On micromechanics-based nonlocal modeling of elastic matrices weakened by voids

Within the mathematical modeling of the mechanical behavior of elastic composites, classical approaches usually treat the material as being macroscopically homogeneous with constant overall properties and develop constitutive equations relating suitable averages of microscopic stress and strain fields over a Representative Volume Element (RVE). As is well known, the application of such local approaches is limited to RVEs sufficiently large compared to the characteristic microstructural length of the material. In this paper the attention is focused on a specific class of elastic two-phase (matrix – heterogeneities) composites consisting of a matrix weakened by a random distribution of voids. The principal objective is to analyze the nonlocal effects of the internal microstructure of such weakened (“damaged”) materials (shape, orientation and distribution of voids) on their macroscopic mechanical properties. In order to do this, the accuracy of a standard local constitutive equation is analyzed by comparison with a nonlocal constitutive equation. From this comparison, quantitative estimates of the minimum RVE size of composite material for which the local model is sensible are also obtained. The formulation of nonlocal modeling here followed is that first developed by the second author together with Willis and more recently generalized by both the authors. Employing a generalization of the Hashin-Shtrikman variational formulation, these authors derived two micromechanics-based, completely explicit nonlocal constitutive equations relating the ensemble averages of stress and strain for random linear elastic two-phase composite materials consisting of an isotropic matrix containing isotropic heterogeneities. In particular, explicit results for the case of isotropic random distributions of heterogeneities (non-overlapping identical spheres and randomly-oriented spheroid-shaped inclusions/voids) resulting in isotropic macroscopic behavior were derived and further generalized to the case of transversely-isotropic distributions of heterogeneities (non-overlapping identical aligned spheroid-shaped inclusions/voids) resulting in more general transversely-isotropic macroscopic behavior. In order to apply the preceding theory to a wider range of practical applications for composites under consideration, including matrices weakened by voids of any shape, the preceding formulation is here specialized to the case of penetrable heterogeneities. From the comparison of previous results with the new set of results obtained, interesting conclusions will be drawn on the nonlocal effects of microstructure on the constitutive response of “damaged” materials.