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Dynamic grain growth in Al–6Ni: Modelling and experiments
Acta Materialia 53 (2005) 4313–4321 www.actamat-journals.com
Dynamic grain growth in Al–6Ni: Modelling and experiments K.B. Hyde, P.S. Bate
*
Manchester Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK Received 16 September 2004; received in revised form 23 May 2005; accepted 27 May 2005 Available online 14 July 2005
Abstract The dynamic grain growth of the eutectic Al–6Ni alloy, processed to give a spheroidal particle morphology, has been investigated. Despite a fine grain size, this material is not superplastic. It shows significant grain growth during deformation, which is essentially strain controlled down to a temperature of about 250 C with a strain rate of 103 s1. It is suggested that the dynamic growth is due to geometric perturbation of the Zener pinned structure, and this has been modelled using the Monte Carlo–Potts method. This modelling gave a reasonably good prediction of the dynamic growth and of the transition temperature to strain controlled growth. Differences between dynamic grain growth in plane strain and uniaxial tension may also be consistent with a geometric mechanism. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth; Monte Carlo; Aluminium; Superplasticity
1. Introduction Grain growth in polycrystals is a very well known phenomenon that has been the subject of much research. It is driven by the energy of grain boundaries, which is lower for a coarser grain size within a given volume of material, its rate is controlled by diffusion and it can be halted by the presence of particles. The coincidence of a particle and a grain boundary gives a configuration with a local minimum of surface energy, and thus a metastable state. This is Zener pinning [1] and, because of the benefits of fine grain sizes, is of great practical importance. One phenomenon that can be exhibited by fine grained materials is superplasticity, where very high ductilities occur as a result of a high sensitivity of the flow stress to strain rate. Conventional superplasticity occurs at high temperatures and slow strain rates, and requires grain sizes of the order of 10 lm or less [2]. Such *
Corresponding author. Tel.: +44 161 200 8842; fax: +44 161 200 3586. E-mail address: [email protected] (P.S. Bate).
materials are invariably Zener pinned to maintain the fine grain sizes at high temperatures, with either a small volume fraction of fine particles or a large volume fraction of coarser particles. Commercial superplastic aluminium alloys fall into the first class, and duplex alloys such as Ti–6Al–4V in the second. In all cases, even though the grain size is relatively stable at the forming temperature in the absence of deformation, grain growth occurs during straining. This is dynamic grain growth. Dynamic grain growth has been recognised for many years, usually in the context of superplasticity [3–8] and it has been suggested [9] that there is an intimate connection between the mechanisms of superplasticity and dynamic grain growth. The mechanisms that have been proposed generally reflect that association, with grain boundary deformation processes being a focus for attention. However, Bate [10] proposed that dynamic growth was a feature of Zener pinned systems, with deformation perturbing the geometry of pinning events and hence the metastable equilibrium. Simple estimates based on curvature and vertex modelling gave reasonably good
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.05.029
K.B. Hyde, P.S. Bate / Acta Materialia 53 (2005) 4313–4321
2. Experimental work 2.1. Material The eutectic Al–6 wt.% Ni alloy was supplied in as-cast and extruded forms by Alcan International, Banbury, UK. The 50 mm thick cast billet was rolled to a thickness of 16 mm, and 15 mm diameter bars were machined from the plate for equal channel angular extrusion (ECAE). These bars were subject to six passes through a 120 ECAE die at 200 C, with rotations of 90 about their long axis between passes. This was found to further break up the original elongated grains and lead to a more homogenous microstructure. The bars were then rolled to 1.5 mm, with the ECAE bar axis parallel to the rolling direction, and machined into flat tensile specimens with the rolling direction parallel to the tensile direction. All specimens were annealed at 500 C for 24 h prior to deformation. Extended annealing at that temperature showed that this treatment gave a suitable Zener pinned structure, consisting of reasonably equiaxed grains with a section diameter of about 6.5 lm and a dispersion of globular Al3Ni particles, which – with the exception of a few large isolated particles – had a mean section diameter of about 0.6 lm. The volume fraction of intermetallic in this alloy is close to 0.1. The extruded section, 10 mm thick and 50 mm wide, did not have a suitable microstructure in the extruded condition, and was hand forged and machined to give 15 mm diameter rods prior to ECAE. This material was used for plane strain compression testing, and was also subject to annealing at 500 C for 24 h before that deformation. This material had a somewhat finer microstructure, with a grain size of about 3.5 lm and a mean intermetallic particle diameter of 0.5 lm. 2.2. Mechanical testing Tensile tests were carried out, in air, at 500 C using specimens with a 9 mm wide parallel gauge, 30 mm long.
Constant strain rates of 104, 5 · 104 and 103 s1 were used in tests for subsequent structural examination. Tests were also conducted with alternating strain rates of 1 · 104 and 2 · 104 s1, and with rates of 0.9 · 103 and 1.1 · 103 s1. Results of these tests are shown in Fig. 1. Stress and strain values were calculated on the assumption that the gauge remained parallel during deformation, which became less valid at the higher strains where significant necking occurred. Interpolation of the high and low rate segments allows the effect of strain rate on stress, quantified as the rate sensitivity index m, to be measured without the influence of gross structural (and geometric) changes occurring during deformation. The values of m were low, at about 0.2. Plane strain compression (PSC) was carried out in a channel die rig. This is shown in Fig. 2, and was designed to allow rapid cooling of the specimen following deformation: specimens could be fully quenched within 5 s of the end of deformation. Hexagonal boron nitride was used as lubricant, and tests were carried
0.5
12
10-3 s-1 10
0.45 0.4 0.35
8
σ
predictions of growth, and the effect of deformational inhomogeneity and particle coarsening was also considered. In fact, dynamic grain growth had been found earlier in two-dimensional Monte Carlo–Potts modelling by Takayama et al. [11], who attributed it to aggregation of particles caused by deformation in their model. There are Zener pinned systems which are not superplastic; one of those, Al–6Ni, is studied here. Some preliminary work [12,13] showed that this material exhibits dynamic grain growth and that it is not superplastic. Results of experimental work on that system and modelling of the dynamic grain growth are given below.
(MPa)
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10-4 s-1
6
0.3 0.25 0.2
4
m
0.15 0.1
2
0.05 0
0
0.2
0.4
0.6
0 0.8
ε Fig. 1. Stress–strain curves from tensile testing at 500 C, with the strain rate systematically perturbed about the mean values given. The strain rate sensitivity, m, values derived from those results are shown as symbols: the slightly higher m values correspond to the faster strain rate test.
