Abstract
We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates,... It turns out that it is wise to choose a third gradient theory for the body. 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. By use of a Galerkin approximation, combined with suitable a priori estimates and a passage to the limit, we prove that there exists a motion which solves a variational formulation of the complete equations of mechanics, at least locally in time. This motion may be interrupted by crushing resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.
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Acknowledgments
The authors thank Professor Franco Maceri and Professor Olivier Maisonneuve for valuable discussions. EB and PC gratefully acknowledge the support of the MIUR-PRIN Grant 2010A2TFX2 “Calculus of variations”. The financial support of PRIN project “Advanced mechanical modeling of new materials and technologies for the solution of 2020 European challenges” Grant F11J12000210001, is gratefully acknowledged by MF.
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Questo articolo è dedicato alla memoria di Enrico Magenes, un grande maestro per più generazioni di studiosi di Matematica e Meccanica.
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Bonetti, E., Colli, P. & Frémond, M. \(\mathbf {2D}\) motion with large deformations. Boll Unione Mat Ital 7, 19–44 (2014). https://doi.org/10.1007/s40574-014-0002-0
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DOI: https://doi.org/10.1007/s40574-014-0002-0