A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities

Abstract

A micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for a matrix containing a random distribution of randomly oriented spheroidal voids or inclusions is derived. The analysis employs J.R. Willis’ generalization of the Hashin–Shtrikman variational formulation to random linear elastic composite materials and builds on that of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359), who derived completely explicit results for the case of isotropic, nonoverlapping identical spherical inclusions/voids. The model of impenetrable particles employed consists of identical particles with fixed spheroidal shape and random orientation. To facilitate a manageable statistical description, the spheroids are placed within concentric hard “security” spheres. The paper derives three main new results: (i) it is proved within the assumptions just outlined that the effects of inclusion shape and their spatial distribution are separable, for arbitrary inclusion shape (not just spheroids) and arbitrary spatial (statistical) distribution of their security spheres when employing up through two-point statistical information; (ii) closed-form analytical results are obtained from the Verlet–Weis improvement of the Percus–Yevick–Wertheim statistical model of a random distribution of nonoverlapping spherical particles/voids, leading to substantial improvements at higher inclusion/void volume fractions in the nonlocal constitutive equations of Drugan and Willis (1996) and Drugan (2000); (iii) approximate analytical nonlocal constitutive equations are derived for composites consisting of a matrix containing randomly oriented oblate or prolate spheroidal inclusions/voids, using the Verlet–Weis statistical model for the security sphere distribution. Among the specific implications of these new results, it is found that the minimum representative volume element (RVE) size estimate for composites containing spherical inclusions/voids using the Verlet–Weis improvement is significantly larger at higher inclusion/void volume fractions (≈0.3–0.64) than the estimates of Drugan and Willis (1996) and Drugan (2000), who used the Percus–Yevick–Wertheim model. Also, deviations in inclusion/void shape from spherical are shown to cause significant modifications to the nonlocal constitutive equations, as evidenced by nontrivial changes in predicted minimum RVE sizes.