The objective of this study is to determine stress intensity factors (SIFs) for a crack in a radially graded FGM layer on a substrate. Functionally graded coating with an edge crack perpendicular to the interface and a homogeneous substrate are bonded together. In order to make the problem analytically tractable, geometry is modeled as an FGM strip attached to a homogeneous layer. Introducing the elastic foundation underneath the homogeneous layer, an FGM coating on a thin walled cylinder can be modeled. At first, governing equations are obtained from stress displacement and equilibrium equations. Then using an assumed form of solution in terms of Fourier Transforms for displacements and applying the boundary conditions, a singular integral equation is obtained for the mode-I problem. Solving this singular integral equation numerically, stress intensity factors are obtained as functions of crack length, strip thicknesses and inhomogeneity parameter.

Este trabalho apresenta a aplicação do Método de Otimização Topológica (MOT) considerando restrição de tensão mecânica em dois problemas de Engenharia: o projeto de estruturas mecânicas sujeitas a restrição de tensão e o projeto da distribuição de material em estruturas constituídas por Materiais com Gradação Funcional (MsGF). O MOT é um método numérico capaz de fornecer de forma automática o leiaute básico de uma estrutura mecânica para que esta atenda a um dado requisito de projeto, como o limite sobre a máxima tensão mecânica no componente. Os MsGF são materiais cujas propriedades variam gradualmente com a posição. Este gradiente de propriedades é obtido através da variação contínua da microestrutura formada por dois materiais diferentes. Neste trabalho o MOT foi implementado utilizando o modelo de material Solid Isotropic Microstructure with Penalization (SIMP) e o campo de densidades foi parametrizado utilizando a abordagem Aproximação Contínua da Distribuição de Material (ACDM). O modelo de material e utilizado em conjunto com um localizador de tensões, de modo a representar as tensões nas regiões com densidade intermediária. O projeto de estruturas tradicionais através do MOT possui dois problemas centrais aqui tratados: o fenômeno das topologias singulares, que consiste na incapacidade do algoritmo de otimização de retirar material de certas regiões da estrutura, onde a tensão mecânica supera o limite de tensão quando os valores da densidade tendem a zero, e o problema do grande número de restrições envolvidas, pois que a tensão mecânica é uma grandeza local e deve ser restrita em todos os pontos da estrutura. Para tratar o primeiro problema é utilizado o conceito de relaxação. Para o segundo são utilizadas duas abordagens: uma é a substituição das restrições locais por uma restrição global e a outra é a aplicação do Método do Lagrangeano Aumentado. Ambas foram implementadas e aplicadas para o projeto de estruturas planas e axissimétricas. No projeto da distribuição de material em estruturas constituídas por MsGF é utilizado um modelo de material baseado na interpolação dos limites de Hashin-Shtrikman. A partir deste modelo as tensões em cada fase são obtidas a partir das matrizes localizadoras de tensão. Para tratar o fenômeno das topologias singulares é proposto um índice estimativo de falha, baseado nas tensões de von Mises em cada fase da microestrutura, que evita tal problema. O grande número de restrições é tratado através da restrição global de tensão. Em ambos os problemas as formulações são apresentadas e sua eficiência é discutida através de exemplos numéricos. / This work presents the Topology Optimization Method (TOM) with stress constraint applied to two Engineering problems: the design of mechanical structures subjected to stress constraint and the design of material distribution in structures made of Functionally Graded Materials (FGMs). The TOM is a numerical method capable of synthesizing the basic layout of a mechanical structure accomplishing to a given design requirement, for example the maximum stress in the structure. The FGMs are materials with spatially varying properties, which are obtained through a continuum change of the microstructuremade of two different materials. In this work, the TOM was implemented with Solid Isotropic Microstructure with Penalization (SIMP) material model and the density field was parameterized with the Continuous Approximations of Material Distribution. To obtain the intermediate density stresses, the material model is applied together with a stress localization matrix. The design of mechanical structures through the TOM has two major problems: the singular topology phenomenon, which is characterized by the optimization algorithm impossibility of removing material from certain regions, where the stress overpasses the limiting stress when the density goes to zero, and the large number of constraints, once the stress is a local value that must be constrained everywhere in the structure. To deal with the first problem, it is applied the \"-realaxation concept, and for the second one two approaches are considered: one is to change the local stress constraint into a global stress constraint and the other is to apply the Augmented Lagrangian Method. Both approaches were implemented and applied to the design of plane and axisymmetric structures. In the design of material distribution in structures made of FGMs a material model based on Hashin-Shtrikman bounds is applied. From this model, stresses in each phase are obtained by the stress localization matrix. To deal with the singular topology phenomenon it is proposed a modified von Mises failure criteria index that avoids such problem. A global stress constraint is applied to deal with the large number of constraints. In both problems formulations are presented and their performance are discussed through numerical examples.

