# Reflection and Transmission Phenomena in Poroelastic Plate Sandwiched between Fluid Half Space and Porous Piezoelectric Half Space

Authors: Vishakha Gupta; Anil K. Vashishth
Journal: Smart Materials Research
 (2013)

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## 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.


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## Body

## 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. Alvarez-Arenas 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 Alvarez-Arenas 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 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 [8] 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 [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 fluid-loaded 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 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 thicknessh overlying a porous piezoelectric half space and underlying a fluid half space. An elastic wave from the fluid half space making an angle θ with x3 axis is assumed to strike the interface x3=0 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(1)[σ1σ2σ3σ4σ5σ6(c11-c12)2σ*D1D2D3D1*D2*D3*]=[c11c12c13000m1100-e3100-ζ31c12c11c13000m1100-e3100-ζ31c13c13c33000m3300-e3300-ζ33000c440000-e1500-ζ1500000c4400-e1500-ζ15000(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)2(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)20(c11-c12)2m11m11m33000R00-ζ~300-e3*0000e1500ξ1100A1100000e150000ξ1100A110e31e31e33000ζ~300ξ3300A330000ζ1500A1100ξ11*00000ζ150000A1100ξ11*0ζ31ζ31ζ33000e3*00A3300ξ33*][ε1ε2ε32ε42ε52ε6(c11-c12)2ε*E1E2E3E1*E2*E3*].Here,cij are the elastic stiffness tensor components of porous bulk material. R is elastic constant corresponding to fluid phase. e31, e33, e15, ζ31, ζ33, ζ15, e3*, and ζ~3 are piezoelectric coefficients. ξ11, ξ33, A11, A33, ξ11*, and ξ33* are the dielectric coefficients. m11 and m33 are elastic coupling coefficients. σij, Di,  and Ei 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(2)σij,j=ρ11u¨j+ρ12U¨j,σ,i*=ρ12u¨j+ρ22U¨j,Di,i=0,Di,i*=0,ρ11, ρ12, and ρ22 are the dynamical coefficients.For a harmonic plane wave, the associated physical quantities can be represented as(3)(ui,Ui,ψ,ψ*)=(Ai,Bi,G,H)exp(ιω(1cx1+px3-t)),(i=1,3),
where p is, unknown parameter and (A1,A3,B1,B3,G,H) are the amplitudes associated with the harmonic wave. c is the apparent phase velocity. ψ and ψ* are the electric potentials corresponding to solid and fluid phases.Equations (1)–(3) imply
(4)c1p10+c2p8+c3p6+c4p4+c5p2+c6=0,
where coefficients ci (i=1,2,…,6) are listed in the appendix. pi (i=1,2,…,5) corresponds to the roots whose real part are positive while pi (i=6,…,10) corresponds to the roots whose real parts are negative. p1, p2, and p3 correspond to slowness of quasi-P1 mode, quasi-S1 mode, and quasi-P2 mode, and p4 and p5 correspond to electric potential modes. These roots of (4) are arranged such that
(5)pi+5=-pi,(i=1,2,…,5).
Thus, we can write
(6)(u1,u3,U1,U3,ψ,ψ*)=∑i=110(1,u1i,u2i,u3i,u4i,u5i)A1iexp(ιω(1cx1+pix3-t)).
Using (1) and (6), we obtain
(7)(σ31,σ33,σ*,D3,D3*)=∑i=110(d1i,d2i,d3i,d4i,d5i)A1iexp(ιω(1cx1+pix3-t)),
where dki, (k=1,2,…,5) are given in the appendix.Similarly, the corresponding expressions in the porous piezoelectric half space are(8)(u1hf,u3hf,U1hf,U3hf,ψhf,ψ*hf)=∑i=15(1,u1ihf,u2ihf,u3ihf,u4ihf,u5ihf)B1i×exp(ιω(1cx1+pihfx3-t)),(σ31hf,σ33hf,σ*hf,D3hf,D3*hf)=∑i=15(d1ihf,d2ihf,d3ihf,d4ihf,d5ihf)B1i×exp(ιω(1cx1+pihfx3-t)).
The normal displacement and normal stress in the fluid half space can be written as
(9)u3f=ρfc[U1fexp(ιω(1cx1+pfx3-t))-U2fexp(ιω(1cx1-pfx3-t))],σ33f=ιωρfc[U1fexp(ιω(1cx1+pfx3-t))-U2fexp(ιω(1cx1-pfx3-t))],
where pf=(1/c)c2/(cf)2-1.Here,cf and ρf are the longitudinal incident wave velocity and density in the fluid medium.The boundary conditions at the interfacex3=0 are
(10)σ33=(1-f)σ33f,σ*=fσ33f,(1-f)u˙3+fU˙3=u˙3f,D3=0,D3*=0,σ13=0.The boundary conditions at the interfacex3=h are
(11)u1=u1hf,u3=u3hf,U3=U3hf,ψ=ψhf,ψ*=ψ*hf,σ33=σ33hf,σ*hf=σ33hf,σ13=σ13hf,D3=D3hf,D3*=D3*hf.Making use of (6)–(9) into (10) and (11), we obtain the following nonhomogeneous system:
(12)AX=B,
where(13)A=[e1⋯e10e1hf⋯⋯⋯e5hf0u11e1⋯u1,10e10u11hfe1hf⋯⋯⋯u15hfe5hf00u31e1⋯u1,10e10u31hfe1hf⋯⋯⋯u35hfe5hf00u41e1⋯u4,10e10u41hfe1hf⋯⋯⋯u45hfe5hf00u51e1⋯u5,10e10u51hfe1hf⋯⋯⋯u55hfe5hf00d11e1⋯d1,10e10d11hfe1hf⋯⋯⋯d15hfe5hf00d21e1⋯d2,10e10d21hfe1hf⋯⋯⋯d25hfe5hf00d31e1⋯d3,10e10d31hfe1hf⋯⋯⋯d35hfe5hf00d41e1⋯d4,10e10d41hfe1hf⋯⋯⋯d45hfe5hf00d51e1⋯d5,10e10d51hfe1hf⋯⋯⋯d55hfe5hf00d21⋯d2,1000000ι(1-f)ωρfcd31⋯d3,1000000ιfωρfc(1-f)u11+fu31⋯(1-f)u1,10+fu3,1000000ρfcd41⋯d4,10000000d51d11⋯⋯d5,10d1,10000000000000],B=[0000000000ι(1-f)ωρfcιfωρfcρfc000]T,X=[A11A12⋯A1,10B11⋯B15R′]T,ei=exp(ιωpih),(i=1,2,…,10),eihf=exp(ιωpihfh),(i=1,…,5).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. AR represents the reflected amplitude ratio. A1, A2, and A3 represent transmitted amplitude ratios corresponding to quasi-P1, quasi-S1 and quasi-P2 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.Box 1:Elastic constants, piezoelectric constants, and dielectric constants of  PZT crystal.
c
11
=
148.0
e
9,  c12=76.2e9, c13=74.2e9, c33=131.0e9, c44=25.3e9, m11=8.8e9,
m
33
=
5.2
e
9, R=20e9, e15=9.3, e31=-2.324, e33=10.99, ζ15=2.72, ζ31=-0.48,
ζ
33
=
5.32, e3*=-3.6, ζ~3=-7.5, ξ11=3.984e-9, ξ33=2.081e-9, ξ11*=11.8e-9,
ξ
33
*
=
13.9
e
-
9, A11=12.8e-9, A33=15.1e-9, f=0.2, cf (m/s) = 1500,
ρ (kg/m3) = 7700, ρ11=0.66×7700, ρ12=-0.15×7700, ρ22=0.66×7700.Box 2:Elastic constants, piezoelectric constants, and dielectric constants of  BaTiO3 crystal.
c
11
=
148.0
e
9, c12=76.2e9, c13=74.2e9, c33=131.0e9, c44=25.3e9, m11=8.8e9,
m
33
=
5.2
e
9, R=20e9, e15=9.3, e31=-2.324, e33=10.99, ζ15=2.72, ζ31=-0.48,
ζ
33
=
5.32, e3*=-3.6, ζ~3=-7.5, ξ11=3.984e-9, ξ33=2.081e-9, ξ11*=11.8e-9,
ξ
33
*
=
13.9
e
-
9, A11=12.8e-9, A33=15.1e-9, ρ (kg/m3) = 5700.Figure1 shows the behaviour of amplitude ratios relative to incidence angle. The reflected amplitude ratio becomes unity after critical angle θ=42∘. After this critical angle, all the transmitted modes become nonpropagating. Before θ=13∘, all the transmitted modes propagate in the medium. Less amount of incident energy gets transmitted into the medium.Figure 1
Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency.Figure2 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.Figure 2
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 Figure3. 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.Figure 3
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 Figure4. 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-P1 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.Figure 4
Effects of thickness of plate on the behaviour of amplitude ratios with frequency.


## 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.



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