Comptes Rendus
no. 9
Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties
[Sur une classe de potentiels élastiques stochastiques pour les matériaux hyperélastiques incompressibles isotropes]
Brian Staber1  ; Johann Guilleminot1
1 Université Paris-Est–Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex, France
Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 503-514.

Dans cette Note, on s'intéresse à la construction d'une classe de modèles stochastiques pour des matériaux hyperélastiques incompressibles. La méthodologie de construction repose sur le principe du maximum d'entropie, formulé à partir de contraintes induites par les théorèmes d'existence en élasticité non linéaire. Plus précisément, des contraintes associées à la polyconvexité et à la cohérence avec l'élasticité linéarisée sont introduites, et éventuellement couplées avec une contrainte relative à la fonction moyenne. Deux modèles probabilistes paramétriques pour les densités d'énergie considérées sont par suite proposés dans le cas isotrope, et reposent notamment sur un conditionnement vis-à-vis du module de cisaillement aléatoire. Des simulations numériques de Monte Carlo pour des potentiels classiques (e.g., Néo-Hookéen ou Mooney–Rivlin) sont ensuite conduites afin d'illustrer les capacités du modèle. Une identification inverse basée sur des résultats expérimentaux est enfin présentée.

In this Note, we address the construction of a class of stochastic Ogden's stored energy functions associated with incompressible hyperelastic materials. The methodology relies on the maximum entropy principle, which is formulated under constraints arising in part from existence theorems in nonlinear elasticity. More specifically, constraints related to both polyconvexity and consistency with linearized elasticity are considered and potentially coupled with a constraint on the mean function. Two parametric probabilistic models are thus derived for the isotropic case and rely in part on a conditioning with respect to the random shear modulus. Monte Carlo simulations involving classical (e.g., Neo-Hookean or Mooney–Rivlin) stored energy functions are then performed in order to illustrate some capabilities of the probabilistic models. An inverse calibration involving experimental results is finally presented.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2015.07.008
Keywords: Maximum entropy principle, Stochastic modeling, Hyperelasticity, Stored energy functions, Polyconvexity, Uncertainties
Mot clés : Principe du maximum d'entropie, Modélisation stochastique, Hyperélasticité, Densités d'énergie élastiques, Polyconvexité, Incertitudes
Brian Staber 1 ; Johann Guilleminot 1

1 Université Paris-Est–Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex, France 
@article{CRMECA_2015__343_9_503_0,
     author = {Brian Staber and Johann Guilleminot},
     title = {Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {503--514},
     publisher = {Elsevier},
     volume = {343},
     number = {9},
     year = {2015},
     doi = {10.1016/j.crme.2015.07.008},
     language = {en},
}
TY  - JOUR
AU  - Brian Staber
AU  - Johann Guilleminot
TI  - Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 503
EP  - 514
VL  - 343
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2015.07.008
LA  - en
ID  - CRMECA_2015__343_9_503_0
ER  - 
%0 Journal Article
%A Brian Staber
%A Johann Guilleminot
%T Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties
%J Comptes Rendus. Mécanique
%D 2015
%P 503-514
%V 343
%N 9
%I Elsevier
%R 10.1016/j.crme.2015.07.008
%G en
%F CRMECA_2015__343_9_503_0
Brian Staber; Johann Guilleminot. Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties. Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 503-514. doi : 10.1016/j.crme.2015.07.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.07.008/

[1] G. Chagnon; M. Rebouah; D. Favier Hyperelastic energy densities for soft biological tissues: a review, J. Elasticity. (2014), pp. 1-32

[2] T.I. Zhodi Propagation of microscale material uncertainty in a class of hyperelastic finite deformations stored energy functions, Int. J. Fract., Volume 112 (2001), pp. 13-17

[3] A. Clément; C. Soize; J. Yvonnet Computational nonlinear stochastic homogenization using a non-concurrent multiscale approach for hyperelastic heterogeneous microstructures analysis, Int. J. Numer. Methods Eng., Volume 91 (2012) no. 8, pp. 799-824

[4] A. Clément; C. Soize; J. Yvonnet Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials, Comput. Methods Appl. Mech. Eng., Volume 254 (2013), pp. 61-82

[5] R.G. Ghanem; P. Spanos Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991

[6] C. Truesdell; W. Noll The Non-Linear Field Theories of Mechanics, Springer, Berlin, 2004

[7] P.G. Ciarlet Mathematical Elasticity, vol. I: Three-Dimensional Elasticity, Elsevier Science Publishers, North-Holland, Amsterdam, 1988

[8] R.W. Ogden Non-Linear Elastic Deformations, Dover Publications, Mineola, New York, 1997

[9] G.A. Holzapfel Nonlinear Solid Mechanics – A Continuum Approach for Engineering, John Wiley & Sons Ltd, the Atrium, Southern Gate, Chichester, England, 2000

[10] R.W. Ogden Large deformation isotropic elasticity – On the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond. A, Volume 328 (1972) no. 1575, pp. 567-583

[11] R.W. Ogden Large deformation isotropic elasticity – On the correlation of theory and experiment for compressible rubberlike solids, Proc. R. Soc. Lond. A, Volume 326 (1972) no. 1567, pp. 565-584

[12] J. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403

[13] J.E. Marsden; T.J.R. Hughes Mathematical Foundations of Elasticity, Dover, 1983

[14] S. Müller; T. Qi; B.S. Yan On a new class of elastic deformations not allowing for cavitation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 11 (1994) no. 2, pp. 217-243

[15] L.R.G. Treloar Stress-strain data for vulcanised rubber under various types of deformation, Trans. Faraday Soc., Volume 241 (1943) no. 39, pp. 59-70

[16] M. Giaquinta; G. Modica; J. Soucek Cartesian currents and variational problems for mappings into spheres, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 16 (1989) no. 3, pp. 393-485

[17] C.E. Shannon A mathematical theory of communication, Bell Syst. Tech. J., Volume 27 (1948), pp. 379-423 (623–659)

[18] E.T. Jaynes Information theory and statistical mechanics i, Phys. Rev., Volume 108 (1957) no. 2, pp. 171-190

[19] E.T. Jaynes Information theory and statistical mechanics ii, Phys. Rev., Volume 106 (1957) no. 4, pp. 620-630

[20] C. Soize Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial derivative operators, Comput. Methods Appl. Mech. Eng., Volume 195 (2006), pp. 26-64

[21] J. Guilleminot; C. Soize Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model, Int. J. Numer. Methods Eng., Volume 90 (2012) no. 5, pp. 613-635

[22] J. Guilleminot; C. Soize On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, J. Elasticity., Volume 111 (2013), pp. 109-130

[23] J. Guilleminot; C. Soize Prior modeling and generator for non-Gaussian tensor-valued random fields with symmetry properties, SIAM Multiscale Model. Simul., Volume 11 (2013), pp. 840-870

[24] C. Soize A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probab. Eng. Mech., Volume 15 (2000) no. 3, pp. 277-294

[25] M. Abramowitz; I.A. Stegun Handbook of Mathematical Functions: With Formulas Graphs and Mathematical Tables, Dover, New York, 1970

[26] S. Kotz; N. Balakrishnan; N.L. Johnson Continuous Multivariate Distributions. Volume 1: Models and Applications, Wiley, 2000

[27] L. Devroye Non-Uniform Random Variate Generation, Springer Verlag, New York, 1986

[28] A.K. Gupta; L. Cardeño; D.K. Nagar Matrix-variate Kummer–Dirichlet distribution, J. Appl. Math., Volume 1 (2001) no. 3, pp. 117-139

[29] J. Guilleminot; C. Soize ISDE-based generator for a class of non-Gaussian vector-valued random fields in uncertainty quantification, SIAM J. Sci. Comput., Volume 36 (2014), pp. 2763-2786

[30] S. Das; R.G. Ghanem A bounded random matrix approach for stochastic upscaling, SIAM Multiscale Model. Simul., Volume 8 (2009) no. 1, pp. 296-325

[31] R.W. Ogden; G. Saccomandi; I. Sigura Fitting hyperelastic models to experimental data, Comput. Mech., Volume 34 (2004), pp. 484-502

[32] A.-S. Caro-Bretelle; P. Ienny; R. Leger Constitutive modeling of a SEBS cast-calender: large strain, compressibility and anisotropic damage induced by the process, Polymer, Volume 54 (2013), pp. 4594-4603

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Exact solution of the Dirac–Weyl equation in graphene under electric and magnetic fields

Mahdi Eshghi; Hosein Mehraban

C. R. Phys (2017)

Jump conditions and dynamic surface tension at permeable interfaces such as the inner core boundary

Frédéric Chambat; Sylvie Benzoni-Gavage; Yanick Ricard

C. R. Géos (2014)

Remarks on the canonical metrics on the Cartan–Hartogs domains

Enchao Bi; Zhenhan Tu

C. R. Math (2017)