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Analysis of cracked transversely isotropic and inhomogeneous solids by aspecial BIE formulation

C.Y. Dong a, X. Yang a, E. Pan b,n

a Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, Chinab Department of Civil Engineering, University of Akron, Akron, OH, USA

a r t i c l e i n f o

Article history:

Received 8 March 2010

Accepted 2 June 2010Available online 24 August 2010

Keywords:

BEM

BIE formulation

Transverse isotropy

Inhomogeneity

Crack

Stress intensity factor

a b s t r a c t

In this paper, a special boundary integral equation (BIE) formation is proposed to analyze the fracture

problem in transversely isotropic and inhomogeneous solids. In this formulation, the single-domain

boundary element method (BEM) is utilized to discretize the cracked matrix and the displacement BEM

to the surface of the embedded inhomogeneity. The two regions are then connected through the

continuity conditions along their joint interface. The conventional and three special nine-node

quadrilateral elements are utilized to discretize the inhomogeneity–matrix interface and the crack

surface. From the crack-opening displacements on the crack surface, the mixed-mode stress intensity

factors (SIFs) are calculated, using the well-known asymptotic expression in terms of the Barnett–Lothe

tensor. In the numerical analysis, the distance between the inhomogeneity and the crack as well as the

orientation of the isotropic plane of the transversely isotropic media is varied to show their influences

on the mixed-mode SIFs along the crack fronts.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Mechanical behaviors of heterogeneous materials such ascomposites, rock structures, porous and cracked media have beenwidely investigated, using various boundary integral-relatedmethods. Bush [1] investigated the interaction between a crackand a particle cluster in composites, using the boundary elementmethod (BEM). Also applying the BEM, Knight et al. [2] analyzedthe effects of the constituent material properties, fibre spatialdistribution and microcrack damage on the localized behavior offibre-reinforced composites. Dong et al. [3,4] presented a general-purpose integral formulation in order to study the interactionbetween the inhomogeneity and crack embedded in two-dimen-sional (2D) and three-dimensional (3D) isotropic matrices. Basedon a symmetric-Galerkin BEM, Kitey et al. [5] investigated thecrack growth behavior in materials embedded with a cluster ofinhomogeneities. Phan et al. [6] used the symmetric-GalerkinBEM to calculate the stress intensity factors (SIFs) for the 2Dcrack-inhomogeneity interaction problem. Lee and Tran [7]applied the Eshelby equivalent inclusion method to carry outthe stress analysis, when a penny-shaped crack interacts withinhomogeneities and voids. Interface cracks in two or more

isotropic materials were also studied by Sladek and Sladek [8] andLiu and Xu [9].

So far, however, only a few studies exist when the inhomo-geneous material is of anisotropy, e.g., transverse isotropy. Bergerand Tewary [10] studied the interface crack problems in 2Danisotropic bimaterials. Huang and Liu [11] used the eigenstrainmethod to obtain the elastic fields around the inclusion andfurther studied the interactive energy in the system. Pan andYuan [12] investigated the fracture mechanics problems in3D anisotropic solids, using the combined displacement andtraction integral representations (i.e., the single-domain BEM).Ariza and Dominguez [13] obtained the boundary tractionintegral equation for cracked 3D transversely isotropic bodies, inwhich explicit expressions for the fundamental traction deriva-tives were presented. Yue et al. [14] calculated the 3D SIFs of aninclined square crack within a bimaterial cuboid, using thesingle-domain BEM. Chen et al. [15,16] studied the fracturebehavior of a cracked transversely isotropic cuboid also using 3DBEM. Benedetti et al. [17] presented a fast dual BEM for cracked3D problems.

While the interaction between the inhomogeneities and cracksembedded in a transversely isotropic medium is important, thereis no existing literature on this topic. Therefore, in this paper, theeffect of a spherical inhomogeneity on the SIFs of a square-shapedcrack, both being embedded in a transversely isotropic matrix, isstudied using a special BIM formulation. The influence of thedistance between the inhomogeneity and the square-shaped

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Engineering Analysis with Boundary Elements

0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enganabound.2010.08.005

n Corresponding author.

