Abstract
A constitutive equation for viscoelastic behavior containing time derivatives of stress and strain to fractional order is obtained from a fractal rheological model. Equivalence between tree and ladder fractal models at long times is demonstrated. The fractional differential equation is shown to be equivalent to ordinary differential formulations in the case of a simple power-law response; the adequacy of such formulations to describe non-linearity has been demonstrated previously. The model gives a good description of viscoelastic behavior under all stress modes and will be extended in future to include aging effects.
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References
Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201–210
Bauwens J-C (1988) Tentative de corrélation entre l'équation de Williams-Watts et la topologie viscoélastique. Deppos IX (Dèformation plastique des polymères solides, 9th annual meeting), ENSAM, Paris, 18th–19th October 1988
Bauwens J-C (1992a) Two nearly equivalent approaches for describing the non-linear creep behavior of glassy polymers. Colloid Polym Sci 270:537–542
Bauwens J-C (1992b) A deformation model of polycarbonate extended to the loss curve in the α transition range. Plastics, Rubber and Composites Processing and Applications 18:149–153
Friedrich Ch (1991a) Relaxation functions of theological constitutive equations with fractional derivatives: thermodynamical constraints. In: Casas-Vázquez J, Jou D (eds) Lecture nodes in physics: vol 381, theological modelling: thermodynamical and statistical approaches. Springer-Verlag, Berlin Heidelberg, 321–330
Friedrich Ch (1991b) 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
Heymans N, Hellinckx S, Bauwens J-C, Analytical and fractal descriptions of non-linear mechanical behaviour of polymers. Accepted for publication in J Non-Crystalline Solids
Le Mehauté A, Crepy G (1983) Introduction to transfer and motion in fractal media: the geometry of kinetics. Solid State Ionics 9, 10:17–30
Mandelbrot BB (1982) The fractal geometry of nature. W. H. Freeman, New York, pp 34–41, 349–350
Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, London
Schiessel H, Blumen A (1993) Hierarchical analogues to fractional relaxation equations. J Phys A: Math Gen 26:5057–5069
Schneider WR, Wyss W (1989) Fractional diffusion and wave equations. J Math Phys 30:134–144
Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behaviour. Springer-Verlag, Berlin Heidelberg
Wyss W (1986) The fractional diffusion equation. J Math Phys 27:2782–2785
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Heymans, N., Bauwens, J.C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheola Acta 33, 210–219 (1994). https://doi.org/10.1007/BF00437306
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DOI: https://doi.org/10.1007/BF00437306