Annealing twins

7.8.5 Annealing twins

A prominent feature of the microstructures of most Figure 7.56 Grain growth during secondary

annealed fcc metals and alloys is the presence of recrystallization .

many straight-sided bands that run across grains. These

244 Modern Physical Metallurgy and Materials Engineering

Figure 7.58 Formation and growth of annealing twins (from Burke and Turnbull, 1952; courtesy of Pergamon Press) .

grain, since the grain boundary configuration will then have a lower interfacial energy. If this happens the twin will then be able to grow in width because one of

Figure 7.57 Relation between grain size, deformation and its sides forms part of the boundary of the growing temperature for aluminium (after Buergers, courtesy of

grain. Such a twin will continue to grow in width Akademie-Verlags-Gesellschaft) .

until a second mistake in the positioning of the atomic layers terminates it; a complete twin band is then

formed. In copper and its alloys twin / gb is low and bands have a twinned orientation relative to their

hence twins occur frequently, whereas in aluminium neighbouring grain and are referred to as annealing

the corresponding ratio is very much higher and so twins (see Chapter 4). The parallel boundaries usually

twins are rare.

coincide with a ⊲1 1 1⊳ twinning plane with the Twins may develop according to the model shown in structure coherent across it, i.e. both parts of the twin

Figure 7.59 where during grain growth a grain contact hold a single ⊲1 1 1⊳ plane in common.

is established between grains C and D. Then if the As with formation of deformation twins, it is

orientation of grain D is close to the twin orientation believed that a change in stacking sequence is all

of grain C, the nucleation of an annealing twin at that is necessary to form an annealing twin. Such

the grain boundary, as shown in Figure 7.60d, will

a change in stacking sequence may occur whenever lower the total boundary energy. This follows because

a properly oriented grain boundary migrates. For example, if the boundary interface corresponds to a

the twin/D interface will be reduced to about 5% of ⊲ 1 1 1⊳ plane, growth will proceed by the deposition of

the normal grain boundary energy, the energies of the additional ⊲1 1 1⊳ planes in the usual stacking sequence

C/A and twin/A interface will be approximately the ABCABC . . . . If, however, the next newly deposited

same, and the extra area of interface C/twin has only a layer falls into the wrong position, the sequence

ABCABCB is produced which constitutes the first layer of a twin. Once a twin interface is formed, further growth may continue with the sequence in reverse order, ABCABCjBACB . . . until a second accident in the stacking sequence completes the twin band, ABCABCBACBACBABC . When a stacking error, such as that described above, occurs the number of nearest neighbours is unchanged, so that the ease of

Figure 7.59 Nucleation of an annealing twin during grain formation of a twin interface depends on the relative

growth .

value of the interface energy. If this interface energy is low, as in copper where twin <

20 mJ/m 2 twinning

occurs frequently while if it is high, as in aluminium, the process is rare.

Annealing twins are rarely (if ever) found in cast metals because grain boundary migration is negligible during casting. Worked and annealed metals show considerable twin band formation; after extensive grain growth a coarse-grained metal often contains twins which are many times wider than any grain that was present shortly after recrystallization. This indicates that twin bands grow in width, during grain growth, by migration in a direction perpendicular to the ⊲1 1 1⊳ composition plane, and one mechanism whereby this can occur is illustrated schematically in Figure 7.58.

Figure 7.60 Combination of transient and steady-state This shows that a twin may form at the corner of a

creep .

Mechanical behaviour of materials 245 very low energy. This model indicates that the number

form the remarkably strong quasi-single-crystal cube of twins per unit grain boundary area only depends

texture. The percentage of cubically aligned grains on the number of new grain contacts made during

increases with increased deformation, but the sharp- grain growth, irrespective of grain size and annealing

ness of the textures is profoundly affected by alloying temperature.

additions. The amount of alloying addition required to suppress the texture depends on those factors which

7.8.6 Recrystallization textures

affect the stacking fault energy, such as the lattice mis- fit of the solute atom in the solvent lattice, valency

The preferred orientation developed by cold work often etc., in much the same way as that described for the changes on recrystallization to a totally different pre-

transition of a pure metal deformation texture. ferred orientation. To explain this observation, Bar-

In general, however, if the texture is to be altered rett and (later) Beck have put forward the ‘oriented

a distribution of second-phase must either be present growth’ theory of recrystallization textures in which

before cold rolling or be precipitated during anneal- it is proposed that nuclei of many orientations ini-

ing. In aluminium, for example, the amount of cube tially form but, because the rate of growth of any

texture can be limited in favour of retained rolling given nucleus depends on the orientation difference

texture by limiting the amount of grain growth with between the matrix and growing crystal, the recrystal-

a precipitate dispersion of Si and Fe. By balancing lized texture will arise from those nuclei which have

the fastest growth rate in the cold-worked matrix, i.e. the components, earing can be minimized in drawn those bounded by large-angle boundaries. It then fol-

aluminium cups. In aluminium-killed steels AlN pre- lows that because the matrix has a texture, all the

cipitation prior to recrystallization produces a higher nuclei which grow will have orientations that differ by

proportion of grains with f1 1 1g planes parallel to 30–40 ° from the cold-worked texture. This explains

the rolling plane and a high R value suitable for why the new texture in fcc metals is often related to

deep drawing. The AlN dispersion affects sub-grain the old texture, by a rotation of approximately 30–40 °

growth, limiting the available nuclei and increasing around h1 1 1i axes, in bcc metals by 30 ° about h1 1 0i

the orientation-selectivity, thereby favouring the high- and in hcp by 30 ° about h0 0 0 1i. However, while

energy f1 1 1g grains. Improved R-values in steels in it is undoubtedly true that oriented growth provides

general are probably due to the combined effect of

a selection between favourable and unfavourable ori- particles in homogenizing the deformed microstructure ented nuclei, there are many observations to indicate

and in controlling the subsequent sub-grain growth. that the initial nucleation is not entirely random. For

The overall effect is to limit the availability of nuclei instance, because of the crystallographic symmetry one

with orientations other than f1 1 1g. would expect grains appearing in a fcc texture to be

related to rotations about all four h1 1 1i axes, i.e. eight orientations arising from two possible rotations about

7.9 Metallic creep

each of the four h1 1 1i axes. All these possible orien- tations are rarely (if ever) observed.

7.9.1 Transient and steady-state creep

To account for such observations, and for those Creep is the process by which plastic flow occurs cases where the deformation texture and the annealing

when a constant stress is applied to a metal for a texture show strong similarities, oriented nucleation

prolonged period of time. After the initial strain ε 0 is considered to be important. The oriented nucle-

which follows the application of the load, creep usually ation theory assumes that the selection of orientations

exhibits a rapid transient period of flow (stage 1) is determined in the nucleation stage. It is generally

before it settles down to the linear steady-stage stage accepted that all recrystallization nuclei pre-exist in the

2 which eventually gives way to tertiary creep and deformed matrix, as sub-grains, which become more

fracture. Transient creep, sometimes referred to as ˇ- perfect through recovery processes prior to recrystal-

creep, obeys a t 1/3 law. The linear stage of creep is lization. It is thus most probable that there is some

often termed steady-state creep and obeys the relation selection of nuclei determined by the representation of

the orientations in the deformation texture, and that the (7.49) oriented nucleation theory should apply in some cases.

In many cases the orientations which are strongly rep- Consequently, because both transient and steady-state resented in the annealing texture are very weakly rep-

creep usually occur together during creep at high resented in the deformed material. The most striking

temperatures, the complete curve (Figure 7.60) during example is the ‘cube’ texture, (1 0 0) [0 0 1], found in

the primary and secondary stages of creep fits the most fcc pure metals which have been annealed follow-

equation

ing heavy rolling reductions. In this texture, the cube axes are extremely well aligned along the sheet axes,

ε D ˇt 1/3 C (7.50) and its behaviour resembles that of a single crystal. It is thus clear that cube-oriented grains or sub-grains

extremely well. In contrast to transient creep, steady- must have a very high initial growth rate in order to

state creep increases markedly with both temperature

246 Modern Physical Metallurgy and Materials Engineering and stress. At constant stress the dependence on tem-

increasing deformation, the sub-grain angle increases perature is given by

while the dislocation density within them remains con-

ε P ss D (7.51)

stant. The climb process may, of course, be important in several different ways. Thus, climb may help a glis-

where Q is the activation energy for steady-state creep, sile dislocation to circumvent different barriers in the while at constant temperature the dependence on stress

structure such as a sessile dislocation, or it may lead to (compensated for modulus E) is

the annihilation of dislocations of opposite sign on dif- ε P ss D n

ferent glide planes. Moreover, because creep-resistant materials are rarely pure metals, the climb process may

Steady-state creep is therefore described by the also be important in allowing a glissile dislocation to equation

get round a precipitate or move along a grain bound- ε P ss

ary. A comprehensive analysis of steady-state creep,

based on the climb of dislocations, has been given by The basic assumption of the mechanism of steady-

Weertman.

state creep is that during the creep process the rate The activation energy for creep Q may be obtained experimentally by plotting ln Pε ss versus 1/T, as shown sufficiently fast to balance the rate of work hardening

in Figure 7.61. Usually above 0.5T m , Q corresponds hD⊲

To prevent work hardening, both the screw and edge parts of a glissile dislocation loop must be able to escape from tangled or piled-up regions. The edge dislocations will, of course, escape by climb, and since this process requires a higher activation energy than cross-slip, it will be the rate-controlling process in steady-state creep. The rate of recovery is governed by the rate of climb, which depends on diffusion and stress such that

where D is a diffusion coefficient and the stress term arises because recovery is faster, the higher the stress level and the closer dislocations are together. The work-hardening rate decreases from the initial rate h

with increasing stress, i.e. h D h 0 q

, thus

ε P ss

n D (7.55)

where B⊲D A/h 0 ⊳ is a constant and n⊲D p C q⊳ is the stress exponent.

The structure developed in creep arises from the simultaneous work-hardening and recovery. The dis-

tion network gets finer, since dislocation spacing is 1/2 . At the same time, the dislo-

cations tend to reduce their strain energy by mutual annihilation and rearrange to form low-angle bound- aries and this increases the network spacing. Straining then proceeds at a rate at which the refining action just balances the growth of the network by recov-

Figure 7.61 Log Pε versus 1/T for (a) Ni –Al 2 O 3 , ery; the equilibrium network size being determined by

(b) Ni –67Co–Al 2 O 3 , showing the variation in activation the stress. Although dynamical recovery can occur by

energy above and below 0 .5 T m (after Hancock, Dillamore cross-slip, the rate-controlling process in steady-state

and Smallman, 1972) .

creep is climb whereby edge dislocations climb out of their glide planes by absorbing or emitting vacancies;

1 Sub-grains do not always form during creep and in some the activation energy is therefore that of self-diffusion.

metallic solid solutions where the glide of dislocations is Structural observations confirm the importance of the

restrained due to the dragging of solute atoms, the recovery process to steady-state creep. These show that

steady-state substructure is essentially a uniform distribution sub-grains form within the original grains and, with

of dislocations.

Mechanical behaviour of materials 247

Figure 7.62 Variation in activation energy Q with temperature for aluminium .

Figure 7.64 Schematic diagram showing influence of stress on diffusion-compensated steady-state creep .

they may (1) slide past each other or (2) create vacan- cies. Both processes involve an activation energy for diffusion and therefore may contribute to steady-state creep.

Grain boundary sliding during creep was inferred initially from the observation of steps at the boundaries, but the mechanism of sliding can be

demonstrated on bi-crystals. Figure 7.65 shows a good (b) Ni –67 Co–Al 2 O 3 (after Hancock, Dillamore and

example of grain boundary movement in a bi-crystal Smallman, 1972) .

