13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Inserting Eq. (24) into Eq. (22),and then taking σ ζ = , results in ( ) ( ) ( ) ( ) 0 0 , σ σ ω σ ω σ ∞ ∞ ⎡ ⎤ + =− + ⎣ ⎦ Bf Bf Bc Bc (27) where σ is the point on the unit circle, and ( ) ( ) ( ) 0 0 σ ω σ = f f is defined. Multiplying Eq. (27) by ( ) 2 i d σ π σ ζ − ⎡ ⎤ ⎣ ⎦, where ζ is an arbitrary point inside the unit circle, and performing the Cauchy integral along the unit circle τ in the anticlockwise direction, we have ( ) ( ) ( ) 0 1 1 2 i 2 i d d τ ω σ ω σ ζ σ σ π σ ζ π σ ζ ∞ ∞ =− − − − ∫ ∫ Bf Bc Bc (28) ( ) ω ζ is analytic inside the unit circle, except for a simple pole at 0=ζ . Since ( ) ω ζ is analytic inside the unit circle, except for the simple poles at 0=ζ , im ζ= , im ζ=− . By the residue theorem in complex variable function, one has ( ) ( ) 1 1 i , 2 i 2 a d ω σ σ ω ζ ρ π σ ζ ζ = − − ∫ (29) ( ) ( ) ( ) 2 1 1 i i . 2 i 2 2 a a d m m τ ω σ ζ σ ω ζ ρ π σ ζ ζ ρ ζ = + − − + ∫ (30) The following result can be obtained by Eq.(18),Eq.(19),Eq.(20) { } 32 32 31 31 1 , i , , 2 T T H H σ σ ∞ ∞ ∞ ∞ ∞ ⎡ ⎤ ⎡ ⎤ = + ⎣ ⎦ ⎣ ⎦ Bc (31) { } 32 32 31 31 1 , - i , , 2 T T H H σ σ ∞ ∞ ∞ ∞ ∞ ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦ ⎣ ⎦ Bc (32) Substituting Eqs. (31) and (32) into Eq. (28), then differentiating the obtained results with respect to ζ leads to ( ) ( ) ( ) 2 2 31 32 0 2 2 2 2 2 2 31 32 1 1 1 1 4 4 1 1 a m a m m m H H m m σ σ ρ ζ ρ ζ ζ ρ ζ ρ ζ ∞ ∞ ∞ ∞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥ = + − − − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ BF (33) ( ) ( ) 0 0 / d d ζ ζ ζ = F f . ( ) ζ ω′ and ( ) ω ζ ′ ⎡ ⎤ ⎣ ⎦ can be given by Eq. (24) as follows ( ) ( ) 2 2 2 2 2 1 1 i , 2 1 a m m m ζ ω ζ ρ ζ ρ ζ ⎡ ⎤ − ⎢ ⎥ ′ = − − ⎢ ⎥ + ⎣ ⎦ (34)
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