13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( ) ( ) ' ' 0 1 2 2 20 z z z γ γ + Ψ = +Γ +Ψ (12) where ( ) 1 1 1 k k i B γ μ π κ =− ⎡⎣ + ⎤⎦ , ( ) ' 2 2 1 k k i B γ μ π κ =− ⎡⎣ + ⎤⎦ ( ) 0,1 k = , ( ) 1 4 xx yy σ σ ∞ ∞ Γ = + , ( ) 2 2 2 yy xx xy i σ σ σ ∞ ∞ ∞ Γ = − + ( xxσ∞ , yyσ∞ and xyσ∞ are the remote stresses). ( ) 10 z Φ , ( ) 10 z Ψ , ( ) 20 z Φ and ( ) 20 z Ψ are holomorphic and the first two can be expanded in Laurent series ( ) 10 0 1 k k k k k k z a z b z ∞ ∞ − = = Φ = + ∑ ∑ (13) ( ) 2 2 10 0 1 k k k k k k z c z d z ∞ ∞ − − − = = Ψ = + ∑ ∑ (14) where the unknown coefficients ka , kb , kc and kd could be determined from the boundary conditions (1)-(3). According to the work of Fang and Liu [9] and Zhao et al. [10], by a sufficient number of calculations, the explicit expressions of two complex potentials ( ) 1 z Φ and ( ) 1 z Ψ in the matrix can be given ( ) 0 1 1 0 1 0 1 k k k k k k z a z b z z z z z γ γ ∞ ∞ − = = Φ = + + + − − ∑ ∑ (15) ( ) ( ) ( ) 2 2 0 0 0 1 1 1 1 2 2 0 1 0 1 0 1 k k k k k k z z z c z d z z z z z z z z z γ γ γ γ ∞ ∞ − − − = = Ψ = + + + + + − − − − ∑ ∑ (16) 3. Critical stress for dislocation emission According to Hirth and Lothe [11] and Peach-Koehler formula, the image force acting on the dislocation can be written as ( ) ( ) ( ) ( ) ( ) ( ) 0 ' 0 0 0 0 0 0 0 0 x y y x y x f if b ib z z b ib z z z ⎡ ⎤ ⎡ ⎤ − = + Φ +Φ + − ⎣ Φ +Ψ ⎦ ⎣ ⎦ (17) where xf and yf are the force acting on the edge dislocation with Burgers vector 0B in the x andy directions, respectively. ( ) 0 0z Φ and ( ) 0 0z Ψ are the perturbation complex potentials in the matrix. According to Qaissaunee and Santare [12], the perturbation complex potentials are calculated as follows: ( ) ( ) ( ) 2 1 2 1 0 0 01 02 1 1 21 22 2 1 21 22 2 3 3 1 k k k k k k z a a a z a a z a z b z b b z b z z z γ ∞ ∞ − − − = = Φ = ++Γ++ +Γ + + + − Γ + − ∑ ∑ (18) ( ) ( ) ( ) ( ) 2 1 2 3 4 2 1 1 1 0 0 01 02 1 1 21 22 2 1 21 22 2 2 3 3 1 1 k k k k k k z z c c z c z c c c z d z d d z d z z z z z γ γ ∞ ∞ − − − − − − − = = Ψ = + + − Γ + + + Γ + + + − Γ + − − ∑ ∑ (19) Based on Stagni [13], the primary physical interest lies on the component of the force along the Burgers vector direction (glide force) which are given by ( ) ( ) cos sin g x y f f f θ ϕ θ ϕ = + + + (20) Adopting the criterion from Lubada et al. [14], it is assumed that the dislocation with Burgers
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