EFFECT OF INITIAL STRESS ON THE REFECTION OF MAGNETO-ELECTRO-THERMO-ELASTIC WAVES FROM AN ISOTROPIC ELASTIC HALF-SPACE

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1 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar EFFECT OF INITIAL STRESS ON THE REFECTION OF MAGNETO-ELECTRO-THERMO-ELASTIC WAVES FROM AN ISOTROPIC ELASTIC HALF-SPACE *Rajneesh Kakar Department of Engineering & Technology, GNA University, Phagwara, India *Author for Correspondence ABSTRACT The present work illustrates a theoretical study of the refection of magneto-electro-thermo-elastic plane waves from an isotropic pre-stressed elastic half-space in generalized thermoelasticity under GL-theory. Magnetic field and electric field acts initially parallel to the plane boundary of the elastic half-space. Refection of magneto-electro-thermoelastic waves under generalized thermoelasticity theory is used to study the refection of plane waves from a semi-infinite elastic solid in a vacuum. The expressions for the refection coefficients are obtained mathematically. Results are plotted with MATLAB software to show the effect of temperature, magnetic field, electric field, relaxation time and initial stresses on the reflection of incident SV-wave. Mathematics Subject Classification: 74F5; 74H5; 8D4. Keywords: Reflection Coefficients; Magnetic Field; Electric Field; Relaxation Time; GL- theory INTRODUCTION Problem related to magneto-electro-thermoelastic plane wave deals with the interactions among strain, temperature, and electromagnetic fields in transversely isotropic and anisotropic medium has many applications in geophysics, optics, electrical power engineering and seismology. The modelling of surface waves dispersion effects has become of growing interest to geotechnical engineers and geophysicists. The seismic waves usually studied in seismology and seismic surveying are those produced by earthquakes, explosions, or impacts. These waves are complex vibrations of limited duration having the nature of an impulse. The problem of propagation velocity of these complex vibrations requires additional research. When the wavelength of the harmonic component is significantly small compared with the heterogeneity, such as thickness of layers, the oscillations are propagated following the laws of geometrical optics. By knowing the reflection and refraction, the magneto-thermal elastic plane waves are the useful source for imagining the interior of the Earth. According to the conventional heat conduction theory, the thermal disturbances travel at infinite velocities. However, from the physical point of view, the above concept is unrealistic in the situation of very low temperature near absolute zero. The hyperbolic equations of motion are applicable in such cases and the elastic disturbances propagate with finite speeds. Thus, generalized thermoelasticity theories are proposed to examine modified thermoelastic models involving a hyperbolic type of heat equation. Shekhar and Parvez (3) purposed plane waves propagating in transversely isotropic dissipative half space under the effect of rotation, magnetic field and stress. Othman and Song () discussed reflection of magneto-thermo-elastic wave by using generalized theory of elasticity. Mehditabar et al., (4) investigated magneto -thermo-elastic functionally graded conical shell. Othman () studied electro-magneto-thermoelastic thermal shock plane waves for a finite conducting half-space. Niraula and Noda () purposed non-linear electro-magneto-thermo-elasticity by deriving material constants. Niraula and Wang (6) studied the property of magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading. Kaur and Sharma () discussed reflection and transmission of thermoelastic plane waves at liquid-solid interface. Fractional order generalized electro-magneto-thermo-elasticity was given by Ya et al., (3). Propagation of plane waves at the interface of an elastic solid half-space and a microstretch thermoelastic diffusion solid half-space was investigated by Kumar et al., (3). Chakraborty (3) Copyright 4 Centre for Info Bio Technology (CIBTech) 79

