Superplasticity Size Effects

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GRAIN SIZE DISTRIBUTION EFFECTS INSUPERPLASTICITYA. K. GHOSH and R. RAJt

Rockwell International Science Center. Thousand Oaks, CA 91360. U.S.A.(Rrceiced I5 Sepremher 1980)

Abstra&-The influence of grain sixe distribution on the stress-strain rate behavior of superplasticmetals has been inv~ti~ted for steady-state as well as transient toading situations. The model, whichconsiders a distribution ofnternal stresses based on grain size distribution, is a detailed development ofone presented ekewhere[I]. Deformation is assumed to occur by a combination of grain boundarycreep (sliding accommodated by diffusion) and power law creep, constrained by equal strain rate in allgrains during steady state. The analytical results of the model based on realistic grain size distributionssimulate steady-state behavior, as well as loading and load relaxation behaviors of superplasticaluminum and titanium alloys. Several other features of material response are also explained.R&um&-Now avons ttudit linfluence de ta repartition des tailles de grains sur le comportement desm&aux superplastiques, aussi bien pour des regimes stationnaires que pour des rtgimes transitoires.Notre mod& qui tient compte dune repartition des contraintes interns reposant sur une repartitiondes tailles de grains, est un dCveloppement d&aillC dun mod&k present& ailleurs [ I]. Nous supposonsque la dtiormation r6sulte de la combmaison du fluage intergranulaire (glissement accommode pardiffusion) et du fluage en loi de puissance, avec ia contrainte dune vitcssc de dtformation tgak damtous ks grains au tours du regime stati~n~~. L.es r&hats ~al~qu~ de cc mod?k, avec unerepartition rcriste de ia taille des grains, simule ie comportement en regime stationnaire, ainsi que kscomportements en charge et en relaxation de charge, dam te CBS es alliages daluminium et de titanesuperplastiqucs. NOW expliquons tgalement plusicurs autres CaractCristiques de la response dumattriau.Zusamm&amamg-Der EinfluD der KomgWenverteilung auf das Spannungs De@eschwindigkeitverhalten eines superphu&schen Metalks wird Nr den stationgren Fall end fUr U~~~~~t~guntersucbt. Das h&dell betracbtet eine Verteihmg von auf der K~~~~~lung beruhendeninneren Spamnmgen und stellt eine ausftihrlicbe Entwicklung tines anderweitig verHfentiichtenMod&s Cl] dar. Es wird angenommen, dag die Verformung mit einer Kombmation von Korngtenzk-riechen (durch Diffusion angepaBte Gleitung) und Potenzgesetxkriechen ablpuft, die im station&enZustand durch dieselbe Debngeachwindigkeit in &bntlichen KImem eingeschrgnkt ist. Die analytischenErg&&se des auf realistisehen Komgr&nverteihmgen aufbeuenden Modells simu&ren das station&eVerhalten vcm superpiastiscben Aluminium- und Titanlegierungen. Auikrdem werden weitere Eigenschaften des Mat~~~a~~s e&&t.

1. INTRODUCTIONLarge tensile elongation exhibited by certain metals atelevated temperature is broadly classified as super-plasticity. The phenomenon might arise from morethan one mechauism of which the importance of grainboundary sliding process is well recognized andaccepted. It is believed that sliding along gram boun-daries with auntie by diilkional and/or dis-location processes can lead to large plastic strainswithout significant grain shape change E2-43. The re-quirement of a fine grain structure for superplasticityalso seems to be tied to this sliding mechanism, since

t On leave from the Department of Materials scienmand Engineering, Cornell University, Ithaca, NY 14853,U.S.A.$The threshold stress at extremely low strain-ratesassumed in this model, appears to bc somewhat unrealisticand controversial.

the average grain boundary area per unit volume in-creaseswithad- ingrainsize.

One attractive meduinism for superplastic d&r-mation, even though not entirely acceptable in all itsdetails,S is that by Ashby and Verrall[2]. The grainboundary sliding process is diffusion-accommodatedin this model, and is the dominant contributor to thematerials shape change below a certain strain rate.Above this rate, however, increasing degrees of graincreep (Rower law) occur with increasing strain rate,leading to grain elongation. The behavior is mixed inthis strain-rate range, and above this range the gramcreep dominates. Grain boundary sliding in thismodel, as well as in Cable creep[Sj and GiRtinsmantle theory [4], assume a Newtonian viscous be-havior (o a


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608 GHOSH AND RAJ: GRAIN SIZE IN SUPERPLASTICITY

(b

Fig 1, Schematic u w i and m vs P curves for two discrete grain sizes based on equation (1). Dashedlines suggest thrcsbold stress type effcn

The overall equation ~mb~g the ~~buti~sfrom grain boundary prorzsses and power law creepmay be given by

where R = atomic volume, d = 2 x Burgers vector,D# = boundary diflbsion coefficient, k = Boltzmannsconstant, T= absolute temperature, d = grain size,K = constant for power law creep (containing depen-dencies on temperature and shear modulus),n = creep exponent, u = stress and E = strain rate.Note that the first term on the right side of equation(1) is the Coble boundary sliding term, having inversecube dependence on grain size, while the second termfor power law creep has no grain size dependence.The question may be asked as to what form equa-tion (1) should take when, in reality, a grain size dis-tribution exists in all polycrystaliine materials ofinterest, This has been addressed elsewhere for asimpIe biiodal grain size distribution El]. Beforeanalyzing this problem in any detail, the case of singlegrain sizes is shown in Fig. l(a) The smaller grain sizeshows departure from the power law Mavior at a.:*Ahigher strain rate. In each case, the compostte beha&ior shows a curved transition region bounded bylinear regimes for each mechanism. However, thetransition region is extremely narrow for single grainsizes, a result which is not usually observed. Athreshold stress-like behavior is also added inFig. t(a) (dashed) as suggested often in the literature,

t Finer grain size also gives rise to a higher value of peakn. This effed is discussed later.

and not as a result of equation (1). The ~~~n~~gslopes, m = (3 In u/3 in& shown in Fig, I@), indicatethat tiner grain size yields higher m at higher strain-rates.t While this behavior is always observed, theapproach to a low m at lower strain rates aribing froma sigmoidal u - ci curve is observed only in somecasea In this paper, this topic is analyzed with par-ticular reference to grain size dis~buti~ effect onsuperplasticity, and new insight has been generated. Itis shown that this distribution has a strong influenceon the shape of the c vs 4 curve in steady state as wellas on the transient loading and relaxation behavior ofmetafs.

2. THE MODEL2.1 Steadystate deformation

Fine scribed-lines are often used in the measure-ment of grain boundary offsets caused by the bound-ary sliding process. Ma~~~pi~lly speaking,parallel scribed lines are, however, found to remainparallel and straight even after large superplasticextensions. Thus any representative volume compris-ing grains of different sixes (as shown in Fig 2) issub&&d to the overall specimen strain rate. The finegrains in this element are constrained to strain at thesame rate as the coarse grains However, as seen fromFig. 1, coarse grains can support higher stress thanline grains at the same overall strain rate, and there-fore a nonuniformity in stress develops in the deform-ing solid

In this model, strain rate constraint is imposed overall grains having a certain diameter, d,, rather thanover each individual grain in a representative volume


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GHOSH AND RAJ: GRAIN SIZE IN SUPERPLASTICITY 609

DISPLACEMENT

APPLIEDLOAD

Fig, 2. Cutaway of a specimen under applied load indicat-ing uniform displacement across all grains.

element. Each grain may deform at a rate slightlydifferent from the average; however, the averagestrain rate, h,, for grains of diameter d,, is assumed tobe

zj = z (2)where i = imposed strain rate. If uf is the averagestress supported by these grains, 2 may be expressed,according to equations (I ) and (2), as

where K1 = (lSOR//G) 2&b. The overall stress, 0, isthen given by

@= F&a8 (4)wheref* = volume (or average area) fraction of grainshaving diameter, dI. For a grain size distributionfunction, p,(d,j, fi may be derived from

where pi denotes either number or density of grains ofsize 4. Thus. for a given p,(d,), a can be computed foreach input value of i from equations (3-S).

