Numerical approximation of variational fracture through variable finite elements with gaps

The work is concerned with the numerical implementation of a variational model for quasi-static brittle fracture. Essentially the analysis is based on the formulation of Francfort and Marigo the main difference being the fact that we rely on local rather that on global minimization. Propagation of fracture is obtained by minimizing in a step by step process a form of energy that is the sum of bulk and interface terms. Recent attempts of producing numerical codes for variational fracture are based on the approximation of the energy, in the sense of Gamma-convergence, by means of elliptic functionals . Here instead we adopt discontinuous finite elements and search for the minima of the energy through descent methods. In constructing such a numerical model there are two main questions that must be answered: (1) The variational model for fracture requires the ability to accurately approximate the location of cracks, as well as their length. Cracks may not be restricted to propagate along the skeleton of a fixed finite element mesh. To overcome this difficulty our mesh is made variable in the sense that node positions in the reference configuration of the body are considered as further variables. (2) On adopting a Griffith type interface energy the initiation of fracture in a previously virgin body is always brutal, that is the system cannot proceed along neutral equilibrium paths or, in other words, descent directions toward local minima do not exist. To reach local minima the system must have the ability to surmount small energy barriers and then proceed through descent paths. Here we use a sort of mesh dependent relaxation of the interface energy to get out of small energy wells. The relaxation consists in the adoption of a carefully tailored cohesive type interface energy, tending to the Griffith limit as the mesh size tends to zero.