Second-order stress relations for hyperelastic constrained materials

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.