A non-linear thermoelastic constitutive model for isotropic materials based on Gibbs free energy

Abstract

This paper presents a Gibbs free energy-based constitutive framework for large thermoelastic deformations of isotropic materials. The independent constitutive variables are the Kirchhoff stress tensor and the temperature, while the dependent variables are the spatial logarithmic Hencky strain tensor and the entropy. The approach relies on the multiplicative decomposition of the deformation gradient, which naturally leads to an additive decomposition of the Hencky strain into uncoupled deviatoric, volumetric, and thermal parts. The Gibbs potential is likewise additively decomposed into elastic and thermal parts. The elastic part of the Gibbs free energy per unit intermediate volume can be taken as any of the complementary energy potentials developed for isothermal, non-linear elastic deformations, with the additional assumption that the material coefficients are temperature-dependent. The thermal part of the Gibbs free energy depends on the spherical component of the Kirchhoff stress tensor and the temperature. General forms of the constitutive equations for the Hencky strain and entropy are derived for compressible materials. In the case of incompressible materials, the inherently implicit constitutive model yields explicit relations between the Hencky strain and Cauchy stress components. Special forms of the thermoelastic constitutive equations are derived and investigated for two cases: (i) a modified Hencky-type model suitable for moderately large strains, and (ii) a power-law form of the elastic Gibbs free energy expressed in terms of stress invariants. The predictive capabilities of these models, particularly with respect to the thermoelastic inversion effect and structural heating in rubber-like materials, are evaluated through parameter fitting to experimental data. Comparisons are also made with predictions from a thermoelastic extension of Ogden’s constitutive model.