V. A. Babeshko, O. V. Evdokimova, O. M. Babeshko, M. V. Zaretskaya, V. S. Evdokimov
{"title":"On Contact Problems with a Deformable Punch and Variable Rheology","authors":"V. A. Babeshko, O. V. Evdokimova, O. M. Babeshko, M. V. Zaretskaya, V. S. Evdokimov","doi":"10.1134/s1063454123040027","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>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. The solutions to the 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. A contact problem with a deformable punch admits the construction of a solution if it is possible to solve the contact problem for an absolutely rigid punch and construct a solution to the boundary problem for a deformable punch. In earlier works of the authors, the deformable stamp was described by a separate Helmholtz equation. In this paper, we consider a contact problem on the action of a semiinfinite stamp on a multilayer base, described by the system of Lame equations. One of the methods of transition to other rheologies is shown when describing the properties of a deformable stamp in contact problems.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"117 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454123040027","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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. The solutions to the 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. A contact problem with a deformable punch admits the construction of a solution if it is possible to solve the contact problem for an absolutely rigid punch and construct a solution to the boundary problem for a deformable punch. In earlier works of the authors, the deformable stamp was described by a separate Helmholtz equation. In this paper, we consider a contact problem on the action of a semiinfinite stamp on a multilayer base, described by the system of Lame equations. One of the methods of transition to other rheologies is shown when describing the properties of a deformable stamp in contact problems.
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
