In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, we connect these large-scale estimates with local regulartity results at the microscopic-level to achieve interior Lipschitz regularity at every scale.

Modern engineering analysis requires accurate, reliable and efficient evaluation of outputs of interest. These outputs are functions of "input" parameter that serve to describe a particular configuration of the system, typical input geometry, material properties, or boundary conditions and loads. In many cases, the input-output relationship is a functional of the field variable - which is the solution to an input-parametrized partial differential equations (PDE). The reduced-basis approximation, adopting off-line/on-line computational procedures, allows us to compute accurate and reliable functional outputs of PDEs with rigorous error estimations. The operation count for the on-line stage depends only on a small number N and the parametric complexity of the problem, which make the reduced-basis approximation especially suitable for complex analysis such as optimizations and designs. In this work we focus on the development of finite-element and reduced-basis methodology for the accurate, fast, and reliable prediction of the stress intensity factors or strain-energy release rate of a mode-I linear elastic fracture problem. With the use of off-line/on-line computational strategy, the stress intensity factor for a particular problem can be obtained in miliseconds. The method opens a new promising prospect: not only are the numerical results obtained only in miliseconds with great savings in computational time; the results are also reliable - thanks to the rigorous and sharp a posteriori error bounds. The practical uses of our prediction are presented through several example problems. / Singapore-MIT Alliance (SMA)

A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.

A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.

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A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

En mécanique des matériaux et des structures, l’interface entre deux composants matériels ou deux éléments structuraux est traditionnellement et le plus souvent supposé parfaite. Au sens mécanique, une interface parfaite est une surface à travers laquelle le vecteur de déplacement et le vecteur de contrainte sont tous les deux continus. L’hypothèse des interfaces parfaites est inappropriée dans de nombreuses situations en mécanique. En effet, l’interface entre deux corps ou deux parties d’un corps est un endroit propice aux réactions physico-chimiques complexes et favorable à l’endommagement mécanique. L’intérêt pour les interfaces imparfaites devient depuis quelques années grandissant avec le développement des matériaux et structures nanométriques dans lesquels les interfaces et surfaces jouent un rôle prépondérant. A partir de la configuration de base où une interphase de faible épaisseur sépare deux phases, ce travail établit trois modèles d’interface imparfaite généraux qui permettent de remplacer l’interphase par une interface imparfaite dans les cas de la conduction thermique, de l’élasticité linéaire et de la piézoélectricité sans perturber les champs en questions à une erreur fixée près. La dérivation de ces modèles est basée sur le développement de Taylor et sur une approche originale de géométrie différentielle indépendante de tout système de coordonnées. Les trois modèles généraux permettent non seulement de mieux appréhender certains modèles phénoménologiques d’interface imparfaite mais aussi de décrire les effets d’interface que les modèles existants ne sont pas en mesure de prendre en compte. Les modèles d’interface imparfaite établis sont appliqués dans la détermination des propriétés effectives thermiques, élastiques et piézoélectriques d’un matériau composite constitué d’une matrice renforcée par des particules ou fibres enrobées d’une interphase. La méthode utilisée pour rendre compte des effets des interfaces imparfaites sur les propriétés effectives repose sur une condition d’équivalence énergétique qui ramène un matériau hétérogène avec interfaces imparfaites à un matériau hétérogène avec interfaces parfaites / In mechanics of materials and structures, the interface between two material components or two structural elements is traditionally and the most often assumed to be perfect. In mechanics, a perfect interface is a surface through which the displacement and stress vectors are continuous. The assumption of the perfect interfaces is inappropriate in many situations in mechanics. Indeed, the interface between two bodies or two parts of a body is a place propitious to complex physicochemical reactions and vulnerable to mechanical damage. The interest in imperfect interfaces has become for a few years growing with the development of nanometric materials and structures in which the interfaces and surfaces play a preponderant role. Starting from the basic configuration where an interphase of thin thickness separates two phases, this work establishes three general models of imperfect interface which make it possible to replace the interphase by an imperfect interface in the cases of thermal conduction, linear elasticity and piezoelectricity without disturbing the fields in questions to within a fixed error. The derivation of these models is based on the development of Taylor and an original coordinate-free approach of differential geometry. The three general models make it possible not only to get a better understanding of certain phenomenological models of imperfect interface but also to describe the effects of interface which the existing models are not able to take into account. The established models of imperfect interface are applied to determining the thermal, elastic and piezoelectric effective properties of composite materials consisting of a matrix reinforced by particles or fibers coated with an interphase. The method used to account for the effects of imperfect interfaces on the effective properties rests on an energy equivalency which brings back a heterogeneous material with imperfect interfaces to a heterogeneous material with perfect interfaces

In this work we study peridynamics, a non-local model in continuum me- chanics introduced by Silling (2000). The non-locality is reflected in the fact that points at finite distance exert a force upon each other. If, however, these points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We compare peridynamics with elasticity, especially in the limit of small horizon. We restrict ourselves, concerning this vanishing non-locality, to variational formulation of time- independent processes. We compute a Γ-limit for homogeneous and isotropic solid in linear peridynamics. In some cases this Γ-limit coincides with linear elasticity and the Poisson ratio is equal to 1 4. We conclude by clarifying why in some situation the computed Γ-limit can differ from the linear elasticity. 1

Différentes applications de l'élasticité linéaire en géologie structurale sont présentées dans cette thèse à travers le développement de trois types de codes numériques. Le premier utilise la modélisation directe pour étudier les déplacements et champs de contraintes autour de zones faillées complexes. On montre que l'ajout de contraintes inégalitaires, telles que la friction de Coulomb, permet d'expliquer l'angle d'initiation des dominos dans les relais extensifs. L'ajout de matériaux hétérogènes et d'optimisations, telles la parallélisation sur processeurs multi-coeurs ainsi que la réduction de complexité des modèles, permettent l'étude de modèles beaucoup plus complexes. Le second type de code numérique utilise la modélisation inverse, aussi appelée estimation de paramètres. L'inversion linéaire de déplacements sur les failles ainsi que la détermination de paléo-contraintes utilisant une approche géomécanique sont développées. Le dernier type de code numérique concerne la restoration de structures complexes plissées et faillées. Il est notamment montré qu'une telle méthode permet de vérifier l'équilibre de coupes géologiques, ainsi que de retrouver la chronologie des failles. Finalement, nous montrons que ce même code permet de lisser des horizons 3D faillés, plissés et bruités en utilisant la géomécanique. / Different applications of linear elasticity in structural geology are presented in this thesis through the development of three types of numerical computer codes. The first one uses forward modeling to study displacement and perturbed stress fields around complexly faulted regions. We show that incorporating inequality constraints, such as static Coulomb friction, enables one to explain the angle of initiation of jogs in extensional relays. Adding heterogeneous material properties and optimizations, such as parallelization on multicore architectures and complexity reduction, admits more complex models. The second type deals with inverse modeling, also called parameter estimation. Linear slip inversion on faults with complex geometry, as well as paleo-stress inversion using a geomechanical approach, are developed. The last type of numerical computer code is dedicated to restoration of complexly folded and faulted structures. It is shown that this technique enables one to check balanced cross-sections, and also to retrieve fault chronology. Finally, we show that this code allows one to smooth noisy 3D interpreted faulted and folded horizons using geomechanics.