单裂纹问题的虚边界无网格伽辽金法分析

Analysis of Single Crack Problem Based on Virtual Boundary Meshless Galerkin Method

使用含裂纹复变基本解,虚边界无网格伽辽金法被进一步推广应用于弹性材料的单裂纹问题求解.为了清晰地说明单裂纹问题的虚边界元法实现过程,单裂纹问题的虚边界元法示意图、复变坐标平面下含裂纹问题的复变位移和复变面力基本解示意图被展示.含裂纹复变基本解,因自动满足裂纹处边界条件,故使用虚边界无网格伽辽金法计算单裂纹问题,无需在裂纹处布置节点或单元,给出含裂纹复变基本解中的φ′(x)的详细表达式、裂纹左右裂尖应力强度因子的虚边界无网格离散公式,方便了其他学者使用本方法计算裂纹问题。数值计算两端受拉长方形钢板中心含有裂纹的应力强度因子的算例,计算结果证明了本方法的精确性与稳定性。

Using the complex fundamental solution containing a crack,the virtual boundary meshless galerkin method is further extended to solve the single crack problem of elastic material.Schematic diagram of virtual boundary element method for single crack problem,and schematic diagram of the fundamental solutions of complex displacement and complex surface force with crack in complex coordinate plane,are given to clearly illustrate the process of the virtual boundary element method for the single crack problem.The complex fundamental solution containing a crack,can automatically satisfy the boundary conditions of the crack.Therefore,no nodes or elements exist on the crack for the single crack problem by the virtual boundary meshless galerkin method.Detailed expression of the complex fundamental solution containing a crack,and the virtual boundary meshless discrete formulas for crack stress intensity factors of left and right crack tips,are presented to conveniently calculate the crack problem for other scholars.The stress intensity factors of the center crack for the rectangular steel plate with tension at both ends,are numerically calculated.The calculation results prove the accuracy and stability of the proposed method.