An Application of a Boundary Perturbation Method to Some Problems of Elasticity

Abstract

There are a lot of problems of great importance in continuum mechanics, which can be solved by means of the perturbation method. In many cases, this method relates to a perturbation of a reference boundary for which a similar boundary problem has a relatively simple solution or can be easy solved in a closed form. The weakness of all cited works is that they are confined to constructing the first order perturbation solutions and the perturbation technique is carried out in different manners even for the 2-D problems. At the same time, using Kolosov’s complex potentials, a unified perturbation method for the 2-D problem of elasticity has been developed. Based on this method and algorithm constructed, one can easy compute any-order accurate solution of a wide range of boundary problems. The aim of the present paper is to illustrate the general approach to constructing an algorithm of the unified boundary perturbation method. A brief review of the works devoted to the application of the perturbation technique to the crack and interface problems is presented. A unified boundary perturbation method based on Goursat-Kolosov’s complex potentials and Muskhelishvili’s complex variable representations is expounded for the cases of a slightly curved interface and an interfacial crack slightly deviating from a straight one. An algorithm of deriving the complex potentials of anyorder accurate perturbation solution in a closed form has been developed for the problems under consideration. Explicit results are given for the first-order solutions when local deviations of the boundary from the straight one are described by power functions. Characteristics of a stress field are analyzed for locally curved interfaces and curvilinear interfacial cracks. These studies have been partly described in works [1, 2, 3].