# Mathematical Modeling on Rotational Magneto-Thermoelastic Phenomenon under Gravity and Laser Pulse considering Four Theories

Authors: Hammad Alotaibi; S. M. Abo-Dahab; E. M. Khalil; S. Abdel-Khalek; Emad E. Mahmoud; M. Higazy
Journal: Complexity
 (2021)

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## Abstract

The aim of this investigation is making mathematical model for the variation in laser pulse, rotational gravity, and magnetic fields on the generalized thermoelastic homogeneous isotropic half-space. The governing dynamical system equations have been formulated considering the four thermoelastic models: coupled (CT) model, Lord and Shulman (LS) model, Green and Lindsay (GL) theory, and Green and Naghdi (GN III) model. Normal mode analysis technique is used to obtain the analytical expressions for the displacement components, temperature, and mechanical and Maxwell’s stresses distribution. The effect of laser pulse, gravity, and magnetic field is studied by numerical examples and displayed graphically. A comparison has been made between the theories as well as the present results and agreement with it as a special case from this study. The results predict the strong effect of magnetic field, laser pulse, and gravity field on the wave propagation phenomenon.


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## Body

## 1. Introduction

In last year, the magnetic field, thermal field, elastic field, and rotation interaction received more attention due to the various applications in astronomy, geophysics, engineering, structures, medicine, etc. The topic of thermoelasticity with magnetic field has received the attention of many researchers because of the applications, especially the practical in labs. Biot [1] is the first to discuss the developed thermoelastic theories to overcome the propagation of thermal signals in infinite speed as predicted by the classical thermoelastic coupled dynamical theory. The theories of generalized thermoelasticity have been developed to remove the confliction of the infinite speed of thermal signals in coupled thermoelasticity, which is a physically impossible phenomenon (see [2, 3]). Green et al. [4, 5] formulated three models of thermoelasticity named GN (types I, II, and III). The gravitational effect on the wave propagation in solids in an elastic globe was investigated by Bromwich [6]. Othman et al. [7] investigated the rotation and gravitational effect on the generalized thermoelastic medium in the context of model of a dual-phase lag. The surface wave propagation in a nonhomogeneous medium under the parameter of gravitational is investigated by Das et al. [8]. Abd-Alla et al. [9] discussed the thermoelastic wave propagation in an isotropic homogeneous half-space of a material under gravity field. The Stoneley, Love, and Rayleigh waves in anisotropic fibre-reinforced general viscoelastic media of higher order have been pointed out [10] considering the rotational effect. The Earth’s material nature generally is magnetoelastic which may affect the propagation of waves. The thermal stress and magnetic field effect in thermoelasticity neglecting the dissipation of the energy is pointed out by Abo-Dahab et al. [11]. Abo-Dahab et al. [12] pointed out S-waves propagation in anisotropic nonhomogeneous medium with magnetic field initial stress, gravity field, and rotation. Some recent works on two-temperature effect with or without rotation are discussed in [13, 14]. Recently, Abo-Dahab et al. [15] investigated the two-dimensional magnetic field and rotation in generalized thermoelasticity. Lotfy et al. [16] introduced the problem of semiconducting response in an infinite medium with two-temperature and photothermal excitation because of the laser pulse. Abo-Dahab [17] discussed the generalized magneto-thermoelastic reflection waves under two-temperature, thermal shock, and initial stress. Recently, Saroj et al. [18] investigated the Love waves transference in prestressed PZT-5H material stick. The authors in [19, 20] discussed waves propagation considering new external parameters and types of waves. Abo-Dahab et al. [21] investigated the problem of magneto-thermoelasticity under influence of laser pulse and gravity field in the context of four theories. The authors in [22, 23] arrived to new results considering the magnetization and nonlinear heat on the flow phenomenon. Othman et al. [24] discussed the thermal loading due to laser pulse influence on a generalized micropolar thermoelastic solid with different theories.In this paper, we suggest mathematical modeling considering the effect of variation in laser pulse, magnetic field gravity, and rotation on a generalized thermoelastic model in an isotropic homogeneous half-space. The governing dynamical system equations have been formulated considering the four thermoelastic models. The four thermoelastic theories are coupled (CT), Lord and Shulman (LS), Green and Lindsay (GL), and Green and Naghdi (GN III) theories. The technique of normal mode technique is used to analyze the expressions for the displacements, temperature, and mechanical and Maxwell’s stresses. The effect of gravity, rotation, laser pulse, and magnetic field is studied and displayed graphically. A comparison has been made between the theories as well as the present results and the previous results concluded by the othersand agreement with it as a special case from this study. The results obtained have a significant rule in engineering, astronomy, aircrafts, dynamical system reactors, and aircrafts.


