Multidimensional stability analysis of the phase-field method for fracture with a general degradation function and energy split.

A local physical stability criterion for multidimensional fracture problems modeled by the phase field method is developed and studied. Stability analysis provides a rigorous mathematical way to determine the onset of an unstable damage growth and fracture of the structure. In this work, stability is determined by examining the roots of a characteristic equation that arise when a linear perturbation technique is applied to the instantaneous partial differential equation system in a general viscoplastic material. It is shown that such analysis is not limited to a particular degradation function or energy split and could therefore be applied to a wide range of cases. Numerical results are presented to verify the theoretical predictions assuming quadratic and cubic degradation functions. Additionally we show that this stability criterion can be directly expanded to 2D with robust mesh-insensitive predictive capabilities with respect to crack nucleation and path. Several numerical examples are presented to verify these results.