Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution

Acta Mech 223, 293–308 (2012)DOI 10.1007/s00707-011-0559-y

Kun Zhou

Elastic field and effective moduli of periodic compositeswith arbitrary inhomogeneity distribution

Received: 31 May 2011 / Published online: 20 October 2011© Springer-Verlag 2011

Abstract This paper develops a semi-analytic model for periodically structured composites, of which eachperiod contains an arbitrary distribution of particles/fibers or inhomogeneities in a three-dimensional space.The inhomogeneities can be of arbitrary shape and have multiple phases. The model is developed using theEquivalent Inclusion Method in conjunction with a fast Fourier Transform algorithm and the Conjugate Gra-dient Method. The interactions among inhomogeneities within one computational period are fully taken intoaccount. An accurate knowledge of the stress field of the composite is obtained by setting the computationalperiod to contain one or more structural periods of the composite. The effective moduli of the composite arecalculated from average stresses and elastic strains. The model is used to analyze the stress field and effectivemoduli of anisotropic composites that have cubic symmetry. It shows that the bulk and shear moduli predictedby the present model are well located within the Hashin-Shtrikman bounds. The study also shows that thestress field of the composite can be significantly affected by the distribution of inhomogeneities even thoughthe effective moduli are not affected much.

1 Introduction

Particle/fiber constituents in a composite material have material properties different from those of the matrixconstituent and are generally referred to as inhomogeneities. Inhomogeneities interact with each other, and theinteraction becomes strong as they are closely packed. The presence of inhomogeneities and their interactiondisturb the stress field of the composite material when it is subject to external loading and significantly affectsits mechanical property at the local scale and the global scale.

Eshelby [1] pioneered the works on inhomogeneities by proposing the Equivalent Inclusion Method (EIM)that models an inhomogeneity as a homogenous inclusion with properly selected equivalent eigenstrain.Eshelby’s remarkable finding is that the equivalent eigenstrain is uniform for a single ellipsoidal inhomo-geneity embedded in an isotropic infinite space subject to remote loading. However, the equivalent eigenstrainis not uniform anymore when the inhomogeneity interacts with its neighboring ones.

Extensive studies have been done to estimate the overall or effective mechanical properties of compositematerials. These studies can be categorized into two types. One type is the prediction of the upper and lowerbounds for the effective moduli of composite materials (see, e.g., [2–5]). The bounds depend on the relativevolume ratios of the constituents and reflect limited or no information about their microstructures and geom-etries. The other type is the estimation of the effective moduli based on constituent geometries and micro-structures. A few prominent methods were developed for such an estimation, which include the differential

K. Zhou (B)School of Mechanical and Aerospace Engineering, Nanyang Technological University,50 Nanyang Avenue, Singapore 639798, SingaporeE-mail: [email protected].: +65 6790 5499Fax: +65 6792 4062

294 K. Zhou

method [6,7], the self-consistent method [8,9], the Mori-Tanaka method [10–12], the generalized self-con-sistent method [13,14], and the double-inclusion method [15], among others. The EIM was utilized mostly whenresearchers applied the self-consistent method, the Mori-Tanaka method, and the double-inclusion method.

The differential method, the self-consistent method, and the Mori-Tanaka method are based on an isolatedinhomogeneity embedded in an infinite domain, while the generalized self-consistent method and the double-inclusion method are based on an inhomogeneity embedded in a finite domain surrounded by an infinite space.The latter two methods approximately take into account the interactions between the inhomogeneities to largerextents than the former three methods. For instance, when the EIM was utilized, the self-consistent method [9]and the Mori-Tanaka method [10] treated the equivalent eigenstrain for each non-isolated inhomogeneity asuniform, while the double-inclusion method [15] was able to realize the treatment of non-uniform equivalenteigenstrain. A detailed comparison of these methods was given by Christensen [16], Nemat-Nasser and Hori[17], and Hu and Weng [18].

To account for the interaction effect more accurately, researchers have paid significant attention to com-posites with periodic structures. In such composites, the inhomogeneities are periodically distributed, and eachperiod or periodic unit cell contains a definite distribution of inhomogeneities. Therefore, the microstructurein each unit cell can be described exactly. Due to the periodicity, the Fourier series can be used to repre-sent the disturbance fields, such as displacements or strains/eigenstrains, caused by inhomogeneities so that aproblem may be solved analytically [19]. On the basis of a Fourier series approach, the EIM was extensivelyused to calculate the effective moduli of composites with periodically distributed single-phase or multiple-phase inhomogeneities of various shapes, e.g., spherical voids [20], spherical and circular-cylindrical voids[21], ellipsoidal particles [22], elliptic cracks [23], and eccentrically multi-coated ellipsoidal particles [24,25].In these studies, the unit cell is regularly structured, and most of them considered the unit cell as consisting ofa single inhomogeneity.

