Scattering of elastic waves by a small inclined surface-breaking crack

I we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A and W , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A and W , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in , and concluding remarks are made about the general method as well as the results presented here.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

Scattering of plane, elastic waves by a plane crack of finite width

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a crack of finite width is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered field together with the constitutive equation, it is shown that the resulting integral equations uncouple for this kind of obstacle. In them, the amount by which the components of particle displacement jump across the crack occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Scattering of monochromatic elastic waves on a planar crack of arbitrary shape

Scattering of monochromatic elastic waves on an isolated planar crack of arbitrary shape is considered. The 2D-integral equation for the crack opening vector is discretized by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem have forms of standard one-dimensional integrals that can be tabulated. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz structure, and the corresponding matrix–vector products can be calculated by the fast Fourier transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Examples of calculations of crack opening vectors, dynamic stress-intensity factors, and differential cross-sections of circular (penny-shaped) and non-circular cracks for various incident wave fields are presented. For a penny-shaped crack and longitudinal incident waves normal to the crack plane, an efficient semi-analytical method of the solution of the scattering problem is developed. The results of both methods are compared in a wide frequency region of the incident field.     

Scattering of antiplane shear waves by a submerged crack

The scattering of time harmonic antiplane shear waves by a crack situated parallel to the surface of a homogenious elastic half-space is considered. The problem is investigated for both clamped and stress-free conditions at the surface of the half-space. Singular integral equations are derived for each case and treated numerically by the Gauss-Chebyshev technique. Harmonic stress intensity factors and crack opening displacements are computed as functions of dimensionless frequency, submerged depth-to-crack width ratio, and angle of incidence. Results are obtained for a moderately wide range of frequencies and are set out in graphical form.     

Scattering of love waves by an interface crack between a piezoelectric layer and an elastic substrate

The scattering of Love waves by an interface crack between a piezoelectric layer and an elastic substrate is investigated by using the integral transform and singular integral equation techniques. The dynamic stress intensity factors of the left and the right crack tips are determined. It is found from numerical calculation that the dynamic response of the system depends significantly on the crack configuration, the material combination and the propagating direction of the incident wave. It is expected that specifying an appropriate material combination may retard the growth of the crack for a certain crack configuration. Project supported by the National Natural Science Foundation of China (No. 19891180), the Fundamental Research Foundation of Tsinghua University (JZ 2000.007) and the Fund of the Education Ministry of China.     

Scattering of elastic waves by non-planar cracks

A modification of the null field approach is used to study the scattering of elastic waves by non-planar cracks. A fictitious surface is added to the crack so that a convenient closed surface is obtained and the surface fields on this closed surface are expanded in vector spherical harmonics. The edge conditions are introduced into these expansions and this is shown to be essential for the numerical convergence. Total cross sections and backscattering amplitudes as functions of frequency are computed numerically for rotationally symmetric cracks which are part of spherical or spheroidal surfaces. By integration in frequency backscattered pulses are also computed. Some cases with two cracks are also considered.     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

Scattering of elastic waves by three-dimensional surface topographies

Diffraction of plane harmonic waves by three-dimensional surface irregularities is investigated through the use of an indirect boundary integral equation method. The irregularity of an arbitrary shape is embedded in an elastic half-space and subjected to incident P, SV, SH, and Rayleigh waves. The material of the half-space is assumed to be linear, weakly anelastic, homogeneous and isotropic.

The accuracy of the method is demonstrated through comparison of the results with existing axisymmetric solutions. Several numerical examples for non-axisymmetric canyons are presented. The resulting amplification patterns exhibit strong sensitivity on type and angle of the incident waves and on the location of the observation point. Systematic comparisons of three-dimensional and corresponding two-dimensional models demonstrate similarity of the amplification pattern. The amplification is larger in some three-dimensional models than in two-dimensional ones. Strong coupling between SH and P-SV modes is observed for off-azimuthal incident waves. This phenomenon is specially pronounced for incident SH waves and it is intrinsic to three-dimensional scattering.     

流体饱和弹性半空间裂隙对平面P_I波的衍射

采用间接边界元法,求解了饱和半空间裂隙对平面PI波的二维衍射问题。基于单层位势理论,将边界离散并直接在边界单元上施加虚拟荷载(水平作用力、竖向作用力和流量源的叠加)以构造散射波场,并由边界条件确定虚拟荷载密度,总波场由自由波场和散射波场共同组成。通过参数分析研究了入射波频率、入射倾角、埋深、孔隙率、边界渗透条件等因素对饱和半空间中裂隙对平面PI波衍射的影响规律。结果表明:裂隙随埋深增大,地表位移谱振荡加剧,峰值有所降低;随着入射频率增加,孔隙率影响逐渐增大;垂直入射时,水平位移的放大区域主要分布在裂隙两端,斜入射时,主要集中在裂隙正上方地表附近;透水和不透水两种情况下的地表位移幅值和相位差别较小,但干土情况与饱和情况下的位移幅值相差较大。     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

Scattering of plane, elastic waves by a plane, rigid strip

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a rigid strip is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered wave, it is shown that the resulting integral equations of the first kind uncouple for this kind of obstacle. In them, the amounts by which the shearing stress and the tensile stress jump across the strip occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).     

Scattering of sound waves by smooth convex elastic cylindrical shells

A plane acoustic wave travelling through an infinite inviscid fluid is scattered by an empty thin shell. A simple approximate but explicit formula is proposed to describe the behaviour of the acoustic pressure scattered by the shell, at an observation point situated in the far field, for backscattering. Only the contributions to the scattered field of specularly reflected, refracted and re-radiated wave are taken into account. The domain of validity of the proposed approximation is given. An example of the frequency dependent calculation is presented for the case of an Armco iron shell with elliptical cross shape immersed in water.     

Rayleigh waves emitted by a propagating crack in a strain-rate dependent elastic medium

A simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate. The surface deformation around the crack tip, due to the high stress concentration there, propagated as a surface wave after fracture of this zone. On the other hand, the moving singularity of the crack tip created a dynamic stress field of varying intensity with time all over the specimen. This dynamic stress field resulted in a significant change of the mechanical properties of a strain-rate dependent material and therefore it influenced the velocity of propagation of fracture-Rayleigh wavefronts. An analysis of this varying dynamic strain field explained the non-circular form of Rayleigh waves, accompanying the propagating crack. For the experimental evaluation of the K₁-factor the method of dynamic caustics was used in conjunction with the high-speed photography technique.     

Scattering of harmonic waves by a disk-shaped rigid inclusion in a three-dimensional elastic matrix

We consider a three-dimensional problem on the interaction of harmonic waves with a thin rigid movable inclusion in an infinite elastic body. The problem is reduced to solving a system of two-dimensional boundary integral equations of Helmholtz potential type for the stress jump functions on the opposite surfaces of the inclusion. We propose a boundary element method for solving the integral equations on the basis of the regularization of their weakly singular kernels. Using the asymptotic relations between the amplitude-frequency characteristics of the wave farzone field and the obtained boundary stress jump functions, we determine the amplitudes of the shear plane wave scattering by a circular disk-shaped inclusion for various directions of the wave incident on the inclusion and for a broad range of wave numbers.     

Scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid

A scattering or T-matrix approach is presented for studying the scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid. A Kelvin-Voigt model is used to obtain the complex elastic moduli of the viscoelastic solid. The T-matris formulation is somewhat complicated because the wave equations and fields are quite different in the solid and fluid regions and are coupled by continuity conditions at the interface. We have obtained fairly extensive numerical results for prolate and oblate spheroids for a variety of scattering geometries. The backscattering, bistatic, absorption and extinction cross-section are presented as a function of the frequency of the incident wave.     

Diffraction of torsional elastic waves by a peripheral edge crack around a spherical cavity

This paper deals with the problem of finding the stress distribution in the neighborhood of a peripheral edge crack in a spherical cavity. The crack is excited by a torsional standing wave.The problem is solved by using integral transforms and is reduced to the solution of a singular integral equation. The solution of this equation is obtained numerically by the method due to Erdogan, Gupta, and Cook, and the stress intensity factors are displayed graphically.     

Scattering of Rayleigh and Stoneley waves by two adhering elastic wedges

The method of Sommerfeld integrals is used to study propagation of Rayleigh and Stoneley waves in a system of two bonded solid wedges with a common vertex. Numerical results are obtained for configurations with a wide range of angles of steel and aluminum wedges.