Abstract This paper is devoted to the derivation of improved bounds for the effective behavior of linear elastic matrix-inclusion composites based on a strategy which is inspired by both the works of Huet (1990) and Danielsson et al. (2007) . As shown by the former author, the effective properties of random linear composites can be bounded by ensemble averages of their apparent elastic moduli defined on square (or cubic) volume elements (VEs) and computed with either affine displacement Boundary Conditions (BC) or uniform traction BC. However, in the case of a large contrast of the constituents, the discrepancy between the upper and lower bounds remains significant, even for large values of the VE size. This occurs because the contribution to the total potential (or complementary) energy of the particles (or pores) which intersect the edges of the VE becomes unphysically very large when uniform BC are directly applied to the particles. To avoid such limitations, we considerer non-square (or non-cubic) VEs consisting in simply connex assemblages of cells, each cell being composed of an inclusion surrounded by the matrix, thus forbidding any direct application of BC to the particles. Such VEs are generated by extending the scheme proposed by Danielsson et al. (2007) in the context of periodic random microstructures to fully random microstructures. By applying the classical energy bounding theorems to the non-square VEs, new bounds for the effective behavior are derived. Their application to a two-phase composite composed of an isotropic matrix and aligned identical fibers randomly distributed in the transverse plane leads to sharper bounds which converge quickly with the VE size, even for infinite contrasts. Introduction Most materials, either natural (minerals, biological tissues, etc.) or designed by man (composites, steels, concrete, etc.) are heterogeneous. Efficient modeling approaches are required to determine in a deductive way the global behavior of such a material, to be used for structural calculations at a larger scale, from the local behavior of its various constitutive phases and their spatial distribution. For that, one implicitly or explicitly assumes the following separation of length scales: Introduction where the microscale d is the typical size of the heterogeneities of the constituents, L struct is the typical length of the fluctuations of the mechanical fields generated in the macroscopic structure under consideration if it would be made of a homogeneous material, and the mesoscale ' is some intermediate length, over which these macroscopic mechanical fields can be considered as uniform, and at the same time, such that volume elements (VE) of size ' are representative of all microstructural features of the material. This assumption allows to define an effective mechanical behavior such that structural calculations, performed on a fictitious structure with same macroscopic geometry and boundary conditions but made of a homogeneous material exhibiting this behavior, provide, at a much lower numerical cost, macroscopic strain and stress fields which coincide, at least to some accuracy, with the local mechanical fields developed inside the actual heterogeneous structure averaged over volume elements of size '. The smallest volume element ensuring such a property is called the Representative Volume Element (RVE). Many methods to determine the effective behavior from the properties of the constitutive phases and the microstructure within a RVE have been proposed and can be split into two families. The first one groups so-called homogenization methods or mean field theories. They are described in many reference extended papers or textbooks among which Willis (1981) , Sanchez-Palencia and Zaoui (1987) , Nemat-Nasser and Hori (1993) , Ponte Castañeda and Suquet (1997) and Bornert (2001) . Such methods have now reached a high level of efficiency, especially for linear materials (e.g. Beran, 1968; Willis, 1977; Christensen and Lo, 1979 ; Ponte Cas-0020-7683/$ -see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2012.01.018 tañeda and Willis, 1995; Bornert et al., 1996) but also for nonlinear elastic or viscoplastic composites (e.g. Ponte Castañeda and Suquet, 1997; Ponte Castañeda, 2002; Lahellec and Suquet, 2004) as well as for composites with nonlinear 'hereditary' behaviors, which exhibit a dependence of their current response on the loading path (e.g. Masson and Zaoui, 1999; Lahellec and Suquet, 2007; Doghri et al., 2010) . Mean field methods provide either rigourous bounds or estimates of the effective behavior. Usually, bounds can be derived when statistical informations about the phase distribution are available such as some of the nth order moments of the characteristic functions v r -defined in Section 2.1 -which serve to describe the microstructure. For linear elastic random heterogeneous materials which are focused on in this paper, such informations allow to derive the well-known Voigt (1889) and Reuss (1929) first-order bounds, the Hashin and Shtrikman (1963) second-order bounds and the Beran (1968) higher-order bounds. If the informations about the microstructure are rather of 'morphological' type -e.g. when one phase plays a particular role as it is the case for matrixinclusion composites or when all the phases have the same influence as for granular composites -direct estimates of the effective properties can be proposed such as the well-known Mori and Tanaka (1973) or self-consistent (Kroner, 1958) estimates. When the available informations pertain to both the statistics of the phase distribution and the morphology of the composite, very efficient estimates or bounds for the effective behavior such as the Ponte Castañeda and Willis (1995) estimate or the Morphologically Representative Pattern (MRP) bounds and self-consistent estimates (Stolz and Zaoui, 1991; Bornert et al., 1996) are available. Introduction However, all mean field theories are based on random composites with an infinite extension, for which the condition (1) is exactly satisfied. Mean field estimates or bounds are therefore asymptotic estimates or bounds. Furthermore, they are well suited to predict the behavior of composites with simple ideal model