[Décomposition polaire tensorielle des tenseurs 2D d'ordre 4]
On étudie la structure des tenseurs 2D symétriques d'ordre 4, c'est-à-dire : ayant aussi bien la symétrie indicielle mineure que la symétrie majeure. La décomposition polaire de Verchery est réécrite sous forme tensorielle nommée décomposition polaire tensorielle. Le résultat principal est que tout tenseur 2D symétrique d'ordre 4 peut s'écrire à l'aide de tenseurs d'ordre 2 uniquement dans une décomposition faisant apparaître explicitement les invariants et les classes de symétrie. Le lien avec la décomposition harmonique est fait en utilisant la décomposition de Kelvin de son terme harmonique.
One studies the structure of 2D symmetric fourth-order tensors, i.e. having both minor and major indicial symmetries. Verchery polar decomposition is rewritten in a tensorial form entitled Tensorial Polar Decomposition. The main result is that any 2D symmetric fourth-order tensor can be written in terms of second-order tensors only in a decomposition that makes explicitly appear invariants and symmetry classes. The link with harmonic decomposition is made thanks to Kelvin decomposition of its harmonic term.
Accepté le :
Publié le :
Mots-clés : Décomposition polaire, Invariants, Décomposition harmonique, Décomposition de Kelvin
Boris Desmorat 1, 2 ; Rodrigue Desmorat 3
@article{CRMECA_2015__343_9_471_0, author = {Boris Desmorat and Rodrigue Desmorat}, title = {Tensorial {Polar} {Decomposition} of {2D} fourth-order tensors}, journal = {Comptes Rendus. M\'ecanique}, pages = {471--475}, publisher = {Elsevier}, volume = {343}, number = {9}, year = {2015}, doi = {10.1016/j.crme.2015.07.002}, language = {en}, }
Boris Desmorat; Rodrigue Desmorat. Tensorial Polar Decomposition of 2D fourth-order tensors. Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 471-475. doi : 10.1016/j.crme.2015.07.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.07.002/
[1] Harmonic decomposition of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math., Volume 46 (1993) no. 3, pp. 391-418
[2] On the polynomial invariants of the elasticity tensor, J. Elasticity, Volume 34 (1994) no. 2, pp. 97-110
[3] Symmetry classes for elasticity tensors, J. Elasticity, Volume 43 (1996) no. 2, pp. 81-108
[4] On anisotropic polynomial relations for the elasticity tensor, J. Elasticity, Volume 115 (2014) no. 1, pp. 77-103
[5] Invariant properties of composite materials (S.W. Tsai; J.C. Halpin; N.J. Pagano, eds.), Composite Materials Workshop, Technomic Publishing Co., Lancaster, PA, USA, 1968, pp. 233-253
[6] Les invariants des tenseurs d'ordre 4 du type de l'élasticité, France, Villard-de-Lans, 1979, Éditions du CNRS, Paris (1982), pp. 93-104
[7] Plane anisotropy by the polar method, Meccanica, Volume 40 (2005), pp. 437-454
[8] Structure of the space of 2D elasticity tensors, J. Elasticity, Volume 111 (2013), pp. 21-39
[9] An alternative to the Kelvin decomposition for plane anisotropic elasticity, Math. Methods Appl. Sci., Volume 38 (2015), pp. 164-175
[10] A unified approach to invariants of plane elasticity tensors, Meccanica (2014) | DOI
[11] On Hooke's law, Prikl. Mat. Meh., Volume 48 (1984), pp. 303-314
[12] Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl. Math., Volume 43 (1990), pp. 15-41
[13] Détermination des symétries matérielles de matériaux anisotropes, Université Paris-6, 1995 (PhD thesis)
[14] Nantes, France ( September 2014 ) http://matsymat.sciencesconf.org
- A hybrid Newton’s method for solving tensor square root problem, Journal of Applied Mathematics and Computing (2025) | DOI:10.1007/s12190-024-02351-6
- A Direct Approach to the Polar Representation of Plane Tensors, Journal of Elasticity, Volume 156 (2024) no. 4-5, p. 1065 | DOI:10.1007/s10659-024-10085-6
- Anisotropic damage state modeling based on harmonic decomposition and discrete simulation of fracture, Engineering Fracture Mechanics, Volume 293 (2023), p. 109669 | DOI:10.1016/j.engfracmech.2023.109669
- Computation of minimal covariants bases for 2D coupled constitutive laws, International Journal of Engineering Science, Volume 191 (2023), p. 103880 | DOI:10.1016/j.ijengsci.2023.103880
- Continuous anisotropic damage as a twin modelling of discrete bi-dimensional fracture, European Journal of Mechanics - A/Solids, Volume 89 (2021), p. 104285 | DOI:10.1016/j.euromechsol.2021.104285
- Euromech 579 Arpino 3–8 April 2017: Generalized and microstructured continua: new ideas in modeling and/or applications to structures with (nearly)inextensible fibers—a review of presentations and discussions, Continuum Mechanics and Thermodynamics, Volume 30 (2018) no. 5, p. 1011 | DOI:10.1007/s00161-018-0654-6
- Micromechanics based framework with second-order damage tensors, European Journal of Mechanics - A/Solids, Volume 69 (2018), p. 88 | DOI:10.1016/j.euromechsol.2017.11.014
- Harmonic Factorization and Reconstruction of the Elasticity Tensor, Journal of Elasticity, Volume 132 (2018) no. 1, p. 67 | DOI:10.1007/s10659-017-9657-y
- Second order tensorial framework for 2D medium with open and closed cracks, European Journal of Mechanics - A/Solids, Volume 58 (2016), p. 262 | DOI:10.1016/j.euromechsol.2016.02.004
Cité par 9 documents. Sources : Crossref
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier