The theory of elasticity is primary a theory for isotropic, linearly elastic materials subjected to small deformations. All governing equations in this theory are linear partial differential equations, which means that the principle of superposition may be applied: The sum of individual solutions to the set of equations is also a solution to the equations. The theory of elasticity has a theorem of uniqueness of solution and a theorem of existence of solution. The theorem of uniqueness insures that if a solution of the pertinent equations and the proper boundary conditions for a particular problem is found, then this solution is the only solution to the problem. The theorem is presented and proven in Sect. The theorem of existence of a solution is fairly difficult and complicated to prove and perhaps not so important, as from a practical view point we understand that a physical solution must exist.
Temperature changes in a material result in strains and normally also in stresses, so-called thermal stresses.
Analytical solutions are, apart from being of importance as solutions to practical problems, also serving as test examples for numerical solution procedures like finite difference methods and finite element methods.