Probabilistic Engineering Mechanics 21 (2006) 112–132 www.elsevier.com/locate/probengmech

Material spatial randomness: From statistical to representative volume element*

Martin Ostoja-Starzewski *

Department of Mechanical Engineering, and McGill Institute for Advanced Materials, McGill University, Montreal, Que. H3A 2K6, Canada

Received 4 November 2004; received in revised form 30 June 2005; accepted 27 July 2005 Available online 20 October 2005

Abstract

The material spatial randomness forces one to re-examine various basic concepts of continuum solid mechanics. In this paper we focus on the Representative Volume Element (RVE) that is commonly taken for granted in most of deterministic as well as in stochastic solid mechanics, although in the latter case it is called the Statistical Volume Element (SVE). The key issue is the scale over which homogenization is being carried out—it is called the mesoscale, separating the microscale (level of microheterogeneities) from the macroscale (level of RVE). As the mesoscale grows, the SVE tends to become the RVE. This occurs in terms of two hierarchies of bounds stemming from Dirichlet and Neumann boundary value problems on the mesoscale, respectively. Since generally there is no periodicity in real random media, the RVE can only be approached approximately on finite scales. We review results on this subject in the settings of linear elasticity, finite elasticity, plasticity, viscoelasticity, thermoelasticity, and permeability. q 2005 Elsevier Ltd. All rights reserved.

Keywords: Random media; Representative volume element; Statistical volume element; Scale effects

d! 1. Introduction / ; gL Lmacro (1.1) d/ 1.1. Separation of scales and introduces three scales: Continuum mechanics hinges on the concept of a – the microscale d, such as the average size of grain (or Representative Volume Element (RVE) playing the role of a inclusion, crystal, etc.) in a given microstructure; we mathematical point of a continuum field approximating the true initially assume the microstructures to be character- material microstructure. The RVE is very clearly defined in ized by just one size d; two situations only: (i) unit cell in a periodic microstructure, – the mesoscale L, size of the RVE (if so justified—see and (ii) volume containing a very large (mathematically below); infinite) set of microscale elements (e.g. grains), possessing – the macroscale Lmacro, macroscopic body size. statistically homogeneous and ergodic properties. The approach via unit cell is, strictly speaking, restricted to In (1.1) on the left we do admit two options, because the materials displaying periodic geometries. When we consider inequality d!L may be sufficient for microstructures with case (ii) we intuitively think of a medium with microstructure weak geometric disorder and weak mismatch in properties; so fine we cannot see it—naturally then we envisage a otherwise a much stronger statement d/L applies. Note also homogeneous deterministic continuum in its place. This that the first inequality in (1.1)1 could even be a weak one situation, as suggested by Fig. 1, is called a separation of scales because we may be considering a microstructure with a nearly periodic geometry, though possessing some randomness on the level of the unit cell. * Keynote lecture at International Conference on Heterogeneous Material In any case, the issue of central concern is the trend—either Mechanics, China, 2004. rapid, moderate, or slow—of mesoscale constitutive response, * Corresponding author. Tel.: C1 514 398 7394; fax: 1 514 398 7365. with L/d increasing, to the situation postulated by Hill [1]:“a E-mail address: [email protected]. sample that (a) is structurally entirely typical of the whole 0266-8920/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. mixture on average, and (b) contains a sufficient number of doi:10.1016/j.probengmech.2005.07.007 inclusions for the apparent overall moduli to be effectively M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 113

Tensor notation. Clearly, upon taking the spatial average, the dependence on position x vanishes, while upon taking the ensemble average, the dependence on u vanishes. A tensor of, say, second order is denoted symbolically by t and, in index notation, by tij. Both notations are used interchangeably, as are the scalar product operations t$n; t : a; C : a; (1.5) and, respectively,

tijnj; tijaij; Cijklakl; (1.6)

Fig. 1. A macroscopic body of size Lmacro with a mesoscale window of size L,in when the need arises. which a microstructure of size d is shown. d-Dependence of apparent moduli of a disk-matrix composite at contrasts C(i)/C(m)Z10 (a) and C(i)/C(m)Z1000 (b), at volume fraction 35%, under boundary conditions (2.11)–(2.16). 1.3. The RVE postulate independent of the surface values of traction and displacement, The random material on mesoscale, such as shown in Fig. 1, so long as these values are macroscopically uniform.” In is denoted BL=d ZfBL=dðuÞ; u2Ug with BL/d(u) being one essence, (a) is a statement about the material’s statistics, while realization. Properties on mesoscale are also described by an (b) is a pronouncement on the independence of effective adjective apparent [2], as opposed to effective. The latter term constitutive response with respect to the boundary conditions. pertains to the limit L/d/N as it connotes the passage to the Both of these are issues of mesoscale L of the domain of RVE, while any finite mesoscale involves statistical scatter random microstructure over which smoothing (or homogeniz- and, therefore, describes some Statistical Volume Element ation) is being done relative to the microscale d and macroscale (SVE). Note here that the RVE postulate as well as the Lmacro. These considerations, however, are not rigorous, separation of scales are also known in the solid mechanics because neither spatial statistics nor mechanics (or physics) literature as the MMM principle [3]. definitions of properties have yet been introduced. In the following, it will be convenient to describe the mesoscale by a nondimensional parameter 1.2. Basic concepts d Z L=d (1.7)

A note on determinism. In principle, any realization B(u)of in the range [1, N], so that BL=d will be written Bd, etc. the composite BZfBðuÞ; u2Ug, while spatially disordered The setting is one of quasi-static loading, so that the body is (i.e. heterogeneous), follows deterministic laws of mechanics. governed locally by the equilibrium equation The most preferred approach, dictated by stochastic mechanics, s Z ; would be to first ascertain what happens to each and every B(u) ij;j 0 (1.8) of B, starting from a certain random microstructure model, and where sij is the Cauchy stress, body forces being disregarded. then pass to ensemble setting, by taking the averages or higher For a mesoscale body Bd(u) of volume Vd, such as the moments as the need arises. In most situations, however, this microstructure shown in Fig. 1, we define volume average may generate enormous amounts of perhaps not very useful stress and strain information. ð ð Ensemble versus volume averaging. We reserve the overbar 1 1 sdðuÞ Z sðu; xÞdV 3dðuÞ Z 3ðu; xÞdV: (1.9) for spatial (volume or areal) averages, and h$i for ensemble Vd Vd averages. That is, if we have a random (n-component, Assuming we deal with a linear elastic microstructure, the real valued) field Q defined over some probability space {U, problem is to pass from the random field of stiffness with K F, P} F being a s-field and P a probability measure—over fluctuations on the microscale some domain X in RD of volume V s Z Cðu; xÞ : 3; u2U; x2B (1.10) Q : U!X/Rn; (1.2) to some effective Hooke’s law the said averages are ð ð s Z Ceff : 3 ; (1.11) 1 d d QðuÞ h Qðu; xÞdV hQðxÞi h Qðu; xÞdP: (1.3) V whereby (a) the dependence on u (i.e. randomness) would be v U removed, (b) the dependence on x (i.e. spatial fluctuations) of We assume the conditions necessary for the fulfillment of strain and stress fields would also vanish, and (c) the commutativity of both operations to be satisfied (i.e. independence of response with respect to boundary conditions requirements of Fubini’s theorem) so that would be attained. It is intuitively expected that d needs to be large, but exactly how large it should be, is the key hQi Z hQi: (1.4) question. 114 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

0 Suppose we are tackling a boundary value problem of a where the stress s3j is prescribed. Here (Bd denotes the body on macroscopic length scales. In general situations we boundary of Bd. have to use computational mechanics—such as, say, a finite Fig. 2(a) treats the situation of no mismatch in the material element meshing of the body—and this approach convention- properties: C(i)/C(m)Z1. and so we we can interpret it as either ally assumes that every single finite element is at least as large a uniform displacement field on the boundary (Bd under 0 Z 0 ; 0 0 Z as the RVE, although this is rarely verified. Thus, there arises a 3 ð331 332Þc, with 332 0, resulting in a uniform stress field need to know the rate of approach of SVE to RVE in function 0 Z 0 ; 0 on (Bd, or a uniform stress field on (Bd under s ðs31 s32Þ, of d, phases’ mismatch, phases’ microgeometry, etc. and, 0 Z with s32 0, resulting in a uniform displacement field on (Bd. whether that rate is too slow in a given problem. If the latter is Evidently, both problems are perfectly interchangeable the case, one then needs to set up a stochastic finite element because the microstructure is trivially homogeneous. This (SFE) scheme to account for the microstructural noise on the then is the trivial situation of the RVE. mesoscale of any given finite element, e.g. [4]. In this paper we Both boundary value problems become much more will not discuss such SFE, but only focus on the passage from interesting when C(i)/C(m)s1. In Fig. 2 (b–d) we decrease SVE to RVE. Let us note here that the typical recipes of solid mechanics (e.g. [5]) vaguely say that d of RVE should be about 10–100, but the analyses reported below show that it strongly depends on the type of problem studied. A review of results on the subject of scaling from SVE up to RVE in the setting of single-scale (not multi-scale) hetero- geneous media, defines the goal of this paper; it updates the results collected in [6]. The review is not complete, principally because other definitions of the RVE than that employed here—without the concept of SVE—have been introduced in the literature, e.g. [7,8,72,73]. Let us end this section with an observation that in problems of high strain/stress gradients, the length scale defined by the deformation field may easily become of the order of d (i.e. d/ 1), and the homogenization to RVE in the sense proposed here would then not be recommended, as too much detail would be lost. Second-order hamogenization is then needed [74–76].

2. Volume averaging

2.1. A paradigm of boundary conditions effect

Clearly, the attainment of the RVE is a function of the scale d as well as the mismatch in properties of inclusions versus matrix. To illustrate this point, let us consider boundary distributions of displacement u3 and stress traction t3 in two boundary value problems of the mesoscale window Bd(u)of matrix-inclusion specimen of Fig. 1 (now shown in Fig. 2)in anti-plane elasticity. The material is piecewise uniform with perfectly bonded, isotropic phases, so that the governing equation is

CðpÞV2u Z 0 p Z m; i: (2.1) Here m and i denote matrix and inclusion respectively, and (p) C3i3jZC dij (dij is the Kronecker delta) is the phase stiffness. Two boundary value problems are considered: one of Dirichlet type Z 0 2 ; u3ðxÞ 33ixi cx vBd (2.2) Fig. 2. Anti-plane elastic responses of a matrix-inclusion composite, with 0 nominal 35% volume fraction of inclusions, at decreasing contrasts: (a) where the strain 33j is prescribed, and the other of Neumann C(i)/C(m)Z1, (b) C(i)/C(m)Z0.2, (c) C(i)/C(m)Z0.05, (d) C(i)/C(m)Z0.02. For (b– type d), the first figure shows response under Dirichlet boundary conditions, while 0 Z 0 c 2 ; the second shows response under Neumann boundary conditions with s equal t3ðxÞ s3jnj x vBd (2.3) to the volume average s of stress computed in the Dirichlet problem. M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 115 the mismatch by first setting it to 0.2, then 0.05, and finally we need 0.02. In each case, we first solve the Dirichlet problem under 0 0 Z : 0 Z 0 ; s : 3 0 (2.9) 3 ð331 0Þ, and find t(x). Next, we compute the volume K 0 average t of t(x)on(Bd, and set t Zt to run the Neumann Relation (2.8) is called the Hill condition in the (conven- problem. We keep 30 identical in all four cases (a–d). tional) volume average form ([1]; see also [10–15]). Some Let us now define an ‘apparent stiffness’ Cd in the authors (e.g. [16]) call it the Hill-Mandel macrohomogeneity displacement controlled (d) problem (2.2) via equation condition, after J. Mandel. For an unbounded space domain (d/N), (2.8) is trivially s Z Cd : 30; (2.4) satisfied, but for a finite body it requires that the body be loaded t t and ‘apparent compliance’ S in the Neumann ( ) problem (2.3) in a specific way on its boundary (Bd. Following Hazanov & via equation Amieur [17], by the Green-Gauss theorem, we find the necessary and sufficient condition for (2.8) to hold 3 Z St : s0: (2.5) ð The latter allows us to define ‘apparent stiffness’ in the s : 3 Z s : 35 ðtKs$nÞ$ðuK3$xÞdS Z 0: (2.10) tZ t K1 traction controlled problem as C (S ) . vBd The following points are noteworthy: This is satisfied by three different types of boundary

(i) the volume average displacement of the resulting u3(x) conditions on the mesoscale: uniform displacement (also called distribution in the problem (2.2) differs from that in the kinematic, essential, or Dirichlet) boundary condition (dd) problem (2.3); uðxÞ Z 30$x cx2vB ; (2.11) (ii) the ‘apparent stiffness’ in one boundary value problem d is different from that in the other one; this should not be uniform traction (also called static, natural, or Neumann) surprising given the preceding observation; boundary condition (tt) (iii) the degree to which Cd is different from (St)K1 may be tðxÞ Z s0$n cx2vB ; (2.12) regarded as an indication of the departure from the d effective moduli Ceff in separation of scales; as a uniform displacement-traction (also called orthogonal-mixed) measure of their closeness to Ceff one might use the boundary condition (dt) deviation of the product Cd: St from unity; ðtðxÞKs0$nÞ$ðuðxÞK30$xÞ Z cx2vB : (iv) while the governing partial differential equation is linear 0 d (2.13) 0 0 (and we could even replace (2.1) by (Ciju3,j),iZ0), the Here we employ 3 and s to denote constant tensors, resulting property is nonlinear as a function of actual prescribed a priori, and note, from the strain average and stress realization u,scaled,mismatchC(i)/C(m), and type of average theorems: 30 Z3 and s0 Zs. boundary conditions (i.e. Dirichlet or Neumann) [9]. It is highly popular in mechanics of random media to assume some finite scale periodicity of the microstructure, while still keeping spatial disorder. In that case, one can specify 2.2. Hill Condition periodic boundary conditions (pp)

0 2.2.1. Mechanical versus energy definitions uðx CLÞ Z uðxÞ C3 ,xtðx CLÞ ZKtðxÞ Let us consider a body B (u) with a given microstructure, in (2.14) d c 2 ; which, as a result of some boundary conditions and in the x vBd absence of body forces, there are stress and strain fields s and 3. where LZLn. If we represent them as a superposition of the means (s and 3) 0 0 Two other possibilities of mixed boundary conditions [18] with zero-mean fluctuations (s and 3 ) include combinations of (2.11), (2.12) and (2.14), namely sðu; xÞ Z s Cs0ðu; xÞ 3ðu; xÞ Z 3 C30ðu; xÞ; (2.6) displacement-periodic boundary condition (dp) Z 0 c 2 d we find for the volume average of the energy density over uðxÞ 3 x x vBd Bd(u) C Z C 0 C ZK c 2 p ð uðx LÞ uðxÞ 3 xtðx LÞ tðxÞ x vBd 1 1 U h sðu; xÞ : 3ðu; xÞdV Z s : 3 (2.15) 2V 2 traction-periodic boundary condition (tp) BdðuÞ (2.7) Z 0 c 2 t 1 1 tðxÞ s n x vBd Z s : 3 C s0 : 30 : 2 2 C Z C 0 C ZK c 2 p uðx LÞ uðxÞ 3 xtðx LÞ tðxÞ x vBd Thus, we see that for the average of a scalar product of stress (2.16) and strain fields to equal the product of their averages Each of these boundary conditions results in a different s : 3 Z s : 3; (2.8) mesoscale (or apparent, after Huet) stiffness, or compliance 116 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 tensor. Either of these terms is used to make a distinction from case [21], which was followed by Huet [13]. Later on, Hazanov the macroscale (or effective, global, overall,.) properties that and Huet [22] extended it to the following eff * are typically denoted by or , see, e.g. [19]. t K1 dt d ½SdðuÞ %Cd ðuÞ%Cd ðuÞ; (2.20) For a given realization Bd(u) of the random medium Bd, taken as a linear elastic body (sZC(u, x): 3), on some that is, the modulus of Bd(u) obtained under the mixed dt- mesoscale d, dispalcement boundary condition (2.11) yields an conditions (2.13) always lies between the moduli obtained d apparent random stiffness tensor Cd ðuÞ—sometimes denoted under the tt-conditions (2.12) and the dd-conditions (2.11). e CdðuÞ—with the constitutive law Other consequences of (2.20)—especially, in the context of orthotropic materials—were discussed by [17]. s Z CdðuÞ : 30: (2.17) d Another important result, also due to [13], is that

On the other hand, the traction boundary condition (2.12) t K1 eff d t hSdðuÞi %C ðuÞ%hCd ðuÞi: (2.21) results in an apparent random compliance tensor SdðuÞ— n eff sometimes denoted SdðuÞ—with the constitutive law being That is, the effective modulus C (u) always lies between stated as the harmonic average of moduli obtained under the Neumann t 0 boundary conditions on the ensemble Bd and the arithmetic 3 Z SdðuÞ : s : (2.18) average of moduli obtained under the Dirichlet conditions on The third type of boundary condition, (2.13), evidently the same ensemble. involves a combination of (2.11) and (2.12); it results in a Various quantitative estimates of d-dependence—or, dt stiffness tensor Cd ðuÞ. In fact, this condition may best what is called finite-size scaling in condensed matter represent actual experimental setups. For example, it may physics—were computed for many different materials by signify displacement boundary conditions on two parallel Huet and co-workers, this author and co-workers; see also sides, and traction-free boundary conditions on the remaining [26], and [27]. This paper reviews more recent results on two parallel sides. Or, it may signify pure shear loading this subject. through boundary conditions (see Fig. 3) 0 ZK 0 0 Z : 2.3. Apparent properties 311 322 s12 0 (2.19)

2.2.2. Order relations dictated by three types of loading In general, if we consider a body Bd(u) of volume V Note from the discussion above that the use of (2.8) assures subjected to a volume average strain by the boundary condition the equivalence of the properties from the mechanical (2.11), i.e. standpoint—i.e. via apparent Hooke’s law (2.11) or (2.12)— u ðxÞ Z 30 x cx2vB ; (2.22) with the properties from the energy standpoint. i ij j d Both approaches are equivalent for a homogeneous given the continuity of displacements throughout, we have the material—i.e. the RVE—but not necessarily so for a average strain theorem ð ð heterogeneous one, Bd(u), of size finite relative to the 1 1 microscale heterogeneity (d!u). In fact, a following relation 3 Z u n dS Z 3 x n dS Z 30 : (2.23) ij V i j V ik k j ij ordering the Neumann and Dirichlet apparent moduli holds: V V ½St ðuÞK1 %CdðuÞ. A proof of the above in the framework of d d The apparent stiffness Cd (hCd)ofB (u) with a linear functional analysis has been given in [20], and another one in klij d elastic microstructure may now be defined by the framework of general convex analysis applied to nonlinear d 0 sij Z Cijkl3kl; (2.24) where sg is the volume average stress. Alternatively, we may consider the volume average energy density in B (u) ð ð d 1 1 U Z s 3 dV Z s u ; dV 2V ij ij 2V ij i j V vV ð ð 1 1 1 Z s u n dS Z s 30 dV Z s 30 2V ij i j 2V ij ij 2 ij ij vV V 1 Z 30 Cd 30 ; (2.25) 2 kl klij ij

where the equilibrium sij,jZ0 was used. Thus, the apparent d stiffness Cklij may be defined either from the mean stress s or from the mean energy density U if the boundary condition Fig. 3. Possible loading under the orthogonal-mixed boundary condition (2.13). (2.11) is imposed; see also [69]. M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 117

On the other hand, considering the traction boundary covariance depends only on the shift h from x to xCh condition (2.12), i.e. hQðxÞi hm; t ðxÞ Z s0 n ðxÞ cx2vB ; (2.26) i ij j d h½QðxÞKhQðxÞi½Qðx ChÞKhQðx ChÞii (3.3) we first have the average stress theorem ð ð hKQðhÞ!N: 1 1 s Z s dV Z s0 x n dS Z s0 : (2.27) ij V ij V ik k j ij Clearly, SSS implies WSS whenever the first-order V V distribution F1 yields a finite second moment. It is through We can now define the apparent compliance St from this the covariance function KQ(h) that we introduce the concept of relation correlation distance lc, which, in turn, gives an indication of decay of correlations between two different points. If this decay Z t 0 ; 3ij Sijklsij (2.28) is such that Q(x)andQ(xCh) become asymptotically uncorrelated according to limjhj/NKQ(h), then, using lc we or from the volume average energy density in Bd(u) ð ð would rewrite the separation of scales (1.1) as 1 1 U Z sij3ijdV Z sijui;jdV l ! 2V 2V c / : gL Lmacro (3.4) V vV l / ð ð c D 1 1 0 1 0 In other words, the RVE of size LZV1/ (DZ1,.,3) should Z sijuinjdS Z sijui;jdS Z sij3ij 2V 2V 2 be at least larger than lc, and the material could be taken as vV V homogeneous beyond L. This concept can also be applied to the 1 SSS fields. Z s0 St s0 : (2.29) 2 kl klij ij There also exist two more general classes of spatially t homogeneous fields: This shows that the apparent compliance Sklij may be defined either from 3 or from U if the boundary condition – intrinsically stationary (locally homogeneous) ran- (2.12) is imposed; see also [23–25]. dom fields, – quasi-stationary random fields. 3. Spatial statistics These types of random fields become relevant when there is 3.1. Stationarity of spatial statistics no hope of establishing even the WSS property. In those situations, there is no hope of having the RVE in the sense In the following we revisit the key concepts of spatial employed in this paper, and the SVE is needed. In this paper we homogeneity (stationarity) and ergodicity from the standpoint confine our attention to fields which have the SSS or WSS of what is required by Hill’s definition of RVE. We shall do this property. in terms of a material property (or a vector of properties) Q entering the heterogeneous medium model as a random field 3.2. Ergodicity of spatial statistics over the D-dimensional physical space Theoretical considerations. To estimate any random field of D Q : U!R /R: (3.1) material properties via observations on a specific sample Q(u), In the case of an r-phase material microstructure, Q is one needs some type of ergodic concept and/or ergodic assumption. First, recall that the classical ergodic property itself described by a random indicator function cr of all the phases. All that follows can then be generalized to a vector having its origin in dynamical systems of statistical physics is or tensor random field Q, as may be the case with a linear concerned with the random process parametrized by time t U elastic microstructure represented by a random stiffness field wandering almost surely through the entire space .This means that a sample average converges C. Now, let us recall two well-known classes of spatial ð homogeneity: 1 QðuÞ h lim Qðu; tÞdt: (3.5) Strict-sense stationary (SSS) random fields. This requires T/N T that all n-order probability distributions Fn are invariant with V 0 respect to arbitrary shifts x , and for any n and any choice of The issue of actual convergence—i.e. the mathematical xi’s, they satisfy existence of the limit (3.5)—is the subject of Birkhoff’s ; .; ; .; theorem, according to which Q must be SSS and hQ(t)iZm Fnðq1 qn; x1 xnÞ must be finite. In general, however, QðuÞ is a random variable, Z ; .; C 0; .; C 0 : Fnðq1 qn; x1 x xn x Þ (3.2) and of interest is finding conditions under which it equals the constant ensemble average Wide-sense (or weak-sense) stationary (WSS) random fields. The ensemble mean is constant and its finite-valued QðuÞ Z hQðtÞi: (3.6) 118 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

The above is the basis of the engineering concept of the we need fourth-order moments, for any h0, ergodic property, a quite different issue from the classical one ð above. In general, the ‘engineering’ ergodicity focuses on the 1 K h 2 C C K Z : lim 1 ½KQðhÞ KQðh h0ÞKQðh h0Þdx 0 question: Under what conditions do statistics in the form of V/N V V time averages equal ensemble averages? Recall from prob- V ability theory that, one may establish ergodicity with respect to (3.13) a certain parameter and in a certain sense (e.g. in quadratic mean), but this does not necessarily guarantee ergodicity with Measurements on finite domains. The above holds only with /N respect to another parameter. some accuracy, and the limiting process V cannot truly be When transcribed from a temporal process to a spatial one in carried out. In practice, the left and right hand sides of (3.7) are one dimension, (3.6) takes the form replaced by a spatial (or volume) average from a finite number of sampling points n taken over one realization on a finite QðuÞ Z hQðxÞi: (3.7) domain (!)

XM Thus, for a random field, the spatial average involves 1 QðuÞ h Qðu; x Þ; (3.14) measurements on a ray in a given direction x, whereby the m M mZ1 volume VZL. Perhaps the most basic result is that and an ensemble average from a finite number of realizations u ð taken at one sampling point 1 limV/N Qðu; xÞdx Z m (3.8) V XN V 1 hQðxÞi h Qðun; xÞ: (3.15) N nZ1 is equivalent to ð 1 The fundamental question therefore is: What size V of lim hQðu; xÞidx Z m and the domain is large enough to attain the limits such as in V/N V V (3.9) with a certain precision? A closely related issue is the ð ð (3.9) actual choice between l !L and l ZL in the separation of 1 c c ð ; Þ Z : scales (1.1). Quantitative answers can be found with the lim 2 KQ x1 x2 dx1dx2 0 V/N V help of mechanics problems elaborated in Sections 4 and 5 V V below. Note that this does not require Q to be even weak-sense Ergodic response. Any given boundary value problem for stationary (!), and so this result may be used for intrinsically an elastic body BZfBðuÞ; u2Ug is defined by the equili- stationary and quasi-stationary random fields. In those cases, brium Eq. (1.8), Hooke’s law, compatibility conditions, and ergodicity is replaced by local-ergodicity and quasi-ergodicity. some boundary conditions. Now, subject B(u) to a uniform For example, the quasi-ergodic random field would require boundary condition such as (2.11) or (2.12) and find the spatial averaging over domains smaller than lc but larger than ensemble of solutions LQ. f3ðu; xÞ; sðu; xÞ; uðu; xÞ; u2U; x2Vg; (3.16) When a WSS property is introduced, KQ(x1, x2)ZKQ(h), hZjx1Kx2j and (3.9)2 is replaced by which would represent a complete solution of the stochastic ð 1 x mechanics problem. Then, at any point x, we may define lim 1K KQðhÞdx Z 0: (3.10) eff V/N V V effective moduli C (x)as V hsðxÞi Z hCðxÞ : 3ðxÞi Z Ceff ðxÞ : h3ðxÞi: (3.17) This, in turn, is assured providing Note: lim KQðxÞ Z 0: (3.11) jxj/N (i) For any given boundary conditions, Ceff is, in general, a It follows that to estimate the mean (i.e. a first-order function of x. statistic) the covariance (i.e. a second-order statistic) is needed. (ii) Ceff also generally depends on the boundary conditions This suggests a pattern revealed upon consideration of a applied. condition for estimation of the covariance function itself from a (iii) Assuming B is ergodic in the sense that the field C is single realization Q(u). Namely, to ensure ergodic, the explicit dependence on location in (3.17) ð vanishes and we can write 1 lim /N Qðu; x ChÞQðu; xÞdx Z K ðhÞ; (3.12) V V Q V h3ðxÞi Z 3 or hsðxÞi Z s; (3.18) M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 119 providing (2.11), respectively (2.12), is applied and the homogeneity and ergodicity of the material, ensemble aver- d d dr displacement and stress fields are continuous; consequently, aging of (4.4) allows us to replace Cd ðuÞ by hCd i, and Cd ðuÞ,by d hC 0 i, so that s Z Ceff : 3: (3.19) d d % d c 0 Z = : hCd i hCd0 i d d 2 (4.5) (iv) Providing B is ergodic in the sense that the field C is By applying this inequality to ever larger windows ad / d ergodic, and mesoscale windows tend to macroscale d infinitum we get a hierarchy of bounds on hCNi from above N, write the Hill condition (2.8) as d d d d V h i%/%h i%h 0 i%/%h i h Z : CN Cd Cd C1 C h3 : si h3i : hsi (3.20) (4.6) 0 Of course, by analogy to (2.9), we have cd Z d=2: h30 : s0i Z 0; (3.21) In fact, we have

d eff eff which means that stresses and strains are statistically hCNi Z CN Z C (4.7) uncorrelated. for the macroscopically effective response (d/N) because, by the ergodicity argument, it must be deterministic. On the upper end, the hierarchy stops at the scale of a single 4. Hierarchies of mesoscale bounds for linear elastic heterogeneity. Now, since the single heterogeneity—like an microstructures inclusion or a crystal—is homogeneous, the uniform strain is true strain, so that ensemble averaging gives the Voigt 4.1. Basic results V bound C . The above derivation suggests an analogous procedure We first sketch the proof of a hierarchy of bounds on the eff for proving a hierarchy bounding the effective stiffness macroscopically effective stiffness tensor C , i.e. the RVE’s eff eff K1 C (h(S ) or the effective compliance) from below. The response. We take a square-shaped 2D (or a cubic-shaped 3D) proof is analogous, providing we first replace the displacement body B (u) to be described everywhere by the local stress– d by traction boundary conditions (2.12) and use the minimum strain relations sZC(u, x):3. We call (2.11) an unrestricted complementary energy principle for statically admissible fields uniform displacement boundary condition. We also define a ðs~; 3~Þ in Bd(u) restricted displacement boundary condition ð ð ð ð 1 1 urðxÞ Z 30 : x cx2vB s Z 1; .; 4; (4.1) s : 3 dV K t$u dS% s~ : 3~ dV K t~$u dS: (4.8) ds 2 2 u u Bd vBd Bd vBd actingonapartitionofBd(u) into four square-shaped 4 subdomains B such that B ZgZ B . In 3D the partition ds d s 1 ds Using the Hill condition and ensemble averaging, leads then involves eight domains, but we work in 2D for the sake of to clarity. The superscript r in Eq. (4.1) indicates a ‘restriction’. t % t c 0 Z = : We recall the minimum potential energy principle for the Sd Sd0 d d 2 (4.9) fields ðs~; 3~Þ in Bd(u) (e.g. [28]) ð ð ð ð By applying this inequality to ever larger windows ad t $ K1 % $ K1 ; infinitumwe get a hierarchy of bounds on hiSN from above (i.e. t u~dS s~ : 3~dV t udS s : 3dV (4.2) d 2 2 on CN from below) t t vB Bd vB Bd d d t %/% t % t %/% t h R SN Sd Sd0 S S where the tilde indicates kinematically admissible fields. This (4.10) allows us to prove that the solution fields ðs~r; 3~rÞ under the cd0 Z d=2; restricted condition (4.1) are admissible with respect to the unrestricted condition (2.11) (but not vice versa), so that where % r r: t eff eff eff K1 s3 s~ 3~ (4.3) SN Z SN Z S Z ðC Þ : (4.11) This in turn implies a weak inequality between the mesoscale Combining relations (4.6) with (4.10), we arrive at a scale- stiffness tensors obtained under unrestricted ðCdðuÞÞ and d dependent hierarchy of bounds on the macroscopically restricted ðCdrðuÞÞ conditions d effective moduli 1 X4 d % dr Z d c 0 Z = : t K1 t K1 t K1 eff Cd ðuÞ Cd ðuÞ Cd ðuÞ d d 2 (4.4) S %/% S 0 % S %/%CN %/ 4 s d d sZ1 % d % d %/% d That is, the effective stiffness of a partitioned domain Cd Cd0 C (4.12) subjected to (4.1) involves respective stiffnesses of four c 0 Z = : subdomains. Now, in view of the tacitly assumed statistical d d 2 120 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

These hierarchies were first derived by Huet [13]. A more random (i.e. deterministic) constant Fhom such that, for all u rigorous proof using techniques of homogenization and with probability one, we will have probability theories was given by Sab [15], see Section 4.2 lim jjFðB; QÞ Z Fhom cB; u: (4.15) below. The decrease of the upper (displacement-controlled) L=d/N bound with increasing scale appears to had first been demonstrated on planar random networks of Delaunay with the bound topology by Ostoja-Starzewski and Wang [29] the uniform infðFðB; QÞÞ Z Fhom cB; u: (4.16) traction conditions, however, could not be applied in a unique L=d way to such a disordered discrete system, see also [30]. This limit is understood in the sense of the homogenization Spatial statistics aspects. Considering that the hierarchy theory: x/F3ðxÞZFðx=3Þ hF3ðyÞ, where, x and y are the so- (4.12) is stated in terms of the ensemble averages, it suffices to called slow (macroscopic) and fast (microscopic) variables, choose the setting of material properties specified via wide- respectively, and 3 is a small parameter, reciprocal of our dZ sense stationary random fields. Furthermore, note that in the L/d. The limit d/N should be taken here in such a way that L eff case of C being an isotropic random field, C should become is kept finite so as to keep the energy finite. If F represents the eff eff an isotropic tensor involving two Lame´ constants l and m . volume average elastic energy density (or complementary) eff On the other hand, for an orthotropic random field C, C energy density, then, Q stands for the stiffness C (respectively, should become an orthotropic tensor. compliance S) tensor of the domain B, and from (4.16) one has The hierarchy (4.12) has been shown to hold for K K commensurate partitions on mesoscale, i.e. d0Zd/2. An sup hiS 1 Z ðShomÞ 1 Z Chom Z inf hiS : (4.17) = L=d extension of these inequalities to arbitrary pairs of mesoscales L d d!d is also true [31]. where (Shom)K1ZChom is the macroscopic (effective) stiffness tensor Ceff in the sense of Hill [1]. However, (4.17) does not K1 4.2. Homogenization theory viewpoint assert that the averages hSi and hCi are monotonic functions of L/d (Zd); this property, resulting in a scale-dependent hom We assume the composite—such as that of Fig. 2—to be hierarchy of the apparent properties convergent to C was made of a finite number of r phases each of which is linear obtained above, following a procedure originally due to Huet elastic and elliptic: 0!e:C:e!N, ces0. Next, following Sab [13]; see also [32,33]. [15], let F(B, Q) represent a real functional of the open bounded domain B of volume V and random field Q with the Finally we note that, for the RVE Bd/N(u), in the notation following five properties: of homogenization theory, we have

SðuÞ Z s /NðuÞ EðuÞ Z 3 /NðuÞ: (4.18) 1. F is a property of the medium invariant with respect to any d d translation in the material domain. 2. F or any partition of the domain B into n disjoint 4.3. Apparent moduli in in-plane elasticity subdomains, F satisfies a subadditivity property Xn When considering the apparent constitutive law Vi n FðB; QÞ% FðB ; QÞ B ZgZ B : (4.13) i i 1 i s Z C ðu; xÞ3 i; j; k; l Z 1; 2; (4.19) iZ1 V ij ijkl kl

of a planar elastic material Bd(u), we must, in general, deal 3. F is a measurable mapping with respect to the sample space with an arbitrary anisotropy. Thus, to determine six unknown d t U of outcomes u. Cijkl’s (or Sijkl’s) for Bd(u) we need six tests, Fig. 4. When d 4. Q is a statistically homogeneous, ergodic random field. seeking Cijkl’s, each test is run by applying the affine 5. F is uniformly bounded in B and u in the sense that there displacements on B so that the strain energy density is exists a real b, such that h Z V 0 Z V 0 0 Z V 0 2 C 0 2 jjFðB; QÞ %0 cB; u: (4.14) Ud sij3ij 3ijCijkl3kl C1111 311 C2222 322 2 2 2 Let us now take B to be a square-shaped domain, with side C 0 2 C 0 0 C 0 0 C1212l 312 2311C1122322 2322C2212312 of length L, and which contains some microstructure of C 30 30 : characteristic microscale d, see, e.g. Fig. 2. With the conditions 2 12C1211 11 1–5 satisfied, we can adopt the result that there exists a non- (4.20)

Fig. 4. Six tests—#1, #2,.,#6 from left to right—to determine the six unknowns of the in-plane stiffness tensor Cijkl. M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 121

In each and every test, separately, the energy is found by computational mechanics, and this is set equal to a corresponding special form of (4.20). For example, in test #1, V 2 U Z C 30 ; (4.21) d 2 1111 11 from which we infer C1111. Similarly, from the test #2 we find C2222, while test #3 gives C1211. We are then in a position to find C1122 from hi V 2 2 U Z C 30 CC 30 C230 C 30 ; (4.22) d 2 1111 11 2222 22 11 1122 22 and then C2212 as well as C1211 in an analogous fashion. In practice, one can proceed by truly carrying out six tests, or by carrying out only the three tests #1–#3 of Fig. 4, and then combining their results through a superposition. In any case, the solution necessarily involves some computational mech- anics method for discretization of the composite, such as spring networks, finite elements, or boundary elements. In the case of traction boundary conditions, determination of the apparent compliances follows the same type of approach and one works with the complementary energy, that is h V V V * Z 0 Z 0 0 Z 0 2 C 0 2 Ud sij3ij sijSijklskl S1111 s11 S2222 s22 2 2 2 2 CS s0 C2s0 S0 s0 C2s0 S0 s0 1212l 12 11 1122 22 22 2212 12 C 0 0 0 : : 2s12S1211s12 ð4 23Þ

An extension of this methodology to three dimensions is straightforward. Examples of bounds obtained in this way are given in [35,41].

4.4. Examples of hierarchies of mesoscale bounds

4.4.1. Random chessboards and Bernoulli lattices In this section we focus on anti-plane elasticity of random microstructures, which, by virtue of mathematical analogies recalled in Appendix, also gives information on various other physical problems of material systems having the same morhpology. Let us now consider the Bernoulli lattice process Fp,a on a Cartesian lattice of spacing a with each point of this lattice being of type 1 (or 2) with probability p (respectively, Fig. 5. Mesoscale bounds on tr(Ceff) for a random disk-matrix at contrasts 10 qZ1Kp) independently of all the other points. Evidently, p e n K1 Z (a), 100 (b), and 1000 (c), showing htrCdi and htrSdi at d 4, 10 and 20; also H H and q define the volume fractions of both types of phases (1 and shown, by dashed lines, are Hashin bounds Cu and Cl ; after [34]. 2). Clearly, the local stress and strain concentrations cannot be H Z C = K C = K1 resolved, but the statistics of such a simple system gives an Cu C2 f1½1 ðC1 C2Þ f2 2C2 indication of the statistics of random media because this, (4.24) H Z C = K C = K1: perhaps, is the simplest setup in which to investigate the scale Cl C1 f2½1 ðC2 C1Þ f1 2C1 and volume fraction dependence of the ensemble average H H estimates based on the essential (e) and natural (n) boundary Observe that Cu and Cl are outside the windows of size 4. conditions. The notation e and n is equivalent to d and t, In other words, relatively very small windows can give tighter respectively. In three Fig. 5(a–c) taken from [34] we give traces mesoscale bounds than those of Hashin. e n K1 Z of Cd and Sd at d 4, 10, and 20 (for the highest Note that the problem of scale dependence, especially in the contrast) and their comparison with Hashin (-Shtrikman) upper setting of such binary systems, is akin to the so-called finite- and lower bounds size scaling in statistical physics, but the attention in that area 122 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

at these large scales. But, (4.25)–(4.26) give an idea of the scaling laws for other volume fractions. Now, the Bernoulli lattice at volume fraction below, say, 30% can also be interpreted as a very crude model of a disk matrix composite—again with one degree of freedom per disk. Given the fact that a more realistic spring network model requires several (at least five) lattice spacings per disk, a lattice of some 5000!5000 nodes (25!106 degrees of freedom) would have to be run. Thus, the above scaling laws provide the best available indication of finite-size scaling of both bounds, e n Cd and Sd, of disk-matrix composites.

4.4.2. Disk-matrix composites The hierarchy of bounds (4.12) is now illustrated on the example of a planar disk-matrix composite [41].These microstructures have been generated via the planar Poisson Fig. 6. Bounds for a random disk-matrix composite at contrast C(i)/C(m)Z0.01, point process for disk centers, with sequential inhibition rule that showing mesoscale bounds under uniform displacement (dd) and traction (tt) prevents any two Poisson points from coming closer than 110% of Z boundary conditions on mesoscales d 6 and 48, as well as the Voigt, Reuss, diameter, so as to avoid the numerically and analytically difficult and Hashin(-Shtrikman) upper and lower bounds; after [41]. problem of very narrow necks between disks. First, in Fig. 6 (in analogy to Fig. 5 above) we give bounds stemming from uniform has always been focused on the phase transition problems [36]. displacement and uniform traction boundary conditions for the The approach to such a transition at about 2/3 volume fraction entire range of volume fractions at contrast C(i)/C(m)Z10 and of the soft phase is shown in the last of three figures here—at mesoscales dZ6 and 48. These results are compared to those of contrast 1000. However, in contradistinction to the terminol- Voigt and Reuss bounds, as well as Hashin(-Shtrikman) upper ogy of phase transitions, we now have a different tool to and lower bounds e.g. [19]. To further illustrate the scaling trends Z Z describe the scale dependence. The particular case of p q of all the boundary conditions (2.11)–(2.16), in Fig. 7 we 0.5 has been studied in [34], and it was found that reproduce two graphs—at contrasts C(i)/C(m)Z10 and C(i)/C(m)Z e Km n Kn 1000, respectively—at volume fraction 35%.Clearly, the larger is Cd Z exp½Kd Sd Z exp½d ; (4.25) the contrast in the composite, the larger is the mesoscale window where m and n are functions of the contrast a necessary to attain the RVE within, say, 10%. The mixed : : boundary conditions give responses intermediate between those m Z 3:8a0 14 n Z 2:4a0 59: (4.26) have purely kinematic and purely traction type. However, at high These results were obtained from computations over a range contrasts (Fig. 7(b)) their use tends to disappear. of scales 1–1000. While the smallest scale can be calculated We next compare responses of the disk-matrix explicitly as the Voigt and Reuss bounds, the largest involved a composite with relatively hard disks versus one with lattice of 1000!1000 nodes, i.e. having 106 degrees of relatively soft disks, both at volume fraction 20%. In the freedom. The parameter space of contrast and volume fraction first case, we have a soft matrix (C(m)Z1) and inclusions is vast, and therefore only select cases can be run numerically C(i)Z102 in Fig. 8(a), and C(i)Z104 in Fig. 8(b). Clearly,

