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PLASTIC ANISOTROPY IN A SUPERPLASTIC DUPLEX STAINLESS STEEL
A Pergamon PII: S1359-6454(96)00396-5
mater.Vol.45,No.7,pp.2747-2757, 1997 (j 1997ActaMetallurgicInc. PublishedbyElsevierSci;nceLtd Printedin GreatBritain,Allrightsreserved 1359-6454/97 $17.00+ 0.00
PLASTIC ANISOTROPY IN A SUPERPLASTIC DUPLEX STAINLESS STEEL J. L. SONG and P. S. BATE
IRC in Materialsfor HighPerformanceApplicationsand Schoolof Metallurgyand Materials, The Universityof Birmingham,BirminghamB152TT,U.K. (Received21 August 1996;accepted6 November1996)
Abstract—Measurements of the plasticanisotropyin uniaxialtensionof the duplexstainlesssteel, SAF2304, havebeenmadeat roomtemperatureandunderconditionswherethematerialwassuperplastic. Therewassignificantplasticanisotropyin both typesof deformationand thereweresomesimilarities betweenthe lowand hightemperaturevariationswithtensileaxisorientation.Althoughit waspossible to modelthehightemperatureanisotropyusinga grainboundaryslidingmodel,theassumeddistribution of slidingboundarieswasconsideredto beunrealistic.This,togetherwithaspectsofmicrostructuraland texturaldevelopment,indicatedthat deformationwas principallyoeeurringby intragranularslipwith a significantcontributioncausedby mechanicalinhomogeneity in the two-phasematerial.0 1997Acta Metallurgic Inc.
1. INTRODUCTION
superplastic conditions, and at room temperature, and its anisotropy analysed.
There have been severalreports of plastic anisotropy in superplastic materials. Johnson et al. [1]observed 2. EXPERIMENTAL METHODS that the circular cross-sections of Zn–Al test pieces became elliptical after superplastic deformation, and 2.1. Material concluded that slip was responsible for this effect. The material used was the duplex stainless steel Later work on a variety of alloys has not necessarily SAF2304,manufactured by Avesta Sheffield,with the led to that conclusion.Melton et al. [2], workingwith chemical composition: P&Sn, and Bricknelland Edington [3],working with Cr Ni Mo N c Fe A1–Cu–Zr plate, both concluded that mechanical Element Wt.percent 22.8 4.8 0.3 0.092 0.017 balance fibring was responsible for plastic anisotropy in the superplastic regime. In particular, Bricknell The material was hot rolled and solution treated at and Edington showed that the plastic strain ratios 108O”Cbefore being cold rolled to 2 mm thickness, observed at slow strain rates were lower than those with an intermediate and a final solution treatment. predicted by slip-based models, but that the differencesin strain ratio between specimensparallel and perpendicular to the rolling direction of the 2.2. Tensile testing Uniaxial tensile testing, both at room and elevated plate remained more or less the same as at higher strain rates, where slip was assumed to occur. The temperature, was carried out using a screw drive mechanical fibring argument is closely allied to the machine controlled via a microcomputer, which also common assumption that grain boundary sliding is recorded the test data. Tests were carried out on the principal deformation mechanism in superplastic specimensprepared using electro-dischargemachinflow, and has been taken to be the cause of plastic ing such that the tensile axeswere at angles of 0°, 45° anisotropy in several instances, for example and 90° to the rolling direction of the sheet. Room temperature tests used specimens with a McDarmaid et al. [4] in Ti–Al–V. However, the plastic anisotropy observed in A1–Li–Mg–Cu–Zrby parallel gauge section 12.5MMwide and 50mm long. Bate [5] showed a dominant four-fold anisotropy Electronicextensometerswere used to monitor width which was thought to be due to slip-based as well as length during the tests, and in addition deformation. In the work presented here, a material scribe marks on the gauge portion were measured has been used which has a microstructure which is before and after testing to allow total strains in likely to show significant mechanical fibring effects those directions to be calculated. These tests were and possessesa sufficientlywell developed crystallo- terminated before the natural maximum load graphic texture to give a reasonable degree of condition to minimise necking effects on the total slip-related anisotropy. It has been tested under strain measurements. --. , L147
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Elevated temperature testing was carried out, in air, at a temperature of 970”Cusing specimenswith a gauge length of 35 mm and gauge width of 9.5 mm. The cross-head was controlled to give, on the assumptions that the gauge section remained parallel and that its volume was constant, a strain rate, ;, which alternated between values of 0.9 and 1.1x 10-3s-’ every 0.025 of strain. The resulting steps in the stress, rr, were interpolated to give two curves for the high and low rates which allowed the strain rate sensitivity index, m, to be approximated as: m = d in a/d in; = A in a/A in 1.
(1)
The relatively small perturbations in strain rate and small strain interval means that the average stress develops with strain in very much the same way as in a constant rate test. In the case of the elevated temperature tests, metallographic specimens from the gauge were prepared such that they could be measured using a traveling microscope to give total strains in the width and thickness directions in addition to measurements made using a micrometer of the cross sections before and after deformation. As well as the influence of tensile axis orientation on flow stress, plastic anisotropy in the uniaxial tensile tests was quantified using the contraction ratio, q, determined in terms of total strains in the width, EW, and the length, G, directions by: q = –&w/&l.
(2)
This is related to the more frequently used r-value measure of anisotropy by: r = .5w/6h = q/(1 —q),
(3)
where ~h is the strain in the thickness direction. The isotropic value of q for deformation at constant volume is 0.5, corresponding to r = 1. Because extensometers were used for the room temperature tests, incremental contraction ratios, effectivelythe ratios of strain rates rather than total strains, could also be measured.
diffraction with MoL radiation. Reflections used were 002, 112 and 123 for the ferrite, and 002, 022 and 113 for the austenite. In order to give through-thicknessaverage measurements, composite specimens were made. Slices about 1mm thick were cut from specimen gauges after deformation and from undeformed material using a slow-speed diamond saw. Thesewere then glued onto glassslides with epoxy resin and the exposed faces ground, polished and etched to an optical metallographic standard. Alignment errors were unavoidable with this procedure and the small pieces involved, but these were less than 5° in all cases. Becauseonly three incompletepole figuresfor each phase were measured, the even-order coefficients of the harmonic series expansion describing the texture werecalculatedto a truncation levelof L = 16 rather than the more usual L = 22. These coefficients were transformed to a basis with the Z-axis in the sheet normal direction and the X-axis in the tensile direction. These coefficients were used to predict anisotropy and to produce maps of the distribution of (001) directions, equivalent to recalculated OOh pole figures, and inverse pole figures for the tensile directions. The pole figure type representation is adequate for the textures involved. 3. RESULTS 3.1. Tensile test results
Testing at room temperature gave the stress–strain results shown in Fig. 1. There was a marked flow stress anisotropy, with the highest flow stress where the tensile axis was in the sheet transverse direction and the lowest when it was at 45° to the rolling direction. Because there was significant strain hardening, taking levels of flow stress at a constant strain will not give an adequate representation of anisotropy for equivalent work hardened states. It is better to consider the flow stresses at some given amount of plastic work as this will, in simple
2.3. Metallography For conventionaloptical metallography,specimens were electrolyticallyetched in an aqueous solution of 56 wt% KOH at 5 V. This darkened the ferrite (d) and left the austenite (y) light. It was not possible, however, to reveal the grain structure with optical metallography and so scanning electron microscopy, using electron channeling contrast, was used for that purpose. It was necessary to identify the phase of individualgrains by examination of the backscattered Kikuchi patterns. Quantitative measures of the microstructure were made using commercial image processing and analysis software. 2.4. Texture determination The crystallographic texture was determined from partial reflection pole figures measured using X-ray
Sool-
>
IL____ o
0.05
0.1
0.15
2
&
Fig. 1.Uniaxialtensileflowstressdevelopmentwithstrain at roomtemperaturewiththeanglesbetweenthetensileaxes and sheetrollingdirectionindicated.The dashedlineis a hyperbolicsegmentused to give values of stress for comparisonwithpredictions,as describedin the text.
SONGand BATE: PLASTICANISOTROPY slip-based models of plasticity at least, represent a constant amount of accumulated slip strain. The idea that the degree of plastic work should determine the hardening state has been discussed by Hill [6] in a more general sense. Rather than integrating the stress–strain curves, if it can be assumed that the stress–strain curves are similar in form, i.e. a simple scalingof stress and strain can make them coincident, then levels of flow stress at some constant value of o.~ will represent the flow stress anisotropy by approximately compensating for the increased work dissipatedwith a higher flowstress.Such a hyperbolic contour is shown in Fig. 1, and the variation of flow stress with tensile axis angle derived on this basis is shown in Fig. 2, along with the contraction ratio results, q. Those contraction ratio results, shownwith scalesof both q and r-value, showa marked four-fold anisotropy, with low values at 0° and 90°. The hot tensile behaviour is shown in Fig. 3. There was significant strain rate sensitivity of the flow stress, and in all directions values of the rate sensitivity index, m, were close to 0.5 throughout the tests which indicates superplastic behaviour. As would be expected with this high rate sensitivity, the gaugesof specimensdeformedto a strain of unity were very uniform except for the transition at the gripped ends. The flow stress was anisotropic, and there were differences in its evolution with strain in the three directions. Most noticeable was the relativelyhigh flow stress at small strains for tension along the rolling direction, which became almost equal to that for transverse direction tension at a strain of 1. The flow stress at 45° to the rolling direction was always lowest at a given strain. The strain hardening rates were low, and the hyperbolic
2749
40
30
c1
0.6
(MP,) 20 0.5 ~
.
