STONELEY WAVES IN A NON-HOMOGENEOUS ORTHOTROPIC GRANULAR MEDIUM UNDER THE INFLUENCE OF GRAVITY

The aim of this paper is to investigate the Stoneley waves in a non-homogeneous orthotropic granular medium under the influence of a gravity field. The frequency equation obtained, in the form of a sixth-order determinantal expression, is in agreement with the corresponding result when both media are elastic. The frequency equation when the gravity field is neglected has been deduced as a particular case.


Introduction
Problem of Stoneley waves play an important role in the earthquake science, optics, geophysics, and plasma physics.Many authors such as Abd-Alla and Ahmed [1,2], El-Naggar et al. [8], Das et al. [6], and others studied the effect of gravity of the propagation of surface waves (Stoneley waves, Rayleigh waves, and Love waves) in an elastic solid medium.Goda [9] studied the effect of inhomogeneity and anisotropy on Stoneley waves.
The study of granular medium has been necessiated by its possible application in soil mechanics, geophysical prospecting, mining engineering, and so forth.The theoretical outline of the development of the subject from the mid-1930s was given by Paria [13].The present paper, however, is based on the dynamics of granular media as propounded by Oshima [11,12].
The medium under consideration is discontinuous such as one composed numerous large or small grains.Unlike a continuous body, each element or grain cannot only translate but also rotate about its centre of gravity.This motion is the characteristic of the medium and has an important effect upon the equation of motion to produce internal friction.It is assumed that the medium contains so many grains that they will never be separated from each other during the deformation and that the grain has perfect elasticity.The propagation of Rayleigh waves in granular medium was given by many authors such as Bhattacharyya [5], El-Naggar [7], Ahmed [4], and others.In [3], Ahmed discussed the influence of gravity on the propagation of Rayleigh waves in granular medium.
This paper is devoted to the study of the effect of granular body and also of the gravity field in the propagation of Stoneley waves.The wave velocity equation has been derived in the form of a sixth-order determinant.The roots of this equation are in general complex and the imaginary part of an appropriate root measures the attenuation of the waves.It is shown that the frequency of Stoneley waves contains terms involving the acceleration due to gravity and so the phase velocity changes with respect to this acceleration due to gravity.When the gravity field is neglected, the frequency equation has been deduced as a particular case.Also when both media are elastic, the frequency equation reduces to the corresponding result obtained by Abd-Alla and Ahmed [2] in the form of a fourth-order determinant.

Formulation of the problem
Let M 1 and M 2 be two non-homogeneous orthotropic granular media.They are perfectly welded in contact and are under the influence of gravity.These two media extend to infinitely great distance from the origin and are separated by a plane horizontal boundary and M 2 is to be taken above M 1 .Let Ox 1 x 2 x 3 be a set of orthogonal Cartesian coordinates, the origin O being any point on the plane boundary, x 3 -axis is vertically downwards into the medium M 1 .
We consider the possibility of a type of wave traveling in the direction Ox 1 , in such a manner that the disturbance is largely confined to the neighborhood of the boundary which implies that the wave is a surface wave.
Notice that at any instant all particles in any line are parallel to Ox 2 , having equal displacement, therefore all partial derivatives with respect to u 2 are zero and there is no propagation of displacement u 2 [2].
The state of deformation in the granular medium is described by the displacement vector U(u 1 ,0,u 3 ) of the centre of gravity of a grain and the rotation vector ξ(ξ,η,ζ) of the grain about its centre of gravity.There exist a stress tensor and a stress couple and are non-symmetric, that is, (2.1) The stress tensor τ i j can be expressed as the sum of symmetric and anti-symmetric tensors where 3) The symmetric tensor σ i j = σ ji is related to the symmetric strain tensor e i j = e ji = 1 2 by the Hook's law.

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The anti-symmetric stress σ i j are given by where F is the coefficient of fraction.
The stress couple M i j is given by where M is the third elastic constant, where If g is the acceleration due to gravity, then the components of body forces are X = 0, Z = g.Assuming that the initial stress field due to gravity is hydrostatic, the states of initial stress τ i j are [10] where τ is a function of depth Ox 3 only.The equilibrium conditions of the initial stress field are [10] ∂τ ∂x where ρ is the density of the material medium.
The six equations of motion are [2,5] (2.10) These equations, when the stresses are substituted, take the forms where C i j are elastic constants.

Solution of the problem
We assume that the non-homogeneities are of the form where a i j , ρ 0 , F 0 , M 0 , and m are constants.

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Substituting from (3.1) into (2.11),we get We assume that the displacements u 1 and u 3 are derivable from the displacement potentials φ(x 1 , x 3 , t), ψ(x 1 ,x 3 ,t) by the relations ) ) where S 0 = F 0 /M 0 .Eliminating η from (3.6) and (3.8), we get where D ≡ d/dx 3 .Equations (3.13) and (3.14) must have exponential solutions in order that φ 1 , ψ 1 will describe surface waves; they must become vanishingly small as x 3 → ∞.Hence, for the medium M 1 , where the constants A j are related with the constants B j , respectively, by means of (3.13).

25) and (3.22), one can obtain
We use the symbols with a bar for the upper medium (except x 3 , L, b, g) and the functions ξ1 , ζ1 , η1 , φ1 , and ψ1 must vanish as x → −∞.
For the upper medium M 2 , we have

Boundary conditions and frequency equation
The boundary conditions on the interface where From the boundary conditions (iii), (v), (vi), and (vii), we get whence The other significant boundary conditions are responsible for the following relations: (i) (iLn j + λ j )B j = (iL nj − λj ) Bj , (ii) (iL − n j λ j )B j = (iL + nj λj ) Bj , (iv) Eliminating the constants B j , Bj ( j = 3,4,5), we obtain the wave velocity equation in the form of a sixth-order determinantal equation, where 3) is the frequency equation of Stoneley waves in a non-homogeneous orthotropic granular medium under the influence of gravity, this equation depends on the particular values of λ j and λj creating a dispersion of the general wave form.Moreover, the wave velocity C (= b/L) depends on the gravity field, the non-homogeneous of the material medium and the granular rotations.
From (3.18), (3.19), and (4.3), we can assert that when L is Large, so that the length of the wave is small, the effect of gravity is sufficiently small, that is, the wave length of the wave is large, the effect of gravity is no longer negligible and plays an important role on the determination of the wave velocity C.
If we neglect the gravity field, we obtain the wave velocity equation for Stoneley waves in a non-homogeneous orthotropic granular medium which is the same equation as (4.3) with where λ j are the roots of the equation = 0. (4.9) Equation (4.9) determines the wave velocity equation for Stoneley wave in a nonhomogeneous orthotropic elastic medium under the influence of gravity and is in complete agreement with that obtained by Abd-Alla and Ahmed [2].