Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

A measurement of inclusive quasi-elastic electron scattering cross sections at high momentum transfer

We have measured inclusive electron-scattering cross sections for targets of ^(4)He, C, Al, Fe, and Au, for kinematics spanning the quasi-elastic peak, with squared, four­ momentum transfers (q^2) between 0.23 and 2.89 (GeV/c)^2. Additional data were measured for Fe with q^2's up to 3.69 (GeV/c)^2 These cross sections were analyzed for the y-scaling behavior expected from a simple, impulse-approximation model, and are found to approach a scaling limit at the highest q^2's. The q^2 approach to scaling is compared with a calculation for infinite nuclear matter, and relationships between the scaling function and nucleon momentum distributions are discussed. Deviations from perfect scaling are used to set limits on possible changes in the size of nucleons inside the nucleus.

Methods of multiparameter inversion of seismic data using the acoustic and elastic born approximations

This thesis presents two different forms of the Born approximations for acoustic and elastic wavefields and discusses their application to the inversion of seismic data. The Born approximation is valid for small amplitude heterogeneities superimposed over a slowly varying background. The first method is related to frequency-wavenumber migration methods. It is shown to properly recover two independent acoustic parameters within the bandpass of the source time function of the experiment for contrasts of about 5 percent from data generated using an exact theory for flat interfaces. The independent determination of two parameters is shown to depend on the angle coverage of the medium. For surface data, the impedance profile is well recovered.

The second method explored is mathematically similar to iterative tomographic methods recently introduced in the geophysical literature. Its basis is an integral relation between the scattered wavefield and the medium parameters obtained after applying a far-field approximation to the first-order Born approximation. The Davidon-Fletcher-Powell algorithm is used since it converges faster than the steepest descent method. It consists essentially of successive backprojections of the recorded wavefield, with angular and propagation weighing coefficients for density and bulk modulus. After each backprojection, the forward problem is computed and the residual evaluated. Each backprojection is similar to a before-stack Kirchhoff migration and is therefore readily applicable to seismic data. Several examples of reconstruction for simple point scatterer models are performed. Recovery of the amplitudes of the anomalies are improved with successive iterations. Iterations also improve the sharpness of the images.

The elastic Born approximation, with the addition of a far-field approximation is shown to correspond physically to a sum of WKBJ-asymptotic scattered rays. Four types of scattered rays enter in the sum, corresponding to P-P, P-S, S-P and S-S pairs of incident-scattered rays. Incident rays propagate in the background medium, interacting only once with the scatterers. Scattered rays propagate as if in the background medium, with no interaction with the scatterers. An example of P-wave impedance inversion is performed on a VSP data set consisting of three offsets recorded in two wells.

A measurement of the proton elastic form factors for 1 ≤ Q^2 ≤ 3 (GeV/c)^2

We report measurements of the proton form factors, G^p_E and G^p_M, extracted from elastic electron scattering in the range 1 ≤ Q^2 ≤ 3 (GeV/c)^2 with uncertainties of <15% in G^p_E and <3% in G^p_M. The results for G^p_E are somewhat larger than indicated by most theoretical parameterizations. The ratio of Pauli and Dirac form factors, Q^2(F^p_2/F^p_1), is lower in value and demonstrates less Q^2 dependence than these parameterizations have indicated. Comparisons are made to theoretical models, including those based on perturbative QCD, vector-meson dominance, QCD sum rules, and diquark constituents to the proton. A global extraction of the form factors, including previous elastic scattering measurements, is also presented.

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.

Part I - The radiation of elastic waves from a spherical cavity in a half space. Part II - Precision determination of focal depths and epicenters of earthquakes.

Part I: The dynamic response of an elastic half space to an explosion in a buried spherical cavity is investigated by two methods. The first is implicit, and the final expressions for the displacements at the free surface are given as a series of spherical wave functions whose coefficients are solutions of an infinite set of linear equations. The second method is based on Schwarz's technique to solve boundary value problems, and leads to an iterative solution, starting with the known expression for the point source in a half space as first term. The iterative series is transformed into a system of two integral equations, and into an equivalent set of linear equations. In this way, a dual interpretation of the physical phenomena is achieved. The systems are treated numerically and the Rayleigh wave part of the displacements is given in the frequency domain. Several comparisons with simpler cases are analyzed to show the effect of the cavity radius-depth ratio on the spectra of the displacements.

Part II: A high speed, large capacity, hypocenter location program has been written for an IBM 7094 computer. Important modifications to the standard method of least squares have been incorporated in it. Among them are a new way to obtain the depth of shocks from the normal equations, and the computation of variable travel times for the local shocks in order to account automatically for crustal variations. The multiregional travel times, largely based upon the investigations of the United States Geological Survey, are confronted with actual traverses to test their validity.

