Farid Reza Biglari,1 Alireza Rezaeinasab,1 Kamran Nikbin 2 and Iradj Sattarifar 1

Finite Element Simulation of Dynamic Crack Propagation Without Remeshing

ABSTRACT: Simulation of the crack growth for complex geometries is presented in this paper. Determi-nation of the crack propagation direction under mixed mode conditions is one of the most important pa-rameters in fracture mechanics. There are several criteria that have been developed to predict crack growthand its direction using linear elastic fracture mechanics LEFM, many of which have recently been incor-porated into finite element codes. These criteria are commonly adopted in the prediction of crack propa-gation in simple geometries and in straight crack paths. In more complex geometries, a more accuratedetermination of the crack propagation path, using remeshing methods can be employed. However, theremeshing technique usually suffers from the loss of strain energy density that can occur at the tip of thecrack during the interpolation of field solutions. In this research work, the crack growth simulation is pre-sented which allows for crack path deviation without the use of remeshing of the model. This method dealswith a nonstraight crack growth path, is based on a node releasing technique and appropriate fracturecriteria. The maximum principal stress and maximum strain energy release rate criteria is used in this paperexclusively. The results of simulation have been compared with experimental results as well as with nu-merical works of others that have been found in the recently published literature.KEYWORDS: finite element method, crack propagation, node releasing, maximum energy releaserate, strain energy density

Introduction

An important problem in linear elastic fracture mechanics LEFM is the prediction of crack growth andits direction in a component. Since the first introduction of LEFM theories in 1920s substantial researchin developing new theories and applications have been carried out. For example, several fatigue crackpropagation and crack growth criteria under impact loading have been developed. Essentially, there havebeen two strategies for solving these problems. These are to

Use an analytical approach for calculation of stress intensity factors SIFs as a function of cracklength.

Use general numerical methods such as the finite element method FEM and boundary elementmethod BEM which can be applied for different boundary conditions and more complex geom-etries.

Using the FEM method the crack is denoted as a discontinuity displacement field that consists of twofree surfaces during the separation. The change of mesh topology is one of the methods for discontinuitymodeling. Many researchers have employed the mesh topology modification technique that uses the noderelease 14 element kill 5,6 and element split methods. Because of sudden variations due to stressgradient and complex stress distribution on the crack tip, the application of a fine mesh is necessary to beemployed in order to achieve acceptable results. To prevent the excessive increase of computation timewhen using a fine mesh, the crack tip remeshing techniques have been introduced by Bittencourt et al. 3and Bouchard et al. 4. Moreover, the crack tip remeshing can help to modify distorted elements locatedright at the crack tip. However, for the cases with large deformation, the mesh distortion at the crack tipinvolves a loss of accuracy during the interpolation of field variables.

Other techniques can model crack discontinuity without using remeshing. Belytschko et al. 7 haveintroduced a meshless method where the discretisation is achieved by a model which consists of only

Manuscript received May 23, 2005; accepted for publication April 17, 2006. Presented at ASTM Symposium on Fatigue andFracture Mechanics: 35th Volume on 1820 May 2005 in Reno, NV; R. E. Link and K. M. Nikbin, Guest Editors.1 Mechanical Engineering Dept., Amirkabir University of Technology, Tehran, Iran.2 Mechanical Engineering Dept., Imperial College, London, SW7 2AZ.

Journal of ASTM International, Vol. 3, No. 7Paper ID JAI13218

Available online at www.astm.org

Copyright 2006 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

nodes. Rashid 8 has developed the arbitrary local mesh replacement method based on two distinctmeshes. First, localized mesh moves with the crack tip and then fills the rest of the domain. Recently, theextended finite element method for modeling of discontinuity in mesh has been developed by otherresearches 9,10. Sukumar and Prevost 10 have implemented a discontinuous function and the near-tipasymptotic functions which are added to the finite element approximation using the framework of partitionof unity. This permits the crack to be represented without explicitly meshing the crack surfaces and crackpropagation simulations can be conducted without the need for any remeshing.

Because of the recent development of general FEM codes, they are becoming more popular formodeling of crack growth for difficult geometries and complex mechanical conditions such as contactsurfaces in addition to convenient output data manipulation during the advanced post processing. In thispaper a program has been adapted to the general purpose finite element code ABAQUS3 by reviewingthe different criteria for determination of kinking angle, crack propagation by node release techniquewithout using remeshing. In the following sections some examples have been implemented and the resultshave been compared with experimental results and numerical results of other published works.

Criteria Used in Crack Initiation and GrowthIn order to extend the crack through the mesh in the modeled component during the FEM process, criteriafor crack extension need to be in place concurrent with the FEM calculation. Furthermore if the directionof the crack extension needs to be determined the criteria for crack extension needs to be further improvedto deal with this, Therefore for modeling of crack growth, in each time step three procedural items must bechecked:

1 The critical loading condition for crack to initiate.2 The direction that the crack is to propagate.3 The extent to which the crack needs to grow.For the evaluation of these criteria, the stress intensity factors SIFs and strain energy release rate

needs to be calculated at every increment. For the prediction of crack initiation a critical value of SIFs isoften used. Therefore it is possible to calculate values of SIFs for each time step and compare it to criticalvalues of the material failure properties and if the values of SIFs are greater than critical values, the firstcondition is satisfied.

For an elastic-plastic material, crack tip opening displacement criterion is also used. This was origi-nally introduced by Wells in 1961 11. In each time step, crack tip opening displacement value iscalculated and compared with critical value. Also according to the Griffith and Irwin theory, crack initia-tion can be predicted by the calculation of the strain energy release rate G. Potential energy of the elasticbody is defined as below

= U F 1

where U is the strain energy stored in body and F is the work done by external forces. G is the rate ofpotential energy variations with respect to crack area that has been shown in Eq 2.

G =

A2

According to this definition, Griffith criterion for crack growth is written as below

G Gf a = 0 no propagationG = Gf a 0 propagation could start 3where Gf is critical strain energy release rate, and a is crack length and a is crack length variation 12.

Fortino and Bilotta 13 has introduced an algorgithm for evaluation of crack growth extension intwo-dimensional 2D LEFM problems. The mentioned method is based on the energetic formulation ofthe coupled displacement-crack propagation problems. In addition, the G-R curve can be used to calculate

3 Hibbitt, Karlsson and Sorensen Ltd. ABAQUS, version 6.3.1, 2003.

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crack length. For example, in virtual finite element VFE method introduced by Gerken and Smith14,15, the calculation of virtual crack length variation between two elements in each step time uses theG-R curve as a criterion.

For the determination of crack kinking angle, researchers have introduced several criteria. Some ofthese criteria determine the crack growth direction based on stresses and strains field at the crack tip. Thesecriteria generally give acceptable results for LEFM, such as maximum principal stress, maximum circum-ferential stress 16, and maximum strain 17. However, for NLFM more complicated methods areneeded such as criteria that determine the crack growth direction based on energy distribution on a crackedbody. The most commonly used criterion is the maximum strain energy release rate 18. Some othercriteria are based on the nature of crack creation such as criteria that use microvoid continuum damage fordetermination of crack growth direction. In these theories, the crack growth is controlled by the creationand propagation of microvoids in the vicinity of the crack tip. Therefore, the crack propagates in thedirection that most of the voids have been nucleated 19,20.

In the next section criteria relating to maximum principal stresses, maximum circumferential stress,minimum strain density energy, and maximum strain energy release rate for crack growth direction aredescribed in further detail. However, the maximum principal stress and maximum strain energy releaserate criteria is used in this paper exclusively.

Maximum Principal Stress CriterionAccording to this criterion crack propagates in the direction that is perpendicular to the maximum principalstress. To derive more accurate results from this method, the size of the mesh must be fine compared toother regions. Also for LEFM, it is better to use singular elements. Bouchard et al. 21 have studied theeffect of mesh size and element type used in the crack tip. The accuracy of this method depends on themesh size and for achieving accurate results, therefore, the application of remeshing is necessary. Theapproach that the present paper takes is described below:

1 A ring of elements around the crack tip is selected as shown in Fig. 1a.2 Stress tensor of each Gaussian point of ring elements is calculated by FEM.3 Depending on the distance of crack tip from Gaussian point a weight function for stress tensor is

considered wi.4 Stress tensor of crack tip is calculated according to Gaussian points stress tensor and weight

function of that point Eq 4.5 The directions of principal stresses of crack tip stress tensor and direction perpendicular to maxi-

mum principal stresses is calculated.

