MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/671783 671783 Research Article Rotation and Magnetic Field Effect on Surface Waves Propagation in an Elastic Layer Lying over a Generalized Thermoelastic Diffusive Half-Space with Imperfect Boundary Abo-Dahab S. M. 1, 2 Lotfy Kh. 3 Gohaly A. 1 Xie Gongnan 1 Mathematics Department Faculty of Science Taif University, Taif 888 Saudi Arabia tu.edu.sa 2 Mathematics Department Faculty of Science SVU Qena 83523 Egypt svu.edu.eg 3 Mathematics Department Faculty of Science Zagazig University Zagazig 44519 Egypt zu.edu.eg 2015 1282015 2015 07 04 2014 14 10 2014 1282015 2015 Copyright © 2015 S. M. Abo-Dahab et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of the present investigation is to study the effects of magnetic field, relaxation times, and rotation on the propagation of surface waves with imperfect boundary. The propagation between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay (GL) model is studied. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness, and then deduced for normal stiffness, tangential stiffness and welded contact. The amplitudes of displacements, temperature, and concentration are computed analytically at the free plane boundary. Some special cases are illustrated and compared with previous results obtained by other authors. The effects of rotation, magnetic field, and relaxation times on the speed, attenuation coefficient, and the amplitudes of displacements, temperature, and concentration are displayed graphically.

1. Introduction

The foundations of magnetoelasticity were presented by Knopoff  and Chadwick  and developed by Kaliski and Petykiewicz . An increasing attention is devoted to the interaction between magnetic field and strain field in a thermoelastic solid due to its many applications in the fields of geophysics, plasma physics, and related topics. All papers quoted above assumed that the interactions between the two fields take place by means of the Lorentz forces appearing in the equations of motion and by means of a term entering Ohm’s law and describing the electric field produced by the velocity of a material particle, moving in a magnetic field. The most ideal interface model, as it is known, is called perfect bond interface where the displacement and traction are continuous across the interface. However, interfaces are seldom perfect. Therefore, various imperfect models such as three-phase and linear models like spring models have been introduced by Yu et al. , Yu , and Benveniste . Perhaps the most frequently studied imperfect interface model is the smooth bond interface, where the normal components of the displacements and traction are continuous across the interface, while the shear traction components are zero on the interface. Lord and Shulman  formulated a generalized theory of thermoelasticity with one thermal relaxation time, who obtained a wave equation by postulating a new law of heat conduction instead of classical Fourier’s law. Green and Lindsay  developed a temperature rate-dependent thermoelasticity that includes two thermal relaxation times and does not violate the classical Fourier’s law of heat conduction, when the body under consideration has a center of symmetry. Hetnarski and Ignaczak  introduced a review and presentation of generalized theories of thermoelasticity. Diffusion can be defined as the random walk of an assemble of particles from regions of high concentration to that of low concentration. Nowadays, there is a great deal of interest in the study of phenomena due to its applications in geophysics and electronic industry. In integrated circuit fabrication, diffusion is used to introduce “depants” in controlled amounts into semiconductor substance. In particular, diffusion is used to form the base and emitter in bipolar transistors, integrated resistors, and the source/drain in metal oxide semiconductor (MOS) transistors and polysilicon gates in MOS transistors. In most of the applications, the concentration is calculated using Fick’s law. This is simple law which does not take into consideration the mutual interaction between the introduced substance and the medium into which introduced. Study of the diffusion phenomenon is used to improve the conditions of oil extractions. These days’ oil companies are interested in the process of thermoelastic diffusion for more efficient extraction of oil from oil deposits. Until recently, thermodiffusion in solids, especially in metals, was considered as a quantity that is independent of body deformation. Practice, however, indicates that the process of thermodiffusion could have a very considerable influence on the deformation of the body. Thermodiffusion in elastic solid is due to the coupling of temperature, mass diffusion, and strain in addition to the exchange of heat and mass with the environment. Nowacki  developed the theory of thermoelastic diffusion by using coupled thermoelastic model. This implies infinite speed of propagation of thermoelastic waves. Olesiak and Pyryev  investigated the theory of thermoelastic diffusion and coupled quasistationary problems of thermal diffusion for an elastic layer. They studied the influence of cross effects arising from the coupling of the fields of temperature, mass diffusion, and strain due to which the thermal excitation results in additional mass concentration and generates additional fields of temperature. Sherief et al.  developed the generalized theory of thermoelastic diffusion with one relaxation time which allows finite speeds of propagation of waves.

