13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 0 0 0 0 Γ Γ 1 e e q q M M q M M M M M J d d . (26) The second integral can be calculated analytically as 0 Γ 0 Γ 1 log( ) | e e M M M M M d . (27) For the first integral in Eq.(26), with the substitution of local coordinate , the detailed formulation for q=1 is 2 1 10 3 0 2 Γ Γ 1 3 1 [ ( ) ( )][( ) ( )(2 )] ( ) [ ( ) ( ) ] e e M M M M M M M c d a b d d a b c , (28) where 2 M M a c a , 2 ( ) M M M b a b d , 2 2 0 2 0 ( ) ( ) ( ) M M M M c a b d x x y y . (29) i(i=1,2,3) are the local coordinates of the collocation points on the element. With a linear transformation , Eq. (28) can be transformed into 2 1 10 3 0 2 Γ Γ 1 3 1 [ ( ) ( )][ (2 )] ( ) ( ) e e M M M M M M M c c d a a b d d a b c . (30) Then nonlinear transformation m proposed by Luo et al. [4] can be added to Eq. (30), so that it can be calculated using the standard Gaussian quadrature formula. For MJ , second Taylor series are essential, and the equation can be separated into three parts as 0 0 0 0 0 0 0 2 0 2 0 Γ Γ Γ ( ) ( ) ( ) e e e q q q M q q M q M M M M M M M M M M J d d d . (31) The third part can be evaluated as in Eq. (27), and the second part can also be evaluated by 0 2 0 Γ Γ 1 1 ( ) e e M M M M M d . (32) With linear transformation, the first part when 1 q can be transformed into 2 1 10 10 3 0 2 2 2 Γ Γ 1 3 1 ( ) [ ( ) ( )][1 (2 ) / ] ( ) ( ) ( ) e e M M M M M M M M M M c d a b a b d d a b c , (33) nonlinear transformation can also be applied. For the cases 2 q and 3 q , the integrals MJ and MJ can be evaluated in the same way as the case 1 q . For MJ , with the substitution of 1 0 2 0 1 2 ( ) ( ) ( ) ( ) M q q M n n n n x x x x , (34) it can be changed into 0 Γ 1 e M q M M M J d , (35) which has a similar form to MJ . So it can be evaluated with the advantage of the first Taylor-expansion series of q. The formulation is so complicated that it isn’t listed here.
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