tip =i=1

Elements

j=1

IntP

wiijM

i=1

IntP

wi

4

The results of this method are independent of the crack tip mesh structure and it gives the same resultsfor meshes of different sizes. However, the method used by Bouchard et al. 21 is strongly depended on

FIG. 1Kinking angle of crack propagation based on maximum principal stress criterion.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 3

the mesh structure and to get acceptable results remeshing is needed in their case. Acceptable results undera linear elastic condition have been achieved by the present method which implements four-node elementswith four Gaussian points without considering the effect of singularity.

Maximum Circumferential Stress CriterionThis criterion which was introduced by Erdogan and Sih 16 for elastic materials uses the maximumcircumferential stress around the crack tip to allow propagation of the crack. Stresses on a circle with aradius centered at the crack tip in polar coordinates as shown in Fig. 2 are expressed as 22

r =1

2rcos/2KI1 + sin2/2 + 32KII sin 2KII tan/2 =

12r

cos/2KI cos2/2 32KII sinr =

12r

cos/2KI sin + KII3 cos 1 5

These equations are valid for both plane stress and plane strain cases. According to maximum circum-ferential stress criterion there is an angle that is maximum. To find maximum , the derivativeequation of is equated by zero.

=

12r

KI sin + KII3 cos 1 6

Solving Eq 6 gives the corresponding to the maximum

= 2 arctan14 KIKII 14 KIKII2

+ 8 7The result that gives the sign as opposite to sign of KII is the correct one. Using Eq 6 in mode I loading

KII=0, the crack propagation angle is zero. In mode II loading, by solving the equation KII3 cos1=0, the crack propagation angle is 70.5. So the maximum range of the crack propagation angleunder LEFM is limited to an angle range of 70.5 to 70.5. Figure 3 illustrates the plots of and rversus propagation angle for arbitrary KI and KII. It can be seen that when the circumferential stress ismaximum, the shear stress tends to become zero.

Minimum Strain Energy Density CriterionThis method which was introduced by Sih and MacDonald 23 in 1974 suggests that the crack propagatesin the direction in which strain energy density is minimum. Sih and MacDonald supposed that the factor

FIG. 2Crack tip stresses according to polar coordinate system.

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preventing crack growth is strain energy We. Therefore, crack extension is more probable to occur in thedirection that the preventing factor is minimal. Strain energy density for those points ahead of the crack tipcan be written as a function of angle Fig. 2 is 1

S = a11KI2 + 2a12KIKII + a22KII2 8where the above factors a11, a12, and a22 are defined as below

a11 =1

161 + Cos k Cos

a12 =1

162 Cos k 1

a22 =1

161 Cos k + 1 + 1 + Cos 3 Cos 1 . 9

To obtain the kinking angle of the crack, the minimum value of S for different values of have tobe found.

Maximum Strain Energy Release Rate CriterionStrain energy release rate G is the required energy to increase the length of the crack one unit ahead. Thecrack propagates in the direction that G is maximum. Therefore, crack propagation direction will bederived from the following equations:

dGd =0 = 0

d2Gd2 =0 0 10To use this criterion, first G is calculated by the use of J integral. This is a common method to

calculate G in fracture mechanics. Under LEFM conditions, the J integral is equal to G. J integral isintroduced by Rice 24 as

J = Udy tiuix dS 11

where U is strain energy density, ti is the traction vector perpendicular to path , ui is displacement vector,and dS is an element on the contour .

If no forces are applied on the crack surfaces, J value for quasistatic and isothermal condition isindependent of path. Another method of calculating G is using a surface integral that is more accurate. deLorenzi 25, derived a method to calculate G using the formulation of continuum mechanics of virtualcrack extension. According to this method for 2D condition, G value is

FIG. 3Typical circumferential and shear stresses around crack tip versus angle in mixed mode.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 5

G =1a

Aij ujx1 wi1 x1xi dA 12

where A is the surface between paths 0, 1 as shown in Fig. 4. x1 is the virtual crack propagation value.This method is more accurate compared to contour integral computation especially for elastic-plasticconditions.

