chapter 3 cracks in electroelastic...
TRANSCRIPT
Chapter 3
Cracks in electroelastic materials
3.1 Introduction
With increasingly wide application of piezoelectric materials in practical
engineering, the study of crack problems in piezoelectric solids has received
much interest. Early in 1975, Barnett and Lothe [1] generalized Stroh’s six-
dimensional framework [2] to an eight-dimensional formalism to treat
dislocations and line charges in anisotropic piezoelectric insulators. Deeg [3]
developed a method of analysing piezoelectric cracks by generalizing a
distributed dislocation method to include piezoelectric effects. Zhou et al. [4]
proposed a multipole function representation and used an analogy theorem to
obtain electroelasto-magnetic field equations for a finite piezoelectric body with
defects. Sosa and Pak [5] studied a three-dimensional eigenfunction analysis of a
semi-infinite crack in a piezoelectric material. Based on the complex potential
approach, Sosa [6] obtained asymptotic expressions for coupled electroelastic
fields at the tip of a crack in an infinite piezoelectric solid. The asymptotic
expressions were obtained as the limiting case of an elliptical hole. Dunn [7]
derived the electroelastic fields in and around an elliptical inclusion in a
piezoelectric solid through use of the equivalent inclusion idea of Eshelby [8].
McMeeking [9] derived a path-independent integral for the problem of a crack
filled with a conducting medium in an elastically deformable dielectric material
that is subjected to both mechanical and electric loads. Suo et al. [10] studied
cracks both in piezoelectric materials and on interfaces between piezoelectric and
other materials such as metal electrodes or polymer matrices. Recently, Qin and
Yu [11] studied the logarithmic singularities of crack-tip fields. Fracture and
damage behaviours of a cracked piezoelectric solid under coupled thermal,
mechanical and electrical loads have also been reported [12-16]. Here, we
provide only a preliminary insight into the research in this field and do not
provide a complete review. The readers may refer to the book [17] and articles
[10,18,19] for more information in this field.
In this chapter, we consider some typical fracture problems in the
mathematical theory of electroelasticity relating to cracks occurring commonly in
practical engineering. The crack-tip singularities of a semi-infinite crack are
investigated first. Then, solutions are provided for a semi-infinite crack, a Griffith
crack opened under constant traction-charge, anti-plane crack, cracks in half-
plane, cracks in bimaterials, interface cracks with a closed crack-tip model,
macro- and micro-crack interactions. Examination of the problems of crack kink
and crack deflection follows. Finally, the three-dimensional problem of elliptic
cracks is discussed in detail. Some numerical results are presented to illustrate the
application of the formulations discussed above.
3.2 Singularity of crack-tip fields for a semi-infinite crack
3.2.1 A general solution for crack-tip fields
The singularity of stress and electric displacement near the tip of a semi-infinite
crack has been studied by Ting [20,21] for anisotropic elasticity, Qin and Yu [11]
for electroelastic problems, and Yu and Qin [12] for thermoelectroelastic
problems. In this section we follow the results given in [11].
Consider a semi-infinite crack along the negative x-axis. The SED
singularities at tips of the crack can be determined by assuming the form of f in
eqns (1.52) and (1.57) in the following form [20]
1)(
1
JJ
zzf (3.1)
where =a+ib is a complex constant with a and b being two real constants.
Substituting eqn (3.1) into eqns (1.52) and (1.56) yields
1
Re2 1 qAU z , (3.2)
qB
z Re22 . (3.3)
If we use the polar coordinate system (r, ) defined in Fig. 2.2, the complex
variable z becomes
)sin(cos prz . (3.4)
We see that with the assumption of eqn (3.1) the SED given by eqn (3.3) is of the
order r . It is obvious that the SED is singular if the real part of , i.e. a, is
positive. For the potential energy to be bounded at the crack tip, we require that
a<1. So we focus our attention on the internal 0< a<1. Using the traction-charge
free condition on the crack surfaces and noting that z=r when =0 and
whenirez , we know that
0)()(2 qBBq
iibiiba ererr , (3.5)
0)()(2 qBBq
iibiiba ererr , (3.6)
or in matrix form
0)( QX (3.7)
where Q={Bq qB }T. To obtain a nontrivial solution for Q we should let the
determinant of X vanish, i.e.
0X (3.8)
where the symbol denotes the determinant, which leads to
b=0, 0)1( 44 aie . (3.9)
The solution of eqn (3.9) reads
.,2 ,1 ,0 ,2
)1(
n
na (3.10)
Hence, to satisfy 0<a<1, we should take n=0, which is a fourfold root of eqn
(3.9). The elastic displacement and electric potential, U, and stress and electric
displacement, 2, may now be written in their asymptotic forms by combining
eqns (3.2), (3.3) and (3.10) as
])sin(cosRe[4 2/12/1qAU pr , (3.11)
])sin(cosRe[2 2/12/1
2 qB
pr . (3.12)
3.2.2 Anisotropic degeneracy
The analyses presented so far tacitly assume that the eigenvalues p’s are distinct.
When one of the p’s is a double root, one may or may not have four independent
functions in eqns (3.11) and (3.12), and a set of additional solutions is required. It
is not difficult to see that if eqns (3.11) and (3.12) are the solutions corresponding
to the double root pi, so are
qAU })sin(cos{Re4 2/12/1)2( p
dp
dr
i
, (3.13)
qB })sin(cos{Re2 2/12/1)2(
2 pdp
dr
i
(3.14)
where ii dpddpd / and / BA can be obtained by differentiating eqns (1.54) and
(1.58) with respect to pi:
0}{ Daidp
d, (3.15)
}){( aTRb
pdp
d
dp
d T
ii
(3.16)
where TRRQD2)( pp T is a 44 matrix. The new solutions (3.13) and
(3.14) exist if the following equation holds true [22]:
MNndp
d
ippn
i
n
,0D (3.17)
where N and M are the order and rank of D, respectively. However, it is found
that the order of singularity is not changed in the presence of the new solution
(3.14).
3.2.3 New solutions for being a multiple root
If is a multiple root of eqn (3.8), the components of q may not be unique and
one has to find other independent solutions. For a root of multiplicity m, the new
solutions are given by
1 ,,2 ,1 ],} {Re[2 )*(
4
)*(
3
)*(
2
)*(
1
)(
2 mizzzz Tiiiii B (3.18)
where
qzzz
i
i ln )(* . Likewise, new solutions exist if
Mnd
dn
n
8 ,02/1
X (3.19)
holds true. Here M is the rank of matrix X. Since =1/2 is a fourfold root (see eqn
(3.9)), the SED singularities at the tips of a semi-infinite crack must occur in one
of the following cases
0./ satisfying ),ln(
0,/ satisfying ),ln(
0,/ satisfying ),ln(
0, satisfyingonly ),(
)(
2/1
3332/1
2/1
2222/1
2/1
2/1
2/1
2
ddrrO
ddrrO
ddrrO
rO
r
X
X
X
XQ
(3.20)
For a semi-infinite crack in an anisotropic piezoelectric medium, it is, therefore,
shown that both stress and electric displacement at the crack tip may be in the
order of ,2/1r or ,ln2/1 rr ,ln 22/1 rr ,ln 32/1 rr as 0r , where r is the
distance from crack tip to field point, depending on which boundary conditions
are satisfied.
3.3 Reduction of a crack problem to a Hilbert problem
The crack problems discussed in the previous chapter can be reduced to the
solution of a Hilbert problem. For a description of the Hilbert problem, see
Muskhelishvili [23].
Suppose that we have a series of colinear cracks occupying the segments
nLLL ,, , 21 of the x-axis X, and that the cracks are opened by application of
traction on the crack faces. If we write
nLLLL 21 , (3.21)
then the boundary conditions are the same as those of eqns (2.32)-(2.34) except
that cx 1 should be replaced by LXx and cx 1
replaced by L. As was
done in Section 2.4, using the subscripts “(1)” and “(2)” to identify variables
associated with the domains x2>0 and x2<0, respectively, the continuity of
traction, t(x), across the whole x-axis requires that
)()()()( )2()2()2()2()1()1()1()1( xxxx fBfBfBfB . (3.22)
Noting that one of the important properties of holomorphic functions used in the
method of analytic continuation is that if the function f(z) is holomorphic in the
region y>0 (or y<0), then )(zf is holomorphic in the region y<0 (or y>0). In
using this property, the above is rearranged in the form
)()()()( )1()1()2()2()2()2()1()1( xxxx fBfBfBfB . (3.23)
Thus, the left-hand side of eqn (3.23) is the boundary value of a function analytic
in the upper half-plane, and the right-hand side is the boundary value of another
function analytic in the lower half-plane. Hence, both functions can be
analytically continued into the entire plane. Since both vanish at infinity to
conform to a zero far-field, they must vanish at any point z from Liouville’s
theorem. Thus,
),()( )1()1(
1
)2()2( zz fBBf y>0, (3.24)
)()( )2()2(
1
)1()1( zz fBBf , y<0 (3.25)
where the one-complex-variable approach presented by Suo [24] has been used.
Suo introduced a vector function, ,)}()( )({)( 321
Tzfzfzfz f ,21 pxxz
Im(p)>0, in his analysis of anisotropic elastic material, and then showed that the
solution to any given boundary value problem can be expressed in terms of such a
vector, regardless of the precise value of p. Once a solution of f(z) is obtained, the
variable z in fi(z) should be replaced with the argument zi, to compute the related
fields. This approach allows standard matrix algebra to be used in conjunction
with the techniques of analytic functions of one variable, and thus bypass some
complexities arising from the use of four complex variables.
From eqns (1.64) and (3.24), the jump across the x-axis can be written as
)]()([)( )2()2()1()1( xxxi fBfBCU (3.26)
where the matrix C was defined in eqn (2.42). Continuity of the displacement and
electric potential across the x-axis, as inferred from eqn (3.26), implies that a
function defined as
0),(
,0),()(
)2()2(
)1()1(
yz
yzz
fB
fB (3.27)
is analytic in the entire plane except on the crack. The traction-charge boundary
condition (2.33) leads to a non-homogeneous Hielbert problem
Lxxx ,)()( 2 . (3.28)
This equation has many possible solutions. The auxiliary conditions that render a
unique solution are: (z)=O(z) as z , (x) is square root singular at the
crack tips, and the net Burgers vector for every finite crack vanishes, i.e.,
nidxxxiL
,,2 ,1 ,0)]()([
. (3.29)
As an illustration, we reconsider the Griffith crack problem described in Section
2.4. The solution is
]1)[(2
1)( 2/122
2 czdzz . (3.30)
Then, the functions f1(z) and f2(z) are obtained from definition (3.27). The field
variables are obtained by replacing z with z in eqns (1.64) and (1.65). The results
are the same as those given in eqns (2.49)-(2.52).
3.4 Solution of a semi-infinite crack
In Section 3.2, the singularity of stress and electric displacement at the tip of a
semi-infinite crack was discussed. As a complement, we present a full-field
solution of the same problem as in this section. For piezoelectric materials, Sosa
and Pak [5] have considered the distribution of SED in the neighbourhood of a
semi-infinite crack by way of a three-dimensional eigenfunction approach. Gao
and Yu [25] studied the same problem based on the Stroh’s formalism and
Green’s function approach. For the present, however, we restrict our attention to
the problems of Gao and Yu [25].
The problem considered by Gao and Yu is an infinite piezoelectric solid with
a semi-infinite crack L, which lies along the positive direction of the x1-axis. The
solid is subjected to a line force-charge q0 at an arbitrary point z0 (x10, x20), and the
crack faces are assumed to be free of traction and external charge, but filled with
air. In this case, the boundary conditions can be stated as
,31 , ,0 ,0 1222 jLxxjj (3.31)
LxxEE
12112424 ,0 , , (3.32)
where the symbols “+” and “” stand for the upper and lower crack faces.
We start by seeking a complex potential function f(z) with the form
)()ln()( 00 zzzz fqf (3.33)
where 20100 xpxz , q is a constant vector (which is shown to be ATq0 in
Section 3.8), and f0(z) is a holomorphic function vector (with four components)
up to infinity. Define the functions
)()( ),()( ),()( 0000 zzzzzz FGfFfF . (3.34)
Then, in a similar manner to that in Section 3.3, we have
000 GBBF . (3.35)
Similarly, from eqn (3.32)2,
0)]()([ 04
4
1
04
JJJ
J
JJJ zGAzFA . (3.36)
From eqns (3.35) and (3.36), we know that
)(Im
)Im(
1)( 0
3
1
4
1
44
1
44
40 zXA
A
zX K
K
JK
J
J
J
JJ
(3.37)
where 1 B , and )()( 0
4
1
0 KK
K
JKJ zFBzX
is a holomorphic function which
will be used later.
To find the function XJ0, rewrite the stress boundary condition (3.31) in the
form
)31( , ,0)(Re2 1
4
1
1
JLxxFBK
KJk . (3.38)
Furthermore, introduce a function
2
kkz , )( kk rt (3.39)
where rk are eigenvalues in the -plane, which maps the region SK (SK is obtained
from the x1-x2 plane cut along L, by the affine transformation, 21 xpxz KK )
onto the upper half-plane in the -plane. Meanwhile, the crack faces are mapped
on the real axis, t, in the -plane. Thus, eqn (3.38) can be expressed in the form
)31( , ),()(~
Re2 0
4
1
0
JtthtFB J
K
KJK (3.40)
where )]([)(~
00 KKKKK zFF , and
00
00
4
1
0 ,11
2Re2)( KK
KKK
KJKJ z
ttt
qBth
. (3.41)
Multiplying both sides of eqn (3.40) by dt/(t-), and integrating along the real
axis t, one obtains
)(~
)(~
00
4
1
IJ
J
IJ XFB , J=1, 2, 3 (3.42)
where
00
4
1
02
1)(
~
J
JIJ
J
JIJ
J
I
qBqBX , J=1, 2, 3. (3.43)
From eqns (3.37) and (3.39) and integrating eqn (3.42) with respect to Kz , we
obtain
)(Im
)Im(
1)( 0
3
1
4
1
44
1
44
40 KM
M
JM
J
J
J
JJ
K zYA
A
zY
(3.44)
where
)()()( 0
4
1
00 KK
K
JKKKJJ zfBdzzXzY
, J=1-4. (3.45)
Substituting eqn (3.39) into eqn (3.43), and then into eqn (3.45) leads to
.31 , )ln()ln()(
4
1
000
JzzqBzzqBzYK
KKKJKKKKJKKJ
(3.46)
Solving eqn (3.45) for fK0, one obtains
4
1
00 )()(J
KJKJKK zYzf . (3.47)
Thus Green’s function for a semi-infinite crack in a piezoelectric material
subjected to a line force-charge q0 can be obtained by substituting eqn (3.47) into
eqn (3.33), and then into eqns (1.64) and (1.65) as
)](Re[2)ln(Im[1
000 zzz TAfqAAU
, (3.48)
)](Re[2)ln(Im[1
000 zzz TBfqAB
. (3.49)
With this solution, the SED intensity factors K can be expressed as
0 ,2lim},,,{ 11,101
xxKKKKx
T
DIIIIII K (3.50)
where K={KII, KI, KIII, KD}T, in which KI, KII, KIII are the usual stress intensity
factors, and KD is the so-called “elastic displacement intensity factor”.
3.5 Griffith crack opened under constant traction-charge
A Griffith crack under constant traction-charge has been discussed in Section 3.3.
The derivation is based on the following boundary conditions on the crack
surface:
022 DD , (3.51)
which means that the impermeable assumption was used due to its much simpler
mathematical treatment and the fact that the dielectric constants of a piezoelectric
material are much larger than those of a vacuum (generally between 1000 and
3500 times larger). This is equivalent to assuming that the dielectric constant of
the air vanishes. However, Hao and Shen [26] argued that even if the permittivity
of a vacuum (or air) is quite small, the flux of electric fields can propagate
through the crack gap, and the electric displacement should not be zero. They
suggested therefore that the boundary condition (1.36) should be used. In this
section we summarize the solution obtained by Hao and Shen [26] under such a
crack face condition.
