I. Rigid body penetration into brittle materials. II. Phase change effect on shock wave propagation

Part I.

We have developed a technique for measuring the depth time history of rigid body penetration into brittle materials (hard rocks and concretes) under a deceleration of ~ 10⁵ g. The technique includes bar-coded projectile, sabot-projectile separation, detection and recording systems. Because the technique can give very dense data on penetration depth time history, penetration velocity can be deduced. Error analysis shows that the technique has a small intrinsic error of ~ 3-4 % in time during penetration, and 0.3 to 0.7 mm in penetration depth. A series of 4140 steel projectile penetration into G-mixture mortar targets have been conducted using the Caltech 40 mm gas/ powder gun in the velocity range of 100 to 500 m/s.

We report, for the first time, the whole depth-time history of rigid body penetration into brittle materials (the G-mixture mortar) under 10⁵ g deceleration. Based on the experimental results, including penetration depth time history, damage of recovered target and projectile materials and theoretical analysis, we find:

1. Target materials are damaged via compacting in the region in front of a projectile and via brittle radial and lateral crack propagation in the region surrounding the penetration path. The results suggest that expected cracks in front of penetrators may be stopped by a comminuted region that is induced by wave propagation. Aggregate erosion on the projectile lateral surface is < 20% of the final penetration depth. This result suggests that the effect of lateral friction on the penetration process can be ignored.

2. Final penetration depth, P_max, is linearly scaled with initial projectile energy per unit cross-section area, e_s , when targets are intact after impact. Based on the experimental data on the mortar targets, the relation is P_max(mm) 1.15e_s (J/mm² ) + 16.39.

3. Estimation of the energy needed to create an unit penetration volume suggests that the average pressure acting on the target material during penetration is ~ 10 to 20 times higher than the unconfined strength of target materials under quasi-static loading, and 3 to 4 times higher than the possible highest pressure due to friction and material strength and its rate dependence. In addition, the experimental data show that the interaction between cracks and the target free surface significantly affects the penetration process.

4. Based on the fact that the penetration duration, t_max, increases slowly with e_s and does not depend on projectile radius approximately, the dependence of t_max on projectile length is suggested to be described by t_max(μs) = 2.08e_s (J/mm² + 349.0 x m/(πR²), in which m is the projectile mass in grams and R is the projectile radius in mm. The prediction from this relation is in reasonable agreement with the experimental data for different projectile lengths.

5. Deduced penetration velocity time histories suggest that whole penetration history is divided into three stages: (1) An initial stage in which the projectile velocity change is small due to very small contact area between the projectile and target materials; (2) A steady penetration stage in which projectile velocity continues to decrease smoothly; (3) A penetration stop stage in which projectile deceleration jumps up when velocities are close to a critical value of ~ 35 m/s.

6. Deduced averaged deceleration, a, in the steady penetration stage for projectiles with same dimensions is found to be a(g) = 192.4v + 1.89 x 10⁴, where v is initial projectile velocity in m/s. The average pressure acting on target materials during penetration is estimated to be very comparable to shock wave pressure.

7. A similarity of penetration process is found to be described by a relation between normalized penetration depth, P/P_max, and normalized penetration time, t/t_max, as P/P_max = f(t/t_max, where f is a function of t/t_max. After f(t/t_max is determined using experimental data for projectiles with 150 mm length, the penetration depth time history for projectiles with 100 mm length predicted by this relation is in good agreement with experimental data. This similarity also predicts that average deceleration increases with decreasing projectile length, that is verified by the experimental data.

8. Based on the penetration process analysis and the present data, a first principle model for rigid body penetration is suggested. The model incorporates the models for contact area between projectile and target materials, friction coefficient, penetration stop criterion, and normal stress on the projectile surface. The most important assumptions used in the model are: (1) The penetration process can be treated as a series of impact events, therefore, pressure normal to projectile surface is estimated using the Hugoniot relation of target material; (2) The necessary condition for penetration is that the pressure acting on target materials is not lower than the Hugoniot elastic limit; (3) The friction force on projectile lateral surface can be ignored due to cavitation during penetration. All the parameters involved in the model are determined based on independent experimental data. The penetration depth time histories predicted from the model are in good agreement with the experimental data.

