  In this paper we prove the existence of quasistatic evolutions for a cohesive
fracture on a prescribed crack surface, in small-strain antiplane elasticity.
The main feature of the model is that the density of the energy dissipated in
the fracture process depends on the total variation of the amplitude of the
jump. Thus, any change in the crack opening entails a loss of energy, until the
crack is complete. In particular this implies a fatigue phenomenon, i.e., a
complete fracture may be produced by oscillation of small jumps.
  The first step of the existence proof is the construction of approximate
evolutions obtained by solving discrete-time incremental minimum problems. The
main difficulty in the passage to the continuous-time limit is that we lack of
controls on the variations of the jump of the approximate evolutions. Therefore
we resort to a weak formulation where the variation of the jump is replaced by
a Young measure. Eventually, after proving the existence in this weak
formulation, we improve the result by showing that the Young measure is
concentrated on a function and coincides with the variation of the jump of the
displacement.
