Dislocation source operation

6.4.3 Dislocation source operation

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, etc.) and the strain rate imposes a

particular velocity on the mobile dislocation population. In a crystal of dimensions L 1 ×L 2 × 1 cm, shown in Figure 6.19, a dislocation with velocity v moves through the crystal in time t =L 1 / v and produces a shear strain b/L 2 , i.e. the strain rate is bv/L 1 L 2 . If the density of glissible dislocations is ρ , the total number of dislocations which become mobile in the crystal is ρL 1 L 2 and the overall strain rate is thus given by

(6.6) L 2 L 1

ρ L 1 L 2 = ρbv.

Mechanical properties I 307 (iv)

(iv) (iii)

/m

(iii) (ii)

/m

(ii) Yield stress

Yield stress

(i) (i) T/T m

Strain rate

(b) Figure 6.17 Variation of yield stress with: (a) temperature and (b) strain rate for crystals

(a)

with (i) fcc, (ii) bcc, (iii) ionic-bonded and (iv) covalent-bonded structure.

Dislocation loop expanding in a non-basal plane

Non-basal plane

Partial dislocation in the basal plane

Basal plane

Stacking fault

Figure 6.18 Dissociation in the basal plane of a screw dislocation moving on a non-basal glide plane.

Figure 6.19 Shear produced by gliding dislocations.

308 Physical Metallurgy and Advanced Materials

Node Node Node Node A B A B A B A B A B

(5) Figure 6.20 Successive stages in the operation of a Frank–Read source. The plane of the paper

is assumed to be the slip plane. At conventional strain rates (e.g. 1 s −1 ) the dislocations would be moving at quite moderate speeds of a

few cm s −1 if the mobile density ≈10 7 cm −2 . During high-speed deformation the velocity approaches the limiting velocity. The shear strain produced by these dislocations is given by

γ = ρb¯x, (6.7) where ¯x is the average distance a dislocation moves. If the distance x ≃ 10 −4 cm (the size of an average

sub-grain) the maximum strain produced by ρ ≈ 10 7 is about (10 7 × 3 × 10 −8 × 10 −4 ), which is only

a fraction of 1%. In practice, shear strains >100% can be achieved, and hence to produce these large strains many more dislocations than the original ingrown dislocations are required. To account for the increase in number of mobile dislocations during straining the 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 6.20 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 ( ≈4 eV per atom plane threaded by a dislocation) the dislocation possesses

a line tension tending to make it shorten its length as much as possible (position 1, Figure 6.20). This line tension T is roughly equal to αμb 2 , where μ is the shear modulus, b the Burgers vector and α 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 line to decrease its radius of curvature further until it becomes semicircular (position 2). Beyond this point it has no equilibrium position so it will expand rapidly, rotating about the nodes and taking up the succession of forms indicated by 3, 4 and 5. Between stages 4 and 5 the two parts of the loop below AB meet and annihilate each other to form a complete dislocation loop, which expands into the slip plane and

a new line source between A and B. The sequence is then repeated and one unit of slip is produced by each loop that is generated. To operate the Frank–Read source the force applied must be sufficient to overcome the restor- ing force on the dislocation line due to its line tension. Referring to Figure 6.21, this would be 2T dθ/2 > τbldθ/2, and if T

∼ μb 2 / 2 the stress to do this is about μb/l, where μ and b have their usual meaning and l is the length of the Frank–Read source; the substitution of typical

values (μ = 4 × 10 10 Nm −2 ,b = 2.5 × 10 −10 m and l = 10 −6 m) into this estimate shows that a critical shear stress of about 10 MPa is required. This value is somewhat less than, but of the same order as, that observed for the yield stress of virgin pure metal single crystals. Another source mechanism involves multiple cross-slip, as shown in Figure 6.22. It depends on the Frank–Read principle but does not require a dislocation segment to be anchored by nodes. Thus,

Mechanical properties I 309

du l/2 Figure 6.21 Geometry of Frank–Read source used to calculate the stress to operate.

Primary slip planes

Figure 6.22 Cross-slip multiplication source.

if part of a moving screw dislocation undergoes double cross-slip the two pieces of edge dis- location on the cross-slip plane effectively act as anchoring points for a new source. The loop expanding on the slip plane parallel to the original plane may operate as a Frank–Read source, and any loops produced may in turn cross slip and become a source. This process therefore not only increases the number of dislocations on the original slip plane, but also causes the slip band to widen.

The concept of the dislocation source accounts for the observation of slip bands on the surface of deformed metals. The amount of slip produced by the passage of a single dislocation is too small to be observable as a slip line or band under the light microscope. To be resolved it must be at least 300 nm in height and hence ≈1000 dislocations must have operated in a given slip band. Moreover, in general, the slip band has considerable width, which tends to support the opera- tion of the cross-glide source as the predominant mechanism of dislocation multiplication during straining.

Worked example

A Frank–Read source is operated by an applied stress of magnitude 10 −4 μ , where μ is the shear modulus. If the limiting speed of a dislocation is 10 3 ms −1 , show that the source could nucleate a slip band which is observed in the light microscope to form in about 10 −6 s.

Solution

To observe a slip band in the light microscope, which has a resolution of approximately the wavelength of light, requires about 1000 dislocations to have emanated from the source.

310 Physical Metallurgy and Advanced Materials

No upper

yield point B

Stress

immediately

First test

Unload and retest

Unload, age and retest

Strain

(b) Figure 6.23 Schematic representation of strain ageing (a) and Lüders band formation (b). The stress to operate the source is τ = μb/ℓ, where μ is the shear modulus, b the Burgers vector

(a)

and ℓ the source length. Time to produce one dislocation loop takes t ∼ ℓ/v, where v is the dislocation velocity. Thus, to nucleate a slip band of 1000 dislocations requires total time