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Multiscale modeling and numerical methods in nonlinear elasticity

Abstract : The most important part of this work deals with the mathematical analysis of
numerical methods for the homogenization of multiple integrals widely used in nonlinear elasticity.
These methods couple, at the mesoscopic scale, a heterogeneous hyperelastic material or a network
of interacting bonds with, at the macroscopic scale, a nonlinear elasticity model. The macroscopic
constitutive law is obtained by solving mesoscopic problems, either continuous or discrete. In
chapters 1, 2, and 3, we introduce the mechanical models and mathematical tools we use in the
sequel. In chapters 5, 6, and 7, we present a direct method for the numerical solution of the
homogenized behavior of a periodic composite material at finite strains, and a general framework
to study numerical homogenization methods. We prove the convergence of such methods within
general hypotheses and provide a numerical corrector convergence result. We also extend the
analysis to cover the cases of oversampling and windowing. In chapters 8, 9, and 10, we consider
a mesoscopic model based on discrete systems of bonds. We first study a G-closure problem for
a network of conductances. In the next chapter, we prove an integral representation result for a
system of interacting spins. We then address the rigorous derivation of a continuous hyperelastic
model starting from a stochastic network of interacting points. We apply this result to prove
the convergence of discrete models for rubber developed in mechanics. In the last chapter, we
introduce a new solution method for fluid structure interaction problems with three dimensional
shell elements to describe the structure.
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Contributor : Antoine Gloria <>
Submitted on : Thursday, August 2, 2007 - 5:31:10 PM
Last modification on : Tuesday, April 24, 2018 - 1:21:56 PM
Long-term archiving on: : Thursday, April 8, 2010 - 9:06:49 PM

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  • HAL Id : tel-00166171, version 1

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Antoine Gloria. Multiscale modeling and numerical methods in nonlinear elasticity. Modeling and Simulation. Ecole des Ponts ParisTech, 2007. English. ⟨tel-00166171⟩

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