This study aims to develop a semi-analytic model for periodically structured composites, of which eachunit cell contains an arbitrary distribution of inhomogeneities. The inhomogeneities can be of arbitrary shapeand have different materials among themselves (multiple phases). Such a periodic composite structure hasmore generality, compared with most previously studied periodic structures. This model will be constructedbased on a solution for multiple interacting inhomogeneities in an isotropic infinite body, which was recentlyobtained by Zhou et al. [26]. Their solution fully takes into account the interactions among all the inhomo-geneities and was also extended to study inclusions beneath a frictional surface under elastic indentation [27]and a film-substrate system under elastic-plastic indentation by considering the film as an inclusion [28].

The knowledge of the stress field of a composite is the prerequisite for the study of crack nucleation, plasticdeformation and interfacial debonding within the composite. However, so far few studies have been on theelastic field of a composite because of difficulties in taking into account the interactions of inhomogeneitiesand thus providing an accurate description of the elastic field. This study aims not only to predict the effectivemoduli of a composite material but also to obtain its stress field.

2 Modeling periodically distributed inhomogeneities

Figure 1 illustrates a periodically structured composite material of which each period or unit cell containsan arbitrary distribution of inhomogeneities in a three-dimensional (3D) space. The periodicity occurs alongeach axis direction in an x–y–z Cartesian coordinate system, and the projection on the x–y plane is shown inFig. 1. The inhomogeneities can have arbitrary shape and various materials. One inhomogeneity can also becontained in another; for example, inhomogeneity �t is surrounded by inhomogeneity �s embedded in thematrix material (Fig. 1).

When two inhomogeneities interact with each other, their distance plays a critical role. The interactiondramatically decays as the distance increases, and it becomes negligible beyond a certain distance. Zhou et al.[26] demonstrated that the interaction becomes almost negligible when the centers of two identical cuboidalinhomogeneities of length size a are distanced by 4a. Therefore, it is sufficiently accurate to set a cutoff dis-tance for the interaction between two inhomogeneities, beyond which their interaction is considered negligible.In fact, this is a general way to account for the interactions among atoms in molecular dynamic simulation [29].For instance, the potential energy between two atoms beyond the cutoff distance is neglected in the atomisticsimulation of the relaxation mechanisms of a disclinated nanowire [30–33]. A computational period is definedas consisting of one or more structural periods of a composite. The proposed model sets the range of thecomputational period as the cutoff distance. Let us suppose the computational period consists of one structuralperiod in Fig. 1 and take inhomogeneity �r for an example. The model not only describes the interaction

Elastic field and effective moduli of periodic composites 295

O x

y

(z)

rΩtΩvΩuΩ

sΩ

wΩ

D

Fig. 1 Projection on the x–y plane of periodically random distribution of inhomogeneities in a composite material

Fig. 2 Discretization of a domain that contains multiple inhomogeneities into many small cuboidal elements

between�r and the other inhomogeneities within D but also the interactions between�r and inhomogeneities�s, �t , �u and �v in the neighboring periods because the five of them are located within one period rangeof �r . If the interaction between �w in the lower left corner and �r needs to be considered, a simple way isto set the computational period to consist of two structural periods. Therefore, the proposed model would beable to provide an accurate description of the interactions between all the inhomogeneities through setting asuitable computational period.

Figure 2 illustrates a computational domain D for the periodic structure in a 3D view. Domain D containsn number of arbitrarily distributed inhomogeneities�ψ (ψ = 1, 2, . . . , n), which may form one or multiplestructural periods. It is noted that the term “computational domain” rather than “unit cell” is used in the presentmethod, considering that a computational domain can contain a few structural units, while a unit cell generallyrefers to one structure unit. The elastic moduli of each�ψ are denoted by CΨ in a 6×6 matrix form, and thoseof the matrix material denoted by C in the same form. The stress, elastic strain, and total strain at any pointare denoted by σ , εe and ε in a 6 × 1 matrix form, respectively. The total strain ε equals the elastic strain εe

in the absence of inelastic strain. Since there is no inelastic strain within �ψ , the stress σ at any point within�ψ is given by σ = CΨ ε according to Hooke’s law. Using the EIM, each �ψ is simulated by a homogeneousinclusion �I

ψ that has the same material as the matrix but contains unknown equivalent eigenstrain ε∗ to bedetermined. The equivalent eigenstrains ε∗ function to represent the material dissimilarities of the inhomo-geneities, the interactions among them and the matrix, and their responses to external loading. As the totalstrain ε within �I

ψ consists of elastic strain εe and equivalent eigenstrain ε∗, the stress σ within �Iψ is given

296 K. Zhou

by σ = C(ε − ε∗). On the other hand, σ can be decomposed into σ = σ ∗ + σ 0 where σ ∗ is the eigenstresscaused by the equivalent eigenstrains ε∗ in all the homogeneous inclusions�I

ψ, and σ 0 the applied stress dueto external loading. The combination of the three stress expressions leads to the establishment of the followinggoverning equation:

(CψC−1 − I)σ ∗ + Cψε∗ = (I − CψC−1)σ 0 (1)

where I is a unit matrix and the equivalent eigenstrains ε∗ are unknowns. Equation (1) cannot be solved untilthe relationship between the eigenstress σ ∗ and the eigenstrain ε∗ is determined.

Domain D is discretized into Nx × Ny × Nz small cuboidal elements of the same size, which are eachindexed by [α, β, γ ](0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1) (Fig. 2). Each �ψ is geo-metrically approximated by the arrangement of many cuboidal inhomogeneities within it. The accuracy of theapproximation can be achieved by increasing the fineness of the domain discretization [34]. Accordingly, eachcuboidal inhomogeneity is simulated by a cuboidal homogeneous inclusion with equivalent eigenstrain. Givena fine discretization, another approximation is made by considering the equivalent eigenstrain as uniformin each cuboidal inclusion. As a result, for one cuboidal inclusion, there is only one unknown ε∗ (or sixunknown components) to be determined, and thus only one observation point within the cuboid is required forinvestigating its stress and strain status. The governing equation (1) is rewritten as

(Cα,β,γC−1 − I)σ ∗α,β,γ + Cα,β,γ ε∗

α,β,γ = (I − Cα,β,γC−1)σ 0α,β,γ ,(

0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1),

(2)

where Cα,β,γ denotes the elastic moduli of cuboid [α, β, γ ] within�ψ ; σ ∗α,β,γ and σ 0

α,β,γ denote eigenstressand applied stress at the observation point of cuboid [α, β, γ ], respectively. The observation points are set atthe center of all the cuboids.

In a periodic 3D space, domain D has 26 neighboring domains. Let us imagine that the same discretizationis applied to them. The eigenstress σ ∗

α,β,γ within D can be decomposed into

σ ∗α,β,γ = U∗

α,β,γ + V∗α,β,γ , (3)

where U∗α,β,γ and V∗

α,β,γ are the eigenstresses caused by the equivalent eigenstrains in all the cuboidal inclu-sions within D and in the neighboring domains, respectively. In the neighboring domains, only those cuboidalhomogeneous inclusions that are located within one period range of cuboid [α, β, γ ] are considered to haveeffect on V∗

α,β,γ .

The eigenstress U∗α,β,γ can be obtained by summing the eigenstress contributions of all the cuboidal inclu-

sions in domain D:

U∗α,β,γ =

Nz−1∑

ϕ=0

Ny−1∑

ς=0

Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕε∗ξ,ς,ϕ,

(0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1),

(4)

where Bα−ξ,β−ζ,γ−ϕ is the 6×6 matrix form of the influence coefficients that relate the eigenstress U∗α,β,γ within

cuboid [α, β, γ ] to the uniform eigenstrain ε∗ξ,ς,ϕ in cuboid [ξ, ζ, ϕ]. The expression of Bα−ξ,β−ζ,γ−ϕ was given

by Chiu [35] in solving the problem of a cuboid containing uniform eigenstrain in an infinite space. The summa-tions are performed on the whole domain D, although many cuboids do not contain equivalent eigenstrain. Thesummations demonstrate a discrete convolution of the influence coefficients and the equivalent eigenstrains.Liu et al. [36] developed a discrete convolution-fast Fourier transform (DC-FFT) algorithm to significantlyaccelerate such summations. The FFT algorithm can reduce the summations over multiplication operations inEq. (4) from the order of O((Nx × Ny × Nz)

2) to the order of O(Nx × Ny × Nz × ln(Nx × Ny × Nz)). Theapplication of FFT algorithm requires that the domain be equally discretized, which explains why domain Dis discretized into small cuboidal elements of the same size. In the present model, the DC-FFT algorithm isused not only to efficiently calculate the summations in Eq. (4) but also to extend them to incorporate V∗

α,β,γ

in Eq. (3) at the same time. The procedure is explained as follows.


The convolution in Eq. (4) is first converted into a cyclic convolution. Domain D is double expanded intoa 2Nx × 2Ny × 2Nz domain denoted by T, and upon domain T the following summations are established:

M∗α,β,γ =

2Nz−1∑

ϕ=0

2Ny−1∑

ς=0

2Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕε∗ξ,ς,ϕ,

(0 ≤ α ≤ 2Nx − 1, 0 ≤ β ≤ 2Ny − 1, 0 ≤ γ ≤ 2Nz − 1).

(5)

Equation (5) contains non-valued influence coefficients and equivalent eigenstrains, as can been seen from theranges of their subscripts. The non-valued equivalent eigenstrains ε∗

ξ,ς,ϕ in domain T are filled by ε∗ξ,ς,ϕ in

the original domain D in a periodic manner along each axis direction. The non-valued influence coefficientsBα−ξ,β−ς,γ−ϕ in Eq. (5) are filled by Bα−ξ,β−ς,γ−ϕ in Eq. (4) using the technique of wrap-around order [36].In an infinite space, the stress effect of the source cuboid [ξ, ζ, ϕ] with uniform eigenstrain on the observationpoint of cuboid [α, β, γ ] depends only on the relative position of the two cuboids, i.e., (α− ξ, β − ς, γ − ϕ),as this is an infinite space problem. One may imagine that two cuboids move together in domain T whileremaining in the same relative position. As they move, their index numbers [α, β, γ ] and [ξ, ζ, ϕ] change,but the coefficients Bα−ξ,β−ς,γ−ϕ that relate them remain the same. If two cuboids in domain T are distancedbeyond one period range, the coefficients Bα−ξ,β−ς,γ−ϕ related to them in Eq. (5) are assigned zero valueaccording to the rule of cutoff distance. Therefore, compared with Eq. (4), Eq. (5) extends to incorporate theeigenstress contributions of cuboidal inclusions that are located within the 7 neighbors of domain D but stillwithin one period range of an observation point in domain D.

If the summations in Eq. (5) are further extended over periodic distributions of domain T along each axisdirection, an FFT and a subsequent inverse FFT on the periodic summations would result in the final eigen-stress field as given by Eq. (3). The periodic summations fully take into account the eigenstress contributionsof cuboidal inclusions that are located within the 26 neighbors of domain D but still within one period rangeof an observation point in domain D. The cuboidal inclusions in non-neighboring domains of domain D wouldnot have any effect on the eigenstress field in domain D due to the cutoff distance. Also, it is necessary tomention that the Fourier transform automatically satisfies the periodic boundary conditions for the elastic fieldof the composite.

The FFT operation makes the convolution summation in the space domain of Eq. (5) replaced by a multi-plication operation in the frequency domain:

FFT

⎡

⎣2Nz−1∑

ϕ=0

2Ny−1∑

ς=0

2Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕε∗ξ,ς,ϕ

⎤

⎦ = FFT[Bα,β,γ

]FFT

[ε∗α,β,γ

],

(0 ≤ α ≤ 2Nx − 1, 0 ≤ β ≤ 2Ny − 1, 0 ≤ γ ≤ 2Nz − 1),

(6)

where

FFT[Bα,β,γ ] =2Nz−1∑

m=0

2Ny−1∑

l=0

2Nx −1∑

k=0

ei π mγ /Nz ei π lβ/Ny ei π kα/Nx Bk,l,m, (7)

FFT[ε∗α,β,γ ] =

2Nz−1∑

m=0

2Ny−1∑

l=0

2Nx −1∑

k=0

ei π mγ /Nz ei π lβ/Ny ei π kα/Nx ε∗k,l,m . (8)

The inverse FFT operation changes the frequency domain back to the space domain, leading to the finaleigenstress field:

σ ∗α,β,γ = U∗

α,β,γ + V∗α,β,γ = FFT−1

(FFT[Bα,β,γ ]FFT[ε∗

α,β,γ ]),

(0 ≤ α ≤ 2Nx − 1, 0 ≤ β ≤ 2Ny − 1, 0 ≤ γ ≤ 2Nz − 1).(9)

The computational efficiency of the FFT and inverse FFT and their advantage over direct summations areexplained by Zhou et al. [34]. In Eq. (9), the eigenstress field is given for the entire domain T, but only the

298 K. Zhou

eigenstress field in domain D is needed for use. The substitution of Eq. (9) into (2) results in the final governingequation:

(Cα,β,γC−1 − I)FFT−1(

FFT[Bα,β,γ ]FFT[ε∗α,β,γ ]

)+ Cα,β,γ ε∗

α,β,γ = (I − Cα,β,γC−1)σ 0α,β,γ ,

(0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1).(10)

Equation (10) seems complicated, but essentially it is a linear equation having the simple form of Gε∗ = a.The Conjugate Gradient Method (CGM) [37], a well-established method in solving linear equations, is used tosolve Eq. (10) by means of iteration. The CGM ensures absolute convergence provided that G is a symmetricpositive matrix. Nevertheless, even though G is not a symmetric positive matrix, the CGM is still capable ofreaching a deep convergence to the solution before divergence or oscillation occurs. This advantage has earnedthe CGM wide applications (see, e.g., [26–28,38,39]).

Once the equivalent eigenstrain ε∗α,β,γ in each cuboidal inclusion within domain D is determined, the entire

stress field of the domain is obtained by

σ α,β,γ = σ ∗α,β,γ + σ 0

α,β,γ = σ ∗α,β,γ +

Nz−1∑

ϕ=0

Ny−1∑

ς=0

Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕ ε∗ξ,ς,ϕ, (11)

and the elstic strain field is given by

εeα,β,γ = C−1

α,β,γ σ α,β,γ = C−1α,β,γ

⎛

⎝σ ∗α,β,γ +

Nz−1∑

ϕ=0

Ny−1∑

ς=0

Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕ ε∗ξ,ς,ϕ

⎞

⎠ within �ψ (12)

εeα,β,γ = C−1σα,β,γ = C−1

⎛

⎝σ ∗α,β,γ +

Nz−1∑

ϕ=0

Ny−1∑

ς=0

Nx −1∑

ξ=0

Bα−ξ,β−ς,γ−ϕ ε∗ξ,ς,ϕ

⎞

⎠ within matrix (13)

(0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1).

The elastic strain is determined according to Hooke’s law from the stress and the elastic moduli. Therefore,the elastic moduli Cα,β,γ of a cuboidal inhomogeneity are used in Eq. (12) to obtain the elastic strain withinthe inhomogeneity, while the elastic moduli C of the matrix are used in Eq. (13) to obtain the elastic strainwithin the matrix.

It is highlighted that it is through the determination of the equivalent eigenstrains that the material dissim-ilarity of inhomogeneities, the interactions between every two inhomogeneities within one period range andbetween the inhomogeneities and the matrix, and the response of the inhomogeneities to external loading areaccounted for.

The average stress and strain fields in a representative volume element of a composite are given by

σi j = 1

V

∫

V

σi j dV , (14)

εei j = 1

V

∫

V

εi j dV (15)

where V denotes the volume of the element (see., e.g., [19,40]). In the present model, the computational domainD is a representative volume element. The observation points of all the cuboidal elements in the discretizeddomain D are selected to be equally spaced. Thus, for a fine discretization, the averages of the stresses andelastic strains at these observation points can represent the average stress and elastic strain of the composite,i.e.

σ = 1

Nx Ny Nz

Nz−1∑

γ=0

Ny−1∑

β=0

Nx −1∑

α=0

σ α,β,γ , (16)

εe = 1

Nx Ny Nz

Nz−1∑

γ=0

Ny−1∑

β=0

Nx −1∑

α=0

εeα,β,γ , (17)


(0 ≤ α ≤ Nx − 1, 0 ≤ β ≤ Ny − 1, 0 ≤ γ ≤ Nz − 1).

Finally, the effective moduli C of a composite material can be obtained by its average stress σ and elastic strainεe:

C = σ (εe)−1. (18)

3 Numerical calculation and discussion

3.1 Composites with single-phase inhomogeneities

Figure 3 illustrates the projection on the x–y plane of a computational domain D that contains 3 × 3 × 3cuboidal inhomogeneities embedded in a matrix material. Domain D has the dimensions of lx × ly × lz , andthe lengths are set to be lx = ly = lz = 6a. All the cuboidal inhomogeneities are identical and have thedimensions of ax × ay × az . The inhomogeneities within one domain are equally spaced by dx , dy , and dzalong the three axis directions, and the spacings are set to have dx = dy = dz = d .

The configuration in Fig. 3 first represents the periodic structure of a WC/steel composite. The isotropicWC inhomogeneity has the Young’s modulus Ei = 450 GPa and Poisson’s ratio vi = 0.25; the isotropic steelmatrix has Em = 210 GPa and vm = 0.3.

The effect of the inhomogeneity spacing d on the entire stress field is studied. The lengths of thecuboids are set to have ax = ay = az = a. Thus, the volume ratio of the inhomogeneities to the com-posite is f1 = 0.125. A unidirectional stress σ0 is applied along the x-direction on the composite. Figures 4, 5and 6 present the contours of the normal stress components on a central plane for the inhomogeneity spacingsd = a and d = 0.25a. The stresses are normalized by σ0. The central plane is defined as parallel to the x–yplane and passing through the centers of cuboids. When d = a, all the inhomogeneities in the composite areequally spaced, and domain D consists of three structural periods of the composite with the period being 2aalong each axis direction. This periodicity is reflected by the stress contours in Figs. 4a, 5a and 6a, thus demon-strating that the interactions between neighboring inhomogeneities across the boundaries of the computationaldomain are taken into account. When d �= a, domain D consists of only one structural period of the composite.Figures 4b, 5b and 6b show that the stress field is significantly changed when the inhomogeneities crowd intothe center of domain D at d = 0.25a, compared with that at d = a.

Figure 7 further presents the normalized stress σ11 along a central line for different inhomogeneity spac-ings. The central line is parallel to the x-axis and through the centers of cuboids. It shows that the arrangementof inhomogeneities strongly affects the distribution and magnitudes of σ11. As the inhomogeneities get closer,the variation of σ11 along the central line becomes stronger. At d = a, the peak value of σ11inside the inho-mogeneities is about 2.8% larger than the minimum value of σ11outside the inhomogeneities. In contrast, at

yd

xd

xaya

yl

xl

x

y

O (z)

Fig. 3 Schematic of the computational domain that contains 3 × 3 × 3 equally spaced cuboidal inhomogeneities along three axisdirections

300 K. Zhou

(b) 011 /σσ(a) ad = ad 25.0=

Fig. 4 Contours of the normal stress component σ11 in a central plane for two different inhomogeneity spacings d

(b)(a) ad = 022 /σσad 25.0=


(a) (b)ad = 033 /σσad 25.0=


0 1 2 3 4 5 61.0

1.1

1.2

1.3

1.4

1.5

ad 25.0=

ad =

ad 5.0=

ad 75.0=

x/a

σ 11/

σ0

Fig. 7 Normal stress components σ11 along a central line for four different inhomogeneity spacings d


(c) (d)

ad = ad 75.0=

ad 5.0= ad 25.0=

0/σσ v

(a) (b)

Fig. 8 Contours of von Mises stress field in a central plane for four different inhomogeneity spacings d

d = 0.25a, the peak value of σ11 is more than 22% larger than the minimum value of σ11. The peak value ofσ11 for d = 0.25a increases more than 8% compared with that for d = a.

Figure 8 presents the normalized von Mises stresses on the central plane for different inhomogeneity spac-ings. It shows that stress concentration occurs around the corners of the cuboidal inhomogeneities and increasesas they get closer to each other. Such stress concentration may cause cracks to initiate at the interface betweenthe inhomogeneities and the matrix or cause yielding to occur, thus affecting the performance of compositematerials. The growth of cracks may eventually lead to material failure. Thus, it is of significance to select aproper arrangement of the inhomogeneities in the manufacturing of composite materials.

The effective moduli of the composite are also calculated. The composite structure in Fig. 3 is anisotropicand has a cubic symmetry. The material of cubic symmetry has three independent elastic constants C11,C12and C44. The relationship between the stress and elastic strain is given by

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

σ11

σ22

σ33

σ12

σ13

σ23

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

C11 C12 C12 0 0 0

C12 C11 C12 0 0 0

C12 C12 C11 0 0 0

0 0 0 2C44 0 0

0 0 0 0 2C44 0

0 0 0 0 0 2C44

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

εe11

εe22

εe33

εe12

εe13

εe23

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

. (19)

When a unidirectional normal stress σ11 is applied on such a composite, the calculation of the stress field inthe composite shows that the average shear stress and shear elastic strain components are all approximatelyzero, and σ22 ≈ σ33, εe

22 ≈ εe33. Therefore, the effective elastic constants C11 and C12 can be obtained by

C11 = εe11σ11 + εe

22σ11 − 2εe22σ22

(εe11)

2 + εe11ε

e22 − 2(εe

22)2 ,

C12 = εe11σ22 − εe

22σ11

(εe11)

2 + εe11ε

e22 − 2(εe

22)2 .

(20)

302 K. Zhou

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

600

f1

Cij

(GPa

)44C

11C

12C

Fig. 9 Dependence of the elastic constants of the composite on the volume ratio of the inhomogeneities

Similarly, the application of a unidirectional shear stress σ12 on the composite reveals the effective elasticconstant C44:

C44 = σ12

2εe21. (21)

Figure 9 presents the dependence of the three effective elastic constants on the volume ratio of the inhomo-geneities. The inhomogeneity spacings are all set to be d = 0.25a for different volume ratios. It shows thatelastic constants monotonically increase as the volume ratio increases. At f1 = 0, there is only steel andno inhomogeneities; at f1 = 1, there is only WC material. The isotropic WC material has C12 = C44, ascalculated from its Young’s modulus and Poisson’s ratio.

The calculations show that the change of the inhomogeneity spacings in the composite does not signifi-cantly affect the effective elastic constants C11, C12 and C44 of the composite for the same volume ratio ofthe inhomogeneities. For instance, for the volume ratio f1 = 0.125, the values of C11, C12 and C44 changeless than 1% when the inhomogeneity spacings vary from d = 0.25a to d = a. However, as demonstrated inFigs. 4, 5, 6, 7, and 8, the stress field experiences significant change, though the overall elastic moduli are notaffected much.

Hashin and Shtrikman [2] gave the upper and lower bounds for the effective bulk modulus and shearmodulus of an isotropic composite of two constituents. In order to compare with the Hashin–Shtrikman (H–S)bounds, the three effective elastic constants of the anisotropic composite obtained by the present method areaveraged to give the effective bulk and shear moduli. The Voigt model is used to average the three effectiveelastic constants as [41]

K = 1

3(C11 + 2C12), (22)

G = 1

5(C11 − C12 + 3C44). (23)

The H–S bounds for the effective bulk modulus and shear modulus of an isotropic composite of twoconstituents are given as [2]

K HS± = K1 + f2

(K2 − K1)−1 + f1(K1 + 4G1/3)−1 , (24)

GHS± = G1 + f2

(G2 − G1)−1 + 2 f1(K1 + 2G1)/[5G1(K1 + 4G1/3)]−1 . (25)

The parameters K1 and K2 denote the bulk moduli of individual constituents 1 and 2, respectively, and G1and G2 denote their respective shear moduli. Their respective volume ratios in the composite are denoted byf1 and f2.

Upper and lower bounds are calculated by interchanging the subscripts for the constituents. Generally,when the stiffer one of the two constituents, which has a larger Young’s modulus, is subscripted 1, the expres-sions give the upper bounds; otherwise, the expressions give the lower bounds. When the difference between


0.0 0.2 0.4 0.6 0.8 1.0150

200

250

300

H-S upper bound

H-S lower bound

Present solution

f1

K (

GPa

)

Fig. 10 Dependence of the bulk modulus of the WC/steel composite on the volume ratio of WC

0.0 0.2 0.4 0.6 0.8 1.050

100

150

200

H-S upper bound

H-S lower bound

Present solution

f1

G (

GPa

)

Fig. 11 Dependence of the shear modulus of the WC/steel composite on the volume ratio of WC

the Young’s moduli of two constituents is small, the upper and lower H-S bounds become close to each other,leading to a good estimate of the effective moduli of the composite.

Figures 10 and 11 present the effect of the volume ratio of WC inhomogeneities on the bulk modulus andshear modulus of the WC/steel composite, respectively. Since the Young’s modulus of WC is only about twotimes larger than that of steel, the gap between the upper and lower H–S bounds is very small. It shows thatthe results obtained by the present method are located well within the narrow gap between the two bounds,which validates the present method. This validation also shows that the present model can provide an accurateknowledge of the elastic field for a composite, as the bulk and shear moduli are directly calculated from theelastic field.

The narrow gap between the upper and lower bounds also explains the reason that the effective moduli arenot affected much when the inhomogeneity spacing changes. The effective moduli always lie within the gapno matter how the configurations are changed as long as the volume ratio is fixed.

Furthermore, it can be observed that the present solution is close to the lower bound. There is a physicalexplanation for this. A space is filled by numerous different sizes of spheres of material 1, each of which iscoated by a spherical shell of material 2. When material 1 is stiffer or a stiffer material forms the core, the lowerbound is realized; when material 2 is stiffer or a stiffer material forms the shell, the upper bound is realized[42]. In this case, the stiffer WC inhomogeneities are like the core in the composite. Thus, the present solutionfalls close to the lower bound.

A new case is investigated by exchanging the roles of steel and WC in the composite. The new compos-ite has steel inhomogeneities embedded in the WC matrix. Figures 12 and 13 present the solutions for thebulk modulus and shear modulus of the steel/WC composite, respectively. The variable f1 still represents thevolume ratio of WC to the composite. Therefore, the upper- and lower-bound curves in Figs. 12 and 13 arethe same as those in Figs. 10 and 11. It shows that the solution obtained by the present method falls close tothe upper bound. This is because the stiffer WC matrix now becomes like the shell in the composite.

304 K. Zhou

0.0 0.2 0.4 0.6 0.8 1.0150

200

250

300

f1

H-S upper bound

H-S lower bound

Present solution

K (

GPa

)

Fig. 12 Dependence of the bulk modulus of the steel/WC composite on the volume ratio of WC

0.0 0.2 0.4 0.6 0.8 1.050

100

150

200

f1

H-S upper bound

H-S lower bound

Present solution

G (

GPa

)

Fig. 13 Dependence of the shear modulus of the steel/WC composite on the volume ratio of WC

Consider now the composite that has the same structure as shown in Fig. 3 but consists of SiC and Mg.The constituents SiC and Mg in the composite have E = 410 GPa, v = 0.16 and E = 44 GPa, v = 0.35,respectively. Figures 14 and 15 present the effect of the volume ratio of SiC on the bulk modulus and shearmodulus of the composite, respectively. It shows that the gap between the upper and lower bounds becomeslarge compared with those for the composites of steel and WC. The large gap is due to the large difference(nearly 10 times) between the Young’s moduli of the two constituents SiC and Mg. Both Mg-matrix andSiC-matrix composites are studied. For the SiC/Mg composite, the stiffer SiC is the inhomogeneity, and thesolutions obtained from the present method fall close to the lower bounds. For the Mg/SiC composite, SiCbecomes the matrix, and the present solutions fall close to the upper bounds.

The structure in Fig. 3 is one of the numerous structures that a composite of two constituents can havefor a given volume ratio. For instance, the inhomogeneities can have different sizes, shapes, orientations, andunequal spacings. Therefore, it is predicted that the effective moduli will fall in the middle between the twobounds for certain structures or configurations, although results presented here do not demonstrate this.

