It is generally accepted that a linear stability theory, together with a slowly diverging base flow, can describe many of the characteristics of coherent structures in free shears flow. In this dissertation we model these two-dimensional instability waves as they travel in a slightly inhomogeneous steady and viscous unstable base flow. These unstable and inviscid wave packets are analysed using linear stability theory. The analysis is performed by separating the physical flow in two parts. In the first part, the instability waves are evolving in a parallel mixing layer and their solution serves of initial conditions for the second part of the flow. The parallel flow analysis leads to the receptivity of the flow to both pulse-type and periodic excitations. This part of the study is done by solving the initial-value problem completely and studying its long-time behaviour, which is a wave packet. We then repeat the same analysis with some modifications and arrive at the receptivity of the flow for sinusoidal excitations. We find that a shear layer is very receptive to high-frequency disturbances that are generated near the center line of the layer. The second part of the solution is concerned with the evolution of the wave packets on longer space-time scales which are associated with non-parallel effects arising from the spreading of the mixing layer. The solution in this part of the physical flow is handled by extending Whitham's kinematic wave theory, and the ray equations for instability waves are derived for physical and propagation spaces using a WKB J expansion. Our high-frequency ansatz also leads us to the derivation of a very simple complex amplitude equation. While the rays obtained represent characteristics in the complex plane along which the complex frequency of our disturbances is conserved (steady base flow), the amplitude equation expresses the conservation of the volume integrals of a complex wave action density subject to a certain flux and a source term. The amplitude equation was rendered easily tractable due to a transformation of our dependent variables and their practical projections on the cross and propagation spaces. Different methods, (steepest descent, ray-tracing, and fully numerical solution) are used to solve the ray equations, and comparisons are made among them. The results presented are obtained for the piecewise linear profile of Rayleigh and the general tanh profile. The very good agreement among all the methods of solution reveals the validity of the method of characteristics in the complex plane, (ie complex rays). Finally we perform some calculations for spatially varying shear layers and and study their implications in the development of spatial instability modes. We discover that when starting with a convectively unstable base velocity profile it is possilble to interrupt the development of spatial instability modes by allowing the base velocity profile to vary slowly and become absolutely unstable. However the reverse is not true. That is to say that in a base flow that is initially absolutely unstable, one does not observe spatial modes, even after the base flow is permitted to assume slowly a convectively unstable profile.

The effect of gravity modulation on the instability onset in an infinite horizontal layer of double-diffusive fluid is investigated in this dissertation. The spectral-Galerkin method is used to transform the linearized perturbation equations to the system of time-periodic ordinary differentiate equations. The Chebyshev expansion method (Sinha and Wu, 1991) is applied to calculate the foundamental matrix which is used to determine the stability of the system according to the Floquet theory. Fluids of Prandtl number Pr = 0.01, 1, and 7 are investigated. The instability onsets in one of three modes, synchronous, subharmonic, and quasi periodic mode. In the synchronous mode, the instability oscillates at the same frequency as the gravity modulation O, in the subharmonic mode, the instability oscillates at O/2, while in the quasi-periodic mode, the oscillation frequency of the instability is different from the above two. The quasi-periodic mode onsets at the same thermal Rayleigh number, RT as that of instability onset under steady gravity. The subharmonic mode onsets with wave numbers in the neighboring region of k where the oscillation frequency of the instability onset in the steady-g case, o, equals to half of the modulation frequency, O/2. Similarly, the synchronous mode onsets at the neighboring of k where the o equals to O. The onset RT for quasi-periodic mode is not changed by the modulation frequency O and the relative amplitude of the modulation, h. For the synchronous and subharmonic modes, destabilization increases with increasing h. If h is large enough, the subharmonic mode will be more unstable than the synchronous and quasi periodic mode, so the instability mode will be switched by increasing h. For a given h with varying O, the resonance effect occurs in the neighborhood of O ≈ 2o cr, i.e, twice the critical oscillation frequency of the instability in the steady-g case associated with the critical RT. The resonant phenomena is found for fluids with Pr = 0.01, 1, and 7, and the effect diminishes as the Prandtl number increases. The effect of gravity modulation is asymptotic to zero when the modulated frequency O approaches zero and infinitely large. For the case of Prandtl number, Pr, = 0.01, it is found that the critical thermal Rayleigh number RT is reduced from the steady-g value of 2183 by 4%, 41%, and 86% as h is increased from 0.01, 0.1 and 0.2. In fact when h = 0.22737, the layer of fluid is destabilized at O = 9.1 with RT = 0, i.e., without heating from below. This is analogous to the case in the research of Gresho and Sani (1970) that a horizontal layer being heated from above can be destabilized by the oscillation of the layer. In this dissertation the stabilization effect caused by the modulation is found at some cases of O, which is analogous to the stability of motion of the pendulum with pivot in oscillation as discussed in Gresho and Sani (1970).

A two-dimensional finite element model for simulation of dendritic solidification of binary alloys is developed. The model solves the coupled time-dependent temperature and solute concentration equations on two independent meshes: a fixed mesh for the temperature and an adaptive interface-conforming mesh for the concentration. The temperature is solved on the whole domain while the concentration is solved only on the liquid region because diffusion in the solid is much smaller than that in the liquid and can be neglected. Temperature and concentration are coupled at the interface through the generalized Gibbs-Thompson relation. The solid-liquid interface is explicitly tracked with a set of marker points that defines its position at all times. Latent heat of fusion, interfacial energy, kinetic mobility and crystalline anisotropy are taken into account. The adaptive mesh is generated at every time step as the interface position changes. The model is easy to use in the sense that it works with physical variables as opposed to those based on the phase-field variable and level-set method. The model is very accurate as demonstrated by a series of calculations that compare to exact solutions or predictions by solidification theories. In simulations of solidification of pure materials, where only the energy equation is solved, the model produces very complicated dendritic structures that are in close agreement with experimental observations, and the computation is very efficient. Calculations under a variety of conditions show that the undercooling and surface tension are the main factors that determine the final dendritic structures. In simulations of alloy solidification, where both the energy and solutal concentration equations are solved, the results for the onset of interface instability and the prediction of the most unstable wavelength are in good agreement with linear stability theory. When applied to simulations of directional solidification of Pb-Sb alloys, the model generates dendritic structures and solute segregation similar to those observed experimentally, and the interface-development from a planar form to cells to dendritic structures is clearly demonstrated. These simulations are the first of their kind.

This study presents two new, generic methods to modeling planar elastically coupled rigid body systems using Geometric Algebra. The two methods are twist-based potential energy function method and twistor-based potential energy function method. In this research, the rigid body motion in the plane is modeled as a twist or twistor motion in which the rotational motion and translational motion happen simultaneously. The twist is denoted as a bivector using Geometric Algebra which facilitates the notation and computation. A twistor is defined in an intermediate frame half way between two displacement frames. The twistor parameters intuitively represent the relative displacement between two frames. Both twist-based and twistor-based potential energy functions are shown to be frame-independent and body-independent. The kinematics is studied using twist and twistor parameters. The constitutive equations are derived in which the wrench exerted by a pair of elastic bodies is computable given twist or twistor displacements. To analyze large displacements, this study also provides two higher order polynomial potential energy functions of twist parameters and twistor parameters. The polynomial potential energy functions are also shown to be frame-independent and body-independent. They are generally applicable to analyze large displacements of elastically coupled rigid body systems. Several case studies are provided in this research to demonstrate the utility of the presented modeling methods. A micropositioning stage device is modeled as a flexural mechanism with 6 rigid bodies and 7 flexural joints. Simulation is performed using Scilab software. The simulation results show good agreement with actual experimental data. The methods are also applied to simulate the displacement of flexural four-bar linkages with various geometry and various flexural hinges. This case study shows that the presented methods in this research are generic and case-independent. In another case study, the higher order polynomial function method is applied to fit some randomly generated data which demonstrates the generality of the method and the applicability of the method in cases when only experimental data is available without knowing the geometry parameters of a mechanism. The case study of modeling electrostatic potential energy between liquid water molecules using polynomial function of twistor shows the potential utility of the method in the analysis of large displacement. The methods presented in this research have been shown to be generic, easily applicable, and easily computable.

A heat transfer subroutine has been implemented into an existing finite element code developed in theCivil Engineering and Engineering Mechanics Department at the University of Arizona by Dr. Desai and students. The code is capable of performing non linear material and dynamic analysis. The heat transfer subroutine has been implemented such that any inelastic material behavior induced by a temperature increment is captured at every time step in a loading cycle. With the addition of the heat transfer routine, both thermal sources and sinks can be modeled. For example, power generating chips and power dissipating heat sinks, respectively. This will allow a more realistic representation of electronic packages under operational conditions. A 313 ball PBGA staggered area array package was used in all the analyses performed in this dissertation. The calibration of the models was based on research performed by the JPL consortium which included members such as Raytheon, Boeing and Xilinx. The focus of this dissertation was to determine the thermal and vibration fatigue lifetimes of electronic packages using the Disturbed State Concept. To achieve this goal, numerous analyses were performed, representing different test cases. The different test cases included thermal test chamber cycling (TCT), power cycling (PCT), vibration, thermal test chamber cycling with voids in solder balls, vibration with voids in solder balls, and coupled temperature with vibration. Based on the results of these analyses, the Disturbed State Concept was found to predict the fatigue lifetimes of the 313 PBGA package with excellent accuracy, when test results were available for comparison.

This dissertation describes the development of a coupled model for the analysis of a novel polymer/metal composite (IPMC) actuator under large external voltage. A general continuum model describing the transport and deformation processes of solid polymer electrolyte is proposed. The formulation is based on global integral postulates for the mass conservation, charge conservation, momentum equilibrium, the first law of thermodynamics, and the second law of thermodynamics. The global equations are localized in the volume and on the material surfaces bounding the polymer. The model is simplified to a three-component system comprised of a fixed negatively charged polymeric matrix, protons, and free water molecules within the polymer matrix. Among these species, water molecules are considered as the dominant specie responsible for the deformation of the IPMC actuators. In this work, the electrochemical process occurring at both electrodes is analyzed as boundary conditions during the deformation of actuator in the regime of large voltage (over 1.2 V). These are used in the framework of overpotential theory to develop boundary conditions for the water transport in the bulk of polymer. The proposed coupled model successfully captures the stress relaxation phenomenon due to water redistribution governed by diffusion. The fabrication process are described, and experiments including the role of initial water content on the electro-mechanical response of the actuator are also discussed. Comparison of simulations and experimental data showed good agreement.

Metal fatigue is generally considered to be the most important failure mode in structural and mechanical design. But there are general large uncertainties in the design factors. Therefore it is recommended that fatigue analysis and design be based on probabilistic and/or statistical methods. To date, most of the work on reliability based design of high-cycle fatigue critical components subjected to random stresses has been based on the characteristic S-N approach. Reliability analysis employing the fracture mechanics crack growth model is still in its infancy. A comprehensive model for fatigue reliability assessment based on a fracture mechanics approach is presented. The overall goal of this study is to develop a practical model and computational procedure for obtaining the statistical distribution of the time-to-failure of a single fatigue critical component under general non-zero mean and wide band stationary Gaussian stress processes. The theme of this proposed research is to merge and extend recently developed technologies on crack closure fatigue crack growth modeling, time-variant reliability and first-passage concepts, and advanced structural reliability computational methods giving full consideration to engineering applications. A first passage approach using an advanced crack growth model and an efficient structural reliability computational procedure is proposed. Failure is defined as the event that the first time the random stress process exceeds the strength of the component. But strength degrades with time as the fatigue crack grows. Assuming a stationary Gaussian stress process, the barrier upcrossing is modeled as a Poisson clumping process. And the threshold level of residual strength is derived using a crack-closure fatigue crack growth model. Fatigue design factors are modeled as random variables. An implicit expression for time-to-fatigue-failure T is derived in terms of the random fatigue design factors. The cumulative distribution function of T can be estimated efficiently and accurately by an advanced mean value procedure together with a log transformation on select variables. Contributions of this study include: (1) Development of a time-variant first-passage fatigue reliability model employing the Poisson upcrossing approach thereby combining the synergistic effects of fatigue and fracture failure modes. (2) The introduction of a refined fatigue crack growth model, which includes the crack opening stress concept and the rainflow cycle counting principle, to predict crack growth under wide band random stresses. The model is capable of accommodating effects of mean stress and bandwidth factor of the stress process. (3) Development and demonstration of methods and strategies of structural reliability analysis to accommodate large variances of the design factors. (4) Gaining an understanding of the physical process of fatigue fracture failure behavior by identifying the important design factors through a sensitivity study. (5) Close comparison of mathematical forms between various fatigue/fracture limit states.

The objective of this dissertation is to present the analytical solutions to the heat spreading problems that arise due to a flux specified circular heat source on a finite thickness substrate with isothermal or convective surfaces. The solutions to heat spreading resistance of these problems are obtained for the first time by the exact treatment of the mixed boundary conditions present on the substrate at the heat source side. In the case of heat spreading through a substrate with isothermal surfaces the solution method utilizes the two-dimensional axisymmetric equation of thermal conduction allowing for the convective cooling over source region. In the absence of convection over the source, it is shown that the total thermal resistance is composed of spreading resistance of an otherwise isothermal substrate and a correction due to inhomogeneous substrate thermal boundary condition. The application of the method of superposition elucidates the exact definition of source adiabatic temperature that takes care of the correction due to inhomogeneous substrate thermal boundary condition. In the case of heat spreading through a substrate with convective surfaces it is also shown that the expression for the total thermal resistance can be decomposed into a base solution and a correction. Thus the effects of the unequal heat sinks are consolidated in an approximate way to an equivalent or effective heat sink, Stheta1 that contributes the correction of Stheta1 to the base resistance of the homogeneous solution where the upper and lower heat sink temperatures are the same.

In this investigation an attempt was made to understand the dominant mechanisms of heat and momentum transport in an externally excited wall jet. The mean and fluctuating characteristics of this flow were experimentally evaluated for a constant wall temperature boundary condition. Temperature and streamwise velocity profiles were obtained through simultaneous hot and cold wire measurements in air. Selective forcing of the flow at the most amplified frequencies produced profound effects on the temperature and velocity fields and hence the time-averaged wall heat transfer and shear stress. Large amplitude excitation of the flow (up to 2% of the velocity measured at the jet exit plane) at a high frequency resulted in a reduction in the maximum skin friction by as much as 60% with an increase in the maximum wall heat flux as high as 30%. The skin friction and wall heat flux were much less susceptible to low frequency excitation. These profound effects on the skin friction and heat transfer present a breakdown in the Reynolds analogy. The Reynolds analogy factor increased significantly relative to the unforced case by as much as 200% for high frequency forcing at a large excitation level. Thus, the ability to predict the heat transfer in a wall jet from a known hydrodynamic solution is restricted in the presence of large amplitude disturbances. The amplitude and phase distributions for the fluctuating streamwise velocity and temperature demonstrate, that for large excitation levels, a sub-harmonic wave experiences substantial growth in the measurement domain. Significant distortion of the sub-harmonic component of the fluctuating temperature provides evidence that these large scale structures are responsible for the significant widening of the boundary layer and the transport of energy and momentum away from the surface. This motion may also explain the increase in the temperature gradient near the surface since the unsteady upward coherent transport is increased compared to diffusive transport in this region. Temperature and velocity profiles were also acquired at different spanwise locations. Consistent with previous flow visualization studies, it was found that the transition process (including the coherent transport of the sub-harmonic wave) is two-dimensional in nature.

This study presents the node annexation method for modeling kinematic joints between rigid and flexible bodies of rigid-flexible multibody systems. Each node of a flexible body is assumed to have lumped mass and three translational degrees of freedom, resulting in a diagonal mass matrix. Based on the node annexation method, both the nodal- and the modal-coordinate formulations for rigid-flexible multibody dynamics are developed. Conventionally rigid-to-flexible-body joints are treated as kinematic constraints using the Lagrange multiplier method. The formulations based on kinematic constraint method yield coupled equations of motion which have the difficulties associated with modal truncation. On the other hand, the node annexation method transfers the inertia and force effect of connected nodes of a flexible body to the connected rigid body. The mass matrix of the resultant equations of motion consists of two different kind of sub-matrices: one is rigid-body sub-system matrix containing the inertia of both rigid bodies and connected nodes of the flexible body and another is flexible-body sub-system matrix containing the inertia of free nodes of the flexible body. Since there is no off-diagonal terms coupling the sub-matrices, the node annexation method allows the division of the equations of motion into smaller sub-system equations. The node annexation method not only provides computational efficiency but also fundamentally eliminates any kinematic error at rigid-to-flexible-body joints. In addition, the node annexation method preserves the uncoupled nature of modal coordinates, allowing a mathematically justified modal truncation. Computer simulations are performed using a vehicle model with a flexible car body. The simulation results show computational advantage over the kinematic constraint method.