13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Because the Poisson’s ratio has less affected by oxygen vacancy, vηcan be set zero. Once Vτ is obtained, it can be plugged into diffusion and mechanical equations[8]. The distribution of oxygen vacancy and stress field under coupling effect can be calculated. In the present work, the electrolyte is 0.8 0.2 1.9 Ce Gd O (20GDC), the corresponding non-stoichiometry chemical formula is 0.8 0.2 1.9 Ce Gd O δ− ( δrepresents the VρΔ in formula (8)). The parameters in the coupling theory can be calculated by molecular dynamics simulations: 0 236.036GPa E = , 0 0.267 v = , 0.0728 η= , 0.9571 Eη =− [9]. 2.2. The coupling effect of the planar electrolyte In this section, a planar electrolyte is selected to study the coupling effect, as shown in Fig.1. Figure 1. Schematic of electrolyte membrane The thickness of the membrane is h, and it is completely fixed alongy and z directions. The displacement boundary conditions are: 1 2 3 ( ), 0 u u x u u = = = (10) where, / x x h = , and the range of values allowed for it is 0 : 1. The strain tensors are: 1 1 1 1 1 ( ) ( ) ij i j x du x dx ε δ δ = (11) Stress tensor can be obtained: ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 11 1 11 1 1 ( ) (1 ) ( ) 1 1 ( ) 1 1 c V x K v x v v x v v σ ε η ρ ε = − − + − Δ − + − (12) ( ) ( ) ( ) 0 22 1 33 1 11 1 1 0 0 0 0 ( ) ( ) ( ) ( ) 1 1 c V x x Kv x x v v v v σ σ ε η ρ ε = = − Δ − + + (13) where 0(1 ) E K K α η ρ = + Δ , and cεrepresents constant strain. Here, pressure is low and no shearing strain exists, so 11σ can be assumed zero. According to formula (12), 11 1 ( ) x ε can be obtained: ( ) ( ) ( ) ( ) 11 1 0 0 0 0 1 ( ) 1 1 1 1 ( ) c V x v v v v x ε ε η ρ = + − + + − Δ (14) Then, the stress tensors can be rewritten as: ( ) 22 33 1 1 ( ) 1 ( ) c V E V E E x x σ σ η ρ ε η ρ η η = =− Δ + + Δ (15)
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