Relations de dispersion pour chaînes linéaires comportant des interactions harmoniques auto-similaires

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)).
Liste complète des métadonnées

Littérature citée [13 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-00420287
Contributeur : Thomas Michelitsch <>
Soumis le : mercredi 30 septembre 2009 - 14:19:05
Dernière modification le : jeudi 13 décembre 2018 - 01:32:02
Document(s) archivé(s) le : mercredi 16 juin 2010 - 00:10:39

Fichiers

michelitsch-et-al-fractal-fran...
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-00420287, version 1
  • ARXIV : 0910.0728

Citation

Thomas Michelitsch, Gérard Maugin, Franck Nicolleau, Andrzej Nowakowski, Shahram Derogar. Relations de dispersion pour chaînes linéaires comportant des interactions harmoniques auto-similaires. Version française abrégée de l'article Michelitsch et al. Phys. Rev. E 80, 011135 (2009). 2009. 〈hal-00420287〉

Partager

Métriques

Consultations de la notice

455

Téléchargements de fichiers

157