13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) ( ) 4 1 ln 8π 1 k k k k k k z z z z z s z z z z μω Φ ν Δ = ⎛ ⎞ − − = − ⎜ ⎟ − ⎝ − − ⎠ ∑ ( ) ( ) ( ) ( ) ( ) ( ) 4 0 1 0 0 1 1 ln ln 8π 1 k k k k k k k k k k z z X z s z z z z z z X z X z z z μω ν = ⎛ ⎞ ⎛ ⎞ − + − − + − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ − ⎠ ⎝ ⎠ ∑ , (19) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 4 0 2 2 1 1 0 0 3 3 1 1 8π 1 8π 1 k k k k k k k k k k k k k k k k k k z z z z z z z z z z z z z s X z s z z z z X z z z X z z z z z z z μω μω Ψ ν ν Δ = = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − − − ⎜ ⎟ = − − − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − − − − − − − − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∑ ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 0 1 0 0 1 1 ln ln 8π 1 k k k k k k k k k z z X z s z z z z z z z z z X z X z z z μω ν = ⎛ ⎞ ⎛ ⎞ − ′ − − − − − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ − ⎠ ⎝ ⎠ ∑ , (20) where ( ) ( )( ) 0 1 X z z a z c = − − , ( ) ( ) 0 0 d d X z X z z ′ = . Substituting Eqs. (19) and (20) into formulae (4), (5) and (6), we obtain the stress fields due to the cooperative grain boundary sliding and migration. 3. The emission force of lattice dislocations Let us consider the emission of lattice dislocations from the crack tip. For simplicity, we focus on the situation where the dislocations are of edge character and their Burgers vectors lie along the slip plane that makes an angle θ with the x-axis. For the first dislocation located at 0 i 0 0e z r θ = in the coordination system, the elastic fields can be evaluated by using the complex potentials ( )z Φ⊥ , ( )z Ψ⊥ and ( )z Ω⊥ . Referring to the work in Fang et al. [35-37], the complex potentials can be expressed as: ( ) ( ) ( ) * 0 z z z Φ Φ Φ ⊥ ⊥ ⊥ = + , ( ) ( ) ( ) * 0 z z z Ψ Ψ Ψ ⊥ ⊥ ⊥ = + , where ( ) 0 0 w z z z Φ⊥ = − and ( ) ( ) 0 0 2 0 0 wz w z z z z z Ψ⊥ = + − − . Using the same method in the section 2, we can obtain ( ) ( ) ( ) 0 0 2 0 0 0 1 2 w z z w w z z z z z z z Φ⊥ ⎛ − ⎞ = ⎜ − − ⎟ ⎜ ⎟ − − − ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 w z z wz z z w w X z X z Xzzz Xzzz Xz z z z z ⎛ ⎞ − − + + + + ⎜ ⎟ ⎜ ⎟ − − − − ⎝ ⎠ , (21) ( ) ( ) ( ) 0 0 2 0 0 0 1 2 w z z w w z z z z z z z Ω⊥ ⎛ − ⎞ = ⎜ − − ⎟ ⎜ ⎟ − − − ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 w z z wz z z w w X z X z Xzzz Xzzz Xz z z z z ⎛ ⎞ − − − + + + ⎜ ⎟ ⎜ ⎟ − − − − ⎝ ⎠ , (22) ( ) ( ) ( ) ( ) z z z z z Ψ Φ Φ Ω⊥ ⊥ ⊥ ⊥′ =− − − , (23) where ( ) ( ) i 4π 1 y x w b b μ ν = − − . The above complex potentials ( ) 0 z Φ⊥ and ( ) * z Φ ⊥ are consistent with the complex functions dφ ′ and ( ) i z φ ′ in Zhang and Li [38]. Then, the force acting on the edge dislocation consists of three parts: the image force, the force produced by the cooperative grain boundary sliding and migration and the external force. Firstly, using the Peach-Koehler formula [39], the image force can be written as ) ( ) ) ( ) ) ( ) ) ( ) 0 0 0 0 i i xy yy xx xy x y x y x y f f f z b z b z b z b σ σ σ σ ⊥ ⎡ ⎤ ⎡ ⎤ = − = + + + ⎣ ⎦ ⎣ ⎦
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