E-mail address: [email protected] (E. Pan).

Engineering Analysis with Boundary Elements 35 (2011) 200–206


crack and the material orientation on the SIFs of the crack fronts isdiscussed.

2. Boundary integral equations

We consider the general case where a transversely isotropicinhomogeneity is embedded in a cracked infinite matrix oftransverse isotropy. In order to study the effect of the inhomo-geneity on the SIFs of the crack, a special BIE formulation ispresented. In our formulation, the displacement and tractionboundary integral equations [12]

bijujðySÞ ¼

ZS

UijðyS,xSÞtjðxSÞdSðxSÞ�

ZS

TijðyS,xSÞujðxSÞdSðxSÞ

�

ZGþ

TijðyS,xGþ Þ½ujðxGþ Þ�ujðxG� Þ�dGðxGþ Þþu0i ðySÞ ð1Þ

½tlðyGþ Þ�tlðyG� Þ�=2þnmðyGþ Þ

ZS

clmikTij,kðyGþ ,xSÞujðxSÞdSðxSÞ

þnmðyGþ Þ

ZG

clmikTij,kðyGþ ,xGþ Þ½ujðxGþ Þ�ujðxG� Þ�dGðxGþ Þ

¼ nmðyGþ Þ

ZS

clmikU�ij,kðyGþ ,xSÞtjðxSÞdSðxSÞþ½t0l ðyGþ Þ�t0

l ðyG� Þ�=2

ð2Þ

are applied to the cracked matrix. In Eqs. (1) and (2), bij arecoefficients that depend only on the local geometry of theinhomogeneity–matrix interface S at yS. A point on the positive(or negative) side of the crack is denoted by xGþ (or xG� ), and onthe inhomogeneity–matrix interface S by both xS and yS; nm is theunit outward normal of the positive side of the crack surface atyGþ ; clmik is the fourth-order stiffness tensor of the material; u0

i ðysÞ

is the displacement component along the i-direction at point yS

caused by a given uniform remote loading, and t0l ðyGþ Þ and t0

l ðyG� Þ

are the corresponding traction components along l-direction atpoints yGþ and yG� and ti are the displacements and tractions onthe inhomogeneity–matrix interface S (or the crack surface G); Uij

and Tij are the Green’s functions of the displacements andtractions; Uij,k and Tij,k are, respectively, the derivatives of theGreen’s displacements and tractions with respect to the sourcepoint. The displacement and traction Green’s functions are takenfrom Pan and Chou [18], whilst their derivatives are taken fromPan and Yuan [12].

The displacement integral equation is applied to the surface ofthe inhomogeneity as follows:

bijujðySÞ ¼

ZS

UijðyS,xSÞtjðxSÞdSðxSÞ�

ZS

TijðyS,xSÞujðxSÞdSðxSÞ ð3Þ

Eqs. (1)–(3) then can be used to investigate the effect of theinhomogeneity on the SIFs of the crack embedded in atransversely isotropic matrix. In discretization of these equations,we apply nine-node quadrilateral curved elements as shown inFig. 1 to the inhomogeneity–matrix interface and the cracksurface with the crack front being discretized by special elements.For any point within each element on the inhomogeneity–matrixinterface, the global coordinates, displacements and tractions canbe expressed, in terms of the element type I (Fig. 1), as [12,15,16]

xi ¼X9

k ¼ 1

fkxki , ui ¼

X9

k ¼ 1

fkuki , ti ¼

X9

k ¼ 1

fktki , i¼ 1,2,3 ð4Þ

where the subscript i is the Cartesian coordinate component; thesuperscript k is the nodal number; fk(k¼1�9) are the shapefunctions (of the local coordinates x1 and x2), which are given in

Pan and Yuan [12]; xki , tk

i , uki are, respectively, the coordinates,

tractions and displacements at nodal point k.Similarly, the crack-opening displacements (CODs) Duið ¼

uiðxGþ Þ�uiðxG� ÞÞ on the crack surface can be expressed as

Dui ¼X9

k ¼ 1

fkDuki , i¼ 1,2,3 ð5Þ

where Duki are the crack-opening displacements at nodal point k.

