Phase-field modelling of cohesive fracture

In phase-field models the damage evolution problem is considered as a minimisation problem of a Griffith like energy functional, governed by the principles of irreversibility, stability and energy balance. Herein, we consider phase-field models characterised by different degradation and energy dissipation functions. With a proper choice for the characteristic functions in phase-field models, it is possible to reproduce cohesive fracture in a one-dimensional setting. We consider a one-dimensional bar with stress softening, which exhibits homogeneous deformations provided that the length of the bar length is below a state-dependent critical value. Otherwise, the bar will lose stability and show a localised response. It appears that the phase-field method can partially reproduce the response of a cohesive zone model, for instance the traction-separation law, but not all aspects of the model, like the dissipated energy. For a one-dimensional problem, the crack nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion for different lengths of the bar. We have compared the one-dimensional results with the numerical solutions in a two-dimensional setting, which yielded a very good agreement.