  In this paper we propose a notion of irreversibility for the evolution of
cracks in presence of cohesive forces, which allows for different responses in
the loading and unloading processes, motivated by a variational approximation
with damage models. We investigate its applicability to the construction of a
quasi-static evolution in a simple one-dimensional model. The cohesive fracture
model arises naturally via Gamma-convergence from a phase-field model of the
generalized Ambrosio-Tortorelli type, which may be used as regularization for
numerical simulations.
