Skip to main content
Log in

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

  • Published:
Mechanics of Time-Dependent Materials Aims and scope Submit manuscript


Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  • Alcoutlabi, M., Martinez-Vega, J.J.: Modeling of the viscoelastic behavior of amorphous polymers by the differential and integration fractional method: the relaxation spectrum H(τ). Polym. 44, 7199–7208 (2003)

    Article  Google Scholar 

  • Atanackovic, T.M.: A modified Zener model of a viscoelastic body. Continuum Mech. Thermodyn. 14, 137–148 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    Article  MATH  ADS  Google Scholar 

  • Friedrich, C.: Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol. Acta. 30, 151–158 (1991a)

    Article  Google Scholar 

  • Friedrich, C.: Relaxation functions of rheological constitutive equations with fractional derivatives: thermodynamical constraints. In Casas-Vazquez, J., Jou, D. (eds.) Rheological Modelling: Thermodynamical and Statistical Approaches, Lecture notes in Physics, vol. 381, pp. 321–330. Springer, Berlin, Heidelberg, New York (1991b)

    Google Scholar 

  • Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1965)

    Google Scholar 

  • Glöckle, W.G., Nonnenmacher, T.F.: Fractional integral operators and fox functions in the theory of viscoelasticity. Macromol. 24(24), 6426–6434 (1991)

    Article  Google Scholar 

  • Heymans, N.: Constitutive equations for polymer viscoelasticity derived from hierarchical models in cases of failure of time-temperature superposition. Signal Process. 83, 2345–2357 (2003)

    Article  MATH  Google Scholar 

  • Mathai, A.M., Saxena R.K.: The H-function with Applications in Statistics and Other Disciplines. Wiley Eastem, New Delhi (1978)

    MATH  Google Scholar 

  • Metzler, R., Nonnenmacher, T.F.: Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plast. 19, 941–959 (2003)

    Article  MATH  Google Scholar 

  • Nonnenmacher, T.F., Metzler, R.: On the Riemann-Liouville fractional calculus and some recent applications. Fractals 3(3), 557–566 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Nonnenmacher, T.F., Glöckle, W.G.: A fractional model for mechanical stress relaxation. Philos. Mag. Lett. 64(2), 89–93 (1991)

    ADS  Google Scholar 

  • Park, S.W.: Rheological Modeling of Viscoelastic Passive Dampers, Smart Structures and Materials 2001: Damping and Isolation. In Inman, D.J. (ed.), Proceedings of SPIE, vol. 4331 (2001)

  • Podlubny, I.: The Laplace transform method for linear differential equations of the fractional order. UEF-02-94, Institute of Experimental Physics, Slovak Acadamy of Science, Kosice (1994)

  • Pritz, T.: Five-parameter fractional derivative model for polymeric damping materials. J. Sound Vib. 265, 935–952 (2003)

    Article  ADS  Google Scholar 

  • Rossikhin, Yu. A., Shitikova, M.V.: Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders. SVD 36(1), 3–26 (2004)

    MathSciNet  Google Scholar 

  • Sakakibara, S.: Relaxation properties of fractional derivative viscoelasticity models. In: Third World Congress of Nonliear Analysis 47, 5449–5454 (2001)

    MATH  MathSciNet  Google Scholar 

  • Schiessel, H., Metzler, R., Blumen, A., Nonnenmacher, T.F.: Generalized viscoelastic models: their fractional equations with solutions. J. Phys. A: Math. Gen. 28, 6567–6584 (1995)

    Article  MATH  ADS  Google Scholar 

  • Surguladze, T.A.: On certain applications of fractional calculus to viscoelasticity. J. Math. Sci. 112(5), 4517–4557 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Welch, S.W.J., Rorrer, R.A.L., Ronald G.D., Jr.: Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech. Time-Depend. Mater. 3, 279–303 (1999)

    Article  Google Scholar 

  • Xu, M.Y., Tan, W.C.: Theoretical analysis of the velocity field, stress field and vortex sheet of a generalized second order fluid with fractional anomalous diffusion. Sci. China, Ser. A 44(11), 1387–1399 (2001)

    Article  MATH  Google Scholar 

  • Xu, M.Y., Tan, W.C.: Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions. Sci. China, Ser. G 46(2), 145–157 (2003)

    Article  Google Scholar 

  • Zhu, X.H., et al.: Study on viscoelasticity of human cranial bone. Chin. J. Biomed. Eng. 12(1), 35–42 (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jia Guo Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, J.G., Xu, M.Y. Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions. Mech Time-Depend Mater 10, 263–279 (2006).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: