Depuis 1984, les matériaux à gradients de fonction (FGM) permettent de former une barrière thermique et réduire les fortes discontinuités des propriétés entre deux matériaux de nature différente. Ces multi-matériaux, qui consistent en une variation intentionnelle de la composition chimique entrainant par conséquent une modification des propriétés microstructurales, chimiques, mécaniques et thermiques, permettent de lisser la distribution des contraintes thermiques. L’élaboration in situ de ces alliages sur mesure est rendu possible grâce à l’utilisation de procédés de fabrication additive tel que le procédé par dépôt de poudres DED-CLAD®. Ces procédés connaissent un essor considérable depuis les années 1980 et sont idéaux dans la fabrication de FGM. Dans le cadre de cette thèse CIFRE, des développements techniques ont été effectués pour adapter le procédé DED-CLAD® et permettre la réalisation de FGM. Grâce à plusieurs collaborations industrielles, une étude complète a été réalisée sur les alliages titane-molybdène et titane-niobium. Ces alliages permettent dans le premier cas de réaliser des pièces résistantes à de fortes sollicitations thermiques (secteur spatial), et dans le second cas d’associer les propriétés mécaniques et la biocompatibilité (secteur biomédical). L’originalité de cette thèse repose sur l’étude d’un gradient complet, c’est-à-dire que l’ajout en élément d’alliage varie de 0% à 100%. En effet, les études reportées dans la littérature ne font pas mentions des alliages titane-matériaux réfractaire pour des taux élevés en élément réfractaire. Les analyses microstructurale (DRX, structure cristallographique par EBSD, microstructure), chimique (EDS) et mécanique (microdureté, tests de traction et essais d’indentation instrumentée) ont mis en évidence une évolution des propriétés le long du gradients de composition. La caractérisation mécanique des échantillons par indentation instrumentée s’est par ailleurs révélée particulièrement pertinente dans les cas de ces multi-matériaux / Since 1984, the Functionally Graded Material (FGM) allow to create a thermal barrier and to reduce the strong discontinuities of properties between two materials of different composition. These multimaterials,whose consist of an intentional variation in the chemical composition and, consequently, modify the microstructural, chemical, mechanical and thermal properties, lead to a smooth distribution of the thermal stress. The in-situ development of these custom-made alloys is made possible by the use of additive manufacturing processes such as the DED-CLAD® powder deposition process. These processes have grown substantially since the 1980s and are optimal for the manufacture of FGM. During this industrial thesis, technical developments have been carried out to adapt the DED-CLAD® process and to allow the manufacturing of FGM. Thanks to two industrial collaborations, a full study was carried out on titanium-molybdenum and titanium-niobium alloys. These alloys make it possible, in the first case, to produce parts resistant to strong thermal stress (space sector), and in the second case to combine mechanical properties and biocompatibility (biomedical sector). The originality of this thesis rests on the study of a complete gradient, that is the addition in alloy element varied from 0% to 100%. In fact, studies reported in the literature do not mention titanium-refractory material for high levels of refractory element. Microstructural (XRD, crystallographic analysis by EBSD technique), chemical (EDS) and mechanical (microhardness, tensile test and instrumented indentation) analyses revealed an evolution of the properties along the chemical gradient. The mechanical characterization of the sample by instrumented indentation has also proved particularly relevant in the case of these multi-materials

The main aim of this study is to utilize a Jk-integral based computational method in order to calculate crack tip parameters for orthotropic functionally graded materials (FGMs). The crack is subjected to mixed mode thermal loading. Mixed mode thermal fracture analysis requires the calculation of mode-I and mode-II stress intensity factors (KI ,KII ). In addition to stress intensity factors, energy release rate and T-stress are calculated by means of Jk-integral. Jk-integral is defined as a line integral over a vanishingly small curve. Since it is difficult to deal with a line integral on a vanishing curve , Jk-integral is converted to a domain independent form containing area and line integrals by the help of plane thermoelasticity constitutive relations. Steady-state temperature distribution profiles in FGMs and the components of the Jk-integral are computed by means of the finite element method. In both thermal and structural analyses, finite element models that possess graded isoparametric elements are created in the general purpose finite element analysis software ANSYS. In the formulation of Jk-integral, all required engineering material properties are assumed to possess continuous spatial variations through the functionally graded medium. The numerical results are compared to the results obtained from Displacement Correlation Technique (DCT). The domain independence of Jk-integral is also demonstrated. The results obtained in this study show the effects of crack location and material property gradation profiles on stress intensity factors, energy release rate and T-stress.

