Fracture energy in phase field models

The phase field approach to brittle fracture is based on smeared energetic representation of sharp fracture into surface. The passage between damaged and undamaged zones is influenced by an internal length scale parameter. In the present paper the approximation of fracture energy in phase field models is studied. Firstly, the diffusion equation of the phase field is numerically investigated. It is demonstrated through simple paradigmatic 2d and 3d cases that the fracture energy during crack initiation and propagation phenomena, such as crack branching and bifurcation, is strictly correlated with the internal length parameter. Moreover, it is shown that for finite value of the internal length parameter the dissipated energy does not depend only on the crack extension but on the geometrical configuration of fracture differently from the Griffith sharp approach. In particular, it is demonstrated that 3d cracks with same area may be characterized by different values of approximated fracture energy.