13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- ( ) ( ) 4 2 ' ' 1 1 2 ( ) v f z f z α α α α α α α α α ρ μ = ⎡ ⎤ = ℜ − ⎢ ⎥ ⎣ ⎦ ∑ t g a b (38) ( ) 4 ' 2 1 2 f z α α α α= ⎡ ⎤ = ℜ ⎢ ⎥ ⎣ ⎦ ∑ t b (39) It is obvious that the distributions of the electroelastic fields near the crack tip are of great interest to us. By introducing a polar coordinate system (r, θ) with the origin at the crack right tip, we have ( ) cos sin z a r α α θ μ θ − = + (40) When r is small compared to the half-length a of the crack, z a α ≈ . Eqs. (30d), (32d) and (34b) can be expressed as ' 0 1 1 ( ) ( 1) 2 2 cos sin a f z r α α θ μ θ = − + (41a) ' 2 2 1 1 ( ) ( 1) 2 b a f z i b a = − − (41b) 1 log ( 2tan ) z i i a z i π − + = − − (41c) From Eq. (39), the stress in front of the crack tip on the x-axis is calculated 4 * 2 2 2 2 44 j j j s x t D x a Λ σ Λ ∞ = + − j=1, 2, 3 (42) By using the definition of dynamic intensity factor vector 2 [ , , , ] 2 ( ) lim T D x a K K K K x a π → = = − Ⅱ Ⅰ Ⅲ K t (43) 4 Crack perpendicular to the poling axis, anti-plane problem In this situation, the infinite plate only subjects to 23σ ∞ and 2D ∞. The present authors have studied the anti-plane moving PS model using the continuous distribution dislocation method. In this article, some results will be verified using the complex function method. From Eq. (A.13), we obtain 15 23 2 11 [ , ] [ , 0] T T D e K K a D π σ ε ∞ ∞ = = + Ⅲ K (44) where KⅢ is independent of the crack propagation velocity. From Eqs. (15), (16) and (A.13), we have
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