A. K. Gautesen

Ray Analysis of Surface-Wave Interaction with an Edge Crack

Abstruct-A ray-theory approach is presented to analyze scattering of Rayleigh surface waves by a surface-breaking crack. The two-dimensional problem of normal incidence on an edge crack of depth d in an elastic half-space is discussed in detail. The basic diffraction mechanisms in the high-frequency range at the mouth and the edge of the crack are investigated one by one on the basis of elastodynamic ray theory. The results are then superimposed to yield simple expressions for the backscattered and forward-scattered Rayleigh surface waves and for the elastodynamic stress-intensity factors, in terms of reflection, transmission, and diffraction coefficients. These approximate results are compared with exact numerical results. Good agreement is observed for d/A > l, where A is the wavelength of the incident surface wave. A simple formula for the inverse problem is presented, which relates the periodicity of the amplitude modulation in the high-frequency range directly to the depth d of the crack. PROMISING method for the detection of a surfacebreaking crack, and for the subsequent determination of its size, shape, and orientation, is based on the scattering of Rayleigh surface waves. For a two-dimensional configuration of a line crack normal to the free surface of a half-space, an exact solution to the direct scattering problem has only recently been obtained by Mendelsohn et al. [l] . This exact solution does, however, involve a substantial computational effort, since it is based on the numerical solution of two singular integral equations with kernels that are themselves complicated integrals. Because of this numerical aspect, little guidance to the inverse problem is obtained from the exact solution to the direct problem. The results of [l] are, however, very useful for the testing of approximate methods of analysis. In this paper we present a simple approximate approach to scattering of Rayleigh surface waves by surface-breaking cracks, which is valid in the high-frequency range. Solutions are shown to agree well with the results of [l] for wd/cR > 6, where W is the circular frequency, d is the depth of the crack, and cR is the velocity of Rayleigh surface waves. The method of analysis which is based on elastodynamic ray theory can potentially be A

Geometrical theory of diffraction for three‐D elastodynamics

Keller’s geometrical theory of diffraction is applied to three‐dimensional elastodynamics, in particularly to diffraction of longitudinal waves by a crack. The theory provides useful approximations for large frequencies and/or large distances from the edge of the crack. For the class of problems considered in this paper, the canonical solution is provided by the fields describing diffraction by a semi‐infinite crack of a plane longitudinal wave which is incident under an arbitrary angle with the edge of the crack. The formal solution to the canonical problem is obtained by means of integral transform techniques in conjunction with an application of the Wiener–Hopf method to a set of coupled equations. The pertinent asymptotic expressions for the diffracted field are evaluated, and the diffraction coefficients which enter the geometrical theory are determined. As an example, the three‐dimensional problem of diffraction of a point‐source field by a semi‐infinite crack is worked out in detail.

Surface‐wave rays in elastodynamic diffraction by cracks

A geometrical diffraction theory has been worked out to analyze the fields generated by diffraction of high‐frequency waves by cracks. The theory accounts for curvature of incident wavefronts, curvature of crack edges, and finite dimensions of the crack by providing first‐order corrections to the results for a semi‐infinite crack. The diffracted fields include direct diffractions from the crack edges as well as well as diffractions of signals which travel via the crack faces. On the faces of the crack the main contributions to the diffracted fields come from rays of surface waves. The directions of these surface‐wave rays, and the amplitudes, wavelengths, and phases of the associated surface‐wave motions have been related to the corresponding quantities of the incident body‐wave rays. Reflection and diffraction of surface‐wave rays by the edge of a crack have also been analyzed. As an example, diffraction by a penny‐shaped crack of a plane longitudinal wave under normal incidence has been considered in some detail. Explicit expressions are given for the diffracted fields. In these expressions a correction was introduced to extend the validity of the results to the normal axis through the center of the crack, which is a caustic axis. A simple expression for the scattering cross section is presented.

A Ray Theory for Elastodynamic Stress-Intensity Factors

Elastodynamic stress intensity factors generated by the intersection of wave motions with a crack are analyzed. It is shown that in an asymptotic approximation, which is valid for high frequencies, the stress intensity factors at the edge of a crack are related to the fields of incident rays by a matrix of stress intensity factor coefficients, which can be computed from canonical solutions. The canonical solutions are provided by the fields describing diffraction by a semi-infinite crack of plan body waves and plane surface waves, which are incident under an arbitrary angle with the edge of the crack. Several applications of the theory are presented. For cracks of finite length, the contributions due to the travelling back and forth of rays between the two crack tips is taken into account in a simple manner, to yield results which are in excellent agreement with numerical results obtained by other authors.

Elastic surface waves guided by the edge of a slit

Abstract Surface waves propagating along the free surface of a homogeneous, isotropic, linearly elastic half-space, are shown to have the property that the normal displacement component at the free surface is governed by a reduced wave equation. This suggests a “membrane analogy”, and a corresponding family of surface waves. Of particular interest is a three-dimensional surface wave, whose displacement components in the sagittal plane vary linearly with the co-ordinate normal to that plane, while the displacement component in the direction normal to the sagittal plane is uniform in that direction. This new wave arises when surface waves propagate along the free surfaces of a semi-infinite slit, parallel to the edge of the slit, with the classical Rayleigh wave velocity. It is also shown that a semi-infinite slit cannot support surface waves which decay with the distance from the edge of the slit.

On the Existence of Surface Waves Guided by a Slit

Guidance of elastic surface waves by a stress free crack of finite width is studied. A criterion which shows when a crack cannot act as a waveguide for surface waves is derived. Available asymptotic results satisfy this criterion.