Comptes Rendus
Linear bending of star-like pyramids
Comptes Rendus. Mécanique, Volume 331 (2003) no. 12, pp. 805-810.

A family of polyhedra possessing unusual deformation properties is found. On one hand, models of these polyhedra admit free continuous large reversible bending without visible distortions of the material. On the other hand, the polyhedra themselves are rigid and do not admit continuous bending in the sense of the Cauchy definition. The found polyhedra are called model flexors in order to distiguish them from theoretical flexsors of Connelly. Bendings of the models are asymptotically exactly approximated by linear bendings of polyhedra. They represent a nonrigid, soft or slow, loss of stability that corresponds to the loss of stability in small accordingly to Euler. This new phenomenon in mechanics of deformable solid bodies may be considered as an original geometric machine of catastrophe.

On exhibe une famille de polyèdres qui possèdent des propriétés inhabituelles de déformations. D'une part, les modèles de ces polyèdres admettent des flexions libres continues, grandes, réversibles, sans distorsions visibles du matériel. D'autre part, les polyèdres sont rigides et n'admettent pas des flexions continues dans le sens de la définition de Cauchy. Les polìedres décris sont appelés des flexors modèles pour les distinguer des flexors théoriques de Connelly. Des flexions de ces modèles sont approximées asymptotiquement par des flexions linéaires des polyèdres. Elles représentent une perte de stabilité, douce, qui correspond à la perte de stabilité « in small » conformement à la définition de Euler. Ce nouveau phénomène dans la mécanique de corps solides déformables peut être considéré comme l'origine d'un processus de catastrophe géométrique.

Published online:
DOI: 10.1016/j.crme.2003.09.010
Keywords: Dynamics of rigid or flexible systems, Rigidity and bendings of polyhedra, Linear bending, Nonrigid, soft or slow, loss of stability, Supercritical deformations of shells
Mot clés : Dynamique des systèmes rigides ou flexibles, Rigidité et flexions de polyèdres, Flexion linéaire, Perte de stabilité non-rigide, douce ou ralentie, Déformations supercritiques d'enveloppes

Anatoliy D. Milka 1

1 Verkin Institute for Low Temperature Physics, 47, Lenin avenue, 61103 Kharkiv, Ukraine
     author = {Anatoliy D. Milka},
     title = {Linear bending of star-like pyramids},
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Anatoliy D. Milka. Linear bending of star-like pyramids. Comptes Rendus. Mécanique, Volume 331 (2003) no. 12, pp. 805-810. doi : 10.1016/j.crme.2003.09.010.

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