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The present paper concentrates on the study of reflection and refraction phenomena of waves in pyroelectric and piezo-electric media under initial stresses and two relaxation times influence by apply suitable conditions. The generalized theories of linear piezo-thermoelasticity have been employed to investigate the problem. In two-dimensional model of transversely isotropic piezothermoelastic medium, there are four types of plane waves quasi-longitudinal (qP), quasi-transverse (qSV), thermal wave (T-mode), and potential electric waves (φ-mode) The amplitude ratios of reflection and refraction waves have been obtained. Finally, the results in each case are presented graphically.

Piezoelectricity is the phenomenon of electricity produced by the squeezing or stretching of certain materials. The propagation of waves in piezoelectric materials is one of the richest fields for scientists because it has many applications in piezoelectric: ﬁlters, resonators, transducers, sensors and other devices. This kind of devices represent a great challenge in the industry, and as a result it has been an object of different investigations in the last decades, because these devices are to operate under various piezoelectric-thermo-mechanical conditions over a broad spectrum, in view of its importance to industry applications. The theory of thermo-piezoelectricity was first proposed by Mindlin [

Deresiewicz [

Consider a homogeneous, anisotropic, generalized piezothermoelastic medium of hexagonal type. The origin is taken on the thermoelasticity and stress-free plane surface and z-axis is directed normally into the half-space which is represented by. Let the wave motion in this medium be characterized by: the displacement vector , the electric potential function, all these quantities being dependent only on the variables x, z, t. (see

The governing field equations of generalized hexagonal piezothermoelastic for two dimensional motion in the plane are [

• The coupled constitutive relations can be written in the forms:

• The strain-displacement relation and the electric field according to the quasi-static approximation have the forms as:

(2)

• The equations of motion under initial stress, Gauss’s divergence equation, and heat conduction can be written as (3).

where;, , and are the mechanical displacement, electric potential and absolute temperature, respectively;, and are the strain, stress and thermal elastic coupling tensors, respectively;, are the electric field and electric displacement, respectively; is the elastic parameters tensor;, and are the piezoelectric, dielectricpyroelectric moduli, respectively; is the relaxation time; and are the initial stress tensor and mass density, respectively; are the heat conduction tensor, reference temperature, Kronecker delta, specific heat at constant strain, respectively. The constitutive relations (1) of the hexagonal (6 mm) crystals symmetry given by

Substituting Equations (4)-(5) into Equation (3), we get (6).

We will consider a transversely isotropic piezoelectric half space (see

sent hexagonal crystals (transversely isotropic materials), we will consider the motion in the plane (plane). According to Achenbach [

where

where n = 0 represent the incidence of qP wave, n = 1, 2, represent the reflected waves, n = 3, 4 represent the refracted waves.

Consider the problem of two bounded semi-infinite pie-- zothermoelastic materials with the interface z = 0 subjected to a harmonic incident wave of frequency ω with an incident angle as shown in

1) The free mechanical boundary conditions:

2) The electrical condition:

3) The thermal condition:

Substituting Equations (2), (4), and (7) into Equations (8)-(10), we obtain the following set of equations:

where

Equations (11)-(14) must be valid for all values of t and x, hence

From the above relations, we get

Furthermore, we should now use the equations of motion of the media, i.e., Equation which will give us additional relations between amplitudes.

where.

So, substituting from Equation (7) (when z = 0) into Equation (17) for the incident (qP) wave, the reflected and refracted waves, we get

where

By using Equation (7) into Equation, we get

where

By using Equation (7) into Equation, we get

where

From Equations (11)-(14), it is easy to see that

where

Solving Equation (21), we can determine the reflection and refraction coefficients as:

where

By using Equations (18)-(20) we get:

The material chosen for the purpose of numerical calculations is (6 mm class) Cadmium Selenide (CdSe) for upper medium and Lead Zirconate Titanate ceramics (PZT-5A) for lower medium, which are transversely isotropic materials. The physical data for a single crystal of CdSe material and PZT-5A ceramics are given as [6,21]:

Here the thermal relaxation time is estimated its value about and is taken proportional to. The variations of phase velocities computed from

where

The real and imaginary values of the amplitude ratios corresponds to qP, qSV, T, -mode for incident qP wave are computed for various angle of incidence (in degrees) under various of initial stresses, in the context of Green and Lindsay theory (G-L) of generalized thermoelasticity [

.

The reflection and refraction coefficients have been presented on curves in Figures 2-21 which have the following observations:

•

•

•

•

from the figures that the effect caused by the relaxation time on is very slight.

• Figures 6-8 represent the relation between the electric potential coefficients with the angle of incidence, as well as the relaxation time effect.

• Figures 9-11 represent the relation between the thermal coefficients with the angle of incidence , as well as the relaxation time effect.

• Figures 12-21 show the initial stress effect on relative reflection and refraction, thermal, and electric potential coefficients when , In the period that shows the initial stress effect, we note that inverse relationship between the initial stress and reflection coefficients (and ) and the opposite what happens with the relative refraction coefficients (and).

• Equations (22)-(23) show the existence proportionality relations between the reflection coefficients of the quasi-longitudinal wave falling and reflection coefficients at the fall of the other two types of waves (T-mode), (-mode). The constants of proportionality for these relations are functions of angle of incidence, relaxation times, and piezoelectric.

• It can get some previous studies as a special case through neglect the thermal effects and the relaxation times as [