2 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar discussed reflection of plane elastic waves in half-space subjected to temperature and initial stress. Singh and Yadav () discussed the reflection of plane waves in a rotating transversely isotropic magneto-thermoelastic solid half-space. Singh and Bala () purposed the reflection of P and SV waves from the free surface of a two-temperature thermoelastic solid half-space. Kakar and Kakar (3) theoretical discussed electro-magneto-thermo viscoelastic surface waves in non-homogenous solid medium under the effect of gravity, compression and couple-stress In this paper, we have discussed the problem of reflection of magneto-electro-thermoelastic waves under initial stress in transversely isotropic solid half space. Biot s equations are modified in terms of Green and Lindsay s theory of thermoelasticity. We find that P-wave is affected due to the presence of thermal and magnetic field whereas SV-wave remains unaffected which is in accordance with the GL-theory since the temperature and magnetic field in an infinite space results only in irrotational changes. The governing equations are solved in light of modified heat equation to obtain reflection coefficients for P-wave, thermal wave and SV-wave. The effect of temperature, magnetic field, electric field, relaxation time and initial stresses on reflection coefficients of incident SV-wave are plotted under certain practical assumptions. Governing Equations The governing equations of linear, isotropic and homogenous magneto-electro-thermoelastic solid with initial stress are a. The stress-strain-temperature relation: sij P( ij ij ) epp ij eij ( ) ij, () k where, T sij are the components of stress tensor, P is initial pressure, ij is the Kronecker delta, ij are the components of small rotation tensor,, are the counterparts of Lame parameters, e ij are the components of the strain tensor, α is the volume coefficient of thermal expansion, k T is the isothermal compressibility, is small temperature increment, is the absolute temperature of the medium, is the reference uniform temperature of the body chosen such that The displacement-strain relation: u u i j eij () xj x i where, u i are the components of the displacement vector. The small rotation-displacement relation: u u i j ij (3) xj x i where, u i are the components of the displacement vector. b. The modified Fourier s law: hi a* h i K (4) xi where, K is the thermal conductivity, aa, * are the thermal relaxation times c. The heat conduction equation : 3 3 u v u v K c γ p ij x y t t xt yt xt yt (5) Copyright 4 Centre for Info Bio Technology (CIBTech) 8.

3 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar where, K is the thermal conductivity, cp is specific heat per unit mass at constant strain, is the first relaxation time, is second relaxation time, ij is the Kronecker delta, is density and T is the incremental change of temperature from the initial state of the solid half space. Moreover the use of the relaxation times, and a parameter ij marks the aforementioned fundamental equations possible for the three different theories: Classical Dynamical theory:, ij Lord and Shulman s theory:,, ij Green and Lindsay s theory:, ij d. Maxwell s equations:,,, e e (6) t t where,, B, e and e are electric field, magnetic field, permeability and permittivity of the medium. e. The components of electric and magnetic field:,, h,,, e (7) where, h is the perturbed magnetic field over and e is the perturbed electric field over E. f. Maxwell stress components for electro-magneto-thermoelastic medium: Tij e Hiei H jej Hkek ij ] e[ Eiei E jej Ekek ij where i, j, k =,,3 (8) where, H, H, H are the components of primary magnetic field, E, E, E are the components of i primary electric field, j k e i, e j, e k are the stress components acting along X-axis, Y-axis, Z-axis respectively and ij is the Kronecker delta. Using Eq. (8), we get u v T u v eh ee x y x y and T (9) The dynamical equations of motion for the propagation of wave have been derived by Biot (965) and in two dimensions these are given by s s u P Bx () x y y t i j k s s v P By () x y x t where, s, s and s are incremental thermal stress components. The first two are principal stress components along x- and y-axes, respectively and last one is shear stress component in the x-y plane, is the density of the medium and u, v are the displacement components along x and y directions respectively, B is body force and its components along x and y axis are B and B respectively. is the v u rotational component i.e. x y and P s s. The body forces along x and y axis under constant primary magnetic field H parallel to z-axis are given by x y Copyright 4 Centre for Info Bio Technology (CIBTech) 8

4 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar u v u v Bx eh ee x xy x xy () u v u v By eh ee xy x xy x (3) where, e and e are permeability and permittivity of the medium. Following Biot (965), the stress-strain relations with incremental isotropy are s ( P) exx ( P) eyy exx x (4) s exx ( ) eyy x (5) s e (6) xy where, u xx, v e eyy, e v u xy (7) x x x y where, e xx and eyy are the principle strain components and exy is the shear strain component, (3 ) t, t is the coefficient of linear expansion of the material, are Lame s constants, is the incremental change of temperature from the initial state and is second relaxation time. Formulation of the Problem We consider a transversely isotropic, homogeneous elastic half space under constant magnetic and electric field acting along z-axis and initial compressive stress P acting along x-axis at absolute temperature (Figure ). y-axis Atmosphere y x-axis θ θ θ Elastic half space Incident SV-wave θ θ Reflected SV-wave Reflected P-wave Reflected Thermal wave Figure : Reflection of magneto-electro-thermo-elastic plane waves Copyright 4 Centre for Info Bio Technology (CIBTech) 8