microstructures. For heterogeneous materials with more complex microstructures, when detailed microstructural informations are available in the form of appropriate morphological models or representative images of the microstructure, full-field simulations provide an alternative and nowadays efficient way to evaluate more accurately the effective behavior. This approach consists in computing numerically the apparent behavior of the composite on different VEs of same finite size and approximating the effective behavior as the ensemble average of the apparent behavior. Such an estimate of the effective behavior crucially depends on the size of the VEs, the number of realisations used to evaluate ensemble averages and the prescribed boundary conditions, in addition to the properties of the constitutive phases and the microstructure. Its convergence towards the effective behavior is obtained, at some accuracy, when the size of the samples reaches the size of the RVE, this latter depending on the investigated effective property, the material characteristics such as the contrast between the phases, the volume fractions, the spatial distribution of the phases, etc. -and on the expected accuracy. Accordingly, in order to obtain a precise estimate of the effective behavior it becomes essential to evaluate the size of the RVE associated with a particular mechanical or physical property for a given material. However, this size can be very large for many materials, thus leading to excessive computational expense to evaluate the effective behavior from samples of that size. For these materials, we need rigorous criteria able to evaluate how far ensemble averages of the apparent behavior are from the actual effective behavior of the random media when the numerical simulations are performed on samples the size of which is smaller than the RVE size. This problem has been first tackled by Huet (1990) and later by Sab (1992) . By making use of different arguments, both these authors independently showed that the arithmetic ensemble average of the apparent stiffness associated with Affine Displacement Boundary Conditions (ADBC) and the harmonic ensemble average of the apparent stiffness associated with Uniform Traction Boundary Conditions (UTBC), both applied to square or cubic shaped samples of size L, are upper and lower bounds of the effective behavior, respectively (see Eq. (14)). Further, they proved that the upper bounds (resp. the lower bounds) are decreasing (resp. increasing) functions of the nondimensional size d ¼ L=d of the VE. The gap between the upper and lower bounds allows to evaluate how far these bounds are from the actual effective behavior. It has been checked on different applications, investigated for instance by Ostoja-Starzewski (1998) and Ostoja-Starzewski (2006) , that the bounding property (14) is able to provide good estimates of the effective behavior for relatively low values of d but only for weak to medium contrast between the constituents of the composite. For composites with large contrast, e.g. rigid particles or pores, simulations performed on planar linear elastic disk-matrix composites have revealed that the discrepancy between the upper and lower bounds remains significant even for large values of d. Later on, numerical simulations carried out by Jiang et al. (2001) again on planar linear elastic disk matrix composites showed that the ensemble average of the apparent behavior associated with periodic BC converges much faster towards the effective behavior than with affine displacement or uniform traction BC. This property has been used by Kanit et al. (2003) who developed an efficient method to evaluate the effective properties based on the notion of integral range and confidence interval. This method allows to predict the number of samples of size d necessary to reach a given level of accuracy for the wanted effective property and conversely to predict the relative accuracy for a given number of simulations. However, as pointed out by these authors, the method is efficient only if the size of the samples is larger than the RVE size associated with periodic BC, this latter being defined as the size for which the mean value of the apparent properties of a given material for periodic BC does not evolve any more with the size d of the samples. For smaller sizes, the ensemble averages of the apparent properties are potentially biased estimates of the effective properties. As noticed by the authors, this bias is limited by the difference between Huet's bounds, which might however not be accurate for composites with high contrasts, as mentioned above. Introduction The main objective of this work is to provide new sharper bounds of the effective behavior derived from the apparent behaviors as initiated by Huet (1990) and Sab (1992) but by removing the aforementioned limitations observed for uniform strain or stress BC, that is the blow up of the upper bound for rigidly reinforced composites and the vanishing of the lower bound for porous materials. The key point is to notice that this blow up (resp. vanishing) occurs because the part of the strain energy (resp. complementary energy) due to the rigid particles (resp. pores) which intersect the edges of the cell becomes very large when ADBC (resp. UTBC) are directly applied to these rigid particles (resp. pores). To avoid such limitations, we consider non-square (or non-cubic for 3D microstructures) VEs made of assemblages of cells composed of an inclusion located strictly inside the cell and surrounded by the matrix (e.g. Voronoï cells), thus forbidding any direct application of boundary conditions to particles. Such VEs can be generated by extending the scheme proposed by Danielsson et al. (2007) in the context of periodic random microstructures to fully random microstructures. By making use of the classical bounding theorems for linear elasticity and appropriate averaging procedures, sharper bounds of the effective behavior are derived from ensemble averages of the apparent behaviors associated with non-square (or non-cubic) VEs. Introduction The structure of this paper is as follows. The approach initiated by Huet to derive upper and lower bounds for the effective properties from ensemble