Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

Interaction of multiple inhomogeneous inclusions beneath a surface

Kun Zhou a,b,⇑, Leon M. Keer b,⇑, Q. Jane Wang b, Xiaolan Ai c, Krich Sawamiphakdi c, Peter Glaws c,Myriam Paire d, Faxing Che e

a School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singaporeb Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USAc The Timken Company, Canton, OH 44706, USAd Department of Mechanics, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, Francee Institute of Microelectronics, 11 Science Park Road, Singapore 117685, Singapore

a r t i c l e i n f o

Article history:Received 22 October 2011Received in revised form 7 January 2012Accepted 10 January 2012Available online 18 January 2012

Keywords:Inhomogeneous inclusionInhomogeneityHalf space3DEquivalent inclusion method

0045-7825/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.cma.2012.01.006

⇑ Corresponding authors at: School of MechanicalNanyang Technological University, 50 Nanyang Avenpore. Tel.: +65 6790 5499; fax: +65 6792 4062 (K. Zho+1 847 491 3915 (L.M. Keer).

E-mail addresses: kzhou@ntu.edu.sg (K. Zhou(L.M. Keer).

a b s t r a c t

This paper develops a numerical method for solving multiple three-dimensional inhomogeneous inclu-sions of arbitrary shape in an isotropic half space under external loading. The method considers interac-tions between all the inhomogeneous inclusions and thus could provide an accurate stress field for theanalysis of material strength and reliability. In the method, the inhomogeneous inclusions are first brokenup into small cuboidal elements, which each are then treated as cuboidal homogeneous inclusions withinitial eigenstrains plus unknown equivalent eigenstrains using Eshelby’s equivalent inclusion method.The unknown equivalent eigenstrains are introduced to represent the material dissimilarity of the inho-mogeneous inclusions, their interactions and their response to external loading, and determined by solv-ing a set of simultaneous constitutive equations established for each equivalent cuboidal inclusion. Themethod is validated by the finite element method and then applied to investigate a cavity-containedinhomogeneous inclusion and a stringer/cluster of inhomogeneities near a half-space surface. This solu-tion may have potentially significant application in addressing challenging material science and engi-neering problems concerning inelastic deformation and material dissimilarity.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Materials scientists and engineers have paid considerable atten-tion to inhomogeneities in the study of mechanical properties andbehaviors of materials, e.g., fracture and fatigue [1–3]. On the onehand, inhomogeneities are often ubiquitous in materials in theform of micro-defects such as voids and precipitates where cracksoften nucleate and grow to cause material failure. On the otherhand, inhomogeneities may be intentionally introduced to designfunctional materials and structures like composites and coatedmaterials. Inhomogeneities are defined as having different elasticmoduli than the host material or matrix. Inclusions are character-ized as having the same material as the matrix but contain eigen-strains such as thermal expansion, phase transformation, plasticstrain and misfit strain [4]. Inhomogeneities may also containeigenstrains, and if that is the case, according to Mura [4], theyare termed inhomogeneous inclusions.

ll rights reserved.

and Aerospace Engineering,ue, Singapore 639798, Singa-u), tel.: +1 847 491 4046; fax:

), l-keer@northwestern.edu

Many phenomena such as crack nucleation, martensite forma-tion, thermal expansion and plastic deformation often occur inthe vicinity of a surface. Such phenomena, either detrimental orbeneficial to the materials, can be investigated with inhomogenei-ties or inhomogeneous inclusions in a half space. For instance, acoated surface subject to thermal or plastic strain can be simulatedby a half space with a layer of inhomogeneity containing eigen-strains, i.e., with an inhomogeneous inclusion. Therefore, the stud-ies of inhomogeneous inclusions in a half space would provide aversatile and unified framework for addressing many challengingmaterials science and engineering problems.

Although the problems related to inhomogeneities or inhomoge-neous inclusions are of significant interest, due to their complexity,only a limited number of investigations on them have been reportedin the literature. Some early studies include a circular rigid discembedded in partial contact with an incompressible elastic halfspace and subjected to a normal force [5], an embedded ellipsoidalinhomogeneity under all-around tension parallel to the surface [6],and a hemispheroidal inhomogeneity at the half-space surface witheither all around tension at infinity or uniform eigenstrains intro-duced in the inhomogeneity [7]. Recent research includes the stressanalysis of a half space containing a prolate spheroidal inhomoge-neous inclusion subjected to a uniform shear eigenstrain [8], thedisplacement analysis of an elastic half space containing a circular

26 K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33

rigid disc under a uniform surface load of finite extent [9], and theinvestigation of the stress concentration caused by two concentricspherical inhomogeneities under uniform compression located neara half-space surface [10]. These problems were solved by using thePapkovich–Neuber or the Boussinesq displacement potentials.More recently, Kuo [11,12] studied the stress disturbances causedby a single and then multiple two-dimensional (2D) arbitrarily-shaped inhomogeneities in a half space subjected to contact loadingusing the boundary element method.

