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Mathematics and Mechanics of Solids
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A Proposal Concerning the Physical Rate of Dissipation of Materials with Memory: The Non-isothermal Case

J.M. Golden

School of Mathematical Sciences, Dublin Institute of Technology, Dublin 8, Ireland

It has been known for several decades that the free energy and entropy of a material with memory is not in general uniquely determined, nor is the rate of dissipation. The objective of the present work is to propose a formula for the physical free energy and rate of dissipation of a material with memory, described by a constitutive equation with linear memory terms and non-linear equilibrium contributions, under non-isothermal conditions. This is a generalization of work recently presented for the linear scalar case. This formula follows from a new physical hypothesis of Maximum Parametric Symmetry, which states that the physical free energy has the closest possible level of symmetry among the parameters of the theory to that of the work function. The final formula proposed can be expressed in simple, closed form.

It is shown that non-trivial equivalence classes of states, in the sense of Noll, exist only if the material has a relaxation function derivative, the Fourier transform of which has only isolated singularities in the complex frequency plane. The members of the family of free energies used to determine the physical free energy are all functions of such an equivalence class.

Key Words: Materials with memory • viscoelasticity • thermodynamics • free energy • non-isothermal • dissipation

Mathematics and Mechanics of Solids, Vol. 12, No. 4, 403-449 (2007)
DOI: 10.1177/1081286505061450
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