static process is a sufficiently slow process whereby enough time remains for the system to adjust itself internally. It is basically a process during which the system is in equilibrium at all times; the contributions due to dynamical quantities are negligible. Heat is the form of (thermal) energy that is transferred between a system and its surroundings (or between two systems) by virtue of a temperature gradient. The term 'heat' is understood to mean heat transfer in thermodynamics.

Thermal power. Let the thermal (non-mechanical) power or the rate of thermal work be denoted by Q and defined by

f

rd'U =

== -q(x, t) · n

or

= -Q(X, t) · N

or

qn (x, t, n)

QN(X, t, N)

qnds +

f

f

QNdS + ./ Rdlf , (4.118) an n ano no which is represented in the spatial and material descriptions, respectively. The time-dependent scalar functions qn and QN denote heat fluxes, determining heat per unit time and per unit current and reference surface area, respectively. The total heat fluxes ./~n qnds and fano QNdS measure the rate at which heat enters (inward normal flux) the body across the current and the reference boundary surfaces 80. and ano, respectively. The time-dependent scalar fields r = r(x, t) and R = R(X, t) in eq. (4.118) denote heat sources per unit time and per unit current and reference volume, respectively (see Figure 4.3 ). A heat source is a reservoir that supplies energy in the form of heat. The total heat sources Jn rdv and fno Rdlf measure the rate at which heat is generated (or destroyed) into a certain region of a body. The counterpart of Cauchy's stress theorem (3.3) in continuum mechanics is the Stokes' heat flux theorem in thermodynamics. It postulates that the scalar functions q11 and QN are linear functions of the outward unit normals so that

Q(t) =

qn

= -q n" 0

,

}

(4.119)

The time-dependent spatial vector field q = q(x, t) is the so-called Cauchy heat flux (or true heat flux) defined per unit surface area in n, and n is the outward unit at the current position x. normal to an infinitesimal spatial surface element ds E The Piola-Kirchhoff heat flux (or nominal heat flux) and the outward unit normal to an infinitesimal material surface element dS E 0 at X are denoted by the vectors Q = Q(X, t) and N, respectively. The time-dependent material vector field Q determines the heat flux per unit surface area in no. The negative signs in eqs. (4.119) are needed (see Figure 4.3) because the unit vectors n and N are outward normals to an and an 0 , respectively. However, we claim in (4.118) that heat enters the body (inward normal flux).

an

an

4.5


163

n

Figure 4.3 Heal flux vectors q, Q and heat sources r, R.

In order to relate the contravariant vectors q and Q to one another, we equate the total heat fluxes J~no QNdS and Jan qnds. Hence, with the fundamental Stokes' heat flux theorem (4.119), Nanson 's formula (2.54) and manipulations according to ( 1.95) we obtain

.!

Q · NdS

=

Dno

.! .!

q · nds =

q · JF-TN
ano

an

=

.!

.JF- 1q · NdS .

(4.120)

ano Comparing the left and right-hand sides of (4.120) we conclude that the Pio la-Kirchhoff heat flux Q may be related to the Cauchy heat flux q. We have, by analogy with (3.8),

Q = JF- 1q = Jx; 1(qP)

or

CJA =

JF;\:q(l '

(4.121)

which may be viewed as the pull-back operation on the contravariant vector q~ by the inverse motion x- 1 scaled by the volume ratio J. Such a transformation we called the Pio/a transformation, here relating the heat flux vectors Q and q (compare with the definition on p. 84 ).

164


First law of thermodynamics in spatial description. Here we consider the case in which thermal power is added to .a thermodynamic continuum. Thus, the rate of work done on the continuum body, i.e. the sum of the rate of internal nzechanical work Pint and the rate of thermal work Q, equals the rate of internal energy e. Consequently, eq. (4.92) becomes (4.122)

which is often referred to as the balance of thermal energy. By substituting eq. (4.122) into (4.85) we deduce the important identity, namely

\. . D. . · ·-"· · · · · · · · · · · ·i;-······ . . ·.·. · · · . · ·-· . . . . . . . . . . . . . .

~·--

········. · · · · · ·.:

\ n/C(t) + n/(t) = 'Pext(t) + Q(t}_j .,_

.. . . ....

...

..· .,.,, ... ,, , ....• ~ ..... , ,.

..... ..... ,

, ...,.,,,.,-.; ~·· ~,,,: :,.,,~,,,,.,,

.....

.....___.,~

.... ,,,,

~

...... ,,,,, .... ·: .... ·', .. ,

_.,.,

(4.123)

~·.

Using the explicit expressions for JC,£,, which are given by eqs. (4.83), (4.91), and for 'Pext, Q, which are given by . eqs. (4.82), (4.1 l8)i, we may write the global form

!(.1

D · 2pv-·> + ec)
n

1· (t ·

=.

oo

v

+q

11

!

)cls +. (b · v

+ r)dv

.

(4.124)

n

It is a thermodynamic extension of eq. (4.94) and postulates balance of energy (mechanical and thermal), a fundamental axiom in mechanics, known as the ·first ..Jaw of thermodynamics. In particular, eq. (4.1.24) is the first law of thermodynamics in the spatial description. It states that the rate of change of total energy (kinetic IC and .internal energy f.) of a thermodynamic system equals the rate at which external mechanical work (external mechanical power) Pext is done on that system by surface tractions and body fore-es .plus the rate at which thermal work Q is done by heat fluxes and heat sources. The first law of thermodynamics governs the transformation from one type .of energy involved .in a thermodynamic process into another, but it never governs the direction of that energy transfer. In order to express eq. (4.122) more -explicitly we rewrite the thermal power Q first, i.e. (4.1.18) 1 • By means of the Stokes' heat flux theorem (4.119) 1 and the dive-rgenc-e theore.m .according to eq. (l.293), we may deduce that Q(t) = fn(-divq + r)dv. By re-calling the rate of internal mechanical work Pint and the internal energy E from eqs. (4.84) 1 and (4.91), we obtain

D ;· ecdv = ;· (er : d - clivq + r)dv , -· D't, n

~

(4.125)

n

which is a reduced global form of balance of energy (4.124) in the spatial description.

4.5

Balance of Energy ·in Continuum Thermodynamics

165

First law of thermodynamics in material description. In order to rewrite balance of energy (4.124) in terms of material coordinates, we recall the equivalent forms of the total energy, the externa] mechanical power 'P(~xt and the thermal power Q, that are given by eqs. (4.95), {4.105), and (4.82), (4.98) and (4. I 18), respectively. Thus, the first law of thermodynamics in the material description reads

gt/(~poV2 +e)dF= /(T·V+QN)ds+/(B·V+R)dF flo

Oho

.

(4.126)

flu

Following arguments analogous to those which led to (4.125), we find the reduced global form of balance of ·energy in the material description, i.e.

g1· t ~

ed1/ := ;·(P: .

-!lo

F-

DivQ + R)cnr .

{4.127)

no

I.n order to achieve the local form we must rewrite the term on the left-hand side of eq. (4.127). Since the reference volume 1/ is independent of time we may write D /Dt J~20 e dl/ = J~ 20 edl. . . Note that the volume is arbitrary, leading to the local form of the balance of -energy .in the =material description

e= P : F -

DivQ + R

or

(4.128)

presented in symbolic and index notation, .respectively. The first-order partial differential ·equation ·is due to Kirclzho.fl and holds at any particle of the body for all times.

EXERCISES

l. Assume a continuum body not subjected to body forces and heat sources. Establish the balance equations

!( 2

D Dt.

l r>v-.,

n

1·

!

+ ec)clv = . (uv -

D (2'p l 0 V:..> + e)dV Dt. no

)

q · nds ,

an

1·

=.

(P"I' V - Q) · NdS .

/Jiln

These relations -equate the rate of -change of the total energy of a continuum body occupying a certain region with the total flux out of the boundary surfaces of that region. The vectors uv - q and pTv - Q are often referred to as the energy flux vectors with respect to the -current and .reference configurations, respectively.

166


2. Starting from (4 . .124), show that the local spatial form of the balance -equation has the form

a ( 2pvi ,, . + ec )· = chv[uv -

Dt

1 ., . q - (2'pv- + ec)v1+ b. v + r .

Hint: Use eqs. (2.50), (2.25), (2.175)n and the rule .according to (I .287) for the.

left-hand side of (4.1.24). 3. Derive the "local form of the balance equation in the spatial description -corresponding to eq. (4.128) and note that we have used the internal energy per unit volume rather than per unit mass (see eq. (4.91)).

4.6

Entropy Inequality Principle

The first law of thermodynamics governs the energy transfer within a thermodynamic process, but is insensitive to the direction of the energy transfer. In the .following we derive a much finer principle, that is the second law of thermodynam.ics, which is respo11sible for the direction of an -energy transfer process. We introduc·e the concept of entropy and discuss some of the consequences of the second law of thermodynamics. Second -law of thermodynamics. Physical observations show that heat always flows from the warmer to the colder region of a body (free from sources o:f heat), not vice versa; mechanical energy can be transformed into heat by friction, and this can never be converted back into mechanical energy. We introduce a fundamental state variable, the entropy (c01ning from the Greek words i11 and Tpomx» meaning 'in' and 'turning, direction', respectively). It is an important thermodynamic property first described in the works of Clausius .in the second half of the nineteenth century. The entropy can be viewed as the quantitative measure of microscopic randomness and disorder (see, for example, CALLEN [1985, Chapter 17]) . A physical interpretation ·is provided by the subject of statistical mechanics. For a more detailed physical interpretation the reader is referred to Section 7. l .of this text. We introduce the notations T/c = T/c(X, t) and "7/ = ·11(X, t) for the entropy per unit cur.rent and reference volume, respectively, at a certain point and time t. I.n other texts the entropy is frequently designated by s or S . The entropy possessed by a continuum body occupying a certain region, denoted by S, is defined to be

S(t) =

f n

·11c(x, t)c:lv

=

f no

11(X, t)dV ,

(4.129)

l67

4.. 6 Entropy Inequality Principle

with -17(X, t) = J(X, t)rJc(x, t), which should be compared with the analogues of (4.1.03) and (4.104), respectively. This definition .is commonly introduced in continuum mechanics and differs from the definitions used in statistical mechanics . Let the rate ·of entropy input into a certain reg.ion of a continuum body consist of the value of entropy transferred across its boundary surface and the entropy generated (or destroyed) inside that region. We denote it by Q and write

j h · ncls + ./ hlv = -

Q(t} -:-- -

on

n

./ H · NdS + ./ R
(4.1.30)

no

The time-dependent scalar fields f r(x, t) and R = .R(X.., t) denote entropy sources per unit time and per unit current and reference volume, respectively. The total entropy sources J~ 'fdv and ./~ 20 Rdv· measure the rate at which entropy is generated (or destroyed) into a region of a body. The time-dependent vector fields h == h(x, t) and H = H(X, t) determine the Cauchy entropy flux (or true entropy Hux) defined per unit current surface area .in .n and the Piola-Kirchhoff entropy Hux (or nominal entropy flux) per unit reference surface area in :.n 0 , respectively. As usual, the ·Outward unit normals to the infinitesimal surface elements ds E (JO, at x and dS E DQ 0 at X are denoted by n and N, respectively. Since we have defined the unit vectors n and N to be outward normals to 80 and an 0 and since we compute the rate of entropy inplll (entropy entering the body}, we need the negative signs in relation (4.130). The .difference between the rate of change of entropy S and the rate of entropy input Q into a body determines the total production of entropy per unit time, which we denote by r. We postulate th.at the total entropy production for all thermodynamic processes is never negative, following the mathematical expression

r(t)

D

.

= n-/(t) -

Q(t) > o ,

(4. l.3l)

which is known as the second la''T of thermodynamics. Unlike the considerations in previous sections, it is an inequality rather than an equation often referred to as the ·entropy inequality principle. ·More explicitly, with eqs. (4.129) 1 and (4.130) 1 we find from {4.131) the global spatial form of the second law of thermodynamics in the notation of continuum mechanics, Le.

r(t)

= -0D

1·

t '

n

·1Jc(X, t)dv

+

1·

' lJn

h · nds -

1·

Pdv > 0 .

(4.132)

'

n

These relations assert clearly a trend in time by describing the dire.ction of the energy transfer and postulating irreversibility of various thermodynamic processes.

.168


The second law of thermodynamics is not a balance principle. It indicates a trend in both living and inanimate systems, where the situation r < 0 never occurs (it would mean that molecules organize themselves globally). Unlike the mass or the energy, in general, the entropy is not a conserved quantity, i.e. r > 0. A thermodynamic process is called reversible if it .is not accompanied by any entropy production, i.e. r = 0. For each cycle the material response returns to its initial state. A reversible process is a very useful idealized .limit of a real process. The associated state of a reversible process is in equilibrium; thus, the equal signs hold in (4~ 13 l) and (4.132). Reversible processes belong to the realm of equilibrium thermodynamics, also known as reversible thermodynamics which is (classical) thermostatics . A real process js irreversible and characterized by (4.13 l) and (4 ..132) in which the strict inequalities hold. This indicates that the rate of change of entropy Sis always greater than the rate of entropy input Q. .Irreversible processes are always associated with dissipation of energy studied in the realm of non-equilibrium .thermodynamks, also known as .irreversible thermodynamics. For the concept of different thermodynamic theories the reader is referred to the review article by HUTTER [1977]. Clausius .. Duhem inequality. The rate of entropy input "is often closely related to the rate of thermal work (the total heat fluxes and sources). Very often the -entropy fluxes h, Hand the entropy sources f ~Rare assumed to be re.lated to the heat nuxes q, Q and heat S·OUrces r, R by the proportional factor 1/8. Thereby e = 8(x, t) > 0 denotes a time-dependent scalar field known as the absolute temperature. The corresponding·· unit of temperature is called a Kelvin, denoted by K, which always has pos·itive values. However, in practice 8 is often measured in 'Celsius temperature' or 'Fahrenheit temperature' for which the unit .is called the degree Celsius °C or the degree Fahren.. hcit °F, respectively. lt is important to note that from the thermodynamic point of view the Celsius scale and the Fahrenheit scale are not true temperature scales at all. The temperatures on these scales can be negative, the zero is incorre·ct and temperature ratios are .inconsistent with those required by the thermodynamic principles. We postulate -

r

r= -

8

-

and

R

R= 8 ,

(4..133)

but note, however, that these relations are~ for example, inconsistent with the consequences of the kinetic theory of ideal gases· and therefore .not valid for the thermody . . namic.s of diffusion. With re-lations (4.1.33) and eqs. (4J.29) and (4.130) we find from the second law of thermodynamics in the form of (4.1.32) that

n;·

r(t) = Dt. TJc
~ . ncls - ./& ~ dv > 0 n

'

(4.134)

4.6

r(t) ==

169

Entropy l.nequality ·principle

gt / T]dV + .! ·~ .NdS - .! ~cw > 0 no

ano

1

(4.135)

no

which is known as the Claus.-ius-Duhem inequality here presented in the spatial and material descriptions, respectively. The Clausius-Duhem :inequality is widely used in modern thermodynamic research initiated by TRUESDELL and TOUPIN [1960] and first clearly studied in the work of COLEMAN and NO.LL [1963.]. In order to derive the local :form, which we only present in the material description, we convert the surface integral in -eq. (4.135) to a volume integral according to the divergence theorem (1 . 297) . By the .product rule (.1.287) we obtain

/ ano

~ · NdS =/Div(~) dll = / ( ~DivQ- ~2 Q ·Grade) dV no

, (4.136)

no

where Grade denotes the material gradient of the smooth temperature field 8. By substituting eq. (4.136)2 back into (4.135) and noting that the reference volume ll .is arbitrary and .independent of time, the local.form of the Clausius-Du hem inequality in the material description reads, in symbolic and index notation,

i7 - e R + .!.oivQ - _!_Q e e 2 ·Grade>- o or

. . .R q- e

aQ .4 1 Q ae > 0 + eax. . - e 2 AaxA -

,

(4.137)

i

·

An alternative local version results, for example, from (4.137) by elimination of the heat source R by means of eq. (4 . .128)~ leading to 1

~

e+ e,; - e Q · GradE> > o ,

P :F -

or

PaA r-cin.4

-

.

e+

e· - TJ -

o . ei QA aae X..1 >

(4.138)

The last term in eq. (4.1 J8) determines entropy production by conduction of heat.

Clausius-Planck inequality and heat conduction. .Based on physical observations heat flows from the warmer to the co.Ider region of a body (free :from sources of heat), not vice versa. Hence, entropy production by conduction of heat must be non-negative, i.e. -(1/8)Q ·Grade > 0 (see the last term in (4.138)). The spatial version of this condition reads -(1/B}q ·grade > 0, where grade de·notes the spatial gradient of the smooth temperature field e. Knowing that 8 is non-negative, we have q · -grad8

<0

or

ae Qa a'l'

~'a

Q ·Grade<

o

or

<0 '

. ae < 0 QA8':' J'\. .-'1

(4.139)

(4. .140)

170

4 ·Balance Principles .

which .is known as the classic.al heat conduction inequality, here presented in terms of spatial and material coordinatesT respectively. The heat conduction inequaHty expresses that heat does ·flow against a temperature gradient It imposes an essential restriction on the heat flux vector. Obviously the Cauchy heat flux -q depends on the spatial temperature gradient. With restriction (4.l 39) we may deduce that q(grad8) · grade < 0 holds identically if grade = o. Therefore, the dot product takes -on a local maximum at grade = o and its derivative must vanish. Consequently, q = o is implied, which means that there is no heat flux without a temperature gradient. According to restriction (4.1.40), the Clausius-Duhem .inequality (4.138) leads to .an alternative strongerform of the second law of thermodynamics, often referred to as the Clausius . . PJanck inequality, i.e. or with the inte~al diss~pation or local production of -entropy 'Dint > 0, which is required to be non-negative at any particle of .a body for all times. The non-negative internal dissipation V 1nt consists of three terms: the work conjugate pair P : F, i.e. the rate of internal mechanical work (or stress-power) per unit. reference volume, then the rate uf internal energy, e, and the absolute temperature ·multiplied by the rate of entropy, The internal .d.iss.ipation is zero for reversible processes while ~he inequality holds for .irreversible processes. By means of the Clausius-Planck inequality (4.141 )" the local form of the balance of energy (4.128) can be rephrased in the convenient entropy form

-e.,;.

8i1

= - DivQ + 'Dint + R

aq~l e-r7. = - a ~-,. + Vint + R , _•\. ...4

or

(4.1.42)

in which the local evolutia.n (change) of the entropy 11 appears explicitly. A suitable constitutive assertion which relates the Cauchy heat ftux q = q(x, t) in x = x(X, t) to the spatial temperature gradient grad8 = grad8(x, t) .is furnished by

q = --K,grad8

or

(4.143)

This classical (phenomenological) law is ·motivated by -experimental observations and is known .as Duhame-J's law of :heat conduction, here presented in the spatial descrip--

tion. The symmetric second-order tensor "' denotes the -spatial therma:J -conduct,ivi-ty tensor; its components Kab = fb·ba are e.ither-cons-tants or functions of deformation and temperature. Substituting Duhamel's law of heat conduction (4.143) into restriction (4..l 39) we obtain K-grad8 · _grad8 > 0. This implies that"" is a positive semi-definite tensor at each x.

4.6

·EXAMPLE 4.6 ordinates,

171


Derive Duhamel's law of heat conduction in terms of material co-

Q\,'A --

or

-

p- .l ..._~

. AJ3'"'0 !JC

p-1

ae

DC !.."l v·

U ..·"\J)

'

(4~144)

where F- 1 is the inverse of the deformation gradient F according to eq. (2.40) and Ko = J K denotes the positive semi-definite material thermal conductivity tensor~ The volume ratio is J = detF > 0.

Solution.

In order to show ·eq. (4.144) we ne·ed an expression between the spatial and the material temperature gradients. By using the absolute temperature 8 instead of in relation (2.4 7) we find that (4.145)

Subsequently, by recalling the Piola transformation (4.121 _) for the heat ·flux vector we obtain from e-q. (4.143)

(4.146)

which after recasting gives the desired result {4J 44). Sinc.e

~o

.is a positive semi-

definite tensor the imposed restriction on the inequality (4 . .I 40) is satisfied.

•

lf K is an isotropic tensor we say that the material is thermally isotropic (no preferred direction for the heat conduction). For such a type of material the -conductivity tensors become "" = k I and ~ 0 = k0I. These relations substituted into (4.143) and (4.144) give

= -kgTad8

or

CJa =

Q = -koC -"J·Grade

or

Q;t =

q

c-

ae

(4.]47)

-k-a ' 1:a

ae

-koC;tkav· , "'·"\. fJ

(4.148)

1 where = F- 1F-T characterizes the .inverse of the right Cauchy...;Green tensor C. The scalars k > 0 and k 0 = Jk > 0 denote coefficients -of thermal conductivity (constants or, in general, deformation and temperature-dependent) and .are naturally associated with the current and reference configurations of a body, respectively. The conditions k > 0 and k 0 > -0 imply that heat .is conducted in the direction of decreasing temperature. The relations (4.147) and (4.148) .are well-known as Fourfor's law of heat conduction which are basically constitutive relations.

172


Types of thermodynamic processes. Finally, we formulate special cases of thermodynamic processes and introduce some common terminology. If the heat flux qn = -q·n across the surface.an of a certain region n (QN = -Q·N on 8f2 0 ), and the he.at source r in that region n (R in r2 0 ) vanish for all points of a body at ·each time, then a thermodynamic process is said to be adiabatic (coming from the Greek word a&u5f3o:roc;·, which means impassable). This definition is based on the work of TRUESDELL and TOUPIN [1960, Section 258]. However, in engineering thermodynamics an adiabatic process is frequently defined as a process which cannot involve any heat transfer, qn = 0 (or CJN = 0). This means that the heat source r (or R) need not be zero . .In our case an adiabatic process is based on the condition that thermal energy can neither cross (enter or leave) the boundary surface nor be generated or destroyed within the body, so that the thermal power Q is zero (compare with eq. (4.118)). Additionally, under the assumption of relations (4.133) the rate of entropy input Q is also zero (compare with eq. (4.130)). Consequently, the second law of thermodynamics .in the form of (4.131) reads r = S > 0, wbich means that the total entropy S cannot decrease in an adiabatic process. Nevertheless, the entropy TJ at all points and times is not necessarily non-decreasing. For an adiabatic process the energy balance equation, for example, in the entropy form (4.142), degenerates to (4.149) .If, in addition, an adiabatic process is .reversible (no entropy is produced entailing that S is zero), the balance equation (4J 49) reduces further to

er = 0), (4.150)

which bas· some important applications in thermodynamics. If the absolute temperature e .during a .thermodynamic process remains constant (8 = 0)~ the process is said to be isothermal and if the entropy .S possessed by a body ,. remains ·constant (S = 0), the .process is said to be isentropic. For an isentropic process the second law of thermodynam.ics (4.131) takes on the form r = - Q > 0. Hence, the local form of the Clausius-Duhem inequality in the material description (4.137) in symbolic and index notation degenerates to

-R+DivQor

aQ.4

-R+ .a~_.,. ..l\.A

1

8 i

8

Q·Grad8? 0 ,

ae.

QA ,,...

8..."\A

(4.15.1)

>o .

If an is·entropic process is reversible, the rate of entropy input Q (and the thermal power Q via assumption (4.133)) is zero since there is no entropy production r anymore. For this special case the .equal signs in (4.15.1) hold. Consequently, it is im-

4.. 6


173

portant to .note that adiabatic processes and isentropic processes are identical for the case in which both are reversible . .In discussing deformations of elastic materjals (see Chapter 6_) it is convenient to work with the strain-energy function \JI introduced in eq. (4 ..116)2. However, for the case in which W is us-ed within the thennodynamic regime incorporating thermal variables such as 8 or 'f/, then is commonly referred to as the ·uelmholtz free-energy function or referred to briefly as the free energy . .Next, we express the Helmholtz free-energy .function .in terms of the internal energy e and the entropy rJ. Having in mind this aim, we apply the Legendre transformation, generally defining a procedure which replaces one or some variables with the conjugate variables_, particularly used in analytical mechanics (see, for example, ABRAHAM and MARSDEN [1978]). We may write

w

\JI

=e-

8r1 .

(4 ..152)

Note that all three quantities W, e, J} are introduced so that they refer to a unit volume of the reference configuration. By using the material time derivative of the free energy, i.e. W= Dw /Dt, we may write .the Clausius-Planck inequality (4.141) in the convenient form (4.153) often employed when the absolute temperature is used as an independent variable. Note that for the case of a purely mechanical theory, that is, if thermal effects are "ignored (8 and r1 are om.itted), inequality (4..15J) degenerates to ~

Vint

=

\Vint -

\JI

>0

1

(4.154)

and the free energy \JI coine:ides with the internal energy e (see the Legendre transformation (4.] 52)). For a reversible .process for which the internal dissipation Vint is zero (no -entropy production, = 0), we conclude .from (4J54) that the rate of internal mechanical work (or stress-power) Wint = P : :F per unit reference volume (see eq. (4.113)) equals w.

r

EXERCISES

J. Recall relation (4.145) and Piola transformation (4.121) for the heat flux vector. Show the equivalence of relations (4.139) and (4.140) with manipulations according to identity ( 1.81 ). 2. Starting from (4.132), show that the local spatial form of the second .law of thermodynamics is

Bric at >- -div(h + ·n·.1c v) + f (compare with Exercise 2 on p. 166).

174

4 Balance Pr-incip:fes

3. Derive the local forms of the Clausius-Duhem inequality and the Clausius-Planck inequality in the spatial description associated with (4.137), (4.138) and (4."141.), (4 ..142), respectively. Note the fact that the .internal energy and the entropy possessed by a body were introduced per unit volume rather than per unit mass (see relations (4.91.) and (4.129)). Hint:

Use Reynold's transport theorem in the form of (4.26).

4. Show that if the absolute temperature E> is merely a function of time, the ClausiusDuhem inequality and the Claus.ius-Planck ine·quality coincide, leading to the local material fo.rm Eh}> R- DivQ

or

. DQ4 E~r1 > R- -~-~ D..\A

also valid .for isothermal processes.

4. 7 Master Ba·Jance Principle We recognize from previous sections that conservation of mass, the momentum balance principles and the two fundamental laws of thermodynamics are all of the same mathematical structure. To generalize these relations .is the objective of the following section. Master balance principle .in the global form. The present status of a set of particles occupying an arbitrary region n of a continuum body B with boundary surface an at time t may be characterized by a tensor-valued function I(t) = J~ .f dv (compare w.ith eq. (4.22)). In the following let f = f (x, t) be a smooth spatial tensor field .per unit current volume of order -n. In particular, f may characterize some physical scalar, vector or tensor quantity such as density, linear and angular momenlllm, total energy and so forth. A change of these quantities may now be expressed as the following master balance principle here presented in the global spatial form

o;·

Dt. f (x, t)dv n

;· . /...

=.

t/>(x, t, n)ds +. E(x, t)d'IJ . oo n

(4.155_)

The first term on the right-hand side is the integral of the so-ca] led surface density ¢> = .cp(x, ·t, .n), which is a tensor nf order n. The surface density is defined per unit current are.a and distributed over the boundary surface an. Note that c/J depends not only on position x and time t, but also on the orientation of the infinitesimal spatial

175

IVlaster Balance Principle

4. 7 Ba·lance principles

.f

II

Mass (4.8}.?

p

0

0

Linear momentum (4.44)

(JV

t

b

rxt

rxb

t · V + q11

b . v+r

Angular momentum (4.45)

r

First law of thermodynamics (4.124)

(JV~

X (JV

'J/ 2 -1- ec

Identified quantities for Lhe maste=r balance principle.

Table 4.1

surface ·element ds E an characterized by the unit vector fie.Id n normal to 8r2 at x. Thus, (/1 is not a tensor fie.ld. The remaining term on the right-hand side is the integral of the so-called volume density E := ~(x, t), which consists of external and internal sources. It is a spatial tensor field of order n defined per unit current vofome and distributed over region n. The spatial quantities

If equality (4.155) is replaced by an .inequality .in the form

it j

.f (x, f;)dv >

j rjJ(x, t, n)d.
n

,

(4..156)

n

we may refer to this as the master inequality :principle, essential in thermodynamics. Hence, in regard to the second law of the1modynamics in the glob.al spatial form of (4.132 ), the fields f ~
f =

'f}c

'

-
= -b · n

,

~=f

.

We. now introduce Cauchy's stress theorem, as presented by eq. (3.3)., in a more general setting. There exists a unique tensor .field cp == cl»(x, ·t) (with respect to current volume) .on fl of order n + 1 so that

efJ(x, t, n) (the proof is omitted). the surface density .
Thus~

= ti>(x, t}n

(4."158)

176

4

Balance Princi.ples

With the fundamental relation (4.158) and the divergence theorem by analogy with (1.294) (and (1..293)), the surface integral in the master.balance p.ri.ncip.Ie (4.155) transforms into a volume integral according to

J

J

J

an

an

n

.rfa(x, t,n)ds =

<)(x, t)nd.o; =

(4.159)

div<)(x, t)dv .

The spatial field -I>, which we have assumed to ·be smooth, characterizes the Cauchy .flux out of the boundary surface 8!1. Hence, the integral over the surface density els, describes the total flux of . To be consistent with the notation, if <) is a vector field the tenn ·
Master balance principle in the local form.

We are now interested in the equivalent local spatial forms ·of master balance principle (4.155) and master inequality principle

(4.156).

By the material time derivative of function J(t) = Jn f dv~ recall relation {4.33h, and by .means of ·eq. (4,.159)2, the master balance principle (4.155) takes on the form

r

. . . [Df 8t +div(!® v)]dv = ./' div<)(x, t)d11 +.;· I:(x, t)d-v

n

r2

.

(4.-.t6q)

n

Since·equation (4.160) holds for any current volume v, we conclude that

d"IV~+ . ~ -8/ = at y,.

LJ

with

'11=~-f®v,

(4.161)

which is the master balance principle in the spatial description at a point x in region n and at time t, also called the master field equation. Here, '11 = '11 (x, t} is a tensor .field of order n + 1. For the case that f is a scalar field, the notation f 0 v in eqs . (4.160) and (4.161) should be understood as .fv. For example, recaH Cauchy's first ·equation of motion in the form of (4.55) which may be .identified by setting

.f

= pv ·'

'11 = u - pv ·0 v ,

(4.162)

Equation (4.16.1 ), which is replaced by an inequality of the form

fJf . -· > div(tl> - f 0 v) + ~ at ·

'

gives the associated local spatia"J form of master inequality principle (4 . .156).

(4.163)

4.7

Master Balance Principle

177

EXE.RCISES

1. Find the Cauchy flux '11., as derived in (4.16.1 }, for each of the balance principles (see Table 4.1.). 2. Define the Piola-Kirchhoff tlux q,R = R(X, t) so that the corresponding surface density per unit reference area is .PRN, where N is the unit outward normal to the boundary surface 80 0 • Find the maste.r balance principle in the global and local material .fonns equivalent to eqs. (4.155) and (4.161), respectively.

5

Some Aspects of Objectivity

.If laws of nature discovered at different places and times were not the same, scientific work would have to be redone at every new place and at each time. We know that .the laws of nature we ~iscover have to take :the same form, however we .are oriented or we set our clock; there is no difference whether we measure distances relative to, for example~ east or west, or we date events from, for examp.Ie, the birth of Christ or the death of Newton. Qualitative and quantitative descriptions of physical .Phenomena have to remain unchanged even if we make any changes jn the point of view from which we observe the·m. Thus, physical processes do not depend on the change of observer. To ancient natural philosophers this was not so obvious. The mathematical representation of physical .phenomena must re'Hect this invariance. The following chapter has the task to express tMs fundamental finding with t.he concept of objectivity, or frame-indifference, which constitutes an essential part in nonlinear continuum mechanics. We introduce the terminology of an observer, consider changes of observers and apply the concept of objectivity to tensor fields. Transformation rules for various kinematical, stress and stress rate quantities under changes of observers are also derived . .It is obvious to claim that material properties must be invariant under changes of observers. This fundamental requirement is expressed through the principle of material frame-indifference. To show how this objectjvity requirement restricts elastic material response is the aim of the last section. Additional information is found in the same monographs which have a1ready been suggested for Chapter 2 (see the reference list on p. 56). Of course, the short list does not contain a comprehensive review of the large number of papers and books avaHable on this subject.

.5.1

·Chan.ge of Observer, and Objective Tensor Fields

Before examining specific constitutive equations for some elastic materials it is first necessary to present a mathematical .foundation for the change of an .observer and to

.179

180

5


introduce the concept of objectivity for tensor fields. We study further how velocity and acceleration fields behave under .changes of observers. The description of a physical process Observer and Euclidean transformation. .is related directly to the choice of an observer, which we denote subsequently by 0. An arbitrarily chosen observer in the three-dimensional Euclidean space and in time is equipped to measure (i) .relative positions of points in space (with a ruler), and

(ii) instants of time (with a clock).

An event is notked by an observer in tenns of position (place x) and time t. Consider two arbitrary events in the Euclidean space characterized by the pairs (x0 , t 0 ) and (x, t) (compare with OGDEN [1997]). We .assume that (x0 , t 0 ) is 'frozen'

as long as the event (x, t) occurs. An observer records that the pair of points .in space is separated by the distance fx - x0 l, and that the time interval (lapse) between the events under observation is t - t 0 • In the following we let the pairs (x0 , t 0 ) and (x., t) map to (xt, tt) and {x+, t+) so that both the distance Ix - x0 I and the time interval t - .t0 are preserved (see 'Figure 5.1).

,/,,,..--...,,!\

( :~•'·· ..

,.,··(·-·,,

(.

J

,,__,_,,,,,.

)

']

·-~·

'

·'~-'"':''' __ ..• ; ..

(x, t)

Distance:

Ix -

Q

xo I

Ix+ - x(j(

Time interval: t - to

.t+ -

=

Ix - xol

tri = t -

to

Figure 5.1 Map of two points preserving distance and time interval.

A spatial mapping which satis:fi.es the requirements above may be represented by the time-dependent transformation

x+ - x[j

= Q(t)(x -

xo) .

(5.l)

5.1

Change of Observer, and Objective Tensor ·Fields

181

The po.int differences x+ - x·t and x- x0 can be interpreted as vectors which are related through the orthogonal tensor Q(t), with the well-known property Q-1.(t) = Q(t)T (compare with Section 1..2, p. 16). In order to maintain orientations we admit only rot.ati on, consequently, Q is assumed to be proper orthogonal (detQ = + 1). Hence, with (5.1), we may write the following mathematical -expression

x+

= c(t) + Q(t)x

,

t+

= t + lt ' .

(5.2)

where in regard to x+ and x we think of position vectors characterizing two _points. For eq. (5.2) we have introduced a vector c(i), and a real number o· denoting the time-sh(ft, which are defined to be

c(t)

= xt - Q(t)xn

,

(t

=

t(j - to .

{5.3)

Note that both c and Q are continuous .functions of time which, for convenience, are assumed to be continuously differentiable . The one-to-one mapping of the form (5.2) connecting the pair (x, t) with its corresponding pair (x+, t+) .is fre-quently referred to as a Euclidean t.ranstormation.

Change .of observer.

We assert, for -example, that macroscop.ic properties of materials are not affected by the choice of an observer, a fundamental principle of physics. However, the general aim is to ensure that the stress state in a body and any physical -quantity with an .intrinsic feature must be invariant relative to a particular change of observer. Therefore, we ·expect from a change of observer that distances between arbitrary pairs of points in space and time inten1als between events are preserved. In other words_, we require that a different observer o+ monitors the same relative distances of points and the same time intervals between events under observation. We can show that the spatial transformation (5.1) or (5.2) 1, combined with the timeshift (5.2)2, denote the most general time-dependent change of observer from 0 too+. The event at place x and at time t recorded by observer 0 is the same event as that recorded by a different observer o+ at x+ and t+. Note that we are now considering one event recorded by two (different) observers 0 and o+ who are moving .relative to each other. In orde:r to describe a physical process in the (three-dimensional) Euclidean space and on the real time axis we .assign to each of the observers a rectangular Cartesian coordinate system, which we characterize by a set of fixed basis vectors, i.e. {Ca} and {e~·} relative to 0 and o+, respectively. We call them reference frames of the observers. Hence, any points x and x+ may be represented by the position vectors x = ~r 0 e,1 and x+ = :l:te.~, with :1:a and :c:% denoting rectangular Cartesian coordinates, as usual. The shift in the time scale between the observer 0 and o+ is t+ = t +a. In order to fonnulate the change of observer in index notation_, we identify Q(t) as the relative .rotation of the reference frames of the observers. Hence, e;t = Q(-t)ea

182

5 Some As.peels of Objectivity

(comp.are with eq. (.l..1.82) 1_). Multiplying eqs. {5.2) 1 and (5.3)1 bye; and using identities (1.81), QTQ =I and the relation according to ( l.23.h, we find that '1.+

•·'(1

= x+ · e+ = c+(t) + ,.1· t1

(1

(5.4)

•1(1'

The mathematical express.ion (5.4)i states that the observers 0 and o+ assign, with the the same coordinates to the corresponding points ·x and x+. exception of the shift The reader should be aware that the introduced mapping, i.e. a change of observer, affects the points .in space-ti1ne and not the coordinates of points. However, a .change of reference frame changes the coordinates of points (and not the points themselves) and is simply governed by a coordinate transformation, as introduced in Section 1.5.

ct,

EXAMPLE 5.l Consider two .arbitrary points of a continuum body identified by their position vectors x and y at time t. The events (x., t) and (y, t;) are recorded by. an observer 0 with the reference frame ea·· A second observer o+ with the reference frame c~· records the same events at the associated points x+ and y+ at time r 1·. Compute the transformation of the spatial vector field u = y - x ·= ·1t(l.ea into its and determine the components ua (of u) and counterpart u+ = y+ - x+ = (of u+), recorded by the two (different) observers 0 and o+ respectively.

u.;

u;-et

T

Solution.

Using (5.2), u transforms according to

u+ = y+ - x+

= ·c(t) + Q(t)(y ·- x) -

c(t)

= Q(t)u·

.

(5.5)

With referen~e to eqs. ( 1.23 h and { 1.2 .l ), the components of the spatial vector fields u+ and u relative to one and to the other reference frames of the observers read u+ ·= .u+ · e+ ll (I.

= ·(·11+ •· 11

:r+)c+ U a · e+ (}.

= ·11+ .Ill

~c+ · ·U

'

(5.6) (5~7)

By means of (5.4) 1 we deduce .from (5.6)3 that (5~8)

With (5. 7h this shows that u; = ua, signifying lhat both observers measure the same. distance. •

Any spatial vector field u that transforms according to eq. (5.5h, i.e . u+

= Q(t)u

,

(5.9)

is said to be objective or equivalently frame-indifferent. According to Example 5.1.,

5..1

183

Change of Observer, and Objective Tensor Fields

o+

transformation (5.9) implies that two observers and 0, mov.ing relative to each other, record the same coord.inatesT u; = ·u.a, which is the meaning of an objective spatial vector field. In general, if a physical quantity is objective then it is independent of an observer. Consider now a motion x = x(X, t) of a continuum body ~s seen by an arhitrary observer 0 in space. The motion specifies the place x at cmTent time "t of a certain mate.rial point .initially at X. A second observer o+ monitors the same motion at the place x+ and at current time t\ we write x+ = x+(.x, t+). Note that the reference configuration (and any referential position X) is fixed and therefore independent of the change of observer. Hence, the motion x+ == x+ (X_, t+) is related to x = x(X, t) by the Euclidean transformation (5 .2), i.e.

x+(.x, t+)

= c(t) + ·Q(.t)x(X, t)

,

t+ =

t + lt

'

(5.10)

for each point X and time "t.

Veloc.ity .and acceleration fields under changes of observers. fo general, observers are located at different places in space and move relative to eac.h other, as implied by the ti.me-dependence of c(t) and Q(t). Therefore_, the descriptions of motions depend on the observers and, consequentl:y, the velocity and .acceleration of motion are, in general, not objective (frame-indifferent), as shown in the following. To begin with, we introduce

.(

=

v(x, t-) = X X, t")

Bx(X,·tj Dt ~ (5.1 l)

v

+( + ct-) _ . +(x t+) _

x ·'

I

-

x

'

.(

I

a (_.x, t ) = v x, :t a+·(x+ t+) J

= v+(x+

)

.

-

ax+(x, t+) i)t+

(x, t) = Dv Dt

t+) '

=

av+(x+ t+) . ' EJt+

(5. l 2J

which are the spat.ia"I velocities and accelerations of a certain point as -observed by

0 and o+, respectively. Before examining the transformation rules for the velocity and acceleration fields under chan_ges .of observers it is first necessary to formulate the inverse relation of (5.2) 1 and its material time derivative. Hence, with Q(t)TQ(t) = I, we deduce from (5.2)1

x = Q(t)T[x+ - c(t)] ,

{5.1.3)

and using (2.28) 1, the product rule of differentiation .and (5. l lh, we find that

V{X, t) = Q(t)T[x+ - c(t))

+ Q(t)T[v+ -

c(t)) .

(5.14)

184

5


The overbar covers the quantity to which the time differentiation is applied. Further, we define the skew tensor

n(t) = Q(t)Q(t)T = -n(tfr

(5.15)

with the property

The tens.or n :represents the spin of the .reference frame of observer 0 relative to the refere nee frame of observer o+. Hence, material time differentiation of the spatial part of (5.10.) gives, using (5.. .l 1) and the .product rule,

v+.(x+.1 t+)

= c(t) + Q(t)x + Q(t)v(x, t;)

.

(5.17)

From (5.17) we find, with the aid of (5.13) and (5.15) 1, the transformation law for the spatial velocity field, namely

v+ = Qv + c+ n{x+ - c)

(5."18)

(sup.pressing the arguments of functions). We deduce from relation (.5.18) that the spatial velocity field v is not objective u~.de.r changes of observers following the arbitrary transformation x+ = c(t) + Q(t)x. Since the extra terms ·C and O(x+ - c) are present, the .requirement for obj-ectiv.ity, i.e. eq. (5.9), is not satisfied. Hence, the velocity field v is only objective if

c+ fl(x+ -

c) = o ,

(5.19)

implying a change of observer according to the fo11owing time-independent trans.formation with

Co= 0

(5.20)

,

referred to as a time-.independent rigid t-ranst'ormation. Therein, the vector c0 and the orthogonal tensor·Q0 .are assumed to be time-independent (constant) quantities. For a time-independent rigid transformation the magnitudes of v and v+ are equal, so that v+ = Qv, as would be require.d for an objective vector field~ The material time differentiation of (5.1. 8) gives, by means of the product rule and eqs. (5.12) and (5.11 h,

a+

= Qa + c + Qv + n(x+ -

c)

+ Sl(v+ -

c)

(5.21)

(suppressing the arguments of functions). Finally, by means of (5.14) and eqs. (5.15) 1 , (5.16h, we obtain from (5.21) the

5.1

185

Change of Observer, and Objective Tensor Fields

trans.formation a+= ·Qa + c +(.fl

- 0 2 )(x+ -

c)

+ 20.(v+ - c)

(5.22)

for the spatial acceleration field a. Like for the spatial velocity field, the acceleration field is not objective for a general change of observer. The terms O(x+ - c) and -!1 2 (x+ - c) are called the .Euler .acceleration and centrifugal acceleration, while the last term in eq. (5.22), i.e. 2S'l(v+ - c), represents the Coriolis acceleration. An acceleration fi.eld is objective under all changes of observer if and 011/y (f the lengths of a+ and a are equal, i.e. a+ (x+, t+) = Q( t )a{ x, t), requiring that

c+ (n -

0 2 )(x+ - c)

Consequently, this implies that

+ 2n(v+ -

c) = 0

(5.23)

.

c is constant and that the orthogonal tensor Q is

also constant. A change of observer from 0 to o+ of this .type, for which the spatial acceleration is objective, is ca~led a Galilean transformation, and is governed by with

c(t)

=o

(5.24)

,

for all times t. Here, Q0 denotes the time-independent orthogonal tensor, and c(t) v0 t + c0 , with the initial (constant) quantities for c0 and the velocity v0 •

=

Objective highe-r.. order tensor fields. A spatial tensor field of order n, n = 1, 2, .... , .i.e. u 1 ® · · ·®Un, is called objective or equivalently frame-·indifferent, if, during any change of observer, u1 0 · · · 0 Un transforms according to (5.25)

which holds for every tensor Q and every vector UnBy introducing a spatial second-order_tensor field A(x, t) to be u 1 (x, t) 0 u2 (x, t) {n = 2), we find, using (5.9), the ·important relation A +(x+, t+)

= (u1 (x, t) ® u2(x, t)]+ = Q(t)u 1(x, t) 0 Q(t)u2(x, t) = Q{t)[u1(x, t) 0 u2(x_, t)]Q(tfr = Q(t)A(x t)Q{t)T 1

.

(5.26)

For any spatial vector field u (n = .1), eq. (5.25) reduces to u+(x+, t+) = Q(t)u(x, t), which we found through eq. (5.9). In particulart for n = 0 we have a scalar field. It is obvious that any spatial scalar .field cJ>(x, t), recorded by 0, is unaffected by a change of observer. Hence, a spatial scalar field is obj-e·ctive if, under all Euclidean transformations (5.2), transforms

according to (5.27)

5

186


where + is the corresponding scalar field recorded by observer o+. In summary: the requirement .of objectivity means that tensor, vector and scalar fields transform under changes of observers according to the laws

Q(t)A(x, t)Q(t)T (.5.28)

{xJ)

,

where x+ and x are related by the Euclidean transformation (5.2) .

.EXAM.PLE 5.2 Show that the spatial gradient of an objective vector field u transforms according to

= u(x, t) (5.29)

where (gradu)+ = au+ /ox+ denotes the spatial gradient of the vector u+ recorded by an observer o+. Note that in view of (5.28) 1 the second-order tensor field gTadu retains the obJectivity property. A vector field u remains unchanged during any change of observer if u+(x+,t+) = Q{t)u(x,t). Hence, by the chain rule,

Solution.

au+ EJ-x.+ ax+ {)x

au

= Q.ax .

(5.30)

(the arguments of the functions have been omitted). With the aid of transformation (5. l 0) 1 we obtain the desired result. Note that, in general, the gradient of an objective tensor .field of order n is also objective. •

EXERCISES

L Consider objective scalar, vector and tensor fields (x, t), u(x~ ·t) and A(x, t_), .respectively. (a) Show that grad~!>, divu and div A are objective fields during any change of -observer. (b) Show that the Lie time derivative of a -contravar.iant spatial vector field u, as determined in e.q. (2.192), is objective.

187

5.2 Superi"mposed Rigid-body Motions

2.. Using eqs. (5.13)-(5...15) obtain the a1ternative relation for the spatial acceleration field (5.2.1) in terms of x, v and a in the form a+ = Qa +

c+ Qx + 2Qv

,

and show that its gradient is g·iven by (grada)+ =

(Q + 2Ql + Qgra
,

(5.3 I)

where:) = gnulv denotes the spatial velocity gradient 3. Assume an objective transformation of the body· force (per unit volume), i.e. b+(x+, t+) = Q(t)b(x, t). Show that the local form o:f Cauchts first equation of motion in the spatial description, i.e. (4.53), is only objective under a Galilean transformation.

5.2 Superimposed Rigid~body Motions In the following section we show that a change of observer may equivalently be viewed as certain rigid-body motions superimposed on the current configuration. We apply this concept to various kinematical quantities and to some stress tensors of importance. Rigid-body motion. As noted, the fundamental relationship (5.2) desc~ibes a change of observer, preserving both the distances between arbitrary pairs of points in space;, and ti.me intervals between -events under observation. It .is essential to introduce an important equivalent mechanical statement of the specification (5.2): for this purpose we consider a motion x+ = x+ (X, t+) of a continuum body whic.h differs from another motion x = x(X, t) of the same body by a superimposed (possibly time-dependent) rigid-body motion and by a time-shift, as depicted in Figure 5 .2. We emphasize that, in contrast to the considerations of the .Jast section, x+ = x+(x, t+) .and x = x(X, t) are motions of two events recorded by .a s·ingl·e observer 0. The rigid-body motion moves the region n in space occupied by the body at time t, defined by the ,motion x = x(X, t), to a new region n+ occupied by the same body at t+, which is given by x+ = x+ (X, t+). Here and elsewhere we will employ the symbol -( • )+ to designate quantities associated with the new region n+. According to the principle of relativity, the description .nf a single motion monitored by two (different) obse~vers, as described in -the last section, is equivalent to the description of two (different) .motions monitored by a single observer. Hence, the pairs (x, t) and ( x+, t+), which are defined on regions n and n+, are precise]y related by the Euclidean transformation (5.2), i.e. x+ = c(t) + Q(t)x .and t+ = t + o:.

188

5 Some Aspects of Objectivity

c_,Q

time t+ time ·t

,... time ·t

...

=0 ·v

i)•

.J'\.]' .... 1'

x·+l

Figure 5.2 Two motions x+ and x of a body (monitored by .a single observer) which differ by .a superimposed rigid .. body motion and by a time .. shift. A spatial vector field u transforms into u+ = Qu, with length ju+j = fuj. Within this context the vector c describes a superimposed (tim-e-dependent, pure) rigid-body translation for which any material po.int moves an identical distance, with the same magnitude and direction at time t. Since Q is .a proper orthogonal tensor (detQ = +1), the orientation .is preserved and Q describes a superimposed (time~ dependent, pure) rigid-body rotation. For a pure rigid-body rotation, transformation (5.2) reduces to x+ = Q(t)x. Hence, at each instant of time a rigid-body motion is the composition of .a rigidbody translation c and a rigid-body rotation Q about an ax.is of rotation, combined with a time-shift a = t+ - t. The material points occupy the same relative position in each motion (the angle between two arbitrary ·vectors and their lengths remain constant). We now recall Example 5.1 and apply the described concept to the spatial vector fie:Id u = y - x located at region Q (see Figure 5.2). Hence, a rigid-body .motion maps the points x, y to the associated points x+., y+ located inn+ and the spatial vector u = y - x to u+ = y+ - x+. With (5.2) we may conclude that the -distance between the two points y+ and x+ remain unchanged. Namely, y+ -· x+ = Q(t)(y - x) (compare with eq . (5."I)), which immediately implies, on use of definition ( 1.15), identity (l .81) and the orthogonality condition Q(t)TQ(t) = I, that ly+ - x+f = Jy - xf. Consequently the lengths of the vectors u+ and u are equal, i.e. ju+ I = juj. We say that the spatial vector field u is objective during the rigid-body motion.

5.2

Superimposed Rig-id .. body Motions

189

It is trivial but worthy of mention that any material field F(X, t) of some physical scalar, vector or tens.or quantity, which is characterized as a function of the referential pos.ition X and time t, is unaffected by a ri_gid-body motion superimposed on n. Hence, J=-+(X, t+) = :F(X, t).

Euclidean transformation of various ki:nematical quantities. The follow·ing discuss.ion is concerned with the behavior of various .kinematical quantities during a super.imposed rig.id-body motion. To begin with, we consider the deformation gradient at the point x E associated point x+ E .n+, i.e.

F(x

.) = ·ax(X, t)

't

ax

'

F +(x ,t+)

= ax+(x_, t+) ax ·

n and its (5.32)

Differentiating (5.2) with respect to X gives the transformation rule

F+ = ax+ = Q Bx = QF

ax

ax

or

for the deformation gradient (for convenience, we will not indicate subsequently the dependence for the above functions on space and time). Note .that the second-order tensor F is objective ·even though (5.33) does not coincide with the fundamentaJ (objectivity) requirement (5.28)i. However, recall that the deformation gradient F is a two-point tensor field, in which one index describes material coordinates ..X.4 which are intrinsically independent of the observer. That is why the deformation gradient .transforms like a vector according to (5.28)2 and why .F is regarded as objective. ·Moreover, let .J = detF and ,J+ = detF+. Since the tensor Q is proper orthogonal (detQ = +l)T eq. (5.33h implies, through the property (J.101), that

.J+ = J

>0 .

(5.34)

Hence., the scalar field ,J remains unaltered by a superimposed rigid-body motion. AJso the sign of the volume ratio J is preserved, since detQ = +1. Next, we recall the unique polar decomposition of the deformatio·n gradient at x E !l and x+ E n+, i.e.

F =RU= vR ,

(5.35)

Apply.ing (5.33h to (5.35), we arrive at the representations (5.3~)

Since the tensor QR is orthogonal it follows from (5.36) 1 that the transformation roles for the rotation tensor R and the right strekh tensor U are

u+ = u .

(5.37)

.190

5

Some Aspects of Obje-ctivity

By analogy with the deformation gradient, R is an objective two-point tensor field. The right stretch tensor is defined with respect to the reference configuration. Hence, U remains unaltered by a superimposed rigid-body motion and U is therefore also objective. From eq. (5.36h, we obtain, using result (5.37) 1 and the orthogonality condition RTR = I, the transformation rule (5.38)

Clearly, the left stretch tensor vis objective. Next, we discuss the spatial velocity gradient according to (2.139) 4 , i.e. I·= FF- 1 • In an analogous manner, the spatial velocity gradient generated by the motion x+ reads

(5.39)

By der.iving-eq. (5.33h with respect to time, using the product rule, i.e. F+ = QF +QF,".·. and the inverse relation of (5.33):1, i.e. -(F+_)- 1 = F- 1QT, the spatial velocity gradient follows from (5.39h as (5.40) with the skew tensor n = QQT. Sinc·e ·n is present, the spatial velocity .gradient I fails to satisfy the obj-ectivity requirement (I+ :f QIQ1 \ Hence, the kinematical quantity l is not a suitable candidate for formulating constitutive equations, which must be objective.

Euclidean transformation of stress tensors. Let dynamical processes be given by the pairs (u+, x+) and (u, x), where x+ and x are related through (5.10). ·we now want to show how the Cauchy stress tensors .u+ and .u are re.lated. We recall the Cauchy trac~ion vector t = un with the unit vector n at point x directed along the outward norma·1 to the boundary surface of an arbitrary region at time t. A superi.mposed rigid-body motion trans~~~nns region Q lo a new region n+ which is bounded by the associated boundary surface an+ at a later time ·t.+ = t + (\;. The Cauchy traction vector transforms tot+ = u+n+ with the unit vector n+ at point x+ normal to an+. By taking note that the vectors t and n transform according to the objectivity requirement (528)2, we obtain that Qt = u+Qn. A comparison with t = crn gives the fundamental transformation rule

.an

u+

= QuQT

n

(5.41)

for the stress tensor. This means that the Cauchy stress tensor is objective.. In order to describe the first Piola-Kirchhoff stress tensor which is generated by the motion x+ = x+ (X, t+) . we may write -the .Piola transformation (3.8) as p+ (F+yr =

191

Objective .l~a.tes

5.3

,J+ u+. Knowing Lhat the scalar J is objective according to eq. (5.34), and using (5.33h and (5.4·1 ), we find, with the help of (3.8), that

p+(QF)T = JQuQT ' p+ (FTQT) = p+

Q J uQT

= QP

= QP(FTQT)

,

.

(5.42)

Since the two-point stress tensor field P transforms .like .a vector fie.Id according to the objectivity requirement (5.28:h, P is objective. The second Piola-Kirchhoff stress tensor Sis parameterized by material coordinates only. Therefore, the material tensor field does not depend on any superimposed rigid-body motion~ and hence S = s+. Note that all the stress tensors u, P and S discussed are suitable candidates for the description of material response, which fundamentally .is required to be independent of the observer.

EXERCISES

1. Recal1 the kinematic relations (2.63), (2.67) and (2.77), .(2.81 ). Using the transformation ru1e F+ = QFt show that the material strain tensors C and E are unaffected by any possible rigid-body motion., i.e.

c+=c '

(5.43)

.

and that the spatial strain tensors band e transform according to the rules

b+ == QbQT ,

e+

= QeQT

.

(5.44)

Note that .all these kinematical quantities are objective, since C and .E are defined with respect to the .reference configuration and the second-order tensor fields b and e conform with the requirement of objectivity given in eq. (5.28) 1• 2. By eqs. (2.146) and (2.147), the spatial velocity gradient :J = .F.F-t. is recalled to be ·the sum of the rate of deformation -tensor d = dT and the spin tensor w =-\VT. Show that rigid-body motions involve the transformations

d+ = QdQT ,

(5.45)

where d is objective. Note that w, which is expressed through the skew tensor n, is affected by rigid-body motions, and hence w .is not objective.

192

5


5.3 ·Objective Rates One aim of this section is to perform objective time derivatives, which are essential in order to fonnulate constitutive equations in the .rate form. We focus attention on some important objective .stress rates associated with the names Oldroyd, Green, Naghdi, Jaumann, Zaremba or Truesdell. Objective rates. The material time derivatives of the objective vector field u = u(x, t) and the objective second-order tensor field A = A(x, t), which transform according to eqs. (5.28) 1 and (5.28)2, are given by means of the product rnle of differentiation as ..

A+ = QAQT + QAQT + QAQT

.u+ = Qu + Qu , neither ti nor .A retains

.

(5.46)

the objectivity requirements (5.28) (u+ =A . .Q~. and A+ f- QA.QT). Note that material time derivatives of objective spatial tensor fields will not, in general, be objective and they are not, therefore, suitable -quantities.for formulating constitutive equations in the .rate form. This motivates the introduction of objective time derivatives cal.led objective rates, which are basically modified material tim-e derivatives. Before proceed.ing to examine objective rate forms it is .first necessary to express the material time derivatives of Q a~d QT from relation (5.45h~ With definition (5.15)i and property w = -wT for the spin tensor we find that Clearly,

Q

= w+Q-Qw

(5.47)

,

Hence, substituting (5.4 7}1 into (5.46) 1, we find immediately, by analogy with the transformation rule (5.28.h, that

(u - wu)+ (u)+

= Q(ti = ·Qu

wu)

(5..48)

l

,

(5.49)

where we have introduced the definition for the co .. rotational rate of the -objective vector field u, i.e. ~

U

= U. -

WU

(5.50)

•

"

In general, we denote co-rotational rates with the accent (• ). By analogy with the above we introduce the co-rotational rate -of the ·objective second-order tensor .field A. With the help of eqs. (5.47) and (5.28) 1 we find from (5.46 )2 after some straightforward recasting that

(A - wA + Awft- = Q(A - wA + Aw)Q 1-~ (A)+

= Q.AQT '

,

(5.5l) (.5.52)

5.3 Objective Rates

193

where we have introduced the definition 0

•

A = A - wA

+ Aw

·~

(5.53)

known as the Jaumann~Zaremba rate, which is often used in plasticity theory. Obviously, in regard to eqs. {5.49) and (5.52), the co-rotational rates of u and A are indeed objective. If A is a symmetric tensor we can easily show an interesting property connecting the Jaumann-Zaremba rate and the material time derivative of A. Using (5.53) and the property of double contraction according to (1.95), i.e. A : wA =A: Aw, we obtain 2A : A = 2A : A - 2A : wA + 2A : Aw = A : A .

(5.54)

Finally we define the convected rates . of u and A. These are the ot?jective .fields A

T

u=u+I u,

(5.55) A

~

where the accent (•) indicates convected rates. The rate A is also called the CotterRivlin rate .

Objective stress rates.

We now focus attention on some of the infinitely ·many possible ·objective stress rates that may be defined. The choke of suitable, .i.e. objective, stress rates is essential in the formulation of .constitutive rate equations, which must be objective. The ·Oldroyd stress rate of a spatial stress field .is defined to be the Lie tim.e derivative of that field. We shall indicate Oldroyd stress rates by the abbreviation Oldr. By recalling the concept of Lie time derivatives from Section 2.8, in particular, rule (2.186) 1, the Lie time derivative of the contravariant Cauchy stress tensor u is given by

.

.

= F(F- 1 uF-T + F- 1 i.TF-T + F- 1 uF-T)FT ,

(5.56)

where the transformations (2.85) and the product rule of differentiation are to be used. Hence, using the identities (2. l43h and (2 . 144h, we conclude that the Oldroyd stress rate of the Cauchy stress u is .:=·

Olg_r(cr)

= ir -

,..

lu - ul · ,

(5.57)

where i:r .denotes the m~~etial time derivative -of the Cauchy stress tensor. We now show h~)\i/the Oldroyd stress rate, Oldr(u), generated by the m·otion x, is related to its cquriterpart Oldr( u )+, generated by x+. Considering Oldr(.u) + =

.5

194

Some As:pects of Objectivity

o-+ - J+ u+ -

u+1+T and using transformations (5.4.1) and (5.40.h in combination with eqs. (5.15) and (5.57), we ·obtain

= QuQT - (f! + QIQT)QcrQT = Q(u - Ju - ufr)QT = Q Dldr( er) QT .

Oldr(u)+

QuQT(n

+ QIQT)'-r (5.58J

Hence, the Oldroyd stress rate is objective. In general, we can prove th.at Lie time derivatives of objective spatial tensor fields yield objective spatial tensor fields. By analogy with (5.57), the Oldroyd stress ·rate of the Kirchhoff stress r .is

Oldr(r)

= 1- -

·Jr - Tfr .

Adopting rule (2.186) for the contravariari.t Kirchhoff stress tensor relation (3.63) 1, we obtain the important equation

r a·nd using

(5.60) relating the Lie time derivative of r, i.e. Oldr{ r), and the material time dedvative Sof the second Piola-Kirchhoff stress S according to the push-forward operation (2.85) 1 •

EXAMPLE 5.3

Cons.ider the definitions

u = ir -

w.u + uw

(5.6.1.)

u

of-objective stress rates, where & is··called the Green .. Naghdi stress rate and theJ.auma.nn-Zaremba stress rate (compare with the Jaumann-Zaremba rate (5.53) for any objective second-order tensor field_). The spin tensor w is _given by definition (2.147).

u

Show that both & and are special cases of the Oldroyd stress rate of er in the sense that ~ corresponds to the Lie time derivative (5.56), with ·F replaced by the rotation tensor R, and is the Lie time derivative (5.56), with the rate of deformation tensor d set to zero. Discuss the case in which the Green-Naghdi stress rate and the Jaumann-Zaremba stress rate coincide.

.u

Setting F = R in relation (5.56) and emp]oyin.g .identities (2.143}1 and (2. 144).1 (change F to R) with the orthogonality condition RTR = I., we obtain

Solution.

1\(u~)IF=R

=u-

. R.R- u - uR-TRT == ir - RRTu + uRRT , 1

where RRT is a skew tensor according to (2.158).

(5~62)

5.3 ObJective l(ates

195

However, setting d = 0 :in the Lie time derivative (5.56), which equivalently means that I ·== w, we deduce from (5~57) that

l

l

(.

• v CT

tt)I·d=O -_ O'.

-

WU

+ UW

,

(5~63)

with the skew tensor w = -wT. The Green-Naghdi stress rate and the Jaumann-Zaremba stress rnte clearly coin .. cide for w = RRT. This is the case for a rigid-body rotation (recall Example 2.12~ eq. (2.160)). IJ

The Truesdell stress rate of the Cauchy stress, denoted by Trues(u), is defined as the Piola transformation of S. Thus,

Trues{u)

= J-lFSFT

1

(5.64)

that .is the push-forward .of Sscaled by the inverse of the volume ratio, J- 1 == (cletF)- 1 .. Hence, using the Piola transfonnation (3.65h and the product rule of differentiation we find from (5.64) that

Trues(u)

= .l

-t

-1

F

.:[D(JF uF ..

Dt

-T

)] T .• F

.

• -L -T = J _ 1F(.JF uF .

+JF- 1 uF-T + JF- 1 &F-T + JF- 1 uF-T)FT

(5.65)

Using relations (2.176) 2 and (2.143)2, (2.144)2 we,,, deduce from {5.65h that ............

(5.66)

By comparing eqs. (.5.57) and (~. ~.59)° with (5.66) we may easily deduce the relation.,..· ships ....... Oldr(.u)

= Trues (d) -

Oldr(r)

utrd ,

= JTrues(u)

(5.67) (5.68)

between the Oldroyd stress rate and the Truesdell stress rate.

EXA1"1PLE 5.4 Suppose that the transformation .(5.68) .is given. Express u as J- 1r and derive the relation (5.67) by simply applying the product rule of differentiation to the Oldroyd stress rate. Solution. With u = J- 1 r and the fact that the directional derivative (in our case the Lie time derivative) satisfies the common rules of differentiation, for example, .the

196

5

Some Aspec.ts of Objectivity

.product ruleT we may write (5.69)

Accord.in_g to considerations of Section 2.8, the Lie time derivative of a scalar fie!~ is equal to the ·material time -derivative of that scalar fieldt and hence £ v{ ,J- l) = J- l. Therefore, with the chain rule and eqs. (2. l 76h and (5 ..68), relation (5.6.9)2 leads to the desired result, ..

Oldr(u)

where the relation u .,, .. ::,., ..,,,, .. ,.,, .. ,.. ,,,,,., .. ,.,., .. , ..

,.~

= J- Oldr(-r) + J- 1,,- = = Trues(u) - otrd , 1

= J- 1 r

is to be used again.

,J- 10.ldr(r) - J- 1rtrd

(5.70) •

... ,,,., .. ,,.:,•,.:,·,,... ,.,,.,,.,,,,.,.,,,:•':'•'.. ''"•"'•"''''·'"'"''"'"'":".''"'"''···,.,... ,.. ,,,.,,,., .... ,, .... , ... ,, ..,.,..,.,,,.,.,... , ... , .. .,,.,,, .. .,, .. .,,, .. ,.. ,,,..,,.,.,., ... ,,_,,., ... ,,,. .. , ..... ,.... - .. -... ,, .....' .•. .,.,.,.::'."''···:•.·•'."•,'(•·:'·"."·"'''·"........ , ... ,.:.,,.:,,,.,., ... ,,,.. ,., .. ,,,, ........... ,,,.... ,..

EXERCISES .u

I.. Consider the Cotter-Rivlin rate A defined by (5.55) 2 • ~

(a) Show that A .is the Lie time derivative of a cnvariant tensor field A . (b) With (5.53) and I = d + w show that the connection between the CotterRivlin and Jaumann-Zaremba rates is ~,)

6

A =A+dA+Ad

!

~

2. Recall the convected .rates ~and A, as defined in eq. (5_55). Using eqs. (5.40h, (5 . 15) and (5.46)~ (5.28), show thatthey are objective according to

5..4 Invariance of Elastic Material Response In this section we introduce the principle of material fr.ame-indifferenc~ which states basically that material properties do not d~pend on the change of observer.. .In particular, we show how it restricts the response of elastic materials and derive objective constitutive equations which are defined to be invariant for all changes of observer. This principle is crucial when constitutive theories such as the theory of elasticity or plasticity are considered.

5.4

.Invariance of Elastic Material Response

197

In the following we consider only the isothermal case for which the absolute temperature e remains constant during the process. Cauchy-elastic materials. A material is called Cauc.hy-elastic or elastic if the stress field at time t depends only on the state of defonnation (and the state of temperature) at this time 't and not on the deformation history (and temperature history). Hence, the stress field of a Cauchy-elastic material is indepe.ndent of the deformation path (independent .of the time). However, note that the actual work done by the stress field on a Cauchy-elastic material does, in general, depend on the deformation path. A constitutive equation (or equation of state) represents the intrinsic physical properties of a continuum body. It determines generally the state of stress at any point of that body to any arbitrary motion at time t. A constitutive equation is either regarded as mathematically generalized (ax.iomatic) or is based upon experimental data (empirical). The constitutive equation of an is.othermal elastic body relates the Cauchy stress tensor u = u(x_, t) at -each place x = x(X, t) with the deformation gradient F = F(X, t). We may express the constitutive equation in the general form

u(x, t) == g(F(X, t), X)

1

(5.71)

where g is referred to as the response function assoc~ .ated with the Cauchy stress tensor u. . . . ._. , . . In equation (5.7.l), .u was allowed to depend up.on the referential position X E n0 in addition .to F. Hence, the stress response varie~.:·'from one particle to the other. However, for subsequent introductory treatments, .~r. ·is convenient to restrict our atten.tion on continuum bodies, in which both the CaucJ1.y stress tensor u and the reference mass density p0 are independent of the position-~;'' such bodies are called homogeneous. Hence, .instead of (5.71), we write _thf(constitutive equation in the form (5.72)

which determines the stresses u from the given deformation _gradient F. From the mechanical point of view g characterizes the material properties of a (isothermal) Cauchyelastic material.., while from the mathematical point of view g is a tensor-valued function of.one tensor variable F. The concept of tensor functions, as we will use it here, is explained in Section 1. 7. A constitutive equation of the type of (5. 72) .is often referred to as a stress relation. Nole that for homogeneous deformations the corresponding stresses are constant (since F has the same value at every point of the body) and, interestingly enough, Cauchy's equation of equilibrium (4.54) trivially .reduces to divu = o. For this case the body force bis zero, which means that a homogeneous deformation of a continuum body occurs without body force.

198 Principle of material

5 Some Aspects of Objectivity frame~indifference.

As already mentioned .in previous sec-

tions, constitutive equations must be objective (frame-indifferent) with respect to the. Euclidean transformation (5.2). ln other words, if a constitutive equation is satisfied for a dynamical process (u, x) then it must also be satisfied .for any associated (equiv~ alent) dynamical process (u+., x+) which is generated by the transformations {5.41) and (5. l 0). This is a fundamental ax.iom of mechanics which is known as the pri.ndple of mater"ial f.rame-.indifference or the :prindplc of material objectivity or simply as obJectivity (see TRUESDELL and NOLL [1992, Sections .l.9, 19AJ). If this principle -is violated, the constitutive equations are affected by rigid--body motions and meaningless results are obtained. To begin with, the material frame-indifference of the stres·s relation (5.72) impos·es certain .restrictions on the response function g. We consider a mo.~ion.x+ which differs from x by a rigid-body motion superimposed on the current co~figura:tion (compare with Figure 5.2). The rigid-body motion maps the region n to a new r,¢gion n+ and the stress relation (5.72) to .u+ = g(F+). We demand that both .regions/namely n and n+, are associated with the same function D because it is for the same elastic material. Hence, using (5.33)a on the one hand and (5..41 ), {5..72) on the other hand, we arrive at (5.73)

Combining (5.73)i and (5.73h, we .find the restriction Qg(F)QT

= g(Q.F)

(5.74)

on g for every nonsingular F and orthogonal Q. In other words, constitutive equation .(5.72) is independent of the observer if the.response function g satisfies the invariance relation (5.74). Employing the right polar decomposition F = RU on the right-hand side nf (5. 74), we may write Qg(F)QT = g(QRU), where R -is the orthogonal rotation tensor and U the right stretch tensor. Since the latter relation holds .for all proper orthogonal tensors Q, it also holds for the special choice Q =RT. Hence, using the orthogonality condition RTR = I, we obtain a corresponding reduced form of eq. (5.74), Le.

g(F)

= Rg(U)RT

,

(5.75)

for the function .9 and for every F and R. Therefore, the associated stress relation reads u = Rg(U}RT ·'

(5~76)

which shows that the properties of an elastic material are independent of the rotational part of F =RU, characterized by R. Note that the reduced constitutive equation (5.76) is compatible with the principle of material frame . . indifterence which can be shown

5. 4

199

Invariance of Elastic Material Response

as follows. By analogy with (5.76), let u+ ·== R+g(U+){R+)'r and use eq. (5..37) in order to obtain u+ = QRg(U)RTQT. Hence, by use of (5.76), we obtain once more u+ = QuQT (compare with (5.41 )). An alternative form of constitutive equation (5.72) follows from Piola transforma. t.ion (3.8). With (5.72) and the volume ratio J = det.F_, we obtain

P = Ju·F-T = detFg(F)F-T = 6(F) ,

(5.77)

where we have de.fined the tensor-valued tensor function ~ assoc-iated with the first Pio la- Kirchhoff stress tensor P. By analogy with the above we may now show the material frame-indifference of the stress relation {5 . 77)a. Considering p+ = 6(F+) and, using (5.33h on the one hand and (5.42_) and (5.77h on the other hand, we find that p+ = QP

= Q~_.(F)

.

(5. 78)

'· ~ ........ ·

Equating (5. 78) 1 and (5. 78h., we find the invariant re1atio_p,,,· ·Q~(F) =

6(QF)

(5.79)

and for every F and Q. Re.lat~ons (5.74) and (5.79) are necessary and .sufficient conditions for the constitutive eq.~_~tions (5.72) and (5.77h to satisfy the principle of material frame-indifference . A reduced form of constitutive equation (5. 77.h is obtained from restriction (5.79). Setting Q = RT and rep-lacing F by its right polar decomposition RU on the right-hand side of (5.79), we obtain, us.ing RTR = t for the function

~

~(F)

·= R6(U) .

(5.80)

The restriction on <5 expressed through eqs. (5. 79) and (5.80) are equivalent to the restriction on g, as given in eqs. (5.74) and (5.75_).

Another alternative form of Lhe constitutive equation which turns out to be very useful in the theory of elasticity follows from relation (3.65)i. With the volume ratio J = detU, i.e. (2.94h, the .polar decomposition F = RU, the stress relation (5.76), and the fact that the right stretch tensor U is symmetric, we obtain (5.81)

By recalling that U is the unique square root of the right Cauchy-Green tensor C, we may write c 1/ 2 in the place of U. Defining a tensor-valued tensor function .S) we may ·introduce finally the second Pio la-Kirchhoff stress tensor S in the form S = SJ(C) .

(5.82)

200

.5


Since the reference configuration is unaffected by superimposed ri.gid-body m·otions . we know that the second Piola-Kirchhoff stress tensor and the right CauchyGreen tensor simply transform according to s+ = S and c+ = C. We conclude that the stress relation (5.82) is independent of the observer.

EXAMPLE ·5.·5

Investigate if the elastic material given by u

= J(E)

,

(5.83)

associated with .motion x, satisfies .the principle of material frame-indifference. Solu.tion. For another motion x+ (recorded by a single obsery~_r.·O.), assumption (5.83) implies u+ = J(E+). By recalling the transformations for the Cauchy stress tensor u and the Green-Lagrange strain tensor E, which satisfy eqs. (5.41 )..:and (5.43ht and knowing that J is the same function for the two (different) :motions x and x+ (it is for the same elastic material), we conclude that (5.84) Note that this relation is only true for Q = I; thus,. constitutive equation (5.83) does not satisfy the principle of material frame-indifference. II

We assume that the Cauchy stress tensor u depends on the left Cauchy-Green tensor b = FFT. The constitutive equation (5.72) may then be written .in the alternative form Isotropic Cauchy-elastic materials..

er

= IJ(b)

,

(5.85)

where ~ is a tensor-valued function of the symmetric second-order tensor b associated with the Cauchy stress tensor tr. ln order to "find the restriction imposed on the response function ~ by the assumption of material...frame-:indifference, let .u·+ = ~ (b+), where the response function lJ is the same for the two motions x and x+. We now use (5.44h on the one hand and (5.41), {5.85) on the other hand .in order to obtain (5.86) Combining eqs. (5.86) 1 and (5 ..86)2 we find the fundamental :invariance relation (5.87) for the function ~ and for every tensor band orthogonal tensor Q. Hence, constitutive

5.4

Invarfance of Elastic Material Response

201

equation (5.85) is independent of the observer if~ satisfies restriction (5.87)~ A specific elastic material which may be described by the constitutive equation in the form (5.85), with property (5.87), is said to be isotropic. A tensor-valued function such as ~ (b) is said to be isotropic if it satisfies relations of type (5 ..87). He-nee, we refer to ~ (b) as a tensor-valued isotropic tensor .function of one variable b. From the physical point of view the condition o.f :isotropy is expressed by the property that the material exhibits no preferred directions. In fact, the stress response of an isotropic elastic material is not .affected by the choice of the reference configuration. For a piece of wood, for example, which .is of cellular structure, th_~_.propefries in the direction of the grain differ from those in .other directions, so th~.--riiaterial certainly is not isotropic.

_/./

The isotropic tensor function ~(.b), which satisfies (5.~7)", may be represented in the explicit form

for each b, which is known as the Rivlin-Ericksen representation theorem. ·For a proof of this crucial .relation see RIVLIN and ERICKSEN :[1955], TRUESDELL and NOLL [1992, Section 12], SPENCER .[1980, Appendix] and GURTIN [198la, pp. 233-

235]. Note that the representation theorem (5.8-8) represents a fundamental requirement for the mathematical form of the stress relation. Here, aa, a = 0, 1, 2, are three scalar functions called response coefficients or material functions. He-nee, in general, for an isotropic material only three parameters are needed in order to describe the stress state. The scalar functions aa depend on the three invariants of tensor b and therefore on the current deformation state. The invariants are defined with respect to eqs. (1.170)-( 1.172) as .

I1(b)

= .trb. = ,\21 + A22 + ~\12 ,

J2 (b) =

~ ((trb) 2 -

tr(b2 )] = trb- 1detb =Ai A~+ Ai A~+,\~,\~ ,

(5.89) (5.90) (5.91)

where /\~ are the three eigenvalues of the -symmetric spatial tensor b, see eq. (2.117). In eq. (5.9.1), relation (2.79) 2 was used. Representation (5.88) is the most general form of .a constitutive -equation for isotropic elastic materials also known as the first representation theorem for isotropic tensor functions. From the constitutive equation (5.88) we deduce that the principal directions of the Cauchy stress tensor u and the left Cauchy-Green tensor b coincide. Hence, for isotropic elastic materials the two symmetric tensors u and b are said to be coaxial in eve.ry configuration.

5

202


By analogy with (5.S-8), the constitutive equation ·CT

2 = -1·+-d+-d no 0:2 0~1

(5.92)

_,

characterizes the behavior of a viscous fluid, in particular, of a so-called ·Reiner-RivUn Huid. Therein, p and d are the spatial mass density and the rate of defonnation tensor, while ctn, a = 0, 1, 2, are scalar functions of the .invariants Ia, a = 1., 2_, ·3, given .in (5.89)-(5.91) with d replacing b. ln order to ·find .an alternative explicit re.presentation for (5."88), we recall the Cayley.. Hamilton. equation ( 1..174). Since any tensor satisfies its own characteristic equation, we may write Cl. 174) as ba - 11b 2 + l 2 b - 1:11 = 0 and find, by multiplying this equation with b- 1, that (5."93)

Eliminating b 2 from (5.88) in favor of b- 1 we obtain an alternative represe~tation of a constitutive equation for isotropic -elastic materials, i.e .

where /3<"' .a = O_, l~ -1, are three scalar func~ions (response coefficients) which, in tenns of the three invariants of b_, are expressed as

/3-1

= Ian2

.

(5.95)

Representation (5.94) .is also known as the second representation theorem for isotropic tensor functions, see, for example, GURTIN [1981.a, p. 235]. Incomp·ressible Cauchy-elastic -materials. .If the Cauchy-elastic material is incompressible, then the stress relation .is detem1ined only up to an arbitrary scalar JJ which can be identified as a pressure-like -quantity. The constitutive equations (5. 72), (5. 76) and (5.82) are then replaced by

u =-pl+ g(F) =-pl+ Rg(U).RT ,

(5.96)

= -pc- 1 + .sJ(C)

(5.97)

S

·'

where the tensor-valued tensor functions 11(F) 1 g(U) and j](C) need only be defined for the. kinematic constrain.ts detF = 1, det:U = 1 and detC = 1, respective.Iy. The indeterminate tenns -pl and -pc- 1 are known as reaction stresses, which do no work in any motion compatible with above constraints. For incompressible materials the (indeterminate) scalar p required to maintain incompressibility may only be found by means of the equilibrium conditions and the boundary conditions and is not specified

5.4

203

Invariance of Elastic M.aterial Response

by a constitutive equation. Note that the scalar p must always be included in a stress relation of an incompressible material. In an incompressible and isotropic Cauchy-elastic material we replace constitutive equations (5.88) 1 and (5.94) 1 by u

= -pl+ n1 b + n2b2

·U=-pl+f31b+/J_1b-l,

_,

(5.98)

respectively, where the response coefficients n 1, n 2 and /3 1, (3_ 1 depend no~.--dnly on the two scalar invariants 1 1 and 12 (since Ia ·== cletb == 1). Note th(.1t--the. -~-~alars p in eqs. (5.98) differ by the term o:2 h For an incompressible an~Jis6fr~Pic Cauchy-elastic material the stresses given in (5.9.8) are determined ?.~}JY..-·t.fp top. The scalar functions n 0 and /30 multiplying I in (5.88) 1 and (5.94}1 are absorbed into the reaction stresses. The response coefficients in (5.98) are related by 0:2

=

#-1

(5.99)

.

Two special cases result directly from (5.98h, i.e. the so-called M·ooney-RivJ.in model for incompressible materials, for which /3L and (1_ 1 are constants, and the neo . . Hookean model for .incompressible materials, for which /3 1 is -constant and (3_ 1 = 0. For a study of these types of material see more in Section 6.5.

EXERCISES

1. Consider the classical Newtonian fluid, for which the viscous stress depends linearly on the rate of deformation tensor d. It is the simplest model for a viscous fluid and is given by the constitutive equation

u

= [-p(p) + "\(p)trd]I + ·21J(p)d

,

characterizing (low molecular weight) liquids and gases such as water, oil or air~ Therein, the function p(p) depends on the spatia.l mass density p, and A and ·17 are two parameters characterizing the vis·cosity of the Newtonian fluid. (a) Show that the response of this type of fluid conforms to the principle .of material frame-indifference. (b) Apply the Newtonian constitutive equation to a motion which causes simple shear deformation {see eq. (2.3)). Take a constant viscosity 'IJ and show that the only non-vanishing shear stress a 12 ·is o-12

= a~n = 'TJC

'

with the shear rate i~ (compare with Exercise 4(a),(b) on p. 105).

(5.100)

204

5

Some Aspec.ts of Objectivity

Note that the Newtonian viscous fluid is simply a special case of the Reiner-. Rivlin fluid (5.92) obtained by choosing the response coefficients eta, a= 0, 1, 2, as a 0 = -v(p) + A(p)trd, a 1 = 217(p) and a 2 = 0, respectively. However., by setting a 0 = -p(p) and o:·1 = a 2 = 0 we obtain the constitutive equation u = -p(p)I characterizing the material properties of an elastic ·fluid. 2. By recalling the transfonnation rules (5.41), (5.31) and (5.40h, (5.45h (from Sections 5. l and 5.2), show that the constitutive equations

ir

= Ju + ulT + ad

are acceptable forms which conform to the principle of material frame-indifference, where a is a material parameter and a_, I and d denote the spatial acceleration field, the spatial ·velocity gradient and the .rate of deformation tensor, respectively. ······ ..

3. Consider a material which is isotropic and Cauchy-elastic. Let a uniform extension (or compression) in all three directions according to relation (2·~ 129), which -corresponds to a triaxial stress state, ·be given.

By detenninin.g the deformation gradient F and the left Cauchy-Green tensor b (with respect to a set of some orthonormal basis vectors ea) find the most general representation for the Cauchy stress tensor CT. 4. By means of a pull-back of (5.88) to the reference configuration .and a scaling with the .inverse volume ratio J- 1 , show that for isotropic elastic materials the second Piola-Kirchhoff stress tensor S is coaxial with the .right Cauchy-stress tensor C if.and only ~"{ u is coaxial with b.

6

Hyperelastic Materials ,

.....' '

··'·"'·

..... ., ...··

·····'·'' __ _,..,..,,, ...' ' J .......... :··

The fundamental equations .introduced in Chapters 2-4 are essential to characterize kine-matics, stresses and balance .principles, and hold for any continuum body for all times. However, they do not distinguish one material from another and remain val.id in all branches of continuum mechanics. For the case of deformable bodies the equations mentioned are certainly not suffkient on their own to detern1ine the material response. Hence, we .must establish additional equations in the :form of appropriate constitlllive laws which are furnished to specify the ideal material .in question. A constitutive law should approximate the observed physical behavior of a real material under specific conditions of interest. Generally we use a functional relationship as a .constitutive equation and this enables us :to specify the stress components .in terms of other field functions such as strain and temperature. A constitutive equation determines the state of stress at any point x of a continuum body at time t and .is ne·cessari.ly different for different types of continuous bodies. Each field of continuum mechanics .deals with certain .continuous media including fluids, which are liquids or gases (such as water, oil, .air etc.) and solids (such as .rubber, metal, ceramics., wood, living tissue etc.). If the constitutive equations are valid for physical objects such as fluids we call the field of continuum mechanics -Ouid mechanics. Another important field in which constitutive equations are valid for solids is known as solid mechanics. Note that fluid and solid mechanics differ only with respect to constitutive equations, but they share the same set of field equations. The main goal of the next two chapters is to study various constitutive equations within the field of solid mechanics appropriate for approximation techniques such as the finite ele·ment .met.hod. For the most part we follow the so-called phenomenological approach, describing the macroscopic nature of materials as continua. The phenomenological .approach is mainly concerned with fitting mathematical equations to experimental data and is particularly successful in solid 1nechanics (such as classical elastoplasticity). However, phenomenological modeling is not capable of relating the mechanism of deformation to the underlying physical (microscopic) structure of .the material. 205

.206

6 Hyperelastic l\tlaterials

ln Chapter 6 we discuss phenomenological constitutive equations which interrelate the stress components and the strain components within a non.Ii.near regime. Since we are studying so-called purely mechanical theories, thermodynam.ic variables such as the ·entropy and the temperature are ignored. However, they are taken into account and elaborated on within Chapter 7. No attempt is made to present a comprehensiv·e list of the various important contributions on constitutive modeling to date. This text .discusses some selected material models essential in science, industrial engineering practice and in the :field of biomech.anics, where structures exhibit large strain behavior, very often within the coupled thermodynam.ic regime. Sections 6.1-6.8 consider hyperelastic materials in general, and cover a wide range of important types of material such as isotropic and transversely isotropic materials., incompressible and compressible hyperelastic materials and composite materials, in particular. Some important specifications for rubber-like (or other) materials are also presented. The remaining Sections 6. 9-6 . 11 focus attention on .inelastic materials . Based on the concept of .internal variables., viscoelastic materials and isotropic hyperelastic materials with damage at finite strains are introduced. Plastic and viscoplastic mate~ rials which have the ability to undergo irreversible or .permanent deformations are not co.nsiderd in this text.

6.1

Gen·eral Remarks on Constitutive Equations

It is the aim of constitutive .theories to develop maLhemat.ical models for representing the real behavior of matter. Constitutive theories of materials are very important but they are a difficult subject in modern nonlinear continuum mechanics. We make no attempt to conduct a comprehensive review of the large number of constitutive theories. For more on formulating nonlinear-constitutive theories see, for example, TRUES.DELL and NOLL [1992] and the excellent contributions by Rivlin in the 1940s and 1950s collected by BARENBLATT and JOSEPH [1997],. In particular, we present a nonlinear constitutive theory suitable to describe a wide variety of physical phenomena in which the strains may be large, i.e. finite. For the case of a (hyper)elastic material the resulting theory is called finite (hyper)efasticity theory or just finite (hyper)elasticity for which nonlinear continuum mechanics is the fundamental basis (see GREEN and ADKINS [1970] for an analytic.a] treatment .and inter alia LE TALLEC [1994] for numerical solution techniques). Constitutive equations for hyperelastic materials. A so-called hyperelastic material (or in the literature often called a Green-elastic ·material) postulates the existence of a Helmholtz free ..energy function \JI, which is defined per unit reference volume rather than per unit mass.

6.1

207

General Remarks on Constitutive Equations

For the case in which \JI = W(F) is solely .a function of ·F or some strain tensor, as introduced in Sect.ion 2.4, the Helmholtz free-energy function is referred to as the strain-energy function or stored-energy funcUon (see Section 4.4, p. :159). Subsequently, we often use the common terminology ·strain energy or stored energy. The strain-energy function \II = \JI (F) is a typical example of a scalar-valued function of one tensor variable F, which we assume to be .continuous. A concept of importance in elasticity is polyconvexity of strain-energy functions. The global existence theory of solutions, for example, is based on the condition of po:lyconvex.ity of strain-energy functions. For an extensive discussion on the underlying issue, see BALL [ 1977], CIARLET [.1988, Chapters 4, 7], "MARSDEN and HUGHES f.1.994, Section 6.4] and SlLHAVY [1997, Sections ]7.5, 18.5]. We now restrict attention to homogeneous materials in which the distributions of the internal constituents are assumed to be uniform on the continuum scale. For this type of ideal material the strain-energy function ~1' depends only upon the deformation gradient F. Of course, for so-called heterogeneous materials (a material that is not homogeneous) \JI will depend additionally upon the position of a po.int in the medium. A hypereh1stic material is defined as a subclass of an elastic material, as given in eqs. (5.77)a and (5.72), whose response functions 6 and g have physical express.ions of the form

p == ,l5 (F) ·= 8\JI (F)

or

DF

Dw

and by use of relation (3.9) for the symmetric Cauchy stress tensor, i.e. u T

·U

(6.1.)

PciA = DF.a A ' ==

,J- 1PFT =

'

(T

= g(F)

or

= .J _

lf11b -

_ aw(F).

DF F

I

J

_1

FAti

T

= .l

-l

(aw(F)) · F DF T

aw -_ J -.• FaA aw ~

BF

Ah

(6.2)

!-)

UlTbA

These types of equation we already know as (purely mechanical) constitutive equations (or equations of state). They establish .an axiomatic or empirical model as the basis for approximating the behavior of a real materiaL Such a model we call a material model or a constitutive model. As is clear from the ·Constitutive equations (6.1) and (6.2) the stress response of hypere:lastic materials is derived from a given scalar-valued ene1~gy function, which implies that hyperelasticity has a conservative structure. The derivative of the s.calar-va:lued function \}I with respect to the tensor variable F determines the gradient of w and -is understood according to the definition introduced in ( 1.239) ..It is a second-order tensor which we know as the first Pio la-Kirchhoff stress tensor P,. The derivation .requires that the component function w(F;.A) is differentiable with respect to .all components Fa...i.

208

6 Hyperelastic ·Materials

A so-ca1led perfectly elastic material is by definition a material which produces

locally no entropy (see TRUESDELL and NOLL [.1992, Section 80]). In other words, we use subsequently the term 'perfectly' for a certain class of materials which has the. special merits that for every admissible process the internal dissipation Vint .is zero (naturally, damage, viscous mechanisms and plastic deformations are excluded). We will consider perfectly elastic materials up to Section 6.9. We may derive the constitutive equation (6.1) directly from the Clausius-Planck form of the second Jaw of thermodynamics (4.154) whkh degenerates to an equality for the class of perfectly elastic materials. With the expression (4 . 154.) for .Vint the time differentiation of the strain-energy function, i.e. qi (F) = aw (F) /8F : F' gives 'Dint

= .P : F.· -

.

\JI =

( p - .8\Jt(F)) · 8F :F

=0

,

(6.3)

at every point of the continuum body and for all times during the process. As F and hence F can be chosen arbitrarily, the expressions in parentheses must be zero. Therefore, as a consequence of the second law of thermodynam.ics, the physical expression (6. I) holds. We often say that P is the thermodynamic "force lvork conjugate to F. This procedure goes back to COLEMAN and NOLL [1963] and COLEMAN and GURTIN [ 1967], and in the literature is sometimes referred to as the Coleman-Noll :procedure. For convenience, throughout this text we require that the strain-energy function vanishes in the reference configuration, .i.e. where F = I. We express this assumption by the normalization condition

W= \JI (I)

=O

.

(6.4)

From the physical observation we know that the strain-energy function \ll inpreases with deformation. In addition to (6.4), we therefore require that

\JI = \JI (.F) > O ,

(6.5)

which restricts the ranges of adm.issible functions occurriQg in expressions for the strain energy. Tbe strain-energy function \{1 attains its global minimum for F = I at .thermodynamic equilibrium (in fact, fro.m (6.4), ·w(I) is zero). We assume that \JI has no other stationary points in the strain space. Relations (6.4) and (6.5) ensure that the stress in the reference configuration, which we call the .residual stress, is zero. We say that the reference configuration is stressMfree. For the behavior at finite strains we require additionally that the scalar-valued function '1i must satisfy so-called growth conditions. This implies that \JI tends to +oo if

6,.1

either J


209

= detF .approaches +oo or o+, i.e. w{F) ->- +oo

as

detF -+ +oo

w(F)-+ +oo

as

detF

4

, }

(6.6)

o+

Physically, that means that we would require an .infinite amount of strain energy in order to expand a continuum body to the infinite range or to compress it to a point with vanishin_g volume.. For further discuss.ions see the books by, for example, CIARLET [1988] and OGDEN [19.97].

E-quivalent forms of .the strain-energy func.tion. In order to illustrate \JI w.e imagine a stretched (rubber) band with a certain amount of energy stored. The strain energy w{F) _generated by the motion x = x(X, t) is assumed to be objective. This means, after a (possibly time-dependent) translation and rotation of the stretched (rubber) band in space, that the amount of energy stored is unchanged. Hence, the strain energy w(F) must be equal to the strain energy \ll(F+) generated by a second motion x+ = x+(x, t+) which differs from by a superimposed rigid-body motion (recall Section 5.2). Employing the transformation rule for the deformation gradient (5.33)a~ we see that \JI cannot be an arbitrary function of F. In .particular, it must obey the restriction

x

w(F) = w(F+) = ·w-( QF) for all tensors F, with detF

>

(6.7)

0, and for all orthogonal tensors Q, since F transforms

into QF, i.e. Fd;1 = QabFbA· In order to obtain equiva]ent formulations of (6.7) we take a special choice for Q, namely the transpose of the proper orthogonal rotation tensor, RT, and use the right polar decomposition (2. 91) 1 • Then, from (6. 7), we find that \JI (.F) = '11 (RTF) = w(.RTRU), and finally,

w(Jr)

= w(U) ,

(6.8)

which ho"lds for arbitrary F. From (6.8) we learn that \JI is .independent of the rotational part of F = RU. We conclude that a hyperelastic material depends only on the stretching part of F, i.e. the symmetric right stretch tensor U. It is important to note that the relation F) = \JI (U) specifies the necessary and suffic~ent condition for the strain energy to be objective during superimposed rigid-body motions. Since the right Cauchy-Green tensor and the Green-Lagrange strain tensor are given by C = U2 and E (U 2 - I) /2, we may express \JI as a function of the six components CAIJ., EAJJ of the symmetric material tensors C, E, respectively. Hence, we may write

w(

=

w(F)

= w(C) = w(E)

.

(6.9)

210

6

Hyperelastic IVlatcri.als

For notational simplicity, here and elsewhere we wiJl use the same Greek letter \II for different strain-energy functions. Reduced forms of constitutive equations. In the following we present some re·duced forms of constitutive equations for hyperelastic materials at finite strains. Consider the derivative of the strain-energy function ·w (F) = \JI ( C) with respect to time t. By means of the chain rule of differentiation, property (.l .93) 1 and the comb.i .. nation of (2.166) i. and (2.163) J., we obtain the .expressions

. = [(aw(F))T ·]; "[(Dw(C)) DF · F· = DC · C·] .

\JI

tr

= tr

tr

(-T-

~[8\Jl(C) DC F F + FT")J: F . = 2tr (aw(C) . ac FT•) F

!

(6...10)

which must be val.id for arbitrary tensors F. Since C is a syn1metric second-order tensor, the gradient of the scalar-valued tensor function w(C), used in (6.10), is also symmetric. From (6.10) we deduce immediately that

(

D\Jl(F))T = ?Dlll(C) BF

....

T

ac

.F

(6.1 l)

'

which, when substituted back into (6.2)3, gives an important reduced form of the con .. stitutive equation for hyperelastic materia.ls, namely u = J

or

_1

·r (aw(F)) · = 2.J _F Dw(c) F · F

DC

DF

_

aub -

T

1

J

~i

.

aw _ ~J

FaA DI'bA L? .

-

...

J

_1

~~·--

.

Fn ..tFbu

(6.12)

aw

ac.rAJJ.

Alternative expressions may be obtained for the Piola-Kirchhoff stress tensors ·p (which is non-symmetric) and S (which is symn1etric). From (3.8) and (3.65) 1 we .find, by means of the stress relation (6. l2h, the chain rule and 2E = C - I, that

P=2F or

D\Jl(C) ·

ac

,

aw

Pa;1 = 2F'i111 DC. . 1

AIJ

8\J! (C) 2 S = · DC

=

O\JI (E) DE

D\JJ

SAFI== 2 DC

·-'All

a\v

(6.13)

DE.~111

Note that the response function occurring in the general constitutive equation (5.82) is with reference to (6.13h determined by SJ(C) = 2()\:fJ{C)/DC.

6.1


.211

The Es.helby tensor and the tensor of chemical potential. The Eshclby tensor (or the (e.Jastic) energy .. momentum tensor) is a crucial quanti-ty infi·acture mechanics and the continuum theoJ)' <~l dislocatimzs (which are not discussed in this text). However, for completeness we present the (isothermal) ·refe·rential Eshelby tensor G, which is, in general, non-symmetric and is de.fined as

(·q!(F)) G= - JF·T!_ DF J f

or

G An

.

= -J.Fr,/-r DFan/J ( \(~,')

(6~:14)

(see ESHELBY [1.975] and CHADWICK [1975]}, where J = detF denotes the volume ratio4 The physical dimension of the Eshelby tensor is the same .as that of the strain energy. The symmetric tensor of chemkal potential k is related to G according to (6.15)

or

see, for example, BOWEN [l976a]. The spatial tensor of chemical potential is required in the theory of diffusing mix.. tures (see TRUESDELL [ 1984] which contains more details). Work done on hyperelastic mater"ials. We consider a dynamical process within some dosed time interval denoted by [t 1 , t 2 ] .in which the two arbitrary instants t 1 and t 2 are .elements of the interval. The dynamical process is given by a motion x = x(X, i;) and the stress u with the corresponding Cauchy traction vector t and the body force ·b. During the process the body deforms according to the deformation gradient F = F(t), with t E [t1, t2J. We say that a dynamical process is closed if F 1 = F 2. We introduce the definitions F 1 = F(t 1) and ·F2 = F(t2 ) of the deformation gradients in the initial configuration and the final configuration of the dynamical process, respectively. Our nex.t step is to determ.ine the work done by the stress field on a continuum body of unit volume during a certain time interval ft t, t 2 ]. Consider a body whose material properties .are hype.relastic according to the general constitutive equation given in (6. l ). Hence, from (4.1.0.l ) 1 and the above definitions we find by m-eans of the chain rule that f•l

'

.

/-: Dw(F)

tJ

tl

. P:Fdt=.

/

l1

·t·J

t·.•)

/-; .a~u .

DF:Fdt=.

Dt

which_, for a closed dynamical process with F 1

dt=\ll(F2)-W(F1),

(6.16)

= F 2 , reduces to

I.',!

Ip: lJ

Fdt = W(F2) - \J!(Fi)

=0

.

(6.17)

212

·6

Hyperelastic Materials

Thus, as distinct from Cauchy-elastic materials (see Section 5 .4 ), the actual work done by the stress field on a hyperelastic material during a certain (closed) time interval depends only on the initial and.final configurations (path independent). In fact, the work is zero in closed dynamical processes. This important result also holds for continuum bodies which may undergo inhomogeneous deformations, in which W = \Jl(F, X) and Po

= Po(X), with F = F(X, t). EXERCISES

l . Expand the strain-energy function \JI (F) in the form of tensorial :polynomials in the vicinity of the reference configuration, i.e. for F = I,

W(F)

= W(I} + (F -

I) :

a~r) + · ·-> 0

for all F - I. Using relations (6.4) and (6.5) show that the stress in the reference configuration is zero. 2. Recall the definitions of the referential Eshelby tensor G and the tensor of chemical potential k. (.a) Show that the forms

G ·= w{F)I - JFTuF-T ,

·:k

= w(F)I -

are equivalent to those given in (6.14) and (6.15), where stress tensor, as defined in eq. (3.62).

T

r

(6.18)

is the Kirchhoff

(b) By applying requirement (3.10), show with (6.18) 1 that

GC

= CGT

with the right Cauchy~Green tensor C

'

= FTF, as defined in eq. (2.64).

6.2 Isotropic Hyperelastic Materials We now restrict the strain-energy function by a particular .property that the material may possess, namely isotropy. This property is based on the physical idea that the response of the material, when studied .in a stress-strain experiment, is the same in all directions. ·One example of an (approximately) isotropic material with a wide range of applications is rubber. In this section we are concerned with the mathematical formulation of isotropy within the context of hyperelasticity.

6.2

213

Isotropic Hyperelastic Materials

Scalar. . valued isotropk tensor function.

We consider an arbitrary po.int X of an elastic ·continuum body occupying the region 0 0 (reference configuration) at time t = 0. A motion X may carry this point X E fl 0 to a place x = x(X, t) specifying a location in the region fl (current ·Configuration) at time t. We now study the effect of a rigid-body ·motion superimposed on the reference co1rfiguration. We postulate that the body occupying the region !10 is tmnslated by the vector c and rotated by the orthogonal tensor Q according to

X*

= c+QX

,

(6.19)

no

whic'h moves no to a new region (new reference configuration), and the arbitrary point with position vector X to a new location identified by the position vector X* E f2 0 {see Figure 6.1).

' Xr*3 ' :i.3 '"• X 3'

time t

=0 time ·t

x(X, t) = x*(X*, t)

Figure 6.1 Rigid .. body motion superimposed on the reference configuration.

We now demand that configuration so that

n

a different motion x = x* (X*, t) moves x

= x(X, t) = x*(X*, t)

,

n~ to the current

(6.20)

mapping X* to place x. By the chain rule and relation (6. l 9) the deformation gradient

214

6

Hyperelas.tic Materials

F may be expressed as F= or

-1

_

Fa.4 -

i; = :~Q=F*Q

fJ:ru _ D:ra (l . a·· ~~ - a·· ~-:"'* ·1t JJA ...·\.A _:\.. ll

(6.21)

F* ) .

_ -

a.11(" JJA

where F* = .Dx/Dx*· is defined to be the deformation gradient relative to the reg.ion 0 0. From (6 . 21 )a we find the important transformation, namely or

* -FaA

I;

/"""} . .

r u/J'-·t: AB

..

(6.22_)

Hence, we say that a hyperelastic .material is isotro.pic relative to the reference configuration n0 if the values of the strain -energy \JI (F) and \JI (Fk) are the sa.me for all orthogonal tensors ·Q. With (6.22) we may write \ll(F)

= w(F*) = w(FQT)

.

(6 ..23)

I.n other words, if we can show that a mot.ion of an elastic body superimposed on any particularly translated and/or rotated reference configuration leads to the same strain-energy function at time t, then the material is said to be .isotropic. However, if a superimposed rigid-body motion changes the strain-energy function in the sense that (6.23) is not satisfied (W(F) ~ w(F*)) the hypere.lastic material is said to be :anisotropk (see, for example, TRUESDELL and NOLL [1992, .Section 33] and ·OGDEN .[ 1997, Section 4.2~5]).. lt is important to mention that .relation (6.23) is .fimdamentally distinct fro.m requirement {6.7), which says that the strain energy .must be objective during rig.id-body motions, i.e. independent of an observer~ The later condition hoJds for all materials {it is a fundamental physical requirement and must be satisfied) while the condition of isotropic response (6.23) holds only for some mate.rials (it -~s a material-dependent .requirement and may or may not be satisfied), namely for .isotropic materials, wh.ich ,makes a crucial difference. I.n addition, it is important to note that for the material-dependent requirements (6.22) and (6.23) it is the reference configuration that has been translated and rotated. For that case the deformation gradient Fis multiplied ·on the right b_y ·QT, whereby Q acts with material coordinates, i.e. CJAH· However, for the objectivity requirements (5.33h and (6.7), it is the current configuration that has been translated and rotated, and F .is multiplied on the left by Q, acting with spatial coordinates, i.e. Q 011 • We now suppose that during .motion x == x(X, t) the strain-energy function may adopt the form w(F) = w(C); recall eq. (6 . 9)i. If we restrict the hyperelastic material to isotropic hypere.lastic response we require that '11 (C) = \JI ( C* ), with C* = F*'rF*. With reference to (6.22) we conclude that (6.24)

6.2

lsotro.pic Hyperelastic Materials

implying with the right Cauchy-Green tensor C

215

= FTF, that (6.25)

If the requirement for -isotropy (6.25) holds for all symmetric tensors ·C and orthogonal tensors Q, we say that the strain-energy function \V (C) is a scalar-valued isotropic tensor function of one variable C or simply an .invariant of the symmetric tensor C. We can show, if the strain-energy function is an invariant, then its gradient .is a tensor-valued isotropic tensor function.

EXAMPLE 6.1 Assume that the hyperelastic material is restricted to isotropic re . . sponse. Show that the strain ene(gy may be expressed by the identity

\ll(C)

= W(b)

,

(6.26)

where W(b) characterizes an isotropic function of the (spatial) left Cauchy .. Green tensorb =FFT . Solution. We substitute for Qin condition (6.25) the proper orthogonal rotation ten . . sor .R. Then, kinematic relation (2. l07):i implies (6.26), which holds for any isotropic • deformation.

Constitutive equations in terms of invariants. If a scalar-valued tensor function is an .invariant under a rotation, according to (6.25), it may be expressed in terms of the principal invariants of its argument (for example, C orb), which is a fundamental result for .isotropic scalar functions, known as the representation theorem for inva.riants (for a proof see, for exam.pie, GURTIN [.198.la, p. 231] or TRUESDELL and NOLL [l 992, Sect.ion l OJ) . Having this in mind, the strain energies, as established in eq. (6.26), may be expressed as a set of independent strain invariants of the symmetric Cauchy-Green ten~ sors C and :b, namely, through ! 0 = Ia(C) .and J,1 = Iu(b), a = 1, 2, 3, respectively. With reference to {6.26), we may write equivalently

(627)

Again, (6.27) is exclusively valid for isotropic hyperelast.ic materials satisfying condition (6.25) for all orthogonal tensors·Q. Since C and b have the same eigenvalues, which are the squares of the principal stretches,,\?,., a = 1., 2, 3, we conclude that (6.28) where the three principal .invariants are explicitly given in accordance with .eqs. (5.89)-

216

6 .Hyperelastic Materials

(5.91). Note that for the stress-free reference configuration, the strain-energy functions (6.27), with (5.89)-(5.91), ·must satisfy the normalization condition (6.4), i.e. '11 = 0, for 11 = 12 = 3 and / 3 = 1. The representation in the form of invariants was established in the classical work of RIVLIN [1948]. In order to determine constitutive equations for isotropic hyperelastic materials in terms of strain invariants, consider a differentiation of w{C) = \J!(Ii, 12, 13 ) with respect to tensor C. We assume that \ll(C) has continuous derivatives with respect to the principal invariants Ia, a = 1, 2, 3. By means of the chain rule of differentiation we find 3 aw(C) = aw OI1 + aw aI2 + aw 8!3 = I: aw Bla

811 8C

8C

812 8C

8Ia 8C

a=I

.

8Ia 8C

(6.29)

The derivative of the first invariant / 1 with respect to C, as needed for (6.29), gives with (6.28_)i, (5 ..89) 1 and the .property (1.94) of double contraction

811

ac

= 8trC = ac

8(1 : C)

ac

=1

8/i

or

acA.n

8

=

AB

•

(6 .30)

The derivatives of the remaining two invariants with respect to C follow from eqs. (5.90) 1 and (5.9l)i, by means of (6.28), (6.30h, the chain rule and relations ( 1.252)2, ( 1.241) (use the symmetric tensor C in the place of A), and have the forms

812

ac

or

= ! (2 2

2

tr

CI- 8tr(C

812

8cAB

ac

=

I

.r

}U ..4B

-

))

C

=I1 l- C

'AB

·'

'

8/3 ac· AB

ala= I 3 c-i

ac

=.::

I

:1

(6.31)

c-1 AB

.

Substituting (6.29)-(6.31) into constitutive equation (6.13) 2 gives the most general form of a stress relation in terms of the three strain invariants, which characterizes isotropic hyperelastic materials at finite strains, i.e.

s=

9 BW(C) ... 8C

=?.[(aw 1 aw) 1 _aw c 1 aw c-i.] 811 + 812 812 + 81 1

3

(6.32)

3

The .gradient of the invariant w{C) = \J!(li., 12 , / 3 ) has the simple representation {6.32), which is a fundamental relationship in the theo.ry of finite hyperelasticity. Note that (6.32) is a general representation for three dimensions, in which \II may adopt .any scalar-valued isotropic function of one symmetric second-order tensor variable. Multiplication of (6.32h by tensor C from the right-hand side or from the left-hand side leads to the same result. We say that 8\f!(C)/BC commutes (or, is coaxial_) with C in the sense that

aw(qc = caw(C) ac ac ' which is an essential consequence of isotropy.

(6.33)

6.2

217

Isotropic Hyperelastk Materials

Next, we present the spatial counterpart of constitutive equation (6.32). According to relation (3.66), the Cauchy stress u follows from the second Piola-Kirchhoff stress S by the Piola transformation u = J- 1FSFT. By multiplying the tensor variables 1 I., C, with F fro-m the left-hand side and with FT fron1 the right-hand side, we may write by means of the left Cauchy-Green tensor b = FFT, that FIFT = FFT = b, FCFT = (F-FT) 2 ·= b 2 , Fc- 1 FT = (FF- 1){F-TFT) =I. With (6.32) we deduce from u = J- 1 FSFT that

c-

_ 'J

u - ~J

_ 1 ·[ .

aw

Ia 8!3 I+

( aw

aw ) . _ aw

811 + 11 812 b

2]

(6.34)

8/2b

Following arguments analogous to those which led from (5.88) to (5.94), we find an alternative form to (6.34), namely

u

= 2J-1.

((12 8!2 aw +

/3

aw) 1 + aw h - Ia awh- 1] 8/3 8!1 . 812

(6.35)

By comparing (6.34) and (6.35) with (5.88) and (5.94) we ·may derive the response coefficients aa, .a= -0, 1, 2, and {30 , a= 0, 1, -1, in terms of the strain-energy function (6.27). We obtain the forms

a'll + Ir 8I2 aw)

_1 ( o:i = 2J 8I1

' (6.36)

1 aw /3 1 = ?.J_ .... 811 ,

(6.37)

which specify the first and the second representation theorem for isotropic tensor functions, .i.e. (5.88) and (5.94), respectively. Note that in order to formulate constitutive equations which are not restricted to isotropic response and which satisfy the objectivity requirement, it is ~ppropriate to use ..quantities which are referred to the reference configuration. Within the context of hyperelasticity it is obvious to use the right Cauchy-stress tensor C and its work co~jugate stress .field, i.e. the second Piola-Kirchhoff stress tensor S.

Constitutive equations derived from '11 (b) and '!I (v).

If the strain-energy function depends on the (symmetric) left Cauchy-Green tensor b, then the isotropic hyperelastic response is

u

= ?J-1 ·B\J1(b)b = ?.J-1b8\Jl(b) ~

8b

..

8b

or

(6.3.8)

218

6


(see also TRUESDELL and NOLL [1992, Section 85, p. 313]), where w{b) is .a scalarvalued isotropic function of the tensor variable b = FFT. This constitutive equation plays an essential part in isotropic finite hyperelastici.ty. On comparison with eq. (5.85) we deduce that the right-hand side of eq. (6.38) corresponds to the response function

IJ(b). Proof

In order to obtain the constitutive equation (6.38), we start by differentiating the postulated stra:in-ene.rgy function \V (b) with respect to time t. Considering symmetries, a straightforward algebraic manipulation gives, by means of the chain rule, relution (2.169) and the property (1.95) of double contraction,

¢ = D\Jt(b) . b = D\J!(b) . (lb .

Bb

.

.ab

.

+

_ aw (b) . lb _ ')aw (h) h . - -9 ob . - ... Db . . 1 .

bfr).

1

(639)

where 1 is the spatial velocity gradient, .in general a non-sym1netric tensor. With respect to eqs. (6.32) and (6.33), taking b .in place of C, we deduce that D\JI (b) / 8b commutes with the symmetric s·econd-order tensor b, in the sense that

aw(b)

ob

implying the sym·metry of tensor

aw(b)

(6.40)

b = b ah · ,.

(aw (b) /Db) h~

However, from (4.154), we know that for perfectly elastic mater"ials (for which Vint = 0) the stress power Wint per unit reference volume equals \JI. The combination of identity (4J 13)h i.e. Wint = .Ju : d, and eq. (6.39) 4 with the requirement (6.40) for isotropy implies ~Ju

.d .

= ?aw{b)b ~ d = ?baw(b) - Db . ~ ob

.d . '

(6.41)

where the rate of deformation tensor d is the symmetric part of I. For this relation we used the fact that the double ·Contraction of a symmetric tensor and a skew tensor is

zero. Simple arguments ·reduce .relation (6.41) to the desired fundamental constitutive • equation (6.38) for ·isotropic response. Since the ·1eft stretch tensor v = b 112 is the unique square root of b, the strain energy may also be expressed as an isotropic function of v; thus, relation (6.26) may be extended to 'P (C) = \JI (b) = '1T (v). Note that for notational convenience, we do not distinguish between different strain-energy functions \Ji. With .respect to constitutive equation (6.38) we derive an equivalent form in terms of the left stretch tensor v. Analogous to the procedure which was used to establ.ish

6.2

219


eq. (6.1 l) we find that 9

.8\Jl(b) _ aw(v) _1

~ 8b

-

av

v

(6.42)

'

so that ( 6 .3 8) reads

_ J_ 1 8\Jl(v) _. J-J. 8\ll(v) ua··v v-- v 8 v ,

(6.43)

which is another important stress relation characterizi~g the behavior of isotropic hyperelastic materials at finite strains. Note that, since v is the unique square root of b, aw /8b also com:mutes with v. Constitutive equations -in terms of principal stretches. If the strain-energy function \JI is an invariant, we may regard \JI as a function of the principal stretches ,\a, a = 1, 2, 3. In the place of (6.27), we .may rep.resent W in the form

(6.44) For the stress-free reference configuration the normalization condition (6.4) takes on the form '11(1, 1, 1) = 0. Consider the left stretch tensor v = b 112 describing the -defonned state of an isotropic hyperelastic material. .From the e-igenvalue problem (2. l l 6h we know that ,,\a denote the three principal stretches (the real eigenvalues) of v. Since the principal directions of v coincide with those of b (compare with eqs . (2..116.h and (2 . .l l7h) they also coincide with the principal directions of the Cauchy stress tensor u (recall representation (5.88)). Consequently, with respect to (6.43) the principal Cauchy stresses aa, .a = 1, .2, 3, simply result in

aw _, "o.

a a = J- Ao a\ 1

a = 1, 2, 3 ,

(6.45)

with the volume ratio (6.46) according to (.5.9.lh. In addition to (6.45), we introduce equivalent relations for the three principal PiolaKirchhoff stresses Pn and Sa, namely ~,

aw = a,..,,1 \ ,

1 aw s,l = ,An a·''a \ ,

a

= 1, .2, 3

<6.41)

(compare with the fo.llowing Example 6.2)., which may be expressed in terms of the Cauchy stresses (6.45) as a

= 1, 2, 3

..

(6.48)


220

Constitutive relations (6.45) and (6.47) show that principal stresses in an isotropic hyperelastic material depend only upon the principal stretches. They are simply obtained by differentiating the strain-energy function with res.pect to the corresponding principal stretches. -~,,~,,-~-·· ----·'""''''"'''•:"''''"'''''"'"'"'"''"'"'J."o.. •''''''''''•·'-"•"''Y•'•','•."'-';,,,,,, •• ,• .. •,,,,.,-- ..•"'~':"V'~'"'""" _ _ _ ,.,,,,•.,,, .... ':Y'''''"'••-,••'"'.''-'"':':''~-·'--'''''"'"''":'''''''' ... '•-'•' .. '•'''-.','-''•"''':•'"''"'•'''•""'•'•''•'•'•'•"'''''•''•'''''••'•'•'''''""''','•:'•'''."•'•'•••"'"•"•'•'""''•'"""•'•""'

EXAMPLE 6.2 Consider the strain energy w(C) stitutive equations in the spectral fonns

= w(A 1 , .1\ 2 , A3 ).

Obtain the con-

:1

O'

=L

(6.49)

aaDa ®Ila ,

a=.l

:1

3

P ==

L Pciiia ® Na a=l

,

S=

L SaNa ® Na

,

(6.50)

n=l

where Ga and Pc.., Sa, a = 1, 2, 3, are the principal values of the Cauchy stress tensor er and the two Piola-Kirchhoff stress tensors P, .S according to the expressions (6.45) and (6..47)., respectively. The orthonormal vectors Na and Du RNa, a = 1, 2, 3, denote t~e principal referential and spatial directions (axes qf stretch), respectively.

=

Thes.e constitutive equations describe isotropic response of byperelastic materials and hold if and only if ;\1 '# ..\2 # .Ai1 # /\1.

Solution.

We start with constitutive equation (6.50) 2 which is expressed in tenns of second Pio.la-Kirchhoff stresses Sa, a = 1, 2, 3. We compute the derivative of the isotropic function w{C) with respect to the symmetric tensor C. By means of the chain rule and kinematic relation (2 . .123), we obtain for .the general case ~\ 1 # A2 # .1\~ =f )q, (6.51.)

N

In (6.51 )2, .,,\~ are the eigenvalues (the squares of the principal stretches) and 0 the corresponding eigenvectors (principal .referential directions) of C (compare with the eigenvalue problem (2.115)). With (6.5 lh and the chain rule we find from (6.1.3) 2 that

(.6.52)

which gives the desired .results (6.47h and (6.50h. By the use of ( 6.52) 2 , the relation according to ( 1.58.) and eq. (2 . .132) 1, i.e. FN 0 = Aufi,u a = 1, 2, 3, the spectral fonn of .the first Piola-Kirchhoff stress tensor ·p may be

6.2

221


found from transformation (3.67) as

(6.53)

Similarly, having in mind the results (6.53) 4 and (2.132) 1, trans.formation (3.9) gives, using (3.10), the spectral form "' = J

-·1 FPT= .J

-1

~ aw ,. - T) = ~ -1. aw ,. . ~ f);\ (na 0 Na) ~ J ;\a[)).. Ila 0 Ila

(

F

a='l

(6.54)

lt=l~

a

of the Cauchy stress tensor, where the property 'F(na ®Na)'r = (FNa) C8H1a, a= l, 2, 3, • was used (compare with relations which are analogous to (1.85).and (.1.58)). __ •..-:·-· -·,.,,..... ...... _,,,.,,,,,...,,.,,.. .. ... -,.- -,,. ... .......... ..... _..... .... __ ____ ___ ___ . __ .. - .. -,-- ... --,-·-,, - .... . --,,,-,,... ,,, ...

...

--·-·-·-,·~·----'~,····"'~

.... ,,._,,,, ... ;,,,:,,, ... ;,,:,,,,,,:.,.,:,,,,:,,,,..,,, .... ,,.,, ... ,,

~,,,.,,,,,..,,,_.:.•

,

.... , ...

_._,,..,.,,.,,,,., ...

~,,.,,,,,,.

,,..,._

,

,

,

,,

,

... _.,

...

·.'~···

-:: ...

EXERCISES

l. By analogy with the procedure which led to (6.11 ), obtajn the eq. (6.42) and relation

(

8\Jt(F))T = 2FT8W(b)

aF

ab

(6.55)

·

2. Rewrite .the spectral representations of constitutive equations (6.49) and (6.50) for a given strain-energy function of the particular form w = '11(ln.A 1, ln . \. 2 , ln,\ 3 ). Consider the general case A1 # /\2 # /\3 # /\ l • 3. Take the strain energy \JI as a function of the principal stretches characterizing the behavior of isotropic hyperelastic materials. Let at least one principal stretch be equal to the other. (a) For the case in which we have two equal principal stretches, namely --\ 1 = ,\ 2 f. .A:1, obtain the constitutive equations u

= ·,

-1.,

.111

aw

P = DAi 1

aw(,..

D>..i

~ (n1

. . . . ) _ , aw + n2 0 n2 + .1 1 ;\3 a>..3 03 0 A

0

-

. ,. aw " + 02 0 N2) + a>..3 o 3 0 N3

0 N1

aw . .

,.

n1

n1

....

n:i

A

"

A

..

1

.aw .

,

'"'

S = ---(N1 ·®Ni+ N2 ® N2) + ---N3 0 Na . .A1 8A1 /\3 8,\i

,

222

6


(b) Using the property (l .65)2 for the second-order unit tensor "I and relation (2. l 22h for the rotation tensor R, show that for A1 = ,.,\ 2 = ~.\:i == A 3

u = a

L Da 0 Da = al

l

Cl= 1

~i

:J

P

=PL ·Da 0 Nu = PR

,

S=

SL N, 0 Na = SI 1

,

a=J

a="I

with the scalar-valued scalar functions er 8,T! I DA, s ·= ,\- 18\J! /D,\.

=

J- 1;\8'11 / [)/\ and with P

6.3 Incompressible Hyperelastic Materials Numerous polymeric mate.rials can sustain finite strains without noticeable volume changes. Such types of material may be regarded as incompressible so that only .isochoric motions are possible. For many cases, this is a common idealization and accepted assumption often invoked in continuum and computational mechanics. In this sect.ion we present the constitutive foundation of incompressible hyperelastic materials. 1-ncompressible hyperelastidty. Materia]s which keep the volume constant throughout a motion are characterized by the incompressibility .constraint

J = 1 '

(6.56)

or hy some other equivalent expressions according to (2.1.77) (recall the expression (2.5 l) for Lhe volume ratio J). In general, a material which is subJected to an internal constraint, of which incompressibility is the most common, is referred to .as a .constrained material. In order to derive general constitutive equations for incompressible hyperelastic materials, we may postulate the strain-energy function \JI= w(F) - p(J - l) ,

(6.57)

where the strain energy \II is defined for J = det"F = 1. The scalar JJ introduced .in (6.57) serves as an indeterminate Lagrange .m.ultiplier, which can be identified .as a hydrostatic pressure. Note that the scalar v may only be determined from the equilibrium equations and the boundary conditions. It repres·ents a workless reaction to the kinematic constraint on the deformatio.n field.

6.3 .Incompressible .Hyperelast.ic Materials

223

Differentiating eq. (6.57) with respect to the defonnation gradient F and using identity (2.174), we .arrive at a general constitutive equation for the first Piola-Kirchho.ff stress tensor P. Hence, eq. (6..1) may be adopted in the form

p

= -11F-T + _ol_Jl_(F_) . DF

.

(6.58)

An alternative derivation of (6.58) is obtained by reference to the express.ion (6.3) 2 • For :incompressible hyperelasticity, F is not arbitrary anymore and the e.xp.ressio.ns .in parentheses of (6.3)2 need not be zero. Howeverl (63}2 must be satisfied for every F which is governed by the incompressibility constraint in the form of J = :F-T : F = 0 (recall (2.177);.1). Consequently, adding the zero term to (6.3h, we find that

-T) F· = 0 .

8\JJ (F) . ( .P - DF + pF ·

:

(6.59)

·with standard arguments, the Coleman-Noll procedure implies physic.al expression (6.58) . .Multiplying eq. (6.58) by F- 1 .from the .left-hand side, we conclude from (3.·65h that the second Piola-Kirchhoff stress tensor Stakes on the form (6.60) where the inverse of relation (2.63), i.e. c- 1 = F- l F-T, and identity (6.11) are to .be used. However, multiplying eq. (6. 58) by FT from the right-hand side, we conclude from (3.9) that the symmetric Cauchy stress tensor
= -1JI. + 8\J!(F) DF F:I = -JJI n

(J'

.

_L

r

F.

(ow(F))T DF

(6.61)

The :fundamental constitutive equations (6.58), (6.·60) and (6.61) are the most general forms used to define .incompressible hyperelastic materials at finite strains. Equations (6.60h and (6 . 61) are associated with (5.97) and (5.96) . Note that the response functions SJ(C) and g(F) occurring in (5.97) and (.5.96) 1 are identified by jj(C) = 2D\J!(C)/8C and g(F) = (O\Jl(F)/8.F)FT = F(8'1r(.F)/8Ffr, i.e. constitutive equation (6.2h for J = 1. Incompressible isotropic .hyperelasticity. For the case ·of isotropy we have already pointed out that the dependence of 'It on the Cauchy-Green tensors C orb may be ex .. pressed by their three strain invariants (see eq. (6.27)). However, for the incompressible case we consider the kine.matic constraint, namely .la = detC = detb = l. Therefore,

the two principal invariants / 1 and 12 are the only independent deformation variables.

224


For a review on the theory of incompressible isotropic hyperelasticity see, for -example, OGDEN [l.982, 1986]. A suitable strain-energy function for incompressible isotropic hyperelastic materials is, in view of ( 6.27), given by

'1t = W[I1(C),l2(C)] -

~p(Ia -1) =

W[I1(b)J2(b)] -

~p(h -1)

,

(6.62)

where p/2 serves as an indeterminate Lagrange multiplier. In order to examine the associated constitutive equation in terms of the two principal strain invariants / 1 ., / 2 ., we proceed by deriving (6.62)i with respect to tensor C. Analogous to the procedure which led to (6.32) we find, using the chain rule, eqs. (6.30)J, (6.31) and the constraint ! 3 = 1, that

s = .9aw(I1.,I2) ...

ac

a"{p{J3 -1)] = _ c-1 '>(aw

_

ac

aw) 1 _9awc

+ - a11 + 11ar2

P

-ar2

<

' 663 · )

which is basically constitutive equation (6.32), in which the term / 3 { 8\J! / 813 ) is substituted by -p /2. A push-forward operation of (6.63h and an elimination of b 2 in favor of b- 1 (see relation (5.93)) yields two alternative forms of u_, corresponding to eqs. (6.34) and (6.35), namely

a\J!

aw ) 11 DI2 + 1

u = -pl+ 2 ( 01 -

(6.64)

. + ?-b aw - ?-b aw -1

,,,. - -pl

.v

aw "

b - 2 012 b~ '

~

.

811

(6~65)

- 8I2

Note that the scalars pin eqs. (6.64) and (6.65) differ by the term 212 (8\f! /812 ). By· comparing (6.64) and (6.65) with (5.98) we obtain explicitly the response coefficients

a1

aw aw ) = 2 ( O!i + f 1 8!2 _ 9 aw f3 1 -

""'8/1

'

aw.

' /J

_

P-l -

a2

= -2 8I2

_

aw

9 *-'

8/2

.

'

(.6.66) (6..67)

In order to find a constitutive equation for incompressible materials which is associated with (6.38), we recall the transformation (6.55). Then, (6.61) gives the-constitutive equation

8\J!(b) 8\ll(b) .u =-pl+ 2 Db b =-pl+ 2b ab ·

or

a ab =

. -pc)ub

aw

+ 2bac -al Jc(J

(6.68)

225

6.3 Incompressible Hyperelastic Materials

in tenns of the spatial strain variable b. This is only valid for .incompressible .isotropic hyperelastic materials. If we express \JI as .a function of the .three principal stretches Aa we write \JI = '11("\ 11 A2 , A3 ) - p(J - 1) in the place of (6.57), with the-indeterminate Lagrange multiplier JJ. Using 8.JI oAa = J)_-;;1_, a = 1, 2, 3, which is relation (2 ..1.74) expressed in principal stretches, eqs. (6.45) and (6.47) are then replaced by 0-ci

Pa = -

= -p

aw

+ >.a{),\a

1 {)\J! \JJ + l'\a

a, ,

'

a - 17 ?._., 3 T

(6.69)

'

a

= 1, 2, 3

,

(6.70)

.1\a

with the three principal Cauchy stresses 0:0 and the Piola-Kirchhoff stresses Pa, Sa. These stress relations -incorporate the unknown scalar p, which .must be determined from the equilibrium equations and the boundary conditions. The incompressibility constraint J = 1 takes on the form

(6.71) .leaving two independent stretches as the deformation measures. Expressing the first and second Piola-Kirchhoff stresses in terms of the Cauchy stresses (6.69), we obtain, by analogy with (6.48), ~1 = A;1cru and 8 0 = /\; 2 cra, a = 1, 21 3.

EXAMPLE 6.3 Consider a thin sheet of .incompressible hyperelastic material which is embedded in a reference frame of (.right-handed) coordinate axes with a fixed set of orthononnal basis vectors ea, a = 1, .2, 3. Suppose that the .axes are aligned with the major faces of the sheet. A deformation created by the stretch ratios ,.\ 1 , /\ 2 along the directions -e 1 , e2 results in a (lw11u~ge11eous) biaxial deformation with the kinematic .relation (2.130). The assodated stress state is assumed to be plane throughout the sheet so that the Cauchy stress ·components a 1 :~, 0"231 a 33 are equal to zero which is in accordance with (3.59). Show that the biaxial stress state of the homogeneous problem is of the form (6.72)

(6.73) (see RIVLIN [1948, eq. {6.5)]), with the principal .invariants 1 1 = ~W

I2 ·= ,,\i A~ + /\1 2 + /\i··2 and l:i

= 1.

+ ..-\~ + .1\) 2 ,,\22 ,

226•

6.

Hyperelastic J\tlaterials

Solution. Since the tensors u and ·b are coaxial for :isotropic elastic materials (recall p. 20 l), the principal stresses follow from (6.65), a

== 1, 2, 3 ,

(6.74)

where ,\~ are the three eigenvalues of the left Cauchy-Green tensor b (see the eigenvalue problem (2. l l 7h,). This relation was ·first presented by RIVLIN fl 948]. With the condition of incompressibility (6.7.l) in the form of A;1 = ()q.1\ 2 )- 1 and the boundary condition a~J = 0 we may determine p explicitly. For a = 3~ we deduce from (6.74) that .(73

·= 0

--+

(6.75)

This result substituted back into eq. (6.74) leads to the nonzero stress components a 1 and a 2 •

•

EXERCISES

l. Consider a thin sheet of incompressible hyperefast.ic materia.l (.Ta = 1) with the same setting as formulated in Example 6.3. (~)

Consider a simple tension for w·hkh Ai = .A. Then, obeying incompress-· ibility constraint .1\ 1,.\~/\ 3 = 1, the equal stretch ratios in the transverse directions are, by symmetry, A2 = .1\3 = .1\- 1/ 2 • Show that for this .mode of deformation the homogeneous stress state reduces to a 1 = a, a 2 = aa = 0, with

w"here the invariants are / 1 = 2.1\- 1 + A2 , / 2

= /\- 2 + 2.A.

As a special case of the biaxial deformation, as discussed in Example 6.3, consider an equibia.xial deformation for which ,,\ 1 = ;\2 = ,,\, ,\i = ,,\ - 2 and a1

.==

-a2

. hII

W 1t

= a, a:i = 0. Show that

= ?_. )\\ .-, + \ -•1 A

T , J2

= A + ?. . . ;\., -'>-. \"

6.4

Compressible Hypcrc)astic ·rv1.aterials

227

(b) Consider a ho.mogeneous pure shear deformation with the kinematic relation ,,\ 1 = ,\, --\ 2 = 1, Ai = l/ A. (compare with eq. (2.131 )). Show that the nonzero Cauchy stress components are

(6. 76)

(6. 77) .I Il

W:lt 1

= J.2 =

\ 2 + .,\, _ •~.,

.1\

+ 1.

2. Consider a th.in sheet of incompressible hyperelastic material with the same setting as formulated in Example 6.3 but subjected to a homogeneous simple shear deformation which is caused by a motion in the form of (2.3) (compare also with Exercis-e 2 -on p . 93). (a) Show that the associated stress state is completely defined by au

') aw a,T! = -p + 2(1 + c~) BI1 - 2·aI6. 2

~

(6. 78) where Jt > 0, called the shear .modulus, is a measure of resistance to distortion and p is a scalar to be determined from the boundary conditions. (b) Consider a plane stress state throughout the sheet in the sense

that the face

of the body nonnal to the direction e;1 is free of surface tractions, i.e. a 1;i o 2a

= a3 :~

=O.

=

Show that the nonzero Cauchy .stress components ure 0-.12

= JJ.C

•

6.4 Compressible Hyperelastic Materials A .material which can undergo changes of volume is said to be compressible. Foamed elastomers~ for example, .are able to sustain finite strains with volume changes. The only restriction on this class of materials is that the vo]ume ratio J must be positive.

228

6 .Hyperelastic Materials

In this section we introduce suitable constitutive equations .in order to characterize compressible hyperelastic materials, and we discuss isotropy as a special case. Compressible hyperelasti-dty. Since some materials behave .quite differently in bulk and shear it is beneficial to split the deformation locaUy into a so-called volumetric part and an isochoric part, origina11y proposed by FLORY [1961] and successfully applied within the context of isothermal finite strain elasticity by, for example, LUBL.INER [1.985], SIMO and TAYLOR [199.la], OGDEN [1997] and within the context ·of finite strain elastoplasticity by, for ex.ample, SIMO et al. [ 1985] amon.g many others. In .particular, we consider the deformation gradient F .and the corresponding strain measure C = FTF. Rather than dealing directly with F and C we perform a multiplicative decomposition ofF into volume-changing (dilational) and volume-preserving (distortiona/) parts, often used in elastopl.asticity (see, for example, LEE [1969]). We write

mos.t

(6.79.)

The terms J 113 I and J 2 !=11 are associated with volume-changing defomiations, while F and C = F1~F are associated with volume-preserving deformations of the materia'I, with

and

detC = (detF) 2

=1

,

(6.80)

where \

\ ''a -- J-1/:J A.u

,

.a= 1, 2, 3

(6.8 I)

characterize the so-called modified principal stretches. We call F and C the ·modified deformation grad.font and the modified right Cauchy-Green tensor, respectively. A material for which dilational changes require a much higher exterior work than vo.lumepreserv.iQg changes is called a nearly .incompressible (or sUghtly compressible) material, for which the compressibility effects are small. The concept of the multiplicative decomposition of F is supported additionally by the field of computational mechanics. For example, to avoid numerical complications in the fi~ite element analysis of slightly compressible materials it is often advantageous to separat'e numerical treatments of the volumetric and isoc.hor.ic pa.rts of the deformation gradient F; this will be ·discussed in Sections .8.5 and 8.6. Before proceeding to exa.inine constitutive equations for compressible .hyperelastic materials it is first necessary to stick to kinematics and to compute the derivative of the modified right Cauchy-Green tensor C relative to the symmetric tensor C. By means of (5.91.)2, we obtain from (6.31 }2, 8J 2 I ac = J 2 c-). Using the cha.in .rule we arrive at

(6.82)

6.4 Compressible Hyperelastic Materials

229

Finally, according to the inverse.of relation (6.79)2, property (l .256).and relation (6 . 82h, we obtain the fourth-order tensor

~~

a(J~~ C) 3

=

= .r213 {n -

'

= .r2;3

(ll + J2 ac 0a.~~ 1

~c ® c-1) 3

y

= .r21apT ,

13

) (6.83)

,,

pT in which pT defines the transpose of the fourth-order tensor JP governed by the identity ( 1.1.57). We call Jr the projection tensor with respect to the reference configuration, therefore expressed thro~gh C. With the associated property (1.159), the relation for the projection tensor JP> reads, with reference to (6.83)a, as

1 1 JP>=Il--C®C , 3

(6.84)

where E denotes the .fourth-order unit tensor, as defined in eq. ( l. l 6Q) 1, with representation ( l ~ 161). Earli~r we agreed to study the purely mechanical theory. To characterize processes within an isothermal situation at constant temperature, we postulate a unique decoup_led . rep.resentat:~on of the strain-energy function \JI == \Jl(C) (per unit reference vo:lume). It is based ·on kinematic assumption (6.79h and of the specific form

---

'\w

(F) '

"~'.Y:~cD·; ~i~~CSJ' 'j1 --=---------

,.,_

c~- Pc~)

.........,~~-.......---·

where \ltv0 1(J) and Wiso(C) are given scalar-valued functions of J and C which are assumed to be obje·ctive. They describe the so-called volumetric (or dilation.al) elastic response and the isochoric (or distortional) elastic response of the .material, re~pec tively. Additionally, we .require that 'W vol ·is a strictly convex function taking on its unique minimum at J = 1 (for formal definitions of strictly convex functions see, for example, OGDEN [1997~ Appendix l]). With reference to normalization condition (6.4) we daim that Wvor(.l) = 0 .and Wiso(C) = 0 ho.Id if and only if J = 1 and C =I, respectively. We now determine constitutive equations for compressible hyp.erelastic materials. In ·order to particularize the second law of thermodynamics through the ClausiusPlanck inequality (4.154) to the specific strain energy (6.85) at hand we determine the derivative of '11 with respect to time t first. By means of the chain rule we obtain from (6 ..85)

(6.86) Hence, we need to compute j and C, which, with eqs.(6.82}1 and (6.83),1, simply

230

6

Hyperelastic M·aterials

J = a.JI DC : -t = Jc-.l : C/2 and c

= 2(DC/ ac) : C/2 = 2J- 2 /:JIPT~ :" C/2. Having this in mind, with the stress power Wint = S : C/2 per unit reference volume and relation (6.86}, we may deduce from (4.154) that

results in

V .lilt

=

(·s _1d'11vo-1(J) c-1 _, dJ

,-2;:.hm. 9 8'1tiso(C)) . C =CJ Jr • ac .2 ,

,

(6.87)

where the identity ( 1.157) is to be used. Since we consider perfectly elastic materials the internal dissipation Vint. must vanish. The standard Coleman-Noll procedure leads to constitutive equations .for compressible hyperelastic materials, in which the stress response constitutes an additive split of (6. I 3h, i.e. S = 28\II( C) /DC. In particular, the second P.iofa .. Kirchhoff stress S consists of a purely vo.lumetric contribution and a purely isochoric contribution, i.e. Svol and Si 80 , respectively. We write

S

=2

D\J! (C) .DC · == Svol + .Siso

(6.88)

.

This split is based on the definitions

Svol -_

9 0Wvo1(J) ~

ac

(6.89_)

t

8\JI· (C)

·1;·J

1. ··.

-

= r- "(JI - ~c- 1 ® C) : S 1- 2/a DevS = J-'1/~ip : S ,

SisCJ = 2 =

_ J··c-1. p ,

-

;)~ ·

{6.90)

with the constitutive equations for the hydrostatic pressure p and the fictitious second Piola-Kirchhoff stress S defined by

v=

d \P rnl ( J) dJ

and

(6.91)

lt is important to note that in contrast to incompressible materials, the scalar function vis specified by a constitutive equation. The projection tensor-IP= Il-tc- 10C in (6.90) .furnishes the physically correct deviatoric operator in the Lagrangian description, Le. Dev(•) = (•) - (1/3)[( •) : C ]c- 1, so lhat

DevS: C

=0

.

(6."92)

The characterization of the stress response in the material description in terms of the projection tensor IP leads to a convenient sh01t-hand notation (see also, for example, HOLZAPFEL l1996a]).

6.4

231

Compressible Hypcrelastic l\tlatcrials

EXAMPLE 6.. 4 Consider the decoupled strain . . energy function (6.85) with the associated stress relation (6.88). Perform a Piola transformation according to (3.66) and obtain the additive decomposition lT

= Uvol +
(6.93)

of the Cauchy stress tensor u, where the purely volumetric and isochoric stress contributions are de.fined by Uvol

=pl '

Uiso

= J -1~F(IP. : -S)Fy

(.6.94)

,

F being the mod-ified deformation gradient. The ·constitutive equations for the hydrostatic pressure p and the fictitious second Piola-Kirchhoff stress Sare given by (6 . 91). Solution. A push-forward operation on the second Pioia-Kirchhoff stress tensor S to the current configuration and a scaling with the inverse of the volume ratio transforms (6.88) 1 to

u

= 1-1 t

x*

(S~) = l)J-1F (8\ll.vo1(J) + D'I!iso(C)) F·r

.-

ac

ac

·'

(6.95)

where the decoupled form (6.85) is to be used. Hence, considering the first term on the 1 right~hand s.ide, we obtain, using (6.89.) 2 and = F- 1F-T,

c-

(6.96) which is the volumetric Cauchy stress contribution

Uvol

defining a hydrostatic stress

state, as discussed on p. 125. Considering the second term on the right-hand side of eq. (6.95h, we obtain, using (6.90) 4 and the kinematic assumption (6.79)i, (6.97)

which is the .isochoric Cauchy stress contribution O"iso·

II

Compressible isotropic hyperelasticity. A suitable decoupled representation of the strain-energy function for compressible isotropic hyperelastic materials is, by analogy with assumption (6.85), given by

\JI (b) = Wvol ( ./)

+ '11 isu (.b)

~

(6.98)

232

6

Hyperelas.tic Materials

with the multiplicative split of the left Cauchy-Green tensor b

= F'FT in the form (6.99)

(compare with eq. (6.79) 2). The terms J 213 I and b = FFT represent the volumechanging (dilational) and volume-preservjng contributions to the deformation. We call b the modified "left Caucby . .Green tensor, with detb 1. The derivative of the volume ratio J = (detb) 112 and the modified left CauchyGreen tensor b relative to bis given by (6.82) 1 and (6.83h (with C replaced by b). We obtain

=

8b 8b

= J-2/J(Il - !b b-1) 3 ®.

.

-(6.100)

Following arguments analogous to those which led to eqs. (6.88)-(6.91), we obtain from (6.98) the spatia·1 version of constitutive equations which are expressed in terms of J and b. Given entirely in the spatial description and characterizing the isotropic behavior of -compressible hyperelastic materials, we have
')

~J

-1

b

8'1t(b) - ~ -i-8'11(b) Ob - 2J Ob b -

+ O'iso

,

(6.101)

where the stress contributions are defined by (6.102)

(6.103) Use has been made of eqs. (6.99) and (6.100) ·and properties (1.95) and (1.155). The constitutive equation for the hydrostatic pressure pis given in (6.91 ) 1 and the fictitious Cauchy stress tensor u is defined to be (6.104) In (6. l 03) 4 we have introduced, additionally, the projection tensor

1 3

.P~Il--1@1

(6.105)

6.4

233

Compressible Hyperelastic 1"1.aterials

which .furnishes the physically correct deviatoric operator in the Eulerian description, i.e. dev (•) = (•) - (1/3 )[ ( •) : I ]I, so that devu .: I= 0 .

(6.106)

For the characterization of the stress response in the spatial description in terms of the projection tensor=n:- see also, for example., the work of MIEHE [1994]. Note the sbnilar structure of the stress relations in the Lagrangian description (6. 88)-(6. 91) to those presented in (6.101)-(6.104 ).

Compressible isotropic hyperelastic.ity in terms of invariants. We now introduce a strain-energy function for compressible isotropic hyperelastic m.ateria]s in terms of strain invariants. By analogy with the decoupled representation (6.85), or (6.98), we write (6.107) with the first two strain invariants f 1 and 12 of the symmetric modified Cauchy .. Qreen tensors. Since C and b have the same eigenvalues, we deduce that (6. I 08)

The strain invariants Ia, a = 1, 2, 3, are referred to as the modified invariants and are defined by

11.

= trC := trb ,

12 = ~

.f1 =

[(trC) 2

-

(6. l09)

tr(C

2 ))

=

~ [(trb)2- tr(b2 )]

,

(6.110)

(6.1:11)

detC = detb .

With the kinematic assumption (6.79.h,-or (6 . 99), and properties (L92) and .(l.101), we conclude from (6.109)-(6.111) with reference to (5.89)-(5.91) that r _

l:1·- -- J -2/3 11 '

12 -

J-4/~JJ

2

la=

'

1 .

(6.112)

Finally, we formulate the associated constitutive equation in terms of the volume ratio J and the modified invariants lt. 1 12 , which reads in the material description as S

aw (c) = 2 ac = Svot + Stso

with the volumetric contribution

S\'ol

,

(6.113)

to the second Piola-Kirchhoff stress, i.e. (6.89),

234

6 sh~ot

and the isochoric contribution S ..ISO

U:ypereJastic J\tlaterials

defined by

== ?8W1so(l1, 12) = 1 _ 2;:Jllll. ·S ac lf •

(6.1J4)

•

The isochoric second Piola-Kirchhoff stress tensor Siso is J- 2 /:J multiplied by the double contraction of the fourth-order projection tensor Il1\ see eq.(6.84), with the fictitious second Pio.la-Kirchhoff stress tensor S, which is here defined as

-S -'J aw·. · 1so (11 , J.,) __ --i+-c 1'1 /2 -

ac

·M

-

-

(6.115)

'

with the two response coefficients given by

The details are left to be supplied as an exercise by the .reader. For the stress response of compressible isotropic hypere]astic materials in terms of the volume ratio J and two of the modified principal stretches An as independent variables, see the study by OGDEN [1.997, Section 7.2.3].

EXERCISES

l. Consider the modified right Cauchy-Green tensor C according to (6. 79h and properties (.1..159), (1.134). For the four.th-order projection tensor Ir obtain the identities 1 _1 / 0

r = .Il - --c 3

1--1.

.

c = Il - --c 3

®c

-.- --·-1 T = n- -31 (-C 0 c )·

t

where n .is a .positive integer. 2. Show that the properties (6.92) and (6. l 06) hold.. 3. Consider the strain-energy function Wiso[l1 (C), l 2 (C)] in terms of the modified invariants ! 1 = J- 2 / 3 Ii and 12 = J-~ 1 1 3 12 • (a) Show that the derivatives of ft and 12 with respect to tensor Care

811 =I

ac

'

at)

-

ac

·

-

-=/11-C

(6.117)

(b) Use the cha.in rule and the results (6.117) and (6.83):1 in order to obtain the constitutive equation (6..l 14h (with e-qs. (6.115) and (6.116)) in the material description.

6.5

Some Forms ·of Strain-energy Functions

235

4. Consider- the strain-energy function \JI = \IT vur( .J) + \JI iso [f1..(b), l2 (b)] with the .associated constitutive equation for compressible isotropic hyperelastic materials in the spatial description, i.e. u = u vol + O'jso· (a) By analogy with the above Exercise 3(b)., obtain the constitutive equation for the isochoric (Cauchy) stress contribution ui~o = P : .u, where TI!\ is the fourth-order projection tensor, i.e. eq.(6.105), and u the fictitious Cauchy stress tensor, defined as (6.118) The response

coefficients~,. 1, "'l'2

are equivalent to those given in (6.116).

2

(b.) Elitninate b from (6.] IR) in favor of b -• and derive the equivalent form -u

- )b-.1 = '1- 1(_·-b J(l + ~12 .

with

for the fictitious Cauchy stress tensor, which .is responsible for volumepreserv.ing deformations. The response coefficient ')' 2 is given by (6.1.l 6h .

Hint.: Recall .identity (5.93).

6.5 Some Forms of Strain-energy Functions

ma-

From previous sections we have learnt that the stress response of hyperelastic . terials is derived from the given strain-energy function \JI. Numerous specific forms of strain-energy functions to describe the elastic properties of incompressible as well compressible materials have been proposed in the literature and more or less efficient new specific forms are published on a daily basis. The aim of this section is to specify some forms of strain-energy functions which are well tried within the constitutive theory of finite elasticity and frequently employed in the literature.. In particular, we present a se-lection ·Of representative examples of '1t known from rubber elasticity describing isotropic hyperelastic materials within the .isothermal regime (for a collection of constitutive models for rubber see the book edited by DORFMANN and MUHR [1999]). We start by presenting suitable strain-energy functions for incompressible materials and continue with some particular forms which are able to describe compressibility.

