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 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 elastic waves by a surface-breaking crack

Scattering of incident surface waves and incident body waves by a surface-breaking crack is investigated in a two-dimensional geometry. By decomposing the scattered fields into symmetric and antisymmetric fields with respect to the plane of the crack, two boundary value problems for a quarter-plane have been obtained. The formulation of each boundary-value problem has been reduced to a singular integral equation which has been solved numerically. For incident surface waves the back-scattered and forward-scattered surface waves have been plotted versus the dimensionless frequency. Curves are also presented for the scattered displacement fields in the interior of the body generated by incident body waves, both versus the angle of incidence and versus the dimensionless frequency.     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

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.     

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.     

Harmonic waves in three-dimensional elastic composites

For a periodic elastic composite which consists of a matrix and fibers with finite dimensions (i.e. a three-dimensional problem), here are given estimates for eigenfrequencies and eigenfunctions. Calculations are based on a new quotient which has been proposed by Nemat-Nasser. The periodic character of the eigenfrequencies is pointed out, and illustrative examples are given.     

Diffraction of surface waves by floating elastic plates

Based on the dynamical theories of water waves and Mindlin thick plates, the diffraction of surface waves by a floating elastic plate is presented by using the Wiener–Hopf technique. Firstly, the problem is related to a wave guide in water of finite depth, which is analysed to determine the poles. The resulting hybrid boundary value problem is reduced to solving an infinite system of linear algebraic equations. The results obtained are compared with those calculated by an alternative analysis, and with experimental data. Finally, the effects of the geometric and physical parameters on the distribution of deflection and bending moments in plates are analysed and discussed.     

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.     

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 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 elastic waves by elastically transparent obstacles (integral-equation method)

A formulation of elastodynamic diffraction problems for sinusoidally in time varying disturbances in a linearly elastic medium is presented. Starting with the elastodynamic reciprocity relation, an integral representation for the particle displacement is derived. In it, the particle displacement and the traction at the boundary of the obstacle occur. From the integral representation, an associated integral equation is obtained by letting the point of observation approach the boundary of the obstacle. The “obstacle” may be either a rigid body, a void, or a body with elastic properties differing from those of its environment, or a combination of these. The integral equation thus obtained is well-suited for numerical treatment, when obstacles up to a few wavelengths in maximum diameter are considered.     

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 pulsed Rayleigh surface waves by a cylindrical cavity

A pulsed Rayleigh surface wave of prescribed shape is incident on a cylindrical cavity which is parallel to both the plane free surface and the plane wave front. Multiple reflections at the cylindrical and plane free surface are considered and the resulting displacements and stress components are calculated in the surrounding of the cavity by approximately summing infinite double sums. Use is made of the stationary loading case simulated by a periodic train of wave pulses and its time Fourier series representation and of expansions of all incident and reflected waves in terms of cylindrical wave functions. For reflection, the free surface of the half-space is approximated by a fictitious convex (or concave) cylindrical surface of “large” radius. The wave pattern due to a single pulse loading is constructed from the stationary solution by enforcing homogeneous initial conditions in the half-space ahead of the single loading pulse and by prescribing a wide spacing in the periodically set-forth train of pulses. The numerical results for stresses and dynamic stress magnification factors are especially useful for the interpretation of recent measurements in dynamic photoelasticity.     

Scattering of elastic waves in simple and complex polycrystals

The propagation of elastic waves in polycrystals is a classical topic with a rich history of research, with primary focus on attenuation in single phase materials with randomly oriented, equiaxed grains. Over the last decade, the need to nondestructively evaluate the degree of damage of engineering components has led to extension of the classical understanding to a number of more complex cases. These motivations include the desire to understand how the noise backscattered from microstructure, and limiting flaw detectability, is controlled by the measurement configuration and microstructure of the material, the desire to use the understanding of attenuation and backscattering in designing improved inspections and in assessing their capability as quantified by probability of detection, and the desire to develop improved procedures for characterizing microstructures. This paper provides an overview of this work. A brief review of the classical understanding of how elastic waves are attenuated and backscattered by scattering from grain boundaries in randomly oriented polycrystals is first presented. This is followed by the results of recent experiments and analysis concerning how these phenomena change in engineering materials with more complex microstructures. For single phase polycrystals, the paper presents results verifying the classical theories in copper, showing how these theories can be used to determine single crystal elastic constants from measurements in alloy polycrystals, demonstrating this technique on nickel-base superalloys, and providing evidence of multiple scattering effects that are not accounted for in the classical, first-order theory. Additional results are presented in titanium alloys having duplex microstructures that demonstrate the existence of fluctuations of beam amplitude and phase, and a simple two-dimensional theory is presented which qualitatively explains the results. The paper concludes with the presentation of some pitch–catch (bi-static) experiments that clearly illustrate the role of multiple scattering.     

Scattering attenuation of elastic waves due to low-contrast inclusions

Seismic scattering attenuation due to random lithospheric heterogeneity has been theoretically modeled using two approaches. One approach is the Born approximation theory (BAT), which is primarily used to treat weak continuous heterogeneity, and the other approach is the Foldy approximation theory (FAT), which deals with sparsely distributed discrete inclusions. We apply the BAT to elastic wave scattering due to inclusions having low contrast with the matrix, and compare the results with those predicted by the FAT. We thus investigate the valid wavenumber range of the BAT based on a reasonable assumption that the inclusions are distributed so sparsely that the FAT is effectively correct for any wavenumber. For simplicity, we consider a specific type of round inclusion, which is either two- or three-dimensional and has a two-valued wave velocity and/or mass density. Both theories are confirmed to yield essentially equivalent results below a certain wavenumber limit, depending on the contrast. This is known as the Rayleigh-Gans scattering regime. Beyond the wavenumber limit, the BAT overestimates the attenuation for common-mode scattering due to wave-velocity contrast, but remains valid with respect to the attenuation for scattering due to mass-density contrast and/or conversion scattering. These conclusions are independent of the spatial dimensions of the media as well as the modes of the elastic waves (P or S). Some advantages of the BAT over the FAT for application to low-contrast inclusions are discussed.     

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 in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

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.