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.

Parmi les différents phénomènes responsables de la déformation à court et long terme des roches poreuses, on s’intéresse dans ce travail à l’endommagement d’un calcaire oolitique en régime semi-fragile et sous différentes conditions hydriques. Dans ce but, nous avons élaboré un dispositif expérimental permettant de mesurer les vitesses des ondes ultrasonores et les déformations ainsi que d’écouter les émissions acoustiques sur le même échantillon, sous chargement mécanique et sous différentes conditions hydriques. En particulier, le dispositif expérimental permet de mesurer simultanément, les vitesses des ondes ultrasonores selon 3 directions de propagation (axiale, latérale et hors axe) et 3 polarisations différentes (P et 2 S perpendiculaires), ceci quasi-instantanément, ce qui in fine permet d’enregistrer les évolutions des vitesses des ondes ultrasonores en continu pendant le chargement. Ces mesures de vitesses permettent d’évaluer le tenseur acoustique à tout moment et donc de suivre en continu l’évolution de l’état d’endommagement de la roche. Le comportement endommageable à court et long terme du calcaire est modélisé à l’aide d’un modèle phénoménologique qui est basé sur une généralisation macroscopique des mécanismes microscopiques mis en évidence par les études expérimentaux (glissement-ouverture des fissures existantes, nucléation de nouvelles fissures, propagation et coalescence de fissures). Le modèle à court terme reproduit assez bien le comportement expérimental instantané de la roche (courbes contrainte-déformations et évolution des modules élastiques). Le modèle de comportement à long terme, découplé dans sa formulation du modèle à court terme, permet de reproduire qualitativement les courbes de fluage expérimentales / Among the different phenomena responsible for the short and long term deformation of porous rocks, we have studied in this work the damage of an oolithic limestone in the semi-brittle regime and under different hydrous conditions. For this purpose, we have developed an experimental device allowing the simultaneous and continuous measurement of strains and elastic wave velocities, as well as acoustic emissions, on the same sample under mechanical loading and under different hydrous conditions. Particularly, the experimental setup allows simultaneous and continuous measurement of the five elastic wave velocities in 3 different directions of propagation (axial, lateral and off-axis) and 3 different directions of polarization (P and 2 perpendicular S), this almost instantaneously. These velocity measurements allow to assess the acoustic tensor at any time and thus to continuously monitor the evolution of the damage of the rock. The short and long term damage behavior of the limestone is modelled thanks to a phenomenological model which is based on a macroscopic generalization of the microscopic mechanisms highlighted by the experimental study (sliding-opening of existing cracks, nucleation of new cracks, propagation and coalescence of cracks). The short-term model reproduces very well the instantaneous behavior (stress-strain curves and evolution of elastic moduli). The long-term model, whose formulation is uncoupled from the short-term model, allows reproducing qualitatively the experimental creep curves

The time-dependent elastic wave equation is the foundation of seismology. It is very useful in studying the wave propagation in elastic solids. Simulation of Rayleigh waves, which is governed by the equation, requires high accuracy solutions. Finite difference method have been widely used for the simulation of Rayleigh waves. However, it is not obvious how to effectively impose the free surface boundary condition on a curved surface. On the other hand, discontinuous Galerkin methods are more flexible in handling complex geometries. / In this thesis, we develop a class of discontinuous Galerkin methods for time-dependentelastic wave equation in isotropic homogeneous medium. This method is explicit, locally and globally energy conserving. Also, the L² convergence of this method is optimal and the convergence in energy norm is independent of Poisson's ratio. / Besides, we apply our method to simulate Rayleigh waves on curved free surfaces. We also impose a perfectly matched layer to absorb the outward waves. Numerical examples show that our method can accurately capture features of Rayleigh waves even in a domain with high Poisson's ratio. / 時間依賴型彈性波動方程」是地震學的基礎。這組方程對於波在彈性固體中傳播的研究非常有用。雷利波是由這個方程所控制。模擬雷利波須要有非常準確的解。有限差分法廣泛地應用在雷利波的模擬上，可是如何有效地施加自由表面邊界條件於曲面上的方法並不明顯。另一方面，間斷伽遼金方法能更靈活地處理複雜的幾何形狀。 / 在這篇論文中，我們發展了一類間斷伽遼金方法去求「在均勻各向同性的介質上的時間依賴型彈性波動方程」的解。我們將表明，這種方法是顯式的，局部及全域能量守恆的，而它的收斂是最優的和獨立於泊松比的。 / 除此之外，我們運用這個方法來模擬雷利波在具有起伏的自由表面的半空間模型的傳播。我們會使用完美匹配層去吸收朝外的波動。數值算例反映，即使在高柏松比的介質中，這個方法也可以準確地捕捉雷利波的特徵。 / Lam, Chi Yeung. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 44-47). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

In this dissertation we describe a coordinate scaling technique for the numerical approximation of solutions to certain problems posed on unbounded domains in two and three dimensions. This technique amounts to introducing variable coefficients into the problem, which results in defining a solution coinciding with the solution to the original problem inside a bounded domain of interest and rapidly decaying outside of it. The decay of the solution to the modified problem allows us to truncate the problem to a bounded domain and subsequently solve the finite element approximation problem on a finite domain. The particular problems that we consider are exterior problems for the Laplace equation and the time-harmonic acoustic and elastic wave scattering problems. We introduce a real scaling change of variables for the Laplace equation and experimentally compare its performance to the performance of the existing alternative approaches for the numerical approximation of this problem. Proceeding from the real scaling transformation, we introduce a version of the perfectly matched layer (PML) absorbing boundary as a complex coordinate shift and apply it to the exterior Helmholtz (acoustic scattering) equation. We outline the analysis of the continuous PML problem, discuss the implementation of a numerical method for its approximation and present computational results illustrating its efficiency. We then discuss in detail the analysis of the elastic wave PML problem and its numerical discretiazation. We show that the continuous problem is well-posed for sufficiently large truncation domain, and the discrete problem is well-posed on the truncated domain for a sufficiently small PML damping parameter. We discuss ways of avoiding the latter restriction. Finally, we consider a new non-spherical scaling for the Laplace and Helmholtz equation. We present computational results with such scalings and conduct numerical experiments coupling real scaling with PML as means to increase the efficiency of the PML techniques, even if the damping parameters are small.

Forward modelling of seismic waves is an essential tool in the determination of the underlying structure of the Earth using inversion techniques. Despite recent advances in computer power and memory resources, full 3-D elastic wave modelling continues to place a heavy burden on a typical personal computer. 2.5-D modelling reduces the computational burden while maintaining 3-D wavefield characteristics. In this thesis I present 2.5-D frequency-domain equations of motion for elastic wave modelling in anisotropic media. The reduced set of equations for vertical transversely isotropic media and tilted transversely isotropic media are presented separately. Using the spectral-element method, I develop the equations of motion into readily implemented sub-equations by identifying simple 1-D and 2-D patterns. Some aspects of my computational implementation are unique, in particular the use of a system of dynamically growing binary trees to serve as a system matrix. Using this system, the matrix is automatically stored in compressed row format. I investigate the use of both distributed memory and shared memory super-computers for 3-D modelling and compare the resource use of various matrix solvers. In this thesis I adapt recently developed Perfectly Matched Layer formulations to the 2.5-D elastic case, and find them to be adequate in most situations. I investigate the possiblity of instability in the absorbing layers. Observation of 2.5-D modelling results in the frequency wavenumber domain uncovers polelike behaviour at critical wavenumbers within the spectrum. I demonstrate how this behaviour threatens the accuracy of the inverse Fourier transformed frequency-domain solution. However for inhomogeneous media, under certain conditions the only medium that exhibits pole-like behaviour is the medium containing the source. Further study of the phenomenon shows that in homogeneous, transversely isotropic media, the critical wavenumber values are not dependent on the receiver position, but rather can be predicted using the maximum phase velocities of the media. The recommended strategy for wavenumber sampling is to use dense even spacing of values, to adequately capture the behaviour close to the critical wavenumbers. A further recommendation it to introduce slight attenuation through the use of complex velocities (or elastic constants) to eliminate any pole-like behaviour at the critical values. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1385923 / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2010

The goal of elastography is to map the mechanical properties of soft tissues associated with health and disease. The mechanical property of interest in this work is the complex shear modulus, composed of a real part, the storage modulus, which is a measure of elasticity, and an imaginary part, the loss modulus, which is a measure of viscosity. Together, they determine the speed and attenuation of shear waves in the medium. Elastography techniques based on either ultrasound imaging or MRI can image shear wave propagation and thus are capable of measuring shear wave speed and attenuation. Dispersion, or the frequency-dependence of material parameters, is a primary confounding factor when comparing measurements between different shear wave elastography implementations. Prior attempts at quantifying this frequency-dependence suffered from inaccurate modeling assumptions and low signal-to-noise ratios (SNR). To overcome these limitations, a high-fidelity forward model of shear wave propagation in homogeneous media was developed. The model is an exact semi-analytical solution of Navier's equation and is well-suited for acoustic radiation force impulse shear wave elastography (ARFI-SWE) because it does not require precise knowledge of the strength of the source, nor its spatial or temporal distribution. Unlike models used in ARFI-SWE heretofore, it accounts for the vector polarization of shear waves and exactly represents geometric spreading of the shear wavefield, whether spherical, cylindrical, or neither. Furthermore, it is material-model independent, i.e. it makes no assumption about the frequency-dependence of material parameters. It overcomes the problem of low SNR through spatial averaging and enables estimation of the frequency-dependent complex shear modulus over a wider frequency range than has hitherto been possible. This improved ARFI-SWE was named Shear Wave Rheometry (SWR). By combining SWR with a novel torsional vibration rheometry, dispersion in tissue-mimicking gels was quantified from 1--1800 Hz. The measurements show sizable frequency-dependent variation in the shear modulus of gelatin, a material often assumed to be non-dispersive based on narrow-band measurements. SWR measurements in ex vivo bovine liver tissue yielded complex shear modulus estimates from 25--250 Hz and showed that liver tissue exhibits significant dispersion in this frequency range: a factor of 4 increase in the storage modulus and a factor of 10 increase in the loss modulus. Quality metrics showed that liver tissue can be reasonably approximated as homogeneous and isotropic for ARFI-SWE measurements in this frequency range. Results demonstrate that accounting for dispersion is essential for meaningful comparisons of measurements between systems. Moreover, improved tissue characterization enabled by SWR may have clinical relevance, for example, in the diagnosis and monitoring of chronic liver disease.

This dissertation contains research directed at investigating the behavior and properties of a class of composite materials known as phononic crystals. Two categories of phononic crystals are explicitly investigated: (I) elastic phononic crystals and (II) nano-scale phononic crystals. For elastic phononic crystals, attention is directed at two-dimensional structures. Two specific structures are evaluated (1) a two-dimensional configuration consisting of a square array of cylindrical Polyvinylchloride inclusions in air and (2) a two-dimensional configuration consisting of a square array of steel cylindrical inclusions in epoxy. For the first configuration, a theoretical model is developed to ascertain the necessary band structure and equi-frequency contour features for the realization of phase control between propagating acoustic waves. In contrasting this phononic crystal with a reference system, it is shown that phononic crystals with equifrequency contours showing non-collinear wave and group velocity vectors are ideal systems for controlling the phase between propagating acoustic waves. For the second configuration, it is demonstrated that multiple functions can be realized of a solid/solid phononic crystal. The epoxy/steel phononic crystal is shown to behave as (1) an acoustic wave collimator, (2) a defect-less wave guide, (3) a directional source for elastic waves, (4) an acoustic beam splitter, (5) a phase-control device and (6) a k-space multiplexer. To transition between macro-scale systems (elastic phononic crystals) and nano-scale systems (nano-phononic crystals), a toy model of a one-dimensional chain of masses connected with non-linear, anharmonic springs is utilized. The implementation of this model introduces critical ideas unique to nano-scale systems, particularly the concept of phonon mode lifetime. The nano-scale phononic crystal of interest is a graphene sheet with periodically spaced holes in a triangular array. It is found through equilibrium molecular dynamics simulation techniques, that phonon-boundary collision effects and coherent phononic effects (band-folding) are two competing scattering mechanisms responsible for the reduction of acoustic and optical phonon lifetimes. Conclusions drawn about the lifetime of thermal phonons in phononic crystal patterned graphene are linked with the anharmonic, one-dimensional crystal model.

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.

Thesis (M. S.)--Mechanical Engineering, Georgia Institute of Technology, 2010. / Committee Chair: Dr. Michael Leamy; Committee Member: Dr. Karim Sabra; Committee Member: Dr. Aldo Ferri. Part of the SMARTech Electronic Thesis and Dissertation Collection.