Quasi-static damage evolution and homogenization: A case study of non-commutability

Abstract

In this paper we consider a family of quasi-static evolution problems involving oscillating energies ${\mathcal{E}}^{\epsilon }$ and dissipations ${\mathcal{D}}^{\epsilon }$. Even though we have separate $\Gamma$-convergence of ${\mathcal{E}}^{\epsilon }$ and ${\mathcal{D}}^{\epsilon }$, the $\Gamma$-limit $\mathcal{F}$ of the sum does not agree with the sum of the $\Gamma$-limits. Nevertheless, $\mathcal{F}$ can still be viewed as the sum of an internal energy and a dissipation, and the corresponding quasi-static evolution is the limit of the quasi-static evolutions related to ${\mathcal{E}}^{\epsilon }$ and ${\mathcal{D}}^{\epsilon }$. This result contributes to the analysis of the interaction between $\Gamma$-convergence and variational evolution, which has recently attracted much interest both in the framework of energetic solutions and in the theory of gradient flows.