Yield points and crystal structure

7.4.5 Yield points and crystal structure

The characteristic feature of discontinuous yielding is that at the yield point the specimen goes from

Mechanical behaviour of materials 213

a condition where the availability of mobile dislo- cations is limited to one where they are in abun- dance, the increase in mobile density largely aris- ing from dislocation multiplication at the high stress level. A further feature is that not all the dislo- cations have to be immobilized to observe a yield drop. Indeed, this is not usually possible because specimen handling, non-axial loading, scratches, etc. give rise to stress concentrations that provide a small local density of mobile dislocations (i.e. pre-yield microtrain).

For materials with a high Peierls–Nabarro (P–N) stress, yield drops may be observed even when they possess a significant mobile dislocation density. A common example is that observed in silicon; this is an extremely pure material with no impurities to lock dislocations, but usually the dislocation density is quite

modest ⊲10 7 m/m 3 ⊳ and possesses a high P–N stress. When these materials are pulled in a tensile test the overall strain rate P imposed on the specimen by the machine has to be matched by the motion of

Figure 7.25 Yield point in a copper whisker .

attained at a high stress level (the upper yield stress) because of the large P–N stress. As the dislocations glide at these high speeds, rapid multiplication occurs and the mobile dislocation density increases rapidly.

average velocity of dislocations is then required to maintain a constant strain rate, which means a lower glide stress. The stress that can be supported by the specimen thus drops during initial yielding to the lower yield point, and does not rise again until the disloca- tion–dislocation interactions caused by the increased

produce a significant work-hardening. In the fcc metals, the P–N stress is quite small and

the stress to move a dislocation is almost indepen- dent of velocity up to high speeds. If such metals are to show a yield point, the density of mobile disloca-

Figure 7.26 Calculated stress–strain curves showing tions must be reduced virtually to zero. This can be

influence of initial dislocation density on the yield drop in achieved as shown in Figure 7.25 by the tensile test-

iron for n D 35 with (i) 10 1 cm , (ii) 10 3 cm , ing of whisker crystals which are very perfect. Yielding

and (iv) 10 7 cm (after Hahn, 1962; begins at the stress required to create dislocations in the

(iii) 10 5 cm

courtesy of Pergamon Press) .

perfect lattice, and the upper yield stress approaches the theoretical yield strength. Following multiplica- tion, the stress for glide of these dislocations is several

It is evident that discontinuous yielding can be orders of magnitude lower.

produced in all the common metal structures provided Bcc transition metals such as iron are intermediate

the appropriate solute elements are present, and cor- in their plastic behaviour between the fcc metals and

rect testing procedure adopted. The effect is particu- diamond cubic Si and Ge. Because of the significant

larly strong in the bcc metals and has been observed P–N stress these bcc metals are capable of exhibit-

in ˛-iron, molybdenum, niobium, vanadium and ˇ- ing a sharp yield point even when the initial mobile

brass each containing a strongly interacting interstitial dislocation density is not zero, as shown by the cal-

solute element. The hexagonal metals (e.g. cadmium culated curves of Figure 7.26. However, in practice,

and zinc) can also show the phenomenon provided the dislocation density of well-annealed pure metals

interstitial nitrogen atoms are added. The copper- is about 10 10 m/m 3 and too high for any significant

and aluminium-based fcc alloys also exhibit yielding yield drop without an element of dislocation locking

behaviour but often to a lesser degree. In this case it is by carbon atoms.

substitutional atoms (e.g. zinc in ˛-brass and copper in

214 Modern Physical Metallurgy and Materials Engineering aluminium alloys) which are responsible for the phe-

APB-locking model will give rise to sharp yield- nomenon (see Section 7.4.7).

ing because the energy required by the lead dislo- cation in creating sharp APB is greater than that released by the trailing dislocation initially moving

across diffuse APB. Experimental evidence favours Discontinuous yield points have been observed in a

7.4.6 Discontinuous yielding in ordered alloys

the APB-model and weak-beam electron microscopy wide variety of A 3 B-type alloys. Figure 7.27 shows

(see Figure 7.28) shows that the superdislocation sep-

aration for a shear APB corresponds to an energy of The addition of Al speeds up the kinetics of ordering

the development of the yield point in Ni 3 Fe on ageing.

48 š 5 mJ/m 2 , whereas a larger dislocation separation and therefore the onset of the yield point. Ordered

corresponding to an APB energy of 25 š 3 mJ/m 2 was materials deform by superdislocation motion and the

observed for a strained and aged Cu 3 Au. link between yield points and superdislocations is con-

firmed by the observation that in Cu 3 Au, for example,

7.4.7 Solute –dislocation interaction

a transition from groups of single dislocations to more Iron containing carbon or nitrogen shows very marked randomly arranged superdislocation pairs takes place

yield point effects and there is a strong elastic interac- at ¾S D 0.7 (see Chapter 4) and this coincides with

tion between these solute atoms and the dislocations. the onset of a large yield drop and rapid rise in work

The solute atoms occupy interstitial sites in the lat- hardening.

tice and produce large tetragonal distortions as well Sharp yielding may be explained by at least two

as large-volume expansions. Consequently, they can mechanisms, namely (1) cross-slip of the superdislo-

interact with both shear and hydrostatic stresses and cation onto the cube plane to lower the APB energy

can lock screw as well as edge dislocations. Strong effectively pinning it and (2) dislocation locking by

yielding behaviour is also expected in other bcc met- rearrangement of the APB on ageing. The shear APB

als, provided they contain interstitial solute elements. between a pair of superdislocations is likely to be

On the other hand, in the case of fcc metals the energetically unstable since there are many like bonds

arrangement of lattice positions around either intersti- across the interface and thermal activation will mod-

tial or substitutional sites is too symmetrical to allow a ify this sharp interface by atomic rearrangement. This

solute atom to produce an asymmetrical distortion, and

Figure 7.27 Development of a yield point with ageing at 490 ° C for the times indicated. (a) Ni 3 Fe, (b) Ni 3 Fe C 5 % Al; the tests are at room temperature .

Mechanical behaviour of materials 215

Figure 7.28 Weak-beam micrographs showing separation of superdislocation partials in Cu 3 Au . (a) As deformed, (b) after ageing at 225 ° C (after Morris and Smallman, 1975) .

the atmosphere locking of screw dislocations, which requires a shear stress interaction, would appear to be impossible. Then by this argument, since the screw

the expression

dislocations are not locked, a drop in stress at the yield point should not be observed. Nevertheless, yield V points are observed in fcc materials and one reason

⊲ 7.10⊳ for this is that unit dislocations in fcc metals dissoci- ate into pairs of partial dislocations which are elasti-

This is the interaction energy at a point whose polar cally coupled by a stacking fault. Moreover, since their

coordinates with respect to the centre of the dislocation Burgers vectors intersect at 120 ° there is no orienta-

tion of the line of the pair for which both can be pure screws. At least one of them must have a substantial

⊲ v > 0⊳, and negative on the lower side, which edge component, and a locking of this edge component

agrees with the qualitative picture of a large atom being by hydrostatic interactions should cause a locking of

repelled from the compressed region and attracted into the pair although it will undoubtedly be weaker.

the expanded one.

In its quantitative form the theory of solute atom It is expected that the site for the strongest bind- locking has been applied to the formation of an

ing energy V max

atmosphere around an edge dislocation due to hydro- r 0 static interaction. Since hydrostatic stresses are scalar

equation (7.10) we obtain A ' 3 ð 10 Nm 2 and quantities, no knowledge is required in this case of

V max ' 1 eV for carbon or nitrogen in ˛-iron. This the orientation of the dislocation with respect to the

value is almost certainly too high because of the limi- interacting solute atom, but it is necessary in calcu-

tations of the interaction energy equation in describing

conditions near the centre of a dislocation, and a more have shown that if the introduction of a solute atom

lating shear stresses interactions. 1 Cottrell and Bilby

realistic value obtained from experiment (e.g. internal causes a volume change v at some point in the lattice

friction experiments) is V max ' 1 to 2 3 4 eV. For a sub- where the hydrostatic pressure of the stress field is p,

stitutional solute atom such as zinc in copper v is the interaction energy is

not only smaller but also easier to calculate from lat- tice parameter measurements. Thus, if r and r⊲1 C ε⊳

are the atomic radii of the solvent and solute, respec- where K is the bulk modulus and  is the local dilata-

V D pv D Kv

tively, where ε is the misfit value, the volume change 

3 ε and equation (7.10) becomes

7.11⊳ with a screw dislocation since there is no dilatation around

⊲ To a first approximation a solute atom does not interact

the screw; a second-order dilatation exists however, which gives rise to a non-zero interaction falling off with distance

b D 2.55 ð 10

m, r 0

2 and ε D 0.06, we find A '

Nm , which gives a much lower binding anisotropic elasticity will lead to first-order size effects even

from the dislocation according to 1/r 2 . In real crystals,

energy, V

max D 8 eV.

with screw dislocations and hence a substantial interaction The yield phenomenon is particularly strong in iron is to be expected.

because an additional effect is important; this concerns

216 Modern Physical Metallurgy and Materials Engineering the type of atmosphere a dislocation gathers round

and from equation (7.13) it can be shown that a 0.1 at. itself which can be either condensed or dilute. Dur-

% alloy has a condensation temperature T c D 250 K. ing the strain-ageing process migration of the solute

Copper-based alloys, on the other hand, usually form atoms to the dislocation occurs and two important

extensive solid solutions, and, consequently, concen- cases arise. First, if all the sites at the centre of the

trated alloys may exhibit strong yielding phenomena. dislocation become occupied the atmosphere is then

The best-known example is ˛-brass and, because said to be condensed; each atom plane threaded by the

V max ' 1 8 eV, a dilute alloy containing 1 at. % zinc dislocation contains one solute atom at the position

has a condensation temperature T c ' 300 K. At low of maximum binding together with a diffuse cloud of

zinc concentrations (1–10%) the yield point in brass other solute atoms further out. If, on the other hand,

is probably solely due to the segregation of zinc atoms equilibrium is established before all the sites at the

to dislocations. At higher concentrations, however, it centre are saturated, a steady state must be reached

may also be due to short-range order. in which the probability of solute atoms leaving the

centre can equal the probability of their entering it. The steady-state distribution of solute atoms around

7.4.8 Dislocation locking and temperature

the dislocations is then given by the relation The binding of a solute atom to a dislocation is short

C Dc 0 exp [V / kT ] range in nature, and is effective only over an atomic distance or so (Figure 7.29). Moreover, the dislocation

where c 0 is the concentration far from a dislocation, k line is flexible and this enables yielding to begin by is Boltzmann’s constant, T is the absolute temperature

throwing forward a small length of dislocation line, and c the local impurity concentration at a point near

only a few atomic spacings long, beyond the position the dislocation where the binding energy is V. This

marked x 2 . The applied stress then separates the rest is known as the dilute or Maxwellian atmosphere.

of the dislocation line from its anchorage by pulling Clearly, the form of an atmosphere will be governed

the sides of this loop outward along the dislocation by the concentration of solute atoms at the sites of

line, i.e. by double kink movement. Such a breakaway maximum binding energy, V max and for a given alloy

process would lead to a yield stress which depends (i.e. c 0 and V max fixed) this concentration will be

sensitively on temperature, as shown in Figure 7.30a.

c It is observed, however, that k y , the grain-size depen-

V max Dc 0 exp ⊲V max / k T⊳

dence parameter in the Hall –Petch equation, in most

annealed bcc metals is almost independent of tem- depends only on the temperature, and as the temper-

as long as c V max is less than unity. The value of c V max

perature down to the range (<100 K) where twinning ature is lowered c V max will eventually rise to unity.

occurs, and that practically all the large temperature- By definition the atmosphere will then have passed

i (see Figure 7.30b). To explain from a dilute to a condensed state. The temperature at

this observation it is argued that when locked disloca- which this occurs is known as the condensation tem-

tions exist initially in the material, yielding starts by perature T c , and can be obtained by substituting the

unpinning them if they are weakly locked (this corre- value c V max

D 1 in equation (7.12) when sponds to the condition envisaged by Cottrell –Bilby), but if they are strongly locked it starts instead by

T c DV max /k ln⊲1/c 0 ⊳

(7.13) Substituting the value of V max for iron, i.e. 1 2 eV in this

equation we find that only a very small concentration of carbon or nitrogen is necessary to give a condensed atmosphere at room temperature, and with the usual concentration strong yielding behaviour is expected up to temperatures of about 400 ° C.

In the fcc structure although the locking between a solute atom and a dislocation is likely to be weaker, condensed atmospheres are still possible if this weak- ness can be compensated for by sufficiently increasing the concentration of the solution. This may be why examples of yielding in fcc materials have been mainly obtained from alloys. Solid solution alloys of alu- minium usually contain less than 0.1 at. % of solute element, and these show yielding in single crystals only at low temperature (e.g. liquid nitrogen tempera-

C) whereas supersaturated alloys show evi- dence of strong yielding even in polycrystals at room

Figure 7.29 Stress–displacement curve for the breakaway temperature; copper dissolved in aluminium has a mis-

of a dislocation from its atmosphere (after Cottrell, 1957; fit value ε ' 0.12 which corresponds to V max

4 eV,

courtesy of the Institution of Mechanical Engineers) .

Mechanical behaviour of materials 217

Figure 7.30 Variation of lower yield stress with (a) temperature and (b) grain size, for low-carbon steel and niobium; the curve for nickel is shown in (a) for comparison (after Adams, Roberts and Smallman, 1960; Hull and Mogford, 1958) .

the creation of new dislocations at points of stress solvent matrix and the region near a solute. Such an concentration. This is an athermal process and thus

inhomogeneity interaction has been analysed for both k y is almost independent of temperature. Because of

a rigid and a soft spherical region; the former cor- the rapid diffusion of interstitial elements the con-

responds to a relatively hard impurity atom and the ventional annealing and normalizing treatments should

latter to a vacant lattice site. The results indicate that commonly produce strong locking. In support of this

the interaction energy is of the form B/r 2 where B is a theory, it is observed that k y is dependent on tempera-

constant involving elastic constants and atomic size. It ture in the very early stages of ageing following either

is generally believed that the inhomogeneity effect is straining or quenching but on subsequent ageing k y

small for solute–dislocation interactions but dominates becomes temperature-independent. The interpretation

the size effect for vacancy–dislocation interaction. The of k y therefore depends on the degree of ageing.

kinetics of ageing support this conclusion. Direct observations of crystals that have yielded show that the majority of the strongly anchored dis-

7.4.10 Kinetics of strain-ageing

locations remain locked and do not participate in the yielding phenomenon. Thus large numbers of dislo-

Under a force F an atom migrating by thermal agi- cations are generated during yielding by some other mechanism than breaking away from Cottrell atmo-

addition to its random diffusion movements) in the spheres, and the rapid dislocation multiplication, which

direction of the F, where D is the coefficient of diffu- can take place at the high stress levels, is now con-

sion. The force attracting a solute atom to a dislocation sidered the most likely possibility. Prolonged ageing

is the gradient of the interaction energy dV/dr and tends to produce coarse precipitates along the disloca-

2 ⊳ . Thus atoms originally at a tion line and unpinning by bowing out between them

distance r from the dislocation reach it in a time given should easily occur before grain boundary creation.

approximately by

This unpinning process would also give k y independent of temperature. 3 kT/AD

7.4.9 Inhomogeneity interaction

After this time t the number of atoms to reach unit length of dislocation is

A different type of elastic interaction can exist which arises from the different elastic properties of the

2 c 0 0 [⊲AD/kT⊳t] 2/3

218 Modern Physical Metallurgy and Materials Engineering where c 0 is the solute concentration in uniform solution

deforms as if it were an unconstrained single crys- in terms of the number of atoms per unit volume. If

tal. This is not the case, however, and the fact that

the aggregate does not deform in this manner is indi- fraction of the original solute which has segregated to

is the density of dislocations ⊲cm/cm 3 ⊳ and f the

cated by the high yield stress of polycrystals compared the dislocation in time t then,

with that of single crystals. This increased strength

hardness of a grain caused by the presence of the grain This expression is valid for the early stages of ageing,

of polycrystals immediately poses the question–is the

boundary or by the orientation difference of the neigh- and may be modified to fit the later stages by allowing

bouring grains? It is now believed that the latter is for the reduction in the matrix concentration as ageing

the case but that the structure of the grain boundary proceeds, such that the rate of flow is proportional to

itself may be of importance in special circumstances the amount left in the matrix,

such as when brittle films, due to bismuth in copper

or cementite in steel, form around the grains or when ⊲ 2/3⊳t 2/3 ⊲

the grains slip past each other along their boundaries which when integrated gives

during high-temperature creep. The importance of the orientation change across a grain boundary to the pro-

1/3 g (7.15)

cess of slip has been demonstrated by experiments on ‘bamboo’-type specimens, i.e. where the grain bound-

This reduces to the simpler equation (7.14) when the aries are parallel to each other and all perpendicular to exponent is small, and is found to be in good agree-

the axis of tension. Initially, deformation occurs by slip ment with the process of segregation and precipitation

only in those grains most favourably oriented, but later on dislocations in several bcc metals. For carbon in

spreads to all the other grains as those grains which ˛ -Fe, Harper determined the fraction of solute atom

are deformed first, work harden. It is then found that still in solution using an internal friction technique

each grain contains wedge-shaped areas near the grain

boundary, as shown in Figure 7.31a, where slip does not operate, which indicates that the continuance of

this slope at a series of temperatures allows the acti- slip from one grain to the next is difficult. From these vation energy for the process to be determined from

observations it is natural to enquire what happens in a an Arrhenius plot. The value obtained for ˛-iron is

completely polycrystalline metal where the slip planes

84 kJ/mol which is close to that for the diffusion of must in all cases make contact with a grain bound- carbon in ferrite.

ary. It will be clear that the polycrystalline aggregate The inhomogeneity interaction is considered to be

must be stronger because, unlike the deformation of the dominant effect in vacancy–dislocation interac-

2 where B is a constant; this com- bamboo-type samples where it is not necessary to raise the stress sufficiently high to operate those slip planes

which made contact with a grain boundary, all the slip

be appropriate for the interstitial –dislocation interac- planes within any grain of a polycrystalline aggregate tion. It is convenient, however, to write the interaction

make contact with a grain boundary, but, nevertheless,

have to be operated. The importance of the grain size lowing the treatment previously used for the kinetics

and hence, fol-

of strain-ageing, the radial velocity of a point defect towards the dislocation is

V D ⊲D/kT⊳⊲nA/r nC 1 ⊳

The number of a particular point defect specie that reach the dislocation in time t is

0 [ADn⊲n C 2⊳/kT] 2/⊲nC2⊳ t 2/⊲nC2⊳

and when n D 2 then n⊲t⊳ / t 1/2 , and when n D 1, n⊲t⊳ / t 2/3 . Since the kinetics of ageing in quenched copper follow t 1/2 initially, the observations confirm the importance of the inhomogeneity interaction for vacancies.