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The reflection and transmission of elastic waves in porous piezoelectric plate, overlying a porous piezoelectric half space and underlying a fluid half space, is studied. The constitutive and governing equations are formulated for porous piezoelectric materials. The expressions for the mechanical displacements, electric displacements, stresses, and electric potentials are derived for porous piezoelectric plate, porous piezoelectric half space, and fluid half space. The boundary conditions are described for the studied model. The behaviour of reflected and transmitted amplitude ratios relative to frequency, incident angle, thickness, and porosity is observed numerically. The impedance mismatching problem between the dense piezoelectric materials and the surrounding medium can be solved by the inclusion of porosity in dense piezoceramics.

The field of smart materials (piezoelectric) has advanced rapidly due to an increasing awareness about capabilities of such materials, the development of new materials and transducer designs, and increasingly stringent design and control specifications in aerospace, aeronautics, industrial, automotive, biomedical, and nanosystems. Piezoelectric materials are brittle in nature which leads to the failure of devices. It is commonly found that 5% porosity exists in piezoelectric materials which are considered as manufacturing defects. Instead of considering it as a manufacturing defect, such materials can be modelled as porous piezoelectric materials. Porous piezoelectric materials are widely used in ultrasonic transducers, hydrophones, and pressure sensors. Porous ceramics are of interest for ultrasonic transducer applications. Porosity allows to decrease the acoustic impedance, thus improving transfer of acoustic energy to water or biological tissues. For underwater applications, the figure of merit can also be improved as compared to dense material. The surface impedance of porous piezoelectric materials is less as compared to dense piezoelectric materials. Alvarez-Arenas and De Espinosa [

The influence of piezoelectricity on the reflection-transmission phenomena in fluid-loaded piezoelectric half space and fluid-loaded piezoelectric plate was studied by Nayfeh and Chien [

The fabrication of piezoelectric transducers for high resolution medical imagining applications requires a backing material to damp the piezoelectric resonance, resulting in a short time resonance, that is, improved resolution. Thus, the choice of such a substrate must be made according to its acoustical properties. For ultrasonic transducer model, a thick film of porous piezoelectric material has to be deposited on a porous piezoelectric substrate. Levassort et al. [

Motivated the theoretical and experimental models developed to study the properties of the integrated structures for ultrasonic imaging applications and transducer application, a theoretical model is developed in this paper to study the effects of porosity, thickness, and incidence angle on the reflection-transmission phenomena. In the present paper, the effects of piezoelectricity, frequency, porosity, incidence angle, and thickness on the amplitude ratios corresponding to reflected and transmitted waves in fluid-loaded porous piezoelectric plate are studied analytically as well as numerically for a particular model. The results obtained in this paper can be used to improve the properties of porous piezoelectric materials.

Let us consider a porous piezoelectric plate of thickness

The constitutive equations for transversely isotropic porous piezoelectric material are

Here,

The equations of motion are

For a harmonic plane wave, the associated physical quantities can be represented as

Equations (

Similarly, the corresponding expressions in the porous piezoelectric half space are

Here,

The boundary conditions at the interface

The boundary conditions at the interface

Making use of (

On solving nonhomogeneous system (

The reflected and transmitted amplitude ratios are calculated numerically for a particular model. The material of the porous piezoelectric plate is taken as PZT and that of porous piezoelectric half space is taken as BaTiO_{3}.

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Figure

Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency.

Figure

Behaviour of reflected amplitude ratio with porosity at fixed angle of incidence and fixed frequency.

The behaviour of amplitude ratios with frequency in the frequency range 1 MHz–11 MHz is observed in Figure

Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence.

The effect of thickness of the porous piezoelectric plate sandwiched between fluid half space and porous piezoelectric half space on the behaviour of amplitude ratios relative to frequency is observed in Figure

Effects of thickness of plate on the behaviour of amplitude ratios with frequency.

The reflection and transmission phenomenon in porous piezoelectric plate sandwiched between fluid half space and porous piezoelectric half space is studied. The behaviour of reflected and transmitted amplitude ratios relative to frequency, angle of incidence, layer thickness, and porosity is observed numerically for a particular model. The amplitude ratios are found to be sensitive with respect to frequency and incidence angle. The number of maxima/minima increases with increase in the thickness of the layer. With increase in porosity of the medium, less amount of incident energy gets reflected back.

We have the following: