Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2011; 87:617–638Published online 21 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3117

Semi-analytic solution for multiple interacting three-dimensionalinhomogeneous inclusions of arbitrary shape in an infinite space

Kun Zhou∗,†,‡, Leon M. Keer and Q. Jane Wang

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston,

IL 60208, U.S.A.

SUMMARY

This paper develops a semi-analytic solution for multiple arbitrarily shaped three-dimensional inhomoge-neous inclusions embedded in an infinite isotropic matrix under external load. All interactions betweenthe inhomogeneous inclusions are taken into account in this solution. The inhomogeneous inclusions arediscretized into small cuboidal elements, each of which is treated as a cuboidal inclusion with initialeigenstrain plus unknown equivalent eigenstrain according to the Equivalent Inclusion Method. All theunknown equivalent eigenstrains are determined by solving a set of simultaneous constitutive equationsestablished for each equivalent cuboidal inclusion. The final solution is obtained by summing up theclosed-form solutions for each individual equivalent cuboidal inclusion in an infinite space. The solutionevaluation is performed by application of the fast Fourier transform algorithm, which greatly increasesthe computational efficiency. Finally, the solution is validated by taking Eshelby’s analytic solution of anellipsoidal inhomogeneous inclusion as a benchmark and by the finite element analysis. A few sampleresults are also given to demonstrate the generality of the solution. The solution may have potentiallysignificant applications in solving a wide range of inhomogeneity-related problems. Copyright � 2011John Wiley & Sons, Ltd.

Received 17 July 2009; Revised 12 November 2010; Accepted 27 November 2010

KEY WORDS: inhomogeneous inclusion; inhomogeneity; arbitrary shape; three-dimensional; equivalentinclusion method; fast Fourier transform

1. INTRODUCTION

The study of inhomogeneities, which have different elastic moduli than their surrounding materialor matrix, is of prime significance as many engineering materials contain inhomogeneities, e.g.reinforcing fibers/particles, precipitates, and voids/cracks. The presence of inhomogeneities greatlyinfluences the mechanical and physical properties of such materials at the local and global scales[1, 2]. In order to better understand the material behavior, the knowledge of the entire elastic fieldcaused by the inhomogeneities is required.

An inhomogeneity may also contain eigenstrain, which refers to non-elastic strain, such asthermal expansion, phase transformation, plastic strain, or misfit strain [3]. Following Mura’s [3]definition, such an inhomogeneity is called an inhomogeneous inclusion. In contrast, an inclusionis of the same material as the matrix but contains eigenstrain. A few analytical solutions have beenobtained for inclusions of simple geometry embedded in an infinite isotropic medium, such as the

∗Correspondence to: Kun Zhou, Department of Mechanical Engineering, Northwestern University, 2145 SheridanRoad, Evanston, IL 60208, U.S.A.

†E-mail: [email protected]‡Present address: School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang

Avenue, Singapore 639798, Singapore.

Copyright � 2011 John Wiley & Sons, Ltd.

618 K. ZHOU, L. M. KEER AND Q. J. WANG

ellipsoidal inclusion [4], the cuboidal inclusion [5], and the cylindrical inclusion [6, 7]. A compre-hensive report about the inclusion problems can be found in two review papers by Mura [8, 9].

In Eshelby’s [4] pioneering work, an ellipsoidal inhomogeneous inclusion in an infinite spacewas solved using the Equivalent Inclusion Method (EIM), which assumes that an inhomogeneousinclusion can be simulated as an inclusion with initially prescribed eigenstrain, plus equivalenteigenstrain. In the ellipsoid, the equivalent eigenstrains and the stresses were found to be uniformif the initial eigenstrains are uniform. Since then, the EIM has been used intensively to addresstwo-dimensional (2D) and three-dimensional (3D) infinite space problems. Johnson et al. [10, 11]determined the strain field due to a cuboidal inhomogeneity by expressing the equivalent eigenstrainin a polynomial form. The stress field due to an arbitrarily shaped 2D inhomogeneity was studiedby establishing the equations in triangular polar coordinates to avoid the 1/r singularity [12]and by means of the regularized domain integral formulation in conjunction with the subdividetechnique to address the same singularity [13]. In addition to the isotropic problems mentioned, aspheroidal inhomogeneity with arbitrary orientation in a transversely isotropic medium was studiedby Kirilyuk and Levchuk [14].

Since in most cases inhomogeneities are not present as solitary ones, the EIM was alsoused to investigate interactions between them. The seminal work was done by Moschovidis andMura [15], who studied two non-intersecting ellipsoidal inhomogeneities by approximating theequivalent eigenstrains with a Taylor series. Other works that concern non-intersecting inho-mogeneities include the studies of two spherical inhomogeneities [16], a penny-shaped crackinteracting with a spherical inhomogeneity [17, 18], and multiple spherical inhomogeneities [19].The double-inhomogeneity problem in which one inhomogeneity is surrounded by another wasinvestigated to simulate a single coated particle/fiber embedded in a matrix [20]. Periodicallydistributed double-inhomogeneities in a composite were also studied by Shodja and Roumi [21].In addition, a composite with multiple interacting inhomogeneities was simulated by multiplenon-interacting double inhomogeneities embedded in an infinite medium having the average prop-erties of the composite [22–24]. The average elastic field due to an infinite number of periodicallydistributed elliptic, spherical and cylindrical cracks was also investigated with the EIM and Fourierseries [25, 26].

A few other approaches offer alternatives to the EIM for the study of inhomogeneities in aninfinite space. Using Muskhelishvili’s [27] complex potentials, multiple planar cracks or circularholes [28] and elliptic inhomogeneities [29] were investigated. The multipole expansion techniquewas proposed to study aligned spheroidal voids as well as the interaction between spheroidalvoids and penny-shaped cracks by adding up the solution for each single-inhomogeneity problemin the local spherical coordinate system [30, 31]. The Taylor series expansion was used to solveintegral equations for the strain field in a cuboidal inhomogeneity [32] and in a cuboidal inhomo-geneous inclusion [10, 11]. The volume integral equation technique was used to investigate twocoated cylindrical inhomogeneities and multiple planar cracks by Lee and Mal [33, 34]. Using thesame technique, Dong et al. [35] investigated the stress field of single spherical, tetrahedral, andhexahedral inhomogeneities and the interaction between two identical spherical inhomogeneities.

