Upper bounds estimates of the distance to cubic or orthotropic elasticity

Rodrigue Desmorat; Boris Kolev

doi:10.5802/crmeca.246

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Upper bounds estimates of the distance to cubic or orthotropic elasticity
[Estimations de bornes supérieures de la distance à l’élasticité cubique ou orthotrope]

Rodrigue Desmorat ¹ ; Boris Kolev ¹

¹ Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Comptes Rendus. Mécanique, Volume 352 (2024), pp. 169-200.

Résumé
Abstract

Nous abordons le problème, non pas de la détermination – qui nécessite généralement des méthodes numériques – mais d’une estimation analytique précise de la distance d’un tenseur d’élasticité brut à la symétrie cubique et à l’orthotropie. Nous soulignons qu’il n’y a pas un mais plusieurs tenseurs du second ordre qui portent le système de coordonnées cubique/orthotrope probable du tenseur brut. Étant donné que tous les covariants du second ordre d’un tenseur d’élasticité (exactement) cubique sont isotropes, les estimations de distance basées uniquement sur ces covariants ne sont pas précises. Nous étendons à la symétrie cubique et à l’orthotropie la technique récemment suggérée par Klimeš pour l’isotropie transverse : résoudre analytiquement un problème auxiliaire de minimisation quadratique dont la solution est un tenseur qui porte le système de coordonnées cubique probable. Des exemples numériques sont fournis, sur lesquels nous évaluons la précision de différentes estimations de bornes supérieures de la distance à la symétrie cubique ou orthotrope.

We address the problem, not of the determination–which usually needs numerical methods–but of an accurate analytical estimation of the distance of a raw elasticity tensor to cubic symmetry and to orthotropy. We point out that there are not one but several second-order tensors that carry the likely cubic/orthotropic coordinate system of the raw tensor. Since all the second-order covariants of an (exactly) cubic elasticity tensor are isotropic, distance estimates based only on such covariants are not always accurate. We extend to cubic symmetry and to orthotropy the technique recently suggested by Klimeš for transverse isotropy: solving analytically an auxiliary quadratic minimization problem whose solution is a second-order tensor that carries the likely cubic coordinate system. Numerical examples are provided, on which we evaluate the accuracy of different upper bounds estimates of the distance to cubic or orthotropic symmetry.

Reçu le : 2023-04-28
Révisé le : 2024-03-02
Accepté le : 2024-03-04
Publié le : 2024-06-14

DOI : 10.5802/crmeca.246

Classification : 74B05, 74E10, 15A72, 58D19
Keywords: Symmetry class, Elasticity, Distance to a symmetry class, Cubic symmetry, Orthotropy, Upper bounds
Mot clés : Classe de symétrie, élasticité, distance à une classe de symétrie, symétrie cubique, orthotropie, bornes supérieures

Affiliations des auteurs :

Rodrigue Desmorat ¹ ; Boris Kolev ¹

¹ Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Upper bounds estimates of the distance to cubic or orthotropic elasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {169--200},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {352},
     year = {2024},
     doi = {10.5802/crmeca.246},
     language = {en},
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Rodrigue Desmorat; Boris Kolev. Upper bounds estimates of the distance to cubic or orthotropic elasticity. Comptes Rendus. Mécanique, Volume 352 (2024), pp. 169-200. doi : 10.5802/crmeca.246. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.246/

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