Comparatively, more works have been reported on the prob-lems of inclusions in a half space since they do not require the con-sideration of material dissimilarity. Analytical solutions have beenobtained for a regularly-shaped three-dimensional (3D) inclusionwith uniform eigenstrains in a half space, e.g., a spheroidal inclu-sion [13], a cuboidal inclusion [14], an ellipsoidal inclusion [15],a hollow cylindrical inclusion [16], a solid cylindrical inclusion[17], and a hemispherical inclusion [18]. Semi-analytical solutionsfor multiple 3D arbitrarily-shaped inclusions with non-uniformeigenstrains were also obtained by discretizing the inclusions intomultiple small cuboids and then superimposing the solutions foreach cuboidal inclusion [19–21].

Eshelby [22] proposed that an inhomogeneous inclusion can betreated as an inclusion with initially prescribed eigenstrains plusequivalent eigenstrains, and the approach is termed equivalentinclusion method (EIM).

Using the EIM, Eshelby [22] found uniform stresses in an ellip-soidal inhomogeneous inclusion subjected to uniform initial eigen-strains in an infinite space. Since Eshelby’s [22] pioneering work,the EIM has been intensively used to address many problems ofinhomogeneities or inhomogeneous inclusions embedded in aninfinite space. The works include the studies of a single inhomoge-neity [23,24], the interaction between two inhomogeneities[25–29], and multiple inhomogeneities [30–36]. It is necessary topoint out that those studies of multiple inhomogeneities did notfully take into account the interactions between all the inhomoge-neities except the recent work by Zhou et al. [36]. The finite ele-ment method (FEM) was also applied to inhomogeneityproblems. However, the FEM is not an ideal tool for analyzingproblems involving infinite dimension because an accuratedescription of boundary conditions to model far fields is requiredas well as a large computation time.

The current research is devoted to solving the problem ofmultiple 3D arbitrarily-shaped inhomogeneous inclusions in a halfspace subject to surface load by using the technique of EIM. Theinhomogeneous inclusions may contain uniform or non-uniformeigenstrains. In the problem, all the interactions between the inho-mogeneous inclusions are taken into account. Zhou et al. [21] solvedmultiple homogeneous inclusions in a half space which containsonly one material, i.e., the half space is homogenous. Compared withtheir work, this research has significant advancement by consider-ing material dissimilarities in a half space, i.e., the half space is inho-mogeneous. Due to the generality of the formulated problem, thesolution has potentials in solving engineering problems that involvematerial dissimilarity and inelastic deformation, e.g., multi-layeredcoatings subject to thermal or plastic deformation.

The paper is organized into four sections. Section 2 describesthe formulation and numerical solution of the problem. Section 3presents numerical results for validation and applications, and Sec-tion 4 concludes the paper.

2. Problem formulation and solution

2.1. Governing equation

Consider a half space bounded by the plane surface z = 0 in anx–y–z Cartesian coordinate system, as illustrated in Fig. 1a. The half

space contains n arbitrarily-shaped subdomains Xw ðw ¼ 1; 2; . . . ;

nÞ, each of which has different material property than the remain-ing part of the half space, the matrix. The elastic moduli of the ma-trix and the subdomains Xw are denoted by Cijkl and Cw

ijld,respectively. Each subdomain Xw can be an isolated one or sur-rounded by others. As defined in Section 1, the subdomain Xw iscalled an inhomogeneity of the half space or an inhomogeneousinclusion, provided that it contains initial eigenstrains. For general-ity, we assume that the half space is subjected to an external loadand that Xw contains initial eigenstrain ep

ij. The external load can bea surface load F (concentrated or distributed) acting on the surfaceplane z = 0 or a remote load at the infinity. For convenience, thestress at any point in the half space caused by the external loadis called applied stress and denoted by r0

ij.Using the EIM, each inhomogeneous inclusion Xw can be simu-

lated by an inclusion in the matrix with the eigenstrain epij plus an

equivalent eigenstrain e�ij. The problem in Fig. 1a is thus convertedinto that in Fig. 1b. The final governing equation can be establishedas

CwijklC

�1klmqr

�mq � r�ij þ Cw

ijkle�kl ¼ rp

ij þ r0ij � Cw

ijklC�1klmqðrp

mq þ r0mqÞ;

ðw ¼ 1; 2; . . . ;n; i; j; k; l;m; q ¼ 1; 2; 3Þ; ð1Þ

where r�ij is the eigenstress caused by the equivalent eigenstrains e�ijin all the inclusions Xw, rp

ij is the eigenstress caused by the initialeigenstrains ep

ij in all the Xw, and r0ij is the applied stress due to

external load. The total stress at any point is given byrij ¼ r�ij þ rp

ij þ r0ij. Eq. (1) takes the same form as the governing

equation for solving inhomogeneous inclusions in an infinite space[36], however the relationships between r�ij and e�ij and between rp

ij

and epij are different from those obtained by Zhou et al. [36]. Eq. (1)

cannot be solved until the eigenstrain–eigenstress relationships aredetermined for the half space.

