Thermodynamically consistent, maximum dissipation, time-dependent models for non-equilibrium behavior

Abstract

A thermodynamically consistent construction yields evolution equations for the non-equilibrium behavior of a body. Given 2 n thermodynamic variables divided into control variables y 1 , … , y n and state variables x 1 , … , x n and a thermodynamic function of the controls φ ˆ ( y 1 , … , y n ) such that x i = ∂ φ ˆ / ∂ y i for i = 1 , … , n define the distinguished states of the system, a generalized thermodynamic function is defined such that the distinguished states are determined by a zero-gradient condition. The remaining states are the non-equilibrium states. The construction of objective, time-dependent, non-equilibrium evolution equations for the thermodynamic state variables is based on the Lie time derivative and on a novel maximum dissipation criterion that supplements the second law of thermodynamics. If φ ˆ is the free energy, the evolution tends to the long-term states distinguished by φ ˆ and represents viscoelastic or viscoplastic behavior. If φ ˆ is the entropy production, then the construction gives the non-steady evolution of thermodynamic fluxes to steady states distinguished by φ ˆ and produces physically realistic finite velocity thermal and mass transport. The combination of these two sets of objective evolution equations and the balance laws is the constitutive model that defines the behavior of a non-equilibrium process. The construction is inspired by Gibbs thermodynamics rather than continuum thermodynamics, but the Clausius–Duhem inequality is deduced from a Gibbs one-form that defines admissible processes. The construction is not restricted to processes near equilibrium and reproduces, as validation, several well-known constitutive models as maximum dissipation processes in the sense used here.