As we saw in Chapter 11, the stress required to make a crystalline material deform plastically is that needed to make the dislocations in it move. Their movement is resisted by (a) the intrinsic lattice resistance, and (b) the obstructing effect of obstacles (e.g., dissolved solute atoms, precipitates formed with undissolved solute atoms, or other dislocations). Diffusion of atoms can “unlock” dislocations from obstacles in their path, and the movement of these unlocked dislocations under the applied stress is what leads to dislocation creep.

How does this unlocking occur? Figure 23.1 shows a dislocation that cannot glide because a precipitate blocks its path. The glide force τb per unit length is balanced by the reaction f₀ from the precipitate. But unless the dislocation hits the precipitate at its mid-plane (an unlikely event) there is a component of force left over. It is the component τb tan θ, which tries to push the dislocation out of its slip plane.

The dislocation cannot glide upward by the shearing of atom planes—the atomic geometry is wrong—but the dislocation can move upward if atoms at the bottom of the half-plane are able to diffuse away (Figure 23.2). We have come across Fick’s law in which diffusion is driven by differences in concentration. A mechanical force can do exactly the same thing, and this is what leads to the diffusion of atoms away from the “loaded” dislocation, eating away its extra half-plane of atoms until it can clear the precipitate. The process is called “climb,” and since it requires diffusion, it can occur only when the temperature is above 0.3T_M or so. At the lower end of the creep régime (0.3–0.5T_M) core diffusion tends to be the dominant mechanism; at the higher end (0.5T_M–0.99T_M) it is bulk diffusion (Figure 23.2).

Climb unlocks dislocations from the precipitates that pin them and further slip (or “glide”) can then take place (Figure 23.3). Similar behavior takes place for pinning by solute, and by other dislocations. After a little glide, of course, the unlocked dislocations bump into the next obstacles, and the whole cycle repeats itself. This explains the progressive, continuous nature of creep. The role of diffusion, with diffusion coefficient

D=D0e–Q/RT

The dependence of creep rate on applied stress σ is due to the climb force: the higher σ, the higher the climb force τb tan θ, the more dislocations become unlocked per second, the more dislocations glide per second, and the higher is the strain rate.

Diffusion creep (giving linear-viscous creep)

As the stress is reduced, the rate of power-law creep (Equation (23.1)) falls quickly (remember n is between 3 and 8). But creep does not stop; instead, an alternative mechanism takes over. As Figure 23.4 shows, a polycrystal can extend in response to the applied stress, σ, by grain elongation; here, σ acts again as a mechanical driving force, but this time atoms diffuse from one set of the grain faces to the other, and dislocations are not involved. At high T/T_M, this diffusion takes place through the crystal itself, that is, by bulk diffusion.

The rate of creep is then obviously proportional to the diffusion coefficient D (refer to data in Table 22.1) and to the stress σ (because σ drives diffusion in the same way that dc/dx does in Fick’s law). The creep rate varies as 1/d² where d is the grain size (because when d gets larger, atoms have to diffuse further). Assembling these facts leads to the constitutive equation

ɛ˙ss=CDσd2=C′σe–Q/RTd2

Deformation mechanism diagrams

This competition between mechanisms is conveniently summarized on deformation mechanism diagrams (e.g., Figure 23.5). They show the range of stress and temperature in which we expect to find each sort of creep (they also show where plastic yielding occurs, and where deformation is simply elastic). Diagrams like these are available for metals and ceramics.

Sometimes creep is desirable. Extrusion, hot rolling, hot pressing, and forging are carried out at temperatures at which power-law creep is the dominant mechanism of deformation. Then raising the temperature reduces the pressures required for the operation. The change in forming pressure for a given change in temperature can be calculated from Equation (23.1).

Creep fracture

Diffusion gives creep. It also gives creep fracture. If you stretch anything for long enough, it will break. You might think that a creeping material would—like toffee—stretch a long way before breaking in two but, for crystalline materials, this is very rare. Indeed, creep fracture (in tension) can happen at unexpectedly small strains, often only 2 to 5%, by the mechanism shown in Figure 23.6. Voids appear on grain boundaries that lie normal to the tensile stress. These are the boundaries to which atoms diffuse to give diffusional creep, coming from the boundaries that lie parallel to the stress. But if the tensile boundaries have voids on them, they act as sources of atoms too, and in doing so, they grow. The voids cannot support load, so the stress rises on the remaining intact bits of boundary, the voids grow more and more quickly, until finally they link and fracture takes place.