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浙江大学学报（理学版）
浙江大学学报（理学版）  2017, Vol. 44 Issue (5): 548-554    DOI: 10.3785/j.issn.1008-9497.2017.05.009
数学与计算机科学     
断裂或接触力学问题中第二类柯西奇异积分方程的一种解析方法
金晓清1, 吕鼎1, 张向宁1, 李璞1, 周青华2, 胡玉梅1
1. 重庆大学 机械传动国家重点实验室, 重庆 400044;
2. 四川大学 空天科学与工程学院, 四川 成都 610065
An analytical method for solving Cauchy singular integral equations of the second kind with applications to fracture and contact analyses
JIN Xiaoqing1, LYU Ding1, ZHANG Xiangning1, LI Pu1, ZHOU Qinghua2, HU Yumei1
1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China;
2. School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China
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摘要： 第二类柯西奇异积分方程因涉及复奇异因子往往造成求解困难，而适用第一类奇异积分方程的高效数值方法并不能推广至第二类奇异积分方程，即便是第二类奇异积分方程，其数值解法仍是一个难题. 为此提出了构造第二类奇异积分方程解析解的一种新方法. 通过分解柯西奇异项，并利用雅克比多项式的正交性，推导针对右端载荷项为单项式（monomial）的递推解析解，进而借助级数展开的方法推广至一般的载荷问题. 提出的基于递推的解析解构造方案，能完美地结合maple软件编程，从而提供一种方便、快捷、有效的算法. 由给出的算例可见，本方法适用于处理界面断裂或接触分析问题中含复数奇异因子的复杂情形，从而为研究该类典型力学问题提供了一种可供选择的方法.
关键词： 第二类奇异积分方程;  柯西主值积分;  复数奇异因子;  界面裂纹    
Abstract: Due to the presence of complex singularity, solutions to the singular integration equation (SIE) of the second kind are still under development. As a matter of fact, numerical methods for SIE of the first kind are hardly applicable to SIE of the second kind. With the assistance of maple programming, this paper presents a novel approach to formulate an analytical solution to a typical SIE of the second kind. By splitting the Cauchy kernel, and taking advantage of the orthogonality of Jacobi polynomials, we derive an analytical solution corresponding to the monomial loading case. Furthermore, the solution to a general loading case may be obtained via series expansion. The present method appears efficient and convenient, providing an effective tool for treating tangentially loaded contact analyses and interface crack problems.
Key words: SIE of the second kind    Cauchy principal value integration    complex singularity    interface crack
收稿日期: 2016-04-13 出版日期: 2017-05-01
CLC:  O343.3  
基金资助: 国家自然科学基金资助项目（51475057）；中央高校基本科研业务费专项（106112017CDJQJ328839）.
作者简介: 金晓清(1974-),ORCID:http://orcid.org/0000-0002-8836-3505,博士,研究员,博士生导师,主要从事断裂疲劳、细观力学、摩擦学等研究,E-mail:jinxq@cqu.edu.cn.
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引用本文:

金晓清, 吕鼎, 张向宁, 李璞, 周青华, 胡玉梅. 断裂或接触力学问题中第二类柯西奇异积分方程的一种解析方法[J]. 浙江大学学报（理学版）, 2017, 44(5): 548-554.

JIN Xiaoqing, LYU Ding, ZHANG Xiangning, LI Pu, ZHOU Qinghua, HU Yumei. An analytical method for solving Cauchy singular integral equations of the second kind with applications to fracture and contact analyses. Journal of ZheJIang University(Science Edition), 2017, 44(5): 548-554.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2017.05.009    或    https://www.zjujournals.com/sci/CN/Y2017/V44/I5/548

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