Recently, Sherief and Saleh  investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Singh  discussed the reflection phenomena of waves from free surface of a thermoelastic diffusion with one relaxation time and with two relaxation times in . Aouadi  investigated different problems in thermoelastic diffusion. Sharma and Walia [22, 23] discussed the effect of rotation on Rayleigh waves in the piezothermoelastic half-space. Kumar and Kansal  discussed the propagation of Rayleigh waves on free surface in transversely isotropic thermoelastic diffusion. Kumar and Kansal  derived the basic equations for generalized thermoelastic diffusion and discussed the Lamb waves. Dawn and Chakraborty  studied Rayleigh waves in Green-Lindsay’s model of generalized thermoelastic media. Kumar and Chawla  investigated the effect of rotation and stiffness on surface waves propagation in an elastic layer lying over a generalized thermodiffusive elastic half-space with imperfect boundary. New contributions on waves propagation in thermoelastic media have been discussed .

In this paper, linear model is adopted to represent the imperfectly bonded interface conditions. The linear model is simplified and idealized situation of imperfectly bonded interface, where the discontinuities in displacements at interfaces have a linear relationship with the interface stresses. Taking these applications into account, the surface waves propagation at imperfect boundary between an isotropic elastic layer and isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay theory is investigated. The phase velocity and attenuation coefficients of wave propagation have been computed from the secular equations. The amplitudes of displacements, temperature, concentration, and specific loss are computed and depicted graphically to make clear the influence of magnetic field, rotation, stiffness, relaxation times, and diffusion on the phenomena and compare with the practical results.

2. Basic Equations

The basic governing equations for homogenous generalized thermodiffusive solid in the absence of heat and mass diffusion sources are as follows (Singh ).

Constitutive relations are(1)σji=2μεji+λεkk-β1T+τ1T˙-β2C+τC˙δji.

Equation of motion in the rotating frame of reference is (2)μui,jj+λ+μuj,ij-β1T+τ1T˙,i-β2C+τC˙,i+Fi=ρu¨+Ω×Ω×u+2Ω×u.i,

where Ω×Ω×u is the centripetal acceleration due to the time varying motion only and 2Ω×Ω. is the Coriolis acceleration:(3)Ω×Ω×u=-Ω2u,0,-Ω2w=-Ω2u,0,w2Ω×u.=2Ωw˙,0,-Ωu˙=2Ωw˙,0,-u˙,where(4)F=J×B.Consider that the medium is a perfect electric conductor; we take the linearized Maxwell equations governing the electromagnetic field, taking into account absence of the displacement current (SI): (5)curlh=J,curlE=-μeht,divh=0,divE=0,where (6)h=curlu×H0,where we have used(7)H=H0+hx,z,t,H0=0,H,0(8)B=μeH=μeHJ;then(9)Fx=BJz=μeHH2ux12+2wx1x3=μeH22ux12+2wx1x3,Fz=BJx=μeHH2ux3x1+2wx32=μeH22ux3x1+2wx32.

Equation of heat conduction is(10)ρcET˙+τ0T¨+β1T0ε˙kk+aT0C˙+τ0C¨=KT,ii.

Equation of mass diffusion is(11)Dβ2εkk,ii+DaT+τ1Tt,ii+C˙-DbC+τCt,ii=0.

Here, the medium is rotating with angular velocity Ω=Ωn_, where n_ is the unit vector along the axis of rotation and this equation of motion includes two additional terms, namely,

the centripetal acceleration Ω×Ω×u due to time-varying motion,

the Carioles acceleration 2Ω×u.,

where β1=3λ+2μαt and β2=3λ+2μαc, λ and μ are Lame’s constants, αt is the coefficient of linear thermal expansion, ρ and CE are, respectively, the density and specific heat at constant strain, a, b are, respectively, coefficient describing the measure of thermoelastic diffusion effects and of diffusion effects, T0 is the reference temperature assumed to be such that T/T01, τ0,τ1 are thermal relaxation times with τ1τ0>0 and τ0, τ are diffusion relaxation times with ττ0>0, and ui are components of displacement vector. T(x,y,z) is the temperature change and C is the concentration; σij=σji, εij=ui,j+uj,i/2 are, respectively, the components of stress and strain tensor.

The symbols correspond to partial derivative and time derivative, respectively.

Following Bullen , the equations of motion and constitutive relations in isotropic elastic medium are given by(12)λe+μeuj,ije+μeui,jje+Fie=ρeu¨ie,(13)σjie=λeθeδij+2μeεije,i=j=1,2,3,where(14)Θe=uk,ke,(15)εije=ui,je+uj,ie2,i=j=1,2,3,and ue=u1e,u2e,u3e is the displacement vector, ρe is the density of the isotropic medium and λe,μe are Lame’s constants, σjie=σije are components of stress tensor, and δij is the Kronecker delta.

3. Formulation of the Problem

As shown in Figure 1, we consider an isotropic elastic layer (Medium M1) of thickness H overlaying a homogeneous, isotropic, generalized thermodiffusive elastic half-space in rotating frame of reference (Medium M2). The origin of the coordinate system (x,y,z) is taken at any point on the horizontal surface and x2-axis in the direction of wave propagation and x3-axis taking vertically downward into half-space, so that all particles on a line parallel to x2-axis are equally displaced. Therefore, all the field quantities will be independent of x2-axis coordinate. The interface between isotropic elastic layer and thermodiffusive elastic half-space with rotation has been taken at an imperfect boundary. The displacement vector u, temperature T, concentration C, and rotation for medium M2 are taken as(16)u=u1,0,u3,Tx1,x3,t,Cx1,x3,t,Ω=0,Ω,0and displacement vector ue for the layer (Medium M1) is taken as(17)ue=u1e,0,u3e.We define the dimensionless quantities(18)xi=ω1xiv1,ui=ω1uiv1,t=ω1t,τ1=ω1τ,T=β1Tρv12,C=β2Cρv12,τ0=ω1τ0,τ0=ω1τ0,τ=ω1τ,σij=σija1T0,Ω=Ωω1,Kn=v1knβ1T0ω1,K1=v1k1β1T0ω1,v12=λ+2μ+μeH2ρ,ω1=ρcEv12k1.

Schematic of the problem.

Upon introducing the quantities in (1)–(3), (8), and (14)–(16), after suppressing the primes, with the aid of (17) and (18), we obtain(19)u1,11+δ2u3,13+δ1u1,33-τΘT.1-τcC,1=u¨1-Ω2u1+2Ωu˙3,δ1u3,11+δ2u1.31+u3,33-τθT.3-τcC,3=u¨3-Ω2u3-2Ωu˙1,2T=τθ0T˙+χ1τf0C˙+χ2e˙q12e+q2τθ2T-q3τc2+C˙=0,(20)(u1,11e+u3,13e)δ32+2u1eδ42=u¨1e,(u1,13e+u3,33e)δ32+2u3eδ42=u¨3e,where(21)δ1=μ(λ+2μ+μeH),δ2=λ+μ+μeH(λ+2μ+μeH),χ1=aT0v12β1ω1kβ2,χ2=T0β12ρkω1,q1=Dω1β22ρv12,q2=Dω1bv12,q3=Dω1β2aβ1v12,2=2x2+2Z2,e=ux+wz,δ42=ρev12μe,δ32=ρev12μeH2+λe+μe,δ52=ρev12λe+2μeμeH2,τθ=1+τ1t,τθ0=1+τ0t,τc=1+τt,τf0=1+τ0t,RH2=μeHλ+2μ.

For an isotropic elastic layer, we introduce potential functions Φ and Ψ through the relations(22)u1e=Φx1-Ψx3,u3e=Φx3+Ψx1.Substituting from (22) into (20), we have(23)Φ,11+Φ,33-δ52Φ¨=0,(24)Ψ,11+Ψ,33-δ42Ψ¨=0.

4. Solution of the Problem

To solve (19), (22), and (23), we assume the solution in the form(25)u1,u2,T,C,Φ,Ψ=1,W,S,R,P1,P2Uexpiξx1+mx3-ct,where c=ω/ξ is the nondimensional phase velocity, ω is the frequency, m is still parameter 1, and W, S, and R are, respectively, the amplitude ratio of u1, u3, T, and C with respect to u1. Substituting the values of u1, u3, T, and C from (25) in (19), we obtain(26)1+δ1m2-c21+Λ2+δ2m-2iΛc2W+icω-1τ11S+τ21R=0,δ2m-2Λic2+δ1+m2-c21+Λ2W+icω-1mτ11S+τ21R=0,χ2C+χ2CmW-iω-1c2χ1τ20R-iω-1iω1+m2+c2τ10S=0,q11+m2+q11+m2mw-iq21+m2τ11cω-1S+iq31+m2τ21cω-1+c3ω-2R=0,where(27)Λ=Ωω-1,τ11=1-iωτ1,τ21=1-iωτ,τ10=1-iωτ0,τ20=1-iωτ0.The system of (26) has a nontrivial solution if the determinant of the coefficient [1,W,S,R]T vanishes, which yield to the following polynomial characteristic equation:(28)m8+Am6+Bm4+Cm2+D=0.

