The orthogonalized harmonic decomposition of symmetric fourth-order tensors (i.e. having major and minor indicial symmetries, such as elasticity tensors) is completed by a representation of harmonic fourth-order tensors by means of two second-order harmonic (symmetric deviatoric) tensors only. A similar decomposition is obtained for non-symmetric tensors (i.e. having minor indicial symmetry only, such as photo-elasticity tensors or elasto-plasticity tangent operators) introducing a fourth-order major antisymmetric traceless tensor . The tensor is represented by means of one harmonic second-order tensor and one antisymmetric second-order tensor only. Representations of totally symmetric (rari-constant), symmetric and major antisymmetric fourth-order tensors are simple particular cases of the proposed general representation. Closed-form expressions for tensor decomposition are given in the monoclinic case. Practical applications to elasticity and photo-elasticity monoclinic tensors are finally presented.
La décomposition harmonique orthogonalisée des tenseurs symétriques d'ordre quatre (ayant les symétries majeures et mineures, tels que le tenseur d'élasticité) est complétée par une représentation des tenseurs harmoniques d'ordre quatre à l'aide de deux tenseurs harmoniques (symétriques déviatoriques) d'ordre deux. Une décomposition similaire est obtenue pour les tenseurs non symétriques (ayant uniquement la symétrie mineure, tels que ceux rencontrés en photo-élasticité et en élasto-plasticité), introduisant un tenseur antisymétrique majeur à traces nulles . Le tenseur est représenté par deux tenseurs d'ordre deux, le premier harmonique et le second antisymétrique. Les représentations des tenseurs d'ordre quatre complètement symétriques (rari-constants), symétriques et antisymétriques majeurs sont des cas particuliers simples de la représentation proposée. Les expressions analytiques de la décomposition correspondante dans le cas monoclinique sont obtenues et appliquées à l'élasticité et à la photo-élasticité monocliniques.
Accepted:
Published online:
Mots-clés : Décomposition harmonique, Anisotropie, Élasticité, Photo-élasticité, Tenseur rari-constant, Feldspar, Taurine
Rodrigue Desmorat 1; Boris Desmorat 2, 3
@article{CRMECA_2016__344_6_402_0,
author = {Rodrigue Desmorat and Boris Desmorat},
title = {3D extension of {Tensorial} {Polar} {Decomposition.} {Application} to (photo-)elasticity tensors},
journal = {Comptes Rendus. M\'ecanique},
pages = {402--417},
publisher = {Elsevier},
volume = {344},
number = {6},
year = {2016},
doi = {10.1016/j.crme.2016.01.006},
language = {en},
}
TY - JOUR AU - Rodrigue Desmorat AU - Boris Desmorat TI - 3D extension of Tensorial Polar Decomposition. Application to (photo-)elasticity tensors JO - Comptes Rendus. Mécanique PY - 2016 SP - 402 EP - 417 VL - 344 IS - 6 PB - Elsevier DO - 10.1016/j.crme.2016.01.006 LA - en ID - CRMECA_2016__344_6_402_0 ER -
Rodrigue Desmorat; Boris Desmorat. 3D extension of Tensorial Polar Decomposition. Application to (photo-)elasticity tensors. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 402-417. doi : 10.1016/j.crme.2016.01.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.006/
[1] Symmetry classes for elasticity tensors, J. Elast., Volume 43 (1996) no. 2, pp. 81-108
[2] Classi di simmetria in elasticita piana, Florence (2005)
[3] Symmetry classes and harmonic decomposition for photoelasticity tensors, Int. J. Eng. Sci. (1997), pp. 1317-1326
[4] Tensor Analysis for Physicists, Clarendon Press, 1954
[5] A geometrical picture of anisotropic elastic tensors, Rev. Geophys., Volume 8 (1970) no. 3, pp. 633-671
[6] A note on the decomposition of tensors into traceless symmetric tensors, Int. J. Eng. Sci., Volume 8 (1970), pp. 475-481
[7] Harmonic decomposition of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math., Volume 46 (1993) no. 3, pp. 391-418
[8] Properties of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math., Volume 42 (1989), pp. 249-266
[9] Lectures notes of Summer School MatSyMat, http://matsymat.sciencesconf.org, N. Aufray, M. Coret, M. François, B. Kolev, G. de Saxcé, J.F. Witz, Nantes, September 2014.
[10] Effective properties of elastic materials that contain penny shaped voids, Int. J. Eng. Sci., Volume 22 (1984), pp. 1013-1021
[11] Singularities and Groups in Bifurcation Theory, vol. 2, Springer-Verlag, New York–Berlin, 1985
[12] Representations for transversely hemitropic and transversely isotropic stress–strain relations, J. Elast., Volume 37 (1995), pp. 243-280
[13] Symmetry classes of piezoelectric solids, C. R. Math. Acad. Sci. Paris, Sér. I, Volume 335 (2002), pp. 847-852
[14] On the polynomial invariants of the elasticity tensor, J. Elast., Volume 34 (1994) no. 2, pp. 97-110
[15] Integrity basis for a second-order and a fourth-order tensor, Int. J. Math. Math. Sci., Volume 5 (1981), pp. 87-96
[16] Group Theory and Physics, Cambridge University Press, 1994
[17] Theory of representations for tensor functions—a unified invariant approach to constitutive equations, Appl. Mech. Rev., Volume 47 (1994) no. 11, pp. 545-587
[18] On anisotropic polynomial relations for the elasticity tensor, J. Elast., Volume 115 (2014) no. 1, pp. 77-103
[19] Invariant properties of composite materials, St. Louis, Missouri, 1967 (S.W. Tsai; J.C. Halpin; N.J. Pagano, eds.), Technomic Publishing Company, USA (1968), pp. 233-253
[20] Les invariants des tenseurs d'ordre 4 du type de l'élasticité, Villard-de-Lans, 1979, Éditions du CNRS, Paris (1982), pp. 93-104
[21] Plane anisotropy by the polar method, Meccanica, Volume 40 (2005), pp. 437-454
[22] Structure of the space of 2D elasticity tensors, J. Elast., Volume 111 (2013), pp. 21-39
[23] A unified approach to invariants of plane elasticity tensors, Meccanica, Volume 49 (2014) no. 9, pp. 2000-2012
[24] Tensorial Polar Decomposition of 2D fourth-order tensors, C. R. Mecanique, Volume 343 (2015) no. 9, pp. 471-475
[25] Anisotropy of plane complex elastic bodies, Int. J. Solids Struct., Volume 47 (2010), pp. 1154-1166
[26] A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, 1905
[27] The evolution of Hooke's law due to texture development in FCC polycrystals, Int. J. Solids Struct., Volume 38 (2001), pp. 9437-9459
[28] Elasticity of minerals, glesses and melts (T.J. Ahrens, ed.), Mineral Physics and Crystallography: A Handbook of Physical Constants, AGU, Washington, DC, 1995, pp. 45-63
[29] CRC Handbook of Chemistry and Physics (W.M. Haynes; D.R. Lide; T.J. Bruno, eds.), CRC Press, Boca Raton, London, New York, 2016 http://www.hbcpnetbase.com
Cited by Sources:
Comments - Policy