Strain aging and strain hardening in NiC alloys

Acla metal/. Vol. 33. No. 4. Printed in Great Britain

pp.623-638.

STRAIN

ooal-6160’85 53.00+0.00 Pcrgamon Press Ltd

1985

AGING AND STRAIN HARDENING IN Ni-C ALLOYS7

U. F. KOCKSS, R. E. COOK and R. A. MULFORDg Materials Scimce and Technology Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (Receiued 2 October 1984) Abstract-Two nickel+rbon alloys and pure nickel were investigated by compression in the temperature range where strain aging pbmomena occur. The effect of carbon on the dynamic-recovery part of strain hardening is severe, especially at higher temperatures. Static strain aging increases strongly with strain. This dependence persists even when the time and concentration dependmce of the aging stress increment have saturated; it can therefore not be due to deformation-generated vacancies. The strain dependmce can be well expressed by a linear relation with flow stress, up to the beginning of dynamic recovery, where it saturates. Tbis behavior, as well as dynamic strain-aging results, are very similar to observations in substitutional alloys. It is concluded that vacancies are rarely if ever responsible for tbe strain dependence of aging phenomena, and that this is instead due to a strong interaction between solute bardming and strain hardening. R&m&Nous avons dCformC m compression deux alliages de nickel et de carbone et du nickel pur, dans le donmine de temfiratures od se produit le phenombne Port&n-LeCbatelier. L’influence du carbonc sur Ia partie de i%crouissage B restauration dynamiquc est trts grandc, surtout P baute temp&ature. Le vieilbssement statique croit fortemmt avee la deformation. Cette influmce per&e mime lorsque ies variations de I’incr&nmt de la contrainte de vieillissemmt m fonction du temps et de la concentration arrivmt i saturation; elle ne peut done pas Btre due P da lacunes produites par la diformation. On peut exprimer cette variation en fonction de la difonnation par unc relation h&ire avee la contrainte d’ecoulemmt, jusqu’au debut de la restauration dynamiquc ou clle se sature. Cc comportemmt est, comme les risultats du vieillissemmt dynamique, semblable aux observations dans les a&ages substitutionnels. En conclusion, les lacunes sent raremmt responsables de la variation du vicillisscmmt m fonction de la &formation; cette variation at due a une forte interaction entre le durcissemmt par les solutes et le durcissemmt par la diformation. Zuammmfmauag-Zwei Nickel-Koblmstoff-Legierungm und rcines Nickel wurdm im Druckversuch in dem Temperaturbereicb verformt, in dem Erschcinungen der Reckalterung auhretm. Der Einflu5 da Kohlmstoffes auf den B&rag der dynamischm Erbolung bei der Verfestigung ist basonders bei hbherm Temperaturm betrfchtlicb. Statische Reckalterung nimmt mit da Debnung stark N. Dime Abhiingigkeit ist such vorhandm, wmn die Zeit- und Konxmtrationsabhiingigkeit da durcb die Alterung auftretmden Spannungsanstieges in der Siittigung ist. Sic kann daher nicht von den wiibrmd der Verfortnung produxiertm Leerstellm hcrriihrm. Die Dehnungsabhiingigkeit kann mit einem linearm Zusammmhang mit der PIieEtspannung crkllrt werden, da bis xum Rinsetxen der dynamischen Erholung gilt und dart in die S&tigung gebt. Dieses verhalten, und aucb die Ergebnisse xur dynamischm Reckalterung, iihneln dmjmigm sebr, die an substitution&n Legierungm erhaltm wordm aind. Es wird gefolgert, da5 die Lmmtellm kaum, wmn tiberbaupt, fur die Debnungsabhiingigkeit der Alterungseracbeinungen veratwortbch sind, und da5 diese vielmebr wcgm eines starken Zusamm mbanges zwiscbmMiscbkristallbPr-

tung turd Verfcstigungauftretm. 1. INTRODUCTlON Nickel-carbon alloys have attracted a fair amount of attention [I-IO], for a number of reasons. From a practical point of view, carbon is a common impurity even in “purr? nickel such as INC0270, and certainly in the many commercial nickel alloys. Since its effects are profound, as we shall cot&m in this investigation, even small amounts of carbon are of interest. Work supported by tbe U.S. Departmmt of Energy. ?Now at the Center for Materials Scimce, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. @. E. Corporate Research and Developmmt Center, now with G. E. Knolls Atomic Power Laboratory, Schcnectady, NY 12301, U.S.A.

From a scientific point of view, the nickel-carbon system is one of the few interstitial alloys in the face-centered cubic (f.c.c.) lattice structure. The interstitials in this structure have cubic symmetry rather than tetragonal symmetry as in the more common body-centered cubic (b.c.c.) interstitial alloys. As one consequence of this symmetry, the individual solute atoms in Ni-C have only a weak interaction with dislocations, much like substantial solutes [l 1). From this reasoning alone, it is surprising that relatively small amounts of carbon have the large effects that are observed. One clue to the reason for the strong influence of carbon on the mechanical behavior of nickel comes from the observation of jerky tlow [l-S, 7.91, which indicates some degree of mobility of the solute atoms,

623

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KOCKS et al.:

STRAIN AGING AND HARDENING IN Ni-C ALLOYS

at least near the dislocation cores. As a consequence of this mobility, an attractive interaction between solute atoms and dislocations may lead to an increased concentration of solutes in the vicinity of dislocations at the time a rate-controlling step occurs, thus explaining the strong solution hardening. The solute mobility itself is, however, a problem inasmuch as bulk diffusion is not expected in the temperature range in which jerky flow is observed; namely, near room temperature. This discrepancy is a common finding in strain-aging alloys and has been explained, since Cottrell [12], on the basis of deformation-generated vacancies. Vacancies, however, are not necessary for the diffusion of intersririul solutes; in fact, their influence is often assumed absent u priori in discussions of aging mechanisms [IO, 131. In at least one case, tests were performed to demonstrate this lack of an effect [5]. Even if one allowed that vacancies might aid interstitial diffusion to some extent (or else impede it through trapping of the solutes) one would certainly expect profound differences in the behavior of substitutional and interstitial alloys in their response to straining. Such differences have indeed been observed-but only when the interstitial alloy was body-centered cubic (and thus also had a much stronger solute/dislocation binding). In order to separate effects of the interaction strength from those of vacancies, experiments on an interstitialalloy in the f.c.c. lattice structure were deemed essential [14]. While most of the previous investigations of strain aging in Ni-C alloys were primarily concerned with its temperature and concentration dependence, we will here address the strain dependence of dynamic strain aging, which has been a point of particular controversy. This dependence has classically been rationalized on the basis of, again, deformationgenerated vacancies and of a strain-dependent mobile dislocation density [15-191. In this view, then, any strain dependence of the aging characteristics in Ni-C alloys could only be attributed to a change in the density of mobile dislocations with strain [IO, 131. There is a way to eliminate any influence of the mobile dislocation density [20,14]: namely, to investigate static strain aging, since here the aging time is known and need not be derived from the strain rate (assuming a given mobile density). Thus, sroric aging experiments in interstitial alloys should show no strain dependence at all according to these theories. This is approximately true in b.c.c. alloys [20-221; is it also true in f.c.c. interstitial alloys? Mulford and Kocks [23] and Wycliffe et al. [20] have proposed that the strain dependence of both static and dynamic strain aging is linked to the strain-hardening behavior rather than vacancy gener-

