13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- vector 0B will be emitted from the surface of the void if its equilibrium distance ρfrom the surface of the void is equal to the dislocation core cut-off radius 0ρ(one half of the dislocation width, which represents the extent of the dislocation core spreading). In the equilibrium dislocation position, the glide force vanishes, namely 0 gf = . In the present study, we consider the remote applied critical stress is the stress required to keep dislocation with Burgers vector 0B in equilibrium position. A lower stress would suffice to keep the dislocation in the equilibrium at the distance greater than 0ρ, i.e., the equilibrium position of the dislocation is unstable, and the dislocation would be driven away from the void indefinitely, or until it is blocked by an obstacle. The angle cr θ θ = at which the dislocation is emitted from nanovoid corresponds to the minimum value of the applied stress min crσ . So by letting 0 ρ ρ = specifies the stress required to emit the dislocation from the surface of the nanovoid. When considering the effect of the remote axial loading, we suppose that xxσ σ ∞ = , 1 yy j σ σ ∞ = , 2 xy j σ σ ∞ = , it yields ( ) 1 1 1 4 j σ Γ = + , ( ) 2 1 2 1 2 2 j ij σ Γ = − + . The following expression for the critical stress crσ can be expressed as follows: ( ) ( ) [ ] ( ) [ ] ( ) Re cos Im sin Im sin Re cos img img cr f f M M θ ϕ θ ϕ σ θ ϕ θ ϕ ⎡ ⎤ ⎡ ⎤ + − + ⎣ ⎦ ⎣ ⎦ = + − + (21) where ( ) ( ) ( ) ( ) ( ) ( ) 0 ' 0 0 0 0 0 0 0 0 d img y x d d y x d f b ib z z b ib z z z ⎡ ⎤ ⎡ ⎤ = + Φ +Φ + − ⎣ Φ +Ψ ⎦ ⎣ ⎦ ( ) 2 1 2 1 0 0 01 1 21 1 21 3 3 1 k k d k k k k z a a z a z a z b z b z b z z z γ ∞ ∞ − − − = = Φ = + + + + + + + − ∑ ∑ ( ) ( ) 2 1 2 3 4 2 1 1 1 0 0 01 1 21 1 21 2 3 3 1 1 k k d k k k k z z c z c z c c z d z d z d z z z z z γ γ ∞ ∞ − − − − − − − = = Ψ = + + + + + + + + − − ∑ ∑ ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 02 02 1 22 1 2 0 22 1 2 0 2 2 22 1 2 0 22 1 2 0 3 2 22 0 0 1 2 22 0 0 1 2 02 0 1 4 22 1 2 22 0 1 2 1 4 1 2 2 1 2 2 1 2 2 1 2 2 1 2 1 2 1 4 1 2 2 1 2 2 y x y x a a j a j ij z b j ij z M b ib a j ij z b j ij z a z z j ij b z z j ij c z j b ib c j ij d z j ij − − − − − ⎡ ⎤ + + + − + − − − ⎢ ⎥ = + ⎢ ⎥ + − − − − + ⎣ ⎦ ⎡ ⎤ − + + − − + + + − ⎢ ⎥ + − + − − − ⎢ ⎥ ⎣ ⎦ It is found that the remote critical stress is independent of location from which the dislocation will be emitted, which is accord with the result of Zeng et al. [15]. 4. Condition for dislocation emission The critical stress required to emit the dislocation from the surface of the nanovoid can be determined accurately and explicitly given by Eq. (21). In this section considerable attention has be paid to elaborating the influence of the nanovoid size, the surface effect, nanovoid content and uniform distribution density of neighboring nanovoids in the effective medium on the critical condition required for dislocation emission from nanovoid surface. In this paper, we suppose that the normalized critical stress for the edge dislocation emitted from nanovoid surface by the shear modulus of the matrix 0 1 cr cr σ σ μ = , the intrinsic lengths of the nanovoid surface 0 1 1 α μ μ = , 0 1 1 β λ μ = and 0 1 1 δ τ μ = , the ratio of the shear modulus of the matrix and the
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