On a new high order phase field model for brittle and cohesive fracture: numerical efficiency, length scale convergence and crack kinking

Being able to seamlessly deal with complex three dimensional crack patterns like branching and merging, phase field models (PFMs) are promising in the computational modelling of fracture of solids. Regarding the damage field, if only its second order derivative is present in the governing equation we have a second order PFM and if its fourth order derivative is also present we have a fourth order PFM. Previous studies have demonstrated that fourth order PFMs, with its smoother damage profile, are better in convergence and able to model strong anisotropic fracture energies. However, previous fourth order PFMs were developed only for brittle fracture and do not embed the material strength directly in the formulation. This paper presents a fourth order phase field regularised cohesive zone model (fourth order PF-CZM) for both brittle fracture and quasi-brittle fracture. We present a semi-analytical approach to compute the coefficients of the rational degradation function used in the fourth order PF-CZM. The proposed model is tested with multiple benchmark problems for mode I and mixed-mode fracture, and the results demonstrate that the fourth order PF-CZM: (1) provides results independent of the length scale and (2) is more efficient than its second order counterpart. The proposed model was then applied to modelling strong anisotropic fracture energies. A detailed study on sawtooth like crack patterns, which is a signature of strongly anisotropic fracture, is carried out with focus on periodicity and length scale sensitivity.