tThe etching was done sequentially in two solutions: (a) 100 ml of 70 wt% HNO, with 20 g CuSO,. SH,O; and (b) 7pdHN0,

and glacial acetic acid in equal parts,

ation. This view was based on observations in a variety of nickel and aluminum alloys and gains some support from the well-known reverse effect, namely that strain hardening is (sometimes profoundly) influenced by the presence of strain aging [2,4,23-271. If this model is correct, both static and dynamic strain aging in Ni-C should be depend on strain much as other alloy systems do (except b.c.c. interstitial alloys in which there is a strong interaction of individual solute atoms with dislocations) [14]. Thus, experiments on Ni-C could provide a distinguishing test between the two views. In this investigation, two Ni-C alloys were tested between room temperature and 280°C. The strainhardening behavior, the rate sensitivity of both flow stress and strain hardening, and dynamic as well as static strain aging behavior were studied. After outlining the experimental procedure (section 2) we report and discuss our findings on these various effects (section 3). In section 4, we summarize, and partly re-evaluate, other experimental and theoretical work briefly. Our conclusions are summarized in section 5: we emphasize the remarkable similarity between many observations ‘and propose a general phenomenology for static and dynamic strain aging. The presence or lack of mechanistic understanding in the various regimes is assessed.

AND EVALUATION PROCEDURE

2. EXPERIMENTAL

Two alloys of pure nickel (INC0270) with carbon were prepared by vacuum casting and hot working, in the form of slabs of approximate dimensions 1 x 5 x 10 cm. Compression specimens were machined from these slabs and then spark-planed to final dimensions of approx. 4 x 4 x 6 mm (the long edge being in the compression direction and perpendicular to the large face of the original slab). Specimens of similar size were also prepared from spectrographically pure Ni (99.995%) by arc melting and rolling into 4 x 4 x 150 mm bars and spark-cutting into 6 mm long pieces. The aspect ratio of all samples was kept near 1.4, except in a few early specimens (C13, C14, Dl5-D24), in which it was about 1.8. All specimens were etchedt until about 0.05mm had been removed from each face. To ensure carbon solution [28], the alloys were annealed for 1 hr at about 1000°C in a purified He atmosphere and then quenched in water. The pure Ni was annealed for 1 hr at 700°C in vacua to obtain a similar grain size, about 0.2 f 0.1 mm. Compression tests were performed between platens of Carpenter 610 tool steel, lubricated with molybdenum disulfide. The specimen shape after deformation was satisfactorily square proving the efficacy of the lubrication. Tests were done at room temperature or in a three-zone split furnace. At 180°C a flow of about SOOcc/min of nitrogen was main-

KOCKS ef al.:

STRAIN AGING AND HARDENING

(a)

(b) .2%

I 10 MPa

/ ho,

V 1

(cl I

I

1

.2%

Fig. 1. Some load vs displacement recorder traces, showing types of scrrations observed, and evaluation procedure. In (a) and (b), the strain rate was increased by a factor 10 (specimen D21, 295 K); in (c), the specimen is aged at the test temperature in a partially unloaded state and restrained at the previous strain rate @37, 453 K, aging time 110 s). The initial aspect ratio of specimen D21 was 1.8; this gives rise to the double-jerks in (a) through heterogeneous dcformation. tamed; at 280°C about double as much.1 In some experiments, the temperature varied by 24” with position in the specimen or with time. In later tests (specimen numbers above 36), heating was accomplished by two resistive cartridges (1OOW Dalton Watt-Flex) in each of the compression platens. The cross-head displacement was measured by an LVDT and used as input for an X/Y plot, together with the load-cell output, to a resolution (1 mm on the graph) of about lo-’ in strain and 0.5MPa in stress. The effective machine compliance was determined from the reloading curve after completion of each test; it was approx. 2 x lo-I’mm/N. All data were digitized from the X/Y-recorder traces. Differentiations were performed on the midpoint of a cubic polynomial fitted to five points (adjacent 5point sequences overlapping). Spacimen-tospecimen reproducibility of the restthing stress strain curves was excellent. $In a few rpecimms, namely DlS, D16, D32 and D36, a Nz-lOO/$I,mixture was used for part of the test. The hydrom is not expected [9] and was not found, to have any effect. #The Partial unloading procedure prevents any measurable strain from occurring during the aging-which may be important [29]. Most previous tests have been done during stress relaxation [7,10,20].

IN Ni-C ALLOYS

625

Most tests were run under predominantly constant conditions of strain rate and temperature, with multiple small excursions: typically, for about 1% strain at intervals of about 3%. One type of excursion was to a higher strain rate (usually times 10, sometimes times 4 or 5). Figures l(a) and l(b) illustrate the evaluation procedure on an actual recorder trace, for the nontrivial case that jerky flow occurs both before and after the change. Only the flow stress change upon leaving the standard conditions was used in the evaluations. The other type of excursion consisted of unloading to about 70% of the current flow stress, aging for various times (usually at the test temperature), and reloading (usually at the same strain rate&Q Figure l(c) shows our standard evaluation procedure for this case, again when flow is jerky (or wavy). The aging time was defined as the total time elapsed from the beginning of unloading to the yield peak. Figure I(a)-(c) also demonstrate the various types of “‘jerks” or “serrations” and “waviness” on the stress strain curves. The carbon content was analyzed in a LECO gas detenninator both before and, on a number ofspecimens, after the test. The “C-alloy” contained 0.28 at.% carbon in the original slab, and 0.15 k 0.5’; after the tests; the “D-alloy” contained 0.9 f 0.1 at.?, C both before and after the tests; and the pure Ni contained ~0.05 at.% C. In a spectrographic analysis after the tests, the only measurable impurities were about 0.004 at.% Al and Mg each. The oxygen plus nitrogen content before the tests was less than 0.02 at.%. All results in this paper will be labelled with the specimen number: a prefix of N, C and D will respectively indicate pure Ni and the “C” and “D” alloys. 3. RESULTS AND DISCUSSION

3.1. Strain hardening and dynamic recovery Figure 2 shows the true stress vs true strain curves for the pure Ni and the two Ni-C alloys, taken at a temperature of 453 K at a constant cross-head displacement rate of 0.05 mmimin (a nominal strain rate of about 1.5 x lo-‘s-l). It is apparent that even an atomic concentration of less than 0.2 at.% C (“alloy C”) causes a significant increase in strain hardening at the larger strains [2]; the higher “ahoy D” (-0.9 at.% C) causes a doubting of the flow stress in the 30-40x strain range. In the latter alloy, it is worth noting that the high stress level at large strains seems to come about through the existence of a long, almost straight, initial portion of the stress strain curvoof essentially the same slope as that at the beginning of both the pure Ni and the C-ahoy. This observation suggests that the influence of solute concentration on strain hardening is primarily due to an effect on the rate of dynamic recovery [30]. Note that the influence of carbon concentration on the yield stress is small compared to that on strain hardening.

