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.

切割有限元方法 (CutFEM) 属于允许边界/界面切割元素的方法类别，可避免任何网格划分/重新划分网格问题。从实用的角度来看，这非常方便，尤其是在考虑非平稳界面时，例如固体中的相边界，因为界面可以独立于网格移动。与 CutFEM 相关的研究方向有很多，其中之一侧重于固体力学方程。最初，发展集中在线性弹性上，在作者之前的出版物中，该方法已扩展到大变形和任意本构关系，而重点是固体中的相边界和耦合力学中的局部化学反应前沿-扩散-反应系统。在本文中，该方法进一步扩展到更复杂的界面物理场 - 断裂，即在当前配置中将界面分离为两个表面，以及分离的表面之间的接触。考虑了几种情况 - 线性和非线性牵引分离的断裂，没有和有粘连的接触。该方法的每个增量概括都包含作为特定情况的先验方法，例如，相边界问题是断裂问题的特定情况。接触问题以无偏的方式处理——弱形式关于积分接触面的选择是对称的。弱形式来自总能量泛函。