Page 1

Love wave propagation in functionally graded piezoelectric

material layer

Jianke Du*, Xiaoying Jin, Ji Wang, Kai Xian

Piezoelectric Device Laboratory, Department of Mechanics and Engineering Science, School of Engineering, Mechanics and Materials

Science Research Center, 818 Fenghua Road, Ningbo University, Ningbo, Zhejiang 315211, China

Available online 13 October 2006

Abstract

An exact approach is used to investigate Love waves in functionally graded piezoelectric material (FGPM) layer bonded to a semi-

infinite homogeneous solid. The piezoelectric material is polarized in z-axis direction and the material properties change gradually with

the thickness of the layer. We here assume that all material properties of the piezoelectric layer have the same exponential function dis-

tribution along the x-axis direction. The analytical solutions of dispersion relations are obtained for electrically open or short circuit

conditions. The effects of the gradient variation of material constants on the phase velocity, the group velocity, and the coupled electro-

mechanical factor are discussed in detail. The displacement, electric potential, and stress distributions along thickness of the graded layer

are calculated and plotted. Numerical examples indicate that appropriate gradient distributing of the material properties make Love

waves to propagate along the surface of the piezoelectric layer, or a bigger electromechanical coupling factor can be obtained, which

is in favor of acquiring a better performance in surface acoustic wave (SAW) devices.

? 2006 Elsevier B.V. All rights reserved.

Keywords: Love waves; Functionally graded piezoelectric material; Surface acoustic waves; Dispersion relation

1. Introduction

A new-style material called functionally graded material

(FGM) was proposed to solve problems in the thermal-

protection systems of aerospace structures in 1980s. Since

then, FGM has attracted interest of investigators from

many engineering disciplines. Today, FGM can be used

not only in thermal-protection systems but also in elec-

tronic and many other fields. The results obtained for the

FGM layered structures lead us to consider that the

FGM may be applicable to surface acoustic wave (SAW)

devices, if functionally graded piezoelectric material

(FGPM) can be properly manufactured, as known from

recent techniques for fabricating FGPMs [1].

Since White [2] invented the interdigital transducers

(IDTs) utilized for transmitting and receiving SAW signals

in 1965, SAW are adopted successfully in the electronic

industry with filters, delay lines, resonators, and oscillators

for signal processing applications. With the development of

the material technology, FGPMs can be manufactured and

used in SAW devices to improve the efficiency and other

features. Hence, the research of wave propagation behav-

iors and characteristics in FGPM has become a topic of

practice importance [2–11]. Liu and Tani investigated sur-

face waves in FPGM plates with the application of strip

element method [2–8]. Han et al. introduced a hybrid

numerical method (HNM) to analyze characteristics of

waves and transient responses in FGM cylinders [9,10].

Recently Han and Liu investigated the frequency and

group velocity dispersion behaviors, and characteristic sur-

faces of waves in FGPM cylinders using an analytical

numerical method [11]. Li et al. studied the behaviors of

Love waves in a layered functionally graded piezoelectric

structure using WKB method [12].

0041-624X/$ - see front matter ? 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultras.2006.09.004

*Corresponding author. Tel.: +86 574 87600074; fax: +86 574

87608358.

E-mail address: dujianke@nbu.edu.cn (J. Du).

www.elsevier.com/locate/ultras

Ultrasonics 46 (2007) 13–22

Page 2

The layered structures with inhomogeneous boundary

conditions, for example, a thin film on a substrate, are cur-

rently adopted to achieve high performance for SAW

devices. It is well known that phase velocities and disper-

sion relations of acoustic waves are important for applica-

tions. In 1911, Love [13] analyzed a layered structure

consisting of an isotropic elastic layer on an isotropic sub-

strate with perfect bonding at the interface. He concluded

that shear surface waves propagate in the layer and atten-

uate along thickness of the substrate if the velocity of the

bulk shear wave in the layer is less than that in the sub-

strate. These shear surface waves are now known as the

Love waves and their polarization is perpendicular to the

sagittal plane formed by both the normal to the interface

of a medium and the wavevector in the direction of wave

propagation [14]. Numerous investigations have been

undertaken for the analysis of Love waves in piezoelectric

media [15–18].

In this paper, we analyze the propagation of Love waves

over a half space of elastic solid covered by a piezoelectric

layer of finite thickness. The piezoelectric material is polar-

ized in z-axis direction and the material properties change

gradually with the thickness of the layer. We here assume

that all material properties of the piezoelectric layer have

the same exponential function distribution along the x-axis

direction. The analytical solution of dispersion relations

can be obtained for electrically open or short circuit

conditions.

