Research Article  Open Access
Reflection and Transmission Phenomena in Poroelastic Plate Sandwiched between Fluid Half Space and Porous Piezoelectric Half Space
Abstract
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.
1. Introduction
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. AlvarezArenas and De Espinosa [1] made a study related to characterization of porous piezoelectric ceramic. The experimental study on wave propagation in composite structures was done by AlvarezArenas et al. [2]. Different authors [3, 4] developed models related to synthesis, fabrication, and processing of porous piezoelectric materials. Kumar et al. [5] investigated the effects of porosity and pore forming agents in porous piezoceramics.
The influence of piezoelectricity on the reflectiontransmission phenomena in fluidloaded piezoelectric half space and fluidloaded piezoelectric plate was studied by Nayfeh and Chien [6, 7]. Darinskii and Weihnacht [8] studied theoretically the propagation and the generation of leaky waves in a fluid layer enclosed between the halfinfinite lithium niobate substrate coated with a thin silicon oxide layer and the halfinfinite 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 [9]. Vashishth and Gupta [10] 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 [11]. The general theorems of elasticity were extended for the linear theory of elasticity for porous piezoelectric materials by Vashishth and Gupta [12]. The reflection and transmission of waves from fluidloaded porous piezoelectric half space was studied by Vashishth and Gupta [13].
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. [14] and Marechal et al. [15] 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 reflectiontransmission 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 fluidloaded 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
Making use of (6)–(9) into (10) and (11), we obtain the following nonhomogeneous system: where
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 BaTiO_{3}. 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 [16], 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.
4. Conclusion
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.
Appendix
We have the following:
References
 T. E. G. AlvarezArenas and F. M. De Espinosa, “Characterization of porous piezoelectric ceramics: the length expander case,” Journal of the Acoustical Society of America, vol. 102, no. 6, pp. 3507–3515, 1997. View at: Publisher Site  Google Scholar
 T. E. G. AlvarezArenas, A. J. Mulholland, G. Hayward, and J. Gomatam, “Wave propagation in 03/33 connectivity composites with complex microstructure,” Ultrasonics, vol. 38, no. 9, pp. 897–907, 2000. View at: Publisher Site  Google Scholar
 E. Roncari, C. Galassi, F. Craciun, C. Capiani, and A. Piancastelli, “A microstructural study of porous piezoelectric ceramics obtained by different methods,” Journal of the European Ceramic Society, vol. 21, no. 3, pp. 409–417, 2001. View at: Publisher Site  Google Scholar
 C. R. Bowen, A. Perry, A. C. F. Lewis, and H. Kara, “Processing and properties of porous piezoelectric materials with high hydrostatic figures of merit,” Journal of the European Ceramic Society, vol. 24, no. 2, pp. 541–545, 2004. View at: Publisher Site  Google Scholar
 B. P. Kumar, H. H. Kumar, and D. K. Kharat, “Study on poreforming agents in processing of porous piezoceramics,” Journal of Materials Science, vol. 16, no. 10, pp. 681–686, 2005. View at: Publisher Site  Google Scholar
 A. H. Nayfeh and H.T. Chien, “The influence of piezoelectricity on free and reflected waves from fluid loaded anisotropic plates,” Journal of the Acoustical Society of America, vol. 91, no. 3, pp. 1250–1261, 1992. View at: Publisher Site  Google Scholar
 A. H. Nayfeh and H.T. Chien, “Wave propagation interaction with free and fluidloaded piezoelectric substrates,” Journal of the Acoustical Society of America, vol. 91, no. 6, pp. 3126–3135, 1992. View at: Google Scholar
 A. N. Darinskii and M. Weihnacht, “Acoustic waves guided by a fluid layer on a piezoelectric substrate,” Journal of Applied Physics, vol. 104, no. 5, Article ID 054904, 9 pages, 2008. View at: Publisher Site  Google Scholar
 J. Liu and S. He, “Properties of Love waves in layered piezoelectric structures,” International Journal of Solids and Structures, vol. 47, no. 2, pp. 169–174, 2010. View at: Publisher Site  Google Scholar
 A. K. Vashishth and V. Gupta, “Vibrations of porous piezoelectric ceramic plates,” Journal of Sound and Vibration, vol. 325, no. 45, pp. 781–797, 2009. View at: Publisher Site  Google Scholar
 A. K. Vashishth and V. Gupta, “Wave propagation in transversely isotropic porous piezoelectric materials,” International Journal of Solids and Structures, vol. 46, no. 20, pp. 3620–3632, 2009. View at: Publisher Site  Google Scholar
 A. K. Vashishth and V. Gupta, “Uniqueness theorem, theorem of reciprocity, and eigenvalue problems in linear theory of porous piezoelectricity,” Applied Mathematics and Mechanics, vol. 32, no. 4, pp. 479–494, 2011. View at: Publisher Site  Google Scholar
 A. K. Vashishth and V. Gupta, “Reflection and transmission of plane waves from a fluidporous piezoelectric solid interface,” Journal of the Acoustical Society of America, vol. 129, no. 6, pp. 3690–3701, 2011. View at: Publisher Site  Google Scholar
 F. Levassort, J. Holc, E. Ringgaard, T. Bove, M. Kosec, and M. Lethiecq, “Fabrication, modelling and use of porous ceramics for ultrasonic transducer applications,” Journal of Electroceramics, vol. 19, no. 1, pp. 125–137, 2007. View at: Publisher Site  Google Scholar
 P. Marechal, F. Levassort, J. Holc et al., “Electromechanical properties of piezoelectric integrated structures on porous substrates,” Ferroelectrics, vol. 371, no. 1, pp. 89–97, 2008. View at: Publisher Site  Google Scholar
 R. KarGupta and T. A. Venkatesh, “Electromechanical response of porous piezoelectric materials,” Acta Materialia, vol. 54, no. 15, pp. 4063–4078, 2006. View at: Publisher Site  Google Scholar
Copyright
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.