The properties of composite materials in which spherical inclusions are embedded in a matrix of some kind, have been studied for many decades and many analytical models have been developed which measure these properties. There has been a steady progression in the complexity of models over the years, providing greater insight into the nature of these materials and improving the accuracy in the measurement of their properties. Some of the properties with which this thesis is concerned are, the elastic, thermal and electrical properties of such composites. The size of the spherical inclusion which acts as the reinforcing phase, has a major effect on the overall properties of composite materials. Once an inclusion is embedded into a matrix, a third region of different properties between the inclusion and matrix is known to develop which is called the interphase. It is well known in the composite community that the smaller the inclusion is, the larger the interphase region which develops around it. Therefore, with the introduction of nanoparticles as the preferred reinforcing phase for some composites, the interphase has a major effect on its properties. It is the aim of this thesis to consider the role of the interphase on the properties of composites by modeling it as an inhomogeneous region. There is much scientific evidence to support the fact that the interphase has an inhomogeneous nature and many papers throughout the thesis are cited which highlight this. By modeling the inhomogeneous properties by arbitrary mathematical functions, results are obtained for the various properties in terms of these general functions. Some specific profiles for the inhomogeneous region are considered for each property in order to demonstrate and test the models against some established results.

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.

This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A). In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration. In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  −→ A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.

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Permeable friction course (PFC), or new generation open-graded friction course (OGFC) mixtures, are hot mix asphalt (HMA) characterized by high total air voids (AV) content (minimum 18 %) as compared to the most commonly used dense-graded HMA. The high AV content confers to PFC mixtures both high permeability and noise reduction effectiveness. These characteristics and the high values of surface friction exhibited by PFC mixtures, as compared to dense-graded HMA, lead to improvements in safety and the environment, which make PFC one of the safest, cleanest, and quietest alternatives currently available for surface paving. The main objective of this study was improving the current PFC mix design method and construction practices in terms of compaction control. Corresponding results were integrated in an improved mix design method that is based on the guidelines of the current mix design method used by the Texas Department of Transportation. The improved mix design included modified computation of the inputs required to determine mixture density (or corresponding total AV content). These changes led to a proposed modification of the density specification for mix design from 78-82 % to 76-80 %. In addition, the water-accessible AV content was proposed as a surrogate of the total AV content for mix design and evaluation. The improved mix design method also includes verification of drainability, durability, and stone-on-stone contact. Computation of the expected value of permeability (E[k]) and measurement of the water flow value were recommended, respectively, for verification of drainability in the laboratory (using specimens compacted in the Superpave Gyratory Compactor (SGC)) and in the field. The Cantabro loss test conducted in both dry- and wet-conditions was suggested for assessing mixture durability. Improved criteria were proposed for verification of stone-on-stone contact based on the evaluation of the AV content in the coarse aggregate fraction of the mixture. In addition, comparison of the internal structure of field-and laboratory-compacted mixtures supported recommendation of a field-compaction control. Recommendations to reduce the horizontal heterogeneity of AV encountered in PFC specimens included using road cores with a minimum 152.4 mm diameter and coring SGC specimens from 152.4 to 101.6 mm in diameter

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.

In this study, it is aimed to investigate the mixed-mode fracture behavior of orthotropic functionally graded material (FGM) coatings bonded to a homogeneous substrate through a homogeneous bond-coat. Analytical and computational methods are used to solve the embedded cracking problems under mechanical or thermal loading conditions. It is assumed that the material property gradation of the FGM coating is in the thickness direction and cracks are parallel to the boundaries. The principal axes of orthotropy are parallel and perpendicular to the boundaries. A single embedded crack in the orthotropic FGM coating is investigated analytically assuming that crack surfaces are subjected to either uniform normal or uniform shear stresses. Using Fourier transformations, the problem is reduced to a couple of singular integral equations that are solved numerically to obtain the mixed-mode stress intensity factors, energy release rate and crack opening displacements. To investigate the analytically untractable problems without restrictive assumptions, a computational approach is employed. The adopted computational approach is based on finite element method and displacement correlation technique. Using the computational approach, fracture parameters are obtained considering single and periodic embedded cracking conditions in the orthotropic FGM coatings under mechanical or thermal loads. The results obtained in this study show the effects of material nonhomogeneity, material orthotropy and geometric variables on the fracture behavior of the structure.

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.

The effects of geometry parameters of graded-index polymer optical fiber (GI-POF) components on the coupling efficiency and signal mixed proportion are studied in this thesis. Simulation and experimental approaches are used to investigate the effects of light sources on the coupling efficiency of misalighment, Y-couplers and V-groove couplers. Two different light sources are employed in this study: Laser diode (LD) and vertical-cavity surface-emitting laser (VCSEL). The optimum coupling angle and refractive index of filler in the Y-coupler are studied with a light-emitting diode (LED) light source. A good agreement between the simulation and the experiment results is shown in this work. Furthermore, two V-groove array arrangements, i.e. the parallel V-groove array and the skew V-groove array, are proposed in this study to mix multi-light-sources. The optimum parameters of V-groove are designed to achieve the highest coupling efficiency. The performances of different V-groove array arrangements have also been demonstrated for multi-signal mixing.