Precipitation strengthening of binary AlLi alloys by δ′ precipitates

Materials Science and Engineering, A104 (1988) 149-156

149

Precipitation Strengthening of Binary AI-Li Alloys by

Precipitates

J. C. HUANG* and A. J. ARDELL

Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024 (U.S.A.) (Received January 29, 1988; in revised form May 24, 1988)

Abstract

The published data on the strengthening of binary A I - L i alloys by d' precipitates are analyzed according to a recent version of the theory of order strengthening. The results are found to agree with the data provided that the antiphase boundary energy ~/apb on {Ili} is assigned a value in the range 0.140-0.160 J m -2. The present results are in essentially perfect agreement with a recent analysis of the contribution of 5' precipitates to the strengthening of ternary A l - L i - C u alloys, which contain T1 precipitates in varying proportions," in that study, )tapb was found to be 0.150 J m -2. The success of the current analysis indirectly supports the validity of the generalized superposition rule used to extract the 5' contribution from the overall strength of the ternary alloys. The results of this study show that the d' contribution can be characterized uniquely by the order-strengthening mechanism. It is also shown that with this value of V,pb the theoretically predicted dislocation pair spacings and average particle sizes at peak strength are in very good agreement with most of the available experimental results. 1. Introduction

A1-Li-base alloys have recently attracted considerable attention in the aircraft industry because their light weight is accompanied by improved modulus and strength. The strength levels achieved are intimately related to the various precipitates which develop during aging. Among these, the spherical coherent ordered 5' phase (A13Li; L I : crystal structure) is ubiquitous and therefore regarded as one of the most important of all the microstructural constituents. Because the d' particles have the ordered C u 3 A u *Present address: Materials Science and Technology Division, Mail Stop K765, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. 0921-5093/88/$3.50

crystal structure and paired dislocations are found in aged and deformed A1-Li alloys, the strengthening imparted by the 6' particles has been attributed to the order-hardening mechanism [1-6]. This mechanism has also been quite successful in explaining the strengthening of nickel-base alloys containing y' (Ni3Al-type) precipitates [7], which are isomorphous with the 6' phase. We have recently investigated microstructural evolution and strengthening mechanisms in two ternary A1-Li-Cu alloys and analyzed the strengthening contributed specifically by the d' precipitates [5, 6]. It was concluded that the order-hardening mechanism accounts entirely for the 6' contribution [6]. However, because other precipitates are also present in the aged alloys, especially T 1 particles, it was necessary to use a certain type of superposition rule to extract the individual strengthening contribution of the 6' particles. The addition rule used assumed that the superposition of the 6' and T 1 contributions was neither linear nor Pythagorean but instead involved an exponent q with a value of 1.4. Quite clearly, there are uncertainties inevitably associated with such an analysis, but these disappear when 6' particles are the sole precipitates present in the microstructure, as is the case for aged binary AI-Li alloys. In this paper, we analyze published data on the precipitation hardening of binary AI-Li alloys, using the identical version of the theory of order hardening employed for the analysis of 6' strengthening of the ternaries. Logically, if the superposition rule used in our previous papers is correct, the value of the antiphase boundary energy ~/apb on { 111} required for quantitative agreement between theory and experiment should be nearly the same for the binary and ternary A1-Li-Cu alloys. This follows from the fact that Yapb is independent of the copper concentration of the alloys because the 5' precipitates do not dissolve copper [8]. © Elsevier Sequoia/Printed in The Netherlands

150

2. Theoreticalbackground

or

For underaged alloys with a small volume fraction f~, of 6' precipitates, the theoretical critical resolved shear stress (CRSS) re0 predicted by the theory of order strengthening is given either by [7] re0 = ~ 2b

( B 1/2 - f ~ , )

1

(4b)

K~apb B 1/2

_Gb2[ 32F / 1/2 Ky~2 13~ap~r)f6t]

(la)

or z-c0= ~FapbB1/2 2b

Gb 2 O = - -

(lb)

where K = 1 for screw dislocations and 1 - v for edges, and the distinction between eqns. (4a) and (4b) is as before. The other parameter is the average radius (r)max of the precipitates at peak strength which is determined from the following formula for initially pure screw dislocations [10]:

where

4File' {lq 0"7 }

B = 33r2~apbj°"r/¢°(~

(2)

32F b is the Burgers vector of the dislocations of line tension F and (r) is the average radius of the precipitates. Equation (la) applies when the trailing dislocation in the pair is straight, whereas eqn. (1 b) obtains when the trailing dislocation is pulled through the already sheared 6' precipitates immediately on encountering them. It is possible to modify the expressions above to allow for larger values of f~,, using the formalism of Ardell e t al. [9], but we shall see that the scatter in the extant data does not justify this refinement. When the precipitates are relatively large and strong and the alloy is near the peak aged condition, the appropriate expressions for rc0,ma~are

[7] ~cO,ma~= 0.81 ~ -

or

lj22b

(3b)

The distinction between eqns. (3) is identical with that between eqns. ( 1 ). The theory also predicts two other important experimentally measurable parameters. One of them is the average spacing D between the dislocation pairs given by either [7] Gb 2 D = - -

KYl~tapb

1 B1/z

+f6'

(4a)

(r)max = yg~/apb(~-~v)l/2

ln(~F0i

(5)

where A and r 0 are outer and inner cut-off distances in the calculation of the line energy or line tension of the dislocation, and /3c' is equal to cos(~0/2), where 73 is the critical cusp angle subtended by the arms of the dislocation in the model of the Orowan stress of Bacon e t al. [11]; tic' is equal to about 0.85. The preceding formulae require carefully considered estimates of the line tension, expressions for which differ depending on how much the dislocation is forced to bow out between the precipitates. The generalized formula for the line tension of a straight dislocation is given by [12, 13] F-

4~-

1-v

(6a)

where ~ is the angle between b and the dislocation line. Equation (6a) is a good approximation when the precipitates are relatively weak, i.e. for underaged alloys, in which case the best estimate of A is L F, the so-called Friedel spacing [13]. When the precipitates are strong, however, a better approximation for F is [14] r - 4~(1

-

1,')1/2 In

(6b)

with A given by either 2(rs) or {(2(rs))-l+ Ls-1}-1, where (r~) is the average planar radius of the precipitates and L is the square lattice spacing. The distinction getween the two options for A depends on the volume fraction of the precipitates; iff~, is not too large (smaller than about 0.1, say), the simpler choice L = 2(r~) is adequate.

151

3. Analysis of data The character of the paired dislocations in binary A1-Li single crystals has been found to be predominantly of the screw type [4, 15]. Therefore the dislocation pairs in the following analyses are taken to be screw in character. From eqn. (6a), the line tension for screw dislocations, with v = 0.339 [15, 16], can be expressed as F = 2.026 Gb2 In (7~./ 4:r

(7)

where r 0 = 2b and A = L F. All the subsequent calculations make use of the parameters G = 30 GN m -2 [17] and b = 0.2864 nm. The experimentally measured contribution of the d' precipitates is denoted by Ara, (or AZ'a, max ) = r t -- Trn, where zt is the CRSS of the precipitation-strengthened alloy and rm is the CRSS of the precipitate-free matrix. For polycrystalline samples, both of these quantities are calculated from oy/M, where % is the yield stress and M is the Taylor factor. 3.1. Underaged condition Furukawa et al. [1] used polycrystalline 3.1 wt.% Li alloys and conducted quantitative measurements of f a,, (r) and the yield strength of alloys aged at 200°C. They reported a Taylor factor of 2.5. The volume fraction varied from 0.03 to 0.14 during aging, a pronounced variation. They did not consider the variation in the strength r m of the matrix solid solution with aging time, but we believe it is necessary to take this into account. The data needed for the analysis of

TABLE 1

order strengthening are summarized in Table 1. rm was evaluated using the values of f6, and the data on the variation in oy of the solid solution vs. lithium content reported by Furukawa et al. [1]. Sainfort and Guyot [2] tested polycrystalline 2.5 wt.% Li alloys at 200°C. They did not measure fa, but assumed that it is constant throughout aging, its value being 0.15, a drastically different result from that obtained by Furukawa et al. [1]. Because of the constancy of fa,, rm was taken as constant. The value of M reported by Sainfort and Guyot was 2, which seems rather low. Their data are also included in Table 1. Miura et al. [4] performed experiments on monocrystalline samples containing approximately 2 wt.% Li aged at 200 °C. As was the case for the polycrystalline samples of Furukawa et al. [1], f6, varied considerably, from 0.01 to 0.07. The variation in rm with solute content was therefore taken into account. Cassada et al. [18] investigated the strengthening of a 2.22 wt.% Li alloy aged at 200 °C. They claimed that f6, increased significantly during the early stages of aging, ultimately reaching a saturation value. M was not reported in their paper, and we have assumed it to be 3.1 for the current analysis. Figure 1 illustrates the results of these four investigations, plotted as Ara, vs. ((r)fa,/F) 1/2 in keeping with the predictions of eqns. (1) and (2). On analysing the sets of data individually according to eqns. (la) or (lb) and (2), the slopes of the least-squares fitted curves produce the following values for 7apb: Furukawa et al. [1], 7,pb = 0.154 J

Data on the underaged AI-Li alloys

(r) (nm)

fa,

rt (MPa)

rm (MPa)

A "ra, (MPa)

F (nN)

(F) (nN)

Reference

2.10 6.25 9.00 11.56 14.44 5.72 7.81 9.59 2.00 3.50 5.00 8.00 15.00 6.00 8.50 13.00

0.030 0.067 0.090 0.120 0.140 0.150 0.150 0.150 0.010 0.018 0.025 0.033 0.070 0.017 0.058 0.059

20.0 56.0 72.0 88.0 98.0 91.0 100.0 104.5 10.0 17.0 30.0 42.5 58.0 39.4 111.3 132.9

16.0 15.0 14.0 13.5 13.0 22.0 22.0 22.0 7.0 6.5 6.0 5.5 5.0 21.0 20.0 20.0

4.0 41.0 58.0 74.5 85.0 69.0 78.0 82.5 3.0 10.5 24.0 37.0 53.0 18.4 91.3 112.9

1.74 1.80 1.81 1.80 1.82 1.62 1.68 1.72 1.95 1.95 1.95 1.98 1.97 2.06 1.89 1.97

1.73 1.71 1.63 1.50 1.36 1.56 1.55 1.52 1.94 1.92 1.89 1.82 1.47 1.97 1.72 1.56

[ 1] [ 1] [1] [ 1] [ 1] [2] [2] [2] [4] [4] [4] [4] [4] [ 18] [18] [ 18]

.

1161

l

o’/’ 0.0

0.2

- 0.4 ’ -

’

0.6

-

’

0.6

(r)fss I/* r

t-1

- ’

1.0

-

’

1.2

- ’

1.4

’

0.6

-

’

0.6

(m/N)“*

Fig. 1. Data on the underaged precipitation-hardened AI-Li alloys.

-

’

1.0

-

’

1.2

-

’

1.4

(m/N)“* binary

rne2; Sainfort and Guyot [2], yap,,= 0.130 J rnm2; Miura et al. [4], yapb= 0.177 J rnm2; Cassada et al. [18], yap,,= 0.200 J rnm2. Excluding the data of Cassada et al. for the sole reason that they disagree considerably with the others, the overall value of yapb of the binary alloys is calculated to be 0.151 J rnm2. The analysis above does not allow for the bowing-out of the initially straight pure screw dislocations. Since some edge character is acquired when this occurs, the average line tension of the dislocations that shear the precipitates is lowered. The simplest way to account for this effect is to replace r by the average line tension (r) which is generally slightly smaller than r, the difference between the two quantities increasing as the dislocation bows out more. (I’) can be evaluated from the general expression [7]

(8) where /3, is a measure of the strength of the obstacle, given by n(r)y,,,/4r. Equation (8) reduces to the formula (r)=r(i-0.253Bc2)

o14* - 0.4’ 0.0 0.2

(9)

on substitution of the appropriate constants. The calculation of(r) is iterative, requiring initial trial values of yapb and r, but the result quickly converges. Figure 2 shows the variation in Az~,,,,~~as a function of the new quantity, (f6,(r)/(r))1/2. The initial trial values of yapb and r were 0.16 J m-*

Fig. 2. The data in Fig. 1 replotted tension (IJ instead of r.

using the average line

and 1.8 x 10m9 N respectively. The individual values of yapb extracted from the slopes are slightly smaller than before (0.148 J m-*, 0.125 J rnm2, 0.173 J mm2 and 0.190 J m-* respectively). The value of yapbfor the binary alloys (excluding the data of Cassada et al. [ 181 for the same reason as before) is now 0.143 J rnm2, compared with 0.151 J mV2 obtained when the effect of bowing of the dislocations is excluded. 3.2. Peak-aged condition Since the CRSS at peak strength is theoretically neither a function of the particle size nor a function of the dislocation line tension, the comthe experimental data of parison between and the theoretical predictions of eqns. are more straightforward. The only experiP3;b’max mental parameter needed is fs,. Noble et al. [3] reported data on four alloys of different lithium contents (hence different fs,), all aged at 170 “C for 24 h, which they regarded as the strength in the peak-aged condition. They did not perform a series of tensile tests to find the exact peak strengths but assumed that the alloys were aged to the peak strength condition because the particle sizes in their alloys exceeded lY/Yapb, which is the minimum size for the peak-aged condition discussed by Brown and Ham [ 131. The exact peak-aged condition is near 48 h at 170 “C, according to the aging curves determined by hardness testing. Noble et al. noted that the aging curves were quite flat near peak aging and considered that the peak strength obtained by tensile tests should not be very sensitive to the aging

153

time. In contrast, we have found in our previous studies [5, 19] that ay is much more strongly dependent on the aging time than are the hardness curves, which approach peak hardness well before the maximum value of ay is attained. Other available experimental results on peak strength are those published by Tamura et al. [15] and Miura et al. [4] on single crystals, and by Sainfort and Guyot [2], Furukawa et al. [1] and Cassada et al. [18] on polycrystals. It should be noted that the value of f6, (0.22) estimated by Tamura et al. for their 2.6 wt.% Li alloy is obviously an overestimate, as noted by Furukawa et al. [1 ]. We used a more reasonable value based on the 6' solvus at 200°C (f6,=0.12)[20]. Additionally, the value of f6, reported by Miura et al. for their 2 wt.% Li alloy (re, = 0.10) is probably also slightly higher. These effects are incorporated into the data presented in Table 2. From Fig. 3, it is evident that all the data are in good agreement except for the data of Cassada et al. [18]; their alloys appear to be much stronger than the others, as was the case for the underaged material. The value of ~apb extracted from the slope of the straight line (excluding those of Cassada et al.) is 0.144 J m-2; when their data are included, Yapbis 0.154 J m -2.

perimentally measured spacings was made using eqn. (4b) with Yapb = 0.150 J m -2. This value was chosen as representative of the range over which ~apb varies (0.140-0.160 J m -2) according to the analyses in Sections 3.1 and 3.2. The number of reported measurements of this quantity is quite limited, and the results are summarized in Table 3. With the exception of the data of De Hosson et al. [23] the agreement between theory and experiment is exceptionally good. The use of eqn. (4b) instead of eqn. (4a) cannot remove the disparity because the theoretically predicted value of D is already far too small. The most likely reason for the discrepancy is that De Hosson et al. appear to have measured the spacing of only one dislocation pair; hence the statistical sampling was poor. 3.4. The average particle size at peak strength In comparing the theoretical predictions with experimental results, it is important to define the manner of measuring (r)max , since the transition from shearing to looping of the precipitates is not always apparent in the age-hardening curves [10], and it is impossible to measure (r)max microstructurally. In general (r)max is not measurable with the same degree of reliability as the other quantities.

120 3.3. Dislocation pair spacings A comparison between the predicted dislocation pair spacings in underaged alloys with ex-

~ 100 Q" >¢

TABLE 2

f0,

Data on the peak-aged AI-Li alloys

Reference

S ~-

"

~

80 60

40

J

~

7

+ x

[2] =' [3] . [4]

t't

Tm

A ~'6l, max

(MPa)

(MPa)

(MPa)

0.180 0.150 0.004 0.043 0.128 0.220 0.100 0.120 0.070

112 112 45 75 100 112 70 100 134

12 22 32 28 25 22 4 10 20

100 90 13 47 75 90 66 90 114

TABLE 3

Comparison of experimentally measured dislocation pair spacings D with the predictions of eqn. (4b)

/

<~ [1] [2] [3] [3] [3] [3] [4] [15] [18]

0

0.0

|

0.1

|

0.2

[1]

|

0.3

|

0.4

|

0.5

|

0.6

Fig. 3. Data on the peak-aged precipitation-hardened binary Al-Li alloys.

Amount of Li (wt.%)

(r) (nm)

fa,

F (nN)

DCxp (rim)

Dtheor (nm)

Reference

3.0 2.5 2.2

3.20 9.59 7.50

0.04 0.15 0.03

1.74 1.72 1.98

50.0 17.4 96.0

51.7 15.3 41.6

[21] [2, 22] [23]

154 TABLE 4 Comparison between the experimentally measured values (r),.. x of the particle size associated with the transition from shearing to looping with those theoretically predicted by eqn. (5)

Amount of Li

F

(r)...... p

(nN)

(nm)

(r}..... heor (nm)

Reference

(wt.%) 3.10 2.50 1.50 1.90 2.80 3.90 2.00 2.22

1.10 0.88 1.01 0.94 0.89 0.93 1.12 0.99

26.0 10.0 13.0 11.0 10.0 13.0 25.0 14.0

12.1 9.7 8.8 9.5 9.7 10.6 12.0 10.4

[1] [2] [3] ]3] [3] [3] [4] [18]

The values of F in this table were calculated using eqn. (6b).

The identification of @)maxwith the particle size at peak strength or hardness Was used by us previously [10] to calculate 7apb~'0.125 J m -2 from the experimental data according to eqn. (5). Here we use 7apb= 0.150 J m-2 to calculate @)max from eqn. (5), with the results presented in Table 4. From Table 4, it can be seen that the theoretically calculated values of @)max slightly underestimate those measured experimentally, except for the data of Miura et aL [4] and Furukawa et al. [1 ], for which the disagreement is quite large. 4. Discussion

From the various analyses of the data on the binary A1-Li alloys presented in the foregoing, the most reasonable value of 7apb lies between 0.140 and 0.160 J m -2. So long as 7apb is somewhere in this range, all aspects of precipitation strengthening in this alloy system can be acceptably accounted for by the order-strengthening mechanism, with self-consistent agreement between theory and experiment throughout. If the effect of the finite volume fraction of 6' were taken into account using the procedure of Ardell et al. [9] 7apb would be slightly larger, but the difference is less than 5%. The intercept in Fig. 1 has a small negative value, but those in Figs. 2 and 3 are essentially equal to zero within experimental error. In no case is it possible to argue that eqns. (la) and (3a) are preferable to eqns. (lb) and (3b). This result provides additional evidence that the trailing dislocations are most probably pulled through the already sheared ordered precipitates immediately on shearing them. The greatest source of uncertainty among all the data analyzed are the values of M reported by the various workers, which we have little choice but to accept. For all the underaged alloys, M is

less than 3 (M = 2.5 in the work of Furukawa et al. [1] and M = 2 in the work of Sainfort and Guyot [2]), and in no instance was the method used to evaluate M reported. This would not be cause for concern, were it not for the fact that accurate values of M are crucial to the analysis. We believe it unlikely that M could be lower than reported, but would not be surprised if it were larger. This would have the effect of lowering the values of ~apb derived from the data in question by as much as 20%-25%. However, in the absence of confirmed information on this point, further speculation is unwarranted. It is also worth pointing out that the parameters derived by us are frequently in disagreement with those derived by the researchers themselves. For example, Sainfort and Guyot [2, 22], Tamura et al. [21] and De Hosson et al. [23] derived Yapb values equalling 0.150 J m -E, 0.195 J m -2 and 0.140 J m -2 respectively from measurements of D. However, they used equations different from eqn. (4b) in their calculations. Sainfort and Guyot [2, 22] derived )'apb=0.150 J m -2 from their measurement of @)max but used a simpler and less accurate equation than eqn. (5) for their calculation. In a different vein, Furukawa et aL [1] used the value 0.053 J m -2 in an analysis of their data on strengthening of underaged alloys, but it is clear [5, 24] that this number is incorrect. It is important to note that the success of the analysis implies that eqns. (lb), (3b) and (4b) rather than eqns. (la), (3a) and (4a) correctly predict the precipitation-hardening behavior of aged AI-Li alloys, in the same way that their counterparts for edge dislocations correctly predict the precipitation-hardening behavior of Ni-A1 alloys strengthened by y' precipitates [7]. It is simply impossible to attain the consistent level of agreement observed through the use of the latter set of

155

equations. This further implies that there is a universality concerning the way that the trailing dislocation in the pair attacks the ordered precipitates. Whether or not it is initially edge or screw in character, it must be pulled nearly completely through the already sheared particles immediately on encountering them rather than remaining straight. Finally, we address the issue of whether the magnitude of 7apb is reasonable in the light of recent calculations suggesting that it should be significantly lower, in the neighborhood of either 0.072 J m -2 [24] or 0.057 [25] J m -2. We have already commented [10] on the theoretical and experimental foundation for the calculations and analyses of Glazer and Morris [25], who used a variation of eqn. (5) to derive ~apb : 0.057 J m - 2 and shall not repeat them here. In the context of this paper, we point out that the only way such low values of 7apb could ever be consistent with the data on A r~, is to assume that edge dislocations control the flow stress, in contradiction with microstructural observations [4, 15]. Even then it is easy to show, using eqn. (6a) in conjunction with eqns. (lb) and (2) and the appropriate physical constants, that 7apb would have to lie between 0.090 and 0.100 J m -2 t o achieve a good fit with the data; this still considerably exceeds the value proposed by Glazer and Morris [25]. Furthermore, the predictions of eqns. (3) and (4) would then disagree hopelessly with the experimental results on A r6,.max and D, irrespective of whether the theory is formulated for edge dislocations or screw dislocations with 7 a p b = 0 . 0 5 7 o r 0.072 J m 2. 5. Conclusions It is shown that the precipitation strengthening imparted by 6' particles in binary AI-Li alloys is excellently predicted by the theory of order strengthening, using the recent version suggested by Ardell [7]. This analysis yields results consistent with those of Huang and Ardell [6] on the 6' contribution to strengthening of ternary A1Li-Cu alloys. This implies indirectly that the generalized superposition rule used by Huang and Ardell [6] to calculate the separate contributions of the 6' and T~ precipitates is essentially correct. The foregoing analysis shows that Vapb on {111 } of the 6' phase lies between 0.140 and 0.160 J m 2, which is in excellent agreement with

the average value of 0.151+0.008 J m -2 obtained from the analysis of the 6'-strengthening contribution in ternary A1-Li-Cu alloys. Using ~'apb=0.150 J m -2 the theory also successfully predicts values of the dislocation pair spacings and average particle sizes at peak strength which are in very good agreement with most of the experimental data available. The success of the present approach implies that the theory of order strengthening can be used reliably to estimate the contribution of ordered precipitates to the strength of precipitationhardened alloys and that the trailing dislocation in the pair is pulled through the precipitates which are already sheared by the leading dislocation, immediately on encountering them.

Acknowledgment The authors are grateful to the Alcoa Corporation for their continuing financial assistance in support of this research.

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23 24 25.

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