A micromechanics-based nonlocal model for isotropic composites containing non-spherical inclusions

A generalization of the Hashin-Shtrikman variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for random linear elastic composite materials. The analysis builds on previous works by the second author, who derived completely explicit results for the case of an isotropic matrix containing a random distribution of isotropic, non-overlapping identical spheres. Here, it is shown how to derive an explicit nonlocal constitutive equation for a matrix containing a random distribution of non-spherical voids, cracks or inclusions. The model of impenetrable particles considered consists of identical particles with fixed spheroidal shape and random orientation. A convenient statistical description of the microstructure is obtained by placing the particles within security spheres. This restricts the analysis to composites having macroscopically isotropic behavior. A new approach is applied to separate the effects of inclusion shape and spatial distribution. Completely explicit expressions for terms related to inclusion shape are derived for the case of spheres; whereas terms describing effects of spatial distribution are obtained explicitly for both a standard and an improved statistical model. Finally, the new constitutive equation is used to estimate the minimum size of a material volume element over which standard local constitutive equations provide a sensible description of the constitutive response of a matrix reinforced/weakened by spherical particles/voids, when spatially-varying average strain is considered.