板条内分叉裂纹的Ⅲ型应力强度因子

Mode Ⅲ Stress Intensity Factors for Branch Crack in a Strip

利用复变函数和奇异积分方程方法,求解板条内的分叉裂纹问题。首先给出了反平面弹性情况下,边界(即板条下边界)自由的半平面内单分叉裂纹问题的复势函数。通过用一个长的二分叉裂纹来代替板条上边界,以满足板条的上边界自由,将问题转化为半平面内的多分叉裂纹来处理。根据边界条件建立了以集中位错强度和分布位错密度为未知函数的Cauchy型奇异积分方程,然后,利用半开型积分法则求解该奇异积分方程,得到了各分支尖端的应力强度因子。最后,给出数值算例。

The anti-plane branch crack problems of strip were solved by using complex variable function and singular integral equation approach. Firstly, complex potential of single branch crack of half-plane which satisfied the traction-free condition along the boundary （ also the lower boundary of the strip） was given. The problem was converted to the multiple branch cracks problem in half-plane by replacing the upper boundary of the strip with a two equal branch crack to satisfy the traction-free condition along the upper boundary. Next by matching the traction along the cracks, Cauchy singular integral equations were obtained, in which the point dislocation and the distributed dislocation density served as the unknown function. Finally, by using a semi-open quadrature rule, the singular integral equations were solved. Thus, the SIF values at the crack tips were calculated. At last, Two numerical examples were given.