Convergence analysis for a smeared crack approach in brittle fracture

Our analysis focuses on the mechanical energies involved in the propagation of fractures: the elastic energy, stored in the bulk, and the fracture energy, concentrated in the crack. We consider a finite 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 define in this way a discrete free energyGh (h being the element size) which accounts for both elastic displacements and fractures. Our main interest is the behaviour of Gh as h ! 0. We prove that, 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