Asymptotic homogenization analysis for damage amplification due to singular interaction of micro-cracks

The paper investigates the overall damage amplification effect due to micro-crack interaction in a framework of two-scale modeling. A homogenization method based on asymptotic expansions is employed to deduce the macroscopic damage equations. The damage model completely results from energy-based micro-crack propagation laws. We consider a locally periodic microstructure with periods containing pairs of micro-cracks separated by small ligaments. The asymptotic solution in the ligament region allows the study of the effect of micro-crack interaction on the effective coefficients. The local macroscopic response expresses the collective coalescence of a periodic microstructure with interacting micro-cracks. We show that the slope of the homogenized coefficients is inversely proportional to the square root of the distance between the tips of the interacting micro-cracks, accounting for the singularity in the stress fields as the micro-cracks approach each other. This leads to damage amplification as the result of the interaction of micro-cracks.