On the integrable case of a Riemann-Hilbert boundary value problem for two functions and the solutions of certain mixed problems for a composite elastic plane

Abstract

A method for solving the Riemann-Hilbert boundary value problem with piecewise-constant coefficients is generalized /1/. It is shown that the following static problems of a composite elastic plane with three kinds of connection conditions allow of exact solutions: 1) the splicing line is weakened by a system of loaded slots and a transverse shear crack or the edges of one of the slots are partially contacting, or one of the slots is cleaved by a rigid insert; 2) the splicing line is reinforced by a system of thin rigid inclusions and there is one arbitrarily located delamination zone; 3) the elastic half-planes are contacting (with slip) on a certain section of their boundaries, and mixed boundary conditions in the displacements and stresses are given on the rest of the boundaries. In the general case the Riemann-Hilbert boundary value problem for many functions reduces to the problem of a linear conjugation, and then to Fredholm integral Eqs./2/. Closed solutions are obtained in certain special cases /3–5/. For applications we mention the papers /6, 7/, where problems are considered concerning slits at the interface of two elastic media with two kinds of physical boundary conditions taken into account simultaneously.