All real crystals contain some dislocations except in the case of tiny, carefully pre­pared whiskers. Nevertheless, dislocations are not produced like intrinsic point defects such as vacancies or self-interstitials. The net free energy change due to the presence of dislocations is positive. Dislocations in freshly grown crystals are formed for various reasons: (a) Dislocations that may appear as preexisting in the seed crystal to grow the crystal. (b) “Accidental nucleation” — (i) heterogeneous nucleation of dislocations due to internal stresses generated by impurity, thermal contraction, and so on, (ii) impingement of different parts of the growing interface, and (iii) formation and subsequent movement of dislocation loops formed by the collapse of vacancy platelets. When a dislocation is created in a region of the crystal that is free from any defects, the nucleation is called “homogeneous.” This occurs only under extreme conditions and requires rupturing of atomic bonds, which need very high stresses. Nucleation of dislocations at stress concentrators is important — generation of prismatic dislocations at precipitates/inclusions (Figure 4.22a), misfit dislocations during coherency loss, dislocation generation at other surface irregularities and cracks. As a matter of fact, grain boundary irregu­larities such as grain boundary ledges/steps (Figure 4.22b) are considered to be important sources of dislocations, especially in the early stages of deformation.

Regenerative multiplication of dislocations is key to sustaining large plastic strains. Frank-Read (F-R)-type sources and multiple cross-glide can participate as a multiplication mechanism of dislocations. It requires a preexisting dislocation (DD0) pinned down at two ends (dislocation intersections or nodes, composite jogs, precipitates, etc.) with distance between pinning points being L, as illustrated in Figure 4.23. An applied resolved shear stress (t) makes the dislocation bow out and the radius of curvature R depends on the stress according to Eq. (4.17). When R

reaches a minimum value of L/2 and taking a = 0.5, the stress required to produce the following configuration is given by

Gb

t = = rFR — (4-19)

At this point in semicircular configuration, the segment becomes unstable and further increase in the dislocation loop does not require any more additional stress. As the loop expands, the dislocation at places such as m and n in Figure 4.23a(iv) attracts and annihilates leaving the loop expanding further while the original dislo­cation segment L is recovered. Under the applied stress (tFR), the dislocation seg­ment expands again resulting in another loop, while the first loop keeps on expanding due to the applied stress. The process repeats itself until the lead disloca­tion loop gets stuck at some obstacle such as a grain boundary or another similar dislocation loop on a parallel glide plane. Since the dislocation loops generated are all alike, the second loop will be repelled by the first one, while the third loop will experience repulsive force arising from both the first and second loops, and so on, and finally the force due to all the dislocation loops in this “pileup” results in the original dislocation segment unable to produce any further dislocation loops until the lead dislocation climbs out of the glide plane. This is the Frank-Read source that is responsible for dislocation multiplication, and the stress (tFR) required for F-R source to operate is given by Eq. (4.19).

Total dislocation length can also be increased by climb: (a) the expansion of pris­matic loops and (b) spiraling of a dislocation with a predominantly screw character. A regenerative multiplication known as the “Bardeen-Herring” source (Figure 4.24) can occur by climb in a way similar to the Frank-Read mechanism.