Convergence of non-local finite element energies for fracture mechanics

Usually, smeared crack techniques are based on the following features: the fracture is represented by means of a band of finite elements and by a softening constitutive law of damage type. Often, these methods are implemented with nonlocal operators that control the localization effects and reduce the mesh bias. We consider a nonlocal smeared crack energy defined for a finite element space on a structured grid. We characterize the limit energy as the mesh size h tends to zero and we establish a precise link between the discrete and continuum formulations of the fracture energies, showing the correct scaling and the explicit form of the mesh bias.