626

KOCKS er al.: STRAIN AGING AND HAVENING 1000

1

IN Ni-C ALLOYS

,

Ni-0.9

9o,, _ 453 K i=l.?xWW EOO-

010

-

4-

r/o

C

i, a 1.7 x 10-W

700 600 -

3-

2-

l* 0. 0

1 0.3

I 1 01 0.3 0.4 0.5 0.2 4 5 1 2 3 0 STRAIN r/g x IO’ Fig. 2. Typical true-stress-true-strain curves for Ni (specimen N36) and Ni-C alloys; “C”-specimenscontain Fig. 4. The strain-hardening rate # as a Function of the 0.2 f 0.1 at.%C, “I)“-specimens 0.9 + 0.1 at??C. The shear flow stress T (both normalized by the shear modulus r) for the D-alloy (DI6, D17, D24, D38) at two temmarked points refer to TEM observations. peratures, for the standard strain rate (with excursions) and a IO-times higher strain rate (continuous test).

This inte~retat~on is confirmed by an analysis of the temperature dependence of strain hardening. Figure 3 displays the behavior of all three materials at both a lower temperature (295 K) and a higher ~m~rature (555 K) than in Fig, 2; this time, it is plotted [31] in terms of a parameter, 70 that is proportional to the rate of dislocation storage with strain, dp fdy: 7 is the flow stress in terms of crys~llo~ap~c shear (obtained by di~ding u by the average Taylor factor 3.06) and is assumed proportional to Jp ;0 is the rate of shear strain hardening, dr/dy, and p is the appropriate shear modulus (79 GPa at R. T., 74 GPa at 453 K, 71 GPa at 555 K). In such a plot, the athcrmal hardening rate (an extension [30] of the concept of “stage II hardening” 10

I

1

t

o 295 K

9 I 7 v * 6 3 $4 3 2 1 0

0

1

2 3 fl#Jx 18

4

Fig. 3. The ordinate is proportional to the net dislocation storage rate: the linear increasecorrespondsto athermal,

statisticalstorage,the later downwarddeviationsto superposed dynamic recov:ry. r and c arc the average cays~0~~~~1~ resoIvcd shear flow stress and the shear modulus, respectively, 8 is the shear hardening rate. Nickel (N) and Ni-C alloys (C and D) at two temperatures.The

arrowsmark the beginningof dynamicncovery.

[32] exhibits itself as an initial straight line {not necessarily through the origin), which is temperature insensitive [31]. Such behavior is indeed observed: the initial slope in Fig. 3 is quite similar for the two widely different temperatures (though it increases with concentration). Its average magnitude corresponds to a normalized athermal hardening rate f?& z 0.003; a very typical value for many f.c.c. metals and alloys. The rate of dynamic recovery exhibits itself in plots of the kind shown in Fig. 3 as a departure from the straight athe~al-harde~ng portion. The arrows in Fig. 3 mark the approximate points for this beginning of dynamic recovery (conventionally called rll, in single crystals 1321).Note that for pure Ni, there is more dynamic recovery at the higher temperature, as expected; but in the Ni-C alloys, dynamic recovery is delayed to higher strains at 555 K than at room temperature. This inversion of the normal pattern can only be due to solute mobility. A similar reversal is observed in the strain-rate dependence of strain hardening. Figure 4 shows the B(s)-dependence of the high alloy (D) at both 453 K, for two strain rates: about lo-’ and 10W3s-‘; this time, the data are shown as shear hardening rate 6 vs shear flow stress 7, both normalized by cc.It is evident that the lower strain rate causes more hardening. At the highest temperature, this is obviously due to a &lay in the beginning of dynamic recovery, causing a plateau in the hardening rate. At 453 K, the difference between the two strain rates is more in the nature of a proportional increase in both stress and strain-hardening rate [33]. At room temperature (not shown), the rate dependence of the continuous stress strain curves is negligible: an unusual situation if this were a pure material. In summary, one may express the concentration dependence of Bow stress and strain hardening in

KOCKS ef al.:

STRAIN AGING AND HARDENING

Ni-C as follows u = uxc) + a(c)pbJp

(1)

and da Z = a(c).{kp

-R(c)}.

The friction stress o, is only mildly concentration dependent. The mild concentration dependence of the athermal hardening rate demands a mild influence of c on the dislocation “interaction strength” a [33]. The strongest effect is on the dislocation rearrangement rate through the term 6, (k is a constant). A number of transmission electron micrographs were taken for both alloys at various temperatures. The general appearance of the dislocation structure is in accord with classical patterns. In all specimens that were strained into a regime in which the strainhardening rate is still high and athermal, regardless of temperature and solute content, the dislocation arrangement is more or less random (with some alignment presumably along slip planes); an example is shown in Fig. 5(a), which comes from a D-specimen in the state marked by a solid circle on the stress strain curve in Fig. 2. On the other hand, all specimens strained into a regime of lower (and rate sensitive) strain hardening display the cell structure characteristic of dynamic recovery; this is shown in Fig. 5(b), which corresponds to a D-specimen in the state marked by an open circle in Fig. 2; a C-specimen strained to the same slope in the stress strain curve (characterized by an open square) exhibited a similar pattern. One should expect the total dislocation density to be different between the last two cases, because of the different flow stresses, but this could not be observed with sufficient accuracy. 3.2. Rate sensitivity and dynamic strain aging Solute mobility tends to raise the flow stress-even dynamically, i.e. during deformation. This general phenomenon may be identified as “dynamic strain aging” [23]. Since the rise in flow stress increases as the time allowed for the segregation of solutes to dislocations increases, this mechanism leads to a negative contribution to the strain-rate sensitivity. This negative contribution often increases with strain; in fact, it appears to be generally absent at the yield point. When the total rate sensitivity of the flow stress becomes negative, jerky flow may [34,35] and usually does [23,36] set in. The development of the strain-rate sensitivity fl E (&r/a In i ), with increasing strain can also be expressed as a dependence on the flow stress. Such “Haasen plots” [37] are shown in Fig 6(a)-(c) for Ni and the two Ni-C alloys, at three temperatures. The tThe process of (local) strain hardening itself is the cause for stopping the instability at the bottom of the “jerk” [35]. (Note that the machine stiffness is usually irrelevantunless it is especially low.)