2.2. Numerical method for solving governing equation

A cuboidal domain D in a half space is chosen to contain the narbitrarily-shaped inhomogeneous inclusions Xw ðw ¼ 1; 2; . . . ; nÞunder investigation, with the origin Oð0; 0; 0Þ of the x–y–z Cartesiancoordinate system being set at one corner of the domain D (Fig. 2).The domain D is discretized into Nx � Ny � Nz cuboidal elementsand each of them is indexed by a sequence of three integers[a,b,c] (0 6 a 6 Nx � 1; 0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1). Each Xw

within D can therefore be approximated by a collection of cuboidalelements. It is apparent that the smaller the cuboidal element orthe finer the discretization, the more accurate the geometricapproximation will be.

Using the EIM, each cuboidal inhomogeneous inclusion insideXw is also simulated by a cuboidal homogeneous inclusion (equiv-alent cuboidal inclusion) with initial eigenstrain plus equivalenteigenstrain. Provided that the size of each cuboid is small enough,the eigenstrain in each cuboidal inhomogeneous inclusion may beapproximated as constant. The accuracy of the domain discretiza-tion and constant eigenstrain approximation was demonstratedby Zhou et al. [21]. Thus, each equivalent cuboidal inclusion willhave only one unknown equivalent eigenstrain e⁄. Accordingly,Eq. (1) is established at the observation point of each cuboidalinhomogeneous inclusion as

ðCa;b;c � IÞC�1r�a;b;c þ Ca;b;ce�a;b;c ¼ ðI� Ca;b;cC�1Þðrp

a;b;c þ r0a;b;cÞ;

ðCa;b;c 2 Cw ðw ¼ 1; 2; . . . ; nÞ; 0 6 a 6 Nx � 1;0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1Þ;

ð2Þ

where I is a unit matrix, e�a;b;c denotes the constant equivalent eigen-strain in the cuboid [a, b, c], r�a;b;c rp

a;b;c and r0a;b;c denote the stresses

at the observation point within cuboid [a,b,c], caused by their

a b

Fig. 1. Each inhomogeneous inclusion with initial eigenstrain epij in (a) is treated as an inclusion with ep

ij plus equivalent eigenstrain e�ij (b).

1Ω

ΨΩ

2Ω

z

D

x

yO

Surface plane z = 0Cuboid [0, 0, 0]

Cuboid [Nx-1, Ny-1, Nz -1]

Fig. 2. Discretization of the domain D into Nx � Ny � Nz cuboids. The domain Dcontains n arbitrarily-shaped inhomogeneous inclusions Xw ðw ¼ 1; 2; . . . ; nÞ in anisotropic half space bounded by the surface z = 0.

K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33 27

respective sources, and Ca,b,c denotes the elastic moduli of thecuboidal inhomogeneous inclusion [a,b,c]. In order to solve Eq.(2), the relationships need to be determined between rp

a;b;c andthe initial eigenstrains ep

a;b;c in all the cuboidal inclusions andbetween r�a;b;c and the equivalent eigenstrains e�a;b;c in all theinclusions.

Zhou et al. [21] obtained the approximate solution of multiplecuboidal homogeneous inclusions in a half space by decomposingthe solution into three parts and approximating the calculationof the third part: (1) multiple cuboidal inclusions in an infinitespace, (2) their image counterparts in the same infinite space,and (3) a normal traction distribution rpn on a half-space surface.The approximate solution is given as

rpa;b;c ¼

XNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�uepn;f;u

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþuepmn;f;u

�XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;crpnn;f;0;

ð0 6 a 6 Nx � 1; 0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1Þ: ð3Þ

In the first and second parts of Eq. (3), Ba�n,b�f,c�u and Ba�n,b�f,c+urelate the stress rp

a;b;c at the observation point (xa,yb,zc) in the cu-boid [a,b,c] to the uniform eigenstrain ep

n;f;u in the cuboid [n,f,u]and the eigenstrain ePm

n;f;u in its mirror counterpart in an infinitespace, respectively. The relationship between ep in a cuboidal inclu-sion and epm in its image counterpart is given by ep ¼ ðep

