International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pure Appl. Sci. Technol., 3() (0), pp International Journal of Pure and Applied Sciences and Technology ISSN Available online at Research Paper Reflection of Quasi Vertical Transverse Waves in the Thermo-Piezoelectric Material under Initial Stress (Green- Lindsay Model) Fatimah A. Alshaikh Department of Mathematics, Science College, Jazan University, Jazan, Saudi Arabia Corresponding author, (Received: 7-9-; Accepted: 9-0-) Abstract: The purpose of this research is to study the reflection transverse waves in the Green- Lindsay theory for kind of smart materials under initial stress and relaxation times effect. It is found that, for two dimensional model of thermo-piezoelectric material, there exists four waves namely, quasi-transverse wave (QS-mode), quasi-longitudinal wave (QP-mode), thermal wave ( -mode), and potential electric waves ( ). The reflection coefficients of these plane waves are computed and depicted graphically along direction of wave propagation. Some special cases of interest are also discussed. Keywords: Thermo-piezoelectric, Green-Lindsay theory, Vertical transverse waves, Reflection coefficients, Initial stresses, Relaxation time.. Introduction: In the classical theory of thermoelasticity the velocity of heat propagation is assumed to be infinitely large. The different generalizations of the classical theory of thermoelasticity are developed to remove this paradox. For example, Lord and Shulman (L-S) theory [], in this theory a modified law of heat conduction including both the heat flux and its time derivative replaces the conventional Fourier s law. The heat equation associated with this a hyperbolic one and, hence, automatically eliminates the paradox of infinite speeds of propagation inherent in both the uncoupled and the coupled theories of thermoelasticity. However, for many problems involving steep heat gradients, and when short time effects are sought, this theory gives markedly different values than those predicted by any of the other theories. This is the case encountered in many problems in industry especially inside nuclear reactors where very high heat gradients act for very short times. Green and Lindsay theory [] is another example of the generalized thermoelasticity, by including temperature rate among the constitutive

2 Int. J. Pure Appl. Sci. Technol., 3() (0), variables, developed a temperature-rate-dependent thermoelasticity that does not violate the classical Fourier law of heat conduction, when body under consideration has center of symmetry and this theory also predicts a finite speed for heat propagation. Sinha et al, and Sinha et al [3,,5] investigated the reflection of thermoelastic waves from the free surface of a solid half-space and at the interface between two semi-infinite media in welded contact, in the context of generalized thermoelasticity. Abd-Alla and Al Dawy [6] discussed the reflection phenomena of SV waves in a generalized thermoelastic medium. Sharma et al. [7] studied the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of a solid half-space in the context of different theories of generalized thermoelasticity. The thermo-piezoelectric material response entails an interaction of three major fields, namely, mechanical, thermal and electric in the macro-physical world. One of the applications of the thermopiezoelectric material is to detect the responses of a structure by measuring the electric charge, sensing or to reduce excessive responses by applying additional electric forces or thermal forces, actuating. If sensing and actuating can be integrated smartly, a so-called intelligent structure can be designed. The thermopiezoelectric materials are also often used as resonators whose frequencies need to be precisely controlled. Because of the coupling between the thermoelastic and pyroelectric effects, it is important to quantify the effect of heat dissipation on the propagation of wave at low and high frequencies. The theory of thermopiezoelectricity was first proposed by Mindlin [8]. The physical laws for the thermo-piezoelectric materials have been explored by Nowacki [9,0,]. Chandrasekharaiah [,3] has generalized Mindlin s theory of thermo-piezoelectricity to account for the finite speed of propagation of thermal disturbances. Many researchers have studied the propagation of waves in bodies that are made of thermo-piezoelectric materials such as Sharma and Kumar [], Sharma et al [5], Sharma and Walia [6,7], Kuang and Yuan [8], Alshaikh [9]. Initial stresses are developed in the medium due to many reasons, resulting from difference of temperature, process of quenching, shot pinning and cold working, slow process of creep, differential external forces, gravity variations, etc. The Earth is supposed to be under high initial stresses. It is therefore of great interest to study the effect of these stresses on the propagation of stress waves. During the last decades considerable attention has been directed toward this phenomenon [0,,, 3,, and 5]. In the present paper, the reflection phenomenon of quasi-vertical transverse (QS) wave is employed to study the smart medium under initial stresses effect and thermal relaxation time values. The boundary conditions are formulated to obtain the expression of reflection coefficients, both theoretically and as well as numerically. The numerical results are shown graphically to show the effect of initial stress and thermal relaxation times upon the reflection coefficients.. Governing Equations and Boundary Conditions: Following Green-Lindsay [] and Sharma and Kumar [], the constitutive relations and hexagonal thermopiezoelectric equations under initial stress effect for two dimensions motion are given by: =, +, () = ( + ) (), +, = (3) Ω, =, +,, +, + + (), =0, = + + (5) Where,,, =,,3. =, is the electric field,

