Free Energies and Minimal States for Scalar Linear Viscoelasticity

详细信息    查看全文

• 作者：Giovambattista Amendola ; Mauro Fabrizio ; John Murrough Golden
• 关键词：Viscoelasticity ; Thermodynamics ; Memory effects ; Free energy ; 74D05 ; 80A17
• 刊名：Journal of Elasticity
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：123
• 期：1
• 页码：97-123
• 全文大小：1,272 KB
• 参考文献：1. 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) CrossRef MathSciNet MATH
  2. Amendola, G., Fabrizio, M., Golden, J.M.: Thermodynamics of Materials with Memory: Theory and Applications. Springer, New York (2012) CrossRef
  3. Amendola, G., Fabrizio, M., Golden, J.M.: Algebraic and numerical exploration of free energies for materials with memory. Electr. J. Diff. Eq. 72, 1–42 (2015) MathSciNet
  4. Berti, V., Gentili, G.: The minimum free energy for isothermal dielectrics with memory. J. Non-Equilib. Thermodyn. 24, 154–176 (1999) CrossRef ADS MATH
  5. Coleman, B.D.: Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 17, 1–45 (1964)
  6. Coleman, B.D., Owen, D.: A mathematical foundation for thermodynamics. Arch. Ration. Mech. Anal. 54, 1–104 (1974) CrossRef MathSciNet MATH
  7. Del Piero, G., Deseri, L.: On the analytic expression of the free energy in linear viscoelasticity. J. Elast. 43, 247–278 (1996) CrossRef MATH
  8. Del Piero, G., Deseri, L.: On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. Anal. 138, 1–35 (1997) CrossRef MATH
  9. Deseri, L., Di Paola, M., Zingales, M.: Free energy and states of fractional-order hereditariness. Int. J. Solids Struct. 51(18), 3156–3167 (2014). doi:10.​1016/​j.​ijsolstr.​2014.​05.​008 CrossRef
  10. Deseri, L., Gentili, G., Golden, J.M.: An explicit formula for the minimum free energy in linear viscoelasticity. J. Elast. 54, 141–185 (1999) CrossRef MathSciNet MATH
  11. 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) CrossRef MathSciNet MATH
  12. Deseri, L., Golden, J.M.: The minimum free energy for continuous spectrum materials. SIAM J. Appl. Math. 67, 869–892 (2007) CrossRef MathSciNet MATH
  13. Deseri, L., Zingales, M., Pollaci, P.: The state of fractional hereditary materials (FHM). Differ. Equ. Dyn. Syst. B 19(7), 2065–2089 (2014). doi:10.​3934/​dcdsb.​2014.​19.​2065 MathSciNet MATH
  14. Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia (1992) CrossRef MATH
  15. Fabrizio, M., Golden, J.M.: Maximum and minimum free energies for a linear viscoelastic material. Q. Appl. Math. 60, 341–381 (2002) MathSciNet MATH
  16. Fabrizio, M., Gentili, G., Golden, J.M.: Free energies for linear non-isothermal materials with memory. Math. Comput. Model. 39, 219–253 (2004) CrossRef MathSciNet MATH
  17. Golden, J.M.: Free energies in the frequency domain: the scalar case. Q. Appl. Math. 58, 127–150 (2000) MathSciNet MATH
  18. Golden, J.M.: A proposal concerning the physical rate of dissipation in materials with memory. Q. Appl. Math. 63, 117–155 (2005) CrossRef MathSciNet MATH
  19. Golden, J.M.: A proposal concerning the physical dissipation of materials with memory: the non-isothermal case. Math. Mech. Solids 12, 403–449 (2007) CrossRef MathSciNet MATH
  20. Golden, J.M.: Constructing free energies for materials with memory. Evol. Equ. Control Theory (Special Edition in Honour of Professor M. Fabrizio) 3, 447–483 (2014) CrossRef MathSciNet MATH
  21. Golden, J.M.: Free energies for materials with memory in terms of state functionals. Meccanica 49, 2207–2235 (2014) CrossRef MathSciNet MATH
  22. Golden, J.M.: The minimum and other free energies for non-linear materials with memory. Quart. Appl. Math. (2015, to appear)
  23. Golden, J.M.: Unique characterization of materials with memory. Quart. Appl. Math. (to appear)
  24. Graffi, D.: Sull’expressione analitica di alcune grandezze termodinamiche nei materiali con memoria. Rend. Semin. Mat. Univ. Padova 68, 17–29 (1982) MathSciNet MATH
  25. Graffi, D., Fabrizio, M.: Sulla nozione di stato materiali viscoelastici di tipo ‘rate’. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 83, 201–208 (1990) MathSciNet
  26. Graffi, D.: Ancora sull’espressione analitica dell’energia libera nei materiali con memoria. Atti R. Accad. Sci. Torino 120, 111–124 (1986) MathSciNet
  27. Noll, W.: A new mathematical theory of simple materials. Arch. Ration. Mech. Anal. 48, 1–50 (1972) CrossRef MathSciNet MATH
  
• 作者单位：Giovambattista Amendola (1)
  Mauro Fabrizio (2)
  John Murrough Golden (3)
  
  1. Dipartimento di Matematica, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy
  2. Dipartimento di Matematica, University of Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy
  3. School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin, 8, Ireland
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-2681

文摘

The concept of a minimal state was introduced in recent decades, based on earlier work by Noll. The property that a given quantity is a functional of the minimal state is of central interest in the present work. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this result and recent work on constructing free energy functionals, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a functional of the minimal state. Keywords Viscoelasticity Thermodynamics Memory effects Free energy