A model of concentric circular dislocation loops is used to examine the Hall-Petch relation. For a given critical stress concentration at the grain boundary, the Hall-Petch slope produced by this model is higher than that from either single or double-ended pile-ups of infinite straight dislocations. The minimum stress required to maintain a certain number of circular dislocations is compared with the stress necessary for the continuous distribution of these dislocations in a circular slip zone.
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Li et al. (1967) studied this question.
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