3 Int. J. Pure Appl. Sci. Technol., 3() (0), ,, and are the mechanical displacement, electric potential and absolute temperature, is the electric displacement,, are the stress and initial stress tensors, is the strain tensor, is the thermal elastic coupling tensors is the density of the medium, is the elastic parameters tensor, is the piezoelectric constants, is the dielectric moduli, is the pyroelectric moduli,, are thermal relaxation times, Ω is the heat conduction tensor, is the reference temperature, are the specific heat at constant strain, is the Kronecker delta. The appropriate mechanical, electrical, and thermal boundary conditions at the interface z=0 half space are as follows: ( ) ( ) ( ) + + =0, (6) ( ) ( ) ( ) + + =0, (7) ( ) + ( ) + ( ) =0, (8) ( ) + ( ) + ( ) =0 (9) 3. Reflection of Plane Waves under Initial Stresses Effect: Figure : Schematic diagram for the problem

4 Int. J. Pure Appl. Sci. Technol., 3() (0), We consider a thermopiezoelectric plane wave QS propagating under initial stress through the medium, which we identify as the region 0 and falling at the plane z = 0, with its direction of propagation making an angle with the normal to the surface. Corresponding to each incident wave, we get the waves in medium as reflected QP-, QS-, - and - waves. The complete geometry of the problem is shown in Figure. Let the wave motion in this medium be characterized by the displacement (,0, ), the temperature, and the electric potential function, all these quantities being dependent only on the variables,,. We assume solution of the form [6]: ( ) = cos, ( ) = sin, ( ) =, ( ) =. (0) ( ) = sin, ( ) = cos, ( ) =, ( ) =. () ( ) = cos, ( ) = sin, ( ) =, ( ) =. () Where = ( sin + cos ), = ( sin cos ), = ( sin cos ). = /, = /, = / are the velocity of incident QS, reflected QP, and reflected QS waves. ( ) ( ) Substituting in equations 6-9, the values of,, ( ) and ( ) (=,,,3,) from equations 0 using equations and, one may have ( ) cos + cos (( ) ) exp + ( sin + cos ) cos (( ) ) exp + ( ) cos cos (( ) ) exp (3) cos sin exp + sin sin exp + cos sin exp =0 () exp + exp + exp =0 (5) exp + exp + exp =0 (6) Equations. 3-6 must be valid for all values of t and x, hence From the above relations, we get = = sin = sin = sin = = = (7) =, =, = Ω=, = Ω (8)

5 Int. J. Pure Appl. Sci. Technol., 3() (0), Furthermore, we should now use the equations 3-5 of the media, i.e., using will give us additional relations between amplitudes. equations and which ( + ( ) ), +( + ( ) ( ) ( ) ), +( + ), +( + ), ( ( ) + ( ) ), = ( ) (9) ( ) ( ) Ω, +Ω (( ), + ( ) ( ) ( ) )=, +, ( ) ( ) ( ) ( ) +, +,, +, (0) ( ) ( ) ( ) ( ) ( ) ( + ), +, +,,, + ( ( ) + ( ) ), =0 () where =0,,. So, substituting from equations 0- (when z = 0 ) into equations 9- for the incident QS wave, the reflected waves, we get + + =0, + + =0, () + + =0 + + =0, + + =0, (3) + + =0 + + =0, + + =0, () + + =0 where = cos ( + )sin ( + )cos +( + )sin = ( + )sin cos, = ( )sin =sin ( + )sin ( + + )cos =( + )sin cos, = ( )sin =, =, = = ( )( )sin cos, = ( )cos, = ( ) (Ω sin +Ω cos ) = ( )( sin + cos ), = ( )cos, = ( ) (Ω sin +Ω cos ), =, =, =,

6 Int. J. Pure Appl. Sci. Technol., 3() (0), = ( + )sin cos sin, = sin + cos, = ( )cos = ( + )sin cos + cos = sin + cos, = ( )cos, =, =, = From equations 3-6 and -, It is easy to see that ( + ) =h, ( + ) =h, ( + ) =h, ( + ) =h (5) Where =, =, =, =, h =h = =Ω sin + cos cos ( ), =Ω( sin sin ), = ( )sin cos + cos ( ), =( cos sin ), =( ) ( ), =( ) ( ) =, =( ) ( ) =( ) ( ), = Solving equations 5, we can determine the reflection coefficients as: = ( + ), =( ) ( + ), (6) = ( )+, = ( )+ (7) =( ), = ( )+ (8). Numerical Results The material chosen for the purpose of numerical calculations is Lead Zirconate Titanate ceramics (PZT-5A) of hexagonal symmetry (6 mm class), which is transversely isotropic material. The physical data for a single crystal of PZT-5A material is given as [7]: =3.9 0 Nm, = Nm, =7.5 0 Nm, =.3 0 Nm, =.56 0 Nm, =7750 Kgm, = 6.98 Cm, =3.8 Cm, = 3. Cm, =98 K,

