On the three stress tensors for linearly elastic constrained materials

In this paper we obtain the constitutive equation for the second Piola-Kirchhoff stress tensor according to the linearized finite theory of elasticity for hyperelastic constrained materials. We show that in such a theory the three stress tensors (Cauchy stress tensor, first and second Piola-Kirchhoff stress tensor) differ by terms that are first order in the strain, while in classical linear theory of elasticity they are indistinguishable to first order of approximation both for unconstrained and constrained materials. Moreover we show that the constitutive equations for the three stress tensors usually adopted in classical linear elasticity are not correct to first order in the strain. Finally we provide an example for a particular material symmetry and for a particular constraint in which the three stress tensors coincide, while in general they are different.