Skip to main content
SpringerLink
Log in

An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Breuer and E.T. Onat, On the maximum recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964) 12-21.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Breuer and E.T. Onat, On the determination of free energy in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964) 184-191.

    Article  MATH  MathSciNet  Google Scholar 

  3. B.D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964) 1-45.

    MathSciNet  ADS  Google Scholar 

  4. B.D. Coleman and V.J. Mizel, Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1967) 87-123.

    MathSciNet  ADS  Google Scholar 

  5. B.D. Coleman and V.J. Mizel, On the general theory of fading memory, Arch. Rational Mech. Anal. 29 (1968) 18-31.

    MATH  MathSciNet  ADS  Google Scholar 

  6. B.D. Coleman and D.R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974) 1-104.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. W.A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23 (1970) 1-15.

    MATH  MathSciNet  Google Scholar 

  8. G. Del Piero and L. Deseri, Monotonic, completely monotonic and exponential relaxation functions in linear viscoelasticity, Quart. Appl. Math. 53 (1995) 273-300.

    MATH  MathSciNet  Google Scholar 

  9. G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity 43 (1996) 247-278.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997) 1-35.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. E.H. Dill, Simple materials with fading memory, Continuum Physics II. A.C. Eringen (ed.), Academic Press, New York (1975).

    Google Scholar 

  12. M. Fabrizio, Existence and uniqueness results for viscoelastic materials, Crack and Contact Problems for Viscoelastic Bodies, G.A.C. Graham and J.R. Walton (eds), Springer-Verlag, Vienna (1995).

    Google Scholar 

  13. M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125 (1994) 341-373.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. M. Fabrizio, C. Giorgi and A. Morro, Internal dissipation, relaxation property, and free energy in materials with fading memory, J. Elasticity 40 (1995) 107-122.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity 19 (1988) 63-75.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  17. I.C. Gohberg and M.G. Krein, Systems of integral equations on a half-line with kernels depending on the difference of arguments, Am. Math. Soc. Transl. Ser. 2 14 (1960) 217-287.

    MATH  MathSciNet  Google Scholar 

  18. J.M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. to appear.

  19. D. Graffi, Sull'espressione dell'energia libera nei materiali viscoelastici lineari, Ann. di Mat. Pura e Appl. (IV) 98 (1974) 273-279.

    Article  MATH  MathSciNet  Google Scholar 

  20. D. Graffi, Sull'espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova 68 (1982) 17-29.

    MATH  MathSciNet  Google Scholar 

  21. D. Graffi, Ancora sull'espressione analitica dell'energia libera nei materiali con memoria, Atti Acc. Scienze Torino 120 (1986) 111-124.

    MathSciNet  Google Scholar 

  22. D. Graffi and M. Fabrizio, Sulla nozione di stato materiali viscoelastici di tipo 'rate', Atti Accad. Naz. Lincei 83 (1990) 201-208.

    MathSciNet  Google Scholar 

  23. M.E. Gurtin and I. Herrera, On dissipation inequalities and linear viscoelasticity, Quarth. Appl. Math. 23 (1965) 235-245.

    MATH  MathSciNet  Google Scholar 

  24. P.R. Halmos, Finite-dimensional Vector Spaces, Springer-Verlag, New York (1972).

    MATH  Google Scholar 

  25. A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover Publications Inc. (1975).

  26. A. Morro, Wave solutions in linear viscoelastic materials, Crack and Contact Problems for Viscoelastic Materials, G.A.C. Graham and J.R. Walton (eds), Springer-Verlag, Vienna (1995).

    Google Scholar 

  27. A. Morro and M. Vianello, Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. 4A (1990) 45-55.

    MathSciNet  Google Scholar 

  28. N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen (1953).

    MATH  Google Scholar 

  29. W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48 (1972) 1-50.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York (1972).

    MATH  Google Scholar 

  31. G. Talenti, Sulle equazioni integrali di Wiener-Hopf, Boll. U.M.I. 7(4), Suppl. Fasc. 1 (1973) 18-118.

    MATH  MathSciNet  Google Scholar 

  32. E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon, Oxford (1937).

    MATH  Google Scholar 

  33. V. Volterra, Energia nei fenomeni elastici ereditari, Acta Pontificia Accad. Scient. 4 (1940) 115-128.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Ferrara, 44100, Ferrara, Italy; e-mail

    Luca Deseri & Giorgio Gentili

  2. School of Mathematics, Statistics and Computer Science, Dublin Institute of Technology, Dublin, Ireland; e-mail: jmgolden@maths1.kst.dit.ie

    Murrough Golden

Authors
  1. Luca Deseri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Gentili
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Murrough Golden
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deseri, L., Gentili, G. & Golden, M. An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity. Journal of Elasticity 54, 141–185 (1999). https://doi.org/10.1023/A:1007646017347

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007646017347

Search

Navigation