PSI - Issue 39

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

740

6

( ) f θ ′′ for mode II crack (plane stress conditions).

Table 4. The new computed eigenvalues λ and

n

λ

(0) f ′′′

2 3 4 5 6 7 8 9

-0.229025 -0.211788 -0.231524 -0.254723 -0.272662 -0.284944 -0.292723 -0.297559 -0.301350

-1.984444 -1.874896 -1.553342 -1.431444 -1.408239 -1.403140 -1.399880 -1.396527 -1.393349

10

Fig. 1 . Circumferential distributions of the stress components in the vicinity of Mode I crack tip (plane strain conditions).

Fig. 2. Circumferential distributions of the stress components in the vicinity of Mode I crack tip (plane strain conditions). Thus, the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues leading to nontrivial solutions of Eqs. (10) or (12) satisfying the traction-free boundary conditions. Therefore, the order of the stress singularity is the eigenvalue and the angular variations of the field quantities correspond to the eigenfunctions. When we consider mode I loading or mode II loading conditions symmetry or antisymmetry requirements of the problem with respect to the crack plane at θ are utilized. Due to the symmetry (or antisymmetry) the solution is sought for one of the half-planes. In analyzing the crack problem under mixed-mode loading conditions the symmetry or antisymmetry arguments can’t be used and it is necessary to seek for the solution in the whole plane π θ π − ≤ ≤ . To find the numerical solution one has to take into account the value of the mixity parameter p M )

   

( , ( , r r θθ σ θ σ θ

0) 0)

2 arctan lim

= 

(15)

p

M

=

  

π

=

0

r

→

r

θ

For this purpose Eqn. (10) is numerically solved on the interval 0

θ π ≤ ≤ and the two-point boundary value problem

is reduced to the initial problem with the initial conditions reflecting the value of the mixity parameter