Modern Uniqueness Theorems in Three-Dimensional Elastostatics

Abstract

As outlined in the previous chapter, uniqueness theorems for the standard boundary value problems of classical linear isotropic elasticity when the elasticities are within the “physical” range, were found by Kirchhoff over a century ago. In addition, the Cosserats, almost forty years later, discovered that in the displacement boundary value problem of isotropic elasticity, uniqueness held for an extended range of values of the elasticities. In general, uniqueness theorems established by early workers made use of an appropriate energy identity. For the most part we continue to use this device in the present chapter in discussing the uniqueness of either weak or classical solutions to various standard boundary value problems in anisotropic and isotropic elasticity. We shall establish conditions sufficient (and in some cases also necessary) for uniqueness in these problems and make special mention of results peculiar to spherical regions.