Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Structure of the space of 2D elasticity tensors

Pages: 1525 - 1537, Volume 6, Issue 6, December 2013      doi:10.3934/dcdss.2013.6.1525

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Géry de Saxcé - Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, bâtiment Boussinesq, Cité Scientifique, 59655 Villeneuve d'Ascq cedex, France (email)
Claude Vallée - Institut Pprime, UPR CNRS 3346, Bd M. et P. Curie, téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil cedex, France (email)

Abstract: In this paper, we present a geometric representation of the 2D elasticity tensors using the representation theory of linear groups. We use Kelvin's representation in which $\mathbb{O}(2)$ acts on the 2D stress tensors as subgroup of $\mathbb{O}(3) $. We present the method in the simple case of the stress tensors and we recover Mohr's circle construction. Next, we apply it to the elasticity tensors. We explicitly give a linear frame of the elasticity tensor space in which the representation of the rotation group is decomposed into irreducible subspaces. Thanks to five independent invariants choosen among six, an elasticity tensor in 2D can be represented by a compact line or, in degenerated cases, by a circle or a point. The elasticity tensor space, parameterized with these invariants, consists in the union of a manifold of dimension $5$, two volumes and a surface. The complet description requires six polynomial invariants, two linear, two quadratic and two cubic.

Keywords:  Linear elasticity, anisotropy, representation theory, applications of Lie groups to physics.
Mathematics Subject Classification:  Primary: 74B05, 74E10; Secondary: 20G05, 22E70.

Received: June 2012;      Revised: September 2012;      Available Online: April 2013.