On micromechanics-based nonlocal modeling of elastic matrices containing non-spherical heterogeneities

A micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for a matrix containing a random distribution of non-spherical voids, cracks or inclusions but having macroscopically isotropic behavior is derived. The model of impenetrable particles considered consists of identical particles with fixed spheroidal shape and random orientation. It is shown how the effects of inclusion shape and spatial distribution can be separated. Terms related to inclusion shape reduce to certain intergrals which can be evaluated analytically only in special cases. Terms describing effects of spatial distribution can be obtained explicitly for different statistical models, within the framework of up through two-point statistics. As verification of the formulation, completely explicit expressions are derived for the limiting case of spherical inclusions and for a standard statistical model on the basis of results found in the literature. The new constitutive equation can be used to produce quantitative estimates of the minimum size of a material volume element over which standard local constitutive equations provide a sensible description of the macroscopic constitutive response of the material.