13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where 0 0 / (1 ) E E v = − . Bringing formula (15) into (9), we can obtain: ( ) ( ) { } 2 3 1 0 1 1 2 1 3 1 ( ) ( ) ( ) ( ) V V V V x E P P x P x P x τ ρ ρ ρ = + Δ + Δ + Δ (16) where: ( ) 2 2 2 0 1 2 3 2 3 2 3 ( ) ( ) 2 (1 3 ) (3 2) 6 3 3 , , c c c c c c E E c E E E P P P P ηε η ε η ε η ε ηη ε ε η η ε η η ηη = + + = + + + = + + = ⎧⎪ ⎨ ⎪⎩ (17) Besides, the oxygen partial pressure is ( ) ( ) ( ) 2 2 O O logP 0/P 1 20 x x = = =− , and the chemical boundary conditions can be obtained based on experimental datas[10]: ( 0) 0.1407 x δ = = , ( 1) 0 x δ = = .Until now, all necessary parameters are obtained: 0 236.036GPa E = , (0) 0.1407 VρΔ = , 0 0.1 ρ= , 0.0728 η= , 0.9571 Eη =− , 0 0.267 v = . 3. Description of the multi-scale method Here, the multi-scale method is defined a combined method of MD simulation with finite element method (FEM). As illustrated in Fig.2.(1), the center-cracked specimen can be decomposed into three parts: in the far field, it is governed by the linear elastic fracture mechanics, at the crack tip, the region is modeled by MD simulation, and the between is called transition region[11]. (1) (2) Figure 2. Illustration of the multi-scale model As seen in Fig.2.(2), an intemediate model is introduced, its size is far smaller than the initial model. During the calculation process, the stress intensity factor (SIF) IK of the initial model is calculated firstly by FEM. Then the external load 2σfor the intermediate model can be computed with the same IK . For the intermediate model, the element size around crack tip is controlled as the same as the lattice constant of CeO2, as seen in Fig.2.(2) (c). After the second FEM calculation is completed, the displacements of boundary nodes can be obtained. And then, these displacements are applied to the MD model as boundary conditions for MD simulation. In MD simulation, NPT ensemble is selected, temperature is controlled at 1100K, time step is 1fs (1e-15s), and the in-home potential is used[9]. 4. Simulated results and discussion The electrolyte membrane with a crack is presented in Fig.3.(a), the length of the crack is 0.04mm,
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