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.
In this paper the second-order stress relations for hyperelastic internally constrained materials are derived, both for the Cauchy stress and the two Piola-Kirchhoff stresses. In our approach the constitutive equations are obtained by the corresponding finite constitutive equations by means of suitable expansions. In contrast to the classical approach, our method guarantees the accuracy required by a second-order theory. For incompressible isotropic materials explicit stress relations are derived and compared with those used in classical theory, in order to show that only our constitutive equations are accurate to second order of approximation.
The problem of nonuniqueness of static axisymmetric solutions for a non-linearly elastic cylindrical shell in which the ends are pulled apart with a constant traction while retaining the radii of its ends fixed is studied. In the elastic case, we prove the existence of buckled states and the possibility of necking. In the hyperelastic case a global existence and nonuniqueness theorem is proved, via the energy criterion.
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