Applying the method of virtual elements to solving contact problems of shells of revolution interacting with surfaces of inconsistent shape

Abstract

An approach based on the method of virtual elements is used to solve the contact problem for a thin shell of revolution lying on a rigid foundation. In this case, the surface of the base has a shape inconsistent with the surface of the shell. The method makes it possible to determine the contact area and the contact pressure from the contact area unknown in two coordinate directions. Since the contact area is not known in advance, the problem is structurally nonlinear. A thin isotropic shell is described by the classical theory based on the Kirchhoff–Love hypotheses. The base is taken absolutely rigid, but with the presence of an elastic gasket. The shell equations are integrated by S. K. Godunov’s method of discrete orthogonalization. To determine the forces of interaction between the shell and the base, a mixed method of structural mechanics is used. With this aim in view, the maximum possible contact area is discretized by virtual rectangular elements. A constant value of the contact pressure is assumed on each element obtained in this area, and the contact pressure is assumed to be zero on the elements in the area where the shell leaves the base. Based on the assumptions, a system of linear algebraic equations is constructed, which determines the contact pressure and deflection of the shell circumference axis. Since the shell can move away from the base, iterative procedures are applied to search for the real contact area, which depends on the geometric and elastic parameters of the shell and the magnitude of the external load. As an example, the contact interaction of a cylindrical shell (part of the shell of a tank car) lying on a rigid base with a gasket is considered. It is shown how the contact area and contact pressure change depending on the rigidity of the gasket and the difference between the radii of the shell and the base (inconsistency in the shape of the surfaces).