Dislocation behaviour during plastic deformation now been made in some crystals by means of the etch

7.4 Dislocation behaviour during plastic deformation now been made in some crystals by means of the etch

pitting technique; the results of such an experiment are

7.4.1 Dislocation mobility

shown in Figure 7.16. Edge dislocations move faster than screws, because of the frictional drag of jogs on

The ease with which crystals can be plastically screws, and the velocity of both varies rapidly with deformed at stresses many orders of magnitude less

remarkable, and due to the mobility of dislocations. Figure 7.15a shows that as a dislocation glides through the lattice it moves from one symmetrical lattice position to another and at each position the dislocation is in neutral equilibrium, because the atomic forces acting on it from each side are balanced. As the dislocation moves from these symmetrical lattice positions some imbalance of atomic forces does exist, and an applied stress is required to overcome this lattice friction. As shown in Figure 7.15b, an intermediate displacement of the dislocation also leads to an approximately balanced force system.

The lattice friction depends rather sensitively on the dislocation width w and has been shown by Peierls and Nabarro to be given by

for the shear of a rectangular lattice of interpla-

The friction stress is therefore often referred to as the Peierls–Nabarro stress. The two opposing factors affecting w are (1) the elastic energy of the crystal, which is reduced by spreading out the elastic strains, and (2) the misfit energy, which depends on the num- ber of misaligned atoms across the slip plane. Metals with close-packed structures have extended disloca- tions and hence w is large. Moreover, the close-packed planes are widely spaced with weak alignment forces between them (i.e. have a small b/a factor). These metals have highly mobile dislocations and are intrin- sically soft. In contrast, directional bonding in crystals tends to produce narrow dislocations, which leads to intrinsic hardness and brittleness. Extreme examples are ionic and ceramic crystals and the covalent mate- rials such as diamond and silicon. The bcc transition

Figure 7.16 Stress dependence of the velocity of edge and metals display intermediate behaviour (i.e. intrinsically

screw dislocations in lithium fluoride (from Johnston and ductile above room temperatures but brittle below).

Gilman, 1959; courtesy of the American Institute of Physics) .

208 Modern Physical Metallurgy and Materials Engineering

Figure 7.17 (a) Correlation between stress to cause dislocation motion and the macro-yield stresses of crystals. (b) Edge dislocation motions in Fe-3% Si crystals (after Stein and Low, 1960; courtesy of the American Institute of Physics) .

of moving dislocations to hinder their motion. Such

strengthening mechanisms increase the stress neces- speed and n is an index which varies for different

0 is the stress for unit

sary to produce a given finite dislocation velocity in a materials. At high stresses the velocity may approach

similar way to that found by lowering the temperature.

the speed of elastic waves ³10 3 m/s. The index n is

usually low ⊲<10⊳ for intrinsically hard, covalent crys-

7.4.2 Variation of yield stress with

tals such as Ge, ³40 for bcc crystals, and high ⊲³200⊳

temperature and strain rate

for intrinsically soft fcc crystals. It is observed that a critical applied stress is required to start the disloca-

The high Peierls–Nabarro stress, which is associated tions moving and denotes the onset of microplasticity.

with materials with narrow dislocations, gives rise to a

A macroscopic tensile test is a relatively insensitive short-range barrier to dislocation motion. Such barriers measure of the onset of plastic deformation and the

are effective only over an atomic spacing or so, hence yield or flow stress measured in such a test is related

thermal activation is able to aid the applied stress in not to the initial motion of an individual dislocation but

overcoming them. Thermal activation helps a portion to the motion of a number of dislocations at some finite

of the dislocation to cross the barrier after which glide velocity, e.g. ¾10 nm/s as shown in Figure 17.17a.

then proceeds by the sideways movement of kinks. Decreasing the temperature of the test or increasing the

(This process is shown in Figure 7.29, Section 7.4.8.) strain-rate increases the stress level required to produce

Materials with narrow dislocations therefore exhibit the same finite velocity (see Figure 7.17b), i.e. dis-

a significant temperature-sensitivity; intrinsically hard placing the velocity–stress curve to the right. Indeed,

materials rapidly lose their strength with increasing hardening the material by any mechanism has the

temperature, as shown schematically in Figure 7.18a. same effect on the dislocation dynamics. This obser-

In this diagram the (yield stress/modulus) ratio is plot- vation is consistent with the increase in yield stress

ted against T/T m to remove the effect of modulus with decreasing temperature or increasing strain-rate.

which decreases with temperature. Figure 7.18b shows Most metals and alloys are hardened by cold working

that materials which exhibit a strong temperature- or by placing obstacles (e.g. precipitates) in the path

dependent yield stress also exhibit a high strain-rate

Mechanical behaviour of materials 209

Figure 7.18 Variation of yield stress with (a) temperature, (b) strain-rate for crystals with (i) fcc, (ii) bcc, (iii) ionic-bonded, (iv) covalent-bonded structure .

sensitivity, i.e. the higher the imposed strain rate, the higher the yield stress. This arises because thermal activation is less effective at the faster rate of defor- mation.

In bcc metals a high lattice friction to the move- ment of a dislocation may arise from the dissocia- tion of a dislocation on several planes. As discussed in Chapter 4, when a screw dislocation with Burgers vector a/2[1 1 1] lies along a symmetry direction it can dissociate on three crystallographically equivalent planes. If such a dissociation occurs, it will be nec- essary to constrict the dislocation before it can glide in any one of the slip planes. This constriction will

be more difficult to make as the temperature is low- ered so that the large temperature dependence of the yield stress in bcc metals, shown in Figure 7.18a and also Figure 7.30, may be due partly to this effect. In fcc metals the dislocations lie on f1 1 1g planes, and although a dislocation will dissociate in any given ⊲ 1 1 1⊳ plane, there is no direction in the slip plane

along which the dislocation could also dissociate on Figure 7.19 Dissociation in the basal plane of a screw dislocation moving on a non-basal glide plane . other planes; the temperature-dependence of the yield stress is small as shown in Figure 7.18a. In cph metals the dissociated dislocations moving in the basal plane will also have a small Peierls force and be glissile with low temperature-dependence. However, screw dislo- cations moving on non-basal planes (i.e. prismatic and pyramidal planes) may have a high Peierls force because they are able to extend in the basal plane as shown in Figure 7.19. Hence, constrictions will once again have to be made before the screw dislocations

can advance on non-basal planes. This effect con- tributes to the high critical shear stress and strong temperature-dependence of non-basal glide observed in this crystal system, as mentioned in Chapter 4.