  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.
