Downloads & Free Reading Options - Results

Shear transformations (i.e., localised rearrangements of particles resulting in the shear deformation of a small region of the sample) are the building blocks of mesoscale models for the flow of disordered solids. In order to compute the time-dependent response of the solid material to such a shear transformation, with a proper account of elastic heterogeneity and shear wave propagation, we propose and implement a very simple Finite-Element (FE) -based method. Molecular Dynamics (MD) simulations of a binary Lennard-Jones glass are used as a benchmark for comparison, and information about the microscopic viscosity and the local elastic constants is directly extracted from the MD system and used as input in FE. We find very good agreement between FE and MD regarding the temporal evolution of the disorder-averaged displacement field induced by a shear transformation, which turns out to coincide with the response of a uniform elastic medium. However, fluctuations are relatively large, and their magnitude is satisfactorily captured by the FE simulations of an elastically heterogeneous system. Besides, accounting for elastic anisotropy on the mesoscale is not crucial in this respect. The proposed method thus paves the way for models of the rheology of amorphous solids which are both computationally efficient and realistic, in that structural disorder and inertial effects are accounted for.

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.

Numerical simulation of wave propagation in an infinite medium is made possible by surrounding a finite region by a perfectly matched layer (PML). Using this approach a generalized three-dimensional (3D) formulation is proposed for time-domain modeling of elastic wave propagation in an unbounded lossless anisotropic medium. The formulation is based on a second-order approach that has the advantages of, physical relationship to the underlying equations, and amenability to be implemented in common numerical schemes. Specifically, our formulation uses three second-order equations of the displacement field and nine auxiliary equations, along with the three time histories of the displacement field. The properties of the PML, which are controlled by a complex two-parameter stretch function, are such that it acts as near perfect absorber. Using finite element method (FEM) 3D numerical results are presented for a highly anisotropic medium. An extension of the formulation to the particular case of a Kelvin-Vogit viscoelastic medium is also presented.

Knowledge of the elastic properties of composite materials can be an invaluable tool for both the quality assurance of manufacturing techniques and design verification. Recent advancements in ultrasonic velocity measurements have demonstrated the ability to recover elastic properties in anisotropic laminates. A simplified experimental setup was investigated to recover the elastic properties based on the flexural wave propagation in anisotropic laminates. The initial objective of this thesis was to verify flexural wave propagation in composite laminates through the comparison of experimental and theoretical phase velocities. In the second part of this thesis, the experimental phase velocities were used to calculate the elastic properties of the material by inverting the governing equations. The initial method used to recover elastic constants was successful in the recovery of a partial set of the bending and extensional stiffnesses. The inability to recover all bending stiffnesses dictated the investigation of a second method. This method used an iterative method based on a nonlinear Newton's method to recover the bending stiffnesses. This method did not converge due to ill conditioning of the solution matrix. Although this method did not converge, it is believed that other more robust methods suggested herein would converge to the proper solution.

This thesis presents a model which simulates the scattering from a fluid loaded I-beam and the resultant behavior due to fluid-structure interaction. Chapter I gives an overview of the problem and describes the characteristics of the solid and fluid, the aspects of periodicity, boundary conditions and the coupling of the two media. The governing equations of motion are scaled in Chapter II. In Chapter III, the finite difference formulae for these equations are derived, as is the non-local radiation boundary condition. Difference formulas for typical boundary points of the solid and corner nodes are also derived. All finite difference formulae used are presented in Appendix C. Chapter IV contains numerical results. Conclusions are drawn and areas of the problem that would require further study are in Chapter V. Finite difference approximation of irregularly shaped domains; wave propagation in solids; wave propagation in fluids; fluid structure interaction; finite difference approximations of a nonlocal radiation boundary condition.

Elastic wave propagation in two dimensions using theory of characteristics

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

Thesis advisors, Clyde Scandrett and V.E. Henson

Thesis advisors, Clyde Scandrett and V.E. Henson

Waveguide Meshes are efficient and versatile models of wave propagation along a multidimensional ideal medium. The choice of the mesh geometry affects both the computational cost and the accuracy of simulations. In this paper, we focus on 2D geometries and use multidimensional sampling theory to compare the square, triangular, and hexagonal meshes in terms of sampling efficiency and dispersion error under conditions of critical sampling. The analysis shows that the triangular geometry exhibits the most desirable tradeoff between accuracy and computational cost.

