Damage models have been successfully employed for many decades in the modelling of tensile failure, where the crack surfaces separate as a crack grows. The advantage of this approach is that crack trajectories can be computed simply and efficiently on a fixed finite element mesh without explicit tracking. The development of damage models for shear failure in compression, where the crack faces slide over each other subject to friction, is a non-trivial extension of this approach. A major difference is that part of the material stiffness in the damaged region must be retained to avoid interpenetration of the crack faces. This problem is resolved here by employing an anisotropic modification to the elastic stiffness tensor in the damaged region. This has the benefit of driving frictional cracks into the correct orientation, according to the Mohr–Coulomb failure criteria, but three issues remain. The first is that the shear discontinuity introduces some spurious stress perturbations around crack regions that are narrow (less than 3-4 elements wide). This is ameliorated by Helmholtz smoothing to allow efficient simulation on a coarse mesh. The second is that the complementarity of shear stress means that the shear stiffness is removed normal to the crack interface as well as parallel to it, and the third is that there are two potentially active failure planes at each point. Both these latter two issues are resolved by the introduction of a novel failure plane selection variable, which regulates either single plane or dual plane failure, and prevents the growth of erroneous cracks normal to the crack face. Both local and non-local models are investigated for linear and exponential strain softening responses. Unlike the non-local model, the local model demonstrates some mesh-size dependence, but it still retains some properties of interest, in that it supports narrower cracks and more rapidly forms a preference for the growth of a single crack when there are a number of competing cracks. The model is implemented in commercial finite element package COMSOL Multiphysics v5.5 and validated against two benchmark simulations: biaxial compression and the failure of a 45° slope. The correct crack angles, stipulated by the Mohr–Coulomb friction angle, are correctly reproduced, as is the post-failure residual frictional force in biaxial compression. The effect of the shear fracture energy on the force–displacement response is investigated, demonstrating successful simulation of the range of material behaviour expected in geological samples, from broad ranged gradual collapse to sharp, almost instantaneous failure.

几十年来，损伤模型已成功用于拉伸破坏的建模，其中裂纹表面随着裂纹的增长而分离。这种方法的优点是可以在固定的有限元网格上简单有效地计算裂纹轨迹，而无需显式跟踪。压缩剪切破坏损伤模型的开发，其中裂纹面在摩擦下相互滑动，是这种方法的重要扩展。一个主要区别是必须保留损坏区域的部分材料刚度以避免裂纹面相互渗透。这个问题在这里通过使用各向异性修改损伤区域的弹性刚度张量。根据 Mohr-Coulomb 失效准则，这有利于将摩擦裂纹驱动到正确的方向，但仍然存在三个问题。首先是剪切不连续性在狭窄的裂纹区域（小于 3-4 个单元宽）周围引入了一些虚假应力扰动。这通过亥姆霍兹平滑得到改善，以允许在粗网格上进行有效模拟。二是剪切应力的互补性意味着剪切刚度垂直于裂纹界面并平行于裂纹界面移除，第三个是每个点有两个潜在的活动失效平面。后两个问题都通过引入新的失效平面选择变量来解决，该变量调节单平面或双平面失效，并防止垂直于裂纹面的错误裂纹的生长。研究了局部和非局部模型的线性和指数应变软化响应。与非局部模型不同，局部模型表现出一些网格大小的依赖性，但它仍然保留了一些特性有趣的是，它支持更窄的裂缝，并且当存在多个竞争裂缝时，它会更快地形成对单个裂缝生长的偏好。该模型在商业有限元软件包 COMSOL Multiphysics v5.5 中实现，并针对两个基准模拟进行了验证：双轴压缩和 45° 斜坡的失效。由莫尔-库仑摩擦角规定的正确裂纹角被正确再现，双轴压缩中的失效后残余摩擦力也是如此。研究了剪切断裂能对力-位移响应的影响，证明了对地质样品中预期材料行为范围的成功模拟，从大范围的逐渐坍塌到急剧的、几乎瞬时的破坏。