Towards Stochastic Continuum Thermodynamics

J. Non-Equilib. Thermodyn. 2002 Vol. 27 pp. 335±348 Towards Stochastic Continuum Thermodynamics M. Ostoja-Starzewski Department of Mechanical Engineering, McGill University, MontreÂal, Canada Registration Number 934 Abstract We explore the relation of a line of studies over the last dozen years on mesoscale material response below the Representative Volume Element with the line of studies on mesoscopic continuum physics. The objective is the development of stochastic continuum thermodynamics able to grasp random microstructural features absent in deterministic continuum theories. We rst compare two lines of studies ± those dealing with the response on a scale smaller than the Representative Volume Element (RVE) vis-aá-vis those on mesoscopic continuum physics ± and observe the dilemma of choice of the random eld approximation. As a common basis for connecting both approaches, we propose a mesoscale Statistical Volume Element (SVE) and consider bounding its response via two admissible loadings. Employing the paradigm of thermal conductivity, we also present the relevant Legendre transformations. The latter are then generalized to the more general situation of elds governed by a quartet of Legendre transformations. 1. Introduction In recent years there has been a growing interest in introducing statistical microstructural features into continuum thermodynamics models and theories. Here we identify two lines of such studies: I. Studies on the (thermo)mechanical material response below the Representative Volume element (RVE) by Huet and his co-workers (e.g. [1±4]), and this author and his co-workers (e.g. [5±7]); II. Studies on mesoscopic continuum physics by Muschik and his coworkers (e.g. [8±10]). This communication starts out by exploring the relation of these two lines of studies, which might perhaps be combined to construct a stochastic continuum thermodynamics. Relevant challenges are identi ed along the way. The reason for constructing such thermodynamics is that both I and II deal with the problem of # Copyright 2002 Walter de Gruyter Berlin New York

336 M. Ostoja-Starzewski homogenization when the separation of scales tacitly assumed in conventional, deterministic continuum thermomechanics does not hold. The latter typically requires that d L L macro ; 1:1 when d is the microscale, such as the average size of grain in a given microstructure (or molecule in a polymeric solid, etc.); L is the mesoscale; L macro is the macroscale of the macroscopic body. Note, however, that d L may by replaced by d < L, or even by d L when it is periodic with noise on the level of the unit cell. 2. Comparison of Approaches I and II 2.1. I: Material response below the RVE One situation lacking separation of scales (1.1) is depicted with the help of Figure 1. It shows: (a) a realization B! of a random microstructure BˆfB! ;! 2 g; (b) a Fig. 1. (a) Germ-grain ber model (modeling ber structures on microscale) [11]; (b) hardcore Boolean random function (modeling cellular/biological tissue on microscale) [12]; (c) diffusion random function (modeling smoothly inhomogeneous media on mesoscale); (d) macroscopic body.

Towards stochastic continuum thermodynamics 337 smoothing mesoscale continuum; and (c) a macroscopic body. B! is given by a eld of material property (or properties) of the heterogeneous medium, being the space of all elementary events (realizations). B may be described by some scalar (or vector, tensor,...) n-component random eld over the D-dimensional physical space (R D ; D ˆ 1;...; 3) and time (R t ) : R D R t! R n : 2:1 Another alternative, however, actually shown in Figure 1, (a) and (b), is to use mathematical morphology models. Thus, (a) shows a germ-grain ber model modeling ber structure [11], while (b) depicts a hard-core Boolean random function modeling a cellular/biological tissue; see [12] and references therein for an introduction to such modeling tools. Evidently, the full description of a complicated material microstructure such as (a) or (b) involves a tremendously large amount of information, and some smoothing operation on a larger scale is desired/necessary, so as to make the model more tractable. Such smoothing involves a passage from the level of (a-b) in Figure 1 to that of (c) ± it is displayed here by a realization of a so-called diffusion random function, which is suited to modeling smoothly inhomogeneous media. The smoothing is effected by a Statistical Volume Element on scale L, that transforms the highly heterogeneous eld information of (a or b) into a smoother one (c) with continuous realizations L : R D R t! R n : 2:2 It is intuitively clear that, as the window size tends to in nity, the SVE becomes the classical RVE and the uctuations vanish, thus allowing one to recover the conventional, deterministic continuum mechanics. However, the L=d!1 limit can either be untenable or undesirable (due to the loss of microstructural information), and so the basic issues are: At what mesoscale ˆ L=d can RVE be approximately set up? What needs to be done below this scale? How can we work with a stochastic eld theory on scales between L (c) and L macro (d)? Consider now the steady-state, in-plane thermal conductivity in a body with random microstructure, that is, where the thermal conductivity tensor K ˆ K ij is a random eld K : R D! R 3 : 2:3 In order to effect smoothing on any realization B! such as in Figure 1(a-b), in the vein of (2.2), K L : R D! R n ; 2:4

338 M. Ostoja-Starzewski we need to make either the temperature gradient =T ˆ T ;1 ; T ;2 or the heat ux q ˆ q 1 ; q 2 controllable. Thus, if @B denotes the boundary of B!, ˆ L=d being a dimensionless parameter of the mesoscale, we have two options: (i) uniform essential ( e ) boundary condition on the mesoscale window (a) T x ˆ=T 0 x 8x 2 @B ; 2:5 where =T 0 T;i 0 is a constant gradient, resulting in a mesoscale conductivity Ke! ; or (ii) uniform natural ( n ) boundary condition on the mesoscale window (a) q x ˆq 0 n x 8x 2 @B ; 2:6 where q 0 q 0 i is a constant heat ux and n is the outer unit normal vector, resulting in a mesoscale resistivity R n!. (See the Appendix for de nition of those mesoscale properties.) We observe that (2.5) and (2.6) satisfy the Hill condition for the thermal conductivity problem: the volume average (denoted by overbar) of a scalar product of temperature gradient and heat ux elds must equal the product of their volume averages =T q ˆ =T q: 2:7 There is also a third boundary condition which satis es (2.7), namely a uniform mixed m one T x =T 0 xš q x q 0 n x Š ˆ 0 8x 2 @B ; 2:8 and here the resulting conductivity is K m. It might look at rst sight, that Km! is the best choice for the SVE, especially given the fact that it stabilizes much more rapidly with the increasing mesoscale ˆ L=d than the other two, see e.g. [7] for a discussion of the mathematically analogous problem in anti-plane elasticity. Unfortunately, there is more than one way of setting up the boundary condition (2.8), although, no matter what this setup is, all three responses ± under (2.5), (2.6), and (2.8) ± are connected by the order relation R t! Š 1 K m! Kd! : 2:9 All that has been said above pertains to a single and deterministic realization B! of the microstructure B, and hence the explicit dependence on!. When we pass to the ensemble, providing the microstructure is statistically homogeneous and ergodic in space R D, invoking variational principles for elliptic equations, we have a hierarchy of bounds on the effective macroscopic tensor K eff [1, 2, 5, 6] hr n 1 i 1 hr n 0i 1 hr n i 1 K eff hk e ihke 0i hk e 1 i 80 <=2: 2:10

Towards stochastic continuum thermodynamics 339 The tensor K eff satis es two requirements of Hill's de nition [13] of the RVE: (a) statistical homogeneity and ergodicity of the material; (b) some scale L of the material domain, suf ciently large relative to the microscale d so as to ensure the independence of response from the boundary conditions. Clearly, the attainment of the RVE is a function of the scale as well as the mismatch in properties of a given microstructure. Among others, reference [6] gives illustrations of such -dependence for several material systems in anti-plane strain, which, of course, is mathematically equivalent to in-plane thermal conductivity. 2.2. II: Mesoscopic continuum physics The line of studies II [8±10] introduces the concept of a statistical element as a ``mesoscopic distribution function (MDF) f m; x; t generated by the different values of the mesoscopic variables in a volume element f m; x; t f m; x; t 2MR D R t :'' 2:11 The MDF is de ned on the mesoscopic space MR D R t describing the distribution of m in a volume element at x; t, normalized by f m; x; t dm ˆ1, as it should. The macroscopic properties are obtained via averaging with respect to the distribution f m; x; t. Thus, when trying to integrate both lines I and II, we rst recognize M to be analogous to the random eld L on mesoscale L. In terms of our elastic microstructure problem above, f m; x; t is analogous to a probability distribution of the mesoscale stiffness. On one hand, this analogy necessarily forces one to choose L in mesoscopic continuum physics, and on the other, it shows that the mesoscale continuum thermodynamics approximation is non-unique for we have three possible choices stemming from three boundary conditions: either (2.5) or (2.6), or (2.8). This appears to be a serious dilemma and perhaps no single choice may be universally good for all the applications. Additionally, we see from studies I that an assumption of af ne temperature eld can only provide an upper bound (of Voigt type: hk e 1 i)on the mesoscale, or macroscale, response; the lower bound (of Reuss type: hk n 1 i 1 )is provided by a uniform heat ux eld. 3. Dilemma of Non-Unique Mesoscale Response The scalar products in (2.7) are of the form of the thermal entropy production th ˆ q =T, where, for simplicity, we drop the T 1 term. Clearly, going to a mesoscale model reduces the enormous number of degrees of freedom in a microscale (usually discrete) model, recall Figure 1(a-b). The mesoscale description smoothes this `noise' with the help of some statistics that becomes causal in the L=d!1limit. Now, it has recently turned out that one aspect of the statistics on mesoscale ± namely, the coef cient of variation of the second invariant (measuring the anisotropy) of the mesoscale K ij ± is independent of the window size, the strength of microscale uctuations, the type of uniform boundary conditions and even the

340 M. Ostoja-Starzewski speci c kind of microstructure, providing it originates from a Poisson-type point eld [14]. Basically then, one may only require the speci cation of statistics of the rst invariant plus the mean of the second invariant to set up the mesoscale random eld K. While on microscale, pointwise ( th and th being functions of location), we have a Legendre transformation th =T th q ˆ q =T; 3:1 on mesoscale we encounter either =T 0 ;! q! ˆ q =T0 ; 3:2 when the uniform temperature-controlled boundary condition (2.5) is applied, or =T! q0 ;! ˆ q 0 =T; 3:3 when the uniform heat- ux-controlled boundary condition (2.6) is applied. We have now replaced the subscript th by to explicitly point out the scale dependence. Upon ensemble averaging, (3.2) and (3.3) become (Fig. 2(a)), respectively, h =T 0 i hqi ˆ hqi=t0 ; 3:4 and (Fig. 2(c)) h=ti h q0 i ˆ q 0 h=ti: 3:5 The above holds also for a nonlinear situation providing that, for each specimen B! 2B, (i) =T 0 ;! is star-shaped and convex, (ii) q0 ;! is star-shaped, convex, and both functions are homogeneous. Finally, we have a reversible Legendre transformation =T 0! q0 ;! ˆ q 0 =T 0 3:6 in the case of uniform mixed boundary conditions on mesoscale T x =T 0 xš q x q 0 xš ˆ0, 8x 2 @B L, although there remains a non-unique choice of the actual setup of =T 0 ; q 0 Š loading. As the mesoscale increases inde nitely L=d!1 ± or, in other words, as the SVE turns into the RVE ± the relations (3.4) and (3.5) should coincide and turn into the classical statement of a deterministic continuum theory (Fig. 2(b)) =T q ˆ q =T; 3:7 whereby the distinction between =T 0 and =T, as well as between q 0 and q, vanishes since we are dealing with the RVE situation. Accordingly, we have dropped the subscript in the above.

Towards stochastic continuum thermodynamics 341 Fig. 2. Thermodynamic orthogonality in: (a) the spaces of prescribed temperature gradient =T 0 and ensemble average heat ux hqi on mesoscale ; (b) the spaces of =T and q on macroscale (RVE level), where the scatter is absent; (c) the spaces of ensemble average temperature gradient h=ti and prescribed heat ux q 0 on mesoscale. In all three cases, surfaces of dissipation functions and respective duals, ensemble averaged and explicitly parametrized by on mesoscale, are shown. 4. Legendre Transformations in Thermomechanics of Random Media 4.1. Legendre transformations in deterministic thermomechanics The version of thermomechanics we adopt here belongs to the category of thermodynamics with internal variables (T.I.V.) as developed and expounded, among others, in [15±17]. As is well known, the RVE response in T.I.V. is described by the free energy and dissipation function, both of which are scalar products ˆ 1 2 r ee ˆ th intr th ˆ qrt=t intr ˆ Y _a ˆ r d p A _a 4:1

