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PIRA #AER200903012 Copyright 2009 by Lockheed Martin Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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XFEM Toolkit for Delamination Failure Prediction of

Composite Bonded Pi Joints

Jim Lua1

Global Engineering and Materials, Inc., Princeton, NJ, 08530

Barrett D. Flansburg2 and Stephen P. Engelstad

3

Lockheed Martin Aeronautics Company, Marietta, Georgia, 30063

This paper presents an automatic delamination onset and residual strength prediction for

composite bonded pi joints with multiple delamination cracks. The toolkit is developed based

on the eXtended Finite Element Method (XFEM) in ABAQUS via a user-defined element

and Python script using ABAQUS’ CAE. The resulting XFEM for ABAQUS (XFA) enables

the ability to arbitrarily insert an initial delamination crack that is independent of the finite

element mesh and characterize the crack growth without re-meshing. The XFA toolkit

features: 1) generation of three-dimensional (3D) pi-joint models from user-defined model

parameters; 2) representation of a delamination crack via level set description in XFEM

based on the user-defined crack configuration; 3) automatic generation of ABAQUS input

files for the base and delaminated pi-joint; and 4) computation of energy release rate along

the crack front based on the VCCT method. Verification studies were performed for a pi

joint with one-crack and two-crack using double node fracture mechanics models. Both the

versatility and the high computational efficiency made the XFA an ideal solution module for

the probabilistic damage tolerance analysis where the initial defect configuration and joint

geometry can be treated as random variables without user intervention.

Nomenclature

XFEM = Extended Finite Element Method

XFA = XFEM for ABAQUS

VCCT = Virtual Crack Closure Method

SERR = Strain Energy Release Rate

),( txφ = Level Set Function for a Crack Surface

),( txψ = Level Set Function for a Crack Front

GI = Strain Energy Release Rate for Mode I

GII, GIII = Strain Energy Release Rate for Mode II, and III

I. Introduction

The delamination failure mode is particularly significant in the damage tolerance of advanced composites, since

manufacturing flaws and in-service damage most often manifest themselves as interlaminar cracks. Aircraft

structures are frequently subjected to low/high velocity impact during maintenance and operation. The fact that

some of these impact events may take place without leaving any visible evidence on the surface of an impacted

laminate makes it even more important to account for them in the design process. A fracture mechanics approach

was used here to account for and characterize the discrete delamination cracks in bonded composite joint structure.

To assess the criticality of an initial delamination flaw and its impact on the residual strength and life, a finite

element analysis coupled with VCCT has to be performed for various crack configurations in terms of the crack’s

1 Senior Principal Scientist, GEM-NJ Office, NJ, 08540, AIAA Member.

2 Senior Staff Engineer, Advanced Development Programs, MZ 0663, AIAA Senior Member.

3 LM Fellow, Advanced Development Programs, MZ 0663, AIAA Associate Fellow.

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Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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location, shape, and size as shown in Fig. 1. The standard finite element approach uses a set of coincident (double)

nodes to represent a crack where the crack has to be located along the element boundary. Since the mesh constructed

in the standard finite element method has to conform to the assumed damage configuration, any change in the

assumed damage configuration will force an analyst to re-build a finite element mesh. The standard finite element

method requires remeshing of the cracks as they grow, which costs additional computer time. This can be very

burdensome in 3D and for cracks with very complex geometries. The use of XFEM will enable the analyst to assess

various damage tolerance scenarios easily without needing the existing finite element mesh to conform to the crack

configuration.

II. Testing

Figure 1. Potential Locations of an Initial Delamination Crack in a Joint Structure

The fracture mechanics based discrete damage model has been used extensively to comply with the current damage

tolerance practice. An initial delamination flaw induced from either fabrication or impact damage is inserted at

predetermined critical locations of the pi-preform joint. However, under normal operating conditions, local failures such

as matrix cracks, fiber breakage, and fiber matrix debonding may develop in the bonded composite joint structures. The

presence of these micro-damage may reduce the material stiffness and fracture properties. The coupling effects between

the diffuse in-plane damage and discrete out-of-plane delamination have not been adequately addressed to date [1]. The

stiffness degradation from the continuum in-plane damage will have a direct impact on the inter-laminar stress and the

resulting delamination initiation and propagation. In addition, the intensified stress field in the vicinity of a moving crack

tip will initiate new micro-damage that will accelerate the stiffness degradation. The synergistic interaction between the

continuum and discrete damage is explored via its application to a sandwich beam subjected to a low velocity impact [2].

However, the lack of experimental data on the diffuse damage induced stiffness degradation makes the fracture

mechanics based approach dominant in current damage tolerance design and certification.

A delamination crack may not continuously grow within the same ply or material interface. An active delamination

crack can stop its growth while a new delamination crack can be initiated at an adjacent ply interface and start to

propagate. In addition, multiple delamination cracks can be initiated under the impact loading as shown in Fig. 2. The

presence of multiple cracks makes the problem more challenging in prediction of the final fracture pattern and

exploration of an optimal way to arrest the crack growth. Since both the crack initiation and propagation cannot be

determined prior to the analysis, an adaptive mesh has to be used in a conventional approach where the crack plane is

located along the element interface. Despite the fact that a cohesive model can simulate the crack initiation, the

computational task associated with creation and damage monitoring of cohesive layers along all element interfaces will

be prohibitively large to simulate the multiple crack growth in a composite structure.

To greatly reduce the computational burden associated with insertion of an arbitrary crack into an existing model

and simulate the crack growth without re-meshing, an XFEM (eXtended Finite Element Method) has been

developed by Ted Belytschko and his research group at Northwestern University. XFEM allows modeling both a

stationary and growing crack in a homogeneous material subjected to static, dynamic, and fatigue loading [3-7].

Under Air Force sponsored Phase I and Phase II Small Business Innovative Research (SBIR) programs, Global

Engineering and Materials, Inc. (GEM) has implemented the XFEM technology in ABAQUS for both 2D and 3D

delamination onset and growth prediction [8-9]. The architecture of XFA (XFEM for ABAQUS) is shown in Figure


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3. The implementation of the user interface is achieved via scripting in both the model database (MDB) and output

database (ODB) and a customized GUI (graphics user interface) within ABAQUS CAE (ABAQUS pre- and post-

processor software).

Figure 2. Example of two delaminations in a pi joint after subjected to impact and mechanical loading

Figure 3. Display of the Architecture of XFA

A 2D XFEM toolkit along with its application to a delamination problem is given in Reference [8-9] and its

extension to a 2D curvilinear crack growth is given in Reference [10] for an aluminum weldment with a material

heterogeneity and residual stress field. In the study addressed in this paper, a 3D XFEM toolkit for multiple

delamination cracks in a pi-joint is developed by integrating our existing XFA with a Python script via ABAQUS’

CAE. Both displacement jump at the delamination wake and tip enrichment along the crack front are included in the

element formulation via additional degrees of freedom. Two orthogonal level set functions are used to represent the

delamination plane and crack front location. To define an arbitrary joint and crack geometry, a Python script for a

parameterized 3D pi-joint was developed for a given set of user-defined mesh density parameters. Using the level-

set description of the crack geometry, a user-defined enrichment zone is automatically defined along with its

specification of enrichment type associated with each node within the enrichment zone. ABAQUS input files of both

the base and cracked joint are automatically generated by the Python script. After execution of XFA, the tip enriched

displacement and nodal traction components are used in a post VCCT solution module to determine the strain

energy release rates (SERRs) along the crack front. The developed XFEM based pi-joint delamination analyzer has

been integrated with a probabilistic analysis framework by LM Aero [11] to perform reliability based design of


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0=ψ

crack front

:crack surface

surface of solid

0=φ

( ) 0t =,xφ ( ) 0t <,xψ crΓon

( ) 0t =,xφ ( ) 0t =,xψ

( ) 0t >,xψ does not intersect the crack

is crack front

The crack surface is given by

( )t,xφ ( )t,xψand are constructed to be orthogonal: 0=ψφ ∇∇∇∇∇∇∇∇ .For convenience,

0=ψ

crack front

:crack surface

surface of solid

0=φ

0=ψ

crack front

:crack surface

surface of solid

0=φ

( ) 0t =,xφ ( ) 0t <,xψ crΓon

( ) 0t =,xφ ( ) 0t =,xψ

( ) 0t >,xψ does not intersect the crack

is crack front

The crack surface is given by

( )t,xφ ( )t,xψand are constructed to be orthogonal: 0=ψφ ∇∇∇∇∇∇∇∇ .For convenience,

composite bonded pi joints. An analysis flow chart along with its integration with a probabilistic analyzer

(NESSUS) via the limit state function is depicted in Fig. 4 using a representative 2D pi-joint model.

Figure 4. Display of XFA Analysis Flow Chart along with Its Integration with NESSUS

II. Overview of 3D XFEM Analyzer

Different from the 2D XFEM [8-9], a 3D crack is defined by two almost orthogonal level sets (signed distance

functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space (φ), and the

second is used to describe the one-dimensional crack front (ψ), which is the interaction of the two level sets as

shown in Figure 5. With this implicit description of the crack surface by level sets, arbitrary evolving cracks can be

modeled without any explicit representation of the crack surface. In the level set formalism, the surface of the crack

is represented by a signed distance function, so that the evolution of the discontinuity can be accomplished by the

solution of the governing hyperbolic equations or geometric updates of the signed distance function via a fast

marching method [12].

Figure 5. Level Sets (Implicit Functions) Description of a 3D Crack

As shown in Figure 6, a delamination crack is located on the bi-material interface. A set of double nodes on the

crack surface has to be used in the standard FEM where its finite element discretization needs to conform to the


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Enriched Node of “H” Type

Enriched Node of “C” Type

Standard Node

Mat 1

Mat 2

TipCrack Front

Mat 1

Mat 2



Standard Node



Standard Node

Mat 1

Mat 2

TipMat 1

Mat 2

TipCrack Front

Mat 1

Mat 2

delamination crack. In contrast, using XFEM a delamination can be easily introduced into an existing finite element

model without necessity to reconstruct the finite element mesh. As shown in Fig. 6, both the Heaviside enrichment

(H) and the crack tip enrichment (C) are introduced at the nodes of the original mesh without the crack. The tip can

terminate at any position within the existing mesh. The enriched displacement approximation u(x) can be expressed

as

i

Ji

i

k

i

Nk

k

k

Ci

ii

m

i

i

hbxfHxNaxxNuxNxu ))(()()()()()(

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∑∑∑∑∈=∈=

++= γ (1)

where )()()(1

i

m

i

i xfxNxf ∑=

= , Ni is the interpolation functions as used in the formulation of standard FEM, m

is the number of nodes in the finite element, C denotes the node set considering the asymptotic solution, J denotes

the discontinuity of displacement near a crack, ui, kia , bi denote the vectors of freedom assigned to each node, γi

(i=1, Nk) are the basis for expressing the asymptotic solution, H(x) is the Heaviside function used to express the

displacement jump across the crack face, and f(x) is a level set function introduced to express the shape of the crack

using nodal information.

Figure 6. XFEM Representation of a Delamination Crack along a Bi-Material Interface

To identify the location of these enriched nodes, a signed distance function is used as the level set function

as ))~()~((~min)(~

xxxnsignxxxfx

−⋅−=Γ∈

, where Γ represents the crack surface in 3D, x~ is a point on Γ

which is closest to x. At x~ , we can construct the tangential and normal vector to the crack line, es, and en, with the

orientation of en, taken such that ex x en=es. This local coordinate system is used in a 3D problem along the crack

front to define the local response parameters for computing SERRs via VCCT. The numerical implementation of the

XFEM for the interface crack follows the same procedure as derived by Belytschko et al. [4]. To ensure that the

displacement approximation reduces to the actual asymptotic fields adjacent to the crack tip, the additional

coefficients (i.e. kia in Eq. (1)) corresponding to the enrichments at the nodes of the elements surrounding the crack

tip are forced to be equal. Inserting Equation (1) into the virtual work equation at the element level, an element

stiffness matrix can be derived in terms of normal (ui) and additional (kia , bi) degrees of freedom.

The numerical implementation of the XFEM for a 3D crack is illustrated in Fig. 7 via a circular delamination

crack embedded in the base skin of the pi-joint. Based on the geometric configuration of a 3D joint and the

embedded delamination crack, an XFEM region in the base model is identified as a nodal enrichment zone. An

XFEM input file for the user-defined elements (UELs) is generated. In addition, an initialization via the XFEM pre-

processor is performed to determine the initial level-set values that are used to characterize the location, size, and

shape of the 3D delamination. As shown in Fig. 7, a circular crack is defined on a delamination plane which is

discretized by an arbitrary non-structural mesh. Two level set functions are used to define the delamination crack

plane and crack front. To accurately extract the traction force along a circular crack front for VCCT calculation, a

ray mesh can be generated using the Python script. After execution of the XFEM solver, the deformed shape of the

cracked body and the SERRs along the crack front can be visualized using ABAQUS’ CAE.


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Figure 7. Level-Set Description of a Circular Delamination in a Pi-Joint in XFEM

III. Python Script for a Parameterized 3D Pi-Joint

In order to automatically generate the XFEM input files for an arbitrary joint and crack configuration, a Python

script is developed for a 3D pi-joint with two delamination cracks as shown in Fig. 2. Figure 8 describes the two

delamination crack model within the XFEM computational framework. It is assumed that the manufacture induced

flaw (Crack 1) is rectangular in shape and traverses the width of the model (see Figure 8). Since the primary failure

mode is driven by the rapid propagation of Crack 1, the strain energy release rates (SERRs) along its front is used as

the key XFEM output for the reliability analysis. Given the simple geometry of the rectangular crack, the variation

of the mesh resulting from the size change of Crack 1 is small. However, the re-meshing effort from the variation of

the size and location of the circular crack (Crack 2) would be quite extensive in the conventional double node finite

element method due to 1) the introduction of the ray mesh to conform to the circular crack front; and 2) intersection

of the ray mesh and the rectangular mesh due to the distinct geometry of these two cracks.

Figure 8. Description of a Two-Crack Model in XFA

The challenge from the re-meshing is circumvented from using the XFEM approach. The mesh independent

nature of the XFEM can allow the user to insert an arbitrary circular crack into an existing rectangular mesh as

shown in Figure 8. The front of Crack 2 can cut through the element boundary arbitrarily without forcing the nodes


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to be located along the front. To uniquely define these two cracks, a set of new geometric parameters has to be

defined in the parameterized model input.

In the two-crack model shown in Figures 9 and 10, all the geometric parameters for the 3D pi-joint remained the

same as described in Reference [11]. The edge delamination crack has its fixed width of L and a user-defined crack

length of a. The circular delamination crack (Crack 2) is centered along the width of the pi-joint. Additional user-

defined crack geometric parameters include the radius, horizontal (x2), and vertical (y2) location of Crack 2.

Figure 9. Display of a Parameterized Model for a Two-Crack Model

Figure 10. Illustration of Crack Parameters for Two Delamination cracks

To optimize the mesh of the two-crack model, a set of mesh parameters can be defined by a user. The one-crack

model (Crack 1 only) can be easily generated by re-defining the number of cracks in the data input file. Using the

one-crack model, the circular delamination is automatically removed from the XFEM input file. With the help of the

one-crack model, the effect of the presence of the circular delamination crack on the edge delamination crack is

explored by comparing the SERR predictions from the one-crack and two-crack model.


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IV. Demonstration and Verification of XFA

In order to verify the XFEM-based two-crack model, several analyses are performed to examine: 1) the effect of

mesh density on the XFEM model prediction; 2) the accuracy of the XFEM solution from its comparison with a

conventional double node model; and 3) the effect of inclusion of contact on the SERR prediction. Several double

node models with different mesh densities have been developed (see Figure 11). As shown in Figure 11, a ray mesh

has to be employed in the double node model to conform to the circular crack front. Different mesh seeds along the

width direction (L) are used to discretize the model. Orthotropic materials are used for pi pre-form, web, skin

laminate, and adhesive interface. The joint is simply supported with a pull load distributed on the surface of the web

(see Fig. 9). The edge crack has the size of 0.03 in while the diameter of the circular delamination crack is 1.9 in.

Figure 11. Display of Double Node Modes with Different Mesh Densities

Using the 180-seed meshing, the deformed shape and the bending stress distribution is given in Figure 12. The

failure mode under this pull-off load is Mode I dominant and a higher stress concentration zone has shown in the

vicinity of the crack front. Further, using the model without a contact interface, a slight interface penetration

between the upper and lower crack surface is shown in Figure 13. To determine the effect of interface penetration on

the SERR along the edge crack front, a new analysis is performed by embedding a contact interface between the

delamination planes ahead of the crack front. As shown in Figure 14, the interface penetration has been completely

eliminated using a large penalty stiffness via the interface approach.

Figure 12. Distribution of Bending Stress and Display of the Crack Opening


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Figure 13. Interface Penetration from the Double Node Model without Contact Interface

Figure 14. Re-Analysis of the 120-seed Mesh Using a Contact Interface

While the interface penetration will have a larger impact on the SERR along its circular crack front, its effect

on the edge delamination is relatively small. In Figure 15, the SERRs for Mode I, II, and III are plotted for the case

with and without contact. The small distinction between these curves has indicated that the effect from including the

Figure 15. Comparison of SERRs Using the Double Node Model with and without Contact for the 120 Seed

Mesh


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contact has a negligible impact on the variation of the SERR along the edge delamination crack. Given the higher

computational cost associated with the contact and a large number of repeated analyses in the probabilistic analysis,

the contact interface was not included in the XFEM solution module in this study.

To verify the XFEM toolkit for the two-crack model, both double node and XFEM models are used for the joint

with a mesh seed of 20 along the width direction. A comparison of Mode I, II, and III SERRs is shown in Figure 16.

An excellent agreement has been observed from the verification study. As shown in Figure 16, the failure mode is

driven by Mode I and the variation of GI against tip location has several peaks and valleys. To fully understand this

variation, a closer look at the XFEM prediction of the deformed crack shape in shown in Figure 17. The edge crack

Figure 16. A Comparison of SERRs Using XFEM and Double Node Model

Figure 17. A Closer Look at the XFEM Prediction at the Intersection Between the Circular and Straight

Crack Front

has its largest opening in the middle of the crack front. In addition, an intensified stress zone has been observed in

the vicinity of the cut zone where the circular crack front crosses the straight crack front of the edge crack. Since the

interface penetration has been ignored in the XFEM solver, the intrusion of the upper and lower circular


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delamination crack surface can be seen in Figure 17. The similar crack opening profile and stress variation have

been observed in the double node model shown in Figure 18.

Figure 18. Deformed Crack Shape and Stress Distribution From the Double Node Model

To demonstrate the convergence and sufficiency of using the 20 seed mesh for probabilistic analysis, the effect

of mesh density on the variation of SERR is examined using mesh seed values of 20 and 30. A comparison of

SERRs using these two mesh densities is shown in Figure 19. Both the profile and its maximum values of SERRs

are almost identical for the 20- and 30- seed models.

Figure 19. Demonstration of solution convergence using the 20 and 30 seed models

Finally, to explore the effect of the presence of the circular delamination crack on the edge delamination, a one-

crack model is used by removing the circular delamination crack from the analysis. As shown from Fig. 14, the edge

and the delamination crack is closely spaced. A comparison of GI and GII for the one-crack and two-crack models is

depicted in Figures 20 and 21, respectively. As can be seen from both Figures 20 and 21, the presence of a circular


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delamination spaced very close to the edge delamination has a large impact on the variation of SERRs along the

edge delamination crack. For the one-crack model, the SERR is almost constant along the front and its slight

variation is mainly attributed to the plane stress condition on the surface and plane strain condition at the mid-

section.

Figure 20. A Comparison of GI for the One-crack and Two-Crack Models

Figure 21. A Comparison of GII for the One-Crack and Two-Crack Models

V. Conclusions

An automatic delamination onset and residual strength prediction toolkit was developed for composite bonded pi

joints with multiple delamination cracks. A 3D XFEM solution module was implemented in ABAQUS to enable the

analyst to assess various damage tolerance scenarios easily without needing the existing finite element mesh to

conform to the crack configuration. Both the versatility and the high computational efficiency made the XFEM for


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ABAQUS (XFA) an ideal solution model for the probabilistic damage tolerance analysis where an arbitrary initial

defect size and location can be incorporated into an existing mesh without user intervention. To further enhance its

computational efficiency and facilitate its integration with a probabilistic analyzer, a Python script was developed

for a parameterized pi-joint with multiple delamination cracks.

Both the applicability and validity of the 3D XFA are demonstrated via the use of computational fracture models

based on a double node approach. Numerical results have revealed a strong elastic interaction between two closely

spaced delamination cracks. Mesh sensitivity, solution convergence, and the effect of contact have been examined

using both the XFEM and double node models.

References

1. Yang Q., Cox B. Cohesive models for damage evolution in laminated composites. International Journal o

Fracture, 2005; 133: 107-137.

2. Lua, J., W. Gregory, and J. Sankar, 2006. “Multi-Scale Dynamic Failure Prediction Tool for Marine

Composite Structures,” J. of Material Science, 41 (20), 6673-6692.

3. Belytschko T, Fish J, Englemann B., “A finite element method with embedded localization zones,”

Computer Methods in Applied Mechanics and Engineering, 70,, 1988, pp. 59-89.

4. Belytschko T, Black T., “Elastic crack growth in finite elements with minimal re-meshing,” International

Journal for Numerical Methods in Engineering, 45, 1999, pp. 601-620.

5. Moës, N, Dolbow, J, Belytschko, T. A, “Finite element method for crack growth without re-meshing,”

International Journal for Numerical Methods in Engineering, 46, 1999, pp. 131-150.

6. Sukumar, N, Moës, N, Moran, B, Belytschko T., “Extended finite element method for three-dimensional

crack modeling,” International Journal for Numerical Methods in Engineering, 48, 2000, pp. 1549-1570.

7. Belytschko, T., N. Moes, S. Usui, and C. Parimi, “Arbitrary Discontinuities in Finite Elements”,

International Journal for Numerical Methods in Engineering, 50; 2001, pp. 993-1013.

8. Shi, J., Lua, J., Waisman, H. and Belytschko, T., (2008). XFEM for ABAQUS Toolkit for Automated

Crack Onset and Growth Prediction. ABAQUS Users’ Conference, Providence, RI, May 2008.

9. Shi, J., J. Lua, and P. Liu, “Delamination Failure Prediction in Pi-Joints Using XFEM”. 49th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Schaumburg, IL,

April 7 – 10, 2008.

10. Lua, J., Shi, J., Liu, P., and Collette, M., “Curvilinear crack growth and remaining life prediction of

aluminum weldment using XFEM”, 49th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and

Materials Conference, Schaumburg, IL, April 7 – 10, 2008.

11. Flansburg, B., Engelstad, S., and Lua, J., “Robust Design of Composite Bonded Pi Joints”. 50th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs,

CA, May 4 – 7, 2009.

12. Sukumar, N., D.L. Chopp, and B. Moran, “Extended finite element method and fast marching method for

three-dimensional fatigue crack propagation”. Engineering Fracture Mechanics 70, 29-48, 2003.

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