RESEARCH OF THE MATHEMATICAL MODEL OF THE CONTACT OF ELASTIC HALF-SPACES AND A RING STAMP WITH INITIAL STRESSES

Abstract

A study of the mathematical model of the contact of two prestressed half-spaces pressing on an elastic ring cylinder with initial stresses is presented. It is assumed that the surfaces outside the contact boundary remain free from the influence of external forces, and at the contact boundary displacements and stresses are continuous. The study was performed in a general form for compressible (incompressible) bodies for the theory of large (final) initial deformations and two variants of the theory of small initial deformations with an arbitrary structure of the elastic potential using the relations of the linearized theory of elasticity without taking into account frictional forces. We assume that the initial states of the elastic cylindrical die and the elastic bases (half-spaces) are homogeneous and equal. The study is carried out in the coordinates of the initial deformed state, which are related to the Lagrangian coordinates (of the natural state). In addition, the influence of the cylindrical stamp causes small perturbations of the corresponding values of the main stress-strain state. The problem is solved for the case of equal roots of the characteristic equation and is formulated as a solution of triple integral equations. They are reduced to one integral equation by substitution. Since the problem is axisymmetric, the kernel of the integral equation depends on the product of three Bessel functions. For the solution, a formula representing the product of two Bessel functions in a series was used. This made it possible to reduce the problem to a functional equation. This equation relates the die displacement to the unknown contact stress distribution coefficients. In turn, the obtained functional equation was reduced to an infinite system of linear algebraic equations. We solve this system by the method of reduction. When a load is applied to the ring dies, the distribution of contact stresses is found in the form of a series of products of the connected Legendre functions. The numerical analysis is presented graphically for the case of harmonic potential. It is important to note that taking into account the initial (residual) stresses within the linearized theory of elasticity significantly changes the formulation and significantly complicates the solution of the contact problem. The method proposed in the article made it possible to reveal the influence of initial stresses on the contact characteristics of bodies and contribute to increasing the reliability and durability of engineering structures and structures.