A general study is given of plane and cylindrical wave propagation of combined stresses in an elastic-plastic medium. The coefficients of the governing differential equations, when written in matrix notation, are symmetric matrices and can be divided into sub-matrices each of which has a special form. The relations between the stresses on both sides of an elastic-plastic boundary are derived. Also presented are the restrictions on the speed of an elastic- plastic boundary.

A general study is given of plane and cylindrical wave propagation of combined stresses in an elastic-plastic medium. The coefficients of the governing differential equations, when written in matrix notation, are symmetric matrices and can be divided into sub-matrices each of which has a special form. The relations between the stresses on both sides of an elastic-plastic boundary are derived. Also presented are the restrictions on the speed of an elastic- plastic boundary.

Weight-adjusted inner products are easily invertible approximations to weighted $L^2$ inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time-domain method for wave propagation which is low storage, energy stable, and high order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight-adjusted DG (WADG) methods to the case of matrix-valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind-like dissipation incorporated through simple penalty fluxes. A semi-discrete convergence analysis is given, and numerical results confirm the stability and high order accuracy of WADG for several problems in elastic wave propagation.

This study is aimed at development and application of a new wave propagation and modeling method for regional waves in heterogeneous crustal waveguides using one-way wave approximation. As the first step, we solve the 2D SH-wave problem. A half-space GSP (Generalized Screen Propagator) is formulated for the SH half space problem. We apply our method to simulating regional wave propagation in different types of complex crustal waveguides including those with small-scale random heterogeneities and random rough interfaces of sedimentary layers. Synthetic seismograms and snapshots are shown to facilitate the study of path effects of Lg waves. Slowness domain analysis is especially useful in investigating energy transfer in crustal waveguides and for determining which part of the energy can be trapped in the waveguide. The influence of crustal heterogeneities and rough interfaces on Lg amplitude attenuation and Lg coda formation are studied.

This study is aimed at development and application of a new wave propagation and modeling method for regional waves in heterogeneous crustal waveguides using one-way wave approximation. As the first step, we solve the 2D SH-wave problem. A half-space GSP (Generalized Screen Propagator) is formulated for the SH half space problem. We apply our method to simulating regional wave propagation in different types of complex crustal waveguides including those with small-scale random heterogeneities and random rough interfaces of sedimentary layers. Synthetic seismograms and snapshots are shown to facilitate the study of path effects of Lg waves. Slowness domain analysis is especially useful in investigating energy transfer in crustal waveguides and for determining which part of the energy can be trapped in the waveguide. The influence of crustal heterogeneities and rough interfaces on Lg amplitude attenuation and Lg coda formation are studied.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

Solution of the wave equation using techniques such as finite difference or finite element methods can model elastic wave propagation in solids. This requires mapping the physical geometry into a computational domain whose size is governed by the size of the physical domain of interest and by the required resolution. This computational domain, in turn, dictates the computer memory requirements as well as the calculation time. Quite often, the physical region of interest is only a part of the whole physical body, and does not necessarily include all the physical boundaries. Reduction of the calculation domain requires positioning an artificial boundary or region where a physical boundary does not exist. It is important however that such a boundary, or region, will not affect the internal domain, i.e., it should not cause reflections that propagate back into the material. This paper concentrates on the issue of constructing such a boundary region.

This document presents the results of a detailed research investigation on a fundamental problem in wave mechanics: the propagation of stress waves in heterogeneous elastic solids. this phenomenon is fundamental to many disciplines in engineering and mathematical physics: seismology, earthquake engineering structure acoustics, composite materials, and many other areas. The theory and methodologies of hierarchical modeling of heterogeneous materials are extended to elastodynamic cases to make possible the control of the modeling error in the local average stress. One-dimensional steady state and transient applications are given.

Literature on longitudinal wave propagation in cylindrical shells is reviewed.

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wavefront followed by random oscillations made of multiply scattered waves. We find that the coherent wavefront is insensitive to details of the packing: force chains do not play an important role in determining this wavefront. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse-propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wavefront scales with pressure as $p^{1/6}$; we compare this result with experimental data on various granular systems where deviations from the $p^{1/6}$ behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

Knowledge of the elastic properties of composite materials can be an invaluable tool for both the quality assurance of manufacturing techniques and design verification. Recent advances in ultrasonic velocity measurements have demonstrated the ability to recover elastic properties in anisotropic laminates. A simplified experimental setup was investigated to recover the elastic properties based upon the flexural wave propagation in anisotropic laminates. The initial objective of this thesis was to verify flexural wave propagation in composite laminates through the comparison of experimental and theoretical phase velocities. In the second part of this thesis, the experimental phase velocities were used to calculate the elastic properties of the material by inverting the governing equations. The initial method used to recover elastic constants was successful in the recovery of a partial set of the bending and extensional stiffnesses. The inability to recover all bending stiffnesses dictated the investigation of a second method. This method used an iterative method based upon a nonlinear Newton's method to recover the bending stiffnesses. This method did not converge due to the ill conditioning of the solution matrix. Although this method did not converge, it is believed that more robust methods suggest herein would converge to the proper solution.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

The field due to a general point source of energy in an isotropic, elastic solid with a free surface is investigated. The development of new plane wave representations for the fundamental solutions of elastodynamics is reported. There are two types of situation involved; one is the simpler type involved in the case of a steady point source which moves steadily with any constant velocity in an elastic medium, this type involves superposition of plane waves with respect to a single parameter; the other is the more complicated transient problem in which a point source is set up at a given moment, and thereafter moves at constant velocity, without change of strength. The new representation for the field of a steadily moving source and for the transient source is used in the calculation of fields and displacements in the presence of a free surface. The application of the new approach to the case of a vertical load, to a horizontal load, and to a couple of arbitrary orientation, and the singularities to be expected for the general point source are discussed.

Develop elastic parabolic equation (EPE) method capabilities in order to characterize effects of elastic propagation mechanisms such as elastic interface scattering, conversion from elastic propagation to acoustic propagation, and intense interface waves on underwater acoustic environments with elastic bottoms or elastic ice cover.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

This thesis presents a model which simulates the scattering from a fluid loaded I-beam and the resultant behavior due to fluid-structure interaction. Chapter I gives an overview of the problem and describes the characteristics of the solid and fluid, the aspects of periodicity, boundary conditions and the coupling of the two media. The governing equations of motion are scaled in Chapter IL In Chapter III, the finite-difference formulae for these equations are derived, as is the non-local radiation boundary condition. Difference formulas for typical boundary points of the solid and corner nodes are also derived. All finite difference formulae used are presented in Appendix C. Chapter IV contains numerical results. Conclusions are drawn and areas of the problem that would require further study are in Chapter V.

The resonance modes and the related effects to the transmission of elastic waves in a two dimensional phononic crystal formed by periodic arrangements of a two blocks unit cell in one direction are studied. The unit cell consists of two asymmetric elliptic cylinders coated with silicon rubber and embedded in a rigid matrix. The modes are obtained by the semi-analytic method in the least square collocation scheme and confirmed by the finite element method simulations. Two resonance modes, corresponding to the vibration of the cylinder along the long and short axes, give rise to resonance reflections of elastic waves. One mode in between the two modes, related to the opposite vibration of the two cylinders in the unit cell in the direction along the layer, results in the total transmission of elastic waves due to zero effective mass density at the frequency. The resonance frequency of this new mode changes continuously with the orientation angle of the elliptic resonator.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

A higher-order parabolic equation (PE) based on a Pade series and an elastic PE are applied to wave propagation in the ocean. In contrast to the standard PE models of underwater acoustics, the higher-order PE provides accurate solutions for problems involving arbitrarily long ranges, propagation nearly normal to the preferred direction, and large variations in sound speed. The most important applications of the higher-order PE are for problems involving elastic ocean bottoms. A new numerical approach based on centered differences is applied to handle interface conditions. The accuracy of the elastic PE is demonstrated with benchmark calculations. The elastic PE is applied to a range-dependent propagation problem. Reprints. (jhd)

Thesis advisor,Michael R. Gorman

Elastic wave propagation in actual fact is the transport of energy in a medium that acts as a sink for this energy. Engineering and physical applications rely upon an understanding of the wave propagation process to model systems and describe the physical world, and, as such one desires to broaden his knowledge of the actual propagation process in real inhomogeneous media. In this work, on examines the vector wave equation developed for an anisotropic, inhomogeneous lossy medium with a set of 'effective' Lame' constants derived from microscopic considerations of the medium and develops a composite loss function, L, to account for the energy loss mechanisms that occur as the wave propagates. This loss function is incorporated into the general equations of motion and series solutions for bulk, and surface waves in the inhomogenous lossy medium are derived. The displacement fields are examined over the frequency regime in the near and far field for the waves. The effective Lame' parameters are included in the solution, and their contribution, as well as the frequency contribution to the wave field is seen. Higher orders of scattering and diffraction are seen as well as the an harmonicites and nonlinear effects of losses. These effects may be observed in the general wavefield description. Also the effect of losses on the dispersion behavior is observed as a frequency shift of the wave.

The long-term goal is to create codes accurately model wave propagation and scattering in shallow water, and to quantify effects of poroelastic, stratified sediment and elastic mode conversion. Also to relate these effects to sonar system behaviour and optimization-based inversion of sediment acoustic and elastic properties.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

This report presents a review of the theory of elastic wave propagation in any number of fluid or solid layered media. Suitable wave equations are derived, and boundary conditions determined by the nature of the media and the character of the interfaces are applied, yielding a formal solution for the transmitted and reflected, longitudinal and transverse wave potentials. Computer-generated graphs are presented which illustrate properties of the solutions in materials of interest in nondestructive evaluation. A tabulation of acoustic-velocity data for media commonly encountered in ultrasonic work is also included.

Develop a new method and the code to simulate 3D acoustic/elastic wave propagation and interaction with the ocean water and ocean bottom environment. The method will be applied to numerical simulations and imaging to study the wave/sea-bottom interaction, energy partitioning, scattering mechanism and other problems that are crucial for many ocean bottom-surveying techniques. Our understanding of shallow water acoustic wave propagation and its interaction with sediments can be improved.

A total of nine investigators completed 32 research projects yielding 22 publications, 10 accepted or submitted, and 5 in preparation. Subject areas were structural analysis (12), wave propagation (13), fluid dynamics (5), and transport theory (5). Topics included higher order buckling in the presence of imperfections, dynamic instability, composite material, wave propagation through nonlinear and random media, vortex motion in fluids, the small mean free path approximation to the transport equations, etc. Asymptotic, perturbation and bifurcation techniques predominated.

It is shown that information transmission inside a cell can occur by means of mechanical waves transmitted through DNA. The propagation of the waves is strongly dependent on the shape of the DNA thus proteins that change the shape of DNA can alter signal transmission. The overall effect is a method of signal processing by DNA binding proteins that creates a "cellular communications network". The propagation of small amplitude disturbances through DNA is treated according to the mechanical theory of elastic rods. According to the theory four types of mechanical waves affecting extension(compression), twist, bend or shear can propagate through DNA. Each type of wave has unique characteristic properties. Disturbances affecting all wave types can propagate independently of each other. Using a linear approximation to the theory of motion of elastic rods, the dispersion of these waves is investigated. The phase velocities of the waves lies in the range 5-8 angstroms/ps using constants suitable for a description of DNA. The dispersion of all wave types of arbitrary wave length is investigated for a straight, twisted rod. Based on these findings, we propose all-atom numerical simulations of DNA to investigate the propagation of these waves as an alternative measure of the wave velocity and dispersion analysis.

Engineering materials have long been known to exhibit marked differences in their mechanical behavior under conditions of impact and high rates of loading as compared to their behavior under quasi-static loading conditions.. Thus, the proper utilization of these materials in dynamic environments requires a knowledge of their behavior under dynamic conditions.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By approximating the matrix exponential with a product formula, an unconditionally stable algorithm is derived that conserves the total elastic energy density. By expanding the matrix exponential in Chebyshev polynomials for a specific time instance, a so-called ``one-step'' algorithm is constructed that is very accurate with respect to the time integration. By formulating the conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD) in matrix exponential form, the staggered-in-time nature can be removed by a small modification, and higher order in time algorithms can be easily derived. For two different seismic events the accuracy of the algorithms is studied and compared with the result obtained by using the conventional VS-FDTD algorithm.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

A theoretical and experimental study is described for the optical birefringence associated with the propagation of a plane shear wave in a viscoelastic medium. The shear wave propagation is characterized by a complex propagation constant from which the complex coefficient of shear viscosity of the medium may be derived. The optical bire fringence is related to the mechanical action of the shear wave by a complex mechano-optic coefficient. In the experiment the birefringence due to the shear wave is analyzed by transmitting circularly polarized light through the material at right angles to the direction of shear wave propagation and to the direction of particle displacement. The modified circularly polarized light is then analyzed by a plane analyzer and a photomultiplier detector. The mechanical and mechano-optic constants for the medium are determined for an aqueous milling yellow solution and for a polystyrene solution and are compared with independent determinations of these same factors from other experiments.

A numerical method is described by which theoretical formulas governing the propagation of symmetric and antisymmetric waves in elastic plate theory may be evaluated. Vital to evaluating the formulas is the availability of a means for solving transcendental equations. A generalized iterative rootfinder that was developed for this purpose is described. This rootfinder provides the investigator having access to computing facilities with a reliable means for solving transcendental equations. Dispersion curves for both the flexural and extensional waves in rubber, steel, and beryllium plates are calculated and plotted. These curves approach a real value k for the ratio of the Rayleigh to shear wave speeds. Values of this quantity calculated from the exact elasticity theory were compared with those obtained from an approximate formula given by Victorov. In addition, the inflection and the maximum points of the strain distribution throughout a free plate were calculated. It was found that inflection points in the strain distribution do not exist at all frequencies. In addition to a description and flow chart of the iterative rootfinder method, the significant graphs, equations, and computer programs that arose when computing the dispersion curve calculations are included. (Author)

The problem of wave propagation in an infinite, fluid-loaded, homogeneous, transversely isotropic cylinder is studied within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of the cylinder are formulated using the constitutive equations of a transversely isotropic material with a preferred material direction collinear with the longitudinal axis of the cylinder. The equations of motion of the internal and external fluids are formulated using the constitutive equations of an inviscid fluid. Displacement potentials are used to solve the equations of motion of the cylinder and the fluids. The frequency equation of the coupled system, consisting of the cylinder and the internal and external fluids, is developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces. This frequency equation is general in axial wavenumber k, circumferential wavenumber n, cylinder wall thickness h, and radial frequency. Cut-off frequencies and frequency spectra are computed for the n=1 modes in hollow cylinders, hypothetical fluid columns, fluid-filled cylinders, and cylinders that are fluid filled and immersed in fluid. Numerical results are obtained for two isotropic cylinders (composed of steel and soft (linear) and for a highly anisotropic, fiber-reinforced cylinder. (AN)

This paper is concerned with the role of supplementary conditions such as the entropy inequality at shock waves or kinetic relations at phase boundaries in the selection of physically appropriate solutions to systems of quasi-linear differential equations describing wave propagation. The differences in this respect among various materials are illustrated by contrasting the behavior of waves in linear, bilinear and trilinear bars.

Laboratory tests were conducted on selected granular materials to determine the velocities of propagation of the shear and compression waves and to evaluate the internal damping. Test variables included the confining pressure, amplitude of vibration, void ratio of the material, saturation, and grain size. Resonant column tests were used for wave velocity evaluation, and the vibration decay method and static torsion tests were used to determine damping. The granular materials used were Ottawa standard sand, two sizes of glass spheres, and a crushed quartz. Confining pressure had the most significant effect on velocities of wave propagation, with velocities increasing about with the 1/4 power of the confining pressure. Damping determined from the decay o steady state vibrations behaved like viscous damping. The values of logarithmic decrement varied from about 0.02 to 0.20 for these materials and test conditions. Higher values of logarithmic decrement were found in the static torsion tests because tresses up to 75 per cent of the failure conditions were used.

We compare two high order finite-difference methods that solve the elastic wave equation in two dimensional domains with curved boundaries and material discontinuities. Two numerical experiments are designed with focus on wave boundary interaction, the response of a pressure wave impinging on a circular cavity and the wave field generated by an explosive impact on the wall an underground tunnel. Qualitative comparisons of the results are made where similarities and differences are pointed out.

In the present work, we experimentally implement, numerically compute with and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally-resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly-localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but are also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, for example, to achieve strong attenuation and modulation of high-amplitude impacts without relying on damping in the system.