When a single crystal is continued to be deformed in the plastic range beyond the yield point, the shear stress required to produce further deformation continues to increase. The increase in the stress required to continue the deformation because of the previous plastic deformation is called as strain-hardening, or work-hardening.
As the dislocation density also increases simultaneously with strain (dislocation density of well annealed crystal is 105 to 106cm-2, and of cold worked metal is 1010 to 1012cm-2), the reasons for the increase in flow stress is to be sought in various interactions occurring between dislocations themselves and with barriers. As strain-hardening occurs in metals, means that dislocations experience increased resistance in moving through the lattice.
Some of the reasons for this are:
As the dislocations pile-up on slip planes at the barriers in the crystal, the pile-up produces a ‘back-stress’ which opposes the applied stress on the slip plane. The presence of the back stress can be experimentally proved such as by plastically straining a HCP crystal, which is so oriented that slip occurs only on the basal plane, up to a point X, Fig. 6.30.
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The crystal is unloaded and again reloaded but in a direction opposite to the first slip direction (i.e., slip direction 180° to original slip direction). The HCP crystal now yields at a lower stress than earlier (Fig. 6.30). The back-stress produced by the pile-up of dislocations, developed during the first loading, gets added up with the applied stress when the slip direction is reversed during second loading in opposite direction.
Also, the same Frank-Read source generates now opposite-signed dislocations, when the direction of slip is reversed. Dislocations of opposite signs attract and annihilate each other resulting in the softening of the crystal. This phenomenon of decreased yield strength when the deformation in one direction is followed by deformation in opposite direction is called Bauschin-ger effect. The strain-hardening effect produced by the pile-up of dislocations, occurs over longer distances, and thus, is independent of temperature and strain rate.
The barriers for the pile-up of dislocations could be microscopic precipitates or even foreign atoms in impure single crystals of metals. But in a pure metal, the barrier could be a sessile dislocation-alike a Lomer-Cottrell barrier in FCC metals. Even the lead-dislocation may combine with another dislocation on an intersecting plane to produce a new dislocation which is not in a slip direction, i.e., results in a sessile-dislocation.
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Another barrier could be a jog in a screw dislocation produced by the intersecting screw dislocations. Such a jog impedes the motion of dislocations, or may create vacancies or interstitials if jogs are forced to move non-conservatively. Such jogs may stop the cross-slip of screw dislocations.
The result is the increased rate of strain hardening (the stress-strain curve has higher slope). Jogs exert only short-range forces over 5 to 10 interatomic distances, and thus, can be overcome at finite temperatures, i.e., such barriers are dependent on temperature and strain rate.
The typical strain-hardening curves (resolved shear stress versus resolved shear strain) for single crystals of three common types of crystal structures -BCC, FCC and HCP-are illustrated in Fig. 6.31. Copper single crystal normally exhibits three distinct stages. These stages are also observed in Al, Ag, (FCC), Nb, Ta (BCC), Zn, Cd, (HCP), Ge (diamond cubic).
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A generalised flow curve for pure metal single crystal of FCC can be schematically divided into three stages, Fig. 6.32. The important points about these three hardening stages are summarised below. It should however be emphasized that the three stages are not always present in the stress-strain curve of a crystal.
One or more stages may disappear if the testing conditions are varied. The variables which affect the deformation behaviour include the crystal orientation, temperature of deformation, purity and crystal structure. For example, as the temperature of deformation is lowered, the extent of stage I hardening is increased, whereas the stage III tends to predominate at high temperatures.
Stage I:
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The region of easy glide follows immediately the yield point. It is linear in which the crystal undergoes little strain-hardening. During stage I, slip always occurs on only one slip-system, and for this reason, slip in this stage is also called laminar flow. The density of dislocations increases with plastic strain without increasing the number of the intersection processes, and that is why, dislocations move large distances without encountering barriers.
The rate of hardening is thus insensitive to temperature. The little linear hardening in stage I in cubic crystals is of the same order as that for HCP metals, where only one glide plane operates. The length of the stage I depends on orientation, purity and size of the crystals.
In order to obtain a curve, Fig. 6.32 with all the three stages, it is essential that the crystal be oriented initially such that the resolved shear stress on one slip system exceeds the resolved shear stress in any other slip system. In such orientation, the deformation initially occurs by slip on only one set of parallel slip planes. Stage I is characterised by long (~ 1 mm) fine slip lines in the microstructure as most of dislocations escape from crystal at surface.
The value for the glide stress in stage I depends markedly on the purity of the crystal. For example. Zinc of 99.96% purity exhibits a glide stress of 0.096 kg/mm2 at the onset of stage I compared to 0.0184 kg/mm2 for 99.999% purity. The hardening rate is very low, generally below 10-3 G per unit strain.
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The slip on easy glide plane ends and the turbulent flow begins, when the crystal reaches an orientation for which two or more slip systems are equally stressed.
Stage II:
It is the linear-hardening region where strain-hardening increases rapidly. The slope (though constant) of the curve is approximately independent of applied stress, temperature, orientation, or alloy purity. The characteristic feature of this stage is that slip occurs on more than one set of planes and intersection of dislocations occurs, and that is why short slip lines are formed, and the length of the slip lines decreases with the plastic strain.
The number of Lomer-Cottrell barriers increases with the strain. Such dislocations act as formidable barriers to subsequent dislocation motion. As their number multiply, further movement of dislocations becomes more and more difficult, and thus the distance a dislocation glides decreases resulting in decreased length of slip lines.
The obstacles are in the form of ribbons of high densities of dislocations which like, pile-ups, tend to form sheets. As stage II is independent of the temperature, the chief strain-hardening mechanism is the pile-up of dislocations.
