Indian Journal of Engineering & Materials Sciences Vol. 10, October 2003, pp. 371-380

Propagation of Lamb waves in transversely isotropic piezoelectric elastic plate

J N Sharma & Mohinder Pal

Department of Applied Sciences, National Institute of Technology, Hamjrpur 177 OOS , India

Received 20 May 2002; revised received 28 lilly 2003

The propagation of Lamb waves in a homogeneous, transversely isotropic, piezoelectric plate subjected to charge free and electrically shorted, stress free boundary conditions is investigated. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and anti symmetric wave mode propagation in completely separate terms are derived. It is shown that motion of purely transverse (SH) mode gets decoupled from rest of the motion and remains unaffected due to piezoelectric effects. At short wavelength limits the secular equation for symmetric and skew-symmetric waves in both charge free and electrically shorted, stress free plate reduce to Rayleigh surface wave frequency equations because a finite plate in such situation behave like a semi-infinite medium. The amplitudes of displacement and electrical potential have also been computed in case of symmetric and skew-symmetric motions of the plate. Finally, numerical solution is carried out for cadmium-selenide (6 mm class) material. The dispersion curves and amplitudes of displacements and electrical potential for symmetric and antisymmetric wave modes are presented graphically in order to illustrate and compare the analytical results. The theory and numerical computations is found to be in close agreement. The various wave characteristics are found to be more stable and realistic in the presence of piezoelectric effects than in the absence of such a effect thereby making such materials more viable for practical importance and use.

The piezoelectric effect in piezoelectric materials induces electric field due to the application of strain. The effect is used to control the deflection and stresses in intelligent plate structures. The piezoelectric material response entails an interaction of two major fields namely mechanical and electric in the macro physical world. Further they can be used to detect the responses of a structure by measuring the electric charge, sensing or to reduce the excessive response by applying additional electric forces, actuating. If actuating and sensing can be integrated smartly, a so-called intelligent structure can be designed. The piezoelectric materials are often used as resonators whose frequency needs to be precisely controlled. In recent years, piezoelectric materials have been integrated with structural systems to form a class of "smart stuctures". The piezoelectric materials are capable of altering the structure's response through sensing, actuation and control. By integrating surface-bonded and embedded actuators into structural systems, desired localized strains may be induced by applying the appropriate voltage to the actuators. In order to successfully incorporate piezoelectric actuators into structures, the mechanical interaction between the actuators and the base structure must be fully understood. Mechanical

*For correspondence [Email:jns@reham.ernet.in]

models were developed by Crawley and de Luis I, 1m and Atluri2

, Crawley and Anderson3 for piezoelectric patches mounted to top and/or bottom surface of a beam. Lee4 developed a theory for laminated plates with distributed piezoelectric layers based on the classical lamination theory . A coupled first-order shear deformation theory for multilayered piezoelectric plates was presented by Huang and Wu5

.

Mitchell and Reddy's6 coupled higher-order theory is based on an equivalent single-layer theory for the mechanical displacement and layer-wise discretization of the electric potential .

The propagation of Rayleigh-Lamb waves in homogeneous isotropic elastic plates has been

discussed in detail by Graff7. Recently, resurgent interest in Lamb waves was partially initiated by its application of multisensorss- IO

• Schoch" derived the dispersion relation for leaky Lamb waves for an isotropic plate and an inviscid liquid. Incidentally, the dispersion equations also have an interface wave solution whose velocity is slightly less than the bulk sound velocity in the liquid and most of energy is in the liquid. Watkins et al. 12 calculated the attenuation of Lamb waves in the presence of an inviscid liquid using an acoustic impedance method. Wu and Zhu 13

studied the propagation of Lamb waves in a plate bordered with inviscid liquid layers on both sides. The

