Yield points and crystal structure

6.4.5 Yield points and crystal structure

The characteristic feature of discontinuous yielding is that at the yield point the specimen goes from a condition where the availability of mobile dislocations is limited to one where they are in abundance, the increase in mobile density largely arising from dislocation multiplication at the high stress level.

A further feature is that not all the dislocations 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 microstrain).

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 mm −3 ) and possesses a high P–N stress.

When these materials are pulled in a tensile test, the overall strain rate ˙γ imposed on the specimen by the machine has to be matched by the motion of dislocations according to the relation ˙γ = ρbv.

Mechanical properties I 313

Stress (kg mm

Stress (GN m

0 2 4 6 8 10 Percentage elongation

Figure 6.24 Yield point in a copper whisker.

However, because ρ is small the individual dislocations are forced to move at a high speed v, which is only 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. Because of the increased value of the term ρ, a lower 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 dislocation–dislocation interactions caused by the increased ρ 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 independent of velocity up to high speeds. If such metals are to show a yield point, the density of mobile dislocations must be reduced virtually to zero. This can be achieved as shown in Figure 6.24 by the tensile testing of whisker crystals which are very perfect. Yielding begins at the stress required to create dislocations in the perfect lattice, and the upper yield stress approaches the theoretical yield strength. Following multiplication, the stress for glide of these dislocations is several orders of magnitude lower.

Bcc transition metals such as iron are intermediate in their plastic behavior between the fcc metals and diamond cubic Si and Ge. Because of the significant P–N stress these bcc metals are capable of exhibiting a sharp yield point, even when the initial mobile dislocation density is not zero, as shown by the calculated curves of Figure 6.25. However, in practice, the dislocation density of well-annealed

pure metals is about 10 10 mm −3 and too high for any significant yield drop without an element of dislocation locking by carbon atoms.

It is evident that discontinuous yielding can be produced in all the common metal structures provided the appropriate solute elements are present and correct testing procedure adopted. The effect is particularly strong in the bcc metals and has been observed in α-iron, molybdenum, niobium, vanadium and β-brass, each containing a strongly interacting interstitial solute element. The hexagonal metals (e.g. cadmium and zinc) can also show the phenomenon provided interstitial nitrogen atoms are added. The copper- and aluminum-based fcc alloys also exhibit yielding behavior, but often to

a lesser degree. In this case it is substitutional atoms (e.g. zinc in α-brass and copper in aluminum alloys) which are responsible for the phenomenon (see Section 6.4.7).

314 Physical Metallurgy and Advanced Materials

Stress (kg mm 5 Stress (GN m

Plastic strain 0 0.5%

Figure 6.25 Calculated stress–strain curves showing influence of initial dislocation density on the yield drop in iron for n

= 35 with: (i) 10 1 cm −2 , (ii) 10 3 cm −2 , (iii) 10 5 cm −2 and (iv) 10 7 cm −2 (after Hahn, 1962).

Worked example

A low-carbon steel exhibits a yield point when tensile tested at a strain rate of 1 s −1 . If the density of mobile dislocations before and after the yield phenomenon is 10 11 and 10 14 m −2 respectively, estimate:

(i) The dislocation velocity at the upper yield point, and (ii) The magnitude of the yield drop.

(Take the lattice parameter of the alloy to be 0.28 nm and the stress dependence of the dislocation velocity to have an exponent n = 35.)

Solution

(i) Plastic strain rate is √ ˙ε = φbρ m v. Thus, at the upper yield point (uyp), √

1 = 0.5 × 3 2 × 2.8 × 10 −10 × 10 11 × v, since b = ( 3/2)a and φ ∼ 0.5. ∴ v

= 8.06 × 10 −2 ms −1 . (ii) The average velocity v = (σ/σ o ) n . Since ˙ε is the same at both upper and lower yield points, then

0.5 11 /σ ) 35 14 × 10 35 × (σ uyp o = 0.5 × 10 × (σ lyp /σ o ) (σ uyp /σ lyp ) 35 = 10 3 .

The ratio of upper to lower yield points, σ uyp /σ lyp