Crack-path selection in phase-field models for brittle fracture

This work presents a critical overview of the effects of different aspects of model formulation on crack path selection in quasi-static phase field fracture. We consider different evolution methods, mechanics formulations, fracture dissipation energy formulations, and forms of the irreversibility condition. The different model variants are implemented with common numerical methods based on staggered solution of the phase-field and mechanics sub-problems via FFT-based solvers. These methods mix standard approaches with novel elements, such as the use of bound-constrained conjugate gradients for the phase field sub-problem and a heuristic method for near-equilibrium evolution. We examine differences in crack paths between model variants in simple model systems and microstructures with randomly heterogeneous Young's modulus. Our results indicate that near-equilibrium evolution methods are preferable for quasi-static fracture of heterogeneous microstructures compared to minimization and time-dependent methods. In examining mechanics formulations, we find distinct effects of crack driving force and the model for contact implicit in phase field fracture. Our results favor the use of a strain-spectral decomposition for the crack driving force but not the contact model. Irreversibility condition and fracture dissipation energy formulation were also found to affect crack path selection, but systematic effects were difficult to deduce due to the overall sensitivity of crack selection within the heterogeneous microstructures. Our findings support the use of the AT1 model over the AT2 model and irreversibility of the phase field within a crack set rather than the entire domain. Sensitivity to these differences in formulation was reduced but not eliminated by reducing the crack width parameter $\ell$ relative to the size scale of the random microstructures.