Gloria, Antoine (2007) Modélisation et méthodes numériques multi-échelles en élasticité non linéaire. PhD thesis, ENPC p.256.
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.
| Item Type: | PhD Thesis (PhD) |
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| Thesis Supervisor: | Le bris, Claude and Gerbeau, jean-Frédéric |
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| Date: | 20 June 2007 |
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| Board of examiners: | Le Dret, Hervé and Efendiev, Yalchin and Le Tallec, Patrick and Allaire, Grégoire and Francfort, Gilles and Sab, Karam and Le bris, Claude and Gerbeau, Jean-Frédéric |
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| Collection (Fonds): | ENPC |
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| Institution: | ENPC |
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| Subjects: | 1. Mathematics and Applications |
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| Uncontrolled Keywords: | Homogénéisation numérique, élasticité non linéaire, Méthodes variationnelles, équations aux dérivées partielles, Γ-convergence, Passage discret continu, Interaction fluide structure, Numerical homogenization, Nonlinear elasticity, Variational methods, Partial differential equations, Γ-convergence, Discrete to continuum, Fluid structure interaction |
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