Michelitsch, T., Maugin, G., Nicolleau, F. et al. (2 more authors) (2010) Relations de dispersion pour des chaînes linéaire comportant des interactions harmoniques auto-similaires. In: Actes du 10ème Congrès Français d'Acoustique. 10ème Congrès Français d’Acoustique, 12-16 Apr 2010, Lyon, France. Soci´et´e Fran¸caise d’Acoustique - SFA , Lyon, France
Abstract
Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, etc. and look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity? How can we describe the mechanics of self-similar structures in the static and dynamic framework? Physical systems with self-similarity as a symmetry property require the introduction of non-local particle-particle interactions and a (quasi-) continuous distribution of mass. We construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D'Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals and which yields in the low-frequency regime a power law frequency dependence for the oscillator density. For details of the present model we refer to our recent paper (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)).
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2010 The Authors. |
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) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 19 Apr 2016 14:48 |
Last Modified: | 19 Dec 2022 13:33 |
Published Version: | https://hal.archives-ouvertes.fr/CFA2010/hal-00530... |
Status: | Published |
Publisher: | Soci´et´e Fran¸caise d’Acoustique - SFA |
Identification Number: | hal-00530290 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:97733 |