13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- -(3) can be expressed in an expanded tensor notation respectively as IJ IJKL KL C σ ε = , (7) , , 1 ( ) 2 IJ I J J I u u ε = + , (8) , 0 IJ I σ = . (9) 2.2. Definition of the interaction integral 2.2.1. Interaction integral A 2D linear MEE solid with an electrically and magnetically impermeable crack is considered. The interaction integral is the 'cross term' in the J-integral by superimposing the actual fields ( Iu , IJσ , IJε ) and some known auxiliary fields ( aux Iu , aux IJσ , aux IJε ). As shown in Fig. 1, the J-integral for MEE media is [2] 1 ,1 ,1 ,1 0 lim ( ) i ij j i i i J F u D B nd ε ε δ σ φ ϕ Γ → Γ = − − − Γ ∫ . (10) where ( )/2 jk jk j j j j F D E B H σ ε = − − is the electro-magnetic enthalpy density for linear MEE media, ijδ is Kronecker delta and in is the unit outward normal vector to the contour ε Γ . According to Section 2.1.2, the J-integral can also be expressed as 1 ,1 0 1 lim ( ) 2 JK JK I IJ J I J u n d ε ε σ ε δ σ Γ → Γ = − Γ ∫ . (11) Superposition of an actual state and a auxiliary state leads to another equilibrium state (state S ) for which the J-integral is ( ) 1 ,1 ,1 0 1 lim [ ( )( ) ( )( )] 2 S aux aux aux aux JK JK JK JK I IJ IJ J J I J u u n d ε ε σ σ ε ε δ σ σ Γ → Γ = + + − + + Γ ∫ . (12) By expanding Eq. (12) as ( )S aux J J J I = + + where J and aux J are respectively the J-integral corresponding to the actual state alone and the auxiliary state alone, one obtains the interaction integral as 1 ,1 ,1 0 1 lim [ ( ) ( )] 2 aux aux aux aux JK JK JK JK I IJ J IJ J I I u u n d ε ε σ ε σ ε δ σ σ Γ → Γ = + − + Γ ∫ . (13) Figure 1. Integral contours around the crack tip 2.2.2. Auxiliary fields
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