Fig. 2. Image of the channel die PSC (plane strain compression) rig together with a schematic of the tooling. The arrangement shown on the right is closed by a hydraulic ram (A) at 50 kN between cartridge heated plattens to give the actual constrained channel. Insulation is by partially stabilised zirconia. The tool is activated by the water cooled platten (B) attached to the load cell.
K.B. Hyde, P.S. Bate / Acta Materialia 53 (2005) 4313–4321
out at a strain rate of 103 s1 using a variety of temperatures. Using compression testing overcame the problems associated with necking in tension, which becomes severe at temperatures less than about 400 C. 2.3. Microstructures All microstructural characterisation was carried out using a Philips XL30 scanning electron microscope (SEM) equipped with a field emission gun. Grain size data was derived from orientation maps obtained from electron backscattered diffraction (EBSD) data. Surfaces for examination were prepared by grinding and mechanical polishing. Electropolishing led to problems, especially with EBSD, because of differential polishing producing high surface relief. Samples from tensile specimens were taken from regions of the gauge where the local strain, determined by cross-sectional dimensions, was equal to the overall tensile strain. Incidentally, both transverse strains were approximately equal, i.e., the deformation was close to axisymmetric tension. Micrographs of undeformed and deformed material are shown in Fig. 3. It is clear that dynamic growth has occurred, together with some degree of grain elongation in the tensile direction. Grain sizes were measured in a variety of ways. In all cases, the EBSD data was processed to remove non-indexed points and grains defined using the criterion that a grain boundary had a misorientation angle greater than 15. As well as simple mean linear intercepts, grain sections were fitted to ellipses (of arbitrary orientations). This was selected as the preferred quantification method. About 1500 grains were measured in each of the three principal sections of each specimen. Grain dimensions following tensile testing are shown in Fig. 4, and the significant degree of dynamic growth can be seen. This was not sufficient to maintain a near-equiaxed microstructure, however. The development of the mean planimetric
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(circular equivalent section) diameter is shown, along with the sheet plane aspect ratio, as functions of strain in Fig. 5. There is no significant difference between results for the different strain rates, and the dynamic growth is again clearly shown. To a first order approximation, the growth would be expected to depend only on current grain size, D, and strain rate, e_ D_ ¼ kD_e;
ð1Þ
where k is the dynamic growth coefficient [7,9]. This integrates to give an exponential dynamic growth rate D ¼ D0 expðkeÞ;
ð2Þ
where D0 is the initial grain size. The value from data given in Fig. 5 is k 0.4. Although the aspect ratio has increased, that increase is much less than would happen without dynamic boundary migration. Using, again, an exponential fit a ¼ a0 expðjeÞ
ð3Þ
gives a coefficient of j 0.45, compared to a value of 1.5 expected if no boundary migration were occurring. Boundary migration occurs by diffusion, and so a sufficiently fast strain rate at a given temperature would mean that the intrinsic rate of boundary migration would be insufficient for dynamic boundary migration – and so growth – to occur. Plane strain compression was used to investigate this transition. The grain growth results for different temperatures are shown in Fig. 6. There is a fairly clear transition from the regime where dynamic growth is occurring to one where the boundary mobility is insufficient for strain controlled growth to occur, i.e., the growth is boundary mobility controlled. This occurs at about 250 C with the strain rate used (103 s1). The dynamic growth coefficients at the higher temperatures are significantly higher than those measured in tensile testing, at k 0.9. A possible reason for that difference is discussed below.
Fig. 3. Composite micrographs of Al–6Ni after annealing for 24 h at 500 C (left) and then subsequent tensile elongation to a strain of 0.7 at a rate of 103 s1, also at 500 C. Micrographs are from EBSD orientation images.
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Fig. 4. Mean grain ellipse dimensions, K, in the principal specimen directions, as functions of strain for tensile deformation at 500 C with the strain rates indicated. Error bars are ±1 standard deviation, derived from all equivalent results. The fitting lines are best fit exponential functions to the data irrespective of strain rate, and the dashed lines indicate the change in dimensions that would occur as a geometric consequence of deformation without any migration of the original grain boundaries.
Fig. 5. Mean planimetric grain diameter, D, and the aspect ratio, a, of grain dimensions in the tensile and transverse directions, as functions of strain for tensile deformation at 500 C with the strain rates indicated.
3. Modelling The dynamic grain growth was modelled using the Monte Carlo–Potts (MCP) method. In the initial investigation using a vertex model [10], a homogeneous Zener drag field was used. This approximation may be adequate when the pinning particles are much smaller than the grain size, but that is not true in the material used here. It is possible to modify that vertex model such that the Zener drag is concentrated in localised regions, and Cauchy functions were used for the results given by Humphreys and Bate [12]. Doing this increased the dynamic growth coefficient from k 0.4 to k 0.7.
Vertex modelling is computationally efficient, but it is difficult to include particle pinning effects in a locally realistic manner, though this has been done by Weygand et al. [14]. It is also difficult to perform this modelling in three dimensions, and three-dimensional modelling may be necessary for realistic modelling of particle pinning events [15,16]. In view of this, MCP modelling was used here. This has been used extensively for static grain growth, and is adequately described elsewhere [17–19]. For dynamic growth, the domain must be deformed and simple shear is the most efficient way of doing this. This was done in two dimensions by Takayama et al. [11], and by Mackenzie and Bate [20], but as noted
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Fig. 6. Grain growth in plane strain compression, showing the effect of temperature on the mean grain diameter (left), and the effect of temperature on the dynamic growth coefficient, k (right).
Fig. 7. Isometric views of MCP domains of 2003 cells containing 0.1 fraction of r = 3 particles, in (a) at the static Zener limit, in (b) after shearing that structure to c = 1 at a fast rate and in (c) after the same shear but at a slow rate. The sense of the simple shear is indicated.
