L'objectif principal du travail réalisé au cours de la thèse consistera à proposer une démarche théorique rigoureuse visant à intégrer les éléments microstructuraux et les gradients de déformation dans une approche micromécanique. La thèse comportera deux volets : - Dans une première partie, on s'attachera à établir un cadre théorique rigoureux permettant d'intégrer les effets du gradient de la déformation et les longueurs caractéristiques de la microstructure sur le comportement effectif des matériaux composites - L'approche par développement asymptotique conduit à la résolution d'une succession problèmes d'élasticité tridimensionnelle posée sur une cellule élémentaire du milieu périodique. La résolution de ces problèmes d'élasticité, et par conséquent la détermination des propriétés effectives du composite, nécessite la mise en œuvre d'une méthode de résolution numérique. Dans cette seconde partie du travail, il s'agira de proposer une méthode de résolution basée sur la transformée de Fourier rapide (Méthode FFT) / The main objective of the thesis is to provide a rigorous theoretical approach to integrating the microstructural effects and deformation gradients in a micromechanical approach. The thesis has two components:- In the first part, we will focus on establishing a rigorous theoretical framework for integrating the effects of the deformation gradient and the characteristic lengths of the microstructure on the effective behavior of composite materials - The asymptotic expansion approach leads to the resolution of a series of three-dimensional elasticity problems posed on a unit cell of the periodic medium. Solving these problems of elasticity, and therefore the determination of the effective properties of the composite, requires the implementation of a numerical method. In this second part of the work, it will propose a resolution method based on fast Fourier transform (FFT method)

Uma matriz `A IND.N´ de ordem N ´e normal se e somente se comuta com sua adjunta. Nesta tese investigamos a estatística dos autovalores (no plano complexo) de ensembles de matrizes aleatórias normais quando a ordem N destas tende a infinito. A função distribuição de probabilidade no espaço das matrizes normais atribui, como na mecânica estatística, um peso de Boltzmann `e POT.-NF(`A IND.N´)´ a cada realização `A IND.N´ destas matrizes, onde F é uma função a valores reais invariante por transformações unitárias. Realizando uma mudança de variáveis (das variáveis de entrada para as variáveis espectrais), escrevemos a distribuição marginal conjunta dos autovalores `{`z IND.i´} POT.N´ `IND.i=1´, bem como a função de n-pontos correspondente a vários ensembles, como o determinante de um núcleo integral associado. A partir deste formalismo bem estabelecido na literatura, apresentaremos nesta tese dois tipos de resultados: Primeiramente, explorando a semelhança da distribuição conjunta dos autovalores a um problema variacional sobre as medidas de equilíbrio eletrostático de cargas sujeitas a um potencial externo V : C ? R (escolhendo F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), podemos aplicar a teoria de potenciais logarítmicos para obter a única medida de equilíbrio coincidente com a função de 1-ponto destes ensembles. Com base nesta teoria, propomos nesta tese um método de interpolação analítica capaz de projetar a medida de equilíbrio dos ensembles normais em medidas de equilíbrio dos ensembles hermitianos e unitários correspondentes. Ilustramos o procedimento com várias aplicações. O segundo tipo de resultados utiliza o método de ponto de sela ao nícleo integral da família de ensembles de matrizes normais com potenciais `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Analogamente ao que foi demonstrado em ensembles hermitianos por Deift, estabelecemos por intermédio desta expansão um conceito similar de universalidade para esta família, fazendo uso de mapas conformes e a teoria de espaços de Segal-Bargmann. Sobre o sentido de universalidade definido por G. Oas, mostramos que a afirmação de universalidade neste sentido por este autor é incorreta quando a cauda desta probabilidade é levada em conta. / A matrix `A IND.N´ of order N is normal if and only if it commutes with its adjoint. In the present thesis we investigate the eigenvalues statistics (in the complex plane) of ensembles of normal random matrices when their order N tends to infinite. The probability distribution function in the space of normal matrices attributes, as in statistical mechanics, a Boltzmann weight `e POT.-NF(`A IND.N´)´ at each matrix realization `A IND.N´, where F is a real-valued function invariant by unitary transformations. By performing a change of variables (from entry variables to spectral variables) we write the marginal joint distribution of eigenvalues {`z IND.i´} POT.N´ `IND.i=1´, as well as the n-points functions corresponding to several ensembles, as the determinant of an associated integral kernel. From this formalism well-established in the literature, we shall present in this thesis two types of results: Firstly, exploiting the similarity of joint distribution of eigenvalues to a variational problem on electrostatic equilibrium measures of charges subjected to an external potential V : C - > R (by choosing F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), we can apply the theory of logarithmic potentials to obtain the unique equilibrium measure coinciding with the 1-point function of these ensembles. Based on this theory, we propose in this thesis a method of analytical interpolation capable of projecting the equilibrium measure of normal ensembles in equilibrium measures of corresponding Hermitian and unitary ensembles. We give several applications of this procedure. The second type of results utilizes the saddle point method applied to integral kernel of a family of normal matrix ensembles with potentials `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Similarly to what has been shown in hermitian ensembles by Deift, we established by mean of this expansion a similar concept of universality for this family, making use of conformal maps and theory of Segal-Bargmann space. Concerning the universality defined by G. Oas, we show that the universality claimed by this author is incorrect when the tail of this probability is taking into account.

Wir beginnen mit der Herleitung der asymptotischen Entwicklung der Spur des Wärmeleitungskernes, $\tr e^{-t\Delta}$, für $t\to0+$, wobei $\Delta$ der Laplace-Beltrami-Operator auf einer Mannigfaltigkeit mit Kegel-Singularitäten ist; dabei folgen wir der Arbeit von Brüning und Seeley. Dann untersuchen wir, wie die Koeffizienten der Entwicklung mit der Geometrie der Mannigfaltigkeit zusammenhängen, insbesondere fragen wir, ob die (mögliche) Singularität der Mannigfaltigkeit aus den Koeffizienten - und damit aus dem Spektrum des Laplace-Beltrami-Operators - abgelesen werden kann. In wurde gezeigt, dass im zweidimensionalen Fall ein logarithmischer Term und ein nicht lokaler Term im konstanten Glied genau dann verschwinden, wenn die Kegelbasis ein Kreis der Länge $2\pi$ ist, die Mannigfaltigkeit also geschlossen ist. Dann untersuchen wir wir höhere Dimensionen. Im vier-dimensionalen Fall zeigen wir, dass der logarithmische Term genau dann verschwindet, wenn die Kegelbasis eine sphärische Raumform ist. Wir vermuten, dass das Verschwinden eines nicht lokalen Beitrags zum konstanten Term äquivalent ist dazu, dass die Kegelbasis die runde Sphäre ist; das kann aber bisher nur im zyklischen Fall gezeigt werden. Für geraddimensionale Mannigfaltigkeiten höherer Dimension und mit Kegelbasis von konstanter Krümmung zeigen wir weiter, dass der logarithmische Term ein Polynom in der Krümmung ist, das Wurzeln ungleich 1 haben kann, so dass erst das Verschwinden von mehreren Termen - die derzeit noch nicht explizit behandelt werden können - die Geschlossenheit der Mannigfaltigkeit zur Folge haben könnte. / We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with one conic singularity, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley. Then we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmic term, its vanishing is a necessary condition for smoothness of the manifold. It is shown in the paper by Bruening and Seeley that in the two-dimensional case this implies that the constant term of the expansion contains a non-local term that determines the length of the (circular) cross section and vanishes precisely if this length equals $2\pi$, that is, in the smooth case. We proceed to the study of higher dimensions. In the four-dimensional case, the logarithmic term in the expansion vanishes precisely when the cross section is a spherical space form, and we expect that the vanishing of a further singular term will imply again smoothness, but this is not yet clear beyond the case of cyclic space forms. In higher dimensions the situation is naturally more difficult. We illustrate this in the case of cross sections with constant curvature. Then the logarithmic term becomes a polynomial in the curvature with roots that are different from 1, which necessitates more vanishing of other terms, not isolated so far.

Uma matriz `A IND.N´ de ordem N ´e normal se e somente se comuta com sua adjunta. Nesta tese investigamos a estatística dos autovalores (no plano complexo) de ensembles de matrizes aleatórias normais quando a ordem N destas tende a infinito. A função distribuição de probabilidade no espaço das matrizes normais atribui, como na mecânica estatística, um peso de Boltzmann `e POT.-NF(`A IND.N´)´ a cada realização `A IND.N´ destas matrizes, onde F é uma função a valores reais invariante por transformações unitárias. Realizando uma mudança de variáveis (das variáveis de entrada para as variáveis espectrais), escrevemos a distribuição marginal conjunta dos autovalores `{`z IND.i´} POT.N´ `IND.i=1´, bem como a função de n-pontos correspondente a vários ensembles, como o determinante de um núcleo integral associado. A partir deste formalismo bem estabelecido na literatura, apresentaremos nesta tese dois tipos de resultados: Primeiramente, explorando a semelhança da distribuição conjunta dos autovalores a um problema variacional sobre as medidas de equilíbrio eletrostático de cargas sujeitas a um potencial externo V : C ? R (escolhendo F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), podemos aplicar a teoria de potenciais logarítmicos para obter a única medida de equilíbrio coincidente com a função de 1-ponto destes ensembles. Com base nesta teoria, propomos nesta tese um método de interpolação analítica capaz de projetar a medida de equilíbrio dos ensembles normais em medidas de equilíbrio dos ensembles hermitianos e unitários correspondentes. Ilustramos o procedimento com várias aplicações. O segundo tipo de resultados utiliza o método de ponto de sela ao nícleo integral da família de ensembles de matrizes normais com potenciais `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Analogamente ao que foi demonstrado em ensembles hermitianos por Deift, estabelecemos por intermédio desta expansão um conceito similar de universalidade para esta família, fazendo uso de mapas conformes e a teoria de espaços de Segal-Bargmann. Sobre o sentido de universalidade definido por G. Oas, mostramos que a afirmação de universalidade neste sentido por este autor é incorreta quando a cauda desta probabilidade é levada em conta. / A matrix `A IND.N´ of order N is normal if and only if it commutes with its adjoint. In the present thesis we investigate the eigenvalues statistics (in the complex plane) of ensembles of normal random matrices when their order N tends to infinite. The probability distribution function in the space of normal matrices attributes, as in statistical mechanics, a Boltzmann weight `e POT.-NF(`A IND.N´)´ at each matrix realization `A IND.N´, where F is a real-valued function invariant by unitary transformations. By performing a change of variables (from entry variables to spectral variables) we write the marginal joint distribution of eigenvalues {`z IND.i´} POT.N´ `IND.i=1´, as well as the n-points functions corresponding to several ensembles, as the determinant of an associated integral kernel. From this formalism well-established in the literature, we shall present in this thesis two types of results: Firstly, exploiting the similarity of joint distribution of eigenvalues to a variational problem on electrostatic equilibrium measures of charges subjected to an external potential V : C - > R (by choosing F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), we can apply the theory of logarithmic potentials to obtain the unique equilibrium measure coinciding with the 1-point function of these ensembles. Based on this theory, we propose in this thesis a method of analytical interpolation capable of projecting the equilibrium measure of normal ensembles in equilibrium measures of corresponding Hermitian and unitary ensembles. We give several applications of this procedure. The second type of results utilizes the saddle point method applied to integral kernel of a family of normal matrix ensembles with potentials `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Similarly to what has been shown in hermitian ensembles by Deift, we established by mean of this expansion a similar concept of universality for this family, making use of conformal maps and theory of Segal-Bargmann space. Concerning the universality defined by G. Oas, we show that the universality claimed by this author is incorrect when the tail of this probability is taking into account.

Obtemos uma expansão assintótica da função de distribuição sob a hipótese nula da estatística gradiente para testar hipóteses nulas compostas na presença de parâmetros de perturbação. Esta expansão é derivada utilizando uma rota Bayesiana baseada no argumento de encolhimento descrito em Ghosh e Mukerjee (1991). Usando essa expansão, propomos uma estatística gradiente corrigida por um fator de correção tipo-Bartlett, que tem distribuição qui-quadrado até um erro de ordem o(n-1) sob a hipótese nula. A partir disso, determinamos fórmulas matriciais e algébricas que auxiliam na obtenção da estatística gradiente corrigida em modelos lineares generalizados com dispersão conhecida e desconhecida. Simulações de Monte Carlo são apresentadas. Finalmente, discutimos a obtenção de regiões de credibilidade via inversão da estatística gradiente. Caracterizamos as densidades a priori, matching priors, que asseguram propriedades de cobertura frequentista acuradas para essas regiões. / We obtain an asymptotic expansion for the null distribution function of the gradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argument described in Ghosh and Mukerjee (1991). Using this expansion, we propose a Bartlett-type corrected gradient statistic, which has a chi-square distribution up to an error of order o(n1) under the null hypothesis. Also, we determined matrix and algebraic formulas that assist in obtaining Bartett-type corrected statistic in generalized linear models with known and unknown dispersion. Monte Carlo simulations are presented. Finally, we obtain credible regions based by the inversion of gradient statistic. We characterize priori densities, matching priors, that ensure accurate frequentist coverage properties for these regions.

Part I: The present work is to calculate dimensionless three-dimensional universal phase and lift force diagrams for a microbubble or pendant drop in a static liquid on a solid surface or orifice. Studying microbubble dynamics is important due to its controlling mass, momentum, energy and concentration transfer rates encountered in micro- and nano-sciences and technologies. In this work, dimensionless phase and force diagrams are presented by applying an equation for microbubble shape to accuracy of the second order of small Bond number provided by O¡¦Brien (1991). Two dimensionless independent parameters, Bond number and contact angle (or base radius), are required to determine dimensionless phase and force diagrams governing static and dynamic states of a microbubble. The phase diagram divides the microbubble surface into three regions, the apex to inflection, inflection to neck, and neck to the edge of microbubble. The growth, collapse, departure and entrapment of a microbubble on a surface thus can be described. The lift forces include hydrostatic buoyancy, difference in gas and hydrostatic pressures at the microbubble base, capillary pressure and surface tension resulted from variation of circumference. The force to attach the microbubble to solid surface is the surface tension resulted from variation of circumference, which is not accounted for in literature. Adjusting the base radius to control static and dynamic behaviors of a microbubble is more effective than Bond number. Part II: Controlling states and growth of a microscale bubble (or pendant drop) in a static liquid on a surface by introducing general phase diagrams is proposed. Microbubbles are often used to affect transport phenomena in micro- and nano-technologies. In this work, a general phase diagram is provided by applying a perturbation solution of Young-Laplace equation for bubble shape with truncation errors of the second power of small Bond number. The three-dimensional phase diagram for a given Bond number is uniquely described by the dimensionless radius of curvature at the apex, contact angle and base radius of the microbubble. Provided that initial and end states are chosen, adjusting two of them gives the desired states and growth, decay and departure of the bubble described by path lines in the phase diagram. A universal three-dimensional phase diagram for a microbubble is also introduced.

Grundlage der vorliegenden Arbeit sind mathematische Aspekte des Kapillarflächenproblems. Die Arbeit ist in zwei Teile gegliedert. Im ersten Teil werden existierende glatte Lösungen des klassischen Kapillarflächenproblems betrachtet. Diese sind unbeschränkt, wenn das Definitionsgebiet Spitzen enthält. Es wird eine Vielzahl von asymptotischen Formeln hergeleitet. Der zweite Teil der Arbeit beschäftigt sich mit verallgemeinerten Lösungen des allgemeinen Kapillar- flächenproblems. Es wird die Existenz dieser Lösungen unter sehr schwachen Voraussetzungen bewiesen. Für konstante Gravitationspotentiale und Benetzungsverhalten werden verallgemeinerte Lösungen näher untersucht und z. T. sogar explizit konstruiert. Die Eigenschaften einer speziellen Klasse solcher Lösungen könnte einen Beitrag zur Erklärung des Wasseranstiegs in Bäumen liefern. / The present paper is based on mathematical aspects of the capillary surface problem. It is divided into two parts. In the first part we consider the classical capillary surface problem, for which smooth solutions exist. These solutions are unbounded if the domain of definition contains cusps. We prove a large variety of asymptotic formulas. The second part is concerned with generalized solutions of the general capillary problem, for which there is not always a smooth solution. We prove existence of generalized solutions under very weak preconditions. We can construct some generalized solutions for zero-gravity and constant wetting-behaviour explicitly. These solutions have a very restricted geometry and could be of interest for the understanding of water lift in trees.