9. Based on planar impact and previous quasi-static experimental data, the strain rate dependence of the mortar compressive strength is described by σ_f/σ⁰_f = exp(0.0905(log(έ/έ_0) ^1.14, in the strain rate range of 10^-7/s to 10³/s (σ⁰_f and έ are reference compressive strength and strain rate, respectively). The non-dispersive Hugoniot elastic wave in the G-mixture has an amplitude of ~ 0.14 GPa and a velocity of ~ 4.3 km/s.

Part II.

Stress wave profiles in vitreous GeO₂ were measured using piezoresistance gauges in the pressure range of 5 to 18 GPa under planar plate and spherical projectile impact. Experimental data show that the response of vitreous GeO₂ to planar shock loading can be divided into three stages: (1) A ramp elastic precursor has peak amplitude of 4 GPa and peak particle velocity of 333 m/s. Wave velocity decreases from initial longitudinal elastic wave velocity of 3.5 km/s to 2.9 km/s at 4 GPa; (2) A ramp wave with amplitude of 2.11 GPa follows the precursor when peak loading pressure is 8.4 GPa. Wave velocity drops to the value below bulk wave velocity in this stage; (3) A shock wave achieving final shock state forms when peak pressure is > 6 GPa. The Hugoniot relation is D = 0.917 + 1.711u (km/s) using present data and the data of Jackson and Ahrens [1979] when shock wave pressure is between 6 and 40 GPa for ρ₀ = 3.655 gj cm³ . Based on the present data, the phase change from 4-fold to 6-fold coordination of Ge⁺⁴ with O^-2 in vitreous GeO₂ occurs in the pressure range of 4 to 15 ± 1 GPa under planar shock loading. Comparison of the shock loading data for fused SiO₂ to that on vitreous GeO₂ demonstrates that transformation to the rutile structure in both media are similar. The Hugoniots of vitreous GeO₂ and fused SiO₂ are found to coincide approximately if pressure in fused SiO₂ is scaled by the ratio of fused SiO₂to vitreous GeO₂ density. This result, as well as the same structure, provides the basis for considering vitreous Ge0₂ as an analogous material to fused SiO₂ under shock loading. Experimental results from the spherical projectile impact demonstrate: (1) The supported elastic shock in fused SiO₂ decays less rapidly than a linear elastic wave when elastic wave stress amplitude is higher than 4 GPa. The supported elastic shock in vitreous GeO₂ decays faster than a linear elastic wave; (2) In vitreous GeO₂ , unsupported shock waves decays with peak pressure in the phase transition range (4-15 GPa) with propagation distance, x, as α 1/x^-3.35 , close to the prediction of Chen et al. [1998]. Based on a simple analysis on spherical wave propagation, we find that the different decay rates of a spherical elastic wave in fused SiO₂ and vitreous GeO₂ is predictable on the base of the compressibility variation with stress under one-dimensional strain condition in the two materials.

The intergalactic and circumgalactic medium surrounding star-forming galaxies at redshifts 2 < z < 3

We present measurements of the spatial distribution, kinematics, and physical properties of gas in the circumgalactic medium (CGM) of 2.0<z<2.8 UV color-selected galaxies as well as within the 2<z<3 intergalactic medium (IGM). These measurements are derived from Voigt profile decomposition of the full Lyα and Lyβ forest in 15 high-resolution, high signal-to-noise ratio QSO spectra resulting in a catalog of ∼6000 HI absorbers.

