Some uniqueness theorems of solutions for the problemsof elasticity

Abstract

In order to construct the well-posed mathematical models of the elastic problems, it is necessary to study the mathematical properties of solutions, including existence, uniqueness, and stability (the continuous dependence on boundary conditions). The uniqueness theorem of solutions provides the methods for solving the problem. The uniqueness of the solutions is one of the most basic and important issues of elasticity theory. In the three-dimensional elasticity theory, unconditional uniqueness is not expected. Therefore, the uniqueness theorem of solutions is established under certain conditions, such as the restrictions on the elasticity tensor, the strain energy function and elastic deformation range. This study reviews the background and history of the uniqueness theorem in elasticity. The focuses are on the uniqueness theorems of solutions in the boundary-value problems of the linear elasticity theory, nonlinear elasticity theory of finite deformation, and elasticity theory with initial stress field. The classical proofs of uniqueness theorems are also given. We also present some unsolved problems on the solution uniqueness in elasticity.