Effective Elastic Moduli of Two-Dimensional Brittle Solids With Interacting Microcracks, Part I: Basic Formulations

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Abstract
Statistical micromechanical formulations are presented to investigate effective elastic moduli of two-dimensional brittle solids with interacting slit microcracks. The macroscopic stress-strain relations of elastic solids with interacting microcracks are micromechanically derived by taking the ensemble average over all possible realizations which feature the same material microstructural geometry, characteristics, and loading conditions. Approximate analytical solutions of a two-microcrack interaction problem are introduced to account for microcrack interaction among many randomly oriented and located microcracks. The overall elastic-damage compliances of microcrack-weakened brittle solids under uniaxial and biaxial loads are also derived. Therefore, stationary statistical micromechanical formulation is completed. Moreover, some special cases are investigated by using the proposed framework. At variance with existing phenomenological continuum damage models, the proposed framework does not employ any fitted “material parameters. ” “Cleavage 1” microcrack growth and “evolutionary damage models” within the proposed context will be presented in Part II of this series. It is emphasized that microstructural statistical informations are already embedded in the proposed ensemble-averaged equations and, therefore, no Monte Carlo simulations are needed.

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