International Journal of Fracture 118: 285–301, 2002.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Conducting cracks in dissimilar piezoelectric media
H. G. BEOM1,∗ and S. N. ATLURI2
1 Department of Mechanical Engineering, College of Engineering, Chonnam National University, 300
Yongbong-dong, Gwangju, 500-757, Korea (∗ Author for correspondence: E-mail: hgbeom@chonnam.ac.kr)
2 Center for Aerospace Research & Education, 7704 Boelter Hall, School of Engineering and Applied Science,
University of California at Los Angeles, Los Angeles, CA 90095-1600, USA
Recieved 30 May 2002; accepted in revised form 27 November 2002
Abstract. Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic
piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown
that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of realvalued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for
basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral
associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors
due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the
interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.
Key words: Analytic functions, conservation integrals, electromechanical fracture.
1. Introduction
Piezoelectric ceramics are being widely used in various electromechanical devices. Active
components composed of piezoelectric ceramics are used in intelligent material systems.
The reliability issues associated with the active components such as actuators and sensors
are becoming important. Crack growth under electrical or electromechanical loading is responsible for failure of many electroceramic systems. The subject of cracks in piezoelectric
materials, for various failure modes, has thus received considerable attention (Parton, 1976;
Pak, 1990; Suo et al., 1992; Zhang et al., 2002). Among the various failure modes, the dielectric breakdown associated with growth of conducting cracks has received considerable
attention. Conducting cracks in linear piezoelectrics (Suo, 1993; Ru and Mao, 1999) and in
electrostrictive materials (Beom, 1999a, b) have been studied in the literature. Experimental
investigations to study the fracture criteria for conducting cracks in piezoelectric ceramics
have been carried out by Heyer et al. (1998) and Fu et al. (2000).
Interfacial fracture between piezoelectric ceramic layers has been identified as a major
failure mode. Considerable effort has been made to understand the mechanism of delamination
of layers in piezoelectric devices, under electromechanical loadings. Suo et al. (1992) examined the problem of an insulating crack between dissimilar anisotropic piezoelectric media.
They found that the interfacial insulating crack-tip field consists of two pairs of singularities;
0
0
r −1/2±iε and r −1/2±κ at distance r from the crack tip, where ε 0 and κ 0 are real numbers
depending on the material constants. Recently, Wang and Han (1999), and Gao and Wang
(2000) analyzed collinear permeable cracks between dissimilar piezoelectric materials. These
previous works have focused either on insulating interface cracks, or on permeable interface
286 H. G. Beom and S. N. Atluri
cracks. However, conducting cracks on interfaces between dissimilar piezoelectric materials
have not been examined at all.
It is the purpose of this study to investigate the problem of a conducting crack at an interface between dissimilar piezoelectric media. The problem is formulated using the complex
representation derived in this paper. Two generalized parameters for an anisotropic piezoelectric bimaterial, which are the only bimaterial parameters needed to describe the stress
field and electric field, for problems wherein tractions and electric field are prescribed at
the boundary, are proposed. The general form of the near tip fields for the interface crack
between dissimilar anisotropic piezoelectric materials is derived here for the first time using
an analysis based on analytic functions. A new type of singularity around conducting interface
crack tips is discovered. Specially, the singularities, in general, form two pairs: r −1/2±iε and
r −1/2±iκ at distance r from the crack tip, where ε and κ are real numbers depending on the
bimaterial constants. New definitions of real-valued stress and electric field intensity factors
are proposed. The mutual integral, defined in terms of the J integral proposed here, is applied
to determine the stress and electric field intensity factors, for a semi-infinite crack, as well as
for a finite crack, at the interface between dissimilar anisotropic piezoelectric media.
2. Formulation
Consider a generalized two-dimensional deformation of a linear anisotropic piezoelectric solid
in which the three components of displacement and the electric potential depend only on
the in-plane coordinates, x1 and x2 . A general solution for the generalized two-dimensional
problem may be written in terms of four analytic functions, as (Barnett and Lothe, 1975):
4
4
0
0
0
0
AJ M fM (zM ) ,
ψJ = −2 Re
BJ M fM (zM ) .
(2.1)
vJ = 2 Re
M=1
Here
vJ0 =
M=1
uj ,
J = 1, 2, 3,
φ,
J = 4,
(2.2)
where uj is the displacement, φ is the electric potential. ψJ0 is the generalized stress potential
defined by
0
0
= ψJ,2
,
1J
0
0
2J
= −ψJ,1
,
(2.3)
in which the subscript comma (,) denotes a partial derivative with respect to the Cartesian
coordinates, and
σij ,
J = 1, 2, 3,
0
(2.4)
=
iJ
J = 4,
Di ,
where σij is the stress and Di is the electric displacement. Re denotes the real part, and fM (zM )
are analytic in their arguments, zM = x1 + pM x2 ; and pM are four distinct complex numbers
with positive imaginary parts. In this paper, the repetition of an index in a term denotes a
summation with respect to that index over its range 1 to 3 for a lowercase script, and 1 to 4
for an uppercase script, unless indicated otherwise; and boldfaced symbols represent vectors
or matrices. The general solution (2.1) can be rewritten as
Conducting cracks in dissimilar piezoelectric media 287
4
4
AJ M fM (zM ) ,
ψJ = −2Re
BJ M fM (zM ) .
