Stoneley wave generation by an external seismic point source in an infinite fluid-filled borehole embedded in a transversely isotropic formation: Basic relationships

The method of integral transformations is used to obtain a long-wave solution to the problem of tube wave excitation by an external point source in an infinite fluid-filled borehole embedded in a transversely isotropic formation. The external field that occurs in the formation gives rise to waves in the borehole fluid. The waves generated in the borehole include the lowest mode of the Stoneley wave (tube wave), which is the borehole eigenmode, and the qP-and qSV-waves. It is shown that the Stoneley wave is determined by the contributions of two poles in the complex plane of horizontal slowness. According to the asymptotic solution, the Stoneley wave can be described by one of three different waveforms depending on the relationship between the elastic parameters of the surrounding anisotropic formation and the borehole fluid. An analysis of the results of calculations shows that the shape and polarity of the Stoneley wave strongly depend on the sign of the nonellipticity parameter of the elastic medium, which offers a possibility of estimating the anisotropy of the borehole environment from observations of the waveform of the Stoneley wave.     

Excitation of a tube wave in a borehole by an external isotropic point source

The excitation of a tube wave in an infinite fluid-filled borehole by an external isotropic point source is considered. The solution to the problem is obtained in the form of a double integral with respect to the ray parameter (slowness) and frequency. The integral with respect to the slowness is transformed to a contour integral in the complex slowness plane and then reduced to the integral over the edges of the cut of the vertical slowness function and the semiresidues at the poles. An asymptotic expression for the wave field in the borehole is obtained with allowance for the radiation condition at infinity. It is shown that, when a longitudinal spherical wave is incident on the borehole, only one low-frequency Stoneley wave is excited and not two, as was assumed earlier [1].     

横向各向同性固体材料中含定向非均匀体的有效弹性模量

针对 含定向非均匀体的横向各向同性复合材料(即TI介质), 采用球形有效体散射等效的方法, 根据TI材料下的D, N_ij表达式, 对横向各向同性条件下Eshelby 张量的积分通用表达式进行化简, 推导出了复合材料的具有横向各向同性特性的有效弹性模量的表达式, 并依此进行了数值分析. 计算结果表明: 利用本方法计算的有效模量随非均匀体含量的增大而减小; 定向排列的非均匀体影响横向各向同性介质的固有各向异性, 水平指向的非均匀体会增大材料的横向各向同性, 模拟结果对评价含非均匀体各向异性介质的特征具有指导意义.     

Simulation of the vector wave field of a low-frequency sound source in the ocean: Some important features

An approach to the simulation of low frequency vector wave fields in stratified media (mainly in the ocean) is considered. The approach is characterized by an improved stability with respect to dividing the medium into many layers of arbitrary thickness. The model for the sound field of a point source is based on an integral representation of two-dimensional, cylindrically symmetric vector wave fields in inhomogeneous media, so that the contributions of all types of waves are included automatically. The model medium is subdivided into N horizontally homogeneous layers for which 4(N?1) equations are formulated to satisfy the boundary conditions between adjacent layers. The method of the generalized Schmidt matrix is used to obtain the coefficients of the equations; these coefficients are substituted into the expressions (of the Fourier-Bessel integral type) for the local parameters of the field. The latter are calculated according to the numerical procedure, and the results are used to model the distributions of the acoustic pressure and the horizontal and vertical components of the particle velocity in liquid and elastic media. The instability of the calculation procedure may result in a disagreement between the model and the exact solution. However, the disagreement is shown to occur mainly in models containing excessively thick layers. A way for improving the stability of the numerical model is suggested. The simulation results are compared with the exact analytical solution for the simplest example and with the results obtained according to the commonly used generalized matrix procedure (the benchmark problem). The examples of the practical application of the model for investigating more complex seismoacoustic wave fields in the ocean are presented.     