## 2. Formulation of the Problem and Basic Equations

A coordinate of Cartesian systemx,y,z is considered on the surface y=0 and pointing z-axis vertically into the half-space medium x≥0, as shown in Figure 1.Figure 1
Schematic of the problem.The dynamic displacement vector in two dimensionu=u,0,w, and consider all the quantities are dependent on (t, x, z).The fundamental equations of linear rotational generalized thermoelasticity, whereΩ⟶=0,Ω,0 considering Coriolis component and magnetic field H⟶=0,Ho,0 in the absence of heat sources, take the following form:(1)μui,jj+λ+μuj,ij−γ1+ϑ0∂∂tT,i++Fi+Gi=ρ∂2u∂t2−Ω2u,kT,ii+k∗T˙,ii=ρCen1∂∂t+τ0∂2∂t2T+γT0n1∂∂t+n0τ0∂2∂t2∇.u−ρQ˙,σij=λuk,k−1+ϑ0∂∂tTδij+2μeij,i,j,k=1,2,3,eij=12ui,j+uj,i,i,j=1,2,3.The laser pulse is given as(2)Q=I0γ¯2πr2exp−z2r2−γ¯xft.The temporal functionft takes the following form:(3)ft=tt02exp−tt0.Assuming a homogeneous, thermally, and conducting electrically elastic solid, the electrodynamics system linear equations of slowly moving in a medium are(4)curlh⟶=J⟶,curlE⟶=−μ0h⟶,divh⟶=0,divE⟶=0,E⟶=−μ0u⟶×H⟶.Considering the laser pulse and gravitational field, the governing fundamental equations of a linear homogeneous isotropic thermoelastic medium take the following form:(5)μ∇2u+λ+μ∂e∂x−γ1+ϑ0∂∂t∂T∂x+ρg∂w∂x−μ0H0∂h∂x=ρ∂2u∂t2−Ω2u,(6)μ∇2w+λ+μ∂e∂z−γ1+ϑ0∂∂t∂T∂z−ρg∂u∂x−μ0H0∂h∂z=ρ∂2w∂t2−Ω2w,(7)k∇2T+k∗∂∂t∇2T=ρCen1∂∂t+τ0∂2∂t2T+γT0n1∂∂t+n0τ0∂2∂t2∇.u−ρ∂Q∂t.In the context of the thermoelastic models, equations (5)–(7) are applicable to the following:(i)
CT model:n0=n1=1, τ0=ϑ0=0, k∗ = 0(ii)
L-S model:n0=n1=1, τ0>0, ϑ0=0, k∗ = 0(iii)
G-L model:n0=0, n1=1, ϑ0≥τ0>0, k∗ = 0(iv)
G-N II model:n0=1, n1=ϑ0=0, τ0=1, k∗ > 0We know that the CT model considered the coupling between the thermal and strain; assume one and two relaxation times for L-S and G-L models, respectively, but take into account absence of energy dissipation for G-N II.The previous quantities in the dimensionless variables are as follows:(8)x′,z′=ω∗c0x,z,ϑ0′=ω∗ϑ0,t′=ω∗t,τ0′=ω∗τ,Q′=Qw∗T0Ce,Ω′=Ωω∗,u′,w′=ρc0ω∗υT0u,w,T′=TT0,δij′=δijυT0,g′=gc0w∗,h′=hH0,where(9)ω∗=ρCEc02K,c02=λ+2μρ.Equations (5)–(7) in the nondimensional form with dropping primes for convenience take the following form:(10)∇2u+b1∂e∂x−b21+ϑ0∂∂t∂T∂x+b3∂w∂x−Rh∂h∂x=b2∂2u∂t2−Ω2u,(11)∇2w+b1∂e∂z−b21+ϑ0∂∂t∂T∂z−b3∂u∂x−Rh∂h∂z=b2∂2w∂t2−Ω2w,(12)ε3∇2T+ε2∂∂t∇2T=ε4n1∂∂t+τ0ω∗∂2∂t2T+ε1n1∂∂t+n0τ0∂2∂t2e−∂Q∂t,where(13)ε1=γ2T0w∗c02ρCe,ε2=k∗w∗ρc02Ce,ε3=kρc02Ce,ε4=1ω∗,b1=λ+μμ,b2=ρc02μ,b3=ρgc02μ,Rh=μ0H02μ.The components of displacementux,z,t and wx,z,t considering the rigid body take each of the functions of potential ψ1x,z,t and ψ2x,z,t in the dimensionless forms as follows:(14)u=∂ψ1∂x−∂ψ2∂z,(15)w=∂ψ1∂z+∂ψ2∂x,with(16)e=∇2ψ1,∂u∂z−∂w∂x=∇2ψ2.Using (14) and (15) into (10)–(12), we get(17)1+b1−Rh∇2−b2∂2∂t2−b2Ω2ψ1+b3∂∂xψ2−b21+θ0∂∂tT=0,(18)−b3∂∂xψ1+∇2−b2∂2∂t2+b2Ω2ψ2=0,(19)−ε1n1ω∗∂∂t+n0τ0∂2∂t2∇2ψ1+ε3+ε2∂∂t∇2T−ε4n1∂∂t+τ0ω∗∂2∂t2T=−∂∂tQ.The stress-strain (constitutive) relation gives the following form:(20)σxx=∂u∂x+L∂w∂z−1+ϑ0∂∂tT,(21)σyy=Le−1+ϑ0∂∂tT,(22)σzz=∂w∂z+L∂u∂x−1+ϑ0∂∂tT,(23)σxz=1b2∂u∂z+∂w∂x,σxy=σyz=0.(24)τzz=G¯∂u∂x+∂w∂z,where(25)L=λλ+2μ,G¯=μeH02λ+2μ.


