466 J.S. Kuang, Y.H. Wang
satisfied by the BCM and least square technique. Numerical investigations for two cracking cases, two-equal- cracks and one crack, under biaxial or uniaxial loading are given and the results are presented in the related
figures and discussed.
2. Basic formulation
As shown by Muskhelishvili 1953, the representations of the in-plane stresses in an isotropic and homogeneous plate by two complex functions, 8x and ωx, lead to a general method of solving two-
dimensional plane problems. The stresses and displacements which are related to those functions can be expressed as
σ
x
+ σ
y
= 4Re[8z], 1
σ
y
− iτ
xy
= 8z + z + z − ¯z
′
z, 2
2Gu + iv = ηφz − ωz − z − ¯z8z, 3
where 8z = φ
′
z, z = ω
′
z, 4
η =
3 − ν 1 + ν
plane stress, 3 − 4ν plane strain,
5
G = E
21 + ν .
6 As shown figure 1, two cracks emanating from a hole are situated along the interface of a bi-material plate.
Both materials in the upper and lower regions of the plate are assumed to be isotropic and homogeneous. From the analysis of an infinite bi-material plate with a central crack Woo et al., 1992, the stress functions
for the upper region y 0 can be assumed as φ
1
z =
p
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
+
M
X
k=−N
F
k
z
k
, 7
ω
1
z = e
2π ε
p
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
−
M
X
k=−N
F
k
z
k
and for the lower region y 0 φ
2
z = e
2π ε
p
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
+ µ
M
X
k=−N
F
k
z
k
, 8
ω
2
z =
p
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
− µ
M
X
k=−N
F
k
z
k
,
Analysis of interfacial cracks emanating from a hole in a bi-material plate 467
Figure 1.
Crack emanating from a hole in a bi-material plate.
where E
k
and F
k
are the complex coefficients to be determined; ε is defined as the bi-elastic constant, given by ε =
1 2π
ln η
1
G
1
+ 1
G
2
.
η
2
G
2
+ 1
G
1
9 and
µ = G
2
G
1
η
1
+ 1 η
2
+ 1 ,
10 where η
1
, η
2
, G
1
and G
2
are the material constants and defined by Eqs 5 and 6. For the assumed stress functions, the stress singularity at the crack tips has been satisfied. With the stress
functions φ
i
z and ω
i
z , the functions 8
i
z and
i
z are obtained by using Eq. 4,
8
1
z = 1
√ z − az + b
z + b z − a
iε M
X
k=−N
E
k
z
k−1
z − a − b
2 − a + biε + z − az + b
k − 1 z
+
M
X
k=−N
kF
k
z
k−1
, 11
1
z = e
2π ε
1 √
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
z − a − b
2 − a + biε + z − az + b
k − 1 z
+
M
X
k=−N
kF
k
z
k−1
, and
468 J.S. Kuang, Y.H. Wang
8
2
z = e
2π ε
1 √
z − az + b z + b
z − a
iε M
X
k=−N
E
k
z
k−1
z − a − b
2 − a + biε + z − az + b
k − 1 z
+ µ
M
X
k=−N
kF
k
z
k−1
, 12
2
z = 1
√ z − az + b
z + b z − a
iε M
X
k=−N
E
k
z
k−1
z − a − b
2 − a + biε + z − az + b
k − 1 z
− µ
M
X
k=−N
kF
k
z
k−1
. The displacement can then be determined by Eq. 3. It is obvious that the displacements are single-
value in both the upper and lower regions. In addition, there are equal displacements of the upper and lower regions along the interface. Although the plate in figure 1 is an internal multi-connection one, the single-value
displacement condition is still satisfied automatically with the assumed stress functions.
Therefore, it is a only need for the boundary conditions on the hole and external boundaries of the plate to be considered, and they can be approximately satisfied by the BCM. For a boundary point in the upper or lower
region, two stress equations with the complex coefficients E
k
and F
k
can be developed. If the total number of points taken on the hole and the external boundary is L, 2L equations are developed. Thus, if the term number
of the summation of the stress functions is M + N, 4M + N real unknowns need to be determined. When the number of equations is equal to that of the unknowns, E
k
and F
k
can be determined. Usually, in order to obtain better results more boundary points are taken, and the least square technique is employed to solve the
equations. Once the complex coefficients E
k
and F
k
are determined, the stress intensity factors will be calculated conveniently Rice and Sih, 1965. For the right crack tip z = a,
K
I
− iK
II
= 2 √
2π lim
z→a
e
π ε
z − a
12+iε
8
1
z 13
and for the left crack tip z = −b K
I
− iK
II
= 2 √
2π e
−iπ
lim
z→−b
e
π ε
z + b
12−iε
8
1
z. 14
By substituting 8
1
z into Eqs 13 and 14, the stress intensity factors for the right and left crack tips can
be expressed respectively as K
I
− iK
II
= 2 √
2π e
π ε
a + b
iε
√ a + b
M
X
k=−N
E
k
a
k−1
a + b 2
− a + biε ,
15
K
I
− iK
II
= 2 √
2π e
π ε
−1a + b
iε
√ a + b
M
X
k=−N
E
k
−b
k−1
− b a − b
2 − a + biε
. 16
Analysis of interfacial cracks emanating from a hole in a bi-material plate 469
Figure 2. Two equal cracks emanating from a circular hole in a bi-material rectangular plate.
3. Numerical investigations
