On contact problems with a deformable punch and variable rheology

Abstract

The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary value problems for systems of partial differential equations. With its help, solutions of complex vector boundary value problems for systems of differential equations can be decomposed into solutions of scalar boundary value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. Solutions to scalar boundary value problems are represented as fractals, self-similar mathematical objects first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of academician B. G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome, and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I. I. Vorovich.