Discontinuous yielding

6.4.4 Discontinuous yielding

In some materials the onset of macroscopic plastic flow begins in an abrupt manner with a yield drop in which the applied stress falls, during yielding, from an upper to a lower yield point. Such yield behavior is commonly found in iron containing small amounts of carbon or nitrogen as impurity. The main characteristics of the yield phenomenon in iron may be summarized as follows.

6.4.4.1 Yield point

A specimen of iron during tensile deformation (Figure 6.23a, curve 1) behaves elastically up to

a certain high load A, known as the upper yield point, and then it suddenly yields plastically. The important feature to note from this curve is that the stress required to maintain plastic flow immediately after yielding has started is lower than that required to start it, as shown by the fall in load from A to

B (the lower yield point). A yield point elongation to C then occurs, after which the specimen work hardens and the curve rises steadily and smoothly.

6.4.4.2 Overstraining

The yield point can be removed temporarily by applying a small preliminary plastic strain to the specimen. Thus, if after reaching the point D, for example, the specimen is unloaded and a second

Mechanical properties I 311 test is made fairly soon afterwards, a stress–strain curve of type 2 will be obtained. The specimen

deforms elastically up to the unloading point D, and the absence of a yield point at the beginning of plastic flow is characteristic of a specimen in an overstrained condition.

6.4.4.3 Strain-age hardening

If a specimen which has been overstrained to remove the yield point is allowed to rest, or age, before retesting, the yield point returns, as shown in Figure 6.23a, curve 3. This process, which is accompanied by hardening (as shown by the increased stress, EF, to initiate yielding) is known as strain ageing or, more specifically, strain-age hardening. In iron, strain ageing is slow at room temperature but is greatly speeded up by annealing at a higher temperature. Thus, a strong yield point returns after an ageing treatment of only a few seconds at 200 ◦

C, but the same yield point will take many hours to develop if ageing is carried out at room temperature.

6.4.4.4 Lüders band formation

Closely related to the yield point is the formation of Lüders bands. These bands are markings on the surface of the specimen which distinguish those parts of the specimen that have yielded, A, from those which have not, B. Arrival at the upper yield point is indicated by the formation of one or more of these bands and, as the specimen passes through the stage of the yield point elongation, these bands spread along the specimen and coalesce until the entire gauge length has been covered. At this stage the whole of the material within the gauge length has been overstrained, and the yield point elongation is complete. The growth of a Lüders band is shown diagrammatically in Figure 6.23b. It should be noted that the band is a macroscopic band crossing all the grains in the cross-section of a polycrystalline specimen, and thus the edges of the band are not necessarily the traces of individual slip planes. A second point to observe is that the rate of plastic flow in the edges of a band can be very high, even in an apparently slow test; this is because the zones, marked C in Figure 6.23b, are very narrow compared with the gauge length.

These Lüders bands frequently occur in drawing and stamping operations, when the surface mark- ings in relief are called stretcher strains. These markings are unsightly in appearance and have to be avoided on many finished products. The remedy consists of overstraining the sheet prior to pressing operations, by means of a temper roll, or roller leveling, pass so that the yield phenomenon is elimi- nated. It is essential, once this operation has been performed, to carry out pressing before the sheet has time to strain-age; the use of a ‘non-ageing’ steel is an alternative remedy.

These yielding effects are influenced by the presence of small amounts of carbon or nitrogen atoms interacting with dislocations. The yield point can be removed by annealing at 700 ◦

C in a wet hydrogen atmosphere, and cannot subsequently be restored by any strain-ageing treatment. Conversely, expos- ing the decarburized specimen to an atmosphere of dry hydrogen containing a trace of hydrocarbon at 700 ◦

C for as little as one minute restores the yield point. The carbon and nitrogen atoms can also be removed from solution in other ways – for example, by adding to the iron such elements as molybdenum, manganese, chromium, vanadium, niobium or titanium, which have a strong affinity for forming carbides or nitrides in steels. For this reason, these elements are particularly effective in removing the yield point and producing a non-strain-ageing steel.

The carbon/nitrogen atoms are important in the yielding process because they interact with the dislocations and immobilize them. This locking of the dislocations is brought about because the strain energy due to the distortion of a solute atom can be relieved if it fits into a structural region where the local lattice parameter approximates to that of the natural lattice parameter of the solute. Such a condition will be brought about by the segregation of solute atoms to the dislocations, with

312 Physical Metallurgy and Advanced Materials large substitutional atoms taking up lattice positions in the expanded region and small ones in the

compressed region; small interstitial atoms will tend to segregate to interstitial sites below the half- plane. Thus, where both dislocations and solute atoms are present in the lattice, interactions of the stress field can occur, resulting in a lowering of the strain energy of the system. This provides a driving force tending to attract solute atoms to dislocations and if the necessary time for diffusion is allowed,

a solute atom ‘atmosphere’ will form around each dislocation. When a stress is applied to a specimen in which the dislocations are locked by carbon atoms the

dislocations are not able to move at the stress level at which free dislocations are normally mobile. With increasing stress, yielding occurs when dislocations suddenly become mobile, either by breaking

away from the carbon atmosphere or by nucleating fresh dislocations at stress concentrations. At this high stress level the mobile dislocation density increases rapidly. The lower yield stress is then the stress at which free dislocations continue to move and produce plastic flow. The overstrained condition corresponds to the situation where the mobile dislocations, brought to rest by unloading the specimen, are set in motion again by reloading before the carbon atmospheres have time to develop by diffusion. If, however, time is allowed for diffusion to take place, new atmospheres can re-form and immobilize the dislocations again. This is the strain-aged condition when the original yield characteristics reappear.

The upper yield point in conventional experiments on polycrystalline materials is the stress at which initially yielded zones trigger yield in adjacent grains. As more and more grains are triggered, the yield zones spread across the specimen and form a Lüders band.

The propagation of yield is thought to occur when a dislocation source operates and releases an avalanche of dislocations into its slip plane, which eventually pile up at a grain boundary or other obstacle. The stress concentration at the head of the pile-up acts with the applied stress on the dislocations of the next grain and operates the nearest source, so that the process is repeated in the next grain. The applied shear stress σ y at which yielding propagates is given by

(6.8) where r is the distance from the pile-up to the nearest source, 2d is the grain diameter and

=σ 1/2 i + (σ c r )d −1/2 ,

σ c is the stress required to operate a source which involves unpinning a dislocation τ c at that temperature. Equation (6.8) reduces to the Hall–Petch equation σ y =σ i +k y d −1/2 , where σ i is the ‘friction’ stress term and k y the grain size dependence parameter ( =m 2 τ c r 1/2 ) discussed in

Section 6.4.11.