Contact problems for an inhomogeneous elastic wedge with variable Poisson’s ratio

Abstract

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.