Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year

The journal celebrates now its 20 years!

IMPACT FACTOR 2016: 2.034
5-year IMPACT FACTOR: 2.359

CiteScore 2016: 2.18

SCImago Journal Rank (SJR) 2016: 1.372
Source Normalized Impact per Paper (SNIP) 2016: 1.492

Mathematical Citation Quotient (MCQ) 2016: 0.61

See all formats and pricing
More options …

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


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


  • [1]

    G.S. Agarwal, D.N. Pattanayak, E. Wolf, Electromagnetic fields in spatially dispersive media. Phys. Rev. B 10, No 4 (1974), 1447–1475.Google Scholar

  • [2]

    T.M. Atanacković, B. Stanković, Generalized wave equation in nonlocal elasticity. Acta Mech. 208, No 1-2 (2009), 1–10.Web of ScienceGoogle Scholar

  • [3]

    B. Banerjee, An Introduction to Metamaterials and Waves in Composites. CRC Press, London (2011).Google Scholar

  • [4]

    D.K. Banerjee, Y.H. Pao, Thermoelastic waves in anisotropic solids. J. Acoust. Soc. Am. 56, No 5 (1974), 1444–1454.Google Scholar

  • [5]

    A.B. Bhatia, Ultrasonic Absorption: An Introduction to the Theory of Sound Absorption and Dispersion in Gases, Liquids and Solids. Dover Publications, New York (2012).Google Scholar

  • [6]

    M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2 (2015), 73–85.Google Scholar

  • [7]

    J.M. Carcione, A generalization of the Fourier pseudospectral method. Geophysics 75, No 6 (2010), A53–A56.Web of ScienceGoogle Scholar

  • [8]

    A. Carpinteri, P. Cornetti, A. Sapora, A fractional calculus approach to nonlocal elasticity. Eur. Phys. J.-Spec. Top. 193, No 1 (2011), 193–204.Google Scholar

  • [9]

    Á. Cartea, D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, No 4 (2007), 041105.Google Scholar

  • [10]

    G. Casula, J.M. Carcione, Generalized mechanical model analogies of linear viscoelastic behaviour. B. Geofis. Teor. Appl. 34, No 136 (1992), 235–256.Google Scholar

  • [11]

    N. Challamel, D. Zorica, T.M. Atanacković, D.T. Spasić, On the fractional generalization of Eringen’s nonlocal elasticity for wave propagation. Cr. Mecanique 341, No 3 (2013), 298–303.Web of ScienceGoogle Scholar

  • [12]

    S.C. Chapra, R.P. Canale, Numerical Methods for Engineers.McGraw-Hill, New York (2009).Google Scholar

  • [13]

    W. Chen, S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115, No 4 (2004), 1424–1430.Google Scholar

  • [14]

    A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity. Int. J. Eng. Sci. 10, No 3 (1972), 233–248.Google Scholar

  • [15]

    A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, No 5 (1972), 425–435.Google Scholar

  • [16]

    A.C. Eringen, Vistas of nonlocal continuum physics. Int. J. Eng. Sci. 30, No 10 (1992), 1551–1565.Google Scholar

  • [17]

    A.C. Eringen, Nonlocal Continuum Field Theories. Springer, (2002).Google Scholar

  • [18]

    A. Hanyga, M. Seredy`nska, Spatially fractional-order viscoelasticity, non-locality, and a new kind of anisotropy. J. Math. Phys. 53, No 5 (2012), 052902-1–052902-21.Web of ScienceGoogle Scholar

  • [19]

    S. Holm, S.P. Näsholm, F. Prieur, R. Sinkus, Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations. Comput. Math. Appl. 66, No 5 (2013), 621–629.Web of ScienceGoogle Scholar

  • [20]

    S. Holm, S.P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography. Ultrasound Med. Biol. 40, No 4 (2014), 695–703.Web of ScienceGoogle Scholar

  • [21]

    H.A.H. Jongen, J.M. Thijssen, M. van den Aarssen, W.A. Verhoef, A general model for the absorption of ultrasound by biological tissues and experimental verification. J. Acoust. Soc. Am. 79, No 2 (1986), 535–540.Google Scholar

  • [22]

    D. Klatt, U. Hamhaber, P. Asbach, J. Braun, I. Sack, Noninvasive assessment of the rheological behavior of human organs using multi-frequency MR elastography: a study of brain and liver viscoelasticity. Phys. Med. Biol. 52, No 24 (2007), 7281–7294.Web of ScienceGoogle Scholar

