GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

Journal for Multiscale Computational Engineering, 13 (1): 11–53 (2015)

GENERAL INTEGRAL EQUATIONS OFMICROMECHANICS OF HETEROGENEOUSMATERIALS

Valeriy A. Buryachenko

Civil Engineering Department, University of Akron, Akron, Ohio 44325-3901, USA andMicromechanics and Composites LLC, 2520 Hingham Lane, Dayton, OH 45459, USA;E-mail: [email protected]

One considers a linear composite medium, which consists of a homogeneous matrix containing either the periodic orrandom set of heterogeneities. An operator form of the general integral equation (GIE) is obtained for the general casesof local and nonlocal problems, static and wave motion phenomena for composite materials with periodic and random(statistically homogeneous and inhomogeneous, so-called graded) structures containing coated or uncoated inclusionsof any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled,homogeneous or inhomogeneous external fields of different physical nature. The GIE, connecting the driving fields andfluxes in a point being considered and the fields in the surrounding points, are obtained for both the random and periodicfields of heterogeneities in the infinite media. The new GIE is presented in a general form of perturbations introduced bythe heterogeneities and taking into account a possible imperfection of interface conditions. The mentioned perturbationscan be found by any available numerical method which has advantages and disadvantages and it is crucial for theanalyst to be aware of their range of applications. The method of obtaining of the GIE is based on a centering procedureof subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliaryassumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods.One proves the absolute convergence of the proposed GIEs which are presented in two equivalent forms for both thedriving fields and fluxes. Some particular cases, asymptotic representations, and simplifications of proposed GIE arepresented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfectinterfaces, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocalelastic properties of constituents, and the wave propagation in composites with electromagnetic, optic, and mechanicalresponses.

KEY WORDS: microstructures, inhomogeneous material, thermoelastic material, interface effects, integralequations

1. INTRODUCTION

For random structure composites, in general, the problem of estimation of effective properties cannot be solved ex-actly and is usually treated in an approximate manner. The sketch of micromechanics of random structure compos-ites can be subdivided on computational micromechanics and analytical micromechanics (this classification as anyother classification cannot be perfect and reflects just one from the possible points of views). Computational mi-cromechanics contains both the analytical and numerical solutions for deterministic (in particular, periodic) fieldsof heterogeneities in the infinite homogeneous matrix (see the comprehensive review by Ghosh, 2011; Sejnoha andZeman, 2013). Such methods are based on periodization of random media (generated by the Monte Carlo simula-tions) with forthcoming numerical micromechanical analysis for each random realization of multiparticle interactionsof microinhomogeneities; see for details Buryachenko (2007) (pp. 11, 12, 335) where some limitations of computa-tional micromechanics are shortly described. According to Willis (1983) (see also Buryachenko, 2007, pp. 12–14), thenumerous methods in analytical micromechanics can be classified into four broad categories: perturbation methods

1543–1649/15/$35.00 c⃝ 2015 by Begell House, Inc. 11

12 Buryachenko

(Beran and McCoy, 1970; Lomakin, 1970), variational methods (Hashin and Shtrikman, 1962, 1963; Milton, 2002;Willis, 1977, 1981), self-consistent methods of truncation of a hierarchy (Buryachenko, 2007; Dvorak, 2013; Hill,1965; Kanaun and Levin, 2008; Kroner, 1958; Sejnoha and Zeman, 2013; Shermergor, 1977; Torquato, 2002; Willis,1983), and the model methods (Christensen, 1979; Hill, 1965; Kroner, 1958; Nielsen, 1974), among which there areno rigorous boundaries.

In particular, the multiparticle effective field method (MEFM, belonging to the group of self-consistent methods)put forward and developed by the author (see for references Buryachenko, 2007) is based on the theory of functions ofrandom variables and Green’s functions. Within this method one constructs a hierarchy of statistical moment equationsfor conditional averages of the stresses in the inclusions. The hierarchy is then cut by introducing the notion of aneffective field. This way the interaction of different inclusions is taken into account. Thus, the MEFM does notmake use of a number of hypotheses which form the basis of the traditional one-particle methods. Buryachenko(2007) demonstrated that the MEFM includes in particular cases the well-known methods of mechanics of stronglyheterogeneous media [such as the method of effective field (MEF), effective medium method (EMM), Mori–Tanakamethod (MTM), differential methods, and some others].

The most popular methods of analytical micromechanics are based just on a few basic concepts. The effectivefield hypothesis (EFH, also called theH1a hypothesis; see p. 253 in Buryachenko, 2007) is apparently the most fun-damental and most exploited concept of analytical micromechanics (see Buryachenko, 2007, where other referencescan be found). This concept has directed a development of analytical micromechanics over the last 60 years and madea contribution to their progress incompatible with any another concept. The idea of this concept dates back to Poisson(1824) and Faraday (1838) (in the electricity and magnetism context), Mossotti (1850) and Clausius (1879) (in thedielectric context), Lorenz (in the refractivity context), and Maxwell (in the conductivity context), who introduced theeffective field concept as a local homogeneous field acting on the inclusions and differing from the applied macro-scopic one. Landauer (1978), Markov (1999), and Scaife (1989) presented comprehensive reviews of the 150 yearhistory of this concept accompanied by some famous formulas with extensive references. Walpole (1966) pioneeredthe application of the concept to the static of composites under the name uniform image field. Effective field techniquewas intensively applied in analytical micromechanics of random and periodic structure composites (see for references,e.g., Morse and Feshbach, 1953; Buryachenko, 2007; Kanaun and Levin, 2008) as well as in analytical micromechan-ics of multiple interacting cracks addressed as traction or pseudo-load (Hori and Nemat-Nasser, 1987). Buryachenko(2007) has drawn the conclusion that the effective field concept is used (either explicitly or implicitly) in most popularmethods of analytical micromechanics such as, e.g., the methods of self-consistent fields and effective fields. The con-cept of the EFH (even if this term is not mentioned) in combination with subsequent assumptions totally dominates(and creates the fundamental limitations) in all four groups of analytical micromechanics in physics and mechanicsof heterogeneous media: model methods, perturbation methods, self-consistent methods, and variational ones (see forreferences and details Buryachenko, 2007).

However, EFH serves the dual function of investigations in analytical micromechanics. The EFH is acting as abackground of micromechanical models that is particularly striking in the group of model methods (see, e.g., Hill,1965; Kroner, 1958). The EFH is just a tool exploiting for approximate solution of a so-called general integral equa-tion (GIE). The GIE is the exact integral equation connecting the random fields at the point being considered andthe surrounding points. The method of obtaining this GIE that first comes to mind is the modeling of perturbationsproduced by the surrounding particles by the Green function technique. However, realization of a direct superpositionscheme of summations of these perturbations leads to the integrals which are not absolutely convergent at infinity.There is a very long and dramatic history (which looks like a scientific thriller) of attempts to modify these con-ditionally convergent integrals resulted by the long-range interactions. In particular, an idea of so-called method ofrenormalization goes back to Lord Rayleigh (1892), who apparently was the first to encounter the mentioned difficultyin consideration of effective conductivity of periodic system containing a square array of cylinders or cubic lattice ofspheres (see for details Jeffrey, 1973; O’Brian, 1979). In parallel with the renormalization technique (introduced withthe different intuitive levels of rigor), a similar assignment is performed by “noncanonical regularization” proposedby Kroner (1974) and attributed to Kanaun (1977) (see also for details Kanaun and Levin, 2008; Buryachenko, 2007,2010a) and based on an intuitive introduction of an operation (which was unknown before) of corresponding gener-alized functions on a constant symmetric tensor. At last almost 90 years after Lord Rayleigh (1892), several authors

Journal for Multiscale Computational Engineering

General Integral Equations of Micromechanics 13

Shermergor (1977), Khoroshun (1978), O’Brien (1979) rigorously proved the correctness of GIE previously proposedat the intuitive level. The centering method proposed by Shermergor (1977) for statistically homogeneous media wasgeneralized by Buryachenko (2007) to the case of statistically inhomogeneous media. The mentioned proofs weregenerally recognized as the rigorous ones and apart from their own significance, it gives the answer on the philosoph-ical question, which is primarily either EFH or GIE. Thus, it was proved that the GIE is the primary while the EFHis just a tool used for an approximate solution of this GIE, and, therefore, GIE can be considered as a background ofanalytical micromechanics.

Unfortunately, this stable period of understanding of the essence of analytical micromechanics was interrupted byBuryachenko (2010a,b), who proved that so-called exact GIEs are only approximate ones and these approximations arethe intrinsic features of these GIEs, which were implicitly assumed even before obtaining of these GIEs. Moreover,the mentioned assumptions are in fact some sort of EFH that defines a primogeniture of the EFH with respect tothe GIEs. However, a primariness of the GIEs was vindicated very soon by proposal of a new really exact GIE byBuryachenko (2010a,b). The new GIE was obtained by centering procedure of subtraction from both sides of a newinitial integral equation, their statistical averages obtained without any auxiliary assumptions such as EFH, which isimplicitly exploited in the known centering methods. Moreover, one showed that the EFH is a central one and otherconcepts play a satellite role providing the conditions for application of the EFH. The new GIEs were proposed for thelocal and nonlocal constitutive equations of thermoelasticity. Exploiting of GIEs allows us to abandon some popularunnecessary concepts of analytical micromechanics. In particular, estimated inhomogeneities of effective fields leadto the detection of fundamentally new effects for the local stresses inside heterogeneities. It is expected to get a largerdifference (which can reach infinity with the change of the sign of predicted local stresses) between the results obtainedby the use of either the new GIE or old one for composites reinforced by heterogeneities demonstrating greaterinhomogeneity of stress distributions inside the heterogeneities (see for details Buryachenko, 2010a,b, 2011a,b,c,2013, 2015; Buryachenko and Brun, 2011, 2012).

The outline of the paper is as follows. In Section 2 we present the statistical description of the composite mi-crostructure and the basic field equations in the general operator form of linear constitutive equations covering boththe nonlocal and coupled phenomena. In Section 3 the new operator form of the GIE connecting the driving fieldsand fluxes in a point being considered and the fields in the surrounding points, are obtained for the random fields ofheterogeneities in the infinite media. The new GIE is presented in a general form of perturbations introduced by theheterogeneities and taking into account a possible imperfection of interface conditions. Infinite systems of the operatorcoupled GIEs are obtained for the driving fields and fluxes. In Section 4 some particular cases, asymptotic represen-tations, and simplifications obtained in the framework of the popular micromechanical hypotheses and concepts areconsidered in parallel with analyses of its connection with the known GIEs. Section 5 is dedicated to obtaining of thegeneral operator form of the GIE for composites of periodical structures. In Section 6 one considers the GIEs for theparticular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces, conduc-tivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic propertiesof constituents, and the wave propagation in composites with electromagnetic, optic, and mechanical responses.

The current paper is dedicated to generalization of the mentioned results (Buryachenko, 2010a,b, 2011b,c, 2013;Buryachenko and Brun, 2011, 2012) of analytical micromechanics to its operator form of GIEs (rather than to solutionof these equations) for the general cases of local and nonlocal problems, static and wave motion phenomena forcomposite materials with periodic and random (statistically homogeneous and inhomogeneous, so-called graded)structures containing both the coated or uncoated inclusions of any shape and orientation with perfect and imperfectinterfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneous external fields ofdifferent physical nature. There are many other topics in micromechanics, of much industrial and scientific importance,that are either treated schematically or only mentioned. In particular, the homogenization theory of periodic structures,the geometrically and physically nonlinear problems, flow in porous media, viscoelasticity problems, cross-propertyrelations, and multiscale discrete modeling are not considered at all (or only mentioned) in the current paper. Particularsimplified and asymptotic cases of GIEs mentioned are considered and qualitatively compared with the known GIEsfor the different specific constitutive equations. For lack of space, at the consideration of the particular cases of GIEsin Sections 4, 5, and 6, the readers are referred only to the references where these GIEs were already analyzed andwhere additional references (with the corresponding numerical and experimental data) can be found. The particular

Volume 13, Number 1, 2015

14 Buryachenko

methods of analytical micromechanics are mentioned just as the examples where one or another GIE was used. For anisolated inclusion, the major results of computational mechanics obtained without any GIEs are assumed to be knownand not considered.

2. PRELIMINARIES

2.1 Statistical Description of the Composite Microstructure

Let a full spaceRd with a space dimensionalityd (d = 2 andd = 3 for 2D and 3D problems, respectively) containa homogeneous matrixv(0) and, in general, a statistically inhomogeneous setX = (vi) of heterogeneitiesvi withindicator functionsVi and bounded by the closed smooth surfacesΓi := ∂vi (i = 1, 2, . . .) defined by the relationsΓi(x) = 0 (x ∈ Γi), Γi(x) > 0 (x ∈ vi), andΓi(x) < 0 (x ∈ vi). It is assumed that the heterogeneities can begrouped into components (phases)v(q)(q = 1, 2, . . . , N) with identical mechanical and geometrical properties (suchas the shape, size, orientation, and microstructure of heterogeneities).

It is assumed that the representative macrodomainw contains a statistically large number of realizationsα (pro-viding validity of the standard probability technique) of heterogeneitiesvi ∈ v(k) of the constituentv(k) (i =1, 2, . . . ; k = 1, 2, . . . , N). A random eventα belongs to a sample spaceA, over which a probability densityp(x,α) is defined [see, e.g., Willis (1981)]. For any givenα, any random functiong(x,α) (e.g.,g = V, V (k)) isdefined explicitly as one particular member, with labelα, of an ensemble realization. Then, the mean, or ensembleaverage is defined by the angle brackets enclosing the quantityg

⟨g⟩(x) =∫Ag(x,α)p(x,α)dα. (2.1)

No confusion will arise below in notation of the random quantityg(x,α) if the labelα is removed. One treats twomaterial length scales [see, e.g., Torquato (2002)]: the macroscopic scaleL, characterizing the extent ofw, andthe microscopic scalea, related with the heterogeneitiesvi. Moreover, one supposes that applied field varies on acharacteristic length scaleΛ. The limit of our interests for both the material scales and field one is

L ≫ Λ ≥ a. (2.2)

All the random quantities under discussion are described by statistically inhomogeneous random fields. For thealternative description of the random structure of a composite material, let us introduce a conditional probabilitydensityφ(vi,xi|v1,x1, . . . , vn,xn), which is a probability density for finding a heterogeneity of typei with the centerxi in the domainvi, given that the fixed heterogeneitiesv1, . . . , vn are centered atx1, . . . ,xn (see, e.g., Willis, 1978).The configuration(vi,xi) is completely described by a detailed marked density functionφ(vi,xi|v1,x1, . . . , vn,xn)of the centers of an inclusion with markvi (which can contain information about the inclusions such as the shape,size, orientation, and material properties) being placed atxi (see for details Section 5.3.1 in Buryachenko, 2007). Thenotationφ(vi,xi|; v1,x1, . . . , vn,xn) denotes the casexi = x1, . . . ,xn. In the case of statistically inhomogeneousmedia with homogeneous matrix (for so-calledFunctionally Graded Materials, FGM, see, e.g., Markworth et al.,1995; Mortensen and Suresh, 1995) the conditional probability density is not invariant with respect to translation

φ(vi,xi|v1,x1, . . . , vn,xn) = φ(vi,xi + x|v1,x1 + x, . . . , vn,xn + x), (2.3)

i.e., the microstructure functions depend upon their absolute positions. In particular, a random field is called statisti-cally homogeneous in a narrow sense if its multipoint statistical moments of any order are shift-invariant functions ofspatial variables

φ(vi,xi|v1,x1, . . . , vn,xn) = φ(vi,xi + x|v1,x1 + x, . . . , vn,xn + x) (2.4)

for ∀x ∈ Rd and∀n ∈ Z. Of course,φ(vi,xi|; v1,x1, . . . , vn,xn) = 0 (since heterogeneities cannot overlap)for values ofxi placed inside some area∪v0mi (m = 1, . . . , n) called “excluded volumes,” wherev0mi ⊃ vm withindicator functionV 0

mi is the “excluded volumes” ofxi with respect tovm (it is usually assumed thatv0mi ≡ v0m),



andφ(vi,xi|; v1,x1, . . . , vn,xn) → φ(vi,xi) as |xi − xm| → ∞, m = 1, . . . , n (since no long-range order isassumed).φ(vi,x) is a number density,n(k) = n(k)(x) of componentv(k) ∋ vi at the pointx andc(k) = c(k)(x)is the concentration, i.e., volume fraction, of the componentvi ∈ v(k) at the pointx: c(k)(x) = ⟨V (k)⟩(x) =vin

(k)(x), vi = mesvi (k = 1, 2, . . . , N ; i = 1, 2, . . .), c(0)(x) = 1− ⟨V ⟩(x).Hereafter, if the pair distribution functiong(xi−xm) ≡ φ(vi,xi|; vm,xm)/n(k) depends onxm−xi only through

|xm − xi| it is called the radial distribution function (RDF, see for references and details Buryachenko, 2007). Addi-tionally to the average⟨(.)⟩(x), the notation⟨(.)|v1,x1; . . . ; vn,xn⟩(x) will be used for the conditional average takenfor the ensemble of a statistically inhomogeneous setX = (vi) at the pointx, on the condition that there are hetero-geneities at the pointsx1, . . . ,xn andxi = xj if i = j (i, j = 1, . . . , n). The notations⟨(.)|; v1,x1; . . . ; vn,xn⟩(x)are used for the additional conditionx /∈ v1, . . . , vn. We will distinguish macro-coordinatex used above (with“resolution” equal toΛ) and micro-coordinatez ∈ vi ⊂ v(k) So, the notation⟨(·)⟩i(x, z) at z ∈ vi ⊂ v(k) meansthe average over an ensemble realization of surrounding heterogeneities (but not over the volumevi of a particularinhomogeneity, in contrast to⟨(·)⟩(i)) at the fixedvi.

It should be mentioned that the equality for the field (e.g., stress, strain, and other) parameterg

⟨g⟩ =N∑q=1

c(q)⟨g⟩q (2.5)

is fulfilled only for statistically homogeneous media subjected to the homogeneous boundary conditions; here thesummation in the right-hand side is performed over the volume of the representative inclusionsvq ∈ v(q) (q =1, . . . , N). If any of these conditions are broken then it is necessary to consider two sorts of conditional averages(see for details Buryachenko, 2007). At first, the conditional statistical average in the inclusion phase⟨g⟩(q)(x) (atthe condition that the point with the macro-coordinatex is located in the inclusion phasex ∈ v(q)) can be foundas⟨g⟩(q)(x) = ⟨V (q)⟩−1

(x)⟨gV (q)⟩(x) where the averages⟨(.)⟩(x) are defined by Eq. (2.5). Usually, it is simplerto estimate the conditional averages of these tensors in the concrete point with the micro-coordinatez of the fixedinclusionz ∈ vq: ⟨g|vq,xq⟩(x, z) ≡ ⟨g⟩q(x, z). Although in a general case

⟨g⟩(x) ≡N∑q=1

c(q)(x)⟨g⟩(q)(x) =N∑q=1

c(q)(x)⟨⟨g|vq,xq⟩(x, z)⟩(q), (2.6)

wherevq ∈ v(q), it can be easy to establish a straightforward relation between these averages for the ellipsoidalinclusionsvq with the semi-axesaq = (a1q, . . . , a

dq)

⊤. Indeed, at first we built some auxiliary setv1q (x) formed bythe centers of translated ellipsoidsvq(0) around the fixed pointx. We constructv1q (x) as a limitv0kq → v1q (x) ifa fixed ellipsoidvk is shrinking to the pointx. Then we can get a relation between the mentioned averages [x =(x1, . . . , xd)

⊤]:

c(q)(x)⟨g⟩(q)(x) =∫v1q(x)

n(q)(y)⟨g|vq,y⟩(x,y − x) dy. (2.7)

Formula (2.7) is valid for any material inhomogeneity of inclusions of any concentration and any shape. In thisrelation (2.7), the total probability is expressed in term of conditional probabilities over portioned probability space.Obviously, the general Eq. (2.7) is reduced to Eq. (2.5) for both the statistically homogeneous media subjected tohomogeneous boundary conditions and statistically homogeneous fieldsg (e.g.,g = σ, ε). However, in a generalcaseg(vq,y)(x) ≡ f(x,y)g1(vq,y) [g1(vq,y) is a statistically homogeneous field andf(x,y) is a function ofx,y],

Eq. (2.7) is not reduced to Eq. (2.5). For statistically homogeneous fields⟨g⟩(q)(x) ≡ ⟨g⟩(q) = const. andv1q (x) ≡ v1q= const., while in general⟨g⟩q(x, z) ≡ ⟨g⟩q(z) ≡ const. at the micro-coordinatez ∈ vq. Because of this, no

confusion will arise below in notations of the average⟨g⟩(q) and conditional average⟨g⟩q(z) (at the fixed inclusionvq) which are the functions of macro-coordinate (with resolution equal toΛ) and micro-coordinatez ∈ vq (used inthe case of fixed inclusionvq), respectively.


16 Buryachenko

2.2 Basic Equations

Following Rogula (1982), we consider a linear-response problem, or a basic constitutive equation

σ(x) = (L ∗ ε)(x) + α(x), (2.8)

which physical meaning needs not be prescribed here and the current notations (2.8) usually used in elasticity theorycan have a different sense. In particular, we can consider elasticity withε – displacement,σ – external forces, or withε – strains,σ – stresses,α – eigenstresses, or electrostatics withε – electric field,σ – free charge, orε – electric field,σ – electric induction, etc. In the case of a nonlocal theory (see for references Buryachenko, 2011b,c), the operatorLis represented by the integral operator with the kernelL(x,y)

L ∗ (·) =∫

L(x,y)(·)(y)dy, α(x) =

∫m(x,y)∆T (y)dy, (2.9)

while the transformation field is described by the functionα(x) corresponding to an assumption (in the case ofthermoelasticity) that a current temperatureT is uniform in the space and the action of the nonlocal thermal expansioncoefficient is reduced to the transformation strain tensorα(x). In the case of the local theory, the operatorL is reducedto the tensor

L(x,y) = L(x)δ(x− y), (2.10)

whereδ(x− y) is the Delta function. In the framework of strongly nonlocal (integral type) elasticity (Kroner, 1967;Edelen and Laws, 1971), we follow a simplified theory for linear (macroscopically) homogeneous isotropic elastic-ity (see for references Eringen, 1999, 2002; Bazant and Jirasek, 2002), which differs from the classical one in thestress–strain constitutive relation (2.8) and (2.9) only, whereas the equilibrium and compatibility equations remainunaltered. The current interest in nonlocal simulation is driven largely by a practical need in the design, fabrication,and characterization of nanocomposites containing heterogeneities with at least one size of nano length scale.

In the event of the coupled multifield linear response, the application of driving fieldsε ≡ (ε1, . . . , εM ) tothe material induces in it “fluxes”σ ≡ (σ1, . . . ,σM ) that are related linearly to the driving fields, the latter beingderivable from potentialsu = (u1, . . . ,uM ) of different tensorial ranksεα = ∇uα (α = 1, . . . ,M), e.g., a vectorpotential in elasticity and scalar potential in conductivity (i, j = 1, . . . , d)

εij =1

2

( ∂ui

∂xj+

∂uj

∂xi

), εi =

∂u

∂xi, (2.11)

respectively. A symmetry and positive definiteness of theM × M response matrixL = (Lαβ) of submatricesLαβ = (Liαjβ) was considered by Milgrom (1999), Milgrom and Shtrikman (1989) by the use of analysis of someenergy function and Onsager reciprocity relations. For the sake of definiteness, in the 2D case we will consider a planedriving field problem. At first no restrictions are imposed on the elastic symmetry of the phases or on the geometry ofheterogeneities.1

The reciprocal constitutive equation is given by a similar representation

ε(x) = M∗σ(x) + β(x) (2.12)

with obvious correlations between the parameters involved into Eqs. (2.8) and (2.12)

L∗M = M∗L = I, M∗α = −β, L∗β = −α, (2.13)

which are reduced to the algebraic relations for the local properties

L∗M = M∗L = I, M∗α = −β, L∗β = −α, (2.14)

1For example, it is known that for 2D elastic problems the plane–strain state is only possible for material symmetry no lower thanorthotropic (see, e.g., Lekhnitskii, 1963) that will be assumed hereafter in 2D case.



whereI = diag(I1, . . . , IM ) (2.15)

is the unit tensor combined from the unitIα (α = 1, . . . ,M ) fourth-order and second-order tensors for the vector(2.111) and scalar (2.112) potentials, respectively.

We introduce acomparison body, whose mechanical propertiesgc (g = L,M,α,β) denoted by the upper indexc andgc will usually be taken as uniform overw, so that the corresponding boundary value problem is easier tosolve than that for the original body. All tensorsg (g = L,M,α,β) of material properties are decomposed as

g ≡ gc + g1(x) = gc + g(m)1 (x). The upper index(m) indicates the components and the lower indexi indicates

the individual inclusions;v(0) = w\v, v ≡ ∪v(k) ≡ ∪vi, V (x) =∑

V (k)(x) =∑

Vi(x), andV (k)(x) is anindicator function ofv(k) equals 1 atx ∈ v(k) and 0 otherwise,(m = 0, k; k = 1, 2, . . . , N ; i = 1, 2, . . .).V δ(x) ≡

∑i δ(x − xi) is the delta function of random set of heterogeneities centers (called also density field, see

Stratonovich, 1963; Ponte Castaneda and Willis, 1995). The introduction of jumps of material properties allows oneto define the fluxτ and drivingη polarization tensors by two equivalent ways(x ∈ Rd)

τ(x) = L1 ∗ ε(x) + α1(x), η(x) = M1 ∗ σ(x) + β1(x),τ = −Lc ∗ η, (2.16)

τ(x) = σ(x)− [Lc ∗ ε(x) + αc(x)], η(x) = ε(x)− [Mc ∗ σ(x) + βc(x)], (2.17)

which are simply a notational convenience and vanish inside the matrixτ(x) ≡ η(x) ≡ 0 (x ∈ v(0)) if

Lc = L(0), αc = α(0). (2.18)

HereafterM1(y) andβ1(y) are the jumps of the operatorM(k) and of the eigenstrainβ(k) inside the componentv(k) (k = 0, . . . , N) with respect to the operatorMc andβc, respectively. It is interesting, that for the local elasticity(2.10) withLc = L(0), α(x) ≡ 0, the popular polarization tensors (2.16) (see, e.g., Willis, 1981) are attributed toHashin and Shtrikman (1962) and Hill (1963b) who introduced, in fact, these notions in the equivalent form (2.17).For both local and nonlocal elasticity theory, Eqs. (2.16) and (2.17) are also equivalent; however, the form (2.17) ispreferable for subsequent manipulation because Eq. (2.17) does not explicitly use the constitutive equation (2.8) forthe heterogeneitiesx ∈ v.

Let the fieldsbegoverned by the linear elliptic partial differential operator equation (x ∈ Rd)

Lu(x) = −f(x), e.g. L = ∇L ∗ ∇+ b, (2.19)

with the functionsb(x) andf(x). The fundamental solutionG(x,y) for the whole homogeneous spaceRd definedthrough the operatorL

c= ∇Lc ∗ ∇+ bc by the equality

LcG(x,y) = −δδ(x− y) (2.20)

has the orderO(∫|x|1−dd|x|) as|x| → ∞; hereδ = diag(δ1, . . . ,δM ) is the unit tensor combined from the unitδα

(α = 1, . . . ,M ) second-order tensorsδij and scalar1 for the vector (2.111) and scalar (2.112) potentials, respectively.We additionally introduce (similarly to the momentum polarization tensor by Willis, 1980) a polarization tensor

π(x) = b1(x)u(x) + f1(x), (2.21)

whichvanishesinside the matrixπ(x) ≡ 0 (x ∈ v(0)) if bc = b(0) andf c = f (0).The interfaces between the constituent phases of CM are classically assumed to be perfect. If the phases are

perfectly bonded, the potentialu and the tractiont(x) ≡ σ(x)·n(x) components are continuous across the interphaseboundaries, i.e.,

[[u(x)]] = 0, [[σ(x)]] · n(x) = 0, (2.22)

on the interface boundaryx ∈ Γ = ∪Γi (i = 1, . . .) (assumed to be sufficiently smooth) wheren is the outwardnormal vector onΓ from v to v(0) and [[(.)]] =(out)−(in) is the jump operator. We will also consider a case of animperfect interface


18 Buryachenko

[[u(x)]] = 0, x ∈ Γui , (2.23)

[[σ(x)]] · n(x) = 0, x ∈ Γσi , (2.24)

with the jumps (2.23) and (2.24) at theΓui andΓσ

i (Γui ,Γ

σi ⊂ Γi), respectively, described by the different models of

interface imperfections. One usually assumes that either

Γui = Γi, Γ

σi = ∅ or Γσ

i = Γi, Γui = ∅. (2.25)

It is assumed, that the boundary conditions prescribed at the infinity (|x| → ∞) generate the field (x ∈ Rd)

σ0(x) = (Lc ∗ ε0)(x) + αc (2.26)

in a homogeneous medium

Lc(y + x, z+ x) = Lc(y, z), αc(x) ≡ αc = const., bc(x) ≡ bc = const., f c(x) ≡ bc = const. (2.27)

without heterogeneities (V (x) ≡ 0, ∀x,y, z ∈ Rd).For the homogeneous loading, Eq. (2.26) is presented by the constant parameters

σ0 = Lcε0 + αc, σ0, ε0 ≡ const., (2.28)

whereLc(x) ≡∫Lc(x,y)dy = const. In such a case, independently on both the local and nonlocal nature of the

constitutive law (2.8) in the micro (nano) points of a statistically homogeneous medium, the field macrovariables⟨σ⟩and⟨ε⟩ are related by the local material effective constant tensors

⟨σ⟩ = L∗⟨ε⟩+ α∗, ⟨ε⟩ = M∗⟨σ⟩+ β∗. (2.29)

3. GENERAL INTEGRAL EQUATIONS

3.1 General Integral Equation in Terms of the Green’s Functions

Substituting Eq. (2.11) and the constitutive equation (2.8) into the governing equation (2.19), we obtain a differentialequation with respect to the potentialu

∇[(L∗∇u)(x) + α(x)] + b(x)u(x) + f(x) = 0. (3.1)

In the framework of the traditional scheme, we introduce a homogeneous “comparison” body (2.27) with the solutionσ0(x), ε0(x) = ∇u0(x) corresponding to the same boundary-value problem. So, Eq. (3.1) can be transformed tothe Navier-like governing equation with respect to the potential fieldu

∇[(Lc∗∇u)(x) + αc] + bcu(x) + f c = −∇τ(x)− π(x), (3.2)

with a fictitious random “body force” in the right-hand side of the equation, whereτ(x) andπ(x) are defined byEqs. (2.161), (2.171), and (2.21), respectively. Then Eq. (3.2) can be reduced to a symmetrized integral form:

ε(x) = ε0(x) +∇∫

G(x− y)∇τ(y) + π(y)

dy, (3.3)

whereG is the infinite-homogeneous-body Green’s function of the Navier-like equation with a response operatorL

c(2.19). The Green’s functionG is assumed to be known for any operatorL

c(2.19) being considered. The

deterministic functionε0(x) is the driving field (2.26) that would exist in the medium with homogeneous properties(2.27) and appropriate boundary conditions at the infinity|x| → ∞.



The improper integral over the whole spaceRd (3.3) is defined as a limit (p → ∞) of definite integrals over anincreasing sequencewp filling Rd

ε(x) = ε0(x) + limwp→Rd

∫wp

∇xG(x− y)∇yτ(y) + π(y)

dy, (3.4)

where the limit exists independently on the particular choice of closed bounded regionswp in Rd such that anywp iscontained in everywp1 for whichp1 > p and for any bounded regionw′ in Rd there isp2 such thatw′ ⊂ wp2 . Afterintegration of Eq. (3.4) by parts, it is found that

ε(x) = ε0(x) + limwp→Rd

∫wp

U(x− y)τ(y) dy +

∫wp

∇G(x− y)π(y) dy

+

∫Γ0p

∇G(x− s)τ(s)n(s) ds

, (3.5)

which used that∇y = −∇x, ∇ = ∇x, and the surface integration is taken over the external surfaceΓ0p of the

mesodomainwp containing a statistically large number of inclusion realizations, and the integral operator kernelU isan even homogeneous generalized function of degree−d defined as

Ukljl(x) = [∇k∇lGij(x)](ki)(jl), Ukl(x) = ∇k∇lG(x) (3.6)

for the vector (2.111) and scalar (2.112) potentials, respectively. For construction of the regularization of a generalizedfunction of the type of derivatives of homogeneous regular function, we use a scheme proposed in Gelfand and Shilov(1964) according to which the tensorU(x) is split into the singularUs(x) and formalUf (x) parts

U(x) = Us(x) +Uf (x), (3.7)

whereUs(x) = δ(x)Us

(Us≡ const.) is a singular function associated with some infinitely small exclusion region

andUf (x) ≡ 1/r−dUf(n) (x = |x|n) is a formal function. Both of these terms depend on the shape of an exclusion

region being prescribed, while their sum, being the left-hand side of (3.7), is defined uniquely.Now we center Eq. (3.5) by the use of statistical averages, i.e., from both sides of Eq. (3.5) their statistical averages

are subtracted

ε(x) = ⟨ε⟩(x) + limwp→Rd

∫wp

⟨⟨U(x− y)τ⟩⟩(y)dy +

∫wp

⟨⟨∇G(x− y)π(y)⟩⟩(y)dy+ IΓϵ, (3.8)

where one introduces a centering operation

⟨⟨h(x,y)g(y)⟩⟩(y) ≡ h(x,y)g(y)− ⟨h(x,y)g(y)⟩(y), (3.9)

and in the right-hand side of Eq. (3.8), the integral over the external surfaceΓ0p := ∂wp

IΓϵ ≡ limwp→Rd

∫Γ0p

⟨⟨∇G(x− s)τ(x)⟩⟩(s)n(s)ds, (3.10)

can be dropped out, because this integral vanishes at sufficient distancex from the boundaryΓ0p:

a ≪ |x− s|, ∀s ∈ Γ0p. (3.11)

This means that if|x− s| is large enough for∀s ∈ Γ0p, then at the portion of the smooth surfaceds ≈ |x− s|d−1dωs

with a small solid angledωs the tensor∇G(x−s)|x−s|d−1 depends only on the solid angleωs variables and slowlyvaries on the portion of the surfaceds; in this sense the tensor∇G(x − s) is called a “slow” variable of the solid


20 Buryachenko

angleωs while the expressionτ(s) is a rapidly oscillating function onds and is called a “fast” variable. Thereforewe can use a rigorous theory of “separate” integration of slow and fast variables, according to which (freely speaking)the operation of surface integration may be regarded as averaging [see for details, e.g., Filatov and Sharov (1979) andits applications in elasticity Shermergor (1977)]. In consequence of established separation of slow and fast variables,no confusion is possible in understanding of estimations of averaged items⟨(.)⟩(S) depending on macro-coordinateS ∈ Γ0

p while the corresponding unaveraged items depend on micro-coordinatess ∈ Γ0p. Indeed, the slow variable

∇G(x − s) can be consistent (after the first separation of the fast and slow variables) with the macro-coordinateS ∈ Γ0

p as∇G(x − S), while the fast variableτ(s) is in agreement with the micro-coordinates ∈ Γ0p that makes

it possible to present the integrand in the integral (3.10) in the formal form∇G(x − S)[τ(s) − ⟨τ⟩(S)

]leading

(after additional separation of the fast and slow variables) to vanishing of integral from the term in the square bracketsfor statistically large number of random eventsα. If (as we assume) there is nolong-rangeorder and the functionφ(vj ,xj |; vi,xi) − φ(vj ,xj) decays at infinity sufficiently rapidly,2 then it leads to a degeneration of the surfaceintegral.

To express Eq. (3.8) in terms of fluxes, we use the identities:

L1∗(ε− β) = −Lc∗[M1∗σ

], (3.12)

ε = [Mc∗σ+ βc] + [M1∗σ+ β1]. (3.13)

Substituting (3.12) and (3.13) into the right-hand side and the left-hand side of (3.8), respectively, and contractingwith the operatorLc gives the general integral equation for the stresses:

σ(x) = ⟨σ⟩(x) + limwp→Rd

∫wp

⟨⟨Γ(x− y)∗η(y)⟩⟩(y) dy (3.14)

+

∫wp

Lc∗⟨⟨∇G(x− y)π(y)⟩⟩(y) dy+ IΓσ.

where we define

IΓσ = − limwp→Rd

∫Γ0p

Lc∗⟨⟨∇G(x− s)Lc∗η(s)⟩⟩(s)n(s) ds. (3.15)

If we assume no long-range order, then the tensorIΓσ degenerates analogously to the tensorIΓϵ (3.9) and can bedropped. The integral operator kernel

Γ(x− y) = −Lc∗(Iδ(x− y) +ULc∗) (3.16)

is called the Green flux tensor [see the local elasticity case (2.10) in Kroner, 1977, 1990].For a statistically inhomogeneous fieldX the dependence of the statistical averages⟨(.)⟩(y) (3.9) in Eqs. (3.8) and

(3.14) on the current coordinatey is of fundamental importance. But even in this case, the expressions in the averagingbrackets⟨⟨(·)⟩⟩(y) in Eqs. (3.8), (3.14) are of the orderO(|x− y|−2d) as|x− y| → ∞, and the integrals in Eqs. (3.8)and (3.14) with the kernelsU andΓ, respectively, converge absolutely. In a similar manner, the integrals with the bodyforce density converge absolutely. In fact,⟨⟨∇G(x− y)π(y)⟩⟩(y) tends to zero with|x− y| → ∞ (x ∈ vi, y ∈ vj)asO(|x−y|−d+1)

[φ(vj ,xj |; vi,xi)−φ(vj ,xj)

]. For nolong-rangeorder, an absolute convergence of the integrals

involved is assured if the functionφ(vj ,xj |; vi,xi)−φ(vj ,xj) decays at infinity sufficiently rapidly. Therefore, forx ∈ wp considered in Eqs. (3.8), (3.14) and removed far enough from the boundaryΓ0

p, the right-hand-side integralsin (3.8) and (3.14) do not depend on the shape and size of the domainwp, and they can be replaced by the integralsover the whole spaceRd that defines the general integral equations (GIEs)

ε(x) = ⟨ε⟩(x) +∫

⟨⟨U(x− y)τ(y)⟩⟩(y)dy +

∫⟨⟨∇G(x− y)π(y)⟩⟩(y)dy, (3.17)

σ(x) = ⟨σ⟩(x) +∫

⟨⟨Γ(x− y)∗η(y)⟩⟩(y) dy +

∫Lc∗⟨⟨∇G(x− y)π(y)⟩⟩(y) dy. (3.18)

2Exponential decreasing of this function was obtained by Willis (1978) for spherical inclusions; Hansen and McDonald (1986) andTorquato and Lado (1992) proposed a faster decreasing function for aligned fibers of circular cross section.



Thus, there are no difficulties connected with the asymptotic behavior of the functionsU, Γ and∇G at infinity (as|x− y|−d and|x− y|−d+1, respectively).

Particular linear elastic cases [for both local and nonlocal constitutive Eq. (2.8)] of Eqs. (3.17) and (3.18) were alsoproved by Buryachenko (2010a, 2011a) for a bounded domainw with prescribed boundary conditions at the boundaryΓ0 := ∂w which is far removed fromx ∈ w as in Eq. (3.11). However, the method used is more cumbersome thanproposed in this paper because it involves the additional ”fast” variables atΓ0 where boundary data[u0(s), t0(s)]not prescribed by the boundary conditions depend on perturbations introduced by all inhomogeneities, and, thereforeε0(x) = ε0(x,α), σ0(x) = σ0(x,α). The boundary problem becomes even more complicated for nonlocal consti-tutive Eq. (2.1) corresponding to different nonstandard boundary conditions (see, e.g., Polizzotto, 2003). The surfaceintegrals overΓ0 related with the boundary conditions onΓ0 vanish similar to the integrals (3.10) and (3.15) but theproof of this vanishing takes some additional efforts which are absent in the method proposed in the current paper andbased on the analysis of integrals over the increasing sequencewp filling Rd (3.5).

3.2 General Integral Equation in the Operator Form

The disadvantage Eqs. (3.17) and (3.18) is defined by the limited opportunities of the perfect interface conditions(2.22) while the real ones can describe the jump of both the flux[[σ]] · n and potential[[u]] (see for details Section6.1) at the interfacesΓi (i = 1, . . .). We will present an operator form of Eqs. (3.17) and (3.18) generalized to thecase of imperfect interface conditions. Namely, analysis of Eqs. (3.5) and (3.8) for one heterogeneityvk in the infinitematrix (2.18) generates an introduction of general notions of theperturbatorsof the driving fieldLτ

k(x− xk, τ), theflux Lη

k(x− xk, η), and the potentialLuk(x− xk, τ)

Lτk(x− xk, τ) ≡ ε(x)− ε0(x), (3.19)

Lηk(x− xk, η) ≡ σ(x)− σ0(x), (3.20)

Luτk (x− xk, τ) ≡ u(x)− u0(x), (3.21)

which produce the perturbation of the fieldsε(x), σ(x), andu(x), respectively, in the pointx due to insertionof the heterogeneity center into the pointxk. τ, η are the symbolic notations of dependance ofLτ

k(x − xk, τ),Lη

k(x − xk, η), andLuτk (x − xk, τ) onτ(x), η(x) (x ∈ vk), respectively, andf1(x) (x ∈ vk), [[u(x)]] (x ∈ Γu

k),and[[σ(x)]] · n (x ∈ Γσ

k ). Obviously, the perturbatorsLτk(x − xk, τ), Lη

k(x − xk, η), andLuτk (x − xk, τ) are the

generalizations of multiplications of corresponding Green’s functions and some polarizations of a point defect to theimpacts of the heterogeneityvk of the finite size with possible imperfection of boundary conditions. In particular, theperturbators can be presented through the Green functions for the perfect interface conditions [see Eq. (3.5)] and canbe modified for the case of imperfect interface in the following manner (see for details Buryachenko, 2013):

Lτk(x− xk, τ) =

∫vk

U(x− y)τ(y)dy +

∫vk

∇G(x− y)π(y)dy

−∫Γuk

∇T⊤(x− s)[[u(s)]]ds−∫Γσk

∇G(x− s)[[σ(s)]] · n(s)ds, (3.22)

Lηk(x− xk, η) =

∫vk

Γ(x− y)η(y)dy +

∫vk

Lc∇G(x− y)π(y)dy

−∫Γuk

Lc∇T⊤(x− s)[[u(s)]]ds−∫Γσk

Lc∇G(x− s)[[σ(s)]] · n(s)ds, (3.23)

Luτk (x− xk, τ) =

∫vk

∇G(x− y)τ(y)dy +

∫vk

G(x− y)π(y)dy

−∫Γuk

T⊤(x− s)[[u(s)]]ds−∫Γσk

G(x− s)[[σ(s)]] · n(s)ds, (3.24)

The perturbatorsLτk(x−xk, τ), Lη

k(x−xk, η), andLuτk (x−xk, τ) can be found from joint solutions of Eqs. (3.19)–

(3.21) and (3.22)–(3.24), respectively, and the contact conditions at the interfacesΓuk (3.23) andΓσ

k (3.24) (which are


22 Buryachenko

considered in Section 6.2 for linear elastic case in more detail). The volume integrals in Eqs. (3.22)–(3.24) areestimated over the open domains where the boundariesΓu

i , Γσi are excluded.