Functionally graded materials (FGMs) are composites with continuous material property variations. The distinct interfaces between the reinforcement and the matrix in classical composites are potential damage initiation sites. The concept of FGM aims at avoiding the material mismatch at the interfaces. Functionally graded materials originated from the need for a material that has high-toughness at very high operating temperatures that occur in rocket nozzles and aeroplane engines. One of the early applications of graded materials can be thus found in thermal barrier coatings of gas turbine blades. Recent applications of FGMs include optoelectronics, ballistic impact resistance structures, wear resistant coatings and others. Although the manufacturing and applications of FGMs are well developed the basic mechanics of failure is not well understood, which is important in developing engineering design methodologies. Modern day design practice uses the concepts of fracture mechanics and the fracture properties of graded materials is not well understood. Most studies in the literature have assumed that the material response of the bulk functionally graded material to be elastic even though the constituents are nominally ductile. Some asymptotic analysis available in the literature have described the effect of ductility on the fracture parameters. However, these analysis are not complete in the sense that they have some undetermined constants. The present thesis aims at performing whole-field finite element (FE) simulations of a crack embedded in a ductile functionally graded material subjected to an anti-plane shear (mode-3) loading. A J2-deformation theory based power-law hardening nonlinear material response is assumed. The material property variation is assumed to be in the radial-direction (r-FGM), tangential to the crack (x-FGM), normal to the crack plane (y-FGM) and also at an arbitrary angle to the crack-plane (xy-FGM). Yet another power law described the material property variation. The competition between the indices of the hardening and material property variation is understood by performing a parametric analysis by varying both systematically. Our results indicate that the first most singular term of the asymptotic series remains unaffected. For some values of the material property variation index, the second asymptotic term is affected. The semi-closed form solutions available in the literature were unable to decipher the relative range of dominance of the first and second terms. From the present whole-field FEM analysis were able to extract this relative range of dominance. Our results indicate the range of dominance of the first term is least for FGMs when the material property variation is in the direction to the crack (x-FGM), and it is more for y-FGM.

Functionally graded materials (FGMs) are composites that have continuously varying material properties, which eliminate undesirable stress concentrations that might otherwise occur in layered composites. The concept of inhomogeneously varying properties is observed in nature; examples include bones, teeth, shells and timber. Modern engineering applications of FGMs include thermal barrier coatings, wear-resistant coatings, biomedical implants and MEMS devices. Syntactic foams, particle filled nano-composites are examples of inhomogeneous materials of current interest. Analyses and experiments available in the literature have focused on characterizing the inhomogeneous material modulus and density variations. Common techniques employed are nano-indentation and wave propagation studies. There are a few fracture mechanics analyses and experiments available in the literature; most of which are devoted to measuring the fracture toughness of graded materials. A few fracture analyses of graded materials are devoted to deriving asymptotic stress, strain and displacement fields around stationary and steadily growing cracks in inhomogeneous materials. Only a few studies exist that deal with understanding the effect of material property inhomogeneity on the spontaneous crack propagation. In the present thesis the effect of material property inhomogeneity on the dynamic fracture mechanics of cracks in FGMs is described. Numerical analysis of the elastodynamic initial boundary value problem is performed using a spectral scheme. Spectral scheme is a special numerical technique developed to simulate spontaneous, planar crack propagation in a variety of materials. The method is numerically efficient as it can be implemented on parallel machines with ease. The numerical scheme is versatile and can handle any state-and rate-dependent traction-separation laws (cohesive zone models) or frictional laws. Spectral scheme has successfully been used in simulating intersonic crack propagation, earthquake slip dynamics and also direct silicon wafer bonding process used in realizing 3D MEMS structures. In the present work, the spectral formulation accounts for the inhomogeneous variation in the material wave speeds in the medium. The effect of inhomogeneity on spontaneous crack propagation due to in-plane mixed-mode loading is also addressed here. A parametric study has been performed by varying the inhomogeneity length scales independently in the top and bottom half-spaces. The effect of inhomogeneity in shear wave speed on the dynamic stress intensity factors (SIFs) of a crack propagating in a quasi-steady-state along the interface between the two functionally graded half-spaces is studied. A symmetric hardening FGM offers the maximum fracture resistance, while the fracture resistance is minimum for a symmetric softening FGM. Our simulation shows that increasing the inhomogeneity in the wave speed leads to eliminate the overshoot in the dynamic stress intensity factor. The magnitude of the steady-state (long-time) SIF increases indicating an increase in the fracture resistance. The effect of the inhomogeneous wave speed on the mode-3 crack propagation characteristics is demonstrated by taking snapshots of the crack opening at a time interval. The magnitude of the crack sliding displacement is found to increase with increase in the inhomogeneity. The effect of the material property inhomogeneity on the mode-1 crack propagation is simulated to track the crack opening displacements. The inhomogeneity is assumed to be symmetric about the weak-plane. Our spectral scheme developed here for functionally graded material with exponential variation in the material properties is capable of simulating independent bimaterial combinations. When the graded material becomes progressively stiffer and denser (hardening), the crack opening displacement in reduced, indicating an increase in the fracture resistance. On the other hand, for the softening FGMs the crack opening displacement increases indicating a reduction in fracture toughness. It is noted that the cohesive fracture resistance on the weak-plane remains same in all the FGMs.

Les procédés existant pour la fabrication de sous-couches et de systèmes barrière thermique pour les aubes mobiles des turbomachines sont complexes, onéreux, et, de leur reproductibilité dépend la durée de vie de ces systèmes. Cette étude montre la faisabilité d'obtention de sous-couches γ-Ni + γ'-Ni3Al enrichies en platine et de systèmes barrière thermique complets, par l'utilisation du procédé de Spark Plasma Sintering (SPS). Les paramètres du procédé SPS (pression, durée, température et nombre de paliers) ont été ajustés afin de fabriquer les différents types de systèmes. Des améliorations ont été apportées à l'outillage, en vue notamment de mieux appréhender la température de la pièce lors du cycle thermique et, d'empêcher la formation de carbures. Les propriétés microstructurales des revêtements obtenus sont caractérisées et mises en relation avec les paramètres d'élaboration. De plus, les propriétés d'usage (cinétique d'oxydation, résistance au cyclage thermique et à la corrosion par les aluminosilicates fondus, CMAS, conductivité thermique) ont été évaluées. Un large domaine de composition de sous-couches a pu être exploré, notamment par l'addition, par pulvérisation cathodique, d'éléments réactifs (Hf, Y, Si) et d'autres éléments (Ag, Au, Cu) ayant un effet sur le domaine de stabilité de la phase γ'. La possibilité de réaliser des systèmes barrière thermique mono et bi-couches céramiques en une seule étape par le procédé SPS est également démontrée. Un mode d'endommagement spécifique des systèmes barrière thermique élaborés par SPS a été mis en évidence lors d'essais de cyclage thermique et interprété avec l'aide de simulations numériques par éléments finis. Les nombreuses compositions et architectures réalisées au cours de cette étude, sont un encouragement à poursuivre les améliorations apportées au procédé pour l'obtention de systèmes plus complexes et plus fiables. / Fabrication of bond coatings and thermal barrier coating systems for aircraft engine turbine blades and vanes, rely on complex and costly processes, and the lifetime of the systems highly depends on their reproducibility. This work demonstrates the feasibility of platinum rich γ-Ni + γ'-Ni3Al bond coatings and complete thermal barrier coating systems using the Spark Plasma Sintering (SPS) process. Processing parameters (pressure, number, temperature and duration of dwells) have been finely tuned to fabricate the different types of systems. Some necessary adjustments to the equipments have been made to prevent the specimens from pollution, and to better control the samples temperature, and are discussed. The microstructure characteristics and their relation with process parameters have been investigated. Properties such as oxidation kinetics, thermal cycling resistance and CMAS (molten aluminosilicate) corrosion are evaluated. Physical vapor deposition have been used for bond coat doping with reactive elements (Hf, Y, Si) and elements that extend the γ' phase stability domain (Au, Ag, Cu). The possibility to fabricate complete thermal barrier coating systems with a mono or a bi-ceramic top coat layer in a single step is demonstrated. SPS thermal barrier coating systems exhibit a singular spalling behavior during thermal cycling. A finite-element numerical model has been developed and allows its understanding. The versatility of the SPS process has allowed the fabrication of a large number of bond coat compositions and thermal barrier coatings architectures. There is a strong incentive in developing this process for fabricating more reliable and competitive systems.

O objetivo precípuo deste estudo é a implementação computacional de modelos constitutivos elásticos e elastoplásticos para materiais com gradação funcional em regime de grandes deslocamentos e elevadas deformações. Para simular numericamente um problema estrutural, são empregados aqui elementos finitos sólidos (tetraédrico e hexaédrico) com ordem de aproximação polinomial qualquer. Grandezas da Mecânica Não Linear do Contínuo, como deformação e tensão, são utilizadas na formulação deste estudo. Para reproduzir os grandes deslocamentos, é empregada a análise não linear geométrica. A descrição adotada aqui é a Lagrangiana total, e o equilíbrio da estrutura é expresso pelo Princípio da Mínima Energia Potencial Total. Com relação à resposta elástica do material, são usadas leis constitutivas hiperelásticas, nas quais a relação tensão-deformação é obtida a partir de um potencial escalar. O comportamento elastoplástico do material é definido pela decomposição da deformação nas parcelas elástica e plástica, pelo critério de plastificação de von-Mises, pela lei de fluxo associativa, pelas condições de consistência e de complementaridade, pelo parâmetro de encruamento isotrópico e pelo tensor das tensões inversas, relacionado ao encruamento cinemático. Duas formulações elastoplásticas são utilizadas aqui: a de Green-Naghdi, na qual a deformação é decomposta de forma aditiva; e a hiperelastoplástica, em que o gradiente é decomposto de forma multiplicativa. É empregado também o conceito de material com gradação funcional (GF), a qual é definida como a variação gradual (contínua e suave) das propriedades constitutivas do material. A solução numérica do equilíbrio de forças é feita via método iterativo de Newton-Raphson. Para satisfazer o critério de plastificação, são utilizadas as estratégias de previsão elástica, e de correção plástica via algoritmos de retorno. Basicamente foram desenvolvidos cinco programas computacionais: o gerador automático das funções de forma; o gerador de malhas de elementos finitos sólidos; o código para análise de materiais em regime elástico; o código para análise de materiais em regime elastoplástico; e o programa de pós-processamento. Além desses, o aluno teve contato com os programas EPIM3D e DD3IMP ao longo de seu estágio de doutorado na Universidade de Coimbra (Portugal). Os programas EPIM3D e DD3IMP são empregados para analisar, respectivamente, materiais em regime elastoplástico, e processos de conformação de metais. Para o problema da barra sob tração uniaxial uniforme, são descritas equações e soluções analíticas para materiais homogêneos e com GF em regime elastoplástico. Para reduzir o tempo de simulação, foi empregada a programação em paralelo. De acordo com os resultados das simulações numéricas, as principais conclusões são: o refinamento da malha de elementos finitos melhora a precisão dos resultados para materiais em regimes elástico e elastoplástico; as formulações elastoplásticas de Green-Naghdi e hiperelastoplástica parecem ser equivalentes para pequenas deformações; a formulação hiperelastoplástica é equivalente ao modelo mecânico dos programas EPIM3D e DD3IMP para materiais em regime de pequenas deformações elásticas; foram constatados ganhos significativos, em termos de tempo de simulação, com a paralelização dos códigos computacionais de análise estrutural; e os programas desenvolvidos são capazes de simular - com precisão - problemas complexos, como a membrana de Cook e o cilindro fino transversalmente tracionado. / The main objective of this study is the computational implementation of elastic and elastoplastic constitutive models for functionally graded materials in large deformation regime. In order to numerically simulate a structural problem, the finite elements used are solids (tetrahedric and hexahedric) of any order of approximation. Entities from Nonlinear Continnum Mechanics, as strain and stress, are used in the present formulation. To reproduce the finite displacements, the geometrically nonlinear analysis is employed. The description adopted here is the total Lagrangian, and the structural equilibrium is expressed by means of the Principal of Minimum Total Potential Energy. Regarding the elastic material response, hyperelastic constitutive laws are used, in which the stress-strain relation is obtained from a scalar potential. The elastoplastic material behavior is defined by the strain decomposition in the elastic and plastic parts, by the von-Mises yield criterion, by the associative flow law, by the consistency and complementarity conditions, by the isotropic hardening parameter, and by the backstress tensor, related to the kinematic hardening. Two elastoplastic formulations are used here: the Green-Naghdi one, in which the strain is additively decomposed; and the hyperelastoplasticiy, in which the gradient is multiplicatively decomposed. The concept of functionally graded (FG) material, in which the constitutive properties vary gradually (continuous and smoothly), is also used. The numerical solution of the forces equilibrium is obtained via Newton-Raphson iterative procedure. In order to satisfy the yield criterion, the strategies of elastic prediction and plastic correction (via return algorithms) are used. Basically, five computer codes have been developed: the automatic shape functions generator; the solid mesh generator; the code for analysis of materials in the elastic regime; the code for analysis of materials in the elastoplastic regime; and the post-processor. Besides these, the student had contact with the programs EPIM3D and DD3IMP during his doctoral stage in the University of Coimbra (Portugal). The programs EPIM3D and DD3IMP are employed to analyze, respectively, materials in the elastoplastic regime, and sheet-metal forming processes. For the problem of the bar under uniform uniaxial tension, equations and analytical solutions are described for homogeneous and FG materials. To reduce the simulation time, the parallel programming has been employed. According to the numerical simulation results, the main conclusions are: the results accuracy is improved with mesh refinement for materials in the elastic and elastoplastic regimes; the Green-Naghdi elastoplastic formulation and the hyperelastoplasticity appear to be equivalent for small strains; the hyperelastoplastic formulation is equivalent to the mechanical model of the programs EPIM3D and DD3IMP for materials the small elastic strains regime; simulation time reduction has been obtained with the parallelization of the computer codes for structural analysis; the developed programs are capable of simulating, precisely, complex problems, such as the Cook\'s membrane and the pulled thin cylinder.

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No. of bitstreams: 2 Dissertação - Roger Otávio Pires Montes - 2015.pdf: 8169771 bytes, checksum: e580ffb280dfa5136f41ab38cf0aec4e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-05-25 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This master’s thesis analyses the free and forced nonlinear vibrations of a simply supported functionally graded cylindrical shell which the material’s properties are described by gradient’s law along the shell’s thickness. The nonlinear equations of motion are obtained using nonlinear theories Donnell and Sanders, where the field displacements and field strain of nonlinear Donnell’s shallow shell theory is obtained as a simplification of the nonlinear Sanders’s formulation. The effects of the internal fluid, that is incompressible, irrotational and inviscid and it has been described as a potential velocity to consider the fluid-structure interaction, and the influence of a thermal field in the nonlinear dynamic behavior of the functionally graded cylindrical shell will be investigated. It is developed a low-dimensional model, wherein the shell of the system equilibrium equations is solved by an analytical procedure, which yields the longitudinal and circumferential displacement field as a function of transverse displacement, satisfying the boundary conditions problem. The determination of transverse displacement is obtained by the perturbation techiniques, which enables the achievement of the main nonlinear modes that should be present in the displacement fields of the functionally grade cylindrical shell. To analyze the nonlinear free vibration, it is applied the Galerkin-Urabe method to obtain the system of non-linear algebraic equations, and then resolved by the Newton-Raphson method. The results show the influence of functional gradation, geometry, the effect of the internal fluid, considering a fluid-filled shell, and the thermal action of the nonlinear free vibrations of the shell by the frequency-amplitude relations. Finally, a parametric analysis to study the nonlinear forced vibrations of the cylindrical shell subjected to a harmonic loading side for some geometric relations is conducted. In this case the system of ordinary differential equations of second order in time is obtained from the application of the Galerkin method and integrated over time from the Runge-Kutta fourth order method. The results evaluates the influence of the internal fluid and the thermal effects in the nonlinear oscillation of functionally graded cylindrical shell, using the resonances’ curves, the basins’ attraction, time responses and the phase portraits. / Nesta dissertação são analisadas as vibrações, livres e forçadas, não lineares de uma casca cilíndrica simplesmente apoiada feita com um material com gradação funcional, que as propriedades dos materiais constituintes são descritas por determinadas leis de gradação ao longo da espessura. As equações não lineares de movimento são obtidas utilizando-se as teorias não lineares de Donnell e de Sanders, sendo que os campos de deslocamentos e as deformações referentes à teoria não linear de Donnell para cascas abatidas podem ser obtidos como uma simplificação da formulação não linear de Sanders. Serão investigados os efeitos da presença de um fluido interno, incompressível, não viscoso e irrotacional, sendo descrito a partir de um potencial de velocidade, considerando a interação fluido-estrutura, além da influência de um campo térmico no comportamento dinâmico não linear da casca cilíndrica com gradação funcional. É desenvolvido um modelo de baixa dimensão, em que o sistema de equações de equilíbrio da casca é resolvido através de um procedimento analítico, o qual permite obter os campos de deslocamento axial e circunferencial em função dos deslocamentos transversais, além de atender as condições de contorno do problema. A determinação dos deslocamentos transversais é feita a partir do método da perturbação, o qual possibilita a obtenção dos principais modos não lineares que devem estar presentes nos campos de deslocamentos da casca cilíndrica. Para analisar as vibrações livres não lineares, aplica-se o método de Galerkin-Urabe para se obter o sistema de equações algébricas não lineares, sendo, em seguida, resolvido a partir do método de Newton-Raphson. Os resultados mostram a influência da gradação funcional, da geometria, do efeito do fluido interno, considerando uma casca totalmente preenchida, e da ação térmica nas vibrações livres não lineares da casca por meio das relações frequência-amplitude. Por fim, é feita uma análise paramétrica das vibrações forçadas não lineares da casca cilíndrica submetida a um carregamento lateral harmônico para algumas relações geométricas. Neste caso o sistema de equações diferenciais ordinárias de segunda ordem no tempo é obtido a partir da aplicação do método de Galerkin e integrado ao longo do tempo a partir do método de Runge-Kutta de quarta ordem. Da mesma forma avalia-se a influência do fluido interno e dos efeitos térmicos nas oscilações não lineares da casca cilíndrica com gradação funcional, utilizando-se as curvas de ressonância, as bacias de atração, as respostas no tempo e os planos fase.

[pt] O Método Híbrido dos Elementos de Contorno foi introduzido em 1987. Desde então, o método foi aplicado com sucesso a diferentes tipos de problemas de elasticidade e potencial, inclusive problemas dependentes do tempo. Esta Tese apresenta uma tentativa para consolidar a formulação simplificada do Método Híbrido dos Elementos de Contorno para a análise geral da resposta dinâmica de sistemas elásticos. Baseado em um método de superposição modal, um conjunto acoplado de equações diferenciais de movimento de alta ordem é transformado em um conjunto desacoplado de equações diferenciais de segunda ordem que podem ser integradas normalmente por meio de procedimentos conhecidos. Este método também é uma extensão de uma formulação introduzida por J. S. Przemieniecki, para a análise de vibração livre de barras e elementos de viga baseada em uma série de freqüências. O método trata estruturas restringidas, com condições iniciais não homogêneas dadas como valores nodais e também através de campos prescritos no domínio, assim como forças genéricas de massa (além de forças inerciais). Esta tese também tem por objetivo estabelecer a consolidação conceitual da aplicação da versão simplificada do Método Híbrido dos Elementos de Contorno a materiais com gradação funcional. São obtidas várias classes de soluções fundamentais para problemas de potencial dependentes e independentes do tempo, para a análise no domínio da freqüência combinada com uma técnica avançada (mencionada acima) de superposição modal baseada em séries de freqüências. Com isso, consegue- se a utilização de integrais somente no contorno mesmo para materiais heterogêneos. Apresenta-se um grande número de resultados numéricos de problemas bidimensionais, para validação dos desenvolvimentos teóricos realizados. / [en] The hybrid boundary element method was introduced in 1987. Since then, the method has been successfully applied to different problems of elasticity and potential, including time-dependent problems. This thesis presents an attempt to consolidate a formulation for the general analysis of the dynamic response of elastic systems. Based on a mode- superposition technique, a set of coupled, higher-order differential equations of motion is transformed into a set of uncoupled second order differential equations, which may be integrated by means of standard procedures. The first motivation for these theoretical developments is the hybrid boundary element method, a generalization of T. H. H. Pian`s previous achievements for finite elements, which, requiring only boundary integrals, yields a stiffness matrix for arbitrary domain shapes and any number of degrees of freedom. The method is also an extension of a formulation introduced by J. S. Przemieniecki, for the free vibration analysis of bar and beam elements based on a power series of frequencies. It handles constrained and unconstrained structures, non-homogeneous initial conditions given as nodal values as well as prescribed domain fields and general domain forces (other than inertial forces). This thesis also focuses on establishing the conceptual framework for applying the simplified version of the hybrid boundary element method to functionally graded materials. Several classes of fundamental solutions for steady-state and time-dependent problems of potential are derived for a frequency-domain analysis combined with an advanced mode superposition technique based on a power series of frequencies. Thus, the boundary-only feature of the method is preserved even with such spatially varying material property.Several numerical examples are given in terms of an efficient patch test for irregular bounded, unbounded and multiply connected regions submitted to high gradients.

El presente trabajo, tiene como objetivo el estudio de la vibración libre de vigas Timoshenko aplicados a materiales funcionalmente graduados; esto se resuelve utilizando el método de elementos finitos (MEF) , al implementarlo en el software MATLAB, con el fin de obtener las frecuencias fundamentales y las gráficas de los modos de vibración para cada caso. Se define el campo de desplazamientos según la teoría Timoshenko considerando tres variables fundamentales; asimismo, se utiliza el Principio de Hamilton para obtener las vibraciones libres del elemento. Para el desarrollo de las relaciones constitutivas se usa la La ley de potencia, el cual describe como varían las propiedades de un material heterogéneo e isotrópico FGM (FGM por sus siglas en inglés) en el peralte de la viga. Los resultados obtenidos se compararon con otros estudios de la literatura validados por revistas como SCOPUS y SPRINGER, demostrando que el modelo es bastante preciso y satisfactorio. / The present work aims to study the free vibration of Timoshenko beams applied to functionally graduated materials; This is solved using the finite element method (FEM), when implemented in the MATLAB software, in order to obtain the fundamental frequencies and the graphs of the vibration modes for each case. The field of displacements is defined according to the Timoshenko theory considering three fundamental variables; likewise, the Hamilton Principle is used to obtain the free vibrations of the element. For the development of the constitutive relationships, the power law is used, which describes how the properties of a heterogeneous and isotropic material FGM vary in the heightn of the beam. The results obtained were compared with other literature studies validated by journals such as SCOPUS and SPRINGER, showing that the model is quite accurate and satisfactory. / Tesis

<p>Naturally-occurring biological materials with stiff mineralized reinforcement embedded in a ductile matrix are commonly known to achieve excellent balance between stiffness, strength and ductility. Interestingly, nature offers a broad diversity of architectural motifs, exemplify the multitude of ways in which exceptional mechanical properties can be achieved. Such diversity is the source of bio-inspiration and its translation to synthetic material systems. In particular, the helicoid and the “brick and mortar” architectured materials are two key architectural motifs we are going to study and to synthesize new bio-inspired materials. </p> <p>Due to geometry mismatch(misorientation) and incompatibilities of mechanical properties between fiber and matrix materials, it is acknowledged that misoriented stiff fibers would rotate in compliant matrix beneath uniaxial deformation. However, the role of fiber reorientation inside the flexible matrix of helicoid composites on their mechanical behaviors have not yet been extensively investigated. In the present project, fiber reorientation values of single misoriented laminae, mono-balanced laminates and helicoid architectures under uniaxial tensile are calculated and compared. In the present work, we introduce a Discontinuous Fiber Helicoid (DFH) composite inspired by both the helicoid microstructure in the cuticle of mantis shrimp and the nacreous architecture of the red abalone shell. We employ 3D printed specimens, analytical models and finite element models to analyze and quantify in-plane fiber reorientation in helicoid architectures with different geometrical features. We also introduce additional architectures, i.e., single unidirectional lamina and mono-balanced architectures, for comparison purposes. Compared with associated mono-balanced architectures, helicoid architectures exhibit less fiber reorientation values and lower values of strain stiffening. The explanation for this difference is addressed in terms of the measured in-plane deformation, due to uniaxial tensile of the laminae, correlated to lamina misorientation with respect to the loading direction and lay-up sequence.</p> <p>In addition to fiber, rod-like, reinforced laminate, platelet reinforced composite materials, “brick and mortar” architectures, are going to be discussed as well, since it can provide in-plane isotropic behavior on elastic modulus that helicoid architecture can offer as well, but with different geometries of reinforcement. Previous “brick and mortar” models available in the literature have provided insightful information on how these structures promote certain mechanisms that lead to significant improvement in toughness without sacrificing strength. In this work, we present a detailed comparative analysis that looks at the three-dimensional geometries of the platelet-like and rod-like structures. However, most of these previous analyses have been focused on two-dimensional representations. We 3D print and test rod-like and tablet-like architectures and analyze the results employing a computational and analytical micromechanical model under a dimensional analysis framework. In particular, we focus on the stiffness, strength and toughness of the resulting structures. It is revealed that besides volume fraction and aspect ratio of reinforcement, the effective shear and tension area in the matrix governs the mechanical behavior as well. In turns, this leads to the conclusion that rod-like microstructures exhibit better performance than tablet-like microstructures when the architecture is subjected to uniaxial load. However, rod-like microstructures tend to be much weaker and brittle in the transverse direction. On the other hand, tablet-like architectures tend to be a much better choice for situations where biaxial load is expected.</p> <p>Through varying the geometry of reinforcement and changing the orientation of reinforcement, different architectural motifs can promote in-plane mechanical properties, such as strain stiffening under uniaxial tensile, strength and toughness under biaxial tensile loading. On the other hand, the various out-of-plane orientation of the reinforcement leads to functionally graded effective indentation stiffness. The external layer of nacre shell is composed of calcite prisms with graded orientation from surface to interior. This orientation gradient leads to functionally graded Young’s modulus, which is confirmed to have higher fracture resistance than homogenous materials under mode I fracture loading act.</p> <p>Similar as graded prism orientation in calcite layer of nacre, the helicoid architecture found in nature exhibits gradients on geometrical parameters as well. The pitch distance of helicoid architecture is found to be functionally graded through the thickness of biological materials, including the dactyl club of mantis shrimp and the fish scale of coelacanth. This can be partially explained by the long-term evolution and selection of living organisms to create high performance biological materials from limited physical, chemical and geometrical elements. This naturally “design” procedure can provide us a spectrum of design motifs on architectural materials. </p> <p>In the present work, linear gradient on pitch distance of helicoid architectures, denoted by functionally graded helicoid (FGH), is chose to be the initial pathway to understand the functionality of graded pitch distance, associated with changing pitch angle. Three-point bending on short beam and low-velocity impact tests are employed in FEA to analyze the mechanical properties of composite materials simultaneously. Both static(three-point bending) and dynamic(low-velocity impact) tests reveal that FGH with pitch angle increasing from surface to interior can provide multiple superior properties at the same time, such as peak load and toughness, while the helicoid architectures with constant pitch angle can only provide one competitive property at one time. Specifically, helicoid architectures with smaller pitch angle, such as 15-degree, show higher values on toughness, but less competitive peak load under static three-point bending loading condition, while helicoid architectures with middle pitch angle, larger than or equal to 22.5-degree and smaller than 45-degree, exhibit less value of toughness, but higher peak load. The explanation on this trend and the benefits of FGH is addressed by analyzing the transverse shear stresses distribution through the thickness in FEA, combined with analytical prediction. In low-velocity impact tests, the projected delamination area of helicoid architectures is observed to increase when the pitch angle is decreasing. Besides, laminates with specific pitch angles, such as 45-degree, classical quasi-isotropic laminate, 60-degree, specific angle ply, and 90-degree, cross-ply, are designed to compare with helicoid architectures and FGH.</p>