A.M.Abd-Alla,S.M.Abo-Dahab,M.A.Abdelhafez and A.M.Farhan

1Department of Mathematics,Faculty of Science,Sohag University,Sohag,Egypt

2Department of Mathematics,Faculty of Science,South Valley University,83523,Egypt

3Department of Physics,Faculty of Science,Jazan University,Jazan,Saudi Arabia

4Department of Physics,Faculty of Science,Zagazig University,Zagazig,Egypt

Abstract: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 analytically 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 comparisons 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.

Keywords:Magnetic field;micropolar;thermoelastic;rotation;ramp parameter;normal mode analysisThis work is licensed under a Creative Commons Attribution 4.0 International License,which permits unrestricted use,distribution,and reproduction in any medium,provided the original work is properly cited.

Nomenclature

1 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.

2 Constitutive Relations and Field Equations

3 Formulating the Problem

4 Normal Mode Analysis

5 Applications

Take into account a magneto-thermoelastic micropolar solid having a half-space that rotatesz≥0.M′ns,as constants,are defined by imposing the proper boundary settings.

5.1 Thermal Boundary Conditions

The authors apply a heat shock of the ramp kind to the isothermal boundary of the planez=0,as follows

in whichδ(x)represents Dirac-delta function andφ?represents a constant temperature.Moreover,H(t)represents a Heaviside function defined as

In this equation,t0represents a ramp parameter that shows the time required for raising the heat.

5.2 Mechanical Boundary Conditions

The boundary planez= 0 is free of traction.In terms of mathematics,the boundary conditions take the following form

Applying Eqs.(16)–(27)make the former boundary conditions represented,as follows

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)as follows

The expressions ofMn,(n = 1,2,3,4)attained by resolving the structure(56),when replaced in Eq.(38)and Eqs.(51)–(55),give these expressions of field variables

in which

6 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]:

Taking into account the previous physical data,non-dimensional field variables are estimated,and the findings are presented graphically at various positions ofzatt=0.1 and x = 1.0.The motion range is 0 ≤z≤1.0.Figs.1–3 show variations,correspondingly.

Figure 1:Various values of the magnitude of u,w,σzz,τzx,mzy,θ for various values of H0 in the presence of distance 0 ≤z ≤1

Figure 2:Various values of the magnitude of u,w,σzz,τzx,mzy,θ for various values of Ω in the presence of distance 0 ≤z ≤1

Figure 3:Various values of the magnitude of u,w,σzz,τzx,mzy,θ for various values of t0 in the presence of distance 0 ≤z ≤1

Fig.1:displays the various values of the components of displacement |u|and |w|,stressas well as temperature |θ|in the presence of distancezfor the various values of the magnetic fieldH0.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 componentsand temperaturewith spatial coordinateszfor various values of rotationΩare 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 aboutz=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 atz=1.

In Fig.3,we have depicted displacement componentsstress componentsand temperature |θ| in the presence of distancezto explore the impacts of ramp parametert0.The ramp parameter’s highest impact zone is aboutz=0.3.Furthermore,all curves share a corresponding zero value stating point to satisfy the boundary conditions.

7 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.

Funding Statement:The authors express their gratitude to the Academy of Scientific Research and Technology,Egypt for funding the present study under Science Up Grant No.(6459).

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