13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Lubarda [1, 2] have modified previous analysis by using the expression for the image force on a dislocation emitted from the surface of the void. In this case, the stress fields of an edge dislocation emitted from the surface of the void correspond to the imposed displacement discontinuity along the cut from the surface of the void to the center of the dislocation. The geometrical structure is shown in Fig. 1. One edge dislocation with Burgers vector 0B was emitted from the surface of the circular nanovoid to the point 0z in the matrix phase, and ( ) 0 1 i i z R e e θ ϕ ρ = + . The rest edge dislocation with Burgers vector 1B is located at the surface of the circular nanovoid, and 1 1 i z Re ϕ = . They are both assumed to be straight and infinite along the direction perpendicular to the xy -plane, and ( ) 0 1 i x y z B B b ib b e ϕ θ+ =− = + = , 2 2 z x y b b b = + . Fig.1 Dislocation emitted from the nanovoid surface in generalized self-consistent model For the current problem, the elastic strain and stress in the two materials produced by lattice mismatch and dislocations can easily be calculated using the theory of elasticity. For nanovoid surface, surface stress resulting from a surface free energy and a constant residual stress was suggested in the Gurtin-Murdoch model [3-5]. So according to Sharma et al. [6], the equilibrium equation and the constitutive relations on the surface 1Γ and the interface 2Γ can be expressed as ( ) ( ) ( ) ( ) 1 1 0 0 1 1 1 1 rr r t t i t t i R θθ θθ θ σ σ σ σ θ − ⎡ ⎤ ∂ + = − ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ∂ ⎢ ⎥ ⎣ ⎦ (1) ( ) ( ) ( ) ( ) 2 2 1 1 0 rr r rr r i i θ θ σ ζ σ ζ σ ζ σ ζ − + + − + = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ (2) ( ) ( ) ( ) ( ) 0 0 1 1 2 2 r r r u iu u iu u iu θ θ θ ζ ζ ζ ζ + − + − + = + ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ (3) where ru anduq are the displacement components, rr s and rq s are the stress components in the polar coordinates, 0 ru and 0u θare the displacements induced by growth or shrink of neighboring voids. In addition, 1 t R= , 2R ζ = . The symbols 1R and 2R are the inner and outer radii of the intermediate annular region (the matrix phase). 2μ 2υ 1μ 1υ 1Γ 2Γ xx σ ∞ yy σ ∞ xx σ ∞ yy σ ∞ x y 0 1R 2R 0ε 0z 1z ϕ θ
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