Thermodynamic consistency of objective rate equations

The paper addresses the modelling of solids via rate equations for the heat flux and the stress. As with any constitutive property, the rate equations are required to be both objective and consistent with the second law of thermodynamics. Upon a review of a connection between objective time derivatives, attention is restricted to rate equations where the time derivatives are those named after Jaumann, Green and Naghdi, Oldroyd, and Truesdell. Equations involving the Truesdell rate are investigated within the material description thanks to the identity between the time derivative of the material fluxes and the Truesdell rate in the current configuration. The remaining equations are examined within the spatial description. The occurrence of a skew tensor in the objective derivative results in a further restriction on the constitutive properties. The thermodynamic requirements are found to be satisfied and the corresponding free energy is determined by direct integrations.