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Rheological models containing fractional derivatives

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Summary

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.

The possibility of using fractional derivatives in the construction of rheological models is demonstrated. The behaviour of a simple fractional derivative model is calculated in a number of experiments, often performed on fibres.

The model is checked against the results of some measurements on nylon 6 and PETP fibres, which leads to the addition of an elastic term.

Finally, in broad outline, the relation between fractional derivative models and the principal expressions of the theory of linear viscoelasticity is indicated.

Zusammenfassung

In diesem Beitrag wird auf einige Aspekte rheologischer Modelle, die partielle Differentialquotienten enthalten, eingegangen.

Nach einem kurzen historischen Überblick über die Anwendungen dieser Differentialquotienten in der Rheologie wird eine mathematische Formulierung gegeben. Auch wird gezeigt, wie diese Differentialquotienten in rheologische Modelle eingeführt werden können; das dynamische Verhalten eines einfachen Modells wird mit Hilfe verschiedener Experimente, wie sie oft an viskoelastischen Stoffen durchgeführt werden, berechnet.

Das Modell wird anhand der Ergebnisse einiger Messungen an Nylon 6- und PETP-Fasern geprüft; dies veranlaßte die Einführung eines elastischen Terms.

Schließlich wird die Beziehung zwischen partielle Differentialquotienten enthaltenden Modellen und den wichtigsten Gleichungen der Theorie der linearen Viskoelastizität in großen Zügen angegeben.

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Smit, W., de Vries, H. Rheological models containing fractional derivatives. Rheol Acta 9, 525–534 (1970). https://doi.org/10.1007/BF01985463

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  • DOI: https://doi.org/10.1007/BF01985463

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