11; ep22;

ep33; e

p12; e

p13; e

p23Þ

T and epm ¼ ðep11; e

p22; e

p33; e

p12; e

p13; e

p23Þ

T. The 3D sum-mations are to calculate the stress contributions of all the cuboidalinclusions in the domain D. In the third approximate part, rpn

n;f;0 isobtained by evaluating the first and second parts to obtain thestresses at the observation points on the surface. The matrixMa–n,b–f,c relates the stress ra,b,c at (xa,yb,zc) to the constant normaltraction rn

n;f;0 in the surface area of the cuboid [n,f,0] and the 2Dsummations are used to calculate the subsurface stress due torn

n;f;0 in each discretized surface area. The expressions of Ba–n,b–f,c–u(or Ba�n,b�f,c+u) and Ma�n,b�f,c can be seen in the work by Zhouet al. [21] and can also refer to Chiu’s [37] original solution on acuboidal homogeneous inclusion in an infinite space.

Similarly, the eigenstress r⁄ at ðxa; yb; zkÞ due to the equivalenteigenstrains in all the equivalent cuboidal inclusions inside Xw

can be obtained by

r�a;b;c ¼XNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�ue�n;f;uþXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþue�mn;f;u

�XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;cr�nn;f ð06a6Nx�1; 06 b6Ny�1; 06 c6Nz�1Þ:

ð4Þ

Eqs. (3) and (4) give the eigenstress–eigenstrain relationships re-quired for solving Eq. (2).

In Eq. (2), the applied stress r0a;b;c at the observation point in the

cuboid [a,b,c] due to external surface load F is obtained by the fol-lowing formula:

r0a;b;c ¼

XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;cpn;f;0 þXNy�1

f¼0

XNx�1

n¼0

Ta�n;b�f;cfn;f;0;

ð0 6 a 6 Nx � 1; 0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1Þ: ð5Þwhere pn,f,0 and fn,f,0 represent a constant normal pressure and aconstant tangential traction on the discretized surface area of thecuboid, respectively. Eq. (5) takes a similar form as the third termin Eq. (3) or Eq. (4) and its explanation refers to that for Eq. (3).The expressions of Ta�n,b�f,c refer to the solution of subsurfacestresses induced by uniform tangential traction pressure acting ona rectangular area of a half-space surface [38]. A 2D fast fouriertransform (FFT) algorithm is used to accelerate the calculation ofapplied stresses.

28 K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33

The substitution of Eqs. (3)–(5) into Eq. (2) results in

ðCa;b;cC�1 � IÞXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�ue�n;f;u

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþue�mn;f;u �XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;cr�nn;f

!r�a;b;c

þ Ca;b;ce�a;b;c ¼ ðI� Ca;b;cC�1Þ

XNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�uepn;f;u

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþuepmn;f;u �

XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;crPnn;f

þXNy�1

f¼0

;XNx�1

n¼0

Ma�n;b�f;cpn;f;0 þXNy�1

f¼0

XNx�1

n¼0

Ta�n;b�f;cfn;f;0

!

ðCa;b;c 2 Cwðw ¼ 1; 2; . . . ; nÞ; 0 6 a 6 Nx � 1;0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1Þ: ð6Þ

As mentioned for rpn in Eq. (3), the normal tractions r⁄n in Eq. (6)can be determined only after the equivalent eigenstrains e⁄ areknown, which however are to be solved. Nevertheless, this willnot be a problem if Eq. (6) is solved iteratively. At each iterationstep, the values e⁄ are known and the values r⁄n can thus bedetermined.

In order to apply the FFT algorithm to efficiently calculate the2D and 3D multi-summations in Eq. (6), all the cuboids are set tohave the same size 2Dx � 2Dy � 2Dz and their bulk centers selectedto be the observation points [21]. However, this leads to that theobservation points in the cuboids of the surface layer, at whichthe normal surface tractions rpn

n;f;0 and r�nn;f;0 are calculated, will belocated on the plane z = Dz rather than at the surface z = 0. To avoidthis dilemma, we particularly treat each cuboid 2Dx � 2Dy � 2Dz inthe surface layer as composed of a virtual half above the surfaceand a real half beneath the surface. In light of the decompositionmethod mentioned, this treatment does not influence the compu-tational results even though the cuboids in the surface layer maycontain eigenstrains as do their virtual halves. According to thework by Zhou et al. [21], a combination of 3D and 2D FFTalgorithms can be used to evaluate the summations in Eq. (6) toimprove computational efficiency.

