[Extension 3D de la Décomposition Polaire Tensorielle. Application aux tenseurs de (photo-)elasticité]
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 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 . Le tenseur 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.
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 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 . The tensor 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.
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Mot clés : Décomposition harmonique, Anisotropie, Élasticité, Photo-élasticité, Tenseur rari-constant, Feldspar, Taurine
Rodrigue Desmorat 1 ; Boris Desmorat 2, 3
@article{CRMECA_2016__344_6_402_0,
author = {Rodrigue Desmorat and Boris Desmorat},
title = {3D extension of {Tensorial} {Polar} {Decomposition.} {Application} to (photo-)elasticity tensors},
journal = {Comptes Rendus. M\'ecanique},
pages = {402--417},
publisher = {Elsevier},
volume = {344},
number = {6},
year = {2016},
doi = {10.1016/j.crme.2016.01.006},
language = {en},
}
TY - JOUR AU - Rodrigue Desmorat AU - Boris Desmorat TI - 3D extension of Tensorial Polar Decomposition. Application to (photo-)elasticity tensors JO - Comptes Rendus. Mécanique PY - 2016 SP - 402 EP - 417 VL - 344 IS - 6 PB - Elsevier DO - 10.1016/j.crme.2016.01.006 LA - en ID - CRMECA_2016__344_6_402_0 ER -
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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