纵向剪切问题中圆边界上分叉裂纹的应力强度因子

STRESS INTENSITY FACTOR FOR BRANCH CRACKS ON CIRCULAR BOUNDARY IN LONGITUDINAL SHEAR PROBLEM

采用位错配置法,研究弹性纵向剪切情况下圆边界上的分叉裂纹问题。在给出无限大域中点位错复势的基础上引入补充项,以满足圆边界自由的条件,得到圆边界上分叉裂纹问题的基本解。再由裂纹边界条件,建立奇异积分方程。然后利用半开型数值积分公式,把奇异积分方程化为代数方程,通过数值计算,直接得到裂纹端的应力强度因子值。这是一种解析数值相结合求解应力强度因子的方法,充分利用解析方法精度高和数值方法适用性广的特点,各裂纹位置可以是任意的。特例的计算结果和保角变换结果是一致的。文中算例给出远处作用纵向载荷时圆孔边缘上分叉裂纹的若干应力强度因子,以及圆柱边上作用纵向集中力时柱边缘处分叉裂纹的若干应力强度因子,讨论裂纹各分支之间的相互影响,所得的图表可以应用于工程实际。

The problems of branch cracks on circular boundary in elastic longitudinal shear are investigated by placing distributed dislocations along cracks. Based on the complex potential of a point dislocation in an infinite region, a complementary term is introduced to satisfy the traction-free condition along the circular boundary, and then the elementary solution for edge branch crack problems on circular boundary is obtained. By matching the traction along the cracks, singular integral equations are derived. By using a semi-open quadrature rule, the singular integral equations are transformed to algebra equations, and finally the stress intensity factors at the crack tips can be obtained. Two kinds of problems are studied. One is for the branch crack on the edge of a hole with remote stress, and the other is for the branch crack on the edge of a cylinder with concentrate forces acting on the boundary. The position of cracks can be arbitrary. The interaction between crack branches is discussed quantitatively. The calculated results of several examples are in agreement with those by conformal mapping method. The semi-analytical method proposed is more accurate than a purely numerical method.