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Free energies for incompressible viscoelastic fluids
Author:
G. Amendola
Journal:
Quart. Appl. Math. 68 (2010), 349-374
MSC (2000):
Primary 74D05, 76A10, 30E20
Posted:
February 19, 2010
MathSciNet review:
2663004
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Abstract: In this work we consider some expressions for the free energy, already proposed and studied for viscoelastic solids, and adapt them to incompressible viscoelastic fluids. The internal dissipation corresponding to each of these various forms of the free energy is also evaluated. In particular, the form of the minimum free energy for the discrete spectrum model is also considered in order to show its equivalence to some classical free energies.
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(50 #3744)
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Marshall
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Rational Mech. Anal. 68 (1978), no. 1, 1–18. MR 0503004
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C. Truesdell, The meaning of viscometry in fluid dynamics, Ann. Rev. Fluid Mech. 6, 111-146 (1974).
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V. Volterra, Theory of Functional and of Integral and Integro-Differential Equations, Blackie & Son Limited, London, 1930.
- 1.
- G. Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Meth. Appl. Sci. 29, 2201-2223 (2006). MR 2273157 (2008a:76010)
- 2.
- G. Amendola and M. Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Diff. Int. Eq. 20 (4), 445-466 (2007). MR 2307142 (2008c:74018)
- 3.
- G. Amendola, M. Fabrizio and J.M. Golden, Maximum recoverable work and free energies for incompressible viscoelastic fluids, Ukrainian Mathematical Journal,to appear.
- 4.
- S. Breuer and E.T. Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15, 12-21 (1964). MR 0178644 (31:2901)
- 5.
- S. Breuer and E.T. Onat, On the determination of free energy in linear viscoelastic solids, Z. Angew. Math. Phys. 15, 184-191(1964). MR 0178645 (31:2902)
- 6.
- B.D. Coleman and W. Noll, Foundation of linear viscoelasticity, Rev. Modern Phys. 33, 239-249 (1961). MR 0158605 (28:1828)
- 7.
- B.D. Coleman and R.D. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54, 1-104 (1974). MR 0395502 (52:16299)
- 8.
- W.A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-14 (1970). MR 0273881 (42:8757)
- 9.
- 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 (97g:73042)
- 10.
- L. Deseri, M. Fabrizio and J.M. Golden, The concept of a minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Rational Mech. Anal. 181 (1), 43-96 (2006). MR 2221203 (2009a:74024)
- 11.
- 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). MR 1728444 (2001i:74012)
- 12.
- E.D. Dill, Simple materials with fading memory, in:Continuum Physics II, Academic Press, Berlin, 1972.
- 13.
- M. Fabrizio, Free energies in the materials with fading memory and applications to PDEs, in: Proceedings WASCOM 2003, World Scientific, Singapore, 2004. MR 2089846
- 14.
- M. Fabrizio and C. Giorgi, Sulla termodinamica dei materiali semplici, Boll. Un. Mat. Ital. (6) 5-B, 441-464 (1986).
- 15.
- 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 (95j:73012)
- 16.
- 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 (96g:73008)
- 17.
- M. Fabrizio and J.M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. LX (2), 341-381 (2002). MR 1900497 (2003b:74013)
- 18.
- M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Int. Eq. 6 (3), 491-504 (1993). MR 1202554 (94b:76009)
- 19.
- M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992. MR 1153021 (93a:73034)
- 20.
- G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math. LX (1), 153-182 (2002). MR 1878264 (2002m:74011)
- 21.
- J.M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58, 127-150 (2000). MR 1739041 (2001e:74025)
- 22.
- D. Graffi, Sull'espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova 68, 17-29 (1982).
- 23.
- D. Graffi, Ancora sull'espressione dell'energia libera nei materiali con memoria, Atti Acc. Scienze Torino 120, 111-124 (1986). MR 958166 (90f:73030)
- 24.
- D. Graffi and M. Fabrizio, Non unicità dell'energia libera per materiali viscoelastici, Atti Accad. Naz. Lincei 83, 209-214(1990). MR 1142460 (93a:73037)
- 25.
- D.D. Joseph, Slow motion and viscometric motion; stability and bifurcation of the rest state of a simple fluid, Arch. Rational Mech. Anal. 56, 99-157 (1974). MR 0351255 (50:3744)
- 26.
- D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids , Springer Verlag, New York, 1990. MR 1051193 (91d:76003)
- 27.
- A. Morro and M. Vianello, Minimal and maximal free energy for materials with fading memory, Boll. Un. Mat. Ital. A 4, 45-55 (1990). MR 1047512 (91b:73012)
- 28.
- N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953. MR 0355494 (50:7968)
- 29.
- M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal. 62, 303-332 (1976). MR 0416245 (54:4320)
- 30.
- M. Slemrod, An energy stability method for simple fluids, Arch. Rational Mech. Anal. 68, 1-18 (1978). MR 0503004 (58:19873)
- 31.
- C. Truesdell, The meaning of viscometry in fluid dynamics, Ann. Rev. Fluid Mech. 6, 111-146 (1974).
- 32.
- V. Volterra, Theory of Functional and of Integral and Integro-Differential Equations, Blackie & Son Limited, London, 1930.
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Additional Information
G. Amendola
Affiliation:
Dipartimento di Matematica Applicata “U. Dini”, via F. Buonarroti 1c, 56127-Pisa, Italy
Email:
amendola@dma.unipi.it
DOI:
http://dx.doi.org/10.1090/S0033-569X-10-01185-3
PII:
S 0033-569X(10)01185-3
Keywords:
Viscoelastic fluid,
fading memory,
free energy
Received by editor(s):
June 5, 2008
Received by editor(s) in revised form:
March 12, 2009
Posted:
February 19, 2010
Additional Notes:
This work was performed under the support of C.N.R. and M.I.U.R
Dedicated:
In memory of Tristano Manacorda, Member of the Lincei Academy, Emeritus Professor of the Faculty of Engineering of the University of Pisa, Italy (Firenze, October 24, 1920 – Pisa, May 20, 2008)
Article copyright:
© Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.
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