IJCRR - Vol 04 Issue 24, December
Date of Publication: 22-Dec-2012
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S-WAVE PROPAGATION IN A NON-HOMOGENEOUS INITIAL STRESSED ELASTIC MEDIUM UNDER THE EFFECT OF MAGNETIC FIELD
Author: Rajneesh Kakar, Shikha Kakar
Abstract:The propagation of magneto shear waves in a non-homogeneous, anisotropic, incompressible and initially stressed medium has been discussed in this study. The problem has been solved analytically using linear inhomogenities and the exact solution of frequency equations has been obtained. In fact, these equations are in agreement with the corresponding classical results when the medium is isotropic. The graphs have been plotted for frequency equations with MATLAB. It is observed that the shear waves have dependence on the direction of propagation, the anisotropy, magnetic field, non-homogeneity and the initial stress of the medium.
Keywords: Incompressible, initial-stress, anisotropic, shear-wave, magnetic field.
Problem of shear waves in an orthotropic elastic medium is been very important for the possibility of its extensive application in various branches of Science and Technology, particularly in Optics, Earthquake science, Acoustics, Geophysics and Plasma physics. The term “Initial stress” is meant by stresses developed in a medium before it is being used for study. The earth is an initially stressed medium. Due to presence of external loading, slow process of creep and gravitational field, considerable amount of stresses (called pre-stresses or initial stresses) remain naturally present in the layers. These stresses may have significant influence on elastic waves produced by earthquake or explosions and also in the stability of the medium. The propagation of surface waves is well documented in the literature, Abd-Alla and AboDahab  investigated time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation. Abd-Alla  studied the effect of initial stress and orthotropy on the propagation waves in a hollow cylinder. Abd-Alla et al.  presented Rayleigh waves in a magnetoelastic half-space of orthotropic material under an influence of initial stress and gravity field. Abd-Alla and Mahmoud  solved the magneto-thermoelastic problem in rotating a non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, and investigated the effect of the rotation on propagation of thermoelastic waves in a nonhomogeneous infinite cylinder of isotropic material. Abd-Alla et al.  Kakar  has investigated surface wave in non-homogeneous, general magneto thermo, viscoelastic Media. The propagation of Rayleigh waves in granular medium was given by many authors such as Bhattacharyya , El-Naggar , Ahmed , and others. In , Ahmed discussed the influence of gravity on the propagation of Rayleigh waves in granular medium. The Edge wave propagation in an incompressible anisotropic initially stressed plate of finite thickness has been studied by Dey et al. . Addy et al.  have studied Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field. Liu et al.  have demonstrated the propagation characteristics of converted refracted wave and its application in static correction of converted wave. Moczo et al.  provided mathematical modeling of seismic wave propagation using the FiniteDifference time-domain method. Huber  has explained the physical meaning of a nonlinear evolution equation of the fourth order relating to locally and non-locally supercritical waves in his work. Duan1 et al. [16-17] have investigated heterogeneous fault stresses from previous earthquakes and the effect on dynamics of parallel strike-slip faults and non-uniform pre-stress from prior earthquakes and the effect on dynamics of branched fault systems. Zhou and Chen  have studied the influence of seismic cyclic loading history on small strain shear modulus of saturated sands. Sharma et al.  discussed about the wave velocities in a pre-stressed anisotropic elastic medium. Selim et al.  have demonstrated the propagation and attenuation of seismic body waves in dissipative medium under initial and couple stresses. Seismology is the study of progressive elastic wave. But most of this studies and investigations do not include very important factor viz, the influence of initial stress, anisotropy and non-homogeneity present in the body. In this paper, the propagation of shear waves in a non-homogeneous anisotropic incompressible initially stressed medium under the influence of magnetic field is discussed. The frequency equation that determines the velocity of the shear wave has been obtained. The dispersion equations have been obtained, and investigated for different cases. Also, when the non-homogeneity are neglected, the frequency equation is in well agreement with the corresponding classical result.
This paper aims to present an account of the theory of wave propagation in non-homogeneous elastic media. The treatment necessarily involves considerable mathematical analysis. The pertinent mathematical techniques are, however, discussed at some length. The basic equations are the problem is dealing with magnetoelasticity. Therefore the basic equations will be electromagnetism and elasticity. The Maxwell equations of the electromagnetic field in vacuum are
where H, B, E, J , D denote the magnetic field intensity, magnetic induction, electric field intensity current density vector and displacement current vector respectively, c is the velocity of light in vacuum. The Gaussian units have been used.
FORMULATION OF THE PROBLEM
Most materials behave as incompressible media and the velocities of longitudinal waves in them are very high Initial stress arises in the earth due to factors like external pressure, slow process of creep, difference in temperature, manufacturing processes, nitriding, pointing etc., of rocks. Owing to the variation of elastic properties and the presence of these initial stresses, the medium becomes isotropic as well. We consider an unbounded incompressible anisotropic medium under initial stresses ? 11 and ? 22 along the x, y directions, respectively. When the medium is slightly disturbed, the incremental stresses ? 11, ? 12 and ? 22 are developed, and the equations of motion in the incremental state become from (**) and (*)
Where's is the magnetic permeability and H0 , the intensity of the uniform magnetic field, parallel to x-axes, also, τij is incremental stresses, ? ?is the rotational component about z-axis. The incremental stress-strain relation for an incompressible medium may be taken as
The effect of a non-homogeneous anisotropic incompressible, magnetic field and initially stressed respectively on shear wave velocity c with respect to depth b is as shown in figures [1-4]. It is obvious that shear wave velocity increases with the increasing of the depth b. The velocity of propagation also depends on the inclination of the direction of propagation; an increase in the inclination angle decreases the velocity in the beginning, takes a minimum value before increasing
1. The anisotropy, magnetic field, inhomogeneity of the medium, the initial stress, the direction of propagation and the depth have considerable effect on the velocity of propagation of shear wave. 2. The velocity of the shear waves depends on the parameters associated with anisotropy. 3. The velocity of the shear waves increases with the increase in magnetic field and vice versa. 4. The shear wave velocity decreases with the increase of initial stress.
The authors are thankful to the referees for their valuable comments.
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