13th International Conference on Fracture June 16–21, 2013, Beijing, China 3 1 1 2 1 1 1 1 2 1 1 ( , ) ( , ) 1 1 1 1 2 2 ( , ) ( , ) 1 1 1 2 1 ˆ( , , ) ( , ) ( , ) ( , ) ( , ) d , 2 1 ˆ( , , ) ( , ) ( , ) d , 2 p y p y i x p y p y i x u x y p G p C p e G p C p e e v x y p C p e C p e e (7) 3 1 4 1 1 3 1 4 1 1 ( , ) ( , ) 1 1 3 3 4 4 ( , ) ( , ) 1 1 3 4 1 ˆ( , , ) ( , ) ( , ) ( , ) ( , ) d , 2 1 ˆ( , , ) ( , ) ( , ) d , 2 p y p y i x p y p y i x u x y p G p C p e G p C p e e v x y p C p e C p e e (8) where ( , )( 1 4) j C p j are unknown functions, and ( , )( 1 4) j p j are the roots of the characteristic equation 4 3 2 1 2 3 4 0, (9) where 2 2 1 2 2 0 1 2 2 1 2 0 3 2 , 2 2 2 , 1 1 p i (10) 2 2 2 0 2 1 3 2 0 8 2 2 , 1 1 p i (11) 2 2 2 2 2 2 4 2 4 2 4 3 2 2 0 1 0 0 0 4 1 1 2 0 0 0 3 2 2 2 . 1 1 1 1 p i p p p i (12) Here, ( , )( 1 4) j G p j can be expressed as 1 2 2 2 2 2 1 0 0 [2 (3 )] ( 1) ( , ) . ( 1) ( 1) ( )( 1) ( 1) / j j j j i i G p i p (13) 4. Singular integral equation Now, we define the following new unknown functions 1 1 1 1 1 1 2 1 1 1 1 1 ˆ ˆ ˆ ( , ) [( ,0, ) ( ,0, )], , ˆ ˆ ˆ ( , ) [( ,0, ) ( ,0, )], . g x p u x p u x p a x b x g x p v x p v x p a x b x (14) From (3-6), we obtain 1 1 1 1 2 1 11 1 1 12 1 2 1 0 1 2 21 1 1 22 1 2 1 0 ˆ ( , ) ( 1) ˆ ˆ d ( , , ) ( , )d ( , , ) ( , )d , 2 ˆ ( , ) ( 1) ˆ ˆ d ( , , ) ( , )d ( , , ) ( , )d , 2 b b b x a a a b b b x a a a g t p t K x t p g t p t K x t p g t p t t x pe g t p t K x t p g t p t K x t p g t p t t x pe (15) where 2 11 1 11 11 1 11 11 1 0 2 1 11 11 1 0 2( 1) 1 ( , , ) ( )cos[ ( )] ( )cos[ ( )] 4( 1) (1 ) 2( 1) cos[ ( )] ( ) sin[ ( )] , (1 ) { } A c c A c A K x t p k k t x d k k t x d t x d k k i t x d (16)
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