The elastic properties of aluminum at high temperatures

J. Phys. Chem. Solids Vol. 48, No. 7, pp. 603-606, 1987 Printed in Great Britain.

0022-3697/87 $3.00 +0.00 © 1987 Pergamon Journals Ltd.

THE ELASTIC PROPERTIES OF A L U M I N U M AT HIGH TEMPERATURES REX B. MCLELLAN a n d TOMAZ ISHIKAWA Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, Texas 77251, U.S.A.

(Received 23 September 1986; accepted 4 December 1986) Ala~tract--Thin-line ultrasonic techniques have been used to measure the Young's modulus and shear modulus of polycrystallinealuminum up to 850 K. In both cases, large modulus defects are found at high temperatures. A simple model for grain boundary relaxation is used to analyze the measured elastic behavior. The value of the activation energy for grain boundary diffusion extracted from the comparison of the data with the model is close to previous determinations of this energy.

Keywords: Elasticity, modulus defect, grain boundary, relaxation, diffusion.

1. INTRODUCTION In a recent report [1], it was shown that deviations found from simple classical behavior [2] in the elastic moduli of polycrystalline palladium were explainable in terms of a grain boundary relaxation phenomenon. A comparison between the elastic behavior of the material at high temperatures (i.e. in the region of the relaxation) and the behavior extrapolated upwards from temperatures at which the elastic behavior appears to be classical, enables values for the grain boundary diffusivity to be deduced from the experimental elastic data. Unfortunately values for Qe, the grain boundary diffusivity, are difficult to measure and values are scant. The difficulties involved in the experimental determination of Qa have been outlined by Peterson [3]. Frost and Ashby [4] have given the value Qa = 84 k J/tool, which is deduced by rationalizing the available data so that they are in mutual accord with the deformation-mechanism map for aluminum. This Qa-value is, however, very close to the value of 75.35 kJ/mol determined by Rozenberg and Epshtein [5] from measurements of grain boundary displacement during the creep of polycrystalline AI. Now the values of E, the Young's modulus of polycrystalline A1, extend up to 773 K. They indicate a small deviation from the essentially linear behavior found at low temperatures. The values currently available are illustrated in Fig. 1 and were taken from Hearman [6](®), Alexandrov and Ryzhova [7]([]) and Sutton [8] (A). It can be seen that the degree of mutual accord is good. Fortunately, single crystal elastic data are also available for AI. Values of C11 and Cu up to 900 K are shown in Fig. 2. The data are due to Sutton [8] (©), Alexandrov and Ryzhova [7](I-q), Vallin, Mongy, Salama, and Beckman [9] (O), and the data evaluation of Wawra [10](®). It can be seen that the data are

in good mutual accord and behave in a linear manner above 300 K. Now A1 was the metal studied by internal friction techniqes by K~ [11] in his attempt to confirm Zener's classical calculations of grain boundary relaxation in metals [12]. K~ did detect a large "softening" effect in the shear modulus of polycrystalline AI at 500 K. Thus, it is not surprising that the data of Fig. 1 begin to show a departure from linear behavior at high temperatures. The object of the present work is to extend the elastic measurements of polycrystalline Al to higher temperatures and to analyze the data in terms of the model deduced previously [1].

2. EXPERIMENTAL PROCEDURE

MARZ-grade aluminum wires of length 10-1m and diameters of 1.41, 1.67 and 2.0 x 10 -3 m were studied by the thin-line ultrasonic technique using longitudinal waves of 1 MHz frequency. The details of the method have been given in previous reports [13, 14]. Grain sizes were in the range (1-4) x 10-7/m 2. The AI wires were suspended in an argon atmosphere (99.999% pure) and the transit time of the waves measured in the temperature range 273-850 K both upon increasing and decreasing the temperature. The scatter in the data was found to be small. The transit time could be measured to within 0.5#s, which leads to an uncertainty of + 3% in the elastic constants. The extraction of E-data from the wave velocity measurements [14] requires values for the density of the material. Both the density (p) and length (1) of the wires were calculated for each experimental temperature from published values for the thermal expansion coefficient and the values of p and 1 at 273 K [151.

