THERMOELASTIC LAMB WAVES IN ELECTRICALLY SHORTED TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE

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THERMOELASTIC LAMB WAVES INELECTRICALLY SHORTED TRANSVERSELYISOTROPIC PIEZOELECTRIC PLATEJ. N. Sharma a , Mohinder Pal a & Dayal Chand ba Department of Applied Sciences National Institute of Technology ,Hamirpur, Indiab Department of Physics Beant College of Engineering & Technology ,Gurdaspur, IndiaPublished online: 17 Aug 2010.

To cite this article: J. N. Sharma , Mohinder Pal & Dayal Chand (2004) THERMOELASTIC LAMB WAVESIN ELECTRICALLY SHORTED TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE, Journal of ThermalStresses, 27:1, 33-58, DOI: 10.1080/01495730490255709

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THERMOELASTIC LAMB WAVES

IN ELECTRICALLY SHORTED TRANSVERSELY

ISOTROPIC PIEZOELECTRIC PLATE

J. N. Sharma and Mohinder Pal

Department of Applied SciencesNational Institute of Technology

Hamirpur, India

Dayal Chand

Department of PhysicsBeant College of Engineering & Technology

Gurdaspur, India

The propagation of Lamb waves in a homogeneous, transversely isotropic, piezo-thermoelastic plate, which is stress free, electrically shorted, and thermally insulated(or isothermal), is investigated. Secular equations for the plate in closed form andisolated mathematical conditions for symmetric and antisymmetric wave mode pro-pagation are derived in completely separate terms. It is shown that the motion of thepurely transverse shear horizontal (SH) mode gets decoupled from the rest of themotion and remains unaffected due to piezoelectric, pyroelectric, and thermal effects.The secular equations for stress-free piezoelectric, thermoelastic, and elastic plates arededuced as special cases in the current analysis. At short wavelength limits the secularequations for symmetric and skew symmetric modes reduce to Rayleigh surface wavefrequency equation, because a finite-thickness plate in such a situation behaves like asemi-infinite medium. The amplitudes of dilatation, electrical potential, and tempera-ture change are also computed during the symmetric and skew symmetric motion of theplate. Finally, numerical solutions of various secular equations and other relevant re-lations are carried out for cadmium selenide (6mm class) material. The dispersioncurves, attenuation coefficients and amplitudes of dilatation, temperature change, andelectrical potential for symmetric and antisymmetric wave modes are presented gra-phically to illustrate and compare the analytical results. The theory and numericalcomputations are found to be in close agreement. The coupling between the ther-mal=electric=elastic fields in piezoelectric materials provides a mechanism for sensingthermomechanical disturbances from measurements of induced electric potentials andfor altering structural responses via applied electric fields. Therefore, the analysis will

Received 10 October 2002; accepted 9 April 2003.

The authors thankfully acknowledge the financial support from the Council of Scientific and

Industrial Research, New Delhi, via project grants No. 25 (0115)=01=EMR-II.

Address correspondence to J. N. Sharma (JNS), Department of Applied Sciences, National Institute

of Technology, Hamirpur, 177005 India. E-mail: [email protected]

Journal of Thermal Stresses, 27: 33–58, 2004

Copyright # Taylor & Francis Inc.

ISSN: 0149-5739 print/1521-074X online

DOI: 10.1080/01495730490255709

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be useful in the design and construction of Lamb wave sensors, temperature sensors, andsurface acoustic wave filter devices.

Keywords acoustic wave devices, electrically shorted, Lamb waves, Rayleigh waves,sensors

Mindlin [1] first proposed a thermopiezoelectricity theory. He also derived governingequations of a thermopiezoelectric plate [2]. The physical laws for thermopiezo-electric materials have been explored by Nowacki [3–5]. Chandrasekharaiah [6, 7]generalized Mindlin’s theory of thermopiezoelectricity to account for the finite speedof propagation of thermal disturbances. Several investigators [8–14] have studied thepropagation of waves in plates, cylinders, and general three-dimensional bodies thatare made of thermopiezoelectric materials. Tauchert [15] recently applied thermo-piezoelectricity theory to composite plates. Tang and Xu [16] derived the generaldynamic equations, which include mechanical, thermal, and electric effects, based onthe anisotropic composite laminated plate theory. They also obtained analyticaldynamical solutions for the case of general forces acting on a simply supportedpiezothermoelastic laminated plate; harmonic responses to temperature variationand electric field have been examined as a special case. The piezothermoelasticmaterials can be used as sensors to detect the response of a structure by measuringthe electric charge, or as actuators to reduce the excessive response by additionalelectric or thermal forces. These materials are also often used as resonators whosefrequencies need to be precisely controlled. Because of the coupling between thethermoelastic and pyroelectric effects, it is important to quantify the effect of heatdissipation on the propagation of waves at low and high frequencies. Yang andBatra [17] studied the effect of heat conduction on shift in the frequencies of a freelyvibrating linear piezoelectric body with the help of two perturbation methods.It is shown that the first-order effect on frequencies is to shift them by a smallimaginary number, thereby signifying that the effect of energy dissipation due to heatconduction is to reduce the amplitude of vibration.

Recently, resurgent interest in Lamb waves was partially initiated by theapplication of multisensors [18–20]. Schoch [21] derived the dispersion relation forleaky Lamb waves for an isotropic plate immersed in an inviscid liquid. Incidentally,the dispersion equations also have an interface wave solution whose velocity isslightly less than the bulk sound velocity in the liquid, and most of the energy is inthe liquid. It is often called the Scholte wave [22]. Watkins et al. [23] calculated theattenuation of Lamb waves in the presence of an inviscid liquid using an acousticimpedance method. Wu and Zhu [24] studied the propagation of Lamb waves in aplate bordered with inviscid liquid layers on both sides. The dispersion equations ofthis case were derived and solved numerically. Zhu and Wu [25] derived thedispersion equations of Lamb waves of a plate bordered with a viscous liquid layeror half-space viscous liquid on both sides. Numerical solutions of the dispersionequations related to sensing applications were obtained. Sharma and Pathania [26]studied thermoelastic Lamb waves in a homogeneous isotropic plate bordered withlayers of inviscid liquid in the context of the coupled theory of thermoelasticity.

