Nonisothermal free energies for linear theories with memory

Abstract

Explicit expressions for minimum, maximum, and intermediate free energies have recently been given for isothermal linear theories. In this paper, these results are generalized to a nonisothermal, linear model. Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within a linear theory. In particular, an integral equation for the continuation associated with the maximum recoverable work from a given state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, generalizing a result previously derived in the isothermal case. A new variational method developed for the isothermal case is extended to nonisothermal conditions. In the time domain, this approach yields integral equations for both the minimum and maximum free energies associated with a given state. In the frequency domain, explicit forms of a family of free energies, associated with a given state of a discrete spectrum material, are derived. This includes both maximum and minimum free energies. These latter developments, given previously for the isothermal scalar case, are generalized (with some restrictions) in this work to the tensor case. The physical relevance of various results are discussed.