PSI - Issue 39
L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Pro edia 00 (2019) 000–000
743 9
(
) ( 0 f
) 2
2 2 1 λ λ ′′ + + + − = 2 1 f
0 IV f
0,
(19)
0
0
0
with the boundary conditions ( ) ( 0, f f θ π = ± =
)
θ π ′
0.
(20)
= ± =
0
0
The solution of (39) has the form ( ) ( ) ( cos 1 sin B B θ λ θ = − +
)
(
)
(
) 1 .
(21)
1 λ θ
cos
1 λ θ
sin
f
B
B
− +
+ +
λ θ +
0
1
0
2
0
3
0
4
0
For the function 1 ( ) f θ one can derive ( ) ( ) 2 2 2 1 0 1 0 1 2 1 1 IV f f f λ λ ′′ + + + − = −
(
)
0 0 0 x f x w + IV
0
1 f C f C x ′′ ′′ + − + + 1 2 λ
2
,
n
0 0 0 a f λ
(22)
1
0 0
1 0
2 0
g
0
where
( ) 1 , + − 1 0 n λ
′′
2 2 4 , f λ ′ 0 0
2 1 , λ
2 = +
1
,
4 2
a
x a f
f
0 2 λ 0 0 0 f f g x
C
0 λ λ
= −
0 0 = +
=
0
0
0
0
0
1
( ) 1 . = + − 0 λ 1 0 2 1 n
2 w x a x f ′ = + 0 0 0 ′′ 0 0
′′
, ′ ′′′
2 2
1 2
4
4
f
C
+
+
λ
0 0
The boundary conditions for this equation are ( ) ( ) 1 1 0, 0. f f θ π θ π ′ = ± = = ± = (23) The results are presented in Table 5. Angular distributions of the Airy stress function for different values of the mixity parameter p M are shown in Figs. 6 – 10. Table 5. The calculated values of coefficients 1 n and 2 n for 0 1 / 2 λ = and 0 1 / 2 λ = − . 0 1 / 2 λ = 0 1 / 2 λ = − p M 1 n 2 n 1 n 2 n 0.0 p M = 4.00000 8.0000000 1.3333333 3.0123456 0.1 p M = 4.00000 7.9997562 1.3333333 3.0269835 0.2 p M = 4.00000 7.9986923 1.3333333 3.0701926 0.3 p M = 4.00000 7.9955185 1.3333333 3.1397530 0.4 p M = 4.00000 7.9867700 1.3333333 3.1397530 0.5 p M = 4.00000 7.9630625 1.3333333 3.1397530 0.6 p M = 4.00000 7.8971870 1.3333333 3.4551500 0.7 p M = 4.00000 7.7042370 1.3333303 3.5674200 0.8 p M = 4.00000 7.1181770 1.3334023 3.6780000 0.9 p M = 4.00014 5.5650000 1.3518183 20.130000 1 p M = 4.00000 8.0000000 1.3333333 1.1380240
4. Co
p M =
p M =
0.1
0.2
1 / 2
1 / 2
λ = −
λ = −
Fig. 6. Eigensolutions for
and
(left) and for
and
(right) (plane strain).
0
0
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