Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites

Abstract

A generalization of the Hashin–Shtrikman variational formulation to random composites, due to J.R. Willis, is employed to derive micromechanics-based variational estimates for a higher-order nonlocal constitutive equation relating the ensemble averages of stress and strain, for a class of random linear elastic composite materials. We analyze two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), within a formulation accounting for one- and two-point probabilities, and derive an explicit nonlocal constitutive equation that includes terms up through the fourth gradient of average strain. The analysis is carried out first for an arbitrary comparison medium. Then, a new approach is outlined and applied which employs the nonlocal correction to determine the optimal choice of comparison medium, and hence the optimal effective modulus tensor (as well as the optimal tensor coefficients of the nonlocal terms) for the amount of statistical information employed. The new higher order analysis provides a highly accurate nonlocal constitutive equation, valid down to quite small volume size scales and to rather strong variations of average strain with position. Among several applications illustrated, it permits accurate analytical assessment of the remarkably small predictions derived by Drugan and Willis (1996. Journal of the Mechanics and Physics of Solids 44, 497–524) of the minimum representative volume element (RVE) size needed for accuracy of the standard constant-effective-modulus macroscopic constitutive equation for elastic matrix-inclusion composites that have spherical inclusions/voids. It also affords an analytical assessment of the improved (i.e., reduced) minimum RVE size scale, compared to a standard constant-effective-modulus constitutive equation, to which the leading-order nonlocal constitutive equation derived by Drugan and Willis applies. This improvement is shown to be dramatic in some example cases.