fracture analysis of a functionally graded piezoelectric strip

Composite Structures 69 (2005) 294–300

www.elsevier.com/locate/compstruct

Fracture analysis of a functionally graded piezoelectric strip

Li Ma a,b,*, Lin-Zhi Wu a, Zhen-Gong Zhou a, Li-Cheng Guo a

a Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, PR Chinab School of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, PR China

Available online 11 August 2004

Abstract

In the present paper, the electroelastic behavior of a Griffith crack in a functionally graded piezoelectric strip is investigated. It is

assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric strip vary

continuously as an exponential function, and that the strip is under out of plane mechanical loading and in plane electrical loading.

By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as

unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displace-

ment and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples

are provided to show the effects of material properties on the stress and electric displacement intensity factors.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded piezoelectric materials (FGPMs); Crack; Stress intensity factor (SIF); Electric displacement intensity factor (EDIF);

Schmidt method

1. Introduction

In the design of piezoelectric materials, it is important

to take into account imperfections, such as cracks, thatare often pre-existing or are generated by external loads

during the service life. The crack problems of piezoelec-

tric materials have received much attention in recent

decades. On the other hand, the development of func-

tionally graded materials (FGMs) has demonstrated

that they have the potential to reduce the stress concen-

tration and increase the fracture toughness. Conse-

quently, the concept of FGMs can be extended to thepiezoelectric materials to improve the reliability of pie-

zoelectric materials and structures. These new kinds

of materials with continuously varying properties may

be called functionally graded piezoelectric materials

0263-8223/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2004.07.007

* Corresponding author. Address: Center for Composite Materials,

Harbin Institute of Technology, P.O. Box 1247, Harbin 150001, PR

China. Tel.: +86 451 86402323.

E-mail address: [email protected] (L. Ma).

(FGPMs). The advantages of using a device wholly

made of the FGPMs or using the FGPMs as a transit

layer instead of the bonding agent are that no discerni-

ble internal seams or boundaries exist. And no internalstress peaks are caused when the voltage is applied

and the failure from internal de-bonding or stress peaks

developed in conventional bimorphs can be avoided

[1,2]. Because the FGPMs are just emerging class of pie-

zoelectric materials, researches on their fracture mecha-

nical behaviors are still very few. Wang and Noda [3]

studied the crack problem of FGPMs under the thermal

loading, but they considered the FGPM layer as multi-layered homogeneous media. Recently, the Mode-III

crack problem of FGPMs has been considered by Li

and Weng [4]. And Wang [5] investigated the similar

problem by using a somewhat different approach.

In the present paper, stress and electric displacement

intensity factors are determined in a functionally graded

piezoelectric strip containing a Griffith crack under out

of plane mechanical loading and in plane electrical load-ing. The Fourier transform technique is used to reduce

mailto:[email protected]

L. Ma et al. / Composite Structures 69 (2005) 294–300 295

the mixed boundary value conditions to a set of dual

integral equations. The equations are solved by expand-

ing the differences of crack surface displacements and

electric potential in a series of Jacobi polynomials. The

unknown coefficients in the series are then determined

using the Schmidt method [6–10] and the stress and elec-tric displacement intensity factors are numerically calcu-

lated for the functionally graded piezoelectric strip.

2. Formulation of the problem

Consider a functionally graded piezoelectric strip that

has the infinite length and contains a crack of length 2awith reference to the rectangular coordinate system

(x,y), as shown in Fig. 1. The piezoelectric boundary

value problem can considerably be simplified for the

case of out of plane displacement and in plane electric

fields, i.e.

u ¼ v ¼ 0; w ¼ wðx; yÞ ð1Þ

Ex ¼ Exðx; yÞ; Ey ¼ Eyðx; yÞ; Ez ¼ 0 ð2Þwhere u, v, w are displacement and Ex, Ey, Ez are com-

ponents of the electric field vector. Assume the material

properties vary along the y-axis, the constitutive equa-

tions can be written as

sxz ¼ c44ðyÞowox

þ e15ðyÞo/ox

; syz ¼ c44ðyÞowoy

þ e15ðyÞo/oy

Dx ¼ e15ðyÞowox

� e11ðyÞo/ox

; Dy ¼ e15ðyÞowoy

� e11ðyÞo/oy

ð3a–dÞwhere c44(y), e15(y) and e11(y) are the shear modulus, the

piezoelectric constant and dielectric constant, sxz and syzare the shear stress components, Dx and Dy are the com-

ponents of the electric displacement vector, respectively.

And the electric field components can be related to the

electric potential / by using the relation

Ex ¼ � o/ox

; Ey ¼ � o/oy

ð4Þ

a a

1

2

yeβ)(44 yc , )(11 ye )(11 yε

x

y

h1

h2

Fig. 1. Geometry of the crack problem in a functionally graded

piezoelectric strip.

In order to overcome the complexity of mathematics,

we will focus the present study on a special class of

FGPMs in which the variations of the material proper-

ties have the same proportion [5,11,12]. Therefore, we

assume

ðc44; e15; e11Þ ¼ ðc0; e0; e0Þeby ð5Þ

Under the above consideration, the governing equa-

tions can be simplified to the following form:

c0o2wox2

þ o2woy2

� �þ e0

o2/ox2

þ o2/oy2

� �þ bc0

owoy

þ be0o/oy

¼ 0

e0o2wox2

þ o2woy2

� �� e0

o2/ox2

þ o2/oy2

� �þ be0

owoy

� be0o/oy

¼ 0

ð6a;bÞ

3. Solution of the problem

For the present problem, it is convenient to divide the

functionally graded piezoelectric strip into two regions,

namely material 1 in the upper strip with thickness h1and material 2 in the lower strip with thickness h2.

The governing equations (6) are solved by using theFourier transform technique. The general expressions

for displacement components and electric potentials

can be written as

w1ðx; yÞ ¼2

p

Z 1

0

A1ðsÞe�c1y þ A2ðsÞe�c2y½ � cosðsxÞds;

0 6 y 6 h1

w2ðx; yÞ ¼2

p

Z 1

0

A3ðsÞe�c1y þ A4ðsÞe�c2y½ � cosðsxÞds;

� h2 6 y 6 0

ð7a;bÞ

/1ðx; yÞ ¼2

p

Z 1

0

B1ðsÞe�c1y þ B2ðsÞe�c2y½ � cosðsxÞdsþ e15e11

w1ðx; yÞ;

0 6 y 6 h1

/2ðx; yÞ ¼2

p

Z 1

0

B3ðsÞe�c1y þ B4ðsÞe�c2y½ � cosðsxÞdsþ e15e11

w2ðx; yÞ;

� h2 6 y 6 0

ð8a;bÞwhere

c1 ¼b þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ 4s2

q2

; c2 ¼b �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ 4s2

q2

ð9a;bÞ

Now consider the boundary and continuity conditions

(see Fig. 1). As discussed by Chen et al. [12], Shindo et al.

[13], Yu and Chen [14], Narita and Shindo [15], the con-

tinuity and boundary conditions of the present problem

are as follows:

296 L. Ma et al. / Composite Structures 69 (2005) 294–300

sð1Þyz ðx; 0þÞ ¼ sð2Þyz ðx; 0

�Þ ¼ �s0; jxj 6 a

sð1Þyz ðx; 0þÞ ¼ sð2Þyz ðx; 0

�Þ; jxj > a

w1ðx; 0þÞ � w2ðx; 0�Þ ¼ 0; jxj > a

/1ðx; 0þÞ � /2ðx; 0�Þ ¼ 0; jxj > a

syzðx; h1Þ ¼ Dyðx; h1Þ ¼ 0; jxj 6 1syzðx;�h2Þ ¼ Dyðx;�h2Þ ¼ 0; jxj 6 1Dð1Þ

y ðx; 0þÞ ¼ Dð2Þy ðx; 0�Þ ¼ �D0; jxj 6 a

Dð1Þy ðx; 0þÞ ¼ Dð2Þ

y ðx; 0�Þ; jxj > a

ð10a–hÞ

In this paper, we only consider that s0 and D0 are

positive. To solve the present problem, the jumps of

the displacement and the electric potential across the

crack surface can be defined as follows:

f1ðxÞ ¼ w1ðx; 0þÞ � w2ðx; 0�Þ; jxj 6 a ð11Þ

f2ðxÞ ¼ /1ðx; 0þÞ � /2ðx; 0�Þ; jxj 6 a ð12Þ

Applying the Fourier cosine transform and the

boundary condition, it can be obtained

l½c1A1ðsÞ þ c2A2ðsÞ� þ e0½c1B1ðsÞ þ c2B2ðsÞ�¼ l½c1A3ðsÞ þ c2A4ðsÞ� þ e0½c1B3ðsÞ þ c2B4ðsÞ�

c1B1ðsÞ þ c2B2ðsÞ ¼ c1B3ðsÞ þ c2B4ðsÞl½c1A1ðsÞe�c1h1 þ c2A2ðsÞe�c2h1 � þ e0½c1B1ðsÞe�c1h1

þ c1B2ðsÞe�c2h1 � ¼ 0

c1B1ðsÞe�c1h1 þ c2B2ðsÞe�c2h1 ¼ 0

l½c1A3ðsÞec1h2 þ c2A4ðsÞec2h2 � þ e0½c1B3ðsÞec1h2

þ c1B4ðsÞec2h2 � ¼ 0

c1B3ðsÞec1h2 þ c2B4ðsÞec2h2 ¼ 0

A1ðsÞ þ A2ðsÞ � A3ðsÞ � A4ðsÞ ¼ �f 1ðsÞ

B1ðsÞ þ B2ðsÞ þe0e0½A1ðsÞ þ A2ðsÞ� � ½B3ðsÞ þ B4ðsÞ�

� e0e0½A3ðsÞ þ A4ðsÞ� ¼ �f 2ðsÞ

ð13a–hÞ

where

l ¼ c44 þe215e11

ð14Þ

A superposed bar indicates the Fourier cosine trans-form in this paper. It is defined as follows:

�f ðsÞ ¼Z 1

0

f ðxÞ cosðsxÞdx;

f ðxÞ ¼ 2

p

Z 1

0

�f ðsÞ cosðsxÞdsð15a;bÞ

By solving equations (13) with eight functions, wehave

A1ðsÞ ¼c2

c2 � c1

ec1h1 ec1h2 � ec2h2ð Þec1ðh1þh2Þ � ec2ðh1þh2Þ

�f 1ðsÞ

A2ðsÞ ¼c1

c1 � c2


�f 1ðsÞ

A3ðsÞ ¼c2

c1 � c2


�f 1ðsÞ

A4ðsÞ ¼c1

c2 � c1


�f 1ðsÞ

B1ðsÞ ¼c2

c2 � c1


�f 2ðsÞ

� c2c2 � c1


e0e0�f 1ðsÞ

B2ðsÞ ¼c1

c1 � c2


�f 2ðsÞ

� c1c1 � c2


e0e0�f 1ðsÞ

B3ðsÞ ¼c2

c2 � c1


�f 2ðsÞ

� c2c2 � c1


e0e0�f 1ðsÞ

B4ðsÞ ¼c1

c2 � c1


�f 2ðsÞ

� c1c2 � c1


e0e0�f 1ðsÞ

ð16a–hÞ

Therefore, the present problem reduces to the deter-

mination of eight unknown functions Ai(s) and Bi(s),

i = 1,2,3,4. The boundary conditions can be applied to

yield two pair of integral equations

2

p

Z 1

0

�f 1ðsÞ cosðsxÞds ¼ 0; jxj > a

2

p

Z 1

0

ec1h1 � ec2h1ð Þ ec1h2 � ec2h2ð Þec1ðh1þh2Þ � ec2ðh1þh2Þ½ �

c1c2c2 � c1

�f 1ðsÞ cosðsxÞds

¼ 1

ls0 þ

e0D0

e0

� �; jxj 6 a

ð17a;bÞ

2

p

Z 1

0

�f 2ðsÞ cosðsxÞds ¼ 0; jxj > a

2

p

Z 1

0

ec1h1 � ec2h1ð Þ ec1h2 � ec2h2ð Þ e0�f 1 � e0�f 2

� �ec1ðh1þh2Þ � ec2ðh1þh2Þ½ �e0

c1c2c2 � c1

cosðsxÞds ¼ �D0

e0; jxj 6 a

ð18a;bÞDue to the symmetry of the geometry and the applied

loading on the y-axis, it is sufficient to consider the prob-

lem for 0 6 x 6 1 only. From the nature of the dis-

placement and the electric potential along the crack, it

can be found that the jumps of the displacement and

electric potential across the crack surface are finite, dif-


ferential and continuous functions. Therefore, the jumps

of the displacement and electric potential across the

crack surface can be expanded as the following series:

f1ðxÞ ¼X1n¼1

anPð1=2;1=2Þ2n�2

xa

1� x2

a2

� �1=2

; for 0 6 x 6 a

f1ðxÞ ¼ 0; for x > a

ð19a;bÞ

f2ðxÞ ¼X1n¼1

bnPð1=2;1=2Þ2n�2

xa

1� x2

a2

� �1=2

; for 0 6 x 6 a

f2ðxÞ ¼ 0; for x > a

ð20a;bÞwhere an and bn are unknown coefficients to be deter-

mined and P ð1=2;1=2Þn ðxÞ is a Jacobi polynomial. The Fou-

rier transformation of Eqs. (19) and (20) are [16]