3.2 Composites with two-phase inhomogeneities

Figure 16 illustrates the x–y plane projection of a computational domain D that contains 12 eccentrically coatedcuboidal inhomogeneities in a matrix material. The 12 coated inhomogeneities are arranged in the way thatevery four of them are arbitrarily structured in the x–y plane, and this pattern is repeated three periods along thez-direction. The interior cuboid of each coated inhomogeneity has the dimensions of 0.8a × 0.8a × 0.8a, theexterior cuboid has 1.6a × 1.6a × 1.6a, and the centers of the two cuboids are displaced by 0.4a. Apparently,the coating layer possesses a complicated geometry. Domain D has the dimensions of 6a × 6a × 6a.


0.0 0.2 0.4 0.6 0.8 1.00

60

120

180

240

f1

K (

GPa

)

H-S upper bound

H-S lower bound

SiC/Mg

Mg/SiC

Fig. 14 Dependence of the bulk modulus of the composite on the volume ratio of SiC

0.0 0.2 0.4 0.6 0.8 1.00

60

120

180

f1

G (

GPa

)

H-S upper bound

H-S lower bound

SiC/Mg

Mg/SiC

Fig. 15 Dependence of the shear modulus of the composite on the volume ratio of SiC

x

y

O (z)

1.6a

6a

0.8a

6a

Fig. 16 Schematic of the computational domain that contains 12 eccentrically coated inhomogeneities

The configuration in Fig. 16 is used to represent the periodic structure of the steel-matrix composite withalumina-coated Titania particles. Titania has Young’s modulus E1 = 230 GPa and Poisson’s ratio v1 = 0.27;Alumina has E2 = 370 GPa and v2 = 0.22. The composite is subjected to applied stress σ0 along the x and ydirections.

306 K. Zhou

(a) 011 /σσ 022 /σσ

033 /σσ 0/σσ v

(b)

(d)(c)

Fig. 17 Contours of the normal stress components and von Mises stresses in a central plane

Figure 17 presents the contours of the normalized normal stress components and von Mises stresses ona central x–y plane through the centers of cuboids. It shows that the stress fields are strongly distorted bythe eccentrically coated particles and do not demonstrate any symmetry due to their arbitrary presence. Stressconcentration occurs near the interfaces between the coatings and the matrix and between the coatings and theparticles.

Upon the descriptions of the elastic fields of the composite in response to applied stresses, its effectiveelastic constants can be conveniently obtained according to Hooke’s law, as given in Eq. (18).

This sample study demonstrates that the present model is capable to handle composites with complicatedconfigurations. In fact, the present model can handle multiple inhomogeneities of arbitrary shape and variousmaterials as conveniently as handle a single inhomogeneity of regular shape, without increasing any compu-tational complexity. The reason is that regardless of the geometry and material, any inhomogeneity within acomposite is first decomposed into multiple cuboidal inhomogeneities, which are then modeled by cuboidalhomogeneous inclusions that have the same material as the matrix but contain unknown equivalent eigen-strains. The unknown equivalent eigenstrains in all the cuboidal inclusions are determined simultaneously bymeans of iteration.

4 Conclusions

In this paper, a semi-analytic model is developed for periodically structured composite materials using theEIM in conjunction with the CGM and DC-FFT techniques. Each structural period of the composite containsan arbitrary distribution of inhomogeneities, which can be of any geometry and have multiple phases. Such aperiodic structure demonstrates a certain generality, compared with most previously studied periodic structuresthat contain regularly distributed inhomogeneities of simple geometries. A computational period is selectedto contain one or multiple structure periods, and the interactions between inhomogeneities within it are fullytaken into account. An accurate description of the stress fields of composites can be obtained by setting asuitable computational period. The effective moduli of the composite are calculated from the average stressesand elastic strains according to Hooke’s law.

The stress field and effective moduli of anisotropic composites of cubic symmetry are analyzed. It showsthat the bulk and shear moduli predicted by the present model are well located within narrowly gapped H–S


bounds, thus validating the model. The study also shows that the stress fields of the composite can be signifi-cantly affected by the distribution of inhomogeneities even though its effective moduli are not affected much.This highlights the importance of a proper distribution of inhomogeneities in the manufacturing of compositematerials.

Finally, it is necessary to note that solving the stress field of composites is a major challenge for the studyof composites. When the stress field of a composite is known, its elastic moduli can be conveniently calculated.It is in order to verify the stress field of a composite obtained by the current approach that the effective modulipredicted by it are compared with the H–S bounds. It is not the intention of the paper to predict the elasticmoduli using a complicated way when simple bound prediction methods are available. In fact, this method canalso predict the elastic moduli of a composite containing multiple phases of inhomogeneities, which is beyondthe capability of simple bound prediction methods.

Acknowledgments The author would like to acknowledge the Start-up Grant from Nanyang Technological University,Singapore.

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Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution