ON CONTACT PROBLEMS WITH DEFORMABLE STAMP
В. А. Бабешко, О. В. Евдокимова, О. М. Бабешко
Abstract
This paper presents one of the methods for studying the behavior of deformable stamps on a deformable base. It is based on a new universal modeling method previously published by the authors, used in boundary value problems for systems of partial differential equations. Its advantage is the possibility of avoiding the need to solve complex boundary value problems for systems of differential equations by replacing them with separate differential equations, among which the Helmholtz equations are the simplest. With the help of combinations of solutions of boundary value problems for this equation, it is possible to describe the behavior of complex solutions of multicomponent boundary value problems, including for cracks of a new type formed by objects on a deformable base, and models of nano particles located on deformable multicomponent bases. However, without the ability to solve contact problems for deformable stamps, these models are not built. The mixed problem is reduced to the solution of the Wiener–Hopf integral equation. Two cases are considered: the case of a large-width strip stamp and the case of a semi-infinite stamp. A packed block element is accepted as a deformable stamp, as a solution of the Helmholtz equation in the specified area. Mechanically, it can be imitated as a membrane that is located on a multilayer medium occupying the contact area. A combination of such objects can be used to describe solutions to the contact problem for flat deformable objects of more complex rheology, as well as for three-dimensional ones. Along with the proof of constructing an exact solution to the contact problem under consideration, the appearance of unknown functionals is noted in the course of the study. In problems with an absolutely solid stamp, they do not occur. The paper finds a way to determine them and an analytical representation of them is obtained. It is suggested that their appearance will be typical for solving other contact problems with deformable stamps. The features of the method and the results obtained are discussed.