34 J. N. SHARMA ET AL.

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Sharma and Kumar [27] studied the propagation of plane harmonic waves inpiezothermoelastic materials. No problem that analyzes analytically the propagationof Lamb-type waves in piezothermoelastic materials is currently available in theliterature at the time of this writing.

The coupling between the thermal=electric=elastic fields in piezoelectric materialsprovides amechanism for sensing thermomechanical disturbances frommeasurementsof induced electric potentials and for altering structural responses via applied electricfields. In view of this phenomenon, an attempt has been made in this article toinvestigate the propagation of Lamb waves in a piezothermoelastic, transverselyisotropic elastic plate that is stress free, thermally insulated=isothermal, andelectrically shorted. The Rayleigh–Lamb-type dispersion relations have been obtainedfor both symmetric and skew symmetric modes of wave propagation in the plate. Thedilatation, electric potential, and temperature change are also computed. The analy-tical results have been verified and computed numerically for a cadmium selenide(CdSe)material plate and are found to be in close agreement with the analytical results.

FORMULATION OF THE PROBLEM

We consider an infinite homogeneous, transversely isotropic, piezothermoelasticplate of thickness 2d initially at uniform temperature T0 and electric potential f0.We take the origin of the coordinate system ðx1; x2; x3Þ on the middle surface ofthe plate. The x1–x2-plane is chosen to coincide with the middle surface and thex3-axis normal to it along the thickness. The surfaces x3 ¼ �d are subjected todifferent boundary conditions. We take x1–x3 as the plane of incidence andassume that the solutions are explicitly independent of x2; however, implicitdependence is there, so the component u2 of the displacement vector is non-vanishing. The basic governing equations for homogeneous transversely isotropicpiezothermoelasticity, in the absence of charge density, heat sources, and bodyforces, are as follows:

c11u1;11 þ c44u1;13 þ c13 þ c44ð Þu3;13 þ ðe15 þ e31Þf;13 � b1T;1 ¼ r€uu1 ð1Þ

c66u2;11 þ c44u2;33 ¼ r€uu2 ð2Þ

ðc13 þ c44Þu1;13 þ c44u3;11 þ c33u3;33 þ e15f;11 þ e33f;33 � b3T;3 ¼ r€uu3 ð3Þ

e15 þ e31ð Þu1;13 þ e15u3;11 þ e33u3;33 � E11f;11 � E33f;33 þ p3T;3 ¼ 0 ð4Þ

K11T;11 þ K33T;33 � rCe_TT ¼ T0½b1 _uu1;1 þ b3 _uu3;3 � p3 _ff;3� ð5Þ

where b1 ¼ c11 þ c12ð Þa1 þ c13a3; b3 ¼ 2c13a1 þ c33a3; a1 and K11 are coefficients oflinear thermal expansion and thermal conductivity, respectively; in the directionorthogonal to axes of symmetry, a3 and K3 are the corresponding quantities alongthe axis of symmetry; r and Ce are, respectively, the mass density and specific heat atconstant strain; cij are the isothermal elastic parameters; eij are the piezoelectricconstants; E11; and E33 are electric permittivities; and p3 is the pyroelectric constant.The superposed dot denotes time differentiation and comma notation is used for

LAMB WAVES IN ELECTRICALLY SHORTED THERMOPIEZOELECTRIC PLATE 35

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spatial derivatives. In Eqs. (1)–(5), u! ¼ u1; u2; u3ð Þ is the displacement vector, f is

the electric potential, and T x1; x2; x3; tð Þ is the temperature change. Equations (1)–(5)can be written in nondimensional form as follows:

u1;11 þ c2u1;33 þ c3u3;13 þ Epe1f;13 � T;1 ¼ €uu1 ð6Þc2u2;33 þ c4u2;11 ¼ €uu2 ð7Þc3u1;13 þ c2u3;11 þ c1u3;33 þ Ep e2f;11 þ f;33

� �� bbT;3 ¼ €uu3 ð8Þ

e1u1;13 þ e2u3;11 þ u3;33 � EZ �EEf;11 þ f;33

� �þ pT;3 ¼ 0 ð9Þ

T;11 þ �KKT;33 � _TT ¼ Eð _uu1;1 þ �bb _uu3;3 � Epp _ff;3Þ ð10Þ

where we define the following quantities:

x 0i ¼

o�xivP

t 0 ¼ o�t u 0i ¼

ro�vpuib1T0

T 0 ¼ T

T0f 0 ¼ f

f0

o� ¼ Cec11K11

E ¼ T0b21

rCec11

vP ¼ffiffiffiffiffiffic11r

rp ¼ p3c11

b1e33c1 ¼

c33c11

c2 ¼c44c11

c3 ¼c13 þ c44

c11c4 ¼

c11 � c122c11

e1 ¼e15 þ e31

e33

e2 ¼e15e33

�EE ¼ E11E33

EP ¼ o�e33f0

vPb1T0

EZ ¼ Z3 EP Z3 ¼E33c11e233

o 0 ¼ oo� c 0 ¼ c

vP

d 0 ¼ o�d

vPh 0 ¼ hvP

o� x 0 ¼ xvPo�

s 0ij ¼

sijb1T0

�bb ¼ b3b1

�KK ¼ K33

K11

ð11Þ

where E is the thermoelastic coupling constant, o� is the characteristic frequency ofthe medium, Ep is the piezothermoelastic coupling constant, and vP is the longitudinalwave velocity in the medium. The primes have been suppressed for convenience.

Boundary Conditions

To study the effect of heat dissipation on propagation of waves due to coupling betweenthermoelastic and pyroelectric fields, the surfaces x3 ¼ �d of the plate are assumed to bestress free, thermally insulated (or isothermal), and electrically shorted. Then thenondimensional boundary conditions at the surfacesx3 ¼ �d of the plate are as follows.

1. Mechanical conditions (stress-free surfaces):

s33 ¼ 0 s13 ¼ 0 s23 ¼ 0 ð12aÞ2. Thermal conditions:

T;3 þ hT ¼ 0 ð12bÞ


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where h is the surface heat transfer coefficient, h ! 0 denotes thermallyinsulated, and h ! 1 denotes isothermal surfaces.

3. Electrical conditions (electrically shorted surfaces):

f ¼ 0 ð12cÞwhere f is the electric potential.

