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Elastic moduli of two phase aluminum alloys
Scripta
METALLURGICA
Vol. 15, pp. 5 2 3 - 5 2 4 , 1 9 8 1 P r i n t e d in the U . S . A .
Pergamon P r e s s Ltd. All r i g h t s r e s e r v e d
ELASTIC MODULI OF TWO PHASE ALUMINUM ALLOYS
M. E. Fine Dept. of Materials Science & Engineering and Materials Research Center Northwestern University, Evanston, IL 60201 USA (Received
February
23,
1981)
In two phase alloys, the Young's modulus or shear modulus of the composite is, of course, related to the moduli of the individual phases. Two simple methods of averaging have been proposed: Voight (i) averaging which assumes uniform strain in both phases and Reuss (2) averaging which assumes uniform stress in both phases. The former gives equation (i) which is the linear law of mixtures, the latter gives equation (2) which predicts a smaller modulus than the linear law of mixtures.
Mv MR
=
~A + ~% X
MA
I +
(1) (2)
Y
MB
In the above, X and Y are the volume fractions of the two phases; M A and ~ moduli.
are their
Figure I plots equations (i) and (2) versus volume fraction of the A phase for several values of MA/M B. As this ratio deviates further from one, the difference between the results of the two methods of averaging increases. Recent interest in attempting to develop further or con~nercialize high modulus-low density aluminum alloys (3) led us to examine the literature on moduli of two phase aluminum alloys to gain better insight into how the moduli of the alloys may be determined from the moduli of the pure phases. A rather complete set of data of Young's modulus for cast and annealed AI-Mn alloys was found for compositions to approximately 50 vol.% MnAIs(4). These data, converted from wt. pct. Mn to vol. pct. MnAI s, are plotted in Fig. 2. A linear (Voight) relation is a rather good approximation and an average Young's modulus of 130 Gpa is predicted for MnAIs. MnAI6 is orthorhombie and, of course, Young's modulus is expected to be highly anisotropic. Reuss averaging would obviously not fit the data nearly as well as Voight averaging. Lyle (5) measured Young's modulus of AI and Al~Os mixtures prepared by sintering and forging mixed powder compacts. The average Young's modulus of AI~O 3 (6) is approximately six times that for AI so a very large difference between the Reuss and Voight curves are predicted, as shown in Fig. 3. The one data point plotted is much closer to the Reuss curve than to the Voight curve. Why Voight averaging seems to hold for AI-MnAI6 while Reuss averaging is imation for Ai-AleOs is an interesting question. The bonding between the two factor. This may be poor between AI20 m and AI so that both stress and strain A ~ 0 3 , i.e., the stress and strain are not completely transferred from the AI AleO 3.
a better approxphases may be a are lower in the matrix to the
The main candidates for high modulus-low density AI base alloys are AI-AI3Li (6 ~ phase) mixtures modified by addition of other elements such as Mg and Cu (3). Young's modulus of an AI-2.84 wt. pct. Li alloy (density is 2.49 g/cm 3) aged to peak strength is 80.6 Gpa which may
523 0036-9748/81/050523-02502.00.0 Copyright (c) 1 9 8 1 P e r g a m o n Press
Ltd.
524
ELASTIC
HODULI
OF A1 A L L O Y S
Vol.
15,
No.
be compared to 65 Gpa for pure aluminum. According to Ceresara et. al. (7), the solubility limit of Li in AI with respect to metastable 6' is 6.2 at.% Li at 20~C, the aging temperature. On this basis, an average Young's modulus of 140 Gpa was computed for A13Li assuming the Voight linear law of mixtures. Acknowledgement This research was supported by the U. S. Air Force Office of Scientific Research, Office of Aerospace Research, Grant No. AFOSR 78-3732B. References W. Voight, Lehrbuch der Kristallphysik Leipzig, Teubner, p. 716 (1928). A. Reuss, Z. Angew. Math. Mech. 9, 55 (1929). T. H. Sanders and E. S. Balmuth, Metals Progress 113, No. 3, 32 (1978). N. Dudzinski, J. R. Murray, B. W. Mett and B. Chalmers, J. Inst. Metals 74, 291 (1947-48). J. P. Lyle, Jr. in "Aluminum", ed. K. R. Van Horn (ASM, Metals Park, Ohio, 1967) Vol. I, Chap. i0, pp. 337-358. N. Soga and O. L. Anderson, J. Am. Ceram. Soc. 49, 355 (1966). S.Ceresara, G. Cocco, G. Fagherazzi and L. Schiffini, Phil. Mag. 35, 373 (1977).
i.
2. 3. 4. 5. 6. 7.
MA
'
RE'USS . . . . . . . .
...... M A : F 5 M B --
3MA *MB
/.:1
: 5MB
1
125
=o
17
.E_
16
15
O
IOO 13
3 O >-
j¢" IO # 04
06
'
0'8
'
0
XA VOLUME FRACTION A
Voight and Reuss averages for Young's or shear modulus for mixtures of A and B phases versus volume fraction of A phase. 60
i
i
i
_
o .¢_
g {.n
AIzO 3 -AI
40
i
I
0.4
0.6
0.8
IO
M n ,6,1s
FIG. 2 Young' s modulus of aluminum-MnAl s alloys versus volume fraction of MnAIs.
¢.o c
/
30
-d
2oo
~: FIG.
g.
20 /
g
>-
I
0.2
400
3oo
(3.
o
M u r r a y and C h a l m e r s I
Cornhill
/
Predicted Moduli 50
9
75
From Dudzinski, Mort,
Volume F r a c t i o n
FIG. i
/ >-
Io
• Lyle Alcoa, P/M Extrusion o
I o
0 2
L
I
I
0.4
0.6
0.8
Volume Fraction AIzO 3
1,0
~;
14
_u) E
02
I
CL
/
0 ~;
0
i
I
/.~'/
0
MB
I
19
~.;/I
M~ : ZMB
----MA
20
' -,~
3
Young's modulus of AI203 -AI composite compared to Voight and Reuss averaging for AI203- AI mixtures.
5
METALLURGICA
Vol. 15, pp. 5 2 3 - 5 2 4 , 1 9 8 1 P r i n t e d in the U . S . A .
Pergamon P r e s s Ltd. All r i g h t s r e s e r v e d
ELASTIC MODULI OF TWO PHASE ALUMINUM ALLOYS
M. E. Fine Dept. of Materials Science & Engineering and Materials Research Center Northwestern University, Evanston, IL 60201 USA (Received
February
23,
1981)
In two phase alloys, the Young's modulus or shear modulus of the composite is, of course, related to the moduli of the individual phases. Two simple methods of averaging have been proposed: Voight (i) averaging which assumes uniform strain in both phases and Reuss (2) averaging which assumes uniform stress in both phases. The former gives equation (i) which is the linear law of mixtures, the latter gives equation (2) which predicts a smaller modulus than the linear law of mixtures.
Mv MR
=
~A + ~% X
MA
I +
(1) (2)
Y
MB
In the above, X and Y are the volume fractions of the two phases; M A and ~ moduli.
are their
Figure I plots equations (i) and (2) versus volume fraction of the A phase for several values of MA/M B. As this ratio deviates further from one, the difference between the results of the two methods of averaging increases. Recent interest in attempting to develop further or con~nercialize high modulus-low density aluminum alloys (3) led us to examine the literature on moduli of two phase aluminum alloys to gain better insight into how the moduli of the alloys may be determined from the moduli of the pure phases. A rather complete set of data of Young's modulus for cast and annealed AI-Mn alloys was found for compositions to approximately 50 vol.% MnAIs(4). These data, converted from wt. pct. Mn to vol. pct. MnAI s, are plotted in Fig. 2. A linear (Voight) relation is a rather good approximation and an average Young's modulus of 130 Gpa is predicted for MnAIs. MnAI6 is orthorhombie and, of course, Young's modulus is expected to be highly anisotropic. Reuss averaging would obviously not fit the data nearly as well as Voight averaging. Lyle (5) measured Young's modulus of AI and Al~Os mixtures prepared by sintering and forging mixed powder compacts. The average Young's modulus of AI~O 3 (6) is approximately six times that for AI so a very large difference between the Reuss and Voight curves are predicted, as shown in Fig. 3. The one data point plotted is much closer to the Reuss curve than to the Voight curve. Why Voight averaging seems to hold for AI-MnAI6 while Reuss averaging is imation for Ai-AleOs is an interesting question. The bonding between the two factor. This may be poor between AI20 m and AI so that both stress and strain A ~ 0 3 , i.e., the stress and strain are not completely transferred from the AI AleO 3.