IN Ni-C ALLOYS

621

described trend of a decreasing rate sensitivity is observed in the initial parts of the alloy curves at the two lower temperatures. After more deformation, a countertrend begins to dominate: the pure Ni curves, as well as those of all the alloys, bend up. This has been associated with an effect of dynamic recovery [30,23], and such a correlation can be observed here also: the arrows on the curves indicate the beginning of dynamic recovery, as determined from curves like those shown in Fig. 3. It appears that the bending-up in Fig. 6 occurs somewhat earlier than suggested by the arrows-but the correlation is certainly there. An interesting observation should be emphasized at the highest temperature for the D-alloy: the instantaneous rate sensitivity is never negative and becomes strongly positive at larger strains; yet the continuous stress strain curve for the higher strain rate is lower, especially at large strains. In other words, the rate sensitivity of strain hardening is negative. This is an unusually clear demonstration of the fact that kinetic effects on stress strain curves are due at least in part (and often in a dominating part) to the kinetics of dislocation structure evolution, not to the kinetics of the flow stress at constant structure. This is the reason why a comparison of flow stresses at the same strain is generally meaningless. In cases where the instantaneous rate sensitivity turns negative, jerky flow was observed: to the extent that a “critical strain” can be defined, it should be the strain at this point. We found it more meaningful to attempt an approximate quantification of “jerkiness” by measuring the jerk height Atr, [as defined in Fig. l(a)] as a function of flow stress. Figure 7 shows a comparison of Aa, with b for two D-specimens at room temperature (at two different base strain rates). The jerk height after a strain-rate increase was different from that before: sometimes larger, usually smaller; this is indicated by the endpoints of the lines connected to each Au,-point. The main observation, confirmed on other specimens (also at 453 K), is that there is a gradual increase in jerk height which correlates well with the decrease in rate sensitivity. (The jerks never start immediately at the yield point.) At large strains, however, the. jerkiness ceases while the rate sensitivity is still negative. It is, of course, true that a negative rate sensitivity does not need to give rise to jerky flow: the instability caused by it may be damped out by other mechanisms. The presence of second-phase particles has been shown to have such an impeding effect [36]; it may be that the presence of a cell structure [Fig. 5(b)] as opposed to the more open tangle structure [Fig. 5(a)] also serves to stop a developing instable glide process. t The types of serrations observed were of three kinds, which sometimes occurred in sequence during the progress of a single test. The sharpest and the mildest serrations were shown in Fig. l(a) and l(c), respectively; in an intermediate case, these alternated with each other. Note that all the experiments re-

628

KOCKS er ai.: STRAIN AGING AND HARDENING

IN Ni-C ALLOYS

Fig. 5. TEM-micrographs (100 kV) of deformed Ni-0.9 at.TOCalloys: {a) specimen D36, strained I St0at 453 K (solid circle in Fig. 2); (b) specimen D32, strained 319; at 453 K (open circle in Fig. 2).

KOCKS er

al.:

STRAIN AGING AND HARDENING

629

IN Ni-C ALLOYS

4

662 K

3

0

200

400

a00

a00

1000

Flow stf.66 (66f’O] Fig. 7. Strain-rate sensitivity fi (as Fig. 6) and jerk height 0

100406

6-

606

606

D24

c25 .

I-

1000

_

0

::u

0200400600

Au,vs flowstressu, strainedat the standardbasestrainrate (circles,D21) and IO-times faster (squares, D26). concentration at the dislocation during the aging interval I,, and this increase in concentration must, under the governing statistics [ 141,be proportional to the bulk concentration c itself. Furthermore, it should be a function of the product D . r,, where D is the bulk diffusivity (or some other, more appropriate measure of the relevant mobility). Thus, we can write with fair generality Aa,=c.K(D.r,).S

600

loo0

FLOW STRESS [MFr]

Fig. 6. Strain rate sensitivity /I vs flow stress u for Ni (N) and two Ni-C alloys (C and D), at three temperatures (P c 1.7 x lo-‘s-l). The arrows mark the beginning of dynamic recovery.

here were done in compression on relatively squat specimens. As a consequence, the periodicity of the jerk amplitude sometimes observed in tension and associated with “hopping Liiders bands” did not occur, except with a period of two jerks only [Fig. l(b)] in those specimens where the aspect ratio was larger than 1.5. Jerks on compression stress strain curves are, then, apparently an exhibition of the intrinsic behavior of materials, rather than of nonuniformities in the tension test. ported

3.3. Static strain-aging A primary consequence of solute mobility is that dislocations which have been mobile are locked by aging. The yield stress upon reloading is therefore increased; however, it drops again to the steady-state value for mobile dislocations. This happens even when the annealed material yields smoothly. In fact, the difference between “upper” and “lower” yield point increases with strain: that is why the phenomenon has been called strain aging. The stress increment Au, [defined in Fig. l(c)] should be proportional to the increase in solute

(3)

where the proportionality constant S may depend on the current structure (such as the dislocation density) and through it on the strain history. The (pre)strain E may, in addition, enter directly into D through the influence of deformation-generated vacancies; in classical treatments [I S-191,this is assumed to be the only effect of history. Wycliffe et al. [20], on the other hand, concluded from the dynamic strain-aging results of Mulford and Kocks [23] on nickel alloys that the entire strain dependence could be well expressed by a function S(a), i.e. of the current flow stress (a parameter of the current state and only indirectly of the history) and that this function appeared to be linear at small to intermediate strains. They postulated a close connection to strain hardening rather than to vacancies. Figure 8 shows our results for aging of the D-alloy at room temperature, plotted as a function of the flow stress minus the yield stress. (A constant equal to 1 MPa was substracted from the ordinate: it seemed to be a time-independent reloading yield point.) The results are from two specimens, each aged after every few percent strain, alternating between two different times. It is seen that the initial history dependence is in fact well expressed as linear in the current flow stress; but at larger strains, Au, saturates. Also, there is a clear increase of the effect with aging time. In Fig. 9, we plotted the best-fit initial slope in Fig. 8 (forced through the origin of that figure) for the four aging times against (z.)*” [IS, 381 (even though this

630

KOCKS et 01.: STRAIN AGING AND HARDENING I

Nl-0.0r/o C

11 10

IN Ni-C ALLOYS

296 K A

i p 1.7 x 10-w

A

0

A

6oor

120 L 00 QO Q

A0 o

‘0

0

o

206

0 A “00

1

b

o” 0

Ol 100

I

1

1

1

1

200

200

400

500

600

700

10

20

0 - oy[MPa]

Fig. 8. Static-aging stress-increment Au, (- 1 MPa) vs flow stress u (minus yield stress a,) at room temperature for various aging times. Specimens D23 and D25. The arrow at the top marks the beginning of dynamic recovery.

proportionality is rarely obeyed [21,22, 391): it is obviously not obeyed in the present experiments. A better approximation would be to postulate an exponential form [40]that saturates after about 10min. In summary, the present experiments do not allow any conclusion about the quantitative effect of time; they do confirm a qualitative increase. Similar conclusions can be drawn from the data for the C-alloy at room temperature (Fig. lo), except that here the scatter is even larger (presumably due to the greater uncertainty of the concentration in each specimen). The ordinate in Fig. 10 was multiplied by a factor of 4 as compared to Fig. 8, with the result that the initial slopes in both figures are very similar. The factor 4 is close enough to being the ratio of the two concentrations, as expected from equation (3) (although a factor of 5 or 6 might have fit the average of the concentrations measured ufier the test even better). All in all, then, we have at least qualitative agreement with equation (3), plus a confirmation of Scco for small to intermediate strains. The situation is qualitatively and importantly different at 453 K. Figures 11(a) and (b) display the results for both alloys, on the same scales. It is apparent that all concentration and time dependence has vanished, except for the saturation value of Au, at large flow stresses in the lower alloy. (If anything, the concentration dependence is slightly reversed.) The absence of a time effect may put in question the basic tenet that these are aging effects. Of course, the aging effect may have saturated out. For this reason, we attempted some aging experiments at even shorter times. In this case, the aging had to be performed during stress relaxation, whereas all other aging tA comparison at longertimes between aging during stress relaxation and aging under partial unloading demonstrated that Au, Was slightly higher after stress relaxation, as observed previously [29].

20

40

50

60

70

80

(Aging limo [c])~"

Fig. 9. Initial slope of Fig. 8 as a function of aging time.

experiments had been done under partial unloading.+ The few points for I, between 5 and 10 set in Fig. 1l(a) Seem to indicate that we are still dealing with a result of aging. The independence of the measured aging at 453 K from t, must mean an independence from D, according to equation (3), and the insensitivity to c would suggest the same. Yet there still is a clear dependence on strain (still well expressed by a proportionality to flow stress). We are forced to conclude that the strain dependence cannot have anything to do with the solute mobility: for example, it cannot be due to any deformation generated vacancies. Static strain aging depends on one additional variable, which is not explicitly contained in equation (3): namely, on the strain rate. This has been seen previously in Ni-C [7], and also in Au-G [41]alloys. In our experiments, it is even true in the regime where Ao, has saturated with time. In Fig. 12, we have plotted the stress increment Aa, vs the flow stress for two different base strain rates: it is more pronounced at the higher strain rate.

‘( 2 4

bA

a=

dkAdA

QQ AA

l-

.AQ

O0

QQ&%~* Qlti.

ooo

Q O” 0 0, 0

2oB

0

4hA I

I

100 200

1

I

I

I

300

400

100

600

0 -

: lo

or (MPa]

Fig. 10. Same as Fig. 8, for specimens Cl3 and C23. The ratio of the ordinates is approximately equal to that of the concentrations.

KOCKS e/ al.:

STRAIN AGING AND HARDENING

IN Ni-C ALLOYS

631

A

1

6 - Ni-0.O r/o C a-

453 K i s 1.7 x 10-w

A 04 Q bg

ric

6-

!i

s-

A?

20-600 8 ’ h ~00

8”

F

0

200

4

NI-0.8 alo C

2

453 K 0 i a 1.7 x t 0-w n i P a.7 X10-+-r

"O

400

600

600

or

I

1000

Flow Stress (MPa]

(a)

0

100200300400500aoo700Boo9001000

Fktw Strom [MPa]

Fig. 12. Correlation of stress increment due to static aging (positive) and that (negative) due IO a strain-rate increase by a factor 10 (from two base strain rates). The arrow marks the beginning of dynamic recovery. (DI 6, D29, D30, D32A. D47).

was defined, as before, as the peak stress upon reloading minus the last stress reached during prestraining. If it had been defined as the stress drop afler the peak, the two sets of symbols would be essentially

0

200

400

600

600

1000

Flow Strom [MPa] (b) Fig. 11. Static-aging stress-increment Au, vs flow stress. (a) Alloy D at 453 K (D15, D29, D32): for aging times larger than about 10s, there is little aging-time dependence. yet a strong, essentially linear, flow-stress dependcncc.(Theshorr aging was undertaken during stress relaxation.) (b) The C-alloy (C21, C3 I), plotted to the same scale, shows that the time-independent initial rise also shows little concentration dependence. The arrow marks the beginning of dynamic recovery.

Fig. 12 also shows the effect of strain rate on the strain-rate sensitivity of the flow stress, fi, under similar conditions. This is well-known and easy to understand, since here it is the strain rate that controls the aging time of the mobile dislocations. But why does this correlate so well with Au,, where the aging time is externally prescribed? We have conducted a few experiments in an effort to elucidate this effect. They were designed to ascertain whether it is an effect of the strain rate just before aging, or of the strain used to probe Au,,. One specimen was strained (in the regime of aging-time independence) alternately at two strain rates by similar, significant amounts; between any two straining periods, the specimen was unloaded to 70% of the stress and aged by abolit 2 mins. Figure 13(a) shows the results, demonstrating the strain-rate effect without any doubt, and it upppars to be one of rate hisrory: for example, the two points for which the strain rate was slow borh before and after the aging, fall in with the one for low-rate htitory. There is, however, a question of evaluation. For Fig. 13(a), Au,

’

:,

’

’

’

Nb0.8r/o C 442 K, (. - 2 mln t a I.? i 6.7 x 10-w ’ ’ ’ ’ ’

0100260200400640aoo700Mo2001ooa

(W

Fig. 13. (a) Static-aging stress-incrementAu, for a single specimen (D46) strained at 453 K at two alternating strain mtes (factor 4), vs the Bow stress at the standard strain rate. Symbols: the line to the left of the circle is low when the slrain rate &fore agingwas low, high when the strain rate before aging was high; the line to the right of the circle similarly indicates the strain rate &er aging. Aa, was mecuured as the excess of the peak in the transient after aging over the flow stress last reached before aging (at that strain rate). (b) Schematic of stress-strain behavior.

632

ICOCKS et al.: STRAIN AGING AND HAFlDENING IN Ni-C ALLOYS

. \

TEMPERATURE

“C

0

\

NOT SERRATED

tee

SERRATED

1*,.

n \

0

‘,a

.\,e

*,n

l

bee00

c

\

2 .o

4.0

2.5 I/T

(x 1O-3)

Fig. 14. Regimes of strain rate and temperature studied by Nakada and Keh [S](circles), van den Beukel (squares), and in this investigation (bars}.

reversed, and one would conclude that the drop is conslunntof proportionality; see Fig. 12). Now the controlled by the current strain rate. static-aging stress-increment becomes T&e apparent con&t is due to the rate sensitivity Aa, = F(a.,)+{KG,) - K(s&)3. (7) /I of the flow stress itself. Note that the difference between the two sets of symbols in Fig. 13(a) is just 3.4. The activation energy equal to the rate sensitivity /I, multiplied with Figure 14 shows, in a diagram of log g vs l/T, the In(ci&); since the strain-rate ratio was 4 in Fig. 13, tests that were performed by Nakada and Keh IS] on the data from Fig. 12 (lower half) must be multiplied Ni-0.77 at,%C (solid and open circles), as well as by ln4/ln 10 to verify this very good correlation An those reported here (bars) and those by van Haastert unexpectedly simple explanation results [Fig 13(b)]: and van den Beukel [lo] on Ni-O.55 at.%C (to be the ~~olure value of the peak stress is insensitive to discussed in detail later-squares). Nakada and Keh strain rate (and only a function of strain); the rate showed solid symbols for “jerky flow”, open ones for sensitivity of the stress increment Aa, is due to that of “smooth flow”. The demarkation lines between those the previous steady-state stress (and that of the stress two regions (shown as dashed lines) have classically drgp due to that of the following steady-state stress). been used to derive “activation energies for jerky This state of affairs can be expressed by the follow- flow” (anywhere on the stress-strain curve). Alternaing formulae, to be used instead of equation (3). The tively, a connection is established between the “critflow stress is decomposed into three parts (somewhat ical strain” for the beginning (and that for the end) similar to Ref. [42]) of serrations on the stress-strain curve and the strain rate; then, the slope in a diagram of the logarithm of a =tr,fud+u, (4) the critical strain against l/T is interpreted as being where the additivity of the friction stress of and the proportional to the relevant activation energy. ~sl~tion-interaction stress a, is presumed but of We have seen that the jerk height varies considno concern here, whereas nothing is assumed, in erably with strain (Fig. 7). Thus, the “critical strain” general, about the independence of the aging stress a, may be lookad upon as the strain at which the jerk from the other two. In fact, van den Beukal considers height first becomes measurable, Alternatively, one it a part of the friction stress, whereas we consider it may compare measurable jerk heights at a given a function of crd,initially linear. Then, we may write strain [8,39); then, a more general “activation energy of jerky flow” might be obtained from a comparison a,, = S(u,) *K(Dt,,, c) (3 of tests under different stein-rate~~m~raturc condifor the effect of static aging (where the kinetic term tions which give rise to the same (finite) jerk height. K becomes a constant at large aging times, and the Most appropriately, such tests should be done as structure term S becomes a constant after large “instantaneous-change” tests, i.e. short excursions prestrains); and from a long test at standard conditions into conditions of simultaneously increased strain rate and a, = S(a&K(D -ci#, c) (6) temperature, chosen by trial and error such as to give for the effect of dynamic strain-aging during a the same appearance of the serrations on the stress waiting time a& (where Q, is proportional to the strain curve. mobile dislocation density). We have assumed the We have done such tests, both near room temfunctions S and K to be the same, because that is perature and near 453 K, on our D-alloys. They were what our experiments indicate (except for a possible only sufficient to put some very broad bounds on

