  This paper addresses the question of the interplay between relaxation and
irreversibility through quasi-static evolutions in damage mechanics, by
inquiring the following question: can the quasi-static evolution of an elastic
material undergoing a rate-independent process of plastic deformation be
derived as the limit model of a sequence of quasi-static brittle damage
evolutions? This question is motivated by the static analysis performed by
Babadjian-Iurlano-Rindler, where they have shown how the brittle damage model
introduced by Francfort and Marigo can lead to a model of Hencky perfect
plasticity. Problems of damage mechanics being rather described through
evolution processes, it is natural to extend this analysis to quasi-static
evolutions, where the inertia is neglected. We consider the case where the
medium is subjected to time-dependent boundary conditions, in the
one-dimensional setting. The idea is to combine the scaling law considered by
Babadjian-Iurlano-Rindler with the quasi-static brittle damage evolution
introduced by Francfort and Garroni, and try to understand how the
irreversibility of the damage process will be expressed in the limit evolution.
Surprisingly, the interplay between relaxation and irreversibility is not
stable through time evolutions. Indeed, depending on the choice of the
prescribed Dirichlet boundary condition, the effective quasi-static damage
evolution obtained may not be of perfect plasticity type.
