PHENOMENOLOGY of SUPERPLASTIC DEFORMATION

Contents

• Introduction
• Non-Superplastic Flow
  • Region III - Power Law Creep
  • Region 0 - Diffusion Creep
• Superplasticity as a unique deformation process - Region II
  • Geometric - Topological Models
  • Dislocation Models
  • Diffusion Models
  • Grain Boundary Sliding

Introduction

Several plausible hypotheses as to the micromechanical origin of superplasticity in fine grain materials have been suggested yet none has been found capable of accurately describing both the mechanical and microstructural features of superplastic deformation. The strain rate at which a material will deform at elevated temperature is defined by eqn.(1) which can be stated in an expanded temperature dependent form as:

(1)

where p and n are characteristic of the micromechanism of deformation, D is a diffusion coefficient which is dependent on the rate controlling process and defines the temperature dependence of the strain rate at constant stress and structure, d is the grain size and b the burgers vector or characteristic dimension. Deformation is driven by the deviatoric (shear) component of the applied stress field, characterised by stress in eqn.(1) and G is the shear modulus. It should be stressed that eqn.(1) is phenomenological rather than being derived from first principals.

A logarithmic plot of the dependence of flow stress on strain rate for a typical superplastic material is shown in Figure 4.1 and can be divided into 4 regions. The slope of the plot is the strain rate sensitivity, m, where

(2)

At high strain rates, i.e. Region III, the strain rate sensitivity is low (m=0.2 to 0.3) and deformation is by recovery controlled dislocation creep. The activation energy for flow in Region III is similar to that for lattice diffusion and the strain rate is relatively insensitive to the grain size (p=0). At intermediate strain rates, Region II, the material deforms superplastically, with grain boundary sliding being a major feature of the flow process. The strain rate sensitivity is high (m > 0.5). The measured activation energy for flow in Region II is often similar to that for grain boundary diffusion and the flow stress is grain size dependent (p=2 to 3).

At low strain rates, Region I, the slope of the stress - strain rate plot varies. In many superplastic alloys the strain rate sensitivity in region I is low and this has been interpreted as evidence for some form of threshold stress for superplastic flow since dislocation activity is not normally observed at such low strain rates. However, in other materials the slope at low strain rates has been observed to increase, tending toward unity. This has been cited as evidence for diffusion controlled flow with the activation energy for flow in Region I being similar to that measured for volume diffusion. This is contrary to that which might normally be expected in a fine grain material, i.e. the activation energy would be that for grain boundary diffusion. Moreover, different studies on the Zn-Al eutectoid, have shown both types of Region I behaviour.

To add further to the controversy, it has been reported that the strain rate sensitivity can initially decrease as the strain rate is reduced but further reductions in the strain rate result in a transition to a slope of unity, termed Region 0, where true diffusion creep dominates (Vale et.al.). Unfortunately, it is difficult to resolve the arguments concerning Region I as the materials examined were microstructurally unstable. Grain growth during slow strain rate deformation or prolonged periods of primary creep have both been cited as the reasons for the apparent discrepancies between different sets of experimental data. These factors coupled to the different testing techniques that have been used to obtain stress - strain rate data at very low strain rates have complicated the interpretation of the low strain rate regime (Langdon).

Non-superplastic flow

Region III

At high strain rates, i.e. Region III, the strain rate sensitivity is low and the material is deforming by conventional recovery controlled dislocation creep. Strain is accumulated by the glide of dislocations within the grains. Dislocation glide is opposed by the microstructure of the material, for example, on a near atomic scale by other dislocations, solutes, even the lattice itself, and on a larger scale by precipitates and dislocation arrays within the grains. Glide is therefore dependent on the rate at which the obstacles can be by-passed. At elevated temperatures, thermal activation enables dislocations to either absorb or emit vacancies at a realistic rate and to climb from one glide plane to another thereby overcoming the obstacles. Similarly, if the dislocation is trapped as a consequence of jogs formed by the intersection with forest dislocations then the absorption or emission of vacancies will enable the gliding dislocation to be released. Those dislocations which become trapped in the sub-boundaries will climb and either be annihilated in the boundary or escape. The strain rate is therefore controlled by the rate at which dislocations are made available for glide, the glide time being negligible with respect to the time spent at pinning points.

Several theoretical treatments, each of which assume that dislocation climb in one form or another is the rate controlling process have led to values for the stress exponent, n, ranging from 3 to 5 when volume diffusion is the dominant mode of vacancy supply. At lower temperatures vacancy diffusion along the dislocation cores predominates. The rate at which vacancies are supplied to or removed from the climbing dislocations is then dependent on the dislocation density which is itself a function of the applied stress. Stress exponents of 5 to 7 (n+2) are then predicted. As a result of continued slip on a limited number of active slip systems, the grains become elongated parallel to the tensile axis during deformation. Grain boundary sliding, whilst operative in Region III, is only a minor flow mechanism. Since the distances over which dislocations move between pinning points are small when compared with the grain size, then the strain rate is virtually independent of the grain size. The phenomenological equations defining this region are discussed in detail elsewhere, but can be approximated to

(3)

where

(4)

where A_III is a constant, D_eff is the effective lattice diffusion coefficient, D_v the volume diffusion coefficient, D_c the dislocation core diffusion rate, a_c the dislocation core cross-sectional area. The presence of a deviatoric component in the applied stress field will result in a difference between the normal stresses on some grain boundary surfaces, introducing a gradient in chemical potential which in turn induces diffusional mass transfer to reduce those differences. Diffusional transport both through and around the grains results in a shape change and hence creep.

Region 0

The strain rate for true diffusion creep, Region 0, is given by

(5)

(6)

where A₀ is a constant, D_gb is the grain boundary diffusion coefficient, d, the grain boundary width and d, the grain size.

At sufficiently low stresses, diffusion creep will dominate over dislocation creep. At low temperatures, the activation energy for diffusion creep will be that for grain boundary diffusion with a grain size exponent p=3, while at elevated temperatures, lattice diffusion will dominate with p=2. If the vacancy creation or absorption processes rather than vacancy motion are rate controlling, as might occur when solutes or precipitates are present in the grain boundaries, then the deformation rate may become interface controlled. The strain rate would then vary as the square of the applied stress.

Superplastic flow - Region II

The Conventional Argument - Superplastic Flow is a Distinct Deformation Mechanism.

At intermediate strain rates, Region II, the flow process is less well understood, although there is agreement on the microstructural features associated with it. Strain is accumulated by the motion of individual grains or clusters of grains relative to each other by sliding and rolling. Grains are observed to change their neighbours and to emerge at the free surface from the interior. During deformation the grains remain equi-axed, or, if they were not equi-axed prior to deformation, become so during superplastic flow. Textures become less intense as a result of deformation in Region II, while the converse is normally observed in Region III. The motion of individual grains is dependent on both the normal and shear stresses acting on their grain boundaries and is therefore dependent on the shape and orientation of the grains. Translation and rotation are thus stochastic in nature, occurring in different directions, to different extents at different locations.

Many attempts have been made to develop deformation models that are capable of predicting both the mechanical and topological features of superplastic deformation. However, none have yet been completely successful. Unlike recovery controlled dislocation creep, where dislocation climb is accepted as the rate controlling process, several rate limiting mechanisms have been proposed for superplastic deformation (see reviews by Mukherjee , Kashyap and Mukherjee , Langdon and Gifkins ).

If grain boundary sliding was to occur in a completely rigid system of grains then voids would develop in the microstructure (Geckinli) see above. The holes or cavities would expand or contract as grains, moving in three dimensions, approached or receded from them. However, many superplastic materials do not cavitate. Grain boundary sliding is therefore accommodated. Even when cavities are observed, their distribution is far from homogeneous and while they would accommodate sliding, cavitation is not as likely an accommodation mechanism as either diffusion or dislocation activity. If the accommodation processes are sufficiently rapid at the deformation temperature, then grain boundary sliding itself could be the rate controlling mechanism (Beere). Alternatively, if grain boundary sliding was intrinsically rapid then the accommodating processes would be rate limiting.

Microstructural studies have found only limited evidence of dislocation activity within the grains of materials deformed superplastically, although as the strain rate approaches that of the transition to Region III they become increasingly apparent (Mukherjee, Samuelson et.al., Melton & Edington, Bricknall & Edington, Edington et.al.). By way of contrast, large grained non-superplastic microstructures deforming under the same conditions of strain rate and temperature as those associated with Region II show extensive dislocation activity, often with a well defined sub-structure present. The dislocation density in the sub-grains themselves is, however, generally very low.

The fine grain size of superplastic materials, coupled with low flow stresses at a given strain rate and temperature, ensures that the equilibrium sub-grain size is greater than the grain size. It might the be argued that the material is deforming by conventional dislocation creep and that the dislocations are not observed as most of them are trapped within the grain boundaries. However, the absence of dislocations within the grains may also be cited as evidence in support of superplastic flow accommodated solely by diffusion.

In examining the models which have been proposed to explain superplastic flow, it is important to distinguish between cause and effect. Often, it is assumed that grain boundary sliding occurs as a result of the applied stress and that the transient stresses generated as a consequence of the attempt by the grains to slide are relaxed by a physical rearrangement of matter. It is the rate at which the latter processes proceed that is often thought to govern the rate at which strain is accumulated.

Dislocation Models

When grain boundaries slide, stress concentrations develop wherever that sliding is obstructed. Relaxation of the stress concentrations by the emission of dislocations from one grain boundary and their absorption by another can be limited by the rate at which the dislocations are emitted (source control), the rate at which they can cross the grains (glide or lattice climb control) or the rate at which they are absorbed into the boundaries (grain boundary climb control). The glide process has been assumed to occur relatively rapidly in superplasticity since there is a lack of either 'strong' obstacles or significant solute drag effects within the grains at the deformation temperature (Ball and Hutchison, Mukherjee) . Pile-ups of dislocations adjacent to the grain boundaries are thought to develop and provide a back stress against which the sliding grain(s) would have to work to emit further dislocations along a particular slip plane. Climb of the leading dislocation from the pile-up into the boundary would allow another dislocation to be emitted and enable a small increment of grain boundary sliding (i.e. strain) to be accumulated . Stress exponents and grain size dependencies of 2 would be predicted together with an activation energy commensurate with that of grain boundary diffusion. If only one source per grain were activated then flow would be boundary climb controlled. By allowing grain boundary ledges to act as dislocation sources, the absolute magnitude of the predicted strain rate can vary by upwards of two orders of magnitude and is therefore source dependent. The deformation rate is given by

(7)

An alternative view of dislocation accommodated flow arises if grain boundary sliding is accommodated by dislocation climb and glide within the grain boundaries themselves (Gifkins, Falk et.al). Pile-ups of grain boundary dislocations could form in the grain boundaries at triple points. Dissociation of the leading dislocation into either lattice dislocations, or boundary dislocations which could glide in the other boundaries intersecting the triple point, would then be the rate controlling process.

Several arguments have been raised against dislocation based models of superplastic flow, namely:

• The dislocation pile-up models do not predict a threshold stress for superplastic deformation, i.e. Region I
• There is no implicit mechanism by which the crystal lattice of either the sliding or accommodating grains could rotate in the lattice pile-up model of Ball and Hutchison . Grain elongation is implicit in any model involving dislocation glide/climb on a limited number of slip systems
• Dislocation pile-ups are not observed experimentally. Furthermore, at the high temperatures at which deformation takes place pile-ups would not be expected since the average stress is low.

If grain rotation was accepted as resulting from a non-balanced system of unrelaxed grain boundary shear stresses then grain elongation would not be observed. Random variations in the direction and magnitude of rotation would cause slip to switch from one slip system to another. Since the rotations can be large (>30°) no nett change in grain shape would then be apparent.

Grain rotation would, on the other hand, be an implicit feature of the grain boundary dislocation model. Again the shear stresses acting on the grain boundaries would be able to move dislocations causing a shear within the boundary zone. The resultant torque on the grains would re-orientate the crystal lattice of the grain and an oscillatory grain motion would be observed. Figure 4.4 shows the effect of superplastic strain on the microstructure of a Pb-62Sn eutectic alloy. It is clearly apparent from the micrographs that the grains not only slide and rotate, but also move perpendicular to the free surface (Vastava et.al.). Measurements of the orientation of specific grains has shown that the rotations can vary by as much as 40° (Geckinli & Barrett).