7 Int. J. Pure Appl. Sci. Technol., 3() (0), =.5 0 NK m, =.53 0 NK m, = 5 0 CK m,ω = Ω =.5 Wm K, = C N m, =5.7 0 C N m, =0 J Kg K, =. 0 s. The variations of phase velocities computed from ( λ λ ) C = C = C + C sin + C cos v ρ, T T 33 ( ϑ ϑ ) C = C + C sin + C cos + v ρ, L 33 Where ( ) ( ) ( ) = v C C sin λ C C cos λ C C sin λ ( ) ϑ ( ) ϑ ( ) ϑ. v = C C sin + C C 33 cos + C3 + C sin Equations (6-8) show the existence relations between the reflection coefficients of QS wave incident and reflection coefficients at the incident of the other two types of waves ( -mode),( mode). The constants of proportionality for these relations are functions of angle of incidence, initial stress, relaxation times, and piezoelectric constants. We have focused in the study on the period 0 90, this is because it is found that all the curves almost coincide with each other, as well as the effect of initial stresses and relaxation times not clear on the reflection coefficients in the previous period. In this study we have two groups of graphs as follows: The first group (Figures -7) displays the initial stresses effect ( =(6,7,8,9) 0, =0) on the imaginary (Im) and real (Re) values of the reflection coefficients /, /, / (i=,) in the context of Green and Lindsay (G-L) theory of generalized thermoelasticity where δ=0, = 0, =6, which have the following observations: Figures display the association of the coefficient / in the angle of incidence and the initial stresses effect, which appears in the period 0 5 for imaginary part, then this effect disappear in the following period. We also note that the effect of initial stresses on the coefficient ( / ) where it increases when the initial stress value increases. Figures 3 display the association of ( / ) and ( / ) of the coefficient of reflection of an incident of QS wave in the angle of incidence, where the imaginary part of the coefficient of reflection increases as long as the angle of incidence increases until it reaches the maximum value at =5. Then its value decreases to the minimum point at =90, We also note the effect of initial stress on ( / ) and ( / ) where the coefficient value increases as long as the initial stress value increases. Figures (-7) display the initial stress effect on relative electric potential ( / / ), and thermal ( / / ) coefficients. The second group (Figures 8-3) displays the relaxation times effect where δ=0, = 0, =,( =5,6,7,8) on the real and imaginary values of the reflection coefficients /, /, / (i=,) in the context of Green and Lindsay theory of generalized

8 Int. J. Pure Appl. Sci. Technol., 3() (0), thermoelasticity when the initial stresses are ( =5 0, =0), which have the following observations: Figs. (8-9) display the relaxation time effect on / and / coefficients for an incident of QS wave. We note that ( / ) and ( / ) values decrease as long as increases, and ( / ) and ( / ) values increase as long as increases. We also note the relaxation time effect on /, /, / and / in the figures (0-3). It can get some previous studies as a special case through neglect the thermal influence as [5]. =0, =0, =6 =6 0 = = =9 0 6 Figures. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model..5 =0, =0, =6 =6 0 = = =9 0 Figures 3. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model.

9 Int. J. Pure Appl. Sci. Technol., 3() (0), Figures. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model =0, =0, =6 =6 0 = = =9 0 =0, =0, =6 =6 0 = = =9 0 Figures 5. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model..5 =0, =0, =6 =6 0 = =.5 =9 0 Figures 6. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model.

10 Int. J. Pure Appl. Sci. Technol., 3() (0), =0, =0, =6 =6 0 = = =9 0.5 Figures 7. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle under effect of different values of the initial stress for (G-L) model Figures 8. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model. =5 0, =0,δ=0, =0 =5 =6... =8 =7.5 Figures 9. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model..5 =5 0, =0,δ=0, =0 =5 =6... =8 =7

11 Int. J. Pure Appl. Sci. Technol., 3() (0), =5 0, =0,δ=0, =0 =5 =6... =8 =7 Figures 0. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model. 3.5 =5 0, =0,δ=0, =0 =5 =6... =8 =7 Figures. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model..5 =5 0, =0,δ=0, =0 =5 = =8 =7 Figures. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model.

