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 EεEε and dissipations DεDε. Even though we have separate Γ  -convergence of EεEε and DεDε, the Γ  -limit FF of the sum does not agree with the sum of the Γ  -limits. Nevertheless, FF 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 EεEε and DεDε. 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.