A unified variational framework for phase-field fracture and third-medium contact in finite deformation hyperelasticity

This paper presents a theoretical and computational framework that integrates phase-field fracture and third-medium contact within finite deformation hyperelasticity. Both crack and contact are treated through regularization: the sharp crack topology is regularized into a diffuse damage field, while the discrete contact interface is regularized by a compliant fictitious medium with auxiliary fields. Integrating these two methods into a single formulation is not a mere combination of existing ingredients but requires robust numerical treatment of several challenges. The framework is validated through two-dimensional three-point bending and two-dimensional Brazilian disk test simulations, demonstrating stable load–displacement response and the interplay between contact-induced stress concentration and crack nucleation/propagation. In particular, the Brazilian disk simulation naturally reproduces secondary crushing-type fracture zones near the contact regions – a phenomenon consistently observed in experiments yet inaccessible to simplified loading models – highlighting the capability of the proposed finite element framework to resolve complex fracture mechanisms driven by evolving contact areas.