Convergence of alternate minimization schemes for phase field fracture and damage

We consider time-discrete evolutions for a phase field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized $BV$-evolutions, introducing a suitable family of ``intrinsic energy norms''. Further, we show that the limit evolution satisfies Griffith's criterion, for a phase field energy release, and that the irreversibility constraint is thermodynamically consistent.