General dissipative materials for simple histories
Authors:
G. Amendola, M. Fabrizio and J. M. Golden
Journal:
Quart. Appl. Math. 76 (2018), 627-672
MSC (2010):
Primary 74D05
DOI:
https://doi.org/10.1090/qam/1500
Published electronically:
February 13, 2018
MathSciNet review:
3855825
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Abstract:
A material with memory typically has a set of many free energy functionals associated with it, all members of which yield the same constitutive relations. An alternative interpretation of this set is explored in the present work.
Explicit formulae are derived for the free energy and total dissipation of an arbitrary material in the cases of step function and sinusoidal/exponential histories. Expressions for the fraction of stored and dissipated energy are deduced. Also, various formulae are given for discrete spectrum materials.
For materials with relaxation function containing one decaying exponential, the associated Day functional is the physical free energy. For more general materials, we seek a best fit of the relaxation function with one decaying exponential to that chosen for the general case. The free energy, total dissipation and fractions of stored and dissipated energies relating to the Day material are derived for the various histories. Similar data, in the case of the general material, are explored for the minimum and maximum free energies and also for a centrally located free energy given in the literature. Various plots of aspects of this data, including comparisons between the behaviour for general and Day materials, are presented and discussed.
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
- 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
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864
- 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, The thermodynamics of materials with memory, in Materials with Memory, ed. D. Graffi, Liguori, Naples, 1979.
- 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
- 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
- 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
- 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, G. Gentili, and J. M. Golden, Nonisothermal free energies for linear theories with memory, Math. Comput. Modelling 39 (2004), no. 2-3, 219–253. MR 2037395, DOI https://doi.org/10.1016/S0895-7177%2804%2990009-X
- Mauro Fabrizio, Claudio Giorgi, and Vittorino Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 189–232. MR 2679371, DOI https://doi.org/10.1007/s00205-010-0300-3
- J. D. Ferry, Viscoelastic Properties of Polymers, 3rd edn, John Wiley, New York, 1980.
- J. M. Golden, A molecular theory of adhesive rubber friction, J. Phys. A 39 (1975), 53–70.
- 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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1007/s11012-014-9967-0
- J. M. Golden, Unique characterization of materials with memory, Quart. Appl. Math. 74 (2016), no. 2, 361–374. MR 3505608, DOI https://doi.org/10.1090/S0033-569X-2016-01428-4
- J. M. Golden, Free energies for singleton minimal states, Contin. Mech. Thermodyn. 28 (2016), no. 6, 1821–1845. MR 3549910, DOI https://doi.org/10.1007/s00161-016-0512-3
- J. M. Golden and G. A. C. Graham, Boundary value problems in linear viscoelasticity, Springer-Verlag, Berlin, 1988. MR 958684
- 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
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
- Giovambattista Amendola, Mauro Fabrizio, and John Murrough Golden, Thermodynamics of materials with memory, Springer, New York, 2012. Theory and applications. MR 2856615
- 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
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 0171419
- Bernard D. Coleman and David R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974), 1–104. MR 0395502
- W. A. Day, The thermodynamics of materials with memory, in Materials with Memory, ed. D. Graffi, Liguori, Naples, 1979.
- 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
- 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
- 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
- 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
- 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
- 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
- M. Fabrizio, G. Gentili, and J. M. Golden, Nonisothermal free energies for linear theories with memory, Math. Comput. Modelling 39 (2004), no. 2-3, 219–253. MR 2037395
- Mauro Fabrizio, Claudio Giorgi, and Vittorino Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 189–232. MR 2679371
- J. D. Ferry, Viscoelastic Properties of Polymers, 3rd edn, John Wiley, New York, 1980.
- J. M. Golden, A molecular theory of adhesive rubber friction, J. Phys. A 39 (1975), 53–70.
- J. M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 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
- 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
- John Murrough Golden, Constructing free energies for materials with memory, Evol. Equ. Control Theory 3 (2014), no. 3, 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
- J. M. Golden, Unique characterization of materials with memory, Quart. Appl. Math. 74 (2016), no. 2, 361–374. MR 3505608
- J. M. Golden, Free energies for singleton minimal states, Contin. Mech. Thermodyn. 28 (2016), no. 6, 1821–1845. MR 3549910
- J. M. Golden and G. A. C. Graham, Boundary value problems in linear viscoelasticity, Springer-Verlag, Berlin, 1988. MR 958684
- 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
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Additional Information
G. Amendola
Affiliation:
Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127-Pisa Italy
MR Author ID:
25450
Email:
amendola@dma.unipi.it
M. Fabrizio
Affiliation:
Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127-Bologna Italy
MR Author ID:
64800
Email:
fabrizio@dm.unibo.it
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
Received by editor(s):
November 9, 2017
Published electronically:
February 13, 2018
Article copyright:
© Copyright 2018
Brown University