Search

The layered multiple-scattering method is based on an approximate solution for infinite gratings. In this method, an array of regularly arranged scatterers is viewed as comprising of layers of infinite grating and treated as a multiple transmission-reflection process in a multilayer panel. In this paper, this method is evaluated by comparing with exact solutions obtained by other means. One is a multiple-scattering solution. Another is the exact solution for an infinite grating, which is obtained by combining the T-matrix formulation of the multiple-scattering theory and an alternative representation of the Schlomilch series. The infinity nature enables the waves due to a planar incident wave to be expressed as planar waves and divided into propagating and evanescent modes. The layered multiple-scattering method accounts only for the propagating modes. Details of these modes are analyzed for a single grating, and it is concluded that only the first evanescent modes would have significant presence in a lim...

An exact analytical solution for the scattering of antiplane elastic waves by a layered elastic circular cylinder is obtained. The solution and its degenerate cases are compared with other simpler models of circular cylindrical scatterers. The effects of the geometrical and physical properties of the interphase are studied. Numerical results confirm the existence of a resonance mode in which the scatterer's core undergoes a rigid-body motion when the outer layer of the scatterer is very compliant. This resonance mode has been attributed [Liu et al., Science 289, 1734 (2000)] to a new mechanism for the band gap formed in the extremely low frequency range for phononic crystals made of layered spherical scatterers. Numerical results also show the existence of a similar resonance mode when the outer layer of the scatterer has very high mass density.

When a guided wave is propagated in a layer containing elastic cylinders, the presence of the cylinders gives rise to both amplitude and phase changes of the wave, and thus the variations in the dispersion relation for guided wave propagation in the layer. Using the reflection and transmission coefficients of an antiplane shear wave propagating across an array of periodically spaced cylinders in a full space, the effects of periodically spaced cylinders on the dispersion relation for guided antiplane shear wave propagation in a layer and a layer resting on a half‐space (Love waves) have been investigated. This study is aimed at ultrasonic testing of continuously reinforced laminae or reinforced concrete slabs. Although only SH waves are studied, the results obtained should be relevant for other wave modes and lay the foundation for solutions of localized ultrasonic sources as well. It is found that the volume fraction of cylinders determines the influence of the cylinders on the dispersion relation. Thus ...

The antiplane shear, SH or scalar problem of scattering from an interface characterized by a narrow region of heterogeneity is considered. An approximate integral representation of the scattered wave fields in terms of a Green’s function for a reference problem, and the stress and momentum polarizations averaged through the thickness of the interface is derived. Providing certain material contrast parameters are O(1), the error in this expression is formally O(kh)2, where k is the wave number and 2h is the thickness of the interface. To find the unknown polarizations we specialize to a model configuration that is periodic over a length scale much greater than the characteristic length scales of the heterogenities in the interface. This allows an exact formulation in terms of an infinite system of algebraic equations. Expressions modeling the reflection and transmission measured by a coaxial, confocal pair of transducers are then derived so that the effects of the transducer can be included in the modeling...

A mathematical model describing how a confocal arrangement of two focused ultrasonic transducers is used to interrogate a complex interface between two materials by scanning the focal point across the interface is outlined. A complex interface is one that has roughness and partial contact at several length scales, many of which are equal to or smaller than the compressional or shear wavelength in the material. When the focused ultrasound strikes such an interface, though the focal region be small, multiple scattering takes place among the scatterers within and adjacent to the focal region, making it unclear exactly how the interface is being sampled. To clarify this issue a specific interface model, consisting of a planar array of multiple small cracks having arbitrary lengths and spacings, is used. This interface is interrogated by a focused, antiplane shear wave. While the antiplane assumption makes it scalar, the model shows that what is measured are the multiply scattered signals averaged over the aperture of the transducer, and that the dominant contribution comes from the scattered signals that phase match to the interrogating signal. Explicit expressions relating the modeled reflected and transmitted signals to the convolution of the incident wave field with the crack‐opening displacements at the focal region are given. Numerical examples are worked out for similar and contrasting materials on each side of the interface.

The imaging of thin solid films or interfaces by a confocal arrangement of ultrasonic transducers is modeled in both a two‐dimensional and a three‐dimensional context. In the first model, an antiplane shear, focused beam is directed across an interface comprised of an array of cylinders, with contrasting material properties. Its periodicity is destroyed by removing cylinders arbitrarily. In the second, a thin solid film, having quite general mechanical properties, is immersed in a fluid and scanned by a three‐dimensional acoustic beam. This last work is only partially completed. An approximate integral representation of wave fields scattered by these thin regions of heterogeneity in terms of a Green’s function for the host media, and the stress and momentum polarizations averaged through the thickness of the film or interface is the starting point. The stress polarization measures the change in elastic properties and the momentum polarization that in inertial properties, both changes being with respect to...

Following a well-established formula used by many researchers, the scattering of an anti-plane shear wave by an infinite elastic cylinder of arbitrary relative radius centered in a traction-free two-dimensional isotropic plate has been examined. The plate is divided into three regions by introducing two imaginary planes located symmetrically away from the surface of the cylinder and perpendicular to surfaces of the plate. The wave field is expanded into cylinder wave modes in the central bounded region containing the cylinder, while the fields in the other two outer regions are expanded into plate wave modes. A system of equations determining the expansion coefficients is obtained according to the traction-free boundary conditions on the plate walls and the stress and displacement continuity conditions across the imaginary planes. By taking an appropriate finite number of terms of the infinite expansion series and a few selected points on the two properly chosen virtual planes and the surfaces of the plate through convergence and precision tests, a matrix equation to numerically evaluate the expansion coefficients is found. The method of how to choose the locations of the imaginary planes and the terms of the expansion series as well as the points on each respective boundary is given in Sec. III in detail. Curves of the reflection and transmission coefficients against the relative radius of the cylinder in welded and slip or cracked interfacial conditions are shown. Analysis on the contrast variations of the reflection and transmission coefficients for a cylinder in bonded and debonded interfacial situations is made. The relative errors estimated by the deviation of the numerical results from the principle of the conservation of energy are found to be less than 2%.

The calculation of the three‐dimensional scattered field caused by the insonification of two‐dimensionally rough interfaces in horizontally stratified elastic media is discussed. In general, the scattered field may be calculated either deterministically for a realization of surface roughness, or stochastically as scattered covariances for a surface with a specified roughness power spectra. The perturbation method of Kuperman and Schmidt [J. Acoust. Soc. Am. 86, 1511–1522 (1989)] is used in the full three‐dimensional formulation to model an isotropically rough elastic ice plate floating over a fluid half‐space. Scattered field realizations are calculated for the point source configuration, exhibiting the antiplane shear (SH) coupling unique to the 3‐D model. The ability of explosive sources in the water column to generate SH waves in ice has been observed experimentally and it is argued that rough surface scattering is the likely mechanism that facilitates this mode conversion. [Work supported by ONR.]

Reflection and transmission of an antiplane shear wave by an infinite array of randomly oriented cracks in an isotropic elastic medium are investigated. The problem has been formulated for an averaged scattered field, and a ‘‘periodization’’ technique has been developed to derive the governing singular integral equation for the conditionally averaged crack‐opening displacement. The singular integral equation has been solved by splitting the kernel into a singular and a regular part. A point scatterer approximation was introduced for the part containing the regular kernel. The approximation has been checked by comparison with exact results for a deterministic periodic system. By using this approximation, the coherent part of the averaged reflection and transmission coefficients of zeroth order has been calculated for normal incidence, a completely random crack orientation, and various values of the wavenumber and the ratio of crack length and crack‐center spacing. The problem was formulated in the context of antiplane shear waves. The results are, however, also applicable to reflection and transmission of acoustic or electromagnetic waves by an array of screens.

Elastodynamic stress intensity factors are computed for diffraction of antiplane shear waves by a crack in a layered composite. The crack is parallel to the layering. The problem is formulated by means of integral transforms, and reduced to the solution of a singular integral equation. The special case when the direction of the wave motion is perpendicular to the layering is studied numerically, and stress intensity factors are obtained for several values of frequency, material constants, and geometrical parameters.Subject Classification: 40.55; 20.30.

Elastodynamic stress intensity factors are computed for diffraction of antiplane shear waves by a crack in a layered composite. The crack is normal to and bisected by the midplane of the layer. Both cases of a partially broken layer and a completely broken layer are studied. Integral transform techniques are used to formulate the problem as a singular integral equation. The propagation of symmetric modes (Love waves) is studied numerically, and stress intensity factors are obtained for several values of frequency, geometrical parameters, and material constants. The amplitude ratios of the incident waves to incident plus scattered waves at a large distance from the crack are also calculated.

The impact response of a crack in a layered composite which is subjected to antiplane shear deformation is considered in this study. The geometry of the composite consists of a finite layer that is sandwiched between two half‐planes made of a different material. The finite layer contains a central crack which is oriented normally to the interfaces. Laplace and Fourier transforms are used to reduce the problem to a pair of dual integral equations. The solution to the dual integral equation is expressed in terms of a Fredholm integral equation of the second kind. Dynamic stress intensity factor is obtained as a function of the material properties, the geometry parameters, and time. It is found that the amplitude of the local stresses reaches a peak very quickly and then decreases in amplitude oscillating about the corresponding static value. Depending on the stiffness ratio between the materials, this peak value can be either higher or lower than the value for a homogeneous isotropic material.

The problem of the scattering of transient antiplane shear electroelastic waves from a crack in a piezoelectric medium possessing 6‐mm symmetry is solved by determining the electric dipole density induced on the crack by the incident wave. An integral equation satisfied by the Laplace‐transformed dipole distribution is derived by appealing to the piezoelectric reciprocity relation. The solution of the integral equation is accomplished by expanding the dipole distribution in a series of first‐order spline functions with unknown coefficients and then applying the method of moments to transform the problem to a vector‐matrix one, which is readily handled by a computer. Inversion back into the time domain is effected by an efficient numerical method of inverting Laplace transforms. Results are given for the cases of normal and oblique incidence of the wave onto the crack.

The scattering of polarized harmonic shear waves by a sharp crack of finite length under antiplane strain is considered. Use is made of integral transforms, which reduce the problem to the evaluation of a system of coupled integral equations. Special emphasis is placed on obtaining the detailed structure of the crack‐front stress and displacement fields, which control the instability behavior of cracks in brittle materials. While the dynamic stresses around the singular crack point are found to be qualitatively the same as those encountered under statical loading, they differ quantitatively in that the intensity of the dynamical stress field, which may be regarded as a measure of the force tending to cause crack propagation, depends on the incident wavelength. At certain wavelengths, this intensification is shown to be larger than the static case. The method of solution in this paper applies equally well to boundary value problems in electromagnetic and acoustic theory.