Free energies for incompressible viscoelastic fluids

Amendola, G.

doi:10.1090/S0033-569X-10-01185-3

Author: G. Amendola
Journal: Quart. Appl. Math. 68 (2010), 349-374
MSC (2000): Primary 74D05, 76A10, 30E20
DOI: https://doi.org/10.1090/S0033-569X-10-01185-3
Published electronically: February 19, 2010
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References

Giovambattista Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Methods Appl. Sci. 29 (2006), no. 18, 2201–2223. MR 2273157, DOI https://doi.org/10.1002/mma.769
Giovambattista Amendola and Mauro Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Differential Integral Equations 20 (2007), no. 4, 445–466. MR 2307142

G. Amendola, M. Fabrizio and J.M. Golden, Maximum recoverable work and free energies for incompressible viscoelastic fluids, Ukrainian Mathematical Journal, to appear.

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 Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 0158605, DOI https://doi.org/10.1103/RevModPhys.33.239

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

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

Luca Deseri, Mauro Fabrizio, and Murrough Golden, The concept of minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Ration. Mech. Anal. 181 (2006), no. 1, 43–96. MR 2221203, DOI https://doi.org/10.1007/s00205-005-0406-1

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.D. Dill, Simple materials with fading memory, in: Continuum Physics II, Academic Press, Berlin, 1972.

Mauro Fabrizio, Free energies in the materials with fading memory and applications to PDEs, “WASCOM 2003”—12th Conference on Waves and Stability in Continuous Media, World Sci. Publ., River Edge, NJ, 2004, pp. 172–184. MR 2089846, DOI https://doi.org/10.1142/9789812702937_0022

M. Fabrizio and C. Giorgi, Sulla termodinamica dei materiali semplici, Boll. Un. Mat. Ital. (6) 5-B, 441-464 (1986).

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

Mauro Fabrizio and Barbara Lazzari, On asymptotic stability for linear viscoelastic fluids, Differential Integral Equations 6 (1993), no. 3, 491–505. MR 1202554

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

Giorgio Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math. 60 (2002), no. 1, 153–182. MR 1878264, DOI https://doi.org/10.1090/qam/1878264

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

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, Nonuniqueness of free energy for viscoelastic materials, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 83 (1989), 209–214 (1990) (Italian, with English summary). MR 1142460

Daniel D. Joseph, Slow motion and viscometric motion; stability and bifurcation of the rest state of a simple fluid, Arch. Rational Mech. Anal. 56 (1974/75), 99–157. MR 351255, DOI https://doi.org/10.1007/BF00248137

Daniel D. Joseph, Fluid dynamics of viscoelastic liquids, Applied Mathematical Sciences, vol. 84, Springer-Verlag, New York, 1990. MR 1051193

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

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

Marshall Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal. 62 (1976), no. 4, 303–321. MR 416245, DOI https://doi.org/10.1007/BF00248268

Marshall Slemrod, An energy stability method for simple fluids, Arch. Rational Mech. Anal. 68 (1978), no. 1, 1–18. MR 503004, DOI https://doi.org/10.1007/BF00276175

C. Truesdell, The meaning of viscometry in fluid dynamics, Ann. Rev. Fluid Mech. 6, 111-146 (1974).

V. Volterra, Theory of Functional and of Integral and Integro-Differential Equations, Blackie & Son Limited, London, 1930.