Work hardening

6.6.2 Work hardening

6.6.2.1 Theoretical treatment

The properties of a material are altered by cold working, i.e. deformation at a low temperature relative to its melting point, but not all the properties are improved, for although the tensile strength, yield strength and hardness are increased, the plasticity and general ability to deform decreases. Moreover, the physical properties such as electrical conductivity, density and others are all lowered. Of these many changes in properties, perhaps the most outstanding are those that occur in the mechanical properties; the yield stress of mild steel, for example, may be raised by cold work from 170 up to 1050 MN m −2 .

Such changes in mechanical properties are, of course, of interest theoretically, but they are also of great importance in industrial practice. This is because the rate at which the material hardens during deformation influences both the power required and the method of working in the various shaping operations, while the magnitude of the hardness introduced governs the frequency with which the component must be annealed (always an expensive operation) to enable further working to

be continued. Since plastic flow occurs by a dislocation mechanism, the fact that work hardening occurs means that it becomes difficult for dislocations to move as the strain increases. All theories of work hardening depend on this assumption, and the basic idea of hardening, put forward by Taylor in 1934, is that some dislocations become ‘stuck’ inside the crystal and act as sources of internal stress, which oppose the motion of other gliding dislocations.

One simple way in which two dislocations could become stuck is by elastic interaction. Thus, two parallel-edge dislocations of opposite sign moving on parallel slip planes in any sub-grain may become stuck, as a result of the interaction discussed in Chapter 3. G. I. Taylor assumed that dislocations become stuck after traveling an average distance, L, while the density of dislocations reaches ρ, i.e.

Mechanical properties I 333

Cu, Ag 4

10 Cu–Al

⫻ /m 2 t

0 4 8 b √r ⫻ 10 3

Figure 6.37 Dependence of flow stress on (dislocation density) 1/2 for Cu, Ag and Cu–Al.

work hardening is due to the dislocations getting in each other’s way. The flow stress is then the stress necessary to move a dislocation in the stress field of those dislocations surrounding it. This stress τ is quite generally given by

τ = αμb/l, (6.23) where μ is the shear modulus, b the Burgers vector, l the mean distance between dislocations, which is

≈ρ −1/2 , and α a constant; in the Taylor model α = 1/8π(1 − v) where ν is Poisson’s ratio. Figure 6.37 shows such a relationship for Cu–Al single crystals and polycrystalline Ag and Cu.

In his theory Taylor considered only a two-dimensional model of a cold-worked metal. However, because plastic deformation arises from the movement of dislocation loops from a source, it is more appropriate to assume that, when the plastic strain is γ, N dislocation loops of side L (if we assume for convenience that square loops are emitted) have been given off per unit volume. The resultant plastic strain is then given by

γ = NL 2 b (6.24) and l by l

≈ [1/ρ 1/2 ] = [1/4LN ] 1/2 . (6.25) Combining these equations, the stress–strain relation τ

= const · (b/L) 1/2 γ 1/2 (6.26) is obtained. Taylor assumed L to be a constant, i.e. the slip lines are of constant length, which results

in a parabolic relationship between τ and γ. Taylor’s assumption that during cold work the density of dislocations increases has been amply verified, and indeed the parabolic relationship between stress and strain is obeyed, to a first approxima- tion, in many polycrystalline aggregates where deformation in all grains takes place by multiple slip. Experimental work on single crystals shows, however, that the work or strain hardening curve may deviate considerably from parabolic behavior, and depends not only on crystal structure but also on other variables such as crystal orientation, purity and surface conditions (see Figures 6.38 and 6.39).

