Comptes Rendus
On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity
[Variable duale du tenseur des contraintes de Cauchy dans le cas des matériaux hyperélastiques isotropes]
Comptes Rendus. Mécanique, Volume 336 (2008) no. 11-12, pp. 851-855.

Les matériaux élastiques sont régis par une loi de comportement reliant le second tenseur des contraintes de Piola–Kirchhoff Σ et le tenseur de Cauchy–Green droit C=FTF. Les matériaux élastiques isotropes sont les seuls matériaux pour lesquels le tenseur des contraintes de Cauchy σ ne dépend que du tenseur des déformations B=FFT. Dans cette Note nous revisitons la propriété suivante des matériaux isotropes hyperélastiques : si la loi de comportement reliant Σ et C dérive d'un potentiel ϕ, alors σ et lnB sont reliés par une loi de comportement dérivant du potentiel composé ϕexp. Nous donnons une preuve nouvelle et concise qui est basée sur une formule intégrale explicite exprimant la dérivée de l'exponentiel d'un tenseur.

Elastic materials are governed by a constitutive law relating the second Piola–Kirchhoff stress tensor Σ and the right Cauchy–Green strain tensor C=FTF. Isotropic elastic materials are the special cases for which the Cauchy stress tensor σ depends solely on the left Cauchy–Green strain tensor B=FFT. In this Note we revisit the following property of isotropic hyperelastic materials: if the constitutive law relating Σ and C is derivable from a potential ϕ, then σ and lnB are related by a constitutive law derived from the compound potential ϕexp. We give a new and concise proof which is based on an explicit integral formula expressing the derivative of the exponential of a tensor.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2008.10.003
Keywords: Finite strain, Isotropic hyperelasticity, Dual variables, Logarithmic strain, Hencky strain tensor, Constitutive law
Mot clés : Déformation finie, Hyperélasticité isotrope, Variables duales, Déformation logarithmique, Tenseur des déformations de Hencky, Loi de comportement

Claude Vallée 1 ; Danielle Fortuné 1 ; Camelia Lerintiu 1

1 Laboratoire de Mécanique des Solides, UMR CNRS 6610, Université de Poitiers, SP2MI, téléport 2, boulevard Marie et Pierre Curie, B.P. 30179, 86962 Futuroscope-Chasseneuil cedex, France
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Claude Vallée; Danielle Fortuné; Camelia Lerintiu. On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity. Comptes Rendus. Mécanique, Volume 336 (2008) no. 11-12, pp. 851-855. doi : 10.1016/j.crme.2008.10.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.10.003/

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