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

摘要

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.%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.