  Due to its computational robustness and versatility, the phase field fracture
model has become the preferred tool for predicting a wide range of cracking
phenomena. However, in its conventional form, its intrinsic tension-compression
symmetry in damage evolution prevents its application to the modelling of
compressive failures in brittle and quasi-brittle solids, such as concrete or
rock materials. In this work, we present a general methodology for decomposing
the phase field fracture driving force, the strain energy density, so as to
reproduce asymmetrical tension-compression fracture behaviour. The generalised
approach presented is particularised to the case of linear elastic solids and
the Drucker-Prager failure criterion. The ability of the presented model to
capture the compressive failure of brittle materials is showcased by
numerically implementing the resulting strain energy split formulation and
addressing four case studies of particular interest. Firstly, insight is gained
into the capabilities of the model in predicting friction and dilatancy effects
under shear loading. Secondly, virtual direct shear tests are conducted to
assess fracture predictions under different pressure levels. Thirdly, a
concrete cylinder is subjected to uniaxial and triaxial compression to
investigate the influence of confinement. Finally, the localised failure of a
soil slope is predicted and the results are compared with other formulations
for the strain energy decomposition proposed in the literature. The results
provide a good qualitative agreement with experimental observations and
demonstrate the capabilities of phase field fracture methods to predict crack
nucleation and growth under multi-axial loading in materials exhibiting
asymmetric tension-compression fracture behaviour.
