A micromechanics-based nonlocal constitutive equation and minimum RVE size estimates for random elastic composites containing aligned spheroidal heterogeneities

Abstract

An explicit micromechanics-based nonlocal constitutive equation is derived that relates the ensemble averages of stress and strain for macroscopically transversely isotropic elastic composites consisting of a matrix containing a random distribution of aligned impenetrable spheroidal inclusions or voids. The analysis builds on and generalizes previous models developed by the authors [Drugan, W.J., Willis, J.R., 1996. J. Mech. Phys. Solids 48,1359–1387; Monetto, I., Drugan, W.J., 2004. J. Mech. Phys. Solids 52, 359–393) wherein elastic composites containing spherical and randomly oriented spheroid-shaped inclusions were analyzed. One advance involves derivation of a closed-form statistical characterization of the aligned spheroidal microstructure that is valid for the full physically attainable range of inclusion volume fractions. This was accomplished by applying a simple scale transformation to a sensible model of the microstructure, without the use of rigid “security spheres”. An explicit micromechanics-based nonlocal constitutive equation is derived by employing this statistical characterization. This constitutive equation is then employed to explore the nonlocal effects of both shape and orientation of inclusions on the macroscopic response of the composite. Finally, quantitative estimates of the characteristic internal length of the random material are derived for the extreme cases of spheroidal voids and rigid particles; this length is indicative of the minimum representative volume element (RVE) size needed for a standard local composite constitutive equation sensibly to characterize the composite's overall response. Among the results, it is found that this minimum volume element size must be substantially larger for accurate description of composites containing aligned stiff spheroids as compared with those containing randomly oriented spheroids or spherical particles.