Complex-valued wavenumber, reflection and transmission in an elastic solid containing a cracked slab region

Abstract

Propagation of ultrasonic or seismic waves in an elastic solid where slit cracks are randomly distributed is investigated. The distribution has a uniform probability density in a slab region, the cracks are parallel to the boundaries of the slab, and the solid is uncracked on either side of the slab. When an antiplane wave is normally incident on the cracks, it is shown that the average (coherent) motion is governed by two coupled integral equations. These equations are solved assuming that the average exciting stress near a fixed crack is equal to the average total stress. Inside the slab, where multiple scattering occurs, there is a forward motion and a backward motion. The velocity and attenuation of the two motions are shown to be given by simple formulae that depend on frequency and crack density. Simple formulae, which depend on frequency, crack density, and slab thickness, are also obtained for the wave amplitudes outside the slab. Plots of the velocity, attenuation, reflected amplitude, and transmitted amplitude vs the frequency are presented for several values of crack density and slab thickness. Low and high frequency limits of these quantities are examined analytically and numerically.