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Convergence in time of discrete evolutions generated by alternate minimizing schemes

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Mechanics of deformable solids; Calculus of variations and optimal control; optimization; Material science

video/mp4

Author affiliation: University of Pavia

2017-02-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59680

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/