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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 763429, 47 pages
Rayleigh Waves in Generalized Magneto-Thermo-Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress
1Mathematics Department, Faculty of Science, Taif University, Taif 21974, Saudi Arabia
2Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
3Mathematics Department, Faculty of Science, Umm Al-Qura University, P.O. Box 10109, Makkah 13401, Saudi Arabia
Received 4 December 2010; Revised 14 January 2011; Accepted 25 February 2011
Academic Editor: Ezzat G. Bakhoum
Copyright © 2011 A. M. Abd-Alla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The surface waves propagation in generalized magneto-thermo-viscoelastic granular medium subjected to continuous boundary conditions has been investigated. In addition, it is also subjected to thermal boundary conditions. The solution of the more general equations are obtained for thermoelastic coupling. The frequency equation of Rayleigh waves is obtained in the form of a determinant containing a term involving the coefficient of friction of a granular media which determines Rayleigh waves velocity as a real part and the attenuation coefficient as an imaginary part, and the effects of rotation, magnetic field, initial stress, viscosity, and gravity field on Rayleigh waves velocity and attenuation coefficient of surface waves have been studied in detail. Dispersion curves are computed numerically for a specific model and presented graphically. Some special cases have also been deduced. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field is very pronounced.
The dynamical problem in granular media of generalized magneto-thermoelastic waves has been studied in recent times, necessitated by its possible applications in soil mechanics, earthquake science, geophysics, mining engineering, and plasma physics, and so forth. The granular medium under consideration is a discontinuous one and is composed of numerous large or small grains. Unlike a continuous body each element or grain cannot only translate but also rotate about its center of gravity. This motion is the characteristic of the medium and has an important effect upon the equations of motion to produce internal friction. It was assumed that the medium contains so many grains that they will never be separated from each other during the deformation and that each grain has perfect thermoelasticity. The effect of the granular media on dynamics was pointed out by Oshima . The dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress has been illustrated by El-Naggar . Rayleigh wave propagation of thermoelasticity or generalized thermoelasticity was pointed out by Dawan and Chakraporty . Rayleigh waves in a magnetoelastic material under the influence of initial stress and a gravity field were discussed by Abd-Alla et al.  and El-Naggar et al. .
Rayleigh waves in a thermoelastic granular medium under initial stress on the propagation of waves in granular medium are discussed by Ahmed . Abd-Alla and Ahmed  discussed the problem of Rayleigh wave propagation in an orthotropic medium under gravity and initial stress. Magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model is discussed by Abd-Alla and Mahmoud . Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section is discussed by Venkatesan and Ponnusamy . Some problems discussed the effect of rotation of different materials. Thermoelastic wave propagation in a rotating elastic medium without energy dissipation was studied by Roychoudhuri and Bandyopadhyay . Sharma and Grover  studied the body wave propagation in rotating thermoelastic media. Thermal stresses in a rotating nonhomogeneous orthotropic hollow cylinder were discussed by El-Naggar et al. . Abd-El-Salam et al.  investigated the numerical solution of magneto-thermoelastic problem nonhomogeneous isotropic material.
In this paper, the effect of magnetic field, rotation, thermal relaxation time, gravity field, viscosity, and initial stress on propagation of Rayleigh waves in a thermoelastic granular medium is discussed. General solution is obtained by using Lame’s potential. The frequency equation of Rayleigh waves is obtained in the form of a determinant. Some special cases have also been deduced. Dispersion curves are computed numerically for a specific model and presented graphically. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field are very pronounced.
2. Formulation of the Problem
Let us consider a system of orthogonal Cartesian axes, Oxyz, with the interface and the free surface of the granular layer resting on the granular half space of different materials being the planes and , respectively. The origin is any point on the free surface, the -axis is positive along the direction towards the exterior of the half space, and the -axis is positive along the direction of Rayleigh waves propagation. Both media are under initial compression stress along the -direction and the primary magnetic field acting on -axis, as well as the gravity field and incremental thermal stresses, as shown in Figure 1. The state of deformation in the granular medium is described by the displacement vector of the center of gravity of a grain and the rotation vector of the grain about its center of gravity. The elastic medium is rotating uniformly with an angular velocity , where is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional terms, is the centripetal acceleration due to time varying motion only, and is the Coriolis acceleration, and .
The electromagnetic field is governed by Maxwell equations, under the consideration that the medium is a perfect electric conductor taking into account the absence of the displacement current (SI) (see the work of Mukhopadhyay ): where where is the perturbed magnetic field over the primary magnetic field vector, is the electric intensity, is the electric current density, is the magnetic permeability, is the constant primary magnetic field vector, and is the displacement vector.
The stress and stress couple may be taken to be nonsymmetric, that is, , . The stress tensor can be expressed as the sum of symmetric and antisymmetric tensors where
The symmetric tensor is related to the symmetric strain tensor The antisymmetric stress are given by where is the coefficient of friction between the individual grains. The stress couple is given by where is the third elastic constant, , and so forth, are the components of the resultant acting on a surface.
The non-symmetric strain tensor is defined as where .
The dynamic equation of motion, if the magnetic field and rotation are added, can be written as  The heat conduction equation is given by  where is density of the material, is thermal conductivity, s is specific heat of the material per unit mass, are thermal relaxation parameter, is coefficient of linear thermal expansion, and are Lame’s elastic constants, is the absolute temperature, , is reference temperature solid, is temperature difference , is the mechanical relaxation time due to the viscosity, and .
The Maxwell's electro-magnetic stress tensor is given by which takes the form
The dynamic equations of motion are where is the Earth's gravity and
Also, then and, from (2.16), we have where
3. Solution of the Problem
By Helmholtz's theorem , the displacement vector can be written in the displacement potentials and form, as which reduces to
For a plane harmonic wave propagation in the -direction, we assume
The solution of (3.20) takes the form where the constants and are related to the constants and in the form Substituting (3.22) into (3.11), we obtain and values of displacement components and are where , and are taken to be the complex roots of the following equation where
With the lower medium, we use the symbols with primes, for , and , for ,
4. Boundary Conditions and Frequency Equation
In this section, we obtain the frequency equation for the boundary conditions which are specific to the interface , that is,(i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix)(x), (xi), (xii), (xiii).
The boundary conditions on the free surface are(xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx).
From conditions (iii), (v), (vi), and (vii), we obtain Hence,
The other significant boundary conditions are responsible for the following relations:(i)(ii)(iv)(viii)(ix)(x)(xii)(xiii)(xvi)(xvii)(xviii)(xx)
5. Special Cases and Discussion
5.1. The Magnetic Field, Initial Stress, and Thermal Relaxation Time Are Neglected
In this case (i.e., , and ), (3.26) tends to where
Also, Using the boundary conditions, we obtain where
5.2. The Magnetic Field, Initial Stress, Rotation, and Thermal Relaxation Time Are Neglected and in Viscoelastic Medium
In this case (i.e., , , and), the previous results obtained as in Abd-Alla et al. .
5.3. Absence of the Gravity Field
The solution of (5.6) take the form where and and are taken to be the complex roots of equation where