Archive for Rational Mechanics and Analysis

, Volume 198, Issue 1, pp 189–232 | Cite as

A New Approach to Equations with Memory

Article

Abstract

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function \({G : \mathbb{R}^+ \to \mathbb{R}^+}\) such that
$$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $$
we consider an abstract version of the evolution equation
$$\partial_{tt}{\varvec u}({\varvec x}, t) - \Delta\left[G(0){\varvec u}({\varvec x}, t) + \displaystyle\int_0^\infty G'(s){\varvec u}({\varvec x}, t - s){\rm d}s\right] = 0$$
arising from linear viscoelasticity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boltzmann L.: Zur Theorie der elastischen Nachwirkung. Wien. Ber. 70, 275–306 (1874)Google Scholar
  2. 2.
    Boltzmann L.: Zur Theorie der elastischen Nachwirkung. Wied. Ann. 5, 430–432 (1878)Google Scholar
  3. 3.
    Breuer S., Onat E.T.: On recoverable work in linear viscoelasticity. Z. Angew. Math. Phys. 15, 13–21 (1964)CrossRefGoogle Scholar
  4. 4.
    Chepyzhov V.V., Pata V.: Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 50, 269–291 (2006)MATHMathSciNetGoogle Scholar
  5. 5.
    Coleman B.D.: Thermodynamics of materials with memory. Arch. Rational Mech. Anal. 17, 1–45 (1964)MathSciNetADSGoogle Scholar
  6. 6.
    Coleman B.D., Mizel V.J.: Norms and semi-groups in the theory of fading memory. Arch. Rational Mech. Anal. 23, 87–123 (1967)MathSciNetADSGoogle Scholar
  7. 7.
    Coleman B.D., Mizel V.J.: On the general theory of fading memory. Arch. Rational Mech. Anal. 29, 18–31 (1968)MATHMathSciNetADSGoogle Scholar
  8. 8.
    Coleman B.D., Noll W.: Foundations of linear viscoelasticity. Rev. Modern Phys. 33, 239–249 (1961)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Day W.A.: Reversibility, recoverable work and free energy in linear viscoelasticity. Quart. J. Mech. Appl. Math. 23, 1–15 (1970)MATHCrossRefGoogle Scholar
  11. 11.
    Day W.A.: The Thermodynamics of Simple Materials with Fading Memory. Springer, New York (1972)Google Scholar
  12. 12.
    Del Piero G., Deseri L.: Monotonic, completely monotonic and exponential relaxation functions in linear viscoelasticity. Quart. Appl. Math. 53, 273–300 (1995)MATHMathSciNetGoogle Scholar
  13. 13.
    Del Piero G., Deseri L.: On the concepts of state and free energy in linear viscoelasticity. Arch. Rational Mech. Anal. 138, 1–35 (1997)MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Deseri L., Fabrizio M., Golden M.J.: The concept of minimal state in viscoelasticity: new free energies an applications to PDEs. Arch. Rational Mech. Anal. 181, 43–96 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Deseri L., Gentili G., Golden M.J.: An explicit formula for the minimum free energy in linear viscoelasticity. J. Elasticity 54, 141–185 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fabrizio M., Giorgi C., Morro A.: Minimum principles, convexity, and thermodynamics in linear viscoelasticity. Continuum Mech. Thermodyn. 1, 197–211 (1989)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Fabrizio M., Giorgi C., Morro A.: Free energies and dissipation properties for systems with memory. Arch. Rational Mech. Anal. 125, 341–373 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Fabrizio M., Giorgi C., Morro A.: Internal dissipation, relaxation property and free energy in materials with fading memory. J. Elasticity 40, 107–122 (1995)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Fabrizio M., Golden M.J.: Maximum and minimum free energies for a linear viscoelastic material. Quart. Appl. Math. 60, 341–381 (2002)MATHMathSciNetGoogle Scholar
  20. 20.
    Fabrizio M., Golden M.J.: Minimum free energies for materials with finite memory. J. Elasticity 72, 121–143 (2003)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Fabrizio M., Lazzari B.: On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rational Mech. Anal. 116, 139–152 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Fabrizio M., Lazzari B.: Stability and free energies in linear viscoelasticity. Matematiche (Catania) 62, 175–198 (2007)MATHMathSciNetGoogle Scholar
  23. 23.
    Fabrizio M., Morro A.: Viscoelastic relaxation functions compatible with thermodynamics. J. Elasticity 19, 63–75 (1988)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. SIAM Studies in Applied Mathematics No. 12. SIAM, Philadelphia, 1992Google Scholar
