# Initial Stress and Gravity on P-Wave Reflection from Electromagneto-Thermo-Microstretch Medium in the Context of Three-Phase Lag Model

Authors: F. S. Bayones; S. M. Abo-Dahab; A. M. Abd-Alla; S. H. Elhag; A. A. Kilany; M. Elsagheer
Journal: Complexity
 (2021)

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## Abstract

The present paper studied the reflection of thermo-microstretch waves under the generalized thermoelasticity theory which is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid under the effect of the electromagnetic field, initial stress, and gravity. The formulation is applied under the thermoelasticity theory with three-phase lag, and the reflection coefficient ratio variations with the angle of incidence under different conditions are obtained. Numerical results obtained from the present study are presented graphically and discussed. It is observed that the initial stress, gravitation, and electromagnetic field exert some influence in the thermo-microstretch medium due to reflection of P-waves.


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## Body

## 1. Introduction

In recent decades, the influences of electromagnetic field, initial stress, gravitation, and thermal field have been much pronounced in diverse fields, especially, engineering, geophysics, geology, acoustics, plasma, and physics because of its utilitarian aspects in these fields. The generalized theories of thermoelasticity, which admit the finite speed of a thermal signal, have been the center of interest of active research. Allam et al. [1] discussed the Green–Lindsay model of reflection of P and SV waves from the free surface of thermoelastic diffusion, solid under influence of the electromagnetic field, and initial stress. Abo-Dahab and Kilicman [2] illustrated the reflection and transmission of P- and SV-wave phenomena at the interface between solid and liquid media with magnetic field and two thermal relaxation times. Othman and Song [3] developed the reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation. Abd-Alla et al. [4] studied SV wave’s incidence at the interface between solid and liquid media under electromagnetic field and initial stress in the context of three thermoelastic theories. Aboueregal and Abo-Dahab [5] discussed the dual-phase model on magneto-thermoelasticity infinite nonhomogeneous solid having a spherical cavity. Zhou et al. [6] developed the reflection and transmission of plane waves at the interface of pyroelectric bimaterials. Singh [7] illustrated the reflection of P and SV waves from the free surface of an elastic solid with generalized thermodiffusion. Sinha and Sinha [8–10] studied the reflection of thermoelastic waves at a solid half-space with thermal relaxation. Angel and Achenbach [11] discussed the reflection and transmission of elastic waves by a periodic array of crack. Kilany et al. [12] developed photothermal and void effect of a semiconductor rotational medium based on the Lord–Shulman theory. Quintanilla and Racke [13] studied a note on stability of three-phase lag heat conduction. Kothari et al. [14] developed the fundamental solutions of generalized thermoelasticity with three-phase lags. Banik and Kanoria [15] illustrated generalized thermoelastic interaction in a functionally graded isotropic unbounded medium due to varying heat source with three-phase lag effect. Chou and Yang [16] developed two-dimensional dual-phase lag thermal behavior in single-/multilayer structures using the space-time conservation element and solution element (CESE) method. Liu [17] studied numerical analysis of dual-phase lag and heat transfer in a layered cylinder with nonlinear interface boundary conditions. Marzougui et al. [18] used a computational analysis of heat transport irreversibility phenomenon in a magnetized porous channel. Abo-Dahab et al. [19] developed MHD Casson nanofluid flow over nonlinearly heated porous medium in the presence of extending surface effect with suction/injection. Zaim et al. [20] used Galerkin finite element analysis of magneto-hydrodynamic natural convection of Cu-water-nanoliquid in a baffled U-shaped enclosure. Mebarek-Oudina et al. [21] illustrated magneto-thermal convection stability in an inclined cylindrical annulus filled with a molten metal. Ezzat and Othman [22] investigated the plane waves in an electromagneto-thermoelastic perfect conductivity medium with two relaxation times.The objective of the present investigation is to determine the reflection of P waves from the thermo-electro-magneto-microstretch medium in the context of three-phase lag model under the effect of rotation, electromagnetic field, and gravity with initial stress. The reflection coefficient ratios of various reflected waves with the angle of incidence have been obtained from the 3PHL model and LS theory. Also, the effect of the electromagnetic field and gravity is discussed numerically and illustrated graphically. The physical quantities are obtained and tested by a numerical study using the parameters of Cu as a target and presented graphically. The distribution of these quantities is represented graphically in the presence and absence of electromagnetic field, initial stress, and gravity field.


## 2. Formulation of the Problem

The equations of generalized thermo-microstretch in a rectangular coordinate system (x, y, z) with z−axis directed in the media are used. Constant magnetic field intensity H=0,H0,0 is taken as the direction of the y-axis. We begin our consideration with linearized equations of electrodynamics of slowly moving media.(1)J=curlh−ε0E˙,curlE=−μ0h˙,E=−μ0u˙×H,∇.h=0.The constitutive equation in the presence of gravitational and electromagnetic field is(2)σji,i+Fi+Gi=ρu¨i,(3)Fi=μ0J×Hi,Gi=ρg∂w∂x,0,−∂u∂x.From the above equations, we can obtain(4)E=μ0H0w˙,0,−u˙,h=0,−H0e,0,J=−hz−ε0μ0H0w¨,0,hx+ε0μ0H0u¨.From equations (3) and (4), we obtain(5)F=Fx,Fy,Fz=μ0H02ex−ε0μ02H02u¨,0,μ0H02ez−ε0μ02H02w¨.So, the displacement vectoru has the components ux=ux,z,t,uy=0,anduz=wx,z,t.(6)σij=λ0φ∗−P+λuk,kδij+μ+kuj,i+μui,j−kεijkφk−γ^Tδij−Pωij,mij=αφk,kδij+βφi,j+γφj,i,λi=α0φi∗,where ωij=1/2uj,i−ui,j.In component form, the basic governing equations become(7)μ+k−P2∂2u∂x2+∂2u∂z2+λ+μ+P2∂2u∂x2+∂2w∂x∂z−k∂φ2∂z+λ0∂φ∗∂x−γ^∂T∂x+F1+ρg∂w∂x=ρ∂2u∂t2,(8)μ+k−P2∂2w∂x2+∂2w∂z2+λ+μ+P2∂2u∂x∂z+∂2w∂z2+k∂φ2∂x+λ0∂φ∗∂z−γ^∂T∂z+F3−ρg∂u∂z=ρ∂2w∂t2,(9)α+β+γ∇∇.φ−γ∇×∇×φ+k∇×u−2kφ=ρj∂2φ∂t2,(10)α0∇2φ∗−13λ1φ∗−13λ0∇.u+13γ^1T=32ρj∂2φ∗∂t2,(11)K∗∇2T+τν∗∇2T˙+KτT∇2T¨=1+τq∂∂t+τq22∂2∂t2ρCeT¨+γ^T0e¨+γ^1T0φ¨∗,such that τν∗=K+K∗τν,∇2=∂2/∂x2+∂2/∂z2.The constitutive relation can be written as(12)σxx=λ+2μ+k∂u∂x+λ∂w∂z+λ0φ∗−γ^T−P,(13)σzz=λ+2μ+k∂w∂z+λ∂u∂x+λ0φ∗−γ^T−P,(14)σxz=μ+P2∂u∂z+μ+k−P2∂w∂x+kφ2,σzx=μ+P2∂w∂x+μ+k−P2∂u∂z−kφ2,(15)mxy=γ∂φ2∂x,mzy=γ∂φ2∂z,(16)λx=α0∂j∗∂x,λz=α0∂j∗∂z.To facilitate the