A dispersive wave equation using nonlocal elasticity

Noël Challamel; Lalaonirina Rakotomanana; Loïc Le Marrec

doi:10.1016/j.crme.2009.06.028

A dispersive wave equation using nonlocal elasticity
[Une équation des ondes dispersive utilisant l'élasticité non-locale]

Présenté par : Jean-Baptiste Leblond

Noël Challamel ¹ ; Lalaonirina Rakotomanana ² ; Loïc Le Marrec ²

¹ Université européenne de Bretagne, INSA de Rennes – LGCGM, 20, avenue des buttes de Coësmes, 35043 Rennes cedex, France
² Université européenne de Bretagne, IRMAR – université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France

Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 591-595.

Résumé
Abstract

La mécanique des milieux continus non-locaux permet de prendre en compte des effets d'échelle qui peuvent être significatifs lorsque l'on s'intéresse aux structures à faible échelle (micro ou nano-structures). Cette Note s'intéresse à un modèle de propagation d'ondes dans un milieu élastique non-local. Nous montrons qu'une équation d'ondes dispersive est obtenue à partir d'une loi constitutive non-locale, basée sur une combinaison des déformations locales et non-locales. Le modèle comprend à la fois le modèle au gradient classique et le modèle intégral d'Eringen. Les propriétés dynamiques du modèle sont discutées et corroborent des résultats récents permettant d'unifier les approches au gradient en régime statique et dynamique. De plus, ce modèle permet de décrire de manière très précise les courbes de dispersion du modèle de Born–Kármán.

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This Note investigates a model of wave propagation in a nonlocal elastic material. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. This model comprises both the classical gradient model and the Eringen's integral model. The dynamic properties of the model are discussed, and corroborate well some recent theoretical studies published to unify both static and dynamics gradient elasticity theories. Moreover, an excellent matching of the dispersive curve of the Born–Kármán model of lattice dynamics is obtained with such nonlocal model.

Reçu le : 2009-04-22
Accepté le : 2009-06-22
Publié le : 2009-07-18

DOI : 10.1016/j.crme.2009.06.028

Keywords: Continuum mechanics, Nonlocal model, Gradient model, Wave equation, Elasticity, Heterogeneous material, Dispersive properties, Born–Kármán model
Mots-clés : Milieux continus, Modèle non-local, Modèle au gradient, Équation des ondes, Élasticité, Matériau hétérogène, Dispersion, Modèle de Born–Kármán

Affiliations des auteurs :

Noël Challamel ¹ ; Lalaonirina Rakotomanana ² ; Loïc Le Marrec ²

¹ Université européenne de Bretagne, INSA de Rennes – LGCGM, 20, avenue des buttes de Coësmes, 35043 Rennes cedex, France
² Université européenne de Bretagne, IRMAR – université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France

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     pages = {591--595},
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     year = {2009},
     doi = {10.1016/j.crme.2009.06.028},
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Noël Challamel; Lalaonirina Rakotomanana; Loïc Le Marrec. A dispersive wave equation using nonlocal elasticity. Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 591-595. doi : 10.1016/j.crme.2009.06.028. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.06.028/

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