Maximum recoverable work, minimum free energy and state space in linear viscoelasticity
Author:
Giorgio Gentili
Journal:
Quart. Appl. Math. 60 (2002), 153-182
MSC:
Primary 74D05; Secondary 45E10, 74A15
DOI:
https://doi.org/10.1090/qam/1878264
MathSciNet review:
MR1878264
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The various formulations of the maximum recoverable work used in the literature are proved to be equivalent. Then an explicit formula of the minimum free energy is derived, starting from the formulation of the maximum recoverable work given by Day. The resulting expression is equivalent to that found by Golden and other authors. However, the particular formulation allows us to prove that the domain of definition of minimum free energy is the whole state space. Finally, the maximum recoverable work is shown to be put as the basis of the thermodynamics of viscoelastic materials under isothermal conditions. In this context the usual relation between the Clausius-Duhem inequality and the dissipation of the material is restored.
- 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
- Shlomo Breuer and E. Turan Onat, On the determination of free energy in linear viscoelastic solids, Z. Angew. Math. Phys. 15 (1964), 184–191 (English, with German summary). MR 178645, DOI https://doi.org/10.1007/BF01602660
- 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
- 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
- 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
- William Alan Day, The thermodynamics of simple materials with fading memory, Springer-Verlag, New York-Heidelberg, 1972. Springer Tracts in Natural Philosophy, Vol. 22. MR 0366234
- 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
- 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
- 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
E. H. Dill, Simple materials with fading memory, in Continuum Physics II, A.C. Eringen ed., Academic Press, New York, 1975
- 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
- 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
- M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60 (2002), no. 2, 341–381. MR 1900497, DOI https://doi.org/10.1090/qam/1900497
- 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
- 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
- I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on a half line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. (2) 14 (1960), 217–287. MR 0113114
- 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
- Dario Graffi, Sull’espressione dell’energia libera nei materiali viscoelastici lineari, Ann. Mat. Pura Appl. (4) 98 (1974), 273–279 (Italian). MR 345497, DOI https://doi.org/10.1007/BF02414027
D. Graffi, Sull’espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova 68, 17-29 (1982)
- Dario Graffi, More on the analytic expression of free energy in materials with memory, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 120 (1986), no. suppl., 111–124 (1987) (Italian). Writings in mathematical physics in honor of the ninetieth birthday of Cataldo Agostinelli (Italian). MR 958166
- 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
- M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23 (1965), 235–245. MR 189346, DOI https://doi.org/10.1090/S0033-569X-1965-0189346-9
- A. Morro and M. Vianello, Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. A (7) 4 (1990), no. 1, 45–55 (English, with Italian summary). MR 1047512
- 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
- Jan C. Willems, Dissipative dynamical systems. I. General theory, Arch. Rational Mech. Anal. 45 (1972), 321–351. MR 527462, DOI https://doi.org/10.1007/BF00276493
S. Breuer and E. T. Onat, On the maximum recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15, 12-21 (1964)
S. Breuer and E. T. Onat, On the determination of free energy in linear viscoelasticity, Z. Angew. Math. Phys. 15, 184-191 (1964)
B. D. Coleman and D. R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54, 1-104 (1974)
W. A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-15 (1970)
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 and Appl. Math. 23, 469-479 (1970)
W. A. Day, The Thermodynamics of Simple Materials with Fading Memory, Springer Tracts in Natural Philosophy, vol. 22, Springer-Verlag, New York, 1972
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity 43, 247-278 (1996)
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)
L. Deseri, G. Gentili, and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. of Elasticity 54, 141-185 (1999)
E. H. Dill, Simple materials with fading memory, in Continuum Physics II, A.C. Eringen ed., Academic Press, New York, 1975
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, C. Giorgi, and A. Morro, Internal dissipation, relaxation property, and free energy in materials with fading memory, J. Elasticity 40, 107-122 (1995)
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., to appear
M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity 19, 63-75 (1988)
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992
I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on a half-line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. Ser. 2, 14, 217-287 (1960)
J. M. Golden, Free energies in the frequency domain: The scalar case, Quart. Appl. Math. 58, 127-150 (2000)
D. Graffi, Sull’espressione dell’energia libera nei materiali viscoelastici lineari, Ann. di Mat. Pura e Appl. (IV), 98, 273-279 (1974)
D. Graffi, Sull’espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova 68, 17-29 (1982)
D. Graffi, Ancora sull’espressione analitica dell’energia libera nei materiali con memoria, Atti Acc. Scienze Torino 120, 111-124 (1986)
D. Graffi and M. Fabrizio, Sulla nozione di stato materiali viscoelastici di tipo ’rate’, Atti Accad. Naz. Lincei 83, 201-208 (1990)
M. E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235-245 (1965)
A. Morro and M. Vianello, Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. 4A, 45-55 (1990)
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1-50 (1972)
J. C. Willems, Dissipative dynamical systems. Part I: General theory, Arch. Rational Mech. Anal. 45, 321-351 (1972)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74D05,
45E10,
74A15
Retrieve articles in all journals
with MSC:
74D05,
45E10,
74A15
Additional Information
Article copyright:
© Copyright 2002
American Mathematical Society