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Dispersion relations and wave operators in self-similar quasicontinuous linear chains

Michelitsch, T.M., Maugin, G.A., Nicolleau, F.C.G.A., Nowakowski, A.F. and Derogar, S. (2009) Dispersion relations and wave operators in self-similar quasicontinuous linear chains. Physical Review -Series E, 80 (1). Art no.011135. ISSN 1539-3755

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Abstract

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function that exhibits self-similar and fractal features. We also derive a continuum approximation, which relates the self-similar Laplacian to fractional integrals, and yields in the low-frequency regime a power-law frequency-dependence of the oscillator density.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009 American Physical Society. This is an author produced version of a paper subsequently published in Physical Review E. Uploaded in accordance with the publisher's self-archiving policy.
Academic Units: The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Mechanical Engineering (Sheffield)
The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Civil and Structural Engineering (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 01 Sep 2009 10:28
Last Modified: 08 Feb 2013 16:59
Published Version: http://dx.doi.org/10.1103/PhysRevE.80.011135
Status: Published
Publisher: American Physical Society
Identification Number: 10.1103/PhysRevE.80.011135
URI: http://eprints.whiterose.ac.uk/id/eprint/9269

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