SOLUTION OF THE PROBLEM

We assume a solution of the form

u1; u2; u3;f;Tð Þ ¼ 1; �uu2;V;W;S;ð ÞU eixðx1 sin yþmx3�ctÞ ð13Þ

where x is the wave number; o is the angular frequency; c ¼ o=x is the phase velocityof the wave; y is the angle of inclination of a normal wave with axes of symmetry(x3-axis); m is still an unknown parameter; and �uu2;V;W, and S are, respectively, theamplitude ratios of displacement u2, u3; electric potential f, and temperature T tothat of displacement u1. The use of Eqs. (13) in Eqs. (6)–(10) leads to a system ofcoupled equations for the amplitudes 1;V;W;S; �uu2½ �T:

s2þ c2m2� c2 c3ms Epe1ms �s 0

c3ms c2s2þ c1m

2� c2 Ep e2s2þm2

� ��bbm 0

e1ms e2s2þm2 �EZ �EEs2þm2

� �pm 0

c2Es c2E �bbm �c2EEppm c2þ z s2þ �KKm2� �

0

0 0 0 0 c2m2þ c4s

2� c2

26666664

37777775

�

1

V

W

S

�uu2

26666664

37777775¼

0

0

00

0

2666664

3777775 ð14Þ

where z ¼ io; s ¼ sin y.The system of equations (14) has a nontrivial solution if the determinant of

coefficients of 1;V;W;S; �uu2½ �T vanishes, which leads to following polynomial char-acteristic equation:

m8 þ a1 þs 2

�KKþ Fc2

z �KK

� �m6 þ a2 þ a1

s 2

�KKþ Fc2

�KKzA1

� �

m4 þ a3 þ a2s 2

�KKþ Fc2

�KKzA2

� �m2 þ a3

s 2

�KKþ Fc2A3

�KKz

� �¼ 0 ð15Þ

c2m2 þ c4s

2 � c2 ¼ 0 ð16Þ


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where the coefficients ai and Ai (i ¼ 1; 2; 3) are given in the Appendix. Equation (16)corresponds to a purely transverse shear horizontal (SH) wave mode that decoupledfrom the rest of the motion and is not affected by either the thermal variations,pyroelectric, or piezoelectric effects and, hence, will not be considered in thefollowing analysis. Equation (15), being a quartic in m2, admits eight solutions for mthat have the following properties:

m2 ¼ �m1 m4 ¼ �m3 m6 ¼ �m5 m8 ¼ �m7

For each mq ðq ¼ 1; 3; . . . ; 8Þ, the amplitude ratios V;W, and S can be expressedas

Vq ¼R1ðmqÞRðmqÞ

Wq ¼R2ðmqÞRðmqÞ

Sq ¼R3ðmqÞRðmqÞ

ð17Þ

where RðmqÞ and RiðmqÞ (i ¼ 1,2,3) are given in the Appendix. Combining Eqs. (17)with stress, strain, electric displacement, and temperature relation gives

s33 ¼ c3 � c2ð Þu1;1 þ c1u3;3 þ EPf;3 � �bbT

s13 ¼ c2 u1;3 þ u3;1� �

þ EPe2 f;1 D3 ¼ e1 � e2ð Þu1;1 þ u3;3 � EZf;3 þ pT ð18Þ

We rewrite the formal solution for displacements, temperature, and electricpotential as

u1; u3;f;Tð Þ ¼X8q¼1

1;Vq;WqSq

� �Uqe

ixðx1 sin yþmqx3�ctÞ ð19Þ

The stresses, electric potential, and temperature gradient are obtained as follows:

ðs33; s13;f;T;3Þ ¼X8q¼1

ixðD1q;D2q;Wq;mqSqÞUqeixðx1 sin yþx3mq�ctÞ ð20Þ

where

D1q ¼ ðc3 � c2Þ sin yþ c1mqVq þ EpmqWq ��bbixSq ð21aÞ

D2q ¼ c2mq þ c2 sin yVq þ Epe2 sin y Wq q ¼ 1; 2; 3; . . . ; 8 ð21bÞ

DERIVATION OF THE SECULAR EQUATIONS

Stress-Free, Electrically Shorted, and Isothermal Boundary Conditions(h !11)

By invoking stress-free, isothermal, and electrically shorted boundary conditions(12a), (12b), and (12c) at the plate surfaces x3 ¼ �d we obtain a system of eight


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simultaneous linear equations in amplitudes Uq; q ¼ 1; 2; . . . ; 8 as follows:

X8q¼1

D1qEqUq ¼ 0X8q¼1

D2qEqUq ¼ 0X8q¼1

WqEqUq ¼ 0X8q¼1

D4qEqUq ¼ 0 ð22Þ

where D4q ¼ Sq and Eq ¼ e�ixmqd (q ¼ 1; 2; 3; . . . ; 8). The system of equations (22) hasa nontrivial solution if the determinant of the coefficients of Uqðq ¼ 1; 2; . . . ; 8Þvanishes, which leads to a characteristic equation for the propagation of modifiedguided waves in the plate. The characteristic equation for the piezothermoelasticwaves in this case, after applying lengthy algebraic reductions and manipulations,leads to the following secular equations:

G57

G37

T1T3

T5

� ��1

þG35

G37

T1T7

T5

� ��1

þG15

G37

T3T7

T5

� ��1

¼ � T�11 þ G17

G37T�13 þ G13

G37T�17

� �ð23Þ

where

T1 ¼ tanðgm1Þ T3 ¼ tanðgm3Þ T5 ¼ tanðgm5Þ T7 ¼ tanðgm7Þ g ¼ xd ð24Þ

G13 ¼ ðD21W3 �W1D23ÞðD15D47 �D17D45ÞG57 ¼ ðD25W7 �W5D27ÞðD11D43 �D13D41ÞG15 ¼ ðD21W5 �W1D25ÞðD17D43 �D13D47ÞG37 ¼ ðD23W7 �W3D27ÞðD15D41 �D11D45ÞG17 ¼ ðD21W7 �W1D27ÞðD13D45 �D15D43ÞG35 ¼ ðD23W5 �W3D25ÞðD11D47 �D17D41Þ

ð25Þ

where D1q;D2q are given in Eqs. (21) and D4q ¼ Sq. In Eq. (23) the superscript –1corresponds to the symmetric and þ 1 refers to the skew symmetric mode of wavepropagation in the plate.

Stress-Free, Thermally Insulated, Electrically Shorted Boundary Conditionsðh ! 0ÞUpon invoking the stress-free, thermally insulated, and electrically shorted boundaryconditions (12a), (12b), and (12c) at the plate surfaces x3 ¼ �d, we obtain the secularequation

T1

T5

� ��1

�D13G3

D11G1

T3

T5

� ��1

�D17G7

D11G1

T7

T5

� ��1

¼ �D15G5

D11G1ð26Þ

G1 ¼D23 D25 D27

W3 W5 W7

D43 D45 D47

�� G3 ¼

D21 D25 D27

W1 W5 W7

D41 D45 D47

��

G5 ¼D21 D23 D27

W1 W3 W7

D41 D43 D47

�� G7 ¼

D21 D23 D25

W1 W3 W5

D41 D43 D45

��

ð27Þ


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where D1q;D2q are defined in Eqs. (21) and D4q ¼ mqSq. The secular equations (23)and (26) are the transcendental equations that contain complete information aboutthe phase velocity, wave number, and attenuation coefficient of the plate waves. Ingeneral, wave number and, hence, the phase velocity of the waves are complexquantities; therefore, the waves are attenuated in space. If we write

c�1 ¼ v�1 þ io�1q ð28Þ

so that x ¼ Rþ iq; R ¼ o=v, where v and q are real, the exponent in the plane wavesolution (13) becomes

�qðx1 sin yþmx3Þ � io v�1ðx1 sin yþmx3Þ � t� �

ð29Þ

This shows that v is the propagation velocity and q is the attenuation coefficient ofthe wave. Upon using Eq. (28) in Eqs. (23) and (26) the values of v and q for differentmodes can be obtained.

SPECIAL CASES

Case 1: Uncoupled Thermoelasticity (Piezoelectric Plate)

If we set E ¼ 0 ¼ p, the motion corresponding to thermal wave (T-mode) decouplesfrom the rest of the motion and the various results reduce to those of a stress-freepiezoelectric elastic plate. The secular equations for symmetric and skew symmetricmodes for the rest of the motion under different boundary conditions take thefollowing forms:

T1

T5

� ��1

�D 013F3

D 011F1

T3

T5

� ��1

¼ �D 015F5

D 011F1

ð30Þ

where

F1 ¼ D 023W

05 �D 0

25W03 F2 ¼ D 0

21W05 �D 0

25W01 F3 ¼ D 0

21W03 �D 0

23W01

D 01q ¼ ðc3 � c2Þ sin yþ c1mqV

0q þmqW

0q

D 02q ¼ c2mq þ c2 sin yV 0

q þ e2 sin yW 0q

V 0q ¼ �R 0

1ðmqÞR 0ðmqÞ

W 0q ¼

R 02ðmqÞ

R 0ðmqÞ

ð31Þ

R 01ðmqÞ ¼ ½c2e3m4

q þ ðe3 þ c2e2 � e1c3Þs 2 � e3c2

� �m2

q þ e2s2ðs 2 � c2Þ�

R 02ðmqÞ ¼ c1c2m

4q þ ðPs 2 � Jc2Þm2

q þ ðs 2 � c2Þðc2s 2 � c2ÞR 0ðmqÞ ¼ mqs½ðc3e3 � c1e1Þm2

q þ ðc3e2 � e1c2Þs 2 þ e1c2�

ð32Þ


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where mqðq ¼ 1; 2; 3; 4; 5; 6Þ are the roots of the equation

m6 þ a1m4 þ a2m

2 þ a3 ¼ 0 ð33Þ

and where a1; a2; a3 are defined in the Appendix.

Case 2: Stress-Free Thermoelastic Plate

In the absence of piezoelectric effect, we have EP ¼ 0 ¼ p; the various results reduceto those of a stress-free thermoelastic plate. The secular equations (23) and (26) forsymmetric and skew symmetric modes for the rest of the motion under variousboundary conditions take the following form:

V �q ¼

mqaq= sin y; q ¼ 1; 2; 7; 8

�ðc2m2q þ sin2 y� c2 þ c3 sin yS �

q Þ=cemq sin y; q ¼ 5; 6

(

S �q ¼

½ðc2 þ c3aqÞm2q þ sin2 y� c2�= sin y; q ¼ 1; 2; 7; 8

½ðc1m2q þ c2 sin

2 y� c2Þðc2m2q þ sin2 y� c2Þ � c23m

2q sin

2 y �sin y½ðc1 � c3b Þm2

q þ c2 sin2 y� c2�

; q ¼ 5; 6

8>><>>:

ð34Þ

where

G�1 ¼ D�

25D�47 �D�

27D�45 G�

5 ¼ D�21D

�47 �D�

27D�41 G�

7 ¼ D�21D

�45 �D�

25D�41

D�1q ¼ ðc3 � c2Þ sin yþ c1mqV

�q �

�bb Sq

ix

ix

D�2q ¼ c2mq þ c2 sin yV�

q

D�4q ¼

S�q isothermal

mqS�q thermally insulated

(ð35Þ

V�q ¼

mqaqsin y

q ¼ 1; 2; 7; 8

� sin ymqaq q ¼ 5; 6

8><>:

S�q ¼

½ðc2 þ c3aqÞm2q þ s 2 � c2�=s q ¼ 1; 2; 7; 8

½c2m2q þ 1� c3=aq

� �s 2 � c2�=s q ¼ 5; 6

(

aq ¼ �bb½c2m2

q þ 1� c3=�bb� �

s 2 � c2�ðc1 � c3�bb Þm2

q þ c2s 2 � c2

ð36Þ

where mqðq ¼ 1; 2; 5; 6; 7; 8Þ are roots of the equation obtained by equating thedeterminant of Eq. (14) to zero after ignoring third and fifth rows and columns.Equation (34) has been solved and discussed in detail by Sharma et al. [28] fortransversely isotropic plates and by Sharma [29] for isotropic plates.


