13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- with the thickness of membrane is 2mm. When the model is loaded, the vacancy concentration near crack tip will be influenced significantly by the high stress field, the interesting phenomenon is called local stress effect. (a) (b) Figure 3. (a) Cross-section of the electrolyte membrane with crack (b) Distribution of concentration of oxygen vacancy when no local stress effect is considered Because the stress field is just affected by oxygen vacancy, so only oxygen vacancy is considered here for studying the local stress effect. For comparison, the coupling field without local stress effect is investigated firstly. The diffusion equation of oxygen vacancy is[8]: ( ) ( ) m J D V D RT ρ ρ τ =− ∇ − ∇ (18) Based on the deduction in section 2.2, the diffusion potential τcan be described as: ( ) 3 3 2 3 2 2 3 3 2 E E E E E E τ ηη ρ η η η ρ η ρ = Δ + + Δ + Δ (19) Where, ( ) 0 0 1 E E ν = − . Equation (19) is plugged into equation (18), the diffusion equation can be obtained: ( ) { } ( ) 3 2 2 3 3 0 2 3 2 0 ˆ 9 2 6 18 2 3 m E E E m E d V d J E E E E dx RT dx EV d RT dx ρ ρ ρ ηη ρ η η η ηη ρ ρ ρ ρ η η η ρ η =− − + + − ⎡ ⎤ + + − ⎣ ⎦ (20) where, ( ) ˆJ Jh D = , x x h = , h is the thickness of the membrane. The implicit solution can be given out by integrating the equation with chemical boundary conditions: 4 3 2 2.8751 6.8512 1.1028 0.6189 0.4431 0 x ρ ρ ρ ρ + − − + − = (21) The numerical solution can be calculated by programing the equation, the result is shown in Fig.3.(b). In order to study the local stress effect, the stress field of the crack-tip (where 0 θ= ) is applied to the region around the crack tip directly. After the external stress ( cσ) is applied, the stress tensor near the crack tip ( 0 θ= ) can be described as:
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