13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 1 1 2 2 11 1 2 22 1 2 1 2 2 2 2 2 2 I I I I I K K x x h K E x K K x E E x h h h σ π π σ η ρ π η ρ η ρ π π − − − − − = = =− Δ + ⎛ ⎞ =− Δ + ⎜ ⎟ =− Δ + ⎝ ⎠ (22) Where, xis normalized. Similarly, τ can be deduced according to formula (9): ( ) ( ) ( ) 3 3 2 2 2 2 2 2 1 3 3 3 1 3 3 1 3 2 1 1 3 2 E E E c E c c E E c c v E E v v E E v E τ ηηρ ηη η ηησ ρ ηη σ ησ η η η ρ η η σ ησ 3 − ⎡ ⎤ = Δ + + + Δ ⎢ − ⎥ ⎣ ⎦ − ⎡ ⎤ + + + + Δ ⎢ ⎥ − ⎣ ⎦ + + − (23) For convenience, ( ) ( ) ( ) 3 2 2 2 2 2 3 1 3 , 3 3 1 3 3 1 3 2 1 1 3 2 E E E c E c c E E c c v A E B E v v C E E v D E η η η η η η η σ ηη σ ησ η η η η η σ ησ − = = + + − − = + + + − = + − (24) Formula (24) is plugged into equation (18), we can obtain: ( ) ( ) { } { } 2 2 0 0 0 3 2 ˆ 3 2 6 3 2 1 m d V d J A B A A B C dx RT dx d A B C dx ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ =− − + − + − + ′ ′ ′ =− + + + (25) where, ( ) ( ) ( ) ( ) ( ) 2 0 0 0 3 , 2 6 , 3 2 m m m AVRTAB BAVRTC A B CVRT ρ ρ ρ ′ ′ ′ = = − = − + (26) By integrating equation (25), the implicit solution can be obtained: 4 3 2 ˆ 3 4 6 12 12 M A B C J ρ ρ ρ ρ ′ ′ ′ + + + =− + (27) where M is a arbitrary constant generated during integration. Figure 4. Distribution of concentration of oxygen vacancy near crack tip when stress field of crack tip is applied
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