O objetivo principal deste trabalho é o desenvolvimento de uma formulação e sua implementação computacional para se analisar, via Método dos Elementos Finitos (MEF),o comportamento dinâmico não linear geométrico de sólidos bidimensionais. Trata-se o comportamento geometricamente não linear através de uma formulação posicional classificada como lagrangeana total com cinemática exata. No estudo do comportamento dinâmico utiliza-se um algoritmo de integração temporal baseado na família de integradores temporais de Newmark. Para a consideração do impacto adota-se uma técnica que utiliza como integrador temporal o algoritmo de Newmark, modificado de forma a garantir sua estabilização, e limita-se a posição de cada nó da estrutura que por ventura sofra impacto. O código computacional desenvolvido é validado através de exemplos tradicionais da literatura científica. Analisam-se exemplos com comportamento apenas não linear geométrico e não linear geométrico dinâmico com ou sem impacto / The main goal of this work is the development of a formulation and its computational implementation, based on the finite element method (FEM), to analyze the dynamic geometrically nonlinear behavior of bidimensional solids. The geometrically nonlinear behavior is treated with a positional formulation classified as total Lagrangean with exact kinematics. In the study of the dynamic behavior, a time integration algorithm based on the family of time integrators of Newmark is applied. In order to consider the impact, a technique based on the time integrator of Newmark, modified to assure its stabilization, is used. This technique limits the position of each node that suffers impact. The developed computational code is validated through benchmarks of scientific literature. Examples with static geometrically nonlinear and dynamic geometrically nonlinear behavior, with or without impact, are analyzed

O composto magnético geometricamente frustrado Gd2Ti2O7 da família dos pirocloros apresenta um comportamento bastante interessante, sendo que a natureza da fase magnética em baixas temperaturas se encontra ainda sob intenso debate. Este material entra em um estado antiferromagnético parcialmente ordenado à temperatura T1N ~ 1 K, apresentando outra transição de fase em T2N ~ 0.7 K. Neste trabalho é investigada a física de baixas temperaturas de amostras de Gd2Ti2O7 com defeitos estruturais tais como vacâncias de oxigênio e diluição de ítrio. Amostras policristalinas com composição Gd2Ti2O7 e Gd2-xYxTi2O7 foram sintetizadas em diferentes condições por uma rota alternativa conhecida como método sol-gel. O refinamento de um modelo para os dados de difração de raios X mostra que vacâncias de oxigênio são os principais defeitos estruturais neste material. As vacâncias de oxigênio resultam numa ligeira diminuição de T1N e numa redução da magnetização de saturação. A diluição da rede com ítrio leva a uma clara diminuição de T1N e da temperatura de Curie-Weiss. Medidas de calor específico evidenciaram as duas transições T1N e T2N no composto com menor grau de vacâncias de oxigênio. A análise da contribuição magnética ao calor específico Cm em baixas temperaturas, 0.39 K < T < 0.68 K, revelou um comportamento proporcional a T^2 previamente discutido na literatura. Entretanto, verificamos que uma dependência T^3, usualmente encontrada em antiferromagnetos convencionais, descreve igualmente bem nossos dados experimentais resultando em uma velocidade de magnons consistente com a apresentada por outros pirocloros. / The geometrically frustrated compound Gd2Ti2O7 of the pyrochlore family displays such an interesting behaviour that the nature of the ordered magnetic phase at low temperatures is still under intense discussion. This material enters in a partially ordered magnetic state at a temperature T1N ~ 1 K, and there is another phase transition at T2N ~ 0.7 K. In this thesis we study the low temperature physics of Gd2Ti2O7 with structural defects such as oxygen vacancies and yttrium dilution. Polycrystalline samples of Gd2Ti2O7 and Gd2-xYxTi2O7 were synthesized in different conditions by an alternative route known as the sol-gel method. The refinement of a model for the X-ray diffraction data reveal that the oxygen vacancies are the leading defects in this material. The oxygen vacancies result in a slight decrease of T1N and in a reduction of the saturation magnetization. The yttrium dilution of the lattice leads to a clear reduction of T1N and of the Curie-Weiss temperature. Specific heat measurements display both transitions T1N and T2N in the compound with lower degree of oxygen vacancies. The analysis of the magnetic contribution to the specific heat Cm at low temperatures, 0.39 K < T < 0.68 K, reveals a behaviour proportional to T^2 previously discussed in the literature. However, we verify that a dependence T^3, usually found in standard antiferromagnets, describes similarly well our experimental data resulting in a velocity of magnons consistent with the ones exhibited for another pyrochlores.

Nesta tese demonstramos, entre outras coisas, a existência de innitas geodésicas fechadas em good orbifolds Riemannianos M/, onde é um grupo de isometrias virtualmente Abeliano. No caso particular onde é um produto semi-direto de um grupo nito por um grupo Abeliano, concluimos a existência de uma família de geodésicas fechadas com comprimentos tendendo a innito. / In this PhD theses we prove, among other things, the existence of innity many (geometric distinct) closed geodesics on good Riemannian compact orbifolds M/, where is a virtual abelian group of isometries. In the particular case where is a semi-direct product of a nite group with an abelian group, we also assure that there isa family of closed geodesics for which the lengths tend to innity.

The elastic properties of novel transition metal oxides have been investigated, using a powerful technique known as Resonant Ultrasound Spectroscopy (RUS). Two sets of transition metal oxides have been studied. One is the ruthenate Ca2-xSrxRuO4 series with a layered perovskite structure, a Mott transition system that connects the Mott insulator Ca2RuO4 with the unconventional superconductor Sr2RuO4. The other set contains geometrically frustrated materials, including vanadium spinels AV2O4 (A = Zn, Mn and Fe) and titanate pyrochlores A2Ti2O7 (A= Y, Tb, Yb, Ho and Dy). The elastic response of five Ca2-xSrxRuO4 single crystals (x = 2.0, 1.9, 0.5, 0.3 and 0.2) has been measured. For 2.0 ≥ x ≥ 0.5, a dramatic softening over a wide temperature range is observed upon cooling, caused by the rotational instability of RuO6 octahedra (for x = 2.0 and 1.9) or the static rotation of the octahedra (for x = 0.5). For the Ca-rich samples (x = 0.3 and 0.2), the softening occurs in a very narrow temperature range, corresponding to the structural phase transition from high-temperature-tetragonal to low-temperature-orthorhombic symmetry. Elastic softening in ZnV2O4 is observed near the cubic-to-tetragonal structural phase transition at 50 K. The elastic response of MnV2O4 is quite unusual, displaying a softening over a wide temperature range with decreasing temperature. Upon cooling, C’ of FeV2O4 becomes so soft that it drops to almost zero around 140 K, where the cubic-to-tetragonal structural transition occurs. For Y2Ti2O7, all three elastic constants show normal “Varshni” behavior. For spin liquid Tb2Ti2O7, all three elastic constants show a pronounced softening below 50 K, indicative of a possible Jahn-Teller, cubic-to-tetragonal transition at very low temperatures. It is also found that the application of a magnetic field suppresses the elastic softening in this compound. Another spin liquid Yb2Ti2O7 shows no elastic softening. The elastic moduli of the spin-ice compounds, Ho2Ti2O7 and Dy2Ti2O7, show a broad “dip” around 100 K, which is believed to be caused by the strong crystal field effect in those two compounds.

The goal of this research work is to further the understanding of crystal plasticity, particularly at reduced structural and material length scales. Fundamental understanding of plasticity is central to various challenges facing design and manufacturing of materials for structural and electronic device applications. The development of microstructurally tailored advanced metallic materials with enhanced mechanical properties that can withstand extremes in stress, strain, and temperature, will aid in increasing the efficiency of power generating systems by allowing them to work at higher temperatures and pressures. High specific strength materials can lead to low fuel consumption in transport vehicles. Experiments have shown that enhanced mechanical properties can be obtained in materials by constraining their size, microstructure (e.g. grain size), or both for various applications. For the successful design of these materials, it is necessary to have a thorough understanding of the influence of different length scales and evolving microstructure on the overall behavior. In this study, distinction is made between the effect of structural and material length scale on the mechanical behavior of materials. A length scale associated with an underlying physical mechanism influencing the mechanical behavior can overlap with either structural length scales or material length scales. If it overlaps with structural length scales, then the material is said to be dimensionally constrained. On the other hand, if it overlaps with material length scales, for example grain size, then the material is said to be microstructurally constrained. The objectives of this research work are: (1) to investigate scale and size effects due to dimensional constraints; (2) to investigate size effects due to microstructural constraints; and (3) to develop a size dependent hardening model through coarse graining of dislocation dynamics. A discrete dislocation dynamics (DDD) framework where the scale of analysis is intermediate between a fully discretized (e.g. atomistic) and fully continuum is used for this study. This mesoscale tool allows to address all the stated objectives of this study within a single framework. Within this framework, the effect of structural and the material length scales are naturally accounted for in the simulations and need not be specified in an ad hoc manner, as in some continuum models. It holds the promise of connecting the evolution of the defect microstructure to the effective response of the crystal. Further, it provides useful information to develop physically motivated continuum models to model size effects in materials. The contributions of this study are: (a) provides a new interpretation of mechanical size effect due to only dimensional constraint using DDD; (b) a development of an experimentally validated DDD simulation methodology to model Cu micropillars; (c) a coarse graining technique using DDD to develop a phenomenological model to capture size effect on strain hardening; and (d) a development of a DDD framework for polycrystals to investigate grain size effect on yield strength and strain hardening.

abstract: Recent studies of the occurrence of post-flutter limit cycle oscillations (LCO) of the F-16 have provided good support to the long-standing hypothesis that this phenomenon involves a nonlinear structural damping. A potential mechanism for the appearance of nonlinearity in the damping are the nonlinear geometric effects that arise when the deformations become large enough to exceed the linear regime. In this light, the focus of this investigation is first on extending nonlinear reduced order modeling (ROM) methods to include viscoelasticity which is introduced here through a linear Kelvin-Voigt model in the undeformed configuration. Proceeding with a Galerkin approach, the ROM governing equations of motion are obtained and are found to be of a generalized van der Pol-Duffing form with parameters depending on the structure and the chosen basis functions. An identification approach of the nonlinear damping parameters is next proposed which is applicable to structures modeled within commercial finite element software. The effects of this nonlinear damping mechanism on the post-flutter response is next analyzed on the Goland wing through time-marching of the aeroelastic equations comprising a rational fraction approximation of the linear aerodynamic forces. It is indeed found that the nonlinearity in the damping can stabilize the unstable aerodynamics and lead to finite amplitude limit cycle oscillations even when the stiffness related nonlinear geometric effects are neglected. The incorporation of these latter effects in the model is found to further decrease the amplitude of LCO even though the dominant bending motions do not seem to stiffen as the level of displacements is increased in static analyses. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2015

O objetivo principal deste trabalho é o desenvolvimento de uma formulação e sua implementação computacional para se analisar, via Método dos Elementos Finitos (MEF),o comportamento dinâmico não linear geométrico de sólidos bidimensionais. Trata-se o comportamento geometricamente não linear através de uma formulação posicional classificada como lagrangeana total com cinemática exata. No estudo do comportamento dinâmico utiliza-se um algoritmo de integração temporal baseado na família de integradores temporais de Newmark. Para a consideração do impacto adota-se uma técnica que utiliza como integrador temporal o algoritmo de Newmark, modificado de forma a garantir sua estabilização, e limita-se a posição de cada nó da estrutura que por ventura sofra impacto. O código computacional desenvolvido é validado através de exemplos tradicionais da literatura científica. Analisam-se exemplos com comportamento apenas não linear geométrico e não linear geométrico dinâmico com ou sem impacto / The main goal of this work is the development of a formulation and its computational implementation, based on the finite element method (FEM), to analyze the dynamic geometrically nonlinear behavior of bidimensional solids. The geometrically nonlinear behavior is treated with a positional formulation classified as total Lagrangean with exact kinematics. In the study of the dynamic behavior, a time integration algorithm based on the family of time integrators of Newmark is applied. In order to consider the impact, a technique based on the time integrator of Newmark, modified to assure its stabilization, is used. This technique limits the position of each node that suffers impact. The developed computational code is validated through benchmarks of scientific literature. Examples with static geometrically nonlinear and dynamic geometrically nonlinear behavior, with or without impact, are analyzed

The micro-electromechanical systems (MEMS) industry has grown incredibly fast over the past few years, due to the irresistible character and properties of MEMS. MEMS devices have been widely used in various fields such as aerospace, microelectronics, and the automobile industry. Increasing prominence is given to the development and research of MEMS; this is largely driven by the market requirements. Multi-physics coupled fields are often present in MEMS. This makes the modelling and analysis o such devices difficult and sometimes costly. The coupling between electrostatic and mechanical fields in MEMS is one of the most common and fundamental phenomena in MEMS; it is this configuration that is studied in this thesis. The following issues are addressed: 1. Due to the complexity in the structural geometry, as well as the difficulty to analyze the behaviour in the presence of coupled fields, simple analytical solutions are normally not available for MEMS. The finite element method (FEM) is therefore used to model electrostaticmechanical coupled MEMS. In this thesis, this avenue is followed. 2. In order to capture the configuration of the system accurately, with relatively little computational effort, a geometric non-linear mixed assumed stress element is developed and used in the FE analyses. It is shown that the developed geometrically non-linear mixed assumed stress element can produce an accuracy level comparable to that of the Q8 element, while the number of the degrees of freedom is that of the Q4 element. 3. Selected algorithms for solving highly non-linear coupled systems are evaluated. It is concluded that the simple, accurate and quadratic convergent Newton-Raphson algorithm remains best. To reduce the single most frustrating disadvantage of the Newton method, namely the computational cost of constructing the gradients, analytical gradients are evaluated and implemented. It is shown the CPU time is significantly reduced when the analytical gradients are used. 4. Finally, a practical engineering MEMS problem is studied. The developed geometric nonlinear mixed element is used to model the structural part of a fixed-fixed beam that experiences large axial stress due to an applied electrostatic force. The Newton method with analytical gradients is used to solve this geometrically nonlinear coupled MEMS problem. / Dissertation (MEng (Mechanical))--University of Pretoria, 2007. / Mechanical and Aeronautical Engineering / unrestricted

This study focuses on the response of flat unsymmetric laminates to an inplane compressive loading that for symmetric laminates are of sufficient magnitude to cause bifurcation buckling, postbuckling, and secondary buckling behavior. In particular, the purpose of this study is to investigate whether or not the concept of bifurcation buckling is applicable to unsymmetric laminates. Past work by other researchers has suggested that such a concept is applicable for certain boundary conditions. The study also has as an objective the determination of the response of flat unsymmetric laminates if bifurcation buckling does not occur. The finite-element program ABAQUS is used to obtain results, and a portion of the study is devoted to becoming familiar with the way ABAQUS handles such highly geometrically nonlinear problems, particularly for composite materials and particularly when instabilities and dynamic behavior are involved. Familiarity with the problem, in general, and with the use of ABAQUS, in particular, is partially gained by considering semi-infinite unsymmetrically laminated cross- and angle-ply plates, a one-dimensional problem that can be solve in closed form and with ABAQUS by making the appropriate approximations for the infinite geometry. In this portion of the study it is found that semi-infinite cross-ply laminates with clamped boundary conditions and semi-infinite angle-ply plates with simple-support boundary conditions remain flat under a compressive load until the load magnitude reaches a certain level, at which time the out-of-plane deflection become indeterminate, essentially an eigenvalue problem as encountered with classic bifurcation buckling analyses. Obviously, a linear analysis of such problems would not reveal this behavior and, in fact, there are other revealed significant differences between the predictions of linear and nonlinear analyses. Transversely-loaded and inplane-loaded finite isotropic plates are studied by way of semi-closed form Rayleigh-Ritz-based solutions and ABAQUS in a step to approaching the problem with unsymmetric laminates. A method to investigate the unloading behavior of postbuckled finite isotropic plates is developed that reveal multiple plate configurations in the postbuckled region of the response, and this method is then extended to the study of finite inplane-loaded unsymmetric laminates. To that end, two specific laminates, a symmetric and an unsymmetric cross-ply laminates, and a variety of boundary conditions are used to study the response of inplane-loaded unsymmetric laminates. The symmetric laminate is included to provide a familiar baseline case and a means of comparison. Plates with all four edges clamped and a variety of inplane boundary conditions are studied. Of course the symmetric cross-ply laminate exhibits bifurcation behavior, and when the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained, secondary buckling behavior occurs. For the unsymmetric cross-ply laminate, bifurcation buckling behavior does not occur unless the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained, or the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are free. If either of these conditions are not satisfied, the unsymmetric cross-ply laminate exhibits what could be termed 'near-bifurcation' behavior. In all cases rather complex behavior occurs for high levels of inplane load, including asymmetric postbuckling and secondary buckling behavior. For clamped loaded edges and simply-supported unloaded edges, bifurcation buckling behavior does not occur unless the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained. For this case, rather unusual asymmetric bifurcation and associated limit point behavior occur, as well as secondary buckling. This is a very interesting boundary condition case and is studied further for other unsymmetric cross-ply laminates, including the use of a Rayleigh-Ritz-based solution in attempt to quantify the problem parameters responsible for the asymmetric response. The overall results of the study have led to an increased understanding of the role of laminate asymmetry and boundary conditions on the potential for bifurcation behavior, on the response of the laminate for loads beyond that level. / Ph. D.

Ultra-fine grained materials with sub-micrometer grain size exhibit superior mechanical properties when compared with conventional fine-grained material as well as coarse-grained materials. Severe plastic deformation (SPD) techniques have been shown to be an effective way to modify the microstructure in order to improve the mechanical properties of the material. Crystalline materials require dislocations to accommodate plastic strain gradients and maintain lattice continuity. The lattice curvature exists due to the net dislocation that left behind in material during deformation. The characterization of such defects is important to understand deformation accumulation and the resulting mechanical properties of such materials. However, traditional techniques are limited. For example, the spatial resolution of EBSD is insufficient to study materials processed via SPD, while high dislocation densities make interpretations difficult using conventional diffraction contrast techniques in the TEM. A new technique, precession electron diffraction (PED) has gained recognition in the TEM community to solve the local crystallography, including both phase and orientation, of nanocrystalline structures under quasi-kinematical conditions. With the assistant of precession electron diffraction coupled ASTARÔ, the structure evolution of equal channel angular pressing processed commercial pure titanium is studied; this technique is also extended to two-phase titanium alloy (Ti-5553) to investigate the existence of anisotropic deformation behavior of the constituent alpha and beta phases.