above, three-dimensional modelling is preferred for Zener pinning situations. A cubic array of 2003 cells were used, with translational boundary conditions, i.e., sites were taken to repeat with a period of 200 in each dimension. The transition probability of a cell was determined with reference to its first, second and third nearest neighbours in the ‘‘cubic 123’’ scheme [18], and transition probabilities determined using the Metropolis method [19]. Following Miodownik et al. [16], the ÔtemperatureÕ factor was set to unity, and a radius of 2 cells was used for the particles, meaning that each particle occupied 33 cells. No account was taken of boundary crystallography in determining energies and mobilities, both of those parameters having values of unity irrespective of misorientation. Deformation was by simple shear effected by slip. Cells above a given plane were shifted sideways by one cell, that plane being taken from a predetermined sequence which was designed to make the shear as uniform as possible at any given time. Particles were not sheared: matrix cells could either be effectively consumed by a particle, or a duplicate cell created in the wake of a slip event. The energy between particle and matrix cells was unity, and the mobility zero. Some preliminary results of this model and further background were given by Bate [21]. The approach to
the Zener limit was asymptotic, and agreed with the model proposed by Hillert [22]. About 105 Monte Carlo steps (MCS, the average time taken for a potential transition of each cell of the domain to be tested) gave an adequate approximation to the Zener limit. Example views of model domains at the Zener limit, and subsequently deformed at high and low strain rates to a simple shear of unity, are shown in Fig. 7. Sufficiently slow deformation clearly gives significant grain growth, and this is shown in the graph of mean grain diameter as functions of shear for a slow and for a fast shear rate given in Fig. 8. At the relatively small shears involved, the simple shear can be taken to be a plane strain of half the magnitude, and the predicted dynamic growth coefficient at slow rates was k 0.8. The aspect ratio increased, with a coefficient j 0.9, significantly lower than the value of about 2 that would result with no boundary migration for this deformation mode.
4. Discussion The Al–6Ni alloy was chosen for several reasons. The solubility of nickel in aluminium is very low, leading to slow particle coarsening via Ostwald ripening. No
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Fig. 8. Graph of the mean grain diameter, D, in units of MCP cells, as functions of shear, c, for low and high shear rates in deforming 3D MCP models. The fitting lines shown are exponential functions.
significant change in particle size occurred during the tests reported here. It also gives a suitable microstructure for comparison with MCP modelling, as it is not feasible to model systems containing very small particles compared with the grain size using that method. In practice, the material was not ideal. The most significant problem was in generating a reasonably homogenous microstructure. Conventional processing, by rolling or extrusion, gave large colonies of similarly oriented grains presumably corresponding to the as-cast eutectic cells. ECAE proved effective at reducing this feature of the material, though there were still minor indications of long range heterogeneity even after this processing. Despite these issues, the experimental evidence for dynamic grain growth, of essentially the same form as that observed in superplastic aluminium alloys, is very clear. In view of the results from both vertex and MCP modelling, there can be very little doubt that the geometric effect of deformation on a Zener pinned microstructure will lead to dynamic grain growth, as long as the intrinsic boundary mobility is sufficiently rapid. Whether it is the only mechanism for dynamic grain growth is an open question. There are certainly other factors in the development of microstructure in systems such as the Al–6Ni and superplastic aluminium alloys. One test of the mechanism is its ability to predict the transition to a strain controlled dynamic grain growth as the boundary mobility increases with temperature. 4.1. Transition from mobility controlled to strain controlled growth The overall type of dynamic grain growth predicted by the MCP modelling, as with the vertex models reported previously, is similar to that observed experi-
mentally. According to the model, the dynamic growth is controlled by the effect of geometric changes to the particle/grain boundary configuration, and strain rate (or temperature) will have little effect, provided that the intrinsic, diffusion controlled, boundary mobility is sufficiently rapid. This transition from strain controlled growth to boundary mobility controlled growth is clear from the experimental results shown in Fig. 6: at temperatures less than about 250 C in plane strain compression at a rate of 103 s1, the boundary mobility is too slow for the dynamic growth mechanism to give strain controlled growth. To compare that result with model predictions, a scaling must be introduced to make the effects of boundary mobility comparable. In the absence of particles, normal grain growth is driven by curvature; the pressure P on a boundary segment is given by twice the surface tension, c, multiplied by the mean curvature. The mean curvature will be related to grain radius and so diameter, D, which means that P ¼ b2c=D;
ð4Þ
where b is a numerical factor relating grain diameter to driving curvature. The intrinsic rate of migration is the pressure multiplied by the boundary mobility, l, and so the relative rate of this grain growth is _ e_ ¼ D=D ¼ b2v=D2 ;
ð5Þ
where v = lc is the kinematic mobility. This normalised growth rate has the dimensionality of strain rate, and can be used to scale model results; the strain rate normalised by the intrinsic relative growth rate is g ¼ e_ D2 =ð2bvÞ.
ð6Þ
For model results, values of bv can be obtained from simulations of normal grain growth, and for both vertex and MCP models these values correspond to b 0.5. The results for both MCP modelling and vertex modelling using localised Zener drag fields [12] are shown in Fig. 9. There are some differences between the results of the two types of modelling, with the transition being more rapid for the MCP predictions and the strain controlled (low rate) values of dynamic growth coefficient being higher. However, both sets of results indicate that there will be little dynamic growth for g > 10 and that growth will become fully strain controlled at g < 0.01. To evaluate the experimental results, the kinematic boundary mobility as a function of temperature needs to be known. Using the experimental data given by Humphreys and Huang [23], the average kinematic mobility of high angle boundaries in aluminium was taken to be v 60 exp(17,700/T) m2 s1. Using that formula and Eq. (5) gives the values shown in Fig. 10. These will then correspond to g = 1 and define the transition range. At 250 C and a strain rate of 103 s1, the
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Fig. 9. Graph of the dynamic growth exponent, k, as a function of the normalised strain rate, g, for 2D vertex and 3D MCP models.
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and that the strain rate is sufficiently slow. The issue of strain rate, and temperature, was dealt with above, and at 500 C with any of the strain rates involved, tensile testing was clearly carried out in the regime of strain-controlled growth. There was no indication of particle coarsening, at least of a magnitude that would contribute noticeably to the difference in dynamic growth rates. Boundary curvature plays a very important role in grain growth. While it is local events that are likely to be of critical importance when dealing with the Zener pinned state, some insight can perhaps be gained by considering the effect of deformation on average grain curvatures. In this context, it is interesting to note that the average maximum mean curvature of ellipsoids best fitting grains in the MCP simulations remained almost constant during dynamic grain growth. This is because, although the mean grain size increased, the aspect ratio also increased. Some comments about curvature in this context were made previously [10]. In an idealised case, the maximum arithmetic mean curvature of an ellipsoid, Æqæ, resulting from the straining of a unit sphere is hqimax ¼ expð2eÞ coshðj2q 1jeÞ;
ð7Þ
where q is the ratio of the negative of the minimum and the maximum (tensile) principal strains, i.e., q = 0.5 for axisymmetric tension and 0 or 1 for plane strain states. The minimum mean curvature is hqimin ¼ expð2ðq 1ÞeÞ coshððq þ 1ÞeÞ Fig. 10. The critical strain rates corresponding to the normalised rate of curvature driven grain growth (Eq. (4)), as functions of temperature for different grains sizes.
