Strain gradient solution for the Eshelby-type polyhedral inclusion problem · Strain gradient...

Strain gradient solution for the Eshelby-type polyhedralinclusion problem

X.-L. Gao a,n, M.Q. Liu b

a Department of Mechanical Engineering, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080-3021, USAb Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

a r t i c l e i n f o

Article history:

Received 22 July 2011

Received in revised form

27 October 2011

Accepted 28 October 2011Available online 6 November 2011

Keywords:

Eshelby tensor

Polyhedral inclusion

Size effect

Eigenstrain

Strain gradient

a b s t r a c t

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an

infinite homogeneous isotropic elastic material is analytically solved using a simplified

strain gradient elasticity theory (SSGET) that contains a material length scale para-

meter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a

general analytical form in terms of three potential functions, two of which are the same

as the ones involved in the counterpart Eshelby tensor based on classical elasticity.

These potential functions, as volume integrals over the polyhedral inclusion, are

evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with

each duplex and the associated local coordinate system constructed using a procedure

similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of

the three volume integrals is first transformed to a surface integral by applying the

divergence theorem, which is then transformed to a contour (line) integral based on

Stokes’ theorem and using an inverse approach different from those adopted in the

existing studies based on classical elasticity. The newly derived SSGET-based Eshelby

tensor is separated into a classical part and a gradient part. The former contains

Poisson’s ratio only, while the latter includes the material length scale parameter

additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-

based Eshelby tensor reduces to that based on classical elasticity when the strain

gradient effect is not considered. For homogenization applications, the volume average

of the new Eshelby tensor over the polyhedral inclusion is also provided in a general

form. To illustrate the newly obtained Eshelby tensor and its volume average, three

types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are

quantitatively studied by directly using the general formulas derived. The numerical

results show that the components of the SSGET-based Eshelby tensor for each of the

three inclusion shapes vary with both the position and the inclusion size, while their

counterparts based on classical elasticity only change with the position. It is found that

when the inclusion is small, the contribution of the gradient part is significantly large

and should not be neglected. It is also observed that the components of the averaged

Eshelby tensor based on the SSGET change with the inclusion size: the smaller the

inclusion, the smaller the components. When the inclusion size becomes sufficiently

large, these components are seen to approach (from below) the values of their classical

elasticity-based counterparts, which are constants independent of the inclusion size.

& 2011 Elsevier Ltd. All rights reserved.

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Journal of the Mechanics and Physics of Solids

0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jmps.2011.10.010

n Corresponding author. Tel.: þ1 972 883 4550; fax: þ1 972 883 4659.

E-mail address: [email protected] (X.-L. Gao).

Journal of the Mechanics and Physics of Solids 60 (2012) 261–276

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14. ABSTRACT The Eshelby-typeproblemofanarbitrary-shapepolyhedralinclusionembeddedinaninfinitehomogeneousisotropicelasticmaterialisanalyticallysolvedusingasimplified straingradientelasticitytheory(SSGET)thatcontainsamateriallengthscalepara-meter.TheEshelbytensorforapolyhedralinclusionofarbitraryshapeisobtainedinageneralanalyticalformintermsofthreepotentialfunctions,twoofwhicharethesame astheonesinvolvedinthecounterpartEshelbytensorbasedonclassicalelasticity.Thesepotentialfunctions,asvolumeintegralsoverthepolyhedralinclusion,areevaluatedbydividingthepolyhedralinclusiondomainintotetrahedralduplexes,with eachduplexandtheassociatedlocalcoordinatesystemconstructedusingaproceduresimilartothatemployedbyRodin(1996.J.Mech.Phys.Solids44,1977?1995).Eachof thethreevolumeintegralsisfirsttransformedtoasurfaceintegralbyapplyingthedivergencetheorem,whichisthentransformedtoacontour(line)integralbasedon Stokes?theoremandusinganinverseapproachdifferentfromthoseadoptedintheexistingstudiesbasedonclassicalelasticity.ThenewlyderivedSSGET-basedEshelby tensorisseparatedintoaclassicalpartandagradientpart.TheformercontainsPoisson?sratioonly,whilethelatterincludesthemateriallengthscaleparameteradditionally,therebyenablingtheinterpretationoftheinclusionsizeeffect.ThisSSGET-basedEshelbytensorreducestothatbasedonclassicalelasticitywhenthestraingradienteffectisnotconsidered.Forhomogenizationapplications,thevolumeaverage ofthenewEshelbytensoroverthepolyhedralinclusionisalsoprovidedinageneral form.ToillustratethenewlyobtainedEshelbytensoranditsvolumeaverage,three typesofpolyhedralinclusions?cubic,octahedralandtetrakaidecahedral?arequantitativelystudiedbydirectlyusingthegeneralformulasderived.Thenumerical resultsshowthatthecomponentsoftheSSGET-basedEshelbytensorforeachofthethreeinclusionshapesvarywithboththepositionandtheinclusionsize,whiletheircounterpartsbasedonclassicalelasticityonlychangewiththeposition.Itisfoundthat whentheinclusionissmall,thecontributionofthegradientpartissignificantlylarge andshouldnotbeneglected.ItisalsoobservedthatthecomponentsoftheaveragedEshelbytensorbasedontheSSGETchangewiththeinclusionsize:thesmallertheinclusion,thesmallerthecomponents.Whentheinclusionsizebecomessufficiently large,thesecomponentsareseentoapproach(frombelow)thevaluesoftheirclassical elasticity-basedcounterparts,whichareconstantsindependentoftheinclusionsize.

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1. Introduction

Eshelby’s (1957, 1959) solution for the problem of an infinite homogeneous isotropic elastic material containing anellipsoidal inclusion prescribed with a uniform eigenstrain is a milestone in micromechanics. The solution for the dynamicEshelby ellipsoidal inclusion problem was obtained by Michelitsch et al. (2003), which reduces to the Eshelby solution inthe static limiting case. Both of these solutions are based on classical elasticity. Recently, the Eshelby ellipsoidal inclusionproblem was solved by Gao and Ma (2010b) using a simplified strain gradient elasticity theory, which recovers Eshelby’s(1957, 1959) solution when the strain gradient effect is not considered.

A remarkable property of Eshelby’s (1957) solution is that the Eshelby tensor, which is a fourth-order straintransformation tensor directly linking the induced strain to the prescribed uniform eigenstrain, is constant inside theinclusion. However, this property is true only for ellipsoidal inclusions (and when classical elasticity is used), which isknown as the Eshelby conjecture (e.g., Eshelby, 1961; Rodin, 1996; Markenscoff, 1998a,b; Lubarda and Markenscoff, 1998;Liu, 2008; Li and Wang, 2008; Gao and Ma, 2010b; Ammari et al., 2010).

For non-ellipsoidal polyhedral inclusions, Rodin (1996) provided an algorithmic analytical solution and showed thatEshelby’s tensor cannot be constant inside a polyhedral inclusion, thereby proving the Eshelby conjecture in the case ofpolyhedral inclusions. The expressions of Eshelby’s tensor for two-dimensional (2-D) polygonal inclusions were includedin Rodin (1996). The explicit expressions of the Eshelby tensor for three-dimensional (3-D) polyhedral inclusions werelater derived by Nozaki and Taya (2001), where an exact solution for the stress field inside and outside an arbitrary-shapepolyhedral inclusion was obtained and numerical results for five regular polyhedral inclusion shapes and three othershapes of the icosidodeca family were presented. Both Rodin (1996) and Nozaki and Taya (2001) made use of an algorithmdeveloped by Waldvogel (1979) for evaluating the Newtonian (harmonic) potential over a polyhedral body. A morecompact form of the Eshelby tensor than that presented in Nozaki and Taya (2001) for a polyhedral inclusion in an infiniteelastic space was proposed by Kuvshinov (2008) using a coordinate-invariant formulation, where problems of polyhedralinclusions in an elastic half-space and bi-materials were also investigated. In addition, specific analytical solutions havebeen obtained for polyhedral inclusions of simple shapes such as cuboids (e.g., Chiu, 1977; Lee and Johnson, 1978; Liu andWang, 2005) and pyramids (e.g., Pearson and Faux, 2000; Glas, 2001; Nenashev and Dvurechenskil, 2010). Also, illustrativeresults have been provided for dynamic Eshelby problems of cubic and triangularly prismatic inclusions along withspherical and ellipsoidal ones by Wang et al. (2005) using their general solution for the dynamic Eshelby problem forinclusions of various shapes.

However, these existing studies on polyhedral inclusion problems are all based on the classical elasticity theory, which doesnot contain any material length scale parameter. As a result, the Eshelby tensors obtained in these studies and the subsequenthomogenization methods cannot capture the inclusion (particle) size effect on elastic properties exhibited by particle–matrixcomposites (e.g., Vollenberg and Heikens, 1989; Cho et al., 2006; Marcadon et al., 2007). Solutions for polyhedral inclusionproblems are also important for describing interpenetrating phase composites reinforced by 3-D networks (e.g., Poniznik et al.,2008; Jhaver and Tippur, 2009) and for understanding semiconductor materials buried with quantum dots that are typicallypolyhedral-shaped (e.g., Kuvshinov, 2008; Nenashev and Dvurechenskil, 2010). These materials often exhibit microstructure-dependent size effects whose interpretation requires the use of higher-order continuum theories.