During initial loading of a superplastic material,the strain rate constraint among different grains is notestablished immediately. When strain rate is not large,the very fine grains are able to relax any slight loadrapidly by diffusional flow, and therefore loading upis a slow process. During loading as well as relax-ation, the stress-rate for each grain size group is givenby

&I = E(Z, - &) (61where E = elastic modulus, and & = applied totalstrain rate, which is zero for Ioad relaxation. If load-ing is started from an extremely small value of pre-stress, at which finer grains have a greater a, than

coarser grains. the elastic loading rate of coarse grainswould be larger since (it - i,) in equation (6) is larger.In spite of this, the strain rate for coarse grains in-creases more slowly because of a larger extent ofpower law regime. Similarly, the unloading of thecoarse grains would be slower during relaxation. (Theeffect of power law on loading and relaxation timesare discussed in Ref [ 13)Thus, greater stress is alwayssupported by coarser grains.

During computation. the average stress. ai, for eachgrain size group is obtained from

Oi = a: + h,At (7)where a: = previous value of al, and At = time incre-ment. The overall stress is then calculated from cqua-tion (4), and inelastic strain rate is given by

where a* = previous value of a. Based on the new aifrom equation (7b new GI is again calculated fromequation (3), which is substituted in equation (6) andthe process is repeated.23 Grain size distribution

The grain size dist~bution for metals determinedby linear intercept is generally described by a log-normal function [6]. Even though there are a largenumber of grains near the mean, a long tail is oftenshown at large intercepts. Bimodal distributions andgrain size banding are quite common in materials forhigh temperature use. In fact, when onlys few largegrains are present in an otherwise fine grain material,the statistics of metallographic sectioning and grainintercepts could give the impression that distributionat large grain sizes is a continuous function as inlog-normal distribution. As illustrated in Fig 3, amixture of three discrete grain sizes, can give rise to amuch smoother dist~buti~ of grain intercept lengths.Especially when some spread is present about eachdiscrete grain size, as shown by a series of triangulardistributions, the intercept lengths can approach thecommonly suggested log-normal distribution. A largedegree of bimodal distribution cannot be masked bysuch statistical effects; however, a very small fractionof grains having undergone exaggerated grain growthmight go unnoticed in an intercept distribution plot.To simplify analytical procedure, grain size distri-bution has been selected as triangular in this work.The sides of the triangle may be varied to approxi-mate different types of dist~butions. Discrete grainsizes may also be simulated by narrowing the grainsize spread within a reasonably small range. At leasttwo such triangular distributions were needed tocharacterize bimodal distributions. In, the computerprogram to calculate stresses and strain rates, threesuch distributions with variable height (number) andbase (grain size spread) were used for g&a&r flexibi-lity. The Appendix describes a method for calculating


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610 GHOSH AND RAJ: GRAIN SIZE IN SUPERPLASTICITY

I I IGRAIN DIAMETER

INTERCEPT LENGTH INTERCEPT LENGTHFig. 3. Effect of grain sizedistribution on intercept kngth distribution: (a) for three discrete sizes, (b) fortriangular spreads around three discrete sizes. Solid lines-overall distribution, dashed lines- density ofindividual distributions

the volume fraction of any grain size within such adistribution.3. RESULIS AND DISCUSSIONS

A preliminary evaluation of the model can be madeby considering a mixture of two discrete grain sizes,dt and d2 (d, < d2) as in Ref. [l]. As shown in Fig 4,at a volume fractionJ,, of the fine grains the compo-site curve approaches a power law behavior given by(1 -f,)o on the str%s scale, bDundcd betweendilbional regimes of u = (df/K,) 0 and u = (1 -h)(d#,) 2. The transition regions are curved, due to theinteraction of two mechanisms,and show a change inslope, m = d lo8 u/d ti, as a function of strain rate.At strain rates above A, u vs i curve is sigmoidal andrate sensitivity index, m, exhibits a maximum-a be-havior generally associated with superplastic defor-

