Thesis (Ph. D.)--California Institute of Technology, 1977. UM #77-24,050. / Advisor names found in the Acknowledgments pages of the thesis. Title from home page (viewed 03/09/2010). Includes bibliographical references.

Bloch analysis was originally developed by Felix Bloch to solve Schrödinger's equation for the electron wave function in a periodic potential field, such as that found in a pristine crystalline solid. His method has since been adapted to study elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In this thesis, we detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. We show conclusively that translational invariance is not a proper justification for invoking the existence of a "propagation constant," and that in nonlinear media this results in a flawed analysis. Next, we propose a general framework for applying Bloch analysis in damped systems and investigate the effect of damping on dispersion curves. In the context of Schrödinger's equation, damping is absent and energy is conserved. In the damped setting, application of Bloch analysis is not straight-forward and requires additional considerations in order to obtain valid results. Results are presented in which the approach is applied to example structures. These results reveal that damping may introduce wavenumber band gaps and bending of dispersion curves such that two or more temporal frequencies exist for each dispersion curve and wavenumber. We close the thesis by deriving conditions which predict the number of wavevectors at each frequency in a dispersion relation. This has important implications for the number of nearest neighbor interactions that must be included in a model in order to obtain dispersion predictions which match experiment.

The analysis of wave propagation in solids with complex microstructures, and local heterogeneities finds extensive applications in areas such as material characterization, structural health monitoring (SHM), and metamaterial design. Within continuum mechanics, sources of heterogeneities are typically associated to localized defects in structural components, or to periodic microstructures in phononic crystals and acoustic metamaterials. Numerical analysis often requires computational meshes which are refined enough to resolve the wavelengths of deformation and to properly capture the fine geometrical features of the heterogeneities. It is common for the size of the microstructure to be small compared to the dimensions of the structural component under investigation, which suggests multiscale analysis as an effective approach to minimize computational costs while retaining predictive accuracy. This research proposes a multiscale framework for the efficient analysis of the dynamic behavior of heterogeneous solids. The developed methodology, called Geometric Multiscale Finite Element Method (GMsFEM), is based on the formulation of multi-node elements with numerically computed shape functions. Such shape functions are capable to explicitly model the geometry of heterogeneities at sub-elemental length scales, and are computed to automatically satisfy compatibility of the solution across the boundaries of adjacent elements. Numerical examples illustrate the approach and validate it through comparison with available analytical and numerical solutions. The developed methodology is then applied to the analysis of periodic media, structural lattices, and phononic crystal structures. Finally, GMsFEM is exploited to study the interaction of guided elastic waves and defects in plate structures.

D. Tech. Mathematical Technology. / Investigates hyperbolic and pseudohyperbolic equations and the results are applied to higher-order rod approximations for the propagation of the longitudinal stress waves in elastic rods. The main objectives of this thesis are: 1. Provide a unified approach to the derivation of the families of one-dimensional hyperbolic differential equations simultaneously with the associated natural and essential boundary conditions describing longitudinal vibration of finite length rods. 2. Establish a new theoremto shorten the derivation of equations of motion and the corresponding boundary conditions, modelling longitudinal wave propagation in the rod. 3. Prove that, when deriving the higher-order rod equations, the lower-order are still included, thus increasing the number of deformations in the rod or the accuracy of the model. 4. Provide mathematical tools for the classification of the obtained equations. 5. Compare the accuracy of the above-mentioned vibration theories in elastic rods based on the investigation of their frequency spectrums which are not available in the literature. 6. Show how two of the above vibration theories, the Rayleigh-Bishop and Mindlin-Herrmann theories, can be applied to predict wave propagation in a piezoelectric circular cylinder and isotropic conical rod. In both cases a numerical example is given as a simulation of the solution.7. Find general methods for solving problems of longitudinal vibration of finite length rods for all of the above-mentioned theories.

This study extends a recently-developed [1] cellular automata (CA) elastodynamic modeling approach to arbitrary two-dimensional geometries through development of a rule set appropriate for triangular cells. The approach is fully object-oriented (OO) and exploits OO conventions to produce compact, general, and easily-extended CA classes. Meshes composed of triangular cells allow the elastodynamic response of arbitrary two-dimensional geometries to be computed accurately and efficiently. As in the previous rectangular CA method, each cell represents a state machine which updates in a stepped-manner using a local "bottom-up" rule set and state input from neighboring cells. The approach avoids the need to develop partial differential equations and the complexity therein. Several advantages result from the method's discrete, local and object-oriented nature, including the ability to compute on a massively-parallel basis and to easily add or subtract cells in a multi-resolution manner. The extended approach is used to generate the elastodynamic responses of a variety of general geometries and loading cases (Dirichlet and Nuemann), which are compared to previous results and/or comparison results generated using the commercial finite element code, COMSOL. These include harmonic interior domain loading, uniform boundary traction, and ramped boundary displacement. Favorable results are reported in all cases, with the CA approach requiring fewer degrees of freedom to achieve similar or better accuracy, and considerably less code development.

