13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Fig. 1a and b show the following parameters: - SB and GB orientation related ones: m ρ ( (1,0, 1) 2 1 − = ), n ρ ( (1,1,1) 3 1 = ), GB n ρ , SBα , GBα and θ denote respectively slip direction vector, normal vector to slip plane, normal vector to the GB, angle between slip plane and loading direction (m x ρ ρ , ), angle between the GB normal and loading direction (n x GB ρ ρ , ) and an angle between the SB and the GB, given by: θ GB SB o α α = − + 90 . (1) - SB size and loading parameters: t, L, 0Σ and f are respectively SB thickness, SB length, macroscopic applied tensile stress and Schmid factor. In addition, for further assumptions, let nnσ , nmσ , 0τ and r be respectively the GB normal stress, the GB shear stress, SB yield shear stress and the distance to the SB along the GB. It is worth to note that the developments involved in the current paper concern the close fields configuration which means that one focuses on stress evolution near the intersection of the SB and the GB (at a distance r such as 0 < r << t). The point located at the intersection of SB and GB corresponds to r = 0. GB stresses singularity is the same as the crack one in the LEFM framework, leading to an exponent of 0.5 of the stress expansion. [22 ; 23], the GB normal stress field induced by one edge dislocation pile-up is given by: ( ) ∞ − − Σ − +Σ = n pile up pile up n h f r L r ) ( ) / ( 2 3 ( , ) 0 0 1/2 θ τ θ σ , (2) where ( ) sin cos( / 2) θ θ θ = h , Σ = ∞ n cos ( ) 2 0 GBα Σ and /2 L L pile up = − . The absence of any term accounting for the SB thickness, t, in Eq. 2 is noticeable. Indeed, slip is assumed to occur on one atomic plane only as mentioned earlier. However, experimental observations have shown that slip may occur in many atomic plane and lead to the question of taking into account SB thickness. This implies the existence of two characteristic lengths: SB length and thickness, in the new problem of finite thickness. It is also proved [17] that the driving force ) ( 0 0τ Σ − f is proportional to the macroscopic shear stress. These two points make our problem be similar to the case of a crack with a V-noch tip in an elastic matrix even the stress singularity is induced by a slip localization in ours. That is why, following the theory of matching expansions [24 ; 29], we perform a modeling of the GB normal and shear stress close fields with respect to the SB length, L and the SB thickness, t: ( ) ( ) ) / ( / ( ) 0 0 0.5 τ σ α Σ − = f r A L t t r nn nn , (3) and ( ) ( ) ) ( / / ( ) 0 0 0.5 τ σ α Σ − = f r A L t t r nm nm . (4) α is the singularity exponent nn A and nm A are model parameters. The subscript “nn” corresponds to the GB normal stress and “nm” to the GB shear stress. It is worth to highlight that this model assumes a linear dependence of GB stresses on the driving shear stress, T 0 0 τ = Σ − f , and the same singularity exponent is valid for both shear and normal stress components and whatever L and t. The main difference between this model and the pile-up one is that the finite SB thickness, t, is taken into account. The stress singularity is assumed to be weaker in the proposed model than in the pile-up case, 0.5 <α as it will be probably shown.
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