13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- = , A I = i 0 B B (14) where I is a 2×2 unit matrix. Eqs. (12) and (13) are the general solutions of anti-plane deformations. It is seen that the stresses and stains of the phonon field and the phason field can be obtained from Eqs. (7) and (8) if the complex potential vector ( )z f is available. 3. Stress fields and stress intensity factors We consider a lip-shape crack in a 1D hexagonal quasicrystal solid infinitely large. It is assumed that the quasi-periodic direction of 1D hexagonal QCs is along the positive direction of 3x axis. The solid is subjected to uniform remote anti-plane shear loadings of the phonon field and the phason field, as shown in Fig. 1 2a is the crack length and 2h is the crack height. For the current case, we will study the complex potentials and the stress intensity factors under anti-plane shear loadings of the phonon field and the phason field at infinity. In this case, the complex function ( )z f has the following form [21] ( ) ( ) 0 , z z z ¥ = + f c f (15) c∞ is a complex constant related to the remote loading conditions, and ( ) 0 z f is an unknown complex function vector, which vanishes at infinity, i.e., ( ) 0 ? 0 f . Differentiating Eqs. (12) and (13) with respect to 1x , we have ( ) ( ) ,1 , z z = + u AF AF (16) ( ) ( ) ,1 , z z = + BF BF ∑ (17) where ( ) ( ) z d z dz = F f . Substituting Eq. (15) into Eqs. (16) and (17), and then taking →∞ z , results in ,1 , ∞ ∞ ∞ = + u Ac Ac (18) ,1 , ∞ ∞ ∞ = + Bc Bc ∑ (19) ,1 32 32 , , T H σ ∞ ∞ ∞ ⎡ ⎤ = ⎣ ⎦ ∑ ,1 31 32 , . T w ε ∞ ∞ ∞ ⎡ ⎤ = ⎣ ⎦ u (20) The boundary along the surfaces of cracks is ( ) ( ) , s s z z t ds + =∫ Bf Bf [ ] 3 3 , , T st t h = (21)
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