Fracture Lecture of Abaqus

Basic Concepts of Fracture Mechanics Lecture 1

L1.2

Overview • Introduction

• Fracture Mechanisms • Linear Elastic Fracture Mechanics

• Small Scale Yielding • Energy Considerations

• The J-integral • Nonlinear Fracture Mechanics • Mixed-Mode Fracture • Interfacial Fracture • Creep Fracture

• Fatigue

Modeling Fracture and Failure with Abaqus

L1.3

Overview • This lecture is optional.

• It aims to introduce the necessary fracture mechanics concepts and quantities that are relevant to the Abaqus functionality that is presented in the subsequent lectures. • If you are already familiar with these concepts, this lecture may be omitted.


Introduction

L1.5

Introduction • Fracture mechanics is the field of solid mechanics that deals with the behavior of cracked bodies subjected to stresses and strains. • These can arise from primary applied loads or secondary selfequilibrating stress fields (e.g., residual stresses).


L1.6

Introduction • Objective of fracture mechanics

• The objective of fracture mechanics is to characterize the local deformation around a crack tip in terms of the asymptotic field around the crack tip scaled by parameters that are a function of the loading and global geometry.


Fracture Mechanisms

L1.8

Fracture Mechanisms • For engineering materials, such as metals, there are two primary modes of fracture: brittle and ductile. • Brittle fracture • Cracks spread very rapidly with little or no plastic deformation.

• Cracks that initiate in a brittle material tend to continue to grow and increase in size provided the loading will cause crack growth. • Ductile fracture

• Three stages: void nucleation, growth, and coalescence. • The crack moves slowly and is accompanied by a large amount of plastic deformation. • The crack typically will not grow unless the applied load is increased. Modeling Fracture and Failure with Abaqus

L1.9

Fracture Mechanisms • Brittle fracture in polycrystalline materials displays either cleavage (transgranular) or intergranular fracture. • This depends upon whether the grain boundaries are stronger or weaker than the grains .

Cleavage fracture


L1.10

Fracture Mechanisms • Ductile fracture has a dimpled, cup-and-cone fracture appearance .

• Ductile fracture surfaces have larger necking regions and an overall rougher appearance than a brittle fracture surface.


L1.11

Fracture Mechanisms • Fracture process zone

• The fracture process zone is the region around the crack tip where dislocation motions, material damage, etc. occur. • It is a region of nonlinear deformation. • The fracture process zone size is characterized by

• a number of grain sizes for brittle fracture or • either inclusion or second phase particle spacings for ductile fracture. • Different theories have been advanced to describe the fracture process in order to develop predictive capabilities • LEFM

• Cohesive zone models • EPFM

• Etc. Modeling Fracture and Failure with Abaqus

Linear Elastic Fracture Mechanics

L1.13

Linear Elastic Fracture Mechanics • Fracture modes

• Linear Elastic Fracture Mechanics (LEFM) considers three distinct fracture modes: Modes I, II, and III • These encompass all possible ways a crack tip can deform. • Mode I: • The forces are perpendicular to the crack, pulling the crack open. • This is referred to as the opening mode.


L1.14

Linear Elastic Fracture Mechanics • Mode II:

• The forces are parallel to the crack. • One force pushes the top half of the crack back and the other pulls the bottom half of the crack forward, both along the same line. • This creates a shear crack: the crack slides along itself.

• This is referred to as the in-plane shear mode. • The forces do not cause out-ofplane deformation.


L1.15

Linear Elastic Fracture Mechanics • Mode III:

• The forces are transverse to the crack. • This causes the material to separate and slide along itself, moving out of its original plane • This is referred to as the out-of-plane shear mode. • The objective of LEFM is to predict the critical loads that will cause a crack to grow in a brittle material.


L1.16

Linear Elastic Fracture Mechanics • Stress intensity factor

• For isotropic, linear elastic materials, LEFM characterizes the local crack-tip stress field in the linear elastic (i.e., brittle) material using a single parameter called the stress intensity factor K.

• K depends upon the applied stress, the size and placement of the crack, as well as the geometry of the specimen.

• K is defined from the elastic stresses near the tip of a sharp crack under remote loading (or residual stresses).

• K is used to predict the stress state ("stress intensity") near the tip of a crack. • When this stress state (i.e., K) becomes critical, a small crack grows ("extends") and the material fails. • This critical value is denoted KC and is known as the fracture toughness (it is a material property; discussed further later).


L1.17

Linear Elastic Fracture Mechanics • Asymptotic crack tip solutions

• The stress and strain fields in the vicinity of the crack tip are expressed in terms of asymptotic series of solutions around the crack tip. • They are valid only is a small region near the crack tip. • This size of this region is quantified by small scale yielding assumptions (discussed later). • The stress intensity factor is the parameter that relates the local crack-tip fields with the global aspects of the problem.


L1.18

Linear Elastic Fracture Mechanics • The leading-order terms of the asymptotic solution are:

 ij (r , ) 

KI K K fijI ( )  II fijII ( )  III fijIII ( ), 2 r 2 r 2 r

x2

r

 x1

where

r is the distance from the crack tip,  = atan(x2/x1),

KI is the Mode I (opening) stress intensity factor, KII is the Mode II (in-plane shear) stress intensity factor, KIII is the Mode III (transverse shear) stress intensity factor, and the

fija define the angular variation of the stress for mode a.


L1.19

Linear Elastic Fracture Mechanics • Crack-tip singularity

• The predicted stress state at the crack tip in a linear elastic (brittle) material possesses a square-root singularity:



1 . r

• In reality, the crack tip is surrounded by the fracture process zone where plastic deformation and material damage occur. • Inside this zone, the LEFM solution is not valid.

• Outside of this zone (i.e., sufficiently "far" from the fracture process zone), the LEFM is accurate provided the plastic/damage zone is “small enough.” • This is called small-scale yielding (discussed further later).


L1.20

Linear Elastic Fracture Mechanics • Some comments on fracture toughness

Fracture toughness

• Fracture toughness is strongly dependent on temperature.

Temperature

• The brittle-ductile transition temperature range depends on the material.

• For many common metals it may lie within the reasonable operating temperature range for the design, so the temperature dependence of the fracture toughness must be considered.


L1.21

Linear Elastic Fracture Mechanics • Experimentally, the fracture toughness KC is a function of specimen thickness. • Since plane strain gives the practical minimum value of KC …

• The plane strain value is usually the value that is determined experimentally.

Fracture toughness

• However, if the application is fracture of thin sheets of material, KC values somewhere between the plane stress and plane strain values may be appropriate.

KC Thickness

→


L1.22

Linear Elastic Fracture Mechanics • Aside from temperature and thickness, the fracture toughness is also a function of the crack extension. • The fracture toughness as a function of crack extension is called the resistance curve (shown below). ductile

Variation in fracture toughness with crack growth is Kr(Da):

Kr(0)= KC brittle

• The resistance curve is used to predict crack growth stability.


L1.23

Linear Elastic Fracture Mechanics • Crack growth and stability

• The condition for continued crack growth for a crack length a + Da is

Kapplied  K R (Da). • The condition for stable continued crack growth is

K applied a

 load

dK R . d Da


Small-Scale Yielding

L1.25

Small-Scale Yielding • Small-scale yielding (SSY) means the region of inelastic deformation at the crack tip is contained well within the zone dominated by the LEFM asymptotic solution. • For LEFM to be valid, there must be an annular region around the crack tip in which the asymptotic solution to the linear elasticity problem gives a good approximation to the complete stress field.

Plastic zone

K-dominated zone

Transition zone


L1.26

Small-Scale Yielding • The size of the process zone and the plastic region must be sufficiently small so that this is true. Typical shapes of plastic zones follow:

plane strain

plane stress (diffuse)


plane stress (Dugdale)

L1.27

Small-Scale Yielding • We can estimate the plastic zone size, rp, by setting 22 = 0 in the LEFM asymptotic solution, where 0 is the yield stress. This gives (for Mode I)

rp 

1 2

2

2

 KI  1  KI       . 6  0   0 

• Since the tractions across the boundary of the plastic zone have no net force or moments (St. Venant’s principle), the effect on the elastic field surrounding the plastic zone decays rapidly with distance from the boundary, becoming negligible at ~3rp. • LEFM predicts infinite stress at the crack tip—obviously this is unrealistic. • But we can use LEFM results if the region of inelastic deformation near the crack tip is small enough that there is a finite zone outside this region where the LEFM asymptotic solution is accurate.


L1.28

Small-Scale Yielding • If a is a characteristic dimension in the problem, such as remaining ligament size or thickness or crack length, then, to have a finite zone rK in which the K-field dominates, we need

1  K IC  a / 5  rK  3rp    2  0  or

2

2

 K IC  a  2.5   .  0 

ASTM Standard for validity of LEFM

• This is the limit on specimen size in ASTM Standard E-399 for a valid KIC test.

• KIC is KC (the fracture toughness) in Mode I. • The fracture toughness represents the critical value of K required to initiate crack growth. Modeling Fracture and Failure with Abaqus

L1.29

Small-Scale Yielding • For some typical metal materials rp is calculated by matching the yield stress to the Mises stress of the K field and the minimum characteristic length is calculated using the ASTM standard limit. • For materials with high fracture toughness the size of the specimen for a valid fracture test is very large. Characteristic dimension (mm)

T

0

KIC

rp

(ºC)

(MPa)

(MN/m3/2)

(mm)

A061-T651 (Al)

20

269

33

5

38

A075-T651 (Al)

20

620

36

0.35

8.4

AISI 4340 (Steel)

0

1500

33

0.05

1.2

A533-B (Steel)

93

620

200

11

260

Material


Energy Considerations

L1.31

Energy Considerations • Energy principles play an important role in studying crack problems.

• This is motivated by the fact that crack propagation always involves dissipation of energy. Sources of energy dissipation include: • Surface energy, plastic dissipation, etc. • By considering fracture from an energetic point of view, crack growth criteria can be postulated in terms of energy release rates. • This approach offers an alternative to the K-based fracture criteria discussed earlier and reinforces the connection between global and local fields in fracture problems. • The energy release rate is a global parameter while the stress intensity factor is a local crack-tip parameter.


L1.32

Energy Considerations • The energy available to grow a crack is defined as

G-

 ( PE ) , a Loads

where PE is the potential energy and G is the Energy Release Rate. • We consider the difference in the energy for two essentially identical specimens, one with crack length a, the other with crack length a + Da. • The area under the loaddisplacement curve gives -PE for elastic materials.


L1.33

Energy Considerations • For isotropic linear elastic materials, one can show that

1 - v2 2 G K for plane strain E and

K2 G for plane stress. E • In a three-dimensional body under general loading that contains a crack with a smoothly changing crack-tip line, the energy release rate (assuming linear elasticity) per unit crack front length is

1 - v2 2 1 2 G ( K I  K II2 )  K III . E 2G

• Thus, we see the stress intensity factors are directly related to the energy release rate associated with infinitesimal crack growth in an isotropic linear elastic material. Modeling Fracture and Failure with Abaqus

L1.34

Energy Considerations • Initiation of crack growth in SSY

• The necessary condition for crack growth expressed in terms of the energy release rate is G  GC.

• GC is a material property and represents the energy per unit crack advance going into:

• the formation of new surfaces, • the fracture process, and

• plastic deformation. • As noted earlier, for linear elastic materials, G and K are related.

• This leads to an alternative condition for K  KC. • Recall KC is the fracture toughness of the material.


The J-integral

L1.36

The J-integral • The J-integral is used in rate-independent quasi-static fracture analysis to characterize the energy release associated with crack growth. • It can be related to the stress intensity factor if the material response is linear. • As will become apparent in the next section, it also has the advantage that it provides a method for analyzing fracture in nonlinear materials.


L1.37

The J-integral • J is defined as follows:

  u J   Wn1 - i  ij n j  ds x1  



x2

x1 • It is path independent when contours are taken around a crack tip. • The definition of J assumes: • The material is homogeneous in the crack direction.

• The material is elastic. • For linear elastic materials, the value of J is equal to the energy release rate associated with crack advance:

J G Modeling Fracture and Failure with Abaqus

L1.38

The J-integral • J in small-scale yielding • Choose , the contour for J, to fall entirely within the annular region in which the K fields dominate.

3rp • The integrand for J can be evaluated directly in terms of the (known) K fields. Direct calculation for Mode I in a linear elastic material gives

1 - v2 2 J G  K I for plane strain and E 1 J  G  K I2 for plane stress. E Modeling Fracture and Failure with Abaqus

Nonlinear Fracture Mechanics

L1.40

Nonlinear Fracture Mechanics • LEFM applies when the nonlinear deformation of the material is confined to a small region near the crack tip. • For brittle materials, it accurately establishes the criteria for failure. • However, severe limitations arise when the region of the material subject to plastic deformation before a crack propagates is not negligible. • Nonlinear fracture mechanics attempts to extend LEFM to consider inelastic effects.

• The theory is sometimes called Elastic-Plastic Fracture Mechanics (EPFM). • However, the theory is not based on an elastic-plastic material model, but rather a nonlinear elastic material.


L1.41

Nonlinear Fracture Mechanics

n

• Consider a material that has a power-law hardening form, n

  e a  , e0  0  where 0 is the effective yield stress, e0 = 0 / E is the associated yield strain, E is Young's modulus, and a and n are chosen to fit the stressstrain data for the material. Modeling Fracture and Failure with Abaqus

L1.42

Nonlinear Fracture Mechanics • For such a material, Hutchinson, Rice, and Rosengren (extended to mixed mode loading by Shih) showed that the near-tip fields have the form Loading parameter is J



 ij   0 

J

1  n1

 a e I r  0 0 n 



e ij  e 0 

J

n  n1

 a e I r  0 0 n 

 ij ( ),

eij ( ), n  n1

 J ui - uî  ae 0 r   a e I r  0 0 n 

ui ( ).

• Here ui - uî is the displacement relative to the displacement of the crack tip, uî . These fields are commonly referred to as the HRR crack-tip fields.


L1.43

Nonlinear Fracture Mechanics • Why not elastic-plastic?

• The HRR field assumes a nonlinear elastic power law material: n

  e a   e0 0  • Under monotonic loading, this nonlinear elastic material can be matched to the behavior of an elastic-plastic material whose hardening behavior is accurately modeled by a power law.

• Thus, evaluating J allows us to characterize the strength of the singularity in the crack-tip region in an elastic-plastic material subjected to monotonic loading. Modeling Fracture and Failure with Abaqus

L1.44

Nonlinear Fracture Mechanics • In unloading situations, the HRR fields do not describe the state around the crack tip, and hence J does not characterize the strength of the stress state ahead of a crack tip for plastic materials. Use caution when: • The loading is not monotonic and an incremental plasticity material is used • Crack growth occurs under monotonic loading (individual material particles may unload even when the overall structure is being loaded).

• The HRR solution: • Gives the leading term in an asymptotic expansion of the deformation around the crack tip for a power law material; and • Does not take into account finite-strain effects.


L1.45

Nonlinear Fracture Mechanics • Some comments on the HRR fields

• The HRR fields, thus, describe the near-tip crack fields in terms of J.

• J gives the strength of the near-tip singularity in any power-law material (nonlinear elastic or plastic) solid • Recall that in LEFM K plays this role in linear elastic materials.

• J-based fracture mechanics is applied in much the same way as LEFM. • Crack growth initiates when J reaches a critical value: J  JC . • To apply the theory, must ensure conditions for J-dominance are satisfied (discussed next).


L1.46

Nonlinear Fracture Mechanics • J-dominance

• J-dominance refers to situations when J can be used as a method of predicting fracture.

• In general, J is an adequate characterization when there exists a state of high triaxial tension (high triaxiality) ahead of the crack tip. • High triaxiality ahead of the crack tip leads to low fracture toughness. • Examples: states of small-scale and well-contained yielding (where the plastic zone is surrounded by an elastic zone): • Deeply notched bend specimen c «d

d c


L1.47

Nonlinear Fracture Mechanics • In some situations the crack-tip stress field does not exhibit high triaxiality. • Example: large-scale yielding (the plastic zone extends to the free boundaries of the body): • Fully plastic flow of single-edge cracked specimens under tension loading

• Shallow cracks under bending • Center-cracked panel

• A two-parameter approach can be used to extend the fracture characterization to such cases (discussed next).


L1.48

Nonlinear Fracture Mechanics • Two-parameter fracture mechanics • The Williams’ expansion of the Mode I stress field about a sharp crack in a linear elastic body with respect to r, the distance from the crack tip, is

 ij (r , ) 

KI fij ( )  T 1i1 j  O(r1/2 ). 2 r

• The T-stress thus represents a stress parallel to the crack faces. • The magnitude of the T-stress affects the size and shape of the plastic zone and the region of tensile triaxiality ahead of the crack tip. • For positive T-stress, J-dominance exists and a single parameter J can be used for a fracture criterion. • For negative T-stress, a two-parameter approach (J, T) is required to characterize the stress fields.


Mixed-Mode Fracture

L1.50

Mixed-Mode Fracture • Under general loading almost all theories for the direction of crack growth assume or predict that the continued crack growth will be with KII = 0. • Can assume that macroscopic cracks growing with continuously turning tangents will advance straight ahead, presumably under Mode I conditions.

• The crack curvature will evolve in such a way as to maintain this in response to the loading. • If the loading changes such that the local crack-tip stress field experiences a large change in local stress intensities, mixed-mode fracture will occur.


L1.51

Mixed-Mode Fracture • Different criteria for homogeneous, isotropic linear elastic materials have been proposed, including:

• The maximum tangential stress criterion. • The maximum energy release rate criterion. • The KII = 0 criterion. • Although all three imply that KII = 0 as the crack extends, they predict slightly different angles for crack initiation. Comparison of predictions of crack propagation direction for different ratios of KII / KI


Interfacial Fracture

L1.53

Interfacial Fracture • Many engineering applications involve bonded materials.

• Examples: • adhesive joints;

• protective coatings; • composite materials;

• etc. • Engineers must be able to predict the strength of the bond. • Interfacial fracture mechanics provides a method by which to do this. • It extends LEFM to predict the behavior of cracks between two linear elastic materials.


L1.54

Interfacial Fracture • Once a crack has started to grow in an isotropic, homogeneous material, it generally does so in an opening mode; that is, in Mode I. • A crack lying on an interface can kink off the interface and grow under Mode I conditions, or it can grow along the interface under mixed mode conditions. • Whether the crack kinks off the interface or propagates along it is frequently determined through energy considerations.


L1.55

Interfacial Fracture • If the crack kinks off the interface, the fact that there is an interface is important only in how it influences the stress and strain fields. • If the crack grows along the interface, it grows under mixed mode conditions due to material asymmetry and possibly (though not necessarily) under mixed remote loading conditions. • In such situations the conditions for crack growth depend on the interface properties. It is not sufficient to define crack initiation and growth criterion based on the conventional fracture toughness, KC.

• Specifically KC = KC (). • Toughness depends strongly on the mode mixity .


L1.56

Interfacial Fracture • Asymptotic fields

• The asymptotic stress field for an interfacial crack between linear elastic materials is given by

 K * ie   ij  Re  r  ij ( , e )   2 r  where K* = K1  iK2 is the complex stress intensity factor (i.e., it has real and imaginary parts) and  ij  , e  is a complex function of the angle and material mismatch parameter e :

e

 ( - 1) - 2 (1 - 1) 1 1-  log , where   1 2 , and 2 1  1 ( 2  1)  2 (1  1)

 3 -     1  3 - 4

for plane stress for plane strain, axi, 3D


L1.57

Interfacial Fracture • The complex exponent rie indicates that the stresses will oscillate near the crack tip:

• Both the stresses and crack opening displacements will oscillate wildly as the crack tip is approached. • At some distance ahead of the crack tip, the fields settle down. • The fracture criterion should be measured at this point. Provided the location of this point is the same in different specimens, a fracture criterion is valid.


Creep Fracture

L1.59

Creep Fracture • High-temperature fracture

• For temperatures above 0.3M (where M is the melting temperature on an absolute scale), metals will typically creep. • In plastics creep can occur even at room temperature. • There are typically two mechanisms that are active in creep fracture:

• Blunting of the crack tip due to a relaxing stress field. • This tends to retard crack growth.

• Accumulation of creep damage (microcracks, void growth, and coalescence). • This enhances crack growth. • Steady-state creep crack growth occurs when the two effects balance one another.


L1.60

Creep Fracture • The stress state around a crack tip in a material that can creep is more complicated than for the corresponding plasticity problem. • Because of the time-dependent effects there is no one parameter that can characterize the stress state around the crack tip for all possibilities.

• This makes measuring the relevant parameters more difficult. • Hence, creep fracture is not as well established as elastic-plastic fracture. Initially, the crack-tip field is the elastic field. Stationary crack: O(e cr )  O(e el ) around the el cr crack tip (RR field); around this field O(e )  O(e ) (K field). el cr Growing crack: region develops where O(e )  O(e ) (HR field), which is in turn surrounded by the RR field. Eventually the HR field envelops the RR field (which ultimately disappears).


L1.61

Creep Fracture • Contour integrals

• The contour integral for creep fracture is called the C(t)-integral. • It plays an analogous role to the J-integral in the context of timedependent creep fracture. • Its development assumes a power law creep material:

e  e el  e cr

     e0   E  0 



n

• The C(t)-integral is proportional to the rate of growth of the crack-tip creep zone for a stationary crack under small-scale creep conditions:

u j   n C (t )   ijeij n1 - ni  ij ds.    r 0  n  1 x1 



• Under steady-state creep conditions, when creep dominates throughout the specimen, C(t) becomes path independent and is known as C*.


L1.62

Creep Fracture • Asymptotic fields for stationary crack

• The near tip stress and strain fields were obtained by Riedel and Rice in terms of C(t). They are known as the RR fields and are analogous to the HRR fields in power law hardening plasticity. C(t) acts like a time-dependent loading parameter

1  n1

 C (t )   e  I r  0 0 n 

 ij   0  eijcr

n  n1

 C (t )  e0    e  I r  0 0 n 

 ij ( , n)

Crack tip fields are similar to those for an elastic-plastic material

eij ( , n)

Here In is a function of n and the magnitude of  ij ( , n) is approximately 1.


L1.63

Creep Fracture • Small-scale vs. extensive creep

• For the case of no crack growth the loading parameters that characterize the crack-tip fields are reasonably well understood. • Under small-scale creep conditions with secondary creep, K is the loading parameter characterizing the crack-tip field.



Small-scale creep

• For extensive secondary creep C* is a loading parameter characterizing the crack-tip field upon which a fracture criterion may be based. • Suitable criteria for crack extension that will predict an initiation time for crack growth for general cases are not yet available. Modeling Fracture and Failure with Abaqus

K  ( ) r

creep zone

Extensive creep

Fatigue

L1.65

Fatigue • Fatigue is a special kind of failure in which cracks gradually grow under a prolonged period of subcritical loading. • It is the single most common cause of failure in metallic structures.

Damage at the ball grid array (BGA) in a solder joint after 2700 thermal loading cycles

• The Paris Law can be used to predict crack growth as a function of cycles (or time):

da  C (DK ) n , where dN DK  K max - K min


L1.66

Fatigue • Abaqus offers a direct cyclic low-cycle fatigue capability based on the Paris Law. • Models progressive damage and failure both in bulk materials and at material interfaces for a structure subjected to a sub-critical cyclic loading. • For more advanced fatigue analysis capabilities, consult www.safetechnology.com. • fe-safe is a suite of fatigue analysis software that has a direct interface to Abaqus.


Modeling Cracks Lecture 2

L2.2

Overview • Crack Modeling Overview • Modeling Sharp Cracks in Two Dimensions • Modeling Sharp Cracks in Three Dimensions • Finite-Strain Analysis of Crack Tips • Limitations Of 3D Swept Meshing For Fracture • Modeling Cracks with Keyword Options


Crack Modeling Overview

L2.4

Crack Modeling Overview • A crack can be modeled as either

• Sharp • Small-strain analysis

• Singular behavior at the crack tip • Requires special attention

• In Abaqus, a sharp crack is modeled using seam geometry • Blunted • Finite-strain analysis

• Non-singular behavior at crack tip • In Abaqus, a blunted crack is modeled using open geometry • For example, a notch


L2.5

Crack Modeling Overview • Mesh refinement

• Crack tips cause stress concentrations. • Stress and strain gradients are large as a crack tip is approached.

• The finite element mesh must be refined in the vicinity of the crack tip to get accurate stresses and strains. • The J-integral is an energy measure; for LEFM, accurate J values can generally be obtained with surprisingly coarse meshes, even though the local stress and strain fields are not very accurate. • For plasticity or rubber elasticity, the crack-tip region has to be modeled carefully to give accurate results.


L2.6

Crack Modeling Overview • The crack-tip singularity in small-strain analysis

• For mesh convergence in a small-strain analysis, the singularity at the crack tip must be considered.

• J values are more accurate if some singularity is included in the mesh at the crack tip than if no singularity is included. • The stress and strain fields local to the crack tip will be modeled more accurately if singularities are considered. • In small-strain analysis, the strain singularity is: • Linear elasticity   r -½ • Perfect plasticity   r -1 • Power-law hardening   r -n/(n+1)


Modeling Sharp Cracks in Two Dimensions

L2.8

Modeling Sharp Cracks in Two Dimensions • In two dimensions…

• The crack is modeled as an internal edge partition embedded (partially or wholly) inside a face. • This is called a seam crack • The edge along the seam will have duplicate nodes such that the elements on the opposite sides of the edge will not share nodes. • Typically, the entire 2D part is filled with a quad or quad-dominated mesh. • At the crack tip, a ring of triangles are inserted along with concentric layers of structured quads. • All triangles in the contour domains must be represented as degenerated quads. Modeling Fracture and Failure with Abaqus

L2.9

Modeling Sharp Cracks in Two Dimensions • Example: Slanted crack in a plate

• In Abaqus/CAE a seam is defined by through the Crack option underneath the Special menu of the Interaction module. • The seam will generate duplicate nodes along the edge.

Seam

Create face partition to represent the seam; assign a seam to the partition.


L2.10

Modeling Sharp Cracks in Two Dimensions • To define the crack, you must specify • Crack front and the crack-tip • Normal to the crack plane or the direction of crack advance • The crack advance direction is called the q vector.

Select the vertex at either end as the crack front. (Repeat for the other end.)

Crack tip same as crack front in this case

The crack extension direction (q vector) defines the direction in which the crack would extend if it were growing. It is used for contour integral calculations.


L2.11

Modeling Sharp Cracks in Two Dimensions • Other options for defining the crack front and crack tip Crack front for a geometric instance

Crack tip for an orphan mesh

Crack front may be: Vertex/Node Edges/Element edges Faces/Elements Geometric Instances

Orphan Mesh

Crack tip may be:

Vertex/Node

Geometric Instances

Orphan Mesh


L2.12

Modeling Sharp Cracks in Two Dimensions • Example: crack on a symmetry plane

• If the crack is on a symmetry plane, you do not need to define a seam. • This feature can be used only for Mode I fracture.

Crack normal

Crack tip


L2.13

Modeling Sharp Cracks in Two Dimensions • Modeling the crack-tip singularity with second-order quad elements

• To capture the singularity in an 8-node isoparametric element: • Collapse one side (e.g., the side made up by nodes a, b, and c) so that all three nodes have the same geometric location at the crack tip. • Move the midside nodes on the sides connected to the crack tip to the ¼ point nearest the crack tip.


L2.14

Modeling Sharp Cracks in Two Dimensions • If nodes a, b, and c are free to move independently, then

A B   as r  0 r r everywhere in the collapsed element. • If nodes a, b, and c are constrained to move together, A = 0: • The strains and stresses are square-root singular (suitable for linear elasticity). • If nodes a, b, and c are free to move independently and the midside nodes remain at the midsides, B = 0 : • The singularity in strain is correct for the perfectly plastic case.

• For materials in between linear elastic and perfectly plastic (most metals), it is better to have a stronger singularity than necessary. • The numerics will force the coefficient of this singularity to be small.


L2.15

Modeling Sharp Cracks in Two Dimensions • Usage:

Quarter-point midside nodes on the sides connected to the crack tip

The crack tip nodes are independent: r -1 singularity

3

The crack tip nodes are constrained: r -½ singularity

4

2

1, 2 1,2,3,4

3

1

1,1,2,3 Modeling Fracture and Failure with Abaqus

L2.16

Modeling Sharp Cracks in Two Dimensions • Aside: Controlling the position of midside nodes for orphan meshes • Singularity controls cannot be applied to orphan meshes. • Use the Mesh Edit tools to adjust their position.


L2.17

Modeling Sharp Cracks in Two Dimensions • If the side of the element is not collapsed but the midside nodes on the sides of the element connected to the crack tip are moved to the ¼ point: • The strain is square root singular along the element edges but not in the interior of the element. • This is better than no singularity but not as good as the collapsed element.

nodes moved to ¼ points Modeling Fracture and Failure with Abaqus

L2.18

Modeling Sharp Cracks in Two Dimensions • Angular resolution

• We need enough elements to resolve the angular dependence of the strain field around the crack tip. • Reasonable results are obtained for LEFM if typical elements around the crack tip subtend angles in the range of 10 (accurate) to 22.5 (moderately accurate).

• Nonlinear material response usually requires finer meshes.


L2.19

Modeling Sharp Cracks in Two Dimensions • Modeling the crack-tip singularity with first-order quad elements

• Collapsing the side of a first-order quadrilateral element with independent nodes on the collapsed side gives

A   as r  0. r


L2.20

Modeling Sharp Cracks in Two Dimensions • Example: Slanted crack in a plate

• To enable the creation of degenerate quads, you must create swept meshable regions around the crack tips (using partitions) and specify a quad-dominated mesh. 24 elements around crack tip: 15 angles

Quarterpoint nodes

CPE8R elements; typical nodal connectivity shows repeated node at crack tip: Quad-dominated mesh + swept technique for the circular regions surrounding the crack tips

Quadratic element type assigned to part

8, 8, 583, 588, 8, 1969, 1799, 1970

All crack-tip elements repeat node 8 in this example (nodes are constrained).


L2.21

Modeling Sharp Cracks in Two Dimensions • Example (cont’d): Alternate meshes • No degeneracy:

• Degenerate with duplicate nodes:

With swept meshable region: CPE6M elements at crack tip — cannot be used for fracture studies in Abaqus.

With arbitrary mesh, singularity only along edges connected to crack tip.

CPE8R elements at crack tip but no repeated nodes: 1993, 1992, 583, 588, 2016, ...

Coincident nodes located at crack tip


L2.22

Modeling Sharp Cracks in Two Dimensions • Example (cont’d): Deformed shape

Focused mesh; deformation scale factor = 100

Arbitrary mesh; deformation scale factor = 100


Modeling Sharp Cracks in Three Dimensions

L2.24

Modeling Sharp Cracks in Three Dimensions • In three dimensions…

• The seam crack is modeled as a face partition that is either partially or totally embedded into a solid body. • This can be done by partitioning or using a cut (Boolean) operation.

• The face along the seam will have duplicate nodes such that the elements on the opposite sides of the face will not share nodes.

Penny-shaped seam crack: Full model

Quarter model

Wedge elements

Meshed model

• Wedge elements must be created along the crack front. • Generally, this will require partitioning.


L2.25

Modeling Sharp Cracks in Three Dimensions • Options for defining the crack front and crack line Crack front for a geometric instance

Crack line for an orphan mesh

Crack front may be: Edges/Element edges Faces/Element faces

Crack line may be: Edges/Element edges

Cells/Elements Geometric Instances

Orphan Mesh

Geometric Instances


Orphan Mesh

L2.26

Modeling Sharp Cracks in Three Dimensions • Specifying the crack growth direction in three dimensions

• In 3D you can specify either the • normal to the crack plane (only when the crack is planar)

or the • virtual crack extension direction (the q vector).

• Only a single q vector can be defined for geometric instances. • The implications of this will be discussed shortly.


L2.27

Modeling Sharp Cracks in Three Dimensions • Modeling the crack-tip singularity in three dimensions

• 20-node and 27-node bricks can be used with a collapsed face to create singular fields. midplane

C3D20(RH)

edge plane

2 nodes collapsed to the same location

crack line

3 nodes collapsed to the same location

midside nodes moved to ¼ points


L2.28

Modeling Sharp Cracks in Three Dimensions • On an edge plane (orthogonal to the crack line): Double-edge notch specimen (symmetry model)



A as r  0 r



A B  as r  0 r r



B as r  0 r

Crack line

Edge plane nodes displace independently


Edge plane nodes displace together

L2.29

Modeling Sharp Cracks in Three Dimensions • On a midplane for 20-node bricks:

• If the two nodes on the collapsed face at the midplane can displace independently,   r -1 at the midplane (i.e., element interior).

• If on each plane there is only one node along the crack line, no singularity is represented within the element. • In either case the interpolation is not the same on the midplane as on an edge plane. • This generally causes local oscillations in the J-integral values along the crack line. Modeling Fracture and Failure with Abaqus

L2.30

Modeling Sharp Cracks in Three Dimensions • On a midplane for 27-node bricks with all the extra nodes on the element faces: midplane

C3D27(RH)

edge plane

3 nodes collapsed to same location

centroid

crack line 3 nodes collapsed to same location


L2.31

Modeling Sharp Cracks in Three Dimensions • If all midface nodes and the centroid node are included and moved with the midside nodes to the ¼ points, the singularity can be made the same on the edge planes and midplane. • Abaqus does not allow the centroid node to be moved from the geometric centroid of the element. • Therefore, the behavior at the midplane will never be the same as at the edge planes. • This usually causes some small oscillation of the crack fields along the crack line. • The midface node marked “A” is frequently omitted.

• This creates differences in interpolation between the midplane and the edge planes and, hence, causes further oscillation in the cracktip fields. • These oscillations are minor in most cases.


L2.32

Modeling Sharp Cracks in Three Dimensions • Example: Conical crack in a halfspace • A conical crack in an infinite halfspace is considered. • Only the aspects related to the geometric modeling are considered here. • The results of this analysis (J-integral values, etc) will be considered in the next lecture. • The modeling procedure is outlined next.


L2.33

Modeling Sharp Cracks in Three Dimensions 1 Example (cont’d): Create the basic geometry

• Because of symmetry, only a quarter model is created

a = 15 r = 10

q

Large solid block (300 × 300 × 300) used to represent the half-space.

= 45º

Conical shell of revolution (revolved 90º); this will be used to cut the block.


L2.34

Modeling Sharp Cracks in Three Dimensions 2 Example (cont’d): Merge the block and cone

• This will create the edges and surface necessary to define the seam and the crack. Instance and merge the two parts to create a new part. The instance must be independent.


L2.35

Modeling Sharp Cracks in Three Dimensions 3 Example (cont’d): Define the seam and the crack front/line

Only one q vector can be defined for geometry. The q vectors will be adjusted at the end of the modeling process by editing an orphan mesh. Modeling Fracture and Failure with Abaqus

L2.36

Modeling Sharp Cracks in Three Dimensions 4 Example (cont’d): Partition the block for meshing The regions surrounding the crack front are partitioned to permit structured meshing.

A small curved tube is centered at the crack tip; this region is meshed with a single layer of wedge elements. This mesh is swept along the length of the tube.


L2.37

Modeling Sharp Cracks in Three Dimensions • Aside: Why is the small curved tube needed? The swept meshing technique sweeps a mesh through a cross section. For the curved tube, this implies the sweep direction is along its length. In order for Abaqus to automatically create a focused mesh at the crack tip, however, it would need to sweep around the circumference. To overcome this, two concentric tubes are used; the smaller one is meshed with a single layer of wedge elements (which is then swept along the length of the tube). If only a single curved tube was created (shown at right), the mesh around the crack tip would be arbitrary—not focused (wedge elements not created).


L2.38

Modeling Sharp Cracks in Three Dimensions • Aside: What about the seam?

• After all the partitions are created for meshing purposes, the definition of the seam remains intact.

Mesh seam


L2.39

Modeling Sharp Cracks in Three Dimensions 5 Example (cont’d): Mesh the part

• Specify appropriate edge seeds to create a focused mesh around the crack front with minimal mesh distortion.


L2.40

Modeling Sharp Cracks in Three Dimensions 6 Example (cont’d): Adjust the

q vectors

• As noted earlier, only a single q vector can be defined for geometry. As seen in the figure, the vector that was defined is only accurate at the left end of the crack line. • Individual q vectors can be defined on an orphan mesh, however. Thus, either… • Create a mesh part (Mesh module)

or

To take advantage of the input file approach, define a set that contains the conical region before writing the input file. Then you will be able to easily create a display group based on this set when manipulating the orphan mesh.

• Write an input file and import the model • This approach has the advantage that it preserves attributes (sets, loads, etc). Modeling Fracture and Failure with Abaqus

L2.41

Modeling Sharp Cracks in Three Dimensions • For the orphan mesh, adjust each vector individually

To redefine this particular vector, select these nodes as the start and end points of the vector.


L2.42

Modeling Sharp Cracks in Three Dimensions • For all elements, the singularities are modeled best if the element edges are straight. • In three dimensions the planes of the element perpendicular to the crack line should be flat. • If they are not, when the midside nodes are moved to the ¼ points, the Jacobian of the element at some integration points may be negative. • One way to correct this is to move the midside nodes slightly away from the ¼ points toward the midpoint.


L2.43

Modeling Sharp Cracks in Three Dimensions • Example: Conical crack model


Finite-Strain Analysis of Crack Tips

L2.45

Finite-Strain Analysis of Crack Tips • Finite-strain analyses:

• Singular elements should not be used (normally). • The mesh must be sufficiently refined to model the very high strain gradients around the crack tip if details in this region are required. • Even if only the J-integral is required, the deformation around the crack tip may dominate the solution and the crack-tip region will have to be modeled with sufficient detail to avoid numerical problems.


L2.46

Finite-Strain Analysis of Crack Tips • Physically, the crack tip is not perfectly sharp, and such modeling makes it difficult to obtain results. • Instead, we model the tip as a blunted notch, with a suggested radius  10-3rp. • Here, rp is the size of the plastic zone (discussed in Lecture 1).

• The notch must be small enough that under the applied loads, the deformed shape of the notch no longer depends on the original geometry. • Typically, the notch must blunt out to more than four times its original radius for this to be true.


L2.47

Finite-Strain Analysis of Crack Tips • Geometric modeling of blunt cracks

• In 2D, the geometry of a blunted (or open) crack is modeled as a cut having a significant thickness. • Meshing is done in the usual way. • A very fine mesh is required at the crack tip. • This can be achieved by simply assigning small element sizes to the notch.


L2.48

Finite-Strain Analysis of Crack Tips • 3D open cracks can be created in Abaqus/CAE in one of two ways: • Adding a Cut feature in the Part module.

Penny shaped open crack: Full model

• Subtracting a flaw from the original part with a Boolean operation in the Assembly module. • Hex meshing more difficult due to irregular geometry.

Quarter model

Meshed model

• Creating a fine mesh at the crack front generally requires many partitions.

Partitions to control mesh Modeling Fracture and Failure with Abaqus

Refined mesh

L2.49

Finite-Strain Analysis of Crack Tips • The size of the elements around the notch must be about 1/10 th the notch-tip radius. Biased edge seeds can reduce the size of the mesh by focusing small elements towards the crack tip. SEN specimen

crack-tip mesh

rnotch

10% of rnotch Modeling Fracture and Failure with Abaqus

L2.50

Finite-Strain Analysis of Crack Tips • For J-integral evaluation, the region on the surface of the blunted notch should be used to define the crack front.

Crack tip region

q vector Crack surface is detected automatically

The blunted notch surface is the crack front region

Symmetry plane

• For the J- and Ct-integrals to be path independent, the crack surfaces must be parallel to one another (or parallel to the symmetry plane). • If this is not the case, Abaqus automatically generates normals on the crack surface. • If the notch radius shrinks to zero, all nodes that would be at the crack tip should be included in the crack-tip node set. Modeling Fracture and Failure with Abaqus

L2.51

Finite-Strain Analysis of Crack Tips • If the mesh is so coarse that the integration points nearest the crack tip are far from the tip, most of the details (accurate stresses and strains) of the finite-strain region around the crack tip will be lost. • However, accurate J values may still be obtained if cracks are modeled as sharp.


L2.52

Finite-Strain Analysis of Crack Tips • Example: SEN specimen

Deformed shape

Moderate blunting Undeformed shape

Severe blunting

Deformed vs Undeformed Shapes Modeling Fracture and Failure with Abaqus

Contours of PEEQ

L2.53

Finite-Strain Analysis of Crack Tips • In situations involving finite rotations but small strains, such as the bending of slender structures, a small keyhole around the crack tip should be modeled.

crack-front region

• The region defining the crack front for the contour integral consists of the region on the keyhole.

• The elements should not be singular. Modeling Fracture and Failure with Abaqus

Limitations Of 3D Swept Meshing For Fracture

L2.55

Limitations Of 3D Swept Meshing For Fracture • For curved regions cannot generate wedges at the center using a hexdominated approach and then sweep along the length of the region. • This was discussed earlier in the context of the conical crack problem. • To create a focused mesh in this case, embed a small tube within a larger concentric tube. Mesh the smaller tube with a single layer of wedge elements; the surrounding regions are meshed with hex elements. Sweep direction


L2.56

Limitations Of 3D Swept Meshing For Fracture • Partition for a penny-shaped crack

• Illustrates the limitation that the path for the partition must be perpendicular to its bounding surfaces; thus, cannot properly partition along the arc of a circle as shown in this example:

Tangent direction of arc arc (not a semi-circle as in previous example)

Cross-sectional view of block

Partition by sweeping circular edge along arc


L2.57

Limitations Of 3D Swept Meshing For Fracture • The workaround is to partition the face with circular arcs, and then partition the cell using the n-sided patch technique.

Face partition

Note that the cross-sectional area of the swept region is not constant along its length because the tangents at the ends are not perpendicular to the block (generalized sweep meshing)


n-sided patch

Resulting mesh around the crack front using wedge elements

Modeling Cracks with Keyword Options

L2.59

Modeling Cracks with Keyword Options • Defining a crack with keyword options:

• The *CONTOUR INTEGRAL option is used to define both, the crack itself and the fracture output, in an Abaqus input ( .inp) file. • In this section, we focus solely on the crack-specific parameters of this option. • These include: *CONTOUR INTEGRAL, SYMM, NORMAL

• In the next lecture, we discuss the output-specific parameters of this option. • As noted earlier, the main requirements in defining a crack are: • Defining the crack front

• Defining the crack extension direction


L2.60

Modeling Cracks with Keyword Options • Crack symmetry *CONTOUR INTEGRAL, SYMM

• The crack lies on a plane of symmetry and only half the structure is being modeled • This feature should only be used for Mode I problems.


L2.61

Modeling Cracks with Keyword Options • Crack extension *CONTOUR INTEGRAL, NORMAL

• The NORMAL parameter is used to define the normal to the crack plane when the crack is planar. • Usage: *contour integral, normal nx, ny, nz nodeSet1, nodeSet2, ...

• In this case, give a list of the node set names defining the crack front from one end to the other end, in sequential order, without missing any points on the crack line. • In two-dimensional cases, only one node set is needed.

These sets define the crack front; the first node in each set defines the crack tip node for that set.

(An optional CRACK TIP NODES parameter is available to specify the crack tip nodes directly).


L2.62

Modeling Cracks with Keyword Options • Example: Penny-shaped crack in an infinite space *Contour integral, symm, normal, ... 0.0, 1.0, 0.0 Crack-Front-1, Crack-Front-2, Crack-Front-3, ...

Crack-Front-1


L2.63

Modeling Cracks with Keyword Options • If the NORMAL parameter is omitted, we must give the crack-tip node set name, and the crack propagation direction q, at each node set defining the crack front. • Usage: *contour integral, ... nodeSet1, (qx)1, (qy)1, (qz)1 nodeSet2, (qx)2, (qy)2, (qz)2 :

• Data must start with the node set at one end and be given for each node set defining the crack line sequentially until the other end of the crack is reached. • The first node in each set is the crack tip node for that set unless the CRACK TIP NODES parameter is used.

• This format allows nonplanar cracks to be analyzed.


L2.64

Modeling Cracks with Keyword Options • Example: conical crack in an infinite half-space *Contour integral, ... Crack-Front-1, 0.707107, -0.707107, 0. Crack-Front-2, 0.705994, -0.707107, 0.0396478 Crack-Front-3, 0.702661, -0.707107, 0.0791708

Crack-Front-1


L2.65

Modeling Cracks with Keyword Options • Generating a focused mesh with keyword options

• Example: DEN specimen • The focused mesh shown in the figure will be generated with the use of keyword options. • The options include

*NODE *NGEN

*NFILL *ELEMENT

*ELGEN


L2.66

Modeling Cracks with Keyword Options • Node definitions *node 1, 16001, 101, 4101, 12101, 16101, *ngen, 1, *ngen, 101, 4101, 12101,

Start node

0.0125, 0.0000 0.0125, 0.0000 0.0250, 0.0000 0.0250, 0.0125 0.0000, 0.0125 0.0000, 0.0000 nset=tip 16001, 1000 nset=outer 4101, 1000 12101, 1000 16101, 1000

End node

8101

12101

4101

14101

2101

16101

101

Increment in node number

tip *NGEN generates nodes incrementally between any two previously defined nodes. In this example, 17 crack-tip nodes are created (contained in the set tip); the 17 nodes on the outer boundary are contained in set outer.


L2.67

Modeling Cracks with Keyword Options • Quarter-point nodes *nfill, singular=1 tip, outer, 10, 10

Start set: first bound

End set: second bound

This parameter generates quarterpoint nodes; the 1 indicates the first bound represents the crack tip

Node number increment

Number of intervals between bounding nodes

8021 4021

4011 11

2021 1021 21

31

*NFILL generate nodes for a region of a mesh by filling in nodes between two bounds. In this example, 10 rows of nodes are generated between each tip node and its corresponding outer node. Modeling Fracture and Failure with Abaqus

L2.68

Modeling Cracks with Keyword Options • Element definitions *element, type=cps8r 1, 1, 21, 2021, 2001, 11, 1021, 2011, 1001 *elgen, elset=plate 1, 5, 20, 10, 8, 2000, 1000 First row of elements

Total number of rows

Nodes 1, 1001, and 2001 are coincident

1

2021 1021

21

11 1

*ELGEN generates elements incrementally.

In this example, 5 elements form the first row (extending radially outward from the tip); a total of 8 rows of elements (based on the first row) are created around the crack tip.


L2.69

Modeling Cracks with Keyword Options • Crack-tip nodes

• If the crack-tip nodes are permitted to behave independently, the strength of the strain-field singularity is   r -1. • The crack-tip nodes can be constrained using equations, multi-point constraints, using repeated nodes in the element definition, etc. For example, to constrain the crack-tip nodes with a multi-point constraint: *nset, nset=constrain, generate 1, 15001, 1000 *mpc tie, constrain, 16001

• Only node 16001 is independent in this case. • The strain-field singularity is   r -½.


Fracture Analysis Lecture 3

L3.2

Overview • Calculation of Contour Integrals • Examples • Nodal Normals in Contour Integral Calculations

• J-Integrals at Multiple Crack Tips • Through Cracks in Shells • Mixed-Mode Fracture

• Material Discontinuities • Numerical Calculations with Elastic-Plastic Materials

• Workshop 1 • Workshop 2


Calculation of Contour Integrals

L3.4

Calculation of Contour Integrals • Abaqus offers the evaluation of J-integral values, as well as several other parameters for fracture mechanics studies. These include: • The KI, KII, and KIII stress intensity factors, which are used mainly in linear elastic fracture mechanics to measure the strength of local crack tip fields; • The T-stress in linear elastic calculations; • The crack propagation direction: an angle at which a preexisting crack will propagate; and • The Ct-integral, which is used with time-dependent creep behavior.

