In this paper, a non-standard finite element (FE) method, coupled with a mathematical programming procedure, is presented to simulate initiation and propagation of in-plane tensile cracks in quasi-brittle material. Potential crack lines are defined and limited by the FE edges. The cracks are considered as localized plastic deformations imposed at the nodes along the potential crack paths. Crack opening is detected at each node by a limit value of tensile force normal to the crack direction. When the limit value is reached, a cohesive crack starts, following a linear softening branch. The mechanical model is such that no remeshing neither the introduction of interfaces is necessary to describe the discontinuity between elements, as usually required in standard FE approaches. The non-linear solution of the structural problem is obtained using a mathematical programming procedure based on the solution of a parametric linear complementarity problem (PLCP). The main advantage of the PLCP formulation is its ability to manage structural configurations in which instability and multiplicity of solutions are possible, providing one solution even at points characterized by a not-positive-semidefinite Hessian matrix. Simple numerical simulations are presented with the aim to test the method and the solution procedures.

在本文中，提出了一种非标准有限元 (FE) 方法，结合数学编程程序，用于模拟准脆性材料中面内拉伸裂纹的萌生和扩展。潜在的裂纹线由有限元边缘定义和限制。裂纹被认为是沿潜在裂纹路径施加在节点上的局部塑性变形。通过垂直于裂纹方向的张力极限值在每个节点检测裂纹开口。当达到极限值时，内聚裂纹开始，跟随线性软化分支。机械模型是这样的，不需要重新划分接口，也不需要引入接口来描述元素之间的不连续性，这在标准 FE 方法中通常是必需的。结构问题的非线性解是使用基于参数线性互补问题 (PLCP) 解的数学规划程序获得的。PLCP 公式的主要优点是它能够管理结构配置，其中可能存在不稳定性和多种解，即使在以非正半定 Hessian 矩阵为特征的点上也能提供一种解。给出了简单的数值模拟，目的是测试该方法和求解程序。即使在以非正半定 Hessian 矩阵为特征的点上也能提供一种解决方案。给出了简单的数值模拟，目的是测试该方法和求解程序。即使在以非正半定 Hessian 矩阵为特征的点上也能提供一种解决方案。给出了简单的数值模拟，目的是测试该方法和求解程序。