Rotational Effect on the Propagation of Waves in a Magneto-Micropolar Thermoelastic Medium

: The present paper aims to explore how the magnetic field, ramp parameter, and rotation affect a generalized micropolar thermoelastic medium that is standardized isotropic within the half-space. By employing normal mode analysis and Lame’s potential theory, the authors could express analyt-ically the components of displacement, stress, couple stress, and temperature field in the physical domain. They calculated such manners of expression numerically and plotted the matching graphs to highlight and make compar-isons with theoretical findings. The highlights of the paper cover the impacts of various parameters on the rotating micropolar thermoelastic half-space. Nevertheless, the non-dimensional temperature is not affected by the rotation and the magnetic field. Specific attention is paid to studying the impact of the magnetic field, rotation, and ramp parameter of the distribution of temperature, displacement, stress, and couple stress. The study highlighted the significant impact of the rotation, magnetic field, and ramp parameter on the micropolar thermoelastic medium. In conclusion, graphical presentations were provided to evaluate the impacts of different parameters on the propagation of plane waves in thermoelastic media of different nature. The study may help the designers and engineers develop a structural control system in several applied fields.


Introduction
Because it is relevant in several applications, researchers have paid due attention to generalized thermoelasticity. Its theories include governing equations of the hyperbolic type and disclose the thermal signals' finite speed. In a standardized thermoelastic isotropic half-space, Abd-Alla et al. [1] investigated how wave propagation is affected by a magnetic field. Abouelregal [2] discussed the improved fractional photo-thermoelastic system for a magnetic field-subjected rotating semiconductor half-space. Singh et al. [3] studied reflecting plane waves from a solid thermoelastic micropolar half-space with impedance boundary circumstances. Said [4] studied the propagation of waves in a two-temperature micropolar magnetothermoelastic medium for a three-phase-lag system. The authors of [5] explored the impact of rotation on a two-temperature micropolar standardized thermoelasticity utilizing a dual-phase lag system. Rupender [6] studied the impact of rotation on a micropolar magnetothermoelastic medium because of thermal and mechanical sources. Bayones et al. [7] investigated the impact of the magnetic field and rotation on free vibrations within a non-standardized spherical within an elastic medium. Bayones et al. [8] discussed the thermoelastic wave propagation within the half-space of an isotropic standardized material affected by initial stress and rotation. Kalkal et al. [9] investigated reflecting plane waves in a thermoelastic micropolar nonlocal medium affected by rotation. Lotfy [10] explored two-temperature standardized magneto-thermoelastic responses within an elastic medium under three models. Zakaria [11] discussed the effect of hall current on a standardized magnetothermoelasticity micropolar solid under heating of the ramp kind. Ezzata et al. [12] studied the impacts of the fractional order of heat transfer and heat conduction on a substantially long hollow cylinder that conducts perfectly. Morse et al. [13] employed Helmholtz's theorem. The authors of [14] investigated the transient magneto-thermoelasto-diffusive interactions of the rotating media with porous in the absence of the dissipation of energy influenced by thermal shock. Kumar et al. [15] studied the axisymmetric issue within a thermoelastic micropolar standardized half-space. The authors of [16] studied the free surface's reflection of the thermoelastic rotating medium reinforced with fibers with the phase-lag and two temperatures. Deswal et al. [17] studied the law of fractional-order thermal conductivity within a two-temperature micropolar thermoviscoelastic. Using radial ribs, the authors of [18] examined the improved rigidity of fused circular plates. The authors of [19] investigated the motion equation for a flexible element with one dimension utilized for the dynamical analysis of a multimedia structure.
The paper aims to study the thermoelastic interactions within an elastic standardized micropolar isotropic medium in the presence of rotation affected by a magnetic field. The authors could obtain accurate solutions of the measured factors using the Lame's potential theory and the normal mode analysis. They could obtain and present in graphs the numerical findings of the distributions of stress, displacement, and temperature, for a crystal-like magnesium material.

Constitutive Relations and Field Equations
Constitutive relations and field equations are considered within a generalized micropolar magneto-thermoelastic medium in the presence of rotation as: (ii) Motion's Stress Equation (iii) Motion's Couple Stress Equation (iv) The Equation of Thermal Conductivity For these equations, the authors use the summation convention and utilize the comma to signify material derivative.

Formulating the Problem
Take into account an isotropic standardized micropolar generalized magneto-thermoelastic in the presence of rotation. The rectangular Cartesian coordinate scheme (x, y, z) is used in which the half-space surface plays the role of the plane z = 0 and the z−axis points vertically inwards. There are two extra terms for the displacement motion equation within the rotating frame, i.e., in which → u represents the displacement vector, → H represents the magnetic field, μ e represents the magnetic permeability, → J represents the density of the electric current, → E represents the electric intensity, and → h represents the perturbed magnetic field over the principal magnetic field. We apply the initial magnetic field vector where → f is the Lorentz force.

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The following form expresses Eq. (5) of thermal conductivity given expression (5a) Substituting (5a) into (1) and (2), we can express the arising stress as where These variables that are non-dimensional are employed to change the previous equations into non-dimensional structures where Eqs. (8)- (11) concerning previous the variables that are non-dimensional decline to (dropping the primes) Utilizing Helmholtz decomposition [13] and providing the potential φ and → ψ through the following equation Using Eq. (5a), the components of displacement u and w take the following form where φ (x, z, t) and ψ (x, z, t) represent scalar potential functions and

Normal Mode Analysis
For resolving the governing equations, the authors measured physically the decomposed variable concerning the normal modes in the following form: in which ω represents the complex time constant (frequency), i represents the imaginary unit, m represents the wave number in the x-direction, and u * , w * , φ * , ψ * , θ * , σ * ij and φ * 2 represent the functions' amplitudes.

Applications
Take into account a magneto-thermoelastic micropolar solid having a half-space that rotates z ≥ 0. M n s, as constants, are defined by imposing the proper boundary settings.

Thermal Boundary Conditions
The authors apply a heat shock of the ramp kind to the isothermal boundary of the plane z = 0, as follows in which δ (x) represents Dirac-delta function and φ * represents a constant temperature. Moreover, H (t) represents a Heaviside function defined as In this equation, t 0 represents a ramp parameter that shows the time required for raising the heat.

Mechanical Boundary Conditions
The boundary plane z = 0 is free of traction. In terms of mathematics, the boundary conditions take the following form Utilizing the non-dimensional quantities identified in (16), expressing stress Eqs. (12)-(15) and displacement Eq. (22), as well as the relations (44)-(46) becomes: in which A four-equation non-homogeneous structure is yielded by the boundary conditions (47)-(50), supported by expressions (51)-(55) and (38) in which

Numerical Results and Discussion
With an aim to highlight the theoretical findings of the previous sections, the authors provide some numerical findings using MATLAB. The material selected for this goal of magnesium crystal whose physical data resemble those provided in [17]:     Fig. 1: displays the various values of the components of displacement |u| and |w|, stress |σ zz |, |τ xz | , m zy , as well as temperature |θ| in the presence of distance z for the various values of the magnetic field H 0 . A zero value is the beginning. It agrees entirely with the boundary conditions. The magnetic field demonstrates significant growing and declining impacts on the components of stress and displacement. These impacts vanish when moving away from the point of the application of the source. The temperature is not affected by the magnetic field. Nevertheless, the qualitative performance is almost equal in the two values.
Variations in displacement components |u| , |w|, stress components |σ zz |, |τ xz | , m zy and temperature |θ| with spatial coordinates z for various values of rotation are shown in Fig. 2 that begins with the value of zero, which agrees totally with the boundary conditions. The highest impact zone of rotation is about z = 0.3. Nevertheless, the temperature is not affected by the rotation. The components of temperature, stress, and displacement increase numerically for 0 ≤ z ≤ 1. Then, they decrease and moves to minimum value at z = 1.
In Fig. 3, we have depicted displacement components |u| , |w|, stress components |σ zz |, |τ xz | , m zy and temperature |θ| in the presence of distance z to explore the impacts of ramp parameter t 0 . The ramp parameter's highest impact zone is about z = 0.3. Furthermore, all curves share a corresponding zero value stating point to satisfy the boundary conditions.

Conclusion
The study concludes that: a. The magnetic field and rotation play an influential part in distributing the physical quantities whose amplitude varies (rise or decline) as the rotation and magnetic field increase.
With the rotation and the magnetic field, the quantities to surge near or away from the application source point are restricted. b. All physical quantities satisfy the boundary conditions. c. The theoretical findings of the study can give stimulating information for seismologists, researchers, and experimental scientists who are interested in the field.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.