Rayleigh Waves in Generalized Magneto-Thermo-Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress

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.


Introduction
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 1 .The dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress has been illustrated by El-Naggar 2 .Rayleigh wave propagation of thermoelasticity or generalized thermoelasticity was pointed out by Dawan and Chakraporty 3 .Rayleigh waves in a magnetoelastic material under the influence of initial stress and a gravity field were discussed by  .
Rayleigh waves in a thermoelastic granular medium under initial stress on the propagation of waves in granular medium are discussed by Ahmed 6 .Abd-Alla and Ahmed 7 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 8 .Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section is discussed by Venkatesan and Ponnusamy 9 .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 10 .Sharma and Grover 11 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. 12 .Abd-El-Salam et al. 13 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.

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 z K and z 0, respectively.The origin O is any point on the free surface, the z-axis is positive along the direction towards the exterior of the half space, and the xaxis is positive along the direction of Rayleigh waves propagation.Both media are under initial compression stress P along the x-direction and the primary magnetic field − − → H 0 acting on y-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 − → U u, o, w of the center of gravity of a grain and the rotation vector − → ξ ξ, η, ζ of the grain about its center of gravity.The elastic medium is rotating uniformly with an angular velocity Ω Ωn, where n 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, Ω × Ω × u is the centripetal acceleration due to time varying motion only, and 2 u is the Coriolis acceleration, and Ω 0, Ω, 0 .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 14 : where where − → h is the perturbed magnetic field over the primary magnetic field vector, − → E is the electric intensity, − → J is the electric current density, μ e is the magnetic permeability, − − → H 0 is the constant primary magnetic field vector, and − → u is the displacement vector.
The stress and stress couple may be taken to be nonsymmetric, that is, τ ij / τ ji , M ij / M ji .The stress tensor τ ij can be expressed as the sum of symmetric and antisymmetric tensors where

2.4
The symmetric tensor σ ij σ ji is related to the symmetric strain tensor e ij e ji 1 2 The antisymmetric stress σ ij are given by where F is the coefficient of friction between the individual grains.The stress couple M ij is given by where, M is the third elastic constant, M 11 , M 13 , M 33 , and so forth, are the components of the resultant acting on a surface.where ω 2 1/2 ∂u/∂z − ∂w/∂x .The dynamic equation of motion, if the magnetic field and rotation are added, can be written as 15 The heat conduction equation is given by 16 where ρ is density of the material, K is thermal conductivity, s is specific heat of the material per unit mass, τ 1 , τ 2 are thermal relaxation parameter, α t is coefficient of linear thermal expansion, λ and μ are Lame's elastic constants, θ is the absolute temperature, γ α t 3λ 2μ , T 0 is reference temperature solid, T is temperature difference θ − T 0 , τ 0 is the mechanical relaxation time due to the viscosity, and τ m 1 τ 0 ∂/∂t .The components of stress in generalized thermoelastic medium are given by σ 13 τ m μ ∂u ∂z ∂w ∂x .

2.11
If we neglect the thermal relaxation time, then 2.11 tends to Nowacki 17 and Biot 18 .The Maxwell's electro-magnetic stress tensor τ ij is given by 12 which takes the form

2.13
The dynamic equations of motion are

2.17
Substituting 2.17 into 2.14 and 2.16 tends to

2.19
Also, and, from 2.16 , we have where

Solution of the Problem
By Helmholtz's theorem 19 , the displacement vector − → u can be written in the displacement potentials φ and ψ form, as where Substituting 3.2 into 2.10 , we obtain From 3.3 and 3.7 , by eliminating T, we obtain From 3.4 and 3.5 by eliminating η, we obtain For a plane harmonic wave propagation in the x-direction, we assume

3.19
Substituting 3.11 into 3.8 and 3.10 , we obtain where

3.21
The solution of 3.20 takes the form C j e ikN j z D j e −ikN j z , E j e ikN j z F j e −ikN j z ,

3.22
where the constants E j and F j are related to the constants C j and D j in the form E j m j C j , F j m j D j , j 1, 2, 3, 4,

3.23
Substituting 3.22 into 3.11 , we obtain C j e ikN j z D j e −ikN j z e ik x−ct , ψ 4 j 1 E j e ikN j z F j e −ikN j z e ik x−ct ,

3.24
and values of displacement components u and w are 1 − N j m j C j e ikN j z 1 N j m j D j e −ikN j z e ik x−ct , w ik N j m j C j e ikN j z m j − N j D j e −ikN j z e ik x−ct ,

3.25
where N 1 , N 2 , N 3 , and N 4 are taken to be the complex roots of the following equation where 3.27 3.28 Mathematical Problems in Engineering 3.29

3.31
Using 3.22 and 3.11 into 3.3 , we obtain 2 − ikgm j C j e ikN j z D j e −ikN j z e ik x−ct .

3.32
With the lower medium, we use the symbols with primes, for ξ 1 , ζ 1 , η 1 , T, φ, ψ, and q, for z > K, D j e −ikN j z e ik x−ct , ψ 4 j 1 F j e −ikN j z e ik x−ct . 3.33

Boundary Conditions and Frequency Equation
In this section, we obtain the frequency equation for the boundary conditions which are specific to the interface z K, that is, From conditions iii , v , vi , and vii , we obtain Hence, The other significant boundary conditions are responsible for the following relations: i 4 j 1 1 − N j m j C j e ikN j K 1 N j m j D j e −ikN j K − 1 N j m j D j e −ikN j K 0, 4. 4.14

The Magnetic Field, Initial Stress, and Thermal Relaxation Time Are Neglected
In this case i.e., H 0 0, p 0, and τ 1 τ 2 0 , 3.26 tends to where

5.2
Also, F j e −ikV j z ,

5.3
Using the boundary conditions, we obtain where 5.5

The Magnetic Field, Initial Stress, Rotation, and Thermal Relaxation Time Are Neglected and in Viscoelastic Medium
In this case i.e., H 0 0, P 0, Ω 0, and τ 0 τ 1 τ 2 0 , the previous results obtained as in Abd-Alla et al. 20 .

Absence of the Gravity Field
In this case, we put g 0, then 3.20 becomes where E * j e ikX j z F * j e −ikX j z e ik x−ct ,

5.8
where and X 1 , X 2 , X 3 , and X 4 are taken to be the complex roots of equation where * j e ikX j z D * j e −ikX j z e ik x−ct .

5.12
With the lower medium, we use the symbols with primes, for ξ 1 , ζ 1 , η 1 , T, φ, ψ, and q, for z > K, F * j e −ikX j z e ik x−ct .

5.13
From conditions iii , v , vi , vii , we get the same equations 4.1 and 4.2 : the other significant boundary conditions are responsible for the following relations: i

5.19
Mathematical Problems in Engineering 25 xii

The Gravity Field, Initial Stress, and Magnetic Field Are Neglected and There Is Uncoupling between the Temperature and Strain Field
In this case g 0, P 0, H 0 0, and θ 0, we obtain q 22 q 23 q 24 q 30 q 31 q 32 −q 38 −q 39 −q 40 q 50 q 51 q 52 q 62 q 63 q 64 0.

5.31
From 5.30 , we can determine by numerical effects the initial stress, gravity field, friction coefficient, magnetic field, and rotation, for a computation using the maple program; we use sandstone as a granular medium and nephiline as a granular layer taking into consideration that the relaxation times τ 0 0.1, τ 1 0.4, and τ 2 0.5, the friction coefficient F 0.4, and the third elastic constant M 0.2.
i Effects of the initial stress, gravity field, friction coefficient, magnetic field, relaxation time, and rotation are discussed in Figures 2 and 3.
ii From 5.30 , if the initial stress are neglected, we can discuss the effects of the gravity field, friction coefficient, magnetic field, relaxation time, and rotation, and the discussion is clear up from Figure 4.
iii From 5.30 , if the initial stress and magnetic field are neglected, we can discuss the effects of the gravity field, friction coefficient, relaxation time and rotation, and the discussion is clear up from Figure 5. iv From 5.30 , if the initial stress, magnetic field, and gravity field are neglected, we can discuss the effects of the friction coefficient, relaxation time, and rotation, and the discussion is clear up from Figure 6.
v From 5.30 , if the initial stress, magnetic field, and gravity field are neglected and there is uncoupling between the temperature and strain field, we can discuss the effects the friction coefficient, relaxation time, rotation, and the discussion is clear up from Figure 7.

Numerical Results and Discussion
In order to illustrate theoretical results obtained in the proceeding section, we now present some numerical results.The material chosen for this purpose of Carbon steel, the physical data is given 21 as follows: 6.1
Figure 3 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under effect of initial stress, gravity field, friction coefficient, magnetic field, relaxation time and rotation with respect to the wave number, we find that the velocity of Rayleigh waves Re and attenuation coefficient Im decreased and increased with increasing values of H 0 , respectively, and the velocity of Rayleigh waves Re and attenuation coefficient Im increased and decreased with increasing values of Ω and g, respectively; also, the values of Re and Im increased with increasing values of K, while the values of Re and Im take one curve at another value of the relaxation time τ 1 , decreased with increasing values of wave number k.     coefficient Im decreased with increasing values of H 0 and Ω, while that contrary with increasing values of g; also, the values of Re and Im increased and decreased with increasing values of K, respectively, while the values of Re and Im take one curve at another value of the relaxation time τ 1 decreased, then increased with increasing values of wave number k.   values of Re and Im take one curve at another value of the relaxation time τ 1 , increased, then decreased with increasing values of wave number k.

