Asymptotic analysis of thin plates under normal load and horizontal edge thrust

We consider the radially symmetric nonlinear von Kármán plate equations for circular or annular plates in the limit of small thickness. The loads on the plate consist of a radially symmetric pressure load and a uniform edge load. The dependence of the steady states on the edge load and thickness is studied using asymptotics as well as numerical calculations. The von Kármán plate equations are a singular perturbation of the Fӧppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions.

We give evidence for the existence of a singular compressive solution for the circular membrane and show by a singular perturbation expansion that the nonsingular compressive solution approach this singular solution as the radial stress at the center of the plate vanishes. In this limit, an infinite number of folds occur with respect to the edge load. Similar behavior is observed for the annular membrane with zero edge load at the inner radius in the limit as the circumferential stress vanishes.

We develop multiscale expansions, which are asymptotic to members of this family for plates with edges that are elastically supported against rotation. At some thicknesses this approximation breaks down and a boundary layer appears at the center of the plate. In the limit of small normal load, the points of breakdown approach the bifurcation points corresponding to buckling of the nondeflected state. A uniform asymptotic expansion for small thickness combining the boundary layer with a multiscale approximation of the outer solution is developed for this case. These approximations complement the well known boundary layer expansions based on tensile membrane solutions in describing the bending and stretching of thin plates. The approximation becomes inconsistent as the clamped state is approached by increasing the resistance against rotation at the edge. We prove that such an expansion for the clamped circular plate cannot exist unless the pressure load is self-equilibrating.

Black holes in the early universe, in compact binaries, and as energy sources inside solar-type stars

This thesis consists of three separate studies of roles that black holes might play in our universe.

In the first part we formulate a statistical method for inferring the cosmological parameters of our universe from LIGO/VIRGO measurements of the gravitational waves produced by coalescing black-hole/neutron-star binaries. This method is based on the cosmological distance-redshift relation, with "luminosity distances" determined directly, and redshifts indirectly, from the gravitational waveforms. Using the current estimates of binary coalescence rates and projected "advanced" LIGO noise spectra, we conclude that by our method the Hubble constant should be measurable to within an error of a few percent. The errors for the mean density of the universe and the cosmological constant will depend strongly on the size of the universe, varying from about 10% for a "small" universe up to and beyond 100% for a "large" universe. We further study the effects of random gravitational lensing and find that it may strongly impair the determination of the cosmological constant.

In the second part of this thesis we disprove a conjecture that black holes cannot form in an early, inflationary era of our universe, because of a quantum-field-theory induced instability of the black-hole horizon. This instability was supposed to arise from the difference in temperatures of any black-hole horizon and the inflationary cosmological horizon; it was thought that this temperature difference would make every quantum state that is regular at the cosmological horizon be singular at the black-hole horizon. We disprove this conjecture by explicitly constructing a quantum vacuum state that is everywhere regular for a massless scalar field. We further show that this quantum state has all the nice thermal properties that one has come to expect of "good" vacuum states, both at the black-hole horizon and at the cosmological horizon.

In the third part of the thesis we study the evolution and implications of a hypothetical primordial black hole that might have found its way into the center of the Sun or any other solar-type star. As a foundation for our analysis, we generalize the mixing-length theory of convection to an optically thick, spherically symmetric accretion flow (and find in passing that the radial stretching of the inflowing fluid elements leads to a modification of the standard Schwarzschild criterion for convection). When the accretion is that of solar matter onto the primordial hole, the rotation of the Sun causes centrifugal hangup of the inflow near the hole, resulting in an "accretion torus" which produces an enhanced outflow of heat. We find, however, that the turbulent viscosity, which accompanies the convective transport of this heat, extracts angular momentum from the inflowing gas, thereby buffering the torus into a lower luminosity than one might have expected. As a result, the solar surface will not be influenced noticeably by the torus's luminosity until at most three days before the Sun is finally devoured by the black hole. As a simple consequence, accretion onto a black hole inside the Sun cannot be an answer to the solar neutrino puzzle.

Parylene-C as a new piezoelectric material

The goal of this thesis is to develop a proper microelectromechanical systems (MEMS) process to manufacture piezoelectric Parylene-C (PA-C), which is famous for its chemical inertness, mechanical and thermal properties and electrical insulation. Furthermore, piezoelectric PA-C is used to build miniature, inexpensive, non-biased piezoelectric microphones.

These piezoelectric PA-C MEMS microphones are to be used in any application where a conventional piezoelectric and electret microphone can be used, such as in cell phones and hearing aids. However, they have the advantage of a simplified fabrication process compared with existing technology. In addition, as a piezoelectric polymer, PA-C has varieties of applications due to its low dielectric constant, low elastic stiffness, low density, high voltage sensitivity, high temperature stability and low acoustic and mechanical impedance. Furthermore, PA-C is an FDA approved biocompatible material and is able to maintain operate at a high temperature.

To accomplish piezoelectric PA-C, a MEMS-compatible poling technology has been developed. The PA-C film is poled by applying electrical field during heating. The piezoelectric coefficient, -3.75pC/N, is obtained without film stretching.

The millimeter-scale piezoelectric PA-C microphone is fabricated with an in-plane spiral arrangement of two electrodes. The dynamic range is from less than 30 dB to above 110 dB SPL (referenced 20 µPa) and the open-circuit sensitivities are from 0.001 – 0.11 mV/Pa over a frequency range of 1 - 10 kHz. The total harmonic distortion of the device is less than 20% at 110 dB SPL and 1 kHz.

Zirconocenes as models for homogeneous Ziegler-Natta olefin polymerization catalysts

Using density functional theory, we studied the fundamental steps of olefin polymerization for zwitterionic and cationic Group IV ansa-zirconocenes and a neutral ansa- yttrocene. Complexes [H₂E(C₅H₄)₂ZrMe]ⁿ (n = 0: E = BH₂ (1), BF₂ (2), AlH₂(3); n = +: E = CH₂(4), SiH₂(5)) and H₂Si(C₅H₄)₂YMe were used as computational models. The largest differences among these three classes of compounds were the strength of olefin binding and the stability of the β-agostic alkyl intermediate towards β-hydrogen elimination. We investigated the effect of solvent on the reaction energetics for land 5. We found that in benzene the energetics became very similar except that a higher olefin insertion barrier was calculated for 1. The calculated anion affinity of [CH₃BF₃]^- was weaker towards 1 than 5. The calculated olefin binding depended primarily on the charge of the ansa linker, and the olefin insertion barrier was found to decrease steadily in the following order: [H₂C(C₅H₄)₂ZrMe]⁺ > [F₂B(C₅H₄)₂ZrMe] ≈ [H₂B(C₅H₄)₂ZrMe] > [H₂Si(C₅H₄)₂ZrMe]⁺ > [H₂Al(C₅H₄)₂ZrMe].

We prepared ansa-zirconocene dicarbonyl complexes Me₂ECp₂Zr(CO)₂ (E = Si, C), and t-butyl substituted complexes (t-BuCp)₂Zr(CO)₂, Me₂E(t-BuCp)₂Zr(CO)₂ (E = Si, C), (Me₂Si)₂(t-BuCp)₂Zr(CO)₂ as well as analogous zirconocene complexes. Both the reduction potentials and carbonyl stretching frequencies follow the same order: Me₂SiCp₂ZrCl₂> Me₂CCp₂ZrCl₂> Cp₂ZrCl₂> (Me₂Si)₂Cp₂ZrCl₂. This ordering is a result of both the donating abilities of the cyclopentadienyl substituents and the orientation of the cyclopentadiene rings. Additionally, we prepared a series of analogous cationic zirconocene complexes [LZrOCMe₃][MeB(C₆F₅)₃] (L = CP₂, Me₂SiCp₂, Me₂CCP₂, (Me₂Si)₂Cp₂) and studied the kinetics of anion dissociation. We found that the enthalpy of anion dissociation increased from 10.3 kcal•mol^-1 to 17.6 kcal•mol^-1 as exposure of the zirconium center increased.

We also prepared series of zirconocene complexes bearing 2,2-dimethyl-2-sila-4-pentenyl substituents (and methyl-substituted olefin variants). Methide abstraction with B(C₆F₅) results in reversible coordination of the tethered olefin to the cationic zirconium center. The kinetics of olefin dissociation have been examined using NMR methods, and the effects of ligand variation for unlinked, singly [SiMe₂]-linked and doubly [SiMe₂]-linked bis(cyclopentadienyl) arrangements has been compared (ΔG‡ for olefin dissociation varies from 12.8 to 15.6 kcal•mol^-1). Methide abstraction from 1,2-(SiMe₂)₂(η⁵-C₅H₃)₂Zr(CH₃)-(CH₂CMe₂CH₂CH = CH₂) results in rapid β-allyl elimination with loss of isobutene yielding the allyl cation [{1,2-(SiMe₂)₂(η⁵-C₅H₃)₂Zr(η³-CH₂CH=CH₂)]⁺.

Ancillary ligand effects : from zirconium(IV)-catalyzed homogeneous propylene polymerization to platinum(II)-mediated C-H bond activation

A series of C_s- and C₁-symmetric doubly-linked ansa-metallocenes of the general formula {1,1'-SiMe₂-2,2'-E-('ƞ⁵-C₅H₂-4-R¹)-(ƞ⁵-C₅H-3',5'-(CHMe₂)₂)}ZrC₂ (E = SiMe₂ (1), SiPh₂ (2), SiMe₂ -SiMe₂ (3); R¹ = H, CHMe₂, C₅H₉, C₆H₁₁, C₆H₅) has been prepared. When activated by methylaluminoxane, these are active propylene polymerization catalysts. 1 and 2 produce syndiotactic polypropylenes, and 3 produces isotactic polypropylenes. Site epimerization is the major pathway for stereoerror formation for 1 and 2. In addition, the polymer chain has slightly stronger steric interaction with the diphenylsilylene linker than with the dimethylsilylene linker. This results in more frequent site epimerization and reduced syndiospecificity for 2 compared to 1.

C₁-Symmetric ansa-zirconocenes [1,1 '-SiMe₂-(C₅H₄)-(3-R-C₅H₃)]ZrCl₂ (4), [1,1 '-SiMe₂-(C₅H₄)-(2,4-R₂-C₅H₂)]ZrCl₂ (5) and [1,1 '-SiMe₂-2,2 '-(SiMe₂-SiMe₂)-(C₅H₃)-( 4-R-C₅H₂)]ZrCl₂ (6) have been prepared to probe the origin of isospecificity in 3. While 4 and 3 produce polymers with similar isospecificity, 5 and 6 give mostly hemi-isotactic-like polymers. It is proposed that the facile site epimerization via an associative pathway allows rapid equilibration of the polymer chain between the isospecific and aspecific insertion sites. This results in more frequent insertion from the isospecific site, which has a lower kinetic barrier for chain propagation. On the other hand, site epimerization for 5 and 6 is slow. This leads to mostly alternating insertion from the isospecific and aspecific sites, and consequently, a hemi-isotactic-like polymers. In comparison, site epimerization is even slower for 3, but enchainment from the aspecific site has an extremely high kinetic barrier for monomer coordination. Therefore, enchainment occurs preferentially from the isospecific site to produce isotactic polymers.

A series of cationic complexes [(ArN=CR-CR=NAr)PtMe(L)]⁺[BF₄]⁺ (Ar = aryl; R = H, CH₃; L = water, trifluoroethanol) has been prepared. They react smoothly with benzene at approximately room temperature in trifluoroethanol solvent to yield methane and the corresponding phenyl Pt(II) cations, via Pt(IV)-methyl-phenyl-hydride intermediates. The reaction products of methyl-substituted benzenes suggest an inherent reactivity preference for aromatic over benzylic C-H bond activation, which can however be overridden by steric effects. For the reaction of benzene with cationic Pt(II) complexes, in which the diimine ligands bear 3,5-disubstituted aryl groups at the nitrogen atoms, the rate-determining step is C-H bond activation. For the more sterically crowded analogs with 2,6-dimethyl-substituted aryl groups, benzene coordination becomes rate-determining. The more electron-rich the ligand, as reflected by the CO stretching frequency in the IR spectrum of the corresponding cationic carbonyl complex, the faster the rate of C-H bond activation. This finding, however, does not reflect the actual C-H bond activation process, but rather reflects only the relative ease of solvent molecules displacing water molecules to initiate the reaction. That is, the change in rates is mostly due to a ground state effect. Several lines of evidence suggest that associative substitution pathways operate to get the hydrocarbon substrate into, and out of, the coordination sphere; i.e., that benzene substitution proceeds by a solvent- (TFE-) assisted associative pathway.

Extracting material response from simple mechanical tests on hardening-softening-hardening viscoplastic solids

Compliant foams are usually characterized by a wide range of desirable mechanical properties. These properties include viscoelasticity at different temperatures, energy absorption, recoverability under cyclic loading, impact resistance, and thermal, electrical, acoustic and radiation-resistance. Some foams contain nano-sized features and are used in small-scale devices. This implies that the characteristic dimensions of foams span multiple length scales, rendering modeling their mechanical properties difficult. Continuum mechanics-based models capture some salient experimental features like the linear elastic regime, followed by non-linear plateau stress regime. However, they lack mesostructural physical details. This makes them incapable of accurately predicting local peaks in stress and strain distributions, which significantly affect the deformation paths. Atomistic methods are capable of capturing the physical origins of deformation at smaller scales, but suffer from impractical computational intensity. Capturing deformation at the so-called meso-scale, which is capable of describing the phenomenon at a continuum level, but with some physical insights, requires developing new theoretical approaches.

A fundamental question that motivates the modeling of foams is ‘how to extract the intrinsic material response from simple mechanical test data, such as stress vs. strain response?’ A 3D model was developed to simulate the mechanical response of foam-type materials. The novelty of this model includes unique features such as the hardening-softening-hardening material response, strain rate-dependence, and plastically compressible solids with plastic non-normality. Suggestive links from atomistic simulations of foams were borrowed to formulate a physically informed hardening material input function. Motivated by a model that qualitatively captured the response of foam-type vertically aligned carbon nanotube (VACNT) pillars under uniaxial compression [2011,“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech.Phys. Solids, 59, pp. 2227–2237, Erratum 60, 1753–1756 (2012)], the property space exploration was advanced to three types of simple mechanical tests: 1) uniaxial compression, 2) uniaxial tension, and 3) nanoindentation with a conical and a flat-punch tip. The simulations attempt to explain some of the salient features in experimental data, like
1) The initial linear elastic response.
2) One or more nonlinear instabilities, yielding, and hardening.

The model-inherent relationships between the material properties and the overall stress-strain behavior were validated against the available experimental data. The material properties include the gradient in stiffness along the height, plastic and elastic compressibility, and hardening. Each of these tests was evaluated in terms of their efficiency in extracting material properties. The uniaxial simulation results proved to be a combination of structural and material influences. Out of all deformation paths, flat-punch indentation proved to be superior since it is the most sensitive in capturing the material properties.

Horizontal Bloch line motion in magnetic bubble materials

The purpose of this work is to extend experimental and theoretical understanding of horizontal Bloch line (HBL) motion in magnetic bubble materials. The present theory of HBL motion is reviewed, and then extended to include transient effects in which the internal domain wall structure changes with time. This is accomplished by numerically solving the equations of motion for the internal azimuthal angle ɸ and the wall position q as functions of z, the coordinate perpendicular to the thin-film material, and time. The effects of HBL's on domain wall motion are investigated by comparing results from wall oscillation experiments with those from the theory. In these experiments, a bias field pulse is used to make a step change in equilibrium position of either bubble or stripe domain walls, and the wall response is measured by using transient photography. During the initial response, the dynamic wall structure closely resembles the initial static structure. The wall accelerates to a relatively high velocity (≈20 m/sec), resulting in a short (≈22 nsec ) section of initial rapid motion. An HBL gradually forms near one of the film surfaces as a result of local dynamic properties, and moves along the wall surface toward the film center. The presence of this structure produces low-frequency, triangular-shaped oscillations in which the experimental wall velocity is nearly constant, v_s≈ 5-8 m/sec. If the HBL reaches the opposite surface, i.e., if the average internal angle reaches an integer multiple of π, the momentum stored in the HBL is lost, and the wall chirality is reversed. This results in abrupt transitions to overdamped motion and changes in wall chirality, which are observed as a function of bias pulse amplitude. The pulse amplitude at which the n^th punch- through occurs just as the wall reaches equilibrium is given within 0.2 0e by H_n = (2v_sH'/γ)^1/2 • (nπ)^1/2 + H_sv), where H' is the effective field gradient from the surrounding domains, and H_sv is a small (less than 0.03 0e), effective drag field. Observations of wall oscillation in the presence of in-plane fields parallel to the wall show that HBL formation is suppressed by fields greater than about 40 0e (≈2πM_s), resulting in the high-frequency, sinusoidal oscillations associated with a simple internal wall structure.

Large plane deformations of thin elastic sheets of neo-hookean material

Large plane deformations of thin elastic sheets of neo-Hookean material are considered and a method of successive substitutions is developed to solve problems within the two-dimensional theory of finite plane stress. The first approximation is determined by linear boundary value problems on two harmonic functions, and it is approached asymptotically at very large extensions in the plane of the sheet. The second and higher approximations are obtained by solving Poisson equations. The method requires modification when the membrane has a traction-free edge.

Several problems are treated involving infinite sheets under uniform biaxial stretching at infinity. First approximations are obtained when a circular or elliptic inclusion is present and when the sheet has a circular or elliptic hole, including the limiting cases of a line inclusion and a straight crack or slit. Good agreement with exact solutions is found for circularly symmetric deformations. Other examples discuss the stretching of a short wide strip, the deformation near a boundary corner which is traction-free, and the application of a concentrated load to a boundary point.

Small scale turbulence in high Karlovitz number premixed flames

The purpose of this thesis is to characterize the behavior of the smallest turbulent scales in high Karlovitz number (Ka) premixed flames. These scales are particularly important in the two-way coupling between turbulence and chemistry and better understanding of these scales will support future modeling efforts using large eddy simulations (LES). The smallest turbulent scales are studied by considering the vorticity vector, ω, and its transport equation.

Due to the complexity of turbulent combustion introduced by the wide range of length and time scales, the two-dimensional vortex-flame interaction is first studied as a simplified test case. Numerical and analytical techniques are used to discern the dominate transport terms and their effects on vorticity based on the initial size and strength of the vortex. This description of the effects of the flame on a vortex provides a foundation for investigating vorticity in turbulent combustion.

Subsequently, enstrophy, ω² = ω • ω, and its transport equation are investigated in premixed turbulent combustion. For this purpose, a series of direct numerical simulations (DNS) of premixed n-heptane/air flames are performed, the conditions of which span a wide range of unburnt Karlovitz numbers and turbulent Reynolds numbers. Theoretical scaling analysis along with the DNS results support that, at high Karlovitz number, enstrophy transport is controlled by the viscous dissipation and vortex stretching/production terms. As a result, vorticity scales throughout the flame with the inverse of the Kolmogorov time scale, τ_η, just as in homogeneous isotropic turbulence. As τ_η is only a function of the viscosity and dissipation rate, this supports the validity of Kolmogorov’s first similarity hypothesis for sufficiently high Ka numbers (Ka ≳ 100). These conclusions are in contrast to low Karlovitz number behavior, where dilatation and baroclinic torque have a significant impact on vorticity within the flame. Results are unaffected by the transport model, chemical model, turbulent Reynolds number, and lastly the physical configuration.

Next, the isotropy of vorticity is assessed. It is found that given a sufficiently large value of the Karlovitz number (Ka ≳ 100) the vorticity is isotropic. At lower Karlovitz numbers, anisotropy develops due to the effects of the flame on the vortex stretching/production term. In this case, the local dynamics of vorticity in the strain-rate tensor, S, eigenframe are altered by the flame. At sufficiently high Karlovitz numbers, the dynamics of vorticity in this eigenframe resemble that of homogeneous isotropic turbulence.

Combined, the results of this thesis support that both the magnitude and orientation of vorticity resemble the behavior of homogeneous isotropic turbulence, given a sufficiently high Karlovitz number (Ka ≳ 100). This supports the validity of Kolmogorov’s first similarity hypothesis and the hypothesis of local isotropy under these condition. However, dramatically different behavior is found at lower Karlovitz numbers. These conclusions provides/suggests directions for modeling high Karlovitz number premixed flames using LES. With more accurate models, the design of aircraft combustors and other combustion based devices may better mitigate the detrimental effects of combustion, from reducing CO₂ and soot production to increasing engine efficiency.

I. Fundamental limitations in microelectronics. II. Power Schottky diode design and comparison with the junction diode. III. Permittivity of strontium titanate

Part I

The physical phenomena which will ultimately limit the packing density of planar bipolar and MOS integrated circuits are examined. The maximum packing density is obtained by minimizing the supply voltage and the size of the devices. The minimum size of a bipolar transistor is determined by junction breakdown, punch-through and doping fluctuations. The minimum size of a MOS transistor is determined by gate oxide breakdown and drain-source punch-through. The packing density of fully active bipolar or static non-complementary MOS circuits becomes limited by power dissipation. The packing density of circuits which are not fully active such as read-only memories, becomes limited by the area occupied by the devices, and the frequency is limited by the circuit time constants and by metal migration. The packing density of fully active dynamic or complementary MOS circuits is limited by the area occupied by the devices, and the frequency is limited by power dissipation and metal migration. It is concluded that read-only memories will reach approximately the same performance and packing density with MOS and bipolar technologies, while fully active circuits will reach the highest levels of integration with dynamic MOS or complementary MOS technologies.

Part II

Because the Schottky diode is a one-carrier device, it has both advantages and disadvantages with respect to the junction diode which is a two-carrier device. The advantage is that there are practically no excess minority carriers which must be swept out before the diode blocks current in the reverse direction, i.e. a much faster recovery time. The disadvantage of the Schottky diode is that for a high voltage device it is not possible to use conductivity modulation as in the p i n diode; since charge carriers are of one sign, no charge cancellation can occur and current becomes space charge limited. The Schottky diode design is developed in Section 2 and the characteristics of an optimally designed silicon Schottky diode are summarized in Fig. 9. Design criteria and quantitative comparison of junction and Schottky diodes is given in Table 1 and Fig. 10. Although somewhat approximate, the treatment allows a systematic quantitative comparison of the devices for any given application.

Part III

We interpret measurements of permittivity of perovskite strontium titanate as a function of orientation, temperature, electric field and frequency performed by Dr. Richard Neville. The free energy of the crystal is calculated as a function of polarization. The Curie-Weiss law and the LST relation are verified. A generalized LST relation is used to calculate the permittivity of strontium titanate from zero to optic frequencies. Two active optic modes are important. The lower frequency mode is attributed mainly to motion of the strontium ions with respect to the rest of the lattice, while the higher frequency active mode is attributed to motion of the titanium ions with respect to the oxygen lattice. An anomalous resonance which multi-domain strontium titanate crystals exhibit below 65°K is described and a plausible mechanism which explains the phenomenon is presented.