PSI - Issue 37
L.V. Stepanova et al. / Procedia Structural Integrity 37 (2022) 908–919 Author name / Structural Integrity Procedia 00 (2019) 000 – 0 0
915
8
Fig. 5. Circumferential distributions of the stress components in the vicinity of mixed mode crack tip (plane stress conditions).
4. Approximate solution of the nonlinear eigenvalue problem The analytical expression for the eigenvalue as a function of the material nonlinearity parameter eigenvalue 0 corresponding to the linear problem ( (perturbation method). This approach is based on the presentation 0 = + . (16) Together with (16), the material nonlinearity parameter n and the function that describes the angular distribution of the Airy stress function ( ) f are presented in the following form where 0 ( ) f is the solution of the undisturbed linear problem ( 1 n = ): 1 n = ) can be found by methods of the asymptotic theory 1 n = and the
2 n n n = = + + + + = 3 1 2 3 1 ...
j
, n n
1,
n
=
(17)
0
j
0
j
+ =
2 ( ) = + + + ( ) ( ) ( ) f f f f 3
( ) j j f
( ) ...
f
.
(18)
0
1
2
3
0
j
=
For the function 0 ( ) f one can easily obtain the linear ordinary differential equation ( ) ( ) 2 2 2 0 0 0 0 0 2 1 1 0, IV f f f + + + − =
(19)
with the boundary conditions ( ) ( 0, f f = =
)
0.
= =
(20)
0
0
The solution of (39) has the form ( ) ( ) ( cos 1 sin B B = − +
)
(
)
(
) 1 .
(21)
1
cos
1
sin
f
B
B
− +
+ +
+
0
1
0
2
0
3
0
4
0
For the function 1 ( ) f one can derive ( ) ( ) 2 2 2 1 0 1 0 1 2 1 1 IV f f f + + + − = −
(
)
0 0 0 x f x w + IV
0
1 f C f C x + − + + 1 2
2
,
n
0 0 0 a f
(22)
1
0 0
1 0
2 0
g
0
where
( ) 1 , + − 1 0 n
2 2 4 , f 0 0
2 1 ,
2 = +
1
,
4 2
a
x a f
f
0 2 0 0 0 f f g x
C
0
= −
0 0 = +
=
0
0
0
0
0
1
( ) 1 . = + − 0 1 0 2 1 n
2 w x a x f = + 0 0 0 0 0
,
2 2
1 2
4
4
f
C
+
+
0 0
The boundary conditions for this equation are
Made with FlippingBook Ebook Creator