Convergence analysis for a smeared crack approach in brittle fracture

Abstract

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 energy $G_h$ (being $h$ the element size) which accounts both for elastic displacements and fractures. Our main interest is the behaviour of $G_h$ as $h\to 0$. We prove that, only for a suitable choice of the (mesh dependent) constitutive law, $G_h$ converges to a limit functional $G_{\varphi }$ 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.