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

We consider an evolution in phase field fracture which combines, in a system of {sc pde}s, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic $L^2$-penalty in the phase-field variable (i.e.~an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of {sc pde}s. Then, we show that the vanishing viscosity limit is a quasi-static (parametrized) $BV$-evolution. All these solutions are described both in terms of energy balance and, equivalently, by {sc pde}s within the natural framework of $W^{1,2} (0,T; L^2)$