�f 1ðsÞ ¼X1n¼1

anGn1

sJ 2n�1ðsaÞ ð21Þ

�f 2ðsÞ ¼X1n¼1

bnGn1

sJ 2n�1ðsaÞ ð22Þ

with

Gn ¼ 2ffiffiffip

pð�1Þn�1 Cð2n� 1=2Þ

ð2n� 2Þ! ð23Þ

where C(x) and Jn(x) are Gamma and Bessel functions,

respectively. Substituting Eqs. (21) and (22) into Eqs.

(17) and (18), it can be shown that Eqs. (17a) and

(18a) are automatically satisfied. After integration with

respect to x in [0,x], Eqs. (17b) and (18b) reduce to

X1n¼1

anGn

Z 1

0

1

s2uðsÞ c1c2

c2 � c1J 2n�1ðsaÞ sinðsxÞds

¼ ps02l

ð1þ jÞx ð24Þ

X1n¼1

bn �e0e0

an

� �Gn

Z 1

0

1

s2uðsÞ c1c2

c2 � c1J 2n�1ðsaÞ

sinðsxÞds ¼ � pD0

2e0x ð25Þ

where j is the combination parameter, defined as

j ¼ e0D0

e0s0and uðsÞ ¼ ec1h1�ec2h1ð Þ ec1h2�ec2h2ð Þ

ec1ðh1þh2Þ�ec2ðh1þh2Þ. From the

relation [17]

Z 1

0

1

sJnðsaÞ sinðbsÞds ¼

sin nsin�1ðb=aÞ� �

n; a > b

an sinðnp=2Þn bþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � a2

ph in; b > a

8>>>><>>>>:

ð26Þ

The semi-infinite integral in Eqs. (24) and (25) can be

modified asZ 1

0

1

s2uðsÞ c1c2

c2 � c1J 2n�1ðsaÞ sinðsxÞds

¼ 1

2ð2n� 1Þ sin ð2n� 1Þsin�1 xa

h i

þZ 1

0

1

suðsÞs

c1c2c2 � c1

� 1

2

� �J 2n�1ðsaÞ sinðsxÞds

ð27Þ

where lims!11s uðsÞ

c1c2c2�c1

¼ 12. The semi-infinite integral in

Eqs. (27) can be evaluated directly, so coefficients an and

bn in Eqs. (24) and (25) can now be solved by the Sch-

midt method [6–10]. For brevity, Eq. (24) can be rewrit-

ten as (Eq. (25) can be solved using a similar method as

follows)

X1n¼1

anEnðxÞ ¼ UðxÞ; 0 < x < a ð28Þ

where En(x) and U(x) are known functions and coeffi-

cients an are unknown and will be determined. A set

of functions Pn(x) which satisfy the orthogonality condi-

tionZ a

0

PmðxÞPnðxÞdx ¼ Nndmn; Nn ¼Z a

0

P 2nðxÞdx ð29Þ

can be constructed from the function En(x), such that

PnðxÞ ¼Xn

i¼1

Min

MnnEnðxÞ ð30Þ

where Min is the cofactor of the element din of determi-

nant Dn, which is defined as

Dn ¼

d11 d12 . . . d1n

d21 d22 . . . d2n

. . . . . . . . . . . .

dn1 dn2 . . . dnn

��

��; dij ¼

Z a

0

EiðxÞEjðxÞdx

ð31ÞFrom Eqs. (28)–(31), we obtain

an ¼X1j¼n

qj

Mnj

Mjjð32Þ

where

qj ¼1

Nj

Z a

0

UðxÞPjðxÞdx ð33Þ

4. Crack tip field intensity factors

Since coefficients an and bn are known, the entire

perturbation stress field and electric displacement

can be obtained. In fracture mechanics, however, it is

6

8

10

κ =0.5

κ =0.25

κ =0.0

κ =-0.25


important to determine the perturbation stress syz andthe perturbation electric displacement Dy in the vicinity of

crack tips. syz and Dy along the crack can be expressed as

sð1Þyz ðx; 0Þ ¼ � 2

p

X1n¼1

ðc0an þ e0bnÞGn

Z 1

0

uðsÞ

c1c2c2 � c1

1

sJ 2n�1ðsaÞ cosðsxÞds ð34Þ

Dð1Þy ðx; 0Þ ¼ 2

p

X1n¼1

ðe0bn � e0anÞGn

Z 1

0

uðsÞ c1c2c2 � c1

1

sJ 2n�1ðsaÞ cosðsxÞds ð35Þ

Observing the expressions in Eqs. (34) and (35), the

singular portion of the stress field and electric displace-

ment can be obtained, respectively, from the relation

[17].

Z 1

0

JnðsaÞ cosðbsÞds

¼

cos nsin�1ðb=aÞ� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � b2

p ; a > b

� an sinðnp=2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � a2

pbþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � a2

ph in; b > a

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

ð36Þ

The singular parts of the stress field and electric dis-

placement field can be expressed, respectively, as fol-

lows:

s ¼ � 2

p

X1n¼1

1

2ðc0an þ e0bnÞGnHn ð37Þ

D ¼ � 2

p

X1n¼1

1

2ðe0an � e0bnÞGnHn ð38Þ

where

Hn ¼ð�1Þna2n�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2

pxþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2

ph i2n�1ð39Þ

Thus, we obtain the stress intensity factor KrIII and elec-

tric displacement intensity factor KDIII as

0.0 0.2 0.4 0.6 0.8 1.0

2

4

k

h/a

κ =-0.5

Fig. 2. Normalized SIFs versus h/a with different j at b = 0.

KrIII ¼ lim

x!aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðx� aÞ

ps

¼ 2ffiffiffia

pX1n¼1

ðc0an þ e0bnÞCð2n� 1=2Þð2n� 2Þ! ð40Þ

KDIII ¼ lim

x!aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðx� aÞ

pD

¼ 2ffiffiffia

pX1n¼1

ðe0an � e0bnÞCð2n� 1=2Þð2n� 2Þ! ð41Þ

5. Results and discussions

It is observed from Eqs. (40) and (41) that the forms

of the stress intensity factor and the electric displace-

ment intensity factor in the FGPMs are similar to those

in the homogeneous piezoelectric material. Their values,however, are different because an and bn depend on the

properties of FGPMs. In addition, from Eq. (25), let

bn � e0e0an ¼ cn, so that cn is independent of the mechan-

ical loading. Substituting this relation into Eq. (35), we

can obtain

Dy ¼ � 2

p

X1n¼1

e0cnGn

Z 1

0

uðsÞ k1k2

k2 � k1

1

sJ 2n�1ðsaÞ

cosðsxÞds ð42Þ

Since cn is independent of the mechanical loading, the

electric displacement Dy is also independent of the

mechanical loading.

Fig. 2 illustrates the variation of the normalized stress

intensity factor (SIF) k3 ¼ KrIII=s0

ffiffiffiffiffiffipa

pas a function of the

strip thickness to crack-length ratio h/a (h = h1 = h2) fordifferent values of the combination parameter j at

b = 0. Note that when h/a increases the stress intensity

factor decreases. The presence of positive electric dis-

placement D0 leads to an increase in the stress intensity

factor. In contrast, the stress intensity factor decreases

as the electric displacement D0 increases in the negative

direction.

Fig. 3 shows the variation of the normalized stressintensity factor with h/a for different gradient parameter

ba at j = 0. It is evident that the normalized stress inten-

sity factor increases with the increasing of ba. And with

the increasing of h/a, the values of normalized stress

intensity factor at the crack tip will decrease.

Fig. 4 shows the variation of the stress intensity fac-

tor with the location h1/h2 of the crack for different

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

k

h1 /h2

βa=0.0βa=0.5βa=1.0βa=2.0βa=5.0

Fig. 5. Normalized SIFs versus h1/h2 with different ba at j = 0.

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

6

k

h1 /h2

κ =0.0κ =0.25κ =-0.25κ =-0.5κ =0.5

Fig. 4. Normalized SIFs versus h1/h2 with different j at b = 0.