Line-integral solution for the stress and displacement fields of an arbitrary dislocation segment in isotropic bi-materials in 3D space

The solutions for the stress and displacement fields due to an arbitrary dislocation segment in an isotropic bi-material medium consisting of joined three-dimensional (3D) half spaces are derived and expressed in terms of line integrals, integrands of which are given in an exact analytical form that, in turn, can also be integrated to yield analytical expressions for the stress–displacement field. The solution is constructed by employing a general solution derived by Walpole [Int. J. Eng. Sci. 34 (1996) p.629] for any elastic singularity in joined isotropic half space, and combining it with Mura's integral formula for the displacement gradient of an arbitrary dislocation segment in homogeneous medium. The resulting new solution provides a framework for deriving analytical expressions for stress and displacement fields of dislocation curves of arbitrary shapes and orientations. The benefit of the method developed, as compared with other methods found in the literature, is that the new solution presented is naturally divided into two components, a homogenous component representing the field of a dislocation in an infinitely homogenous medium, and an image component. This makes it easy and straightforward to modify existing dislocation dynamics codes that already include the homogenous part. To illustrate the accuracy of the method, the stress field expressions of an edge dislocation with Burgers vector perpendicular to the bi-material interface are derived as a degenerate case of the general result. It is shown that our solution is identical to that found in the literature for this case.     

Modeling the quasicompressional wave field of a rectangular transducer in a transversely isotropic solid

A fast model for simulating the transient quasicompressional wave field of a rectangular ultrasonic transducer directly coupled to a transversely isotropic elastic half-space of general orientation is developed. The so-called two-tier asymptotic approach and the uniform stationary phase method are used to derive the high-frequency asymptotics of time-harmonic displacements. Then, transient fields are modeled by means of harmonic synthesis. In geometrical regions, the formulas involve elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral), and are applicable in the radiating near field. The asymptotics elucidate the physics in terms of various arrivals and give explicit dependence of the radiated waves upon model parameters. The asymptotic code is tested against a direct numerical solution. It is at least a thousand times faster but describes accurately both arrival times and amplitudes of various pulses radiated by the transducer.     

Asymmetric and axisymmetric vibrations of finite transversely isotropic circular cylinders

This paper presents an approach for obtaining the exact frequency equations of axisymmetric and asymmetric free vibrations of transversely isotropic circular cylinders. The solution method is based on the three dimensional theory of linear elasticity and uses potential functions. Using this approach, the frequency spectra and vibration mode shapes are plotted for a number of transversely isotropic cylinders. The proposed approach introduces a number of merits compared to earlier approximate and exact solution methods. First, unlike numerically complicated series methods that provide approximate solutions, the proposed approach is exact. Second, combination of scalar functions employed for representing the displacement field is consistent with the physics of the problem. One scalar potential function has been considered for each component of the wave field inside the elastic cylinder. As a result, the solution is systematically divided into coupled and decoupled equations. In addition, by using this approach, there is no need to guess the final of the solution a priori. These merits make the proposed approach suitable for other vibration problems of anisotropic materials.     

Electromagnetic fields in dispersive chiral media generated by modulated nonuniformly moving sources

A representation for the fields generated by moving sources in chiral media in the form of double time-frequency oscillating integrals is obtained by using quaternionic analysis methods. Some additional assumptions concerning the source allow us to introduce a large dimensionless parameter λ > 0 which characterizes simultaneously the slowness of variations of the amplitude and of the velocity of the source. Application of the two-dimensional stationary phase method to the integral representation of the field leads to asymptotic formulas for the electromagnetic field for large λ > 0, and efficient formulas for the frequency and the time Doppler effects in dispersive chiral media. As an application of the proposed method, we consider the Vavilov-Cherenkov radiation in chiral dispersive media.     

