Work-hardening in polycrystals on one particular secondary plane predominates. In

7.6.2.3 Work-hardening in polycrystals on one particular secondary plane predominates. In

niobium, a metal with high SF , the dislocation dis- The dislocation structure developed during the defor- tribution is rather similar to copper. In Mg, typical of

mation of fcc and bcc polycrystalline metals follows cph metals, Stage I is extensive and the dislocations

the same general pattern as that in single crystals; are mainly in the form of primary edge multipoles,

primary dislocations produce dipoles and loops by but forest dislocations threading the primary slip plane

interaction with secondary dislocations, which give do not appear to be generated.

rise to local dislocation tangles gradually develop- From the curve shown in Figure 7.40 it is evident

ing into three-dimensional networks of sub-boundaries. that the rate of work-hardening decreases in the later

The cell size decreases with increasing strain, and the stages of the test. This observation indicates that at

structural differences that are observed between var-

a sufficiently high stress or temperature the disloca- ious metals and alloys are mainly in the sharpness of the sub-boundaries. In bcc metals, and fcc metals

tions held up in Stage II are able to move by a process with high stacking-fault energy, the tangles rearrange which at lower stresses and temperature had been sup-

into sharp boundaries but in metals of low stacking pressed. The onset of Stage III is accompanied by

fault energy the dislocations are extended, cross-slip is cross-slip, and the slip lines are broad, deep and con-

restricted, and sharp boundaries are not formed even at sist of segments joined by cross-slip traces. Electron

large strains. Altering the deformation temperature also metallographic observations on sections of deformed

has the effect of changing the dislocation distribution; crystal inclined to the slip plane (see Figure 7.41d)

lowering the deformation temperature reduces the ten- show the formation of a cell structure in the form

dency for cell formation, as shown in Figure 7.42. For of boundaries, approximately parallel to the primary

a given dislocation distribution the dislocation density slip plane of spacing about 1–3 µ m plus other bound- aries extending normal to the slip plane as a result of cross-slip.

the form

(7.27) the experimental observations is that the screw disloca-

The simplest process which is in agreement with

tions held up in Stage II, cross-slip and possibly return where ˛ is a constant at a given temperature ³0.5; to the primary slip plane by double cross-slip. By this

0 for fcc metals is zero (see Figure 7.38). The mechanism, dislocations can bypass the obstacles in

workhardening rate is determined by the ease with their glide plane and do not have to interact strongly

which tangled dislocations rearrange themselves and with them. Such behaviour leads to an increase in slip

is high in materials with low , i.e. brasses, bronzes distance and a decrease in the accompanying rate of

and austenitic steels compared to Al and bcc met- work-hardening. Furthermore, it is to be expected that

als. In some austenitic steels, work-hardening may screw dislocations leaving the glide plane by cross-

be increased and better sustained by a strain-induced slip may also meet dislocations on parallel planes and

phase transformation (see Chapter 8).

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 forma- tion, 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, , on the stress–strain curve is evident from its importance to the process of cross-slip. Low values of 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 aluminium the stacking fault ribbon width is very small because has a high value, and cross-slip occurs at room temperature. Stage II is, therefore, poorly developed unless testing is carried

Figure 7.42 Influence of deformation strain and out at low temperatures. These conclusions are in

temperature on the formation of a cell structure in ˛-iron .

Mechanical behaviour of materials 231 Grain boundaries affect work-hardening by acting as

the strain gradient which arises because one component barriers to slip from one grain to the next. In addition,

deforms plastically more than the other, determine the the continuity criterion of polycrystals enforces com-

work-hardening. A determination of the average den- plex slip in the neighbourhood of the boundaries which

sity of dislocations around the particles with which the spreads across the grains with increasing deformation.

primary dislocations interact allows an estimate of the This introduces a dependence of work-hardening rate

work-hardening rate, as initially considered by Ashby. on grain size which extends to several per cent elonga-

Thus, for a given strain ε and particle diameter d the tion. After this stage, however, the work-hardening rate

number of loops per particle is is independent of grain size and for fcc polycrystals is

n ¾ εd/b

is roughly comparable with that found in single crys- tals deforming in multiple slip. Thus from the relations

and the number of particles per unit volume

on a slip plane is rather less than half the applied tensile stress, and the average shear strain parallel to the slip

The total number of loops per unit volume is nN v and plane is rather more than twice the tensile elongation.

The polycrystal work-hardening rate is thus related to v The stress–strain relationship from equation (7.27)

the single-crystal work-hardening rate by the relation

is then

(7.29) For bcc metals with the multiplicity of slip systems

and the work-hardening rate and the ease of cross-slip m is more nearly 2, so that

the work-hardening rate is low. In polycrystalline cph

0 1/2 ⊲b/ε⊳ 1/2 (7.30) metals the deformation is complicated by twinning,

but in the absence of twinning m ³ 6.5, and hence the Alternative models taking account of the detailed struc- work-hardening rate is expected to be more than an

ture of the dislocation arrays (e.g. Orowan, prismatic order of magnitude greater than for single crystals, and

and secondary loops) have been produced to explain also higher than the rate observed in fcc polycrystals

some of the finer details of dispersion-hardened mate- for which m ³ 3.

rials. However, this simple approach provides a useful working basis for real materials. Some additional fea-

7.6.2.4 Dispersion-hardened alloys tures of dispersion-strengthened alloys are discussed On deforming an alloy containing incoherent, non-

in Chapter 8.

deformable particles the rate of work-hardening is much greater than that shown by the matrix alone