Unique characterization of materials with memory
Author:
J. M. Golden
Journal:
Quart. Appl. Math. 74 (2016), 361-374
MSC (2010):
Primary 80A17, 74A15
DOI:
https://doi.org/10.1090/qam/1428
Published electronically:
March 16, 2016
MathSciNet review:
3505608
Full-text PDF Free Access
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Abstract:
In general, materials with linear memory constitutive relations are characterized by a relaxation function. This leads to a situation where the free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. An alternative procedure is proposed which characterizes a material by the kernel of the rate of dissipation functional. Using some recent results, we find that a unique free energy and relaxation function may then be deduced.
An example is given for discrete spectrum materials. Also, the new results are used to show that a previously derived general representation of rate of dissipation and free energy functionals is not complete, in the sense that there are valid functionals which cannot be described by this general formula.
References
- 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
- 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, Algebraic and numerical exploration of free energies for materials with memory, Electron. J. Differential Equations (2015), No. 72, 43. MR 3337849
- G. Amendola, M. Fabrizio, and J. M. Golden, Free energies and minimal states, submitted for publication.
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI 10.1007/BF00283864
- 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
- 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
- John Murrough Golden, Constructing free energies for materials with memory, Evol. Equ. Control Theory 3 (2014), no. 3, 447–483. MR 3296513, DOI 10.3934/eect.2014.3.447
- 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
- J. M. Golden, The minimum and other free energies for non-linear materials with memory, Quart. Appl. Math., to appear.
- Dario Graffi, Analytic expression of some thermodynamic quantities in materials with memory, Rend. Sem. Mat. Univ. Padova 68 (1982), 17–29 (1983) (Italian). MR 702143
- 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
References
- G. Amendola, M. Fabrizio, and J. M. 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
- G. Amendola, M. Fabrizio, and J. M. 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
- G. Amendola, M. Fabrizio, and J. M. Golden, Algebraic and numerical exploration of free energies for materials with memory, Electron. J. Differential Equations (2015), No. 72, 43. MR 3337849
- G. Amendola, M. Fabrizio, and J. M. Golden, Free energies and minimal states, submitted for publication.
- B. D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 0171419 (30 \#1650)
- L. Deseri, G. Gentili, and M. 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
- M. Fabrizio and A. 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
- 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 (2003b:74013)
- J. M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 1739041 (2001e:74025)
- 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 (2005k:74002), DOI 10.1177/1081286506061450
- 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 (2008i:74008), DOI 10.1177/1081286505061450
- J. M. Golden, Constructing free energies for materials with memory, Evolution Equations and Control Theory (Special Edition in Honour of Professor M. Fabrizio) 3 (2014), 447-483. MR 3296513
- 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
- J. M. Golden, The minimum and other free energies for non-linear materials with memory, Quart. Appl. Math., to appear.
- D. Graffi, Analytic expression of some thermodynamic quantities in materials with memory, Rend. Sem. Mat. Univ. Padova 68 (1982), 17–29 (1983) (Italian). MR 702143 (84i:73093)
- D. 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)
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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,
rate of dissipation kernel,
unique free energy functional
Received by editor(s):
January 15, 2015
Published electronically:
March 16, 2016
Article copyright:
© Copyright 2016
Brown University