5 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar A plane SV-wave is incident at an angle at y, such that it get reflected and giving three waves namely reflected SV at an angle, thermal-waves at an angle and P-wave at an angle respectively as shown in the diagram Boundary Conditions The following boundary conditions are supplemented at y : i. f s Pe, x xy ii. f s, y iii. T ht. (8) y Solution of the Problem From Eq. (), Eq. (3), Eq. (4), Eq. (5), Eq. (6) and Eq. (7), we get u P v P u u v x xy y x xy P eh ee u t x tx v P u P v u v y xy x xy y eh ee Eq. (5) can be modified as u v u v K = cp +γ ij t t t x y t x y Eq. (9) and Eq. () can be solved by choosing potential functions and as u and v x y x y From Eq. (9) and (), we get ( H E P) t ( H E P) t e e e e v t y ty P t From Eq. () and Eq. (), we get ( H E ) t ( H E ) t P t e e e e (5) (9) () () () (3) (4) (6) Copyright 4 Centre for Info Bio Technology (CIBTech) 83

6 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Eq. (3) and Eq. (5) represent magneto-electro-thermo compression waves along x- axis and y- axis respectively, whereas Eq. (4) and Eq. (6) represent magneto-electro-thermo distortional waves along x- axis and y- axis respectively. For initial stress along x- axis, the four equations (3)-(6) reduced to two equations as T T (7) c t ( H E P) t e e (8) c t where, P ( eh ee P) c and c (9) c is known as P-wave velocity and c is called SV-wave velocity. Also, for P-wave v and for SV-wave u. Now, from Eq. () and (), we get K = c p +γ ij t t t t (3) where, x y From equations (7), (8) and (3), we conclude that P-wave depends on the presence of magnetic, electric and thermal field whereas SV-wave remains unaffected which is in accordance with the GL-theory. The solution of Eq. (7), Eq. (8) and Eq. (3) is plane harmonic waves travelling perpendicular to the x-y plane, which is given as exp[ i { k ( x sin y cos ) t }] (3) exp[ i { l ( x sin y cos ) t }] (3) exp[ i { k ( x sin y cos ) t }] (33) where, k and l are compression and rotational wave numbers, is angular frequency. From Eq. (7), Eq. (9) and Eq. (33), we get c k i (34) c From Eq. (3), Eq. (3) and Eq. (33), we get i i k ic i k (35) ij p In order to satisfy Eq. (34) and Eq. (35), the determinant of the coefficients of both Eq. (34) and Eq. (35) will be zero, therefore c k i c (36) ij p i i k ic i k Copyright 4 Centre for Info Bio Technology (CIBTech) 84

7 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Expanding Eq. (36), we get 4 i i (37) where, γ kc c c i c c,,, p p i ij i and i Eq. (37) is biquadratic in, it means that P-wave and thermal wave travel with different velocities. Therefore, on striking the SV-wave at y = making an angle in the solid half space it will have one reflected SV-wave making an angle, P-wave and thermal wave at an angle and (figure ). Therefore from the above discussion we can take displacement potential and perturbation temperature in the following form exp[ i{ k ( xsin ycos ) t}] exp[ i{ k ( xsin ycos ) t}] (39) exp[ i{ l( xsin ycos ) t}] exp[ i{ l( xsin ycos ) t}] (4) exp[ i{ k ( xsin ycos ) t}] exp[ i{ k ( xsin ycos ) t}] (4) where,, represent amplitudes of the P-wave and thermal wave, represents amplitude of incident SV wave and is the amplitude of reflected SV-waves respectively. Also,, and are related to respective wave numbers as k sin k sin l sin (4) Eq. (4) can be written in terms to Snell s law as sin sin sin (43) where, c and, c, and kc kc are the roots of equation i c c i, and are given as 4 (38) (44) Introducing Eq. (3) and Eq. (33) in into Eq. (3), we get i ij i i i i i and i ij i i i i i (45) Copyright 4 Centre for Info Bio Technology (CIBTech) 85

8 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar i ij i exp[ i{ k( xsin y cos ) t} i i i i i ij i exp[ i{ k( xsin y cos ) t}] (46) i i i i Introducing Eq. (4), Eq. (5), Eq. (6) and Eq. () in the first boundary condition of Eq. (8), we get P xy x y (47) Introducing Eq. (4), Eq. (5), Eq. (6) and Eq. () in the second boundary condition of Eq. (8), we get eh ee eh ee (48) y y xy y ty Since we have taken the upper layer is thermally insulated, therefore from the third boundary condition of Eq. (8), we get T (49) y Substituting Eq. (39) and Eq. (4) in Eq. (47) and with the help of Eq. (4) and Eq. (4), we get cos sin sin cos (5) Similarly, substituting Eq. (39), Eq. (4) and Eq. (4) in Eq. (48) and with the help of Eq. (9), Eq. (3), Eq. (4) and Eq. (43), we get ' sin sin ( sin ) i (5) sin ( sin ) ' sin i P P where, and c c Substituting Eq. (46) in Eq. (49) and with the help of Eq. (44), we get cos cos (5) i i Eliminating from Eq. (5) and Eq. (5), we get i i cos i cos cos sin cos sin Copyright 4 Centre for Info Bio Technology (CIBTech) 86 (53)

9 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Eliminating from Eq. (5) and Eq. (5), we get i cos sin sin sin cos cos i cos sin cos i sin ' Equating Eq. (53) and Eq. (54), we get R P Equating Eq. (54) and Eq. (55), we get R SV Equating Eq. (5) and Eq. (55), we get C RP where, A, B, C and H are given as i i i i cos [ i { cos( ) ( )cos cos sin ' sin sin } cos ] cos [ { cos( ) ( )cos cos sin ' sin sin } cos ] (58) cos cos sin ' (59) C cos cos sin ' (6) cos [ { cos( ) ( )cos cos sin ' sin sin } cos ] cos [ i { cos( ) ( )cos cos sin ' sin sin } cos ] (6) R SV reflection coefficients of plane SV-wave, R P ( c ) represents reflection coefficients of thermal wave and R P ( c ) reflection coefficients of P- wave. Copyright 4 Centre for Info Bio Technology (CIBTech) 87 (54) (55) (56) (57)

10 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Numerical Analysis and Calculations Approximate expression for reflection coefficients is obtained by assuming practical values of and for elastic materials. Solving Eq. (38) and retaining only first degree terms of and, we get T and 3 Eq. (43) can be written as sin sin,sin sin, i (6) T i cos sin sin, cos sin sin. The equations (58-6) can be written by using Eq. (63) as RP, R SV and RP (64) where, pq( ') r scos, cos sin ( ')( T ) Copyright 4 Centre for Info Bio Technology (CIBTech) 88, (63) T q sin T pq( ') r scos, p 4 sin cos T, s sin, 3 T r cos T cos sin T cos i sin. (65) From Eq. (64), we conclude that there is no P-wave in the reflection when practical values and for elastic materials are assumed. Various graphs are plotted between R, R, R and for taking S.,.3,.5,.7,.9 for SV P( c) P( c) tensile stress and S.,.3,.5,.7..9 for compressional stress. R SV, RP ( c) and RP ( c) are calculated by taking parameters for copper alloy (Table ). The results are compared with purposed model and standard model from approximation and are illustrated graphically with the help of MATLAB software. The results are closed to the standard model. The various curves are plotted by approximating the Eq. (55), Eq. (56) and (57) by considering relaxation factor and coupling factor very small, we observed that no thermal P-wave is reflected while both reflected SV-wave and reflected P-wave is seen. Figure to Figure 6 are plotted without using approximation, Figure 7 to Figure are plotted after using approximation for Eq. (64) at various incident angles. A graphical view is taken for variation of angle of incidence θ from to 4 and from to 5 ; two series are taken while plotting the graphs i.e. taking S.,.3,.5,.7,.9 for tensile stresses and S.,.3,.5,.7..9 for P compressional stresses. Here S is known as stress parameter. It is observed that SV-wave is greatly affected by the presence of magnetic field and temperature of the solid half space. Figure shows that the maxima and minima of reflection coefficients of SV-wave are in the range 3 for S.,.3,.5,.7,.9. The variation of reflection coefficients for SV reflected,