0.0 0.2 0.4 0.6 0.8 1.0

2

4

6

k

h/a

βa=0.0βa=0.5β a=1.0β a=2.0β a=5.0

Fig. 3. Normalized SIFs versus h/a with different ba at j = 0.


combination parameter j at b = 0. It can be found that

the stress intensity factor at the crack tip decreases with

the increasing of h1/h2. These results indicate the free

surface of the strip has a more significant influence on

the normalized stress intensity factor and can easy be

understand.Plotted in Fig. 5 is the variation of the normalized

SIFs with the location h1/h2 of the crack for different

normalized gradient parameter ba when j = 0. Similar

phenomenon to Fig. 4 can be observed. When the ratio

h1/h2 decreases, the stress intensity factor at the crack tip

increases.

6. Conclusions

The electroelastic behavior of the functionally graded

piezoelectric strip with a Griffith crack has been analyzed

theoretically. The analysis is based upon an integral

transform technique. The Fredholm integral equation

is solved by using the Schmidt method. The present

method is applied to illustrate the fundamental behaviorof a crack in the functionally graded piezoelectric strip

the under out of plane mechanical loading and in plane

electrical loading. The numerical results show that the

gradient of the material property has a considerable ef-

fect on the fracture behavior of the functionally graded

piezoelectric strip.

Acknowledgements

The present work is supported by National Science

Fund for Distinguished Young Scholars under grant

no. 10325208, Trans-Century Training Programme

Foundation for the Talents by the Ministry of Educa-

tion, the Foundation of Heilongjiang Province for Out-

standing Talents, the Multidiscipline Scientific ResearchFoundation of Harbin Institute of Technology and the

Foundation of Fundamental Research in Advance for

National Defence.

References

[1] Wu CM, Kahn M, Moy W. Piezoelectric ceramic with functional

gradients: a new application in material design. J Am Ceram Soc

1996;79:809–12.

[2] Zhu X, Xu J, Meng Z, et al. Microdisplacement characteristics

and microstructures of functionally gradient piezoelectric ceramic

actuator. Mater Design 2000;21:561–6.

[3] Wang BL, Noda N. Thermally induced fracture of a smart

functionally graded composite structure. Theor Appl Fract Mech

2001;35:93–109.

[4] Li C, Weng GJ. Antiplane crack problem in functionally graded

piezoelectric materials. ASME J Appl Mech 2002;69:481–8.

[5] Wang BL. A mode-III crack in functionally graded piezoelectric

materials. Mech Res Commun 2003;30:151–9.


[6] Morse PM, Feshbach H. Methods of theoretical physics, vol. 1.

New York: McGraw-Hill; 1958.

[7] Yau WF. Axisymmetric slipless indentation of an infinite elastic

cylinder. SIAA J Appl Math 1967;15:219–27.

[8] Itou S. Three dimensional waves propagation in a cracked elastic

solid. ASME J Appl Mech 1978;45:807–11.

[9] Zhou Z, Wang B. The scattering of harmonic elastic anti-plane

shear waves by a Griffth crack in a piezoelectric material plane by

using the non-local theory. Int J Eng Sci 2002;40:303–17.

[10] Ma L, Wu LZ, Guo LC. Dynamic behavior of two collinear anti-

plane shear cracks in a functionally graded layer bonded to

dissimilar half planes. Mech Res Commun 2002;29:207–15.

[11] Jin B, Zhong Z. A moving mode-III crack in functionally graded

piezoelectric materials: permeable problem. Mech Res Commun

2002;29:217–24.

[12] Chen J, Liu Z, Zou Z. Electromechanical impact of a crack in a

functionally graded piezoelectric medium. Theor Appl Fract

Mech 2003;39:47–60.

[13] Shindo Y, Narita K, Tanaka K. Electroelastic intensification near

anti-plane shear crack in orthotropic piezoelectric ceramic strip.

Theor Appl Fract Mech 1996;25:65–71.

[14] Yu SW, Chen ZT. Transient response of a cracked infinite

piezoelectric strip under anti-plane impact. Fatigue Eng Mater

Struct 1998;21:1381–8.

[15] Narita K, Shindo Y. Scattering of anti-plane shear waves by a

finite crack in piezoelectric laminates. Acta Mech 1999;134:27–43.

[16] Erdelyi A, editor. Table of the integral transforms, vol. 1. New

York: McGraw-Hill; 1954.

[17] Gradshteyn IS, Ryzhik IM. Table of integral, series and

products. New York: Academic Press; 1980.

fracture analysis of a functionally graded piezoelectric strip

Documents