1 Introduction

Smart materials are different from the usual materials and can sense their environment and respond. Their properties can be significantly altered in a controlled fashion by external stimuli such as stress,temperature,electric,and magnetic fields. Such characteristics enable technological applications across a wide range of sectors including electronics,construction, transportation,agriculture,food and packaging,health care,space,defence,etc. A lot of studies have been done in this field,and details are sketched in many texts^{[1, 2, 3, 4, 5, 6]}.

The problems of reflection and refraction of elastic waves have been discussed by many authors. Knott^[7] discussed the problem of reflection and refraction of elastic waves with seismological applications. Crampin and Taylor^[8] studied surface wave propagation in anisotropic media. Crampin^[9] calculated the particle motion of surface waves propagating in particular symmetry directions in anisotropic media. A full account of reflection and refraction of elastic waves were given by Achenbach^[10]. Pal and Chattopadhyay^[11] studied the reflection phenomenon of plane waves at a free boundary in a pre-stressed elastic half-space. The three-dimensional reflection and transmission coefficients for plane waves in isotropic elastic media were derived by Borejko^[12],where an extensive survey of literature pertinent to the problem of reflection and transmission of plane waves was also given. Ogden and Sotirropoulos^{[13, 14]} investigated the problems of reflection in both incompressible and compressible elasticity. Chattopadhyay and Rogerson^[15] studied reflection of plane waves at an incrementally traction-free boundary of a half-space composed of a linearly incompressible elastic material. Chattopadhyay^[16] studied the wave reflection in a triclinic crystalline medium,and Deschamps^[17] studied the problem of reflection and refraction of the plane wave.

The coupled theory of thermoelasticity was extended by Lord and Shulman^[18] and Green and Lindsay^[19] by incorporating the thermal relaxation time in constitutive relations. Some problems on reflection in thermoelastic solid have been discussed by Deresiewicz^[20],Sinha and Sinha^[21],and Singh^[22]. Sinha and Elsibai^[23] discussed the reflection of thermoelastic waves at a free surface of solid half-space with two kinds of relaxation time. Abd-Alla and Al-Dawy^[24] investigated the reflection phenomenon of the shear vertical (SV) waves in a generalized thermoelastic medium. Sharma et al.^[25] studied the reflection of generalized thermoelastic waves from the stress free as well as the rigidly fixed thermally insulated or isothermal boundary of a thermoelastic solid half-space by considering the case of P and SV incidence waves. Deschamps and Cheng^[26] studied the reflection problem in a liquid-thermoviscoelastic solid interface.

A number of problems,which are related to the phenomenon of reflection and refraction of plane waves for piezoelectric materials,can be found in the literature^{[2, 27]}. In designing acoustic transducers,knowledge and control of the reflection and transmission of ultrasonic waves at the boundary between piezoelectric materials and water are important,which,in most of cases,are piezoelectric ceramics for use in under water imaging^{[28, 29]}. Nayfeh and Chien^{[30, 31]} derived the analytical expressions for the reflected and transmitted amplitude ratios for the fluid loaded piezoelectric plate and fluid loaded piezoelectric half-space in order to study the influence of piezoelectricity on such waves.

A thermo-piezoelectricity theory was first proposed by Mindlin^[32],who later derived the governing equations of a thermo-piezoelectric plate^[33]. The physical laws for the thermopiezoelectric materials were explored by Nowacki^[34]. Chandrasekharaiah^[35] has generalized Mindlin’s theory of thermo-piezoelectricity to take into account the finite speed of propagation of thermal disturbances. The effect of anisotropy on wave propagation in piezo-thermoelastic media has also been analyzed in some recent studies^{[36, 37, 38]}. Sharma^[39] solved the mathematical model to calculate the complex phase velocities of four attenuated waves. Sharma et al.^[40] discussed the reflection of piezo-thermoelastic waves from the charge and stress free boundary of a transversely isotropic half-space. The theoretical and experiemental studies of reflection and refraction of piezo-thermoelastic waves are of considerable interest in many engineering and industrial applications such as sensors,actuators,intelligent structures,radio,computer technology,and ultrasonic. A number of problems of reflection and refraction at fluid-solid interfaces have been undertaken by many authors^{[41, 42, 43]},but the problem of reflection and refraction at fluid-piezothermoelastic interfaces is still left.

