Prediction of elongation of superplastic materials — influence of grain growth and cavitation

Scripta

METALLURGICA

V o l . 18, pp. Printed in

333-336, 1984 the U.S.A.

Pergamon P r e s s Ltd. All rights reserved

PREDICTION OF ELONGATION OF SUPERPLASTIC MATERIALS-INFLUENCE OF GRAIN GROWTH AND CAVITATION M. Su6ry G~nie Physique et M~canique des Mat~riaux, Equipe de Recherche associ@e au CNRS no. 1024 Institut National Poly~echnique de Grenoble, I.E.G. Domaine Universitaire - B.P.46 38402 - Saint Martin D'H~res C~dex - France (Received (Revised

October 28, February 7,

1983) 1984)

Introduction Very large elongations are considered as one of the main characteristics of superplasticity which is usually observed in materials with a fine-grained structure. Such elongations are the result of high plastic stability during a tensile test and a necessary condition for this stability is a high value of the strain-rate sensitivity coefficient m. This condition is however not sufficient and several alloys with a fine-grained structure exhibit only a few hundred percent elongation ; steels , bronzes and brasses are in this case and this is due to extensive cavitation during straining which leads to earlier fracture. Grain growth can also limit the elongations of superplastic materials. Indeed rapid coarsening of the structure can lead to a change in the deformation mechanism from superplasticity (domain II) to power-law creep (domain III) with lower m values and then with lower plastic stability. The aim of this study is the prediction of the elongation achieved in the superplastic range during a tensile test at constant strain-rate for a material with grain size evolution and cavity growth, m/~ brass is taken as a reference for this study because of the extensive work already done on this alloy. Analysis Prediction of elongation of superplastic materials : influence of grain growth. The basis of the analysis concerning the prediction of the elongation for a superplastic material with grain coarsening can be easily understood by considering the diagram of fig.l. This figure shows a typical Logo vs. Log~ plot for a superplastic alloy of grain size D. at a given temperature. The plot is divided into two regions : the superplastic region II which is assumed to be well represented by the following constitutive equation : = K on/(mi )a where n (= i/m), described by :

a and K are

constants,

and

the power-law n' = K' o

(i) creep

region

III

which

may be

(2)

where K' and n' are constants. This law was generally found to be independent of the grain size of the alloy. It is furthermore assumed that region I, which is usually observed on the Logo Log~ curve, is only the result of structural evolution (i) ; consequently it is not shown on a plot corresponding to constant grain size. Let us consider now a specimen with an initial grain-size D. being strained at constant strain-rate Eo" During straining, the structure coarsens and after a given time t, the grain size is D, corresponding to a flow stress given by: o =(~

D a /K) I/n

(3)

o It is assumed here that no other structural evolution takes place. Transition from region II to region III will occur when the grain size will be such that the stress, as given by relation (3) is equal to that obtained by extrapolation of region III to a strain-rate equal to ~ . For the transition, we then have : o

333 0036-9748/84 $ 3 . 0 0 + .00 Copyright (c) 1 9 8 4 P e r g a m o n Press

Ltd.

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(~o(Df)a/K)i/n

MATERIALS

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(4)

= (~o/K') I/n'

or : Df = (K/K 'n/n')I/a (~o)(n-n')/n'a

Log (7 ........ -..~% •..............

E.

Log

f, Fig.

!

Log o vs. Log e for a superplastic alloy of initial grain size D. deformed at £ , showing the • 1 . o . stress Increase due to graln coarsening. L~

(5)

Only few studies concerning grain-size evolution during superplastic deformation have been performed and the law for the kinetics of growth is not yet firmly established. Clark and Alden have compared the grain coarsening during annealing to that during superplastic deformation in the quasi-single phased Sn-l% Bi alloy (2). They have shown that the same type of law applies, with however an enhancement of growth during deformation, being due to the production of an excess of vacancies during grain boundary sliding, which represents the main deformation mechanism in this material. In the case of ~/8 brasses, experimental results reported later show that the grain size of the p-phase is well related to the deformation time in the superplastic range through the relation :

= L~i + kt p

(6)

where k and p are constants depending on the phase proportion in the alloy. In the following, it will be assumed that the same type of law applies for i.e : D = D. + ktp 1 The superplastic

strain £

sp

is then given by : e = ~ sp o

Where At is the time interval necessary

~t

grain coarsening,

(7) (8)

for the grain size to grow from D.I

to Df

Df = D.l + k (At) p

(S)

It is clear from figure 1 that g = 0 for eo = ~l where e. is the strain-rate at the transition between region II and r e ~ o n III for the graln size D . } Combining equations (5), (8) and (9) gives : i asp = 6o k-i/P Numerical

calculations

of £

sp

((K/K,n/n ' (~o) (n-n')/n'

show that e

sp

if p > (n'-n)/n'a ; for p < (n'-n)/n'a, o max For p~(n'-n)/n'a, we have : o max

exhibits a maximum e e

sp

sp max

increases continually

(ii) and (12) show that e

((n'-n)/n'pa)i/P

o max

and e

sp

max

as 6

o

decreases.

