13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- partial differentiation; ijσ , 3j ε , 3u are the stress, strain and displacement of the phonon field, respectively; ij H , 3j w , 3v are the stress, strain and displacement of the phason field; 44 C and 2K are the elastic constants of the phonon field and the phason field, respectively; 3R is the phonon-phason coupling elastic constant. Substituting Eq. (3) into Eq. (2) ,then we can obtain the following result 2 0 , ? B u 0 (4) where 2∇ indicates the two-dimensional Laplace operator, and [ ] 3 3 , , T u v u= 44 3 0 3 2 , C R R K B 轾 犏= 犏 臌 (5) where the superscript T represents the transpose. Because 2 44 2 3 0, C K R− ≠ 1 0B- exists and thus Eq. (4) is equivalent to 2 . ? u 0 (6) The general solution to Eq. (6) is ( ) ( ), z z u f f = + 1 2 i , z x x = + (7) where ( )z f is an unknown complex vector; and ( )z f stands for the conjugate of ( )z f . To introduction Stroh-type formalism for anti-plane deformation, we take a generalized stress function vector∑, such that [21,22] [ ] 31 31 ,2 , T s H = - å [ ] 32 32 ,1 , T s H = å (8) Inserting Eq. (3) into Eq. (8) results in ,2 0 0 1 1 x x 抖 - = + 抖 f f å B B (9) ,1 0 0 2 2 x x 抖 = + 抖 f f å B B (10) Eq. (9) or Eq. (10) gives ( ) ( ) 0 0 i i f z z = - B f B å (11) Eqs. (7) and (11) can be rewritten as ( ) ( ), z z = + u Af Af (12) ( ) ( ), z z + =Bf Bf å (13) where
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