KOCKS er al.: STRAIN

AGING

AND

these “activation energies”: about 70 + 10 kJ mol at average an temperature of 312K. about 150 f 20 kJ/mol near 440 K. The first of these is in agreement with that measured by Nakada and Keh (who did not report one for another temperature). The second is close to the quoted [43] activation energy for diffusion of carbon in nickel. We presume that the latter correlation is entirely accidental inasmuch as a single atomic jump at this temperature would take of the order of JO’sec (using Do = 0.1 cm’jsec [43]), whereas the aging effects are completed after about 20 set [Fig. 1l(a)]. In fact, the strong dependence of the measured “activation energy” on temperature, without any apparent change in mechanism, may suggest that the jerk height is not directly related to any single thermally activated process. Finally, the meaning of the measured activation energies is impaired by the fact that the temperature dependence of the shear modulus (or other entropy effects) have not been taken into account; this can lead to serious errors when the rate sensitivity is very small or negative. A more appropriate test may be one that keeps the static aging stress-increment, measured under standard conditions of strain rate and temperature, the same after aging for different times at different temperatures. Such tests were performed on our D-alloy, slightly above room temperature, and they gave the same activation energy as that quoted above. Near 443 K, the “activation energy” was again much larger (even larger than for the jerk heights); this is expected in view of the fact that aging had effectively saturated and become time-independent at the times used during temperature changes. Finally, we saw (Fig. 7) that the curve of rate sensitivity versus flow stress depends on strain rate. We measured it for slightly different temperatures also, at different strain rates. Again, we found the results consistent with an activation energy of about 70 f 10 kJ/mol near room temperature. Near 443 K, this “activation energy” would have come out negative, since the rate sensitivity is still negative, but the temperature dependence of the flow stress is in the normal direction (i.e. negative); apparently, at this upper limit of the strain-aging temperature regime, it is not possible to determine meaningful activation energies. In summary, the concept of an activation energy for strain aging seems problematical [39]. If one restricts oneself to the low-temperature end of the strain-aging regime, a variety of different methods, relating to the flow stress itself or to static or dynamic strain aging. appears to give the same approximate value of the activation energy: 70 f 10 kJ/mol (about half the activation energy for bulk diffusion of C in Ni [43]). 3.5. INCONEL

600

The interest in the present work was originally generated by the work on INCONEL 600 and some

IN Ni-C ALLOYS

HARDENING

633

other commercial nickel alloys by Mulford and Kocks [23]. It has been suggested [8] that the strainaging phenomena in these alloys are primarily due to the diffusion of carbon. which typically has a concentration of about 0.3 at.:& in INCONEL 600, rather than the plentiful substitutional atoms present (primarily Fe and Cr). While the observed “activation energies” are similar, one then needs to explain why the temperature regime in which the effects are observed in INCONEL (about 550 K) is so much above the temperature where the aging effects in pure Ni-C have saturated out (even though it is indeed the temperature where bulk diffusion of C becomes easy). To clarify the effect of carbon, we have conducted a few tests on a model alloy with a composition much like INCONEL 60&but decarburized to 0.05 at.% C (measured both before and after the tests). The oxygen plus nitrogen content (before the tests) was about 0.03 at.:/,. The behavior of this alloy was identical to that of commercial INCONEL 600. (This contradicts the assertion that jerky flow does not occur in interstitial-free nickel alloys [4].) We must conclude that the substantial solutes control the aging behavior of INCONEL 600. Furthermore, complexes involving C, which have been postulated to be responsible for jerky flow in austenitic steels [44], cannot be responsible in the present case either. Finally, we complemented the former work on dynamic strain-aging in INCONEL 600 by some static aging experiments. The results are shown in Fig. 15. They again demonstrate an approximate linearity between Ao, and u; furthermore, in this case, the initial slope depends on aging time in the way usually suggested [ 15,381, namely on the 213 power. 3.6. Summar~~ qf experimental results It is convenient to distinguish different regimes of strain. At large strains, both the static-aging stressincrement and the negative rate sensitivity saturate or 14 -

1600

-

12 10 3

78 s

8-

t a=

35s

6

-

42-

1

OY

0 0

111

200

”

” 400

”

600

’

800

loo0

FLOW STRESS IMPa] Fig. 15. Static-aging stress-increment vs flow stress for INCONEL 600 aged at the test temperature for times

indicated. i 2 1.7 x IO-‘s-‘.

634

KCKXS er al.: STRAIN AGING AND HARDENING

IN Ni-C ALLOYS

decrease slightly in magnitude, and there is no longer any jerky Aow. The beginning of this regime correlates well with the beginning of dynamic recovery, especially for static aging. Since dynamic recovery itself is strongly delayed by the solutes, these “‘large” strains are in excess of 10% (at room temperature) to 30% (at 18tYC). The “smaller-strain” regime is quite extensive in some of our results-and the longer this regime gets the more clearly does it exhibit its remarkable simplicity. The principal observations for this regime are as follows. (a) There is a clear strain dependence of aging phenomena, both static and dynamic, over the entire temperature regime in which these phenomena occur. 0 [MPI] This in clear distinction to the behavior of mild steel, Fig 16. Work of den Haastert and van den bukel IlO] on a bodycentered cubic interstitial alloy [20]. Nd0.55 a&XC at 243 K: Sress increment after increase in (b) This strain dependence of aging can be well strain rate by factor 10(solid circles) and after various aging described as being linear in the flow stress. (This times during stress relaxation fall other symbols), From a single specimenI linearity was also shown to hold for INCONEL 600; but the aging species is here probably nor carbon.) strain-aging phenomena and agreed on two critical (@The strain dependence of static strain-aging experiments. One was concerned with static strainpersists irk a temperature and time regime (around aging in an f.cs. intetstitial alloy; itl Esponse, both 450 EC,at times larger than about 20 set) where the the present investigation and a short study by van aging process itself has saturated; thus, the strain Haastert and van den Beukel IlO] were undertaken. dependenct cannot be due to a kinetic e&Z. In The latter authors tested a Ni-OXa%C alloy (right particular, it cannot be due to deformation-generated between our two) in tension at low temperatures vacancies, even if these contributed in any essential (213-C-248 IQ. Thus, their results should complement way to the mobility of carbon in~rs~tials (which has ours. Their static aging was done during stress relaxalready been disproved by Nakada and Keh [5]). ation, however, and their excursions were much more (d) Bulk diffusion of carbon is not likely to be the closely spaced than ours (thus perhaps making evalrate controlling process in these strain aging phenomuation less clear-cut). ena. To the extent that a meaningful activation Van Haastcrt and van den Beukel (“HB”[lO]) energy can be determined at all, it is about half of pIott& their results as a function of strain, since that that for the diffusion of C in Ni, during roomis the variable used in their theory. Figure 15 replots temperature aging; at 450 EC,it is of the same order these data as a function of flow stress, for comparison as the diffusion energy, but the jump frequency with our results and assessment in terms of our views. should still be negligible at that temperature. It The point emphasized by HB is the constancy of Aa, appears that ““activationenergy” determinations canover a significant regime of strains. This observation not easily bc used to identify the dominant mechis entirely in agreement with our data at larger anism in the case of strain aging. strains-but the regime of smaller strains is ignored {efThe flow stress shows a clear negative rate by HB; it extends to strains bctwecn 2 and 7%. In sensitivity in the jerky-flow regime. The static-aging comparison with our data, saturation is achieved stress-increment exhibits a rate sensitivity only bemore quickly. cause the flow stress is rate sensitive: the peak stress No statements can be made about the form of the itself appears to be a function of structure only, not stress or strain dependence of 66, at the smaller of rate. strains: tht number of points is too small, because HB @Measured jerk heights correlated well with the investigated five different aging times on a single magnitude of the negative strain-rate sensitivity of sample. The point WC wish to emphasize is the the flow stress (in this small-strain regime). The jerk qualitative fact of a substantial increase in A@,,with heights also depended on strain rate, sometimes increasing (as in Fig. 7) and sometimes decreasing (as strain-an observation incompatible with van den also observed by [g] and [39]). We could not find a Beukers theory. The other critical experiment suggested by van den way to abstract this behavior into a simple statement. 3eukrl and Kocks [14]concerned the curvature in the Haasen plot, in the lower half of Fig. i6. In our 4. O’IHFiR WORK simple description, this curvature should be zero; it 4.1. Vwz den Betrket on Ni-C could be convex upward if the mobile dislocation Van den Beukel and Kocks 11418s$essed their density increased with strain. So van den 3eukel’s overlaps and differences in the interpretation of description, it is expected to be concave upward. The

KOCKS er al.: 12 11

*u-14rloCu 273K

STRAIN AGING AND HARDENING

+

IN Ni-C ALLOYS

635

+

1,[min] = 30 l

C

5 a

6-

ti4T

32-

IMPa]

Fig. 18. Stress increment on NaCl+ 10e4 Cd:- single crystal after aging at 363 K for the stated times. under small load 1451.Compression at room temperature. The lines are Brown’s interpretation (generalizing from further data).

1 ---

Fig. 17. Static-aging stress increment for Au--14 at.?, Cu testedat273K.dr1.7~ 10-4s-‘byvandenBeuke]erc/. [41]: aging times in minutes.

latter statement is in fact true: for the data of Fig. 16, as well as for all of ours. However, we have interpreted the curvature in Haasen plots to be due instead to the gradual onset of dynamic recovery, in agreement with the behavior in pure materials [30], but that a straight-line description holds until then. It would appear that one can make both the static and dynamic strain-aging results of HB compatible with our views, if one assumed that dynamic recovery starts earlier at their lower temperatures-which is the same trend as in our alloy data. 4.2. Van den Beukel on Au-Cu Another paper of van den Beukel and coworkers that has appeared since our joint work [ 141must also be briefly mentioned, since it is claimed again to be cupportive of their views and incompatible with ours. In this paper [41], they have complemented their earlier work on dynamic strain-aging in Au-01 alloys by static aging experiments. Figure 17 reproduces the data points from their Figs 5 and 6, omitting their interpretive line. A dotted line is added at the level Au = 1 MPa. Below this tine is a regime that van den Beukel er al. place much emphasis on; we choose to de-emphasize it as a “small-effect regime”: in addition to the stress increment being of the order of 1% of the total stress, either the aging time is small or the total accumulated strain (which is less than 1% for the set of points at the lowest stress level). Note that a Au of this order has been observed near the yield point in all other tests also. Above this “small-effect regime”, the data are well described by straight lines. While we expect deviations at larger AM 334-G

strains when dynamic recovery becomes important, such deviations are not yet evident in Fig. 17. Van den Beukel’s description, on the other hand, which describes the “small-effect regime” quite well. demands a new mechanism at strains which are well within this plot. In summary, our difference is more one of emphasis than of fact. 4.3. Brown on NoCl Probably the first work that investigated the relation between the static-aging stress-increment and the flow stress was that by Brown [45] on NaCl single crystals containing a concentration of 0.5 to 2 x 10e4 Cd?’ ions. He found that it was linear up to about 7% strain and then saturated; Fig. 18 reproduces his low-concentration case, compressed at room temperature, with aging at 370 K under a small load. When the concentration was stronger and the yield stress higher, the straight lines intercept the abscissa at about the yield stress. The similarity to Ni-C goes even beyond these macroscopic observations. In NaCI. it was known at the time [46] that vacancies are-not generated during deformation until after a strain of about 5”,,-i.e. when most of the strain dependence of strain aging has already taken place. As a consequence, Brown called this “one of the most mysterious effects described in this thesis”. 4.4. Schwarz’s model Finally, one other recent work will be briefly mentioned. In a theory primarily aimed at describing discontinuous front propagation in tension, and tested on an Al alloy [471, Schwarz [48] has proposed that the strain dependence of aging effects is mainly due to the strain dependence of the strain-hordening rote, 0. Specifically, he suggests the formula I /Aa =o+b.O

(8)

636

KCKKS a of.: STRAIN AGING AND HARDENING

for both the static-aging stress increment Au, and the jerk height Aa,. Even though the mode1 should perhaps not be applicable to ~~~~r~~~jo~ experiments (in squat specimens), we did evaluate our data in terms of equation (8). Since the strain-hardening rate varies little (Fig. 4) in the regime where Aa, varies most (Figs 7,8, 10,1I), we needed to choose a imge value of b for a reasonable empirical fit-and this required the constant u to be negarire. Both of these results are incompatible with the model. (The constant b is, according to Schwa&s theory, proportional to the effective machine compliance, which in our case is an order of magnitude smaller than in his and should therefore make the second term small compared to the first.) We conclude that the strain dependence of aging phenomena in compression of Ni-C cannot be described by Schwarr’s model. 5. ASSESSMENT

IN NX

ALLOYS

FLOW STRESS, o-

Fig. 19. Solid lines: 3-retime model of static strain-aping. Dashed lines: Strain-rate sensitivity. of the ffow stress iFSo;= fl in&/s,)] should have a negative contribution proportional to do,, (thin dashed line). modified by effects of mobile dislocation density and dynamic recovery (heavy dashed line, observed). Dmed lines: the jerk hetpht Au, could be proportional to the negative rate sensitivity (thin dotted line); this correlation is observed at small strains, but not after dynamic recovery sets in (heavy dotted line). The arrow marks the beginning of dynamic recovery.

The experimental results summarized in sections 3.6 and 4, as well as in reference [20], show that there is a remarkable similarity of behavior in many materials: whether the alloys are substitutional or interThe transition from regime 2 to regime 3 correlates stitial, metallic or ionic,. binary or complex; and whether testing is done in tension or compression, in quite well with the beginning of dynamic recovery, i.e. with the transition from “stage I!” to **stageIll” creep or in a hard machine, and in the lower or upper of strain hardening [32]. This correlation was studied part of the temperature range in which strain aging occurs. The single exception so far seems to be in detail for all the present experiments. and it holds well. To our knowledge, there has been no attempt to interstitial b,c.c. alloys [20-22,423. This universality explain regime 3, and we can offer no viable hypothseems to rule out mechanisms in which deformationesis here: this is one of two important questions we generated vacancies [15-193, the mobile dislocation perceive in the field right now. density (the same references plus [49]), the formation of solute complexes [4.10,44, SO], or front propagaFor regime 2, on the other hand, a mechanism has tion [48,49] play a prominent role. been postulated i20] and later generalized [14] and On the other hand, this universality of behavior interpreted [27]. It says that would seem to allow the formulation of a genera1 b, i= (@ + ~aJ*bC(c, D,) (9) pbenomenology of static and dynamic strain aging. This will be done in the following, using in particular Here, a, is used instead of Aa,, for the reason our results on nickel-carbon alloys, since these explained in equation (7): u, is with respect to a showed many of the salient traits of strain-aging nonaging case, Aa, with respect to a rate-dependent behavior with particular clarity. We will also try to d~a~i~ll~a~ng case (and tberefore rate dcpcnassess where and how a connection of the phtnomdent). The term AC in equation (9) is the change in ecology to physical processes has been established (or concentration at the dislocation (which may saturate even attempted) and where not. and become independent of all its arguments). The We start with static strain aging: in many ways the term f~ is due to a change in the fricrion stress, and purest manifestation of the effects, The solid he in is the one emphasized by van den Beukel. It is Fig. 19 sketches the general behavior. There is an proportional to the normalized interaction strengthj initial finite value of Aa, at the yield stress (of the between individual solutes and mobile dislocations, order of i MPa), followed by a short curved (convex which is large for tetragonal defects, small for or concave) portion: this first regime is labelled “1”. spherical ones ill]; thus it is expected to be strongest in “regime 2”, Au, is linear in u, and in “regime 3”, in b.c.c. interstitial alloys 1141.By itself, it does not it tends to saturate (or decreases slightly). When give any strain dependen*and b.c.c. interstitial regime 2 is short, the division may seem articificiak alloys in fact exhibit no essential strain dependence of it could be a single sigmoidal curve. However, in staric strain aging [20,21,42]. some cases, especially in the present investigation, a The strain dependence of static strain aging, where very extended regime 2 has been observed; we thcre- found, is usually presumed to reside in the strain fore consider it physically meaningful. It is the regime dependence of the diffusion coefficient D. We have in which the essenrial strain dependence of “strain shown that this cannot be true for M-C, particularly aging” is found; and it is linear in the flow stress. since the strain dependence persists when AC is

KICKS

er al.:

STRAIN AGING AND HARDENING

saturated at high temperature; it was also ruled out by Brown [45] in his NaCl experiments. Yet these two behave like all the other f.c.c. alloys investigated. We conclude that vacancies cannot provide a general explanation for the strain dependence, but that such a general explanation can be provided by the term proportional to u., in equation (9). The interpretation of this effect, albeit qualitative, is that the solute concentration along the dislocations, at the time they bred;. dway from forest dislocations, affects the junction :.cngth [51,271. Hr ,‘U turn our attention to the strain-rate deper$. j ,Lof the flow stress. With equation (4) it is

As in equation (4). it is not implied that the terms are independent of each other. Let us, however, for the moment assume that the first two terms are constant (and positive). With the assumption that the dynamic aging time is proportional to k/C, the last term in equation (10) follows, except for a (rate dependent) scaling constant, as the negative of equation (9). Then, /? should mirror 6, in Fig. 19; this is shown as the thin dashed line. For generality, it is started at a positive value of /I at the yield stress, even though in the particular experiments reported here, this was essentially zero. One would, however, expect deviations from this simple behavior for two reasons: first, the mobile dislocation density enters into the term &, above, and it may depend on strain; second. the rate sensitivity of the dislocation interaction stress a,, i.e. the second term in equation (lo), begins to increase significantly as dynamic recovery sets in [30]. This leads qualitatively to the line sketched in heavy dashes in Fig. 19-which is a fair representation of the generally observed behavior. An influence of dynamic recovery on r&r line would therefore be expected [23] even if a,(a) were straight. Since, however, a number of effects get into this line, the beginning of nonlinearity, and its correlation with the beginning of dynamic recovery, are less well defined. Finally, to the jerk height. If jerky flow is a consequence of a negative total rate sensitivity [23,34,36], one might expect a quantitative correlation between the jerk height Au, and the value of -/I (so long as /3 < 0). This is sketched as the thin &red line in Fig. 19. While the experimental observations show this general character, jerkiness stops (in the compression of Ni-C) rather abruptly right at the beginning of dynamic recovery, while /? is still clearly negative. This is the other important question we perceive in the current status of strain-aging theory. Note, however, that the correlation still holds generally that jerky flow does not occur unless the rate sensitivity is negative. There is an interesting aside with respect to the point where this firs! becomes true: it depends not only on the contribution from dynamic strain aging [the third term in equation

637

IN Ni-C ALLOYS

(IO), which is negative], but also on the constant positive contribution from the friction stress. The latter increases with the addition of other alloying elements; thus, the more complex the alloy (e.g. INCONEL 600 or stainless steel) the stronger the aging contribution must become before it dominates. This, rather than diffusion, may be the principal reason for the temperature dependence of the “critical strain” for jerky flow in commercial alloys. In summary, the regime where the main strain dependence of static and dynamic strain aging occurs (regime 2 in Fig. 19) can be at least qualitatively understood in terms of an interaction between solute strengthening and forest hardening: that is why the various aging stress increments are proportional to the flow stress. The small effects near the yield stress (regime 1) may well be a consequence of frictionstress effects (and so is strain aging in b.c.c. interstitial alloys). The major problem of interpretation appears to us to lie in the behavior at large strains (regime 3), and its strong

correlation

to dynamic

recovery.

Acknowledgements-F~itful discussions with R. B. Schwarr, H. Wiedcrsich, A. van den Beukcl, M. F. Ashby and J. D. Embury are gratefully acknowledged. The’ alloys were prepared by W. Moore of the General Electric Research and Development Center’s Metals Processing Facility. This work was supported by the U.S. Department of Energy. