Stability analysis of the phase-field method for fracture with a general degradation function and plasticity induced crack generation

The phase-field method is a popular technique for modeling crack initiation, bifurcation and coalescence without the need to explicitly track the crack surfaces. In this framework, the governing equations are derived from thermodynamic principles and cracks are modeled as continuous entities, whose width is defined by a small process zone parameter. Propagation of cracks is governed by the partition of strain energy that contributes to fracture and may be stable, in which case additional energy is required to form cracks, or unstable, where cracks advance with no additional input energy. In this work we propose a stability framework, based on a linear perturbation analysis, to determine the onset of unstable crack growth. The derivations lead to an analytical, energy based criterion for the phase field method in linear elastic and visco-plastic materials. The stability criterion is valid for a general degradation function and accounts for fracture induced by cold-work. Numerical results on linear elastic materials show that the proposed criterion not only recovers the critical stress value reported in the literature but also provides a stability criterion for visco-plastic materials with a general degradation function. The criterion is tested on one dimensional problems and a two dimensions homogeneous example, successfully predicting the instability point. (C) 2017 Elsevier Ltd. All rights reserved.