3.5.1 The extended Lekhnitskii formalism
We begin by assuming that
ii D and are independent of x3. From Table 1.1, we
know that
),( 213333,3 xxDDgfu jj , j=1-6, =1-3. (3.52)
Since the fields ii D and are independent of x3, there exist the relations
DgfuuxxWxxDxu jj 4443,22,32102133 ),,(),( . (3.53)
From eqn (3.53), one obtains
),(2
)( 2102,032,
2
34432 xxVWxD
xDgfxu jj
. (3.54)
Similarly
),(2
)( 2101,031,
2
35531 xxUWxD
xDgfxu jj
. (3.55)
Because i is independent of x3 by assumption, we have
02
)(
3,
1,011,0311,
2
31,5533,1
UWxDx
Dgfxu jj , (3.56)
which leads to
0)( and 0 1,1,05511, WDgfD jj . (3.57)
In a similar way, we know that
.0)()( ,0
,0)( ,0
1,2,0442,1,05512,
2,2,04422,
WDgfWDgfD
WDgfD
jjjj
jj
(3.58)
From eqns (3.55)-(3.58), we have
, ,
, ,
6661,02,01,22,12222,02,2
1111,01,132211
DgfVUuuDgfVu
DgfUuAxAxAD
jjjj
jj
DgfxAWuu
DgfxAWuu
jj
jj
4441142,02,33,2
5552241,01,33,1 , (3.59)
where A1-A4, 1 and 2 are integral constants to be determined. Following eqns
(3.52), (3.59) and the constitutive relation in Table 1.1, we have
3,333
3333322113 ,/)(
DgE
fDgfAxAxA
ii
iijj
(3.60)
where the dumb index j=1, 2, 4, 5, 6, and i=1-3. Since E3 is independent of x3, we
obtain
2 1, ,),( ,0,33,21033 ExExxEx . (3.61)
Thus, we have
0,3 E . (3.62)
This results in E3=constant. For convenience we set E3=0. Then, 3 and D3 can be
expressed as
DJHAxAxAKD
DGFAxAxAL
jj
jj
)(
,)(
322113
322113
(3.63)
where the dumb index j=1, 2,
4,
5,
6, and =1, 2, with
. , ,
, , , ,1
33
33233
33
333
3
33
333
33
3
33
33
333
3
33
3
2
3333
3333
g
fgMJf
g
gfMH
f
ggM
f
gG
MfKf
fggM
f
fF
f
gMMgL
j
j
j
j
j
j
j
(3.64)
Substituting eqn (3.63) into eqn (3.59), one obtains
)(
),(
),(
),(
),(
322116662,01,0
322114442,0
322115551,0
322112222,0
322111111,0
AxAxADUV
AxAxADW
AxAxADW
AxAxADV
AxAxADU
jj
jj
jj
jj
jj
(3.65)
where the dumb index j=1, 2,
4,
5,
6, and =1,
2, with
jijiij
jijiji
ijij gfgfg
ggf
f
ffgM
f
fff
)( 3333
33
3333
33
3333
33
33, (3.66)
KgLfJgGfg iiiiiii 3333 , . (3.67)
Substituting eqn (3.63) into the constitutive relation in Table 1.1, we have
6 ,5 ,4 ,2 ,1 1,2;, ),( 32211 jAxAxADhE jj (3.68)
where
KLgJGgHFggh iiijijiijijjijiijij 333333 , , . (3.69)
Having obtained the above relations, we can now introduce a new shear function
, such that
2,51,4 , , (3.70)
and use the stress function F and induction V defined in eqn (1.94) for obtaining
the final solution.
Substituting eqns (1.94) and (3.70) into eqn (3.65) and eliminating U0 and
V0, we obtain
41524623
534
2
,0
AAAVLLFL
VLLFL (3.71)
where
.)(
,)()(
,2)2(2
,)()(
,2
2
1
2
42
21
2
41522
2
2
516
3
1
3
22
2
2
1
3
62212
21
3
61123
2
3
115
4
2
4
113
21
4
162
2
2
1
4
6612
2
3
1
4
264
1
4
224
3
2
3
152
21
3
5614
2
2
1
3
46263
1
3
243
2
2
2
55
21
2
452
1
2
442
xxxxL
xxxxxxL
xxxxxxxxL
xxxxxxL
xxxxL
(3.72)
From eqns (1.94), (3.55) and (3.70), and considering E2,1-E1,2=0, and then
eliminating Ei, we have
2112987 AAVLLFL (3.73)
where
.)(
,)(
,)()(
2
1
2
22
21
2
12212
2
2
119
2
1
2
24
21
2
14252
2
2
158
3
1
3
22
2
2
1
3
26122
21
3
21163
2
3
117
xxxxL
xh
xxhh
xhL
xh
xxhh
xxhh
xhL
(3.74)
Equations (3.71) and (3.73) can be further written as
0)( 763752835
2
39249486 FLLLLLLLLLLLLLLLLL . (3.75)
As discussed in Section 1.8 for two-dimensional problems, eqn (3.75) can be
solved by assuming a solution of )( izf , where zi=x1+pix2. Thus the most general
real solution is obtained from a linear combination of four arbitrary functions
4
1
)(Re2j
jj zfF . (3.76)
Substituting eqn (3.76) into eqn (3.71), we obtain
4
1
4
1
)(Re2 ),(Re2j
jjj
j
jjj zfcVzfa . (3.77)
Let ii r
jjj
r
jijji dzzfdplzfL /)()()]([ with ri being the order of Li such that
./)()()(])([
)()]([
22
9221221
2
11
2
1
2
22
21
2
12212
2
2
119
jjjjjjjj
jjj
jj
dzzfdplzfpp
x
f
xx
f
x
fzfL
(3.78)
Using this notation, aj and cj can be expressed as
.)()()()(
)()()(
,)(
)(
)()()()(
)()()(
)(
)(
6325
2
324
5
4
6325
2
324
3
5
jjjj
jjj
j
j
j
jjjj
jjj
j
j
j
plplplpl
plplplc
pl
pl
plplplpl
plplpl
pl
pla
(3.79)
Thus, one obtains
. , , , ,
,1 , ,)(Re2 ,)(Re2
21654
22
1
4
1
4
1
jjjjjjjjjjjj
jjj
j
jjj
j
jjiji
bdpcdpkpakak
kpkzfdDzfk
(3.80)
To obtain the explicit expression of ),( and )( nn uu we begin by
considering the electric potential on the crack faces. From eqns (3.68) and (3.80),
we have
0 ),()(2)(Re2 2
4
1
11111
4
1
11
xCAxxfdxfkhEj
jjj
j
rjr (3.81)
where r=1, 2,
4,
5,
6, =1,
2.
Noting that 1,1 E , we have
0 ),2/()(2)(Re2 2
4
1
1111111
4
1
1
xCCAxxxfdxfkhj
jjj
j
rjr .
(3.82)
The displacement u2 on the crack surface can be obtained by using eqns (3.59),
(3.65) and (3.80) as
0 2; 1, ;6 ,5 ,4 ,2 ,1 ),(/)()(Re2 21
*
1
4
1
222
xrxCpxfdku j
j
jjrjr
(3.83)
where C *(x1) is an arbitrary function of x1.
3.5.2 The solution of the Griffith crack
Consider a Griffith crack as shown in Fig. 2.1, subjected to uniform load:
DDii , , i=1, 2,
4,
5,
6; =1,
2 at infinity, and on the crack surface
cxxFF 1212,1,11,642 ,0 on ,0 i.e. ,0 . (3.84)
We begin by seeking a complex potential function fj(zj) with the general form
)()(ln)()( 021 jjjjjjjjjj zfziCBziXXzf (3.85)
where X1j, X2j, Bj and Cj are real constants, and
1
00 /)(n
n
jnjjjj zaazf (3.86)
is a holomorphic function up to infinity. Single-valuedness of fj on the crack
surface under conditions (3.84) leads to (X1j+iX2j)=0. Next, the constants Bj and Cj
are determined by means of the far-field electro-mechanical loads:
.2 ,1 ;6 ,5 ,4 ,2 ,1 ,)(Re2 ,)(Re24
1
4
1
iDiCBdiCBkj
jjj
j
ijjij
(3.87)
Because of the arbitrariness of the rotation at infinity [27], we can let C1=0. Thus
the number of equations in eqn (3.87) are enough to determine the constants Bj
and Cj.
Integrating the boundary condition (3.84) and neglecting some constants
which are of no use to the present problem, we obtain
cxxFF 122,1, ,0 on ,0 . (3.88)
Noting that )/()( 22
uua is equal to a constant D0 [26], considering that
D2=-V1, one has
cxxxDV 1210 ,0 on , . (3.89)
Substituting eqns (3.76) and (3.77) into eqns (3.88) and (3.89), yields
c. 0, ,)(Re2
,0)(Re2 ,0)(Re2 ,0)(Re2
12
4
1
101
4
1
1
4
1
1
4
1
1
xxxDxfc
xfaxfpxf
j
jj
j
jj
j
jj
j
j
(3.90)
Now the four functions fj0 can be derived by means of eqn (3.90) together with the
following conformal transformation which maps the exterior of the crack gap onto
the exterior of a unit circle
)(2
1 kkk
cz . (3.91)
Through some algebraic manipulation we obtain the complex potentials as
41 ,1
)()( 22
0
kczzep
cziCBzf jjkjkjjjjj (3.92)
where
)(2
1 ,
2
1 ,
2
1 ,
2
1024436221 DDcececece
(3.93)
and 0 is the determinant of matrix ][ ijp
4321
4321
4321
1111
][
cccc
aaaa
pppppij (3.94)
while pij is the algebraic complement of ][ ijp with the relations
04
4
1
03
4
1
02
4
1
01
4
1
, , ,
i
j
jiji
j
jiji
j
jiji
j
ji pcpappp . (3.95)
From eqn (3.92), we have
cxxkxcepc
izfzf kjkjjjj
12
2
1
2
0
,0 ,41 ,2
)()( . (3.96)
Therefore, the quantities )( and )( 22
uu can be expressed as
,2
)(Re2 2
1
2
0
4
1
11 xcepc
idkh kjk
j
jrjr
(3.97)
2
1
2
0
4
1
2222
2)(Re2 xcep
c
idkuu kjk
j
jrjr
(3.98)
for k=1-4, x2=0 and cx 1 . With the usual definition, the stress and electric
displacement intensity factors are given by
ap
a
excxK
j
jkk
cxI
4
1
2
0
121
2)()(2lim
1
, (3.99)
app
a
excxK
j
jkjk
cxII
4
1
6
0
161
2)()(2lim
1
, (3.100)
apa
a
excxK
j
jkjk
cxIII
4
1
4
0
141
2)()(2lim
1
, (3.101)
aDDpc
a
exDcxK
j
jkjk
cxD
4
1
02
0
121 )(2
)()(2lim1
. (3.102)
Apparently, D0 has reduced the value of KD, but has no influence on stress
intensity factors, except that it affects the value of ij near the crack tip.
3.6 Solution of elliptic hole problems
The fracture properties of a Griffith crack may be investigated based on an
analysis of elliptic hole problems in piezoelectric media. In fact, a crack of length
2c can be obtained by letting the minor axis, say b in this section, of the ellipse
approach zero. The problem of determining the SED fields in the neighbourhood
of an elliptic hole in a piezoelectric plate has been studied by Sosa and
Khutoryansky [28], Later, Lu et al. [29] and Qin [30] extended the analysis to
thermopiezoelectric problems. Qin [30] used the exact boundary condition at the
rim of an elliptic hole to avoid the common assumption of electric
impermeability, and solved the problem by means of Lekhnitskii’s formalism.
The problem has also been considered by Gao and Fan [31] and others, but we
shall follow the results given in [28,30] here.
3.6.1 Problem statement
Consider an infinite two-dimensional piezoelectric material containing an elliptic
hole, where the material is transversely isotropic and coupling between in-plane
stress and in-plane electric fields takes place. The contour (see Fig. 3.1) of the
hole is represented by
sin ,cos 21 bxcx (3.103)
where is a real parameter and c, b are the half-length of the major and minor
axes respectively of the ellipse (see Fig. 3.1). The hole is assumed to be filled
with a homogeneous gas of dielectric constant 0, and is free of traction and
surface-charge density. On the hole boundary the normal component of electric
displacement and the electric potential are continuous. We, therefore, have
boundary conditions on the contour :
cc
nnmnnn
D
, ,0 ,0 0 (3.104)
where n and m, respectively, represent the normal and tangent to the hole
boundary (Fig. 3.1). nn and nm are the normal and shear stresses along the
boundary, Dn is the normal component of electric displacement vector,
superscript “c” indicates the quantity associated with the hole medium.
When the piezoelectric material is subjected to a uniform remote SED
),,( 21
where TT DD } { ,} { 2222120112111
for the genera-
lized plane strain problem under consideration, the SED condition at infinity is
Fig. 3.1: Geometry of an ellipse.
x1
x2
n m
c
0
c
b
2
2
2
1 when , xx . (3.105)
From the definition in eqn (1.94), we know that
dsDVdstFdstFs
n
ss
0
0 12,
0 21, , , (3.106)
where 21 and tt are the rectangular Cartesian components of surface traction
vector t, and ds is an element of arc length on . Substituting eqn (1.106) into
eqn (3.104) and noting eqn (1.106), we have
.}{Re2 ,}{Re2
,0}{Re2 ,0Re2
3
1
*
0 0
3
1
3
1
3
1
c
k
kk
sc
k
kk
k
kk
k
k
tdsn
p
(3.107)
Noting that
m
ixxi
n
ixx cc
)()( 2121 , (3.108)
equation (3.107)3 can be further simplified as
)}(Re{2}{Re2 *
0
3
1
zFik
kk
(3.109)
where z=x1+ix2 throughout this and the next sections, and
c(z)= )()( ** zFzF . (3.110)
3.6.2 General solutions
Since conformal mapping is a fundamental tool used to find complex potentials,
the transformation
k
kk
kk
bipcbipcz
1
22 (3.111)
will be used to map the region, , occupied by piezoelectric material onto the
outside of a unit circle in the -plane, since it has been shown that all the roots of
equation 0/ kk ddz for eqn (3.111) are located inside the unit circle 1k
[32]. To find the single-valued mapping of the region c, occupied by a vacuum
(or air), let
1
22
bcbcz . (3.112)
The roots of 0/ ddz are e
e
1
12,1 , where e=b/c. Thus, mapping of the
region c can be achieved by excluding a straight line 0 along x1 and of length
212 ea from the ellipse (Fig. 3.1). In this case, the function (3.112) will
transform and 0 into the ring of outer and inner circles with radii 1outr and
e
erin
1
1, respectively.
The problem is now to find the complex potentials, k (k=1-3) in the region
and the function F* in the region c. To this end we assume a general solution
of the form
)(ln)( 00 kkkkkkkk czcz (3.113)
where
1
00 )(i
i
kkikkk dd (3.114)
is a holomorphic function up to infinity with constants dki. The boundary
conditions (3.107) require that ck0=0 for k to be single valued. The constants ck
are determined from the far field loading conditions, as described at the end of
this subsection.
Furthermore, the solution for F*(z), is assumed as
1
0
* )(k
k
ksszF (3.115)
for a crack, and
k
k
kszF )(* (3.116)
for other situations, whose coefficients have the following relation [28]:
k
k
ink srs 2 . (3.117)
Substituting eqns (3.111) and (3.113) into the left-hand side of eqn (3.107) yields
the boundary conditions on the unit circle, namely
1
110
3
1
0 )()(
llk
k
k , (3.118)
1
220
3
1
0 )()(
llpp k
k
kkk , (3.119)
])()([)()( **
0
1
330
3
1
0
FFillk
k
kkk , (3.120)
)()()()( **1
440
3
1
*
0
*
FFlltt k
k
kkk (3.121)
where ie stands for the point located on the unit circle in the k-plane, is a
polar angle, and
3
1
**
3
1
43
3
1
2
3
1
21
)].Re()[Re( )],Re()[Re(
)],Re()[Re( )],Re()[Re(
k
kkkkk
k
kkkkk
k
kkkk
k
kkk
tpcietcalpciecal
pciepcalpciecal
(3.122)
The substitution of eqns (3.113) and (3.115) [or (3.116)] into eqns (3.118)-(3.121)
provides a new system of three equations in the unknowns dki:
3
1
31
3
1
21
3
1
11 )( , ,k
ikikikiki
k
ikik
k
iki ldtdldpld (3.123)
where ij is the Kronecker delta, and
4033
*
0 ,0 , lilltti kikkki , (3.124)
41
3
1
* ldts iki
k
ii
(3.125)
for a crack, and
],)1(2[1
,1
2 ,)1(
1
4
4
4
2
4
033
4
*2
0*4
4
0
lrlrr
ill
r
trittr
r
i
i
in
i
ini
in
i
in
k
i
inkik
i
ini
in
kki
(3.126)
)()(1
14
2
41
**23
14
lrldtdtrr
s i
inikikkik
i
in
ki
in
i (3.127)
for other situations. Thus the unknowns dij can be solved from eqn (3.123). Once
the constants dij are obtained, the complex potentials, k(zk), can be written in the
form
])([)( 22221 bpczzbipc
dzcz kkk
k
kkkkk
, (3.128)
through use of the relation (3.112) and the fact that all dij equal zero for j>1.
To determine SED, the derivative of eqn (3.128) with respect to zk is
evaluated, which results in
)(1)(
2222
1
bpcz
z
bipc
dcz
kk
k
k
kkkk
. (3.129)
To solve for ck (k=1-3), one must invoke the remote electromechanical loading
conditions. In the most general case, three mechanical and two electrical variables
can be enforced. If these loads are the stress and electric displacement
components, by making use of eqns (1.106), (3.105) and (3.129) we can generate
a system of five equations for the six unknowns involved in the real and
imaginary part of the constants ck. For the reason stated in Section 3.5, one can
arbitrarily set one of these constants equal to zero. We will let Im(c1)=0 in our
analysis. Thus, using eqns (1.106), (3.105) and (3.129) and considering kz ,
yields
,1Re2
3
1
2
12
22
11
k
k
k
k
c
p
p
k
k k
kk
cp
D
D
3
12
1Re2 (3.130)
which, compared with eqn (3.122), yields
).(2
),(2
),(2
1231112212221
ieDDc
liec
liec
l (3.131)
As was pointed out by Sosa and Khutoryansky [28], an electric field will be
induced by the applied load at infinity in the piezoelectric material, whose value
is obtained by inserting eqn (3.130) into eqn (1.91) giving
3
1
*
0
2
0
11
Re2k
kk
k
ctpE
E (3.132)
which, compared with eqn (3.122), yields
)(2
0
2
0
14 ieEEc
l . (3.133)
Thus all unknown constants in eqn (3.129) have been completely determined.
3.6.3 Cracks
A crack of length 2c can be formed by letting the minor axis b of the ellipse
approach zero. The solutions for a crack in an infinite piezoelectric plate can then
be obtained from the formulation in subsection 3.6.2 by setting b=0. In this case,
eqn (3.128) can be simplified to
)()( 221 czzc
dzcz kk
kkkkk . (3.134)
Thus, the asymptotic form of SED, 2, ahead of the crack tip along the x1
axis can be given by
22
1
*222122
1} {
cxD T
(3.135)
where
3
1
11Re2k
k
k
k
d
p
. (3.136)
With the usual definition, the SED intensity factors are given by
a
axax
*21 )(2lim
1
K . (3.137)
3.7 Solutions of the anti-plane crack problem
Using linear piezoelectricity theory, the stress and electric fields around a crack
under anti-plane loads are studied in this section.
Consider a mode III fracture problem for which a finite crack of length 2c is
embedded in an infinite piezoelectric medium subjected to far-field mechanical
and electric loads (see Fig. 3.2). This problem has been investigated by Pak [33],
Zhang and Tong [34], and others. Pak [33] considered four possible cases of
boundary conditions at infinity. The first case is a uniform shear traction,
zy , applied with a uniform electric displacement, DDy ; the second is a
uniform shear strain, zy , with a uniform electric field, EEy ; the third is
a uniform shear traction with a uniform electric field; and the fourth is a uniform
shear strain with a uniform electric displacement. The boundary conditions on the
upper and lower surfaces of the crack are to be free of surface traction and surface
charge for all cases of far-field loading conditions, i.e.,
0 , on ,0 ,0 ycxDyzy . (3.138)
Examination of eqn (1.46) reveals that the field equations are satisfied if uz and
are harmonic functions. This can be achieved by letting uz and be the imaginary
part of some analytic functions such that
(z)]Im[ )],(Im[ zXuz (3.139)
where z=x+iy. The SED can then be obtained from these functions
)()( ),()( 11151544 zzXeiDDzezXci xyzxzy . (3.140)
Taking a semi-inverse approach by assuming X(z) and (z) to be
2/1222/122 )()( ,)()( czBzczAzX , (3.141)
we see that the governing equations (1.46) and the boundary conditions (3.138)
are satisfied. In these equations, A and B are real constants. Substituting eqn
(3.141) into eqn (3.140), we arrive at
.
,
2211
2215
2215
2244
cz
Bz
cz
AzeiDD
cz
Bze
cz
Azci
xy
zxzy
(3.142)
By applying the far-field loading conditions, the constants A and B are obtained as
follows:
Case 1:
22 as and , yxDDyzy
2
151144
1544
2
151144
1511 ,ec
eDcB
ec
DeA
. (3.143)
Case 2:
22 as and , yxEEyzy
EBA , . (3.144)
Case 3:
22 as and , yxEEyzy
EB
c
EeA ,
44
15 . (3.145)
Fig. 3.2: Far-field loads.
Case 4:
22 as and , yxDDyzy
11
15 ,
eDBA . (3.146)
Thus, the near-tip fields are given by
22
22
2 ,
2
,2
,2
i
Dxy
i
Exy
i
zxzy
i
zxzy
er
KiDDe
r
KiEE
er
Kie
r
Ki
(3.147)
where cAK is the “strain intensity factor”, cBKE the “electric field
factor”, EKeKcK 1544 the “stress intensity factor”, and ED KKeK 1115
the “electric displacement intensity factor”. For this particular piezoelectric
fracture problem, all the field variables have the same crack-tip behaviour as in
the classical mode III fracture problem.
The path-independent integral given in eqn (2.14) can now be evaluated to
obtain the energy release rate for the mode III piezoelectric fracture problem
under consideration. Using the solution obtained above, evaluating the integral in
eqn (2.14) on a vanishingly small contour at a crack tip results in
2
EDKKKKJ
. (3.148)
,
c2
1x2x
ED ,
Examining eqn (3.148) one can see that J may have negative values depending on
the direction, magnitude, and type of electric load. Solving the roots of the
quadratic equations in (3.148), the crack extension force can be shown to be
positive when
,ee
:1 Case44
154411
2
15
44
154411
2
15
c
ecD
c
ec
(3.149)
Case 2: ,ee
11
154411
2
15
11
154411
2
15
ecEec
(3.150)
,e
1
e
1 :3 Case
4411
2
154411
2
15 c
E
c
(3.151)
4411
2
154411
2
15 ee :4 Case cD
c
. (3.152)
This shows that, at certain ratios of electric load to mechanical load, crack arrest
can be observed. It is also interesting to note that in the absence of mechanical
load the crack extension force is always negative, indicating that a crack would
not propagate under these conditions.
As an example, consider an anti-plane crack problem in PZT-5H material,
with material properties as given by [33]: ,N/m1053.3 2
1044 c 15e
,C/m 0.17 2 C/(Vm) 1051.1 811
, N/m 0.5crJ , where Jcr is the critical
energy release rate. The energy release rate for PZT-5H material with a crack of
length 2c=0.02 m is plotted in Figs 3.3 and 3.4 as a function of electrical load
with the mechanical load fixed such that J=Jcr at zero electric load. As can be
seen from Figs 3.3 and 3.4, the energy release rate can be made negative by
setting electric loads outside the ranges given by (3.149)-(3.152).
3.8 Solution of half-plane problems
In this section, the problem of a crack embedded arbitrarily in a half-plane
piezoelectric solid with traction-charge loading on the crack faces is considered.
The boundary condition on the infinite straight boundary of the half-plane is free
of surface traction-charge. Using Stroh’s formalism, simple explicit expressions
of Green’s functions for a piezoelectric half-plane subjected to line force-charge
and line dislocations are derived at first. Then, a system of singular integral
equations for the unknown dislocation densities defined on the crack faces is
formulated based on the Green’s function. The stress and electric displacement
intensity factors are evaluated by numerically solving these singular integral
equations.
Fig. 3.3: Energy release rate in PZT-5H for cases 1 and 4.
Fig. 3.4: Energy release rate in PZT-5H for cases 2 and 3.
3.8.1 Green’s function for infinite spaces
For an infinite domain subjected to a line force-charge 0q and a line dislocation b
both located at z0(x10, x20), the solution in the form of eqns (1.64) and (1.65) is [35]
-4 -2 0 2 4 6 8
-3
-2
-1
0
1
2
)C/m 10( 23
D
26 N/m 102.4
:1 Case
5106.7
:4 Case
crJJ /
-4 -3 -2 -1 0 1 2 3-4
-3
-2
-1
0
1
2
265 N/m 104.3 105.9
:3 Case :2 Case
V/m) (105 E
crJJ /
},)ln(Im{1
0 qAU
zz })ln(Im{1
0 qB
zz (3.153)
where q is a complex vector to be determined. Since )ln( 0 zz is a multi-
valued function we introduce a cut along the line defined by x2=x20 and 101 xx .
Using the polar coordinate system (r, ) with its origin at z0(x10, x20) and =0
being parallel to the x1-axis, the solution (3.153) applies to
, r>0. (3.154)
Therefore
4-1for at ,ln)ln( 0 irzz . (3.155)
Due to this relation, eqn (3.153) must satisfy the conditions
0 )()( ,)()( qbUU (3.156)
which lead to
0)Re(2 ,)Re(2 qBqbAq . (3.157)
This can be written as
0q
b
q
q
BB
AA. (3.158)
It follows from eqn (1.75) that
0q
b
AB
AB
q
q
TT
TT
. (3.159)
Hence
bBqAqTT 0
. (3.160)
This result has been used in Sections 2.3 and 3.4 for eqns (2.15), (2.16) and
(3.48).
3.8.2 Green’s function for half-spaces
Let the material occupy the region 02 x , and a line force-charge 0q and a line
dislocation b apply at z0 (x10, x20). To satisfy the boundary conditions on the
infinite straight boundary of the half-plane, the solution should be modified as
})ln(Im{1
})ln(Im{1
0
4
1
0
qAqAU zzzz , (3.161)
})ln(Im{1
})ln(Im{1
0
4
1
0
qBqB zzzz (3.162)
where q is given in eqn (3.160) and q are unknown constants to be determined.
Consider first the case in which the surface 02 x is traction-charge free so
that
0at 0 2 x . (3.163)
Substituting eqn (3.162) into eqn (3.163) one obtains
0})Im{ln(1
})ln(Im{1
4
1
1
BqqB zzzx . (3.164)
Noting that )Im()Im( ff , we have
})ln(Im{})ln(Im{ 0101 qBqB zxzx , (3.165)
and
I
4
1
001 )ln()ln( zzzx , (3.166)
where
],,,[diag 4321 iiiiii I . (3.167)
Equation (3.164) now yields
)( 0
11bBqAIBBqIBBq
TT
. (3.168)
If the boundary 02 x is a rigid surface, then
0at ,0 2 xU . (3.169)
The same procedure shows that the solution is given by eqns (3.161) and (3.162)
with
)( 0
1bBqAIAAq
TT
. (3.170)
3.8.3 Inclined crack in a half-plane plate
Consider the problem of a straight crack with length 2c embedded in a
piezoelectric half-plane plate. The geometrical configuration of the problem to be
solved is depicted in Fig. 3.5. The orientation of the crack is denoted by and the
depth of the centre of the crack from the infinite straight boundary of the half-
plane is denoted by d. Uniform traction-charge is applied on both faces of the
crack while the straight boundary of the half-plane is assumed to be traction-
charge free. The mathematical statement of the boundary conditions of this
problem can be stated more precisely as follows:
0 , ,)( 0 cT t , (3.171)
02 , 02 x , (3.172)
0 , 2
2
2
1 xx , (3.173)
where t is a traction-charge vector, (, , x3) the local coordinate system with
the origin centred at the middle point of the crack, and the direction of is
parallel to the crack faces (Fig. 3.5). The traction applied on the lower crack face
is denoted by TD } ,0 , ,{ 0 0 00
where the components are expressed in
terms of the local coordinates (, , x3). The 44 matrix, ),( whose compo-
nents are the cosine of the angle between the local coordinates and the global
coordinates, is in the form [14]
1000
0100
00cossin
00sincos
)( . (3.174)
Note that defined here is positive when measured from the positive direction of
the x1-axis to the positive direction of the -axis, as shown in Fig. 3.5.
c
-c
d
o
1x
2x
Fig. 3.5: Geometry of the inclined crack.
The boundary condition given by eqn (3.171) can be satisfied by redefining
the discrete dislocation function b in eqns (3.160) and (3.168) in terms of
function b() defined along the crack line, ,*kkk tzdpz ,*
0 kkk zdpz
where
sincos*kk pz . In this case, the load parameter q0 appearing in eqns
(3.160) and (3.168) should be zero. Enforcing satisfaction of the applied traction-
charge conditions on the crack faces, a singular integral equation for the
dislocation function b is obtained as
)()(),( 1)(
2
0 tdt
t
dc
c
c
c
tbKbL
, ct (3.175)
where
})({1
),( 11
0
4
1
0
Tzzt BIBBBK
(3.176)
is a kernel function of the singular integral equations and is Hölder-continuous
along cc .
For single valued displacements and electric potential around a closed
contour surrounding the whole crack, the following conditions must also be
satisfied
c
cd
0)( b . (3.177)
For convenience, normalize the interval (-c, c) by the change of variables,
i.e., cscst ,0 . If we retain the same symbols for the new functions caused
by the change of variables, eqns (3.175) and (3.177) can be rewritten as
)()(),( 1)(
200
1
1
1
1 0
0
sdssssss
dss
tbK
bL, cs 0 , (3.178)
1
1 0)( dssb . (3.179)
The singular integral equation (3.178) for the dislocation density b combined with
eqn (3.179) can be solved numerically [36]. Since the solution for the functions,
)(sb , has a square root singular at both crack tips, it is more efficient for
numerical calculation to let
2
1
2 1
)(
1
)()(
s
sT
s
s
m
k
kk
E
b
(3.180)
where (s) is a regular function defined in a closed interval cs , Ek are the real
unknown constants, and Tk(s) the Chebyshev polynomials. Thus the discretized
form of eqns (3.178) and (3.179) may be written as [36]
m
k
k
rk
m
k
kr
kr
s
ssssssn
1
0
1
00
0
0)(
),()( ),()(2
1tK
L
(3.181)
where
.1,,2 ,1 ,cos ,,,2 ,1 ,2
)12(cos 0
mr
m
rsmk
m
ks rk (3.182)
Equation (3.181) provides a system of 4m linear algebraic equations to determine
(sk). Once the function (sk) has been found from eqn (3.181), the stresses and
electric displacements, ),(tn in a coordinate system local to the crack line can be
expressed in the form
dtt
dt
c
c
c
c)(),(
1)(
2)()()(
0 bK
bLt . (3.183)
Using eqn (3.181) we can evaluate the SED intensity factors K={KII, KI,
KIII,
KD}
T
at the tips, e.g., at the right tip (=c) of the crack by following definition:
)()(2lim
cc
K . (3.184)
Combining this with the results of eqn (3.183), one then obtains
)1()(4
* LK
c
. (3.185)
Thus the solution of the singular integral equation enables direct determination of
the stress intensity factors.
