Rayleigh Waves in Generalized Magneto-Thermo-Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress
A. M. Abd-Alla,1S. M. Abo-Dahab,1,2and F. S. Bayones3
Academic Editor: Ezzat G. Bakhoum
Received04 Dec 2010
Revised14 Jan 2011
Accepted25 Feb 2011
Published02 Jun 2011
Abstract
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.
1. 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 Abd-Alla et al. [4] and El-Naggar et al. [5].
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.
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 [14]):
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 [15]
The heat conduction equation is given by [16]
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 components of stress in generalized thermoelastic medium are given by
If we neglect the thermal relaxation time, then (2.11) tends to Nowacki [17] and Biot [18].
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
From (2.3)–(2.8) and (2.11), we have
Substituting (2.17) into (2.14) and (2.16) tends to
then
Substituting (3.11) into (3.8) and (3.10), we obtain
where
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. [20].
5.3. Absence of the Gravity Field
In this case, we put , then (3.20) becomes
where
and , and are as in (3.21).
The solution of (5.6) take the form
where
and and are taken to be the complex roots of equation
where
With the lower medium, we use the symbols with primes, for , and , for ,
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)(ii)(iv)(viii)(ix)(x)(xii)(xiii)(xvi)(xvii)(xviii)(xx)
where
Elimination of , and gives the wave velocity equation in the determinant form
This equation has complex roots: the real part (Re) gives the Rayleigh wave velocity, and the imaginary part (Im) gives the attenuation coefficient due to the friction of the granular nature of the medium, where the nonvanishing of the twelfth-order determinant of is given by
5.4. The Gravity Field, Initial Stress, and Magnetic Field Are Neglected and There Is Uncoupling between the Temperature and Strain Field
In this case , and , we obtain
Multiplying the rows 10, 11, and 12 of the determinant by and then taking , (5.28) reduces, after some computation, to the following ninth-order determinant equation:where
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 , and , the friction coefficient , and the third elastic constant .
(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.
6. 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. Effects of the Initial Stress, Gravity Field, Friction Coefficient, Magnetic Field, Relaxation Time, and Rotation
Figure 2 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under the effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with respect to the initial stress; we found that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) increased with increasing values of and , and the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased and increased with increasing values of and , respectively; while the values of (Re) and (Im) take one curve at another value of the relaxation time increased with increasing values of initial stresse .
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 , respectively, and the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) increased and decreased with increasing values of and , respectively; also, the values of (Re) and (Im) increased with increasing values of , while the values of (Re) and (Im) take one curve at another value of the relaxation time , decreased with increasing values of wave number .
6.2. If the Initial Stresses Are Neglected
Figure 4 shows 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 coefficient (Im) decreased with increasing values of and , while that contrary with increasing values of ; also, the values of (Re) and (Im) increased and decreased with increasing values of respectively, while the values of (Re) and (Im) take one curve at another value of the relaxation time decreased, then increased with increasing values of wave number .
6.3. If the Initial Stresses and Magnetic Field Are Neglected
Figure 5 shows 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 , and the values of (Re) and (Im) increased with increasing values of g, while that contrary with increasing values of ; also, the values of (Re) and (Im) take one curve at another value of the relaxation time , increased, then decreased with increasing values of wave number .
6.4. 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 , and the values of (Re) and (Im) decreased with increasing values of , while the values of (Re) and (Im) take one curve at another value of the relaxation time , decreased and increased with increasing values of the wave number , respectively.
6.5. 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 , respectively, while that contrary with increasing values of ; finally, the values of (Re) and (Im) take one curve at another value of the relaxation time , decreased with increasing, the values of the wave number .
7. 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 , 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 , 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 , respectively, while that contrary if , and are neglected and with respect to the wave number; finally, if , and are neglected and with respect to the wave number, the values of (Re) and (Im) increased with increasing values of .
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 , 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 .
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 , 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 , and are neglected with respect the wave number, the values of (Re) and (Im) decreased with increasing values of ; finally, if , and are neglected and with respect to the wave number, the values of (Re) and (Im) decreased and increased with increasing values of .
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 magneto-thermoelastic granular medium.
References
N. Oshima, “A symmetrical stress tensor and its application to a granular medium,” in Proceedings of the 3 rd Japan National congress for Applied Mechanics, vol. 77, pp. 77–83, 1955.
A. M. El-Naggar, “On the dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress,” Astrophysics and Space Science, vol. 190, no. 2, pp. 177–190, 1992.
N. C. Dawan and S. K. Chakraporty, “On Rayleigh waves in Green-Lindsay model of generalized thermoelastic media,” Indian Journal Pure and Applied Mathematics, vol. 20, pp. 276–283, 1998.
A. M. Abd-Alla, H. A. H. Hammad, and S. M. Abo-Dahab, “Rayleigh waves in a magnetoelastic half-space of orthotropic material under influences of initial stress and gravity field,” Applied Mathematics and Computation, vol. 154, no. 2, pp. 583–597, 2004.
A. M. El-Naggar, A. M. Abd-Alla, and S. M. Ahmed, “Rayleigh waves in a magnetoelastic initially stressed conducting medium with the gravity field,” Bulletin of the Calcutta Mathematical Society, vol. 86, no. 3, pp. 243–248, 1994.
S. M. Ahmed, “Rayleigh waves in a thermoelastic granular medium under initial stress,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 9, pp. 627–637, 2000.
A. M. Abd-Alla and S. M. Ahmed, “Rayleigh waves in an orthotropic thermoelastic medium under gravity and initial stress,” Earth, Moon and Planets, vol. 75, pp. 185–197, 1998.
A. M. Abd-Alla and S. R. Mahmoud, “Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model,” Meccanica, vol. 45, no. 4, pp. 451–462, 2010.
M. Venkatesan and P. Ponnusamy, “Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section immersed in a fluid,” International Journal of Mechanical Sciences, vol. 49, no. 6, pp. 741–751, 2007.
S. K. Roychoudhuri and N. Bandyopadhyay, “Thermoelastic wave propagation in a rotating elastic medium without energy dissipation,” International Journal of Mathematics and Mathematical Sciences, no. 1, pp. 99–107, 2005.
J. N. Sharma and D. Grover, “Body wave propagation in rotating thermoelastic media,” Mechanics Research Communications, vol. 36, no. 6, pp. 715–721, 2009.
A. M. El-Naggar, A. M. Abd-Alla, M. A. Fahmy, and S. M. Ahmed, “Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder,” Heat and Mass Transfer, vol. 39, no. 1, pp. 41–46, 2002.
M. R. Abd-El-Salam, A. M. Abd-Alla, and H. A. Hosham, “A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method,” Applied Mathematical Modelling, vol. 31, no. 8, pp. 1662–1670, 2007.
S. Mukhopadhyay, “Effects of thermal relaxations on thermoviscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading on the boundary,” Journal of Thermal Stresses, vol. 23, no. 7, pp. 675–684, 2000.
A. M. Abd-Alla, “Propagation of Rayleigh waves in an elastic half-space of orthotropic material,” Applied Mathematics and Computation, vol. 99, no. 1, pp. 61–69, 1999.
Masa. Tanaka, T. Matsumoto, and M. Moradi, “Application of boundary element method to 3-D problems of coupled thermoelasticity,” Engineering Analysis with Boundary Elements, vol. 16, no. 4, pp. 297–303, 1995.
W. Nowacki, Thermoelasticity, Addison-Wesley, London, UK, 1962.
M. A. Biot, Mechanics of Incremental Deformations. Theory of Elasticity and Viscoelasticity of Initially Stressed Solids and Fluids, Including Thermodynamic Foundations and Applications to Finite Strain, John Wiley & Sons, New York, NY, USA, 1965.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, NY, USA, 1953.
A. M. Abd-Alla, S. M. Abo-Dahab, H. A. Hammad, and S. R. Mahmoud, “On generalized magneto-thermoelastic rayleigh waves in a granular medium under the influence of a gravity field and initial stress,” Journal of Vibration and Control, vol. 17, no. 1, pp. 115–128, 2011.
J. N. Sharma and M. Pal, “Rayleigh-Lamb waves in magneto-thermoelastic homogeneous isotropic plate,” International Journal of Engineering Science, vol. 42, no. 2, pp. 137–155, 2004.