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Fractional Calculus and Applied Analysis

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Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity

Vikash Pandey / Sven Peter Näsholm / Sverre Holm
Published Online: 2016-05-04 | DOI: https://doi.org/10.1515/fca-2016-0026

Abstract

We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kröner-Eringen’s model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.

MSC 2010: Primary 26A33, 74B20, 74D10; Secondary 26A30, 42A38, 65H04

Key Words and Phrases: fractional calculus; acoustic wave equations; Eringen model; nonlocal elasticity; tempered fractional calculus; fractional Zener model; spatial dispersion; anomalous attenuation

Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 498–515, DOI: 10.1515/fca-2016-0026

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About the article

Received: 2015-07-15

Revised: 2016-01-28

Published Online: 2016-05-04

Published in Print: 2016-04-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0026.

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