In this paper,the problem of reflection and transmission of plane waves from a fluid-piezothermoelastic solid interface is considered. A piezo-thermoelastic half-space (PTHS),having 6mm symmetry,loaded with fluid half-space (FHS),is considered,and the expressions for the amplitude ratios and the energy ratios corresponding to reflected and transmitted waves are derived analytically using boundary conditions. Some reduced cases are discussed,and the effects of the angle of incidence,the frequency,the specific heat,the relaxation time,and the thermal conductivity on the reflected and transmitted energy ratios are observed for a particular model of cadmium selenide (CdSe) and water. Some special cases of interest are also discussed. 2 Basic equations

Consider a homogeneous,anisotropic,thermally conducting,piezoelectric elastic medium of the density ρ at a uniform temperature T_o in an undisturbed state. The basic equations are as follows. The strain-displacement relations are

the stress-strain-temperature and electric field relations are the equations of motion are the heat conduction equation is and the Gauss equation is where σ_ij are the stress tensors,c_ijkl are the elastic constants,e_kl are the strain tensors, ui are the displacement components in the medium,ηijk_ijk and ε_ij represent the piezoelectric constants and the dielectric permittivity coefficients,respectively,β_ij are the thermoelastic coupling tensors,and K_ij correspond to the thermal conductivities. ρ and C_e are the density and the specific heat at constant strain,respectively. The Lord and Shulman (LS) theory involves the thermal relaxation time τ_o,which represents the time lag needed to establish steady state heat conduction in an element of volume when a temperature gradient is suddenly imposed on that element. The Green and Lindsay (GL) theory involves the thermal relaxation time τ_o and τ₁ such that τ_o>τ₁>0. p_i are the pyroelectric constants,E_k are the components of the electric field vector,and φ represents the electric potential. The convention of repeated index for summation is adopted. The dot and comma notations denote (partial) differentiation with respect to time and space,respectively. The symbols δ_1k and δ_2k denote the Kronecker delta. Here,κ=1 for the LS theory and κ = 2 for the GL theory. The parameters in Eqs. (1)-(5) are assumed to satisfy the following conditions:

(i) The thermal conductivity K_ij and the dielectric permittivity ε_ij are symmetric and positive definite.

(ii) The pyroelectric coefficients are positive.

(iii) The piezoelectric modulus tensors η_ijk are symmetric in indices j and k.

(iv) The thermoelastic coupling tensors β_ij are non-singular.

(v) The specific heat C_e at constant strain is positive.

(vi) The isothermal linear elastic coefficients are positive definite in the sense that c_ijkle_ije_kl > 0.

The constitutive equations of piezo-thermoelastic material in the x₁x₃-plane in the absence of body forces and free charge density are

where

The harmonic plane waves,propagating in the x₁x₃-plane at a given angular frequency (ω), can be expressed as

where q is the unknown slowness parameter,and c is the apparent phase velocity. U,A,B, and C are the amplitudes associated with the harmonic wave. Then,Eqs. (3)-(5),along with Eqs. (6) and (7),reduce to a system where S = [U,A,B,C]^T,and Γ is a 4×4 matrix whose elements are listed in Appendix A. This system is consistent if

The desired expressions of the coefficients T_j (j = 1,2,· · · ,5) are given in Appendix B. q₁,q₂,q₃,and q₄ correspond to the roots of this equation whose imaginary parts are positive, and q₅,q₆,q₇,and q₈ denote those whose imaginary parts are negative. Here,q₄ corresponds to the electric potential component wave mode (PE₁),and q₁,q₂,and q₃ correspond to the propagating quasi P (qP) mode,quasi S (qS) mode,and quasi T (qT) mode of wave propagation, respectively. For each q_i (i = 1,2,· · · ,8),the corresponding eigen vectors U_i,A_i,B_i,and C_i can be written as

where and c(Γ_ij)qi denotes the cofactor of Γ_ij to the eigen value q_i. The amplitudes (U,A,B,and C) of the plane harmonic waves decrease as these waves progress in a piezo-thermoelastic medium. The amplitudes of the plane harmonic waves propagating in a piezo-thermoelastic medium also depend on the frequency. The formal solution for the mechanical displacement and the electric potential becomes

The dynamical equations for the fluid are

where σ_ij^f and u_i^f are the stress tensors and displacements in the FHS,respectively,and ρ_f and λ_f are the density and the bulk modulus of the fluid,respectively.

The normal stresses are related to the dilatation by the relation

The solution to Eq. (14) can be expressed as where is the unknown slowness parameter,U^f and A^f are the associated amplitudes,and c_f is the longitudinal wave velocity in the fluid medium. 3 Reflection and transmission coefficients 3.1 Amplitude ratios

Consider a PTHS,having 6mm symmetry,loaded with the FHS. The PTHS occupies the region x₃ ≥ 0,and the FHS occupies the region x₃ ≤ 0,as shown in Fig. 1. A plane wave, making an angle θ with the x₃-axis,becomes incident at the interface. This wave results in one reflected wave in the FHS and four transmitted modes in the PTHS. The transmitted wave modes are represented by qP,qS,qT,and the fourth mode PE₁ corresponds to the electric potential wave mode. The formal solutions for the mechanical displacements,the temperature, the electric potential,the stress components,and the electric displacements in a PTHS are

The coefficients D_1i,D_2i,and D_4i have been calculated and are given in Appendix C. Substituting Eq. (16) into Eq. (14) gives the displacements of incident and reflected waves as

where p = 1 corresponds to the incident wave,p = 2 corresponds to the reflected wave,and

Accordingly,the normal stress becomes

The boundary conditions at the interface x₃ = 0 are as follows:

(i) The mechanical boundary conditions are

(ii) The electrical boundary conditions are

(iii) The thermal boundary conditions are

Equations (17)-(19) and the above boundary conditions result in a non-homogeneous system

where

and the elements of 5×5 matrix A are given in Appendix C. After solving the system (21),the transmitted and reflected amplitude ratios are obtained as

and where the expressions for A′,Y ,and dt_i (i = 1,2,· · · ,4) are given in Appendix C. 3.2 Energy ratios

At the interface x₃ = 0,the distribution of energy between different reflected and transmitted waves across a surface element of unit area is considered. Following Ikeda^[4],Scott^[44],and Kuang and Yuan^[45],the normal acoustic flux P in a piezo-thermoelastic solid is

The average energy fluxes of the incident and reflected waves are

and and the average energy fluxes of the transmitted waves are derived as The energy ratios of the reflected and transmitted waves are defined as The interaction energy ratios,which account for the interaction between different fields and displacements corresponding to different transmitted waves,are described as where The energy is conserved if where is the resultant interaction energy between the transmitted waves. 4 Reduced cases

Reflection and transmission of plane waves are studied for the following types of the fluidPTHS interface:

(i) Open and isothermal interface

By invoking the stress free,isothermal,and charge free boundary conditions (20a)-(20c), (20d),and (20f) at the surface x₃ = 0,we obtain a non-homogeneous system

where

and the matrix A₁ is given as

By solving this system,we obtain the values of transmitted and reflected amplitude ratios. Then,using these values of amplitude ratios and equations from (23)-(28),we can calculate the energy ratios for different waves.

(ii) Closed and isothermal interface

Upon invoking the stress free,isothermal,and electrically shorted boundary conditions (20a)-(20c),(20e),and (20f) at the surface x₃ = 0,we obtain a non-homogeneous system

where

and the matrix A₂ is given by

(iii) Open and adiabatic interface

Upon invoking the stress free,adiabatic,and charge free boundary conditions (20a)-(20c), (20d),and (20g) at the surface x₃ = 0,we obtain a non-homogeneous system

where

and the matrix A3 is given as

(iv) Closed and adiabatic interface

Upon invoking the stress free,adiabatic,and electrically shorted boundary conditions (20a)-(20c),(20e),and (20g) at the surface x₃ = 0,we obtain a non-homogeneous system

where

and the matrix A4 is given by

where the expressions of D_ij (i = 1,2,· · · ,7; j = 1,2,· · · ,4) are given in Appendix C.

