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: 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.
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A. J. Staverman and F. Schwarzl, Thermodynamics of viscoelastic behaviour, Proc. Konink. Nederl. Akad. Wettensch. B55, 474-485 (1952)
D. Bland, The Theory of Linear Viscoelasticity, Pergamon, London, 1960
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)
S. Breuer and E. T. Onat, On recoverable work in linear viscoelasticity, Z. Aneew. Math. Phys. 15, 13-21 (1964)
S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. Angew. Math. Phys. 15, 185-191 (1964)
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17, 1-46 (1964)
B. D. Coleman and V. J. Mizel, A general theory of dissipation in materials with memory, Arch. Rational Mech. Anal. 27, 255-274 (1967)
B. D. Coleman and D. R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54, 1-104 (1974)
B. D. Coleman and D. R. Owen, On thermodynamics and elastic-plastic materials, Arch. Rational Mech. Anal. 59, 25-51 (1975)
W. A. Day, Thermodynamics based on a work axiom, Arch. Rational Mech. Anal. 31, 1-34 (1968)
W. A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-15 (1970)
W. A. Day, The thermodynamics of materials with memory, in Materials with Memory, ed. D. Graffi, Liguori, Naples, 1979
M. Fabrizio and A. Morro, Thermodynamic restrictions on relaxation functions in linear viscoelasticity, Mech. Res. Comm. 12, 101-105 (1985)
M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity 19, 63-75 (1988)
D. Graffi and M. Fabrizio, Sulla nozione di stato materiali viscoelastici di tipo “rate", Atti Accad. Naz. Lincei 83, 201-208 (1990)
D. Graffi and M. Fabrizio, Non unicità dell’energia libera per materiali viscoelastici, Atti Accad. Naz. Lincei 83, 209-214 (1990)
A. Morro and M. Vianello, Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. 4A, 45-55 (1990)
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992
M. Fabrizio, C. Giorgi, and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125, 341-373 (1994)
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
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)
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
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)
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity 43, 247-278 (1996)
M. E. Gurtin and W. J. Hrusa, On energies for nonlinear viscoelastic materials of single-integral type, Quart. Appl. Math. 46, 381-392 (1988)
J. M. Golden and G. A. C. Graham, Boundary Value Problems in Linear Viscoelasticity, Springer-Verlag, Berlin, 1988
N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235-245 (1965)
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon, Oxford, 1937
I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953
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