### A quasistatic evolution model for the interaction between fracture and damage

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**Abstract:** This paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture. In both cases, the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process, the minimality at each time, and the energy balance. From a mechanical point of view, it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium. The brutal character of the damage process is then replaced by a progressive one, where the original damage internal variable, i.e. the characteristic function of the damaged part, is replaced by the local volume fraction. The analysis rests on a locality property for mixtures which enables to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the G-closure set.

**Mots Clés:** *Gamma-convergence; Relaxation; Homogenization; Free discontinuity problems; Functions of bounded variation; Fracture mechanics; Damage; Rate independent processes*

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