(2.5)
vJ = 2 Re
M=1
Here
vJ =
ψJ =
AJ M =
M=1
uj ,
J = 1, 2, 3,
ψ40 ,
J = 4,
ψj0 ,
J = 1, 2, 3,
(2.6)
φ,
J = 4,
0
Aj M ,
J = 1, 2, 3,
BJ M =
J = 4,
0
,
−B4M
Bj0M ,
J = 1, 2, 3,
−A04M ,
J = 4.
We employ this formulation to analyze crack problems since it simplifies greatly the analysis
of conducting cracks. Some useful properties and identities can be derived from those existing
in the formulation (2.1). The stress and electric fields are given in terms of the potential ψJ ,
as:
1J = ψJ,2 ,
in which
1J =
2J = −ψJ,1 ,
σ1j ,
J = 1, 2, 3,
−E2 ,
J = 4,
(2.7)
2J =
σ2j ,
J = 1, 2, 3,
E1 ,
J = 4,
(2.8)
where Ei is the electric field.
The matrices A and B in (2.5) are not unique in the sense that any arbitrary constant can be
multiplied to the eigenvectors (the column vectors of A and B). Normalizing the eigenvectors
according to 2AI J BI J = 1 (no sum on J ), we can define three real matrices H, L and S,
which will appear subsequently in this paper, as (see Appendix A for details):
H = 2iAAT ,
L = −2iBBT ,
S = i(2ABT − I),
(2.9)
where H and L are symmetric matrices, I is the identity matrix, and superscript T indicates
the transpose of a matrix. According to Suo (1993), the matrix L is positive-definite. Using
the result of Barnett and Lothe (1975), the matrices H, L and S can be calculated directly from
the material constants. The three real matrices are not entirely independent, but are related by
the following identities:
LS + ST L = 0,
HST + SH = 0,
HL − SS = I.
(2.10)
Making use of (2.9) together with (2.10), we have the following relation
iAB−1 = L−1 − iM,
−iBA−1 = H−1 + iN,
(2.11)
where M and N are the anti-symmetric matrices defined as M = SL−1 and N = H−1 S,
respectively.
For convenience, we will present our solutions through the vector function, f(z), defined as
f(z) = (f1 (z)f2(z)f3 (z)f4 (z))T ,
(2.12)
288 H. G. Beom and S. N. Atluri
Figure 1. Region near crack tip along piezoelectric bimaterial interface.
where the argument has the generic form z = x1 + px2 (Im p > 0). This one-complex-variable
approach has been originally introduced by Suo (1990). Once the solution of f(z) is obtained
for a given boundary value problem, a replacement of z1 , z2 , z3 or z4 should be made for each
component function, to calculate the field quantities.
3. Near tip stress and electric fields
Consider a crack lying along the interface between two dissimilar, anisotropic, homogeneous
linear piezoelectric materials, with material 1 above and material 2 below as shown in Figure 1. The crack tip lies on the plane x2 = 0 at x1 = 0, and the crack is traction-free and
conductive. We seek the form of solution in some region (= (1) + (2) ) surrounding the
tip of a traction-free and conductive interface crack. Continuity of 2J across all the interface,
both the bonded and cracked portions, in requires that
(2) (2)
B(1)f(1) (x1 ) − B f
(1) (1)
(x1 ) = B(2) f(2) (x1 ) − B f
(3.1)
(x1 ),
where the superscripts 1 and 2 in the parentheses indicate that the quantities are for the
materials 1 and 2, respectively, and prime ( ) implies the derivative with respective to the
associated argument. By the standard analytic continuation arguments, we see from (3.1) that
(2) (2)
B(1)f(1) (z) − B f
(1) (1)
(z) = B(2)f(2) (z) − B f
(z) = 2h(z),
(3.2)
where h(z) is analytic throughout , including points along all the interface. With the same
arguments, the continuity of vJ across the bonded interface gives an analytic continuation of
different linear combinations of f (z) and f (z) across the interface, such that
(2) (2)
A(1)f(1) (z) − A f
(1) (1)
(z) = A(2)f(2) (z) − A f
(3.3)
(z),
(1) (1)
holds everywhere in except on the crack line. We may express the function B f
terms of B(1)f(1)(z) and h(z) from (3.2) and (3.3)
(1) (1)
B f
(z) = (I + iβ)−1 (I − iβ)B(1) f(1) (z) − 2(I + iβ)−1 (I + α)h(z),
(z) in
(3.4)
Conducting cracks in dissimilar piezoelectric media 289
where
−1
,
α = L(1) − L(2) L(1) + L(2)
−1 (1)
M − M(2) .
β = L(1)−1 + L(2)−1
(3.5)
The bimaterial matrices α and β defined by (3.5) are two generalized matrices, pertinent to
the problem of a piezoelectric bimaterial, subjected to tractions and electric field prescribed
on its boundary. Two bimaterial matrices α and β are the only bimaterial parameters needed to
describe the stress field and electric field, for problems wherein tractions and electric field are
prescribed at the boundary. Another version of such generalized parameters for an anisotropic
piezoelectric bimaterial subjected to tractions and electric displacement prescribed on its
boundary has been proposed by Beom and Atluri (1996). The bimaterial matrix β has the
following properties
tr(β) = 0,
tr(β 2 ) ≤ 0,
tr(β 3 ) = 0,
β ≥ 0,
[tr(β 2 )]2 − 16β ≥ 0, (3.6)
where · denotes the determinant of a matrix. Details for the derivation of (3.6) are presented
in Appendix B. The traction-free and conductive condition on the surface of the crack leads
to a homogeneous Hilbert problem
(1) (1)−
B(1)f(1)+ (x1 ) + B f
(x1 ) = 0,
x1 < 0.
