  Motivated by models of fracture mechanics, this paper is devoted to the
analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional,
where unilaterality comes from an irreversibility constraint on the fracture
density. In the spirit of gradient flows in metric spaces, such evolutions are
defined in terms of curves of maximal unilateral slope, and are constructed by
means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah
regime is also carried out. It shows the convergence towards a generalized heat
equation outside a time increasing crack set.
