Dislocation mobility

6.4.1 Dislocation mobility

The ease with which crystals can be plastically deformed at stresses many orders of magnitude less than the theoretical strength (τ t = µb/2πa) is quite remarkable, and due to the mobility of dislocations. Figure 6.14a 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 6.14b, 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

τ ≈ μ exp[−2π w/b] (6.5) for the shear of a rectangular lattice of interplanar spacing a with w = μb/2π(1 − ν)τ t = a/(1 − ν).

The friction stress is therefore often referred to as the Peierls–Nabarro stress. The two opposing

304 Physical Metallurgy and Advanced Materials

Slip plane

(b) Figure 6.14 Diagram showing structure of edge dislocation during gliding from equilibrium

(a)

(a) to metastable position (b). 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 number of misaligned atoms across the slip plane. Metals with close-packed structures have extended dislocations 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 intrinsically 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 materials such as diamond and silicon. The bcc transition metals display intermediate behavior (i.e. intrinsically ductile above room temperatures but brittle below).

Direct measurements of dislocation velocity v have now been made in some crystals by means of the etch-pitting technique; the results of such an experiment are shown in Figure 6.15. Edge dislocations move faster than screws, because of the frictional drag of jogs on screws, and the velocity of both

varies rapidly with applied stress τ according to an empirical relation of the form v = (τ/τ 0 ) n , where τ 0 is the stress for unit speed and n is an index which varies for different materials. At high stresses the velocity may approach the speed of elastic waves

≈ 10 3 ms −1 . The index n is usually low (<10) for intrinsically hard, covalent crystals such as Ge, ≈40 for bcc crystals and high (≈200) for intrinsically

soft fcc crystals. It is observed that a critical applied stress is required to start the dislocations moving and denotes the onset of microplasticity. A macroscopic tensile test is a relatively insensitive measure of the onset of plastic deformation and the yield or flow stress measured in such a test is related not to the initial motion of an individual dislocation, but to the motion of a number of dislocations at some finite velocity, e.g. ∼10 nm s −1 , as shown in Figure 6.16a. Decreasing the temperature of the test or increasing the strain rate increases the stress level required to produce the same finite velocity (see Figure 6.16b), i.e. displacing the velocity–stress curve to the right. Indeed, hardening the material by any mechanism has the same effect on the dislocation dynamics. This observation is consistent with the increase in yield stress with decreasing temperature or increasing strain rate. Most metals and alloys are hardened by cold working or by placing obstacles (e.g. precipitates) in the path of moving dislocations to hinder their motion. Such strengthening mechanisms increase the stress necessary to produce a given finite dislocation velocity in a similar way to that found by lowering the temperature.