Mechanisms of precipitation hardening

7.2.3 Mechanisms of precipitation hardening

7.2.3.1 The significance of particle deformability

The strength of an age-hardening alloy is governed by the interaction of moving dislocations and precipitates. The obstacles in precipitation-hardening alloys which hinder the motion of dislocations may be either (1) the strains around GP zones, (2) the zones or precipitates themselves, or both. Clearly, if it is the zones themselves which are important, it will be necessary for the moving dislocations either to cut through them or go round them. Thus, merely from elementary reasoning, it would appear that there are at least three causes of hardening, namely: (1) coherency strain hardening, (2) chemical hardening, i.e. when the dislocation cuts through the precipitate, or (3) dispersion hardening, i.e. when the dislocation goes round or over the precipitate.

The relative contributions will depend on the particular alloy system but, generally, there is a critical dispersion at which the strengthening is a maximum, as shown in Figure 7.7. In the small- particle regime the precipitates, or particles, are coherent and deformable as the dislocations cut through them, while in the larger-particle regime the particles are incoherent and non-deformable as the dislocations bypass them. For deformable particles, when the dislocations pass through the particle, the intrinsic properties of the particle are of importance and alloy strength varies only weakly with particle size. For non-deformable particles, when the dislocations bypass the particles, the alloy strength is independent of the particle properties but is strongly dependent on particle size and dispersion strength, decreasing as particle size or dispersion increases. The transition from deformable to non-deformable particle-controlled deformation is readily recognized by the change in microstructure, since the ‘laminar’ undisturbed dislocation flow for the former contrasts with the turbulent plastic flow for non-deformable particles. The latter leads to the production of a high density of dislocation loops, dipoles and other debris, which results in a high rate of work hardening. This high rate of work hardening is a distinguishing feature of all dispersion-hardened systems.

7.2.3.2 Coherency strain hardening

The precipitation of particles having a slight misfit in the matrix gives rise to stress fields which hinder the movement of gliding dislocations. For the dislocations to pass through the regions of

Mechanical properties II – Strengthening and toughening 395

Non-deformable particles

Particle size Figure 7.7 Variation of strength with particle size, defining the deformable and non-deformable

particle regimes.

internal stress, the applied stress must be at least equal to the average internal stress, and for spherical particles this is given by

τ = 2μεf , (7.2) where μ is the shear modulus, ε is the misfit of the particle and f is the volume fraction of precipitate.

This suggestion alone, however, cannot account for the critical size of dispersion of a precipitate at which the hardening is a maximum, since equation (7.2) is independent of L, the distance between particles. To explain this, Mott and Nabarro consider the extent to which a dislocation can bow round

a particle under the action of a stress τ. Like the bowing stress of a Frank–Read source, this is given by r = αμb/τ,

(7.3) where r is the radius of curvature to which the dislocation is bent, which is related to the particle

spacing. Hence, in the hardest age-hardened alloys, where the yield strength is about μ/100, the dislocation can bend to a radius of curvature of about 100 atomic spacings, and since the distance between particles is of the same order it would appear that the dislocation can avoid the obstacles and take a form like that shown in Figure 7.8a. With a dislocation line taking up such a configuration, in order to produce glide, each section of the dislocation line has to be taken over the adverse region of internal stress without any help from other sections of the line – the alloy is then hard. If the precipitate is dispersed on too fine a scale (e.g. when the alloy has been freshly quenched or lightly aged), the dislocation is unable to bend sufficiently to lie entirely in the regions of low internal stress.

As a result, the internal stresses acting on the dislocation line largely cancel and the force resisting its movement is small – the alloy then appears soft. When the dispersion is on a coarse scale, the dislocation line is able to move between the particles, as shown in Figure 7.8b, and the hardening is again small.

