Propagation and bifurcation of cracks based on implicit surfaces and discontinuous velocities

Cracks and their propagation with a given kinematic velocity are described as zero-level sets of a non-negative scalar function satisfying a transport equation. For smooth velocities this description is equivalent to a crack parameterization where the moving crack is obtained as the image of an initial reference crack under a coordinate transformation. Based on the implicit formulation, bifurcation type phenomena such as branching and merging of the crack, which cannot occur in the parameterized situation, are investigated numerically. Analytical and computational examples of the crack evolution with continuous as well as discontinuous velocities are presented in 2D and 3D domains.