372 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

dispersion equations of this case were derived and solved numerically. Zhu and WU l4 derived the dispersion equations of Lamb waves of a plate bordered with viscous liquid layer or half space viscous liquid on both sides. Numerical solutions of the dispersion equations related to sensing applications are obtained. Sharma et ai. 15

•l6 studied

the propagation of thermoelastic Rayleigh-Lamb waves in homogeneous isotropic plates subjected to stress free insulated, stress free isothermal, rigidly fixed insulated and rigidly fixed isothermal boundary conditions in the context of Conventional-Coupled (CT), Lord-Schulman (LS), Green-Lindsay (GL) and Green-Nagdh i (GN) theories of thermoelasticity . The secular equations for the symmetric and anti­symmetric wave modes in the plate have been derived in the compact form and solved numerically. The propagation of plane harmonic waves in piezothermoelastic materials has been studied by Sharma and Kumar17 .

In the present paper, an attempt has been made to investigate the propagation of Lamb waves in piezoelectric, transversely isotropic elastic plate which is subjected to (i) stress free and electrically shorted and (ii) stress free and charge free boundary conditions. The Rayleigh-Lamb type dispersion relation has been obtained in both cases, for symmetric and skew-symmetric modes of wave propagation in the plate. The displacement components and electric potential are also computed. The analytical results have been verified and computed numerically for cadmium-selenide (Cd-Se) material plate, which are found to be in close agreement with the analytical results. The analysis will be useful in the design and construction of Lamb wave sensors and surface acoustic wave (SAW) filter devices , which are appli ed extensively to many technical fields, such as television, telecommuni­cation, non-destructive evaluation, radar and health monitoring systems.

Formulation of the Problem and its Solution We consider an infinite homogeneous transversely

isotropic piezoelectric elastic plate of thickness 2d.

We take otigon of the co-ordinate system (Xl' X 2 ' X))

on the middle surface of the plate as shown in Fig. 1. The "1- X1 plane is chosen to coincide with the

middle surface and x-axis normal to it along the thickness. The surfaces .XJ =± d are assumed to be

stress and charge free or stress free and electrically shorted.

We take X l - X ) as the plane of incidence and

assume that the solutions are explicitly independent of

x 2 but implicit dependence is there so that the

component u2 of the displacement is non-vani shing.

The basic governing equations fo r homogeneous transversely isotropic piezoelectric elasticity, in the absence of charge density and body forces, are given by:

CIIULII +C44 UI.33 +(CI3 +c44 )~.I 3

+(els +e31 )¢.13 = PUI

C66U 2•11 + C44 U 2.)) = pil 2

(C13 +c44 )U1.I 3 +C44~.11 +C33 ~.33

+eIS¢.I1 +e33¢133 =p~

(els +e31 )~.I 3 + t;S~ .1 1 +e33 ~.33

-Ell ¢.II-E 33 ¢ .33 =0

(1)

(2)

(3)

00' (4)

where U =(ul ,u2 ,u3 ) are components of mechanical

displacement and ¢ is electric potential; cij ' eij' Eij

are respectively the elastic, piezoelectric and dielectric constants, and p is the density of the

material. The relation CII =C12 + 2c66 holds for material

with transverse isotropy . In the above equations comma notation is used for spatial detivative and dot over the quantity denotes differentiation with respect to time.

Boundary conditions

The plate is subjected to following two types of

boundary condition at .XJ =±d . (i) Stress free and charge free boundary conditions

given by:

00 . (5)

x) =-d

Y---------Y.. .. _ .. ..... X I

Fig. l--Geomeuy of the problem

SHARMA & PAL: PROPAGATION OF LAMB WAVES IN TRANSVERSELY ISOTROPIC PIEZOELECTRlC PLATE 373

(ii) Stress free and electrically shorted boundary conditions are:

... (6)

We assume solution of the form

(UI ,u2 ,u3 ,</»=(1,S,v ,W)

U exp{i~ (~sine + m-X) -ct)} ... (7)

where ~ is the wavenumber, W IS the angular

frequency and c =w/~ is the phase velocity of the

wave. Here, e is the angle of inclination of wave

normal with axes of sy mmetry (x) - axis), 'm' is

still an unknown parameter; S, V and W are respectively the amplitude ratios of the displacements

u 2 ' u) and potential </> to that of displacement

amplitude u l • The use of Eq. (7) in Eqs (1), (3) and

(4) leads to the following coupled equations

sin'O+c,I1( -rt SlIIsinO /i lllsinO 0 0

s msinO c,sin'O+c,nl-rt e,sin' O+f':Jnf 0 V =0

/i lllsinO e,sin' O+f':Jnf -11 (sin' O+€nr) 0 W 0

0 0 0 c,nf+c.sin' O-rt S 0

... (8) where

c c c =~ c =~, cJ I '2

CII CII

CII + C I2 _ C66

2cII CII

The system of Eq. (8) has a non-trival solution if the

determinant of the co-efficients of [1,S,v ,wt vanishes, which leads to the following polynomial equations:

(9a)

(9b)

where

... (10)

p' =CI +C2C2 -C3C3 , J' =CI +C2' p" =C\'E\ +C2C; -C; ,

J"=C\' +c2

CI' =cI +e; 11]1 E, C; =c2 +e2e3 11]1 E, C; =c3 +e\e3 11]1 E ,

£1 =1 +e\2 11]1 E ... (11)

The Eq. (9a) corresponds to purely transverse (SH) modes, which decoupled from rest of the motion and are independent of piezoelectric effects. This equation provides us:

The Eq. (9b) corresponds to the coupled longitudinal, shear vartical (SV) and electrical potential in plane motion. This equation being cubic in m2 admits six solutions for m which also have the property ~ =-n~ , m4 =-~ , m6 =-rrIs . For each

m'l ,q=1,2,3 .... .. 6 the amplitude ratios V and W can

be expressed as:

Vq =-RI (mq)lmqsR(mq)'

Wq =[ c3RI (mq)-(c2m~ + .• / -1]2)R(mq)]lel mqsR(mq)

... (12)

where R(mq) and R\ (m,,) are given by:

... (13)

374 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

Combining Eq. (12) with stress-strain and electric displacement relations given below:

Gi33 = C I3U I.I + C33 U 3.3 + e33¢.3'

Gi l3 = c 44 (uJ.3 + U 3.1 ) + els¢.1

0-2) =C44 U 2 .3 D3 =t;IUI.I+t;3~,3 -E 33 ¢,3 ... (14)

We rewrite the formal solution for the displacements, electric potential, stresses and electric displacement as :

6

(u U .+0) = ~ (1 V W)U i~ (xi sinll+lllqx)-cr) I. 3 ' '1' £..J ' q' q qe

q=1

8

( ) ~ (5 l';:'C D )U i~(xl sinll +lII"xJ-cr ) U2 ' a 23 = L.J q ' ':> I I 3q (/

q=7

... (15) 6

(a a D )=",;J:. C (D D D)U i~(xlsi nl.l+ "'qxJ -(·') 33. 13' 3 £..J'=> 11 1(1' 2'1' 4'1 'Ie

'1=1

... (16)

where DI =(c3-c2)sin8+cl nt V +P·nt W q q q J q q

D2q =czntq +cz sin8 V:I +e2 sin8 Wq

D3q =c2 m q 5q , D4q=(el-ez)sin8+t;ntqV:I -113ntqWq

Derivation of Secular Equation Stress and charge free plate

... (17)

By invoking the stress free and electrically charge free boundary conditions (5) at plate surfaces

x) = ±d, we abtain a system of six simultaneous

linear equations in amplitude UJ, U2, U) , U4 , Us, U6

as:

666

L DlqEqUq =0, L D2q EqUq =0, L D4q EqUq =0, '1=1 q=1 q=1

8

q~7 D3q E/fq =0 (18)

where E =e±i ~ mqd I 2 38Th t f Eq q , q= , , ...... e sys .em 0 s

(18) have a non-trivial solution if the determinant of the co-efficients of Uq ,q=1,2,3 .......... 8 vanishes,

which leads to the characteristic equation for the

propagation of modified guided waves in the plate. The characteristic equation for the piezo electric elastic waves in this case, after applying lengthy algebraic reductions and manipulations, leads to the following secular equations,

sin(2y~)=0 ... (19)

where

G3 =DII D4S -D4I DI5, Gs =DII D43 -D41D1 3

In Eq. (20) the superscript -1 corresponds to symmetric and + I corresponds to anti-symmetric modes of wave propagation in the plate.

Stress free and electrically shorted plate Invoking the stress free and electrically shorted

boundary conditions (6) at the plate surfaces .XJ =±d, we obtain the secular equations as below:

sin(2yr~)=0

with GI'=D23WS -DZ5~'

G; = DZI ~ - DZ3 ~

-DlsG;

DIIGI'

... (21)

... (22)

where again the superscript -1 and + 1 correspond to symmetric and skew-symmetric modes of wave propagation, respectively.

Stress free elastic plate In the absence of piezoelectric effects, the secular

Eqs (20) and (22) reduce to:

[ , 'JI tan(y~) ' DISD21

tan(y~) D;p~ . .. (23)

Here the roots m l and ms are given by:

. . . (24)

SHARMA & PAL: PROPAGATION OF LAMB W A YES IN TRANSYERSEL Y ISOTROPIC PlEZOELECTRlC PLATE 375

where D,'q =(c) -c2) sinO +clmq~, '

D;q =c2mq +c2 sinO Vq , q = 1,5

Here

q=1,5

... (25)

C2m~ +sin2 0 _TJ2

c)mqsinO

The superscript + 1 correspond to skew-symmetric and -I corresponds to symmetric modes respectively. The waves governed by secular Eqs (20), (22) and (23) are referred as modified guided Lamb type waves rather than Lamb waves whose properties were derived by Lamb in 1917 for isotropic solids in elastokinetics. The secular Eqs (20) and (22) govern the symmetric and skew-symmetric motion of the plate with stress and charge free boundaries and stress free electrically shorted boundaries of the plate, respectively. The Eq. (23) governs the symmetric and skew-symmetric motion of the homogeneous transversely isotropic elastic plate in the absence of piezoelectric effect. The Eqs (19) and (21) are the secular equations of decoupled SH modes of wave propagation in the plate and will not be considered in the following analysis.

Waves at short wavelength

Some information on the asymptotic behaviour is obtainable by letting ~ ----7 00. If we take

~>(J)/VH' v:, = C44 / P it follows that ~>(J) and

C < v H ' then we replace ~, ~ and m5 in the secular

equation by i m / and im; and i m;. Hence for

): tanh(y~) 1 tanh(y~) 1 '" ----7 00

, ----7 , , so that the tanh(yms ) tanh(yms )

secular Eqs (20) and (22) respectively, reduce to:

(27)

for symmetric and antisymmetric cases. In the absence of piezoelectric effects, the Eq. (23) reduces to:

... (28)

The Eqs (26), (27) and (28) are merely Rayleigh surface wave equations in piezoelectric elasticity for charge free and electrically shorted surfaces and elastokinetics respectively. The Rayleigh results enter here since, for such small wavelengths, the finite thickness plate appears as a semi-infinite medium. Hence, vibrational energy is transmitted mainly along the surface of the plate.

Displacement and electrical potential amplitudes Using Eqs (7) and (18), the amplitudes

(u l )s,V ' (u) ) ,y , (¢),Y and (ul )llS:' (~){/sy ' (¢ )llSY may be

calculated as:

B j ~(xi sinO ·cl) Ie

(~)" Y ={~ cos(~~X:J)+ V) ~ COS(~I11:lX:J )+ Vsl.z cos(~rn., X:J)}

B j ~(xlsinO oct) Ie

(qJ )sy = {WI cos(gmlX:!)+ W)~ COS(g 11l:JX:! )+ WsLz cos(gmSx) }

where

~ DISD21SICS -DIID25CISS

D n D 2Sc)sS - DISD2)S)CS '

l--z = D" D2)s)cl - D21 Dl3c)s,

DI)D25C)SS - D'SD2)s)cS

DISD2ISSCI-DIID2SCSSI

D n D 2S cSs) -D'SD2)sSc) ,

... (29)

... (30)

376 INDIAN 1. ENG. MATER. SCI., OCTOBER 2003

If the piezoelectric effects are ignored, then the expressions for displacement amplitudes

(u, )s," (u3 ),\, and (u, ),," , (l.l:l t,y are given by:

(u, ).,\' ={cos('; n~ -'1 ) + N, cos('; I1ls-'J) }A/~(XI si nO - cl )

(U3

)"Y ={~ COS('; 111., A3 ) + N, V, caS('; I11.sX:J)} A eiS(XIS inO - cl )

(u3 ) "'-' = {V, si n(,; m,X:! ) + N 2 Vs sin (,; f115X:!) }B,ei~ ( XIS in O -{/)

.. . (3 1)

Results and Discussion The material chosen fo r the purpose of numerical

calculations is cadmium selenide (6 mm class) of hexagonal symmetry wh ich is transveresly isotropic material. The physical data' 8, '9 for a single crystal of CdSe material is given in Table I.

The roots mi , i = 1,2,3,4,5,6 of Eq . (9) have

been obtained numerically by using Cardon's method, which are then used in various subsequent relevant relations and secular equations. The secu lar Eqs (20), (22) and (23) are solved by iteration method to obtain the phase veloci ties of symmetric and skew­symmetric modes of wave propagation. The sequence of the values of phase velocity has been allowed to iterate approximately 100 times to make it converge in order to achieve the desired level of accuracy, viz., up to seven decimal pl aces here. The phase velocities of symmetric and skew-symmetric modes of wave Table I- Physical data for a sing le crystal of cadmium selenide

Quantity Numerical value p 5504 kgm-3

C,' 7.44xldo Nm-2

Cl2 4.52x IO'O Nm-2

c13 3.93x ldo Nm-2

e33 8.36xldo Nm'z

r 1.32x 10'o Nm-2 ~-14

e' 3

-O.160cm-2

i?:J3 0.347 cm-2

e51 -0.138 cm-2

Ell 8.26x lO- 1I C2N" m' Z

EJ3 9.03xlO- 1I C2N·'m-2

propagation have been computed for various values of wave number from dispersion relations (20), (22) and (23) for different boundary conditions. The corrosponding dispersion curves for Rayleigh-Lamb type modes are presented in Figs 2, 3 and 4 respectively. The phase velocities of different modes of propagation of waves in a charge free and stress free surfaces of piezoelectric elastic plate are plotted in Fig 2. The corresponding phase velocities for elastic plate (without piezoelectic effects) of same material are represented in Fig. 4.

The comparison of Figs 2 and 4 shows that the phase velocity of acoustic skew··symmetric (Ao) mode

2.5 .

o ~ 2 >

~ .r; a.

~ 1.5 · o

. ~

.§

--8=30 - - - 8=45 ···· · ·8=60 -. _. A=QO

"9 I · ~ r-----~~---•• ----~p-.----.-----~----~

0.5 curves with ball-symmetric

curves with(lul ball-skewsymmetric

O~----~----~------------~----~----~

o Non-dimensional wave number

Fig. 2-Variation of non-dimensional phase velocity wi th non· dimensiona l wave. number in piezoelectric plate with stress and charge free surfaces

2.5

0.5

I',. n=l " '.

