13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- of grain size will contain, say, three dislocations. For two dislocations, say, to be in the pile-up the grain size must decrease to a pre-determined size. Joining the tips of the “steps” gives the envelope of possible values for the flow stress. Small grain sizes and low numbers of dislocations in them also relate to the improbability of cleavage in fine grained Fe as described by Armstrong and Antolovich [6] as discussed below. 3. Strain Concentration and Fracture An updated model for plastically-induced cleavage fracturing of bcc and related metals and alloys is shown in Fig. 2. The basic model fracture had been developed originally by Cottrell [7]. Figure 2. Cottrell model for fracture in BCC ( ) Fe [7] In this model, parallel dislocations on intersecting slip systems (i.e. {110}-type planes) react to form a sessile dislocation on a non-slip {100} plane thus providing a cleavage crack nucleus. Repetition of this reaction promotes growth of the microcrack until it is large enough to satisfy a Griffith-type criterion and spread by cleavage on a {100} type plane. The reaction shown in Fig. 2 is vectorily correct and energetically favourable. Furthermore, careful observations showed that fracture occurred only after some plastic deformation as would be required by this model. The model has been applied to explaining crystallographic aspects of cleavage fracture in other crystalline structures, for example, at indentations on MgO (001) crystal surfaces [8]. However, not all reactions such as that shown in Fig. 2 actually result in good predictions of cleavage. Antolovich and Kip [9] pointed out that not only must the reaction be energetically favourable but one must also consider the force barrier to be overcome. An analogy would be two parallel dislocations on parallel planes. Their lowest energy position is when they are lined up vertically forming a finite segment of a low angle tilt boundary. However, to bring the dislocations into the position shown, a force barrier must be overcome which is a maximum when a line joining the two dislocations is inclined at 22.5o to the slip planes. Once aligned there is a restoring force directed against separation. Antolovich and Kip considered: (1) the effect of angular orientation of the two dislocations on the forces that must be overcome to form a crack nucleus and (2) the forces required to bring an additional dislocation to within a Burgers vector of the crack nucleus. The problem that they considered is illustrated in Fig. 3. The results were illuminating. First of all, with respect to BCC structures and -Fe in particular, the calculations showed that for almost all angular orientations as defined above, glide dislocations are attracted to a common junction implying that overcoming the Peierls force would be sufficient to form a crack nucleus. This was in opposition to most other crystal structures (e.g. intermetallics, FCC crystals) in which such attractive ranges were rare. However, once formed, it becomes increasingly difficult to grow the crack since the back stress due to the nucleus requires additional force to move a dislocation into the vicinity of the nucleus (i.e. to grow the incipient crack). Based on these detailed calculations, one would expect
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