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Case 3: Stress-Free Elastic Plate

In the absence of piezoelectric, pyroelectric, and thermal effects, the secular equa-tions (23), (25), and (27) reduce to

tanðgm1Þtanðgm5Þ

¼ D0015D

0021

D0011D

0025

� ��1

ð37Þ

where

D001q ¼ ðc3 � c2Þ sin yþ c1mqV

00q D00

2q ¼ c2ðmq þ sin yV 00q Þ q ¼ 1; 5

V 00q ¼ � c3mqs

c1m2q þ c2s 2 � c2

¼ �c2m

2q þ s 2 � c2

c3mqsq ¼ 1; 5

The roots m1 and m5 are given by

m21 þm2

5 ¼�ðPs 2 � Jc2Þ

c1c2m2

1m25 ¼

ðs 2 � c2Þðc2s 2 � c2Þc1c2

Equation (37) has also been obtained and discussed by Sharma et al. [28] fortransversely isotropic plates and by Sharma [29] in the case of isotropic materials.

WAVES AT SHORT WAVELENGTH

Some information on the asymptotic behavior is obtainable by letting x ! 1. If wetake x > o=vH,v2H ¼ c44=r, it follows that c < vH; then we replace m1;m3;m5, and m7

in the secular equations with im 01, im

03, im

05, and im 0

7, respectively. Hence, for x ! 1;

tanhðgm1Þtanhðgm5Þ

! 1tanhðgm3Þtanhðgm5Þ

! 1tanhðgm7Þtanhðgm5Þ

! 1

so the secular equations (23), (26), (30), (34), and (37), respectively, reduce to

G13 þ G15 þ G17 þ G35 þ G37 þ G57 ¼ 0 ð38aÞD11G1 �D13G3 þD15G5 �D17G7 ¼ 0 ð38bÞ

D 011F1 �D 0

13F3 þD 015F5 ¼ 0 ð38cÞ

D�11G

�1 �D�

17G�7 þD�

15G�5 ¼ 0 ð38dÞ

D0011D

0025 �D00

15D0021 ¼ 0 ð38eÞ


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for both symmetric and antisymmetric cases. Equations (38) are merely Rayleighsurface wave equations in piezothermoelasticity, piezoelectricity, thermoelasticity,and elastokinetics, respectively, in the case of electrically shorted surfaces of theplate. The Rayleigh results enter here since, for such small wavelengths, the finitethickness plate appears as a semi-infinite medium. Hence, vibration energy istransmitted mainly along the surface of the plate.

AMPLITUDES OF DILATATION, TEMPERATURE,AND ELECTRIC POTENTIAL

The amplitudes of dilatation, electric potential, and temperature change duringsymmetric modes of vibration are obtained as follows:

e ¼ @u1@x1

þ @u3@x3

� �¼ c 01ðsin yþm1v3Þ þ Lc 03ðsin yþm3v3Þ þMc 05ðsin yþm5v5Þ þNc 07ðsin yþm7v7Þ½ �

�A1ðixÞeixðx1 sin y�ctÞ ð39aÞ

f ¼ W1c01 þW3Lc

03 þW5Mc 05 þW7Nc 07

� �A1ðixÞeixðx1 sin y�ctÞ ð39bÞ

T ¼ S1c01 þ S3Lc

03 þ S5Mc 05 þ S7Nc 07

� �A1ðixÞeixðx1 sin y�ctÞ ð39cÞ

where L;M;N are given by

L ¼ ½D11c�1ðD25D37 �D35D27Þs�5s�7 �D15c

�5ðD21D37 �D31D27Þs�1s�7

þD17c�7ðD21D35 �D31D25Þs�1s�5�=D

M ¼ ½D13c�3ðD21D37 �D31D27Þs�1s�7 �D11c

�1ðD23D37 �D33D27Þs�3s�7

þD17c�7ðD23D31 �D33D21Þs�3s�1�=D

N ¼ ½D13c�3ðD25D31 �D35D21Þs�5s�1 �D15c

�5ðD23D31 �D33D21Þs�3s�1

þD11c�1ðD23D35 �D33D25Þs�3s�5�=D

D ¼ �½D13c�3ðD25D37 �D35D27Þs�5s�7 �D15c

�5ðD23D37 �D33D27Þs�3s�7

þD17c�7ðD23D35 �D33D25Þs�3s�5�

c�q ¼ cos xmqd s�q ¼ sin xmqd c 0q ¼ cos xmqx3 s 0q ¼ sin xmqx3

q ¼ 1; 2; 3; . . . ; 8

The amplitudes of dilatation, electric potential, and temperature change for anti-symmetric modes of vibration are obtained from Eq. (39) by interchanging c 0q with s 0qand L;M;N with L 0;M 0;N 0, respectively, where L 0;M 0;N 0 are obtained fromL;M;N by replacing s�q with c�q.


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NUMERICAL RESULTS AND DISCUSSION

The material chosen for the purpose of numerical calculations is cadmium selenide(6mm class) of hexagonal symmetry, which is transversely isotropic material. Thephysical data for a single crystal of CdSe material is given as follows:

c11 ¼ 7:41� 1010 Nm�2 c12 ¼ 4:52� 1010 Nm�2 c13 ¼ 3:93� 1010 Nm�2

c33 ¼ 8:36� 1010 Nm�2 c44 ¼ 1:32� 1010 Nm�2 b1 ¼ 0:621� 106NK�1m�2

b3 ¼ 0:551� 106NK�1m�2 e13 ¼�0:160 Cm�2 e33 ¼ 0:347Cm�2

e51 ¼�0:138Cm�2 E11 ¼ 8:26� 10�11 C2N�1m�2 E33 ¼ 9:03� 10�11 C2N�1m�2

Ce ¼ 260Jkg�1K�1 p3 ¼�2:94� 10�6 CK�1m�2 Yr ¼ 4:48� 1010 Nm�2

ar ¼ 4:4� 10�6 K�1 a1 ¼ 3:92� 10�12 CN�1 K1 ¼K3 ¼ 9Wm�1 K�1

r¼ 5;504Kgm�3 T0 ¼ 298K

The roots mi ði ¼ 1; 2; 3; . . . ; 8Þ of the biquadratic equation (15) are obtainednumerically by using Descartes’ procedure and are then used in various subsequentrelevant relations and secular equations. The secular equations (23) and (26) aresolved via the iteration method to obtain the phase velocity of symmetric andantisymmetric modes of vibrations. The sequence of the values of phase velocitieshas been allowed to continue for approximately 100 iterations to make it convergeand to achieve the desired level of accuracy (viz. four decimal places). An infinitenumber of roots exist for a given value of frequency, which can be obtained by givinga value for the wave number from the secular equations (23) and (26). Each rootrepresents a propagating mode. Note that care must be taken in the root-findingprocedure, because the transcendental functions change their values rapidly. Thephase velocity ðc= ffiffiffiffiffi

