PSI - Issue 52
466 12
T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000
T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471
1
( r E r
)
1 = = 1 0 D D
,
0
(39)
=
1/2
2
r
and from the boundary condition (36) 2 it follows
K
1
f
( ) ,0
(
)
E
5 3 5 A l
E r
C
12 3 4 + + + = A A
=
−
(40)
1
1
r
1/2
1/2
(
)
2
a
2
r
2
r
After substituting expression for the amplitude factors 1,…,4 from Eq. (25) one arrives at ( ) ( ) ( ) ( ) ( ) ( ) 2 1 12 2 2 1 2 1 1/2 12 2 3 5 2 1 1 5 1 12 7 5 12 1 2 6 8 1 8 9 6 2 2 l fK r C l aEl − + + − − + − − + − − +
K
E
(41)
+ − =
1/2
( ) 2 r
0
from which the constant C 1 can be expressed as
(
)
) (
)
2
2 3 5 2 l +
1
+ −
K
fK
5 6
1
(
(
)
12
2
2
1
( ) 2 (42) It is convenient to introduce a non-dimensional constant 4 instead of K E , see also (Mao & Purohit, 2015) , which is a linear combination of C 11 , C 12 (see Eq. (28)) and K E . It is defined as ( ) 4 0 lim , , I r r K D r l → = (43) ( ) 2 ( ) 1 1/2 1/2 12 0 12 7 5 12 1 8 1 8 9 6 − − + 2 E C l aEl = +− + − − + + −
( ) ,
( ) ,
( ) ,
D r
E r
P r
=
+
where
and the asymptotic form of the electric polarization is
0
1
1 2
1 1 2 2 +
5
1 2
f
sin
cos
5 9 f
sin
P
C
D
f + −
A
=
−
1
1
1
2
1
1/2
a
2
ar
(44)
1 2
3 1 2 2
1 2
f
f
(
)
(
)
12 33 5 l
12 13 27 l
sin
3 + − − f f
sin .
f
f A
A
+
+ − + +
1
2 3
1
2
4
a
a
Substituting (38) 2 and (44) into (43) while considering D 1 = 0, the expression for C 1 in Eq. (42), a relation connecting 4 and K E is obtained. Formally, considering the definition of C 11 , C 12 , this relation can be written as: ( ) 2 12 4 4 12 2 1 2 . 2 2 I I E l K C K K C − = − + − − (45) = √2 [ 12 − 4 + 1 − 2 2 − 2 ( 2 4 − 12 )] The J – integral for Mode I (pure conductive crack) mixed with Mode E ( I & E ) then reads
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