The effects of grain size on yield strength
A change in grain size affects the yield strength due to the dislocations interacting with the grain boundary as they move. The boundaries act as obstacles, hindering the dislocation glide along the slip planes. As subsequent dislocations move along the same slip plane the dislocations pile-up at the grain boundaries.
The dislocations repel each other, so as the number of dislocations in the pile-up increases the stress on the grain boundary increases. In fact, if there are n dislocations in the pile-up, the stress at the grain boundary will be n times the applied stress.
If the grain boundary in a sample gives way at a stress τ, there needs to be a stress of τ/n applied to the sample in order to cause the boundary to collapse.
In a larger grain there will be more dislocations within the grain, so there will be more dislocations in the pile-up. Therefore a lower applied stress is required to produce a local stress great enough to cause the grain boundary to collapse. Accurate modelling is difficult, but it is found that the tensile yield stress, σy, is related to grain diameter, d, by the Hall-Petch equation:
where σi is the 'intrinsic' yield stress, and k is a constant for a particular material.
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