Effect of a Magnetic Field on the Propagation of Waves in a Homogeneous Isotropic Thermoelastic Half-Space

The prime objective of the present paper is to analyze the propagation of thermoelastic waves in a homogeneous isotropic elastic semi-infinite space that is exposed to a magnetic field at initial temperature T0 and whose boundary surface is subjected to the moving heat source and load moving with finite velocity. Temperature and stress distribution occurring due to heating or cooling have been determined using certain boundary conditions. Numerical results indicate that the effect of the magnetic field is very pronounced. Comparison is made with the results predicted by the theory of thermoelasticity in the absence of a magnetic field. Apart from geophysical applications, the consequences of the present study offer a better platform to design a surface wave sensor by means of its established results. The obtained results may be also used for acquiring a better performance in surface acoustic wave devices and waveguides.


INTRODUCTION
Thermoelastic problems play an important part in different branches of technical sciences. In engineering practice, most structures contain internal interfaces when heat flow in the structure is disturbed by some defects, such as holes and cracks, the local temperature gradient around the defects is increased and the temperature field is often discontinuous across the defects. Thermal disturbances of this type may produce material failure. Therefore, thermal analysis of such structures is very important. As far as loading is concerned, generally two types of loadingmechanical and thermalare applied to the structure surface. Mechanical loading consists of pressure and shear tractions, and thermal loading is caused by frictional heating and results in thermoelastic stress. The knowledge of the thermoelastic stress field in the structure is essential for failure prevention and life prediction because the total stress consists of the thermoelastic and elastic stresses. Due to this phenomenon, frictional heating significantly influences the failure of components in contact under relative motion, e.g. thermocracking of breaks and face sears, and scuffing in gear. The linear dynamic theory of thermoelasticity introduced by Biot [1] holds that the governing equations for displacement and temperature fields consist of the two coupled partial differential equations. He and Cao [2] presented a valuable investigation based on the generalized thermoelastic theory with thermal relaxation in the context of Lord and Shulmans theory, which is used to investigate the magneto-thermoelastic problem of a thin slim strip placed in the magnetic field and subjected to the moving plane of the heat source. Deformation of a rotating generalized thermoelastic medium subjected to ramp-type heating and loading with a hydrostatic initial stress was studied by Ailawalia and Narah [3]. Disturbances in a homogeneous isotropic elastic medium subjected to the moving source with generalized thermoelastic diffusion were considered by Deswal and Choudhary [4]. Abd-Alla and Ahmed [5] investigated the influence of both the gravity field and initial stress on the propagation of Rayleigh waves in an orthotropic thermoelastic medium. A linear temperature ramp function used to model more realistically thermal loading of the half-space surface subjected to the moving heat source was considered by Amin et al. [6]. Abouelregal [7] investigated the induced temperature, displacement, and stress field in an infinite transversely isotropic unbounded medium with a cylindrical cavity due to the moving heat source and harmonically varying heat. Abd-Alla et al. [8] studied the influence of rotation and initial stress on the propagation of Rayleigh waves in an orthotropic elastic half-space with gravity. The microrotation effect of a load applied normal to the boundary and moving at a constant velocity along one of the coordinate axes in the generalized thermoelastic half-space was discussed by Kumar and Deswal [9]. Abd-Alla et al. [10] solved the problem of transient coupled thermoelasticity of an annular fin using the finite difference method. Abd-Alla et al. [11] studied the influence of the magnetic and gravity fields on the generalized thermoelastic Rayleigh waves in a granular medium with initial stress. Ailawalia and Narah [12] obtained expressions for displacement, force, stress and temperature distributions in a rotating generalized thermoelastic medium with a hydrostatic initial stress by applying Laplace and .ourier transforms subjected to ramp-type heating and loading. Kumar and Gupta [13] studied the general plane strain problem of an orthotropic micropolar thermoelastic half-space with one relaxation time. The effect of the load velocity, nonlocal parameter, and ramping-time parameter on the dynamic deflection, temperature, and bending moment of the nanobeam was investigated by Abouelregal and Zenkour [14]. Abd-Alla et al. [15] established the influence of the thermal stress and magnetic field in the thermoelastic half-space without energy dissipation. Abo-Dahabet et al. [16] investigated the effect of rotation and gravity on the reflection of P-waves from thermo-magnetomicrostretch medium in the context of the three-phase-lag model with initial stress. Xia et al. [17] solved the generalized thermoelastic problem of an isotropic semi-infinite plate subjected to the moving heat source using the finite element method directly in the time domain. Wang and Dong [18] studied the magneto-thermodynamic stress and perturbation of the magnetic field vector in an inhomogeneous thermoelastic cylinder. Sherief and Allam [19] investigated the electromagnetic interaction of a two-dimensional generalized thermoelastic problem for an infinitely long solid conducting circular cylinder.
In contrast to all these investigations, the present paper studies the propagation of a thermoelastic wave in the half-space made of homogeneous isotropic mate-rials exposed to the magnetic field and in contact with the coordinate system moving with the load point by shifting the origin to the position of the load point. Here the Lame potential method is proposed to analyze the problem and to obtain numerical solutions for displacement components, stresses, and temperature distribution. The effects of the magnetic field, moving heat source, and load moving with finite velocity on the components of displacement, stress, tangential stress, and temperature are considered. The results obtained in this investigation are more general in the sense that some earlier published results are obtained from our result as special cases. The results obtained are presented graphically with concluding remarks.
where h is the perturbed magnetic field over the primary magnetic field, E is the electric intensity, J is the electric current density, e µ is the magnetic permeability, H is the magnetic field, and u is the displacement vector. A substitution of the initial magnetic field vector Let us consider the half-space 0 z ≥ initially at a temperature 0 T and in the stress-free state. Temperature, displacement, and stress fields will vary due to external loading. On this assumption, the displacement occurs along the x and z axes and is a function of the spatial coordinates x, z and time t.
The dynamic equations of motion in the magnetic field are given by The stressdisplacement relations with incremental isotropy are given by Here t (3 2 ) , γ = λ + µ α T is the temperature, e c is the specific heat, ρ is the density, t α is the coefficient of thermal expansion, 0 τ is the relaxation time, K is the thermal conductivity, x f and z f are the components of the Lorentz force: We will consider a moving heat source applied to the half-space surface with the following initial conditions: Along with the moving heat source, we consider the moving load with the following initial conditions: Here h is the surface heat transfer coefficient, f and g are the arbitrary functions, and L is the velocity of both source and load. Now, we will use the following dimensionless variables to transform the above equations into dimensionless forms: where 2 1 ( 2 ) c = λ + µ ρ and 1 e ( ).
In terms of the dimensionless variables in (9), Eqs. (3)(6) take the following forms: The stress components in the dimensionless forms are ( 1 2 ) , In the subsequent discussions we omit primes for convenience.

SOLUTION O. THE PROBLEM
Using Helmholtzs theorem [20], the displacement components can be written as where φ(x, z, t) and ψ(x, z, t) are the scalar functions, and (0, , 0) = ψ ø is the vector potential function. The substitution of Eqs. (14) into Eqs. (10)(12) gives where 2 ( ), c = µ λ + µ ( 2 ) ( 2 ). C H = λ + µ +µ λ + µ We use the coordinate system moving with the load point by shifting the origin to the load point is the dimensionless loading speed, and the coordinates x′′ and z′′ move in the positive direction of the x axis with the speed m. A substitution of Eq. (16) into Eqs. (15) gives Replacing 1 ∇ by ∇ to and x′′ by x results in We see that Eqs. (20) and (21) are coupled with T and φ, while Eq. (22) is independent on ψ. In order to solve the coupled Eqs. (20) and (21), we eliminate T or φ to obtain the following equation To solve Eqs. (22) and (23), we take , , where A and B are the constants, and k, σ, and ′ σ are the unknown quantities to be determined.
A substitution of φ and ψ into (22) and (23) gives .or ′ σ to be real, A B and k C are the arbitrary constants, which will be determined by using certain boundary conditions.
The stress components in terms of φ and ψ are In the first equation in (38), we assume ) .
.ig. 1. Case I. Variation of T, u, w, , xx σ , zz σ and xz τ at different relaxation times 0 τ with respect to horizontal distance z.

CASE II: MOVING LOAD SOURCE
Let us consider the boundary conditions for the moving load source given by Eq. (8) to determine the arbitrary constants , , k k A B ′ ′ and k C′ for the medium surface with z = 0: where h is the coefficient of heat transfer. The load function in (43) will be given by We find the solution of Eqs. (45)(47) by Cramers rule (Appendix II).

NUMERICAL RESULTS AND DISCUSSION
To illustrate the analytical procedure presented earlier, we now consider a numerical example for which computational results are given.
The material chosen for this purpose is stainless steel, the physical data for which are the following: λ = 9. The graphical results for the displacement components, normal stress, tangential stress, and temperature are shown in .igs. 16.
.igure 1 shows variations of the absolute values of temperature , T displacement components , , u w normal stresses , , xx zz σ σ and shear stress xz τ with respect to the z axis at different relaxation times 0 τ due to the moving heat source. It is obvious that the absolute values of displacement component , w normal stress , zz σ temperature and shear stress xz τ increase with increasing relaxation time, while the absolute value of normal stress and shear stress have an oscillatory behavior for generalized thermoelastic material in the whole range of axis z due to the moving heat source. We observe larger oscillation in the left side and smaller oscillation in the right side. There is no effect of the relaxation time on the absolute value of the displacement component , u and it is noticed that the temperature , T shear stress , xz τ and normal stress zz σ satisfy the boundary conditions. As a and w have an oscillatory behavior due to the moving heat source. We see larger oscillation in the left side and smaller oscillation in the right side. Here we find that the magnetic field causes the displacement components to oscillate, and it increases with increasing axis z in the whole range of axis z. It is also seen that the stress distribution has a local extremum at the position of the magnetic field increase.
. load source and moving heat source reveals that the relaxation time play a significant role in all the analyzed quantities.
Variations of temperature, displacement components, and stress components depend on the nature of this moving heat source and the moving load source.
The magnetic field, relaxation time, and wave number of the half-space give the same effect in case I and case II, as mentioned above in the results.
The present theoretical results may provide interesting information for experimental scientists, researchers, and seismologists working on this subject.