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Rheological models containing fractional derivatives
W. Smit,H. De Vries +1 more
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In this paper, a short historical review of the application of fractional derivatives in rheology, a mathematical formulation of these derivatives is given, and some aspects of rheological models containing fractional derivative are shown.Abstract:
In this contribution some aspects of rheological models containing fractional derivatives are shown. After a short historical review of the application of fractional derivatives in rheology, a mathematical formulation of these derivatives is given.read more
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TL;DR: In this article, the authors review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics.
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Variable Order and Distributed Order Fractional Operators
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Fractional calculus in the transient analysis of viscoelastically damped structures
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TL;DR: In this article, the authors used fractional calculus to model the viscoelastic behavior of a damping layer in a simply supported beam and analyzed the beam by using both a continuum formulation and a finite element formulation to predict the transient response to a step loading.
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Relaxation and retardation functions of the maxwell model with fractional derivatives
TL;DR: In this article, a four-parameter Maxwell model with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition is used to determine the relaxation and retardation functions.
References
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Book
Differential and integral calculus
TL;DR: The Integral and Differential Calculus: The Definite Integral, The Derivative, The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus.
Journal ArticleDOI
A Method of Analyzing Experimental Results Obtained from Elasto‐Viscous Bodies
TL;DR: In this article, the transient equation for relaxation processes can be deduced from the experimental impedance formula, and relaxation experiments are in agreement with the deduced transient equation, and the relation of this equation to the well-known Maxwell equation is discussed.
Journal ArticleDOI
XLV. On fractional differentials
TL;DR: In this paper, the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 25, No. 168, pp. 540-549, and XLV.
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Limitations of the Newtonian Time Scale in Relation to Non-Equilibrium Rheological States and a Theory of Quasi-Properties
TL;DR: The behaviour of complex materials under stress is described in terms of entities which are not strictly ‘physical properties’, and the use of the unmodified Nutting equation is often adequate even when the addition of a second term would significantly improve individual curves.