若干含幂函数类对称曲线裂纹平面弹性问题的解析解

Analytical Solutions for the Elastic Plane Problem with Symmetric Power Function Cracks

研究含幂函数类对称曲线裂纹平面弹性问题,与解决孔口问题类似,采用传统的复变函数保角映射法,给出适当的保角变换公式,将裂纹外的区域映射到一个复平面的单位圆内,得到了含幂函数类对称曲线裂纹尖端Ⅰ-Ⅱ型应力强度因子的解析表达式.该解在特殊极限条件下可解析地退化到穿透型直线裂纹的经典解.参数分析表明,幂函数类对称曲线裂纹尖端的应力强度因子与裂纹的尺寸和形状有关.

Problems of an elastic plane having symmetric power function cracks were discussed. Classical complex variable method was followed as in solving problems about hole. Some new conformal mapping formulae were proposed so that the exterior part of the symmetric power function cracks could be mapped into a unit circle. Analytical solutions for the mode Ⅰ -Ⅱ stress intensity factors at the crack tips of symmetric power function cracks were thus obtained. These solutions analytically reduced to the classical solutions for the line crack in a special case. It was found that size and shape of the symmetric cubric curve crack affected the mode Ⅰ -Ⅱ stress intensity factor at the crack tips.