A cut finite-element method for fracture and contact problems in large-deformation solid mechanics

Cut finite-element methods (CutFEMs) belong to the class of methods that allow boundaries/interfaces to cut through the elements, which avoids any meshing/remeshing problems. This is highly convenient from a practical point of view, especially when non-stationary interfaces are considered, e.g. phase boundaries in solids, as the interfaces can move independently of the mesh. There are many research directions related to CutFEM, one of which focuses on the equations of solid mechanics. Initially, the developments centred on linear elasticity and, in the previous publication by the authors, the method has been extended to large deformations and arbitrary constitutive relations, while the focus has been on phase boundaries in solids and on localised chemical reaction fronts in coupled mechanics–diffusion–reaction systems. In this paper, the method is further extended to more complex physics of the interfaces — fracture, i.e. separation of the interface into two surfaces in the current configuration, and contact between the separated surfaces. Several cases are considered — fracture with linear and non-linear traction separation, contact without and with adhesion. Each incremental generalisation of the approach contains a prior approach as a particular case, e.g. the phase boundary problem is a particular case of the fracture problem. The contact problem is treated in an unbiased way — the weak form is symmetric with respect to the choice of the contact surfaces for the integration. The weak forms are derived from the total energy functional. The proposed method has been tested computationally for the case of linear elements and passed the so-called patch tests and the convergence rate tests demonstrating the asymptotically optimal rates.