Abstract:
The effects of reduction in the strength and deviation from the Hall–Petch relationship under plastic deformation of specimens with micro- and nano-grained structures with decreasing size of their cross section have been considered theoretically. The analysis is based on the kinetic equation for the dislocation density, which takes into account that the surface of the specimen serves as both the source and the sink for dislocations, whereas the grain boundaries are barriers limiting the mean free path of dislocations. It has been found that, when the ratio of the transverse size of the specimen $D$ to the grain size $d$ becomes less than 3, in the dependence of the yield stress on the size of the specimen there appears a minimum as a result of the increase in the number of near-surface grains that exhibit a weak resistance to plastic deformation due to the withdrawal of dislocations through the external surface of the fine-dimensional specimen. The minimum of the strength in the range $d < D < 3d$ is a consequence of the competition and nonlinear interaction of the size factors $D$ and $d$.
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