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Houssam Abdoul-Anziz; Pierre Seppecher
Homogenization of periodic graph-based elastic structures
(Homogénéisation de structures élastiques basées sur un graphe périodique)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 259-288, doi: 10.5802/jep.70
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Class. Math.: 35B27, 35J30
Mots clés: Homogénéisation périodique, Gamma-convergence, modèle de second gradient

Résumé - Abstract

Nous étudions, dans le cadre de la $\Gamma $-convergence et de l’homogénéisation périodique de matériaux fortement contrastés, des structures cylindriques constituées d’un unique matériau élastique linéaire et de vide. L’intérêt actuel pour l’homogénéisation à fort contraste est important mais en général des hypothèses ad hoc sont faites de manière à obtenir un modèle limite qui reste dans le cadre de l’élasticité classique. Nous cherchons, au contraire, à obtenir des énergies homogénéisées prenant en compte des effets de second gradient du déplacement (ou, de manière équivalente, de gradient de la déformation). Nous montrons d’abord que l’étude des structures considérées peut se réduire à l’étude de systèmes discrets correspondant à des réseaux périodiques de nœuds liés par des interactions élastiques. Notre étude de tels réseaux diffère de celles que l’on peut trouver dans la littérature par le fait que nous prenons en compte la différence d’ordre de grandeur des raideurs à l’extension et à la flexion des éléments élancés qui relient les nœuds du réseau. Cela nous permet de traiter des structures qui auraient été mobiles si l’on avait négligé les raideurs en flexion et complètement rigides si l’on avait considéré qu’elles étaient du même ordre de grandeur que les raideurs en extension. À notre connaissance, cette étude est le premier résultat rigoureux d’homogénéisation dans lequel l’énergie limite peut dépendre de toutes les composantes du second gradient du déplacement.

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