The crystal structure is important (see Figure 6.38) in that single crystals of some hexagonal metals slip only on one family of slip planes, those parallel to the basal plane, and these metals show a low rate of work hardening. The plastic part of the stress–strain curve is also more nearly linear than parabolic,

334 Physical Metallurgy and Advanced Materials

fcc (Cu)

bcc (Nb)

Shear stress hcp

(Mg)

Shear strain

Figure 6.38 Stress–strain curves of single crystals (after Hirsch and Mitchell, 1967; courtesy of the National Research Council of Canada).

with a slope which is extremely small: this slope (dτ/dγ) becomes even smaller with increasing temperature of deformation. Cubic crystals, on the other hand, are capable of deforming in a complex manner on more than one slip system, and these metals normally show a strong work-hardening behavior. The influence of temperature depends on the stress level reached during deformation and on other factors which must be considered in greater detail. However, even in cubic crystals the rate of work hardening may be extremely small if the crystal is restricted to slip on a single slip system. Such behavior points to the conclusion that strong work hardening is caused by the mutual interference of dislocations gliding on intersecting slip planes.

Many theories of work hardening similar to that of Taylor exist but all are oversimplified, since work hardening depends not so much on individual dislocations as on the group behavior of large numbers of them. It is clear, therefore, that a theoretical treatment which would describe the complete stress–strain relationship is difficult, and consequently the present-day approach is to examine the various stages of hardening and then attempt to explain the mechanisms likely to give rise to the different stages. The work-hardening behavior in metals with a cubic structure is more complex than in most other structures because of the variety of slip systems available, and it is for this reason that much of the experimental evidence is related to these metals, particularly those with fcc structures.

6.6.2.2 Three-stage hardening

The stress–strain curve of a fcc single crystal is shown in Figure 6.39 and three regions of hardening are experimentally distinguishable. Stage I, or the easy glide region, immediately follows the yield

point and is characterized by a low rate of work hardening θ 1 up to several per cent glide; the length of this region depends on orientation, purity and size of the crystals. The hardening rate (θ 1 /μ ) ∼ 10 −4 and is of the same order as for hexagonal metals. Stage II, or the linear hardening region, shows

a rapid increase in work-hardening rate. The ratio (θ 11 /μ ) = (dτ/dγ)/μ is of the same order of magnitude for all fcc metals, i.e. 1/300, although this is ≈1/150 for orientations at the corners of the stereographic triangle. In this stage short slip lines are formed during straining quite suddenly,

mean length of the slip lines, L ≈ 25 µm, decreases with increasing strain. Stage III, or the parabolic hardening region, the onset of which is markedly dependent on temperature, exhibits a low rate of work hardening, θ 111 , and the appearance of coarse slip bands. This stage sets in at a strain which

Mechanical properties I 335

t III

II Stress

Strain g

Figure 6.39 Stress–strain curve showing the three stages of work hardening. increases with decreasing temperature and is probably associated with the annihilation of dislocations

as a consequence of cross-slip. The low stacking-fault energy metals exhibit all three work-hardening stages at room temperature, but metals with a high stacking-fault energy often show only two stages of hardening. It is found, for example, that at 78 K aluminum behaves like copper at room temperature and exhibits all three stages, but at room temperature and above, Stage II is not clearly developed and Stage III starts before Stage II becomes at all predominant. This difference between aluminum and the noble metals is due not only to the difference in melting point, but also to the difference in stacking-fault energies, which affects the width of extended dislocations. The main effect of a change of temperature of deformation is, however, a change in the onset of Stage III; the lower the temperature of deformation, the higher is the stress τ 111 corresponding to the onset of Stage III.

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 6.12). 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 observations have shown that the slip lines on the surface are very long ( ≈1 mm) and closely spaced, and that the slip steps are small corresponding to the passage of only a few dislocations. This behavior obviously depends on such variables as sample size and oxide films, since these influence the probability of dislocations 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 accom- modate 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.

336 Physical Metallurgy and Advanced Materials The characteristic feature of deformation in Stage II is that slip takes place on both the primary

and secondary slip systems. As a result, several new lattice 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 dislocations cutting through source dislocations. Consequently, the flow stress τ may be identified, in general 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 resistance 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 pile-up 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 structures with increasing strain. The most detailed observations have been made for copper crystals, and these are summarized below to illustrate the general pattern of behavior. In Stage I, bands of dipoles are formed (see Figure 6.40a), elongated normal to the primary Burgers vector direction. Their formation is associated with isolated forest dislocations and individual dipoles are about 1 µm in length and ˜>10 nm wide. Different patches are arranged at spacings of about 10 µm 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 6.40b) 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 primary 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 dislocations formed by interactions (e.g. Lomer–Cottrell dislocations in fcc crystals and a Dislocation barriers are thus formed surrounding the original sources.

In Stage II (see Figure 6.40c) 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 on the effective radius of the obstacles, and this has been considered in detail by Hirsch and co-workers and shown to be a constant fraction k of the discrete pile-up length on the primary slip system. In general, the

work-hardening rate is given by θ 11 = kμ/3π and, for an fcc crystal, the small variation in k with orientation and alloying element is able to account for the variation of θ 11 with those parameters. The dislocation arrangement in metals with other structures is somewhat similar to that of copper, with differences arising from stacking-fault energy. In Cu–Al alloys the dislocations tend to be confined more to the active slip planes, the confinement increasing with decreasing γ SF . In Stage I dislocation multipoles are formed as a result of dislocations of opposite sign on parallel nearby slip planes ‘pairing up’ with one another. Most of these dislocations are primaries. In Stage II the density of secondary dislocations is much less

3 than that of the primary dislocations. The secondary slip occurs in bands and in each band slip on one particular secondary plane predominates. In niobium,

Mechanical properties I 337

(d) Figure 6.40 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, HM Stationery Office).

a metal with high γ SF , the dislocation distribution is rather similar to copper. In Mg, typical of cph metals, Stage I is extensive and the dislocations are mainly in the form of primary edge multipoles, but forest dislocations threading the primary slip plane do not appear to be generated.

From the curve shown in Figure 6.39 it is evident that the rate of work hardening decreases in the later stages of the test. This observation indicates that at a sufficiently high stress or temperature the dislocations held up in Stage II are able to move by a process which at lower stresses and temperature had been suppressed. The onset of Stage III is accompanied by cross-slip, and the slip lines are broad, deep and consist of segments joined by cross-slip traces. Electron metallographic observations on

338 Physical Metallurgy and Advanced Materials sections of deformed crystal inclined to the slip plane (see Figure 6.40d) show the formation of a cell

structure in the form of boundaries, approximately parallel to the primary slip plane of spacing about 1–3 µm, plus other boundaries extending normal to the slip plane as a result of cross-slip.

The simplest process which is in agreement with the experimental observations is that the screw dislocations held up in Stage II cross-slip and possibly return to the primary slip plane by double cross- slip. By this mechanism, dislocations can bypass the obstacles in their glide plane and do not have to interact strongly with them. Such behavior leads to an increase in slip distance and a decrease in the accompanying rate of work hardening. Furthermore, it is to be expected that screw dislocations leaving the glide plane by cross-slip may also meet dislocations on parallel planes and be attracted by those of opposite sign. Annihilation then takes place and the annihilated dislocation will be replaced, partly at least, from the original source. This process, if repeated, can lead to slip-band formation, which is also an important experimental feature of Stage III. Hardening in Stage III is then due to the edge parts of the loops which remain in the crystal and increase in density as the source continues to operate.

The importance of the value of the stacking-fault energy, γ, on the stress–strain curve is evident from its importance to the process of cross-slip. Low values of γ give rise to wide stacking-fault ‘ribbons’, and consequently cross-slip is difficult at reasonable stress levels. Thus, the screws cannot escape from their slip plane, the slip distance is small, the dislocation density is high and the transition from Stage

II to Stage III is delayed. In aluminum the stacking-fault ribbon width is very small because γ has a high value, and cross-slip occurs at room temperature. Stage II is therefore poorly developed unless testing is carried out at low temperatures. These conclusions are in agreement with the observations of dislocation density and arrangement.

6.6.2.3 Work hardening in polycrystals

The dislocation structure developed during the deformation of fcc and bcc polycrystalline metals follows the same general pattern as that in single crystals: primary dislocations produce dipoles and loops by interaction with secondary dislocations, which give rise to local dislocation tangles gradually developing into three-dimensional networks of sub-boundaries. The cell size decreases with increasing strain, and the structural differences that are observed between various metals and alloys are mainly in the sharpness of the sub-boundaries. In bcc metals, and fcc metals with high stacking-fault energy, the tangles rearrange into sharp boundaries but in metals of low stacking-fault energy the dislocations are extended, cross-slip is restricted and sharp boundaries are not formed even at large strains. Altering the deformation temperature also has the effect of changing the dislocation distribution; lowering the deformation temperature reduces the tendency for cell formation, as shown in Figure 6.41. For a given dislocation distribution the dislocation density is simply related to the flow stress τ by an equation of the form:

τ =τ 1/2 0 + αμbρ , (6.27) where α is a constant at a given temperature ≈0.5; τ 0 for fcc metals is zero (see Figure 6.37). The

work-hardening rate is determined by the ease with which tangled dislocations rearrange themselves and is high in materials with low γ, i.e. brasses, bronzes and austenitic steels, compared to Al and bcc metals. In some austenitic steels, work hardening may be increased and better sustained by a strain-induced phase transformation (see Chapter 7).

Grain boundaries affect work hardening by acting as barriers to slip from one grain to the next. In addition, the continuity criterion of polycrystals enforces complex slip in the neighborhood of the boundaries, which spreads across the grains with increasing deformation. This introduces a dependence of work-hardening rate on grain size which extends to several percent elongation. After this stage, however, the work-hardening rate is independent of grain size and for fcc polycrystals is

Mechanical properties I 339

⫹ 50 0 Cell

formation No cell

formation

Deformation temperature

Figure 6.41 Influence of deformation strain and temperature on the formation of a cell structure in α-iron.

about μ/40, which, allowing for the orientation factors, is roughly comparable with that found in single crystals deforming in multiple slip. Thus, from the relations σ = mτ and ε = γ/m, the average resolved shear stress on a slip plane is rather less than half the applied tensile stress, and the average shear strain parallel to the slip plane is rather more than twice the tensile elongation. The polycrystal work-hardening rate is thus related to the single-crystal work-hardening rate by the relation

dσ/dε =m 2 dτ/dγ. (6.28) For bcc metals with a multiplicity of slip systems the ease of cross-slip m is more nearly 2, so that the

work-hardening rate is low. In polycrystalline cph metals the deformation is complicated by twinning, but in the absence of twinning m ≈ 6.5, and hence the work-hardening rate is expected to be more than an order of magnitude greater than for single crystals, and also higher than the rate observed in fcc polycrystals, for which m ≈ 3.

6.6.2.4 Dispersion-hardened alloys

On deforming an alloy containing incoherent, non-deformable particles, the rate of work hardening is much greater than that shown by the matrix alone (see Figure 7.10). The dislocation density increases very rapidly with strain because the particles produce a turbulent and complex deformation pattern around them. The dislocations gliding in the matrix leave loops around particles either by bowing between the particles or by cross-slipping around them; both these mechanisms are discussed in Chapter 7. The stresses in and around particles may also be relieved by activating secondary slip systems in the matrix. All these dislocations spread out from the particle as strain proceeds and, by intersecting the primary glide plane, hinder primary dislocation motion and lead to intense work hardening. A dense tangle of dislocations is built up at the particle and a cell structure is formed with the particles predominantly in the cell walls.

At small strains ( $ < 1%), work hardening probably arises from the back-stress exerted by the few Orowan loops around the particles, as described by Fisher, Hart and Pry. The stress–strain curve is reasonably linear with strain ε according to

i + αμf ε ,

340 Physical Metallurgy and Advanced Materials with the work hardening depending only on f , the volume fraction of particles. At larger strains

the ‘geometrically necessary’ dislocations stored to accommodate the strain gradient which arises because one component deforms plastically more than the other determine the work hardening. A determination of the average density of dislocations around the particles with which the primary dislocations interact allows an estimate of the work-hardening rate, as initially considered by Ashby.

Thus, for a given strain ε and particle diameter d the number of loops per particle is n ∼ εd/b and the number of particles per unit volume N v

= 3f /4πr 2 , or 6f /πd 3 .

The total number of loops per unit volume is nN v and hence the dislocation density ρ = nN v π d = 6f ε/db. The stress–strain relationship from equation (6.27) is then

i + αμ( fb/d) ε 1/2 (6.29) and the work-hardening rate dσ/dε =α ′ μ ( f /d) 1/2 (b/ε) 1/2 .

(6.30) Alternative models taking account of the detailed structure of the dislocation arrays (e.g. Orowan,

prismatic and secondary loops) have been produced to explain some of the finer details of dispersion- hardened materials. However, this simple approach provides a useful working basis for real materials. Some additional features of dispersion-strengthened alloys are discussed in Chapter 7.

6.6.2.5 Work hardening in ordered alloys

A characteristic feature of alloys with long-range order is that they work-harden more rapidly than in the disordered state. θ 11 for Fe–Al with a B2 ordered structure is ≈μ/50 at room temperature, several times greater than a typical fcc or bcc metal. However, the density of secondary dislocations in Stage II is relatively low and only about 1/100 of that of the primary dislocations. One mechanism for the increase in work-hardening rate is thought to arise from the generation of anti-phase domain boundary (APB) tubes. A possible geometry is shown in Figure 6.42a; the superdislocation partials shown each contain a jog produced, for example, by intersection with a forest dislocation, which are non-aligned along the direction of the Burgers vector. When the dislocation glides and the jogs move non-conservatively a tube of APBs is generated. Direct evidence for the existence of tubes from weak-beam electron microscope studies was first reported for Fe–30 at.% Al. The micrographs show faint lines along expected to be weak, since the contrast arises from two closely spaced overlapping faults, the second effectively canceling the displacement caused by the first, and are visible only when superlattice reflections are excited. APB tubes have since been observed in other compounds.

Theory suggests that jogs in superdislocations in screw orientations provide a potent hardening mechanism, estimated to be about eight times as strong as that resulting from pulling out of APB tubes on non-aligned jogs on edge dislocations. The major contributions to the stress to move a dislocation are (1) τ s , the stress to generate point defects or tubes, and (2) the interaction stress τ i with dislocations on neighboring slip planes, and τ s

4 α s μ (ρ f /ρ p )ε. Thus, with α s = 1.3 and provided ρ f /ρ p is constant and small, linear hardening with the observed rate is obtained.

+τ 3 i =

Mechanical properties I 341

Dissociated screw

APB SS tube

Figure 6.42 Schematic diagram of superdislocation: (a) with non-aligned jogs which, after glide, produce an APB tube and (b) cross-slipped onto the cube plane to form a Kear–Wilsdorf (K–W) lock.

In crystals with A 3 B order only one rapid stage of hardening is observed compared with the normal three-stage hardening of fcc metals. Moreover, the temperature dependence of θ 11 /μ increases with temperature and peaks at ∼0.4T m . It has been argued that the APB tube model is unable to explain why anomalously high work-hardening rates are observed for those single-crystal orientations favorable for single slip on {1 1 1} planes alone. An alternative model to APB tubes has been proposed based on cross-slip of the leading unit dislocation of the superdislocation. If the second unit dislocation cannot follow exactly in the wake of the first, both will be pinned.

For alloys with L1 2 structure the cross-slip of a screw superpartial with b 1 = 2 [¯1 0 1] from the primary (1 1 1) plane to the (0 1 0) plane was first proposed by Kear and Wilsdorf. The two 1 2 [¯1 0 1] superpartials, one on the (1 1 1) plane and the other on the (0 1 0) plane, are of course dissociated into

a Kear–Wilsdorf (K–W) lock and is shown in Figure 6.42b. Since cross-slip is thermally activated, the number of locks and therefore the resistance to (1 1 1) glide will increase with increasing temperature. This could account for the increase in yield stress with temperature, while the onset of cube slip at

elevated temperatures could account for the peak in the flow stress. Cube cross-slip and cube slip has now been observed in a number of L1 2 compounds by TEM. There is some TEM evidence that the APB energy on the cube plane is lower than that on the (1 1 1) plane (see Chapter 8) to favor cross-slip, which would be aided by the torque, arising from elastic anisotropy, exerted between the components of the screw dislocation pair.