三维切口/裂纹结构的扩展边界元法分析

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性. 

ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD

According to the theory of linear elasticity, the conventional numerical methods are difficult to calculate the singular stress fields of three dimensional V-notched/cracked structures because of the stress singularity in the V-notch/crack tip region. In this paper, the extended boundary element method (XBEM) is first proposed to calculate the whole displacement and stress fields of three dimensional V-notch/crack structures. Firstly, the three dimensional V-notched/cracked structure is divided into two parts, which are a small sectoral column around the notch/crack tip and the outer region without the tip sectorial column. The displacement and stress components in the small sector column are expressed as the asymptotic series expansions with respect to the radial coordinate from the tip. The stress singular orders and the associated displacement and stress eigen-functions in the tip region are determined by the interpolating matrix method. The amplitude coefficients in the asymptotic series expansions are taken as the basic unknowns. Secondly, the boundary element method is used to analyze the three dimensional V-notched/cracked structure removed the small sector column. Hence, the whole displacement and stress fields of both the tip region and outer region are obtained by combining the boundary element analysis and the asymptotic series expansions of the displacement and stress fields in the notch/crack tip region, where the XBEM has the characteristics of the semi-analytic approach. The XBEM is suitable for the displacement and stress analysis of the three dimensional V-notched/cracked structures, and its solution can accurately describe the displacement and stress fields from the notch/crack tip to the whole region of the V-notched/cracked structures. Finally, two typical examples are given to demonstrate the effectiveness and accuracy of the extended boundary element method.