For coherency strain hardening, the flow stress depends on the ability of the dislocation to bend and thus experience more regions of adverse stress than of aiding stress. The flow stress therefore depends on the treatment of averaging the stress, and recent attempts separate the behavior of small and large coherent particles. For small coherent particles the flow stress is given by

τ = 4.1με 3/2 f 1/2 (r/b) 1/2 , (7.4)

396 Physical Metallurgy and Advanced Materials

Stress field of precipitate Dislocation line

loop left

Figure 7.8 Schematic representation of a dislocation: (a) curling round the stress fields from precipitates and (b) passing between widely spaced precipitates (Orowan looping).

which predicts a greater strengthening than the simple arithmetic average of equation (7.2). For large coherent particles

τ 1/2 (εb 3 / r 3 ) = 0.7μf 1/4 . (7.5)

7.2.3.3 Chemical hardening

When a dislocation actually passes through a zone, as shown in Figure 7.9, a change in the number of solvent–solute near-neighbors occurs across the slip plane. This tends to reverse the process of clustering and, hence, additional work must be done by the applied stress to bring this about. This process, known as chemical hardening, provides a short-range interaction between dislocations and precipitates, and arises from three possible causes: (1) the energy required to create an additional

particle/matrix interface with energy γ 1 per unit area which is provided by a stress τ

(7.6) where α is a numerical constant, (2) the additional work required to create an anti-phase boundary

1 ( fr) 1/2 /μ b 2 ,

inside the particle with ordered structure, given by τ

(7.7) where β is a numerical constant, and (3) the change in width of a dissociated dislocation as it passes

through the particle where the stacking-fault energy differs from the matrix (e.g. Al–Ag, where SF ∼ 100 mJ m −2 between Ag zones and Al matrix), so that

τ SF / b. (7.8)

Mechanical properties II – Strengthening and toughening 397 Ordered

particle

Interface

Slip plane

Figure 7.9 Ordered particle (a) cut by dislocations (b) to produce new interface and APB.

Usually γ 1 <γ APB and so γ 1 can be neglected, but the ordering within the particle requires the dislocations to glide in pairs. This leads to a strengthening given by

τ = (γ APB / 2b)[(4γ APB rf /πT ) 1/2 − f ], (7.9) where T is the dislocation line tension.

7.2.3.4 Dispersion hardening

In dispersion hardening it is assumed that the precipitates do not deform with the matrix and that the yield stress is the stress necessary to expand a loop of dislocation between the precipitates. This will

be given by the Orowan stress τ = αμb/L,

(7.10) where L is the separation of the precipitates. As discussed above, this process will be important

in the later stages of precipitation when the precipitate becomes incoherent and the misfit strains disappear. A moving dislocation is then able to bypass the obstacles, as shown in Figure 7.8b, by moving in the clean pieces of crystal between the precipitated particles. The yield stress decreases as the distance between the obstacles increases in the overaged condition. However, even when the dispersion of the precipitate is coarse a greater applied stress is necessary to force a dislocation past the obstacles than would be the case if the obstruction were not there. Some particle or precipitate strengthening remains, but the majority of the strengthening arises from the dislocation debris left around the particles, giving rise to high work hardening.

7.2.3.5 Hardening mechanisms in Al–Cu alloys

The actual hardening mechanism which operates in a given alloy will depend on several factors, such as the type of particle precipitated (e.g. whether zone, intermediate precipitate or stable phase), the magnitude of the strain and the testing temperature. In the earlier stages of ageing (i.e. before overage- ing) the coherent zones are cut by dislocations moving through the matrix, and hence both coherency strain hardening and chemical hardening will be important, e.g. in such alloys as aluminum–copper, copper–beryllium and iron–vanadium–carbon. In alloys such as aluminum–silver and aluminum– zinc, however, the zones possess no strain field, so that chemical hardening will be the most important contribution. In the important high-temperature creep-resistant nickel alloys, the precipitate is of the

398 Physical Metallurgy and Advanced Materials

True stress (MN m 80

Strain (%)

Figure 7.10 Stress–strain curves from single crystals of aluminum–4% copper containing GP[1] zones, GP[2], zones, θ ′ precipitates and θ precipitates respectively (after Fine, Bryne and Kelly, 1961; courtesy of Taylor & Francis).

Ni 3 Al form, which has a low particle/matrix misfit and hence chemical hardening due to dislocations cutting the particles is again predominant. To illustrate that more than one mechanism of hardening is in operation in a given alloy system, let us examine the mechanical behavior of an aluminum–copper alloy in more detail.

