Crack nucleation and propagation in the phase-field cohesive zone model with application to Hertzian indentation fracture

Hertzian indentation fracture (Hertz, 1896) is an intriguing problem with no pre-defects. Recent studies indicated that those popular phase-field models for brittle fracture might be restrictive in dealing with this problem — crack nucleation can be considered only for the failure strength within a rather limited range. In this work, this problem is analyzed by the phase-field cohesive zonde model (PF-CZM), focusing on the whole failure process of crack nucleation and propagation. Analytical and numerical results show that for any realistic failure strength, the ‘spontaneous’ crack nucleation in an initially defect-free surface, the subsequent small but rapid vertical extension and finally the stable cone-shaped crack propagation, etc., can all be qualitatively and quantitatively captured by the PF-CZM with no modification. This competence is credited to treating the failure strength and the fracture energy (toughness) as two independent material properties. Accordingly, the phase-field length scale is a numerical parameter of no further constraint, which can be made as small as possible such that the convergence to the Barenblatt (1959) cohesive zone model is guaranteed even for short cracks. Moreover, the seamless incorporation of the strength-based crack nucleation criterion and the energy-based crack propagation criterion endows the PF-CZM with the capability of tackling crack nucleation and propagation in pristine solids.