Contact problems for an elastic inhomogeneous body with a cylindrical cavity

An axially symmetric elastic equilibrium problem is investigated for a continuously inhomogeneous space with a cylindrical cavity when Poisson’s ratio is being an arbitrary fairly smooth function with respect to radial coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the radial coordinate. A general solution is suggested which leads us to a vector Laplace equation and a scalar Poisson equation whose right-hand side depends on Poisson’s ratio. As a result, exact general solutions of the Laplace and Poisson equations are constructed in integral forms with the help of Fourier transformations. Then integral equations of two axially symmetric contact problems are derived on the interaction between the cavity surface and a rigid cylindrical insert fitted with interference. In the first problem the contact is supposed to be absolutely ideal, a singular asymptotical method is used here to solve the integral equation of the first kind with respect to the contact pressure, which is effective for fairly long inserts. In the second problem the mine surface is supposed to be rough simulated by an extra Winkler type, a collocation method is used for solving the integral equation of the second kind, which is effective for fairly short inserts. The contact pressure has typical square root singularities at end-points in the first problem while it takes finite values at those points in the second problem. For a homogeneous material, the integral characteristics of the contact pressures in both problems are close for small coefficients of roughness for some values of the insert relative length. It is shown that the rough surface distributes contact pressures more uniformly removing effect of nonhomogeneity. The calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the cavity.