# American Institute of Mathematical Sciences

December  2013, 6(6): 1525-1537. doi: 10.3934/dcdss.2013.6.1525

## Structure of the space of 2D elasticity tensors

 1 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 2 Institut Pprime, UPR CNRS 3346, Bd M. et P. Curie, téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil cedex, France

Received  June 2012 Revised  September 2012 Published  April 2013

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.
Citation: Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525
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