Chapter 2 of this thesis focuses on HI surrounding high-z star-forming galaxies drawn from the Keck Baryonic Structure Survey (KBSS). The KBSS is a unique spectroscopic survey of the distant universe designed to explore the details of the connection between galaxies and intergalactic baryons within the same survey volumes. The KBSS combines high-quality background QSO spectroscopy with large densely-sampled galaxy redshift surveys to probe the CGM at scales of ∼50 kpc to a few Mpc. Based on these data, Chapter 2 presents the first quantitative measurements of the distribution, column density, kinematics, and absorber line widths of neutral hydrogen surrounding high-z star-forming galaxies.

Chapter 3 focuses on the thermal properties of the diffuse IGM. This analysis relies on measurements of the ∼6000 absorber line widths to constrain the thermal and turbulent velocities of absorbing "clouds." A positive correlation between the column density of HI and the minimum line width is recovered and implies a temperature-density relation within the low-density IGM for which higher-density regions are hotter, as is predicted by simple theoretical arguments.

Chapter 4 presents new measurements of the opacity of the IGM and CGM to hydrogen-ionizing photons. The chapter begins with a revised measurement of the HI column density distribution based on this new absorption line catalog that, due to the inclusion of high-order Lyman lines, provides the first statistically robust measurement of the frequency of absorbers with HI column densities 14 ≲ log(N_HI/cm^-2) ≲ 17.2. Also presented are the first measurements of the column density distribution of HI within the CGM (50 <d < 300 pkpc) of high-z galaxies. These distributions are used to calculate the total opacity of the IGM and IGM+CGM and to revise previous measurements of the mean free path of hydrogen-ionizing photons within the IGM. This chapter also considers the effect of the surrounding CGM on the transmission of ionizing photons out of the sites of active star-formation and into the IGM.

This thesis concludes with a brief discussion of work in progress focused on understanding the distribution of metals within the CGM of KBSS galaxies. Appendix B discusses my contributions to the MOSFIRE instrumentation project.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Randomness-efficient curve sampling

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the sampled curve is still low-degree. This property is often used in combination with the sampling property and has found many applications, including PCP constructions, local decoding of codes, and algebraic PRG constructions.

The randomness complexity of curve samplers is a crucial parameter for its applications. It is known that (non-explicit) curve samplers using O(log N + log(1/δ)) random bits exist, where N is the domain size and δ is the confidence error. The question of explicitly constructing randomness-efficient curve samplers was first raised in [TU06] where they obtained curve samplers with near-optimal randomness complexity.

In this thesis, we present an explicit construction of low-degree curve samplers with optimal randomness complexity (up to a constant factor) that sample curves of degree (m log_q(1/δ))^O(1) in F_q^m. Our construction is a delicate combination of several components, including extractor machinery, limited independence, iterated sampling, and list-recoverable codes.

High-strain composites and dual-matrix composite structures

Most space applications require deployable structures due to the limiting size of current launch vehicles. Specifically, payloads in nanosatellites such as CubeSats require very high compaction ratios due to the very limited space available in this typo of platform. Strain-energy-storing deployable structures can be suitable for these applications, but the curvature to which these structures can be folded is limited to the elastic range. Thanks to fiber microbuckling, high-strain composite materials can be folded into much higher curvatures without showing significant damage, which makes them suitable for very high compaction deployable structure applications. However, in applications that require carrying loads in compression, fiber microbuckling also dominates the strength of the material. A good understanding of the strength in compression of high-strain composites is then needed to determine how suitable they are for this type of application.

The goal of this thesis is to investigate, experimentally and numerically, the microbuckling in compression of high-strain composites. Particularly, the behavior in compression of unidirectional carbon fiber reinforced silicone rods (CFRS) is studied. Experimental testing of the compression failure of CFRS rods showed a higher strength in compression than the strength estimated by analytical models, which is unusual in standard polymer composites. This effect, first discovered in the present research, was attributed to the variation in random carbon fiber angles respect to the nominal direction. This is an important effect, as it implies that microbuckling strength might be increased by controlling the fiber angles. With a higher microbuckling strength, high-strain materials could carry loads in compression without reaching microbuckling and therefore be suitable for several space applications.

A finite element model was developed to predict the homogenized stiffness of the CFRS, and the homogenization results were used in another finite element model that simulated a homogenized rod under axial compression. A statistical representation of the fiber angles was implemented in the model. The presence of fiber angles increased the longitudinal shear stiffness of the material, resulting in a higher strength in compression. The simulations showed a large increase of the strength in compression for lower values of the standard deviation of the fiber angle, and a slight decrease of strength in compression for lower values of the mean fiber angle. The strength observed in the experiments was achieved with the minimum local angle standard deviation observed in the CFRS rods, whereas the shear stiffness measured in torsion tests was achieved with the overall fiber angle distribution observed in the CFRS rods.

High strain composites exhibit good bending capabilities, but they tend to be soft out-of-plane. To achieve a higher out-of-plane stiffness, the concept of dual-matrix composites is introduced. Dual-matrix composites are foldable composites which are soft in the crease regions and stiff elsewhere. Previous attempts to fabricate continuous dual-matrix fiber composite shells had limited performance due to excessive resin flow and matrix mixing. An alternative method, presented in this thesis uses UV-cure silicone and fiberglass to avoid these problems. Preliminary experiments on the effect of folding on the out-of-plane stiffness are presented. An application to a conical log-periodic antenna for CubeSats is proposed, using origami-inspired stowing schemes, that allow a conical dual-matrix composite shell to reach very high compaction ratios.

Mean-speed measurements in two-dimensional, incompressible, fully-developed turbulent channel flow

Mean velocity profiles were measured in the 5” x 60” wind channel of the turbulence laboratory at the GALCIT, by the use of a hot-wire anemometer. The repeatability of results was established, and the accuracy of the instrumentation estimated. Scatter of experimental results is a little, if any, beyond this limit, although some effects might be expected to arise from variations in atmospheric humidity, no account of this factor having been taken in the present work. Also, slight unsteadiness in flow conditions will be responsible for some scatter.

Irregularities of a hot-wire in close proximity to a solid boundary at low speeds were observed, as have already been found by others.

That Kármán’s logarithmic law holds reasonably well over the main part of a fully developed turbulent flow was checked, the equation u/u_t = 6.0 + 6.25 log₁₀ y^ut/v being obtained, and, as has been previously the case, the experimental points do not quite form one straight line in the region where viscosity effects are small. The values of the constants for this law for the best over-all agreement were determined and compared with those obtained by others.

The range of Reynolds numbers used (based on half-width of channel) was from 20,000 to 60,000.

Observation and interpretation of tectonic strain release mechanisms

In four chapters various aspects of earthquake source are studied.

Chapter I

Surface displacements that followed the Parkfield, 1966, earthquakes were measured for two years with six small-scale geodetic networks straddling the fault trace. The logarithmic rate and the periodic nature of the creep displacement recorded on a strain meter made it possible to predict creep episodes on the San Andreas fault. Some individual earthquakes were related directly to surface displacement, while in general, slow creep and aftershock activity were found to occur independently. The Parkfield earthquake is interpreted as a buried dislocation.

Chapter II

The source parameters of earthquakes between magnitude 1 and 6 were studied using field observations, fault plane solutions, and surface wave and S-wave spectral analysis. The seismic moment, M_O, was found to be related to local magnitude, M_L, by log M_O = 1.7 M_L + 15.1. The source length vs magnitude relation for the San Andreas system found to be: M_L = 1.9 log L - 6.7. The surface wave envelope parameter AR gives the moment according to log M_O = log AR₃₀₀ + 30.1, and the stress drop, τ, was found to be related to the magnitude by τ = 0.54 M - 2.58. The relation between surface wave magnitude M_S and M_L is proposed to be M_S = 1.7 M_L - 4.1. It is proposed to estimate the relative stress level (and possibly the strength) of a source-region by the amplitude ratio of high-frequency to low-frequency waves. An apparent stress map for Southern California is presented.

Chapter III

Seismic triggering and seismic shaking are proposed as two closely related mechanisms of strain release which explain observations of the character of the P wave generated by the Alaskan earthquake of 1964, and distant fault slippage observed after the Borrego Mountain, California earthquake of 1968. The Alaska, 1964, earthquake is shown to be adequately described as a series of individual rupture events. The first of these events had a body wave magnitude of 6.6 and is considered to have initiated or triggered the whole sequence. The propagation velocity of the disturbance is estimated to be 3.5 km/sec. On the basis of circumstantial evidence it is proposed that the Borrego Mountain, 1968, earthquake caused release of tectonic strain along three active faults at distances of 45 to 75 km from the epicenter. It is suggested that this mechanism of strain release is best described as "seismic shaking."

Chapter IV

The changes of apparent stress with depth are studied in the South American deep seismic zone. For shallow earthquakes the apparent stress is 20 bars on the average, the same as for earthquakes in the Aleutians and on Oceanic Ridges. At depths between 50 and 150 km the apparent stresses are relatively high, approximately 380 bars, and around 600 km depth they are again near 20 bars. The seismic efficiency is estimated to be 0.1. This suggests that the true stress is obtained by multiplying the apparent stress by ten. The variation of apparent stress with depth is explained in terms of the hypothesis of ocean floor consumption.

SteelConverter and Caltech VirtualShaker: Rapid Nonlinear Cloud-Based Structural Model Conversion and Analysis

STEEL, the Caltech created nonlinear large displacement analysis software, is currently used by a large number of researchers at Caltech. However, due to its complexity, lack of visualization tools (such as pre- and post-processing capabilities) rapid creation and analysis of models using this software was difficult. SteelConverter was created as a means to facilitate model creation through the use of the industry standard finite element solver ETABS. This software allows users to create models in ETABS and intelligently convert model information such as geometry, loading, releases, fixity, etc., into a format that STEEL understands. Models that would take several days to create and verify now take several hours or less. The productivity of the researcher as well as the level of confidence in the model being analyzed is greatly increased.

It has always been a major goal of Caltech to spread the knowledge created here to other universities. However, due to the complexity of STEEL it was difficult for researchers or engineers from other universities to conduct analyses. While SteelConverter did help researchers at Caltech improve their research, sending SteelConverter and its documentation to other universities was less than ideal. Issues of version control, individual computer requirements, and the difficulty of releasing updates made a more centralized solution preferred. This is where the idea for Caltech VirtualShaker was born. Through the creation of a centralized website where users could log in, submit, analyze, and process models in the cloud, all of the major concerns associated with the utilization of SteelConverter were eliminated. Caltech VirtualShaker allows users to create profiles where defaults associated with their most commonly run models are saved, and allows them to submit multiple jobs to an online virtual server to be analyzed and post-processed. The creation of this website not only allowed for more rapid distribution of this tool, but also created a means for engineers and researchers with no access to powerful computer clusters to run computationally intensive analyses without the excessive cost of building and maintaining a computer cluster.

In order to increase confidence in the use of STEEL as an analysis system, as well as verify the conversion tools, a series of comparisons were done between STEEL and ETABS. Six models of increasing complexity, ranging from a cantilever column to a twenty-story moment frame, were analyzed to determine the ability of STEEL to accurately calculate basic model properties such as elastic stiffness and damping through a free vibration analysis as well as more complex structural properties such as overall structural capacity through a pushover analysis. These analyses showed a very strong agreement between the two softwares on every aspect of each analysis. However, these analyses also showed the ability of the STEEL analysis algorithm to converge at significantly larger drifts than ETABS when using the more computationally expensive and structurally realistic fiber hinges. Following the ETABS analysis, it was decided to repeat the comparisons in a software more capable of conducting highly nonlinear analysis, called Perform. These analyses again showed a very strong agreement between the two softwares in every aspect of each analysis through instability. However, due to some limitations in Perform, free vibration analyses for the three story one bay chevron brace frame, two bay chevron brace frame, and twenty story moment frame could not be conducted. With the current trend towards ultimate capacity analysis, the ability to use fiber based models allows engineers to gain a better understanding of a building’s behavior under these extreme load scenarios.

Following this, a final study was done on Hall’s U20 structure [1] where the structure was analyzed in all three softwares and their results compared. The pushover curves from each software were compared and the differences caused by variations in software implementation explained. From this, conclusions can be drawn on the effectiveness of each analysis tool when attempting to analyze structures through the point of geometric instability. The analyses show that while ETABS was capable of accurately determining the elastic stiffness of the model, following the onset of inelastic behavior the analysis tool failed to converge. However, for the small number of time steps the ETABS analysis was converging, its results exactly matched those of STEEL, leading to the conclusion that ETABS is not an appropriate analysis package for analyzing a structure through the point of collapse when using fiber elements throughout the model. The analyses also showed that while Perform was capable of calculating the response of the structure accurately, restrictions in the material model resulted in a pushover curve that did not match that of STEEL exactly, particularly post collapse. However, such problems could be alleviated by choosing a more simplistic material model.

Entrainment of fine sediments by turbulent flows

A study was made of the means by which turbulent flows entrain sediment grains from alluvial stream beds. Entrainment was considered to include both the initiation of sediment motion and the suspension of grains by the flow. Observations of grain motion induced by turbulent flows led to the formulation of an entrainment hypothesis. It was based on the concept of turbulent eddies disrupting the viscous sublayer and impinging directly onto the grain surface. It is suggested that entrainment results from the interaction between fluid elements within an eddy and the sediment grains.

A pulsating jet was used to simulate the flow conditions in a turbulent boundary layer. Evidence is presented to establish the validity of this representation. Experiments were made to determine the dependence of jet strength, defined below, upon sediment and fluid properties. For a given sediment and fluid, and fixed jet geometry there were two critical values of jet strength: one at which grains started to roll across the bed, and one at which grains were projected up from the bed. The jet strength K, is a function of the pulse frequency, ω, and the pulse amplitude, A, defined by

K = Aω^-s

Where s is the slope of a plot of log A against log ω. Pulse amplitude is equal to the volume of fluid ejected at each pulse divided by the cross sectional area of the jet tube.

Dimensional analysis was used to determine the parameters by which the data from the experiments could be correlated. Based on this, a method was devised for computing the pulse amplitude and frequency necessary either to move or project grains from the bed for any specified fluid and sediment combination.

Experiments made in a laboratory flume with a turbulent flow over a sediment bed are described. Dye injection was used to show the presence, in a turbulent boundary layer, of two important aspects of the pulsating jet model and the impinging eddy hypothesis. These were the intermittent nature of the sublayer and the presence of velocities with vertical components adjacent to the sediment bed.

A discussion of flow conditions, and the resultant grain motion, that occurred over sediment beds of different form is given. The observed effects of the sediment and fluid interaction are explained, in each case, in terms of the entrainment hypothesis.

The study does not suggest that the proposed entrainment mechanism is the only one by which grains can be entrained. However, in the writer’s opinion, the evidence presented strongly suggests that the impingement of turbulent eddies onto a sediment bed plays a dominant role in the process.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Solubility and Diffusivity of Water in Lunar Basalt, and Chemical Zonation in Olivine-Hosted Melt Inclusions

I report the solubility and diffusivity of water in lunar basalt and an iron-free basaltic analogue at 1 atm and 1350 °C. Such parameters are critical for understanding the degassing histories of lunar pyroclastic glasses. Solubility experiments have been conducted over a range of fO₂ conditions from three log units below to five log units above the iron-wüstite buffer (IW) and over a range of pH₂/pH₂O from 0.03 to 24. Quenched experimental glasses were analyzed by Fourier transform infrared spectroscopy (FTIR) and secondary ionization mass spectrometry (SIMS) and were found to contain up to ~420 ppm water. Results demonstrate that, under the conditions of our experiments: (1) hydroxyl is the only H-bearing species detected by FTIR; (2) the solubility of water is proportional to the square root of pH₂O in the furnace atmosphere and is independent of fO₂ and pH₂/pH₂O; (3) the solubility of water is very similar in both melt compositions; (4) the concentration of H₂ in our iron-free experiments is <3 ppm, even at oxygen fugacities as low as IW-2.3 and pH₂/pH₂O as high as 24; and (5) SIMS analyses of water in iron-rich glasses equilibrated under variable fO₂ conditions can be strongly influenced by matrix effects, even when the concentrations of water in the glasses are low. Our results can be used to constrain the entrapment pressure of the lunar melt inclusions of Hauri et al. (2011).

Diffusion experiments were conducted over a range of fO₂ conditions from IW-2.2 to IW+6.7 and over a range of pH₂/pH₂O from nominally zero to ~10. The water concentrations measured in our quenched experimental glasses by SIMS and FTIR vary from a few ppm to ~430 ppm. Water concentration gradients are well described by models in which the diffusivity of water (D^*_water) is assumed to be constant. The relationship between D^*_water and water concentration is well described by a modified speciation model (Ni et al. 2012) in which both molecular water and hydroxyl are allowed to diffuse. The success of this modified speciation model for describing our results suggests that we have resolved the diffusivity of hydroxyl in basaltic melt for the first time. Best-fit values of D^*_water for our experiments on lunar basalt vary within a factor of ~2 over a range of pH₂/pH₂O from 0.007 to 9.7, a range of fO₂ from IW-2.2 to IW+4.9, and a water concentration range from ~80 ppm to ~280 ppm. The relative insensitivity of our best-fit values of D^*_water to variations in pH₂ suggests that H₂ diffusion was not significant during degassing of the lunar glasses of Saal et al. (2008). D^*_water during dehydration and hydration in H₂/CO₂ gas mixtures are approximately the same, which supports an equilibrium boundary condition for these experiments. However, dehydration experiments into CO₂ and CO/CO₂ gas mixtures leave some scope for the importance of kinetics during dehydration into H-free environments. The value of D^*_water chosen by Saal et al. (2008) for modeling the diffusive degassing of the lunar volcanic glasses is within a factor of three of our measured value in our lunar basaltic melt at 1350 °C.

In Chapter 4 of this thesis, I document significant zonation in major, minor, trace, and volatile elements in naturally glassy olivine-hosted melt inclusions from the Siqueiros Fracture Zone and the Galapagos Islands. Components with a higher concentration in the host olivine than in the melt (MgO, FeO, Cr₂O₃, and MnO) are depleted at the edges of the zoned melt inclusions relative to their centers, whereas except for CaO, H₂O, and F, components with a lower concentration in the host olivine than in the melt (Al₂O₃, SiO₂, Na₂O, K₂O, TiO₂, S, and Cl) are enriched near the melt inclusion edges. This zonation is due to formation of an olivine-depleted boundary layer in the adjacent melt in response to cooling and crystallization of olivine on the walls of the melt inclusions concurrent with diffusive propagation of the boundary layer toward the inclusion center.

Concentration profiles of some components in the melt inclusions exhibit multicomponent diffusion effects such as uphill diffusion (CaO, FeO) or slowing of the diffusion of typically rapidly diffusing components (Na₂O, K₂O) by coupling to slow diffusing components such as SiO₂ and Al₂O₃. Concentrations of H2O and F decrease towards the edges of some of the Siqueiros melt inclusions, suggesting either that these components have been lost from the inclusions into the host olivine late in their cooling histories and/or that these components are exhibiting multicomponent diffusion effects.

A model has been developed of the time-dependent evolution of MgO concentration profiles in melt inclusions due to simultaneous depletion of MgO at the inclusion walls due to olivine growth and diffusion of MgO in the melt inclusions in response to this depletion. Observed concentration profiles were fit to this model to constrain their thermal histories. Cooling rates determined by a single-stage linear cooling model are 150–13,000 °C hr^-1 from the liquidus down to ~1000 °C, consistent with previously determined cooling rates for basaltic glasses; compositional trends with melt inclusion size observed in the Siqueiros melt inclusions are described well by this simple single-stage linear cooling model. Despite the overall success of the modeling of MgO concentration profiles using a single-stage cooling history, MgO concentration profiles in some melt inclusions are better fit by a two-stage cooling history with a slower-cooling first stage followed by a faster-cooling second stage; the inferred total duration of cooling from the liquidus down to ~1000 °C is 40 s to just over one hour.

Based on our observations and models, compositions of zoned melt inclusions (even if measured at the centers of the inclusions) will typically have been diffusively fractionated relative to the initially trapped melt; for such inclusions, the initial composition cannot be simply reconstructed based on olivine-addition calculations, so caution should be exercised in application of such reconstructions to correct for post-entrapment crystallization of olivine on inclusion walls. Off-center analyses of a melt inclusion can also give results significantly fractionated relative to simple olivine crystallization.

All melt inclusions from the Siqueiros and Galapagos sample suites exhibit zoning profiles, and this feature may be nearly universal in glassy, olivine-hosted inclusions. If so, zoning profiles in melt inclusions could be widely useful to constrain late-stage syneruptive processes and as natural diffusion experiments.

Bicyclo[3.2.0]hepta-1,4,6-triene: synthesis, thermal rearrangement, and anion formation

In order to determine the properties of the bicycloheptatrienyl anion (Ia) (predicted to be conjugatively stabilized by Hückel Molecular Orbital Theory) the neutral precursor, bicyclo[3. 2. 0] hepta-1, 4, 6-triene (I) was prepared by the following route.

Reaction of I with potassium-t-butoxide, potassium, or lithium dicyclohexylamide gave anion Ia in very low yield. Reprotonation of I was found to occur solely at the 1 or 5 position to give triene II, isolated as to its dimers.

A study of the acidity of I and of other conjugated hydrocarbons by means of ion cyclotron resonance spectroscopy resulted in determination of the following order of relative acidities:

H₂S ˃ C₅H₆ ˃ CH₃NO₂ ˃ 1, 4- C₅H₈ ˃ I ˃ C₂H₅OH ˃ H₂O; cyclo-C₇H₈ ˃ C₂ H₅OH; фCH₃ ˃ CH₃OH

In addition, limits for the proton affinities of the conjugate bases were determined:

350 kcal/mole ˂ PA(C₅ H₅^-) ˂ 360 kcal/mole

362 kcal/mole ˂ PA(C₅H₇^-, Ia, cyclo-C₇H₇^-) ˂ 377 kcal/mole PA(фCH₂^-) ˂ 385 kcal/mole

Gas phase kinetics of the trans-XVIII to I transformation gave the following activation parameters: E_a = 43.0 kcal/mole, log A = 15.53 and ∆S^ǂ (220°) = 9.6 cu. The results were interpreted as indicating initial 1,2 bond cleavage to give the 1,3-diradical which closed to I. Similar studies on cis-XVIII gave results consistent with a surface component to the reaction (E_a = 22.7 kcal/mole; log A = 9.23, ∆S^ǂ (119°) = -18.9 eu).

The low pressure (0.01 to 1 torr) pyrolysis of trans-XVIII gave in addition to I, fulvenallene (LV), ethynylcyclopentadiene (LVI) and heptafulvalene (LVII). The relative ratios of the C₇H₆ isomers were found to be dependent upon temperature and pressure, higher relative pressure and lower temperatures favoring formation of I. The results were found to be consistent with the intermediacy of vibrationally excited I and subsequent reaction to give LV and LVI.