Full text of DOI:10.1007/s004190000119

Archive of Applied Mechanics 70 ꢀ2000) 707±714 Ó Springer-Verlag 2000
Interface crack between piezoelectric and elastic strips
J. H. Kwon, K. Y. Lee
707
Summary An interface crack between piezoelectric and elastic strips is analyzed using the
theory of linear piezoelectricity. The combined out-of-plane mechanical and in-plane electrical
loads are applied to the layered strip. Fourier transforms are used to reduce the problem to a
pair of dual integral equations, which is then expressed in terms of a Fredholm integral
equation of the second kind. The stress intensity factor is determined, and numerical analysis is
performed and discussed.
Key words piezoelectricity, layered strip, interface crack, anti-plane shear, stress intensity factor
1
Introduction
Studies on piezoelectric materials were reported actively during the last decade. Since Curie
discovered the electro-mechanical coupling phenomenon of piezoelectric material in 1880, the
piezoelectric materials are used in various applications of the electro-mechanical devices such
as actuators, sensors and transducers. In practical structures, many piezoelectric devices
consist of both piezoelectric and elastic layers, and the failure of the piezoelectric layers, due to
their brittle nature, can in¯uence the performance of such devices. In [1] it was reported that
the brittle fracture of the piezoelectric composite structures could be initiated from the
interface of dissimilar materials. Parton [2] considered the problem of a ®nite interface crack
between two piezoelectric materials subjected to a far ®eld uniform tension. A three-dimen-
sional analysis of a semi-in®nite crack in a piezoelectric material was performed in [3] by
means of the eigenfunction expansions method. More recently, a complete form of local stress
and electric displacement ®elds was derived in [4] for an interfacial crack between two dis-
similar anisotropic piezoelectric media. In [5], the solution for an anti-plane shear crack
between two dissimilar in®nite piezoelectric materials was proposed. Paper [6] considered a
piezoelectric medium bonded between homogeneous half-space elastic materials with an anti-
plane shear interface crack. Solutions for an interface crack in piezoelectric and elastic mate-
rials are relatively rare in the literature.
A crack problem for the body with a piezoelectric strip bonded between two dissimilar elastic
strip materials containing an interface ®nite crack under combined out-of-plane mechanical and
in-plane electrical loads is considered in this paper. Layered strips consisting of piezoelectric and
elastic materials are most widely adopted in many practical devices, and the present model is
easily matched with the structure of shear-type piezoelectric accelerometers, [7]. Fourier
transforms are used to reduce the problem to a pair of dual integral equations, which is then
expressed as a Fredholm integral equation of the second kind. The stress intensity factor 4SIF) is
determined, and the numerical analysis is prepared and discussed.
Received 22 September 1999; accepted for publication 3 May 2000
J. H. Kwon, K. Y. Lee 4&)
Department of Mechanical Engineering, Yonsei University,
Seoul 120±749, Korea
phone & fax: 82-2-2123-2813,
e-mail: fracture@yonsei.ac.kr
This work was supported by the Brain Korea 21 Project, and the
authors are also grateful for the support provided by a grant from
the Korea Science & Engineering Foundation 4KOSEF) and Safety
and Structural Integrity Research Center at the Sungkyunkwan
University.

2
Problem statement
Consider an interface crack of a piezoelectric medium that is bonded between two dissimilar
elastic strip layers as shown in Fig. 1. A Grif®th interface crack of length 2a is located between
the piezoelectric layer of thickness h1 and the lower elastic layer 1 of thickness h2. The thickness
of the upper elastic layer 2 is h3. The cartesian coordinates ꢀx; y; z† is set at the center of a crack
for reference. The piezoelectric medium is considered to be transversely isotropic with hex-
agonal symmetry, which has an isotropic basal plane at the xy-plane and a poling direction of
the z-axis. The three-layered strip is subjected to a combined constant shear stress s0 at the
edges, which contains the effects of both mechanical and electrical loads due to the electro-
mechanical coupling phenomenon of the piezoelectric material.
