扩展边界元法分析切口和裂纹结构应力场的准确性

在线弹性理论中，切口/裂纹结构尖端区域存在奇异应力场，数值方法不易求解。本文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离 的渐近级数展开式表达，其级数项的幅值系数作为基本未知量，而外部区域采用常规边界元法离散方程。两者方程联立求解可获得切口和裂纹结构完整的位移和应力场。扩展边界元法具有半解析法特征，适用于一般的切口和裂纹结构应力场分析，其解可精细描述从尖端区域到整体结构区域的应力场。作者研制了扩展边界元法程序，文中给出了两个算例，通过计算结果分析，表明扩展边界元法求解切口和裂纹结构应力场的准确性和有效性。

Accuracy of the extended boundary element method analyzing the stress fields of V-notched/cracked structures

According to the theory of linear elasticity, the numerical methods are difficult to calculate the singular stress field in a V-notch/crack tip region. In this paper, the extended boundary element method (XBEM) is proposed to determine the singular stress field near a V-notch/crack tip. Firstly, a small sector around the V-notch/crack tip is dug out from the V-notch/crack structures. The displacement and stress components in the small sector are expressed as the asymptotic series expansions with respect to the radial coordinate from the tip. The amplitude coefficients in the series expansions are taken as the basic unknowns. Secondly, the boundary element method is used to analyze the V-notch/crack structure removed the small sector. Consequently, the complete displacement and stress fields of the V-notched/cracked structure are solved by combining the boundary element analysis and the asymptotic series expansions of the stress field, where the extended boundary element method has the characteristics of the semi-analytic approach. The XBEM is suitable for the stress analysis of general V-notched/cracked structures, and its solution can accurately describe the displacement and stress field from the notch/crack tip to the whole region. Finally, two numerical examples are given to demonstrate the effectiveness and accuracy of the XBEM.