Concentration versus Oscillation Effects in Brittle Damage

This work is concerned with an asymptotic analysis, in the sense of Gamma-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order epsilon, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at epsilon fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as epsilon -> 0, concentration and saturation of damage are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at epsilon fixed) to being of linear-growth type (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. In particular, the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far. We explicitly identify the Gamma-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky-plasticity type. We further consider the regime where the divergence remains square-integrable in the limit, which leads to a Tresca-type model. (c) 2020 Wiley Periodicals LLC

Automated summary

This work focuses on the asymptotic analysis, in the sense of Gamma-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study reveals that concentration and saturation of damage are favored as epsilon tends to zero, resulting in a degeneration of the growth of the elastic energy. The interaction of homogenization effects with singularity formation requires new analysis methods.