13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- sliding is effectively accommodated, nanocrystalline solids show enhanced ductility and/or superplasticity [4, 15, 29]. And it is effectively realized that accommodation of grain boundary sliding occurs through lattice dislocation emission from triple junctions, diffusion, and rotational deformation [4, 15, 29]. There has been a new way to accommodate grain boundary sliding through grain boundary migration suggested and theoretically analyzed [20, 26]. Grain boundary migration is another toughening micromechanism [30] and specific deformation mode in nanocrystalline materials [3, 22]. The new accommodation mode we discussed above involving the cooperative grain boundary sliding and stress-driven grain boundary migration, servers as a special deformation mode enhanced compared to pure grain boundary sliding in nanocrystalline materials [20]. In the cooperative process, defects created by grain boundary sliding are, in part, accommodated by defects created by grain boundary migration. The cooperative grain boundary sliding and migration has been theoretically described as a deformed mode operation in crack-free nanocrystalline materials in Bobylev et al. [20]. And it is theoretically revealed that the mode enhances the ductility of nanocrystalline solids in wide ranges of their structural parameters. In the work of Ovid’ko et al. [26], the cooperative grain boundary sliding and migration process near crack tips has been described, and its effect on the fracture toughness of nanocrystalline materials has been theoretically analyzed. The results show that the mechanism considerably enhances the fracture toughness of nanocrystalline materials. The previous works have lots of experimental and theoretical results suggest that the cooperative grain boundary sliding and migration serving as a special deformation contributes to the toughening of nanocrystalline materials [20, 26]. But the effect of the cooperative grain boundary sliding and migration on the emission of lattice dislocations from crack tips has not been well quantitatively studied. Considering nanocrystalline solids with cracks, if the stress intensity near the crack tip is large enough, the crack can induce plastic shear through the emission of lattice dislocations from the crack tip. The emission of dislocations from cracks causes effective blunting of cracks, thus suppresses their growth, improves the toughness of nanocrystalline materials. So the crack blunting and growth processes are controlled by dislocation emission from crack tips. In the context discussed, there is large interest in the effect of the cooperative mode on lattice dislocation emission from crack tips. The main aim of the paper is to study the effect of the cooperative grain boundary sliding and migration on lattice dislocation emission from the crack tip using a theoretical mode. 2. Model and problem formulation Let us consider a deformed nanocrystalline solid with a crack under remote mode I loadings and remote mode II loadings. The solid is supposed to be an elastically isotropic solid characterized by the shear modulus μ and Poisson’s ratio ν. For definiteness, referring to the approximation in Ovid’ko et al. [26], we will focus our analysis on the situation where the crack is flat and plane, and characterized by the length L. For simplicity, we assume that the defect structure of the solid is the same along the coordinate axis z perpendicular to the xy plane. This assumption will allow us to simplify the mathematical analysis of the problem in our study, reducing it to the consideration of a two-dimensional structure. At the same time, the two-dimensional description definitely reflects the key aspects of the problem. The applied loadings and high stress concentration near the crack tip can induce both grain boundary sliding and migration near this crack tip. And the geometry of the cooperative grain boundary sliding and migration deformation is schematically presented in Fig. 1. Fig. 1a depicts a deformed nanocrystalline solid with a flat crack. The formation of two disclination dipoles CD and BE results from the cooperative grain boundary sliding and migration process and the dislocation emission is shown in Fig. 1b [26]. Within the model, the vertical grain boundary AB is assumed to be normal to the crack growth direction and make an angle ϕ with the grain boundaries 1 AA and 2 BB . Let the triple junction
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