Plastic deformation in single crystals is guided by relatively simple laws. However, in polycrystalline materials, it becomes complicated. Strength is inversely related to the dislocation mobility. That is, the dislocation movement can be obstructed by various factors that can, in turn, strengthen (or harden) the material. Grain bound­aries can act as such an obstacle to the dislocation motion (refer to Section 2.2 for more information on the grain boundaries). The grain boundary ledges are a com­mon feature — these steps can be produced by external dislocation joining the grain boundary or packs of dislocations in the grain boundary. Thus, the density of ledges increases with increasing misorientation angle. Grain boundary ledges act as an effective source of dislocations. Hence, before we introduce the topic of grain size strengthening, we need to know some fundamentals of the grain size effect on the plastic deformation in polycrystals. Grain size strengthening is not possible in a single crystal (i. e., single grain) for the obvious reasons.

In a polycrystal unlike a single crystal, during plastic deformation the continuity must be maintained between the deforming crystals. If not maintained, it will read­ily form flaws leading to fast failure. Although it is true that each of the grains attempts to deform as homogeneously as possible in concert with the overall defor­mation of the material, the geometrical constraints imposed by the continuity requirement cause significant differences between neighboring grains and even in

parts of each grain. As the grain size decreases, the deformation becomes more homogeneous. That is why decreasing grain size in conventional materials gener­ally improves ductility. Due to the constraints imposed by the grain boundaries, slip occurs on several systems even at low strains. As the grain size is reduced, more of the effects of grain boundaries will be felt at the grain center. Thus, the strain hard­ening of a fine grain size material will be greater than that of a coarse-grained poly­crystalline material. Theoretically, it has been shown that the strain hardening rate (da/de) for an FCC polycrystalline material is ~9.6 times that of (dt/dy) for a sin­gle crystal as noted in Eq. (4.34):

da —2 dt

de — MdC ’

where M is called Taylor’s factor (for FCC lattice, it is taken as 3.1). This has also been verified by experiments.

Von Mises [27] showed that for a crystal to undergo a general change of shape by slip requires the operation of five independent slip systems. Crystals having less than five slip systems are never ductile in polycrystalline form even though small elonga­tions can be obtained through twinning or a favorable preferred orientation. Gener­ally, cubic metals easily satisfy this requirement (higher ductility), whereas HCP and other low symmetry metals do not (lower ductility). At elevated temperatures, other slip systems can get activated and thus increase the number of slip systems to at least five.

At temperatures above ~0.5Tm (Tm is the melting point), deformation can occur by sliding along grain boundaries. Grain boundary sliding becomes more promi­nent with increased temperature and decreasing strain rate. A general way to find­ing out when grain boundary sliding may start operating is the use of equicohesive temperature concept. Above this temperature, the grain boundary is weaker than grain interior and strength increases with increasing grain size. Below this tem­perature, the grain boundary region is stronger than the grain interior and strength increases with decreasing grain size. The grain size strengthening mech­anism to be discussed below is thus applicable only to lower homologous temper­ature regimes.

A mathematical description of the grain size strengthening can be expressed in terms of a general relationship between yield stress and grain size, first proposed by Hall (1951) and later extended by Petch (1953):

ay — ai ^ (4.35)

where ay is the yield stress, ai is the friction stress (also can be defined as yield strength at an infinite grain size), ky is the unlocking parameter (that measures relative hardening contribution ofthe grain boundaries), and d is the grain diame­ter. The Hall-Petch relation has also been found to be applicable for a wide variety of situations, such as the variation of brittle fracture stress on grain size, grain size- dependent fatigue strength, and even to the boundaries that are not such grain boundaries as found in pearlite, mechanical twins, and martensite plates.

Figure 4.33 A schematic of a dislocation pileup is shown to form by the dislocations originating from a Frank-Read source in the center of Grain-1. The leading edge of the dislocation pileup produces enough stress concentration to create dislocations in Grain-2.

The Hall-Petch relation was developed based on the concept that grain bounda­ries act as barriers to the dislocation movement (as noted earlier during discussion of dislocation pileups). As the grain orientation changes at the grain boundary, the slip planes in a grain get disrupted at the grain boundary. This implies that slip planes are not continuous in a polycrystalline material from one grain to another. Thus, the dislocations gliding on a slip plane cannot burst through the grain boundary. Instead, they get piled up against it. In a larger grain, the number of dislocations in a pileup is higher. The magnitude of the stress concentration at the leading edge of the pileup (as shown in Figure 4.33) varies as the square root of the number of dislocations in the pileup. Hence, a greater stress concentration would be created at a larger grain. However, this event can trigger dislocation sources in the next grain quite readily. The situation is schematically shown in Figure 4.33. Now if the grain size is smaller, the stress concentration in Grain-1 will not be enough to produce slip in the next grain readily, and thus fine grain sized material will exhibit higher yield strength.

4.4.3