On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

Claude Vallée; Danielle Fortuné; Camelia Lerintiu

doi:10.1016/j.crme.2008.10.003

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]

Présenté par : Philippe G. Ciarlet

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

Comptes Rendus. Mécanique, Volume 336 (2008) no. 11-12, pp. 851-855.

Résumé
Abstract

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 $C = F^{T} F$ . Les matériaux élastiques isotropes sont les seuls matériaux pour lesquels le tenseur des contraintes de Cauchy σ ne dépend que du tenseur des déformations $B = F F^{T}$ . 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 $\ln B$ sont reliés par une loi de comportement dérivant du potentiel composé $ϕ ○ \exp$ . 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 $C = F^{T} F$ . Isotropic elastic materials are the special cases for which the Cauchy stress tensor σ depends solely on the left Cauchy–Green strain tensor $B = F F^{T}$ . 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 $\ln B$ are related by a constitutive law derived from the compound potential $ϕ ○ \exp$ . 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 : 2008-05-05
Accepté le : 2008-10-06
Publié le : 2008-10-28

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

Affiliations des auteurs :

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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     author = {Claude Vall\'ee and Danielle Fortun\'e and Camelia Lerintiu},
     title = {On the dual variable of the {Cauchy} stress tensor in isotropic finite hyperelasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {851--855},
     publisher = {Elsevier},
     volume = {336},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crme.2008.10.003},
     language = {en},
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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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