critical value should occur with a grain size of about 10 lm. This is of the correct order, and reinforces the view that the transition to strain controlled growth occurs when the Zener pinning becomes the limiting factor, that Zener pinning then being affected by the deformation to give the dynamic growth. The three (decimal) orders of magnitude in strain rate over which the transition occurs in the models corresponds to a temperature interval of about 100 C, which is consistent with the experimental results. 4.2. Curvature and growth There was a large difference between the experimental dynamic growth coefficients for uniaxial tension and plane strain. There were differences in both strain state and the material microstructure of those experiments which could have contributed to a difference in growth. In principle, the scale of microstructure, which was smaller for the material used in the plane strain experiments, should make no difference to the dynamic growth, provided that there is no particle coarsening
ð8Þ
for q 6 0.5. There is a fairly large effect of strain state on these values, with both the maximum curvature, and especially the ratio of maximum to minimum curvatures, increasing more rapidly with strain for states near plane strain than axisymmetric tension. For example, in the absence of boundary migration at a strain of 0.5, the maximum curvature in plane strain is about 12 per cent greater than that in axisymmetric tension, and the ratio of curvatures is 43 per cent greater. This effect of strain state on the development of curvature could make a significant contribution to the difference in growth rates between plane strain and uniaxial tension that was observed in the Al–6Ni. The curvature estimates given above indicate that plane strain has a greater effect on the geometry of the grain structure than axisymmetric tension. The dynamic growth coefficients predicted by the MCP modelling were similar to those measured (at high temperature) in plane strain, and that modelling was effectively plane strain as well. It is unfortunate that a way of MCP modelling in other strain states is not known at this time. 4.3. Grain boundary sliding and grain growth Another reason for using Al–6Ni is that it is not superplastic. It has a low strain rate sensitivity of flow
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stress, and in tension fails with considerable strain localisation (necking) prior to fracture, giving quite modest tensile elongations. Apart from the dynamic grain growth, the material also exhibited boundary sliding, another behaviour usually taken to be characteristic of superplastic materials. To show this, a tensile specimen was mechanically polished on the sheet plane surface after annealing, and was then scratched using a cloth deliberately contaminated with a fine dispersion of 6 lm diamond particles. This produced fine, parallel scratches either parallel with or perpendicular to the tensile direction, in different regions of the polished surface. The specimen was then deformed at 500 C with a strain rate of 104 s1 to a strain of 0.3. Examination using SEM showed significant deviations of the scratches associated with localised shear at grain boundaries, as shown in Fig. 11. Such features have been observed in Al–4Mg at a temperature and strain rate where it was also not superplastic [24], and must not, as with dynamic grain growth, be associated exclusively with superplastic deformation. The models used for the Zener pinning mechanism of dynamic grain growth assume approximately homogeneous deformation, although heterogenous deformation is actually inherent in the MCP model and a variable
deformation field predicted by a simplified crystal plasticity finite element model was used in some of the original vertex modelling [10]. A significant amount of boundary sliding could conceivably make a contribution to dynamic grain growth. In overall terms, however, the sliding will still have its main effect in changing the geometry of pinning events and so destabilising the local, metastable, equilibrium. In this respect, a certain amount of boundary sliding would not affect the basic mechanism examined in this paper. 4.4. Summary MCP modelling has shortcomings, including the computational effort when simulating Zener pinned systems [16,25]. The total computer time for the model results used in Fig. 9 was about four months. However, it does appear to give reasonably good simulations of grain growth, and is the only feasible way of carrying out three-dimensional simulations. The results given here are for relatively small strains. Grain growth to much larger strains has been measured in superplastic materials, and the aspect ratio, although changing as in the simulations given above, does not carry on increasing. Results from Li et al. [26], for example, could be interpreted as showing that new grains are forming during deformation. Formation of subgrains in the larger grains of the structure and an increase in the misorientation of the subgrain boundaries would be a mechanism for this within a framework where intragranular dislocation glide was the main deformation mechanism. It must be remembered that in many cases, particle coarsening is occurring during deformation and this will make a further contribution to dynamic grain growth.
5. Conclusions
Fig. 11. Secondary electron SEM images of the surface of a tensile test piece which had been deliberately scratched prior to deformation to a strain of 0.3. The displacements at grain boundaries of scratches initially parallel with the tensile axis (top) and perpendicular to it (bottom) are clearly shown, as are surface relief steps associated with boundaries.
Experiments using uniaxial tension and plane strain compression have shown that Al–6Ni processed to give a fine grain size, statically stabilised by a distribution of globular intermetallic particles, undergoes dynamic grain growth of the type reported in superplastic materials. This alloy is not superplastic, having a strain rate sensitivity exponent of about 0.2 and limited ductility. The main features of the dynamic growth can be ascribed to geometric perturbation of the metastable Zener pinned state. This mechanism was simulated using Monte Carlo–Potts modelling, which gave good predictions of the growth characteristics in plane strain deformation, including the temperature of transition from boundary mobility controlled growth to strain controlled growth. The dynamic growth in tension was somewhat less, and this could be due to the effect of
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straining state on the evolution of grain shape and curvature.
Acknowledgements We thank the EPSRC for support via Grant GR/ R52695, and Alcan International for materials. Special thanks go to Peter Dean for construction of the plane strain compression rig.
References [1] Zener C, quoted in Smith CS. Trans Met Soc AIME 1948;175:15. [2] Pilling J, Ridley N. Superplasticity in crystalline solids. London: The Institute of Metals; 1989. [3] Lee D, Backofen WA. Trans Met Soc AIME 1966;293:1034. [4] Watts BM, Stowell MJ. J Mater Sci 1971;6:228. [5] Clark MA, Alden TH. Acta Metall 1973;21:1195. [6] Suery M, Baudelet B. J Mater Sci 1973;8:363. [7] Wilkinson DS, Ca´ceres CH. Acta Metall 1984;32:1335. [8] Seidensticker JR, Mayo MJ. Acta Mater 1998;46:4883.
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[9] Seidensticker JR, Mayo MJ. Scripta Mater 1998;38:1091. [10] Bate P. Acta Mater 2001;49:1453. [11] Takayama Y, Tozawa T, Kato H, Furushiro N. Mater Sci Forum 1996;204–206:319. [12] Humphreys FJ, Bate PS. Mater Sci Forum 2001;357–359:477. [13] Hyde KB, Bate PS. Mater Sci Forum 2004;447–448:79. [14] Weygand D, Bre´chet Y, Le´pinoux J. Acta Mater 1999;47:961. [15] Miodownik M, Martin JW, Cerezo A. Philos Mag A 1999;79:203. [16] Miodownik M, Holm EA, Hassold GN. Scripta Mater 2000;42:1173. [17] Srolovicz DJ, Grest GS, Anderson MP. Acta Metall 1985;33:2233. [18] Rollett AD. Prog Mater Sci 1997;42:79. [19] Raabe D. Acta Mater 2000;48:1617. [20] Mackenzie L, Bate P. In: Gottstein G, Molodov DA, editors. Recrystallization and Grain Growth. Berlin: Springer; 2001. p. 269. [21] Bate P. In: Palmiere EJ, Mahouf M, Pinna C, editors. Thermomechanical Processing: Mechanics, Microstructure and Control. Sheffield: Department of Engineering Materials, The University of Sheffield; 2003. p. 79. [22] Hillert M. Acta Metall 1965;13:227. [23] Humphreys FJ, Huang Y. Acta Mater 1999;47:2259. [24] Bate P, Blackwell P. Scripta Mater 1998;38:901. [25] Bate P. In: Gottstein G, Molodov DA, editors. Recrystallization and Grain Growth. Berlin: Springer; 2001. p. 39. [26] Li F, Bae DH, Ghosh AK. Acta Mater 1997;45:3887.
Dynamic grain growth in Al–6Ni: Modelling and experiments K.B. Hyde, P.S. Bate
*
Manchester Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK Received 16 September 2004; received in revised form 23 May 2005; accepted 27 May 2005 Available online 14 July 2005
Abstract The dynamic grain growth of the eutectic Al–6Ni alloy, processed to give a spheroidal particle morphology, has been investigated. Despite a fine grain size, this material is not superplastic. It shows significant grain growth during deformation, which is essentially strain controlled down to a temperature of about 250 C with a strain rate of 103 s1. It is suggested that the dynamic growth is due to geometric perturbation of the Zener pinned structure, and this has been modelled using the Monte Carlo–Potts method. This modelling gave a reasonably good prediction of the dynamic growth and of the transition temperature to strain controlled growth. Differences between dynamic grain growth in plane strain and uniaxial tension may also be consistent with a geometric mechanism. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth; Monte Carlo; Aluminium; Superplasticity
1. Introduction Grain growth in polycrystals is a very well known phenomenon that has been the subject of much research. It is driven by the energy of grain boundaries, which is lower for a coarser grain size within a given volume of material, its rate is controlled by diffusion and it can be halted by the presence of particles. The coincidence of a particle and a grain boundary gives a configuration with a local minimum of surface energy, and thus a metastable state. This is Zener pinning [1] and, because of the benefits of fine grain sizes, is of great practical importance. One phenomenon that can be exhibited by fine grained materials is superplasticity, where very high ductilities occur as a result of a high sensitivity of the flow stress to strain rate. Conventional superplasticity occurs at high temperatures and slow strain rates, and requires grain sizes of the order of 10 lm or less [2]. Such *
Corresponding author. Tel.: +44 161 200 8842; fax: +44 161 200 3586. E-mail address: [email protected] (P.S. Bate).
materials are invariably Zener pinned to maintain the fine grain sizes at high temperatures, with either a small volume fraction of fine particles or a large volume fraction of coarser particles. Commercial superplastic aluminium alloys fall into the first class, and duplex alloys such as Ti–6Al–4V in the second. In all cases, even though the grain size is relatively stable at the forming temperature in the absence of deformation, grain growth occurs during straining. This is dynamic grain growth. Dynamic grain growth has been recognised for many years, usually in the context of superplasticity [3–8] and it has been suggested [9] that there is an intimate connection between the mechanisms of superplasticity and dynamic grain growth. The mechanisms that have been proposed generally reflect that association, with grain boundary deformation processes being a focus for attention. However, Bate [10] proposed that dynamic growth was a feature of Zener pinned systems, with deformation perturbing the geometry of pinning events and hence the metastable equilibrium. Simple estimates based on curvature and vertex modelling gave reasonably good
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.05.029
K.B. Hyde, P.S. Bate / Acta Materialia 53 (2005) 4313–4321
2. Experimental work 2.1. Material The eutectic Al–6 wt.% Ni alloy was supplied in as-cast and extruded forms by Alcan International, Banbury, UK. The 50 mm thick cast billet was rolled to a thickness of 16 mm, and 15 mm diameter bars were machined from the plate for equal channel angular extrusion (ECAE). These bars were subject to six passes through a 120 ECAE die at 200 C, with rotations of 90 about their long axis between passes. This was found to further break up the original elongated grains and lead to a more homogenous microstructure. The bars were then rolled to 1.5 mm, with the ECAE bar axis parallel to the rolling direction, and machined into flat tensile specimens with the rolling direction parallel to the tensile direction. All specimens were annealed at 500 C for 24 h prior to deformation. Extended annealing at that temperature showed that this treatment gave a suitable Zener pinned structure, consisting of reasonably equiaxed grains with a section diameter of about 6.5 lm and a dispersion of globular Al3Ni particles, which – with the exception of a few large isolated particles – had a mean section diameter of about 0.6 lm. The volume fraction of intermetallic in this alloy is close to 0.1. The extruded section, 10 mm thick and 50 mm wide, did not have a suitable microstructure in the extruded condition, and was hand forged and machined to give 15 mm diameter rods prior to ECAE. This material was used for plane strain compression testing, and was also subject to annealing at 500 C for 24 h before that deformation. This material had a somewhat finer microstructure, with a grain size of about 3.5 lm and a mean intermetallic particle diameter of 0.5 lm. 2.2. Mechanical testing Tensile tests were carried out, in air, at 500 C using specimens with a 9 mm wide parallel gauge, 30 mm long.