10
I 0
0.2
0.4
0.6
0.s
1
&
Fig. 3. Uniaxial tensile flow stress developmentwith strain at 970”C and a strain rate alternating between 0.9 and 1.1 x 10-3s-’, with a strain period of O.025at each rate. The m values, derived by interpolation of the flow stress at the two rates, are shown by symbols below the stress–strain results. The anglesbetweenthe tensileaxes and sheet rolling direction are indicated.
line method used for the room temperature results was redundant. The flow stresses at constant strains together with the contraction ratios are shown as functions of the tensile axis angle in Fig. 4. The contraction ratio showed a somewhat similar variation to that exhibited at room temperature, and as with the flow stress anisotropy, decreased with increasing strain. Perhaps the most notable difference between the room temperature and elevated temperature results is the increased level of two-fold anisotropy in the superplastic tension. This is most obvious in the flow stress results but also occurs to a lesser extent in the variation of contraction ratio with tensile axis orientation.
36 34
o
32
u (ME+
750 -
30 2s
- 1.2 - 1.0
0.5 -
0.5 L
-11.0
0.4 q 0.3
o,,~ 0“
45’ a
90”
~woo”z 0“
Fig.2. Thevariationof flowstress,U,at constantU.C,and a contractionratio, q, followinga strainof about0.15,with anglebetweenthe tensileaxisand the rollingdirectionfor Fig. 4. The variation of flow stress, at a rate of testsat roomtemperature.Therelatedr-valuescaleisgiven 0.9 x 10-3s-’, and total contraction ratio with angle for the contractionratioresults,and theinterpolatinglines betweenthe tensile axis and the rollingdirection for tests at 970°C.The values are for the total strains indicated. are froma Fouriercosineseries.
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3.2. Microstructure
The microstructure of the material comprised about equal fractions of ferrite (6) and austenite (y) phases at 970°C and in the condition tested at room temperature. This banding was also apparent in sections normal to the rolling direction. The initial microstructure consisted of bands of alternate 6 and y, with elongated grains in both phases, as shown in Fig. 5. The mean random linear intercept lengths of ferrite and austenite grains were 6.1 and 5.4 Km, respectively. The microstructure coarsened with straining at elevated temperature, for example, the mean random intercept lengths of ferrite and austenite grains following a strain of 0.3 at a strain rate of 10-3S-l at 970°C were 6.9 and 6.1 pm respectively.Following the work of Siegmund et al. [7] on thermal cycling of duplex stainless steel, the contiguity parameters CJ and C?were calculated as c’= 2s$’/(2s:’ + St’) c’= 2YJ/(2sy + S$’) (4)
Fig. 6. Optical micrograph of material deformed to a strain of 1, at 970”Cwith a strain rate of 10–3S–l,with the tensile axis parallel with the rolling direction. As well as the austenite inclusionshapes,this showsremnants of the initial structural banding (indicated between arrows).
where S“ is the surface area per unit volume of the grain boundary types identifiedby the superscripts,in practice the boundary length per unit area on sections were used. These parameters describe the relative likelihood of boundaries being between grains of the same and different phases. In addition, the cluster parameters r~ and r’ are defined as r* = NS/(Na+ NY) rY= NY/(N8+ NY)
(a)
(b) Fig. 5. The microstructure of the as-receivedmaterial. The optical micrograph, (a), shows the banded structure of alternate ferrite (dark) and austenite (light) phases. The scanningelectron micrograph (b) showsthe grain structure, which is elongated in the rolling direction.
(5)
where N refers to the number of isolated grain clusters of the relevant phase, identified by the superscript.In the initial microstructure, the contiguities were Ca= 0.23 and CY= 0.18, and the cluster parameters were r’ = 0.43 and r’ = 0.57. In that condition then, the microstructure cannot be said to consist of inclusionsof either phase in a matrix of the other. However, followinga strain of 0.3 at a strain rate of 10-3S-l at 970°C,the contiguity of the ferrite had increased to Cs = 0.44 and that of the austenite had reduced to C’ = 0.09, and the cluster parameters had changed significantly to give values close to rs = O and ry = 1. The austenite is then present as inclusionsin a ferrite matrix. Although the austenite broke up into separated inclusions in hot deformation at quite small strains, those inclusionstended to remain arranged in layers parallel with the sheet plane, and some of the initial banding was still discernibleafter large strains, as shown in Fig. 6. The individual y inclusionswere irregular, as can be seen in Fig. 6 and also in the sheet plane section from material deformed to the same strain, in Fig. 7. 3.3. Crystallographic textures Pole figure type representations of the austenite and ferrite textures are shown in Figs 8 and 9, respectively.In both cases,heating to 970”Cfor times comparable with tensile testing durations did not affect the preferred orientation to any significant extent. The initial austenite texture was quite weak, and can be approximately described as {011}<112) with limited amounts of other orientations lying on
SONG and BATE: PLASTICANISOTROPY
Ih
2751
/a
E.(M
Fig. 7. Optical micrograph of material deformedto a strain of 1, as in Fig. 6, showing the irregular shapes of the austenite inclusions in the sheet plane.
the /3fibre of orientations, between {011}(112) and {225}(544), whichis typical of the rolling textures of f.c.c. metals. The initial ferrite texture was stronger than the austenite and was dominated by orientations near {001}(110). The textures changed significantly with hot deformation: in general they became somewhat weaker but there were noticable changesinthe type oftexture, whichwillbediscussed below.
E=]
—
90”
w
w
“-”””
v
“-’
Fig. 9. The texturesof the ferrite phase presentedin the same way as the austenite textures of Fig. 5. In the case of the texture for a strain of 1 at 45°, the dashed lines indicate intermediate levels of 0.5 and 1.5 times random density.
4. DISCUSSION
The most important result of this work was the measurement of the form of the plastic anisotropy. There was a significantvariation of both flow stress and contraction ratio with tensile axis orientation, both at room temperature and at a temperature and strain rate where the material was behaving in a superplastic manner. There are both similarities and differencesin the anisotropy exhibited in those two regimesand whether the similaritiesare fortuitous or indicate fundamental similarities in the deformation mechanisms at room and elevated temperature, and whether the differences are of significance,can be assessedby comparing the experimental results with approximate models. Some of the other features observed, specifically the development of microstructure and texture, are also useful in attempting to distinguish between possible mechanisms of deformation in the stainless steel. 4.1. Anisotropy due to slip
Fig. 8. The textures of the austenite phase, presented as (001) axis distributions with respect to specimenaxes, i.e. recalculated OOhpole figures, with orthotropic symmetry assumed.The strains and tensiledirectionsindicatedrefer to deformation at 970°C. The centre of the stereographic projectionquadrants correspondsto the sheet plane normal, and the tensile axis directionson the projectionsare inclined appropriately. For the zero strain condition,the upper quadrant is for material held at 970”C for 103s, and the lower quadrant is for as-receivedmaterial. The levels are multiples of random density.
The use of the harmonic series expansion to describe the texture makes the prediction plastic anisotropy in textured polycrystals simple if the strain is taken to be uniform and slip is the deformation mechanism, i.e. as assumed in the Taylor [8] model. Sincethe early work by Bunge and
Roberts [9] and Kallend and Davies [10], many results using this approach have been published. The extensionto rate sensitiveslip is straightforward, and this was used by Bate [5] to model plastic anisotropy in a superplastic aluminium alloy. In that case, quite reasonable correlation between the
2752
SONG and BATE: PLASTIC ANISOTROPY
predicted form of plastic anisotropy and that observed in practice was found, and this approach has been applied to the present results. In all cases octahedral glidewas assumed,i.e. {111}<110) for the austenite and {110}(111) for the ferrite. Pencil glide, {W}
1.9 ------
1.8
Y
----
------/
6
~l’co 1.7 k 1.6
E = 0.3 t
1.9 ‘ ----
1.8
6< -----
Y----
1.7 E=l
1.6 b
I
I
,
I
,
0
15
30
45
w
75
90
a
Fig. 11.The predictedvariation of flowstress relative to slip stress at an equal rate of shear to uniaxial tension, TO, calculated for the two phases with the textures for different tensile strains and directions using the rate-sensitive slip model, The results are interpolated using a Fourier cosine series.
often observed, and is due to the small radii of curvature whichcan occur on yieldsurfacespredicted using the Taylor method. This makes contraction ratio resultsvery sensitiveto variations in stress state. 3.5 The presence of even quite small rate sensitivitiesof slip makes a significant difference to these radii of 3.25 curvature, but as this work is more concerned with the anisotropy in hot deformation no further work was carried out on the room temperature anisotropy. % 3 Rate sensitive slip, obeying a power law relationship betweenslip stressand slip strain rate with a rate 2.75 sensitivity index of 0.5, was used for modelling the hot tensile anisotropy. Predictions were made for 0.8 angles between the tensile axis and rolling direction 1.0 of 0°, 45° and 90° using data from the initial and 0.7 !.0 deformed experimental textures of both phases 0.6 measured in the appropriate tensile test pieces. The .0 0.5 results are givenin Figs 11and 12.In addition to the ~ o,~ uniform strain rate calculations, a uniform stress, t5 0.3 lower bound [12]calculation was carried out using 0.2 the initial ferrite texture, for comparison with the 0.1 upper bound result, and the contraction ratio I 0 predictions from this are shown in Fig. 12. The 15 30 45 60 75 90 o anisotropy predicted is of the same form but is of a slightly reduced magnitude. Fig. 10. The variation with tensile axis angle, a, of flow Although the dominant four-fold component of stress relative to slip stress, T,and contraction ratio/r-value predicted for the ferrite (6) and austenite (y, dashed) using anisotropy, manifest in both flow stress and the rate insensitiveTaylor model with the initial textures. contraction ratio results, is reasonably consistent
SONG and BATE: PLASTIC ANISOTROPY 0.6
1.4 1.2 1.0
0.5
0.8 0.4
2753
distribution of plane normals and sliding directions, the latter of which was assumed to be random in the present case. The strain rate due to each slidingmode was given by:
0.6
0.3
0.4
SEC) t ’36 ~
0.3
,.4
0.4
E =0.3
t
Cti= ~ ~(~iVj+
~iVj)
(6)
i
i
where y is the sliding rate for each system, P is the slide direction vector and v is the grain boundary normal vector; these being perpendicular.The sliding rate was determined by the resolved shear stress on the slidingsystem,~GBs, whichis related to the applied stress, a, by 7G~s=
~iVjOij.
(7)
So far, then, this is analogous to the Sachs[12]type of lowerbound model of polycrystalplasticitywith grain boundariesreplacingcrystal slipplanes and a random distribution of directions lyingin the grain boundary planes replacing crystal slip directions. The distribution of grain boundary normals was describedby a 0.3 ,=, -JOA I seriesof surfaceharmonics,in a manner analogous to I t , , , I I pole figures, with orthotropic symmetry. The grain 15 30 45 60 75 90 o boundary normal distribution function was then used a to weightthe summationfor total strain rate. The final Fig. 12. The values of contraction ratio/r-value calculated part of the model was the relationship between zG~~ using the rate-sensitive slip model, corresponding to the stress predictions of Fig. 8. The short-dashed line for zero and y. In the simplestcase, this was assumed to be a strain is based on the lower bound, uniform stress, simple power law with a rate sensitivityexponent of assumption. 0.5, the same as the overalldeformation. In addition, modelling was carried out in which the relationship betweenslidingstressand rate weredeterminedby slip with Nip predictions, both the average level of accommodation. For each slidingmode, a set of 1000 contraction ratio and the two-fold component of grain orientations was employedand the rate sensitive anisotropy apparent in Fig. 4 reveal deficiencies. crystallographicslipmodelused to calculateyin terms In all cases the overall level of contraction ratio of ~@.,essentiallyfrom equating the work rate given predicted is greater than that measured. Transverse bytheirproduct with that givenby the minimumwork stresses due to curvature of the specimen gauge rate of the slip required for accommodation of the would tend to have a greater effectin reducingwidth grain boundary sliding strain. The slip contributions reduction than in reducing thickness reduction, from the grain boundary/grain orientation combibecause they depend, to a first-order approximation, nations were weightedby the orientation probability on the dimension divided by the radius of curvature. of each accommodating grain as well as the grain At the mid-gaugesof the hot tensile specimens,radii boundary normal probability and summed for the of curvature were measured to be about 2.5 m, total strain rate. and the transverse stresses in the width direction It was possible, by adjusting the distribution would only be about 10-3of the applied stress,which function of grain boundary normals, to mimic the is negligible. The predicted two-fold component observed anisotropy very well without involvingslip consistently has the opposite sense to that observed. accommodation in a textured polycrystal. This is Before discussing a possible reason for these shown for two sets of harmonic coefficients, differences,an alternative approach to modelling the optimised to give the correct form of anisotropy plastic anisotropy in hot deformation will be observed at strains of 0.3 and 1, in Fig. 13. These predictions are close to being in exact agreement explored. with the experimental data. The anisotropy pre4.2. Anisotropy due to grain boundary sliding dicted using a random distribution of sliding grain The plastic anisotropy observed can potentially boundary normals with accommodating slip deterbe explained using grain boundary sliding as the mined by the initial ferrite texture-which would principal deformation mechanism rather than rate give the greatest anisotropy—is also shown. In that sensitive slip, and an approximate model based on case, though there is flow stress anisotropy of a this mechanism was constructed. In this, the strain reasonable magnitude, there is almost no anisotropy rate of the specimen was given by the sum of strain of contraction ratio. Calculations with both a rates due to sliding on boundaries with a specified non-random distribution of sliding grain boundary 0’
~1.4
SONGand BATE: PLASTICANISOTROPY
2754 1.2
Grain boundary
=2
contraction ratios in tensile deformation along the rolling and along the transverse directions. Becauseit 1.1 Grain boundary ,/ notionally represents the amounts of grain boundary distribution B / sliding surface in different spatial orientations, it is ‘\ / \ ~/G I / \ tempting to equate the shape of this body with the [ \ \ +. shape of the austenite inclusions,with the assumption 0.9 / that sliding occurs predominantly on the 6–Y Random grain boundary boundaries and all such boundaries offer equal distribution -slip controlled resistanceto shear. Certainly, flatteningin the rolling 0.5 1.0 plane is common but there is no evidence of the ---C‘iG-&”m~mTnGR&-– distribution -slip controlled 45°corners, and that latter feature is necessaryto give - 0.8 the dominant four-fold character of the anisotropy. 0.4 Because of the complications associated with 0.6 r q re-entrant inclusion shapes, i.e. inclusions with concave regions of their surfaces, the possibility that 0.3 0.4 6–6 and y–y grain boundaries are involved and / Grain Lwmdary probable inequalitiesin slidingresistancedue to grain distribution A , , t I boundary crystallographyetc., no attempt was made 0.2 o 15 30 45 60 75 90 to measure the grain boundary normal distribution. a Becauseof this, a slidinggrain boundary distribution Fig. 13. The predicted variations of flow stress normalised of the correct form cannot be ruled out, but on the to an average of unity, and contraction ratio/r-value made basis of observation the features which would lead to using the grain boundary sliding model, with either nonrandom distributions of sliding surfaces (full lines) or a the four-fold component of anisotropy are not random distribution with rate sensitiveslip accommodation thought likely to exist in the microstructure. /
distribution A
I
(dashed lines).
planes and rate sensitive slip accommodation determined by the texture give results which show that the contraction ratio is dominated by the sliding plane distribution, whereas the relative flow stress variation with tensile axis angle is intermediate betweenpredictionswith random boundaries and slip accommodation and with no slip accommodation and a non-random distribution of sliding planes. The simplegrain boundary slidingmodel givesvery good correlation with experimentalresults. However, how reasonable is the distribution of sliding grain boundary planes which give that good correlation? To test this, it is possible to construct convex shapes which have the same distribution of surface normals as used in the model. These “average slidingbodies” give a reasonable representation of the relative proportions of sliding grain boundary orientations which would need to be present to give the required anisotropy. The one corresponding to the sliding plane distribution optimised for correlation with results at a strain of 0.3 is shownin Fig. 14.The shape of this “average sliding body” is prejudiced by the use of a limited harmonic series to describe the surface distribution and other shapes could well give surface distributions which would generate adequate predictions of anisotropy. Nevertheless, some features are necessary. The body must be flattened in the rolling plane, and must have corners in the 450-type directions. That latter feature is important because it is necessary to give the limited amount of grain boundary sliding surface with normal directions bisectingthe rolling and transverse directions of the sheet, and so give the relativelylow
4.3. Anisotropy due to mechanical inhomogeneity Leaving grain boundary sliding aside for the moment and returning to deformation by intragranular slip, a featurewhichmust now be considered is the mechanical inhomogeneity of the material. Hutchinson et al. [11] showed that the room temperature anisotropy of a duplex stainless steel was due to crystallographic texture and not phase disposition. At room temperature, their microhardness measurementsindicated that the flow stressesof the two phases are likely to be quite similar. At elevated temperature, however, there will be a large difference between the flow stresses of the phases. Chubb [13] reported that the hot hardness of austenite was three times that of ferrite for pure iron in the vicinity of the transformation at 920”C. The 1
RD
N
TD
BE ND
R
TD
eE3
Fig. 14.Projectionsof the shape of a body which givesthe distribution of slidingsurfaces used for the grain boundary distribution type (A) predictions shown in Fig. 10.
SONGand BATE: PLASTICANISOTROPY work of Rossard and Blain, reviewedby Jonas et al. [14], gives flow stresses of FGO.25?40Cwhich are between twice and four times that of Fe–25%Cr at 1100”C with strain rates between about 10-3 and 2 S-’, which is consistent with Chubb’s hardness values. Because of the phase distribution and y inclusion shape in the stainlesssteel, this differencein flow stress can affect the plastic anisotropy. Under conditions of principally intragranular deformation, this difference in flow stress will lead to a system of internal stress. Deviations from a spherical average shape of the austenite inclusions will give rise to a contribution to the anisotropy which can be estimated to a first approximation on the basis of Eshelby’sanalysis of the elastic field in an ellipsoidalinclusion[15].In this, the elastic stresses inside an ellipsoid, o‘, due to a “transformation” strain, 6T,are given by: ~,~= C~kl(&tn81. —Sl+n.& ) ,
(8)
where C are the elastic stiffness modulii and S, the “Eshelby tensor”, is determined by the shape of the ellipsoid. The elasticity is isotropic and homogeneous. Now, although both phases are deforming plastically, the difference in their flow stresses will mean that a small difference in the plastic strains inside and outside the inclusion will exist which, by acting as the “transformation” strain in equation (8), will give rise to stresses inside and outside the inclusion which will then ensure stress equilibrium in this situation of unequal flow stresses. This type of approach, where a plastic strain mismatch is treated as the “transformation” strain, has been used in connection with models of strain hardening in materials with heterogeneous dislocation distributions, for example see Argon and Haasen [16]. The “transformation” strain, equivalent to the plastic strain mismatch in the present case, is assumed to have the sameproportions as the overallstrain rate and a magnitude which shall be treated, for present purposes, as a free parameter. There will also be an associated mean elastic stress outside the inclusion and these internal and external stressesrepresent the differencesbetween the applied stress and the actual flow stress of each phase, i.e. the forward and back stressesgenerated by the mechanical inhomogeneity. The austenite of the inclusions is relativelyisotropic, and with that assumption as well, the effective increase in flow stress can be calculated as some multiple of
2755
“:F 0.5
1.0
< 0.7,0.5> 0.8 0.4 0.6 r 0.3
0.4
<0.5,0.35>
0.’~ o
15
30
45
60
75
90
a
Fig. 15. Predictions of normalised flow stress and contraction ratio/r-value made using the rate-sensitive slip model augmented by stresses predicted using the Eshelby ellipsoidalinclusionmodel.The valuepairs indicatedare the ratios of the ellipsoid axial lengths in the transverse sheet direction and the sheet plane normal, respectively,to that in the rollingdirection. The ferrite textures used were averages of the initial and 0.3 strain for the <0.5,0.35) ellipsoid,and averages of those for strains of 0.3 and 1 in the case of the (0.7, 0.5) ellipsoid.
rate are determined. Various weightings for the contribution of a+ and values of the ratio of ellipsoid semi-axeswere tried, to evaluate the potential for the mechanicalinhomogeneityto explain the anisotropy. Making the stress contribution u+ about two-thirds of the total with modest ellipsoid axial ratios gave encouraging results, as shown in Fig. 15. It is certainly possible to reduce the mean level of contraction ratio predicted to a similar value to that observed,and to givethe correct senseof the two-fold component of anisotropy. As with the “equivalent sliding body” given above, it is not known how well the mean ellipsoid shapes (used to give the results of Fig. 15)relate to the real microstructure. No attempt was made to make a correlation because it is by no means clear how the quite complex shapes of individual y inclusions and their spatial distribution would contribute in reality, and it is most unlikely that any simple averaging of geometric parameters would be adequate. However, despite this, it is clear that a combination of intragranular slip, crystalloa+ = &;/J(&:n), (9) graphic texture and internal stresses resulting from which represents, then, the elastic strain energy the plastic deformation of a mechanically inhomodensity in the inclusion divided by the effective geneous system has the potential for giving the “transformation” strain, which is proportional to the correct type of anisotropy. overall effective strain. This can be weighted and added to the flow stress, relative to slip stress at 4.4. The evolution of microstructure and texture Two quite different approaches can be made to unit strain rate, of the ferrite calculated by the rate-sensitive slip model, and so the relative flow explainingthe plastic anisotropy of the stainless steel stress and the contraction ratio for minimum work under conditions of superplastic tension. In both of
2756
SONG and BATE: PLASTICANISOTROPY
these, geometric features of the microstructure have been simplifiedfor convenienceand dealt with as free parameters in order to assess the potential of those approaches. There is a major difficultycaused by the complexity of the real microstructure and how that can be adequately represented in a simple way, and it is therefore not possible to determine whether slip or grain boundary sliding was the principal deformation mechanism during superplastic tension on the basis of plastic anisotropy measurements in the material used in this work. The modelling indicates that grain boundary sliding controlled by rate sensitiveaccommodating slip is not appropriate, and although the anisotropy can be predicted very well using a grain boundary sliding model, the distribution of sliding surfaces required to give, particularly, the four-foldcomponent of anisotropy is probably unrealistic. Other observations are more consistent with a slip-based mechanism than with grain boundary sliding. The most obvious is the preservation of banding in the structure to high strains, as shown in Fig. 6. The preservation of linear alignment,either of grain structure [17, 18] or of internal markers [19], over significant length scales is very difficult to account for with the type of relative grain translation which would occur with grain boundary sliding. Although they are lessobvious,the changesin texture are probably more consistent with slip-based deformation as well. The evolution of crystallographic texture with deformation is not consistentwith homogeneousslip, particularly in the ferrite. There is a notable aspect of the texture development, however, which is best shown by inverse pole figures in the tensile axis directions. These are given in Fig. 16 for the ferrite
OOAAA
0.3 &—
1
w“ Fig. 17.Inversepole figuresin tensile axis directions for the ferrite following deformation to the strains indicated at 970°Cat a mean rate of 10-3s-’. Levelsare in multiples of random density.
and Fig. 17 for the austenite. Despite large initial differencesin the distributions of crystal directions in the tensile directions, these become quite similar after deformation. In the ferrite a <111) alignment develops,which is most marked in the a = 90° case. In the austenite, a (011) + near (111) alignment develops which is similar to the initial rolling direction alignment. The ferrite texture is most interesting, as tension would tend to give (011) alignmentwith homogeneousslip,and that alignment is clearly depleted here. A deformation mode which does give the (1 11) alignment is simple shear on planes containing the tensile direction. This is consistent with inhomogeneous deformation due to the 6 – y flow stress difference. Rather than this differencegiving rise to a simple strain mismatch at the phase boundaries, assumed for simplicity in the application of the Eshelby model given above, the mismatch in this case becomes more distributed through the ferrite, in particular, giving rise to considerable amounts of shear in the vicinity of the austenite inclusions.This does not actually make the application of the Eshelby method invalid, in fact the final part of the paper dealing with the ellipsoidal inclusion [15] makes reference to the methods applicabilityto the case of inclusionsin viscousfluids, using the analogy between linear elastic and linear viscous behaviors. However, the relationship with the plastic anisotropy becomes more complex and would require, say, finite element modelling to solve in a more satisfactory manner. Despite the complex nature of deformation at elevated temperature in this material, several of the experimentalobservationsare probably more consistent with slip being the principal deformation mode. They include the significant four-fold component of plastic anisotropy, the preservation of aligned
::::!;! ;::”,: ]4soAAA WOAAA o 0.3
1
&—
0.511
.522.5
Fig. 16.Inverse pole figuresin tensile axis directionsfor the austenite followingdeformation to the strains indicated at 970”Cat a mean rate of 10-3s-’. Levelsare in multiples of random density.
SONG and BATE: PLASTIC ANISOTROPY
microstructure to a reasonably large strain and the changes in texture. Although the texture changes in the ferrite were not those which would occur with homogeneous deformation, and involved a general reduction in intensity, they were consistent in a way which would be very difficultto account for if grain boundary sliding were the dominant mechanism. Although, as Padmanabhan [20]has argued, crystallographicallydifferentboundaries would be expected to slideat differentrates and so affecttexture changes in superplastic flow, it is very difficultto see how this could account for the present experimental results. It was suggested previously [5] that superplastic deformation is likely to be inhomogeneousbetween grains and that this could, in principle, lead to deviations from predictions of texture changes based on homogeneous slip, whilst anisotropy, being an averaged property, may not be affectedto any great extent by the inhomogeneity. In this context, the results presented here are useful because they are from a material with an intrinsic inhomogeneity of large magnitude, rather than that which might occur in nominally single-phasesuperplastic materials. 5. CONCLUSIONS The plastic anisotropy of the duplex stainless steel SAF2304, measured as both flow stress and contraction ratio variations with tensile axis orientation, shows similar four-fold anisotropy in both room temperature and superplastic deformation. However, general levels of contraction ratio and the two-fold component of anisotropy are different in those two regimes of deformation. The plastic anisotropy could be modelled using grain boundary sliding,but the distribution of sliding surface required is somewhat unrealistic. The four-fold component of the anisotropy seems likely to be due to crystallographic texture, particularly in the ferrite, but slip anisotropy associated with this texture only gives the four-fold anisotropy of contraction ratio if intragranular slip, rather than slip accommodating grain boundary sliding, is assumed. It is possible, in an approximate way, to account for the difference between slip induced anisotropy and that actually measured by including an
2757
anisotropic contribution related to the flow stress differencebetween the austenite and ferrite, coupled with the spatial anisotropy of the austenite morphology and distribution. Preservation of banding in the microstructure to large strains and systematic changes in crystallographic texture indicate that intragranular slip, with a significantinhomogeneous component, is probably the dominant mechanism in superplastic deformation of this material. Acknowledgements—Thanks are due to the Committee of Vice-chancellorsand Principals of the Universities of the United Kingdom for an ORS award for one of us (J.L.S.), to W. B. Hutchinsonfor material and to P. L. Blackwellfor his comments and advice.
REFERENCES 1. Johnson,R. H., Packer,C. M., Anderson,L. and Sherby,O. D., Phil. Mug., 1968,18, 1309. 2. Melton, K. N., Cutler, C. P. and Edington, J. W., Scriptametall., 1975,9, 515. 3. Bricknell, R. H. and Edington, J. W., Acta metall., 1979,27, 1313. 4. McDarmaid, D. S., Bowen,A. W. and Partridge, P. G., J. Mater. Sci., 1984,19, 2378. 5. Bate, P. S., Meta[l. Trans., 1992,A23, 1467. 6. Hill, R., The Mathematical Theory of Plasticity. Oxford University Press, Oxford, 1950,pp. 25-26. 7. Siegmund,T., Werner, E. and Fischer, F. D., Mater. Sci. Engng, 1993, A169, 125.
8. Taylor, G. 1,, J. Inst. Metals., 1938, 62, 307. 9. Bunge, H.-J. and Roberts, W. T., J. appl. Crystall., 1969, 2, 116. 10. Kallend, J. S, and Davies, G. J., J. Inst. Metals, 1971, 99, 257.
11. Hutchinson, Ushioda, K. and Runnsjo, G., Mater. Sci. Technol., 1985, 1, 728.
12. Sachs, G., Z. Ver. Dtsch. Ing., 1928, 72, 734. 13. Chubb, W., Trans. AIME J. Metals, 1955, 203, 189. 14. Jonas, J. J., Sellars, C. M. and McG. Tegart, W. J., Metall. Rev., 1969, 14, 1. 15. Eshelby, J. D., Proc. R. Soc., 1957,241A,367. 16. Argon,A.andHaasen,P., Acta metall. mater., 1993,41, 3289.
17. Blackwell,P. L. and Bate, P. S., Metall. Trans., 1993, A24, 1085.
18. Blackwell,P. L. and Bate, P. S., in Superplasticity: 60 years after Pearson, ed. N. Ridley. The Institute of Materials, Book no. 618, London, 1995,pp. 183–192. 19. Blackwell,P. L. and Bate, P. S., Metall. Mater. Trans., 1966, A27, 3747. 20. Padmanabhan, K. A., Metal Sci., 1980,14, 506.
mater.Vol.45,No.7,pp.2747-2757, 1997 (j 1997ActaMetallurgicInc. PublishedbyElsevierSci;nceLtd Printedin GreatBritain,Allrightsreserved 1359-6454/97 $17.00+ 0.00
PLASTIC ANISOTROPY IN A SUPERPLASTIC DUPLEX STAINLESS STEEL J. L. SONG and P. S. BATE
IRC in Materialsfor HighPerformanceApplicationsand Schoolof Metallurgyand Materials, The Universityof Birmingham,BirminghamB152TT,U.K. (Received21 August 1996;accepted6 November1996)
Abstract—Measurements of the plasticanisotropyin uniaxialtensionof the duplexstainlesssteel, SAF2304, havebeenmadeat roomtemperatureandunderconditionswherethematerialwassuperplastic. Therewassignificantplasticanisotropyin both typesof deformationand thereweresomesimilarities betweenthe lowand hightemperaturevariationswithtensileaxisorientation.Althoughit waspossible to modelthehightemperatureanisotropyusinga grainboundaryslidingmodel,theassumeddistribution of slidingboundarieswasconsideredto beunrealistic.This,togetherwithaspectsofmicrostructuraland texturaldevelopment,indicatedthat deformationwas principallyoeeurringby intragranularslipwith a significantcontributioncausedby mechanicalinhomogeneity in the two-phasematerial.0 1997Acta Metallurgic Inc.
1. INTRODUCTION
superplastic conditions, and at room temperature, and its anisotropy analysed.
There have been severalreports of plastic anisotropy in superplastic materials. Johnson et al. [1]observed 2. EXPERIMENTAL METHODS that the circular cross-sections of Zn–Al test pieces became elliptical after superplastic deformation, and 2.1. Material concluded that slip was responsible for this effect. The material used was the duplex stainless steel Later work on a variety of alloys has not necessarily SAF2304,manufactured by Avesta Sheffield,with the led to that conclusion.Melton et al. [2], workingwith chemical composition: P&Sn, and Bricknelland Edington [3],working with Cr Ni Mo N c Fe A1–Cu–Zr plate, both concluded that mechanical Element Wt.percent 22.8 4.8 0.3 0.092 0.017 balance fibring was responsible for plastic anisotropy in the superplastic regime. In particular, Bricknell The material was hot rolled and solution treated at and Edington showed that the plastic strain ratios 108O”Cbefore being cold rolled to 2 mm thickness, observed at slow strain rates were lower than those with an intermediate and a final solution treatment. predicted by slip-based models, but that the differencesin strain ratio between specimensparallel and perpendicular to the rolling direction of the 2.2. Tensile testing Uniaxial tensile testing, both at room and elevated plate remained more or less the same as at higher strain rates, where slip was assumed to occur. The temperature, was carried out using a screw drive mechanical fibring argument is closely allied to the machine controlled via a microcomputer, which also common assumption that grain boundary sliding is recorded the test data. Tests were carried out on the principal deformation mechanism in superplastic specimensprepared using electro-dischargemachinflow, and has been taken to be the cause of plastic ing such that the tensile axeswere at angles of 0°, 45° anisotropy in several instances, for example and 90° to the rolling direction of the sheet. Room temperature tests used specimens with a McDarmaid et al. [4] in Ti–Al–V. However, the plastic anisotropy observed in A1–Li–Mg–Cu–Zrby parallel gauge section 12.5MMwide and 50mm long. Bate [5] showed a dominant four-fold anisotropy Electronicextensometerswere used to monitor width which was thought to be due to slip-based as well as length during the tests, and in addition deformation. In the work presented here, a material scribe marks on the gauge portion were measured has been used which has a microstructure which is before and after testing to allow total strains in likely to show significant mechanical fibring effects those directions to be calculated. These tests were and possessesa sufficientlywell developed crystallo- terminated before the natural maximum load graphic texture to give a reasonable degree of condition to minimise necking effects on the total slip-related anisotropy. It has been tested under strain measurements. --. , L147
SONG and BATE: PLASTICANISOTROPY
2748
Elevated temperature testing was carried out, in air, at a temperature of 970”Cusing specimenswith a gauge length of 35 mm and gauge width of 9.5 mm. The cross-head was controlled to give, on the assumptions that the gauge section remained parallel and that its volume was constant, a strain rate, ;, which alternated between values of 0.9 and 1.1x 10-3s-’ every 0.025 of strain. The resulting steps in the stress, rr, were interpolated to give two curves for the high and low rates which allowed the strain rate sensitivity index, m, to be approximated as: m = d in a/d in; = A in a/A in 1.
(1)
The relatively small perturbations in strain rate and small strain interval means that the average stress develops with strain in very much the same way as in a constant rate test. In the case of the elevated temperature tests, metallographic specimens from the gauge were prepared such that they could be measured using a traveling microscope to give total strains in the width and thickness directions in addition to measurements made using a micrometer of the cross sections before and after deformation. As well as the influence of tensile axis orientation on flow stress, plastic anisotropy in the uniaxial tensile tests was quantified using the contraction ratio, q, determined in terms of total strains in the width, EW, and the length, G, directions by: q = –&w/&l.
(2)
This is related to the more frequently used r-value measure of anisotropy by: r = .5w/6h = q/(1 —q),
(3)
where ~h is the strain in the thickness direction. The isotropic value of q for deformation at constant volume is 0.5, corresponding to r = 1. Because extensometers were used for the room temperature tests, incremental contraction ratios, effectivelythe ratios of strain rates rather than total strains, could also be measured.
diffraction with MoL radiation. Reflections used were 002, 112 and 123 for the ferrite, and 002, 022 and 113 for the austenite. In order to give through-thicknessaverage measurements, composite specimens were made. Slices about 1mm thick were cut from specimen gauges after deformation and from undeformed material using a slow-speed diamond saw. Thesewere then glued onto glassslides with epoxy resin and the exposed faces ground, polished and etched to an optical metallographic standard. Alignment errors were unavoidable with this procedure and the small pieces involved, but these were less than 5° in all cases. Becauseonly three incompletepole figuresfor each phase were measured, the even-order coefficients of the harmonic series expansion describing the texture werecalculatedto a truncation levelof L = 16 rather than the more usual L = 22. These coefficients were transformed to a basis with the Z-axis in the sheet normal direction and the X-axis in the tensile direction. These coefficients were used to predict anisotropy and to produce maps of the distribution of (001) directions, equivalent to recalculated OOh pole figures, and inverse pole figures for the tensile directions. The pole figure type representation is adequate for the textures involved. 3. RESULTS 3.1. Tensile test results
Testing at room temperature gave the stress–strain results shown in Fig. 1. There was a marked flow stress anisotropy, with the highest flow stress where the tensile axis was in the sheet transverse direction and the lowest when it was at 45° to the rolling direction. Because there was significant strain hardening, taking levels of flow stress at a constant strain will not give an adequate representation of anisotropy for equivalent work hardened states. It is better to consider the flow stresses at some given amount of plastic work as this will, in simple
2.3. Metallography For conventionaloptical metallography,specimens were electrolyticallyetched in an aqueous solution of 56 wt% KOH at 5 V. This darkened the ferrite (d) and left the austenite (y) light. It was not possible, however, to reveal the grain structure with optical metallography and so scanning electron microscopy, using electron channeling contrast, was used for that purpose. It was necessary to identify the phase of individualgrains by examination of the backscattered Kikuchi patterns. Quantitative measures of the microstructure were made using commercial image processing and analysis software. 2.4. Texture determination The crystallographic texture was determined from partial reflection pole figures measured using X-ray
Sool-
>
IL____ o
0.05
0.1
0.15
2
&
Fig. 1.Uniaxialtensileflowstressdevelopmentwithstrain at roomtemperaturewiththeanglesbetweenthetensileaxes and sheetrollingdirectionindicated.The dashedlineis a hyperbolicsegmentused to give values of stress for comparisonwithpredictions,as describedin the text.
SONGand BATE: PLASTICANISOTROPY slip-based models of plasticity at least, represent a constant amount of accumulated slip strain. The idea that the degree of plastic work should determine the hardening state has been discussed by Hill [6] in a more general sense. Rather than integrating the stress–strain curves, if it can be assumed that the stress–strain curves are similar in form, i.e. a simple scalingof stress and strain can make them coincident, then levels of flow stress at some constant value of o.~ will represent the flow stress anisotropy by approximately compensating for the increased work dissipatedwith a higher flowstress.Such a hyperbolic contour is shown in Fig. 1, and the variation of flow stress with tensile axis angle derived on this basis is shown in Fig. 2, along with the contraction ratio results, q. Those contraction ratio results, shownwith scalesof both q and r-value, showa marked four-fold anisotropy, with low values at 0° and 90°. The hot tensile behaviour is shown in Fig. 3. There was significant strain rate sensitivity of the flow stress, and in all directions values of the rate sensitivity index, m, were close to 0.5 throughout the tests which indicates superplastic behaviour. As would be expected with this high rate sensitivity, the gaugesof specimensdeformedto a strain of unity were very uniform except for the transition at the gripped ends. The flow stress was anisotropic, and there were differences in its evolution with strain in the three directions. Most noticeable was the relativelyhigh flow stress at small strains for tension along the rolling direction, which became almost equal to that for transverse direction tension at a strain of 1. The flow stress at 45° to the rolling direction was always lowest at a given strain. The strain hardening rates were low, and the hyperbolic
2749
40
30
c1
0.6
(MP,) 20 0.5 ~
.
10
I 0
0.2
0.4
0.6
0.s
1
&
Fig. 3. Uniaxial tensile flow stress developmentwith strain at 970”C and a strain rate alternating between 0.9 and 1.1 x 10-3s-’, with a strain period of O.025at each rate. The m values, derived by interpolation of the flow stress at the two rates, are shown by symbols below the stress–strain results. The anglesbetweenthe tensileaxes and sheet rolling direction are indicated.
line method used for the room temperature results was redundant. The flow stresses at constant strains together with the contraction ratios are shown as functions of the tensile axis angle in Fig. 4. The contraction ratio showed a somewhat similar variation to that exhibited at room temperature, and as with the flow stress anisotropy, decreased with increasing strain. Perhaps the most notable difference between the room temperature and elevated temperature results is the increased level of two-fold anisotropy in the superplastic tension. This is most obvious in the flow stress results but also occurs to a lesser extent in the variation of contraction ratio with tensile axis orientation.
36 34
o
32
u (ME+
750 -
30 2s
- 1.2 - 1.0
0.5 -
0.5 L
-11.0
0.4 q 0.3
o,,~ 0“
45’ a
90”
~woo”z 0“
Fig.2. Thevariationof flowstress,U,at constantU.C,and a contractionratio, q, followinga strainof about0.15,with anglebetweenthe tensileaxisand the rollingdirectionfor Fig. 4. The variation of flow stress, at a rate of testsat roomtemperature.Therelatedr-valuescaleisgiven 0.9 x 10-3s-’, and total contraction ratio with angle for the contractionratioresults,and theinterpolatinglines betweenthe tensile axis and the rollingdirection for tests at 970°C.The values are for the total strains indicated. are froma Fouriercosineseries.
2750
SONG and BATE: PLASTICANISOTROPY
3.2. Microstructure
The microstructure of the material comprised about equal fractions of ferrite (6) and austenite (y) phases at 970°C and in the condition tested at room temperature. This banding was also apparent in sections normal to the rolling direction. The initial microstructure consisted of bands of alternate 6 and y, with elongated grains in both phases, as shown in Fig. 5. The mean random linear intercept lengths of ferrite and austenite grains were 6.1 and 5.4 Km, respectively. The microstructure coarsened with straining at elevated temperature, for example, the mean random intercept lengths of ferrite and austenite grains following a strain of 0.3 at a strain rate of 10-3S-l at 970°C were 6.9 and 6.1 pm respectively.Following the work of Siegmund et al. [7] on thermal cycling of duplex stainless steel, the contiguity parameters CJ and C?were calculated as c’= 2s$’/(2s:’ + St’) c’= 2YJ/(2sy + S$’) (4)
Fig. 6. Optical micrograph of material deformed to a strain of 1, at 970”Cwith a strain rate of 10–3S–l,with the tensile axis parallel with the rolling direction. As well as the austenite inclusionshapes,this showsremnants of the initial structural banding (indicated between arrows).
where S“ is the surface area per unit volume of the grain boundary types identifiedby the superscripts,in practice the boundary length per unit area on sections were used. These parameters describe the relative likelihood of boundaries being between grains of the same and different phases. In addition, the cluster parameters r~ and r’ are defined as r* = NS/(Na+ NY) rY= NY/(N8+ NY)
(a)
(b) Fig. 5. The microstructure of the as-receivedmaterial. The optical micrograph, (a), shows the banded structure of alternate ferrite (dark) and austenite (light) phases. The scanningelectron micrograph (b) showsthe grain structure, which is elongated in the rolling direction.
(5)
where N refers to the number of isolated grain clusters of the relevant phase, identified by the superscript.In the initial microstructure, the contiguities were Ca= 0.23 and CY= 0.18, and the cluster parameters were r’ = 0.43 and r’ = 0.57. In that condition then, the microstructure cannot be said to consist of inclusionsof either phase in a matrix of the other. However, followinga strain of 0.3 at a strain rate of 10-3S-l at 970°C,the contiguity of the ferrite had increased to Cs = 0.44 and that of the austenite had reduced to C’ = 0.09, and the cluster parameters had changed significantly to give values close to rs = O and ry = 1. The austenite is then present as inclusionsin a ferrite matrix. Although the austenite broke up into separated inclusions in hot deformation at quite small strains, those inclusionstended to remain arranged in layers parallel with the sheet plane, and some of the initial banding was still discernibleafter large strains, as shown in Fig. 6. The individual y inclusionswere irregular, as can be seen in Fig. 6 and also in the sheet plane section from material deformed to the same strain, in Fig. 7. 3.3. Crystallographic textures Pole figure type representations of the austenite and ferrite textures are shown in Figs 8 and 9, respectively.In both cases,heating to 970”Cfor times comparable with tensile testing durations did not affect the preferred orientation to any significant extent. The initial austenite texture was quite weak, and can be approximately described as {011}<112) with limited amounts of other orientations lying on
SONG and BATE: PLASTICANISOTROPY
Ih
2751
/a
E.(M
Fig. 7. Optical micrograph of material deformedto a strain of 1, as in Fig. 6, showing the irregular shapes of the austenite inclusions in the sheet plane.
the /3fibre of orientations, between {011}(112) and {225}(544), whichis typical of the rolling textures of f.c.c. metals. The initial ferrite texture was stronger than the austenite and was dominated by orientations near {001}(110). The textures changed significantly with hot deformation: in general they became somewhat weaker but there were noticable changesinthe type oftexture, whichwillbediscussed below.
E=]
—
90”
w
w
“-”””
v
“-’
Fig. 9. The texturesof the ferrite phase presentedin the same way as the austenite textures of Fig. 5. In the case of the texture for a strain of 1 at 45°, the dashed lines indicate intermediate levels of 0.5 and 1.5 times random density.
4. DISCUSSION
The most important result of this work was the measurement of the form of the plastic anisotropy. There was a significantvariation of both flow stress and contraction ratio with tensile axis orientation, both at room temperature and at a temperature and strain rate where the material was behaving in a superplastic manner. There are both similarities and differencesin the anisotropy exhibited in those two regimesand whether the similaritiesare fortuitous or indicate fundamental similarities in the deformation mechanisms at room and elevated temperature, and whether the differences are of significance,can be assessedby comparing the experimental results with approximate models. Some of the other features observed, specifically the development of microstructure and texture, are also useful in attempting to distinguish between possible mechanisms of deformation in the stainless steel. 4.1. Anisotropy due to slip
Fig. 8. The textures of the austenite phase, presented as (001) axis distributions with respect to specimenaxes, i.e. recalculated OOhpole figures, with orthotropic symmetry assumed.The strains and tensiledirectionsindicatedrefer to deformation at 970°C. The centre of the stereographic projectionquadrants correspondsto the sheet plane normal, and the tensile axis directionson the projectionsare inclined appropriately. For the zero strain condition,the upper quadrant is for material held at 970”C for 103s, and the lower quadrant is for as-receivedmaterial. The levels are multiples of random density.
The use of the harmonic series expansion to describe the texture makes the prediction plastic anisotropy in textured polycrystals simple if the strain is taken to be uniform and slip is the deformation mechanism, i.e. as assumed in the Taylor [8] model. Sincethe early work by Bunge and
Roberts [9] and Kallend and Davies [10], many results using this approach have been published. The extensionto rate sensitiveslip is straightforward, and this was used by Bate [5] to model plastic anisotropy in a superplastic aluminium alloy. In that case, quite reasonable correlation between the
2752
SONG and BATE: PLASTIC ANISOTROPY
predicted form of plastic anisotropy and that observed in practice was found, and this approach has been applied to the present results. In all cases octahedral glidewas assumed,i.e. {111}<110) for the austenite and {110}(111) for the ferrite. Pencil glide, {W}
1.9 ------
1.8
Y
----
------/
6
~l’co 1.7 k 1.6
E = 0.3 t
1.9 ‘ ----
1.8
6< -----
Y----
1.7 E=l
1.6 b
I
I
,
I
,
0
15
30
45
w
75
90
a
Fig. 11.The predictedvariation of flowstress relative to slip stress at an equal rate of shear to uniaxial tension, TO, calculated for the two phases with the textures for different tensile strains and directions using the rate-sensitive slip model, The results are interpolated using a Fourier cosine series.
often observed, and is due to the small radii of curvature whichcan occur on yieldsurfacespredicted using the Taylor method. This makes contraction ratio resultsvery sensitiveto variations in stress state. 3.5 The presence of even quite small rate sensitivitiesof slip makes a significant difference to these radii of 3.25 curvature, but as this work is more concerned with the anisotropy in hot deformation no further work was carried out on the room temperature anisotropy. % 3 Rate sensitive slip, obeying a power law relationship betweenslip stressand slip strain rate with a rate 2.75 sensitivity index of 0.5, was used for modelling the hot tensile anisotropy. Predictions were made for 0.8 angles between the tensile axis and rolling direction 1.0 of 0°, 45° and 90° using data from the initial and 0.7 !.0 deformed experimental textures of both phases 0.6 measured in the appropriate tensile test pieces. The .0 0.5 results are givenin Figs 11and 12.In addition to the ~ o,~ uniform strain rate calculations, a uniform stress, t5 0.3 lower bound [12]calculation was carried out using 0.2 the initial ferrite texture, for comparison with the 0.1 upper bound result, and the contraction ratio I 0 predictions from this are shown in Fig. 12. The 15 30 45 60 75 90 o anisotropy predicted is of the same form but is of a slightly reduced magnitude. Fig. 10. The variation with tensile axis angle, a, of flow Although the dominant four-fold component of stress relative to slip stress, T,and contraction ratio/r-value predicted for the ferrite (6) and austenite (y, dashed) using anisotropy, manifest in both flow stress and the rate insensitiveTaylor model with the initial textures. contraction ratio results, is reasonably consistent
SONG and BATE: PLASTIC ANISOTROPY 0.6
1.4 1.2 1.0
0.5
0.8 0.4
2753
distribution of plane normals and sliding directions, the latter of which was assumed to be random in the present case. The strain rate due to each slidingmode was given by:
0.6
0.3
0.4
SEC) t ’36 ~
0.3
,.4
0.4
E =0.3
t
Cti= ~ ~(~iVj+
~iVj)
(6)
i
i
where y is the sliding rate for each system, P is the slide direction vector and v is the grain boundary normal vector; these being perpendicular.The sliding rate was determined by the resolved shear stress on the slidingsystem,~GBs, whichis related to the applied stress, a, by 7G~s=
~iVjOij.
(7)
So far, then, this is analogous to the Sachs[12]type of lowerbound model of polycrystalplasticitywith grain boundariesreplacingcrystal slipplanes and a random distribution of directions lyingin the grain boundary planes replacing crystal slip directions. The distribution of grain boundary normals was describedby a 0.3 ,=, -JOA I seriesof surfaceharmonics,in a manner analogous to I t , , , I I pole figures, with orthotropic symmetry. The grain 15 30 45 60 75 90 o boundary normal distribution function was then used a to weightthe summationfor total strain rate. The final Fig. 12. The values of contraction ratio/r-value calculated part of the model was the relationship between zG~~ using the rate-sensitive slip model, corresponding to the stress predictions of Fig. 8. The short-dashed line for zero and y. In the simplestcase, this was assumed to be a strain is based on the lower bound, uniform stress, simple power law with a rate sensitivityexponent of assumption. 0.5, the same as the overalldeformation. In addition, modelling was carried out in which the relationship betweenslidingstressand rate weredeterminedby slip with Nip predictions, both the average level of accommodation. For each slidingmode, a set of 1000 contraction ratio and the two-fold component of grain orientations was employedand the rate sensitive anisotropy apparent in Fig. 4 reveal deficiencies. crystallographicslipmodelused to calculateyin terms In all cases the overall level of contraction ratio of ~@.,essentiallyfrom equating the work rate given predicted is greater than that measured. Transverse bytheirproduct with that givenby the minimumwork stresses due to curvature of the specimen gauge rate of the slip required for accommodation of the would tend to have a greater effectin reducingwidth grain boundary sliding strain. The slip contributions reduction than in reducing thickness reduction, from the grain boundary/grain orientation combibecause they depend, to a first-order approximation, nations were weightedby the orientation probability on the dimension divided by the radius of curvature. of each accommodating grain as well as the grain At the mid-gaugesof the hot tensile specimens,radii boundary normal probability and summed for the of curvature were measured to be about 2.5 m, total strain rate. and the transverse stresses in the width direction It was possible, by adjusting the distribution would only be about 10-3of the applied stress,which function of grain boundary normals, to mimic the is negligible. The predicted two-fold component observed anisotropy very well without involvingslip consistently has the opposite sense to that observed. accommodation in a textured polycrystal. This is Before discussing a possible reason for these shown for two sets of harmonic coefficients, differences,an alternative approach to modelling the optimised to give the correct form of anisotropy plastic anisotropy in hot deformation will be observed at strains of 0.3 and 1, in Fig. 13. These predictions are close to being in exact agreement explored. with the experimental data. The anisotropy pre4.2. Anisotropy due to grain boundary sliding dicted using a random distribution of sliding grain The plastic anisotropy observed can potentially boundary normals with accommodating slip deterbe explained using grain boundary sliding as the mined by the initial ferrite texture-which would principal deformation mechanism rather than rate give the greatest anisotropy—is also shown. In that sensitive slip, and an approximate model based on case, though there is flow stress anisotropy of a this mechanism was constructed. In this, the strain reasonable magnitude, there is almost no anisotropy rate of the specimen was given by the sum of strain of contraction ratio. Calculations with both a rates due to sliding on boundaries with a specified non-random distribution of sliding grain boundary 0’
~1.4
SONGand BATE: PLASTICANISOTROPY
2754 1.2
Grain boundary
=2
contraction ratios in tensile deformation along the rolling and along the transverse directions. Becauseit 1.1 Grain boundary ,/ notionally represents the amounts of grain boundary distribution B / sliding surface in different spatial orientations, it is ‘\ / \ ~/G I / \ tempting to equate the shape of this body with the [ \ \ +. shape of the austenite inclusions,with the assumption 0.9 / that sliding occurs predominantly on the 6–Y Random grain boundary boundaries and all such boundaries offer equal distribution -slip controlled resistanceto shear. Certainly, flatteningin the rolling 0.5 1.0 plane is common but there is no evidence of the ---C‘iG-&”m~mTnGR&-– distribution -slip controlled 45°corners, and that latter feature is necessaryto give - 0.8 the dominant four-fold character of the anisotropy. 0.4 Because of the complications associated with 0.6 r q re-entrant inclusion shapes, i.e. inclusions with concave regions of their surfaces, the possibility that 0.3 0.4 6–6 and y–y grain boundaries are involved and / Grain Lwmdary probable inequalitiesin slidingresistancedue to grain distribution A , , t I boundary crystallographyetc., no attempt was made 0.2 o 15 30 45 60 75 90 to measure the grain boundary normal distribution. a Becauseof this, a slidinggrain boundary distribution Fig. 13. The predicted variations of flow stress normalised of the correct form cannot be ruled out, but on the to an average of unity, and contraction ratio/r-value made basis of observation the features which would lead to using the grain boundary sliding model, with either nonrandom distributions of sliding surfaces (full lines) or a the four-fold component of anisotropy are not random distribution with rate sensitiveslip accommodation thought likely to exist in the microstructure. /
distribution A
I
(dashed lines).
planes and rate sensitive slip accommodation determined by the texture give results which show that the contraction ratio is dominated by the sliding plane distribution, whereas the relative flow stress variation with tensile axis angle is intermediate betweenpredictionswith random boundaries and slip accommodation and with no slip accommodation and a non-random distribution of sliding planes. The simplegrain boundary slidingmodel givesvery good correlation with experimentalresults. However, how reasonable is the distribution of sliding grain boundary planes which give that good correlation? To test this, it is possible to construct convex shapes which have the same distribution of surface normals as used in the model. These “average slidingbodies” give a reasonable representation of the relative proportions of sliding grain boundary orientations which would need to be present to give the required anisotropy. The one corresponding to the sliding plane distribution optimised for correlation with results at a strain of 0.3 is shownin Fig. 14.The shape of this “average sliding body” is prejudiced by the use of a limited harmonic series to describe the surface distribution and other shapes could well give surface distributions which would generate adequate predictions of anisotropy. Nevertheless, some features are necessary. The body must be flattened in the rolling plane, and must have corners in the 450-type directions. That latter feature is important because it is necessary to give the limited amount of grain boundary sliding surface with normal directions bisectingthe rolling and transverse directions of the sheet, and so give the relativelylow
4.3. Anisotropy due to mechanical inhomogeneity Leaving grain boundary sliding aside for the moment and returning to deformation by intragranular slip, a featurewhichmust now be considered is the mechanical inhomogeneity of the material. Hutchinson et al. [11] showed that the room temperature anisotropy of a duplex stainless steel was due to crystallographic texture and not phase disposition. At room temperature, their microhardness measurementsindicated that the flow stressesof the two phases are likely to be quite similar. At elevated temperature, however, there will be a large difference between the flow stresses of the phases. Chubb [13] reported that the hot hardness of austenite was three times that of ferrite for pure iron in the vicinity of the transformation at 920”C. The 1
RD
N
TD
BE ND
R
TD
eE3
Fig. 14.Projectionsof the shape of a body which givesthe distribution of slidingsurfaces used for the grain boundary distribution type (A) predictions shown in Fig. 10.
SONGand BATE: PLASTICANISOTROPY work of Rossard and Blain, reviewedby Jonas et al. [14], gives flow stresses of FGO.25?40Cwhich are between twice and four times that of Fe–25%Cr at 1100”C with strain rates between about 10-3 and 2 S-’, which is consistent with Chubb’s hardness values. Because of the phase distribution and y inclusion shape in the stainlesssteel, this differencein flow stress can affect the plastic anisotropy. Under conditions of principally intragranular deformation, this difference in flow stress will lead to a system of internal stress. Deviations from a spherical average shape of the austenite inclusions will give rise to a contribution to the anisotropy which can be estimated to a first approximation on the basis of Eshelby’sanalysis of the elastic field in an ellipsoidalinclusion[15].In this, the elastic stresses inside an ellipsoid, o‘, due to a “transformation” strain, 6T,are given by: ~,~= C~kl(&tn81. —Sl+n.& ) ,
(8)
where C are the elastic stiffness modulii and S, the “Eshelby tensor”, is determined by the shape of the ellipsoid. The elasticity is isotropic and homogeneous. Now, although both phases are deforming plastically, the difference in their flow stresses will mean that a small difference in the plastic strains inside and outside the inclusion will exist which, by acting as the “transformation” strain in equation (8), will give rise to stresses inside and outside the inclusion which will then ensure stress equilibrium in this situation of unequal flow stresses. This type of approach, where a plastic strain mismatch is treated as the “transformation” strain, has been used in connection with models of strain hardening in materials with heterogeneous dislocation distributions, for example see Argon and Haasen [16]. The “transformation” strain, equivalent to the plastic strain mismatch in the present case, is assumed to have the sameproportions as the overallstrain rate and a magnitude which shall be treated, for present purposes, as a free parameter. There will also be an associated mean elastic stress outside the inclusion and these internal and external stressesrepresent the differencesbetween the applied stress and the actual flow stress of each phase, i.e. the forward and back stressesgenerated by the mechanical inhomogeneity. The austenite of the inclusions is relativelyisotropic, and with that assumption as well, the effective increase in flow stress can be calculated as some multiple of
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“:F 0.5
1.0
< 0.7,0.5> 0.8 0.4 0.6 r 0.3
0.4
<0.5,0.35>
0.’~ o
15
30
45
60
75
90
a
Fig. 15. Predictions of normalised flow stress and contraction ratio/r-value made using the rate-sensitive slip model augmented by stresses predicted using the Eshelby ellipsoidalinclusionmodel.The valuepairs indicatedare the ratios of the ellipsoid axial lengths in the transverse sheet direction and the sheet plane normal, respectively,to that in the rollingdirection. The ferrite textures used were averages of the initial and 0.3 strain for the <0.5,0.35) ellipsoid,and averages of those for strains of 0.3 and 1 in the case of the (0.7, 0.5) ellipsoid.
rate are determined. Various weightings for the contribution of a+ and values of the ratio of ellipsoid semi-axeswere tried, to evaluate the potential for the mechanicalinhomogeneityto explain the anisotropy. Making the stress contribution u+ about two-thirds of the total with modest ellipsoid axial ratios gave encouraging results, as shown in Fig. 15. It is certainly possible to reduce the mean level of contraction ratio predicted to a similar value to that observed,and to givethe correct senseof the two-fold component of anisotropy. As with the “equivalent sliding body” given above, it is not known how well the mean ellipsoid shapes (used to give the results of Fig. 15)relate to the real microstructure. No attempt was made to make a correlation because it is by no means clear how the quite complex shapes of individual y inclusions and their spatial distribution would contribute in reality, and it is most unlikely that any simple averaging of geometric parameters would be adequate. However, despite this, it is clear that a combination of intragranular slip, crystalloa+ = &;/J(&:n), (9) graphic texture and internal stresses resulting from which represents, then, the elastic strain energy the plastic deformation of a mechanically inhomodensity in the inclusion divided by the effective geneous system has the potential for giving the “transformation” strain, which is proportional to the correct type of anisotropy. overall effective strain. This can be weighted and added to the flow stress, relative to slip stress at 4.4. The evolution of microstructure and texture Two quite different approaches can be made to unit strain rate, of the ferrite calculated by the rate-sensitive slip model, and so the relative flow explainingthe plastic anisotropy of the stainless steel stress and the contraction ratio for minimum work under conditions of superplastic tension. In both of
2756
SONG and BATE: PLASTICANISOTROPY
these, geometric features of the microstructure have been simplifiedfor convenienceand dealt with as free parameters in order to assess the potential of those approaches. There is a major difficultycaused by the complexity of the real microstructure and how that can be adequately represented in a simple way, and it is therefore not possible to determine whether slip or grain boundary sliding was the principal deformation mechanism during superplastic tension on the basis of plastic anisotropy measurements in the material used in this work. The modelling indicates that grain boundary sliding controlled by rate sensitiveaccommodating slip is not appropriate, and although the anisotropy can be predicted very well using a grain boundary sliding model, the distribution of sliding surfaces required to give, particularly, the four-foldcomponent of anisotropy is probably unrealistic. Other observations are more consistent with a slip-based mechanism than with grain boundary sliding. The most obvious is the preservation of banding in the structure to high strains, as shown in Fig. 6. The preservation of linear alignment,either of grain structure [17, 18] or of internal markers [19], over significant length scales is very difficult to account for with the type of relative grain translation which would occur with grain boundary sliding. Although they are lessobvious,the changesin texture are probably more consistent with slip-based deformation as well. The evolution of crystallographic texture with deformation is not consistentwith homogeneousslip, particularly in the ferrite. There is a notable aspect of the texture development, however, which is best shown by inverse pole figures in the tensile axis directions. These are given in Fig. 16 for the ferrite
OOAAA
0.3 &—
1
w“ Fig. 17.Inversepole figuresin tensile axis directions for the ferrite following deformation to the strains indicated at 970°Cat a mean rate of 10-3s-’. Levelsare in multiples of random density.
and Fig. 17 for the austenite. Despite large initial differencesin the distributions of crystal directions in the tensile directions, these become quite similar after deformation. In the ferrite a <111) alignment develops,which is most marked in the a = 90° case. In the austenite, a (011) + near (111) alignment develops which is similar to the initial rolling direction alignment. The ferrite texture is most interesting, as tension would tend to give (011) alignmentwith homogeneousslip,and that alignment is clearly depleted here. A deformation mode which does give the (1 11) alignment is simple shear on planes containing the tensile direction. This is consistent with inhomogeneous deformation due to the 6 – y flow stress difference. Rather than this differencegiving rise to a simple strain mismatch at the phase boundaries, assumed for simplicity in the application of the Eshelby model given above, the mismatch in this case becomes more distributed through the ferrite, in particular, giving rise to considerable amounts of shear in the vicinity of the austenite inclusions.This does not actually make the application of the Eshelby method invalid, in fact the final part of the paper dealing with the ellipsoidal inclusion [15] makes reference to the methods applicabilityto the case of inclusionsin viscousfluids, using the analogy between linear elastic and linear viscous behaviors. However, the relationship with the plastic anisotropy becomes more complex and would require, say, finite element modelling to solve in a more satisfactory manner. Despite the complex nature of deformation at elevated temperature in this material, several of the experimentalobservationsare probably more consistent with slip being the principal deformation mode. They include the significant four-fold component of plastic anisotropy, the preservation of aligned
::::!;! ;::”,: ]4soAAA WOAAA o 0.3
1
&—
0.511
.522.5
Fig. 16.Inverse pole figuresin tensile axis directionsfor the austenite followingdeformation to the strains indicated at 970”Cat a mean rate of 10-3s-’. Levelsare in multiples of random density.
SONG and BATE: PLASTIC ANISOTROPY
microstructure to a reasonably large strain and the changes in texture. Although the texture changes in the ferrite were not those which would occur with homogeneous deformation, and involved a general reduction in intensity, they were consistent in a way which would be very difficultto account for if grain boundary sliding were the dominant mechanism. Although, as Padmanabhan [20]has argued, crystallographicallydifferentboundaries would be expected to slideat differentrates and so affecttexture changes in superplastic flow, it is very difficultto see how this could account for the present experimental results. It was suggested previously [5] that superplastic deformation is likely to be inhomogeneousbetween grains and that this could, in principle, lead to deviations from predictions of texture changes based on homogeneous slip, whilst anisotropy, being an averaged property, may not be affectedto any great extent by the inhomogeneity. In this context, the results presented here are useful because they are from a material with an intrinsic inhomogeneity of large magnitude, rather than that which might occur in nominally single-phasesuperplastic materials. 5. CONCLUSIONS The plastic anisotropy of the duplex stainless steel SAF2304, measured as both flow stress and contraction ratio variations with tensile axis orientation, shows similar four-fold anisotropy in both room temperature and superplastic deformation. However, general levels of contraction ratio and the two-fold component of anisotropy are different in those two regimes of deformation. The plastic anisotropy could be modelled using grain boundary sliding,but the distribution of sliding surface required is somewhat unrealistic. The four-fold component of the anisotropy seems likely to be due to crystallographic texture, particularly in the ferrite, but slip anisotropy associated with this texture only gives the four-fold anisotropy of contraction ratio if intragranular slip, rather than slip accommodating grain boundary sliding, is assumed. It is possible, in an approximate way, to account for the difference between slip induced anisotropy and that actually measured by including an
2757
anisotropic contribution related to the flow stress differencebetween the austenite and ferrite, coupled with the spatial anisotropy of the austenite morphology and distribution. Preservation of banding in the microstructure to large strains and systematic changes in crystallographic texture indicate that intragranular slip, with a significantinhomogeneous component, is probably the dominant mechanism in superplastic deformation of this material. Acknowledgements—Thanks are due to the Committee of Vice-chancellorsand Principals of the Universities of the United Kingdom for an ORS award for one of us (J.L.S.), to W. B. Hutchinsonfor material and to P. L. Blackwellfor his comments and advice.
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8. Taylor, G. 1,, J. Inst. Metals., 1938, 62, 307. 9. Bunge, H.-J. and Roberts, W. T., J. appl. Crystall., 1969, 2, 116. 10. Kallend, J. S, and Davies, G. J., J. Inst. Metals, 1971, 99, 257.
11. Hutchinson, Ushioda, K. and Runnsjo, G., Mater. Sci. Technol., 1985, 1, 728.
12. Sachs, G., Z. Ver. Dtsch. Ing., 1928, 72, 734. 13. Chubb, W., Trans. AIME J. Metals, 1955, 203, 189. 14. Jonas, J. J., Sellars, C. M. and McG. Tegart, W. J., Metall. Rev., 1969, 14, 1. 15. Eshelby, J. D., Proc. R. Soc., 1957,241A,367. 16. Argon,A.andHaasen,P., Acta metall. mater., 1993,41, 3289.
17. Blackwell,P. L. and Bate, P. S., Metall. Trans., 1993, A24, 1085.
18. Blackwell,P. L. and Bate, P. S., in Superplasticity: 60 years after Pearson, ed. N. Ridley. The Institute of Materials, Book no. 618, London, 1995,pp. 183–192. 19. Blackwell,P. L. and Bate, P. S., Metall. Mater. Trans., 1966, A27, 3747. 20. Padmanabhan, K. A., Metal Sci., 1980,14, 506.