12 Int. J. Pure Appl. Sci. Technol., 3() (0), =5 0, =0,δ=0, =0 =5 =6... =8 =7 Figures 3. Imaginary and real parts of reflection coefficient ( / ) as a function of incidence angle for various values of the relaxation times for (G-L) model. Conclusion In this paper, the reflection phenomena in thermo-piezoelectric media under various initial stresses are studied. We apply the generalized thermoelasticity theory with two relaxation time developed by Green and Lindsay theory []. The numerical results of the two dimensional model is given. The coefficients of reflected waves are also computed and presented graphically. References [] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solid, 5(967), [] A.E. Green and K.A. Lindsay, Thermoelasticity, Journal of Elasticity, () (97), -7. [3] A.N. Sinha and S.B. Sinha, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, Journal of Physics of the Earth, () (97), 37-. [] S.B. Sinha and K.A. Elsibai, Reflection of thermoelastic waves at a solid half-space with two thermal relaxation times, Journal of Thermal Stresses, 9(996), [5] S.B. Sinha and K.A. Elsibai, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two thermal relaxation times, Journal of Thermal Stresses, 0() (997), 9-6. [6] A.N. Abd-Alla and A.A.S. Al-Dawy, The reflection phenomena of SV waves in a generalized thermoelastic medium, International Journal of Mathematics and Mathematical Sciences, 3(8) (000), [7] J.N. Sharma, V. Kumar and D. Chand, Reflection of generalized thermoelastic waves from the boundary of a half-space, Journal of Thermal Stresses, 6(0) (003), [8] R.D. Mindlin, On the equations of motion of piezoelectric crystals, In: N.I. Muskilishivili, Problems of Continuum Mechanics (70th Birthday Volume), SIAM, Philadelphia, 96. [9] W. Nowacki, Some general theorems of thermo-piezoelectricity, Journal of Thermal Stresses, () (978), 7-8. [0] W. Nowacki, Foundation of linear piezoelectricity, H. Parkus, Editor, Interactions in Elastic Solids, Springer, Wein, 979. [] W. Nowacki, Mathematical models of phenomenological piezo-electricity, New Problems in Mechanics of Continua, University of Waterloo Press, Waterloo, Ontario, 983. [] D.S. Chandrasekharaiah, A temperature-rate-dependent theory of thermopiezoelectricity, J. Thermal Stresses, 7(98),

13 Int. J. Pure Appl. Sci. Technol., 3() (0), [3] D.S. Chandrasekharaiah, A generalized linear thermoelasticity theory for piezoelectric media, Acta Mechanica, 7(988), [] J.N. Sharma and M. Kumar, Plane harmonic waves in piezo-thermoelastic materials, Indian Journal of Engineering and Material Science, 7(000), 3-. [5] J.N. Sharma, M. Pal and D. Chand, Three-dimensional vibration analysis of a piezothermoelastic cylindrical panel, International Journal of Engineering Science, (5-6) (005), [6] J.N. Sharma and V. Walia, Further investigations on Rayleigh waves in piezothermoelastic materials, Journal of Sound and Vibration, 30(-) (007), [7] J.N. Sharma, V. Walia and S.K. Gupta, Reflection of piezo-thermoellastic waves from the charge and stress free boundary of a transversely isotropic half space, International Journal of Engineering Science, 6() (008), 3-6. [8] Z.-B. Kuang and X.-G. Yuan, Reflection and transmission of waves in pyroelectric and piezoelectric materials, Journal of Sound and Vibration, 330(6) (0), -0. [9] F. Alshaikh, The mathematical modelling for studying the influence of the initial stresses and relaxation times on reflection and refraction waves in piezothermoelastic half-space, J. Applied Mathematics, 3(8) (0), [0] A. Chattopadhyay, S. Bose and M. Chakraborty, Reflection of elastic waves under initial stress at a free surface, J. Acoust. Soc. Am., 7(98), [] R.S. Sidhu and S.J. Singh, Comments on reflection of elastic waves under initial stress at a free surface, J. Acoust. Soc. Am., 7(98), [] S. Dey, N. Roy and A. Dutta, Reflection and refraction of P-waves under initial stresses at an interface, Indian Journal of Pure and Applied Mathematics, 6(985), [3] A.N. Abd-Alla and F.A. Alsheikh, Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses, Archive of Applied Mechanics, 79(9) (009), [] A.N. Abd-Alla and F.A. Alsheikh, The effect of the initial stresses on the reflection and transmission of plane quasi-vertical transverse waves in piezoelectric materials, Proceedings of World Academy of Science, Engineering and Technology, 38(009), [5] A.N. Abd-Alla, F.A. Alsheikh and A.Y. Al-Hossain, The reflection phenomena of quasivertical transverse waves in piezoelectric medium under initial stresses, Meccanica, 7(3) (0), [6] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company- Amsterdam, 973. [7] J.N. Sharma, M. Pal and D. Chand, Three-dimensional vibration analysis of a piezothermoelastic cylindrical panel, International Journal of Engineering Science, (5-6) (005),