Les phénomènes de diffraction et de diffusion des ondes jouent un rôle important dans l'interprétation de la coda des sismogrammes. Par conséquent, une compréhension approfondie des mécanismes de diffraction et de leurs influences sur la propagation des ondes est une étape fondamentale vers l'identification des propriétés statistiques d'un milieu aléatoire. Cette thèse porte sur la diffraction des ondes élastiques dans des milieux aléatoirement hétérogènes avec un comportement local isotrope. On s'intéresse au régime où: La longueur d'onde est du même ordre de grandeur que la longueur de corrélation, la longueur d'onde est petite comparé à la distance de propagation (haute-fréquence) et l'amplitude des fluctuations est petite. Une approche cinétique basée sur les équations de transfert radiatif des ondes élastiques est adoptée. La première partie de cette thèse décrit une analyse détaillée de l'influence de la structure de corrélation sur les paramètres de diffraction et sur l'établissement d'un régime de diffusion. La seconde partie présente les simulations éléments spectraux à grande échelle des ondes élastiques afin d'observer numériquement l'apparition d'un régime d'équipartition. Des analyses théoriques ainsi que des simulations montrent également une nouvelle approche pour l'identification des propriétés statistiques du milieu. / Scattering and diffusion phenomena play a crucial role in the interpretation of the coda part ofseismograms. Consequently, a profound understanding of scattering mechanisms and their effectson wave propagation is a fundamental step towards the identification of the statistical propertiesof random media. The focus of this work is on the scattering of elastic waves in a randomly heterogeneousmedia with locally isotropic material behavior. The weakly heterogeneous regime isconsidered, in which the wave length is similar to the correlation length, the wave length is smallcompared to the propagation length (high frequency) and the amplitude of the heterogeneities issmall. A kinetic framework based on the transport equations of elastic waves is adopted. Thefirst part of the thesis describes a detailed analysis of the influence of the correlation structure onthe scattering parameters and on the arising of the diffusion regime. The second part presentslarge scale spectral element simulations of elastic waves to observe numerically the onset of theequipartitioning regime. The theoretical analyses and simulations also reveal a novel approach toidentify local properties of the heterogeneous medium.

The present research is concerned with developing analysis methods for analyzing and exploring finite-amplitude elastic wave propagation through periodic media. Periodic arrangements of materials with high acoustic impedance contrasts can be employed to control wave propagation. These systems are often termed phononic crystals or metamaterials, depending on the specific design and purpose. Design of these systems usually relies on computation and analysis of dispersion band structures which contain information about wave propagation speed and direction. The location and influence of complete (and partial) band gaps is a particularly interesting characteristic. Wave propagation is prohibited for frequencies that correspond to band gaps; thus, periodic systems behave as filters, wave guides, and lenses at certain frequencies. Controlling these behaviors has typically been limited to the manufacturing stage or the application of external stimuli to distort material configurations. The inclusion of nonlinear elements in periodic unit cells offers an option for passive tuning of the dispersion band structure through amplitude-dependence. Hence, dispersion analysis methods which may be utilized in the design of nonlinear phononic crystals and metamaterials are required. The approach taken herein utilizes Bloch wave-based perturbation analysis methods for obtaining closed-form expressions for dispersion amplitude-dependence. The influence of material and geometric nonlinearities on the dispersion relationship is investigated. It is shown that dispersion shifts result from both self-action (monochromatic excitation) and wave-interaction (multi-frequency excitation), the latter enabling dynamic anisotropy in periodic media. A particularly novel aspect of this work is the ease with which band structures of discretized systems may be analyzed. This connection enables topology optimization of unit cells with nonlinear elements. Several important periodic systems are considered including monoatomic lattices, multilayer materials, and plane stress matrix-inclusion configurations. The analysis methods are further developed into a procedure which can be implemented numerically with existing finite-element analysis software for analyzing geometrically-complex materials.

This work is a numerical and analytical study of wave motion through dynamic materials (DM). This work focuses on showing several results that greatly extend the applicability of the checkerboard focusing effect. First, it is shown that it is possible to simultaneously focus dilatation and shear waves propagating through a linear elastic checkerboard structure. Next, it is shown that the focusing effect found for the original €œperfect€� checkerboard extends to the case of the checkerboard with smooth transitions between materials, this is termed a functionally graded (FG) checkerboard. With the additional assumption of a linear transition region, it is shown that there is a region of existence for limit cycles that takes the shape of a parallelogram in (m,n)-space. Similar to the perfect case, this is termed a €œplateau€� region. This shows that the robustness of the characteristic focusing effect is preserved even when the interfaces between materials are relaxed. Lastly, by using finite volume methods with limiting and adaptive mesh refinement, it is shown that energy accumulation is present for the functionally graded checkerboard as well as for the checkerboard with non-matching wave impedances. The main contribution of this work was to show that the characteristic focusing effect is highly robust and exists even under much more general assumptions than originally made. Furthermore, it provides a tool to assist future material engineers in constructing such structures. To this effect, exact bounds are given regarding how much the original perfect checkerboard structure can be spoiled before losing the expected characteristic focusing behavior.

We are concerned with the high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveforms of the medium's response to drive the inverse problem. Specifically, we use a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to the proposed inversion schemes. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). The PML is a buffer zone that surrounds the domain of interest, and enforces the decay of outgoing waves. In order to resolve the forward problem, we present a hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. We discuss several time-integration schemes, including an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We report numerical results demonstrating stability and efficacy of the forward wave solver, and also provide examples attesting to the successful reconstruction of the two Lame parameters for both smooth and sharp profiles, using synthetic records. We also report the details of two field experiments, whose records we subsequently used to drive the developed inversion algorithms in order to characterize the sites where the field experiments took place. We contrast the full-waveform-based inverted site profile against a profile obtained using the Spectral-Analysis-of-Surface-Waves (SASW) method, in an attempt to compare our methodology against a widely used concurrent inversion approach. We also compare the inverted profiles, at select locations, with the results of independently performed, invasive, Cone Penetrometer Tests (CPTs). Overall, whether exercised by synthetic or by physical data, the full-waveform inversion method we discuss herein appears quite promising for the robust subsurface imaging of near-surface deposits in support of geotechnical site characterization investigations.

[pt] É notório que a resistência global de uma estrutura composta por várias camadas coladas depende diretamente da qualidade da adesão entre as mesmas. Imperfeições ao longo das interfaces de adesão comprometem significativamente a performance da estrutura. A caracterização não destrutiva destas imperfeições é geralmente tarefa muito difícil. O principal objetivo deste trabalho é apresentar um método analítico-numérico que permite modelar o campo acústico resultante da interação entre ondas ultra-sônicas e interfaces imperfeitas, auxiliando na escolha de parâmetros para o emprego de métodos ultra-sônicos de inspeção. No trabalho, a aproximação quase-estática, proposta por Thompson em meados da década de oitenta,é combinada com o método das perturbações para a modelagem de defeitos localizados ao longo da camada de adesão. O método desenvolvido admite que as camadas sejam feitas de materiais anisotrópicos, e permite a modelagem em altas frequências. Três simulações são apresentadas para exemplificar a aplicação do método. Resultados destas simulações onde feixes ultra-sônicos são incidentes em placas imersas em água revelam frequências e ângulos de incidência para os quais efeitos de espalhamento, que permitem a caracterização de defeitos localizados, são mais significativos. / [en] It is well known that the global strength of multi-layered composite structures strongly depends on the quality of the adhesion between its constituent elements. Imperfections along interfaces of adhesion can strongly compromise structure s performance. The characterization of such defects is a very difficult task. The main goal of this study is the development of an analytic-numerical method to simulate the acoustic field resulting from the interaction between ultrasonic waves and imperfect interfaces, helping in selection of parameters for ultra-sonic inspecting methods. The Quasistatic-approximation (QSA), introduced by Thompson in 1982, is applied together with the perturbation method to allow modelling of interfacial localized aws. A solution algorithm for the problem is developed with the aid of the invariant embedding method. It is applicable to solve wave propagation problems in arbitrarily anisotropic layered plates and it is stable for high frequencies. Three simulations of multi-layered plates immersed in acoustic uid are presented as illustration of the application of the developed method. Results of those simulations indicate the frequencies and angles of incidence where the scattering effects, which allow the characterization of localized defects, are more significant.