Anti-plane analysis of Francfort-Marigo model for quasi-static brittle fracture

The study and prediction of crack behavior in materials have always been central to classical theories of solids. Since Griffith's 1921 seminal paper, all subsequent literature on brittle fracture has built on his work. In this thesis, we focus on the extended variational setting of Francfort and Marigo, which aims at studying quasi-static brittle fracture by incorporating an interplay between elastic and fracture energies and a minimization over admissible crack evolutions. The assumptions and limitations of the model are clearly stated, as are the similarities and differences with Griffith's model. Some existence results are needed for an appropriate formalization in the context of the calculus of variations. By focusing on two-dimensional anti-plane cracks, we can avoid setting the problem in spaces of (special) functions of bounded variation or deformation, but this comes at the price of requiring some results on the fine properties of Sobolev functions and sets in metric space, as well as approximation results for the crack set. The existence of the quasi-static evolution is then obtained by discretizing the crack evolution and letting the time step tends to 0. We prove that Griffith's criteria can be obtained from this setting, as well as certain results on the regularity of the energy release rate and the stress intensity factor.