PSI - Issue 39

L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Procedia 00 (2019) 000–000

742

8

Fig. 4. Circumferential distributions of the stress components in the vicinity of mixed mode crack tip (plane stress conditions).

Fig. 5. Circumferential distributions of the stress components in the vicinity of mixed mode crack tip (plane stress conditions).

4. Approximate analytical solution of the nonlinear eigenvalue problems The analytical expression for the eigenvalue λ as a function of the material nonlinearity parameter eigenvalue 0 λ corresponding to the linear problem ( (perturbation method). This approach is based on the presentation 0 λ λ ε = + . (16) Together with (16), the material nonlinearity parameter n and the function that describes the angular distribution of the Airy stress function ( ) f θ are presented in the following form where 0 ( ) f θ 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 ε ε ε = = + + + + = 3 1 2 3 1 ...

j

, n n

1,

n

ε

=

(17)

0

j

0

j

∞

+ = ∑

2 ( ) θ ε θ ε θ ε θ + + ( ) ( ) f f f 3

( ) j j f ε θ

( ) θ

( ) ...

f

f = +

.

(18)

0

1

2

3

0

j

=

For the function 0 ( ) f θ one can easily obtain the linear ordinary differential equation