3.8.4 Direction of crack initiation
The strain energy density criterion [37] as well as the formulation presented
above can be used to predict the direction of crack initiation in half-plane
piezoelectric materials. To make the derivation tractable, the crack-tip fields are
first studied. In doing this, a polar coordinate system (r, ) centred at a crack tip,
say the right tip of the crack, ,cos(),( 21 cxx )sin cd and =0 along the
crack line is used. Then, the variable zk becomes
)]cos()[cos()sin(cos kkkk prpcdpz . (3.186)
With this coordinate system, SED fields near the crack tip can be evaluated by
taking the asymptotic limit of eqn (3.183), and using expressions (1.56) and
(3.162) as 0r
),(2
1)1(
)()(Im
2
1),( 1
**1
XBBcrzz
p
crr T (3.187)
)(2
1)1(
)()(
1Im
2
1),( 2
**2
XBBcrzzcr
r T (3.188)
where xpxxz sincos)(*
.
For a piezoelectric material, the strain density factor S() can be calculated
by considering the related electroelastic potential energy W. The relationship
between the two functions is as follows:
2
1
21 }{2
)(
F
TTrrWS (3.189)
where the matrix F is the inverse of stiffness matrix E, and is given in eqn (1.40).
The substitution of eqns (3.187) and (3.188) into eqn (3.189) leads to
2
1
21 }{4
1)(
X
XFXX
TT
cS . (3.190)
As shown in [37], the strain energy density criterion states that the direction of
crack initiation coincides with the direction of the strain energy density factor Smin,
i.e., the necessary and sufficient condition of crack growth in the direction 0 is
that
.0 and 000 2
2
SS (3.191)
Substituting eqn (3.190) into eqn (3.191)1 yields
0}{
2
1
21
X
XFXX
TT. (3.192)
Solving eqn (3.192), several roots of may be obtained. The fracture angle will
be the one satisfying eqn (3.191)2.
3.9 Solution of bimaterial problems
We now consider a bimaterial piezoelectric plate containing a crack either lying
along the interface of the bimaterial or embedded in any one material, for which
the upper half-plane )0( 2 x is occupied by material 1, and the lower half-plane
)0( 2 x is occupied by material 2. They are rigidly bonded together so that
0at , , 2
)2()1()2()1( xUU (3.193)
where the superscripts (1) and (2) label the quantities relating to materials 1 and
2. The equality of traction-charge continuity comes from the relation t s/ ,
where t is the surface traction-charge vector on a curve boundary and s is the arc
length measured along the curve boundary. When points along the crack faces are
considered, integration )2()1(tt provides ,)2()1( since the integration
constants can be neglected, which correspond to rigid motion. Furthermore,
assume that an arbitrary and self-equilibrated loading is specified on the crack
faces. This problem has been considered by many researchers [10,14,16]. Kuo
and Barnett [38] investigated the character of the singularity at interface crack
tips of piezoelectric material. Suo et al. [10] showed the possibility of the
existence of real as well as oscillating singularities at an interface crack tip. Qin
and Yu [39] presented a solution for an arbitrarily oriented crack terminating at
the interface of a piezoelectric bimaterial. Qin and Mai [14,16] generalized these
results to the case of thermoelectroelastic problems. In this section, we follow the
results presented in [14,16].
3.9.1 Green’s function for bimaterials
For a bimaterial plate subjected to a line force-charge 0q and a line dislocation b
both located in the upper half-plane at z0(x10, x20), the solution may be assumed, in
a similar way as treated in the half-plane problem, in the form
})ln(Im{1
})ln(Im{1 )1()1(
0
)1()1(
4
1
)1(
0
)1()1()1(
qAqAU zzzz , (3.194)
})ln(Im{1
})ln(Im{1 )1()1(
0
)1()1(
4
1
)1(
0
)1()1()1(
qBqB zzzz (3.195)
for material 1 in x2>0 and
})ln(Im{1 )2()1(
0
)2()2(
4
1
)2(
qAU zz , (3.196)
})ln(Im{1 )2()1(
0
)2()2(
4
1
)2(
qB zz (3.197)
for material 2 in x2<0. The value of q is given in eqn (3.160) and )2()1( , qq are
unknown constants which are determined by substituting (3.194)-(3.197) into
(3.193). Following the derivation in subsection 3.8.2, we obtain
,)1()2()2()1()1(qIAqAqA qIBqBqB )1()2()2()1()1( . (3.198)
Solving eqn (3.198) yields
qIBLMMIBq
)1(1)1(11)2(1)1(1)1()1( ])(2[ , (3.199)
qIBLMMBq
)1(1)1(11)2(1)1(1)2()2( )(2 , (3.200)
where 1)()()( jjj i ABM is the surface impedance matrix.
3.9.2 Solution for an inclined crack in bimaterials
The geometrical configuration of the problem to be solved is depicted in Fig. 3.6,
showing a crack with an orientation angle and length 2c near an interface
between material 1 and material 2. The boundary conditions of the problem are
given by eqns (3.171), (3.173) and (3.193). The crack can be modelled by the
Green’s functions of eqn (3.195). To this end, the discrete dislocation b
represents generalized distributions along a given crack line. Expressing the crack
line as *)1()1(
0
)1(
kkk zzz , where 20
)1(
10
)1(
0 xpxz kk and sincos )1(*)1(
kk pz ,
the dislocation function b becomes b(). At this point, b() is unknown, and will
be determined such that the surface traction-charge is equal to the given value.
Thus, the boundary condition (3.171) requires that
)()(),( 1)(
2
0
)1(
tdtt
dc
c
c
c
tbKbL
, c (3.201)
where
})({1
),( )1(*1)1(
0
)1(
4
1
)1(
0
Tzzt BIBBK
, (3.202)
with
)1(1)1(11)2(1)1(1)1(* ])(2[ BLMMIBB . (3.203)
The singular integral equation (3.201) together with eqn (3.177) can be used
to determine the unknown b, and then to evaluate the SED intensity factors.
3.9.3 Multiple cracks in bimaterials
In what follows, formulations will be derived for an infinite piezoelectric plate of
bimaterial with N arbitrary located cracks of length 2ci ),,2 ,1( Ni in the
material 1
material 2interface
c
-c
crack
),( 2010 xx
Fig. 3.6: Geometry of a crack in a bimaterial solid.
Fig. 3.7: Geometry of the multiple crack system.
plane ),( 21 xx and subjected to remote traction-charge 20. The configuration of
the crack system is shown in Fig. 3.7.
Assume that all cracks are located in material 1 (upper half plane) and that
material 2 has no cracks. These assumptions are made for simplifying the ensuing
writing; extension to the case of cracks in the whole plane is straightforward. The
central point of the ith crack is denoted as ),( 21 ii xx and the orientation angle is
denoted as i (see Fig. 3.7). The cracks are initially assumed to remain open and
hence be free of tractions and charges. The corresponding boundary conditions
are, then, as follows:
material 2
material 1
interface
i
),( 21 ii xx
crack i
ic
ic
On the faces of each crack i:
0cossin )1(
2
)1(
1
)1( iini t , ),,2 ,1( Ni . (3.204)
At infinity:
2021 ,0 , (3.205)
where n stands for the normal direction to the lower face of a crack, and )1(
nit is
the surface traction and charge vector acting on the ith crack.
It is convenient to represent the solution as the sum of a uniform remote
SED loading in an unflawed solid and a corrective solution in which the boundary
conditions are
On the faces of each crack i:
iiini coscossin 20
)1(
2
)1(
1
)1( t , ),,2 ,1( Ni . (3.206)
At infinity:
021 . (3.207)
Using the principle of superposition [40], the boundary value problem defined
above is decomposed into N sub-problems, each of which contains only one
crack. The boundary conditions (3.205) can be satisfied by replacing the discrete
dislocation vector b by the distributing dislocation density )(b defined along a
given crack line such as crack i, *0)1(kiikiki ztzz , *0)1(
0 kiikiki zzz , where
ikiki xpxz 2)1(
10 , ikiki pz sincos )1(* , and i is shown in Fig. 3.7. Enforcing
the satisfaction of the applied traction-charge conditions on each crack face, a
system of singular integral equations for the dislocation density )(b is obtained
as
)1(
,
0
,
0
)1(
)(),(),( 1)(
2nijjjji
N
ijj
ijji
c
c
c
c
N
ijj
ij
ii
iii dttt
di
i
i
i
tbGKbL
,
Nict ii ,,2 ,1 , (3.208)
where
T
jjjiiiijiij zzztzzt )1(1*0*0*)1(
0 )(),( BBK
, (3.209)
T
jjjiiiijiij zzztzzt )1(*1*0*0*
4
1
)1(
0 )(),( BIBBG
. (3.210)
For single valued displacements and electric potential around a closed
contour surrounding the whole ith crack, the following conditions must also be
satisfied
i
i
c
ci d
0)( b , ),,2 ,1( Ni . (3.211)
If we express the function )( ii b in the form
22
)()(
i
i
c
b , (3.212)
then, in a similar manner to that in Section 3.8, we have
)()(4
)( )1(*
iii
i
i cc
c LK
, (3.213)
and the near-tip fields 0 as r
)()()(
Im2
1),( )1(
**
)1()1(
1 ii
T
iii
czz
p
rcr
BB , (3.214)
)()()(
1Im
2
1),( )1(
**
)1(
1 ii
T
iii
czzrc
r
BB (3.215)
where )(* xz and the coordinate system (r, ) are defined in subsection 3.8.4.
3.9.4 SED jump across interface
It should be mentioned that for interface problems the normal stress, 11, in the
x1-direction and the horizontal component of electric displacement are
discontinuous across the interface. This implies that it is not possible to have the
parallel stress and electric displacement vanish in both materials: perpendicular
loading will give rise to parallel SED. The SED that should be used in an infinite
region analysis in a finite body with uniform (typically zero) parallel loading is
thus not as obvious as in the simple case of homogeneous materials. Thus, it is
necessary to distinguish 111)( and 11 )( D in the region x2>0, i.e., material 1, from
211)( and 21 )( D in the region x2<0. Following the procedure of Rice and Sih
[41], they may be derived by considering the equilibrium of a differential element
occupying both regions x2>0 and x2<0, x2=0 being the interface. The stress 11
and the electric displacement D1 are taken to be discontinuous across the interface
and the strain 11 and the electric field E1 to be continuous along such an
interface, i.e.,
2111211111 )()( ,)()( EE . (3.216)
It follows from the relation (1.91) that
].)()([1
)(
],)()()([1
)(
12
)1(
13
)2(
1321
)2(
11)1(
11
11
2
)1(
21
)2(
2122
)1(
12
)2(
12211
)2(
11)1(
11
111
ppDD
Dppffff
(3.217)
It is obvious from eqn (3.217) that there is a jump for 11 and D1 across the
interface.
3.10 A closed crack-tip model for an interface crack
In this section, a closed crack-tip model for analysing an interface crack is
presented to treat the oscillatory singularity of SED crack-tip fields. Previous
studies [42,43] have indicated that the full open crack-tip model will cause stress
oscillatory singularity at crack tips. As was pointed out by England [43], a
physically unreasonable aspect of the oscillatory singularities is that they lead to
overlapping near the ends of the crack. To correct this unsatisfactory feature,
Comninou [44] introduced a closed crack-tip model for analysing interface cracks
in isotropic elastic materials. Qin and Mai [45] extended this model to the case of
thermopiezoelectric materials. We follow the formulations presented in [45] in
this section.
3.10.1 Problem description
Consider a crack of length 2c lying in the interface of dissimilar anisotropic
piezoelectric media. Let the crack be partially closed with frictionless contact in
the intervals (-c, -a) and (b,
c) and opened in the interval (-a,
b) (see Fig. 3.8). We
assume that the media is subjected to far-field mechanical and electric loads, say 2 . The surface of the crack is traction- and charge-free. For the present
problem, it suffices to consider the associated problem in which the crack surface
is subjected to the conditions:
4 ,3 ,1 , ),0 ,()0 ,()( 1111 KcxxUxUxb KKK , (3.218)
bxaxUxUxb
1121212 ),0 ,()0 ,()( , (3.219)
,2
)2(
2
)1(
2
KKK 4 ,3 ,1 , 1 Kcx , (3.220)
,22
)2(
22
)1(
22
bxa 1 (3.221)
and the continuous conditions on the interface excluding the interval [-c, c] are
given by eqn (3.193).
3.10.2 Full open crack-tip model
In a manner similar to that discussed in Section 3.3, define
2, material),(
1, material),()(
22
11
zz
zzz
fBM
fMB (3.222)
where
21 MMM . (3.223)
Fig. 3.8: Geometry of a partially closed interface crack.
It should be pointed out that the physical meaning of is quite obvious and can
be seen by noting the dislocation vector b defined by
)(1
UUb
x, (3.224)
and checking eqn (3.222), with which we have
)( b . (3.225)
The Plemelj-Sokhotskii formulae for the Cauchy integral yields
1
12
1)( dx
zxiz
c
c
b. (3.226)
Using the results (3.24), (3.25) and (3.222), the conditions (3.220) and (3.221)
can be expressed as
)()()( 1211 xxx NN (3.227)
a bc c
1x
2xmaterial 1
material 2
interface
where 1 MN , for a full open interface crack model, whose solution is [45]
)()(])([2
)()( 10
1
0
20 zzdsszsi
zz
c
c
pXXX
(3.228)
where p1(z) is a vector of linear function of z, and X0(z) a matrix of the basic
Plemelj function defined as
)()()( ],,,,diag[ ),()(
)1(
43210 czczzzzX , (3.229)
while and are the eigenvalues and eigenvectors of
0)( 2 MM e . (3.230)
The explicit solution of eqn (3.230) has been obtained by Suo et al. [10] as
2
1 ,
2
13,42,1 i , (3.231)
where
2/12/1212/12/121 ])[(tan
1 ,])[(tanh
1rsrrsr
, (3.232)
with
WDWD121 ],)tr[(
4
1 cb , (3.233)
where “tr” stands for the trace of the matrix, and
1
22
1
11
1
2
1
1 )Im( ,)Re( LSLSMWLLMD (3.234)
are two real matrices.
3.10.3 Closed crack-tip model
Introducing the following symbols:
24
23
21
*
4
3
1
*
444341
343331
141311
* , ,
NNN
NNN
NNN
N , (3.235)
}~
~
~
{ },~
~
~
{ 24232102423210 DDDWWW DW , (3.236)
)()( ),()( 220**0*2220220*
DWDWf ifi TT
(3.237)
and using the boundary conditions (3.220) and (3.221), eqn (3.227) can be
rewritten as
cx
1****** ,fNN , (3.238)
bxa fNN
1*222222222 , (3.239)
where the real matrices
DW~
and ~
are defined by [35]
IWDWD ))(~~
( ii . (3.240)
Equation (3.238) is a generalized Hilbert problem if 2 is taken as a known
function. The Hilbert problem can be solved by way of the technique of Clements
[46]. In doing this, multiplying eqn (3.238) by Yi and summing over i results in
cxfYNYNY iiijijijiji
1****** ),( . (3.241)
The Yi are chosen such that
jijijiji VNYVNY ** , . (3.242)
Eliminating Vj from eqn (3.242), one has
0)( ** YNN . (3.243)
For a non-trivial solution it is necessary that
0** NN . (3.244)
Let )3 ,2 ,1( denote the roots of eqn (3.244) and the corresponding values
of Yi and Vi be expressed by . and ii VY Equation (3.241) may then be recast as
cxxfxYxYxY iiiiiii
11*1*1*1* )},()({)()( . (3.245)
Equation (3.245) is a standard Hilbert problem and its solution may be given by
c
c
b
ak
jkj
k
jkjk
k
iki dxzxxX
xfYdx
zxxX
xY
i
zXz
**3
1
1
*))((
)(
))((
)(
2
)()()(
V (3.246)
where
kkki
czczzX kk
ln2
1 ,)()()(
1. (3.247)
In eqn (3.246) the branches of Xk(z) are chosen such that zXk(z) 1, as z ,
and the argument of k is selected to lie between 0 and 2. Substitution of eqn
(3.246) into eqn (3.239) leads to
)( )(1),(1
)(),(1
)()(
22
2 xDddt
t
tb
x
xCdttb
xt
txBxbxA
b
a
b
a
b
a
(3.248)
where
ik
i k
ik WNxA 2
*
4,3,1
3
1
1 ~)()(
V , (3.249)
22
~),( DtxB
4,3,1
3
1
2
*
2
**1 ~
)(
)(~)(
i k
ik
k
kikik DN
tX
xXWNV , (3.250)
),(xC ik
k
k
i k
ik DNtX
xX2
**
4,3,1
3
1
1 ~
)(
)()(
V , (3.251)
22)( xD
c
ck
ikijkj
i k
ikxttX
dtDxXWY
22*
4,3,1
3
1
1
))((
~)(
1~)( V (3.252)
with
233222211
**
233222211
* ~~~ ,
~~~DYDYDYNWYWYWYN kkkkkkkk . (3.253)
For convenience, normalizing the interval (-a, b) by the change of variables;
.22
,22
,22
0
s
ababs
ababxs
ababt (3.254)
If we retain the same symbols for the new functions, eqn (3.248) can be rewritten
as
)()(1),(1
)(),(1
)()( 0
1
1
1
1
2
0
02
1
1 0
0020 sDdsds
ss
sb
ss
ssCdssb
ss
ssBsbsA
.
(3.255)
In addition to eqn (3.255), the single-valuedness of elastic displacements and
electric potential around a closed contour surrounding the whole crack requires
1
1 2 0)( dssb . (3.256)
Finally, the separation condition and the condition of unilateral constraint require
that
. ,for ,0)0 ,(
,for ,0)0,()0 ,(
11122
11
)2(
21
)1(
2
cxbaxcx
bxaxuxu
(3.257)
From eqn (3.255) we know that
dssbss
ssBsbsA )(
),(1)()( 2
1
1 0
002022
)( )(1),(1
0
1
1
1
1
2
0
0 sDdsdsss
sb
ss
ssC
(3.258)
for cxbaxc 11 and . Since the contact is smooth, 22 should be equal to
zero at s0=1, which provides two conditions for determining the unknowns, a
and b. Using eqn (3.180), the discretized form of eqns (3.255) and (3.256) may be
written as [36]
),()())((
)1)(,(1),(1)()( 02
1 1 0
2
0
0
0020 rk
n
k
m
j jkrj
jjr
rk
krrr sDs
ssss
sssC
mss
ssB
nssA
,0)(1
2
n
k
ks (3.259)
where
. ,,,2 ,1 ,2
)12(cos
,1,,2 ,1 ,cos ,,,2 ,1 ,2
)12(cos 0
nmmkm
js
nrn
rsnk
n
ks
j
rk
(3.260)
Equation (3.259) provides a system of n linear algebraic equations to determine
).(2 ks Once the function 2(sk) has been found from eqn (3.259), the stresses
and electric displacements, ),( 12 x can be given from eqn (1.56) in the form
2
1
1 1
12
)(~
~)(
sx
dssx
bDbW . (3.261)
Thus, we can evaluate the SED intensity factors by the following equation:
)(2lim 1
*
1
cxcx
K
1
1 1
)(~
~
sx
dssbDbW . (3.262)
The energy release rate can be evaluated by considering the relative
displacements U, which are obtained from the definition in eqn (3.226) as
m
k k
kk
kk
m
k
kk
kjkj
k
iki
k
EcxT
k
EN
cxTk
ExcNYxU
1 ,3,1
211
2**
1
1
*
1222*
*
3
1
1
1 )/(
1
)/(2)()(
V
(3.263)
where )(* xT j are Chebyshev polynomials of the second kind. The energy release
rate G can then be evaluated by substituting eqns (3.261) and (3.263) into eqn
(2.17).
3.11 Interaction between macro- and micro-cracks
The effect of microcracks on the stress and electric displacement near the tips of a
main crack has been considered by Qin [47]. In that study, the Green’s function
solution and principle of superposition are used to formulate a system of singular
integral equations for solving the unknown dislocation density of elastic
displacement-electric potential. The residual stress and electric displacement on
the microcrack are evaluated directly from the near-tip field of the main crack.
c0 c0
ci
ci
i
d i
( , )x xi i1 2
Fig. 3.9: Geometry of the crack-microcrack problem.
3.11.1 Statement of the problem
The geometric configuration of the problem is depicted in Fig. 3.9, showing a
main crack of length 2c0 and the N microcracks of length 2ci (i=1, 2 ),, N near
the main crack tip. The central point of the ith microcrack is denoted by
),( 21 ii xx . The orientation angle is i. The cracks are initially assumed to remain
open with free traction-charge. The corresponding boundary conditions are
For the ith crack:
0cossin 21 iini t , (i=0, 1,
2 ),, N . (3.264)
At infinity:
0 , 1202 . (3.265)
The main crack is assumed to be much larger than the ith microcrack. Moreover,
the distance between the right tip of the main crack and the centre of the ith
microcrack is such that ii dccc 00 , (c0, ci and di are shown in Fig. 3.9).
Under this assumption, the residual stress and electric displacement are calculated
approximately by the near-tip solutions of the main crack.
3.11.2 Singular integral equations
The problem shown in Fig. 3.9 can be considered as a superposition of N+1
single crack problems with unknown dislocation densities on each crack face. The
boundary conditions for each single problem are still defined by eqns (3.264) and
(3.265), except that
0
21 cossin niiini t , (i=1, 2 ),, N (3.266)
where 0ni is the residual stress and electric displacement, which is evaluated by
the near-tip fields of the main crack. For a crack of length 2c0 subjected to the far-
field 20 , the near-tip fields are given in eqns (2.12) and (2.13). With these two
equations, the residual SED 0ni can be expressed as
}sincos{Re2
)()( 1
4
1
020
0
iiJK
i
JKni pBBr
c
(3.267)
where 2
2
2
01
2 )( iii xcxr .
Using the solution (3.153) and the principle of superposition [40], the
electroelastic solution to the problem shown in Fig. 3.9 can be expressed as
follows:
i
T
i
N
i
zz bBBP ])ˆ([Im1 1
0
1
, (3.268)
i
T
i
N
i
zz bBB ])ˆ([Im1 1
0
2
(3.269)
where
)sin(cosˆ21
*0
iiiiiii pxpxzzz . (3.270)
Using eqns (3.268) and (3.269), the boundary condition (3.266) can be
formulated as
0)(),( Im1)(
2
0
0
0
1
0
dd
i
c
c
c
ci
N
i
i
i
bKbL
, (3.271)
)()(),( Im1)(
2
0
,0
nij
c
c
c
cij
N
ijj
i ddi
i
j
j
bKbL
(3.272)
where
T
jiiij zzz BBK1* )ˆ(),(
(3.273)
with
0* kikiki zzz . (3.274)
Moreover, the single-valued condition requires that
i
i
c
ci d
0)( b , ),,2 ,1 ,0( Ni . (3.275)
The singular integral equations (3.271) and (3.272) together with eqn (3.275) can
be used to determine the unknown b, and then to evaluate the SED intensity
factors.
3.12 Anti-plane crack in piezoelectric ceramic strip [48,49]
This section deals with the determination of stress and electric displacement
fields of a piezoelectric ceramic strip containing an anti-plane crack. This
problem has been considered by Shindo et al. [48, 49] and Narita and Shindo [50]
by way of the Fourier transform method. These researchers considered a
piezoelectric body in the form of an infinitely long strip containing a finite crack
subjected to external loads as shown in Fig. 3.10. A set of Cartesian coordinates
(x, y, z) is attached to the centre of the crack for reference purposes. The poled
piezoelectric ceramic strip with z as the poling axis occupies the region
( ) , hyhx , and is thick enough in the z-direction to allow a state
of anti-plane shear. The crack is situated along the plane (-c<x<c, y=0). Because
of the assumed symmetry in geometry and loading, it is sufficient to consider the
problem for ,0 x hy 0 only. Four possible cases of loading
conditions at the edges of the strip are considered. They are
Case 1: 00 ),( ,),( DhxDhx yyz , (3.276)
Case 2: 00 ),( ,),( EhxEhx yyz , (3.277)
Case 3: 00 ),( ,),( EhxEhx yyz , (3.278)
Case 4: 00 ),( ,),( DhxDhx yyz , (3.279)
where 0 is a uniform shear traction, 0 a uniform shear strain, D0 a uniform
electric displacement and E0 a uniform electric field. The boundary conditions are
given by
0)0 ,( xyz , cx 0 , (3.280)
0)0 ,( xuz , xc , (3.281)
0)0 ,( x , xc , (3.282)
)0 ,()0 ,( xExE cxx , cx 0 , (3.283)
)0 ,()0 ,( xDxD cyy , cx 0 (3.284)
where the superscript “c” stands for the electric quantities in the crack gap.
In a vacuum, the constitutive equation (1.45) and the governing equation
(1.46) become
yyxx EDED 00 , , (3.285)
Fig. 3.10: A piezoceramic strip with a crack. Definition of geometry and loading.
2h
oc - c
,
x
y
ED ,
0,, c
yy
c
xx (3.286)
where 0 is the dielectric constant of gas.
The solution for this problem can be obtained by way of the Fourier
transform technique. To this end, a Fourier transform is applied to eqn (1.46) with
fb3=qb=0, and the results are
yadxyAyAyxuz 02
0 1 )cos()}sinh()()cosh()({
2),(
, (3.287)
ybdxyByByx 02
0 1 )cos()}sinh()()cosh()({
2),(
(3.288)
where )2 ,1( )( ),( iBA ii are the unknowns to be determined and 00 , ba are
real constants, which will be determined from the edge loading conditions. A
simple calculation leads to
,)cos()]cosh()}()({
)sinh()}()([{ 2
),(
0215244
115
0 144
cdxyBeAc
yBeAcyxyz
(3.289)
0211215
111
0 115
)cos()]cosh()}()({
)sinh()}()([{ 2
),(
ddxyBAe
yBAeyxDy
(3.290)
where
01101500150440 , baedbeacc . (3.291)
Application of a Fourier transform to eqn (3.286) yields
dxyCyxc )cos()sinh()( 2
),(
0 , cx 0 (3.292)
where )(C is also an unknown function.
By applying the edge loading conditions, the constants
00 and ba are
obtained as
2
151144
01504402
151144
0150110 ,
ec
eDcb
ec
Dea
, for case 1, (3.293)
0000 , Eba , for case 2, (3.294)
00
44
01500 , Eb
c
Eea
, for case 3, (3.295)
11
0150000 ,
eDba , for case 4. (3.296)
The boundary conditions of eqns (3.276)-(3.279) lead to the following relations
between unknown functions
)()tanh()( ),()tanh()( 1212 BhBAhA . (3.297)
Satisfaction of the four mixed boundary conditions (3.280)-(3.283) leads to two
simultaneous dual integral equations of the following form
, ,0)cos()(
,0 ,2
)cos()()tanh(
0 1
44
0
0 1
xcdxA
cxc
cdxAh
(3.298)
. ,0)cos()(
,0 ,0)sin()(
0 1
0 1
xcdxB
cxdxB
(3.299)
The set of two simultaneous dual integral eqns (3.298) and (3.299) may be solved
by using new functions )(1 and )(2 defined by
dcJc
BdcJc
A )()(2
)( ,)()(2
)(1
0 02
2
1
1
0 01
2
1 (3.300)
where J0() is the zero order Bessel function of the first kind. The function
)(1 is the solution to the following Fredholm integral equation of the second
kind:
d
c
cK
c
c 1
0 1
0
441
0
44 )(),()( (3.301)
where the kernel function ),( K is expressed as
dcJcJhK )()(}1){tanh(),( 0
1 0 . (3.302)
The function )(2 =0.
Thus, the singular parts of the SED fields in the neighbourhood of the crack
tip are
,2
cos2
,2
sin2 4444
rc
K
rc
K IIIyz
IIIxz (3.303)
0 yx EE , (3.304)
,2
cos2
,2
sin2
r
K
r
K IIIyz
IIIxz (3.305)
2cos
2 ,
2sin
2 44
15
44
15
rc
KeD
rc
KeD III
yIII
x (3.306)
where r and are the polar coordinates defined in Fig. 2.2, and KIII is the so-
called stress intensity factor defined as
)1()0 ,()(2lim 144
ccxcxK yzcx
III . (3.307)
With the expressions (3.303)-(3.306), the generalized path-independent J-integral
(2.14) can be evaluated as
2
442
1IIIK
cJ . (3.308)
Writing the energy release rate expression J in terms of the mechanical and
electric loads, we obtain
)1(2
2
1
2
0
044
cc
cJ , for cases 1 and 3, (3.309)
)1(2
2
1
2
0
01504444
c
Eecc
cJ , for case 2, (3.310)
)1()(
2
2
1
2
011
0150
2
15114444
c
Deecc
cJ , for case 4. (3.311)
It can be seen from eqn (3.309) that the electric fields have no effect on the value
of J in cases 1 and 3, and thus they do not affect crack propagation.
To illustrate the effect of electroelastic interactions on the stress intensity
factor and the energy release rate, the numerical results obtained by Shindo et al.
[49] are presented here. In the analysis, they take PZT-5H, PZT-6B and BaTiO3
ceramics as an example, for which the engineering material constants are listed in
Table 3.1.
Table 3.1: Material properties of piezoelectric ceramics
Ceramics PZT-5H PZT-6B BaTiO3
)N/m10( 210
44 c 3.53 2.71 4.3
)C/m( 2
15e 17.0 4.6 11.6
)C/Vm10( 10
11
151.0 36.0 112.0
Critical energy release rate Jcr (N/m) 5.0 5.0 4.0
0
2.5
5
-2 0 2 4 6
c/h=1.8
1.0
0.0
crJ
J/
5
0 105.9
)V/m10( 5
0
E
c=0.01m
Fig. 3.11: Energy release rate of PZT-5H (Case 2).
Fig. 3.12: Energy release rate of piezoelectric ceramics (Case 2).
-0.2 1.6 3.4 5.2 7-20
2
4
5
0 105.9 PZT-5H
PZT-6B3BaTiO
4101.1 5107.7
c/h=1.5, c=0.01m
m)/V10( 5
0
E
The normalized energy release rate J/Jcr of a PZT-5H ceramic is shown in
Fig. 3.11 as a function of the applied electric field E0 and the c/h ratio for a crack
length of 2c=0.02 m and 5
0 105.9 (case 2). In case 2, as the magnitude of
the electric load is increased from zero, J can be made to increase or to decrease
depending on the directions of the load. However, once the minimum value of J is
reached, further increase in the electric load will monotonically increase J. Fig.
3.12 displays the normalized energy release rate J/Jcr of the above-mentioned
three types of piezoelectric ceramics against the applied electric field E0 for case
2 and c/h=1.5. The mechanical loads are fixed such that J=Jcr at zero electric
loads for c/h0. J is found to depend on the piezoelectric material constants.
3.13 Crack kinking problem
The problems of crack kinking in a bimaterial system with various material
combinations is treated in this section. The combinations may be piezoelectric-
piezoelectric (PP), piezoelectric-anisotropic conductor (PAC), piezoelectric-
isotropic dielectric (PID), or piezoelectric-isotropic conductor (PIC). The focus is
on the effect of electric field on the path selection of crack extension.
There are many analytical studies of crack kinking in solid materials, for
example, the work of Atkinson [51], Lo [52], Hayashi and Nemat-Nasser [53], He
and Hutchinson [54] for isotropic materials; Miller and Stock [55], Atkinson et
al. [56] for anisotropic solids, and Qin and Mai [57] for piezoelectric materials.
However, we restrict our attention here to the formulations and numerical results
presented in [57].
Fig. 3.13: Geometry of the kinked crack system.
The geometrical configuration analysed is depicted in Fig. 3.13. A main
crack of length 2c is shown lying along the interface between two dissimilar
piezoelectric materials, with a branch of length a and angle kinking upward into
a
material 1
material 2
interface1x
2x
2c
material 1 or downward into material 2. The length a is assumed to be small
compared to the length 2c of the main crack. To solve the branch problem, we
first derive the Green’s function satisfying the traction-charge free conditions on
the faces of the main crack by way of the Stroh formalism and the solution for
bimaterials due to an edge dislocation. Competition between crack extension
along the interface and kinking into the substrate is investigated with the aid of
the integral equations developed and the maximum energy release rate criterion.
The formulations for the energy release rate and the stress and electric
displacement (SED) intensity factors to the onset of kinking are expressed in
terms of dislocation density function and branch angle. The numerical results for
the energy release rate versus loading condition and branch angle are presented to
study the relationships among the kinking angle, loading condition and energy
release rate.
3.13.1 Crack-dislocation interaction in various bimaterial systems
(a) Crack-dislocation interaction in a homogeneous material (HM)
Consider the problem of a piezoelectric branched crack in a two-dimensional
infinite medium (see Fig. 3.13). The interaction between crack and dislocation
can be analysed by first studying the Green function due to a dislocation line,
since a crack may be viewed as a continuous distribution of dislocation
singularities. For an infinite plane subject to a line dislocation, say b, located at
z0(x10, x20), the electroelastic solution is [58]
bBfT
i
zzz
2
)ln()( 0
0 . (3.312)
The expression for f0(z) is obtained by imposing zero net traction-charge in a
circuit around the dislocation, and by demanding that the jumps in EDEP be
given by T321 } { uuub , where u is the jump for the quantity u across
the dislocation line. In the presence of crack, however, the solution for an edge
dislocation is no longer given by f0(z) alone. In addition to the conditions imposed
at z0, the boundary conditions for the crack must be satisfied. This can be
accomplished by evaluating the traction-charge on the crack face due to the
dislocation singularity and introducing an appropriate function fI(z) to cancel the
traction-charge. Substituting for f0(z) from eqn (3.312) into eqn (1.65), and then
into eqn (1.56), the SED on the crack surface induced by the edge dislocation
alone at z0 is of the form
bBBbBBfBfBtTT
zxizxixxx
00
00
1
2
11
2
1)()()( . (3.313)
The prescribed traction-charge, -t(x1), leads to the following Hilbert problem
[23]:
)())((
)(
2
)()(
z
zxx
dxx
i
zz
c
cI
ct
fB (3.314)
where (z) is the basic Plemelj function defined as
2/122 )()( czz . (3.315)
Using contour integration one is finally led to
4 ,),(
2
1),(
2
1)( 0*
1
0*
LbcbBFBBbBFf
TT
I zzi
zzi
z (3.316)
where
)(
)(1
1
2
1),(
00
0*z
z
zzzzF . (3.317)
The functions defined in eqns (3.312) and (3.316) together provide a Green’s
function for the crack problem:
bBFBBbBFfffTT
I zzzzzzi
zzz ),(),(1
2
1)()()( 0*
1
0*
0
0 .
(3.318)
It is obvious that the singularity is contained in f0(z) and the crack interactions are
accounted for by fI(z).
(b) Crack-dislocation interaction in a PP bimaterial
The solution for this bimaterial group has been provided in Section 3.9. Using
those solutions, the functions )( )(
0
izf are expressed as
bBBf
4
1
*)1(
0
)1(
*
)1()1(
*0
)1(
*
)1()1(
0 )ln()ln(2
1)(
k
kk
T zzzzi
z , (3.319)
for material 1 (x2=y>0), and
bBf**
4
1
)1(
0
)2()2()2(
0 )ln(2
1)( k
k
kzzi
z
, (3.320)
for material 2 (y<0), where
T
kk
)1()1(1)1(11)2(1)1(1)1(* ])(2[ BIBLMMIBB , (3.321)
T
kk
)1()1(1)1(11)2(1)1(1)2(** )( BIBLMMBB . (3.322)
The interface SED induced by the dislocation b, then, is of the form
bFFfBfBt )]()([)()()( ****
)2(
0
)2()2(
0
)2( xxxxx (3.323)
where
4
1
)1(
0
**)2(
**2
1)(
k k
k
zxix
BBF . (3.324)
This traction-charge vector, t(x) can be removed by superposing a solution
with the traction-charge, -t(x), induced by the potential fI(z). The solution for fI(z)
can be obtained by setting
2, materialin ),(
1, materialin ),()(
)2()2(1
)1()1(
z
zz
I
I
fBHH
fBh (3.325)
where )2()1(MMH .
The function h(z) defined above together with prescribed traction-charge,
t(x), on the crack faces, yields the following non-homogeneous Hilbert problem:
)()()( 1 xxx tHhHh . (3.326)
Writing the quantities h(x) and t(x) in their components in the sense of the
eigenvectors w, w3 and w4 as [10]:
,)()()()()(
,)()()()()(
4433
443321
wwwwt
wwwwh
xtxtxtxtx
xhxhxhxhx
(3.327)
one obtains
)())((
)(
2
)()(
,)())((
)(
2
)()(
),())((
)(
2
)()(
),())((
)(
2
)()(
4
4
444
3
3
333
2
2
1
1
zczxx
dxxt
i
zzh
zczxx
dxxt
i
zzh
zczxx
dxxt
i
zzh
zczxx
dxxt
i
zzh
c
c
c
c
c
c
c
c
(3.328)
where w, w3 and w4 are eigenvectors and have been defined in [10], and
.)()()(
,)()()(
,)()()(
2/12/1
4
2/12/1
3
2/12/1
czczz
czczz
czczz ii
(3.329)
The functions hi and ti are evaluated by taking inner products with w, w3 and w4,
respectively, i.e.
.)(
,)(
,)(
,)(
)( ,)(
)( ,)(
)( ,)(
)(
43
34
34
43
43
34
34
4321
Hww
Htw
Hww
Htw
Hww
Htw
Hww
Hhw
Hww
Hhw
wHw
Hhw
Hww
Hhw
T
T
T
T
T
T
T
T
T
T
T
T
T
T
xt
xt
xt
xxh
xxh
xxh
xxh
(3.330)
The contour integration of eqn (3.328) provides
bh )()( * zzh ii, (i=1,
2,
3,
4), (3.331)
)],,,,,,(),,,,,([)( **)2()1(
0
4
1
**)2()1(
0
*
1 kk
k
kkT
T
zzzzz BBFBBFHww
Hwh
(3.332)
)],,,,,,(),,,,,([)( **)2()1(
0
4
1
**)2()1(
0
*
2 kk
k
kkT
T
zzzzz BBFBBFwHw
Hwh
(3.333)
)],,,,,,(),,,,,([)( **)2()1(
03
4
1
**)2()1(
03
34
4*
3 kk
k
kkT
T
zzizziz BBFBBFHww
Hwh
(3.334)
)],,,,,,(),,,,,([)( **)2()1(
04
4
1
**)2()1(
04
43
3*
4 kk
k
kkT
T
zzizziz BBFBBFHww
Hwh
(3.335)
iT
T
iT
T
T
T
T
T
ec
ec
ec
ec
2
*
43
342
*
34
43
2
*22
*1
1 ,
1
,1
,1
X
Hww
HwX
Hww
Hw
X
wHw
HwX
Hww
Hw
(3.336)
where
.)(
,1
)(
)(1
)1(),,,,,(
**)2(4
1
**)2(
*
00
20
bBBBBX
XYYXF
k
k
k
kk
rkzzz
z
ezzr
(3.337)
Once hi(z) has been obtained, the functions f I
j z( ) ( ) can be evaluated through
use of eqns (3.325) and (3.327).
It should be pointed out that the above solutions do not apply when the
dislocation is located on the interface. For an edge dislocation b located on the
interface, say x0, the solution can be obtained by setting
.0for ,)ln()(
,0for ,)ln()(
20
)2()2(
0
10
)1()1(
0
yxzz
yxzz
qf
qf
(3.338)
The continuity condition on the interface and the condition related to the
dislocation give
1
)1(1)2(
2
)1(1)2()2()1(
1 ,)(4
1qBBqbBBAAq
i
. (3.339)
The corresponding interface traction-charge vector is
bYYqBqBt )(1
)(1
)( **
0
1
)1(
1
)1(
0
xxxx
x (3.340)
where ).(4
)1(1)2()2()1()1(
* BBAAB
Y
i
As was done previously, let
)(1
)( 4433
0
wwwwt ttttxx
x
. (3.341)
Similarly, the related functions hi(x) are as follows:
),,,,,,()( **0
*
1 YYIFHww
Hwh xzz
T
T
(3.342)
),,,,,,()( **0
*
2 YYIFwHw
Hwh xzz
T
T
(3.343)
),,,,,,()( **03
34
4*
3 YYIFHww
Hwh xziz
T
T
(3.344)
),,,,,,()( **04
43
3*
4 YYIFHww
Hwh xziz
T
T
(3.345)
,1
)( ,
1
)(2
**22
**1
ec
ec
T
T
T
TbYY
wHw
HwbYY
Hww
Hw (3.346a)
.1
)( ,
1
)(2
**
43
342
**
34
43
iT
T
iT
T
ec
ec
bYY
Hww
HwbYY
Hww
Hw (3.346b)
(c) Crack-dislocation interaction in a PAC bimaterial
In (b) the solutions for a PP bimaterial due to a single dislocation were presented.
Here we extend them to the case of a PAC bimaterial. When one of the two
materials, say material 2, is an anisotropic conductor, the interface condition
(3.193) should be changed to
)2(
3
2
1
)1(
4
3
2
1
)2(
3
2
1
)1(
3
2
1
0
,
0
u
u
u
u
u
u
. (3.347)
Further inside the conductor, we have
0)2()2( ii ED . (3.348)
The formulations presented in (b) are still available if the following modifications
are made:
10
0 ,
10
0 )33()2()33()2(B
BA
A , (3.349)
0} { , )2(
3
)2(
2
)2(
1
(2))2(
4 qqqip q , (=1, 2,
3,
4) (3.350)
where A(33) and B(33) are well-defined material matrices for anisotropic elastic
materials (see [35], for example).
(d) Crack-dislocation interaction in a PID bimaterial
In this subsection we consider a PID bimaterial for which the upper half-plane
(x2>0) is occupied by piezoelectric material (material 1), and the lower half-plane
(x2<0) is occupied by isotropic dielectric material (material 2). For simplicity a
plane strain deformation only is considered in this subsection. Thus the related
interface condition is reduced to
)2(
3
2
1
)1(
3
2
1
)2(
2
1
)1(
2
1
,
u
u
u
u
, at x2=0. (3.351)
(d1) General solutions for isotropic dielectric material
The general solutions for isotropic dielectric material can be assumed in the form
21 )],(Re[2 )],(Re[2 pxxzzfhzfvu iiii (3.352)
where v(={vi}) and h(={hi}) are determined by
.)(
,0])([ 2
vTRh
vTRRQ
p
pp
T
T
(3.353)
For isotropic dielectric material, the 33 matrices Q, R and T are in the form
00 00
020
00
,
000
00
00
,
00
00
002
G
G
GG
G
T R Q (3.354)
where and G are the Lamé constants [59] defined by
)1(2
,)21)(1(
EG
E.
Since p=i is a double root, a second independent solution can be written as
)]([Re2 ,)]([Re2 zfh
pzfv
pu iiii , (3.355)
where
0} 1 { 0}, 1{2
1ii
Gpp hh vv (3.356)
for plane elastic deformation, and
} 0 0{ 1}, 0 0{1
0
iee
hhvv (3.357)
for electric field. Following the method of Ting and Chou [59], the complete
general solutions can be expressed as
),(2
)( ),(2
)( 22 zfqi
zzzfzfq
i
zzzf pp
hBqvAqU (3.358)
i
i
i
G
ki
ik
G00
01
00
,
200
02
0)21(1
2
1
1
0
BA (3.359)
where 0 is the dielectric constant, q={q1 q2 q3}T are constants to be determined,
and
G
Gk
2. (3.360)
(d2) Green’s function for homogeneous isotropic materials
For an infinite domain subject to a line dislocation b located at )( 20100 ixxz ,
the function f(z) in eqn (3.358) can be assumed as
)ln()( 0zzzf . (3.361)
The condition
0
point theenclosing curve closedany for , zCdC
bU (3.362)
yields
bAqq*
30201002
1} {
iqqq T
, (3.363)
where
)41(00
022
0)42(4
41
1
0
*
k
GiG
iGkkG
kA . (3.364)
(d3) Green’s function for PID bimaterials
(d3.1) z0 located in isotropic dielectric material (material 2). Consider a PID
bimaterial for which a line dislocation b is applied at ,( 20100 ixxz x20<0). The
interface condition (3.193) suggests that the solution can be assumed in the form:
]ln(Im[1
],ln(Im[1
10
)1((1)
10
)1()1(qBqAU zzzz
(3.365)
for material 1 in x2>0 and
)ln()(
)()ln(Im
1
,)ln()(
)()ln(Im
1
02
)2(
0
20
0
*
2
)2(
02
)2(
0
20
0
*)2()2(
zzzz
qzzzz
zzzz
qzzzz
p
p
qBh
bAB
qAv
bAAU
(3.366)
for material 2 in x2<0, where q1 and q2 are unknowns to be determined. The
substitution of eqns (3.365) and (3.366) into eqn (3.193), yields
bABqBqBbAAqAqA*)2(
2
)2(
1
)1(*)2(
2
)2(
1
)1( , . (3.367)
Solving eqn (3.367), we have
bBqbBqba 21 , , (3.368)
where
.)()(
,)()(
*)2(1)1()2(1)1(1)2(1)1()2(1)1(
*)2(1)2()2(1)2(1)1(1)2()1(1)2(
AAABBAABBB
AAABBAABBB
b
a
(3.369)
(d3.2) z0 located in piezoelectric material (material 1). When the single
dislocation b is applied at z0 with 020 x , the electroelastic solution can be
assumed in the form
]ln([Im1
]ln(Im[1 )1(
0
)1(
3
1
)1(
0
)1()1(
qAbBAU zzzz T, (3.370)
]ln([Im1
]ln(Im[1 )1(
0
)1(
3
1
)1(
0
)1()1(
qBbBB zzzz T (3.371)
for material 1 in x2>0 and
]ln([Im1 )2(
0
)2(
3
1
)2(
qAU zz (3.372)
]ln([Im1 )2(
0
)2(
3
1
)2(
qB zz (3.373)
for material 2 in x2<0. Substituting eqns (3.370)~(3.373) into eqn (3.193), we find
the solution has the same form as that given in (b) except that all matrices here
are 33 matrices.
The interface SED induced by the dislocation b, then, is of the form
bBBfBt
a
zxx )1(
0
)1(
0
)1((1)
,1
1Im
1]Re[2 )( (3.374)
for z0 in material 2 and
bFFqBfBt )]()([1
Im1
]Re[2)( ****
)2(
0
3
1
)2()2(
0
)2()2(
1, xxzx
x
.
(3.375)
for z0 in material 1.
Similarly, t(x) can be removed by superposing a solution with the traction-
charge, -t(x), induced by the potential fI(z). The solution for fI(z) can also be
obtained by setting
2. materialin ),(
1, materialin ),()(
)2()2(1
)1()1(
z
zz
I
I
fBHH
fBh (3.376)
The subsequent derivation is the same as that in (b) except that
2/12/1
3
2/12/1
)()()(
,)()()(
czczz
czczz ii
(3.377)
where is determined in a similar manner to that of Suo et al. [10].
(e) Crack-dislocation interaction in a PIC bimaterial
As treated in (c), the solutions for a PIC bimaterial due to a single dislocation can
be obtained based on the results given in (d). When one of the two materials, say
material 2, is an isotropic conductor, the interface condition (3.193) should be
changed to
)2(
2
1
)1(
3
2
1
)2(
2
1
)1(
2
1
0
,
0
u
u
u
u
, at x2=0. (3.378)
The formulations presented in subsection (d) are still available if the following
modification is made:
0} { 22212 qqq , (=1, 2,
3). (3.379)
3.13.2 Singular integral equations
For the sake of conciseness, all formulations developed in this section are only
for the material combinations HM, PP and PAC. The related singular integral
equations for PID and PIC bimaterials can be obtained straightforwardly, as it is
necessary only to choose the appropriate dislocation density function.
(a) Electroelastic fields induced by remote uniform load
To satisfy the boundary conditions for the branch crack, it is necessary to know
the electroelastic fields induced by the remote uniform load 2 , as the Green’s
functions derived above do not satisfy the condition of uniform SED at infinity.
Consequently, set
)()()()( 0 zzzz I ffff (3.380)
where )(zf is the solution corresponding to the remote load. It is convenient to
represent the solution due to 2 as the sum of a uniform SED in an unflawed
solid and a corrective solution in which the main crack is subject to .2 The
solution to the latter is [10]:
2
12/122 ]1)([2
1)( Bf czzz (3.381)
for homogeneous piezoelectric material, and
))()()()(()(
),)()()()(()(
443321
11)2()2(
443321
1)1()1(
wwwwHHBf
wwwwBf
zhzhzhzhz
zhzhzhzhz
(3.382)
for PP or PAC bimaterials, where
.1)2()(1
1)(
,1)2()(1
1)(
,1)2()(1
1)(
,1)2()(1
1)(
43
232/122
24
34
242/122
23
22/122
22
22/122
21
Hww
Hw
Hww
Hw
wHw
Hw
Hww
Hw
T
T
i
T
T
i
T
Ti
T
Ti
czczcz
cz
ezh
czczcz
cz
ezh
iczczcz
cz
ezh
iczczcz
cz
ezh
(3.383)
So far the function f(z) satisfies the conditions at infinity and on the main
crack face. It remains only to ensure that the traction-charge free conditions for
the branch crack be met.
(b) Singular integral equations and solution methods
The remaining boundary conditions for the branch can be satisfied by redefining
the discrete dislocation, b, in terms of distributed dislocation densities, b(),
defining along the line, z=c+z*, z0=c+z
*, where z
*=cos +psin emanating
from the main crack tip. For simplicity we first consider the case of a branch in a
homogeneous solid. Enforcing satisfaction of traction-charge free condition on
the branch, a system of singular integral equations for the dislocation b() is
obtained as
0)()(),(1)(
2
0 0
0
QbY
bLd
d aa
(3.384)
where Y0(, ) is a kernel matrix function of the singular integral equation and is
Holder-continuous along ,0 a and )(Q is a known function vector
corresponding to the SED field induced by the external load. Y0(, ) and )(Q
are expressed as
}),(),(Im{),( 0*
*
0*
*
0
TT zzzzzz BFBBFBY , (3.385)
2
1
2/122
* 1)(
Re)( BBQcz
zz , (3.386)
in which sincos* pz .
For the purpose of numerical solution, the following normalized quantities
are introduced
a
as
a
as
2 ,
20 . (3.387)
If we retain the same symbols for the new functions caused by the change of
variables, the singular integral equation (3.384) can then be rewritten in the form
0)()(),(1)(
200
1
1 0
1
1 0
sdssss
ss
dssQbY
bL. (3.388)
Similarly, for a branch in a PP or PAC bimaterial solid, the corresponding
singular integral equation can be obtained as
0)()},()(),({1)(
20010
1
1 0
1
1 0
)1(
sdssssss
ss
dssQYbY
bL (3.389)
where
4
1
*
*)1(*)1(
0
*)1()1(
00)1()1(
Im),(k
k
kzszs
zss BBY , (3.390)
)},(Re{),( 0
1
1
*)1()1(
01 sszass hBBY
, (3.391)
)}(Re{2)( 0
)1(*)1()1(
0 szs fBQ . (3.392)
(c) Numerical scheme
Both the integral equations (3.388) and (3.389) can be solved numerically using
a method developed by Erdogan and Gupta [36]. First we write the unknown
density function, b(s) in eqn (83) in the form [54]:
2
1
2 1
)(ˆ
1
)(ˆ)(
s
sT
s
ss
m
j
jj
bb
b (3.393)
where )(ˆ sb is a regular function vector defined in the interval 1s , and
0)1(ˆ b [54], since the SED behaves like 2/1d , where d measures distance
from the branched crack tip, and the SED at the vertex behaves as d , (<1/2)
for a SED-free wedge with a wedge angle less than . jb̂ are the real unknown
constant vectors, and Tj(s) the Chebyshev polynomials of first kind. Thus the
discretized form of eqn (3.388) (or (3.389)) together with the condition,
0)1(ˆ b , may be written as [36]
,0)1(ˆ
,0)()(ˆ),()(2
1
1
1
000
0
m
k
kk
m
k
rkkr
kr
T
ssssssn
b
QbYL
(3.394)
for a homogeneous piezoelectric solid, and
,0)1(ˆ
,0)(),(ˆ)(ˆ),()(2
1
1
1
00100
0
)1(
m
k
kk
m
k
rkrkkr
kr
T
ssssssssn
b
QYbYL
(3.395)
for a PP or PAC bimaterial solid, where
2
11
1
),(ˆ),(
s
yxyx
YY , (3.396)
).1,2, 1,=( ),/cos(
),,2, 1,=( ,2
)12(cos
0
mrmrs
mkm
ks
r
k
(3.397)
Equation (3.394) (or (3.395)) provides a system of 4m equations for the
determination of the 4m-values of
kjb̂ . Once the function
)(ˆ sb has been found
from eqn (3.394) (or (3.395)), the SED, )()1( n , in a coordinate local to the
crack branch line can be expressed in the form
)()(),()(
2)()( 0
1
1
1
1 00
0
0
)1( sdssssdsss
ssn QbY
bL (3.398)
for a homogeneous solid, and
)()},()(),({1)(
2)()( 0010
1
1 0
1
1 0
)1(
0
)1( sdssssssss
dsssn QYbY
bL
(3.399)
for a PP or PAC bimaterial solid.
(d) SED intensity factors and energy release rate
The SED intensity factors at the right tip of the branch crack are of interest and
can be derived by first considering the traction and surface charge (TSC) on the
direction of the branch line and considering the TSC very near the tip ( 10 s )
which is given by eqn (3.398) (or (3.399)) as
)1(ˆ)(4
1
)1(8
)1(ˆ)()( *
0
*)1(
bLbL
r
a
srn (3.400)
where r is a distance ahead of the branch tip, L*=L for a homogeneous material,
and L*=L
(1) for a bimaterial.
Using eqn (3.400) we can calculate the SED intensity factors at the right tip
of the branch by the following usual definition:
)(2lim} { )1(
0rrKKKK n
r
T
DIIIIII
K . (3.401)
Combining with the results of eqn (3.400), one obtains
)1(ˆ)(8
*bLK
a. (3.402)
The energy release rate G* can be computed by the closure integral [60]:
drrxrx
G n
xT
nx
)()(2
1lim
0
)1(
0
* U (3.403)
where x is the assumed crack extension, U is a vector of elastic displacement
and electric potential jump across the branch crack. Noting that
darr
)( )(
0
bU , (3.404)
we have
drx
rxr
xG n
xT
nx
)()(
2
1lim
0
)1(
0
* U (3.405)
where
r
nn dxxr
0
)1()1( )()( . (3.406)
During the derivation of eqn (3.405), the following conditions are employed:
0)( ,0)0()1( r=xn rxU . (3.407)
Substituting eqn (3.393) into eqn (3.404), and eqn (3.400) into eqn (3.406),
later into eqn (3.405), one obtains
)1(ˆ)]([)1(ˆ16
2**bLb
TTa
G . (3.408)
3.13.3 Criterion for crack kinking
An interface crack can advance either by continued growth in the interface or by
kinking out of the interface into one of the adjoining materials. This competition
can be assessed by comparing the ratio of the energy release rate for kinking out
of the interface and for interface cracking, Gkink/Gi, to the ratio of substrate
toughness to interface toughness, is / . In essence, if
i
s
i
kink
G
G
(3.409)
kinking is favoured. Conversely, if the inequality in eqn (3.409) is reversed, the
flaw meets the condition for continuing advance in the interface at an applied
load lower than that necessary to advance the crack into the substrate, where
is and are substrate and interface toughness, respectively. For a given load
condition, 2 , the fracture toughness, , is defined as the energy release rate G
at the onset of the crack growth and has been discussed in detail elsewhere [61],
Gkink and Gi are the energy release rates for kinking out of the interface and for
interface cracking, respectively, in which Gi can be derived by setting =0 in the
previous sections, and Gkink is related to SED intensity factors, K, by
KDK2
1kinkG (3.410)
where )()(1*
TT LD .
3.13.4 Numerical results
Since the main purpose of this section is to outline the basic principles of the
proposed method, assessment is limited to a PP bimaterial plate with an interface
crack of length 2c, a branch crack of length a and its interface coinciding with the
x1-axis as shown in Fig. 3.13. In all calculations, the ratio of a/c is taken to be 0.1.
The upper and lower materials are assumed to be PZT-5H and PZT-5 [39],
respectively. The material constants for the two materials are as follows:
(1) material properties for PZT-5H:
C/Vm. 10151 ,C/Vm 10130 ,C/m 17
,C/m 3.23 ,C/m 5.6 GPa, 3.35 GPa, 55
,GPa 5.35 ,GPa 126 ,GPa 53 ,GPa 117
10
3322
10
11
2
2635
2
11
2
1312665523
443322131211
ee
eeeccc
cccccc
(2) material properties for PZT-5:
C/Vm. 107.81 ,C/Vm 1046.73 ,C/m 3.12
,C/m 8.15 ,m/C 4.5 GPa, 1.21 GPa, 4.75
,GPa 8.22 ,GPa 121 ,GPa 2.75 ,GPa 111
10
3322
10
11
2
2635
2
11
2
1312665523
443322131211
ee
eeeccc
cccccc
The ratio ikink GG / versus kink angle with 223 D =0 is shown in Fig. 3.14
for a number of loading combinations measured by )/(tan 1 III KK , where
KI and KII are the conventional mode I and mode II stress intensity factors. The
curve clearly shows that the maximum energy release rate occurs at different kink
angle ̂
for different loading phases , where ̂ is the critical kink angle for
which the maximum energy release rate will occur under a given loading
condition. For example, 0ˆ for curves 4 and 5,
11ˆ for curve 3, 13ˆ
for curve 2 and 18ˆ for curve 1.
Figure 3.15 shows the ratio of ikink GG / as a function of kink angle for
several values of remote loading 2D with 26
22 N/m104 and 02312 .
The calculation indicates that the energy release rate can have negative value
depending on the direction and magnitude of the remote loading 2D . For this
particular problem, the energy release rate Gi will be positive when 23
223 C/m 108.1C/m 108.1 D . Figure 3.15 also shows that the critical
kink angle is affected by the load 2D . Generally, the critical kink angle will
increase along with the increase of absolute value of 2D .
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
12345
)degree(
ikink GG /
0 5
25 4
45 3
90 2
65 1
Fig. 3.14: Variation of ikink GG / with kink angle for several loading
combinations measured by )/(tan 1 III KK .
Fig. 3.15: Variation of ikink GG / with kink angle for several loadings
2D ( 26
22 N/m 104 ).
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1
234
5
ikink GG /
)C/m10( ,0 5
5.0 4 ;1 3
5.0 2 ;1 1
23
22
22
22
DD
DD
DD
(degree)
Fig. 3.16: Variation of 0/GGkink
with remote loading
2D for several kink angles
( 26
22 N/m 104 ).
Figure 3.16 shows the variation of
0/GGkink
as a function of
2D
for several
values of kink angle and again for 26
22 N/m 104 and
.02312 Here
0G is the value of G at zero electric loads. It can be seen from this figure that the
ratio of 0/GGkink will mostly decrease along with an increase in the magnitude of
electric load.
The applied electric displacement can either promote or retard crack
extension. This conclusion may be useful in designing piezoelectric components
with an interface.
3.14 Crack deflection in bimaterial systems
In the previous section, crack kinking problems were considered in piezoelectric
materials in which the main crack is assumed to lie along the interface. When the
main crack is arbitrarily oriented and terminated at the interface, the formulation
developed above cannot be used to solve this kind of problem. In this section, we
show how to solve this crack problem, namely, the problem of crack deflection.
The geometrical configuration analysed is depicted in Fig. 3.17. An inclined finite
crack of length 2c in material 2 is shown impinging on an interface between two
dissimilar piezoelectric materials, with a branch of length a and angle 1 between
the branch and the interface. The length a is assumed to be small compared to the
length 2c of the inclined finite crack. Based on the Green’s functions developed
in the previous section, the branch crack problem is modelled by applying
distributed dislocation along the cracks and placing a concentrated dislocation at
the joint of the finite crack and the branch. Then, a set of singular integral
equations is developed and solved numerically. Generally, a crack impinging on
an interface joining two dissimilar materials may arrest or may advance by either
penetrating the interface or deflecting into the interface. Competition between
deflection and penetration is investigated using the integral equations developed
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1
2
3
)C/m10( 23
2
-D
0/GGkink
o
o
o
0 3
20 2
40 1
and the maximum energy release rate criterion. Numerical results are presented to
demonstrate the role of remote electroelastic loads on the path of crack extension.
3.14.1 Singular integral equations
For the sake of conciseness, all formulations developed in this section are for the
material combinations PP or PAC. The related singular integral equations for PID
and PIC bimaterials can be obtained straightforwardly, by choosing the
appropriate dislocation density functions.
2c
a
material 1
material 2
o
interface1x
1
2
0
0
Fig. 3.17: Geometry of the branch crack.
(a) Boundary conditions
Consider a finite crack with length 2c and a branch with length a (a<<c)
embedded in a bimaterial plane subjected to the remote load 0 (see Fig. 3.17).
The cracks are initially assumed to remain open and hence to be free of tractions
and charges. The corresponding boundaries are, then, as follows:
On the faces of each crack i:
2) 1,( 0cossin )(2
)(1
)( - iii
iii
n t . (3.411)
At infinity:
0 , 102 , (3.412)
where n is the normal direction to the lower face of a crack, and )(i
nt is the surface
traction and charge vector acting on the faces of the ith crack.
To study the effect of cracks on fracture behaviour, it is convenient to
represent the solution as the sum of a uniform SED in an unflawed solid and a
corrective solution in which the boundary conditions are
On the faces of each crack i:
2) 1,( cos cossin 0)(
2)(
1)( -- iii
ii
iin t . (3.413)
At infinity:
012 . (3.414)
(b) Singular integral equations
The singular integral equations for the problem described above can be
formulated by considering the stress and electric displacement fields on the
prospective site of the branch crack configuration due to the distributed
dislocation caused by all the cracks. The assumed distribution of dislocation
consists of two parts: (i) a concentrated dislocation with intensity, ,B̂ applied at
the joint of the finite crack and the branch (the origin is shown in Fig. 3.17); (ii)
distributed dislocation densities ; , , ,( Diziyixii bbbbb i=1,2) applied along the ith
crack.
Using the principle of superposition [40], the problem shown in Fig. 3.17 is
decomposed into two subproblems, each of which contains only one single crack.
The boundary conditions in eqn (3.413) can be satisfied by redefining the discrete
Green’s function bi given in Section 3.13 in terms of distributing Green’s
function bi() defined along each crack line, , , )*()(0
)*()( ik
ik
ik
ik zzzz
ii
kz cos)*(i
ikp sin)( , (i=1, 2). Enforcing satisfaction of the applied stress
conditions on each crack face, a system of singular integral equations for Green’s
function is obtained as
102
2
0 121
0 11
)1(
cosˆ)(),( )( ),()(2
-
- BV
bKbKL
ddca
,
(3.415)
201
0 212
2
0 22
)2(
cosˆ)(),( )( ),()(2
-
-
- BV
bKbKL
ddac
(3.416)
where Kij are 44 known kernel matrices and are regular within the related
interval, which can be derived from eqns (1.56), (3.194)-(3.197). They are
-
4
1
)1(*
)1(*)1(*
)1*()1(
11 Im T
zz
zBIBBK , (3.417)
-
4
1
**
)2(*)1(*
)1*()1(
12 Im a
zz
zBBK , (3.418)
-
4
1
**
)1(*)2(*
)2*()2(
21 Im BBKzz
z, (3.419)
-
4
1
)2(*
)2(*)2(*
)2*()2(
22 Im Ta
zz
zBIBBK , (3.420)
)](Im[2
1 )1(1)2()2()1()1(BBAABV
-- , (3.421)
)1(1)2(11)1(1)2(1)2(* ])(2[ BLMMIBB----- -a ,
.)( )2()2(1)2(11)1(1)2(1)1(** TaBIBLMMBB
-----
(3.422)
The single-valued displacements around the closed contour surrounding the
branch crack configuration require that
c2
0 2
0 1
ˆ)()( Bbb dda
. (3.423)
The singular integral equations (3.415), (3.416) and (3.423) are too complicated
to be evaluated exactly and consequently a numerical evaluation is necessary. To
this end, let
)()( *
2/1
ii
ii
iii
d
-
Bb (3.424)
where )(* ii B are regular functions defined on the related interval ii d0
and d1=a, d2=2c.
Employing the semi-open quadrature rule [62], eqns (3.415), (3.416) and
(3.423) can be rewritten as
),(ˆ
)(),()(),(
11
1
1
22*2211211*11111
11
)1(
kn
k
M
m
mmmkmmmk
mk
AA
-
tBV
BKBKL
(3.425)
),(ˆ
)(),()(),(
22
2
1
11*1122122*22222
22
)2(
kn
k
M
m
mmmkmmmk
mk
AA
-
-
tBV
BKBKL
(3.426)
BBB ˆ)()(1
22*211*1
M
m
mmmm AA (3.427)
where
)1,,2 ,1 ;2 ,1( ,2
sin2 -
MmiM
m
M
dA i
im ,
M
dA
M
dA MM
2 ,
2
22
11
,
,2
sin2
M
md iim
),,2 ,1 , ;2 ,1( ,
2
)5.0(sin2 Mkmi
M
kdiik
- .
Equations (3.425)-(3.427) provide a system of 4(2M+1) algebraic equations
to determine B̂ , )( and )( 22*11* mm BB . Once these unknowns have been found
from eqns (3.425)-(3.427), the SED, )()1( n , in a coordinate local to the crack
branch line can be expressed in the form
. cosˆ)(),(
)( ),()(2
)(1
)(
102
2
0 12
1
0 11
)1(
1
)1(
-
BV
bK
bKL
d
d
c
a
n
(3.428)
3.14.2 SED intensity factors and energy release rate
The SED intensity factors at the right tip of the branch crack are of interest and
can be derived by first considering the traction and surface charge (TSC) on the
direction of the branch line and considering the TSC very near the tip
),( a
which is given, from eqn (3.428), as
)()(2
1
)(2
)()()( 1*
)1(
11*
)1(
1
)1( ar
a
a
aarn BL
BL
- (3.429)
where r is a distance ahead of the branch tip.
Using eqn (3.429) we can calculate the SED intensity factors at the right tip
of the branch by the following usual definition
)(2lim} { )1(
0rrKKKK n
r
T
DIIIIII
K . (3.430)
Combining with the results of eqn (3.429), one obtains
)()(2
1*
)1(
1 aa
BLK
. (3.431)
The energy release rate G* can be computed by the closure integral
drrxrx
G n
xT
nx
)()(2
1lim
0
)1(
0
* - U (3.432)
where x is the assumed crack extension, and U is a vector of elastic displacement
and electric potential jump across the branch crack. Noting that
- darr
)( )(
0 1bU , (3.433)
we have
drx
rxr
xG n
xT
nx
-
)()(
2
1lim
0
)1(
0
* U , (3.434)
where )()1( rn is defined in eqn (3.406).
Substituting eqn (3.424) into eqn (3.433), and eqn (3.428) into eqn (3.406),
later into eqn (3.434), one obtains
)()]([)(4
1*
2
1
)1(
1*
* aaa
G TTBLB
. (3.435)
3.14.3 Criterion for crack propagation A crack impinging an interface joining two dissimilar materials may advance by
either penetrating the interface or deflecting into the interface. This competition
can be assessed by comparing the ratio of the energy release rate for penetrating
the interface and for deflecting into the interface, Gp/Gd, to the ratio of the mode I
toughness of material 1 to the interface toughness, icc / . In essence, if
ic
c
d
p
G
G
(3.436)
the impinging crack is likely to penetrate the interface, since the condition for
penetrating across the interface will be met at a lower load than that for
propagation in the interface. Conversely, if the inequality in eqn (3.436) is
reversed, the crack will tend to be deflected into the interface, where cic and
are the interface toughness and mode I toughness of material 1, respectively. For
a given load condition, say
2 , the fracture toughness, , is defined as the
energy release rate G at the onset of crack growth and has been discussed in detail
elsewhere [61], Gd and Gp are the energy release rates for deflecting into the
interface and for penetrating the interface, respectively.
3.14.4 Numerical results
Since the main purpose of this section is to outline the basic principles of the
proposed method, the assessment has been limited to a PP bimaterial plate with
an inclined crack of length 2c, a branch crack of length a and its interface
coinciding with the x1-axis as shown in Fig. 3.17. In all calculations, the ratio of
a/c is taken to be 0.1. The upper and lower materials are assumed to be PZT-5H
and PZT-5 [39], respectively. The material constants for the two materials are
given in Section 3.13.
The ratio dp GG / versus kink angle 1, with o902 - and 223 D =0, is
shown in Fig. 3.18 for a number of loading combinations measured
by )/(tan 1 - III KK , where IK and
IIK are the conventional mode I and
mode II stress intensity factors. The curve clearly shows that the maximum
energy release rate occurs at a different kink angle ̂ for different loading phases
, where ̂ is the critical kink angle for which the maximum energy release rate
will occur under a given loading condition. For example, o43ˆ for curve 1;
o34ˆ for curve 2; o28ˆ for curve 3;
o25ˆ for curve 4 and o22ˆ for
curve 5. It is also evident from Fig. 3.18 that the ratio dp GG / will increase along
with the increase of phase angle , which means that the possibility of the crack
penetrating the interface will increase.
To study the effect of remote loading 2D on the path selection of crack
propagation, the numerical results of the ratio
dp GG / have been obtained and are
plotted in Fig. 3.19 as a function of kink angle 1 for several values of remote
loading 2D with o1802 ,
2622 N/m104
and 02312 . From the
Fig. 3.18: Variation of dp GG / with kink angle 1 for several loading
combinations specified by )/(tan 1 - III KK .
0 20 40 60 80 1000.4
0.6
0.8
1
1.2
1.4
1.6
12345
dp GG /
o
oo
oo
0 5
30 4 ,45 3
60 2 ,90 1
degree)(1
degree)(1
0 20 40 60 80 1000.6
0.8
1
1.2
1.4
1.6
1.8
2
oo
oo
90 4 ,60 3
45 2 ,30 1
22
22
dp GG /1
2
3
4
Fig. 3.19: Variation of dp GG / with kink angle 1 for several angles
2 ( ,C/m 10 232
- D 2622 N/m 104 ).
calculations we find that the energy release rate may have a negative value
depending on the direction and magnitude of the remote loading 2D . For this
particular problem, the energy release rate Gi will be positive when 23
223 C/m 108.1C/m 108.1 -- - D and negative for other values of
2D .
3.15 Problems of elliptic cracks
In this section, we consider three-dimensional problems in the mathematical
theory of electroelasticity relating to elliptic cracks. In the analysis, we find that,
unlike in the two-dimensional case, the method of curvilinear coordinates and the
method of complex potential functions play central roles.
Based on the potential functions developed by Wang and Zheng [63], Zhao
et al. [64,65] presented some Green’s function solutions for penny-shaped cracks.
The problem of determining SED intensity factors for an elliptic crack has been
considered by Wang and Huang [66,67]. They considered an infinite transversely
isotropic piezoelectric medium containing an elliptic crack with semi-axes a and
b (a>b), as shown in Fig. 3.20. The crack plane lies in the xy-plane, being parallel
to the plane of isotropy, while the z-axis is directed normal to the crack.
Furthermore, equal and opposite uniform pressure 0p and electric displacement
0q are applied to the upper and lower crack surfaces. The boundary conditions of
the problem can be stated as follows:
0 and 1 , ,2
2
2
2
00 - zb
y
a
xqDp zz , (3.437)
,0 w 0 and 1 2
2
2
2
zb
y
a
x, (3.438)
Fig. 3.20: Polar coordinates at crack tip.
0 ,0 zzxyz (3.439)
where w is the elastic displacement in the z-direction, and all disturbances vanish
as )( 222 zyx . That is,
0 . (3.440)
To find solutions for the problem defined above, the general solutions expressed
by the potential functions are adopted (see Section 2.5). Considering the variety
of eqn (2.62), namely
2
2
2
2
2
2
j
jjj
zyx
-
(3.441)
where zj=x3j throughout this section, and using the expressions (2.63), zz D and
can be given as
3
1
2
23
1
2
2
,j j
j
jz
j j
j
jzz
CDz
A (3.442)
where
2
23313331
2
23313313 )( ,)( jjjjjjjj srreeCsrerccA --- , j=1-3. (3.443)
Without loss of generality, 4 (= here) may be set to zero on account of
symmetry. The boundary conditions (3.439) give
0
3
1
2
23
1
02
2
, qz
Cpz
Aj j
j
j
j j
j
j -
-
, 0 and 1 2
2
2
2
zb
y
a
x, (3.444)
J
r
x
y
while eqns (3.438) and (3.439) yield the respective equations
0 ,0
3
1
2
2
2
3
1
2
2
1
j j
j
jj
j j
j
jjz
srz
sr , 0 and 1 2
2
2
2
zb
y
a
x (3.445)
and
0 ,0
3
1
23
1
2
j
j
j
j
j
jzx
Bzy
B , z=0 (3.446)
where jjj rercB 215144 )1( . It can be found from eqn (3.446) that
3
10
jz
jjB . (3.447)
This implies that 321 and , are not completely independent. Let
03322011 )(
-zz
BBB . (3.448)
Substituting eqn (3.448) into eqn (3.444) yields
1221
0101
0
2
3
3
2
1221
0202
0
2
2
2
2
,NMNM
qMpN
zNMNM
qMpN
zzz
-
--
-
-
, (3.449)
where
1
3132
1
2121
1
3132
1
2121 , , ,
B
BCCN
B
BCCN
B
BAAM
B
BAAM ---- . (3.450)
It is expedient to introduce the ellipsoidal coordinates (
), which are related
to the Cartesian coordinates (x, y,
z) as
-
-
2222222222
2222222
),)()(()(
),)()(()(
zbabbbyabb
aaaxbaa (3.451)
where are some roots v of the ellipsoidal equation
012
2
2
2
2
-
v
z
vb
y
va
x, (3.452)
and
-- 022 ba , (3.453)
while z=0, =0 indicates points inside the ellipse 1// 2222 byax and z=0,
=0 represents points outside the ellipse. It can be proved that the function [67]
3 ,2 ,)(
12
1),,(
2
2
2
2
2
-
k
vQ
dv
v
z
vb
y
va
xHzyx
k
k
k
k
kk
kkkk
(3.454)
satisfies eqn (2.62), where
))(()( 22
kkkk vbvavvQ (3.455)
and )3 ,2( kHk stand for
))((2
)( ,
))((2
)(
1221
1010
2
3
1221
2020
2
2NMNMkE
NpMqabH
NMNMkE
NpMqabH
-
--
-
- . (3.456)
The complete elliptic integral of the second kind is denoted by E(k) with the
argument
2
22
a
bak
- . (3.457)
Differentiation of eqn (3.454) involves the quantities
)(
12
1
2
2
2
2
2
k
k
k
k
kk
kvQ
dv
v
z
vb
y
va
xI
k
-
, (3.458)
whose derivatives are
--
--
---
j
jj
j
jjjjjj
jjjjjjjj
j
j
dnu
cnusnuuE
ab
baba
bababa
z
I
)(2
)())()((
])()()([2
2
2/122/1222
2222222/1
2
2
(3.459)
where u, snu, cnu and dnu are all elliptic functions. With j=2 and j=3 in eqn
(3.459), and making use of eqn (2.62) with the recognition that if =0, then
2/32 uu and hence )( 1uE )( 2uE =E(k), which is the complete elliptic
integral of the second kind with argument 2/12))/(1( abk - , and
0/ jjj dnucnusnu , H2 and H3 in eqn (3.456) are determined. Owing to the fact
that
)(jj
j
j
j
j
vQ
dvz
z
I, (3.460)
while z=0, 0jz , it is evident from eqn (3.454) that the boundary conditions
(3.445) are satisfied.
To sum up, when eqn (3.456) is substituted into eqn (3.454), the solutions of
the potential functions j are fully determined.
The stress and electric displacement intensity factors at the crack tip can be
found by referring to a system of local coordinates (r, ) as indicated in Fig. 3.20
[66]. The local coordinate system is related by
,sin
,sin cos sin
,cos cos cos
J
J
rz
rby
rax
(3.461)
or
,2
sin)cossin(
2
,2
cos)cossin(
2
)cossin(
2
2/12222
2
2/12222
2222
JJ-
JJ
JJ-
ba
abr
ba
abr
ba
(3.462)
where J is the angle in the parametric equations of the ellipse and is defined as
the angle between the outer normal of the elliptical crack border and x-axis.
Substituting the above-mentioned solutions of potential functions (3.454) and eqn
(3.459) into eqn (3.442) with the help of eqns (3.461) and (3.462), yields
),,,,,( ),,,,,,( 321321 CCCrFDAAArF zz JJ , (3.463)
.2
cos})]()()[(
)]()()[(
)]()()[({
)]}()()
()[(2}{)cossin{),,,,,(
2/1
301221012213
2/1
203113031132
2/1
103223023321
1
322313223132
231
4/122221
321
--
--
---
---
JJJ
-
-
-
-
-
rsqBABApCBCBA
rsqBABApCBCBA
rsqBABApCBCBA
BABACACACBCB
CBAkEbabayyyrF
(3.464)
Thus, the SED intensity factors can be evaluated from the conventional
definitions (3.99) and (3.102) as
),,,1,,0(2 ),,,,1,,0(2 321321 CCCFKAAAFK DI JJ . (3.465)
The remaining stress and electric displacement components near the crack tip can
be found similarly, which we do not intend to elaborate here. In order to indicate
application of the solutions obtained, a numerical example for piezoelectric
ceramic PZT-6B is considered, whose physical constants of material are [66]
.)Vm(C 1034 ,)Vm(C 1036 ,Cm 1.7
,Cm 9.0 ,Cm 6.4 ,Nm 100.6
,Nm 1071.2 ,Nm 103.16 ,Nm 108.16
110
33
110
11
2
33
2
31
2
15
210
1312
210
44
210
33
210
11
-----
---
---
-
e
eecc
ccc
Once the eigenvalues i (i=1-3) are found from eqn (2.59), si can be calculated by
the relation 2/12 - iis . They are ,505.01 s is 475.002.12 - , 02.13 s
+0.475i. With these results, the SED intensity factors KI and KD can be further
expressed as
])1055.6(28.1[)(
)cossin(0
9
0
4/12222
qpa
b
kE
baK I
JJ , (3.466)
]7.0)1064.1[()(
)cossin(00
114/12222
qpa
b
kE
baKD -
JJ -
. (3.467)
0 20 40 60 800.2
0.4
0.6
0.8
1
1.2
1.4
b/a=0.1
0.2
1
1
J Angle
Fig. 3.21: Variation of normalized stress intensity factor with angle J.
Displayed graphically in Fig. 3.21 is a plot of the normalized intensity factor 1
)(
)cossin(28.1
4/1222
0
1kEbp
K I JJ
-
, (3.468)
with the parametric angle J for q0=0 and several values of , where =b/a. It can
be seen from Fig. 3.21 that as the angle J increases from oo 90 to0 , 1 tends to
increase with the lowest value at the border intersecting the major axis to a
maximum at the minor axis. The rate of increase becomes more pronounced for
small ratios of corresponding to narrower elliptical cracks.
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