averages of the apparent properties is briefly recalled in Section 2. The new procedure allowing to derive improved bounds for the effective properties by extending Huet's approach to non-square (or non-cubic) VEs for linear matrix-inclusion random composites is presented in Section 3. Special attention is given to the fact that the considered random VEs have a non constant volume. Its applications to a two-phase composite composed of a matrix and aligned identical cylindrical fibers randomly distributed in the transverse plane is depicted in Section 4. The results are presented in Section 4 and compared to the classical Huet's bounds for several contrasts ranging from rigid particles to porous ones and for different inclusion volume fractions. Conclusions are summarized in Section 6. Introduction The tensor notation used herein is a fairly standard one. The orders of the tensors are clear when taken in context. Products containing dots denote summation over repeated latin indices. For example, L : e ¼ L ijkl e kl e i e j and E 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4 F ¼ E ijkl F klij where ðe i ; i ¼ 1; 3Þ is a orthonormal cartesian basis and denotes the tensor product. On probabilistic aspects, all the random quantities (variables, fields) considered in this work are assumed to be defined on the same probability space H; F; P f g , where H is a sample space, F is a r-algebra of subsets of H and P is a probability on F. Moreover, the following notations will be used: if n ¼ ðnðxÞ; x 2 HÞ is a random variable defined on fH; F; Pg, for any fixed x 2 H; nðxÞ is a x-realization of n (i.e. the realization of n associated with the particular event xÞ. For the sake of simplicity, and if no mistake in interpretation is to be feared, the notation nðxÞ will also be used to denote such a random variable. In the same way, if w ¼ ðwðx; xÞ; x 2 D; x 2 HÞ is a random field defined on fH; F; Pg and indexed on D # R d ðd 2 N Ã Þ, for any fixed x 2 H, the function x ! wðx; xÞ is a x-realization of w and, to simplify the notations, wðx; xÞ will be used to denote just as well this realization as the random field itself. Finally, the statistical expectation and standard deviation of any random variable n will be denoted EðnÞ and rðnÞ, respectively. Classical bounds for the apparent and effective behaviors In this section we briefly recall some important and classical results on the apparent and effective properties of random linear elastic composites. Definition of the apparent behavior Let A be a microstructure characterized by a set of stochastic fields B 2 v r 2 0; 1 f g Definition of the apparent behavior where K is the number of phases, d ¼ 2 or 3 is the space dimension and v r the random characteristic functions, being 1 if the position vector x is in phase r, and 0 otherwise. In what follows, for the sake of simplicity, every microstructure (e.g. A) will be identified with the set of functions which characterizes it (i.e. B). Furthermore, only stationary and ergodic microstructures will be considered. From the microstructure B we obtain for each x 2 H a realization BðxÞ of the composite. It is worth noting that for two-phase composites only one characteristic function, e.g. the matrix one's v M , is enough to characterize the microstructure. In order to define square VEs as considered in Huet's approach, let us introduce, in 2D-space, square domains X s d;X of centers X ¼ ðx 1 ; x 2 Þ and side length L ¼ dd given by Definition of the apparent behavior The same notation can be used for cubic VEs in 3D-space, except that X ¼ ðx 1 ; x 2 ; x 3 Þ and X Definition of the apparent behavior In what follows, because of the stationarity of the random set B, the spatial dependence with respect to the square or cubic window centers X will not be specified unless required. Further, @A will denote the outer surface of a set A and nðxÞ its outward normal at x 2 @A. Definition of the apparent behavior When the VE is smaller than the RVE, the behavior of the composite at the mesoscopic scale -namely the behavior of the VE B Definition of the apparent behavior In Eqs. (3) and (4) Definition of the apparent behavior The apparent moduli C For ergodic and stationary materials, it has been shown (Papanicolaou and Varadhan, 1978; Sab, 1992) Definition of the apparent behavior thus defining the effective behavior C eff which does not depend on either the BC or the realization x. In practice, Eq. (8) Definition of the apparent behavior Similarly, by making use of the complementary energy principle applied to problem (6) (for details see Ostoja-Starzewski (2006)), a hierarchy of upper bounds for the effective compliance is obtained and reads Definition of the apparent behavior Combining Eq. (12) with (13), it follows Definition of the apparent behavior The scale-dependant hierarchy of lower and upper bounds for the effective behavior (14) has been first derived by Huet (1990) and later by Sab (1992) but in a different manner. This relation shows that the discrepancy between the upper and lower bounds decreases as d increases, until tending towards zero for infinite d, where the value of the effective behavior is attained. The notations Definition of the apparent behavior will be used to refer to these bounds based on apparent behaviors of 'Square Cells'. Discussion As mentioned in Section 1, numerical applications of relation (14) provide good evaluations of the effective properties for situations with small to medium contrast between the phases (Ostoja-Starzewski, 1998). However, for large contrast -e.g., rigid particles or pores -the discrepancy between the upper and lower bounds remains significant even for large value of d. Illustration of this fact is given in Fig. 2 which reports the evolution of the Huet's bounds of the in-plane effective bulk modulus as functions of d for a rigidly-reinforced material and a porous material composed of an isotropic linear elastic matrix and aligned identical cylindrical fibers with diameter d isotropically distributed in the transverse plane. Details on the generation of such microstructures will be given in Section 4. For all numerical applications presented here and later, we have set l M ¼ 1. The unphysically large (resp. low) values