c2p Þ profiles of the first three modes are obtained in different

directions of propagation for an electrically shorted, isothermal=thermally insulatedstress-free plate for various values of nondimensional wave number ðRdÞ. The velocityprofiles of symmetric and skew symmetric modes are given in Figures 1a and 1b foran isothermal plate and in Figures 2a and 2b for thermally insulated plates,respectively. The attenuation coefficient for the fundamental mode of wave pro-pagation has also been computed and is represented in Figures 4 and 5 for cases ofisothermal and thermally insulated boundaries of the plate. We also computed thedilatation, electric potential, and temperature change for the fundamental mode inthe case of stress-free isothermal, electrically shorted plates, which are plotted inFigures 6, 7, and 8.

From Figures 1 and 2 the velocity profiles of fundamental symmetric and skewsymmetric modes for y¼ 75� in both isothermal and thermally insulated plates arenoted to be almost straight lines, indicating that these are nearly nondispersive alongthis direction of propagation. All the modes show dispersive behavior in otherconsidered directions of wave propagation. It is also seen that there are crossover


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Figure 1. (a) Variation of phase velocity of symmetric modes of wave propagation with wave number in

various directions (isothermal boundary condition); (b) Variation of phase velocity of skew symmetric

modes of wave propagation with wave number in various directions (isothermal boundary condition).


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Figure 2. (a)Variation of phase velocity of symmetric modes of wave propagation with wave number in

various directions (isothermal boundary condition); (b) Variation of phase velocity of skew symmetric

modes of wave propagation with wave number in various directions (thermally insulated boundary

condition).


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points between various curves corresponding to the same mode in different directionsof propagation. The crossover phenomenon physically indicates that at particularwavelength the mechanical energy may be exchangeable between the correspondingdirections of wave propagation in the same mode. However, unlike elastic, ther-moelastic, and piezoelectric plate cases where the symmetric and skew symmetricmodes are clearly distinguishable, it is no longer possible to very clearly define thesymmetric and skew symmetric modes in a piezothermoelastic plate. It can be seenthat as the wave number increases, the phase velocity of each mode decreases in alldirections of wave propagation. When wave number becomes indefinitely large, thecurves asymptotically approach the Rayleigh wave velocity, because in such asituation a finite-thickness plate behaves like a half-space and the transportation ofenergy takes place mainly across the free surface of the plate. For low wave number,the phase velocity of the lowest mode in a piezothermoelastic plate follows closelythat of the rotational wave speed of cadmium selenide material. Except for thefundamental mode, all higher modes have phase velocities greater than the shearwave speed in the considered range of wave number along all the directions of wavepropagation. It is also observed that as the thickness of the piezothermoelastic plateincreases, the phase velocity decreases in all directions of wave propagation. This canbe explained by the fact that as the thickness of the plate increases, the coupling

Figure 3. Variation of phase velocity of symmetric and skew symmetric modes of wave propagation with

wave number in various directions for stress-free, electrically shorted piezoelectric plate.


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Figure 4. (a) Variation of attenuation coefficient of symmetric mode of wave propagation with wave

number in various directions (isothermal boundary condition); (b) Variation of attenuation coefficient of

skew symmetric mode of wave propagation with wave number in various directions (isothermal boundary

condition).


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Figure 5. (a) Variation of attenuation coefficient of symmetric mode of wave propagation with wave

number in various directions (thermally insulated boundary condition); (b) Variation of attenuation

coefficient of skew symmetric mode of wave propagation with wave number in various directions

(thermally insulated boundary condition).


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effect of various interacting fields also increases, resulting in lower phase velocity.It can also be observed that the Rayleigh wave velocity is reached at lower wavenumber as the thickness increases, because the transportation of energy mainlytakes place in the neighborhood of the free surfaces of the plate in this case. FromFigures 2a and 2b for y¼ 75�, there is almost negligible change in the behavior offundamental (acoustical) and higher (optical) modes of propagation compared tothat seen from Figures 1a and 1b in this direction of propagation. Althoughsignificant changes and shifts are observed in the mode shapes along the otherconsidered directions of propagation, the most affected velocity profiles are the onesalong y¼ 30� and y¼ 45�, in this case. Thus, the effect of isothermal and thermallyinsulated plate boundaries is also observed to be significant in cadmium selenidematerial. The presence of dips in various curves shows the existence of the dampingphenomenon, which is noticed to be more prominent for 0 < y � 45� than for45� � y < 90�. However, the damping effect is quite significant along the directionsfor which y satisfies the inequalities 30� � y � 45�. No such dips are noticed to bepresent in the phase velocity profiles of fundamental (acoustical) and higher (optical)modes along y ¼ 75� in the cases of both stress-free isothermal and stress-freethermally insulated, electrically shorted plates. Thus, the various modes of wavepropagation are monotonic, though attenuated=damped, along this direction ofwave propagation. Also, these dips are not observed in any of the curves for variousmodes in a piezoelectric plate in the absence of thermal and pyroelectric effects,which are shown in Figure 3. The oscillatory behavior is also not observed for athermoelastic plate in the absence of piezoelectric and pyroelectric effect. Clearly thepresence of these dips in various dispersion curves for a piezothermoelastic plate isattributed to the coupling between thermal and electric fields, that is, pyroelectriceffects. The reduction in the amplitude of vibrations compared to that of elastic andpiezoelectric plates signifies the impact of energy dissipation due to heat conductionand pyroelectric effects.

