On the subsonic stationary motion of stamps and flexible cover-plates on the boundary of an elastic half-plane and a composite plane☆

Abstract A mixed dynamic problem for an elastic half-plane on different sections of whose boundary shear and normal stresses and displacements are given simultaneously in four fundamental combinations is considered. It is assumed that all the sections move at an identical constant subsonic velocity along the half-plane boundary and their number and mutual arrangement are arbitrary. An analogous problem on the interaction of two half-planes of different materials (a composite plane) is examined under the formulation of six kinds of contact conditions simultaneously in two modifications. The solutions are constructed in quadratures on the basis of new representations of the complex Galin potentials /1/. The first problem is reduced to a scalar combined Hilbert-Riemann boundary value problem /2/ for a plane with slots, and the second to unrelated Hilbert-Riemann and Hubert problems for the same domain. Both problems of the theory of analytic functions are solved by a new method different from /2/. The problem of the wedging of a composite plane by a finite stamp moving at a sub-Rayleigh velocity /3/, and the problem of the motion of a stamp and a flexible cover plate over a half-plane boundary at subsonic velocity are examined as examples. The exact solutions of stationary contact problems for a half-plane with two kinds of boundary conditions were first obtained by Galin /1/. The problem was formulated for a composite plane with three kinds of boundary conditions, whose solution is obtained in quadratures in the case of one slipping section /4/. However, as shown in /3, 5/, the method described in /4/ does not result in an exact solution for a large number of sections.