Computation of the crack opening displacement in the phase-field model

The phase-field model treats discrete cracks in a smeared sense by a regularisation technique. It holds attractive properties: there is no need to consider cracks as geometric discontinuities, and it avoids remeshing around crack tips. The method has been employed in the analysis of brittle and cohesive fracture problems. In the brittle fracture model, a Griffith-like energy functional is used in the simulation, while in the cohesive fracture model, the fracturing problem exploits a displacement jump governed energy functional. Obviously, the displacement jump is crucial in the cohesive fracture model and in certain other applications, e.g. for hydraulic fracturing. In the current study, the approximated form of the crack opening displacement is derived for the brittle and cohesive fracture models. In both models, the crack opening displacement is associated with a line integral normal to the crack, but different factors in front of the integral apply. The derived integrals are verified analytically in a one-dimensional setting and numerically in multi-dimensional set-ups, featuring straight and curved cracks.