The free surfaces admit a Rayleigh-type surface wave with complex wavenumber and, hence, phase velocity. Consequently, the surface wave propagates withattenuation due to the radiation of energy into the medium. This radiated energy willbe reflected back to the center of the plate by the lower and upper surfaces.Consequently, the attenuated surface wave on the lower surface is enhanced by thisreflected energy to form a propagation wave. In fact, the multiple reflections betweenthe upper and lower surfaces of the plate form caustics at one of the free surfaces anda strong stress concentration arises, which causes the wave field to becomeunbounded in the limit d ! 1. The unbounded displacement field is characterizedby the singularities of circular tangent functions. From the curves in Figure 4a, it isobserved that the value of the attenuation coefficient is observed to be quitesignificant at small values of the wave number along the directions 0� < y � 45�

compared to those satisfying 45� � y < 90� for a stress-free, isothermal electricallyshorted plate during the acoustic mode of wave propagation. The attenuationcoefficient increases as the angle of inclination of wave normal with x3-axisprogresses from 0 to 90�. Moreover, the damping oscillatory effect is more promi-nent in the considered range 0 � Rd � 10 of the wave number for 0� < y � 45� thanfor 45� � y < 90�. For example, the damping effect is quite high, in the range


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Figure 6. (a) Variation of symmetric dilatation with plate thickness in various directions (isothermal

boundary condition); (b) Variation of skew symmetric dilatation with plate thickness in various directions

(isothermal boundary condition).


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Figure 7. (a) Variation of amplitude of symmetric electric potential with plate thickness in various di-

rections (isothermal boundary condition); (b) Variation of amplitude of skew symmetric electric potential

with plate thickness in various directions (isothermal boundary condition).


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Figure 8. (a) Variation of amplitude of symmetric temperature change with plate thickness in various

directions (isothermal boundary condition); (b) Variation of amplitude of skew symmetric temperature

change with plate thickness in various directions (isothermal boundary condition).


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2 < Rd � 7 for y ¼ 30� and 3 < Rd � 10 for y ¼ 45�, as observed from Figure 4a.From Figure 4b, it is noticed that the attenuation coefficient for the skew symmetricmode increases logistically with nondimensional wave number along y ¼ 75�. Thisquantity performs damped oscillations for other directions of propagation in theconsidered range of the values of nondimensional wave number. From Figures 5aand 5b (for thermally insulated plate) only a negligibly small change in the behaviorof the attenuation coefficient of the fundamental (acoustic) mode is noticed alongy ¼ 75� compared to that observed for an isothermal plate at low wave number. Thischange is, however, quite significant at higher values of nondimensional wavenumber. The behavior of this quantity along other considered directions of wavepropagation is observed to be significantly departed from their counterparts inFigures 4a and 4b and is subjected to the presence of peaks and dips in shiftedintervals in this case. Thus, the acoustic and higher modes are noticed to besignificantly attenuated in space and damped with time for both the cases of stress-free isothermal and stress-free, thermally insulated electrically shorted plates alongy ¼ 30�; 45�; 60�. Along the direction of propagation, y ¼ 75�, the acoustic as well ashigher modes are almost free of the damping effect. This result agrees with thecorresponding result obtained earlier from the phase velocity profiles.

The amplitude of dilatation for fundamental symmetric and skew symmetricmodes is represented in Figures 6a and 6b, respectively. It is seen that there is largeamplitude of skew symmetric dilatation near the upper and lower surfaces of theplate in all directions, which is primarily due to the presence of free surfaces. Becausethe upper and lower free surfaces give rise to the caustic effect, this results in large-displacement amplitude and, consequently, volumetric deformation (dilatation). Fora relatively thicker plate, the caustic effect becomes less important and the funda-mental mode becomes a Rayleigh-type surface wave on the upper or lower freesurface. In this case one may still call the fundamental mode a Rayleigh-type surfacewave on the free surface, because its amplitude on the free surface is several timesthat found elsewhere inside the plate. However, along the directions y ¼ 45� and 60�

there is negligible variation in dilatation with plate thickness. In the case of thesymmetric mode there is maximum volumetric deformation at the center of the plateand less deformation takes place at the upper and lower surfaces of the plate. In thecase of the skew symmetric mode, significant amplitude development of dilatation isobserved only along the directions y ¼ 30� and 75�. The symmetric and skew sym-metric dilatation along the directions y ¼ 45� and 60� of wave propagation is noticedto be almost nondispersive in character.

The electric potential change along the plate thickness for fundamentalsymmetric and skew symmetric modes is plotted in Figures 7a and 7b. No change inelectric potential is noticed at lower and upper surfaces of the plate, which isconsistent with the boundary conditions. The change in electric potential is mainlyobserved in the directions y ¼ 30� and 45� of wave propagation. There is almostnegligible change in electric potential when the wave propagates along the directiony ¼ 60� in both symmetric and skew symmetric modes. The symmetric electricpotential change is significant in the directions y ¼ 30� and 45�. Thus, a significantamount of mechanical and thermal energy gets converted into electrical energy alongy ¼ 30� and 45�. Also, curves in these directions intersect at two points, which are


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symmetrically located at x3 ¼ �0:5 from the center of the plate. Between these pointsthe magnitude of electric potential is quite large for y ¼ 30� and beyond these pointsthe electric potential change is higher in magnitude along y ¼ 45�. At these points theelectric potential is exchangeable between these two directions at particular wave-length. As observed from Figures 8a and 8b, the amplitude of the temperaturechange significantly varies with plate thickness only along y ¼ 30� for both sym-metric and skew symmetric modes of wave propagation. Therefore, for the wavespropagating in this direction, some portion of the mechanical energy is convertedinto thermal energy giving significant rise in the amplitude of temperature change inthis case compared to other directions of wave propagation. In all other directionsthere is negligible temperature change with plate thickness and the phenomenon isalmost nondispersive in character.

