ABSTRACT CONTACT MECHANICS OF GRADED MATERIALS WITH TWODIMENSIONAL MATERIAL PROPERTY VARIATIONS G&ouml / kay, Kemal M.S., Department of Mechanical Engineering Supervisor: Asst. Prof. Dr. Serkan Dag September 2005, 62 pages Ceramic layers used as protective coatings in tribological applications are known to be prone to cracking and debonding due to their brittle nature. Recent experiments with functionally graded ceramics however show that these material systems are particularly useful in enhancing the resistance of a surface to tribological damage. This improved behavior is attributed to the influence of the material property gradation on the stress distribution that develops at the contacting surfaces. The main interest in the present study is in the contact mechanics of a functionally graded surface with a two &ndash / dimensional spatial variation in the modulus of elasticity. Poisson&rsquo / s ratio is assumed to be constant due to its insignificant effect on the contact stress distribution [30]. In the formulation of the problem it is assumed that the functionally graded surface is in frictional sliding contact with a rigid flat stamp. Using elasticity theory and semi-infinite plane approximation for the graded medium, the problem is reduced to a singular integral equation of the second kind. Integral equation is solved numerically by expanding the unknown contact stress distribution into a series of Jacobi polynomials and using suitable collocation points. The developed method is validated by providing comparisons to a closed form solution derived for homogeneous materials. Main numerical results consist of the effects of the material nonhomogeneity parameters, coefficient of friction and stamp size and location on the contact stress distribution.

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.

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High Speed Machining (HSM) is associated with higher cutting velocities and table feedrates and higher material removal rate, lower cutting forces in contrast to conventional machining. HSM can be undertaken dry or near dry and hence it is considered as environmentally friendly machining in relation to the use of cutting fluids. A key challenge in HSM is that, the thermal loads generated during the cutting process can be a major driver of thermally activated wear mechanism and hence affect machining performance. The ability of cutting tools to act as thermal barrier can be a highly desirable property for dry and HSM. Recently, research work has been conducted on laboratory based coated cutting tools to model and understand the fraction of heat that enters the cutting tool. These studies have shown the potential for TiN and TiAlN coated tools in reducing heat partition to the cutting tool when compared to uncoated tools. This PhD extended this work to modelling and characterising the heat partition for new generation commercial coated cutting tools considering tools from major insert manufactures. For this study commercial coated carbide tools were classified into two groups. In one group were coatings uniformly applied on both rake and flank faces of the insert (SERIES). The second group were tools that had different top coats for the rake and flank faces (Functionally Graded). This concept of functional grading is used to tailor the coating selection to the conditions that exist on a tool face. Moreover, the issue of restricted chip contact was modelled and clarified in terms of its impact on heat partition. This chip breaker design is of particular importance to inserts used for machining ductile materials. Thus the PhD has applied research methods to industrial cutting tools and helped elucidate the important aspects relating to the design, layout and selection of multilayer coatings. The heat partition was quantified by using a combined Finite Element (FE) and experimental approach. This methodology was applied by taking into consideration the appropriate friction phenomena during HSM i.e. sticking and sliding. A restricted contact length with groove profile geometry was considered for the application of heat load in the FE model. Orthogonal and external turning of AISI/SAE 4140 medium carbon alloy steel was conducted over a wide range of cutting speeds. An infrared thermal imaging camera was used to measure cutting temperatures. The results show that the layout of the coating can significantly affect the heat distribution into the cutting tool, specifically; the top coat can alter the friction conditions between the tool-chip contact. The distribution of heat (heat partition) into the cutting tool insert with the thickest layer of Al2O3 as a top coating is the lowest in the entire range of cutting speeds tested i.e. 10.5% at lower cutting speed and reduced to 3.4% at highest cutting speed. Investigations were also conducted to quantify the contribution of heat from the primary and secondary deformation zones using a combination of finite element modelling, analytical modelling and experimental data. The results deduced that the primary deformation zone heat source contributes 9.1% (on average) to the heat partition into the cutting tool. The contribution of the Thesis should be of interest to those who design, manufacture and coat cutting tools. It defines heat partition values for commercial coated carbide tools, assesses the requirements for multilayer design of thermally insulating cutting tools, the selection of coating top layer coats and the role of contact phenomenon on heat partition in dry and HSM of steels.

The challenges in the deposition of dissimilar materials are mainly related to the large differences in the physical and chemical properties of the deposited and substrate materials. These differences readily cause residual stresses and intermetallic phases. This has led to the development of functionally graded materials which exhibit spatial variation in composition. Laser direct metal deposition due to its flexibility, it offers wide variety of dissimilar and functionally graded materials deposition. Despite considerable advances in process optimization, there is a rather limited understanding of the role of metallurgical factors in the laser deposition of dissimilar and functionally graded alloys. The aim of this work is to understand and explain mechanisms occurring in diode laser deposition of dissimilar materials and functionally graded materials. The first part of this work addressed diode laser deposition of Inconel 718 nickel alloy to Ti-6Al-4V titanium alloy. Here, the effect of laser pulse parameters and powder mass flow rates on the stress formation and cracking has evaluated by experiment and numerical techniques. Results showed that the clad thickness was an important factor affecting the cracking behaviour. In the second part of this study, an image analysis technique has been developed to measure the surface disturbance and the melt pool cross section size during laser direct metal deposition of Inconel 718 on a Ti-6Al-4V thin wall. It was noted that under tested conditions the overall melt pool area increased with the increase in powder flow rate; the powder carrier gas flow rates also seemed to play important roles in determining the melt pool size. In the third part of this study, a parametric study on the development of Inconel 718 and Stainless steel 316L continuously graded structure has been carried out. Results suggested that microstructure and other mechanical properties can be selectively controlled across the deposited wall. The results presented in this dissertation can be used as a metallurgical basis for further development of dissimilar and functionally graded manufacturing using LDMD technique, guiding future manufacturing engineers to produce structurally sound and microstructurally desirable laser deposited samples.

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El texto completo de este trabajo no está disponible en el Repositorio Académico UPC por restricciones de la casa editorial donde ha sido publicado. / In this research, an improved first order formulation is presented to study the critical buckling load in functionally graded beams. The formulation has five independent variables in comparison with the Timoshenko theory that has three. The Trefftz criterion is utilized with incremental and fundamental states to define the stability analysis. Virtual work statements are derived for the finite element model where the field variables are interpolated by Lagrange polynomials. The numerical results are compared and verified with other formulations found in literature. Parametric studies are also carried out for buckling behavior due to different slenderness ratios, power-law indices and boundary conditions. Applications of the model to functionally graded materials show the validity of the present approach.

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