2. Problem formulation

Consider an anisotropic semi-infinite elastic substrate

covered by a functionally graded piezoelectric material

layer as illustrated in Fig. 1. The piezoelectric material is

polarized in z-axis direction and the material properties

change gradually with the thickness of the layer. For the

piezoelectric layer, the equilibrium equations of elasticity

without body forces and the Gauss’ law of electrostatics

without free charge are given as follows:

rji;j¼ q€ ui;

where rij is the stress tensor, Di is the electric dis-

placement, and q is the mass density of the piezoelectric

material.

On the assumption that the Love waves propagate in the

y direction, so the total out-of-plane displacement and the

electric potential are expressed as

Di;i¼ 0;

i;j ¼ 1;2;3;

ð1Þ

u ¼ v ¼ 0;

w ¼ wðx;y;tÞ;

/ ¼ /ðx;y;tÞ:

ð2Þ

Substituting Eq. (2) into Eq. (1), we can obtain

oszx

oxþosyz

oy

¼ qo2w

ot2;

oDx

oxþoDy

oy

¼ 0:

ð3Þ

The constitutive equations of the functionally gradient pie-

zoelectric materials can be written as

syz¼ c44ðxÞow

szx¼ c44ðxÞow

Dx¼ e15ðxÞow

Dy¼ e15ðxÞow

oyþ e15ðxÞo/

oxþ e15ðxÞo/

ox? e11ðxÞo/

oy? e11ðxÞo/

oy;

ox;

ð4Þ

ox;

ox:

ð5Þ

Substituting Eqs. (4) and (5) into Eq. (3), the governing

equations of the piezoelectric layer are obtained as

follows:

oc44ðxÞ

ox

þ c44ðxÞo2w

ow

oxþ c44ðxÞo2w

oy2þ e15ðxÞo2/

ow

oxþ e15ðxÞo2w

þ e15ðxÞo2w

ox2þoe15ðxÞ

oy2¼ qo2w

ox2?oe11ðxÞ

ox

o/

oxþ e15ðxÞo2/

ox2

ot2;

ox? e11ðxÞo2/

ð6Þ

oe15ðxÞ

ox

ox

o/

ox2

oy2? e11ðxÞo2/

oy2¼ 0:

ð7Þ

We here assume that all material properties of the piezo-

electric layer as Fig. 1 have the same exponential function

distribution along the x-axis direction. Though these

material constants distributions are unrealistic, it would

allow us to comprehend the influence of material gradient

upon the characteristics of wave propagation, and make

use of it for designing more effective devices in practice.

In the mean time, we can obtain one analytical and exact

resolution with the assuming of exponential function dis-

tribution, which can be used to verify the accuracy of

other numerical methods. The material properties are gi-

ven as

c44ðxÞ ¼ c0

e11ðxÞ ¼ e0

44ebx;

11ebx;

e15ðxÞ ¼ e0

qðxÞ ¼ q0ebx;

15ebx;

ð8Þ

Table 1

Material constants

c44(109N/m2)

e15(C/m2)

e11(10?9F/m)

q(103kg/m3)

PZT-5H

SiO2

23

31.2

17

0

157.5

2.20.033

z

x

Substrate

Layer h

y

Fig. 1. An elastic half-space covered by a piezoelectric layer.

14

J. Du et al. / Ultrasonics 46 (2007) 13–22

Page 3

where b is the exponential factor characterizing the degree

of the material gradient in the x-direction, and the

superscript 0 is attached to indicate the x-direction factors

in the material coefficients. It is obvious that b = 0 corre-

sponds to the homogenous material case.

The governing equations are rewritten as

e0

15bow

oxþ e0

15r2w ¼ e0

11bo/

oxþ e0

11r2/;

15r2/ ¼ q0o2w

ð9Þ

c0

44bow

oxþ c0

44r2w þ e0

15bo/

oxþ e0

ot2:

ð10Þ

The governing equations for the substrate are given as

follows:

cm

44r2wm¼ qmo2wm

ot2;

r2/m¼ 0;

ð11Þ

where cm

the displacement of the host medium in the z direction, /m

is the electric potential, and $2is the Laplacian operator

given by r2¼ ðo2=ox2Þ þ ðo2=oy2Þ. The superscript m

indicates the properties in the substrate. The shear stress

can be written as

44is the shear modulus, qmis the mass density, wmis

sm

xz¼ cm

44

owm

ox;

sm

yz¼ cm

44

owm

oy:

ð12Þ

Since the dielectric constant of the air is far less than that of

the piezoelectric medium, the air can be regarded as vac-

uum and the electric potential in the vacuum is determined

by the Laplacian equation

r2/a¼ 0:

The stress and electric displacement of the substrate tend to

be zero in points far from the FGPM layer, i.e.,

ð13Þ

x ! þ1;

The electric potential in the vacuum tend to zero in points

far from layer along the negative x-direction

wm¼ 0;

/m¼ 0:

ð14aÞ

x ! ?1;

The mechanical and electrically open conditions at the free

surface can be given as

/a¼ 0:

ð14bÞ

sxzð?h;yÞ¼0;

/ð?h;yÞ¼/0ð?h;yÞ;

Dxð?h;yÞ¼D0ð?h;yÞ:

ð15Þ

The mechanical and electrically short conditions at the free

surface are expressed as follows:

sxzð?h;yÞ ¼ 0;

The continuity conditions at the interface between the

piezoelectric layer and the substrate are written as

/ð?h;yÞ ¼ 0:

ð16Þ

wð0;yÞ ¼ wmð0;yÞ;

/ð0;yÞ ¼ /mð0;yÞ;

sxzð0;yÞ ¼ sm

Dxð0;yÞ ¼ Dmð0;yÞ:

xzð0;yÞ;

ð17aÞ

ð17bÞ

3. Analytical solution

The solutions of wmand /mfor the wave propagation in

the y direction can be expressed as

wm¼ fmðxÞexp½ikðy ? ctÞ? ¼ Ame?kbmxexp½ikðy ? ctÞ?;

/m¼ Bme?kxexp½ikðy ? ctÞ?;

where bm¼

velocity of the shear waves in the isotropic substrate. The

derivation of Eqs. (18) and (19) is under the assumption

c < cm

fracted waves carrying energy from the layer. Such a wave

system would quickly lose its energy and not be of signifi-

cance at any distance, and thus is beyond the scope of our

attention.

By assuming

w ¼ / ?e15

ð18Þ

ð19Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 ? c2=ðcm

sh

p

Þ2

and cm

sh¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cm

44=qm

p

is the

sh. When c > cm

sh, such waves would represent re-

e11w;

ð20Þ

Eqs. (9) and (10) can be rewritten as follows:

bow

oxþ r2w ¼ 0:

1

q0

c0

44þe0

15

e0

11

2

!

bow

oxþ r2w

??

¼o2w

ot2;

ð21Þ

ð22Þ

Assuming the solutions of Eqs. (21) and (22) are

w ¼ wðxÞexp½ikðy ? ctÞ?;

w ¼ wðxÞexp½ikðy ? ctÞ?:

The ordinary-differential equations can be obtained from

(21) as follows:

ð23Þ

ð24Þ

w00ðxÞ þ bw0ðxÞ ? k2wðxÞ ¼ 0;

w00ðxÞ þ bw0ðxÞ þ

?

wðxÞ ¼ A1er1xþ A2er2x;

where r1;2¼

of Eq. (26) is obtained as

wðxÞ ¼ C1esxcoskx þ C2esxsinkx;

where s = ?b/2, k ¼1

Substituting Eqs. (27) and (28) into Eq. (20), the electric

potential can be obtained as

ð25Þ

c2

c2

sh

? 1

e0

15

e0

11

?

44þ

?

. The solution of Eq. (25) is

k2wðxÞ ¼ 0;

ð26Þ

where c2

sh¼1

q0

c0

2

?

ð27Þ

?b?

ffiffiffiffiffiffiffiffiffiffiffi

b2þ4k2

2

p

. When c2> c2

shð1 þb2

4k2Þ, the solution

ð28Þ

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4k2ðc2

c2

sh? 1Þ ? b2

q

.

/ðx;y;tÞ ¼ A1er1xþ A2er2xþe0

? exp½ikðy ? ctÞ?;

when c2> c2

can be given by Eq. (13) as follows:

15

e0

11

C1esxcoskx þ C2esxsinkx

ðÞ

ð29Þ

shð1 þb2

4k2Þ. The electric potential in the vacuum

/aðx;y;tÞ ¼ Aaekxexp½ikðy ? ctÞ?:

ð30Þ

J. Du et al. / Ultrasonics 46 (2007) 13–22

15

Page 4

4. Phase velocity equations

4.1. Solution for the electrically open conditions at the free

surface

Substituting Eqs. (18) and (19) into Eqs. (12) and (5), the

shear stresses and the electric displacement of the substrate

can be obtained as follows:

sm

sm

xz¼ ?cm

yz¼ icm

Dm

44kbmAme?kbmxexp½ikðy ? ctÞ?;

44kAme?kbmxexp½ikðy ? ctÞ?;

x¼ em

ð31Þ

ð32Þ

ð33Þ

11kBme?kxexp½ikðy ? ctÞ?:

Substituting Eqs. (28) and (29) into Eqs. (4), (5), we can

obtain the stresses and the electric displacements of FGPM

layer

syz¼ c0

44ebxikðC1esxcoskx þ C2esxsinkxÞexp½ikðy ? ctÞ?

þ e0

?

szx¼ c0

þ C2kesxcoskxÞexp½ikðy ? ctÞ?

þ e0

15ebxik A1er1xþ A2er2xþe0

15

e0

11

ðC1esxcoskx

?

þC2esxsinkxÞ

exp½ikðy ? ctÞ?;

ð34Þ

44ebxðC1sesxcoskx ? C1kesxsinkx þ C2sesxsinkx

15ebxA1r1er1xþ A2r2er2xþe0

15

e0

11

ðC1sesxcoskx

?

?C1kesxsinkx þ C2sesxsinkx þ C2kesxcoskxÞ

? exp½ikðy ? ctÞ?;

Dx¼ ð?A1r1e0

Dy¼ ð?ike0

?

ð35Þ

ð36Þ

ð37Þ

11ebxþr1x? A2r2e0

11ebxþr1x? ike0

11ebxþr2xÞexp½ikðy ? ctÞ?;

11ebxþr2xÞexp½ikðy ? ctÞ?:

From the boundary conditions (15) and continuity condi-

tions (17), we can obtain the following algebraic equations

to determine the unknown constants C2,C1,A1,A2,A0,

Am,Bm

c0

44e?bx?shscoskh þ c0

44e?bh?shksinkh:

þe0

15

e0

11

2

se?bh?shcoskh þe0

15

e0

11

2

ke?bh?shsinkh

!

C1

þ

c0

44e?bx?shkcoskh ? c0

44se?bh?shsinkh:

þe0

15

e0

11

15r1e?bh?rh

e?shC1coskh ?e0

þ A1e?r2h? A0e?kh¼ 0;

2

ke?bh?shcoskh ?e0

15

e0

11

2

se?bh?shsinkh

!

C2

þ e0

15

e0

11

1A1þ e0

15r2e?bh?r2hA2¼ 0;

e?shC2sinkh þ A1e?r1h

ð38aÞ

e0

15

e0

11

ð38bÞ

A1r1e0

Am¼ C1;

44s þe0

11e?bh?r1hþ A2r2e0

11e?bh?r2h? A0ke0e?kh¼ 0;

ð38cÞ

ð38dÞ

c0

15

e0

11

2

s

!

C1þ

c0

44k þe0

15

e0

11

2

k

!

C2þ A1r1e0

15

þ A2r2e0

A1þ A2þe0

A1r1e0

15þ Amcm

C1? Bm¼ 0;

11þ Bmkem

44kbm¼ 0;

ð38eÞ

15

e0

11

ð38fÞ

ð38gÞ

11þ A2r2e0

In order to obtain the nontrivial solutions of unknown

constants C2,C1,A1,A2,A0,Am,Bm, the determinant of

the coefficient matrix of linear algebraic Eqs. (38a)–

(38g) have to equal zero, so we can obtain the phase

velocity equation

11¼ 0:

k2ehðsþbÞe0em

11ðehr1?ehr2Þðe0

2e0em

?cosðhkÞehsðehr1þehr1ÞÞ?2kðr2ehr2?r1ehr1Þ

?sinðhkÞehsðem

?r1r2ehsðehr1?ehr2Þe0

þr1r2sinðhkÞe0

?ehðsþbÞ=ðe0

!

þr2ðehðsþbþr2Þe0þehðsþr1Þem

þðr1kehðsþbþr2Þ?r2kehðsþbþr1ÞÞe0

15

2þc0

44e0

11Þðs2þk2ÞsinðhkÞ

þk2e0

15 11kðr2?r1Þehbððe2hsþehðr2þr1ÞÞ

11þehbe0Þðe0

15

2þc0

11ðehbe0þem

11?kem

44e0

44e0

11Þe0

11ÞkcosðhkÞ

11ðs2þk2Þ

15

2ke0

15

4ðse0

11Þke0ðehr1?ehr2Þ

11Þþr1r2sinðhkÞehsðehr1?ehr2Þe0

15

2þc0

11

?

c0

44e0

11

2ðs2þk2Þþe0

15

2ðs2e0

11?ksem

11þe0

11k2Þ

þehbc0

ðe0

44se0

2þc0

11e0

15

2ke0

1544e0

11Þ

þcm

44ke0

11bmððr1ðehðsþbþr1Þe0þehðsþr2Þem

11Þ

11ÞÞke0

11ðsinðhkÞs?kcosðhkÞÞ

2ke0em

1511sinðhkÞ=ðe0

15

2þc0

44e0

11Þ

?ehsðehr1?ehr2Þðr

e0

11

1

2

?ehbk2e0em

!

11ÞðsinðhkÞs?kcosðhkÞÞ

?r1r2ehbe0

11e0

2þc0

15

2ke0sinðhkÞ

44e0

ðe0

1511Þ

¼0:

ð39Þ

4.2. Solution for the electrically short conditions at the free

surface

From the boundary condition (16) and continuity

conditions Eq. (17a) and (17b), we can obtain the

following algebraic equations of the unknown constants

A1,A2,C1,C2,Am,Bm

under

condition

A1e?r1hþ A2e?r2hþe0

e0

11

theelectricallyshort

15

C1e?shcoskh ?e0

15

e0

11

C2e?shsinkh ¼ 0:

ð40Þ

According to Eqs. (40), (38a) and (38d)–(38g), we can ob-

tain the following phase velocity equation for the electri-

cally short case:

16

J. Du et al. / Ultrasonics 46 (2007) 13–22

Page 5

? ðr2? r1Þðehsðehr1þ ehr2ÞcosðhkÞ ? ðe2hsþ ehðr2þr1ÞÞÞ

? e0

þ ðr2? r1ÞehssinðhkÞehr1ðe0

!

? ðs2þ k2ÞsinðhkÞ þ cm

? ð?sinðhkÞs þ kcosðhkÞÞ þ ehsðr1ehr1? r2ehr2Þe0

? ð?sinðhkÞs þ kcosðhkÞÞ þ ehsðr2ehr1? r1ehr2Þ

? e0

? e0

15

2kem

11k þ ðr2þ r1ÞehssinðhkÞðehr1? ehr2Þe0

15

2ksem

11

15

2þ c0

44e0

11Þe0

11ðs2þ k2Þ

þ r1ehsðehr1? ehr2Þe0

15

2

r2e0

11ðsinðhkÞs þ kcosðhkÞÞ

?r2e0

15

2kem

2þ c0

11sinðhkÞ

44e0

ðe0

1511Þ

? ehsðehr1? ehr2Þkðe0

15

2þ c0

44e0

11Þem

11

44ke0

11bmðehsðehr1? ehr2Þkem

11

11

15

2kem

11sinðhkÞ=ðe0

11sinðhkÞ=ðe0

15

2þ c0

2þ c0

44e0

11Þ þ ehsr1r2ðehr1? ehr2Þ

11ÞÞ ¼ 0:

15

2e0

15 44e0

ð41Þ

Eqs. (39) and (41) are the phase velocity equation of Love

wave propagating in the layered piezoelectric structure for

the electrically open and short cases, respectively. It is

readily seen that the phase velocity c is related to the

wavelength k, layer thickness h, elastic constant c44, dielec-

tric coefficient e, piezoelectric constant e15, and decaying

factor b.

5. Solution of the stress fields

For the electrically open circuit case, we can obtain from

Eqs. (38a)–(38g) that

Fig. 2. Phase velocity of the first mode for the electrically short case with

h = 0.0002.

Fig. 3. Phase velocity of the first mode for the electrically open case with

h = 0.0002.

-1 -0.9-0.8 -0.7 -0.6-0.5-0.4 -0.3 -0.2-0.10

x 10

-4

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

x 10

-7

Coordinate in thickness direction of piezoelectric layer

Displacement W(m)

beta=0

beta=2000

beta=3000

beta=4000

Fig. 4. Displacement of the first mode for electrically open with

h = 0.0001.

-1-0.9

Coordinate in thickness direction of piezoelectric layer

-0.8 -0.7 -0.6-0.5 -0.4-0.3 -0.2-0.10

x 10

-4

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-7

Displacement W(m)

beta=0

beta=2000

beta=3000

beta=4000

Fig. 5. Displacement of the second mode for electrically open with

h = 0.0001.

J. Du et al. / Ultrasonics 46 (2007) 13–22

17