It is notable that the abovementioned works addressed inhomogeneity problems that were limitedto certain categories. First, most studies focused on regularly shaped inhomogeneities and only afew were for arbitrarily shaped 2D inhomogeneities. Practically, no solutions for inhomogeneitiesof arbitrary 3D shape are given. Second, since there is tremendous difficulty in handling inter-actions among three or more inhomogeneities, most investigations concerned only one or twoinhomogeneities. Third, the interactions between every two inhomogeneities for multiple inhomo-geneities in composites were not accounted for. Fourth, either prescribed initial eigenstrains ininhomogeneities or external loading acting on the matrix was considered and few investigationsdealt with a combination of them. Fifth, the applied stresses due to external loading in most caseswere assumed to be uniform. However, the applied stresses are non-uniform in many practicalcases; for example, in the study of the mechanical response to remote loading of an inhomogeneityembedded in a medium, the applied normal stresses are assumed to be linear through the thicknessof the inhomogeneity. This investigation is devoted to solving a comprehensive inhomogeneityproblem that goes beyond the above limitations, by using the technique of EIM. Here, the solution

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2011; 87:617–638DOI: 10.1002/nme

SEMI-ANALYTIC SOLUTION FOR INHOMOGENEOUS INCLUSIONS 619

to multiple interacting arbitrarily shaped 3D inhomogeneous inclusions embedded in an infinitematrix and subjected to uniform/non-uniform applied stresses is developed. Owing to the gener-ality of the formulated problem, the solution has potentially significant applications in solving awide range of inhomogeneity-related problems.

The paper is organized as follows: the problem formulation and solution are given in Section 2,the numerical results are presented in Section 3, and the conclusions are drawn in Section 4.

2. PROBLEM FORMULATION AND SOLUTION

This section is divided into two subsections. In Section 2.1, the governing equation for solvingthe inhomogeneity problem is formulated. In Section 2.2, the numerical method for solving thegoverning equation is described.

2.1. Governing equation

Consider n arbitrarily shaped 3D subdomains �� (�=1,2, . . . ,n) with elastic moduli C�ijkl (i, j,k, l =

1,2,3) embedded in an infinite matrix with elastic moduli Cijkl, with respect to an x − y−z Carte-sian coordinate system (Figure 1(a)). Each subdomain �� can be an isolated one or surroundedby others. For generality, the subdomains �� are assumed to contain initial eigenstrain ε

pij and are

therefore considered as inhomogeneous inclusions. The inhomogeneous inclusions �� may alsobe subjected to uniform or non-uniform applied stresses �0

ij due to the external load acting on thematrix. The external load can be a force P, moment M, or a combination of them.

Using the EIM, each inhomogeneous inclusion �� can be simulated by a (homogeneous) inclu-sion in the matrix with initial eigenstrain ε

pij plus equivalent eigenstrain ε∗

ij. In this way, theinhomogeneous inclusion problem in Figure 1(a) is equivalently converted into the (homogeneous)inclusion problem in Figure 1(b). The equivalent eigenstrains ε∗

ij are introduced to represent thematerial differences between the inhomogeneous inclusions and their surrounding matrix, the inter-actions among them, and their response to the external load and the initial eigenstrains ε

pij they

contain. When the equivalent eigenstrains ε∗ij are non-uniform in an equivalent (homogenous) inclu-

sion, one may determine the equivalent eigenstrains pointwise within the equivalent (homogenous)inclusion.

In the original inhomogeneous inclusion problem (Figure 1(a)), the total strain εij at any pointwithin the inhomogeneous inclusions �� contains two parts: the elastic strain εe

ij and an initial

eigenstrain εpij , where the elastic strain is given by εe

ij =εij −εpij . According to Hooke’s law, the

following equation can be established for any point inside the inhomogeneous inclusions ��:

�ij =C�ijkl(εkl −ε

pkl) or εij =C�−1

ijkl �kl +εpkl (�=1,2, . . . ,n; i, j,k, l =1,2,3). (1)

z

xy

O

Ω,C

,C

,C

Ω

Ω

C

z

xy

O

Ω

,,C

Ω

Ω

C

,,C

,,C

PP

M M

(a) (b)

Figure 1. (a) Schematic of multiple arbitrarily shaped 3D inhomogeneous inclusions ��(�=1,2, . . . ,n)with elastic moduli C�

ijkl and initial eigenstrains εpij embedded in an infinite matrix with elastic moduli

Cijkl (i, j,k, l =1,2,3) under external load and (b) Using EIM, each inhomogeneous inclusion �� in (a)is equivalently treated as an inclusion with ε

pij plus equivalent eigenstrain ε∗

ij.



It is necessary to note that Equation (1), which is in tensor form, contains six equations due tothe fact that �ij =�ji (i, j =1,2,3) as do the subsequent relative equations.

In the equivalent (homogenous) inclusion problem (Figure 1(b)), the elastic strains εeij in the

equivalent inclusions are obtained by εeij =εij −ε∗

ij −εpij . Therefore, according to Hooke’s law, the

following equation is also established for any point in each equivalent inclusion ��:

�ij =Cijkl(εkl −ε∗kl −ε

pkl), (i, j,k, l =1,2,3). (2)

Substituting Equation (1) into Equation (2) in

C�ijkl(εkl −ε

pkl) = Cijkl(εkl −ε∗

kl −εpkl) or �ij =Cijkl(C

�−1klmn�mn −ε∗

kl)

(�=1,2, . . . ,n; i, j,k, l,m,n =1,2,3). (3)

The second formula in Equation (3) gives the relationship between the stress �ij and equivalenteigenstrain ε∗

ij in ��.On the other hand, the stress �ij at any point in each equivalent inclusion �� (Figure 1(b)) can

be obtained by

�ij =�∗ij +�p

ij +�0ij (i, j =1,2) (4)

where �∗ij is the eigenstress caused by the equivalent eigenstrains ε∗

ij in all the inclusions ��, �pij

the eigenstress caused by the initial eigenstrains εpij in all the ��, and �0

ij the applied stress dueto external load. The substitution of Equation (4) into the second formula in Equation (3) and thesubsequent arrangement result in

�∗ij −Cijkl(C

�−1klmq �∗

mq −ε∗kl) = CijklC

�−1klmq (�p

mq +�0mq)−�p

ij −�0ij

(�=1,2, . . . ,n; i, j,k, l,m,q =1,2,3). (5)

The right-hand side of Equation (5) contains the applied stress due to external load and theeigenstress due to the initial eigenstrain, while the left side contains the equivalent eigenstrain andthe eigenstress due to it. This arrangement for Equation (5) reflects the intrinsic characteristics ofinhomogeneities, i.e. it is in response to external load and/or initial eigenstrain that stress occursin the inhomogeneities. From Equation (5) it is apparent that if the external load and the initialeigenstrain both are zero, then the equivalent eigenstrain and the eigenstress due to it would bezero; consequently, the total stress in �� would be zero.

Equation (5) is the governing equation for solving the inhomogeneous inclusion problem, inwhich the equivalent eigenstrains ε∗

ij are unknowns. However, it is not solvable until the relationshipsbetween �∗

ij at a given point and the equivalent eigenstrains ε∗ij in all the equivalent inclusions ��

and between �pij at a given point and the initial eigenstrains ε

pij in all the �� are found. Therefore,

a numerical method for determining such relationships and then solving Equation (5) is introducedin the next section.

2.2. Numerical method for solving the governing equation

A cuboidal domain D is chosen which contains the n arbitrarily shaped inhomogeneous inclusions�� (�=1,2, . . . ,n) in an infinite space under investigation, as illustrated in Figure 2. The originO(0,0,0) of the x − y−z Cartesian coordinate system is set at one corner of the domain D. Thedomain D is discretized into Nx × Ny × Nz cuboidal elements of the same size 2�x ×2�y ×2�z ,each of which is indexed by a sequence of three integers [�,�,�] with 0��Nx −1, 0��Ny −1, and 0��Nz −1. Each �� (�=1,2, . . . ,n) within D can therefore be approximated by acollection of such cuboidal elements. Apparently, the smaller the cuboidal element or the finer thediscretization, the more accurate the geometric approximation will be.



x

y

O

z

2Ω

Ω

1Ω3Ω

D

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

Cuboid [0, 0, 0]

Figure 2. Discretization of the domain D, which contains n arbitrarily shaped inhomogeneous inclusions��(�=1,2, . . . ,n) in an infinite space, into Nx × Ny × Nz cuboids.

As done in Section 2.1, each cuboidal inhomogeneous inclusion inside �� (�=1,2, . . . ,n) isalso simulated by a cuboidal inclusion with initial eigenstrain plus equivalent eigenstrain. Therefore,according to Equation (5), the following equation should be satisfied for any point in the cuboidsinside ��(�=1,2, . . . ,n):

r∗−C(C�−1r∗−e∗)=CC�−1(rp +r0)−rp −r0 (�=1,2, . . . ,n), (6)

where C, C�, e∗, r∗, rp, and r0 are the matrix forms of the parameters introduced in Equation (5).For simplification, we may treat the eigenstrain in each cuboidal inhomogeneous inclusion

as uniform, provided that the size of each cuboid is small enough. Here, it is noted that theeigenstrains in any �� that is composed of multiple cuboidal inhomogeneous inclusions can stillbe non-uniform. Thus, each equivalent cuboidal inclusion will have only one unknown equivalenteigenstrain e∗ or six unknown equivalent eigenstrain components due to ε∗

ij =ε∗ji (i, j =1,2,3).

In order to solve the unknowns, the number of simultaneous equations to be established shouldequal that of the cuboidal inhomogeneous inclusions inside �� (�=1,2, . . . ,n); i.e. Equation(6) is established only at one point in each cuboidal inhomogeneous inclusion. For convenience,hereafter such a point is called the observation point. In the domain D, the observation pointsare selected to be at the center of each cuboid in order that the fast Fourier transform algorithm(FFT) can be applied for improving computational efficiency. The FFT algorithm requires thatthe observation points should be equally distributed along the direction in which the algorithmwould be applied [36]. As a result, Equation (6) is established at the center of each cuboidalinhomogeneous inclusion as

r∗�,�,�−C(C−1�,�,�r

∗�,�,�−e∗�,�,�)=CC−1

�,�,�(rP�,�,�+r0

�,�,�)−rP�,�,�−r0

�,�,�,

(C�,�,� ∈{C�}(�=1,2, . . . ,n); 0��Nx −1,0��Ny −1,0��Nz −1), (7)

where C�,�,� denotes the elastic moduli of the cuboidal inhomogeneous inclusion [�,�,�] centeredat (x�, y�, z�) and it apparently equals one of the elastic moduli C�(�=1,2, . . . ,n). It should benoted that six equations are contained in the matrix form of Equation (7) for each inhomogeneouscuboidal inclusion.

Chiu [5] obtained a closed-form solution for the eigenstress field due to a cuboidal inclusionwhich contains uniform eigenstrain and lies in an infinite space. In Chiu’s [5] solution, the eigen-stress at an observation point (x, y, z) is related to the eigenstrain in a cuboidal inclusion centeredat the origin (0,0,0) by the influence coefficient Bijkl(x, y, z) as below

�pij (x, y, z)= Bijkl(x, y, z)ε p

kl(0,0,0) (i, j,k, l =1,2,3) (8)



The analytic form of Bijkl(x, y, z) is given in Appendix A. Note that Bijkl contains the informationabout material moduli of the infinite space.

The eigenstress rp at an observation point due to the initial eigenstrains in all the inhomoge-neous inclusions �� (�=1,2, . . . ,n) can be obtained by adding up the contribution of the initialeigenstrain in each cuboidal element in the domain D as

rp�,�,� =

Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�ep,,�

(0��Nx −1,0��Ny −1,0��Nz −1), (9)

where B�−,�−,�−� is the 6×6 matrix form of the influence coefficients Bijkl that relate the stressr

p�,�,� at the observation point (x�, y�, z�) in the cuboidal element [�,�,�] to the initial eigenstrain

ep,,� in the cuboidal element [,,�] centered at (x, y, z�); the subscript �−, �−, �−� means

that the influence coefficient B�−,�−,�−� depends on the relative distance between (x�, y�, z�)and (x, y, z�) along each axis direction. In order to apply the FFT algorithm, summations areperformed on the whole domain D, although many cuboidal elements outside �� (�=1,2, . . . ,n)do not contain initial eigenstrains. Similarly, the eigenstress r∗ at (x�, y�, z�) due to the equivalenteigenstrains in all the equivalent cuboidal inclusions inside ��(�=1,2, . . . ,n) can be obtained by

r∗�,�,� =Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�e∗,,�

(0��Nx −1,0��Ny −1,0��Nz −1). (10)

Equations (9) and (10) give the eigenstress and eigenstrain relationships required for solvingEquation (7). Substituting Equations (9) and (10) into Equation (7) results in

(I−CC−1�,�,�)

Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�e∗,,�+Ce∗�,�,�

= (CC−1�,�,�−I)

(Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�ep,,�+r0

�,�,�

),

(C�,�,� ∈C�(�=1,2, . . . ,n),0��Nx −1,0��Ny −1,0��Nz −1), (11)

where I is a unit matrix. Equation (11) is the final form of the governing equation for solvingthe inhomogeneous inclusion problem. As mentioned, Equation (11) in matrix form contains sixequations for each inhomogeneous cuboidal inclusion, which has six unknown equivalent eigen-strain components. If all the inhomogeneous inclusions �� (�=1,2, . . . ,n) contain Nv cuboidalelements, there would be 6× Nv unknown equivalent eigenstrain components and 6× Nv simulta-neous equations for solving them.