The geometry of the hyperbolic system for an anisotropic perfectly elastic medium

We evaluate the fundamental solution of the hyperbolic system describing the generation and propagation of elastic waves in an anisotropic solid by studying the homology of the algebraic hypersurface defined by the characteristic equation, also known as the slowness surface. Our starting point is the Herglotz-Petrovsky-Leray integral representation of the fundamental solution. We find an explicit decomposition of the latter solution into integrals over vanishing cycles associated with the isolated singularities on the slowness surface. As is well known in the theory of isolated singularities, integrals over vanishing cycles satisfy a system of differential equations known as Picard-Fuchs equations. Such equations are linear and can have at most regular singular points. We discuss a method to obtain these equations explicitly. Subsequently, we use the monodromy properties around the regular singular points to find the asymptotic behavior according to the different types of singularities that may appear on a wave front in three dimensions. This is a method alternative to the one that arises in the Maslov theory of oscillating integrals. Our work sheds new light on how to compute and classify the Cagniard-De Hoop contour in the complex radial horizontal slowness plane; this contour is used in numerical integration schemes to obtain the full time behaviour of the fundamental solution for a given direction of propagation.     

横观各向同性的半无限弹性体的若干问题

一.引言 作者在前文1里求得了横观各向同性体的弹性力学空间问题的通解,并且作了初步应用。本文的目的在应用前文的通解来具体地讨论横观各向同性的半无限体(假定物体的各向同性面与它的表面平行)的若干问题。这些问题是实用上比较重要的问题,包括:(i)已知半无限体表面上的载荷;(ii)已知表面上各点的位移;(iii)已知表面上各点的法应力和切向位移;(iv)已知表面上各     

Focusing of electromagnetic waves into a uniaxial crystal

We derive integral representations suitable for studying the focusing of electromagnetic waves through a plane interface into a uniaxial crystal. To that end we start from existing exact solutions for the transmitted fields due to an arbitrary three-dimensional (3D) wave that is incident upon a plane interface separating two uniaxial crystals with arbitrary orientation of the optical axis in each medium. Then we specialize to the case in which the medium of the incident wave is isotropic and derive explicit expressions for the dyadic Green's functions associated with the transmitted fields as well as integral representations suitable for asymptotic analysis and efficient numerical evaluation. Relevant integral representations for focused 3D electromagnetic waves are also given. Next we consider the special case in which (i) the incident field is a two-dimensional (2D) TM wave and (ii) the optical axis in the crystal lies in the plane of incidence, implying that we have a 2D vectorial problem, and derive dyadic Green's functions, integral representations suitable for asymptotic and numerical treatment, and integral representations for focused TM fields. Numerical results for focused 2D TM fields based on these integral representations as well as corresponding experimental results will be presented in forthcoming papers.     

The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

A computational method for wave propagation from a point load in an anisotropic material

One approach which is employed to solve dynamic point load problems in plates and laminates is to take integral transforms to reduce the governing equations to a system of ordinary differential equations with respect to the depth variable. The solution of this system leads to expressions for the transforms of the displacement and stress components at any level in the plate and the transient response at any location may then be recovered by inversion of the multiple transforms. The formal transform inversion involves a double infinite integral but by making a change of variable this may be replaced by an infinite integral associated with a line source and a finite integral with respect to the orientation of the line. A first attempt at applying this approach to obtain the point load response of quasi-isotropic fibre composite laminate led to a non-causal predicted signal. This paper deals with an investigation of this proposed method applied to the simpler model problem of wave propagation in a two-dimensional anisotropic medium. Results are obtained for two different time histories of point loads, namely: a delta function; and a single period of a sine function. In the case of the delta function source a comparison is made with the analytic solution and the errors arising from the numerical approach are discussed. Graphs are also presented showing the non-causal contributions to the overall response which arise at individual angles of orientation of the line source.     

Radiation of ultrasound into an anisotropic solid

A half-space, x₃ 0, of a transversely isotropic solid whose axis of symmetry makes an angle with the x₃-axis, is subjected to a spatially uniform time-harmonic distribution of normal surface tractions over a circular area of the plane x₃ = 0. The wave motion radiated into the half-space is investigated. Using an integral representation for the displacement components the problem is first reduced to a system of singular integral equations for the displacements on the surface x₃ = 0. This system is solved by the boundary element method over a truncated area, where use is made of recently derived simplified forms of the Green's functions. The results show the skewing of the beam as the angle between the axis of symmetry of the transversely isotropic solid and the normal to the surface of the solid is increased.     

Depolarization of the electromagnetic field scattered by a three-dimensional large-scale irregularity of the lower ionosphere

This paper develops analytical and numerical methods for the solution of three-dimensional problems of radio wave propagation. We consider a three-dimensional vector problem for the electromagnetic field of a vertical electric dipole in a planar Earth-ionosphere waveguide with a largescale local irregularity of negative characteristics at the anisotropic ionospheric boundary. The field components at the boundary surfaces obey the Leontovich boundary conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr≫1, where r is the distance from the source or receiver to the nearest point of the irregularity, k=2π/λ, and λ is the radio wavelength) integration over the direction perpendicular to the ray path, we transform this system to one-dimensional integral equations where integration contours represent the geometric contour of the irregularity. The system is numerically solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and subsequent iterations. The proposed numerical algorithm reduces the computer time required for the solution of this problem and is applicable for studying both small-scale and large-scale irregularities. We obtained novel estimates for the field components that are not excited by the source but result entirely from scattering by the sample three-dimensional ionospheric irregularity.     

Excitation of an elastic half-space by a buried line source of conical waves

A formal solution is obtained to the problem of a buried line source of conical waves propagating at a constant phase velocity c in an isotropic elastic half space. By applying the boundary conditions at the free surface, it is determined that the reflected field, in addition to the incident field, requires addition of a scalar potential and two components of the vector potential. The latter is in contrast to the case of cylindrical waves where only one component of the vector potential is needed. The formal solution for the conical wave source goes over to that for the two dimensional cylindrical wave case in the limit of infinite phase velocity c.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.     

The transient response of a transversely isotropic cylinder under a laser point source impact

The transient response of a transversely isotropic cylinder under a laser point source impact is solved theoretically. The radial displacement generated by the laser under the ablation regime is numerically calculated by introducing Fourier series expansion and two-dimensional Fourier transform. The validity of this theoretical solution is demonstrated on a fiber reinforced composite cylinder with a strong anisotropy. Experimental displacements are detected at the cylinder surface by the laser ultrasonic technique, and are analyzed by the ray trajectories. Corresponding theoretical displacements are calculated numerically and compared to the experimental signals. Good agreement is found. The diffraction effect caused by the cusp is observed in both theory and experiment.     

Analysis of a penny-shaped crack in a magneto-electro-elastic medium

The solution for an ellipsoidal cavity in an infinite transversely isotropic magneto-electro-elastic medium under remotely applied axisymmetric, combined mechanical–electric–magnetic loading is derived. The exact solution for a penny-shaped crack is obtained by letting the minor axis of the ellipsoidal cavity approach zero. The results demonstrate that the stress intensity factor depends only on the applied mechanical loading, but the electric displacement intensity factor and the magnetic induction intensity factor depend on the applied combined loading, as well as the two ratios of β/α and γ/α. Parameter α is the ratio of the minor axis to the major axis of the ellipsoidal cavity, β is the ratio of the dielectric constant of the material in the cavity to the effective dielectric constant of the magneto-electro-elastic medium, and γ is the ratio of the magnetic permeability of the material in the cavity to the effective magnetic permeability of the magneto-electro-elastic medium. The two ratios characterize the permeability of the crack to electric and magnetic fields. Several limiting cases for β/α and γ/α are studied. A self-consistent method is adopted to determine the real crack opening α under the combined mechanical–electric–magnetic loading. The stress, electric displacement and magnetic induction intensity factors of a penny-shaped crack in BaTiO₃–CoFe₂O₄ composites are calculated for different volume fractions and different applied combined loadings.