Study of the axisymmetric contact problem of the wear of a pair consisting of an annular stamp and a rough half-space

The solution of on axisymmetric contact problem of wear in an elastic, rough half-space by an annular stamp, is used to construct a method of investigating a class of two-dimensional integral equations of the second kind containing Fredholm coordinate operators and Volterra time operators. By applying to these equations an analogue of the method of separation of variables, we can reduce the problem to consecutive solutions of integral Fredholm and Volterra equations of the second kind, and in the case of the Fredholm equation the problem solved is that of determining the eigenvalues and eigenfunctions. The method enables us to obtain expansions for the basic characteristics of the phenomenon of contact interaction, valid over the whole range of time changes. In the case of contact problems with wear and smooth bodies, the basic solution equations can be studied using the method of matched asymptotic expansions. The integral equation of the second kind with a logarithmic kernel obtained in the boundary layer, is solved in a semi-infinite interval in closed form by reducing it to a functional difference equation with shear.