mation. Furthermore, these peaks are much broaderin strain rate range than those expected from a singlegrain size. At very low strain rates, i.e., below the ratesnormally investigated in studies on superplasticity. mrises toward unity again. (Measurements at such lowrates might, however, be obscured by concurrentgrain growth effects.) As the fine grain fraction is in-creased to fs (shown in Fig. 4), a decrease in flowstress and an increase in m value peak are found tooccur. As discussed later in the paper, a finer overallgrain size (i.e., smaller dl and d,, or just smaller d,)causes further decrease in flow stress, and increasesin the values of peak m as well as the strain rate atwhich it occurs. Finally, the transition to diffusionalcreep at the low rates, also shown in Fig. 4, moves, oa higher strain rate.A more detailed grain size distribution is nowcon-sidered. Figure 5 shows the influence of three tri-

(f = VOL. FR. OF GRAINS OF SIZE dr)

I I ILOG iFig. 4. The influence of the volume fraction.1; of finer grains on the composite curves of log uvs log iand IV vs log i.


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GHOSH AND RAJ: GRAKN SIZE iN SUPERPLASTICITY 611

n-5

Fig. 5. u vs i =curvesor different trime&l d~st~~ut~on~~t~an~l~ spreads) are contrasted qykst curvesfor discrete and individuat grain sizes.

angular grain size distributions at median values of2.5, 7 and 141m5 respactively. Ihe power law par-ameters and diffusion coef&ient vah~es selected forthese plots are similar to those for TiiAHV alloy at927C. T&e individual cr - i curves for steady-stateare shown by dotted lines for each median grain size,These curves are &a+criz& by $ower law regions(m = i/n = 0.2) at the higher strain rates, Newtonianviscous region (m = 1) at the lower rates, and a tam-sition region spread over a decade and a half in strainrates within which 0.2 < m < 1.0. The solid linesincorporatig various mixtures of these grain sizesexhibit a much larger spread in the Q - Z curves. Thetransition region can now lie over three decades instrain rates, as common& observed in superplasticmetals and not explained by single grain size bchav-ior.While the finest (2Sm) grains are largest innumber, changes in the ratio of other grains can causesignificant changes in the shape of the Q - i curves.For example+with a ratio of numbers 25oo:SO:l ofthe 2.5,7 and 14~ grains, i-e., volume fractions of74,3/, 20.5%attd 5,29/, rcspective~y~ nearly consmtm of 0.55 is exhibited for 6 x IO-) 5 x IO+ s-l. This is often the casewith many literature data on superplastic metals.Finally, for Z < 5 x lo+ s-*, m begins to increaseagain toward unity. This has also been observed inMany instances [7,8].

3.1 Comparisonwith experimentalraults~imu~tions were carried out for an mourn alloy(7475alloy, processed to develop fine grain size) @],which is found to be moderateiy superplastic (elonga-tion -WA) and Ti-6Al-4V alloy which exhibitssu~~l~ti~ty in the a + /I phase Wd {ebtgptkm-5crrrlW~) while the diffbiivrtyof II ii~ Q and /Iphases is quite different (D#- 100 D,), this large diffi-MCC n diffusivities may not be too important indetermining @ain boundary ditisivity as required inthe present model. Upon averaging over diffbrentgrain sizes, and like and unlike phase boundaries, thegrain boundary ~ffu~~ty would tend to be uniformandf~oZany~sirebiasForthearseof&singIephase &m&urn alloy, this problemdoes not exist. It

does, however, contain an elongated grain structure,which is ignored for the present analysis.Typical photomicrographs of the Al and Ti alloysand their grain intercept plots am shown in Fig. 6,Grain intercept plots were developed frownhree suchmicrographsof achmateriat Ear&r studies on grainintcreept length di~~butio~s [1-O, ] generally support the present results. While it ig difficult to describeexactly the nature of grain size distribution thatcauses this, several possible distributious used in thepresent calculation are listed in Table 1.The most important problem regarding grainboundary diffusivities is the Iack of data; this is par-ticularly complicated by the fact that diffusiviti~ aresi~ifi~tly altered by deformation. Several fold in-crease in diffusioncoefficientsare possible under con-ditions of concurrent deformation [K&13]. Because ofthe uncertainty over its value, L&has been used as anadjustable parameter to obtain a match between thetheory and experiment. Table 2 lists the values of DBat the superplastic temperature from literature, alongwith those used in the calculations here. It may be


