Action of a band stamp on a three-dimensional elastic wedge

Abstract

The pressure of a band stamp on a three-dimensional elastic incompressible wedge was considered in [i, 6]. The corresponding integral equations were obtained by applying Fourier and Kontorovich-Lebedev integral transformations at the real axis to the Lame equations and the corresponding boundary conditions, which are greatly simplified when v = 1/2, as noted in [9]. In [5, 8], to solve the first basic boundary problem of elasticity theory for a three-dimensional wedge, integral Fourier and Kontorovich--Lebedev transformations in the complex plane were effectively used. In [5], the first boundary problem for the case of symmetric loading corresponding to slipping attachment of one face of the wedge, with halving of the aperture angle, was reduced to a Fredholm integral equation of the second kind, and it was proven that, if the load is such that the right-hand side of this equation belongs to the space L2(0, ~), the equation may be solved by successive approximation. It was found that, with a point load, the right-hand side of the Fredholm equation does not belong to L2(0, ~), but corresponds to the space of the functions CM(0 , ~), which are continuous and finite at the semiaxis. An analogous situation arises for a mixed boundary problem, in which the load is specified at one face of the wedge and the displacement at the other. In the present work, the norms of the Fredholm integral operators in the space CM(0, ~) are investigated for the first time for different types of boundary conditions at a single face of the wedge, and the compressibility of these operators in a certain range of variation of the Poisson's ratio ~ at different wedge aperture angles is proven, offering the possibility of representing the corresponding Green's functions by integrals of Neumann series with respect to (i 2v). Then the integral equations of the contact problems on the action of a band-shaped stamp on a three-dimensional elastic wedge are studied.