(v) Fluid-piezoelectric solid interface

If we set β₁₁ = β₃₃ = p₃ = K₁₁ = K₃₃ = Ce = 0,the motion corresponding to the thermal wave decouples from rest of motion. The Christoffel equation for the piezoelectric material now becomes

where T₁^P,T₂^P ,T₃^P ,and T₄^P are given in Appendix D. q₁^P,q₂^P,and q₃^P correspond to the roots of this equation whose imaginary parts are positive,and q₄^P ,q₅^P,and q₆^P denote those whose imaginary parts are negative. Here,q₃^P corresponds to the electric potential component wave mode,and q₁^P and q₂^P correspond to the propagating qP mode and qS mode of wave propagation, respectively. The reflection and refraction coefficients for such an interface are found for two sub cases.

Case 1 Open interface

In this case,the boundary conditions (20a)-(20d) are used,which gives A_P1X = B_P1,where X = [U₁,U₂,U₃,U₂^f]^T,B_P1 = [D′₁₄,0,D′₃₄,0]^TU₁^f,and the matrix A_P1 is given as

Case 2 Close interface

When the fluid-piezoelectric interface is short circuited,the boundary conditions (20a)- (20c) and (20e) are applicable,and the corresponding system is obtained as A_P2X=B_P2,where X=[U₁,U₂,U₃,U₂^f]^T,B_P2=[D′₁₄,0,D′₃₄,0]^TU₁^f,and the matrix A_P2 is given by

where the expressions of D′ _ij (i,j = 1,2,· · · ,5) are given in Appendix D.

(vi) Fluid-thermoelastic solid interface

If we set η₃₁ = η₁₅ = η₃₃ = ε₁₁ = ε₃₃ = p₃ = 0,the results of the PTHS reduce to those of the thermoelastic half-space. The Christoffel equation for the thermoelastic material becomes

where T₁^T,T₂^T ,T₃^T ,and T₄^T are given in Appendix E. q₁^T,q₂^T,and q₃^T correspond to the roots of Eq. (35) whose imaginary parts are positive,and q₄^T ,q₅^T,and q₆^T denote those whose imaginary parts are negative. Here,q₃^T corresponds to the thermal mode,and q₁^T and q₂^T correspond to the propagating qP mode and qS mode of wave propagation,respectively. The reflection and refraction coefficients for such an interface are found for two sub cases.

Case 1 Isothermal interface

In this case,the boundary conditions (20a)-(20c) and (20f) are used,which give A_T1X = B_T1,where X = [U₁,U₂,U₃,U₂^f]^T,B_T1 = [M₁₄,0,M₃₄,0]^TU₁^f,and the matrix A_T1 is given as

Case 2 Adiabatic interface

When the fluid-thermoelastic interface is adiabatic (thermally insulated),the boundary conditions (20a)-(20c) and (20g) are applicable,and the corresponding system is obtained as A_T2X = B_T2,where X = [U₁,U₂,U₃,U₂^f]^T,B_T2 = [M₁₄,0,M₃₄,0]^TU₁^f,and the matrix A_T2 is given by

where the expressions of M_ij (i,j = 1,2,· · · ,5) are given in Appendix E.

(vii) Fluid-elastic solid interface

In the absence of piezoelectric,pyroelectric,and thermal effects,Eq.(10) reduces to

where T₁^E,T ₂^E,and T₃^E are given in Appendix F. q₁^E and q₂^E correspond to the roots of this equation whose imaginary parts are positive,and q₃^E and q₄^E denote those whose imaginary parts are negative. Here,q₁^E and q₂^E correspond to the P mode and S mode of wave propagation,respectively. In this case,the boundary conditions (20a)-(20c) are applicable,and then we obtain the non-homogeneous system A_EX = B_E,where X = [U₁,U₂,U₂^f]^T,B_E= [E₁₃,0,E₃₃]^TU₁^f,and the matrix A_E is given by

where the expressions of E_ij (i = 1,2,· · · ,3; j = 1,2,· · · ,4) are given in Appendix F. 5 Numerical discussion

The amplitude and energy ratios for reflected and transmitted waves are calculated for a particular model CdSe and water. The elastic,dielectric,piezoelectric,thermal,and pyroelectric coefficients,followed from Sharma et al.^[40] and Vashishth and Gupta^[46],are listed in Tables 1 and 2. The PTHS is loaded with water.