Fig. 7. Effect of boundary conditions in planar disk-matrix composites: displacement-controlled (dd), traction-controlled (tt), peri-odic (pp), and mixed (displacement-periodic (dp), displacement-traction (dt), and traction-peri-odic (tp)) on the half-trace of the apparent anti-plane stiffness tensor in function of the mesoscale d, at contrasts 10 (a) 1,000 (b). M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 123

Fig. 9. A hierarchy of scale dependent bounds on tr(Ceff) for the disk-matrix composite at contrasts 10K2 (a), and 10K4 (b); after [33]. Fig. 8. A hierarchy of scale dependent bounds on tr(Ceff) of the disk-matrix composite at contrasts 102 (a), and 104 (b); after [33]. In Fig. 9 we show results for the opposite case: soft inclusions the larger is the contrast in the composite, the larger is in a hard matrix, with the same volume fraction 20%. Again, we (i) (m) K2 K4 the mesoscale window necessary to attain the RVE within, consider two cases of contrast—C /C Z10 and 10 — n K1 while keeping the matrix at C(m)Z1. The first case of contrast is say, 10%. Thus, to attain the discrepancy between hSdi e 2 Z shown in Fig. 9(a), and the second one in Fig. 9(b). As before, an and hCdi of about 30% at contrast 10 on mesoscale d 10, one has to take dZ50 at contrast 104. increase in the contrast in the composite has the effect of slowing

Fig. 10. (a) A 1000!1000 window of 2000 randomly placed 10!1 needles with an isotropic distribution; a subwindow of size (dZ50) is indicated. (b) Normalized e n K1 Z Z holes ! overall moduli Cd and Sd ,atd 10 and d 50, and the effective stiffness C for a random field of short (1 10) needles (such as that of Fig. 13), as functions of the volume fraction x. Data were computed only at discrete intervals xZ1.21, 2.42, 3.63, and 4.84; after [31]. 124 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

n e down the convergence of the product Sd : Cd to unity with d 5. Hierarchies of mesoscale bounds for nonlinear elastic increasing, but, by comparison with Fig. 8, this convergence is and/or inelastic materials relatively much slower (!) for extreme contrasts. It follows that one needs to go to very large scales in order to homogenize such 5.1. Nonlinear elastic materials a composite material. This is the principal difference from the case of high contrasts, and is indicative of all the material 5.1.1. General systems with soft inclusions of whatever shape. Both Figs. 8 and It should be clear from the derivation of mesoscale bounds 9 were obtained using a spring network of a much finer scale for linear elastic materials that three properties are required: than the single inclusion [37]. (i) statistical homogeneity and ergodicity (strictly speak- 4.5. Needle and crack systems ing, quasi-ergodicity); (ii) Hill condition leading to admissible boundary con- Since we are interested in planar fields of randomly placed ditions; needles, we consider a random fiber field generated from the (iii) variational principle. Poisson point process—Fig. 10(a) presents its typical realiz- ation B(u) [31]. The process density chosen here is 0.015. As in It follows that the mesoscale bounds can be shown to hold the previous section, the material has two locally isotropic for other types of materials providing these three properties are phases—matrix (m) and inclusions (i)—and we keep C(m)Z1 appropriately generalized. In the following we present some and vary C(i). principal results. Before we proceed further, we note that work on effective Consider physically nonlinear elastic materials in the range moduli of materials with microcracks dates back to [38] and of infinitesimal strain, described by the constitutive law [39], while a comprehensive review was given by Kachanov vwð3Þ vw*ðsÞ [40]. The effective medium theory, however, cannot say s Z sð3Þ Z 3 Z 3ðsÞ Z ; (5.1) v3 vs anything about the finite-size scaling of (mesoscale) moduli and their statistics; nor is it reliable for higher crack densities. where the energy densities are related by w*Zs$3Kw; w is a Note also that these problems—especially in connection with statistically homogeneous and ergodic field. The Hill con- finite-size scaling—can be treated by the lattice, or spring- dition, and its implication for the type of admissible boundary network models [37], bearing in mind two approximations: conditions, is ð needles have a finite thickness, and the contrast is finite although it can be made very close to zero. s : d3 Z s : d35 ðtKs$nÞ$ðduKd3$xÞdS Z 0; (5.2) e n K1 When studying scaling of hCdi and hSdi for needle systems vBd at contrasts 10K2 and 10K4, we found the same type of slow Z 0 Z 0 approach to the RVE as that in soft disk-matrix systems noted where s s and 3 3 . This results in the same three types of in the preceding section. Now, in the studies of effective boundary conditions on the mesoscale as in the linear elastic moduli of heterogeneous materials, the resulting Ceff is case. For a given realization Bd(u) of the random medium Bd typically presented versus the volume fraction x of one of the on some mesoscale d, the first of these yields an apparent phases. For a system of short needles, this is shown in constitutive law holes Z 2 0 Fig. 10(b) in terms of C against x nLeff almost all the way s Z sð3 Þ: (5.3) to the percolation point at w5.9; Choles is computed by the physicist’s mean field method for needle-shaped holes of any Similarly, the traction condition results in an apparent aspect ratio and with arbitrarily strong interactions. In addition, constitutive law we also plot here the Dirichlet and Neumann moduli and at two vwdð30Þ vwt*ðs0Þ Z ð Þ Z d Z ð 0Þ Z d : mesoscales: dZ10 and dZ50. To sum up, this figure displays s s 3 0 3 3 s 0 (5.4) e n K1 v3 vs (i) a very slow approach of hCdi and hSdi to the RVE (i.e. holes holes Next, we have the minimum potential energy principle curve C ), and (ii) a discrepancy between C and the ð ð Dirichlet as well as the Neumann bounds. Note that Choles 1 Pð3Þ% wð3~ÞdV K t~$u~ dS; (5.5) corresponds to an effective (macroscopic) response of a very 2 B t large random system, that is typically computed under periodic d vBd boundary conditions, see also [42]. and the minimum complementary energy principle For the system at hand, there is an interesting result ð ð e 1 concerning the statistics of second invariants of CdðuÞ and % K ~$ : n PðsÞ wðs~ÞdV t u~ dS (5.6) SdðuÞ. Namely, within a few percent, their coefficients of 2 B vBu variation are constant for a given type of boundary conditions d d (either Dirichlet or Neumann), are independent of the From this, the apparent constitutive responses are shown to changing mesoscale d, and independent of the contrast in the be related by material, except, of course the singular and trivial case of (m) (i) d 0; % d 0; c 0 Z = ; C /C Z1 [41,43]. wd ð3 uÞ wd0 ð3 uÞ d d 2 (5.7) M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 125 and elasticity, such as dealt with here. In [46] we assumed this

t* 0 t* 0 0 equivalence to also hold for apparent elastoplastic response, w ð3 ; uÞ%w 0 ð3 ; uÞ cd Z d=2: (5.8) d d and have thus obtained energy bounds on random elastoplastic By passing to the ensemble, we get a hierarchy of bounds composites. In that reference we have also proposed an from above [46] approach via tangent moduli. Finally, the effect of imperfect interfaces on the hierarchies of bounds has been investigated by d %/% d % d %/% d c 0 Z = ; wN hwd i hwd0 i hw1i d d 2 (5.9) Hazanov [45]. and from below

t* t* t* t* 0 5.1.3. Finite elasticity wN %/%hw i%hw 0 i%/%hw i cd Z d=2: (5.10) d d 1 The key assumption of the finite hyperelasticity theory is the t s 0 K d Note: wd s : 3 wd because the mesoscale response existence of a strain energy function j per unit volume of an depends on the type of loading. undeformed body, which depends on the deformation of the object and its material properties. Here we restrict ourselves to the reference configuration, so that, the equation of state of the 5.1.2. Power-law nonlinear elastic materials material takes the form: Consider a random medium governed pointwise by this constitutive law [18] vj P Z : (5.18) ik vF XN vwð3Þ ik s Z sð3Þ Z C : 3n Z n v3 Here P is the first Piola–Kirchhoff stress tensor and F is nZ1 ik ik (5.11) the deformation gradient tensor. XN * n vw ðsÞ In analogy to Section 2.2, consider two types of boundary 3 Z 3ðsÞ Z Sn : s Z ; 0 vs conditions: uniform displacement gradient Fik is prescribed: nZ1 Z 0 K 0 c 2 ; While the above forms are of mechanical form, the uiðxÞ ðFik dikÞxi x vBd (5.19) energetic laws are 0 or uniform traction where Pik is prescribed: XN n XN n 3 s Z 0 2 : ð Þ Z *ð Þ Z : tiðxÞ Piknk cx vBd (5.20) w 3 Cn : C w s Sn : C (5.12) nZ1 n 1 nZ1 n 1 Next, we consider the functional This leads to apparent responses as ð ð Z K 0 ; XN d 0 Pðu~Þ jðu~i;kÞdV ti u~idS (5.21) d 0 n vwd ð3 Þ s Z C dðuÞ : ð3 Þ Z B t n 0 d vBd nZ1 v3 (5.13) where u~ is an admissible displacement field such that u~Zu on XN t vw *ðs0Þ u 3 Z t ðuÞ ðs0Þn Z d : the portion of the boundary vBd where displacement is Snd : 0 0 nZ1 vs prescribed, and ti is the specified boundary traction on the remaining part of (Bd. This is the counterpart of the principle of and minimum potential energy for infinitesimal elastic deformation XN 3n XN sn in finite elasticity in that the functional Pðu~Þ assumes a local Z * Z : wð3Þ Cn : w ðsÞ Sn : (5.14) minimum for the actual solution u if n C1 n C1 nZ1 nZ1 ð v2j The apparent constitutive laws for Bd(u) are next shown to di;kdp;qdV Z 0; (5.22) vu ; vu ; be related by a partition theorem [44,45] i k p q Bd t K1 eff d ðS ðuÞÞ %C ðuÞ%C ðuÞ; (5.15) Z u nd nd nd for all non-zero di such that di 0onvBd [46]. and, in view of the statistical homogeneity and ergodicity of the By proceeding in a fashion analogous to Section 4.1, we get material, we have hierarchies of bounds from above a hierarchy of bounds on the energy density of the RVE 0 (JN(F )) from above d %/% d % d %/% d c 0 Z = ; CnN hCndi hCnd0 i hCn1i d d 2 (5.16) 0 %/% 0 % 0 %/% 0 JNðF Þ hJdðF Þi hJd0 ðF Þi hJ1ðF Þi and from below (5.23) 0 t %/% t % t %/% t c 0 Z = : cd Z d=2; SnN hSndi hSnd0 i hSn1i d d 2 (5.17) It is known that, under proportional monotonic loading, We now turn to the derivation of a reciprocal expression for strain-hardening elastoplastic composites may be treated in the the lower bounds. Since in nonlinear elasticity, the strain- framework of deformation theory of plasticity, which is energy function can be non-convex and therefore non- formally equivalent to physically nonlinear, small-deformation invertible, we employ the functional first proposed by Lee 126 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 and Shield [47] (e.g. [28]) ð ð ds0 vj vj 0 Z ij C vf Z R Qðu~Þ Z u~ik Kj dV K nku~idS: (5.24) d3 ij l dfp whenever fp cpanddf 0 vu~ik vu~ik 2Gp vsij B vBt d d 0 0 ds ij d3 ij Z whenever fp!cp ; Here u~ik is a trial function, which satisfies the following 2Gp conditions: 1K2n Z p Z = Z = d3 ds everywhere ðd3 d3ii 3ds dsii 3Þ 2Gpð1CnpÞ v vj vj 0 t Z 0inBd; nk Z ti on vBd: (5.25) vxk vu~i;k vu~i;k (5.30)

where Gp (shear modulus), np (Poisson’s ratio), and cp (yield It was shown that the functional Q is stationary for u~ik Zuik, limit) form a vector, whose each component (described by its where ui, the actual solution of a problem, assumes a local minimum if indicator function cp) gives rise to a scalar random field, such as ð 2 v j p O Xtot dikdpqdV 0 ; Z ; c 2 : vui;kvup;q Gðu xÞ Gpcpðu xÞ u U (5.31) Bd pZ1 Zf ðuÞ u2Ug for all nonzero dik satisfying the following conditions: The entire body B B ; is described by a random Z vector field Q {G, n, c}. 2 First, note that the results of Section 5.1.1 can be applied to v v j Z ; dpq 0inBd the loading, although not the unloading, response regime. Next, vxk vui;kvup;q (5.26) we can consider tangent moduli. Thus, on the mesoscale we 2 have tangent moduli—CTðuÞ or STðuÞ—of the body B (u), v j Z t : d d d dpqnk 0onvBd which connect stress increments with strain increments applied vui;kvup;q to it Again, by following the path employed for linear elastic s Z CTðuÞ 3 3 Z STðuÞ s: random media, we can derive a scale dependent hierarchy of d d : d d d : d (5.32) 0 lower bounds on the effective property QN(F ) Consequently, the Hill condition, and its implication for the type of admissible boundary conditions, is 0 %/% 0 % 0 %/% 0 QNðF Þ hQdðF Þi hQd0 ðF Þi hQ1ðF Þi ð (5.27) ds : d3 Z ds : d35 ðd tKds$nÞ$ðd uKd 3$xÞdS Z 0; cd0 Z d=2: vBd

Now, noting that vj=vu~ik u~ik can be equivalently expressed (5.33) as P: F, (5.24) becomes where dsZds0 and d3Zd30. Thus, u and 3 should be replaced by du and d3 in both (2.11) and (2.13). Qðu~ ; uÞ Z P : FKjðP0; uÞ Z P0 : F KjðP0; uÞ; (5.28) ik Next, we recall two extremum principles (e.g. [28]): one for kinematically admissible fields where the bar denotes a volume average, and averaging ð ð ð ð theorems [48,70] have been employed. Ultimately, (5.23) and 1 1 dt$du~ dSK ds~ : d3~dV % dt$du dSK ds : d3dV; (5.27) can be combined to result in a hierarchy of mesoscale 2 2 t B t B bounds on the effective energy function of a composite vBd d vBd d material at finite strains [49,50]: (5.34)

0 0 0 0 KhQ ðP Þi%/%KhQ 0 ðP Þi%KhQ ðP Þi%/%QNðP Þ and another for statically admissible fields 1 d d ð ð ð ð 0 0 0 0 Z JNðF Þ%/%hJ ðF Þi%hJ 0 ðF Þi%/%hJ ðF Þi 1 1 d d 1 ds$d3 dV K dt$du dS% ds~ : d3~ dV K dt~$du dS: !c 0 Z = : : 2 2 d d 2 ð5 29Þ u u Bd vBd Bd vBd (5.35) From this, we find 5.2. Elastic–plastic materials Td % Td ; Tt % Tt ; c 0 Z = : Cd ðuÞ Cd0 ðuÞ Sd ðuÞ Sd0 ðuÞ d d 2 (5.36)

Let us now consider a multiphase (pZ1,.,ptot) elastic- Upon ensemble averaging, and applying this to ever larger plastic-hardening material described by an associated flow rule windows ad infinitum we get a hierarchy of bounds on M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 127 T T K1 the macroscopically effective tangent modulus CNZ SN hierarchy of bounds, computed here on two scales, is shown in Tt K1 %/% Tt K1 % Tt K1 %/% T %/% Td Fig. 12. Note: hS1 i hSd0 i hSd i CN hCd i Z Td Td 0 (i) at a smaller scale (d 6), the response under uniform %hC 0 i%/%hC i; cd Z d= d 1 2 (5.37) kinematic (2.11) is much more uniform than that under Tt K1 Td uniform stress boundary conditions (2.12); where hS1 i and hC1 i are recognized as the Sachs and Taylor bounds, respectively. See [71] for a comprehensive (ii) at a larger scale (dZ15), the discrepancy due to reviews of effective (RVE level) properties of nonlinear different types of loading is much smaller, which shows composites. a tendency to homogenize with d tending to N; Indeed, one particular problem we studied involved a (iii) the macroscopic response is better (in fact, quite well) matrix-inclusion composite material [46], with the matrix approximated by the uniform stress than by the uniform phase being elastic–hardening plastic, and the inclusions being strain assumption. This situation would be reversed for elastic and, indeed, of the same modulus as that of the matrix in an elastic–plastic matrix with soft, rather than hard, the elastic range, see also the recent review [51]. Here we inclusions, and helps explain why Taylor bound works display the bounding character of responses computed under well for polycrystals with dislocations. (2.11) and (2.12), respectively, for elastic–plastic materials with both phases being elastic-hardening plastic [52,53]. Another setting where mesoscale tangent moduli of power- Patterns of plastic shear bands under these two boundary law hardening materials have recently been employed is the conditions are displayed in Fig. 11, while the corresponding mechanics of paper [54].

Fig. 11. (a) Two realizations of a random matrix-inclusion composite at mesoscales dZ6 and dZ15. (b) and (c) Contour plots of equivalent plastic strain of the matrix-inclusion composite of Figs. (a, b) under displacement (left) and traction (right) boundary conditions at dZ6 and 15, respectively; after [53]. 128 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

some set

PdðuÞ Z fsdjdsðxÞ; divs Z 0; Fðs; uÞ%0; cx2Bdg: (5.42) hom In contradistinction to P , Pd(u) depends on the configuration u and the mesoscale d. The starting point is provided by the upper bound theorem allowing for discontinuities in the velocity field [55] ð ð ð $ Z ~ C ; t v~ dS s~ : d dV tYj½v~jdS (5.43)

½ * vBd Bd S v where v~ is an arbitrary kinematically admissible velocity field, ½v* ZgM ½v* s~ an associated stress field, and S mZ1Sm is the set of internal surfaces of discontinuity in v*. With this Ansatz, we showed in [51] that ^v 4 ^v c 0 ! ; Pd Pd0 d d (5.44) ^v ^v where Pd and Pd0 denote two domains in stress space bounded, v respectively, by the ensemble average yield surfaces hFdi and v hFd0 i. By induction, this leads to a hierarchy of inclusions 0 Fig. 12. Ensemble average stress-strain responses of matrix-inclusion v 4.4 ^v 4 ^v 4.4 ^v c Z = ; PN Pd Pd0 P1 d d 2 (5.45) composites for different mesoscales d, under different boundary conditions v hom [53]. where PN is the yield locus P of RVE. On the other hand, P^v is the domain bounded by the average yield locus 5.3. Rigid-perfectly plastic materials 1 of a single grain, or the Taylor bound of (5.29), which results from prescribing a uniform deformation rate field Here a random rigid–plastic material BZfBðuÞ; u2Ug, everywhere. cu2U, is defined by stating that, for any grain (i.e. on While an analytical derivation of a lower hierarchy of microscale), bounds under uniform traction boundary conditions is not vFðs ; uÞ vFðs ; uÞ available, our computational results lead us to conjecture that Z ij ; Z ij R : dij l sijdij sijl 0 (5.38) this hierarchy of inclusions holds vsij vsij t J.J ^t J ^t J.J ^t c 0 Z = : Thus, in each grain the admissible stress states lie within the PN Pd Pd P1 d d 2 (5.46) set ^t ^t Here Pd and Pd0 denote domains in stress space bounded by Z ; % h % ; t t P1ðuÞ fsjFðs uÞ 0g fsjseq sY g (5.39) ensemble averaged yield surfaces hFdi and hFd0 i.More information is given in the recent review [51]. where seq is the equivalent stress; sY is the yield stress. For the RVE (d/N), the yield (or extremal) surface of the 5.4. Viscoelastic materials composite BN delimits the set

! Here we give a brief account of research carried out by Huet Phom Z fS2R3 3jdsðxÞ with s Z S; [2,56,57], First, the Hill condition now involves strain rates (5.40) Z ; ; % ; c 2 ; divs 0 Fðs uÞ 0 x Bg s : 3_ Z s : 3_: (5.47) hom and the elementary Taylor and Sachs bounds on P are Translated to the mesoscale, it implies that expressed via a hierarchy of inclusion relations () s : 3_ Z s : 3_0 or s0 : 3_; (5.48) X X Sachs hom 0 0 j % inf sY ðxÞ 3P depending on whether the strain rate ð3_ Þ or stress (s )is x2B eq d prescribed. () On the microscale (i.e. locally) the material is governed by a Taylor X X formula involving a relaxation modulus tensor (r) 3 j %sY : (5.41) eq ðt Z K 0 0 ; On any finite mesoscale—a scale below RVE—there holds sðtÞ rðt t Þ : d3ðt Þdt (5.49) some form of an apparent yield surface Fdðs; uÞ, bounding 0 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 129 or a dual one involving a creep compliance tensor (f)

ðt 3ðtÞ Z fðtKt0Þ : dsðt0Þdt: (5.50)

0 On the mesoscale, under the kinematic boundary condition, the material domain B(u) is governed by a formula involving a relaxation modulus tensor (rd) ðt 0 0 0 sðtÞ Z rdðtKt Þ : d3 ðx; t Þdt; (5.51) 0 and, under the traction boundary condition, a similar one involving a creep compliance tensor (fd) ðt 0 0 0 Fig. 13. Effect of increasing window scale on the hierarchies (5.64) and (5.65) 3ðtÞ Z fdðtKt Þ : ds ðx; t Þdt: (5.52) of lower and upper bounds on the macroscopic thermal expansion (strain) 0 coefficient [61]. On the macroscale the two tensors become dual, and then As shown in the works referenced above, the (macroscopi- we have reff (t) and feff (t). Now, Huet has shown that reff (t)is eff Z eff cally) effective thermal expansion coefficients a ð akl Þ can bounded by the hierarchy eff Z eff Z be derived in terms of the effective moduli C ð Cijkl ðSeff ÞK1Þ and the information on distribution of individual eff %/% % %/% ijkl r ðtÞ hrdðtÞi hrdðtÞi hr1ðtÞi phases. That is, in the case of two phases 1 and 2, (5.53) cd0 Z d=2; ctR0; eff Z ð1Þ K ð2Þ eff K ð2Þ C ð2Þ; aij ðakl akl ÞPklmnðSmnij SmnijÞ aij (5.58) and feff (t) is bounded by the hierarchy where eff %/% % %/% f ðtÞ hfdðtÞi hfdðtÞi hf1ðtÞi (5.54) 1 cd0 Z d=2; ctR0: P ðSeff KS Þ Z I Z ðd d Cd d Þ: (5.59) klmn mnrs mnrs klrs 2 kr ls ks lr A number of related results on that subject are in the papers Alternatively, as pointed out in the aforementioned referenced above. works, bounds on aeff can be obtained using bounds on Ceff, and such a result was produced employing the Hashin– 5.5. Thermoelastic materials Shtrikman bounds. Of course, these derivations provide no information on the SVE and its trend to RVE. In a current Let us now consider a linear thermoelastic composite study [60,61], by taking advantage of (4.12) and using the material, e.g. [58,59]. Each specimen BðuÞ2B is now minimum energy principles of thermoelasticity, the described locally by following hierarchies of bounds have been derived for stationary and ergodic random media: 3ij Z Sijklðu; xÞskl Caijðu; xÞDT or on aN from above (5.55) s Z C ðu; xÞ3 CG ðu; xÞDT; ij ijkl kl ij %/% t % t %/% t c 0 Z = ; aN hadi had0 i ha1i d d 2 (5.60) ; Z K1 ; where Sijklðu xÞ Cijklðu xÞ, DT is a temperature rise, and and on GN from above G ðu; xÞ ZKC ðu; xÞa ðu; xÞ (5.56) ij ijkl kl %/% d % d %/% d c 0 Z = : GN hGd i hGd0 i hG1i d d 2 (5.61) are the thermal stress coefficients computed from the thermal In view of (5.49), this provides a two-sided bounding expansion coefficients akl. Of interest is the derivation of scale eff hierarchy on aN, or, equivalently, on GN. dependent bounds on a haN using the already available eff These hierarchies have been computed for a planar two- bounds on C hCN. Note here that phase composite material with circular shaped inclusions, at 40% volume fraction of inclusions, recall, e.g. Fig. 1.[Herea eff ; ZK eff ; eff ; : Gij ðu xÞ Cijklðu xÞakl ðu xÞ (5.57) finite element mesh, again much finer than the single 130 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

n e Fig. 14. (a) Effect of increasing window scale on the convergence of the permeability/resistance tensor hierarchy. (b) Scale effect on the convergence of R11K11 [59]. inclusion, was employed.] Taking the aluminum for the K fp Z Vp Vp fc Z U $mR$U (5.64) matrix and steel for the inclusions, the contrast in moduli is m D D E(i)/E(m)Z1/3, while the contrast in thermal expansion coefficients is a(i)/a(m)Z0.5. The scale dependencies of Tr 0 (G )andTr (a ), following from the Neumann and Dirichlet Next, with Pp being a constant pressure gradient, and d d 0 e boundary value problems, are displayed in Fig. 13. Clearly, UD being a constant velocity, consider either essential ( )or n this property asymptotes with d increasing, and the RVE is natural ( ) uniform boundary conditions [59]: attained at and beyond, say, dZ10. Z 0$ c 2 ; Du [60] also gives computed physical examples of (5.52, pðxÞ Vp x x vBd 5.53) and obtains related hierarchies on the specific heat (5.65) $ Z 0 $ c 2 : coefficient. UDðxÞ n UD n x vBd

e n 5.6. Permeable materials In the first case we find Kd, and in the second case Rd. Employing the variational principles, and proceeding in the In phenomenological, deterministic continuum mechanics, same fashion as for other types of materials, a hierarchy of eff Z fluid flow in porous media is described by Darcy’s law. With mesoscale bounds on the effective permeability Kd eff K1 the slow flow in a porous medium also considered as ðRd Þ of porous media is obtained as follows incompressible and viscous, Darcy’s law is governed by the e K1 %/% e K1 % e K1 %/% eff K1 equations [62]: hR1i hRd0 i hRdi ðR Þ Z eff %/% n % n %/% n K K hKdi hKd0 i hK1i (5.66) UD ZK $Vp V$UD Z gðxÞ; (5.62) m 0 cd Z d=2: P where UD is the average (or Darcy) velocity, p is the applied Fig. 14(a) illustrates this hierarchy for a 2D flow in a pressure gradient driving the flow, m is the fluid viscosity, and bed of circular disks randomly distributed by a planar K is the permeability, a second-rank tensor which depends on Poisson point process, at porosity 0.6. This figure clearly the microstructure of the porous medium. Eq. (5.62)2 is the shows the convergent trend of the upper and lower bounds continuity equation with g(x) being the source/sink term. By in (5.66) towards macroscopically effective permeability of Z K1 defining the resistance tensor R K for porous media, we the porous medium. Clearly, the RVE is attained with a can rewrite Darcy’s law as high accuracy on the length scales some 80 times larger than the disk size. Overall, the ongoing studies [60] indicate ZK $ : Vp mR UD (5.63) that the higher is the density of random disks—or, equivalently, the narrower are the micro-channels in the We now recall the variational principle stating that, system—the smaller is the size of RVE for Darcy flow. among the admissible solutions of flow field in porous Consequently, the left-side inequality in (1.1) may then be media, the actual solution has a minimum value of simply taken as !. eff Z eff K1 dissipation rate [63,64]. Note here that the potential and According to Kd ðRd Þ , we can expect an identity tensor p c n$ e complementary ‘dissipation energy’ (f and f ) represen- from the product of Rd Kd at the RVE level. Fig. 14(b) shows this tations are, respectively, convergence towards 1 in function of mesoscale d. M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132 131

Physical subject u(Zu3) 3(Z33i) C(ZC3i3j) s(Zs3i) Anti-plane elasticity displacement strain elastic moduli stress Thermal conductivity temperature thermal gradient thermal conductivity heat flux Torsion stress function strain shear moduli stress Electrical conduction potential intensity electrical conductivity current density Electrostatics potential intensity permittivity electric induction Magnetostatics potential intensity magnetic magnetic permeability induction Diffusion concentration gradient diffusivity flux

6. Closure [2] Huet C. Bounds and hierarchies for the overall properties of viscoelastic heterogeneous and composite materials. Arch Mech 1995;47(6):1125–55. As noted at the beginning, the material spatial randomness [3] Hashin Z. Analysis of composite materials—a survey. J Appl Mech 1983; 50:481. forces one to re-examine various basic concepts of continuum [4] Ostoja-Starzewski M. Microstructural disorder, mesoscale finite solid mechanics. In this paper we have focused on the elements, and macroscopic response. Proc Roy Soc Lond A 1999;455: Representative Volume Element (RVE) commonly taken for 3189–99. granted in most of deterministic as well as in stochastic solid [5] Lemaitre J, Chaboche J-L. Mechanics of solid materials.: Cambridge mechanics, although in the latter case it is more appropriately University Press; 1990. called the Statistical Volume Element (SVE). The key issue is [6] Ostoja-Starzewski M. Mechanics of random materials: stochastics, scale effects, and computation. In: Jeulin D, Ostoja-Starzewski M, editors. the scale over which homogenization is being carried out—it is Mechanics of random and multiscale microstructures. CISM courses and called the mesoscale, separating the microscale (level of lectures. Wien: Springer-Verlag; 2001. microheterogeneities) from the macroscale. As the mesoscale [7] Drugan WJ, Willis JR. A micromechanics-based nonlocal constitutive grows, the SVE tends to become the Representative Volume equation and estimates of representative volume element size for elastic Element (RVE). This occurs in terms of two hierarchies of composites. J Mech Phys Solids 1996;44:497–524. [8] Stroeven M, Askes H, Sluys LJ. Numerical determination of representa- bounds stemming from Dirichlet and Neumann boundary value tive volumes for granular materials. Comp Meth Appl Mech Eng 2004; problems on the mesoscale, respectively. Unless there is some 193:3221–38. periodicity in the random medium—and this is practically [9] Markov K. Elementary micromechanics of heterogeneous media. In: never the case—the RVE is not attained exactly. The Markov K, Preciosi L, editors. Heterogeneous media. Basel: Birkha¨user; approximate attainment of the RVE—i.e. the scaling trend 2000. toward it—has to be quantified for any specific random [10] Kro¨ner E. Statistical continuum mechanics CISM courses and lectures. Wien: Springer-Verlag; 1972. material with the help of computational mechanics. [11] Kro¨ner E. The statistical basis of polycrystal plasticity. In: Gittus J, We review results on this subject in the settings of linear and Zarka J, Nemat-Nasser S, editors. Large deformations of solids: physical nonlinear elasticity, plasticity, viscoelasticity, thermoelasticity, basis and mathematical modelling. Amsterdam: Elsevier; 1986. and permeability. The case of thermoelasticity demonstrates [12] Huet C. Universal conditions for assimilation of a heterogeneous material the applicability of our methodology to coupled-field pro- to an effective medium. Mech Res Comm 1982;9(3):165–70. blems, such as piezoelectricity or poroelasticity. Since (almost) [13] Huet C. Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 1990;38:813–41. all constitutive behaviors in continuum mechanics may be [14] Sab K. Principe de Hill et homoge´ne´isation des mate´riaux ale´atoires. C R brought under the single umbrella of thermomechanics [65,66], Acad Sci Paris II 1991;312:1–5. a generalization of respective Legendre transformations of free [15] Sab K. On the homogenization and the simulation of random materials. energy and dissipation functionals to mesoscales in random Europ J Mech, A/Solids 1992;11:585–607. media can also be pursued [67,68]. Finally, let us note that [16] Stolz C. General relationships between the micro and macro scales for the below the RVE scale one has to deal with random fields having non-linear behaviour of heterogeneous media. In: Gittus J, Zarka J, editors. Modelling small deformations of polycrystals. New York: smooth realizations, and this review provides guidance on the Elsevier; 1986. specification of their one-point characteristics. [17] Hazanov S, Huet C. Order relationships for boundary conditions effect in the heterogeneous bodies smaller than the representative volume. J Mech Appendix A. Analogies to anti-plane elasticity Phys Solids 1994;42:1995–2011. [18] Kro¨ner E. Nonlinear elastic properties of micro-heterogeneous media. ASME J Eng Mat Tech 1994;11:325–30. For the sake of completeness, in the table below we collect [19] Nemat-Nasser S, Hori M. Micromechanics: overall properties of various classical analogies of problems locally governed by the heterogeneous solids. Amsterdam: North-Holland; 1993. Laplace equation CT, iiZ0 (or by (CijT, jZ0) in two [20] Suquet PM, Suquet PM. Elements of homogenization for solid mechanics. dimensions, where, for reference, we use the terminology of In: Sanchez-Palencia E, Zaoui A, editors. Homogenization techniques for anti-plane elasticity; see also [3] composite media, lecture notes in physics 272. Springer-Verlag; 1986. p. 193–278. [21] Willis JR, Talbot DSR. In: Maugin GA, editor. Continuum models and References discrete systems. [22] Hazanov S, Amieur M. On overall properties of elastic heterogeneous [1] Hill R. Elastic properties of reinforced solids: some theoretical principles. bodies smaller than the representative volume. Int J Engng Sci 1995;23: J Mech Phys Solids 1963;11:357–72. 1289–301. 132 M. Ostoja-Starzewski / Probabilistic Engineering Mechanics 21 (2006) 112–132

[23] Hill R. The essential structure of constitutive laws for metal composites [50] Khisaeva Z, Ostoja-Starzewski M. Mesoscale bounds in finite elasticity of and polycrystals. J Mech Phys Solids 1967;15:79–95. random composites Proceedings of 20th CANCAM. Montreal, Canada: [24] Mandel J. Plasticite´ classique et viscoplasticite´. CISM courses and McGill University; 2005 pp. 454–455. lectures. vol. 97. Wien-New York: Springer; 1972. [51] Ostoja-Starzewski M. Scale effects in plasticity of random media: status [25] Cristescu ND, Craciun E-M, Soo´s E. Mechanics of elastic composites.: and challenges. Int J Plast 2005;21:1119–60. Chapman & Hall/CRC Press; 2003. [52] Li W. PhD Thesis (in progress). Montreal, Canada: McGill University; [26] Terada K, Hori M, Kyoya T, Kikuchi N. Effective thermal expansion 2005. coefficients and specific heats of composite materials. Int J Solids Struct [53] Li W, Ostoja-Starzewski M. Elastoplasticity of random heterogeneous 2000;37:2285–311. materials 20th Canadian Congress on Applied Mechanics—CANCAM, [27] Kanit T, Forest S, Galliet I, Monoury V, Jeulin D. Determination of 465–466, Montreal, Canada 2005. the size of the representative volume element for random composites: [54] Ostoja-Starzewski M, Castro J. Random formation, inelastic response, statistical and numerical approach. Int J Solids Struct 2003;40: and scale effects in paper. Phil Trans Roy Soc Lond A 2003;361(1806): 3647–79. 965–86. [28] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; [55] Kachanov LM. Foundations of the theory of plasticity. Amsterdam: North-Holland; 1971. 1950. [56] Huet C. An integrated micromechanics and statistical continuum [29] Ostoja-Starzewski M, Wang C. Linear elasticity of planar Delaunay thermodynamics approach for studying the fracture behaviour of networks: Random field characterization of effective moduli. Acta Mech microcracked heterogeneous materials with delayed response. Eng Fract 1989;80:61–80. Mech 1997;58(5–6):459. [30] Huyse L, Maes MA. Random field modeling of elastic properties using [57] Huet C. Coupled size and boundary conditions effects in viscoelastic homogenization. ASCE J Eng Mech 2001;127(1):27–36. heterogeneous and composite bodies. Mech Mater 1999;31:787–829. [31] Ostoja-Starzewski M. Scale effects in materials with random distributions [58] Rosen BW, Hashin Z. Effective thermal expansion coefficients and of needles and cracks. Mech Mater 1999;31(12):883–93. specific heats of composite materials. Int J Eng Sci 1970;8:157–73. [32] Ostoja-Starzewski M. Micromechanics as a basis of random elastic [59] Christensen RM. Mechanics of composite materials.: Krieger Publ. Co; continuum approximations. Probab Eng Mech 1993;8(2):107–14. 1991. [33] Ostoja-Starzewski M. Random field models of heterogeneous materials. [60] Du X. PhD Thesis (in progress). Canada: McGill University;2005. Intl J Solids Struct 1998;35(19):2429–55. [61] Du X, Ostoja-Starzewski M. Mesoscale bounds on effective thermal [34] Ostoja-Starzewski M, Schulte J. Bounding of effective thermal expansion of random composites 6th International congress of thermal conductivities of multiscale materials by essential and natural boundary stresses and related topics. Austria: Vienna; 2005 pp. 755–758. conditions. Phys Rev B 1996;54:278–85. [62] Dullien FAL. Porous media: fluid transport and pore structure.: Academic [35] Amieur M, Hazanov S, Huet C. Numerical and experimental assessment Press; 1979. of the size and boundary conditions effects for the overall properties of [63] Berryman JG, Milton GW. Normalization constraint for variational granular composite bodies smaller than the representative volume. In: bounds on fluid permeability. J Chem Phys 1985;83(2):754–60. Parker DF, England AH, editors. IUTAM Symposium on Anisotropy, [64] Chung TJ. Computational fluid dynamics.: Cambridge University Press; Inhomogeneity and Nonlinearity in Solid Mechanics. Kluwer Academic; 2000. 1995. p. 149–54. [65] Ziegler H. An introduction to thermomechanics. Amsterdam: North- [36] Cardy JL, editor. Finite-size scaling. Amsterdam: North-Holland; 1988. Holland; 1983. [37] Ostoja-Starzewski M. Lattice models in micromechanics. Appl Mech Rev [66] Maugin GA. The thermomechanics of nonlinear irreversible behaviors— 2002;55(1):35–60. an introduction. Singapore: World Scientific; 1999. [38] Vakulenko AA, Kachanov ML. Continuum model of medium with cracks [67] Ostoja-Starzewski M. Microstructural randomness versus representative (in Russian). Mech Solids 1971;4:54–9. volume element in thermomechanics. ASME J Appl Mech 2002;69: [39] Budiansky B, O’Connell RJ. Elastic moduli of a cracked solid. Int J Solids 25–35. Struct 1976;12:81–97. [68] Ostoja-Starzewski M. Towards stochastic continuum thermodynamics. [40] Kachanov M. Elastic solids with many cracks and related problems. J Non-Equilib Thermodyn 2002;27(4):335–48. Advances in Applied Mechanics. vol. 30.: Academic Press; 1993 p. 259. [69] Willis JR. Variational and related methods for the overall properties of [41] Jiang M, Alzebdeh K, Jasiuk I, Ostoja-Starzewski M. Scale and boundary composites. Adv Appl Mech 1981;21:1–78. conditions effects in elastic properties of random composites. Acta Mech [70] Hill R. On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 1972;326:131–47. 2001;148(1–4):63–78. [71] Ponte Castan˜eda P, Suquet PM. Nonlinear composites. Adv Appl Mech [42] Garboczi EJ, Thorpe MF, DeVries MS, Day AR. Universal conductance 1998;34:171–302. curve for a plane containing random holes. Phys Rev A 1991;43:6473–80. [72] Ren ZY, Zheng QS. A quantitative study on minimum sizes of [43] Ostoja-Starzewski M. Universal material property in conductivity of representative volume elements of cubic polycrystals - numerical planar random microstructures. Phys Rev B 2000;62(5):2980–3. experiments. J Mech Phys Solids 2002;50:881–93. [44] Hazanov S. Hill condition and overall properties of composites. Arch [73] Ren ZY, Zheng QS. Effects of grain sizes, shapes and distribution on Appl Mech 1998;68:385–94. minimum sizes of representative volume elements of cubic polycrystals— [45] Hazanov S. On apparent properties of nonlinear heterogeneous bodies numerical experiments. Mech Mater 2004;36:1217–29. smaller than the representative volume. Acta Mech 1999;138:123–34. [74] Kouznetsova V, Geers MGD, Brekelmans WAM. Multi-scale constitutive [46] Jiang M, Ostoja-Starzewski M, Jasiuk I. Scale-dependent bounds on modelling of heterogeneous materials with a gradient-enhanced compu- effective elastoplastic response of random composites. J Mech Phys tational homogenization scheme. Int J Numer Meth Engng 2002;54: Solids 2001;49(3):655–73. 1235–60. [47] Lee SJ, Shield RT. Variational principles in finite elastostatics. ZAMP [75] Kouznetsova V, Geers MGD. Brekelmans WAM, Size of a representative 1980;31:437–53. volume element in a second-order computational homogenization [48] Costanzo F, Gray GL, Andia PC. On the notion of average mechanical framework. Int J Multiscale Comput Eng 2004;2(4):575–98. properties in MD simulation via homogenization. Modell Simul Mater Sci [76] Kouznetsova V, Geers MGD. Brekelmans WAM, Multi-scale second- Eng 2004;12: S333–S345. order computational homogenization of multi-phase materials: a nested [49] Khisaeva Z. PhD Thesis (in progress). Montreal, Canada: McGill finite element solution strategy. Comp Meth Appl Mech Eng 2004;193: University; 2005. 5525–50.