# The Effects of Piezoelectricity on the Interaction of Waves in Fluid-Loaded Poroelastic Half-Space

Authors: Vishakha Gupta; Anil K. Vashishth
Journal: Smart Materials Research
 (2014)

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## Abstract

The effects of piezoelectricity on the interaction of waves at fluid-poroelastic interface are studied. The constitutive equations and governing equations are formulated and their solution is obtained. The boundary conditions are described at fluid-solid interface. The effects of various parameters on the angle of refraction, amplitude ratios, displacements, electric potentials, and vertical component of slowness are studied numerically for a particular model. The results obtained are in agreement with the general laws of physics.


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## Body

## 1. Introduction

It is well known that piezoelectric materials produce an electric field when deformed and undergo deformation when subjected to an electric field. Piezoelectric materials are acting as very important functional components in sonar projectors, fluid monitors, pulse generators, and surface acoustic wave devices. The field of piezoelectric materials 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. Engineering applications in surface acoustic waves (SAW) devices, materials characterizations, and smart structures require the analysis of elastic wave interaction with piezoelectric materials. The problem of reflection-transmission in piezoelectric materials has been attracting unceasing attention in view of its theoretical interest and from application points of view also. The characteristics of the reflected and refracted waves at such boundaries give information about the resolution characteristics of acoustic transducers [1]. The reflection and refraction of plane waves in piezoelectric anisotropic materials have been mentioned in the texts of Auld [2], Dieulesaint and Royer [3], and Parton and Kudryavtsev [4]. Noorbehesht and Wade [5] obtained the analytical expressions for reflection and transmission coefficients of the waves at a boundary between piezoelectric materials and water and studied the effects of angle of incidence and material properties on these coefficients. Auld [6] studied the wave propagation in piezoelectric materials. The reflection of a transverse wave from the surface of a piezoelectric crystal of class 6 was studied by Alshits et al. [7]. Nayfeh and Chien [8] made an analytical study on ultrasonic wave interaction with fluid-loaded anisotropic piezoelectric substrates. The analytical expressions for the reflection and transmission coefficients were derived and propagation characteristics of the leaky wave and free wave are also identified. Every and Neiman [9] analyzed the reflection of electroacoustic waves at the boundary of a piezoelectric half-space. Zinchuk and Podlipenets [10, 11] obtained dispersion equations for acoustoelectric Rayleigh wave in a periodic layer piezoelectric half-space in a study for the 6 mm crystal class. Zaitsev and Kuznetsova [12] performed a detailed analysis of energy characteristics of bulk, surface, and plane acoustic waves in piezoelectric materials and structures. Le Clezio and Shuvalov [13] dealt theoretically and experimentally with the transmission of acoustic waves through a piezoelectric plate. Burkov et al. [14, 15] derived the basic equations describing the wave at the interface between acentric crystals and studied the effects of a uniform external electric field on the reflection and transmission of waves. The role of electromagnetic effects in the reflection and transmission of acoustic waves in piezoelectric materials was discussed by Darinskii et al. [16]. Abd-Alla and Alsheikh [17] studied the reflection-transmission of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties.Porous piezoelectric materials are widely used for applications such as low frequency hydrophones, miniature, accelerometers, vibratory sensors, and contact microphones. Different theoretical models [18–21] were developed to study the effects of porosity and connectivity on the elastic, dielectric, and piezoelectric coefficients of piezoelectric composites. Different experimental studies [22–25] were made on manufacturing of porous microstructures and the properties of such porous piezoelectric materials. Vashishth and Gupta [26, 27] derived the constitutive equations for porous piezoelectric materials and studied the wave propagation in these materials. The general theorems of elasticity were generalized for the linear theory of porous piezoelectric materials by Vashishth and Gupta [28]. Recently, Vashishth and Gupta [29] studied the effects of piezoelectricity on the poroelastic plate sandwiched between fluid half-space and porous piezoelectric half-space. The effects of angle of incidence and frequency on the amplitude ratios are studied therein.Survey of the literature related to porous piezoelectric materials reveals that a lot of experimental work has been done and is being done. Very few authors have established theoretical models for porous piezoelectric materials. A lot of work has also been done on reflection and transmission of waves in piezoelectric materials but theoretical work in porous piezoelectric materials is comparatively less reported and thus finds scopes for further investigations. In the present paper, the effects of piezoelectricity on the interaction of waves in fluid-loaded poroelastic half-space are studied. In the present model the porous piezoelectric plate is sandwiched between two fluid half-spaces. The formal expressions for the mechanical displacements, electric potentials, stresses, and electric displacements are obtained for the porous piezoelectric half-space. The effects of various parameters on the amplitude ratios, displacements, electric potentials, angle of refraction, and vertical component of slowness are also studied numerically for a particular model.


## 2. Geometry of the Problem

Let us consider a poro-piezoelectric half-space loaded with fluid half-space. Let a plane wave from the fluid half-space strike the interface, which results in one reflected wave and five transmitted waves in poro-piezoelectric half-space.The constitutive equations for porous piezoelectric materials, having 6 mm symmetry, are [26](1)[σ11σ22σ33σ23σ13σ12σ*D1D2D3D1*D2*D3*]=[c11c12c13000m1100-e3100-ζ31c12c11c13000m1100-e3100-ζ31c13c13c33000m3300-e3300-ζ33000c440000-e1500-ζ1500000c4400-e1500-ζ150000000(c11-c12)20000000m11m11m33000R00-ζ~300-e3*0000e1500ξ1100A1100000e150000ξ1100A110e31e31e33000ζ~300ξ3300A330000ζ1500A1100ξ11*00000ζ150000A1100ξ11*0ζ31ζ31ζ33000e3*00A3300ξ33*][ε11ε22ε332ε232ε132ε12ε*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*,ζ~3 are piezoelectric coefficients. ξ11,ξ33,A11,A33,ξ11*,ξ33* are the dielectric coefficients.m11,m33 are elastic coupling coefficients. σij,εij,Di, and Ei are components of the stress tensor, strain tensor, electric displacement vector, and electric field vector for the solid phase of the porous aggregate, respectively. σ*,ε*,Di*, and Ei* are the stress, strain, electric displacement components, and electric field components corresponding to the fluid phase of the aggregate.The equations of motion are(2)σij,j=ρ11u¨j+ρ12U¨j*,σ,i*=ρ12u¨j+ρ22U¨j*,Di,i=0,Di,i*=0,(i,j=1,2,3).
Here, ρ11, ρ12, and ρ22 are the dynamical coefficients. ui and Ui* are the mechanical displacement components corresponding to solid and fluid phase.Further,(3)εij=12(ui,j+uj,i),ε*=Uk,k*,Ei=-ψ,i,Ei*=-ψ,i*,
where ψ and ψ* are electric potential functions for solid and fluid phase of the porous aggregate.Let us assume that(4)uj=Ajexp⁡⁡(ıω(pxx1+pzx3-t)),Uj*=Hjexp⁡⁡(ıω(pxx1+pzx3-t)),ψ=Θexp⁡⁡(ıω(pxx1+pzx3-t)),ψ*=Ωexp⁡(ıω(pxx1+pzx3-t)),
where px and pz are the horizontal and vertical components of the slowness vector. ω is the circular frequency of waves. Aj, Hj, Θ, and Ω are the associated amplitudes.Equations (1)–(4) imply
(5)WΖ=0,Ζ=(Α1,Α3,Η1,Η3,Θ,Ω)T.
The system (5) has a nontrivial solution if
(6)det⁡(W)=0.
This leads to
(7)c1p′5+c2p′4+c3p′3+c4p′2+c5p′+c6=0,
where p′=pz2 and the coefficients ci  (i=1,2,…,6) are given in the Appendix.The five values ofp′ correspond to the slowness of five coupled waves termed as quasi P1, quasi S1, and quasi P2 and another two waves due to piezoelectric interaction, propagating in a medium. The waves termed as quasi P1, quasi S1, and quasi P2 will be denoted by qP1,qS1,qP2, respectively. The mechanical displacements and electric potentials in the poro-piezoelectric half-space can be written as
(8)(u1,u3,U1*,U3*,ψ,ψ*)=∑i=15(1,r1i,r2i,r3i,r4i,r5i)Ζ1ieıω(pxx1+pzx3-t).
Using (1) and (8), we obtain
(9)(σ31,σ33,σ*,D3,D3*)=∑iıω(g1i,g2i,g3i,g4i,g5i)Ζ1ieıω(pxx1+pzx3-t),
where
(10)r1i=Ζ2iΖ1i,r2i=Ζ3iΖ1i,r3i=Ζ4iΖ1i,r4i=Ζ5iΖ1i,r5i=Ζ6iΖ1i,g1i=c55pi′+(c55r1i+e15r4i+ζ15r5i)px,g2i=(c33r1i+m33r3i+e33r4i+ζ33r5i)pi′+(c31+m33r2i)px,g3i=(m33r1i+Rr3i+ζ~3r4i+e3*r5i)pi′+(m11+Rr2i)px,g4i=(e33r1i+ζ~3r3i-ξ33r4i-a33r5i)pi′+(e31+ζ~3r2i)px,g5i=(ζ33r1i+e3*r3i-a33r4i-ξ33*r5i)pi′+(ζ31+e3*r2i)px.(Z1i,Z2i,Z3i,Z4i,Z5i,Z6i) is the eigenvector of the system (5) corresponding to the ith value of p′.The displacements(u1f,u3f) and normal stress (σ33f) in fluid half-space are
(11)(u1f,u3f,σ33f)=∑p=1,2(1,Wpf,ıωρfupx)Upfeıω(pxx1+(-1)p+1qfx3-t),
where W1f=pxc and W2f=-pxc. ρfu is the upper fluid density. qf is the vertical slowness of wave propagating in the fluid half-space.The boundary conditions on the interfacex3=0 are
(12)D3=D3*=σ13=0,σ33=(1-f)σ33f,σ*=fσ33f,(1-f)u˙3+fU˙3*=u˙3f.
Making use of (8)–(12), a nonhomogeneous system of order 6 is obtained. Solution of this system gives the reflected and transmitted amplitude ratios.


## 3. Numerical Observations

In this section, the vertical component of slowness, reflected and transmitted amplitude ratios, mechanical displacements, and electric potentials are calculated numerically for a particular model and their variation with different parameters is observed. Following Kar-Gupta and Venkatesh [30], the elastic, piezoelectric, and dielectric constants for PZT are given in Table 1.Table 1
Elastic constants, piezoelectric constants, dielectric constants, and other parameters for PZT crystal.
Elastic constants(GPa)
Piezoelectric constants(C/m2)
Dielectric constants(nC/Vm)
Other parameters
c
11
=
148.0
,
c
12
=
76.2
,
c
13
=
74.2
,
c
33
=
131.0
,
c
44
=
25.3
,
m
11
=
8.8
,
m
33
=
5.2
,
R
=
20
.
e
15
=
9.3
,
e
31
=
-
2.324
,
e
33
=
10.99
,
ζ
15
=
2.72
,
ζ
31
=
-
0.48
,
ζ
33
=
5.32
,
e
3
*
=
-
3.6
,
ζ
~
3
=
-
7.5
.
ξ
11
=
3.984
,
ξ
33
=
2.081
,
ξ
11
*
=
11.8
,
ξ
33
*
=
13.9
,
A
11
=
12.8
,
A
33
=
15.1
.
f
=
0.2
,
c
f
(m/s)
=
1500
,
ρ
(kg/
m
3
)
=
7700
,
ρ
11
(kg/
m
3
)
=
5082
,
ρ
12
(kg/
m
3
)
=
-
1155
,
ρ
22
(kg/
m
3
)
=
4928
.Figure1 shows the variation of angle of refraction with the angle of incidence. The refraction angles corresponding to qP1, qS1, and qP2 modes are denoted by γ1, γ2, and γ3, respectively. The critical angles corresponding to qP1, qS1, and qP2 modes are θ1=13∘, θ2=31∘, and θ3=41∘. Thus qP1 wave becomes evanescent mode after critical angle θ1=13∘ and qS1 wave does not propagate into the medium afterθ2=31∘. qP2 wave also becomes evanescent mode after critical angle θ3=41∘. All the incident energy gets reflected back after angle of incidence θ3=41∘.Variation of angle of refraction(γi) corresponding to propagating modes with the angle of incidence (θ); (a) qP1 wave, (b) qS1 wave, and (c) qP2 wave.
(a)
(b)
(c)Figure2 shows the variation of absolute value of reflected and transmitted amplitude ratios with the horizontal component of slowness. AR corresponds to reflected amplitude ratio and Ai  (i=1,2,…,5) denote the amplitude ratios corresponding to transmitted modes. Major portion of incident energy gets reflected back which signifies the fact that the transmitting medium is much denser. The dips in the reflected amplitude ratio correspond to the excitation of pseudosurface mode. Thus we can find the values of incidence angles at which surface modes will be generated. This type of information is of practical importance for design of piezoelectric transducers. The variation of vertical component of slowness with the horizontal component of slowness is shown in Figure 3. The vertical component of the slowness associated with the qP1 wave is minimum while that associated with the qP2 wave is maximum, which verifies the fact that the velocity of qP1 wave is maximum while that of the qP2 wave is minimum. The value of the vertical component of the slowness helps in deciding the nomenclature of the waves propagating in the medium. It is also clear from this figure that the first of all,qP1, wave becomes nonpropagating.Figure 2
Variation of absolute value of amplitude ratios with the horizontal component of slowness.Variation of vertical component of slowness with horizontal component of slowness; (a)qP1 wave, (b) qS1 wave, and (c) qP2 wave.
(a)
(b)
(c)Figure4 displays the variation of real part of mechanical displacements and electric potentials with x3. It is observed that mechanical displacements and electric potentials decrease with x3 and become zero for large value of x3. Thus the amplitudes of the waves decrease as they travel far away from the interface and die out after a very long distance.Figure 4
Variation of real part of mechanical displacements and electric potentials withx3.


## 4. Conclusion

In the present paper the wave interaction of piezoelastic waves at fluid-porous piezoelectric solid half-space is studied. The analytical expression of the Christoffel equation is derived. The vertical slowness of the five coupled waves propagating in a medium is obtained from the solution of Christoffel equation. The analytical solution of the mechanical displacements, electric displacements, electric potentials, and stress in the porous piezoelectric half-space and fluid half-space is formulated. The behavior of angle of refraction, amplitude ratios, displacements, potentials, and vertical component of slowness relative to various parameters is studied numerically for particular model. It is found that major portion of incident energy gets reflected back in case of studied model which signifies the fact that the transmitting medium is much denser. The displacements and potentials decrease with the increase inx3 and become zero for large value of x3.



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