PSI - Issue 52
J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646
638 14
Author name / Structural Integrity Procedia 00 (2019) 000 – 000
ij
ij
1
11 a a − 12
J a
J
,
.
J
(57)
=
−
=
11 +
21 − 12
ij
22
21
22
2
a
a
a a
a
J
In order to determine the stress intensity factors, we need the variations of displacement and traction need. A static problem as a reference ( , ) x y u u is to be considered with the same configuration of a cracked plate in homogenous media, given by applying reciprocal theory ( ) ( ) 0 0 0 0 0, x x y y x x y y x y u t u t tu tu d uX uYd + − − + + = (58) where X and Y are equivalent body forces defined in Eq. (8) . From Betti’s reciprocal theorem, for the plane strain elasticity, the relationship between stress intensity factor, displacement and stress variation of crack length a is as follows
E
(
) ( 0
) ( 0
) ( 0 t u t x x
)
0 t u d y y
I I K K K K + II II
x x x u u t +
y y y u u t
t − +
y
0 2 2(1 ) −
=
+ +
− +
x
a
(59)
(
) (
)
+ +
,
x
x
y
y
u u X u u Y d + +
where 0 E denotes the Young’s modulus of reference. Since we know that after variation, there is
(60)
0, (,) x y
,
0,
(,) x y
.
x u u
y
t
y t = =
= =
u
x
t
t
a
u
block I
block II
a
block IV
block III
Fig. 2. Blocks surround the crack tip with variation of crack length.
Thus, we obtain
E
(
)
(
)
0
0 u t + − −
0 t u t u d 0 x x y y
.
I I K K K K + II II
u t
x
y
u X u Y d +
(61)
0
=
+
x x
y y
2 2(1 ) −
a
Therefore, the mixed-mode transformed stress intensity factors, for a plane strain problem, yield
E
(
)
(
)
(
)
( )
0
0 u t d y y
0 t u t u d + 0 x x y y
,
u t
x
y
(62)
K s
u X u Y d +
0
=
+
−
+
I
x x
2 2(1 ) −
(1 )
aK
+
I
t
u
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