of the upper (resp. lower) bound for the rigidly-reinforced (resp. porous) material occurs because the part of the strain energy due to the particles which intersect the edges of the cell becomes very large when ADBC (resp. UTBC) are directly applied to rigid particles (resp. pores). Thus, by applying unrealistic BC such as a direct enforcement of affine displacement (resp. uniform traction) on the rigid particles (resp. pores), the bounds obtained by Huet (1990) or Sab (1992) strongly overestimate (resp. underestimate) the actual effective properties. Discussion It is worth noting that the evolution of the upper bound as a function of d depicted in Fig. 2 can be predicted by a simple model described as follows. In order to account for the part of the strain energy due to the particles which intersect the edges of a square cell of size L, the VE B (15) can be interpreted as the apparent stiffness tensors of the boundary layer and the core of the VE, respectively, relative to these specific BC. After taking ensemble averages of this equality and expliciting the volume fractions Discussion In a first approximation, it can be assumed that the statistical expectation of the apparent behavior of the boundary layer with constant thickness d does only very slightly depend on d and that the statistical expectation of the apparent modulus of the core can be evaluated by a modelling approach which does not artificially over-evaluate the effective properties, since it is based on much more realistic boundary conditions. It could for instance be approximated by the bound E C VORþ d proposed in Section 3 of this paper. Combining these considerations with (12) and noting in addition that the contribution of the boundary layer is predominant, i.e. EðC BL Þ ) EðC app d Þ, one gets: Discussion This relation can be further simplified by noting that the dependance with d of the first term of the second member is likely to be much weaker than that of the second (see Section 3), so that the substitution EðC as a function of d for rigidly-reinforced materials is essentially due to the predominant contribution of the boundary layer to the apparent behaviors when ADBC are applied. Of course, the approximate bound defined by Eq. (19) should not be understood as a rigorous result but rather as a simplified approach aiming at evaluating in a very simplified way the contribution of the boundary layer which dominates the dependence of the SC + bounds with d. In this regard, the discrepancy observed for small values of d between the SC + and approximate bounds is probably due to the crude approximation assuming that the apparent behavior of the boundary layer does not depend on d. To this regard, it is worth noting that the evolution of the upper bound K at d ¼ 30. For the in-plane deviatoric behavior, because of the quadratic anisotropy of Huet's upper bound generated by the square shape of the VEs, the simple approach described by Eq. (19) needs to be slightly modified to fully describe the evolution of C SCþ d Discussion . For the sake of clarity, this modified approach will be described in Section 4.5.3 devoted to the specificities induced by square shaped VEs. However, the key point is still that the blow-up of in-plane deviatoric moduli is produced by the particles which intersect the edges of the cells. Note that another closed-form scaling function describing the evolution with respect to d of the apparent tensor of modulinamely a nonlinear combination of K -has been obtained for linear elastic random polycrystals by Ranganathan and Ostoja-Starzewski (2008) but does not apply to the present context. Let us finally mention that making use of the classical Voigt upper bound (Voigt, 1889) to estimate the apparent behavior of the boundary layer, i.e. . Indeed, the assumption made by Voigt's approach which considers that the strain inside the quasi-rigid particles ðc ¼ 10 4 Þ belonging to the boundary layer is uniform and equal to the imposed strain e is too strong and should be relaxed. Only some components e ij ðxÞ of the local strain tensor turn out to be equal to e ij , and that only in a small part of the particles intersecting the edges of the VE. Improved procedure to bound the effective properties of matrix-inclusion random composites To avoid the above-mentioned limitation of Huet's approachnamely the blow up or vanishing for high contrast of the bounds of the effective properties generated by a direct application of uniform strain or stress BC on the particles, we describe in this section a new procedure to bound the effective properties of random composites which relies on the construction of non-square (or non-cubic in 3D) VEs whose boundaries do not intersect the particles. Non-square VEs In order to design VEs forbidding any direct application of boundary conditions to particles, the random microstructures considered in this paper are of matrix-inclusion type only. It is assumed that no contact occurs between the inclusions which can be of any size and any shape. The new VEs are defined as elementary cell assemblages, each elementary cell being composed of a sole inclusion surrounded by matrix. To generate such VEs, we consider a very large square window of the microstructure BðxÞ -namely a Representative Volume Element (RVE) -and perform a subdivision (e.g. a Voronoï subdivision) of this microstructure into M non-overlapping elementary cells where M is the number of inclusions in this window , such that the union of these cells coincides with the large window. Then, n non-overlapping square windows X s d;X k (n being the number of realizations) of size d and centers X k ðk ¼ 1 . . . nÞ are extracted from the RVE. They are used to generate n non-square domains X d;X k ðxÞ made of the set of elementary cells the centers of which belong to X s d;X k , the center of each elementary cell being defined as the center of mass of its inclusion. Each VE B d;X ðxÞ is therefore a restriction of a realization of the composite BðxÞ to its corresponding non-square domain X d;X ðxÞ derived from the subdivision into elementary cells. Note that the upperscript s introduced in Section 2.1 to denote square VEs is left over for non-square VEs. A classical manner to generate