CONCLUSION

The propagation of thermoelastic Lamb waves in a transversely isotropic piezo-electric plate, which is (i) electrically shorted, stress free, and isothermal, or(ii) electrically shorted, stress free, and thermally insulated at the surfaces has beendiscussed. Secular equations for symmetric and skew symmetric modes of vibrationhave been derived. It is found that a purely transverse (SH) wave gets decoupledfrom the rest of the motion and remains independent of piezoelectric, pyroelectric,and thermal effects. The analysis for elastic, piezoelectric, and thermoelastic plateshas been deduced as particular cases from current study. At short wavelength limitsthe secular equations for symmetric and skew symmetric modes reduce to Rayleighsurface wave frequency equations. This is because a finite-thickness plate in such asituation behaves like a semi-infinite medium. Dispersion curves for symmetric andantisymmetric modes of vibration in piezothermoelastic plates are shown graphicallyfor cadmium selenide (6mm class) material. In addition to this, attenuationcoefficients of waves are also computed and plotted with wave number. It is observedthat a decrease in phase velocity with wave number is oscillatory in character in mostof the directions except in the neighborhood of the direction y ¼ 75�. In all otherdirections the decrease in velocity fluctuates like a damped oscillator and dies outwith the increasing wave number. This effect is not observed in stress-free piezo-electric, thermoelastic, and elastic plates, where the variation of phase velocityðc= ffiffiffiffiffi

c2p Þ with wave number ðRdÞ is monotonic. The attenuation coefficients are found

to increase with increasing wave number; however, this increase is also oscillatory.The oscillatory behavior of dispersion curves and attenuation coefficient is attributedto the coupling between thermal and electric fields (viz. pyroelectric effect). Theamplitude of vibrations is reduced due to the heat conduction and pyroelectriceffects, thereby signifing the energy dissipation in cadmium selenide material plate.The analytical and numerical results are found to be in close agreement. The resultsprovide evidence that piezoelectric elements can be used effectively as temperaturesensors for use in intelligent structural systems. This study will be useful in the designand construction of Lamb wave sensors, temperature sensors, and surface acousticwave filter devices.


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APPENDIX

The coefficients ai;Ai ði ¼ 1; 2; 3Þ in Eq. (15) andRðmqÞ;RiðmqÞ ði ¼ 1; 2; 3Þ in Eq. (17)are given as follows:

a1 ¼Ps 2 � Jc2 þ �EEc1c2s 2 þ �EEZ½ð2c2e2 � 2c3e1 þ e21c1 þ 1Þs 2 � c2�

c1c2 þ �EEZc2

a2¼�EEðPs2�Jc2Þþðs2�c2Þðc2s2�c2Þþ�EEZ ðe21c2þe22c2�2c3e1e2Þs4�e21s

2c2þ2e2s2 s2�c2� ��

c1c2þ�EEZc2

a3 ¼s 2ðs 2 � c2Þ �EEðc2s 2 � c2Þ þ �EEZe22s

2� �c1c2 þ �EEZc2

A1 ¼E1ðP 0s2 � J 0�cc2Þ þ EC1C2s

2 þ�EEZ ð2C2�ee2 � ðEþ 1Þ�ee1C3 þ �ee21C1 þE2Þs2 �E2�cc2� �� C1C2E1 þ�EEZE2C2

A2 ¼

Es 2ðP 0s 2 � J 0�cc2Þjþ E1ðs 2 � �cc2ÞðC2s2 � �cc2Þ

þ�EEZ ð�ee21C2 þ �ee22C2 � 2C3�ee1�ee2Þs4 þ 2E�ee2ðs 2 � �cc2Þs 2 � �ee21s2�cc2

� �( )

C1C2E1 þ �EEZE 2C2

A3 ¼s 2ðs 2 � �cc2Þ �EEZ�ee22s 2 þ EðC2s

2 � �cc2Þ� �

C1C2E1 þ �EEZE 2C2

RðmqÞ ¼ �EPs

½c3ð�bb�EEZ � pÞ þ ð�bbþ pc1Þe1 � c1�EEZ � 1�m4q

þ½c3ð�bb�EEZ�EE� pe2Þ þ e1ðpc2 þ �bbe2Þ � 2e2 � �EEZð�EEc1 þ c2Þs 2

�ðpe1 � �EEZÞc2�m2q

�s 2½ð�EEZ�EEc2 þ e22Þs 2 � �EEZ�EEc2�

8>>>><>>>>:

9>>>>=>>>>;

R1ðmqÞ ¼ EPmq

c2ð�bb�EEZ � pÞm4q

þf½�bb�EEZð�EEc2 þ 1Þ þ e21�bb � pð1þ c2e2 � c3e1Þ � ð�EEZc3 þ e1Þ�s 2

�ðb�EEZ � pÞc2gm2q

þs 2½ðs 2 � c2Þð�bb�EEZ�EE� pe2Þ � ðc3�EEZ�EEþ e1e2Þs 2�

8>>><>>>:

9>>>=>>>;

R2ðmqÞ¼mq

ðpc1þ�bb Þc2m4qþ pðPs2�Jc2Þþ�bb ðs2�c2Þþðe1c1�c3þ�bb e2c2��bb c3e1Þs2

� �m2

q

þðs2�c2Þ ðpc2þ�bb e2Þs2�pc2� �

þðe1c2�c3e2Þs4�e1c2s2

( )


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R3ðmqÞ¼EP

c2ð1þ�EEZc1Þm6qþ

� c3ðc3�EEZþe1Þþe1ðc3�e1c1Þ� �

s2þð�EEc1þ1Þðs2�c2Þþc2 ðc1�EEZ�EEþc2�EEZþ2e2Þs2��EEZc2

� �" #

m4q

�c3ðc3�EEZ�EEþe1e2Þþe1ðe1c2�e2c3Þ� �

s4þc2s2 �EEZ�EEðc2s2�c2Þþe22s

2� �

þðs2�c2Þ ðc1�EEZ�EEþc2�EEZþ2e2Þs2��EEZc2� �

�e21c2s2

" #m2

q

þðs2�c2Þs2 �EEZ�EEðc2s2�c2Þþe22s2

� �

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

where

C1 ¼c1 þ E�bb

2� �

1þ EC2 ¼

c21þ E

C3 ¼c3 þ E�bb1þ E

�ee1 ¼e1 � Ep1þ E

�ee2 ¼e2

1þ E

E ¼1� E�bb p� �

1þ EE ¼ �EE

1þ EE1 ¼

1� Ep2=Z3� �

1þ E�cc2 ¼ c2

1þ E

P ¼ c1 þ c22 � c23 J ¼ c1 þ c2 P 0 ¼ C1 þ C22 � C2

3 J 0 ¼ C1 þ C2

F ¼ ð1þ EÞ3C2ðC1E1 þ �EEZE 2Þc2ðc1 þ �EEZÞ


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THERMOELASTIC LAMB WAVES IN ELECTRICALLY SHORTED TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE