Scattering of antiplane surface waves by an embedded crack in a layered elastic solid

The scattering of an incident antiplane surface wave (Love-wave) by an embedded crack normal to the free surface in a layered half-space is investigated in this study. The crack is first assumed to be entirely contained in the surface layer, then the case in which the crack breaks through the interface is considered. The total displacement and stress fields are analyzed as the superposition of the incident fields in an uncracked half-space and the scattered fields in the cracked half-space. A general solution for the scattered displacement and stress fields in the cracked half-space is obtained by using Fourier sine and cosine transforms techniques. The mixed displacement and stress boundary value problem is reduced to a singular integral equation for the density of displacement discontinuity across the crack faces (dislocation density). The singular integral equation is approximated by a linear system of equations by using a Gaussian method. Further, the amplitudes of the reflected and transmitted displacement fields in the cracked half-space at some distance away from the crack plane are evaluated. It is shown that these displacement fields are the superposition of a finite number of Love-wave modes. In the case when the embedded crack breaks through the interface, the dislocation density function is shown to be discontinuous across the interface between the two solids, and the magnitude of the discontinuity is related to the ratio of the shear moduli. The numerical results for the reflection coefficients of the first three modes as well as for the transmitted coefficient of the first mode are presented for three different layer-embedded cracks and for four different interface-breaking cracks. These coefficients depend strongly on the position of the upper tip and the width of the crack. The results, when the upper tip is very close to the free surface, are compared with those for the surface-breaking crack configurations that are available in the literature. Good agreement is observed