For the crack elements away from the crack front, theshape functions fk(k¼1�9) are the same as those in Eq. (4).However, for the crack element near the crack front, thecorresponding shape functions need to be modified. In otherwords, the shape functions near the crack front should bemultiplied by suitable weight functions to represent the near-field behavior of the crack. For the element type II shown in Fig. 1,the CODs can be expressed as

Dui ¼X9

k ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þx2

pfkDuk

i , i¼ 1,2,3 for type II ð6Þ

For the element types III and IV shown in Fig. 1, the CODs havethe following expressions

Dui ¼X9

k ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þx1Þð1þx2Þ

pfkDuk

i , i¼ 1,2,3 for type III ð7aÞ

Dui ¼X9

k ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�x1Þð1þx2Þ

pfkDuk

i , i¼ 1,2,3 for type IV ð7bÞ

We point out that element types II–IV are in general callednon-conforming elements, employed to better approximate thefield behavior. The concept of this type of elements wasintroduced and discussed in [19–23]. We further mention thatwhile in this paper, the concerned nodes are fixed at 2/3, otherlocations, such as the quarter point, could be selected with equalefficiency.

Taking each node in turn as the collocation point andperforming the involved integrals, we finally obtain the compactforms of the discretized equations from Eqs. (1)–(3) as

H11 H12

H21 H22

" #Um

DUc

" #þ

B1

B2

" #¼

G11 G12

G21 G22

" #Tm

Tc

" #ð8Þ

and

HiUi ¼ GiTi ð9Þ

where the subscripts i and m represent, respectively, theinhomogeneity and matrix; H and G are, respectively, the

9

1

7 8

4 5 6

2 3

(1,1)

(-1,-1)

2/3

2ξ2ξ9

1

7 8

4 5 6

2 3

2/3

2/3

2ξ9

1

7 8

4 5 6

2 3

Type III

Type II Type I

Type IV

2/3

2/3

9

1

7 8

4 5 6

2 3

2ξ

1ξ1ξ

1ξ1ξ

Fig. 1. Four types of elements employed for the discretization of the crack surface

[12], where the dash line represents the crack front.

C.Y. Dong et al. / Engineering Analysis with Boundary Elements 35 (2011) 200–206 201


influence coefficient matrices containing integrals of the funda-mental Green’s function solutions; B1 and B2 are, respectively, thedisplacement and traction vectors induced by the remote loading;Um(Ui) and Tm(Ti) are, respectively, the node displacement andtraction vectors on the matrix side (inhomogeneity side) of theinhomogeneity–matrix interface; DUc and Tc are, respectively, thediscontinuous displacement and traction vectors over the cracksurface. In this paper, we assume that the tractions on both sidesof the crack are equal and opposite. Therefore Tc is equal to zero.

Using the continuity condition of the displacement andtraction vectors along the interface, i.e., Um¼Ui and Tm¼ �Ti,between the inhomogeneity and matrix, we can combine Eqs. (8)and (9) into

H11þG11G�1i Hi H12

H21þG21G�1i Hi H22

" #Um

DUc

( )¼�

B1

B2

( )ð10Þ

Therefore, once the unknowns Um and DUc are solved, the SIFs(KI, KII, KIII) along the crack front can be evaluated, using thefollowing asymptotic expression [12]

Du1

Du2

Du3

8><>:

9>=>;¼ 2

ffiffiffiffiffi2r

p

rL�1

KII

KI

KIII

8><>:

9>=>; ð11Þ

where r is the distance behind the crack front; L is the Barnett–Lothe tensor [24], which depends only on the anisotropicproperties of the solid in the local crack-front coordinates; Du1,Du2 and Du3 are the relative CODs in the local crack-frontcoordinates.

3. Numerical examples

We study the effect of a spherical inhomogeneity on the SIFsalong the crack fronts of a square-shaped crack. Both theinhomogeneity and crack are embedded in an infinite matrix,which is under a far-field stress sN

¼1.0 GPa in the z-direction.The side length of the square is 2a (¼2.0 m). The radius of thesphere is R¼1.0 m and it is made of transversely isotropic marblewith the following elastic properties: EX¼90 GPa, EZ¼55GPa,nXY¼nYZ¼0.3, GYZ¼21 GPa [15,16]. The matrix materialproperties are EX¼12 GPa, EZ¼4 GPa, nXY¼nYZ¼0.3, GYZ¼1.6 GPa.We should point out that all these coefficients are with respect tothe local material coordinates with X, Y and Z being, respectively,along the longitudinal, transverse and normal directions of theX–Y plane of isotropy. The space-fixed global coordinates (x, y, z)can be related to (X, Y, Z), using the orientation and inclined anglesb and C between them. In other words, the transformationrelation between the local (X, Y, Z) and global (x, y, z) coordinatesis as follows [25]

x

y

z

264375¼

�coscsinb cosb sincsinb�cosccosb �sinb sinccosbsinc 0 cosc

264

375

X

Y

Z

264

375 ð12Þ

In the numerical analysis, 24 nine-node quadrilateral elementswith 98 nodes (Fig. 2a) and 100 nine-node quadrilateral elementswith 441 nodes are employed to discretize the inhomogeneity–matrix interface and the square-shaped crack surface (Fig. 3below), respectively. A refined mesh with 386 nodes (96elements, Fig. 2b) is also used to discretize the inhomogeneity–matrix interface to check the accuracy of the numerical solution.It is found that SIFs from both refined and coarse meshes arenearly the same (to the third decimal number) and therefore, onlythe results from the coarse mesh are discussed. We consider two

different relative orientations of the inhomogeneity and crack,and they are discussed below separately.

3.1. The spherical inhomogeneity and square-shaped crack are both

in the x–y plane, separated by a distance d in the x-direction

The relative locations and orientations of the sphericalinhomogeneity and square-shaped crack are shown in Fig. 3.For varying distance d but fixed b¼01 and C¼01 for boththe inhomogeneity and the matrix, the normalized SIF KI¼

KI=ðs1ffiffiffiffiffiffipapÞ along the crack fronts AB, BC, CD and DA of the

square is shown in Fig. 4 (The crack fronts AB, BC, CD and DA aredenoted, respectively, by (�1,1), (1,3), (3,5) and (5,7) in all SIFplots). It is obvious that as d decreases, the SIF along the crack

Fig. 2. Discretization of a spherical inhomogeneity–matrix interface with 24 nine-

node quadrilateral elements (98 nodes) in (a) and with 96 elements (386 nodes)

in (b).

R=1m

x

y

d

A B

CD

L=2m R=1m

x

z

d

σ zz=1GPa

Fig. 3. A spherical inhomogeneity and a square-shaped crack within an infinite

matrix under a far-field stress. The distance between the inhomogeneity and crack

is d in the x-direction. The x–z plane view in (a) and the x–y plane view in (b). The

crack fronts AB, BC, CD and DA are denoted, respectively, by (�1,1), (1,3), (3,5) and

(5,7) in the SIF plots.

C.Y. Dong et al. / Engineering Analysis with Boundary Elements 35 (2011) 200–206202


front DA (which closes to the inhomogeneity) is significantlydecreased, while the SIFs along the other crack fronts (i.e., AB, BCand CD) are nearly insensitive to d.

For fixed distance d¼0.5 m, fixed b¼01 and C¼01 for theinhomogeneity, but different angles b and C for the matrix, thenormalized SIF KI¼ KI=ðs1

ffiffiffiffiffiffipapÞ along crack fronts AB, BC, CD and

DA of the square crack is shown in Fig. 5. It is observed that withincreasing angle C, the SIF KI along the crack fronts AB and CDdecreases, while it increases along the crack fronts BC and DA. Themaximum SIF KI appears in the middle of the crack front BC,approximately equal to 0.9, whilst the minimum KI appears in themiddle of the crack fronts AB and CD, approximately equal to 0.6.

Fig. 6 shows the effect of the material orientations b and c ofthe inhomogeneity on the SIF KI along the crack fronts AB, BC, CDand DA of the square crack. In this example, the distance is fixedat d¼0.5 m and the orientations of the matrix are fixed at b¼01and C¼01. Contrary to Fig. 5, where the SIF KI is very sensitive tothe matrix anisotropy, here the SIF KI is nearly independent of theinhomogeneity anisotropy.

For fixed d¼0.5 m, fixed b¼01 and C¼01 for the inhomo-geneity and fixed b¼01 and C¼451 for the matrix, the normalizedSIFs KII¼ KII=ðs1

ffiffiffiffiffiffipapÞ and KIII¼ KIII=ðs1

ffiffiffiffiffiffipapÞ along the crack

fronts AB, BC, CD and DA of the square crack is shown in Fig. 7. It is

observed that the variation of the SIFs KII and KIII along the crackfront is more complicated than the SIF KI.

The effect of material anisotropy on the SIFs is further studiedby comparing to the corresponding isotropic case. Shown in Fig. 8

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01KI

5.0E-01

4.0E-01

3.0E-01

-1 1 3x

d=0.3md=0.5md=0.7md=1.0md=1.2m

d=0.1m

5 7

Fig. 4. The normalized SIF KI along the square-shaped crack fronts (�1,1), (1,3),

(3,5) and (5,7) for different sphere-square distance d with fixed material

orientations b¼01 and C¼01 for both the spherical inhomogeneity and matrix

(Hereafter, (�1,1), (1,3), (3,5) and (5,7) denote, respectively, the crack fronts AB,

BC, CD and DA).

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01KI

5.0E-01

4.0E-01

3.0E-01

2.0E-01-1 1 3

x5 7

β=0°, ψ=0°β=0°, ψ=45°β=0°, ψ=90°


(3,5) and (5,7) for different material orientations b and C of the matrix with fixed

distance d¼0.5 m, and fixed angles b¼01 and C¼01 for the inhomogeneity.

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01KI

5.0E-01

4.0E-01

3.0E-01

2.0E-01-1 1 3

x5 7

β=0°, ψ=0°

β=0°, ψ=45°

β=0°, ψ=90°


(3,5) and (5,7) for different material orientations b and C of the inhomogeneity

with fixed distance d¼0.5 m, and fixed angles b¼01 and C¼01 of the matrix.

1.0E-01

8.0E-02

6.0E-02

-8.0E-02

-6.0E-02

Stre

ss in

tens

ity fa

ctor

s

4.0E-02

-4.0E-02

2.0E-02

-2.0E-02

0.0E+00

-1 1 3x

5 7

KIIIKII

Fig. 7. The normalized SIFs KII and KIII along the square-shaped crack fronts

(�1,1), (1,3), (3,5) and (5,7) for fixed distance d¼0.5 m, fixed angles b¼01 and

C¼01 of the inhomogeneity, and fixed angles b¼01 and C¼451 of the matrix.

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01KI

5.0E-01

4.0E-01

3.0E-01

2.0E-01-1 1 3

x5 7

Iso(m) - Iso(i)

Iso(m) - Tr(i)

Tr(m) - Iso(i)

Tr(m) - Tr(i)


(3,5) and (5,7) for fixed distance d¼0.1 m, but with different material anisotropy

pairs for the inhomogeneity and matrix.



