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This paper addresses the question of the interplay between relaxation andirreversibility through quasi-static evolutions in damage mechanics, byinquiring the following question: can the quasi-static evolution of an elasticmaterial undergoing a rate-independent process of plastic deformation bederived as the limit model of a sequence of quasi-static brittle damageevolutions? This question is motivated by the static analysis performed byBabadjian-Iurlano-Rindler, where they have shown how the brittle damage modelintroduced by Francfort and Marigo can lead to a model of Hencky perfectplasticity. Problems of damage mechanics being rather described throughevolution processes, it is natural to extend this analysis to quasi-staticevolutions, where the inertia is neglected. We consider the case where themedium is subjected to time-dependent boundary conditions, in theone-dimensional setting. The idea is to combine the scaling law considered byBabadjian-Iurlano-Rindler with the quasi-static brittle damage evolutionintroduced by Francfort and Garroni, and try to understand how theirreversibility of the damage process will be expressed in the limit evolution.Surprisingly, the interplay between relaxation and irreversibility is notstable through time evolutions. Indeed, depending on the choice of theprescribed Dirichlet boundary condition, the effective quasi-static damageevolution obtained may not be of perfect plasticity type.

Perfect plasticity versus damage: an unstable interaction between irreversibility and $\Gamma$-convergence through variational evolutions