PSI - Issue 45

Koji Fujimoto et al. / Procedia Structural Integrity 45 (2023) 74–81 Koji Fujimoto / Structural Integrity Procedia 00 (2019) 000 – 000

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̅( , )≡ − 2 ( + 2 + − 2 , + 2 + − 2 ). Here, the integral kernel ̅( , )= 0 − + ̅ ∗ ( , ), ̅( ) = ( √1− ) 2 , ∙∙∙∙∙(7) where the known function ̅ ∗ ( , ) and the unknown function ( ) have no singularities within the interval −1< , < 1 and 0 is a known constant. Here, ( ) is called weight function and we must obtain this function. Next, the weight function ( ) is approximated as the expansion of Chebyshev polynomials of the first kind ( ) = cos ( = cos ) with finite number of terms as follows. ( )≃∑ ( ) =0 =∑ cos =0 ∙∙∙∙∙(8) From the condition (3), 0 must vanish. Then, the singular integral equation (5) is rewritten as the following equation in which ( = 1, 2, 3, ⋯ , ) are unknown. ∑[ { 0 ∫ ( ) ( − )√1 − 2 1 −1 +∫ ̅ ∗ ( , ) ( ) √1− 2 1 −1 }] =1 = ̅( ) (−1< <1) ∙∙∙∙∙(9) Further, the transformations of variables; =cos , =cos (0< , < ) make the equation (9) into ∑[ { 0 ∫ cos cos − cos 0 +∫ ̅ ∗ (cos , cos ) cos 0 }] =1 = ̅(cos ) (0< < ). ∙∙∙∙∙(10) Letting the above equation (10) satisfied at the collocation points = ≡ 2 2− 1 ( =1,2,3,⋯, ) ∙∙∙∙∙(11) and using the following simple trapezoidal rule ∫ ̅ ∗ (cos , cos ) cos 0 ≅ ℎ 2 ,1 cos 1 +ℎ∑ , cos =2 + ℎ 2 , +1 cos +1 ( , ≡ ̅ ∗ (cos , cos ), ≡ − 1 , ℎ ≡ ) , ∙∙∙∙∙(12) then we can obtain the following linear algebraic equations in which ( = 1, 2, ⋯ , ) are unknown. ∑[ {− 0 sin sin + ℎ 2 ,1 cos 1 +ℎ∑ , cos =2 + ℎ 2 , +1 cos +1 }] =1 = ̅(cos ) ( =1,2,3,⋯, ) ∙∙∙∙∙(13) Here, the following important mathematical formula is considered. ∫ cos cos − cos 0 = − sin sin (0< < ) ∙∙∙∙∙(14) By solving the linear algebraic equations (13), we can obtain the coefficients of the Chebyshev polynomials; ( = 1, 2, 3, ⋯ , ) in the equation (8). In the case of a mode I crack problem, stress intensity factors IB at the crack tip = can be calculated using ( = 1, 2, 3, ⋯ , ) as follows. ∙∙∙∙∙(6) ̅( , ) and the dislocation density ̅( ) have the following singularities.