We consider the quasi-static evolution of a prescribed cohesive interface: dissipative under loading and elastic under unloading. We provide existence in terms of parametrized BV-evolutions, employing a discrete scheme based on local minimization, with respect to the $H^1$-norm, of a regularized energy. Technically, the evolution is fully characterized by: equilibrium, energy balance and Karush–Kuhn–Tucker conditions for the internal variable. Catastrophic regimes (discontinuities in time) are described by gradient flows of visco-elastic type.
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