In this paper, the Eshelby-type inclusion problem of a polyhedral inclusion prescribed with a uniform eigenstrain and auniform eigenstrain gradient and embedded in an infinite homogeneous isotropic elastic material is solved using asimplified strain gradient elasticity theory (SSGET) (e.g., Gao and Park, 2007), which contains a material length scaleparameter and can describe size-dependent elastic deformations. The Eshelby tensor is analytically obtained in terms ofthree potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based onclassical elasticity. These potential functions, as three volume integrals over the polyhedral inclusion, are evaluated bydividing the polyhedral inclusion domain into tetrahedral duplexes. Each duplex and the associated local coordinatesystem are constructed using a procedure similar to that developed by Rodin (1996) based on the algorithm proposed inWaldvogel (1979). Each of the three volume integrals is first transformed to a surface integral by applying the divergencetheorem, which is then transformed to a contour (line) integral based on Stokes’ theorem and using an inverse approachdifferent from those employed in the existing studies for evaluating the two integrals involved in the classical elasticity-based Eshelby tensor for a polyhedral inclusion.

The rest of this paper is organized as follows. In Section 2, the general form of the Eshelby tensor for a 3-D arbitrary-shape inclusion based on the SSGET is presented in terms of three potential functions (volume integrals). The expressionsof the SSGET-based Eshelby tensor for a polyhedral inclusion of arbitrary shape are analytically derived in Section 3, whichis separated into a classical part and a gradient part. The averaged Eshelby tensor over the inclusion volume is alsoanalytically evaluated there. Numerical results are provided in Section 4 to quantitatively illustrate the position andinclusion size dependence of the newly obtained Eshelby tensor for the polyhedral inclusion problem. The paper concludesin Section 5.

2. Eshelby tensor based on the SSGET

The SSGET is the simplest strain gradient elasticity theory evolving from Mindlin’s pioneering work. It is also known asthe first gradient elasticity theory of Helmholtz type and the dipolar gradient elasticity theory (e.g., Gao and Ma, 2010a).

X.-L. Gao, M.Q. Liu / J. Mech. Phys. Solids 60 (2012) 261–276262

According to the SSGET (Gao and Park, 2007; Gao and Ma, 2010a), the Navier-like displacement equations of equilibrium aregiven by

ðlþmÞui,ijþmuj,kk�L2½ðlþmÞui,ijþmuj,kk�,mmþ f j ¼ 0 in O, ð1Þ

and the boundary conditions have the form:

sijnj�ðmijknkÞ,jþðmijkn

knlÞ,lnj ¼ ti or ui ¼ ui

mijknjnk ¼ qi or ui,lnl ¼@ui

@n

9=; on @O, ð2Þ

where l and m are the Lame constants in classical elasticity, L is a material length scale parameter, ui are the components of thedisplacement vector, fi are the components of the body force vector (force per unit volume), sij are the components of the totalstress, r¼sijei�ej, mijk are the components of the double stress, l¼mijkei�ej�ek, and ti and qi are, respectively, the componentsof the Cauchy traction vector and double stress traction vector. Also, in Eqs. (1) and (2), O is the region occupied by theelastically deformed material, @O, is the smooth bounding surface of O, ni is the outward unit normal vector on @O, and theoverhead bar represents the prescribed value. In addition,

mijk ¼ L2tij,k ¼ mjik, sij � tij�mijk,k ¼ tij�L2tij,kk ¼ sji, ð3Þ

where tij are the components of the Cauchy stress, s¼tijei�ej.When the strain gradient effect is not considered (i.e., L¼0), mijk¼0 and sij¼tij (see Eq. (3)), and Eqs. (1) and (2) reduce

to the governing equations and the boundary conditions in terms of displacement in classical elasticity (e.g., Timoshenkoand Goodier, 1970; Gao and Rowlands, 2000).

Note that the standard index notation, together with the Einstein summation convention, is used in Eqs. (1), (2) and (3)and throughout this paper, with each Latin index (subscript) ranging from 1 to 3 unless otherwise stated.

Solving Eq. (1), subject to the boundary conditions of ui and their first, second and third derivatives vanishing at infinity,gives the fundamental solution and Green’s function based on the SSGET. By using 3-D Fourier and inverse Fouriertransforms, the fundamental solution of Eq. (1) has been obtained as (Gao and Ma, 2009)

uiðxÞ ¼ZZZ þ1�1

Gijðx�yÞf jðyÞdy, ð4Þ

where x is the position vector of a point in the infinite 3-D space, y is the integration point, and Gij( � ) is Green’s function(a second-order tensor) given by

GijðxÞ ¼1

16pmð1�vÞ½AðxÞdij�BðxÞ,ij�, ð5Þ

with

AðxÞ � 4ð1�vÞ1

x1�e�x=L� �

, BðxÞ � xþ2L2

x�

2L2

xe�x=L: ð6Þ

When L¼0, Eqs. (5) and (6) reduce to the Green’s function for 3-D problems in classical elasticity (e.g., Li and Wang, 2008). In Eqs.(5) and (6), v is Poisson’s ratio, which is related to the Lame constants l and m through (e.g., Timoshenko and Goodier, 1970)

l¼Ev

ð1þvÞð1�2vÞ, m¼ E

2ð1þvÞ, ð7Þ

where E is Young’s modulus.By using the Green’s function method entailing Eqs. (4), (5) and (6), the general expressions of the Eshelby tensor based

on the SSGET can then be obtained, as summarized below.Consider the problem of a 3-D inclusion of arbitrary shape embedded in an infinite homogenous isotropic elastic body.

The inclusion is prescribed with a uniform eigenstrain en and a uniform eigenstrain gradient jn. There is no body force orany other external force acting on the elastic body. The disturbed strain, eij, induced by en and jn can be shown to be (Gaoand Ma, 2009, 2010b)

eij ¼ SijklenklþTijklmkn

klm, ð8Þ

where Sijkl is the fourth-order Eshelby tensor having 36 independent components, and Tijklm is a fifth-order Eshelby-liketensor with 108 independent components. Eq. (8) shows that e (¼eijei�ej) is solely linked to e* in the absence of jn (i.e.,the classical case) and is fully related to jn when en¼0. The fourth-order Eshelby tensor has been obtained as

Sijkl ¼ SCijklþSG

ijkl, ð9aÞ

SCijkl ¼

1

8pð1�vÞ½F,ijkl�2vL,ijdkl�ð1�vÞðL,ljdikþL,kjdilþL,lidjkþL,kidjlÞ�, ð9bÞ

SGijkl ¼

1

8pð1�vÞ½2vG,ijdklþð1�vÞðG,jldikþG,ildjkþG,jkdilþG,ikdjlÞþ2L2

ðL,ijkl�G,ijklÞ�, ð9cÞ

X.-L. Gao, M.Q. Liu / J. Mech. Phys. Solids 60 (2012) 261–276 263

where SCijkl is the classical part, SG

ijkl is the gradient part, dij is the Kronecker delta, and

FðxÞ ¼ZO9x�y9dy, ð10aÞ

LðxÞ ¼ZO

1

9x�y9dy, ð10bÞ

GðxÞ ¼ZO

e�9x�y9=L

9x�y9dy ð10cÞ

are three scalar-valued potential functions that can be obtained analytically or numerically by evaluating the volumeintegrals over the domain O occupied by the inclusion, with 9x9¼x¼(xkxk)1/2 and y (AO) being the integration variable.Note that the first two potential functions given in Eqs. (10a) and (10b) are the same as the ones involved in the Eshelbytensor based on classical elasticity (e.g., Mura, 1987; Nemat-Nasser and Hori, 1999; Li and Wang, 2008), whereas the thirdone defined in Eq. (10c) results from the use of the SSGET. It should be mentioned that F(x) in Eq. (10a) and L(x) inEq. (10b) are, respectively, known to be a biharmonic potential and a Newtonian potential (e.g., Li and Wang, 2008), whilea variant of G(x) in Eq. (10c) is called the Yukawa potential in physics (e.g., Rowlinson, 1989).

Eqs. (10a), (10b) and (10c) show that among the three potential functions, only the third one, G(x), involves the lengthscale parameter L. It then follows from Eqs. (9c), (10b) and (10c) that SG

ijkl depends on L, while SCijkl, expressed in terms of

L(x) and F(x) only according to Eqs. (9b), (10a) and (10b), is independent of L. Also, it is seen from Eqs. (9c), (10b) and(10c) that SG

ijkl¼0 when L¼0 (and thus G(x)�0), thereby giving Sijkl ¼ SCijkl (from Eq. (9a)). That is, the Eshelby tensor derived

using the SSGET reduces to that based on classical elasticity when the strain gradient effect is not considered.The fifth-order Eshelby-like tensor T, which relates the eigenstrain gradient jn to the disturbed strain e in the elastic

body (see Eq. (8)), can be shown to be

Tijklm ¼L2

8pð1�vÞ½2vC,ijmdklþð1�vÞðC,lmidjkþC,lmjdikþC,kmidjlþC,kmjdilÞ�P,ijklm�, ð11Þ

where

CðxÞ �L�G, PðxÞ �Fþ2L2ðL�GÞ, ð12a;bÞ

with the scalar-valued potential functions F(x), L(x) and G(x) defined in Eqs. (10a), (10b) and (10c). It is clear fromEq. (11) that T vanishes when L¼0. Then, with Sijkl ¼ SC

ijkl as discussed above, Eq. (8) simply becomes eij ¼ SCijklenkl when L¼0,

which is the defining relation for the Eshelby tensor based on classical elasticity (Eshelby, 1957), as expected.Eqs. (9a), (9b), (9c) and (11) provide the general formulas for determining Sijkl ð ¼ SC

ijklþSGijklÞ and Tijklm for an inclusion of

arbitrary shape in terms of the potential functions L(x), F(x) and G(x) defined in Eqs. (10a), (10b) and (10c). For the casesof a spherical inclusion, a cylindrical inclusion and an ellipsoidal inclusion in an infinite elastic medium, analyticalexpressions have been obtained for L(x), F(x) and G(x) and thus for the Eshelby tensor (Gao and Ma, 2009, 2010b; Ma andGao, 2010). The more complex case of a polyhedral inclusion of arbitrary shape, for which L(x), F(x) and G(x) are difficultto evaluate analytically, is examined in this study.

3. Polyhedral inclusion

The problem of an arbitrary-shape polyhedral inclusion in an infinite elastic body has been analytically studied by Rodin(1996), Nozaki and Taya (2001) and Kuvshinov (2008) using classical elasticity. The Eshelby tensor for this problem is derivedhere using the SSGET-based general formulas and a new method for evaluating the potential functions L(x), F(x) and G(x).

3.1. Eshelby tensor

Consider an arbitrarily shaped polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material.The polyhedral inclusion has p faces and is prescribed with a uniform eigenstrain e* and a uniform eigenstrain gradient jn.

The p-faced polyhedral domain occupied by the inclusion can be divided into tetrahedral duplexes originated from a chosen(arbitrary) point x (Waldvogel, 1979; Rodin, 1996). Each duplex can be further divided into two simplexes, each of which is atetrahedron with three of its four faces being right triangles (see Fig. 1). The four vertices of each of the duplexes are, respectively,the projection point of x on a polyhedral surface (i.e., xI), two adjacent vertices on this surface (i.e., VþJI and V�JI ), and the point xitself. For each of these duplexes, a local Cartesian coordinate system is constructed, with point x being set as the origin. The three

orthogonal axes of the local coordinate system are denoted by l, Z and z, respectively. The coordinates of the two vertices

VþJI and V�JI on the Jth edge of the Ith surface are, respectively, given by ðbJI ; lþ

JI ; aIÞ and ðbJI ,l�

JI ,aIÞ, as shown in Fig. 2.

In Fig. 2, k0JI , g0

JI and f0I are the unit vectors associated with the local coordinates lJI, ZJI and zI, y is an arbitrary point on

the Jth edge of the Ith surface, r is the position vector of y relative to the origin x (i.e., r¼y�x), and rsI is the projection of

r on the Ith surface. The usual Cartesian coordinates (x1, x2, x3) are used in the global coordinate system having (e1, e2, e3)as the associated base vectors.


To obtain the Eshelby tensor for the polyhedral inclusion using Eqs. (9a), (9b) and (9c), the three potential functionsF(x), L(x) and G(x) defined in Eqs. (10a), (10b) and (10c) are first evaluated over the polyhedral domain O using anapproach different from those employed in Rodin (1996), Nozaki and Taya (2001) and Kuvshinov (2008) for evaluatingF(x) and L(x) involved in the classical elasticity-based Eshelby tensor, as shown next.

For a sufficiently smooth function M(x�y), the use of the divergence theorem gives

@

@xk

ZZZO

MdVðyÞ ¼�ZZZ

O

@M

@yk

dVðyÞ ¼�Xp

I ¼ 1

ðz0I Þk

ZZ@OI

MdSðyÞ, ð13Þ

where p is the number of surfaces of the polyhedron, and ðz0I Þk is the kth component of the unit outward normal vector on

the Ith surface @OI, f0I .

To transform the surface integral in Eq. (13) to a contour (line) integral, let

M¼ ðr �mÞUf0I , ð14Þ

where m is a yet-unknown vector located on the Ith surface of the polyhedron, and r�m denotes the curl of m. Using theStokes theorem then yields, upon applying Eq. (14),

ZZ@OI

MdSðyÞ ¼Xq

J ¼ 1

ZCJI

mUg0JIdl, ð15Þ

where g0JI is the unit vector along the Jth boundary edge CJI of the Ith surface.

Now, write

m¼ f0I �

rSI

rSI

gðrÞ

" #, ð16Þ

x

x

xI

xI

VJI

VJI

VJI

VJI�

��

Fig. 1. A polyhedron represented by duplexes: (a) a polyhedron (with five duplexes shown); (b) a duplex and the associated local coordinate system

constructed from an arbitrary point x.

ee

x

x r

a

b

y

η

λ ζ

V

V

x

x

e

Fig. 2. A duplex with its base on the Ith surface and one local coordinate axis (Z) along the Jth edge of the Ith surface.


where rSI ð ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffir2�a2

I

q¼ 9rS

I 9Þ is the length of the projection of r on the Ith surface, and g(r) is a function of r (¼9r9) yet to be

determined. Substituting Eq. (16) into Eq. (15) leads toZZ@OI

MdSðyÞ ¼Xq

J ¼ 1

bJI

ZCJI

gðrÞ

rSI

dl, ð17Þ

where bJI � rSI Uk

0JI is the distance from point xI (the projection of point x on the Ith surface) to the Jth edge CJI (see Fig. 2). For

each specific function M, a different expression of g(r) and thus m can be determined, as shown next for the three casesrepresenting the integrands of the potential functions F(x), L(x) and G(x) defined in Eqs. (10a), (10b) and (10c).

For M¼r¼9y�x9 (corresponding to F(x)), Eqs. (14) and (16) gives

r¼ gðrÞ rUrS

I

rSI

!þ½rgðrÞ�U

rSI

rSI

, ð18Þ

where r is the gradient operator, and use has been made of the identity: a�b� c¼(a � c)b�(a �b)c, with a, b, c beingarbitrary vectors and ‘‘� ’’, ‘‘ � ’’ representing the cross, dot products, respectively. After carrying out the differentiation anddot product operations, Eq. (18) can be further simplified to

r¼gðrÞ

rSI

þg0ðrÞrS

I

r, ð19Þ

where g0(¼dg/dr) is the first derivative of g with respect to r. The solution of Eq. (19) reads

gFðrÞ ¼r3�a3

I

3rSI

, ð20Þ

where aI ð ¼ rUf0I Þ is the distance from point x to the Ith surface (see Fig. 2), and gF(r) denotes the function g(r) for the case

with M¼r.Similarly, it can be shown that

gLðrÞ ¼rS

I

rþaIð21Þ

when M¼1/r¼1/9y�x9 (corresponding to L(x)), and

gGðrÞ ¼Lðe�aI=L�e�r=LÞ

rSI

ð22Þ

when M¼ e�r=L=r¼ e�9y�x9=L=9y�x9 (corresponding to G(x)).Using Eqs. (13), (17), (20), (21) and (22) in Eqs. (10a), (10b) and (10c) then leads to, with the local coordinate axis Z

being along the Jth edge,

F,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞibJI

Z lþJI

l�JI

ða2I þb2

JIþZ2Þ3=2�a3

I

3ðb2JIþZ2Þ

dZ, ð23Þ

L,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞibJI

Z lþJI

l�JI

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JIþZ2

qþaI

dZ, ð24Þ

G,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞibJI

Z lþJI

l�JI

Lðe�aI=L�e�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

Iþb2

JI þZ2p

=LÞ

b2JIþZ2

dZ, ð25Þ

where lþJI and l�JI are, respectively, the coordinates of the two vertices VþJI and V�JI on the Jth edge, with lþJI being positive andl�JI negative (see Fig. 2).

The integrals in Eqs. (23) and (24) can be exactly evaluated by direct integration to obtain the following closed-formexpressions:

F,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I Þi

lþJI bJI

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JIþðl

þ

JI Þ2

qþ

a3I

3tan�1

aIlþ

JI

bJI


I þb2JIþðl

þ

JI Þ2

q264

375

8><>:

�a3

I

3tan�1

lþJIbJI

!þ

3a2I bJIþb3

JI

6ln

lþJI þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JIþðl

þ

JI Þ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JI

q264

375


�l�JI bJI

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JIþðl

�

JI Þ2

q�

a3I

3tan�1

aIl�

JI

bJI


I þb2JIþðl

�

JI Þ2

q264

375þ a3

I

3tan�1

l�JIbJI

� ��

3a2I bJIþb3

JI

6ln

l�JI þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

I þb2JIþðl

�

JI Þ2


I þb2JI

q264

3759>=>;

��Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞiF

JI1 ¼�

Xp

I ¼ 1

Xq

J ¼ 1

ðz0I Þi½ðF

JI1 Þþ�ðFJI

1 Þ��, ð26Þ

L,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I Þi aI tan�1

aIlþ

JI

bJI


I þb2JIþðl

þ

JI Þ2

q264

375�aI tan�1

lþJIbJI

!8><>:

þbJI lnlþJI þ


I þb2JIþðl

þ

JI Þ2


I þb2JI

q264

375�aI tan�1

aIl�

JI

bJI


I þb2JIþðl

�

JI Þ2

q264

375þaI tan�1

l�JIbJI

� ��bJI ln


I þb2JIþðl

�

JI Þ2


I þb2JI

q264

3759>=>;

��Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞiL

JI1 ¼�

Xp

I ¼ 1

Xq

J ¼ 1

ðz0I Þi½ðL

JI1 Þþ�ðLJI

1 Þ��, ð27Þ

where FJI1 , ðFJI

1 Þþ , ðFJI

1 Þ�, LJI

1 , ðLJI1 Þþ and ðLJI

1 Þ� are functions defined by

FJI1 ¼FJI

1 ðaI ,bJI ,lþ

JI ,l�JI Þ ¼ ðFJI1 ÞþðaI ,bJI ,l

þ

JI Þ�ðFJI1 Þ�ðaI ,bJI ,l

�

JI Þ,

ðFJI1 Þþ¼

lþJI bJI

6


I þb2JIþðl

þ

JI Þ2

qþ

a3I

3tan�1

aIlþ

JI

bJI


I þb2JIþðl

þ

JI Þ2

q264

375� a3

I

3tan�1

lþJIbJI

!

þ3a2

I bJIþb3JI

6ln


I þb2JIþðl

þ

JI Þ2


I þb2JI

q264

375,

ðFJI1 Þ�¼

l�JI bJI

6


I þb2JIþðl

�

JI Þ2

qþ

a3I

3tan�1

aIl�

JI

bJI


I þb2JIþðl

�

JI Þ2

q264

375� a3

I

3tan�1

l�JIbJI

� �

þ3a2

I bJIþb3JI

6ln


I þb2JIþðl

�

JI Þ2


I þb2JI

q264

375,

LJI1 ¼LJI

1 ðaI ,bJI ,lþ

JI ,l�JI Þ ¼ ðLJI1 ÞþðaI ,bJI ,l

þ

JI Þ�ðLJI1 Þ�ðaI ,bJI ,l

�

JI Þ,

ðLJI1 Þþ¼ aI tan�1

aIlþ

JI

bJI


I þb2JIþðl

þ

JI Þ2

q264

375�aI tan�1

lþJIbJI

!þbJI ln


I þb2JIþðl

þ

JI Þ2


I þb2JI

q264

375,

ðLJI1 Þ�¼ aI tan�1

aIl�

JI

bJI


I þb2JIþðl

�

JI Þ2

q264

375�aI tan�1

l�JIbJI

� �þbJI ln


I þb2JIþðl

�

JI Þ2


I þb2JI

q264

375: ð28Þ

Note that e�r/L can be written as a power series:

e�r=L ¼X1n ¼ 0

ð�1Þn

n!

r

L

� �n

: ð29Þ

Using Eq. (29) in Eq. (25) then leads to

G,i ¼�Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞiL e�aI=L tan�1

lþJIbJI

!�tan�1

l�JIbJI

� �" #(

þX1n ¼ 0

ð�1Þnþ1

Lnn!

lþJI ½a2I þb2

JIþðlþ

JI Þ2�n=2

bJI1þðlþJI Þ

2

a2I þb2

JI

" #�n=2

F11

2,�

n

2,1,

3

2,�ðlþJI Þ

2

a2I þb2

JI

,�ðlþJI Þ

2

b2JI

" #

�X1n ¼ 0

ð�1Þnþ1

Lnn!

l�JI ½a2I þb2

JIþðl�

JI Þ2�n=2

bJI1þ

ðl�JI Þ2

a2I þb2

JI

" #�n=2

F11

2,�

n

2,1,

3

2,�ðl�JI Þ

2

a2I þb2

JI

,�ðl�JI Þ

2

b2JI

" #9=;

��Xp

I ¼ 1

Xq

J ¼ 1

ðz0I ÞiG

JI1 ¼�

Xp

I ¼ 1

Xq

J ¼ 1

ðz0I Þi½ðG

JI1 Þþ�ðGJI

1 Þ��, ð30Þ


where GJI1 , ðGJI

1 Þþ and ðGJI

1 Þ� are functions defined by

GJI1 ¼GJI

1 ðaI ,bJI ,lþ

JI ,l�JI Þ ¼ ðGJI1 ÞþðaI ,bJI ,l

þ

JI Þ�ðGJI1 Þ�ðaI ,bJI ,l

�

JI Þ,

ðGJI1 Þþ¼ L e�aI=L tan�1

lþJIbJI

!þX1n ¼ 0

ð�1Þnþ1

Lnn!

lþJI ½a2I þb2

JIþðlþ

JI Þ2�n=2

bJI1þðlþJI Þ

2

a2I þb2

JI

" #�n=28<:

F11

2,�

n

2,1,

3

2,�ðlþJI Þ

2

a2I þb2

JI

,�ðlþJI Þ

2

b2JI

" #),

ðGJI1 Þ�¼ L e�aI=L tan�1

l�JIbJI

� �þX1n ¼ 0

ð�1Þnþ1

Lnn!

l�JI ½a2I þb2

JIþðl�

JI Þ2�n=2

bJI1þ

ðl�JI Þ2

a2I þb2

JI

" #�n=28<:

F11

2,�

n

2,1,

3

2,�ðl�JI Þ

2

a2I þb2

JI

,�ðl�JI Þ

2

b2JI

" #), ð31Þ

and F1 is the first Appell hypergeometric function of two variables given by

F1 a,b,c,d; x,yð Þ ¼X1

m,n ¼ 0

ðaÞmþnðbÞmðcÞnðdÞmþnm!n!

xmyn, ð32Þ

with (f)m being the Pochhammer symbol representing the following rising factorial:

ðf Þm ¼Gðf þmÞ

Gðf Þ¼ f ðf þ1Þðf þ2Þ � � � ðf þm�1Þ: ð33Þ

Note that Eqs. (26), (27) and (30) are applicable to both the interior case with x being inside the polyhedral inclusion(i.e., xAO) and the exterior case with x being outside the inclusion (i.e., xeO). For the former aI is a positive value, while forthe latter aI is a negative value. The similarity and difference identified here between the interior and exterior cases can beseen from Fig. 3, where how the duplex in each case is constructed is schematically shown.

It should be mentioned that no attempt is made here to obtain the expressions of the potential functions F(x), L(x) andG(x) from Eqs. (26), (27) and (30), since only the second and/or fourth derivatives of these functions are involved in thegeneral expressions of the Eshelby tensor given in Eqs. (9a), (9b) and (9c).

Note that for a smooth function FðxÞ � FðaI ,bJI ,lþ

JI ,l�JI Þ ¼ Fþ ðaI ,bJI ,lþ

JI Þ�F�ðaI ,bJI ,l�

JI Þ the use of chain rule gives

F ,i �@F

@xi¼@F

@aI

@aI

@xiþ@F

@bJI

@bJI

@xiþ@Fþ

@lþJI

@lþJI@xi�@F�

@l�JI

@l�JI@xi

, ð34Þ

where the parameters aI, bJI, lþJI , l�JI are related to x through

aI ¼ ðvþ

k �xkÞðz0I Þk, ð35aÞ

bJI ¼ ðvþ

k �xkÞðl0JIÞk, ð35bÞ

lþJI ¼ ðvþ

k �xkÞðZ0JIÞk, ð35cÞ

Vl

l

b

x

a

x

η

λζ

V

l

aηl

b

x

x

λ

ζ

V

V

Fig. 3. Duplex and parameters aI, bJI, lþJI , l�JI for (a) xAO and (b) xeO.


l�JI ¼ ðv�k �xkÞðZ0

JIÞk, ð35dÞ

where xk, vþk and v�k are, respectively, the coordinates of the points x, VþJI and V�JI in the global coordinate system, and ðz0I Þk,

ðl0JIÞk and ðZ0

JIÞk are the components of the unit base vectors f0I , k0

JI and g0JI in the global coordinate system.

It then follows from Eqs. (34), (35a), (35b), (35c) and (35d) that

F ,i ¼�@F

@aIðz0

I Þi�@F

@bJIðl0

JIÞi�@F þ

@lþJI�@F�

@l�JI

!ðZ0

JIÞi, ð36aÞ

F ,ijk ¼�@3F

@a3I

ðz0I Þiðz

0I Þjðz

0I Þk�

@3F

@b3JI

ðl0JIÞiðl

0JIÞjðl

0JIÞk�

@3Fþ

@ðlþJI Þ3�@3F�

@ðl�JI Þ3

" #ðZ0

JIÞiðZ0JIÞjðZ

0JIÞk

�@3F

@a2I @bJI½ðl0

JIÞiðz0I Þjðz

0I Þkþðz

0I Þiðl

0JIÞjðz

0I Þkþðz

0I Þiðz

0I Þjðl

0JIÞk��

@3Fþ

@a2I @lþJI�@3F�

@a2I @l�JI

" #½ðZ0

JIÞiðz0I Þjðz

0I Þkþðz

0I ÞiðZ

0JIÞjðz

0I Þk

þðz0I Þiðz

0I ÞjðZ

0JIÞk��

@3F

@aI@b2JI

½ðz0I Þiðl

0JIÞjðl

0JIÞkþðl

0JIÞiðz

0I Þjðl

0JIÞkþðl

0JIÞiðl

0JIÞjðz

0I Þk��

@3Fþ

@b2JI@lþJI

�@3F�

@b2JI@l�JI

!½ðZ0

JIÞiðl0JIÞjðl

0JIÞk

þðl0JIÞiðZ

0JIÞjðl

0JIÞkþðl

0JIÞiðl

0JIÞjðZ

0JIÞk��

@3F þ

@aI@ðlþ

JI Þ2�

@3F�

@aI@ðl�

JI Þ2

" #½ðz0

I ÞiðZ0JIÞjðZ

0JIÞkþðZ

0JIÞiðz

0I ÞjðZ

0JIÞkþðZ

0JIÞiðZ

0JIÞjðz

0I Þk�

�@3F þ

@bJI@ðlþ

JI Þ2�

@3F�

@bJI@ðl�

JI Þ2

" #½ðl0

JIÞiðZ0JIÞjðZ

0JIÞkþðZ

0JIÞiðl

0JIÞjðZ

0JIÞkþðZ

0JIÞiðZ

0JIÞjðl

0JIÞk�

�@3F þ

@aI@bJI@lþJI�

@3F�

@aI@bJI@l�JI

!½ðz0

I Þiðl0JIÞjðZ

0JIÞkþðz

0I ÞiðZ

0JIÞjðl

0JIÞkþðl

0JIÞiðz

0I ÞjðZ

0JIÞk

þðl0JIÞiðZ

0JIÞjðz

0I ÞkþðZ

0JIÞiðz

0I Þjðl

0JIÞkþðZ

0JIÞiðl

0JIÞjðz

0I Þk�: ð36bÞ

Using Eqs. (26), (27), (30), (36a) and (36b) in Eqs. (9b) and (9c) will lead to the final expressions of the Eshelby tensorfor the p-faced polyhedral inclusion as

