On some mathematical problems in fracture dynamics

This thesis is devoted to the study of several mathematical problems in fracture mechanics for brittle materials. In the first part, we prove existence, uniqueness, and continuous dependence on the data for the solutions of the elastodynamics system in time-dependent cracked domains. Then, we compute the mechanical energy (elastic + kinetic) of the solutions corresponding to a sufficiently regular crack evolution in the antiplane case, and we characterize the pairs of crack and solution which satisfy the dynamic energy-dissipation balance of Griffith. Next, we consider a dynamic model of fracture for viscoelastic materials with a possibly degenerate viscosity term, and we show existence and uniqueness under suitable assumptions on the cracks. We also provide an example of a solution in the antiplane case for which the crack can grow while balancing the energy. Finally, we study a phase-field model of dynamic fracture with a dissipative term related to the speed of the crack-tips. Under suitable assumptions on the dissipation, we prove the existence of a dynamic phase-field evolution that satisfies the energy-dissipation balance of Griffith.