13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- [ ]T 1 1 2 2 3 3 4 4 ( ) (), (), (), () z f z f z f z f z α = f (19) Then Eqs. (14) and (16) can be rewritten as ( ) 2 zα ⎡ ⎤ = ℜ ⎣ ⎦ U Af , ( ) 2 zα Φ ⎡ ⎤ = ℜ ⎣ ⎦ Bf (20) Eqs. (15) and (20) together with the relations given by Eq. (17) are the main results of this section. In these expressions, the only unknown is the function vector f(zα). The appropriate form of f(zα) depends on the boundary conditions. In order to obtain the function vector f(zα), following the same procedure [11, 12], the continuity of t(x) on the whole real axis is ' ' ' ' ( ) ( ) ( ) ( ) x x x x − + + − + = + Bf B f Bf B f , x −∞< <+∞ (21) A new complex function vector ( )z h is defined as ' ' ( ) ( ) ( ) z z z = = h Bf B f (22) And it should satisfy the boundary conditions (10) along the crack faces 2 ( ) ( ) x x + − ∞ + =− h h t , x a< (23) We also have ' ' ' ( ) ( ) ( ) i x x x + − = − δ HBf HBf (24) where ' 1 1 2 2 3 3 ( ) { , , , } x u u u u u u φ φ + − + − + − + − = − − − − δ is the generalized opening displacement, In addition, other two matrices are defined by 1 2 [ ] i − = ℜ H AB , 1− = H Λ (25) Introduce a new complex function vector ' ( ) ( ) z z = g HBf (26a) ' 1 1 ( ) ( ) z z − − = f B H g (26b) And the next task is to determine the unknown complex function vector g(z). The g1(z), g2(z) and g3(z) are holomorphic functions in whole z plane with a cut (-a, a). g4(z) is holomorphic in whole z plane with a cut (-c, c). Thus, using the Eqs. (23) and (10), we have the following equations 1 44 4 4 21 ( ( ) ( )) ( ( ) ( )) j j j g x g x g x g x Λ Λ σ + − + − ∞ + + + =− , x a< (27a) 2 44 4 4 22 ( ( ) ( )) ( ( ) ( )) j j j g x g x g x g x Λ Λ σ + − + − ∞ + + + =− , x a< (27b) 3 44 4 4 23 ( ( ) ( )) ( ( ) ( )) j j j g x g x g x g x Λ Λ σ + − + − ∞ + + + =− , x a< (27c) 4 44 4 4 2 ( ( ) ( )) ( ( ) ( )) j j j g x g x g x g x D Λ Λ + − + − ∞ + + + =− , x a< (27d)
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