13th International Conference on Fracture June 16–21, 2013, Beijing, China -10- Fig. 10 Dependences of the critical normalized SIFs 0 IC K on the distance between crack tip and coordinate origin p′ ( 0 π 6 θ = , 2π 3 ϕ= , 300nm L= , 15nm d = , 0 p = , 0.3 x d = , 0.1 y d = ) 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 K0 IC x/d p′= 0nm p′= 20nm p′=500nm Fig. 11 Dependences of the critical normalized SIFs 0 IC K on the ratio x d with different distances p′ ( π 6 ω= , 0 π 6 θ = , 2π 3 ϕ= , 100nm L= , 15nm d = , 0 p = , 0.1 y d = ) 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 K0 IC y/d p′= 0nm p′= 20nm p′=500nm Fig. 12 Dependences of the critical normalized SIFs 0 IC K on the ratio y d with different distances p′ ( π 6 ω= , 0 π 6 θ = , 2π 3 ϕ= , 100nm L= , 15nm d = , 0 p = , 0.3 x d = ) If the crack tip is not at the coordinate origin, we define the horizontal distance between the crack tip and the coordinate origin is p′ . Fig. 10 plots the dependences of the critical normalized SIFs on the distance between the crack tip and the coordinate origin p′ with different disclination strengths. The critical normalized SIFs decrease and tend to constants with increasing of the distance p′ . The variations of the critical normalized SIFs on the ratios x d and y d with different distances p′ are depicted in Fig. 11 and Fig. 12, respectively. Fig. 11 indicates that the critical normalized SIFs first increase then decrease with the increment of the ratio x d when the crack tip is at the coordinate origin ( 0 p′ = ), and the change is remarkable. In other cases, the critical normalized SIFs just increase with increasing of the ratio, and the distances p′ have slight influence on them. The variation tendency of the critical normalized with respect to the radio y d is just opposite to that to the radio x d . 5. Concluding remarks Thus, the interaction of the cooperative grain boundary sliding and migration with the crack is investigated by the complex variable method. We have theoretically described the effects of the two wedge disclination dipoles produced by the cooperative grain boundary sliding and migration on the emission of edge dislocations from the crack tip in deformed nanocrystalline metals and ceramics. The critical stress intensity factors for the first dislocation emission are calculated. The influence of the disclination strength, the grain size, the location of the two disclination dipoles and the crack length on the critical stress intensity factors is discussed in detail. Some conclusions are summarized as follows. (1) The cooperative grain boundary sliding and migration can promote the dislocation emission from the crack tip, which causes effective blunting of the crack, thus suppresses its growth, and improves the toughness of nanocrystalline materials. (2) There is a critical disclination strength value making the critical stress intensity factor equal to zero and the dislocation can emit from the crack tip without any external loadings. And the mode II loadings are easier than the mode I loadings to make the dislocation emit form the crack tip.
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