Fractional Rheological Constitutive Equations

This Demonstration shows the simplest two-element fractional rheological constitutive models. Choose between the constitutive model, the type of rheological experiment, and the plot type. Use the sliders to vary the fractional power-law exponents of the two springpots independently ( and ), and their corresponding quasi-properties ( and ). The convention for the exponents used here is .
Variable definitions:
, : springpot exponents
, : quasi-properties
: relaxation modulus
: creep compliance
, : storage and loss modulus
: time-dependent stress
: time-dependent strain


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More details about fractional constitutive equations and power-law rheology can be found in the references.
[1] A. Jaishankar and G. H. McKinley, "Power-Law Rheology in the Bulk and at the Interface: Quasi-Properties and Fractional Constitutive Equations," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2149), 2013. doi.10.1098/rspa.2012.0284.
[2] C. Friedrich, "Relaxation and Retardation Functions of the Maxwell Model with Fractional Derivatives," Rheologica Acta, 30(2), 1991 pp. 151–158.
[3] H. Schiessel, R. Metzler, A. Blumen, and T. Nonnenmacher, "Generalized Viscoelastic Models: Their Fractional Equations with Solutions," Journal of Physics A: Mathematical and General, 28(23), 1995 pp. 6567–6584.
[4] I. Podlubny, Fractional Differential Equations (Mathematics in Science and Engineering, Vol. 198), San Diego: Academic Press, 1999.
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