SCijkl ¼

1

8pð1�vÞ

Xp

I ¼ 1

Xq

J ¼ 1

fðSC1ÞJIðz

0I Þiðz

0I ÞjdklþðS

C2ÞJI½ðz

0I Þiðz

0I Þkdjlþðz

0I Þiðz

0I Þldjk�

þðSC3ÞJI½ðz

0I Þkðz

0I Þjdilþðz

0I Þlðz

0I Þjdik�þðS

C4ÞJIðz

0I Þiðl

0JIÞjdklþðS

C5ÞJI½ðz

0I Þiðl

0JIÞkdjlþðz

0I Þiðl

0JIÞldjk�

þðSC6ÞJI½ðl

0JIÞkðz

0I Þjdilþðl

0JIÞlðz

0I Þjdik�þðS

C7ÞJIðz

0I ÞiðZ

0JIÞjdklþðS

C8ÞJI½ðz

0I ÞiðZ

0JIÞkdjlþðz

0I ÞiðZ

0JIÞldjk�

þðSC9ÞJI½ðZ

0JIÞkðz

0I ÞjdilþðZ0

JIÞlðz0I Þjdik�þðS

C10ÞJI½ðz

0I Þiðl

0JIÞjðz

0I Þkðz

0I Þlþðz

0I Þiðz

0I Þjðl

0JIÞkðz

0I Þlþðz

0I Þiðz

0I Þjðz

0I Þkðl

0JIÞl�

þðSC11ÞJI½ðz

0I ÞiðZ

0JIÞjðz

0I Þkðz

0I Þlþðz

0I Þiðz

0I ÞjðZ

0JIÞkðz

0I Þlþðz

0I Þiðz

0I Þjðz

0I ÞkðZ

0JIÞl�þðS

C12ÞJI½ðz

0I Þiðz

0I Þjðl

0JIÞkðl

0JIÞl

þðz0I Þiðl

0JIÞjðz

0I Þkðl

0JIÞlþðz

0I Þiðl

0JIÞjðl

0JIÞkðz

0I Þl�þðS

C13ÞJI½ðz

0I Þiðz

0I ÞjðZ

0JIÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðz

0I ÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðZ

0JIÞkðz

0I Þl�

þðSC14ÞJI½ðz

0I Þiðl

0JIÞjðZ

0JIÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðl

0JIÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðZ

0JIÞkðl

0JIÞl�þðS

C15ÞJI½ðz

0I ÞiðZ

0JIÞjðl

0JIÞkðl

0JIÞl

þðz0I Þiðl

0JIÞjðZ

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðl

0JIÞkðZ

0JIÞl�þðS

C16ÞJI½ðz

0I Þiðz

0I Þjðl

0JIÞkðZ

0JIÞlþðz

0I Þiðz

0I ÞjðZ

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðZ

0JIÞkðz

0I Þl

þðz0I Þiðl

0JIÞjðz

0I ÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðz

0I Þkðl

0JIÞlþðz

0I ÞiðZ

0JIÞjðl

0JIÞkðz

0I Þl�g, ð37aÞ

where

ðSC1ÞJI ¼

1

3

@3FJI1

@a3I

�2v@LJI

1

@aI, ðSC

2ÞJI ¼1

3

@3FJI1

@a3I

�ð1�vÞ@LJI

1

@aI, ðSC

3ÞJI ¼�ð1�vÞ@LJI

1

@aI, ðSC

4ÞJI ¼1

3

@3FJI1

@b3JI

�2v@LJI

1

@bJI,

ðSC5ÞJI ¼

1

3

@3FJI1

@b3JI

�ð1�vÞ@LJI

1

@bJI, ðSC

6ÞJI ¼�ð1�vÞ@LJI

1

@bJI, ðSC

7ÞJI ¼1

3

@3ðFJI1 Þþ

@ðlþJI Þ3�@3ðFJI

1 Þ�

@ðl�JI Þ3

" #�2v

@ðLJI1 Þþ

@lþJI�@ðLJI

1 Þ�

@l�JI

" #,

ðSC8ÞJI ¼

1

3

@3ðFJI1 Þþ


1 Þ�

@ðl�JI Þ3

" #�ð1�vÞ

@ðLJI1 Þþ

@lþJI�@ðLJI

1 Þ�

@l�JI

" #, ðSC

9ÞJI ¼�ð1�vÞ@ðLJI

1 Þþ

@lþJI�@ðLJI

1 Þ�

@l�JI

" #,

ðSC10ÞJI ¼

@3FJI1

@a2I @bJI�

1

3

@3FJI1

@b3JI

, ðSC11ÞJI ¼

@3ðFJI1 Þþ

@a2I @lþJI

�@3ðFJI

1 Þ�

@a2I @l�JI

�1

3

@3ðFJI1 Þþ


1 Þ�

@ðl�JI Þ3

" #, ðSC

12ÞJI ¼@3FJI

1

@b2JI@aI

�1

3

@3FJI1

@a3I

,

ðSC13ÞJI ¼

@3ðFJI1 Þþ

@ðlþJI Þ2@aI

�@3ðFJI

1 Þ�

@ðl�JI Þ2@aI

" #�

1

3

@3FJI1

@a3I

, ðSC14ÞJI ¼

@3ðFJI1 Þþ

@ðlþJI Þ2@bJI

�@3ðFJI

1 Þ�

@ðl�JI Þ2@bJI

" #�

1

3

@3FJI1

@b3JI

,

ðSC15ÞJI ¼

@3ðFJI1 Þþ

@b2JI@lþJI

�@3ðFJI

1 Þ�

@b2JI@l�JI

�1

3

@3ðFJI1 Þþ


1 Þ�

@ðl�JI Þ3

" #, ðSC

16ÞJI ¼@3ðFJI

1 Þþ

@aI@bJI@lþJI�@3ðFJI

1 Þ�

@aI@bJI@l�JI, ð37bÞ


and

SGijkl ¼

1

8pð1�vÞ

Xq

J ¼ 1

Xp

I ¼ 1

fðSG1 ÞJIðz

0I Þiðz

0I ÞjdklþðS

G2 ÞJI½ðz

0I Þiðz

0I Þkdjlþðz

0I Þiðz

0I Þldjk�þðS

G3 ÞJI½ðz

0I Þkðz

0I Þjdilþðz

0I Þlðz

0I Þjdik�

þðSG4 ÞJIðz

0I Þiðl

0JIÞjdklþðS

G5 ÞJI½ðz

0I Þiðl

0JIÞkdjlþðz

0I Þiðl

0JIÞldjk�þðS

G6 ÞJI½ðl

0JIÞkðz

0I Þjdilþðl

0JIÞlðz

0I Þjdik�

þðSG7 ÞJIðz

0I ÞiðZ

0JIÞjdklþðS

G8 ÞJI½ðz

0I ÞiðZ

0JIÞkdjlþðz

0I ÞiðZ

0JIÞldjk�þðS

G9 ÞJI½ðZ

0JIÞkðz

0I ÞjdilþðZ0

JIÞlðz0I Þjdik�

þðSG10ÞJI½ðz

0I Þiðl

0JIÞjðz

0I Þkðz

0I Þlþðz

0I Þiðz

0I Þjðl

0JIÞkðz

0I Þlþðz

0I Þiðz

0I Þjðz

0I Þkðl

0JIÞl�þðS

G11ÞJI½ðz

0I ÞiðZ

0JIÞjðz

0I Þkðz

0I Þl

þðz0I Þiðz

0I ÞjðZ

0JIÞkðz

0I Þlþðz

0I Þiðz

0I Þjðz

0I ÞkðZ

0JIÞl�þðS

G12ÞJI½ðz

0I Þiðz

0I Þjðl

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðz

0I Þkðl

0JIÞlþðz

0I Þiðl

0JIÞjðl

0JIÞkðz

0I Þl�

þðSG13ÞJI½ðz

0I Þiðz

0I ÞjðZ

0JIÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðz

0I ÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðZ

0JIÞkðz

0I Þl�þðS

G14ÞJI½ðz

0I Þiðl

0JIÞjðZ

0JIÞkðZ

0JIÞl

þðz0I ÞiðZ

0JIÞjðl

0JIÞkðZ

0JIÞlþðz

0I ÞiðZ

0JIÞjðZ

0JIÞkðl

0JIÞl�þðS

G15ÞJI½ðz

0I ÞiðZ

0JIÞjðl

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðZ

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðl

0JIÞkðZ

0JIÞl�

þðSG16ÞJI½ðz

0I Þiðz

0I Þjðl

0JIÞkðZ

0JIÞlþðz

0I Þiðz

0I ÞjðZ

0JIÞkðl

0JIÞlþðz

0I Þiðl

0JIÞjðZ

0JIÞkðz

0I Þlþðz

0I Þiðl

0JIÞjðz

0I ÞkðZ

0JIÞl

þðz0I ÞiðZ

0JIÞjðz

0I Þkðl

0JIÞlþðz

0I ÞiðZ

0JIÞjðl

0JIÞkðz

0I Þl�g, ð37cÞ

where

ðSG1 ÞJI ¼

2L2

3

@3ðLJI1�G

JI1 Þ

@a3I

þ4vð1�vÞ

ð1�2vÞ

@GJI1

@aIþ

2vL2

ð1�2vÞ

@3ðLJI1�G

JI1 Þ

@a3I

þ@3ðLJI

1�GJI1 Þ

@aI@b2JI

þ@3½ðLJI

1 Þþ�ðGJI

1 Þþ�

@aI@ðlþ

JI Þ2

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@aI@ðl�

JI Þ2

),

(

ðSG2 ÞJI ¼

2L2

3

@3ðLJI1�G

JI1 Þ

@a3I

þð1�vÞ@GJI

1

@aI; ðSG

3 ÞJI ¼ ð1�vÞ@GJI

1

@aI;

ðSG4 ÞJI ¼

2L2

3

@3ðLJI1�G

JI1 Þ

@b3JI

þ4vð1�vÞ

ð1�2vÞ

@GJI1

@bJIþ

2vL2

ð1�2vÞ

@3ðLJI1�G

JI1 Þ

@a2I @bJI

þ@3ðLJI

1�GJI1 Þ

@b3JI

þ@3½ðLJI

1 Þþ�ðGJI

1 Þþ�

@bJI@ðlþ

JI Þ2

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@bJI@ðl�

JI Þ2

),

(

ðSG5 ÞJI ¼

2L2

3

@3ðLJI1�G

JI1 Þ

@b3JI

þð1�vÞ@GJI

1

@bJI; ðSG

6 ÞJI ¼ ð1�vÞ@GJI

1

@bJI;

ðSG7 ÞJI ¼

2L2

3

@3½ðLJI1 Þþ�ðGJI

1 Þþ�

@ðlþJI Þ3

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@ðl�JI Þ3

( )þ

4vð1�vÞ

ð1�2vÞ

@ðGJI1 Þþ

@lþJI�@ðGJI

1 Þ�

@l�JI

" #

þ2vL2

ð1�2vÞ


1 Þþ�

@a2I @lþJI

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@a2I @l�JI

þ@3½ðLJI

1 Þþ�ðGJI

1 Þþ�

@b2JI@lþJI

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@b2JI@l�JI

(

þ@3½ðLJI

1 Þþ�ðGJI

1 Þþ�

@ðlþJI Þ3

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@ðl�JI Þ3

),

ðSG8 ÞJI ¼

2L2

3


1 Þþ�

@ðlþJI Þ3

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@ðl�JI Þ3

( )þð1�vÞ

@ðGJI1 Þþ

@lþJI�@ðGJI

1 Þ�

@l�JI

" #,

ðSG9 ÞJI ¼ ð1�vÞ

@ðGJI1 Þþ

@lþJI�@ðGJI

1 Þ�

@l�JI

" #; ðSG

10ÞJI ¼ 2L2 @3ðLJI1�G

JI1 Þ

@a2I @bJI

�1

3

@3ðLJI1�G

JI1 Þ

@b3JI

" #,

ðSG11ÞJI ¼ 2L2 @3½ðLJI

1 Þþ�ðGJI

1 Þþ�

@a2I @lþJI

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@a2I @l�JI

�1

3


1 Þþ�

@ðlþJI Þ3

þ1

3

@3½ðLJI1 Þ��ðGJI

1 Þ��

@ðl�JI Þ3

),

(

ðSG12ÞJI ¼ 2L2 @3ðLJI

1�GJI1 Þ

@b2JI@aI

�1

3

@3ðLJI1�G

JI1 Þ

@a3I

" #; ðSG

13ÞJI ¼ 2L2 @3½ðLJI1 Þþ�ðGJI

1 Þþ�

@aI@ðlþ

JI Þ2

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@aI@ðl�

JI Þ2

�1

3

@3ðLJI1�G

JI1 Þ

@a3I

( ),


1 Þþ�ðGJI

1 Þþ�

@bJI@ðlþ

JI Þ2

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@bJI@ðl�

JI Þ2

�1

3

@3ðLJI1�G

JI1 Þ

@b3JI

( ),


1 Þþ�ðGJI

1 Þþ�

@b2JI@lþJI

�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@b2JI@l�JI

�1

3


1 Þþ�

@ðlþJI Þ3

þ1

3

@3½ðLJI1 Þ��ðGJI

1 Þ��

@ðl�JI Þ3

),

(


1 Þþ�ðGJI

1 Þþ�

@aI@bJI@lþJI�@3½ðLJI

1 Þ��ðGJI

1 Þ��

@aI@bJI@l�JI

" #( ): ð37dÞ

In Eqs. (37b) and (37d), FJI1 , ðFJI

1 Þþ , ðFJI

1 Þ�, LJI

1 , ðLJI1 Þþ , ðLJI

1 Þ�, GJI

1 , ðGJI1 Þþ and ðGJI

1 Þ� are defined in Eqs. (28) and (31).

It should be mentioned that the classical part SCijkl in Eqs. (37a) and (37b) depends only on Poisson’s ratio v and cannot

account for the inclusion size effect, noting that FJI1 , ðFJI

1 Þþ , ðFJI

1 Þ�, LJI

1 , ðLJI1 Þþ and ðLJI

1 Þ� involved in Eq. (37b) do not contain

the material length scale parameter L (see Eq. (28)). However, the gradient part SGijkl in Eqs. (37c) and (37d) can capture the

inclusion size effect, since Eqs. (37c) and (37d) as well as the expressions of GJI1 , ðGJI

1 Þþ and ðGJI

1 Þ� (see Eq. (31)) contain the


parameter L in addition to Poisson’s ratio v. Clearly, when L¼0 (i.e., in the absence of the strain gradient effect), SGijkl � 0

according to Eqs. (37c), (37d) and (31), thereby resulting in Sijkl ¼ SCijkl from Eq. (9a). That is, the SSGET-based Eshelby tensor

reduces to its counterpart based on classical elasticity when the strain gradient effect is not considered. Also, it is seen thatthe expressions of the classical elasticity-based Eshelby tensor in Eqs. (37a), (37b) and (28) derived here are more compactthan those given in Nozaki and Taya (2001).

The expressions of the Eshelby tensor Sijkl in Eqs. (9a), (37a), (37b), (37c) and (37d) are derived for a p-faced polyhedralinclusion of arbitrary shape. For simple-shape inclusions, more explicit expressions can be obtained for Sijkl.

3.2. Averaged Eshelby tensor

The volume average of the position-dependent Eshelby tensor, Sijkl, is given by

Sijkl ¼1

VO

Xp

M ¼ 1

Xn

N ¼ 1

Xp

I ¼ 1

Xq

J ¼ 1

ZZZONM

ðSNMÞijklðaI ,bJI ,lþ

JI ,l�JI ÞdV , ð38Þ

where (SNM)ijkl is the Eshelby tensor at point x inside ONM presented in Eqs. (9a), (37a), (37b), (37c) and (37d), VO is thevolume of the polyhedral inclusion O, ONM is the region occupied by the duplex formed by the origin (point O) of the globalcoordinate system, the projection of point O onto the Mth polygonal surface (i.e., OM) and two vertices on the Nth edge ofthe Mth surface (i.e., VþNM and V�NM), and n is the number of edges on the Mth surface. Note that this duplex ONM is differentfrom that formed by point x, its projection onto the Ith polygonal surface (i.e., xI) and two vertices on the Jth edge of the Ithsurface (i.e., VþJI and V�JI ), as shown in Fig. 4.

For the NMth duplex ONM originated from point O, the local Cartesian coordinate system (lNM, ZNM, zM) can be chosen in away similar to what was done earlier (see Fig. 2). Then, the coordinates of the vertices of the duplex OJI on the Jth edge of the Ith

surface and of an arbitrary point x within the NMth duplex ONM in the (lNM, ZNM, zM) local coordinate system can be identified

as ðvJINMþ1 , vJINMþ

2 , vJINMþ3 Þ, ðvJINM�

1 , vJINM�2 , vJINM�

3 Þ and ðxJINM1 , xJINM

2 , xJINM3 Þ, respectively. Also, the base vectors k0

JI , g0JI and f0

I

of the local coordinate system attached to the duplex OJI originated at x can be expressed in terms of the base vectors k0NM , g0

NM

and f0M . It then follows that the parameters for the duplex OJI can be determined as

aI ¼ ðvJINMþk �xJINM

k Þðz0I Þ

NMk ¼ ðv

JINM�k �xJINM

k Þðz0I Þ

NMk , ð39aÞ

bJI ¼ ðvJINMþk �xJINM

k Þðl0JIÞ

NMk ¼ ðv

JINM�k �xJINM

k Þðl0JIÞ

NMk , ð39bÞ

JIb

JIlJIl

0JIλ

Ix

x

Ia

0NMλ

0Mζ

η

O

O

ζ

η

VNM

VNM

VJI

VJI

+

++

-

- -

Fig. 4. Duplexes and the corresponding local coordinate systems constructed from an arbitrary point x and from the origin O of the global coordinate

system, respectively.


lþJI ¼ ðvJINMþk �xJINM

k ÞðZ0JIÞ

NMk , ð39cÞ

l�JI ¼ ðvJINM�k �xJINM

k ÞðZ0JIÞ

NMk , ð39dÞ

where ðl0JIÞ

NMk , ðZ0

JIÞNMk and ðz0

I ÞNMk represent, respectively, the kth components of the unit vectors k0

JI , g0JI and f0

I in the local

coordinate system (lNM, ZNM, zM) with the base vectors k0NM , g0

NM and f0M .

Using Eqs. (39a), (39b), (39c) and (39d) in Eq. (38) yields

Sijkl ¼1

VO

Xp

M ¼ 1

Xn

N ¼ 1

Xp

I ¼ 1

Xq

J ¼ 1

ZZZONM

ðSNMÞijkl½aIðlNM ,ZNM ,zMÞ, bJIðlNM ,ZNM ,zMÞ, lþJI ðlNM ,ZNM ,zMÞ, l�JI ðlNM ,ZNM ,zMÞ�dlNMdZNMdzM : ð40Þ

This general formula can be used for a polyhedral inclusion of arbitrary shape.For a polyhedral inclusion that is symmetric about the global coordinate axes x1, x2 and x3, only one eighth of the

inclusion needs to be considered and the global coordinate system can be used in all computations. The one-eighthpolyhedral domain can be divided into several sub-polyhedra with their top and bottom surfaces parallel to the x1x2-plane,and the volume integral over each sub-domain can be evaluated by direct integration using the global coordinate system.Also, only the global coordinates of all vertices need to be determined, and the unit vectors k0

JI , g0JI and f0

I in the localcoordinate system can be expressed in terms of the base vectors (i.e., e1, e2, e3) in the global coordinate system. As a result,Eq. (40) can be simplified to

Sijkl ¼8

VO

Xt

T ¼ 1

Xp

I ¼ 1

Xq

J ¼ 1

ZZZOT

ðST Þijkl½aIðx1,x2,x3Þ, bJIðx1,x2,x3Þ, lþJI ðx1,x2,x3Þ, l�JI ðx1,x2,x3Þ�dx1dx2dx3, ð41Þ

where t is the number of sub-polyhedra in the one eighth of the polyhedral inclusion, and (ST)ijkl is the Eshelby tensor atpoint x inside OT given in Eqs. (9a), (37a), (37b), (37c) and (37d).

4. Numerical results

To illustrate the general formulas of the Eshelby tensor for a p-faced polyhedral inclusion of arbitrary shape derived inSection 3, three types of polyhedral inclusions (i.e., cubic, octahedral and tetrakaidecahedral) shown in Fig. 5 arequantitatively studied in this section. Cuboids are the first polyhedral inclusions investigated using classical elasticity (e.g.,Chiu, 1977; Lee and Johnson, 1978; Liu and Wang, 2005). A tetrakaidecahedron can be generated by uniformly truncatingthe six corners of an octahedron and is known to be the only polyhedron that can pack with identical units to fill space andnearly minimize the surface energy (e.g., Li et al., 2003). Tetrakaidecahedral cells have been frequently used to representfoamed materials and interpenetrating phase composites (e.g., Li et al., 2003, 2006; Jhaver and Tippur, 2009).

Two components, S1111 and S1212, of the Eshelby tensor at any point x inside each polyhedral inclusion with varioussizes are evaluated using Eqs. (9a), (37a), (37b), (37c), (37d), (28) and (31) and plotted to demonstrate how thecomponents change with the position and inclusion size. Also, how the average Eshelby tensor component S1111 varieswith the inclusion size is presented here, which is computed using Eq. (41). For illustration purposes, Poisson’s ratio v istaken to be 0.3 and the material length scale parameter L to be 17.6 mm in the current numerical analysis, as was doneearlier (Gao and Ma, 2009, 2010a,b; Ma and Gao, 2010, 2011).

The distributions of S1111 for the cubic, octahedral, and tetrakaidecahedral inclusions along the x1 axis predicted by thecurrent model are shown in Figs. 6–8, where the values of SC

1111 are also displayed for comparison.It can be seen from Figs. 6–8 that the classical part SC

1111 (based on classical elasticity) varies with the position of xwithin each polyhedral inclusion rather than uniform, which shows that the Eshelby conjecture is true for the threepolyhedral inclusion shapes considered here. Also, it is found that for each of the three inclusion shapes SC

1111 at a givenvalue of x1/R is the same for all values of R/L, confirming the inclusion size-independence of the classical part of theEshelby tensor, which is noted near the end of Section 3.1. In addition, for all three polyhedral inclusion shapes considered,it is observed from Figs. 6–8 that when the characteristic inclusion size R (see Fig. 5) is small (compared to the length scale

x1

x2

x3

o

2R

x1

x2

x3

o

2R

x3

x2

x12R o

Fig. 5. Three types of polyhedral inclusions: (a) cubic, (b) octahedral and (c) tetrakaidecahedral.


parameter L, e.g., R/L¼2), the strain gradient part SG1111, which is the difference between S1111 and SC

1111 (i.e., SG1111 ¼

S1111�SC1111) and is displayed as the vertical distance between the SC

1111 curve and each S1111 curve in Figs. 6–8, is significantand should not be neglected. However, as the inclusion size becomes larger, the values of S1111 are all getting closer tothose of SC

1111. This means that the inclusion size effect is less significant and may be ignored for large inclusions in somecases, which agrees with the general trend observed experimentally (e.g., Cho et al., 2006).

The change of S1212 with the position and inclusion size is illustrated in Figs. 9–12 together with a comparison withSC

1212 for the three types of polyhedral inclusions. Clearly, SC1212 varies with the position of x inside each polyhedral

inclusion, which differs from that in an ellipsoidal inclusion and supports the Eshelby conjecture. But the classical partSC

1212 at a given value of x1/R remains the same for all inclusion sizes, as expected from the discussion in Section 3.1.

Fig. 7. Variation of S1111 along the x1 axis inside the octahedral inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the edge length

(see Fig. 5(b)).

Fig. 6. Variation of S1111 along the x1 axis inside the cubic inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the edge length (see Fig. 5(a)).

Fig. 8. Variation of S1111 along the x1 axis inside the tetrakaidecahedral inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the cell height

(see Fig. 5(c)).

Fig. 9. Variation of S1212 along the x1 axis inside the cubic inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the edge length (see Fig. 5(a)).


The gradient part SG1212, as the difference between S1212 and SC

1212 (i.e., SG1212 ¼ S1212�SC

1212), is seen to be significantly largefor small inclusions (e.g., R/L¼2) and becomes insignificant for large inclusions for all three types of polyhedral inclusions.More specifically, it is observed from Fig. 9 that for the cubic inclusion the strain gradient effect, as measured by the valueof SG

1212, is large for all inclusion sizes when x1/R40.6. For the octahedral inclusion, the strain gradient effect isinsignificant and can be neglected when R/L44, as illustrated in Fig. 10. For the tetrakaidecahedral inclusion, Fig. 11 showsthat the strain gradient effect is also small, especially in the region away from the square faces.

The component S1111 of the averaged Eshelby tensor varying with the inclusion size is shown in Fig. 12 for the threeinclusion shapes, where S

C

1111 is also displayed for comparison. The values of S1111 shown in Fig. 12 are obtained using Eqs.(41) and (39a), (39b), (39c) and (39d), which are also applied to get the values of S

C

1111 with L-0.It can be seen from Fig. 12 that for each polyhedral inclusion S

C

1111 (based on classical elasticity) is a constant

independent of the inclusion size R. However, S1111 predicted by the current model based on the strain gradient elasticitytheory does vary with the inclusion size: the smaller the inclusion, the smaller the Eshelby tensor component. In

particular, when the inclusion is small, the strain gradient effect, as measured by SG

1111ð ¼ S1111�SC

1111Þ, is significantly large

and should not be ignored. As the inclusion becomes large, S1111 approaches SC

1111 from below, indicating that the strain

gradient effect gets small and may be neglected for very large inclusions.The observations made here are also true for the other components of the Eshelby tensor Sijkl in Eqs. (9a), (37a), (37b),

(37c), (37d), (28) and (31) and its volume average Sijkl in Eq. (41).From the numerical results presented above, it is clear that the newly obtained Eshelby tensor based on the SSGET can

capture the inclusion size effect at the micron scale, while the Eshelby tensor based on classical elasticity does not havethis capability.

5. Conclusions

An analytical solution is provided for the Eshelby-type problem of an arbitrarily shaped polyhedral inclusion embeddedin an infinite elastic matrix using a simplified strain gradient elasticity theory (SSGET) that contains one material lengthscale parameter in addition to two classical elastic constants. The SSGET-based Eshelby tensor for a polyhedral inclusion ofarbitrary shape is analytically derived in a general form in terms of three potential functions, two of which are the same asthe ones involved in the Eshelby tensor based on classical elasticity. These potential functions, as three volume integralsover the inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes. Each of the threevolume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformedto a contour (line) integral based on Stokes’ theorem and using an inverse approach that differs from those employed inthe existing studies based on classical elasticity. The newly obtained Eshelby tensor is separated into a classical part and agradient part. The classical part depends only on Poisson’s ratio of the matrix material, while the gradient part depends on

Fig. 10. Variation of S1212 along the x1 axis inside the octahedral inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the edge length (see

Fig. 5(b)).

Fig. 11. Variation of S1212 along the x1 axis inside the tetrakaidecahedral inclusion: (a) R¼2L, (b) R¼4L and (c) R¼6L, with R being half of the cell height

(see Fig. 5(c)).


both Poisson’s ratio and the material length scale parameter that enables the explanation of the inclusion size effect. ThisSSGET-based Eshelby tensor reduces to its counterpart based on classical elasticity when the strain gradient effect is notconsidered. A general form of the volume averaged Eshelby tensor over the polyhedral inclusion is also obtained, whichcan be used in homogenization analyses of composites containing polyhedral inclusions.

To demonstrate the newly derived Eshelby tensor, three types of polyhedral inclusions, cubic, octahedral and tetra-kaidecahedral, are analyzed by directly applying the general formulas. The numerical results reveal that for each of the threeinclusion shapes the components of the new Eshelby tensor change with the position and inclusion size, whereas their classicalelasticity-based counterparts only vary with the position. When the inclusion is small, the gradient part is seen to contributesignificantly and should not be ignored. Also, it is found that the smaller the inclusion size is, the smaller the components of thevolume-averaged Eshelby tensor are. These components approach from below the values of their classical elasticity-basedcounterparts as the inclusion size becomes large. Hence, the inclusion size effect may be neglected for large polyhedralinclusions in some cases.

Acknowledgments

The work reported here is funded by a Grant from the U.S. National Science Foundation (NSF), with Dr. Clark V. Cooperas the program manager, and by a MURI Grant from the U.S. Air Force Office of Scientific Research (AFOSR), with Dr. DavidStargel as the program manager. The support from these two programs is gratefully acknowledged. The authors also wouldlike to thank Professor Huajian Gao and two anonymous reviewers for their encouragement and comments on the paper.

Fig. 12. Variation of S1111 with the inclusion size: (a) cubic, (b) octahedral and (c) tetrakaidecahedral.


References

Ammari, H., Capdeboscq, Y., Kang, H., Lee, H., Milton, G.W., Zribi, H., 2010. Progress on the strong Eshelby’s conjecture and extremal structures for theelastic moment tensor. J. Math. Pures Appl. 94, 93–106.

Chiu, Y.P., 1977. On the stress field due to initial strains in a cuboid surrounded by an infinite elastic space. ASME J. Appl. Mech. 44, 587–590.Cho, J., Joshi, M.S., Sun, C.T., 2006. Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles. Compos. Sci.

Technol. 66, 1941–1952.Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London A 241, 376–396.Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. London A 252, 561–569.Eshelby, J.D., 1961. Elastic inclusions and inhomogeneities. In: Sneddon, I.N., Hill, R. (Eds.), Progress in Solid Mechanics, vol. 2. North-Holland,

Amsterdam, pp. 89–140.Gao, X.-L., Ma, H.M., 2009. Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mech. 207, 163–181.Gao, X.-L., Ma, H.M., 2010a. Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite

spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797.Gao, X.-L., Ma, H.M., 2010b. Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. A 466, 2425–2446.Gao, X.-L., Park, S.K., 2007. Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled

cylinder problem. Int. J. Solids Struct. 44, 7486–7499.Gao, X.-L., Rowlands, R.E., 2000. Hybrid method for stress analysis of finite three-dimensional elastic components. Int. J. Solids Struct. 37, 2727–2751.Glas, F., 2001. Elastic relaxation of truncated pyramidal quantum dots and quantum wires in a half space: an analytical calculation. J. Appl. Phys. 90,

3232–3241.Jhaver, R., Tippur, H., 2009. Processing, compression response and finite element modeling of syntactic foam based interpenetrating phase composite

(IPC). Mater. Sci. Eng. A 499, 507–517.Kuvshinov, B.N., 2008. Elastic and piezoelectric fields due to polyhedral inclusions. Int. J. Solids Struct. 45, 1352–1384.Lee, J.K., Johnson, W.C., 1978. Calculation of the elastic strain field of a cuboidal precipitate in an anisotropic matrix. Phys. Status Solidi (a) 46, 267–272.Li, K., Gao, X.-L., Roy, A.K., 2003. Micromechanics model for three-dimensional open-cell foams using a tetrakaidecahedral unit cell and Castigliano’s

second theorem. Compos. Sci. Technol. 63, 1769–1781.Li, K., Gao, X.-L., Subhash, G., 2006. Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell

foams. J. Mech. Phys. Solids 54, 783–806.Li, S., Wang, G., 2008. Introduction to Micromechanics and Nanomechanics. World Scientific, Singapore.Liu, L.P., 2008. Solutions to the Eshelby conjectures. Proc. R. Soc. A 464, 573–594.Liu, S., Wang, Q., 2005. Elastic fields due to eigenstrains in a half-space. ASME J. Appl. Mech. 72, 871–878.Lubarda, V.A., Markenscoff, X., 1998. On the absence of Eshelby property for non-ellipsoidal inclusions. Int. J. Solids Struct. 35, 3405–3411.Ma, H.M., Gao, X.-L., 2010. Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory. Acta Mech.

211, 115–129.Ma, H.M., Gao, X.-L., 2011. Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical

inclusion in a finite elastic matrix. Int. J. Solids Struct. 48, 44–55.Marcadon, V., Herve, E., Zaoui, A., 2007. Micromechanical modeling of packing and size effects in particulate composites. Int. J. Solids Struct. 44,

8213–8228.Markenscoff, X., 1998a. On the shape of the Eshelby inclusions. J. Elasticity 49, 163–166.Markenscoff, X., 1998b. Inclusions with constant eigenstress. J. Mech. Phys. Solids 46, 2297–2301.Michelitsch, T.M., Gao, H., Levin, V.M., 2003. Dynamic Eshelby tensor and potentials for ellipsoidal inclusions. Proc. R. Soc. London A 459, 863–890.Mura, T., 1987. Micromechanics of Defects in Solids, second ed. Martinus Nijhoff, Dordrecht.Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Properties of Heterogeneous Materials, second ed. Elsevier Science, Amsterdam.Nenashev, A.V., Dvurechenskil, A.V., 2010. Strain distribution in quantum dot of arbitrary polyhedral shape: analytical solution. J. Appl. Phys. 107,

064322-1–064322-8.Nozaki, H., Taya, M., 2001. Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems. ASME J. Appl. Mech. 68, 441–452.Pearson, G.S., Faux, D.A., 2000. Analytical solutions for strain in pyramidal quantum dots. J. Appl. Phys. 88, 730–736.Poniznik, Z., Salit, V., Basista, M., Gross, D., 2008. Effective elastic properties of interpenetrating phase composites. Comput. Mater. Sci. 44, 813–820.Rodin, G.J., 1996. Eshelby’s inclusion problem for polygons and polyhedra. J. Mech. Phys. Solids 44, 1977–1995.Rowlinson, J.S., 1989. The Yukawa potential. Physica A 156, 15–34.Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity, third ed. McGraw-Hill, New York.Vollenberg, P.H.T., Heikens, D., 1989. Particle size dependence of the Young’s modulus of filled polymers: 1. Preliminary experiments. Polymer 30,

1656–1662.Waldvogel, J., 1979. The Newtonian potential of homogeneous polyhedra. Z. Angew. Math. Phys. 30, 388–398.Wang, J., Michelitsch, T.M., Gao, H., Levin, V.M., 2005. On the solution of the dynamic Eshelby problem for inclusions of various shapes. Int. J. Solids Struct.

42, 353–363.


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