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Structure of the space of 2D elasticity tensors. (English) Zbl 1345.74009

Summary: 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.

MSC:

74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics
20G05 Representation theory for linear algebraic groups
22E70 Applications of Lie groups to the sciences; explicit representations
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