PSI - Issue 45

Koji Fujimoto et al. / Procedia Structural Integrity 45 (2023) 74–81 Koji Fujimoto / Structural Integrity Procedia 00 (2019) 000 – 000 IB = 2√2 + 1 lim → − {√ − ∙ ( )} = √2 ( − ) + 1 (1) = √2 ( − ) + 1 ∑ =1 In the case of a mode II crack, stress intensity factors can be obtained by a similar equation too. The value of B , T -stress at the tip ( = ) of a mixed mode crack (modes I and II; including a single mode) can be obtained using the stress component in the vicinity of the crack tip expressed in the equation (1) as B = 1 2 lim → − ([ ] = +[ ] =− ). ∙∙∙∙∙(16) In a simple case, we can describe the stress on the crack surface by the following equation. [ ] =± = ∫ ( , ) ( ) ∓ 4 + 1 ( ) ∙∙∙∙∙(17) where double-sign corresponds and ( , ) is at the position ( ,0) induced by a single dislocation with a unit Burgers vector at the position ( ,0) . The second term (non-integral term) in the right hand side of the equation (17) vanishes in the case of a mode I crack. We can rewrite ( , ) as follows considering its singularity. ( , )= 1 − + ∗ ( , ) ∙∙∙∙∙(18) Here, 1 is a constant ( 1 =0 in the case of a mode II crack) and ∗ ( , ) has no singularities within the interval < , < . Then, the equation (16) can be rewritten as B = lim → − [∫ { 1 − + ∗ ( , )} ( ) ]= 1 lim → − [∫ ( ) − ]+∫ ∗ ( , ) ( ) ∙∙∙∙∙(19) and the first and the second terms of the right hand side of the equation can be approximated using ( = 1, 2, 3, ⋯ , ) as follows. lim → − [∫ ( ) − ]= lim →1− [∫ ( ( ) − )√1 − 2 1 −1 ]≅ lim →0+ ∑[ ∫ cos cos − cos 0 ] =1 =lim →0+ ∑[ × (− sin sin )] =1 = − ∑[ ] =1 ∙∙∙∙∙(20) ∫ ∗ ( , ) ( ) = − 2 ∫ ̅ ∗ ( ) ( √1 − ) 2 1 −1 ( ̅ ∗ ( ) ≡ ∗ ( , + 2 + − 2 )) ≅ − 2 ∑[ ∫ ̅ ∗ (cos ) cos 0 ] =1 ≅ − 2 ∑[ ℎ∑{ ̅ ∗ (cos ) cos } =1 ] =1 = − 2 ℎ∑[ ̅ ∗ (cos )∑( cos ) =1 ] =1 = − 2 ℎ∑[ ̅ ∗ (cos ) ∙ ] =1 ( ≡ 2 2− 1 = ( − 12) ℎ, ℎ ≡ , ≡∑( cos ) =1 ) ∙∙∙∙∙(21) Thus, we can calculate T -stress using the coefficients of the Chebyshev polynomials; ( = 1, 2, 3, ⋯ , ) obtained by solving the linear algebraic equations (13). In more complicated cases where there are multiple unknown functions (dislocation density functions), basically we can solve the problems by extending the abovementioned method simply. ∙∙∙∙∙(15) 77 4