The Boussinesq Potentials

Abstract

The Galerkin and Papkovich-Neuber solutions have the advantage of presenting a general solution to the problem of elasticity in a suitably compact notation, but they are not always the most convenient starting point for the solution of particular three-dimensional problems. If the problem has a plane of symmetry or particularly simple boundary conditions, it is often possible to develop a special solution of sufficient generality in one or two harmonic functions, which may or may not be components or linear combinations of components of the Papkovich-Neuber solution. For this reason, it is convenient to record detailed expressions for the displacement and stress components arising from the several terms separately and from certain related displacement potentials.

The first catalogue of solutions of this kind was compiled by Boussinesq¹ and is reproduced by Green and Zerna², whose terminology we use here. Boussinesq identified three categories of harmonic potential, one being the strain potential of §20.1, already introduced by Lamé and another comprising a set of three scalar functions equivalent to the three components of the Papkovich-Neuber vector, Ψ. The third category comprises three solutions particularly suited to torsional deformations about the three axes respectively. In view of the completeness of the Papkovich-Neuber solution, it is clear that these latter functions must be derivable from equation (20.17) and we shall show how this can be done in §21.3 below³.