PMM U.S.S.R.,Vol.50,No.2,pp.231-239, 1986. Printed in Great Britain 0021-8928/86 $~O.CO+O.OO 0 1987 Pergamon Journals Ltd. zyxwvutsr PROPAGATION OF ELASTIC WAVES THROUGH MEDIA WITH THIN CRACK-LIKEINCLUSIONS* S.K. KANAUN and V.M. LEVIN 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 /l-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-ll/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. 1. A model of a thin inclusion in an elastic medium. Let an unbounded homo- geneous elastic medium with the tensor of elastic moduli Cijkl and density p contain a region V with elastic characteristics C;jkl and density p'.We shall assume that one of the character- istic dimensions of this region, namely h, is small compared with the other two, and that the moduli of elasticity of the inclusion are appreciably smaller than those ofthe medium. We shall choose, at every point x of the middle surface 61 of the region V, a local coordinate system Y,,Y,, 6rs with.the axis y, directed along the normal n(x) to the surface 9. We denote by h (x) the transverse dimension of the region V along the y, axis. tensor C' can be represented in the form The function h(x) and h (5) = 611 (z)* Cijtl = %Cijkl (1.1) where 6, and 6, are small dimensionless parameters, Z(z) is of the order of the largest linear dimension of the region V, and the components of the tensor C"are of the same order as the moduli of elasticity of the basic medium. Below, we shall assume that h(x) is a fairly smooth function satisfying the condition Iah 141 everywhere on Q, with the exception of a small neighbourhood of the contour r of the boundary Q. Here the symbol a denotes the grad operation along the surface 8 a, = VI - ni (2) nj (x)V,, vi = a/ax,,x E 8 (1.2) Let us consider the problem of the propagation of elastic stationary waves of frequency 0 through a medium with a thin defect. Using the smallness of the transverse dimension of this defect, we can replace the initial problem by a boundary value problem for a medium with the boundary conditions at the surface a, which approximately models the presence of an inclusion. Such boundary conditions were formulated in /l, 2/ inthestaticcase (o = 0). In ndealingwiththeproblemofthe stationaryoscillationofamediumwitha thindefect, theseconditions canbegeneralizedinanaturalwasas follows. Ifwedenotetheamplitude valuesofthedisplacement vectorandstresstensorby n(z) and a(~), then for xE 51 we have *Prikl.I4atem.MekhBn. ,50,2,309-319,1986 231