Qualitative and quantitative results in ...
Document type :
Habilitation à diriger des recherches
Title :
Qualitative and quantitative results in stochastic homogenization
English title :
Qualitative and quantitative results in stochastic homogenization
Author(s) :
Gloria, Antoine [Auteur]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Département de Mathématique [Bruxelles] [ULB]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Département de Mathématique [Bruxelles] [ULB]
Thesis director(s) :
Denis Talay
Defence date :
2012-02-24
Jury president :
Bob Kohn (rapporteur)
Stefano Olla (rapporteur)
George Papanicolaou (rapporteur)
Christophe Besse (examinateur)
Albert Cohen (examinateur)
Thierry Goudon (examinateur)
Claude Le Bris (examinateur)
Felix Otto (examinateur)
Stefano Olla (rapporteur)
George Papanicolaou (rapporteur)
Christophe Besse (examinateur)
Albert Cohen (examinateur)
Thierry Goudon (examinateur)
Claude Le Bris (examinateur)
Felix Otto (examinateur)
Jury member(s) :
Bob Kohn (rapporteur)
Stefano Olla (rapporteur)
George Papanicolaou (rapporteur)
Christophe Besse (examinateur)
Albert Cohen (examinateur)
Thierry Goudon (examinateur)
Claude Le Bris (examinateur)
Felix Otto (examinateur)
Stefano Olla (rapporteur)
George Papanicolaou (rapporteur)
Christophe Besse (examinateur)
Albert Cohen (examinateur)
Thierry Goudon (examinateur)
Claude Le Bris (examinateur)
Felix Otto (examinateur)
Accredited body :
Université des Sciences et Technologie de Lille - Lille I
English keyword(s) :
stochastic homogenization
nonlinear elasticity
quantitative results
nonlinear elasticity
quantitative results
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Analyse numérique [math.NA]
Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Analyse numérique [math.NA]
English abstract : [en]
The issue of establishing the status of nonlinear elasticity theory for rubber with respect to the point of view of polymer physics is at the heart of this manuscript. Our aim is to develop mathematical methods to describe, ...
Show more >The issue of establishing the status of nonlinear elasticity theory for rubber with respect to the point of view of polymer physics is at the heart of this manuscript. Our aim is to develop mathematical methods to describe, understand, and solve this multiscale problem. At the level of the polymer chains, rubber can be described as a network whose nodes represent the cross-links between the polymer chains. This network can be considered as the realization of some stochastic process. Given the free energy of the polymer network, we'd like to derive a continuum model as the characteristic length of the polymer chains vanishes. In mathematical terms, this process can be viewed as a hydrodynamic limit or as a discrete homogenization, depending on the nature of the free energy of the network. In view of the works by Treloar, by Flory, and by Rubinstein and Colby on polymer physics, and in view of the stochastic nature of the network, stochastic discrete homogenization seems to be the right tool for the analysis. Hence, in order to complete our program we need to understand the stochastic homogenization of discrete systems. Two features make the analysis rich and challenging from a mathematical perspective: the randomness and the nonlinearity of the problem. The achievement of this manuscript is twofold: - a complete and sharp quantitative theory for the approximation of homogenized coefficients in stochastic homogenization of discrete linear elliptic equations; - the first rigorous and global picture on the status of nonlinear elasticity theory with respect to polymer physics, which partially answers the question raised by Ball in his review paper on open problems in elasticity. Although the emphasis of this manuscript is put on discrete models for rubber, and more generally on the homogenization of discrete elliptic equations, we have also extended most of the results to the case of elliptic partial differential equations --- some of the results being even more striking in that case.Show less >
Show more >The issue of establishing the status of nonlinear elasticity theory for rubber with respect to the point of view of polymer physics is at the heart of this manuscript. Our aim is to develop mathematical methods to describe, understand, and solve this multiscale problem. At the level of the polymer chains, rubber can be described as a network whose nodes represent the cross-links between the polymer chains. This network can be considered as the realization of some stochastic process. Given the free energy of the polymer network, we'd like to derive a continuum model as the characteristic length of the polymer chains vanishes. In mathematical terms, this process can be viewed as a hydrodynamic limit or as a discrete homogenization, depending on the nature of the free energy of the network. In view of the works by Treloar, by Flory, and by Rubinstein and Colby on polymer physics, and in view of the stochastic nature of the network, stochastic discrete homogenization seems to be the right tool for the analysis. Hence, in order to complete our program we need to understand the stochastic homogenization of discrete systems. Two features make the analysis rich and challenging from a mathematical perspective: the randomness and the nonlinearity of the problem. The achievement of this manuscript is twofold: - a complete and sharp quantitative theory for the approximation of homogenized coefficients in stochastic homogenization of discrete linear elliptic equations; - the first rigorous and global picture on the status of nonlinear elasticity theory with respect to polymer physics, which partially answers the question raised by Ball in his review paper on open problems in elasticity. Although the emphasis of this manuscript is put on discrete models for rubber, and more generally on the homogenization of discrete elliptic equations, we have also extended most of the results to the case of elliptic partial differential equations --- some of the results being even more striking in that case.Show less >
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