A Riemannian approach to strain measures in nonlinear elasticity

Patrizio Neff; Bernhard Eidel; Frank Osterbrink; Robert Martin

doi:10.1016/j.crme.2013.12.005

A Riemannian approach to strain measures in nonlinear elasticity

Patrizio Neff ¹ ; Bernhard Eidel ²; Frank Osterbrink ¹; Robert Martin ¹

¹ Chair for Nonlinear Analysis and Modelling, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
² Institute of Mechanics, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany

Comptes Rendus. Mécanique, Volume 342 (2014) no. 4, pp. 254-257.

Abstract
Résumé

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set $SO (n)$ of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group $GL (n)$ . Objectivity requires the Riemannian metric to be left- $GL (n)$ -invariant, isotropy requires the Riemannian metric to be right- $O (n)$ -invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of $GL (n)$ . Surprisingly, the final result is basically independent of the chosen parameters.

In deriving the result, geodesics on $GL (n)$ have to be parameterized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved.

L'énergie isotrope de Hencky est une mesure naturelle de la distance du gradient de déformation à l'ensemble des rotations rigides $SO (n)$ dans la métrique riemanienne canonique du groupe linéaire $GL (n)$ . Le principe d'indifférence matérielle exige que la métrique soit $GL (n)$ -invariante à gauche, et l'isotropie implique son invariance à droite par $O (n)$ . Ces deux conditions sont uniquement satisfaites par une famille à trois paramètres de métriques riemaniennes sur l'espace tangent à $GL (n)$ . On note cependant que le résultat final se révèle, en essence, indépendant des paramètres choisis. Pour obtenir ce résultat, on effectue une paramétrisation des géodésiques de $GL (n)$ et l'on résout un problème de minimisation qui fait intervenir le logarithme de matrices non symétriques.

Received: 2013-05-10
Accepted: 2013-12-16
Published online: 2014-02-28

DOI: 10.1016/j.crme.2013.12.005

Keywords: Nonlinear elasticity, Geodesic distance, Hencky strain, Polar decomposition
Mot clés : Élasticité non linéaire, Distance géodésique, Déformation de Hencky, Décomposition polaire

Author's affiliations:

Patrizio Neff ¹; Bernhard Eidel ²; Frank Osterbrink ¹; Robert Martin ¹

¹ Chair for Nonlinear Analysis and Modelling, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
² Institute of Mechanics, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany

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     author = {Patrizio Neff and Bernhard Eidel and Frank Osterbrink and Robert Martin},
     title = {A {Riemannian} approach to strain measures in nonlinear elasticity},
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Patrizio Neff; Bernhard Eidel; Frank Osterbrink; Robert Martin. A Riemannian approach to strain measures in nonlinear elasticity. Comptes Rendus. Mécanique, Volume 342 (2014) no. 4, pp. 254-257. doi : 10.1016/j.crme.2013.12.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.12.005/

References
Cited by

[1] H. Hencky Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen, Z. Tech. Phys., Volume 9 (1928), pp. 215-220

[2] P. Neff; J. Lankeit; A. Madeo On Grioli's minimum property and its relation to Cauchy's polar decomposition, Int. J. Eng. Sci. (2014) (in press)

[3] P. Neff; I. Münch Curl bounds Grad on $SO (3)$ , ESAIM: Control Optim. Calc. Var., Volume 14 (2008) no. 1, pp. 148-159

[4] A. Mielke Finite elastoplasticity, Lie groups and geodesics on $SL (d)$ (P. Newton; P. Holmes; A. Weinstein, eds.), Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 61-90

[5] P. Neff, R. Martin, Minimal geodesics on $GL (n)$ for left invariant Riemannian metrics which are right invariant under $O (n)$ , in preparation.

[6] P. Neff; Y. Nakatsukasa; A. Fischle A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm, SIAM J. Matrix Anal. (2014) (in press) | arXiv

[7] P. Neff, B. Eidel, F. Osterbrink, R. Martin, The isotropic Hencky strain energy ${‖ \log U ‖}^{2}$ measures the geodesic distance of the deformation gradient $F \in {GL}^{+} (n)$ to $SO (n)$ in the unique left-invariant Riemannian metric on ${GL}^{+} (n)$ , which is also right $O (n)$ -invariant, in preparation.

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