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612 GHOSH AND RAJ : GRAIN SIZE IN SUPERPLASTICITY

Ti-6AI-4V ALLOY

200r

GRAIN INTERCEPT LENGTH; pm

-J 60GRAIN INTERCEPT LENGTH, pm

7476 Al, ALLOYFig 6. Photomicrographs ofli and Al alloys and corresponding distributions of grain intercept lengths.

Table 1. Combinations of triangular grain size distributions for simulation

Material Combination Range of Peak Value Range of Peak Value Range of Peak ValueNo. Base (pm) (at d pm) B= (cun) (at d rm) Base l.p) (at d C(T)1 610 lo(&) 18-20 1(19) - -I7415 Al 2 8-12 lOoU0) 28-30 f(29) - -1 2-8 W5) 12-15 5(14) 28-30 I(29)

TiiAl-W2 2-8 5W5) 13-15 5(14) 33-35 1(34)

Table 2. Material data used in equation (I)Available Diffusion Assumed D#!iMaterial T(R) R(d) d 5 th (cm) Coeficient (cm*/s) (cm2/s) K(s-(MPa)- n

Aluminum 790 1.62 x IO-* 6 x 10-s DB = 2.45 x lo-*t 5 x lO-h 3.015 5.883Ti-6Al-W 1200 1.7 x IO- 6 x IO- D, = 10-9, Di = IO-$ 3.6 x IO- 5.92 x lo- 5.0t Grain boundary diffusion coefficient from Ref. [ 143.1 Bulk diffusion coetbcients for z and p phases (Refs. 15 and 16).$ Based on cnhancemcnt of diNusivity due to grain boundary migration erects (Refs. 12 itnd 1.1).


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l f

zE0

0.

GHOSH AND RAJ: GRAIN SIZE IN SUPERPLASTICITV 613

Ti 6AI 4VT-12000~K =2.44x l o-" MPa-"S- 1

o-" ' 5Kd = 1. 45x 10s21 MPa- ' rn3S-1

dbni

IO

I5YD

1.1

3. 01

-I

_I

Fig. 7. o vs + curves calculated for Ti-6A I-4V (solid lines) compared to experimental data (open circle).

noted that the values of DB used for dynamic con-ditions is not measurable in view of the results in Ref.12. All other parameters are also listed in Tabk 2.

Calculations of steady-state Q vs i curves based ongrain size distributions of Table 1 are compared withexperimental results in Figs 7 and 8. The experimentalu vs i data for Ti alloy is from Ref [17j. and that forAl alloy (similar to Ref [18].) has recently been devel-oped in our laboratory. Good agreement in stresslevel and slope between the model and experimentaldata is observed in both cases.3.2 Transient behavior

According to the rationale developed in section 22,simulations were carried out for loading a super-plastic material at a slow strain rate (total strain rate,El = lo- s-l). Figure 9 shows calculated stress and

strain rate distribution among the various grain sixesduring the course of loading up toward steady state.The tie grains are found to attain the applied strainrate rather rapidly, while the strain rate in coarsegrains increases slowly, thereby leading to a slow timehardening transient behavior of the average u vs rcurve[Fig. 9(a)]. As the coarse grains continue todeform by power law, this may be part)y responsiblefor the observed strain hardening in aluminumalloys [ 183. It is noteworthy that a decrease in elasticmodulus or an increase in the size and volume fraotion of larger grams can significantly increase strainto reach saturation stress. It is obvious that, whenimposed strain rate is larger, the transknt region isreduced because of a more rapid loading of the coarsegrams. However, a larger fraction of coarse grains canlead to a long transient region.

l o-

aEDl -

O. l y

5YD91012 2930 1475 Al

T=7!XPKK 14. 11 x l o- l oMPa' "Ssln- 5. 883

. l

Kd = 1.45x o- l 9MPa-' n' t3S- ', . . . . . ..*I . I #....I1 * . . . . . ..I . . . . . . J O. Ol

10-G 10-5 10-4 10- 3 10-2 10- lt bc-' )

Fig. 8. g vs a curves calculated for 7475 AI alloy (solid times) compared to experimental data (opencircles).


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614 GHOSH ASD RAJ: GRAIN SIZE IN SUPERPLASTICITY

I 1 1400/I6- ,/( - 200

d*34pm ?/: K = 2.44 x 10-l MPe- S-II-5

,ooo

6- //Kd = 1.45 x lo-*O MPa- tn3 S

z E - 3.44 GPaz_800 z

; // iIs 4- ! :/ r!__-

-606 &

! 25814 34!

d hml - 4002-

/ AVERAGE, 200

APPLIED C3.

0 2 4 8 8 10 12TIME (min)

Fig. 9. Rise in stress and inelastic strain rate in coarse and fine grains as well as For the average materialduring loading up at a constant &.

The presence of such a long transient has interest-ing implications on instantaneous rate sensitivityindices obtained from strain rate perturbations car-ried out at various strain levels [18]. Since fine grainscan .readily attain imposed strain rate through dif-fusional assist, their influence (m * 1) on instanta-neous rate sensitivity index is greater when coarsegrains are supporting less stress, i.e., at short times. Atlonger times, the stress supported by tine grainsbecomes a smaller fraction of the overall stress, andinstantaneous rate sensitivity may approach that forpower law creep (m 5 l/n) within a certain strain raterange. This kind of drop in instantaneous m as a funotion of strain has been recorded for aluminumalloys[18]. Supporting observations have also beenreported in which dislocation structure develops incertain grains, while other grains remain relativelydislocation free 1191.

A large number of superplastic materials, includingTi, Pb, Al alloys, etc., exhibit significant grain growthduring deformation. Concurrent grain growth leads toadditional hardening during deformation. whichmight last over much larger strain levels than thatindicated in Fig 9. This happens primarily becausethe fraction of finer grains, as they begin to grow,start to support larger stresses. In the case ofTiiAl-4V alloy[17]. this is believed to be thedominant hardening mechanism. Grain growth alsoleads to a decrease in the instantaneous rate sensi-

tivity index, because the coarser grains approachpower law behavior.

The results of simulation of a load relaxation testare described in Fig 10(a) The behavior is the reverseof the loading behavior. In this case, the higherstresses supported by coarser grains begin to diminishmore rapidly than finer grains can shed their stresses(or strain rates). This causes a steep drop in stress foronly a slight decrease in strain rate. Subsequently, asthe strain rate in coarse grains decreases significantly,and that in 8ne grains also begins to decrease, theoverall strain rate drops at a greater rate in compari-son to the stress drop. The slope of the Q vs i curvebecomes less than that for steady-state, and continuesin this manner to lower strain rates. Analytical resultsare shown in Fig 10(b); the experimental data haverecently been developed in our laboratory. Thus the Qvs i curves obtained from loading and relaxationtransients have different characteristics from thesteady-state behavior because of different stress andstrain-rate distributions associated with each case.The low m values over a large strain rate range ob-served in relaxation tests are reminiscent of similarobservations by Woodford [20], Hart [21]. Li [22]and others for high temperature deformation. Thissuggests that the grain size distribution effect mayhave a much broader applicability in creep thansimply in superplasticity. Based on the understandingdeveloped here, more attention should now be placed


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GHOSH AND RAJ: GRAIN SIZE IN SlJPERPLASflCITY 615

100 -EE

Pt-l

RELAXATION

1 I I IK = 2.44 x 10-l Mh- s-1 (b) loo

n-5 I100 - -P Kd 1.45 1O-2ox MPa- m3 S- -10 c.l

E = 3.44 GRiiE 10 -

l-/; i

d (amII I I I10-5 10-5 10-4 10-3 10-2 to-

STRAIN RATE WlFig. 10. (a) Experimental u vs i curves from strain rate jump and load relaxation tests. (b) simulations of

the same through present modelon such observed differences, as well as on the role ofgrain size distributions.

