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We characterize quasi-static rate-independent evolutions, by means of their graph parame- trization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals F of class C 1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals Fn and its Γ-limit F we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics