Hindawi Publishing CorporationSmart Materials ResearchVolume 2013 Article ID 767019 8 pageshttpdxdoiorg1011552013767019

Research ArticleReflection and Transmission Phenomena inPoroelastic Plate Sandwiched between Fluid Half Space andPorous Piezoelectric Half Space

Vishakha Gupta1 and Anil K Vashishth2

1 Department of Mathematics Dyal Singh College Karnal 132 001 India2Department of Mathematics Kurukshetra University Kurukshetra 136 119 India

Correspondence should be addressed to Vishakha Gupta vi shu85yahoocoin

Received 25 April 2013 Accepted 22 June 2013

Academic Editor Chris Bowen

Copyright copy 2013 V Gupta and A K VashishthThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

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

1 Introduction

The field of smart materials (piezoelectric) has advancedrapidly due to an increasing awareness about capabilities ofsuch materials the development of new materials and trans-ducer designs and increasingly stringent design and controlspecifications in aerospace aeronautics industrial automo-tive biomedical and nanosystems Piezoelectric materialsare brittle in nature which leads to the failure of devices Itis commonly found that 5 porosity exists in piezoelectricmaterials which are considered as manufacturing defectsInstead of considering it as a manufacturing defect suchmaterials can be modelled as porous piezoelectric materialsPorous piezoelectric materials are widely used in ultrasonictransducers hydrophones and pressure sensors Porousceramics are of interest for ultrasonic transducer applicationsPorosity allows to decrease the acoustic impedance thusimproving transfer of acoustic energy to water or biologicaltissues For underwater applications the figure of meritcan also be improved as compared to dense material Thesurface impedance of porous piezoelectric materials is lessas compared to dense piezoelectricmaterials Alvarez-Arenas

and De Espinosa [1] made a study related to characterizationof porous piezoelectric ceramic The experimental studyon wave propagation in composite structures was done byAlvarez-Arenas et al [2] Different authors [3 4] developedmodels related to synthesis fabrication and processing ofporous piezoelectric materials Kumar et al [5] investigatedthe effects of porosity and pore forming agents in porouspiezoceramics

The influence of piezoelectricity on the reflection-transmission phenomena in fluid-loaded piezoelectric halfspace and fluid-loaded piezoelectric plate was studied byNayfeh andChien [6 7] Darinskii andWeihnacht [8] studiedtheoretically the propagation and the generation of leakywaves in a fluid layer enclosed between the half-infinitelithium niobate substrate coated with a thin silicon oxidelayer and the half-infinite domain occupied by silicon rubberLove waves propagating in a layered structure with an elasticlayer deposited on a piezoelectric substrate were analyti-cally investigated by Liu and He [9] Vashishth and Gupta[10] derived the constitutive equations for porous piezo-electric materials The characteristics of waves propagatingin porous piezoelectric materials were studied analytically

2 Smart Materials Research

and numerically by Vashishth and Gupta [11] The generaltheorems of elasticity were extended for the linear theory ofelasticity for porous piezoelectric materials by Vashishth andGupta [12] The reflection and transmission of waves fromfluid-loaded porous piezoelectric half space was studied byVashishth and Gupta [13]

The fabrication of piezoelectric transducers for highresolution medical imagining applications requires a backingmaterial to damp the piezoelectric resonance resulting ina short time resonance that is improved resolution Thusthe choice of such a substrate must be made according toits acoustical properties For ultrasonic transducer model athick film of porous piezoelectricmaterial has to be depositedon a porous piezoelectric substrate Levassort et al [14] andMarechal et al [15] developed experimental models to studythe properties of integrated structures for ultrasonic imagingapplication purpose These integrated structures contain aporous piezoelectric layer overlying a porous piezoelectricsubstrate

Motivated the theoretical and experimental modelsdeveloped to study the properties of the integratedstructures for ultrasonic imaging applications and transducer

application a theoretical model is developed in this paper tostudy the effects of porosity thickness and incidence angleon the reflection-transmission phenomena In the presentpaper the effects of piezoelectricity frequency porosityincidence angle and thickness on the amplitude ratioscorresponding to reflected and transmitted waves in fluid-loaded porous piezoelectric plate are studied analyticallyas well as numerically for a particular model The resultsobtained in this paper can be used to improve the propertiesof porous piezoelectric materials

2 Formulation of the Problem

Let us consider a porous piezoelectric plate of thickness ℎoverlying a porous piezoelectric half space and underlyinga fluid half space An elastic wave from the fluid half spacemaking an angle 120579 with 119909

3axis is assumed to strike the

interface 1199093= 0 which results into one reflected wave in fluid

half space and five transmitted waves in porous piezoelectrichalf space

The constitutive equations for transversely isotropicporous piezoelectric material are

[[[[[[[[[[[[[[[[[[[[

[

1205901

1205902

1205903

1205904

1205905

1205906

120590lowast

1198631

1198632

1198633

119863lowast

1

119863lowast

2

119863lowast

3

]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[[

[

11988811

11988812

11988813

0 0 0 11989811

0 0 minus11989031

0 0 minus12057731

11988812

11988811

11988813

0 0 0 11989811

0 0 minus11989031

0 0 minus12057731

11988813

11988813

11988833

0 0 0 11989833

0 0 minus11989033

0 0 minus12057733

0 0 0 11988844

0 0 0 0 minus11989015

0 0 minus12057715

0

0 0 0 0 11988844

0 0 minus11989015

0 0 minus12057715

0 0

0 0 0 0 0(11988811minus 11988812)

20 0 0 0 0 0 0

1198981111989811119898330 0 0 119877 0 0 minus120577

30 0 minus119890

lowast

3

0 0 0 0 11989015

0 0 12058511

0 0 11986011

0 0

0 0 0 119890150 0 0 0 120585

110 0 119860

110

11989031

11989031

11989033

0 0 0 1205773

0 0 12058533

0 0 11986033

0 0 0 0 12057715

0 0 11986011

0 0 120585lowast

110 0

0 0 0 120577150 0 0 0 119860

110 0 120585

lowast

110

12057731

12057731

12057733

0 0 0 119890lowast

30 0 119860

330 0 120585

lowast

33

]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[

[

1205761

1205762

1205763

21205764

21205765

21205766

120576lowast

1198641

1198642

1198643

119864lowast

1

119864lowast

2

119864lowast

3

]]]]]]]]]]]]]]]]]]]]

]

(1)

Here 119888119894119895are the elastic stiffness tensor components of porous

bulk material 119877 is elastic constant corresponding to fluidphase 119890

31 11989033 11989015 12057731 12057733 12057715 119890lowast3 and 120577

3are piezoelectric

coefficients 12058511 12058533 11986011 11986033 120585lowast11 and 120585lowast

33are the dielectric

coefficients11989811and119898

33are elastic coupling coefficients 120590

119894119895

119863119894 and 119864

119894are the stress tensor components electric displace-

ments and electric field vector for the solid phases of theporous aggregate respectively The corresponding quantitieswith superscript lowast are associated with the fluid phase

The equations of motion are

120590119894119895119895= 12058811119895+ 12058812119895

120590lowast

119894= 12058812119895+ 12058822119895

119863119894119894= 0

119863lowast

119894119894= 0

(2)

12058811 12058812 and 120588

22are the dynamical coefficients

For a harmonic plane wave the associated physicalquantities can be represented as

(119906119894 119880119894 120595 120595lowast)

= (119860119894 119861119894 119866119867) exp(120580120596 (1

1198881199091+ 1199011199093minus 119905)) (119894 = 1 3)

(3)

where 119901 is unknown parameter and (1198601 1198603 1198611 1198613 119866119867)

are the amplitudes associatedwith the harmonic wave 119888 is theapparent phase velocity 120595 and 120595lowast are the electric potentialscorresponding to solid and fluid phases

Equations (1)ndash(3) imply

119888111990110+ 11988821199018+ 11988831199016+ 11988841199014+ 11988851199012+ 1198886= 0 (4)

Smart Materials Research 3

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 119891 = 02 119888

119891(ms) = 1500

120588 (kgm3) = 7700 12058811= 066 times 7700 120588

12= minus015 times 7700 120588

22= 066 times 7700

Box 1 Elastic constants piezoelectric constants and dielectric constants of PZT crystal

where coefficients 119888119894(119894 = 1 2 6) are listed in the appendix

119901119894(119894 = 1 2 5) corresponds to the roots whose real part

are positive while 119901119894(119894 = 6 10) corresponds to the roots

whose real parts are negative 1199011 1199012 and 119901

3correspond

to slowness of quasi-1198751mode quasi-119878

1mode and quasi-119875

2

mode and 1199014and 119901

5correspond to electric potential modes

These roots of (4) are arranged such that

119901119894+5= minus119901119894 (119894 = 1 2 5) (5)

Thus we can write

(1199061 1199063 1198801 1198803 120595 120595lowast)

=

10

sum

119894=1

(1 1199061119894 1199062119894 1199063119894 1199064119894 1199065119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(6)

Using (1) and (6) we obtain

(12059031 12059033 120590lowast 1198633 119863lowast

3)

=

10

sum

119894=1

(1198891119894 1198892119894 1198893119894 1198894119894 1198895119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(7)

where 119889119896119894 (119896 = 1 2 5) are given in the appendix

Similarly the corresponding expressions in the porouspiezoelectric half space are

(119906ℎ119891

1 119906ℎ119891

3 119880ℎ119891

1 119880ℎ119891

3 120595ℎ119891 120595lowastℎ119891)

=

5

sum

119894=1

(1 119906ℎ119891

1119894 119906ℎ119891

2119894 119906ℎ119891

3119894 119906ℎ119891

4119894 119906ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(120590ℎ119891

31 120590ℎ119891

33 120590lowastℎ119891 119863ℎ119891

3 119863lowastℎ119891

3)

=

5

sum

119894=1

(119889ℎ119891

1119894 119889ℎ119891

2119894 119889ℎ119891

3119894 119889ℎ119891

4119894 119889ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(8)

The normal displacement and normal stress in the fluid halfspace can be written as

119906119891

3= 120588119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

120590119891

33= 120580120596120588

119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

(9)

where 119901119891 = (1119888)radic1198882(119888119891)2 minus 1Here 119888

119891and 120588119891 are the longitudinal incident wave

velocity and density in the fluid mediumThe boundary conditions at the interface 119909

3= 0 are

12059033= (1 minus 119891) 120590

119891

33 120590

lowast= 119891120590119891

33

(1 minus 119891) 3+ 1198913= 119891

3 119863

3= 0

119863lowast

3= 0 120590

13= 0

(10)

The boundary conditions at the interface 1199093= ℎ are

1199061= 119906ℎ119891

1 119906

3= 119906ℎ119891

3 119880

3= 119880ℎ119891

3

120595 = 120595ℎ119891 120595

lowast= 120595lowastℎ119891 120590

33= 120590ℎ119891

33

120590lowastℎ119891= 120590ℎ119891

33 120590

13= 120590ℎ119891

13 119863

3= 119863ℎ119891

3

119863lowast

3= 119863lowastℎ119891

3

(11)

4 Smart Materials Research

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 120588 (kgm3) = 5700

Box 2 Elastic constants piezoelectric constants and dielectric constants of BaTiO3 crystal

Making use of (6)ndash(9) into (10) and (11) we obtain thefollowing nonhomogeneous system

119860119883 = 119861 (12)

where

119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

1198901

sdot sdot sdot 11989010

119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890

ℎ119891

50

119906111198901

sdot sdot sdot 11990611011989010

119906ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

15119890ℎ119891

500

119906311198901

sdot sdot sdot 11990611011989010

119906ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

35119890ℎ119891

500

119906411198901

sdot sdot sdot 11990641011989010

119906ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

45119890ℎ119891

500

119906511198901

sdot sdot sdot 11990651011989010

119906ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

55119890ℎ119891

500

119889111198901

sdot sdot sdot 11988911011989010

119889ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

15119890ℎ119891

500

119889211198901

sdot sdot sdot 11988921011989010

119889ℎ119891

21119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

25119890ℎ119891

500

119889311198901

sdot sdot sdot 11988931011989010

119889ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

35119890ℎ119891

500

119889411198901

sdot sdot sdot 11988941011989010

119889ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

45119890ℎ119891

500

119889511198901

sdot sdot sdot 11988951011989010

119889ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

55119890ℎ119891

500

11988921

sdot sdot sdot 119889210

0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888

11988931

sdot sdot sdot 119889310

0 0 0 0 0 120580119891120596120588119891119888

(1 minus 119891) 11990611+ 11989111990631sdot sdot sdot (1 minus 119891) 119906

110+ 119891119906310

0 0 0 0 0 120588119891119888

11988941

sdot sdot sdot 119889410

0 0 0 0 0 0

11988951

11988911

sdot sdot sdot

sdot sdot sdot

119889510

119889110

0

0

0

0

0

0

0

0

0

0

0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

119861 = [0 0 0 0 0 0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888 120580119891120596120588

119891119888 120588119891119888 0 0 0]

119879

119883 = [1198601111986012sdot sdot sdot 119860

11011986111sdot sdot sdot 119861151198771015840]119879

119890119894= exp (120580120596119901

119894ℎ) (119894 = 1 2 10)

119890ℎ119891

119894= exp (120580120596119901ℎ119891

119894ℎ) (119894 = 1 5)

(13)

On solving nonhomogeneous system (12) the reflected andtransmitted amplitude ratios can be obtained

3 Numerical Study

The reflected and transmitted amplitude ratios are calculatednumerically for a particular model The material of theporous piezoelectric plate is taken as PZT and that of porouspiezoelectric half space is taken as BaTiO

3 A119877represents

the reflected amplitude ratio 1198601 1198602 and 119860

3represent

transmitted amplitude ratios corresponding to quasi-P1

quasi-S1and quasi-P

2waves The frequency thickness of

layer incidence angle and porosity are assumed 1MHz0001 5∘ and 02 respectively unless specified otherwiseFollowing Gupta and Venkatesh [16] the elastic piezo-electric dielectric coefficients and other parameters usedin the study for these ceramics are given in Boxes 1 and2

Smart Materials Research 5

0 20 40 60 80 1000

02

04

06

08

A2

120579

0 20 40 60 80 10004

05

06

07

08

09

1A

R

120579

0 20 40 60 80 1000

005

01

015

02

A1

120579

0 20 40 60 80 1000

001

002

003

004

A3

120579

Figure 1 Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency

01 02 03 04 05 06 07 08 090760780808208408608809092094

Porosity

AR

Figure 2 Behaviour of reflected amplitude ratio with porosity atfixed angle of incidence and fixed frequency

Figure 1 shows the behaviour of amplitude ratios relativeto incidence angle The reflected amplitude ratio becomesunity after critical angle 120579 = 42∘ After this critical angleall the transmitted modes become nonpropagating Before120579 = 13

∘ all the transmitted modes propagate in the mediumLess amount of incident energy gets transmitted into themedium

Figure 2 depicts the effects of porosity on the reflectedamplitude ratio It is observed that the reflected amplituderation decreases with increase in porosityThus as the poros-ity of the piezoelectric materials increases more amount ofincident energy gets transmitted into the medium

The behaviour of amplitude ratios with frequency inthe frequency range 1MHzndash11MHz is observed in Figure 3The behaviour of reflected amplitude ratio with frequency is

oscillatory and repeats itself in small frequency pockets Thepattern of variation of reflected amplitude ratio is symmetricwhile that of transmitted amplitude ratios is not symmetricThe frequencies at which reflected amplitude ratio is maxi-mum or minimum give important information in the designof transducers

The effect of thickness of the porous piezoelectric platesandwiched between fluid half space and porous piezoelectrichalf space on the behaviour of amplitude ratios relative tofrequency is observed in Figure 4The thickness of the plate isassumed 1mm 2mm 3mm and 4mm in respective figuresIt is observed that change in the thickness of the plate doesnot affect the behaviour corresponding to quasi-P

1waveThe

number of maximaminima increases with increase in thethickness of the plate Thus with increase in the thickness ofthe plate we can find more frequencies at which reflection isvery low or very high

4 Conclusion

The reflection and transmission phenomenon in porouspiezoelectric plate sandwiched between fluid half space andporous piezoelectric half space is studied The behaviourof reflected and transmitted amplitude ratios relative tofrequency angle of incidence layer thickness and porosity isobserved numerically for a particular model The amplituderatios are found to be sensitive with respect to frequency andincidence angle The number of maximaminima increaseswith increase in the thickness of the layer With increase inporosity of the medium less amount of incident energy getsreflected back

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

Submit your manuscripts athttpwwwhindawicom

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Smart Materials Research

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 Smart Materials Research

and numerically by Vashishth and Gupta [11] The generaltheorems of elasticity were extended for the linear theory ofelasticity for porous piezoelectric materials by Vashishth andGupta [12] The reflection and transmission of waves fromfluid-loaded porous piezoelectric half space was studied byVashishth and Gupta [13]

The fabrication of piezoelectric transducers for highresolution medical imagining applications requires a backingmaterial to damp the piezoelectric resonance resulting ina short time resonance that is improved resolution Thusthe choice of such a substrate must be made according toits acoustical properties For ultrasonic transducer model athick film of porous piezoelectricmaterial has to be depositedon a porous piezoelectric substrate Levassort et al [14] andMarechal et al [15] developed experimental models to studythe properties of integrated structures for ultrasonic imagingapplication purpose These integrated structures contain aporous piezoelectric layer overlying a porous piezoelectricsubstrate

Motivated the theoretical and experimental modelsdeveloped to study the properties of the integratedstructures for ultrasonic imaging applications and transducer

application a theoretical model is developed in this paper tostudy the effects of porosity thickness and incidence angleon the reflection-transmission phenomena In the presentpaper the effects of piezoelectricity frequency porosityincidence angle and thickness on the amplitude ratioscorresponding to reflected and transmitted waves in fluid-loaded porous piezoelectric plate are studied analyticallyas well as numerically for a particular model The resultsobtained in this paper can be used to improve the propertiesof porous piezoelectric materials

2 Formulation of the Problem

Let us consider a porous piezoelectric plate of thickness ℎoverlying a porous piezoelectric half space and underlyinga fluid half space An elastic wave from the fluid half spacemaking an angle 120579 with 119909

3axis is assumed to strike the

interface 1199093= 0 which results into one reflected wave in fluid

half space and five transmitted waves in porous piezoelectrichalf space

The constitutive equations for transversely isotropicporous piezoelectric material are

[[[[[[[[[[[[[[[[[[[[

[

1205901

1205902

1205903

1205904

1205905

1205906

120590lowast

1198631

1198632

1198633

119863lowast

1

119863lowast

2

119863lowast

3

]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[[

[

11988811

11988812

11988813

0 0 0 11989811

0 0 minus11989031

0 0 minus12057731

11988812

11988811

11988813

0 0 0 11989811

0 0 minus11989031

0 0 minus12057731

11988813

11988813

11988833

0 0 0 11989833

0 0 minus11989033

0 0 minus12057733

0 0 0 11988844

0 0 0 0 minus11989015

0 0 minus12057715

0

0 0 0 0 11988844

0 0 minus11989015

0 0 minus12057715

0 0

0 0 0 0 0(11988811minus 11988812)

20 0 0 0 0 0 0

1198981111989811119898330 0 0 119877 0 0 minus120577

30 0 minus119890

lowast

3

0 0 0 0 11989015

0 0 12058511

0 0 11986011

0 0

0 0 0 119890150 0 0 0 120585

110 0 119860

110

11989031

11989031

11989033

0 0 0 1205773

0 0 12058533

0 0 11986033

0 0 0 0 12057715

0 0 11986011

0 0 120585lowast

110 0

0 0 0 120577150 0 0 0 119860

110 0 120585

lowast

110

12057731

12057731

12057733

0 0 0 119890lowast

30 0 119860

330 0 120585

lowast

33

]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[

[

1205761

1205762

1205763

21205764

21205765

21205766

120576lowast

1198641

1198642

1198643

119864lowast

1

119864lowast

2

119864lowast

3

]]]]]]]]]]]]]]]]]]]]

]

(1)

Here 119888119894119895are the elastic stiffness tensor components of porous

bulk material 119877 is elastic constant corresponding to fluidphase 119890

31 11989033 11989015 12057731 12057733 12057715 119890lowast3 and 120577

3are piezoelectric

coefficients 12058511 12058533 11986011 11986033 120585lowast11 and 120585lowast

33are the dielectric

coefficients11989811and119898

33are elastic coupling coefficients 120590

119894119895

119863119894 and 119864

119894are the stress tensor components electric displace-

ments and electric field vector for the solid phases of theporous aggregate respectively The corresponding quantitieswith superscript lowast are associated with the fluid phase

The equations of motion are

120590119894119895119895= 12058811119895+ 12058812119895

120590lowast

119894= 12058812119895+ 12058822119895

119863119894119894= 0

119863lowast

119894119894= 0

(2)

12058811 12058812 and 120588

22are the dynamical coefficients

For a harmonic plane wave the associated physicalquantities can be represented as

(119906119894 119880119894 120595 120595lowast)

= (119860119894 119861119894 119866119867) exp(120580120596 (1

1198881199091+ 1199011199093minus 119905)) (119894 = 1 3)

(3)

where 119901 is unknown parameter and (1198601 1198603 1198611 1198613 119866119867)

are the amplitudes associatedwith the harmonic wave 119888 is theapparent phase velocity 120595 and 120595lowast are the electric potentialscorresponding to solid and fluid phases

Equations (1)ndash(3) imply

119888111990110+ 11988821199018+ 11988831199016+ 11988841199014+ 11988851199012+ 1198886= 0 (4)

Smart Materials Research 3

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 119891 = 02 119888

119891(ms) = 1500

120588 (kgm3) = 7700 12058811= 066 times 7700 120588

12= minus015 times 7700 120588

22= 066 times 7700

Box 1 Elastic constants piezoelectric constants and dielectric constants of PZT crystal

where coefficients 119888119894(119894 = 1 2 6) are listed in the appendix

119901119894(119894 = 1 2 5) corresponds to the roots whose real part

are positive while 119901119894(119894 = 6 10) corresponds to the roots

whose real parts are negative 1199011 1199012 and 119901

3correspond

to slowness of quasi-1198751mode quasi-119878

1mode and quasi-119875

2

mode and 1199014and 119901

5correspond to electric potential modes

These roots of (4) are arranged such that

119901119894+5= minus119901119894 (119894 = 1 2 5) (5)

Thus we can write

(1199061 1199063 1198801 1198803 120595 120595lowast)

=

10

sum

119894=1

(1 1199061119894 1199062119894 1199063119894 1199064119894 1199065119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(6)

Using (1) and (6) we obtain

(12059031 12059033 120590lowast 1198633 119863lowast

3)

=

10

sum

119894=1

(1198891119894 1198892119894 1198893119894 1198894119894 1198895119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(7)

where 119889119896119894 (119896 = 1 2 5) are given in the appendix

Similarly the corresponding expressions in the porouspiezoelectric half space are

(119906ℎ119891

1 119906ℎ119891

3 119880ℎ119891

1 119880ℎ119891

3 120595ℎ119891 120595lowastℎ119891)

=

5

sum

119894=1

(1 119906ℎ119891

1119894 119906ℎ119891

2119894 119906ℎ119891

3119894 119906ℎ119891

4119894 119906ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(120590ℎ119891

31 120590ℎ119891

33 120590lowastℎ119891 119863ℎ119891

3 119863lowastℎ119891

3)

=

5

sum

119894=1

(119889ℎ119891

1119894 119889ℎ119891

2119894 119889ℎ119891

3119894 119889ℎ119891

4119894 119889ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(8)

The normal displacement and normal stress in the fluid halfspace can be written as

119906119891

3= 120588119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

120590119891

33= 120580120596120588

119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

(9)

where 119901119891 = (1119888)radic1198882(119888119891)2 minus 1Here 119888

119891and 120588119891 are the longitudinal incident wave

velocity and density in the fluid mediumThe boundary conditions at the interface 119909

3= 0 are

12059033= (1 minus 119891) 120590

119891

33 120590

lowast= 119891120590119891

33

(1 minus 119891) 3+ 1198913= 119891

3 119863

3= 0

119863lowast

3= 0 120590

13= 0

(10)

The boundary conditions at the interface 1199093= ℎ are

1199061= 119906ℎ119891

1 119906

3= 119906ℎ119891

3 119880

3= 119880ℎ119891

3

120595 = 120595ℎ119891 120595

lowast= 120595lowastℎ119891 120590

33= 120590ℎ119891

33

120590lowastℎ119891= 120590ℎ119891

33 120590

13= 120590ℎ119891

13 119863

3= 119863ℎ119891

3

119863lowast

3= 119863lowastℎ119891

3

(11)

4 Smart Materials Research

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 120588 (kgm3) = 5700

Box 2 Elastic constants piezoelectric constants and dielectric constants of BaTiO3 crystal

Making use of (6)ndash(9) into (10) and (11) we obtain thefollowing nonhomogeneous system

119860119883 = 119861 (12)

where

119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

1198901

sdot sdot sdot 11989010

119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890

ℎ119891

50

119906111198901

sdot sdot sdot 11990611011989010

119906ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

15119890ℎ119891

500

119906311198901

sdot sdot sdot 11990611011989010

119906ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

35119890ℎ119891

500

119906411198901

sdot sdot sdot 11990641011989010

119906ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

45119890ℎ119891

500

119906511198901

sdot sdot sdot 11990651011989010

119906ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

55119890ℎ119891

500

119889111198901

sdot sdot sdot 11988911011989010

119889ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

15119890ℎ119891

500

119889211198901

sdot sdot sdot 11988921011989010

119889ℎ119891

21119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

25119890ℎ119891

500

119889311198901

sdot sdot sdot 11988931011989010

119889ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

35119890ℎ119891

500

119889411198901

sdot sdot sdot 11988941011989010

119889ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

45119890ℎ119891

500

119889511198901

sdot sdot sdot 11988951011989010

119889ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

55119890ℎ119891

500

11988921

sdot sdot sdot 119889210

0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888

11988931

sdot sdot sdot 119889310

0 0 0 0 0 120580119891120596120588119891119888

(1 minus 119891) 11990611+ 11989111990631sdot sdot sdot (1 minus 119891) 119906

110+ 119891119906310

0 0 0 0 0 120588119891119888

11988941

sdot sdot sdot 119889410

0 0 0 0 0 0

11988951

11988911

sdot sdot sdot

sdot sdot sdot

119889510

119889110

0

0

0

0

0

0

0

0

0

0

0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

119861 = [0 0 0 0 0 0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888 120580119891120596120588

119891119888 120588119891119888 0 0 0]

119879

119883 = [1198601111986012sdot sdot sdot 119860

11011986111sdot sdot sdot 119861151198771015840]119879

119890119894= exp (120580120596119901

119894ℎ) (119894 = 1 2 10)

119890ℎ119891

119894= exp (120580120596119901ℎ119891

119894ℎ) (119894 = 1 5)

(13)

On solving nonhomogeneous system (12) the reflected andtransmitted amplitude ratios can be obtained

3 Numerical Study

The reflected and transmitted amplitude ratios are calculatednumerically for a particular model The material of theporous piezoelectric plate is taken as PZT and that of porouspiezoelectric half space is taken as BaTiO

3 A119877represents

the reflected amplitude ratio 1198601 1198602 and 119860

3represent

transmitted amplitude ratios corresponding to quasi-P1

quasi-S1and quasi-P

2waves The frequency thickness of

layer incidence angle and porosity are assumed 1MHz0001 5∘ and 02 respectively unless specified otherwiseFollowing Gupta and Venkatesh [16] the elastic piezo-electric dielectric coefficients and other parameters usedin the study for these ceramics are given in Boxes 1 and2

Smart Materials Research 5

0 20 40 60 80 1000

02

04

06

08

A2

120579

0 20 40 60 80 10004

05

06

07

08

09

1A

R

120579

0 20 40 60 80 1000

005

01

015

02

A1

120579

0 20 40 60 80 1000

001

002

003

004

A3

120579

Figure 1 Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency

01 02 03 04 05 06 07 08 090760780808208408608809092094

Porosity

AR

Figure 2 Behaviour of reflected amplitude ratio with porosity atfixed angle of incidence and fixed frequency

Figure 1 shows the behaviour of amplitude ratios relativeto incidence angle The reflected amplitude ratio becomesunity after critical angle 120579 = 42∘ After this critical angleall the transmitted modes become nonpropagating Before120579 = 13

∘ all the transmitted modes propagate in the mediumLess amount of incident energy gets transmitted into themedium

Figure 2 depicts the effects of porosity on the reflectedamplitude ratio It is observed that the reflected amplituderation decreases with increase in porosityThus as the poros-ity of the piezoelectric materials increases more amount ofincident energy gets transmitted into the medium

The behaviour of amplitude ratios with frequency inthe frequency range 1MHzndash11MHz is observed in Figure 3The behaviour of reflected amplitude ratio with frequency is

oscillatory and repeats itself in small frequency pockets Thepattern of variation of reflected amplitude ratio is symmetricwhile that of transmitted amplitude ratios is not symmetricThe frequencies at which reflected amplitude ratio is maxi-mum or minimum give important information in the designof transducers

The effect of thickness of the porous piezoelectric platesandwiched between fluid half space and porous piezoelectrichalf space on the behaviour of amplitude ratios relative tofrequency is observed in Figure 4The thickness of the plate isassumed 1mm 2mm 3mm and 4mm in respective figuresIt is observed that change in the thickness of the plate doesnot affect the behaviour corresponding to quasi-P

1waveThe

number of maximaminima increases with increase in thethickness of the plate Thus with increase in the thickness ofthe plate we can find more frequencies at which reflection isvery low or very high

4 Conclusion

The reflection and transmission phenomenon in porouspiezoelectric plate sandwiched between fluid half space andporous piezoelectric half space is studied The behaviourof reflected and transmitted amplitude ratios relative tofrequency angle of incidence layer thickness and porosity isobserved numerically for a particular model The amplituderatios are found to be sensitive with respect to frequency andincidence angle The number of maximaminima increaseswith increase in the thickness of the layer With increase inporosity of the medium less amount of incident energy getsreflected back

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

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Journal ofNanomaterials

Smart Materials Research 3

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 119891 = 02 119888

119891(ms) = 1500

120588 (kgm3) = 7700 12058811= 066 times 7700 120588

12= minus015 times 7700 120588

22= 066 times 7700

Box 1 Elastic constants piezoelectric constants and dielectric constants of PZT crystal

where coefficients 119888119894(119894 = 1 2 6) are listed in the appendix

119901119894(119894 = 1 2 5) corresponds to the roots whose real part

are positive while 119901119894(119894 = 6 10) corresponds to the roots

whose real parts are negative 1199011 1199012 and 119901

3correspond

to slowness of quasi-1198751mode quasi-119878

1mode and quasi-119875

2

mode and 1199014and 119901

5correspond to electric potential modes

These roots of (4) are arranged such that

119901119894+5= minus119901119894 (119894 = 1 2 5) (5)

Thus we can write

(1199061 1199063 1198801 1198803 120595 120595lowast)

=

10

sum

119894=1

(1 1199061119894 1199062119894 1199063119894 1199064119894 1199065119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(6)

Using (1) and (6) we obtain

(12059031 12059033 120590lowast 1198633 119863lowast

3)

=

10

sum

119894=1

(1198891119894 1198892119894 1198893119894 1198894119894 1198895119894) 1198601119894exp(120580120596 (1

1198881199091+1199011198941199093minus 119905))

(7)

where 119889119896119894 (119896 = 1 2 5) are given in the appendix

Similarly the corresponding expressions in the porouspiezoelectric half space are

(119906ℎ119891

1 119906ℎ119891

3 119880ℎ119891

1 119880ℎ119891

3 120595ℎ119891 120595lowastℎ119891)

=

5

sum

119894=1

(1 119906ℎ119891

1119894 119906ℎ119891

2119894 119906ℎ119891

3119894 119906ℎ119891

4119894 119906ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(120590ℎ119891

31 120590ℎ119891

33 120590lowastℎ119891 119863ℎ119891

3 119863lowastℎ119891

3)

=

5

sum

119894=1

(119889ℎ119891

1119894 119889ℎ119891

2119894 119889ℎ119891

3119894 119889ℎ119891

4119894 119889ℎ119891

5119894) 1198611119894

times exp(120580120596 (11198881199091+ 119901ℎ119891

1198941199093minus 119905))

(8)

The normal displacement and normal stress in the fluid halfspace can be written as

119906119891

3= 120588119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

120590119891

33= 120580120596120588

119891119888 [119880119891

1exp (120580120596 (1

1198881199091+ 1199011198911199093minus 119905))

minus119880119891

2exp (120580120596 (1

1198881199091minus 1199011198911199093minus 119905))]

(9)

where 119901119891 = (1119888)radic1198882(119888119891)2 minus 1Here 119888

119891and 120588119891 are the longitudinal incident wave

velocity and density in the fluid mediumThe boundary conditions at the interface 119909

3= 0 are

12059033= (1 minus 119891) 120590

119891

33 120590

lowast= 119891120590119891

33

(1 minus 119891) 3+ 1198913= 119891

3 119863

3= 0

119863lowast

3= 0 120590

13= 0

(10)

The boundary conditions at the interface 1199093= ℎ are

1199061= 119906ℎ119891

1 119906

3= 119906ℎ119891

3 119880

3= 119880ℎ119891

3

120595 = 120595ℎ119891 120595

lowast= 120595lowastℎ119891 120590

33= 120590ℎ119891

33

120590lowastℎ119891= 120590ℎ119891

33 120590

13= 120590ℎ119891

13 119863

3= 119863ℎ119891

3

119863lowast

3= 119863lowastℎ119891

3

(11)

4 Smart Materials Research

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 120588 (kgm3) = 5700

Box 2 Elastic constants piezoelectric constants and dielectric constants of BaTiO3 crystal

Making use of (6)ndash(9) into (10) and (11) we obtain thefollowing nonhomogeneous system

119860119883 = 119861 (12)

where

119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

1198901

sdot sdot sdot 11989010

119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890

ℎ119891

50

119906111198901

sdot sdot sdot 11990611011989010

119906ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

15119890ℎ119891

500

119906311198901

sdot sdot sdot 11990611011989010

119906ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

35119890ℎ119891

500

119906411198901

sdot sdot sdot 11990641011989010

119906ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

45119890ℎ119891

500

119906511198901

sdot sdot sdot 11990651011989010

119906ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

55119890ℎ119891

500

119889111198901

sdot sdot sdot 11988911011989010

119889ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

15119890ℎ119891

500

119889211198901

sdot sdot sdot 11988921011989010

119889ℎ119891

21119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

25119890ℎ119891

500

119889311198901

sdot sdot sdot 11988931011989010

119889ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

35119890ℎ119891

500

119889411198901

sdot sdot sdot 11988941011989010

119889ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

45119890ℎ119891

500

119889511198901

sdot sdot sdot 11988951011989010

119889ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

55119890ℎ119891

500

11988921

sdot sdot sdot 119889210

0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888

11988931

sdot sdot sdot 119889310

0 0 0 0 0 120580119891120596120588119891119888

(1 minus 119891) 11990611+ 11989111990631sdot sdot sdot (1 minus 119891) 119906

110+ 119891119906310

0 0 0 0 0 120588119891119888

11988941

sdot sdot sdot 119889410

0 0 0 0 0 0

11988951

11988911

sdot sdot sdot

sdot sdot sdot

119889510

119889110

0

0

0

0

0

0

0

0

0

0

0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

119861 = [0 0 0 0 0 0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888 120580119891120596120588

119891119888 120588119891119888 0 0 0]

119879

119883 = [1198601111986012sdot sdot sdot 119860

11011986111sdot sdot sdot 119861151198771015840]119879

119890119894= exp (120580120596119901

119894ℎ) (119894 = 1 2 10)

119890ℎ119891

119894= exp (120580120596119901ℎ119891

119894ℎ) (119894 = 1 5)

(13)

On solving nonhomogeneous system (12) the reflected andtransmitted amplitude ratios can be obtained

3 Numerical Study

The reflected and transmitted amplitude ratios are calculatednumerically for a particular model The material of theporous piezoelectric plate is taken as PZT and that of porouspiezoelectric half space is taken as BaTiO

3 A119877represents

the reflected amplitude ratio 1198601 1198602 and 119860

3represent

transmitted amplitude ratios corresponding to quasi-P1

quasi-S1and quasi-P

2waves The frequency thickness of

layer incidence angle and porosity are assumed 1MHz0001 5∘ and 02 respectively unless specified otherwiseFollowing Gupta and Venkatesh [16] the elastic piezo-electric dielectric coefficients and other parameters usedin the study for these ceramics are given in Boxes 1 and2

Smart Materials Research 5

0 20 40 60 80 1000

02

04

06

08

A2

120579

0 20 40 60 80 10004

05

06

07

08

09

1A

R

120579

0 20 40 60 80 1000

005

01

015

02

A1

120579

0 20 40 60 80 1000

001

002

003

004

A3

120579

Figure 1 Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency

01 02 03 04 05 06 07 08 090760780808208408608809092094

Porosity

AR

Figure 2 Behaviour of reflected amplitude ratio with porosity atfixed angle of incidence and fixed frequency

Figure 1 shows the behaviour of amplitude ratios relativeto incidence angle The reflected amplitude ratio becomesunity after critical angle 120579 = 42∘ After this critical angleall the transmitted modes become nonpropagating Before120579 = 13

∘ all the transmitted modes propagate in the mediumLess amount of incident energy gets transmitted into themedium

Figure 2 depicts the effects of porosity on the reflectedamplitude ratio It is observed that the reflected amplituderation decreases with increase in porosityThus as the poros-ity of the piezoelectric materials increases more amount ofincident energy gets transmitted into the medium

The behaviour of amplitude ratios with frequency inthe frequency range 1MHzndash11MHz is observed in Figure 3The behaviour of reflected amplitude ratio with frequency is

oscillatory and repeats itself in small frequency pockets Thepattern of variation of reflected amplitude ratio is symmetricwhile that of transmitted amplitude ratios is not symmetricThe frequencies at which reflected amplitude ratio is maxi-mum or minimum give important information in the designof transducers

The effect of thickness of the porous piezoelectric platesandwiched between fluid half space and porous piezoelectrichalf space on the behaviour of amplitude ratios relative tofrequency is observed in Figure 4The thickness of the plate isassumed 1mm 2mm 3mm and 4mm in respective figuresIt is observed that change in the thickness of the plate doesnot affect the behaviour corresponding to quasi-P

1waveThe

number of maximaminima increases with increase in thethickness of the plate Thus with increase in the thickness ofthe plate we can find more frequencies at which reflection isvery low or very high

4 Conclusion

The reflection and transmission phenomenon in porouspiezoelectric plate sandwiched between fluid half space andporous piezoelectric half space is studied The behaviourof reflected and transmitted amplitude ratios relative tofrequency angle of incidence layer thickness and porosity isobserved numerically for a particular model The amplituderatios are found to be sensitive with respect to frequency andincidence angle The number of maximaminima increaseswith increase in the thickness of the layer With increase inporosity of the medium less amount of incident energy getsreflected back

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Smart Materials Research

11988811= 14801198909 119888

12= 7621198909 119888

13= 7421198909 119888

33= 13101198909 119888

44= 2531198909119898

11= 881198909

11989833= 521198909 119877 = 201198909 119890

15= 93 119890

31= minus2324 119890

33= 1099 120577

15= 272 120577

31= minus048

12057733= 532 119890lowast

3= minus36 120577

3= minus75 120585

11= 3984119890 minus 9 120585

33= 2081119890 minus 9 120585lowast

11= 118119890 minus 9

120585lowast

33= 139119890 minus 9 119860

11= 128119890 minus 9 119860

33= 151119890 minus 9 120588 (kgm3) = 5700

Box 2 Elastic constants piezoelectric constants and dielectric constants of BaTiO3 crystal

Making use of (6)ndash(9) into (10) and (11) we obtain thefollowing nonhomogeneous system

119860119883 = 119861 (12)

where

119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

1198901

sdot sdot sdot 11989010

119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890

ℎ119891

50

119906111198901

sdot sdot sdot 11990611011989010

119906ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

15119890ℎ119891

500

119906311198901

sdot sdot sdot 11990611011989010

119906ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

35119890ℎ119891

500

119906411198901

sdot sdot sdot 11990641011989010

119906ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

45119890ℎ119891

500

119906511198901

sdot sdot sdot 11990651011989010

119906ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119906

ℎ119891

55119890ℎ119891

500

119889111198901

sdot sdot sdot 11988911011989010

119889ℎ119891

11119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

15119890ℎ119891

500

119889211198901

sdot sdot sdot 11988921011989010

119889ℎ119891

21119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

25119890ℎ119891

500

119889311198901

sdot sdot sdot 11988931011989010

119889ℎ119891

31119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

35119890ℎ119891

500

119889411198901

sdot sdot sdot 11988941011989010

119889ℎ119891

41119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

45119890ℎ119891

500

119889511198901

sdot sdot sdot 11988951011989010

119889ℎ119891

51119890ℎ119891

1sdot sdot sdot sdot sdot sdot sdot sdot sdot 119889

ℎ119891

55119890ℎ119891

500

11988921

sdot sdot sdot 119889210

0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888

11988931

sdot sdot sdot 119889310

0 0 0 0 0 120580119891120596120588119891119888

(1 minus 119891) 11990611+ 11989111990631sdot sdot sdot (1 minus 119891) 119906

110+ 119891119906310

0 0 0 0 0 120588119891119888

11988941

sdot sdot sdot 119889410

0 0 0 0 0 0

11988951

11988911

sdot sdot sdot

sdot sdot sdot

119889510

119889110

0

0

0

0

0

0

0

0

0

0

0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

119861 = [0 0 0 0 0 0 0 0 0 0 120580 (1 minus 119891)120596120588119891119888 120580119891120596120588

119891119888 120588119891119888 0 0 0]

119879

119883 = [1198601111986012sdot sdot sdot 119860

11011986111sdot sdot sdot 119861151198771015840]119879

119890119894= exp (120580120596119901

119894ℎ) (119894 = 1 2 10)

119890ℎ119891

119894= exp (120580120596119901ℎ119891

119894ℎ) (119894 = 1 5)

(13)

On solving nonhomogeneous system (12) the reflected andtransmitted amplitude ratios can be obtained

3 Numerical Study

The reflected and transmitted amplitude ratios are calculatednumerically for a particular model The material of theporous piezoelectric plate is taken as PZT and that of porouspiezoelectric half space is taken as BaTiO

3 A119877represents

the reflected amplitude ratio 1198601 1198602 and 119860

3represent

transmitted amplitude ratios corresponding to quasi-P1

quasi-S1and quasi-P

2waves The frequency thickness of

layer incidence angle and porosity are assumed 1MHz0001 5∘ and 02 respectively unless specified otherwiseFollowing Gupta and Venkatesh [16] the elastic piezo-electric dielectric coefficients and other parameters usedin the study for these ceramics are given in Boxes 1 and2

Smart Materials Research 5

0 20 40 60 80 1000

02

04

06

08

A2

120579

0 20 40 60 80 10004

05

06

07

08

09

1A

R

120579

0 20 40 60 80 1000

005

01

015

02

A1

120579

0 20 40 60 80 1000

001

002

003

004

A3

120579

Figure 1 Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency

01 02 03 04 05 06 07 08 090760780808208408608809092094

Porosity

AR

Figure 2 Behaviour of reflected amplitude ratio with porosity atfixed angle of incidence and fixed frequency

Figure 1 shows the behaviour of amplitude ratios relativeto incidence angle The reflected amplitude ratio becomesunity after critical angle 120579 = 42∘ After this critical angleall the transmitted modes become nonpropagating Before120579 = 13

∘ all the transmitted modes propagate in the mediumLess amount of incident energy gets transmitted into themedium

Figure 2 depicts the effects of porosity on the reflectedamplitude ratio It is observed that the reflected amplituderation decreases with increase in porosityThus as the poros-ity of the piezoelectric materials increases more amount ofincident energy gets transmitted into the medium

The behaviour of amplitude ratios with frequency inthe frequency range 1MHzndash11MHz is observed in Figure 3The behaviour of reflected amplitude ratio with frequency is

oscillatory and repeats itself in small frequency pockets Thepattern of variation of reflected amplitude ratio is symmetricwhile that of transmitted amplitude ratios is not symmetricThe frequencies at which reflected amplitude ratio is maxi-mum or minimum give important information in the designof transducers

The effect of thickness of the porous piezoelectric platesandwiched between fluid half space and porous piezoelectrichalf space on the behaviour of amplitude ratios relative tofrequency is observed in Figure 4The thickness of the plate isassumed 1mm 2mm 3mm and 4mm in respective figuresIt is observed that change in the thickness of the plate doesnot affect the behaviour corresponding to quasi-P

1waveThe

number of maximaminima increases with increase in thethickness of the plate Thus with increase in the thickness ofthe plate we can find more frequencies at which reflection isvery low or very high

4 Conclusion

The reflection and transmission phenomenon in porouspiezoelectric plate sandwiched between fluid half space andporous piezoelectric half space is studied The behaviourof reflected and transmitted amplitude ratios relative tofrequency angle of incidence layer thickness and porosity isobserved numerically for a particular model The amplituderatios are found to be sensitive with respect to frequency andincidence angle The number of maximaminima increaseswith increase in the thickness of the layer With increase inporosity of the medium less amount of incident energy getsreflected back

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

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Nano

materials

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Journal ofNanomaterials

Smart Materials Research 5

0 20 40 60 80 1000

02

04

06

08

A2

120579

0 20 40 60 80 10004

05

06

07

08

09

1A

R

120579

0 20 40 60 80 1000

005

01

015

02

A1

120579

0 20 40 60 80 1000

001

002

003

004

A3

120579

Figure 1 Behaviour of reflected and transmitted amplitude ratios with angle of incidence at fixed frequency

01 02 03 04 05 06 07 08 090760780808208408608809092094

Porosity

AR

Figure 2 Behaviour of reflected amplitude ratio with porosity atfixed angle of incidence and fixed frequency

Figure 1 shows the behaviour of amplitude ratios relativeto incidence angle The reflected amplitude ratio becomesunity after critical angle 120579 = 42∘ After this critical angleall the transmitted modes become nonpropagating Before120579 = 13

∘ all the transmitted modes propagate in the mediumLess amount of incident energy gets transmitted into themedium

Figure 2 depicts the effects of porosity on the reflectedamplitude ratio It is observed that the reflected amplituderation decreases with increase in porosityThus as the poros-ity of the piezoelectric materials increases more amount ofincident energy gets transmitted into the medium

The behaviour of amplitude ratios with frequency inthe frequency range 1MHzndash11MHz is observed in Figure 3The behaviour of reflected amplitude ratio with frequency is

oscillatory and repeats itself in small frequency pockets Thepattern of variation of reflected amplitude ratio is symmetricwhile that of transmitted amplitude ratios is not symmetricThe frequencies at which reflected amplitude ratio is maxi-mum or minimum give important information in the designof transducers

The effect of thickness of the porous piezoelectric platesandwiched between fluid half space and porous piezoelectrichalf space on the behaviour of amplitude ratios relative tofrequency is observed in Figure 4The thickness of the plate isassumed 1mm 2mm 3mm and 4mm in respective figuresIt is observed that change in the thickness of the plate doesnot affect the behaviour corresponding to quasi-P

1waveThe

number of maximaminima increases with increase in thethickness of the plate Thus with increase in the thickness ofthe plate we can find more frequencies at which reflection isvery low or very high

4 Conclusion

The reflection and transmission phenomenon in porouspiezoelectric plate sandwiched between fluid half space andporous piezoelectric half space is studied The behaviourof reflected and transmitted amplitude ratios relative tofrequency angle of incidence layer thickness and porosity isobserved numerically for a particular model The amplituderatios are found to be sensitive with respect to frequency andincidence angle The number of maximaminima increaseswith increase in the thickness of the layer With increase inporosity of the medium less amount of incident energy getsreflected back

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

Submit your manuscripts athttpwwwhindawicom

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CeramicsJournal of

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NanoparticlesJournal of

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Journal of

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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BioMed Research International

MaterialsJournal of

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Smart Materials Research

0 2 4 6 8 10 12Frequency

minus092

minus091

minus09

minus089

minus088

minus087

AR

times106

0

05

1

minus05

A2

0 2 4 6 8 10 12Frequency times10

6

0

02

04

06

minus02

A1

0 2 4 6 8 10 12Frequency times10

6

0

A3

minus006

minus005

minus004

minus003

minus002

minus001

0 2 4 6 8 10 12Frequency times10

6

Figure 3 Behaviour of reflected and transmitted amplitude ratios with frequency at fixed angle of incidence

Appendix

We have the following

1198881= 12057311199101minus 120573511991010+ 120573911991017minus 1205731311991026

1198882= 12057321199101+ 12057311199102minus 120573611991010minus 120573511991011+ 1205731011991017

+ 120573911991018minus 1205731411991026minus 1205731311991027

1198883= 12057331199101+ 12057321199102+ 12057311199103minus 120573711991010minus 120573611991011+ 1205731111991017

+ 1205731011991018+ 120573911991019minus 1205731511991026minus 1205731411991027minus 1205731311991028

1198884= 12057341199101+ 12057331199102+ 12057321199103minus 120573811991010minus 120573711991011+ 1205731211991017

+ 1205731111991018+ 1205731011991019minus 1205731611991026minus 1205731511991027minus 1205731411991028

1198885= 12057341199102+ 12057331199103+ 1205731211991018+ 1205731111991019

minus 120573811991011minus 1205731611991027minus 1205731511991028

1198886= 12057341199103+ 1205731211991019minus 1205731611991028

1205731= 119902111991012minus 119902411991014+ 119902711991015

1205732= 119902211991012+ 119902111991013minus 119902511991014+ 119902811991015+ 119902711991016

1205733= 119902311991012+ 119902211991013minus 119902611991014+ 119902911991015+ 119902811991016

1205734= 119902311991013+ 119902911991016 120573

5= 11990211199104minus 11990241199106+ 11990271199108

1205736= 11990221199104+ 11990211199105minus 11990251199106minus 11990241199107+ 11990281199108+ 11990271199109

1205737= 11990231199104+ 11990221199105minus 11990261199106minus 11990251199107+ 11990291199108+ 11990281199109

1205738= 11990231199105minus 11990261199107+ 11990291199109 120573

9= 11990311199104minus 11990341199106+ 11990371199108

12057310= 11990321199104+ 11990311199105minus 11990351199106minus 11990341199107+ 11990381199108+ 11990371199109

12057311= 11990331199104+ 11990321199105minus 11990361199106minus 11990351199107+ 11990391199108+ 11990381199109

12057312= 11990331199105minus 11990361199107+ 11990391199109 120573

13= 11990411199104minus 11990441199106+ 11990471199108

12057314= 11990421199104+ 11990411199105minus 11990451199106minus 11990441199107+ 11990481199108+ 11990471199109

12057315= 11990431199104+ 11990421199105minus 11990461199106minus 11990451199107+ 11990491199108+ 11990481199109

12057316= 11990431199105minus 11990461199107+ 11990491199109

1199021= 1199102211991033minus 1199102411991031

1199022= 1199102211991034+ 1199102311991033minus 1199102511991031minus 1199102411991032

1199023= 1199102311991034minus 1199102511991032 119902

4= 1199102011991033minus 1199102411991029

1199025= 1199102011991034+ 1199102111991033minus 1199102411991030minus 1199102511991029

1199026= 1199102111991034minus 1199102511991030 119902

7= 1199102011991031minus 1199102211991029

1199028= 1199102011991032+ 1199102111991031minus 1199102311991029minus 1199102211991030

1199029= 1199102111991032minus 1199102311991030

1199031= 1199101411991033minus 1199101511991031 119903

2=1199101411991034minus 1199101611991031minus1199101511991032

1199033= minus1199101611991032 119903

4= 1199101211991033minus 1199101511991029

1199035= 1199101211991034+ 1199101311991033minus 1199101611991029minus 1199101511991030

1199036= 1199101311991034minus 1199101611991030 119903

7= 1199101211991031minus 1199101411991029

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Smart Materials Research 7

minus01

minus005

0

minus01

minus005

0

Frequency Frequency Frequency Frequency

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

minus095

minus09

minus085

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0 5 10times10

6

0

05

minus05

0

05

minus05

0

05

minus05

0

05

minus05

0

1

minus1

0

1

minus1

0

1

minus1

0

1

minus1

0

01

minus01

0

01

minus01

A2AR A1 A3

Figure 4 Effects of thickness of plate on the behaviour of amplitude ratios with frequency

1199038= 1199101211991032minus 1199101411991030+ 1199101311991031 119903

9= 1199101311991032

1199041= 1199101411991024minus 1199101511991022 119904

2= 1199101411991025minus 1199101611991022minus1199101511991023

1199043= minus1199101611991023 119904

4= 1199101211991024minus 1199101511991020

1199045= 1199101211991025+ 1199101311991024minus 1199101611991020minus 1199101511991021

1199046= 1199101311991025minus 1199101611991021 119904

7= 1199101211991022minus 1199101411991020

1199048= 1199101211991023minus 1199101411991021+ 1199101311991022 119904

9= 1199101311991023

1199101= 119890331199091+ 120577331199096

1199102=(119890151199091+ 120577151199096)

1198882+(11988844+ 11988813)

119888+ 119890331199092+ 120577331199097

1199103=(119890151199092+ 120577151199097)

1198882 119910

4= 119890331199093+ 120577331199098+ 11988833

1199105=(119890151199093+ 120577151199098+ 11988844)

1198882minus12058811

33 119910

6=119890331199094+120577331199099+11989833

119888

1199107=(119890151199094+ 120577151199099)

1198882 119910

8= 119890331199095+ 1205773311990910+ 11989833

1199109=(119890151199095+ 1205771511990910)

1198882minus 12058812

33 119910

10= 12057731199091+ 119890lowast

31199096

11991011=11989811

119888+ 12057731199092+ 119890lowast

31199097 119910

12= 11989833+ 12057731199093+ 119890lowast

31199098

11991013= minus12058812

33 119910

14=119877

119888+ 12057731199094+ 119890lowast

31199099

11991015= 119877 + 120577

31199095+ 119890lowast

311990910 119910

16= minus12058822

33

11991017= minus119909112058533minus 119909611986033

11991018=(minus119909112058511minus 119909611986011)

1198882+(11989015+ 11989031)

119888minus 120585331199092minus 119860331199097

11991019= minus(119909212058511+ 119909711986011)

1198882 119910

20= 11989033minus 119909312058533minus 119909811986033

11991021=(11989015minus 119909312058511minus 119909811986011)

1198882 119910

22=1205773

119888minus 119860331199099minus 120585331199094

11991023= minus(119909412058511+ 119909911986011)

1198882 119910

24= minus1199091011986033minus 119909512058533+ 1205773

11991025= minus(119909512058511+ 1199091011986011)

1198882 119910

26= minus119909111986033minus 1199096120585lowast

33

11991027=(minus119909111986011minus 1199096120585lowast

11)

1198882+(12057715+ 12057731)

119888minus 119909211986033minus 1199097120585lowast

33

11991028= minus(119909211986011+ 1199097120585lowast

11)

1198882 119910

29= 12057733minus 119909311986033minus 1199098120585lowast

33

11991030=(12057715minus 119909311986011minus 1199098120585lowast

11)

1198882 119910

31=119890lowast

3

119888minus 120585lowast

331199099minus 119860331199094

11991032= minus(119909411986011+ 1199099120585lowast

11)

1198882 119910

33= minus11990910120585lowast

33minus 119909511986033+ 119890lowast

3

11991034= minus(119909511986011+ 11990910120585lowast

11)

1198882

1199091= minus11988844119889119888

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Smart Materials Research

1199092= minus (

11988811

1198882minus 12058811

11)119889119888 minus

(119898111198882minus 12058812

11) 119888

1205773

+(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

1205773

1199093=(11989833(11989015+ 11989031) 119889 minus 119898

33)

1205773

minus (11988813+ 11988844) 119889

1199094= minus (

11989811

1198882minus 12058812

11)119889119888 minus

(1198771198882minus 12058822

11) 119888

1205773

+(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

1205773

1199095=(119877 (11989015+ 11989031) 119889 minus 119877)

1205773

minus 11989811119889 119909

6=119888441205773119889119888

119890lowast3

1199097=(119888111198882minus 12058811

11) 1198891198881205773

119890lowast3

minus(119898111198882minus 12058812

11) (11989015+ 11989031) 119889119888

119890lowast3

1199098=(11988813+ 11988844) 1198891205773

119890lowast3

minus11989833(11989015+ 11989031) 119889

119890lowast3

1199099=(119898111198882minus 12058812

11) 1198891198881205773

119890lowast3

minus(1198771198882minus 12058822

11) (11989015+ 11989031) 119889119888

119890lowast3

11990910=119898111198891205773

119890lowast3

minus119877 (11989015+ 11989031) 119889

119890lowast3

119889 =119890lowast

3

(119890lowast3(11989015+ 11989031) minus 1205773(12057715+ 12057731))

1198891119894= 11988855119901119894+(119888551199031119894+ 119890151199034119894+ 120577151199035119894)

119888

1198892119894= (119888331199031119894+ 119898331199033119894+ 119890331199034119894+ 120577331199035119894) 119901119894+(11988831+ 119898331199032119894)

119888

1198893119894= (119898331199031119894+ 1198771199033119894+ 12057731199034119894+ 119890lowast

31199035119894) 119901119894+(11989811+ 1198771199032119894)

119888

1198894119894= (119890331199031119894+ 12057731199033119894minus 120585331199034119894minus 119886331199035119894) 119901119894+(11989031+ 12057731199032119894)

119888

1198895119894= (120577331199031119894+ 119890lowast

31199033119894minus 119886331199034119894minus 120585lowast

331199035119894) 119901119894+(12057731+ 119890lowast

31199032119894)

119888

(A1)

References

[1] T E G Alvarez-Arenas and F M De Espinosa ldquoCharacteriza-tion of porous piezoelectric ceramics the length expander caserdquoJournal of the Acoustical Society of America vol 102 no 6 pp3507ndash3515 1997

[2] T E G Alvarez-Arenas A J Mulholland G Hayward and JGomatam ldquoWave propagation in 0-33-3 connectivity compos-ites with complexmicrostructurerdquoUltrasonics vol 38 no 9 pp897ndash907 2000

[3] E Roncari C Galassi F Craciun C Capiani and APiancastelli ldquoA microstructural study of porous piezoelectricceramics obtained by different methodsrdquo Journal of the Euro-pean Ceramic Society vol 21 no 3 pp 409ndash417 2001

[4] C R Bowen A Perry A C F Lewis and H Kara ldquoProcessingand properties of porous piezoelectric materials with highhydrostatic figures of meritrdquo Journal of the European CeramicSociety vol 24 no 2 pp 541ndash545 2004

[5] B P Kumar H H Kumar and D K Kharat ldquoStudy on pore-forming agents in processing of porous piezoceramicsrdquo Journalof Materials Science vol 16 no 10 pp 681ndash686 2005

[6] A H Nayfeh andH-T Chien ldquoThe influence of piezoelectricityon free and reflected waves from fluid- loaded anisotropicplatesrdquo Journal of the Acoustical Society of America vol 91 no3 pp 1250ndash1261 1992

[7] A H Nayfeh and H-T Chien ldquoWave propagation interactionwith free and fluid-loaded piezoelectric substratesrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3126ndash31351992

[8] A N Darinskii and M Weihnacht ldquoAcoustic waves guided bya fluid layer on a piezoelectric substraterdquo Journal of AppliedPhysics vol 104 no 5 Article ID 054904 9 pages 2008

[9] J Liu and S He ldquoProperties of Love waves in layered piezoelec-tric structuresrdquo International Journal of Solids and Structuresvol 47 no 2 pp 169ndash174 2010

[10] A K Vashishth and V Gupta ldquoVibrations of porous piezoelec-tric ceramic platesrdquo Journal of Sound and Vibration vol 325 no4-5 pp 781ndash797 2009

[11] A K Vashishth and V Gupta ldquoWave propagation in trans-versely isotropic porous piezoelectric materialsrdquo InternationalJournal of Solids and Structures vol 46 no 20 pp 3620ndash36322009

[12] A KVashishth andVGupta ldquoUniqueness theorem theoremofreciprocity and eigenvalue problems in linear theory of porouspiezoelectricityrdquo Applied Mathematics and Mechanics vol 32no 4 pp 479ndash494 2011

[13] A K Vashishth and V Gupta ldquoReflection and transmission ofplane waves from a fluid-porous piezoelectric solid interfacerdquoJournal of the Acoustical Society of America vol 129 no 6 pp3690ndash3701 2011

[14] F Levassort J Holc E Ringgaard T Bove M Kosec and MLethiecq ldquoFabrication modelling and use of porous ceramicsfor ultrasonic transducer applicationsrdquo Journal of Electroceram-ics vol 19 no 1 pp 125ndash137 2007

[15] P Marechal F Levassort J Holc et al ldquoElectromechanicalproperties of piezoelectric integrated structures on poroussubstratesrdquo Ferroelectrics vol 371 no 1 pp 89ndash97 2008

[16] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of porous piezoelectric materialsrdquo Acta Materialiavol 54 no 15 pp 4063ndash4078 2006

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials