Propagation of elastic waves through media with thin crack-like inclusions

Abstract

Wave propagation in elastic homogeneous media containing a random number of thin inclusions is studied. The material of the inclusions is assumed to be elastic or viscoelastic and appreciably softer than the medium surrounding them. Only the principal terms of the expansion of elastic fields in terms of the small parameters of the problem are considered, namely, the ratio of the characteristic linear dimensions of a typical inclusion and the ratio of the characteristic mdouli of elasticity of the inclusion and the medium. This makes it possible to replace every inclusion by an equivalent singuar model. In the case of statics, analogous models of thin inclusions were given in /1–3/. The model problem of long-wave scattering by a single thin ellipsoidal inclusion is solved explicitly, and the solution is then used to study a medium containing a random number of thin defects. The effective-field method /4, 5/ which takes into account multiple scattering of waves is used to obtain the averaged equation of motion of such a medium (the effective wave operator) in the long-wave approximation. The operator describes the wave propagation in a homogeneous medium with dispersion and attenuation. The velocities of propagation and the attenuation coefficients of various types of elastic waves propagating through materials with randomly oriented inclusions or cracks, and with a system of parallel cracks, are found. The static moduli of elasticity of media with cracks, and hence the velocities of propagation of long waves in such materials, were determined using the effective field method in /6–8/. Other method were used in /9–11/ to find the attenuation coefficients of elastic waves in a medium with cracks, in the Rayleigh approximation. In the case of a medium with cracks, the results of this paper agree with those obtained in the papers listed above.