含椭圆孔有限板应力集中的复变函数直接解法

The Stress Concentration in Finite-Size Plates Containing Elliptical Holes with a Direct Solution of Complex Series

采用复变函数级数展开方法研究了含椭圆孔的有限大矩形板在承受拉伸和剪切载荷时的应力场和应力集中系数,通过直接对边界力的差值进行最小化求取级数中的待定系数,避免了通常采用的将椭圆孔变换成圆孔的保角变换过程,从而极大地简化了求解过程.与有限元计算的对比分析表明,对于承受单向拉伸载荷的含中心椭圆孔(两轴比在0.7至2之间)的有限尺寸矩形板,计算精度高,且较之传统的保角变换法更简单,易于应用.另外,给出了计算含中心椭圆孔(两轴比为0.8)的细长板在拉伸载荷作用下以及含中心圆孔的细长板在面内剪切载荷作用下孔边应力集中系数的经验公式,便于工程应用.

We solve the elasto-static boundary-value problem of finite-size plates containing elliptical holes using the complex series expansion. Instead of using the usual conformal mapping which maps the elliptical hole into a circular hole, we calculate the coefficients of the series using the least square method that minimizes the difference between the calculated boundary traction and the given boundary condition. This method avoids the conformal mapping and thus greatly simplifies the procedure of solution. By comparison with the results of the finite element computation, we show that for a rectangular plate containing a central elliptical hole with a ratio of axes between 0.7 and 2.0, the stress concentration factors given by the complex series are quite accurate, and the method is much simpler than the conventional conformal mapping method and is easy to use. Moreover, we also give empirical formulas that calculate the stress concentration factors for a plate containing an elliptical hole（hole axes ratio 0.8, plate aspect ratio greater than 2） under uniaxial tension and a plate containing a circular hole（plate aspect ratio greater than 1） under shear. These formulas are simple and easy to use.