the elementary cell is to subdivise the RVE into M Voronoï cells, each cell being made of a sole inclusion surrounded by matrix (see Fig. 1(b) ). Such a way to generate non-square VEs by means of Voronoï cell assemblages can be interpreted as an extension of the scheme proposed by Danielsson et al. (2007) for periodic random microstructures to fully random microstructures. The Voronoï subdivision is well adapted to matrix-inclusion microstructures for which the inclusions are of same shape or size, as it will be the case in the numerical applications presented hereafter. However, for inclusions of different sizes or/and shapes, a more general subdivision should be considered. One instance of subdivision can be derived from the distance function dðx; I i Þ between a point x belonging to the RVE and an inclusion i ði ¼ 1; . . . ; MÞ of domain I i defined as dðx; I i Þ ¼ min y2I i dðx; yÞ where dðx; yÞ is the euclidian distance between two points x; y. The elementary cell number i is then defined, as by e.g. Bilger et al. (2005) , by the set X i ¼ fx 2 RVE; 8j -i; dðx; I i Þ 6 dðx; I j Þg, thus characterizing the subdivision thoroughly. Non-square VEs An other example of subdivision suitable for matrix-inclusion composites with particles of different sizes and/or shapes is provided by the Morphologically Representative Pattern -based description of the microstructure developed by Stolz and Zaoui (1991) and Bornert et al. (1996) . In this approach, the heterogeneous medium is described as a set of disjoint subdomains that map the whole RVE. It is assumed that these domains may be grouped into families with identical geometries and mechanical properties within the same family. These families are called ''Morphologically Representative Patterns'' (MRP). The way to define precisely the MRP is reported in Bornert et al. (1996) . An example of pattern could be a particle of given size and shape surrounded by a layer of matrix -several possibilities to define its external shape are conceivable. The classical Hashin Composite Spheres Assemblage (CSA) (Hashin, 1962) is one example of MRP-based subdivision for a ''theoretical'' microstructure consisting of spherical inclusions of different sizes embedded in a matrix such that the union of all the patterns, i.e. the composite inclusions, fills the whole RVE. For the CSA, each composite inclusion consists of an inclusion surrounded by a spherical layer of matrix such that the inclusion volume fraction inside the pattern is equal to the inclusion volume fraction of the RVE. An example of VEs generated by the above-mentioned scheme and issued from a Voronoï subdivision of a matrix-inclusion composite made of randomly distributed fibers of same size and orientation is reported in Fig. 1(b) . In 3D-situations, non-cubic VEs can be obtained by the same methodology. In what follows, for the sake of simplicity, only 2D-situations are considered. However, all the obtained results still hold in 3D-situations since they can be enlarged in a straightforward way by making use of non-cubic VEs. New bounds for linear matrix-inclusion random composites To derive new bounds for the effective properties, a key point is to notice that the shape and volume of the non-square VEs B d ðxÞ derived by the methodology described in the previous section depend on the realization x. Again, because of the stationarity of the random set B, the spatial dependence on X is not specified unless required. Accordingly, all the definitions given in Section 2.1 for deterministic domains X New bounds for linear matrix-inclusion random composites Derivation of (22): For the sake of simplicity, the proof of Eq. (22) is given for a Voronoï subdivision of the microstructure knowing that it still works for more general partitions such as the ones described in the previous section. Similarly to Huet's approach, the VE B d;X ðxÞ of size d and center X is subdivided into four disjoined sub-VEs New bounds for linear matrix-inclusion random composites In view of the assumed stationarity of the material, New bounds for linear matrix-inclusion random composites Averaging inequality (23) and making use of relations (24) yields 8d; New bounds for linear matrix-inclusion random composites Since Eq. (8) implies New bounds for linear matrix-inclusion random composites we have New bounds for linear matrix-inclusion random composites By the same reasoning but using the complementary energy principle applied to problem (6) now defined on B d ðxÞ for UTBC, the following new hierarchy of upper bounds for the effective compliance is obtained New bounds for linear matrix-inclusion random composites Combining Eq. (27) with (28) leads to (22), thus ending the proof. In what follows, for ease of notation, we make use of the definitions New bounds for linear matrix-inclusion random composites for the Voronoï C VORþ=À d type upper and lower bounds. Statistical estimation The ensemble averages used in the new bounding theorem (22) differ from the ones employed in Huet's bounds (14) since a weight Statistical estimation where the subscript n means that the expectation is performed by using n realizations of the couple (C Statistical estimation and thus provides another estimate of the ensemble average in Eq. Statistical estimation . From a theoretical viewpoint, this second approach is more relevant than the first one since we are left with classical expectations where each realization has the same weight. However, from a practical point of view, the expectation, the standard deviation as well as the distribution function of the random variables C Application: transverse response of random linear composites The bounding theorem (22) presented in Section 3.2 holds for two or three dimensional matrix-inclusion random composites. However, in order to reduce the numerical cost required to compute the apparent behaviors for each realization, we consider as a first illustration the case of a composite made of a matrix and aligned fibers which can be addressed by means of low cost 2D numerical simulations. For such composites, it becomes possible to compute the ensemble averages of the apparent behaviors by making use of at least 1000 different realizations for each VE size d with standard computational power. Material description More precisely, the studied material is a two-phase composite consisting of a matrix and aligned identical cylindrical fibers randomly and isotropically distributed in the transverse plane. Each phase is assumed to be isotropic and to follow Hooke's law. The relative accuracy In order to simplify the computations, we only consider the transverse response of the composite. The local problem to be solved can thereby be reduced in plane strain framework to a two-dimensional problem for which the fibers are represented by disks of same diameter d and the VEs are subjected to in-plane loading. Due to the transversally isotropic property of both the effective behavior and its Voronoï type bounds, their in-plane behavior is fully characterized by their in-plane bulk and shear moduli. Microstructure generation The microstructure of the material is generated by the following way. The centers of the disks are randomly implemented by a Poisson process in a large window of size 125d containing approximatively 3000 fibers for f ¼ 15%. A constraint on the minimal distance between the disks of same diameter d is imposed to prevent overlapping and contact. In this process, the disk are generated sequentially: if the current disk the position of which is randomly chosen in the large window does not satisfy the minimal distance requirement, it is rejected and a new one is generated until the distance condition is fulfilled. New disks are added until the prescribed porosity is reached. The overall inclusion volume fraction can be monitored precisely for the size of the microstructure BðxÞ is much larger than the fibers diameter d. Since Finite Element calculations are used in this work to compute the apparent behaviors, a strictly positive minimal distance between the disks of 3:5%d (cherry pit model) has been prescribed to prevent mesh difficulties. It is well known that such a way to generate the microstructure -which is a slightly modified version of the well-known random sequential adsorption (RSA) algorithm -is not able to deal with volume fractions greater than about 35% in 2D. To generate matrix-inclusion microstructures with higher volume fractions, it is needed to use more sophisticated algorithm such as the simulated annealing algorithm (Gusev, 1997; Rintoul and Torquato, 1997) . Such an implementation is left for future work. Lastly, it should be noted that the stochastic process used to generate the microstructure of the material -i.e the modified Poisson process sustaining the modified RSA algorithm -is unique and thoroughly characterized. Accordingly, the effective properties as well as the VOR bounds associated with this specific and completely characterized microstructure are uniquely defined and, as a consequence of Eq. (8), have the same values when d ! þ1. Voronoï subdivision and non-square VEs generation From the large microstructure generated in Section 4.2, a Voronoï subdivision is performed by using a Matlab function. Then, similarly to Section 3.1, n square windows of same size L ¼ dd are extracted from this subdivision and are used to generate n nonsquare VEs made of the set of elementary Voronoï cells the centers of which belong to a same d-sized window. The n square windows of same size d are extracted sequentially by means of a Poisson process providing the centers of the square windows. In order to prevent windows overlapping and possible statistical bias, a minimal distance of 2d between the centers is imposed. Similarly to the implementation of the disk centers, if the current windows center does not satisfy the minimal distance requirement, it is rejected and a new one is generated until the distance condition is fulfilled. Ideally, a single very large microstructure is enough to extract n windows of same size d and generate n non-square VEs. However, the computational time to perform a Voronoï subdivision by Matlab seems to follow an exponential growth function of the size of the microstructure, thus leading to prohibited computational times for very large microstructures. In practice, as mentioned in the previous section, large microstructures of size 125d are generated. For each of these microstructures, mðdÞ square windows are extracted. The parameter mðdÞ is adjusted numerically for each size d once for all: for instance mð2Þ ¼ 25; mð8Þ ¼ 5 and for d larger than 20 mðdÞ ¼ 1. The number MðdÞ of generated microstructures required to built n non-square VEs is given by MðdÞ ¼ bn=mðdÞc þ 1 where bxc denotes the integer part of x. Such a process is performed for each size d considered. For numerical applications, 1000 6 n 6 2000. The generation of both the microstructures and the non-square VEs is performed by means of C++ subroutines. Computation of the apparent behavior 4.4.1. local problem Once the non-square VEs B d ðxÞ are generated, the apparent behaviors for ADBC and UTBC are computed. Since we focus on the transverse response of the composite, the local problem to be solved can be dealt with as a 2D problem in the plane strain framework. Computation of the apparent behavior 4.4.1. local problem In plane strain, second-order tensor fields such as the strain and in-plane components of the stress fields admit the decomposition e ¼ e m i þ e ps e ps þ e ss e ss ; r ¼ r m i þ r ps e ps þ r ss e ss ð33Þ where the second-order tensors i; e ps and e ss are given on the considered cartesian basis by i ¼ e 1 e 1 þ e 2 e 2 ; e ps ¼ e 1 e 1 À e 2 e 2 ; e ss ¼ e 1 e 2 þ e 2 e 1 : Computation of the apparent behavior 4.4.1. local problem The fourth-order tensor J T is the usual spherical projector in the transverse plane while the tensors E ps ; E ss introduced by Willot et al. (2008) are projectors on the subspace of deviatoric second-order tensors spanned by e ps and e ss , respectively. The remaining tensors E (35) in-plane hydrostatic, pure shear and simple shear homogeneous stress components, respectively again in the plane strain framework. Any numerical scheme able to solve the local problem (35) can be employed to compute the apparent behaviors. In the sequel, the FE method will be used. . Exact expressions As explained in Section 4.1, due to rotational invariance along the fiber axis, the effective behavior C eff and the ensemble average of the apparent behavior associated with non-square VEs, i.e. C VORþ=À d . Exact expressions , are transversally isotropic. Accordingly, they read . Exact expressions Formulae (41) Approximate expressions The evaluation of the bounds (40) requires to compute the Approximate expressions In practice, only estimates of the exact value of these ensemble averages are computed according to definition (30). However, the obtained results are not strictly bounds of the effective behavior. They are only numerical approximations of the exact bounds and their accuracy increases with the number of realizations n. The numerical accuracy of these approximations can be quantified by means of the notion of confidence interval (CI). If we consider a random variable XðxÞ and the unbiased estimatorsXðxÞ ¼ Approximate expressions where Approximate expressions Such an interval can be seen as a particular realization of the random interval CI n 1Àa ðEðXÞÞðxÞ (Casella and Berger, 2001 ). Thus, the length tða;nÞSn ffiffi n p is a probabilistic characterization of the relative accuracy of the estimated expectation value X n . The result (42) is theoretically established only in the case of unbiased estimates X n and S 2 n which are defined by giving the same weight to each realization. In the numerical applications, we consider an extension of (42) where X n and S n are now defined by means of different weight Approximate expressions where Approximate expressions where ' Approximate expressions CI ðEðXÞÞ denotes the half length of the (1 À a)-confidence Approximate expressions Note that the ensemble averages X used in Eq. (44) should be understood as weighted ensemble averages as defined in (43) when K þ d;est and K À d;est are derived by means of the Voronoï type approach whereas the same notation X corresponds to the classical equalweighted ensemble average when associated with the classical Huet's approach. The same holds true for the estimate S of the standard deviation. Specificities associated with square VEs For square VEs, the ensemble averages of the apparent behavior, i.e. C SCþ=À d , exhibit quadratic symmetry unlike the bounds C VORþ=À which are (almost) transversally isotropic. Indeed, although we still have rotational invariance along the fiber axis e 3 of both the constitutive relations of the phases and the phase distribution (when averaging over the whole set of realizations), the shape of each square VE whose sides are parallel to the reference axes e 1 ; e 2 leads to a quadratic symmetry with respect to e 1 ; e 2 . Therefore, Specificities associated with square VEs In Eq. (47) as well as in all the expressions associated with the Huet's SC type bounds, the expectations should be understood in a classical sense, i.e. with a same weight ascribed to each realization. Applying pure shear (e ps ) or simple shear (e ss ) macroscopic strain e and stress r loadings to Eq. (14) and making use of relations (47), the following bounds for the in-plane effective shear modulus are obtained sup k Specificities associated with square VEs To the best of the authors knowledge, the bounding property (48) which makes use of the quadratic anisotropy of C SCþ=À d Specificities associated with square VEs has never been used nor noticed before. Indeed, the Huet's upper and lower bounds have been until now considered to be isotropic in the literature. This assumption of isotropy was supported by various numerical simulations performed for some range of parameters such as for instance sufficiently large VE d size or finite contrast. Although we find again in this study that Huet's bounds are isotropic for some simulations -e.g. for reinforced composites, the lower bound is nearly isotropic (up to 1% of accuracy) -their quadratic anisotropy Eq. (46) which is theoretically established at the beginning of this section by symmetry arguments is well confirmed by other numerical simulations. For instance, for rigidly reinforced composites the numerical estimate of the Huet's upper boundevaluated for a inclusion volume fraction f I ¼ 15% -provides a quadratic behavior (k At last, the evolution of Huet's SC shear upper bounds can be predicted by a simple approach inspired from the one presented in Section 2.3 and applied to the in-plane bulk modulus. When this latter approach (Eq. (19) ) is also applied to the in-plane pure-shear modulus, thus leading to k . The reason lies in the fact that the simple-shear uniform strain loading is parallel to the boundaries of the square cell. For such a loading, the nearly rigid particles which intersect the edges of the cell are no longer strained but in fact rotate inside the matrix unlike the case of pure-shear uniform strain solicitations which enforce actual deformations to the boundary particles. Therefore, these particles do not generate the blow-up of the strain energy observed for l SCþ d when simple-shear uniform strain are applied. However, there is one exception to this situation: the particles which intersect one of the four corners of a square VE -more precisely the particles which intersect two different edges -are always deformed even when submitted to simple shear loading. Although not many, these at most 4 nearly rigid particles can explain the blow-up of the strain energy under simple shear deformation. To see if such a conjecture holds true, we make again the reasoning on which relies on the simple approach presented in Section 2.3. The part of the strain energy due to the particles which intersect the cell corners is taken into account by means of a new subdivision of the square VE in two parts. The first part B is the core of the VE, i.e. the part inside which the nearly rigid particles are not enforced to be strained by ADBC in simple shear (see Fig. 5 ). From this subdivision, the ratio V ð1Þ d =V d is now proportional to 1=d 2 instead of ð1=d À 1=d 2 Þ as it was the case for the simple model associated with a pure-shear uniform strain. By reproducing the reasoning introduced in Section 2.3, we get the following closed-form approximate bound for the effective shear modulus Specificities associated with square VEs As shown in Fig. 8 , the agreement between the approximate and