--8=30 n=2 -- - 8=45

. . .. . ,8=60

• curves with ball·symmctric curves without ball -skewsymrnetric

o I------~----~----~-----~ o

Non-dimensionaJ wl.\ve number

Fig. 3-Variation of non-d imensional phase velocity wi th non­dimensional wave number in piezoelectric plate wi th stress free and electrically shorted surfaces

SHARMA & PAL: PROPAGATION OF LAMB WAVES IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE 377

3 .---~--~------------------------~

0 2.5 'g Gl > 2 ~ -a. ~ 1.5 .9

5 .5 '" c o ;Z; 0.5

--30 --- 45 .. -- - -60 -- - - 90

Curves with ball-symmetric

Curves without ball-skewsymmetric

o .-------r-----~-------.-------.--~ o 2 4 6 8

Non-dimensional wave number

Fig. 4--Variation of non-dimensional phase velocity with non­dimensional wave number in elastic plate with stress free surfaces

in all direction increases from zero value at vanishing wave number to become closer to Rayleigh wave velocity at higher values of wave number in elastokinetics whereas in the presence of piezoelectic effect, the phase velocity of this mode is maximum at the vanishing wave number which then decreases and become closer to Rayleigh wave velocity with increas ing wave number. Acoustic symmetric (So) mode in both the cases is observed to be dispersionalless and have similar trend in all directions except the magnitude of phase velocity is slightly amplified in case of piezo-electricity . The behaviour of acoustic skew-symmteric (Ao) mode of wave propagation in case of piezo-electric plate with charge free surfaces is observed to be distinct from that of the elastic plate. While in case of elastic plate this mode disappear at the vanishing wave number, but for piezo-electric plate with charge free surfaces this mode is significant and dominant at vanishing wave number. This distinct behaviour is attributed due to the presence of piezoelectic effect. The phase velocities of optical modes of wave propagation in both symmetric and skew-symmetric motion attain quite large values at the vanishing wave number which sharply slash down to become steady with increasing values of wave number. The optical modes (n ~ 1) of wave propagation are again observed to develop at a rate which is approximately n times the magnitude of the phase velocity of first (n = 1) optical mode. The phase velocities of wave propagation in piezoelectric elastic plate are also computed in case of a electrically shorted surfaces from the dispersion

relation (22) and are plotted in Fig. 3. The symmetric acoustic (So) mode of wave propagation has almost similar characteristic to those of charge free plate except in the direction e =30° . Lowest skew­symmetric acoustic (Ao) mode of wave propagation again shows different behaviour. Both symmetric and skew-symmetric modes start from same value of phase velocity at vanishing wave number but the behaviour of Ao mode is different in different directions. For e =30° the phase velocity of Ao mode is almost constant and for e =45°,60°, 90° , the phase

velocity of Ao mode increases from its finite value at vanising wave number to the maximum value at ~ =1

and then decreases to become closer to Rayleigh wave velocity with increasing wave number. In all the above cases namely, elastic plate and electrically charge free as well as electrically shorted piezoelectic plate, the magnitude of phase velocity increases as wave normal moves away from the axis of elastic symmetry.

The amplitudes of -Xi and .XJ components of

displacement vector and electrical potential are also computed from Eqs (29) to (32) both in case of symmetric and skew-symmetric modes of wave propagation. These are plotted in Figs 5-10. The plate is subjected to stress and charge free boundary conditions on its surfaces. The amplitudes of the x-component of displacement vector for different values of thickness (X3) of the plate in presence as well as absence of the piezo-electric effect are shown in Fig. 5. In the absence of piezo-electric effects, the amplitude is maximum at the surface and minimum at the centre of the plate. Moreover, the amplitude has same values in all the directions at symmetrically located two points, namely X3 = ±0.5. While between these points, the amplitude is maximum in the direction e =30° and minimum along e =90° , these curves interchange their behaviour beyond these points. Such points also exist in the presence of piezo­electric effects, but they are symmetrically shifted towards the surfaces of the plate and observed to be located at X3 = ±0.5. The curves of variation in amplitude of UJ during symmetric mode of wave propagation change their trend from convex to concave which amount to a phase difference of 180° due to presence of piezo-electric effects in all directions as shown in Fig. 5. The similar observation is made for amplitude of symmetric displacement component U3 except the curves in this case, change

378 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

their trends from concave to convex due to piezo­electric effect in all directions, as shown in Fig. 6. For skew-symmetric motion the amplitudes of u1 and U:J

are shown in Figs 7 and 8. The maximum amplitude occurs in a direction perpendicular to the axis of symmetry and minimum along the axis of symmetry ff)r both piezo-electric and elastic plates. The amplitude has maximum value at the surfaces of the r late and zero at the centre of the plate. It is observed that amplitudes of u1 and U:J decrease as the material

is made to behave as piezoelectric for both symmetric and skew-symmetric motion because 10 a

2.5 -r-------------- ----,

\ 2 · \

, \

'. \

1.5 .

0.5

-I

--9=30 - --9.;;f5 ... . ·9=60 -·- ·A:'IO

non-

' \~~:.:-::--::-: ~ ~:::: :~-::7 ....... .-

piezoelectri

-0.5 0 Thickness of plate

/ :

/ /

0.5

Fig. 5- Variation of amplitude of xI-component of di splacement with thickness of plate during symmetric mode of wave propagation

1.2.,--------------·------ ,

/

/ ;'

" "

--9=30 - - - 9=45

_.- . - . - ._. . . -- - ·9=60 --. -: . - -9=90

"

Non-piewelectric "' \ --------------..... ' .. ', ...... -~

o +----~----~----r_---~ -I -0.5 o 0.5

Thickness of plate

Fig. 6--Variation of amplitude of xrcomponent of di splacement with thickness of plate during symmetric mode of wave propagation

piezoelectric material electric dipoles alligned in a particular direction whereas in the absence of piezoelectric effect the electric dipoles are randomly oriented. Due to this orderely allignrnent of dipoles, the amplitudes of vibration decreases in the presence of piezoelectric effect. In the absence of this effect, the amplitude of vibrations has large magnitude due to random orientation of dipoles, Thi s fact can also be ascertained from the comparison of respective quantities from Eqs (29) and (31 ) that the term pertaining to piezoelectric effects tends to decrease the amplitude and affect the behaviour of these functions.

Thus, behaviour of amplitudes of ~ and U:J in case

of symmetric and skew-symmetric modes of vibration is observed to be more stable than that in the absence

2.5.,----------·---------,

2 ---9=30 - - - 9.;;f5 ---- -9=60 --- -9=90 //

./" . '

/ .

o 0.5

Thickness of plate

Fig . 7- Variation of amplitude of x I-component of displacement with thickness of plate during skew-symmetric mode of wave propagation

0.5

0.4 ii E 0.3 ~ ]- 0.2 -0

'0 0.1 ii c 0 &. E 0 -0.1 -~ '0 -0.2 ~ .€ -0.3 c. E < -0.4

-0.5

--9=30 - --9=45 -- -- - 9=60 - - --9=90

----.~ ~,-~:;~ ~ 0

---~;..;:/

---- -"

./" ./"

.,/ '" Non-piezoelcctric

Thickncs:; of pillo

----

0.5

Fig. 8-Variation of amplitude of xrcompone.nt of displacement with thickness of plate during skew-symmetric mode of wave propagation

SHARMA & PAL: PROPAGATION OF LAMB WAVES IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE 379

1.8

.... - .. . . _- _ ........ - .. -- _ ... _- -_ ...... _- ... __ .... - . -1.6

O! 1.4 ·a

i 1.2 u ·c --9-30 B I U - - - f)a4S -0 - - - - - &-60 u ..,

0.8 .~ --- -9=90 C. e 0.6 <

0.4

0.2

0

-I -o.S o.S

Thickness of plate

Fig. 9- Variation of amplitude of electric potential with thickness of plate during symmetric mode of wave propagation

of this effect in all the directions of wave propagation. The variation of amplitudes of electric potential during symmetric and skew-symmetric motion is shown in Figs 9 and 10 respectively. It is found that during symmetric motion the amplitude almost remains constant for different values of thickness of the plate. The amplitude is maximum along the direction perpendicular to axis of symmetry for both symmetric and skew-symmetric motion. The electric potential is observed to attain higher amplitude values at surfaces of the plate than interior of the plate, where it is almost constant. The magnitude of various wave characteristics increases when the wave normal moves from the axis of elastic symmetry which clearly exhibits the effect of anisotropy of the material.

Conclusions The propagation of Lamb waves in a transversely

isotropic piezoelectric plate has been discussed. The plate is subjected to (i) stress and charge free and (ii) stress free and electrically shorted boundary conditions at the surfaces. To study the effect of piezoelectricity, the corresponding results for elastic plate (in absence of piezoelectric effect) are also obtained. Secular equations for symmetric and skew­symmetric modes of wave propagation have been derived. It is found that purely transverse (SH) wave gets decoupled from rest of the motion and remains independent of piezoelectric effects. Dispersion curves for symmetric and skew-symmetric modes of

3.---------------------------------, --9=30

2 - - - 9=045

- ---- 9=60

--- -9=90 :§

~ 1 8.

.~ ~ o +-------~----~~~~----~-------~ o u

1-1 -~

o 0.5

_3L-----____________________________ ~

Thickness of plate

Fig. 1000Variation of amplitude of electric potential with thickness of plate during skew-symmetric mode of wave propagation

wave propagation in piezoelectric and elastic plate are shown graphically for cadmium selenide (6 mm class) material. The skew-symmetric acoustic mode (Ao) in piezoelectric plate exhibits distinct behaviour to that in elastic plate. In case of elastic plate this mode (Ao) disappears at vanishing wave number whereas it is dominant and significant in piezoelectric plate with stress and charge free surfaces at vanishing wave number. This mode is also significant in piezoelectric plate with stress free and electrically shorted surfaces . The behaviour of acoustic symmetric mode (So) is dispersion less and similar in piezoelectric as well as elastic plate. The behaviour and trend of optical symmetric and skew-symmetric modes of wave propagation is also similar in the presence as well as absence of piezoelectric effect. At short wavelength limits the velocity of symmetric and skew-symmetric modes of wave propagation approaches to Rayleigh wave velocity. This is because finite thickness plate in such a situation behaves like a semi-infinite medium.

The amplitudes of displacement components and electric potential are also computed and shown graphically. It is found that the amplitudes of displacement component are observed to change their behaviour from convex to concave or vice-versa when we move !-rom elastic plate to piezoelectric plate. This means that there is a phase difference of 1800

between the amplitudes of displacement component in the presence and absence of piezoelectric effect. The amplitudes of displacement component are observed to be small and decrease in the presence of piezoelectric effect as compared to that in the absence of this effect, because electric dipole aligned in a

380 INDIAN 1. ENG. MATER. SCI., OCTOBER 2003

particular direction in piezoelectric material which have random orientation in non-piezoelectric materials . The various wave characteristics are observed to be significantly affected when the wave normal moves away from the axis of elastic symmetry which clearly depicts the effect of anisotropy of the material. Most of the wave phenomena are found to be more stable and realistic in the presence of piezoelectric effects than in the absence of such effects thereby making such materials more viable for wide practical applications and use.

Acknowledgements The authors are thankful to CSIR, New Delhi, for

providing financial assistance to carry out this work under Project grant No. 2S(011S)/01IEMR-II.

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