13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- where 3t and 3h represent the anti-plane shear traction of the phonon field and the phason field along the boundaries of cracks. In the current case, the surfaces of the cracks are free of external loadings, thus, Eq. (21) becomes ( ) ( ) 0. z z + = Bf Bf (22) Inserting Eq. (15) into Eq. (22) gives ( ) ( ) ( ) 0 0 . z z z z ∞ ∞ + =− + Bf Bf Bc Bc (23) In order to obtain the complex function ( ) 0 z f from Eq. (23), we introduce the following mapping function [23] ( ) ( ) 2 2 1 i , 2 1 a z m m ζ ω ζ ρ ζ ζ ρ ζ ⎡ ⎤ ⎢ ⎥ = = + − + ⎢ ⎥ ⎣ ⎦ (24) in which , 1 1 m− =ρ ( ) ( ) 2 2 2 1 1 1 1 1 2 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − − − + = = m m m m a h β (25) 1<β ,the approximate representations are , 2 4 1 2 β β ρ≈ + + 2 8 2 β β ≈ − m (26) It can be shown that Eq. (24) maps the exterior region of a lip-shape crack in the z plane into the interior of a unit circle in the ζ plane, and the boundary of the lip-shape crack is transformed the unit circle τ, where we take ( ) 1 i, a ω− →− ( ) 1 i a ω− − → , as shown in Fig. 2. Figure 2. Mapping function of lip-shape crack into a unit circle.
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