Constant strain rates of 104, 5 · 104 and 103 s1 were used in tests for subsequent structural examination. Tests were also conducted with alternating strain rates of 1 · 104 and 2 · 104 s1, and with rates of 0.9 · 103 and 1.1 · 103 s1. Results of these tests are shown in Fig. 1. Stress and strain values were calculated on the assumption that the gauge remained parallel during deformation, which became less valid at the higher strains where significant necking occurred. Interpolation of the high and low rate segments allows the effect of strain rate on stress, quantified as the rate sensitivity index m, to be measured without the influence of gross structural (and geometric) changes occurring during deformation. The values of m were low, at about 0.2. Plane strain compression (PSC) was carried out in a channel die rig. This is shown in Fig. 2, and was designed to allow rapid cooling of the specimen following deformation: specimens could be fully quenched within 5 s of the end of deformation. Hexagonal boron nitride was used as lubricant, and tests were carried
0.5
12
10-3 s-1 10
0.45 0.4 0.35
8
σ
predictions of growth, and the effect of deformational inhomogeneity and particle coarsening was also considered. In fact, dynamic grain growth had been found earlier in two-dimensional Monte Carlo–Potts modelling by Takayama et al. [11], who attributed it to aggregation of particles caused by deformation in their model. There are Zener pinned systems which are not superplastic; one of those, Al–6Ni, is studied here. Some preliminary work [12,13] showed that this material exhibits dynamic grain growth and that it is not superplastic. Results of experimental work on that system and modelling of the dynamic grain growth are given below.
(MPa)
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10-4 s-1
6
0.3 0.25 0.2
4
m
0.15 0.1
2
0.05 0
0
0.2
0.4
0.6
0 0.8
ε Fig. 1. Stress–strain curves from tensile testing at 500 C, with the strain rate systematically perturbed about the mean values given. The strain rate sensitivity, m, values derived from those results are shown as symbols: the slightly higher m values correspond to the faster strain rate test.
Fig. 2. Image of the channel die PSC (plane strain compression) rig together with a schematic of the tooling. The arrangement shown on the right is closed by a hydraulic ram (A) at 50 kN between cartridge heated plattens to give the actual constrained channel. Insulation is by partially stabilised zirconia. The tool is activated by the water cooled platten (B) attached to the load cell.
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out at a strain rate of 103 s1 using a variety of temperatures. Using compression testing overcame the problems associated with necking in tension, which becomes severe at temperatures less than about 400 C. 2.3. Microstructures All microstructural characterisation was carried out using a Philips XL30 scanning electron microscope (SEM) equipped with a field emission gun. Grain size data was derived from orientation maps obtained from electron backscattered diffraction (EBSD) data. Surfaces for examination were prepared by grinding and mechanical polishing. Electropolishing led to problems, especially with EBSD, because of differential polishing producing high surface relief. Samples from tensile specimens were taken from regions of the gauge where the local strain, determined by cross-sectional dimensions, was equal to the overall tensile strain. Incidentally, both transverse strains were approximately equal, i.e., the deformation was close to axisymmetric tension. Micrographs of undeformed and deformed material are shown in Fig. 3. It is clear that dynamic growth has occurred, together with some degree of grain elongation in the tensile direction. Grain sizes were measured in a variety of ways. In all cases, the EBSD data was processed to remove non-indexed points and grains defined using the criterion that a grain boundary had a misorientation angle greater than 15. As well as simple mean linear intercepts, grain sections were fitted to ellipses (of arbitrary orientations). This was selected as the preferred quantification method. About 1500 grains were measured in each of the three principal sections of each specimen. Grain dimensions following tensile testing are shown in Fig. 4, and the significant degree of dynamic growth can be seen. This was not sufficient to maintain a near-equiaxed microstructure, however. The development of the mean planimetric
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(circular equivalent section) diameter is shown, along with the sheet plane aspect ratio, as functions of strain in Fig. 5. There is no significant difference between results for the different strain rates, and the dynamic growth is again clearly shown. To a first order approximation, the growth would be expected to depend only on current grain size, D, and strain rate, e_ D_ ¼ kD_e;
ð1Þ
where k is the dynamic growth coefficient [7,9]. This integrates to give an exponential dynamic growth rate D ¼ D0 expðkeÞ;
ð2Þ
where D0 is the initial grain size. The value from data given in Fig. 5 is k 0.4. Although the aspect ratio has increased, that increase is much less than would happen without dynamic boundary migration. Using, again, an exponential fit a ¼ a0 expðjeÞ
ð3Þ
gives a coefficient of j 0.45, compared to a value of 1.5 expected if no boundary migration were occurring. Boundary migration occurs by diffusion, and so a sufficiently fast strain rate at a given temperature would mean that the intrinsic rate of boundary migration would be insufficient for dynamic boundary migration – and so growth – to occur. Plane strain compression was used to investigate this transition. The grain growth results for different temperatures are shown in Fig. 6. There is a fairly clear transition from the regime where dynamic growth is occurring to one where the boundary mobility is insufficient for strain controlled growth to occur, i.e., the growth is boundary mobility controlled. This occurs at about 250 C with the strain rate used (103 s1). The dynamic growth coefficients at the higher temperatures are significantly higher than those measured in tensile testing, at k 0.9. A possible reason for that difference is discussed below.
Fig. 3. Composite micrographs of Al–6Ni after annealing for 24 h at 500 C (left) and then subsequent tensile elongation to a strain of 0.7 at a rate of 103 s1, also at 500 C. Micrographs are from EBSD orientation images.
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Fig. 4. Mean grain ellipse dimensions, K, in the principal specimen directions, as functions of strain for tensile deformation at 500 C with the strain rates indicated. Error bars are ±1 standard deviation, derived from all equivalent results. The fitting lines are best fit exponential functions to the data irrespective of strain rate, and the dashed lines indicate the change in dimensions that would occur as a geometric consequence of deformation without any migration of the original grain boundaries.
Fig. 5. Mean planimetric grain diameter, D, and the aspect ratio, a, of grain dimensions in the tensile and transverse directions, as functions of strain for tensile deformation at 500 C with the strain rates indicated.
3. Modelling The dynamic grain growth was modelled using the Monte Carlo–Potts (MCP) method. In the initial investigation using a vertex model [10], a homogeneous Zener drag field was used. This approximation may be adequate when the pinning particles are much smaller than the grain size, but that is not true in the material used here. It is possible to modify that vertex model such that the Zener drag is concentrated in localised regions, and Cauchy functions were used for the results given by Humphreys and Bate [12]. Doing this increased the dynamic growth coefficient from k 0.4 to k 0.7.
Vertex modelling is computationally efficient, but it is difficult to include particle pinning effects in a locally realistic manner, though this has been done by Weygand et al. [14]. It is also difficult to perform this modelling in three dimensions, and three-dimensional modelling may be necessary for realistic modelling of particle pinning events [15,16]. In view of this, MCP modelling was used here. This has been used extensively for static grain growth, and is adequately described elsewhere [17–19]. For dynamic growth, the domain must be deformed and simple shear is the most efficient way of doing this. This was done in two dimensions by Takayama et al. [11], and by Mackenzie and Bate [20], but as noted
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Fig. 6. Grain growth in plane strain compression, showing the effect of temperature on the mean grain diameter (left), and the effect of temperature on the dynamic growth coefficient, k (right).
Fig. 7. Isometric views of MCP domains of 2003 cells containing 0.1 fraction of r = 3 particles, in (a) at the static Zener limit, in (b) after shearing that structure to c = 1 at a fast rate and in (c) after the same shear but at a slow rate. The sense of the simple shear is indicated.
above, three-dimensional modelling is preferred for Zener pinning situations. A cubic array of 2003 cells were used, with translational boundary conditions, i.e., sites were taken to repeat with a period of 200 in each dimension. The transition probability of a cell was determined with reference to its first, second and third nearest neighbours in the ‘‘cubic 123’’ scheme [18], and transition probabilities determined using the Metropolis method [19]. Following Miodownik et al. [16], the ÔtemperatureÕ factor was set to unity, and a radius of 2 cells was used for the particles, meaning that each particle occupied 33 cells. No account was taken of boundary crystallography in determining energies and mobilities, both of those parameters having values of unity irrespective of misorientation. Deformation was by simple shear effected by slip. Cells above a given plane were shifted sideways by one cell, that plane being taken from a predetermined sequence which was designed to make the shear as uniform as possible at any given time. Particles were not sheared: matrix cells could either be effectively consumed by a particle, or a duplicate cell created in the wake of a slip event. The energy between particle and matrix cells was unity, and the mobility zero. Some preliminary results of this model and further background were given by Bate [21]. The approach to
the Zener limit was asymptotic, and agreed with the model proposed by Hillert [22]. About 105 Monte Carlo steps (MCS, the average time taken for a potential transition of each cell of the domain to be tested) gave an adequate approximation to the Zener limit. Example views of model domains at the Zener limit, and subsequently deformed at high and low strain rates to a simple shear of unity, are shown in Fig. 7. Sufficiently slow deformation clearly gives significant grain growth, and this is shown in the graph of mean grain diameter as functions of shear for a slow and for a fast shear rate given in Fig. 8. At the relatively small shears involved, the simple shear can be taken to be a plane strain of half the magnitude, and the predicted dynamic growth coefficient at slow rates was k 0.8. The aspect ratio increased, with a coefficient j 0.9, significantly lower than the value of about 2 that would result with no boundary migration for this deformation mode.
4. Discussion The Al–6Ni alloy was chosen for several reasons. The solubility of nickel in aluminium is very low, leading to slow particle coarsening via Ostwald ripening. No
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Fig. 8. Graph of the mean grain diameter, D, in units of MCP cells, as functions of shear, c, for low and high shear rates in deforming 3D MCP models. The fitting lines shown are exponential functions.
significant change in particle size occurred during the tests reported here. It also gives a suitable microstructure for comparison with MCP modelling, as it is not feasible to model systems containing very small particles compared with the grain size using that method. In practice, the material was not ideal. The most significant problem was in generating a reasonably homogenous microstructure. Conventional processing, by rolling or extrusion, gave large colonies of similarly oriented grains presumably corresponding to the as-cast eutectic cells. ECAE proved effective at reducing this feature of the material, though there were still minor indications of long range heterogeneity even after this processing. Despite these issues, the experimental evidence for dynamic grain growth, of essentially the same form as that observed in superplastic aluminium alloys, is very clear. In view of the results from both vertex and MCP modelling, there can be very little doubt that the geometric effect of deformation on a Zener pinned microstructure will lead to dynamic grain growth, as long as the intrinsic boundary mobility is sufficiently rapid. Whether it is the only mechanism for dynamic grain growth is an open question. There are certainly other factors in the development of microstructure in systems such as the Al–6Ni and superplastic aluminium alloys. One test of the mechanism is its ability to predict the transition to a strain controlled dynamic grain growth as the boundary mobility increases with temperature. 4.1. Transition from mobility controlled to strain controlled growth The overall type of dynamic grain growth predicted by the MCP modelling, as with the vertex models reported previously, is similar to that observed experi-
mentally. According to the model, the dynamic growth is controlled by the effect of geometric changes to the particle/grain boundary configuration, and strain rate (or temperature) will have little effect, provided that the intrinsic, diffusion controlled, boundary mobility is sufficiently rapid. This transition from strain controlled growth to boundary mobility controlled growth is clear from the experimental results shown in Fig. 6: at temperatures less than about 250 C in plane strain compression at a rate of 103 s1, the boundary mobility is too slow for the dynamic growth mechanism to give strain controlled growth. To compare that result with model predictions, a scaling must be introduced to make the effects of boundary mobility comparable. In the absence of particles, normal grain growth is driven by curvature; the pressure P on a boundary segment is given by twice the surface tension, c, multiplied by the mean curvature. The mean curvature will be related to grain radius and so diameter, D, which means that P ¼ b2c=D;
ð4Þ
where b is a numerical factor relating grain diameter to driving curvature. The intrinsic rate of migration is the pressure multiplied by the boundary mobility, l, and so the relative rate of this grain growth is _ e_ ¼ D=D ¼ b2v=D2 ;
ð5Þ
where v = lc is the kinematic mobility. This normalised growth rate has the dimensionality of strain rate, and can be used to scale model results; the strain rate normalised by the intrinsic relative growth rate is g ¼ e_ D2 =ð2bvÞ.
ð6Þ
For model results, values of bv can be obtained from simulations of normal grain growth, and for both vertex and MCP models these values correspond to b 0.5. The results for both MCP modelling and vertex modelling using localised Zener drag fields [12] are shown in Fig. 9. There are some differences between the results of the two types of modelling, with the transition being more rapid for the MCP predictions and the strain controlled (low rate) values of dynamic growth coefficient being higher. However, both sets of results indicate that there will be little dynamic growth for g > 10 and that growth will become fully strain controlled at g < 0.01. To evaluate the experimental results, the kinematic boundary mobility as a function of temperature needs to be known. Using the experimental data given by Humphreys and Huang [23], the average kinematic mobility of high angle boundaries in aluminium was taken to be v 60 exp(17,700/T) m2 s1. Using that formula and Eq. (5) gives the values shown in Fig. 10. These will then correspond to g = 1 and define the transition range. At 250 C and a strain rate of 103 s1, the
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Fig. 9. Graph of the dynamic growth exponent, k, as a function of the normalised strain rate, g, for 2D vertex and 3D MCP models.