The constants A, B, C, and D are given in Appendix A. The characteristic equation (28) is biquadratic in m2 and hence possesses four roots, mp2; p=1,2,3,4 corresponding to four roots; there exist three types of quasilongitudinal waves and one quasitransverse wave. The formal expression for displacement, temperature, and concentration satisfying the radiation condition Remp0 can be written as(29)u1,u3,T,C=P=141,n1p,n2p,n3pexpiξx+impz-ct.

Substituting the values of Φ and Ψ from (25) in (23) and (24) and with the aid of (22), we obtain(30)u1e=iξB1c5+B2s5+ξm6D1s6-D2c6expiξx-ct,u3e=ξm5B2c5-B1s5+iξD1c6+D2s6expiξx-ct,where(31)m5=c2δ52-1,m6=c2δ42-1,c5=cosξm5x3,c6=cosξm5x3,s5=sinξm5x3,s6=sinξm6x3,where AP  P=1,2,3,4, B1,B2,D1,D2 are arbitrary constants. The coupling constants n1p,n2p,n3p  p=1,2,3,4 are given in Appendix B.

5. Boundary Conditions

In this paper, linear model is adopted to represent the imperfectly bonded interface conditions. The boundary conditions are the vanishing of the normal stress; Maxwell’s electromagnetic stress tensor τij is given by(32)τij=μeHihj+Hjhi-Hkhkδijand tangential stress at free surface. The discontinuities in displacements have linear relations with stresses, continuity of normal, Maxwell’s electromagnetic stress tensor and tangential stress, vanishing of the gradient of temperature, and concentration at the interface between the isotropic elastic layer and isotropic thermodiffusive elastic half-space. Mathematically, these can be written as follows.

Mechanical conditions:(33)σ33eM1+τ33e=0σ31eM1+τ31e=0,ppppppppppppppppppppppppx3=-H,σ33eM1+τ33e=σ33M2+τ33σ31eM1+τ31e=σ31M2+τ31σ33M1+τ33e=knu3M2-u3M1σ31M1+τ31e=ktu3M2-u3M1;ppppppppppppppipipppppppx3=0.

Temperature condition(34)Tx3=0,x3=0.

Concentration condition(35)Cx3=0,x3=0,

where kn and kt are the normal and transverse stiffness of layer which have dimension Nm-3.

6. Derivation of the Secular Equations

Substituting the value of u1,u3,T,C,u1e,u3e from (29)-(30) into (33)–(35), with the aid of (1), (10) and (13)-(14), after simplification we obtain(36)b5tanξm5HΔ1+b5Δ2+b6s6c5Δ3-b6c6c5Δ4=0,where Δ1=Rij7×7 the entries Rij of the determinant are given in Appendix C and Δ2 is obtained by replacing the first column of Δ1 by [R1100R41R51R61R71]T,  Δ3 is obtained by replacing the second column of Δ2 by [R1200R42R52R62R72]T, and Δ4 is obtained by replacing the third column of Δ3 by [R1300R43R53R63R73]T. The entries of ΔP(P=1,2,3,4) are given in Appendix C.

If we write(37)c-1=v-1+iω-1G,then ξ=F+iG where F=ω/v and G are real numbers. Also the roots of characteristic equations are in general complex.

Hence, assume that mp=pp+iqp so that exponent in the plane wave solutions in (25) becomes(38)iFx1-mpAx3-vt-FGFx1+mpRx3,where(39)mpA=GFpp+qp,mpR=pp-iGFqp.

This shows that v is the propagation velocity and G is the attenuation coefficient of the wave. Upon using the representation (37) in secular equation (36), the values of propagation speed v and attenuation coefficient G of wave propagation can be obtained.

7. Particular Cases

(i) Normal Stiffness. In this case, kn0,  kt, and the secular equation (36) remains the same. But the following will be replaced in the values of Δpp=1,2,3,4:(40)R71=-iξβ1T0,R72=0,R73=ξm6β1T0,R74=R75=R76=R77=1,R71=0,R72=-iξ,R73=0.

