In the thesis, the reliability analysis of structural components and structural details subject to random loading and random resistance degradation is addressed. The study concerns evaluation of the probability of failure due to an overload of a component or structural detail, in consideration of random (environmental) loads and their combination, uncertain resistance parameters, statistical and phenomenological uncertainty and random resistance degradation mechanisms. Special attention is devoted to resistance degradation, as it introduces an additional level of difficulty in the solution of time variant reliability problems. The importance of this study arrives from the ageing of existing infrastructure in a world wide scale and from the lack of standards and codes for the ongoing safety management of general structures past their original design lives. In this context, probabilistic-based risk assessment and reliability analysis provide a framework for the safety management of ageing structures in consideration of inherent load and resistance uncertainty, current state of the structure, further resistance degradation, periodic inspections, in the absence of past experience and on an individual basis. In particular, the critical problem of resistance degradation due to fatigue is addressed. The formal solution of time variant reliability problems involves integration of local crossing rates over a conditional failure domain boundary, over time and over random resistance variables. This solution becomes very difficult in the presence of resistance degradation, as crossing rates become time dependent, and the innermost integration over the failure domain boundary has to be repeated over time. Significant simplification is achieved when the order of integrations is changed, and crossing rates are first integrated over the random failure domain boundary and then over time. In the so-called ensemble crossing rate or Ensemble Up-crossing Rate (EUR) approximation, the arrival rate of the first crossing over a random barrier is approximated by the ensemble average of crossings. This approximation conflicts with the Poisson assumption of independence implied in the first passage failure model, making results unreliable and highly conservative. Despite significant simplification of the solution, little was known to date about the quality of the EUR approximation. In this thesis, a simulation procedure to obtain Poissonian estimates of the arrival rate of the first up-crossing over a random barrier is introduced. The procedure is used to predict the error of the EUR approximation. An error parameter is identified and error functions are constructed. Error estimates are used to correct original EUR failure probability results and to compare the EUR with other common simplifications of time variant reliability problems. It is found that EUR errors can be quite large even when failure probabilities are small, a result that goes against previous ideas. A barrier failure dominance concept is introduced, to characterize those problems where an up-crossing or overload failure is more likely to be caused by a small outcome of the resistance than by a large outcome of the load process. It is shown that large EUR errors are associated with barrier failure dominance, and that solutions which simplify the load part of the problem are more likely to be appropriate in this case. It is suggested that the notion of barrier failure dominance be used to identify the proper (simplified) solution method for a given problem. In this context, the EUR approximation is compared with Turkstra’s load combination rule and with the point-crossing formula. It is noted that in many practical structural engineering applications involving environmental loads like wind, waves or earthquakes, load process uncertainty is larger than resistance uncertainty. In these applications, barrier failure dominance in unlikely and EUR errors can be expected to be small. The reliability problem of fatigue and fracture under random loading is addressed in the thesis. A solution to the problem, based on the EUR approximation, is constructed. The problem is formulated by combining stochastic models of crack propagation with the first passage failure model. The solution involves evaluation of the evolution in time of crack size and resistance distributions, and provides a fresh random process-based approach to the problem. It also simplifies the optimization and planning of non-destructive periodic inspection strategies, which play a major role in the ongoing safety management of fatigue affected structures. It is shown how sensitivity coefficients of a simplified preliminary First Order Reliability solution can be used to characterize barrier failure dominance. In the fatigue and fracture reliability problem, barrier failure dominance can be caused by large variances of resistance or crack growth parameters. Barrier failure dominance caused by resistance parameters leads to problems where overload failure is an issue and where the simplified preliminary solution is likely to be accurate enough. Barrier failure dominance caused by crack growth parameters leads to highly non-linear problems, where critical crack growth dominates failure probabilities. Finally, in the absence of barrier failure dominance, overload failure is again the issue and the EUR approximation becomes not just appropriate but also accurate. The random process-based EUR solution of time-variant reliability problems developed and the concept of barrier failure dominance introduced in the thesis have broad applications in problems involving general forms of resistance degradation as well as in problems of random vibration of uncertain structures. / PhD Doctorate