Figure 2 shows the variation of reflected and transmitted amplitude ratios with the angle of incidence (θ) of compressional wave propagating in the fluid medium at the frequency ω = 2π×10¹² Hz and the relaxation time τ_o=10^-12 s and τ₁=3×10^-12 s using two different theories LS and GL of thermoelasticity. Both theories almost show the same behaviour except at θ = 26° in the case of transmitted PE₁ mode,where the value of amplitude ratio is less in the case of the GL theory in comparison to the LS theory. Figure 3 depicts the variation of reflected and transmitted amplitude ratios under isothermal open and isothermal short boundary conditions. In the case of isothermal open boundary conditions,there is a slight increase in the reflected amplitude ratio at θ = 31°,and the amplitude ratio of transmitted qP wave increases from θ = 31° to θ = 59 ° and then decreases up to θ = 81°. The amplitude ratio of transmitted PE₁ mode decreases in the case of isothermal short boundary conditions. Figure 4 shows that by changing the boundary conditions from isothermal to adiabatic,the amplitude ratios are not affected,and Fig. 5 shows that in the case of adiabatic open and adiabatic short boundary conditions,the amplitude ratios show the same behaviour as in the case of isothermal open and isothermal short boundary conditions

From Fig. 6,it is observed that the amplitude ratios are not affected by the change in the values of thermal conductivity from K₁₁ = K₃₃=9 to K₁₁= K₃₃=0.000 1. From Fig. 7,it is observed that,when the value of the specific heat varies from 100 J/(kg·K) to 300 J/(kg·K) and to 800 J/(kg·K),the amplitude ratio of the qP and qT mode decreases. Figures 8 and 9 show that the change in the frequency affects the amplitude ratio slightly,and the change in the relaxation time (τ_◦) affects the amplitudes of qP and qT waves slightly.

Next,we shall observe the effect of thermoelasticity on the reflected and transmitted amplitude ratios with the angle of incidence in Fig. 10. In the absence of piezoelectricity,there are three modes in the transmitting medium instead of four,as expected theoretically. The amplitude ratio of qT mode is very small. In the absence of thermoelasticity,there are also three modes in the transmitting medium instead of four,as expected theoretically. This is observed from Fig. 11,and in Fig. 12,the effect of elasticity on the reflected and transmitted amplitude ratios with the angle of incidence is observed.

The energy ratios of reflected wave,transmitted waves,and interaction energy coefficients are computed using Eqs. (26)-(29). Figure 13 shows the variation of energy ratios with the angle of incidence (θ) of compressional wave propagating in the fluid medium at the frequency=2π× 10¹² Hz. The energy ratios corresponding to reflected and transmitted waves are represented by E_ref and E_s (s = 1,2,· · · ,4),respectively. The total interaction energy ratio between the transmitted waves is denoted by E_int. It is observed that before θ = 25°,all the transmitted wave modes,namely,PE₁,qP,qS,and qT,propagate in the PTHS,and after θ=25°,the transmitted qP wave mode is no longer excited. Between θ = 25° and θ = 76°,two modes, i.e.,qS and qT,propagate in the PTHS,and after θ = 76°,only quasi thermal mode is excited. The energy carried out by the electric potential components PE₁ is very small. The contribution of interaction energy ratio between the transmitted waves is almost negligible except at θ = 26° and from θ = 74° to θ = 86°. For a compressional wave,incident from the fluid medium,there are two critical angles θ₁ = 25° and θ₂ = 76° corresponding to transmitted P wave and S wave in the PTHS,respectively. Beyond the second critical angle,i.e., θ=76°,the incident wave is totally reflected,and the energy reflection coefficient equals unity.