(3.7)
Substituting (3.4) into (3.7), it is found that
B(1)f(1)+ (x1 ) + (I + iβ)−1 (I − iβ)B(1)f(1)− (x1 ) = 2(I + iβ)−1 (I + α)h(x1 ),
x1 < 0. (3.8)
The general solution of (3.8) for f (z) is given by (see Appendix C for details)
1
(I + iβ)Y(ziε , ziκ )g(z) + (I + α)h(z),
B(1)f(1) (z) = √
2 2π z
(3.9)
in which
ε=
1+η
1
ln
,
2π 1 − η
κ=
1
1+ω
ln
,
2π 1 − ω
η = [{( 14 tr(β 2 ))2 − β}1/2 − 14 tr(β 2 )]1/2 ,
(3.10)
ω = [−{( 14 tr(β 2 ))2 − β}1/2 − 14 tr(β 2 )]1/2 .
where tr represents the trace of a matrix. It is seen from (3.6) and (3.10) that ε and κ are
real numbers depending on the real bimaterial matrix β. The matrix function Y(ξ(z), ζ(z)) is
expressed explicitly in terms of the real bimaterial matrix β, as:
ω2 η2 1
− 2
ξ(z) + ξ (z) + 2
ζ(z) + ζ (z) I
Y(ξ(z), ζ(z)) =
2
η − ω2
η − ω2
−iω2 iη2
1
ξ(z) − ξ (z) +
ζ(z) − ζ (z) β
+
2 η(η2 − ω2 )
ω(η2 − ω2 )
(3.11)
1
1
{−[ξ(z) + ξ (z)] + [ζ(z) + ζ (z)]}β 2
+ 2
2 η − ω2
i
1
i
[ξ(z) − ξ (z)] −
[ζ(z) − ζ (z)] β 3 ,
−
2
2
2
2
2 η(η − ω )
ω(η − ω )
290 H. G. Beom and S. N. Atluri
where ξ(z) and ζ(z) are arbitrary functions of z. Y(ξ, ζ ) given by (3.11) can be shown to have
the following properties
Y(1, 1) = I,
Y(ξ1 , ζ1 )Y(ξ2 , ζ2 ) = Y(ξ1 ξ2 , ζ1 ζ2 ).
(3.12)
Substitution of (3.9) into (3.4) yields
g(z) = g(z),
h(z) = −h(z).
(3.13)
Using (3.2) and (3.9), we obtain for the other function f(2) (z)
1
(I − iβ)Y(ziε , ziκ )g(z) + (I − α)h(z).
B(2)f(2) (z) = √
2 2π z
(3.14)
A Williams type expansion of the near-tip field is generated from (2.5), (2.7), (3.9) and
(3.14) by writing g(z) and h(z) in terms of local Taylor series expansions, as
g(z) =
∞
n=0
an zn,
h(z) =
∞
ibn zn ,
(3.15)
n=0
where an and bn are real vectors. Then a0 represents the strength of the crack tip singularity,
which can be defined as an intensity factor of stress and electric field. Since f(1)(z) and f(2) (z)
are determined as above, the complete fields of the stress and the electric field in the vicinity
of the crack tip are evaluated from (2.7).
The singular stress and electric field along the bonded interface near the crack tip is given
by
τ (x1 ) = √
1
Y(x1iε , x1iκ )g(x1 ),
2π x1
(3.16)
where τ = (σ21σ22 σ23 E1 )T . It is noted that the crack-tip singularities for the interfacial conducting crack are different from those for an interfacial insulating crack (Suo et al., 1992;
Beom and Atluri, 1996) and an interfacial permeable crack (Wang and Han, 1999; Gao and
Wang, 2000). The vector of stress and electric field intensity factors which uniquely characterize the singular field can be defined by
(3.17)
k = lim+ 2π x1 Y(x1−iε , x1−iκ )τ (x1 ),
x1 →0
where k = (K2 K1 K3 K4 )T . Since Y(x1−iε , x1−iκ ) and τ (x1 ) are real, k is real. Although k
defined in (3.17) does not have the proper dimension, it provides a unique characterization
of the crack tip state. Stress and electric field intensity factors with the same dimension of
classical intensity factor, denoted by k̂l also can be defined based on the characteristic length
l as suggested by Rice (1988) for the isotropic elastic bimaterial case. k̂l is related to k by
k̂l = Y(l iε , l iκ )k. It is noted that the intensity factor k given in (3.17) for the piezoelectric
bimaterial recovers the classical intensity factor (KI I KI KI I I KE )T as the bimaterial continuum degenerates to be a homogeneous one. In terms of k, the analytic functions generating
the singular part of the interface stress and electric displacement can be expressed as
1
(I + iβ)Y(ziε , ziκ )k,
B(1)f(1) (z) = √
2 2π z
1
(I − iβ)Y(ziε , zκ )k.
B(2)f(2) (z) = √
2 2π z
(3.18)
Conducting cracks in dissimilar piezoelectric media 291
Integrating (3.18), we have
ziε
ziκ
z
(1) (1)
(I + iβ)Y
,
k,
B f (z) =
2π
1 + 2iε 1 + 2iκ
ziε
ziκ
z
(2) (2)
(I − iβ)Y
,
k.
B f (z) =
2π
1 + 2iε 1 + 2iκ
(3.19)
The generalized displacement jump at distance r behind of the crack tip, calculated from
(3.19), is given by
r iκ
r iε
2r (1)−1
(2)−1
(L
,
k, (3.20)
+L
)Y
v(r) =
π
(1 + 2iε) cosh π ε (1 + 2iκ) cosh π κ
where v(r) = v(x1 , 0+ ) − v(x1 , 0− ). The derivative of the generalized displacement with
respect to x2 is discontinuous at the bonded interface (x1 > 0), which is given by
∂v(1) ∂v(2)
1
−
=√
Re G(1) − G(2) + i(G(1) + G(2) )β Y(x1iε , x1iκ )k,
∂x2
∂x2
2π x1
(3.21)
where G = APB−1 and P = diag(p1 p2 p3 p4 ).
4. Conservation integral
The J integral for a linear piezoelectric medium is defined by (Cherepanov, 1979; Pak, 1990)
0 0
) ds.
(4.1)
J v ; = (W 0 n1 − tJ0 vJ,1
0 0
vJ,i , ni is the unit outward
Here W 0 is the electric enthalpy density, given by W 0 = 12 iJ
0
normal vector, tJ is the surface traction and the surface electric displacement, given by tJ0 =
0
, is a path connecting any two points on opposite sides of the crack surface and
ni iJ
enclosing the crack tip and ds is an element of arc length along as shown in Figure 1.
It is well known that the generalized J integral is independent of any path , and has the
physical meaning of energy release rate due to crack extension. We define in this paper a J ∗
integral for a linear piezoelectric medium as
∗
(4.2)
J {v; } = (W n1 − tJ vJ,1 ) ds,
where W is the internal energy density, given by W = 12 iJ vJ,i , tJ is given by tJ = ni iJ .
As noted in the previous section, the matrices A and are not unique. For convenience, we
use the normalized matrices A and B hereafter; f(z) is the normalized function associated with
the normalized matrices A and B. Recently, Beom and Atluri (1996) obtained the complex
form of the J integral. In a similar way, it can be shown that the J ∗ integral is written in the
complex form, for an anisotropic piezoelectric solid, as
4 ∗
0
2
{fJ (zJ )} dzJ ,
(4.3)
j {v; 0 } = J {v ; 0 } = Re
J =1
0
292 H. G. Beom and S. N. Atluri
where 0 is a closed contour. It is noted that J ∗ = J for the crack problem. That is, the J ∗
integral is another form of the J integral. Thus, the J ∗ integral is independent of any path ,
and has the physical meaning of energy release rate due to crack extension.
Since the complete general solutions for the near tip fields are determined as shown in the
previous section, the relation between the J ∗ integral and the intensity factors can be derived
through the complex formula of the J ∗ integral. The J ∗ integral is evaluated with near tip
fields given by (3.18), resulting in
J ∗ {v; δ } = 14 kT U−1 k.
(4.4)
Here U−1 = (L(1)−1 + L(2)−1)(I + β 2 ), and δ is a circle with vanishingly small radius δ as
shown in Figure 1. In obtaining (4.4), the following relations have been used
Y(ziε , ziκ ) = Y(ziε , ziκ ),
YT (ziε , ziκ )U−1 Y(ziε , ziκ ) = U−1 ,
(I + iβ)T (L(1)−1 + L(2)−1 )(I + iβ) = U−1 .
(4.5)
Consider two independent equilibrium states of a piezoelectrically deformed bimaterial
body, with each displacement and charge potential being denoted by v and v, respectively.
The mutual integral for the two states, denoted by M{v, ṽ; } is defined by
i,J vJ,i n1 − tJ vJ,1 − tJ vJ,1 ) ds.
(4.6)
M{v,
v; } = (
M{v,
v; } can be written in terms of the J ∗ integral as
v; } − J ∗ {v; } − J ∗ {
v; }.
M{v,
v; } = J ∗ {v + (4.7)
The M integral satisfies the same conservation law as that of the J ∗ integral. Thus we have
the following conversation law M{v,
v; 0 } = 0. Here an area enclosed by 0 containing
the interface bonded perfectly is assumed to be free from any singularities. This conservation
law will be applied to the direct calculation of stress and electric field intensity factors without
actually solving complicated boundary value problems, which will be shown later. Making use
of the complex form of the J ∗ integral and the relation between J ∗ integral and M integral, it
can be shown that the complex form of the integral is given by
4 fJ (zJ )fJ (zJ ) dzJ ,
(4.8)
M{v,
v; 0 } = 2 Re
J =1
0
where overscript tilde () represents the quantities associated with the equilibrium state v.
5. Interface cracks
Two crack configurations in an infinite medium as shown in Figure 2, which are of particular
importance in the practical application, are considered. First consider a semi-infinite crack at
the interface between two dissimilar anisotropic piezoelectric media as shown in Figure 2(a).
Electromechanical tractions t+ (x1 ) = ts (x1 ) and t− (x1 ) = −ts (x1 ) are applied on the upper
and lower surfaces of the crack, respectively. The boundary condition on the crack surfaces
leads to the following Hilbert problem for the determination of f(1)(z)
(I + iβ)y+ (x1 ) + (I − iβ)y− (x1 ) = −ts ,
−∞ < x1 < 0.
(5.1)
Conducting cracks in dissimilar piezoelectric media 293
Figure 2. Interfacial cracks with electromechanical crack facing loading.
where y(z) = (I + iβ)−1 B(1)f(1) (z). A homogeneous solution X(z) which satisfies the homogeneous Hilbert problem
(I + iβ)X+ (x1 ) + (I − iβ)X− (x1 ) = 0,
−∞ < x1 < 0,
(5.2)
may be written as
1
X(z) = √ Y(ziε , ziκ ).
z
From (5.1) and (5.2), we find
0
−1
1 1
X(z)
(I + iβ)X+ (ξ ) ts dξ.
y(z) =
2π i
−∞ z − ξ
Using (5.3) and (5.4), it can be shown that a solution of f (z) is given by
√
iε iκ 0
1
−ξ −iε −iκ s
(1) (1)
Y ξ0 , ξ0
t dξ,
B f (z) =
√ Y z ,z
2π z
−∞ z − ξ
√
iε iκ 0
1
−ξ −iE −iκ s
(2) (2)
Y ξ0 , ξ0
t dξ,
B f (z) =
√ Y z ,z
2π z
−∞ z − ξ
(5.3)
(5.4)
(5.5)
where ξ0 = −ξ eiπ . The stress and electric field intensity factors are evaluated by using (3.17)
and (5.5), which results in
0
2
dξ
(5.6)
Y ξ0−iε , ξ0−iκ (I + iβ)−1 ts √ .
k=
π −∞
−ξ
In obtaining (5.6), the following relation has been used
(I + iβ)Y(ξ0iε , ξ0iκ ) = (I − iβ)Y(ξ0iε , ξ0iκ ).
(5.7)
294 H. G. Beom and S. N. Atluri
Next, we consider a finite crack, in the interval (−a, a), between dissimilar anisotropic
media as shown in Figure 2(b). Tractions t+ (x1 ) = ts (x1 ) and t− (x1 ) = −ts (x1 ) are applied
on the upper and lower surfaces of the crack, respectively. The solution procedure is similar
to the case of the semi-infinite crack. For a finite crack in interval (−a, a), the boundary
condition on the crack surfaces leads to (5.1). A homogeneous solution X(z) for the finite
crack which satisfies (5.2) may be written as
z − a iε z − a iκ
1
Y
,
.
(5.8)
X(z) = √
z+a
z+a
z2 − a 2
Thus, we find for the finite crack
a
−1
1 1
X(z)
(I + iβ)X+ (ξ ) ts dξ.
y(z) =
2π i
−a z − ξ
(5.9)
From (5.8) and (5.9), it can be shown that a solution of f (z) for the finite crack is
a 2
1
z − a iε z − a iκ
a − ξ2
(1) (1)
Y(ζ0−iε , ζ0−iκ )ts dξ,
Y
,
B f (z) =
√
2
2
z
+
a
z
+
a
z
−
ξ
2π z − a
−a
iε iκ a 2
1
z−a
z−a
a − ξ2
√
Y
,
Y(ζ0−iε , ζ0−iκ )ts dξ.
B(2)f(2) (z) =
z+a
z+a
z
−
ξ
2π z2 − a 2
−a
(5.10)
where ζ0 = {(a − ξ )/(a + ξ )}eiπ . Evaluating the stress and electric intensity field factors by
using (5.10), we have
a
1
∗−iε
∗−iκ
−1 s a + ξ
dξ,
(5.11)
Y(ζ0 , ζ0 )(I + iβ) t
k= √
a−ξ
π a −a
where ζ0∗ = 2a(a − ξ )/(a + ξ )eiπ . For the special case in which ts is a constant vector, (5.9)
reduces to
z
1
z − a iε z − a iκ
(1) (1)
Y
,
− I ts ,
B f (z) = (I + iβ) √
2
z+a
z+a
z2 − a 2
(5.12)
iε iκ z
1
z
−
a
z
−
a
Y
,
− I ts .
B(2)f(2) (z) = (I − iβ) √
2
z+a
z+a
z2 − a 2
The stress and electric intensity field factors for the special case are given by
√
k = π a y (2a)−iε , (2a)−iκ ts ,
(5.13)
6. Intensity factors
Consider a semi-infinite crack, as well as a finite crack, at the interface between two dissimilar
piezoelectric media as shown in Figure 3. Electromechanical tractions t+ (x1 ) and t− (x1 ) are
applied on the upper and lower surfaces of the crack, respectively. Electromechanical singularities q = (q1 q2 q3 q4 )T and b = (b1 b2 b3 b4 )T are embedded in the elastic material 2
at the point z = z0 . q1 , q2 and q3 are the components of a line force and q4 is the electric
Conducting cracks in dissimilar piezoelectric media 295
Figure 3. Interfacial crack with singularities and electromechanical crack facing loading.
Figure 4. Integration contours.
dipole layer. b1 , b2 and b3 are the components of a dislocation and b4 is the electric charge
jump. It will be shown that the stress and the electric field intensity factors of each problem
can be calculated directly by the application of the conservation laws, without actually solving
the boundary value problem. The mutual integral M can be used to determine the individual
stress and electric field intensity factors for the equilibrium state v, if a solution for another
equilibrium state v, called the auxiliary solution, is known.
First consider a semi-infinite crack at the interface between two dissimilar anisotropic
piezoelectric media as shown in Figure 3a. Choosing auxiliary solutions generating vJ for
a semi-infinite crack as
296 H. G. Beom and S. N. Atluri
1
B(1)−1(I + iβ)Y ziε , ziκ êJ ,
fJ (1)(z) = √
2 2π z
J (2)
f
1
B(2)−1(I − iβ)Y ziε , ziκ êJ ,
(z) = √
2 2π z
(6.1)
where êJ (J = 1, 2, 3, 4) is the base vector with the component êJM = δJ M and δJ M is the
Kronecker delta, the stress and electric field intensity factors for the equilibrium state v are
given by
vJ ; δ },
kM = 2UMJ M{v,
(6.2)
where δ is the vanishingly small circular path enclosing the crack tip. Invoking the conservation law of M{v,
v J ; 0 } = 0 for the integration contour as shown in Figure 4a, the stress
and the electric field intensity factors can be calculated directly, which results in
0
dξ
1
k= √
Y(ξ0−iε , ξ0−iκ )(I + iβ)−1 (I − α)t+ − (I + α)t− √
−ξ
2π −∞

 (6.3)
4
2 
1
Y zS0−iε , zS0−iκ (I + iβ)U b + (−iL(2)−1 + M(2) )q  .
Re
−
π
zS0
S=1
In deriving (6.3), we used (4.5), (5.7), (6.1) and the relation
fM (zM ) = AJ M 2J + BJ M vJ,1
(no sum over M),
(6.4)
together with potentials for the singularities near the point z = z0 given by
f(2) (z) =
−i
(B(2)T b + A(2)T q) + f∗ (z),
2π(z − z0 )
(6.5)
where f∗ (z) is analytic at z = z0 .
Next, consider a finite crack, in the interval (−a, a), between dissimilar anisotropic media
as shown in Figure 3b. The solution procedure is similar to the case of the semi-infinite crack.
We invoke the conservation law of M{v, v̂ J ; 0 } = 0 for the contour as shown in Figure 4b.
Choosing auxiliary solutions generating v̂J for a finite crack as
iε iκ 1
z
−
a
z
−
a
z
+
a
B(1)−1(I + iβ)Y
2a
, 2a
êJ ,
f̂J (1)(z) = √
z+a
z+a
4 πa z − a
(6.6)
iε iκ 1
z
−
a
z
−
a
z
+
a
B(2)−1(I − iβ)Y
2a
, 2a
êJ .
f̂J (2)(z) = √
z+a
z+a
4 πa z − a
The individual stress and electric field intensity factors for the finite crack are given by
vJ ; δ },
kM = 2UMJ M{v,
(6.7)
Evaluating the integral M{v, v̂ J ; δ } by using the conservation law of M{v, v̂ J ; 0 } = 0, it
can be shown that
Conducting cracks in dissimilar piezoelectric media 297
a
a+ξ
1
Y(ζ0∗−iε , ζ0∗−iκ )(I + iβ)−1 (I − α)t+ − (I + α)t−
dξ
k= √
a−ξ
2 π a −a
1
− √ Y (2a)−iε , (2a)−iκ (I + β 2 )−1 β(I − α) + (I − α)S(1)T qc
2 πa
4 z0 + a
1
S
Y(ζS0 , ζS0 )(I + iβ)U b + (−iL(2)−1 + M(2))q
− √ Re
0
πa
z
−
a
S
S=1
(6.8)
1
+ √ Y (2a)−iε , (2a)−iκ U b − (L(1)−1S(1)T − βL(1)−1)q ,
πa
a
where ζS0 = (2a(zS0 − a)/(zS0 + a)) and qc = −a {t+ (x1 ) + t− (x1 )} dx1 .
7. Concluding remarks
A complete form of stress and electric fields in the vicinity of the tip of a conducting crack,
between two dissimilar anisotropic piezoelectric media, is obtained in terms of two generalized bimaterial matrices as proposed in this paper. It is shown that the interfacial conducting
crack-tip field consists of two pairs of oscillatory singularities; r −1/2±iε and r −1/2±iκ at distance r from the crack tip, where ε and κ are real numbers depending on only one of the
two generalized bimaterial matrices. New definitions of real-valued stress and electric field
intensity factors are proposed. In defining the intensity factors, a matrix function that plays an
important role in representing the oscillations in the crack-tip fields is introduced. The matrix
functions is shown to be related explicitly to only one of the presently proposed generalized
bimaterial matrices. Exact solutions of the stress and electric fields for a semi-infinite crack
as well as for a finite crack at the interface between two dissimilar piezoelectric media are
obtained. Another form of the J integral is derived, and the mutual integral associated with
the J integral is proposed. The stress and electric field intensity factors associated with an
interfacial crack between two dissimilar anisotropic piezoelectric media are represented by
the mutual integrals. Closed form solutions of the stress and electric field intensity factors
for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar
piezoelectric media are obtained by using the mutual integral. These solutions can be implemented in computational methods for assessing the integrity of smart composite structures
with embedded sensors and actuators, as outlined, for instance, in Atluri (1997).
Acknowledgement
This work was supported by a grant from the Office of Naval Research, with Dr. Y. D. S.
Rajapakse as the cognizant program official. The first author performed a part of this work at
Center for Aerospace Research and Education, University of California at Los Angeles.
Appendix A. The matrices H, L and S
Normalizing the eigenvectors according to 2A0I J BI0J = 1 (no sum on J ), the matrices A0 and
B0 have the following relations (Barnett and Lothe, 1975)
298 H. G. Beom and S. N. Atluri
A0 A0T + A0 A0T = 0,
B0 B0T + B0 B0T = 0,
A0T B0 + B0T A0 = I,
A0T B0 + B0T A0 = 0.
A0 B0T + A0 B0T = I,
(A1)
Because of (A1), we can define:
H0 = 2iA0 A0T ,
L0 = −2iB0 BB0T ,
S0 = i(2A0 B0T − I),
(A2)
where H0 and L0 are real symmetric matrices and S0 is a real matrix. According to Barnett
and Lothe (1975), the matrices H0 , L0 and S0 can be calculated directly from the material
constants. Making use of (2.6) and (A1), the matrices A and B have the following relations
AAT + AAT = 0,
BBT + BBT = 0,
AT B + BT A = I,
AT B + BT A = 0,
ABT + ABT = I,
(A3)
Because of (A3), we can define the three real matrices H, L and S given in (2.9). The matrices
H, L and S are related to the matrices H0 , L0 and S0 by
0
0
0
0T
Lij Si4
Sij −Hi40
Hij0 −Si4
, L=
, S=
.
(A4)
H=
0T
0
0
0
−S4i
−L044
S4i
−H44
L04i S44
Thus, the matrices H, L and S can be calculated directly from the material constants.
Appendix B. Properties of bimaterial matrix β
The bimaterial matrix β can be rewritten as
β = γ ω,
(B1)
where γ is the real symmetric positive-definite matrix given by γ = (L(1)−1 + L(2)−1)−1 and
ω is the anti-symmetric matrix given by ω = M(1) − M(2). Since γ is symmetric and ω is
anti-symmetric,
tr(β) = 0,
tr(β 3 ) = 0.
(B2)
We diagonalize γ as
γ = QQT ,
QQT = I,
(B3)
where is a diagonal matrix with positive diagonal elements γ1 , γ2 , γ3 and γ4 . Then,
tr(β 2 ) = tr(ω∗ ω∗ ),
β = ω∗ ,
where ω∗ = QT ωQ. Let


0 −ω6 ω5 −ω4


0
ω3 −ω2 
 ω6
∗
,

ω =

−ω
−ω
0
ω

5
3
1 
ω4
ω2 −ω1
0
it is readily shown that (B4) reduces to
(B4)
(B5)
Conducting cracks in dissimilar piezoelectric media 299
tr(β 2 ) = −2(γ3 γ4 ω12 + γ2 γ4 ω22 + γ2 γ3 ω32 + γ1 γ4 ω42 + γ1 γ3 ω52 + γ1 γ2 ω62 ) ≤ 0,
β = γ1 γ2 γ3 γ4 (ω1 ω6 − ω2 ω5 + ω3 ω4 )2 ≥ 0.
(B5)
From Schwarz’s inequality follows the inequality for any pair of vectors u and v
(B6)
(uj uj + vj vj )2 − 4(uj vj )2 ≥ 0.
√
√
√
√
√
√
Choosing u = ( γ3 γ4 ω1 γ2 γ4 ω2 γ2 γ3 ω3 )T , v = ( γ1 γ2 ω6 − γ1 γ3 ω5 γ1 γ4 ω4 )T , we get
from (B6)
[tr(β 2 )]2 − 16β ≥ 0.
(B7)
Appendix C. Derivation of (3.9)
Introducing a new function vector y(z) defined by
y(z) = (I + iβ)−1 B(1)f(1)(z),
(C1)
(3.8) is rewritten as
(I + iβ)y + (x1 ) + (I − iβ)y − (x1 ) = 2(I + iβ)−1 (I + α)h(x1 ),
−∞ < x1 < 0.
(C2)
A homogeneous solution χ(z) which satisfies the homogeneous Hilbert problem
(I + iβ)χ + (x1 ) + (I − iβ)χ − (x1 ) = 0,
−∞ < x1 < 0,
(C3)
can be found by considering functions of the form χ(z) = z−1/2+iδ v, where v is a eigenvector.
