MATHEMATICAL MODELING OF CONTACT INTERACTION OF A PRE-STRESSED RING STAMP AND ELASTIC HALF-SPACE WITH INITIAL STRESSES

Abstract

The article is devoted to the mathematical modeling of the contact interaction of a pre-stressed annular die and an elastic half-space with initial (residual) stresses. The solution for annular elastic die is presented, which take into account the influence of initial stresses. The problem is solved in the case of equal roots of the defining equation. The study is presented in general for the theory of large initial deformations and two versions of the theory of small initial deformations in the framework of the linearized theory of elasticity for an arbitrary structure of the elastic potential. It is assumed that the initial states of the elastic annular die and the elastic half-space are homogeneous and equal. The study was carried out in the coordinates of the initial deformed state, which are associated with the Lagrangian coordinates (natural state). In addition, the effect of the annular die causes small perturbations of the corresponding values of the basic stress-strain state. It is also assumed that the elastic annular die and the elastic half-space are made of various isotropic, transversely isotropic, or composite materials. General solutions of the main differential equations of the linearized theory of elasticity in the case of axisymmetric deformation for a finite annular region are presented. As a result, the solutions to the problem posed are presented in the form of infinite series. The coefficients of this series are determined from an infinite system of algebraic equations. The study of the influence of the initial (residual) stresses in the half-space and the annular die on the distribution of contact characteristics in the contact area is carried out. In the case of equal roots and the Bartenev-Khazanovich`s potential, the results of numerical analysis are presented. These results are presented in the form of graphs. They illustrate the rather significant influence of the initial stresses. Therefore, the effect of initial stresses on the stress-strain state of the elastic annular die, which is pressed into the elastic half-space, is that: in the case of compression, the initial stresses in the half-space lead to a decrease in stresses in the elastic die, and in the case of stretching - to their increase. But in the case of displacement is the opposite.