The Reissner-Sagoci problem

Abstract

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates (p, φ, z), the displacement vector has only one non-vanishing component uφ (p, z), and the stress tensor has only two non-vanishing components, σρπ(p, z) and σπz(p, z). The stress-strain relations reduce to the two simple equationswhere μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equationis satisfied provided that the function uπ(ρ, z) is a solution of the partial differential equationThe boundary conditions can be written in the formwhere, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, uπ, σρπ and σπz all tend to zero.