From phase field to sharp cracks: Convergence of quasi-static evolutions in a special setting

We consider the quasi-static evolution of a straight crack within the recent phase field approach and the classical sharp crack approach. We show a strong correlation between the two approaches: energy, minimizers, energy release rate and quasi-static evolutions converge as the internal length parameter of the phase-field approach tends to zero. A crucial point in the proof is a novel representation of the energy release rate, which allows to pass to the limit under weak convergence of the strains.