<p>The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.</p><p>We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode. </p><p>We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.</p><p>The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.</p><p>Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.</p><p>The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.</p> / Dissertation

Ossos trabeculares possuem uma microestrutura porosa e podem ser modelados como um sólido elástico linear, heterogêneo e anisotrópico. A microtomografia tridimensional por raios-x (uCT) tem sido mundialmente utilizada para a caracterização de osso trabecular em pesquisas relacionadas à qualidade óssea e à doenças do metabolismo ósseo como a osteoporose. Na literatura há poucas investigações comparando as análises morfométricas 2D e 3D de osso trabecular. Nesta investigação amostras de osso trabeculares cilíndricas extraídas da cabeça femoral bovina foram utilizadas para a comparação dessas análises por \'mü\'CT. O software CT-Analyser foi utilizado para medidas em três direções (crânio-caudal, lateral-medial e anterior-posterior) de diversos parâmetros da microestrutura trabecular tais como espessura trabecular, separação trabecular, número trabecular e os autovalores do tensor de anisotropia (M). A comparação entre os valores dos parâmetros medidos por análises morfométricas 2D e 3D foi realizada pelo teste-t pareado com nível de significância p < 0,05 e por correlação linear de Pearson. Os autovalores da matriz M mostram que a microestrutura trabecular bovina tem uma tendência para a simetria transversalmente isotrópica. Os resultados obtidos mostram que a microtomografia tridimensional por raios-X é uma técnica de grande potencial para caracterização da qualidade óssea gerando bons parâmetros para o diagnóstico de doenças do metabolismo ósseo. / Trebecular bones have a porous microstructure and can be modeled as a linear elastic solid, heterogenous and anisotropic. The x-ray tridimensional microtomography (\'mü\'CT) has been worldwide used for the assessment of trabecular bone in investigations related to bone quality and aimed to the diagnostic of bone metabolic diseases such as osteoporosis. In the literature few investigations have compared the 2D and 3D morphometric analysis of trabecular bone. In this investigation cylindrical trabecular bone samples were harvested from bovine head femur to carry out that comparison by \'mü\'CT. The CT-Analyser software was used to measure in three directions (superior-inferior, lateral-medial and anterior-posterior) several microstructural trabecular parameters such as trabecular thickness, trabecular separation, trabecular number and the eigenvalues of the fabric tensor (M). The comparison between the parameters values measured by 2D and 3D morphometric analysis was performed by the paired-sample t test with a level of significance p < 0,05 and the Pearson\'s linear correlation. The eigenvalues show that the bovine trabecular microstructure has a tendence to transverse isotropy symmetry. The results show that x-ray tridimensional microtomography (\'mü\'CT) is a technique of great potential for characterization of generating good bone quality parameters for the diagnosis of diseases of bone metabolism.

Ossos trabeculares possuem uma microestrutura porosa e podem ser modelados como um sólido elástico linear, heterogêneo e anisotrópico. A microtomografia tridimensional por raios-x (uCT) tem sido mundialmente utilizada para a caracterização de osso trabecular em pesquisas relacionadas à qualidade óssea e à doenças do metabolismo ósseo como a osteoporose. Na literatura há poucas investigações comparando as análises morfométricas 2D e 3D de osso trabecular. Nesta investigação amostras de osso trabeculares cilíndricas extraídas da cabeça femoral bovina foram utilizadas para a comparação dessas análises por \'mü\'CT. O software CT-Analyser foi utilizado para medidas em três direções (crânio-caudal, lateral-medial e anterior-posterior) de diversos parâmetros da microestrutura trabecular tais como espessura trabecular, separação trabecular, número trabecular e os autovalores do tensor de anisotropia (M). A comparação entre os valores dos parâmetros medidos por análises morfométricas 2D e 3D foi realizada pelo teste-t pareado com nível de significância p < 0,05 e por correlação linear de Pearson. Os autovalores da matriz M mostram que a microestrutura trabecular bovina tem uma tendência para a simetria transversalmente isotrópica. Os resultados obtidos mostram que a microtomografia tridimensional por raios-X é uma técnica de grande potencial para caracterização da qualidade óssea gerando bons parâmetros para o diagnóstico de doenças do metabolismo ósseo. / Trebecular bones have a porous microstructure and can be modeled as a linear elastic solid, heterogenous and anisotropic. The x-ray tridimensional microtomography (\'mü\'CT) has been worldwide used for the assessment of trabecular bone in investigations related to bone quality and aimed to the diagnostic of bone metabolic diseases such as osteoporosis. In the literature few investigations have compared the 2D and 3D morphometric analysis of trabecular bone. In this investigation cylindrical trabecular bone samples were harvested from bovine head femur to carry out that comparison by \'mü\'CT. The CT-Analyser software was used to measure in three directions (superior-inferior, lateral-medial and anterior-posterior) several microstructural trabecular parameters such as trabecular thickness, trabecular separation, trabecular number and the eigenvalues of the fabric tensor (M). The comparison between the parameters values measured by 2D and 3D morphometric analysis was performed by the paired-sample t test with a level of significance p < 0,05 and the Pearson\'s linear correlation. The eigenvalues show that the bovine trabecular microstructure has a tendence to transverse isotropy symmetry. The results show that x-ray tridimensional microtomography (\'mü\'CT) is a technique of great potential for characterization of generating good bone quality parameters for the diagnosis of diseases of bone metabolism.