Structure of polynomial solutions of elasticity equations in stresses

Abstract

For an isotropic elasticity problem in stresses in three-dimensional space minus the origin, we study solutions that have the singularity 1/r2 and, after the multiplication by r2, polynomially depend on the direction cosines. In this polynomial class, for the equilibrium equation we write out the general solution that is a statically admissible (in the sense of Castigliano) solution of the Kelvin problem. We show that if one or several Beltrami equations are not satisfied, then the classical Kelvin solution becomes nonunique. A method for constructing nonunique solutions of this kind is given. The equivalence of various statements of the elasticity problem in stresses is discussed. For the problem on the action of a lumped force at the vertex of an arbitrary conical elastic body, we write out the exact solution in stresses for the case of an incompressible material. The solution for a compressible material is represented in the form of series in a parameter characterizing the deviation of the Poisson ratio from 1/2. We obtain iterative chains of problems in stresses and conditions for the finiteness of these chains. We also analyze the realizability of a linear-fractional dependence of the solution in stresses on the Poisson ratio.