603

604

REx B. McLELLAN and TOMAZ ISHIKAWA

.70 -,~,

"'L~,.~,. "~. '~'....~ %x

.60

"h..~

"s %.

~°

Qx%

Z

A'%

o .E %x

UJ

.50

'400 273

IO0 373

,

2'00 473

t

300 573

,

400 673

.

500 773

600 T eC 87]5 T *K----"

TEMPERATURE

Fig. 1. Young's modulus of polycrystalline aluminum as a function of temperature. The points ( 0 ) are taken from this work and the points (®) from Ref. [6], ([]) from Ref. [7] and (Z~) from Ref. [8].

Shear waves of the same frequency were also transmitted through AI wires of differing diameter so that the shear modulus could be determined. The results are shown in Fig. 3 and compared with the values given at lower temperatures by Sutton [8] ( A symbols) and by K6ster [16] (o-symbols). In the present work, results were taken using 1.41 x 10-3m wire (C), heating; x , cooling), 1.67 x 10-3m wire (heating). These wires were then annealed at 850 K for 5 h and more measurements undertaken. The

eK

TEMPERATURE

300

400

500

700

600

800

900

=,q i ~t

2,6 Z,5 2,4 2,3

o

2,2

.E

(n Z,I ..J

2.5

0.30

Vo

*~,

2,0

'~,,

0 -

~,O z

O,25

o.::

=

U-

°;~

O.ZO

1,5

,<,

1,9

rr

1,8

X; U)

o

1.7 1.6

o~

e

1.5

"%; . . . . . . . . . O.1~

hO

1,4 1.3

,60

=~o

-~6o 4~o ~o TEMPERATURE "C

i

J

TxlO -t K Fig. 3. Temperature dependence of the shear modulus of Fig. 2. Variation with temperature of the elastic constants aluminum. The points (O) ( x ) (*) were taken from this Cll and C,M for aluminum. The data (O) are taken from work (see text for differences) and the data (~7) were Ref. [8], ([]) from Ref. [7] and (<>) from Ref. [9]. The taken from Ref. [8] and those given by the symbols ( e ) from ReC [16]. symbols (®) represent the data evaluation of Wawra [10]. '

The elastic properties of aluminum

13

\

605

corresponding to the relation:

;7 = ;70exp(Q /k T)

(1)

o

then it is easy to show that the relaxed modulus is given by [1]:

12

In ~

~

- 1

= 2 In(ore0) - ~-~,

(2)

C

o\

where ~o is the frequency. Now the value of Q may be compared with Qa, the activation energy for grain boundary diffusion by using an expression derived by Raj and Ashby [17] for the Newtonian viscosity r/B of the grain boundary material. This expression is

\ 12

13

14

104/T K-I Fig. 4. Arrhenius plot of the quantity ~b. points (©) result from the thinner wires (heating mode) and the points (*) from the thicker wires (cooling mode). All the current data are in good mutual accord and also in excellent agreement with those of Krster [16]. The data of Sutton [8] are slightly lower in the low-temperature range. There is, however, no doubt that the shear modulus exhibits a departure from linear behavior which sets in at ~ 670 K. 3. DISCUSSION The data obtained for E are shown in Fig. 1. The points taken upon increasing the temperature are shown by (O)-symbols and those shown by the ( x )-symbols result from measurements made upon cooling from the maximum temperature. It is clear from Fig. 1 that the E-values are in excellent accord with the previously available lowtemperature data and that a large modulus defect occurs with an onset at ~ 650 K. The shear modulus data also exhibit the modulus defect clearly. The solid line in Fig. 1 is a least-squares regression to the data in the range 273-520 K. The data in the range 520-850 K were fitted to a polynomial of the form: E = A + BT + C(T - Tg) 2 + D(T - Tg) 3, where Tg = 520 K. The constants A, B, C and D were then used to calculate the ratio E/E*, where E* denotes the relaxed dynamic modulus and E the values given by the solid line in Fig. 1. In the simple model for grain boundary relaxation previously discussed [1], the temperature dependence of the relaxation was introduced in a simple manner by considering that the aggregate is composed of a series-coupled solid (grains) of Young's modulus E, and a Newtonian liquid (boundary) of viscosity 7. If grain boundary flow is a thermally activated process

