Skip to main content
Log in

On a New Class of Constitutive Equations for Linear Viscoelastic Body

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript


We study a viscoelastic body involving a constitutive equation with distributed order fractional derivatives of complex order. Using a dissipation inequality in a weak form, we derive a sufficient conditions on coefficients of a model that guarantee that the Second law of thermodynamics under isothermal conditions is satisfied. Several known constitutive equations follow from our model as special cases. As an application, a new constitutive equation is related to an equation of motion of a generalized linear oscillator.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. G. Amendola, M. Fabrizio, J.M. Golden, Thermodynamics of Materials with Memory. Springer, 2012.

    Book  Google Scholar 

  2. G. Amendola, M. Fabrizio, J.M. Golden, The minimum free energy in fractional models of materials with memory. Communications in Applied and Industrial Mathematics 6, No 1 (2014), e–488.

    Article  MathSciNet  Google Scholar 

  3. T.M. Atanackovic, On a distributed derivative model of a viscoelastic body. C. R. Mecanique 331, No 10 (2003), 687–692.

    Article  Google Scholar 

  4. T.M. Atanackovic, M. Janev, S. Konjik, S. Pilipovic, D. Zorica, Vibrations of an elastic rod on a viscoelastic foundation of complex fractional Kelvin–Voigt type. Meccanica 50, No 7 (2015), 1679–1692; 10.1007/s11012-015-0128-x.

    Article  MathSciNet  Google Scholar 

  5. T.M. Atanackovic, S. Konjik, S. Pilipovic, D. Zorica, Complex order fractional derivatives in viscoelasticity. Mech Time-Depend Mater. 20, No 2 (2016), 175–195.

    Article  Google Scholar 

  6. T.M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a Viscoelastic body. Acta Mechanica 159, No 1 (2002), 77–86; 10.1007/BF01171449.

    Article  Google Scholar 

  7. T.M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, Fractional Calculus with Application in Mechanics: Vibrations and Diffusion Processes. ISTE, London, John Wiley & Sons, New York, 2014.

    Book  Google Scholar 

  8. T.M. Atanackovic, D. Dolicanin, S. Konjik, S. Pilipovic, Dissipativity and stability for a nonlinear differential equation with distributed order symmetrized fractional derivative. Applied Mathematics Letters 24, No 6 (2011), 1020–1025.

    Article  MathSciNet  Google Scholar 

  9. T.M. Atanackovic, D. Dolicanin, S. Pilipovic, B. Stankovic, Cauchy problems for some classes of linear fractional differential equations. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1039–1059; 10.2478/s13540-014-0213-1;

    Article  MathSciNet  Google Scholar 

  10. T.M. Atanackovic, M. Janev, S. Konjik, S. Pilipovic, Wave equation for generalized Zener model containing complex order fractional derivatives. Continium Mech. Thermodyn.; 10.1007/s00161-016-0548-4.

  11. R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior. Journal of Rheology 30, No 1 (1986), 133–155.

    Article  Google Scholar 

  12. R.L. Bagley, P.J. Torvik, On the existence of the order domain and the solution of Distributed order equations - Part I. Int. Journal of Appl. Mathematics 2, No 8 (2000), 865–882.

    MathSciNet  MATH  Google Scholar 

  13. R.L. Bagley, P.J. Torvik, On the existence of the order domain and the solution of Distributed order equations - Part II. Int. Journal of Appl. Mathematics 2, No 8 (2000), 965–987.

    MathSciNet  MATH  Google Scholar 

  14. M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, No 4 (2001), 421–442.

    MathSciNet  MATH  Google Scholar 

  15. M. Enelund, L. Mähler, K. Runesson, B. Lennart Josefson, Formulation and integration of the standard linear viscoelastic solid with fractional order rate. Internat. J. Solids Structures 36, (1999), 2417–2442.

    Article  Google Scholar 

  16. S. von Ende, A. Lion, R. Lammering, On the thermodynamically consistent fractional wave equation for viscoelastic solids. Acta Mech. 221, No 1 (2011), 1–10.

    Article  Google Scholar 

  17. M. Fabrizio, A. Morro, Mathematical Problems in Linear Viscoelasticity. Society for Industrial and Applied Mathematics, 1987.

    Google Scholar 

  18. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  19. F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, No 4 (2012), 712–717; 10.2478/s13540-012-0048-6;

    Article  MathSciNet  Google Scholar 

  20. S.P. Näsholm, S. Holm, On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 26–50; 10.2478/s13540-013-0003-1;

    Article  MathSciNet  Google Scholar 

  21. M. Reed, B. Simon, Methods of Modern Mathematical Physics. Academic Press, San Diego, 1980.

    MATH  Google Scholar 

  22. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Amsterdam, 1993.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Diana Dolićanin-Đekić.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dolićanin-Đekić, D. On a New Class of Constitutive Equations for Linear Viscoelastic Body. FCAA 20, 521–536 (2017).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: