Bouhala Etal(2013)

Engineering Fracture Mechanics 102 (2013) 51–64

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Engineering Fracture Mechanics

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An XFEM crack-tip enrichment for a crack terminating at abi-material interface

0013-7944/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engfracmech.2013.02.023

⇑ Corresponding author. Tel.: +352 425 991 4574; fax: +352 425 991 555.E-mail address: [email protected] (L. Bouhala).

L. Bouhala a,⇑, Q. Shao a,b, Y. Koutsawa a, A. Younes b, P. Núñez c, A. Makradi a, S. Belouettar a

a Centre de Recherche Public Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourgb Laboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST, CNRS, 1 rue Blessig, 67084 Strasbourg, Francec Departamento de Quı́mica Inorgánica, Universidad de La Laguna, 38200-La Laguna, Tenerife, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 July 2012Received in revised form 12 February 2013Accepted 18 February 2013

Keywords:Crack tip singularityXFEMSIFCharacteristic exponentsEnrichment functions

Crack-tip enrichment functions are determined for a crack terminating at a bi-materialinterface. The stress and displacement components at the crack-tip are expressed consid-ering plane elasticity solution based on Airy functions. The crack-tip characteristic expo-nents and the enrichment functions are determined using the problem boundaryconditions. Single real, double real or complex characteristic exponents are found depend-ing on the crack-angle and the relative mechanical properties of the considered bi-material.Crack-tip enrichment functions are determined for cases where the characteristic exponentis real and complex. To validate the efficiency of the enrichment functions at the crack-tipsingularity, the eXtended Finite Element Method (XFEM) is used to solve a mode-I crackterminating at the bi-material interface. Thanks to two Level-set functions, the crack andthe bi-material interface are located and the nodes belonging to elements cut by these dis-continuities are enriched. The Stress Intensity Factors (SIF) are calculated for a crack per-pendicular and terminating at the bi-material interface using the body force method(BFM). The implemented method was tested and good agreement is found when comparedto other investigations in the literature.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The development of the extended finite element method (XFEM) and the Meshless numerical methods gave new insighton dealing with discontinuities in solid materials, without any needs of re-meshing at the discontinuity but by only usingappropriate enrichment functions. This concept was first introduced by Fix and co-authors [1,2] to deal with finite elementconvergence in presence of singularities. Since then different type of enrichment functions were developed to enforce thediscontinuity of fields and/or discontinuity of field derivatives [3–8]. In application, for instance these enrichment functionsare used to describe the discontinuity of the displacement and the discontinuity of its displacement derivatives through a bi-material interface. In case of cracked materials, the discontinuity of the displacement through the crack is represented byusing either the sign function or the Heaviside function. However, additional enrichment functions are needed to describethe stress singularity at the crack tip. These kinds of enrichment functions can be determined from the asymptotic solutionsof the displacement in the vicinity of the crack tip. In case of crack within a pure homogenous elastic material, the near cracktip displacement and stress are of the form ðrij � r�k;ui � r1�kÞ. The characteristic exponent (k) is a real value equal to 0.5,and the near tip enrichment functions are given by rksin (h/2), rkcos (h/2), rksin (h/2) sin (h) and rkcos (h/2) sin (h). XFEM con-vergence analysis of an enriched crack tip in a homogeneous elastic material can be found in the work of Laborde et al. [9]

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Nomenclature

Ai XFEM nodal displacement unknowns at the crick-tipE1, E2 Young’s modulusNI shape functionsu crack-tip asymptotic displacementU XFEM displacementuI XFEM nodal displacement unknownsuc

I XFEM nodal displacement unknowns at the crack surfaceW, Wb, Wc, Wtip integration domainsx point coordinatesa, b Dundurs parametersf crack angler, h polar coordinatesk crack-tip characteristic exponentskR, kI real and imaginary parts of the crack-tip characteristic exponentl shear modulusm Poisson’s ratiorij stresses/, w Airy functions/c, /i level set functionsuk crack-tip enrichment functions

52 L. Bouhala et al. / Engineering Fracture Mechanics 102 (2013) 51–64

and Béchet et al. [10]. Chahine et al. [11] and Nicaise et al. [12] proposed other variant of enrichment methods for crack tipsingularities in homogeneous elastic materials, where a cutoff function was used to enrich the crack tip area. In case of acrack terminating at a bi-material interface (see Fig. 1), the characteristic exponent (k) can be real single, real double or com-plex depending on the mismatch between the two materials and the angle (f) between the crack and the interface [13–15].Further, the corresponding analytical asymptotic displacement solution in the vicinity of the crack tip, that is needed to deter-mine the enrichment functions, is also dependent on the material properties and the crack angle (f). Not that, the crack-tipenrichment functions can also determined numerically based on the asymptotic displacement solution at the crack-tip[16]. Based on the work of Li et al. [16], numerical enrichment functions are used recently in conjunction with the XFEMby Menk and Bordas [17,18] to enrich bi-material crack tip and polycrystal grains junction.

Crack terminating at a bi-material interface with an oblique angle can be found in engineering application such as het-erogeneous composites, where the distribution and orientation of the fillers is complex. In such composites, the multipleinteraction of the crack-tip with the fillers can lead to the case of a crack terminating at filler/matrix interface with an ob-lique angle [19]. Study of this kind of cracks is reported by several authors, see for instance [20–22]. However, their exploi-tation to feed the well-established XFEM and mesh free numerical methods with appropriate crack tip enrichment functionsis still not clear.

The present paper focuses on the determination of enrichment functions for crack terminating at a bi-material interface.The crack-tip singularity formulation and crack-tip enrichment functions are recalled in Section 2. In Section 3, the XFEM is

Fig. 1. Crack terminating on bi-material interface.

