Some results on cohesive energies: approximation, lower semicontinuity and quasistatic evolution

In this thesis, cohesive fracture is investigated under three different perspectives. First we study the asymptotic behaviour of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter, which forces damage concentration on regions of codimension one. We determine the Gamma-limit as the small parameter tends to zero and show that it contains an energy term involving the crack opening. The second problem we consider is the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrised gradient. We introduce a suitable class of cohesive surface terms, which make the functional lower semicontinuous with respect to L^1 convergence. Finally, 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.