13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 11 12 13 BΓ =Γ +Γ +Γ (8) and tip ijkl S is the compliance tensor at the crack tip. As shown in Fig. 1, there is bi-material interface interface Γ in the domain and the interface is assumed to be perfectly bonded. Thus, the whole integral domain is divided by interface Γ into two parts, i.e., 1A and 2A . In addition, * interface I in Eq. (7) denotes the interface integral and will be discussed below. Fig. 1 Interaction integral domain cut by a material interface The line integral corresponding to the interface can be written as int * (1) (2) (1) (2) (1) (2) int 1 ,1 ,1 ,1 (1) (2) 1 ( ) ( ) ( ) ( ) erface aux aux aux erface jk jk jk i j ij ij ij j j aux j j i i I u u u u u mqd σ ε ε δ σ σ σ ρ ρ δ Γ ⎡ = − − − − − ⎣ ⎤ + − ⎦ Γ ∫ && (9) To simplify the above equation, we firstly build a curvilinear coordinate system, as shown in Fig. 2. Fig. 2 A curvilinear coordinate system originating from the interface Then, we have 2 2 1 1 10 2 20 2 0 ( ) ( ) q x x x x r dl ξ ξ = − + − = =∫ (10) 1 1 1 1 2 2 cos sin m x m x ξ α ξ α ∂ = = ∂ ∂ = = ∂ (11) where 1 2 ( , ) x x and 10 20 ( , ) x x are the global coordinates of the point p and q, respectively. 1 2 ( , ) ξ ξ are the curvilinear coordinates of the point p. 1m and 2m denotes the components of the outward
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