Magneto-Thermoelasticity with Thermal Shock Considering Two Temperatures and LS Model

: The present investigation is intended to demonstrate the magnetic field, relaxation time, hydrostatic initial stress, and two temperature on the thermal shock problem. The governing equations are formulatedin the context of Lord-Shulman theory with the presence of bodily force, two temperatures, thermal shock, and hydrostatic initial stress. We obtained the exact solution using the normal mode technique with appropriate boundary conditions. The field quantities are calculated analytically and displayed graphically under thermal shock problem with effect of external parameters respect to space coordinates. The results obtained are agreeing with the previous results obtained by others when the new parameters vanish. The results indicate that the effect of magnetic field and initial stress on the conductor temperature, thermodynamic temperature, displacement and stress are quite pronounced. In order to illustrate and verify the analytical development, the numerical results of temperature, displacement and stress are carried out and computer simulated results are presented graphically. This study helpful in the development of piezoelectric devices.


Introduction
inserted two relaxation times and advocating finite wave speed thermal in solids by correcting the energy equation and Neuman-Duhamel relation or modifying Fourier's conduction heat equation. Chandrasekharaiah et al. [8] studied the thermoelastic interactions without energy dissipation due to a point heat source. Chandrasekharaiah et al. [9] studied the temperature-rate-dependent thermo-elastic interactions due to a line heat source. The magnetoelastic earth's material nature may affect on the wave propagation, especially surface waves. A literature review of the earliest contributions to the subject has been discussed in details by Puri [10]. Nayfeh et al. [11] who studied the plane wave propagation under the electromagnetic field in a solid medium. Choudhuri et al. [12] discussed the rotation effect on magneto-thermoelastic media in an elastic medium. Ezzat et al. [13] investigated the electromagneto-thermoelastic plane waves with two relaxation times in a medium of perfect conductivity Ezzat et al. [14] studied the electromagnetothermoelastic plane waves with thermal relaxation in a medium of perfect conductivity. Zhuang et al. [15] studied the explicit phase field method for brittle dynamic fracture. Rabczuk et al. [16] investigated the nonlocal operator method for partial differential equations with application to the electromagnetic waveguide problem. Bahar et al. [17] introduced the formulation of state space in thermoelastic problems which also developed in Sherief [18] including the heat sources effectively. Sherief et al. [19] studied the two dimensional generalized thermoelasticity problem for an infinitely long cylinder. Youssef et al. [20] analysis a generalized thermoelastic infinite layer problem with the state space approach considering three models. Formulation of state space for the vibration of gold nano-beam in femtoseconds scale pointed out by Elsibai et al. [21]. Biot [22] clears that under stress free state would be fundamentally different from initial stresses states acoustic propagation and obtained the longitudinal and transverse wave velocities along the coordinate axes only. Chattopadhyay et al. [23] explained the plane wave reflection and refraction in an unbounded medium under initial stresses. Montanaro [24] studied the linear thermoelasticity problem with hydrostatic initial stress. Othman et al. [25] studied the reflection waves in a generalized thermoelastic medium from a free surface under hydrostatic initial stress under different thermoelastic theories. Youssef [26] studied the problem of generalized thermoelastic infinite medium with a cylindrical cavity subjected to a ramp-type heating and loads.
The main purpose of the present investigation is intended to demonstrate the magnetic field, relaxation time, hydrostatic initial stress, and two temperature on the thermal shock problem. The governing equations are formulated in the context of Lord-Shulman theory with the presence of body force, two temperatures, thermal shock, and hydrostatic initial stress. We obtained the exact solution using the normal mode technique with appropriate boundary conditions. The field quantities are calculated analytically and displayed graphically under thermal shock problem with effect of external parameters respect to space coordinates.

Formulation of the Problem
Considering that the medium is a perfect electric conductor and the absence of the displacement current (SI) [10], the linearized Maxwell equations governing the electromagnetic field as the form as shown in Fig. 1 curl in which h is the perturbed magnetic field over the primary magnetic field, J represents the electric current density, μ e represents the magnetic permeability, H 0 represents the constant primary magnetic field, and u represents the displacement vector, H = (0, 0, H 0 ), u = (u, v, 0).

Figure 1: Schematic of the problem
The equation of heat conduction given [15] as The stress-displacement relations for the isotropic material are The Maxwell's equation formulated as The motion equation splits to The heat conduction and dynamical heat related by the form The non-dimensional variables for simplifying gives as where By dropping the dashed for convenience, and substitute Eq. (10), then Eqs. (2) and (9) take the following form where ε = γ ρC E and β = aη 2 c 2 0 Assuming the scalar and vector potential functions (x, y, t) and ψ(x, y, t) in the nondimensional form defined as: By using (15) and (10) in Eqs. (7) and (8), we obtain.

The Solution to the Problem
The solution of the previous physical variables can be decomposed in terms of normal mode technique in the exponential harmonic form where i = √ −1, b be a wave number, ω is the time constant, and u * (x), ϕ * (x), θ * (x) and σ * ij (x) are the amplitudes of the physical field quantities.
Using Eq. (17), into Eqs. (12) and (14)-(16), we obtain where where Similarly, we get which can be factorized to where k 2 n (n = 1, 2) are the roots of the characteristic equation as x → ∞, the solution of Eq. (26) is given by Similarly The solution of Eq. (21) has the form To get the amplitudes of the displacements u and v, which bounded as x → ∞, then Eqs. where Thus

Substitution of Eqs. (35) and (36) into Eqs. (3)-(5), we obtain
where The normal mode analysis is, in fact, to look for the solution in Fourier transformed domain. Assuming that all the field quantities are sufficiently smooth on the real line such that normal mode analysis of these functions exists.

Applications
Now we will obtain the parameters M n (n = 1, 2, 3), we will suppress the positive exponentials that are unbounded at infinity. The constants M 1 , M 2 , M 3 must choose such that the boundary conditions on the surface at x = 0 take the form: 1) Boundary conditions for the thermal at surface under thermal shock 2) Boundary condition for the mechanic at surface under initial stress 3) Boundary condition for the mechanic at the surface is traction free Substitute into the above boundary conditions in the physical quantities, we obtain In the context of the boundary conditions in Eqs. (56)-(58) at the surface x = 0, we get a system of three Algebraic equations, we will apply the inverse matrix method, we will get the three constants M j , j = 1,2,3, after that by substituting into the main expressions to obtain the displacements, temperature and other physical quantity.

Numerical Results and Discussions
To illustrate the analytical variable obtained earlier, we will consider a numerical example consider copper material. The results display the variation of displacements, temperature and stress in the context of LS theory. λ = 7.59 × 10 9 N/m 2 , μ = 3.86 × 10 10 kg/ms 2 , C E = 383.1 J/(kgk), ε = 0.0168 α = −1.28 × 10 9 N/m 2 , ρ = 7800 kg/m 2 , K = 386 N/Ks, τ 0 = 0.02, f * = 1 α t = 1.78 × 10 −5 N/m 2 , a = 1, T 0 = 293 K, ω = ω 0 + iξ , ω 0 = 2, ξ = 1, η = 8886.73 m/s 2 We took the constants: y = −1, time t = 0.1, b = 0.25, H = 10 5 , P = 10 10 , β = 0.1, τ = 0.1 for all computations, and we used for the real part of the displacement u, v, strain e and the stresses (σ xx , σ yy , σ xy ), thermal temperature θ and conductive temperature φ. All the field quantities don't depend only on space x and time t, also on the relaxation time τ and in dimensionless form: The output is plotted in Figs. 2-7. Fig. 2 displays the values of the conductive temperature φ, which decreases with increasing of axial x. They indicate that the medium along the axial x tends to zero as x tends to infinity. The effect of time t, wave number b, longitudinal wave velocity, β, relaxation time τ , magnetic field H and initial stress P on the conductive temperature. The conductive temperature increases with increasing of time and longitudinal wave velocity, while it decreases with increasing of wave number and relaxation time, also, there isn't any effect due to the magnetic field and initial stress on the conductive temperature. Fig. 3 plots the values of the thermodynamic temperature θ, which decreases with increasing of axial x. These figures indicate that the medium along axial x. The effect of time t, wave number b, longitudinal wave velocity β, relaxation time τ , magnetic field H and initial stress P on the conductive temperature. The thermodynamic temperature increases with increasing of time and longitudinal wave velocity, while it decreases with increasing of wave number and relaxation time, as well, there is no effect of magnetic field and initial stress on the thermodynamic temperature. Fig. 4 shows the values of displacement u, which has an oscillatory behavior in the whole range of axial x under the effects of time t, wave number b, longitudinal wave velocity β, relaxation time τ , magnetic field H and initial stress P. In these figures, it is clear that the displacement has a nonzero value only in the bounded region of space, while it increases with increasing of time, wave number and longitudinal wave velocity, as well it decreases with increasing relaxation time and magnetic field, while there is no effect of initial stress on the displacement. Fig. 5 displays the value of displacement v which has an oscillatory behavior in the whole range of axial x under the effects of time t, wave number b, longitudinal wave velocity, β, relaxation time τ , magnetic field H and initial stress P. In these figures, it is clear that the displacement has a non-zero value only in the bounded region of space, while it increases with increasing of time, wave number and longitudinal wave velocity, as well it decreases with increasing relaxation time and magnetic field, while there is no effect of initial stress on the displacement.  Fig. 7 shows the values of the stress σ yy , which increases with increasing of axial x. These figures indicate that the medium along axial x the effect of time t, wave number b, longitudinal wave velocity, β, relaxation time τ , magnetic field H and initial stress P on the stress. The stress decreases with increasing of time, wave number, longitudinal wave velocity and initial stress, while it increases with increasing relaxation time, as well, there is no effect of magnetic field on the stress.

Conclusion
The results and conclusions can be summarized as follows 1) Normal mode analysis of the problem of magneto-thermoelastic solid has been applied and developed.
2) The generalized magneto-thermoelasticity with thermal shock, two temperatures, initial stress described with characteristic by fourth order equation.
3) The role of the initial stress, thermal shock, magnetic field clears strongly on the physical quantities depending on the nature of the medium, horizontal and vertical distances x and y respectively. The nature of forced applied as well as the type of boundary conditions deformation. 4) Finally, it is concluded that all the external parameters affect strongly on the physical quantities of the phenomenon which has more applications, especially, in engineering, geophysics, astronomy, acoustics, industry, structure, and other related topics.