1 d3kT r/s = 132 6 f l D B'

(3)

where d is the grain size, 6 is the effective thickness of the grain boundary diffusion path, f~ the atomic volume and D B is the grain boundary diffusivity: D B = D ° exp( -- QB/kT).

(4)

If the ;7 of eqn (1) is identified with T/s, then it is easy to show [1] that In ~b = In ~

- 1

= K -- k--'T-'

(5)

where K is a constant. A plot of In ~b vs 1/T is shown in Fig. 4. The points are calculated at 50 K intervals from the values of E and E* deduced from the data of Fig. 1 by the fitting processes discussed. It can be seen that the plot of Fig. 4 is reasonably linear. At least-squares regression gives a value of QB corresponding to the "best value" of the slope of Fig. 4 of QB = 78.4 kJ/mol. This value of QB is close to the two values discussed in the introduction to this short report (i.e. 84 kJ/mol [4] and 75.35 k/mol [5]). Since the value of QB derived from the current data is obtained from the slope of an Arrhenius plot which is itself derived by curve fitting techniques, it is not proper to assign error bars to QB. However, if the uncertainty in the values of the calculated points in Fig. 4 is arbitrarily set at zero (i.e. they are considered to be directly measured), then the best value of QB is 78.4 + 4.3 kJ/mol. This coincidence between the three values of QB presented here should, however, not be taken to infer solid evidence for the validity of the model concepts employed in arriving at eqn (5). It should be borne in mind that direct experimental determinations of QB are difficult and the scant data existing often show alarming mutual discrepancies.

Acknowledgement--The authors are indebted to the National Science Foundation for providing support under the Metallurgy Program (Grant No. DMR 78-1306).

606

REx B. MCLELLANand TOMAZ ISHIKAWA

REFERENCES 1. Yoshihara M., McLellan R. B. and Brotzen F. R., Acta metall. 35, 775 (1987). 2. Weiner J. H., Statistical Mechanics of Elasticity. Wiley, New York (1983). 3. Peterson N, L., Int. Metals Rev. 28, 65 (1983). 4. Frost H. J. and Ashby M. F., Cambridge University Engineering Deparment publication CUED/C/ MATS/TR. 87 (1981). 5. Rozenberg V. M. and Epshtein I, A., Fizika Metall. 9, 124 (1960). 6. Hearmon R. F. S., Adv. Phys. 5, 323 (1956). 7. Alexandrov K. S. and Ryzhova T. V., Soviet PhysCryst. 6, 228 (1961).

8. Sutton P. M., Phys. Rev. 91, 816 (1953). 9. Vallin J., Mongy M., Salama K. and Beckman O., J. appl. Phys. 35, 1925 (1964). 10. Wawra H. H., Aluminium 8, 232 (1974). 1!. K~, T.-S., Phys. Rev. 71, 533 (1947). 12. Zener C., Phys. Rev. 60, 906 (1941). 13. Yoshihara M. and McLellan R. B., Acta metall. 29, 1277 (1981). 14. Arnoult W. J. and McLellan R. B., Acta metall. 23, 51 (1975). 15. Aluminum: Properties, Physical Metallurgy and Phase Diagrams (Edited by K. Van Horn), A. S. M. Metals Park, Ohio (1967). 16. Kfster W., Z. Metallk. 39, 1 (1948). 17. Raj R. and Ashby M. F., Acta metall. 73, 425 (1975).