The minimum and other free energies for non-linear materials with memory
Author:
J. M. Golden
Journal:
Quart. Appl. Math. 74 (2016), 137-164
MSC (2010):
Primary 80A17, 74D10
DOI:
https://doi.org/10.1090/qam/1407
Published electronically:
December 4, 2015
MathSciNet review:
3472523
Full-text PDF Free Access
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Additional Information
Abstract: Expressions are obtained for free energies of materials with a certain type of non-linear constitutive relation. In particular, the minimum and related free energies are considered in some detail. Minimal states are defined for these materials, and it is shown that any free energy yielding a linear constitutive equation that is a functional of the minimal state has a counterpart in the non-linear case which is also a minimal state functional in this more general context. These results are explored for simple examples, including discrete spectrum materials.
References
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- Giovambattista Amendola, Mauro Fabrizio, and John Murrough Golden, Thermodynamics of materials with memory, Springer, New York, 2012. Theory and applications. MR 2856615, DOI 10.1007/978-1-4614-1692-0
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI 10.1007/BF00283864
- Bernard D. Coleman and Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 0158605, DOI 10.1103/RevModPhys.33.239
- 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 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 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 10.1023/A:1007646017347
- 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 10.1007/s00205-005-0406-1
- L. Deseri and J. M. Golden, The minimum free energy for continuous spectrum materials, SIAM J. Appl. Math. 67 (2007), no. 3, 869–892. MR 2300315, DOI 10.1137/050639776
- 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, DOI 10.1137/1.9781611970807
- 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 10.1090/qam/1900497
- J. M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 1739041, DOI 10.1090/qam/1739041
- J. M. Golden, A proposal concerning the physical rate of dissipation in materials with memory, Quart. Appl. Math. 63 (2005), no. 1, 117–155. MR 2126572, DOI 10.1090/S0033-569X-05-00958-2
- J. M. Golden, A proposal concerning the physical rate of dissipation of materials with memory: the non-isothermal case, Math. Mech. Solids 12 (2007), no. 4, 403–449. MR 2349161, DOI 10.1177/1081286505061450
- J. M. Golden, Free energies for materials with memory in terms of state functionals, Meccanica 49 (2014), no. 9, 2207–2235. MR 3240290, DOI 10.1007/s11012-014-9967-0
- 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
- 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
- Walter Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48 (1972), 1–50. MR 445985, DOI 10.1007/BF00253367
References
- Giovambattista Amendola, Mauro Fabrizio, and John Murrough Golden, Free energies in a general non-local theory of a material with memory, Math. Models Methods Appl. Sci. 24 (2014), no. 6, 1037–1090. MR 3187191, DOI 10.1142/S0218202513500760
- Giovambattista Amendola, Mauro Fabrizio, and John Murrough Golden, Thermodynamics of materials with memory, Theory and applications, Springer, New York, 2012. MR 2856615 (2012j:74001), DOI 10.1007/978-1-4614-1692-0
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 0171419 (30 \#1650)
- Bernard D. Coleman and Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 0158605 (28 \#1828)
- 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 (97g:73042), DOI 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 (98i:73023), DOI 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 (2001i:74012), DOI 10.1023/A:1007646017347
- 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 (2009a:74024), DOI 10.1007/s00205-005-0406-1
- Luca Deseri and John Murrough Golden, The minimum free energy for continuous spectrum materials, SIAM J. Appl. Math. 67 (2007), no. 3, 869–892 (electronic). MR 2300315 (2008d:74003), DOI 10.1137/050639776
- 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 (93a:73034), DOI 10.1137/1.9781611970807
- Mauro Fabrizio and John Murrough Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60 (2002), no. 2, 341–381. MR 1900497 (2003b:74013)
- John Murrough Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 1739041 (2001e:74025)
- John Murrough Golden, A proposal concerning the physical rate of dissipation in materials with memory, Quart. Appl. Math. 63 (2005), no. 1, 117–155. MR 2126572 (2005k:74002), DOI 10.1177/1081286506061450
- John Murrough Golden, A proposal concerning the physical rate of dissipation of materials with memory: the non-isothermal case, Math. Mech. Solids 12 (2007), no. 4, 403–449. MR 2349161 (2008i:74008), DOI 10.1177/1081286505061450
- John Murrough Golden, Free energies for materials with memory in terms of state functionals, Meccanica 49 (2014), no. 9, 2207–2235. MR 3240290, DOI 10.1007/s11012-014-9967-0
- 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. Natur. (8) 83 (1989), 201–208 (1990) (Italian, with English summary). MR 1142459 (93a:73036)
- Dario Graffi, More on the analytic expression of free energy in materials with memory, Writings in mathematical physics in honor of the ninetieth birthday of Cataldo Agostinelli (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 120 (1986), suppl., 111–124 (1987) (Italian). MR 958166 (90f:73030)
- Walter Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48 (1972), 1–50. MR 0445985 (56 \#4318)
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Additional Information
J. M. Golden
Affiliation:
School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
MR Author ID:
213598
Email:
murrough.golden@dit.ie
Keywords:
Thermodynamics,
memory effects,
free energy functional,
non-linear constitutive equation,
rate of dissipation
Received by editor(s):
June 16, 2014
Published electronically:
December 4, 2015
Article copyright:
© Copyright 2015
Brown University