• Output can be written to the output database ( .odb), data (.dat), and results (.fil) files.


L3.5

Calculation of Contour Integrals • Domain representation of J

• For reasons of accuracy, J is evaluated using a domain integral. • The domain integral is evaluated over an area/volume contained within a contour surrounding the crack tip/line. • In two dimensions, Abaqus defines the domain in terms of rings of elements surrounding the crack tip. • In three dimensions, Abaqus defines a tubular surface around the crack line.


L3.6

Calculation of Contour Integrals • Different contours (domains) are created automatically by Abaqus. • The first contour consists of the crack front and one layer of elements surrounding it.

• Ring of elements from one crack surface to the other (or the symmetry plane).

Contour 1

Contour 2

Contour 3

Contour 4

• The next contour consists of the ring of elements in contact with the first contour as well as the elements in the first contour.

• Each subsequent contour is defined by adding the next ring of elements in contact with the previous contour.


L3.7

Calculation of Contour Integrals • The J-integral and the Ct-integral at steady-state creep should be path (domain) independent. • The value for the first contour is generally ignored. • Examples of contour domains:

2nd contour

2nd contour

1st contour

Crack-tip node

1st contour crack-front nodes Crack-tip node


L3.8

Calculation of Contour Integrals • Usage: *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}

Specifies the number of contours (domains) on which the contour integral will be calculated

This is the output frequency in increments

Note: In this lecture, we focus on the output-specific parameters of the *CONTOUR INTEGRAL option. The crack-specific parameters SYMM and NORMAL were discussed in the previous lecture.


L3.9

Calculation of Contour Integrals • Usage (cont’d): *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}

• J for J-integral output, • C for Ct-integral output. • T STRESS to output T-stress calculations • K FACTORS for stress intensity factor output


L3.10

Calculation of Contour Integrals • Usage (cont’d): *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0} Three criteria to calculate the crack propagation direction at initiation

• Use with TYPE=K FACTORS to specify the criterion to be used for estimating the crack propagation direction in homogenous, isotropic, linear elastic materials: • Maximum tangential stress criterion (MTS) • Maximum energy release rate criterion (MERR) • KII = 0 criterion (KII0) Modeling Fracture and Failure with Abaqus

L3.11

Calculation of Contour Integrals • Output files *CONTOUR INTEGRAL, OUTPUT

• Set OUTPUT=FILE to store the contour integral values in the results (.fil) file. • Set OUTPUT=BOTH to print the values in the data and results files. • If the parameter is omitted, the contour integral values will be printed in the data (.dat) file but not stored in the results (.fil) file.


L3.12

Calculation of Contour Integrals • Loads

• Loads included in contour integral calculations: • Thermal loads.

• Crack-face pressure and traction loads on continuum elements as well as those applied using user subroutines DLOAD and UTRACLOAD. • Surface traction and crack-face edge loads on shell elements as well as those applied using user subroutine UTRACLOAD. • Uniform and nonuniform body forces. • Centrifugal loads on continuum and shell elements. • Not all types of distributed loads (e.g., hydrostatic pressure and gravity loads) are included in the contour integral calculations. • The presence of these loads will result in a warning message.


L3.13

Calculation of Contour Integrals • Other loads not included in contour integral calculations:

• Contributions due to concentrated loads are not included. • If needed, modify the mesh to include a small element and apply a distributed load to the element. • Contributions due to contact forces are not included.

• Initial stresses are not considered in the definition of contour integrals.


Examples

L3.15

Examples • Penny-shaped crack in an infinite space

• Model characteristics • The mesh is extended far enough from the crack tip so that the finite boundaries will not influence the crack-tip solution. • The radius of the penny-shaped crack is 1.

• Two types of loading are considered: • Uniform far-field loading • Nonuniform loading on the crack face: p = Ar n.


L3.16

Examples 20

• Different mesh characteristics:

• Axisymmetric or three-dimensional • Fine or coarse focused meshes

• With or without ¼ point elements • Various element types used:

20

• First- and second-order • With and without reduced integration

Axisymmetric model

Crack tip Focused mesh around crack tip Modeling Fracture and Failure with Abaqus

L3.17

Examples • Fine mesh vs. coarse mesh (axisymmetric and 3D models)

0.08

0.0004

The fine mesh is shown to the left; the coarse mesh above. The length perpendicular to crack line of the crack-tip elements are indicated.

~0.08 Modeling Fracture and Failure with Abaqus

L3.18

Examples • Axisymmetric model: geometry

Symmetry planes

Close up of crack tip region for coarse mesh model (identical for fine mesh model—only the inner semicircular region is smaller)

Model geometry


L3.19

Examples • Axisymmetric model: crack definition Crack tip with extension direction

Set to 0.5 to use midpoint rather than ¼ point elements


L3.20

Examples • 3D model: geometry and mesh

• A 90 sector is modeled because of symmetry.

Fine 3D mesh

Symmetry planes

Additional partition required for swept mesh

On planes perpendicular to the crack front, the mesh is very similar to the axisymmetric mesh Partitions used for coarse mesh model (identical for fine mesh model—only the inner semicircular region is smaller)

In the circumferential direction around the crack line, 12 elements are used.


L3.21

Examples • Why is the additional partition required?

• Without the additional partition, the region shown below would require irregular elements at the vertex located on the axis of symmetry. • This is not supported by Abaqus. Irregular elements required here because revolving about a point

A 7-node element is an example of an irregular element.


L3.22

Examples • 3D model: crack definition

• Orphan mesh created to edit q vectors.


L3.23

Examples • Contour integral output requests (axisymmetric and 3D)

Separate output requests are required for J, K-factors, and the T-stress.


L3.24

Examples • Loads (axisymmetric and 3D) The far-field load is suppressed.


L3.25

Examples • Results

• MISES stress shown below for the axisymmetric fine mesh.

J analytical  J numerical J analytical

100%

Deformation scale factor = 250

Analytical

5.796E-02

Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

5.8169E-02

5.8095E-02

5.8121E-02

5.8104E-02

5.8084E-02

Contour 6

Contour 7

Contour 8

Contour 9

Contour 10

5.8064E-02

5.8044E-02

5.8024E-02

5.8005E-02

5.7985E-02


L3.26

Examples J values from meshes with ¼ point elements (reduced integration) Loading

Analytical result

3-D

Axisymmetric

C3D20R

CAX8R

Coarse

Fine

Coarse

Fine

Uniform far field

.0580

.0578

.0580

.0579

.0581

Uniform crack face

.0580

.0578

.0580

.0579

.0581

Nonuniform crack face (n = 1)

.0358

.0356

.0357

.0356

.0358


.0258

.0256

.0260

.0256

.0258


.0201

.0199

.0206

.0200

.0202

• Abaqus values are based on the average of contours 3−5 in each mesh. Modeling Fracture and Failure with Abaqus

L3.27

Examples J values from meshes with ¼ point elements (full integration) Loading

Analytical result

3-D

Axisymmetric

C3D20

CAX8

Coarse

Fine

Coarse

Fine

Uniform far field

.0580

.0577

.0572

.0578

.0580

Uniform crack face

.0580

.0577

.0572

.0578

.0580


.0358

.0355

.0352

.0356

.0358


.0258

.0255

.0253

.0255

.0258


.0201

.0198

.0197

.0199

.0201


L3.28

Examples J values from meshes without ¼ point elements (reduced integration) 3-D

Loading

Analytical result

C3D20R

Axisymmetric

C3D8R

CAX8R

CAX4R

Coarse

Fine

Coarse

Coarse

Fine

Coarse

Uniform far field

.0580

.0574

.0580

.0563

.0574

.0581

.0562

Uniform crack face

.0580

.0574

.0580

.0563

.0574

.0581

.0562


.0358

.0350

.0357

.0336

.0350

.0358

.0337


.0258

.0250

.0260

.0234

.0250

.0258

.0236


.0201

.0193

.0206

.0177

.0193

.0202

.0179


L3.29

Examples J values from meshes without ¼ point elements (full integration) 3-D

Loading

Analytical result

C3D20

Axisymmetric

C3D8

CAX8

CAX4

Coarse

Fine

Coarse

Coarse

Fine

Coarse

Uniform far field

.0580

.0573

.0572

.0552

.0574

.0580

.0557

Uniform crack face

.0580

.0573

.0572

.0552

.0574

.0580

.0557


.0358

.0350

.0352

.0329

.0350

.0358

.0333


.0258

.0249

.0253

.0229

.0250

.0258

.0232


.0201

.0193

.0197

.0172

.0193

.0201

.0175


L3.30

Examples • Conclusions

• 3D fine meshes with second-order elements are more sensitive to the choice of integration rule when determining J. • The results are still very accurate (within 2% of analytical value). • The inclusion of the singularity helps most in the coarser meshes.

• For mesh convergence in small strain, the singularity must be included.


L3.31

Examples • Conical crack in a half-space

• At each node set along the crack front, the crack propagation direction is different.


L3.32

Examples • Three-dimensional model

• Displaced shape and Mises stress distribution of full threedimensional model.

Deformation scale factor = 1.e6


L3.33

Examples • J values of three-dimensional mesh • There is some oscillation between J values evaluated at corner nodes compared to J values evaluated at midside nodes.

Variation of J with angular position

J-integral

1.338E-07 1.336E-07

3D contour 5

1.334E-07

3D contour 4

1.332E-07

3D contour 3

1.330E-07

3D contour 2

1.328E-07 0

45

90

Angle (degrees)


L3.34

Examples • Axisymmetric model and results

Contours 3-5 have converged

Axisymmetric results are used as reference results.


L3.35

Examples • Comparison of axisymmetric and 3D results Variation of J with angular position Contour 1

Variation of J with angular position Contour 2

1.360E-07

3D

1.340E-07

AXI

1.320E-07

J-integral

J -integral

1.380E-07

1.300E-07 0

45

1.334E-07 1.333E-07 1.332E-07 1.331E-07 1.330E-07 1.329E-07

3D AXI

0

90



1.334E-07

3D

1.332E-07

AXI

1.330E-07

1.328E-07 90

J-integral

J -integral

1.336E-07

45

90

Angle (degrees)

Angle (degrees)

0

45

1.338E-07 1.336E-07 1.334E-07 1.332E-07 1.330E-07 1.328E-07

3D AXI

0

Angle (degrees)


45 Angle (degrees)

90

L3.36

Examples • Since the three-dimensional mesh is quite coarse around the axis of symmetry, these results are considered to be good—the error is less than 0.5% for all but the first contour.

% difference

% difference in J between AXI and 3D results 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Contour 1

Contour 2 Contour 3

Contour 4 Contour 5 0

45

90

Angle (degrees)


L3.37

Examples • Submodeling

• We can use submodeling to create two meshes that are significantly smaller than the full threedimensional model. • The top-right figure is the coarse mesh global model in the vicinity of the crack.

• The bottom-right figure shows the refined submodel mesh overlaid on the global model mesh.


L3.38

Examples % difference in J between AXI and 3D results

• Inaccuracies are introduced by the coarser mesh used in the global model.

% difference

• J values of submodel:

• Errors in J are less than 1%.

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Contour 1 Contour 2 Contour 3 Contour 4 Contour 5 0

45

• CPU time was reduced by a factor of 3.

90

Angle (degrees)

Variation of J with angular position

Variation of J with angular position Contour 5 1.335E-07

3D contour 5

1.324E-07

3D contour 4

1.322E-07

3D contour 3

1.320E-07

3D contour 2

1.318E-07

J-integral

J -integral

1.326E-07

1.330E-07

3D

1.325E-07

AXI

1.320E-07 1.315E-07

0

45

90

0

Angle (degrees)


45 Angle (degrees)

90

L3.39

Examples • Compact Tension Specimen

• This is one of five standardized specimens defined by the ASTM for the characterization of fracture initiation and crack growth. • The ASTM standardized testing apparatus uses a clevis and a pin to hold the specimen and apply a controlled displacement.


L3.40

Examples

Prescribed load line displacement

• Model details

Crack seam

• Plane strain conditions assumed. • The initial crack length is 5 mm. • Elastic-plastic material

• Low alloy ferritic steel

q-vector

1/√r singularity modeled in the crack-tip elements


L3.41

Examples • Results

Small strain analysis

Finite strain analysis


L3.42

Examples

At small to moderate strain levels, the small and finite strain models yield similar results.

Finite strain effects must be considered to represent this level of deformation and strain accurately.


Nodal Normals in Contour Integral Calculations

L3.44

Nodal Normals in Contour Integral Calculations • Sharp curved cracks

• For sharp cracks, if the crack faces are curved, Abaqus automatically determines the normal directions of the nodes on the portions of the crack faces that lie within the contour integral domains. • This improves the accuracy of the contour integral estimation.

Normals to top crack surface nodes

n (normal to crack plane) Normals to bottom crack surface nodes

• The normal is not used at the crack-tip node, however.


q

L3.45

Nodal Normals in Contour Integral Calculations • Example: sharp curved crack

Contour # J without normals J with normals

1

2

3.363 3.600

2.980 3.602

3 2.475 3.605

4 1.888 3.605


5 1.283 3.605

L3.46

Nodal Normals in Contour Integral Calculations • Blunt cracks and notches

• All nodes on the notch should be included in the crack-tip node set. • The J-integral results are more accurate since the q vector is parallel to the crack surface in this case, as illustrated below.

Crack surface

Crack surface

Paths for contour integrals

n q Single node in crack-tip node set; normals calculated on nodes of blunted surface; q not parallel to crack surface.

q All nodes on blunted surface in crack-tip node set; q parallel to crack surface.


J-Integrals at Multiple Crack Tips

L3.48

J-Integrals at Multiple Crack Tips • Abaqus can calculate J (or Ct ) at multiple crack tips • Abaqus/CAE: multiple crack tips and history output requests • Input file: repeated use of the *CONTOUR INTEGRAL option.

• If the domain for one crack tip envelopes the other crack tip, the J value will go to zero (as it should).


Through Cracks in Shells

L3.50

Through Cracks in Shells • Second-order quadrilateral shell elements must be used if contour integral output is requested. • Sides of S8R elements should not be collapsed. If a focused mesh is used, the crack tip must be modeled as a keyhole whose radius is small compared to the other dimensions measured in the plane of the shell.

Shell mesh

Crack-tip mesh for S8R elements


L3.51

Through Cracks in Shells • S8R5 elements can be collapsed and midside nodes moved to the 1/4 points.

Shell mesh

Crack-tip mesh for S8R5 elements

• The q vector must lie in the shell surface.

• It should be tangent to the surface.


L3.52

Through Cracks in Shells • Example: Circumferential through crack under axial load

• Mean radius R = 10.5 in • Wall thickness t = 0.525 in • Crack half-angle q = p / 4

• Longitudinal membrane stress = 100 psi


L3.53

Through Cracks in Shells • Model details • Axial load is applied using a shell edge load • Symmetry used to reduce mode size

Edge loads

symmetry


L3.54

Through Cracks in Shells • Modeling a crack with a keyhole

Crack front

q vector

Crack tip


L3.55

Through Cracks in Shells • Results

Deformed shape—axial loading

J values—axial loading


L3.56

Through Cracks in Shells • In shell element meshes, mechanical loads which act normal to the shell surface and are applied within the contour integral domain are not taken into account in the calculation of the contour integral. • For example, pressure loads are not considered because they act normal to the shell surface • Conversely, axial edge loads are considered because they act in the shell surface. • Two workarounds exist:

• Run successive shell models with differing crack lengths and numerically differentiate the potential energy • Use solid elements (if the response is membrane dominated)


L3.57

Through Cracks in Shells • Using numerical differentiation to obtain J:

 ( PE ) J = a Constant Load =

Potential energy:

PE = ALLSE  ALLWK

PE a Da  PE a Da

. Constant Load

• The PE values should be obtained from two separate analyses, with crack lengths differing by Da. • The values of PE in the Abaqus data (.dat) file are generally not printed to a sufficient number of figures to be useful for this calculation and must be read from the results ( .fil) file. • A similar technique can be used to get Ct at long times.


L3.58

Through Cracks in Shells • Using solid elements:

• If membrane deformation is dominant, the shell can be modeled with a single layer of 20-node bricks since these solid elements include loading contributions to contour integrals.


L3.59

Through Cracks in Shells • To obtain accurate values of J through the shell thickness with solid elements, more than one element should be used in the thickness direction.

J values will show significant path dependence unless averaged. • If only one element is used through the thickness, the values can be averaged by thinking of J as a force per unit length: • The average is calculated as if the J values were equivalent nodal forces:

J

shell

=

J A  4J B  JC . 6


A B C

L3.60

Through Cracks in Shells • Aside: Generating a solid element mesh from a shell mesh.

• A shell mesh can easily be converted to a solid one using the ―Offset Mesh‖ tool. • Creates solid layers from a shell mesh.


L3.61

Through Cracks in Shells • Example: Circumferential through crack in an internally pressurized, closed-end pipe • The same pipe discussed earlier, now subjected to 10 psi internal pressure + axial load (which simulates the closed end).

• Comparison of J values using one layer of C3D20R elements through the thickness : A

J values  100

CONTOUR 1

2

3

4

5

At Node A

2.0965

2.1317

2.1505

2.1557

2.1697

At Node B

3.7396

3.6992

3.7004

3.6968

3.6904

At Node C

5.0226

5.0501

5.0813

5.1471

5.2373

Averaged

3.6796

3.6631

3.6722

3.6817

3.6948


B C

L3.62

Through Cracks in Shells • Example: Circumferential through crack under axial load revisited

• Now we revisit the problem in which the pipe is subjected to an axial load. • Comparison of J values using one layer of C3D20R elements through the thickness: J values  100

CONTOUR

1

2

3

4

5

At Node A

2.2122

2.2524

2.2700

2.2740

2.2850

At Node B

3.7629

3.7202

3.7212

3.7184

3.7136

At Node C

4.9560

4.9893

5.0175

5.0737

5.1492

Averaged

3.7033

3.6871

3.6954

3.7036

3.7148

Analytical

3.7181


L3.63

Through Cracks in Shells • Comparing these results with the shell element results presented earlier: • Errors with respect to the analytical solution for the 3D model are less than 1%.

• Much closer agreement because transverse shear effects are considered in the 3D model. • Only in-plane stress and strain terms are included in the Abaqus J calculations for shells. • Transverse shear terms are neglected.


Mixed-Mode Fracture

L3.65

Mixed-Mode Fracture •

Abaqus uses interaction integrals to compute the stress intensity factors. •

This approach accounts for mixed-mode loading effects.

•

Note that the J- or Ct-integrals do not distinguish between modes of loading.

•

Usage: *CONTOUR INTEGRAL, TYPE=K FACTORS

•

Stress intensity factors can only be calculated for linear elastic materials.


L3.66

Mixed-Mode Fracture • Example: Center slant cracked plate under tension 

Element type

 

  

22.5º

CPE8

0.185 (2.9%)*

0.403 (0.2%)

22.5º

CPE8R

0.185 (2.9%)

0.403 (0.2%)

67.5º

CPE8

1.052 (3.6%)

0.373 (1.0%)

67.5º

CPE8R

1.053 (3.8%)

0.374 (1.3%)

K0 =  p a *Values enclosed in parentheses are percentage differences with respect to the reference solution. See Abaqus Benchmark Problem 4.7.4 for more information.

 = 22.5

 = 67.5 Modeling Fracture and Failure with Abaqus

Material Discontinuities

L3.68

Material Discontinuities • The J-integral will be path independent if the material is homogeneous in the direction of crack propagation in the domain used for the contour integral calculation. • If there is material discontinuity ahead of the crack in this region, the *NORMAL option can be used to correct the calculation of J so that it will still be path independent.

• The normal to the material discontinuity line must be specified for all nodes on the material discontinuity that will lie in a contour integral domain.

n


L3.69

Material Discontinuities • Example: J-integral analysis of a two material plate • As an example, the figure shows a single-edge notch specimen made from two materials in which the material interface runs at an angle to the sides of the specimen.

• The material containing the crack (left) has a Young’s modulus of 2  105 MPa and a Poisson’s ratio of 0.3. • The uncracked material (right) has Young’s modulus of 2  104 MPa and a Poisson’s ratio of 0.1. • The specimen is stretched by uniform displacement at its ends.


L3.70

Material Discontinuities • J-integral analysis of a two material plate (cont’d) • Along the material discontinuity, the normal to the discontinuity is given using the *NORMAL option. • The normal needs to be defined on both sides of the discontinuity. *NORMAL LEFT, NORM, 1.0, 0.125, 0.0 RIGHT, NORM, -1.0, -0.125, 0.0


L3.71

Material Discontinuities • The calculated J-integral values for 10 contours are as follows: Contour

J (N/mm)

Without normals

With normals

1

55681

55681

2

57085

57085

3

57052

57052

4

57058

57058

5

35188

57116

6

31380

57114

7

27536

57114

8

23512

57113

9

19172

57116

10

14181

57094

• The need for the normals on the interface (contours 5–10) is clear. Modeling Fracture and Failure with Abaqus

Numerical Calculations with Elastic-Plastic Materials

L3.73

Numerical Calculations with Elastic-Plastic Materials • For Mises plasticity the plastic deformation is incompressible.

• The rate of total deformation becomes incompressible (constant volume) as the plastic deformation starts to dominate the response. • All Abaqus quadrilateral and brick elements suitable for use in J-integral calculations can handle this rate incompressibility condition except for the ―fully‖ integrated quadrilaterals and brick elements without the ―hybrid‖ formulation. • Do not use CPE8, CAX8, C3D20 elements with these materials. They will ―lock‖ (become overconstrained) as the material becomes more incompressible.


L3.74

Numerical Calculations with Elastic-Plastic Materials • Second-order elements with reduced integration (CPE8R, C3D20R, etc.) work best for stress concentration problems in general and for crack tips in particular. • If the displaced shape plot shows a regular pattern of deformation, this state is an indication of mesh locking. • Locking can be seen in quilt contour plots of hydrostatic pressure for first-order elements—the pressure shows a checkerboard pattern.

• Change to reduced integration elements if you are using fully integrated elements. • Increase the mesh density if you already using reduced integration elements. • If these steps do not help, use hybrid elements. • Hybrid elements must be used for fully incompressible materials (such as hyperelasticity, linear elasticity with n = 0.5).


L3.75

Numerical Calculations with Elastic-Plastic Materials • Results with elastic-plastic materials (and nonlinear materials in general) are more sensitive to meshing than for small-strain linear elasticity. • Meshes adequate for linear elasticity may have to be refined. • The more complex the solution, the more J values tend to be path dependent. • A lack of path dependence can be an indication of a lack of mesh convergence; however, path independence of J does not prove mesh convergence.


Workshop 1

L3.77

Workshop 1 • Crack in a three-point bend specimen

• Two-dimensional geometry • Mesh sensitivity study

• Focus vs. unfocused mesh • Quarter-point vs. mid-side nodes


Workshop 2

L3.79

Workshop 2 • Crack in a helicopter airframe component

• Three-dimensional geometry • Create mesh and evaluate response for cracks at different locations


Material Failure and Wear Lecture 4

L4.2

Overview • Progressive Damage and Failure

• Damage Initiation for Ductile Metals • Damage Evolution

• Element Removal • Damage in Fiber-Reinforced Composite Materials

• Failure in Fasteners • Material Wear and Ablation


Progressive Damage and Failure

L4.4

Progressive Damage and Failure • Abaqus offers a general capability for modeling progressive damage and failure in engineering structures • Material failure refers to the complete loss of load carrying capacity that results from progressive degradation of the material stiffness. • Stiffness degradation is modeled using damage mechanics. • Progressive damage and failure can be modeled for: • Ductile materials

• Continuum constitutive behavior • Fiber-reinforced composites

• Interface materials • Cohesive elements with a traction-separation law

• Damage and failure of cohesive elements are discussed in the next lecture.


L4.5

Progressive Damage and Failure • Two distinct types of ductile material failure can be modeled with Abaqus • Ductile fracture of metals • Void nucleation, coalescence, and growth • Shear band localization • Necking instability in sheet-metal forming • Forming Limit Diagrams • Marciniak-Kuczynski (M-K) criterion • Damage in sheet metals is not discussed further in this seminar.


L4.6

Progressive Damage and Failure • Components of material definition

• Undamaged constitutive behavior (e.g., elastic-plastic with hardening)

Undamaged response



A Damaged response

• Damage initiation (point A) • Damage evolution (path A–B) • Choice of element removal (point B) Keywords

*MATERIAL *ELASTIC

B Multiple damage definitions are allowed

*PLASTIC *DAMAGE INITIATION,CRITERION=criterion

Typical material response showing progressive damage

*DAMAGE EVOLUTION *SECTION CONTROLS, ELEMENT DELETION=YES Modeling Fracture and Failure with Abaqus



Damage Initiation Criteria for Ductile Metals

L4.8

Damage Initiation Criteria for Ductile Metals • Damage initiation defines the point of initiation of degradation of stiffness • It is based on user-specified criteria

• Ductile or shear • It does not actually lead to damage unless damage evolution is also specified • Output variables associated with each criterion • Useful for evaluating the severity of current deformation state

• Output DMICRT DMICRT > 1 indicates damage has initiated

Ductile

Shear

Different damage initiation criteria on an aluminum double-chamber profile


L4.9

Damage Initiation Criteria for Ductile Metals • Ductile criterion:

• Appropriate for triggering damage due to nucleation, growth, and coalescence of voids • The model assumes that the equivalent plastic strain at the onset of damage is a function of stress triaxiality and strain rate. Pressure stress

• Stress triaxiality h = - p / q Mises stress

• The ductile criterion can be used with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models, including equation of state.

Ductile criterion for Aluminum Alloy AA7108.50-T6 (Courtesy of BMW)


L4.10

Damage Initiation Criteria for Ductile Metals • Usage:

• Specify the equivalent plastic strain at the onset of damage as a tabular function of • Stress triaxiality • Strain rate *DAMAGE INITIATION, CRITERION=DUCTILE

 pl , h ,  pl , T , fi Equivalent fracture strain at damage initiation

Temperature and field variable dependence optional

• Output: DUCTCRT (wD)

The criterion for damage initiation is met when wD = 1.


L4.11

Damage Initiation Criteria for Ductile Metals • Shear criterion:

• Appropriate for triggering damage due to shear band localization • The model assumes that the equivalent plastic strain at the onset of damage is a function of the shear stress ratio and strain rate. • Shear stress ratio defined as:

qs = (q + ks p) /tmax • The shear criterion can be used with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models, including equation of state.

ks = 0.3

Shear criterion for Aluminum Alloy AA7108.50-T6 (Courtesy of BMW)


L4.12

Damage Initiation Criteria for Ductile Metals • Usage:

• Specify the equivalent plastic strain at the onset of damage as a tabular function of • Shear stress ratio • Strain rate *DAMAGE INITIATION, CRITERION=SHEAR, KS=ks

 pl , q s ,  pl , T , fi

ks is a material parameter

Temperature and field variable dependence optional

Equivalent fracture strain at damage initiation

• Output: SHRCRT (wS)

The criterion for damage initiation is met when wS = 1.


L4.13

Damage Initiation Criteria for Ductile Metals • Example: Axial crushing of an aluminum double-chamber profile

Cross section

Quasi-static buckling mode


L4.14

Damage Initiation Criteria for Ductile Metals • Model details

• Steel base: • C3D8R elements

• Enhanced hourglass control

Rigid plate with initial downward velocity

• Elastic-plastic material

• Aluminum chamber:

Aluminum chamber

• S4R elements • Stiffness hourglass control • Rate-dependent plasticity • Damage initiation

• General contact • Variable mass scaling


Steel base: bottom is encastred.

L4.15

Damage Initiation Criteria for Ductile Metals • Material definition : Keywords interface Ductile criteria for Aluminum Alloy AA7108.50-

strain at damage initiation

*MATERIAL, NAME=ALUMINUM T6 (Courtesy of BMW) 7 *DENSITY strain rate=0.001/s 6 2.70E-09 strain rate=250/s 5 *ELASTIC 4 7.00E+04, 0.33 *PLASTIC,HARDENING=ISOTROPIC,RATE=0 3 : 2 *DAMAGE INITIATION, CRITERION=DUCTILE 1 5.7268, 0.000, 0.001 0 0 0.2 0.4 0.6 4.0303, 0.067, 0.001 stress triaxiality 2.8377, 0.133, 0.001 : Strain rate,  pl 4.4098, 0.000, 250 2.5717, 0.067, 250 Stress triaxiality, h 1.5018, 0.133, 250 : Equivalent fracture strain at damage initiation, pl




L4.16

Damage Initiation Criteria for Ductile Metals • Material definition : Keywords interface (cont'd)

0.3025, 1.463, 250 0.3323, 1.501, 250 :

strain at damage initiation

*MATERIAL, NAME=ALUMINUM : *DAMAGE INITIATION, CRITERION=DUCTILE 5.7268, 0.000, 0.001 4.0303, 0.067, 0.001 : *DAMAGE INITIATION, CRITERION=SHEAR, KS=0.3 0.2761, 1.424, 0.001 0.2613, 1.463, 0.001 0.2530, 1.501, 0.001 : 0.2731, 1.424, 250 Strain rate,  pl

Shear criteria for Aluminum Alloy AA7108.50-T6 (Courtesy of BMW)



Shear stress ratio,

0.8 0.7 0.6 0.5 0.4 0.3 0.2

strain rate=0.001/s

0.1 0

strain rate=250/s 1.6

1.7

1.8

1.9

shear stress ratio

qs

Equivalent fracture strain at damage initiation,  pl


2

L4.17

Damage Initiation Criteria for Ductile Metals • Material definition : Abaqus/CAE interface

: *DAMAGE INITIATION, CRITERION=DUCTILE 5.7268, 0.000, 0.001 4.0303, 0.067, 0.001 2.8377, 0.133, 0.001 : 4.4098, 0.000, 250 2.5717, 0.067, 250 1.5018, 0.133, 250 :


L4.18

Damage Initiation Criteria for Ductile Metals • Material definition : Abaqus/CAE interface (cont'd)

: *DAMAGE INITIATION, CRITERION=SHEAR, KS=0.3 0.2761, 1.424, 0.001 0.2613, 1.463, 0.001 0.2530, 1.501, 0.001 : 0.2731, 1.424, 250 0.3025, 1.463, 250 0.3323, 1.501, 250 :


L4.19

Damage Initiation Criteria for Ductile Metals • Results (without damage evolution)

Ductile Quasi-static response


Shear

Damage Evolution

L4.21

Damage Evolution • Damage evolution defines the post damage-initiation material behavior. • That is, it describes the rate of degradation of the material stiffness once the initiation criterion is satisfied. • The formulation is based on scalar damage approach:

 = (1 - d )

Stress due to undamaged response

• The overall damage variable d captures the combined effect of all active damage mechanisms. • When damage variable d = 1, material point has completely failed. • In other words, fracture occurs when d = 1.


L4.22

Damage Evolution • Elastic-plastic materials

• For a elastic-plastic material, damage manifests in two forms • Softening of the yield stress • Degradation of the elasticity

• The strain softening part of the curve cannot represent a material property. • The above argument is based on

Undamaged response





(d = 0)

 y0

- d

0 E

• Fracture mechanics considerations • Mesh sensitivity

softening

Degradation of elasticity

E

 0pl

(1 - d ) E

 fpl

Schematic representation of elastic-plastic material with progressive damage.




L4.23

Damage Evolution • To address the strain softening issue, Hillerborg’s (1976) proposal is adopted. • The fracture energy to open a unit area of crack, Gf , is assumed to be a material property. • The softening response after damage initiation is characterized by a stress-displacement response (rather than a stress-strain response) • This requires the introduction of a characteristic length L associated with a material point.


L4.24

Damage Evolution • The fracture energy is written as

Gf =

 fpl



pl 0

L y pl =



u fpl 0

 y u pl

where u pl is the equivalent plastic displacement. • The characteristic length L is computed automatically by Abaqus based on element geometry. • Elements with large aspect ratios should be avoided to minimize mesh sensitivity. • The damage evolution law can be specified either in terms of fracture energy (per unit area) or in terms of the equivalent plastic displacement. • Both approaches take into account the characteristic length of the element. • The formulation ensures that mesh-sensitivity is minimized.


L4.25

Damage Evolution • Displacement-based damage evolution

d

d

d

1

1

1

0 (a) Tabular

u

pl

0

u

pl f

u

pl

(b) Linear

0

u fpl u pl

(c) Exponential

*DAMAGE EVOLUTION,TYPE=DISPLACEMENT, SOFTENING={TABULAR,LINEAR,EXPONENTIAL}


L4.26

Damage Evolution Undamaged response

• Procedure for generating d vs u pl table from tensile test data 1.

2.

3.

4.

Plot true stress,  vs. total displacement u measured over the gauge length L For stress values in the softening branch (i.e. beyond damage initiation), compute damage parameter d from the expression  = (1 - d ) Compute the corresponding plastic displacement u pl as shown in the schematic.

In the absence of intermediate data, choose linear softening and provide value of



d = 0; u = 0



pl

 y0

u

pl f

- d softening

0 E L

E L

(1-d)

E L

u

u pl

d = 1; u = u pl

u

pl f

Schematic representation of tensile test data in stress – displacement space for elastic-plastic materials


pl f

L4.27

Damage Evolution • Energy-based damaged evolution y  y0

u fpl =

y  y0

2G f

 y0

Gf (a) Linear

NOTE: The response is linear or exponential only if the undamaged response is perfectly plastic

Gf

u fpl u pl

(b) Exponential

u pl

*DAMAGE EVOLUTION,TYPE=ENERGY, SOFTENING={LINEAR,EXPONENTIAL}


L4.28

Damage Evolution • Example: Tearing of an X-shaped cross section Tie constraints

Pull and twist this this end

Fix this end

Failure modeled with different mesh densities

*damage initiation, criterion=fld 0.20, *damage evolution, type=displacement, softening=tabular 0.0, 0.0 1.0, 0.003 damage-plastic displacement data pairs Modeling Fracture and Failure with Abaqus

L4.29

Damage Evolution • Comparison of reaction forces and moments confirms mesh insensitivity of the results.


L4.30

Damage Evolution • Example: Axial crushing of an aluminum double-chamber profile

• Dynamic response with damage evolution

*Material, name=Aluminum : *Damage initiation, criterion=Ductile : *Damage evolution, type=displacement 0.1, *Damage initiation, criterion=Shear, ks=0.3 : *Damage evolution, type=displacement 0.1,


L4.31

Damage Evolution • With damage evolution, the simulation response is a good approximation of the physical response.

Simulation without damage evolution

Aluminum double-chamber after dynamic impact


Simulation with damage evolution

Element Removal

L4.33

Element Removal • Abaqus offers the choice to remove the element from the mesh once the material stiffness is fully degraded (i.e., once the element has failed). • An element is said to have failed when all section points at any one integration point have lost their load carrying capacity. • By default, failed elements are deleted from the mesh.


L4.34

Element Removal • Removing failed elements before complete degradation

• The material point is assumed to fail when the overall damage variable D reaches the critical value Dmax. • You can specify the value for the maximum degradation Dmax. • The default value of Dmax is 1 if the element is to be removed from the mesh upon failure.


L4.35

Element Removal • Usage: *SECTION CONTROLS, NAME=Ec-1, ELEMENT DELETION=YES, MAX DEGRADATION=0.9

: ** Refer to the section controls by name on the element section definition. *SOLID SECTION, ELSET=Elset_1, CONTROLS=Ec-1, MATERIAL=Material_1 :


L4.36

Element Removal • Retaining failed elements

• You may choose not to remove failed elements from the mesh. *SECTION CONTROLS, ELEMENT DELETION = NO

• In this case the default value of Dmax is 0.99, which ensures that elements will remain active in the simulation with a residual stiffness of at least 1% of the original stiffness. • Here Dmax represents • the maximum degradation of the shear stiffness (three-dimensional), • the total stiffness (plane stress), or • the uniaxial stiffness (one-dimensional).

• Failed elements that have not been removed from the mesh can sustain hydrostatic compressive stresses. Modeling Fracture and Failure with Abaqus

L4.37

Element Removal • Output Failed elements removed by default when STAUS output is available

• The output variable SDEG contains the value of D. • The output variable STATUS indicates whether or not an element has failed. • STATUS = 0 for failed elements • STATUS = 1 for active elements • Abaqus/Viewer will automatically remove failed elements when the output database (.odb) file includes STATUS.

failed elements

Deactivate status variable to view failed elements Modeling Fracture and Failure with Abaqus

Damage in Fiber-Reinforced Composite Materials

L4.39

Damage in Fiber-Reinforced Composite Materials • Abaqus offers a general capability for modeling progressive damage and failure in fiber-reinforced composites. • Material failure refers to the complete loss of load carrying capacity that results from progressive degradation of the material stiffness. • Stiffness degradation is modeled using damage mechanics. • The model must be used with elements with a plane stress formulation (plane stress, shell, continuum shell, and membrane elements) • Four different modes of failure are considered: • fiber rupture in tension; • fiber buckling and kinking in compression; • matrix cracking under transverse tension and shearing; and • matrix crushing under transverse compression and shearing

Common damage types in composite laminates


L4.40

Damage in Fiber-Reinforced Composite Materials • User interface

• Damage Initiation

*DAMAGE INITIATION, CRITERION=HASHIN, ALPHA= XT,XC,YT,YC,SL,ST


L4.41

Damage in Fiber-Reinforced Composite Materials • Damage Evolution *DAMAGE EVOLUTION, TYPE=ENERGY, SOFTENING=LINEAR

Gft,Gfc ,Gmt,Gmc

• Viscous Regularization *DAMAGE STABILIZATION ηft, ηfc, ηmt, ηmc


L4.42

Damage in Fiber-Reinforced Composite Materials • Output

• Initiation Criteria Variables • HSNFTCRT – tensile fiber Hashin’s criterion

• HSNFCCRT – compressive fiber Hashin’s criterion • HSNMTCRT – tensile matrix Hashin’s criterion

• HSNMCCRT – compressive matrix Hashin’s criterion • Damage Variables • DAMAGEFT – tensile fiber damage • DAMAGEFC – compressive fiber damage • DAMAGEMT – tensile matrix damage

• DAMAGEMC – compressive matrix damage


L4.43

Damage in Fiber-Reinforced Composite Materials • Output (cont'd)

• Status • STATUS – element status (1 – present, 0 – removed)

•

Energies • Damage energy (ALLDMD,DMENER,ELDMD,EDMDDEN)

• Viscous regularization (ALLCD, CENER, ELCD, ECDDEN)


L4.44

Damage in Fiber-Reinforced Composite Materials • Example: Analysis of blunt notched fiber metal laminate

• Fiber metal laminates (FMLs) are composed of: • laminated thin aluminum layers

• Intermediate glass fiber-reinforced epoxy layers


L4.45

Damage in Fiber-Reinforced Composite Materials • Geometry of blunt notched fiber metal laminate (Glare 3 3/2–0.3) Aluminum core and exterior

1/8 part model

a through-thickness hole

• Through-thickness view of the laminate:

glass fiber-reinforced epoxy layers

Example Problem 1.4.6, "Failure of blunt notched fiber metal laminates” Modeling Fracture and Failure with Abaqus

L4.46

Damage in Fiber-Reinforced Composite Materials • Results

damage in matrix

and

damage in fibers

for one of glass fiber-reinforced epoxy layers

Net blunt notch strength (MPa) Test (De Vries, 2001)

446

Abaqus

453


L4.47

Damage in Fiber-Reinforced Composite Materials • Abaqus allows the import of the damage model for fiber-reinforced composites from Abaqus/Explicit to Abaqus/Standard. • Details of the import capability will not be covered in this lecture (please refer to ―Importing and transferring results,‖ Section 9.2 of the Abaqus Analysis User’s Manual). • One typical application is the analysis of Barely Visible Impact Damage (BVID) in composite structures used in aerospace applications. • Non-visible damage to composite structures is a significant concern in the aerospace industry.

from McGowan, D.M., and Ambur, D.R., NASA TM-110303

Damage-Tolerance Characteristics of Composite Fuselage Sandwich Structures With Thick Facesheets


Damage in Fasteners

L4.49

Damage in Fasteners • Connection methodologies—point fasteners

• Fastener (spot weld) compliance and failure are available in Abaqus.

multiple layers

attachment points

radius of influence


L4.50

Damage in Fasteners • Fastener failure

• Model combines plasticity and progressive damage

S  0

– Response depends on loading angle (normal/shear) N  90

– Stages: • Rigid plasticity with variable hardening • Damage initiation

• Progressive damage evolution using fracture energy

Spot weld

F

Plasticity + Damage

0



45 90 Plasticity

damage initiation boundary

Schematic representation of the predicted numerical response Modeling Fracture and Failure with Abaqus

u pl

L4.51

Damage in Fasteners • Example

• Spot-welded hat section of three layers of sheet metals subjected to severe compressive loading Deformable fastener still holding

Failed fasteners

Rigid spot welds

Compliant spot welds with damage


Material Wear and Ablation

L4.53

Material Wear and Ablation • Material wear/erosion in Abaqus/Standard

• Many applications require the modeling of wear/erosion of material at one or more surfaces • Capability enables modeling of material wear/erosion on the surface of the body • Idea is to erode material while receding mesh away from surface (with same number and topology of elements) • Involves remeshing, state mapping—handled through an Arbitrary Lagrangian-Eulerian (ALE) technique • User interface takes advantage of existing adaptive meshing framework to define mesh motion

Adaptive mesh domain for modeling material wear. Wear extent/velocity applied as mesh constraints


L4.54

Material Wear and Ablation • Applications

• Geotechnical • Well bore sand production

• Plastic strain, fluid velocity • Aerospace

• Rocket motor ablation • Pyrolysis, char formation • Solid propellants • Automotive • Tire wear

• Disk brake wear Fluid velocity dependent wear of a well bore

• Manufacturing

• Machining Modeling Fracture and Failure with Abaqus

L4.55

Material Wear and Ablation • User interface *Adaptive mesh, elset=... *Adaptive mesh constraint, type=[velocity|displacement], User *Adaptive mesh controls

• Adaptive mesh constraints define mesh motion (wear extent or velocity)

• Wear criterion • General descriptions possible through user subroutine UMESHMOTION

• User access to solution variables • Nodal

• Material • Contact

• A local surface coordinate system is provided


L4.56

Material Wear and Ablation • Example of wear criterion

• Tire wear • Use of CSLIP, CSHEAR, CPRESS

h =  E Rate of frictional energy dissipation

Rate of recession of tread

Proportionality constant

h


L4.57

Material Wear and Ablation • Example: erosion of material from oil bore hole perforation tunnel • Setup consists of bore hole with perforations, loaded by weight of material above

• Pore pressure gradient leads to flow into perforation • Material wear rate controlled by fluid flux, transport concentration, porosity, sand production coefficient, and the local plastic deformation • Optimum design to minimize wear rate • Example Problem 1.1.22

Perforation tunnel

Bore hole Geometry of oil well

Courtesy of Exxon


L4.58

Material Wear and Ablation • Analysis steps

• Geostatic • Model change removal of well bore and casing (drilling operation)

• Apply pore pressure; establish steady state conditions • Transient soils consolidation (during which the erosion occurs)

• Ablation relation: V = 10 × (PEEQ - 0.028)

Erosion velocity


L4.59

Material Wear and Ablation • Adaptive mesh constraints *Adaptive mesh, elset=Adaptive-Zone, Freq=1, Mesh=40 *Adaptive mesh constraint, constraint type=Lagrangian

Lag *Adaptive mesh constraint, type=velocity, user

Rock-Perf, 1, 1, 1.0

Adaptive-Zone

Cut section of the adaptive mesh domain showing the perforation tunnel


Lag: Nodes on back face of adaptive domain Rock-Perf

L4.60

Material Wear and Ablation • User subroutine

subroutine umeshmotion(uref,ulocal,node,nndof,lnodetype,alocal, $ ndim,time,dtime,pnewdt,kstep,kinc,kmeshsweep,jmatyp,jgvblock,lsmooth)

c include 'aba_param.inc' c parameter parameter dimension dimension dimension dimension dimension

(zero=0.d0, ten=10.d0, peeqCrit=0.028d0) (nelemmax=100) array(1000) ulocal(*) jgvblock(*),jmatyp(*) alocal(ndim,*) jelemlist(nelemmax),jelemtype(nelemmax)

locnum = 0 jtyp = 1 peeq = zero nelems = nelemmax call getNodeToElemConn(node,nelems,jelemlist, $ jelemtype,jrcd,jgvblock) call getVrmAvgAtNode(node, jtyp, 'PE', array, jrcd, $ jelemlist, nelems, jmatyp, jgvblock) peeq = array(7)

When NDIM=3 the 3-direction is normal to the surface

if (peeq .gt. peeqCrit) then ulocal(ndim) = ulocal(ndim)- ten*(peeq - peeqCrit) end if return ulocal passed in as the value end the mesh smoothing algorithm


determined by

L4.61

Material Wear and Ablation • Results

Material wear at bore hole/perforation junction

Total volume lost due to erosion is available with history output variable VOLC


L4.62

Material Wear and Ablation • Mesh smoothing

• Two options • Original configuration projection method • Smoothing performed according to the original configuration Original-configuration smoothing

• Volume-based smoothing

• Either method can include a geometric-based enhancement

Volumetric smoothing


L4.63

Material Wear and Ablation • Smoothing permitted in conjunction with UMESHMOTION constraints

• Enables UMESHMOTION to describe normal mesh motions, while the smoothing algorithm handles the tangential mesh motions.


L4.64

Material Wear and Ablation • Limitations

• Available for a subset of continuum elements • Available only for following procedures using geometric nonlinearity

• Static • Soils

• Coupled Temperature-Displacement • Tracer particles not supported


Element-based Cohesive Behavior Lecture 5

L5.2


• Element Technology • Constitutive Response

• Viscous Regularization • Modeling Techniques

• Examples • Workshop 3 (Part 1) • Workshop 4


L5.3

Overview • Historical perspective

• The concept of a cohesive zone has been around for some time: • Dugdale (1960) and Barenblatt (1962) were the first to apply the concept of a cohesive stress zone to fracture modeling. • Many extensions since then.

• For example, Needleman (1987) recognized that cohesive elements are particularly attractive when interface strengths are relatively weak compared to the adjoining materials. • Examples: composite laminates and parts bonded with adhesives


Introduction

L5.5

Introduction • Cohesive behavior is useful in modeling adhesives, bonded interfaces, and gaskets. • Models separation between two initially bonded surfaces

• Progressive failure of adhesives • Delamination in composites

• Idealize complex fracture mechanisms with a macroscopic “cohesive law,” which relates the traction across the interface to the separation.

T-peel analysis: Cohesive elements are used for modeling adhesive patches

• The cohesive behavior can be: • Element-based • Modeled with cohesive elements • Surface-based • Modeled with contact pairs in Abaqus/Standard and general contact in Abaqus/Explicit

Failed adhesive is red

Rail crush: Cohesive surfaces Modeling Fracture and Failure with Abaqus

(CSDMG = 1)

L5.6

Introduction • Element-based cohesive behavior—cohesive elements

• Cohesive elements allow very detailed modeling of adhesive connections, including • specification of detailed adhesive material properties, direct control of the connection mesh, modeling of adhesives of finite thickness, etc. • Cohesive elements in Abaqus primarily address two classes of problems:

• Adhesive joints • Adhesive layer with finite thickness • Typically the bulk material properties are known • Delamination • Adhesive layer of “zero” thickness

• Typically the bulk material properties are not known Modeling Fracture and Failure with Abaqus

L5.7

Introduction • The constitutive modeling depends on the class of problem:

• Based on macroscopic properties (stiffness, strength) for adhesive joints • Continuum description: any Abaqus material model can be used • Modeling technique is relatively straightforward: cohesive layer has finite thickness; standard material models (including damage). • The continuum description is not discussed further in this lecture.

• Based on a traction-separation description for delamination • Linear elasticity with damage

• Modeling technique is less straightforward: typical applications use zero-thickness cohesive elements; non-standard constitutive law • This application is the primary focus of this lecture


L5.8

Introduction • In addition, the uniaxial response of a laterally unconstrained adhesive patch can also be modeled • This represents the behavior of a gasket. • Limited capability for modeling gaskets with cohesive elements: • The complexity of the response in the thickness direction is not as rich as with gasket elements available in Abaqus/Standard. • Compared to gasket elements, however, cohesive elements:

• are fully nonlinear (can be used with finite strains and rotations); • can have mass in a dynamic analysis; and • are available in both Abaqus/Standard and Abaqus/Explicit.

• The use of cohesive elements for modeling gaskets is not discussed further in this lecture.


L5.9

Introduction • Surface-based cohesive behavior—cohesive surfaces • This is a simplified and easy way to model cohesive connections, using the traction-separation interface behavior. • It offers capabilities that are very similar to cohesive elements modeled with the traction-separation constitutive response. • However, it does not require element definitions.

• In addition, cohesive surfaces can bond anytime contact is established (“sticky” contact) • It is primarily intended for situations in which interface thickness is negligibly small. • It must be defined as a surface interaction property. • Damage for cohesive surfaces is an interaction property, not a material property. • The kinematics of cohesive surfaces is different from that of cohesive elements. • By default, the initial stiffness of the interface is computed automatically. Modeling Fracture and Failure with Abaqus

L5.10

Introduction • Cohesive elements are the focus of this lecture.

• Cohesive surfaces are discussed in the next lecture. • A workshop exercise will allow you to compare and contrast the two cohesive modeling techniques in the context of a simple problem.


Element Technology

L5.12

Element Technology Top face

• Element types*

• 3D elements • COH3D8

• COH3D6

Bottom face

• 2D element

• COH2D4 • Axisymmetric element • COHAX4 • These elements can be embedded in a model via • shared nodes or • tie constraints. *Cohesive pore pressure elements are also available. Modeling Fracture and Failure with Abaqus

L5.13

Element Technology • Element and section definition *ELEMENT, TYPE = COH3D8 *COHESIVE SECTION, ELSET =..., RESPONSE = {TRACTION SEPARATION, CONTINUUM, GASKET },

THICKNESS = { SPECIFIED, GEOMETRY}, MATERIAL = ...

Specify thickness in dataline (default is 1.0)


L5.14

Element Technology • Default thickness of cohesive elements

• Traction-separation response: • Unit thickness

• Continuum and gasket response • Geometric thickness based on nodal coordinates


L5.15

Element Technology • Output variables

• Scalar damage (i.e., degradation) variable • SDEG

• Variables indicating whether damage initiation criteria met or exceeded • Discussed shortly

• Element status flag • STATUS


L5.16

Element Technology • Import of cohesive elements

• The combination of Abaqus/Standard and Abaqus/Explicit expands the range of applications for cohesive elements. • For example, you can simulate the damage in a structure due to an impact event then study the effect of the damage on the structure's load carrying capacity.


Constitutive Response

L5.18

Constitutive Response • Delamination applications

T

• Traction separation law

N

• Typically characterized by peak strength (N) and fracture energy (GTC) • Mode dependent

GT C

• Linear elasticity with damage • Available in both Abaqus/Standard and Abaqus/Explicit

 Typical traction-separation response

• Modeling of damage under the general framework introduced earlier

7 6

• Damage initiation

• Damage evolution

GTC

• Traction or separation-based criterion

Shear mode

5 4

Normal mode

3 2 1 0

• Removal of elements

0

0.2

0.4

0.6

0.8

Mode Mix

Dependence of fracture toughness on mode mix Modeling Fracture and Failure with Abaqus

1

L5.19

Constitutive Response • Linear elasticity with damage

• Linear elasticity • Defines behavior before the initiation of damage • Relates nominal stress to nominal strain • Nominal traction to separation with default choice of unit thickness • Uncoupled traction behavior: nominal stress depends only on corresponding nominal strain

• Coupled traction behavior is more general

*ELASTIC, TYPE = { TRACTION, COUPLED TRACTION }


L5.20

Constitutive Response • The elastic modulus for the traction separation law should be interpreted as a penalty stiffness.

N N max

• For example, for the opening mode:

Kn = Nmax / ninit

• In Abaqus, nominal stress and strain quantities are used for the traction separation law.

Kn 1

• If unit thickness is specified for the element, then the nominal strain corresponds to the separation value.

 ninit

• Elastic response governed by Kn. • If you specify a non-unit thickness for the cohesive element, you must scale your data to obtain the correct stiffness Kn. Example on next slide.

Displacement at damage initiation in normal (opening) mode


 nfail

n

L5.21

Constitutive Response • Example: Peel test model

N = En n

Abaqus evaluates this…

= K n n

…which is equivalent to this

 n =  n / heff  K n = En / heff Assume separation at initiation  ninit = 1e-3 and Nmax = 6.9e9.

A

For model A: use geometric thickness heff = hgeom =1e-3 n =  ninit/heff = 1; init

En=Knheff

Nmax = En = 6.9e9  Kn = 6.9e12 For model B: specify unit thickness heff = 1  n =  n / heff =1e-3; init

init

Nmax = 6.9e9  En = Kn = 6.9e12

B Geometric thickness (based on nodal coordinates) of the adhesive hgeom = 1e-3 Modeling Fracture and Failure with Abaqus

L5.22

Constitutive Response • Damage initiation • Mixed mode conditions • Maximum stress (or strain) criterion:

 n t  s  MAX  , ,  =1  N max Tmax Smax 

n

 n for  n  0 = 0 for  n  0

• Output:

• MAXSCRT • MAXECRT * DAMAGE INITIATION, CRITERION = { MAXS, MAXE }


L5.23

Constitutive Response • For example, for Mode I (opening mode) the MAXS condition implies damage initiates when n = Nmax.

N N max

Damage initiation point

*Damage initiation,criterion=MAXS 290.0E6, 200.0E6, 200.0E6

n Nmax

Tmax

Smax


L5.24

Constitutive Response • Quadratic stress (or strain) interaction criterion:

2

2

2

 n   t    s        =1  N max   Tmax   Smax  • No damage initiation under pure compression • Output: • QUADSCRT

• QUADECRT * DAMAGE INITIATION, CRITERION = { QUADS, QUADE }


L5.25

Constitutive Response • Summary of damage initiation criteria Maximum nominal stress criterion

Maximum nominal strain criterion

     MAX  n , s , t  = 1  N max Smax Tmax 

  n s  t  MAX  max , max , max  = 1 s  t    n

*DAMAGE INITIATION, CRITERION=MAXS

*DAMAGE INITIATION, CRITERION=MAXE

N max , S max , Tmax

Quadratic nominal stress criterion 2

2

2

 nmax ,  smax ,  tmax

Quadratic nominal stress criterion 2

2

2

 n    s   t        =1  N max   Smax   Tmax 

  n    s   t   max    max    max  = 1   n    s   t 

*DAMAGE INITIATION, CRITERION=QUADS

*DAMAGE INITIATION, CRITERION=QUADE

N max , S max , Tmax n: nominal stress in the pure normal mode s: nominal stress in the first shear direction t: nominal stress in the second shear direction Note :  n =

n To

, s =

s To

, t =

t To

 nmax ,  smax ,  tmax

n: nominal strain in the pure normal mode s: nominal strain in the first shear direction t: nominal strain in the second shear direction

where n,

s, and t are components of relative displacement between the top and bottom of the cohesive element; and To

is the original thickness of the cohesive element. Modeling Fracture and Failure with Abaqus

L5.26

Constitutive Response • Damage evolution • Post damage-initiation response defined by:

 = 1 - d 

  -d (1 - d )

• d is the scalar damage variable

K0 (1 - d ) Κ 0

d = 0: undamaged d = 1: fully damaged

d monotonically increases

 K0 Typical damaged response


L5.27

Constitutive Response • Damage evolution is based on energy or displacement

N N max Area under the curve is the fracture energy

• Specify either the total fracture energy or the post damage-initiation effective displacement at failure

GT C

• May depend on mode mix

• Mode mix may be defined in terms of energy or traction

n Displacement at failure  n in normal (opening) mode

fail


L5.28

Constitutive Response • Displacement-based damage evolution

Traction Linear postinitiation response

• Damage is a function of an effective displacement:

=

n

2

  s2   t2

• The post damage-initiation softening response can be either • Linear • Exponential

 init

• Tabular


 fail 

L5.29

Constitutive Response • Keywords interface for displacement-based damage evolution *DAMAGE EVOLUTION, TYPE = DISPLACEMENT, SOFTENING = { LINEAR | EXPONENTIAL | TABULAR }, MIXED MODE BEHAVIOR = TABULAR

• For LINEAR and EXPONENTIAL softening:

• Specify the effective displacement at complete failure fail relative to the effective displacement at initiation init. • For TABULAR softening: • Specify the scalar damage variable d directly as a function of  –init. • Optionally specify the effective displacement as function of mode mix in tabular form.

• Abaqus assumes that the damage evolution is mode independent otherwise.


L5.30

Constitutive Response • Abaqus/CAE interface for displacement-based damage evolution


L5.31

Constitutive Response • Energy-based damage evolution

• The fracture energy can be defined as a function of mode mix using either a tabular form or one of two analytical forms: • Power law 





 GI   GII   GIII        =1  GIC   GIIC   GIIIC  • BK (Benzeggagh-Kenane) 

 Gshear  GIC   GIIC - GIC    = GTC  GT  where Gshear = GII  GIII GT = GI  Gshear Modeling Fracture and Failure with Abaqus

For isotropic failure (GIC = GIIC), the response is insensitive to the value of .

L5.32

Constitutive Response • Keywords interface for energy-based damage evolution *DAMAGE EVOLUTION, TYPE = ENERGY, SOFTENING = { LINEAR | EXPONENTIAL}, MIXED MODE BEHAVIOR = { TABULAR | POWER LAW | BK }, POWER = value

• Specify fracture energy as function of mode mix in tabular form, or • Specify the fracture energy in pure normal and shear deformation modes and choose either the POWER LAW or the BK mixed mode behavior


L5.33

Constitutive Response • Abaqus/CAE interface for energy-based damage evolution


L5.34

Constitutive Response • Example

Normal (opening) mode:

• In the simplest case, Abaqus requires that you input the adhesive thickness heff and 10 material parameters: *Elastic, type=traction

En, Et, Es *Damage initiation, criterion = maxs

Cohesive material law: Traction, Damage Evolution

N max Traction (nominal stress)

• The preceding discussion was very general in the sense that the full range of options for modeling the constitutive response of cohesive elements was presented.

Kn = Kn 1

GIC

En heff

(area under entire curve)

 ninit

 nfail Separation

What do you do when you only have 1 property and the adhesive thickness is essentially zero?

Nmax, Tmax, Smax Diehl, T., "Modeling Surface-Bonded Structures with

*Damage evolution, type=energy, ABAQUS Cohesive Elements: Beam-Type Solutions," mixed mode behavior=bk, power= ABAQUS Users' Conference, Stockholm, 2005. GIC, GIIC , GIIIC Modeling Fracture and Failure with Abaqus

L5.35

Constitutive Response • Example (cont’d)

• Common case: you know GTC for the surface bond. • Assume isotropic behavior

GIC = GIIC = GIIIC = GTC • For MIXED MODE BEHAVIOR = BK, this makes the response independent of  term, so set  = any valid input value (e.g., 1.0)

• Bond thickness is essentially zero • Specify the cohesive section property thickness heff = 1.0  Nominal strains = separation; elastic moduli = stiffness

• Isotropy also implies the following:

En = Et = Es = Eeff

(=Keff since we chose heff = 1.0)

Nmax = Tmax = Smax = Tult Modeling Fracture and Failure with Abaqus

L5.36


• Introduce concept of damage initiation ratio:

ratio= init /fail, where 0  ratio  1. • Use GC and equation of a triangle to relate back to Keff and Tult :

Keff =

2 GTC

 ratio  2fail

Tult =

2 GTC

 fail

• The problem now reduces to two penalty terms: fail and ratio.

• Assume ratio = ½. • Choose fail as a fraction of the typical cohesive element mesh size.

• For example, use fail = 0.050  typical cohesive element size as a starting point.


L5.37


• Thus, after choosing the two penalty terms, a single (effective) traction-separation law applies to all modes (normal + shear):

Effective properties: Cohesive material law: Traction, Damage Evolution

Traction (nominal stress)

Tult

K eff =

Eeff

K eff GTC 1

(area under entire curve)

 fail

 init Separation

heff

*Cohesive section, thickness=SPECIFIED, ... 1.0, : : *Elastic, type=TRACTION Keff, Keff, Keff *Damage initiation, criterion = MAXS Tult, Tult, Tult *Damage evolution, type=ENERGY, mixed mode behavior=BK, power=1 GTC, GTC , GTC


L5.38


• What if the response is dynamic? What about the density? • The density of the cohesive layer should also be considered a penalty quantity. • For Abaqus/Explicit, the effective density should not adversely affect the stable time increment. Diehl suggests the following rule:

eff = Eeff

 D tstable   ft 2 D heff 

  

2

• Dtstable = stable time increment without cohesive elements in the model • ft2D = 0.32213 (for cohesive elements whose original nodal coordinates relate to zero element thickness)

• The Abaqus Analysis User’s Manual provides additional guidelines for determining a cohesive element density that minimizes the effect on the stable time increment in Abaqus/Explicit. Modeling Fracture and Failure with Abaqus

Viscous Regularization

L5.40

Viscous Regularization • Cohesive elements have the potential to cause numerical difficulties in the following cases • Stiff cohesive behavior may lead to reduced maximum stable time increment in Abaqus/Explicit • Potentially addressed through selective mass scaling • Unstable crack propagation may lead to convergence difficulties in Abaqus/Standard • Potentially addressed through built-in viscous regularization option specific to cohesive elements


L5.41

Viscous Regularization • Viscous regularization

• Material models with damage often lead to severe convergence difficulties in Abaqus/Standard • Viscous regularization helps in such cases • Helps make the consistent tangent stiffness of softening material positive for sufficiently small time increments • Similar approach used in the concrete damaged plasticity model in Abaqus/Standard

 = 1 - dv  1 dv =  d - dv 




L5.42

Viscous Regularization • Consistent material tangent stiffness

D = 1 - d  K 0 - f

d   

K0 is the undamaged elastic stiffness

f is a factor that depends on the details of the damage model • Viscous regularization ensures that when

Dt



 0 , D = (1 - d ) K 0

• “Offending” second term is eliminated when the analysis cuts back drastically


L5.43

Viscous Regularization • User interface for viscous regularization *COHESIVE SECTION, CONTROLS = control1 *SECTION CONTROLS, NAME = control1, VISCOSITY = factor

• Add-on transverse shear stiffness may provide additional stability *COHESIVE SECTION *TRANSVERSE SHEAR STIFFNESS • Output • Energy associated with viscous regularization: ALLCD


L5.44

Viscous Regularization • Example: Multiple delamination problem (Alfano & Crisfield, 2001) – Industry standard AlfanoCrisfield nonsymmetric delamination examples

12 layers

2 layers

Initial cracks

Interface elements

• Plies are initially bonded with predefined cracks, then peeled apart in a complex sequence

• Example done in Abaqus/Standard and Abaqus/Explicit

10 layers a

a

a

1

2

2

• Effect of viscous regularization is investigated


L

L5.45

Viscous Regularization

 = 5.e - 4

 = 1.e - 3

=0  = 1.e - 4

 = 2.5e - 4


L5.46

Viscous Regularization • Effect of viscous regularization on convergence of multiple delamination problem: • Significant improvements with small regularization factor

Viscous regularization factor

Total number of increments

0.

375

1.0e-4

171

2.5e-4

153

1.0e-3

164


Modeling Techniques

L5.48

Modeling Techniques • Model problem: double-cantilever beam

• Alfano and Crisfield (2001) • Pure Mode I

• Displacement control

u

• Analyzed using

• 1D (B21), • 2D (CPE4I), and

-u Initial crack

• 3D (C3D8I) elements • Delamination assumed to occur along a straight line • Beams: Orthotropic material

• Cohesive layer: Traction-separation with damage


L5.49

Modeling Techniques • One-dimensional model

• Use tie constraints between the cohesive layer and the beams • Require distinct parts for the beam and cohesive zone geometry

• Geometry


L5.50

Modeling Techniques • One-dimensional model (cont’d)

• Assembly

Create 2 instances of the beam; one of the cohesive zone

Position the parts to leave gaps between them; this will later facilitate picking surfaces


L5.51


• Tie constraints coh-top beam-top

beam-bot

coh-bot Define tie constraints between mating surfaces.

The cohesive side should be the slave surface (because it is a softer material) This approach is required when quadratic displacement elements are used.


L5.52


• Properties: beam


L5.53


• Properties: adhesive


L5.54


• Meshing

1 Cohesive elements can only be assigned to sweep meshable regions Sweep path must be aligned with thickness direction

3 Assign seeds and mesh

Only one element through the thickness


2 Assign cohesive element type to the swept region

L5.55


• Meshing (cont’d)

4

Edit the nodal coordinates of each part instance so that they all have the same 2-coordinate

Toggle this off; otherwise, nodes will project back to their original positions

Final mesh


L5.56

Modeling Techniques • Two-dimensional model

• All geometry is 2D and planar • Properties, attributes, etc. treated in a similar manner to the 1D case presented earlier • Modeling options include: • Shared nodes

• Tie constraints • Similar to the 1D model


L5.57

Modeling Techniques • Two-dimensional model (cont’d)

• Shared nodes 1

Define a finite thickness slit in the beam as shown below

• Use the actual overall thickness of the DCB • The center region represents the cohesive layer 2

Mesh the part:


L5.58


• Shared nodes (cont’d) 3

Edit the coordinates of the nodes along the interface


L5.59


• Tie constraints 1

Create two instances of the beams and position them as shown below.

• Suppress the visibility of the instances to facilitate picking surfaces, etc. 2

Create a finite thickness cohesive layer, position it appropriately in the horizontal direction, define surfaces, etc. • After meshing, adjust the coordinates of all the nodes in the cohesive layer so that they lie along the interface between the two beams.


L5.60

Modeling Techniques • Three-dimensional model

• All geometry is 3D • Solid geometry for beams

• Solid or shell geometry for cohesive layer • Modeling options include

• Shared nodes • Tie constraints


L5.61

Modeling Techniques • Three-dimensional model (cont’d)

• Shared nodes

1 Partition the geometry and define a mesh seam between these two faces


L5.62


• Shared nodes (cont’d)

2

Mesh the part with solid (continuum elements).

3

Create a orphan mesh

Mesh→Create Mesh Part


L5.63

Modeling Techniques

4 Create a single zero-thickness

Tip 1: Remove elements from top region with display groups (select by angle)

solid layer by offsetting from the midplane (selected by angle) of the orphan mesh created in the previous step

Tip 2: Use the selection options tools to facilitate picking. In particular, select from interior entities.

Create a set for the new layer so you can easily assign element type and section properties. Modeling Fracture and Failure with Abaqus

L5.64


• Shared nodes (cont’d)

5 Assign section properties and the element type to the set created in the previous step


L5.65


• Tie constraints • The cohesive region can be defined as • Solid (with finite thickness) • Edit nodal coordinates of cohesive elements as in previous examples

• Shell geometry • Mesh geometry then create orphan mesh • Offset a zero-thickness layer of solid elements from the orphan mesh Define surfaces automatically to facilitate tie constraints


L5.66


• Tie constraints (cont’d)

When defining the tie constraints, query the mesh stack direction to determine when the “top” and “bottom” surfaces should be used

Brown = top

Purple = bottom


L5.67

Modeling Techniques •

What if I don't use Abaqus/CAE?

•

In this case do the following in the preprocessor of your choice: 1. Generate the mesh for the structure and cohesive layer (temporarily assigning an arbitrary element type to the cohesive layer) 2. Position the layer of cohesive elements over the interface 3. Define surfaces on the structure and cohesive layer

4. Write the input file

Surface top-beam

Surface bot-beam


Surface top-coh

Surface bot-coh

L5.68


Edit the input file: 5. Change the element type assigned to the cohesive layer *element, elset=coh, type=coh2d4

6. Assign cohesive section properties *cohesive section, elset=coh, material=cohesive, response=traction separation, stack direction=2, controls=visco 1.0, 0.02 : *material, name=cohesive *elastic, type=traction 5.7e+14, 5.7e+14, 5.7e+14 *damage initiation, criterion=quads 5.7e7, 5.7e7, 5.7e7 *damage evolution, type=energy, mixed mode behavior=bk, power=2.284 280.0, 280.0, 280.0


L5.69


The stack direction defines the thickness direction based on the element isoparametric directions. •

Set STACK DIRECTION = { 1 | 2 | 3 } to define the element thickness direction along an isoparametric direction.

•

2D example (extends to 3D): 2

201

202

201

102

202

2

1 101

1

Thickness direction

101

102

Element connectivity: 101, 102, 202, 201

Element connectivity: 102, 202, 201, 101

Stack direction = 2

Stack direction = 1


L5.70


Edit the input file (cont'd): 7. Define tie constraints between the surfaces

Cohesive surface is the slave

*tie, name=top, adjust=yes, position tolerance=0.002 top-coh, top-beam *tie, name=bot, adjust=yes, position tolerance=0.002 bot-coh, bot-beam

Setting adjust=yes will force Abaqus to move the slave (cohesive element) nodes onto the master surface. By adjusting both the top and bottom cohesive surfaces in this way, a zero-thickness cohesive layer is produced.

The position tolerance should be large enough to contain the slave nodes when measured from the master surface. In this case the overclosure is equal to 0.001 on either side of the interface so a position tolerance of 0.002 is sufficient to capture all slave nodes. 0.001


L5.71

Modeling Techniques • Results


L5.72

Modeling Techniques • Effect of viscous regularization

Viscous regularization factor

Total number of increments

1.e-5

636

2.5e-5

163

5.0e-5

129

1.0e-4

90


L5.73

Modeling Techniques • Effect of mesh refinement

• Typically, you will need to use a much finer mesh (for both the stress/displacement and cohesive elements) than may be necessary for a problem without cohesive elements.


L5.74

Modeling Techniques • Non-planar geometry

• The technique for embedding a layer of solid elements into an orphan mesh is not restricted to planar geometry. • As an example, consider the following fiber-matrix pullout model

matrix Orphan mesh

fiber


L5.75

Modeling Techniques • Failure driven by mismatch in CTEs

View cut of the matrix-fiber interface at 100% of the applied load (magnified 5×)

Failure levels at 38% of the applied load


L5.76

Modeling Techniques • Cohesive elements on a symmetry plane

N

• The traction-separation law is based on N max the separation between the top and bottom faces of the cohesive element.

GC area = 2

• On a symmetry plane, however, the separation that is computed is ½ the actual value. • To account for this, specify:

2Kn 1

• 2 the cohesive stiffness that would be used in a full model. • ½ the fracture toughness that would be used in a full model. • Linear equations between the nodes on the top and bottom faces in the lateral directions.


 ninit

 nfail

2

2

2Kn =

2 En En = heff heff / 2

n

L5.77

Modeling Techniques • Symmetry example

Constraint on lateral displacements

Symmetric model (top) overlaid on full model

Symmetric model

Full model

Constitutive thickness is same as for the full model so double the elastic modulus to double the cohesive stiffness


Examples

L5.79

Examples • Composite components in aerospace structures (Courtesy: NASA) • Stress concentrations around stiffener terminations and flanges

• Residual thermal strains at the interface at room temperature • Analysis of the effects of residual strains on skin/stiffener debonding • Delamination initiation and propagation

Beginning of separation

After separation

Abaqus/Standard simulation of skin/stiffener debonding

Example Problem 1.4.5


L5.80

Examples

Abaqus/Standard simulation of skin/stiffener debonding Modeling Fracture and Failure with Abaqus

L5.81

Examples • Electronic packaging (Courtesy: INTEL) • Solder to motherboard fracture due to static overload • Experiments to assess integrity of solder joints under various loading conditions (e.g., board bending) • Strain in motherboard at which solder joint fails

Ball grid array Modeling Fracture and Failure with Abaqus

L5.82

Examples Debonded solder balls

Damage severity in cohesive layer between motherboard and solder balls Modeling Fracture and Failure with Abaqus

L5.83

Examples • Delamination of a composite

• This model is a representative of composite delamination. • It comprises 3 layers of composite with adhesive layers applied between composite layers. • The composite delaminates under the impact of a heavy mass displayed in light greenish shade in the animation.

