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
[Une approche riemannienne vers des mesures des deformation en élasticité non linéaire]

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.

Résumé
Abstract

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.

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.

Reçu le : 2013-05-10
Accepté le : 2013-12-16
Publié le : 2014-02-28

DOI : 10.1016/j.crme.2013.12.005

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

Affiliations des auteurs :

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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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/

Bibliographie
Cité par

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[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.

Lev Borisov; Andreas Fischle; Patrizio Neff Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 99 (2019) no. 6 | DOI:10.1002/zamm.201800120
Robert J. Martin; Ionel-Dumitrel Ghiba; Patrizio Neff A non-ellipticity result, or the impossible taming of the logarithmic strain measure, International Journal of Non-Linear Mechanics, Volume 102 (2018), pp. 147-158 | DOI:10.1016/j.ijnonlinmec.2018.02.011
Lev Borisov; Patrizio Neff; Suvrit Sra; Christian Thiel The sum of squared logarithms inequality in arbitrary dimensions, Linear Algebra and its Applications, Volume 528 (2017), p. 124 | DOI:10.1016/j.laa.2016.06.026
Patrizio Neff; Bernhard Eidel; Robert J. Martin Geometry of Logarithmic Strain Measures in Solid Mechanics, Archive for Rational Mechanics and Analysis, Volume 222 (2016) no. 2, pp. 507-572 | DOI:10.1007/s00205-016-1007-x
Zdeněk Fiala Geometry of finite deformations and time-incremental analysis, International Journal of Non-Linear Mechanics, Volume 81 (2016), pp. 230-244 | DOI:10.1016/j.ijnonlinmec.2016.01.019
Giuseppe Montella; Sanjay Govindjee; Patrizio Neff The Exponentiated Hencky Strain Energy in Modeling Tire Derived Material for Moderately Large Deformations, Journal of Engineering Materials and Technology, Volume 138 (2016) no. 3 | DOI:10.1115/1.4032749
Patrizio Neff; Robert J. Martin Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics, Journal of Geometric Mechanics, Volume 8 (2016) no. 3, p. 323 | DOI:10.3934/jgm.2016010
Patrizio Neff; Ingo Münch; Robert Martin Rediscovering GF Becker’s early axiomatic deduction of a multiaxial nonlinear stress–strain relation based on logarithmic strain, Mathematics and Mechanics of Solids, Volume 21 (2016) no. 7, p. 856 | DOI:10.1177/1081286514542296
Lev Borisov; Patrizio Neff; Suvrit Sra; Christian Thiel The sum of squared logarithms inequality in arbitrary dimensions, PAMM, Volume 16 (2016) no. 1, p. 665 | DOI:10.1002/pamm.201610321
Robert J. Martin; Patrizio Neff Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices, Archive of Applied Mechanics, Volume 85 (2015) no. 12, pp. 1761-1778 | DOI:10.1007/s00419-015-1017-4
Patrizio Neff; Mircea Bîrsan; Frank Osterbrink Existence Theorem for Geometrically Nonlinear Cosserat Micropolar Model Under Uniform Convexity Requirements, Journal of Elasticity, Volume 121 (2015) no. 1, p. 119 | DOI:10.1007/s10659-015-9517-6
Waldemar Pompe; Patrizio Neff On the generalized sum of squared logarithms inequality, Journal of Inequalities and Applications, Volume 2015 (2015) no. 1 | DOI:10.1186/s13660-015-0593-8
Waldemar Pompe; Patrizio Neff On the generalised sum of squared logarithms inequality, Journal of Inequalities and Applications, Volume 2015 (2015) no. 1 | DOI:10.1186/s13660-015-0623-6
Johannes Lankeit; Patrizio Neff; Yuji Nakatsukasa The minimization of matrix logarithms: On a fundamental property of the unitary polar factor, Linear Algebra and its Applications, Volume 449 (2014), p. 28 | DOI:10.1016/j.laa.2014.02.012
Patrizio Neff; Ionel‐Dumitrel Ghiba; Johannes Lankeit; Robert Martin Rank‐one convexity and polyconvexity of Hencky‐type energies, PAMM, Volume 14 (2014) no. 1, p. 735 | DOI:10.1002/pamm.201410350
Patrizio Neff; Yuji Nakatsukasa; Andreas Fischle A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms, SIAM Journal on Matrix Analysis and Applications, Volume 35 (2014) no. 3, p. 1132 | DOI:10.1137/130909949

Cité par 16 documents. Sources : Crossref, NASA ADS

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