708
The piezoelectric boundary value problem is simpli®ed considerably under the out-of-plane
mechanical displacement and in-plane electric ®elds such that
uꢀxp† ˆ uꢀyp† ˆ 0; uzꢀp† ˆ wꢀp†ꢀx; y† ;
ꢀ1†
ꢀ2†
Ex ˆ Exꢀx; y†; Ey ˆ Eyꢀx; y†; Ez ˆ 0 ;
uꢀxe † ˆ uꢀye † ˆ 0; uzꢀe † ˆ wꢀe †ꢀx; y† ꢀ j ˆ 1; 2† ;
ꢀ3†
j
j
j
j
where the superscripts ꢀ p† and ꢀej† imply the piezoelectric and elastic layers, respectively.
In this case, the basic relations become
rꢀp†ꢀx; y† ˆ c44cꢀp†ꢀx; y† À e15Eiꢀx; y† ;
ꢀ4†
zi
zi
ꢀej†
ꢀej† ꢀej†
r
ꢀx; y† ˆ c44
c
ꢀx; y† ;
ꢀ5†
ꢀ6†
zi
zi
Eiꢀx; y† ˆ À/ꢀx; y†;i ;
Diꢀx; y† ˆ e15cꢀp†ꢀx; y† ‡ ꢀ11Eiꢀx; y† ;
ꢀ7†
zi
where /ꢀx; y† is the electric potential, rziꢀx; y†, cziꢀx; y†, Eiꢀx; y† and Diꢀx; y† ꢀi ˆ x; y† are the
components of the stress, strain, electric ®eld and electric displacement vectors, respectively.
Also, c44, e15 and ꢀ11 are the elastic stiffness of the piezoelectric material measured in a constant
electric ®eld, the piezoelectric constant and the dielectric permittivity measured at a constant
ꢀej†
strain, respectively. And c are the shear moduli of the elastic materials.
44
The anti-plane governing equations are simpli®ed to
2
2
c44r wꢀp†ꢀx; y† ‡ e15r /ꢀx; y† ˆ 0 ;
ꢀ8†
ꢀ9†
2
2
e15r wꢀp†ꢀx; y† À ꢀ11r /ꢀx; y† ˆ 0 ;
2
r wꢀe †ꢀx; y† ˆ 0 ꢀ j ˆ 1; 2† ;
ꢀ10†
j
Fig. 1. A layered piezoelectric strip with an inter-
face crack

2
2
2
where r ˆ ꢀo =ox2† ‡ ꢀo =oy2† is the two-dimensional Laplacian operator.
The following boundary conditions are considered:
ꢀp†
‡
ꢀe1†
r ꢀx; 0 † ˆ r ꢀx; 0À† ˆ 0; jxj < a ;
ꢀ11†
zy
zy
wꢀp†ꢀx; 0† ˆ wꢀe †ꢀx; 0†; jxj ꢁ a ;
ꢀ12†
ꢀ13†
1
ꢀp†
ꢀe1†
r ꢀx; 0† ˆ r ꢀx; 0†; jxj ꢁ a ;
zy
zy
wꢀp†ꢀx; h1† ˆ wꢀe †ꢀx; h1†; jxj < 1 ;
ꢀ14†
ꢀ15†
2
709
rꢀzpy†ꢀx; h1† ˆ rzꢀey †ꢀx; h1†; jxj < 1 ;
2
ꢀ
ꢀe1†
ꢀe2†
r
ꢀx; Àh † ˆ r ꢀx; h4† ˆ s0;
zy
zy
2
4case 1)
4case 2)
jxj < 1 ;
jxj < 1 ;
ꢀ16†
Dyꢀx; 0† ˆ Dyꢀx; h1† ˆ D0;
ꢀ
ꢀe1†
ꢀe2†
r
ꢀx; Àh † ˆ r ꢀx; h4† ˆ s0;
zy
zy
2
ꢀ17†
Eyꢀx; 0† ˆ Eyꢀx; h1† ˆ E0;
where s0 is the constant shear stress combined by the purely mechanical shear stress sÃ0 and
uniform electric displacement D0 or uniform electric ®eld E0.
In this interface crack problem between the piezoelectric and elastic materials, it is not
needed to consider the electrical boundary condition of the crack face, which is under heavy
debate, [2, 8±18].
3
Solution procedure
For the piezoelectric medium, the anti-plane governing Eqs. 48) and 49) give
2
2
r wꢀp†ꢀx; y† ˆ 0; r /ꢀx; y† ˆ 0 :
ꢀ18†
Applying Fourier cosine transforms, [19], to Eqs. 410), 418), and adding the effects of
thickness with respect to loads given at the edges, the solutions of the mechanical displace-
ments and electric potential are formed as follows:
1
Z
2
wꢀp†ꢀx; y† ˆ
fA1ꢀa† expꢀay† ‡ A2ꢀa† expꢀÀay†g cosꢀax†da ‡ a0y ;
ꢀ19†
ꢀ20†
ꢀ21†
ꢀ22†
p
0
1
Z
2
/ꢀx; y† ˆ
p
fB1ꢀa† expꢀay† ‡ B2ꢀa† expꢀÀay†g cosꢀax†da À b0y ;
0
1
Z
2
wꢀe †ꢀx; y† ˆ
fC1ꢀa† expꢀay† ‡ C2ꢀa† expꢀÀay†g cosꢀax†da ‡ c0y ;
1
p
0
1
Z
2
wꢀe †ꢀx; y† ˆ
fD1ꢀa† expꢀay† ‡ D2ꢀa† expꢀÀay†g cosꢀax†da ‡ d0y ‡ e0 ;
2
p
0
where Ajꢀa†; Bjꢀa†; Cjꢀa† and Djꢀa† ꢀj ˆ 1; 2† are the unknown functions, and a0; b0; c0; d0 and
e0 are real constants.
The ®eld components of the piezoelectric and elastic layers can be obtained from a simple
calculation. Then, applying the boundary conditions, the unknown functions and real constants
are determined as follows:

ꢀ11s0 ‡ e15D0
c44ꢀ11 ‡ e215
c44D0 À e15s0
c44ꢀ11 ‡ e215
s0
a0 ˆ
d0 ˆ
;
b0 ˆ
;
c0 ˆ
;
ꢀe1†
c
44
4case 1)
4case 2)
423a)
423b)
ꢀe2†
f
s0
c44 À c44
e15
;
e0 ˆ À
h1s0 ‡
h1D0 ;
ꢀe2†
ꢀe2†
f
c44ꢀ11
f
c
c44c44
44
s0 ‡ e15E0
s0
a0 ˆ
d0 ˆ
;
b0 ˆ E0; c0 ˆ
;
c44
ꢀe1†
c
44
c44 À cꢀe †
2
s0
e
h1s0 ‡ 15 h1E0 ;
44
;
e0 ˆ À
ꢀe2†
c44
c44cꢀe †
2
710
c
44
44
A1ꢀa† ˆ Mꢀa†A2ꢀa† ;
ꢀ24†
ꢀ25†
8
e
15 Mꢀa†A2ꢀa† 4case 1) ,
<
ꢀ11
B1ꢀa† ˆ
B2ꢀa† ˆ
:
0
4case 2) ,
8
e
15 A2ꢀa† 4case 1) ,
<
ꢀ11
ꢀ26†
:
0
4case 2) ,
cÃ44
1 À Mꢀa†
C1ꢀa† ˆ À
A2ꢀa† ;
ꢀ27†
ꢀ28†
ꢀ29†
ꢀ30†
ꢀe1†
1 À expꢀÀ2ah2†
c
44
cÃ44
1 À Mꢀa†
C2ꢀa† ˆ À
expꢀÀ2ah2†
A2ꢀa† ;
A2ꢀa† ;
ꢀe1†
1 À expꢀÀ2ah2†
c
44
2cÃ44
expꢀÀ2ah4†
D1ꢀa† ˆ
ꢀe2†
Ã
1 ‡ expꢀÀ2ah3†
c44 ‡ c44 tanhꢀah3†
2cÃ44
1
D2ꢀa† ˆ
A2ꢀa† ;
ꢀe2†
Ã
1 ‡ expꢀÀ2ah3†
c44 ‡ c44 tanhꢀah3†
where
c4Ã4 À cꢀ4e42† tanhꢀah3†
c4Ã4 ‡ cꢀ4e42† tanhꢀah3†
Mꢀa† ˆ
expꢀÀ2ah1† ;
ꢀ31†
ꢀ32†
ꢀ
e215
ꢀ11
cÃ44
ꢂ
f
c44 ˆ c44
‡
4case 1) ,
4case 2) ,
c44
f
and c44 is the piezoelectric stiffened elastic modulus.
From Eqs. 423)±432), the mixed boundary conditions, Eqs. 411) and 412), lead to a set of dual
integral equations in the form
1
Z
ꢀe1†
p c c0
44
af1 À Mꢀa†gA2ꢀa† cosꢀax†da ˆ
;
jxj < a ;
ꢀ33†
ꢀ34†
2 cÃ44
0
"
#
1
Z
cÃ44
f1 ‡ Mꢀa†g ‡
cothꢀah2†f1 À Mꢀa†g A2ꢀa† cosꢀax†da ˆ 0; jxj ꢁ a :
ꢀe1†
c
44
0

Equations 433) and 434) can be solved by applying the method of Copson [20], and the
solution is found as follows:
R
a nUꢀn†J0ꢀan†dn
0
A2ꢀa† ˆ
;
ꢀ35†
cÃ44
f1 ‡ Mꢀa†g ‡
cothꢀah2†f1 À Mꢀa†g
ꢀe1†
c
44
where J0ꢀ † is the zero-order Bessel function of the ®rst kind.
Inserting Eq. 435) into Eqs. 433) and 434), the new function Uꢀn† should satisfy the Fredholm
integral equation of the second kind in the form
711
a
Z
p cÃ44 ‡ cꢀ4e41†
Uꢀn† ‡ Kꢀn; g†Uꢀg†dg ˆ
c0 ;
ꢀ36†
2
cÃ44
0
where
1
Z
Kꢀn; g† ˆ g a‰Fꢀa† À 1ŠJ0ꢀag†J0ꢀan†da ;
ꢀ37†
ꢀ38†
0
cÃ44 ‡ cꢀ4e41†
1 À Mꢀa†
Fꢀa† ˆ
:
cÃ44
ꢀe1†
c
44
f1 ‡ Mꢀa†g ‡
cothꢀah2†f1 À Mꢀa†g
ꢀe1†
c
44
The following variables and functions are introduced for the numerical analysis:
S
a ˆ
;
n ˆ aN; g ˆ aH;
a
ꢀ39†
p cÃ44 ‡ cꢀ4e41† WꢀN†
p cÃ44 ‡ cꢀ4e41† WꢀH†
p
p
Uꢀn† ˆ
c0
;
Uꢀg† ˆ
c0
:
2
cÃ44
2
cÃ44
H
N
Substituting Eq. 439) into Eqs. 436)±438) and 431), the normalized Fredholm integral
equation of the second kind can be obtained in the form
1
Z
p
WꢀN† ‡ LꢀN; H†WꢀH†dH ˆ N ;
ꢀ40†
ꢀ41†
0
where
1
ꢃ ꢁ ꢂ
ꢄ
Z
p
S
LꢀN; H† ˆ NH S F
À 1 J0ꢀSH†J0ꢀSN†dS :
a
0
In the case of a layered piezoelectric medium bonded between dissimilar half-space elastic
materials 4h2 ! 1 and h3 ! 1), we obtain
ꢁ ꢂ
S
ꢁ ꢂ
1 À M
S
a
cÃ44 ‡ cꢀ4e41†
a
ꢅ
ꢁ ꢂꢆ
ꢅ
ꢁ ꢂꢆ
F
ˆ
;
ꢀ42†
ꢀ43†
cÃ44
ꢀe1†
S
S
c
44
1 ‡ M
‡
1 À M
ꢀe1†
a
a
c
44
ꢁ ꢂ
ꢁ
ꢂ
S
cÃ44 À cꢀ4e42†
h1
M
ˆ
exp À2S
:
cÃ44 ‡ cꢀ4e42†
a
a
ꢀe1†
ꢀe2†
In this case, the numerical solution for c ˆ c
agrees with the earlier one [6].
44
44

The solution WꢀN† of Eq. 440) can be numerically calculated by using Gaussian quadrature
formula.
The mode III SIF is determined by considering Eqs. 423), 435) and 439) as well as the stress
®elds near the crack tip in the form
p
p
KIII ꢂ lim 2pꢀx À a†rꢀzky†ꢀx; 0† ˆ s0 paWꢀ1†; k ˆ p; e1 :
ꢀ44†
x!a
4
Numerical results and discussions
Because the combined shear stress s0 contains the effects of both mechanical and electrical
loads, considering Eqs. 44), 419) and 423), s0 can be divided into two terms
712
8
f
c
e15
>
<
44sÃ0 À
D0 4case 1),
4case 2) ,
c44
ꢀ11
s0 ˆ
ꢀ45†
>
:
sÃ0 À e15E0
where sÃ0 is the purely mechanical shear stress at zero electrical loads.
We de®ne the following normalized electrical loads for the numerical analysis:
c44e15D0
DÃ0 ꢂ
EÃ0 ꢂ
4case 1);
Ã
f
c44ꢀ11s0
e15E0
s0Ã
ꢀ46†
4case 2) ;
where DÃ0 and EÃ0 are the normalized electric displacement and normalized electric ®eld,
respectively.
PZT-5H, aluminium and epoxy are considered as the strip materials. Material properties are
given in Table 1.
p
The normalized SIF KIII=s0 pa vs. the ratio of the crack length to the piezoelectric layer
thickness 2a=h1 is shown in Fig. 2 in the case of h1=h2 ˆ 1 and h1=h3 ˆ 1 under the condition
of case 1. The normalized SIF increases with increase of the normalized crack length 2a=h1. The
structure of epoxy/PZT-5H/epoxy has the largest SIF, and the SIF for aluminium/PZT-5H/
aluminium is smallest. The SIFs for both aluminium/PZT-5H/epoxy and epoxy/PZT-5H/
aluminium are the same in this case.
p
Figure 3 shows the normalized SIF KIII=s0 pa vs. the ratio of the piezoelectric layer
thickness to the lower elastic layer thickness h1=h2 in the case of 2a=h1 ˆ 2, h1=h3 ˆ 1. The
normalized SIF increases with the increase of h1=h2. The SIF depends on the stiffness of the
lower elastic layer containing an interface crack.
p
Figure 4 shows the normalized SIF KIII=s0 pa vs. the ratio of the piezoelectric layer
thickness to the upper elastic layer thickness h1=h3 in the case of 2a=h1 ˆ 2 and h1=h2 ˆ 1. The
ratio h1=h3 affects the SIF less than h1=h2.
p
The SIF normalized by the purely mechanical shear stress, KIII=sÃ0 pa vs. the normalized
electric displacement DÃ0 is shown in Fig. 5. Regardless of the composition and thickness ratios
of the structures, the increase of the positive normalized electric displacement decreases the
SIF. In contrast, the negative normalized electric displacement with opposite direction induces
the increase of the SIF.
The similar results are obtained under the condition of case 2.
5
Conclusions
The three-layered strip with a piezoelectric strip between two elastic strips containing a ®nite
crack at the interface has been considered. In this interface crack problem between the
Table 1. Material properties
ꢀe†
Materials
c44 or c
e15ꢀC=m2†
ꢀ11ꢀÂ10À10 C=Vm†
44
ꢀÂ1010 N=m2†
PZT-5H
Aluminium
Epoxy
2.3
2.65
0.176
17.0
0
0
150.4
±
±

713
p
Fig. 2. Normalized SIF KIII=s0 pa vs. the ratio
of crack length to piezoelectric layer thickness
2a=h1 4case 1)
p
Fig. 3. Normalized SIF KIII=s0 pa vs. the
thickness ratio of piezoelectric layer to lower
elastic layer h1=h2 4case 1)
p
Fig. 4. Normalized SIF KIII=s0 pa vs. the
thickness ratio of piezoelectric layer to upper
elastic layer h1=h3 4case 1)

714
p
Fig. 5. Normalized SIF KIII=sÃ0 pa vs. normalized
electric displacement DÃ0 4case 1)
piezoelectric and elastic materials, it is not needed to consider the electrical boundary
condition of the crack face, and the fracture criterion is determined only by the SIF. The
numerical results show that the SIF increases with the increase of the normalized crack length
and the thickness ratio of piezoelectric layer to the lower elastic layer. The ratio of the
piezoelectric layer thickness to the upper elastic layer thickness little affects the SIF. The SIF
depends also on the magnitudes and directions of electrical loads.
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