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. (C) 2014 Elsevier Masson SAS. All rights reserved