solution, the following dimensionless quantities are introduced:(17)xi′=ω∗c0xi,ui′=ρc0ω∗γ^T0ui,Θ′=γ^ρc02T−T0,t′,τT′,τv′,τq′=ω∗t,τT,τν,τq,σij′=σijγ^T0,(18)m′ij=ω∗c0γ^T0mij,λ′i=ω∗c0γ^T0λi,φ′∗=ρc02γ^T0φ∗,φ2′=ρc02γ^T0φ2,g′=gc0ω∗,ω∗=ρCec02k,h′=h/H0,andc02=λ+2μ+k/ρ,i,j=1,2,3Using equation (18), the governing equations (7)–(16) recast in the following form (after suppressing the primes):(19)2μ+2k−P2ρc02∇2u+2λ+2μ+P2ρc02+RH∂e∂x−kρc02∂φ2∂z+λ0ρc02∂φ∗∂x−ρc02γT0∂Θ∂x+g∂w∂x=β2∂2u∂t2,(20)2μ+2k−P2ρc02∇2w+2λ+2μ+P2ρc02+RH∂e∂z+kρc02∂φ2∂x+λ0ρc02∂φ∗∂z−ρc02γT0∂Θ∂z−g∂u∂z=β2∂2w∂t2,(21)∇2φ2−2kc02γω∗2φ2+kc02γω∗2∂u∂z−∂w∂x=ρjc02γ∂2φ2∂t2,(22)C12c02∇2−C22ω∗2−∂2∂t2φ∗−C32ω∗2e+a0Θ=0,(23)Ck∇2Θ+Cν∇2Θ.+CT∇2Θ..=1+τq∂∂t+τq22∂2∂t2Θ..+ε1e¨+ε2φ¨∗,(24)σxx=ux+a1wz+a2φ∗−a3Θ−a4,(25)σzz=wz+a1ux+a2φ∗−a3Θ−a4,(26)σxz=a5uz+a6wx+a7φ2,(27)σzx=a5wx+a6uz−a7φ2,(28)mxy=a8∂φ2∂x,mzy=a8∂φ2∂z,(29)λx=a9∂φ∗∂x,λz=a9∂φ∗∂z,where(30)RH=μ0H02ρc02,β2=1+ε0μ02H02ρ,C12,C22,C32=29ρj3α0,λ0,λ1,a0=2ρc04γ^19jγ^2T0ω∗2,Ck,Cv,CT=1ρc02CeΚ∗,τv∗,ΚτTω∗,ε1=γ^3T0ρ3c04Ce,ε2=γ^1γ^2T02ρ3c04Ce,a0=2ρc049jγ^T0ω∗2,a4=Pγ^T0+1,a1,a2,a5,a6,a7=1ρc02λ,λ0,μ+P2,μ+k−P2,k,a8,a9=ω∗2ρc04γ,α0.The displacement componentsux,z,t and wx,z,t may be written in terms of potential functions Φx,z,t and Ψx,z,t as(31)u=Φx+Ψz,w=Φz−Ψx.Using equation (31) in equations (19)–(23), we obtain(32)∇2−ζ0∂2∂t2Φ−ζ1∂Ψ∂x−ζ2Θ+ζ3φ∗=0,(33)ζ4∂Φ∂x+∇2−ζ5∂2∂t2Ψ−ζ6φ2=0,(34)kc02γω∗2∇2Ψ+∇2−jρc02γ∂2∂t2−2kc02γω∗2φ2=0,(35)C12c02∇2−C22ω∗2−∂2∂t2φ∗−C32ω∗2∇2Φ+a0Θ=0,(36)Ck∇2Θ+Cν∇2Θ˙+CT∇2Θ¨=1+τq∂∂t+τq22∂2∂t2Θ¨+ε1∇2Φ¨+ε2φ¨∗,where(37)ζ0,ζ1,ζ2,ζ3=11+RHβ2,g,ρc02γT0,λ0ρc02,ζ4,ζ5,ζ6=2ρc022μ+2k−Pg,β2,k.


## 3. Solution of the problem

We assume now that the solution of equations (32)–(36) takes the following form:(38)Φ,Ψ,Θ,φ∗,φ2=Φ¯,Ψ¯,Θ¯,φ∗¯,φ2¯expiξxsinθ+zcosθ−iωt.Substituting equation (38) into equations (32)–(36), we get(39)−ξ2+ζ0ω2Φ¯+iζ1ξsinθΨ¯−ζ2Θ¯+ζ3φ∗¯=0,(40)ξ2−ω2ζ5Ψ¯−ζ6φ2¯=−ιξζ4sinθΦ¯,(41)−p∗ξ2Ψ¯+ξ2−jρc02ω2γ+2kc02γω∗2φ2¯=0,(42)C32ω∗2ξ2Φ¯+a0Θ¯−C12c02ξ2+C22ω∗2−ω2φ∗¯=0,(43)−ε1ω2τq∗ξ2Φ¯+−Ck−iωCv+ω2CTξ2+ω2τq∗Θ¯+ε2ω2τq∗φ∗¯=0,where τq∗=1−iωτq−ω2τq2/2,p∗=kc02/γω∗2.From equations (39)–(43), we get(44)−ξ2+ζ0ω2iζ1ξsinθ−ζ2ζ30iξζ4sinθξ2−ω2ζ500−ζ60−p∗ξ200ξ2−jρc02ω2γ+2kc02γω∗2C32ω∗2ξ20a0−C12C02ξ2−C22ω∗2+ω20−ε1ω2τq∗ξ20−Ck−iωCv+ω2CTξ2+ω2τq∗ε2ω2τq∗0=0.which tends to(45)Aξ10+Bξ8+Cξ6+Dξ4+Eξ2+F=0,where A, B, C, D, E, and F in equation (45) are given in Appendix AFrom equation (45), there are five waves with five different velocities.ThenΦ,Ψ,Θ,φ∗,andφ2 will take the following forms:(46)Φ,Ψ,Θ,φ∗,φ2=1,η0,κ0,χ0,ϑ0A0expiξxsinθ0+zcosθ0−iωt+∑j=151,ηj,κj,χj,ϑjAjexpiξjxsinθj−zcosθj−iωt.From equation (40) and (41), we get ηj,κj,χj,andϑj which are given in Appendix B.


### 3.1. The Boundary Conditions

The boundary conditions for the problem be taken as(47)σzz+τzz=0,σxz+τxz=0,∂T∂z=0,myz=0,λx=0,atz=0,(48)∑j=15a1ϑj−μeH02ξj2−a3ξj2sin2θj+ηj2sin2θj−a2ξj2cos2θj−ηj2sin2θj+kjξj2Aj=a1ϑ1−μeH02ξ12−a3ξ12sin2θ0+η12sin2θ0−a2ξ12cos2θ0−η12sin2θ0+k1ξ12A0,(49)∑j=15a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξjAj=−a4+a5ξ122sin2θ0+a4ξ12η1cos2θ0−a5ξ12η1sin2θ0+ca6ξ1A0,(50)∑j=15kjξjcosθjAj=k1ξ1cosθ0A0,(51)∑j=15ζjξjcosθjAj=ζ1ξ1cosθ0A0,(52)∑j=15ϑjξjcosθjAj=−ϑ1ξ1cosθ0A0.From equations (55)–(60), we get(53)aijZj=Bi,i,j=1,2,...,5,Zj=AjA0,(54)a1j=a1ϑj−ξj2a3sin2θj+ηj2sin2θj−a2cos2θj−ηj2sin2θj+kjξj2+μeH02,(55)a2j=a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξj,(56)a3j=kjξjcosθj,a4j=ζjξjcosθjAj,a5j=ϑjξjcosθj,(57)B1=−a1ϑ1+ξj2a3sin2θ0+η12sin2θ0−a2cos2θ0−η12sin2θ0+kjξj2+μeH02,(58)B2=−a4+a5ξ122sin2θ0−a4ξ12η1cos2θ1+a5ξ12η1sin2θ1−a6ξ1,(59)B3=k1ξ1cosθ0,B4=ζ1ξ1cosθ0,B5=−ϑ1ξ1cosθ0.


## 3.1. The Boundary Conditions

The boundary conditions for the problem be taken as(47)σzz+τzz=0,σxz+τxz=0,∂T∂z=0,myz=0,λx=0,atz=0,(48)∑j=15a1ϑj−μeH02ξj2−a3ξj2sin2θj+ηj2sin2θj−a2ξj2cos2θj−ηj2sin2θj+kjξj2Aj=a1ϑ1−μeH02ξ12−a3ξ12sin2θ0+η12sin2θ0−a2ξ12cos2θ0−η12sin2θ0+k1ξ12A0,(49)∑j=15a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξjAj=−a4+a5ξ122sin2θ0+a4ξ12η1cos2θ0−a5ξ12η1sin2θ0+ca6ξ1A0,(50)∑j=15kjξjcosθjAj=k1ξ1cosθ0A0,(51)∑j=15ζjξjcosθjAj=ζ1ξ1cosθ0A0,(52)∑j=15ϑjξjcosθjAj=−ϑ1ξ1cosθ0A0.From equations (55)–(60), we get(53)aijZj=Bi,i,j=1,2,...,5,Zj=AjA0,(54)a1j=a1ϑj−ξj2a3sin2θj+ηj2sin2θj−a2cos2θj−ηj2sin2θj+kjξj2+μeH02,(55)a2j=a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξj,(56)a3j=kjξjcosθj,a4j=ζjξjcosθjAj,a5j=ϑjξjcosθj,(57)B1=−a1ϑ1+ξj2a3sin2θ0+η12sin2θ0−a2cos2θ0−η12sin2θ0+kjξj2+μeH02,(58)B2=−a4+a5ξ122sin2θ0−a4ξ12η1cos2θ1+a5ξ12η1sin2θ1−a6ξ1,(59)B3=k1ξ1cosθ0,B4=ζ1ξ1cosθ0,B5=−ϑ1ξ1cosθ0.


## 4. Numerical Results and Discussion

The physical constants used are [3](60)i=−1,α0=0.779×10−4,λ0=0.5×1011,λ1=0.5×1011,j=0.2×10−15,ρ=8954,Ce=383.1,k=386,T0=293,λ=7.76×1010,μ=3.86×1010,γ=0.779×10−4,Κ∗=2.97×1013,μ0=0.1,ε0=0.1,ω=0.24,τT=0.002,τv=0.001,τq=0.005.Figure2 shows the variation of the amplitude of reflected P1 wave Z1, amplitude of reflected P2 wave Z2, amplitude of reflected P3 wave Z3, amplitude of reflected P4 wave Z4, and amplitude of reflected P5 wave Z5 with respect to the angle of incidence θ for different values of the initial stress. The amplitude of the reflected wave increases with increasing of the angle of incidence until it reaches a maximum value at θ about 90. The amplitude of reflected P2 wave Z2, the amplitude of reflected P3 wave Z3, the amplitude of reflected P4 wave Z4, and the amplitude of reflected P5 wave Z5 each of them decrease with increasing of the angle of incidence until it becomes a minimum value at θ=90. In addition, all lines begin to coincide when the horizontal angle of incidence θ increases to the maximum. These results obey physical reality for the behavior of reflection of P waves [3].Figure 1
Geometry of the problem.Figure 2
Variation of the amplitudesZii=1,2,...,5 with the angle of incidence of P waves for variation of initial stress.
(a)(b)(c)(d)(e)Figure3 shows the variation of the amplitude of reflected P1 wave Z1, amplitude of reflected P2 wave Z2, amplitude of reflected P3 wave Z3, amplitude of reflected P4 wave Z4, and amplitude of reflected P5 wave Z5 with respect to the angle of incidence θ for different values of the gravitation. The amplitude of reflected P1 wave Z1, the amplitude of the reflected wave Z2, and the amplitude of reflected P5 wave Z5 increases with increasing angle of incidence until it reaches a maximum value at θ about 90. The amplitude of reflected P3wave Z3 and the amplitude of reflected P4 wave Z4 decreases with increasing angle of incidence until it becomes a minimum value at θ=90. It is noticed that the reflection of P waves is strongly affected by the presence of gravity [23].Figure 3
Variation of the amplitudesZii=1,2,...,5 with the angle of incidence of P waves for variation of gravity field.
(a)(b)(c)(d)(e)Figures4 and 5: show the variation of the amplitude of reflected P1 wave Z1, amplitude of reflected P2 wave Z2, amplitude of reflected P3 wave Z3, amplitude of reflected P4 wave Z4, and amplitude of reflected P5 wave Z5 with respect to the angle of incidence θ for different values of the electric field and magnetic field, respectively. The amplitude of reflected P1 wave Z1, amplitude of reflected P2 wave Z2, amplitude of reflected P3 wave Z3, amplitude of reflected P4 wave Z4, and amplitude of reflected P5 wave Z5 increase with increasing electric field and magnetic field. The electric field and magnetic field increase the magnitude of reflection of P waves. This is mainly due to the fact that the electric field and magnetic field correspond to a term signifying a positive force that tends to accelerate the charge carries [2].Figure 4
Variation of the amplitudesZii=1,2,...,5 with the angle of incidence of P waves for variation of magnetic field.
(a)(b)(c)(d)(e)Figure 5
Variation of the amplitudesZii=1,2,...,5 with the angle of incidence of P waves for variation of electric field.
(a)(b)(c)(d)(e)Figure6 shows the variation of the amplitude of reflected P1 wave Z1, amplitude of reflected P2 wave Z2, amplitude of reflected P3 wave Z3, amplitude of reflected P4 wave Z4, and amplitude of reflected P5 wave Z5 with respect to the angle of incidence θ. The change of amplitude of reflected to wave appeared in this figure, where we notice Z3>Z4>Z5>Z2>Z1 of the existence of gravity field gravitation, electric field, magnetic field, and initial stress. These figures are very important to study the dependence of these physical quantities on the vertical amplitude of reflected of distance. The curves obtained are highly depending on the vertical distance from origin, and all the physical quantities satisfy boundary condition and are moving in P waves [24].Figure 6
Variation of the amplitudesZii=1,2,...,5 with the angle of incidence of P waves: P=1010, ε0=0.1, g=0.1,andH0=103.


## 5. Conclusion

We obtain the following conclusions based on the above analysis:(1)
The reflected P waves and amplitude of the reflected P waves depend on the angle of incidence, rotation, initial stress, gravity field, and magnetic field, and the nature of this dependence is different for different reflected P waves.(2)
The rotation, initial stress, gravity field, and electromagnetic field play a significant role, and the effects have the inverse trend for the reflected P waves and amplitude of the reflected P waves.(3)
The rotation, initial stress, gravity field, and electro-magnetic field have a strong effect on the reflected P waves and amplitude of the reflected P waves.(4)
The result provides a motivation to investigate the reflection of P waves from thermo-magneto-microstretch materials as a new class of applications thermoelectric solids. The results presented in this paper should prove to be useful for researchers in material science, designers of new materials, microsystem technologies, physicists, as well as for those working on the development of thermo-electro-magneto-microstretch and in practical situations as in geophysics, optics, acoustics, and geomagnetic and oil prospecting.(5)
The equations were solved by using the wave function assumption which is the basis of the normal mode method and the features of this system. We can find the amplitude of the primary wave under the influence of different variables, whether gravity, primary gain, and electric and magnetic fields, by varying relaxation time.It is observed that the amplitudes of reflected P waves change in the presence of rotation, initial stress, gravity field, and magnetic field. Hence, the presence of rotation, initial stress, gravity field, and electromagnetic field significantly affected the reflection phenomena.



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