Comptes Rendus
3D extension of Tensorial Polar Decomposition. Application to (photo-)elasticity tensors
Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 402-417.

The orthogonalized harmonic decomposition of symmetric fourth-order tensors (i.e. having major and minor indicial symmetries, such as elasticity tensors) is completed by a representation of harmonic fourth-order tensors H by means of two second-order harmonic (symmetric deviatoric) tensors only. A similar decomposition is obtained for non-symmetric tensors (i.e. having minor indicial symmetry only, such as photo-elasticity tensors or elasto-plasticity tangent operators) introducing a fourth-order major antisymmetric traceless tensor Z. The tensor Z is represented by means of one harmonic second-order tensor and one antisymmetric second-order tensor only. Representations of totally symmetric (rari-constant), symmetric and major antisymmetric fourth-order tensors are simple particular cases of the proposed general representation. Closed-form expressions for tensor decomposition are given in the monoclinic case. Practical applications to elasticity and photo-elasticity monoclinic tensors are finally presented.

La décomposition harmonique orthogonalisée des tenseurs symétriques d'ordre quatre (ayant les symétries majeures et mineures, tels que le tenseur d'élasticité) est complétée par une représentation des tenseurs harmoniques d'ordre quatre H à l'aide de deux tenseurs harmoniques (symétriques déviatoriques) d'ordre deux. Une décomposition similaire est obtenue pour les tenseurs non symétriques (ayant uniquement la symétrie mineure, tels que ceux rencontrés en photo-élasticité et en élasto-plasticité), introduisant un tenseur antisymétrique majeur à traces nulles Z. Le tenseur Z est représenté par deux tenseurs d'ordre deux, le premier harmonique et le second antisymétrique. Les représentations des tenseurs d'ordre quatre complètement symétriques (rari-constants), symétriques et antisymétriques majeurs sont des cas particuliers simples de la représentation proposée. Les expressions analytiques de la décomposition correspondante dans le cas monoclinique sont obtenues et appliquées à l'élasticité et à la photo-élasticité monocliniques.

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Published online:
DOI: 10.1016/j.crme.2016.01.006
Keywords: Harmonic decomposition, Anisotropy, Elasticity, Elasto-optics, Rari-constant tensor, Feldspar, Taurine
Mot clés : Décomposition harmonique, Anisotropie, Élasticité, Photo-élasticité, Tenseur rari-constant, Feldspar, Taurine

Rodrigue Desmorat 1; Boris Desmorat 2, 3

1 LMT-Cachan (ENS Cachan, CNRS, Université Paris-Saclay), 94235 Cachan cedex, France
2 Sorbonne Universités, UPMC (Université Paris-6), CNRS, UMR 7190, Institut Jean-Le-Rond-d'Alembert, 75005 Paris, France
3 Université Paris-Sud, 91405 Orsay, France
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Rodrigue Desmorat; Boris Desmorat. 3D extension of Tensorial Polar Decomposition. Application to (photo-)elasticity tensors. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 402-417. doi : 10.1016/j.crme.2016.01.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.006/

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