Governing equation (6) can be solved by the conjugate gradientmethod [39]. In order to handle void inhomogeneities, the modi-fied conjugate gradient method developed by Zhou et al. [36] canalso be used. Once the unknown equivalent eigenstrains e⁄ aresolved, the stress field in the computational domain can be ob-tained as

ra;b;c ¼r�a;b;c þ rpa;b;c þ r0

a;b;c ¼XNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�ue�n;f;u

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþue�mn;f;u �XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;cr�nn;f

!

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;c�uepn;f;u

þXNz�1

u¼0

XNy�1

f¼0

XNx�1

n¼0

Ba�n;b�f;cþuepmn;f;u �

XNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;crpnn;f;0

!

þXNy�1

f¼0

XNx�1

n¼0

Ma�n;b�f;cpn;f;0 þXNy�1

f¼0

XNx�1

n¼0

Ta�n;b�f;cfn;f;0

!

ð0 6 a 6 Nx � 1; 0 6 b 6 Ny � 1; 0 6 c 6 Nz � 1Þ: ð7Þ

It must be noted that the equivalent eigenstrains e⁄ in Eq. (7) con-tain all the information regarding the material dissimilarity of theinhomogeneous inclusions, their interactions and their responseto external loading. It is in solving e⁄ through the governing equa-tion (6) that such information is fully considered in Eq. (7). Thus,Eq. (7) represents more than an explicit superposition of a fewstress terms.

Chiu’s solutions for a cuboidal inclusion containing uniformeigenstrain in an infinite space or half space can capture the stresssingularities at certain edges and corners of the cuboidal inclusion[14,37]. However, when utilizing Chiu’s solutions, the presentmethod only chooses the centers of all the cuboids in a discretizeddomain as stress observation points. In another word, the ‘‘corner’’or ‘‘edge’’ points of an arbitrarily-shaped inclusion considered inthe present method are not really at corners or edges and insteadare near to them at a certain distance that depends on the discret-ization fineness of the domain. Therefore, the present method can-not capture the singularities at the corners or edges of anyinclusions.

In the present method, each cuboidal element is assumed tocontain constant or uniform eigenstrain, but overall eigenstrainin an arbitrarily-shaped inclusion consisting of multiple cuboidsremains non-uniform. Thus, the solution is valid and its accuracydepends on the fineness of discretizing a domain into many cuboi-dal elements. If eigenstrain in each cuboidal element is assumed tobe non-uniform, it is predicted that the solution accuracy wouldsignificantly improve for the same discretization fineness. How-ever, this assumption cannot be done here simply because thereis no analytic solution available for a cuboidal inclusion containingnon-uniform eigenstrain that can be utilized as Chiu’s solution[14,37]. If such an analytic solution exists, more unknowns ratherthan six unknown eigenstrain components should be solved forone cuboidal inclusion element and the eigenstrain compatibilityconditions should also be considered for the interfaces betweenall the cuboidal elements.

3. Validation and applications

The present method is first validated by the FEM for investigat-ing the problem of two interacting inhomogeneities, and then usedto solve a few typical problems concerning a double inhomoge-neous inclusion, a stringer of inhomogeneities and a cluster ofinhomogeneities. In these problems, the inhomogeneities or inho-mogeneous inclusions are in the vicinity of the half-space surface.

3.1. Two inhomogeneities

Fig. 3 shows two cuboidal inhomogeneities beneath a half-space surface, which is bounded by the plane z = 0 in the x–y–zCartesian coordinate system, under normal pressure. The semi-infinite matrix has Young’s modulus E ¼ 210 GPa and Poisson’s ra-tio v = 0.3; both inhomogeneities have EX ¼ 420 GPa and vX = 0.3.The two cuboids are identical and have the dimensions of a � a � a.The left one X1 is centered at ð�0:75a; 0; aÞ and the right one X2 isdistanced by d at ð�0:75aþ d; 0; aÞ; their upper faces are parallelto the surface plane z = 0 with the depth of 0.5a. The normal pres-sure p on the surface is uniformly distributed in a square area of4a � 4a centered at (0,0,0). In this paper, three normal stresscomponents along the x, y and z-axis directions are indicated byrx, ry and rz, respectively.

The subsurface stress field is first studied using both the presentmethod and the FEM. Using the present method, the computationaldomain is discretized into 88 � 88 � 64 = 495,616 cuboidal ele-ments and deep convergence is achieved after six iteration steps.

Fig. 3. Schematic of two cuboidal inhomogeneities underneath a surface subject to a uniform normal pressure.

Fig. 4. Comparison of the subsurface stress components along the x0-axis betweenthe present method and the FEM.