is the normalized SIF KI along the crack fronts AB, BC, CD and DAof the square for fixed d¼0.1 m with various material pairs. In thisfigure, Iso(m)–Iso(i) denotes the case, where both the inhomo-geneity and matrix are of isotropy with E¼4 GPa and n¼0.25;Tr(m)–Iso(i) denotes the case, where the matrix is of transverseisotropy with EX¼12GPa, EZ¼4GPa, nXY¼nYZ¼0.3, GYZ¼1.6GPa,whilst the inhomogeneity is of isotropy with E¼4 GPa andn¼0.25; Iso(m)–Tr (i) denotes the case, where the matrix is ofisotropy with E¼4 GPa and n¼0.25, whilst the inhomogeneity isof transverse isotropy with EX¼12 GPa, EZ¼4 GPa, nXY¼nYZ¼0.3,GYZ¼1.6 GPa; Tr(m)–Tr(i) denotes the case, where both theinhomogeneity and matrix are of transverse isotropy withEX¼12GPa, EZ¼4GPa, nXY¼nYZ¼0.3, GYZ¼1.6GPa. The effect ofmaterial anisotropy on the SIF KI can be clearly observed fromFig. 8, where the SIF KI corresponding to material pair Tr(m)–Tr(i)(i.e., both the inhomogeneity and matrix are of transverseisotropy) is smaller than those corresponding to other materialpairs. Particularly along the crack front AD, even the behavior ofthe SIF KI variation for the material pair Tr(m)–Tr(i) is different, asalso observed in Fig. 4.

3.2. The spherical inhomogeneity and square-shaped crack are in the

x–y plane, separated by a distance d in the z-direction.

The relative locations and orientations of the sphericalinhomogeneity and square-shaped crack are shown in Fig. 9. Allthe material parameters, mesh size and remote loading are thesame as those in the first case (see Section 3.1) (Fig. 3). Fordifferent distance d and fixed b¼01 and C¼01 of both theinhomogeneity and the matrix, the normalized SIFKI¼ KI=ðs1

ffiffiffiffiffiffipapÞ along crack fronts AB, BC, CD and DA of the

square is shown in Fig. 10 (again, the crack fronts AB, BC, CD andDA are denoted, respectively, by (�1,1), (1,3), (3,5) and (5,7) in allSIF plots). It is observed from Fig. 10 that the SIF KI distribution ofthe crack fronts AB, BC and CD is symmetrical with respect to themiddle point of each crack front, as expected. Also for this case,different to the first case (see Section 3.1), the normalized SIFs

KII¼ KII=ðs1ffiffiffiffiffiffipapÞ and KIII¼ KIII=ðs1

ffiffiffiffiffiffipapÞ along the crack fronts

AB, BC, CD and DA of the square are nonzero, as shown in Fig. 11.For fixed d¼0.5 m, fixed b¼01 and C¼01 of the inhomogene-

ity and different angles b and C of the matrix, the SIF

L=2m

R=1m

x

z

d

=1GPa

x

y

BA

CD

σ zz

Fig. 9. A spherical inhomogeneity and a square-shaped crack within an infinite

matrix under a far-field stress. The distance between the inhomogeneity and the

crack is d in the z-direction. The x–z plane view in (a) and the x–y plane view in (b).

The crack fronts AB, BC, CD and DA are denoted, respectively, by (�1,1), (1,3), (3,5)

and (5,7) in the SIF plots.

1.2E+00

1.1E+00

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01

KI

5.0E-01

4.0E-01

3.0E-01-1 1 3

x

d=0.3md=0.5md=0.7md=1.0md=1.2m

d=0.1m

5 7


(3,5) and (5,7) for different distance d and fixed material orientations b¼01 and

C¼01, for both the inhomogeneity and matrix.

4.0E-02

2.0E-02

3.0E-02

1.0E-02

Stre

ss in

tens

ity fa

ctor

s

-2.0E-02

-3.0E-02

-1.0E-02

0.0E+00

-1 1 3x

5 7

KIIIKII

Fig. 11. The normalized SIFs KII and KIII along the square-shaped crack fronts

(�1,1), (1,3), (3,5) and (5,7) for different distance d and fixed material orientations

b¼01 and C¼01, for both the inhomogeneity and matrix.

β=0°, ψ=0°β=0°, ψ=45°β=0°, ψ=90°

1.1E+00

1.0E+00

9.0E-01

8.0E-01

7.0E-01

6.0E-01

KI

5.0E-01

4.0E-01

3.0E-01

2.0E-01-1 1 3

x5 7



distance d¼0.5 m, and fixed angles b¼01 and C¼01 of the inhomogeneity.



KI¼ KI=ðs1ffiffiffiffiffiffipapÞ along the crack fronts AB, BC, CD and DA of the

square is shown in Fig. 12. It is observed that the distribution ofthe SIF KI is similar to that in the first case (see Section 3.1)(Fig. 5). In other words, with increasing angle C, the SIF KI alongthe crack fronts AB and CD decreases, whilst the SIF KI along thecrack fronts BC and DA increases. The maximum value of KI

appears in the middle of the crack fronts BC and DA and isapproximately equal to 1.0, whilst the minimum value of KI

appears in the middle of the crack fronts AB and CD, with a valueequal to 0.65. For fixed d¼0.5 m, fixed b¼01 and C¼01 of theinhomogeneity and different values of b and C of the matrix, thenormalized SIFs KII¼ KII=ðs1


ffiffiffiffiffiffipapÞ along

the crack fronts AB, BC, CD and DA of the square are shown inFigs. 13 and 14. It is obvious that relatively larger SIFsKII¼ KII=ðs1


ffiffiffiffiffiffipapÞ are observed for fixed

b¼01 and C¼451 of the matrix.

4. Conclusions

A special BIE formulation is developed for the study of thefracture problem in a transversely isotropic and heterogeneous

medium. In this formulation, the single-domain BEM is applied tothe cracked matrix, whilst the displacement BEM to the surface ofthe inhomogeneity. The continuity conditions along the inhomo-geneity–matrix interface are then used to derive the final systemof equations. In the numerical analysis, four sets of nine-nodequadrilateral elements are applied to discretize the inhomogene-ity–matrix interface and the square-shaped crack surface. Themixed-mode SIFs are calculated from the solved discontinuousdisplacements on the crack surface. The effect of the distancebetween the inhomogeneity and the crack as well as the materialanisotropy on the SIFs of crack fronts is investigated. It is observedthat accurate SIFs can be obtained with 24 nine-node quad-rilateral elements to the spherical surface and 100 elements to thesquare-shaped crack surface. It is believed that the proposedformulation could be applied to study more complicated interac-tion problems between inhomogeneities and cracks in 3Danisotropic media.

Acknowledgements

The support of the National Natural Science Foundation ofChina under Grant no. 10772030 is gratefully acknowledged. Theauthors would also like to thank the reviewers for theirconstructive comments.

References

[1] Bush MB. The interaction between a crack and a particle cluster. Int J Fract1997;88:215–32.

[2] Knight MG, Wrobel LC, Henshall JL. Fracture response of fibre-reinforcedmaterials with macro/microcrack damage using the boundary elementtechnique. Int J Fract 2003;121:163–82.

[3] Dong CY, Cheung YK, Lo SH. An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain. Comput Mech2002;4–5:313–21.

[4] Dong CY, Lo SH, Cheung YK. Numerical analysis of the inclusion–crackinteractions using an integral equation. Comput Mech 2003;30:119–30.

[5] Kitey R, Phan AV, Tippur HV, Kaplan T. Modeling of crack growth throughparticulate clusters in brittle matrix by symmetric-Galerkin boundaryelement method. Int J Fract 2006;141:11–25.

[6] Phan AV, Gray LJ, Kaplan T. On some benchmark results for the interaction ofa crack with a circular inclusion. ASME J Appl Mech 2007;74:1282–4.

[7] Lee HK, Tran XH. On stress analysis for a penny-shaped crack interacting withinclusions and voids. Int J Solids Struct 2010;47:549–58.

[8] Sladek J, Sladek V. Boundary element analysis for an interface crack betweendissimilar elastoplastic materials. Comput Mech 1995;16:396–405.

[9] Liu YJ, Xu N. Modeling of interface cracks in fiber-reinforced composites withthe presence of interphases using the boundary element method. Mech Mater2000;32(12):769–83.

[10] Berger JR, Tewary VK. Boundary integral equation formulation for interfacecracks in anisotropic materials. Comput Mech 1997;20:261–6.

[11] Huang JH, Liu HK. On a flat ellipsoidal inclusion or crack in three-dimensionalanisotropic media. Int J Eng Sci 1998;36:143–55.

[12] Pan E, Yuan FG. Boundary element analysis of three-dimensional cracks inanisotropic solids. Int J Num Methods Eng 2000;48:211–37.

[13] Ariza MP, Dominguez J. Boundary element formulation for 3D transverselyisotropic cracked bodies. Int J Num Methods Eng 2004;60:719–53.

[14] Yue ZQ, Xiao HT, Pan E. Stress intensity factors of square crack inclined tointerface of transversely isotropic bi-material. Eng Anal Boundary Elem2007;31:50–60.

[15] Chen CS, Chen CH, Pan E. Three-dimensional stress intensity factors of acentral square crack in a transversely isotropic cuboid with arbitrary materialorientations. Eng Anal Boundary Elem 2009;33:128–36.

[16] Chen CS, Chen CS, Pan E, Tseng HC, Yu PS. Boundary element analysis ofmixed-mode stress intensity factors in an anisotropic cuboid with an inclinedsurface crack. Eng Comput 2009;26:1056–73.

[17] Benedetti I, Milazzo A, Aliabadi MH. A fast dual boundary element methodfor 3D anisotropic crack problems. Int J Num Methods Eng 2009;80:1356–78.

[18] Pan YC, Chou TW. Point force solution for an infinite transversely isotropicsolid. J Appl Mech 1976;43:608–12.

[19] Krishnasamy G, Rudolphi TJ, Schmerr LW, Rizzo FJ. Hypersingular boundaryintegral equations: some applications in acoustic and elastic wave scattering.J Appl Mech 1990;57:404–14.

β=0°, ψ=0°β=0°, ψ=45°β=0°, ψ=90°

8.0E-02

1.2E-01

4.0E-02

KII

2.0E-02

6.0E-02

1.0E-01

-2.0E-02

-4.0E-02

-6.0E-02

0.0E+00

-1 1 3x

5 7

Fig. 13. The normalized SIF KII along the square-shaped crack fronts (�1,1), (1,3),



β=0°, ψ=0°β=0°, ψ=45°β=0°, ψ=90°

5.0E-02

KIII

1.5E-01

1.0E-01

-1.0E-01

0.0E+00

-1 1 3x

5 7-1.5E-01

-5.0E-02

Fig. 14. The normalized SIF KIII along the square-shaped crack fronts (�1,1), (1,3),





[20] Krishnasamy G, Rizzo FJ, Rudolphi TJ. Continuity requirements for densityfunctions in the boundary integral equation method. Comput Mech 1992;9:267–84.

[21] Liu YJ, Rizzo FJ. A weakly singular form of the hypersingular boundaryintegral equation applied to 3-D acoustic wave problems. Comput MethodsAppl Mech Eng 1992;96:271–87.

[22] Krishnasamy G, Rizzo FJ, Liu YJ. Boundary integral equations for thin bodies.Int J Numer Methods Eng 1994;37:107–21.

[23] Liu YJ, Rizzo FJ. Scattering of elastic waves from thin shapes in three

dimensions using the composite boundary integral equation formulation. J

Acoust Soc Am 1997;102((2) (Pt.1):926–32.

[24] TCT Ting. Anisotropic elasticity: theory and applications. New York: Oxford

University Press; 1996.

[25] Pan E, Amadei B. A 3-D boundary element formulation of anisotropic elasticity

with gravity. Appl Math Modelling 1996;20:114–20.


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