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| We consider rate-independent evolutions for energy functionals $\F$ of class $C^1$ in infinite dimensional reflexive separable Banach spaces. Evolutions are characterized by means of their graph parametrization (as in Efendiev \& Mielke) in terms of a couple of equations which give stationarity and energy balance. Similarities and differences with the formulation of energetic and BV evolutions by Mielke will be shown. The proofs of existence are based on incremental problems based on the Euler scheme (both backward and forward), they share common features with the theory of minimizing movements for gradient flows and make reference to a suggested problem by De Giorgi. In the spirit of Sandier \& Serfaty, considering a sequence of functionals $\F_n$ and its $\Gamma$-limit $\F$ we provide a convergence result for the associated quasi-static evolutions. An application to phase-field models for brittle fracture will be presented. |
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