Modélisation de tissus biologiques en hyperélasticité anisotrope – Étude théorique et approche éléments finis

François Peyraut; Christine Renaud; Nadia Labed; Zhi-Qiang Feng

doi:10.1016/j.crme.2009.03.007

Modélisation de tissus biologiques en hyperélasticité anisotrope – Étude théorique et approche éléments finis

Présenté par : Jean-Baptiste Leblond

François Peyraut ¹ ; Christine Renaud ² ; Nadia Labed ¹ ; Zhi-Qiang Feng ²

¹ Laboratoire M3M, Université de technologie de Belfort-Montbéliard, 90010 Belfort, France
² Laboratoire LME-Evry, Université d'Évry – Val d'Essonne, 91020 Évry, France

Comptes Rendus. Mécanique, Volume 337 (2009) no. 2, pp. 101-106.

Résumé (VO)
Abstract

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 $λ_{2}$ 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 $λ_{2}$ 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).

Reçu le : 2008-12-19
Accepté le : 2009-03-13
Publié le : 2009-02-01

DOI : 10.1016/j.crme.2009.03.007

Mots-clés : Biomécanique, Hyperélasticité anisotrope, Modèle HGO, Éléments finis
Keywords: Biomechanics, Anisotropic hyperelasticity, HGO model, Finite element

Affiliations des auteurs :

François Peyraut ¹ ; Christine Renaud ² ; Nadia Labed ¹ ; Zhi-Qiang Feng ²

¹ Laboratoire M3M, Université de technologie de Belfort-Montbéliard, 90010 Belfort, France
² Laboratoire LME-Evry, Université d'Évry – Val d'Essonne, 91020 Évry, France

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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/

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