L. Bouhala et al. / Engineering Fracture Mechanics 102 (2013) 51–64 53

implemented and effects of the crack-tip enrichment functions on rate of convergence are investigated. For validation, theStress Intensity Factors are calculated for a crack perpendicular to the bi-material interface and compared with ones from theliterature in Section 4.

2. Crack tip singularity formulation

In the absence of body forces the stress and displacement solution of plane elasticity governing equations can be ex-pressed in terms of the Airy functions uj(r,h) as follow:

rjr ¼ r�2/jðr; hÞ;hh þ r�1/jðr; hÞ;r ; ð1Þ

rjh ¼ /jðr; hÞ;rr ; ð2Þ

rjrh ¼ �ðr�1/jðr; hÞ;hÞ;r ; ð3Þ

2ljujr ¼ �/jðr; hÞ;r þ ð1� mjÞrwjðr; hÞ;h; ð4Þ

2ljujh ¼ �r�1/jðr; hÞ;h þ ð1� mjÞr2wjðr; hÞ;r; ð5Þ

where j, (j = 1, 2, 3) indicates the regions as reported in Fig. 1, and the functions /j and wj are related by: r2/ = (rw,h),r.The solution of the Airy functions in Eqs. (1)–(5) depends on the crack angle f as well as the material properties at the

vicinity of the crack tip. These functions are determined following the Williams’s approach [23,24] considering a parametricsolution (Eq. (8)) whose parameters are determined using the problem boundary conditions. In case of crack terminating at abi-material interface, the boundary conditions describing the continuity at the bi-material interface and the traction freecrack faces are given by:

Continuity boundary conditions at the bi-material interface:

r1h ðr; h ¼ 0Þ ¼ r3

hðr; h ¼ 0Þ; ð6:aÞ

r2h ðr; h ¼ pÞ ¼ r3

h ðr; h ¼ pÞ; ð6:bÞ

r1rhðr; h ¼ 0Þ ¼ r3

rhðr; h ¼ 0Þ; ð6:cÞ

r2rhðr; h ¼ pÞ ¼ r3

hðr; h ¼ pÞ; ð6:dÞ

u1r ðr; h ¼ 0Þ ¼ u3

r ðr; h ¼ 0Þ; ð6:eÞ

u2r ðr; h ¼ pÞ ¼ u3

r ðr; h ¼ pÞ; ð6:fÞ

u1hðr; h ¼ 0Þ ¼ u3

hðr; h ¼ 0Þ; ð6:gÞ

u2hðr; h ¼ pÞ ¼ u3

h ðr; h ¼ pÞ; ð6:hÞ

Traction free boundaries at the crack surfaces:

r1h ðr; h ¼ �fÞ ¼ 0; ð6:iÞ

r2h ðr; h ¼ 2p� fÞ ¼ 0; ð6:jÞ

r1rhðr; h ¼ �fÞ ¼ 0; ð6:kÞ

r2rhðr; h ¼ 2p� fÞ ¼ 0; ð6:lÞ

Here (r,h) are the polar coordinates at the crack tip (see Fig. 1). Depending on the crack angle (f) and the mismatch of the bi-material mechanical properties, the resulted characteristic exponents (kk) could be real single (kk,k = 1), real double(kk,k = 1,2) or complex (k1 = kR + ikI,k2 = kR � ikI), where kR and kI are the real and imaginary components of the complex val-ues and i ¼

ffiffiffiffiffiffiffi�1p

. The Airy functions, /j (r,h), can be defined as a superposition of different terms, where each term verifiesthe Eqs. (1)–(6).

/jðr; hÞ ¼X

k

rðkkþ1ÞFkj ðhÞ ð7Þ

where

Fkj ðhÞ ¼ ak

j sinðkþ 1Þhþ bkj cosðkþ 1Þhþ ck

j sinðk� 1Þhþ dkj cosðk� 1Þh ð8Þ

On what follows, only one term of the Airy functions /j (r,h) will be considered for solving the system of Eqs. (1)–(8), thenthe final solution can be found by superposition of the individual solutions. Therefore considering only one term and ignor-ing the superscript (k = 1), Eqs. (7) and (8) become:

/jðr; hÞ ¼ rkþ1FjðhÞ ð9ÞFjðhÞ ¼ aj sinðkþ 1Þhþ bj cosðkþ 1Þhþ cj sinðk� 1Þhþ dj cosðk� 1Þh ð10Þ


Substituting Eq. (9) into Eqs. (1)–(5), the stresses and displacement components are given by:

rjr ¼ rðk�1ÞðF 00j ðhÞ þ ðkþ 1ÞFjðhÞÞ ð11Þ

rjh ¼ ðkþ 1Þkrðk�1ÞFjðhÞ ð12Þ

rjrh ¼ �krðk�1ÞF 0jðhÞ ð13Þ

2ljujr ¼ k�1rk½ð�kðkþ 1Þ þ ð1� mjÞð1þ kÞ2ÞFjðhÞ þ ð1� mÞF 00j ðhÞ� ð14Þ

2ljujh ¼ rk �F 0jðhÞ þ k�1ð1� mjÞðk� 1Þ F 0jðhÞ þ ðkþ 1Þ2ð�ajðkþ 1Þ�1 cosðkþ 1Þhþ bjðkþ 1Þ�1 sinðkþ 1Þh

�h�cjðk� 1Þ�1 cosðk� 1Þhþ cjðk� 1Þ�1 sinðk� 1ÞhÞ

�ið15Þ

Here F 0j and F 00j are, respectively, the first and the second derivatives of the function Fj relative to h. The details on the deri-vation of the displacement components (Eqs. (14) and (15)) are given in Appendix A.

To determine the characteristic exponent (k), Eqs. (9) and (10) are substituted into Eqs. (11)–(15), then the boundary con-ditions (6) are written as function of (k,aj,bj,cj,dj,j = 1,2,3). The resulted system of twelve algebraic linear equations ðRx ¼ 0Þ isgiven in Appendix B, where the vector (x) contains the unknowns (aj,bj,cj,dj, j = 1, 2, 3). For nontrivial solution, the determi-nant of the matrix ðRÞ must be equal to zero. Therefore the characteristic exponent solution (k) is calculated numericallyusing a MATLAB iterative solver: fsolve, by solving the following equation,

ðdetðRÞ ¼ 0Þ ð16Þ

2.1. Characteristic exponent k

For the investigation of the characteristic exponent’s dependence on the crack angle (f) and on the bi-material mechanicalproperties mismatch, the crack is considered lying in the material-1 and terminating at the interface between the two mate-rials (see Fig. 1).

Fig. 2 shows the dependence of the characteristic exponent on crack angle (f) and the mechanical mismatch through thebi-material interface. The characteristic exponent is real single in case of a crack perpendicular to the interface then becomesreal double once the crack is oblique. Once the crack goes to zero or to p the characteristic exponent becomes complex. Thepassage from real double to complex values depends on both the crack angle (f) and the mechanical mismatch (E2/E1). Ar-rows on Fig. 3 show the value of (E2/E1) at which the real double value of k becomes complex for each crack angle.

2.2. Enrichment functions

Substituting Eq. (10) and its derivatives into Eqs. (14) and (15), the displacement components in polar coordinates writes:

Fig. 2. Characteristic exponents’ (k1,k2) vs E2/E1 for different crack angles.

Fig. 3. Re{k1}, Re{k2} vs E2/E1; arrows indicate the passage from real to complex values of the characteristic exponents.


ujr ¼

rk

2lj½�ajðkþ 1Þ sinðkþ 1Þh� bjðkþ 1Þ cosðkþ 1Þhþ cjð3� k� 4mÞ sinðk� 1Þhþ djð3� k� 4mÞ cosðk� 1Þh� ð17Þ

ujh ¼

rk

2lj½�ajðkþ 1Þcosðkþ 1Þhþ bjðkþ 1Þ sinðkþ 1Þhþ cjð�3� kþ 4mÞ cosðk� 1Þhþ djð3þ k� 4mÞ sinðk� 1Þh� ð18Þ

On what follows, the enrichment functions will be deduced from Eqs. (17) and (18) by considering separately the casewhere the characteristic exponent (k) is real and the case where it is complex.

2.2.1. Real characteristic exponentCombining the terms of Eqs. (17) and (18), the displacement components (ur,uh) can be rearranged as follow:

ur ¼X

k

Ak1

� �rrkk cosðkk þ 1Þhþ Ak

2

� �rrkk sinðkk þ 1Þhþ Ak

3

� �rrkk cosðkk � 1Þhþ Ak

4

� �rrkk sinðkk � 1Þh

uh ¼X

k

Ak1

� �hrkk cosðkk þ 1Þhþ Ak

2

� �hrkk sinðkk þ 1Þhþ Ak

3

� �hrkk cosðkk � 1Þhþ Ak

4

� �hrkk sinðkk � 1Þh;

ð19Þ

where ðAki¼1...:4Þr;h are constants for a given characteristic exponent (k) and bi-material mechanical properties. These constants

do not need to be determined analytically since in XFEM and meshless methods analysis they are considered as unknownsand calculated numerically. In Eq. (19), k is the number of the characteristic exponents: k = 1 in case where k = k1 is real sin-gle, and k = 2 in case where k = (k1,k2) is real double.

Considering Eq. (19), the crack tip enrichment functions for a crack terminating at a bi-material interface in case of realsingle or double characteristic exponents are given by:

uj¼1;...4�k ¼ ðrkk cosðkk þ 1Þh; rkk sinðkk þ 1Þh; rkk cosðkk � 1Þh; rkk sinðkk � 1ÞhÞ ð20Þ

Therefore, the displacement components approximation within XFEM and meshless methods analysis [24–28] of a cracklying in one material and terminating at the bi-material interface is given by:

UðxÞ ¼XI2W

NIðxÞuI þXI2Wc

NIðxÞHðxÞucI þ

XI2Wtip

NIðxÞX4�k

J¼1

ðAJÞIuJðr; h; kÞ ð21Þ

Here uI is the continuous nodal displacement vector in the integration zone (W), and ucI is the discontinuous nodal dis-

placement vector in the zone (Wc) enriched by the Heaviside function H(x). Then in the last term of Eq. (21), the unknowns AJ

are introduced to enrich the crack tip singularity in the crack tip zone Wtip. NI are the standard shape functions.

