PSI - Issue 13

Masayuki Arai et al. / Procedia Structural Integrity 13 (2018) 131–136 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 { ̅ ̂ ̂ ( ̂ ) = 2 1 ( 1 + 1) ∫ [ 11 ( ̂ , ̂ ) ̂ ( ̂ ) + 12 ( ̂ , ̂ ) ̂ ( ̂ )] ̂ ̅ ̂ ̂ ( ̂ ) = 2 1 ( 1 + 1) ∫ [ 21 ( ̂ , ̂ ) ̂ ( ̂ ) + 22 ( ̂ , ̂ ) ̂ ( ̂ )] ̂ ̅ ̂ ̂ ( ̂ ) = 2 1 ( 1 + 1) ∫ [ 31 ( ̂ , ̂ ) ̂ ( ̂ ) + 32 ( ̂ , ̂ ) ̂ ( ̂ )] ̂ where ( ̂ , ̂ ) is the influence function, and ̂ ( ̂ ) and ̂ ( ̂ ) are the dislocation densities.

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3

(1)

Fig. 2 Edge dislocations and cracks near the inclusion

Fig. 3 Curved crack near the inclusion

Consider that the curved crack near the inclusion is subjected to a uniform tensile loading as shown in Fig. 3. In order to calculate the stress intensity factors at the endpoint of such a curved crack, Eq. (1) is adapted to each segmented crack into which the curved crack was divided. Thus, the tractions on the curved crack that correspond to problem A become: { ̅ ̂ ̂ ( ̂ ) ̅ ̂ ̂ ( ̂ ) } = ∑[ ( − )] =1 { ̅ ̂ ̂ ( ̂ ) ̅ ̂ ̂ ( ̂ ) ̅ ̂ ̂ ( ̂ ) } ( = 1,2, … , ) (2) where the transformation matrix [ ( )] is: [ ( )] = [ sin 2 cos 2 sin 2 1 2 sin 2 − 1 2 sin 2 cos 2 ] For problem B, the tractions along the virtual crack line can be listed easily as the following: ̃ ̂ ̂ = 2 [1 + 2 2 cos 2 + 2 2 { 1 (1 − 2 ) sin 2 − 2 (1 + 1 cos 2 )} 2 2 − 1 + 2 1 + {cos 2 + 2 4 ( 1 − 2 )(3 2 cos 2 + 4 12 sin 2 ) 1 2 + 1 } cos 2( + ) + {cos 2 + 2 14 ( 1 − 2 )(3 2 cos 2 + 4 12 sin 2 ) 1 2 + 1 } cos 2( + )]