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Comptes Rendus Mathématique
Volume 352, n° 3
pages 183-187 (mars 2014)
Doi : 10.1016/j.crma.2014.01.007
Received : 25 October 2013 ;  accepted : 16 January 2014
The 3D motion of a solid with large deformations
Mouvement en dimension 3 d'un solide en grandes déformations
 

Elena Bonetti a , Pierluigi Colli a , Michel Frémond b
a Laboratorio Lagrange, Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italy 
b Laboratorio Lagrange, Dipartimento di Ingegneria Civile, Università di Roma “Tor Vergata”, Via del Politecnico, 1, 00133 Roma, Italy 

Abstract

We study in dimension 3 the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, shells, etc. Therefore, it is wise to choose a third gradient theory for the body. It is known that the stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint, which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time for it may be interrupted by a crushing, which entails a discontinuity of velocity with respect to time, i.e., an internal collision.

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Résumé

On étudie en dimension 3 le mouvement d'un solide en grandes déformations. Ce solide peut être chargé par des pointes, des fils, des poutres, des coques… Cela nous conduit à retenir une théorie du troisième gradient dans le solide. La matrice d'élongation qui apparaît dans la décomposition polaire doit être symétrique. Cette liaison interne introduit une contrainte de réaction qui contribue à la contrainte de Piola–Kirchhoff–Boussinesq. On montre alors qu'il existe un mouvement qui satisfait toutes les équations de la Mécanique dans un cadre variationnel convenable. Ce mouvement est local en temps, car il peut être interrompu par un écrasement provoquant une discontinuité de vitesse par rapport au temps, c'est-à-dire une collision interne.

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