2.2.2. Imaginary characteristic exponentIn case of imaginary characteristic exponents, both k = kR + ikI and its conjugate �k ¼ kR � ikI are solutions of the Eq. (16). To

determine the enrichment functions in case where the characteristic exponent is complex, the displacement components arecombined as follows:


ujr � iuj

h ¼rk

2lj½�ajðkþ 1Þðsinðkþ 1Þh� i cosðkþ 1ÞhÞ � bjðkþ 1Þðcosðkþ 1Þhþ i sinðkþ 1ÞhÞ

þ cjð3� k� 4mÞ sinðk� 1Þhþ djð3� k� 4mÞ cosðk� 1Þh� ð22Þ

The Eq. (22) can be rearranged to Eqs. (23) and (24), respectively for k and �k:

ujr � iuj

h ¼rkR rikI

2lj½ð�ajðkþ 1Þi�1 � bjðkþ 1ÞÞeiðkRþ1Þhe�kIh � ðcji

�1 þ djÞkeiðkR�1Þhe�kIh

þ ðcjð�5þ 4mÞi�1 þ djð3� 4mÞÞe�iðkR�1ÞhekIh� ð23Þ

ujr � iuj

h ¼rkR r�ikI

2lj½ð�ajð�kþ 1Þi�1 � bjð�kþ 1ÞÞeiðkRþ1ÞhekIh � ðcji

�1 þ djÞkeiðkR�1ÞhekIh þ ðcjð�5þ 4mÞi�1

þ djð3� 4mÞÞe�iðkR�1Þhe�kIh� ð24Þ

Substituting the following expressions in Eqs. (23) and (24):

rikI ¼ eikI logðrÞ ¼ cosðkI logðrÞÞ þ i sinðkI logðrÞÞeiðkRþ1Þhe�kIh ¼ ðcosðkR þ 1Þhþ i sinðkR þ 1ÞhÞe�kIh

eiðkR�1Þhe�kIh ¼ ðcosðkR � 1Þhþ i sinðkR � 1ÞhÞe�kIh

e�iðkR�1Þhe�kIh ¼ ðcosðkR � 1Þh� i sinðkR � 1ÞhÞe�kIh

The displacement vector u = (ur,uh) can be expressed as follows:

u ¼ rkR ekIh½A1 cosðkI logðrÞÞ cosðkR þ 1Þ þ A2 cosðkI logðrÞÞ sinðkR þ 1Þ þ A3 sinðkI logðrÞÞ cosðkR þ 1Þþ A4 sinðkI logðrÞÞ sinðkR þ 1Þ þ A5 cosðkI logðrÞÞ cosðkR � 1Þ þ A6 cosðkI logðrÞÞ sinðkR � 1Þþ A7 sinðkI logðrÞÞ cosðkR � 1Þ þ A8 sinðkI logðrÞÞ sinðkR � 1Þ� þ rkR e�kIh½A9 cosðkI logðrÞÞ cosðkR þ 1Þþ A10 cosðkI logðrÞÞ sinðkR þ 1Þ þ A11 sinðkI logðrÞÞ cosðkR þ 1Þ þ A12 sinðkI logðrÞÞ sinðkR þ 1Þþ A13 cosðkI logðrÞÞ cosðkR � 1Þ þ A14 cosðkI logðrÞÞ sinðkR � 1Þ þ A15 sinðkI logðrÞÞ cosðkR � 1Þþ A16 sinðkI logðrÞÞ sinðkR � 1Þ� ð25Þ

Here each vector Ai=1,. . .,16 have two components: (Ai)r and (Ai)h. In XFEM or meshless methods analysis, the coefficientsA1, . . . A16 are considered as displacements unknowns. Therefore the enrichment functions in case of a complex characteristicexponent for a crack tip terminating at a bi-material interface are given by:

uk¼1;...;16 ¼ rkR ekIh½cosðkI logðrÞÞ cosðkR þ 1Þ; cosðkI logðrÞÞ sinðkR þ 1Þ; sinðkI logðrÞÞ cosðkR þ 1Þ; sinðkI logðrÞÞsinðkR þ 1Þ; cosðkI logðrÞÞ cosðkR � 1Þ; cosðkI logðrÞÞ sinðkR � 1Þ; sinðkI logðrÞÞ cosðkR � 1Þ;sinðkI logðrÞÞ sinðkR � 1Þ�; rkR e�kIh½cosðkI logðrÞÞ cosðkR þ 1Þ; cosðkI logðrÞÞ sinðkR þ 1Þ; sinðkI logðrÞÞcosðkR þ 1Þ; sinðkI logðrÞÞ sinðkR þ 1Þ; cosðkI logðrÞÞ cosðkR � 1Þ; cosðkI logðrÞÞ sinðkR � 1Þ; sinðkI logðrÞÞcosðkR � 1Þ; sinðkI logðrÞÞ sinðkR � 1Þ� ð26Þ

Then, the displacement at a gauss point (x) is written as follows:

UðxÞ ¼XI2W

NIðxÞuI þXI2Wc

NIðxÞHðxÞucI þ

XI2Wtip

NIðxÞX16

J¼1

ðAJÞIuJðr; h; kR; kIÞ ð27Þ

All terms in Eq. (27) are already defined in Eq. (21).

3. Implementation of the method

3.1. Displacement approximation

In order to take into account the strong discontinuity at the crack surface, the weak discontinuity at the bi-material inter-face and the crack-tip singularity the displacement field is approximated as follows:

UðxÞ ¼XI2W

NIðxÞuI þXI2Wc

NIðxÞHIðxÞucI þ

XI2Wb

NIGðxÞubI þ

XI2Wtip

NIðxÞX

J

ðAJÞIuJ ð28Þ

Here, terms in Eq. (28) are same as the ones of Eqs. (27) and (21) except for the third term, which is introduced to enforcethe weak discontinuity at the bi-material interface in order to allow for the no-alignment of the mesh to the interface. Thefunction G(x) is the gradient jump enrichment function (at the interface).


The enrichment functions corresponding to the jumps in displacement and in the gradient are given respectively in Eqs.(29) and (30) by:

Table 1Normal

l2/l

0.000.0423.0138.

