13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- A lie at a vertical distance p and a horizontal distance p′ from the crack tip and the length of all grain boundaries in the initial state be denoted as d . The disclination dipole CD of wedge disclinations is characterized by the strength magnitude ω and the arm x y − . The disclination dipole BE is characterized by the strength magnitude ω and the arm y . And x, y denote the distances of the grain boundary sliding and migration. Let us introduce a Cartesian system ( ) ,x y and a polar coordinate system ( ) ,r θ with the origins at the point O. Then, the coordinates of the disclinations located at the points D, C, E and B can be described as 1z ( ) 1 i 1er θ = , 2z ( ) 2 i 2er θ = , 3z ( ) 3 i 3er θ = and 4z ( ) 4 i 4er θ = , respectively. And the coordinates ( ) ,j j r θ are calculated as follows: ( ) 2 2 1 2 cos r y y p yp ϕ = + − , ( ) 2 2 2 2 cos r x x p xp ϕ = + − , ( ) ( ) ( ) 2 2 3 2 cos r y y p d y p d ϕ = + + − + , 4r p d = + ; ( ) ( ) 1 1 arccos sin y y r θ ϕ =− , ( ) ( ) 2 2 arccos sin x x r θ ϕ =− , ( ) ( ) 3 3 arccos sin y y r θ ϕ =− , 4 π 2 θ =− . crack A1 A2 A O D B E C B1 B2 ω ω− ω ω− ϕ d x y p p′ θ r 0z (b) (a) crack A B a c c Fig. 1 The cooperative grain boundary sliding and migration in a deformed nanocrystalline solid with a crack (a) General view. (b) The formation of two disclination dipoles CD and BE results from the cooperative grain boundary sliding and migration process, and the dislocation emission from the crack tip. Now, let us calculate the stress fields produced by the cooperative grain boundary sliding and migration in the deformed nanocrystalline solid with a flat crack. For the plane strain problem, stress fields ( xxσ , xyσ and yyσ ) and displacement fields ( xu and yu ) may be expressed in terms of two Muskhelishvili’s complex potentials ( ) Φ z and ( ) Ψ z in the complex plane i z x y = + : [31] ( ) ( ) ( ) 2 xx yy z z σ σ Φ Φ + = + , (1) ( ) ( ) ( ) ( ) i yy xy z z z z z σ σ Φ Φ Φ Ψ ′ − = + + + , (2) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 4 x y u u z z z z z μ ν Φ Φ Φ Ψ ′ ′ ′ + = − − − − , (3) where x x u u x ′ =∂ ∂ , y y u u x ′ =∂ ∂ , ( ) ( ) d d z z z Φ Φ ′ = ⎡ ⎤ ⎣ ⎦ , the over-bar represents the complex conjugate. And the stress fields can be written as ( ) ( ) ( ) Re 2 xx z z z z σ Φ Φ Ψ ′ = − − ⎡ ⎤ ⎣ ⎦ , (4) ( ) ( ) Im xy z z z σ Φ Ψ ′ = + ⎡ ⎤ ⎣ ⎦ , (5) ( ) ( ) ( ) Re 2 yy z z z z σ Φ Φ Ψ ′ = + + ⎡ ⎤ ⎣ ⎦ . (6) The boundary condition of the crack for the present problem can be expressed as ( ) ( ) i 0 yy xy t t σ σ − = , crack t ∈ . (7) According the Romanov and Vladimirov [32], the elastic stress fields produced by a wedge disclination characterized by strength ω, located at the point kz ( i k k x y = + ) in an infinite homogeneous medium may be expressed as follows:
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