The driving field perturbatorLτk(x− xk, τ) and the flux perturbatorLη

k(x− xk, η) are linked by the relation

Lηk(x− xk, η) = τ(x)Vk(x) +Lc∗Lτ

k(x− xk, τ), (3.25)

which can be established by the substitution of both the constitutive law (2.8) and Eqs. (3.12), (3.13) into Eq. (3.20).Thus we have proved that Eqs. (3.19) and (3.20) are equivalent and the perturbatorsLη

k(x−xk, η) andLτk(x−xk, τ)

are related by the equality (3.25) independently of the concrete representations of these operators.Let us consider an arbitrary random realizationα of inclusions in the increasing sequencewp

ε(x,α) = ε0(x,α) + limwp→Rd

∫wp

Lτk(x− xk, τ)V

δk (xk,α)dxk

+

∫Γ0p

∇G(x− s)τ[[u]](s,α)n(s)ds

, (3.26)

σ(x,α) = σ0(x,α) + limwp→Rd

∫wp

Lηk(x− xk, η)V

δk (xk,α)dxk

−∫Γ0p

Lc∗∇G(x− s)Lc∗η[[u]](s,α)n(s)ds

, (3.27)

where the parametersτ[[u]](s,α) andη[[u]](s,α) are introduced in the surface integrals in (3.26) and (3.27) appearingto the use of the Gauss theorem analogously to Eq. (3.5) and taking into account the possibility of intersection of somepart of heterogeneitiesvi with the macroboundaryΓ0

p whereτ[[u]](s,α),η[[u]](s,α) ≡ 0 (s ∈ Γ0p). The parameters

τ[[u]](s,α) andη[[u]](s,α) are the summations of the known eigenfieldsτ(s,α) andη(s,α) (s ∈ vi) as in Eq. (3.10)and (3.15), respectively, and some additional items corresponding, e.g., to a possible imperfection of interfacesΓi

(see for details a particular elastic case in Buryachenko, 2013). However, the surface integrals in Eqs. (3.26) and(3.27) will be eliminated in subsequent transformations of Eqs. (3.26) and (3.27), and, because of this, the parametersτ[[u]](s,α) andη[[u]](s,α) are not considered here in more detail.

Now, we apply the centering method of subtraction from both sides of Eqs. (3.26) and (3.27), their statisticalaverages obtained without any auxiliary assumptions (such as, e.g., EFH). It leads to the following new equation forstatistically inhomogeneous media:

ε(x,α) = ⟨ε⟩(x) + limwp→Rd

∫wp

[Lτk(x− xk, τ)V

δk (xk,α)− ⟨Lτ

k(x− xk, τ)⟩(xk)]dxk +LΓϵ, (3.28)

σ(x,α) = ⟨σ⟩(x) + limwp→Rd

∫wp

[Lηk(x− xk, η)V

δk (xk,α)− ⟨Lη

k(x− xk, η)⟩(xk)]dxk +LΓσ. (3.29)

In the right-hand-side of Eqs. (3.28) and (3.29), one introduced the integrals over the external surfaceΓ0p

LΓϵ ≡ lim

wp→Rd

∫Γ0p

[∇G(x− s)τ[[u]](s,α)− ⟨∇G(x− s)τ[[u]]⟩(s)

]n(s)ds, (3.30)

LΓσ ≡ − lim

wp→RdL(0)

∫Γ0p

[∇G(x− s)L(0)η[[u]](s,α)− ⟨∇G(x− s)L(0)η[[u]]⟩(s)

]n(s)ds, (3.31)

which vanish at the sufficient distancex (assumed hereafter) from the boundaryΓ0p (3.11), when the validity of

separation of length scale Eq. (2.2) holds. Vanishing of the integrals (3.30) and (3.31) can be proved analogously tothe analyses of Eqs. (3.10) and (3.15).

Thus, the remaining integrals in Eqs. (3.28) and (3.29) depend only on the perturbatorsLτk(x−xk, τ) andLη

k(x−xk, η) presenting notations of the numerical solutions (3.19) and (3.20) which are quite general and not related with a



concrete numerical method. We only need to know thatLτk(x− xk, τ) andLη

k(x− xk, η) behave asO(|x− xk|−d)at the infinity|x− xk| → ∞. The volume integrals in (3.28) and (3.29) converge absolutely for both the statisticallyhomogeneous and inhomogeneous random setX of heterogeneities. Indeed, even for the FGMs, the integrands in thesquare brackets in Eqs. (3.28) and (3.29) are of orderO(|x− y|−2d) as|x− y| → ∞, and the integrals in Eqs. (3.28)and (3.29) converge absolutely. Therefore, Eqs. (3.28) and (3.29) can be reduced [analogously to Eqs. (3.17) and(3.18)] to the new GIEs

ε(x,α) = ⟨ε⟩(x) +∫[Lτ

k(x− xk, τ)Vδk (xk,α)− ⟨Lτ

k(x− xk, τ)⟩(xk)]dxk, (3.32)

σ(x,α) = ⟨σ⟩(x) +∫

[Lηk(x− xk, η)V

δk (xk,α)− ⟨Lη

k(x− xk, η)⟩(xk)]dxk, (3.33)

where the integration domainRd will be omitted hereafter for simplicity of notations.Equations (3.32) and (3.33) are equivalent and can be obtained one from the other by the use of the identities

(3.12), (3.13), and (3.25). This equivalence leads to the equivalence of the dual effective laws with reciprocallyinverse effective properties. Furthermore, although Eqs. (3.32) and (3.33) depend only on the perturbators (3.19) and(3.20), their derivation was performed by the use of the Green function [see Eqs. (3.26) and (3.27)], and, therefore,the linear response (2.8) for the matrix was assumed. However, the linear response (2.8) was not presumed for theheterogeneitiesvi and their interfacesΓi (i = 1, 2, . . .). Therefore, Eqs. (3.32) and (3.33) can be applied to a lotof problems with nonlinear properties of heterogeneities, such as, e.g., accumulation of plastic deformations in thecoating surrounding heterogeneities (see the localized model of plasticity of composites Buryachenko, 2007, Chapter16), and cohesive zone model (CZM, see, e.g., Tan et al., 2007a,b).

The mentioned perturbators (3.19)–(3.21) can be found by any available numerical method, such as, e.g., thevolume integral equation (VIE), boundary element method (BEM), FEM, hybrid FEM–BEM, multipole expansionmethod, complex potential method, and other (see for references Buryachenko, 2007; Ghosh, 2011; Liu et al., 2011).Each method has advantages and disadvantages and it is crucial for the analyst to be aware of their range of appli-cations. In particular, the VIE method for the ideal contact conditions enables one to restrict discretization to theinclusions only (in contrast to the finite element analysis, FEA), and an inhomogeneous structure of inclusions (see,e.g., Chen et al., 1990; Jayaraman and Reifsnider, 1992; You et al., 2006) presents no problem in the frameworkof the same numerical scheme (compared to the standard BEM). The VIE method (see Buryachenko, 2010b) haswell-developed routines for the solution of integral equations (such as, e.g., the iteration method and the quadra-ture schemes) and allows to analyze arbitrary inhomogeneous fieldsε0(x) andσ0(x). However, the VIE method isquite time-consuming and no optimized commercial softwares exist for its application. From other side, the FEMis supported by well-developed commercial software and gives strong advantages in term of CPU time. The FEA isespecially effective for estimations of perturbators at the constant fieldsε0,σ0 = const. (see, e.g., Buryachenko andBrun, 2011, 2012).

3.3 Infinite Coupled System of General Integral Equations

Let the inclusionsv1, . . . , vn be fixed and we define two sorts of effective fields for the driving fieldεi(x), ε1,...,n(x),and fluxσi(x), σ1,...,n(x) (i = 1, . . . , n; x ∈ v1, . . . , vn) by the use of the rearrangement of Eq. (3.32) and (3.33) inthe following operator forms (analogous particular case of these manipulation approach for linear elasticity problemwith perfect interface is given in Buryachenko, 2010a):

ε(x) = εi(x) +Lτi (x− xi, τ),

εi(x) = ε1,...,n(x) +n∑

j =i

Lτj (x− xj , τ),

ε1,...,n(x) = ⟨ε⟩(x)+∫[Lτ

k(x− xk, τ)Vδ(xk|; v1,x1; . . . ; vn,xn)

− ⟨Lτk(x− xk, τ)⟩(xk)]dxk, (3.34)


24 Buryachenko

and

σ(x) = σi(x) +Lηi (x− xi, η),

σi(x) = σ1,...,n(x) +

n∑j =i

Lηj (x− xj , η),

σ1,...,n(x) = ⟨σ⟩(x)+∫[Lη

k(x− xk, η)Vδ(xk|; v1,x1; . . . ; vn,xn)

− ⟨Lηk(x− xk, η)⟩(xk)]dxk, (3.35)

respectively, forx ∈ vi, i = 1, 2, . . . , n; hereV δ(xk|; v1,x1; . . . ; vn,xn) =∑

m δ(xk−xm)−∑n

i=1 δ(xk−xi) is arandom delta function of heterogeneity centersxm (m = 1, 2, . . .) under the condition thatxk = xi, xi = xj if i = j(i, j = 1, . . . , n). The definitions of the effective fieldsεi(x), ε1,2,...,n(x) andσi(x), σ1,2,...,n(x) as well as theirstatistical averages⟨εi⟩(x), ⟨ε1,2,...,n⟩(x) and⟨σi⟩(x), ⟨σ1,2,...,n⟩(x) are nothing more than a notation conveniencefor different terms of the infinite systems (3.34), and (3.35), respectively.

Then, considering some conditional statistical averages of the general integral equations (3.32) and (3.33) leads toan infinite system ofnew integral equations(n = 1, 2, . . .)

⟨ε | v1,x1; . . . ; vn,xn⟩(x)−n∑

i=1

⟨Lτi (x− xi, τ)|v1,x1; . . . ; vn,xn⟩i

= ⟨ε⟩(x) +∫

⟨Lτj (x− xj , τ)|; v1,x1; . . . ; vn,xn⟩jφ(vj ,xj |; v1,x1, . . . , vn,xn)

− ⟨Lτj (x− xj , τ)⟩(xj)

dxj , (3.36)

⟨σ | v1,x1; . . . ; vn,xn⟩(x)−n∑

i=1

⟨Lηi (x− xi, η)|v1,x1; . . . ; vn,xn⟩i

= ⟨σ⟩(x) +∫

⟨Lηj (x− xj , η)|; v1,x1; . . . ; vn,xn⟩jφ(vj ,xj |; v1,x1, . . . , vn,xn)

− ⟨Lηj (x− xj , η)⟩(xj)

dxj . (3.37)

Sincex ∈ v1, . . . , vn in the nth line of the system can take the values inside the inclusionsv1, . . . , vn, the nthline actually containsn equations. Statistical averaging⟨(·)⟩i stands for the averaging over the all surrounding het-erogeneities at the fixedvi, while the average⟨(·)⟩(xj) implies the averaging over all possible location ofvi withpossible dependence of this average on the macrocoordinatexi as for FGMs.

Note that substitutions of particular linear case of representations (forf1, [[u]], [[σ]] · n = 0) Lτk(x − xk, τ)

andLηk(x − xk, η) represented through Eqs. (3.5), (3.19) and (3.14), (3.20), respectively, into the new equations

(3.32), (3.33), and (3.34)–(3.37) reduce them to the known counterparts based on the Green’s functions (see, e.g.,Buryachenko, 2010a)

⟨ε|v1,x1; . . . ; vn,xn⟩(x)−n∑

i=1

∫U(x− y)⟨Vi(y)τ|v1,x1; . . . ; vn,xn⟩(y)dy

= ⟨ε⟩(x) +∫

U(x− y)⟨τ|; v1,x1; . . . ; vn,xn⟩(y)− ⟨U(x− y)τ⟩(y)dy, (3.38)

⟨σ|v1,x1; . . . ; vn,xn⟩(x)−n∑

i=1

∫Γ(x− y)⟨Vi(y)η|v1,x1; . . . ; vn,xn⟩(y)dy

= ⟨σ⟩(x) +∫

Γ(x− y)⟨η|; v1,x1; . . . ; vn,xn⟩(y)− ⟨Γ(x− y)η⟩(y)dy. (3.39)

However, comparison of Eqs. (3.341), (3.351) and (3.19), (3.20), respectively, shows that the operatorsLτk(x−xk, τ)

andLηk(x − xk, η) have the physical interpretation of perturbations introduced by a single heterogeneityvk in the



infinite homogeneous matrix (the infinite dimensions of the matrix can be approximated with the length of 40 inclusiondiameters; see Chapter 4 in Buryachenko, 2007) subjected to the effective fieldsεi(x) andσi(x), respectively, whereat first no restrictions are imposed on the inhomogeneities of effective fields.

This fact defines a fundamental advantage of representations (3.32), (3.33), and (3.34)–(3.37) with respect to theirknown counterparts (see Buryachenko, 2010a), because the mentioned perturbators (3.19)–(3.21) can be found byany available numerical method (see Section 3.2) for a wide class of static (b1 ≡ 0) and wave motion phenomena(b1 ≡ 0), coupled and uncoupled problems with perfect and imperfect interface conditions between the constituentsdescribed by local (2.10) and nonlocal constitutive laws (2.8) which arelinearfor the matrix and, perhaps, nonlinearfor the heterogeneities.

Introduction of the effective fields (3.341) and (3.351) makes it possible to define another sort of the perturbators

Lϵk(x− xk, ε) ≡ ε(x)− ε(x), (3.40)

Lσk (x− xk,σ) ≡ σ(x)− σ(x). (3.41)

Strictly speaking, the perturbatorsLτk(x− xk, τ) (3.19),Lη

k(x− xk, η) (3.20), andLuτk (x− xk, τ) (3.21) and their

integral representations (3.22)–(3.24) are just the notations of some problem which is destined to be solved while theperturbatorsLϵ

k(x − xk, ε) (3.40) andLσk (x − xk,σ) (3.41) are the solutions of this problem. In the case of the

integral representation (3.22)–(3.24), this solutions should be found from the joint solutions of Eqs. (3.19)–(3.21) and(3.22)–(3.24), respectively, and the contact conditions at the interfacesΓu

k andΓσk (linear elastic case is considered in

Section 6.2 where other methods are presented if the integral representations (3.22)–(3.24) are not used). In such acase, the GIEs (3.36) and (3.37) can be transformed to the new GIEs in terms of effective fields (x ∈ vk, k = 1, 2, . . .)

⟨ε | v1,x1; . . . ; vn,xn⟩(x)−n∑

i =k

⟨Lϵi (x− xi, ε)|v1,x1; . . . ; vn,xn⟩i

= ⟨ε⟩(x) +∫

⟨Lϵj (x− xj , ε)|; v1,x1; . . . ; vn,xn⟩jφ(vj ,xj |; v1,x1, . . . , vn,xn)

− ⟨Lϵj (x− xj , ε)⟩(xj)

dxj , (3.42)

⟨σ | v1,x1; . . . ; vn,xn⟩(x)−n∑

i =k

⟨Lσi (x− xi,σ)|v1,x1; . . . ; vn,xn⟩i

= ⟨σ⟩(x) +∫

⟨Lσj (x− xj ,σ)|; v1,x1; . . . ; vn,xn⟩jφ(vj ,xj |; v1,x1, . . . , vn,xn)

− ⟨Lσj (x− xj ,σ)⟩(xj)

dxj . (3.43)

4. SOME PARTICULAR CASES, ASYMPTOTIC REPRESENTATIONS, AND SIMPLIFICATIONS

4.1 Particular Cases

The subsequent analysis of Eqs. (3.36) and (3.37) can be performed for any response matrixLc of the comparisonmedium, which necessarily leads to some additional assumptions for the structure of the fields in the matrix. Equa-tions (3.36) and (3.37) are much easier to solve when it is necessary to estimate the fieldsσ andε only inside theheterogeneities. There are two fundamentally different approaches to ensuring it.

In the first one we postulate

Lc ≡ L(0), βc ≡ β(0), bc ≡ b(0), f ≡ f (0). (4.1)

Then the integrands with the argumentsy in Eqs. (3.36) and (3.37) vanish aty ∈ v(0). However, it does not remove thenecessity of estimating the field distributions in the matrix in the general cases of both the inhomogeneous inclusionsand inhomogeneous boundary conditions. Fortunately, this domain of the matrix is only located in the vicinity of arepresentative inhomogeneityvq (see elastic case for details, Buryachenko, 2010b).


26 Buryachenko

In the second one we chooseLc, βc, bc, f c quite arbitrarily, and analyze Eq. (3.37) [Eq. (3.36) can be consideredanalogously]. Equation (3.37) being exact for any⟨η⟩0(x) can be simplified with the additional assumption that thestrain polarization tensor in the matrixη(x), (x ∈ v(0)) coincides with its statistical average in the matrix

η(x) ≡ ⟨η⟩0(x), x ∈ v(0). (4.2)

In so doing, the assumption (4.1) is more restricted in the sense that the assumption (4.1) yields the assumption (4.2)(the opposite is not true) and, moreover, in such a case the exact equalityη(x) ≡ ⟨η⟩0(x) ≡ 0, x ∈ v(0) holds. Forconvenience in the presentation that follows, we will recast Eq. (3.37) in another form, for which we introduce theoperationλ1(x) = λ(x)− ⟨λ⟩0(x) for the random functionλ (e.g.,λ = σ, ε, u, τ, η, f ) with statistical average inthe matrix⟨λ⟩0(x). Then, Eq. (3.37) atn = 1 can be rewritten in the form

⟨σ | vi,xi⟩(x)− ⟨Lηi (x− xi, η

1)|vi,xi⟩i

= ⟨σ⟩(x) +∫

⟨Lηj (x− xj , η

1)|; vi,xi⟩jφ(vj ,xj |; vi,xi)− ⟨Lη

j (x− xj , η1)⟩(xj)

dxj , (4.3)

where in the case of the perturbation representationLηj (x−xj , η

1) through the Green’s functions we need to replace in

Eq. (3.23)η(y) → η(y)−⟨η⟩0(xj) andf1(y) → f1(y)−⟨f1⟩0(xj). In such a case, the perturbatorsLηj (x−xj , η

1)

only depend on the stress and displacement distributions inside both the heterogeneities and interfaces. However, forestimation of the effective fieldsσ (3.352) (which are necessary for assessment of effective properties) we need toevaluate the field distributions in the matrix in the vicinity of a moving inhomogeneityvi. For both assumptions (4.1)and (4.2) only in the case of asymptotic approximations of Eqs. (4.37) and (4.38) considered in Section 4.2, one canestimate the effective properties through the evaluation of the field distributions only inside the heterogeneities and atthe interfaces.

In parallel with the correct representations (3.32) and (3.33), the different particular cases of the following equa-tions for the driving field and flux are popular:

ε(x,α) = ε0(x) +

∫Lτ

k(x− xk, τ)Vδk (xk,α)dxk, (4.4)

σ(x,α) = σ0(x) +

∫Lη

k(x− xk, η)Vδk (xk,α)dxk. (4.5)

These equations contain the improper volume integrals while the surface integrals (3.26) and (3.27) are lost. However,Eqs. (4.4) and (4.5) are only correct for the fieldX bounded in one direction such as a laminated structure of some realFGM (see Plankensteiner et al., 1996, 1997, and Chapter 12 in Buryachenko, 2007) where, e.g., the heterogeneitiescentersxi = (x1

i , . . . , xdi ) satisfy the equality

x− < x1i < x+ (4.6)

for somei andx−, x+ = const. Indeed, the surface integral in (3.5) over a “cylindrical” surface (with the surface areaproportional toρd−2, whereρ = |x− s|, s ∈ Γ0

p) tends to zero with|x− s| → ∞ asρ−1 simply because the function∇G(x − s) in Eq. (3.26) behaves asO(ρ−d+1) at ρ → ∞. Therefore, for infinite media the surface integral (3.5)vanishes, and Eq. (3.26) atwp → Rd is reduced to Eq. (4.4), or, alternatively, to Eq. (4.5) presented in terms of fluxes.Clearly, in the considered case ofX bounded in one direction, Eqs. (4.4) and (4.5) are exact, and the right-hand-sideintegrals in (4.4) and (4.5) converge absolutely. A particular case of the laminated structure (4.6) was considered byTorquato (1997, 2002) for the homogeneous boundary conditions generated the fields (4.5) and for the inclusion fieldX with a constant concentration of inclusions within an ellipsoidal domain included in the infinite matrix. AlthoughEqs. (3.32) and (3.33) are more complicated than Eqs. (4.4) and (4.5), respectively, nevertheless they provide practicaladvantages because their integrands decay at infinity faster than the integrands involved in Eq. (4.4) and (4.5).

However, if the fieldX does not exhibit similar properties (4.6) then Eqs. (4.4) and (4.5) are incorrect, although itsparticular cases are of frequent use in practice for statistically homogeneous fieldX (see Sections 6.1 for referencesand details of particular examples). It such a case the improper integrals in Eqs. (4.4) and (4.5) do not absolutely



converge at infinity where the integrands behave asO(|x − y|−d) andO(|x − y|−d+1) [see Eq. (3.5)]. There are afew ways known how to overcome these difficulties (see Section 6.1 for details) and none of them can be recognizedas a mathematically rigorous method.

Analysis of incorrectness of Eqs. (4.4) and (4.5) for a general fieldX (e.g., for statistically homogeneous fieldX) was performed for a general case of external loading generating the fieldsε0(x), σ0(x) (2.26) in the comparisonmedium (2.27). However, we can consider a special loadingε0(x), σ0(x) ≡ 0 with deterministic fictitious bodyforce f1(x) ≡ 0 that [together with randomb1(x)] decays sufficiently rapidly at infinity (|x| → ∞). In such a casethe surface integrals in Eqs. (3.26) and (3.27) vanish and Eqs. (3.26) and (3.27) are reduced to Eqs. (4.4) and (4.5),respectively. However, correctness of Eqs. (4.4) and (4.5) for the special loading mentioned does not rule out itsdependence on an arbitrary deterministic functionf1(x). Nevertheless, ensemble averaging of Eqs. (4.4) and (4.5)and subtracting the results from the initial Eqs. (4.4) and (4.5) lead to Eqs. (3.32) and (3.33), which are not dependon both any boundary conditions and fictitious body forcef1(x). The last statement can be visually demonstrated forthe particular case of integral representations (3.22)–(3.24) with[[u]], [[σ]] · n,b1 ≡ 0 when, e.g., Eq. (4.5) can bepresented in a form

σ(x) =

∫Γ(x− y)η(y)dy +

∫∇G(x− y)f1(y)dy, (4.7)

where the last integral can be denoted by a known deterministic functionσ0′(x) that makes Eq. (4.7) identical in formto Eq. (4.5). Then executing of the mentioned centering operation to Eq. (4.57) yields analog of Eq. (3.33) presentedthrough Green’s function which does not depend on the prescribed deterministic functionf1(x) [the last scheme wasproposed by Drugan and Willis (1996) for a pure elastic caseβ(x) ≡ 0].

4.2 Effective Field Hypothesis and Asymptotic Approximations

The effective field hypothesis (EFH,H1) which is apparently the most fundamental and most exploited concept ofmicromechanics (see Introduction and Buryachenko, 2007, where other references can be found). The EFH is usuallyformulated as a combination of two hypotheses where the second one is currently presented in an operator form:

Hypothesis H1a). Each heterogeneityvi has an ellipsoidal form and is located in the fields (3.342) and (3.352)

εi(y) ≡ ε(xi), σi(y) ≡ σ(xi) (y ∈ vi), (4.8)

which are homogeneous over the inclusionvi.Hypothesis H1b)The perturbations introduced by the ellipsoidal heterogeneityvi at the pointx ∈ Rd are defined

by the relationsLτ

i (x− xi, τ) = Lτi (x− xi, τ(i)), Lη

i (x− xi, η) = Lηi (x− xi, η(i)). (4.9)

Here a volume average ofg over the volume ofvi denoted asg(i) (g = τ, η) is considered below. It is interestingthat hypothesisH1a is equivalent to

Lϵi (x− xi, ε) = Lϵ

i (x− xi, ε(i)), Lσi (x− xi,σ) = Lσ

i (x− xi,σ(i)). (4.10)

Only for a homogeneous ellipsoidal heterogeneityvi with b(x), f(x) ≡ 0 (x ∈ Rd), the assumption (4.10) yields theassumption (4.9), otherwise Eqs. (4.9) define the additional assumption.

For termination of the hierarchy of statistical moment equations (3.34) and (3.35) we will use the closing effectivefield hypothesis:Hypothesis H2a)For a sufficiently largen, we complete the systems (3.36) and (3.37) by the assumption

⟨ε1,...,j,...,n+1(x)⟩i = ⟨ε1,...,n(x)⟩i, ⟨σ1,...,j,...,n+1(x)⟩i = ⟨σ1,...,n(x)⟩i, (4.11)

where the right-hand side of the equality does not contain the indexj = i (i = 1, . . . , n; j = 1, . . . , n+ 1; x ∈ vi).The hypothesisH2a rewritten in terms ofσ(x) andε(x) (x ∈ vi) is a standard closing assumption (see, e.g.,

Khoroshun 1978, 1987; Willis, 1981) degenerating to the “quasi-crystalline” approximation (QCA) by Lax (1952) at


28 Buryachenko

n = 1 which is denoted asH2b. Among a few hypotheses used by Mossotti (1850), one of the most important oneswas precisely the QCA proposed 100 years later by Lax (1952) in a modern concise form. However, in the QCA thehypothesisH2b atn = 1 (4.11) implies acceptance of the equalities for the perturbators (3.42) and (3.43)

⟨Lϵj (x− xj , ε)|; v1,x1⟩j = ⟨Lϵ

j (x− xj , ε)⟩j ,

⟨Lσj (x− xj ,σ)|; v1,x1⟩j = ⟨Lσ

j (x− xj ,σ)⟩j ,

while in the MEFM, the hypothesis (4.11) atn = 1 is applied to the fieldsε1,j(x) andσ1,j(x) after estimations ofthe perturbatorsLϵ

j (x− xj , ε) andLσj (x− xj ,σ), respectively, for the considered fixed inclusionsv1 andvj placed

in the fieldsε1,j(x) andσ1,j(x). What seems to be only a formal trick yields to the discovery of fundamentally newnonlocal effects in the theory of FGMs (see for details Buryachenko, 2007).

To make further progress, the hypothesis of “ellipsoidal symmetry” for the distribution of inclusions attributed toWillis (1977) (see also Khoroshun, 1972, 1974, 1978; Ponte Castaneda and Willis, 1995) is widely used:Hypothesis H3, “ellipsoidal symmetry.” The conditional probability density functionφ(vj ,xj |; vi,xi) depends onxj − xi only through the combinationρ = |(a0ij)−1(xj − xi)|:

φ(vj ,xj |; vi,xi) = h(ρ), (4.12)

where the matrix(a0ij)−1 (which is symmetric in the indexesi andj, a0ij = a0ji) defines the ellipsoid excluded volume

v0ij = x : |(a0ij)−1x|2 < 1.The destination of the hypothesisH3 is directed toward providing conditions for applying the hypothesisH1

(see for details Buryachenko and Brun, 2011, 2012). The use of the satellite hypothesisH3 has no sense without thehypothesisH1. If hypothesesH1, H2b, andH3 hold for the statistically homogeneous material and the field parameter(that corresponds to the method of effective field, MEF, see for details Buryachenko, 2007) then the effective propertiesL∗ andα∗ do not depend on the size of the correlation holev0i and the binary correlation functionφ(vj ,xj |; vi,xi).

We consider the hypothesisH1 in most details. For the comparison medium properties coinciding with the matrixproperties (4.1), we decomposeτ(i) andη(i) in such a manner that the perturbators (4.9) of the volume average ofpolarization tensors are described as

Lτi (x− xi, τ(i)) =

∫vi

U(x− y)dyτf(i) +

∫vi

∇G(x− y)dyπ(i), (4.13)

Lηi (x− xi, η(i)) =

∫vi

Γ(x− y)dyηf(i) +

∫vi

Lc∇G(x− y)dyπ(i), (4.14)

Luτi (x− xi, τ(i)) =

∫vi

∇G(x− y)dyτf(i) +

∫vi

G(x− y)dyπ(i), (4.15)

whereτf(i) and ηf

(i) do not depend onb1(x), f1(x) and are defined by the averages of Eqs. (2.17) over the closeddomainvi ∪ Γi taking into account the jumps (2.23), (2.24) at the interfaceΓi

τf(i) = σ(i) − [Lc ∗ ε(i) + αc] +

1

vi

∫Γσi

([[σ]] · n)⊗ sds− 1

2viLc∗

∫Γϵi

([[u]]⊗ n+ n⊗ [[u]])ds, (4.16)

ηf(i) = ε(i) − [Mc ∗ σ(i) + βc]− 1

viMc∗

∫Γσi

([[σ]] · n)⊗ sds+1

2vi

∫Γϵi

([[u]]⊗ n+ n⊗ [[u]])ds, (4.17)

while the averagesε(i) andσ(i) in (4.16) and (4.17) are estimated over the opened regionvi (see, e.g., Duan andKarihallo, 2007; Hashin, 1991b).

For a homogeneous ellipsoidal inclusionvi the classical assumption (4.8) (see, e.g., Buryachenko, 2007; Kanaunand Levin, 2008) yields the assumptions (4.13)–(4.15) for the basic problems of micromechanics (4.1), otherwise theformulas (4.13)–(4.15) define the additional assumptions. Due to arbitrary inhomogeneity of the heterogeneityvi, theassumptionH1a for the ellipsoidal shape ofvi can be relaxed and we can assume any shape ofvi in the hypothesis



H1a. In so doing, the ellipsoidal shape of heterogeneitiesvi is essential for subsequent application of the hypothesisH1 if we are going to work in the framework of the effective field homogeneity (see for details Buryachenko, 2010a).

For b1(x), f1(x) ≡ 0, according to the hypothesisH1a and in view of the linearity of the problem, there existconstant tensorsRϵ

i (x),Ri(x),Fϵi (x),Fi(x), such as (x ∈ Rd)

viτf(x) = Rϵ

i (x)ε(xi) + Fϵi (x), (4.18)

viηf(x) = Ri(x)σ(xi) + Fi(x). (4.19)

Representations (4.13)–(4.15) can be recast in the new notations(x ∈ Rd)

Lτi (x− xi, τ(i)) = viT

ϵ(x− xi)τf(i) + viT

∇G(x− xi)π(i), (4.20)

Lηi (x− xi, η(i)) = viT

σ(x− xi)ηf(i) + viLc∗T∇G(x− xi)π(i), (4.21)

Luτi (x− xi, τ(i)) = viT

∇G(x− xi)τf(i) + viT

G(x− xi)π(i), (4.22)

where one introduces the tensors

Tεi (x−xi) =

−(vi)

−1Pi(x) for x ∈ vi,

(vi)−1 ∫ U(x− y)Vi(y)dy for x ∈ vi,

, (4.23)

Tσi (x−xi) =

−(vi)

−1Qi(x) for x ∈ vi,

(vi)−1 ∫ Γ(x− y)Vi(y)dy for x ∈ vi,

, (4.24)

T∇Gi (x−xi) =

−(vi)

−1P∇Gi (x) for x ∈ vi,

(vi)−1 ∫ ∇G(x− y)Vi(y)dy for x ∈ vi,

, (4.25)

TGi (x−xi) =

−(vi)

−1PGi (x) for x ∈ vi,

(vi)−1 ∫ G(x− y)Vi(y)dy for x ∈ vi,

, (4.26)

with the known point approximation

Tεi (x− xi) = U(x− xi), Tσ

i (x− xi) = Γ(x− xi), (4.27)

T∇Gi (x− xi) = ∇G(x− xi), TG

i (x− xi) = G(x− xi). (4.28)

HereTεi (x−xi) (4.23) andTσ

i (x−xi) (4.24) have analytical representations inside [Pi(x) ≡ −∫U(x−y)Vi(y)dy,

Qi(x) ≡ −∫Γ(x − y)Vi(y)dy, x ∈ vi] and outside (x ∈ vi) of an ellipsoidal inclusionvi presented through the

Eshelby (1961) tensorSi and considered in Buryachenko (2007) for some particular problems in more detail. It isobvious that Eqs. (4.20)–(4.22) are only asymptotically fulfilled at|x−xi| → ∞, and it is possible to propose a formalcounterexample where an error of any approximations (4.20)–(4.22) equals infinity, e.g. ifg(x) ≡ 0 and⟨g⟩(i) = 0

(g = τ,η, x ∈ vi). The most popular point approximations (4.27) and (4.28) (which are simultaneously the mostcrude) was implicitly used by many authors for some particular problems (see for early references, e.g., Beran andMcCoy, 1970) including some sort of the centered Eq. (3.5) (see Zeller and Dederichs, 1973). A quantitative analysisof results obtained by the use of the some representations (4.20)–(4.22) was performed by Buryachenko (2010b) forlocal uncoupled elasticity.


30 Buryachenko

Thus, in the framework of the hypothesesH1a andH1b, Eqs. (3.36) and (3.37) atn = 1 are reduced to (x ∈ v1)

⟨ε | v1,x1⟩(x)− ⟨Lτ1(x− x1, τ(1))|v1,x1⟩1

= ⟨ε⟩(x) +∫

⟨Lτj (x− xj , τ(j))|; v1,x1⟩jφ(vj ,xj |; v1,x1)− ⟨Lτ

j (x− xj , τ(j))⟩(xj)dxj , (4.29)

⟨σ | v1,x1⟩(x)− ⟨Lη1 (x− x1, η(1))|v1,x1⟩1

= ⟨σ⟩(x) +∫

⟨Lηj (x− xj , η(j))|; v1,x1⟩jφ(vj ,xj |; v1,x1)− ⟨Lη

j (x− xj , η(j))⟩(xj)dxj . (4.30)

where the argument⟨(·)⟩(xk) means a possibility of functional graded effects. For statistically homogeneous mediasubjected homogeneous boundary conditions, this argument can be dropped.

The use of Eqs. (3.42) and (3.43) for the effective field rather than for the polarization tensors (3.36) and (3.37) ispreferable for construction of its counterparts in the framework of the hypothesisH1. Indeed, exploiting only of thehypothesisH1a without the hypothesisH1b [as in Eqs. (4.29) and (4.30)] makes it possible to reduce Eq. (3.42) and(3.43) averaged over the volume of the fixed inclusionv1 atn = 1 to (v1, vj ⊂ v(1))

⟨ε⟩(1) = ⟨ε⟩(1) +∫

⟨Lϵj (x− xj , ε(j))|; v1,x1⟩j(1)φ(vj ,xj |; v1,x1)− ⟨Lϵ

j (x− xj , ε(j))⟩(1)(xj)dxj , (4.31)

⟨σ⟩(1) = ⟨σ⟩(1) +∫

⟨Lσj (x− xj ,σ(j))|; v1,x1⟩j(1)φ(vj ,xj |; v1,x1)− ⟨Lσ

j (x− xj ,σ(j))⟩(1)(xj)dxj , (4.32)

which leaves room for a subsequent simplification in the framework of the hypothesisH2b

⟨ε⟩(1) = ⟨ε⟩(1) +∫

⟨Lϵj (x− xj , ε(1))⟩j(1)[φ(vj ,xj |; v1,x1)− n(j)]dxj , (4.33)

⟨σ⟩(1) = ⟨σ⟩(1) +∫

⟨Lσj (x− xj ,σ(1))⟩j(1)[φ(vj ,xj |; v1,x1)− n(j)]dxj , (4.34)

where⟨σ(1)⟩ = ⟨σ⟩(1).For the comparison medium properties coinciding with the matrix properties (4.1), the perfect interface (2.22),

and the integral representations of the perturbators (3.22) and (3.23), Eqs. (3.32) and (3.33) can be presented in theforms

ε(x) = ⟨ε⟩(x) +∫

U(x− y)[τ(y)− ⟨τ⟩(y)]dy +

∫∇G(x− y)[π(y)− ⟨π⟩(y)]dy, (4.35)

σ(x) = ⟨σ⟩(x) +∫

Γ(x− y)∗[η(y)− ⟨η⟩(y)]dy +L(0)∗∫

∇G(x− y)[π(y)− ⟨π⟩(y)]dy, (4.36)

where one used the point approximations (4.27) and (4.28) which in combination with (4.9) are equivalent to theequality

⟨U(x− y)τ⟩(y) = U(x− y)⟨τ⟩(y), (4.37)

⟨Γ(x− y)∗η⟩(y) = Γ(x− y)∗⟨η⟩(y), (4.38)

⟨∇G(x− y)π⟩(y) = ∇G(x− y)⟨π⟩(y). (4.39)

Buryachenko (2013) has estimated the error of the approximations (4.37) in some model example and proved an incor-rectness of a popular justification (with an intuitive level of rigor) of Eqs. (4.37)–(4.39) asserting that the deterministicfunctionsU(x − y), Γ(x − y), and∇G(x − y) can be always carried out from the brackets⟨(·)⟩(y) of statisticalaverage.



Additional assumptions of the statistical homogeneity of media subjected to the homogeneous boundary condi-tions reduce Eqs. (4.35) and (4.36) to the following ones:

ε(x) = ⟨ε⟩(x) +∫

U(x− y)[τ(y)− ⟨τ⟩]dy,+∫

∇G(x− y)[π(y)− ⟨π⟩]dy, (4.40)

σ(x) = ⟨σ⟩(x) +∫

Γ(x− y)∗[η(y)− ⟨η⟩]dy +L(0)∗∫

∇G(x− y)[π(y)− ⟨π⟩]dy. (4.41)

For the statistically homogeneous material (2.4) and field (2.24) parameters and the uncoupled (M = 1) local elasticityconstitutive equation, Buryachenko (2013) proved that substitution of both the approximations (4.23)–(4.26) and(4.27) and (4.28) into Eqs. (4.20) and (4.21) leads to the equivalent final results presented in the form of Eqs. (4.40)and (4.41). This equivalence proof can be generalized in a straightforward manner to the currently considered case ofcoupled nonlocal constitutive equations (2.8). However, in the case of violating the assumptions of either the statisticalhomogeneity of media or homogeneity of boundary conditions, the mentioned equivalence is not true.

Moreover, for a particular case of the fieldX bounded in one direction, Eqs. (4.4) and (4.5) are reduced to theequations

ε(x) = ε0 +

∫U(x− y)τ(y)dy +

∫∇G(x− y)π(y)dy, (4.42)

σ(x) = σ0 +

∫Γ(x− y)∗η(y)dy +L(0)∗

∫∇G(x− y)π(y)dy, (4.43)

which particular cases were also widely used for statistically homogeneous media subjected to the homogeneousboundary conditions [incorrectness of Eq. (4.42) and (4.43) in the last case is considered in more detail in Section 6.1,see also Buryachenko, 2007, 2010a].

Following Beran and McCoy (1970), we introduce a projector operatorP called the deviation operator whichprojects out the deviation part of the random function standing on its right side in the form of a single or productexpressions:Pf ≡ f − ⟨f⟩, Pfg ≡ fg − ⟨fg⟩, andP2 = P. The equations (3.17) and (4.40) can be presented in theoperator forms

ε(x) = ⟨ε⟩(x) + (PU ∗ τ)(x) + (P∇G ∗ π)(x), (4.44)

ε(x) = ⟨ε⟩(x) + (U ∗ Pτ)(x) + (∇G ∗ Pπ)(x), (4.45)

with the analogous recasting of Eqs. (3.18) and (4.43), respectively. Thus, the operatorsP andU,∇G are notcommutative:PU = UP, P∇G = ∇GP. It is obviously that the projector operatorP is a particular case of thecentering operationPf ≡ ⟨⟨f⟩⟩ for statistically homogeneous media (2.4), (2.28). In a similar manner, Eqs. (4.44),(4.45) are the particular case of Eqs. (3.17), (4.35), respectively, presented here just for its recasting in terms of thepopular projector operatorP.

4.3 Some Comments about Approximate GIEs

In the light of the analyzed approximations (4.9) and (4.37)–(4.39), it is possible to observe an additional unex-pected advantage of the operator form of the GIEs (3.32), (3.33) with respect to its Green’s function counterparts(3.17), (3.18), respectively. At first, we should indicate some ambiguous interpretation related with separation of themicro-coordinatey (3.17), (3.18) (and (3.38), (3.39)) and macro-coordinate (with resolution equal toΛ). It is notcritical in the case of using of rough methods making it possible to estimate only average fields inside inclusions [e.g.,⟨σ⟩i(y) = ⟨σ⟩(k), y ∈ vi ⊂ v(k)]. However, more advanced methods (see, e.g., Section 6.1) require an advancedtechnique adapted for separate manipulations with both the micro- and macro-coordinates. It is interesting that therequired separation feature is intrinsic in the introduced unique objectLτ

k(x − xk, τ) (or Lηk(x − xk, η)) because

the macro-coordinatexj is presented, [e.g., in Eqs. (3.36), (3.37)] while the micro-coordinatey ∈ vj and possibleimperfect interface conditions are completely concealed insideLτ

k(x − xk, τ) (or Lηk(x − xk, η)). Moreover, the


32 Buryachenko

second advantage of Eqs. (3.36), (3.37) is a psychological one and appearing because the concrete forms of operatorrepresentations of the perturbators (3.19)–(3.21) area priori unknown and may be different from their representationsthrough the Green’s functions (3.22)–(3.24), and, therefore, there is no temptation to separate the deterministic func-tionsU,Γ and the random onesτ andη, respectively, as in the approximations (4.37)–(4.39) ironically recognized asthe exact equations.

Moreover, the approximations (4.9) and (4.37)–(4.39) are not just the abstract hypotheses considered for estab-lishment of the difference between the new GIEs (3.17), (3.18) [or (3.32), (3.33)] and the known ones (4.35), (4.36).Equations (4.9) and (4.37)–(4.39) can be considered as some sort of a test site for discovery of fundamentally newresults in the following sense. If the assumption (4.9) is exactly fulfilled in the framework of both the other hypothesesand methods used, then GIEs (3.17), (3.18) and (4.35), (4.36), respectively, lead to identical estimations of effectivematerial and field parameters. Otherwise, violation of the approximation (4.9) can lead to the large difference betweenthe new (3.17), (3.18) and old (4.35), (4.36) approaches with the possible change of sign of the predicted local statisti-cal average fields⟨σ⟩i(x) (x ∈ vi, see for details Section 6.1). For example, in the framework of the hypothesesH1a,H2b, andH3, Buryachenko (2010a,b) proved equivalence of Eqs. (3.17), (3.18) and old ones (4.35), (4.36), respec-tively, for the locally elastic composites with homogeneous ellipsoidal inclusions. However, Buryachenko (2011a)replaced the accepted hypothesisH2b by the hypothesisH2a (n = 2) (4.11), providing direct estimation of binaryinteractions of inclusions that made it possible to evaluate inhomogeneity of local statistical average stresses inside thecircle inclusions, and, therefore, to detect a violation of the assumption (4.9) and, as a direct consequence, a dramaticdifference between the results obtained by the use of Eqs. (3.17), (3.18) and (4.35), (4.36), respectively.

Some popular approaches are related the GIEs at least in a sense that these approaches constitute the methods ofsolutions of the GIEs. Indeed, the effective medium method (EMM, see Kroner, 1958; Hill, 1965), also called theself-consistent method, is based on the following hypothesis: each inclusionx ∈ vi in the composite material behavesas an isolated one in a homogeneous medium whose properties coincide with the effective properties of the wholecomposite:

Mc = M∗, ⟨σ⟩(x) ≡ ⟨σ⟩. (4.46)

The essential assumption in the Mori and Tanaka (1973) method (MTM) (see also Benveniste, 1987; Buryachenko,2007; Weng, 1990) states that each inclusionvi behaves as an isolated one in the infinite matrix and subjected to someeffective field (3.361) coinciding with the average field in the matrix:

Mc = M(0), ⟨σ⟩(x) ≡ ⟨σ⟩(0). (4.47)

Thus, we see that acceptance of either assumption (4.46) or (4.47) is equivalent to truncation of the GIE (3.43) atn = 1 of the integral item depending on the perturbatorLσ

j (x−xj ,σ). This truncation is compensated in subsequentmanipulations by correction of eitherMc (4.46) or⟨σ⟩(x) (4.47). In so doing, the final representations of the effectiveproperties (2.29) do not depend on a method which can be used for estimations of truncated averaged perturbators inthe GIE (3.43). It means that the assumptions (4.46) or (4.47) are so rough that it does not matter which background ofmicromechanics [either the old (4.41) or new (3.18) one] was used. Acceptance of either assumption (4.46) or (4.47)gives no chance to realize a fundamental advantage of GIEs (3.17), (3.18) over (4.35), (4.36), respectively, lying in thefact that the renormalized items⟨U(x− y)τ⟩(y) and⟨Γ(x− y)η⟩(y) directly depend on the inhomogeneity of fielddistributions inside inclusions rather than only on its volume averages as in the renormalized itemsU(x− y)⟨τ⟩(y)(4.35) andΓ(x− y)⟨η⟩(y) (4.36).

5. COMPOSITES OF PERIODICAL STRUCTURE

We considerd periodic structures with linear-independent vectors of the principal period ofΛ ⊂ Rd determining aunit cellΩ of volume eitherΩ = |e1 · (e2 ⊗ e3)| or Ω = |e1 · e2|, for d = 3 or d = 2, respectively. We can representany nodem ∈ Λ in the form

xm =

d∑i=1

miei. (5.1)



If, for example, the basisei is orthonormal, and the coefficientsm = (m1,m2,m3) are the integer setZ 3, independentof one another,Λ defines a simple cubic (SC) packing; in the case where the coefficientsmi (i = 1, 2, 3) are eitherall even or odd, we have a body-centered cubic structure (BCC); a cubic face-centered structure (FCC) is obtained inthe case where the coefficientsmi are either all even or two are odd, while the third is even. The method of assigningthe latticeΛ is also possible where several nodes are located within the limits of a cell, and the coefficientsmi are theinteger setZ 3, independent of one another, see, e.g., Kuznetsov (1991).

Ford− 1 periodic structures

xm =d−1∑i=1

miei + fd(md)ed, (5.2)

wherefd(md) − fd(md + 1) = const. In the planef(md) = const. the composite is reinforced by periodic arraysΛmd

of inclusions in the direction of thee1 axis and theed−1 axis. The type of latticeΛmdis defined by the law

governing the variation in the coefficientsmi (i = 1, d− 1), and also by the magnitude and orientation of the vectorsei (i = 1, d − 1). In the functionally graded directioned the inclusion spacing between adjacent arrays may vary[fd(md)− fd(md + 1) ≡ const.]. For a doubly periodic array of inclusions in a finite ply containing2ml + 1 layersof inclusions we havef(md) ≡ 0 at |md| > ml; in the more general case of doubly periodic structuresf(md) ≡ 0 atmd → ±∞. To make the exposition more clear we will assume that the basisei is an orthogonal one and the axesei(i = 1, 2, d) are directed along axes of the global Cartesian coordinate system (these assumptions are not obligatory).

The composite material is constructed using the building blocks or cells:w = ∪Ωm, vm ⊂ Ωm. Hereafter thenotationfΩ(x) will be used for the average of the functionf over the cellx ∈ Ωi with the centerxΩ

i ∈ Ωi:

fΩ(x) = fΩ(xΩi ) ≡ n(x)

∫Ωi

f(y) dy, x ∈ Ωi, (5.3)

n(x) ≡ 1/Ωi is the number density of inclusions in the cellΩi.Let Vx be a “moving averaging” cell (or moving-window, see Graham et al., 2003) with the centerx and charac-

teristic sizeaV =3√V, and for the sake of definiteness letξ be a random vector uniformly distributed onVx whose

value atz ∈ Vx is φξ(z) = 1/Vx andφξ(z) ≡ 0 otherwise. Then we can define the average of the functionf withrespect to translations of the vectorξ:

⟨g⟩x(x− y) =1

Vx

∫VX

f(z− y) dz, x ∈ Ωi. (5.4)

Among other things, moving averaging cellVx can be obtained by translation of a cellΩi and can vary in size andshape during motion from point to point. Clearly, contracting the cellVx to the pointx occurs in passing to the limit⟨f⟩x(x − y) → f(x − y). To make the exposition more clear we will assume thatVx results fromΩi by translationof the vectorx − xΩ

i ; it can be seen, however, that this assumption is not mandatory. For homogeneous boundaryconditions (2.26),σΩ(x) (5.3) is an invariant with respect to the cell numberi andσΩ(x) = ⟨ξ⟩x(x) = const.,∀x ∈ Ωi.

For the periodic structures mentioned above, Buryachenko (2013) proposed the general integral equations forthe particular static problem (b ≡ 0) with uncoupled [M = 1, see (2.15)] local constitutive equation (2.8), (2.10).Straightforward generalization of these equations to the wave motion problem for the composites with the couplednonlocal properties of constituents leads to the following new GIEs:

ε(x) = ⟨ε⟩x(x) +∫⟨⟨Lτ

k(x− xk, τ)⟩⟩xdxk, (5.5)

σ(x) = ⟨σ⟩x(x) +∫⟨⟨Lη

k(x− xk, η)⟩⟩xdxk, (5.6)

which are exact for linear functionsε0(x) andσ0(x).


34 Buryachenko

In an analogy with Section 4.2, we consider the ideal interface (2.22), and integral representations for the pertur-bators (3.22) and (3.23). Then Eqs. (5.5) and (5.6) are reduced to

ε(x) = ⟨ε⟩x(x) +∫[⟨⟨U(x− y)τ(y)⟩⟩x + ⟨⟨∇G(x− y)π(y)⟩⟩x]dy, (5.7)

σ(x) = ⟨σ⟩x(x) +∫[⟨⟨Γ(x− y)η(y)⟩⟩x +L(0)⟨⟨∇G(x− y)π(y)⟩⟩x]dy. (5.8)

In the case of asymptotic approximations (4.13)–(4.15), Eqs. (5.7) and (5.8) are reduced to the known ones

ε(x) = ⟨ε⟩x(x) +∫[⟨⟨U(x− y)⟩⟩xτ(y) + ⟨⟨∇G(x− y)⟩⟩xπ(y)dy, (5.9)

σ(x) = ⟨σ⟩x(x) +∫[⟨⟨Γ(x− y)⟩⟩xη(y) +L(0)⟨⟨∇G(x− y)⟩⟩xπ(y)dy, (5.10)

which were analyzed in detail and compared with other known equations by Buryachenko (2007) (see also Kanaunand Levin, 2008) for the particular static problem (b ≡ 0) with the uncoupled [M = 1, see (2.15)] local constitutiveequation (2.8), (2.10).

6. GENERAL INTEGRAL EQUATIONS AND THEIR SIMPLIFICATIONS IN SOME PARTICULARPROBLEMS

We will consider in this section the particular cases of basic equations considered in Section 2.2 and its correspondinggeneral integral equations. Some of these equations were obtained by the author before and shortly summarized inthis section for clarification of unified presentations of particular problems.

6.1 Thermoelastic Composites with Perfect Interface

We will consider the local basic equations of thermoelastostatics(b(x) ≡ 0) of composites with no body forces acting(f(x) ≡ 0) and the perfect interface conditions (2.22)

∇σ(x) = 0, (6.1)

σ(x) = L(x)ε(x) + α(x), or ε(x) = M(x)σ(x) + β(x), (6.2)

ε(x) = [∇u+ (∇u)⊤]/2, ∇× ε(x)×∇ = 0, (6.3)

where× is the vector product. It is assumed that the properties of both the comparison medium and the matrixcoincide (4.1). In such a case, only the first integrals in Eqs. (3.22)–(3.25) do not vanish and

Lτk(x− xk, τ) = Lτ

k(x− xk,τ), Lηk(x− xk, η) = Lη

k(x− xk,η), Luτk (x− xk, τ) = Luτ

k (x− xk,τ). (6.4)

Then Eqs. (3.17) and (3.18) defining the new background of micromechanics are reduced to

ε(x) = ⟨ε⟩(x) +∫[U(x− y)τ(y)− ⟨U(x− y)τ⟩(y)] dy, (6.5)

σ(x) = ⟨σ⟩(x) +∫[Γ(x− y)η(y)− ⟨Γ(x− y)η⟩(y)] dy, (6.6)



while its classic counterparts (4.35)–(4.43) are reduced to the following equations:

ε(x) = ⟨ε⟩(x) +∫

U(x− y)[τ(y)− ⟨τ⟩(y)] dy, (6.7)

σ(x) = ⟨σ⟩(x) +∫

Γ(x− y)[η(y)− ⟨η⟩(y)] dy, (6.8)

ε(x) = ⟨ε⟩(x) +∫

U(x− y)[τ(y)− ⟨τ⟩] dy, (6.9)

σ(x) = ⟨σ⟩(x) +∫

Γ(x− y)[η(y)− ⟨η⟩] dy, (6.10)

ε(x) = ε0 +

∫U(x− y)τ(y) dy, (6.11)

σ(x) = σ0 +

∫Γ(x− y)η(y) dy, (6.12)

respectively, which are widely used for both the functional graded materials [(2.3), (6.7), (6.8)] (see for referencesBuryachenko, 2007) and statistically homogeneous ones [(2.4), (6.9), (6.10)] (see Buryachenko, 2007; Khoroshun,1978; O’Brian, 1979; Shermergor, 1977) as well as [(2.4), (6.11), (6.12)] (see Buryachenko, 2007; Kanaun and Levin,2008). HereU andΓ are the linear elastic counterparts of the Greens functionsU (3.6) andΓ (3.16), respectively.

Equations (6.11) and (6.12) are correct for a finite array of inclusions while their use is not justified for the limitingcase of a statistically homogeneous field of an infinite number of inhomogeneous in the whole spacew = Rd. Thisunjustified generalization leads to well known convergence difficulties becauseU(x−y),Γ(x−y) are homogeneousgeneralized functions of degree−d and the integrals in (6.11) and (6.12) are only conditionally convergent, i.e., onesdepend on the shape of the integration domains [see for details Buryachenko (2007, 2010a)] analyzed a few knownways how to avoid the difficulties mentioned above and none of them can be recognized as a mathematically rigorousmethod.

One way of modifying such a conditionally convergent integral is the so-called method of normalization (orrenormalization, in analogy to its use in quantum field theory) achieved by subtracting from (6.11) the conditionallyconvergent behavior which is asymptotically closed toU(x− y)τ(y) at |x− y| → ∞ (see for details and referencesBatchelor, 1972; Jeffrey, 1973; Willis and Acton, 1976; Buryachenko, 2007, 2010a). For example, the renormalizeditems

[I+ g(x− y)]U(x− y)⟨τ⟩, [I+ g(x− y)]Γ(x− y)⟨η⟩ (6.13)

provide absolute convergence of integrals (6.11) and (6.12), respectively, for anyg(x − y) [rather than only forg(x − y) ≡ 0] such asg(|x − y|) → 0 at |x − y| → ∞ that can lead to the different values of integrals estimated.Rigorous justification of the renormalization itemU(x−y)⟨τ⟩ was proposed by O’Brien (1979) (see also Khoroshun,1978) by applying the divergence theorem to the boundary integral in the equation analogous to Eq. (3.5) that leadsto Eq. (6.9) with⟨ε⟩ ≡ ε0, where the operation of “separate” integration of slow∇G(x − s) and fastτ(s) (s ∈ Γ0

p)variables was performed at the intuitive level of justification at the assumptionτ(s) ≡ ⟨τ⟩. The last assumptioncombined with (4.37) were also used by Shermergor (1977) in his more general method of centering of particular caseof Eq. (3.5) leading to Eq. (6.9). This centering method was generalized to the case of statistically inhomogeneousmedia by Buryachenko (2007).

Comparing nonrenormalized (6.11), (6.12) and renormalized (6.9), (6.10) equations, respectively, McCoy (1979,1981) suggested that one can formally remove the conditionally convergent term appearing in the former by simplysetting them equal to zero. There are well-known noncanonical regularizations proposed by Kroner (1974) (see alsoKroner, 1984) and independently reproposed by Kanaun (1977) (see also for references Kanaun and Levin, 2008)∫

U(x− y)h dy = 0, or

∫Γ(x− y)h dy = Lch,∫

U(x− y)h dy = Mch, or

∫Γ(x− y)h dy = 0, (6.14)


36 Buryachenko

for the first (6.141,2) and the second (6.143,4) boundary-value problem, respectively;h is an arbitrary constant sym-metric second-order tensor. Thus, in the light of the note by McCoy (1979, 1981), the “noncanonical regularizations”(6.14) can be considered as some sort of renormalization method. Buryachenko (2007) proved that the correctness ofthese regularizations (6.14) is questionable.

Another widely used simple technique (see Levin, 1975, 1976; Willis, 1983; Sevostianov et al., 1998; and the ref-erences in Kanaun and Levin, 2008) is worthy of notice. This method is based on a statistical averaging of Eqs. (6.11)and (6.12) with subsequent excluding of the fieldsε0 andσ0 from both the obtained averaged equations and the initialEqs. (6.11) and (6.12). Of course, this technique is exactly the centering method (the initial version of this methodwas proposed by Shermergor, 1977) exploited for transforming of Eqs. (6.11) and (6.12) into the correct approximateEqs. (6.9) and (6.10) [compare with the exact Eqs. (6.5) and (6.6)], rather than for excluding of the deterministicconstant known functionsε0 andσ0, respectively. Indeed, denoting Eq. (6.11) asg(x) = 0 leads the mentionedtechnique to the correct approximate Eq. (6.152) [or Eq. (6.9)]

g(x) = 0 ⇒ g(x)− ⟨g⟩ = 0, (6.15)

while the initial Eq. (6.151) [or Eq. (6.11)] is questionable. This apparent simplicity of the mentioned techniqueis explained by the absence of necessity to prove [as in the correct Eqs. (3.8) and (3.14)] vanishing of the surfaceintegrals (3.10) and (3.15) which are lost in Eqs. (6.11) and (6.12).

A physically based choice of renormalizing terms is critical for the values of integrals involved in Eqs. (6.5)–(6.12). Buryachenko and Brun (2012) considered a degenerate problem which makes it possible to obtain the exactestimations of statistically averaged local stresses by the use of Eq. (6.6) [rather than (6.10)] without any assumptionsand compared this solution with evaluations obtained by Eq. (6.12). Namely, it was analyzed elastically homogeneousmediaL(x) = const. with statistically homogeneous field of identical aligned inclusions (N = 1) with stress-free strainα1(x) ≡ 0 whenη(x) ≡ α1(x) (x ∈ Rd), and⟨σ⟩ = 0. It was demonstrated that for a particular 2D case of alignedhomogeneous noncanonical inclusions, the approximative Eq. (6.10) leads to the error 40% in comparison with theexact inhomogeneous effective field⟨σ⟩i(x) (6.6).

The current paper is dedicated to obtaining general integral equations and its simplifications due to some as-sumptions (e.g., hypothesesH1a and/orH1b) rather than to finding of solutions of these equations. However, justfor demonstration of potency of our efforts in this direction of procurement of general equations, we will shortlyreproduce some particular results obtained (or expected) for thermoelastic problems considered in this Section 6.1

Result 6.1.1. For homogeneous loading (2.28) of statistically homogeneous media (2.4) with aligned identicalhomogeneous ellipsoidal heterogeneities, hypothesesH2b andH3 lead to equivalence of the new Eqs. (6.5), (6.6) andclassic Eqs. (6.9), (6.10), respectively (see Buryachenko, 2010a, 2010b), with the identical effective material (2.29)and field [⟨ε⟩i, ⟨σ⟩i, see (3.36), (3.37)] properties.

Result 6.1.2. The effective material and field properties estimated by the use of the new Eqs. (6.5), (6.6) dependon the field distribution in the vicinity of heterogeneities [even for the conditions ofResult 6.1.1, in contrast withEqs. (6.9), (6.10)] in the basic auxiliary one-particle problem (3.19), (3.20) as well as on the size of the excludedvolumev0i and the RDF even in the framework of hypothesisH2b and H3 (see for details Buryachenko, 2010b,Buryachenko and Brun, 2011, 2012).

Result 6.1.3. For both the homogeneous nonellipsoidal heterogeneities (Buryachenko and Brun, 2011, 2012) andcircle inclusions with continuously inhomogeneous coating (Buryachenko, 2010b), inhomogeneity of field distribu-tions inside heterogeneity in the auxiliary one-particle problem (3.19), (3.20) forε0(x),σ0(x) ≡ const. can lead tothe different signs of local stresses⟨σ⟩i(z) (z ∈ vi) estimated by the use of Eqs. (6.5), (6.6) and Eqs. (6.9), (6.10),respectively, while its corresponding effective material properties (2.29) are not too different from one another.

Result 6.1.4. Taking into account binary interaction of spherical heterogeneities in CM satisfying conditions ofResult 6.1.1(with replacement of hypothesisH2b on hypothesisH2a atn = 2) can lead to the different signs of localstresses⟨σ⟩i(z) (z ∈ vi) and average particle ones⟨σ⟩i estimated by the use of Eqs. (6.6) and (6.10), respectively,while its corresponding effective material properties (2.29) are not too different from one another (see for detailsBuryachenko, 2011a).

Buryachenko (2013) has indicated that it is expected to get a larger difference (which can reach infinity with thechange of the sign of predicted local statistical average fields) between the results obtaining the use of either Eqs. (6.5),



(6.6) or (6.9), (6.10) for composites reinforced by heterogeneities demonstrating greater inhomogeneity of field distri-butions inside heterogeneities. The last inhomogeneity can be produced by any one (or a few) reasons including suchas, e.g., nonellipsoidal shape of heterogeneities, inhomogeneity of its properties (e.g., laminated structure or imperfectinterface), binary interaction of heterogeneities, inhomogeneity of external loading (2.26), statistical inhomogeneityof composite structure (2.3), nonlocal constitutive laws (2.8), dynamic effects, effect of size boundness of CM. Inso doing, a difference of effective properties estimated by the use of either Eqs. (6.5), (6.6) or (6.9), (6.10) is notsignificant.

We mention a few meanings of the term generality in the context of evaluation of significance of the new approach.The first meaning is that the operator form of GIE (3.32) and (3.33) generalizes the corresponding GIEs for the differ-ent particular cases of constituent equations considered in Sections 6.1–6.6. The second meaning of generality of thenew approach is demonstrated even for both the particular constitutive equations and partial perturbator representa-tions. For example, for the locally elastic phenomena (6.1)–(6.3) with Green’s function representations (3.22)–(3.24)of the perturbators, the new GIEs (6.5) and (6.6) are more universal than the old ones (6.7)–(6.10) obtained at theassumptionsH1b (4.37), (4.38) (see Result 6.1.3). Moreover, the Green function technique for the mentioned case isjust a particular tool for representation of a solution for the inclusion interaction problem. In so doing, a more generalexpression for this solution in terms of perturbators (3.19)–(3.21) with a subsequent use in Eqs. (3.32) and (3.33) hasthe additional interesting advantage with respect to Eqs. (6.7)–(6.10) because there is no reason for the questionablecaring out from the statistical brackets of some deterministic function as in Eq. (4.37) and (4.38). However, both thegenerality and novelty of GIEs obtained [see Eqs. (3.32), (3.33), (3.36), (3.37), (3.42), (3.43)] provide not only thetheoretical interest but also the critical practical benefits.

Indeed, let us consider (see for details Buryachenko and Brun, 2011) a statistically homogeneous 2D problem forCM containing isotropic phases with the Young moduliE(1) = 100, E(0) = 1 and Poisson ratioν(1) = ν(0) = 0.45(β(x) ≡ 0, x ∈ R2). The aligned inclusions have noncanonical shape (schematically depicted in Fig. 1) withaspect ratio 0.32 andc(1) = 0.7. The componentsB∗

i|2211(x2) of the stress concentration factorB∗i (x) (⟨σ⟩i(x) =

B∗i (x)⟨σ⟩) in a cross sectionx = (0, x2)

⊤ at ⟨σ[ij]⟩ = δi1δj1 are presented in Fig. 1. Curves 4 and 5 obtained by theMTM [used Eq. (4.47)] and MEF (used the assumptionsH1, H2b, andH3), respectively, are invariant with respectto the binary correlation functionφ(vq,xq|vi,xi). The curves 1–3 are estimated for the different binary correlationfunctionsφ(vq,xq|vi,xi) and excluded volumesv0i by the new approach (NA) based on solution of the new GIE

FIG. 1: B∗i|2211(x2) vsx2 estimated by NA (1–3), MTM (4), and MEF (5)


38 Buryachenko

(6.6) at the assumptionsH1a, H2b, andH3. As can be seen, the local stresses predicted by NA (used less restrictedassumptions) and classical methods are distinguished by a sign that is critical for a wide class of nonlinear phenomena(such as, e.g., strength and plasticity).

However, the superiority of NA over the classical approaches can be more visually demonstrated for the problemwhich can be solved by NA exactly without any assumption. Namely, let us analyze (see for details Buryachenkoand Brun, 2012a) the 2D problem with the same geometrical parameters considered in the previous paragraph andmechanical propertiesE(1) = E(0) = 1, ν(1) = ν(0) = 0.45, andβ(1)

[ij] = δij , β(0) = 0, (⟨σ⟩ = 0). For this

elastically homogeneous medium with random eigenstrains, we haveη(x) ≡ β1(x), and, therefore, the new GIE (6.6)can be exactly solved (with a numerical accuracy of integral evaluations) without any assumptions for any prescribedφ(vq,xq|vi,xi) andv0i . In Fig. 2 one presents the normalized effective fieldsσ22 = ⟨σ22⟩i/σ

ellips22 , whereσellips ≡

−Qellipsβ(1)1 is an analytical representation for the residual stresses in an isolated elliptical inclusionvellips with the

same aspect ratio 0.32. Curves 4 and 5 in Fig. 2 obtained by the MEF and MTM, respectively, are also invariant withrespect toφ(vq,xq|vi,xi) andv0i while the curves 1–3 are estimated for the differentφ(vq,xq|vi,xi) andv0i by NA.Curves 4 and 2 estimated by the MEF and the exact NA, respectively, demonstrate an essential difference (40%).

Thus, one demonstrated the quantitative numerical difference and superiority of NA over the classical approaches(such as the MEF and MTM) in some concrete examples.

6.2 Thermoelastic Composites with Imperfect Iinterface

In a considered case of statistically homogeneous thermoelastic composites (2.4) and (2.28) with imperfect interface,Eqs. (6.1)–(6.3) should be complemented by the boundary conditions at the imperfect interface (2.23) and (2.24). Inthe literature, three kinds of models are often used to simulate the properties of interface regions. The first kind ofmodel can be referred to as interface models,in which the traction is continuous across the interfacex ∈ Γu

i whilethe displacement is discontinuous atx ∈ Γu

i [these problems are, in general, solved numerically, see, e.g., Yuan andFish (2009), Fries and Belytschko (2010)]. So, inlinear spring model(LSM, see, e.g., Dvorak, 2013; Dvorak andBenveniste, 1992; Hashin, 1991a, 2002; Huang et al., 1993; Zhong and Meguid, 1997)

[[σ]] · n = 0, Ps · [[u]] = asts +Ps · βi, Pn · [[u]] = ant

n +Pn · βi, (6.16)

FIG. 2: σ22 vsx2 estimated by NA (1–3), MEF (4), and MTM (5)



wherets ≡ Ps · σ · n andtn ≡ Pn · σ · n represents the shear and the normal traction at the interface, respectively,andPn = n⊗ n andPs = δ−Pn are the two orthogonal complementary projection operators of the second order.a = ann⊗ n+ ass⊗ s+ att⊗ t, wherean, as, at (as = at) represent the interface compliance parameters in thenormal and tangential directions, respectively, ands andt are two orthogonal unit vectors in the tangent plane of theinterface.βi = βi

nn+ βiss+ βi

tt denotes an interface eigenstrain, andβin, β

is, β

it (βi

s = βit) represent the interface

eigenstrain parameters in the normaln and tangentials, t directions. For example, the case wherea = 0 andβi = 0implies a perfect bonding interface (2.22), while a casea = 0 andβi ≡ 0 corresponds to thedislocation-like model(Somigliana, 1886; Asaro, 1975), and for a casean = 0 andas = 0, only interfacial sliding takes place with normalcontact remaining intact. Furthermore, the case wherean = 0 andas → ∞ represents thefree sliding model(see,e.g., Jasiuk et al., 1987; Mura and Furuhashi, 1984; Mura et al., 1985, 1996). The conditions (6.16) were generalizedto acohesive zone model(CZM), where nonlinear springs with a specific traction-displacement law are considered atthe interface. Cohesive zone model originated by Barenblatt (1962) in fracture mechanics (see also Needleman, 1990;Ortiz and Pandolfi, 1999) has received wide development in micromechanics of CM (see, e.g., Tvergaard, 1990; Tanet al., 2007, 2007b; Othmani et al., 2011).

The second kind of interface model (calledinterface stress model, ISM, or coherent interface model), which canbe viewed as dual with respect to the linear spring-layer model, was in general considered by Gurtin and Murdoch(1975) (see also Povstenko, 1993; Ibach, 1997; Gurtin, et al., 1998) while its applications to the nanoparticles withextension of Eshelby formalism were considered in Sharma and Ganti (2004), Duan et al. (2005), He and Li (2006),Chen et al. (2007), Maranganti and Sharma (2007), and Wang et al. (2011). In the ISM the displacement vector fieldu (2.221) and the tangential strain fieldεs ≡ PsεPs are continuous[[u(s)]] = 0 and[[εs]] = 0, respectively, acrosss ∈ Γσ

i while the strain fieldε is in general discontinuous acrossΓσi . 2D constitutive equation at the interfaceΓσ

i isdefined as

σs = τ0Is + 2(µs − τ0)ε

s + (λs + τ0)Tr(εs)Is, (6.17)

where isotropic interface considered is characterized by surface Lame constantsλs, µs, and surface tension,τ0; Is

and Tr represent the 2×2 unity tensor and the trace operation, respectively. Coupling of the bulk and surface tensorsat the interfaceΓσ

i is governed by the generalized Young–Laplace equation

[[σ · n]] +∇s · σs = 0, (6.18)

where the action of the surface gradient∇s on a vectorv is defined through the usual 3D gradient∇ operator:∇sv ≡ ∇vPs. In the case of absence of surface terms denoted by the upper indexs, Eq. (6.18) reduces to the usualtraction continuity equation of classical elasticity (2.24). At last, the third kind is the interphase model which describesthe interface region as a layer, called an interphase, perfectly contacted (2.22) with the heterogeneity and matrix (seefor references Buryachenko, 2007).

All authors of the LSM, CZM, and ISM mentioned above did not reveal the GIEs considered in Sections 3 and 4which is not to say that these equations were not really used. Indeed, all these authors implicitly exploited at least thehypothesesH1a, H1b, H2b, andH3 defining so-called one-particle approximation of the MEFM (called the methodof the effective field, MEF, see for details Buryachenko, 2007). It is equivalent to exploiting of Eqs. (4.29) and (4.30)additionally simplified by the hypothesisH2b when

⟨Lτj (x− xj , τ(j))|; v1,x1⟩ = ⟨Lτ

j (x− xj , τ(j))⟩, (6.19)

⟨Lηj (x− xj , η(j))|; v1,x1⟩ = ⟨Lη

j (x− xj , η(j))⟩. (6.20)

However, in Section 8.3.3 in Buryachenko (2007) it was proved that for the same ellipsoidal shape ofv0i andvi ofaligned identical heterogeneities (N = 1) of any microstructure, the solutions of Eqs. (4.29), (4.30) and (6.19),(6.20) are identical to the solutions obtained by the MTM. Precisely MTMtotally dominatesin micromechanics ofcomposites with imperfect interface (see Duan et al., 2007; Chen et al., 2007; Sharma and Wheeler, 2007; Tan etal., 2007, 2007b) described by either the LSM, CZM, or ISM although some other methods which are also just theparticular cases of the MEFM can be applied; e.g., a self-consistent method follows from (4.29), (4.30) and (6.19),(6.20) atLc = L∗ (see for details Buryachenko, 2007). Thus, we can conclude that all cited authors researching CM


40 Buryachenko

with imperfect interface have used actually Eqs. (4.29), (4.30) and (6.19), (6.20) which leaves room for improvementin a few directions of their results obtained. Indeed, solution of Eqs. (4.29), (4.30) and (6.19), (6.20) implies previousestimation of the averaged tensorsRϵ

i , Ri, Tϵi , andTi involved into a solution for a single heterogeneity with

imperfect interface inside the infinite matrix (4.18) and (4.19). The tensorsRϵi andRi were found by the different

methods such as, e.g., FEA, generalized Eshelby formalism, multipole expansion, and complex variables method.Independently of the method exploited, the tensorsRϵ

i andRi can be taken as being known and we can substituteit into the classical version of the MEFM based on the hypothesesH1a, H1b, H2a (rather thanH2b), andH3. It isinteresting that the most advanced model of composites with the CZM interface was developed for the modeling ofthe heterogeneous solid propellants (HSPs) in the framework of the MTM (see for references Tan et al., 2007a,b).However, the distinguishing feature of the HSPs is a high volume fraction (>90%) of spherical stiff particles withmultimodal distribution (see for references Buryachenko, 2013) when the MTM is totally second to the MEFM evenfor the perfect interface (2.22) (see for details Buryachenko, 2013). A direct incorporation of the tensorsRϵ

i (4.18)andRi (4.19) into the advanced MEFM taking into account a binary interaction of particles with multimodal sizedistribution is straightforward (see for details Buryachenko, 2007, 2013).

The use of the new background (3.36) and (3.37) does not invite further investigation (additionally to researchalready performed by the cited authors) for one particle solution (3.19) and (3.20) with either the LSM, CZM, orISM interface. Indeed, the use of averaged tensorsRϵ

i (4.18) andRi (4.19) implies that the tensorsRϵi (x) and

Ri(x) are previously estimated, which in its turn means that the tensorsLϵ(x − xi, ε) (3.19) andLσ(x − xi,σ)(3.20) are found. Therefore, we can immediately exploit either Eqs. (4.33), (4.34) (used hypothesesH1a andH2b) orEqs. (4.31), (4.32) (used hypothesesH1a andH1b atn = 2). Subsequent improvement of the mentioned methods isrelated with abandonment of hypothesisH1a when variation of the effective fields⟨ε⟩i(y) and⟨σ⟩i(y) (y ∈ vi) canbe estimated by the iteration method analogously to the counterpart problem for CM with a perfect interface (see fordetails Buryachenko, 2010b, 2013).

Thus, we can formulate the results in the following manner:Result 6.2.1. It is well known that for CM reinforced by the homogeneous ellipsoidal particles and satisfying

conditions of Result 6.1.1 but with imperfect interface (either LSM, CZM, or ISM), the stress distribution inside iso-lated particles is in general inhomogeneous. Therefore, the effective material (2.29) and field (3.36), (3.37) propertiesestimated by the use of new Eqs. (6.5), (6.6) and classic Eqs. (6.9), (6.10), respectively, are different (see for detailsBuryachenko, 2013).

Imperfection of interface conditions (with either LSM, CZM, or ISM) leads to additional opportunity for stressinhomogeneity inside the particles, and, therefore, after addition of imperfect interface conditions into the Results6.1.2–6.1.4, its can be considered as expected results for CM with imperfect interface.

6.3 Conductivity of Composites

It is well known that the equations describing the quasi-static conditions of such processes as heat and mass transfer,electric conductivity and permittivity, and filtration of a Newtonian liquid in undeformable cracked-porous media (see,e.g., Batchelor, 1974; Halle, 1976; Landauer, 1978; Shvidler, 1985; Furmanski, 1997) are mathematically equivalent.For reasons of mathematical analogy, we consider the basic equations of steady-state transfer process:

∇q = 0, q = −κ∇T, (6.21)

whereT is a potential (e.g., temperature),κ is a conductivity tensor,q is a flux vector, and the first Eq. (6.211) is atransfer equation (Fourier, Ficks, Ohm, Darcy, etc.). Equations (6.21) are simpler field equations than Eqs. (6.1)–(6.3),which simplifies its analysis.

The immediate consequence of Eqs. (3.17), and (4.35), (4.40), and (4.42), are

∇T (x) = ⟨∇T ⟩(x) +∫[U(x− y)κ1(x)∇T (y)− ⟨U(x− y)κ1∇T ⟩(y)]dy, (6.22)



∇T (x) = ⟨∇T ⟩(x) +∫

U(x− y)[κ1(y)∇T (y)− ⟨κ1∇T ⟩(y)]dy, (6.23)

∇T (x) = ⟨∇T ⟩+∫

U(x− y)[κ1(y)∇T (y)− ⟨κ1∇T ⟩]dy, (6.24)

∇T (x) = ∇T 0 +

∫U(x− y)κ1(y)∇T (y)dy (6.25)

whereU(x − y) is defined by Eq. (3.62) and the interface conditions (2.22) are assumed to be perfect. The totalmajority of methods investigating conductivity problem of CMs (see for references and details Buryachenko, 2007)use (either implicitly or explicitly) EFH, and, therefore, imply Eqs. (6.23) and (6.24). The basic historical steps of ob-taining (6.23) and (6.24) (see, e.g., Buryachenko, 2007; Kanaun and Levin, 2008) in fact duplicate the correspondingdevelopment stages of Eqs. (6.7) and (6.10) analyzed in Section 6.1. In the analogous ways, incorrect uses Eq. (6.25)for both statistically homogeneous (Pfeil and Klingenberga, 2004) and inhomogeneous (Yin et al., 2007) compositeswere performed with the same questionable justifications critically analyzed in Section 6.1 for elastic problem (seealso Chapt. 12 in Buryachenko, 2007).

The cases of imperfect interface conditions deserve further comments. Thin interphase layer between two mediais modeled in terms of imperfect interface conditions by Benveniste and Miloh (1986), while Torquato and Rintoul(1995) and Markov (1999) treat nonideal contact of a spherical inclusion of the radiusa in two limiting cases of boththe large and small conductivityκint = κint0 δ of interphase with a small thicknessδ ≪ a (the cases of an ellipsoidalheterogeneity with these nonideal contacts were analyzed in Miloh and Benveniste, 1999). In the firstκint0 → ∞ andsecondκint0 → 0 case, the limits

C =κ(0)0

alim

1/δ,κint0 →∞

δκint0 , R =κ(1)0

alim

δ,κint0 →0

δ

κint0

, (6.26)

respectively, remain finite (see for details Markov, 1999; Torquato and Rintoul, 1995). For the conditions of “super-conducting” interface (6.261), the temperature field is continuous at the interface, but the heat flux suffers a jump thatcorresponds to the ISM (6.17) and (6.18), whereas for “resisting” conditions (6.262), the heat flux remains continuous,but the temperature suffers a jump on the interface that parallels to the CZM (6.16). The next step is an estimationof the tensorRt in a solution for one isolate particle in the infinite matrixvi⟨κ1∇T ⟩i = Rt∇T corresponding tothe general problem (4.18) and subsequent substitution into Eq. (4.29) obtained from Eq. (3.37) [or (6.22)] in theframework of hypothesisH2b.

Thus, the conductivity problems of CMs with both perfect and imperfect interface coincides with the elasticityproblems considered in Sections 6.1 and 6.2 with a precision of notations. In a similar manner we can obtain theintegral equations and its solutions corresponding to the abandonment from either any one or a few hypothesesH1a,H1b, H2b, H3 in a general integral Eq. (3.36). Therefore, after the replacements of the word “elasticity” for the word“conductivity,” the Results 6.1.1, 6.1.2, 6.1.3, 6.2.1 for elasticity problems can be considered as the expected resultsfor conductivity ones.

6.4 Coupled Problems of Composites

We will consider the general integral equations for the coupled problems at the example of thermoelectroelasticityequations for CMs. For notational convenience the elastic and electric variable will be treated on equal footing, andwith this in mind we recast the local linear constitutive relations of thermoelectroelasticity for this material (see, e.g.,Maugin, 1988; Parton and Kudryavtsev, 1988; Kanaun and Levin, 2008) in the notation introduced in Barnett andLothe, 1975). Then the basic steady-state equations can be presented in the form analogous to Eqs. (6.1)–(6.3):

DΣ = 0, E = DU , (6.27)

E = MΣ+Λ, Σ = L(E −Λ), (6.28)


42 Buryachenko

where

E =

∥∥∥∥ ε

E

∥∥∥∥ , Σ =

∥∥∥∥ σ

D

∥∥∥∥ , U =

∥∥∥∥ uϕ

∥∥∥∥ , D =

∥∥∥∥ def 00 grad

∥∥∥∥ , (6.29)

M =

∥∥∥∥ M d⊤

d − b

∥∥∥∥ , L =

∥∥∥∥ L e⊤

e − k

∥∥∥∥ , Λ =

∥∥∥∥ β

qθ

∥∥∥∥ , (6.30)

whereD andE are the vectors of induction and electric field intensity,θ is a deviation of stationary temperaturefield from a given value,k andb are the tensors of dielectric permeability and impermeability,q is the pyroelectriccoefficients,e andd are the piezoelectric moduli, andϕ is the electric potential. To obtain a symmetric matrix ofcoefficients we replaced the electric fieldE by−E, and the tensorsk andb by−k and−b on the right-hand sides of(6.28). It is assumed that the properties of both the comparison medium and the matrix coincide (4.1).

We suppose that the contact between components is ideal, and the normal components of the stress tensor andthe electric induction vector are continuous, as well as the tangential components of the strain tensor and electricalfield intensity. Except for notations, these equations coincide with the equations of linear thermoelasticity (6.1)–(6.3).Because of this, the theory of piezoelectric CM (PCM) retraces at a particular instant the path of development of thetheory of microinhomogeneous elastic media, exhibiting substantial progress [see for references, e.g., Buryachenko(2007), and Kuznetsov and Fish (2012) for random and periodic structure, respectively]. In light of the analogymentioned in our brief survey we will not consider in detail the GIE and average schemes of PCM, one may referinstead to the appropriate schemes of Section 6.1 (see also Buryachenko, 2007; Kanaun and Levin, 2008).

For the basic equations (6.27) and (6.28), the GIEs (3.18), (4.41), and (4.43) are reduced to the following equationsfor Σ (corresponding equations forE can be presented in a similar manner):

Σ(x) = ⟨Σ⟩(x) +∫[Γ(x− y)η(y)− ⟨Γ(x− y)η⟩(y)] dy, (6.31)

Σ(x) = ⟨Σ⟩(x) +∫

Γ(x− y)[η(y)− ⟨η⟩] dy, (6.32)

Σ(x) = Σ0 +

∫Γ(x− y)η(y) dy, (6.33)

respectively, where the polarization tensorη(x) = M1(x)Σ(x) +Λ1(x), and the coupled flux Green’s tensorΓ(x)is defined by Eqs. (3.16), (3.6), and (2.19). The absence of coupled field effect for the matrix essentially simplifiesthe micromechanical problems because in such a case the Green’s functionΓ has a block-diagonal structure definedby the Green’s functions of uncoupled fields. Here Eq. (6.32) holds only for statistically homogeneous material (2.4)and field (2.28) problems while Eq. (6.33) holds true only for the fieldsX bounded in one direction (4.6).

A comprehensive review with numerous references related with universal relations in piezoelectric (and thermo-magneto-electro-elastic) composites can be found in Chapter 17 in Buryachenko (2007) (see also Qin and Yang,2008), where one also presented the references dedicated to straightforward generalization of the known methods ofmicromechanics of linear elastic static composites: the multi-inclusion model, conditional moment method, general-ized singular approximation, effective medium method, one-particle approximation of the MEFM (called MEF), andthe MEFM to the corresponding methods with coupled and wave propagation effects. These and other methods canbe classified with respect to the GIE as two groups of methods. The methods of the first one initially accept veryrough hypotheses [as in EMM (4.46) and MTM (4.47)] which allow us to obtain the final results without using GIE.Analogously to the locally elastic composites considered in Section 6.1, the coupled counterparts of both the EMM(4.46) and MTM (4.47) (see, e.g., Dinzart and Sabar, 2011; Levin et al., 2011; Lu et al., 2011; and references inBuryachenko, 2007) are, in fact, the approximate solutions of the GIE (3.43) [or (6.31)–(6.33)]. In so doing, thesesolutions are so rough that it does not enable one to use all opportunities of the different GIEs [see the comments afterEq. (4.47)]. The methods of the second group use one or another GIE but in many cases a choice of this GIE as wellas its subsequent solution is performed at some additional weakly justified assumptions. For example, for a statisti-cally homogeneous infinite inclusion field, popular versions of the MEF are based on the initial incorrect exploiting



of Eq. (6.33), which is transformed to the correct Eq. (6.32) by the use of the scheme (6.15) (see, e.g., Levin et al.,2002; Chen and Shen, 2007).

Thus, to the best of the author’s knowledge, the mentioned approaches and the most part of other ones eitherexplicitly or implicitly use either Eq. (6.32) or Eq. (6.33), with the different level or rigor. In so doing, the new exactGIE (6.31) holds true for any statistical inhomogeneities of material (2.4) and field (2.28) parameters, and directlytakes into account dependence of the renormalized item⟨Γ(x − y)η⟩(y) on the inhomogeneity of the polarizationtensorη(y) (y ∈ v) inside heterogeneities. Because of this, the Results 6.1.1–6.1.5, and 6.2.1 can be considered asthe expected results for the coupled problems.

6.5 Composites with Nonlocal Elastic Properties of Constituents

We consider a nonlocal uncoupled elastic constitutive equation (2.8) (M = 1) with material parameters subjected tothe following symmetry regulations (K = L,M; γ = α,β): Kijkl = Kijlk = Kjikl, Kijkl(x,y) = Kklij(y,x),γij = γji [see, e.g., Kunin (1967), Kroner (1970), Kroner and Datta, 1970]. The operator kernelK(x,y) is presentedin the form

K(x,y) = K(x)λ(x,y), (6.34)

where the scalar attenuation functionλ(x,y) (called also the nonlocal weight function or the nonlocal averagingfunction) defined for a homogeneous full spacew = Rd (d = 2, 3) can also be adjusted for a finite domainw (see,e.g., Jirasek and Rolshoven, 2003).λ(x,y) depends only on the distancer = |x − y| and can be expressed asλ(x,y) = λ∞(|x− y|), whereλ∞ is an even function satisfying the normalizing condition∫

w

λ∞(x,y)dy = 1, ∀x ∈ w = Rd (6.35)

that gives toK the physical meaning of elastic stiffness and compliance under uniform straining and stressing, respec-tively. In the limiting caseλ∞(x) → δ(x), a nonlocal operatorK is reduced to the local oneK(x,y) → K(x)δ(x−y)(2.10). In particular, among a few functionsλ(x,y) analyzed by Polizzotto, (2001), we consider (following Bury-achenko, 2011b,c; Fish et al., 2012a) the bell-shaped attenuation function

λbell∞ (r) = 3/(πa2λ)(1− (r/aλ)2)2, (6.36)

where the Macauley brackets· denote the positive part, defined asx = max(0, x). Higher-grade continuummodels can be derived by truncation of the corresponding Taylor approximation of integral ones.

In the case of the nonlocal constitutive Eq. (2.8), the GIEs (3.18), (4.41), and (4.43) are reduced to the followingequations forσ (corresponding equations forε can be presented in a similar manner):

σ(x) = ⟨σ⟩(x) +∫[Γ(x− y)∗η(y)− ⟨Γ(x− y)∗η⟩(y)] dy, (6.37)

σ(x) = ⟨σ⟩(x) +∫

Γ(x− y)∗[η(y)− ⟨η⟩] dy, (6.38)

σ(x) = σ0 +

∫Γ(x− y)∗η(y) dy, (6.39)

respectively, and the Green’s tensorΓ(x) is defined by Eqs. (3.16), (3.6), and (2.19). Here Eq. (6.38) and (6.39)hold only for statistically homogeneous material (2.4) and field (2.28) problems, and for the fieldsX bounded in onedirection (4.6), respectively. Equations (6.37) and (6.38) obtained by Buryachenko (2011c) were not yet used by otherresearchers.

The popular approaches of analysis of composites with nonlocal constitutive properties are usually based oncoupling of two problems. At first, the problem for one heterogeneity (either the particle, flake, or fiber) insidean infinite matrix subjected to the homogeneous remote field is solved for the different constitutive laws of boththe heterogeneity and matrix (see, e.g., a comprehensive reviews in Gutkin, 2006; Maranganti and Sharma, 2007,


44 Buryachenko

and also Buryachenko, 2011c). We mention a computationally efficient staggered nonlocal multiscale model byFish et al. (2012a) employing current information for the point under considerationx and past information fromits local neighborhood from the “source” pointy taken from the previous load increment (or time step). It wasdemonstrated (see for references Maranganti and Sharma, 2007; Buryachenko, 2011c) that for the different nonlocalmodels (involving either spatial integrals or gradients of strains), the strains are nonuniform inside the homogeneousellipsoidal inclusion even for the homogeneous remote loading. The detected inhomogeneity indicates a violation ofthe hypothesis (4.9) and, as a consequence, instills confidence (see Section 4.3) in the expected large differences ofeffective material and field parameters estimated by the use of Eqs. (6.37) and (6.38).

The second problem is incorporating the mentioned solution for one isolated inclusion into one of the knownmicromechanical schemes. Buryachenko (2011b) has formulated a theory of thermoelastic composites with nonlocalproperties of constituents at the similar level of generality as Hill (1963a) and Dvorak and Benveniste (1992) (see alsoDvorak, 2013) whose micromechanical theories for composites contain only constituents with local elastic properties.In the framework of the EFHH1, Buryachenko (2011b,c) proposed the nonlocal counterpart of the famous Levin(1967) formula for composites with identical inhomogeneous heterogeneities and presented the nonlocal analogs ofthe EMM (4.46), MTM (4.47), MEF, and MEFM. It was demonstrated that the effective propertiesL∗ andM∗ (2.29)depend on both the stress and strain concentrator factors averaged over the volume of the isolated inclusion ratherthan only on one of them. However, the use of the MTM (4.47) totally dominating in micromechanics of compositeswith nonlocal properties of constituents (see e.g., Sharma and Dasgupta, 2002; Xun et al., 2004; Zhang and Sharma,2005) implicitly implies acceptance of the hypothesisH1b providing an equivalence of GIEs (6.37) and (6.38) aswell as truncation of the corresponding GIEs either (6.37) or (6.38) presented in the form (3.43) [see Section 4.3 andEqs. (4.47)]. Thus, MTM is in fact a solution of the truncated version of GIE (6.38).

We summarize some numerical results obtained by Buryachenko (2011c) for 2D composites with the locallyelastic matrix (2.10) containing the identical circle homogeneous inclusions with the propertiesaλ/a = 0.5 (6.34),(6.36).

Result 6.5.1. In the framework of the hypothesesH1, H2b, andH3 for the classical GIE (6.38), the local statisticalaverage stresses⟨σ⟩(x) are strongly inhomogeneous function of coordinatesx ∈ vi ⊂ v (for any volume fraction ofinclusionsc), and do not depend (asL∗) on both the RDFs and the size of excluded volumev0i . These estimationscoincide with the corresponding MTM’s evaluations.

Result 6.5.2. In the framework of the hypothesesH2b, andH3 for GIE (6.37) withc = 0.65, the local statisticalaverage stresses⟨σ⟩(x) depend on both the RDFs and the size of excluded volumev0i and can differ by the sign fromthe analogous estimations in Result 6.5.1. In so doing, the difference of the effective moduliL∗ estimated by the useof GIEs (6.37) and (6.38) is not significant.

Fundamentally new effects (compare Results 6.5.1 and 6.5.2) were detected at the consideration of one sort ofstress inhomogeneity caused by nonlocal constitutive law (6.34), (6.36). It is expected that investigation of otheradditional sources of stress inhomogeneities inside the inclusions (such as, e.g., nonellipsoidal shape of inclusions, itsbinary interactions corresponding to the hypothesisH2a, imperfect interface conditions, violation of the hypothesisH3) will lead to discovery of additional new effects in the case of the use of the new GIE (6.37) instead of the old GIE(6.38).

6.6 Wave Propagation in Composites

We study a monochromatic elastic wave of frequencyω that propagates in a composite medium with the local elas-tic propertiesL(x), α(x) and the mass densityρ(x) (2.8), (2.10) whilef(x), [[u(x)]], [[σ(x)]] · n(x) ≡ 0. Ifthe dependence of timet is defined by the factorexp(iωt), the displacement fieldui in the medium has the formui(x, t) = ui(x)exp(iωt), and amplitudeui(x) of this field satisfies the following wave motion equation (see, e.g.,Willis, 1980):

Lω(x)u(x) = 0, Lωik(x) = ∇jLijkl(x)∇l + δikρ(x)ω

2 (6.40)



with the corresponding time-reduced wave Green’s tensor satisfying the reduced wave equation for an infinite homo-geneous comparison body

Lωc(x)g(x,y) = −δδ(x− y), Lωc = ∇Lc∇+ δρcω2, (6.41)

that is, one assumes that the coefficientb(x) = δρ(x)ω2 (2.20).The perturbatorsLτ

k(x− xk, τ) andLuτk (x− xk, τ) are presented by Eqs. (3.22) and (3.24), respectively, where

the surface integrals are omitted. We consideru andε as independent variables, and introduce a symbolic notationε = (u, ε)⊤ for which the coupled forms of the GIEs lead from the GIEs (3.18), (4.41), and (4.43)

ε(x) = ⟨ε⟩(x) +∫[Uω(x− y)τ(y)− ⟨Uω(x− y)τ⟩(y)] dy, (6.42)

ε(x) = ⟨ε⟩+∫

Uω(x− y)[τ(y)− ⟨τ⟩] dy, (6.43)

ε(x) = ε0 +

∫Uω(x− y)τ(y) dy, (6.44)

respectively, where

ε =

(uε

), τ =

(π

τ

), Uω=

(g ∇g

∇g ∇∇g

). (6.45)

Here Eqs. (6.43) (see Willis, 1980; see also for references Buryachenko, 2007; Kanaun and Levin, 2008) and (6.44)hold only for statistically homogeneous material (2.4) and field (2.28) problems, and for the fieldsX bounded in onedirection (4.6), respectively, while the new exact GIE (6.42) is valid for any statistically inhomogeneous medium. Fishet al. (2012b) proposed a dispersive nonlinear theory for periodic media and estimated the dispersion and micro-inertiaeffects in either explicit or implicit integration scheme.

Fundamentally new effects of using of the static counterparts of GIEs (6.42) and (6.43) were detected (see Sec-tions 6.1 and 6.2) for any problem and method providing estimation of inhomogeneous stress distributions insideinclusions (such as, e.g., nonellipsoidal shape of inclusions, its binary interactions corresponding to the hypothesisH2a, imperfect interface conditions, violation of the hypothesisH3). All these sources of the stress inhomogeneitiesand their impacts on the estimated effective material and field parameters are also valid for GIEs (6.42) and (6.43).However, there is an additional reason of stress inhomogeneity inside inclusions which is absent in all static problemsand intrinsic in basic wave motion phenomena (6.40) and (6.41) even for a single (c = 0) homogeneous ellipsoidalinclusionx ∈ vi subjected to the homogeneous effective fieldε(x) ≡ const. (x ∈ vi ). Namely, even in such a case,the fields of the strainε(x) and displacementu(x) are inhomogeneous inside the inclusion (see for details, Kanaunand Levin, 2008), and, therefore, for the composite materials considered, the hypothesisH1b is not valid in generaleven in the framework of the hypothesesH1a, H2b, andH3. Thus, a wave motion counterpart ofResult6.1.1 doesnot hold in general.

A popular transformation of analogs of Eq. (6.44) (widely used also for statistically homogeneous media) tothe correct approximate Eq. (6.43) is performed in a spirit of the scheme (6.15) (see, e.g., Willis, 1980; Fokin andShermergor, 1989; Kanaun and Levin, 2005, 2008; Chen and Shen, 2007). A wave motion counterpart of the EMM(4.46) (see for references Kanaun and Levin, 2008, where the different versions of the EMM also can be found)along with its static case (4.46) can also be considered as a truncation of the corresponding GIE (6.44) [or (6.43)].Furthermore, dealing only with one type of scalar waves (e.g., electric waves) propagating significantly simplifiesanalysis of corresponding equations which coincide within notations with Eqs. (6.40), (6.41) where the tensors ofthe fourth (L) and first (u,δ) orders should be formally replaced by the tensors of the second order and the scalar,respectively (see for references Kanaun and Levin, 2008). All comments mentioned with respect to Eqs. (6.42)–(6.44)for elastic waves propagation also remain in force for the scalar wave problems. Generalization of GIEs (6.42)–(6.44) to the consideration of coupled effects (e.g., electromagnetic waves, electromagnetic and elastic waves inthermo-magneto–electro-elastic composites, see, e.g., Chen and Shen, 2007; Lu et al., 2011, Levin et al., 2011) isstraightforward: we only need to extend the block’s structures (6.45) by some further blocks corresponding to theappropriate coupled phenomena (6.29), (6.30).


46 Buryachenko

An upsurge in interest in wave propagation phenomena has been prompted by the recent proliferation of meta-materials (in both the theoretical and practical senses) which are no more than the composites exhibiting exceptionalfrequency electromagnetic, optic and mechanical responses not readily observed in nature. Metamaterials are artifi-cially engineered periodic or random microstructures displaying strong resonance behavior at frequencies at whichthe inclusions themselves and the distances between them are small, which can possess simultaneously negative di-electric constant and magnetic permeability in the context of electromagnetics, and negative effective mass densityand negative elastic constants in the case of elasticity. The original design of metamaterials is the combination ofmetallic split ring resonators and metallic rods, which realize the negative permeability and the negative permittivity,respectively, while in succeeding years it was performed experimental or/and computational investigations of differentinclusion shapes (e.g., silver dendritic cells, U-shaped nanostructures, sphere-rod structures, split hollow spheres) andits microstructure (e.g., coated structures), see Ding and Zhao (2011), Gong and Zhao (2012), Gordon and Ziolkowski(2008), Liu et al. (2008), Varadan and Kim (2012). The most popular methods of effective properties estimations arenumerical homogenization of periodic structures and EMM [see Section 4.3, (4.46)] of random structures of sphericalcoated nanoparticles (e.g., Gordon and Ziolkowski, 2008; Kussow et al., 2008; Wani et al., 2012). However, appli-cation of the new GIE (6.42) instead of a classical one (6.43) opens the new avenue of attack on the metamaterial’sinvestigations. Namely, metamaterials exhibit unique opportunity for simultaneous display of a few sources of fieldinhomogeneities inside the inclusions (coated structure, noncanonical shape, nanoscale size, imperfect interface, wavemotion phenomena) violating the hypothesisH1b even in the framework of the hypothesisH1a. In such a case allresults mentioned in the preceding Sections 6.1–6.5 are expected to be manifested for the metamaterials even moredramatically than for the conventional composites considered before.

7. CONCLUSION

The operator forms of the GIEs (3.32) and (3.33) are proposed for the general cases of local and nonlocal problems,static and wave motion phenomena for composite materials with periodic and random (statistically homogeneous andinhomogeneous, FGM) structures containing coated or uncoated inclusions of any shape and orientation with perfectand imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneousexternal fields of different physical nature. The operator forms (3.32) and (3.33), apart from their generality, eliminatesany opportunity to accept the asymptotic approximations (4.37)–(4.39) which were mistakenly considered by scientificsociety as the exact equalities. Precisely an approval of the Eqs. (4.37)–(4.39) [which are just a particular case of thehypothesisH1b (4.9)] lead to the known approximate Eqs. (6.7), (6.8) instead of the exact new Eqs. (6.5), (6.6).Equations (3.32), (3.33), and its particular cases Eqs. (6.5), (6.6), were obtained by proposed modified centeringmethod without any auxiliary assumptions such as the EFH, which is implicitly exploited in the known centeringmethods. Thus, a primariness of the operator forms of the GIE with respect to the EFH is rigorously proved.

However, besides a theoretical interest to the mentioned relation between GIE and EFH, Eqs. (3.32), (3.33) providethe critical practical benefits. The use of Eqs. (3.32), (3.33) allow one to completely abandon the hypothesesH1 andH3 while the hypothesisH2 can be used for multiparticle generality. It opens a dramatic extension of opportunitiesfor exploiting in analytical micromechanics the tools of computational mechanics explosively progressing (especiallyin front of nanotechnology challenges). Moreover, modification of the renormalized terms [compare Eqs. (6.7), (6.8)with Eqs. (6.5), (6.6)] makes it possible to capture the fine effects (e.g., a field inhomogeneity inside inclusions)which only can be estimated by the advanced computational methods. For example, it is expected that the greaterinhomogeneity of the field concentrator factor for a single heterogeneity inside infinite matrix leads to the greaterdifference between the new (3.32), (3.33) and old (6.7), (6.8) approaches with the possible change of sign of predictedlocal fields in composite materials (see for details Buryachenko, 2010a,b, 2011b,c, 2013; Buryachenko and Brun,2011, 2012). Thus the new GIEs (3.32), (3.33), in fact, form a new background of micromechanics [which is thesecond; the first one based on the EFH was proposed by Mossotti (1850), and others, see Introduction] offering theopportunities for a fundamental jump in multiscale research of composites and nanocomposites.

A fundamental limitation of GIEs proposed in this paper is defined by the differential form of basic equationsof continuum mechanics presented in Section 2.1. In contrast to these classical local and nonlocal theories, the peri-dynamic equation of motion introduced by Silling (2000) (see for references and details Silling and Lehoucq, 2010)



is free of any spatial derivatives of displacement (in the case of elasticity theory counterpart). The basic feature ofthe peridynamic model is a continuum description of a material behavior as the integrated nonlocal force interactionsbetween infinitesimal particles. Generally in peridynamics, the state-based approach describes a material behaviorwhen every material point interacts simultaneously with all other material points within its finite radius horizon viaa response function that completely describes the interaction. This might be an attractive feature especially for theproblems involving discontinuities in the deformation process. Unfortunately, the background concepts of analyticalmicromechanics (see, e.g., the current paper for references) such as the effective moduli, effective fields, and espe-cially the GIEs are not yet defined in the theory of random structure peristaltic CMs. Buryachenko (2015) attemptedto define these notions, concepts, and the corresponding equations. In so doing, the GIEs obtained in both Section 3and Buryachenko (2015) are formally very similar, which opens a way for straightforward expansion of analyticalmicromechanics tools to the new area of random structure peridynamic CMs. However, more detailed considerationof this prospective direction is beyond the scope of the current paper.

ACKNOWLEDGMENTS

This work was partially funded by the US Office of Naval Research (Dr. William M. Mullins, contract: N00014-14-P-1255). The author would like to acknowledge stimulating and fruitful suggestions by Professor Jacob Fish fromColumbia University.

REFERENCES

Asaro, R. J., Somigliana dislocations and internal stresses; with application to second phase hardening,Int. J. Eng. Sci., vol. 13,pp. 271–286, 1975.

Barenblatt, G. I., The mathematical theory of equilibrium of cracks in brittle fracture,Adv. Appl. Mech., vol. 7, pp. 55–129, 1962.

Barnett, D. M. and Lothe, J., Dislocation and line charges in anisotropic piezoelectric insulators,Phys. Stat. Solids (b),vol. 67, pp.105–111, 1975.

Batchelor, G. K., Sedimentation in a dilute dispersion of spheres,J. Fluid Mech., vol. 52, pp. 245–268, 1972.

Batchelor, G. K., Transport properties of two-phase materials with random structure,Ann Rev. Fluid Mech., vol. 6, pp. 227–255,1974.

Bazant, Z. P. and Jirasek, M., Nonlocal integral formulations of plasticity and damage: Survey of progress,J. Eng. Mech., vol. 128,pp. 1119–1149, 2002.

Benveniste, Y., A new approach to application of Mori–Tanaka’s theory in composite materials,Mech. Mater., vol. 6, pp. 147–157,1987.

Benveniste, Y. and Miloh, T., The effective conductivity of composites with imperfect contact at constituent interfaces,Int. J. Eng.Sci., vol. 24, pp. 1537–1552, 1986.

Beran, M. J. and McCoy, J. J., Mean field variations in a statistical sample of heterogeneous linearly elastic solids,Int. J. SolidStruct., vol. 6, pp. 1035–1054, 1970.

Buryachenko, V. A.,Micromechanics of Heterogeneous Materials, Springer, New York, 2007.

Buryachenko, V. A., On the thermo-elastostatics of heterogeneous materials. I. General integral equation,Acta Mech., vol. 213, pp.359–374, 2010a.

Buryachenko, V. A., On the thermo-elastostatics of heterogeneous materials. II. Analyze and generalization of some basic hypothe-ses and propositions,Acta Mech., vol. 213, pp. 375–398, 2010b.

Buryachenko, V. A., Inhomogeneity of the first and second statistical moments of stresses inside the heterogeneities of randomstructure matrix composites,Int. J. Solids Struct., vol. 48, pp. 1665–1687, 2011a.

Buryachenko, V. A., On thermoelastostatics of composites with nonlocal properties of constituents. I. General representations foreffective material and field parameters,Int. J. Solids Struct., vol. 48, pp. 1818–1828, 2011b.

Buryachenko, V. A., On thermoelastostatics of composites with nonlocal properties of constituents. II. Estimation of effectivematerial and field parameters,Int. J. Solids Struct., vol. 48, pp. 1829–1845, 2011c.


48 Buryachenko

Buryachenko, V. A., General integral equations of micromechanics of composite materials with imperfectly bonded interfaces,Int.J. Solids Struct., vol. 50, pp. 3190–3206, 2013.

Buryachenko, V. A., Some general representations in thermoperistatics of random structure composites,Int. J. Multiscale Comput.Eng., 2015 (in press, doi: 10.1615/IntJMultCompEng.2014011358).

Buryachenko, V. A. and Brun, M., FEA in elasticity of random structure composites reinforced by heterogeneities of noncanonicalshape,Int. J. Solid Struct., vol. 48, pp. 719–728, 2011.

Buryachenko, V. A. and Brun, M., Random residual stresses in elasticity homogeneous medium with inclusions of noncanonicalshape,Int. J. Multiscale Comput. Eng., vol. 10, pp. 261–279, 2012.

Chen, P. and Shen, Y., Propagation of axial shear magneto-electro-elastic waves in piezoelectric–piezomagnetic composites withrandomly distributed cylindrical inhomogeneities,Int. J. Solids Struct., vol. 44, pp. 1511–1532, 2007.

Chen, T., Dvorak, G. J., and Benveniste, Y., Stress fields in composites reinforced by coated cylindrically orthotropic fibers,Mech.Mater., vol. 9, pp. 17–32, 1990.

Chen, T., Dvorak, G. J., and Yu, C. C., Solids containing spherical nano-inclusions with interface stresses: Effective properties andthermal-mechanical connections,Int. J. Solids Struct., vol. 44, pp. 941–955, 2007.

Christensen, R. M.,Mechanics of Composite Materials, Wiley Interscience, New York, 1979.

Clausius, R.,Die Mechanische Behandlung der Electricitat, Vieweg, Braunshweig, 1879.

Dinzart, F. and Sabar, H., Magneto-electro-elastic coated inclusion problem and its application to magnetic-piezoelectric compositematerials,Int. J. Solids and Structures, vol. 48, pp. 2393–2401, 2011.

Ding, C.-L. and Zhao, X.-P., Multi-band and broadband acoustic metamaterial with resonant structures,J. Phys. D: Appl. Phys.,vol. 44, 215402 (8pp.), 2011.

Drugan, W. J. and Willis, J. R., A micromechanics-based nonlocal constitutive equation and estimates of representative volumeelements for elastic composites,J. Mech. Phys. Solids, vol. 44, pp. 497–524, 1996.

Duan, H. L. and Karihaloo, B. L., Thermo-elastic properties of heterogeneous materials with imperfect interfaces: GeneralizedLevin’s formula and Hill’s connections,J. Mech. Phys. Solids, vol. 55, pp. 1037–1052, 2007.

Duan, H. L., Wang, J., Huang, Z. P., and Karihaloo, B. L., Eshelby formalism for nano-inhomogeneities,Proc. R. Soc., vol. A461,pp. 3335–3353, 2005.

Duan, H. L., Yi, X., Huang, Z. P., and Wang, J., A unified scheme for prediction of effective moduli of multiphase composites withinterface effects: Part II. Application and scaling laws,Mech. Materials, vol. 39, pp. 94–103, 2007.

Dvorak, G. J.,Micromechanics of Composite Materials, Springer, Dordrecht, 2013.

Dvorak, G. J. and Benveniste, Y., On the thermomechanics of composites with imperfectly bonded interfaces and damage,Int. J.Solids. Struct., vol. 29, pp. 2907–2919, 1992.

Edelen, D. G. B. and Laws, N., On the thermodynamics of systems with nonlocality,Arch. Ration. Mech. Anal., vol. 43, pp. 36–44,1971.

Eringen, A. C.,Microcontinuum Field Theories. I: Foundations and Solids, Springer-Verlag, New York, 1999.

Eringen, A. C.,Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.

Eshelby, J. D., Elastic inclusion and inhomogeneities, in Sneddon, I. N., Hill, R. (eds.),Progress in Solid Mechanics, North-Holland, Amsterdam, vol.2, pp. 89–140, 1961.

Faraday, M., Experimental researches on electricity,Philos. Trans. R. Soc. London, II Ser. 1ff, 1838.

Filatov, A. N. and Sharov, L. V.,Integral Inequalities and the Theory of Nonlinear Oscillations, Nauka, Moscow (in Russian),1979.

Fish, J., Jiang, T., and Yuan, Z., A staggered nonlocal multiscale model for a heterogeneous medium,Int. J. Numer. Meth. Eng.,vol. 91, pp. 142–157, 2012a.

Fish, J., Filonova, V., and Kuznetsov, S., Micro-inertia effects in nonlinear heterogeneous media,Int. J. Numer. Meth. Eng., vol.91, pp. 1406–1426, 2012b.

Fokin, A. G. and Shermergor, T. D., Theory of propagation of elastic waves in nonhomogeneous media,Mekhan. Kompoz. Mate-rialov, vol. 25, no. 5, pp. 821–832 (in Russian Engl. Transl.Mech. Compos. Materials, vol. 25, pp. 600–609, 1990), 1989.

Fries, T. P. and Belytschko, T., The extended/generalized finite element method: An overview of the method and its applications,



Int. J. Numer. Meth. Eng., vol. 84, pp. 253–304, 2010.

Furmanski P., Head conduction in composites: Homogenization and macroscopic behavior,Appl. Mech. Rev., vol. 50, pp. 327–356,1997.

Gelfand, I. A. and Shilov, G.,Generalized Functions, vol. I , Academic Press, New York, 1964.

Ghosh, S.,Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method (ComputationalMechanics and Applied Analysis), CRC Press, Boca Raton, 2011.

Gong, B. and Zhao, X., Three-dimensional isotropic metamaterial consisting of domain-structure,Physica, vol. B407, pp. 1034–1037, 2012.

Gordon, J. A. and Ziolkowski, R. W., CNP optical metamaterials,Opt. Expr., vol. 16, pp. 6692–6716, 2008.

Graham-Brady, L. L., Siragy, E. F., and Baxter, S. C., Analysis of heterogeneous composites based on moving-window techniques,J. Eng. Mech., vol. 129, pp. 1054–1064, 2003.

Gurtin, M. E. and Murdoch, A. I., A continuum theory of elastic material surfaces,Arch. Rational Mech. Anal., vol. 59, pp.291–323, 1975.

Gurtin, M. E., Weissmuller, J., and Larche, F., The general theory of curved deformable interfaces in solids at equilibrium,Philos.Mag., vol. A78, pp. 1093–1109, 1998.

Gutkin, M. Y., Elastic behavior of defects in nanomaterials: I. Models for infinite and semi-infinite media,Rev. Adv. Mater. Sci.,vol. 13, pp. 125–161, 2006.

Halle, D. K., The physical properties of composite materials,J. Mater. Sci.,vol. 11, pp. 2105–2141, 1976.

Hansen, J. P. and McDonald, I. R.,Theory of Simple Liquids, Academic Press, New York, 1986.

Hashin, Z., The spherical inclusion with imperfect interface,J. Appl. Mech., vol. 58, pp. 444–449, 1991.

Hashin, Z., Thermoelastic proties of particulate composites with imperfect interface,J. Mech. Phys. Solids, vol. 39, pp. 745–762,1991.

Hashin, Z., Thin interphase imperfect interface in elasticity with application to coated fiber composites,J. Mech. Phys. Solids, vol.50, pp. 2509–2537, 2002.

Hashin, Z. and Shtrikman, S., On some variational principles in anisotropic and nonhomogeneous elasticity,J. Mech. Phys. Solids,vol. 10, pp. 335–342, 1962.

Hashin, Z. and Shtrikman, S., A variational approach to the theory of the behavior of multiphase materials,J. Mech. Phys. Solids,vol. 11, pp. 127–140, 1963.

He, L. H. and Li, Z. R., Impact of surface stress on stress concentration,Int. J. Solids Struct., vol. 43, pp. 6208–6219, 2006.

Hill, R., Elastic properties of reinforced solids: Some theoretical principles,J. Mech. Phys. Solids,vol. 11, pp. 357–372, 1963.

Hill, R., New derivations of some elastic extremum principles,Prog. in Appl. Mechanics, The Prager Anniversary Volume, Macmil-lan, New York, pp. 99–106, 1963.

Hill, R., A self-consistent mechanics of composite materials,J. Mech. Phys. Solids, vol. 13, pp. 212–222, 1965.

Hori, M. and Nemat-Nasser, S., Interacting microcracks near the tip in the process zone of a macrocrack,J. Mech. Phys. Solids,vol. 35, pp. 601–629, 1987.

Huang, J., Furuhashi, R., and Mura, T., Frictional sliding inclusions.J. Mech. Phys. Solids, vol. 41, pp. 247–265, 1993.

Ibach, H., The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures,Surf. Sci. Rep.,vol. 29(5–6), pp. 193–263, 1997.

Jasiuk, I., Tsuchida, E., and Mura, T., The sliding inclusion under shear,Int. J. Solids Struct., vol. 23, pp. 1373–1385, 1987.

Jayaraman, K. and Reifsnider, K. L., Residual stresses in a composite with continuously varying Young’s modulus in thefiber/matrix interphase,J. Comp. Mater., vol. 26, pp. 770–791, 1992.

Jirasek, M. and Rolshoven, S., Comparison of integral-type nonlocal plasticity models for strain-softening materials,Int. Eng. Sci.,vol. 41, pp. 1553–1602, 2003.

Jeffrey, D. J., Conduction through a random suspension of spheres,Proc. R. Soc. London, vol. A335, pp. 355–367, 1973.

Ju, J. W. and Tseng, K. H., Improved two-dimensional micromechanical theory for brittle solids with randomly located interactingmicrocracks,Int. J. Damage Mech., vol. 4, pp. 23–57, 1995.

Kanaun, S. K., Self-consistent field approximation for an elastic composite medium,Zhurnal Priklad. Tehknich. Fiziki, vol. 18(2),


50 Buryachenko

pp. 160–169 (in Russian Engl. Transl.J. Appl. Mech. Techn. Physics, vol. 18, pp. 274–282), 1977.

Kanaun, S. K. and Levin, V. M., Propagation of shear elastic waves in composites with a random set of spherical inclusions(effective field approach),Int. J. Solids Struct., vol. 42, pp. 3971–3997, 2005.

Kanaun, K. K. and Levin, V. M.,Self-Consistent Methods for Composites, vol. 1, 2, Springer, Dordrecht, 2008.

Khoroshun, L. P., Elastic properties of materials reinforced by uni-directional short fibers,Prikladnaya Mekhanika, vol. 8(12), pp.86–92 (in Russian Engl. Transl.Soviet Appl. Mech., vol. 8, pp. 1358–1363), 1972.

Khoroshun, L. P., Prediction of thermoelastic properties of materials strengthened by unidirectional discrete fibers,PrikladnayaMekhanika, vol. 10(12), pp. 23–30 (in Russian Engl. Transl.Soviet Appl. Mech., vol. 10, pp. 1288–1293), 1974.

Khoroshun, L. P., Random functions theory in problems on the macroscopic characteristics of microinhomogeneous media,PrikladMekh., vol. 14(2), pp. 3–17 (in Russian Engl. Transl.Sov. Appl. Mech., vol. 14, pp. 113–124), 1978.

Khoroshun, L. P., Conditional-moment method in problems of the mechanics of composites,Priklad. Mekh., vol. 23(10), pp.100–108 (in Russian Engl. Transl.Sov. Appl. Mech., vol. 23, pp. 989–998), 1987.

Kroner, E., Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanstanten des Einkristalls,Z. Physik., vol. 151,pp. 504–518, 1958.

Kroner, E., Elasticity theory of materials with long range cohesive forces,Int. J. Solids Struct., vol. 3, pp. 731–742, 1967.

Kroner, E., The problem of non-locality in the mechanics of solids: Review of present status, in: Simmons, J., de Wit, R., Bul-lough, R., (Eds.),Fundamental Aspects of Dislocation Theory, Nat. Bur. Stand. (US), Washington, vol.II , pp. 729–736, 1970.

Kroner, E., On the physics and mathematics of self-stresses, in: Zeman, J. L. and Ziegler, F. (Eds.),Topics in Applied ContinuumMechanics, Springer-Verlag, Wien, pp. 22–38, 1974.

Kroner, E., Bounds for effective moduli of disordered materials,J. Mech. Phys. Solids, vol. 25, pp. 137–155, 1977.

Kroner, E., Statistical modeling, in: Gittus, J. and Zarka, J. (Eds.),Modeling Small Deformations of Polycrystals, Elsevier, London,pp. 229–291, 1986.

Kroner, E., Modified Green function in the theory of heterogeneous and/or anisotropic linearly elastic media,Micromechanics andInhomogeneity. The Toshio Mura 65th Anniversary Volume, pp. 197–211, eds., Weng, G. J., Taya, M., and Abe, H., Springer-Verlag, New York, 1990.

Kroner, E. and Datta, B. K., Non-local theory of elasticity for a finite inhomogeneous medium, a derivation from lattice theory,in: Simmons, J., de Wit, R., and Bullough, R. (Eds.),Fundamental Aspects of Dislocation Theory, Nat. Bur. Stand. (US),Washington, vol.II , pp. 737–746, 1970.

Kunin, I. A., Inhomogeneous elastic medium with nonlocal interaction,Zhurnal Prikl. Mekhan. Tekhn. Fiziki, vol. 8, pp. 60–66 (inRussian Engl. Transl.,J. Appl. Mech. Techn. Phys., vol. 8, pp. 41–44), 1967.

Kussow, A.-G., Akyurtlu, A., and Angkawisittpan, N., Optically isotropic negative index of refraction metamaterial,Phys. Stat.Sol., vol. b245, pp. 992–997, 2008.

Kuznetsov, S. V., Microstructural stress in porous media,Priklad. Mech., vol. 27(11), pp. 23–28 (in Russian Engl. Transl.Sov.Appl. Mech., vol. 27, pp. 750–755), 1991.

Kuznetsov, S. and Fish, J., Mathematical homogenization theory for electroactive continuum,Int. J. Numer. Methods Eng., vol. 91,pp. 1199–1226, 2012.

Landauer, R., Electric conductivity in inhomogeneous media, in: Garland, J. C. and Tanner, D. B. (Eds.),Electric, Transport andOptical Properties of Inhomogeneous Media, American Institute of Physics, New York, pp. 2–43, 1978.

Lax, M., Multiple scattering of waves II. The effective fields dense systems,Phys. Rev., vol. 85, pp. 621–629, 1952.

Lekhnitskii, A. G.,Theory of Elasticity of an Anisotropic Elastic Body, Holder Day, San Francisco, 1963.

Levin, V. M., Thermal expansion coefficient of heterogeneous materials,Izv. AN SSSR, Mekh Tverd Tela, (2), pp. 88–94 (in RussianEngl. Transl.Mech Solids, vol. 2(2), pp. 58–61), 1967.

Levin, V. M., Determination of effective elastic moduli of composite materials,Docl Akad Nauk SSSR, vol. 220, pp. 1042–1045(in Russian Engl. Transl.,Sov. Phys. Docl., vol. 20, pp. 147–148), 1975.

Levin, V. M., Determination of composite material elastic and thermoelastic constants,Izv AN SSSR, Mech Tverd Tela, No. 6, pp.137–145, (in Russian Engl. Transl.,Mech. Solids, vol. 11(6), pp. 119–126), 1976.

Levin, V. M., Michelitsch, T. M., and Gao, H., Propagation of electroacoustic waves in the transversely isotropic piezoelectric



medium reinforced by randomly distributed cylindrical inhomogeneities,Int. J. Solids Struct., vol. 39, pp. 5013–5051, 2002.

Levin, V. M., Valdiviezo-Mijangos, O., and Sabina, G. J., Propagation of electroacoustic axial shear waves in a piezoelectricmedium reinforced by continuous fibers,Int. J. Eng. Sci., vol. 49, pp. 1232–1243, 2011.

Liu, B., Zhao, X., Zhu, W., Luo, W., and Cheng, X., Multiple pass-band optical left-Handed metamaterials based on randomdendritic cells,Adv. Funct. Mater., vol. 18, pp. 3523–3528, 2008.

Liu, Y. L., Mukherjee, S., Nishimura, N., Schanz, M., Ye, W., Sutradhar, A., Pan, E., Dumont, N. A., Frangi, A., and Saez, A.,Recent advances and emerging applications of the boundary element method,Appl. Mech. Rev., vol. 64, 031001 (38 pages),2011.

Lomakin, V. A.,Statistical Problems of the Mechanics of Solid Deformable Bodies, Nauka, Moscow (in Russian), 1970.

Lu, X. Y., Li, H., and Wang, B., Theoretical analysis of electric, magnetic and magnetoelectric properties of nano-structuredmultiferroic composites,J. Mech. Phys. Solids, vol. 59, pp. 1966–1977, 2011.

Maranganti, R. and Sharma, P., Strain field calculation in embedded quantum dots and wires,J. Comput. Theoret. Nanoscience,vol. 4, pp. 715–738, 2007.

Markov, K. Z., Elementary micromechanics of heterogeneous media, in: Markov K., Preziosi L. (Eds.),Heterogeneous Media.Micromechanics, Modelling, Methods, and Simulations, Birkhauser, Boston, pp. 1–162, 1999.

Markworth, A. J., Ramesh, K. S., and Parks, W. P., Review. Modelling studies applied to functionally graded materials,J. Mater.Sci., vol. 30, pp. 2183–2193, 1995.

Maugin, G. A.,Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam, 1988.

McCoy, J. J., On the calculation of bulk properties of heterogeneous materials,Quarterly of Appl. Math., vol. 36, pp. 137–149,1979.

McCoy, J. J., Macroscopic response of continue with random microstructure, in: Nemat-Nasser, S. (Ed.),Mech. Today, PergamonPress, Oxford, vol.6, pp. 1–40, 1981.

Milgrom, M., Linear response of general composite systems to many coupled fields,Phys. Rev., vol. B41, pp. 12484–12494, 1990.

Milgrom, M. and Shtrikman, S., Coupled multifield linear response of two-phase composites: Exact universal relations,Phys. Rev.Lett., vol. 62, pp. 1979–1981, 1989.

Miloh, T. and Benveniste, Y., On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conductinginterfaces,Proc. R. Soc. London, vol. A455, pp. 2687–2706, 1999.

Milton, G. W., The Theory of Composites, Applied and Computational Mathematics, vol.6, Cambridge University Press, Cam-bridge, 2002.

Mori, T. and Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions,Acta Metall,vol. 21, pp. 571–574, 1973.

Mortensen, A. and Suresh, S., Functionally graded metals and metal–ceramic composites: Part 1. Processing,Int. Mater. Rev.,vol.40, pp. 239–265, 1995.

Morse, P. M. and Feshbach, H.,Methods of Theoretical Physics, Parts I and II. McGraw-Hill, Maidenhead, 1953.

Mossotti, O. F., Discussione analitica sul’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’electricita allasuperficie di piu corpi elettrici disseminati in eso,Mem. Mat. Fis. della Soc. Ital. di Sci. in Modena,vol. 24, pp. 49–74, 1850.

Mura, T. and Furuhashi, R., The elastic inclusion with a sliding interface,J. Appl. Mech., vol. 51, pp. 308–310, 1984.

Mura, T., Jasiuk, I., and Tuschida, B., The stress field of a sliding inclusion,Int. J. Solids Struct., vol. 21, pp. 1165–1179, 1985.

Mura, T., Shodia, H. M., and Hirose, Y., Inclusions problems,Appl Mechan Rev., vol. 49, Part 2, pp. S118–S127, 1996.

Needleman, A., An analysis of decohesion along an imperfect interface,Int. J. Fract., vol. 42, pp. 21–40, 1990.

Nielsen, L. E.,Mechanical Properties of Polymers and Composites, Marcel Dekker, Inc., New York, 1974.

O’Brian, R. W., A method for the calculation of the effective transport properties of suspensions of interacting particles,J. Fluid.Mech., vol. 91, pp. 17–39, 1979.

Ortiz, M. and Pandolfi, A., Finite-deformation irreversible cohesive element for three-dimensional crack-propagation analysis,Int.J. Numer. Methods Eng., vol. 44, pp. 1267–1282, 1999.

Othmani, Y., Delannay, L., and Doghri, I., Equivalent inclusion solution adapted to particle debonding with a non-linear cohesivelaw, Int. J. Solids Struct., vol. 48, pp. 3326–3335, 2011.


52 Buryachenko

Parton, V. Z. and Kudryavtsev, B. A.,Electromagnetic Elasticity of Piezoelectric and Electrically Conductive Bodies, Nauka,Moscow, 1988.

Pfeil, K. and Klingenberga, D. J., Nonlocal electrostatics in heterogeneous suspensions using a point-dipole model,J. Appl. Phys.,vol. 96, pp. 5341–5348, 2004.

Plankensteiner, A. F., Bohm, H. J., Rammerstorfer, F. G., and Buryachenko, V. A., Hierarchical modeling of the mechanicalbehavior of high speed steels as layer–structured particulate MMCs,J. Phys. IV, vol. 6, pp. C6-395–C6-402, 1996.

Plankensteiner, A. F., Bohm, H. J., Rammerstorfer, F. G., Buryachenko, V. A., and Hackl, G., Modelling of layer–structured highspeed tool steel,Acta Metall. Mater., vol. 45, pp. 1875–1887, 1997.

Poisson, S., Memoire sur la theorie du magnetisme,Mem de l’ Acad R de France, vol. V, pp. 247–338, 1824.

Polizzotto, C., Nonlocal elasticity and related variational principles,Int. J. Solids Struct., vol. 38, pp. 7359–7380, 2001.

Polizzotto, C., Gradient elasticity and nonstandard boundary conditions,Int. J. Solids Struct., vol. 40, pp. 7399–7423, 2003.

Ponte Castaneda, P. and Willis, J. R., The effect of spatial distribution on the effective behavior of composite materials and crackedmedia,J. Mech. Phys. Solids, vol. 43, pp. 1919–1951, 1995.

Povstenko, Y. Z., Theoretical investigation of phenomena caused by heterogeneous surface tension in solids,J. Mech. Phys. Solids,vol. 41, pp. 1499–1514, 1993.

Qin, Q. and Yang, Q.-S.,Macro-Micro Theory on Multifield Coupling Behavior of Heterogeneous Materials, Springer, Berlin,2008.

Rayleigh, L., On the influence of obstacles arranged in rectangular order upon the properties of a medium,Philos. Mag., vol. 34,pp. 481–502, 1892.

Rogula, D., Introduction to nonlocal theory of material media, in: D. Rogula (Ed.),Nonlocal Theory of Material Media, Springer-Verlag, Berlin, pp. 125–222, 1982.

Scaife, B. K. P.,Principle of Dielectrics, Oxford University Press, Oxford, UK, 1989.

Sejnoha, M. and Zeman, J.,Micromechanics in Practice, WIT Press, Southampton, UK, 2013.

Sevostianov, I. B., Levin, V. M., and Pompe, W., Evolution of the mechanical properties of ceramics during drying,Phys. Stat. Sol.,vol. (a) 166, pp. 817–828, 1998.

Sharma, P. and Dasgupta, A., Average elastic field and scale-dependent overall properties of heterogeneous micropolar materialscontaining spherical and cylindrical inhomogeneities,Phys. Rev., vol. B66, 224110, pp. 1–10, 2002.

Sharma, P. and Ganti, S., Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies,J. Appl. Mech., vol. 71, pp. 663–671, 2004.

Sharma, P. and Wheeler, L. T., Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension,J. Appl. Mech., vol. 74, pp. 447–454, 2007.

Shermergor, T. D.,The Elasticity Theory of Microheterogeneous Media, Nauka, Moscow (in Russian), 1977.

Shvidler, M. I.,Statistical Hydrodynamics of Porous Media, Nauka, Moscow (in Russian), 1985.

Silling, S., Reformulation of elasticity theory for discontinuities and long-range forces,J. Mech. Phys. Solids, vol. 48, pp. 175–209,2000.

Silling, S. A. and Lehoucq, R. B., Peridynamic theory of solid mechanics,Adv. Appl. Mech., vol. 44, pp. 73–168, 2010.

Somigliana, C., Sopra l’equilibrio di un corpo elastico isotropo,II Nuovo Cimento, vol. 19, pp. 84–90, 1886.

Stratonovich, R. L.,Topics in the Theory of Random Noise, Gordon and Breach, New York, 1963.

Tan, H., Huang, Y., Liu, C., Inglis, H. M., Ravichandran, G., and Geubelle, P. H., The uniaxial tension of particle-reinforcedcomposite materials with nonlinear interface debonding,Int. J. Solids Struct., vol. 44, pp. 1809–1822, 2007.

Tan, H., Huang, Y., Liu, C., Ravichandran, G., and Paulino, G. H., Constitutive behaviors of composites with interface debonding:The extended Mori–Tanaka method for uniaxial tension,Int. J. Fract., vol. 146, pp. 139–148, 2007.

Torquato, S., Effective stiffness tensor of composite media. I. Exact series expansion,J. Mech. Phys. Solids, vol. 45, pp. 1421–1448,1997.

Torquato, S.,Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, 2002.

Torquato, S. and Lado, F., Improved bounds on the effective elastic moduli of random arrays of cylinders,J. Appl. Mech., vol. 59,



pp. 1–6, 1992.

Torquato, S. and Rintoul, M. D., Effect of the interface on the properties of composite media.Phys. Rev. Lett.,vol. 75, pp. 4067–4070, 1995.

Varadan, V. V. and Kim, K., Fabrication of 3-D metamaterials using LTCC techniques for high-frequency applications,IEEE Trans.Components, Packaging Manufac. Technol., vol. 2, pp. 410–417, 2012.

Walpole, L. J., On the bounds for the overall elastic moduli of inhomogeneous system, I.J. Mech. Phys. Solids, vol. 14, pp.151–162, 1966.

Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., and Wang, T., Surface stress effect of nanostructuredmaterials,Acta Mech. Solida Sin., vol. 24, pp. 53–82, 2011.

Wani, S. N., Sangani, A. S., and Sureshkumar, R., Effective permittivity of dense random particulate plasmonic composites,J. Opt.Soc. Am., vol. B29, pp. 1443–1455, 2012.

Weng, G. J., The theoretical connection between Mori–Tanaka’s theory and the Hashin–Shtrikman–Walpole bounds,Int. J. Eng.Sci., vol. 28, pp. 1111–1120, 1990.

Willis, J. R., Bounds and self-consistent estimates for the overall properties of anisotropic composites,J. Mech. Phys. Solids, vol.25, pp. 185–203, 1977.

Willis, J. R., Variational principles and bounds for the overall properties of composites, in: Provan, J. W. (Ed.),Continuum Modelsof Disordered Systems, University of Waterloo Press, Waterloo, pp. 185–215, 1978.

Willis, J. R., A polarization approach to the scattering of elastic waves. II: Multiple scattering from inclusions,J. Mech. Phys.Solids, vol. 28, pp. 307–326, 1980.

Willis, J. R., Variational and related methods for the overall properties of composites,Adv. Appl. Mech., vol. 21, pp. 1–78, 1981.

Willis, J. R., The overall elastic response of composite materials,J. Appl. Mech., vol. 50, pp. 1202–1209, 1983.

Willis, J. R. and Acton, J. R., The overall elastic moduli of a dilute suspension of spheres,Q. J. Mech. Appl. Math., vol. 29, pp.163–177, 1976.

Xun, F., Hu, G., and Huang, Z., Effective in plane moduli of composites with a micropolar matrix and coated fibers,Int. J. SolidsStruct., vol. 41, pp. 247–265, 2004.

Yin, H. M., Paulino, G. H., Buttlar, W. G., and Sun, L. Z., Micromechanics-based thermoelastic model for functionally gradedparticulate materials with particle interactions,J. Mech. Phys. Solids, vol. 55, pp. 132–160, 2007.

You, L. H., You, X. Y., and Zheng, Z. Y., Thermomechanical analysis of elastic–plastic fibrous composites comprising an inhomo-geneous interphase,Comput. Mater. Sci., vol. 36, pp. 440–450, 2006.

Yuan, Z. and Fish, J., Multiple scale eigendeformation-based reduced order homogenization,Comput. Methods Appl. Mech. Eng.,vol. 198, pp. 2016–2038, 2009.

Zeller, R. and Dederichs, P. H., Elastic constants of polycrystals,Phys. Stat. Sol., vol. a55, pp. 831–842, 1973.

Zhang, X. and Sharma, P., Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems,Int. J. Solids Struct., vol. 42, pp. 3833–3851, 2005.

Zhong, Z. and Meguid, S. A., On the elastic field of a spherical inhomogeneity with an imperfectly bonded interface,J. Elast., vol.46, pp. 91–113, 1997.


Date post:	16-Nov-2023
Category:	Documents
Upload:	independent
View:	0 times
Download:	0 times

GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

Documents