342 M. Ostoja-Starzewski where the Clausius-Duhem inequality expresses the Second Law of thermodynamics with th being the thermal dissipation and intr the intrinsic dissipation. The latter quantity is a scalar product of the dissipative force Y with the velocity _a (rate of the state variable a). As an example, intr is taken to involve viscous, plastic and internal effects. Thus, r is the Cauchy stress tensor, A is force associated to an internal dissipative process, e e is elastic strain, d p is the plastic deformation rate, _a is the rate of internal parameters, q is the heat ux, and T is the temperature. An important role in thermomechanics is played by Legendre transformations. Already when the material is purely hyperelastic, there is such a well known transformation between stress and strain ± depending on which one is assumed as a controllable variable ± and leading to a relation of potential (Helmholtz) and complemetary (Gibbs) energies. With reference to (4.1), these are versions of. For example, when the material is thermoelastic, this is generalized to a quartet of partial Legendre transformations linking internal energy, enthalpy, Gibbs and Helmholtz energies; see e.g. [18] and Figure 3 here. When the material is dissipative, we have further Legendre transformations involving the dissipation function, because the latter one may be set up either in the space of (controllable) kinematic or force variables. As pointed out by Collins and Houlsby [19], when involves internal variables, we deal also here with a quartet of partial Legendre transformations, see also [20]. All these transformations are set up in the context of a homogeneous deterministic continuum ± that is, for the RVE mentioned in connection with equation (4.1). Shown in the above mentioned references are several similar versions of these partial Legendre transformations. Fig. 3. A quartet of deterministic partial Legendre transformations for pairs x i $ y i and i $ i for the functional X x i ; i [20].

Towards stochastic continuum thermodynamics 343 4.2. Legendre transformations in stochastic thermomechanics Recall that, in connection with (4.1), the RVE postulate in the sense of Hill [13] hinges on two basic requirements (for a review see [21]): (a) statistical homogeneity and ergodicity of random material microstructure; these two properties assure the RVE to be statistically representative of the macroresponse; (b) existence of some scale L of the material domain, suf ciently large relative to the microscale d in (1) so as to ensure an identical constitutive response irrespective of whether kinematic or traction boundary conditions are being applied. In the following we focus on the case of (b) being not satis ed when the RVE is simply too small for the inequality d L in (1) to hold, see Figure 1. (One may also consider the case of (a) not being satis ed.) As pointed out in various papers written in the nineties (e.g. [4, 21]), there are numerous cases when this situation needs to be considered. For example, the RVE size may be too large and too expensive to be handled in the laboratory and/or in the computations; or the RVE size may be too large for the right-hand inequality in (1.1) to hold. There is yet another possibility: the RVE may be undesirable when it misses the full micromechanics of the problem at hand. In all the above cases, we are forced to deal with the SVE in place of the RVE. Consequently, we need to explicitly account for the dependence of SVE's response on the boundary conditions. In general, we have three types of boundary conditions which satisfy the Hill condition (recall 2.7) e r ˆ e r; 4:2 which is necessary for the mechanical and energetic de nitions of response to be consistent: (i) uniform displacement (also called kinematic, essential or Dirichlet) boundary condition (d) u x ˆe 0 x 8x 2 @B ; 4:3 (ii) uniform traction (also called static, natural, or Neumann) boundary condition (t) t x ˆr 0 n x 8x 2 @B ; 4:4 (iii) uniform displacement-traction (also called mixed) boundary condition (dt) u x e 0 xš t x r 0 n x Š ˆ 0 8x 2 @B : 4:5 Suppose for the moment that we deal with a linear elastic microstructure. Then, each of these boundary conditions results in a different stiffness, or compliance tensor. Huet [2] uses here the term `apparent' to distinguish the mesoscale properties from the `effective' (macroscopic, global, or overall) ones that are typically denoted by eff. Thus, condition (4.3) yields a mesoscale random stiffness C d!. On the other hand,