11 Reflection Coefficient SV-Wave SV-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar wave in magneto-thermal medium is same for both compressional and tensile stress; the only difference is the reversal of the stress (Figure 3). Figure 4 is plotted for the reflection coefficients of P-wave, the reflection coefficient is minimum at and in figure 5, the maxima and minima of reflection coefficients of SV-wave are in the range 3. Figure 6 is plotted for reflection coefficient of thermal wave for various values of stress S.,.3,.5,.7,.9, from this figure it is clear that the maximum value of reflection coefficients of thermal waves occur at an angle of incident without using approximation. However in figure 7, the range of maxima and minima for reflection coefficients of thermal waves is 4 when approximation is used. Figure 8 shows that the maximum of reflection coefficients of SV-wave is at an angle 3 for S.,.3,.5,.7..9 after using approximation. Table Parameter Numerical Value kg / m, H 9 e.4 N/ m N/ m E 9 e 9.5 N/ m.5 6 t 6.6 K, K 4 W / ( m. K ) c.39 KJ / Kg K p 4.5 N/ m 8 Tensile Stress (Without Approximation) 6 4 S=. S=.3 S=.5 S=.7 S= Figure Copyright 4 Centre for Info Bio Technology (CIBTech) 89

12 Reflection Coefficient P-Wave Reflection Coefficient SV-Wave SV-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Similarly, Figure 9 to Figure is plotted for reflected SV, reflected P and reflected thermal wave after using approximation. Figure to Figure 4 are plotted for reflected SV, reflected P and reflected thermal wave at various values of magnetic field and electric field, keeping initial stress at.5. It is observed that in the presence of magneto-electro field, there is remarkable variation in these curves. Figures (5 7) are plotted between the reflection coefficients of P, T and SV waves against angle of incidence θ for relaxation times τ = s,.4 s,.8 s at constant H = and S =.5, it is clearly observed that the effect of relaxation time is prominent for τ =.4 s and the effect is more for increase in relaxation time. Figure 8 to Figure are plotted for reflected SV, reflected P and reflected thermal wave at various values of electric field keeping initial stress at.5. It is observed that in the presence of electric field there is remarkable variation in these curves.. Compressional Stress (Without Approximaton).8.6 S= -. S= -.3 S= -.5 S= -.7 S= Figure 3 6 Tensile Stress (Without Approximation) S=. S=.3 S=.5 S=.7 S= Figure 4 Copyright 4 Centre for Info Bio Technology (CIBTech) 9

13 Reflection Coefficient Thermal Wave Reflection Coefficient P-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar 5 Compressional Stress (Without Approximation) S= -. 5 S= -.3 S= -.5 S= -.7 S= Figure 5 5 Tensile Stress (Without Approximation) 5 S=. S=.3 S=.5 S=.7 S= Figure 6 Copyright 4 Centre for Info Bio Technology (CIBTech) 9

14 Reflection Coefficient SV-Wave Reflection Coefficient SV-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Tensile Stress (After Using Approximation) S=. S=.3 S=.5 S=.7 S= Figure S= -. S= -.3 S= -.5 S= -.7 S= -.9 Compressional Stress (After Using Approximation) Figure 8 Copyright 4 Centre for Info Bio Technology (CIBTech) 9

15 Reflectin Coefficient P-Wave Reflection Coefficient P-Wave P-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Tensile Stress (After Using S=. S=.3 S=.5 S=.7 S= Incident Figure Compessional Stress (After Using Approximation) S= -. S= -.3 S= -.5 S= -.7 S= Figure Copyright 4 Centre for Info Bio Technology (CIBTech) 93

16 Reflection Coefficient SV-Wave Reflection Coefficient Thermal Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar 5 5 Tensile Stress (After Using Approximation) S=. S=.3 S=.5 S=.7 S= Figure S=.5 H= H= 4 H= 6 H= Figure Copyright 4 Centre for Info Bio Technology (CIBTech) 94

17 Reflection Coefficient Thermal Wave Reflection Coefficient P-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar S=. H= H= 4 H= 6 H= Figure S=.5 H= H= 4 H= 6 H= Figure 4 Copyright 4 Centre for Info Bio Technology (CIBTech) 95

18 Reflection Coefficient P-Wave Reflection Coefficient Thermal Wave Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Figure Figure 6 Copyright 4 Centre for Info Bio Technology (CIBTech) 96

19 Reflection Coefficient Thermal Wave Reflection Coefficient SV-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar Figure S=.5 E= E= 4 E= 6 E= Figure 8 Copyright 4 Centre for Info Bio Technology (CIBTech) 97

20 Reflection Coefficient P-Wave Reflection Coefficient SV-Wave International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar S=.5 E= E= 4 E= 6 E= Figure S=. E= E= 4 E= 6 E= Figure Conclusion The purpose of this study is to show the combined effect of temperature, magnetic field, electric field, relaxation time and stress on the propagation of elastic plane SV-waves through a solid isotropic elastic half space. It has been observed that in case of free space, very small energy is reflected, however in case of magneto-electro-thermal medium, the SV incident wave is greatly modifies in the presence of stress as well as magnetic field of the medium. The results are compared with purposed model and standard model using from approximation. It is clearly observed that the effect of relaxation time on reflection coefficients of P, T and SV waves is prominent for τ =.4 s and the effect is more for increase in relaxation time. The results are closed to the standard model. This model is useful to study the problems involving heat change, electromagnetic field, mechanical stress applied at the boundary of the surface. The results presented in this paper may be useful for geophysicists to analyze material structures and rocks through nondestructive testing. The solution of such problems also affects different geomagnetic cases. Copyright 4 Centre for Info Bio Technology (CIBTech) 98

21 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 (4) October-December, pp /Kakar ACKNOWLEDGMENT The authors are thankful to unknown reviewers for their valuable comments. REFERENCES Biot MA (965). Mechanics of Incremental Deformations (John Wiley and Sons, Inc.) New York. Chakraborty N (3). Reflection of plane elastic waves at a free surface under initial stress and temperature field, TEPE () Kakar R and Kakar S (3). Theoretical analysis of electro-magneto-thermo viscoelastic surface waves in non-homogenous solid medium subjected to gravity, compression and couple-stress, International Journal of Modern Applied Physics () Kaur R and Sharma JN (). Study of reflection and transmission of thermoelastic plane waves at liquid- solid interface, Journal of International Academy of Physical Sciences 6() 9 6. Kumar R, Garg SK and Ahuja S (3). Propagation of plane waves at the interface of an elastic solid half-space and a microstretch thermoelastic diffusion solid half-space, Latin American Journal of Solids and Structures (6) 8 8. Mehditabar A, Akbari Alashti R and Pashaei MH (4). Magneto -thermo-elastic analysis of a functionally graded conical shell, Steel and Composite Structures, an International Journal 6() Niraula OP and Wang BL (6). A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading, Acta Mechanica 87(-4) Niraula OP and Noda N (). Derivation of material constants in non-linear electro-magneto-thermo-elasticity, Journal of Thermal Stresses 33() -34. Othman MIA (). Generalized electro-magneto-thermoelasticity in case of thermal shock plane waves for a finite conducting half-space with two relaxation times, Mechanics and Mechanical Engineering 4() 5-3. Othman MIA and Song Y (). Reflection of magneto thermo-elastic waves from a rotating elastic half-space in generalized thermoelasticity under three theories, Mechanics and Mechanical Engineering 5() 5-4. Shekhar S and Parvez IA (3). Effect of rotation, magnetic field and initial stresses on propagation of plane waves in transversely isotropic dissipative half space, Applied Mathematics Singh B and Bala K (). Reflection of P and SV waves from the free surface of a two-temperature thermoelastic solid half-space, Journal of Mechanics of Materials and Structures 7() Singh B and Yadav AK (). Reflection of plane waves in a rotating transversely isotropic magneto-thermoelastic solid half-space, Journal of Theoretical and Applied Mechanics 4(3) Ya J, Xiao G and Tian JL (3). Fractional order generalized electro-magneto-thermo-elasticity, European Journal of Mechanics Copyright 4 Centre for Info Bio Technology (CIBTech) 99