Volume 2. number 4B MATERIALS LETTERS May 1984

THE STRESS EXPONENT OF STEADY-STATE CREEP IN AI-Mg ALLOYS

D.O. NORTHWOOD * and 1.0. SMITH Department of Mining and Metallurgical Engineering, University of Queensland, St. Lucia, Queensland 4067, Australia

Received 19 December 1983

The stress exponent of steady-state creep at 300°C for Al-1.77, 4.20 and 7.72 at% Mg alloys varies from 3.37 to 3.98

with increasing Mg content when determined using the applied stress, o. However the stress exponent remains constant at

=3 when calculated using the effective stress, cr,. The back or friction - stress, oO, determined from stress-dip tests in-

creased significantly with increasing 0.

1. Introduction

Under creep conditions at temperatures above =0.4T,, where T, is the absolute melting tempera- ture of the material, the steady-state creep rate, is, is

usually related to the applied stress, u, through an

equation of the form

is =.4un, (1)

where A is a constant which incorporates the depen-

dence on temperature, and n is termed the stress ex- ponent. The stress exponent, n, has been found to be ~4 for pure metals and a wide range of relatively simple alloys. However the empirical fit to eq. (1) of creep data for microstructurally complex alloys has

yielded values of n which differ from 4, with values up to 40 for precipitation or dispersion hardened

materials [ 11. Several authors [2-5] have proposed that creep

in such complex a!loys, and indeed also in simpler

material, is determined by an effective stress ue,

which differs from the applied stress by a quantity u. that is termed either a friction, or back-stress by ad- vocates of recovery-controlled and activated slip models of creep respectively. This leads to a modified equation for steady-state creep where

* On leave from Department of Engineering Materials, Univer-

sity of Windsor, Windsor, Ontario, Canada N9B 3P4.

is = A *(a - uo)n’ (2) and it is the value of n’, not FZ, that is indicative of

the creep process. As part of an ongoing program into the origins of

back stress in high-temperature creep, a series of Al-

Mg alloys of varying Mg content were creep tested at

300°C and both the steady-state creep rate, is, and

the effective stress, u,, were measured. The results confirm the importance of determining the stress ex-

ponent of steady-state creep from the effective stress,

u,, rather than from the applied stress, u, since values of n (derived from u) indicate a change in mechanism

of steady-state creep with increasing % Mg whereas the values of n’ show the creep mechanism to be the same for all Mg contents.

2. Experimental details

Aluminium alloys containing 1.77, 4.20 and 7.72

at% Mg were vacuum cast using high-purity metals. The alloys were swaged and drawn to 2.95 mm diam- eter wire with appropriate intermediate annealing treatments. The wire specimens were annealed for f h at 400°C and quenched into water to obtain a com- plete a-phase solid solution prior to creep testing.

All creep tests were performed in air at 300°C with a temperature gradient along the 76 mm gauge

0 167-557x/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 2, number 4B MATERIALS LETTERS May 1984

length of <l “C. The uniaxial constant stress creep

machine with high sensitivity was used. The experi- mental arrangement is described in more detail in ref.

[6]. Steady-state creep rates down to lop8 s-l could be measured with confidence and reproducibility.

The effective stress, ue, was measured using the stress- dip transient method where once the specimen is

creeping at the steady-state rate, part of the applied stress is removed and the nature of the strain transient is observed. The sign of the strain rate just after the reduction depends on the magnitude of the stress

change with positive creep rates being obtained after small changes and negative transients being recorded for large reductions [2,7,8]. At one particular stress

reduction, the creep rate after the instantaneous drop

is zero, and there is an incubation period before creep resumes in a forward manner. It is this particular

stress reduction to cause zero creep that is considered

to be ue, the effective stress.

3. Experimental results

Log-log plots of the steady-state creep rate, is, against the applied stress are presented in fig. 1. Al- though there is some scatter of the experimental

data points (particularly for the 4.2 and 7.72 at% Mg alloys) all plots can be adequately described as show- ing linear relationship between log ls and log CJ with

a slope which is equal to the stress exponent n. The data for the M-7.72 at% Mg alloy could possibly in- dicate a transition in rate with a lower n value for the lower stresses. Table 1 summarises the results obtain- ed by linear-regression analysis for the stress exponent n_ The correlation coefficients, despite the scatter in

experimental data, are generally high (X.992) and give one reasonable confidence that the measured changes in stress exponent with changing Mg content

are real. The value of n increases with increasing Mg content, with the value for the 1.77 at% Mg alloy being representative of a viscous glide process in creep (n x 3) and that for the 7.72 at% Mg alloy being more representative of a recovery creep process such as found in pure aluminium where n = 4.0-4.5 [9].

The value of the effective stress, cre, varied with the level of the applied stress with ue being a higher percentage of the applied stress for larger values of u, fig. 2. Using these values for u,, the stress exponent

340

lo-~~ 1 5 10 50

Applied Stress : MPa

Fig. 1. Variation of steady-state creep rate, E’s, with applied stress for AI-Mg alloys creep tested at 300°C.

Table 1 Summary of results of stress exponents, n and n’ of steady- state creep for Al-Mg alloys

0

Mg Eq. (1) Eq. (2) content (at%) stress (correlation stress (correlation

exponent n coefficient) exponent n’ coefficient)

1.77 3.37 (0.999 1) 3.04 (0.9934) 4.20 3.68 (0.9928) 3.06 (0.9738) 7.72 3.98 (0.9922) 3.05 (0.9920)

Volume 2, number 4B MATERIALS LETTERS May 1984

1.0

. AI-1.74at %MEJ

o Al-4.2at %hlg

I t / I _ 5 10 15 20

Applied Stress, o : MPa

Fig. 2. Variation of effective stress (a,>, expressed as a percentage of the applied stress, with the applied stress.

can be recalculated using eq. (2). Values of n’ are sum- marised in table 1 where it can be seen that the stress exponent, n’, remains constant at ~3.0, which is

characteristic of creep by a viscous glide process 191. The other point of interest is to look at a0 rather

than ue and see how it varies with the applied stress. This is shown in fig. 3 for all three alloys. Within the experimental scatter all three alloys show an approx-

imately linear dependence of cro on the applied stress with slopes varying from 0.55 to 0.60. The Al-4.20 at% Mg alloy has a stress dependence of a0 similar to that for the Al-l.77 at% Mg alloy at low applied stress-

es (<9 MPa) whereas the Al-7.72 at% Mg alloy shows a stress dependence of u. similar to that for the Al-

1.77 at% Mg alloy at high applied stresses (>9 MPa). In general there is an increase in a0 with increase in

I 1.77at % Mg . 4.2oat%hlg . 7.72 at % Mg

t 5

I 1 t

10 15 20 AF+xI Stmss : WPa

Fig. 3. Variation of back stress (00) with the applied stress.

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Volume 2, number 4B MATERIALS LETTERS May 1984

Mg content, although a0 for the h-4.2 at% Mg and Al-7.72 at% Mg alloys are very similar. (This parallels the steady-state creep behaviour where the $ versus u relationship was similar for the two alloys.)

4. Discussion

The present data for Al-Mg alloys emphasise again

most strongly that use of the applied stress, u, in de-

termining the stress exponent of steady-state creep

according to eq. (1) must be viewed with consider-

able caution. The calculated stress exponent can vary from material to material suggesting a change in rate- controlling mechanism. As has been advised by nu-

merous researchers [2-51 this anomalous behaviour (apparent change in stress exponent) can be rationaliz- ed using an effective stress ue (= (u - uO) where a0 is

the friction- or back (internal) stress) in eq. (2). Using such an analysis for the present results a stress expo-

nent, n’, of ~3 is obtained. Such a value for the stress exponent (~3) is characteristic of a viscous glide pro-

cess [9,10] and has previously been found for the

steady-state creep of other Al-Mg alloys [9,1 l-161.

forces which oppose glide can be considered a part of the total deformation resistance. In alloy creep where dislocation glide is viscous because of solute drag, e.g. for the present case of the Al-Mg alloys, there are frictional contributions of solute drag and the back- stress contribution of dislocation-substructure inter- actions to the total measured back stress [21]. If we

assume that the back stress u. varies linearly with applied stress u (which is suggested in fig. 3 for the higher Mg content alloys) and extrapolate the results back to u = 0 then we find that the intercept, u.

(u = 0), increases with increasing Mg content having values of 0.25, 0.70 and 1.40 MPa for Mg contents

1.77, 4.20 and 7.72 at% respectively. The exact phys- ical significance of u. (u = 0) is uncertain since one

would expect that any back stress would be zero if the applied stress was zero since no glide was occurring

and hence no dislocation substructure was developing.

However u. (a = 0) might represent some base level

of friction stress which would be expected to increase

with increasing amount of Mg in solid solution.

The parameter, uo, varies with stress, temperature

and microstructure Q as [5 ] :

u. = (u, T, S). (3)

In our case where T remains constant at 300°C and the initial structure is a solid solution for all alloys,

a0 is shown to vary with the applied stress, u (fig. 3).

This is not to say that none of the variation in u is

due to structure (,!j) because there could be material

heterogeneities due to differently oriented grains and/

or variations in compositions within grains and from grain to grain. Moreover many alloys are intrinsically

unstable exhibiting significant changes in both the

chemistry and morphology of phase during testing (e.g. the higher Mg alloy (7.72 at% Mg) could have

exhibited some precipitation of AlMg, during strain- ing at 3OO’C). The fact that u. varies with the applied stress u agrees with previous work on Cu [S], Cu-

4.04 wt% Co [S], Al-5.5 at% Mg [ 171 and Al-3.0 at% Mg [4]. A fairly constant u. with varying u has how-

ever been found for Nimonic 80A [ 181 and a nickel- based superalloy IN738LC [19].