## 3. Technique of Normal Mode Analysis

The physical quantities solution in terms of the normal mode technique as an amplitude functions is multiplied by an exponential function as follows:(26)ψ1,ψ2,Tx,z,t=ψ1∗,ψ2∗,T∗xexpωt+iaz,where ψ1∗,ψ2∗,T∗x are the physical quantities amplitudes and i=−1.Substituting equation (26) into equations (17)–(19), we get(27)D2−B1ψ1∗+B2Dψ2∗−B31+θ0iωT∗=0,(28)−b3Dψ1∗+D2−B2ψ2∗=0,(29)B5D2−a2ψ1∗+D2−a2T∗−B6T∗=B7∂∂tQ,where(30)B1=a2−b2ω2+b2Ω21+b1−RH,B2=b31+b1−RH,B3=b21+b1−RH,B4=a2−b2ω2−b2Ω2,B5=ε1ωn1i−n0τ0ω∗ωε3+ε2iωB6=−ε4ω−n1i+τ0ω∗ωε3+ε2iω,B7=−1ε3+ε2iω,D=ddx.Eliminatingψ1∗,ψ2∗, and T∗ from equations (27)–(29) gives the following differential equations:(31)D6−B8D4+B9D2−B10ψ1∗=B111−tt0exp−z2r2+tt0+γ¯x+iωt+iaz,(32)D6−B8D4+B9D2−B10T∗=B121−tt0exp−z2r2+tt0+γ¯x+iωt+iaz,(33)D6−B8D4+B9D2−B10ψ2∗=B131−tt0exp−z2r2+tt0+γ¯x+iωt+iaz,where(34)B8=B1+B4+B6+B2b3−B3B5−B3B5ϑ0i+ω+a2,B9=a2B1+a2B4+a2B2b3−a2B3B5−B3B5B4+B1B4+B6B4+B1B6+b3B2B6+B3B5a2iωϑ0+B3B4B5iωϑ0,B10=a2B1B4−a2B3B5B4+B1B4B6−B3B4B5a2iωϑ0,B11=B3B7γ¯2+iωϑ0γ¯2−B4−B4iωϑ0I0γ¯2πr2t02,B12=B7γ¯2−B1γ¯2−B4+γ¯2B2b3I0γ¯2πr2t02,B13=−B3B7b31+iϑ0ωI0γ¯22πr2t02.Equation (31) can be taken in the following form:(35)D2−k12D2−k22D2−k32ψ1∗=B111−tt0exp−z2r2+tt0+γ¯x+iωt+iaz,where kn2n=1,2,3 are the characteristic equation roots of the homogeneous equations (31)–(33).The general solutions of equations (31)–(33) bounded as x⟶∞ are given as follows:(36)ψ1x,z,t=∑n=13Rnexp−knx+iωt+iaz+L1B11f1,(37)ψ2x,z,t=∑n=13H1nRnexp−knx+iωt+iaz+L1B13f1,(38)Tx,z,t=∑n=13H2nRnexp−knx+iωt+iaz+L1B12f1.Here,(39)H1n=−b3knkn2−B4,n=1,2,3,H2n=kn2−B1−B2H1nknB3,n=1,2,3,L1=−1γ¯6−B8γ¯4+B9γ¯2−B10,f1=1−tt0exp−z2r2−tt0−γ¯x,where Rnn=1,2,3 are the coefficients.To get the displacement vector components, substituting equations (36) and (37) into equation (14), we obtain(40)ux,z,t=∑n=13M1nRnexp−knx+iωt+iaz−γ¯I1+2zI2r2exp−z2r2−tt0−γ¯x,(41)wx,z,t=∑n=13M2nRnexp−knx+iωt+iaz+−γ¯I2+2zI1r2exp−z2r2−tt0−γ¯x,where M1n=−kn−iaH1n,M2n=ia−knH1n,n=1,2,3.To get the stress tensor components, substituting equations (38)–(41) into equations (20)–(24), we get(42)σxxx,z,t=∑n=13H3nRnexp−knx+iωt+iaz+I4exp−z2r2−tt0−γ¯x,σyyx,z,t=∑n=13H4nRnexp−knx+iωt+iaz+I5exp−z2r2−tt0−γ¯x,σzzx,z,t=∑n=13H5nRnexp−knx+iωt+iaz+I6exp−z2r2−tt0−γ¯x,σxzx,z,t=∑n=13H6nRnexp−knx+iωt+iaz+I7exp−z2r2−tt0−γ¯x,τxx=∑n=13H7nexp−knx+iωt+iaz+I8exp−z2r2−tt0−γ¯x.Here,(43)H3n=−M1nkn+LiaM2n−H2n−iωθ0H2n,H4n=−knM1nL+iaLM2n−H2n−iωθ0H2n,H5n=iaM2n−LknM1n−H2n−iωθ0H2n,H6n=1b2iaM1n−knM2n,H7n=iaG¯M2n−knG¯M1n,I1=−B10L11−tt0,I2=B12B10I1,I3=B11B10I1,I4=γ¯γ¯I1−2zr2I2+2zr2γ¯I2+2zI1r2L−I3−θ0t0I3,I5=E1γ¯γ¯I1−2zI2r2+2zr2Lγ¯I2+2zI1r2−I3−θ0t0I3,I6=2zr2γ¯I2+2zI1r2+Lγ¯γ¯I1−2zI3r2−I3−θ0t0I3,I7=1b22zr2γ¯I1−2zI3r2+γ¯γ¯I2−2zI1r2,I8=γ¯G¯γ¯I1−2zI2r2+2zr2G¯γ¯I2+2zI1r2.


## 4. Boundary Conditions

We will obtain the constantsRnn=1,2,3, so the boundary conditions considered and should suppress the positive exponentials to avoid them at infinity unboundedness. The chosen coefficients R1,R2,andR3 will be obtained from the boundary conditions on the surface at x=0 as follows.(i)
The mechanical boundary conditions:(44)σzz+τzz=−p1expωt+iaz,σxz=0.(ii)
On the surface of the half-space, the thermal boundary condition is(45)∂T∂x=0.From the above boundary conditions, we get(46)∑n=13H5n+H7nRn=−p,∑n=13H6nRn=0,∑n=13−knH2nRn=0.Using the method of matrix, we get the coefficient values ofRnn=1,2,3:(47)R1R2R3=H51+H71H52+H72H53+H73H61H62H63−k1H21−k2H22−k3H23−1−p100.After that, we obtain the displacements, temperatures, and stresses.