The procedure to calculate J integral for a four-noded element with linear shape function according tothe contour integral method is given in 26.

If N1 , . . . ,N4 is considered as shape function for nodes of a four-noded element, then

N1 = 1/41 1 N2 = 1/41 + 1 N3 = 1/41 + 1 + N4 = 1/41 1 +

13Jacobian matrix of this element is

Jace = x

y

y

y = i=1

rNi

e

xi

e i=1

rNi

e

yi

e

i=1

rNi

e

xi

e i=1

rNi

e

yi

e 14Jace1 =

x

x

y

y = 1Je

y

y

x

x

15

Using these two equations, the Jacobian mapping of the isoparametric elements are defined for the areaor line integration of the J integral.

If Jie is considered as the J integral value for the part of the path that passes from element i Fig. 5,

FIG. 4Calculation of J using surface integral method.

FIG. 5Contour path along direction of constant value [26].

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then the total value of J integral for whole path is

JContour = i=1

n

Jie 16

where n is the number of elements that contour integral passes through.To calculate Ji

e, the known values of stress, strain, and displacement of element Gaussian point, will

be used. The Jie

value for both =constant and =constant values will be calculated and added together.The following equation shows the method of calculating Ji

e:

Je

= q=1

NGAUS

Ip,qWq 17

where Je is for the part of path that =constant and NGAUS is the number of the integration point on this

path. Wq is the weight factor related to q.

Je

= p=1

NGAUS

Ip,qWp 18

where Je is for the part of the path that =constant and NGAUS is the number of the integration point

on this path. Wp is the weight factor related to p.

Je = Je + J

e 19The strain energy is

U = Ue + UP 20

Ue =12ijije UP =

0

pdP 21

where Ue is the elastic strain energy and UP is the plastic strain energy.The following equations give the traction vector perpendicular to path:

ti = xxn1 + xyn2xyn1 + yyn2

22nT = n1,n2 = E1E12 + E22 , E2E12 + E22 23

E = E1E2 = y y

x

x

y

x

x

y

y

x

For = P 24

dx =x

d, dy =

y

d For = P 25

ds = x

2 + y

2d For = P 26

u,vx

= i=1

nNi

e

xui,vi,

u,vy

= i=1

nNi

e

yui,vi 27

Jie

= e

Udy tiuix dS = 1

1Id For = P 28

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 7

1

1Id =

p=1

NGAUS

Ip,qWp = P=1

NGAUS 12 xxxxe + xyxye + yyyyey + UPy xxn1 + xyn2ux + xyn1 + yyn2vx x

2 + y

2PWP 29

For =P equations are the same as before except for x / and y / should be replaced with x /and y /.

If crack propagates self-similarly under mixed mode loading, stress intensity factor and strain energyrelease rate are related as following equations:

G =k + 1

8KI

2 + KII2 30

k = 3 4 For plane straink = 3 1 + For plane stress 31where is the Poisson ratio, is the shear modulus of material and KI, KII are the stress intensity factors.

When a part is under mixed mode loading, kinking angle is not zero, so Eq 30 does not stand. Nuismer27, has developed the stress intensity factor in a branched crack tip with angle as shown in Fig. 6

K I =12

cos

2KI1 + cos 3KII sin 32

K II =12

cos

2KI sin + KII3 cos 1 33

Therefore, the G value for the present crack tip is

G =k + 1

8K I

2 + K II2 34

Substituting K I and K II in Eq 34 gives

G =k + 1

32cos2

2KI

21 + cos 2 + 9KII2 sin2 6KIKII sin 1 + cos 35

According to the maximum strain energy release rate criterion, crack propagates in the direction thatG is maximum. To find the maximum G , the derivative of G is equated to zero G /=0. Thisequation gives the crack propagation angle

cos =3KII2 + KIKI2 + 8KII2

KI2 + 9KII2

36

FIG. 6Geometry and coordinate systems for the branched crack [27].

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Equation 36 is equivalent to Eq 7. On the other hand, under LEFM conditions, maximum circumfer-ential stress criterion and maximum strain energy release rate criterion are the same. When crack is notself-similar, the maximum strain energy release rate under mixed mode loading can be defined by substi-tuting Eq 36 in Eq 35.

