Gamma-convergenza per energie coesive in spazi BV

In this thesis, we investigate the Gamma-convergence of discrete functionals to a sharp cohesive fracture energy. In particular, we study a family of discrete energy functionals defined on one-dimensional finite element spaces with a non-uniform mesh. These functionals, which couple the displacement field with the damage variable, represent a phase-field model. We analyze the asymptotic behavior of the discrete functionals as the mesh size h and the internal length \epsilon_{h} tend to zero. We require that h=o(\epsilon_{h}). Our main result proves that the discrete model Gamma-converges to a functional defined on the space BV of functions of bounded variation. We show that the Gamma-limit energy is the sum of three terms: a bulk elastic energy, a Cantor part and a cohesive energy concentrated at the jump set of the displacement. The contribution of this work is the generalization of the Gamma-convergence result presented in [MNVD26]. Specifically, we consider a non-uniform mesh, a general decreasing and Lipschitz continuous degradation function a, an even density function f with quadratic-linear behavior and we remove the non-interpenetration condition. These changes are useful to extend the proof of the main theorem to the two-dimensional anti-plane setting, as a future development.