Michelitsch, T.M., Maugin, G.A., Nicolleau, F.C.G.A. et al. (2 more authors) (2009) Dispersion relations and wave operators in self-similar quasicontinuous linear chains. Physical Review -Series E, 80 (1). Art no.011135. ISSN 1539-3755
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.
Metadata
Authors/Creators: |
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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. |
Dates: |
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Institution: | The University of Sheffield |
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: | https://doi.org/10.1103/PhysRevE.80.011135 |