On the generalized Mindlin problem

Abstract

The classical Mindlin paper [1] published in 1936 consists of first and second sections concerning concentrated forces perpendicular and parallel to the boundary of the elastic half-space, respectively. In paper [2], the fundamental solutions obtained by Lord Kelvin and Mindlin (the first section of [1]) are generalized for the case of a three-dimensional elastic wedge where a concentrated force is perpendicular to its edge under different types of boundary conditions at its sides. Below, we present expressions for three Papkovich–Neuber harmonic functions for a wedge that has unstressed sides and where a concentrated force parallel to its edge acts in its middle half-plane. When the opening angle of the wedge corresponds to a halfspace, the expressions for elastic displacements and stresses coincide with formulas from the second section of [1]. Displacements at the wedge edge are calculated as well. The solutions obtained by Boussinesq and Cherutti for an elastic wedge with one side subjected to normal and tangential loads and the other side free of stress are generalized in [3]. The problems of a threedimensional wedge are solved by presenting a harmonic function as the complex Fourier–Kontorovich– Lebedev integral and by reducing the three-dimensional problem of elasticity theory to the Hilbert boundary value problem generalized in the sense of Vekua [4–6].