13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Figure 3. Two parallel reacting dislocations in BCC crystal structure. The axis system for each dislocation as well as angular variables are defined in (a). The signs of the forces on dislocations 1 and 2 that cause attraction to a common junction are shown in (a–d) for all possible angular orientations. cleavage to be prevalent at low temperatures, which is observed in -Fe, and some plasticity to precede cleavage, which is also observed. The pile-up model is currently being used in to study near theoretical limiting stress levels reached for nanopolycrystals on a Hall-Petch basis. Antolovich and Armstrong considered possible modification of the Hall-Petch effect for cleavage of -Fe if only a small number of dislocations were involved [6]. They computed the dimensional scale at which a stress concentration from the pile-up could duplicate that of a cleavage crack. As shown above, the pile-up approach can be applied at rather small dimensions and can also be used to explain “scatter” of strength data. Importantly they showed that for small numbers of dislocations the slip band generated stress computed directly for small numbers of dislocations can never reach the crack-like stress (computed using continuous distributions of dislocations) implying that at small grain sizes (and small numbers of dislocations) cleavage is improbable. The results of their calculations are shown in Fig. 4. Figure 4. Dimensionless shear stress vs. dimensionless distance from crack or slip band tip (normalized to the crack length) [6]. For five free dislocations, the stress ahead of the slip band falls below that of a crack and cleavage is unlikely. Note further the crack like behaviour near the tip of the slip band (slope of -½) and the traditional slip band like behaviour (slope -1) for multiple Burgers vector representation. 0.1 1 10 100 0.1 1 10 100 Demensionless Crack Length (r/a*) Dimensionless Stress (23/inf ) . Pile-ups Cracks Upper lines = 49 free dislocations Lower lines = 5 free dislocations 1 1 1 2 49 5 Dimensionless Distance from Tip (r/a*)
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