PSI - Issue 43
Jelena M. Djoković et al. / Procedia Structural Integrity 43 (2023) 95– 100 Author name / Structural Integrity Procedia 00 (2022) 000 – 000
97
3
( ( 2 1 2 1
) ) + − + + 1 1
( ( 1 2 1 2
) )
( ( 2 1 2 1
) ) − − + + 1 1
( ( 1 2 1 2
) )
1 1
1 1
+ +
− +
,
,
(1)
=
=
where: κ m = 3 − 4 ν m for the plane strain state and κ m = (3− ν m )/(1+ ν m ) for the plane stress state.
Fig. 1. Crack attacking the interface at an arbitrary angle
For the interface crack, the stress field in the vicinity of the crack tip is of the form:
1
(
) ( ) I ,
(
) ( ) II ,
,
(2)
i
i
Re Kr
Im Kr
=
+
ij
2
r
where: r and θ are the polar coordinates, ( ) I ,II ij , are the angular functions that correspond to the tensile forces and in-plane shear across the interface. For material 1, those functions have the form, Rice, Suo and Wang (1990), Nikolic and Veljkovic (2004):
( )
− −
sinh (
) − 3
e
2
( ) I rr
cos
cos (1 sin +
sin )
= −
+
+
cosh
2
2
2
ch
( )
− −
sinh (
) − 3
e
2
I
( )
cos
cos (cos
sin )
=
+
−
cosh
2
2
2
ch
( )
− −
sinh (
) 3 sin −
e
2
I
( ) r
sin (cos
sin )
=
+
−
cosh sh ( cosh
2
2
2
ch
( )
− −
) sin 3 −
e
sin ) −
2
II rr
sin (1 cos +
( ) co =
(3)
−
2
2
2
ch e
( )
− −
cosh (
) 3 sin −
2 2 2
( ) II
sin (sin
sin )
= −
−
+
cosh
2
ch
( )
− −
cosh (
) − 3
e
2 2 2
( ) II r
cos
cos (sin
sin )
=
+
+
cosh
2
ch
, I II
, I II
, I II
, I II = ( )
, I II
(
) 0, =
( ) = + ( ( ) ( )) .
zz
rr
rz
z
For material 2, the angular functions have the form analogous to equations (3); one has everywhere simply to change – to and vice-versa. The parameter is a characteristics of the interface, called the bielastic constant or the oscillatory index, defined by Rice (1988) as:
1 1 2 1 ln
− + .
(4)
=
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