The Wavelet Finite Element Method (WFEM) involves combining the versatile wavelet analysis with the classical Finite Element Method (FEM) by utilizing the wavelet scaling functions as interpolating functions; providing an alternative to the conventional polynomial interpolation functions used in classical FEM. Wavelet analysis as a tool applied in WFEM has grown in popularity over the past decade and a half and the WFEM has demonstrated potential prowess to overcome some difficulties and limitations of FEM. This is particular for problems with regions of the solution domain where the gradient of the field variables are expected to vary fast or suddenly, leading to higher computational costs and/or inaccurate results. The properties of some of the various wavelet families such as compact support, multiresolution analysis (MRA), vanishing moments and the “two-scale” relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) presented in this study, conceptually based on previous works, are constructed using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet families. These two wavelet families possess the desired properties of multiresolution, compact support, the “two scale” relations and vanishing moments. The rod, beam and planar bar WFEs are used to study structural static and dynamic problems (moving load) via numerical examples. The dynamic analysis of functionally graded materials (FGMs) is further carried out through a new modified wavelet based finite element formulation using the Daubechies and BSWI wavelets, tailored for such classes of composite materials that have their properties varying spatially. Consequently, a modified algorithm of the multiscale Daubechies connection coefficients used in the formulation of the FGM elemental matrices and load vectors in wavelet space is presented and implemented in the formulation of the WFEs. The approach allows for the computation of the integral of the products of the Daubechies functions, and/or their derivatives, for different Daubechies function orders. The effects of varying the material distribution of a functionally graded (FG) beam on the natural frequency and dynamic response when subjected to a moving load for different velocity profiles are analysed. The dynamic responses of a FG beam resting on a viscoelastic foundation are also analysed for different material distributions, velocity and viscous damping profiles. The approximate solutions of the WFEM converge to the exact solution when the order and/or multiresolution scale of the WFE are increased. The results demonstrate that the Daubechies and B-spline based WFE solutions are highly accurate and require less number of elements than FEM due to the multiresolution property of WFEM. Furthermore, the applied moving load velocities and viscous damping influence the effects of varying the material distribution of FG beams on the dynamic response. Additional aspects of WFEM such as, the effect of altering the layout of the WFE and selection of the order of wavelet families to analyse static problems, are also presented in this study.

No description available.

In this study the plane linear elastic problem of a functionally graded layer which contains periodic cracks is considered. The main objective of this study is to determine the thermal stress intensity factors for edge cracks. In order to find an analytic solution, Young&rsquo / s modulus and thermal conductivity are assumed to be varying exponentially across the thickness, whereas Poisson ratio and thermal diffusivity are taken as constant. First, one dimensional transient and steady state conduction problems are solved (heat flux being across the thickness) to determine the temperature distribution and the thermal stresses in a crack free layer. Then, the thermal stress distributions at the locations of the cracks are applied as crack surface tractions in the elasticity problem to find the stress intensity factors. By defining an appropriate auxiliary variable, elasticity problem is reduced to a singular integral equation, which is solved numerically. The influence of such parameters as the grading, crack length and crack period on the stress intensity factors is investigated.

No description available.

The main objective of this study is to evaluate mixed mode stress intensity factors for inclined embedded cracks in functionally graded materials. Fracture analysis of inclined cracks requires the calculation of both Mode I and Mode II stress intensity factors ( I K , II K ). In this study, k J -integral is used to calculate I K and II K . Equivalent domain integral approach is utilized to evaluate the k J - integral around the crack tip. The present study aims at developing a finite element model to study inclined crack problems in graded media under thermomechanical loading. A two dimensional finite element model is developed for inclined cracks located in a functionally graded medium. Structural and thermal problems are solved using two dimensional finite elements namely 8- noded triangles. Material properties are sampled directly at the integration points of the elements, as required by the numerical integral evaluation. The main results of the study are the stress intensity factors at the crack tip for functionally graded materials subjected to thermomechanical loading.

In this thesis, the Generalized Finite Difference (GFD) method is applied to analyze the elastoplastic deformation behavior of a long functionally graded (FGM) tube subjected to internal pressure. First, the method is explained in detail by considering the elastic response of a rotating FGM tube. Then, the pressurized tube problem is treated. A long FGM tube with fixed ends (axially constrained ends) is taken into consideration. The two cases in which the modulus of elasticity only and both the modulus of elasticity and the yield limit are graded properties are analyzed. The plastic model here is based on incremental theory of plasticity, Tresca&#039 / s yield criterion and its associated flow rule. The numerical results are compared to those of analytical ones. Furthermore, the elastic response of an FGM tube with free ends is studied considering graded modulus of elasticity and Poisson&#039 / s ratio. The results of these computations are compared to those of Shooting solutions. In the light of analyses and comparisons stated above, the applicability of the GFD method to the solution of similar problems is discussed. It is observed that, in purely elastic deformations the accuracy of the method is sufficient. However, in case of elastic-plastic deformations, the discrepancies between numerical and analytical results may increase in determining plastic displacements. It is also noteworthy that the predictions for tubes with two graded properties, i. e. the modulus of elasticity and the yield limit, turn out to be better than those with one graded property in this regard.

An elastic FGM layer bonded to a semi-infinite linear elastic, isotropic, homogeneous half plane is considered. The half plane contains periodic cracks perpendicular to the interface. Mechanical loading is applied through crack surface pressure, resulting in a mode I crack problem. The plane elasticity problem described above is formulated by using Fourier transforms and Fourier series. A singular integral equation is obtained for the auxiliary variable, namely derivative of the crack surface displacement. Solution is obtained, and stress intensity factors are calculated for various values of crack period, crack length, crack location, layer thickness and material gradation.

Les propriétés idéales d’un matériau de blindage sont la combinaison d’une extrême dureté pour casserles noyaux des projectiles et d’une grande ductilité pour résister à l’impact et arrêter les fragments du projectile. Or cettecombinaison de propriétés est incompatible avec un matériau unique. Pour pallier ce problème, les concepteurs de blindageassocient un matériau dur (céramique) à un matériau ductile (métal). Une autre solution serait de réaliser un matériauprésentant un gradient de propriétés mécaniques : dans le cas présent, d’une très grande dureté de la face avant à une grandeductilité de la face arrière. Les technologies non conventionnelles de frittage telles que le Spark Plasma Sintering (SPS)permettent d’assembler ou de fritter/assembler des matériaux aux caractéristiques aussi différentes et complémentaires. Ils’agit donc d’étudier les conditions d’assemblage ou de cofrittage de tels matériaux (dans le cas présent, Al2O3 et Ti) ainsique l’influence de la microstructure résultante de l’ensemble sur sa performance balistique.La première partie de ce travail a porté sur la caractérisation de l’alumine et du titane. Cinq poudres d’alumines ontété étudiées d’un point de vue comportement au frittage. Trois d’entre elles sont retenues en raison de leurs microstructuresintéressantes, proches en termes de densité et de taille de grains. Ces alumines ont été caractérisées mécaniquement (dureté,ténacité, résistance à la rupture) et balistiquement pour n’en garder qu’une dans la deuxième partie du travail. Le titane, frittédans les mêmes conditions que l’alumine, a montré qu’il n’avait malheureusement pas les propriétés attendues (absence deductilité).La seconde partie du travail a montré que l’obtention de MGFs sains à partir de Al2O3 et Ti uniquement est délicate,que ce soit avec un intercalaire sous forme de monocouche ou de multicouche. La forte affinité du titane avec l’oxygène(formation d’oxyde ou en insertion) et le carbone (formant des carbures), ainsi que sa réactivité avec l’alumine (produisantdes intermétalliques) rend le MGF fragile et incapable d’accommoder les contraintes résiduelles d’élaboration. L’insertiond’une faible proportion de nickel (plus ductile et moins réactif vis-à-vis de l’oxygène que le titane) dans les composites apermis d’obtenir des MGFs sains, dont le comportement balistique a pu être évalué. / The objective is to improve ballistic performance of armors. A perfect armor combines ductility to resistto the impact and high hardness to stop projectile’s fragments. However, such an association of properties is inconsistent witha single material. The solution is to perform a functionally graded material (FGM) with a ductile metal at the back side of thesample and a hard ceramic on the top side. Non-conventional technologies like Spark Plasma Sintering allow joining orsintering all types of materials with different and additional properties. Furthermore, with this technique, high heating ratescan be achieved, limiting grain growth and resulting in a fine microstructure. The goal is to study joining conditions or cosinteringof such materials (in this case, Al2O3 and Ti), as well as the resulting microstructure on the ballistic efficiency.The first part of the study focused on the characterization of alumina and titanium. Five powders of alumina werestudied from a sintering point of view. Three of which were selected because of their interesting microstructures, close indensities and grain sizes. These ceramics have been characterized mechanically (hardness, toughness and strength) andballistically. One of them is adopted to realize FGM. Titanium, sintered with the same conditions, unfortunately, doesn’t haveexpected properties (absence of ductility).The second part of the work showed that the preparation of FGM without cracks from Al2O3 and Ti only ischallenging, with an interlayer with one or more layers. The strong affinity of Ti with oxygen (formation of oxides orinsertion) with C (forming carbides) and its reactivity with alumina (forming intermetallics) make the FGM brittle and enablethe release of residual stresses during the process. By adding a low amount of nickel (more ductile and less reactive withoxygen and titanium) in composites, FGMs almost without cracks were obtained. The latter were evaluated ballistically.