[pt] O presente trabalho propõe um método estatístico computacionalmente eficiente (chamado Green-FOSM) para consideração de incertezas em problemas geomecânicos, com o objetivo de melhorar o processo de tomada de decisão ao analisar problemas associados com o processo de injeção ou depleção de fluídos. A novidade do método proposto está associada com a utilização do método das funções de Green (GFA), que, com o auxílio do método estatístico first-order second-moment (FOSM), é utilizado para propagar as inerentes incertezas associadas às propriedades mecânicas do material para o campo de deslocamento da formação geológica. Além disso, através dos conceitos de grid estocástico e função de autocorrelação, o método proposto permite a consideração da variabilidade espacial de variáveis aleatórias de entrada que representam essas propriedades mecânicas. O GFA utiliza as soluções fundamentais da mecânica clássica (solução fundamental de Kelvin, solução fundamental de Melan, entre outras) e o teorema da reciprocidade para determinar o campo de deslocamento de uma formação geológica com geometria irregular e diferentes tipos de materiais. A grande vantagem deste método em relação ao clássico método dos elementos finitos (MEF) é que ele não requer a imposição de condições de contorno e a análise do problema pode ser realizada considerando apenas o domínio do reservatório ou outras regiões de interesse. Esta estratégia de modelagem diminui os graus de liberdade do modelo e o tempo de processamento da análise. Desta forma, como o GFA requer menos esforço computacional, este método torna-se ideal para ser utilizado na propagação de incertezas em problemas geomecânicos. Inicialmente, baseado no método das funções de Green original proposto por Peres et al. (2021), foi proposto uma versão iterativa do método Green-FOSM, que apresenta resultados estatísticos semelhantes aos encontrados através da clássica simulação de Monte Carlo (SMC). Nesta versão original, o campo de deslocamento é PUC-Rio - Certificação Digital Nº 1912634/CA calculado usando um esquema numérico iterativo que diminui o desempenho computacional do método e pode gerar problemas de convergência. Tais limitações tem dificultado a aplicação do GFA original e do método Green-FOSM iterativo em problemas reais. Assim, o presente trabalho desenvolveu uma nova versão do GFA que utiliza um esquema numérico não-iterativo. Para os problemas de validação analisados, o método não-iterativo demonstra ser até 17.5 vezes mais rápido do que a versão original. Além disso, esta versão demonstra ser capaz de expandir a aplicabilidade do GFA, pois os problemas de convergência foram eliminados e os resultados obtidos por este método, ao analisar um perfil geológico representativo do pré-sal brasileiro, são semelhantes aos encontrados via MEF. Por fim, a partir do GFA não-iterativo foi proposta uma versão não-iterativa do método Green-FOSM. Esta versão não-iterativa é capaz de analisar probabilisticamente formações geológicas complexas, como é o caso das formações geológicas do présal brasileiro. Utilizando os mesmos recursos computacionais, o método GreenFOSM não-iterativo é no mínimo 200 vezes mais rápido que o método iterativo. De forma geral, os resultados encontrados nas análises realizadas (determinísticas e probabilísticas) são próximos dos resultados obtidos pelo método de referência (MEF e SMC, respectivamente). / [en] The present work proposes a computationally efficient stochastic statistical method (called Green-FOSM) that considers uncertainties in geomechanical problems, with the objective of improving the decision-making process related to problems associated with the process of fluid injection or depletion. The novelty of the method lies in the use of the Green s function approach (GFA), which, together, with the first-order second-moment statistical method (FOSM), is used to propagate uncertainties associated with the mechanical properties of material to the displacement field of the geological formation. Furthermore, using the concepts of stochastic grid and autocorrelation function, the proposed method allows the consideration of the spatial variability of random variables that represent these mechanical properties. The GFA uses the fundamental solutions of classical mechanics (Kelvin fundamental solution, Melan fundamental solution, among others) and the reciprocity theorem to calculate the displacement field of a geological formation with irregular geometry, and different types of materials. The great advantage of this method compared to the classical finite element method (FEM) is that it does not require the imposition of boundary conditions and the analysis of the problem can be performed considering only the reservoir or other regions of interest. This modeling strategy decreases the degrees of freedom of the model and the CPU time of the deterministic analysis. In this way, as the GFA requires less computational effort, this approach becomes ideal for propagating the uncertainties in geomechanical problems. Initially, an iterative version of the Green-FOSM method was proposed, which presents statistical results similar to those found through the classic Monte Carlo simulation (MCS). In this initial version, the displacement field is calculated using an iterative numerical scheme, which decreases the computational performance of the method and can generate convergence problems. Such limitations would restrict the application of the PUC-Rio - Certificação Digital number 1912634/CA original GFA and the iterative Green-FOSM method in real problems. Thus, the present work also developed a new version of the GFA, which uses a non-iterative numerical scheme. For the proposed validation problems, the non-iterative method proved to be up to 17.5 times faster than the original version. This version is able to expand the applicability of the GFA, since the convergence problems were eliminated and the results obtained by this method, when analyzing a representative geological profile of the Brazilian pre-salt, are similar to those found via FEM. Finally, based on the non-iterative GFA, a non-iterative version of the Green-FOSM method was proposed. This non-iterative version is capable of probabilistically analyzing complex geological formations, such as the Brazilian pre-salt geological formations. Using the same computational resources, the non-iterative GreenFOSM method is at least 200 times faster than the iterative Green-FOSM method. In general, the results found in the investigated analyzes (deterministic and probabilistic) are close to the results obtained by the reference method (FEM and MCS, respectively).

