Contact problems of an inclusion in a plane elastic wedge

Plane contact problems are considered for an isotropic homogeneous elastic wedge with a thin rigid inclusion of a finite length located on its bisector. The outer faces of the wedge are subject to rigid or sliding fixation. The problems are symmetric with respect to the bisector of the wedge. The inclusion is completely coupled with the elastic medium in the contact region. A tangential force is applied to the inclusion, under the action of which it is displaced along the bisector by a given value. Using the Mellin integral transform, the contact problems are reduced to integral equations with respect to tangential contact stresses, from which the integral equations of the corresponding problems for an elastic strip can be obtained by limiting passages. Special cases also include problems with one or two inclusions in an elastic plane. The main dimensionless geometric parameter is introduced, which characterizes the relative distance of the inclusion from the wedge apex. Three methods are used to solve the integral equations. The first method consists of obtaining a closed solution based on a special approximation of the kernel symbol. The second method, regular asymptotic, involves expansion of the solution in powers of a small parameter and is effective for inclusions relatively distant from the wedge apex. The third method, singular asymptotic, involves expansion of the solution into several parts and solution of the Wiener - Hopf integral equations. A degenerate solution and a superposition of boundary layer solutions are taken. This method works for inclusions located relatively close to the wedge apex. Using the three methods, a numerical analysis is performed for different types of boundary conditions, values of the wedge angle, Poisson's ratio, and the main dimensionless parameter.