On thermomechanical restrictions of continua

The central idea proposed here is that, in entropy-producing processes, a specific choice from among a competing class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production. When attention is restricted to quadratic forms for the rate of entropy production, the assumption leads to results that are fully in keeping with linear phenomenological relations that satisfy the Onsager relations. In other words, the usual linear evolution laws such as Fourier's law of heat conduction, Fick's law, Darcy's law, New-ton's law of viscosity, etc., all corroborate this assumption. We clarify the difference between the maximum rate of entropy production criterion that characterizes choices among constitutive relations and the minimum entropy production theorem due to Onsager (1931) that characterizes steady states for special choices of the rate of entropy production. We then show that for other forms of entropy production that are not quadratic for which the Onsager relations and related theorems cannot be applied, we can use the procedure described here to obtain nonlinear laws. We demonstrate by means of an example that even yield-type phenomena can be accommodated within this framework, while they cannot within the framework of Onsager. We also discuss issues concerning constraints, especially in thermoelasticity within the context of our ideas.