Free energies in the frequency domain: the scalar case

Author:
J. M. Golden

Journal:
Quart. Appl. Math. **58** (2000), 127-150

MSC:
Primary 74D05; Secondary 30E20, 74A20

DOI:
https://doi.org/10.1090/qam/1739041

MathSciNet review:
MR1739041

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Abstract | References | Similar Articles | Additional Information

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 one-parameter family of free energies is constructed, ranging continuously from the maximum to the minimum free energies.

**[1]**A. J. Staverman and F. Schwarzl,*Thermodynamics of viscoelastic behaviour*, Proc. Konink. Nederl. Akad. Wettensch.**B55**, 474-485 (1952)**[2]**D. Bland,*The Theory of Linear Viscoelasticity*, Pergamon, London, 1960 MR**0110314****[3]**S. C. Hunter,*Tentative equations for the propagation of stress, strain and temperature fields in viscoelastic solids*, J. Mech. Phys. Solids**9**, 39-51 (1961) MR**0165780****[4]**S. Breuer and E. T. Onat,*On recoverable work in linear viscoelasticity*, Z. Aneew. Math. Phys.**15**, 13-21 (1964) MR**0178644****[5]**S. Breuer and E. T. Onat,*On the determination of free energy in viscoelastic solids*, Z. Angew. Math. Phys.**15**, 185-191 (1964)**[6]**B. D. Coleman,*Thermodynamics of materials with memory*, Arch. Rational Mech. Anal.**17**, 1-46 (1964) MR**0171419****[7]**B. D. Coleman and V. J. Mizel,*A general theory of dissipation in materials with memory*, Arch. Rational Mech. Anal.**27**, 255-274 (1967) MR**0220474****[8]**B. D. Coleman and D. R. Owen,*A mathematical foundation for thermodynamics*, Arch. Rational Mech. Anal.**54**, 1-104 (1974) MR**0395502****[9]**B. D. Coleman and D. R. Owen,*On thermodynamics and elastic-plastic materials*, Arch. Rational Mech. Anal.**59**, 25-51 (1975) MR**0381526****[10]**W. A. Day,*Thermodynamics based on a work axiom*, Arch. Rational Mech. Anal.**31**, 1-34 (1968) MR**1553518****[11]**W. A. Day,*Reversibility, recoverable work and free energy in linear viscoelasticity*, Quart. J. Mech. Appl. Math.**23**, 1-15 (1970) MR**0273881****[12]**W. A. Day,*The thermodynamics of materials with memory*, in*Materials with Memory*, ed. D. Graffi, Liguori, Naples, 1979**[13]**M. Fabrizio and A. Morro,*Thermodynamic restrictions on relaxation functions in linear viscoelasticity*, Mech. Res. Comm.**12**, 101-105 (1985)**[14]**M. Fabrizio and A. Morro,*Viscoelastic relaxation functions compatible with thermodynamics*, J. Elasticity**19**, 63-75 (1988) MR**928727****[15]**D. Graffi and M. Fabrizio,*Sulla nozione di stato materiali viscoelastici di tipo ``rate"*, Atti Accad. Naz. Lincei**83**, 201-208 (1990) MR**1142459****[16]**D. Graffi and M. Fabrizio,*Non unicità dell'energia libera per materiali viscoelastici*, Atti Accad. Naz. Lincei**83**, 209-214 (1990)**[17]**A. Morro and M. Vianello,*Minimal and maximal free energy for materials with memory*, Boll. Un. Mat. Ital.**4A**, 45-55 (1990) MR**1047512****[18]**M. Fabrizio and A. Morro,*Mathematical Problems in Linear Viscoelasticity*, SIAM, Philadelphia, 1992 MR**1153021****[19]**M. Fabrizio, C. Giorgi, and A. Morro,*Free energies and dissipation properties for systems with memory*, Arch. Rational Mech. Anal.**125**, 341-373 (1994) MR**1253168****[20]**M. Fabrizio,*Existence and uniqueness results for viscoelastic materials*, in*Crack and Contact Problems for Viscoelastic Bodies*, eds. G. A. C. Graham and J. R. Walton, Springer-Verlag, Vienna, 1995 MR**1411685****[21]**M. Fabrizio, C. Giorgi, and A. Morro,*Internal dissipation, relaxation property and free energy in materials with fading memory*, J. Elasticity**40**, 107-122 (1995) MR**1364749****[22]**A. Morro,*Wave solutions in linear viscoelastic materials*, in*Crack and Contact Problems for Viscoelastic Bodies*, eds. G. A. C. Graham and J. R. Walton, Springer-Verlag, Vienna, 1995 MR**1411687****[23]**G. Del Piero and L. Deseri,*On the concepts of state and free energy in linear viscoelasticity*, Arch. Rational Mech. Anal.**138**, 1-35 (1997) MR**1463802****[24]**G. Del Piero and L. Deseri,*On the analytic expression of the free energy in linear viscoelasticity*, J. Elasticity**43**, 247-278 (1996) MR**1415545****[25]**M. E. Gurtin and W. J. Hrusa,*On energies for nonlinear viscoelastic materials of single-integral type*, Quart. Appl. Math.**46**, 381-392 (1988) MR**950610****[26]**J. M. Golden and G. A. C. Graham,*Boundary Value Problems in Linear Viscoelasticity*, Springer-Verlag, Berlin, 1988 MR**958684****[27]**N. I. Muskhelishvili,*Singular Integral Equations*, Noordhoff, Groningen, 1953 MR**0355494****[28]**M. E. Gurtin and I. Herrera,*On dissipation inequalities and linear viscoelasticity*, Quart. Appl. Math.**23**, 235-245 (1965) MR**0189346****[29]**E. C. Titchmarsh,*Introduction to the Theory of Fourier Integrals*, Clarendon, Oxford, 1937**[30]**I. N. Sneddon,*The Use of Integral Transforms*, McGraw-Hill, New York, 1972**[31]**P. M. Morse and H. Feshbach,*Methods of Theoretical Physics*, McGraw-Hill, New York, 1953 MR**0059774**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1739041

Article copyright:
© Copyright 2000
American Mathematical Society