EXACT SOLUTION OF CONTACT PROBLEMS IN A FINITE-WIDTH BAND ON A MULTILAYER MEDIUM

Abstract

In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.