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.

中文翻译：

---

关于可变形冲头和可变流变学的接触问题

摘要

本文首次提出了一种研究和解决需要改变印模材料流变性能的变形印模接触问题的方法。它基于作者之前发布的一种新的通用建模方法，该方法用于偏微分方程组的边值问题。借助它，微分方程组的复向量边值问题的解可以分解为各个微分方程的标量边值问题的解。其中，亥姆霍兹方程最为简单。标量边值问题的解被表示为分形，即自相似的数学对象，首先由美国数学家 B. Mandelbrot 提出。分形的作用是由填充块元素来执行的。从偏导数微分方程组到单独方程的转变是利用 BG Galerkin 院士的变换或势的表示来实现的。众所周知，具有复杂流变性的可变形印模的动态接触问题的求解十分繁琐，研究也一直是困难的。由于此类问题中存在离散谐振频率，问题变得更加复杂，这一点曾经被院士 II Vorovich 发现过。如果能够解决绝对刚性冲头的接触问题并构造可变形冲头边界问题的解，则可变形冲头的接触问题允许构造解。在作者的早期作品中，可变形印模是通过单独的亥姆霍兹方程来描述的。在本文中，我们考虑了半无限印模在多层基底上作用的接触问题，由拉梅方程组描述。在描述接触问题中可变形印模的特性时，显示了过渡到其他流变学的方法之一。