State III:
The parabolic region of the curve exhibits decreasing rate of strain-hardening. This stage is probably associated with the annihilation of dislocations as a consequence of cross-slip, i.e., dynamic recovery occurs in this stage. This stage is strongly dependent on temperature, and coarse slip-bands start appearing. At such sufficiently high stress or temperature, the dislocations held up in stage II are now able to move by a process which at lower stresses and temperature had been suppressed.
Large-scale cross-slip and even double cross-slip of piled up dislocations occurs in stage III, as reflected by the appearance of coarse slip-bands too, because then the dislocations can circumvent obstacles, and thus the strain-hardening rate subsequently decreases. In the absence of such mechanism, higher stress would be required to enable the dislocations to overcome the barriers.
Moreover, the cross-slipping-screw-dislocations often encounter screw dislocations of opposite sign on the adjacent planes, and thus, the two annihilate each other to reduce the dislocation density. Therefore, the rate of increase of the flow stress is lower in stage III as compared to stage II.
Although, cross-slip also occurs in stage I, it is then on a very fine scale being induced by mutual attraction between the screw dislocations of opposite sign, so that at least one of them cross-slips and they annihilate each other. This process is inhibited in stage II, when the secondary dislocations intervene and prevent the primary screw dislocations from approaching close enough to annihilate.
Since, the applied stress at the beginning of the stage III is quite high, the cross-slip presumably takes place with the aid of the applied stress without needing the internal stress from screw dislocations of opposite sign. Hardening in stage III is then due to the edge parts of the loops, which remain in the crystal and increase in their density as the source continues to operate.
Since, the cross-slip is activated by the recombination of partial dislocations, it is a thermally activated process. The stress and strain at which the stage III is initiated consequently decreases with increasing temperature. The stage can be eliminated by deforming a sample close to 0 °K.
Stacking-fault energy has a profound effect on the onset of stage III. If the stacking-fault energy is low, the dislocations are widely extended. Since, recombination must occur before cross-slip can occur, a higher stress is required for cross-slip of a material with low stacking-fault energy. Aluminium, thus, with a higher stacking-fault energy exhibits stage III earlier than copper and silver (both with low SFE). Addition of cobalt to nickel lowers the stacking-fault energy and raises the stress required for the onset of stage III.
It has been experimentally observed that the flow stress of a deformed crystal is proportional to the square root of the dislocation density as:
where, 0 is a material constant, and is the stress required to glide a dislocation in absence of other dislocations, and A is a constant which takes values ranging from 0.3 to 0.6. This relationship holds good for various stages of stress-strain curves of single crystals as well as for polycrystalline materials. Thus, strain-hardening is related to the density of the dislocations in the metal.
Face-Centred Cubic Metals:
Slip normally occurs by the movement of a/2 <110> type dislocations gliding on fill} planes. The metallographic examination of deformed crystal in the easy glide region (stage I) reveals long, fine slip lines which cover the crystal uniformly. The rate of strain-hardening (θ1) in stage I is quite low typically of the order of G/1000. In stage II, the strain-hardening coefficient (θ11) is nearly an order of Magnitude higher than θ1.
The easy glide region generally does not exceed a strain of 20% even in the most favoured orientation. Stage II then sets in. This is due to large number (12) of slip systems in the crystal. In stage III, the slip lines cluster together to from slip bands on account of cross-slip.
The generalized curve, Fig. 6.32 shows deviations. For example, metal like aluminium, having high stacking-fault energy, generally exhibits only a very small stage II region at room temperature, as cross slip starts earlier. The shape and the magnitude of resolved shear stress-strain curve depend on the orientation, temperature of testing, purity of metal and strain rate.
Actually, the stage I is much more prominent in HCP crystals than in FCC metals. This region is more when slip occurs on single system, with metals of high purity, at low temperature, for a favourable orientation and in absence of surface oxide film. Fig. 6.33 illustrates the effect of orientation on shape of the curve to ultimately resemble as for a polycrystalline material.
As the temperature is increased, both stage I and stage II decrease until at high temperatures, the curve shows entirely the parabolic stage III behaviour. Aluminium of low purity may not exhibit first stage. Small inclusion and second phase particles present in impure crystals encourage localised slip on other than the primary slip plane, and the flow stress rises rapidly. This eliminates stage I.
Body Centered Cubic Metals:
Three-stage stress-strain curve can also be observed in BCC crystals, and the pattern of deformation in the deformation in the three stages is very similar to that for FCC crystals. In contrast to FCC crystals, the yield stress of BCC crystals, the yield stress of BCC crystals is however markedly temperature dependent.
Two probable reasons put forward to explain this sensitivity to temperature are the presence of interstitial impurities, and to the temperature-dependence of Peierls-Nabarro stress in BCC structure. Since, it has not been possible to remove the interstitial impurities completely from ay BCC metal, it has not been possible to ascertain the actual mechanism. For example, in a high purity iron containing one part per billion of carbon, the yield stress is very low, but there is still a substantial temperature-dependent contribution to the flow stress.
The effect of orientation on the stress-strain curve is shown in Fig. 6.34. The extent of stage I is markedly affected, and in specimen C, the stage I is completely absent due to equal resolve stress on more than one slip system. As the stress rises rapidly in stage II, the stage III also sets in after relatively small strains in the specimen.
Hexagonal Close-Packed Metals:
Three stage stress-strain curves are also observed in hexagonal close-packed crystals. However, in crystals oriented for glide along (0001) plane—the basal plane—the stage I exists with a low rate of strain hardening value of G/50000 up to large strains. This region of easy glide, instead of extending only several percent as in FCC and BCC crystals, may exceed strain-values of 100%.
This is due to relatively fewer slip systems in HCP crystals, so that in suitably oriented crystals, the resolved shear stress is high only on one slip system and multiple slip can be avoided easily.