Influence of grain boundaries on plasticity

7.4.11 Influence of grain boundaries on plasticity

It might be thought that when a stress is applied Figure 7.31 (a) Grain-boundary blocking of slip. to a polycrystalline metal, every grain in the sample

(b) Blocking of a slip band by a grain boundary .

Mechanical behaviour of materials 219 on a strength is emphasized by Figure 7.30b, which

of a blocked slip band. The slip plane of the sources

will not, in general, lie in the plane of maximum shear diameter, 2d, for low-carbon steel. The smaller the

y , with grain

max will need to be such that the grain size, the higher the yield strength according to a

c required to operate the new source relation of the form

must be generated in the slip plane of the source. In general, the local orientation factor dealing with the

orientation relationship of adjacent grains will differ

i is a lattice friction stress and k a constant from the macroscopic factor of slip plane orientation usually denoted k 0 y to indicate yielding. Because of

max D 1 0 2 m c . For the difficulties experienced by a dislocation in moving

simplicity, however, it will be assumed m Dm and from one grain to another, the process of slip in

hence the parameter k in the Petch equation is given

a polycrystalline aggregate does not spread to each

by k D m 2 c r 1/2 .

grain by forcing a dislocation through the boundary. It is clear from the above treatment that the parame- Instead, the slip band which is held up at the boundary

ter k depends essentially on two main factors. The first gives rise to a stress concentration at the head of

is the stress to operate a source dislocation, and this the pile-up group of dislocations which acts with the

depends on the extent to which the dislocations are applied stress and is sufficient to trigger off sources

anchored or locked by impurity atoms. Strong locking

c and hence a large k; the converse is could sustain if there were no resistance to slip across

i is the stress a slip band

true for weak locking. The second factor is contained in the parameter m which depends on the number of

higher stress sustained by a slip band in a polycrystal, available slip systems. A multiplicity of slip systems

i ⊳ represents the resistance offered by the enhances the possibility for plastic deformation and so boundary, which reaches a limiting value when slip

implies a small k. A limited number of slip systems is induced in the next grain. The influence of grain

available would imply a large value of k. It then fol- size can be explained if the length of the slip band

lows, as shown in Figure 7.32, that for (1) fcc metals, is proportional to d as shown in Figure 7.31(b). Thus,

which have weakly locked dislocations and a multi- since the stress concentration a short distance r from

plicity of slip systems, k will generally be small, i.e.

the end of the slip band is proportional to ⊲d/4r⊳ 1/2

there is only a small grain size dependence of the flow the maximum shear stress at a distance r ahead of a

stress, (2) cph metals, k will be large because of the limited slip systems, and (3) bcc metals, because of

i ⊳ [d/4r] 1/2 and lies in the plane of the strong locking, k will be large. the slip band. If this maximum stress has to reach a

Each grain does not deform as a single crystal in max to operate a new source at a distance r then

simple slip, since, if this were so, different grains

i ⊳ [d/4r] 1/2

would then deform in different directions with the result that voids would be created at the grain bound-

max

or, rearranging, aries. Except in high-temperature creep, where the grains slide past each other along their boundaries, this

max 2r 1/2 ⊳d 1/2 does not happen and each grain deforms in coherence which may be written as

with its neighbouring grains. However, the fact that Ck s d i 1/2

It then follows that the tensile flow curve of a poly- crystal is given by

i Ck s d 1/2 ⊳

where m is the orientation factor relating the applied

a single crystal the m-factor has a minimum value of 2 as discussed, but in polycrystals deformation occurs in less favourably oriented grains and some- times (e.g. hexagonal, intermetallics, etc.) on ‘hard’ systems, and so the m-factor is significantly higher.

i and

k D mk s . While there is an orientation factor on a macroscopic scale in developing the critical shear stress within the

Figure 7.32 Schematic diagram showing the grain various grains of a polycrystal, so there is a local ori-

size-dependence of the yield stress for crystals of different entation factor in operating a dislocation source ahead

crystal structure .

220 Modern Physical Metallurgy and Materials Engineering the continuity of the metal is maintained during plas-

tic deformation must mean that each grain is deformed into a shape that is dictated by the deformation of its neighbours. Such behaviour will, of course, require the operation of several slip systems, and von Mises has shown that to allow this unrestricted change of shape of

a grain requires at least five independent shear modes. The deformation of metal crystals with cubic structure easily satisfies this condition so that the polycrystals of these metals usually exhibit considerable ductility, and the stress–strain curve generally lies close to that of single crystals of extreme orientations deforming under multiple slip conditions. The hexagonal metals do, however, show striking differences between their single crystal and polycrystalline behaviour. This is because single crystals of these metals deform by a process of basal plane slip, but the three shear sys- tems (two independent) which operate do not pro- vide enough independent shear mechanisms to allow unrestricted changes of shape in polycrystals. Conse- quently, to prevent gaps opening up at grain boundaries during the deformation of polycrystals, some addi- tional shear mechanisms, such as non-basal slip and mechanical twinning, must operate. Hence, because the resolved stress for non-basal slip and twinning is greater than that for basal-plane slip, yielding in a polycrystal is prevented until the applied stress is high enough to deform by these mechanisms.