JournalsaihpcVol. 33, No. 2pp. 309–328

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

  • Andrea Braides

    Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, via della Ricerca Scientifica 1, 00133 Roma, Italy
  • Biagio Cassano

    Dipartimento di Matematica ‘G. Castelnuovo’, ‘Sapienza’ Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
  • Adriana Garroni

    Dipartimento di Matematica ‘G. Castelnuovo’, ‘Sapienza’ Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
  • David Sarrocco

    Dipartimento di Matematica ‘G. Castelnuovo’, ‘Sapienza’ Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
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Abstract

In this paper we consider a family of quasi-static evolution problems involving oscillating energies Eε\mathscr{E}^{\varepsilon } and dissipations Dε\mathscr{D}^{\varepsilon }. Even though we have separate Γ-convergence of Eε\mathscr{E}^{\varepsilon } and Dε\mathscr{D}^{\varepsilon }, the Γ-limit F\mathscr{F} of the sum does not agree with the sum of the Γ-limits. Nevertheless, F\mathscr{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\mathscr{E}^{\varepsilon } and Dε\mathscr{D}^{\varepsilon }. 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.

Cite this article

Andrea Braides, Biagio Cassano, Adriana Garroni, David Sarrocco, Quasi-static damage evolution and homogenization: A case study of non-commutability. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 309–328

DOI 10.1016/J.ANIHPC.2014.10.003