A Three-Dimensional Statistical Micromechanical Theory for Brittle Solids with Interacting Microcracks

Abstract

A three-dimensional statistical micromechanical theory is presented to in vestigate effective elastic moduli of brittle solids with many randomly located, penny- shaped microcracks. The macroscopic constitutive relations are statistically and microme chanically derived by taking the ensemble average over all possible realizations which feature the same statistical distribution of microcracks. Approximate analytical solutions of a two-microcrack interaction model are presented to account for pairwise microcrack interaction among many randomly located, aligned microcracks. Therefore, the ensemble- averaged stress perturbations due to microcrack interaction can be constructed in closed- form. The overall effective compliances of microcrack-weakened brittle solids are derived by further taking the volume average of the ensemble-averaged stress-strain relations over the entire mesostructural domain of a representative volume element. Some numerical ex amples are given to illustrate the behavior of the proposed method. Comparison with some existing methods is also appended. Finally, a higher-order ensemble-average formulation of microcrack interaction is briefly discussed. The proposed framework is fundamentally different from existing "effective medium methods" which do not depend on microcrack locations and configurations. It is emphasized that no Monte Carlo simulations are neces sary in the proposed framework.