In the case of a perfect elastic medium,i.e.,for a lossless medium,the transmitted wave decays with distance from the interface for supercritical incidence^[47]. However,for a medium with loss,it is possible to have transmitted waves whose amplitude grows with distance from the interface for some angles of incidence beyond critical angle as in viscoelasticity,critical angles for transmitted wave are isolated^{[47, 48]}. For anelastic reflection-refraction problems,contrasts in anelastic absorption at a boundary give rise to inhomogeneous waves for all angles of incidence as opposed to the elastic reflection-refraction problems where inhomogeneous wave exists only beyond critical angles^[49]. The major portion of incident energy is reflected back,which signifies the fact that the transmitting medium is much denser. The results are in agreement with the law of conservation of energy. There is no null in reflection coefficient which reveals the effect of transversely anisotropy^[50].

Figure 14 shows the variation of energy ratios for isothermal short and isothermal open boundary conditions. The angles of incidence,where the energy ratios for the isothermal short boundary are greater than those for the open boundary,are 76°,71°,and 39°,respectively. For all other angles of incidence,no noticeable difference between the results for these two cases is observed. The energy ratio of PE₁ mode increases after θ = 31° in the case of isothermal open boundary conditions,and the modulus of interaction energy increases in the case of isothermal short boundary conditions. Figure 15 depicts that,when we compare isothermal and adiabatic conditions,the energy ratio of qT wave and the interaction energy ratio increase in the case of adiabatic conditions. Now,in comparison with adiabatic open and adiabatic short conditions in Fig. 16,we find that the energy ratios of qP and qT waves and the interaction energy ratio increase in the case of adiabatic short boundary conditions,and the energy ratio of PE₁ mode decreases in the case of adiabatic short boundary conditions. Figure 17 depicts that the thermal conductivity has no effect on the energy ratios,while the interaction energy ratio slightly increases with the increase in the values of thermal conductivity constants.

Figure 18 depicts that the increase in C_e shifts the critical angle of qP wave,and the energy ratio of PE₁ mode decreases with the increase in the values of C_e. From Fig. 19,we can say that the change in the value of ω slightly affects the energy ratios of qT and PE₁ wave modes and the interaction energy ratios. Figure 20 gives the information that the energy ratio of PE₁ mode and the interaction energy ratio increase with the increase in the thermal relaxation time. Figure 21 gives the information about what happens when the piezo-thermoelastic model changes to the thermoelastic model,and from Fig. 22,we note that the energy ratio of PE₁ mode is zero when the piezo-thermoelastic model changes to the piezoelectric model. Figure 23 gives the same result as in Ref. ^[51],when we take data for aluminium as in this paper.

6 Conclusions

Some observations from the numerical example lead to the following conclusions.

(i) The reflection and transmission of waves from an interface separating the FHS and the PTHS are studied in the present paper. The analytical expressions for the reflected and transmitted amplitudes and the energy ratios are derived.

(ii) The incident wave is totally reflected,and the energy reflection coefficient equals unity after θ = 76°.

(iii) The resultant interaction energy flux and the energy flux corresponding to the electric potential wave mode are less significant in comparison to those of propagating qP,qS,and qT modes. The results are in agreement with the law of conservation of energy.

(iv) There is no null in the reflection coefficient as expected in the case of transversely isotropy.

(v) The change in frequency does not affect significantly the value of critical angle. Appendix A

where κ = 1 for the LS theory,and κ = 2 for the GL theory of thermoelasticity. Appendix B

For the LS theory,i.e.,for κ = 1,

where in which and For κ=2,i.e.,for the GL theory, where α₁,α₂,· · · ,α₁₀ are the same as in Eq. (B2),and in which y₁,y₂,· · · ,y₁₂ and x₁,x₂,· · · ,x₆ are the same as in Eqs. (B3) and (B4),respectively,and Appendix C

where Appendix D

where in which Also, Appendix E

where in which Also, Appendix F

where Also,