G max =k + 1

16KI

4 + 24KII4 + 12KI

2KII2 + KIKI

2 + 8KII2 3/2

KI2 + 9KII2

37

Implementation of Automatic Node Release in ABAQUSUser subroutines are used in the ABAQUS code to introduce new and nonconventional required numericalconditions. In crack propagation problems, because of dynamic boundary condition updates, it is necessaryto change the boundary condition crack surfaces during each solving time step. Therefore, node releasetechnique implemented into the multipoint constraint subroutine MPC has been used. In this methodinstead of conventional meshing that uses one node at each element corner, four nodes are located at anidentical coordinates. Figure 7 shows this mesh. Nodes a, b, c, and d of four elements A, B, C, and D arecoupled together. This means that their displacements are forced to be the same.

ua = ub = uc = ud 38To create a crack surface, it is necessary for these nodes to be separated following the crack growth

direction. Consequently equal displacement of remaining attached nodes is estimated. Figure 8 shows threedifferent possible situations that the direction of a crack can take.

If for a point that contains nodes a, b, c, and d the constraint equation can be written as

k11ua + k21ub + k31uc + k41ud = 0

k12ub + k22uc + k32ud = 0

k13uc + k23ud = 0 39The above-mathematical representation of multipoint constraint for nodes a, b, c, and d can be

FIG. 72D mesh with multinode construction.

FIG. 8Representation of three possible node release situation for creation of crack surfaces: (a) Crackextension is straight ua=uc and ub=ud; (b) crack extension is right direction ua=ub=uc; (c) crack exten-sion is left direction ua=ub=ud.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 9

reexpressed in the ABAQUS input file format as*MPC

a ,a ,b ,c ,db ,b ,c ,dc ,c ,d

According to crack propagation direction, the factors in Eq 39 will take the values of 0, 1, or 1. Table1 shows the factors in Eq 39 for three different situations, shown in Fig. 8.

This strategy can only be used to model three possible situations of crack propagation, which differsapproximately by 90 from each other. Any crack propagation angle less than 90 between each case isextrapolated to one of the above cases. This mean that some localized error in the evaluation of the crackgrowth direction can reduce the accuracy in short term. In Ref. 2, a method to solve this problem hasbeen introduced. This is shown in Fig. 9. The nearest node ahead of the crack growth direction path canbe moved in a way that one of the element faces is aligned in the new crack direction.

The method that has been used in this paper is as follows:1 Amount of crack propagation in each step is calculated by the program or defined by user a.

According to crack propagation angle and a value, the new candidate of crack tip point will bedefined t.

2 The nearest nodes to the tt line will be defined see Fig. 10.3 The nodes around the tt line will be released according to the given algorithm.The element size used in this method should be at least five times smaller than a. So in each step of

crack propagation modeling, more than one node can be released. Therefore, it is possible to model everycrack propagation angle with a good degree of approximations by releasing the nodes around propagationline. Figure 11 shows the algorithm of crack propagation simulation.

TABLE 1Factors of Eq 39 for three possible node release situation.

Crack extension is left handed Crack extension is right handed Crack extension is straightk11=k12=1 k11=k12=1 k11=k12=1k21=k32=1 k21=k22=1 k31=k32=1k31=k41=k22=k13=k23=0 k31=k41=k32=k13=k23=0 k21=k41=k22=k13=k23=0

FIG. 9Automatic remeshing strategy for crack propagation: (a) previous mesh; (b) current; mesh [2].

FIG. 10Method of approximation of crack trajectory with node release technique.

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Numerical ExamplesTo model crack propagation using maximum principal stress criterion and maximum strain energy releaserate criterion, different programs have been written in VISUAL-FORTRAN language and linked to theABAQUS 6.3.1 code. This program has been implemented for four-noded linear elements with four Gaussianpoints successfully. The examples have been selected from Refs. 3,4 to compare the results with thosethat used remeshing techniques.

Example 1: A Rectangular Block with an Oblique PrecrackFigure 12 shows the geometrical properties of a plate with the rectangular shape that contains an obliqueprecrack. The dimensions are in millimeter. The material is purely elastic with a Young modulus of E=98000 MPa and a Poisson ratio of v=0.3.

In Fig. 13 the mesh for the rectangular plate has been shown and the plane strain condition was usedwith four-noded elements containing four integration points. The element type used in this analysis isCPE4. A total number of 20 975 elements and 83 900 nodes have been used in this example. The element

FIG. 11Flowchart used for crack propagation modeling.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 11

size around crack tip is about 40 m. The crack propagation length in each step has been numericallytested for a range of 415 elements and similar results were obtained. Therefore, a length of sevenelements was selected arbitrarily. Figure 14 shows crack propagation trajectory according to the maximumprincipal stress criterion. To use the maximum strain energy release rate criterion, it is necessary tocalculate J integral and stress intensity factors around the crack tip. Figure 15 shows G values for example1 around the crack tip during crack propagation.

Figure 16 shows the results of the crack propagation path modeling using maximum strain energyrelease rate criterion for the rectangular part with an oblique crack.

Figure 17 illustrates the comparison of the result from the present work with the remeshing techniqueresults published in 4.

FIG. 12Rectangular block with an oblique precrack [4].

FIG. 13Mesh used for modeling of crack propagation of rectangular part with an oblique precrack.

FIG. 14Crack trajectory based on maximum principal stress criterion.

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Example 2: Drilled Plate with Two Supporting PointThe geometrical properties and loading condition of example 2 are shown in Fig. 18. The material of thispart is polymethyl methacrylate PMMA. It has a preliminary notch with a defined length in a particulardistance from centerline. The experimental results of this model have been studied at Cornel Universityand published in Ref. 3. The experimental results have also been compared with a finite element mod-eling result using the remeshing technique. Elastic modulus of material used were E=2.94 GPa and thePoisson ratio is =0.3 28. Because of the holes in the model acting as stress concentrators, the shape ofthe crack propagation path is seen to be as a curvature. Figure 19 shows the mesh of this model. Theelement type is the same as example 1 but because of the boundary condition, the plane stress element hasbeen used CPS4 from the ABAQUS library. A total number of 17 657 elements and 70 628 nodes havebeen used in this example and the element size around the crack tip is 0.1524 mm.

Figure 20 shows the crack propagation path using maximum principal stress criterion for this model.Figure 21 compares the result from this model using maximum principal stress and maximum strainenergy release rate criteria with experimental results and also results from Ref. 3.

FIG. 15J integral histories for example 1.

FIG. 16Crack trajectory based on maximum strain energy release rate criterion.

FIG. 17Comparison of crack trajectory based on maximum principal stress and maximum strain energyrelease rate criteria with results of remeshing.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 13

ConclusionsThe simulation of the crack growth for complex geometries using LEFM methods is presented in thispaper. A program which deals with assessing the stress/strain distribution local to the crack tip in 2D has

FIG. 18Initial geometry of specimen. All dimensions in millimeters [3].

FIG. 19Mesh used for modeling of crack propagation of rectangular part.

FIG. 20Crack trajectory based on maximum principal stress criterion.

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been developed to link with a general purpose finite element program. This development allows for themodeling of complex geometries in the engineering applications. The maximum principal stress andmaximum strain energy release rate criteria have been used to find the extent of crack growth, its direction,and length. Since the remeshing technique usually suffers from the loss of strain energy density that canoccur at the tip of the crack during the interpolation of field solutions, a regular quad mesh with noremeshing has been adopted in this research work. It has been shown that the presented method canconveniently deal with a nonstraight crack growth path, which is based on a node releasing technique. Theresults of the simulation have been favorably compared with published experimental and numerical resultsfrom the literature. The presented results showed a good agreement with the published experimental dataand results from numerical simulations that employed remeshing techniques. Since the present techniqueis sufficiently robust and adaptable, it is also suggested that this method can be used to simulate crackgrowth for more ductile materials that undergo elastic-plastic and creep deformation.

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FIG. 21Comparison of crack trajectory based on maximum principal stress and maximum strain energyrelease rate criteria with results of remeshing.

BIGLARI ET AL. ON DYNAMIC CRACK PROPAGATION 15

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