Mathematical Problems in Mechanics/Calculus of Variations
The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 467-471.

We consider the variational problem which consists in minimizing the compliance of a prescribed amount of isotropic elastic material placed into a given design region when it is subjected to a given load. We perform the asymptotics of this problem when the design region is a straight cylinder with infinitesimal cross section. The results presented in this Note concern the pure torsion regime and state the existence of optimal shapes for the limit problem. When the filling ratio tends in turn to zero, these optimal shapes concentrate on the boundary of the Cheeger set of the section of the design region.

On considère le problème d'optimisation suivant : une quantité fixée d'un matériau élastique isotrope donné doit être placée dans un cylindre droit de manière à maximiser sa résistance à un chargement donné tendant à provoquer un mouvement de torsion. Lorsque le rayon et le taux de remplissage du cylindre tendent tous deux vers zéro, on montre que la distribution optimale de matière se concentre dans chaque section sur le bord de l'ensemble de Cheeger.

Accepted:
Published online:
DOI: 10.1016/j.crma.2010.01.006

Guy Bouchitté 1; Ilaria Fragalà 2; Pierre Seppecher 1

1 Laboratoire IMATH, université de Toulon et du Var, 83957 La Garde cedex, France
2 Dipartimento di Matematica, Politecnico, Piazza L. da Vinci, 20133 Milano, Italy
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Guy Bouchitté; Ilaria Fragalà; Pierre Seppecher. The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 467-471. doi : 10.1016/j.crma.2010.01.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.006/`

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