(Ii)

(l-(n'-n)/n'pa)-i/P

are respectively

(i0)

at a given strain-rate

= £I (l-(n'-n)/n'pa)n'a/(n'-n)

esp max = go max (Di/k)i/P

Relations

) i/a _D i ) i/p

(12)

proportional

to

D -an'/(n'-n) (i/p-an'/(n'-n)) and D . This means that the maximum superplastic strain increases 1 as D i decreases and I it is shifted to higher strain-rate. This conclusion is in agreement with some experimental observations which have shown that a superplastic material usually exhibits a maximum elongation at an intermediate strain-rate, this maximum being associated with the maximum of the strain-rate sensitivity coefficient (3). It is however shown here that, even if m is constant and there is no region I, there is a maximum in the elongation of a superplastic

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material and this maximum is due to the evolution of the structure. In the case where grain coarsening is small (p < (n'-n)/n'a), the superplastic strain increases continually as strainrate decreases. It is necessary to remember here that e is the strain achieved in the superplastic range for an alloy with grain size evolutionSP it is not the maximum tensile elongation of the superplastic material. As soon as deformation takes place in region III, however, it is highly unstable with rapid strain localization and rupture. Consequently e can be considered as the maximum uniform elongation which is possible to reach in the sdperpY~stic range. Prediction of elongation of superplastic materials : influence of cavitation. Most of the known superplastic alloys cavitate but only some of them, which are composed of phases having significantly different hardnesses and deformation characteristics are particularly prone to this phenomenon. In these materials, failure occurs by cavity interlinkage after extensive cavity growth has taken place. Several studies have been performed on this problem in various alloys and the main observations have recently been reviewed by Stowell (4). However no failure criterion has been firmly established or properly tested in various superplastic materials. These criteria generally assume a critical dimension for the cavity size at failure, so that cavity growth models have to be developed in order to predict the variation with strain of cavity volume fraction and cavity size during superplastic deformation. Based on experimental observations in two-phase u/8 brasses, a simple geometrical model was proposed recently (5) and later mathematically derived (6) assuming that cavity growth is controlled by the strain of the softer phase. The model enables the prediction of the variation of cavity size with strain and of the onset of cavity coalescence by considering the evolution with strain of the cavitation level. The predictions of this model will be used later for the case of u/~ brasses. Application

of the analysis to u/8 brasses

Experimental results concerning superplastic behaviour of u/8 brasses with different phase proportions were reported in different papers (7-9). The following conclusions were obtained : - the constitutive equation in the superplastic range takes the form :

(13)

= K(u,T) o2/L82 where K(u,T) depends on a-phase volume fraction and temperature - the constitutive law in region III is : 4 = K'(u,T) c

and L~

is the

~ grain-size.

(14)

Fig.2 shows plots of (L# -L^ i) versus deformation time on a Log-Log scale for t h r ~ u/~ brasses of different composition. The LA values were obtained from specimens deform2d at ~00 ° C ~ither at the same strain-rate of 1.67 x i0--- sec -~ up to different strains or at different strain-rates up to the same strain of 0.4. The plots can be well approximated by straight lines, leading to the following equation for R-grain growth during superplastic deformation: L~ = L~i + ktp (15)

~IP b.,,, 20

10

5

2

Table 1 gives all the experimental data derived from fig. 2.According to the values of n, a and n', respectively 2, 2 and 4, (n'-n)/n'a is equal to 1/4. For each alloy, p is then greater than (n'- n)/n'a (see table I) and equations Fig.2 (ii) and (12) can be used in order to calculate L^-L^.51 vs. deformatlon" time for s/8 and E Table 1 gives also the results o 2 ~ max" b~asses with different s-phase volcalcu~ions. It is then shown, for the ~/~ brasses used ume fractions. in this investigation, that superplastic strain_4must exhibit a maximum at a strain-rate of about 2 x i0 m~ sec .The calculated maximum strains seem quite low compared to other superplastic materials, but they are in roughly good agreement with experimental observations for the alloy with low a-phase volume fraction, the fracture strain for the Cu-42~ Zn alloy being about I.i at the same strain-rate (I0). ~0

I0

I SO

I

|

IOD

lO0

UlN,

m~.