-2 -1 0 1 2-0.4

-0.3

-0.2

-0.1

0.0

x/a

σ x/p

d = 0.1a

d = 0.5ad = ∞ (single)

Fig. 5. Comparison of the normal stress component rx for different values of thedistance a between two inhomogeneities X1 and X2.

x

y

z

O

R

1Ω2Ω

C xc

yc

zc

Surface plane

Fig. 6. Schematic of a hemispherical double inhomogeneous inclusion, embeddedin a semi-infinite matrix, consisting of a cuboidal cavity X2 and its surroundinginhomogeneous inclusion X1.

K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33 29

The FEM takes 814,128 8-node elements using ANSYS software.The two methods mesh the domain of interest to about the samefineness, but the FEM takes much more elements in total becauseit needs extra elements to model the far field of the half space.

Fig. 4 illustrates the normal stresses along the x0-axis, which is par-allel to the x-axis and through the centers of the two cuboids, forthe case of d = 0.5a. The magnitudes of the stresses are normalizedby the surface pressure p. It shows that the results obtained by thetwo methods agree well, thus validating the present method. Thestresses are non-uniform within the cuboidal inhomogeneities.The stress component rx normal to the interfaces between inho-mogeneities and the matrix is continuous across them, while thecomponents ry and rz tangential to the interfaces are discontinu-ous across them.

The normal stress component rx is further studied for differentvalues of the distance d between X1 and X2 using the presentmethod, as shown in Fig. 5. The case of d =1means that only a sin-gle inhomogeneity X1 is present beneath the loaded surface. Itshows that due to the presence of X2 (d = 0.5a and d = 0.1a) orthe interaction between X1 and X2, the stresses within X1 are sig-nificantly changed. As the two inhomogeneities get closer, theirinteraction becomes stronger. Since rx is continuous across theinterfaces, the smaller the distance d becomes, the less the stressrx would change from the region between X1 and X2 to the regionwithin X1. When they are close to touch each other, they begin tofunction like a larger inhomogeneity. It is necessary to note thatthe stress filed for d = 0.1a is not symmetric about the plane x = 0because X1 and X2 are not symmetric about it.

30 K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33

3.2. A cuboidal cavity in a hemispherical inhomogeneous inclusion

Fig. 6 shows a hemispherical double inhomogeneous inclusion,which consists of a cuboidal cavity X2 and its surrounding inhomo-geneous inclusion X1 with initial eigenstrain, beneath the surfaceof a half space. The hemisphere has radius R and the spherical cen-ter is located at Oð0; 0; R=4Þ with the hemispherical plane parallelto the half-space surface, bounded by the plane z = 0 in the x–y–zCartesian coordinate system. The cuboidal cavity has the dimen-sion cx = cy = R/4, cz = R/2, and is centered at ð0; 0; 3R=4Þ with itsedges parallel or perpendicular to the z-axis. It is apparent thatthe inhomogeneous inclusion X1 possesses a complicated geomet-ric shape. The semi-infinite matrix has Young’s modulus E ¼210 GPa and Poisson’s ratio v = 0.28; the inhomogeneous inclusionX1 has E1 = 420 GPa and v1 = 0.24. Assuming that X1 contains uni-form dilatation eigenstrains ex = ey = ez = aT with T and a being thetemperature and thermal expansion, respectively, we investigatethe stress field inside and outside the double inhomogeneousinclusion. The computational domain is discretized into 116 �116 � 68 = 915,008 cuboids and deep convergence is achievedafter more than 20 iteration steps. Large material discrepancydue to the presence of the civility causes early divergence in theCGM computation. To avoid the early divergence, the modifiedCGM functions is used to slow down the convergence speed and re-sults in more iteration steps.

Fig. 7a presents the normal stress at the surface along the x-axis.The magnitudes of the stress components are normalized byro = EaT/3(1 � v) and show that the surface is free of normal trac-tion rz, thus satisfying the free surface boundary condition. Thestress components rx and ry are symmetric about the line x = 0and have their maximum value at the surface point (0,0,0) abovethe hemispherical center, where rx and ry become identical. It isnoted that there is a sign change for rx along the x-axis, indicating

Fig. 7. (a) Surface stresses along the x-axis, and (b) subsurface stresses along the z-axis (the depth direction).

that the surface experiences the transition from tensile stress tocompressive stress. The subsurface stresses are also investigated,as shown in Fig. 7b, which plots the normal stress along the z-axisand shows that the cuboidal cavity is free of stress and the pres-ence of the cavity and that the half-space surface strongly distortsthe stress field inside and outside X1. The component rz is contin-uous across the interfaces between X1 and the matrix, thus satis-fying the continuity condition, while rx and ry which areidentical there, have a jump across the interfaces. As the depth in-creases, the stresses tend to zero. This study demonstrates that thepresent method is capable of investigating half-space problemsconcerning cavities and inhomogeneous inclusions of arbitraryshape as well as the interaction between them.