Huet's bounds after a fitting of the constant l ð2Þ at d ¼ 30 is quite good. This shows that the blow-up of the strain energy under simple shear uniform strain solicitations is due to the predominant contribution of the nearly rigid particles intersecting the cell corners to the apparent behavior. Numerical implementation For both the Voronoï and Square Cells approach, the apparent behaviors are computed by making use of the FE software CAST3M (CEA, 2003) according to the procedure presented in Section 4.4. However, the meshing is performed differently according to the shape (square or non-square) of the VEs. Proposed approach For the Voronoï approach, a meshing procedure based on the following principles has been defined. First, we make use of the Voronoï subdivision on which rely the non-square VEs introduced in Section 3.1. All Voronoï cells of a given non-square VE are individually meshed and then gathered together to rebuilt the whole VE, following a methodology very similar to the one introduced by Danielsson et al. (2007) . The meshing of each Voronoï cell is easily performed by making use of the standard meshing command of CAST3M. The way to mesh a Voronoï cell mainly relies on the following considerations. The first one aims at providing accurate estimates of the local fields even in areas where they are subjected to strong gradients, especially the ones located between close inclusions. This can be easily realized by partitioning each Voronoï cell into isosceles triangular subcells the apices of which consist of both the center and two adjacent apices of the Voronoï cell. For the whole Voronoï cells, we ascribe to the matrix part of each Voronoï subcell the same finite element discretization number along the radial directions of its inclusion. Such a procedure leads to fine (resp. crude) radial meshes inside the matrix phase for each triangular Voronoï subcell the inclusion of which is close (resp. far) to its outer boundary. Note that the fact for an inclusion of a Voronoï cell to be close (resp. far) from one segment of its outer boundary amounts to being close (resp. far) from the inclusion of the adjacent Voronoï cell sharing the same segment as boundary. The second consideration is based on the Eshelby solution of a heterogeneous inclusion inside an infinite matrix, both phases being linear elastic (Eshelby, 1957) . The local strain associated with this problem is heterogeneous in the matrix. Its spatial variations although quite significant near the matrix-inclusion interface decrease quickly in the radial direction of the inclusion and the strain field becomes homogeneous far enough of the inclusion. In view of the radial decrease of the local strain solution of the Eshelby problem inside the matrix, we build a mesh whose ascribed densities vary in inverse proportion to the distance from the borders of the inclusions. Similarly, as the strain is homogeneous inside the inclusion of the Eshelby problem, the density of the mesh is reduced in the center of the inclusions. Finally, in order to make compatible the meshes of two adjacent cells, the discretization along a segment of a Voronoï cell relies on a number of elements which depends only on the length of this segment and the distance to the center of the cell, which is the same for adjacent cells. A detailed explanation of the meshing algorithm is given in Salmi (2011) . Fig. 6(a) provides an example of obtained mesh for a non-square VE consisting of two Voronoï cells. Proposed approach In order to quantify the numerical errors generated by the proposed numerical scheme, we studied the influence of the number of finite elements on the apparent behaviors associated with ADBC and UTBC. The numerical study is performed on a non-square VE of size d ¼ 12 comprised of 27 rigid inclusions (c ¼ 10 4 Þ in proportion f I ¼ 15% and submitted to ADBC or UTBC, respectively. Eightnoded quadratic elements are used to carry out the calculation. The results are depicted in Fig. 6(b Fig. 6 (b) are computed for n 1 e ¼ 25000 since it has been numerically noticed that these moduli do not evolve any longer beyond. In the following, all the computations are carried out with a mesh size corresponding to a relative precision of 0:1%. Classical approach For the classical Square Cell approach, the geometries of the VEs are more complex because of the inclusions intersecting the VE boundaries. The standard routines in CAST3M turn out not to be appropriate to generate such meshes. That's why the specialized mesh generator Netgen (Schoberl et al., 2003) specifically designed for random microstructures is preferred and the obtained meshes are exported to CAST3M to compute the apparent behaviors with the same procedures as for the new bounds. It should however be noticed that this generic mesh procedure is not optimized with respect to the particular microstructure under consideration. It generates meshes with almost uniform densities, and, as a consequence, much larger numbers of elements for a given accuracy. This is illustrated by the results depicted in Fig. 6(c) which gives the evolution of the relative accuracy for a square VE, again of size d ¼ 12 and made of rigid inclusions in proportion f I ¼ 15% for ADBC or UTBC, respectively. The results are now computed with n 1 e ¼ 80000. One can observe that to reach a relative accuracy of 0:1% with Netgen requires to use five times more elements than with the Voronoï-type mesh. This shows that the new bounding procedure is not only more efficient from a theoretical point of view, but also from a numerical one as it allows to generate meshes in a much easier and more efficient way. Fig. 7 . For all the curves reported in this paper, the error bars refer to the symmetric confidence inter-val of the considered expectation. As shown in Fig. 7(a) Fig. 7(b) Results Numerical For the rigidly reinforced composite, it should be noted that the pure-shear and simple-shear SC lower bounds are the same unlike their corresponding upper bounds which exhibit a pronounced quadratic anisotropy. The classical
{'paper_id': '137399133', 'title': 'Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior', 's2_url': 'https://api.semanticscholar.org/CorpusID:137399133', 'source': 'S2ORC'}