A MICROMECHANICS-BASED NONLOCAL CONSTITUTIVE EQUATION AND ESTIMATES OF REPRESENTATIVE VOLUME ELEMENT SIZE FOR ELASTIC COMPOSITES

A variational formulation is employed to derive a micromechanics-based , explicit nonlocal constitutive equation relating the ensemble averag es of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistica lly uniform distribution of phases (which themselves may have arbitrar y shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term pr oportional to the second gradient of the ensemble average of strain. T his nonlocal consitutive equation is derived in explicit closed form f or isotropic material in the one case in which there exists a well-fou nded physical and mathematical basis for describing the material's sta tistics: a matrix reinforced (or weakened) by a random dispersion of n onoverlapping identical spheres. By assessing, when the applied loadin g is spatially-varying, the magnitude of the nonlocal term in this con stitutive equation compared to the portion of the equation that relate s ensemble average stresses and strains through a constant ''overall'' modulus tensor, we derive quantitative estimates for the minimum repr esentative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous ''effective modulus'' constituti ve models for composites can be expected to apply. Remarkably, for a m aximum error of 5% of the constant ''overall'' modulus term, we show t hat the minimum RVE size is at most twice the reinforcement diameter f or any reinforcement concentration level, for several sets of matrix a nd reinforcement moduli characterizing large classes of important stru ctural materials. Such estimates seem essential for determining the mi nimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mec hanics theory for composites. Finally, we relate our nonlocal constitu tive equation explicitly to the ensemble average strain energy, and sh ow how it is consistent with the stationary energy principle.