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Développement de méthodes numériques multi échelle pour le calcul des structures constituées de matériaux fortement hétérogènes élastiques et viscoélastiques

Abstract : Concretes are composite materials having complex microstructure and consisting of phases whose physical and mechanical properties can exhibit high contrast. Difficulties arise when macroscopic approaches are used to evaluate their effective behaviors such as the creeping one. Despite thesedifficulties, EDF has to be endowed with tools allowing to model in a predictive way the evolution of concrete structures in service or to prescribe the specifications of concrete for new facilities. Aimed to contribute to solving this problem, this thesis develops multi-scale numerical methods for the computation of structures made of highly heterogeneous elastic or viscoelastic materials. More precisely, this thesis comprises three parts. In the first part, we focus on a composite consisting of an elastic matrix reinforced by elastic inclusions which may be of any geometric shapes and whose volume fraction can be significant. To model this composite material, a first numerical approach which combines the standard extended finite element method (XFEM) and the classical level-set method (LSM) is used. We show that this numerical approach, which appears natural, leads in fact to several numerical artefacts having not been reported in the literature, giving rise in particular to non-optimal convergence with respect to the fineness of the mesh. In view of this, we develop a new numerical approach based on the description of the interfaces with multiple level sets and augmented enrichment so as to account for multiple interfaces in a single finite element. We demonstrate through examples and benchmarks that the proposed method significantly improves convergence compared to the first numerical approach. In the second part, we elaborate a new method to compute the creeping of structures formed of linearly viscoelastic heterogeneous materials. Unlike the approaches proposed up to now, our method operates directly in the time space and allows to sequentially extract the homogenized properties of a linearly viscoelastic heterogeneous material. Precisely, the components of the effective relaxation tensor of the material is first obtained from a representative volume element and sampled over time. An interpolation technique and an implicit algorithm are then used to numerically evaluate the time-dependent response of the material through a convolution product. The creeping of structures is finally calculated by the classical finite element method. Various tests are performed to assess the quality and effectiveness of the proposed method, showing that it can give a gain of time of the order of several hundreds compared to approaches such as multi-level finite element. The third part of the thesis is devoted to the study of the structure of containment of a nuclear reactor. We consider four scale levels associated with cement paste, mortar, concrete and a pre-stressed concrete structure with steel cables. The numerical method of homogenization developed in the second part is applied to construct the constitutive equations for each of the first three scale levels. The results thus obtained are useful for solving some practical problems posed by EDF
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Anh Binh Tran. Développement de méthodes numériques multi échelle pour le calcul des structures constituées de matériaux fortement hétérogènes élastiques et viscoélastiques. Autre. Université Paris-Est, 2011. Français. ⟨NNT : 2011PEST1123⟩. ⟨tel-00657270⟩

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