Cohesive layers


L5.84

Examples • Impact of moving mass with a stationary wall

• Brick wall modeled with adhesives applied to each face of each brick. • Simulating damage of the (stationary) wall from high velocity impact with a heavy mass • Analysis performed in Abaqus/Explicit. • This model is a representative of several problems that can be modeled using cohesive elements

• Hydroplaning • Machining

Section of the model illustrating the application of cohesive layers around the bricks.

• Oil Drilling • Excavation • Effect of explosion on a building. Modeling Fracture and Failure with Abaqus

L5.85

Examples • Deformation sequence


Workshop 3 (Part 1)

L5.87

Workshop 3 (Part 1) • Crack growth in a three-point bend specimen using element-based cohesive behavior • Generate cohesive element mesh • Define/assign traction-separation behavior and damage properties

Layer of cohesive elements


Workshop 4 (Optional)

L5.89

Workshop 4 (Optional) • Crack growth in a helicopter airframe

• Use the mesh offset tool to create a layer of cohesive elements • Impose symmetry conditions on the cohesive elements using linear equations

Cohesive element thickness shrunk to zero


Surface-based Cohesive Behavior Lecture 6

L6.2

Overview • Surface-based Cohesive Behavior

• Element- vs. Surface-based Cohesive Behavior • Workshop 3 (Part 2)


Surface-based Cohesive Behavior

L6.4

Surface-based Cohesive Behavior • Surface-based cohesive behavior provides a simplified way to model cohesive connections with negligibly small interface thicknesses using the traction-separation constitutive model.

• It can also model “sticky” contact (surfaces can bond after coming into contact). • The cohesive surface behavior can be defined for general contact in Abaqus/Explicit and contact pairs in Abaqus/Standard (with the exception of the finite-sliding, surface-to-surface formulation). • Cohesive surface behavior is defined as a surface interaction property. • To prevent overconstraints in Abaqus/Explicit, a pure master-slave formulation is enforced for surfaces with cohesive behavior.


L6.5

Surface-based Cohesive Behavior • User interface

Abaqus/CAE

Abaqus/Standard *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR ... *CONTACT PAIR, INTERACTION=cohesive surface1, surface2

Abaqus/Explicit *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR ... *CONTACT *CONTACT PROPERTY ASSIGNMENT surface1, surface2, cohesive


L6.6

Surface-based Cohesive Behavior • The formulae and laws that govern surface-based cohesive behavior are very similar to those used for cohesive elements with traction-separation behavior: traction • linear elastic traction-separation, • damage initiation criteria, and

GC

• damage evolution laws.

separation

• However, it is important to recognize that damage in surface-based cohesive behavior is an interaction property, not a material property. • Traction and separation are interpreted differently for cohesive elements and cohesive surfaces: Cohesive elements Relative displacement () between the top and bottom of the cohesive layer separation Nominal strain () = Initial thickness (To) traction Nominal stress ()

Cohesive surfaces

Contact separation () Contact stress (t) =


Contact force (F) Current area (A) at each contact point

L6.7

Surface-based Cohesive Behavior • Linear elastic traction-separation behavior

• Relates normal and shear stresses to the normal and shear separations across the interface before the initiation of damage. • By default, elastic properties are based on underlying element stiffness. • Can optionally specify the properties.

• Recall this specification is required for cohesive elements. • The traction-separation behavior can be uncoupled (default) or coupled.

*COHESIVE BEHAVIOR, TYPE= { UNCOUPLED, COUPLED} Optional data line to specify Knn, Kss, Ktt


L6.8

Surface-based Cohesive Behavior • Controlling the cohered nodes

• The slave nodes to which cohesive behavior is applied can be controlled to define a wider range of cohesive interactions: Can include: • All slave nodes • Only slave nodes initially in contact

• Initially bonded node set 1• Applying cohesive behavior to all slave nodes (default)

• Cohesive constraint forces potentially act on all nodes of the slave surface. • Slave nodes that are not initially contacting the master surface can also experience cohesive forces if they contact the master surface during the analysis. *COHESIVE BEHAVIOR, ELIGIBILITY = CURRENT CONTACTS


L6.9

Surface-based Cohesive Behavior 2 Applying cohesive behavior only to slave nodes initially in contact

• Restrict cohesive behavior to only those slave nodes that are in contact with the master surface at the start of a step. • Any new contact that occurs during the step will not experience cohesive constraint forces. • Only compressive contact is modeled for new contact.

*COHESIVE BEHAVIOR, ELIGIBILITY = ORIGINAL CONTACTS


L6.10

Surface-based Cohesive Behavior 3 Applying cohesive behavior only to an initially bonded node set

(Abaqus/Standard only) • Restrict cohesive behavior to a subset of slave nodes defined using *INITIAL CONDITIONS, TYPE=CONTACT. • All slave nodes outside of this set will experience only compressive contact forces during the analysis. • This method is particularly useful for modeling crack propagation along an existing fault line.

*COHESIVE BEHAVIOR, ELIGIBILITY = SPECIFIED CONTACTS


L6.11

Surface-based Cohesive Behavior • Example: Double cantilever beam (DCB)

• Analyze debonding of the DCB model using the surface-based cohesive behavior in Abaqus/Standard. • To model debonding using surface-based cohesive behavior, • you must define: 1• contact pairs and initially bonded crack surfaces;

2• the traction-separation behavior; 3• the damage initiation criterion; and 4• the damage evolution.

• You may also 5• specify viscous regularization to facilitate solution convergence u in Abaqus/Standard.

• Note: Steps 3, 4, and 5, will be covered later in this lecture.

-u Initial crack

Note: Only the Keywords interface is illustrated in the example; the Abaqus/CAE interface is illustrated in the workshop exercise. Modeling Fracture and Failure with Abaqus

Cohesive interface

L6.12

Surface-based Cohesive Behavior 1 • Define contact pairs and initially bonded crack surfaces

• The initially bonded portion of the slave surface (i.e., node set bond) is identified with the *INITIAL CONDITIONS, TYPE=CONTACT option.

bond

TopSurf

BotSurf

Note: Frictionless contact is assumed.

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=cohesive TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond slave surface master surface


a list of slave nodes that are initially bonded

L6.13

Surface-based Cohesive Behavior 2• Define traction-separation behavior

• In this model, the cohesive behavior is only enforced for the node set bond. • Use the ELIGIBILITY=SPECIFIED CONTACTS parameter to enforce this behavior.

• Recall the default elastic properties are based on underlying element stiffness. Here we specify the properties. bond

TopSurf

BotSurf

t Kn (Ks , Kt)

1

 Kn, Ks, and Kt: normal and tangential stiffness components

... *CONTACT PAIR, INTER=cohesive TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS 5.7e14, 5.7e14, 5.7e14 Optional

Kn

Ks


Kt

L6.14

Surface-based Cohesive Behavior • Damage modeling for cohesive surfaces • Damage of the traction-separation response for cohesive surfaces is defined within the same general framework used for cohesive elements. • The difference between the two approaches is that for cohesive surfaces damage is specified as part of the contact interaction properties.

t



tnmax tsmax , ttmax





 nmax  smax ,  tmax





 nf  sf ,  t f





tnmax , tsmax , and ttmax : peak values of the contact stress

 nmax ,  smax , and tmax : peak values of the contact separation

 nf ,  sf , and  t f : separations at failure


L6.15

Surface-based Cohesive Behavior • User interface

Abaqus/CAE

Abaqus/Standard *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR *DAMAGE INITIATION *DAMAGE EVOLUTION *CONTACT PAIR, INTERACTION=cohesive surface1, surface2

Abaqus/Explicit *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR *DAMAGE INITIATION *DAMAGE EVOLUTION *CONTACT *CONTACT PROPERTY ASSIGNMENT surface1, surface2, cohesive


L6.16

Surface-based Cohesive Behavior • Damage initiation criteria Maximum stress criterion

Maximum separation criterion

 tn ts tt  MAX  max , max , max  1 ts tt   tn

  n s  t  MAX  max , max , max  1 s  t    n

*DAMAGE INITIATION, CRITERION=MAXS

*DAMAGE INITIATION, CRITERION=MAXU

tnmax , tsmax , ttmax

Quadratic stress criterion 2

2

2

 nmax ,  smax , tmax

Quadratic separation criterion 2

2

2

 tn   ts   tt   max    max    max   1  tn   ts   tt 

  n    s   t   max    max    max   1   n    s   t 

*DAMAGE INITIATION, CRITERION=QUADS

*DAMAGE INITIATION, CRITERION=QUADU

tnmax , tsmax , ttmax tn: normal contact stress in the pure normal mode ts: shear contact stress along the first shear direction tt: shear contact stress along the second shear direction

 nmax ,  smax , tmax

n: separation in the pure normal mode s: separation in the first shear direction t: separation in the second shear direction

Note: Recall the damage initiation criteria for the cohesive elements: if the initial constitutive thickness To = 1, then  = /To = . In this case, the separation measures for both approaches are exactly the same. Modeling Fracture and Failure with Abaqus

L6.17

Surface-based Cohesive Behavior • Example: Double cantilever beam 3• Define the damage initiation criterion

• The quadratic stress criterion is specified for this problem.

bond

TopSurf

BotSurf

... *CONTACT PAIR, INTER=cohesive TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS 5.7e14, 5.7e14, 5.7e14 *DAMAGE INITIATION, CRITERION=QUADS 5.7e7, 5.7e7, 5.7e7

tnmax

tsmax


ttmax

L6.18

Surface-based Cohesive Behavior • Damage evolution

• For surface-based cohesive behavior, damage evolution describes the degradation of the cohesive stiffness. • In contrast, for cohesive elements damage evolution describes the degradation of the material stiffness. • Damage evolution can be based on energy or separation (same as for cohesive elements). • Specify either the total fracture energy (a property of the cohesive interaction) or the post damage-initiation effective separation at t failure.

• May depend on mode mix



tnmax tsmax , ttmax



• Mode mix may be defined in terms of energy or traction

GTC









 nf  sf ,  t f





L6.19

Surface-based Cohesive Behavior • Separation-based damage evolution

• Damage is a function of an effective separation:



n

2

t



tnmax tsmax , ttmax

Linear postinitiation response



  s2   t2

• As with cohesive elements, the post damage-initiation softening response can be either: • Linear • Exponential • Tabular









 nf  sf ,  t f





L6.20

Surface-based Cohesive Behavior • Separation-based damage evolution (cont’d)

• Usage:

*DAMAGE EVOLUTION, TYPE = DISPLACEMENT, SOFTENING = { LINEAR | EXPONENTIAL | TABULAR }, MIXED MODE BEHAVIOR = TABULAR


L6.21

Surface-based Cohesive Behavior • Energy-based damage evolution

• As with cohesive elements, the energy-based damage evolution criterion can be defined as a function of mode mix using either a tabular form or one of two analytical forms: Power law 



Benzeggagh-Kenane (BK) 

 GI   GII   GIII        1  GIC   GIIC   GIIIC 



 Gshear  GIC   GIIC - GIC     GTC G  T  where Gshear  GII  GIII GT  GI  Gshear


L6.22

Surface-based Cohesive Behavior • Energy-based damage evolution (cont’d)

• Usage:

*DAMAGE EVOLUTION, TYPE = ENERGY, SOFTENING = { LINEAR | EXPONENTIAL}, MIXED MODE BEHAVIOR = { TABULAR | POWER LAW | BK }, POWER = value


L6.23

Surface-based Cohesive Behavior • Example: Double cantilever beam 4 • Define damage evolution

• The energy-based damage evolution based on the BK mixed mode behavior is specified.

GIC   GIIC



G  - GIC   shear   GTC  GT 

bond

TopSurf

BotSurf

... *CONTACT PAIR, INTER=cohesive TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS 5.7e14, 5.7e14, 5.7e14 *DAMAGE INITIATION, CRITERION=QUADS 5.7e7, 5.7e7, 5.7e7 *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=2.284 280.0, 280.0, 280.0



GIC

GIIC


GIIIC

L6.24

Surface-based Cohesive Behavior • Viscous regularization

• Can be specified to facilitate solution convergence in Abaqus/Standard for surface-based cohesive behavior when stiffness degradation occurs. • Output: • Energy associated with viscous regularization: ALLCD

*DAMAGE STABILIZATION


L6.25

Surface-based Cohesive Behavior • Example: Double cantilever beam 5 • Specify a viscosity coefficient for

the cohesive surface behavior

bond

TopSurf

BotSurf

... *CONTACT PAIR, INTER=cohesive TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS 5.7e14, 5.7e14, 5.7e14 *DAMAGE INITIATION, CRITERION=QUADS 5.7e7, 5.7e7, 5.7e7 *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=2.284 280.0, 280.0, 280.0 *DAMAGE STABILIZATION 1.e-5

viscosity coefficient,




L6.26

Surface-based Cohesive Behavior • Example: Double cantilever beam

• Summary of the input for the traction-separation response Cohesive elements *COHESIVE SECTION, MATERIAL=cohesive, RESPONSE=TRACTION SEPARATION, ELSET=coh_elems, CONTROLS=visco , 0.02 *MATERIAL, NAME=cohesive *ELASTIC, TYPE=TRACTION 5.7e14, 5.7e14, 5.7e14 *DAMAGE INITIATION, CRITERION=QUADS 5.7e7, 5.7e7, 5.7e7 *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=2.284 280.0, 280.0, 280.0 *SECTION CONTROLS, NAME=visco, VISCOSITY=1.e-5

Cohesive surfaces *SURFACE INTERACTION, NAME=cohesive *COHESIVE BEHAVIOR, ELIGIBILITY=SPECIFIED CONTACTS 5.7e14, 5.7e14, 5.7e14 *DAMAGE INITIATION, CRITERION=QUADS 5.7e7, 5.7e7, 5.7e7 *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=2.284 280.0, 280.0, 280.0 *DAMAGE STABILIZATION 1.e-5


L6.27

Surface-based Cohesive Behavior • Results

u2 = 0.006

Cohesive elements

Failed cohesive elements

u2 u2 = 0.006

Cohesive surfaces

u2


Element- vs. Surface-based Cohesive Behavior

L6.29

Element- vs. Surface-based Cohesive Behavior Preprocessing

• Cohesive elements • Gives you direct control over the cohesive element mesh density and stiffness properties. • Constraints are enforced at the element integration points. • Refining the cohesive elements relative to the connected structures will likely lead to improved constraint satisfaction and more accurate results.

• Cohesive surfaces

Integration points on an 8-node cohesive element

• Are easily defined using contact interactions and cohesive interaction properties. • A pure master-slave in formulation is used.

• Constraints are enforced at the slave nodes. • Refining the slave surface relative to the master surface will likely lead to improved constraint satisfaction and more accurate results .


L6.30

Element- vs. Surface-based Cohesive Behavior Initial configuration:

• Cohesive elements • Must be bonded at the start of the analysis.

• Once the interface has failed, the surfaces do not re-bond. • Cohesive surfaces

• Can bond anytime contact is established (i.e., “sticky” contact behavior). • Cohesive interface need not be bonded at the start of the analysis. • You can control whether debonded surfaces will stick or not stick if contact occurs again. • By default, they do not stick.


L6.31

Element- vs. Surface-based Cohesive Behavior Constitutive behavior:

• Cohesive elements • Allow for several constitutive behavior types:

• Traction-separation constitutive model • Including multiple failure mechanisms

• Continuum-based constitutive model • For adhesive layers with finite thickness • Uses conventional material models

• Uniaxial stress-based constitutive model • Useful in modeling gaskets and/or single adhesive patches

• Cohesive surfaces • Must use the traction-separation interface behavior. • Intended for bonded interfaces where the interface thickness is negligibly small.

• Only one failure mechanism is allowed. Modeling Fracture and Failure with Abaqus

L6.32

Element- vs. Surface-based Cohesive Behavior Influence on stable time increment (Abaqus/Explicit only):

• Cohesive elements

 Le  t    c d  

• Often require a small stable time increment.

• Cohesive elements are generally thin and sometimes quite stiff. • Consequently, they often have a stable time increment that is significantly less than that of the other elements in the model. • Cohesive surfaces • Cohesive surface behavior with the default cohesive stiffness properties is formulated to minimally affect the stable time increment. • Abaqus uses default contact penalties to model the cohesive stiffness behavior in this case. • You can specify a non-default cohesive stiffness values. • However, high stiffnesses may reduce the stable time increment.


L6.33

Element- vs. Surface-based Cohesive Behavior Mass:

• Cohesive elements • The element material definitions include mass.

• Cohesive surfaces • Do not add mass to the model.

• Indented for thin adhesive interfaces; thus, neglecting adhesive mass is appropriate for most applications. • However, nonstructural mass can be added to the contacting elements if necessary.


L6.34

Element- vs. Surface-based Cohesive Behavior Summary:

• Cohesive elements • Are recommended for more detailed adhesive connection modeling.

• Additional preprocessing effort (and often increased computational cost) is compensated for by gaining: • Direct control over the connection mesh • Additional constitutive response options • E.g., model adhesives of finite thickness

• Cohesive surfaces • Provides a quick and easy way to model adhesive connections. • Negligible interface thicknesses only • Surfaces can bond anytime contact is established (“sticky” contact) • Model contact adhesives, Velcro, tape, and other bonding agents that can stick after separation.


Workshop 3 (Part 2)

L6.36

Workshop 3 (Part 2) • Crack growth in a three-point bend specimen using surface-based cohesive behavior • Repeat the element-based exercise using surface-based behavior • Use default traction-separation elastic properties • Compare with element-based results


Virtual Crack Closure Technique (VCCT) Lecture 7

L7.2


• VCCT Criterion • Output

• VCCT Plug-in • Comparison with Cohesive Behavior

• Examples • Workshop 5


Introduction

L7.4

Introduction • Motivation is aircraft composite structural analysis • To reduce the cost of laminated composite structures, large integrated bonded structures are being considered.

• In primary structures, bondlines and interfaces between plies are required to carry interlaminar loads. • Damage tolerance requirements dictate that bondlines and interfaces carry required loads with damage. Modeling debonding along skin-stringer interface


L7.5

Introduction • Analysis requirements for composite damage

• Apply Linear Elastic Fracture Mechanics (LEFM) to bondlines and interfaces • 2D and 3D delaminations • Propagation

• Mode separation • Multiple cracks

• Non-linear behavior (e.g., postbuckling) • Composite structure

• Practical (CPU time, minimum set of models)


L7.6

Introduction • VCCT uses LEFM concepts

• Based on computing the energy release rates for normal and shear crack-tip deformation modes. • Compare energy release rates to interlaminar fracture toughness. • See Rybicki, E. F., and Kanninen, M. F., "A Finite Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral," Engineering Fracture Mechanics, Vol. 9, pp. 931-938, 1977.

Pure Mode I Modified VCCT

Node numbers are shown

Nodes 2 and 5 will start to release when: 1 v1,6 Fv,2,5  GI  GIC Mode II treated 2 bd similarly where GI  mode I energy release rate GIC  critical mode I energy release rate b  width Fv ,2,5  vertical force between nodes 2 and 5 v1,6  vertical displacement between nodes 1 and 6


VCCT Criterion

L7.8

VCCT Criterion • The debond capability is used to perform the crack propagation analysis for initially bonded crack surfaces. • The crack propagation analysis allows for five types of fracture criteria: 1 • Critical stress criterion

2 • Crack opening displacement criterion 3 • Crack length vs. time criterion 4 • VCCT criterion 5 • Low-cycle fatigue criterion

• Defining case 4, “VCCT criterion,” is the subject of this lecture.

• The details of cases 1, 2, and 3 are not discussed here. Please consult the Abaqus Analysis User’s Manual for more details. • The details of case 5 will be discussed later in Lecture 8 “Low-cycle Fatigue.”


L7.9

VCCT Criterion • When using VCCT to model crack propagation,

• you must: 1• define contact pairs for potential crack surfaces; 2• define initially bonded crack surfaces; 3• activate the crack propagation capability; and 4• specify the VCCT criterion.

• you also may: • define spatially varying critical energy release rates; • use viscous regularization, contact stabilization, and/or automatic stabilization to overcome convergence difficulties for unstable propagating cracks;

• use a linear scaling technique to accelerate convergence for VCCT.


L7.10

VCCT Criterion • Defining the VCCT criterion is not currently supported in Abaqus/CAE.

• However, the VCCT plug-in is available and allows you to interactively define the debond interface(s). • The details of the VCCT plug-in will be discussed later in this lecture. • Downloaded from “VCCT plug-in utility,” SIMULIA Answer 3235.


L7.11

VCCT Criterion • Example: Double cantilever beam (DCB)

• Analyze debonding of a DCB model using the VCCT criterion. • Steps required for setting up the model include: • Define slave (TopSurf) and master (BotSurf) surfaces along the debond interface. • Define a set (bond) containing the initially bonded region (part of TopSurf in this example).

• The Keywords interface is illustrated in this example. bond

TopSurf BotSurf


L7.12

VCCT Criterion 1• Define contact pairs for potential crack surfaces

• Potential crack surfaces are modeled as slave and master contact surfaces. • Any contact formulation except the finite-sliding, surface-to-surface formulation can be used. • Cannot be used with self-contact. bond

TopSurf

BotSurf

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf

Note: The frictionless interaction property is assumed. slave surface Modeling Fracture and Failure with Abaqus

master surface

L7.13

VCCT Criterion 2• Define initially bonded crack surfaces

• The initially bonded contact pair is identified with the *INITIAL CONDITIONS, TYPE=CONTACT option.

bond

TopSurf

BotSurf

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond slave surface

master surface


a list of slave nodes that are initially bonded

L7.14

VCCT Criterion • The unbonded portion of the slave surface will behave as a regular contact surface. • If the node set that includes the initially bonded slave nodes is not specified, the initial contact condition will apply to the entire contact pair. • In this case, no crack tips can be identified, and the bonded surfaces cannot separate. • For the VCCT criterion, the initially bonded nodes are bonded in all directions.


L7.15

VCCT Criterion 3• Activate the crack propagation capability

• The DEBOND option is used to activate crack propagation in a given step. • The SLAVE and MASTER parameters identify the surfaces to be debonded. *NSET, NSET=bond, GENERATE

bond

TopSurf

BotSurf

1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf


L7.16

VCCT Criterion 4• Specify the VCCT criterion

• The BK law model is used in this example. BK law:

GequivC  GIC   GIIC



 GII  GIII   GIC     GI  GII  GIII 

bond

TopSurf

BotSurf

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK 280.0, 280.0, 0.0, 2.284

GIC

GIIC


GIIIC



L7.17

VCCT Criterion • The crack-tip node debonds when the fracture criterion, f,

f 

Gequiv GequivC

,

reaches the value 1.0 within a given tolerance, ftol:

1  f  1  ftol . where

Gequiv is the equivalent strain energy release rate, and GequivC is the critical equivalent strain energy release rate calculated based on the user-specified mode-mix criterion and the bond strength of the interface. • For the VCCT criterion, the default value of ftol is 0.2. • Use following option to control ftol: *FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=ftol Modeling Fracture and Failure with Abaqus

L7.18

VCCT Criterion • In the DCB model, the tolerance is set to 0.1.

bond

TopSurf

BotSurf

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK, TOLERANCE=0.1 280.0, 280.0, 0.0, 2.284


L7.19

VCCT Criterion • In addition to the BK law model, Abaqus/Standard also provides two other commonly used mode-mix criteria for computing GequivC: the Power law and the Reeder law models. • An appropriate model is best selected empirically. • Power law Gequiv GequivC

 G   I   GIC 

am

 G    II   GIIC 

an

 G    III   GIIIC 

ao

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=POWER GIC, GIIC, GIIIC, am, an, ao

• Reeder law • Applies only to three-dimensional problems   GIII GequivC  GIC   GIIC  GIC   GIIIC  GIIC    GII  GIII 

   GII  GIII    Gi  





   

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=REEDER GIC, GIIC, GIIIC,  Modeling Fracture and Failure with Abaqus

L7.20

VCCT Criterion • Spatially varying critical energy release rates

• The VCCT criterion can be defined with varying energy release rates by specifying the critical energy release rates at all nodes on the slave surface. • In this case, the critical energy release rates should be interpolated from the critical energy release rates specified at the nodes with the *NODAL ENERGY RATE option. • However, the exponents (e.g., ) are still read from the data lines under the *FRACTURE CRITERION option. *NODAL ENERGY RATE node ID1, GIC, GIIC, GIIIC model data node ID2, GIC, GIIC, GIIIC ... *STEP *STATIC ... *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK, NODAL ENERGY RATE GIC, GIIC, GIIIC,  Modeling Fracture and Failure with Abaqus

L7.21

VCCT Criterion • Viscous regularization for VCCT • Can be used to overcome some convergence difficulties for unstable propagating cracks. • Example: DCB

• Set the value of the viscosity coefficient to 0.1. bond

TopSurf

BotSurf

*NSET, NSET=bond, GENERATE 1, 121, 1 *SURFACE, NAME=TopSurf _TopBeam_S1, S1 *SURFACE, NAME=BotSurf _BotBeam_S1, S1 *CONTACT PAIR, INTER=... TopSurf, BotSurf *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VISCOSITY=0.1 *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK, TOLERANCE=0.1 280.0, 280.0, 0.0, 2.284


L7.22

VCCT Criterion • In addition, contact and automatic stabilization that are not specific to VCCT can be also used to aid convergence. • They are built into Abaqus/Standard and are compatible with VCCT. • Note that the crack propagation behavior may be modified by the damping forces. • Therefore, monitor the damping energy (ALLVD or ALLSD) and compare it with the total strain energy in the model (ALLSE) to ensure that the results are reasonable in the presence of damping.

• ALLVD stores the damping energy generated from viscous regularization. • ALLSD stores the damping energy generated from contact stabilization and automatic stabilization.


L7.23

VCCT Criterion • Linear scaling to accelerate convergence for VCCT

• Abaqus provides a linear scaling technique to quickly converge to the critical load state. This reduces the solution time required to reach the onset of crack growth. • This technique works best for models in which the deformation is nearly linear before the onset of crack growth. • Once the first crack-tip node releases, the linear scaling calculations will no longer be valid and the time increment will be set to the default value.

• Usage: *CONTROLS, LINEAR SCALING



where  is the coefficient of linear scaling.

• For details of linear scaling to accelerate convergence for VCCT, see “Crack propagation analysis,” Section 11.4.3 of the Abaqus Analysis User’s Manual. Modeling Fracture and Failure with Abaqus

L7.24

VCCT Criterion • Tips for using the VCCT criterion

• Crack propagation problems using the VCCT criterion are numerically challenging. • To help you create a successful model, several tips for using the VCCT criterion are provided: • The master debonding surfaces must be continuous. • The tie MPCs should NOT be used for the slave debonding surface to avoid overconstraints. • A small clearance between the debonding surfaces can be specified to eliminate unnecessary severe discontinuity iterations during incrementation as the crack begins to progress. …… • Note: More tips are provided in “Crack propagation analysis,” Section 11.4.3 of the Abaqus Analysis User’s Manual.


Output

L7.26

Output • The following output options are provided to support the VCCT criterion:

• Abaqus/CAE supports the surface output requests for VCCT.

*OUTPUT, FIELD, FREQUENCY=freq *CONTACT OUTPUT, MASTER=master, SLAVE=slave *OUTPUT, HISTORY, FREQUENCY=freq *CONTACT OUTPUT, [(MASTER=master, SLAVE=slave)|(NSET=nset)]


L7.27

Output • The following bond failure quantities can be requested as surface output:

DBT

The time when bond failure occurred

DBSF

Fraction of stress at bond failure that still remains

DBS

Stress in the failed bond that remains

OPENBC

Relative displacement behind crack.

CRSTS

Critical stress at failure.

ENRRT

Strain energy release rate.

EFENRRTR

Effective energy release rate ratio.

BDSTAT

Bond state (=1.0 if bonded, 0.0 if unbonded)

• All of the above variables can be visualized in Abaqus/Viewer.

• The initial contact status of all of the slave nodes is printed in the data (.dat) file.


L7.28

Output • Example: DCB

• Request surface output:

bond

... *INITIAL CONDITIONS, TYPE=CONTACT TopSurf TopSurf, BotSurf, bond BotSurf *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VISCOSITY=0.1 *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK, TOLERANCE=0.1 280, 280, 280, 2.284 ... *OUTPUT, FIELD, VAR=PRESELECT field output *CONTACT OUTPUT, SLAVE=TopSurf, MASTER=BotSurf DBT, DBS, OPENBC, CRSTS, ENRRT, BDSTAT *OUTPUT, HISTORY history output *CONTACT OUTPUT, SLAVE=TopSurf, MASTER=BotSurf, NSET=bond DBT, DBS, OPENBC, CRSTS, ENRRT, BDSTAT *NODE OUTPUT, NSET=tip U2, RF2 *END STEP Modeling Fracture and Failure with Abaqus

L7.29

Output • Results

VCCT


VCCT Plug-in

L7.31

VCCT Plug-in • VCCT plug-in

• provides an interactive interface to define the debond interface(s). • supports the following keyword options required for VCCT analysis: *INITIAL CONDITIONS, TYPE=CONTACT *DEBOND, SLAVE=slave, MASTER=master, OUTPUT=[fil|dat|both], VISCOSITY=  *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=[BK|POWER|REEDER], TOLERANCE=ftol, NODAL ENERGY RATE *NODAL ENERGY RATE *CONTROLS, LINEAR SCALING

• For details please refer to “VCCT plug-in utility,” SIMULIA Answer 3235.


L7.32

VCCT Plug-in • Example: Double Cantilever Beam (DCB)

• The VCCT plug-in is discussed in the context of the Keywords interface presented earlier.

bond

TopSurf

BotSurf

initially bonded region master surface slave surface


L7.33

VCCT Plug-in 1• Define contact pairs for potential crack surfaces

• Frictionless contact is assumed.

*SURFACE INTERACTION, NAME=IntProp-1 1. *FRICTION 0.0 *CONTACT PAIR, INTERACTION=IntProp-1 TopSurf, BotSurf

bond

TopSurf

BotSurf


L7.34

VCCT Plug-in 2• Define the VCCT criterion 2a • Select the fracture criterion, viscosity

coefficient, and cutback tolerance. ... *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VICOSITY=0.1 *FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=0.2, MIXED MODE BEHAVOIR=BK 280, 280, 280, 2.284 bond

TopSurf

BotSurf


L7.35

VCCT Plug-in 2b • Specify critical strain energy release rates

... *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VICOSITY=0.1 *FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=0.2, MIXED MODE BEHAVOIR=BK 280, 280, 280, 2.284 bond

TopSurf

BotSurf


L7.36

VCCT Plug-in • The VCCT plug-in also supports defining spatially varying critical energy release rates. • Click mouse button 3 to manage the table.

*NODAL ENERGY RATE node ID1, GIC, GIIC, GIIIC node ID2, GIC, GIIC, GIIIC ... *STEP *STATIC ... *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK, NODAL ENERGY RATE GIC, GIIC, GIIIC, 


L7.37

VCCT Plug-in 3• Define the VCCT bonded interface

• Select the initially bonded region, the crack propagation output file and frequency, and the debond initiation step.

• Note: The VCCT plug-in allows specification of linear scaling. *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NAME=Step-1 *STATIC, NLGEOM ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VISCOSITY=0.1 *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK 280, 280, 280, 2.284


L7.38

VCCT Plug-in • The relevant keywords will be generated when Abaqus/CAE writes the input file.

surface interaction initial contact conditions

debond

fracture criterion

field output

history output Modeling Fracture and Failure with Abaqus

Comparison with Cohesive Behavior

L7.40

Comparison with Cohesive Behavior • VCCT and cohesive behavior are very similar in their application and formulation. • Both theories • are used to model interfacial shearing and delamination crack propagation and failure, • use an elastic damage constitutive theory to model the material's response once damage has initiated, and • dissipate the same amount of fracture energy between damage initiation and complete failure.


L7.41

Comparison with Cohesive Behavior • The fundamental difference between VCCT and cohesive behavior is in the way crack propagation is predicted. • In VCCT an existing flaw is assumed. • VCCT is appropriate for brittle crack propagation problems. • However, cohesive behavior can model damage initiation.

• Damage initiation in cohesive behavior is based strictly on the predefined ultimate (normal and/or shear) stress/strain limit. • Cohesive behavior can be used for both brittle and ductile crack propagation problems.


L7.42

Comparison with Cohesive Behavior • VCCT may be viewed as more fundamentally based on fracture mechanics. • The damage initiation and damage evolution are both based on fracture energy, whereas cohesive behavior use the fracture energy only during damage evolution. • Applicability of VCCT is limited to “self-similar” crack propagation analyses. • This implies a steady-state running crack.

• Difficult to reproduce in practice.


L7.43

Comparison with Cohesive Behavior • Summary: Complementary techniques for modeling of debonding VCCT

Cohesive behavior

Use the debond framework (surface based)

Interface elements (element based) or contact (surface based)

Assumes an existing flaw

Can model crack initiation

Brittle fracture using LEFM occurring along a well defined crack front

Ductile fracture occurring over a smeared crack front modeled with spanning cohesive elements or cohesive contact

Requires GI, GII, and GIII

Requires E, σmax, GI, GII, and GIII

Crack propagates when strain energy release rate exceeds fracture toughness

Crack initiates when cohesive traction exceeds critical value and releases critical strain energy when fully open

Crack surfaces are rigidly bonded when uncracked.

Crack surfaces are joined elastically when uncracked.

Available only in Abaqus/Standard

Available in Abaqus/Standard and Abaqus/Explicit

• Both are needed to satisfy general fracture requirements Modeling Fracture and Failure with Abaqus

Examples

L7.45

Examples • Verification problems

• DCB • SLB

• ENF • Alfano-Crisfield

• Alfano, G., and M. A. Crisfield, “Finite Element Interface Models for the Delamination Analysis of Laminated Composites: Mechanical and Computational Issues,” International Journal for Numerical Methods in Engineering, vol. 50, pp. 1701–1736, 2001. • Also available as Abaqus Benchmark Problem 2.7.1 with cohesive elements • NASA Panel • Reeder, J.R., Song, K., Chunchu, P.B., and Ambur, D.R., “Postbuckling and Growth of Delaminations in Composite Plates Subjected to Axial Compression,” AIAA 2002-1746. Modeling Fracture and Failure with Abaqus

L7.46

Examples • Compression Buckling/Delamination Single Disbond (Unreinforced) Multiple crack tips Buckling driven delaminations

30000 Euler buckling 25000

Load (lb)

20000 FEA

15000

closed form

10000 5000 0 0

0.01

0.02

0.03

0.04

0.05

Displacement (in)


L7.47

Examples


L7.48

Examples • Compression Buckling/Delamination Multiple Disbonds (Unreinforced)

Multiple cracks can also be addressed


L7.49

Examples


L7.50

Examples • T-Joint Pull–off Model


L7.51

Examples • Postbuckling Behavior of Skin-Stringer Panels

• VCCT can be applied to determine the global strength and failure mode for typical aerospace composite structures like this skin/stringer panel

Courtesy Boeing


L7.52

Examples

Displacement imposed at corner nodes

Contact surfaces defined for region of fracture


L7.53

Examples

Crack tip

Initially debonded nodes Initially bonded nodes


L7.54

Examples

The Abaqus Tech Brief on skin/stringer bonded joint analysis can be downloaded from www.simulia.com


L7.55

Examples


Workshop 5

L7.57

Workshop 5 • Crack growth in a three-point bend specimen using VCCT

• Repeat the cohesive-based exercises using VCCT and compare results


Low-cycle Fatigue Lecture 8

L8.2


• Low-cycle Fatigue in Bulk Materials • Low-cycle Fatigue at Material Interfaces


Introduction

L8.4

Introduction • Low-cycle fatigue analysis is a quasi-static analysis of a structure subjected to sub-critical cyclic loading. • It can be associated with thermal as well as mechanical loading. • In Abaqus can simulate low-cycle fatigue in: • bulk ductile materials

• material interfaces


L8.5

Introduction • Low-cycle fatigue analysis uses the direct cyclic procedure to directly obtain the stabilized cyclic response of the structure. • The direct cyclic procedure combines a Fourier series approximation with time integration of the nonlinear material behavior to obtain the stabilized cyclic solution iteratively using a modified Newton method.

• You can control the number of Fourier terms, the number of iterations, and the incrementation during the cyclic time period to improve the accuracy. • Within each loading cycle, it assumes geometrically linear behavior and fixed contact conditions. • Geometric nonlinearity can be included only in any general step prior to a direct cyclic step • For more details, please see “Low-cycle fatigue analysis using the direct cyclic approach,” Section 6.2.7 of the Abaqus Analysis User’s Manual.


L8.6

Introduction • Defining low-cycle fatigue analysis *DIRECT CYCLIC, FATIGUE, [CETOL=tolerance, DELTMX=max] t0, T, tmin, tmax, n0, nmax, n, imax Nmin, Nmax, N, Dtol where t0: initial time increment T: time of a single loading cycle tmin: minimum time increment allowed

controls the incrementation

tmax: maximum time increment allowed

n0: initial number of terms in the Fourier series nmax: maximum number of terms in the Fourier series

controls the Fourier series representations

n: increment in number of terms in the Fourier series imax: maximum number of iterations allowed in a step

controls the iteration

N: total number of cycles allowed in a step Nmin: minimum increment in N over which the damage is extrapolated forward Nmax: maximum increment in N over which the damage is extrapolated forward

Dtol: damage extrapolation tolerance


controls damage extrapolation in the bulk material

Low-cycle Fatigue in Bulk Materials

L8.8

Low-cycle Fatigue in Bulk Materials • Abaqus/Standard offers a general capability for modeling the progressive damage and failure of ductile materials due to stress reversals and the accumulation of inelastic strain energy when the material is subjected to sub-critical cyclic loadings.

• Damage in low-cycle fatigue is defined within the same general framework of modeling progressive damage and failure (continuum damage approach): • a constitutive behavior of undamaged ductile materials;

• a damage initiation criterion; and • a damage evolution response.

• The damage initiation and evolution are characterized by the stabilized accumulated inelastic hysteresis strain energy per stabilized cycle. • Note: Damage initiation and evolution for low-cycle fatigue analysis is currently not supported in Abaqus/CAE.


L8.9

Low-cycle Fatigue in Bulk Materials • Example: Thermal cycling failure of solder joint

• Solder joint reliability analysis of automotive electronics under cyclic thermal loading.

The crack propagates forward Modeling Fracture and Failure with Abaqus

L8.10

Low-cycle Fatigue in Bulk Materials electronic chip

• Quarter-symmetry model:

• Solder material (63Sn/37Pb)

solder joints

gullwing leads

• Modeled using temperaturedependent elasticity and power-law creep.

printed circuit board

• Low-cycle fatigue analysis run for 801 cycles.

Quarter-symmetry model

• Each thermal cycle is 1920 seconds. • Define the low-cycle fatigue analysis step *STEP, INC=800 *DIRECT CYCLIC, FATIGUE 60., 1920.,,, 29, 29,, 100 50, 100, 801, 1.1 Temperature load in once cycle


L8.11

Low-cycle Fatigue in Bulk Materials • Damage initiation criterion for ductile damage in low-cycle fatigue

• The onset of damage in low-cycle fatigue is characterized by the accumulated inelastic hysteresis energy per cycle, w, in a material point when the structure response is stabilized in the cycle. • The cycle number (N0) in which damage is initiated is given by

N0  c1wc2 where c1 and c2 are material constants. • Note: c1 depends on the system of units in which you are working; care is required to modify c1 when converting to a different system units. • The initiation criterion can be used in conjunction with any ductile material. • Damage initiation criterion output: CYCLEINI

Number of cycles to initialized the damage


L8.12

Low-cycle Fatigue in Bulk Materials • Defining damage initiation criterion

• Example: Thermal cycling failure of solder joint

c1

*MATERIAL, NAME=SOLDERF *ELASTIC 31976, 0.4, 273 Quarter-symmetry model N0  c1wc2 20976, 0.4, 398 *EXPANSION, ZERO=273 21E-6, *CREEP,LAW=USER *DAMAGE INITIATION, CRITERION=HYSTERESIS ENERGY c2 33.3, -1.52 ... *STEP, INC=800 *DIRECT CYCLIC, FATIGUE solder joint 60., 1920.,,, 29, 29,, 100 50, 100, 801, 1.1 bond pad underneath solder joint


L8.13

Low-cycle Fatigue in Bulk Materials • Damage evolution for ductile damage in low-cycle fatigue

• Once the damage initiation criterion is satisfied at a material point, the damage state is calculated and updated based on the inelastic hysteresis energy for the stabilized cycle. • The rate of the damage (dD/dN) at a material point per cycle is given by

dD c3wc4  dN L where c3 and c4 are material constants, L is the characteristic length associated with the material point, and D is the scalar damage variable.

• The details of choosing characteristic length will be discussed later. • Note: c3 depends on the system of units in which you are working; care is required to modify c3 when converting to a different system units.


L8.14

Low-cycle Fatigue in Bulk Materials • Defining damage evolution

• Example: Thermal cycling failure of solder joint

c3

*MATERIAL, NAME=SOLDERF *ELASTIC dD c3wc4 31976, 0.4, 273  20976, 0.4, 398 dN L *EXPANSION, ZERO=273 21E-6, *CREEP,LAW=USER *DAMAGE INITIATION, CRITERION=HYSTERESIS ENERGY 33.3, -1.52 *DAMAGE EVOLUTION, TYPE=HYSTERESIS ENERGY c4 9.88E-4, 0.98 ... *STEP, INC=800 *DIRECT CYCLIC, FATIGUE 60., 1920.,,, 29, 29,, 100 50, 100, 801, 1.1


Quarter-symmetry model

L8.15

Low-cycle Fatigue in Bulk Materials • Results

Damage initiation at joint toe Cycle number 199

Damage evolution Cycle number 749


Damage evolution Cycle number 801

L8.16

Low-cycle Fatigue in Bulk Materials • Characteristic length associated with an integration point

• The characteristic length implemented in the damage evolution model is based on the element geometry and formulation:

Element type

Characteristic length used in the damage evolution model

first-order element

typical length of a line across the element

second-order element

half of the typical length of a line across the element

beam and truss

characteristic length along the element axis

membrane and shell

characteristic length in the reference surface

axisymmetric element

characteristic length in the rz plane only

cohesive element

the constitutive thickness


L8.17

Low-cycle Fatigue in Bulk Materials • The characteristic length is used because the direction in which fracture occurs is not known in advance. • Therefore, elements with large aspect ratios will have rather different behavior depending on the direction in which the damage occurs. • Some mesh sensitivity remains because of this effect, and elements that are as close to square as possible are recommended.

• However, since the damage evolution law is energy based, mesh dependency of the results may be alleviated.


L8.18

Low-cycle Fatigue in Bulk Materials • Difficulties associated with element removal and LCF

• When elements are removed from the model, their nodes remain in the model even if they are not attached to any active elements. • When the solution progresses, these nodes might undergo nonphysical displacements in Abaqus/Standard. • For example, applying a point load to a node that is not attached to an active element will cause convergence difficulties since there is no stiffness to resist the load.

• It is the user’s responsibility to prevent such situations.


Low-cycle Fatigue at Material Interfaces

L8.20

Low-cycle Fatigue at Material Interfaces • Delamination growth in composites due to sub-critical cyclic loadings is a widespread concern for the aerospace industry. • The low-cycle fatigue criterion available in Abaqus models progressive delamination growth at interfaces in laminated composites subjected to sub-critical cyclic loadings. • The interface along which the delamination (or crack) propagates must be indicated in the model. • The onset and growth of fatigue delamination at the interfaces are characterized by the relative fracture energy release rate • The fracture energy release rates at the crack tips in the interface elements are calculated based on the VCCT technique.


L8.21

Low-cycle Fatigue at Material Interfaces • The onset and fatigue delamination growth at the interfaces are characterized by using the Paris Law, which relates crack growth rates da/dN to the relative fracture energy release rate G,

G = Gmax – Gmin where Gmax and Gmin correspond to the strain energy release rates when the structure is loaded up to Pmax and Pmin, respectively. • The Paris regime is bounded by Gthresh and Gpl. • Below Gthresh, there is no fatigue crack initiation or growth. • Above Gpl, the fatigue crack will grow at an accelerated rate.

a: crack length N: number of cycles G: strain energy release rate Gthresh: strain energy release rate threshold Gpl: strain energy release rate upper limit GequivC: critical equivalent strain energy release rate


L8.22

Low-cycle Fatigue at Material Interfaces • GequivC is calculated based on the user-specified mode-mix criterion and the bond strength of the interface. • This was discussed in Lecture 7 “VCCT.” • Onset of fatigue delamination • The fatigue crack growth initiation criterion is defined as: f 

N  1.0, c2 c1G

where c1 and c2 are material constants. • The interface elements at the crack tips will not be released unless the above equation is satisfied and Gmax  Gthresh.



L8.23

Low-cycle Fatigue at Material Interfaces • Fatigue delamination growth

• Once the delamination growth criterion is satisfied at the interface, the crack growth rate da/dN can be calculated based on G.

• da/dN is given by the Paris Law if Gthresh< Gmax< Gpl,

da  c3G c4 dN

da  c3G c4 dN where c3 and c4 are material constants.



L8.24

Low-cycle Fatigue at Material Interfaces • Fatigue crack growth governed by the Paris Law If Gthresh < Gmax < Gpl

G = Gmax(Pmax) – Gmin(Pmin)

1

a: crack length N: number of cycles N: incremental number of cycles c1, c2 , c3, c4: material constants

2 Crack initiation: No  c1G c2

Calculate the relative fracture energy release rate, G, when the structure is loaded between its maximum and minimum values.

Crack evolution:

aN N  aN  Nc3G c 4

da  c3G c4 dN If N + N > No

N + N

3

Release the most critical element

Damage extrapolation: Calculate the incremental number of cycles, N, for each crack tip and find minimum cycles to fail, Nmin

• Repeat the above process until the maximum number of cycles is reached or until the ultimate load carrying capability is reached.


L8.25

Low-cycle Fatigue at Material Interfaces • The syntax used to define the low-cycle fatigue criterion and the corresponding output requests is similar to those used for the VCCT criterion except the following: • For the low-cycle fatigue criterion, set TYPE=FATIGUE on the *FRACTURE CRITERION option: *FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=[BK|REEDER]

c1, c2, c3, c4, Gthresh/GequivC, Gpl/GequivC, GIC, GIIC GIIIC, , , fv *FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=POWER

c1, c2, c3, c4, Gthresh/GequivC, Gpl/GequivC, GIC, GIIC GIIIC, am, an, ao, , fv

• By default, Gthresh/GequivC = 0.01 and Gpl/GequivC = 0.85. • Note: Defining the low-cycle criterion is not currently supported in Abaqus/CAE. Modeling Fracture and Failure with Abaqus

L8.26

Low-cycle Fatigue at Material Interfaces • Example: Low-cycle fatigue prediction for the DCB model

• This case consists of the following steps: • Step 1: VCCT analysis

• This step can be used to check whether the peak loading leads to static crack propagation. • Step 2: Low-cycle fatigue analysis • This step assesses the fatigue life of the DCB model subjected to sub-critical cyclic loading. u2

bond

u2

 =0.001 TopSurf BotSurf 0

0

0.5

1

t

displacement loading in one cycle

u2 Modeling Fracture and Failure with Abaqus

L8.27

Low-cycle Fatigue at Material Interfaces • Partial input: ... *CONTACT PAIR, SMALL SLIDING TopSurf, BotSurf Model data *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC Step 2: ... Fatigue *DEBOND, SLAVE=TopSurf, analysis MASTER=BotSurf *FRACTURE CRITERION, TYPE=VCCT, Step 1: MIXED MODE BEHAVIOR=BK VCCT 280, 280, 280, 2.284 analysis *OUTPUT, FIELD *CONTACT OUTPUT, SLAVE=TopSurf, MASTER=BotSurf BDSTAT, DBT, DBS, OPENBC, CRSTS, ENRRT *END STEP

*STEP, INC=5000 *DIRECT CYCLIC, FATIGUE 0.25,1,,,25,25,,5 ,,1000 *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=BK 0.5,-0.1,4.8768E-6,1.15,,,280,280 280,2.284 *OUTPUT, FIELD *CONTACT OUTPUT BDSTAT, DBT, DBS, OPENBC, CRSTS, ENRRT ... bond *END STEP

TopSurf


BotSurf

L8.28

Low-cycle Fatigue at Material Interfaces • The procedure to complete the DCB model through the first step (the VCCT analysis) is exactly the same as that discussed in Lecture 7 “VCCT.” 1• Define contact pairs for potential

crack surfaces 2• Define initially bonded crack

surfaces 3• Activate the crack propagation

capability in the first step 4• Specify the VCCT criterion in the

first step (a static, general step)

• The details of defining the low-cycle fatigue analysis (the second step) will be discussed next.

... *CONTACT PAIR, SMALL SLIDING model TopSurf, BotSurf data *INITIAL CONDITIONS, TYPE=CONTACT TopSurf, BotSurf, bond *STEP, NLGEOM *STATIC ... *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK Step 1: 280, 280, 280, 2.284 VCCT *OUTPUT, FIELD analysis *CONTACT OUTPUT BDSTAT, DBT, DBS, OPENBC, CRSTS, ENRRT ... bond *END STEP

TopSurf


BotSurf

L8.29

Low-cycle Fatigue at Material Interfaces 5• Define the low-cycle fatigue analysis • The following data are used to define this low-cycle fatigue analysis:

• Initial time increment: 0.25 sec

... *STEP, INC=5000 Low-cycle Fatigue Analysis *DIRECT CYCLIC, FATIGUE 0.25,1,,,25,25,,5 ,,1000

• Time of a single loading cycle: 1 sec

• Initial number of terms in the Fourier series: 25 • Maximum number of terms in the Fourier series: 25 • Maximum number of iterations allowed in the step: 5 • Total number of cycles allowed in the step: 1000 • Default values are used for all other entries.

bond

TopSurf


BotSurf

L8.30

Low-cycle Fatigue at Material Interfaces 6• Activate the crack propagation capability

• Similar to the VCCT analysis, the *DEBOND option is used to activate the crack propagation in the low-cycle fatigue analysis step.

• The SLAVE and MASTER parameters identify the surfaces to be debonded.

... *STEP, INC=5000 Low-cycle Fatigue Analysis *DIRECT CYCLIC, FATIGUE 0.25,1,,,25,25,,5 ,,1000 *DEBOND, SLAVE=TopSurf, MASTER=BotSurf

bond

TopSurf


BotSurf

L8.31

Low-cycle Fatigue at Material Interfaces 7• Specify the low-cycle fatigue criterion

• In this model, the material constants are assumed to be the following:

• c1 = 0.5,

f 

• c2 = –0.1 • c3 = 4.8768E–6

N  1.0 c1G c2

da  c3G c4 dN

• c4 = 1.15 • Note: The values of these material constants should be determined experimentally.

... *STEP, INC=5000 Low-cycle Fatigue Analysis *DIRECT CYCLIC, FATIGUE 0.25,1,,,25,25,,5 ,,1000 *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=BK 0.5,-0.1,4.8768E-6,1.15,,,280,280 280,2.284

GIC GIIC

GIIIC 

bond

• The BK model (default) is used. TopSurf


BotSurf

L8.32

Low-cycle Fatigue at Material Interfaces 8• Request output

• The output options for the low-cycle fatigue criterion are same as those for the VCCT criterion.

... *STEP, INC=5000 Low-cycle Fatigue Analysis *DIRECT CYCLIC, FATIGUE 0.25,1,,,25,25,,5 ,,1000 *DEBOND, SLAVE=TopSurf, MASTER=BotSurf *FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=BK 0.5,-0.1,4.8768E-6,1.15,,,280,280 280,2.284 *OUTPUT, FIELD *CONTACT OUTPUT BDSTAT, DBT, DBS, OPENBC, CRSTS, ENRRT bond

TopSurf


BotSurf

L8.33

Low-cycle Fatigue at Material Interfaces • Results initially bonded nodes

delamination

N=1

N=11

N=21

N=51 N is the number of cycles Modeling Fracture and Failure with Abaqus

L8.34

Low-cycle Fatigue at Material Interfaces • More results

delamination growth after 100 loading cycles

crack length vs. cycle number Modeling Fracture and Failure with Abaqus

Mesh-independent Fracture Modeling (XFEM) Lecture 9

L9.2


• Basic XFEM Concepts • Damage Modeling • Creating an XFEM Fracture Model

• Example 1 – Crack Initiation and Propagation • Example 2 – Propagation of an Existing Crack • Example 3 – Delamination and Through-thickness Crack Propagation

• Modeling Tips • Current Limitations • Workshop 6

• References Modeling Fracture and Failure with Abaqus

Introduction

L9.4

Introduction • The fracture modeling methods discussed so far only permit crack propagation along predefined element boundaries • This lecture presents a technique for modeling bulk fracture which permits a crack to be located in the element interior • The crack location is independent of the mesh


L9.5

Introduction • This modeling technique…

• Can be used in conjunction with the cohesive zone model or the virtual crack closure technique • Delamination can be modeled in conjunction with bulk crack propagation • Can determine the load carrying capacity of a cracked structure • What is the maximum allowable flaw size for safe operation?

• Applications of this technique include the modeling of bulk fracture and the modeling of failure in composites • Cracks in pressure vessels or engineering structures • Delamination and through-thickness crack modeling in composite plies


L9.6

Introduction • Some advantages of the method:

• Ease of initial crack definition • Mesh is generated independent of crack

• Partitioning of geometry not needed as when a crack is represented explicitly • Nonlinear material and nonlinear geometric analysis • Arbitrary solution-dependent crack initiation and propagation path

• Crack path does not have to be specified a priori • Mesh refinement studies are much simpler

• Reduced remeshing effort • Improved convergence rate for the finite element solution (stationary crack) • Due to the use of singular crack tip enrichment


L9.7

Introduction •

Mesh-independent Crack Modeling – Basic Ingredients

1. Need a way to incorporate discontinuous geometry – the crack – and the discontinuous solution field into the finite element basis functions •

eXtended Finite Element Method (XFEM)

2. Need to quantify the magnitude of the discontinuity – the displacement jump across the crack faces •

Cohesive zone model (CZM)

3. Need a method to locate the discontinuity •

Level set method (LSM)

4. Crack initiation and propagation criteria

•

•

At what level of stress or strain does the crack initiate?

•

What is the direction of propagation?

These topics will be discussed in this lecture


Basic XFEM Concepts

L9.9

Basic XFEM Concepts • eXtended Finite Element Method (XFEM) Background • XFEM extends the piecewise polynomial function space of conventional finite element methods with extra functions • The solution space is enriched by the extra “enrichment functions” • Introduced by Belytschko and Black (1999) based on the partition of unity method of Babuska and Melenk (1997)

• Can be used where conventional FEM fails or is prohibitively expensive • Appropriate enrichment functions are chosen for a class of problems

• Inclusion of a priori knowledge of partial differential equation behavior into finite element space (singularities, discontinuities, ...) • Applications include modeling fracture, void growth, phase change ... • Enrichment functions for fracture modeling • Heaviside function to represent displacement jump across crack face • Crack tip asymptotic function to model singularity


L9.10

Basic XFEM Concepts • XFEM Displacement Interpolation Heaviside enrichment term H(x)

Heaviside distribution

aI

Nodal enriched DOF (jump discontinuity)

NG

Nodes belonging to elements cut by crack

  4   h a u (x)   N I (x)  u I  H (x )a I   Fa (x)b I     a 1 I N    I N G   I N  uI

Nodal DOF for conventional shape functions NI

Crack tip enrichment term Fa(x)

Crack tip asymptotic functions

baI

Nodal DOF (crack tip enrichment)

NG

Nodes belonging to elements containing crack tip


L9.11

Basic XFEM Concepts • The crack tip and Heaviside enrichment functions are multiplied by the conventional shape functions • Hence enrichment is local around the crack • Sparsity of the resulting matrix equations is preserved • The crack is located using the level set method (discussed shortly)

• Heaviside function • Accounts for displacement jump across crack

H(x) = 1 above crack

n

 1 if (x  x* )  n  0 H ( x)    1 otherwise 

s x* x

H(x) = 1 below crack

Here x is an integration point, x* is the closest point to x on the crack face and n is the unit normal at x*


L9.12

Basic XFEM Concepts • Crack Tip Enrichment Functions (Stationary Crack Only)

• Account for crack tip singularity • Use displacement field basis functions for sharp crack in an isotropic linear elastic material









[ Fa ( x), a  1 - 4]  [ r sin , r cos , r sin  sin , r sin  cos ] 2 2 2 2 Here (r,  ) denote coordinate values from a polar coordinate system located at the crack tip


L9.13

Basic XFEM Concepts • Phantom Node Approach (Crack Propagation Implementation)

• Implementation of XFEM fitting into the framework of conventional FEM • Discontinuous element with Heaviside enrichment is treated as a superposition of two continuous elements with phantom nodes • Does not include the asymptotic crack tip enrichment functions

• Introduced by Belytschko and coworkers (2006) based on the superposed element formulation of Hansbo and Hansbo (2004)


L9.14

Basic XFEM Concepts • Level Set Method for Locating a Crack • A level set (also called level surface or isosurface) of a real-valued function is the set of all points at which the function attains a specified value • Example: the zero-valued level set of f (x, y) : x2  y2  r2 is a circle of radius r centered at the origin • Popular technique for representing surfaces in interface tracking problems • Two functions F and Y are used to completely describe the crack • The level set F = 0 represents the crack face

• The intersection of level sets Y = 0 and F = 0 denotes the crack front • Functions are defined by nodal values whose spatial variation is determined by the usual finite element shape functions (example follows) • Function values need to be specified only at nodes belonging to elements cut by the crack Modeling Fracture and Failure with Abaqus

L9.15

Basic XFEM Concepts • Calculating F and Y • The nodal value of the function F is the signed distance of the node from the crack face • Positive value on one side of the crack face, negative on the other • The nodal value of the function Y is the signed distance of the node from an almost-orthogonal surface passing through the crack front • The function Y has zero value on this surface and is negative on the side towards the crack Y=0 F=0 Node

F

Y

1

0.25

1.5

2

0.25

1.0

3

0.25

1.5

4

0.25

1.0

1

2

3

4

0.5

1.5 Modeling Fracture and Failure with Abaqus

Damage Modeling

L9.17

Damage Modeling • Damage modeling is achieved through the use of a traction-separation law across the fracture surface • It follows the general framework introduced in earlier lectures • Damage initiation • Damage evolution

• Traction-free crack faces at failure • Damage properties are specified as part of the bulk material definition Damage initiation

Failure


L9.18

Damage Modeling • Damage Initiation

• Two criteria available at present • Maximum principal stress criterion (MAXPS)

 max f  0  max

• Initiation occurs when the maximum principal stress reaches critical value • Maximum principal strain criterion (MAXPE)

f 

 max 0  max

• Initiation occurs when the maximum principal strain reaches critical value • Crack plane is perpendicular to the direction of the maximum principal stress (or strain) • Crack initiation occurs at the center of the element

• However, crack propagation is arbitrary through the mesh • The damage initiation criterion is satisfied when 1.0 ≤ f ≤ 1.0 + ftol where f is the selected damage criterion and ftol is a user-specified tolerance value Modeling Fracture and Failure with Abaqus

L9.19

Damage Modeling • Damage Evolution

• Any of the damage evolution models for traction-separation laws discussed in the earlier lectures can be used • However, it is not necessary to specify the undamaged tractionseparation response


L9.20

Damage Modeling • Damage Stabilization

• Fracture makes the structural response nonlinear and non-smooth • Numerical methods have difficulty converging to a solution

• As discussed in the earlier lectures, using viscous regularization helps with the convergence of the Newton method • The stabilization value must be chosen so that the problem definition does not change • A small value regularizes the analysis, helping with convergence while having a minimal effect on the response • Perform a parametric study to choose appropriate value for a class of problems


L9.21

Damage Modeling • Damage stabilization can currently be defined in Abaqus/CAE only through the keyword editor


Creating an XFEM Fracture Model

L9.23

Creating an XFEM Fracture Model • Steps

1. Define damage criteria in the material model 2. Define an enrichment region (the associated material model should include damage) •

Crack type – stationary or propagation

3. Define an initial crack, if present 4. If needed, set analysis controls to aid convergence

• Steps will be illustrated later through examples

•

•

Crack initiation and propagation in a plate with a hole

•

Propagation of an existing crack

•

Delamination and through-thickness crack propagation in a double cantilever beam

The next few slides describe step-dependent enrichment activation and postprocessing Modeling Fracture and Failure with Abaqus

L9.24

Creating an XFEM Fracture Model • Step-dependent Enrichment Activation

• Crack growth can be activated or deactivated in analysis steps *STEP . . . *ENRICHMENT, NAME=Crack-1, ACTIVATE=[ON|OFF]

1

2


L9.25

Creating an XFEM Fracture Model • Output Quantities

• Two output variables are especially useful • PHILSM

• The signed distance function F used to represent the crack surface • Needed for visualizing the crack • STATUSXFEM

• Indicates the status of the element with a value between 0.0 and 1.0 • A value of 1.0 indicates that the element is completely cracked, with no traction across the crack faces • Any other output variable available in the static stress analysis procedure


L9.26

Creating an XFEM Fracture Model • Postprocessing

• The crack location is specified by the zero-valued level set of the signed distance function F • Abaqus/CAE automatically creates an isosurface view cut named Crack_PHILSM if an enrichment is used in the analysis • The crack isosurface is displayed by default • Contour plots of field quantities should be done with the crack isosurface displayed • Ensures that the solution is plotted from the active parts of the overlaid elements according to the phantom nodes approach • If the crack isosurface is turned off, only values from the “lower” element are plotted (corresponding to negative values of F) • Probing field quantities on an element currently returns values only from the “lower” element (on the side with negative values of F)


Example 1 – Crack Initiation and Propagation

L9.28

Example 1 – Crack Initiation and Propagation • Model crack initiation and propagation in a plate with a hole

• Crack initiates at the location of maximum stress concentration • Half model is used taking advantage of symmetry


L9.29

Example 1 – Crack Initiation and Propagation 1 Define the damage criteria

• Damage initiation

*MATERIAL . . . *DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05

Damage initiation tolerance (default 0.05)


L9.30

Example 1 – Crack Initiation and Propagation 1 Define the damage criteria (cont’d)

• Damage evolution *DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05 *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=POWER LAW, POWER=1.0 2870.0, 2870.0, 2870.0


L9.31

Example 1 – Crack Initiation and Propagation 1 Define the damage criteria (cont’d)

• Damage stabilization Keyword interface *DAMAGE STABILIZATION 1.e-5 Coefficient of viscosity m

• Abaqus/CAE interface currently not available

• The keyword editor may be used to add stabilization through Abaqus/CAE.


L9.32

Example 1 – Crack Initiation and Propagation 2 Define the enriched region

Pick enriched region Propagating crack

Specify contact interaction (frictionless small-sliding contact only)


L9.33

Example 1 – Crack Initiation and Propagation 2 Define the enriched region (cont’d)

Keyword interface *ENRICHMENT, TYPE=PROPAGATION CRACK, NAME=CRACK-1, ELSET=SELECTED_ELEMENTS, INTERACTION=CONTACT-1

Frictionless small-sliding contact interaction

3 No initial crack definition is needed

• Crack will initiate based on specified damage criteria


L9.34

Example 1 – Crack Initiation and Propagation 4 Set analysis controls to improve convergence behavior

•

Set reasonable minimum and maximum increment sizes for step

•

Increase the number of increments for step from the default value of 100

*STEP *STATIC, inc=10000 0.01, 1.0, 1.0e-09, 0.01 . . .


L9.35

Example 1 – Crack Initiation and Propagation 4 Set analysis controls to improve convergence behavior (cont’d)

•

Use numerical scheme applicable to discontinuous analysis

*STEP *STATIC, inc=10000 0.01, 1.0, 1.0e-09, 0.01 . . . *CONTROLS, ANALYSIS=DISCONTINUOUS


L9.36

Example 1 – Crack Initiation and Propagation 4 Set analysis controls to improve convergence behavior (cont’d)

•

Increase value of maximum number of attempts before abandoning increment (increased to 20 from the default value of 5)

*STEP *STATIC, inc=10000 0.01, 1.0, 1.0e-09, 0.01 . . . *CONTROLS, ANALYSIS=DISCONTINUOUS *CONTROLS, PARAMETER=TIME INCREMENTATION , , , , , , , 20

8th field


L9.37

Example 1 – Crack Initiation and Propagation • Output Requests

• Request PHILSM and STATUSXFEM in addition to the usual output for static analysis


L9.38

Example 1 – Crack Initiation and Propagation • Postprocessing

• Crack isosurface (Crack_PHILSM) created and displayed automatically • Field and history quantities of interest can be plotted and animated as usual


Example 2 – Propagation of an Existing Crack

L9.40

Example 2 – Propagation of an Existing Crack • Model with crack subjected to mixed mode loading

• Initial crack needs to be defined • Crack propagates at an angle dictated by mode mix ratio at crack tip


L9.41

Example 2 – Propagation of an Existing Crack 1 Define damage criteria in the material model as described in Example 1 2 Specify the enriched region as in Example 1

3 Define the initial crack

• Two methods are available to define initial crack in Abaqus/CAE 1. Create a separate part representing the crack surface or line and assemble it along with the part representing the structure to be analyzed

2. Create an internal face or edge representing the crack in the part •

Method 1 is preferred as it takes full advantage of the meshindependent crack representation possible using XFEM •

•

Meshing is easier using this method

Method 2 will create nodes on the internal crack face •

Element faces/edges are forced to align with the crack


L9.42

Example 2 – Propagation of an Existing Crack 3

Define the initial crack (cont’d)

The crack location can be an edge or a surface belonging to the same instance as the enriched region or to a different instance (preferred)

** Model data *INITIAL CONDITIONS, TYPE=ENRICHMENT 901, 1, Crack-1, -1.0, -1.5 901, 2, Crack-1, -1.0, -1.4 901, 3, Crack-1, 1.0, -1.4 901, 4, Crack-1, 1.0, -1.5 Element Number

Enrichment Name

Relative Node Order in Connectivity

F

Y


L9.43

Example 2 – Propagation of an Existing Crack • The other steps are as described in Example 1 and are in line with those necessary for the usual static analysis procedure


Example 3 – Delamination and Through-thickness Crack Propagation

L9.45

Example 3 – Delamination and Through-thickness Crack • Model through-thickness crack propagation using XFEM and delamination using surface-based cohesive behavior in a double cantilever beam specimen • Interlaminar crack grows initially • Through-thickness crack forms once interlaminar crack becomes long enough and the longitudinal stress value builds up due to bending • The point at which the through-thickness crack forms depends upon the relative failure stress values of the bulk material and the interface


L9.46

Example 3 – Delamination and Through-thickness Crack • This model is the same as the double cantilever beam model presented in the surface-based cohesive behavior lecture except: • Enrichment has been added to the top and bottom beams to allow XFEM crack initiation and propagation


Modeling Tips

L9.48

Modeling Tips • General Information

• Averaged quantities are used in an element for determining crack initiation and the propagation direction • The integration point principal stress or strain values are averaged • A new crack always initiates at the center of the element

• Within an enrichment region, a new crack initiation check is performed only after all existing cracks have completely separated • This may result in the abrupt appearance of multiple cracks • Complete separation is indicated by STATUSXFEM=1

• Cracks cannot initiate in neighboring elements • Crack propagates completely through an element in one increment

• Only the initial crack tip can lie within an element


L9.49

Modeling Tips • The enrichment region must not include “hotspots” due to boundary conditions or other modeling artifacts • Otherwise, unintended cracks may initiate at such locations • Damage initiation tolerance • A larger value may result in multiple cracks initiating in a region

• Small value results in small increment size and convergence difficulty • Damage stabilization

• As mentioned earlier, judicious use of viscous regularization can aid in convergence • Initial crack should bisect elements if possible • Convergence is more difficult if crack is tangential to element boundaries

• Use displacement control rather than load control • Crack propagation may be unstable under load control


L9.50

Modeling Tips • Limit maximum increment size and start with a good guess for initial increment size • In general, this is a good approach for any non-smooth nonlinearity • Analysis controls • Can help obtain a converged solution and speed up convergence

• Contour plots of field quantities should be done with the crack isosurface displayed • Ensures that the solution is plotted from the active parts of the overlaid elements according to the phantom nodes approach • If the crack isosurface is turned off, only values from the “lower” element are plotted (on the side with negative values of F)


L9.51

Modeling Tips • When defining the crack using Abaqus/CAE, extend the external crack edges beyond base geometry • This helps avoid incorrect identification of external edges as internal due to geometric tolerance issues

Top View

Defining a through-thickness crack in a cylindrical vessel Modeling Fracture and Failure with Abaqus

Current Limitations

L9.53

Current Limitations • Implemented only for the static stress analysis procedure

• Can use only linear continuum elements • CPE4, CPS4, C3D4, C3D8 and their reduced integration/incompatible counterparts • Element processing is not done in parallel

• On SMP machines, only the solver runs in parallel • Cannot run in parallel on DMP machines

• Contour integrals for stationary cracks not currently supported • Cannot model fatigue crack growth

• Intended for single or a few non-interacting cracks in the structure • Shattering cannot be modeled • An element cannot be cut by more than one crack • Crack cannot turn more than 90 degrees in one increment • Crack cannot branch Modeling Fracture and Failure with Abaqus

L9.54

Current Limitations • The first signed distance function F must be non-zero

• If the crack lies along an element boundary, a small positive or negative value should be used • This slightly offsets the crack from the element boundary • Only frictionless small-sliding contact is considered

• The small-sliding assumption will result in nonphysical contact behavior if the relative sliding between the contacting surfaces is indeed large • Only enriched regions can have a material model with damage • If only a portion of the model needs to be enriched define an extra material model with no damage for the regions not enriched • Probing field quantities on an element currently returns values only from the “lower” element (corresponding to negative values of F)


Workshop 6

L9.56

Workshop 6 • In this workshop, you will continue with the analysis of a cracked beam subjected to pure bending using XFEM

• This workshop demonstrates: • The ease of meshing and initial crack definition compared to the techniques presented in earlier lectures • The use of analysis controls


References

L9.58

References 1. I. Babuska and J. Melenk, Int. J. Numer. Meth. Engng (1997), 40:727-758

2. T. Belytschko and T. Black, Int. J. Numer. Meth. Engng (1999), 45:601-620 3. A. Hansbo and P. Hansbo, Comp. Meth. Appl. Mech. Engng (2004), 193:3523-3540 4. J. H. Song, P. M. A. Areias and T. Belytschko, Int. J. Numer. Meth. Engng (2006), 67:868-893


Fracture Lecture of Abaqus

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