CONSTRUCTION OF SOLUTIONS OF THE BASIC EQUATIONS OF THE LINEARIZED THEORY OF ELASTICITY FOR RING BODIES

Abstract

A method of constructing solutions of the basic equations of the linearized theory of elasticity for bodies (stamps) of ring shape with an arbitrary cross-sectional contour is presented. The solutions are written in the system of circular cylindrical coordinates for an elastic finite ring stamp with initial (residual) stresses for the case of axisymmetric deformation relative to the geometric axis of the body. The presented method of construction can be used in the study of spatial axisymmetric static contact problems of the linearized theory of elasticity in the coordinates of the initial deformed state for compressible and incompressible bodies in the case of uniform initial stresses. This will 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. 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 classical contact problems. Therefore, the results proposed in the article can be used to solve similar problems where there are annular or even cylindrical stamps. The obtained solutions can also significantly help in the derivation of analytical dependencies for the components of the stress-strain state of finite ring dies with initial (residual) stresses when the boundary conditions of a specific contact problem are met. The solutions are obtained in the form of harmonic functions that satisfy the Laplace equation. They are derived using the method of separation of variables (Fourier method) and adapted to meet the boundary conditions of specific contact problems. The condition for the existence of a single solution of the basic differential equation of the linearized theory of elasticity for compressible and incompressible bodies is the condition of strong ellipticity of the equations. Taking this into account, we will present the general solutions of ring bodies in two possible variants, namely: 1) in the case of equal roots of the differential equation; 2) in the case of unequal roots. This construction approach made it possible to use the obtained results for numerical studies of the contact interaction of elastic bodies in cases of arbitrary structure of their elastic potential.