Convergence of Nonlocal Finite Element Energies for Fracture Mechanics

Abstract

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.

Keywords:

• Brittle fracture
• Damage
• Non local functionals

AMS Subject Classification:

• 74R05
• 74S05

ACKNOWLEDGMENTS

This work was partially supported by INdAM with the grant “Mathematical Challenges in Nanomechanics” and by IMATI-CNR of Pavia.