13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- where, αρ is the molar fraction of defect α, and 0 ijkl C is the elastic constant of stoichiometric material ( ) 0 α α ρ ρ = . For isotropic material, there are only two independent parameters: E, v. ( ) ( ) ( )( ) (1 )(1 2 ) 2 1 ijkl ij kl ik jl il jk C Ev v v E v δ δ δ δ δ δ = + + + − + (2) Here, α η, Eαη and v αη are introduced for convenience: ( ) ( ) ( ) 0 0 0 3 , , E v dV V d dE E d dv v d α α α α α η η η ρ ρ ρ = = = (3) where, 0 V , 0E and 0v are volume, Young’s modulus and Poisson’s ratio of the stoichiometric material. The real Young’s modulus and Poisson’s ratio are: 0 1 E E E α α α η ρ = + Δ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ (4) 0 1 v v v α α α η ρ = + Δ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ (5) In the next, η∑ , E∑ and v∑ are introduced to represent α α α η ρΔ ∑ , E α α α η ρΔ ∑ and v α α α η ρΔ ∑ . By using these symbols, formula (2) can be rewritten as: ( )( ) 0 0 0 0 0 0 0 2 1 1 (1 )(1 2 ) ijkl ijkl ij kl v E v C C E v v v v v δ δ = + − + + + − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ (6) where ( )( ) ( ) ( ) ( )( ) 0 0 0 0 0 0 0 1 1 2 2 1 ijkl ij kl ik jl il jk C E v v v E v δ δ δ δ δ δ = + − + + + (7) Based on formula (6), the constitutive relation can be defined: ( ) ( ) ( ) ( )( ) * 0 0 0 0 0 0 0 0 0 = 1 1 2 1 1 + 1 2 E ij ijkl kl ijkl kl kl kk ij ij ij kk ij C C v K E v v v v v K v v v σ ε ε ε ε δ ε ηδ ε η δ = = − + − + − − + + − − ⎛ ⎞ ⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎝ ⎠ ∑ ∑ ∑ ∑ ∑ (8) where * ij ij α α α ε η ρ δ = Δ ∑ is the eigenstrain caused by non-stoichiometry effect, α η is called coefficient of compositional expansion (CCE), and ( ) 0 0 0 0 / (1 )(1 2 ) K E v v = + − . The full diffusion potential for isotropic material can be described[7-8]: ( )( ) ( ) ( ) 0 0 0 2 0 0 0 0 0 2 0 0 1 3 1 2 1 3 1 3 + 2 2 v E kk E v ij ij kk ij ij kk v v v E E v v v E E α α α α α α η τ η η σ η η σ σ η σ σ σ σ + + = − + − + + − + − ⎧ ⎫ ⎛ ⎞ ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ ∑ (9)
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