Keywords: continuum thermodynamics, materials with memory, viscoelasticity, minimum free energy, maximum free energy, function of state, minimal state, factorization, frequency domain, analyticity, isolated singularities, branch cuts.

It has been known for many years that the free energy of materials with memory is not necessarily a uniquely defined quantity [1,2,3,4]. In fact, the free energies associated with a given state of such a material generally form a convex set with a minimum and a maximum element. In this work, explicit expressions are given for these quantities, in the context of a specific model.

Definitions are given of the maximum and minimum free energy associated with a given state of a material with memory, and the dissipation associated with a given free energy and work on a given state. Also, the concept of a minimal state is introduced. This is a class of states equivalent to each other in the sense of Noll [5].

These concepts are then explored in detail for a specific isothermal model, where the stress is given by a non-linear elastic part, which is a function of the current strain tensor, and a memory part which is a linear functional of the history of the strain tensor. Small strains are not assumed. This model is more general than those considered in earlier work. It is readily extended to non-isothermal configurations.

For an isothermal model, indeed without the restriction to linear memory terms, we may define the minimal state concept as follows: if two different histories of strain are equal at time and if the associated stresses are equal at this and all subsequent times, then the strain histories are equivalent and belong to the same minimal state.

An expression is presented for the minimum free energy [6,7] using a factorization property of a particular positive-definite tensor. The minimum free energy is a function of the minimal state, which is defined for the models under consideration using a minor extension of the approach of [8,9] and [10].

The main part of the work deals with maximum free energies and their relationship with minimal states. Firstly, it is pointed out that the memory and dissipative properties of the material are characterized by the singularity structure of the Fourier transform of the relaxation tensor derivative in the complex frequency domain. Such singularities may be of three types, in general: isolated singularities, branch cuts and essential singularities. The last category are excluded.

It is shown that the equivalence class of states constituting the minimal state is a singleton except in the case where only isolated singularities occur in the Fourier transform of the relaxation tensor derivative. These correspond to relaxation tensors which consist of sums of exponentials, oscillarory and decaying with time, multiplied by polynomials.

If the minimal state is a singleton then the state is defined by the current value and history of strain. The maximum free energy is simply the work function which is a function of state. If the minimal state is not a singleton, then the maximum free energy is less than the work function and is a function of the minimal state. An explicit expression is given for this quantity, using a variant of the factorization used to determine the minimum free energy.

Special cases of these results were given in [10], based on constrained optimization techniques. The more general results presented here rely on much simpler arguments.

1

A.J. Staverman, F. Schwarzl , "Thermodynamics of viscoelastic behaviour", Proc. Konink. Nederl. Akad. Wettensch. B55, 474-485, 1952.

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W. Noll, "A new mathematical theory of simple materials", Arch. Rational Mech. Anal. 48, 1-50, 1972.

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J.M. Golden, "Free energies in the frequency domain: the scalar case", Quart. Appl. Math., 58, 127-150, 2000.

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L. Deseri, G. Gentili, J.M. Golden, "An explicit formula for the minimum free energy in linear viscoelasricity", J. Elasticity ,54, 141-185, 1999. doi:10.1023/A:1007646017347

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G. Del Piero, L. Deseri, "On the concepts of state and free energy in linear viscoelasticity", Arch. Rational Mech. Anal. 138, 1-35, 1997.

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G. Del Piero, L. Deseri, "On the analytic expression of the free energy in linear viscoelasticity", J. Elasticity 43, 247-278, 1996. doi:10.1007/BF00042503

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M. Fabrizio, J.M. Golden, "Maximum and minimum free energies for a linear viscoelastic material", Quart. Appl. Math., 60, 341 - 381, 2002.

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