  • [23]

    E. Kröner, Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, No 5 (1967), 731–742.Google Scholar

  • [24]

    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010).Google Scholar

  • [25]

    D.A. McQuarrie, Mathematical Methods for Scientists and Engineers. University Science Books, Sausalito (2003).Google Scholar

  • [26]

    M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. Walter de Gruyter GmbH & Co, Berlin (2011).Google Scholar

  • [27]

    T. Meidav, Viscoelastic properties of the standard linear solid. Geophysical Prospect. 12, No 1 (1964), 80–99.Google Scholar

  • [28]

    T.M. Müller, B. Gurevich, M. Lebedev, Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks - A review. Geophysics 75, No 5 (2010), 75A147–75A164.Web of ScienceGoogle Scholar

  • [29]

    S.P. Näsholm, S. Holm, Linking multiple relaxation, power-law attenuation, and fractional wave equations. J. Acoust. Soc. Am. 130, No 5 (2011), 3038–3045.Web of ScienceGoogle Scholar

  • [30]

    S.P. Näsholm, S. Holm, On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 26–50; DOI: 10.2478/s13540-013-0003-1; http://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.Crossref

  • [31]

    S.P. Näsholm, Model-based discrete relaxation process representation of band-limited power-law attenuation. J. Acoust. Soc. Am. 133, No 3 (2013), 1742–1750.Web of ScienceGoogle Scholar

  • [32]

    M. Di Paola, M. Zingales, Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int. J. Solids Struct. 45, No 21 (2008), 5642–5659.Web of ScienceGoogle Scholar

  • [33]

    M. Di Paola, G. Failla, A. Pirrotta, A. Sofi, M. Zingales, The mechanically based non-local elasticity: an overview of main results and future challenges. Philos. T. R. Soc. A 371, No 1993 (2013), 20120433.Google Scholar

  • [34]

    C. Polizzotto, Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38, No 42 (2001), 7359–7380.Google Scholar

  • [35]

    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, New York (2007).Google Scholar

  • [36]

    F. Sabzikar, M.M. Meerschaert, J. Chen, Tempered fractional calculus. J. Comput. Phys. 293 (July 2015), 14–28.Google Scholar

  • [37]

    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993).Google Scholar

  • [38]

    A. Sapora, P. Cornetti, A. Carpinteri, Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach. Commun. Nonlinear Sci. 18, No 1 (2013), 63–74.Google Scholar

  • [39]

    P. Straka, M.M. Meerschaert, R.J. McGough, Y. Zhou, Fractional wave equations with attenuation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 262–272; DOI: 10.2478/s13540-013-0016-9; http://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.Crossref

  • [40]

    V. Sundararaghavan, A. Waas, Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atom-istic simulations. J. Mech. Phys. Solids 59, No 6 (2011), 1191–1203.Web of ScienceGoogle Scholar

  • [41]

    T.L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law. J. Acoust. Soc. Am. 97, No 1 (1995), 14–24.Google Scholar

  • [42]

    V.E. Tarasov, Lattice model with power-law spatial dispersion for fractional elasticity. Cent. Eur. J. Phys. 11, No 11 (2013), 1580–1588.Web of ScienceGoogle Scholar

  • [43]

    Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, No 12 (2005), 124301.Google Scholar

  • [44]

    P.N.T. Wells, Absorption and dispersion of ultrasound in biological tissue. Ultrasound Med. Biol. 1, No 4 (1975), 369–376.Google Scholar

  • [45]

    M. Zhang, P. Nigwekar, B. Castaneda, K. Hoyt, J.V. Joseph, A. Di S. Agnese, E.M. Messing, J.G. Strang, D.J. Rubens, K.J. Parker, Quantitative characterization of viscoelastic properties of human prostate correlated with histology. Ultrasound Med. Biol. 34, No 7 (2008), 1033–1042.Web of ScienceGoogle Scholar

  • [46]

    T. Zhu, J.M. Carcione, Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives. Geophys. J. Int. 196, No 3 (2014), 1787–1795.Web of ScienceGoogle Scholar

  • [47]

    M. Zingales, Wave propagation in 1D elastic solids in presence of long-range central interactions. J. Sound Vib. 330, No 16 (2011), 3973–3989.Web of ScienceGoogle Scholar

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.

Export Citation

© 2016 Diogenes Co., Sofia. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in