Ogden model for incompressible -(rubber-Hke) mater.ials . The only materials undergoing finite strains relative to an equilibrium state are biomateria.ls such as bio . . logictll soft tissues and solid polymers such as rubber-like materials. On the latter we


236

will focus subsequently. If we subject vulcanized rubber to very high hydrostatic pressures, we observe that it undergoes very small volume changes. To change the shape of a piece of rubber is very much easier than to change its volume~ For the purpos.e of computational analyses, .rubber is often regarded as incompressible with the constraint condition J = ..\1A2A3 = 1. A very sophisticated -development for simulating incompressible (rubber-like) materials in the phenomenological con.text is due to OGDEN [1972a, 1982] and [1997, Chapter 7]. The _postulated strain energy is a function of the principal stretches ,\a, .a = 1, 2, 3, is computationally simple_, and plays a crucial role in the theory of ·finite -elasticity. It describes the changes of the principal stretches from the reference to the current configuration and has the form (6.119) On .comparison with the linear theory we obtain the (consistency) condition N

2/L =

L µ,,o:p

p = 1, ... , 1V ,

with

(6.120)

11=1

where the parameter 11 denotes the classical shear modulus in the reference .configuration, known from the linear theory. In equation (6.119), N is a positive integer which determines the number of terms in the strain-energy function, J.Lp are (constant) shear moduli and ap are dimensionless constants, p = 1, ... , 1V. It emerges that only three pairs of constants (JV = 3) are required to give an excellent correlation with experimental stress . . deform~tion data (see TRELOAR ["1944] and TRELOAR [1975]) for shnple tension, equibiaxial tension and pure shear of vulcanized rubber over a very large strain range. Many scientists conside-r the experimental data of TRELOAR [l 944] to be the essential rubber data. For a more detailed discussion of the correlation with the experimental data and for additional sources, see the works by, for example, OGDEN :[ l 972a, 1986, 1987, l 992a, 1997], TRELOAR U975, Section 11.2], TW"lZELL and OGDEN I.1983] and BE.ATTY :[1987, Sections 8-11]. Typical values of the constants lYp, J.tp, p = 1 J 2, 3, are 0!1

= 1.3

= -0.012 · 1n· N;m2 J.l3 = - o.1. . -· ·10 5 .N/ m r: 1

/t2

ll'.3

= -2.-0

')

(6.121)

~

which determine the shear modulus 11 = 4.225 · 105 N/m 2 according to (6.120.h. VA LAN IS and LAND EL [ 1967] have postulated the hypothesis that the strain energy W = w(/\ 1, A2 _, 1\ 3 ) may be written as the sum of three separate functions w(/\a), a

=

6.5 Some Forms of Strain-energy ·Functions

1, 2., 3, which depend on the principal stretches, we write \JI

237

= w(A 1) + w(A.2 ) +.w(,,\1).

This additive decomposition of the strain energy is known as the Valanis-Landel hy-

pothesis. Hence, .in view of the Valanis-Landel hypothesis the strain-energy function due to Ogden may be written in the equivalent form N

3

W(A1, A2, ,\3)

= L: w(1\a)

w(>.a) =

with

L: µp (>.~1•

-

1) .

(6.122)

p=l 0:7}

a=l

According to OGDEN [.1986, 1997], separation (6.122) may also be motivated by data obtained from biaxial experiments of JONES and TRELOAR [1975].

EXAMPLE 6.5 Consider an incompressible hyperelastic membrane under biaxial defonnation with kinematic assumption (2.130). In particular, the two principal stretches ,,,\ 1 and ,,\2 are given. According to the membrane theory assume a plane stress state .and specify the Cauchy stresses in the plane of the membrane by applying Ogden's

strain-energy function. The three principal values a a of the Cauchy stresses are given according to .relation (6.69). Using (6.119) we find, after differentiation, that

Solution.

N

au

= -p + L

µpA~ 1,

,

a = 1, 2, 3 ,

(6 ..123)

p=l

where p is a scalar not specified by a constitutive equation. It is determined from a boundary condition, namely by the requirement that cr3 = 0 which allows p to be expressed explicitly from (6.123)., setting a = 3, as

(6 ..1.24)

Combining (6.124) and (6.123) we obtain the two nonzero stress components N

a1 =

L µp[ A~t,, -

(...\ 1~\2 )-a,,] .,

(6.125)

µ,,[A~1· - (A1A2.}-a1•] ,

(6.126)

p=l

N

a2

=L 11==1

where the incompressibility constraint Aa

= (/\ 1.,\2 )- 1 has been used.

•

2"38

6

Hype-relastic l\tlaterials

Mooney-Rivlin, neo~.."Hookean, Va.rga model for incomp·ressible (rubber-like) materials.. As a special case we obtain from eq. (6..1.l 9) the Mooney-Riv.Jin .model~ the neo-Hookean model and the Varga model (see MOONEY [1940], RIVLIN [l948, 1949a, b], TRELOAR [1943a, b] and VARGA [1966], respectively).

For example, the very useful Mooney-Rivlin. model results from (6.119_) by setting 1\T = 2, n 1 =-= .2, n 2 = -2 . Using the strain invariants 1 1 , !'2 .as presented by (5.89h and (5.90h, with the constraint condition 1:1 == "'f A~;\~ = 1, we find from (6.119) that ·\JI

= CJ(/\~+ ,\~ + ;\~ - 3} + C2 (1\} 2 -1- ,\2 2 + ,,\32 = Ct (/1 - :J) + C2(.f2 - 3) ,

:J) (6.] 27)

with the constants c 1 = p.. 1/2 and c2 = -p. 2 /2.. Adopting (6.120) 1 the shear modulus 11 has the v'ilue· /L·1 - 112. The classical strain energy '11 = W'(J.1, / 2 ) of the Mooney-Rivlin form is often em.ployed in the description of the behavior of isotropic rubber. . li.ke materials. Mooney derived -it on the basis of mathematical arguments employing considerations of symmetry. The neo-Hookean model results from (6 . .l 19) by setting 1V = l, c1: 1 = 2. Using the first principal .invariant / 1, see eq. (5.89)2._, we find from (6.119) that (6.128) with the constant c 1 = -11 1 /2 and the shear modulus p = fl·i according to (.6.120) 1• This strain-energy function involves .a single .parameter only and provides .a mathematicaUy simple and reliable constitutive model for the nonlinear deformation behavior of isotropic rubber-like materials. It relies-on .phenomenological considerations and includes typical effects known from nonlitiear elastidty within the small strain domain. However, the important strain-energy function {6.128) may also be motivated from the statistic.al theory in which vulcanized rubber is regarded as a three-dimens.ional network of long-chain molecules that are connected at a few points.. A brief discussion is given in Section 7.2 on p. 318. Constitutive relations for the Mooney-Rivlin and the neo-Hookean model follow from (6.65) by means of (6.127h and (6.128) 2 • Derivatives of \JI with respect to the strain invariants 1 1 and 12 give the simple assoc.iated stress relations u = -pl+ 2c 1b··2c2h-1 and u = -pl+ 2c 1b, respectively. Compare also the considerations on p. 203. As the last special case of Ogden's model we introduce the model by \q1rga. It results from (6.119) by setting JV = 1, n 1 = 1, i.e. (6.129)

with the constant .c 1

= 1-1. 1 and the shear modulus 11 = J.Li /2 according to (6. 120)

1•

6.5

Some Forms of.Strain-energy Functions

239

Note that of all constitutive approaches given, the Ogden mod.el with .N = 3 excellently replicates the finite strain behavior of rubber-like materials; see, for -example, OGDEN [.1972a] for an analytical treatment or DUFFETT .and REDDY [ 1983], SUSSMAN and BATHE [19.87], SIMO and TAYLOR [1991.a] and M.lEHE [1994] for a numer.ica1 simulation, among many others. The assumptions nmde in the Varga model, the neo-Hookean model (obeying {Gaussian) statistical theory) or .in the Mooney-Rivi.in model are rather simple. Consequently, these types of constitutive model are not able to capture the finite extensibility domain of polymer chains (see TRELOAR f I 976]).

EXAl\llPLE 6.6 Th.is example has the aim of investigating the inflation -of a spherical .( i11c01npressible rubber) balloon with different material models. Analyses of balloon inflations have some applications in producing, for exmnple, meteorologica.1 balloons for high-altitude measurements or balloon-tipped catheters for clinical treatments. Inflation experiments of spherical neoprene balloons were .carried out by ALEXANDER 11971]. In particular, compute the inflation pressure Pi, i.e. the internal pressure in the balloon, and the circumferential Cauchy stre.ss a as a function of the circumferential stretch A of the balloon. Let the initial (zero-pressure) radius of the rubber baUoon be R == 10.0 and the initial thickness of the wall be .If = 0 ..1. For the geometrical situation of the spherical balloon in the reference and current configuration s.ee Figure 6.2. On the basis of the described four prototypes of constitutive .models, that are the Ogden, Mooney-Rivlin, n.eo-Hookecm and Wirga models, study the different mechan.ica.I behavior and cornpare the solutions, drawn in a diagram. Do not consider aspherical modes which clearly develop during the inHation process. The material properties for the Ogden model are .given according to (6.121 ), with the shear modulus p. = 4.225 · 105.N/rn 2 in the reference configuration. For the ·M-ooney-Rivlin. model take c1 ~ o.~t:375p, c2 = 0.0625/L (c 1/ c2 = 7), as sug·gested by ANAND :[1986], for the neo-Hookean model c1 = J..1/2., and for the Varga model take cL = 2.J.L .

Solution. We know from a perfect sphere under inflation pressure p1 that every direction in the plane of the sphere is a_ principa] direction. Hence, .the stretch ratio is ,\ = "\ 1 = A2 which .characterizes ~· equibiaxial deformation. The· associated circumferential Cauchy stress is a ·= a 1 - ~:,. cr2 (while aa = 0 by the assumption of plane stress). Hence_, constitutive equations ·(6.125) and (6..I 26) reduce to a single .relation, name.ly N

a=

L ,,.,J(N.,,, - ,..\-20,,) ,,= 1

(6.130)

240


Current configuration Pi

Balloon

--L~ . .1~_J r

a

. .-. I

--J·<-··~

·r-·r-----·vuf_ •

I

.

x

l_,,.--vr~--1-,····r--·-~-, tu x I

I

t

I

Figure 6.2 Geometry of a spherical balloon in the reference and the current configuration, showing only one hemisphere.

which is plotted in Figure 6.3 for different material parameters. Th~ figure illustrates the relationships between the Cauchy stress a and the circumferential stretch .A of any point of the rubber baUoon for various constitutive .models. By equilibrium we find from Figure ·6.2 (free-body diagram) that r 211pi = 2r7rha, where r and h denote the radius and the wall thickness of the rubber balloon in the current configuration. According to this condition we find that h Pi= 2-a . r

(6.131)

In view of the kinematical situation of the inflated balloon (see Figure 6.2) the stretch >i at a certain point of the balloon is .r/R. Incompressibility requires that the wall volume is conserved, which means that 4rtr 2 h = 4ri R2 H. With this condition we· find that .A3 = h/ H = 1//\2 which denotes the stretch in the direction perpendicular to the .surface of the sphere, indeed _,\ 1 _,\ 2 /\3 = /\ 2 ..-\1 = 1. Using these relations and constitutive equation (6.130) we may find from (6.131) the analytical expression (6.1.32)

6.5

241

Some Forms of Strain-energy Functfons

40

,

,...., C"I ·E

z

~

+0

= 10.0 H = 0.10

R

30

. . . ,_

::

>.

25

"

::

Pi

• •

. .. . 1

''·,

~

,

•

20

i •

•

Cl.:I

#

Q.)

""" tl.l

;

~

>. ..c

#

#

,' #

, ,,

;

,

,

#

#

, ,,

#

#

I

Neo-Hookean model

,'

.#

,,,,,"

15

(.)

;;;

0 C':I

u

,

#

#

..•

..........

, ,,

~'#

#

•

·,..

........ :IJ

,

Ogden model

•

~

~

. . . f

Mooney .. Rivlin model

35

, ,,

10

........·

,...

~--

.... .. .. .. ....

.v

--·--'

;'

,.-' "'" Varga model

0

2

1

3

5 6 Stretch A

4

7

8

10

9

F.igure 6.3 Geometry and Cauchy stress a versus stretch A of any po.int of the balloon .

..

... .......·· Mooney-Rivlin model

60

:·········

• .........•.•. ,. ..... •••••• ::1· max

..... .. ........

Ogden model

~,-"'

____ __ ..

~~

min ___ ___ _

..

-.. -- ....... ._.....

neo-Hookean model ___ _

. . . . . . . . . . . . 1111111111 •

10

Varga model

................

..........

0 1 i 1.38

2

3

4

5 4.32

6

7

8

9

Stretch .,\

Figure 6.4 Intlalion pressure 11-t versus stretch ;\ of any point of the balloon.

10

242

6


for the inflation pressure JJi, which is plotted in Figure 6.4, for JV = 1~ 2, 3, and with material parameters .as given above. The analytical solutions of the six parameter material model proposed by Ogden are in -excellent .agreement with experimental data by TRELOAR [ 1944] (see also NEEDLEMAN [1977]) who solved the problem on the basis of the Ritz-Galerkin method. Experimental data show a very stiff initial stage (as seen with any party balloon) in which the inflation pressure Pi rises steeply with the circumferential st.retch. After the pressure has reached a maximum the rubber balloon will suddenly 'snap through', and a release of the pressure will allow it to 'snap back' (see Figure 6.4, see also the studies by OGDEN fl972a] and BEATTY [1987] among others). This effect is caused by the deformat.ion dependent pressure load, is dynamic in character and known as snap buckling. The pressure-stretch path clearly shows the .existence of a local maximum .and minimum, the maximum and mini.mum pressures are at ,\ = l.38 and ,\ = 4.32, respectively. The curve for the Mooney-Rivlin model, with c 1/c2 = 7, shows the characteristic behavior of a spherical rubber balloon, but, however, the results based on Ogden's model {and Treloar's experimental data on vulcanjzed rubber) are significantly different The neo-Hookean and Varga form of the strain -energy reproduces more or Jess the real behavior of the balloon for small strain ranges. Reasonable correlations for all material models are obtained at the low strain level. However, for finite strains the typical characteristic of the load-deflection curve cannot be reproduced with the. simplified .neo-Hookean and Varga model. Experimental investigations show that the balloon develops a bulge on one side and becomes aspherical (compare with NEEDLEMAN [l 9771). The bifurcations of pressurized elastic spheric.al shells from an analytical po.int of view are studied by HAUGHTON and OGDEN [1978], HAUGHTON [1980] .and ERICKSEN [ 1998, Chapter 5]. Compressibility effects are considered by HAUGHTON [ 1987]. II

Yeoh, Arruda and Boyce model for i.ncompressible (rubber-like) materials. Nearly all practical engi.neer.ing elastomers .contain .reinforcing fil.lers such as carbon black (in natural rubber vulcanizate) or silica (in silicone rubber). These finely distributed fillers, whkh have typical dimensions of the order of 1 ..0 - 2~0 · 10- 12 -rn, form physical and chemical bonds w.ith the polymer chains. The fine filler particles are added to the elastomers in order to improve their physical properties which .are mainly tensile and tear strength, or abrasion resistance. The associated stress-strain behavior is observed to be highly nonlinear (see_, for example, GENT [.1962]). Carbon-black filled rubbers have important applications in the manufacture of automotive tyres and many other engineering components.

6.5

Some Forms of St.rain-energy Functions

243

It turns out that for carbon-black filled rubbers the strain-energy functions described hitherto in this section are not adequate to approximate the observed physical behavior. For example, consider a simple shear deformation of a filler-loaded rubber. Physical observations show· that the shear modulus Jt of the .material varies with deformation in a significant way. To be more specific, JL decreases with increasing deformation initially and then rises again at large deformations (s·ee YEOH .[1990, Figure 2]). The associated relation for the shear stress is clearly nonlinear. Now" taking, for example, the ·Mooney-Rivlin model according to strain-energy function (6. l27h, then, from the explicit expression (6.78h we may specify a shear modulus

aw + aw) = ( 811 812

fl = 2 .

2(c1

+ c2) > 0

.

(6.133)

The relation for the shear stress is, however, .linear with the constant slope 2(c 1 + c2 ), i.e. the shear .modulus. Apparently the .Mooney-Rivlin (and its neo-Hookean spec.ialization) model is too simple for the characterization of the elastic properties of carbon-black ti I led rubber vulcanizates. The phenomenological material model by YEOH [ 1990] is motivated in order to simulate the mechanical behavior of carbon-black filled rubber vulcanizates with the typical stiffening effect in the large strain domain. Published data for filled rubbers (see KAWABATA and KAWAl [1977] and SEKI et al. :(.1987]) sug·gest .that 0'11/8!2 .is numerically c.Iose to zero. Yeah .made a simplifying assumption th.at 8\J! /812 .is equa1 to zero and proposed a three-term strain-energy function where the second strain invariant does not appear. ]t has the specific form (6.134)

where <~.1., c2 , c:1 are material constants which ·mus.t....~atisfy certain restrictions. Since by (6..5) the strain-energy function \JI is either zero (in which case w has only one real root, corresponding to I 1 = 3) .or positive, we must have f 1 > 3 (note that for an incompressible -.n"aterial Jl > 3 with the equality only in the reference configuration). Hence, the (convex) strain-energy function increases monotonically with / 1 and 8\J! / 81 1 = 0 has no real roots. From the discriminants of the respective cubic and quadrntic equations in (11 - .3) the appropriate restrict.ions on the values for c., c2 , c:1 may be determined. Recall. the simple shear deformation example of a filled rubber from above once more. With the strain energy (6.134) we now conclude from eq. (6.133) 1 that (6.135)

The shear modulus p. involves first-order and second-order terms in (JL -3) and approximates the observed nonlinear physical behavior with satisfying accuracy (provided c2 < 0 and c 1 > 0, ca > 0).

6 Hyperelastic Materials·.

244

Another material model for the response of rubber which has a similar structure to (6.134) is due to ARRUDA and BOYCE (1993]. It is, however, a statistical model where the parameters are physically linked to the chain orientations .involved in the deformation of the three-dimensional network structure of the rubber. The molecular network structure is represented by an e.ight-chain model which replaces classical three and four-chain models. The individual polymer chains in the network are described by the non-Gaussian statistical theory and are able to capture the finite extensibility domain. The physically based constitutive model possesses symmetry with respect to the principal stretch space. The strain-energy function .is derived from the inverse Langevin .function (see_, for example, TRELOAR [1975, Chapter 6]) by means of Taylor's expansion (compare with TRELOAR [.1954]). Here we present the first three terms for the strain energy, i.e.

\fl

1 . = Jl [ 2U1 -

3)

1

2

+ 20n U1 -

11

3

9) + 105Qn2 (Ii - 27)

] .

+ ...

(6 ..136)

w:here /L denotes the shear modulus and n is the number of segments (each of the same length) in a chain, freely jointed together at chemical cross-links. For a ·more detailed explanation of the underlying concept of statistical thermodynan1ics the reader is referred to Section 7 .2 of this text. In this .two parameter model the ·first strain invariant I 1 may be linked to the stretch

.in a chain, /\chain, by the express.ion VJ;

= v'3.1\chain. The chain stretch 1\c11ain "is defined

to be the current chain length divided by the· initial chain length. An advantageous feature of the -eight-chain model (6.136) is that all chains stretch -equally under uniform extension and biaxial extension. For further network mode]s which consider chain interactions see, for example, the book by TRELOAR [197.5, Chapter 6] and the .articles by FLORY and ERMAN [ 1982]

and ANAND [19.96]. Ogden ·mode.I for compressible (rubber-like) mater"ials. Rubber-like materials in the 'rubbery' state used in engineering are often slightly compressible and associated with minor dilatational deformations. Compressibility is accounted for by the addition of a strain energy Wv0 1, describing the purely volumetric elastic response (see the framework of compressible hyperelasticity, Section 6.4). For our considerations, in particular, we use the decoupled representation of the strain-energy function W(A1, ,,\2, Aa) = Wvot(l) + Wi 50 (A1, A2, A3) -expressed in terms of principal stretches. For rubber-like materials, OGDEN fl972b] .proposed a volumetric response function in terms of the volum-e ratio J of the following form \{I vol{ J)

for

= ·"'g (J)

with

(6.137)

/3 > 0. The scalar-valued scalar function Q characterizes a strictly convex function,

6.5 Some Forms of Strain..energy Functions

245

and K and (3 denote the constant bulk modulus in the reference configuration and an (empirical) coefficient, respectively. The strain energy (6.137) satisfies the normalization condition, '1'v01 (1) = 0. Note that this empirical function ·meets experimental results with excellent accuracy (see OGDEN -[197.2b, Figure l]), indicating that rubberlike materials are (slightly) compressible. In particular, for /3 = 9, the distribution of the hydrostatic pressure is in good agreement with experimental data of ADAMS and GIBSON [1930] and BRIDGMAN [1945]. An alternative version of (6.137h, due to SIMO and MI.EH.E :.[.1992], is obtained by setting (3 = -2 to give

.g = 1 (J 2 -

1 - 2ln.J ) .

4

(6.138)

The second part of th.e decoupled strain energy, i.e. Wjsq(~ 1 , .,\2 , .,\3 ), describes the purely isochoric elastic response in terms of modified princip~I. stretches Aa 1- 1/ 3 Aa, a= 1, 2, 3. We have 3

Wiso{A1, A21 .,\3)

= L w(Aa)

with

a=l

and with the condition (6.120). ···--·-.--..

---~.

':""·-·--..

..,.,~-. ~··-·-~.

___,

__

~

..

..

~-. ·-~.

'"":'----.

-. ...

-.~. ·""~~.

.-.....~. ~

"EXAMPLE 6.7 Consider the decomposed structure (6.137) .and (6.139) of the strain energy and the additive split of the second Pio.la-Kirchhoff stress tensor (6.88)2. With SpecificatiOOS (6.} ~7h and (6.13.8) find the purely VO/UmetriC COllfribttfiOIJ s\.'O) Of the stress response in the explicit form. In addition, with (6.139h, find the spectral decomposition of Siso' i.e. the purely isochoric contribution.

Solutjon.

In order to particularize the volumetric stress Svot it is only necessary to derive the term d'11vor(J)/dJ (see -eq. (6.9l)i). From eq. (6.89) we .find, using (6.91) 1 and the relation for the purely volumetric elastic response in the :form of ( 6.13 7) 1 , that

svo] --

- pc-t

28'1ivol(J) - J

ac

'

p

=

dw vol ( J) dJ

dQ ( J) = l\i dJ '

(6.1.40)

in which, with the strain-energy functions (6.137h and (6..138), we obtain the specifi-

·Cation dQ(J) = .2dJ f3J

(1 - _.!_) J/1

'

dQ(J) == _!__ (J2 dJ 2J

-1)

.

(6..141)

As a second step we particularize the isochoric stress Siso in respect of the strain energy (6.139). Before proceeding it is first necessary to provide the relation 8Aa/ 8-"b·

246

6

Hyperelastic IVlaterials ·

Recall (6.81), i.e. Aa = J-I/a An, a = 1, 2, 3, and relation (6.82h which, when formulated in principal stretches, reads DJ/ fJ.,\1 = J ;\; 1 . Thus, we have

D.Au. _ 8( J-·i /:i 1\u). _ 8.,.\b .

-

= J

8>.ib -1/a

·

(Oai. -

-

J-1/:J (.·\ •

1-

Uab

_

! ',-1 DJ ''a\ )·

--1

3"\a,\ ) ,

DA,,

3

a, b = 1, 2, 3 ,

(6.142)

which is relation (6.83) expressed through the modified principal stretches /\1 • Hence, by analogy with (6.52), we obtain the isochoric stress response in terms of principal values for the general case /\ 1 =f 1\ 2 # /\1 -I A1, namely (6."143)

where siHCHt' ll = 1, 2, 3~ are the principal values -of the second Piola-Kirchhoff stress tensor Siso and Na, a = 1, 2_, 3, denote the principal referential directions. By use of the chain rule and relation (6 ..J42h, a straightforward computation fron1 (6.143}i gives the explic.it expressions ~...

.....·.···············:····..--.: .......·.····.'···"•""'""···.·--<----·--·····---; .. ---~-··-······-··:·--·--·-....,'<··

~s? (~,, a~{~)_ ;~_ 0.L~-~~a~:". .=- -.~-. i;. :\ba~::J:0 8

a

a=l,2,3

{6.144)

__ ~ . . . . . . . . . · .. ·. . . . .. ~/ .'·-:---,-..-.,.~-~- .· . . . . . .............. .:...........·, .......;,,_. ........",,, .......:.,.,.-<:.~.......:.;....;:...~ .........,'.~---· ·...,.

for the priiici.pal'iSochork Stress-·va"lues,. (eompare_w,l.t.h. . Q.a DEN [ 1997, Section 7 .2.31)~ The summation symbol (which could be omitted) emphasizes that the index b is re~ peated, meaning summation ove.r 1, 2, 3. Howeve~, there is no summation over the index a. Using· the relation for the purely isocho.ric elastic response in the form of {6..139), we achieve finally the term D\J!1so/ [),,\a, a. = 1, 2, 3, in the specific form

a= 1, 2, 3

(6..145)

(s.ee also the derivation by SIMO and TAYLOR [199laJ). The complete stress response, as given through (6..140), (6."141) and the spectral decomposition (6.143)-(6.145), serves as a meaningfu] basis for finite element ana1yses of constitutive models for isotropic hyperelastic materials at finite strains. •

Similarly to the compressible version of Ogden's .model we can reformulate the Mooney-Rivlin, neo-Hookean, Vm:I:a, · Yeoh, Arruda and Boyce .models, .i.e. (6.127_)(6.1.29), (6.134)., (6.I.36), as decoupled representations. We have just to re.place ,\0 Ia by the modified quantities ,.\a, Lu as defined in (6.81 ), (6..l.09)-(6.111.) and to .add a

247

6.5 Some Forms. of Strain-energy Functions

suitable voiumetric response function '11 v01 , for example, (6 . l.37h or (6.13'8). For example, the decoupled strain-energy function for the Mooney-Rivlin model has the form {6.146) However, material mode]s are often presented in a .coupled form. The compressible ·Mooney-Rivlin model, for example, may be given as

(6.147) where c is a material constant and d de.fines a (dependent) parameter with certain restrictions. By recalling the assumption that the reference configuration is stress-free we may deduce from (6.1.47) that d = 2(c 1 + 2c2 ). The first two tenns in (6 ..147) were proposed by C.IARLET and GEYMONAT U 98~] .in a slightly different :form (see also ClARLET [19.88, Section 4.1.0]). · Another example is the coupled form of the compressible neo-Hookean model given by the strain-energy function 1)

{3

= 1-

2v

(6.148)

(see, for ex.amp'le, BLATZ [ 1971]), with the constants c 1 = µ/2 .and /l The mater'ial parameters /t and 11 denote the shear modulus and Po.isson's ratio, respectively.

Blatz and Ko model. For foamed el.as.tamers which cannot be regarded as incompressible BLATZ and Ko .[.1962] and OGDEN [1972b] proposed a strain-e.nergy function which combines theoretical arguments and .experimental data (performed on certain solid polyurethane rubbers and foamed polyurethane e'lastomers). It is based on a coupled function of volumetric and isochoric parts according to

in which µ and 11 denote the shear modulus and Poisson's ratio, and f E [O, 1] is an interpolation .parameter. By means of the incom.pressibility constraint Ia = 1, eq. (6.149) reduces to the Mooney-Rivlin form :introduced in eq. (6.127) (with the constants c1 = f Jt/2 .and c2 = (1 - f)J.i/2). Another specfri'I case of the strain energy (6.149) may be found by taking f = 1, leading to the compressible neo-Hookean model ·introduced in eq. (6.148) (with la = J 2 and the constant J.L/2 = c1 ). An interesting description ·Of the Blatz and Ko mode] was presented by BEATTY and STALNAKER [1986] and BEATTY [1987].

248

6

Hyperelast.ic Materials EXERCISES

1. For the description of isotropic hyperelastic materials at finite strains we recall the important class of strain-energy functions \JI in terms of principal invariants . Study some models suitable to describe compressible materials and particularize the .associated stress relations. (a) Firstly, we consider the coupled form of the compressible Mooney-Rivli11, neo-Hookean, Blatz and Ko models according to the _given strain energies (6.14 7)-(6.149), respectively. By means of (6.32) deduce the stress relation

_ I+ c+ c-1 S -_ 2 aw(Ii,I2,I3) ac - 11 12 1'3 , with the three response -coefficients 'Yi = 2(8\I! /811 + 11 8\J! /8!2 ), /2 = -28w /812 , /'J = 213 8\P /fJI..1 for the second Piola-Kirchhoff stress tensor S as specified in Table 6.1.

Mooney-Rivlin ·model neo-Hookean model (6.147) 1'1

2(c1

+ C2/i)

(6.148) 2c1

Blatz and Ko model (6.149)

µ.f

+ ~!1/2

/2

-2c2

0

. -f,/2

/3

2cJ(J -1) - d

-2c1I; 11

-µf 131J -~(/2 - Jf+I)/2

Table 6.1 Specified coefficients for the constitutive equations of some materials "in the coupled .form .

For notational simplicity we ·have introduced the .non-negative parameter ~ = 2µ (1 - f) / ! 3 • Note .that the response coefficients ')' 1, / 2 , ")'3 for the neoHooke.an model may be found as a special case of the Blatz and Ko model by taking~= 0 (with the constant µf = 2c1). (b) Secondly, we consider the .decoupled fa.rm of the compressible MooneyRivlin model (6.146) and the compressible neo-Hookean model (obtained by setting c2 0 in the Mooney-Rivlin model). In .addition, we consider the decoupled versions of the Yeoh model and the Arruda and Boyce model, .i.e.

=

6.5 Some Forms of Strain-energy Functions

249

Gust replace I 1 in eqs. (6.134), (6.136) by the modified first invariant l~.). Derive the associated· stress relations

S _ 2 8WisoU1J2) _ - I+- C

-

ac

- 'Y1

· 12

for the fictitious second Piola-Kirchhoff stress tensor, with the specified response coefficients 1 1 ., 7 2 (see ·eq. (6.116)) according to Table 6.2 .

Mooney-Rivlin model (6.146)

1'1 'Y2

2(c1

+ c2l1)

-2c2

neo-Hookean model 2c1

0

Yeolz model (6.1.34)

Arnula and .Boyce

+ 4·c~i(l1 -

+6c3(f1 - 3) 2

+ (1/5n)i1. +(11/l 75n2 )l[ + ... ]

0

0

2c1

model (6.136)

3)

µ.(l

Table 6.2 Specified coefficients for the constitutive equations of some materials in the decoupled form.

2. Consider a s.pherical balloon of incompressible hyperelastic material ( Au\ 2 .A 3 . 1), The material is characterized by a strain-energy function in terms of principal stretches according to (6.150)

with material constants tl and a. For n = 2 we obtain the classical neo-Hookean model. (a) Determine the inflation pressure Pi as a .function of the circumferential stretch A in the form

Vi= 2µ

H R (,\a- 3 -

".

3

>,-~n-)

'

where Ii and R are the initial (zero-pressure) thickness and. radius of the spheric.al balloon, .respectively. -(b) Show that the function Pi = Pi(A) has .a relative maximum if 0 and a relative minimum.if -3/2 < a < 0 (see OGDEN [1972a]) .

<

a

<3

Consequently, balloons made of-materials which .are described by the strainenergy function (6.150) will not 'sn~p throu.gh' for a:~ 3. Typical examples for this type of (stab.le) material behavior are biomaterials such as an .artery (for the mechanics of the arterial wall see the excellent survey text by HUMPHREY [1995]) or a -ventricle (see NEEDLEMAN et aL [ 1983]). Specific results on membrane biomechanics including illustrative examples from the literature .are reviewed in the article by HUMPHREY [1998].

250

6


3. Consider a thin sheet of incompressible hyperelastic material with the same setting as formulated in Example 6.3~ The homogeneous stress response of the material is assumed to be .isotropic and based on Ogden's model. Discuss the stress states (which are plane throughout the sheet) for the fol.lowing two modes of deformations (for the .associated kinematic relations, comp.are with Exercise 1 on .P· 226):

(a_) Consider a simple tension for which ,.\.1 = A (,\ 2 = /\ 3 ). Show that the only nonzero Cauchy stress er, in the direction of the applied stretch ,.,\, is N

a= L/.Lp(Aa,. - ,,.\-0,,/2) .. 11=1

In addition, find the stress required to produce a final extended length of .,\ = 2 for each of the Mooney-Rivi.in and the neo-Hookean -models. (b) Consider a homogeneous pure shear deformation and show that the biaxial stress state (aa = 0) of the problem is of the form N

N

al=

LP.p(_,\a1, - A-o,,) '

a2

= .L JL11 (I -

71:;: 1

~\ -·0:1•) .

1•=1

(c) Compute the associated constitutive equations given in (a) and (b) for the Mooney-Rivlin, neo-Hookean and Varga models and compare the results with Ogden's model (plot the relation between the Cauchy stress and the associated stretch ratio for each material model).

4. The so-called Saint.. Venant Kirchhoff model is characterized by the strainenergy function

w(E) == I(trE) 2 + J.LtrE2 2

(6.151)

(see, for example, CtARLET [ l 988T p. 1.55])., in which / > 0 and JL > 0 are the two constants of Lam·e. The Lame constant "Y is usually denoted in the literature by the symbol A. However, in order to avoid confusion with the stretch ratio ,\ we use a different symbol for it. The Saint-Venant Kirchhoff model is a classical nonlinear model for compressible hyperelastic materials ·often used for metals. Note that the volume ratio J does not appear explicitly in this material model. (a) From the given strain energy w(E) derive the second Piola-Kirchhoff stress S, which linearly depends on the Green-Lagrange strain E.

6.5 Some For.ms of Strain-energy Functions

251

(b) Consider the one-dimensional case of the constitutive equation derived hi (a). For a uniform deformation of a rod (with uniform -cross-section), i . e. :i: = AX'", .derive the relation between the nominal stress P and the associated stretch ratio A (which is .a cubic .equation in .,\) and plot the function

P

= P(A).

Show that P(A) .is not monotonic in compression and derive the critical stretch value Acrit = (1/3) 1-/ 2 at which the Saint-Vena~t Kirchhoff model fails (zero stiffness, the tangent of P( A) at Acrit is horizonta]). This failure is not influenced by the material constants / and JL. In addition show that the· material model does not satisfy the growth condition (6.6)2 (in fact for A -7 o+ the stress tends to zero which is physically unrealistic). CIARLET [.1988] showed, with the proof by RAOULT .(1986], that the Saint-Venant Kirchhoff mo~el does not satisfy the requirement of

.polyconvexity either. Note that this material model is suitable for large displacements -i:mt it is not recommended to use it for large -compressive strains. 5. In the first term in eq. (6.151) replace trE by lnJ and / by the bulk modulus Ii > 0 in order to obtain a modifi-ed Salnt-Venant K..ircbhoff model of the form

w(E)

2 = ""(lnJ) + µtrE 2 2

,

(6.152)

where J = detF denotes the volume ratio and JL > 0 is identified as the shear modulus. The proposed material model (6.152) circumvents the serious drawba9ks of the classical Saint-Venant .Kirchhoff model (see Exercise 4) when used for::l~arge compressive strains. (a) From the strain energy (6.152) derive the .second .Piola-Kirchhoff stress S = S{C) as a function of the right Cauchy-Gre-en tensor C. The result is similar to a stress relation proposed by CUR NIER II 994, eq. ·(6..1I3)] which also has the aim of avoiding the defects of the classical Saint-Venant Kirchhoff model occurring at large co:mpressive strains. (b) Consider a one-dimensional problem as described in Exercise 4(b) and obtain the nominal stress Pas a function of A. Discuss the function P-(.1\) for the two regions .,\ > 1, ;\ < 1 and show that the modified Saint-Venant Kirchhoff model satisfies the growth condition in the sense that the stress tends to (minus) .infinity for A --t o+.

252


6.6 Elasticity Tensors In order to obtain solutions of nonlinear (initial boundary-value) problems ·in compu.tational finite elasticity and inelastidty so-called inc.rementaViterative solution techniques of Newton's type are frequently applied to solve a sequence of linearized problems. This strategy requires knowledge of the linearized constitutive equation, here presented in both the material and spatial descriptions. The underlying technique was first introduced in the mechanics of solids and structures by HUGHES and PISTER [.1978]. The process of linearizing constitutive equations is a very important task in computational mechanics and the main -objective of this section. For more on the concept of linearization, which is basically differentiation, see Section 8.4.

Material and spatial representations of the elasticity tensor. Consider the nonlinear second Piola-Kirchhoff stress tensor Sofa point at a certain time t. We look at S as a nonlinear tensor-valued tensor function of one variable. We assume this variable to be the right Cauchy-Green tensor C. First of all we do not assume that the stress tensor is derived from a strain-energy function ·w. According to considerations (1.247) and (1.248) we are now in a position to determine the total differential 1 (6.153) dS = C: -dC . 2 '

in which we have introduced the definition

C = 28S(C)

ac

or

C ABCD

_ -

9~

asA/3

af"' _, vcn

(6.154)

which, by means of the chain rule, reads, in terms of the Green-Lagrange strain tensor E=(C-1)/2,

C = CJS(E) .OE

or

(6.155)

The quantity C characterizes the gradient of function S and relates the work -conjugate pairs of stress and strain tensors. lt measures the -change .in stress which results from a change in strain and is referred to as the elasticity tensor in the material description or the referential tensor of elasticities. It is a tensor of rank four with the four indices

A,B,C,D. The elasticity tensor C is always symmetric in its .first and second slots, i.e. AB, and in its third and fourth slots, Le. CD, CAJJCD

= CaACJJ = c.4BIJC.

{we have, in general, 36 independent components at each strain state).

(6.156)

6.6

253

Elastic-ity Tensors

We say C possesses the minor symmetries. The symmetry condition (6.156) is independent of the existence -of a strain-energy function w and holds for all elastic materials. Note that the minor symmetry of C follows from the symmetries of the right Cauchy-Green tensor C (or equivalently from the Green-Lagrange strain tensor E) and the second Piola-Kircbhoff stress tensor S. If we .assume the existence of a scalar-valued energy function (hyperelastici.ty), then S may be derived from \JI according to S = 28\ll(C)/8C (see (6 ..13h). Hence, using (6.154 ), we arrive at the crucial relation

w

c

= 4 a2w(C)

or

acac

cAac.o

a2 w

= 4 ac acCD . AB

(6.157)

for the elasticities in the material description, with the symmetries or

CA.acn = CcnAB .

(6.158)

We say C possesses the major symmetries. Thus, tensor C has on·Iy 21 independent components at each strain state. The condition (6.158) is a.necessary and sufficient .condition for a material to be hyperelastic. The symmetry condition CAncn = CcnAB is often referred to as the definition of hyperelasticity. Hence, the major symmetry of C is basically equivalent to the existence -of .a strain-energy function. Note that the major symmetry of the elastidty tensor is associated with the sym-metry of the (tangent) stiffness matrix arising in a finite -element discretization proc-edure. The elasticity tensor in the spatial description or the spatial tensor of elasticities, denoted by c, is defined as the push-forward operation of C times a factor· of J- 1 (see MARSDEN and HUGHES [1994, Section 3.4]), in other texts the definition -of c frequently excludes .the factor J- 1 • .It is the Pio/a transformation of C on each large index so that

(6.159) with the n;linor sym-metries Cabcd

= Ct10.ccl = Cabdc

,

(6.160)

and additionally for hyperelasticity we have the major symmetries c = cT or Cabc(f = Ccdab· The fourth-order tensors C and c are crucial within the concept of linearization, .as will become apparent in Chapter 8, particularly in Section 8.4. The spatial representation of eq. (6.153) can be shown to be £v(r~) =Jc: d

(6.161)

(for .a proof see Section 8.4, .p. 398), .in which £v( r·a), d and c denote the objective Oldroyd stress rate (5.59) -of the contravariant .Kirchhoff stress tensor T, the rate of deformation tensor (2.146), and the spatial elasticity tensor, as defined in eq. (6.159),

254

6


respectively. A material is said to be .hy.poelas-tic if the associated rate equations of the form (6 . 161) are not obtained from a (scalar-valued) energy function. For more on hypoelastic materials see the classical and detailed account by TRUESDELL and NOLL .[ 1992, Sections 99-103]. Systematic treatments of the elasticity tensors have been given by, for example, TRUESDELL .and TOUPIN [1960, Sections 246-249], CHADW"ICK and OGDEN [197la, b], HILL [198.1..J, TRUESDELL and NOLL [1992, Sections 45, 82] and 0.GDEN [1997, Chapter 6].

Decoupled representation of the elastidty tensor. Based on the kinematic assumption (6.79h and the decoupled structure of the strain-energy function (6~85) we derive the associated elasticity tensor. We focus attention solely on the material .description of the elasticity tensor. The elasticity tensor (6.154) may be written in the decoupled form 8S(C)

c=

2

ac = Cvol + Ciso

(6.162)

I

which represents the completion -of the additive split of the stress response (6.88). In relation (6.162) we introduced the definitions Cvo.t

=2

8Svol 8C '

tr'.

_

\L,•so -

')

-

DS.iso

ac .

(6.163)

of the purely volumetric contribution Cvot and the purely isochoric .contribution Ciso· By analogy with eq. (6.157) we express the two contributions Cvol and Ciso in terms of the strain-energy function \JI. Before this exploitation we introduce the definition

ac-1

ac

=

-c-•0 c-1

(6.164)

of the fourth-order tensor ac-J I ac, for convenience (recall Example 1.11, p. 43, and Lake A = C in relation (l.249)}, where the symbol 0 has been introduced to denote the tensor product .according to the ru.le

-{c-

1

0

c-• )AT:Jcv

ac-

1 1 = - -(c_4bc;;b + c.=ibcn6,) = · ATJ 2 , ....... ...;;:.'.::~·.::·:::·.:".'.:~:-:.:~·: ·. ·:·: .. : . ._. .: :· :·,· . . , _. , .:. . ,. . . . . ,.,. .,. .
•

(6.165)

Starting with (6.163)i, a straightforward computation yields~ with definition (6.89h, 1 property (1.256), the derivative of J and with respect to C, Le. eqs. (6.82) 1 and ( 6.164 ), and the product rule of differentiation,

c-

~ -= ?8Svo' = ?8(JpC·ol

~

ac .,

= 2c-1 ®

ac

~

1 ) ··.

op )

( rn[J_!_

~~~~.ac_

:~

J!:_:!.lp ..DC

·=.!]Jc~· 0es 2JvC:-i'0,~:-- 1 1

-

ac- l

\ .\

(6.166)

6.6

255

Elasticity Tensors

For convenience, we have introduced the scalar function

l1

dp

=.?) + .J dJ

p, defined by {6.167)

,.,

.. .

':

with the constitutive equation for p given in (6.91) 1• Note that the only values which must be specified for a given material are p and 1J. The foil owing example shows a lengthy but r~presentative derivation of an elasticity tensor.

EXAMPLE 6 ..8 Show the following explicit expression for the second contribution to the elasticity tensor, .i.e. the isochoric part C;~m' as defined .in eq. (6. l63h,

= P : C : pT + ~Tr( r

Cisn

2 3

1 S)IP

- J2 (C ~) ® Siso + Siso ® C - t)

(6.1.68)

(compare also with HOLZAPFEL [ l 996a]), which is based on the definitions

c = 2J-4/3as = 4 r4/a Cf!'11isn{C)

·

ac

ffe

= (•) : C

Tr( e)

acac

(6.169)

,

= c-1.0 c- 1 - ~c- 1 0 c- 1

(6 ..170)

3

of the fourth-order fictltious elasticity tensor C in the material description, the trace Tr(•) and the modified projection tensor JP -of fourth-order.

Solution. Starting from the definition of C 150 , i.e. (6..163)2, we find, using (6.90)4 and property (I. .256), that

_ ') aSiso _ ?. 0 (J -2/'J' JP> : -) S

mo -

...

ac -

= ?(llll . S) 0 -

ac

*.I

1f •

-'>fl

(

DJ .. .

ac +

-)

2 1-2/:i a IP : S

(6.17.1)

De

t

The first term in this equation yields, through property (6.82.h and definition (6.90) . h -.

2(P:S)®

[)J ~2/;'1

BC

') ,,;.,

=-J(J

-2(1

-

·P:S)®C

- l

') ...

=-3Siso®C

-l

,

(6.172)

which gives the last expression of (6.168) we show. Hence, in the following we analyze exclusively the second t~rm in (6.171). With the definition of the projection tensor (6.84), identities (l.160) 1, (l.152) and the chain

256

6


rule, we have

v-21a8(~~ 8) = 21-2;a

:c (

~(c-1 ® c) :

s-

= 2J_2 1a (as_ !a(S: 9c-

ac

1 :

)

.ac

3

s) ac

(6. 173)

ac '

and finally, using the definition of the fourth-order tensor 8C/8C, i.e. (6.83) 4 , definition (6.169) 1 and property (1 ~256),

2J- 21:1°(~~ S)

= [C -

~(A1 + A2}]: pT

'

(6.174)

in which the definitions

At = u-4/Jc-l ® 8(S :..._C) '

ac

AC) 6o

= 2J-4f3(S : C) ac-1. ac

(6.175)

ar.e to be used. In order to study (6.175) in more detail we apply property (1.255), the chain rule, identity (1.160}i, definition (6.169) 1 of (6.175) 1 and definition (6.164) of (6~175h to give

A1 =

c- 1 0 (c: c + 2.1-213 s) = c- 1 0 c: c + 2c-t 0 J- 213 s ,

(6.176)

= -2J- 2 /:1(S: C)c- 10 c- 1

(6.177)

A2

•

Eqs. (6.17 6) and (6. 177) substituted back into (6.174) yield, using identity ( 1.160) i. and definition (6.169h,

v-->/J 8(1P'ac: S) = (JI - 31 c- I ® c) : -c : PT - 32 c- ·1 ® r-'>/3-s : JPT 1

+~nv- 2 13 s)c- 1 0 c- 1 : pT

(6.178)

•

With the definition of the projection tensor (6.84), then with definition (6.90),i, according to identity (1.157), and by means ·Of (6.164) with rule (L254) and (6.170) we find finally

?J-21a8(P: S)

-

ac

=

'"r - ~c-1

a ® S·. +~Tr(r 2 1 3 s)(c- 10 c- 1 - ~c- 1 ® c- 1)

1DJ.

Jr •

• ic-

150

(6. l 79)

,

If» which is identical to the remaining terms in (6.168). -':"""'!"'·-:·....:.--------:. -·---..

--~-=-""

____,....

....... . ;

~.-.·..:~~

Bl :. . :-..-...:.-.-·

-~~·,,

...

,.~·~·

_

_..,.~,·=~,

...:____ _,,_:"=""·

6.6

Elasticity Tensors

25"7

The two tensor expressions (6.166)4 and (6.168) in the material description represent explicit forms which are generally applicable to any compressible hyperelastic material of interest. Since we have already computed the stress relation S = Svol + Sisoir with the terms Svol = Jpc- 1 and Siso = J- 2 /ap : S, it is a straightforward task to set up the associated elasticity tensor
Elasticity tensor in terms of principal stretches. Consider an isotropic hypereIastic material characterized by the strain... energy function '1t = \ll(AL, A2 , /\ 3 ), with the principal stretches ,\ i, "'\2, .,\3. The aim is now to derive the spectral form of the elasticity tensor C in the material description, namely 3

C

=

""' 1 8Sa. ~ " ,. ~ ~ D>. Na ®Na 0 N,, 0 Nb A

a, b--1

b

,,

L ~~ =~; (Na ® N,, ® Na ® Nb + N11 ® Nb ® Nb ® Na) 3

+

a,b=.l a=;;f b

b

a

1

(6.180)

258

6

Hyperelastk Materials

with the principal second Piola-Kirchhoff stresses

s

(1

aw

= 2 ().,\ ~

1 aw = ,\a a,\(1

I

(L

= 1, 2

1

3

!

(6..181)

and the set {Na}, a = 1, 2, .3, of.orthonormal eigenvectors of the right Cauchy-Green tensor C. They define principal referential directions at a point X, with the conditions

INuf = 1 and N.n ·Nb =

<~ab·

The important fourth-order tensor Cina more general setting was given by OGDEN [1997, Section 6.1..4]. Compare a]so the work of CHADWlCK and OGDEN [ 1971 a, b] with some differences :in notation. The proof of representation (6.180) is as follows:

P.rooJ

In order to prove relation (6.180) we follow an approach which takes advantage of isotropy. ·we know from Chapter 5 that for isotropic elastic materials the second Piola-Kirchhoff stress tensor S .is ·coaxial with the right Cauchy-Green tensor C, so that S has the same principal directions as C. For notational convenience we use henceforth a rate formulation rather than an infinitesimal formulation. In particular, we now compute the material time derivat)ves of the stress and strain tensors S and C, and compare them with the rate form of relation (6 ..153), i.e. S= C: C/2, in order to obtain the elasticity tensorC, as defined in (6.154) and specified .in (6. I·SO). To begin with, consider a set of orthonormal basis vectors ea, a = 1, 2, 3, fixed in space. Consequently, the set {Na}, a = 1, 2, 3, of orthonormal eigenvectors may be governed by the transformation law

Na=

Qea ,

a -- ·1 ' '). . . , 3

(6.182)

(compare with eq. (I. I 82), Section 1.5), where Q -denotes an ortho_gonal tensor with components Q al1 == e,.t ·Nb ·= cosO (ea, Nb), representing the cosine of the angle between the fixed basis vectors ea and the ·orthonormal eigenvectors Nb (principal referential directions). Tensor ·Q is ·characterized by the orthogonality condition, i.e. QTQ = QQT = J. Next, we compute the material time derivative of the principal referential directions ~a- Since the bas:is vectors are assumed to be fixed in space (e0 = o), we may write ...

.Nu = Qea, a ==

I_, 2, 3. By expanding this equation with the orthogonality condition and by means of the skew tensor n, as introduced in (5..:15), and transformation (6.182), we may eli 1ninate the basis {ea} and find that

a - 1, '). . . _, 3. . Note that the components of the skew tensor are obtained from (6.183)2 in the form

(6.18:3)

n = -nT with respect to the bas.is {c,,} (6. 184)

6.6 Elasticity Tensors

259

= 0. By me~ns of identity (l .65h, we may deduce from (6. l 83h the repre·1 "' .., sentation n = .E~1= l N,l ® N(t . ... with n,u1

Knowing Na~ we now can determine -the .material time derivative of the spectral representation of C. Converting eq. (2. l 26) to the rate form, we obtain, by means of eq. {6..183h .and the spectral decomposition of the right Cauchy-Green tensor, i.e. 3 9 C = Ea=J ,,\~Nc1 ®Neu ~

~

3

C-

3

L 2,\).,,N" ® Na = L ,\~(N" ® Nil + Nn ® N,i) a.=1

a

=L

,\~(nNa ® Ntl

+ Na ® nNa)

o=l

= nc- en .

(6..185)

From (6.1. 85.h using (6.184) we deduce that 3

3

t =L

2,\a.\.tN(l 0

Na+

L

nab( A~ - A~)N,, 0 Nb '

(6. I 86)

n,b= l

a=L

ofb

where 2Aa.~a =.Na· CN 0 = C.'0 a, a== 1, 2, 3 {compare with eq. (2.127)), denote normal components (diagonal elements) and nab(,\~ - A~) = Na · t:Nb = Cub,, a ¥ b, denote shear components of C (off-diagonal elements) with respect to the basis {Na}· By isotropy, S has the same principal directions as ·"(:~·~····-·Hen~e, recall (6.52h, Le. "' S = E;,.= 1 SnNa ·0 .Na, with Sa = 1/ Aa(D\J! /8/\a.), a = 1, 2,, 3, we obtain, by analogy with (6.186) ~i

A

3

3

s = L SoNa 0 Na+ L

nab{ Sb - Sa)Nu ®

N,,

(6.187)

1

a=l

in which the material time derivative of the principal second Piola-Kirchhoff stresses is defined to be (6.188)

By expanding the numerator and denominator of the second term in (6.187) with ,\~ - /\; and by means ·of (6. l 8.8)2, eq. (6..l 87) .can be rephrased as

s=

aas

:.i

s

0

L ())..: ,\bNa ® Ntl + L nab(A~ - ,\~)~~~=,\~Na® Nb a,b= 1

a,b=1
/J

.

(6.189)

n

·On comparing the derived ·eqs. (6."I 89) and (6..186) with (6.153) in the rate form, i.e. S = C : C/2!f we find by .inspection that the elasticity tensor C emerges, as given by eq. (6.180). II


260

For the case in which two or even all three ei.genvalues /\~ of C {and also of b) are equal, the associated two or three stresses Sa are also equal, by isotropy. HenceT the divided difference (Sb-Sa)/(.Al-A.~) in expression (6.1"80) represents an indeterm.inate form of type However, it can be shown that the divided difference is well-defined as _,\b approaches Aa. Namely, app.lying l'Hopital 's rule, we see simply that

g.

(6.190)

(compare also with the work of CHADWICK and OGDEN [1971Q]). Consequently, the elasticity tensor, as defined in eq. (6. I.80), is valid for the three cases: ,.\ 1 f. .A 2 -:/: A3 # /\i, A.1 = ~\2 '# ~\1 and ,.,\1 = ~\2 = .t\3. Finally, in order to set up the spectral form of the elasticity tensor ·c in the spatial description we use the Piola transformation of C for principal values. According to (6.159), this gives Cabcd = J- 1 AaAb.Ac,\d Cabcd' with the principal stretches /\1, .-\2, ,\3 and the volume ratio J = A1A2 A3 . A straightforward computation leads to

(6J.9l)

with the principal Cauchy stresses aa = J- 1>..·~Sa, a = 1, 2, 3 (see the inverse of eq. (6..48h), and the principal spatial directions a= 1, 2, 3, which are the orthonormal eigenvectors of v (and also ofb ), with .Jiia I = 1 and Da ·nb = Oab· From the property {2.l 18) we know that the two..-point tensor R rotates the principal referential directions Na into the ..Principal spatial directions Da. If /\a = Ab we may conclude that aa = ab, by isotropy. Hence, the divided difference (ab/\~ - aaA.~)/(/\~ - /\~) in expression (6.19]) gives us and must therefore be determined applying l 'Hopitat's rule. Differentiating the numerator and de.nominator by Ab and takin.g the limits Ab -7 /\a, the divided difference becomes

n,z,

*

(6.192)

An alternative version of solution (6.192)2 in terms of the principal se.cond Piofa-Kirchhoff stresses, frequently used in other texts, i~ left as an exercise.

6.6

261

Elasticity Tensors EXERCISES

I. For the description of isotropic hyperelastic materials at .finite strains consider the strain-energy function w = \f!(l.i, r2, /3) in the coupled form, with the principal invariants Io., a= 1, 2, 3. (a) Use the stress relation (6.32.h, the chain rule and the derivatives of the in· variants with respect .to C, i.e ..eqs. (6.30):i .and (6.31 ), in order to obtain the most general form -of the elasticity tensor C in terms of the three principal invariants

c = ? as

-.ac

2

= 4a '1>'{Ji. I2J:1)

acac

= 811®I+02(.I ® c + c

®I)+ 03(1 ® c- 1 + c- 1 ®I)

+ Or,(C 0 c-l. + c- 1® C) +8f>c- 1 ® c- 1 + .c51c- 10 c- 1 + osll , +8.-iC ® c

(6.193)

with the coefficients &1 , ... , 08 defined by

81

02

==

4(

a2 w

8!1811 +

a2 w

211

a2 \J!

aw

<>a =

aw )

'

2

.a2 w

'

a2 w )

8118!3 + 1113 8128/a

fP\f! Os = -•lla 0128h 8\J! 61 = -4/a- , 8/3

2

DI18I2 + 812 +Ii 8128!2

= - 4 ( DI1DI2 + liDI28I2 4 ( 13

') a w )

c58

~ ( 8\JI J6 = 4 13 8/3 8\I! = -4- . 812

<>4 =

4

a2 w OI28I2 '

2 a ·w ) + 13 8/38/3

6 194 )

( •

?

'

c- c-

1 1 The fourth-order tensors 0 and Il in eq. (6.193)3 are defined .according to (6.164) and (1.160), respectively.

(b) Particularize the coefficients Ou, a = 1, ... , 8, (6.194 ), for the compressible Moon~y-R.iv.lin, neo-Hookean, Blatz and Ko models, Le. eqs. (6.147)(6.149). For convenience, summarize the (nonzero) coefficients for the three .material models to fonn the entries of Table 6.3. For notational simplicity we have introduced the abbreviation ~ = 2tt(1 f) /Ia with ~ > 0. Note that the coefficients 6a, a = 1, ... , 8, for the neoHookean ·model are simple those of the Blatz and Ko model obtained by setting ~ = 0 and µf = 2c1. Compare the .corresponding constitutive equations of the three material models of Exercise l(a) on p. 248, with the specified coefficients summarized in Table 6.1.

262

·6

Moon.ev-Rivlin model v

.

{6.]47) c.~ l

Hyperelastic Materials neo-Hookean model (6.148)

Blatz and .Ko model (6.14~)

~

4c2

c5a

~~Ii.

cl5

(.

Jr .. )

2cJ(2J - 1.)

c)-I

-2[2cJ{J - 1) - dJ

,58

-4c2

J JCi. 1-ll :1 .

t. ·/

1 .{_·Ct

J-B :1 .

+ ~(!2 + {Jl~HI) ')-/1-.,. 1-# :i + ~(J. . 2 - pf.J+l) :1

2tt.f(3I:;JJ

-~

Table 6.3 Specified coefficients for the clasticily te.nsors of some materials in the ·coupled form. 2. The strain-energy function '11(.J~ 11, 12 ) = Wvut ( J) + Wisa(lL, 12 ) is given in terms of the volume ratio J and the two modified principal stretches 11, T2 • This type of strain energy in tlecoupled representation is suitable for the characterization

of compressible .isotropic materials at finite strains. (a) The associated decoupled elasticity tensor C is given by (6 . 162), with the volumetric .and .isochoric parts (6.1·66) 4 and (6.168), respectively. Particularize the fictitious elasticity tensor C in the material description, i.e. eq. (6.169.).i, to the specific strain energy at hand. Start with the constitutive equation for the fictitious second Piola-Kirchhoff stress S, as defined in eq.·: (6J.15), and use the derivatives of l., 12 with respect to tensor C., i.e. (6.117), in order to obtain the most general form of C in terms of 11 and 12 in the form

with the fourth-order unit tensor Il defined by ( 1.160) and the coefficients J,H l1. = 1, • .. l 4, by

(6.195)

"£5';'. 1 ·= -4 aw-iso 8h

6.6

263

Elasticity Tensors

(b) To be specific, take the decoupled form of the compressible Mooney-Rivlin model (6.146) and the compressible neo-.Hookean model (set c2 = 0). Additionally, take the .Yeoh mode.I and the Arruda and Boyce model of the forms

\JI ==

\It

Wvol

-

+ ·C1(.l1 -

= \JI vol + /1 [ -91 ( f·-1 .

M

3)

3)

-

+ c2(f1 -

+ ?O1n (I-2I -

9)

... 3)- + C:i(I1 - 3)'

+

11 n n-

-3

1

-

IO~ O ., (JI - 2 I)

.,.·

corresponding to ·eqs. (6.134) and {6.1.36),

~

')

+ · · ·]

respectively~

Specify the coefficients 6n, a ·= 1., ... , 4, .(6.195.), for the four material 1nodels in question and summarize the result in the form of a table.

J. J;J

Mooney-Rivlin model (6.146)

neo-Hookean model

4c·2

0

-4c2

0

Ye oh model (6.134)

8.[c2

+ 3C;J(J1 0

- 3)]

Arruda and Boyce model (6.136.)

/L·[2/5n +(44/175n2 )11

+ ... ]

0

Table 6.4 Specified coeffidents for the efast.icity tensors of some materials in the decoupled form.

Note that the associated constitutive equations, with the specified coeffic.ients as summarized .in. Table 6.2, are presente,d in Exercise I. (b) on p. 248. 3. Consider a compressible isotropic material charad.~rized by the strain-energy function in the decoupled form of \}I (,.\I' ,\2' A:i) = . . \JI vu) ( J) + \{I iso (Xi, ...\2' ,\:i)' with the volume ratio J = >iu\ 2 .,\1 and the modified principal stretches ,.\1 = J-1./:i .,.\a, a = 1, 2, 3. The associated decoupled structure of the elasticity tensor C in the material description is given as C(,.\., .,.\ 2 , ,,\a) = Cvol +
(6.196) where Sison = {8\J!iso/D,\a)/ ,\1' a = 1, 2~ 3, denote the principal values of the second Piola-Kirchhoff stress tensor Siso ~

264

6

Hyperelastic Materfals

(b) In order to specify the elasticity tensor (6.196), take Ogden's model (6.139) and recall the isochoric hyperelastic stress response in terms of the three principal stresses S1soa, a = 1, 2, 3, determined by the closed form expression (6.144h using (6.145) (compare with Example 6.7). For this important class =of ·material models particularize the coefficients of the first .part of the elasticity tensor (6.196) by means of the given strain. · 3 N -op energy function (6.139), 1.e. Wiso(Ab .,.\2, .Xa) = Ea=l E.P=l (J.t1,/ap)("\a 1). Show that (6.197)

for a= b ,

for a~ b . This representation was given by SIMO and TAYLOR [1.99Ia, Example 2.2], with so.me differences in notation. 4. Consider a strain-energy function in terms of principal stretches Aa, a = 1, 2, 3., and principal ·invariants Ia, a = 1, 2, 3, according to \JI = '11(/\i, .,\ 2 , .A 3 ) =

'1!(11) 12, 1:1)~ Write down the associated constitutive equation for the principal second PiolaKirchhoff stresses S
. Sb - S 1IID \? \ '>0 .-\h4.-\a "b - "a

= _2 8\J! 8J

2

_ ?J 8'11 , _2 , _ 2 .-

3 8/3 Aa

;-\b

5. By means of l'Hopital's rule derive the alternative version of (6.192) in terms of the principal second Piola-Kirchhoff stresses S,1' i.e. .. ab;\~ - aa--\i _ _1 ,2 \2 1llll \2 \2 - 1 /\a /\b

..\b~Aa

/\b -

Aa

(asb _asa) {) \ 2 /\b

.~l\ 2

VAb

6. Consider the constitutive equation (6.38) in tenns of the (spatial) left Cauchy. . Green tensor b representing isotropic hyperelastic response. Deduce from the constitutive rate equation (6.161) that the associated elasticity tensor c in the spatial description has the explicit form

. a w(h) 2

Jc= 4b obtJb b .

(6.198)

6. 7 Transversely Isotropic Materials

Hint: stress~

265

Recall relation (3.62), the Oldroyd stress rate (5 . 59) ·of the Kirchhoff kinematic relation (2.169) and use the chain rule.

Representation (6.198) was given by ·MIEHE and STEIN .(1992]. In their work the definition of the elasticity tensor c = x* (C) in the spatial description excludes the factor J- 1 .

7. Consider the additive split of the Cauchy stress (6.101)-(6.104) in terms of J and b, based on the strain-energy function of the form (6.98). Show that the associated elasticity tensor
= Cvol + Ciso

,

with the definitions Jievol

. D2 Wvol(J)

= 4b

obab

2

(-

b = J pl ® I - 2vli)

with

]J

dp

= p + J dJ

1

-

Jr·. = 4b 8 Wiso(b)b ILJso 8b8b

= n•: iC: !l" +

2

2

3tr(r)n• - 3(1 ®Tiso+ Tiso® I}

of the purely volumetric contribution Cvol and the purely j:sochoric contribution >Ciso• the latter being based on the spatial projection tensor p." ][ :. . . ~I ® I and on the definitions T = .lu, Tiso = J Ui50 , with O"iso = n) ; u, as given in (6.103) and {6.104).

In addition, we introduced the .definitions of the fourth-order fictitious elasticity tensor c in the spatial description and the trace tr{•) according to

-82 '11 iso {b )it·= 4b · b , -

tr(•) ·= ( •) : I . Db8b For an explicit derivation, ·see MIEHE [ 1994~ Appendix A].

6. 7 Transversely Isotropic Materials Numerous materials are composed of a ma.tr.ix material (or in the literature often calJed ground substance) and one or more families of fibers. This type of materiat which we cal1 a composite material or fiber.. reinforced com:posite, is heterogeneous in the sense that it has different compositions throughout the body. We consider only composite materials in which the .fibers are continuously arranged in the matrix material. These types of composites have strong directional properties and their mechanical responses are regarded as anisotropic.

266

6


The challenge in the design of fiber-reinforced composites is to -combine the matrix material and the fibers in such a way that the resulting material is most efficient for the desired application. For engineering applications composite mate.rials provide many advantages over monolithic materials such as high stiffness and strength, low weight and thermal expansion and corrosion resistance. However, the drawbacks in using composite materials seem lo be the high costs when compared with those of monolithic (more dassical) materials and_, from the practical point of view, limited knowledge of how to combine these types of material. A material which is re.inforced by only one family of_ fibers has a single prefe.rred direction. The stiffness of this type of composite material .in the fiber direction is typically much greater than in the directions orlhogonal to the fibers. It is the simplest representation of material anisotropy, which we call .transve.rsely isotrop.ic with respect to this preferred direct.ion. The .material response along directions orthogonal to this preferred direction is isotropic. These composite .materials are employed in a variety of .applications in industrial engineering and medicine. For manufacturing and fabrication processes and for typical features and properties of transversely isotropic materials the reader should consult, for example, the textbook by HERAKOV.ICH .[ 1998]. The aim of the following section is to investigate transversely isotropic materials capable of supporting finite elastic strains. As mentioned above, all .-fibers have a single preferred direction. However, the fibers are assumed to be continuous.ly distributed throughout the material. We derive appropriate -constitutive equations which are based solely on a continuum approach (excluding micromechanical considerations). Constitutive equations which mod.el transversely isotropic materials .in the small elastic strain .regime .are well established and may be found, for example, in the textbooks by TSAI and HAHN [1980], DANIEL and .lSHAI [19.94], HERAKOVlCH [1998.] and JONES [ 1999]. Kinematic relation and ·structure of the free energy. We consider a continuum body B which initially occupies atypical region 0 0 at a fixed reference time t = 0. The region is known as a fixed reference cm~figuration of that body B. A point in n0 may be characterized by the position vector X (with material coordinates .Xr.h .A = 1> 2, 3) related to a fixed set of axes. At a subsequent time .t > 0 the continuum body is in .a deformed configuration occupying a region ·n. The associated point .inn is characterized by the position vector .x (with spatial coordinates :c 0 , a = 1., 2, 3) related to the same

fixed set of axes. For more details about the relevant notation .recall Section 2.1. We suppose that the only anisotropic property of the solid comes from the presence of the fibers. To start with, for a material which .is reinforced by only one family of fibers, the stress at a material point depends not only on the deformation gradient ·F but also on that -single preferred direction, which we call the fiber direction. The direction of a fiber at point X E H0 .is defined by a unit vector field a 0(X)T laol = 1, with material

6.7

Transverse·ly Isotropic Materials

267

coordinates .a 0 A. The fiber under a deformation moves with the material points of the continuum body and arrives at the deformed configuration n. Hence, the new fiber direction at the associated point x E !1 is defined by a unit vector field a(x, t), Jal = 1, with spatial coordinates aa. For subsequent use it is beneficial to review the section on material and spatial strain tensors introduced on pp. 76-81.. Allowing length changes of the fibers, we must determine the stretch ~\ of the fiber idong its direction a 0 • It is defined as the ratio between the length of a fiber element in the deformed and reference configuration. By combining eq. (2.60) with (2. 71) we

find that 1\a(x, t)

= F(X, t)a0 (X)

,

(6.199)

whkh relates the fiber directions in the reference and the deformed configurations. Consequently, since lal = t we find the square of stretch ...\following the symmetry 2 _

T

_

,\ - an · F Fao - au· ·Ca0

,

(6.200)

which we already have .introduced in relation (2.62). This ·means, that the fiber stretch depends on the fiber direction of the undeformed configuration, i.e. the unit vector field a 0 , and the strain measure, i.e. the right Cauchy-Green tensor C. We now assume the transversely isotropic material to be hyperelastic, characterized by a Helmholtz free-energy function w per unit reference v·~fli·nl-~. Because of the directional dependence on the deformation., expressed by the unit vector .fie.Jd a 0 , we require that the free energy depends explicitly on both the .right Cauchy-Green tensor c and the fiber direction ao in the reference configuration. Since the sense of a 0 is immaterial, '11 is taken as an even function of a 0 • Hence, by introducing the tensor product a 0 ® a 0 , \JI may be expressed as a function of the two arguments C and a 0 0 a 0 . The tensor a 0 0 a 0 (with Cartesian components o..0 Aao n) is of order two. For the Helmholtz free-energy function we may therefore write \JI

= w(C, ao ® ao)

.

(6.201)

From previous sections we know that the free energy must be independent of the coordinate system; hence \Jl(C, a 0 0 a 0 ) must be ·Objective. Since C and a 0 ® a 0 are defined with respect to the reference configuration (which is fixed), they are unaffected by a rigid-body motion superimposed on the current configuration. Consequently, the principle of material frame-indifference of the postulated free energy w(C 1 a 0 0 a 0 ) is

satisfied trivially.

"EXAM..PLE 6. .9 The free energy W(C, a 0 ® a 0 ) must be unchanged if both the matrix material and the fibers in the reference configuration undergo a rotation around a certain axis described by the proper orthogonal tensor Q.

268


Show that the requirement for transversely isotropic hy.perel.astic materials formally reads (6.202)

which holds for all proper orthogonal tensors Q. Solution. For the solution it is beneficial to review eqs. (6.19)-(6.25) of Section 6.2. A rotation of the reference configuration by tensor Q transforms a typical point X into position X* = QX. Consequently, fiber ..direction a0 transforms into the new fiber direction a 0 = Qa 0 so that a 0®a~ becomes Qa 0 ® a 0 QT. Now, after a subsequent motion of the rotated reference configuration, X* maps into position x. Thus, the deformation gradient F* and the strain measure C* = F*TF"" relative to the rotated reference .configuration are F* = FQT (compare with (6.22)) and C* = F*TF* = QFTF·QT = QCQT, respectively. We say that a hyperelastic material is transversely isotropic relative to a reference configuration if the identity \P (C, a 0 ® a 0 ) = '11 (C*, a 0® a~) is satisfied for all proper orthogonal tensors Q. Hence, restriction (6.202) follows directly. Note that in view of (6.202), \JI may be seen as .a scalar-valued isotropic tensor function of the two tensor variables C and a 0 ® a 0 . II

According to (6.27), an isotropic hyperelastic material may be represented by the

first three invariants 1 1, 12 , 13 of either C o.r .b, characterized in (5.89)-(5.91 ). These invariants can be used to fulfil requirement (6.25), i.e. '1t (C) = \II ( QCQ'1) for all (Q, C). Followjng SPENCER [1971, 1984], two .additional (new) scalars, 1 1 and 15 , are necessary to form the imegrity bases of the tensors C and a 0 ® a 0 and to satisfy relatio.n (6.202). They a.re the so-called .pseudo-invariants -of C and a 0 0 .a 0 , which are given

by (6.203)

The two pseudo-invariants L1, / 5 arise directly from the anisotropy and contribute to the free energy. They describe the properties of the fiber family and its interaction with the other material constituents. Note that invariant / 4 is equal to the square of the stretch A in the fiber direction a 0 (compare with eq. (6.200)) . For the definition of the integrity bases and the related theory of invariants see the lecture notes by SCHUR (1968], the articles by SPENCER .[1971] and ZHENG Il994, and references therein]. For applications .in continuum mechanics the reader is referred to the works by RIVLIN [.1970], BETTEN [1987a, Chapter D and 1987b], TRUESDELL and NOLL [1992] among others. A brief review of the theory of invariants may also ~e fou~d in SCHRODER [1996]. .

269

·6.7 Transversely Isotropic Materials

For a transversely isotropic material, the free energy can finally be written in tenns of the five independent scalar invariants, and eq. (6.27}1, valid for isotropic mate.rial response, and may consequently be expanded according to

\JI

= \JI [!1 (C), I2(C), Ia(C), Li (C, ao), ls(C, ao)]

.

(6.204)

The free energy (6.204) provides a fundamental basis for .deriving the associated constitutive equations.

Constitutive equations 'in terms of invariants. In order to derive the constitutive equations we apply (6.13h. Then, by use of the chain rule, the second Piola-Kirchhoff stress tensor S is given as a function of the five scalar invariants, i.e. =

S

?8W(C, ao ® aa)

ac

~

=?

~ 8W(C, ao ® aa) 8Ia

-~ a=l

a1a

(6.205)

ac '

in which 8!1 /8C and 812 /BC, 8I3 /8C are given by (6.30) and (6.3.l), respectively. The remaining derivatives follow from (6.203) and ·have the forms

8I1

-

ac =ao®ao 8/5

or

ac = ao 0

(6.206)

Cao+ aoC ®·ao (6.207)

c

a~ BC = lloACBcaoc + aoa .4caoc .

·or

AB

Finally, (6.205) reads, with eqs. (6.30), (6.31), (6 ..206) and..(6.207),

S

= 2 .[(~; +

1 1 +1 ~~) I-~; C+h~: ~~ ' . 1 2 3 aw aa ® ao +ah aw (ao ® Ca0 + a0 C ® a 0) ] aJ.

,

(6.208)

1

which extends the constitutive equation (6.32) by ·the addition of the last two tenns. Using arguments similar to those used for the derivation of the spatial version of the stress relation (6.34), namely a push-forward operation on the material stress tensor S by the motion x, we arrive, using (6.199) and (6.203)i, at _ 1 ;[ aw u = 2J , h Bia I+

aw

( 8'1!

aw )

8I1 + 11 OI2

8~

a·w

2

b - 812 b

J

+L1 L a®a+L1 (a®ba+ab®a)'. 8 1 815

(6.209)

Recall that the unit vector a(x, .t) denotes the fiber direction in the deformed ·configuration while b = FFT is the (second-order) left Cauchy-Green tensor. Observe the

6. Hyperelastic ·Materials.

270

sitnilar structure of the last two terms in the stress relation (6 . 209) to that presented in (6.208).

The associated elasticity tensors -in the material and spatial descriptions follow by means of expressions (6.154) and (6.159), respectively (for .an explicit derivation compare with the work of WEISS -et al. [.I 996.J). For implementations of large strain transversely isotropic models in a finite e.lement program see WE.I.SS et aL [ 199-6], SCHRODER [1996]~ BONET and BURTON [1998] among others. Incompressible transverse_ly isotrop.ic materials. ·we now consider transversely .isotropic materials with an incompressible isotropic matrix materh1L Firstly, we study the case in which the embedded fibers are extensible. Since we

assume incompressibility of the isotrop.ic matrix material, i.e. 1:1 = 1, we are able .to postulate a free energy in terms of the .remaining four independent invariants. Be-caus-e of the incompressib.i-1 ity constraint I:i = 1 the free energy ·w is enhanced by an indeterm.inate Lagrange nwltiplier p/2 which is identified as a reaction pressure. In view of (6.62) we have the assumption

'1' = W[I1(C), h(C), I.1(C, aD), l;;(C, a0 )]

-

~p(l:i ....

1) .

(6.2.10)

The associated stress refations given .in the reference and current configurations are basically those presented by (6.63) and (6.64) supplemented by the fourth and fifth ter~ in eqs. (6.208) and (6.209), respe-ctively. Secondly, we study an "incompressible isotropic matrix nlaterial which .is continuously reinforced throughout by inextensible fibers. This means that ..-\ = 1 and, in view -of (6.203). 1, the fourth .invariant is equal to one. With this additional internal constraint., the free energy \JI is a function only o_f It., 12 , which are responsible for the hyperelastic .isotropic matrix material, and fa, which .is responsible for the fibers. By .adding the term q(L1 - 1.)/2 lo the free energy W we obtain the function

1

1

4i.J

.....

\JI= \ll[I1(C), I2(C)~J5(C.1ao)J - 9P(f3 - 1) - 9
(6. 211)

where q /2 is an additionai indeterminate Lagrange multiplier.

The associated stress relations in the Lagrangian and Eulerian descriptions for the transversely isotropic materials with an incompressible isotropic matrix .material and inextensible ·fibers (with direction a 0 ) are the extended constitutive equations (6.63) and (6.65), i.e.

S

a,T!. +/1 -aw ) 1-2-f_-C aw = -pC -1 -qao®ao+2 ( 811 8!2 . 812

,ow

+2 aJ. {ao ®Cao+ aoC ® a 0 ) .J

(6~212)

271

6.7 Transversely Isotropic Materials

Note that the indeterminate terms -qa0 0 a 0 and -qa ~ a are .identified as ·fiber .reaction stresses which respond Lo the inextensibiHty constraint L1 = 1.

EXERCISES

1. Starting from the pseudo-invariants L1 and 15 , i.e. eqs. (6..203), show their derivative with respect to C, eqs. (6.206) and (6.207). 2. We characterize a transversely isotropic material by the decoupled free energy in the form

(6.214) and wi:m are the volumetric and isochor.ic contributions to the hyperelast.ic response (recall Section 6.4). The modified invariants 11, 12 are given according to eqs. (6.109) .and (6.110), while f:s = detC = 1 (note that 11, ! 2 , f, are the modified principal invariants of the modified tensor C = .1- 2tic). The remaining modified pseudo-invariants are ex.pressed by f 1 = J-'2/a I.1 and 15 ·= J-4/:l h. where

(~)

\1'.vol

Having in mind the free energy (6.214) and the derivatives (6.206) .and (6.207), show th.at the constitutive equation S = Jpc- 1 + J- 2 /:ip = S specializes to

s=

-

·-

-

-

5

-

tNhsi, (I 1 J 2, I1 , Ia) = 2 "'\;'"" aw~so EJIa 2 ac ~ a1ll ac . <1= I

-,

ai~:1

= ,. , . 11+72 C + 7,1ao ® ao + ')'5 (ao ®Cao+ aoC 0

ao)

(6.215)

with the response coefficients

·- - ') aw iso !::. -

J.J

0/5

for the fictitious second Piela-Kirchhoff stress S. Note that the coefficients 1 1 and 1 2 re.fleet the isotropic stress response, as given in eqs. (6. I.1.6). (b) By recalling Section 6.6, a closed form expression for the elasticity tensor C in the material description .is given by relation (6.1.62), with contrjbutions Cvoh i.e. {6.166),b and C1so' i.e. (6.168).

272

6 Hyperelastic Materials By use of the important property (1.256) and the constitutive equation for the fictitious second Piola-Kirchhoff stress (6.215) 3 , show that the fictitious elasticity tensor·C "in the material description takes on the form

with the fourth-order unit tensor I defined by ( 1.160) and the coefficients

Ja,a=

5, ... ,12,by

-61 > = 4awiso -1

..

8l5

Note that the coefficients 8a., a = 1,,. ... ., 4, were given previously in relations (6 . 195) and reflect the isotropic contributions.

6.8

Composite Materials with Two Families of Fibers

In the following we discuss appropriate constitutive equations for the .finite elastic response of fiber-reinforced composites in which the matrix -material is reinforced by two families of fibers. We assume that the fibers are continuously distributed throughout the material so that the continuum theory of fiber-reinforced composites is the constitutive theory of choice.

6. 8

Composite Materials with Two Families of ·Fibers

273

There are many different fibers and matrix materials now in use for composite materials. Examples of specific fibers for structural applications .are boron and glass,. The latter is an important engineerin.g fiber with high strength and low cost. Further examples are carbon and graphite (the difference is in the carbon content), the organic .fiber aramid and the ceramic fibers silicon carbide and alumina among others. Many specific matrix materials are available for the use in composites; for example~ thermoplastic polymers, tlzermoset polymers, metals (such as alum.inum, titanium and cop.per) and ceramics. Vast numbers of applications in industry are concerned with .composite materials~ such as the finite elastic response of belts and high pressure tubes, steel reinforced rubber used in tyres, and integrated circuits used in electronic computing devices. Typical medical applications are lightweight wheelchairs and implant devices such as hip joints (see also the textbook by HERAKOVICH [1998, Chapter l]). The five-volume encydopedia of composites edited by LEE .[ 1990, 1991] includes a detailed account of special types of fiber, matrix materials and composites .as engineering materials. Typical engineering properties, manufacturing and fabrication processes and details on how to use composite materials for different applications are also prov.ided. However, it is important to note that numerous organisms such .as the human body, animals and plants are heterogeneous systems of various composite biomaterials. The textbooks by FUNG [ 1990, 1993, 1997] are concerned with the biomechanics of various biomaterials, soft tissues and organs of the human body. One important example of a fibre-reinforced b.iomaterial is the artery. The layers of the arterial wall are composed mainly of an isotropic matrix material (associated with the .elastin) and two families of ·fibers (associated with the collagen) w·hich are arranged in symmetrical spirals (for arterial histology see RHODIN [1980]). For mechanical properties and constitutive equations of arterial walls, see the reviews by, for example, H"'YASHI [1993], HUMPHREY [1995] and the data book edited by AJH~ et al. (1996, /Chapter 2]. A simple finite element simulation of the .orthotropic biomechanical bepavior of the arterial wall is provided by HOLZAPFEL et al. [.1996d, 1996e] and HOLZAPFEL and WEI.ZSACKER [ 199.8]. For a review of finite element models for arterial wall mechanics, see the article by SIMON et al. [1.993]. Free energy and constitutive ·equations. We may now consider a body built up of a matrix material with two families of fibers each of which is unidirectional with preferred direction. The matrix material .is assumed to be hyperelastic. The preferential fi her directions in the reference. and the current con figuration are denoted by the unit vector fields a 0 ., g 0 and a, g, respectively. By analogy with relation.(6.201) we may postulate the free energy (6.216) \JI = \ll(C, Ao, Go) per unit reference volume. For notational simplicity we have introduced the abbrevia-

274

6

Hypere-lastic ·Materials .

tions Ao = a 0 ® a 0 and G 0 = g0 ~ g0 , frequently referred to as structural tensors. The free energy must be unchanged if the fiber-reinforced composite (i.e. a hyperelustic (matrix) material with two families of fibers) in the reference configuration undergoes a rotation described by the proper -orthogonal tensor Q. Using arguments similar to those used for a single fiber family (see the previous Section 6.7), the requirement for this type of ·composite is, in view of (6.202)., given by 1,

T

T

\Jl(C,Ao,Go) = \ll(QCQ· ,QAuQ ,QGoQ ) ,

(6.217)

which holds for all tensors Q (recall Example 6.9) . Here \JI "is a scalar-valued isotropic tensor function of the three tensor variables C, Ao and G0 • According to SPENCER [.1971, 1984], requirement (6.217) is satisfied if \lJ is a function of the set of invariants 1

Ii. (C) '

Ia(C) ,

L1 (C, ao) ,

f::>{C, no) ,

<

h(C, go) =go· Cgo ' Is(C, ao, g0 )

= (ao · g0 )ao · Cg0

= go· C2 go , fg(ao·: g0 ) = (an · ·g0 ) 2

(6.218)

h(C, .go) ,

.

The three invariants 11 , 12 , 1:.1 are identical to those from the isotropic theory pres.ented in eqs. (5,.89)-(5.9.1). The pseudo-invariants J..1, In are given by eq. (6.203) and characterize one family of fibers with direction a 0 • The pseudo-invariants 14 , ••• , .Iu are associated with the anisotropy generated by the two families of fibers. The dot product a 0 • ·g0 is a geometrical constant determining the cosine of the angle between the two fiber directions in the reference configuration. Therefore, the invariant /~> does not depend on the deformation and is subsequently no longer considered. Note that I 1 and ft; are equal to the squares .of the stretch in the fiber directions a 0 and g0 , respectively. The constitutive equation for the second Piola-Kirchhoff stress S follows from the postulated free energy (6.216) by differentiation with respect to C. By means of the chain rule, S is given as a function of the remaining eight scalar .invariants in the form (6.219)

in which DI1 /8C, ... ~ 8I:d8C and 8I.1/8C., Dh/fJC are given by eqs. (6.30), (6.31) and (6.206), (6.207), respectively. The remaining derivatives ·Of the invariants f o.llow from (6.2 l 8)n-( 6.218) 8 and have the forms

Din -G0 DC -

or

Din"

ac:·AH

= .9oA.90B

= Gfl[J

(6.220)

6.8

Composite l\llaterials with 1\vo Families of Fibers

275

(6.221) or

D.Is

1

ac = 2(ao . go)(ao ® &

-1- go® ao)

8/s 1 {)C = '?{
or

AB

-

(6.222)

l

.,p

_,"7'······-.r~·-:

.• ,. .

~

Using arguments similar to those used for deriving the stress relation (6 . 298), we obtain the explicit expressions ·.;.:,;··· ········.·• ,·.:.:·=: ...,,,

<=·· . : ·.. ·· . ..... =:.

·· ...

·. ..

:/

. ~:.....~· ...

. (6.223) The anisotropic stress response (6.223) extends relation (6.208) by the addition of the last three terms. Note that (D\JJ /8/.i)A 0 and (8\JI /ofu)G 0 characterize the (decoupled) stress contributions arising only from the fibers.

·Orthotropic hyperelastic materials. If a 0 · g 0 = 0, the two fami1ies of fibers have orthogonal directions. Then, the material is said to be .ortbotropic in the reference configuration with respect to the p.lanes normal to the fibers and the surface in which the fibers lie. Since now two directions in space (a 0 , g 0 ) are preferred with respect to the mechanical response of the composite, the remaining third direction -orthogonal to the fiber plane is also a preferred direction. The mec~anical response in the third direction is governed by the matr"-ix material. The list of ihyariants (6.218) reduces to the first seven and the free energy has the form \JI = \f!(/1 ,. >~--,/~ ). A .further special case may be found under the assumption that the isotropic matrix material is incompressible, i.e. Ia = 1. Additionally, the families of fibers may also be inextensible in the two fiber directions a 0 and_ _g 0 , consequently I 1 = 1 and In = 1. For this case a suitable Helmholtz free-energy function is g.iven by

'11

= \Jt(/1 (C), l2(C), h(C, au)~ I,( C~ g0)] 1

1

1

~

~

~

- ;;v(Ia - 1) - ?q(I1 - 1) - 9 r(IH - 1) , with the indeterminate Lagrange multipliers p/2, q/2, -r/2.

(6.224)


276

The constitutive equations for an orthotropic material composed of an incompressible isotropic matrix material and inextensible fibers (with directions a 0 and g0 ) depend only on the invariants 1 1 , / 2 and 15 , h. The constitutive equations in the Lagrangian and Eulerian descriptions, extending eqs~ (6.212) and (6.213), follow from (6.224) and are

(6.225)

(6.226) respectively. Herein, the indeterminate terms -vc- 1., -qA 0 , -rG 0 (and -pl,, ·-qA, -rG) are identified as reaction stresses associated with the constraints / 3 = 1, L1 = 1., 16 = 1, with the pressure-like quantity p and the fiber tensions q, r, respectively. In eq. (6226) 2 we have introduced the definitions A = a ®a and G = g@ g of the structural tensors A and G. Recal1 that a(x, t)., g(x, t) denote the ·fiber directions in the defarmed configuration while b = FFT is the spat~al str~in tensor of secono~.order. However, if the two families of fibers are mechanically equivalent - and n·ot necessarily orthogonal - then the material is said to be locally ortbotropic in the reference configuration with respect to the mutually orthogonal planes which bisect the two fiber families (with directions a 0 and g0 ) and the surfa~e in which the fi bets lie. Then, \JI is a function of the first eight invariants listed .in (6.218) and is sy.mm·etric with respect to interchanges of a 0 and g0 • It can finally be shown that for a locally ortlwtropic material \JI can be expressed as a function of the seven invariants

I1(C)

,

I~~(C., ao, g 0 )

I2(C) ,

,

Ia(C)

lw(C_, ao, g0 )

, ,

ls(C,ao,g 0 )

,

}

(6.227.)

Iu (C, ao, g0 )

(see SPENCER :[1984]), with 11.t 12 , ! 3 .and18 given by eqs. (5.89)-(5.91) and (6.218) 8 ,

and the definitions

lu=L1+h, for the remaining three pseudo-invariants.

(6 ..228.)

6.8

277

Composite Materials with 1\vo Families of Fibers EXERCISES

w

l. Consider a locally orthotropic material with the free energy expressed as a function of the invariants presented by (6.227) using (6.228). Assume an incom-

pressible isotropic matrix material and two families of inextensible fibers. Show that the constitutive equation for the Cauchy stress tensor u is given by

er

aw

= -pl -

aw

aiJ!

qA - rG + 2-b -- 2-b- 1 + (a 0 • g0 )-(a ® g + g ®a) .811 812 8/s 8\J! ·:.-.--..~·\ \... -<--- '"·----·-···--··--···---··, . , .,.,. . ,. 1

+2~

1

u 11

(a®ba+ab®a+g®bg+gb®g), .

where the first three terms .characterize reaction stresses. 2. We characterize a compressible composite with two families of fibers by the decoupled representation of the free energy (6.229)

(comp.are also with Exercise 2 on p. 271), with the volumetric and isochoric parts '11vol and W1 80 , and the modified invariants 11, l2 .given by eqs. (6 . .109) and (6.1.10) (13 = detC = 1). The remaining modified pseudo-invariants are L. = J- 213 Ia, ,. ,f._ a -- 4 , 6, 8, an d I-a -- J- 413 1a., a -- o, Use the free energy (6. 229) to particularize the fictitious second Piola-Kirchhoff stress S which appears in .the constitutive equation for S = Jpc- 1 + J- 2 / 3 p: S,. Show that

with the explicit expressions

OI,

Ol1 =I

ac

8[5

--= ac

= .ao 0

Cao+ .aoC ·®no

·

=Ao

ac Ola = G 0 ac

278

6.9

6

Hyperelastic Materials.

Constitutive Models w·ith Internal Variables

Many materials used in the fields of engineering and physics are inelastic. It turns out that the constitutive ·models introduced hitherto are not adequate to describe this class of .materials, for which every admissible process is d:issipative. Within the remaining sections of this chapter we study inelastic materials and, based on the concept of internal variables, we derive constitutive models for viscoelastic materials and hyperelastic mate.rials with isotropic damage. Conc·ept of inte.rna:J variables. The current thermodynamic state of thermoelastic materials can be determined solely by the current values of the deformation gradient F and the temperature 8. Variables such as F or 8 are measurable and collfrollable quantities and .are accessible to direct observation. In practice these type of variables are usually called external variables. The current thermodynamic state of ·materials that involve dissipation can be determined by a finite number of so-called inte.rnal variables, or in the literature sometimes called bidden variables (hidden to the eyes of external .observers). These additional thermodynamic state variables, which we denote collectively by are supposed to describe aspects of the internal structure of materials associated with irreversible (dissipative) effects. Note that strain (stress) and temperature (entropy) depend on these internal variab]es. The evolution of internal variables replicates indirectly the history of the deformation, and hence they are often also termed history variables. Materials that involve dissipative effects we refer to as dissipative mate.rials· or materials with

e,

dissipation. Hence, the concept of internal variables postulates that the current thermodynamic state at a point of a dissipative material is specified by the triple (F, -8, e) (the current thermodynamic state may .be imagined as ajictitious state of thermodynamic equilibrium). Then, the current thermodynamic state is represented in a finitedimensional state space and described by the current v(.llues (and not by their past history) of the deformation gradient, the temperature and the finite number of internal variables. The nature of .internal variables may be physical, describing the physical structure of materials. In the course of phenomenological experiments one may be able to iden~ tify :internal variables; however, they are certainly n.ot controllable or observable. We use the internal variables as phenomeno1ogical variables which are constructed mathematically~ They are mechanical ..(or themmI, or even chem.ical or electrical ... ) state variables describing structural properties within a macroscopic .framework, such as the 'dashpot displacements' in viscoelastic models, damage., inelastic strains, dislocation densities, point-defects and so on. Hence, here we introduce both external and interna1 variables as macroscopic quantities without referring to the internal mi-

279

6. 9 Constitutive Models with Internal Variables

crostructure of the material. The concept of "internal variables serves as a profound basis for the development of constitutive equations for -dissipative materials studied in .the following section. Constitutive equations and internal dissipation. The existence of non-equilibrium states that do evolve with time is an essential feature of inelastic materials. Two typical examples of irreversible processes .known from classical _mechanics which govern non-equilibrium states are relaxation and creep. Relaxation (and creep) is the time-dependent return to the (new) equi.libr~um state after a d.-isturbance. In general, stress will decrease with time at a fixed (constant) strain~ which is referred to as relaxation, while during a creep.ing process strain will increase with time at a fixed (constalll) stress. For an illustration of these two simple processes, see Figure -6.5. The (strain or stress) response of removing a strain or stress is caHed recovery. A viscoelas .. tic behavior of a material is characterized by hyst-eresis. The term 'hysteresis' means that the loading and unloading curves do not coincide. It represents the non-recoverable energy when a material is loaded to .a point and then unloaded.

·;

·. r

CREEP PROCESS . i, . . ,. ,. , . . . . . . . . . . . . . . . ,. . . . . . . . ~:-·''" . . . ,. . . . . . . ,. . ,_, . . . . , . . ,. .

RELAXATION PROCESS

.~

.... ~

c: .,

·a .b

:·.· :· --·· ·.-. ·--·---:·

1

!

Cl) 1

at fixed slrcss

at fixed strain

"J

.··

Time

T

Relaxation Lime

Retardation time

Kelvin-Voigt model

Maxwell model

481(

~

Time

T

)I-

-<

r:;

I >-

Figure 6.5 Max well and Kelvin- Voi_gl models associated with re1axation and creep behavior.

The Maxwell model (a dashpot is arranged in series with a spring) and the Ke.lvinVoigt -model (a dashpot is arranged in pa.ralle.I with a spring), two mechanical ·models known from linear viscoelasticity, are frequently used to discuss relaxation and creep behavior. These models -combine 'viscous' (or fluid-like) with 'e.Jastic' (or sol.id-like)

6

280·


behavior. Under the action of a constant deformation (strain), the Maxwell model is supposed to produce instantaneously a stress response by the spring which is followed by an exponential stress relaxation due to the dashpot. On the other hand, the KelvinVoigt mode·1 is supposed to produce no immediate deformation for a constant load (stress) . However, in a Kelvin-Voigt model a deformation (strain) will be ·created with time according to an exponential function. Within the realm of non-equilibrium thermodynamics the viscoelastic deformation mechanisms of these material models are not reversible. The rate of decay of.the stress and strain in a viscoelastic process is characterized by the so-called relaxation time T E (0, oo), with dimension of time, known from linear viscoelasticity. The parameter r associated with a creeping process is often referred to as the retardation time. The constitutive equations introduced hitherto are no longer sufficient to describe dissipative materials. The vast majority of constitutive models that are used to approximate the physical behavior of real nonlinear inelastic :materials are developed on the basis of internal variables . .In this chapter we remain within an isothermal framework, in which the temperature is assumed to be constant (8 = 8 0 ). Hence, we postulate a Helmholtz free-energy which defines the thermodynamic state .by the observable variable F and function a set of additional internal ·history variables 0' a = 1, ... , m, to be specified for the particular problem. We write

w

e

(6 . 230)

whe.re the second-order tensors

ea, a

=

1, .... , 'm,, represent the '..dissipation .mech-

anism of the material. They are linked to the irreversible relative movement of the material inside the system and describe the deviations from ·equilibrium (see, .for example, VALANIS [1972]). An assumption of the form (6.230) can easily ·be adjusted to describe a rich variety of poroust viscous or plastic materials. The actual number of the phenomenological internal variables needs to be chosen for each different material and may vary from one theory and (boundary) condition to another; for example, the size of the specimen under observation. However, the definition of internal variables should be chosen so :that they somehow replicate the underlying internal microstructure of the material (even though they are introduced as macroscopic quantities) . In general, the internal variables may take on scalm; vector or tensor values. Here the internal variables are all denoted by second-order tensors. In order to particularize the Clausius-Planck inequality of the form (4.154) to the free energy 'lll at hand, we must differentiate (6.230) with respect to time. By means of

e

E:=l aw

the chain rule we obtain ~ (F, 1, ... 'em) = aw I 8F ·: F + I ae.(t : ~n~ and finally, with the expression for the stress power Wint = P : F per unit reference volume,

6.9

ConstUutive 'Models with Internal Vari.ables

281

we find from (4.154) th.at 1). mt

=

(r- 8'-ll(F, E1,OF... 'em)) .. F. - ~ ~ 8\ll(F, E1, ... ,Em) ~ i: aeo . ""°' >

0 . (6.231)

In order to satisfy 'Dint > 0 we apply the Coleman-Noll procedure. For arbitrary choices of the tensor variable F, we deduce a physical expression for the first PiolaKirchhoff stress P and a remainder inequality governing the non-negativeness of the .internal dissipation Vint (required by the second law of thermodynamics). We have. .m

Vint

= L Ba ; e.a > 0 ~

(6.232)

o:=l

which must hold at every point of the continuum body and for all times during a thermodynamic process. In (6.232h we have defined the internal (second-orde~) tensor variables Sa, et = 1, ... , m, which are related (conjugate) to 0 through the internal

e

constitutive equat.ions

...... ..... '-'Q -

8\JI (.F, -

e

1, ••• ,

aea

em)

'

o:=l, ... ,m .

(6.233)

The additional constitutive equations (6.233) restrict the free energy w and relat~ the gradient of the free energy wwith respect to the internal variables to the associated internal variables 8 0 , a = 1, ... , rn. Note that the presence of additional variables in the free energy (6.230) justifies additional constitutive equations. A physical motivation of restriction (6.233) may be given by several examples, one of whic.h, stemming from linear viscoelasticity, is presented on p. 286; in particular, see eq. (6.251 ). In constitutive equations (6.232)J. and (6.233) the tensor variables F and are associated with the thermodynamic forces P and 8 0 , respectively. A constitutive model which is characterized .by the set of equations (6.231)-(6.233) is called an internal variable .model. For the case in which the .internal variables 0 are not needed to characterize the thermodynamic state of a system, then, the internal dissipation Vint in (6.232h is zero (the material is considered to be perfectly ·elastic) and a11 relations from previous sec~ tions of this chapter may be applied. In order to describe materials without dissipative character, the set of equations {6.23.1 )-(6.233) simply reduces to (6.3) and (6.1) 1 .

ea

.ea

e

Evolution equations and thermodynamic equilibrium. The derived set of .equations (6 ..232) and (6.233) must be complemented by a kinetic relation, which describes the evolution of the involved internal variable and the associated dissipation .mechanism. Consequently, suitable equations of evolution (rate equations) are required in order to describe the way an irreversible process evolves. The only restriction on these equations is thermodynamic admissibility, i.e. the

eo

282


satisfaction of the fundamental inequality (6.232h characterizing local entropy production. The m.issing equations for the evolution of the .intern.al variables 0 may be written, .for example, as

e

n = 1, ... , m, .

(6..234)

The evolution of the system is described by Ao, n = 1 1 • • • , rn, which are tensor-valued functions of .1 + m, tensor variables. Every syste·m will tend towards a state of thermodynamic equilibrium, which implies that the observab]e and internal variables reach equilibrium under a prescribed stress or strain; they remain constant at any particle of the system with time. Hence, the behavior at the equilibrimn state may be considered as a limiting case and does not depend upon time. In view of eq. (6.234 ), the definition of an equilibrium state now requires the additional conditions

n

en

= 1, ... , rn

.

(6.235)

e

Hence, may be seen as the rate ofchan.ge with whic:h 0 (t) tends toward its equilibrium. In an elastic continuum, every state .is an equilibrium state. The internal dissipation Vint at equilibrium is zero, which characterizes, for instance., a perfectly elastic material, as pointed out in Section 6 . .1.

6.10 Viscoel.astic Materials at Large Strains Many materials of practical interest appear to behave in a markedly viscoe.lastic manner over a certain range of stresses and times. The mechanical.. behavior of, for example, thermoplastic elastomers (actually rubber-like materials) or some other types of natural and syntheticpolymers are associated with relaxation and/or creep .Phenom·ena, which are important design factors (see, for example, MCCRUM et al. [1997], SPERLING [1992] and WARD and HADLEY :[.1993]). Problems that involve relaxation and/or creep effects determine irreversible process,es and belong to the realm of nonequilibrium thermodynamics. For a detailed introduction of the 1inear and nonlinear theory of viscoelasticity the reader is referred to the book by CHRISTENSEN ['1982]. Experimental investigations are documented by, for example, SULLIVAN [1986], LION [1996] and M.IEHE and KECK [2000]. In the following we characterize the thermodynamic state of such problems explic"itly by means of an internal variable model as introduced .in the previous section. A description sole'ly via external variables is also possible; but it emerges that such types of formulation are not preferred for numerical realizations using the finite element

6.10

Viscoelastic Materials at Large Strains

283

method. Num·erous viscoelastic materials can often not be modeled adequately within limits by means of a linear theory. Here we postulate a three-dimensional v·iscoelastic model suitable for finite strains and small perturbations away from the equilibrium state. In contrast to several theories of viscoelasticity (see, for exam.pie, the pioneering paper· by GREEN and TOBOLSKY [1946]) the present phenomenological approach is not restricted to isotropy. For theories that account for finite perturbations away from the equilibrium state, the reader is referred to, for ·ex.ample, KOH and ERINGEN [.1963], HAUPT fl 993a, b] and REESE and GOVJNDJEE [.1998a]. Additionally we foJJow a phenomenological approac·h that does not consider the .underlying molecular structure of the physical object.

Structure .of the free energy with internal variables.

In particular, we choose an approach which applies the concept of internal variables motivated by S"IMO [1987] and followed by, for example, Gov·1NDJEE and .S-JMO [I 992b, 1993], HOLZAPFEL [1996a], KALISKE and ROTHERT (1997] and SIMO and HUGHES [l.998, Chapter 10]. Our study is based on the theory of compressible hyperelasticity within the is
w(C,l\, ... ,rm)= w:1(J)

+ 'Pi:(C) + LYa(C,r(~) '

(6.236)

o=l

valid for some closed time interval t E [O., T] of interest. We assume that each contribution to the free energy \JI must satisfy the normalization condition (6A), i.e. (6.237) A material which is characterized by the free energy (6.236) for any point and time we call a vis·coclastic material. The first two terms in (6.236), i..e. 1( J) and W~{C), are strain-energy functions per unit reference volume and characterize the ·equilibrium state of the sol.id. They can be identified as the terms presented by eq. (6.85) describing the volumetric elastic response and the isochoric elastic response as t -+ oo, respectively. In fact, the superscript (• ) 00 characterizes functions which represent the hyperelastic behavior of sufficiently slow processes.

w:

6

284

Hypere1astic Materials

The additional third term in (6.236), i.e. the 'dissipative' potential E~~.1 T °' is responsible for the viscoelastic contribution and extends the decoupled strain-energy function (6.85) to the viscoelastic regime. The scalar-valued functions Y ch a: = 1, ... , m, represent the so-called ·configurational free energy of the viscoelastic solid and characterize the .TJ011-equilibrium state, i.e. the behavior of relaxation and creep. Motivated by experimental data we assume a time~dependent chan.ge of the system caused purely by isochoric defonnations. Hence, the volumetric response remains fully elastic and the configurational free energy is a function of the ·modified right Cauchy-Green tensor C and a set of strain-like internal variables (history variables) not accessible to direct observation, here denoted by 0 , a: = 1, ... , m. Each hidden tensor variable r 0 characterizes the relaxation and/or creep behavior of the material. They are ·considered to be (inelastic) strains ·akin to the strain measure C, with a = I, a: = 1,,. .. , rn, at the (stress-free) reference configuration. The viscoelastic behavior is, in particular, modeled by n = 1, ... , m viscoelastic processes with corresponding relaxation times (or retardt1tion times) 'Ta E (0, oo.), a: = 1, ... , 1n . .Note that the set of .1 +1n tensor variables ( C, 1 , .•• ) rm) completely characterizes the isothermal viscoelastic state .

r

r

r

.Decoupl·ed volumetric-.isochork stress response. In order to obtain the associated constitutive -equations describing viscoelastic behavior at finite strains we specify postulate (6.236). Following arguments analogous to those which led from (6.230) to eqs. (6.232) and (6.233), we obtain physical expressions for the (symmetric) second Piola-Kirchhoff stress S and the non-negative internal dissipation (local entr~py production) Vint in the

forms

Starting from the d~~oupled free energy (6.236), .a straightforward computation leads to an additive split of S, as already derived for purely elastic compressible hyperelastic materials (see Section 6.4). We have

S _ ?8'1.i(C, ri, ... , rm) _ 800

- .-

.ac

-

vol

+

S· ISO

'

(6239)

with the definition Tl&

Siso

=

s:, + LQa

(6.240)

o=l

of the isochoric contributions. In eqs. (6.239h and (6.240) the

soo VO]

=

Jdw~1(J)c-• c}j

'

S?'1 JSO

q~antities

= ,-213r. ?aw~(c) ac '

•

J;.J

(6.241)

6.10 Viscoelastic Materials at Large Strains

285

determine volumetric and isochoric contributions, which we take to be fully ·elastic. In 1 relation (6.241.h the .(fourth-order) projection tensor IP = l[ ® C furnishes the deviatoric operator in the Ltigrangian description. Note that for these elastic contributions we may apply the framework of compressible hyperelasticity and adopt relations

ic-

w:

(6.88)-(6.91) by using \11~ 1 and instead of Wvol and Wiso· In (6.240) we have introduced additional internal tensor variables Q0 , a = .1, ... , ·m., which may be interpreted as non .. equiHbrium stresses in the sense of non-equilibrium thermodynamics. Note that the symbol Q has al.ready been used and must not be confused with the orthogonal tensor. As can be seen from (6.240) the isochoric second Piola-Kirc.hhoff stress is decomposed into an equilibrium p.art and a non-equilibrium part characterized by the e·lastic response of the system and the viscoelastic re. 1y. sponse L.,,o:=I ·Q 0:' respective

s:

'°'m

By ·analogy with (6.90) we have defined the relationship

a

= 1, ... , rn

(6.242)

for the second-order tensors ·Qa, with the definition

a= 1, ... , m,

(6.243)

of the so-called fictitious non-equilibrium stresses Q0 • As can be seen from (6242), Q 0 is the deviatoric projection of Q 0 times J- 2/ 3 , with projection tensor JP, Motivated by the (mechanical) equilibrium equations for the linear viscoelastic solid (see the following Example 6.10., in particular, eq. (6.251)), we conclude further that Q0 are variables related (conjugate) to r 0 , a = 1, ... , 11i, with the internal constitutive equations a=l,~

.. ,-m.

(6.244)

These conditions restrict the configurational free energy L::~ 1 To in view of (6.242)1. Hence, the internal dissipation Vint in-eq. (6.238)2 equivalently reads Vint ·= L::=.l Qo: :

r j2 > 0. 0

The condition for thermodynamic equilibrium (compare with eq. (6.235)) implies that fort -->- oo the stresses in eq. (6.240) reach equilibrium, which means that Q0 = -28'I 0 /8I'a .lt·-HX> 0, a= 1, .. . ,m, and hence, Q.Q characterize the current ·'distance from e.quilibrium '. Consequently, the dissipation at equilibrium .is zero as seen from (6.238h and (6.244). In other words, .at thermodynamic equilibrium the material responds as perfectly elastic; general finite elasticity is recovered.

=

286

6 .Hyperelastic .Materials

EXAMPLE 6.10 By using a simple spring-.and-dashpot model find a meaningful rheological .interpretation for the phenomenological viscoelastic constitutive model presented. Start with a one-dimensional and linear approach. Derive physically motivated evolution equati9ns for the internal variables . Solution. To begin with linear geometry consider a rheological model, as illustrated in Figure 6.6. It is a one-dimensional ·generalized .Maxwell model with a free spring on one end and an .arbitrary number rn of Maxwell elements arranged in parallel.

E1 > 0

>0

'T/I

-----+- :

'Yl .,___ _.. -----1)t• a

• • •

111

Em>O

'1/m

> 0:

a= aoo

a:=l

=E e 'lo: = 1Jo: Io a 00

1/m

Tm=

Em> 0

:·._._..Im

+ Lqo~ 00

,

a

=l.

1 • • • ,

n1,

Figure 6 . 6 Rhcologica'l model..

The viscoelastic model in Figure 6.6, which we call temporarily a mechanical device, displays ·both relaxation and cre~p behavior. It is a suitable simple model to represent quantitatively the mechanica] behavior of real viscoelastic ·materials. The mechanical device .is assumed to have unit area and unit length so that stresses and strains are to be interpreted as forces .and extensions (contractions), respectively. We assume that the solid behavior is modeled by a set of springs responding linear elastically according to Hooke's law. The stiffnesses of the free spring-on one end and the spring for the so-caUed .(~··Maxwell element are determined by Young's moduli E00 > 0 and En > 0, o = 1, ... , m.., respectively. The fl.ow behavior is modeled by a Newtonian viscous fluid responding like. a dashpot. The viscous fluid of the a-Maxwell e.lement is specified adequately by the material constant r/o: > 0, called the viscosity. Based on physics all these parameters are positive.

6.10

Viscoelastic l\tlaterials at Large Strains

287

Let a be the total stress applied to the generalized Maxwell model and c be an extern.al variable which measures the total linear strain due to the stress. By equilibrium, the total stress applied to the device .is found to be rn

a= aoo

+ LC/o:

(6.245)

o:=l

(see Figure 6. 6), where the definition of the stress at equilibrium, i.e. a 00 = E 00 e, .is to be used. The internal variables ·qo., o: = 1., ... , rn, are the non . . equilibrium .stresses in the dashpot of the a-Maxwell element characterizing the dissipation mechanism of the viscoelastic model. The stresses qm n: = 1., ... , -ui, .acting -on each dash pot are related to the associated internal variables 1'0 , which we interpret us (inelastic) strains o~ each dashpot. In particular, for a Newtonian viscous fluid, q0 are set to be proportional to the current 'distance from e.quilibrium •, i.e. the strain rates i'a· We adopt the linear constitutive equation by Newton, i.e. q0 = r1n )'0 , n == 1, ... , ni. On the other hand, the stress in the spring of the a-Maxwell element is determined by q0 = E 0 (e - /a) (see Figure ·6.6). Consequently, the stresses (not necessarily at equilibrium) acting on each dashpot .is

a= 1, _... , rn . Hence, time differentiation of (6246)27 i.e. (6.246) 1 the .important evolution equations

(6.246)

cia = E 0 :(e - i'c:t), implies by means of .a= 1, ... ,-m

(6.247)

for the internal variables with.in the one-dimensional and linear regime, where the def:... inition of the relaxation time (or retardation time) Tu: = T/c.J E 0 > 0, a 1, ... , rn, is to be used. Since
=

m

Vint= Lqo:1n o:=.l

m.

= L11o(i'u) 2 > 0 {\':=

,

(6.248)

l

> 0. It disappears at equilibrium. We now ·define the strain energy 'tP(c, /1, ... , 1'111) = 'tA:}lo{e) + 2:~~ 1 va(c, /o), with the quadratic forms 't/?00 (E) = ~E00 t: 2 and Va(e:, J'a) = ~E0 (E - 1 0 ) 2, and the which is always non-negative, since 170

normalization conditions 4100 (0) = 0 and vn(O, 0) = 0., ft = 1, ... , rn. The physically motivated strain energy 't/1 determines the energy stored elastically in the springs of .the device, as illustrated in Figure 6.6. The strain energy v 0 = va(e, /ci:) is responsible

6 Hyperelastic Mat-erials

288

for the viscoelastic contribution and is related to the .a-relaxation (retardation) process with relaxation (retardation) time Ta E (0, oo). Differentiation of '1/J with respect to the total strain c gives the total stress a applied to the device. On comparison with (6.245) we conclude that

-81/l(c, /1, ... , Im) ( ·) Ge = a 00 c

( +~ L..Jqo: c,7

0

)

=a ,

(6.249)

a=l

where the physical -expressions O"oo

= d-1/1:(c)

= Eooc

,

(6.250)

f o.r the stress at equilibrium a.00 ( e;) and the non-equilibrium stresses q0 .( E', ')'a), a:

=

1, ... , rn, are to be used. Finally, the derivative of 'l/J with respect to the internal variables 1 0 gives with (6.250)2 (or (6 ..246)) the associated non-equilibrium stresses q0 in the dashpots. The resulting .internal constitutive equations read

..._.

av

0: (

e'· 'a) .-- E o (£

81cr

-

/o - qa ' I

)

-

a=l, ... ,1n,

(6.251)

which, when substituted into (6 . 248)i, .gives the intern.al dissipation Vint expressed through the strain energy, i.e. Vint = - E:~ 1 (8vo:/81C'rYYa.· Note that the general stress relation (6.239) with definition (6.240) may be identified as the three-dimensiona·.1 and nonlinear version of the linear rheological model (6.249), which, in view of Figure 6.6, decomposes the stresses .in equilibrium and nonequilibrium parts. In .addition, the internal constitutive equations (6.244) and definition (6.242h may be considered as the three-dimensional generalization of (6.251) and (6.250h and also its extension to the finite strain regime. II

Evolution equations and their solutions. In order to describe the way a viscoelastic process evolves it is necessary to specify complementary equations of evolution so that the local entropy production, i.e. the inequality (6.238)2, is satisfied. .In particular, we look for a law which governs the internal variables Q0 , a = 1, ... , 1n, introduced as isoclzoric non-equilibrium stresses. We require that the evolution -equations have a physical basis and provide a good approximation to the observed physical behavior of real materials in the large strain regime. In addition, we require that they are suitable for efficient time integration algorithms that are accessible for use within a finite element procedure. We motivate the evolution equations for the three-dimensional and nonlinear deformation regime by reference to the relationship (6.24 7). Having th.is in mind, an obvious

6.10

Viscoelastic ·Materials at Large Strains

289

choice of appropriate (linear) evolution equations for each of the .internal variables has the form . Qo . Q0 +-=Sisoo, a=l, ... ,m, (6.252) To

where (6.252) is valid for some semi-open time "interval t E (0, T], in which the value 0 is not included in the interval. Here we employ a superposed dot to designate the material time derivative as usual. The values Q0 f t=O = 0, .c.t = .1, ... , m., for the internal variables at initial time t = 0 are .assumed to be zero, since we agreed to start from .a stress-free reference configuration. In the linear differential equations (6.252), the tensors Shma characterize isochoric second Piola-Kirchhoff stresses corresponding to the strain energies Wi.son(C) of the system (with Wison(I) = 0 in the reference configuration) responsible for the a-relaxation (retardation) process with relaxation (retardation) time Ta E (0, oo ), a = 1, ... , rn. The definition of the .material variables Siso 0 is based on structure {6.90) and has the form Shmo:

= J -''/3 - 'IP: ·8

0

S,, = 2 8\llisoa(C)

ac

,

1

a = 1, ... , m. ,

(6.253)

where (6.253h define the constitutive equations for the fictitious second Piola-Kirchhoff stresses Sn. A particufar stress S1soa depends only on the extern.al variable C, i.e. the modified right Cauchy-Green .tensor introduced in (6.79):.. The linear -equations (6.252) are straightforward generalizations of eqs. (6.247), which .are physically based. Both tensor quantities Qa and Siso o: contribute to the iso·choric response -of the system. It emerges that the structure (6.252) introduced here is suitable for efficient time integration algorithms as we discuss later in this section. For the case of non-constant relaxation times 7 0 , the convenient concept of '·modified' time is used in order to obtain linear evolution equations of the type (6.252). Within this concept, r0 is kept fixed during the process; for more details see, for example, KNAUSS and EMRI [1981] and GOVINDJEE and SIMO [1993]. Fairly simple dosed form solutions of the linear evo]ution equations (6.252), which are valid for some semi~open time interval t E (0, T], are given by the con.. v()Jution integrals Q0

= exp(-T/r

0

.!

)Q 00 +

t.=T

+

exp(-(T - t)/ruJSisoa(t)dt ,

Oa o+

(the proof is omitted). The instantaneous response given by

Qa o+

_ -2/am .. ?. 8Ya (Co+, r ao+) - '1o+ Jl o"t" • 8Cu+

'

a = 1, ... , rn

(6.254)

(set of initial conditions) is

a= 1, .. . ,rn

(6.255)

290

6 Hyperelastic ·Materials

kCQ.;

(compare with eqs. (6.242).i, (6.243)), with .10+, C 0 +, r 00 +~ Po+ = ll 0 Co+ defining, respectively, the volume ratio, the modified right Cauchy-Green tensor, the interna] variables and the projection tensor at time t == o+. The evolution of the internal variables is governed mainly by the strain-energy functions Wisoo via relation (6.253). However, .if a viscoelastic medium such as a thermoplastic elastomer is composed of identical polymer chains we .can motivate the assumption that Wisoo is replaceable by the strain-energy function \IJ~)' which is responsible for the isochoric elastic response as t -t oo. We adopt the expression

'11·· Jso a (C) --

00

r-lo JJ

\}r?O (C) um

= 1, ... ·' rn

a:

'

(6.256)

(which is due to GOVINDJEE and SIMO .[1992b ]), where fJ~ E (0, oo) are given nondimensional strain-energy factors associated with To: E (0, oo), n = 1, ~ .. , 1n. Consequently, the stresses S1soc.u a 1, ... , ni, as introduced in (6.253), may be replaced by means of (6.256) and (6.241)2. We write

=

S· 180 0

= ,-2/lp. 2r1ooaw:,(C) = goos?O (·C) '

•

f-• O:

.ac

I-- 0

JSO

'

n = 1, ... ,-m. .

(6.257)

Hence, the material time derivative of the stress tensors, Sison, which .govern evolution equations (6252), are replaced by fl:S';;0 • In summary: the phenomenological viscoelastic model valid over any range of strains is described by constitutive equations (6.239)-(6.243), evolution equations (6.252) with so1utions (6.254.) and re.placements (6.257). Note that with reference to assumption (6.256) the model problem is completely determined by the specification of only two scalar-va1ued functions, namely ·'11~ 1 ( J) and \JI~ ( C), a crucia1 advantage of the introduced finite strain viscoelastic model. It is important to emphasize that the described constitutive mode.I ·fits within the framework of so-called simple materials with memory, which are expressed in the general fom1 by S(t) = ~J:~ 0 +[C(t - s), C(t)] (see, for example, :MALVERN :[1.969, p. 400, eq. (6.7.62)]), with T = t - s, where.~: is a (response) function.al depending on the h.istory of C from T = -oo to T = t. For further discussion of this issue the interested reader is referred to MALVERN [1969, Section 6.7, and references therein]. The total second Piola-Kirchhoff stress ten~or S is computed according to relatio~s (6.239)-(6.241) with the volumetric and isochoric (elastic) response ands:. and the contribution due to the non-equillbriwn stresses E;:~. Q0 as given by the convolution integrals in the form of (6.254)~

Time integration algorithm.

s:1

For the solutions of the crucial Cauchy's equations of motion (see the local fonns (4.53) and (4.68)) the stress tensor .is required. The main goal of the following is to outline an appropriate update algorithm for the total stresses suitable for implementation

6. 10

Viscoelastic ·1\tlaterials at Large Strains

291

in a finite element program. The update procedure is realized in the reference configuration, and .hence the objectivity requirement based on a Euclidean transformation .is trivially satisfied (see Chapter 5). The key of the update algorithm is the 1wmerical integration of the convolution integrals (6.254). ln order to obtain the algorithmic update of the second Piola-Kirchhoff stress S we consider a partition (time discretization) u~·:~oft;n, ·tn+l] of the closed time interval t E [o+, T] of .interest, where o+ = t 0 < ... < tM+I = T. We now concentrate attention on a typical closed time sub..:interval [t 11 , t 11 +iJ, with

At= "t'n+.l

-

tn

(6.258)

characterizing the associated time increment. Assume now that up to a certain time tn the stress Sn satisfies the equilibrium equation and that the displacement field Un, the tensor variables

Fn

= I + Gradun

Cn

=F

·r

11 .F11

.111

= detFn

-2/:1 Cn =Jn Cn

(6.259)

(see (2.45h, {2.51 ), (2.64) 1, (6.79) 2 ), and the stress S11 (determined via the associated -constitutive equation) are spec.ified uniquely by the given motion Xn at time t 11 • ·within a strain-driven type of numerical procedure, the .aim is now to advance the solution to time tn+l ·= !1t + tn and update all relevant quantities. At .first we make an initial guess for Xn-H, known as a trial solution, and update the prescribed loads. Within a classical solution technique, such as Neivton 's method, the new motion Xn+J. at time ln+ 1 is corrected iteratively until the balance principles are satisfied within a given tolerance of accuracy. To check equilibrium al ti.me 'tn+i the tensor variables Jn+t

Cn+l

= detF

11

+1

-2/a

(6.260)

= Jn+ 1 Cn+ I

have to be computed. TMs process is straightforward sjnce the new motion Xn+.t with the updated displacement field Un+i. is considered to be given. The remaining second Piola-Kirchhoff stress at time t 11 + 1 is again determined uniquely ·via the associated constitutive equation. In particular~ the so-called algorithm·ic stress .at time tn+.t re.ads

as

m.

Sn+1

= (S~l + s:) + L

Qn)ln+l .

(6.261)

n=l

Since all required strain measures at t 11 +.1 are known, the first two stres~ contributions, i.e. S~l n+ I and 1 n+l' are determined Via (6.24] ), which, in the present notation, reads as

s:

(6.262)

292

6 Hyperelastic .Materials

The third term in (6.261), which is the viscoelastic stress contribution E:i=l Qan+t based on (6.254), re.mains to be evaluated. The following derivation is related to the approach by SIMO [1987], which bypasses the need for incremental objectivity as proposed by HUGHES and ·WINGET [1980]. Incremental objectivity requires that the algorithmic constitutive equations ·must be objective (frame-indifferent) during a superimposed {time-dependent) rigid-body motion. Incremental objectivity represents the numerical version of the principle of material frame-indifference, as introduced in Section 5.4. We now split the convolution integral (6.254) into the form of

(1n+l (•)clt =

Jo+

rn (•)dt + (1n+l (•)dt . lo+ Jtn

(6.263)

Hence, the internal variables Qa, a = 1, ... ,-m,, at tn+I are given by

.!

t=tn

Q 0 n+i

= exp{-t11+ifrn)Q00+ +

exp[-(tn+t - t)/ro]Sisoo(t)dt

t=O+ ·1.::::t.11+ 1

+ ./

exp[-(tr1+1 - t)/-ro]Sison(t)dt ,

a

= 1, ..

&

,

1n .

( 6.264)

t=tn

In -order to simplify (6.264) we apply relation (6.258) to all three terms. For the first two tenns we use the standard property

exp[-(L.lt + .B)/r0 ]

= ·exp(-~t/r0.)exp(-/3 /r

0 )

(6.265)

