RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM UNDER INITIAL STRESS

We study the effect of initial stress on the propagation of Rayleigh waves in a granular medium under incremental thermal stresses. We also obtain the frequency equation, in the form of a twelfth-order determinantal expression, which is in agreement with the corresponding classical results.


Introduction.
The propagation of thermoelastic waves in a granular medium under initial stress has some applications in soil mechanics, earthquake science, geophysics, mining engineering, etc.The theoretical outline of the development of the subject from the mid-thirties was given by Paria [9].The present paper, however, is based on the dynamics of granular media as propounded by Oshima [7,8].
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 characteristics of the medium and has an important effect upon the equations 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 frequency equation of Rayleigh waves in a granular layer over a granular half-space was given by Bhattacharyya [2].In [4], Elnaggar investigated the influence of initial stress of the waves propagation in a thermoelastic granular infinite cylinder.Recently [1], Ahmed discussed the influence of gravity on the propagation of waves in granular medium.
This paper is devoted to the study of the effect of initial stress on the propagation of Rayleigh waves in a granular medium under incremental thermal stresses.The medium under consideration is granular half-space overlain by a different granular layer and initial stresses present in this medium have considerable effect in the propagation of Rayleigh waves [3].The wave velocity equation has been derived in the form of twelfth-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 Rayleigh waves contains terms involving thermal coefficients and other terms involving initial stress and so the phase velocity changes with respect to this thermal coefficients and the initial stress.When there is no coupling between the temperature and the strain field in the absence of the initial stress, the derived frequency equation reduces to an equation in the form of ninth-order determinant similar to that obtained by Bhattacharyya [2].Also, the classical frequency equation when both media are elastic and the other effects are absent is obtained.

2.
Formulation of the problem.Consider a system of orthogonal cartesian axes x 1 ,x 2 ,x 3 such that the interface and the free surface of the granular layer resting on the granular half-space of different material are the planes x 3 = H and x 3 = 0, respectively, the origin O is any point on the free surface, x 3 -axis is positive in the direction towards the exterior of the half-space, and the x 1 -axis is positive along the direction of Rayleigh waves propagation.
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.
In this problem the stress tensor and the stress couple are nonsymmetric, i.e., τ ij ≠ τ ji and M ij ≠ M ji .The stress tensor τ ij can be expressed as the sum of symmetric and antisymmetric tensors where The symmetric tensor σ ij = σ ji is related to the symmetric strain tensor by Hooke's law.The antisymmetric stresses σ ij are given by where F is the coefficient of friction.
The stress couple M ij is given by where M is the third elastic constant, where ω 2 = 1/2(∂u 1 /∂x 3 − ∂u 3 /∂x 1 ) is the component of rotation.
The heat conduction equation is given by (see [6]) where K is the thermal conductivity, T is the temperature change about the initial temperature T 0 , ρ is the density, C e is the specific heat per unit mass at constant strain, γ is equal to α(3λ + 2µ), α is the thermal expansion coefficient, and λ and µ are Lame's constants and t is the time.
The components of incremental stress in terms of the displacement are given by (see [3,6]) (2.8) The dynamical equations of motion are and (2.10) Equations (2.9) and (2.10) take the forms, when the stresses are substituted, (2.13) 3. Solution of the problem.Let the constants λ, µ, ρ, F , M, γ and λ, µ, ρ, F , M, γ be characteristics of the layer and the half-space, respectively.Let us introduce the displacement potentials φ(x 1 ,x 3 ,t) and ψ(x 1 ,x 3 ,t) which are related to the displacement components u 1 and u 3 by the relations Substituting from (3.1) into (2.11),(2.13), and (2.15), we see that φ and ψ satisfy the wave equations where From (3.1), the heat conduction equation (2.7) becomes Elimination of T from (3.2) and (3.6), gives where Also, η can be eliminated by the use of equations (3.3) and (3.4) as follows: For a plane harmonic wave propagation in the x 1 -direction, we assume where b is real positive and L is in general complex.

Boundary conditions and frequency equation. The boundary conditions on the interface x
The boundary conditions on the free surface where θ is the ratio of the coefficients of heat transfer to the thermal conductivity.
From the boundary conditions (iii), (v), (vi), and (vii), we get Whence The other significant boundary conditions are responsible for the following relations: (xiii) q 11 A 3 e m 3 H + q 12 B 3 e −m 3 H + q 13 A 4 e m 4 H + q 14 B 4 e −m 4 H = q 12 B 3 e −m 3 H + q 14 B 4 e −m 4 H , (xx) q 11 A 3 + q 12 B 3 + q 13 A 4 + q 14 B 4 = 0, (4.5) where where the non-vanishing entries of the twelfth-order determinant of d ij are given by When there is no coupling between the temperature and strain fields, we have θ vanishes, lim where lim γ→0 q 11 = 0, lim γ→0 q 12 = 0, lim γ→0 q 13 = b 2 L, lim γ→0 q 14 = −b 2 L. (5.2) Similar results hold for the lower medium.Multiplying the rows 10, 11 and 12 of the determinant |d ij | by γ and then taking lim γ→0 , equation (4.7) reduces, after some computation, to the following ninth-order determinantal equation: 0 0 q 1 e −n 3 H −q 1 e n 3 H q 2 e −n 4 H −q 2 e n 4 H 0 0 0 q 3 e −m 3 H q 3 e m 3 H q 5 e −n 3 H −q 5 e n 3 H q 6 e −n 4 H −q 6 e n 4 H 0 0 0 q 7 e −m 3 H −q 7 e m 3 H q 9 e −n 3 H −q 9 e n 3 H q 10 e −n 4 H −q 10 e n 4 H 0 0 q 3 q 5 −q 5 q 6 −q 6 −q 3 q 5 q 6 q 7 −q 7 q 9 −q 9 q 10 −q 10 q 7 q 9 q 10 = 0, ( where (5.4) The frequency equation (5.3) determines the wave velocity equation for the Rayleigh waves in a granular medium under initial stress.
When the initial stress is absent, we have Thus, equation (5.3) with the relations (5.5) reduces to the frequency equation obtained by Bhattacharyya [2].
Finally, in the absence of initial stress and when there is no coupling between the temperature and strain fields, as well as the vanishing of granular rotations, equation (5.7) takes the form where Equation (5.8) is identical to [5, equation (4.195)].

. 6 )
Similar results also hold for the lower medium.Multiplying the columns 5, 6 and 11 of the determinant |d ij | by S and then taking lim M→0 lim S→0