Abstract
Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201–210
Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30:133–155
Bendler JT, Shlesinger MF (1988) Generalized Vogel law for glass-forming liquids. J Stat Phys 53:531–541
Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics I. J Chem Phys 9:341–351
Douglas JF, Hubbard JB (1991) Semiempirical theory of relaxation: Concentrated polymer solution dynamics. Macromolecules 24:3163–3177
Ferry JD (1970) Viscoelastic properties of polymers. Wiley, New York
Friedrich Ch (1991) Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta 30:151–158
Glöckle WG, Nonnenmacher TF (1991) Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24:6426–6434
Glöckle WG, Nonnenmacher TF (1993) Fox function representation of non-Debye relaxation processes. J Stat Phys 71:741–757
Glöckle WG, Nonnenmacher TF (1994) Fractional relaxation equations for protein dynamics. In: Nonnenmacher TF, Losa GA, Weibel ER, Fractals in biology and medicine. Birkhäuser, Basel
Koeller RC (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51:299–307
Ngai KL (1987) Evidences for universal behaviour of condensed matter at low frequencies/long times. In: Ramakrishnan TV, Raj Lakshmi M, Non-Debye relaxation in condensed matter. World Scientific, Singapore
Ngai KL, Rajagopal KA, Teitler S (1988) Slowing down of relaxation in a complex system by constraint dynamics. J Chem Phys 88:5086–5094
Nonnenmacher TF (1991) Fractional relaxation equations for viscoelasticity and related phenomena. In: Casas-Vaźquez J, Jou D, Rheological modelling: Thermodynamical and statistical approaches. Springer, Berlin
Nonnenmacher TF, Glöckle WG (1991) A fractional model for mechanical stress relaxation. Phil Mag Lett 64:89–93
Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York
Schiessel H, Blumen A (1993) Hierarchical analogues to fractional relaxation equations. J Phys A: Math Gen 26:5057–5069
Shlesinger MF (1988) Fractal time in condensed matter. Ann Rev Chem Phys 39:269–290
Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. An introduction. Springer, Berlin
Zwanzig RW (1977) In: Oppenheim I, Shuler KE, Weiss GH, Stochastic processes in chemical physics: The master equation. MIT Press, Cambridge
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Glöckle, W.G., Nonnenmacher, T.F. Fractional relaxation and the time-temperature superposition principle. Rheola Acta 33, 337–343 (1994). https://doi.org/10.1007/BF00366960
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00366960