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 Eε and dissipations Dε . Even though we have separate Γ -convergence of Eε and Dε , the Γ -limit F of the sum does not agree with the sum of the Γ -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 Eε and Dε . This result contributes to the analysis of the interaction between Γ-convergence and variational evolution, which has recently attracted much interest both in the framework of energetic solutions and in the theory of gradient flows.