La fatigue induite par chargement cyclique est un mode d'endommagement majeur des métaux. Elle se caractérise par des effets environnementaux et de grandes dispersions de la durée de vie qui doivent être mieux comprises. Les matériaux analysés sont de type cfc : aluminium, cuivre, nickel et argent. Le comportement de marches naturellement créées en surface par le glissement cyclique de dislocations est examiné par simulations en dynamique moléculaire sous vide et sous environnement oxygène pour le cuivre et le nickel. Un phénomène de reconstruction est observé sur les marches en surface, qui peut induire une forte irréversibilité. Trois mécanismes de reconstruction des marches apparues en surface sont observés et décrits. L’irréversibilité de ces marches est ensuite analysé. Elles sont irréversibles pour des chargements expérimentaux, sauf arrivée de dislocations de signe opposé sur un plan de glissement directement voisin.Avec arrivée de dislocations sur des plans non voisins, l'irréversibilité s’accumule cycle par cycle et il est possible de reproduire l’apparition de fissures en surface dont la profondeur augmente graduellement.Un environnement oxygène modifie la surface (début d’oxydation) mais pas l’irréversibilité parce que l’oxygène n’a pas d'influence majeure sur les différents mécanismes liés à l’évolution du relief.Une estimation grossière de l'irréversibilité est faite pour des dislocations coin pures dans une bande de glissement persistante pour les matériaux dits ondulés. On obtient un facteur d’irréversibilité entre 0,5 et 0,75 pour le cuivre, sous vide et sous l’environnement oxygène, en accord avec des mesures récentes en microscopie à force atomique.La propagation de fissures est simulée en environnement inerte. Les fissures peuvent se propager à cause de l'irréversibilité des dislocations générées, liée à leurs interactions allant jusqu’à la création de jonctions. / Fatigue is one of the major damage mechanisms of metals. It is characterized by strong environmental effects and wide lifetime dispersions which must be better understood. Different face centred cubic metals, Al, Cu, Ni, and Ag are analyzed. The mechanical behaviour of surface steps naturally created by the glide of dislocations subjected to cyclic loading is examined using molecular dynamics simulations in vacuum and in air for Cu and Ni. An atomistic reconstruction phenomenon is observed at these surface steps which can induce strong irreversibility. Three different mechanisms of reconstruction are defined. Surface slip irreversibility under cyclic loading is analyzed. All surface steps are intrinsically irreversible under usual fatigue laboratory loading amplitude without the arrival of opposite sign dislocations on direct neighbor plane.With opposite sign dislocations on non direct neighbour planes, irreversibility cumulates cycle by cycle and a micro-notch is produced whose depth gradually increases.Oxygen environment affects the surface (first stage of oxidation) but does not lead to higher irreversibility as it has no major influence on the different mechanisms linked to surface relief evolution.A rough estimation of surface irreversibility is carried out for pure edge dislocations in persistent slip bands in so-called wavy materials. It gives an irreversibility fraction between 0.5 and 0.75 in copper in vacuum and in air, in agreement with recent atomic force microscopy measurements.Crack propagation mechanisms are simulated in inert environment. Cracks can propagate owing to the irreversibility of generated dislocations because of their mutual interactions up to the formation of dislocation junctions.

Metal fatigue is a recurring problem for metallurgists and materials engineers, especially in structural applications. It has been responsible for many disastrous accidents and tragedies in history. Understanding the micro-mechanisms during cyclic deformation and combating fatigue failure has remained a grand challenge. Environmental effects, like temperature or a corrosive medium, further worsen and complicate the problem. Ultimate design against fatigue must come from a materials perspective with a fundamental understanding of the interaction of microstructural features with dislocations, under the influence of stress, temperature, and other factors. This research endeavors to contribute to the current understanding of the fatigue failure mechanisms. Cast aluminum alloys are susceptible to fatigue failure due to the presence of defects in the microstructure like casting porosities, non-metallic inclusions, non-uniform distribution of secondary phases, etc. Friction stir processing (FSP), an emerging solid state processing technique, is an effective tool to refine and homogenize the cast microstructure of an alloy. In this work, the effect of FSP on the microstructure of an A356 cast aluminum alloy, and the resulting effect on its tensile and fatigue behavior have been studied. The main focus is on crack initiation and propagation mechanisms, and how stage I and stage II cracks interact with the different microstructural features. Three unique microstructural conditions have been tested for fatigue performance at room temperature, 150 °C and 200 °C. Detailed fractography has been performed using optical microscopy, scanning electron microscopy (SEM) and electron back scattered diffraction (EBSD). These tools have also been utilized to characterize microstructural aspects like grain size, eutectic silicon particle size and distribution. Cyclic deformation at low temperatures is very sensitive to the microstructural distribution in this alloy. The findings from the room temperature fatigue tests highlight the important role played by persistent slip bands (PSBs) in fatigue crack initiation. At room temperature, cracks initiate along PSBs in the absence of other defects/stress risers, and grow transgranularly. Their propagation is retarded when they encounter grain boundaries. Another major finding is the complete transition of the mode of fatigue cracking from transgranular to intergranular, at 200 °C. This occurs when PSBs form in adjacent grains and impinge on grain boundaries, raising the stress concentration at these locations. This initiates cracks along the grain boundaries. At these temperatures, cyclic deformation is no longer microstructure- dependent. Grain boundaries don’t impede the progress of cracks, instead aid in their propagation. This work has extended the current understanding of fatigue cracking mechanisms in A356 Al alloys to elevated temperatures.

