含有非圆形双孔的无限平板中应力的解析解研究

ANALYTICAL STRESS SOLUTION RESEARCH ON AN INFINTE PLATE CONTIANING TWO NON-CIRCULAR HOLES

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.

The analytical stress solution for an infinite plate containing a single hole of arbitrary shape can be obtained by complex variable method.As to the doubly-connected domain problem that an infinite plate contains two round holes or two elliptical holes,it can also be solved using a variety of methods,such as the bi-polar coordinate method,the stress function method,the complex variable method,and the Schwarz alternating method.The complex variable method combined with conformal mapping is one of importance methods which can be used to obtain analytical stress solution,but it is not yet used to solve the problem of an infinite plate containing a square hole and an elliptical hole.Taking advantage of the conformal mapping method,the problem that an infinite plate contains two specific holes,which far-field uniform stress is applied at infinity and the boundaries of the two holes are subjected to uniform vertical compression,can be solved.The key step of this method is to find the corresponding mapping function with which the considered region can be mapped onto a ring in the image plane.Based on the Riemann mapping theorem,we propose a general form of the mapping function and figure out the concrete mapping function for the specific problem using optimization method.The basic equation set for solving the two analytical functions is established through the stress boundary condition of the two holes.Then the analytical stress solution can be obtained according to the two analytical functions.The analytical stress solution is compared with numerical stress solution of ANSYS finite element method.Effects of separation distance,size of elliptical hole,and the orientation of holes on tangential stress of the boundary is investigated using the newly derived solution.The stress distributions on the line that connects the centers of the two holes under different loads are presented.