Caution should be exercised in attaching too much

significance to the extrapolated values of a0 (u = 0) since given the fact that the u,/u versus u plot (fig. 2) is found to be approximately linear one would expect

the u. versus u plot (fig. 3) to be non-linear and ex- trapolation back to u = 0 is difficult. In fact such non-linearity is suggested by the data points of the

1.77 at% Mg alloy. When the stress dependence of the creep rate is de-

scribed using eq. (2) so that ;s is considered as a function of (a - uo) rather than of the applied stress u, no change in the stress dependence occurs with

change in Mg content and the stress exponent, n’, is constant at e3.0. The value of the stress exponent n determined using eq. (1) varies with Mg content. This

merely reflects the fact that no is determined by the magnitude and stress dependence of uo. If u. is very

low, n and n’ will be virtually the same. However, as is the case for the Al-Mg alloys, a0 is not low, and has different absolute values and stress dependencies for each alloy, then n and n’ will differ.

5. Conclusions

Kocks et al. [20] have attempted to clarify the (1) The results from creep testing Al-Mg alloys

terminology of back stresses by suggesting that all containing 1.77, 4.20 and 7.72 at% Mg at 300°C con-

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Volume 2, number 4B MATERIALS LETTERS May 1984

firm the importance of determining the stress expo- nent of steady-state creep using the effective stress,

o,, rather than the applied stress, u. Values of the stress exponent determined using u change with Mg content whereas those determined using ue remain constant at a value (~3) that is indicative of creep by a viscous glide process.

(2) The value of the back - or friction - stress, uo, determined from stress-dip tests is shown to vary sig-

nificantly with applied stress (a) and possibly to a less- er extent with Mg content, increasing with an increase in either u or Mg content.

Acknowledgement

Financial support from the Australian Research Grants Scheme is gratefully acknowledged. The tech-

nical assistance of Mr. L. Moerner in conducting the

creep experiments is also gratefully acknowledged. One of the authors (DON) would also like to thank

the Natural Sciences and Engineering Research Coun-

cil of Canada for the provision of a Travel Grant.

References

[l] M. McLean, Proc. Roy. Sot. A371 (1980) 279. [2] R. Lagneborg, J. Mat. Sci. 3 (1968) 596. [ 31 P.L. Threadgill and B. Wilshire, in: Proceedings of the

Iron and Steel Institute Conference on Creep Strength

of Steels (I.S.I., Sheffield, 1972) p. 8.

[4] C.N. Ahlquist and W.D. Nix, Acta Metall. 19 (1971)

373.

[51

I61

1

1101

IllI

]I21

I131

iI41

I151

1161

iI71

1181

I191

I201

Pll

B. Wilshire, in: Creep in structures, eds. A.R.S. Porter

and D.R. Hayhurst (Springer, Berlin, 1981) p. 72.

1.0. Smith, Australian Atomic Energy Commission Re-

search Project Progress Report No. 15 (January 1971).

J.C. Gibeling and W.D. Nix, in: Strength of metals and

alloys, Vol. 2, ed. R.C. Gifkins (Pergamon, London,

1982) p. 613.

W. Blum, J. Hausselt and G. Konig, Acta Metall. 24

(1976) 293.

T.G. Langdon and P. Yavari, in: Creep and fracture of

engineering materials and structures, eds. B. Wilshire

and D.R.J. Owen (Pineridge Press, Swansea, 1981) p. 71.

T.G. Langdon, in: Strength of metals and alloys, Vol. 3,

ed. R.C. Gifkins (Pergamon, London, 1982) p. 1105.

H. Oikawa, K. Sugawara and S. Karashima. Trans. Japan.

Inst. Met. 19 (1978) 611.

M. Pahutova and J. Cadek, Phys. Stat. Sol. 56a (1979)

305.

P. Yavari, F.A. Mohamed and T.G. Langdon, Acta Me-

tall. 29 (1981) 1495.

K.L. Murty, F.A. Mohamed and J.E. Dorn, Acta Metall.

20 (1972) 1009.

H. Oikawa, K. Sugawara and S. Karashima, Scripta Met.

10 (1976) 885.

S. Kikuchi, Y. Motoyama and M. Adachi, J. Japan. Inst.

Light Met. 30 (1980) 449.

K. Kucharova, I. Sax1 and J. Cadek, Acta Metall. 22

(1974) 465.

K.R. Williams and B. Wilshire, Met. Sci. J. 7 (1973)

176.

P.J. Henderson and M. McLean, Acta Metall. 31 (1983)

1203.

U.F. Kocks, A.S. Argon and M.F. Ashby, Progr. Mat. Sci.

19 (1975) 1.

J.C. Gibeling and W.D. Nix, Mat. Sci. Eng. 45 (1980)

123.

343