(ii) Tangential Stiffness. In this case kt0, kn, and the secular equation (36) remains the same with the change of values of Δp(p=1,2,3,4) by taking(41)R61=-iξβ1T0,R62=0,R63=ξm61β1T0,R64=n12,R65=n11,R66=n13,R67=n14,R61=0,R62=-ξm51β1T0,R63=-iξ1β1T0.

(iii) Welded Contact. In this case, kn, kt, and the secular equation (36) remains the same, but the value of Δp(p=1,2,3,4) is given by replacing(42)R61=-iξ1β1T0,R62=0,R63=ξm61β1T0,R64=n12,R65=n11,R66=n13,R67=n14,R61=0,R72=-ξm51β1T0,R63=-iξ1β1T0,R61=-iξ1β1T0,R62=0,R63=ξm61β1T0R64=n12,R65=n11,R66=n13,R67=n14,R61=0,R72=-ξm51β1T0,R63=-iξ1β1T0.

8. Special Cases

Case (i). If we take Ω=0, that is, in the absence of rotation effect, the frequency equation (36) will reduce to the frequency equation for an isotropic elastic layer and a homogenous isotropic thermodiffusive elastic half-space without rotation.

Case (ii). If we take H=0,  Ω=0, τ>0, and τ1>0 and in the absence of diffusion effect, that is, b1=b3=a=b=0, (36) will reduce to the frequency equation for Rayleigh wave(43)2-c2δ122m-12+m-22+m-1m-2-1+c2-4m-1m-2m-3m-1+m-2=0.Here, m3=1-c2/δ12 and m-pp=1,2 are the roots of (28), obtained by taking b1=b3=a=b=0 and τ>0, τ1>0. The resulting equation (43) is similar to (22) as given by Dawn and Chakraborty .

Case (iii). In the absence of isotropic elastic layer and thermal and diffusion effects, we obtain the frequency equation corresponding to isotropic elastic half-space by changing the dimensionless quantities into physical quantities as(44)2-c2c222=41-c2c121/21-c2c221/2,where c12=λ+μ+μe/ρ, c22=μ/ρ.

The frequency equation (36) is the same as derived in Ewing et al. .

9. Surface Displacements, Temperature Change, and Concentration

The amplitude of surface displacements, temperature change, and concentration at the surface z=0 during Rayleigh wave propagation in the cases of stress free, vanishing of the gradient of temperature, and concentration of the half-space are(45)u1=E-AexpiFx1-vt,u3=R-AexpiFx1-vt,T=W-AexpiFx1-vt,C=L-AexpiFx1-vt,where(46)A=A1exp-Fx1,E¯=F1-F2+F3-F4F1,R¯=n11F1-n12F2+n13F3-n14F4F1,W¯=n21F1-n22F2+n23F3-n24F4F1,L¯=n31F1-n32F2+n33F3-n34F4F1and FP (P=1,2,3,4) are given in Appendix D.

10. Specific Loss

The specific loss is the ratio of energy (Δw) dissipated in taking a specimen through a stress cycle, to the elastic energy (w) stored in the specimen when the strain is maximum. The specific loss is the most direct method of defining internal friction for a material. For a sinusoidal plane wave of small amplitude, Kolsky  shows that the specific loss Δw/w equals 4π times the absolute value of imaginary part of ξ to the real part of ξ; that is,(47)Δww=4πImξReξ=4πiGF=4πiGω/v=4πivGω=4π1FiG.

11. Numerical Results and Discussion

Following Sherief and Saleh , we take the following values of relevant parameter for the copper material:(48)λ=7.76×1010Kgm-1s-2,μ=3.86×1010Kgm-1s-2,T0=0.293×103Kρ=8.954×103kgm-3,a=1.2×104m2s-2k-1,b=9×105m5s-2gk-1cE=3831×103JKg-1K-1,αt=1.78×10-5k-1,αc=1.98×10-4k-1m3gK=0.383×103Wm-1k-1,D=0.85×10-8Sm-3g-1,τ0=0.07s,τ1=0.03,τ0=0.04,τ1=0.07s.The elastic constants for granite are given by Bullen :(49)λe=μe=2.238×103J·Kgm-1K-1,ρe=2.65×103J·kgm-3.

The concentration change, phase velocity and attenuation coefficient of wave propagation, displacement, stresses, temperature in the context of Green Lindsay (GL) theory of thermoelastic diffusion with variations of magnetic field, and rotation in 2D and 3D have been computed for various values of nondimensional wave number and calculated numerically and represented graphically in Figures 222.

Variation of concentration with respect to the wave number with variation of rotation.

Variation of phase velocity with respect to the wave number with variation of rotation.

Variation of attenuation coefficient with respect to the wave number with variation of rotation.

Variation of the displacement u with respect to the wave number with variation of rotation.

Variation of normal stress σxx with respect to the wave number with variation of rotation.

Variation of shear stress σxy with respect to the wave number with variation of rotation.

Variation of the temperature θ with respect to the wave number with variation of rotation.

Variation of the concentration charge with respect to the wave number with variation of the magnetic field.

Variation of phase velocity with respect to the wave number with variation of the magnetic field.

Variation of the attenuation coefficient with respect to the wave number with variation of the magnetic field.

Variation of the displacement u with respect to the wave number with variation of magnetic field.

Variation of shear stress σxx with respect to the wave number with variation of the magnetic field.

Variation of shear stress σxy with respect to the wave number with variation of the magnetic field.

Variation of the temperature θ with respect to the wave number with variation of magnetic field.

Variation of the concentration charge with respect to the wave number and x-axis.

Variation of the phase velocity with respect to the wave number and x-axis.

Variation of the attenuation coefficient with respect to the wave number and x-axis.

Variation of displacement u with respect to the wave number and x-axis.

Variation of the stress σxx with respect to the wave number and x-axis.

Variation of the stress σxy with respect to the wave number and x-axis.

Variation of the temperature with respect to the wave number and x-axis.

Figure 2 displays the variation of concentration change with different values of rotation with respect to the wave number; it appears that the concentration change increases with an increasing of the wave number and if there is no rotation and small wave number takes larger values than in the presence of rotation and large values of the wave number. Figures 3 and 4 clear the phase velocity and attenuation coefficients variations with respect to the wave number with variation of the rotation; it is concluded that the phase velocity begins from 1.2 but the attenuation coefficient begins from zero if the wave number equals zero and tends to zero if the wave number tends to infinity; also, it is seen that they increase and decrease periodically with an increasing of the wave number and rotation that indicate the interruption of the phase velocity and attenuation coefficient with the largest values of the wave number and rotation that agree with the practical results. Figure 5 shows the variation of the displacement component with respect to the wave number with variation of the rotation; it is concluded that it takes various values for zeros value of the wave number and tends to zero if the wave number tends to infinity; also, it seems that it increases and decreases periodically with an increasing of the wave number and rotation.

Figures 6 and 7 make clear the variation of the stresses components with respect to the wave number with variation of the rotation; it appears that it begins from zero for zeros value of the wave number and tends to zero if the wave number tends to infinity; also, it is shown that it increases and decreases periodically with an increasing of the wave number and rotation. Figure 8 displays the variation of the temperature with various values of the wave number and rotation; one can see that it begins from zero for zeros wave number and tends to zero, decreases and increases tends to zero if the wave number tends to infinity; it is clear that the temperature decreases with increasing rotation values that physically indicates the negative influence of rotation on the temperature that takes into consideration engineering and structures. Also, it is seen that, in the absence of rotation, all values of the concentration change, phase velocity, attenuation coefficient, displacement, stresses components, and temperature take a smooth behavior comparing with the corresponding values in the presence of rotation that take harmonic behavior.

Figure 9 shows the variation of concentration change with various values of magnetic field with respect to the wave number; it appears that the concentration change increases with an increasing of the wave number and decreases and arrives to the unity if the wave number tends to infinity; if the magnetic field is absent, the concentration change smallest than the corresponding values for small values of the wave number and inverses to the largest values for the large values of the wave number. Figures 10 and 11 make obvious the phase velocity and attenuation coefficients variations with respect to the wave number with variation of the magnetic field; it is concluded that the phase velocity begins from 1.2 but the attenuation coefficient begins from zero if the wave number equals zero and tends to zero if the wave number tends to infinity; also, it appears that they increase and decrease periodically with an increasing of the wave number and magnetic field that indicate the interruption of the phase velocity and attenuation coefficient with the largest values of the wave number and magnetic field that agree with the practical results. From Figure 12, it is concluded that the displacement component takes various values for zeros value of the wave number and tends to zero if the wave number tends to infinity; also, it is shown that it increases and decreases with an increasing of the wave number and magnetic field.

Figures 13 and 14 display the variation of the normal and shear stresses with respect to the wave number with variation of the magnetic field; it appears that it begins from zero for zeros value of the wave number and tends to zero if the wave number tends to infinity; also, it is shown that the normal stress decreases and then increases arriving to zero with the large values of the wave number and magnetic field. It is seen that the shear stress increases and periodically decreases with the variation of the wave number and magnetic field and begins from zero at zeros value of the wave number and tends again to zero if the wave number tends to infinity. Figure 15 dispatches the variation of the temperature with various values of the wave number and magnetic field; we can show that it begins from zero for zeros wave number and tends to zero for infinity wave number and increases and decreases with variation of the wave number and magnetic field.

