Composite laminates are being more employed as fundamental structures due to its low weight and high stiffness. An example of this innovation is the primary structures of modern aircraft, which are lighter than the material formerly used. To predict the material response as load gradually increases can be quite demanding due to composite\'s complex failure mechanism. Hence superior computational models should be further investigated to precisely predict the mechanical behavior of composite media. This dissertation addresses an extended finite element procedure based on the layerwise displacement theory to simulate purely mode I delamination failure in composite laminates. The present model has the potential to perform structural analyzes in a pre-delaminated structure and also considering progressive failure. The type of element to be employed at the discretion of the model is the rectangular 4-node iso-parametric homogeneous element whose displacement field is approximated based in the layerwise theory. There are four types of degrees of freedom, one displacement in each direction, and one degree of freedom associated to the strong discontinuity. Numerical examples already solved in the bibliography are suggested in this dissertation, which demonstrate the potential of the model developed to accurately simulate pure mode I delamination in case of the investigation here is further elaborated. In addition, one possibility of future development of this dissertation is the modeling of fracture mode I without the need to discretize the cohesive planes as realized in traditional Cohesive Zone Methods. / Compósitos laminados estão sendo mais empregados como estruturas fundamentais devido ao seu baixo peso e alta rigidez. Um exemplo dessa inovação são as estruturas primárias das aeronaves modernas, que são mais leves do que as os materiais empregados antigamente. Prever a resposta do material à medida que a carga aumenta gradualmente pode ser difícil devido ao complexo mecanismo de falha dos compósitos. Portanto, modelos computacionais mais refinados devem ser investigados a fim de se prever um comportamento mecânico mais preciso. Esta dissertação aborda um procedimento de elementos finitos estendido baseado na teoria de deslocamento layerwise para simular falhas de delaminação modo I em laminados compósitos. O modelo abordado tem potencial para realizar análises em uma estrutura prédelaminada além de falha progressiva. O tipo de elemento a ser empregado na discrição do modelo é o isoparamétrico, homogêneo de 4 nós, retangular, e o campo de deslocamento é baseado na teoria layerwise. Existem quatro tipos de graus de liberdade, um deslocamento em cada direção, e um grau de liberdade associado à forte. Sugere-se nesse trabalho, exemplos, que são comparados com a bibliografia, e que apontam que o modelo desenvolvido nesta dissertação tem o potencial de simular o fenômeno de delaminação em modo I com acurácia, caso o estudo nessa dissertação seja estendido. Além disso, uma possibilidade de desenvolvimento futuro desse trabalho é a modelagem da fratura modo I sem a necessidade de discretizar os planos coesivos entre as lâminas, como realizado em métodos coesivos tradicionais.

Motivados pelas atuais tendências e suportados pelo grande interesse de indústrias do segmento aeronáutico, estudos e desenvolvimentos vêm sendo conduzidos na área de análise estrutural de materiais compósitos. Todavia, mesmo havendo várias contribuições científicas e tecnológicas nesta área, este assunto continua sendo um campo aberto e bastante promissor para novas pesquisas, devido a sua extensa complexidade e imediata aplicação. A ausência de um modelo capaz de projetar com elevada precisão uma estrutura aeronáutica com presença de escalonamento fabricada em material compósito, que pode sofrer modo de falha translaminar, motivou o presente trabalho. É sabido que o método de elementos finitos estendidos (XFEM - eXtendend Finite Element Method, do Inglês) vem sendo usado de maneira robusta para análise de propagação de trincas em elementos estruturais tridimensionais isotrópicos durante os últimos anos, mas não em compósitos. De forma a contribuir com a pequena quantidade de trabalhos científicos referentes a métodos XFEM 3D para análise de estruturas fabricadas em materiais compósitos não convencionais, como estruturas com escalonamento de camadas e laminados espessos, é apresentada uma nova metodologia implementada como uma ferramenta computacional para analisar quase estaticamente este tipo de estrutura. O modelo é baseado no aprimoramento do \"Método da Seção de Ouro\" que é aplicado em conjunto com uma versão aprimorada do critério de falha de Puck, permitindo assim definir com precisão e baixo custo computacional a iniciação e direção de uma trinca. Esta informação é utilizada para iniciar uma rotina baseada em XFEM, que é usada para o enriquecimento dos elementos finitos que vão falhando progressivamente durante a análise. A nova metodologia (implementada computacionalmente) apresenta convergência uma ordem de grandeza maior quando comparada com o algoritmo tradicional, sendo aproximadamente 20 vezes mais eficiente em termos computacionais. O modelo é ainda avaliado quanto a seus resultados em comparação com dados provenientes de ensaios experimentais, demonstrando uma boa convergência entre as previsões computacionais e os resultados obtidos em laboratório. / Supported by current trends and by the great interest of aeronautic industries, studies and developments have been made in the field of high performance composite materials. Nonetheless, even with the scientific and technological contributions, the matter is still a field wide open and promising for new research due to its high complexity and immediate application. The absence of a model capable of universally reproducing mechanical behavior of composite materials tapered structures, which can suffer translaminar failure mode, motivated the present work. It is well known that the eXtended Finite Elements Method (XFEM) has been used robustly for analysis of crack propagation in isotropic tri-dimensional structural elements lately but not for composites. In order to contribute with the scares amount of available works on 3D XFEM application on non-conventional composite material structures, such as tapered structures and thick laminates, a new methodology is presented as a computational tool for quasi-static analysis of this type of component. The model derives from \"Golden Section Method\" that is applied along with an enhanced version of Puck\'s failure criterion, which allows a low computational cost and high precision estimation of crack initiation and direction of propagation. This information is used to trigger an XFEM based routine that is applied for enriching the elements progressively during analysis. The new methodology (computationally implemented) has a convergence rate one order of magnitude greater than traditional implementation, roughly 20 times more efficient in terms of computational processing. Finally, to assure robustness, the model is validated against standardized and specifically developed experiments, showing good convergence between numerical predictions and results obtained in the laboratory.

[pt] Um elevado número de reservatórios de hidrocarbonetos é naturalmente fraturado. Quando sujeitos a estimulação hidráulica, as fraturas naturais podem influenciar a propagação da fratura hidráulica, que pode tomar uma forma geométrica complexa, criando redes de fraturas no reservatório. De forma a melhor entender e simular tais fenômenos, um elemento baseado no Método dos Elementos Finitos Estendidos (XFEM) é proposto. A formulação do elemento inclui interseção e cruzamento entre fraturas, atrito entre as faces das fraturas, comportamento acoplado entre deslocamentos, poro-pressões e pressões do fluido da fratura, absorção de fluído da fratura para o meio poroso (leak-off) e a eventual perda de pressão nas faces da fratura (filter cake). Os fundamentos teóricos e os aspectos relevantes da implementação são apresentados. Um conjunto de análises é realizado de forma a validar em separado as diferentes funcionalidades do elemento implementado. Finalmente, os resultados de quatro aplicações práticas são analisados e discutidos: dois conjuntos de ensaios de laboratório de interseção de fratura, propagação de fratura hidráulica num modelo sintético multi-fraturado e percolação na fundação fraturada de uma barragem. Conclui-se que o código implementado fornece previsões muito boas do comportamento acoplado do meio fraturado e tem capacidade de simular corretamente a interação entre fraturas hidráulicas e naturais. Pode também verificar-se que o comportamento hidráulico dos modelos e a propagação e interseção de fraturas são muito influenciados por parâmetros tais como o diferencial de tensões in-situ, ângulo entre fraturas, a abertura hidráulica das fraturas e a condutividade transversal das faces da fratura. / [en] A large number of hydrocarbon reservoirs are naturally fractured. When subjected to hydraulic fracturing treatments, the natural fractures may influence the propagation of the hydraulic fracture, which can grow in a complicated manner creating complex fracture networks in the reservoir. In order to better understand and simulate such phenomena an element based on the eXtended Finite Element Method is proposed. The element formulation comprises fracture intersection and crossing, fracture frictional behaviour, fully coupled behaviour between displacements, pore and fracture fluid pressure, leak-off from the fracture to the surrounding medium and the eventual loss of pressure due to filter cake. The theoretical background and implementation aspects are presented. A set of analyses is performed in order to validate different features of the implemented element. Finally, the results of four practical applications are analysed and discussed: two laboratory hydraulic fracture tests, hydraulic fracture propagation in a multifractured synthetic model and percolation through a dam fractured foundation. It is concluded that the implemented code provides very good predictions of the coupled fluid-rock fracture behaviour and is capable of correctly simulating the interaction between hydraulic and natural fractures. Moreover, it is shown that the hydraulic behaviour of the models and the intersection between fractures are very sensible to parameters such as differential in-situ stresses, angle between fractures, initial hydraulic aperture and fracture face transversal conductivity.