If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected
Figure 6 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under the effect of friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves Re and attenuation coefficient Im decreased and increased with increasing Ω, and the values of Re and Im decreased with increasing values of K, while the values of Re and Im take one curve at another value of the relaxation time τ 1 , decreased and increased with increasing values of the wave number k, respectively.

If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected and There Is Uncoupling between the Temperature and Strain Field
Figure 7 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under the effect of friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves Re and attenuation coefficient Im increased and decreased with increasing of Ω, respectively, while that contrary with increasing values of K; finally, the values of Re and Im take one curve at another value of the relaxation time τ 1 , decreased with increasing, the values of the wave number k.

Conclusions
The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field, and initial stress is considered, and the frequency equation of the wave motion in the explicit form is derived, by considering various special cases.The numerical results are obtained for carbon-steel material, although the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and friction coefficient is observed to be quite large on wave propagation of Rayleigh wave velocity Re and attenuation coefficient Im .The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field, and initial stress is considered, and the frequency equation of the wave motion in the explicit form is derived, by considering various special cases.The numerical results are obtained for carbon-steel material, although the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and friction coefficient is observed to be quite large on wave propagation of Rayleigh wave velocity Re and attenuation coefficient Im .
It is easy to see that the values of Re and Im with respect to the initial stress are increased with increasing values of Ω, while that contrary if the initial stress are neglected and with respect to the wave number; also, if the initial stress are constant with respect to the wave number the values of Re and Im increased and decreased with increasing values of Ω, respectively, and if the initial stress and the magnetic field are neglected, the values of Re and Im decreased and increased with increasing values of Ω, respectively, while that contrary if P, H 0 , g, θ, and ε are neglected and with respect to the wave number; finally, if P, H 0 , and g are neglected and with respect to the wave number, the values of Re and Im increased with increasing values of Ω.
It is easy to see that the values of Re and Im with respect to the initial stress are decreased and increased with increasing values of g, respectively, while that contrary if the initial stress are constant and with respect to the wave number; also, if the initial stress are neglected and if the initial stress and the magnetic field are neglected with respect to the wave number, the values of Re and Im increased with increasing values of g.
It is easy to see that the values of Re and Im with respect to the initial stress are decreased and increased with increasing values of K, respectively, while that contrary if the initial stress are neglected and with respect to the wave number; also, if the initial stress and the magnetic field are neglected and if P, H 0 , and g are neglected with respect the wave number, the values of Re and Im decreased with increasing values of K; finally, if P, H 0 , g, θ, and ε are neglected and with respect to the wave number, the values of Re and Im decreased and increased with increasing values of K.
Finally, the frequency equation has been discussed under effect of rotation, gravity field, and initial stress and in case of various classical and nonclassical theories of thermoelasticity.The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field is very pronounced.The frequency equations derived in this paper may be useful in practical applications.It is concluded from the above analyses and results that the present solution is accurate and reliable and the method is simple and effective.So it may be as a reference to solve other problems of Rayleigh waves in generalized magnetothermoelastic granular medium.

Figure 1 :
Figure 1: Depiction of the problem.

1 C
and R 3 are as in 3 .The solution of 5.6 take the form φ 4 j * j e ikX j z D * j e −ikX j z e ik x−ct , ψ 4 j 1

Figure 2 :
Figure 2: Effects of H 0 , Ω, g, τ 1 , and p on Rayleigh wave velocity and attenuation coefficient with respect to the initial stress.

Figure 3 :
Figure 3: Effects of H 0 , Ω, g, τ 1 , and K, P on Rayleigh wave velocity and Attenuation coefficient with respect to the wave number.

Figure 4
Figure4shows the velocity of Rayleigh waves Re and attenuation coefficient Im under effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with respect to the wave number, we find that the velocity of Rayleigh waves Re and attenuation

Figure 4 :
Figure4: Effects of H 0 , Ω, g, τ 1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

Figure 5
Figure5shows that the velocity of Rayleigh waves Re and attenuation coefficient Im under effect of gravity field, friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves Re and attenuation coefficient Im decreased and increased with increasing values of Ω, and the values of Re and Im increased with increasing values of g, while that contrary with increasing values of K; also, the Figure 5: Continued.

Figure 5 :
Figure 5: Effects of Ω, g, τ 1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

Figure 6 :Figure 7 :
Figure 6: Effects of Ω, τ 1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.