Convergence analysis for a smeared crack approach in brittle fracture

Our analysis focuses on the mechanical energies involved in the propa- gation of fractures: the elastic energy, stored in the bulk, and the fracture energy, dissipated in the crack. We consider a nite element model based on a smeared crack approach: the fracture is approximated geometrically by a stripe of elements and mechanically by a softening constitutive law. We dene in this way a discrete free energy Gh (being h the element size) which accounts both for elastic displacements and fractures. Our main interest is the behaviour of Gh as h tends to 0. We prove that, only for a suitable choice of the (mesh dependent) constitutive law, Gh converges to a limit functional G with a positive (anisotropic) term concentrated on the crack. We discuss the mesh bias and compute it explicitly in the case of a structured triangulation.