IJCRR

IJCRR - 4(24), December, 2012

Pages: 131-139

Date of Publication: 22-Dec-2012

S-WAVE PROPAGATION IN A NON-HOMOGENEOUS INITIAL STRESSED ELASTIC MEDIUM UNDER THE EFFECT OF MAGNETIC FIELD

Author: Rajneesh Kakar, Shikha Kakar

Category: Healthcare

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.

Full Text:

INTRODUCTION
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 [1] investigated time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation. Abd-Alla [2] studied the effect of initial stress and orthotropy on the propagation waves in a hollow cylinder. Abd-Alla et al. [3] presented Rayleigh waves in a magnetoelastic half-space of orthotropic material under an influence of initial stress and gravity field. Abd-Alla and Mahmoud [4] 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. [5] Kakar [16] 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 [7], El-Naggar [8], Ahmed [9], and others. In [10], 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. [11]. Addy et al. [12] have studied Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field. Liu et al. [13] have demonstrated the propagation characteristics of converted refracted wave and its application in static correction of converted wave. Moczo et al. [14] provided mathematical modeling of seismic wave propagation using the FiniteDifference time-domain method. Huber [15] 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 [18] have studied the influence of seismic cyclic loading history on small strain shear modulus of saturated sands. Sharma et al. [19] discussed about the wave velocities in a pre-stressed anisotropic elastic medium. Selim et al. [20] 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 these 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.

RESEARCH METHODOLOGY
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 , and 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.

where = the incremental stress components, = the components of the displacement vector of the solid, 

Where  and  are the components of the rotational component .

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,  is incremental stresses the rotational component about z-axis. The incremental stress-strain relation for an incompressible medium may be taken as

Since the problem is treated in x-y plane where  = , e_ij is an incremental strain component, and N and Q are the rigidities of the medium. The incompressibility condition e_xx + e_yy = 0 is satisfied by

Substituting from equations (3) and (4) in equations (1) and (2), we get

Assuming non-homogeneities as

Substituting from equation (7) in equations (5) and (6), we get (8)

Eliminating τ from equations (8) and (9), we get

SOLUTION OF THE PROBLEM For propagation of sinusoidal waves in any arbitrary direction, we take the solution of equation (10) as

 Where is the angle made by the direction of propagation with the x-axis, and c₁ and k are the velocity of propagation and wave number, respectively. Using equation (11) in equation (10) and equating real and imaginary parts separately, we get

ANALYSIS OF PROBLEM IN HOMOGENEOUS MEDIUM (i) Analysis of equation (12) obtained by equating the real part of equations of motion: Case I : In this case Q is homogeneous (a ? 0) i.e., rigidity along vertical direction is constant

The velocity along x-direction (cos ? = 1, sin ??= 0, c₁ = c₁₁₎ as

Equation (16) depends on the initial stress and magnetic field. To obtain the velocity of propagation along y-direction Put cos ??= 0, sin ??= 1 and c₁ = c₂₂ in equation (14), we get

Case II : In this case N is homogeneous (b ??0) i.e., rigidity along horizontal direction is constant

Equation (19) depends on depth and magnetic field. The velocity of propagation along ydirection (cos ? = 0, sin ??= 1, c ₁ = c₁₁), is given by

For P > 0, the velocity along y-direction may increase considerably at a distance from free surface and the wave becomes dispersive.

Case III : In this case N, Q and ? are homogeneous (a ??0, b ??0, c ??0)

In the absence of initial stress the velocity equation becomes

In x-direction (cos ??= 1, sin ??= 0, c1 = c11), the velocity is given by

and in y-direction (cos ??= 0, sin ??= 1, c1 = c22), the velocity is given by 

(ii) Analysis of equation (13) obtained by equating imaginary parts of equation of motion.

In absence the initial stress P in equation (13), following three cases have been analyzed.

Case I : In this case Q is homogeneous (a ??0) i.e., rigidity along vertical direction is constant

This allows that velocity of shear wave is always damped. The velocity of wave along x-direction (cos ??= 1, sin ??= 0, c ₁ = c₁₁) is obtained as

This shows that actual velocity in x-direction is damped by (4N₀ b/Q₀ c), and no damping takes place along y-direction.

Case II : In this case N is homogeneous (b ??0), i.e., rigidity along horizontal direction is constant.

The velocity of wave along x-direction (cos ??= 1, sin ??= 0, c ₁ = c₁₁) is given by

The existence of negative sign shows that damping does not take place along x-direction for (b ??0). The velocity along y-direction is given by

Indicating that a damping of magnitude (2a/c) takes place along y-direction. Case III : In this case N and Q are homogeneous (a ? 0, b ? 0) but density is linearly varying with depth :

i.e. no damping takes place.

NUMERICAL ANALYSIS AND DISCUSSION To get numerical information on the velocity of shear wave in the non-homogeneous initially stressed medium we introduce the following nondimensional parameters:

Using these parameters in the equation (2), we obtain

Various graphs are plotted with the help of MATLAB by taking the parameters as

a = 4; c = 0.7; p = 0.5; N = 2.5; H = 0.3; b = 0, 1, 6

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

CONCLUSIONS
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.

ACKNOWLEDGEMENTS
The authors are thankful to the referees for their valuable comments.

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