Ce mémoire de thèse à été réalisée dans le cadre d'une collaboration scientifique avec "La Manufacture Française des Pneumatiques Michelin". Il porte sur l'analyse mathématique et numérique de la convergence et de la stabilité de formulations mixtes ou hybrides de problèmes d'optimisation sous contrainte avec la méthode des multiplicateurs de Lagrange et dans le cadre de la méthode éléments finis étendus (XFEM). Tout d'abord, nous essayons de démontrer la stabilité de la discrétisation X-FEM pour le problème d'élasticité linéaire incompressible en statique. Le deuxième axe, qui représente le contenu principal de la thèse est dédié à l'étude de certaines méthodes de multiplicateur de Lagrange stabilisées. La particularité de ces méthodes est que la stabilité du multiplicateur est assurée par l'ajout de termes supplémentaires dans la formulation faible. Dans ce contexte, nous commençons par l'étude de la méthode de stabilisation de Barbosa-Hughes appliquée au problème de contact unilatéral sans frottement avec XFEM cut-off. Ensuite, nous construisons une nouvelle méthode basée sur des techniques de projections locales pour stabiliser un problème de Dirichlet dans le cadre de X-FEM et une approche de type domaine fictif. Nous faisons aussi une étude comparative entre la stabilisation avec la technique de projection locale et la stabilisation de Barbosa-Hughes. Enfin, nous appliquons cette nouvelle méthode de stabilisation aux problèmes de contact unilatéral en élastostatique avec frottement de Tresca dans le cadre de X-FEM. / This Ph.D. thesis was done in collaboration with "La Manufacture Française des Pneumatiques Michelin". It concerns the mathematical and numerical analysis of convergence and stability of mixed or hybrid formulation of constrained optimization problem with Lagrange multiplier method in the framework of the eXtended Finite Element Method (XFEM). First we try to prove the stability of the X-FEM discretization for incompressible elastostatic problem by ensured a LBB condition. The second axis, which present the main content of the thesis, is dedicated to the use of some stabilized Lagrange multiplier methods. The particularity of these stabilized methods is that the stability of the multiplier is provided by adding supplementary terms in the weak formulation. In this context, we study the Barbosa-Hughes stabilization technique applied to the frictionless unilateral contact problem with XFEM-cut-off. Then we present a new consistent method based on local projections for the stabilization of a Dirichlet condition in the framework of extended finite element method with a fictitious domain approach. Moreover we make comparative study between the local projection stabilization and the Barbosa-Hughes stabilization. Finally we use the local projection stabilization to approximate the two-dimensional linear elastostatics unilateral contact problem with Tresca frictional in the framework of the eXtended Finite Element Method X-FEM.