a better approxphases may be a are lower in the matrix to the
The main candidates for high modulus-low density AI base alloys are AI-AI3Li (6 ~ phase) mixtures modified by addition of other elements such as Mg and Cu (3). Young's modulus of an AI-2.84 wt. pct. Li alloy (density is 2.49 g/cm 3) aged to peak strength is 80.6 Gpa which may
523 0036-9748/81/050523-02502.00.0 Copyright (c) 1 9 8 1 P e r g a m o n Press
Ltd.
524
ELASTIC
HODULI
OF A1 A L L O Y S
Vol.
15,
No.
be compared to 65 Gpa for pure aluminum. According to Ceresara et. al. (7), the solubility limit of Li in AI with respect to metastable 6' is 6.2 at.% Li at 20~C, the aging temperature. On this basis, an average Young's modulus of 140 Gpa was computed for A13Li assuming the Voight linear law of mixtures. Acknowledgement This research was supported by the U. S. Air Force Office of Scientific Research, Office of Aerospace Research, Grant No. AFOSR 78-3732B. References W. Voight, Lehrbuch der Kristallphysik Leipzig, Teubner, p. 716 (1928). A. Reuss, Z. Angew. Math. Mech. 9, 55 (1929). T. H. Sanders and E. S. Balmuth, Metals Progress 113, No. 3, 32 (1978). N. Dudzinski, J. R. Murray, B. W. Mett and B. Chalmers, J. Inst. Metals 74, 291 (1947-48). J. P. Lyle, Jr. in "Aluminum", ed. K. R. Van Horn (ASM, Metals Park, Ohio, 1967) Vol. I, Chap. i0, pp. 337-358. N. Soga and O. L. Anderson, J. Am. Ceram. Soc. 49, 355 (1966). S.Ceresara, G. Cocco, G. Fagherazzi and L. Schiffini, Phil. Mag. 35, 373 (1977).
i.
2. 3. 4. 5. 6. 7.
MA
'
RE'USS . . . . . . . .
...... M A : F 5 M B --
3MA *MB
/.:1
: 5MB
1
125
=o
17
.E_
16
15
O
IOO 13
3 O >-
j¢" IO # 04
06
'
0'8
'
0
XA VOLUME FRACTION A
Voight and Reuss averages for Young's or shear modulus for mixtures of A and B phases versus volume fraction of A phase. 60
i
i
i
_
o .¢_
g {.n
AIzO 3 -AI
40
i
I
0.4
0.6
0.8
IO
M n ,6,1s
FIG. 2 Young' s modulus of aluminum-MnAl s alloys versus volume fraction of MnAIs.
¢.o c
/
30
-d
2oo
~: FIG.
g.
20 /
g
>-
I
0.2
400
3oo
(3.
o
M u r r a y and C h a l m e r s I
Cornhill
/
Predicted Moduli 50
9
75
From Dudzinski, Mort,
Volume F r a c t i o n
FIG. i
/ >-
Io
• Lyle Alcoa, P/M Extrusion o
I o
0 2
L
I
I
0.4
0.6
0.8
Volume Fraction AIzO 3
1,0
~;
14
_u) E
02
I
CL
/
0 ~;
0
i
I
/.~'/
0
MB
I
19
~.;/I
M~ : ZMB
----MA
20
' -,~
3
Young's modulus of AI203 -AI composite compared to Voight and Reuss averaging for AI203- AI mixtures.
5