HIðxÞ ¼ signð/ðxÞÞ � signð/ðxIÞÞ ð29ÞGðxÞ ¼

XNIj/IðxÞj � j

XNI/IðxÞj ð30Þ

where /(x) is the level-set function, in this study we use two level-set functions, the first (/c) is the signed distance to thecrack surface and the second (/i) is the signed distance to the interface [29–31]. Note that the sign function at a given point isshifted by the sign function of the nodes within the element in order to avoid the blending element case (29). In what fol-lows, we consider a case of a crack perpendicular to the interface (f = p/2) and two cases of an oblique crack: (f = p/3) and(f = p/4). The characteristic exponent k is calculated from Eq. (16) using the bi-material mechanical properties given in Table1. The resulted characteristic exponent k is real single in the case of the crack perpendicular to the interface and real doublein both cases of the oblique crack. The enrichment functions (ui) are deducted from Eq. (20). In case of crack perpendicular tothe bi-material interface, the enrichment functions are given by:

uj¼1;4 ¼ ½rk cosðkþ 1Þh; rk sinðkþ 1Þh; rk cosðk� 1Þh; rk sinðk� 1Þh� ð31Þ

While in case of an oblique crack, the enrichment functions are given by:

uj¼1;8 ¼ ½rk1 cosðkþ 1Þh; rk1 sinðkþ 1Þh; rk1 cosðk� 1Þh; rk1 sinðk� 1Þh; rk2 cosðkþ 1Þh; rk2 sinðkþ 1Þh;rk2 cosðk� 1Þh; rk2 sinðk� 1Þh� ð32Þ

3.2. Enrichment strategy

As mentioned before we use in this numerical model two level-set functions: /c and /i. The crack surface is representedby the zero level-set /c = 0 and the elements cut by the crack are located by the change in the sign of the level-set function /c.The nodes of these elements are enriched by the shifted sign function given in Eq. (29). Similarly, the bi-material interface isrepresented by the zero level-set /i = 0 and the elements cut by the interface are located by the change in the sign of thelevel-set /i. The nodes of these elements are enriched by the modified absolute function given in Eq. (30). The singularcrack-tip region is located within a circular domain with diameter d centered at the crack tip. The nodes inside this zoneare enriched by the crack-tip enrichment functions given in the Eq. (31) or (32).

3.3. XFEM formulation

To solve the mechanical problem we consider the weak form of the equilibrium given by the following equation [30]:
ZXeðdUÞ : C : eðUÞdX ¼
ZCt

dU:�tdC; ð33Þ

with the following boundary conditions:

U ¼ U on Cu;rn ¼ �t on Ct ð34Þ

Here, C is the material stiffness tensor, e, r are the deformation and the stress tensors, U;�t are the prescribed displace-ment and loading, and X, Cu, Ct are the studied domain, the prescribed displacement boundaries and the prescribed loadingboundaries.

Using the relation between the deformation and the displacement (e =rsU) by considering small deformations, andsubstituting the XFEM approximation of the displacement (28) we obtain:
Z
XBT

i CBjdX � V ¼Z

Ct

NTI�tdC ð35Þ

Here, V is the generalized vector of unknowns, i. e. the nodal displacement and the added degrees of freedom; B is the strainmatrix which has the following forms:

When the node is not enriched:

ized stress intensity factor KI ¼ffiffiffi2p

KI=r:a1�k by different investigations.

1 m1 m2 Present study Ref. [15] Ref. [34] Ref. [39]

722 0.3 0.35 0.019 0.0192 0.079 0.019633 0.3 0.35 0.091 0.0955 0.074 0.0958 0.35 0.3 4.232 4.232 4.176 4.24146 0.35 0.3 4.988 5.002 4.922 4.978


Bi ¼Ni;x 00 Ni;y

Ni;y Ni;x

264

375 ð36Þ

When the node is enriched by the weak enrichment:

Bi ¼Ni;x 0 Ni;xGðxÞ þ NiGðxÞ;x 00 Ni;y 0 Ni;yGðxÞ þ NiGðxÞ;y

Ni;y Ni;x Ni;yGðxÞ þ NiGðxÞ;y Ni;xGðxÞ þ NiGðxÞ;x

264

375 ð37Þ

The same form of Bi is obtained by substituting G(x) by H(x) in the case of the strong enrichment.When the node is enriched by the crack-tip enrichment functions and the characteristic exponent is real single, B

becomes:

Bi ¼

Ni;x 0 ci1x 0 ci

2x 0 ci3x 0 ci

4x 0

0 Ni;y 0 ci1y 0 ci

2y 0 ci3y 0 ci

4y

Ni;y Ni;x ci1y ci

1x ci2y ci

2x ci3y ci

3x ci4y ci

4x

26664

37775 ð38Þ

where

cikx ¼ Ni;xukðxÞ þ Niuk;xðxÞ and ci

ky ¼ Ni;yukðxÞ þ Niuk;yðxÞ; k ¼ 1;4 ð39Þ

While, when the node is enriched by the crack-tip enrichment functions and the characteristic exponent is real double, Bi isgiven by:

Bi ¼

Ni;x 0 ci1x 0 ci

2x 0 ci3x 0 ci

4x 0 ci5x 0 ci

6x 0 ci7x 0 ci

8x 0

0 Ni;y 0 ci1y 0 ci

2y 0 ci3y 0 ci

4y 0 ci5y 0 ci

6y 0 ci7y 0 ci

8y

Ni;y Ni;x ci1y ci

1x ci2y ci

2x ci3y ci

3x ci4y ci

4x ci5y ci

5x ci6y ci

6x ci7y ci

7x ci8y ci

8x

26664

37775 ð40Þ

where cikx and ci

ky are given by Eq. (39) with k = 1, . . ., ,8.