3.3. A stringer of inhomogeneities

In most cases, inhomogeneities are not present as isolated onesin matrix materials. Instead, they agglomerate into stringers orclusters during their formation process. To demonstrate this fea-ture, Fig. 8 shows the backscattered electron image of a stringerof Al2O3 inhomogeneities formed in steel [40]. Individual

Fig. 8. A stringer of Al2O3 inhomogeneities in steel observed using electronmicroscopy [40].

R

z

x

Normal pressure p(x, y)

xc

yc

zc

Cuboidal inhomogeneity

O (y) h

Friction force f (x, y)

w

1Ω 2Ω 3Ω

iΩ

Surface plane

Fig. 9. Schematic of a stringer of three cuboidal inhomogeneities underneath asurface to which a distributed load is applied.

K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33 31

inhomogeneities may appear in the shape of cuboids. For example,cuboid-shaped Al2O3 and SiC inhomogeneities were observed insteel with scanning electron microscopy [41]. Here, the study con-cerns a stringer of cuboidal inhomogeneities beneath a half-spacesurface under load, as schematized in Fig. 9. The semi-infinite ma-trix is steel with E = 210 GPa and v = 0.28; the cuboidal inhomoge-neities are formed by Al2O3 with EX = 344 GPa and vX = 0.25. Thus,compared with the steel matrix, Al2O3 inhomogeneities are stiff.The half-space surface is bounded by the plane z = 0 in the x–y–zCartesian coordinate system. The normal load p(x,y) on the surfaceis prescribed as a Hertz distribution. The pressure in the circulararea of radius R is given by p(x,y) = po(1 � (x/R)2 � (y/R)2)1/2, wherepo is the maximum pressure at the center located at O(0,0,0). Thethree cuboids Xi (i = 1, 2, 3) are taken to be identical and have thedimension cx = cy = cz = 0.5R and their centers are aligned in they = 0 plane along a line parallel to the x-axis. The upper faces ofthe cuboids Xi are parallel to the surface plane z = 0 and theirdistance to the surface measures the depth h of Xi. The cuboidsXi are equally spaced by w = 0.125R with the center of X2 rightunder the origin O. The computational domain is discretized into

Fig. 10. Comparison of the normalized von Mises stress in the central plane y = 0 fordifferent depths h of the inhomogeneity stringer underneath the surface z = 0subjected to a Hertz pressure in the area of radius R.

164 � 164 � 88 = 2,366,848 cuboids and deep convergence isachieved after about six iteration steps.

The depth effect of the inhomogeneity stringer on the stressfield is first investigated. Fig. 10 presents the von Mises stress rv

normalized by po in the central plane y = 0 for three values of thedepth h. For comparison purposes, the homogenous half-space isrepresented as h =1 in Fig. 10a and shows that, consistent withthe Hertz solution, the maximum normalized von Mises stressrmax

v reaches 0.629 at A1 (0,0,0.47R) located under the acting pointof the maximum Hertz pressure po. This stress field is dramaticallychanged by the presence of the inhomogeneity stringer, as shownin Figs. 10b and c. For the case with h = 0.75R, it is observed thatrmax

v occurs in the inhomogeneities rather than in the matrix,and reaches about 0.718 at point B1 in X1 and point B2 in X3, whichare near the edges of the cuboids and symmetric about the planex = 0 (Fig. 10b). The value of rv at point A1 also increases to0.665. For the depth closer to the surface with h = 0.25R, rmax

v in-creases to 0.827 and appears in multiple locations at points B3,B4, B5, and B6 in X2. All four points are near the edges of X2 andsymmetric about the planes x = 0 and y = 0, and their projectionson y = 0 are denoted in Fig. 10c. It is noted that the surface pressureis prescribed and assumed not be affected by the presence of inho-mogeneities near the surface.

The case with h = 0.25R is further studied by incorporating afriction force acting on the surface. For simplification, the frictionforce is taken to be proportional to the normal pressure, i.e.,f(x,y) = lp(x,y) with l being the friction coefficient. The results ofthe analysis suggest that as the friction force increases, rmax

v in-creases and its location may move towards the surface and thedirection of the friction force. At l = 0.1, rmax

v increases to 0.847at points B5 and B6 (Fig. 11a); at l = 0.2, it reaches 0.868 at twopoints above B5 and B6 in X2; at l = 0.3, it increases to 0.892 atpoints B7 and B8 in X3 (Fig. 11b). Points B7 and B8 are near the edgeof X3 and symmetric about the plane y = 0, and their projections ony = 0 are denoted in Fig. 11b.

