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Risk-averse estimates of effective properties in heterogeneous elasticity
[Estimations averses au risque des propriétés effectives en élasticité hétérogène]
Jeremy Bleyer1 
1 Laboratoire Navier, ENPC, Univ Gustave Eiffel, CNRS, Cité Descartes, 6-8 av Blaise Pascal, 77455 Champs-sur-Marne, France
Comptes Rendus. Mécanique, Volume 351 (2023), pp. 29-42.

Dans ce travail, nous proposons un cadre théorique pour le calcul d’estimations pessimistes et optimistes des propriétés effectives dans le cas de matériaux élastiques hétérogènes avec des propriétés élastiques microscopiques incertaines. Nous nous appuyons sur une mesure d’aversion au risque largement utilisée en finance appelée la valeur conditionnelle au risque (CVaR). La CVaR calcule l’espérance conditionnelle des événements se produisant au-delà d’un niveau de risque donné, caractérisant ainsi les queues extrêmes de la distribution de probabilité d’une variable aléatoire. Dans le contexte des matériaux élastiques, nous proposons d’utiliser la CVaR sur l’énergie libre élastique pour calculer une estimation optimiste de la rigidité globale pour un certain niveau de confiance α. De même, nous utilisons également la CVaR sur l’énergie élastique complémentaire pour calculer une estimation pessimiste de la rigidité globale. Les estimations CVaR obtenues bénéficient d’une formulation par optimisation convexe. Le comportement du matériau résultant est toujours élastique mais plus nécessairement linéaire. Nous proposons également des approximations conduisant à un comportement élastique linéaire. Nous appliquons les formulations proposées aux estimations micromécaniques des propriétés élastiques effectives de matériaux hétérogènes aléatoires.

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In this work, we propose a theoretical framework for computing pessimistic and optimistic estimates of effective properties in the case of heterogeneous elastic materials with uncertain microscopic elastic properties. We rely on a risk-averse measure widely used in finance called the conditional-value at risk (CVaR). The CVaR computes the conditional expectation of events occurring above a given risk level, thereby characterizing the extreme tails of the probability distribution of a random variable. In the context of elastic materials, we propose to use the CVaR on the elastic free energy to compute an optimistic estimate of the global stiffness for some confidence level α. Similarly, we also use the CVaR on the complementary elastic energy to compute a pessimistic estimate of the global stiffness. The obtained CVaR estimates benefit from a convex optimization formulation. The resulting material behavior is still elastic but not necessarily linear anymore. We discuss approximate formulations recovering a linear elastic behavior. We apply the proposed formulations to the micromechanical estimates of effective elastic properties of random heterogeneous materials.

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DOI : 10.5802/crmeca.171
Keywords: Uncertainty, Risk measure, Elasticity, Random materials, Convex optimization
Mot clés : Incertitude, Mesure de risque, Élasticité, Matériaux aléatoires, Optimisation convexe
Jeremy Bleyer 1

1 Laboratoire Navier, ENPC, Univ Gustave Eiffel, CNRS, Cité Descartes, 6-8 av Blaise Pascal, 77455 Champs-sur-Marne, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Risk-averse estimates of effective properties in heterogeneous elasticity},
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     year = {2023},
     doi = {10.5802/crmeca.171},
     language = {en},
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Jeremy Bleyer. Risk-averse estimates of effective properties in heterogeneous elasticity. Comptes Rendus. Mécanique, Volume 351 (2023), pp. 29-42. doi : 10.5802/crmeca.171. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.171/

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