Phase field method for brittle fracture implemented with polygonal finite elements

The aim of this article is to model fracture propagation in brittle materials, such as
rocks and concrete, with the phase field approach. The hybrid formulation of the phase field theory is adopted because it enables using an ad-hoc, or a problem specific, crack driving force, here of Mohr–Coulomb type, to correctly model brittle materials under compression or shear. Hybrid formulations are variationally inconsistent because the crack driving force is not the same as the one used in the underlying energy functional. They are, however, thermodynamically consistent, and computationally cheap since they allow to use a linear balance of momentum
equation within the robust staggered scheme to solve the coupled system for the phase field and the displacement field. The phase field method is implemented with 2D polygonal finite elements based on the Wachspress interpolation functions. As numerical examples, typical test cases of notched samples under mode I and II loadings are simulated. Finally, a slope stability problem is solved as an engineering application.