Contact Problem for Inhomogeneous Cylinders with Variable Poisson’s Ratio

Abstract

In cylindrical coordinates, the system of two elastic-equilibrium differential equations is studied under the assumption of axial symmetry and the assumption that the Poisson’s ratio is an arbitrary, sufficiently smooth, function of the radial coordinate and the modulus of rigidity is constant. It turns out that the elastic coefficient is variable with respect to the radial coordinate in this case. We propose a general representation of the solution of this system, leading to the vector Laplace equation and scalar Poisson equation such that its right-hand side depends on the Poisson’s ratio. Being projected, the vector Laplace equation is reduced to two differential equations such that one of them is the scalar Laplace equation. Using the Fourier integral transformation, we construct exact general solutions of the Laplace and Poisson equations in quadratures. We obtain the integral equation of the axially symmetric contact problem on the interaction of a rigid band with an inhomogeneous cylinder and find its regular and singular asymptotic solutions by means of the Aleksandrov method.