[Variable duale du tenseur des contraintes de Cauchy dans le cas des matériaux hyperélastiques isotropes]
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 . Les matériaux élastiques isotropes sont les seuls matériaux pour lesquels le tenseur des contraintes de Cauchy σ ne dépend que du tenseur des déformations . 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 sont reliés par une loi de comportement dérivant du potentiel composé . 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 . Isotropic elastic materials are the special cases for which the Cauchy stress tensor σ depends solely on the left Cauchy–Green strain tensor . 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 are related by a constitutive law derived from the compound potential . We give a new and concise proof which is based on an explicit integral formula expressing the derivative of the exponential of a tensor.
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Mot clés : Déformation finie, Hyperélasticité isotrope, Variables duales, Déformation logarithmique, Tenseur des déformations de Hencky, Loi de comportement
@article{CRMECA_2008__336_11-12_851_0,
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},
}
TY - JOUR AU - Claude Vallée AU - Danielle Fortuné AU - Camelia Lerintiu TI - On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity JO - Comptes Rendus. Mécanique PY - 2008 SP - 851 EP - 855 VL - 336 IS - 11-12 PB - Elsevier DO - 10.1016/j.crme.2008.10.003 LA - en ID - CRMECA_2008__336_11-12_851_0 ER -
%0 Journal Article %A Claude Vallée %A Danielle Fortuné %A Camelia Lerintiu %T On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity %J Comptes Rendus. Mécanique %D 2008 %P 851-855 %V 336 %N 11-12 %I Elsevier %R 10.1016/j.crme.2008.10.003 %G en %F CRMECA_2008__336_11-12_851_0
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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