13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 4 44 4 4 2 ( ( ) ( )) ( ( ) ( )) j j j s g x g x g x g x D D Λ Λ + − + − ∞ + + + =− + , a x b ≤ ≤ (27e) where the Einstein summation convention for repeated indices is adopted, and j ranges from 1 to 3. Solving the Eqs. (27a), (27b), (27c) and (27d) gives the following equation * * 2 ( ( ) ( )) ij j j i g x g x t Λ + − ∞ + =− , x a< (28) where * 4 4 44 / mj mj m j Λ Λ Λ Λ Λ = − , * 2 2 24 4 44 / j j j t t t Λ Λ ∞ ∞ ∞ = − , m, j=1, 2, 3. Thus, we can obtain [12] * * * ' 2 0 ( ) ( ) z f z Λ ∞ = g t , * * 1 * ' 2 0 ( ) ( ) ( ) z f z Λ − ∞ = g t , x a< (29) where * * 3 3 [ ] mjΛ × = Λ (30a) * 1 2 3 [ (), (), ()] T g z g z g z = g (30b) * * * * 2 21 22 23 [ , , ] T t t t ∞ ∞ ∞ ∞ = t (30c) ' 0 2 2 1 ( ) ( 1) 2 z f z z a = − − (30d) Solving Eqs. (27d) and (27e), we have ' 4 4 24 44 0 44 ( ) { ( ) ( )}/ ( ) / j j b s g z g z t f z D g z Λ Λ Λ ∞ = − + + (31) where g0(z) has the same property as the function g4(z), which is holomorphic in whole z plane with a cut (-c, c). ' 2 2 1 ( ) ( 1) 2 b z f z z b = − − (32a) 2 2 2 2 0 2 2 2 2 2 2 1 1 ( ) log arccos( ) 2 2 z b a i a z a z b g z i b z b a z b i a z b π π ⎧ ⎫ − + ⎪ ⎪ ⎪ ⎪ − = − − ⎨ ⎬ − − ⎪ ⎪ − ⎪ ⎪ − ⎩ ⎭ (32b) Furthermore, g0(z) has the following property on the crack faces 0 0 ( ) ( ) 0 g x g x + − + = , x a< (33a) 0 0 ( ) ( ) 1 g x g x + − + = , a x b ≤ ≤ (33b) Wang [12] gave the method to calculate the function log z i z i + − .
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