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Convergence in time of discrete evolutions generated by alternate minimizing schemes Negri, Matteo


We consider a couple of evolutions for a phase field energy in brittle fracture. Both are obtained by time discretization using, as incremental problem, some alternate minimization scheme. We start from a time-discrete evolution, which resembles the alternate minimization scheme of Bourdin-Francfort-Marigo. Recasting the algorithm as a gradient flow, we provide a time-continuous limit, characterized in terms of a quasi-static evolution (more precisely a parametrized BV-evolution). Mechanically, the time-continuous evolution satisfies a suitable phase-field Griffith's criterion, at least in continuity points, while dissipation is thermodynamically consistent (with respect to the irreversibility constraint). Then, we consider a time-continuous system with an "irreversible Ginzburg-Landau" equation, for the phase field variable, paired with the elasto-static equilibrium equation. We provide existence by means of time-discrete alternate minimizing movement. Next, we study its quasi-static vanishing viscosity limit, again by means of a parametrized BV-evolution. Technically, characterizations are given both in terms of energy balance and by PDEs.

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