Displacement of scratches on Pb-Sn eutectic after increasing SPF strain

The objection raised against dislocation accommodated flow based on the absence of either dislocation activity or dislocation pile-ups can be countered by the following arguments. Firstly the stress generated at the head of the pile-up could not be supported by the grain boundary at the deformation temperature and climb would be rapid. Secondly, on removal of the applied stress there would be virtually nothing within the microstructure at that temperature to hold the dislocations in the piled up configuration and the dislocations would run back to their sources. The grain boundary dislocation model of superplastic flow is often referred to as the 'Core and Mantle' model as the accommodation of grain boundary sliding is assumed to occur only within a viscous mantle around a rigid grain core. If the grains can be thought of as regular hexagons then the predicted width of the mantle is only 0.07 times the grain diameter, below. In a typical superplastic material the mantle would be only 30 to 70nm wide!

Diffusion Models

It has also been envisaged that mass could be redistributed by diffusional flow [147,148]. Driven by differences in the stress dependent chemical potential on adjacent grain boundaries, mass transport from regions of compression to tension would occur. Sliding is accommodated by a gradual change in grain shape as matter is moved by diffusion. Grain boundary migration restores the original equi-axed shape but in a rotated orientation. The retention of an equi-axed grain shape is therefore achieved in the model of Ashby and Verrall. Furthermore, because a transient, but finite increase in grain boundary area results from the shape change then the model predicts a threshold stress for superplastic flow. The strain rate in the superplastic region is that due to diffusion but operating under an apparent stress, s₀. This stress is equal to the applied stress less the stress necessary to create that additional grain boundary energy.

(8)

If the sliding results in grain switching then no matter how small the individual three dimensional shape change steps are, a finite increase in the grain boundary area would still be required. The work done in creating the new grain boundary surface would be some fraction of the instantaneous grain boundary area and would have to be supplied by the external stress. The increased grain boundary energy is used to drive the subsequent grain boundary migration but the energy is lost in the form of heat. The work done per grain would vary as d². The number of grains in unit volume varies as 1/d³ and thus the threshold stress, which is the work done per unit volume, will vary as 1/d. The threshold stress for superplastic flow predicted by the diffusion accommodation model varies as the inverse of the grain size. As with the case of the dislocation based models, several objections have been raised to the accommodation of superplastic flow solely by diffusion, namely,

• The diffusion paths originally proposed by Ashby and Verrall required that diffusion takes place in different directions on opposite sides of the same grain boundary. As diffusion is driven by the stress acting perpendicular to the grain boundary this is physically impossible.
• Deformation in the Ashby-Verrall model is not symmetrical.
• If grain boundary sliding is accommodated solely by diffusion then the lattices of the individual grains cannot rotate. The rotations shown by Ashby and Verrall are only apparent and result from grain boundary migration.
• The strain rates predicted by equation 4.8 are about two orders of magnitude too fast.
• Elongated grains should be apparent in the microstructure.
• The threshold stress is predicted to decrease with increasing grain size, contrary to the experimental evidence. Moreover, the predicted threshold stress is significantly less than that measured.
• The grain switching event can only be invoked once giving a maximum strain of 0.55 (e_max=0.41 for the two dimensional model illustrated in figure 4.6). The diffusion paths were later modified by Spingarn and Nix so that each grain within the cluster underwent the same change, maintaining symmetry of deformation and a more realistic shape transient

The grain size dependence of strain rate, p, given by equation 8 falls between 2 and 3. However, the observation of an m-value of 0.5 arises only as a result of the transition from the threshold stress (Region I), through normal diffusional flow (Region II, m=1) operating under an apparent stress, to conventional dislocation creep (Region III, m=0.2), rather than a distinct superplastic flow process. However, m values in excess of 0.5 are consistent with reported m values as high as 0.75 to 0.8. In view of the independence of both diffusion and dislocation creep processes, it would be more logical if grain boundary sliding was accommodated by both forms of mass transport rather than either in isolation. However, before examining the multi-mechanism models of superplastic flow, the presumption that grain boundary sliding itself is not a rate controlling process is considered next.

Grain Boundary Sliding

To examine the process of grain boundary sliding consider a two-dimensional array of regular hexagons orientated at random with respect to a remotely applied stress field (Fig 4.7). The normal and shear stresses acting on each grain boundary can be balanced by stresses acting throughout the continuum body. The magnitude of the shear stresses or normal stresses can only be evaluated if an assumption is made with regard to one of the types of stress. If it is assumed that the grain boundary shear stresses are fully relaxed i.e.

(9)

then the grain boundary sliding rate will be controlled by the rate at which the normal stresses on the grain boundaries are relaxed. For the case of uniaxial tension, the average stress on a given grain boundary is

(9a)

where s° is the applied stress, and q is the angle between the applied stress and the normal to the grain boundary. If, for example, diffusion relaxes the normal stresses then the strain rate is simply that due to diffusion creep.ie.

(10)

This may be compared with equation 8 except that there is no threshold stress. During diffusional stress relaxation, the local value of the normal stress can vary from 0 at the triple points to anywhere between +2.16 and -0.72 times the applied stress at the centre of the grain boundaries. For uniaxial compression, diffusional mass transport would be predicted to occur from the centre of grain boundaries orientated approximately perpendicular to the applied stress, via the triple point, to the centre of boundaries orientated mainly parallel to the applied stress. If the applied stress was tensile then diffusional flow would occur in the opposite direction. The predicted diffusion paths are in excellent agreement with the modified paths for diffusion accommodated superplastic flow (Fig 4.6), however, the predicted m-value is too high to match those found experimentally.

If on the other hand the normal tractions on the grain boundaries are relaxed by rapid diffusional flow

(11)

then the average shear stress, t, acting parallel to the boundary would be given by

(11a)

and the corresponding strain rate by

(12)

where n the stress exponent characterising the stress relaxation process. If all the boundaries are equally resistant to sliding then the strain rate is independent of the orientation of the grains. If the boundary viscosity varies with, for example, misorientation or precipitate density, then

(13)

where the grain centre shear rate, , is related to the grain boundary shear rate, , and the shear stress acting on that boundary, t_i, through the relationships

(14)

The grain boundary shear rate is only equal to the grain centre shear rate when the grains do not rotate. Grain boundary sliding can, therefore, be accommodated by relaxing the shear stresses in and adjacent to the grain boundaries, and the process can be equated with the "Core and Mantle" model of superplastic deformation.

From the picture above, it can be seen that a given grain centre shear displacement, DS, and hence a macroscopic strain, e, can be achieved not only by sliding a distance c along the boundary, but also by rotation of the boundary through an angle w. If both processes are allowed to operate simultaneously then a given grain centre shear displacement (rate) can be achieved by lesser amounts of rotation and sliding (rates), i.e.

where is the average rotational velocity of the grains on either side of the grain boundary. If no holes appear in the structure then the average strain rate is twice that when rotation is not allowed (Beeré). A given strain rate can therefore be maintained for smaller grain boundary sliding rates and hence at a lower stress. Moreover, for a given grain boundary sliding rate the strain rate would increase as the grain size was reduced. Thus grain boundary sliding and superplastic deformation would be expected to become increasingly apparent in progressively finer grain materials.

The optimum combination of grain boundary sliding and rotation can be evaluated by minimising the rate of doing work, W, where

The variation of the angular rotation rate, normalised against the strain rate has been calculated by Beeré and is shown in figure 4.9 as a function of the grain boundary orientation and relative grain boundary viscosities. The momentary rate of grain rotation can be quite high, though the maximum rotation is seldom greater than 30° (see also figure 4.4). Thus it seems that the kinetics of grain boundary sliding are controlled by the slower of two accommodation processes, and grain boundary sliding would appear not to be a unique mechanism of superplastic deformation but rather a natural consequence of an unbalanced rate of grain boundary shear stress relaxation.

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