for the exponential function, .for any constants ~t., ·r0 and parameter (3 which takes on values t 11 and tn - t. In addition to (6.265) we use the second-order accurate m·id-point rule on the third term ~t·tr~+t ( • )dt of eq. (6.264), which means that the time variable t n is approximated by (tn+l + tn)/2. We deduce from {6.264) that

.!

t=tn

Q,:rn+t

= exp(-b.t/1a)[exp(-t

11

/ro)Qao+

+

exp[-(t11 - t)/ro]Sisoo(t)dt]

t=O+ "l.=l"n+I

+exp(-b.t/210 )

/

Sisoo(t)dt ,

et=

1, ... ,-m. .

(6.266)

·t=tn

Note that the tenns with.in the brackets are Qa: at time tn (compare with eq. (6.254)). By solving the last tenn in (6.266) and by means of assumption (6.257), we may write Qon+.l

= exp(2~o)Qo-n +exp(~o)/3:(s:rn+l - s~n) '

(6.267)

6"10

Viscoelastic Materials at Large Strains

293

for a = 1, ... , rn. After rearranging eqs. (6.267) 1 we arrive finally at a s·econd-order accurate recurrence update formula for the internal stresses in a simple format, namely

ct

= 1, ... , rn

,

(6.268) (6.269)

and with definition (6.267h of the dimensionless parameters ~c-=· In recurrence relation (6.268) we have introduced the (algorithmic) histo1y term 1£011' a: = 1., ... , rn. This term is determined by the internal history Variables Qon .and S~On' Which are knOWil from the previous step serving as an 'initial' data base. The recurrence update formula of the type (6.268) was proposed by TAYLOR et al. [1.970, and references therein]. Instead of the time integration algorithm outlined above we may use ·other algorithmic updates for the total stresses. For .a slightly different structure see, .for example, SIMO [1987] and co-workers, and HOLZAPFEL [1996a]. Elasticity tensor in material description. The importance of consistent linearized tangent m.oduli in the solution of nonlinear problems by incremental/iterative techniques of Newton's type was emphasized in Section 6.6. We now determine the consistent linearization of the constitutive model presented above, noting the algorithmic stress {6.261) with relations (6.262), (6.268), (6.269). In view of definition (6.154) and the decoupled stress relation (6.261), the associated algorithmic elasticity tensor in the material description at time tn+I may be written in the form m

Cn+l =

(c:, + c: + L c~is)ln+l '

(6.270)

.u=I

where the first two contributions to Cn+t are given by (6.163), which, in the present .notation, reads as

coovol n.+ 1 -_ ?.... as:1 c a·

r1+1

n+l

'

""oo

_ 9as:n+1

~ison+l -

-

ac11.+l .

(6.271)

Explicit expressions for (6.271) are found in {6.166) and (6.168). The third (viscoelastic) contribution to Cn+h i.e. E;=l c~is at tn+h is derived using the expression for the internal stresses (6.268). Note that the derivative of the (algorithmic) history term 1£0 71 , o: = 1, ... , ni, (which defines quantities at t11,) with respect to C 11 + 1 is zero. Hence, the third«:ontribution to the elasticity tensor is

a = 1, ... , ni .

(6.272)


294

Remarkably, the viscoelastic contribution to the algorithmic elastic.ity tensor may be expressed as the v.iscoelastic factors 8n, c:t = 1, ~ .. ·> rn, governing the ti me-dependent part, and by C~) n+ 1, which .is associated with .the isochoric elastic response as t --+ oo. Hence, relation (6.270) may be rewritten by me.ans of (6.272) as 71l

8

=Loa .

(6.273)

o=l

If 6

=

2:~~ 1

;1:exp(c;0 ) tends to zero, the al_gorithmic elasticity tensor reduces to (6.271) and compressible finite hype.relasticity is recovered. The implementation of the viscoelastic mode.I described above into a finite element program is based on the derived algorithmic stress (6.261) and the associated algorithmic elasticity tensor (6.273). The algorithmic update is carried out at eac·h Gauss point of a finite ele-ment. The .presented formulation only needs a particularization of the two strain-energy functions '11~ 1 ( J) and lJ!~ (C). If we use a strain energy which is expressed in principal stretches, then the algorithmic update is actualJy performed on the principal v.a)ues. For an effic.ient computational application of the .iterative process, see the works by SIMO [1987], GOVINDJEE and SIMO [1992b] and HOLZAPFEL [1996a].

EXERCISES

I. Using the relations

a= 1, ... , 'tn and integration over time interval t E .(0, T) obtain the -convolution integrals (6.254), which are the closed form solutions of the ·unear differential equations (6.252).

Assume a thin sheet of incompressible material in the undeformed configuration which may undergo viscoelastic deformations .in the large strain domain. The sheet is stretched to /\ 1 = A (A 2 == /\, = .t\- 112 ) in one direction (simple tension) and is fixed subsequently at this elongation. The deformation .is assumed to be homogeneous.

2. Relaxation test.

The viscoe.lastic behavior of the material is based on a phenomenological ·Maxwell-type ·mode.I with a free spring at one end and one Maxwell element arranged in paralle-l (rn = 1). The underlying strain energy is due to Ogden, as introduced in Section 6.5. We assume the six-parameter ·model and use the typical values according to (6.121 ).

6.11


295

(a) For a certain closed time interval I: E (0, T] compute an explicit expression for the evolution of the remaining non-vanishing internal stress q = (J(.,\, t) and the Cauchy stress a = a(,\ t) (the stresses in the transverse directions a.re zero)~ Discuss the internal dissipation Vint along the time interval and determine thermodynamic equilibrium. (b) For t E [O, T] plot the stress -decay (Q, a_) at a stretch ratio A = 5, with the strain-energy factor f3'r = 1 and the relaxation time -r = 10s. Give a physical interpretation of the relaxation time T.

3. Creep test. Consider the specimen and the viscoelastic constitutive model as described in the previous exercise, but now let a Cauchy stress a be .applied :in one direction up to a certain value a 0 and then be held constant. The stress causes a homogeneous deformation characterized by the -stretch ratios .,\ 1 = ,,\, 1 2 ,\ 2 = ,\a = ,\- / (simple tension). The underlying strain energy is of neoHookean type. (a) Derive the stress relation in the form of a nonlinear differential equation of first-order. (b) Solve the differential equation by means of Newton's method and use the derived recurrence update formula, or alternative·ly solve it with the RungeKutta method. For this purpose write a computer program or simply use some commercially available mathe-matical software-package. For a certain time domain t E [O, T] plot the stress evolution (CJ, -a), and the stretch evolution/\ (for a fixed a 0 ), f3.F = 1 and T = 10s-

6.11

Hyperelastic Materials lfith Isotropic Damage

Continuum damage .theories are either micromechanica:J or ·phenomenological in nature. "Microscopic approaches are certainly the best, but necessitate a strong physical background, and models are mathematically complex and often difficult to identify. The aim of this section is to formulate a three~dimensional and rate-independent isotropic damage model for the large strain domain describing the ·Mullins effect. We employ continuum damage mechanics (often abbreviated as CDM) and follow a purely phenomenological approach, which leads to damage models describing the nwc1vscopic constitutive behavior of materials containing distributed microcracks. The material model introduced is suitable for numerical procedures. For additional .information .on the subject of constitutive models within the context of continuum damage mechanicsT see the monographs by, for example, KACHANOV

296


[1986], LEMAITRE [1996] and KRA.JCINOVIC [l996, Chapter 4] and the review article by DE SOUZA NETO et al. [l.998], which also describes techniques for the numerical simulation of (isotropic) internal damage in finitely deformed solids.

Mullins effect.

Many rubber-like materials consist of a cross-linked elastomer substance with a distribution of small carbon particles as fillers (see the account of filled elastomers in MARK and ERMAN [1988, Chapter 20)). A piece of fil1er-loaded rubber subjected .to a series of loadings typically displays pronounced (strain-induced) stress softening associated with damaget known as the MuHins effect (this effect was pointed out in the early pioneering work of MULLINS [1947]; for a detailed description see, for example, JOHNSON and BEATTY (1993a, b]). Nearly a.Jl practical engineerir~g rubbers contain carbon particles as fillers and exhibit a certain degree of .Mullins effect, which is regarded as essentially being caused by the fillers .

.In order to explain the main features of this stress softening phenomenon we ·consider a stra.in-controiled cyclic tension test of a piece of filled rubber with two different strain amplitudes and neglect viscoelastic effects (slow strain rates). The cyclic loading and unloading process starts from its unstressed (initial) virgin state 0 and follows a path A, which we call the primary loading path (see Figure 6.7). After subsequent unloading initiated from any point 1 on the primary loading path the piece of rubber fallows path B and completely returns to the unstressed state 0 (for real rubber this will hardly ever occur). Note that after the test piece has been subjected to a load up to the point 1 the initial properties of the virgin material containing fillers are changed permanently (see MULLINS [1947]). The first loading and unloading cycle involves dissipation which is represented by the area between the curves A and B (hysteresis behavior). The area is a measure for non-recoverable ·energy. When the material is re-loaded the stress-strain behavior follows the path B again and if a strain beyond the point 1 (at which unloading began) is applied, the path D is activated. It is a continuation of the primary loading path A. For additional unloading which begins at any point 2 of the primary loading path the rubber is retraced back to the unstressed state 0 along the path C. Note that the shape of the second stress-strain cycle differs significantly from the first one. Path C retraces the piece of rubber back to 2 on re-loading and the primary loading path is activated again. In summary.: the stiffness of rubbers containing reinforcing fillers such as carbon black decreases as a result of extensional loading and unloading. The material properties associated with the initial extension of rubber compounds may differ significantly from those associated with successive deformation. With reference to Figure 6.7 we recognize that .for a given strain level the stress required on the first unloading and reloading .path B (and also .on the second unloading and re-loading path C) is less than that on the primary loading path A, and is the essential feature of the Mullins effect. There are a few theories in the literature aimed at explaining the microscop.ic dam-

6.11

Hyperelastic Materials ·with Isotropic Damage

0

297

Strain

F-:i.gure 6.7 Cyclic tension Lest displaying Mullins effect.

age -mechanism. One class of theory is based -on the idea that the .internal damage is caused by debonding of rubber :molecular chains attached between the filler partkles (see BUECHE [1960, .1961]). The higher the (macroscopic) deformation of the.rubber the higher the strain-induced damage . .Based on statistical arguments the stress softening effect may be predicted for a ·fixed level ·of strain due to previous higher strains in the material. The other class of theory for explaining the microscopic damage mechanism goes back to MULLINS and TOBIN [1957] and MULLINS [1969]. They proposed that initially the filler-loaded rubber exists in a so-called hard phase which degrades into a soft phase with increasing strain. The trans.ition between the two phases is characterized by a damage parameter which -is associated with a strain-amplification function. JOHNSON and BEATTY [l993a, b] have adopted the two-phase approach, which shows good agreement with experim-ental data in simple tension. The Mullins effect was observed experimenta11y in u.niaxial cyclic extension tests performed by, for examplet MULLINS [1947], ·MULLINS and TOBIN -[1957] (who provided experimental data for loading), .BUECHE [1961 ], ·MULLINS and TOBIN [1965] (experimental data obtained from the elastic and swelling behavior of filler-loaded rubbers), HARWOOD et al. [1965], HARWOOD and PAYNE [1966a, bJ, STERN [1967] .and .MULLINS [1969]. It is important to emphasize that in practice several other (inelastic) effects arise under extensional loading and unloading and we address these in brief. From the experimental observation it .is known that, besides the Mullins effect (which is an idealiz.ed phenomenon), the shape of the stress-strain curves is essentially rate .and temperature dependent. In addition, the shape of a piece of -carbon-black filled rubber after unloading differs significantly from its virgin shape. This interesting effect caused by

298

6


reinforcement leads to residual strains, which are responsible for the change of shape (in the rubber industry also called pernument set). For suitable constitutive models incorporating residual strains see HOLZAPFEL et al. [1999] and OGDEN and ROXBURGH :[1999b]. Additional .inelastic effects such as relaxation and/or creep depend strongly on the content of solid fillers (seeT for exampJe, So and CHEN [ 1991, and references therein]). An experimental investigation of the large strain time-dependent behavior of carbonblack filled Chlo.r~prene rubber subjected to different loading conditions and a constitutive model based on micromechanical considerations is presented by BERGSTROM and BOYCE [1.998]. A series of uniaxial strain-controlled cyclic experiments on cyJindrical specimens of carbon-black filled rubbers between 100% in tension and 30% in compression, including inelastic effects at room temperature, is presented by LION [1996] .and MIEHE and KECK {2000]. Phenomenological material models for filled rubbery polymers are also proposed therein. A cyclic tension process with three different strain ampJitudes and I 2 loading cycles was performed on a virgin specimen. The process shows clearly that the magnitude of the resu1ting stress softening de.pends on the number of loading cycles and the strain amplitude. The -experimental studies also consider relaxation periods in tension and compress.ion at constant strain. The work of MIErlE and KECK =[2000] also provides experimental stress-deformation curves of pre-damaged specimens under monotonous tension and compression and cyclic tension/compression. For an experimental investigation of a filler-loaded tread -compound at different temperature levels see LION [1997a]. Damage model in -coupled material description. Now we are concerned with the continuum formulation of the (ideal) Mullins effect and neglect rate and tem.perature dependency as well as residual strains. Additionally, in the phenomenological model we do not consider the presence of carbon-black fillers. We choose a Lagrangian description and express the relevant equations in .terms of the .right Cauchy .. Green tensor. Consider an isothermal elastic process, and postulate a Hel.mholtz free-energy function \JI in the coupled Jorm

\II

= W(C, () = (1 -

()\l1 0{C) ,

(6.274)

where \Jl 0 is the effective strain-energy function of the hypothetical undamaged -material, with the normalization condition \11 0 (.1) = 0 and the restriction w0 (C) > 0. The factor (1 - .() is known as the reduction factor, first proposed by .KACHANOV Il 958] who modeled the creep rupture of metals as a uniaxial problem (Kachanov actually introduced ( 1 - () as the '.integrity' parameter). Here the internal vczriable ( E [O, 1] is a scalar, referred to us the damage variable. The damage variable describes an (ideally) isotropic damage process and is .related to the ultimate failure of the material. Note that the strain energy \II 0 is assumed to be objective as usual.

6.1 J.

Hyperelastic 1\ilaterials with Isotropic Damage

299

Of course this simple type of damage model has limited use in practice~ but describes both the dissipation mechanisms and the irreversible rearrangements of the strucll.lre . To refine the model more general (tensorial) forms are needed, especially to describe anisotropic damage. In order to obtain the stress relation we differentiate first (6.274) with respect to time. Using the chain rule we find that · D\lto(C) · . .· '1t = {1 - () DC : C - W0 (C)( ,

(6.275)

with the rate of change of the right Cauchy-Green tensor and the damage variable, .i.e. . . C, (2.166), and.(, respectively. As a particularization of the Clausius-Planck .inequality, as given in (4.154 ), we obtain, by means of (6.. 275)~ Vint = ( S - (1 - ()2

8\lto{C)) DC . :

C + 'llo(C)~: > 0 2

,

(6.276)

and therefore the second Piola-Kirchoff stress tensor S and the non-negative internal dissipation Vint. are

S

= (1 -

()So

with

S _ ?8Wo(C) 0

-

-

with

ac

'

(6.277) (6.278)

In constitutive equation (6.277) the quantity Su denotes the effective second PiolaKirchhoff stress tensor. The dissipation inequality (6.278) clearly shows that damage is a dissipative process, the quantity f therein denotes the thermodynamic force which governs the damage evolution. In continuum -damage mechanics the thermodynamic quantity f has the meaning of the effective strain energy \JI 0 released per unit reference volu·me. The thermodynamic force f is related (conjugate) to the intern a.I variable ( accordin 0o- to

f

.

= Wo(C)

aw = - ac, ,

(6.279)

see relation (6.274). Therefore, instead of controlling the damage process by the i.nterM nal variable ( we can equivalently use its conjugate quantity, i..e. the effective strain energy ·wo. The evolution off is given by

. _ ?DW 11 (C) . C _ S . C f.- . . ac · 2 - 0 ·_2'

(6.280)

recognizing that .f characterizes the effective stress power per unit reference volume

300

6

Hypere:Jastic Materials

according to (4J 13). As noted above we assume that the .total damage accumulation is based on Mullins type (stress) softening of the ·material. For a strain-controlled cyclic loading process with a fixed strain amplitude this type of phenomenological damage occurs only within the first eye.le. We adopt the smooth function ( =-((a:) as the damage variable, with conditions ((0) = 0 and ({oo) E [O, l]. The .phenomenological variable a describes the discontinuous damage. A constitutive particularization of the damage variable ( may, according to MIEHE [l 995a], .be given by

( = ( (.o:) = (oo [1 - exp ( -

a/ l,) J

_,

(6.281)

where ( 00 describes the dimensionless maxi-mum {possible) da.rnage and,, is referred to as the damage satu-ration parameter. We now aim to determine the discontinuous damage variable a over the past history up to the current time_, i.e. the history time interval [O, t]. To control a discontinuous drunage process we use the evolution of the effective strain-energy function '11 0 . Within the closed time interval (0, t] we take the phenomenological variable a to be related to the maximum value of lJl 0 and write

n (t)

= sE(O,t) ma..x \JI o(s)

,

(6.282)

where s E [O_, t] denotes the history variable. Thus, it emerges that a is the maximum the-rmodynamic force with the same dimension as the effective strain energy p.er unit reference volume. Definition (6.282) was employed by, for example~ DE SOUZA NETO et al. [1994] and by MIEHE [l 995a], while the rate-independent damage model of SIMO [1987] invoked the principle of strain-equivalence (see the chapter on damage mechanics in the book by LEMAITRE and CHABOCHE [1.990, Chapter 7]). Other strain-based models for the finite strain domain are due to GOVINDJEE and SIMO [1991, l992a, bJ, JOHNSON and BEATTY [1993a, b] among others. Re.Jation (6.282) _generalizes the one-dimensional damage model (for small strains) proposed by GURTIN and FRANCIS [1981b]. It uses the hypothesis that the current state of damage is characterized by the maximum axial strain attained in the history of-defonnation. Here we do not recall the many important works in the area of small strain damage mechanics. A straightforward refinement of the isotropic damage model may be obtained by including damage effects governed by continuous damage accumulation. In particular, the work of MIEHE [l 995~], which is exclusively presented in an Eulerian setting, describes continuous damage accumulation. lt is based on the arc-length of the effective strain-energy function. A fairly simple and efficient energy-based damage model to describe the main features of the Mullins effect in filled rubber was proposed by OGDEN and ROXBURGH

6.ll


301

II99.9a]. The material model is .composed of a (classical) strain-energy function describing the primary loading path from the unstressed virgin .state and an additive dam.age function responsible for the unloading path which .is initiated from any point on the primary loading path. The formulation is based on the concept of pseudo-elasticity in which the material is treated as one elastic material in loading and another elastic material .in unloading. This idea was used by, for example, FUNG et al. [1979] within the context of modelling arterial walls. It has the significant advantage of convenient and simple description of the stress-strain relationships in cyclic loading and their numerical (finite element) realization.

Damage criterion and damage evolution. We now define a damage criterion in the strain space at any time of the loading process in the form

= f (C) ·-a < 0

(.6.283)

,

with the damage function = 0 which characterizes the so-called damage surface with

normal N = 84>(C, a)

ac

= 8J(C)

(6 ..284)

.ac ·

For a fixed a, the damage surface delimits the strain space in which the behavior of the material is considered to be fully elastic (.no damage accumulation occurs). A representation of the damage surface with .the associated normal in the principal strain space is shown in Figure 6.8.

.a= 1, 2, 3

. · · · · . · -. . Damage surface

c/J(Ca,.a)

=0

Figure 6.·s Illustration of the damage criterion in the principal strain space.

Hence, a double contraction of the two tensors N and C gives, using (6.284h, the scalar N : C = 8f (C)/oC : C = j(C). Borrowing the terminology .from strain space

6

302


plasticity {see NAGHDI and TRAPP [1975] and SIMO and HUGHES [1998, p. 84J), at = 0 we must distinguish between

.

l <0 f =0

and

'

(6.285)

'

f >0 describing unloading, neutral loading and loading, respectively,. Finally, the evolution of the maximum thermodynamic force n, i.e. {6.282), is given, with {6.280):h by

·, _ n. -

c

.

{

f =So: -2

if

0

otherwise

~b

= 0

f >0

and

,

{6.286)

.

Advancement of damage only occurs for the cas·e of loading, and the .initial condition for n is zero. The evolution equation (6.286), which dearly shows the discontinuous property of this damage model, corresponds to those given in GURTIN and FRANCIS [198lb], S·tMO ["1987] and ·MIEHE [1995a].

Coupled representation of the elasticity tensor. According to relations (6.153) and (6.154) we derive the symmetric fourth-order elasticity tensor C in the material description. Starting with the constitutive equation (6.277), we have · . . DSo I · : S=(l-()2ac : c-sn~,

2

(6.287)

where the rate of damage ( takes on the form ~:

= ('(n)o!

?( .

('(a} = · Dex

with

(6.288)

Substituting (6288) into (6.287), we obtain, using (6.286), the ·evolution of the stress tensor in the farm

[{1 - ()
S=

·~

if
=0

and .f > 0 (6.289)

C ( 1 - ()Co: .. 2

otherwise

,

with the effective elasticity tensor C 0 in the material description. For an undamaged material, C0 is defined to be tr"'

"--' 0

=

9 -

8So(C)

ac

.

(6.290)

6.11

Hype.relastic ·Materia"ls with Isotropic Damage

303

By -comparing (6.289) 1 with (6.153") we find that the terms within the bracket -characterize the elasticity tensor C in the -material description. The second term in .the rate equation (6.289)i governs the damage that causes stress softening of the material.

Dam.age -model in decoupled materi.al descr.i_ption. To complete our considerations of finite strain elasticity with isotropic damage in the sense of decoupled vo"Jumetric-isochoric response (introduced due to a multiplicative split of the deformation gradient (6.79)), we postulate finally a decoupled representation of lJr = llt(C_, () in accordance with (.6 .85) in the form

'11(C, ()

= Wvoi(J) + (1 -

(6 ..291)

()\Jloiso(C) ·

Here, W\'ol is a strictly convex function (with the minimum at J = 1) which describes the volwnetric elastic response. The second function '\JI 0 iso denotes the isochoric cf. . :fective strain energy of the undamaged material, which describes the isoc/wric elastic response. Hence_, the damage phenomenon is assumed to affect -only the isochoric .part of the deformation, as proposed by, for example, SlMO [I 987]. We require that \Ji vol ( J) = 0 and WcHso( C) = 0 hold if and only if J = l and C =I, respectively. Consider the structure (6.291), the purely volumetric contribution to the stress and the elasticity tensor are pres·en.ted by eqs. (6,.89) and (6.166), respectively. The isoclwric contribution to the stress is, by analogy with (6.277), given by Siso

== {1 - ()So iso

. _ S01so -

with

? -

iJ\Jln jso(C) DC ·.

(6.292)

The isochoric contribution to the elasticity tensor includes damage and is~ by analogy with (6289), given by

.

[(

-· 1 - ~)
-

~

-1 (

Cl'.

)

So iso 0 So i~o ]

•.

· ..

. c (1 - ()Co iso : · 2

?c

if ~b = 0

....

and

f >

0

,

(6.29.3)

otherwise ,

with the isochoric part C0 iso of the e.ffectiv.e elasticity tensor in the material description. For an undamaged material, C 0 iso is defined to be

Coiso

DSu iso

= 2 ac

~

(6.294)

.Explicit forms of (6.292.h and (6.294) are given by (6.90) and (6.168), respectively. By analogy with eq. (6.278"h the thermodynamic force f has here the meaning of the isochoric effective strain energy W0 ;80 ( C) of the undamaged material. Within the decoupled framework of volumetric-isochoric response, eqs. (6.278)2 and (6.280)2 take on the forms

.f = Wo iso(C) > 0

and

. .. -c 01so • I. -s 2

,

(6.295)


304

The applicability of the constitutive damage model thus described is limited to sufficiently slow ·processes (viscous effects are .not considered). However, the damage model may easily be combined with the viscoelastic model as proposed in the last section. A suitable decoupled free energy for characterizing finite-strain viscoelastic damage mechanisms might be given, with reference to (6.236) and (6.291), as

w(C, (, ri, ... , I'm) = W~i(J)+(l-()[WOfso{C)+ L::=l Too(C, r n)]. Here, \llOiso(C) and L~~ 1 T 00 (C, rn) denote, respectively, the strain energy and the configurational free energy (per unit reference volume) for the hyperelastic undamaged material. Both functions are associated with the isochoric response.

EXERCISES

1. Pure shear with isotropic damage. Consider a thin sheet of (incompressible) hyperelastic material which is subjected to a homogeneous pure shear deformation with the kinematic relation ,\ 1 = A, A2 = 11 A3 = 1/ ;\ (compare with Exer~ cise 1(b) on p. 227). The stress state of this mode of deformation is characterized by £Ti, a2 and a 3 = 0 (recall eqs. (6.76), (6.77)). The material is supposed to undergo stress softening of Mullins type. (a) Based on the isotropic damage model introduced, compute the loading path up to A. = 2 followed by the unloading path back to A = 1 . Apply the constitutive particularization of the phenomenological damage variable (=({a), as given in eq. (6.281), and use the strain-energy function of the Mooney-Rivlin form, Le. W = cr(I1-- 3) + c2(/2 - 3), with the ratio of the material constants c 1 / c2 = 7. (b) Plot the loading and unloading path in the form of the two functions £T1 = a 1 (A) and a2 = a2"(_,\) with the Mooney-Rivlin parameters c 1 = 0.4375_tt, c2 = 0.06.25µ and the shear modulus in the reference configuartion, i.e. 2 µ = 4.225·10 5 N/m • I.n addition, take ( 00 = 0.8 for the maximum damage 2 and 1. = 0.3 .. 106 N /rn for the damage saturation parameter.

2. Equibif1xial deformation with isotropic damage. Consider an equibiaxial deformation of an incompressible material, which may be modeled by the strain energy due to Ogden. Recall the stress strain relation derived in Example 6.6 on 20 p. 239, i.e. a = 1 µp(,.\'!11t - ,.\- J• ). During a load cycle,,\= 1 -->- 3 -+ 1 -t 5 --+ 1 .the material accumulates damage of Mullins type.

z:,:=

Plot the load cycle a= a(A.) with the typical values of the constants O'p, µ.P, p = 1,,. .. , 3, for Ogden's model given by (6.121). Assume the maximum damage 2 6 ( 00 = 0.8 and the damage saturation _parameter .1. = 1.0 · .10 N /rn •

7

Thermodynamics of Materials

Thermodynamics is the science of energy, which studies processes .in systems outside the thermodynamic equilibrium .state. The term 'thermodynamics·' comes from the Greek words .(Jipµ17 and O'u11a1u<; meaning 'heat' and ·'force' (or 'power'), respectively. Thermodynamics has long been a fundamental part of engineering. It constitutes a concept of great generality which is based on a few, simple hypotheses. Today the name 'thermodynamics' is interpreted as including all aspects of energy.

There .are two principle ways of dealin.g with thermodynamics: long ago it was recognized that in the real world physical objects are compositions of molecules which are formed by atoms and even smaller subatomic particles. The special field in which the laws of classical mechanics (or quantum mechanics) are applied to large groups of individual particles (molecules, atoms) is called statistical thermodynamks. This approach investigates the correlation between the average behavior of particles and the macroscopic properties of a system on a statistical basis. The traditional classical, or phenomenological, .approach to the study of thenm;:>dynamics, .in which the molecular structure of a physical object is disregarded (the object is considered as continuous matter with no microscopic holes.) .is ·Called class.ica·J thermodynamics or (phenomenological) continuum ther·modynamics. It may be viewed as a unified field theory of nwchanics and thermodynamics in which all thermodynamic state variables depend on space and time. The essential feature of continuum thermodynamics is the derivation of constitutive equations (for the stress tensor, the entropy, the heat .fl.ux vector) from the basic physical principles of thermodynamics representing the individual (mechanical and thennodynamic) material properties of matter. However, for the formulation of constitutive equations we have several possible choices of independent and de.pendent variables. In this chapter the list of .independent variables is supplemented by non-mechanical variables such as temperature, entropy and their gradients. We shall combine the (isothermal) constitutive theory of finite (visco)elastidty, as introduced in Chapter 6, with the theory of heat conduction under transient conditions. Solutions of the resul.ting coupled thermomechanical ,problem are able to describe the interaction between

305

306

7

Thermodynamics of l\ilatcria·rs

the mechanical.field and the thermalfi.eld. In this chapter we study the thennodynamics of continuous media and, in particular, two different dasses of constitutive .1nodels within the nonlinear constitutive theory of fi.ni.te thermoelasticity and finite thermoviscoelasticity. The only materials undergoing finite strains and temperature changes relative to an equilibrium state are biofogical soft tissues and rubber-like materials. As known from statistical thennodynam.ics of rubber elasticity, extended rubber chains tend to return lo a less-ordered curled up-state which is characterized by a higher conformation .entropy. The thermomech.anical behavior of solid polymers is almost entirely based on an entropy concept. Therefore, in this chapter we start out with the aim of reviewing the crucial difference between rubber and metal within a thermodynamic context. In order to describe the three-dimensional network of rubber by means of the Helmholtz free-energy function, some insights .in the statistical thermodynamics of rubber and the (molecular) network theory are presented briefly. We restrict attention to the Gaussian statistical theory, which .enables us to characterize the thermo.elastic behavior of a (molecular) network within small strains. Within this statistical context, the neo-Hookean model, as derived in Section 6.5, is motivated. In the subsequent sections, quite independently of the network theory, we follow an approae.h to a macroscopic continuum formulation of .thermoelastic and thermovisco.elastic materials by making use of continuum particles. We introduce a constitutive .model for the thermoelastic behavior of materials and present a thermodynamic extension o.f the classical strain-energy function originally proposed by Ogden. The material model is set up in order to reproduce the realistic physic.al stress-straintemperature response of rubber-like materials . .Moreover, a study of one-dimensional problems of finite thermoelas.ticity is presented,. Distinctive attention is paid to the so-called thermoelastic inversion phe1wm .. ena, a remarkable property of rubber-like materials. The last section in this chapter is concerned with the study of thermodynamics in terms of .internal variables. A constitutive model for highly deformable media that accounts for several thermomechanical coupling effects is examined. The proposed phenomenological model is capable of describing relaxation and/or creep phenomena within the thermomechanical rei!.ime.

-

7.1

Physical Preliminaries

As a basis for our next studies we present an introductory review of som.e of the interesting physical aspects of the thermoe1astic behavior of amorphous solid polymers, that are chemically cross-linked, for example, by sulphur bridges.

7.1


307

(a)

I I

I

I

r

' \

\

\.

'' ...........

........

........

- __ .....

.....

Tic point

(b)

................. ,.... :..........

:,...

(c)

•

•

1·a =.L .....·............... ,...... ,

........................ .,..

Figure 7.1 Single po'lymer chain lying between two tic .points with various distances r. Two possible conformations are shown in (a). The number of possible conformations decreases with .increasing end-to-end dislancc (b). Fully extended chain showing the limiting case wilh only one poss.ible conformation (c) .

.Statistical concept. Based on several physical techniques we know that amorphous polymers are composed of bundles of long-chain molecules (which may .be imagined .as strings) having a high degree of ·flexibility. Figure 7.1 shows a model for a single polymer chain lyin.g .between successive points of cross-linkage, which we call tie points. The sing'le .polymer chain forms a typical segment in the coherent and threedimensional network of rubber. This model is due to GUTH and MARK [1935] and KUHN fl 938, I 946l The distance between the tie points of the chain molecule, denoted by r, we call subsequently end ... to-end distance (or separation). The distance r is a parameter that characterizes a molecular conformation. The name conformation comes from chemistry and refers to different shapes (arrangements) ·of ..a chain molecule. The most powerful physical technique now available for determining .conformations of ·chain molecu'les is small-angle neutron scattering (for more details see, for example, SPERLING [1992, Section 5.2]).

308

7


Figure 7. l(a) shows a polymer chain with the end-to-end distance r 1 of the tie points, which are assum·ed to be fixed in .space. The distance r 1 is ·much smaller than the contour length L which is the length of the fully -extended chain. Consequently, the chain may take on an enormous range of possible conformations, two of which are shown in Figure 7.l(a). Obviously, the number of possible conformations decreases for a larger -end-to-end distance, and in the limit the number -of possible conformations diminishes to only one if the chain is in its most extended state, as illustrated in Figure 7 .1 (c). Then the value of the end-to-end distance reaches its max.i.mum1 i.e. r 3 , and equals the contour length L, and the chain .is straight. The most crumpled conformation occurs when r tends to zero, and the tie points coincide. Clearly the end-to..-end distance r of a chain characterizes the molecular conformation. The statistical theory of rubber elasticity (see, for example, the notable works by TRELOAR [1943a, b], JAMES and GUTH [1943~ 1949] and FLORY [1953, Section XT3, and references therein]), which is basically set up on these concepts makes use of the idea to express the number of conformations th.at a chain molecule can .assume as an 'en.tropic effect'. The chains occur in random.Jy coiled conformations in the unstretched state, as seen in Figure 7. l(a), and as the chains are -extended the number of conformations and the entropy decrease. We proceed now to model this characteristic behavior and to analyze the conformations of a chain molecule. However, just for clarity, consider first the problem in one dimension and project the conformation on one coordinate axis, say the :I:i.-axis (the chain may be imagined as being constrained .artificially so that the tie points lie on the ~7-: 1 -axis). The conformations of the individual chains are distributed in a random manner. The _probability p(~z:- 1 )dx 1 that the end-to-end distance of a chain ties in the interval between :i: 1 and :r 1 + d.ri 1 is expressed by the Gaussian distribution function b

., ')

p( :i:i)d:r:1 = ,_/I 11..... , exp (-lr:1:1 )cb: 1

,

(7. l)

where v(:c 1) is the probability density (per unit length) and b is a parameter of the model. This parameter is a measure of a representative length, as we see later in Example 7 .1, p. 313. For an explicit derivation of the Gaussian distribution function (7 . :I) for a chain in one dimension the interested reader is referred to the classical book by FLORY [195.3, Appendix A of Chapter Xl The Gaussian function is a bell-shaped curve, which provides numerous applications in engineering practice and statistics. The most probable value of :ri, LeA the maximum of the Gaussian function, may be found by differentiating eq. (7. l) with respect to ~l; 1 and occurs at :r 1 = 0. The probability decreases monotonically as :z:i increases. Note that the entropy .may be interpreted as a quantitative measure of probability (microscopic random.ness and disorder) by using a fundamental finding which is due to Boltzmann and Planck. This will be made clear in Section 7 .2.

7.1


309

For a more comprehensive survey of the general concepts of statistical mechanics, a terminology which was introduced by Gibbs, the reader may be referred to the books by FLORY '[1969], TRELOAR [l975J, WEINER [.1983], CALLEN .[1985, Part II] and. :MARK and ERMAN [1988]; see also the review paper by GUTH [1966].

.Rubber ve.rsus 'hard' solids. One of the remarkable differences between rubber and 'hard' solids, such as metals, glasses, ceramics, crystals, etc. lies in the effect of temperature. The foil owing crucial physical properties, ·explored and quantified in a set of experiments by Joule, exhibit the distinctive behavior of rubber (see JOULE [.1859.t p. 1.05]): (i) a piece of vulcanized rubber subjected to a weight produces a slight .cooling e./fect in the very low strain range and changes to a heating effect by increasing the weight, and

(ii) rubber will contract its .length under tension when .its temperature 'is raised (it is not very known that healthy human and animal arteries a]so shrink upon heating, a phenomenon that was pointed out in the early work of ROY [ .1880-1882] for the first time).

<:~;~~i;_'.21

.;

These results are based on a (previous) simple qualitative observation by GOUGH [1805] that a rapidly . .stretched rubber band (adiabatic straining) brought into slight contact with the lips as a sensitive detector feels warm. On the other hand a stretched rubber band in thermodynamic equilibrium feels cold after releasing the tension. This (thermoelastic) coupling phenomenon entered the literature as the .so-called Gough.Joule etfect. Note that the behavior of a metallic spring .is in striking contrast to a rubber band. A metaIIic spring coo"ls continuously ·On elastic stretching. Th.is is the opposite behavior of a rubber "band which warms on stretching, a remarkable experimental observation (see Box l). The properties of rubber are well-known aboye the glass transition temperature {see, for example, CYR [1988], WARD and HADLEY :[1993]), and are characterized by (i) extremely long~range extensibility, typica.lly 300-500% extension for vulcanized natural rubber (i.e., without carbon black or other reinforcing :fillers) and even more for synthetic rubber (generating low mechanical stresses), accompanied by (ii) full recovery to the initial dimensions without mechanical and thermal hysteresis within the Io\ver temperature domain of the 'rubbery' region.

Below the glass transition temperature, "flexibility .and mobility of the chains are so reduced that rubber behaves like a brittle glass (the material consists of rigid crystals) . The glass transition te·mperature, for ·example, for natural rubber and butyl rubber is - 73°C (see MARK and ERMAN [1.988, Chapter 2]) .

310

7


Based on experimental observations by ANTHONY et al. [1942] (see also TRELOAR fl 975, Chapters 2, 13]) the retractive force in a real rubber is approximately 90% based on entropy (for additional information the reader is referred to the papers by SHEN and CROUCHER [1975] and CHADWICK and CREASY .(19.84]).

Metals

Polymers Entropic elasticity

Energetic elasticity

=> Total stress is entirely caused by => Entropy does not change with dea ~g~ entropy with deformation.

formation at alt

=> Internal energy does not change => Total stress is .internal energy with deformation at all.

driven, which changes rapidly with deformation.

=> Elasticity arises through entropic => EJasticity arises through -energetic : .straightening of a polymer chain., fallowed by recoiling into a conformation of maximum entropy, see Figure 7. I (iso-volumetric phenomena).

·Polymers are entropic

1

increases due to distance changes between atoms against atomic attractive forces, followed by removing the interatomic forces back to its initial dimensions (in general, substantial volume changes accompany deformation).

·Metals are energetic'

=> A piece of rubber warms on stretch- => An elastic metallic spring cools on :_ mg. =}

A rubber band under constant tensite force substantially will shrink upon heating and expand upon cooling.

stretching.

=> An elastic metallic spring under · con_stant tensile force will expand upon heating and shrink upon coolmg.

From a thermodynamic point of view, the work done in elongating a rubber band ·is unlike the work produced by stretching a coiled elastic meta Uic spring.

Box 1 Composition -of the concepts in polymers (ideal rubbers) and 'hard' solids (metals, glasses, ceramics, crystals, etc.) in the elastic -range.

7.2 Thermoelasticity of Macroscopic Networks

311

For an ide.al rubber, for ·which by definition one property is incompressibility (the volume remains constant (locally and ·globally) during a mechanical process), the re-tractive force is, however, .purely determined by changes in entropy and the internal energy does not change withclefomi'ati·oii_iic'a'n, a-signi'fiC'arit"'characteristic of rubber elasticity. We term this type of rubber-like material 'entropic elastic'. However, elasticity of metals, glasses, ·ceramics or crystals arises bas.ically through removing atoms from their equilibrium positions, accompanied by rapid internal energy changes, while the entropy does not change at all (see, for example, HILL [1975] and ERICKSEN (1977]). These materials with a regular atomic structure and usually with high strength are typically called 'energetic elastic'. They exhibit, in general, substantial volume changes on deformations and stand "in sha~p contrast to rubber~like solids. For a general overview, the corresponding concepts in polymers {ideal rubber) and 'hard' solids (metals) are summarized in Box l. Natural rubber (cis-polyisoprene) does not recover completely. In order to achieve di·me.nsional stability and deformations which are completely reversible in the 'rubbery' state, vulcanization of the rubber (typically done for commercial products) is required. Within a vulcanization. process polymer chains are che·mica.lly connected to ·other cha.ins at different :Jocation.s to produce a cross-linked monolithic three-dimensional network (see SPERLING [1992]). On the other hand crystallization occurring in the highly stretched rubber is influenced by vulcanization. Crystallization is lower for a higher concentration of sulfur used in the vulcanization process (see TRELOAR [1975., pp. 1.6-23]). It is important to note that the material properties of highly stretched crystalline rubber become anisotropic and that .the heat of crystallization is much larger than produced by the mentioned thermoelastic Gough-Joule effect. Moreover, due)to frictional losses during the deformation process an additional heat is generated. In addition, it is mentioned that real networks contain defects (see, for example, MULLINS and THOMAS [1960] and SCANLAN [ 1960]). Crystallinity effects and network imperfectioris lie outside the .scope of this text..

·re..

7.2 Thermoelasticity of Macroscopic Networks A typical vulcanized rubber may ·be considered as the assembly of lon.g-chain molecules. Each chain is attached at both ends and thus produces one giant molecule, which we call the (molecular) network. · From the irregular three-dimensional network we draw condusions regarding its material properties. In the following, motivated by statistical thermodynamics, we describe the material properties of rubber through the Helmholtz free-energy function '11.

312


The freely jointed chain in three dimensions. We assume that the molecular network contains 1V chains per unit volume which is often referred to as the -network density. We consider first a representative single polymer ·chain in space detached from the network, which .means that the chain is taken out of the network. Our aim is to compute the entropy of this chain and to study .its thermodynamic behavior. The chain with contour length .L is cross-linked at the tie points 0 and P of the network (see Figure 7.2). One end of the chain is attached to the fixed ·orig.in 0 of the xh :c2, :r: 3 coordinate syste·m. The other end is given :by the end-to-end vector ·r = :c 1e 1 +·x2 e 2 + :z::.1e 3 ., pointing to P and characterizing a certain numb.er of different shapes .

. . . -~

. . . . . ~l

•:·" :.

dv

= &z~ 1 d:1:2 d:r:a

Figure 7.2 A representative single polymer chain OP detached from the network. Further, we assume a so-called Gaussian chain which is defined so that the distance between the tie points (chain ends) 0 and P ~i.e. r = lrl, is considerably Jess than the contour length L, i.e. r~

L .

(7.2)

Hence, we follow the context of the Gaussian statistical theory of elasticity which is valid for problems where only small strains are involved. The contour length L of a single chain is commonly considered to be an assembly of n (statisti~al) segments joined together, ·each of length l so that L = nl,. We suppose t~at .there is no correlation between the directions of the successive segments. Based on this simple mechanical model we may determine the so-called mean valuer of the end-to-end distance r for this freely jointed chain .observed at one instant of time. This

313

7.2 Thermoelasticity of Macroscopic Networks is given by

--., r- = n l',... ,

(7.3)

where the mean-square value r2 denotes the average over r 2 • For a more detailed explanation the interested reader is referred to the textbooks by, for example, :McCRUM et al. :[I 997, Section 2.8] or WARD and HAD.LEY .[.1993, Section 3.3] and left to study Exerc.ise 1 on p. 320. We wish to calculate the probability that the tie point P lies within the infinitesimal volume element of size dv = d:z: 1 dx2 d:c 3 at P (see Figure 7.2). By analogy with eq. (7.1) we may introduce the probability densities p(:.r:.2) and p(x 3 ) which are associ .. ated with the J:raxis and the x 3 -axis, respectively . .In addition, it is possible to show that p(.1: 1 ) depends only on :i: 1 (p(:z: 2 ) only on :i: 2 and p(x:i) only on :c 3 ) provided that n, is large and x 1 , x 2 , x 3 are much smaller than the contour length L of the chain. Generalizing the relation (7. l) to three dimensions we may find the probability density p(x 1, 1;2 , ~'t:1 ), now per unit volume, that the tie point P of the freely jointed chain occurs in the infinitesimal volume element dv (see Figure 7 .2). Under the restrictions considered it is the product of the independent probability densities according to .

.P( :x1, X2, X:i)&z:-1 d:1:2cfa:3

= p( :1:1 )p( X2)JJ( X3 )&i:1 d:1:2dX3 3

= 7i'l~i/r'- exp(-b2 r 2 )dx1 dx2dX:J

,

(7.4)

where r 2 = :i:i + :z:~ + ~1:5 is the square of the distance between the tie points 0 and P for this detached Gaussian chain and parameter b denotes a measure of a representative length. As for the one-dimensional case the maximum of the Gaussian function (7.4) occurs at -r = 0. The Gaussian distribution function, as given in (7.4 ), represents a sufficiently accurate solution to the stochastic problem in question.

EXAMPLE 7.1 In this example consider a freely jointed Gaussian chain with tie points 0 and P and calculate all possible conformations of the chain at any given value of the end-to-end distance r = f rl, irrespective of direction. Furtherm-ore, show that the :measure of the representative length b controls the mean-square value -r 2 according to ~,

lr

=

3

~

2r 2

(7.5)

(compare also with FLORY [1956, Section X-1 b]). The mean-square value r'2 is defined to be j~ r 2p(r)dv/ J;~ p(r)dv, with the probability density p given by eq. (7.4)2. 00

314

7

Thermodynami~

of M·aterials

Spherical shell du

= 47rr2 dr

Figure 7.3 Sphcrica:I shell which defi ncs all possible conformations of a representative Gauss·ian chain OP irrespective of direction .

.Solution. The restriction to a particular direction in space .as cons·idered in the previous analysis is not appropriate .anymore, so we take into account al1 directions of the vector r equally. Doing so, the tie point P does not move within the infinitesimal rectangular block, .rather within an infinitesimal volume ·dv of a spherical shell which IS

(7.6)

The infinitesimal volume is defined between the inner radius rand the outer radius r + dr from the ·Other tie point, which is fixed at the origin 0 of the coordinate system (see Figure 73). The required probability p(r)dr that the chain length lies .in the interval between r and r + dr, is, by means of (7.4h, '·I

b' 2 2 2 p(r)
(7.7)

The function (7.7) represents the Gaussian distribution of the distance r for a set of free chains. In the r-distribution function, no restriction on the direction of the vector r is involved . The maximum of function (7. 7), i.e. the ·most probable value of r, is obtained by differentiation of p( r) with respect to r. This maximum occurs at r = 1/ b. .In order to compute the important mean-square value of r, i.e. r2, we ·find after

7.2

Thermoelasticity of Macroscopic Networks

315

some manipulations the analytical solution 00

r'l.

=

Jr

00

p{r)4trr dr J r"1 ~xp(-b2 r2 )dr _o_ _ _ _ _ _ _o_ _ _ _ __ 2

2

00

00

J 71(r)4.1fr dr

I r 2exp(-b2 r2)dr

0

0

2

3fo/8b 5 = 3fo/4b3

2b2

(7.8) 1

which proves (7.5). We know froin eq. (7.3) that the mean-square value r 2 depends on the (statistical) segments n .and their lengths l (recall that nl is the contour length L of the chains). Hence, we conclude that bis a measure of a representative length. •

The entropy of a single chain.

On the basis of the statistical concept of thermodynamics we now .determine the entropy r}i of a representative single Gaussian chain i whose -ends .are located at specified points in space. The chain "is assumed to be taken out (detached) from the network. We apply Boltzmann's equation (or what Einstein called the Boltz·mann :principle) relating thermodynamic entropy· and the probability of a thermodynamic state (molecular conformations). Hence, from the statistica1 point of view, the entropy r/i of a single chain is defined to be proportional to the logarithm of the probability density v(r) and varies with the end-to-end distance .r according to Boltzmann's ·equation, whic.h is given in the form 1}i

= a + klnp( r)

,

-(7_.9)

where a denotes a constant entropy with respect .to a reference levet which need not be specified here in more detail. The universal constant of proportionality k = 1.38 · 10- 2~JNm/K denotes Boltzmann's constant. The famous relation between entropy and probabi"Jity was published by the Austrian physicist Boltzma1111 in 1877 (at this time he worked as .a professor for physics in Graz). The term 'Boltzmann's constant' fork and the mathematical formulation of the principle in the form of S = klogll' due to to Planck. This form was carved on Boltzmann's gravestone at the ~zentra.lfriedhof' in Vienna in 1933. Substituting (7. 7)2 into (7. 9) we obtain finally 1Ji

=c-

. ')

,,

.kb"r~

,

(7.10)

where the constant c =a+ kln{li1/7ra/ 2 ) .incorporates the constant entropy o.. The measure of the representative length b for this Gaussian chain detached from the network is given by 3 b2=--

·22 rout

(7.11)

·7 Thermodynamics of Materials ·

316

(see eq. (7 .5)), where r 2 out denotes the mean-square value of the end-to-end distance of this un-cross-linked free chain out of the network. It is an intrinsic property of the -chain molecule and is independent of volume changes. As can be seen from expression (7 .10) the entropy tends to its largest value for r --+ 0, as expected (see the considerations in the previous section).

The elasticity of a molecular network .

In order to determine the elasticity of a molecular network, we choose, without loss of generality~ the case of .a homogeneous de.formation state of a rubber block given by the _principal stretches ,\a, a = l, 2, .3. Further, we introduce two crucial assumptions: (i) there is no change in vo1ume on deformation., the material is idealized as totally incompressible (incompressibility assumption), i.e. /\ 1 ,\ 2 ,,.\J = 1, (ii) changes in the length and orientation of lines marked on -chains in a network are identical to changes in lines marked on the corresponding dimensions of the macroscopic rubber sample (affine motion assumption)~

Thus, we refer to Fig':lre 7.4, which shows the .affine motion of a representative ·Gaussian chain with one end at the origin.

. Po (X1.?X2,Xa) :

ro

; ro

lrol

=

o., . -·-.........L._--·7-l·x;;·. · · , . .::::,-. . . x 2, x2 .·

......,,,..

.···~··,'·'·'·=·'

,,~.,,

..... ,,.. :...:-;····

"J._,./

:1:1

~

....

..

I~"'.•,., ... •' • ,.4 II•• • 4 •"' • ·,, • • •·• • • • • • • • •' •I> • '"'·"' ._

.·

.·

,,

> • ...

•<(

Figure 7.4 Undeformed and deformed confi_gurations or a representative Gaussian chain.

In the undeformed configuration of a network the end-to-end distance of the chain OP 0 is -characterized by the vector fti, with material coordinates ..Y..h ~4 = 1, 2, 3, and length r0 = lr0 J. Since we consider the affine motion assumption all jV chains

7.2

Thermoelasticity of Macroscopic Networ,ks

317

defom1 like the representative chain shown in Figure 7.4 and vector r 0 becomes rafter deformation, ·with spatfal coordinates :c 0 , a = 1, 2, 3, and length 1·· = lrl. In the course of the .motion the tie point P 0 is displaced to P. Because of.the affine motion assumption we may write X'>

.-

= /\ .-,.,Y,,...

(7.12)

'

The change in the end-to-end distance of the ·Chain due to the deformation produces a difference of entropy between the state before .deformation (with Aa = 1, a = 1, 2, 3), i.e. c - kb 2 ( .Xf + .:Y:j + JY:i) (see eq. (7 .10)), and the state after deformation, i.e. c - kb2 (,\T.~Y[ +,\~ ...Yi+ t\~ ..Yi) . Hence, the entropy change in the chain caused by.the deformation of that individual chain i, denoted by il'1Ji, is therefore

ilr11 ·= [c - kll(/\i~Y; +,\~.Xi+ ..q..Y;f)] - [c - kb2 (.X~

= -kb2 [(/\i -

1).Xf

+ (-,\~ - l).Xi + (~\~ -

+ .X~ + .X.:~)] (7.13)

l).Xj] .

The constant c has no physical relevance since we are only concerned with the change of entropy. In order to refonn a network all the detached chains are transferred back into the rubber specimen and cross-linked. Of course., the end-to-end distances of (detached) chains out of the network are not the same as the end-to-end distances of (cross-linked) chains in the network. Therefore, we introduce the mean-square value of the end-toend distance r for the whole assembly,of chains in the specimen., denoted by r 2 in· In contrast to the mean-square value r 2 oub r 2-in is not an intrinsic property of the chain molecule. Since some constraints must be applied to the detached chains in order to reform a network, the mean..-.square value r 2 in differs from. r 2out. In particular, the value r 2 it1 depends on the volume of the rubber and ·Changes by heating (or cooling) (compare also with ·McCRUM et al. [1997, Chapter 3]). Our next aim is to compute the entropy change of a .network of such chains generated by the macroscopic .deformation state. It is the sum of the entropy changes of all JV chains in a unit volume in the network, which we denote by a1J. S:ince we have assumed affine motion, all the chains have the same given intrinsic property band imposed Aa, a= 1, 2, 3. With eq. (7.13) we may write N

1i11

=L

~ 1,,i

i-=1

N

= -kb [(,\i 2

i)

N

N

L x~ + (,\~ - 1) Lx~ + (-"~ -1} LxiJ 1

1

.

(7.14)

:

We claim that N ~

'}

L..J rij in l

-

= Nrfi in

(7.15)


318

holds, where rfiin is the mean-square value of the end-to-end distance roin of the assembled chains i1l the specimen in the undeformed state. Since the vector r 0 has no preferred direction in the undeformed state (which is . • ) • " N v2 ~N x,.2 . . ~N v2 . "'"'N x,...2 1sotrop1c .,wemaywnteL......A'l = L.Ji = ~N L...ti a· B utsmce ""i. +L...,, + ~N Ll.1

1

r2 -

""'N

2

.X 3 - L...Jl rOin we deduce that .and finally, from {7.1.5)

2 "'"'N . r2 -

L..J1. "'~ 1 -

x2

~N v2 Lil ""1. 2 -

61 " N ·v2 -

~ 1 .;·\_a -

tx~=tXi=tXi=!Nr~in. 3 l

I

1

1

2

2 1I.3 "'N Lit r 0 ;

11 ,

(7.16)

l

Combining (7.16) and (7.'11) with (7.14) we find the entropy change of the network independent of the parameter b., i.e.

(7.17) The term rfi in/r 2olit accounts for the different end-to-end distances of chains in the network and detached fro.m the network. We have learnt in the previous section that .for an ideal rubber the internal energy e does not change with deformation at all. Hence, from the Legendre transformation (4.152) it follows .that for an .isothermal process the change in the Helmholtz freeenergy function '11 is ~ '11 = -8~r/. As a consequence of the Gaussian statistical theory of a molecular network, using the fundamental expression eq. (7 .17), we find finally that

\JI =

~Nk8 :fiin (,\i + ,\~ + ,\~ ....

3) .

(7.18)

r out

According to (6A) we have assumed that the free energy .is zero in the undeformed configuration. This important result .shows that, within the scope of the Gaussian statistical theory, the only quantities pertaining to the molecular network are the tota1 number of chains N contained .in the network (per unit volume) and rfiin/r 2 out· On comparison with relation (6.128) it emerges that (7.18) represents the s.impJe neo-Hookean model with the physical parameter /J,, known as the shear modulus, which :is proportional to the.concentration of network chains .N, given as ~

µ

= Nk8 ,,.2ro.;n 1

(7.19)

out

(see, in addition, FLORY [1956] or TRELOAR [1975, p. U4]). A simple .method for determining the number of chains N per unit volume is to measure the shear modulus µfor rubber. For high chain extensions the end-to-end distance r .is close to or .equal to the contour :length L. Therefore, condition (7 .2) cannot be satisfied anymore and the Gaus-

·1.2 Thcrmoelasticity or Macroscopic Networks

.319

sian statistical theory becomes increasingly inadequate for the finite strain domain . .In order to account for the finite extensibility of chains some significant refinements must be taken into account (SPERLING [1992, Section 9.10]). The more accurate nonGaussian statistical theory is required (see, for example, TRELOAR [1975, Chapter 6] and MARK and ERMAN [1988, Chapter 13]). Within the non-Gaussian statistical theory the finite extensibility of cha.ins is considered in the form of correction terms leading to a more realistic form of the distribution function which is valid over the whole range of r-values up to the maximum or fully extended length. One example of such a refined theory is based on the Langevin distribution function. The exact treatment of the freely jointed chain is considered by KUHN and GRUN l.1942], JAMES and GUTH [1943], and summarized by FLORY [1953, Appendix B of Chapter X]. In this type =af refined theory the Gaussian distribution function is included as a special case. Other examples are phenomenologically motivated and based on mathematical arguments (see the material models introduced .in Section 6.5).

EXAMPLE 7.2

At a given temperature E> we consider a thermodynamic process in

a closed system within some closed time interval t E "[O, T]~ in which the values 0 and T denQte the initial (reference) and the .final time, respectively. Assume that the closed I . system' is thermally isolated and conservative. During the thermodynamic process a unit cube of ideal (incompressible) rubber deforms homogeneously to a parallelepiped with sides of length /\i., .,\~ and /\:i = {A 1..\ 2 )- 1, i.e. the principal stretches. The cross.linked network of the rubber cube consists of N Gaussian chains per unit volume .. Based on the Gaussian statistica"J theory find the heat .g·enerate·d (or destroyed) .and the induced total entropy change of the .rubber biock due to the homogeneous .deformation. Specify the problem with the values A1 = 2, ,.\2 :;;:;: 3, .-\ 3 = 1/6 and 1V = 3.0 · 1021 m-:1• The temperature is assumed to be 8 = 293.15.K (= 20°C), where K denotes the 'Kelvin temperature' and °C the 'Celsius temperature'. The term =.

·1·8in/r 2out is equal to 1. .Solution. Since rubber is incompressible, the change in internaJ energy, i.e. £, during deformation is zero. Hence, the first law of thermodynamics (4 .. 122) reads

t-(t)

= Piut(t) + Q(t) = O

.

(7 .. 20)

The .expressions for the thermal power Q(t) = .f~ Rdll (the system is thermally 0 isolated, i.e. thermal energy ·can not enter Of leave .the boundary (QN = 0), no heat transfer) and the stress power Pint = D /Dt J~ 0 \lldll (the system is conservative) are adopted from (4. l 18) and (4.1.16), respectively. After integration over time interval t E (0, T] we find from the last equation (7 .20), by means of (7 .18), the particularized


320

first law of thermodynamics, i.e·. (7.21)

We used the fact that according to assumption (6.4) the strain-energy function wfor the unit cube of rubber vanishes in the reference configuration (normalization condition). The term ./~='{' Rdt in relation (7 .21) represents the heat per unit reference volume within the closed time interval t E (0., TJ (thermal work). All work done which appears as the strain energy is transformed to heat. The total entropy change of the network ~·77 induced by the thermodynamic process is, in accord with (7 . .17), given by

il·TJ = -

'11

1

?

e- = - ~iVk 7:oin ("\T + /\~ + ,,\~ ? .,.2 ""' out

3) .

(7.22)

By substituting the given values into -(7.21) and (7.22) we .find using Boltzmann's constant k = 1.38 · 10- 23 Nm/K that Rdt = -[3.0 · 1021 1.38 -10- 23 293~15 (4 + 9 + 2 1/36 - 3)]/2 = ·-60.85N /m • The negative sign means that energy in the form of .heat :is destroyed within the solid body. The entropy change gives 6.11 = -60 .. 85/293.15 = -0.208N/m2 K which shows dearly that entropy is decreasing as the rubber block is deformed. II

J

·EXERCISES

1. A drunken man starts to walk on a -flat field at a ·starting point 0. He makes one step per second each step of length 0.5 m. The path of the walk of ·Course meanders· randomly (the man is drunke·ri), which means that there is no correlation between the directions of successive steps.

By applying relation (7.3) compute the average distance r from point 0 he has .moved after three minutes.

2,. A rubber band of .initial cross-sectional area Ao is applied to a mass m,. At a certain temperature 8 the mass causes a 200% increase in length. Compute the number of chains JV per unit volume for the assumption that the material is modeled as neo-Hookean and rfiin = r 2oui (Boltzmann's constant

k

= 1.38 · 10-23 Nm/K).

3. Two rubber bands. A and B, with identical material compositions and length l are tied together at their ends. Then the assembled band is stretched up to a total

7.3

Thermodynamic Potentials

321

length of 6l and fixed at this pos.ition. By assuming that the rubber band A is at temperature 8 fl, and the rubber band Bat 8 11 , find the displacement of the knot at which the two bands are tied together.

7.3

Th·ermodynamic Potentials

To characterize continuous media within the context of thermodynamics we need to de.fine two material functions., namely (i) the thermodynamic potential characterizing all thermodynamic properties ·of a

system, and {ii) the heat flux vector describing heat transfer.

A thennodynamic potential is a function from which we may derive state variables characterizing a certain thermodynamic state of a system. In the following we define four com.men thermodynamic potentials. All of them are scalar-valued functions and assumed to be objective. In addition, the potentials are supposed to be at least twice ·differentiable with respect to all associated components. For a supplementary account of the relevant topic see the classical work .of TRUESDELL and TOUP.IN [l 960]; see also the texts by, for example, MALVERN .[1969], ZIEGLER [1983] and HAUPT [1993b].

Assoc.iated thermodynamic potentials. One example of a thermodynamic potential is the uniquely defined Helmlzo/tzfree-energyfunction \JI = W(F, 8) ~ measurecfper unit referen<;.e volume (in thermodynamics the Helmholtz free-energy function is frequently denoted by f or F). The value of the free ener:gy is determined by the changes of two independent variables, i.e. the defonnati.on gradient F and a non-mechanical variable given by the temperature e. In the following we consider homogeneous ·materials~ which means that the as~ sociated functions are independent of position in the medium. With the free energy which describes here non-isothermal thermoelastic processes., we may deduce direct~y physical expre.ssions from the Clausius-Planck form of the second law of thermodynamics (4.153). For all admissible thermoelastic processes the identity 'Dint = P : F - q, - 'IJG = 0 holds, which means the internal dissipation Vint is zero. By applying the chain rule, time differentiation of the free energy \JI (F, 8) gives the hypothetical change of the thermodynamic state. We obtain (7 .23)

322

7


which holds at every point of the ·continuum body .and for all times. As usuat the subscripts in (7.23) indicate variables that are being held constant during the partial differentiation of lJI. For convenience, in the following we will sometimes omit the subscripts. The coupled equation (7.23h is known as the Gibbs relation for elastic solids (Gibbs postulated the equation only for the case of a fluid). By comparing terms, we may evaluate physical expressions imposed on requirement (7.23) which must hold for any given (F., 8). Since F and can be chosen arb i trari Iy

e

_·p ··,.

= (OW(F,6)) DR .,

e

and

re= ,.

(

8\Jl(F, B))

ae

1:

'

-(7 .24)

which are the general forms of constitutive equations for the first Piola-Ki.rchhoff stress P and the entropy ·q describing thermoelastic materials. Note that for the case of any isothermal process (8 := const) the free-energy .function '11 .is identified with the isothermal strain-energy function (compare with eq. (6.1)). Consequently, tll(F) = w(F, 8) le==r.onst· By (7.24), the stress and the entropy are determined by the free energy '1!, which has the status of a potential for the stress, the entropy and their respective conjugate thermodynamic varjables. From physical expressions (7.24) we deduce the stress and entropy functions depending on the deformation gradient F and the temperature e, i.e.

P = J>(F,. 8).. , .

I/.=

-r1(F, 6} .

(7.25)

.·.··'·_.··.

Eqs. (7.24) and (7.25), .also known as thermal equations of s.tate, are crucial in specifying material behavior and are complete1y .determined once the free energy \JI = . . . ·\JI (F.1 f)) is given. Alt~~nativ~ constitutive equations for lhe stress and the entropy may be found by analogy with the treatment carried out in Section 6.1. The second Piola-Kirchhoff . . stress S follo~s from .relation (3.65h., by means of (7.24)l and the analogue of (6.1 l ), as

; : ;:

~ =·F-1P = F~i (Dlli~F,8)) ~·

DF

2 (D\Jl(C,8))

DC

B

'

(7.26)

.f)

where D'll(C, B)/DC is a symmetric tensor. From (7.24):h using the relation (6.9)t, we, obtain the alternative constitutive e.qualion for the entropy, Le.

_ -(D'Jl(C,8))

11 -

De

{727)

c.

In order to describe the thermodynamic state of a system by an alternative thermodynamic potential we require that the entropy function (7 . 25)2 is uniquely invertible with respect toe for each fixed F, so that we have loc.al.ly the condition 817/88 # 0.

7.3 Thermodynamic .Potentials We assume th.at the inversion of eq. (7.25h is given by 8 = 8(F, 17) and we postulate an associated thermodynamic .potential e, which is the internal-energy.function per unit reference volume introduced on p. 157. Knowing that the .internal energy is refated to the free energy \P through the Legendre transformation (4. l 52 ), we have the (canonical) representation

e

= e(F

1

= \JI (F, 8(F, 1"/)) + 8(F, 17 )'IJ

''I)

.

.

,

(7.28)

also known as the caloric equation of state. It is .an equation that. determines the inter. nal energy as a function of the deformatio·n gradient .F and a non-mechanical variable, i.e. the entropy TJ. With potential (7 .28) we may deduce fundamental physical expressions from the entropy principle based on the Clausius . . Planck inequality (4.141). For clll .admissible thermoelastic processes the second law of thermodynam.ics reduces to the identity Dint = P : F - + 8iJ = 0. Hence., the Gibbs r~lation is obtained by determin.ing the total rate of change of e = t!{F:, 17) and by use of the chain rule~ Thus,

e

.

"(F ) = p : F.t + e. - 17 = (De(Fi 8F -r1)) : F.. + (De(F, 8· . r1)) 1/. .,

e =· e . ,.,,.

1J

t/

(7.29)

F

whence, for arbitrary choices of F and i/, we have the physical expressions

P== (ae(F, 'f/)) DF

and

e = (De(F,17))· 017

11

(7.30) F

for the first Piola-Kirchhoff stress P .and the temperature 8. Note that for the case of an isentropic process (17 = const) there exists a state function e, whose partial derivative with respect to F gives the corresponding first Piola-Kirchho.ff stress P. Comparing the physical ·expression (6.l)t i.e. P = D\JJ(F)/DF, with eqs. (7.24) 1 and (7.30) 1 we recognize that the strain energy \JI = \Ji(F) serves as the free energy l{I (F)

= \JI (F, 8) le=mmst or as the internal energy e(F) = e(F, 11) l1i=const.~ depending on

the process considered, isothermal or isentropic. Knowing that e = e(F, 11), we deduce from (7.30) the .stress and temperllturejimctions which depend on ·F and 11. obtain the thermal equations of state in .the form

We

P = P(.F, r1) ,

e = 8(F, ·11)

•

(7 .3.1)

The two thermodynamic potentials '11 and e .introduced are commonly applied in solid meclumics. They are suitable for modeling so-called tbe.rmoel.astic materials (no 'memory effects' occur). For the sake ·of completeness two additional potentials .are reviewed briefly. These are the Gibbs free energy (or in the literature sometimes called the Gibbs function or che.mical potential), denoted by fl (or sometimes in the literature by G), .and the

324

7 Thermodynamics of .Materials

enthalpy, denoted by h (or sometimes by If). The two thermodynamic potentials g and hare used frequently in.fluid dynamics.

We .postulate that the Gibbs free energy g = g(P, E>) is a function of the first PiolaKirchhoff stress P and the temperature 8. Performing a Legendre transformation by analogy with (7 .28) we may express the enthalpy h by means of the thermal equation of state (7.3 l has

h = h(P, 'TJ)

= g(P, 8(F, 17)) + 8(F, 77)17

(7.32)

,

which is a function of the first Piola-Kirchhoff stress P and the entropy 17. Here, we have used two definitions of g and h which are associated with the free energy \JI and the internal energy e by the transformations g = g(P, 8)

= \JI -

P :F ,

h = h(P, 11.)

=e-

P: F .

(7.33)

In order to .find the Gibbs relations we determine the total rates of change of these two thermodynamic potentials by applying the chain rule, i.e.

. g

(8g(P, 8)) = g. (P' 8) .. = DP 8 I.i

= .,.l. (P , 17 I

)

p"

:.

= (Bh(P, 11)) . p· 8P .

+ (og.(P, ae ( 8h(P, 8· rJ

+

TJ

8)) e·- '

(7.34)

p

'f/)) r1. .

(7.35)

p

Using the second law .of thermodynamics in the forms of {4.153) and (4.141) (with 'Dint = 0) and the .material time derivatives of transformations (7.33) we arrive simply at g = g(P., 8) = -P : F - 778 and iz. = iz.(P, r1) == -P : F + 81]. Hence, by comparing with Gibbs relations (7 .34) and {7.35) we obtain expressions for the deformation gradient F~ the entropy 17 and the temperature 8. Thus.,

F=-(ag~~e)) F. = _

:··and a

(Dh(P,17))

DP

and

11=-(ag(P,e))

(7.36)

8 = (ah(P, ·11))

(7.37)

ae

8'17

.1]

.1)

p

The Gibbs free energy g and the enthalpy h have the status of a potential from which we may derive F, 17 and F, 8, respectively. In order to characterize the properties of a thermoelastic material, the considered

therm·odynamic potential must be supplemented by a suitable constitutive equation for the Pio la-Kirchhoff heat .flux Q, necessary to determine heat transfer. It may be introduced as a function of the deformation gradient, te.mperature and temperature gradient, . .i.e.

Q = Q(F, 8, Grad8) , ,•:·····'"''

··············

., .... ::... :....... .·.. ·: ,,. .. ·.. :,. ::.. , .. ·, .. , ., ........ ~·=

·~·:

"..

(7.38)

satisfying the dassical he.at conduction inequality (see relation (4.140), .i.e. the version

7.4

Calorimetry

325

.in the material description). For a more specific constitutive assertion see, for example, the phenomenological Duhamel's law of heat conduction (4.144) on p. 1.69 (given in terms of .material coordinates). A material for which the heat flux Q = o .and the heat source R = 0 vanish foi: any point and time is known as an adiabatic material. EXERCISES

1. Using constitutive equations (7 .24), (7 .30) and (7.36), (7 .3 7) obtain four relations combining .P, 17, F, 6 in the forms

BP) = - (811) (ae DF (88) (D"F) Dr1 ,.=-OP j'

1

and

(7.39)

9

oF )· ( 8·11 ) ( .88 I'= 8F 0

and 11

(7.40)

These identities are known as the the·rmodynamic Maxwell (or reciprocal) relations and are very valuable in thermodynamjc analysis.

2. Each of the four (most common) thermodynamic potentials introduced, i.e. '1t, e, g, h, is related to any other by a Legendre transformation.· Show tbat the identity

e ·- \JI - h + g

=0·

is satisfied.

7.4 Calorimetry Two centuries ago calorimetry became a branch of experimental physics. In the present day calorimetry deals with both the 1.neasurement of the amount of heat generated (or destroyed) within a give·n body duri_~g a change of state, and with its formulation with.in the theory of c·ontinuum thermodynamics . Spec-ific heat capacity and latent heat. Firstly, we .introduce the specific heat capacity at constant deformation (F = co.nst) per unit reference volume, which is usually denoted by c.--. It is the energy required to produce unit increase in the

temperature of a unit volume of the body keeping the defonnationfu:ed. The specific heat capacity eF = CF (F' e) > 0 for .all (F' e) is, in general, defined to be a positive function of the form CF=

.

cr(F,0) = -0

(a w(F,e)) 2

i)SDe

.F

>0 .

(7.41)

It is proportional to the second derivative of the free energy \JI. For a general compress-

3.26

7 Thermodynamics of M-nted~ds

ible material l* depends on the deformation gradient Fas well as on the temperature -E>. The positiveness of CF may be related to the stability of the material (see, for .example, SILHAVY [1997, Section 173]). Using (7.24)2 and the Legendre transfonnation w = e - 118., the specific heat capaci~y may be represented by the alternative convenient form CF=

CF(F, 8)

=

= 88rr(F,8)

ae

e.( +

Dr1(F,

ae

8))

.JI

8\Jl(F,e)

ae

. (F 8)

+ ri

'

= (De(F,8)) ae F

<7A 2)

Hence, the specific heat capacity at constant deformation cF may also be expressed through the internal energy e whicht in general, depends on F and 8. Secondly, we introduce the latent heat whkh is denoted by the symmetric tensor v . .It .is a ~pa tial field defined to be

v

== __ .a2 w(F, e) /I.~

e

OF08

_ _. _

F -

BF

(

a2 w(F, e) ) T 8FD8

(7.43)

·Of

Note that the latent heat v is proportional to the mixed second derivative of the free energy ·\JI.

Structural thermoelastic heating (or cooling). We define the general relation for the structural thermoelastic heating (or cooling) 1ie as the double contraction of the latent heat, as given in (7.43) 1, and the symmetric part -of the spatial velocity .gradient I = :FF- 1, i.e. the rate of deformation tensor d. Thus, with property (] . 95)

1 c_'I )

· -\...:/'

r;:t v: d = -e(J2W(F,e)FT: !(rF-1 + (FF-l)rr] ae 2 /.!::l · f-:;1;q,aF(F~er-·-:. --. [_~O.F'88 ·

:F

.

(7.44)

The scalar quantity 1ie represen ·st e thermoelastic coupling effect This so~called Gougl~-J~ule effect occurs, for ex~mple., during an adi_abatic stretching of a rubber band which typically changes its temperature., as pointed out in Section 7. I. In some p.robfoms the thermoelastic coupling effect is neglected due to the fact that this change -of temperature is. s-malL . . For a thermoelastic process (Vint. = 0) the rate of change of the ·entropy, as derived in (4. l42)~ -may be written by m·eans of (7.25h and (7.24h and the chain ru:le as

8) e. r1"(F . •, - .

= - o·ivQ + R.-= eD17(F,8) - ..aF : F. + e8.r1{F,8)h, - ae o a2 w(F, e) . a2 w(F, (-3) . ;:;: -e DFae : F - 8 aeae 8 '

(7.45)

7.4

Calorimetry

327

with DivQ and R denoting the material divergence .of the Piola-Kirchhoff heat flux Q and the heat source per unit time and per unit reference volume, respectively. Hence, from (7.45) we obtain finally the (coupled) energy balance equation in tempe·rature form (the local evolution of the temperature 8 appears explicitly). Using definitions (7.4 lh and (7 .44h we have cFe

= -DivQ -1ie + R

·.

or

cF8

= - ·aDQ.~t ~v·

- 1ic

.i\.j\

+ .R .

(7.46)

On comparison with the associated energy balance equation .in entropy form, that is eq. (4.142), we recognize that the structu.ral thermoelastic heating (or cooling) 1le appears on]y explicitly in the temperature form (7.46). Consider the case of a process during which DivQ vanishes. We deduce from (7 .46) that the heat source R per unit tim·e and per unit .reference volume is given by

R

= C1;-8 + 'He

,

(7.47)

which we may regard ns defining the theory of calorimetly. For a historical study see TRUESDELL [1980, Chapter 2C]~ Within the theory of finite thennoelasticity, in general, we observe three different types ofthermo~echanical coupling effects, nam·e1y

(i) the influence of .a change in temperature on the stress (thermal stress), (ii) structural thermoelastic" heating (or cooling) - Gough-Joule effect, eq. (7A4).,

.and (iii) geometric coupling (influence of a change in deformation on heat conduction)

(see, for example, eq& (4.144).).

EXERCISE

] . By me.ans of physical expres.~ion (7.36h, show an aHernative version of the .energy balance equation in temperature form (7.46), i.e. C1118

= -DivQ -

1icP

+R

or

. Cp8

=-

8Q-l -+ [)"'JlA

-1{.(!)>

+R

where Cp = c11(P, e) > 0 de.notes the specific heat capacity at constant stress (:P . canst), defined to be cp(P, 0) = -8(82 g(P, E>)/8888)h,· In words: ~!pis the energy .required to produce unit increase in the temperature of a unit volume of the body keeping the stress flxed. Alternatively to eq. (7 .44):i the term 1l(~ 1• represents the structural thermoelastic heatfog (or cooling) which is defined to be 11.cr == -e(a2 !J(P, 8)/DP88): i>.

7 Thermodynamics of ·Materia.ls

328

7.5 Isothermal, Isentropic Elasticity Tensors The following presentation of the isothennal and isentropic elasticity tensors is based on the concept introduced in Section 6~6. It .is an extension of the purely mechanical framework to thermodynamics by one thermal variable, i.e. the temperature e or its conjugate quantity, the entropy 'fJ. Isothermal elasticity tensor and stress-temperature t--:n~or. Suppose that a body admits the right Cauchy-Green tensorC = FTF and the temp~raiiii·e 8~Jndepend.eot ~ec_!!anica1 and .. anci. suppose the ·existence or the.fi.e1mh0Itz freeenergy func.tion in the form of \JI = \Jl(C, 8) ·(and equivalently \JI= w{F, 8)). Then, according to relation (7 .2<5)a, we can find the second .Piola-Kirchhoff stress tensor Sofa point at a certain ti-met, which may be seen as a nonlinear tensor-valued tensor function of the two variables C and 8. By analogy with Section 6.6, we now compute the change .in S. According to-considerations (1.247) and (l.248), we obtain [;,,.. ·-=·''.\;=·.. ~:· ('·- ?. . /·· the total differential

1hermaf:·variab1e·s

..

'l

d~ . ····: .i-

~\

= C: -de:~ Td8

~:---:::·:·-·:-···;"~..-...•,.... {t~~-:. . ~\.-~;:/ ,_.,:_,: ;:- :.: -..::.;:::·:::;-~·:·:;

(7.48)

t

.

which gives expressions for a· purely _mechanical part, C, and a (mixed} me-chanicalthermal part, T, in the material description. In the first term .in (7.48) we have .introduced the definition of the fourth-order tensor C, which is proportional to the second partial derivative of W with respect to C. By analogy with eqs . .(6.154) and (6.1.57) we write or

evaluated at (C,·8), with the major symmetries C =
___

.

or· ~

'""'""'----·~~-.-.,..,.-·--:-·-~--·-····

.

(7.50)

..

The spatial counterpart, denoted by- t," results via a standard push-forward (and Piola) transformation t = .1- 1x* (Ttt) of the (contravariant) referential stress-temperature

7.5

tensor T~

329

Isotbermal, lsentropic Elasticity Tensors

= 8S~ /88 by the· motion X·

By analogy ·with relation (3.66) we find, by

means of eq . (7.50)2, that ,;, t = J-lFTFT = 2J-1FCJ2W(C, 8)FT

acae

}. -~-=--

t -

or

;~_. -

(7.51)

. a2 w J-l li" T - 7-1 S~~-:,;~~:;~;:~~~ - · L'a.4I'bB acABae r;i

l;1

ri . - - -

->'.-'..

Note that the symbols T .and t have already been used and must not be confused with the traction vectors. We now express the symmetric spatial stress . . temperature tensor t in terms of the latent heat v' as de-fined in eq. (7.43 ). Knowing the transformation (aw (F' e) I 8F) T ::;: 2(8\J!(C, 8)/ BC)FT, which is in accord with eq. (6.11), we find from (7.51)2 that

t=i-182W{F,8)FT=i-1F.(a2w(F,8))T = __

aFae

aFae

i-1~1 e

'1i·

··-~--~-- ;

(7 . 52)

or Both tensor quantities T and t measure the change of the stress in a process in which the temperature is raised by one unit keeping the deformation fixed. ~ Finally we ~-ompute the change .in the fu-~cti~n f~r th~ entropy 'f} = 17-(C_, 8). ·with constitutive equation (7.24) 2 and the equivalence \Jl(F, 8) = w(C, 8), we obtain

..

d11 =

a2 w(c, e) 1 a2 w(c, e) -2 acae · : dc aeae de 2

.

(7.53)

By .applying the de.finitions of the referential stress-temperature tensor T and the cific heat capacity at.constant deformation as l

d·q

CF,

spe~.

the entropy change may be expressed Cf.

= -T : 2'dC + e d8

(7.54)

where the relations (7.50)2 and (7.4lh are to be used.

Isentropic ·elasticity tensor and stress.. en.tropy tensor. Consider a body which admits. the right Ca11Ch)'~Y!~~n. tensor c_. ~ FTF and the entropy ·~1 ~lS .independent .1nechanical and thermal variables and consider the existence of the internal-energy .function per unh refe~e vohuri.e in the f~rm of -e = e(C, 17) (and equivalently e =

e (F_, -rj)).

..

Theisen.tropic elasticity tensor in the material description or the referential tensor of isentropic elasticities, denoted by cise' .is derived from the internal energy e in the same way that the isothermal elasticity tensor is derived from the free energy W. Hence, by analogy ·with the above, the change in Sis given by

dS.

= Cise , : ~dC 2 . + rscdT/ .

(7.55)


330

.in which we have introduced the definition of the fourth-order tensor cise

= /lS(C, r1) = />2e(C, ·17)

-· a-c

or

· acac

=

cise

Aucn

4

a2\fl

ac.4.nDCcn

(7.56) J

evaluated at {C, TJ). The is-entropic elasticity tensor ciHe is defined by -~.eeping the en_Y:EPY_fixed duringJtt~_p_rpcE_ss. The second term in eq·. (7 .55) denotes the referential stress-entropy tensor Tise, which :is proportional to the mixed second partial deri_vative of e with respect to C and 1J. It is a symmetric .second-order tensor defined as

Tise= 8S(C, r1)

a -e ~,

= 2 Cfle(C, TJ).

or

T

iSl!

2

(7.57)

A.11 = ac ae · 8C817 .·\11 The isentropic elasti-city -tensor in the spatial description cisc = J- 1x* (cise) (or the OT}

spatial tensor oflsentropic clas"ticities) and the (contravari.ant) spatial stress-entro.:py tensor tise = J- 1 x*(Tise~) are derived from chm a~(! by analogy with the above. ln the above expressions, ~se, Tise and tisc are t(etfiert~ibpic quantities analogous to the isothermal quantities C, T and t. ~ ··:''··•-'·····"='" _ _ _ ....,,.----·- ....., ..

.

EXAMPLE 7.3

.

Obtain the ~undamentti~- ..f.~lati.Q.~ship

.

.·_ . . '. :· .· . .· ."· E>

. :"· :·.-.-·._ ·_

crne _-·c + -· T ·® T \ ·. CL" ..

·-~

(7.58)

I'

between the referential tensors· of isentropic and. isothermal elasticiti.es, with the specific heat capacity at constant deformation c1~ and the referential stress-temperatllre tensor T, as defined in eqs·. (7.41 h .and (7.50h, respective·ly. /, ..-Solution. lf we postulate the ex.istence of the Helmholtz free-energy function \JI = 'V(C, 8) we may derive the second Pio.la-Kirchhoff stress tensor S in the differential form according to (7.48 ).. As can be seen from relation (7 .48), changes in stress, dS, are associated with changes in both the deformation and the temperature. A thermodynamic process in ~hich the en~ropy r7 = ·11(C,·8) is ~o~stan_r. _(fixed), dTJ . O,.necessarily implies.a change in temperature. An explicit.expression may be :, . / -deduced from (7.54) in the form\, . . .

d8

= CF8 T: ~dC 2

.

(7.59)

This result substituted back into eq. (7.48) leads to 1 . dS = C: dc-~Td8

2

e ® T) = ( C + -T CF . . . . . . __~:·,;_:...... ··~~.!"'11...~i~;\ \

,

1 : :_de ,

2

(7.60)

7.5

Isothermal, lsenlropic Elasticity Tensors

331.

which furnishes the desired expression .for .the isentropic elasticity tensor in the material description for which the .entropy .is held constant during the thermodynamic process. This important relation expresses the isentropic elasticity tensor cisc in the material description as .a function of the free energy lp = w(C, 8) (compare with eqs. (7.49)2, (7.50h). Note thaUn terms· of the internal-energy function e = e(C, 17), relation (7 . 60) reduces to (7 .5~h. m ··=··: , .......... :, ,.,,,,: ......... ";' ,., .• : , ... ,.,_ ... ,,.: ........... ,, ,., ....._, ... : ··"':, ,, ''··· -··: -'••, ·:-, -·· ..... ·,.·,.:: ............... , '··- ,, ,, .• .,, .. ,._., ......., :,. ': , ... ,; ...... , .....,,, ......,,.' ·'·:: ....., .. ,, '•: ,_ ,,,-·,.: ........ ·.··'··"' ........... ,_ ............,, ........,., ..... ,_ .. ,, ... , ......... ,.,., .. , : , .......,.:,: ... ·: .., ... , ....... · ... ·: '·' ,, .. , .. ·: ··-·· .,_. __ , ..... : ., ,, ...•. : ,~

.,~

,, .

Some. numerical aspects..

The distinction between isothermal and isentropic -elasticities plays a crucial role in the analysis of nonlinear numerical stability.

Coupled thermomechanical p(oblems in solid mechanics may be solved numerically within one .time step. leading· .to simultaneous (monolithic) solutions of all the fields involved in the problem which have the feature of a good stability characteristic. However, this approach leads to large non-symmetric systems which are inefficient to solve and are associated with high computational cost. This typ~ of fundamental numerical solution strategy goes back to NICKELL and SACKMAN. [1968] .and ODEN [l 969, 1972]. Alternatively, the coupled system of nonUnear differential equations is often so"Jved using the classical staggered solution technique (or also known as the fractionalstep method or staggered method) {see, for example~ YANENKO I1971], MARCHUK [1982, and references therein]). In this method, the key idea is to partition the monolithic system of equations into smaller (symmetric) sub-systems by making use of the physical meaning of the problem considered. Within the concept of a staggered solution techri.ique.ithe system can be solved sequentially with much lower computational cost. For each ~sub-system ~~-~an apply existing algorithms and solution strategies. The dassical (merely standard) staggered solution technique for a coupled thermomechanical problem in solid mechanics is. based on the solution of a mec·hanical (isothermal) :pro~!~!Il at a.ft-red temperature of the system (elastodynamic phase), whiCh···· _., ,.,. .- ., .. . invol ves the isothermal elasticity tensor (7.49), followed by the solution of .a heat conduction problem at a fixed configuration in the temperature form (7 .46),...,.This---Cliissf:. ~'-··-...,--.-. ~-~c-

.........,,

cal ·p~ition is referred to as th~

--.isother.mal operator

-spl~t, which was used within the context of .coupled thermomechanical problems (see, for example, ARGYRIS et al. [1979, .1981, 1982], MIEHE [1~8.8] and SIMO and MIEHE (1992]) . However, this type ·Of staggered soluti~n technique is assoc~~ted wit~ the crocia.I restriction of ·Conditional stability (see ARMERO and SIMO .[1992, 1993])~

It e·merges that an alternative .partition of a strongly coupled thermomechanical problem leads .to a so-called unconditionally stable (time-stepping) solution -technique, characterized as indepe·ndent from the chosen time step. This technique allows solutions in an efficient numericaUy·accurate way. The analysis is based on the .solu-

332

7


tion of a l_!!~Ch!ln!.~12roblem .at a fi'ted entropy of the system {elastodynamic phase), which involves the isentrop.ic elasticity tensor (7 .56), followed by the solution of a heat conduction problem at a.fixed configuration (thermal phase) in the entropy,...,,.fOfm (4.142). This alternative methodology is referred to as the isentropic operator split when dissipative materials are "involved (damage, viscous or plastic effects may occur). Likewise for peifectly thermoelastic materials the split is referred to .as the adiabatic ope·r·ator ·split. Within this solution technique it is possible to show that a defined Lyapwzov functional for the coupfod system of evolution equations - regarded as the canonical free-energy .function for thermoelasticity, first introduced by DU.HEM [1911, Vol.2, ......... pp. 220-23 .I], is decreasing along the ft.ows for each. of the two sub-problems involved. This approach was proposed by ARMERO and SIMO [1992] for linear and nonlinear thermoelasticity and by ARMERO .and SIMO [1993] for finite thermoplasticity. For a successful application to rubber the.rmoelasticity see the papers by MIEHE [1995b] and HOLZAPFEL and SIMO [1996b]. It must be emphasized that this class of staggered solution technique can be applied not only to cou.pled therrnomechanical problems in solid mechanics but also .to coupled problems of, for example. fluid flow in a porous medium, magnetohydrodynamics in fluid mechanics or to stress-diffusion problems. All th.at must be· done is to replace the temperature 8 and the ·entropy ·11 by the associated field variables···of the COttpJed problem considered.

---

EXERCISES

1. Recall the spatial stress-temperature tensor (7 .5 I). By means of eq. (3 .66) derive

the alternative expression

Du{C,8) t ==

ae

or

where u denotes the symmetric Cauchy stress tensor. In addition, derive the spatia'l stress-entropy tensor which has the form tise = au{ c, 'T/) I OT/. 2. Suppose that a body admits the (spatial) left Cauchy-Green tensor b = FFT and the temperature e as .independent variables and consider a free-energy function in the form of w = w(b, 8). By analogy with (6.38) we may find the associated constitutive equation .in the form of T = 2b(Dw{b, 8)/8b), where r denotes the symmetric Kirchhoff stress tensor (note that this type of constitutive equation represents isotropic thermoelastic response on.ly). ~

Recall the kinematic relat~on (2.] 69) .and .obtain from the Oldroyd stress rate {5.59) of the Kirchhoff stress, using the chain rule,

.£\( r~) == .J( c: d + te) ,.

(7.6 l)

7.6

Entropic Elastic Materials

333

with the definitions

.

.a2 w{b, e)

Jt - .J - ...;

a2 w(·b ' 8) b -

?b - -

a2 w(h ' e)

(7.62) 8b.88 of the (fourth-order) isothermal elasticity tensor c in the spatial description and the (second~order) spatial stress-temperature tensor t.

Jc

= 4b

8b8b

b ,

8b88

For explicit.derivations of eqs. (7.61) and (7.62) see the work of MIEHE [l995b]. Therein, the definitions of the tensor variables c and .t exdude the factor J.

7.6 Entropic Elastic Materials. We consider so-cal.led entropic elastic materials, which have the property that the change in internal energy w~th deformation is. small or even zero (recall Section 7..l ). The underlying concept of entropic elastkity· is used particularly for the .thermomechan.ic.al description of rubber-like m. ~~~r!H,h~.,,. ~.uch.,.;as . .,.~la~lOJn.~rn. ,J~.ee, for exam-

7

1

r:~·8~~~;A~~~:~ [1~;~:~k~:~~ ~1.,.B:~~:w~~i~~~~~~~;ifi~b~~LLER

From experimental observations it is known that the bulk modulus for rubbe.r-like materials considerably exceeds the shear modulus. For an ideal rubber the internal energy e .is assumed to be a function of the temperature e alone, e = e(8), which is a typi~-~i~h~r~~t~~i;tic nf i~comp~~~~ib1e materiafs· (see~ for ex.ample, TRELOAR [1975, p. 34] and ANTHONY et al. [.1942] for more details). This assumption leads to the .purely (?.r. strictly) . entro.pic theor! of.D~P.~~r .th~rmoela~t~~ity (for a theoretkal treatment see the work of CHADWICK [ 1.974]). Consequently, the change .in internal energy with de.formation .at a. given reference temperature 8 0 is constant. We assume that

r,~~~(F)-:·0···7

c1.63)

l

:I

Here and elsewhere the subs·cripi'{•}cr:charactetizes quantities at a reference tempera-

ture 8 0 so that, for example, e0 (F) = e (F, Go). Allerna~.i~~·~.~?. . . ~.Jh.ermoelastjc material obeys the CQ.9dified entropic. theo~y if its internal energy e .is expressible as the sum of e(8), already known from the purely entropic theory, and the internal energy e0 ( J) = e( J, 8 0 ). The additional contribution en ( J) to the internal energy depends on the ·deiorffia~a.o·n only through the volume ratio J .at a given reference temperature 8 0 (for a theoretical treatment see the work of CHADWICK [1974] and CHADWICK and CREASY [1984]). Consequently, we conclude that

feo(;)··· ~a{:/) ·:~ I

For notational simplicity we ofle~

.. =-

..

ll~~;the sa~e ieiler for different functions.

(7.64)

334

7


Change in temperature, internal energy, entropy. For both the purely and the modified entropic theories of rubber thermoelasticity we may conclude from eq. (7.42).1 that the specific heat capacity CF is a function of the temperature 8 only. Thus, we write

CF(F, e)

= c(-8)

.

(7.65)

Having this in mind we assume a temperature -change

tJ

= e- eo

(7.66)

between a selected reference state with reference temperature 8 0 (the choice is quite arbitrary) and the current state with absolute tem_perature Hence7 by means of (7.65) the change in internal energy for a process· is determined by integrating eq. (7.42) 4 with respect to the temperature, i.e.

e._

e=e e(F, 8) - e0 (F) =

I

c(8)d8 .

(7.67)

Recall from eqs. (7.64) and (7.63) that e 0 (F) = e0 (J) for the ·modified entropic theory while e0 (F) is assumed to be zero for the purely en.tropic theory. Similarly, by means of (7.65), the -entropy change si~ply results from (7.42)2 by the integration e=e "'

77(F, 0) -11o(F) where ~0 (F)

= 11(F, 8 0 )

=

.!

(7.-68)

c(8): ,

0=0o denotes the entropy at a reference temperature 8

0•

General structure of the thermodynamic potential. Using eqs. (7.-67) and (7.68) a.nd the Legendre transformation \J!(F, 8) = e{F, 8) - 8r1(F, 8), the thermodynamic potential in the form of the uniquely defined free energy W may be expressed in terms of the internal -energy e0 , the entropy 1/o and additional function T for the purely

an

thermal contribution. Thus,

w(F, 8)

= eo(F) ·- 8r1o(F) + T(8) El=E>

T(8)

(7.69)

_.,

= - ...... ftf~@'(B - 8) d~ ,.,. .:.:.;. :· · e

.··········· ................. e::::::eo·."" ·····=······.·

,

.

(7.70)

.. ···"'.'"·'"'···· ····•·.···....... ,,,,_,,,....... ,,.~

Another common.ly used alternative form of the thermodynamic potential may be found by considering only state functions which are assumed to characterize the reference state at a given reference temperature 8 0 • These functions are interrefated by means of the Legendre transformation according to

Wo(F) = eo(F) - 8or/o(F) .

(7.71)

7.6

Entropic Elastic Materials

335

Consider the thermodynamic potential (7.69) and substitute for the entropy 7Jo (at a .given temperature 8 0 ) the expression which follows from transformation (7.71). Then, by means of the temperature change (7 .66), we arrive, after some simple algebra, .at the expression

;::-:,:;;,:;;_·~-J;~----···"· ··~· :-.,~·~~::~5?~--- -·1 .-. ·~···;····~· _t,
>.- >.:11L~Er ·, 0

,,~~-""''~)

(7.72)

:' . ~ .... :. . . . :-. . ·:· . . . . . . . . . : . .~!'.'.'..'-':~':::;~.~·~.·,~~;:.:.::.;:;·~;::;.;.·::.;,~·-~,. ,;?.--;.;. ~.;·:;.~~~·i·~.;..:.· . ~:i'~~"~>-•

which is due to cHAoW.ICK [1974]. Instead the. enfrop)'·7J~·(F) this alternative form uses the isothermal free energy \Jl 0 (F), which is the ·c".hange in strain energy for a deformation from the .reference configuration to the current configuration at a fixed (constant) reference temperature 8 0 • The .purely thennal contribution T(8) is given by eq. (7.70). Note that for a thennoelastic material· that obeys the modified ·entropic theory we require e0 (F) = e0 (J) in eqs. (7.69) and (7.72). However, within the purely entropic theory the characte.rization of a specific thermoelastic material is given basically by the entropy 170 (or the isothennal free energy \Jl 0 ) only, since e 0 (:F) = 0. The structure of the free-energy function w(F, 8) introduced in (7.69) and (7.72) is general in the sense that it may be used for the description of any entropic elastic material. A specification is accomplished with the choice of particular functions for the internal energy .e0 and the .entropy 170 (or the isothermal free energy '11 0 ) at a reference temperature 8 0 . In order to perform the integration (7.70) we need an expression for the specific heat capacity as a function of .the temperature. However, for some cases the specific heat capacity is assumed to be a positive coiistant over a given temperature range, and we write this .as c0 > 0. Then, the integrations in eqs. (7.67), (7.68) and (7.70) can be performed ·explicitly. The purely thermal contribution (7. 70), for example, takes on the standard form . ·: , .~·· •' .... : : .....

•... ,,, ·. .. ..

... ". ·.··" .- ' ~~--""' _,....,_...••'\.

' '

.~ "IT·(8)

[ ': = co . ·0 · :

·~

\ ,/,··- •···· -·-··· __

........... ··'

............ '" .....·. . ..

. .

81.n .

( 8 ) ] :~ _:_

eo

. . ..........•..·,.,......~.... '.,.,.,

\~~r

(7.73)

..,.,. ..__..

whkh will enable us to determine the thermal contribution T(8) for .incompressib.1e mate.rials with sufficient accuracy.

EXERCISES

I. An incompressib]e materia1 with a certain volume, ·constant speci fi.c heat capacity Co and with temperature 0 is thrown into a Very la~ge Jake with reference temperature 8 0 • After some time thermodynamic equilibrium is reached . Assume that the lake will .absorb all the heat rejected by the material without any change in its temperaturtr···\

336


Determine the change .in entropy for the material which ·changes its temperature by tJ = e - E>o. 2. Assume that a thermoelastic material obeys the pure·Iy -entropic theory {e0 (F) = 0). The change in the strain energy from the reference to the .current configuration at a given (constant) te-mpera.ture 8 0 is given by w0 (F). (.a) Recall eq . (7.44h and show that for this type of thermoelastic material the structural .thermoelastic heating (or cooling) 1ie is governed by the relation

.

e

fie= -\Jto(F)- .

(7.74)

·-o 8

(b) By adopting the energy balance equation (7.46) and eq. (7.74) show that for every adiabatic process (in which the heat tlux QN = -Q · N and

the heat source Rare zero for all points of the :material and for all times) the temperature evolution from the re.ference to the current -configuration is given explicitly by

e = 8oexp ( W-n(F)) . 8oco

(7.75)

where the strain energy vanishes in the reference configuration according to ~greement (6.4). For convenience assume that the specific heat capacity is a constant c0 • 3. Consider a thermoelastic material with a _given .isothermal free energy '11 0 (F) and a constant specific heat capacity c0 • Assume that the material obeys the purely entropic theory. (a) Recall the physical -expression .(7.:24.h and show that the evolution (change) of the entropy ·17 is given by

e

. Wo(F) r1(F,e) = - e-u +Coe .. . (b) Consider an is.entropic process in which the -entropy possessed by the given

thermoelastic material remains constant. Deduce that

.

co8

.

e

= Wo(F)8

,

:- 0

which gives the same temperature evolution as in (7. 75) . Interpret this result.

7.7

7.7

Thermodynamic Extension of Ogden's Material Model

337

Thermodynamic ·Extension of Ogden's Material Model

In this section we particularize .the general structure of the thermodynamic potential (7 .69) (or its equivalent form. (7. 72)) introduced previously. Since the bulk modulus for rubber. . like .materials greatly exceeds the shear modulus .it is most advantageous to employ the concept of decoupled (volumetric-isochoric) finite (hyper)elasticity, already introduced within the context of isothermal compressible hyperelasticity (see Section 6.4). This concept is based on a multiplicative .split of the deformation gradi·ent (o.r the corresponding right Cauchy-Green tensor) de-fined in eq. (6.79). Our approach is purely phenomenological, providing a set of constitutive equations appropriate for numerical realization using the finite element method. The basic idea of the constitutive model presented for the isotropic thermoelastic behavior of elastomeric'·(rubber-like) materials incorporating large strains is due to CHADWICK [] 974], while aspects for its computational implementation are addressed by MIEHE [1995b], HOLZAPFEL and SIMO {1.9.96b] and REESE and GOVINDJEE [1998b]. Structure of the Helmholtz free-energy function. A useful constitutive model for the isothermal and isotropic behavior of compressible (rubber-like) materials proposed by OGDEN fl972b] was presented in Section 6.5 on .p. 244. Very briefly we recal.1 Ogden's strain-energy function expressed in terms of the volume ratio J, the modified principal stretches Aa = J- 1/:i Aa, a = 1, 2, 3, and a given (fixed) reference temperature 8 0 (typically room temperature). The decoupled representntion of the strain-e~:ffi.t!~-~-~!lQ!}...9-"!.o...:::_Jf.() 1 , A2, ,\i, ~o) reads

.L~~o,,~-,.<~~Z·?:~~~. 1-·_!_isoO, J

(7.76)

where \Yvo)O == Wvo1(J, 80) and \}lisoO = wi~o(/\:i, ,,\21 .,,\3, 80) are assumed to be objective scalar-valued functions characterizing the volumetric elastic response and the isochoric elastic response of the hyperelastic mate.rial. By recalling (6.137)_1 and (6.139) we may specify the two response functions ·wvolO and Wison· Having iffmind the notation -introduced, we write..... ····:····1

t

,, .._......

.

... Wv11;{:;~~~ ~-~(_e_--·:;,~~.;) , .W;ijn(A1, ,\~:·,~:1, ~o) = w(,\a, 80) r \ w(,\ 80) = Jlp(Bo) (,\~'' - 1) a= 1, 2, 3 ----------··· _ 0

,_, ......... ·.·····

t

...•.·........................ ··,·'~--····1·.·· .....

./ L

(7.77)

a.=1

a1, .. .

················

......................... ···

.............

(7.78) .

c

............. _. __ -.·.·-············-)

In addition., we must satisfy the (consistency) condition N

2!10 =

L 11,, (
... '

11=:= l

with

"l\T 1) -- ·1 ~ ... ,.n

(7.79)


338

where the para.meter µo denotes the shear modulus in the reference configuration at 80. The strain energy '11 vol ( J, 8 0 ) is associated with the volumetric elastic response and is, in general 1 decomposed into .a bulk modulus Ii( 8 0 ) at a fixed reference temperature 0 0 and a scalar-valued scalar function Q{J). One example for Q was introduced in eq. (6.137h. The strain energy Wiso{A 1 , A~h /\ 3 , 8 0 ), however, is associated with the isochoric elastic behavior in the space of principal directions of C, i.e. the ValanisLandel hypothesis {see VA LAN.IS and LAND EL [1967]). The parameters .Jl.p(H0 ) denote the (constant) shear moduli at 0 0 ·and lt11 are dimensionless constants, TJ = 1; .... , N. Next, we derive a relatively simple but very efficient thennodynamic potential for rubber-like materials which is basically a thermodynamic extension of Ogden,s model. We employ the modified entropic theory with assumption (7 .64). Applying the thermodynamic .potential .in the form of (7.72) and using the strain energy (7.76) together with relations (7.77) and (7 . 78), we obtain the non-isothermal free energy \JI for .isotropic thermoelastic material response, i.e.

(7.80)

This decoupled structure "is based on the .definitions

a

wiso = L

w( ,,~, e.)

, c1.-s 1>

a=l

fL

= 1, 2, .3

,

(7.82)

(7.83)

. = 1-ip(80)e

µ 1,(8)

-

8 ()

,

11

= 1, ... , N

(7.84)

,

and on the condition (7.79) . .In (7.81}1 the purely thermal contribution T(8) is given in eq. (7.70). The first contribution \II vot( J, 8) to the thermoefastic response defined by (7 .8 l )i is due to volume changes and .purely thermal causes. The bulk modulus n.( e) in (7.81) 1 depends linearly on the absolute temperature 8 (see eq. (7 ..83)). Note that the energetic contribwion. e0 ( J) to the function \JI vor( J, 8) occurs only within a modified entropic theory. An empirical expression was proposed by CH~DWICK [ 1974] and has the form ·--~.

(en (J) \Go '

)_

r

,. _

.

3cY~_K(8o)9(J)

(7.85)

with

.:

"•,.,~,-':·

This relationship is based upon experimental considerations., where ')'

itive non-dimensional parameter and the quantity n 0 = n{00 )

>

0 -is a posdenotes the so-called

1.·1


339

linear ex·pans·ion coefficient relative to a selected reference state with reference temperature 8 0 • The empirical response function Q( J), which is extensively studied by WOOD [1964) and CHADWICK [.1974], is able to fit experimental data which are obtained from isothermal compression tests performed with different temperature values. For an instructive example -on the .basis of "Y = 1 and the neo-Hookean model the reader is .referred to the paper by OGDEN [l992b, Example l] in which the symbol a 0 is used for the volume coefficient of thermal expansion. Observe that within a purely entrop.ic theory the energetic contribution e0 ( J) vanishes (recall Section 7 .6). Consequently, for this case the stress is proportional to the absolute t-emperature El, since

W = Wo(8/8o) + T. The second contri but.ion wiso ( ,,\ 1_, A2 , ;\a, 8) to the thermoelastic response defined by (7 .81 h is due to isochoric deformations. Here, we consider .a representation of Wiso in terms of the modified principal stretches .\ 1 , ci = 1, 2, 3 . The shear moduli. µp(8)., p = 1, ... , 1V, in (7.82) depend linearly on the absolute temperature e (see eq. (7 ..84)). A physical interpretation of this fact was presented within the cont~~t ~f Gauss-ian statistical theory of molecular networks which is valid for the re.gion of small strains (compare with relation (7.19)),, The thermodynamic potential (7.80) describes the stress-strain-temperature behav·ior of rubber-like materials within the finite strain domain. As for the isothermal case, the thermodynamic extension of the Mooney-Rivlin model and the neo-Hookean model r~sults from (7.82) by setting N = 2, n 1 = 2, a2 ;;:: -2 and N = 1, l.t 1 = 2, respectively. ·Observe that for an :isothermal deformation process (8 = 8 0 ) the second and third term in (7 .81_) 1 vanish, the material parameters (7. 83) and (7. 84) change into constants, and consequently the free-energy function \JI = \JI (Xi, ,.\ 2 , /\ 3 , 8) changes _into the ·strain-energy function ll1 0 = \Jl{)q, ,\ 2 , .A:1, 8 0 ), as presented by eqs. (7.76), (7.77) and (7.78).

Consistent Iinearization. Subsequently, we point out .the consistent line.a.rization ---. . ._....,.___ process of the ..ihermodynamic potential given by (7.80). In particular~ as a first step, we compute the thermoelastic stress response, characterized by the second Piola-Kirclzlzo.lf stress tensor S, followed by a second step which determines the isothermal elasticity tensor C in the material description and the referential stress-temperature tensor T. The formulation is presented exclusively within the concept of spectral decomposition and is .characterized by a geometric setting relative to the reference configuration~ In order to -deduce the stress tensor s = 28'11 (.,\ 1' A2_, A:1, 8) I ac we follow the procedure as shown in Example 6.7 on p. 245. By means of the decomposed structure (7 .80) we may find the purely volumetric and purely isoclzoric stress contributions S-()q, A2, "\a, 8) = Svol + Si 50 , which are defined to be -~·...a.

(7.86)

7 · Thermodynamics or· Materials

340

(7.87)

for the general case .A1 # .,\ 2 :j:. A3 i= A1 (see the analogues of eqs. (6.140)i, (6.143)). The constitutive equation for the hydrostatic pressure p, essential for relation (7 .86h, may be specified in terms .of the free energy (7 ;81 )1 as ]J

- 8lll,,.01(J, e) - -(8)dQ(J) - deo(J) !!___ .. 8.J - ~ dJ dJ 80 '

(7.88)

where the term dQ(J)/dJ was particularized in eq. (6.141) 1 or (6.141.) 2 depending on whether the scalar-valued function (6.137h or (6.138) is used. In addition, with free energy (7.81)2 (and (7.82)) we may compute the three principal isochoric stress functions Sisoa, a = 1, 2, 3, in the form

a=l,2,3,

(7 ..89)

which are needed for (7 .87) 2 (compare with the derivation which led to eqs. (6.144)2 and (6.145)).

The derived set of expressions (7 .86)-(7 .89) completely defines the constitutive model for rubber-like materials, allowing thermoelastic deformations with strain changes unrestricted in magnitude. It is a straightforward thermodynamic extension of Ogden's mode.I known from the isothermal regime, i.e. (6.140), (6.14.l) and (6.143)(6.145). Alternative stress measures follow directly from eqs. (7 .86) and (7 .87) by means of suitable transformations. For exam.pie, the stress response expressed by the Cauchy stress tensor u simply results from a push-forward (and Piola) transformation u = J- 1 x*(S~) = J- 1 FSFT of s. As a second step in the consistent linearization process, we compute the change in the stress tensor S, i.e. dS = C : dC/2 + Td8, with the definitions of the isothermal elasticity tensor C in the material description, i.e. (7.49), and the referential stresstemperature tensor T, i.e. (7.50) . .Based on the deco~posed structure of the derived stress response (7 .86) .and (7 .87), we may obtain the decoupled representation ,•.. ·=·=·-...

C .. >

·t.·.Cv~~\)+ Ciso >./

\ : : ; : : : ·..

.

~~ '''''.';':.·.:~ .-:·:.·,'~··'

and

T = Tvol

+ Tiso

(7.90) ~·.

for the tensors C and T, w·here the first expression represents the familiar additive split

7.7

341


of the fourth-order (isothermal) elasticity tensor C (compare with the considerations on p. 254 and subsequently). The second expression consists of second-order tensors only. Analogously to (7.90)i, it is composed of a pure1y volumetric contribution Tvol and a purely isochoric contribution Tjso· The explicit farms of -~,~he isothermal elasticity tensors Cvol and Ciso are adopted from isothermal finite elas~icity and are based on eqs. (6.166)~1 and (6 . .196), respectively. We bear in mind that the underlying free energies ·wvol and W;so depend on both the three principal stretches ,,\ 0 = J 1/ 3 /\a, a = 1, 2, 3, and the temperature 8. The isothermal elasticity tensor Cvoh as given by eq. (6.166).h requires the c~msti tutive equation for the hydrostatic pressure p .(which is given in (7.88)) and the scalar .function 73. With reference to specification (7 .88), we obtain finally from (6.167) the explicit form JJ

= K(S)

(dQ(J) clJ

2

+ 1 d Q(J)) _ (deo(J) + /Pe0 (J)) 31_ . -dJdJ

dJ

cL/clJ

80

(7.91)

Considering the isothermal elasticity tensor Ciso ·in the spectral form, as _given by eq. (6.196), we have just to take care .of the coefficients 1 asiso11

Siso li -

and

A11 a.Ab

,\i -

Siso a

A·~

_,

(7 ~92)

which depend on the .material mode.I in question. The three values Sisoa' a ~ 1, .2, 3, denote the principal values of the second Piola-Kirchhoff stress tensor Siso and are given by eq. (7.89h. Hence, the second term in (7.92) is a"lready determined. In order to determine the first term recall the closed form solution (6.197) and just consider that the shear moduli Jlp, p = .1, ... _, JV., .are temperature dependent according to relation (7.84). Finally we compute the referential stress-temperature tensor T defined by {7 .50) 1 . By recalling the constitutive equations (7.86h and (7 .87h we may compute the purely volumetric and purely isochoric .contributions to the referential stress-temperature tensor. They are defined in the sense. that

T vol

DSvoJ

= ae

/

= JpC

-:I

(7.93)

,

i . ··..

asiso Tiso= ae ·,

:1

/: ·"

,.

~ = L...,, SisoaNa ®Na

(7.94)

a.=1

the values p' and Sfsoa' a = 1, 2, 3, depend on the material model in question. They are the derivatives of the hydrostatic pressure p and the three principal isochoric stress functions Siso 4 , a = 1, 2, 3, with respect to the te·mperature e.

for .X 1

#

,,\~ -:/=

Aa

=f Ai, where

342

·7

Thermodynamics ·Of Materials

From (7 .8·8) 2 and (7 ..89h we find, using the relations for the bulk modulus (7 .83) and the shear moduli (7.84), which are temperature dependent quantities, that I

Qj)

11=-=

1L{0o) dQ(J)

ae

! Sasoa

= 8Sisoa

ae·

_

eo

dJ

Sis9 a

-

deo(J) 1 dJ eo a

e

= 1, 2, 3

' .

(7.95)

(7.96)

;•

Observe the s.imilar structure of"the referential stress-temperature tensors (7. 93) and (7 ~ 94) to the volumetric-isochoric stress response, as defined in relations (7. 86) .and (7 .87). The spatial counterparts c and t of the tensors defined in (7 .90) result via a standard push-forward (and Piola) transformation.

Heat conduction. The considered thennoelastiC problem, for which (Vint = 0).. is governed essentially by Cauchy ,s .first equation of motion (see, for example, the local fonns (4.53) or (4.63)) and by the balance of (mechanical and thermal) energy in entropy or temperature form (see ·eq. {7.45) 1 or (7.46)). Hence, in regard to the energy balance equation we need an additional constitutive equation for the heat ·flux vector governing heat transfer. One example which satisfies the heat conduction inequality .is Duhainel's law of heat conduction (see the considerations on p. 1.70) . For the dass of .thermally isotropic materials we may express the constitutive equation as

Q{C, 8, Grad8)

= -k0 (8)C- 1G:rad8

(7.97)

(i.e. Fourier's Jaw of heat conduction, which should be compared with eq. (4.148)), where ·Q is the Piola-Kirchhoff heat flux and k0 > 0 denotes the coefficient of thermal conductivity associated with the reference configuration. Note that this coefficient is, in general, not a constant. In fact, for vu Icanized elastomers, ko decreases linearly with increasing temperature (see .SI.RCA R and WELLS [ 1981 ]) according to

ko(8)

;:=

ko{8o)[l -

~(8 -

80)] ,

(7.98)

w·here k0 (8 0 ) denotes the coefficient ·Of thermal ·conductivity at the reference temper.ature 8 0 and ~ is a softening parameter. The solution of the coupled thermomechanical problem may be performed by adopting the staggered solution technique. Within a time step this technique leads to a decomposition of the coupled problem (compare with the statements on p. 331). As a result we must solve two smaller, in general., symmetric decoupled sub-problems on a staggered bas.is. For algorithmic aspects of the entropic theory of rubber thermoelasticity see, for example, M.IEHE [1995b] and HOLZAPFEL and SIMO [l996b] .

7.8

S·imple Tension of Entropic Elastic Materi.als

343

EXERCISES

1. Consider the scalar~valued function Q(J) 13- 2.((:JlnJ + J-IJ - 1) accord~ ing .to eq. (6.137h (due to Ogden) and the energetic contribution e0 (J)/8 0 = 3o:0 K( 8 0 )1- 1 ( J 1 - 1), with/ > 0, according to eq. (7.85) (due to Chadwick)~ Use (7 .88h, (7 .9]) and the relation for the bulk modulus (7 .83) in order to obtain the constitutive equations

which completely determine the isothermal elasticity tensor Cval in the material description.

2. Consider the extension of the isothermal Ogden material to the non-isothermal domain (7 ..80)-(7.84) (including the quantity e0 (J) which represents an energetic (volumetric) contribution to the free energy). Recall the definition of the structural thermoelastic heating (or cooling) 1lc~ Le. eq. (7.44h, and show that 1le may be written in the decoupled structure of the form He = He vot + He iso, with the de-finitions

The response functions \JlvoJ o given by (7.77) with (7.78).

=

Wvol (J, 80),

Wisoo

= '11-iso(/\1, .,,\2, /\3, 80)

are

The analogue of the decoupled structure of 1ie was derived by MIEHE fl 995b] . .In his work the structural thermoelastic heating 1le is, however, based on the multiplicative split of the (spatial) left Cauchy-Green tensor b = FFT and is defined with the opposite sign.

7.8 Simple Tension of Entropic Elastic Materials The aim of this section is to illustrate the ability and performance of Ogden's model for the non-isothermal do.main as outlined in the last section. We set up the basic equations required to describe the realistic physical stress-strain-temperature response of rubberlike materials. In particular, here we consider the simple tension of an entropic elastic

344

7 Thermodynamics of Materials·

materiat a class of material defined in Section 7 .6. A re.presentative example conce.med with the adiabatic stretching of a rubber band will contribute to a deeper insight in the coupled thermomechanical phenomena.

Thermoelastic deformation. Before we start our studies with the simple tension of entropic elastic mate.rials it is most beneficial to point out briefly some aspects of the therm-oelastk deformation of a continuum body. Consider a fixed reference configuration of a body with the geometrical region 0 0 correspondin.g to a fixed reference time t = 0. The position of a typical point may be identified by the position vector X (with material coordinates {..Y 1, .X2 , .Xa)) relative to a fixed origin 0 (see Figure 7.5). The reference configuration is assumed to be stress.free and possesses a homogeneous (uniform) reference temperature value 8 0 ( > o.). A map of the reference configuration 0 0 to a current configuration (with the new region) f2 .is characterized by the macroscopic motion x = x(X, t) for all X E Oo and for all times t. The motion carries a typical point .X ·E Oo to a point x E which is characterized by the spatial coordinates (:r 1 , :1: 2 , :r:3 ). As a measure of the thermoelastic deformation we use the deformation gradient F(X, t) and the volume ratio J{X, t) = detF(X, t) > -0. Very often it is convenient to decompose the (local) motion {X, t) = Xivl [Xe (X, t)] into two successive motions XM and Xe according to

n

x

F = Ox(X, t) = FMFe

J = Jr:..rJe > 0 ,

and

ax

(7.99)

with the definitions F ~·I

- BxM(Xe, t) -

axe

F _ Dxe(X,t) 8 DX .,

'

Jr-,.;1

Je

= detF:r\;1 > 0 = detFe > 0

1

.

(7.100)

(7.101)

The multiplicative decomposition (7.99) separates the total thermoelastic deformation into a purely mechanical contribution F ~·i, J~.1 and a purely thermal contribution Fe, J8 , which represents the Duhamel-Neumann hypothesis for the nonlinear theory (see, for example, CARLSON [1972, p. 310]). The two successive motions establish a new intermediate (i.magined) configuration with geometrical region Q 0 , as illustrated"in Figure 7.5. The new configuration is assumed to be isolated from the body so that a thennal stress-ji·ee deformation may occur. Hence, a relative temperature field .f) = 8 - 8 0 causes a (free) thermal expansion (or contraction) about the reference configuration 0 0 characterized by the associated variables Fe and J0 . The intermediate configuration with the region Q 8 is .given by the macroscopic motion Xe = 9 {X, t) which carries points X located at -0 0 to points

x

7.8

Simple Tension of Entropic Elastic Materials

Intennediate configuration

..

,

_.......... .

1'

.......,...~~

:

,,

'x

, 1' ,,,,

\ t•

I

.'\

I'

0

I

'

I

'

ne __,-,,

......

345

I ,,,

I

-----·-

Current

configuration

Reference configuration

Fe,Je

Fivt, JM

F,J

x time

f

t= 0

.

time t )t

/

X1, :c 1

e2

e.1.

Figure 7.5 Multiplicative decomposition of the thermoelastic deformation into a purely me .. clumical contribution Fr...i, JM = detFrv·r and a purely thermal contribution Fe, Je = detFe.

.

.

Xe in the intermediate configuration He. A typical point Xe is characterized by the coordinates (..-Ye i, )(02, ..rYe3).

According to the multiplicative-split (7 .99) we ·have defined, additionally, a macroscopic motion x = XM (Xe, t) at a constant (fixed) temperature e along with the stre~fs producing deformation gradient F:rvt and the volume change JM. A so-called mechanicaHy incompressible material for which JM = detF~.1 = 1, keeps the volume constant during a motion XM . .

~·':""

•:·'·• '···-·-

-·-···-· .... ,,.. ,,.,.,,.....

,: .,

....... ,. -·· ·-····,.,... , .,,,_•.. ,:' ,._,_, ............ , ., ....... ,...... _,,,,.,,,: .. ,,,,.. .... ,,...... -.· -······· ...... , .........

EXAMPLE 7.4

,_~

,,.~:~

.. ,,.... ,_.;, ,.............. .,,, ....... ,.,.,.: ...........: ........ ,_ .. ,_, ....... ,, ,... ,_, ...... , ..... , .....

-....... ,..... ,..... '--:,• ··:·''•"''-'·''''·"''" '·'····, ....... , ··- .,., ... ,..• ,..,,. __ ,_,_ ......, ·-, -

Consider a mechanically incompressible and thermoelastic material under a non-isothermal deformation process. The thermoelastic material is. assumed to be thermally :isotropic so that the deformation gradient F 9 may be given by an isotropic

7 · Thermodynamics of Materials

346·

tensor according to

e

Fe= F(8)1 ,

F(6) =exp[/ a(0) 0 ,

(7.102)

80

where F(E>) is a scalar-valued scalar function determining the volume change relative to the reference configuration. In eq.(7.102h we have assumed a particularization of F(8), where o: = 0:(8).denotes the temperature dependent expansion coefficient. Detennine an expression for the total volume change J of the ~aterial due to the non-isothermal deformation process. Linearize the result by assuming a constant value n 0 = a(8 0 ) for the expansion coeffic.ient at a reference temperature 8 0 .

Solution. Since the considered thermoelastic material is mechanically incompressible (no volume change during an isothermal process) we introduce the constraint condition throu.gh J~.1 = L Thus, the total volume change J within a non-isothermal deformation process from region no ton is characterized by eq. (7.9.9h which degenerates7 using eqs. (7.101 h and (7 .102), to e

J

= Je(8) = detFe =exp[/ 3a(0)d0] > 0

.

(7 . .103)

E>o

With the linear expansion coefficient ao~ relation (7 .103)3 gives

J = exp[3no{8 - 80)] ,

(7 . .104)

and linearization leads to· the approximate solution .1~1+30:0(8-80).

(7.105.)

It defines the total volume change J of a mechanically inc~mpressible and thermally. .isotropic material within the infinitesimal strain theory. The approximate solution (7 .105) is a well-known relation in linear continuum mechanics. It may be viewed as the volume of a unit cube .at temperature e with values .n0 and 8 0 which correspond to a reference state. •

Entropic elasticity for a st.retched piece of rubber. In the following, attention will be ~onfined t~ the the~m.oelasti.c description of isotropic and entropic elastic materials (such as elastomers) at finite st.rains. The stress-de.formation-temperature response of a piece of rubber under simple tension, in particular, is examined. We assume that the material under consideration obeys the modified entropic theory of rubber thermoelasticity. .Furthermore, the material is assumed to be mechanically incompressible,

7.8

Si·mple Tension of Entropic Elastic Materials

~47

which .motivates the use of a multiplicative split of the thermoelastic deformation, ·as introduced above. Consequently, the total volume change is given in (7.103)i, i.e·. J = Je(8) (detF1v1 = 1). · Now we consider a thin sheet of rubber stretched in one direction from the reference (undeformed) state to A.1 = .-\ (s.imple tension). Then, obeying condition J = J 0 (8) · : A1 ,,\ 2 A3 , we deduce by symmefry that ,,\ 2 = ,\1 = (J/A) 112 • Hence, the Helmholtz free-energy function per unit reference volume

(7.106) is given in terms of one independent mechanical variable and one thermal variable, i.e. the stretch ratio~\ and the temperature 8. For notational convenience we use the same Greek letter \JI for different free-energy functions. Since the material is isotropic it is appropriate to use the thermodynamic extension of Ogden's ·model, .as discusse.d in Section 7.7 (see eqs. (7.80)-(7.84)) . Recall that, in general, one contribution to the free energy is due to volume changes and purely thermal causes (comp.are with relation (7.8.1 h). Since we study a mec·hanically incompressible material we need only to consider the purely thermal contribution. We use, without loss of generality, the standard form (7. 73}, i.e. T (e) = Co [19 - eln (eI 80)]' where the specific heat capacity c0 > 0 is a positive constant. Hence, the free energy relative .to the reference configuration, which is stress-free and free of thermal expansion (or contraction), results from eqs. (7.80)-(7.82) and (7. 84 ), using the temperature change {) = 80 and the specified kinematic relations, in the form

e-

\¥(,\, e) = :

t, Jlp~~o) [-'a). + Gf 2

0

+co

l

2 1 3

[(e -Bo) - 0ln (~J J

(7.107)

where the additional condition (7.79) must be enforced. Note that the volume change due to thermal expansion governed by .J = Je (8) is of considerable importance, which will be pointed out in more detail within the Example 7.5 below. · ··.Next, we derive the associated thermal equations of state, namely the stress and eiuropy functions, as given in eqs. (7.24) and (7 . 25). For simple tension we may write

p = 8\¥(,\, E>) 8A

8'11(A, 8).

and

1J = -

ae

'

(7.108)

with given free energy \JI substituted from (7. I.07)~

. (.J) . :. . a,,/21

"Ao,_, -

[

.,\

(7.109)

348

17 = -

7

1

00

(8o) [

~µ.Pap N

)..

0

P

Thermodynamks of Materials

(J)o:,,/'l-.. 3i· +coin ( 8

+ (2 +3aaE>o:p) . ),,

00

) 4

(7.110_)

Relations (7 . ·109) and (7 .110) specify the (non-zero) nominal stress P1 = P (also called the first Piola-Kirchhoff stress) P2 = P3 = 0, and the entropy 'I} whic.h are generated by the stretch ratio...-\ and the temperature 8. In order to describe the coupled thermoinechanical problem of the piece of stretched rubber completely we must add a constitutive equation for the heat flux vector governing heat transfer. For the class of thermally .isotropic materials we may adopt Fourier's law of heat conduction and refer to eq. (7 .97). Having in mind the specified kinematic relations, the inverse of the right Cauchy-Green tensor, c- 1, as needed in eq. (7.97), may be given in the form of its matrix representation

(7.111)

where the diagonal elements are the eigenvalues of_ c- 1 .

EXAMPLE 7.5 Consider the stretching of a mechanically incompressible piece of rubber, for example a rubber band, obeying the modified entropic theory of rubber thermoelasticity. The rubber band is elongated rapidly so that no ti-me remains for isothermal removal of heat. Hence, the homogeneous deformation process is viewed as an adiabatic process for which the heat flux on the boundary surface is zero and, .in addition, heat sources are zero (thermal energy cannot be generated or destroyed within the material). The non-isotherma] deformation process is assumed to be reversible . .In the present example attention .is paid to the effects of structural thermoelastic heating (or cooling) and to the stress-strain-temperature response of the rubber band. In particular, show how the nominal stress P depends on the temperature change 19 = G - 8 0 at a fixed elon_gation, .i.e. a fixed stretch .-\,.and derive the temperature -evolution of the rubber band with stretch. Finally, discuss the results of this classical example of rubber thermoelasticity, which demonstrates one of the great differences betwe-en rubber and 'hard' solids, nam-ely the distinctive effects of temperature~ As known from Section 6.5 the Ogden model exce11ently rep.licates the finite extensibility dom_ain of rubber-like materials. Hence, as a basis for the con~_titutive model take the thermodynamic extension of Ogden's model with three pairs of constants (1V = 3) characterized by the Helmholtz free-energy function (7.107). The constants O:p, Jip(G 0 ), p = 1, 2_, 3, at the reference temperature 8 0 are those given by OGDEN [1972a], listed in eq. (6.121) of this .text. The volume change due to thermal expansion

7.8 Simple Tension of Entropic Elastic Materials

349

(or contraction) J = J 9 {8) is assumed to be governed by relation (7.104). The specific heat capacity and the linear expansion coefficient are given by c0 = 1.83 · 103Nm/kgK a~d -a0 = 22.333 · 10-5 1{- 1 (se.e CYR [1988]), respectively. We assume that for the reference state the specimen and its environment constitute a system, which is in thermodynamic equilibrium. The homogeneous temperature field is given by the value = 293.15K (= 20°C).

e·o

Solution. The relationship between the nominal stress P and the temperature change l) = e - 80 is given explicitly by the theoretical solution (7.109), with volume change (7.104), and by experimental results due to ANTHONY et a.L [1942]. The respective diagram for various fixed values of/\ is depicted in F.igu~~ 7.6. The figure is supplemented by a curve indicating the evolution of the temperature change iJ with respect to the stretch ratio ,.\. Since the adiabatic process is assumed to be reversible the energy balance equation (4.142) reduces to 81) = 0. Consequently, the entropy (r/ = canst) possessed by the ·material remains constant, which indicates that the deformation process is also js~ntropic. The condition rJ = .co.nst gives the relation between the temperature change 19 and the stretch ,.\, which is based upon the constitutive equation for the entropy, as derived in eq . (7. ll 0), with eq. (7 ..l 04). The analytical solution is completed by experimental data due to JOULE [1859]. The qualitative fit to the experimental data given by JOU.LE [1859] and ANTHONY et al. [.1942] is satisfying; nevertheless, the physical properties of the vulcanized rubber strip used in Joule's and Anthony's experiments .are only partly documented and may differ from those given above. One of the very first works reporting simple observations regarding therma] effects due to deflections of vulcanized rubber bands was published by GOUGH [1805]. The :first work which explored experimentally the crucial physical properties of elastic .india-rubber was .presented by JOULE [1859]. The phenomena of structural thermoelastic heating (or cooling) of stressed rubber-Hke materials is called the Gough-Joule effect. A wide range of.publications followed over more than one century, which mostly deal with theoretical formulations ·of more or less approximate .theories. The history of these investigations is discussed in the classical book by FLORY U953, Section XL "I a], an overview is given by TRELOAR [1975, Chapters 2, 5], CHADWICK [1974] and PRICE [.1976, and .references therein]. For a detailed .account of .the relevant results s·~ also ·OGDEN [1992b]. Numerical .analyses for this type of adiabatic (isentropic) simple tension test within the finite ele·ment context were given by MIEHE [1995b] and HOLZAPFEL and SIMO [l 996b]. The algorithmic .solution procedure reduces .merely to one step, which is ~.o~cerned

with solving a mechanical problem with lsentr~pic elasticities as in~i~ated e?CpliCitly through expression (7.58). During the solution process the entropy c~ns:~raint condition 81} = 0 .must be enforced, which means that the entropy at each time-step

350

7 Thermodynamks of Materials

0.08 8 -

0.07

Theoretical solution, see eq. (7. 109) with (7 .104) - - Experimental data by Amlumy et al.

,.-, C'l

E

..........

z

6

i::O

+0

0.06

-

'---'

~

,,\ = 1.72

----- --A= 1.38

4

ti]

VJ

CJ

1-.

~

~

c

0.-05

·9 0

____ _______ -- __- - -________ - - - - -

2

.Z

0

......

I JI c;::,

0.04

.....,...

;\ .= 1.13

...,,,,,._

.__..

© ©

;\ ;:: 1.22

_..------ -- -- -- ---

d

,........, ~ L.......I

.......

~

._.

,,\ :;:: 1.03

0

~

60

30

0

90

Temperature change 1? = E> - 00 [KJ

00

c::

~

..c:

u ....v

-....

0.03

~

d

C1)

0.

Theoretical solution, s·ee eq. (7. l lO) for .T/ = consf~. with (7. I04)

E

~

0.02 ~

Experimental data by Joule

0.01

0.00 .A = I. I 257

-0.01

LO

1.1

Thermoelastic inversion point at .,\inv = 1.0616

1.2

1.3

1.4

1.5

1.6

Stretch A

Figure 7.6 Temperature rise of a rubber band due to adiabatic stretching - showing the ther.. moelastic inversion point - and .nominal stress P versus temperature change 19 = 8 - 8 0 for

various stretch ratios .A for a thrce. . t-erm Ogden elastic material.

7.8

Simple.Tension o.f Entropic Elastic Materials

351

is unchanged. As a direct result the temperature and the displacement field follow according to Figure 7 .6.

Thermoelastic inve.rs·ion point.

For a small stretch ratio .,\ the rubber band indic.ates an initial cool.ing effect (see Figure 7 .-6), which increases first with deformation an.d . changes to a heating effect at a certain . minimum point, which entered .into the literature as the so-called thermoelastic inversion point. The associated extension, 1.abe~ed as Aim.. , characterizes a configuration of the sample in which the acting force is i~-_dependent of the temperature. · · · To gain more insight in the interesting thermoelastic inversion phenomena the fo]lowing observations are emphasized: (i) An analytical investigation of the reversible adiabatic (ise.ntropic) process - characterized by constitutive relation (7 .110), with -r7 = const -- leads, within an adequate linearization process1 to an explicit expression for the stretch at the "inversion point, denoted by Atnv· Specifically, from ae I 8~\ 0 we obtain the approximate solution

(7.112.) (see JAMES and GUTH [1943] among many others). With given data thermoelastic inversion takes place for.an extension of about Ainv = 1.0616 at which the maximum temperature drop occurs (see Fi.gure 7.6). A second point of interest is at ·1? = 0, where a heating process takes place. Nate that this behavior is contrary to that for a metallic spring, which cools continuous Iy on stretching within the ·elastic domain . An approximate solution for A at ·tJ = 0 may be de.rived from eq. (7.110) using an adequate linearization process. We obtain a quadratic equation in.-\ i.e. ,,\ 2 +.A- 2~\~.• v ~ 0 (see also JAMES and GUTH [ 1943]). Remarkably, the stretch .ratio at iJ = 0 is only influenced by the location of the thennoelastic inversion point; with the given data the nu.merical value is/\ :;:: 1.1257 (see Figure 7.6).

(ii) In view of eq . (7 .109) it is crucial to remark that .for a fixed A the stress is a nonlinear function with respect to the temperature. The nonlinearity is clearly caused by the volume thennal expansion term J = J 6 (8) characterized by eq. (7.104). As a direct consequence the term Je governs the change in the slope oP/80 of the stress-temperature curves, as seen in Figure 7 .6. If the elongation is small the slope is negative and the defonnation behavior is dominated by thermal ex·pansion (energetic contribution). For larger strain ranges the slope DP/88 becomes positive. It emerges that the reversal in the slope of the weak non.linear stresstemperature plot .is indicated at ,,\ = /\inv., which can be shown using eq. (7 ..109) along with condition 8P/D8 = 0.

352

7 Thermodynamics of ·Materials

At the thermoelastic inversion point thermal expansion and entropy contraction balance. Hence, thermoelastic inversion is governed clearly by thermal .expansion, which is :expressed by re]ation (7.104), Le. J = exp[3a 0 (8 - -8 0 )]. As can be seen from (7 .112}, thermoelastic inversion depends basically on the linear expansion coefficient n 0 :/; 0. Alternatively, .it is interesting to note that the .inversion pojnt also occurs at the same value of extension by fixing the load of a strip of rubber and increasing the temperature. A connection of the mentioned observations may be found within the context of the thermodynamic Maxwell relations (7 .39) and (7.40), which motivates the inclusion of the two diagrams within one figure (Figure 7 .6) (see also the theoretical study by OGDEN [ l 992b]).

By setting a 0 = 0 (which gives e0 (J) = 0) we may recover the purely entropic theory of rubber thermoelasticity as a special case. With the assumed constant specific heat capacity c0 the temperature evolution is then given by an explicit function, .Le. 8 = 8 0 exp[W 0 /(8 0c0 )] (compare with Exercis·e 2 on p. 336). Here, '11 0 = w(/\, 8 0 ) denotes the strain energy for the deformation at a fixed • reference temperature 8 0., i.e. eq. (7.107) withe= 8.0 . ... ..

."··:,,., , ,.. ,_,.,,,,.,, .•.

,_.~,,,,,,,:,.,

__

~~·,:,,,,,

. ,_,... ,.. ,,.,,..,.,., .. ,.,.,.., ... ,..... ,.,,,,_,,.,.,,..,.,, .... ,.. , .. ,,,,,,_ .... ,,,,,_,_ ....• ,.......,.,,.,.,,,,.,,,,,,. :,·····,···,•:••,,,.,.,.,.,.... ,,,..... ,.,,,,, ...

,,

.... ,...,, ... ,....

........ , .... ,,_.:,.,,,.;, ...... ,.·: ... , ..,, ........ .. ,_, .... ,, ....... ·.. ,_..:,.,.,, ............ , ..... ,..... ,.,.: .. , ...... .

:,

,

One-dimensional finite tbermoelasticity. We consider here the one-dimensional case of finite thermoelasticity. A .rod (with uniform cross-section) is imagined as being stretched to .X in the direction of its (1;- )axis, with the associated kinematic .relation :c = A.X (uniform non-isothermal deformation). The rod capable of supporting finite thermoelastic deformations admits the stretch ratio A and the temperature 8 as inde .. pendent m.echanical and thermal variables. From the Helmholtz free-energy function \JI = w(.A, 8) (measured per unit reference volume), which is assumed to ·exist, we are able to deduce the stress and entropy functions .P = P-(,.\, 8) and r1 = r1(/\, 8), respectively. They are nonlinear scalar. . valued functions which de.pend on the two scalar variables . \. and 8 and have the same form as (7.1.08). Note that the -constitutive relation (7.108).J. determines the nominal Stress p by keeping the temperature 8 fixed While (7.108h determines the entropy T} at a fixed stretch . \. . For .a more profound understanding of the constitutive relations it is beneficial to note the differential mathematical relationship between the nominal stress P and the . entropy 'T/. For the continuous function '11 the well-known property

a (Dw) D/\

88

9

a (aw) 88 ,\

= .a~\

(7.l.13)

holds. Using .e-qs. (7. I 08) 1 and (7 .108) 2 , (7.113) can be rewritten in an ·expression

7.8 Sim.pie Tension of Entropic Elastic Materials

353

which relates the stress and the entropy function according to

(~~)~ = - (::~t

(7.114)

This identity shows that the change in stress P with the temperature 8 of, for exam:.ple, a rod at fixed stretch .,\ is equal .to the negative value of the change in entropy ··17 with the stretch ratio Aof that rod at fixed temperature 8. Relation (7. l 14) characterizes the thermodynamic MaxweH relation .for the one-dimensional case, fundamental in rubber thermoelasticity (see WALL f 1965, p. 314], TRELOAR [ 1975, p. 30] and CARLSON [1972, p. 304]). Compare also the identities (7 .39) and (7.40), valid for the threedimens-ional case. Furthermore, it is important to note that for .polymers in the 'rubbery' state the term on the left-hand side of eq. (7.114) is found to be positive at large extensions A, and negative at s·maU extensions, as seen cleafly in Figure 7.6 (.recall Example 7.5 on p. 348). All that remains is the computation of the isothermal elasticity tensor in the material description (consistent linearized tangent moduli) and the -referential stress-temperature tensor. With this aim in view we derive the change in the second Piola-Kirchhoff stress 1 S = S(,\, 8), defined as S = p = 1\- 1 (0\JI(.,\, 8)/8...\) (compare with relations ·(6.50), with (6.47), val.id for three dimensions). Knowing that the Green-Lagrange strain, denoted by E, is (.,\ 2 - 1) /2 (compare with relation (2.67}, valid for three dime.nsions), we may find

x-

dS

where dE =

"\d~\.

= (~~t dH

(~~)A d0

= CdE+·Td8

,

(7.115)

Here we have introduced .the definitions

and

T

= 88(;\, 8)

ae

(7.J 16)

.c

of the isothermal elasticity tensor in the material description (fixed temperature during the process) and the referential stress-temperature tensor T. The scalar-valued functions (7 .115h and (7 .1.16) are the one-dim.ensional counterparts -of the tensor. . valued functions .(7.48) and (7.49)i, (7.50)i, respectively. Observe that by ·means .of S = "\ - t P and the thermodynamic Max well relation (7 .114) we may find an equivalent of the referential stress-temperature tensor (7. l .16)2 in the form

AT= (aP) ae

= _ (8·q) A

DA e

(7.11.7)

In order to complete our presentation of one-dimensional finite thermoe.lasticity we

354


have to add a constitutive equation for the heat flux. ·For one dimension, Fourier's law of heat conduction in a coupled thennomechanical regime reads

Q.(A,8,8')

= -k0 (8).,\- 2 8'

.

(7..118)

The temperature gradient along the axis of the considered rod is denoted by 8' while Q denotes the Piola-Kirchhoff heat ·flux, which is, in the one-dimensional case, a scalarvalued function. ~

\

EXERCISES

<_ 1J. Consider the Helmholtz free-energy function \JI = \JI( ..\, 0) in terms of the } stretch ratio >. and the temperature 0 characterizing a one-dime1fi.o1zal consti/

crios)

tutive problem of finite thermoelasticity. According to relations we may derive constitutive equations for the nominal stress P = P{A, 8) and the entropy '1J = 11(A, ·8). Assume that the specific heat-capacity is a positive constant c0 > 0. (a) Using the chain rule, show that the change in entropy may be expressed as

d-rJ

= -.TdE + Cue d8

,

(7.119)

where dE = Ad,,\. The referential stress-temperature tensor ·T is given in eq. (7 .116)2 (or by the equivalent of eq. (7 . .117)). (b) Using eqs. (7 .115)2 and (7. l.19) show that the isentropic elasticity tensor cise in the material description (for a fixed entropy 1/ during a process) is governed by the relationship cisc

=C+ 0T2 .

(7.120_)

Co

Note that relations (7.119), (7.120) are the one-dimenional counterparts of -relations (7.54), {7.58). 2. Suppose that a rod (considered as a one-dimensional structure) admits the stretch ratio /\ and the entropy r1 as independent variables and consider the existence of the internal-energy function per unit reference volume in the form of e ·= e(,,\, r1). Derive the second Piola-Kircnhoff stress S = S(.-\, ·17) and show that its change is dS = CisedE + Tised17, with the definitions

C ise = /\·\-1DS(A,17) 8.A '

Tise

= 8S(>\, 1/_) 8TJ

of the isent1vpic .elasticity tensor cise in the material description and the referential stress-entropy tensor Tii&c, evaluated at (A, 'l}) (compare with Section 7 .5 for the three-dimensional case, in particular, with relations (7 .56}1 and (7 .57) 1_).

Simple Tension of Entropic Elastic Materials

7.8

3S5.

3. Consider an adiabatic (isentropic) stretching of a mechanically incompressible rubber band and study the homogeneous deformation process in the large strain domain up to .a stretch ratio /\ = 8. Take the material properties and assumptions according :to Example 7 .5 (see p. 348). As a basis for the ·constitutive model take the thermodynamic extension of Ogden's model, with N = 3, and compare with the coupled thermomechanical M·ooney-Rivlin model, by setting 1V = 2, and the neo-Hookean and Varga models, by setting JV= 1.

~

~ L.....o

Ogden model

8

········••••4•

Mooney-Rivlin model neo.. Hookean model ----..... Varga model

·-----·-

0

© I

6

(j)

u ~

o:"""

~

c..

e

,

;

••

~

-... .. , .. "'"' ... -·~···"

4

~

~

~

,•

# ... .# ••

::::J ....., C"l

"""' ~

,..•.·· .. ,-, .... _.,....·· #

2

-- - -~::···

~·:c···

.. -- -

~

~..

•'

-0 1

2

3

4

5

6

7

8

Stretch . \.

Figure 7.7 Temperature rise ·{) of a rubber band due to adiabatic stretching for the 'large strain .domain. Comparison between four different thermoelastic models.

~J

E ..._

............... -------

6

z

Non-isothermal

c

+ 0

Isolhermal ( E>

= 00)

5

6

0 1

2

3

4

7

8

Stretch.,\

Figure 7'"8 Nominal stress P versus stretch ratio ,,\. Comparison between a coupled thermomcchanical (non-isothermal) and a decoupled (isothermal) defonnation proc·ess.

3S6

7 Thermodynamics of Materials In particular, for 1V = 2, assume a 1 = 2., a 2 = -.2 and µ 1 (8.0 ) = 0.875µ 0 , µ 2 (8.0 ) = -0.125µ 0 , so that Jl 1 (8 0 )/tt2 (8 0 ) = -7 (see ANAND [1986]), with the shear .modulus JLo = 4.2.25 . 105 N /m 2 in the reference ·Con.figuration. For the neo-Hookean model (1V = l) assume a 1 = 2 and /ti (8.0 ) = JLo and for the Varga model (J.V = 1), n1 = 1 andµ .. (80) = 2µ 0 .

(a) Based -on the Ogden, Mooney-Rivlin, neo-Hookean and Varga material models derive the temperature ·evolution tJ = e - -Go of the rubber band due to the stretch ratio /\.

(b) Derive a relationship between the nominal stress P and the stretch ratio A for a non-isothermal deformation process. Show the difference compared with the (classical) solution, which is based on the isothermal theory (set 8 = 8 0 ). Note that for an isothermal deformation process the free energy \JI reduces to the strain energy, as given in (6 ..119) and eqs. (6.1.27)-(6..129). Figures 7.7 and 7.8 show a comparison between the thermoelastic Ogden, Mooney-Rivlin, neo-Hookean and Varga .material .models. Observe the sharp upturn in the stress at high elongations ("\ > 6) for the Ogden material model as indicated in Figure 7 .8. This fact may partly be explained physically within more advanced theories obeying Langevin distribution function (non . . Gaussian statistical theory),. as discussed by, for .example, TRELOAR [1975., Chapter 6]. The observed rise of stress is mainly caused by limited extensibility of the polymer chains themselves and by strain-induced crystallization (see FLORY f.l 976] for further insi.ght). 4. A bicycle wheel with spokes made of .rubber bands is mounted with the axle horizontal. The spokes are in tension so that the rim .is kept in place. An electric heat plate is placed on the right-hand side of the whe.el (see Figure 7.9) so that the spokes on one side of the wheel are heated. As a consequence, the bicycle wheel starts to rotate counterclockwise, as lon.g as heat is induced. Explain this effect. This amusing device may be viewed as a Carnot thermal engine, in which rubber alone constitutes the working substance. The Carnot thermal engine is alternately subjected to two adiabatic and two isothermal processes. Investigate a hypothetic.al thermodynamic Carnot cycle for rubber-like .materials. In particular, discuss the temperature change fJ of a rubber band which occurs during the four processes as a function of the stretch A and the ·entropy 17. For a detaiJed exposition of the relevant results the reader is referred to the paper by HOLZAPFEL

and

SIMO

.[l 996b].

7.9

Thermodynamics with Internal Variables

357

Wheel ··· Electric heat plate

Spokes made of .rubber

Figure 7!'9 A wheel with rubber spokes starts lo side.

ro~ale

counterclockwise when heated on one

7.9 Thermodynam,ics with Internal Variables In this section we link together finite elasticity and non-equilibrium thermodynamics. We consider the thermodynamics of continuous media within the large strain regime and apply the theory of .finite .thermoviscoelasticity.. We use a thermodynamic approach with internal variables which leads to a very general description of materials involving irreversible (dissipative) effects, such as damage, relaxation and/or creep and plastic deformations. It generalizes finite thermoelastic.ity, as outlined in Section 7 .3, in the sense that additional thermodynamic variables {known as internal variables) are incorporated with the aim of representing .the irreversible mechanism of the (inelastic) structural material behavior. A _general discussion of constitutive models with internal variables was -emphasized in Section 6 . 9. A fully coupled three-dimensional thermomechanical model for viscous materials is examined. It is .particularly suited for the thermoviscoelastic behavior of dissipative elastomeric (rubber-like) materials under varying temperatures at finite strains. Constitutive equations for the stress, the entropy and the internal variables are specified. Finite thermoviscoelasticity. We define a Helmholtz free-energy function (measured per unit reference volume) as (7..121)

The .thermodynamic state is completely characterized through the set of independent ~m.), i.e. the deformation gradient F., the absolute temperature variables (F, -8,

ei, ... ,

35"8

Thermodynami-cs of Materials

7

e

n = 1, ... , 1n. For the case of thermoviscoe.Jasticity the tensor variables ea- represent the thermov.iscoelastic contribution to the material response. Note that the variables may also represent damage and/or plllstic mechanisms. A material which is characterized by the free energy {7 .12 l) for any point and time we ca.II a ,thermoviscoelastic material. The behavior of a thermoviscoelastic material is assumed to be governed by a· = l_, ... , ·m. relaxation (retardation) processes with given relaxation (retardation) times r 0 (8) E (O_, oo ), a: = 1, ... , ni, which are, in general, temperature dependent. The next aim is to derive the complete set of the constitutive equations for the first Piola-Kirchhoff stress tensor P and the entropy 1J (per unit reference volume) -in the general form. For that purpose we follow the standard methods presented in Sections 6.9, 7.3 and use the Clausius-Planck form of the second law of thermodynamics (4.153), Le.. the internal dissipation ·inequality Vint = P : F - W- ·r18 > 0 (the Clausius-Planck form assumes non-negative entropy production due to conduction of heat, i.e. -(1/8)Q ·Grade > 0 (compare with Section 4.6)). By .means of the chain rule, time differentiation of the free energy \}I (F., 0, i.' ... , 111 ) gives the hypothetical change in the thermodynamic state., that is the Gibbs relati-on for thermoviscoelastic materials, namely

8 and

rn additional internal variables

0

,

en

e

\;., _ ,i, (F ., ¥ ¥

e /:

e

c ) -- p .· F. _ ·n8. _ .rn. ·1 Vmt

''itt,···,~m

~ ( 8"\J! (F, 8 _,

e

1 , ... ,

ac ·

+ a:=I ~

~ m) )

~o

ea,

.~ F,e

·~a '

(7..122)

where (j' = 1, ... ' m., denote the internal strain rates. Since the rates F and can be chosen arbitrarily we find the constitutive equations

P=

e

·(aw(F,e,e1, ... ,e"')) 8F

, e,eo

for the first Piola-Kirchhoff stress P and the entropy 17, and we deduce a remainder inequality m

Dint=

L.sa:: ea·> 0

'

(7.1.24)

o=l

The inequality Vint > 0 characterizes the internal dissipation in the viscous material which generates heat in an irreversible manner. The defined tensor variables .80:, ft = 1, ... , m., correspond to the internal tensor

7.·9


359

en

variables according to the internal constitutive equations (7 .J 24h, which is the ther.:. modynamic extension of relation (6.233) (or (6.244)) . .By analogy with a linear solid, 8 0 , a = 1, ... , m., are to be interpreted .as (internal) non-equilibrium stresses (compare with Section 6.10). From physical expressions (7.123) and (7.124.h we deduce functions for the first .Piola-Kirchhoff stress P, the entropy 17 and the internal (stress-like) variables S 0 which depend on F, 8, 1, ..• , Thus,

e

em·

(7.1.25) a= 1., ... , -rn .

(7.126)

The fundamental inequality (7 ..124 )1 , which characterizes internal dissipation_, ·must be satis"fied by a suitable set of evo"lution ·equations for the internal strain rates C'.t = 1, ... , ni, described generally in the f orn1

ea,

a=l, ... ,ni.

(7.127)

The equations of evolution (rate equations) (7.127) describe the way .in which an irre-

·versible process evo:Jves. Note that the non-equilibrium stresses characterize the current 'distance from equilibrium' and vanish at the state of thermodynamic equilibrium. In view of (7.1.27) we 'em) = 0, a = .1, ... ,-rn, as time t goes to may write the equations A~(F, e, infinity; further, Bn = -8\I! I lt~oo 0, a = 1, ... , 1Tl. This implies that with .reference to (7.124) 1 the internal dissipation vanishes (Vint = 0). Then equilibrium is "reached and the values for stress and entropy remain constant They are governed by the potential re-lations as derived in Section 7.3 (see eqs. (7.24)). The limiting case of thermodynamic equilibrium .states that the thermodynamic process is reversible and the continuum respondsjitlly .thermoelastically. All associated ~t~ermodynamic potentials, as outlined in Section 7.3 for finite thermoelasticity, are approximated asymptotically. .· :.. In sum-mary.: the response of a thermoviscoelastic material is defined through the :~~~stitutive equations (7.123) and (7.124)2 (or (7.125) and (7.126)), the internal dissipation (7.124) 1 and the evolution equations as outlined generally in (7.127). In addition, these equations are supplemented by a suitable constitutive equation for the heat ·flux vector, necessary to determine heat transfer. The Piola-Kirchhoff heat -Hux Q ._rr,i<:iy be introduced as a function of the deformation gradient, temperature, temperature gradient and internal variables, .i.e.

e1' ... aeo =

(7.12.8) ·and must satisfy the inequality Q · Grade

< 0.


360

Structural thermov-iscoelastic heating -(or cooling). The specific heat capacity CF at constant deformation per unit reference volume was introduced and discussed in Section 7.4. Within .the theory of thermodynamics with internal variables we define the specific heat capacity to have a positive value which depends .on the deformation gradient F, the temperature field ·E> and, additionally, on the internal variables 0 , a = 1, ... , ni. Thus, we write

e

(7.129)

e

for all (F, 8, 1, ••• , em). Here the speci fie heat capacity is the energy required to produce unit increase in the temperature of a unit volume of the body keeping the deformation and the intern.al variables fixed. For notational convenience we shall use the same symbol cF for the specific heat capacity introduced ·in Section 7.4. Recall that within the theory of finite thermoelasticity we observe, in general, three different thermomechanical coupling .effects (see p. 327). In addition, finite thermoviscoelasticity incorporates viscous dissipation according to (7.124) 1 and structural thermoviscoelastic heating {or .cooling), denoted 1iin and defined as v. TLm

m. 82\Tr(F = -8'""

L

¥

o=l

8 ''i.11· ~ c ) .. ,'!;m.

ac ae

'.

• ~

. ~o

(7~130) '!

'!ta

For the case in which the quantities E'n, a = 1, ... , rn, .represent plastic contributions, eq. (7. l 30) defines structural inelastic (plasti.c) heating (or cooling). By analogy with the derivation which led to the energy balance equation in temperature form (7.46) we proceed now by determining the change in ·entropy. With the equations of state (7 .125) 2 and (7 .123 h and by me.ans of the .chain rule we deduce that

.

.a.,, .

'f/(F, 8,,et, ... 'en.) = OF : F +

8-71 .

~ 817

ae 8 + L

.a=l

ae :ea &

O:

(7.13.1) (the .arguments of the functions have been omitted for simplicity). By multiplying this equation with the temperature and using eqs. (7.44)a, (7.129h and (7.130), we find that

e

(7.132)

On ·comparison with relation (4.142) we obtain finally the energy balance equation in temperatureform, i.e. C1:8

= -DivQ +Vint -

'He - 'Hin+ R .

(7.133)

From the energy balance equation (7 A6) we know that, within the theory of finite

7.9

Thermodynamics with Internal Varfables

361

thermoelastic.ity, the evolution of the temperature e is influenced by the material divergence DivQ of.the Piola-Kirchhoff heat flux ·Q, the structural thermoelastic heating 1-l.e and the heat source R. However, relation (7 .133) indicates that due to viscous effects the quantity c_Fe depends additionally on the internal dissipation :Dint and the structural thermoviscoelastic heating 1ihu us defined in eqs. (7 .124) and (7 .130), and this is particularly important ·to the thermomechanical behavior of viscous materials.

A constitutive model for finite thermov-iscoelasticity.

Many materials which bep~ve elasticaily at ordinary (room) temperatures display pronounced inelas~ic char:acteristics at elevated temperatures, solid polymers being important examples. T~e ~:olecular network of vulcanized. rubber exhibits (nearly) no stress relaxation in the low temperature range of the 'rubbery' state. However, in the temperature range of 100 t_q ~50°C stress relaxation-experiments of vulcanized rubber at constant deformation _(s-~e TOBOLSKY et al. [.1944]) indicate a rapid stress-decay which may_ be explained b_y chemica·1 rupture ~f the three-dimensional network. The phenomenon of stress re.l~;
·and is ·:

m.

(7.134) n=l

valid .for some closed time interval t E [O_, T] of interest. ·we require that m

\JI oo(l, 80)

=0

_,

2:Ta(I,8u,I) = 0 , o=l

(7.135)

362

7 The.rmodyna.mks of .Materials

where 8.0 (> 0) is a given homogeneous reference te.mperature relative to a selected stress-free reference configuration. The set of independent variables ( C, 8, r 1, ... , rm), i.e. the (symmetric) right Cauchy-Green tensor C, the absolute tem:Perature 8 and the (symmetric) internal variables r ()') o: = 1, ... , ni (not accessible to direct observation), completely characterizes the thermodynamic state. The internal variables re~ are considered as inelastic (viscous) strains akin to the strain measure C. The first term 00 (C, e) in (7 .1.34) ·characterizes the equilibrium state of the solid. We employ the subscript ( • )00 to designate fonctions which represent the hyperelastic behavior of sufficiently slow processes. An efficient free energy 'II 00 describing the stress-strain-temperature response of rubber-like materials at finite strains, which is based on the concept -of entropic elasticity, .may be adopted from Section 7. 7 (see eq. (7.80)). The second term Ta(C, n) in (7.134) represents the configurational free energy {'dissipative' potential) and characterizes the non-equilibrium state of the solid (relaxation and/or cre-ep behavior). The potential Y 0 has to satisfy the thermodynamic restrictions imposed on the second law of thermodynamics (namely the non .. negativeness of the internal dissipation) for any thennodynamic process. For a general form of Y 0 and a detailed discussion of this issue the reader is referred to HOLZAPFEL and SIMO [1996c]. Following arguments analogous to those which led from (7. l 21) to relations (7. '123) and (7.124) we find, using the property (6.l 1) and S == F- 1p, physical expressions for the (symmetric) second Pio.la-Kirchhoff stress tensor S and the entropy .,, (per unit reference volume) in the forms

w

E·:::1

e, r

1n

+ 2: Q

.S = Scx1(C, 8)

(C, 8, r o) ,

(7.136)

17a(C, 8, r a) '

(7.137)

0

a= 1

m

·11=1/oo{C, 8)

+L a=l

and a remainder inequality (7 .138) where

r °' a = 1, .... , rn, denote the internal strain rates.

non-negativeness of the internal dissipation Vint

The inequality governs the in the thermoviscoelastic .material.

We have introduced the definitions

S

_ '>8Woo(C, 8) 00 -

.....

ac

,

1Joo = -

awoo(C, 8)

ae .

:t

(7 . .139)

7.9

363


(7.140)

of the contributions to the stress and the entropy~ with a = 11 ,. •• ~ n-i. As a result of the mathematical structure .given in (7 .134) the second Pio la-Kirchhoff stress S and the entropy 'f} are decomposed into equilibrium and non-equilibrium parts according to eqs. (7 .136) and (7 .13 7), respectively. The stress contribution 8 00 and the entropy contribution 1Joo are associated with the fully thermoelastic response which V:Je describe within the framework of finite thermolasticity introduced in Sect.ion 7 .3. fo .particular, we may adopt the r~lations (7.26):1 a·nd (7.27)° by using 00 instead of ~ ...According to definitions (7 .140) the second terms. in eqs. (7 . .136) and (7. l 37), i.e. E:~ 1 Q0 and E~~ 1 1Jn, contribute to the stresses and the entropy and are responsible for the viscous response of the material. . . . . !~e variables Q0 , a = 1, .... , .rn, denote n~n-equilibrh.1:.m stresses and are re.J~ted (co~jugate) to the (right Cauchy-Green) strain-like va~iables r 0 • Henc.e, Q0 are internHl .tensor variables with the intenfal constitutive equations

w

a= 1, ... ., ni ,

(7..141)

which restrict the configurational free energy Y 0 .in v.iew of eq. (7. I 40).t (compare also with the djscussion in Section 6.10, in particular, eq. (6.244)). Hence, considering (7.138), the internal dissipation takes on the form Vint = E::~ 1 Q0: : to./2 > 0, and vanishes at the state of thermodynamic equilibrium, since Q0 = -28Y 0 / 8rn lt___,. 00 0; a = 1; . . . , ·1n. ·In order to describe thermoviscoelastic processes the proposed constitutive model must be complemented by suitable equationJ· of evolution (rate equations). In particular, we want to specify the evolution of.the non-equilibrium stresses Qa, a= 1, ... , ·m, involved. A simple set of linear evolution equations for Qn is assumed to have the form

=

.

Qa

Q

.

so ·- Q +-0:

To-

£'tt· 1,1

O:'

= 1, . . . , 'In

,

(7.142)

where (7.142) is valid for some semi-open time .interval t E (0, T] and for small perturbations away from the equilibrium state (for small strain rates). As usual, we start from a stress-free reference configuration which requires the values Qn lt.=o = 0 for the internal variables to be zero at initial time t" = 0. The first-order differential ·equations (7.. 142) require additional data in the form of initial conditions Q0 .0 + at time t = o+. The term Q0 cpl' a = 1, .... , ·m, represents thermomechanical coupling effects which come from temperature dependent material parameters. Note that this term vanishes for purely mechanically based theories (see eq. (6.252)). The second-order tensors

7 . . · Thermodynamics of ·Materials

364

Q0 c1•1 have to be determined such that evolution equations (7.142) are dissipative and com.patible with the internal constitutive equations (7 .141 ). The second Piola-Kirchhoff stresses Sa, l~ = 1, ... , rn, in evolution equations (7 .142) correspond to the free energies \J! a ( C, H) (with '11 a{ I, 8 0 ) = 0 at the reference configuration). They are responsible for the viscoelastic contribution and .are related to the a-relaxation (retardation) process. We characterize the material variables S0 by the constitutive equations a: ::::: 1, ... , m .

(7.143)

The stresses S0 only depend on the external variables C and 8. for the case of solid polymers that are ·composed of identical polymer chains we may replace \JI 0 by the free ·energy w00 and adopt a relation similar to that given by eq. (6.256), namely '11 0 .(C, 8) = /3':'11 00 (C, 8), where E (0., oo) are given non-dimensional freeenergy factors. Note that '1ta must :be related to the configurational free energy Y 0 in such a way that the second law thermodynamics is satisfied for any thermodynamic process (see HOLZAPFEL and SIMO {l996c]). Then-relaxation process is associated with the relaxation time r 0 E (0, oo ), which, in general, depends on the absolute temperature 8 (see TOBOLSKY et al. [1944]). The temperature dependence on the relaxation time may be related to the -activation energy Ea of the relaxation process and expressed according to the Arrhenius equation,

f3r:

of

T

= Aexp

Ea) (ne

'

(7.144)

n

8.31Nm/Kmol where A represents a constant for the reacting substance and denotes the gas constant (or univ-ersal gas constant),. For an explicit derivation of the empirical exponential function (7..144) the reader is referred to SPERLING [1992, Appendix 10.1]. The form of the functional dependence of the relaxation time on temperature predicts the physical observation that viscoelastic effects occur faster as temperature increases. The Arrhenius equation in the form of (7 .144) is representative of most polymer relaxations (LEE et al. [ 1966] and TOBOLSKY et a"l. [ 1944]).

EXAMPLE 7.6 The purpose of this example is to illustrate the introduced phenomenological constitutive model for thermoviscoelastic materials by means of simple particularizations. The .free-energy functions T 0 and Wa are considered .to be of the quadratic forms

Tu·

= tta{8)tr (E [

?

A(t)-J ,

(7.145)

(with o: = 1,. ~·. rn), where µ 0 > 0 is a temperature dependent {Lame-type) shear modulus characterizing the thermoviscoelastic behavior of the a-relaxation process 1

7.9 Thermodynamics with l.nterna1 Variables

365

with given relaxation time r-0 > 0. The functions (7.145) are of Saint .. Venant Kirchhoff.type (compare with ·eq. (6.151)), in which, for simplicity, only the shear moduli are attached to the functions. The elastic strains are described by the (symmetric) Green .. La.grange strain tensor E defined by (2.67), while 1

A0 =9{I' 0 -I),

.a=l, ... ,m,

(7.146)

..;.J

denote the inelastic (viscous) strain measures expressed by the (symmetric) secondorder tensors A 0 • Consider a coupling term the forms

Q-acpl,

a = l, ... ·' 111, and a viscous dissipation

Vint

of

(7.147)

=

2µo rQ' motivated by the linear theory of viscoelasticity. The parameter f1 0 > 0 characterizes the viscosity of the a-relaxation process (usually .~enoted in the literature by the symbol 'f/, but to avoid confusion with the entropy TJ we place an accent over the symbol), while IQ0 J = (Q 0 : Q0 ) 1/ 2 denotes the norm of the

with the relationship ·1]0:

tensor ·Q 0 , which is a non-negative real number. Hence, the non-equilibrium stresses g~nerate a non-negative dissipation such that the inequality is satisfied. Obtain all the relevant thennodynamic relations, in particular the constitutive equations for the stress and the entropy. Specify the evolution equations introduced in (7 .142) and discuss the state of thermodynamic equilibrium. For a nice rheological interpretation of this type of thermoviscoelastic model, given by the thennomechanical description (7.145)-(7..147), the reader is referred to the exercises below in this section.

By adoptin.g (7.140) 1 and particularization (7.145h we may derive the Solution. .non-equilibrium stresses Q0 ~ which characterize the current 'distance from equilibrium'. By means of property (l .252h, relation (2.67) and the chain rule we obtain

= ?.OT0(C, 8, r J = ') (e) 8tr[(E Q .DC -J.Lrz D(E 0

0

-

= 2tta{8)(E -

A0 )

lt

.1

Ao-) 2 ] A0

)

= 1, ... , rn

.

8(E - Aa)

.

DC

.

(7..148) . .

r

It is important to note that a straightforward differentiation of Y 0 with respect to 0 gives the same result as for Qn, as can be seen by recallin.g (7 .141) .and using pro_perty (1.252) 2 , the relation (7.146) and the chain rule. In order to compute the entropy generated by the relaxation process, we use (7 .140h and particularization (7.145) 1• Thus., we obtain 170

=-

fJT a (C, 8, r a)

ae

= -µ.~(8)[tr(E -

. =-

8µa(8)[tr(E - Aa) 2]

A.0 .)2] ,

ae

a= 1, ... , m, ,

(7.149)

366

7


with the common notation µ~(8)

= dtta-(8)/d8.

Hence, the stress and entropy response .at time t follow from eqs. (7.136) and (7.137) as nt

S

=S +L 00

(7 ..150)

2Jta(8)(E ·-An) ,

0:=1

1}

= 1}oo -

L µ~k(8}tr[(E -

A 0 ) 2]

.

(7.151)

a=J.

Before proceeding to examine the evolution equations it is .first necessary to determine the second Piola-Kirchhoff stresses Sa. From (7.143) and (7.145.h we find, by means of property (l..252h, relation (2.67) and the chain rule, that

s(\ =

2

8\l1 0 (C, -8)

ac

= 2110(8)

8tr(E2 )

DE

aE : ac

= 2110(8)E ,

(7.152)

with n = 1, ... , rn. He.nee, from (7.142) and with help of the product rule and assumption (7 .147) 1 we obtain finally the evolution equations for the :non-equilibrium stresses, ... namely ·· .. ,

n=l, ... ,m,,

(7..153)

which are valid for some se·mi-open time interval t ·E · (0, T]. On comparing the given viscous dissipation (7.147)2 with (7.138) (by means of (7..141)), we conclude that Qa : (Q 0 /fJu - 0 /2) = 0. Hence, for the case in which Q0 is different from zero we find that the expressions in parentheses must vanish. Thus, using .(7 .146), we obtain

r

a: = 1, ... , m ,

(7. 154)

which is viewed as the classical Newtonian .constitutive equation for the shear stress applied to simple shear (see eq. (5.100)). In eq. (7.154), Qa may be interpreted as the non-equilibrium shear stress, A 0 .as the shear rate (of a dash pot) and fj0 has the characteristic of the Newtonian shear viscosity. The state of thermodynamic equilibrium requires that Q0 ·= 0 for t --+ oo, and that the internal dissipation Vint vanishes (see eq. (7.147h). In view of (7.154) this implies that A. 0 lt.-too = 0. For this limiting case the thermodynamic process is reversible and the material response is fully thermoelastic. II :., ..._.,,,.,,., ..... ,,_ ......... ,:,,._,,., .. , ... , ... , .. ,., ... ,:.:-·,··'··:, ..... ,., .. , ... , ... ,,,_ ... :·:·.

···:··:·,·:·:··,··.····:·=··'·."···'···.·:'·······, .. , ............ , .. , ........... , ..... ,.,,:.······':,, ..... , ...... , ... :·.·········•.",.··

.............. :,,:., ...... ,,.,.,,,., .............. , .. , ... , .. : ..... ·.·,· .. ·, .. ,_ .. , .. ,,,:·':··:,·...... ,, .... , .... ·.······:''""·'·.":,·-.'::··.····.'."

Note that the vast majority of polymers exhibit the well-known Newtonian sbear thinning phenomenon (pseudoplasticity) which means that with respect to Newtonian characteristics the shear rate cincreases faster than the shear stress .a 12 increases (see,

7.9

Thermodynamics w'ith Internal Variables

for example, BARNES et al. [1989, Chapter 2]). A mode.I for both Newtonian and non-Newtonian materials is the extensively used power Jaw model (see, for example, ROSEN [.1979, .and references therein]). This .empirical mode] is frequently written in the form a12

·=

0"21

. ·n. = rnc

(7.155)

,

which, in this :form, is only valid for simple shear. Here, n is the power law factor (or the flow behavior index) and -rn is a (temperature dependent) parameter called the viscosity index. Shear thinning occurs if n < 1. This model is used for a large number of engineering applications because it can be fitted to experimental results for various .materials and reduces to a Newtonian fluid for ·n equal to 1, in which case rn is known as the viscosity of the Newtonian fluid (compare with eqs. (7.154) or (5.100)). For typical parameter values see BARNES et al. [.1989., p. 22]. For an overview of different types of rheological models the reader is referred to, for example, ROSEN f.1979] and SCHOFF [.1988, p. 455].

EXERCISES

t.

The rheological model as illustrated in Figure 7. I 0 (referred to briefly as .a thermomechanica/ .device) is a suitable sprin.g-and-dashpot model representing quantitatively the mechanical behavior of real thermoviscoelastic materials.

B:.0(8)

E 1(8)

71

.

>O

'i/1 > ()

= -·

E1

>o

qi .

1/t

>0

:

-.. "YI .,..__ _.,. - - - - . i ) i i .

O'

• f.)

!J;)

-WVVVV\r-j}-

Fi.gure 7..10 Rheological model with temperature dependent moduli.

·: ·. · : ,· For simplicity the thermomechanical device is assumed to have unit area and unit length so that stresses and strains are to be interpreted as forces and extensions (contractions), respectively.

368

7 Thermodynamics of Materials It is considered to be a one-dimensional generalized Maxwell model with springs of Hookean type and dashpots of Newtoni·an type.. The temperature dependent Young's moduli and the Newtonian shear viscos.ities are given by E 00 (8) > 0, Eo:(8) > 0 and ·;]a > 0, r~ = 1, ... , ni, respectively. ·we now define the free energy "lfr(c, 8,/1, ... 'I'm)= '.~Joo(c, E>) + E:~l Vo(€, forms

·>' e·) -·tp'00 (c~ '

!E ?

-

00

·(8)c2 ·i...

e, 1'n), with the quadratic (7.1.56)

'

a

= l, ... , rn

~

(7 .157)

and the requirements -'¢J00 (0, 8 0 ) = 0 and va(O, -8 0 , 0) = 0, a = 1, ... , m. The energy function '1/1 depends on the ·external variable £ (measuring the total linear strain), the absolute temperature e and the inelastic (viscous) strains {o:, n ·= .1, ... , 1n. The free energy v 0 is responsible for the a-relaxation process of the a-Maxwell element with given relaxation time Tn > 0. (a) Based on assumptions (7 .156) and (7.l 57) obtain the explicit constitutive equations for the total linear stress er = Bij' / 8£ and the total entropy .,, -8·1/; /.88 in the forms

=

(7.158)

(7 .1.59) with the common notation E:t(8) = dEn:(8)/d8, n - 1, ... , rn, and the physical expressions for the equilibrium parts a 00 = E 00 (8)e, r100 = -E~(8)c 2

/2.

The non-equilibrium stresses q0 act on each dashpot of the .(\'.-·Maxwell element and are re.lated to the associated inelastic (viscous) strains ')'0 • Compute the internal constitutive equations qn = -fJv0 (c, 8, 1'0.)/870 , a = 1., ... , ·rn, and obtain the current 'distance from equilibrium' q0 .as specified in eq. (7 .158). (b) From Figure 7 .10 using equilibrium derive the total linear stress u and establish eq. (7 .158). Interpret the result as a superposition of the equilibrium stress a 00 and the non-equilibrium stresses qn, u = L, ...., ni. The considered thermoviscoelastic model presents a thermodynamic extension of the viscoelastic constitutive model introduced in Example 6.10 (see p. 286). Remarkably, the equilibrium equation (7.158) and the constitutive equation for the

7.9 Thermodynamics with Internal Variables

36~-

=

entropy (7. L59) constitute the one-dimensional linear counterparts of eqs . (7 .150) and (7.1.5.1), respectively. Hence, the purely phenomenological thermoviscoelastic mode.I presented in Example 7 .6 can be viewed as a nonlinear multi-dimensional generalization of the linear rheological model, as illustrated in Figure 7.10. 2. Recall the proposed one-dimensional thermoviscoelastic model from the previ-

ous exercise. The dashpots in the rheological model (Figure 7. l 0) .characterize the dissipation mechanism. According to a Newtonian viscous fluid we may relate qn to the strain rates i'o by the linear constitutive equations q0 = -~er 'Ya, n

= 1, ~ ..., -m,. (a) Consider the time derivative -of the non-equilibrium stresses q0 = E 0 (8) (£-70 ) and obtain the physically based evolution equations for the internal variables, namely

n where the relations 7n ~

= 1, ... , rn

,

(7 .160)

fJa/ E 0 , a= 1, ... _, m, should be used.

(b) Knowing that q0 and i'n are the stresses and the strain rates acting on each dashpot, derive the rate of work dissipated within .the considered thermo.mechanical device and derive the non-negative expression (7.161)

"Discuss the thermostatic limit and, in particular, specify in which parts of the device the stresses and the entropy remain. Note that the evo:Jution equations (7 .160) and the .internal dissipation .(7. l 6l) constitute the one-dimensional line~ _ counterparts of eqs. (7. I 53) and (7 .147h.

8

Variational Principles

The last chapter deals with the formulation of the field equations in the form of variational principles and methods . The variational approach in various forms is often taken as the cornerstone for the development of discretization techniques such as the well ·established finite element methodo"logy. The finite element method is today becoming widely used in i.~.dustrial applications because of its .predictive capability and general effectiveness in providing approximate solutions for the underlying initial boundary-value problems.. On the finite element method, which is one of the most .powerful numerical .techniques, a large amount of literature is avai-lable. See, for example, the books by ODEN [1972], HUGHES [1987], STRANG and FIX [.l988b], ZIENKIEWICZ und TAYLOR [1989, l.991], BREZZI and FORTIN [1991], REDDY [1993], BATHE [1996] and CRISFIELD [199:1, 1997]. For a description of a finite element program solving problems of continua in the nonlinear context (with available computer software) the reader is referred to, for example, ZIENKIEWJCZ and TAYLOR [1991), CRISFIELD [ 1997] and BONET and WOOD [1997]. His pointed out, however, that this reference list is by no means-complete on this subject. Variational principles are particularly powerful tools for the evaluation of continuous bodies and belong to the fundamental principles in mathematics and mechanics. It is ·important to note that the finite element method need not necessarily depend upon the existence of a variational principle. However, .good approximate .solutions are often related to the weak forms of field equations, which a.re consequences of the stationarity condition of a functional. In this chapter we discuss (and compare).the most important variational principles l~~ding to the finite element method. We focus attention sole.ly on isothermal processes, fiir.wh.ich the temperature remains constant. Coupling between mechanical and thermal qu.a.ntities .is not considered; hence, only the mechanical balance principles enter the vlliiational formulations. We .start by explaining the notion of virtual displacements and v.ariation·s and cont.inue with the prinC:iple of virtual work, which is fundamental for a large num.ber of 371

372

8 Variational Principles

efficient finite element formulations. The powerful concept of li.nearization is reviewed brie·fly and the .principle of virtual work in both the material and spatial descriptions is linearized explicitly. We present som·e of the basic ideas of two and three-field variational principles particularly designed to capture kinematic constraints such as incompressibility. The reader who wishes for additional information on the rich area of variational principles should consult the books by TRUESDELL and TOUPIN [1960], VAIN.BERG [1964], DUVAUT and LIONS [1972], ODEN and REDDY (1976] and WASHIZU [1-982].

8.1


Consider a continuum body B with a typical particle P E B at a given instant of time t. As usual, points X E no and x E !1 characterize the positions X and x -of that particle in the reference configuration Q 0 at ti.me t = 0 and the current configuration n .at a subsequent ti.met > 0. In the .following we indicate the displacement vector field of P as u, pointing from the reference configuration of the continuum body into the current configuration, i.e. from X to x (see Figure ·s ..I).

'~

\\

'\\

'

\

\

' ., '' \

time t = 0

f

~-

j?

X1., :c1

~ e2

\

,,l ,, ...._ ____ .,. __ _-;' ~~

I I I

time t

C1

F-igure "8•.1 Virtua".J configuration in Lhe neighborhood of u, given by u

= u + -nv.

Next_, consider some arbitrary and entirely new vector field w at point x which yields a virtual, slightly ·modified deformed configuration in the neighborhood of u. The virtual configuration is characterized by the modified displacement vector field u according to u=u+cw·,

(8.l)

8.1

Virtual Displacements, Varfations

373

where £ is a scalar parameter. The displacement vector field is regarded as a continuous and differentiable function of space and time. It may be written in the spatial or mate.rial form, i.e. u(x, t), U(X, t), as introduced in Section 2.2. ln order to keep the notation as simple as possible, we agree not to use this distinction any longer, we write subsequently u(x, t) = u(X, t). It will be clear from the text if the displace:ment field actually depends on spatial or material coordinates. In addition, within this chapter, the space and time 3:~guments will often be omitted, for convenience.

Virtual displacement field.. Following Lagrange we know that the difference .between two neighboring disp.lace.ment .fields, i.e. u .and u, .is called the (first) variation of the displacement field u, denoted by Ju. We write

8u = u -.. u = cw .

(8.2)

Jn mechanics 8u is also known as the virtual displacement field. The variation of u is assumed to be an arbitrary, il!fmitesimal (since E ~ 0) and a virtual c.hange, i.e. an imaginary (not a 'real') change. Note that du also characterizes an il~finitesimal change o_f. u. However, du refers to an actual change. The variation of the time-dependent ~~splacement vector field u .is always performed at.fixed instant of time. ·.The .virtual displacement field Ju is totally independent of the actual displacement field u and .may be expressed in terms of spatial coordinates or material coordinates. Omitting the time argument t:, we have

c)u(x) = r5u(x(X)) = c5U(X) .

(8.3)

For" simplicity, we agree, by analogy with the relevant notation introduced above, to write c5u(x) = t5u(X) for the virtual displace·me.ncfield . Ju. ·."> We discuss brie.fly the two fundamental commutative properties of the <~-process. For-the gradient of c5u we find by means of relation (8.2) 1 that

grad(c)u) = gradu - gradu .,

(SA)

while, on the other hand, the variation of gradu =Du/ax yields by analogy with (8.2) 1 o(gradu} . ~ gradu - gradu .

(8.5)

b:i{comparing eq . .(8.4) with .(8.5) we find finally the commutative property

t5(gTadu) = grad(<5u) , ~:~ich

(8.6)

shows that the variation of the derivative of a function (•)is equal to the der.iva~;~~:~ 'of the variation of that function (•). In an analogous manner we may derive the

8

374

Varfational Principles

second characteristic commutative property of the 8-process, namely, that the order of variation and definite integral is ·interchangeable. By analogy with the transformation (2.48) we may relate the gradient with respect to the current position of a particle to the material gradient, defined on region 0 0 • For subsequent use we note the relation gradJu

= Grad~u F-.l

o6ua _ DJua p- .t. Ab a"'b, . - DY .••\. .4

or

•

(8.7)

For more details on the variation see COURANT and HILBERT [I 968a, 1968b]. A clearly arranged summary of the calculus of variations may also be found in the book by .FUNG [1965, Chapter 1.0].

Firs.t variation of a function in material description. In the following let :F = F(u) be a smooth (possibly time-dependent) vector function. The single argument of :Fis the displacement vector variable u given in the material description. We agree that the value of :F, which characterizes some physical quantity, is either .a scalar, vector or tensor. (Note the abuse of notation in regard to Section 2.3 where :F = :F(X, t) characterizes a smooth material field). In order to obtain the first variation of the vector function :F we must eva:Juate simply the directional derivative (or Gateaux derivative) of F(u) at any ·fixed u in the direction of
1~F(u, Ou) =

D,foF(u) = _ii F(u + c.ou) l~=o Cc

(8.8)

and say that <>F(u.,,
EXAMPLE 8.1 be expressed as

Show that the first variation c>.F of the deformation gradient F may

8F=Gradou

or

(8.9)

In addition, verify that the first variation c5F- 1 of the inverse of the deformation gradient .F- 1 is given by or

8.1


375

Solution. With the definition (239) of the deformation gradient and rule (8.8), we may compute the directional derivative of F in the -direction of the virtual displacement field c)u at the position u(x) of the current configuration., i.e,.

.oF = D60 F = dd F(u + cOu)j 0 ==a .E

cl

= de (F + cGradOu)l 0 =o = GraMu .

(8.11)

Alternatively, knowing that .the operators 8(•) and 8( •) commute we obtain simp.ly result (8.9) by thinking of c~ ( •) as a linear operator. Applying re]ation (2.45)2 we conclude that

8F = o(G.radu +I.) = J(Gradu) = Grad8u .

(8 ..12)

In order to show eq. (8 . .10) we start with the variation of the identity F- 1F = l which gives c5F- 1 = -F--i8FF- 1 (compare with eq. (2.143)i). Substituting (8,.9) and using transformation (8~7) we find that <~F-J.

= -F-

1

(Gradc~u.F- 1 )

which is the desired expression (8.10).

= -F- 1grad'5u

,

(8.13)

•

Finally we establish the variation-of the Green-Lagrange strain tensor E. By recallin_g the definition (2.67) of E, we obtain with :the _product rule that oE = [<~(FT')F + FT&F]/2. With relation (8.9) and ,property ( 1..84) we arrive at

cm = ~[(FTGradc5u)T + F''Gradc5u] = sym(FTGradc5u)

'

(8.14)

~

or, in index notation,

cSE.411

1(

=2

88ua F,,H OXA

8c5'Ua )

+ FaAOXu

,

(8.15)

~hich

is an important relation used in subsequent studies. The notation sy111( •) is used to indicate the symmetric part of a tensor (compare with -eq. ( 1..112) 1). Since E = (C - J)/2, we find additionally that oE = (oC)/2. variation of a function in spatial description. Let f = f (u) be a smooth (possibly time-dependent) vector function ·in the spatial description . Note that the value of the function f (u) = x.(F(u))~ which we consider as the push-forward of :F, is _either a scalar, vector or tensor. In order to -obtain the first variation off we formally apply the important concept of pull-back and push-fonvard ope.rations introduced in Sect.ion 2.5. The variation of f . is obtained by the following three steps: -~irst

376


(i) compute the pull-back of .f to the reference configuration, which results -in the

associated function .F{u)

= x:- 1 (.f (u) );

(ii) apply the concept of variation to F, as introduced in eq. (8.8}, and (iii) carry out the push-fonvard of the result to the current configuration.

This concept is actually the same as for the computation .of the Lle time derivative introduced in Section 2.8. Instead of the direct.ion v used for the Lie time derivative we take here the virtual displacement field Ju. Consequently, for the first variation of a vector function f given in the spatial description we merely write, with reference to eq. (2.187), (8.16)

Since Dliu:F(u)

= -6F(u, Ju) according to (8.8)i, we obtain (8.17)

f = f (u_) is the push-forward of the first variation of the associated function F(u) = x; 1 (/(u)) in the direction of the virtual displacement field {}u. If .f = .f (u) is a scalar-valued function, then f = f (u) = F{u) and the variation of f coincides with the variation of its associated function F; thus,

Therefore, the first variation of function

t5.f

= 6F.

Note that in our terminology the introduced operator c5 is used for the variation of a function in both the material and spatial descriptions.

EXAMPLE 8.2 Show that the first variation c5e of the Euler-Almansi strain tensor e may be expressed as 8e

= 91 (grad·T8u + grad6u) = sym(grad8u).

.,

(8.18)

,;,J

or, in index notation~ as .c

_

ueab -

88ua)

1 ( 8<5U11

') -a,c·- + -.n,z.•.h

-

-·

(8.19)

U~

••a

Compare Example 2.15 on p. 107. From rule (8.16) we obtain the variation of the spatial tensor e, i.e .

Solution.

· 6e = ·.F-T (D&11 (F·'l' e.F))F

l

· -l = F -T (D ;uE).F 1

,

(8.20)

8.2

P·rinc'iple of Virtual Work

which is the push-forward of the directional derivative of the associated Green-Lagrange strain tensor E = x; 1 ( e) in the direction of 6u. By means of rule (8.8).i, relation (8 . .14) 1 and transfonnation (8.7) we find from (8.20)2 that

8e

= F-T8EF- 1 = ~(F-TGradTOu + Grad8uF- 1 ) =

1

2(grad

T

-

Ou+ gradOu) .

(8.21)

According to eq. ( 1.112) the variation of the spatial tensor e is the symmetric part of the tensor ,grad8u, which gives the desired result (8.18) 2 ~ •

EXERCISES

·· ·1. Show that the .first variations of the volume.ratio J and the inverse right Cauchy. . . Green tensor c- 1 are

8J

= Jdivc5u

·~

<5c- 1 = -F- 1 (gradT8u + grad6u) F-T

.. 2.. Show that the first variations of the spatial line, surface and volume elements are

8(dx) = gradc5udx , 8( ds) = (cliv6u I - gracfr6u)ds ,

c}{dv) = cliv8udv , where I denotes the second-order unit tensor. Compare eqs. (2.173) 3 , (2. I 78) 3 and (2.180)2.

8.2 · Principle of Virtual ·work

In· the following two sections we study variational principles with only one field of unknowns, called single•field variational .principles. In particular, we introduce work ;and stationary principles in which the displacen1ent vecfor u is the only unknown field . . . ::...J'hese principles are fundamental and will become essential in establishing finite .·eiem~nt forinulations. ~nUial

boundary-value problem.. The finite element method requires the formulaJion·ar the balance Jaws in the form of variational principles.

378

8

Variat"ional Prlnciples

As one of the most fundamental balance laws we recall Cauchy's first equation of motion (i.e. balance of mechanical en.ergy)discussed in Section 4.3. Knowing that .the spatial velocity field v .may be expressed as the time rate of change of the displacement field u, we may write Cauchy's first equation of motion, i.e. (4 . 53), as

divu + b

= pii

-(8.22)

.

From the fundamental standpoint adopted in Section 4.3, the Cauchy stress tensor is governed by the symmetry condition u = uT deriving from the balance of .angular momentum. The -spatial mass density of the material is p = J- 1p0 , which describes continuity ofmtis,s. The body force b per unit current volume which acts on a particle -in region n is considered to be a prescribed (given) force while the term pii characterizes the .inertfa force per unit current volume. Note that when we write ·eq. (8.22) we mean divu(x, + b(x, t) ·= p(x, t)ii.(x, l) at every point x E fl and for all times t. In the following we c-onsider boundary conditions and initial conditions for the motion x -= x(X, .t) required to satisfy the second-order -differential equation (8.22). We assume subsequently that the boundary surface an of a continuum body B occupying region is decomposed ·into disjoint :parts so that

n

n

with

anu n DOu = 0 .

{8.23)

Figure 8.2 illustrates the decomposition of the boundary surface 8f2 in a two-dimensional space at time t!'

n

anu

"Figure 8.2 Partition of a boundary surface

an.

We distinguish two classes of boundary conditions, namely the Dirichlet bound... ary conditions'f which correspond to a displacement field u = u(x, t)., and the von Neumann boundary condi.tions, which are identified physically with the surface traction -t = t(x, t, n).

8.2 Principle of Virtual Work

379

We write on

.t

= un = t

an :

on

(1

.(8.24)

'

where the overbars (ii) denote ·pres·cribed (given) functio.ns on the .boundaries an(•) c an of a continuum body occupying the region n. The unit..exterior vector normal to the boundary surface 80.u is characterized by n. The prescribed displacement field u and the prescribed Cauchy traction vector i (force ·measured per unit current. surface area) are specified on a portion Dnu c Dn and on the remainder Dnu, respectively. Note that in the previous section the symbol u also stands for the modified displacement fie:ld . We call the ..prescribed body force b and surface traction i loads. We say that the continuum body -is subjected to holonomic external co.nstraints if u ·= on the boundary surface Dnu. External constraints are nonholonomic if they are given by an inequality. The second-order differential equations (8.22) themselves require additional data in the form of initial condi.tio.ns. The displacement field ult=o and the velocity field itlt=O at initial time t = 0 are specified as

u

u(x, t) lt=o

= uo{X)

u{x, f)f t=u ·= :uo(X) ,

,

(8.25)

where (• )0 denotes a prescribed function in 110 • Since we agreed to consider a stressfree reference configuration at t == 0, the initial values (• )0 are assumed to be zero in our case. However, in dynamics -the configuration at t = 0 is sometimes not chosen as a reference configuration. In order to achieve compatibility of the boundary and .initial conditions we require additional:ly on anti that

u(x, 0) = u0 (X)

u(x ~ 0) = it

~

0 ( X)

.

(.8.26)

Now, the ·problem is to find a motion th.at satisfies eq. (.8.22) with :the prescribed boundary and initial ·conditions (8.24 ),. (.8.25) and co.m=patibility conditions (8.26). This leads to the formulation in the strong form (or dassical form) of the ini . . ·tial boundary-value problem (IBV.P). Given the body force, and the boundary and initial conditions. find the displace·ment field u so that (considering only mechanical variables} divu + b

=po , on

U=U

= un = t u(x., t) l1,=o = uo(X) ti(x, tHt=o = uo(X)

on

t

,

.

(8.27)


380

Note that the S·et (8 . 27) of equations generally defines a nonlinear initial boundaryvalue problem for the unknown displacement field u. In addition, we need a constitutive equation for the stress u which is, in general, .a nonlinear function of the displacement field u. If data depend on time and the acceleration is assumed to vanish, i.e. ii = o, the considered problem is called quasi .. static. For this ·case the equation of motion is subjected to the conditions (8.24), (8.25)i, and the requirement that eq. (8.26) 1 holds for compatibility. If the data are independent of time the proble.m is referred to as static. For this case we consider a body in static equilibrium for which the set (8.27) of equations reduces to the associated nonlinear boundary.value problem (BV~) of elastostatics, i.e.

diva +·b

=o

,

U=U

t=

un = t

on

(8.28)

on

Thus, the solution of a static problem at a point of a continuum body .depends only on the data of the boundary and not on initial conditions (there ·is no need for initial conditions).

Principle of virtual work in spatial description. An ana.lytical solution of the nonlinear initial boundary-va]ue problem des.cribed is on.ly possible for so.me special cases. Therefore, on the basis of variational principles, solution strategies such as the finite element method are often used in order to achieve approximate solutions. In order to develop the principle -of virtual work we start with Cauchy's first equation of ·motion (.4.53) which we multiply with an arbitrary vector-valued function T/ = TJ(X), defined on the current .configuration n of the body. Integration over the region n of the body yields the scalar-valued function

f (u, 71) = /(-divu - b + pii) · 17d·u = 0 .

(8.29)

n For the first argument of function .f we conveniently introduce the displacement vector field u rather then the motion x for a given time t,. The s·econd argument off is a function T/ = TJ(X) = 11(x(X, t}) (at ajixed instant of time t), called a test function (or wei.ghting function). It is a smooth function with 77 ·= o on the boundary surface .OOu. Eq. (8.2.9) is known as the weak form of the equation of motion with respect "to the spatial configuration. Equations in the weak form often remain valid for discontinuous problems such as shocks where most of the variables undergo a discontinuous variation. For this type of problem differential equations are not necessarily appropriate.

8.2

Principle of Virtual Work

38l

Since T/ is arbitrary, the vector equation divu + b = pii on fl is equivalent to the weak form (8.29). The method used to prove th.is important property goes along with the fundamental lemma of the calculus of variations. The solution of the problem in the strong form is identical to the solution in the weak form. For further details see the books by HUGHES [1987] and MARSDEN and HUGHES [1994]. Subsequently, applying the product rule ( l .289) to the term divu · 71, i.e.

diver

·'I]

= div( U'f/) - -er : grad77 ,

(8.30)

.and using the divergence theorem in the form of (.l.299), eq. (8.29) may be written as

/[u :

.f (u, 71) =

grad71 - (b - pii) · 71]dv - / u71 · nds = 0 . an

n

(8.31)

Since 1J vanishes on the part of the boundary surface 8f1 11 where u is prescribed, the surface integral only nee-ds to be integrated over the portion 80a- c an. By use of boundary conditions (8.24) and by formulating the initial conditions (8 . 25) in the weak form, we obtain the foil owing set .of scalar equations known as the variational

problem:

f{u, rJ)

=/

[u : grad71 - (b - pii) · 71]du -

n

J

J

i · 77ds = 0 ,

Dnu

/ u(x, t)lt=o · 71dv = uo(X) · rJdv • n n ti(x, t)l1=0 · 71dv = / ti0(X) · 71d·v

(8.32)

J

n

n

This set of equations characterizes the weak form (or variational form) of the initial boundary-value problem. It is the equivalent counterpart .in the strong form (827) which is satisfied when (8.32) is satisfied. Note that the stress boundary conditions on the portion are part in the weak form (8.32h, so they are often referred to as natural boundary conditions. However, the conditions u = =ii which are prescribed over the boundary surface anu are called essential boundary conditions of the

an(J

variational problem. Hence, variational problems are related to initial boundary-value problems which .are described through differentia] equations .and i~.itial. and boundary conditions. The differential equation is usually called the Euler-Lagrange equation in the weak formulation which in our case is Cauchy·'s first equation of motion (8.27)i. Formulations in the weak form are mathematically helpful for jnvestigations of -existence, uniqueness or stability of solutions (see, for example, MARSDEN and HUGHES [1994, Sections 6.1-6.5]). Note that the test function 'fJ in (8.32) is arbitra1y. If we look upon 'T/ as the vir-

8

382

Variation-a·) Prindples

tual displacement field t5u, defined on the current co11jiguration, then the formulation in the weak form of the initial boundary . . value problem (8.32) leads to the fundamental principle of vlrtual work (or principle of virtual displacement). Considering the symmetry of .u and the variation of the Euler-Almansi strain tensor 8-e, as derived in eq. (8.1 Sh, we arrive at the principle of virtual work in the spatial description expressed in tenns of the virtual displacement, i.e.

f (u, Ou) =

.!

j t·

[u : c5e - (b - pii) · 8u]dv -

n

1fod.'I

=0

,

(8.33)

an~

with the additional initial conditions ./~ 1 u (x, t) "lt::::o · fiuclv = ./~ u0 (X) · c5ucl v and In u(x1 tHt=O . 8ud'V = J~ ito(X) . t5udv. An equation of type (8.33) is typically called a variational equation. The smooth virtual disp"Jacement field 8u is arbitrary over the region fl and over the boundary surface 800 where the traction vector i is prescribed. We require that Ju vanishes on anu, where the displacement field u .is prescribed (see the boundary conditions (8.24) 1). The virtual displacement field is assumed to be infinit.esima/, which is not a requirement for an arbitrary test function. It is an imaginary (not .a ·'rear') change of the continuum which is subjected to the loadings. The principle of virtual work is the simplest variational principle and it states: the virtual stress work u : c5e at fixed u is equal to the work done by the body force b and inertia force pii per unit current volume and the surface traction t pe:r unit current surface along ou removed from the current configuration. The functions

t5Hin1.(u, c5u) =

.! n

CT :

Oedv

.!

1

b"Wext(U, Ou) = / b · 8udv + I· 8uds , n an"

(8.34)

(8.35)

are known as internal (mechanical) virtual work 61'Vint and external (mechanical) virtual work c5lVcx-t. In the first case the stress u does internal work along the virtual strains cfo . .In the second case external work is done by the loads, which are the body Jorce b and the surface traction t, along the virtual displacement 6u about region n and its boundary surface DO, respectively . For vanishing accelerations ii, the internal virtual work equals the external virtual work, i.e. 8lVint =
8.2

Principle or Virtual Work

383

Pressure boundary loading. In the fol.lowing we are concerned with an :irn.portant load case, the pressure boundary loading, which is caused by liquids or gases, for example, water or wind. Pressure loads are deformation dependent .and of crucial interest for finite defonnation problems. We consider a pressure bowulmy condition on the ·Current boundary surface anO' c DH. In particular, we consider a traction vector t = un = pn per unit current surface acting in .the direction of the (pointwise) outward unit vector n = n(x). The unit normal vector is perpendicular to the pressure loaded surface of .the body with region n. Further, we assume that the normal pressure p is a given constant.. An example in whkh a pressure boundary condition typically exists is -inflation of a balloon. The external virtual work done by the constant pressure p along the virtual displacement Du is then defined by

ana

JTVext(U, Ju) =

]J

J

(8.36)

n ·Jud,<; ,

.lmu

where ds denotes an infinitesimal surface element in the current configuration. The external virtual work of the pressure boundary condition is discussed in more

detail by

SCHWEIZE.RHOF

and

RAMM [1984], BUFLER [1984], BONET

and

WOOD

[I 997, Section 6.5] and SIMO et al. [ 199 lb] describing applications to axisymmetric problems. In the following we introduce br.ie'fly a parametrization of the current boundary surface Drlcr which .is very useful for implementation in a finite element program.

plane with region n~ is characterized 'by ~.t and c;2 (see Figure 8.3 ). rhe parametrization .of the surface on which ]J is prescribed is given in the form x = 1(~1, 6, t) c an" (:1:,1 = 1'a(~i, 6, t)) at fixed time t. Hence, the outward unit vector n ~ay. be expressed as the cross product of the displacement dependent vectors Di/ 8~ 1 and ·81 /8(,2 • The ·infinitesimal surface element els follows from eq. (1.32) and the use rh~ .parameter

of the chain rule. We write

a'Y

a,

-xn

==

D6 a'Y 01 -xuf.1

8~1

0/ .-, = .a~ l

(l p

0/ I c dC x a~'2 c. ~ i ~2

•

(8.37)

8{2

These relations enable the external virtual work (8.36) to be expressed as

(8.38)

·~h.ich is appropriate for finite element approximations.

·s

384

Variational Prlnci.ples

p

= const

6 a"'Y ~1

8{.1

:r:a

:1; 1

"

n

L

/

6.

"

an

time t :z:2

e2

C1

.Figure 8.3 Constant pressure boundary loading and parametrization of the pressure loaded surface an(1.

Principle of virtual work in ·material description. Now we are in a position to express the principle of virtual work in terms of material variables. We assume a region .0.0 of the continuum body which is bounded by a reference boundary surface 8.fl 0 • This boundary surface is partitioned into·disjoint parts (compare with the associated partition (8.23)) so that

ano = ano

u

uan[J

(1

with

anou n DHo(J = 0

.

(8.39)

As a point .of departure we recall the equation of motion in the material description (4.63). We use the form

Div.P + B = Poii , corresponding to (8.22). Here,

P~

(8.40)

B .and p 0 ii denote the first Piola-Kirchho.ff stress

tensor, the reference body force .and the inertia force per unit reference volume, respectively. For the Dirichlet and van Neumann boundary conditionst i.e. u = u(.X, t) and T = T(X, t, N), we write, by analogy with (8.24),

U=U

on

annu '

T=PN=T

on

(8.41)

8.2

385

Principle of Virtual Work

where the unit exterior vector normal to the boundary surface onou is characterized by N. The prescribed displacement field u .and the prescribed first Pio Ia-Kirchhoff traction vector T (force measured per unit reference surface area) are specified on the disjoint parts 8rl 0 u and 800 en respectively. The second-order differential equation (8.40) must be supplemented by initial conditions for the displacement field and the velocity field at the instant of time t = 0 (see eq,. (8.25) ). Using the above concept, we may show the princip:le of virtual work in the material description expressed in tenns of the virtual displacement, i.e.

F(u, Ou)

= /[P: GradOu no

(B - p0 ii) · Ou]dlf - ./ T · JudS anoa

=O

,

(8.42)

with the virtual displacement field 6u (here defined on the reference co1~figuration.) satisfying the condition OU = 0 on the part of the boundary surface anou where the displacement field u is prescribed. The surface traction Tacts on the portion 8fl0 er c 80. 0 • .According to relation (3.1), T has the same direction as t, but T ¥ t. It is i~portant to note that the description of the variationa_l equation (8.42) is equivalent to that of (8.33). · ,,,,,, ~.,.,,,~,, .. ,.: ..... ,, .. , . . ·... ·: .... ~··· '·: ... : ········ ''·~····,··~·"'.-,: , ... ,.... ~·,: ~·, .... ~· '· ......·...... ~·, .... ~·········' ~'···' ~· ~ ..... , ~: .•. ,_,, .•. ,.,., '·"' ..... , .·: ·'.: ..... : ... , ,_.,,.: ''·~·~··~··."'··'·' ~ .. ,,,,, ~.~ .. · .. , .. ·: '··· ......... '·'·' .,. ~·····: '·····~··:: '···"'·'·' ~·' ~, .... , .. ,: ... , .. ,,.,,: ... , ... : ..• , ..... ~·· ... , , ~··.. , ..... : ·:

..

: ~·

~

..,, ...

~.,.,.,

'·"

EXAMPLE ·s.3 Show that the material form of the principle of virtual work, as given in (8A2), can be obtained alternatively by a pull-back operation of relation (8.31.) to the reference configuration.

Solution.

In order to show (8.42) we must express the internal and external virtual work cn.v·int and JTVext in eqs. (8.34) and (8.35) and the contribution J~ pii · oudv in terms of material variables. We begin by considering the intental virtual work (8.34). With the help of identities (8~18.h, (2.50) and transformation (8.7) we obtain

/ u: 6edv n

=/

u: gradOudv =/Ju: Gradc5uF- 1 dV , n no

(8.43)

where the symmetry of the Cauchy stress tensor u is to be used. Applying property (l.95) .and Piola transformation (3.8.) we obtain the canonical representation of the material version, i.e..

c5H~ 11 t(u, du) = /

u : Oedv = / P : GradOucJl! . n no

(8.44)

The external ·virtual wo.rk 8ll1~xt in the form of eq. (8.35) may be transformed by means of the relation for the body force b, i.e. b = J- 1 and the change in volume which -is given by d-v = Jdll. In addition, we must show the equivalence of the

.u,

386

8

Variational .Princi.ples

prescribed traction vectors. With relation (3."I) and boundary conditions (8.24h and (8.4.lh we deduce that ids= T.dS. Hence,

J

b · 15udv +

OIVext(u,Ou) =

JI·

r5uds =

ano

n

J

B · 15udV +

no

/

ano

T · OudS . (8.45) ct

The remaining term in eq. (8.42), i.e. the inertia force p0 ii per unit reference volume over the region H 0 , may simply be established from the third tenn in the .associated eq. (8.31) by means of p0 dV'" = pd11, i.e. conservation of mass in the local fonn (4.6). II This result together with (8.44) .and (8.45) leads to the desired re"lation (8.42).

EXERCISE

I. Starting at eq. (8.44 h, show that the internal virtual work cHlint may also be expressed as the contraction of .the symmetric second Piola-Kirchhoff stress tensor .S .and the virtual Green-Lagrange strain tensor ,YE de-fined in eq. (8.14). ln addition,. show that
cHVinl

=

J

p: GradOudF

·no

J

= s: fiEdl" no

=I

:E:

~c- 1 0CdV

(8.46)

no

.

.

8.3 Principle of Stationary Potential Energy In the principle of virtual work, as derived in the last section, the stresses are considered without .their connections to the strains. We have not taken into account a particular material. Io this section we assume a conservative mechanical system {compare with p. 159} for both the stresses and the loads. requiring the existence of an energy functional The assumption of the existence of n is common in many fields .in solid ·mechanics. The loads .may depend on the motion, but they must emanate from a functional. A formulation based on .energy functionals is very useful, for ex.ample., for the development of robust nmnerical algorithms that are based on optimization techniques. In the following we introduce a stationary energy principle in which the displacement vector field u is taken to be the only fundamental unknown.

n

·.···..

8.3

Principle of Stationary Potential Energy

387

. s~·ationary energy ·princi.ple. From now on we assume that the loads do not depend ·6n the motion of the body, which is usually the case, for example, for body forces . .~t. means that the directions of the loads remain paralle.I and their values unchanged ihroughout the deformation of the body. We say that such loads are 'dead'. Instead of a vibrating body we consider a body in static equilibrium under the action :of specified 'dead' loadings and boundary conditions on u and O"' according to .e.q. (.8.41 ). Then the total potential energy IT of the system is .given as the sum of the ·internal and external potential energy, flint and Ilexb i.e.

anu

Il(u)

I1i111(u)

=/

w(F(u))cW ,

= .IT.int(u) + Ilexi(u)

ano

,

(8.47)

Ilcxt(u) = - ./ B · ucW - ./ T · udS , (8.48)

no

no

anou

~·here '11 = 'lJI {F) denotes the strain-energy function per unit reference volume, as introduced .in Section 6.1 . . . "._: .. Si.nee the deformation gradient F depends on the displacement vector fi.eld u by the relation according to eq. (2.45h, i.e. F = Gradu + I, we indicate explicitly the dependence of Fon u and write F = F(u). For.. nh1t and Ilext we will also indicate subsequently the dependence on u. Note that for a rigid body the term TI int is zero. The 'dead' loadings, given by the external forces Band T, are distributed over the volume of the continuum body and its von Neumann boundary, respectively. One main objective of common engineering interests .is to find the state of .equilibrium (the de.formed configuration) for which the potential is stationary. The stationary Pt?Sition of the total potential energy II is obtained by requiring the directional derivative with respect to the displacements .u to vanish in all directions '5u. Compute

OIT(u, t5u) = Do11I1(u)

= ~~ IT(u + E0n)le:::o = 0

,

(8.49)

which is known as the prindple of stationary ·potential energy, another fundamental variational principle in mechanics. In other words, we require that the first variation of the total potential energy, denoted 8Il, vanishes. The variation of fl clearly .is a function of both u and c5u. The arbitrary vector field Ju is consistent with the conditions imposed on the continuum body. Thus, ·C~U = 0 over anu, where .boundary displacements are prescribed. In order to decide if the solution corresponds to a nuitimum, a min inmm or .a saddle point we must determine the second vadation of the total potential energy fl, denoted by cFIT(u, ,~u, ~u) = D~u,uull(u). Here, ~u .is the increment of the displacement field u which will be discussed later in Section 8.4. The quantity .D~u,ilun (u) is obtained from the directional derivative of variational equation (8.49) with respect to u in the direction ~u (i.-e. the second directional derivative of TI with respect to u), which .is

388


.either a maximum (DJu,~uII(u) < 0), a minimum (D~u,~ 0 II(u) > 0), or a saddle point (D~u,uuIT(u) = 0) . .;, ......... ,...,,,,.-:,... --······--,,,,:,·=·---.-,.,,.-:,,,.,, .. ,; _____ , _ _ _ .. ,.,...,,, _____ ._____ .,, ..... ,;,_ .. _.,._~~:·:""'·~,·; .. ,_ .. ,_., .• , ... _':""""':••,_, __ ,_._:"""" ..·'·':·'':.... .... ,._._,_-:_,.,,,,,. _____ ,_,:,;,:~,-· _ _ _ ,,_..,:,,,,,,,_ ... ____ .... - ... ,,,..:_.,,,._.

EXAMPLE 8.4 Show that the directional derivative of the total potential energy IT, as given by (8.47), (8.48), with respect to u in the arbitrary direction on leads to (8.50)

(8.5 l)

and

The internal virtual work tHVint and the external virtual work JlV.ext are given through eqs. (8A4) and (8.45), respectively. Furthermore, show that the stationary position of the total potential energy TI gives the principle of virtual work :for a body in static equilibrium, as given in eq. (8.42).

Solution. Since the loads .B and T do not depend on the deformation of the body~ they do not contribute to the directional derivative. Hence, from (8.49)], we find~ with use of expressions (8.47) and (8.48), that

d D.ruII(u) = d-e II(u + c8u) le=O =

j__!/. \J!(F(u + cJu))dF de~

no

-f

j' B · (u + cJu)dF

.

no T · (u + Clfo)dSJle=O .

(8.52)

Interchanging differentiation and integration and applying the chain rule, we obtain

.aw BFun : I

D.s11 II(u) = .

(F(

;·

(

D.s11 F(u)dll - . B · 6udll - .

Oo

ilo

T · JudS ,

(8.53)

rJno"(T

where D,iuF denotes the directional derivative of the deformation gradient ·Fat u in the direction of 8u, derive~ in e.q. (8.11). In order to specify the first -integral in eq. (8.53) we recall the physical expression (6.1) 1 and use result (8.1.1).1 to obtain

. aw(F(u)) . D .F( ) 11..r = ;· aw(F(u)) . ~F( :·\. )I T/~ DF . dU .u ( . 8F . de u +__,vu e=O(1 l / flo no f

= / P: Grad15ud11 . On

(R54)

8.3 Principle of Stationary Potential Energy

389

By· .reca11.ing definitions (8.44)2 and (8.45)2 a.nd combining (8.53) with (8.54)2 we find :the desired results (8.50) and (8.51). Hence, stationary condition (8.49) yields precisely the principle of virtual work (8.42) for a configuration in static equilibrium which renders' stationary the functional II. We conclude that the total poten6al energy IT is stationary for arbitrary variations c5u, which means evaluating c5TI(u} 8u) = 0 with respect to the displacements., if and pfi/y if the nonlinear variational equation (8.42) (for ii = o) is satis(i-ed (equilibrium ~~ate) •

.:.: · .:· Finally, note that for the purpose of computing the stationary position of TI the ~agnitude of the virtual displacements need not be small, as is sometimes stated in the ..Ii.:~~rature. However, in order to achieve a first-order approximation the magnitude of "_the virtual displacements must be small. Iii -~.

:;:·:

·;--"'-''"''""''"..,,,..,.,,... ,...................................:.... ,.................... :,:' . ····'''···=·'···,, ......................... :..... :····,., .......,, .......... ,., ... ,;··· .......,.,.....,.,...,,; ..,.. , ..:,.•.,..,..,.,,,,,... _,.:, ....... ,.,..,...., .................... , ................... ,,.,..., ..... ,,,.., ...,,.,......,,., ...:··:··: ......... ,..,......,.,., ........,,: .. -··,••,,:.• ............:......

..

.r~.~~lty method for incompressibility. The principle of virtual work is not the ·:~ippropriate variational approach to invoke kinematic constraint conditions such as in~.~#npressibility, contact boundary conditions or Kirchhoff-Love (kinematic) conditions .. o.~·.iplates and shells often occurring in engineering applications . .'. /·\·::-.. A numerical analysis of nearly incompressible and incompressible ·materials ne:~~s~i.tates so-called multi-field variational principles in which additional variables are incorporated. Multi-field variational principles, dealt with in more detail in Sec.lions 8.5 and 8.6~ lead to mixed or hybrid methods for finite .elements. :f ( Nevertheless, a single-field vari;tional approach with the displacement u as the only .~.~W variable is very often used in order to approximate, for example, incompressible ;T~fe.rials. This leads to the so-called penalty method, which is based on the simple {p~ysical) idea of modeling an incompressible material as slighlly .compressible by us:f~galarge value of the bulk modulus. Of course, following this idea, an incompressible r~\~~erial. can be obtained by taking the limit infinity for the bulk modulus. However, .t·~.~.·~esu.lt of this idea from the numerical point of view is that we always work with .a :~Jightly compressible mate.rial since the incompressible limit can never be achieved. -.:.";>::-_·To be more precise, rather than employing the strain-energy function in the form :c>rw:- = \J!(.F), it :is standard to use the unique decoupled representation .of the strain:e.nergy function in the form ...

:

+ Wiso(C)

w(C) = Wvo1(J)

with

Wvo1(J)

= 11~Q(J)

(8.55)

Whh1rit ·energy functional {8.47) takes on the penalty form

·::::._:.::.:: ..

!~'{ i .\;\<-..". .·

IIp(u) =

I

[\Jlvol (J(u))

+ 'liiso( C(u) )]dF + Ilext(u)

no

\yJtfrthe external potential energy Ilext given by eq. (8.48)2.

(8.56)

.8

390

Prindple~

Variational

Here, J = J(u) = (detC) 112 defines the volume ratio and C = C(u) = J- 2 /:sc the corresponding modified right Cauchy-Green tensor, as .introduced in eq,. (6.79h. The strictly convex function Wvol describes the volumetric .elastic response while ·w iso .is associated with the isoclzoric elastic response of the hyperelastic material. We require '11 vol ( J) and '11 iso ( C) to be zero if and only if J = 1 and C = I, ensuring that the reference configuration is stress-free. According to eq. (8.55h .the volumetric contribution '1i vol is characterized by a (positive) penalty parameter ft, > 0 which is independent of the deformation. The parameter ff, may be viewed as the bulk modulus. The function g is motivated mathematically. It is known as the penalty function and may adopt the simpJe form

1 (.J (u) - 1)')- , g ( J) == 2

(8.57)

often used in numerical computations. Consequently, the meaning of the function 'P v-0l as used in eq. (8.55) 1 differs s·ignificantly from its meaning in eq. (6.85), in which Wvo.t .is of physical relevance. We now derive the stationary position of Ilp with respect to the displacement field, which is basically a procedure according to Example 8.4. Starting with the fundamental condition (8.49) and following the steps which have led to eq. (8.54) we find, using decomposition (8.55), that

D TI ( ) Ju P u

= .;· (Dfilvo1(J(u)) ac + D\Jliso(C(u))) DC

. D C( ) ·ll/~

.

,ha

uc

flo

+D15ullexl(u)

=0

~

(8.58)

where D,iuC denotes the directional derivative of the right Cauchy-Green tensor Cat ·u in the direction of 8u, which is 2<5E (see Section 8.1 ). A specification of the integral in eq. (8.58) implies, by means of the chain rule and relation (6.82) 1 , that

a\T/.~ ) D CdV = ,. ( C)'T~Jvol aJ _~_vc_ll (a .f ac + ac · (511 • dJ ac + ac \T/

1so

!1.\T/.

U :I! LSD

•

no

)

.

•

·

D C 'ill ,fo

c.

no =

J(/ ;:,7 c- +2 :~ ):8EdV 1

1

1

8

0

(8.59)

nu

(the arguments of the functions have been omitted for simplicity) . With reference to eq. (6.91}1 the term d'11v01 /dJ defines the hydrostatic pressure p. Hence, recalling definitions (6.8.9h and (6.90}1 we conclude that the terms ·in parenLheses of (8.59b are associated with the volumetric and isochoric stress contributions Svol and SiscH respectively. Using the second Piola-Kirchho:ff stress tensor S, which is based on the additive t

8.3

Principle of Stationary Potential Energy

391:

decomposition (6.88)2, we achieve finally the principle of virtual work (for a configuration in static ·equilibrium) in the form

Douflp(u) =

.!

S : t5Edlf + DJullext(D) = 0 .

(8.60)

no .Note that the integral in eq. (8.60), i.e. the directional derivative of the internal potential energy Hint with respect to u in the arbitrary direction
= d'l!wi;l~(u}) = ~ clQ~:;u)) = ti:(.J(u) _

l) .

(8.6.1)

In..contrast to eqs. (6 . .l.40h and (6J41), this is an artificial constitutive equation for p designed to prevent a significant volumetric response, as already pointed out. The user-specified penalty parameter"' is merely an adj-ustable numerica] parameter which .is often ·chosen -through numerical experiments. Clearly, with increasing .K. the violation of the constraint is reduced. If we take the restriction on the value 11, -> co, ·~~~.constraint condition is exactly enforced, and then eq. (8.56) represents a functiona] for an incompressible material with J = 1. Unfortunately, for an approx·imation technique such as the finite element method, the .stiffness matrix becomes .increasing} y il/ .. conditioned for i ncreasi.ng n. (see, for examp.le, BERTSEKAS fl 982] and LUENBERGER .[ l 984] for detailed studies). For that case the reduced integration method, and the later proposed selective-reduced ·integration method {which are equivalent to certain types of mixed finite element meth9..~~, . as .discussed in MALKUS and HUGHES II 978] and HUGHES [l 987]) ·is often used t() µ~d.erintegrate (weaken) the penalty function . .. ·: . · .;li~wever, penalty methods are attractive because they are based on a simple variational princip.Je with all .its computational advantages .and are very effective to i-m.ple~~nt. in a finite element program. .

EXERCISES

1. The dassical HamUton's variational pdnciple represents a generalization of .the principle of .stationary potential energy (8.49) to continuum dynamics. It is presented by t1

t1

t5(.f L(u)dt) = Dliu(/ L(u)dt) = 0 to

to

where

L(u)

= IT(u) -

IC(u) ,

·s

392

Vari.ational .Princip:Jes

with Ju denoting the variation of the displacement vector field, which is a function of space and time. The functional L (or in the literature sometimes introduced as -.L) is integrated with respect to time t over the closed time interval t E [t 0 , ti]. The two points t 0 and t 1 denote arbitrary .instants of time at which c5u is assumed to be zero at all points of the body, i.e.

c5ult=to

= r>uJ·t·=t1

·= o ,

(8.62)

valid for a region n. The scalar-valued functionals II and K denote the total potential energy (4.114) and the kinetic -energy (4.83) of the moving body, as usual.. The potential energy of the loads -exists and is given by eq. (8.48h.

Show that the vanishing variation of the functional L with the imposed restrictions (8.62) gives the principle of virtual work (8.42) for aIJ t5u which are zero on D!lu throughout the entire closed ti.me interval t E [t:o, ti.). 2. Consider a constant normal pressure p applied to a boundary surface enclosing a certain region. Show that there exists the associated potential

Ilext(u)

= 7J

f n

dv

= ]J

f

J(X, t)dV ,

no

whose variation gives the external virtual work JH".°"ext.• as defined in eq. (8.36), i.e& D6ullcxt (u)

= JH~ext(u, 6u) = p ./~n(T n · ouds.

8.4 Linearization of the Principle of Virtual Work Variational :principles such as the principle of virtual work in the forms of (8 . 33) or (8.42) are generally non1inear .in the unknown displacement vector field u. Typ"icalJy, the nonlinearities are due to geometric and material contributions_, i.e. the kinematics of the body and the constitutive equation of the material, respectively& As mentioned above, one main objective of engineering analysis is to find the unknown field u which is the sofotion of the associated nonlinear boundary-value problem. Usually, (exact) closed-form mathematical solutions of a set of nonlinear partial" differential equations are only available for some special engineering problems; they are rather complicated and often unusable. In order to keep the complexities of engineering problems intact, approximate numerica] solutions, based on, for -example, the finite e:Jement method, are required. A very common and simple numerical technique to solve nonlinear equations is to employ the reliable incremental/iterative solution technique of Newton's type. It is an efficient method with the feature of a quadratic convergence rate near the solution point. This

8.4 Linearization of the Principle of Virtual Work

393

technique requires a consistent lin.earization of all the quantities associated with the .considered nonlinear problem generating efficient recurrence update formulas. The honlinear problem then .is replaced by a sequence of linear problems which are easy to solve at each iteration. Linearization is a systematic process which is based on the concept of directional derivatives, see the pioneering work of HUGHES and PISTER [1978]; for the more generalized concept see the book by ·MARSDEN and HUGHES [1994, Chapter 4], and °for ".an application to rods and plates the work of WRIGGERS Il 98"8] among others. For the concepts of linearization and directional derivative and their applications in nonlinear continuum mechanics see also the textbook by BONET and WOOD [1997]. The following part of this section deals with the Iin·earization of a nonlinear and smooth (possibly time-dependent) .function :F = :F(u) in the material description which is either scalar-valued, vector-valued or tensor-valued. tile single argu·ment of :F .is the dis.placement vector variable u. Consider u, then the fundamental .relationship for the lineari.z.ation of the nonlinear function :F -is based ·On the first-order (Tay"lor's) expansion, which is expressed as Concept of Iinearization.

F{u, ~u)

= :F(u) + ~F(u, ~u) + o(~u)

(8.63)

,

where ~ (•) denotes the linearization operator similar to t5 (•). The operator A (•) is also linear and the usu.a] properties of differentiation are valid. The quantity ~u denotes the .increment of the displacement field u, here expressed in the reference con figuration. The remainder o(~u), characterized by the Landau order symbol o( •),:is a small ·error that tends to zero faster than Liu -1- o, i.e. Hmilu-4o o(~u)/l.6.ul = o. Within the classical solution technique of Ne1vton 's method, Taylor's expansion is .truncated after the first derivative of :F. Hence, the first term in (8.63) is a constant ·part, i.e. an approx.imate solution for a given state u. The second term il:F is the linear change in :F due to ~u at u. It is the directional derivative of :F at given u (fixed) in the direct.ion of the .incremental displacement field ~u, i.e. Ll.F{u, Liu) -:- D~uF(u)

d

= dt F(u + tLlu)lo=O

,

(8.64.)

where the linear Gateaux operator D( •) is with respe·ct to the incremental displacement .~eld ~u . We say that AF{u, ~u) is the Iinearization (or linear approximation) of ·;:at u. Note that in .regard to eq. (8.8) the fi.rst variation D 6µ.:F(u) of a vector function F(u) and the linearization .DA 0 :F(u) of that vector function are based on the same ,concept of directional derivatives. By taking notice of this equivalence of variation and 'tinearization, all relations derived in the previous Section 8.1 can be adopted here.; we just use the symbol ~ ( •) instead of J (•.).

8 Variational Princi.ples

394

For ex.ample, relation (8 . 6) and transformation (8.7) read ~-(gra
== grad (ilu) ,_

= Grad~u F- 1

grad~u

(8.65)

or

(8.66)

.In addition, the linearizations of tensors F, F- 1 , E are ~F

~·F-l =

L).E = Du11 E =

= DAuF = Grad~u , D~u·F- = -.F- 1grad.6.u

(8.67)

1

~[(FTGradL).uf" + FTGradL).u]

,

(8.68)

= sym(FTGraclL).u) ,

(8.69)

w

which are analogous to eqs. (8.9), (8.10), (8.14).

EXAMPLE 8.5 Show th.at the linearizat.ion A6E of the virtual Green-Lagrange strain tensor <>E = sym(FTGrad6u), .as derived in eq. (8 . 14h, may be expressed as ~t5E

= sy1n(GradT~u Grad8u)

.

(8.70)

Solution. According to the rule (8.64), we compute the directional derivative of c)'E in the direction of ~u at u, i.e.

~trE =

Do118E

= (ell€ 6E(u + ci).u)lc=O

d T .. syrn[(F(u +.c-6.u)). GracM.u]ft.:=O = -Cc 1

.

(8.7.1)

Since the virtual displacement field c5u is independent of the disp.lacement~ the term Grad.Su is not affected by the linearization. Knowing that d/de: F(u + e~u)lc==U = Grad~u (see eq. (8.67)),. we find the desired result (.8.70). a

In order to linearize a nonlinear smooth vector function f = f (u) in the spatial description we adopt the concept for the first variation off introduced on p. 375. By analogy with relation (8 . 16), we may write (8 . 72) for the linear-ization (or Hnear approxi·mation) of .f. Since D 611 F(u) = ~F(u, ~u) according to (8.64) we obtain (8.73) which is analogous to eq .. (8.1.7). For notational simplicity, the linearization

operator~

8.4

.Linearization of the Principle of Virtual ·work

395.

is not particularly marked when .applied to a function in the spatial description, .as for the '5-process. Note that the operators required for the Lie time derivatives, .the variations and /inearizatimzs of spatial tensor variables are formally the same. They are based on the concept of directional derivative. For the Lie time derivative the considered direction of the derivative is v, while for the vmiation and Hnearization :it is the virtual displacement fie.Id <~U and the incremental displacement field Liu_, respectively. Compare relations (2.187), (8.1.6) and (8.72). ·•1'."·.··.··:···:··:···,·:·, ........... :··:·· .. ··:·:- .. , ..... , ........ ,, ........................................................... ,, .... , .......... : ... .,,.,,., ................................ , ............................ ·.·:···. ··:·:··:··:· .. ·:····:···."··:···· ...·.·:.···················

............................ ,:··:·

..................... , ..... ;··

EXAMPLE 8.6 Show that the linearization Ll.8e of the virtual Euler-Almansi strain tensor .Je, which is a spatial tensor field according to eq. (8. l8), ·may be expressed as

~-cfo

= sym(gracfrLlugradc~u)

.

(8.74)

Solution. Since we apply the systematic technique of linearization to material quantities, as a first step we pull-back the variation of the Euler-Almansi strain .tensor c5e~ th.at .is the .inversion of eq. (8.20h, which yields the variation lfE of the associated Green. .Lagran_ge strain tensor E. The lineari.zation of c5E .is carried out by analogy with Example 8.5. In the fast step the push-forward operation on ilc5E, as given in (8. 70), is x*(~
= F-Tsym(Grau) ,

where the relations (8.66) and (8.7) should be used. ....... :···:·····:·:···:···:··· .. ····; .. ........ ;·····"·'···:·.······:·

...........................

··············:····."··:······:···

............ , ....... :·:···:·,··............

··························

{8.75)

:• ................ ;.·.··:·:·:·····::·:·":· .. ·::·"·:··.···'···.···:···

.....................................

•••''••'•'.·"·::·::·:··'······

..

Linearization of the principle of virtual work ln material description. In order to linearize the principle of virtual work in the ·material description we recall the nonlinear variational equation {8.42). For simplicity we consider a purely static problem, so that

ii = o. In addition, we assume the loads B and T to be '-dead' (independent of the deformation of the body), so that the corresponding l.inearization of the external virtual work-(8.45) vanishes, i.e. D.6. 0 8l'1lcxt(u, Ju) = 0. This -is certainly not the cas.e for some other types of loads, like the pressure .loads discussed on p. 383 and subsequently. Hence, the linearization of the variational equation (8.42) only affects the internal virtual work 8l'Vint, on which we will focus subsequently. For our purpose we take the material (or Lagrangian) fonn (8.46)2, i.e.

rHVint(u, Ou)

= /s(E(u)) : OE(u)cW

.

(8.76)

On

Note that the Green-Lagrange strain tensor E depends on the displacement field u

396

8 Variational ·Principles

through the relationship (2.89). Now we may adopt rule (8.-64) in order to compute the linearization of (8 . 76), i.e.

DAui5H~nt(u, Ju) = =

d: OTtint(u + £~u)IE=O

!I/

S(E(u + c~u)) : JE(u + £L.~u)dl']le=O .

(8.77)

no Interchanging differentiation and integration and using the product rule results in

DAu
= /rs(E(u}): DAu8E(u) HE(u): DAuS(E(u))jdV

,

(8.78)

no where D~ 0 8E characterizes the directional derivative of c5E at u in the d-.irection of .6.u, i.e. the Iinearization of 6-E according to eq. (8.70). In order to specify the linearization DAuS of the (symmetric) second Piola-Kirchhoff stress tensor S in eq. (8.78), use the chain rule to obtain

DAuS(E(u)) =

DS(E(u)) OE : DAuE(u) = C(u) : DuuE(u) ,

(8.79)

with D~uE denoting the linearization of the Green-Lagrange strain tensor E (see relations (8.69)). It is important to emphasize that the term 8S(E)/8E is precisely the elasticity tensor C in the material description, as defined in eq. (6 . 155). It is a fourthorder tensor which possesses the min.or synunetries C.~\BC'D = CTJACD = CA11DC· Hence, eq. (8.78) may be re-expressed as

Duuc5I'Vint(u,8u) =./[S(E(u)): Dauc5E(u) + t5E(u): lC(u): DuuE(u)]dV . (8.80) no Finally, we use the e~plic.it expression (.8.70) and property (1.95) for the first term of the integral in eq. (8.80) and relations (8.14) and (8.69) .for the second term. Since the stress tensor S is symmetric and the elasticity tensor C has minor symmetries, the linearization of the i~~emal virtual work in the material description leads to the set of linear increments

DD.uc5H~ 11 t(u, Ou) =

.!

(GracMu: GradLiuS

no +FTG.radc5u : C: FTGrad~u)d·V.. ,

(8.8l)

(8.82)

8. 4

Linearization of the Principle of Virtual Work

397

which describes the fully non.linear (finite) deformation case. The terms 6abSnfJ. and Fc1A.F1icCAncv represent the effective elasticity tensor, which has the nature of the (tangent)

st~ffness

matrix.

Relations (8.81) and (8.82) are linear with respect to Ju and ~u depending on X. These relations show a clear mathematical structure in the sense that du and Liu can be interchanged without altering the result of the .integral; thus D il·urHVint (u, c5.u) = DJu{HVint(u, ~u). Relations .(8.81) and (8.82) lead to a symmetric (tangent) stiffness matrix upon discretization. Note that, for example, the set of nonlinear equations associated with nonlinear heat conduction results in a different mathematical structure leading to a non-symmetric stiffness matrix. The first term in eq. (8.81.) comes from the current state of stress and represents the so-called geometrical stress contribution (in the literature sometim·es called the .initial stress contribution) to the linearization. Since SAB is not the initial stress (it is in fact the current stress}, the terminology is misleading. Within an incremental/iterative solution. technique we can think of Siw as the initial stress at every .increment, so the term initial stress contribution has some meaning. The second term in eq. (8.81) represents the so-called material contribution to the Iinearization. The linearized principle of virtual work (8.8]) constitutes the starting point for approximation techniques such as the finite element method, typically leading to the _geometrical (or initial stress) stiffness matrix and to the material s"tiffness matrix. Note that for some cases .it is more convenient to discretize the nonlinear variational equation as a first step and to linearize the result with respect to the positions of the nodal points as a second step. Linearization of "the principle of v.irtual work in spatial description. In order to linearize the principle of virtual work in the spatial description we recall the nonlinear variational equation (8.33 ).

As above we consider the static case (ii = o) and assum.e the loads band t to be independent of the motion of the body. Only the linearization of the internal virtual work c)Hlint in the spatial des-cription remains. We adopt 6lVint in the spatial {or Eulerian) form (8 . 34). The idea is first to pull-back the spatial quantities to the reference configuration, so they correspond with the internal virtual work in the material description. Then they are linearized, as above, and as a last step it is necessary to push-forward the linearized terms. Starting with the equivalence

c5H'i11t(u, c5u)

=I n

o-(u) : r5e(u)clv

=/

S(E(u)) : c5E(u)dV

(8.83)

no

we consider the linerization of the internal virtual work in the material description

398

8


which we have -derived in (8. 78), i.e.

DAu•HViut

= /(s: DAuOE + JE: Du S)dll 0

(8."84)

no (the .arguments have been omitted). Hence, the push-forward operation on the second Piola-Kirchhoff stress tensor S yields, according to (3.-64), the Kirchhoff-stress tensor T, which is related to the Cauchy stress tensor by T = Ju. Pushing forward the linearized variation of the GreenLagrange strain tensor, ~8E, yields the linearized variation of the Euler-Almansi strain tensor, Li.c5e, as discussed in Example 8.6. Computing the push-forward of JE results in 8e'f as introduced. in Section 8.1. Finally, we -derive the push-forward of the linearized second Piola-Kirchhoff stress tensor, i.e. the last term in (8.84 ), which will yield the linearized Kirchhoff stress tensor ~T. We write (8.85)

with .DauS given by (8.79), i.e ..DAuS = C : D~uE. By use of (8.69) and (8.·66), the term F(DuuS)FT in-eq . (8.85h may be written as

(8.86) where we have also employed the minor symmetries of C~ In order to proceed it is more instructive to employ index notation. With the definition (6.159) of the spatial elasticity t~nsor (c)aiu~tl == Cabcd' eq. (8."86) is equivalent to ~u\CABCIJFcc

.a.6. u(~ a~ 'lLc a·.Xd ~rnFl1JJ = FaAFb11F~cF;uJCA/JCD" a·Xd =

8~'llc Jcabcd

a. :1;tl .

.(8 .. 87)

Hence, the 1inear.ization of the spatial Kirchhoff stress tensor, i.e. (8 . 85), gives .the useful relation ~r

=,Jc: gradAu .

(8.88)

Note that the increment tJ.r denotes the linearized tensor-valued function T according to the concept of directional derivative introduced in (8. 72). Replacing the associated direction ~u, used in the directional derivative, by the velocity vector v, Ar and grad~ u result in the Lie ti.me derivative .t v ( r) of T and the spatial velocity gradient I (defined by (2.139) 4 ), respectively. By using the symmetries of c relation (8.88) reads l~v(r) =Jc: d, which proves (6.16l).

8.4

Linearization of the .Principle of Virtual Work

399

Considering the push-forward operations derived we obtain finally~ from: (8~84) with relation dv = Jdl/ and some rearranging, the linearized internal virtual-Work in the spatial description, i.e.

D~uJlVi11t(u,Ju) = /(gradJu: grad~uu + gradJu: c: gradflu)dv

,

(8.89)

n or, in index notation,

(8.90) where <50 cabd + Cabcd represents the effective elasticity tensor in the spatial description. Relations (8.89) and (8.90) are .linear with respect to the terms 6u and Liu. They describe the fully nonlinear (finite) deformation case and have a similar symmetric .structure to the linearized eqs. (8.8 l) and (8.82). Formulations according to (8.81) and (.8 ..89) are in the literature sometimes called .total-·Lagrangian and updated-Lagrangian, respectively. This really means that integrals are calculated over the respective regions of the reference and the current configuration. However, it is important to emphasize that the derived material representations (8.81) and (8.. 82) of the linearized virtual internal work are equivalent to the spatial versions (8.89) and (8.90). The two representations are bas.ed on the use of change of variables, and the resu Its are the same in both cases. In order to recapture the small deformation (but nonlinear elastic) .case (in the "literature often called the materia1ly nonlinear case) we fix the geometry, i.e. we do not distinguish between initial and current geometry. Further~ we do not account for the initial stress contribution Oaeabcl and ignore the quadratic terms in the Green-Lagrange strain tensor. In addition, for the fully linear case, the coefficients of the elasticity

tensor are given and are not functions of strains anymore.

EXAMPLE 8.7 An alternative approach to derive the linearized .internal virtual work (8.89) in the spatial description is to use the formal equivalence of the material time derivative of scalar-valued functions in the spatial description with the directional derivative of these functions in the direction ~u (compare with Section 2.8). Carry out the material time derivative of the internal virtual work <5H'int in the spatial description and use this property to obtain its linearization. This approach circumvents the extensive pull-back and push-forward operations.

400

8


.Solution. We again start with 8lVint in the spatial form (8.34) and change the domain of integration with dv = J d v· so that

OlVi 111(u, cfo)

=j

u(u) : Oe(u)clt1

n

=

f

.lcr(u) : c5e(u)dll

(8.91)

no

where the .Euler-Almansi strain tensor e depends on the displacement field u through the relationshi.P (2.90). The Cauchy stress tensor u also depends on u, here a function of the current position x. Employin.g mate·rial time derivatives and the product rule we find from (8.9.lh that

cll'Vint{u, c5u)

j [r(u) : Je(u} + Oe(u) : r(u)]dF

=

,

(8.92)

no

where additionally the relationship between the symmetric Kirc.hho.ff-stress tensor r and u, i.e. ,,- = Ju, is to be used,. Recall that the superposed dot denotes the material time derivative, as usual. Firstly, we derive the material time derivative of the virtual Eu1er~Almansi strain tensor cfo. By means of (8.7) and (2.14Jh_, we find from (8.18h that

. {fo

.

= syrn(grad6u) == sym(Gra.d6uF- 1) = sym(GradouF- 1) = syrn[Grad8u(-F- 1 f)]

= -syrn(grad&u I)

(8~93)

,

with the spatial ve.1ocity gradient I = gradv (recall definition (2.135)). Secondly, we focus attention on the crucial material time derivative of the Kirchhoff stress tensor T. We start with the relation for the (objective) Oldroyd stress rate of the spatial stress field T. Recall the Oldroyd stress rate of T which is identical to the Lie time derivative of 1, and with reference to eq. (5 ..59) expressed as .l\. (T) = 1- - Ir - rfr. By means of£,. (r) = Jc : d., i.e. eq. (6."161. ), we conclude that ..

+ = Jc : d + IT + Tfl'

,

(8.94)

where re and d = sym(J) are the spatial elasticity tensor and the rate of deformation tensor (compare with definitions (6.159) and (2..146), respectively). Substituting relations (8.93)s, (8.1.8) and (8.94) into (8.92) and using the symmetry of T and the minor symmetries of ·c, i.e. CatJcd = ClJacli = CabcJc, we obtain

cH-Vint(u,Ou)

j

= [r: (-gradOul) +gradc5u: Jc: d no

+ grad8u : Ir + gracMu : rfr]dl,.·,. .

(8.95)

8.4

Linearization of the Principle of Virtual Work

401

By use of property ( 1.95) the sum of the first and last terms in eq. {8.95) vanishes and

we obtain

c5Wint(u, du)= /(gradOu: Jc: d + graddu: lT)dV .

(8.96)

no App.lying now the formal equivalence of the material time derivatives and the· di~ rectional derivatives, and replacing the spatial velocity field v by the linear incr.eme.nt ~u, we may rewrite (8.96) as

D~uJH'int(u, Ou)= /(grndJu: Jc: gradLlu + gradOu: gradLlu T)dV

. (8.97)

no Changing the domain of integration back and using the stress .transformation r == Ju again we arrive finally at the linearized internal virtual work in the spatial description, i.e. (8."8.9). B _ _ _ ,_.,,,:•·'···--•,,·,-···-~·---'-·''·--·--·--··----·· ... -

.... ~'.--~--·~,._,~,~·,.,.~ ...·~,.,,,..,.~,:.,~,~,~,~,.~·~,,,~ ... ,.,:,,,~~~'"''''·~''''··'~"'''·~·=,~~-:''·'~~····-----'·~-· .. ·• ... ~··--'··--· ... -----··--,·--·-~·,.,~·, ..·~,,,~~,. .. ,,,:,,~'"':'•'.-~-,.~~,.:,,: .. -~,..c-·- .

EXERCISES

1. Take the part Oucabd of the effective elasticity tensor due to the -current stresses in the spatial description. (a) Show that this term possesses the major symmetly, that means that nothing chang-es by the manipulation

(b) Further, show that it does not possess the minor symm.etry, i.e.

However, for the case in which the term 6a.cCTbrl is absent, under the assu mption of a hyperelastic material the minor and the major symmetry of C:ubccl hold. Discuss the consequences in regard to finite element discretizations. 2. Show that the linearization of the internal virtual work cH'Vint may also be written

as

D~uc5I·Vint(u, 8u) =

./ GradOu : A: GradAudF ,

no where A is a (mixed) fourth-order tensor useful in numerical implementations. It .is known as the -(fi:rst) elasticity tensor with the definition

A= tJP(F)

BF

or

f 1aA/J.IJ

= 0.Pa4 u b/J £l L1 •

.L''

,

402

.8


where P denotes the first Piola-Ki.rchhoff stress tensor depending on the deformation gradient F. Note that the linearization of dlVint contains one term only (compare with relation (8.81 )). 3. Carry out the linearization of the external virtual work c5lVexb assuming pressure boundary loading, as derived in eq. (8 ..38). Consider a constant .pressure load v and show by means of the product rule and relation (2.5) that D~u6TVuxt(u, 8u)

=p 'I

0/ a~u a,) .(.a~u x - - / 0 8 0 x ac · ·~ i

.

~2

. ~2

~)

c5ud~ 1 d~~

. (8.98)

n~

Note that the two terms in eq. (8.98) are not symmetric in c5u .and .~u. "In finite element discretizations the associated tangent stiffness matrix -is also not, in general, symmetric.

8.5

Two-field Varia.tion.al Principles

So far we have considered single-field variational princip.les such as the principle of virtual work. It is not always the best principle to choose, particularly when constraint conditions are imposed on the deformation. In finite element analyses of problems whkh a.re associated with ·Constraint conditions, significant numerical difficulties must be expected within the context of a Galerkin method, i.e. a standard dfa~placemen.t based method in which only the displacement field is dis.cretized. This ·method exhibits rather poor numerical performances such as penalty sensitivity and ill-conditioning of the stiffness matrix., as is well-known from (i) the numerical analysis of rubber, which is frequently mode.led as a nearly incompressible or incompressible material,

(ii) bending dominated (plate and shell) problems,

(iii) e.Jastoplastic problems that are based on lrft.ow .theory (the plastic flow is isochoric ), and (iv) Stokes' ft ow, which malhematicaHy yields a problem .identical to that for isotropic incompressible elastic.ity.

In the ·computational literature ·these devastating numerical difficulties are referred to .as locking phenomena. Essentially, these locking difficulties arise from the overstiffening of the systetn and are associated with a significant loss of accuracy observed,

8. 5 Two-field Variational Principles

403

in particular, with low-order finite elements (for fundamental studies and for more references see the :books by HUGHES [1.987] and ZIENKIEW"JCZ and TAYLOR [1989, 1991]). To eliminate these difficulties inherent in the conventional single-field variational approach a great deal of research effort by engineers and mathematicians bas been de·voted to the developments of efficient so-called mixed .finite element methods (see HUGHES (1987, and references therein]). For a more mathematically oriented presentation see lhe work of BREZZI and FORTIN [1991 ]. For these types of methods the constraints imposed on the deformation are dealt with within a variational sense resulting in effective multi-field variational principles. This approach has attracted considerable attention in the computational mechanics literature. Besides the usual displacement field, a mixed {finite element) method incorporates one or more additional fields (typically the internal pressure field, the volume ratio field ... ) which are treated as independent variables. The basic idea within the ·mixed finite -element method is to discretize these additional variables independently with the aim of achieving nonlocking .and stable numerical solutions in the incompressible limit .

Lagrange-multiplier .method.

In the subsequent part we focus attention on a suit-

able variational approach which captures nearly incompressible and incompressible hyperelastic material .response. Rubber or rubber-like materials may show a very high resistance to volumetric changes compared with that to isochoric changes. Typically, the ratio of the bulk modulus to the .shear modulus is roughly of four orders of magnitude (exhibiting an almost incompressible response), and a ·very careful numerical treatment is needed. A standard disp"lacement-based method cannot be applied directly to these types of .problems., be·cause locking or =instabilities will occur. Hence, almost al.I hyperelastic materials which show a nearly incompressib1e or incompressible deformation behavior are treated with a mixed finite element formufation. The Lagrange-multi-plier method, in which a constraint is introduced using a scalar parameter called the Lagrange multiplier, is often used to prevent volumetric locking,. Utilizing .P as the Lagrange multiplier to enforce the incompressibility constraint .l = 1, we .may formulate a functional HL in the -decoupled representation

Il1,(u,p)

= Ilint.(u,p) + Ilext(u)

.

(8.99)

Ilint{U,JJ) = / fp(J(u) - 1) + 'lliso(C(u))JdV , (8.100) no where the term p(J(u)-1) denotes the Lagrange-multiplier term, with the volume ratio J = J (u) = (detC) 1/ 2 • The function \JI iso = \]_i iso ( C) c·haracterizes the isochoric elastic response of the hyperelastic material with .the corres_ponding modified right Cauc-hyGreen tensor C = C(u) = J- 2/:ic. The loads do not depend on the motion of the body,

404


so that the external potential energy IIext .is given by the standard relation (8.48) 2 • Note that the Lagrange-multiplier term vanishes for the case of incompressible finite elasticity. In addition, it is important to emphasize that the solution of the L
(8.101)

for all 8u satisfying c5u = 0 on the boundary surface anou and all <5p. Firstly,, we compute the directional derivative of (8.99) in the direction nf an arbitrary virtual displacement ~u. By means of the chain rule and the relations derived in Section 8.1, i.e. DsuJ = 8J = Jdivc5u and DduC = oC = 2oE, we obtain, with reference to (8.JOl)t, the weak form

. . + 2 aw.iso .) . = j~ (.Jpchvdu DC : 6E cH + Douilext = 0 1

(8.102)

flo

(.the arguments have been omitted). The contribution due to the loads is given by eq. (8.5lh as D.suilext = -(5ll'ext' with the external virtual work cHFext' i.e. (8.45). In order to rewrite the term Jpdiv<5u of (8.102h we use the analogue.of eq. (l.279)2 and invoke relation (8.7) and property (l .95) to obtain

=

= I : Grad8u F- 1 F-T: Grad6u = F- 1F-T: FTGrad6u

=

c- 1 : FTGrad8u

divc5u = I : gradJu

'

(8.103)

8.5

Two-field Variational Principles

405

c-

c- 1 is symmetric

1 with the inverse right Cauchy-Green tensor = F- 1F-T. Since we find finally, by analogy with property (l . 115) and with {8. l4h, that

div~u

= c- 1 : .
.

(8.104)

This result substituted back into (8.102)2 gives

Dauih ( u,p)

= ./

' (

.

l

J(u)pC- (u)

( c (u) ) ) + 2 aw isoBC

: 8E(u)dl'

no +D~ullmct(U) = 0 .

(8.105)

By recalling the unique additive decomposition for the stress, i.e. eq. (6 . 88h with definitions (6.89h and (6.90) 1, we recognize that the two terms in parentheses of the variational equation (8. l 05) give precisely the second Piola-Kirchhoff stress tensor S = S(E(u)). Hence, the integral in (8.105) characterizes the volumetric and isochoric contributions to the interna.I virtual work c>H~nt (see eq. (8.46h). Relation (8.105) .is identified as the standard principle of virtual work expressed in the reference configuration (for a configuration in static equilibrium), Le. cHVint 8fl'ext = 0. The associated Euler-Lagrange equation is Cauchy's equation of equilib-

rium. Secondly, we compute the directional derivative of (8.99) in the direction of an arbitrary virtual pressure OJJ. With reference to {8. l 01 h, we obtain the weak form

DavITdu, p)

=/

{.l(u) - l)Jpdlf

=0

.

(8.106)

flo

As can be seen, the pressure variable p maintains the incompressibility constraint and we find the associated Euler-Lagrange equation to ·be J = 1 (see also the study by LE TALLEC [1994]) .

The variational equations (8~.105) and (8 .. 106) provide the fundamental basis for a finite element implementation.

Linearization of the Lagrange-multiplier method.. In order to solve the nonlinear equations (8.105) and (8.106) for the two .independent variables u and IJ on an .incremental/iterative basis, a Newton type method is usually employed (recall Section 8.4 ). In preparation for an incremental/iterative solution technique a systematic lineariza.. tion of (8.105.) and (8.106) with respect to u and p, essentially the second variation of (8.99), is required. In order to linearize variational equation (8.106) in the directions of the increments ~u and ilp we recall that Duul = Jdiv~u. Employing the concept of directional derivative we obtain

D~p,.
.!

no

J(u)div !:lo Jpdlf ,

D~v,~vUL(u,p)

=0

.

(8.107)

406

8


The :Jinearization of the principle of virtual work (8 .105) in the direction of the .increment D,.p .gives

f

Diu,.e.1,Ili,(u,p) = / J(u).6.pC- 1 (u): OE(n)dF = J(u).6.pclivOudl' , (8.108) no no while a linearization process in the direction of Au was already ·carried -out in detail within the last section., leading to

Diu,AuII1,(u,p) = / (GradJu : Grad.6.u S no +FTGrad6u : (
:

FTGrad~u)dV .

(8.109)

Sinc·e (8. l05) is based on the additive decomposition of the stresses S, we obtain the decoupled representation of the ·elasticity tensor C = Cvot + Cj80 , with the definitions

C

_ ./J(J(u)pc- 1 (u)) vol -

~

ac

and

,,,. 'l,, 180

_ , 8 2 \Jliso(C(u)) 1 acac · -:

·

(8. l.1.0)

For an explicit treatment of these expressions recall Section 6.6, in particular relations (6.166) 4 and -(64168). Note that for the considered case the scalar quantity p must be replaced by pin eq. (6 . .166)4 .. Since the structure of the linearized princip.le of virtual work is symmetric and because of the symmetry .betwe.en eqs. (8.l07h and (8.108) a finite ele.ment implementation of this set of equations will lead to a symmetric (tangent) stiffness matrix. In order to use these equations within a finite element regime so-called interpolation functions must be invoked separately for the displacement field u, the pressure field p and their variations and op, respectively (see SUSSMAN and BATHE [1987], ZIENKfEWl.CZ and TAYLOR [198.9] among others). A wen-considered choice of these functions is a crucial task in order to alleviate volumetric locking. It was observed that a discontinuous (constant) pressure and a continuous displacement interpolation over a typical finite element domain is computationally more efficient than with the choice of functions of the same order for u and p.

.au

Perturbed Lagrange..multiplier method.

The Lagrange-multiplier method resuits in a stiffness matrix which is not positive definite for incompressible materials . .In order to overcome the numerical difficulties associated wjth this fact and to avoid ill-conditioning .of the stiffness matrix .associated with the .penalty approach, regularization procedures such as the so-called perturbed Lagrange-multiplier method (in the literature -often referred to as the .augmented Lagrange. .multiplier method) have been introduced successfully (see, for example, GLOWINSKI and LE TALLEC [1.982, 1984]). "It may be viewed as a two-field variational princ.iple in which the functional (8.99)

·s.s

Two-field Variational Principles

407

is perturbed-( or augmented) by .a penalty term. Thus,

llp1,(u,p) = /(p(J(u) - 1) + '11iso(C(u))]dl' ·no

1/.1 -TJ2.1''" ( + IT () 2.

- -

ext. U

Y

no

(8 . .111)

K,

(see the work of CHANG et al (1991]), where the third (penalty) tenn in functional (8.1 :I I) reguJarizes (relaxes) the incompressibility constraint J = 1 involved in the :first term of the integral. The Lagrange multjplier JJ which enforces the constraint no longer has the mean-ing of a pressure, in contrast to the Lagrange-multiplier method .. The positive penalty parameter n, may be viewed as a (constant) bulk modulus. Incompressible materials can be treated by repfacin_g .1 /"" with zero, so the first and t~e third terms in (8.111) vanish. For this incompressible limit the penalty method, the Lagrange-multiplier method and the perturbed Lagrange-multiplier method lead to identical equations. By taking for the term (.J(u) - 1) a more general, sufficiently smooth and strictly convex function Q(J(u)) (so that ·Q(J{u)) = 0 if.and ouly if J = 1) the functional (8.l 11) .is identical to that proposed by BRINK and STEIN [ 1996]. Functional (8. ll I) may also be identified with a special form of the two-fie"Jd variational principle given by ATLURI and REISSNER [ 1989]. This work deals with a general framework for -incorporating volume constraints into multi-field variational principles. In addition, note that the formulation (8. I 11) also reduces from a mixed {finite element) formulation proposed by SUSSMAN and BATHE l1987l EXERCISES

1. Consider the decoupled strain energy formulation proposed ·in (8.55), i.e. w(C) = WvoifJ) + Wiso(C), with '11vo1(J) = tt.9-(J), and treat the displacement u and the hydrostatic .pressure p as independent ·field variables (permitted to be varied). Require that dQ ( .J) / dJ == 0 if and only if .J = I. (a) Derive the stationary condition with respect to u and incorporate the definition of the volumetric stress contribution, i.e. (6.89), with the constitutive equation for the hydrostatic pressure p according to (6."91 ).1• Show that the resulting variational equation is, in accordance with the principle of virtual work, in the form (8.105).

(b) Obtain the additional variational equation .in the form

/ no '

~ (dQ(J(u)) - 1,.) r dll = 0 p. op ' dJ ·

K,

(8.112)

408

8 Variational Principles which is relation (8.61) enforced in a weak sense. Interpret the .result for the case "'

-7

oo.

Note that variational equation (8 . J12) ·cannot simply be obtained by taking the first variation of the energy functional. This type of two-field variational principle was proposed by DE BORST et al. .[1988] and VAN DEN BOGERT -et al. (1991]. 2. Consider the augmented functional (8.111),. (a) Derive the stationary conditions with respect to u and 1J, i.e. D 00 Ilr1.. ( u., p) = 0 and D~pllrL (u, p) = 0, for arbitrary variations ou and 8p, respectively. Show that the Euler-Lagrange equations are Cauchy's -equation of equilibrium and the artificial constitutive equation TJ = Ii ( J (u) - 1). (b) Show that the linearization of the Euler-Lagrange equations in the weak forms gives

and three further equations which are in accord with (8.107) 1, (8 . 108) and the linearized principle of virtual work, which has basically the form of eq. (8 . .109). Note that, thereby, the elasticity tensor
Assume that the constitutive relation P(F) =

aw (F) I 8F' as .introduced in (6.1) l

t

is invertible, which is not a valid assumption, in general. {It is important to emphasize that, in general, there does not exist a unique deformation gradient F corresponding to a given first Piola-Kirchhoff stress tensor P (see OGDEN [1997, Section 6 . 2.2])). Define a complementary straln . . energy function Wc{P) so that a Legendre transformation gives

Wc(P)

=P :F -

\lt(F) ,

where P and F are the first Piola-Kirchhoff stress tensor and the deformation gradient_, respectively. Hence, the functional

Il1rn(u, P)

= /rr: F no

'11c(P)JdV - / B · udV flo

8.6

Three . .fie.fd Variational Principles

-J

T · udS -

J

409

T · {u - u)dS ,

anuu

ano~

which is valid for large strains, is referred to as the Hellinger-Reiss-ner functional. Here, the prescribed loads are B on r2 0 and Ton 80 0 cr and are assumed to be independent of the motion of the body. The third quantity prescribed is the displacement field u acting on the boundary surface 8!1011 • Note the relation T = PN introduced in (3.3) 2 . (a) Invoke the stationarity of 111-m and determine the weak form of the elastic equilibrium equation. Since the principle is based on treating the displacement u and the stress P as independent field variables (permitted to be varied), -evaluate -separately Dfoilnn(u, P)

=0

,

D(ir.11.nR(u, P)

=0

.

The first variations of u and P are arbitrary vector-valued and tensor-valued functions for which the restriction 8u = 0 over the boundary surface 11

ano

holds. (b) Show that the associated Euler-Lagrange equations for the volume of the body and the (Dirichlet and von Neumann) boundary conditions are

D.ivP + B = o , U=U

on

8f2ou ,

F(P) T =PN

= 8'11c(P) DP

=T

' on

an0CT •

The first relation represents Cauchy's equation of equilibrium in the material description, while the second relation denotes the inverse form of the constitutive equation for a hypere.lastic material (in some cases not avail~ able).

(c) Alternatively to the functional stated ubove find IlnR for which displacements and strains are the independent variables (instead of u and P).

8.6 Three-field Variational Pri.nciples We are interested in a suitable variational approach in order to capture nearly incompressible .and incompressible materials. It is recognized that a constant pressure interpo"Iation over the finite element within the framework ·of two-field variational princi-

410


pies leads to unpleasant pressure oscillation. There are some remedies for this prob-lem .in the computational literature, for example, the pressure smoothing technique by HUGHES et al. [1979]. However, a certain improvement is based on the idea of introducing additional independent field variables such as the volume ratio, leading to a more efficient three-field variational principle. Simo. .Taylor-Pister variational princip·l-e. A very efficient variational principle that takes account of nearly .incompressible response was origina11y proposed by SIMO et al. f 1985] and is known as the mixed Jacobian-pressure formulation (for relevant applications to elasto.mers see SIMO [J.987] and SIMO and TAYLOR [199.la.D. :It emanates from a three-field variational principle of Hu "(1955] and WASHIZU [1.955]. Thereby, be.sides the displacement and pressure fields u and p, a third additional kinematic field variab1e1 which we denote by J, is treated independently within finite element dis~ cretizations. The principle is decomposed into volumetric, isochoric and external parts and is defined by the express-ion

flsTP{u,p, J) = /[\J!vo1(J) no

+ p(J(u) -

i)

+ W;so(C(u))]dV+ Ilcx~(u)

. (8.113)

Following Simo-Taylor-Pister the first two terms in the three-field variational principle are responsible for the nearly incompressible behavior of the material. They describe volume-chan_ging (dilational.) deformations and are expressed by J, p and the new variable J. The kinematic variable J enters the functional as a constraint which is ·enforced by the Lagrange multiplier p . The Lagrange multiplier is an independent field variable which may be identified as the hydrostatic pressure . In .addition to the virtual displacement and pressure fields -r5u and ,ov, we introduce an arbitrary smooth (vector) function oJ(x) = oJ(x(X)) = <5](X) for the constraint, which we call the virtual volume change (here defined on the reference confi_guration) . Jn equilibrium, functional (8.113) must be stationary. The necessary conditions for the stationarity of functional IlsTP with respect to the three field ·variables (u, p, i) are evaluated separately. We require

D5uilsTp(u,v, ])

=0

, (8.114)

for all c5u satisfying ou == o on the part of the boundary surface DOnu where displacements u are prescribed and all c5v, 8]. Differentiating functional IlsTP with respect to changes in u gives the weak form of the elastic equilibrium~ i.e . the principle of virtua1 work in the form of (8 . 105). For an explicit derivation recall the manipulations of the last se-ction. A straightforward differentiation of fisTP with respect to changes in the field vari-

8.6

Three-field Variational Principles

ables p, .i gives the weak enforcement of the equivalence between .J and constitutive equation for the volumetric changes, Le.

D.rpilsTP(u,p, J)

=

j (.J(u) - ])i>pcW

= 0 ,

.i; and the. ··

(8.115)

no (8.1.16) For arbitrary :c5p, the variational equation (8.115) results in th~ Euler-Lagrange equation J - } = 0.. It implies that the additional independent variable J equals J == ( detC ( u)) 1/.2 , i.e. the kinematic constraint associated with the volumetric behav·ior. For arbitrary -6J, eq. (8 . .l 16) results in the second constraint condition in the local form~ that is the Euler-Lagrange equation d'1Iv 01 /dJ - p = 0. This is the standard constitutive equation imp··lying the volumetric stresses to be equal to the hydrostatic pressure. A finite element procedure in which the dilatation J and the pressure variab]es pare discretized by the same local .interpolations as for the displace-ment field u would not give any advantage. To prevent volumetric locking an appropriate choice of the i.nterpoJation functions for the volumetric variables p, j and their variations OJJ, 8 j is crucial. A simple formulation arises by discretizi.ng the dilatation and pressure variables over a typica1 finite element domain with the same discontinuous (constant) function which need not be continuous across the finite element boundaries. This approach is known as the me.an dilatation method and is proposed in the notable work of NAGTEGAAL et al. [ 1974] who recognized _the effect of volumetric locking in elastoplastic Jr flow theory. Since the interpolation functions are discontinuous, the volumetric variables p, J can be eliminated on the finite ele.ment level, a process known as static condensation in the computationa] mechanics literature. Therefore, the variational equations (8. I 15) and (8.116) need not be solved on the global level leading back to a reduced displacement-based method. The work of BRINK and STEIN [1996] is a comparative study of various multifield variational principles. It emerges that under certain conditions the above threefield variational principle and some two-field principles yield the same discrete result in each step of the Newton method. EXERCISES

l. Consider the functional (8,. 1l3) with the three independent field variables (u, p, }) and the associated variation equations (8."115), (8.l 16), (8.105), and show that for each step of the Newton type method the problem is completely described by the

412

8


set of linearized equations

D~p,iluTisTP(u,p, .l) =

/

J(u)div~uOvdV

,

no

D~p,o.}IsTP(u,p, ]) =

- /

~.JOpdF

,

no

D~j,a 1 IIsTP(u,p,J) =

- / ~pt5.icIV ,

no

and the linearized principle of virtual work, which has the .form of eqs . {8.109)

and (8. l 08). 2. The described functional IlsTP ( u, p, ]) .takes into account only the volumetric strain and stress components. Study a more general and very powerful type of a Hn-Washizu variational prindpl-e fundamental for various finite-element meth-

ods, i..e.

TI 11 w(u, F, P)

=/

(llt(F) - P: F - B · u - DivP · u)dl'

flu

+/ anocr

u · (T - T)dS - /

emu

T · (u - u)dS , ll

with the three independent variables u, F, P and the prescribed quantities Bon Oo, Ton 80.oCT and u on cK2o u· The loads Band Tare .assumed to be conservative and the first Piola-Kirchhoff traction vector Tis given in eq. (3.3h. (a) With identity (1.289) show that the Hu-Washizu variational principle can be posed as a g-eneralization of the principle of virtual work, i.e.

TIHw(u, F, P)

= TI -

/ P : (F - Gradu)dV - ./ T · (u - u)dS ,

no

anou

where the total potential energy 11 is given in (8.47) and (8.48). (b) Invoke the stationarity of Huw with respect to u, F and ·r. The vectorvalued and tensor-valued functions 6u and c5F 1 c5P are arbitrary with the -conditions 80 = o over the boundary surface 8fl 0 u and 8P = ·O on 80 0 u· Show that the associated Euler-Lagrange equations for the functional Ilnw

413

8.6 Three-field Variational Principles

are p

DivP+ B = o ,

= 8\Jl(F) DF

'

F =Grado ,

with the (Dirichlet and von Neumann) boundary conditions U=U

on

ano

\1

'

for the body under consideration.

T

= PN = T

on

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Index

'back-cab, rule, 9 balance of

absolute temperature, 168; see al~w temperature acceleration

angular momentum, 142, ·144, 147, 149, 150, 175,378

ccnlrifugal, 185

Coriolis, 185 Euler, 185 gravitational, 142 local, 67 acceleration field, 62-65, 67, ·97, 98, 142, 149

energy in continuum the:rmodynamics, 161-166; :;ee also first law .of thermodynamics in entropy form, 170, 172, 327 in material description, 164, 165 in spatial description, 164 in temperature form, 327, 342r 360 linear momentum, 141-144, 149, 150, 175 mass; see con~ervation or mass mechanical energy, 152-160 in material description, 155-157 in spatial description, 153-155 moment of momentum, l 41, 142 rotatfonal momentum, 141, 142 ·thermal c nergy, J64 balance princip·le, master, 174-1 77 in global form, 174-176 in local form, 17 6 basis Cartesian. 3, 10, 34

convective, 67 under changes or llbscrvers, .183-185

activation energy, 364 adiabatic material, 325 adiabatic operator splil, 332 adiabatic process, ·112, 173, 336, 356 reversible, 172, 173, 348-351 affine motion, 71, 3 .l 6 algorithmic elasticity tensor, 293t 294 algorithmi.c stress, 291 293, 294 alternating symbol, 6 angle between vectors, 2, 5, 16, 28, 33, 188, 258 angular momentum, 141, 142 balance of, 142, 144, :147, 149, 150, 175, 't

378

change of, 28 dual, 32.. 35

angular velocity, 59 angular velocity vector, 98-100 anisotropic material, 214 Arrhenius equation, .364 Arruda and Boyce mode~, 244., 248, 249 . 263 artery, 249, 273 atomistic theory, 56, 57 .augmented Lagrangc-mult.ipHcr method, 406 axial vector, 17, 20, 48, 98, I00, ·105

general, 32-37 orthonormal, 6, 12-14, 25, 26, 88, 91 reciprocal, 32, 35. 37 basis vectors, 3-5, 22, 28-32, 40, 82, J 81 contravarianl, 32-34 covariant, 32-34 general, 32-34 .angle between, 33

435

436 basis vectors, general (comd.) length of. 33 orthonormal, 10, 11, 26, 57 't :) 14, 225, 258 reciprocal, 32 Beltrami vorticity equation, 137 biological soft tissue, 235, 273, 306 bjomaterial, 235,, 249, 273 biomechanics, 249, 273 Biol strain tensor, .88 Biol stress tensor, 128, 158 Biot traction vectur, I 28 Blatz and Ko model, 247, 248, 261, 262 body deformable, 59 free, l l 0 homogeneous, 197 incompressible, I 03, l36 uniform dilation of, 92 body force, 142, 147, ·.t48, .197, 378, 382, 385, 387 prescribed, 379 reference't 144, 384 Boltzmann's conslant, 315, 320 Boltzmann's equation, 315 Boltzmann's principle, 315 boundary conditions Dirichlet, 378, 384, 409 essential, 381 natural, 381 pressure, 383 von Neumann, 378, 384, 387, 409 boundary and initial conditions, 378, 379, 381 compatibility of, 379 boundary loading, pressure, 383, 384, 402 boundary surface 52. 131 decomposition of, 378, 384 insulated., l 32 · parametrization or, J83 boundary-value problem, .380 box product, 8 bulk modulus, 245, 337, 338, 389, 3-90., 407

calculus of variations, fundamental .lemma of, 381 caloric equation of slate, 323 caJorimetryl .325-327 carbon-black fill-ed rubber. 242, 243, 297, 298

Index carbon ..black fillers., 298, 361 Carnot thennal engine, 356 Cartesian components for a tensor, 11, 12,, 23, 29, 49 for a vector, 4, 29 Cartesian coordinate system, 28. 124, 125, .J 81 Cartesian tensor, l 0 Cauchy-el as tic maleri al, 197 . · presst-··bl e, .,. . . O""L., "03 mcom ~ isotropic_, 200-202 Cauchy-Green tensor Euclidean transformation of, 19 l left, "81 , 88 mod Hied, 23 2 right, 78, 88 modified, 228 spectral decom_position of. 90 time der.ivalivc of, l 0 l _, 102 Cauchy stress, 1.11 Cauchy stress lensor, 111-115, 123-.127 additive decomposition of, 231, 232. corotated, 1.28 Euclidean trans.fon11ation of, 190 fictitious, 232 Lie time derivative of, 193; see also Oldroyd stress rate spectral decomposition of't 120 symmetry of, 147 Cauchy traction vector, 11 J, l l 3-120, 142, 147, 148 Cauchy'ts.cquation of equilibrium, 145, 197, 405 Cauchy's first equation of motion, 144-.146, 148, 176,342,378-381 .Cauchy's law, 11. I Cauchy's postulate, 11.l Cauchy's second -equation of motion. 147 Cauc.hy's stress theorem~ .111-J 14_, 147, 14a, 150, 154, 175 Cayley-Hamilton equation, 25, 27 44, 89, 202 chain, 239, 242, 244, 290, 307-311. 356, 364 contour length of, 308~ 312, 313, 315. 3 18 entropy change of, 3 :J 7 Gaussian, 3J2...J16, 319 in the network, 317-318 out of (detached from) the network, 3123"18 characteristic equation, 25, 89, 202 -characteristic polynomial, 25 9

437

Index chemical potential, 323 .tensor of, 2·11, 2 .12 circulation of a vector field, 53 Clausius-Duhcrn inequality, 168-J 70, .172 Clausius-Planck inequality, .170, 173, 208. 229, 280,299,321,323,358 dosed system, 1.31-133, 141, 319 coaxial tensors, 20 I, 204, 216 . 226, 258 coefficicnl of lhermal conductivity, 171, 342 Coleman-Noll procedure, 208, 223, 23.0, 281 collage.n,273 complementary strain-energy function, 408 components accc.Jcration, 63 Cartesian. 4, 11, 12, 23. 29, 49 contravariam, 34-39 covar.iant, 34-39 mixed, 36-39 rectangular, 4; see also Cartesian compo .. nents stress, l l 4-1 l6 of a tensor, 11, 20-22 of a vector, 4

velocity, 63 composite mate.rial, 265, 266, 273 with two families of fibers, 272-277 compressible hypere1asticity, 228-231 compressible isotropic hyperelasticity, 231-234 in tenns of invariants, .233, 234 compression pure, 124 uniform (uniaxial), 92, 124 compressive stresses, 117 concept of directional derivative, 46-48 entropic e.lasticity, 333 internal variables, 278, .279, 283, 361 :Jinearization, 393-395 condition of a sys.tern, 1·61 configuration of a continuum body current (de.fom1ed)., 58, 59 final, 211, 212 homogeneous, l 33 initial, 58, 62, 211, 212 intermediate, 128, 344, 345 reference (undeformed), 58, 59 stress-free, 208, 247, 362, 363, 379, 3:90 therm.a] stress-free . 344

virtual, 372 conformation of .molecules, 307, 308, 310, 313315 conjugate pair, work, .159; see also work .conjugate ·Conservation of mass for a closed system, 132-.134 for an open system, 136 conservative system, 159, 160, 319,, 386 conservative vector field, 48 conserved quantity, 132, 142, J54, 160 ·consistency condition, 60, 64, 74, 236, 337 consistent linearization process, 257, 339-342, 393; see also linearization constitutive equations, 16 J, 1.97, 358 derived from \lf(b}, 217, 218 derived from '11(v), 218, 219 general forms of, 197, 201, 216, 223, .322, 358

internal, 2"81, 285, 288, 359, 363, 368 reduced forms of, 198, 199, 210 in spectral forms, 220-222, 246, 340, 341 in terms of principal invariants, 215-217, 224, 233, 234, 248, 249, 269, 270, 274-277 constitutive model, 207; see also model consti.tut~ve Lh.eory, 206 of fi n.ite . eJasti city, 23 5 of finite the.nno(visco)elasticity, 306 constrained material, 222 constraints auxiliary't l 19 external, 379 mcompressi · ·b·1· 11ty, 103 , .,,,., .... __ , ?"3 __ , n5 _... , ?37 ...... , 247,270,403,405,407 internal, I03, 202, 222, 223, 270., 389, 411 interna) kinematic, l-03; see also .constraints, internal continuity mass equation, 134-136 rate form of, ·J 35, 136 continuum, 57 non-polar, 144 po-Jar, 144~ 147, 152 .continuum approach, 56, 266 continuum body, 57; se.e .also body continuum damage mechanics, 295. 299 continuum damage theory, 295 continuum mechanicst 55

438

Index

continuum particle, 56 continuum theory, 56 contraction of tensors, 14, 15, 21-23 contravariant. basis vectors., 32, 34 . 35 conlravariant components, 34-39 contravariant Lcnsor, 36, 83, 127, 193, 194, 253 contravarianl vector, 35, 83, 163, "I 86 controllable quantities, 278 control mass, 13 l .control surface, .132, 136, 149, ·150 control volume, 132. 136, 149, 150

convected rate of a tensor ·field, 193, l 96 of a vector field, l93, 196 convective rate of change, 66 convolution integral, 289-292, 294 numerical integration of, 291 coordinate (co.mponent) expression, 3 coordinates material (referential), 58, 60, 62, 71, 78 .

I H, 127 spatial (current), .58, 60, 62, 71, 81, 111,

127 co .. rolational rate ·of a tensor field, 192, I 96 of a vector field, "l 92 Cotter.. RivHn rate, 193, 196 couple ·body, 143 resultant, l 10, 143, 144, 147, l52 couple stress tensor, ·152 .covariant bash~ vectors, 32-34 covariant components, 34-39 covariant tensor, 36, 83 covarianl vector, 35, 83 creep. 279, 286 creep test, 295 cross product, 5-7 crystallization, strain-induccdt 311, 356 curl material, 65 spatial, 66 of a vector field, 48, 49

curve closed, 53 material (undeformed), 70 spatial (deformed), 70 cycle, 161, 168

damage, isotropic discontinuous, 300, 302 maximum, 300 saluralion parameter, 300. 304 damage accumulation, 300, 30 I damage criterion, 30 I damage model in coupled material description, 298-30 I in decoupled material description, 303., 304 damage surface, 301 damage variable, 298-300, 304 dashpot, 279, 280, 286-288, 366-369 de.formation, 59., 70-73 biaxial, 92, 93! 124, 225, 237 equibiaxial, 92, 124, 226

with isotropic damage, 304 homogeneous, 71 , 316, 319, 348 inhomogeneous, 71, 212 inverse, 59

plane, 92 pure shear, 92, 93, 227, 250, 304 simple shear, 93, 94, 227., 243, 366. 367 thermoelastic, 344-346, 352 uniform, 92., 251, 352 volume-changing (dilational),.228, 232, 410 ~ . (d.1stort10na .• . l") , "" vo Iume-prescrvmg __ 8, ?J? _ ... deformation gradient~ 70-73, 82, 83, 85, 90 detenninant or, 74.~ see also volume ratio Euclidean transformation of, 1·89, 209 first variation of, 374, 375 inverse of, 71 first variation of, 374, 375 lincarization of, 394

time derivative of, 96 lincarization of, 394 modified, 228 time derivative of, 95, 96 volumc .. chan.g.ing (dilational) part of., 228 volume-preserving (distortional) p.art·of, 228 deformation tensor Finger, 8 l Green, 78 Pio"la, 79 rate of; see rate of deformation lensor rotated rate of, l 01, 158 density, 133 network., 3 12 reference mass, "I 33-135, 197

lndex spatial mass, 133-136, 14 I, :142, 148, ·149 t 202 surface, 174-176 volume, 175 density in the motion, 133; see .a/so density, spatial mass derivative direction al; ,\·ee di rec ti on al derivative Gateaux, 46 Lie time, 106-108, 193-196, 376, 400 normal, 46 of a scalar field, 45 of a scalar function, 40 of a tc nsor field, 50 of a tensor .function scalar-valued, 4·1-42 .-tensor-valued, 42-44 of a vector field, 49 description Eu.lcri an (spatial), 60, 61 Lagrangian (material; refercntia-l), 60, 61 determinant., Jacobian, 74; see also volume ratio determinant of a matrix, 7 determinant of a tensor, 15 derivative of, 4 I , 42 deviutor, :) 9 dev.iatoric operator, 19, 230, 233, 285 deviatoric tensor, 19 direct product, I 0 direct.ional derivative, 46-48, ·65, 95, I 0 I, I 06, I 07, 374-40 I, 404, 405 of a vector function, 37~376, 393 Dirichlet bounda~y condition, 378, 384, 409 displacement field, 61, 62 actual change of, 373 increment of. 393-395

prcscribed,379,382,385,409 virtual, 373-376, 382, 385, 394, 404 displacement gradient tensor, 73, 85 dissipation, internal, 170, 173, 28 l-288, 299,

358-363,369 dissipative material, 278-280, 332 distribution function Gaussian, 308, 313, 314, 319 inverse Langevin. 244 Langevin . 319., 356 divergence material,, 65, 74

.. . : . . .. spatial, 66, 74 . . . . :: : :: . .... -of a tensor field, 4.9, 50 . ··. ·.· ··:.:·: .. of a vector field, 48 divergence theorem., Gauss', 52. ·. dot product of tensors, I 2, 13 of vectors, 2, 3, 5 Duhamel-Neumann hypothesis, 344 Duhamel's law of heal conduction, ·110, ·111 . 325,342 dummy "index, 4, 12-14, 35 dyad, 10 dyadic, lO dynamical process, .148, 153, 159., I60,· l90, :

198 dosed, 211, 212 eigenvalues or a tensor, 24-26 eigenvectors of a tensor, 24-26 ·elas-tic fluid, 125, 126, 204 elastic mater"ial, 197; see also Cauchy-elastic ma1crial; .hyperclastic material; thcrrno(visco)ela.stic material elastic ·potential; see model elastic solid_; see. material e.lasticitics; see also elasticity tensor .isentropic rcforcntia·11cnsor of, 329, 330 spatial tensor of, 330 isotherm.al referential tensor of, 328, 330 spatial tensor of, 328 referential tensor of, 25.2 spatial -tensor of, 253 elasticity; see hyperclasticity; thc.rmoclasticity; thcrmoviscoelaslici ty c"lastic.ity .tensor, 252-265, 328-333, 40 I algorithmic, 293, 294 components of, 252, 253 decoupled representation of, 254-257, 265,

303,340-342,406 cffcc:tive, 302, 303 fictitious in material description . 255-257, 262, 272 in spatial description, 265 iscntropic_. 329-331, 354

440

Index

e:lasticity lensor (comd.) isothermal, 328, 330-333. 353 major symmetries,, 253, 328, 40 l in material description, 252, 253 minor symmetries, 253, 396, 398, 400, 40 I numeric al aspects, 33 l , 33 2 in spatial description, 253 spectral form of in material description, 257-260, 263, 264 in spa.tia] description, 260 end-to-end.distance of tic poincs, 307, 308, 312318 mean value of, 312 end-to-end vector, 31.2 'energetic elastic' material., 311 energy, 131, 132 activation. 364 balance o.f; see balance of energy conserved, 160 free, 173; see also Helmholtz free-energy .function internal, 155., 157, 164, 173, 323-326, 329 changein,3J1,319~334 ~netic,

153-157, 160,392 mechanical, 152-155, 166 non-recoverable,, 279, 296 potential; see potential energy strain (stored), 160; see also strain-energy

function thermal, 161, 172 total, 155, 157, 164, 165 energy flux vector, 165 energy functional, 159, 386, 389, 408 penalty form of, 389 energy-momentum tensor, 21 l enthalpy, 324 ~entropic elastic' material, 311, 333-336 simple tension of, 343-356 entropic e'lasticity concept of, 333 for a stretched ..Piece of rubber, 346-351 entropic theory modified, 333-335, 33'8, 346, 348 purely {strictly), 333-336, 339, 352 entropy, 166-168, 315-319. 328, 329, 358-363 changein,310,319,329,334,353

equiHbrium part . 362, 363

non-equHibrium part, 362, 363 of a single chain, 315, 3 16 total production of, I 67 entropy flux Cauchy (true), 167 Piola-Kirchhoff (nominal), 167 entropy function, 322., 347, 359 entropy inequality principle, 167, 361 entropy input, rate of, J 67,, 168, 172 entropy produclion, 167, J 68., 282, 358; see also dissipation, internal by .conduction of heal, 169 entropy source. 167, l 68 equation global (integral) form of, 134 local (differential) form of, l 34 equation of equHihrium, Cauchy~s. 145 . ·191, 405 equation of evolution; see evolution ·equation equation of motion, 144-146 Cauchy's first, ·144-146. 148, 176. 342, 378~381

Cauchy's ·second, 14 7 weak form of. 380 equation of state, 161 197 caloric, 323 thennal, 322-324, 347 equilibrium, thermodynamic (thermal). 161, 208, 282,285,359,363,365,366 equilibrium state, 16 I, 28:?, 283, 359, 362., 363 ·equilibrium thermodynamics, 168 Eshelby tensor, 21.1, 212 Euclidean space'.t 4, 57, 59, I 80, 181 Euclidean transfonnation, 1. 81 of stress tensors. 19.0, 191 of various kincmatical quantities, 189., 190 Euler-Almansi strain tensor, 82, 88 Euclidean transformation of, 191 first variation of, 376, 377, 382 linearization of, 395, 398 mated al time dcri vati ve of, 400 Lie lime derivative of, l07 material lime derivative or, 102 Euler-Lagrange equation, 381, 405 . 408, 409, 411, 412 Eulerian (spatial) description, 60, 6.1 Eulerian form, 60 .event, 180-182, 187 T

Index .evolution equation, 281, 282, 286-290, 302, 359,

363,366,369 expansion coefficient, linear, 339, 346, 349, 352 extension strip .. biaxial, 93 uniform (uni.axial), 92, 124 external constraints holonomic. 379 no.nholonomic, 379 external variables, 278, 282, 287, 289., '364, 368

fiber, 77-79,267,273 direction, ·84, 96, 266-269, 273-276, extensible, 270 inextensibl-e, 270, 275-277

fiber-reinforced composite, 265, 266, 272, 274 fibers, families of, 265, 273-276 field harmonic, 50, 51, 68, 69, .137 material, 64, 65 mechanical, 306 scalar; see scalar field spatial, 64-67 tensor; see tensor fie'ld themml, 306

vector; see vector field Finger deformation tensor, 8 'J finite element method displaccmcnl·.. based, 402, 403, 411 hybrid, 389 Jacobian-pressure fonnulation. 410 n~xed,389,391,403.404

first law of the.nnodynamics, 164-166.,] 75, 319 first Piola-Kirchhoff strcss tensor, l U-114, 127, 128, 199, 207

Euclidean .transformation of, 190, ·191 first Piofa.. Kirchhoff traction veclor, I 11, J 13, 114, .144 in material description, 1·65 in spatial description, l 63 first variation of a function in material description, 374,, 375 in spatial descriplion, 375-377 flow

irrotalional, 149 steady, 149 flow behavior index., 367

441 ftuid, 205 ·elastic, I 25, l 26, 204 Reiner-Rivlin, 202. 204 v.iscous, 202, 203 Newtonian. 203,.204, 286, 287, 367, 369 fluid mechanics, .205 flux 52, 136,

139, 176, 150

entropy; see entropy ·Hux heat; see heat flux force body; see body force contact, ll 'I external, 110, 387

inertia, 142,378,382,384,386 interna·t~ I I 0 resultant, 110., 1.28, "142 retractive, 310, 311 thermodynamic; see the.rmodynamic force forces, system of, .147-149, 15.3 Fourier's Jaw of heat conductio.n, 171, 342, 348, 354 fourth-order tensor, 22-24 transpose of, 23 fractional-step method, 33 l frame-indifferent spatial fields, 182, 185, 186.; see also objectivity free energy, 173, .206, 267-270, 273-275, 280, 298,321-326~328-335,347,357,361,

364; ;ree also Helmho'hz frce-.encrgy function configuration~,284,285,304~362,364

Gibbs,323y324 free-energy factor, 364 free index. 4~

1.s. 34, 43

free vibration, 154 freely jointed chain, 312-315, 319 friction, 160, J 66, 31 I function

convex, strictly, 229, 244, 30.3, 390, 407 linear. 32, 41, J62 nonlinear. 41, 351, 380, 393 penalty, 390, 391 scalar, 40, 43 tensor, 40-43 test, 380-382 vector, 40 weighting, 380

Ind.ex

442 funclional., 119, 290, 391, 392 energy, 159, 386, 389, 408 Hcllinger-Rcissner, 408, 409 Lyapunov,332 .perturbed, 406

stalionary position of, I J9

Galcrkin method, 402 Galilean transformation, .185 gas ·constant, 364 Gateaux derivative, 46; see al.w directional derivative GfiteauX. operator, 47, 374, 393 Gauss' divergence Lheorcm, 52 Gauss·i an chain, 3 12-316, 3 l "9; see a/so chain Gaussian distribution runcLion, 308, 3 I 3, 314, 319

-Gaussian statistical theory, 312, 318, 319, 339 Gibbs free energy, 323, 324 Gibbs function, 323 Gibbs relation, 322-324, 358 glass trnnsi.tion temperature, 309 -global (integral) fonn of.an equation, 134

Gough ..Joule effect, 309, 31 l, 326,, 327, 349 Gradient material, 65, 66, 74

or a scalar field., 45 spatial, 66, 74 of a (.second-order) tensor field, 50 of a tensor function scalar-valued, 41-42

tensor-valued, 42-44 transposed., 49 of u vector field, 49 Green deformation tensor, 78 Grcc.n··elastic material, 206; see t1lso :hyperclas.. tic mate.rial Grccn-Gauss-Ostrogradskii theorem, 53 Green-Lagrange strain tensor, 79, 82, 83, 88, 209,252,253,353,365,366 Euclidean transformalion of, 1·91 first variation of, 375 linearization of, 394 time derivative of, WO, IOl, 107, 158 Grcen-Naghdi stress rate. 194, 195 ground ·subslance, 265 g.rowth conditions,, 208, 251

Hamilton ,,s variational pri.ncip1c, 391, 392 harmonic field, 50, 51,, 68, 69, 137 heat, 162

heat capacity, specific; see specific heat. capacity heat conduction Duhamcl's law of, 170, 171,, 325, 342 Fourier's law of, 171, 342, 348, 354 heat conducti-on inequality, 170, 324, 342 heal Hux Cauchy (true), 162. 163, 170 Piola-Kirchhoff (nominal),, 162, 163, 324, 342,354,359

heat tlux theorem, SLokes~ 16.2-1 ·64 heating (cooling)., structura.l; see inelastic :heating; thermoelastic heating; thermoviscocla~tic .heating heat source, 162-164, 168 heat transfer; see heat Hcllingcr-Reissner functional, 409 Helling.er-Reissner variational principle, 408, 409 Helmhollz free-energy function, l 73, 206; see al.m free energy decoupled representation of, 27 J, 277, 283,, 't

303,304

Hcncky strain tensor~ "88 Hessian operator, 50 heterogeneous material, 207 hidden variable~, 278; see al.w internal variables higher-order lensor, 20-24, 37 hist.ory term, 293 history variables, 278, 300; see also .internal variables homogeneous material, .207, 32 l Hooke's law, 286 Hu-Washizu variational principle, 412 hydrostatic pressure, 125, 222, 245, 390, 391, 404, 410., 411 hydrostatic stress, state of, 125, 126, 23 I hypcrclastic .ma·terial, 205-304, anisotropic, 214 compressible, 227-235 constitutive equations for, 206-208 . . .bl c,.. ??? mcomprcss1 ..........- ??7 __ isochoric (distortional) elastic response of, 229, 245, 246, 283. 290, 294, 303, 337.338,390,403 isotropic, 212-222

Jocally or.thotropic, 276, 277

Index ortholropic., 275, 276 transversely ·isotropic, .265-272 volumetric (dilatio.nal) elastic -response o·C 229, 244., 245, 283, 290., 303, 337 t

338,390 with isotropic damage, 295-304

with two families or fibers,

273~275

work done on, 21.l, 212 hyperelasticity, 206 compressible, 228-234 . . ·b1 c., ...,?') ')')6 mcompress1 -----hypoclastic material, 254 hysteresis,, 279, 296, 309

ideal rubber., 311, 318, 333 identity tensor, lO; see also unit tensor impenetrability of matter, 74 incomprcssibiH-ty constraints,, ·.103, 2.22, 223, 225, 237,247,270,403,405,407 incompressible ·hypcrclasticity, 222, 223 -incompressible .isotropic hyperelastidty, 223-226 incompressible material, 202, 203, 222-227, 333, 389, 391, 402 mechankaJly, 345-348,, 355 near1y, 228, 389, 402, 403, 409

incrcmcntallitcrative solution techniques, 252, 293.,392,397,405 incremental objectivity, 292

index dummy, 4, J 2-14, 35 free, 4, 18, 34, 43

live, 4 summation, 4 index notation, 3-5 inelastic (plastic) heating (cooling), structural,

360 inertia forc·c, 142, 378, 382, 384, 386 inertia tcnsort l 60 initial boundary-value problem, 377-382 strong (classical) form of, 379 weak (variational) form of, 381 initial conditions, 289., 302, 363, 379-382 -integral theorems, 52-54 integration method, (sclcctivc-)reduccd, 391 integrity bases, 268 internal constraints, I03, 202., 222 . 223, 270. 389, 41 ·1

443 internal -energy, 155, 157, 164, 17.3, 323-326, 329

.internal-energy funccion, 3.23t 329, 331, 354 internal variable model, 281, 282 internal variables, 278, 280-284, 287-290, 292, 293,298,299,358 concept of, 278, 279, .283., 361

interpo1ation .functions, 406, 411 .jntr1nsic .angular moment·um, "J 43 invarjants, 25, 89, 20.1-203., 268-270, 274-276 modified, 233 principalt 25 pseudo-, 268, 276 representat-ion theorem for, 215 strain, 215 . 216, 223, 224, 233, 238, 243

stress, 120., J 22 theory of, 268 inverse Langevin func-tion, 244 inverse square law, 51 inverse ·stretch ratio, 81 irreversible process, 168, 170, 279, 281, 282 irrotatiomll flow, 149 irrotational motion, 6.9, 98 :irrotational vector field, 4'8., 50 iscntro.pic elasticity .tensor in material description_, 329-331, 354 in spatial description, 330 isentrop"ic operator split, 332 isentropic process, 172, 173,. 323 reversible, 172

isochoric motion!' 75, 103, ·136 isothermal elastichy tensor in material description, 328, 330, 331, 353 in spatial descriplion, 328, 332, 333 isothcnnal operator split, 331 isothermal process, 172,. 322, 323, 346, 356 isotropic material, 120, 201, 214 Lhermal1y, .J 71, 342, 345, 346, 348

Lransvcniely, .265-272 ·isotropic ·tensor, 30-32 i solrop ic tensor f unet ion, 20 l , 2 "13-21 8_, 220, 268,274

first representation theorem for, 20 I. 217 second representation theorem for, 202, 217

Jacobian determinant, 74; .~·ee also volume rafio Jaumann-Zaremba rate, l 93T 196

444

Index

Jaumann-Zarcmba stress rate, 194, 195

Kelvin-Voigt model, 279, 280 kinetic energy~ J53-157, 160, 392 Kirchhoff stress tensor, 127, 147, 158, 159 Lie time derivative of, 194; .we also QJ .. droyd stress rate linearization of, 398 material time derivative of, 400 Kronecker de:] ta, 5, 7 .mixed, 33, 35

L,agran.ge mu I··tl.p l..1er, 119., ..,.,.., ----, n4 ~- , ""5 -- , "70 - , 275,403,407,410 Lagrange-multiplier method, 119, 403-407 augmented, 406 linearization of, 405, 406

perturbcd,406,407 Lagrangian (material, referentia:J) description, 60, 61 Lagrangian form~ 60 Lame constants, 250 Landau order symbol, 41, 78, 393 Langevin distribulion function, 319, 356 Lapface's equation, 50 Laplacian operator, 50 latent heat, 326, 329 left Cauchy-Green tensor; see Cauchy-Gree.n tensor, left left stretch tensor; see stretch tensor, tefl Legendre transformation, 173, 31.8, 323, 324, 326,334,408 length of a vector~ 2, 5, 35, 188 :]evel surface, 46, 50 Levi-Civita symbol, 6 l'HopitaJ's rule, 260, 264 .Lie Lime derivative, I 06-108, 193-196, 376, 400 Jinear approximation of a function, 3.93, 394 .linear momentum., 14.1., 142 ·balance of, 141-144, 149,, ·1so, 175 linear operator, 9, 152, 175 lincar.transfonnation, 9, 12, 16, 7l, IOO

1inearization, concept of, 393-395 linearization of a function in material descri.ption . 393 in spatial description, 394t 395

linearization operator, 393, 3.94 lincarization process, consistent, 257, 339-342, 393 Hne element compressed, 7 8 ·extended, 78 material (undeformed), 7.1-73., 75-78, 86

spatial (deformed), 7.1-73. 86 first variation of, 377 time derivative ofl I02 unstretched, 78 loads, 1.1l,379, 386-388, 397, 409, 412

'dead', 387, 395 pressure, 242, 395, 402 .local (differential) form of an equalfon., 134 local time derivative, 66 locking phenomena, 402 volumetric, 403, 406, 41 l Lyapunov functional, 332

macroscopic approach, 56 macroscopic quantities, 57, 16 I, 278., 280 macroscopi·c system, 55, 56 Mandel stress tensor, 128, l 58, 38-6 mass, 56, 131-.136, 139, 140, 320 concentrated, 131, 151. conservation .of, 132-134, 116 Lime derivative of, 133. 134, .l 36 lotal, 134 mass center, 150., 15 l motion of, 151 mass density; see density mass element, infinitesimal. 133, 140 mass sink, 132 mass source, 132 master balance princ.ip.Je, .J 74-177 in global form, 174-176 in local form, 1"76 master fic·ld equation, 176 master inequality principle, 175, .176 matcrjal adiabatic, 325 Cauchy-e.lastic; see Cauchy-elastic material composite, 272-277 constrained, 222 dissipative~

278-280, 332 elastic, 1.97; ,r;ee also Cauchy-elastic material

Index 'energetic clastkt, 311 Cntropic -elastict, 31 l, 333-336 heterogeneous, 207 homogeneous, 207, 321 hyperelastic; see hypcrelastic materia:J hypoclastic, .254 incompressible; see incompressible material inelastic, 278-281 isotropic; .vee isotropic material matrix; see matrix material

mean dilatation method, 4·11 .. measurable quantities,, 278 mechanical device, 286 mechanical energy, 152-·t 60 balance of, in material description, 155-.

orLhotropic~

mechanical theory, _purely, 173, 206·. mechanical work, rate of, 153; see also ex=tcrnal mechanical power; stress power metric coefficients, 33, 36-38 metric tensor, 37, 39 microscopic approach, 55, 56, 295 microscopic system, 55 mid . . point rule, i92 mixed components, 36-39 mixed tensor, 36 model Arruda and Boyce, 244, 248, 249_, 263 B-latz and Ko, 247., 248, 261, ·262

4

275-277 perfccLly elastic, .208 ruhber-Hkc, 235; see also .rubber-like ma-

terial simple, 290

thermoelastic, 323 thermoviscoelastic. 358 transversely isotropic, 265 viscoe1astic, 283 material frame-indifference, principle of, 198200, 267, 292 material function, 20 l material mode), 207; see alw model material o~jectivity. principle of, 198 material point, 56 material strain rate tensor, 10-t, 107, 158 materia] strain tensor, 76-79 material time derivative, 64-68, 95, 96, 99-108, 133-141, 192-196, 258, 259, 399--40 l of a material field. 64._, 65 -of a spatial fieJd_, 66-68 material v.e.locity gradient, 95, 96 matrix column, 12 diagonal, 26 inverse of, 34 orthogonal, 29 row, 12 ·square, 12 stress, ll 2, I 17. 120 matrix mat·eriaL 265-267, 272-275 incompressible isotropic,, 270, 275-277

matrix notali on, 1 l , 12 matrix product, 10

Maxwell element, 286, 287, 368 Maxwell model, 279 generalized, 286, 287, 368 Maxwell relation, thermodynamic, 325, 352, 353

157 balance of, in spatial descripLjon, 153-155

mechanical field, 306 mechanical power, external, J53-.156, 159, 160r 164, 165

damage, 298-30 I 303, 304 :I

MaxwcU; see Maxwell model Mooney-Rivlin; see Mooney-Rivlin model neo .. Hookcnn; see neo-Hookean model Ogden; see Ogden model rheological, 286-288, 367-369 Saint-Venant Kirchhoff, 250, 25 .1, 365 spring-and-dashpot, 286, 367

thermoviscoelaslic, 361-3.64

Vurga,238-242,239,356 viscoe:Jastic, 283-290 with internal variables, 278....;282 Yeoh,243 molecular network; see network molecule, 55 long-chain, 307, 311 moment uf momentum, balanc.e of, l 4.1, 142 moment, resultant, 142 momentum angular (moment of; rotational), 141, 142 linear (translational), 141, 142 spin angufar, 143, 152 momentum balance principles, 141-152 for a closed system, I4.1-.149 for an open sys.Lem, 149-150

446

Index

Mooney-Rivlin model, 203, 339 for compressible materia1s, 247 for incompressible materials, 238-243 motion., 59., ·60 affine, 71 , 3 I 6 equation of, .144-.146 inverse, 59, 71, 80, ·163 irrolalional, 69, 98 isochoric, 75, 103, I 36 plane,69, 106., 137

rigid-body; see .rigid . .body motion steady,-6.8 uniform, 68 vo1umc-prcscrving, 75 Mullins effect, 296-298

Nnbla operator, 45 Nanson's formula, 75. 104, l l3, 114., .146, :163 nco-Hooke.an model,, 203, 318., 339 for compressible .matcr"ia.ls, 247 for .incompress.iblc ma.tcriaJs, 23"8-243 network,238,244,307,311,312~361

elasticity of, 316-320 entropy change of, 317-320 network density, 312 .neutron scattering, 307

Newtonian nuid, 203 Newtonian shear :thinning phenomenon . 366 Newtonian viscous fluid., 203, 204. 286 . 287, 367,369 Newton's law of action and ·reaction., I 12

Newton's method, 257, 29], 295, 393 nomjnal stress tensor, 11.l; ,fee al.w first PiolaKirchhoff s.tress tensor nominal traclion ve.ctor., ·I J .1; see al.m first PiolaKirchhoff traction vector non-equilibrium state, 161, 279, 284, 362 non-equilibrium stresses, 285 . 287-290, 35.9, 363.,365.,3-66,368 non-equilibrium thermodynamics, 'l 68, 280, 282, 285 non-Gaussian statisticaJ theory. 244, 3 19, 356 non-pl1lar continuum, 144 norm of a tensor, 15, 365 norm of a vector, 2 normal derivative, 46 normal stresses, .1 ·J 6, 117 maximium and minimum, 119, 120

normalizalion condition, 208., 216, 219 . 229 . 245,283,287,298

notation absolute,, 3 direct, 3 index, 3-5 matrix, I :J, 12 subscript, 4 suffix, 4

symbolic, 3, 5 used in thermodynamics, I·61, l 62 numerical solution, 331, 392, 403 .numerical stabilily, 331

objective rate, "I 92, 193 objective spatial field, .1.82- I 94 objecr1vc stress rate, 193-1.96 objectivi~y

incremental., 292 .principle of material, .198 of a scalar .field, 185, 186 of a tensor field, 185., .186, 190-194 of a two-point t.cnsor field., 189-191 of a vector field, 182-.186, 188, 192 objectivity, requirement of, l 86, ·J 89-192, 214, 217 t 2.91 observer, 180-18.9, .198, 200 change ·Of, 181-183 octahedral plane., 123 Ogden model

for compressible .materials . 244-246 for incompressible malerials,, 235-242 thermodynamic .ex-tension of, 337-343 .

347-351 Oldroyd stress rate., 193-196, 253, 3327 400 open system, 132, 136, 149 ope rator sp Ii t adiabatic, 332 isentropic, 332 ·isothermal, 331 origin~28,57,58.344

orthogonal matrixt 29 orthogona·J tensor~ 16 orthogonality condition., 29 orthogonality of vectors, 3 orthotropic material, 275-.277

Index parallelogram law,, 2 particle, 56 path line, 59 .penalty function, 390. 391 penalty method, 389, 391, 404, 407 .for incompressibility, 389-391 penalty parameter, 390, 391, 407 perfectly elastic material, 208 permutation, even; odd, 6 permutation symbol, 6~ 7, 17, 21 permutation tensor, 21, 22, 31, 54, 147 perturbed functional, 406 perturbed Lagrange-multiplier method~ 406, 407 phenomenological approach, 205, 283, 295 phenomenological variable, 27.8, 300 Piola .dcfonnalion tensor, 79

Piola identi.ty, .146, 151 Piola-K.irchhoff stress tensor first, '111-114, 127, 128, 199,, 207 Euclidean transformation of, 190, 191 second, 127, 199, 21 O; see.also second Piola-Kirchhoff stress tensor Piola stress, 111 Piola transformation, 83, 84, 1 13, 127, 163, 190, 195,253

place, 58

plane motion, 69, l 06, 137 plane strain, stale of, 84, 92 plane stress, state of., 126., 225, 227 ~ 237 point SOUJC·C, 51, 68 points, 58, 74 material, 56 neighboring, 76, 77 distance between, 76, 78, 8.2 relative position of, 180, 188 Poisson's cqualion, 50 Poisson's ratio, 247 polar continuum, 144, l47, 152 polar decomposition, 85-88., 90, 128 left, ·8'6 right, 86 polyconvexity, 207, 251 polymer chain; see chain polymer network; .'iee network position, current, 58 position, referential, 58 position vector, 58, 5.9, 141, ·1s1. I.82~ 266~ 344 potential, 'dissipative', 284, 362

.potential energy external, 159, 387, 389, 404" · · " of external loading., 159 "· · internal, 159, 387- . total, 159. 160., 387-389, 392, 4.12· first vari.ati on of, 387 second variation of, 387. stalionary position or, 187-389 . potential flow, 149 potential of a veclor field, 4.8 .potentials, thermodynamic, 32 l-325:t 338, 339.

359 power mechanical, external9 153-156, 159, 160, 164, 165 stress; see stress power therm.al, 162, 164, .165, 172, 175, 3 J 9 power expended, .theorem of. .153 power law model, .367 pressure hydrostatic; see hydrostatic pressure mean, 126 pressure boundary loading, 383, 384, 402 primary loading path, 29·6, 301 prindpaJ axes of a tensor, 24 principal directions, 24 referenlia·J, 89, 90 spatial, 90 of strain, 89, .90, 201, 219, 258, 259 of stress, 120-126., 201, 219, 220 . . · I .mvan,mts, · .· · · 25 , 'J 7?5 prmc1pa _ 5·' "16 _ , 'J?3 -.-.. --,.... principal planes, 120-122 principal stresses, 120-·:J 22, 21·9, 220, 225, 226, 237,246,258-260,341 principal stretches, 89-94., 21.9-222, 225-227, 236-242t257-260,316-319 modified, 228, 245, 246, 337-340 principa1 values or a tensor, 24, 25 principle of material frame-indifference, 198-

200, 267, 292 principle of material objectivity, '198 principle of stationary potential energy, 386392

principle of strain-equivalence, 300 principle of virtual displacement. 382; see a/so principle of virtual work principle of virtual work . 377-386, 388-392

448

.Index

principle

or virtual work (comd.)

linearization of, 392-402 in material description, 395-397 in spatial description., 397--40 I in material description~ 384-386 in spatia·I description, 380-3'82 probability, 308, 313-315 probabilily density, 308, 3 t 3, 315 process, 148 adiabatic, 172, 173,336,356

reversible, 172, 173, 348-351 dynamic~,148, 153,159, 160, 190~198 closed, 2 I ] , 212

.irreversible, 168, 170, 279, 281, 282 isentropic, 172, 173, 323 reversible, 172 isothermal, 172, 322, 323, 346, 356

quasi-equilibrium, 162 quasi-static, :l 62 reversible, 168,, 170, 172, J 73, 348-35 .I,

359,366 thermodynamic, 161., 164,. 166-168, 172,

319,330 production of entropy, local, 170; see also dissipation, intema:J projection tensor fourth-order, 24, 229-234, 255, 256, 285, 290

modified, 255 second-orde~.,

1.8., 26, 117 projection of a vector, 3 pseudo-elasLidty, 30 I pull-back operation., 82-84, 106, 107 ~ 127, 163,

375-377,385,395,397,399 pure rotation, 85, .86 pure shear, 92, 93, 227, 250. 304 with isotropic damaget 304 pure stretch, 85-87 pure tension, 124 push... forward -operation, 82-84, 106, l 07. 127, 195,253,375-377,395,397~399

quasi-cquH ibrium process, 162 quasi-static problem, 154, 180 quasi .. static process, 162

rate convected, 193, 196 co-rotational, 192, 196 objective, .l 92, 193 rate of deformation tenso~, 97., 99, :101-107, 194, 202,2.18

Euclidean transformation of, I 91 rotated, 10-.1, 158 physical interpretation of, I 04 spectral decomposition of, l 05 entropy i:npul, 167, 168, 172 external mechanical work, 153; see also external mechanical power internal mcchani.cal work, 153; see also stress power rotation tensor, 97 strain tensor, 97; see also .rate of deformation tensor thennal work, l 62; see also thermal power transport, 139 reaction stresses, 202, 27·1, 276 reciprocal basis, 32, 35. 37

reciprocal basis vector, 32 recovery, 279 recurrence update formula, 293, 295, 393 reduction .factor, 298 reference body force, 144, 384 reference frame, 57 of observers, .18 l, 182, 1.84 spin of, 184 reference mass density, 133-135, 197 reference temperature, 333-339, 342, 344-348 reference time, 58, 1l0, 266, 344 referential stress-entropy tensor, 330, 354 refcrentiu-1 strcss-t·emperature tensor, 328-330.,

339-342,353,354 decoupled representation of, 340-342 referential thermal coefficients .of stress, 328 reflection, 16. 20, 28., 31 region., 52, 5.8-60 Reiner-Rivlin fluid, 202. 204 relaxation, 279, 280, 286, 36 l, 3-64 relaxation time, 280, 284, 287-.289, 358 re:Jaxation {retardation) process, 288, 358, 364, 365,368 relaxation tesl, 294

replacement operator, 5

Index representation theorem for invariants . 215 re.presentation theorem for isotropic tensor functions, 201, 202, 217 residual strain, 298 residual stress. 208 response coefficients. 201-204. 217, 224, 234 response function, 197-200, 207, 210, 218, 223 retardation time, 280, 284, 287-289, 358 reversible process? l68,, "170, 172, 173, 348351. 359, 366 Reynolds' transport theorem, 138-140 rheological model, 286-288, 367-369 right Cauchy-Green tensor.; ue Cauchy-Ore-en tensor. right right stretch tensor; see stretch tensor, right rigid-body, 82, 99. 100 ri_gid-body motion, 82, 153 superimposed, 187-191, 198, 200, 209, 213,214

rigid-body rotation_, 86, 99., I00'.' 1.88, J 95 .rigid-body translation, 62, 71. l 8"8 rigid transformation, time-independent, J 84 Rivlin-Ericksen re.presentation theorem, 94, 20 J rotated rate of deformation tensor., I 01, I 58 rotation, 16, 20, 28-3-1, 87, 99, 100, 209, 267 pure., 85, 86 rigid-body, 86. 99, I 00_, 188, 195 rotation -tensor, 86-88, 99-10 I , 1.28, 194, 209 Euc.lidcan transformation of, ·.J 89 rate or, 97 rmatio.na1 momentum, balance of, 14 I, 142 rubber, 309-311. 317-320, 346-349, 356, 357, 36"J carbon-black lilied, 242, 243. 297, 298 ideal, 311, 333 natural. 311 real, 310 rubber balloon, 239-242 ~snap back' of, 242 ·snap through' of, 242, 249 rubberband.309.310,326,348-351,355,356 cooling effect, 309, 351 heating -effect, 309, 351

rubber-like material, 235, 296. 311. 333, 337340, 357, 403 . compressible, 244-247 incompressible, 235-244

449 stress-strain-temperature response of,, 339, 343,362 Saint. . Venant Kirchhoff model, 250, 251, 365 modified, 251 scalar, 1 scalar field, 45-48, 50, 66, 68, 69., 98, I 06, 13·8·140, 149 objective, 185, 186 scaJar function, 40

scalar multip"lication, 2, 10 scaJar product, 2, 32 triple, .8 second law of thermodynamics, .l 66-l68, 170, l72, 175,208

second Piola-Kirchho.ff stress tensor, ·127. 199,

210 effcctive,, 299 equilibrium part, 28.5, 362 . 363 Euclidean transformation of, 191 fictitious,230,231,234,289 isochoric contribution, 230, 234, 245, 246, 303,339 non-equi1ibrium part, 285, 362, 363 volumetric contribution., 230, 233, 245_, 303, 339 second-order tensor, 9-20, 36'.' 37 transpose, 13, 14

scJf-equilibratcd stress fl-cld, 145 separation of tie points, 307; see also -end-toend distance of tie points

shear pure, 92. 93, 227, 250_, 304 simple, 93, 94, 227, 243, 366, 367 uniform, 93 shear direction, 93 shear modulus_, 227, 236, 247 lemperature dependent, 364 shear planes,, 93 shear rate, J 05, 203, 366 shear stresses. I 16, 117 maximium and minimum, 120-.122 Simo-Tay·lor-Piste.r variationaJ principle, 410, 411 simple material, 290 simple shear, 93, 94, 227, 243, 366, 367 simple tension, 226, 250, 294, 295, 343-352 singJe-field variational princi.p1e, 377-39 l, 3.95401

450 snap buckling, 242 softening parameler, 342

solenoidal vector field, 48, 50 solid, 205, 309-3 :l l solid mechanics, 205 soluLion -technique incremental/iterative, 252, 293, 392, 397, 405 staggered, 331, 332, 342 unconditionally stable, 33'1

source enlropy. 167, 168 154, 155 heal, 162-1-64, 168 mass, 132 point, 51, 68 spatial mass density, 133..... J36, 141, 142, 148. 149,202 spatial strain tensor, 79-82 spatial slrcss-entropy tensor, 330, 332 spatial stress-temperature tensor, 328, 329, 332, 333 spatial time derivative, 66-68 . 98

spatial velocity gradient,, 95-97, 99 Eud idean transformation oC 190 specific heat capacity, 325-327 constant, 335, 347-349 at constant defonnation, 325-327, 329, 330, 334,360 at constant stress, 327 spectral decomposition of a tensor, 25, 26 spherical tensor, 19, 3 I spin, 98 spjn angular mome.ntum, .143, 152 spi.n tensor,, 97-IOO, 154, 156, 192-195 Euclidean transformation oC 191 ·physical .imerprelation .of., 105 spring, 279, 280, 286-288, 309, 310, 351, 368 spring-and-dashpol model, 286, 367 square-root theorem, 86 staggered mclhod, 331 staggered solution technique, 33 ·1, 332, 342

state function, thermodynamic, 161 slate of plane strain, 84., 92 state of plane stress, 126, 225, 227, 23 7 stale of stress, 123-126; ~ftee al.w stress state stale variables, thermodynamic, 161, 278, 305, 3.21 static condcnsation9 411

Index static problem9 380, 395 stationary potential energy, principle of, 386392 statistical concept, 306-309 statistical theory, 238, 239. 308 Gaussian, 312, 318., 319, 339

non-Gaussian, 244, 3 I 9, 356 statistical thermodynamics, 305

steady flow9 "149 steady motion, 68 stiffness matrix, 253, 397, 406 geometric.al (initial slress), 397 ill-conditioned, 391, 402, 406 material, 397 Stokes' heat Hux theorem, 162-·l 64 Stokes·~ theorem, 53 stored energy, 207; .me t1l.w strain-energy funcLion

stored-energy .function, 207; see al.w strain-energy function

strain p.lain, 84 .principal directions of, 89, 90, 201, 2 I 9, 258, .259 residual, 298 strain energy, 160, 207-2 ·1 I, 214-2.22, 233~238,

242-247,252-254,283,320-339;see ttl.m strain-energy function total, 159 strain ..cncrgy factor, 290 strain-energy function, 160, 207 complemc~tary, 408 decoupled representation of, 229, 231, 233, .244, 245, 337' 389 effective, 298-300 isochoric, 303 forms of, 209, 2 I0., 235-251 global .minimum of. 208 lime derivative of, 208, 210, 218, 229 strain-equivalence, principle of, 300 strain invariants, 215, 216, 223, 224, 233, 238, 243 strain ratc9 2:87, 358, 359, 362, 369 strain rate tensor., material, I0 I, ·107, 158 strain space plasticity, 301 strain tensor, 76-85 Biol, .88 Cauchy-Green; .\'ee Cauchy-Green t.cnsor

451

Index eigenvalues of, 89-92 eigenvectors of, 89-92 EuJer-Almansi;-see Euler-Almansi strain tensor Green-Lagrange; see Green-Lagrange strain tensor He.ncky, ·ss material, 76-79 rate of. 97 spatial, 79-82

stress; se.e also stresses algorithmic, 291. 293, 294 Cauchy, 111 P.io1a. l 11 residual, 208 state of, 123-126 stress-entropy tensor referential, 330, 354 spatial,. 330, 332 stress-free configuration, 208, 24 7, 362'.' 363, 379,390 stress function, 322, 323, 347 stress matrix, 112, 117, 120 stress ·PI an cs, 121 stress power, 153-l 59, 164, :170, 173, 218, 230, 280,319 effective. 299 stress rate~ 107 Grccn-Naghdi, 194_, 195 Jaumann-Zaremba, 194, 195 objective, 19.3-196 Oldroyd, 193-196,253,332,400 Truesdell, 195 stress relation, 197-202; see also constilulivc equations stress softening, 296-298, 303, 304 stress state, 123-126 .biaxial, 124, J26 cquibiax"ial~ .124 ho1nogcneous, 123 hydrostatic, 125, -126, 23"1

plane,126,225,227,237 pure nonnal, 123, 124 pure shear, 124 pure tangential, 124 tria-xial, 124 uniform shear, J 24

stress-temperature lensor referential, 328-330, 339-342., 353, 354 decoupled representation of, 340-342 spatial, 328, 329, 332,, 333 uniaxial tension, 124 stress tensor, 1 ll-115, 11-9, 120, ·123-129 alternative, 127-129 Biol., 128, 158 Cauchy; .we Cauchy stress tensor -corotatcd, 12 8 eigenvalues -of, 119, 120 eigenvectors of, 11.9 first Piola-Kirchhoff; see first Piola-Kirchhoff stress tensor Kirchhoff; see Kirchhoff stress tensor Mandel, 128,, 158, 386 nominal, 111 second Piola-Kirchho.ff; see second PiolaKirchhoff s-tress tensor true_, 1.J I stress theorem, Cauchy's, I ll-114, 147, .-]48, ·150, 154, 175 stress vector, t 11 ·stresses; see also stress compressive, 117 maximum and minimum, 126 non-equilibrium, 285, 287-290, 359, 363, 365,366,368 normal, 116, .117 maximium and minimum, 119, 120 reaction, 202, 271,, 276 shear; see shear stresses sign convention for, I 15 tangential, I "17 -tensile, 117 stretch, 78 principal; see principal ·stretches pure, 85-87 stretch ratio, 78; see also stretch stretch tensor, 85 Euclidean transformation or, J 89, 190 left (spatial) . 85-88, 218~ 2.19

·right (material), 85-88, 12-8, "I 98, 199. 209 ti me derj vati ve o.f, 99, 15 8 spectral decomposition of,, 90 stretch vcctur, 78.• 8 l, 87 inverse, 81 structural tensors, 274

452 subscript comma, 45 summation convention, 4 surface

boundary; see boundary surface dosed, 52, 53 control, 132, 136, 149, 150 open.,53 surface density, 174-176 surface element, 52, 74, 110-117 . 119, 162, 383 material (undeformed) . 74 spatial (deformed), 74 first variation of. 377 time derivative of, I 04 surface traction, 109-111, 160 surroundings, 131, 161. 162, 175 ~ymbolic notation, 3, 5 symmetries major, 253, 328, 401 minor, 253, 396, J98 . 400, 401 symmetry of Cauchy stress lensor, 147 system,, 13·1, 132 boundary of, 131 closed, 131-133, 141., 319 condition of, l 61 conservative., 159, 160, 319, 386 isolated, 132, 319 macroscopic, 55, 56 microscopic, 55 open, 132, 136, 149 orthonormal, 3 right.-handec.I, 3 wall of, 13 I system of forces, ·147-149, 153 tangent moduli; see also elasticity tensor algorithmic, 257 consistent .linearized, .257, 293 tangent stiffness matrix; see stiffness matrix tangent vector, 53, 70,, 7 l, 74 .materia·I, 70, Tl spatial, 70t 71 Taylor's ·expansion., 41, 42, 45, 77, 80, 244, 393 temperature, 16 8-17 3, 3 .19-3 69 absolute, 1.68 .change in, 334 Celsius, 168, 319 Fahrenheit, ·I ·68 Kelvin, 168, 319

Index reference, 333-339. 342, 344-348 temperature function, 323 temperature gradient, l 70. 171, 324, 354, 359 tensile stresses, 117 tension pure, 124 simple~ 226, 250, 294, 295., 343-352 uni.form (uniaxial), 124 tensor, 9 antisymmetric, 16 Cartesia.n, I 0 of chemical potential, 211, 212 contravariant, 36, 83, ·121., 193, 194, 253 ·covariant, 36, 83 deviato.ric, 19 eigenvalues .of, 24-26 =eigenvectors of, 24--26 fourth-order, 22-24 hjgher-order, 20-24, 37 inertia, l 60 -inverse of, 15, I 6 isotropic, 30-32 metric, 37, 39 mixed., 36 negative definite, 1 I negative semi-definite, l I nonsingular, 15 norm of, 157 365 order (rank) of, 20 orthogonal, 6 positive definite, 11, 25, 78, 81, 85, 87 positive semi·dcfinile, 11, 170, 17 l projection; see projection lcnsor second-order, 9-20, 36, 37 singular, l 5 skew, 16., .17., 98-1 00 spectral decomposition of, 25, 26 spherical, t 9, 31 spin; see spin tensor symmetric, ·16, 17 third-order, 20....;22 trace of, 14 two-point, 71, 82, 86, 90, l 11, I 89-191 tensor field, 45, 49...,.52 convected rate uf, 193, 196 co-rotational rate of, 192~ 196 frame-.indifferem, .185 objective, 185 . 1.86, l 90-194 ".J

Index .tensor function, 40-43 tensor product, 10-12

453 thermoelastic heating (cooling), structural, 326,

test function, 380-382

LhcrmaJ.conductivi.ty, coefficient of, 171, 342 therma.1 conductivity tensor material, 171 spatial, 170 Lhcnnal energy, 161, 172 thennal ·energy, balance of, 164 thermal equation of slate, 322-324, 347 thermal equilibrium, 161; see .a/so thermody-

namic equilibrium thermal expansion, 344, 347, 348" 351, 352

327,336,348,349,361 decoupled representation of, 343 thermoelastic inversion point, 350-352 thennoclastic material, 323

thermoelasticity, finite" 306, 327, 33.2 of macroscopic networks, 3 'l l-321 one-dimensional, 352-354 Lhcrmome~hankaJ

coupling effects., 327 360, 7

363 lhermomcchanical device, 367 lhermomcchani·cal problem, coupled, 305, 331, 332,342,344,348

thermal field, 306 thermal power, 162, 164, 165, 172, 175, 3.19 Lherma:J work, rate of.. 1·62; see al.w thermal

thermostatics . 161, 168 Lhermoviscoefastic heating (cooling), structural,

power thermul.ly isolropic mater.iaJ, 171, 342, 345, 346,

thermoviscoelastic material, 358 thermoviscoelastic mode.I, 3.61-364 Lhermoviscoelasticity, finite,, 306., 357-360 third-order tensor, 20-22

348

thermodynamic continuum, 16.l :. Iynam1c . . eqm·1·b . ·16'1 , ..,08 l hermm I· num, ~ , ...,8,,....., "85 ... , 359,363,365.366 thermodynamic force, 208, 28 l ~ 299, 303 maximum, 300, 302 lhcrmodynamic Maxwell relation, 325 . 352, 353 thermodynamic potentials, 321-325, 338., 33'9,

359 general structure of, 334, 335 thermodynamic process, 161, 164., 166-.168,

172,319,330 irreversibility or, 167 the.nnodynamic reciprocal relation, 325

thre.e.:fi.eld var.iational principle, 409-413 tie point. 307. 308, 312-314, 317 time fina·1, 319 initial, 58, 289,, 363, 379

instant of. 57-59,,

·1

·so, 188, 372, 385

'modi tied', .289 reference, 58., 110, 266, 344 relaxation (retardation), 280, 284, 287-289, 358 time derivative

Lie, 106-108,193-196,376,400 local, ·66

thermodynamics classical, 305 ·continuum, 131, 16 l, 305, 325 equilibrium, 168 irreversible, 168 of materials, 305-369

material; see material time derivative sputial, 66-68, 98 substantial, 64 total, 64

time increment, 291.

non-equilibrium, 168, 280, 282, 285 notati·on used .in, 161, I 62 reversible, 168 statistical, 305 with int.emal variables., 357-369 .thermodynamic ·slate, 16 ·J, 278, 280-282, 321,

322.,357,362 thermodynamic state function, I 61 thermodynamic state variables, .l 61, 278, 32.l

360,.361

time integration algorithm, 290-293 time interval, 180, 18 .t dosed, 211, 212, 283, 291, 300, 3.19, 320, 361.,392

scmi ..open, 289, 363, 366 time-shift, 181, 187, .188

torque 305~

body, 143 pure, I "10

resultant, 142

lnd·ex

454 tOlal differential, 41. 42, 45, 9 l, 252, 328 total-Lagrangian formulation, 399 trace of a tensor, 14 traction vector, I09-11 "8 B-iot, I 2-8 Cauchy(true), .111, 113-"120., 142, 147, 148 coup"led, 143 first Piola-Kirchhoff (nominal), 111, J 13, 114, 144 prescribed, 379, 385, 412 trajectory, 59 transformation law for basis vectorst 28, 29 tensorial, 29, 30 vectorial, 29 lrnnsfonnation .matrix, 29 translation, 28, 62, 209 rigid-body, 62, 71, 188 transport theorem, Reynolds', 138-·l 40 transversely isotropic material., 265-272 triadic .product of vectors, 21 trial solution, 291 triple scalar produc.t, 8 triple vector product, .8, 9, .19 Truesdell stress rate, t 95 true stress tensor, 111; .we a/.w Cauchy stress tensor lrue traction vector, I l l; see also Cauchy traction vector two~ficld variational principle, 402-409 two-point tensor, Tl, 82, 86, 90, 111, 189-·191

uni.form compression, 92, I 24 uniform deformation, .92, 251 352 unifonn extension, 92, 124 uniform motion, 68 uniform tension, 124 unit tensor of fourth-order, 23, 31, 4.2 of second-order, l 0, l .1, 30, 3 .I, 37 unit vector, 3 update algorithm, 290, 291 .updated-Lagrangian formulation, 399 ll

Va1anis-Landel hypothesis,, 237lt 338 Varga model, 23.8-242, 239, 356

variational equation, 382. 385, 38T, 389, 397, 405, 41] variational operator, 374 variational princi.p1e,, 37"1-413 HamHton 's, 391, 392 ·Hcllingcr-Reissner, 408, 409 1-Iu-Washizu, 412 muJti-field,, 389, 403, 407, 4 =11 Simo-Taylor-Pistcr, 410, 41 J single-field, 377-391, 395-40"1 three-field, 409--413 two-field . 402-409 variat-ional problem, 381 var.iation of a fuoctiont first, 374-377 veclort 1; .~ee al.w vectors axial, 17 20, 48, 98, I00, I05 basis; see basis vectors contravariant, 35, 83, l63~ 186 covariant, 35, 83 dual, 17 length of, 2, 5, 35, 188 magnitude of, 2 norm of, 2 projection nf, 3 vector field, 45, 48-54, 61. 68. 83, 98., 106 conservative, 48 convected rate of, 193, 196 co-rotational rate of, 192 curl-free, 48 divergence-free, 48 frame-indifferent, I 82 irrotationa1, 48, 50 objective, 182-186, 188, 192 solenoidal., 48, 50 vector function, 40 vector operator, 45, 48-50 vector product, 5 .triple, .8, 9, 19 vectors, 1; see al.m vector angle between, 2, 5, 16, 28, 33, 18"8,, 258 equal, 2 orthogonality of, 3 parallel, 6 triadic product·of, 21 velocity, angular, 59 velocity field. 62-65., 67, 68, 1.39, 14.1, 149~ ll

378~379

under changes of observers, J 8.3-.185

Index velocity gradient material, 95, 96 spatial, 95-97, 99 Euclidean transformation of, 190

velocity potential., 69, 98 vibration, free, 154 virtua] displacement field,, 3 73-376, 382, 385, 394,,404 virtual pressure field,, 404 virtual work external {mechanica] ), 382, 38.5,, 388, 391, 404 done by constant pressure, 383, 392

linearization of. 395., 402 internal (mechanical) . 382, 385, 386, 388, 391,395-397,405 linearization of, 396-40 I principle of; see principle of vi.rtual work viscoelastic factor, 294 v.iscoelastic material, 283 viscoelastic model, 283-290 viscosity,, 203, 286, 365-367 viscosity index,, 367 viscous fluid, 202. 203.; see also fluid

vo1ume, 56 control, 132,, 136., 149, 150 volume change, 227, 236, 310, 311,, 3 ·16, 317,

345-347 virtual, 41·0 volume-changing deformation, 228, 232, 4.10 volume density9 175 volume element, 52, 74, 313 incremental, 133 material (undeformed), 74 spatial (deformed),, 74 first variation of, 377 time derivative of, .104 volume-preserving dcfo.nnatlon, 228, 232 volume-preserving motion, 75 volume ratio, 74 Euclidean transformation of., 189 first variation oft 377 time dedvative of,, 103 volumetric .locking phenomena, 403, 406, 41.l von Neumann boundary condhion, 378., 384, 387,409 ·vorticity tensor, 97 vo.rticily vector, 98

455 ·wo.rk conjugate,

159~

170, ·208, 2.17, 252 work C·Onjugate pair, 159; see also work conjugate

work rate; see.external mechanical power; stress power; thermal power

Yeo.h model, 243 Young's moduli, 286, 368

-zero lcnsor, I 0, 43 zero vector, 2

Nonlinear Solid Mechanics a Continuum Approach for Engineering

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