Substitution of χ (z) = z−1/2+iδ v into (C3) yields
(β + iλI)v = 0,
(C4)
where λ = tanh π δ. For a nontrival solution of v, we have
β + iλI = λ4 +
1
2
tr(β 2 )λ2 + β = 0,
(C5)
In obtaining (C5), (3.6) has been used. Solving the eigenvalue problem (C4), we have the
four eignevalues, λ1 = η, λ2 = −η, λ3 = ω and λ4 = −ω, and the associated eigenvectors,
v1 , v2 , v3 and v4 . A general expression for the homogeneous solution may be written as
1
VZ(ziε , z−iε , ziκ , z−iκ )V−1 g(z),
χ(z) = √
2 2π z
(C6)
where V = (v1 v2 v3 v4 ) and Z(ζ1 , ζ2 , ζ3 , ζ4 ) = diag(ζ1 ζ2 ζ3 ζ4 ). Defining a matrix function
Y(ζ1 , ζ2 , ζ3 , ζ4 ) as Y(ζ1 , ζ2 , ζ3 , ζ4 ) = VZ(ζ1 , ζ2 , ζ3 , ζ4 )V−1 , it can be shown that
Y(ζ1 , ζ2 , ζ3 , ζ4 ) = I + 12 (ζ1 − ζ2 )X1 + { 12 (ζ1 + ζ2 ) − 1}X2
− 12 (ζ3 − ζ4 )Y1 + { 12 (ζ3 + ζ4 ) − 1}Y2 ,
(C7)
where I1 = diag(1 − 1 0 0), I2 = diag(0 0 − 1 1), X1 = VI1 V−1 , X2 = VI21 V−1 , Y1 =
VI2 V−1 and Y2 = VI22 V−1 . Making use of the following relations
βV = −iV,
β 4 + (η2 + ω2 )β 2 + η2 ω2 I = 0,
= ηI1 − ωI2 ,
(C8)
300 H. G. Beom and S. N. Atluri
where = diag(λ1 λ2 λ3 λ4 ), it can be shown that
ω2
1
η2
− 2
[ζ1 + ζ2 ] + 2
[ζ3 + ζ4 ] I
Y(ζ1 , ζ2 , ζ3 , ζ4 ) =
2
η − ω2
η − ω2
−iω2
iη2
1
−
[ζ1 − ζ2 ] +
[ζ3 − ζ4 ] β
+
2
η(η2 − ω2 )
ω(η2 − ω2 )
1
1
{−[ζ1 + ζ2 ] + [ζ3 + ζ4 ]}β 2
+ 2
2 η − ω2
i
1
i
[ζ1 − ζ2 ] −
[ζ3 − ζ4 ] β 3 .
−
2 η(η2 − ω2 )
ω(η2 − ω2 )
(C9)
Thus, (C6) is rewritten as
1
Y(ziε , ziκ )g(z),
χ(z) = √
2 2π z
(C10)
where Y(ζ1 (z), ζ3 (z)) ≡ Y(ζ1 (z), ζ 1 (z), ζ3 (z), ζ 3 (z)). A particular solution of (C1) is also
given by
y(z) = (I + iβ)−1 (I + α)h(z).
(C11)
Thus the general solution of (C1) for y(z) is
1
Y(ziε , ziκ )g(z) + (I + iβ)−1 (I + α)h(z).
y(z) = √
2 2π z
(C12)
Finally, we get (3.9) from (C1) and (C12).
References
Atluri, S.N. (1997). Structural Integrity and Durability, Tech Science Press, Forsyth, Ga.
Barnett, D.M., Lothe, J. (1975). Dislocations and line charges in anisotropic piezoelectric insulators. Physica
Status Solidi B 67, 105–111.
Beom, H.G. (1999a). Singular behaviour near a crack tip in an electrostrictive material. Journal of the Mechanics
and Physics of Solids 47, 1027–1049.
Beom, H.G. (1999b). Small scale nonlinear analysis of electrostrictive crack problems. Journal of the Mechanics
and Physics of Solids 47, 1379–1395.
Beom, H.G., Atluri, S.N. (1996). Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic
piezoelectric media. International Journal of Fracture 75, 163–183.
Cherepanov, G.P. (1979). Mechanics of Brittle Fracture, McGraw-Hill, New York.
Fu, R., Qian, C.-F., Zhang, T.-Y. (2000). Electrical fracture toughness for conductive cracks driven by electrical
fields in piezoelectric materials. Applied Physics Letters 76, 126–128.
Gao, C.-F., Wang, M.-Z., (2000). Collinear permeable cracks between dissimilar piezoelectric materials. International Journal of Solids and Structures 37, 4969–4986.
Heyer, V., Schneider, G.A., Balke, H., Drescher, J., Bahr, H.-A. (1998). A fracture criterion for conducting cracks
in homogeneously poled piezoelectric PZT-PIC 151 ceramics. Acta Materialia 46, 6615–6622.
Pak, Y.E. (1990). Crack extension force in a piezoelectric material. Journal of Applied Mechanics 57, 647–653.
Parton, V.Z. (1976). Fracture mechanics of piezoelectric materials. Acta Astronautica 3, 671–683.
Rice, J.R. (1988). Elastic fracture mechanics concepts for interfacial cracks. Journal of Applied Mechanics 55,
98–103.
Ru, C.Q., Mao, X. (1999). Conducting cracks in a piezoelectric ceramic of limited electrical polarization. Journal
of the Mechanics and Physics of Solids 47, 2125–2146.
Conducting cracks in dissimilar piezoelectric media 301
Shin, B.-C., Kim, H.-G. (1988). Partial discharge, microcracking, and breakdown in BaTiO3 ceramics. Ferroelectrics 77, 161–166.
Suo, Z. (1990). Singularities, interfaces and cracks in dissimilar anisotropic media. Proceedings of the Royal
Society of London Series A427, 331–358.
Suo, Z. (1993). Models for breakdown-resistant dielectric and ferroelectric ceramics. Journal of the Mechanics
and Physics of Solids 41, 1155–1176.
Suo, Z., Kuo, C.-M., Barnett, D.M., Willis, J.R. (1992). Fracture mechanics for piezoelectric ceramics. Journal of
the Mechanics and Physics of Solids 40, 739–765.
Wang, T.C., Han, X.L. (1999). Fracture mechanics of piezoelectric materials. International Journal of Fracture
98, 15–35.
Zhang, T.-Y., Zhao, M.H., Tong, P. (2002). Fracture of piezoelectric ceramics. Advances in Applied Mechanics 38,
147–289.