nach oben

Erschienen in:

2013 | OriginalPaper | Buchkapitel

Geometrical Picture of Third-Order Tensors

verfasst von : Nicolas Auffray

Erschienen in: Generalized Continua as Models for Materials

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Because of its strong physical meaning, the decomposition of a symmetric second-order tensor into a deviatoric and a spheric part is heavily used in continuum mechanics. When considering higher-order continua, third-order tensors naturally appear in the formulation of the problem. Therefore researchers had proposed numerous extensions of the decomposition to third-order tensors. But, considering the actual literature, the situation seems to be a bit messy: definitions vary according to authors, improper uses of denomination flourish, and, at the end, the understanding of the physics contained in third-order tensors remains fuzzy. The aim of this paper is to clarify the situation. Using few tools from group representation theory, we will provide an unambiguous and explicit answer to that problem.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Shells and Plates with Surface Effects

Nächstes Kapitel Continuum Modelling of Shear-Coupled Grain Boundary Migration

Nur mit Berechtigung zugänglich

The notation \(\otimes ^S\) indicates the symmetric tensor product.

In field of condensed matter physics this decomposition is known since, at least, the 70’ [15].

\(\mathrm{O }(3)\): the orthogonal group, i.e. the group of all isometries of \(\mathbb R ^{3}\) i.e. if \(\mathrm{Q }\in \mathrm{O }(3)\)\(\mathrm{det }(\mathrm{Q })\pm 1\) and \(\mathrm{Q }^{-1}=\mathrm{Q }^{T}\).

To be more precise, \(\mathrm{H }^{k,\tau }\) is the embedding of the \(\tau \)th irreducible component of order \(k\) into a \(n\)-th order tensor.

Even if some authors explicitly construct this isomorphism [10, 13] this step is useless.

The demonstration of theses theorems will be provided in a paper currently under redaction.

This decomposition is sometimes known as the Schur decomposition.

\(\mathrm{GL }(3)\) is the group of all the invertible transformations of \(\mathbb R ^{3}\), i.e. if \(\mathrm{F }\in \mathrm{GL }(3)\) then \(\mathrm{det }(\mathrm{F })\ne 0\).

Another layer can be introduced in this decomposition if one consider also in-plane isometries.

Auffray, N.: Décomposition harmonique des tenseurs -Méthode spectrale-. Cr. Mecanique. 336, 370–375 (2008)MATHCrossRef

Alibert, J., Seppecher, F., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003)

Backus, G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. 8, 633–671 (1970)CrossRef

Baerheim, R.: Harmonic decomposition of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math 46, 391–418 (1993)MathSciNetMATHCrossRef

Eringen, A.C.: Theory of micropolar elasticity. In: Leibowitz, H. (ed.) Fracture, vol. 2, pp. 621–629. Academic Press, New York (1968)

Eringen, A.C., Suhubi, E.S.: Nonlinear Theory of simple microelastic solids. Int. J. Eng. Sci. 2, 189–203 (1964)MathSciNetMATHCrossRef

Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)CrossRef

Fleck, N.A., Hutchinson, J.W.: An assessment of a class of strain gradient plasticity theories. J. Mech. Phys. Solids 49, 2245–2272 (2001)MATHCrossRef

Forest, S., Sab, K.: Continuum stress gradient theory. Mech. Res. Commun. 40, 16–25 (2012)CrossRef

10.

Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elasticity 43, 81–108 (1996)MathSciNetMATHCrossRef

11.

Germain, P.: The method of virtual power in continuum mechanics. Part II: Application to continuum media with microstructure, SIAM. J. Appl Math. 25, 556–755 (1973)

12.

Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II, Springer, New York (1988)

13.

Geymonat, P., Weller, T.: Symmetry classes of piezoelectric solids. CR Aacad. Sci. I 335, 847–852 (2002)

14.

Jerphagnon, J., Chemla, D., Bonneville, R.: The description of the physical properties of condensed matter using irreducible tensors. Adv. Phys. 27, 609–650 (1978)CrossRef

15.

Jerphagnon, J.: Invariants of the third-rank Cartesian Tensor: optical nonlinear susceptibilities. Phys. Rev. B 2, 1091–1098 (1970)CrossRef

16.

Le Quang, H., Auffray, N., He, Q.-C., Bonnet, G.: Symmetry groups and classes of sixth-order strain-gradient elastic tensors tensors. P. Roy. Soc. Lond. A. Mat. (submitted)

17.

Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)MATHCrossRef

18.

Marangantia, R., Sharma, P.: A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. J. Mech. Phys. Solids 55, 1823–1852 (2007)CrossRef

19.

Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)MATHCrossRef

20.

Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)MathSciNetMATHCrossRef

21.

Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)CrossRef

22.

Nix, W.D., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)MATHCrossRef

23.

Olive, M., Auffray, N.: Symmetry classes for even-order tensors. Math. Mech. Compl. Sys. (Accepted) (2013)

24.

Seppecher, P., Alibert, J.-J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser. 319, 12018–12030 (2011)

25.

Smyshlyaev, V.P., Fleck, N.A.: The role of strain gradients in the grain size effect for polycrystals. J. Mech. Phys. Solids 44, 465–495 (1996)MathSciNetMATHCrossRef

26.

Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)MathSciNetMATHCrossRef

27.

Tekoglu, C., Onck, P.R.: Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models. J. Mech. Phys. Solids 56, 3541–3564 (2008)MATHCrossRef

28.

Truesdell, C., Toupin, R.: The Classical Field Theories, Encyclopedia of Physics (FlÃügge, ed.) vol. III/l, pp. 226–793. Springer, Berlin (1960)

29.

Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics, Handbuch der Physik III/3. Springer, New York (1965)

30.

Zou, W., Zheng, Q., Du, D., Rychlewski, J.: Orthogonal irreducible decompositions of tensors of high orders. Math. Mech. Solids 6, 249–267 (2001)MathSciNetMATHCrossRef

Titel: Geometrical Picture of Third-Order Tensors
verfasst von: Nicolas Auffray
Verlag: Springer Berlin Heidelberg
Buch: Generalized Continua as Models for Materials
Print ISBN: 978-3-642-36393-1

Electronic ISBN: 978-3-642-36394-8

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-3-642-36394-8_2

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht