PSI - Issue 13

134 Masayuki Arai et al. / Procedia Structural Integrity 13 (2018) 131–136 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 2 − 2 { 1 (1 − 2 ) + 2 2 1 } 2 2 (2 2 − 1 + 2 1 ) + (1 + 2 4 ( 1 − 2 )(3 2 − 2 2 ) 1 2 + 1 ) cos 2( + )} sin 2 − {1 − 2 4 ( 1 − 2 )(3 2 − 2 2 ) 1 2 + 1 } cos 2 sin 2( + )] ( = 1,2, … , ) (3) where = √ ̂ 2 + ℎ 2 , sin 2 = 2 ̂ ℎ ̂ 2 + ℎ 2 , cos 2 = ̂ 2 − ℎ 2 ̂ 2 + ℎ 2 , sin 2 = ℎ 2 ̂ 2 + ℎ 2 sin 2( + ) = sin2 { + tan −1 ( ℎ ̂ )} , cos 2( + ) = cos2 { + tan −1 ( ℎ ̂ )} Finally, for problem C, the tractions along the virtual crack line can be also listed easily as the following: { ̂ ̂ ̂ = 1 2 { 2 (1 − 1 ) + 1 (1 + 2 )} ( 1 2 ) cos 2 ̂ ̂ ̂ = − 1 2 { 2 (1 − 1 ) + 1 (1 + 2 )} ( 1 2 ) sin 2 in plane stress condition. The superposition of tractions for problem A, B and C leads to the following: { ̅ ̂ ̂ ( ̂ ) + ̃ ̂ ̂ ( ̂ ) + ̂ ̂ ̂ ( ̂ ) = 0 ̅ ̂ ̂ ( ̂ ) + ̃ ̂ ̂ ( ̂ ) + ̂ ̂ ̂ ( ̂ ) = 0 ( = 1,2, … , ) (5) where ≤ ̂ ≤ , ̂ = ℎ . Equation (5) reduces to a simultaneous singular integral equation. Following the standard procedure for dealing with a singular integral equation, the variable conversion as an end-point singular treatment is defined as: { ̂ ( ̂ ) = ( − ̂ ) 1/2 ( ̂ − ) 1/2 ̂ ( ̂ ) ̂ ( ̂ ) = ( − ̂ ) 1/2 ( ̂ − ) 1/2 ̂ ( ̂ ) ( = 1,2, … , ) (6) Owing to the fact that an additional side condition is required in order for the edge dislocation to be distributed continuously along the crack line, the following restriction is needed: ∑ [ cos −sin sin cos ] ∫ { ̂ ̂ } ̂ = { 0 0 } =1 (7) In addition, the continuity of the dislocation density at the point connecting the segmentations together is implied as: { ̂ −1 ( −1 ) ̂ −1 ( −1 ) } = [ cos( − −1 ) −sin( − −1 ) sin( − −1 ) cos( − −1 ) ] { ̂ ( ) ̂ ( ) } ( = 2,3, … , ) (8) Consequently, equations (5), (7) and (8) lead to a simultaneous singular integral equation ( 2 (2 + 1) × 2 (2 + 1) ) including the unknown functions ̂ ( ) , ̂ ( ) ( = 1,2, … , )( = 1,2, … ,2 + 1) . Here, is a division number for one segmentation. After deriving ̂ ( ) , ̂ ( ) , the stress intensity factors ( , ) at the left tip of the crack ( ̂ 1 = 1 ) and ( , ) at the right tip of crack ( ̂ = ) are evaluated based on the following: (4) 4 ̃ ̂ ̂ = − 2 [{ 2 2