would be somewhat random. Therefore, defining a unit cell model containing one or a few particles may be an over simplification of the problem. To accurately model such solids, a unit cell containing many particles should be analyzed. Thus, one needs to build and carry out analyses for a unit cell model having tens and hundreds of particles. Finite element model, which is required for such analyses, would be gigantic. Generating the finite element model as well as solving the problem would pose many problems. However, mechanical interactions between the particles, and between matrix material and particles are fully accounted for, when the finite element method is adopted. On the other hand, methodologies in micromechanics, such as self-consistent method [9,13,14] and Mori-Tanaka theory [9,15] have been presented. For particulate composites, Eshelby’s tensor [8,9], takes a central role (see, for example, [9,16,17]). These methodologies have advantages over the finite element method such that the effective mechanical behaviors of composites can be estimated in a closed or semi-closed form, based on the elastic moduli of matrix and particles, the distribution, orientation and volume fraction of the particles. Large scale computations are not required. However the methods in micromechanics do not account for detailed mechanical interactions between the distributed particles. The mechanical interactions may have significant roles when the particles are densely distributed or when we attempt to account for the damage evolution or meso-structural changes, such as phase transformation of the particles. Boundary element based homogenization formulation, which is developed in this paper, have advantages of both the above mentioned methodologies. Since the method is based on the boundary element method (BEM [1-6]), detailed mechanical interactions between all the material constituents can be accounted for. This nature is similar to that of the finite element method. A unit cell modeled by the boundary element method can contain many particles and the computation is simplified by using analytical solutions for ellipsoidal inclusions, such as Eshelby’s tensor [8,9]. In this regard, proposed method is similar to the methodologies in micromechanics. HOMOGENIZATION FORMULATION FOR PARTICULATE COMPOSITES In this section, equation formulations for boundary element based homogenization analysis for particulate composites are briefly discussed (see [18,19] for homogenization method based on BEM). In homogenization method, the microstructure of solid is assumed to be spatially periodic, as shown in Fig. 2. A unit of the periodic microstructure is modeled by the boundary element method and is called “unit cell”. As shown in Fig.1 in a two dimensional illustration, the size of the unit cell is represented by e . We introduce two different coordinate systems. One is ix coordinate system, which is fixed in space and the other is iy coordinate system, which is scaled by the size e of unit cell, as: i i i c y x = +e (1) where ic are the components of an arbitrary vector. Displacements iu are expressed by the two scale asymptotic expansion, by following Guedes and Kikuchi [10], as: ( ) ( ) ( ) ( ) ( ) L+ + + + = x, y x,y x,y x,y x, y 3 3 2 2 1 i i i o i i u u u u u e e e (2) For small deformation linear elasticity problem, Hooke’s law and the equation of linear momentum balance are written to be: l l x u E k ijk ij ¶ ¶ s = , 0 + = j i ij b x¶ ¶s (3) e y 1 y 2 Unit cell representing a unit of periodic microstructure Global solid composed of N unit cells (a) (b) Y M Y * Figure 2: Two dimensional illustration for a solid whose microstructure is periodic and its unit cell.
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