L'objet de ce projet de recherche est de prédire la durée de vie d'éléments mécaniques qui sont soumis à des phénomènes de fatigue cyclique. L'idée est de développer un schéma numérique novateur pour prédire la rupture de structures sous de tels chargements. Le modèle est basé sur la mécanique des milieux continus qui introduit des variables internes pour décrire l'évolution de l'endommagement. Le défi repose dans le traitement des cycles de chargement pour la prédiction de la durée de vie, particulièrement pour la prédiction de la durée de vie résiduelle de structures existantes. Les approches traditionnelles de l'analyse de la fatigue sont basées sur des méthodes phénoménologiques utilisant des relations empiriques. De telles méthodes considèrent des approximations simplificatrices et sont incapables de prendre en compte aisément des géométries ou des charges complexes associées à des problèmes d'ingénierie réels. Une approche basée sur la description de l'évolution thermodynamique d'un milieu continu est donc utilisée pour modéliser le comportement en fatigue. Cela permet de considérer efficacement des problèmes d'ingénierie complexe et la détérioration des propriétés du matériau due à la fatigue peut être quantifiée à l'aide de variables internes. Cependant, cette approche peut être numériquement coûteuse et, par conséquent, des approches numériques sophistiquées doivent être utilisées.La stratégie numérique sur laquelle ce projet est basé est singulière par rapport aux schémas incrémentaux en temps usuellement utilisés pour résoudre des problèmes élasto-(visco)plastique avec endommagement dans le cadre de la mécanique des milieux continus. Cette stratégie numérique appelée méthode LATIN (Large Time Increment method) est une méthode non-incrémentale qui recherche la solution de manière itérative sur l'ensemble du domaine spacio-temporel. Une importante innovation de la méthode LATIN est d'incorporer une stratégie de réduction de modèle adaptative pour réduire de manière très importante le coût numérique. La Décomposition Propre Généralisée (PGD) est une stratégie de réduction de modèle a priori qui sépare les quantités d'intérêt spacio-temporelles en deux composantes indépendantes, l'une dépendant du temps, l'autre de l'espace, et estime itérativement les approximations de ces deux composantes. L'utilisation de l'approche LATIN-PGD a montré son efficacité depuis des années pour résoudre des problèmes élasto-(visco)plastiques. La première partie de ce projet vise à étendre cette approche aux modèles incorporant de l'endommagement.Bien que l'utilisation de la PGD réduise les coûts numériques, le gain n'est pas suffisant pour permettre de résoudre des problèmes considérant un grand nombre de cycles de chargement, le temps de calcul peut être très conséquent, rendant les simulations de problèmes de fatigue intraitables même en utilisant les techniques LATIN-PGD. Cette limite peut être dépassée en introduisant une approche multi-échelle en temps, qui prend en compte l'évolution rapide des quantités d'intérêt lors d'un cycle et leur évolution lente au cours de l'ensemble des cycles. Une description type « éléments finis » en temps est proposée, où l'ensemble du domaine temporel est discrétisé en éléments temporels, et seulement les cycles nodaux, qui forment les limites des éléments, sont calculés en utilisant la technique LATIN-PGD. Puis, des fonctions de forme classiques sont utilisées pour interpoler les quantités d'intérêt à l'intérieur des éléments temporels. Cette stratégie LATIN-PGD à deux échelles permet de réduire le coût numérique de manière significative, et peut être utilisée pour simuler l'évolution de l'endommagement dans une structure soumise à un chargement de fatigue comportant un très grand nombre de cycles. / The motivation of the research project is to predict the life time of mechanical components that are subjected to cyclic fatigue phenomena. The idea herein is to develop an innovative numerical scheme to predict failure of structures under such loading. The model is based on classical continuum damage mechanics introducing internal variables which describe the damage evolution. The challenge lies in the treatment of large number of load cycles for the life time prediction, particularly the residual life time for existing structures.Traditional approaches for fatigue analysis are based on phenomenological methods and deal with the usage of empirical relations. Such methods consider simplistic approximations and are unable to take into account complex geometries, and complicated loadings which occur in real-life engineering problems. A thermodynamically consistent continuum-based approach is therefore used for modelling the fatigue behaviour. This allows to consider complicated geometries and loads quite efficiently and the deterioration of the material properties due to fatigue can be quantified using internal variables. However, this approach can be computationally expensive and hence sophisticated numerical frameworks should be used.The numerical strategy used in this project is different when compared to regular time incremental schemes used for solving elasto-(visco)plastic-damage problems in continuum framework. This numerical strategy is called Large Time Increment (LATIN) method, which is a non-incremental method and builds the solution iteratively for the complete space-time domain. An important feature of the LATIN method is to incorporate an on-the-fly model reduction strategy to reduce drastically the numerical cost. Proper generalised decomposition (PGD), being a priori a model reduction strategy, separates the quantities of interest with respect to space and time, and computes iteratively the spatial and temporal approximations. LATIN-PGD framework has been effectively used over the years to solve elasto-(visco)plastic problems. Herein, the first effort is to solve continuum damage problems using LATIN-PGD techniques. Although, usage of PGD reduces the numerical cost, the benefit is not enough to solve problems involving large number of load cycles and computational time can be severely high, making simulations of fatigue problems infeasible. This can be overcome by using a multi-time scale approach, that takes into account the rapid evolution of the quantities of interest within a load cycle and their slow evolution along the load cycles. A finite element like description with respect to time is proposed, where the whole time domain is discretised into time elements, and only the nodal cycles, which form the boundary of the time elements, are calculated using LATIN-PGD technique. Thereby, classical shape functions are used to interpolate within the time element. This two-scale LATIN-PGD strategy enables the reduction of the computational cost remarkably, and can be used to simulate damage evolution in a structure under fatigue loading for a very large number of cycles.