反平面弹性中边缘内分叉裂纹的奇异积分方程解法

Singular Integral Equation Approach for Edge Internal Branch Crack in Antiplane Elasticity

利用复变函数和奇异积分方程方法,求解反平面弹性中半平面边缘内分叉裂纹问题。提出了满足半平面边界自由的由分布位错密度表示的半平面中单裂纹的基本解,此基本解由主要部分和辅助部分组成。将半平面边缘内分叉裂纹问题看作是许多单裂纹问题的叠加,建立了以分布位错密度为未知函数的Cauchy型奇异积分方程组。然后,利用半开型积分法则求解奇异积分方程,得到了裂纹端处的应力强度因子。文中给出两个数值算例的计算结果。

The edge internal branch crack problems for half-plane in antiplane elasticity are solved with complex potentials and singular integral equation approach.The elementasy solution of a single crack with distributing the dislocation density of half-plane is proposed,which is expressed as a function of the distributied dislocation density composed of both the principal part and complementary part.The edge internal branch crack of half-plane can be considered as a superposition of many single cracks,thus a set of Cauchy singular integral equations can be formulated,where the distributing dislocation density serves as the unknown function.According to a semi-open quadrature rule,the singular integral equations are solved and the stress intensity factors at the crack tips can be evaluated.Two numerical examples are presented.