Totally, it is clear that the temperature decreases with increasing rotation values that physically indicates the negative influence of rotation on the temperature that takes into consideration engineering and structures.

Figures 1622 display the 3D variation on the concentration change, phase velocity, attenuation coefficient, displacement, stresses components, and temperature, respectively, with respect to the wave number and x-axis if Ho=105 and Ω=0.1.

It is obvious from Figure 16 that the concentration change decreases to arrive to zero with the increased values of the wave number; it increases and decreases to its minimum value and after that increases with an increasing of x-axis. From Figures 17 and 18, it appears that the phase velocity begins from 1.2 but the attenuation coefficient begins from zero if the wave number equals zero and tends to zero if the wave number tends to infinity; also, it appears that it increases and decreases periodically and arrives to zero with an increasing of x-axis and the wave number. Figure 19 shows that the displacement decreases, increases periodically with the variation of the wave number, and tends to zero as the wave number tends to infinity. It appears that the displacement decreases and increases with the increased values of x-axis.

From Figure 20, we concluded that the normal stress decreases and then increases arriving to zero as the wave number tends to zero and increases with the small values of x-axis and after that decreases to tend to zero. Figure 21 shows that the shear stress component increases and decreases periodically with the variation of the wave number but increases and decreases with an increasing of x-axis.

Finally, it appears that the temperature increases and decreases arrive to zero as the wave number tends to infinity but decreases with an increasing of the small x-axis and after that increases.

12. Concluding Remarks

Surface waves at imperfect boundary between isotropic elastic layer of finite thickness and isotropic thermodiffusive elastic half-space with magnetic field, stiffness, and rotation with two thermal relaxation times (GL) model are illustrated. The secular equation in compact form has been derived. The concentration change, phase velocity, attenuation coefficient, displacement, stresses, and temperature are displayed graphically. The amplitudes of displacements, temperature, and concentration are computed at the free plane boundary and presented graphically. Specific loss of energy is obtained and depicted graphically.

The analysis to be carried will be useful in the design and construction of rotating sensors, engineering, structures, and surface acoustic waves devices and the following remarks have been concluded.

If there is no rotation, small wave number takes larger values than in the presence of rotation and large values of the wave number.

The phase velocity, attenuation coefficient, displacement, stresses components, and temperature begin from zero for zeros value of the wave number and tend to zero if the wave number tends to infinity; also, it is seen that they increase and decrease periodically with an increasing of the wave number and rotation and magnetic field.

The temperature decreases with increasing rotation values that physically indicates the negative influence of rotation on the temperature that takes into consideration engineering and structures and acoustic and rotating sensors.

Appendices A. Constant of (<xref ref-type="disp-formula" rid="EEq34">28</xref>)

Consider the following:(A.1)h1=c21+Λ2,h2=iω-1cτ11,h3=iω-1cτ21,h5=χ2c,h7=-iω-1c2χ1τ10l1=2iΛc2,l2=iω-1cτ11,l3=-iω-1cτ21q3,l4=ω-2c3,l1=1-iω-1c2τ20,d1=l1l4-l3+h7l2,d2=l4+h7l2-l3l1-l3,d3=l4h5-h7q1-h5l3,d4=l2h5+l1q1,d5=l3h5+h7q1,d6=l2h5+l1q1+q1,d7=l3(1+RH2)+h3q1,d8=δ2d2+h2d5,d9=h3q1+δ2l3,f1=d2-δ1-h1l3+h2d5-h3d6δ1,f2=δ2d9,f3=1-h1d7-h3q1-h3δ2q1,f4=d1+d2δ1-h1+h2d3-h3d4δ1,f5=δ22(d8-h3d6),f7=h3δ1-h1q1-d6+d6δ2,f8=δ1(δ1-h1)d1,f9=(δ2d1-h2d3-d4h3)δ2,g1=δ2d3+(δ1-h1)d5-d3,g2=(δ1-h1)d6+d4-δ2d4,g3=1-h1δ1-h1,g4=l12d1,g5=h2δ1-h1,g6=h3(δ1-h1),A=-f1+f2+f3δ1d7,B=-1-h1f1+f4δ1-f5-f7-l3l12δ1d7,C=-1-h1f4-f9-h2g1-h3g2+f8+l12d2δ1d7,D=-[d1g3+g4-d3g5-d4g6]δ1d7.

B. Constant of (<xref ref-type="disp-formula" rid="EEq35">29</xref>)

Consider the following:(B.1)n1p=-Ep-1Λ1+iΛ2mp-Λ3mp2-iΛ4mp3ooooo-Λ5mp4-iΛ9mp5×Ep-1,n2p=Λ7+Λ8mp+Λ9mp2-Λ10mp3+Λ11mp4Ep,n3p=-Ep-1Λ12-Λ13mp+Λ14mp2oooooo-Λ15mp3+Λ16mp4-Λ17mp5oooooo+Λ18mp6×Ep-1Ep=G1-G2mp2-G3mp6+Gmp8,p=1,2,3,4,where(B.2)G1=δ1-h1d1,G2=d1+δ1-h1d2-h2d3-h3d4,G3=d2-δ1-h1l3+h2d5-h3d6,G4=d7,Λ1=l1d1,Λ2=δ2+RH2d1-h2d3-h3d4,Λ3=l1d2,Λ4=d8-h3d6,Λ5=-l1l3,Λ6=d9,Λ7=-δ1-h1d3,Λ8=l1d3,Λ9=δ2d3+δ1-h1d5-d3,Λ10=l1d5,Λ11=d5-δ2d5,Λ12=(δ1-h1)d4,Λ13=l1d4,Λ14=d4+(δ1-h1)d6-δ2d4,Λ15=l1d6,Λ16=d6+(δ1-h1)q1-δ2d6,Λ17=q1l1,Λ18=q11-δ2.

C. Determinant Elements of (<xref ref-type="disp-formula" rid="EEq44">36</xref>)

Consider the following:(C.1)R14=R15=R16=R17=R21=R22=R23=R31=R32=R33=R42=R51=R53=0,R11=w8,R12=-y7,R13=w7,R24=b2,R25=b1,R26=b3,R27=b4,R34=q2,R35=q1,R36=q3,R37=q4,R41=b5+b9,R43=-b6,R44=J2,R45=J1,R46=J3,R47=J4,R52=b7,R54=s2,R55=s1,R56=s3,R57=s4,R16=[b5+b9],R62=-iξknβ1T0,R63=-b6,R64=r2,R65=r1,R66=r3,R67=r4,R71=-iξk1β1T0,R72=b7,R73=ξm6k1β1T0,R74=k1,R75=k1,R76=k1,R77=k1,R11=y8,R41=0,R51=b8,R61=-ξm5knβ1T0,R71=-b8,R12=w8,R52=0,R42=b5+b9=R62,R72=-iξk1β1T0,R13=-y7,R43=0,R53=-b7,R63=-iξknβ1T0,R73=b7,w8=b8sinξm5H,w5=b5cosξm5H,w7=b7sinξm6H,w6=b6sinξm6H,b8=2im5Γ1e,b7=1-m62Γ1e,b5=λe+μeHY2(1+m52)+2μem52ξ2a1T0,b6=2iμem6ξa1T0,y7=-b7cosξm6H,y5=b5sinξm5H,y8=b8cosξm5H,y6=b6cosξm6H,Jp=iΓ2ξ-Γ3n2p-Γ4n3p-Γ5ξn1pmp+μeHY2β1T0iξi-n1pmp,bp=n2pmp,qp=n3pmp,rp=knn1p,Γ5=λ+2μa1T0,Γ2=λβ1T0,Γ3=ρv12τ11β1T0,Γ4=ρv12τ21β1T0,Γ1e=μeβ1T0,Γ1=μβ1T0,sp=iξn1p-ξmpΓ1.

D. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M213"><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msubsup><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>P</mml:mi><mml:mo> </mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mo> </mml:mo><mml:mn>1,2</mml:mn><mml:mo mathvariant="bold">,</mml:mo><mml:mn>3,4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula><bold> Given in (<xref ref-type="disp-formula" rid="EEq99">46</xref>)</bold>

Consider the following:(D.1)F1=q2J3s4-J4s3-q3J2s4-J4s2+q4J2s3-J3s2,F2=q1J3s4-J4s3-q3J1s4-J4s1+q4J1s3-J3s1,F3=q1J2s4-J4s2-q2J1s4-J1s2+q4J2s1-J1s2,F1=q1J2s3-J3s2-q2J1s3-J3s1+q3J1s2-J2s3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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