This paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory.

本文考虑了具有边缘裂纹的棱柱梁的对称三点弯曲问题。该解是通过简化的 Toupin-Mindlin 应变梯度弹性理论中的混合有限元方法获得的。应用了位移-应变-应力及其梯度的边值问题的混合变分公式，简化了近似函数的选择。采用能量平衡的概念来计算裂纹长度虚拟增加的能量释放速率。弹性体势能的增加是通过考虑应变和应力梯度贡献来确定的。使用十字型的准均匀三角形网格进行数值计算。在裂缝尖端附近、集中支座和横向力的施加点处进行网格细化，并在梁的其余部分使用均匀的网格划分。对应力集中域中不同长度尺度参数值的连续压缩网格进行细网格分析。给出了长度尺度参数的各种值的裂纹张开位移以及应变和柯西应力的分布。该参数的增加会增加裂纹的刚度，从而导致裂纹张开位移减小，裂纹尖端的表面平滑闭合。此外，考虑比例参数会减少裂纹尖端附近应变和应力的计算值。 根据能量平衡准则，针对网格台阶的不同值确定局部断裂参数，例如裂纹尖端弹性能的释放速率和应力强度因子。数值计算表明所得近似值的收敛性。包括应变梯度贡献在内的解的主要特点是，与经典弹性理论相比，与断裂能量相关的计算参数值减小。