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 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.

Recently, more attentions have been considered by researchers and engineers to the thermoelasticity theory to release the confliction of infinite speed due to the thermal signals, because of the importance in diverse fields as geophysics, acoustics, engineers, plasma physics, and industries. Biot [

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.

Considering that the medium is a perfect electric conductor and the absence of the displacement current (SI) [

The equation of heat conduction given [

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

By dropping the dashed for convenience, and substitute

Assuming the scalar and vector potential functions

By using

Also

The solution of the previous physical variables can be decomposed in terms of normal mode technique in the exponential harmonic form

Using

Solving

Similarly, we get

which can be factorized to

as

Similarly

The solution of

To get the amplitudes of the displacements u and v, which bounded as

From

Thus

Substitution of

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.

Now we will obtain the parameters

Boundary conditions for the thermal at surface under thermal shock

Boundary condition for the mechanic at surface under initial stress

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

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.

We took the constants:

The output is plotted in

The results and conclusions can be summarized as follows

Normal mode analysis of the problem of magneto-thermoelastic solid has been applied and developed.

The generalized magneto-thermoelasticity with thermal shock, two temperatures, initial stress described with characteristic by fourth order equation.

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.

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.

Taif University Researchers Supporting Project Number (TURSP-2020/164), Taif University, Taif, Saudi Arabia.