Theory Of Elastic Wave Propagation Research Articles

Materials

Measurements of the velocity and amplitude of ultrasonic compressional waves propagating through neat and accelerated American Petroleum Institute class G cement pastes undergoing hydration are compared with the predictions of the theory of elastic wave propagation through fluid-saturated porous media. The compressional wave velocity is observed to decrease slightly during the first few hours after mixing, followed by a rapid rise as the solid phase becomes interconnected. It is shown that this change in behavior results from a change in character of the observed wave as hydration proceeds. At early times the observed wave involves essentially motion of the fluid phase while at longer times it involves essentially motion of the solid frame. Ultrasonic waves are therefore sensitive to the point at which the solid phase becomes interconnected. This point is of practical significance since the connectivity of the solid phase is responsible for the load-bearing capacity of set cement. The early-time decrease in velocity results from an increasing tortuosity of the pore space due to the formation of hydration products while the increasing velocity at later times results from a stiffening of the porous frame. Wave propagation at early times involves motion of the fluid phase and is extremely sensitive to the presence of air bubbles. Cement pastes containing a sufficiently large number of air bubbles are found to act as high-pass filters over the frequency range employed. This results from the resonant scattering of ultrasound by the bubbles. At longer times the solid frame becomes interconnected and propagation of the low frequency components becomes possible. The amplitude and center frequency of these components are observed to increase with increasing connectivity of the solid phase.

Seismic Wave Propagation In Stratified Media B. L. N. Kennett, Cambridge University Press, 1985 12.50, $24.95

Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

Wideband acoustic response of fluid‐saturated porous rocks

The complex compressional and shear frequency‐dependent elastic constants of nonporous solids and porous rocks are measured in the frequency range 5 kHz to 1 MHz. The measurements are performed in a continuous wave acoustic transmission bridge using cylindrical samples. Use of waveguided samples prevents problems of diffraction and wavefront spreading, which are difficult to correct over broad frequency ranges. The theory of waveguided elastic wave propagation in media with complex elastic constants (generalization of the Pochammer‐Cree solution) is presented. This theory is utilized in the interpretation of the experimental results from nonporous solids with real and complex frequency‐independent elastic constants, solids with complex frequency‐dependent elastic constants, and rocks with effective complex frequency‐dependent elastic constants. For solids with frequency‐independent complex elastic constants (constant Q) the results reveal the appearance of specific dissipation peaks associated with wavegu...

A Penalty Function Type of Virtual Work Principle for Impact Contact Problems of Two Bodies

This paper shows how an impact contact force can be expressed as a product series of penalty functions and subsidiary contact conditions having relative movement components between two bodies on a contact surface. A penalty function type of virtual work principle for various contact and separate states of two bodies is formulated by using the expression of the impact force. This principle is most effective for solving a general impact response of the alternation between contact and separation, and a stress wave propagation response of cracked structures with open and/or closed states on a cracked surface, because of both the use of the relative components and the relaxation of all the subsidiary conditions in this principle. A finite element method (FEM) based on this principle is applied to a two-dimensional analysis for longitudinal impact of two uniform rods. The mean value of impact force by FEM results coincides well with the value from the theory of one-dimensional elastic stress wave propagation.

A penalty function type of virtual work principle for impact contact problems of two bodies.

This paper presents that an impact contact force can be expressed as the product series of penalty functions and subsidiary contact conditions having relative movement components between two bodies on a contact surface. A penalty function type of virtual work principle for various contact and separate states of two bodies is formulated by using the expression of the impact force. This principle is most effective for solving a general impact response of the alternation between contact and separation, and a stress wave propagation response of cracked structures with open and/or closed states on a cracked surface. This is because of both the use of the relative components and the relaxation of all the subsidiary conditions in this principle. A finite element method (FEM) based on this principle is applied to a two- dimensional analysis for longitudinal impact of two uniform rods. The mean value of impact forced by FEM results coincides well with the value from the theory of one-dimensional elastic stress wave propagation.

A multiple scattering theory for elastic wave propagation in discrete random media

A multiple scattering theory for elastic wave propagation in a discrete random medium is presented. A self-consistent multiple scattering formalism using the T matrix of a single scatterer in conjunction with the quasicrystalline approximation (QCA) and a self-consistent pair correlation function is employed to study the phase velocity and coherent attenuation of elastic waves by a random distribution of cavities and elastic inclusions embedded in an elastic matrix. Both uniform and Gaussian size distributions are assumed. The theoretical results obtained in this study are shown to be in excellent agreement with experimental observations.

General spatial static contact problem for a prestressed elastic half-space

The general spatial static contact problem for an elastic half-space with initial stresses is considered. Exact solutions are constructed for an arbitrary structure of the elastic potential, which are twice continuously differentiate functions of components of Green's strain tensor. The investigation is carried out in general form for compressible and incompressible bodies. Questions associated with contact problems for bodies with initial stresses were examined in /1–4/ for particular forms of the elastic potential. Problems on the vibration of a rigid stamp on the surface of an initially stressed half-space and cylinder were examined from the aspect of the linearized theory of elastic wave propagation /7/ in /5, 6/. Contact problems for bodies with initial stresses were investigated within the framework of the linearized theory /7/ in /8, 9/ for an arbitrary structure of the elastic potential in general form for the theory of large (finite) initial strains and different modifications of the theory of small initial deformations. A formulation is given in /8, 9/ for contact problems for elastic bodies with initial stresses and torsion contact problems are examined. A number of contact problems for an elastic half-plane with initial stresses is examined in /10–12/ by dusing complex potentials of plane static linearized problems /13, 14/. Investigations are performed in general form for compressible and incompressible bodies. Complex potentials are introduced /15–18/ for plane dynamic problems and the plane dynamic contact problem for a prestressed half-plane is solved on their basis /19, 20/, when the initial problem allows transformation to a stationary problem in a moving coordinate system. The complex potentials introduced in the absence of initial stresses reduce to the complex S.G. Lekhnitskii potentials /21/ for an orthotropic linear elastic body in the case of unequal roots of the governing equation, and into the complex Kolosov-Muskhelishvili potentials for an isotropic linear elastic body in the case of equal roots.

Elastic Wave Propagation in Transversely Isotropic Media

In this monograph I record those parts of the theory of transverse isotropic elastic wave propagation which lend themselves to an exact treatment, within the framework of linear theory. Emphasis is placed on transient wave motion problems in two- and three-dimensional unbounded and semibounded solids for which explicit results can be obtained, without resort to approximate methods of integration. The mathematical techniques used, many of which appear here in book form for the first time, will be of interest to applied mathematicians, engeneers and scientists whose specialty includes crystal acoustics, crystal optics, magnetogasdynamics, dislocation theory, seismology and fibre wound composites. My interest in the subject of anisotropic wave motion had its origin in the study of small deformations superposed on large deformations of elastic solids. By varying the initial stretch in a homogeneously deformed solid, it is possible to synthesize aniso tropic materials whose elastic parameters vary continuously. The range of the parameter variation is limited by stability considerations in the case of small deformations super posed on large deformation problems and (what is essentially the same thing) by the of hyperbolicity (solids whose parameters allow wave motion) for anisotropic notion solids. The full implication of hyperbolicity for anisotropic elastic solids has never been previously examined, and even now the constraints which it imposes on the elasticity constants have only been examined for the class of transversely isotropic (hexagonal crystals) materials.

A unified theory for elastic wave propagation in polycrystalline materials

We have developed a unified approach to solve for the attenuation and phase velocity variations of elastic waves in single-phase, polycrystalline media due to scattering. Our approach is a perturbation method applicable for any material whose single-crystal anisotropy is not large, regardless of texture, grain elongation, or multiple scattering. It accurately accounts for the anisotropy of the individual grains. It is valid for time-harmonic waves with all ratios of grain size to wavelength. It uses an autocorrelation function to characterize the geometry of the grains, and thereby avoids coherent artifacts that occur if the grains are assumed to have symmetrical shapes and suggests new methods for characterizing distributions of grains that are irregularly shaped. We have carried out numerical calculations for materials that are untextured and equiaxed, and have cubic-symmetry grains and an inverse exponential spatial autocorrelation function. These calculations agree with the previous calculations which are valid in the Rayleigh, stochastic, and geometric regions, and show the transitions between these regions. The complex transition between the Rayleigh and stochastic regions for longitudinal waves, and the severe limitations of the stochastic region for grains with fairly large anisotropy are of particular interest.

Multiple scattering of elastic waves in discrete random media

A multiple scattering theory for elastic wave propagation in a discrete random media is presented. A self‐consistent multiple scattering formalism using the quasicrystalline approximation and a self‐consistent paircorrelation function is employed to study the phase velocity and coherent attenuation of elastic waves by a random distribution of cavities and elastic inclusions embedded in an elastic matrix. Both uniform and Gaussian size distributions are assumed. The theoretical results obtained in this study are shown to be in excellent agreement with experimental observations.

Response of rocks to impact loading by bars with pointed ends

An experimental investigation was undertaken to study the response of three rocks — shale, limestone and diorite — to dynamic loading modelling hammer drilling. A long transmission rod subjected to pendulum impact at its distal end transmitted stress waves to six different heads whose tips consisted of 60°, 90°, and 120° cones or wedges in initial contact with the rock surface. Force and linear penetration histories monitored near the interface were found to be in good agreement with predicted values based on strain measurements at the center and distal end of the transmission bar and use of a one-dimensional theory of elastic wave propagation. The dynamic indentation measurements were also found to be in good agreement witha posteriori determination of the crater depth. Effects of multiple wave reflections and repeated impact on the contact force and rock deformation were determined as a function of the input energy. It was found that an optimum value of this energy exists for most efficient penetration for each of the heads utilized. Dynamic force-penetration curves were compared with corresponding static values obtained from an Instron testing machine. The force history resulting from the impact of a commercial pick-ax suspended in a pendulum arrangement was measured for both the conical and wedge-shaped tips of the device.

Analytical investigation of the unloading wave: PMM vol. 40, n≗ 2, 1976, pp. 327–336

In the general case, the determination of the unloading wave shape [1] in the theory of elastic plastic wave propagation is reduced to the solution of a functional equation of complex structure. A characteristics method [2] is proposed for the approximate construction of the unloading wave, in particular loading cases formulas are obtained to determine its initial slope [3] and the next derivatives at the initial point [4–6]. An investigation of the general properties of an unloading wave is given in [7]. It is shown that as the load tends to zero asymptotically, the unloading wave at the end of a semi-infinite bar has an asymptote with a slope determined by the elastic wave velocity. An investigation of the functional equation is given in this paper and a method of solution of this equation in the form of a power series is proposed. This approach to the problem permits obtaining both known and some new results. In the general loading case, formulas are obtained to determine the initial slope of the unloading wave and a method of determining the next derivatives at the initial point is indicated. Conditions are found for linear hardening for which the unloading wave is a straight line. The existence of an asymptote different from those mentioned in [7] is proved; it is shown how to continue the solution to adjacent sections by means of some known section, and an unloading wave in a material with delayed yielding is investigated.

The Voigt-type microwave kerr effect in semiconductors with spherical constant energy surfaces

This chapter discusses the theory of linear elastic wave propagation in crystals. Three different linear elastic waves may propagate along any given direction in an anisotropic material. These three waves are usually not pure modes as each wave has particle displacement components both parallel and perpendicular to the wave normal. When one of these components is normally much larger than the other, the wave with a large parallel component is called quasilongitudinal, while the wave with a large perpendicular component is called quasitransverse. In the event that sufficient symmetry prevails such that the direction of propagation is elastically isotropic or if the material is elastically isotropic, then all modes become pure modes. For linear elastic wave propagation in crystals, the direction of the flow of energy per unit time per unit area, the energy-flux vector, does not in general coincide with the wave normal as it does in the isotropic case. An expression for the energy-flux vector can be obtained by considering the time rate of change of the total elastic energy contained in the wave field. Measurement of the linear elastic properties of solid materials is one of the most important applications of ultrasonic techniques. The measurements range from determination of second-order elastic constants, which yield fundamental information about the binding forces among atoms in crystals, to the field of nondestructive testing where macroscopic internal defects are located in structural solids.

Elastic wave propagation in filamentary composite materials

An approximate first order theory for elastic wave propagation in unidirectional, filamentary composite materials is developed. Included are stress equations of motion, boundary conditions and constitutive relations. For waves propagating parallel to the fiber orientation in an extended medium, the motion separates into three distinct types: longitudinal, flexural and torsional. All motions are dispersive and sensitive to changes in relative material stiffnesses and geometry. For propagation perpendicular to the fiber orientation, the motion is dispersive and the frequency spectra show stopping bands typical of periodic media.

Generation of Synthetic Seismograms Using Linear Systems Theory

The linear systems theory for elastic wave propagation in a multilayered crust has been extended to time domain solutions. Attenuation is specifically included. This direct time domain approach allows the computation of synthetic seismograms for P or SV waveforms incident at an arbitrary angle at the base of the crustal section. To demonstrate the utility of the technique, seismograms are computed for various conditions and comparisons made with teleseismic events recorded in central Alberta.

Perturbation theory of nonlinear elastic wave propagation

In this article a perturbation method is given for obtaining uniformly valid approximations to the solution of problems of plane wave propagation in finite elasticity theory. Examples are given showing its application to problems of longitudinal and shear wave propagation, including an explicit numerical example concerning the propagation of a shear wave in a polyurethane foam rubber.

Contribution to the linearized theory of surface wave transmission

This paper is concerned with the scattering of surface waves by small perturbations in the elastic medium. A radiation condition is obtained which does not depend on the rate of decay of the waves at infinity. The representation theorems of elastodynamics are then extended to a multilayered half-space. The first-order perturbation theory of elastic wave propagation, when applied to problems of surface wave transmission, yields results which can be most elegantly stated by introducing the notion of surface wave content at a point. In this manner a one-dimensional transmission model is obtained.

A perturbation method for elastic wave propagation: 1. Nonparallel boundaries

This is the first of a series of papers in which a small-perturbation theory for elastic wave propagation is presented. Using classical perturbation techniques, we obtain boundary conditions for the perturbation of the displacement field which, by means of the integral representation theorems of elastodynamics, permit us to express the solutions to the problems as integrals of known quantities. In this paper the method is formulated for mediums with boundaries which are slightly nonparallel. It is then applied to a study of the propagation of Love waves through a crustal layer.

Advances in the theory of anisotropic elastic wave propagation

The qualitative effects of anisotropy on elastic waves propagating in a solid medium were well known to Lord Kelvin. Having been recognized, however, these effects were neglected as being of secondary importance in the dynamics of elastic mediums. This relegation of anisotropy to a secondary role in the dynamics of elastic mediums was undoubtedly justified, particularly in view of the relatively primitive state of experimental elasticity and seismology during Kelvin's time. It was not until after World War 2 that the effects of anisotropy again received serious attention. This was primarily because of the development of ultrasonic techniques for the measurement of dynamic elastic constants of pure crystals. In such experimental problems anisotropy no longer plays a secondary role. The study of how a disturbance, generated by a transducer on the surface of a crystal, spreads through the crystal led to the discovery, by Musgrave, that the wave surface, which forms the boundary of the spreading disturbance, could have cuspidal singularities. This had not been previously predicted, although it could have been predicted by Kelvin had he been more familiar with algebraic geometry. In another area of research (seismology), the post World War 2 years also saw a rise of interest in anisotropy, particularly in the effect of possible continental anisotropy on the propagation of Rayleigh waves. The increased experimental activity in crystal dynamics and the improvement of experimental seismology to the point where secondary effects became important resulted in a number of theoretical investigations into the propagation of plane, time harmonic waves in anisotropic mediums. By 1959 the state of the theoretical and experimental understanding of anisotropic elastic wave propagation had advanced to the point where rigorous wave theoretical calculations were in order. All the simple, solvable problems of isotropic dynamic elasticity, i.e. the initial value problem for an unbounded homogeneous medium, the mixed initial and boundary value problem for a surface line source on a half‐space, the normal mode problem for an elastic wave guide, etc., can be formulated and solved in detail in the anisotropic case. Furthermore, the solutions can be obtained by extensions of the usual transform methods used in isotropic problems. The results can be physically interpreted by means of classical differential geometry. This review summarizes those anisotropic problems treated since 1959 and the techniques developed to solve them.

Theory of Vibration Prevention

In the old theory of vibration prevention, the earth's crust is treated as an equivalent simple spring-mass system. In this paper, however, the earth's crust is assumed to be an infinite elastic medium, and the problems of insulation and prevention of the vibration of any machine are treated on the basis of the elastic wave propagation theory. Many new results are obtained which cannot be seen in the old theory.