Comptes Rendus
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.

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.

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.

Received:
Accepted:
Published online:
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
@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/

[1] P.G. Ciarlet Mathematical Elasticity, vol. 1, Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988

[2] J.M. Souriau Calcul linéaire, P.U.F, Paris, 1959

[3] C. Vallée Lois de comportement élastique isotropes en grandes déformations, International Journal of Engineering Science, Volume 16 (1978) no. 7, pp. 451-457

[4] C. Sansour On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues, International Journal of Solids and Structures, Volume 38 (2001) no. 50–51, pp. 9221-9232

[5] A. Hoger The stress conjugate to logarithmic strain, International Journal of Solids and Structures, Volume 23 (1987) no. 12, pp. 1645-1656

[6] O.T. Bruhns; H. Xiao; A. Meyers Constitutive inequalities for an isotropic elastic strain–energy function based on Hencky's logarithmic strain tensor, Proceedings of the Royal Society of London, Series A – Mathematical Physical and Engineering Sciences, Volume 457 (2001) no. 2013, pp. 2207-2226

[7] R.W. Ogden; G. Saccomandi; I. Sgura Fitting hyperelastic models to experimental data, Computational Mechanics, Volume 34 (2004) no. 6, pp. 484-502

[8] D. Peric; D.R.J. Owen; M.E. Honnor A model for finite strain elastoplasticity based on logarithmic strains – computational issues, Computer Methods in Applied Mechanics and Engineering, Volume 94 (1992) no. 1, pp. 35-61

[9] T. Sendova; J.R. Walton On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain, International Journal of Non-Linear Mechanics, Volume 40 (2005) no. 2–3, pp. 195-212

[10] H. Xiao; L.S. Chen Hencky's elasticity model and linear stress–strain relations in isotropic finite hyperelasticity, Acta Mechanica, Volume 157 (2002) no. 1–4, pp. 51-60

[11] Z.-Q. Feng; C. Vallée; D. Fortuné; F. Peyraut The 3é hyperelastic model applied to the modeling of 3D impact problems, Finite Elements in Analysis and Design, Volume 43 (2006) no. 1, pp. 51-58

[12] J.A. Blume On the form of the inverted stress–strain law for isotropic hyperelastic solids, International Journal of Non-Linear Mechanics, Volume 27 (1992) no. 3, pp. 413-421

[13] H. Xiao; L.S. Chen Hencky's logarithmic strain and dual stress–strain and strain–stress relations in isotropic finite hyperelasticity, International Journal of Solids and Structures, Volume 40 (2003) no. 6, pp. 1455-1463

[14] H. Xiao; O.T. Bruhns; A. Meyers Explicit dual stress–strain and strain–stress relations of incompressible isotropic hyperelastic solids via deviatoric Hencky strain and Cauchy stress, Acta Mechanica, Volume 168 (2004) no. 1–2, pp. 21-33

[15] O.V. Sadovskaya; V.M. Sadovskii The theory of finite strains of a granular material, Journal of Applied Mathematics and Mechanics, Volume 71 (2007) no. 1, pp. 93-110

Cited by Sources:

Comments - Policy