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Smart Materials Research
Volume 2013 (2013), Article ID 767019, 8 pages
Reflection and Transmission Phenomena in Poroelastic Plate Sandwiched between Fluid Half Space and Porous Piezoelectric Half Space
1Department of Mathematics, Dyal Singh College, Karnal 132 001, India
2Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India
Received 25 April 2013; Accepted 22 June 2013
Academic Editor: Chris Bowen
Copyright © 2013 Vishakha Gupta and Anil K. Vashishth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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  made a study related to characterization of porous piezoelectric ceramic. The experimental study on wave propagation in composite structures was done by Alvarez-Arenas et al. . Different authors [3, 4] developed models related to synthesis, fabrication, and processing of porous piezoelectric materials. Kumar et al.  investigated the effects of porosity and pore forming agents in porous piezoceramics.
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 [6, 7]. Darinskii and Weihnacht  studied theoretically the propagation and the generation of leaky waves in a fluid layer enclosed between the half-infinite lithium niobate substrate coated with a thin silicon oxide layer and the half-infinite domain occupied by silicon rubber. Love waves propagating in a layered structure with an elastic layer deposited on a piezoelectric substrate were analytically investigated by Liu and He . Vashishth and Gupta  derived the constitutive equations for porous piezoelectric materials. The characteristics of waves propagating in porous piezoelectric materials were studied analytically and numerically by Vashishth and Gupta . The general theorems of elasticity were extended for the linear theory of elasticity for porous piezoelectric materials by Vashishth and Gupta . The reflection and transmission of waves from fluid-loaded porous piezoelectric half space was studied by Vashishth and Gupta .
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.  and Marechal et al.  developed experimental models to study the properties of integrated structures for ultrasonic imaging application purpose. These integrated structures contain a porous piezoelectric layer overlying a porous piezoelectric substrate.
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.
2. Formulation of the Problem
Let us consider a porous piezoelectric plate of thickness overlying a porous piezoelectric half space and underlying a fluid half space. An elastic wave from the fluid half space making an angle with axis is assumed to strike the interface which results into one reflected wave in fluid half space and five transmitted waves in porous piezoelectric half space.
The constitutive equations for transversely isotropic porous piezoelectric material are
Here, are the elastic stiffness tensor components of porous bulk material. is elastic constant corresponding to fluid phase. , , , , , , , and are piezoelectric coefficients. , , , , , and are the dielectric coefficients. and are elastic coupling coefficients. , , and are the stress tensor components, electric displacements, and electric field vector for the solid phases of the porous aggregate, respectively. The corresponding quantities with superscript * are associated with the fluid phase.
The equations of motion are , , and are the dynamical coefficients.
For a harmonic plane wave, the associated physical quantities can be represented as where is, unknown parameter and are the amplitudes associated with the harmonic wave. is the apparent phase velocity. and are the electric potentials corresponding to solid and fluid phases.
Equations (1)–(3) imply where coefficients () are listed in the appendix. () corresponds to the roots whose real part are positive while () corresponds to the roots whose real parts are negative. , , and correspond to slowness of quasi- mode, quasi- mode, and quasi- mode, and and correspond to electric potential modes. These roots of (4) are arranged such that Thus, we can write Using (1) and (6), we obtain where , are given in the appendix.
Similarly, the corresponding expressions in the porous piezoelectric half space are The normal displacement and normal stress in the fluid half space can be written as where .
Here, and are the longitudinal incident wave velocity and density in the fluid medium.
The boundary conditions at the interface are
The boundary conditions at the interface are
On solving nonhomogeneous system (12), the reflected and transmitted amplitude ratios can be obtained.
3. Numerical Study
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 BaTiO3. represents the reflected amplitude ratio. , , and represent transmitted amplitude ratios corresponding to quasi-, quasi- and quasi- waves. The frequency, thickness of layer, incidence angle and porosity are assumed 1 MHz, 0.001, 5°, and 0.2, respectively unless specified otherwise. Following Gupta and Venkatesh , the elastic, piezoelectric, dielectric coefficients and other parameters used in the study for these ceramics are given in Boxes 1 and 2.
Figure 1 shows the behaviour of amplitude ratios relative to incidence angle. The reflected amplitude ratio becomes unity after critical angle . After this critical angle, all the transmitted modes become nonpropagating. Before , all the transmitted modes propagate in the medium. Less amount of incident energy gets transmitted into the medium.
Figure 2 depicts the effects of porosity on the reflected amplitude ratio. It is observed that the reflected amplitude ration decreases with increase in porosity. Thus, as the porosity of the piezoelectric materials increases, more amount of incident energy gets transmitted into the medium.
The behaviour of amplitude ratios with frequency in the frequency range 1 MHz–11 MHz is observed in Figure 3. The behaviour of reflected amplitude ratio with frequency is oscillatory and repeats itself in small frequency pockets. The pattern of variation of reflected amplitude ratio is symmetric while that of transmitted amplitude ratios is not symmetric. The frequencies at which reflected amplitude ratio is maximum or minimum give important information in the design of transducers.
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 4. The thickness of the plate is assumed 1 mm, 2 mm, 3 mm, and 4 mm in respective figures. It is observed that change in the thickness of the plate does not affect the behaviour corresponding to quasi- wave. The number of maxima/minima increases with increase in the thickness of the plate. Thus, with increase in the thickness of the plate we can find more frequencies at which reflection is very low or very high.
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:
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