Composite laminates are being more employed as fundamental structures due to its low weight and high stiffness. An example of this innovation is the primary structures of modern aircraft, which are lighter than the material formerly used. To predict the material response as load gradually increases can be quite demanding due to composite\'s complex failure mechanism. Hence superior computational models should be further investigated to precisely predict the mechanical behavior of composite media. This dissertation addresses an extended finite element procedure based on the layerwise displacement theory to simulate purely mode I delamination failure in composite laminates. The present model has the potential to perform structural analyzes in a pre-delaminated structure and also considering progressive failure. The type of element to be employed at the discretion of the model is the rectangular 4-node iso-parametric homogeneous element whose displacement field is approximated based in the layerwise theory. There are four types of degrees of freedom, one displacement in each direction, and one degree of freedom associated to the strong discontinuity. Numerical examples already solved in the bibliography are suggested in this dissertation, which demonstrate the potential of the model developed to accurately simulate pure mode I delamination in case of the investigation here is further elaborated. In addition, one possibility of future development of this dissertation is the modeling of fracture mode I without the need to discretize the cohesive planes as realized in traditional Cohesive Zone Methods. / Compósitos laminados estão sendo mais empregados como estruturas fundamentais devido ao seu baixo peso e alta rigidez. Um exemplo dessa inovação são as estruturas primárias das aeronaves modernas, que são mais leves do que as os materiais empregados antigamente. Prever a resposta do material à medida que a carga aumenta gradualmente pode ser difícil devido ao complexo mecanismo de falha dos compósitos. Portanto, modelos computacionais mais refinados devem ser investigados a fim de se prever um comportamento mecânico mais preciso. Esta dissertação aborda um procedimento de elementos finitos estendido baseado na teoria de deslocamento layerwise para simular falhas de delaminação modo I em laminados compósitos. O modelo abordado tem potencial para realizar análises em uma estrutura prédelaminada além de falha progressiva. O tipo de elemento a ser empregado na discrição do modelo é o isoparamétrico, homogêneo de 4 nós, retangular, e o campo de deslocamento é baseado na teoria layerwise. Existem quatro tipos de graus de liberdade, um deslocamento em cada direção, e um grau de liberdade associado à forte. Sugere-se nesse trabalho, exemplos, que são comparados com a bibliografia, e que apontam que o modelo desenvolvido nesta dissertação tem o potencial de simular o fenômeno de delaminação em modo I com acurácia, caso o estudo nessa dissertação seja estendido. Além disso, uma possibilidade de desenvolvimento futuro desse trabalho é a modelagem da fratura modo I sem a necessidade de discretizar os planos coesivos entre as lâminas, como realizado em métodos coesivos tradicionais.

Dans un contexte de vulnérabilité militaire des sous-marins, les ingénieurs et chercheurs doivent être capables de prédire le comportement des structures fissurées. Ainsi, la modélisation de la transition des changements de modes de propagation de fissure (cisaillement-traction et inversement) des métaux sous sollicitations extrêmes devient un outil incontournable ou essentiel. Des critères tridimensionnels de direction de propagation de fissure développés pour une rupture par cisaillement ou par ouverture sont exposés. Des formules de direction de propagation semi-analytiques et analytiques, fonctions des facteurs d’intensité des contraintes et du coefficient de Poisson, sont ainsi proposées. L’interprétation de ces formules laisse envisager la prise en compte des effets tridimensionnels dans de futures simulations 3D de propagation de fissure. Une étude du problème en deux dimensions est également développée, proposant une formule analytique du critère en cisaillement. De plus un algorithme automatique de transition cisaillement-traction a été implémenté dans le code de calcul de dynamique explicite Europlexus, développé par le CEA. Une méthodologie d’identification des paramètres du modèle pour un matériau donné et pour un cas quasi-statique a été proposée. Confronté à l’interprétation de deux expériences connues de propagation dynamique (expériences de Kalthoff et de Ravichandran), le modèle proposé a montré sa pertinence. De plus, afin de mieux connaître le comportement à rupture de l’acier à Haute Limite Élastique Soudable, deux études expérimentales dédiées au suivi de la propagation dynamique d’un front de fissure ont été développées et validées sur des essais de rupture sous chargement quasi-statique et dynamique de type choc. Cette étude expérimentale a permis d’observer que les branchements de fissures, relevés sur les essais sous chargement quasi-statique, n’apparaissent plus sous chargement dynamique et pour des sollicitations en mode I pur. Les méthodes théoriques et numériques développées dans ces travaux de thèse permettent donc de simuler, automatiquement et avec un unique modèle, les changements de modes de rupture au cours d’une propagation dynamique de fissure. De plus, les protocoles expérimentaux exposés dans ce manuscrit permettent d’appréhender les phénomènes de transition cisaillement-traction en soulevant l’importance de la vitesse de sollicitation et du mode de sollicitation de l’essai. / We propose an approach to the simulation of the shear-tensile transition in dynamic crack growth based on two points: a new crack propagation criterion which is suitable for shear, and an algorithm which is capable of handling the transition from shear mode to tensile mode and back in the same simulation. The new crack propagation criterion for brittle crack growth is based on the maximum shear stress rather than the maximum hoop stress. The shear stress direction becomes the new crack’s direction in which propagation is initiated for shear-type failure. The stress state at the crack’s tip is obtained through a local approach which can be used even in the case of extensive plasticity. Additionally, we propose to control the transition from shear mode to tensile mode during the simulation of crack propagation using an equivalent strain estimated at the crack’s tip. Depending on a threshold strain, the propagation direction is predicted using the maximum shear stress (in the shear case) or the maximum hoop stress (in the tensile case).

Sous l’effet des sollicitations mécaniques répétées induites par les passages des trains, on observe l’apparition de fissures de fatigue de contact dans les rails. Une fois amorcées, celles-ci peuvent se propager et mener à la rupture du rail. Dans un contexte d’intensification du trafic et d’augmentation globale des vitesses de circulation, il devient stratégique pour SNCF d’optimiser sa politique de maintenance. Afin de définir des pas de surveillance adaptés et une planification optimisée des opérations de maintenance, une meilleure connaissance des mécanismes d’endommagement par fatigue du rail s’avère nécessaire. Tendre vers cette stratégie de maintenance prédictive passe par la mise en place d’outils de simulation numérique adaptés. Dans ce contexte, une chaîne d’outils a été développée : détermination des sollicitations transmises au rail, des champs de contraintes et de déformations résiduelles, localisation des zones critiques vis-à-vis du risque de fissuration. L’étape suivante consiste à estimer le risque lié à la présence de fissures et à étudier leurs propagations. Elle constitue une partie des objectifs de ces travaux de thèse. La résolution du problème tri-dimensionnel d’une structure fissurée, avec contact et frottement entre les lèvres, est effectuée grâce à un modèle tri-dimensionnel éléments finis étendus multi-échelles. Ce modèle fait appel à une formulation mixte stabilisée où chaque champ est écrit à l’aide d’enrichissement. La fissure est représentée grâce à une stratégie implicite-explicite. Le problème est résolu à l’aide du solveur non-linéaire LATIN. Une étude empirico-numérique a permis de proposer des formules a priori assurant à la méthode de résolution un taux de convergence proche de l’optimal. La simulation de la propagation des fissures de fatigue est réalisée à l’aide de critères spécifiques, adaptés à un chargement multi-axial et non-proportionnel, et d’une loi de propagation dédiée en mode mixte. La confrontation des résultats de simulation avec des essais réalisés sur une configuration cylindre-plan a validé la stratégie X-FEM/LATIN à deux échelles. Tous ces développements ont été implémentés dans le code de calcul éléments finis CAST3M. Des contraintes résiduelles réalistes, provenant d’un logiciel externe, ont été introduites. Cette étape a requis la mise en place d’une procédure de transfert des champs entre les deux maillages (celui utilisé pour le calcul des contraintes résiduelles et celui utilisé pour la résolution du problème de mécanique élastique linéaire de la rupture). L’étude de la flexion du rail a révelé l’influence de ce phénomène uniquement lors du passage du chargement sur la fissure.Enfin, une étude numérique a montré la très forte influence de l’orientation du chargement tangentiel, des contraintes résiduelles et de la présence de plusieurs fissures sur la direction et les vitesses de propagation des fissures de fatigue. / To optimize the rail grinding strategy, the prediction of crack growth rates has a vital role. Contact, with friction between the crack faces, notably occurs in rolling contact fatigue (RCF) problems. These time-dependent, multi-axial, non proportional loadings may lead to a crack initiation and propagation, and sometimes to the development of very complex 3D crack network. Numerical simulations of frictional fatigue crack are efficiently performed using the eXtended Finite Element Method (X-FEM). Within this method, the mesh does not need to conform to the crack geometry. Most difficulties associated to complex mesh generation around the crack and the re-meshing steps during the propagation are hence avoided. A 3D two-scale frictional contact fa-tigue crack model developed within the X-FEM framework is presented. It allows the use of a refined discretization of the crack interface independent from the underlying finite element mesh and adapted to the frictional contact crack scale. The model is used here to analyze the crack propagation, rate and direction, under rolling contact fatigue. The wheel-rail contact loading is modeled as a traveling hertzian load. The stress intensity factors are computed at the crack tips during the wheel passage. Criteria for determining crack growth direction under multiaxial non proportional conditions and mixed mode Paris’ law are used. Actual residual stresses are accounted for in the simulation. They are determined thanks to a dedicated model used at SNCF in which the asymptotic mechanical state of the rail is computed when submitted to cyclic loads. A non-uniform elastic-plastic stabilized state is calculated and introduced, by projection of the mechanical fields onto the finite element mesh, in the crack propagation simulation. All this strategy has been implemented in CAST3M and is now used to model 3D frictional crack growth under RCF.

La compréhension du comportement de structures jusqu'à leur ruine est nécessaire pour concevoir au mieux ces structures. Selon le matériau et les sollicitations considérées, les mécanismes physiques à l'origine de la rupture changent. Nous nous intéresserons à des matériaux homogènes pour lesquels la ruine passe par le développement de fissures autour desquelles les non-linéarités de comportement n'ont pas un rôle dominant. Ces conditions sont réunies pour les matériaux fragiles pour lesquels la source principale de dissipation est la génération non réversible d'une surface libre, et pour certaines fissures de fatigue. Sur un cycle de chargement, il existe de nombreuses applications pour lesquelles les non-linéarités restent confinées. La théorie de la mécanique linéaire élastique de la rupture est alors un modèle pertinent pour approcher le comportement de la structure. Sous ces hypothèses, le front de la fissure introduit une singularité. L'étude asymptotique de cette singularité dans des situations plane et anti-plane permet de définir les séries de Williams. La singularité est alors d'ordre un demi et elle est quantifiée par les facteurs d'intensité des contraintes (FIC) pour chacun des trois modes de sollicitations. En 3D, la fissure peut avoir une géométrie complexe, et aucune expression générale de la singularité n'existe. Dans cette thèse, les séries de Williams en déplacements sont utilisées et régularisées le long du front au sens des éléments finis. À partir de cette définition 3D des séries asymptotiques en pointe de fissure, une méthode d'extraction directe des FIC (DEK-FEM) est étendue au cas 3D. Le domaine est décomposé en deux domaines, raccordés en moyenne sur l'interface. Au voisinage du front, les champs mécaniques sont approchés par une troncature des champs asymptotiques. La singularité est donc traitée avec des champs adaptés, et les degrés de liberté associés sont directement les coefficients asymptotiques. Parmi ces coefficients asymptotiques, on retrouve les FIC et les T-stresses. Pour des raisons d'efficacité numérique et pour pouvoir relier l'échelle de la structure à l'échelle de la fissure, cette méthode est intégrée dans un contexte multigrilles localisées X-FEM. Ainsi nous montrons que cette approche permet une bonne évaluation des évolutions des FIC et du T-stress. Cette méthode est développée en parallèle d'une stratégie de post-traitement expérimental (mesure de champs de déplacements par corrélation d'images) basée sur les mêmes séries asymptotiques. Les images tridimensionnels d'un essai de fatigue in situ sont obtenues par micro-tomographie à rayons X et reconstruction. La corrélation et la régularisation basées sur les séries asymptotiques permettent d'obtenir la géométrie de la fissure et les FIC pour pouvoir identifier des lois de propagation de fissures 3D en fatigue. L'efficacité de cette méthode en parallèle d'une simulation DEK-FEM est illustrée en 2D. / It is necessary to understand the behavior of structures up to their failure to enhance their design. The mechanisms and phenomena undergoing failure vary according to the considered material and boundary conditions. We consider homogeneous materials for which cracks propagate in a context where behavior nonlinearities are not dominants. These conditions are matched for brittle and quasi-brittle materials and for some fatigue cracks. For the former, the main source of dissipation is the crack propagation which can be seen as the generation of a new free-surface. For the later, there is many applications where, in one loading cycle, the nonlinearities remains confined around the crack tip. The linear elastic fracture mechanics theory is then a pertinent model to approximate the structure behavior. Under such hypotheses, a singularity appears in the crack tip vicinity. The Williams' series expansion is computed from the asymptotic study of plane and anti-plane states. The stress is singular at the crack tip and the order of this singularity is one out of two. The singularity amplitude is quantified by the stress intensity factors (SIF), one for each of the three loading modes. In 3D, the crack shape is potentially complex (front curvature and non-planar crack), and no general asymptotic series expansion exists. In this PhD thesis, the 2D Williams' series in displacements are used and regularized with a finite element evolution along the front. From this 3D definition of the asymptotic fields in the crack tip vicinity, a numerical method for direct estimation of the SIF (DEK-FEM) is extended to 3D. This method is based on domain decomposition, the two domains are bounded in a weak sense on their interface. In the crack tip vicinity, the mechanical fields are approximated by a truncation of the asymptotic series expansion. Therefore, appropriate fields are used to deal with the singularity, and the associated degrees of freedom are directly the asymptotic coefficients. Among these coefficients are the SIF and the T-stresses. To bridge the scales between the structure and the crack front singularity and to increase the numerical efficiency, this method is embedded in a localized X-FEM multigrids approach. The proposed method is shown to provide an accurate evaluation of the SIF and T-stresses evolution. This approach has been developed in combination of an experimental post-processing method (full field displacement measurement through image correlation) based on the same asymptotic series expansion. The 3D images can be obtained for in situ fatigue experiments by X-ray microtomography and reconstruction. The crack geometry and the SIF are then provided by image correlation and regularization based on Williams series expansion. These data can be used for identifying a 3D fatigue crack growth law. The efficiency of the method is illustrated in 2D.

Tato disertační práce vychází z výzkumu v rámci francouzsko-českého programu doktorátu pod dvojím vedením na pracovišti Institut français de mécanique avancée v Clermont-Ferrand a na Ústavu fyziky materiálu AV v Brně. Úvodní výzkumný úkol na brněnském pracovišti se zabýval numerickou analýzou pole napětí v okolí čela trhliny v tenké kovové fólii. Zvláštní pozornost byla zaměřena na vliv speciálního typu singularity v průsečíku čela trhliny s volným povrchem. Těžiště disertační práce spočívá v numerickém modelování a stochastické analýze problémů šíření trhlin se složitou geometrií v dvojrozměrném prostoru. Při analýze těchto problémů se dříve zřídka používaly numerické metody, a to z důvodu vysoké náročnosti na výpočtový čas. V této disertaci je ukázáno, že aplikací moderních metod numerické mechaniky a vhodných technik v analýze spolehlivosti lze tyto problémy řešit s pomocí numerických metod i na PC. Ve spolehlivostní analýze byla využita lineární aproximační metoda FORM. Pro rychlost šíření trhlin se vycházelo z Parisova-Erdoganova vztahu. Pro parametry tohoto vztahu byl použit dvourozměrný statistický model, který postihuje vysokou citlivost na korelaci obou parametrů. Mechanická odezva byla počítána rozšířenou metodou konečných prvků (XFEM), která eliminuje výpočetní náročnost a numerický šum související se změnou sítě v klasické metodě konečných prvků. Prostřednictvím přímé diferenciace bylo odvozeno několik vztahů pro derivace funkce odezvy, čímž se dosáhlo lepší numerické stability a konvergence spolehlivostní analýzy a výrazného zkrácení doby výpočtu. Problém zatížení s proměnou amplitudou byl řešen aplikací transformace zatížení metodou PREFFAS. Využití distribuce výpočtů v síti PC umožnilo další zrychlení analýzy.

