抛物线裂缝反平面弹性问题的解析解

针对含抛物线裂缝的反平面弹性问题,采用复变函数的保角变换方法,将抛物线裂缝外的区域映射到单位圆的外部.提出了边界积分方程以避免变换函数奇异性引起的困难,求得了抛物线裂缝反平面弹性边值问题的复势解.然后,用本文提出的直接用复势计算曲线裂纹应力强度因子的公式得到了抛物线裂纹尖端应力强度因子的解析表达式.该表达式在特殊情况下可蜕化为穿透型直线裂纹反平面问题的经典解.分析表明,应力强度因子的大小依赖于抛物线裂纹的形状以及无穷远处两个方向的切应力载荷之比.

An Analytical Solution for the Antiplane Elastic Problem with a Parabolic Crack

The problem of elastic antiplane with a parabolic crack was investigated.The exterior region of the parabolic crack was mapped out of a unit circle through conformal transformation.A boundary integral method was proposed to avoid the difficulty caused by the singularities of the transform function.The complex potential solution to the elastic antiplane boundary value problem with a parabolic crack was obtained.The stress intensity factor(SIF) at the tip of parabolic crack was thus given by employing a formula proposed in the present article to calculate SIF directly on the basis of complex potentials.The solution can be reduced to the solution for line crack in a special case.The shape of the parabolic crack and the ratio of the shear stress loads of two directions at infinity affect the magnitude of stress intensity factors.