Pour déterminer les déformations et les contraintes au sein de tissus biologiques tels que les ligaments, les tendons ou les parois artérielles, les lois de comportements hyperélastiques anisotropes sont souvent utilisées dans le cadre de la méthode des éléments finis [J.A. Weiss, B.N. Maker, S. Govindjee, Finite element implementation of incompressible, transversely isotropic hyperelasticity, Comp. Meth. Appl. Mech. Engng. 135 (1996) 107–128]. Dans cet article, on se propose de réaliser une telle étude en parallèle avec une analyse analytique. Cette analyse complémentaire permet de comprendre pourquoi la correspondance n'est pas biunivoque entre la déformation principale et le quatrième invariant de la matrice de dilatation pour un modèle usuel tel que celui proposé par Holzapfel, Gasser et Ogden [G.A. Holzapfel, T.C. Gasser, R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elasticity 61 (2000) 1–48; T.C. Gasser, R.W. Ogden, G.A. Holzapfel, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J. R. Soc. Interface 3 (2006) 15–35]. On établit en effet qu'une correspondance non bijective apparaît lorsque l'angle entre les fibres de collagène et la direction circonférentielle dépasse une valeur critique égale à 54,73°. L'importance de cet angle critique a déjà été relevée par Guo et al. (2006).
To determine the strain and stress in the biological soft tissues such as ligaments, tendons or arterial walls, anisotropic hyperelastic constitutive laws are often used in the context of finite element analysis [J.A. Weiss, B.N. Maker, S. Govindjee, Finite element implementation of incompressible, transversely isotropic hyperelasticity, Comp. Meth. Appl. Mech. Engng. 135 (1996) 107–128]. In the present paper, we propose to realize such a study together with a analytical study. This study allows for the understanding of the reason why it does not exist a one-to-one correspondence between the principal stretch and the fourth invariant of the dilatation tensor for the material model proposed by Holzapfel, Gasser and Ogden [G.A. Holzapfel, T.C. Gasser, R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elasticity 61 (2000) 1–48; T.C. Gasser, R.W. Ogden, G.A. Holzapfel, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J. R. Soc. Interface 3 (2006) 15–35]. In fact, the relationship becomes non-bijective when the angle between the collagen fibers and the circumferential direction is greater that a critical angle of 54.73°. Importance of this critical angle was also discussed by Guo et al. (2006).
Accepté le :
Publié le :
Keywords: Biomechanics, Anisotropic hyperelasticity, HGO model, Finite element
@article{CRMECA_2009__337_2_101_0,
author = {Fran\c{c}ois Peyraut and Christine Renaud and Nadia Labed and Zhi-Qiang Feng},
title = {Mod\'elisation de tissus biologiques en hyper\'elasticit\'e anisotrope {\textendash} {\'Etude} th\'eorique et approche \'el\'ements finis},
journal = {Comptes Rendus. M\'ecanique},
pages = {101--106},
publisher = {Elsevier},
volume = {337},
number = {2},
year = {2009},
doi = {10.1016/j.crme.2009.03.007},
language = {fr},
}
TY - JOUR AU - François Peyraut AU - Christine Renaud AU - Nadia Labed AU - Zhi-Qiang Feng TI - Modélisation de tissus biologiques en hyperélasticité anisotrope – Étude théorique et approche éléments finis JO - Comptes Rendus. Mécanique PY - 2009 SP - 101 EP - 106 VL - 337 IS - 2 PB - Elsevier DO - 10.1016/j.crme.2009.03.007 LA - fr ID - CRMECA_2009__337_2_101_0 ER -
%0 Journal Article %A François Peyraut %A Christine Renaud %A Nadia Labed %A Zhi-Qiang Feng %T Modélisation de tissus biologiques en hyperélasticité anisotrope – Étude théorique et approche éléments finis %J Comptes Rendus. Mécanique %D 2009 %P 101-106 %V 337 %N 2 %I Elsevier %R 10.1016/j.crme.2009.03.007 %G fr %F CRMECA_2009__337_2_101_0
François Peyraut; Christine Renaud; Nadia Labed; Zhi-Qiang Feng. Modélisation de tissus biologiques en hyperélasticité anisotrope – Étude théorique et approche éléments finis. Comptes Rendus. Mécanique, Volume 337 (2009) no. 2, pp. 101-106. doi : 10.1016/j.crme.2009.03.007. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.03.007/
[1] Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J. R. Soc. Interface, Volume 3 (2006), pp. 15-35
[2] A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elasticity, Volume 61 (2000), pp. 1-48
[3] A variational approach for materially stable anisotropic hyperelasticity, Int. J. Solids Struct., Volume 42 (2005), pp. 4352-4371
[4] Pseudoelasticity of arteries and the choice of its mathematical expression, Am. J. Physiol., Volume 237 (1979), p. H620-H631
[5] Finite element implementation of incompressible, transversely isotropic hyperelasticity, Comp. Meth. Appl. Mech. Engng., Volume 135 (1996), pp. 107-128
[6] Isotropic polynomial invariants and tensor functions (J.P. Boehler, ed.), Applications of Tensor Functions in Solids Mechanics, CISM Course, No. 282, Springer Verlag, 1987
[7] A polyconvex framework for soft biological tissues. Adjustment to experimental data, Int. J. Solids Struct., Volume 43 (2006), pp. 6052-6070
Cité par Sources :
Commentaires - Politique
From Boussinesq–Love contact to impact between hyperelastic bodies
Zhi-Qiang Feng; Claude Vallée
C. R. Méca (2007)
On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity
Claude Vallée; Danielle Fortuné; Camelia Lerintiu
C. R. Méca (2008)
Vanessa Bouchart; Mathias Brieu; Djimedo Kondo; ...
C. R. Méca (2008)