Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Maximum and minimum free energies for a linear viscoelastic material


Authors: M. Fabrizio and J. M. Golden
Journal: Quart. Appl. Math. 60 (2002), 341-381
MSC: Primary 74D05; Secondary 74A15, 74A20
DOI: https://doi.org/10.1090/qam/1900497
MathSciNet review: MR1900497
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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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DOI: https://doi.org/10.1090/qam/1900497
Article copyright: © Copyright 2002 American Mathematical Society

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