Eigenfracture: An Eigendeformation Approach to Variational Fracture

Abstract

We propose an approximation scheme for a variational theory of brittle fracture. In this scheme, the energy functional is approximated by a family of functionals depending on a small parameter and on two fields: the displacement field and an eigendeformation field that describes the fractures that occur in the body. Specifically, the eigendeformations allow the displacement field to develop jumps that cost no local elastic energy. However, this local relaxation requires the expenditure of a certain amount of fracture energy. We provide a construction, based on the consideration of $\varepsilon$-neighborhoods of the support of the eigendeformation field, for calculating the right amount of fracture energy associated with the eigendeformation field. We prove the $\Gamma$-convergence of the eigendeformation functional sequence, and of finite element approximations of the eigendeformation functionals, to the Griffith-type energy functional introduced in Francfort and Marigo [J. Mech. Phys. Solids, 46 (1998), pp. 1319–1342]. This type of convergence ensures the convergence of eigendeformation solutions, and of finite element approximations thereof, to brittle-fracture solutions. Numerical examples concerned with quasi-static mixed-mode crack propagation illustrate the versatility and robustness of the approach and its ability to predict crack-growth patterns in brittle solids.