Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A proposal concerning the physical rate of dissipation in materials with memory

Author: J. M. Golden
Journal: Quart. Appl. Math. 63 (2005), 117-155
MSC (2000): Primary 74A15, 74D05; Secondary 30E20
DOI: https://doi.org/10.1090/S0033-569X-05-00958-2
Published electronically: January 19, 2005
MathSciNet review: 2126572
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Abstract: 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 are the total dissipation in the material over a given time period and the rate of dissipation. The dissipation in a material element would in particular seem to be a quantity that has immediate physical objectivity. It must be seen therefore as a significant weakness in the thermodynamics of materials exhibiting memory effects, that a quantity as basic as the rate of dissipation cannot be predicted in terms of the constitutive parameters.

The objective of the present work is to propose a formula for the physical free energy of a linear scalar viscoelastic material in terms of a family of free energies, each of which can be regarded as an estimate of the physical quantity. This formula follows from a new physical hypothesis of Maximum Parametric Symmetries, which states that the physical free energy and dissipation have the closest possible level of symmetry among the parameters of the theory to that of the work function. This results in the assignment of explicit weights to all members of the family of free energies, each of these being associated with a particular factorization of a quantity closely related to the loss modulus of the material. It is interesting that the final formula proposed for the physical free energy can be expressed in simple, closed form. Once the free energy is known, the corresponding physical rate of dissipation can also be determined without difficulty.

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. The derivation of their form is a generalization of work reported in Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60, 341 - 381 (2002).

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Additional Information

J. M. Golden
Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
Email: murrough.golden@dit.ie

DOI: https://doi.org/10.1090/S0033-569X-05-00958-2
Received by editor(s): June 18, 2004
Published electronically: January 19, 2005
Additional Notes: My thanks to M. Fabrizio, L. Deseri and G. Gentili (deceased December 2000) for several useful conversations
Article copyright: © Copyright 2005 Brown University

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