  25. 25.
    Fichera, G.: Analytic Problems of Hereditary Phenomena in Materials with Memory. Corso CIME, Bressanone, 1977, pp. 111–169. Liguori, Napoli, 1979Google Scholar
  26. 26.
    Fichera G.: Avere una memoria tenace crea gravi problemi. Arch. Rational Mech. Anal. 70, 101–112 (1979)MATHCrossRefADSGoogle Scholar
  27. 27.
    Gentili G.: Maximum recoverable work, minimum free energy and state space in linear viscoelasticity. Quart. Appl. Math. 60, 152–182 (2002)MathSciNetGoogle Scholar
  28. 28.
    Giorgi C., Muñoz Rivera J.E., Pata V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Golden J.M., Graham G.A.C.: Boundary Value Problems in Linear Viscoelasticity. Springer, New York (1988)MATHGoogle Scholar
  30. 30.
    Graffi D.: Sui problemi della eredità lineare. Nuovo Cimento 5, 53–71 (1928)CrossRefGoogle Scholar
  31. 31.
    Graffi D.: Sopra alcuni fenomeni ereditari dell’elettrologia. Rend. Istit. Lombardo Sc. Lett. 68–69, 124–139 (1936)Google Scholar
  32. 32.
    Graffi D.: Sull’espressione analitica di alcune grandezze termodinamiche nei materiali con memoria. Rend. Sem. Mat. Univ. Padova 68, 17–29 (1982)MATHMathSciNetGoogle Scholar
  33. 33.
    Graffi D.: On the fading memory. Appl. Anal. 15, 295–311 (1983)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Graffi D., Fabrizio M.: Sulla nozione di stato per materiali viscoelastici di tipo “rate”. Atti Accad. Lincei Rend. Fis. 83, 201–208 (1989)MATHMathSciNetGoogle Scholar
  35. 35.
    Grasselli, M., Pata, V.: Uniform attractors of nonautonomous systems with memory. In: Lorenzi, A., Ruf, B. (eds.) Evolution Equations, Semigroups and Functional Analysis, pp. 155–178. Progr. Nonlinear Differential Equations Appl. No. 50. Birkhäuser, Boston, 2002Google Scholar
  36. 36.
    Green A.E., Rivlin R.S.: The mechanics of nonlinear materials with memory. Arch. Rational Mech. Anal. 1, 1–21 (1957-58)Google Scholar
  37. 37.
    Gurtin M.E., Sternberg E.: On the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 11, 291–356 (1962)MATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Hewitt E., Stromberg K.: Real and Abstract Analysis. Springer-Verlag, New York (1965)MATHGoogle Scholar
  39. 39.
    Ianniello, M.G., Israel, G.: Boltzmann’s concept of “Nachwirkung” and the “mechanics of heredity”. In: Battimelli, G., Ianniello, M.G., Kresten, O. (eds.) Proceedings of the International Symposium on Ludwig Boltzmann, Rome, February 9–11, 1989, pp. 113–133. Verlag der Österreichischen Akademie der Wissenschaften, Wien, 1993Google Scholar
  40. 40.
    König H., Meixner J.: Lineare Systeme und lineare Transformationen. Math. Nachr. 19, 265–322 (1958)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Leitman, M.J., Fisher, G.M.C.: The linear theory of viscoelasticity. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/3, pp. 1–123. Springer, Berlin, 1973Google Scholar
  42. 42.
    Liu Z., Zheng S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 54, 21–31 (1996)MATHMathSciNetGoogle Scholar
  43. 43.
    Muñoz Rivera J.E.: Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52, 629–648 (1994)MATHGoogle Scholar
  44. 44.
    Noll W.: A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48, 1–50 (1972)MATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Pata V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64, 499–513 (2006)MATHMathSciNetGoogle Scholar
  46. 46.
    Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)MATHGoogle Scholar
  47. 47.
    Renardy M., Hrusa W.J., Nohel J.A.: Mathematical Problems in Viscoelasticity. Wiley, New York (1987)MATHGoogle Scholar
  48. 48.
    Volterra V.: Sur les équations intégro-différentielles et leurs applications. Acta Math. 35, 295–356 (1912)MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Volterra V.: Leçons sur les fonctions de lignes. Gauthier-Villars, Paris (1913)Google Scholar
  50. 50.
    Widder D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Mauro Fabrizio
    • 1
  • Claudio Giorgi
    • 2
  • Vittorino Pata
    • 3
  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.Dipartimento di MatematicaUniversità di BresciaBresciaItaly
  3. 3.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanItaly

Personalised recommendations