Entropy and temperature gradients thermomechanics: Dissipation, heat conduction inequality and heat equation

An alternative and consistent approach, not appealing to the principle of virtual power and to Coleman–Noll procedure, is used to derive constitutive and governing equations involving temperature or entropy gradients, in thermomechanics of materials. Using the balance of energy, an analysis of the dissipation naturally leads to the definition of the temperature and the entropy as variational derivatives. The approach preserves the classical forms of the equations and yields to consistent form of the second law and heat conduction inequality. The framework of generalized standard materials is then suitable for deriving admissible constitutive laws. The methodology is applied, first using entropy and its gradient as state variables (with internal energy as thermodynamic potential), and second using temperature and its gradient (starting from the free energy)