Invariant measures of the lack of symmetry with respect to the symmetry groups of 2D elasticity tensors. (English) Zbl 1263.74007
Summary: We study the geometric structure of the 2D elasticity tensor space using the representation theory of linear groups. We use Kelvin’s notation system 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 provide a linear frame of the elastic tensor space in which the representation of the rotation group is decomposed into irreductible subspaces. Thanks to five independent invariants chosen among six, an elasticity tensor in 2D can be represented by a closed 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 complete description requires five polynomial invariants, two linear, two quadratic and one cubic. We reveal the physical and geometrical meaning of these invariants and we propose a simple method to determine the elastic behaviour of an anisotropic material of which the symmetry is not known a priori, thanks to invariant measures of the lack of symmetry with respect to class of materials.
MSC:
| 74B05 | Classical linear elasticity |
| 74E10 | Anisotropy in solid mechanics |
| 15A72 | Vector and tensor algebra, theory of invariants |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
Keywords:
representation theory; applications of Lie groups to physics; linear elasticity; anisotropyReferences:
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