13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 ln 2π 1 2 k xx k k k k y y x x y y x x y y μω σ ν ⎛ ⎞ − = − + − + ⎜ ⎟ ⎜ ⎟ − − + − ⎝ ⎠ , (8) ( ) ( )( ) ( ) ( ) 2 2 2π 1 k k xy k k x x y y x x y y μω σ ν − − =− − − + − , (9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 ln 2π 1 2 k yy k k k k x x x x y y x x y y μω σ ν ⎛ ⎞ − = − + − + ⎜ ⎟ ⎜ ⎟ − − + − ⎝ ⎠ . (10) Assume that the elastic fields produced by the cooperative grain boundary sliding and migration in an infinite homogeneous medium can be evaluated by using two complex potentials ( ) 0 z ΦΔ and ( ) 0 z ΨΔ . Substituting Eqs. (8) and (9) into formula (1), Eqs. (9) and (10) into formula (2), referring to the work in Muskhelishvili [31], Fang et al. [33] and Liu et al. [34], the complex potentials ( ) 0 z ΦΔ and ( ) 0 z ΨΔ can be taken in the forms: ( ) ( ) ( ) 4 0 1 ln 4π 1 k k k z s z z μω Φ ν Δ = = − − ∑ , (11) ( ) ( ) 4 0 1 4π 1 k k k k z z s z z μω Ψ ν Δ = =− − − ∑ . (12) where ks ( ) 1,2,3,4 k = denote the sign of a specified disclination and are defined as 1 4 1 s s = = , 2 3 1 s s = =− . From Eqs. (11) and (12) together with formulae (1) and (2), we obtain the stress fields which are identical to the results in Eqs. (8)-(10). Eqs. (11) and (12) are singularity principal parts of complex potentials on the problem of the cooperative grain boundary sliding and migration in an infinite homogeneous medium without the crack. For the problem shown in Fig. 1, the complex potentials ( )z ΦΔ and ( )z ΨΔ can be written an ( ) ( ) ( ) * 0 z z z Φ Φ Φ Δ Δ Δ = + , (13) ( ) ( ) ( ) * 0 + z z z Ψ Ψ Ψ Δ Δ Δ = , (14) where ( ) 0 z ΦΔ and ( ) 0 z ΨΔ indicate the terms due to the presence of the cooperative grain boundary sliding and migration located in infinite medium, and ( ) * z Φ Δ and ( ) * z Ψ Δ refer to the terms resulting from the interaction of the cooperative grain boundary sliding and migration with the crack, which are holomorphic in the region. By using Riemann-Schwarz’s symmetry principle, we introduce a new analytical function ( ) ( ) ( ) ( ) z z z z z Ω Φ Φ Ψ Δ Δ Δ Δ ′ =− − − . (15) The substitution of Eqs. (13) and (14) into Eq. (15) yields ( ) ( ) ( ) ( ) 4 * 1 ln 4π 1 k k k k k z z z s z z z z z μω Ω Ω ν Δ Δ = ⎛ − ⎞ =− − + + ⎜ ⎟ − − ⎝ ⎠ ∑ , (16) where ( ) * z Ω Δ is holomorphic in the region. Considering the above complex potentials, the crack boundary condition (7) can be written an ( ) ( ) ( ) ( ) 0 t t t t Φ Ω Φ Ω + − Δ Δ Δ Δ − + − = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ , crack t ∈ , (17) ( ) ( ) ( ) ( ) 0 t t t t Φ Ω Φ Ω + − Δ Δ Δ Δ − − − = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ , crack t ∈ , (18) where the superscripts “+” and “−” denote the boundary values of the physical quantity as z approached the crack from the upper half plane and the lower half plane. Without loss in generality, we assume that two ends of the crack are located at points a and c on the x-axis. Therefore, the complex potentials ( )z ΦΔ and ( )z ΨΔ have the following forms:
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