Fig. 11. Effect of the friction force on the normalized von Mises stress for the case ofh = 0.25R, compared with the frictionless case in Fig. 10c.

Fig. 13. Normalized von Mises stress in the planes z = 0.25R (a) and z = 0.5R (b)through the inhomogeneity cluster.

32 K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33

The study shows that the presence of stiff inhomogeneitiescauses stress increase in the regions where they are formed. Inmaterials containing stiff inhomogeneities, stress concentrationgenerally occurs within the inhomogeneities, particularly neartheir edges or corners, as demonstrated in Figs. 10 and 11. Thestress concentration increases as the location of the inhomogenei-ties approaches the material surface and as the friction force in-creases. These stress concentrations may cause subsurface cracknucleation or plastic deformation, leading to material failure. Thus,during manufacturing process, it is desired to avoid the formationof stiff inhomogeneities near the material surface as well as reducethe material surface roughness.

Furthermore, the interaction of the inhomogeneities was alsostudied by varying the spacing between two cuboidal inhomogene-ities. The interaction effect was found significant until the spacingbetween two cuboids becomes larger than four times their sidelength. For the spacing exemplified in Figs. 10 and 11, there thusexist strong interactions between the three cuboidal inhomogene-ities. The interaction of inhomogeneities is another factor thatsignificantly affects the location and magnitude of stressconcentration.

3.4. A cluster of inhomogeneities

Fig. 12 shows the schematic of a square cluster of nine cuboidalAl2O3 inhomogeneities underneath the surface of a semi-infinitesteel matrix, bounded by the plane z = 0 in the x–y–z Cartesiancoordinate system. The surface is subjected to the normal pressurep(x,y) in the area of radius R centered at (0,0,0) and the frictionforce f(x,y), as defined in Section 3.2. The nine cuboids Xi

ði ¼ 1; . . . ; 9Þ are identical with cx = cy = cz = 0.5R and equallyspaced by w = 0.125R along the x and y axes. The central cuboidin the square cluster is beneath the surface origin O(0,0,0) andthe depth of the cluster is denoted by h. The computational domainis discretized into 164 � 164 � 80 = 2,151,680 cuboids and deepconvergence is achieved after about six iteration steps.

Fig. 13 plots the normalized von Mises stresses rv in the planesz = 0.25R and z = 0.5R through the inhomogeneity cluster for thecase of h = 0.25R and l = 0.3. The stresses rv are symmetric aboutthe plane y = 0, as predicted from the geometric symmetry of thesquare cluster, but they are not symmetric about the plane x = 0because of the applied friction force along the x direction. The max-imum stress concentration appears at the corner points D1 and D3

in X1 and X3 with rmaxv ¼ 0:898. It is necessary to note that D1 and

D3 correspond to the centers of the cuboids that form the cornersof the inhomogeneity. Together with the results in Fig. 11, this fact

Fig. 12. Schematic of a square cluster of nine cuboidal inhomogeneities underneatha surface subjected to normal pressure and friction force.

further demonstrates that the surface friction force strongly affectsthe location and magnitude of subsurface stress concentration.Since every pair of cuboidal inhomogeneities are within the inter-action range, the stress concentration at any point is an outcome ofall the interactions among the nine cuboids.

4. Conclusion

This paper develops a numerical method for solving multiple,arbitrarily-shaped 3D inhomogeneous inclusions embedded in anisotropic half space using the EIM. The half space can be subjectedto a remote load at infinity or a combination of normal and tangen-tial forces at the half-space surface. The solution takes into accountthe interactions between all the inhomogeneous inclusions andthus could provide an accurate stress field for the assessment ofmaterial strength and reliability.

The merit of this method is that it could handle the interactionsof many multiple inhomogeneous inclusions in the same way asfor a single inhomogeneous inclusion without increasing computa-tional complexity. The key issue in the method is whether equiva-lent unknown eigenstrains can be properly determined. Todemonstrate this, the present method is validated by comparingits results for two interacting inhomogeneous inclusions withthose obtained by finite element method.

K. Zhou et al. / Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 25–33 33

The solution is general and robust, and has been applied toinvestigate the elastic fields of a cavity-contained semi-sphericalinhomogeneous inclusion, a stringer of cuboidal inhomogeneities,and a cluster of cuboidal inhomogeneities embedded in a halfspace. The solution has potential application in addressing chal-lenging material science and engineering problems concerninginelastic deformation and material dissimilarity.

Acknowledgements

The authors would like to acknowledge the Timken Companyfor financial support to the Center for Surface Engineering and Tri-bology at Northwestern University, USA. K.Z. also acknowledgesthe Start-up Grant from the Nanyang Technological University,Singapore.

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