<div>In this thesis, the role of location-specific microstructural features in the fatigue performance of the safety-critical aerospace components made of Nickel (Ni)-base superalloys and linear friction welded (LFW) Titanium (Ti) alloys has been studied using crystal plasticity finite element (CPFE) simulations, energy dispersive X-ray diffraction (EDD), backscatter electron (BSE) images and digital image correlation (DIC).</div><div><br></div><div>In order to develop a microstructure-sensitive fatigue life prediction framework, first, it is essential to build trust in the quantitative prediction from CPFE analysis by quantifying uncertainties in the mechanical response from CPFE simulations. Second, it is necessary to construct a unified fatigue life prediction metric, applicable to multiple material systems; and a calibration strategy of the unified fatigue life model parameter accounting for uncertainties originating from CPFE simulations and inherent in the experimental calibration dataset. To achieve the first task, a genetic algorithm framework is used to obtain the statistical distributions of the crystal plasticity (CP) parameters. Subsequently, these distributions are used in a first-order, second-moment method to compute the mean and the standard deviation for the stress along the loading direction (σ_load), plastic strain accumulation (PSA), and stored plastic strain energy density (SPSED). The results suggest that an ~10% variability in σ_load and 20%-25% variability in the PSA and SPSED values may exist due to the uncertainty in the CP parameter estimation. Further, the contribution of a specific CP parameter to the overall uncertainty is path-dependent and varies based on the load step under consideration. To accomplish the second goal, in this thesis, it is postulated that a critical value of the SPSED is associated with fatigue failure in metals and independent of the applied load. Unlike the classical approach of estimating the (homogenized) SPSED as the cumulative area enclosed within the macroscopic stress-strain hysteresis loops, CPFE simulations are used to compute the (local) SPSED at each material point within polycrystalline aggregates of 718Plus, an additively manufactured Ni-base superalloy. A Bayesian inference method is utilized to calibrate the critical SPSED, which is subsequently used to predict fatigue lives at nine different strain ranges, including strain ratios of 0.05 and -1, using nine statistically equivalent microstructures. For each strain range, the predicted lives from all simulated microstructures follow a log-normal distribution; for a given strain ratio, the predicted scatter is seen to be increasing with decreasing strain amplitude and are indicative of the scatter observed in the fatigue experiments. Further, the log-normal mean lives at each strain range are in good agreement with the experimental evidence. Since the critical SPSED captures the experimental data with reasonable accuracy across various loading regimes, it is hypothesized to be a material property and sufficient to predict the fatigue life.</div><div><br></div><div>Inclusions are unavoidable in Ni-base superalloys, which lead to two competing failure modes, namely inclusion- and matrix-driven failures. Each factor related to the inclusion, which may contribute to crack initiation, is isolated and systematically investigated within RR1000, a powder metallurgy produced Ni-base superalloy, using CPFE simulations. Specifically, the role of the inclusion stiffness, loading regime, loading direction, a debonded region in the inclusion-matrix interface, microstructural variability around the inclusion, inclusion size, dissimilar coefficient of thermal expansion (CTE), temperature, residual stress, and distance of the inclusion from the free surface are studied in the emergence of two failure modes. The CPFE analysis indicates that the emergence of a failure mode is an outcome of the complex interaction between the aforementioned factors. However, the possibility of a higher probability of failure due to inclusions is observed with increasing temperature, if the CTE of the inclusion is higher than the matrix, and vice versa. Any overall correlation between the inclusion size and its propensity for damage is not found, based on inclusion that is of the order of the mean grain size. Further, the CPFE simulations indicate that the surface inclusions are more damaging than the interior inclusions for similar surrounding microstructures. These observations are utilized to instantiate twenty realistic statistically equivalent microstructures of RR1000 – ten containing inclusions and remaining ten without inclusions. Using CPFE simulations with these microstructures at four different temperatures and three strain ranges for each temperature, the critical SPSED is calibrated as a function of temperature for RR1000. The results suggest that critical SPSED decreases almost linearly with increasing temperature and is appropriate to predict the realistic emergence of the competing failure modes as a function of applied strain range and temperature.</div><div><br></div><div>LFW process leads to the development of significant residual stress in the components, and the role of residual stress in the fatigue performance of materials cannot be overstated. Hence, to ensure fatigue performance of the LFW Ti alloys, residual strains in LFW of similar (Ti-6Al-4V welded to Ti-6Al-4V or Ti64-Ti64) and dissimilar (Ti-6Al-4V welded to Ti-5Al-5V-5Mo-3Cr or Ti64-Ti5553) Ti alloys have been characterized using EDD. For each type of LFW, one sample is chosen in the as-welded (AW) condition and another sample is selected after a post-weld heat treatment (HT). Residual strains have been separately studied in the alpha and beta phases of the material, and five components (three axial and two shear) have been reported in each case. In-plane axial components of the residual strains show a smooth and symmetric behavior about the weld center for the Ti64-Ti64 LFW samples in the AW condition, whereas these components in the Ti64-Ti5553 LFW sample show a symmetric trend with jump discontinuities. Such jump discontinuities, observed in both the AW and HT conditions of the Ti64-Ti5553 samples, suggest different strain-free lattice parameters in the weld region and the parent material. In contrast, the results from the Ti64-Ti64 LFW samples in both AW and HT conditions suggest nearly uniform strain-free lattice parameters throughout the weld region. The observed trends in the in-plane axial residual strain components have been rationalized by the corresponding microstructural changes and variations across the weld region via BSE images. </div><div><br></div><div>In the literature, fatigue crack initiation in the LFW Ti-6Al-4V specimens does not usually take place in the seemingly weakest location, i.e., the weld region. From the BSE images, Ti-6Al-4V microstructure, at a distance from the weld-center, which is typically associated with crack initiation in the literature, are identified in both AW and HT samples and found to be identical, specifically, equiaxed alpha grains with beta phases present at the alpha grain boundaries and triple points. Hence, subsequent fatigue performance in LFW Ti-6Al-4V is analyzed considering the equiaxed alpha microstructure.</div><div><br></div><div>The LFW components made of Ti-6Al-4V are often designed for high cycle fatigue performance under high mean stress or high R ratios. In engineering practice, mean stress corrections are employed to assess the fatigue performance of a material or structure; albeit this is problematic for Ti-6Al-4V, which experiences anomalous behavior at high R ratios. To address this problem, high cycle fatigue analyses are performed on two Ti-6Al-4V specimens with equiaxed alpha microstructures at a high R ratio. In one specimen, two micro-textured regions (MTRs) having their c-axes near-parallel and perpendicular to the loading direction are identified. High-resolution DIC is performed in the MTRs to study grain-level strain localization. In the other specimen, DIC is performed on a larger area, and crack initiation is observed in a random-textured region. To accompany the experiments, CPFE simulations are performed to investigate the mechanistic aspects of crack initiation, and the relative activity of different families of slip systems as a function of R ratio. A critical soft-hard-soft grain combination is associated with crack initiation indicating possible dwell effect at high R ratios, which could be attributed to the high-applied mean stress and high creep sensitivity of Ti-6Al-4V at room temperature. Further, simulations indicated more heterogeneous deformation, specifically the activation of multiple families of slip systems with fewer grains being plasticized, at higher R ratios. Such behavior is exacerbated within MTRs, especially the MTR composed of grains with their c-axes near parallel to the loading direction. These features of micro-plasticity make the high R ratio regime more vulnerable to fatigue damage accumulation and justify the anomalous mean stress behavior experienced by Ti-6Al-4V at high R ratios.</div><div><br></div>