Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity

Pandey, Vikash; Holm, Sverre

doi:10.1103/physreve.94.032606

Cited by 68 publications

(56 citation statements)

References 36 publications

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“…55 Second, the term "strain-hardening" which is central to the GS model has been recognized as the property of rheopecty. Such hardening behavior has already been shown to result in the Lomnitz logarithmic creep law, 24 the seismological applications of which can be witnessed in rock physics, 5 marine sediments 56 and earthquake modeling. 57 By establishing the equivalence of wave equations and dispersion relations of the GS model to the respective equations obtained in the fractional framework we have also demonstrated the possibility of modeling time-dependent non-Newtonian fluid properties using fractional derivatives.…”

Section: Discussionmentioning

confidence: 99%

“…14 − 17, 25− 29 On the one hand, fractional derivatives impart greater flexibility to the fitting process, yet their applications have remained restricted due to a lack in their physical interpretation. 34,35 On the other hand, in our recent publication, 24 we have interpreted the order in terms of the parameters of the viscoelastic model. Through this work, we want to contribute to a change in motivation behind the application of fractional derivatives from inductive to deductive.…”

Section: Introductionmentioning

confidence: 99%

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Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

Pandey

Holm

2016

The Journal of the Acoustical Society of America

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The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796-2815(2000] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and the shear wave equation derived from the GS model are shown to take the form of the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation respectively. Therefore, an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the GS model to establish the equivalence of the respective wave equations. Further, a physical interpretation of the characteristic fractional order present in the wave equations is inferred from the GS model. The overall goal is to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials. Rather, it can also be derived from real physical processes as illustrated in this work by the example of grain-shearing.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

Pandey

Holm

2016

The Journal of the Acoustical Society of America

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“…In a recent paper, Zhou et al [12] investigated the time-dependent properties of Glass Fiber Reinforced Polymers (GFRP) composites by means of a generalized Scott-Blair model with time-varying viscosity. Furthermore, it is worth remarking that viscoelastic models featuring a time-varying viscosity have also been analyzed in [13] by Pandey and Holm.…”

Section: Introductionmentioning

confidence: 99%

Scott-Blair models with time-varying viscosity

Colombaro

Garra

Giusti

et al. 2018

Applied Mathematics Letters

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In a recent paper, Zhou et al. [12] studied the time-dependent properties of Glass Fiber Reinforced Polymers composites by employing a new rheological model with a time-dependent viscosity coefficient. This rheological model is essentially based on a generalized Scott-Blair body with a time-dependent viscosity coefficient. Motivated by this study, in this note we suggest a different generalization of the Scott-Blair model based on the application of Caputo-type fractional derivatives of a function with respect to another function. This new mathematical approach can be useful in viscoelasticity and diffusion processes in order to model systems with time-dependent features. In this paper we also provide the general solution of the creep experiment for our improved Scott-Blair model together with some explicit examples and illuminating plots.

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“…An inductive CPE with a positive phase angle will have the same voltage response as a resistor in parallel with a linearly increasing capacitor. The models are inspired by similar ones in linear viscoelasticity [16,17] where they may model viscosity due to a stick-slip motion between grains in a water-saturated sediment [18].…”

Section: Introductionmentioning

confidence: 99%

Simple circuit equivalents for the constant phase element

Holm

Martinsen

2021

PLoS ONE

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The constant phase element (CPE) is a capacitive element with a frequency-independent negative phase between current and voltage which interpolates between a capacitor and a resistor. It is used extensively to model the complexity of the physics in e.g. the bioimpedance and electrochemistry fields. There is also a similar element with a positive phase angle, and both the capacitive and inductive CPEs are members of the family of fractional circuit elements or fractance. The physical meaning of the CPE is only partially understood and many consider it an idealized circuit element. The goal here is to provide alternative equivalent circuits, which may give rise to better interpretations of the fractance. Both the capacitive and the inductive CPEs can be interpreted in the time-domain, where the impulse and step responses are temporal power laws. Here we show that the current impulse responses of the capacitive CPE is the same as that of a simple time-varying series RL-circuit where the inductor’s value increases linearly with time. Similarly, the voltage response of the inductive CPE corresponds to that of a simple parallel RC circuit where the capacitor’s value increases linearly with time. We use the Micro-Cap circuit simulation program, which can handle time-varying circuits, for independent verification. The simulation corresponds exactly to the expected response from the proposed equivalents within 0.1% error. The realization with time-varying components correlates with known time-varying properties in applications, and may lead to a better understanding of the link between CPE and applications.

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Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity

Cited by 68 publications

References 36 publications

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

Scott-Blair models with time-varying viscosity

Simple circuit equivalents for the constant phase element

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