Herein work is researching extremes asymptotic of Pareto random values. Here is analyzing geometrically maximum (minimum) stability tasks, also asymptotically tasks, when succession value is geometrical and geometrically stability of lower extremes. Aim of this work is to check if Pareto distribution values are stable maximum and minimum distributions and to continue researches in the area of lower extremes structures. It was proved that maximum (minimum) distribution (when ) is geometrically stable maximum (minimum) distribution, while others – asymptotically k-stable. When , maximum (minimum) distribution is asymptotically stable, only maximum distribution is also Pareto distribution, but with the displacement, while other - asymptotically k-stable.

This thesis presents a geometrically exact curved beam theory, with the assumption that the cross-section remains rigid, and its finite element formulation/implementation. The theory provides a theoretical view and an exact and efficient means to handle a large range of nonlinear beam problems. A geometrically exact curved/twisted beam theory, which assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal reference frames starting from a 3-D beam theory. The relevant engineering strain measures at any material point on the current beam cross-section with an initial curvature correction term, which are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The Green strains and Eulerian strains are explicitly represented in terms of the engineering strain measures while other stresses, such as the Cauchy stresses and second Piola-Kirchhoff stresses, are explicitly represented in terms of the first Piola-Kirchhoff stresses and engineering strains. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. For the finite element formulation and implementation of the curved beam theory, some basic concepts associated with finite rotations and their parametrizations are first summarized. In terms of a generalized vector-like parametrization of finite rotations under spatial descriptions (i.e., in spatial forms), a unified formulation is given for the virtual work equations that leads to the load residual and tangent stiffness operators. With a proper explanation, the case of the non-vectorial parametrization can be recovered if the incremental rotation is parametrized using the incremental rotation vector. As an example for static problems, taking advantage of the simplicity in formulation and clear classical meanings of rotations and moments, the non-vectorial parametrization is applied to implement a four-noded 3-D curved beam element, in which the compound rotation is represented by the unit quaternion and the incremental rotation is parametrized using the incremental rotation vector. Conventional Lagrangian interpolation functions are adopted to approximate both the reference curve and incremental rotation of the deformed beam. Reduced integration is used to overcome locking problems. The finite element equations are developed for static structural analyses, including deformations, stress resultants/couples, and linearized/nonlinear bifurcation buckling, as well as post-buckling analyses of arches subjected to conservative and non-conservative loads. Several examples are used to test the formulation and the Fortran implementation of the element. / Master of Science

Recentemente, observamos na comunidade científica o despertar de um interesse pelo estudo de lantanídeos com base em titânio; neste ínterim, as amostras Sm2Ti2O7 e Nd2Ti2O7 se destacam por terem sido pouco exploradas, especialmente a baixas temperaturas. Análises cristalográficas de monocristais de Sm2Ti2O7 e Nd2Ti2O7 nos permitiram confirmar inicialmente características de rede como a geometria cúbica e monoclínica, respectivamente, enquanto as análises magnéticas e calorimétricas forneceram a temperatura de transição, até então inédita no caso do Nd2Ti2O7 (aproximadamente T=0,62K), além de ordenamento do tipo antiferromagnético e presença de anisotropia a altas temperaturas, na amostra de Sm2Ti2O7, e por toda a extensão de temperaturas trabalhadas, na amostra de Nd2Ti2O7. Por fim, identificamos também a entropia esperada para spins do tipo Ising em ambas as amostras. / Recently the scientific community has shown interest for titanium lanthanides, with the titanates Sm2Ti2O7 and Nd2Ti2O7 being poorly approached, specially at low temperatures. Crystallographic analysis were performed on the single crystal samples of Sm2Ti2O7 and Nd2Ti2O7, which allowed us to confirm its expected cubic and monoclinic structures, respectively; Magnetic and calorimetric data displayed the transition temperature, not previously seen in case of Nd2Ti2O7, of T=0.35K and T=0.62K, respectively, and the antiferromagnetic behaviour of both samples; Anisotropy was observed at high temperatures on the Sm2Ti2O7 data and along all the temperature range studied for the Nd2Ti2O7 sample. Finally, we have identified the entropy of Ising spins on both samples.

The total wind power capacity installed in the world has substantially grown during the last few years, mainly due to the increasing number of horizontal axis wind turbines (HAWT). Consequently, a big effort was employed to increase HAWT\'s power capacity, which is directly associated to the size of blades. Then, novel designs of blades may lead to very fexible structures, susceptive to large deformation, not only during extreme events, but also for operational conditions. In this context, this thesis aims to compare two geometrically nonlinear structural modeling approaches that handle large deformation of blade structures: 3D geometrically-exact beam and shell finite element models. Regarding the beam model, due to geometric complexity of typical cross-sections of wind turbine blades it is adopted a theory that allows creation of arbitrary multicellular cross-sections. Two typical blade geometries are tested, and comparisons between the models are done in statics and dynamics, always inducing large deformation and exploring the accuracy limits of beam models, when compared to shells. Results showed that the beam and shell models present very similar behavior, except when violations occur on the beam formulation hypothesis, such as when shell local buckling phenomena takes place. / A capacidade total de energia eólica instalada no mundo cresceu substancialmente nos últimos anos, principalmente devido ao número crescente de turbinas eólicas de eixo horizontal. Consequentemente, um grande esforço foi empregado com o intuito de aumentar a capacidade de produção das turbinas eólicas, que está diretamente associada ao tamanho das pás. Assim, surgiram projetos inovadores quanto à concepção de pás de turbinas eólicas levando a estruturas bastante flexíveis, susceptíveis a grandes deslocamentos, não apenas em eventos extremos, mas também em condições normais de operação. Nesse contexto, a presente dissertação tem por objetivo comparar duas abordagens de modelos estruturais geometricamente não-lineares capazes de lidar com grandes deslocamentos de pás de turbinas eólicas: elementos finitos geometricamente exatos 3D de vigas e cascas. Em relação ao modelo de viga, devido à complexidade geométrica das seções transversais típicas de pás de turbinas eólicas, adota-se uma teoria que permite a criação de seções transversais arbitrárias multicelulares. Duas geometrias de pás s~ao testadas e comparações entre os modelos s~ao feitas em análises estáticas e dinâmicas, sempre induzindo grandes deslocamentos e explorando os limites de precisão do modelo de viga, quando comparado ao modelo de cascas. Os resultados indicam que os modelos de viga e casca apresentam comportamento muito similar, exceto quando ocorrem violações em hipóteses do modelo de viga, tal como quando ocorre flambagem local do modelo de casca.

Recentemente, observamos na comunidade científica o despertar de um interesse pelo estudo de lantanídeos com base em titânio; neste ínterim, as amostras Sm2Ti2O7 e Nd2Ti2O7 se destacam por terem sido pouco exploradas, especialmente a baixas temperaturas. Análises cristalográficas de monocristais de Sm2Ti2O7 e Nd2Ti2O7 nos permitiram confirmar inicialmente características de rede como a geometria cúbica e monoclínica, respectivamente, enquanto as análises magnéticas e calorimétricas forneceram a temperatura de transição, até então inédita no caso do Nd2Ti2O7 (aproximadamente T=0,62K), além de ordenamento do tipo antiferromagnético e presença de anisotropia a altas temperaturas, na amostra de Sm2Ti2O7, e por toda a extensão de temperaturas trabalhadas, na amostra de Nd2Ti2O7. Por fim, identificamos também a entropia esperada para spins do tipo Ising em ambas as amostras. / Recently the scientific community has shown interest for titanium lanthanides, with the titanates Sm2Ti2O7 and Nd2Ti2O7 being poorly approached, specially at low temperatures. Crystallographic analysis were performed on the single crystal samples of Sm2Ti2O7 and Nd2Ti2O7, which allowed us to confirm its expected cubic and monoclinic structures, respectively; Magnetic and calorimetric data displayed the transition temperature, not previously seen in case of Nd2Ti2O7, of T=0.35K and T=0.62K, respectively, and the antiferromagnetic behaviour of both samples; Anisotropy was observed at high temperatures on the Sm2Ti2O7 data and along all the temperature range studied for the Nd2Ti2O7 sample. Finally, we have identified the entropy of Ising spins on both samples.

The total wind power capacity installed in the world has substantially grown during the last few years, mainly due to the increasing number of horizontal axis wind turbines (HAWT). Consequently, a big effort was employed to increase HAWT\'s power capacity, which is directly associated to the size of blades. Then, novel designs of blades may lead to very fexible structures, susceptive to large deformation, not only during extreme events, but also for operational conditions. In this context, this thesis aims to compare two geometrically nonlinear structural modeling approaches that handle large deformation of blade structures: 3D geometrically-exact beam and shell finite element models. Regarding the beam model, due to geometric complexity of typical cross-sections of wind turbine blades it is adopted a theory that allows creation of arbitrary multicellular cross-sections. Two typical blade geometries are tested, and comparisons between the models are done in statics and dynamics, always inducing large deformation and exploring the accuracy limits of beam models, when compared to shells. Results showed that the beam and shell models present very similar behavior, except when violations occur on the beam formulation hypothesis, such as when shell local buckling phenomena takes place. / A capacidade total de energia eólica instalada no mundo cresceu substancialmente nos últimos anos, principalmente devido ao número crescente de turbinas eólicas de eixo horizontal. Consequentemente, um grande esforço foi empregado com o intuito de aumentar a capacidade de produção das turbinas eólicas, que está diretamente associada ao tamanho das pás. Assim, surgiram projetos inovadores quanto à concepção de pás de turbinas eólicas levando a estruturas bastante flexíveis, susceptíveis a grandes deslocamentos, não apenas em eventos extremos, mas também em condições normais de operação. Nesse contexto, a presente dissertação tem por objetivo comparar duas abordagens de modelos estruturais geometricamente não-lineares capazes de lidar com grandes deslocamentos de pás de turbinas eólicas: elementos finitos geometricamente exatos 3D de vigas e cascas. Em relação ao modelo de viga, devido à complexidade geométrica das seções transversais típicas de pás de turbinas eólicas, adota-se uma teoria que permite a criação de seções transversais arbitrárias multicelulares. Duas geometrias de pás s~ao testadas e comparações entre os modelos s~ao feitas em análises estáticas e dinâmicas, sempre induzindo grandes deslocamentos e explorando os limites de precisão do modelo de viga, quando comparado ao modelo de cascas. Os resultados indicam que os modelos de viga e casca apresentam comportamento muito similar, exceto quando ocorrem violações em hipóteses do modelo de viga, tal como quando ocorre flambagem local do modelo de casca.

Composite laminates are increasingly being used as primary load bearing members in<br />structures.  However, because of the directional dependence of the properties of<br />composite materials, additional failure modes appear that are absent in<br />homogeneous, isotropic materials.  Therefore, a stress analysis of a composite<br />laminate is not complete without an accurate representation of the transverse<br />(out-of-plane) stresses.<br /><br />Stress recovery is a common method to estimate the transverse stresses from a<br />plate or shell analysis.  This dissertation extends stress recovery to problems<br />in which geometric nonlinearities, in the sense of von K\\\'rm\\\'{a}n,  are<br />important.  The current work presents a less complex formulation for the stress<br />recovery procedure for plate geometries, compared with other implementations,<br />and results in a post-processing procedure which can be applied to data from<br />any plate analyses; analytical or numerical methods, resulting in continuous or<br />discretized data.<br /><br />Recovered transverse stress results are presented for a variety of<br />geometrically nonlinear example problems: a semi-infinite plate subjected to<br />quasi-static transverse and shear loading, and a finite plate subjected to both<br />quasi-static and dynamic transverse loading.  For all cases, the corresponding<br />results from a fully three-dimensional stress analysis are shown alongside the<br />distributions from the stress recovery procedure.  Good agreement is observed<br />between the stresses obtained from each method for the cases considered.<br />Discussion is included regarding the applicability and accuracy of the<br />technique to varying plate geometries and varying degrees of nonlinearity, as<br />well as the viability of the procedure in replacing a three-dimensional<br />analysis in regard to the time required to obtain a solution.<br /><br />The proposed geometrically nonlinear stress recovery procedure results in<br />estimations for transverse stresses which show good correlation to the<br />three-dimensional finite element solutions.  The procedure is accurate for<br />quasi-static and dynamic loading cases and proves to be a viable replacement<br />for more computationally expensive analyses. / Ph. D.

This work discusses the deformations of various unsymmetric composite panels due to thermal and mechanical loads. Chapter 2 focuses on the warpage of large unsymmetric curved composite panels due manufacturing anomalies. These panels are subjected to a temperature change of -280°F to simulate the cooling from the autoclave cure temperature. Sixteen layer quasi-isotropic, axial-stiff, and circumferentially-stiff laminates are considered. These panels are intended to be symmetric laminates, but are slightly unsymmetric due to the manufacturing anomalies. Rayleigh-Ritz and finite-element models are developed to predict the deformations. Initially, to serve as a basis for comparison, warpage effects due to orthotropic thermal expansion properties in perfect panels are investigated and are found to produce deformations not captured in two-dimensional theories. This is followed by the investigation of the effects of ply misalignments. Ply misalignments of 5° are incorporated into the laminate, one layer at a time, to produce unsymmetric laminates. It is found that ply misalignments produce warpages much larger than those induced by orthotropic thermal expansion properties. Next, unsymmetric laminates resulting from ply thickness variations are investigated. Layers 10% thicker than nominal are incorporated into the laminate, one layer at a time, while the remaining layers are of uniform thickness. Due to the change in fiber volume fraction of the thicker layers, corresponding material properties are modified to reflect this change. The results show that ply thickness variations cause warpages of about 25-50% of those induced by ply misalignments. Finally, warpage of panels due to nonuniform cooling due to inplane thermal gradients during cure is investigated. A thermal gradient of 0.1°F/in. is used to construct six inplane distributions. It is found that the warpages induced by thermal gradients are very small. The warpages are negligible with respect to those induced by ply thickness variations or ply misalignments. Deformations induced by thermal gradients depend primarily on the magnitude of the thermal gradient, but not on the pattern of distribution. Overall, ply misalignments cause the most warpage, followed by ply thickness variations. Important variables for these imperfections are, the through-thickness location of the imperfections, the orientation of the layer containing the imperfections, and the lamination sequence. All cases show that geometric nonlinearities are important to accurately predict the deformations induced by these imperfections. Chapter 3 discusses the deformations of composite plates that are intentionally fabricated to be unsymmetric. Such plates, if flat, might be considered in applications where bending-stretching coupling effects can be used to advantage. It is assumed the laminates are cured at an elevated temperature and then cooled 280°F. Significant deformations result because of the high level of asymmetry in the laminate construction. Accordingly, geometric nonlinearities are included in the models. Four cross-ply laminates and three angle-ply laminates are considered. Four-term and 14-term Rayleigh-Ritz models are developed, together with finite-element models to model the deformations. Actual specimens were constructed and the deformations measured to compare with predictions. The results show that agreement between predictions and the experimental results are good. The 14-term Rayleigh-Ritz model is found to be the most useful due to its ability to find multiple solutions, its physical basis, and computational efficiency. Chapter 4 discusses the deformations of initially flat aluminum, symmetric, and unsymmetric composite plates due to axial endshortening under various boundary conditions, the aluminum and symmetric plates serving as a baseline. Seven plates are considered, each with three boundary condition combinations, namely, clamped ends and sides (CL-CL), clamped ends with simply-supported sides (CL-SS), and simply-supported ends and sides (SS-SS). Generally, the boundary conditions play a key role in the deformation characteristics of the plates. The aluminum and symmetric cross-ply plates have no out-of-plane deformations until classic buckling, or primary instability, then each exhibits two stable solutions. Each also exhibits secondary instability that results in two stable solutions. The symmetric laminates show less of a dependence on the boundary conditions compared to the unsymmetric laminates. Unsymmetric laminates show a mixture of characteristics. Some cases exhibit primary instability, other cases do not. Some cases exhibit secondary instability, while some case do not. The unsymmetric cross-ply laminates have only one stable solution after secondary buckling, while most other laminates and boundary condition combinations have two stable solutions. It is interesting to note that for the unbalanced unsymmetric [302/90/0]2T laminate, the boundary conditions controlled the sign of the out-of-plane deflection from the onset of axial endshortening. Generally speaking, the CL-CL cases carry the most load, followed by the CL-SS, and then the SS-SS cases. Like all the problems discussed in Chapter 2 and 3, geometric nonlinearities are found to be important for this case as well. / Ph. D.

Sandwich structures constitute basic components of advanced supersonic/hypersonic flight and launch vehicles. These advanced flight vehicles operate in hostile environments consisting of high temperature, moisture, and pressure fields. As a result, these structures are exposed to large lateral pressures, large compressive edge loads, and high temperature gradients which can create large stresses and strains within the structure and can produce the instability of the structure. This creates the need for a better understanding of the behavior of these structures under these complex loading conditions. Moreover, a better understanding of the load carrying capacity of sandwich structures constitutes an essential step towards a more rational design and exploitation of these constructions. In order to address these issues, a comprehensive geometrically non-linear theory of doubly curved sandwich structures constructed of anisotropic laminated face sheets with an orthotropic core under various loadings for simply supported edge conditions is developed. The effects of the radii of curvature, initial geometric imperfections, pressure, uniaxial compressive edge loads, biaxial edge loading consisting of compressive/tensile edge loads, and thermal loads will be analyzed. The effect of the structural tailoring of the facesheets upon the load carrying capacity of the structure under these various loading conditions are analyzed. In addition, the movability/immovability of the unloaded edges and the end-shortening are examined. To pursue this study, two different formulations of the theory are developed. One of these formulations is referred to as the mixed formulation, While the second formulation is referred to as the displacement formulation. Several results are presented encompassing buckling, postbuckling, and stress/strain analysis in conjunction with the application of the structural tailoring technique. The great effects of this technique are explored. Moreover, comparisons with the available theoretical and experimental results are presented and good agreements are reported. / Ph. D.

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.