A note on the linearized finite theory of elasticity

In this paper we solve a simple dead load problem for an isotropic constrained material according to the linearized finite theory of elasticity. We show that the solution of such a problem can be obtained by linearizing with respect to the displacement gradient the solution of the corresponding problem in finite elasticity for an isotropic material subject to the same constraint, exactly as occurs for the constitutive equations of the two theories. On the contrary, the solution of the same dead load problem provided by the classical linear elasticity for constrained materials can be obtained by the solution of the corresponding problem for the unconstrained linear elastic material for limiting behaviour of suitable elastic moduli. The same applies for the constitutive equations of the classical linear elasticity for constrained materials: they are derived by those of the linear elasticity for unconstrained materials for limiting values of some elastic modulus. Finally we compare the solutions in finite elasticity, linearized finite theory of elasticity, classical linear elasticity for constrained materials and we show that they are in agreement with different hypotheses on the prescribed loads.