Calculus of Variations and Geometric Measure Theory

M. Negri

An $L^2$ gradient flow and its quasi-static limit in phase-field fracture by alternate minimization

created by negri on 09 Feb 2016



Inserted: 9 feb 2016
Last Updated: 9 feb 2016

Year: 2016


We consider an evolution in phase field fracture which combines, in a system of pdes, an irreversible gradient-flow for the phase-field variable with the equilibrium equations for the displacement field. We employ a discretization in time and an alternate minimization scheme with a quadratic penalty in the phase-field variable (i.e. an “alternate minimizing movement”). First, we prove that discrete solutions con- verge to a solution of our system of pdes. Then, we show that the vanishing viscosity limit is a quasi-static BV -evolution. Both our convergence results are formulated in terms integral characterizations, a' la De Giorgi, for gradient flows and parametrized BV -evolutions, from which the pdes follow.