Comptes Rendus
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.

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.

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.

Received:
Accepted:
Published online:
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
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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/

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