–Al 2 O 3 ,

of tin, where the displacement of the straight grain boundary across its middle is indicated by marker

to the activation energy for self-diffusion E SD , in scratches. Grain boundaries, even when specially agreement with the climb theory, but below 0.5T m ,

produced for bi-crystal experiments, are not perfectly Q<E SD , possibly corresponding to pipe diffusion.

straight, and after a small amount of sliding at the Figure 7.62 shows that three creep regimes may be

boundary interface, movement will be arrested by identified and the temperature range where Q D E SD

protuberances. The grains are then locked, and the can be moved to higher temperatures by increasing

rate of slip will be determined by the rate of plastic the strain rate. Equation (7.55) shows that the stress

flow in the protuberances. As a result, the rate of exponent n can be obtained experimentally by plot-

slip along a grain boundary is not constant with ting ln Pε ss

time, because the dislocations first form into piled- n³

4. While n is generally about 4 for dislocation up groups, and later these become relaxed. Local creep, Figure 7.64 shows that n may vary consider-

relaxation may be envisaged as a process in which ably from this value depending on the stress regime;

the dislocations in the pile-up climb towards the at low stresses (i.e. regime I) creep occurs not by dis-

boundary. In consequence, the activation energy for location glide and climb but by stress-directed flow of

grain boundary slip may be identified with that for vacancies.

steady-state creep. After climb, the dislocations are spread more evenly along the boundary, and are thus

7.9.2 Grain boundary contribution to creep

able to give rise to grain boundary migration, when sliding has temporarily ceased, which is proportional

In the creep of polycrystals at high temperatures the

to the overall deformation.

grain boundaries themselves are able to play an impor-

A second creep process which also involves the tant part in the deformation process due to the fact that

grain boundaries is one in which the boundary acts

248 Modern Physical Metallurgy and Materials Engineering

Figure 7.65 Grain boundary sliding on a bi-crystal tin (after Puttick and King, 1952) .

Figure 7.66 Schematic representation of Herring–Nabarro creep; with c T >c c vacancies flow from the tensile faces to the longitudinal faces (a) to produce creep as shown in (b) .

as a source and sink for vacancies. The mechanism On a tensile face AB the stress exerts a force depends on the migration of vacancies from one side

2 2/3 ) on each surface atom and so does of a grain to another, as shown in Figure 7.66, and

2 ð b each time an atom moves forward is often termed Herring–Nabarro creep, after the two

one atomic spacing b (or  1/3 ) to create a vacancy. workers who originally considered this process. If, in

The energy of vacancy formation at such a face is thus reduced to ⊲E f 3 ⊳ and the concentration of transported from faces BC and AD to the faces AB and

vacancies in equilibrium correspondingly increased DC the grain creeps in the direction of the stress. To

f 3 ⊳/kT ]Dc 0 3 /kT⊳ . The transport atoms in this way involves creating vacancies

to c

vacancy concentration on the compressive faces will on the tensile faces AB and DC and destroying them

be reduced to c c Dc 0 3 /kT⊳ . Vacancies will on the other compressive faces by diffusion along the

therefore flow down the concentration gradient, and paths shown.

the number crossing a face under tension to one under

Mechanical behaviour of materials 249 compression will be given by Fick’s law as

v d 2 ⊲c T

c ⊳/˛d where D v is the vacancy diffusivity and ˛ relates

to the diffusion length. Substituting for c T ,c c and

D D ⊲D v c 0 b 3 ⊳ leads to

3 /kT⊳/˛b 3

Each vacancy created on one face and annihilated on

the other produces a strain ε D b 3 /d 3 , so that the creep

3 /d 3 ⊳ . At high temperatures and low stresses this reduces to ε P

D 3 /˛d 2 kT D B

2 kT (7.56)

where the constant B ¾ 10. In contrast to dislocation creep, Herring–Nabarro

creep varies linearly with stress and occurs at T ³ 0.8T m

6 N/m 2 . The temperature range over which vacancy-diffusion creep is significant can be

extended to much lower temperatures (i.e. T ³ 0.5T m ) if the vacancies flow down the grain boundaries rather than through the grains. Equation (7.56) is then mod- ified for Coble or grain boundary diffusion creep, and is given by

ε P Coble DB c D gb 3 (7.57)

where ω is the width of the grain boundary. Under such conditions (i.e. T ³ 0.5 to 0.6T m and low stresses) dif- fusion creep becomes an important creep mechanism in

a number of high-technology situations, and has been clearly identified in magnesium-based canning materi- als used in gas-cooled reactors.