Mathematics

We consider a unilateral $L^2$ gradient flow within the phase-field approach to fracture. We take into account different discrete schemes, in time and/or space, based on alternate minimizing movements; we consider in particular different ways of implementing irreversibility, in terms of monotonicity of the phase-field variable. First we study the continuum limits, in time and/or space, in the framework of evolutions in $H^1 (0,T;L^2)$ and then the quasi-static limits by the vanishing viscosity approach, at least in $H^1 (0,T;H^1)$. Finally, we will compare the numerical simulations obtained with different schemes; multi-step schemes with weak irreversibility constraint will show good adaptivity in catastrophic regimes and robustness, as far as the choice of time discretization.