Obtemos uma expansão assintótica da função de distribuição sob a hipótese nula da estatística gradiente para testar hipóteses nulas compostas na presença de parâmetros de perturbação. Esta expansão é derivada utilizando uma rota Bayesiana baseada no argumento de encolhimento descrito em Ghosh e Mukerjee (1991). Usando essa expansão, propomos uma estatística gradiente corrigida por um fator de correção tipo-Bartlett, que tem distribuição qui-quadrado até um erro de ordem o(n-1) sob a hipótese nula. A partir disso, determinamos fórmulas matriciais e algébricas que auxiliam na obtenção da estatística gradiente corrigida em modelos lineares generalizados com dispersão conhecida e desconhecida. Simulações de Monte Carlo são apresentadas. Finalmente, discutimos a obtenção de regiões de credibilidade via inversão da estatística gradiente. Caracterizamos as densidades a priori, matching priors, que asseguram propriedades de cobertura frequentista acuradas para essas regiões. / We obtain an asymptotic expansion for the null distribution function of the gradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argument described in Ghosh and Mukerjee (1991). Using this expansion, we propose a Bartlett-type corrected gradient statistic, which has a chi-square distribution up to an error of order o(n1) under the null hypothesis. Also, we determined matrix and algebraic formulas that assist in obtaining Bartett-type corrected statistic in generalized linear models with known and unknown dispersion. Monte Carlo simulations are presented. Finally, we obtain credible regions based by the inversion of gradient statistic. We characterize priori densities, matching priors, that ensure accurate frequentist coverage properties for these regions.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-08T11:32:05Z No. of bitstreams: 1 sabrinamascarenhasvargas.pdf: 12731626 bytes, checksum: b1c5ae34c05edd5e0970eb9289e25f3d (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T13:25:40Z (GMT) No. of bitstreams: 1 sabrinamascarenhasvargas.pdf: 12731626 bytes, checksum: b1c5ae34c05edd5e0970eb9289e25f3d (MD5) / Made available in DSpace on 2016-07-13T13:25:40Z (GMT). No. of bitstreams: 1 sabrinamascarenhasvargas.pdf: 12731626 bytes, checksum: b1c5ae34c05edd5e0970eb9289e25f3d (MD5) Previous issue date: 2016-04-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Visto que o esmalte dentário é um tecido não inervado e avascular, que está constantemente sob a in uência de carregamento cíclico (funcional ou parafuncional) e que o mesmo não tem capacidade de regeneração, torna-se importante o estudo sobre as propriedades mecânicas desse tecido. Possui uma microestrutura única, que o faz apresentar propriedades mecânicas excelentes, porém o mesmo se apresenta frágil, com pouca capacidade de suportar deformação plástica antes da sua fratura. Alguns testes experimentais de indentação tentam entender o comportamento mecânico desse compósito, porém a complexidade desse comportamento e as diferenças de técnicas fazem com que os módulos de elasticidade para a hidroxiapatita, a matriz orgânica e o módulo efetivo do esmalte dentário tenham resultados muito variados na literatura. O mesmo se dá para as simulações multiescala de modelos para o esmalte dentário. Diante disso, esse estudo tem como o objetivo utilizar a modelagem multi-escala em 2D para a determinação dos tensores de propriedades mecânicas efetivas do esmalte dentário, através da técnica de homogeneização por expansão assintótica (HEA). Dentre as conclusões do trabalho têm-se que: 1- O esmalte dentário pode ser representado por um meio homogêneo equivalente, uma célula unitária representativa repetitiva; 2- Os modelos propostos nesse estudo têm comportamento ortotrópico; 3- Embora haja limitações relacionadas às simpli cações mecânicas e geométricas adotadas, os resultados obtidos encorajam aplicações mais realistas e estudos mais aprofundados acerca da microestrutura do material em questão. / Whereas tooth enamel is not an innervated neither vascular tissue which is constantly under the in uence of cyclical loading (functional or parafuncional) and that its tissue has no capacity for regeneration, it becomes important to study the mechanical properties of the enamel. It has an unique microstructure, which makes it exhibit excellent mechanical properties, but it appears fragile, with little ability to withstand plastic deformation prior to fracture. Some experimental indentation tests attempt to understand the mechanical behavior of this composite, but the complexity of its behavior and the di erent techniques imply in the modulus of elasticity for the hydroxyapatite, the organic matrix and the e ective modulus of dental enamel showing very di erent results in the literature. The same occurs for multiscale simulations of dental enamel models. Thus, this study aims 2D multi-scale modeling by Asymptotic Expansion Homogenization (AEH) technic to determine the mechanical properties e ective tensor of dental enamel. The conclusions of this study shows: 1- The enamel can be represented by an equivalent homogeneous medium, a repetitive representative unit cell; 2- The models proposed in this study present orthotropic behavior; 3- Although there are some limitations due to the mechanical and geometric simpli cations adopted, the results suggest more realistic applications and further studies on the microstructure of the material in question.

This thesis extends the basic ordinary differential equations (ODE) course, specifically considering perturbations of ODEs. We introduce uniformly asympto- tic approximation and uniformly ordered approximation. We provide a perturba- tion-based method of computing derivatives of ODE solutions with respect to: an initial value, a parameter, and initial time. We present the method of averaging, error estimate, and a theorem about the existence and stability of a periodic so- lution to ODEs in periodic standard form. Furthermore, we apply the method of averaging to determine the period of a periodic solution of Duffing equation without forcing or damping. All the terms and methods of perturbation theory used in the thesis are accompanied with examples. 1