Tensorial Polar Decomposition of 2D fourth-order tensors

Boris Desmorat; Rodrigue Desmorat

doi:10.1016/j.crme.2015.07.002

Tensorial Polar Decomposition of 2D fourth-order tensors

Boris Desmorat ^1,² ; Rodrigue Desmorat ³

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean-Le-Rond-d'Alembert, 75005 Paris, France
² Université Paris-Sud, 91405 Orsay, France
³ LMT-Cachan (ENS Cachan, CNRS, Université Paris-Saclay), 94235 Cachan, France

Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 471-475.

Abstract
Résumé

One studies the structure of 2D symmetric fourth-order tensors, i.e. having both minor and major indicial symmetries. Verchery polar decomposition is rewritten in a tensorial form entitled Tensorial Polar Decomposition. The main result is that any 2D symmetric fourth-order tensor can be written in terms of second-order tensors only in a decomposition that makes explicitly appear invariants and symmetry classes. The link with harmonic decomposition is made thanks to Kelvin decomposition of its harmonic term.

On étudie la structure des tenseurs 2D symétriques d'ordre 4, c'est-à-dire : ayant aussi bien la symétrie indicielle mineure que la symétrie majeure. La décomposition polaire de Verchery est réécrite sous forme tensorielle nommée décomposition polaire tensorielle. Le résultat principal est que tout tenseur 2D symétrique d'ordre 4 peut s'écrire à l'aide de tenseurs d'ordre 2 uniquement dans une décomposition faisant apparaître explicitement les invariants et les classes de symétrie. Le lien avec la décomposition harmonique est fait en utilisant la décomposition de Kelvin de son terme harmonique.

Received: 2015-05-26
Accepted: 2015-07-09
Published online: 2015-08-04

DOI: 10.1016/j.crme.2015.07.002

Keywords: Polar decomposition, Invariants, Harmonic decomposition, Kelvin decomposition
Mots-clés : Décomposition polaire, Invariants, Décomposition harmonique, Décomposition de Kelvin

Author's affiliations:

Boris Desmorat ^{1, 2}; Rodrigue Desmorat ³

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean-Le-Rond-d'Alembert, 75005 Paris, France
² Université Paris-Sud, 91405 Orsay, France
³ LMT-Cachan (ENS Cachan, CNRS, Université Paris-Saclay), 94235 Cachan, France

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     author = {Boris Desmorat and Rodrigue Desmorat},
     title = {Tensorial {Polar} {Decomposition} of {2D} fourth-order tensors},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {471--475},
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     volume = {343},
     number = {9},
     year = {2015},
     doi = {10.1016/j.crme.2015.07.002},
     language = {en},
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Boris Desmorat; Rodrigue Desmorat. Tensorial Polar Decomposition of 2D fourth-order tensors. Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 471-475. doi : 10.1016/j.crme.2015.07.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.07.002/

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