It is shown that several crustal phases provide control enough to obtain good solutions in depth for nuclear explosions, though not all the recording stations are in the region where crustal corrections are considered. The use of the European travel times, to locate the French nuclear explosion of May 1962 in the Sahara, proved to be more adequate than previous work.

A simpler program, with manual crustal corrections, is used to process the Kern County series of aftershocks, and a clearer picture of tectonic mechanism of the White Wolf fault is obtained.

Shocks in the California region are processed automatically and statistical frequency-depth and energy depth curves are discussed in relation to the tectonics of the area.

Fiber-optic integration and efficient detection schemes for optomechanical resonators

With the advent of the laser in the year 1960, the field of optics experienced a renaissance from what was considered to be a dull, solved subject to an active area of development, with applications and discoveries which are yet to be exhausted 55 years later. Light is now nearly ubiquitous not only in cutting-edge research in physics, chemistry, and biology, but also in modern technology and infrastructure. One quality of light, that of the imparted radiation pressure force upon reflection from an object, has attracted intense interest from researchers seeking to precisely monitor and control the motional degrees of freedom of an object using light. These optomechanical interactions have inspired myriad proposals, ranging from quantum memories and transducers in quantum information networks to precision metrology of classical forces. Alongside advances in micro- and nano-fabrication, the burgeoning field of optomechanics has yielded a class of highly engineered systems designed to produce strong interactions between light and motion.

Optomechanical crystals are one such system in which the patterning of periodic holes in thin dielectric films traps both light and sound waves to a micro-scale volume. These devices feature strong radiation pressure coupling between high-quality optical cavity modes and internal nanomechanical resonances. Whether for applications in the quantum or classical domain, the utility of optomechanical crystals hinges on the degree to which light radiating from the device, having interacted with mechanical motion, can be collected and detected in an experimental apparatus consisting of conventional optical components such as lenses and optical fibers. While several efficient methods of optical coupling exist to meet this task, most are unsuitable for the cryogenic or vacuum integration required for many applications. The first portion of this dissertation will detail the development of robust and efficient methods of optically coupling optomechanical resonators to optical fibers, with an emphasis on fabrication processes and optical characterization.

I will then proceed to describe a few experiments enabled by the fiber couplers. The first studies the performance of an optomechanical resonator as a precise sensor for continuous position measurement. The sensitivity of the measurement, limited by the detection efficiency of intracavity photons, is compared to the standard quantum limit imposed by the quantum properties of the laser probe light. The added noise of the measurement is seen to fall within a factor of 3 of the standard quantum limit, representing an order of magnitude improvement over previous experiments utilizing optomechanical crystals, and matching the performance of similar measurements in the microwave domain.

The next experiment uses single photon counting to detect individual phonon emission and absorption events within the nanomechanical oscillator. The scattering of laser light from mechanical motion produces correlated photon-phonon pairs, and detection of the emitted photon corresponds to an effective phonon counting scheme. In the process of scattering, the coherence properties of the mechanical oscillation are mapped onto the reflected light. Intensity interferometry of the reflected light then allows measurement of the temporal coherence of the acoustic field. These correlations are measured for a range of experimental conditions, including the optomechanical amplification of the mechanics to a self-oscillation regime, and comparisons are drawn to a laser system for phonons. Finally, prospects for using phonon counting and intensity interferometry to produce non-classical mechanical states are detailed following recent proposals in literature.

PP backward elastic scattering between 0.70 and 2.37 GeV/c

̄pp backward elastic scattering has been measured for the cos θ_cm region between – 1.00 and – 0.88 and for the incident ̄p laboratory momentum region between 0.70 and 2.37 GeV/c. These measurements, done in intervals of approximately 0.1 GeV/c, have been performed at the Alternating Gradient Synchrotron at Brookhaven National Laboratory during the winter of 1968. The measured differential cross sections, binned in cos θ_cm intervals of 0.02, have statistical errors of about 10%. Backward dipping exists below 0.95 GeV/c and backward peaking above 0.95 GeV/c. The 180˚ differential cross section extrapolated from our data shows a sharp dip centered at 0.95 GeV/c and a broad hump centered near 1.4 GeV/c. Our data have been interpreted in terms of resonance effects and in terms of diffraction dominance effects.

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.

The influence of the NN* channel on elastic NN scattering

The problem of two channels NN and NN*, coupled through unitarity, is studied to see whether sizable peaks can be produced in elastic nucleon-nucleon scattering due to the opening of a strongly coupled inelastic channel. One-pion-exchange (OPE) interactions are calculated to estimate the NN*→NN* and NN→NN* amplitudes. The OPE production amplitudes are used as the sole dynamical input to drive the multichannel ND^-1 equations in the determinental approximation, and the effect on the J = 2+ (¹D₂) elastic NN scattering amplitude is studied as the width of the unstable N* and strength of coupling to the inelastic channel are varied. A cusp-type enhancement appears in the NN channel near the NN* threshold but for the known value of the N* width the cusp is so “wooly” that any resulting elastic peak is likely to be too broad and diminished in height to be experimentally prominent. A brief survey of current experimental knowledge of the real part of the ¹D₂ NN phase shift near the NN* threshold is given, and the values are found to be much smaller than the nearly “resonant” phase shifts predicted by the coupled channel model.

Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution

The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.

The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.

The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.

Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells.

Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed.

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.

Studies of bacteriophage T4 tail fibers and tail fiber genes

The distal half of the bacteriophage T4 tail fiber interacts with the surface of the bacterium during adsorption. The largest polypeptide in this half fiber is the product of gene 37 (P37). During assembly of the tail fiber, P37 interacts with the product of gene 38 (P38). These two gene products are incompatible with the corresponding gene products from the related phage T2. T2 P37 does not interact with T4 P38 and T2 P38 does not interact with T4 P37. Crosses between T2 and T4 phages mutant in genes 37 and 38 have shown that the carboxyl end of P37 interacts with P38 and with the bacterial surface. In the corresponding region of gene 37 and in gene 38 there is no recombination between T2 and T4. In the rest of gene 37 there are two small regions with relatively high recombination and a region of low recombination.

When T2/T4 heteroduplex DNA molecules are examined in the electron microscope four nonhomologous loops appear in the region of genes 37 and 38. Heteroduplexes between hybrid phages which have part of gene 37 from T4 and part from T2 have roughly located gene 37 mutations in the heteroduplex pattern. For a more precise location of the , mutations a physical map of gene 37 was constructed by determining the molecular weights of amber polypeptide fragments on polyacrylamide gels in the presence of sodium dodecyl sulfate. When the physical and heteroduplex maps are aligned, the regions of low recombination correspond to regions of nonhomology between T2 and T4. Regions with relatively high recombination are homologous.

The molecular weight of T2 P37 is about 13,000 greater than that of T4 P37. Analysis of hybrid phage has shown that this molecular weight difference is all at the carboxyl end of P37.

An antiserum has been prepared which is specific for the distal half fiber of T4. Tests of the ability of gene 37 hybrids to block this antiserum show that there are at least 4 subclasses of antigen specified by different parts of P37.

Observations in the electron microscope of the tailfiber - anti- body complexes formed by the gene 37 hybrids and the specific anti- serum have shown that P37 is oriented linearly in the distal half fiber with its N-terminus near the joint between the two half fibers and its C-terminus near the tip of the fiber. These observations lead to a simple model for the structure of the distal half fiber.

The high recombination in T4 gene 34 was also investigated. A comparison of genetic and physical maps of gene 34 showed that there is a gradient of increasing recombination near one end of the gene.

Using Earth Deformation Caused by Surface Mass Loading to Constrain the Elastic Structure of the Crust and Mantle

Surface mass loads come in many different varieties, including the oceans, atmosphere, rivers, lakes, glaciers, ice caps, and snow fields. The loads migrate over Earth's surface on time scales that range from less than a day to many thousand years. The weights of the shifting loads exert normal forces on Earth's surface. Since the Earth is not perfectly rigid, the applied pressure deforms the shape of the solid Earth in a manner controlled by the material properties of Earth's interior. One of the most prominent types of surface mass loading, ocean tidal loading (OTL), comes from the periodic rise and fall in sea-surface height due to the gravitational influence of celestial objects, such as the moon and sun. Depending on geographic location, the surface displacements induced by OTL typically range from millimeters to several centimeters in amplitude, which may be inferred from Global Navigation and Satellite System (GNSS) measurements with sub-millimeter precision. Spatiotemporal characteristics of observed OTL-induced surface displacements may therefore be exploited to probe Earth structure. In this thesis, I present descriptions of contemporary observational and modeling techniques used to explore Earth's deformation response to OTL and other varieties of surface mass loading. With the aim to extract information about Earth's density and elastic structure from observations of the response to OTL, I investigate the sensitivity of OTL-induced surface displacements to perturbations in the material structure. As a case study, I compute and compare the observed and predicted OTL-induced surface displacements for a network of GNSS receivers across South America. The residuals in three distinct and dominant tidal bands are sub-millimeter in amplitude, indicating that modern ocean-tide and elastic-Earth models well predict the observed displacement response in that region. Nevertheless, the sub-millimeter residuals exhibit regional spatial coherency that cannot be explained entirely by random observational uncertainties and that suggests deficiencies in the forward-model assumptions. In particular, the discrepancies may reveal sensitivities to deviations from spherically symmetric, non-rotating, elastic, and isotropic (SNREI) Earth structure due to the presence of the South American craton.