Three-stage hardening The stress–strain curve of a fcc single crystal is shown

7.6.2.2 Three-stage hardening The stress–strain curve of a fcc single crystal is shown

in Figure 7.40 and three regions of hardening are experimentally distinguishable. Stage I, or the easy

Figure 7.39 Stress–strain curves of single crystals (after glide region, immediately follows the yield point and Hirsch and Mitchell, 1967; courtesy of the National

Research Council of Canada) . 1 up to several per cent glide; the length of this region depends on orientation, purity and size of the crystals.

1 10 and is of the same order as for hexagonal metals. Stage II, or the linear hardening region, shows a rapid increase in work-

the same order of magnitude for all fcc metals, i.e. 1/300 although this is ³1/150 for orientations at the corners of the stereographic triangle. In this stage short slip lines are formed during straining quite suddenly,

not grow either in length or intensity. The mean length of the slip lines, L ³ 25 µ m decreases with increasing strain. Stage III, or the parabolic hardening region, the onset of which is markedly dependent on temperature,

111 , and the Figure 7.40 Stress–strain curve showing the three stages of

appearance of coarse slip bands. This stage sets in at work hardening .

a strain which increases with decreasing temperature and is probably associated with the annihilation of dislocations as a consequence of cross-slip.

work-hardening. The plastic part of the stress–strain The low stacking fault energy metals exhibit all curve is also more nearly linear than parabolic with

three work-hardening stages at room temperature, but metals with a high stacking fault energy often show

becomes even smaller with increasing temperature of only two stages of hardening. It is found, for example, deformation. Cubic crystals, on the other hand, are

that at 78 K aluminium behaves like copper at room capable of deforming in a complex manner on more

temperature and exhibits all three stages, but at room than one slip system, and these metals normally show

temperature and above, stage II is not clearly devel-

a strong work-hardening behaviour. The influence of oped and stage III starts before stage II becomes at temperature depends on the stress level reached dur-

all predominant. This difference between aluminium ing deformation and on other factors which must be

and the noble metals is not due only to the difference considered in greater detail. However, even in cubic

in melting point but also to the difference in stacking crystals the rate of work-hardening may be extremely

fault energies which affects the width of extended dis- small if the crystal is restricted to slip on a single

locations. The main effect of a change of temperature slip system. Such behaviour points to the conclusion

of deformation is, however, a change in the onset of that strong work-hardening is caused by the mutual

stage III; the lower the temperature of deformation, the interference of dislocations gliding on intersecting slip

111 corresponding to the onset of planes.

stage III.

Many theories of work-hardening similar to that of Because the flow stress of a metal may be affected Taylor exist but all are oversimplified, since work-

by a change of temperature or strain-rate, it has been hardening depends not so much on individual dislo-

found convenient to think of the stress as made up of cations as on the group behaviour of large numbers of

two parts according to the relation them. It is clear, therefore, that a theoretical treatment

228 Modern Physical Metallurgy and Materials Engineering s is that part of the flow stress which is

dependent on temperature apart from the variation of

g is

a temperature-independent contribution. The relative

g can be studied conveniently by measuring the dependence of flow stress on tem-

on changing the temperature or strain rate, as function of deformation.

The Stage I easy glide region in cubic crystals, with its small linear hardening, corresponds closely to the hardening of cph crystals where only one glide plane operates. It occurs in crystals oriented to allow only one glide system to operate, i.e. for orientations near the [0 1 1] pole of the unit triangle (Figure 7.13). In this case the slip distance is large, of the order of the specimen diameter, with the probability of dislocations slipping out of the crystal. Electron microscope obser- vations have shown that the slip lines on the surface are very long (³1 mm) and closely spaced, and that the slip steps are small corresponding to the passage of only a few dislocations. This behaviour obviously depends on such variables as sample size and oxide films, since these influence the probability of disloca- tions passing out of the crystal. It is also to be expected that the flow stress in easy glide will be governed by the ease with which sources begin to operate, since there is no slip on a secondary slip system to interfere with the movement of primary glide dislocations.

As soon as another glide system becomes activated there is a strong interaction between dislocations on the primary and secondary slip systems, which gives rise to a steep increase in work-hardening. It is reasonable to expect that easy glide should end, and turbulent flow begin, when the crystal reaches an orientation for which two or more slip systems are equally stressed,

i.e. for orientations on the symmetry line between [0 0 1] and [1 1 1]. However, easy glide generally ends before symmetrical orientations are reached and this is principally due to the formation of deformation bands to accommodate the rotation of the glide plane in fixed grips during tensile tests. This rotation leads to a high resolved stress on the secondary slip system, and its operation gives rise to those lattice irregularities which cause some dislocations to become ‘stopped’ in the crystal. The transformation to Stage II then occurs.

The characteristic feature of deformation in Stage II is that slip takes place on both the primary and sec- ondary slip systems. As a result, several new lat- tice irregularities may be formed which will include (1) forest dislocations, (2) Lomer–Cottrell barriers, and (3) jogs produced either by moving dislocations cutting through forest dislocations or by forest dis- locations cutting through source dislocations. Conse-

terms, with a stress which is sufficient to operate a source and then move the dislocations against (1) the internal elastic stresses from the forest dislocations, (2) the long-range stresses from groups of dislocations

piled up behind barriers, and (3) the frictional resis- tance due to jogs. In a cold-worked metal all these factors may exist to some extent, but because a linear hardening law can be derived by using any one of the various contributory factors, there have been several theories of Stage II-hardening, namely (1) the pileup theory, (2) the forest theory and (3) the jog theory. All have been shown to have limitations in explaining various features of the deformation process, and have given way to a more phenomenological theory based on direct observations of the dislocation distribution during straining.

Observations on fcc and bcc crystals have revealed several common features of the microstructure which include the formation of dipoles, tangles and cell struc- tures with increasing strain. The most detailed obser- vations have been made for copper crystals, and these are summarized below to illustrate the general pattern of behaviour. In Stage I, bands of dipoles are formed (see Figure 7.41a) elongated normal to the primary Burgers vector direction. Their formation is associated with isolated forest dislocations and individual dipoles are about 1 µ m in length and ⱚ 10 nm wide. Differ- ent patches are arranged at spacings of about 10 µ m along the line of intersection of a secondary slip plane. With increasing strain in Stage I the size of the gaps between the dipole clusters decreases and therefore the stress required to push dislocations through these gaps increases. Stage II begins (see Figure 7.41b) when the applied stress plus internal stress resolved on the secondary systems is sufficient to activate secondary sources near the dipole clusters. The resulting local secondary slip leads to local interactions between pri- mary and secondary dislocations both in the gaps and in the clusters of dipoles, the gaps being filled with secondary dislocations and short lengths of other dis- locations formed by interactions (e.g. Lomer–Cottrell dislocations in fcc crystals and ah1 0 0i type dislo- cations in bcc crystals). Dislocation barriers are thus formed surrounding the original sources.

In Stage II (see Figure 7.41c) it is proposed that dislocations are stopped by elastic interaction when they pass too close to an existing tangled region with high dislocation density. The long-range internal stresses due to the dislocations piling up behind are partially relieved by secondary slip, which transforms the discrete pile-up into a region of high dislocation density containing secondary dislocation networks and dipoles. These regions of high dislocation density act as new obstacles to dislocation glide, and since every new obstacle is formed near one produced at a lower strain, two-dimensional dislocation structures are built up forming the walls of an irregular cell structure. With increasing strain the number of obstacles increases, the distance a dislocation glides decreases and therefore the slip line length decreases in Stage II. The structure remains similar throughout Stage II but is reduced in scale. The obstacles are in the form of ribbons of high densities of dislocations which, like pile-ups, tend to form sheets. The work-hardening rate depends mainly

Mechanical behaviour of materials 229

Figure 7.41 Dislocation structure observed in copper single crystals deformed in tension to (a) stage I, (b) end of easy-glide and beginning of stage II, (c) top of stage II, and (d) stage III (after Steeds, 1963; Crown copyright; reproduced by permission of the Controller, H.M. Stationery Office) .

on the effective radius of the obstacles, and this has The dislocation arrangement in metals with other been considered in detail by Hirsch and co-workers and

structures is somewhat similar to that of copper with shown to be a constant fraction k of the discrete pile-

differences arising from stacking fault energy. In up length on the primary slip system. In general, the

Cu–Al alloys the dislocations tend to be confined more

11 to the active slip planes, the confinement increasing an fcc crystal the small variation in k with orientation

with decreasing SF . In Stage I dislocation multipoles and alloying element is able to account for the variation

are formed as a result of dislocations of opposite sign

11 with those parameters. on parallel nearby slip planes ‘pairing up’ with one

230 Modern Physical Metallurgy and Materials Engineering another. Most of these dislocations are primaries. In

agreement with the observations of dislocation density Stage II the density of secondary dislocations is much

and arrangement.

less (³ 1 3 ) than that of the primary dislocations. The secondary slip occurs in bands and in each band slip