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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 579-583 (juillet 2013)
Doi : 10.1016/j.crma.2013.06.012
Received : 6 December 2012 ;  accepted : 27 June 2013
The motion of a solid with large deformations
Mouvement dʼun 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, 00163 Roma, Italy 


We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates… Therefore it is wise to choose a third-gradient theory for the body. 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 because it may be interrupted by crushing, resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.

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On étudie le mouvement dʼun solide en grandes déformations. Ce solide peut être chargé par des pointes, des fils, des poutres, des plaques… Cela nous conduit à retenir une théorie du troisième gradient dans le solide. La matrice dʼélongation 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 quʼil 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, cʼest-à-dire une collision interne.

The full text of this article is available in PDF format.

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