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

In this paper we consider a family of quasi-static evolution problems involving oscillating energies epsilon(epsilon) and dissipations D-epsilon. Even though we have separate Gamma-convergence of epsilon(epsilon) and D-epsilon, the Gamma-limit F of the sum does not agree with the sum of the Gamma-limits. Nevertheless, 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 epsilon(epsilon) and 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.