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Free energies for singleton minimal states

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Abstract

It is assumed that any free energy function exhibits strict periodic behavior for histories that have been periodic for all past times. This is not the case for the work function, which, however, has the usual defining properties of a free energy. Forms given in fairly recent years for the minimum and related free energies of linear materials with memory have this property. Materials for which the minimal states are all singletons are those for which at least some of the singularities of the Fourier transform of the relaxation function are not isolated. For such materials, the maximum free energy is the work function, and free energies intermediate between the minimum free energy and the work function should be given by a linear relation involving these two quantities. All such functionals, except the minimum free energy, therefore do not have strict periodic behavior for periodic histories, which contradicts our assumption. A way out of the difficulty is explored which involves approximating the relaxation function by a form for which the minimal states are no longer singletons. A representation can then be given of an arbitrary free energy as a linear combination of the minimum, maximum and intermediate free energies derived in earlier work. This representation obeys our periodicity assumption. Numerical data are presented, supporting the consistency of this approach.

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References

  1. Amendola, G., Fabrizio, M., Golden, J.M.: Thermodynamics of Materials with Memory: Theory and Applications. Springer, New York (2012)

    Book  MATH  Google Scholar 

  2. Amendola, G., Fabrizio, M., Golden, J.M.: Free energies in a general non-local theory of a material with memory. Math. Models Methods Appl. Sci. 24, 1037–1090 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amendola, G., Fabrizio, M., Golden, J.M.: Algebraic and numerical exploration of free energies for materials with memory. Electron. J. Differ. Equ. 2015, 1–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amendola, G., Fabrizio, M., Golden, J.M.: Free energies and minimal states for scalar linear viscoelasticity. J. Elast. (2016). doi:10.1007/s10659-015-9549-y

    MathSciNet  MATH  Google Scholar 

  5. Coleman, B.D.: Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 17, 1–45 (1964)

    MathSciNet  Google Scholar 

  6. Del Piero, G., Deseri, L.: On the analytic expression of the free energy in linear viscoelasticity. J. Elast. 43, 247–278 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Del Piero, G., Deseri, L.: On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. Anal. 138, 1–35 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deseri, L., Gentili, G., Golden, J.M.: An explicit formula for the minimum free energy in linear viscoelasticity. J. Elast. 54, 141–185 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deseri, L., Fabrizio, M., Golden, J.M.: On the concept of a minimal state in viscoelasticity: new free energies and applications to PDE\(_S\). Arch. Ration. Mech. Anal. 181, 43–96 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deseri, L., Golden, J.M.: The minimum free energy for continuous spectrum materials. SIAM J. Appl. Math. 67, 869–892 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  12. Fabrizio, M., Golden, J.M.: Maximum and minimum free energies for a linear viscoelastic material. Q. Appl. Math. 60, 341–381 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Fabrizio, M., Gentili, G., Golden, J.M.: Free energies for linear non-isothermal materials with memory. Math. Comput. Model. 39, 219–253 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Golden, J.M.: Free energies in the frequency domain: the scalar case. Q. Appl. Math. 58, 127–150 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Golden, J.M.: A proposal concerning the physical rate of dissipation in materials with memory. Q. Appl. Math. 63, 117–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Golden, J.M.: A proposal concerning the physical dissipation of materials with memory: the non-isothermal case. Math. Mech. Solids 12, 403–449 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Golden, J.M.: Constructing free energies for materials with memory. Evol. Equ. Control Theory 3, 447–483 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Golden, J.M.: Free energies for materials with memory in terms of state functionals. Meccanica 49, 2207–2235 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Golden, J.M.: The minimum and other free energies for non-linear materials with memory. Q. Appl. Math. 74, 137–164 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Golden, J.M.: Unique characterization of materials with memory. Q. Appl. Math. 74, 361–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Graffi, D.: Sull’expressione analitica di alcune grandezze termodinamiche nei materiali con memoria. Rend. Sem. Mat. Univ. Padova 68, 17–29 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Graffi, D.: Ancora sull’expressione analitica dell’energia libera nei materiali con memoria. Atti Acc. Sci. Torino 120, 111–124 (1986)

    MathSciNet  Google Scholar 

  23. Graffi, D., Fabrizio, M.: Sulla nozione di stato materiali viscoelastici di tipo ’rate’. Atti Accad. Naz. Lincei 83, 201–208 (1990)

    MathSciNet  Google Scholar 

  24. Noll, W.: A new mathematical theory of simple materials. Arch. Ration. Mech. Anal. 48, 1–50 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

    J. M. Golden

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  1. J. M. Golden
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Correspondence to J. M. Golden.

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Communicated by Andreas Öchsner.

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Golden, J.M. Free energies for singleton minimal states . Continuum Mech. Thermodyn. 28, 1821–1845 (2016). https://doi.org/10.1007/s00161-016-0512-3

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  • DOI: https://doi.org/10.1007/s00161-016-0512-3

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