3.4. Numerical integration

The standard Gaussian integration scheme is used to evaluate the weak form integrals of Eq. (35) on linear quadrilateralelements. For integration purpose, the quadrilaterals elements cut crack and the bi-material interface are partitioned into sixsubtriangles that conform to the crack geometry [30,31]. The accuracy of the integration of the crack tip element plays adrastic role in the quality of the global solution. This sensitivity increases in the case of high singularity ranks; thereforethe use of a high-order Gaussian integration is necessary. In our case, sub-elements at the crack-tip are integrated using athirteenth-order Gaussian integration scheme and sub-elements at the crack surfaces are integrated using a sixth-orderGaussian integration scheme. The blending elements are integrated using 16th-order Gaussian integration scheme.

3.5. Convergence study

The convergence study is carried out for the case where the crack is perpendicular to the interface (f = p/2) and the twocases where the crack is oblique: (f = p/4) and (f = p/3). The case of interfacial crack corresponding to an imaginary charac-teristic exponent was already treated by means of the XFEM in [28]. To conduct the convergence study, a bi-material withthe following properties l1 = 206 GPa, m1 = 0.35, l2/l1 = 138.46, m2 = 0.3 (see Table 1) is subjected to a remote load P (seeFig. 4a). The dimensions of the studied domain are [8 mm] � [8 mm] and the crack-tip intersects the bi-material interfaceat its middle (4 mm from the external edge). The problem is first solved using a high mesh density (about 40,000 degreeof freedom) for both the perpendicular and the oblique cracks, where the solution serves as a reference for the convergencestudy. The displacement in the applied load direction and the stress components are given in Fig. 4 in case of a crack per-pendicular to the bi-material interface, and those corresponding to an oblique crack are given in Fig. 5. Figs. 6 and 7 showrespectively the plot of L2 and H1 error norms vs the total number of the elements used in the simulation for different crackangles, f = p/2, f = p/4 and f = p/3. For both the L2 error norm and H1 error norm, the singular enrichment improves the con-vergence. The error is smaller and the slope of the convergence is larger with the use of the singular enrichment compared tothe ones without the singular enrichment. Note that, the basis of the singular enrichment functions are of number of fourwhen the crack is perpendicular to the bi-material interface (f = p/2) and eight in the case of the oblique crack (f = p/4,f = p/3). Fig. 8 shows the variation of the condition number vs the number of degrees of freedom with and without the sin-gular enrichment. To keep the condition number low and to avoid large ill-conditioned Jacobian matrix, the crack-tip en-riched area is taken circular with a radius equal to two times the crack-tip element size. Indeed, in case of a crack alonga bi-material interface, Sukumar et al. [28] showed that the XFEM solution becomes independent of the crack-tip enrichment

(f)

(b) (c)

(e)(d)

(a)

Fig. 4. Bi-material plate under remote loading where the Material-1 (cracked material) is stiffer than the material-2, (a) geometry and boundary conditions,(b) enrichment strategy, (c) vertical displacement, (d) shear stress, (e) vertical stress, and (f) horizontal stress.


area once its radius is two-times the crack tip element size. Further, Laborde et al. [9] used XFEM with fixed enrichment areathat results in better convergence over the standard XFEM where only the nearest nodes to the crack-tip are enriched. Theprocedure of enrichment and the obtained fields are plotted in Figs. 4 and 5.

4. Validation by means of the stress intensity factor

The validation is done by considering a bi-material with a crack lying in one of the two materials (Material-1) and ter-minating on the interface (see Fig. 4a). The stress intensity factors (SIF) are calculated for different combinations of themechanical properties mismatch using the body force method (BFM). A crack perpendicular to the bi-material interface isconsidered here since methods for the calculation of KIand KII exist in the literature. The determination of the SIF KI andKII in the case of cracks terminating at the interface of a bi-material is one of the most difficult post-processing tasks. Notethat, in this case, not all the classical techniques are suitable to compute the SIF due to the complexity of the problem. Usu-ally, the interpolation, the isoparametric finite element method and the BFMs are the most used techniques to compute theSIF for this kind of problems, see [15,32–39].

In this work, the XFEM solution is combined with the BFM, where the stress around the crack is determined numericallyfrom the solution and not prescribed contrarily to Ref. [15] from where the following equations are derived. In the BFM thestresses are replaced by a tension and a shear type force doublets at the crack surface. The specified traction condition on thecrack surface must satisfy the following equation:

R 2a0 RQ

x ðX2; sÞ � RQx ðX1; sÞ

h iQðsÞds ¼

R X2X1

sðsÞdsR 2a0 RP

yðX2; sÞ � RPyðX1; sÞ

h iPðsÞds ¼

R X2X1

rnðsÞds

8><>: with : 0 6 X1 6 X2 6 2a ð41Þ

where 2a is the crack length, (X1,0) and (X2,0) are the bounds of the interval in which varies the point with local coordinates(s,0), i.e. small segments dividing the crack. Rm

i ðX2; sÞ � Rmi ðX1; sÞ

� �is the resultant forces acting at points (s,0), and s(s) and

rn(s) are the shear and the normal stress at points (s,0). The expression of the resultant forces is given as a function of theposition and the bi-material properties, the definition of Rm

i is expressed in [15,38]. While the density functions Q(s) and P(s)are given by:

Weak enrich.

Singular enrich.

(b) (c)

(d) (e) (f)

(a)

Fig. 5. Bi-material plate with inclined crack (f = p/4) under remote loading, (a) geometry and boundary conditions, (b) enrichment strategy, (c) verticaldisplacement, (d) shear stress, (e) vertical stress, and (f) horizontal stress.

Fig. 6. L2 error with respect to the number of elements with and without the singular enrichment using a logarithmic scale for three crack angles.


PðsÞ ¼ skffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2a� sÞ

p� pðsÞ

QðsÞ ¼ skffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2a� sÞ

p� qðsÞ

(ð42Þ

where p(s), q(s) are weight functions that can be calculated by the XFEM interpolation. To solve the Eq. (41), we take advan-tage from the XFEM by considering the segments formed by the intersection between the zero level-set Uc = 0 and the

Fig. 7. H1 error with respect to the number of elements with and without the singular enrichment using a logarithmic scale for three crack angles.

Fig. 8. The condition number evolution with respect to the degrees of freedom number for the perpendicular crack with and without singular enrichment.


element edges adding to that the segment formed between the crack tip element edge and the crack tip, see Fig. 9. Thevariable s takes the values of the Gauss points coordinates generated on these segments. The variables p1, p2, p3, . . . , pn

and q1, q2, q3, . . . , qn are the unknowns at the set of the intersection points d1, d2, d3, . . . , dn (Fig. 9). The integrands in theEq. (41) are calculated on each Gauss point so that, in the left side term appears the unknowns and in the right side term,the stresses are interpolated from the numerical solution. Thus, the following two linear equation systems are obtained:Kp

ij

h ifpjg ¼ Fp

i

� �, Kq

ij

h ifqjg ¼ Fq

i

� �. Once the system of equation is solved, the SIF can be directly obtained as:

KI ¼2ffiffiffiffiffiffi2ap

kð1� aÞð2kb� 1Þðj1 � 1Þðb2 � 1Þðj1 þ 1Þ2 sinðkpÞ

p1 ð43Þ

KII ¼�2

ffiffiffiffiffiffi2ap

kð1� aÞð2kbþ 1Þðb2 � 1Þðj1 þ 1Þ sinðkpÞ

q1 ð44Þ

Here a and b are the Dundurs bi-material parameters [35]. The simulations are carried out for different materials stiffnessratios in the case of plane strain assumptions, where li, mi are respectively the shear modulus and Poisson’s coefficient. Usingan average of 7000 degrees of freedom, the obtained stress intensity factors for different material pairs are summarized inTable 1. The results are in good agreement with the results found in [15,34,40].

Fig. 9. Crack discretization and patterns of unknowns p1, . . . , pn and q1, . . . , qn by the body force method.


5. Conclusion

The singular enrichment functions for a cracked bi-material media where the crack is lying in one of the two materialsand terminating at the bi-material interface are determined. The characteristic exponent depends on the crack angle andthe mismatch of the bi-material’s mechanical properties. XFEM analysis is conducted on a cracked bi-material consideringdifferent mechanical mismatch and crack angle. At the bi-material interface, a weak enrichment is considered to enforce thejump in deformation and allows the mesh elements to be not aligned to the interface. At the crack faces, a strong enrichmentis considered to allow for the jump in displacements through the crack. The singular enrichment functions are used to im-prove the solution at crack-tip zone. Convergence study shows improvement of the XFEM solution using the singular enrich-ment functions over the one without enrichment functions. Validation is conducted also by comparing the calculated StressIntensity Factors using the body force method with those obtained by other methods, in case of crack perpendicular to the bi-material interface. This comparison shows a good agreement between the present results and those found in the literature.

Acknowledgments

The authors acknowledge the financial support of the Fond National de la Recherche (FNR) of Luxembourg via FNR-CORE2010 project OMIDEF (Grant FNR/786 643).

Appendix A

In plane strain, the polar displacement components ur(r,h) and uh(r,h) can be expressed as a function of /j(r,h) and a har-monic function wj(r,h) by:

2lur ¼ �/jðr; hÞ;r þ ð1� mÞrwjðr; hÞ;h ðA1Þ2luh ¼ �r�1/jðr; hÞ;h þ ð1� mÞr2wjðr; hÞ;r ðA2Þ

with

D/jðr; hÞ ¼ ðrwjðr; hÞ;hÞ;r ðA3Þ

/jðr; hÞ ¼ rðkþ1ÞFjðhÞ ¼ rðkþ1Þ½aj sinðkþ 1Þhþ bj cosðkþ 1Þhþ cj sinðk� 1Þhþ dj cosðk� 1Þh� ðA4Þ

The objective is to express ur(r,h) and uh(r,h) function of the unknown parameters:aj, bj, cj and dj. To this end, the first termsin the right hand side (RHS) of Eqs. (A1) and (A2) are expressed as a function of aj, bj, cj and dj by substituting (A4) into (A1)and (A2). Further the second terms in RHS of (A1) and (A2) are also expressed as a function of aj, bj, cj and dj by calculating thefunction wj(r,h) using Eq. (A3).

Substituting (A4) into (A3) followed by integration with respect to the polar coordinate r, led to:


D/jðr; hÞ ¼ rðk�1Þ½ðkþ 1Þ2FjðhÞ þ F 00j ðhÞ� ðA5ÞZ@

@rr@w@h

dr ¼ r

@w@h¼ rk

k½ðkþ 1Þ2FjðhÞ þ F 00j ðhÞ� ðA6Þ

The function wj(r,h) is then calculated from (A6), by integrating with respect to the polar coordinate h.

wjðr; hÞ ¼rðk�1Þ

kF 0jðhÞ þ ðkþ 1Þ2

ZFjðhÞdh

� �ðA7Þ

wjðr; hÞ ¼rðk�1Þ

kF 0jðhÞ þ ðkþ 1Þ2

�aj

kþ1 cosðkþ 1Þhþ bj

kþ1 sinðkþ 1Þh

� cj

k�1 cosðk� 1Þhþ dj

k�1 sinðk� 1Þh

0@

1A

24

35 ðA8Þ

Substituting (A6) and (A4) into (A1), the radial displacement is given by:

2lur ¼ k�1rk ð�kðkþ 1Þ þ ð1� mjÞð1þ kÞ2ÞFjðhÞ þ ð1� mÞF 00j ðhÞh i

ðA9Þ

2lkr�kur ¼ ajð�ðkþ 1ÞÞ sinðkþ 1Þhþ bjð�ðkþ 1ÞÞ cosðkþ 1Þhþ cjð�ðkþ 1Þ þ 4ð1� mjÞÞ sinðk� 1Þhþ djð�ðkþ 1Þ þ 4ð1� mjÞÞ cosðk� 1Þh ðA10Þ

Substituting (A8) and (A4) into (A1), the angular coordinate is given by:

2lkr�kuh ¼ �ajðkþ 1Þ cosðkþ 1Þhþ bjðkþ 1Þ sinðkþ 1Þh� cjððk� 1Þ þ 4ð1� mjÞÞcosðk� 1Þhþ djððk� 1Þþ 4ð1� mjÞÞ sinðk� 1Þh ðA11Þ

Appendix B

1- a1 sinð�ðkþ 1ÞfÞ þ b1 cosð�ðkþ 1ÞfÞ þ c1 sinð�ðk� 1ÞfÞ þ d1 cosð�ðk� 1ÞfÞ ¼ 0

2- a2 sinððkþ 1Þð2p� fÞÞ þ b2 cosððkþ 1Þð2p� fÞÞ þ c2 sinððk� 1Þð2p� fÞÞ þ d2 cosððk� 1Þð2p� fÞÞ ¼ 0

3- a1ðkþ 1Þ cosð�ðkþ 1ÞfÞ � b1ðkþ 1Þ sinð�ðkþ 1ÞfÞ þ c1ðk� 1Þ cosð�ðk� 1ÞfÞ � d1ðk� 1Þ sinð�ðk� 1ÞfÞ ¼ 0

4- a2ðkþ 1Þcosððkþ 1Þð2p� fÞÞ � b2ðkþ 1Þ sinððkþ 1Þð2p� fÞÞ þ c2ðk� 1Þcosððk� 1Þð2p� fÞÞ
� d2ðk� 1Þ sinððk� 1Þð2p� fÞÞ ¼ 0
5- b1 þ d1 ¼ b3 þ d3 ¼ 0

6- a2 sinððkþ 1ÞpÞ þ b2 cosððkþ 1ÞpÞ þ c2 sinððk� 1ÞpÞ þ d2 cosððk� 1ÞpÞ
¼ a3 sinððkþ 1ÞpÞ þ b3 cosððkþ 1ÞpÞ þ c3 sinððk� 1ÞpÞ þ d3 cosððk� 1ÞpÞ
7- a1ðkþ 1Þ þ c1ðk� 1Þ ¼ a3ðkþ 1Þ þ c3ðk� 1Þ

8- a2ðkþ 1Þ cosððkþ 1ÞpÞ � b2ðkþ 1Þ sinððkþ 1ÞpÞ þ c2ðk� 1Þ cosððk� 1ÞpÞ � d2ðk� 1Þ sinððk� 1ÞpÞ
¼ a3ðkþ 1Þ cosððkþ 1ÞpÞ � b3ðkþ 1Þ sinððkþ 1ÞpÞ þ c3ðk� 1Þ cosððk� 1ÞpÞ � d3ðk� 1Þ sinððk� 1ÞpÞ
9- ð2l1Þ�1ð�b1ðkþ 1Þ þ d1ð�ðkþ 1Þ þ 4ð1� t1ÞÞÞ ¼ ð2l3Þ

�1ð�b3ðkþ 1Þ þ d3ð�ðkþ 1Þ þ 4ð1� t3ÞÞÞ


10- ð2l2Þ�1 �a2ðkþ 1Þ sinððkþ 1ÞpÞ � b2ðkþ 1Þcosððkþ 1ÞpÞ þ c2ð�ðkþ 1Þ þ 4ð1� m2ÞÞ sinððk� 1ÞpÞ½

þ d2ð�ðkþ 1Þ þ 4ð1� m2ÞÞ cosððkþ 1ÞpÞ�¼ ð2l3Þ

�1 �a3ðkþ 1Þ sinððkþ 1ÞpÞ � b3ðkþ 1Þcosððkþ 1ÞpÞ þ c3ð�ðkþ 1Þ þ 4ð1� m3ÞÞ sinððk� 1ÞpÞ½þ d3ð�ðkþ 1Þ þ 4ð1� m3ÞÞ cosððk� 1ÞpÞ�

11- ð2l1Þ�1ða1ð�ðkþ 1ÞÞ þ c1ð�ðkþ 1Þ � 4ð1� m1ÞÞÞ ¼ ð2l3Þ

�1ða3ð�ðkþ 1ÞÞ þ c3ð�ðkþ 1Þ � 4ð1� m3ÞÞÞ

12- ð2l2Þ�1 �a2ðkþ 1Þ cosððkþ 1ÞpÞ þ b2ðkþ 1Þ sinððkþ 1ÞpÞ � c2ððk� 1Þ þ 4ð1� m2ÞÞ cosððk� 1ÞpÞ þ d2ððk� 1Þ½

þ4ð1� m2ÞÞ sinððk� 1ÞpÞ� ¼ ð2l3Þ�1 �a3ðkþ 1Þcosððkþ 1ÞpÞ þ b3ðkþ 1Þ sinððkþ 1ÞpÞ � c3ððk� 1Þ½

þ4ð1� m3ÞÞ cosððk� 1ÞpÞ þ d3ððk� 1Þ þ 4ð1� m3ÞÞ sinððk� 1ÞpÞ�

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