, Volume 39, Issue 6, pp 547–561 | Cite as

Viscoelastic Solids of Exponential Type. II. Free Energies, Stability and Attractors

  • Mauro Fabrizio
  • Claudio Giorgi
  • Maria Grazia Naso


The asymptotic behavior for solutions of the semilinear motion equation of a linear viscoelastic solid of exponential type (VSET) is studied and the existence of a global attractor is proved. These results are obtained by means of a suitable class of quadratic free energies defined on the minimal state space and making use of semigroup techniques. This is the second part of a plan which was started in a previous paper [6] by the study of state-space representation, minimality and controllability for VSET.


Free energy Viscoelasticity Attractors Continuum mechanics 


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  1. Coleman, B., Owen, D. 1974’A mathematical foundation of thermodynamics’Arch Rational Mech. Anal.541104Google Scholar
  2. DelPieroG. Deseri, L. 1996’On the analytic expression of the free energy in linear viscoelasticity’J.Elasticity43247278Google Scholar
  3. DelPieroG. Deseri, L. 1997’On the concepts of state and free energy in linear viscoelasticity’Arch. Rational Mech. Anal.138135Google Scholar
  4. Deseri, L., GentiliG. Golden, M. 1999’An explicit formula for the minimum free energy in linear viscoelasticity’J. Elasticity54141185Google Scholar
  5. Fabrizio, M., GiorgiC. Morro, A. 1994’Free energies and dissipation properties for systems with memory’Arch. Rational Mech. Anal.125341373Google Scholar
  6. Fabrizio, M., GiorgiC. Naso, M.G. 2004’Viscoelastic solids of exponential type. I. Minimal representations and controllability’Meccanica39531546Google Scholar
  7. FabrizioM. Golden, J.M. 2000’Maximum and minimum free energies and the concept of a minimal state’Rend. Mat. Appl.20131165Dedicated to the memory of Gaetano Fichera (Italian)Google Scholar
  8. FabrizioM. Lazzari, B. 1991’On the existence and the asymptotic stability of solutions for linearly viscoelastic solids’Arch. Rational Mech. Anal.116139152Google Scholar
  9. FabrizioM. Lazzari, B. 1993’On asymptotic stability for linear viscoelastic fluids’Diff. Integ. Eq.6491505Google Scholar
  10. FabrizioM. Morro, A. 1988’Viscoelastic relaxation functions compatible with thermodynamics’J. Elasticity196375Google Scholar
  11. Fabrizio, M., Morro, A. 1992Mathematical Problems in Linear Viscoelasticity, Society for Industrial and Applied Mathematics(SIAM)PhiladelphiaGoogle Scholar
  12. Gentili, G. 2002’Maximum recoverable work, minimum free energy and state space in linear viscoelasticity’Quart. Appl. Math.60153182Google Scholar
  13. GhidagliaJ.-M. Temam, R. 1987’Attractors for damped nonlinear hyperbolic equations’J. Math. Pure Appl.(9)66273319Google Scholar
  14. Giorgi, C., Mu~nozRiveraJ.E. Pata, V. 2001’Global attractors for a semilinear hyperbolic equation in viscoelasticity’J. Math. Anal. Appl.2608399Google Scholar
  15. Graffi, D. 1982’Analytic expression of some thermodynamic quantities in materials with memory’Rend. Sem. Mat. Univ. Padova681729Google Scholar
  16. Graffi, D. 1986’More on the analytic expression of free energy in materials with memory’AttiAccad. Sci. Torino Cl.Sci.Fis.Mat. Natur.120(suppl.)111124Google Scholar
  17. GraffiD. Fabrizio, M. 1989a’Nonuniqueness of free energy for viscoelastic materials’AttiAccad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.83209214Google Scholar
  18. GraffiD. Fabrizio, M. 1989b’On the notion of state for viscoelastic materials of “rate” type’Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.83201208Google Scholar
  19. Hale, J. 1985’Asymptotic behaviour and dynamics in infinite dimensions’, in: Nonlinear Differential Equations (Granada, 1984)PitmanMass:Boston142Google Scholar
  20. Hale J., (1988). Asymptotic Behavior ofDissipative Systems, American Mathematical Society Providence, RIGoogle Scholar
  21. NasoM.G. Vuk, E. 2002’Uniform attractors for a semilinear evolution problem in hereditary simple fluids’Int. J. Eng. Sci.40727742Google Scholar
  22. Noll, W. 1972’A new mathematical theory of simple materials’Arch. Rational Mech. Anal.48150Google Scholar
  23. PataV. Zucchi, A. 2001’Attractors for a damped hyperbolic equation with linear memory’Adv. Math. Sci. Appl.11505529Google Scholar
  24. Slemrod, M. 1976’A hereditary partial differential equation with applications in the theory of simple fluids’Arch. Rational Mech. Anal.62303321Google Scholar
  25. Slemrod, M. 1978’An energy stability method for simple fluids’Arch. Rational Mech. Anal.68118Google Scholar
  26. Temam, R. 1988Infinite-Dimensional Dynamical Systems in Mechanics and PhysicsSpringer-VerlagNew YorkGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Mauro Fabrizio
    • 1
  • Claudio Giorgi
    • 2
  • Maria Grazia Naso
    • 2
  1. 1.Dipartimento di Matematica, Facoltá di IngegneriaUniversitá di BolognaBolognaItalia
  2. 2.Dipartimento di Matematica, Facoltá di IngegneriaUniversitá di BresciaBresciaItalia

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