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TABLE

Grain growth characteristics : : : :

: : :

: :

Cu - 42 % Zn C u - 40.6 % Zn Cu - 3g.4 % Zn

Vol.

a

:

:

: L~i (am)

:

:

:

: 0.24 : : 0.45 : : 0.63 :

20 13 8.6

18., No.

4

1

for ~/~ brasses and calculated values of :

Alloys

MATERIALS

P

o max

and

:

:

: eo max (sec-l)

:esp max :

sp max"

:

:

: 0.4 : : 0.64 : : 0.55 :

1.8 x i0 -4. 2.2 x 10-4 2.2 x i0

: : :

0.85 1.17 1.09

: : :

As previously mentioned, however, cavitation is taking place in a/~ brasses and it was shown that the cavitation level is well correlated to the strain of the softer 8-phase, a function of the a-phase volume fraction acting as a normalization factor for alloys with different a-phase volume fractions(ll). According to a recently developed model(6), coalescence would be expected to occur at a strain of about 0.7 for Cu-39.4 Zn and 1.7 for Cu-40.6 Zn. No prediction is possible for Cu-42 Zn because of the very low cavitation level observed in this alloy. Comparison between these predicted strains for coalescence and the calculated values of E shows s ax . then clearly that, for ~/~ brass with high a-phase volume fraction, failure o c c u ~ s % y cavlty coalescence without external necking, whereas for a/8 brass with low ~-phase volume fraction, grain growth is predominant and leads to a change in the deformation mechanism with less plastic stability and then rapid necking until rupture. These different rupture modes were clearly observed as shown in fig.9 of ref.(5). The same type of behaviour is expected to occur in other superplastic alloys particularly prone to grain growth and cavity formation due to the presence of phases of significantly different mechanical characteristics. The calculations carried out in this investigation are based on the assumption that the Logo vs.Log~ plot is divided into only two regions characterized by constant m-values, a n ~ i n this case, grain growth can explain the smaller elongations observed at low strain-rates compared to those in the superplastic range. Other phenomena can however also explain these smaller elongations, andparticularly a different deformation mechanism operating at low strain-rate, as has been put forward often. More detailed investigation dealing with physical aspects of superplastic deformation are necessary to clear u p this matter. Conclusion The influence of grain growth and cavitation has been studied in order to predict the elongation of superplastic materials ; the results are the following : - grain growth restricts elongation in the superplastic range. Under conditions where grain coarsening is relatively important, the superplastic elongation is maximum at a given strainrate, this maximum depending closely on the initial grain size of the alloy and on the kinetics of growth. - cavity growth and coalescence have a dominant influence on the failure strain of superplastic alloys composed of phases having significantly different hardnesses such as a/8 brasses. - for a/8 brasses, grain growth is predominant for alloys with low a-phase volume fractions, leading to rupture with pronounced external necking , whereas for alloys with high a-phase volume fractions, failure occurs by cavity interlinkage without appreciable neck development. Acknowledgements The author is grateful

to Professor Baudelet for helpful discussions. References

i. 2. 3. 4. 5. 6. 7. 8. 9. i0.

M. Su~ry and B. Baudelet, Revue Phys.Appl. 13, 53 (1978). M.A. Clark and T.H. Alden, Acta Metall. 21, 1195 (1973). H. Ishikawa, F.A. Mohamed and T.G. Langdon, Phil. Mag. 32, 1260 (1975). M.J. Stowell, Metal Sci. 17, 1 (1983). J.Belzunce and M. Su~ry, Acta Metall. 31, 1497 (1983). H.M. Shang and M. Su~ry, Metal Sci. To be published. M. Su~ry and B. Baudelet, Phil. Mag. 41, 41 (1980). M. Su~ry and B. Baudelet, Res. Mech. 2, 163 (1981). M. Su~ry, J. Mat. Sci. 17, 3074 (1982). M.Su~ry and B. Baudelet, Superplastic Forming of Structural alloys, p.105 N.E. Paton and C.H. Hamilton. The Metallurgical Society of AIME, Warrendale(i982). ii. J. Belzunce and M. Su~ry, Scripta Met. 15, 895 (1981).