In many cases, the treatment of a cavity as an inhomogeneity might be desired, and Equation(11) will experience a singularity caused by C−1

�,�,�. One can address this issue in two ways. Oneway is to assign C�,�,� in Equation (11) nonzero but small enough values compared with the matrixelastic moduli C. The other approach is to use the transformation of Equation (11) as

(C�,�,�C−1 −I)Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�e∗,,�+C�,�,�e

∗�,�,�

= (I−C�,�,�C−1)

(Nz−1∑�=0

Ny−1∑=0

Nx −1∑=0

B�−,�−,�−�ep,,�+r0

�,�,�

),

(C�,�,� ∈C�(�=1,2, . . . ,n),0��Nx −1,0��Ny −1,0��Nz −1), (12)

which is obtained by multiplying the two sides of Equation (11) by C�,�,�C−1.



It is noted that both Equations (9) and (10) are obtained by direct superposition and indicate nointeraction effect. Equation (9) gives the expression for the stress at an observation point due tothe initial eigenstrains in all the inhomogeneous inclusions and Equation (10) gives the expressionfor the stress at an observation point due to the equivalent eigenstrains in all the inhomogeneousinclusions. However, the governing equation (7) takes into account the interactions between all theinhomogeneous inclusions. The substitution of Equations (9) and (10) into governing equation (7)results in the final governing equation (11) (or governing equation (12)) which contains onlyexplicit unknowns e∗�,�,� and thus becomes solvable with interactions accounted for. The solutionsof unknown equivalent eigenstrains ε∗

ij in each equivalent cuboidal inclusion are directly affectedby the interactions between all the inhomogeneous inclusions, thereby affecting the stresses dueto all the equivalent eigenstrains, as given in Equation (10).

Equation (11) or (12) contains the 3D multi-summations as given in Equations (9) and (10).To solve Equation (11) or (12), such summations need to be dealt with first. In Equation (9)or (10), the stress at an observation point is the sum of the stresses produced by the eigenstrainin each cuboidal element regardless of whether the eigenstrain is zero or not. Therefore, for adomain of having Nx × Ny × Nz elements, the number of the multiplication operations required tocalculate the stress at an observation point is on the order of Nx × Ny × Nz , which for convenienceis denoted by O(Nx × Ny × Nz). In order to obtain the stress at all the observation points in thedomain, O((Nx × Ny × Nz)2) operations are required. This process is called direct multi-summationand is costly in computation time if the domain contains a large number of elements. Since thesummations in Equation (9) and Equation (10) are a discrete convolution of B�,�,� and ep�,�,�and a discrete convolution of B�,�,� and e∗�,�,�, respectively, in order to improve the computa-tional efficiency, a technique of discrete convolution and fast Fourier transform (DC-FFT) can beused [36].

To use DC-FFT, the convolution in Equation (9) is first converted into a cyclic convolution [36].The computational domain D is expanded twice into 2Nx ×2Ny ×2Nz , in which the empty influencecoefficients B�,�,� are filled with the coefficients B�,�,� for the original domain Nx × Ny × Nz usingthe technique of wrap-around order and the empty eigenstrain ep�,�,� is filled by zero-padding [36].Accordingly, Equation (9) is converted into

rp�,�,� =

2Nz−1∑�=0

2Ny−1∑=0

2Nx −1∑=0

B�−,�−,�−�ep,,�

(0��2Nx −1,0��2Ny −1,0��2Nz −1). (13)

The Fourier transform can transfer a function defined in a space domain into a function in afrequency domain. For example, the discrete triple Fourier transform of ep,,� in the expandedspace domain is defined as

ep�,�,� =

2Nz−1∑m=0

2Ny−1∑l=0

2Nx −1∑k=0

ei�m�/Nz ei�l�/Ny ei�k�/Nx epk,l,m

(0��2Nx −1,0��2Ny −1,0��2Nz −1). (14)

The discrete Fourier transform in Equation (14) can be computed by the FFT algorithm whichrequires only O(2Nx ×2Ny ×2Nz ×ln(2Nx ×2Ny ×2Nz)) operations [37]. Thus, the symbol ˆhereafter specifically stands for an FFT operation. Since a convolution operation in the spacedomain can be replaced by a multiplication operation in the frequency domain through Fouriertransform, applying discrete FFT to the two sides of Equation (6) yields

r�,�,� = B�,�,�ep�,�,� (0��2Nx −1,0��2Ny −1,0��2Nz −1). (15)



Each element in r�,�,� is simply the multiplication of two corresponding elements in B�,�,� ande

p�,�,�. Finally, the stress r�,�,� is obtained by applying an inverse FFT on r�,�,�:

r�,�,� =FFT−1(r�,�,�) (0��2Nx −1,0��2Ny −1,0��2Nz −1), (16)

where the symbol FFT−1 stands for an inverse FFT operation. Only the stresses in the orig-inal domain D are retained and the others are discarded. The inverse FFT also requiresO(2Nx ×2Ny ×2Nz ×ln(2Nx ×2Ny ×2Nz)) operations. Since the FFT is performed three timesin total in Equations (15) and (16), the required multiplication operations are O(3×2Nx ×2Ny ×2Nz ×ln(2Nx ×2Ny ×2Nz)). Therefore, the computation time cost for the summation in Equation(9) is greatly reduced by applying the DC-FFT technique. For a computational domain, forexample, in the 128×128×128 discretization, the computation time is reduced by O(5×103)times. Equation (10) can be calculated in the same way as Equation (9).

We turn now to solving Equation (11) (or Equation (10)). Regardless of the complexity inform, Equation (11) is essentially a linear equation and resembles the simple form of Ge∗ =a.The Conjugate Gradient Method (CGM) is a well-established method in solving linear equationsGe∗ =a by means of iteration [38]. When the CGM is used, the iteration convergence to the solutionis absolute provided that G is a symmetric positive matrix. Nevertheless, a deep convergence to thesolution before divergence or oscillation may also be obtained through using the CGM although Gis not a symmetric positive matrix. This advantage has made CGM useful for many applications[39, 40]. Based on the conventional CGM [38], the modified CGM algorithm for solving Ge∗ =ais listed as follows:

d(0) = r(0) =a−Ge∗(0), (17)

(i) = �rT

(i)r(i)

dT(i)Gd(i)

, (18)

e∗(i+1) = e∗(i) + (i)d(i), (19)

r(i+1) = r(i) − (i)Gd(i), (20)

�(i+1) = rT(i+1)r(i+1)

rT(i)r(i)

, (21)

d(i+1) = r(i+1)+�(i+1)d(i), (22)

where the integer number in the subscript indicates the iteration step. The vectors d and r arethe conjugate direction and conjugate residual, respectively; the vector rT is the transpose of r;the variable is the step length in each iteration; the factor � which is no greater than one isnewly introduced into the modified CGM, compared with the conventional CGM, to reduce thestep length before divergence or oscillation begins. When the iteration procedure listed in Equations(17)–(22) is used to solve Equation (11), a deep convergence to the solution can be achievedgenerally in 5–10 steps for most cases except for the ones in which cavities are involved. Insolving cases involving cavities, the step length is reduced through the factor � one step before thedivergence or oscillation begins so that the convergence can remain with a relatively slow speed.The term Ge∗ in Equation (17) is obtained by calculating the left side of Equation (12), whilethe term Gd is calculated by replacing e∗ by d. It is noted that the FFT is used to accelerate thecalculation of Gd as well.

