The elastic torus: anomalous stiffness of shells with mixed type

Basile Audoly; Yves Pomeau

doi:10.1016/S1631-0721(02)01479-1

The elastic torus: anomalous stiffness of shells with mixed type
[Le tore élastique : raideur anormale des coques de type mixte]

Basile Audoly ¹ ; Yves Pomeau ²

¹ Laboratoire de modélisation en mécanique, UMR 7607 du CNRS Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France
² Laboratoire de physique statistique, UMR 8550 du CNRS, École normale supérieure, 24, rue Lhomond, 75231 Paris cedex 05, France

Comptes Rendus. Mécanique, Volume 330 (2002) no. 6, pp. 425-432.

Abstract (VO)
Résumé

We investigate the deformation of a thin elastic torus under axisymmetric surface loads. The strain concentrates near the top and bottom parallels and the inner and outer halves essentially undergo rigid-body translations in opposite directions. An analysis of the inner boundary layers is presented, which allows one to compute the effective stiffness of the torus for this loading. This stiffness is found anomalous compared to classical shells. These mechanical properties are interpreted using purely geometrical arguments.

Nous étudions la déformation d'un tore élastique mince sous l'effet de forces de surface axisymétriques. La déformation est concentrée au voisinage des parallèles supérieur et inférieur et les moitiés externe et interne subissent essentiellement une translation uniforme l'une par rapport à l'autre. Une analyse de couche limite permet de calculer la rigidité effective du tore, qui suit une loi anormale. Toutes ces propriétés mécaniques sont interprétées par des arguments purement géométriques.

Reçu le : 2002-04-12
Accepté le : 2002-04-23
Publié le : 2002-01-01

DOI : 10.1016/S1631-0721(02)01479-1

Keywords: solids and structures, elastic shells, boundary layers
Mots-clés : solides et structures, coques élastiques, couches limite

Affiliations des auteurs :

Basile Audoly ¹ ; Yves Pomeau ²

¹ Laboratoire de modélisation en mécanique, UMR 7607 du CNRS Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France
² Laboratoire de physique statistique, UMR 8550 du CNRS, École normale supérieure, 24, rue Lhomond, 75231 Paris cedex 05, France

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     title = {The elastic torus: anomalous stiffness of shells with mixed type},
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     number = {6},
     year = {2002},
     doi = {10.1016/S1631-0721(02)01479-1},
     language = {en},
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Basile Audoly; Yves Pomeau. The elastic torus: anomalous stiffness of shells with mixed type. Comptes Rendus. Mécanique, Volume 330 (2002) no. 6, pp. 425-432. doi : 10.1016/S1631-0721(02)01479-1. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01479-1/

Bibliographie
Cité par

[1] M. Spivak A Comprehensive Introduction to Differential Geometry, Publish or Perish, Houston, 1975

[2] S. Timoshenko; S. Woinowsky-Krieger Theory of Plates and Shells, McGraw-Hill, New York, 1959

[3] A.E. Green; W. Zerna Theoretical Elasticity, Dover, New York, 1992

[4] P. Karamian; J. Sanchez-Hubert; É. Sanchez-Palencia A model problem for boundary layers of thin elastic shells, Math. Modelling Numer. Anal., Volume 34 (2000), pp. 1-30

[5] A.L. Gol'denveizer Theory of Elastic thin Shells, Pergamon Press, New York, 1961 (Published for the American Society of Mechanical Engineers)

[6] W. Flügge Stresses in Shells, Springer-Verlag, Berlin, 1973 (pp. 30 and 94)

[7] B. Audoly Courbes rigidifiant les surfaces, C. R. Acad. Sci. Paris, Série I, Volume 328 (1999), pp. 313-316

[8] B. Audoly, Y. Pomeau, in preparation

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B.H. Sun Geometry-induced rigidity in elastic torus from circular to oblique elliptic cross-section, International Journal of Non-Linear Mechanics, Volume 135 (2021), p. 103754 | DOI:10.1016/j.ijnonlinmec.2021.103754
Salem Al Mosleh; Christian Santangelo Nonlinear mechanics of rigidifying curves, Physical Review E, Volume 96 (2017) no. 1 | DOI:10.1103/physreve.96.013003
Arthur A. Evans; Basanta Bhaduri; Gabriel Popescu; Alex J. Levine Geometric localization of thermal fluctuations in red blood cells, Proceedings of the National Academy of Sciences, Volume 114 (2017) no. 11, p. 2865 | DOI:10.1073/pnas.1613204114
Christian D. Santangelo Nambu–Goldstone modes and diffuse deformations in elastic shells, Soft Matter, Volume 9 (2013) no. 34, p. 8246 | DOI:10.1039/c3sm50476j
B. Audoly Geometric Boundary Layers in Shells With Mixed Type, Theories of Plates and Shells, Volume 16 (2004), p. 13 | DOI:10.1007/978-3-540-39905-6_2

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