## 5. Numerical Results and Discussion

Following Dhaliwal and Singh [25], to calculate the numerical values of the considered variables, the magnesium material was chosen:(48)λ=2.17×1010N/m2,μ=3.278×1010N/m2,K=1.7×102W/mK,ρ=1.74×103kg/m3,,Ce=1.04×103j/kgK,ω∗=3.58×1011/s,μ0=4×π×10−3,T0=298.The laser pulse constants are(49)I0=102j/m2,r=0.2,γ¯=25/m,t0=10.Consider(50)p1=0.25N/m2,k∗=100W/mK,a=0.5,ω=2.9rad/s,z=2m,t=0.9s,g=9.8m/s2,0≤x≤3.5m.Figures2–33 display the distributions calculated with respect to the range x0≤x≤3.5. The variation presented in two-dimensional figures shows the change in behavior of the values of the displacement components u and w, stresses σxx,σzz, and σxz, and also Maxwell’s stress and temperature distribution T with distance x. In the context of G-N III theory, a schematic has been shown in generalized thermoelasticity medium with constants H0=9×105,g=9.8,andt=0.9 with different values of gravity, laser pulse, and magnetic field taking into account two values of rotation. Figures 2–9 present a comparison between the variation in the four thermoelastic models: (i) CT theory: n0=0,n1=1,τ0=0,ϑ0=0; (ii) LS theory: n0=1,n1=1,τ0=0.2,ϑ0=0; (iii) GL theory: n0=0,n1=1,τ0=0.2,ϑ0=0.3; and (iv) GN theory: n0=0,n1=1,τ0=1,ϑ0=0. We show that the displacement, stresses, and temperature demonstrates clearly the difference between the coupled and the generalized theories of thermoelasticity (see Figures 2–9). It implies that applying G-L theory provides good results compared with the remaining theories, and this physically indicates the developed generalized thermoelastic theories in the positive direction. Figures 10–17 display the values of the displacement, stresses, and temperature with respect to x-axis for different values of gravity field g=0,5,7,9.8 considering two values of rotation Ω=0.2,…,Ω=0.7___. It appears that u,σzz,σxz,τzz,σzz+τzz,andT decreased (i.e., affects negatively) with the increased values of the rotation but increased (i.e., affects positively) concerning the remaining distributions, which indicates strong effect with the strong values of rotation. It is observed that the component of displacement u decreases in the interval 0<x<1.3 with an increasing gravity field but increases in the interval 1.3,2.5, while it coincides if x>2.5 tends to zero, as well as the displacement component w decreases with increasing gravity field in the interval 0,0.8, but increases in the interval 0.8<x<1.8 and it coincides if x>1.8. From Figure 12, it obvious that the stresses σxx,σzz,τzz,σzz+τzz and temperature T increase, decrease, and then coincide tending to zero as distance tends to infinity that agrees with the physical meaning of the phenomena (i.e., if the wave is far from the origin, then the distributions tend to infinity “unknown”), but the shear stress σxz decreases with an increasing gravity field. The distributions are shown graphically in Figures 18–25 considering variation in laser pulse: t = 0, 0.3, 0.6, 0.9, and it obvious that the displacement components u and w, stresses σzz,τzz,σzz+τzz, and temperature T increase, decrease, and then coincide tending to zero as distance x tends to infinity, but only the normal stress σxx decreases with an increasing gravity. Figures 26–33 plot the variation in the displacement and stresses components, also the temperature distribution with respect to the distance and various values of the magnetic field and two values of rotation. It shows that the magnetic field has a strong sense on all the distributions; we concluded that, except the normal stress on the x-direction which makes a slight decreasing change with an increasing H0, one can be obvious that u, w, σzz,τzz,σzz+τzz, and T increased with an increasing magnetic field.Figure 2
Horizontal displacementu concerning x under four thermoelastic theories.Figure 3
Vertical displacementw concerning x under four thermoelastic theories.Figure 4
Stressσxx concerning x under four thermoelastic theories.Figure 5
Stressσzz concerning x under four thermoelastic theories.Figure 6
Stressσxz concerning x under four thermoelastic theories.Figure 7
Stressτzz concerning x under four thermoelastic theories.Figure 8
Stressτzz+σzz concerning x under four thermoelastic theories.Figure 9
TemperatureT concerning x under four thermoelastic theories.Figure 10
Horizontal displacementu concerning x under gravity and rotation.Figure 11
Vertical displacementw concerning x under gravity and rotation.Figure 12
Stressσxx concerning x under gravity and rotation.Figure 13
Stressσzz concerning x under gravity and rotation.Figure 14
Stressσxz concerning x under gravity and rotation.Figure 15
Stressτzz concerning x under gravity and rotation.Figure 16
Stressτzz+σzz concerning x under gravity and rotation.Figure 17
TemperatureT concerning x under gravity and rotation.Figure 18
Horizontal displacementu concerning x with laser pulse “t” and rotation.Figure 19
Vertical displacementw concerning x with laser pulse “t” and rotation.Figure 20
Stressσxx concerning x with laser pulse “t” and rotation.Figure 21
Stressσzz concerning x with laser pulse “t” and rotation.Figure 22
Stressσxz concerning x with laser pulse “t” and rotation.Figure 23
Stressτzz concerning x with laser pulse “t” and rotation.Figure 24
Stressτzz+σzz concerning x with laser pulse “t” and rotation.Figure 25
TemperatureT concerning x with laser pulse “t” and rotation.Figure 26
Horizontal displacementu concerning x with magnetic and rotation.Figure 27
Vertical displacementw concerning x with magnetic and rotation.Figure 28
Stressσxx concerning x with magnetic and rotation.Figure 29
Stressσzz concerning x with magnetic and rotation.Figure 30
Stressσxz concerning x with magnetic and rotation.Figure 31
Stressτzz concerning x with magnetic and rotation.Figure 32
Stressτzz+σzz concerning x with magnetic and rotation.Figure 33
TemperatureT concerning x with magnetic and rotation.Physically, the gravity, magnetic field, laser pulse, and thermoelasticity with the model theories have a significant effect on the propagation of the wave phenomenon.Finally, it strong obvious that all variables calculated satisfied the boundary conditions exactly either the calculus or the graphs (the figures display total normal stressσzz+τzz, shear stress σxz, and temperature distribution T).If the gravity and magnetic field considered the thermal loading influence due to laser pulse on generalized micropolar-thermoelastic solid with comparison of different theories, the results obtained have been deduced and are in agreement with the previous results by Othman et al. [7].On the contrary, when the laser pulse and gravity are neglected, the obtained results are deduced as a special case from the present investigation and are in agreement with the results obtained by Abo-Dahab et al. [15].


## 6. Conclusion

In the view of illustrating the numerical results, we concluded the following remarks:(1)
The technique of normal mode technique is used in wide range of applications in diverse fields such as engineering, geophysics, thermodynamics, geology, acoustics, eyes medicine, and engineering.(2)
The magnetic field, rotation, gravity, and laser pulse have a significant role in temperature, displacements, and stresses and all the physical quantities by decreasing or increasing.(3)
The results described for the medium of crystal, which may provide interesting information for the experimental researchers work on this field, satisfy the boundary conditions.(4)
The values of all physical quantities converge to zero with an increasing distancex, and all functions are exactly continuous.(5)
Applying G-L theory has good results compared with the remaining theories, and this physically indicates the developed thermoelastic theories in the positive direction.(6)
The results obtained should be useful for researchers/scientists in designing new materials, material sciences, physicists, and the development of the magneto-thermoelasticity and in practical situations such as in optics, geophysics, petroleum extraction, acoustics, and oil prospecting.Finally, we conclude that the results obtained have a significant rule in engineering, astronomy, aircrafts, dynamical system reactors, and aircrafts.



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