The solution is approximate because the computational domain that contains all the inhomoge-neous inclusions is discretized into many cuboids and the eigenstrain in each cuboid is treated asuniform. However, a good accuracy can be obtained as long as the discretization is fine enough,as demonstrated by the recent work [41] in which the discretization method was used to studymultiple homogeneous inclusions.

The accuracy of the solution is not compromised by the way that the present method treats theinteractions amongst the inhomogeneous inclusions. In fact, the solution fully takes into account



the interactions between all the inhomogeneous inclusions, which is different from the way thataverage interactions of inhomogeneities in composites are treated, e.g. see [22–24].

It is necessary to highlight that the present method can be conveniently applied to handle peri-odically distributed inhomogeneities in composites without adding any computational complexity.This is because when the FFT is used, the computational domain is regarded as periodicallydistributed and so do the inhomogeneous inclusions contained in the domain. In order to isolatethe computational domain from the periodic structure, the zero-padding technique [36] is used inthe present solution, as pointed out before. If the zero-padding technique is not taken, the presentmethod can then be extended to treat a periodic array of computational domains. In the extendedmethod, the homogeneous inclusions in each domain not only have interactions amongst them-selves but also interact with the inhomogeneous inclusions in the neighboring domains. However,there is no interaction between the inhomogeneous inclusions that are located in non-neighboringcomputational domains. The extended method can be used to investigate composites of periodicstructures as the method of periodic cells or subcells (see e.g. [25, 26, 42–44]). However, thereis a difference between them. For instance, Walker et al. [44] first homogenized and obtainedthe overall macroscopic response of the composite, and then calculated the local response in andaround the inhomogeneities in a unit periodic cell. In the calculation of the local elastic field inthe unit cell, an integral equation approach to determine the total strain increment was used andthe Fourier series and Green’s function were used to construct the integral equations. In contrast,the extended method from the present one would solve the overall response of the compositeand the local response in one computational domain simultaneously. The extended method wouldsolve unknown equivalent eigenstrains periodically distributed in the composite by a system ofsimultaneous equations and then superimposes the eigenstress due to all the equivalent eigenstrainsbased on the eigenstress–eigenstrain relationship obtained by Chiu [5].

The novelty of the present method is its application of the concept of the Eshelby’s EIM,although bypassing Eshelby’s tensor [4]. Eshelby’s tensor gives the relationship between the totalstrain and eigenstrain and the difficulty of its determination is increased tremendously when theinteraction of two inhomogeneous inclusions is involved. In our method, the determination ofEshelby’s tensor is avoided.

The present method has the strength to handle the interactions of many multiple inhomogeneousinclusions as conveniently as to deal with a single inhomogeneous inclusion without increasingcomputational complexity. The reason for this improvement is because no matter how many inho-mogeneous inclusions are involved, they are all decomposed into many cuboidal inhomogeneousinclusions at the first step. This approach directly handles the cuboidal inhomogeneous inclusionsand also solves the unknown equivalent eigenstrains within the cuboidal inclusions. Whether thesecuboidal inclusions are decomposed from a single or multiple spaced inhomogeneous inclusionsdoes not create any difference to the solution process. However, the final solution of the equivalenteigenstrains reflects their difference. It is the values of the equivalent eigenstrains that reflect theinteractions of all inhomogeneous inclusions, their material dissimilarity and their response to theexternal load. Therefore, the key issue in our method is whether equivalent unknown eigenstrainsin each cuboidal inclusion can be properly determined.

3. SAMPLE RESULTS

In this section, the developed solution is validated and used to study the problems concerning twospaced inhomogeneities, a double inhomogeneity, a stringer of inhomogeneities, and a cluster ofinhomogeneities, respectively.

3.1. A single ellipsoidal inhomogeneous inclusion

To verify the present method, Eshelby’s [4] analytic solution for an ellipsoidal inhomogeneousinclusion in an infinite matrix was used as a benchmark. Here, we present the verification by



Table I. Comparison between the present numerical solution and Eshelby’s [4] analytic solution for theequivalent eigenstrains in an ellipsoidal inhomogeneous inclusion.

Numerical solution Eshelby’s solution

�=4×10−6m �=2×10−6m �=1×10−6m

ε∗11 −4.1311×10−3 −4.1320×10−3 −4.1336×10−3 −4.1377×10−3

ε∗22 2.4235×10−3 2.4121×10−3 2.4066×10−3 2.4022×10−3

ε∗33 2.4235×10−3 2.4121×10−3 2.4066×10−3 2.4022×10−3

ε∗12 5.9132×10−10 −9.0563×10−12 −2.1951×10−9 0

ε∗13 1.9131×10−10 1.1819×10−10 −1.3271×10−9 0

ε∗23 1.7080×10−10 2.9594×10−10 −1.0484×10−9 0

n 2103 17 071 137 059 N/Ai 5 5 5 N/At 1.2 s 18.6 s 172.1 s N/A

a sample case in which the ellipsoidal inhomogeneous inclusion contains uniform dilatationaleigenstrains ε

p11 =ε

p22 =ε

p33 =0.01 and is embedded in an infinite matrix under uniform strain

ε011 =0.01. The inhomogeneous inclusion and the matrix have elastic moduli E1 =420 GPa, v1 =0.3

and E =210 GPa, v=0.3, respectively, where E denotes Young’s modulus and v Poisson’s ratio.According to Hooke’s law, the matrix under the strain ε0

11 is subjected to stresses �011 =2.8269 GPa,

�022 =�0

33 =1.2115GPa. The applied uniform strain can be directly used as input in Eshelby’s [4]solution, while the present numerical solution takes the applied stresses as input. In the numericalcalculations, the three semi-axes of the ellipsoid are set to have the same length R =3.2×10−5 m.In the computational domain, the cuboidal elements are set to have the same side lengths, i.e.2�x =2�y =2�z =�, in order that identical discretization accuracy can be achieved along eachcoordinate direction. To show the effect of the discretization fineness, the ellipsoid is approximatedby the arrangements of cuboidal elements of three different sizes, i.e. �=4×10−6 m, 2×10−6 mand 1×10−6 m, respectively.

The numerical calculations show that the equivalent eigenstrains are almost uniform among allthe cuboidal elements that compose the ellipsoidal inhomogeneous inclusion. The mean values ofthe equivalent eigenstrains were calculated for each of their six components and found to agreewell with those obtained from Eshelby’s [4] solution, as shown in Table I. The agreement improvesas the size of the cuboidal elements decreases from �=4×10−6 m to �=1×10−6 m, since thearrangement of the cuboids becomes geometrically closer to the ellipsoid as the discretizationbecomes finer.

For each discretization size, Table I lists the number n of the cuboids in the ellipsoid, the numberi of the iteration steps used in the calculations, and the time t cost for one iteration step. Thecomputation time was based on a computer with 2.33 GHz CPU and 3 GB memory. If the numberof cuboidal elements is increased by m times along each direction in the domain discretization, thetotal number of elements will be increased by m3 times. This dramatic increase in the number ofelements is the nature of a numerical method that deals with problems concerning configurationsin a 3D space. If the present method is degenerated to solve a 2D problem, the increase would bejust m2 times.

3.2. Two cuboidal inhomogeneities

The inhomogeneities in a matrix may be present in the shape of a cuboid. For example, thecuboid-shaped SiC and Al2O3 inhomogeneities were observed in steel through scanning electronmicroscopy [45]. Figure 3 shows the schematic of two cuboidal inhomogeneities �1 and �2embedded in an infinite matrix. The two cuboids are set to have the same side length a with thex-axis through their centers and parallel or normal to their faces.



x

y

z

1CO

d

2C

1 2

Figure 3. Two cuboidal inhomogeneities �1 and �2 separated by a distance d .

00.8

1.0

1.2

1.4

1.6

x / a

o

Single

ad 5.0=

ad 5.1= ad 3=

ad 4=

1 2

ad 125.0=

2 4 6 8 10

Figure 4. Comparison of the normal stress component �x for different values of the distance d betweentwo interacting cuboidal inhomogeneities �1 and �2.

The cuboid �1 is set to be centered at (1.5a,0,0), while �2 is distanced from �1 by a separationd between their two neighboring faces. Assuming that the matrix is of steel, we first investigate thecase in which both �1 and �2 contain SiC material. The steel matrix has E =210GPa, v=0.3, andthe two cuboidal SiC inhomogeneities have E1 = E2 =410GPa, v1 =v2 =0.16. Thus, comparedwith the matrix, �1 and �2 represent stiff inhomogeneities. A uniform stress �0

x is assumed to beapplied at infinity. The computational domain is discretized into 192×20×20 cuboids. Figure 4studies the effect of the separation distance d between �1 and �2 on their interaction by comparingthe normal stress components �x along the x-axis for different values of d . For the purpose ofreference, the stress is also plotted for the case in which only a single �1 is present, and a stronginteraction between �1 and �2 is seen when the distance d is less than 1.5a. The interactionbecomes weaker as d increases. When d approaches 4a, the stresses inside �1 are almost thesame as those in the presence of only a single �1, which implies negligible interaction. Such adistance as 4a may be called the critical interaction distance, below which the interaction betweentwo neighboring inhomogeneities has to be taken into account in order to precisely describe theirstress field. The critical interaction distance depends on the shapes and sizes of two neighboringinhomogeneities as well as the materials they contain and the matrix material.

Figure 4 also shows that the presence of the stiff inhomogeneities causes the increase of thestresses in the regions where the inhomogeneities are embedded. In materials containing stiffinhomogeneities, high stress concentrations generally occur within the inhomogeneities or in theirvicinities, especially when there are strong interactions between the inhomogeneities, which can beseen, e.g. from the stress curve ‘d =0.5a’ in Figure 4. These stress concentrations may cause cracknucleation or lead to plastic deformation. Thus, during the manufacturing process, it is desired toavoid the formation of stiff inhomogeneities in materials.

The finite element analysis was also used to validate the present method. Figure 5 comparesthe stress components �x along the x-axis obtained by the two different methods for the caseof d =0.5a, which involves a strong interaction between two inhomogeneities. Figure 6 furthercompares the entire stress fields of the central x − y plane for the same stress component between



00.8

1.0

1.2

1.4

1.6

x / a

o

Present method

Finite element analysis

d = 0.5a

2 4 6 8

Figure 5. Comparison of the stress components �x along the x-axis obtained by the present method andthe finite element analysis, respectively.

Figure 6. Comparison of the entire stress fields of the central x − y plane obtained by the present method(a) and the finite element analysis (b), respectively.

the two methods. It shows that the results obtained by the present method agree very well with thatobtained from the finite element analysis. It is necessary to point out that although the legends inFigures 5(a) and (b) are quite similar, there is still difference in their color gradients. This resultsin minor differences between the presentation of the stress contours in Figure 5(a) and that inFigure 5(b), although their stress magnitudes are the same, as demonstrated in Figure 4. The finiteelement analysis was performed using the software of ANSYS.

We now investigate the case in which �1 is formed by SiC material, but �2 is formed byTi-6Al-4V with the other parameters remaining the same as in the previous case. The Ti-6Al-4V inhomogeneity �2 has E2 =110GPa, v2 =0.34. Thus, compared with the steel matrix, �2is a compliant inhomogeneity. The study is concerned about the influence of the presence ofthe compliant inhomogeneity �2 on the stress field inside the stiff inhomogeneity �1. Figure 7presents the normal stress components �x and �y along the x-axis for a sample case in which �1and �2 are spaced by d =0.5a. For comparison, �x and �y are also plotted for the case with theabsence of the compliant inhomogeneity �2 (dashed lines), and the figure shows that the presenceof a compliant inhomogeneity in the vicinity of a stiff inhomogeneity can cause the decrease ofthe stresses inside the stiff inhomogeneity, which implies that in certain circumstances it may befeasible to use compliant inhomogeneities to soften local stress concentrations. Figure 7 also shows



0-0.4

0.0

0.4

0.8

1.2x

o

x / a

Single

1 2 3 4 5

y

Figure 7. The stress components �x and �y for the case in which a stiff cuboidal inhomogeneity�1 interacts with a compliant one �2. For comparison, �x and �y are also plotted for the case in

which only �1 is present (dashed lines).

x

y

z

O

R

xc

yc

1

2

zc

C

Figure 8. Schematic of an embedded double inhomogeneity of semi-spherical shape consisting of acuboidal cavity �2 and its surrounding inhomogeneity �1.

that a uniaxial tensile stress �x causes compressive normal stress �y inside stiff inhomogeneitiesbut tensile stress �y inside compliant ones.

3.3. A cuboidal cavity in a semi-spherical inhomogeneity

Consider a double inhomogeneity of semi-spherical shape, which consists of a cuboidal cavity �2and its surrounding inhomogeneity �1, embedded in an infinite matrix (Figure 8). The semi-sphereis set to have radius R =6.4×10−5 m and the spherical center is located at O(0,0,0). The cuboidalcavity is set to have the size cx = R/2, cy =cz = R/4 and centered at (R/2,0,0) with two oppositefaces normal to the x-axis. The inhomogeneity �1 apparently possesses a complicated geometricshape. The infinite matrix and the inhomogeneity �1 have E =110GPa, v=0.3 and E1 =220E ,v1 =0.3, respectively. We investigate the stress field of the double inhomogeneity in responseto a uniform stress �0

x =2GPa applied at infinity. The computational domain is discretized into136×136×136 cuboids. Figure 9 presents the normal stress components �x and �y along thex-axis, where �z is identical to �x . The values of �x and �y are normalized by �0

x and show thatthe cuboidal cavity is free of stress, and its presence strongly distorts the stress field in �1. Thecomponent �x is continuous across the interfaces between �1 and the matrix, thus satisfying theboundary continuity condition, while �y always jumps across the interface. As the distance from�1 increases, �x in the matrix tends to become �0

x , and �y tends to zero. This study demonstratesthat the present method is capable of investigating complicated arbitrarily shaped inhomogeneitiesand cavities as well as the interactions between them.



-1.0-1.0

-0.5

0.0

0.5

1.0

1.5

x / R

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

Figure 9. Stress components �x and �y along the x-axis.

y

z

Ox

Aox

dox

Figure 10. A stringer of five cuboidal inhomogeneities embedded in an infinitematrix under uniform stress �0

x .

0-0.5

0.0

0.5

1.0

1.5

2.0

x

x / a

A continuous inhomogeneity bar

2 4 6 8

y

Figure 11. The stress components �x and �y for the case in which a stringer of five cuboidal inhomogeneitiesare embedded in an infinite matrix, compared with those for the case in which these five discrete

inhomogeneities are simulated by a continuous inhomogeneity bar (dashed lines).

3.4. A stringer of cuboidal inhomogeneities

Figure 10 shows the schematic of a stringer of five cuboidal SiC inhomogeneities embeddedin a steel matrix under uniform stress �0

x =2GPa. The five cuboids are assumed to have thesame side length a and equally spaced along the x-axis, which passes the cuboidal centers. Thecomputational domain is discretized into 330×66×66 cuboids. Figure 11 presents the normalstress components �x and �y along the x-axis for the case with d =0.2a. For comparison, �xand �y are also plotted for the case in which the five discrete inhomogeneities are simulated by acontinuous inhomogeneity bar whose volume is larger than the sum of those five discrete volumes.This approach is often used in engineering practice when there is computational difficulty in taking



Contour of von Mises stress

Figure 12. Contours of the normal stress components and von Mises stress (MPa) in the plane normal tothe z-axis and passing through point A (see Figure 10).

y

zO

x

)(x

Figure 13. A stringer of five cuboidal inhomogeneities embedded in an infinite matrixunder linearly varying stress �o

y(x).

into account the interactions between numerous inhomogeneities. However, it can be seen fromFigure 11 that this simplification may lead to a significant discrepancy in describing the real stressfield of an inhomogeneity stringer, thus resulting in an inaccurate prediction of stringer-inducedmaterial failure. Figure 12 further shows the stress field of the cuboidal inhomogeneity stringerby plotting the contours of the normal stress components and von Mises stress in the plane thatcontains the corners and edges of the cuboidal inhomogeneities. It is observed that there are stressconcentrations in the vicinities of the corners and edges.

In Figure 13, a linear far-field stress �oy(x) is applied to the steel matrix that contains the cuboidal

SiC inhomogeneities. The values of the applied stresses at the centers of the leftmost and the



0-0.6

-0.4

-0.2

0.0

0.2

x / a

2 4 6 8 10

Figure 14. The stress component �y along the x-axis for different varying linear stress.

x / a

0-2

0

2

4

2 4 6 8 10

Figure 15. The stress component �y along the x-axis for different varying linear stress.

middle cuboid are denoted by ��oy and �o

y . Thus, the parameter � controls the slope of the linearlyvarying stress field. Figures 14 and 15 show the normal stress components �x and �y , respectively,along the x-axis for �=1.5, �=2, and �=2.5. The values of �x and �y are normalized by �0

y . Itcan be seen that as the slope of the applied linear stress field increases, the induced stress fieldin the inhomogeneities has increased variations, as predicted. This example demonstrates that thepresent solution can easily treat non-uniform applied far-field stress.

3.5. A cluster of cuboidal inhomogeneities

Figure 16 shows the schematic of a square cluster of nine cuboidal SiC inhomogeneities embeddedin a steel matrix under uniform stress �o

x =�oy =2 GPa. The cuboids have the dimensions cx =cy =a,

cz =2a/3 and are equally spaced by d =a/2 along the x- and y-axes. The computational domainis discretized into 288×288×34 cuboids. Figure 17 presents the contours of the normal stresscomponents and von Mises stress in the plane that is normal to the z-axis and passes throughthe centers of the nine cuboidal inhomogeneities, and the stress contours are shown as symmetricabout the central lines of the square in the x − y plane, as predicted from the geometric symmetryof the square cluster. In the matrix, the stresses in the regions neighboring four inhomogeneitiesare very different from the stresses in the gaps between two of them, which demonstrates stronginteractions among the inhomogeneity cluster. Stress concentrations can also be seen around theedges of the cuboids.



d

xy

z

O

cc

c

B

Figure 16. A cluster of nine cuboidal inhomogeneities embedded in an infinite matrixunder uniform stress �o

x and �oy .


Figure 17. Contours of the normal stress components and von Mises stress (MPa) in the plane normal tothe z-axis and passing through point B (see Figure 16).

Figures 18 and 19 further present the stress contours for a cluster of 5×5 and a cluster of 7×7SiC inhomogeneities embedded in a steel matrix, respectively. The computational domains arediscretized into 324×324×26 and 360×360×22 cuboids, respectively. The comparison shows thatthe magnitudes of the stresses around the central inhomogeneity increase about 1% as more inho-mogeneities are involved in the configuration and more interaction effects are therefore included.




Figure 18. Contours of the normal stress components and von Mises stress (MPa) for a cluster of 5×5SiCinhomogeneities embedded in a steel matrix.

It is also predicted that the increase would become larger when the inhomogeneities in the clusterbecome closer to each other.

4. CONCLUSIONS

In this paper, we developed a semi-analytic solution for multiple arbitrarily shaped 3D inhomo-geneous inclusions embedded in an infinite isotropic matrix subject to external load using theEIM. The solution was validated by taking Eshelby’s [4] analytic solution of an ellipsoidal inho-mogeneous inclusion as a benchmark and by using the finite element analysis to study the sameproblem.

This solution is general and robust because of the following five aspects: (i) a multiple numberof inhomogeneous inclusions can be studied, (ii) the interactions between all the inhomogeneousinclusions are taken into account, (iii) each inhomogeneous inclusion can have an irregular 3Dshape, (iv) the material in one inhomogeneous inclusion can be different from that of the other, and(v) the initial eigenstrain and the applied stress can be non-uniform. The generality of the solutionwas demonstrated by applying it to solve problems concerning a double inhomogeneity, two spacedinhomogeneities, a stringer of inhomogeneities, and a cluster of inhomogeneities, respectively.




Figure 19. Contours of the normal stress components and von Mises stress (MPa) for a cluster of 7×7SiCinhomogeneities embedded in a steel matrix.

It shows that the solution has the potential of solving a wide range of inhomogeneity-relatedproblems.

APPENDIX A

The elastic solution of a cuboidal inclusion in an infinite space.Consider a cuboidal inclusion containing uniform eigenstrain ε

pij (i, j =1,2,3) in an infinite

space with Poisson’s ratio v. A Cartesian coordinate system (x1, x2, x3) is attached to the spaceand its origin is set at the center of the cuboid, which has side lengths 2�1, 2�2, and 2�3 alongthe axes x1, x2, and x3, respectively. Given an observation point at Q(1,2,3), we define thevectors which link the corners of the cuboid to this point as follows [5]:

Cm = (�m1 ,�m

2 ,�m3 ) (m =1,2, . . . ,8), (A1)

whereC1 = (1 −�1,2 −�2,3 −�3), (A2)

C2 = (1 +�1,2 −�2,3 −�3), (A3)

C3 = (1 +�1,2 +�2,3 −�3), (A4)

C4 = (1 −�1,2 +�2,3 −�3), (A5)



C5 = (1 −�1,2 +�2,3 +�3), (A6)

C6 = (1 −�1,2 −�2,3 +�3), (A7)

C7 = (1 +�1,2 −�2,3 +�3), (A8)

C8 = (1 +�1,2 +�2,3 +�3). (A9)

If the cuboidal inclusion contains uniform unit normal eigenstrain εp11, i.e. ε

p11 =1 and the other

components are zero, the elastic strain at Q(1,2,3) due to the inclusion is given by

ε1111 = 1

8�3

8∑m=1

[Dm

,1111 + 2−v

1−v(Dm

,1111 + Dm,1133)

]− H (Q), (A10)

ε2211 = − 1

8�3

8∑m=1

−Dm,1122+ v

1−v(Dm

,2222+ Dm,2233), (A11)

ε3311 = − 1

8�3

8∑m=1

−Dm,1133+ v

1−v(Dm

,2233+ Dm,3333), (A12)

ε1211 = 1

8�3

8∑m=1

v

1−vDm

,1112 + 1+v

1−v(Dm

,2221+ Dm,3312), (A13)

ε1311 = 1

8�3

8∑m=1

v

1−vDm

,1113 + 1+v

1−v(Dm

,3331+ Dm,2213), (A14)

ε2311 = 1

8�3

8∑m=1

v

1−v(Dm

,2233 + Dm,3332). (A15)

If the cuboidal inclusion contains constant unit shear eigenstrain εp12, i.e. ε

p12 =ε

p21 =1 and the

other components are zero, the elastic strain at Q(1,2,3) is given by

ε1112 = 1

8�3

8∑m=1

−2v

1−vDm

,1112 +2(Dm,2221+ Dm

,3312), (A16)

ε2212 = 1

8�3

8∑m=1

−2v

1−vDm

,1222 +2(Dm,1112+ Dm

,3312), (A17)

ε3312 = 1

8�3

8∑m=1

−2v

1−vDm

,3312, (A18)

ε1212 = 1

8�3

8∑m=1

[−2v

1−vDm

,1122 + Dm,1111+ Dm

,2222 + Dm,1133+ Dm

,2233

]− H (Q), (A19)

ε1312 = 1

8�3

8∑m=1

−1+v

1−vDm

,1123 + Dm,2223+ Dm

,3332, (A20)

ε2312 = 1

8�3

8∑m=1

−1+v

1−vDm

,2213 + Dm,1113+ Dm

,3331. (A21)

In Equations (A10)–(A21), H (Q)=1 if the point Q is inside the cuboid and H (Q)=0 otherwise.Four of the functions Dm

,i jkl (m =1,2, . . . ,8; i, j,k, l =1,2,3) are defined as follows:

Dm,1111 = 2�2

[tan−1

[�m

2 �m3

�m1 Rm

]− �m

1 �m2 �m

3

2R

(1

(�m1 )2 +(�m

2 )2+ 1

(�m1 )2 +(�m

3 )2

)], (A22)



Dm,1112 = −�2

[sign

(�m

3

)ln

(Rm +∣∣�m

3

∣∣((�m

1 )2 +(�m2 )2

)1/2− (�m

1 )2�m3(

(�m1 )2 +(�m

2 )2)

Rm

)], (A23)

Dm,1122 = �2�m

1 �m2 �m

3((�m

1 )2 +(�m2 )2

)Rm

, (A24)

Dm,1123 = −�2�m

1

Rm, (A25)

where

Rm =√

(�m1 )2 +(�m

2 )2 +(�m3 )2 (m =1,2, . . . ,8). (A26)

The rest of the functions Dm,i jkl are obtained by circular permutation of the subscripts in Equations

(A22)–(A26).The elastic strain at Q(1,2,3) due to the cuboidal inclusion which contains other unit normal

or shear eigenstrains can also be obtained by circular permutation of the subscripts in Equations(A10)–(A15) or Equations (A16)–(A21).

Finally, according to Hooke’s law, the stress at Q(1,2,3) due to the cuboidal inclusion can beobtained from the elastic strain. Therefore, the functions Bijkl() relating the stress at Q(1,2,3)to the uniform eigenstrain in a cuboidal inclusion centered at the origin (0,0,0) in an infinite spaceare determined.

ACKNOWLEDGEMENTS

The authors acknowledge the Timken Company, U.S.A. for financial supports. The authors also acknowl-edge Dr Faxing Che at Institute of Microelectronics, Singapore and Dr Shangwu Xiong at NorthwesternUniversity, U.S.A. who helped to validate this research work using the finite element analysis.

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Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space