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Abstract.A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material in terms of Fourier-transformed quantities. A oneparameter family of free energies is constructed, ranging continuously from the maximum to the minimum free energies.The simplest case of single component stress and strain tensors and a single viscoelastic function is considered in this paper. Explicit formulae are given for the particular case of a discrete spectrum model material response. Introduction.We address the problem of finding general, explicit forms for the free energy of a viscoelastic solid. The investigation of this issue has a long history.Early work on the determination of the free energy of a linear viscoelastic solid by Staverman and Schwarzl [1] involved arguments based on mechanical models. These authors perceived the problem of non-uniqueness and used detailed model assumptions to deal with the issue. Their results were given independently by others (for example

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Abstract.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 the theory of linear viscoelasticity.In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic 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, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

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