Dislocations generated from a Frank-Read source pile up at barriers such as grain boundaries, second phases, and sessile dislocations or a lead dislocation from

another such source on a parallel glide plane. The number of dislocations in the pileup would be directly proportional to the obstacle distance from the source (i. e., the pileup length Lpileup) and the applied stress. From the force balance between the dislocation-dislocation interactions and the external stress, one can show that

where n is the number of dislocations in the pileup with pileup length Lpileup under an applied shear stress tjy.

This leads to a high stress concentration ahead of the pileup that is usually relieved by plastic deformation. Just ahead of the pileup (i. e., r is small and в « 0 in Figure 4.25, which shows a dislocation pileup as a cross section of the loops com­ing out of the paper) the stress t is enhanced by the number of dislocations in the pileup:

At large r but smaller than L (such as at point P in Figure 4.25), there exists a stress concentration due to the pileup:

tat P&J pileu-rxy (true for q < 70°). (4.22)

One can show from the above Eq. (4.22) the relation between the yield stress and grain size (namely, the Hall-Petch relation; refer to, for example, Ref. [1]):

sy = so + -7=. (4.23)

4.3