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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- Walter Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48 (1972), 1–50. MR 445985, DOI https://doi.org/10.1007/BF00253367
- Bernard D. Coleman and David R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974), 1–104. MR 395502, DOI https://doi.org/10.1007/BF00251256
- Mauro Fabrizio, Claudio Giorgi, and Angelo Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125 (1994), no. 4, 341–373. MR 1253168, DOI https://doi.org/10.1007/BF00375062
- J. M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 1739041, DOI https://doi.org/10.1090/qam/1739041
- Luca Deseri, Giorgio Gentili, and Murrough Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity 54 (1999), no. 2, 141–185. MR 1728444, DOI https://doi.org/10.1023/A%3A1007646017347
W. A. Day, The thermodynamics of materials with memory, Materials with Memory, D. Graffi ed., Liguori, Napoli, 1979
- Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021
- Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI https://doi.org/10.1007/s002050050035
- Gianpietro Del Piero and Luca Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity 43 (1996), no. 3, 247–278. MR 1415545, DOI https://doi.org/10.1007/BF00042503
- Shlomo Breuer and E. Turan Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964), 12–21 (English, with German summary). MR 178644, DOI https://doi.org/10.1007/BF01602660
V. Volterra, Theory of functional and of integral and integro-differential equations, Blackie and Son Limited, London (1930)
- W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math. 23 (1970), 469–479. MR 273881, DOI https://doi.org/10.1093/qjmam/23.4.469
- M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity 19 (1988), no. 1, 63–75. MR 928727, DOI https://doi.org/10.1007/BF00041695
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
- J. M. Golden and G. A. C. Graham, Boundary value problems in linear viscoelasticity, Springer-Verlag, Berlin, 1988. MR 958684
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864
- Bernard D. Coleman and Victor J. Mizel, A general theory of dissipation in materials with memory, Arch. Rational Mech. Anal. 27 (1967), 255–274. MR 220474, DOI https://doi.org/10.1007/BF00281714
- Dario Graffi and Mauro Fabrizio, On the notion of state for viscoelastic materials of “rate” type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 83 (1989), 201–208 (1990) (Italian, with English summary). MR 1142459
D. Graffi and M. Fabrizio, Non unicità dell’ energia libera per i materiali viscoelastici, Atti Acc. Naz. Lincei 83, 209-214 (1990)
- M. Fabrizio, C. Giorgi, and A. Morro, Internal dissipation, relaxation property, and free energy in materials with fading memory, J. Elasticity 40 (1995), no. 2, 107–122. MR 1364749, DOI https://doi.org/10.1007/BF00042457
- Shlomo Breuer and E. Turan Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964), 12–21 (English, with German summary). MR 178644, DOI https://doi.org/10.1007/BF01602660
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1-50 (1972)
B.D. Coleman and D.R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54, 1-104 (1974)
M. Fabrizio, C. Giorgi, and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125, 341-373 (1994)
J. M. Golden, Free energies in the frequency domain: The scalar case, Quart. Appl. Math. 58, 127–150 (2000)
L. Deseri, G. Gentili, and J.M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity 54, 141–185 (1999)
W. A. Day, The thermodynamics of materials with memory, Materials with Memory, D. Graffi ed., Liguori, Napoli, 1979
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992
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)
S. Breuer and E.T. Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15, 13-21 (1964)
V. Volterra, Theory of functional and of integral and integro-differential equations, Blackie and Son Limited, London (1930)
W. A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-15 (1970)
M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity 19, 63-75 (1988)
N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and their Application to Mathematical Physics, Noordhoff, Groningen, 1953
J.M. Golden and G.A.C. Graham, Boundary Value Problems in Linear Viscoelasticity, Springer-Verlag, Berlin, 1988
B.D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17, 1-45 (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)
D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo “rate", Atti Acc. Naz. Lincei 83, 201-208 (1990)
D. Graffi and M. Fabrizio, Non unicità dell’ energia libera per i materiali viscoelastici, Atti Acc. Naz. Lincei 83, 209-214 (1990)
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)
S. Breuer and E.T. Onat, On the determination of free energy in viscoelastic solids, Z. Angew. Math. Phys. 15, 185-191 (1964)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74D05,
74A15,
74A20
Retrieve articles in all journals
with MSC:
74D05,
74A15,
74A20
Additional Information
Article copyright:
© Copyright 2002
American Mathematical Society