4 SUMMARYAll models of micrograin superplasticity to datehave ignored the presence ofany distribution in grainsize. An analysis including realistic grain size distribu-tion presented here, reveals a nonuniform internal

stress distribution during superplastic flow. Thepresent approach basically combines diffusional creepwith power law creep as in a number of other studies.It is observed, however, that while finer grains candeform (or slide along boundaries) by difisional processes and support low stmsses, coarse grains at thesame time deforming by power law creep, under iso-strain-rate constraintq can support larger stressesThe result of this distribution is to increase the strainrate over which rate sensitivity index (m) changesbetween the power law value (l/m) and the New-tonian viscous value (l), by several decades. The mixof grain sixes am also produce an apparent approachtoward a threshold stress at low strain rates, therebyyielding a peak in m. However, at even lower strainrates ,this changes back toward Newtonian viscousbehavior again. Using realistic grain size distribu-tions, the experimental cr vs 2 results on Al and Tialloys have been simulated extremely well betweenstrain rate ranges of 10m6 to 10e2 s-l.The model explains the transient behavior of super-plastic metals quite well. During loading up and load

relaxation tests the iso-strain-rate constraint is notimposed instantaneously and the internal stress distri-bution develops gradually. The coarse grains primar-ily dictate transient loading behavior. The slow load-ing up of the coarse grams causes a time hardeningbehavior and also produees a lower slope of the Q vs kcurve in comparison to steady-state results.On the basis of this study, it is recommended thatmore attention be given to the distribution of grainsizes in order to better understand superplastic behav-ior.Acknowlcdgcmenrs-This work was supportedby the Inde-pendent Research and Development Program, OFRockwellInternational.

REFERENCES1. R. Raj and A. K. Ghosh. Acra m&l. 29, (1981).2. M. F. Ashby and R. A. Verrall, Acra metal/. 21, 149(1973).3. J. R. Spingam and W. D. Nix, Acm metal/. 27, 171(1979).4. R. C. Gitkins, Ref. from article by J. W. Edington, K.N. Melton and C. P. Cutler, Prog. Mater. Sci. 21, 61(1976).5. R. L Cable, J. appl. Phys. 34, 1679 (1963).6. R. T. DcHoff and F. N. Rhines, Quanritariue Micro-

scopy. McGraw-Hilt, New York, (1968).7. A. E. Geckinli and G. R. Barrett, Scripta metal/. 8, 115(1974).8. A. K. Mukherjec, An. Reu. Muter. Sri. 8. (1979).9. U.S. Patent 4,092,181. May 30, 1978, N. E. Paton andC. H. Hamilton, Merhod 4 Imparting a Fine GrainSrrucrure ro Aluminum Alloys Having PrecipitatignConsriturnts.

A.M. 29/4--D


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616 GHOSH AND RAJ: GRAIN SIZE IN SUPERPLASTlClTY10. N. E. Paton and C. H. Hamilton, Metoll. Trnns A, 10,241 (1979).Il. P. Feltham. Act o.met ul l. .97 (1957).12. J. W. Cahn and R. W. Bailufti, 13.499 (1979).13. J. W. Cahn, J. D. Pan and R. W. Balhtffi. 13, 503(1979).14. W. D. Jenkins and 1. G. Digges, J. Rex not n. Bur.Stand. 1.272 (1951).IS. J. F. Murdock. and~C. J. McHargue, Acta metol l. 16,493 (1968).16. J. F. Murdock, T. S. Lundy and E. E. Stansbury, Actameta ll . % 1033 1964).17. A. K. Ghosh and C. H. Hamilton. Metaf~ Trans. A. IO.699 (1979).18. A. K. Ghosh and C. H. Hamilton, Proc. Inr. &II onStr ength f M etal s nd Al loy s,Aug. 1979, Aachen, Gcr-many.19. L. C. A. Samuelson. K. N. Melton, and J. W. Edington,Act a metal /. 4, 1017 (1976).20. D. A. Woodford. M etal l . Trans A , 6, 1963 1975).21. E. W. Hart, J. Engng. M at er. Technol . 98, 193 (1976).22. F. H. Huang, F. Y. Ellis and C. Y. Li, M etal l. Trans. A8,699 (1977).

APPENDIXA triangular grain &distribution can be mathematicallyexpressed as

(4 ( d d d,)

(dm

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Superplasticity Size Effects