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and that the strain rate is sufficiently slow. The issue of strain rate, and temperature, was dealt with above, and at 500 C with any of the strain rates involved, tensile testing was clearly carried out in the regime of strain-controlled growth. There was no indication of particle coarsening, at least of a magnitude that would contribute noticeably to the difference in dynamic growth rates. Boundary curvature plays a very important role in grain growth. While it is local events that are likely to be of critical importance when dealing with the Zener pinned state, some insight can perhaps be gained by considering the effect of deformation on average grain curvatures. In this context, it is interesting to note that the average maximum mean curvature of ellipsoids best fitting grains in the MCP simulations remained almost constant during dynamic grain growth. This is because, although the mean grain size increased, the aspect ratio also increased. Some comments about curvature in this context were made previously [10]. In an idealised case, the maximum arithmetic mean curvature of an ellipsoid, Æqæ, resulting from the straining of a unit sphere is hqimax ¼ expð2eÞ coshðj2q 1jeÞ;
ð7Þ
where q is the ratio of the negative of the minimum and the maximum (tensile) principal strains, i.e., q = 0.5 for axisymmetric tension and 0 or 1 for plane strain states. The minimum mean curvature is hqimin ¼ expð2ðq 1ÞeÞ coshððq þ 1ÞeÞ Fig. 10. The critical strain rates corresponding to the normalised rate of curvature driven grain growth (Eq. (4)), as functions of temperature for different grains sizes.
critical value should occur with a grain size of about 10 lm. This is of the correct order, and reinforces the view that the transition to strain controlled growth occurs when the Zener pinning becomes the limiting factor, that Zener pinning then being affected by the deformation to give the dynamic growth. The three (decimal) orders of magnitude in strain rate over which the transition occurs in the models corresponds to a temperature interval of about 100 C, which is consistent with the experimental results. 4.2. Curvature and growth There was a large difference between the experimental dynamic growth coefficients for uniaxial tension and plane strain. There were differences in both strain state and the material microstructure of those experiments which could have contributed to a difference in growth. In principle, the scale of microstructure, which was smaller for the material used in the plane strain experiments, should make no difference to the dynamic growth, provided that there is no particle coarsening
ð8Þ
for q 6 0.5. There is a fairly large effect of strain state on these values, with both the maximum curvature, and especially the ratio of maximum to minimum curvatures, increasing more rapidly with strain for states near plane strain than axisymmetric tension. For example, in the absence of boundary migration at a strain of 0.5, the maximum curvature in plane strain is about 12 per cent greater than that in axisymmetric tension, and the ratio of curvatures is 43 per cent greater. This effect of strain state on the development of curvature could make a significant contribution to the difference in growth rates between plane strain and uniaxial tension that was observed in the Al–6Ni. The curvature estimates given above indicate that plane strain has a greater effect on the geometry of the grain structure than axisymmetric tension. The dynamic growth coefficients predicted by the MCP modelling were similar to those measured (at high temperature) in plane strain, and that modelling was effectively plane strain as well. It is unfortunate that a way of MCP modelling in other strain states is not known at this time. 4.3. Grain boundary sliding and grain growth Another reason for using Al–6Ni is that it is not superplastic. It has a low strain rate sensitivity of flow
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stress, and in tension fails with considerable strain localisation (necking) prior to fracture, giving quite modest tensile elongations. Apart from the dynamic grain growth, the material also exhibited boundary sliding, another behaviour usually taken to be characteristic of superplastic materials. To show this, a tensile specimen was mechanically polished on the sheet plane surface after annealing, and was then scratched using a cloth deliberately contaminated with a fine dispersion of 6 lm diamond particles. This produced fine, parallel scratches either parallel with or perpendicular to the tensile direction, in different regions of the polished surface. The specimen was then deformed at 500 C with a strain rate of 104 s1 to a strain of 0.3. Examination using SEM showed significant deviations of the scratches associated with localised shear at grain boundaries, as shown in Fig. 11. Such features have been observed in Al–4Mg at a temperature and strain rate where it was also not superplastic [24], and must not, as with dynamic grain growth, be associated exclusively with superplastic deformation. The models used for the Zener pinning mechanism of dynamic grain growth assume approximately homogeneous deformation, although heterogenous deformation is actually inherent in the MCP model and a variable
deformation field predicted by a simplified crystal plasticity finite element model was used in some of the original vertex modelling [10]. A significant amount of boundary sliding could conceivably make a contribution to dynamic grain growth. In overall terms, however, the sliding will still have its main effect in changing the geometry of pinning events and so destabilising the local, metastable, equilibrium. In this respect, a certain amount of boundary sliding would not affect the basic mechanism examined in this paper. 4.4. Summary MCP modelling has shortcomings, including the computational effort when simulating Zener pinned systems [16,25]. The total computer time for the model results used in Fig. 9 was about four months. However, it does appear to give reasonably good simulations of grain growth, and is the only feasible way of carrying out three-dimensional simulations. The results given here are for relatively small strains. Grain growth to much larger strains has been measured in superplastic materials, and the aspect ratio, although changing as in the simulations given above, does not carry on increasing. Results from Li et al. [26], for example, could be interpreted as showing that new grains are forming during deformation. Formation of subgrains in the larger grains of the structure and an increase in the misorientation of the subgrain boundaries would be a mechanism for this within a framework where intragranular dislocation glide was the main deformation mechanism. It must be remembered that in many cases, particle coarsening is occurring during deformation and this will make a further contribution to dynamic grain growth.
5. Conclusions
Fig. 11. Secondary electron SEM images of the surface of a tensile test piece which had been deliberately scratched prior to deformation to a strain of 0.3. The displacements at grain boundaries of scratches initially parallel with the tensile axis (top) and perpendicular to it (bottom) are clearly shown, as are surface relief steps associated with boundaries.
Experiments using uniaxial tension and plane strain compression have shown that Al–6Ni processed to give a fine grain size, statically stabilised by a distribution of globular intermetallic particles, undergoes dynamic grain growth of the type reported in superplastic materials. This alloy is not superplastic, having a strain rate sensitivity exponent of about 0.2 and limited ductility. The main features of the dynamic growth can be ascribed to geometric perturbation of the metastable Zener pinned state. This mechanism was simulated using Monte Carlo–Potts modelling, which gave good predictions of the growth characteristics in plane strain deformation, including the temperature of transition from boundary mobility controlled growth to strain controlled growth. The dynamic growth in tension was somewhat less, and this could be due to the effect of
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straining state on the evolution of grain shape and curvature.
Acknowledgements We thank the EPSRC for support via Grant GR/ R52695, and Alcan International for materials. Special thanks go to Peter Dean for construction of the plane strain compression rig.
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