无限平板中任意双孔问题的复变函数解法  

SOLUTION FOR AN INFINITE PLATE WITH TWO ARBITRARY HOLES USING COMPLEX VARIABLE METHOD

作　　者：曾祥太 王少杰 吕爱钟[3] 刘伟平[1] ZENG Xiang-tai;WANG Shao-jie;LYU Ai-zhong;LIU Wei-ping(Department of Civil Engineering,Nanchang University,Nanchang 330031,China;China Nuclear Power Engineering Co.,Ltd,Beijing 100840,China;Institute of Hydroelectric and Geotechnical Engineering,North China Electric Power University,Beijing 102206,China)

机构地区：[1]南昌大学土木工程系,南昌330031 [2]中国核电工程有限公司,北京100840 [3]华北电力大学水电与岩土工程研究所,北京102206

出　　处：《工程力学》2025年第3期32-41,共10页Engineering Mechanics

基　　金：国家自然科学基金项目(41962015,51974124)。

摘　　要：许多工程中都存在含孔洞的结构,外荷载作用下孔边不可避免地产生应力集中,为了保证结构的安全和稳定,准确了解开孔结构的应力分布情况具有重要的工程实践价值。目前对单孔和特定形状双孔问题的研究较多,可以使用一些方法获得解析解,而对任意形状双孔问题的研究较少。复变函数方法对于任意双孔问题是适用的,关键要找到对应的映射函数。该文提出了可将任意无限双连通域保角变换成同心圆环域的映射函数一般形式,建立了一种优化求解具体映射函数的方法。针对无限平面中的任意双孔问题,基于弹性复变函数理论建立了应力边界条件,借助该映射函数将问题变换至易求解的同心圆环域,双解析函数则变换成洛朗级数,最后使用幂级数法将洛朗级数解出获得了问题的应力解析解。获得的应力解析解和ANSYS数值解进行了对比。使用该方法研究了双隧洞问题在不同荷载和不同间距时洞边应力分布的影响规律。Many types of engineering contain structures with holes,which stress concentration occurs inevitably at the hole edge under external loads.To ensure the safety and stability of the structure,it is of great practical value to accurately derive the stress distribution of the perforated structure.At present,most researches are involved with single hole and double special shaped holes with analytical solution,while there is little research on two arbitrarily shaped holes.Complex variable method is applicable to two arbitrary holes if the corresponding mapping function could be found.A general form of mapping function is proposed that could conformally map arbitrary infinite doubly connected domain into a concentric ring,and the mapping function is found by optimization method.Concerning the problem of an infinite plate with two arbitrarily shaped holes,the stress boundary conditions are established based on complex potential theory.The problem is transformed into concentric ring domain by the mapping function and the two analytical functions are transformed into Laurant series simultaneously which is solved by power series method.The analytical stress solution is compared with the ANSYS numerical solution.The effects of load and distance on the stress on the boundary is investigated by the present method.

关 键 词：无限平面 任意双孔 复变函数方法 映射函数 幂级数解法 

分 类 号：O343.1[理学—固体力学]