344 M. Ostoja-Starzewski condition (4.4) results in a mesoscale random compliance S t!. Finally, condition (4.5) results in a mesoscale random stiffness S dt!. The!-dependence points to the explicit dependence of mesoscale response on the actual microstructure (recall Fig. 1) ± that is, the random nature in the ensemble sense ± of the resulting average stress eld and of the apparent stiffness tensor, with the uctuations disappearing in the limit!1. The de nition of mesoscale properties given in the Appendix applies here as well; (A.1) and (A.4) generalize then to the so-called average strain and average stress theorems. Given the randomness of response under either one of these three boundary conditions ± by analogy to developments of Section 3 ± there are two Legendre transformations linking potential e 0 and complementary r 0 energies, depending on whether strain or stress is controllable hri h e 0 iˆhrie 0 ; h r 0 i hei ˆ r 0 hei: 4:6 Thus, in the rst case here, hri is the ensemble average outcome, while in the second case hei is the ensemble average outcome for the SVE. Just like in (3.4-5), let us note the symmetry occurring between (4.6) 1 and (4.6) 2. When the material is thermoelastic, (4.6) is generalized to a quartet of partial Legendre transformations linking internal energy, enthalpy, Gibbs and Helmholtz energies, e.g. [18]. Thus, depending on what we take as the reference, or controllable, loading case, we have either e 0 or r 0 playing the role of X x i ; i in Figures 3 and 4; x x i is e 0 or r 0, and a i is the temperature or entropy. When the material response involves dissipation according to T.I.V., the quartet of ensemble averaged, partial Legendre transformations applies to the random functional x; _a where x x i may be either velocities or conjugate forces, and Fig. 4. A quartet of ensemble averaged, partial Legendre transformations for pairs x i $ y i and i $ i, when the pair x i ; i is controllable and the functional X x i ; i is random.

Towards stochastic continuum thermodynamics 345 _a _ i may be either rates of internal variables or their conjugate forces. A simpler situation not involving internal variables, and focusing on stochastic generalization of thermodynamic orthogonality, has been outlined in [22]. Indeed, in that case, a relation completely analogous to (4.6) holds with e 0 and r 0 being replaced by _a and Y, respectively. Thus, when (4.3) and (4.4) is used, the rst pair provides bounds on quasi-conservative response, while the second one gives bounds on dissipative response. When (4.5) is employed, we obtain an intermediate response ± very likely with much weaker scale effects than the other two. 5. Closure We have outlined a formulation of continuum thermodynamics with internal variables (T.I.V.) of random media, where the intrinsic entropy production intr is a bilinear form of forces Y and uxes _a intr ˆ Y _a: 5:1 In the case where a given problem does not involve internal variables, we have eld variational principles of mechanics ± such as for (thermo)elasticity or plasticity ± to ensure the true extremum character of solution elds and their volume averages under given boundary conditions. For example, a formulation of stochastic damage mechanics of elastic-brittle solids has been advanced in [22], while a mesoscale elasto-plasticity of random media is given in [23]; the latter work also gives more developments on Legendre transformations such as the one discussed here in Section 3. When the rates _a da=dt of internal variables a i are present within _a (with appropriate conjugate forces showing up in the Y vector) in a mesoscale thermodynamics model, eld variational principles of mechanics cannot be straightforwardly obtained. A method for dealing with this challenge has been outlined in Section 4 ± it involves an ensemble averaging performed on the quartet of Legendre transformations for the random functional X x i ; i, which yields bounds on response, respectively as the pair x i ; i, or its dual y i ; i, taken as controllable. Theoretically, as the mesoscale!1, the SVE tends to the RVE, albeit the latter may not always be attainable. An example of this is the phenomenon of localization in a random microstructure, whereby the RVE in the conventional sense of a deterministic eld theory may not exist. Our proposal has also shed some light on the material operator describing response of a mesodomain [24±26]. Clearly, the mesodomain is analogous to the SVE employed here, although we explicitly consider the nite-size scaling property of the SVE and its passage into the RVE, e.g. [6]. Thus, in view of three possible loadings (3.4±4.5), one may not expect a unique material operator but only a range of its responses. As noted above, the latter can be bounded via (4.3) and (4.4). More work is needed to clarify the links of the proposed stochastic continuum thermodynamics to other theories [27]. We end by also noting a challenge posed by a need to quantitatively solve a eld problem on a macroscale. This corresponds to the range described by the L L macro

346 M. Ostoja-Starzewski part of the inequality (1.1). In the case separation of scales does hold, this is just a deterministic continuum mechanics problem under some prescribed boundary conditions. But, when L is the size of an SVE, we have a stochastic boundary value problem, and, if the problem is to be tackled by nite elements, it may well turn out that a certain mesoscale is optimal [28]; this is due to a competition of a modeling error with a numerical error. Acknowledgment This material is based upon work supported by the National Science Foundation (Grant CMS-9713764), the US Department of Agriculture (Grant 99-35504-8672), and the Canada Research Chairs program. Appendix This appendix, giving the basis for the de nition of mesoscale properties in the vein of mechanics of composite materials, is provided for the completeness of the presentation. First, we consider a body B!, a realization of some heterogeneous microstructure of volume V and boundary @B, subjected to a volume average temperature gradient by the essential boundary condition (2.5). Given the continuity of temperature throughout B!, we see that the volume average temperature T ; j equals the prescribed mean temperature gradient T; 0 j T ; j ˆ 1 T ; j dv ˆ 1 Tn j ds ˆ 1 T; 0 k V V V @B V x kn j ds ˆ 1 T; 0 j @B V dv ˆ T0 ; j : @B A:1 Then, the mesoscale conductivity K e ij Ke of B! locally having a `Fourierlinearly' conducting microstructure, under (2.5), may be de ned by q i ˆ K e ij T0 ; j ; A:2 where q q i is the volume average heat ux. Alternatively, we may consider the volume average of a scalar product of q and =T in B! ± i.e. th with the T 1 term dropped ± also under (2.5), th ˆ 1 2V ˆ 1 2V V V q i T ;i dv ˆ 1 q i Tn i ds ˆ 1 q i T; 0 k 2V @B 2V x kn i ds @B q i T 0 ;i dv ˆ 1 2 q irt 0 i ˆ 1 2 rt0 i Ke ij rt0 j ; A:3 where the equilibrium condition q j; j ˆ 0 and (2.5) have been used. Thus, the mesoscale conductivity K e ij may be de ned either from the average =T or from the average of th if the boundary condition (2.5) is imposed.

Towards stochastic continuum thermodynamics 347 On the other hand, considering the natural boundary condition (2.6), we rst see that the volume average heat ux q i equals the prescribed heat ux q 0 i q j ˆ 1 q j dv ˆ 1 q 0 k V V V x j n k ds ˆ q 0 j : A:4 V Now, we can de ne mesoscale resistivity R n either from the boundary conditions (2.10) via T ;i ˆ R n ij q0 j ; A:5 or from the volume average of th in B! th ˆ 1 2V ˆ 1 2V V V q i T ;i dv ˆ 1 q i Tn i ds 2V @V q 0 i T ;i ds ˆ 1 2 q0 i rt ;i ˆ 1 2 q0 i Rn ij q0 j : A:6 This shows that the apparent compliance R n ij may be de ned either from q or from th if the boundary condition (2.6) is imposed. References [1] Huet, C., Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids, 38 (1990), 813±841. [2] Huet, C., Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies, Mech. Mater., 31 (1999), 787±829. [3] Hazanov, S., Huet, C., Order relationships for boundary conditions effect in the heterogeneous bodies smaller than the representative volume, J. Mech. Phys. Solids, 42 (1994), 1995±2011. [4] Hazanov, S., Amieur, M., On overall properties of elastic heterogeneous bodies smaller than the representative volume, Intl. J. Eng. Sci., 33(9) (1995), 1289±1301. [5] Ostoja-Starzewski, M., Micromechanics as a basis of continuum random elds, Appl. Mech. Rev., 47(1, Part 2) (1994), S221±S230. [6] Ostoja-Starzewski, M., Random eld models of heterogeneous materials, Intl. J. Solids Struct., 35(19) (1998), 2429±2455. [7] Jiang, M., Alzebdeh, K., Jasiuk, I., Ostoja-Starzewski, M., Scale and boundary conditions effects in elasticity of random composites, Acta Mech., 148(1±4), (2001) 63±78. [8] Blenk, S., Ehrentraut, H., Muschik, W., Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation, Physica A, 174 (1991), 119±138. [9] Papenfuû, C., Muschik, W., Liquid crystal theory as an example of mesoscopic continuum mechanics, in: Trends in Continuum Physics (1998), 277±291. [10] Muschik, W., Ehrentraut, H., Papenfuû, C., Concepts of mesoscopic continuum physics with application to biaxial liquid crystals, J. Non-Equilib. Thermodyn., 25 (2000), 179± 197. [11] Ostoja-Starzewski, M., Stahl, D.C., Random ber networks and special elastic orthotropy of paper, J. Elast., 60(2) (2000), 131±149. [12] Jeulin, D., Random structure models for homogenization and fracture statistics, in: Mechanics of Random and Multiscale Microstructures, Eds. D. Jeulin, M. Ostoja- Starzewski, CISM Courses and Lectures 430, pp. 33±91, Springer, Wien, 2001.

348 M. Ostoja-Starzewski [13] Hill, R., Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, 11 (1963), 357±372. [14] Ostoja-Starzewski, M., Universal material property in conductivity of planar random microstructures, Phys. Rev. B, 62 (2000), 2980±2983. [15] Ziegler, H., 1983, An Introduction to Thermomechanics, North-Holland, Amsterdam. [16] Germain, P., Nguyen Quoc Son, Suquet, P., Continuum thermodynamics, ASME J. Appl. Mech., 50 (1983), 1010±1020. [17] Maugin, G.A., The Thermomechanics of Nonlinear Irreversible Behaviors ± An Introduction, World Scienti c, Singapore, 1999. [18] Houlsby, G.T., Puzrin, A.M., A thermomechanical framework for constitutive models for rate-independent dissipative materials, Int. J. Plast., 16 (2000), 1017±1047. [19] Collins, I.F., Houlsby, G.T., Application of thermomechanical principles to the modeling of geotechnical materials, Proc. Roy. Soc. Lond., A, 453 (1997), 1975±2001. [20] Sewell, M.J., Maximum and Minimum Principles, Cambridge University Press, Cambridge, 1988. [21] Ostoja-Starzewski, M., Mechanics of random materials: stochastics, scale effects, and computation, in: Mechanics of Random and Multiscale Microstructures, Eds. D. Jeulin, M. Ostoja-Starzewski, CISM Courses and Lectures 430, pp. 93±161, Springer, Wien, 2001. [22] Ostoja-Starzewski, M., Microstructural randomness versus representative volume element in thermomechanics, ASME J. Appl. Mech., 69 (2002), 25±35. [23] Jiang, M., Ostoja-Starzewski, M., Jasiuk, I., Scale-dependent bounds on effective elastoplastic response of random composites, J. Mech. Phys. Solids, 49 (2001), 655±673. [24] Axelrad, D.R., Micromechanics of Solids, Elsevier-PWN, Amsterdam, 1978. [25] Axelrad, D.R., Foundations of the Probabilistic Mechanics of Discrete Media, Springer, Berlin, 1984. [26] Axelrad, D.R., Muschik, W., Constitutive Laws and Microstructure, Springer, Berlin, 1988. [27] Muschik, W., Fundamentals of nonequilbrium thermodynamics, in: Non-Equilibrium Thermodynamics with Applications to Solids, CISM Courses and Lectures 336, pp. 1± 63, Springer, Wien, 2001. [28] Ostoja-Starzewski, M., Microstructural disorder, mesoscale nite elements, and macroscopic response, Proc. Roy. Soc. Lond., A 455 (1999), 3189±3199. Paper received: 2001-12-07 Paper accepted: 2002-08-03 Prof. Martin Ostoja-Starzewski Department of Mechanical Engineering McGill University 817 Sherbrooke Street West MontreÂal, QueÂbec H3A 2K6 Canada Tel: 1 (514) 398-7394 E-mail: martin.ostoja@mcgill.ca