[pt] A Mecânica da Fratura é uma área da mecânica clássica que estuda os processos que podem resultar no surgimento e propagação de fraturas e trincas nos materiais. Seus conceitos podem ser empregados em diversas áreas, como, por exemplo, no desenvolvimento de novos materiais, na área industrial (auxiliando na definição dos parâmetros de criticidade de defeitos) e em projetos estruturais, onde a presença de descontinuidades pode resultar na diminuição da resistência do material, levando-o ao colapso estrutural. As fraturas podem surgir como defeitos básicos nos materiais constituintes dos elementos, podem ser induzidas nos processos de construção ou, ainda, podem ser desenvolvidas durante a vida útil da estrutura. O estudo do comportamento de uma fratura é fundamental para a verificação coerente de tensões e deformações nos projetos estruturais. Alguns autores apresentaram diferentes metodologias para o cálculo de parâmetros importantes associados à propagação de fraturas. Tais parâmetros podem variar de acordo com o comportamento do material, características geométricas, carregamentos, condições de contorno e configuração das trincas. Essas análises também podem ser validadas através de metodologias numéricas. O presente projeto tem por objetivo avaliar o Fator de Intensidade de Tensões a partir de análises numéricas de três casos de fratura com base nos conceitos da Mecânica da Fratura Linear Elástica (MFLE). Para tal, aplicaram-se duas técnicas de modelagem em Elementos Finitos: elementos quarter-point 2D e 3D e o Método dos Elementos Finitos Estendidos (XFEM) 3D. Por fim, os resultados obtidos são comparados com os resultados publicados na literatura. Apesar das vantagens associadas ao uso do XFEM para modelagem de fraturas, o cálculo pela integral de domínio para esta técnica apresenta oscilações nos valores fornecidos para as diferentes solicitações de contornos. A técnica de modelagem com Elementos Quarter-Points 2D e 3D apresenta resultados mais estáveis e próximos das soluções analíticas. / [en] Fracture Mechanics is an area of classical mechanics that studies processes that can result in the creation and propagation of fractures and cracks in materials. Its concepts can be utilized in many areas, such as, for example, the development of new materials, in the industrial area (to assist in the definition of defect criticality parameters) and in structural projects, where the presence of discontinuities can result in decreased material resistance, leading to its structural collapse. The fractures can emerge as basic defects in materials that constitute the structural elements, can be inducted in construction processes or could even be developed during the lifespan of the structure. Studying the behavior of a fracture is fundamental to verifying coherently stress and deformations on structural projects. Some authors presented different methodologies to calculate important parameters associated to the propagation of cracks. These parameters could vary according to the material behavior, geometrical characteristics, loads, boundary conditions and cracking patterns. Those analyses can also be validated through numerical methodologies. This present project aims at evaluating the Stress Intensity Factor from numerical analyses of three cracks cases based on the concepts of Linear Elastic Fracture Mechanics (LEFM). Two techniques of Finite Element modeling were considered: quarter-point elements 2D and 3D and Extended Finite Element Method (XFEM) 3D. In the end, the results obtained are compared with results already published in the available literature. Despite the advantages associated with the use of XFEM for fracture modeling, the calculation of the stress intensity factor by the domain integral for this technique presents oscillations in the values provided for the different contour requests. The modeling technique with 2D and 3D quarter-points elements presents results that are more stable and closer to the analytical solutions.