Figure 7.10 shows the deformation characteristics of single crystals of an aluminum–copper (nominally 4%) alloy in various structural states. The curves were obtained by testing crystals of appro- ximately the same orientation, but the stress–strain curves from crystals containing GP[1] and GP[2] zones are quite different from those for crystals containing θ ′ or θ precipitates. When the crystals con- tain either GP[1] or GP[2] zones, the stress–strain curves are very similar to those of pure aluminum crystals, except that there is a two- or threefold increase in the yield stress. In contrast, when the crys- tals contain either θ ′ or θ precipitates the yield stress is less than for crystals containing zones, but the initial rate of work hardening is extremely rapid. In fact, the stress–strain curves bear no similarity to those of a pure aluminum crystal. It is also observed that when θ ′ or θ is present as a precipitate, defor- mation does not take place on a single slip system but on several systems; the crystal then deforms, more nearly as a polycrystal does and the X-ray pattern develops extensive asterism. These factors are consistent with the high rate of work hardening observed in crystals containing θ ′ or θ precipitates.

The separation of the precipitates cutting any slip plane can be deduced from both X-ray and electron-microscope observations. For the crystals, relating to Figure 7.10, containing GP[1] zones this value is 15 nm and for GP[2] zones it is 25 nm. It then follows from equation (7.3) that to avoid these precipitates the dislocations would have to bow to a radius of curvature of about 10 nm. To do this requires a stress several times greater than the observed flow stress and, in consequence, it must be assumed that the dislocations are forced through the zones. Furthermore, if we substitute the observed values of the flow stress in the relation μb/τ = L, it will be evident that the bowing mechanism is unlikely to operate unless the particles are about 60 nm apart. This is confirmed by electron-microscope observations which show that dislocations pass through GP zones and coherent precipitates, but bypass non-coherent particles. Once a dislocation has cut through a zone, however, the path for subsequent dislocations on the same slip plane will be easier, so that the work-hardening rate of crystals containing zones should be low, as shown in Figure 7.10. The straight, well-defined slip bands observed on the surfaces of crystals containing GP[1] zones also support this interpretation.

If the zones possess no strain field, as in aluminum–silver or aluminum–zinc alloys, the flow stress would be entirely governed by the chemical hardening effect. However, the zones in aluminum– copper alloys do possess strain fields, as shown in Figure 7.4, and consequently the stresses around a

Mechanical properties II – Strengthening and toughening 399

(a)

(b)

Figure 7.11 Cross-slip of edge (a) and screw (b) dislocation over a particle producing prismatic loops in the process.

zone will also affect the flow stress. Each dislocation will be subjected to the stresses due to a zone at a small distance from the zone.

It will be remembered from Chapter 6 that temperature profoundly affects the flow stress if the barrier which the dislocations have to overcome is of a short-range nature. For this reason, the flow stress of crystals containing GP[1] zones will have a larger dependence on temperature than that of those containing GP[2] zones. Thus, while it is generally supposed that the strengthening effect of GP[2] zones is greater than that of GP[1], and this is true at normal temperatures (see Figure 7.10), at very low temperatures it is probable that GP[1] zones will have the greater strengthening effect due to the short-range interactions between zones and dislocations.

The θ ′ and θ precipitates are incoherent and do not deform with the matrix, so that the critical resolved shear stress is the stress necessary to expand a loop of dislocation between them. This corresponds to the overaged condition and the hardening to dispersion hardening. The separation of the θ particles is greater than that of the θ ′ , being somewhat greater than 1 µm, and the initial flow stress is very low. In both cases, however, the subsequent rate of hardening is high because, as suggested by Fisher, Hart and Pry, the gliding dislocation interacts with the dislocation loops in the vicinity of the particles (see Figure 7.8b). The stress–strain curves show, however, that the rate of work hardening falls to a low value after a few percent strain, and these authors attribute the maximum in the strain-hardening curve to the shearing of the particles. This process is not observed in crystals containing θ precipitates at room temperature and, consequently, it seems more likely that the particles will be avoided by cross-slip. If this is so, prismatic loops of dislocation will be formed at the particles, by the mechanism shown in Figure 7.11, and these will give approximately the same mean internal stress as that calculated by Fisher, Hart and Pry, but a reduced stress on the particle. The maximum in the work-hardening curve would then correspond to the stress necessary to expand these loops; this stress will be of the order of μb/r, where r is the radius of the loop, which is somewhat greater than the particle size. At low temperatures cross-slip is difficult and the stress may be relieved either by initiating secondary slip or by fracture.