Inserted: 29 jan 2007
Last Updated: 20 feb 2009
Journal: Math. Models Methods Appl. Sci.
We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasi-static evolution by means of local minimizers of the free energy. We show that this evolution satisfies Griffith's criterion in a suitable form which takes into account both stable and unstable propagation, as well as an energy balance formula which accounts for dissipation in the unstable regime. If the load is monotonically increasing this solution is explicit and almost everywhere unique. For more general loads we construct a solution via time discretization, finding a local minimizer for every incremental problem. Finally, we consider a finite element discretization of the problem and prove convergence of the discrete solutions.