Dislocation source operation

7.4.3 Dislocation source operation

Figure 7.20 Shear produced by gliding dislocations . When a stress is applied to a material the specimen

plastically deforms at a rate governed by the strain rate of the deformation process (e.g. tensile testing, rolling,

dimensions L 1 ð L 2 ð 1 cm shown in Figure 7.20 a etc.) and the strain rate imposes a particular velocity on the mobile dislocation population. In a crystal of

in time t D L 1 and produces a shear strain b/L 2 , i.e.

210 Modern Physical Metallurgy and Materials Engineering

Figure 7.21 Successive stages in the operation of a Frank–Read source. The plane of the paper is assumed to be the slip plane .

1 L 2 . If the density of glissible line to decrease its radius of curvature further until it becomes semi-circular (position 2). Beyond this point

it has no equilibrium position so it will expand rapidly, overall strain rate is thus given by

1 L 2 and the

rotating about the nodes and taking up the succession

b of forms indicated by 3, 4 and 5. Between stages 4

D 1 L 2 (7.6)

and 5 the two parts of the loop below AB meet and L 2 L 1 annihilate each other to form a complete dislocation

1 At conventional strain rates (e.g. 1 s loop, which expands into the slip plane and a new ) the disloca- line source between A and B. The sequence is then

tions would be moving at quite moderate speeds of

7 2 a few cm/s if the mobile density ³10 repeated and one unit of slip is produced by each loop / cm . During

that is generated.

high-speed deformation the velocity approaches the To operate the Frank–Read source the force applied limiting velocity. The shear strain produced by these

must be sufficient to overcome the restoring force on dislocations is given by

the dislocation line due to its line tension. Referring to

where x is the average distance a dislocation moves. and b have their usual meaning and l is the length If the distance x ' 10 cm (the size of an average 7 of the Frank–Read source; the substitution of typical

Nm , b D 2.5 ð 10 m, and ð 3 ð 10 ð 10 ⊳ which is only a

is about ⊲10 7

l D 10 m) into this estimate shows that a critical fraction of 1%. In practice, shear strains >100% can

shear stress of about 100 gf mm is required. This

be achieved, and hence to produce these large strains value is somewhat less than but of the same order as many more dislocations than the original ingrown

that observed for the yield stress of virgin pure metal dislocations are required. To account for the increase

single crystals. Another source mechanism involves in number of mobile dislocations during straining the

multiple cross-slip as shown in Figure 7.23. It depends concept of a dislocation source has been introduced.

The simplest type of source is that due to Frank and Read and accounts for the regenerative multiplication of dislocations. A modified form of the Frank–Read source is the multiple cross-glide source, first proposed by Koehler, which, as the name implies, depends on the cross-slip of screw dislocations and is therefore more common in metals of intermediate and high stacking fault energy.

Figure 7.21 shows a Frank–Read source consisting of a dislocation line fixed at the nodes A and B (fixed, for example, because the other dislocations that join the nodes do not lie in slip planes). Because

of its high elastic energy (³4 eV per atom plane Figure 7.22 Geometry of Frank–Read source used to threaded by a dislocation) the dislocation possesses a calculate the stress to operate .

line tension tending to make it shorten its length as much as possible (position 1, Figure 7.21). This line

shear modulus, b the Burgers vector and ˛ a constant

usually taken to be about 1 2 . Under an applied stress

the dislocation line will bow out, decreasing its radius of curvature until it reaches an equilibrium position in which the line tension balances the force due to the applied stress. Increasing the applied stress causes the

Figure 7.23 Cross-slip multiplication source .

Mechanical behaviour of materials 211 on the Frank–Read principle but does not require a

to a certain high load A, known as the upper yield dislocation segment to be anchored by nodes. Thus, if

point, and then it suddenly yields plastically. The part of a moving screw dislocation undergoes double

important feature to note from this curve is that the cross-slip the two pieces of edge dislocation on the

stress required to maintain plastic flow immediately cross-slip plane effectively act as anchoring points

after yielding has started is lower than that required to for a new source. The loop expanding on the slip

start it, as shown by the fall in load from A to B (the plane parallel to the original plane may operate as

lower yield point). A yield point elongation to C then

a Frank–Read source and any loops produced may occurs after which the specimen work hardens and the in turn cross slip and become a source. This process

curve rises steadily and smoothly. therefore not only increases the number of dislocations

on the original slip plane but also causes the slip band Overstraining The yield point can be removed tem- to widen.

porarily by applying a small preliminary plastic strain The concept of the dislocation source accounts

to the specimen. Thus, if after reaching the point D, for for the observation of slip bands on the surface of

example, the specimen is unloaded and a second test is deformed metals. The amount of slip produced by

made fairly soon afterwards, a stress–strain curve of the passage of a single dislocation is too small to be

type 2 will be obtained. The specimen deforms elasti- observable as a slip line or band under the light micro-

cally up to the unloading point, D, and the absence of scope. To be resolved it must be at least 300 nm in

a yield point at the beginning of plastic flow is char- height and hence ³1000 dislocations must have oper-

acteristic of a specimen in an overstrained condition. ated in a given slip band. Moreover, in general, the slip

band has considerable width, which tends to support Strain-age hardening If a specimen which has been the operation of the cross-glide source as the predom-

overstrained to remove the yield point is allowed to inant mechanism of dislocation multiplication during

rest, or age, before retesting, the yield point returns as straining.

shown in Figure 7.24a, curve 3. This process, which is accompanied by hardening (as shown by the increased

stress, EF, to initiate yielding) is known as strain- In some materials the onset of macroscopic plastic

7.4.4 Discontinuous yielding

ageing or, more specifically, strain-age hardening. In flow begins in an abrupt manner with a yield drop

iron, strain-ageing is slow at room temperature but is in which the applied stress falls, during yielding, from

greatly speeded up by annealing at a higher tempera- an upper to a lower yield point. Such yield behaviour

ture. Thus, a strong yield point returns after an ageing is commonly found in iron containing small amounts

C, but the same of carbon or nitrogen as impurity. The main char-

treatment of only a few seconds at 200 °

yield point will take many hours to develop if ageing acteristics of the yield phenomenon in iron may be

is carried out at room temperature. summarized as follows.

L¨uders band formation Closely related to the yield Yield point

A specimen of iron during tensile defor- point is the formation of L¨uders bands. These bands mation (Figure 7.24a, curve 1) behaves elastically up

are markings on the surface of the specimen which

Figure 7.24 Schematic representation of (a) strain ageing and (b) L¨uders band formation .

212 Modern Physical Metallurgy and Materials Engineering 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 L¨uders band is shown diagrammatically in Figure 7.24b. 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 7.24b, are very narrow compared with the gauge length. These L¨uders bands frequently occur in drawing and stamping operations when the surface markings in relief are called stretcher strains. These markings are unsightly in appearance and have to be avoided on many finished products. The remedy consists in overstraining the sheet prior to pressing operations, by means of a temper roll, or roller levelling, pass so that the yield phenomenon is eliminated. 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 pres- ence of small amounts of carbon or nitrogen atoms interacting with dislocations. The yield point can

be removed by annealing at 700 °

C in wet-hydrogen

atmosphere, and cannot subsequently be restored by any strain-ageing treatment. Conversely, exposing the decarburized specimen to an atmosphere of dry hydro- gen 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 solu- tion 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 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 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 disloca- tions are not able to move at the stress level at which free dislocations are normally mobile. With increas- ing stress yielding occurs when dislocations suddenly become mobile either by breaking away from the car- bon 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 over- strained 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 diffu- sion 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 L¨uders 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

y at which yielding propagates is given by

yi

c r 1/2 ⊳d (7.8) where r is the distance from the pile-up to the nearest

c is the stress required to operate a source which involves unpinning

c at that temperature. Equation (7.8)

i Ck y d ,

i is the ‘friction’ stress term and k y the grain size dependence parameter ⊲D m 2 c r 1/2 ⊳ discussed in Section 7.4.11.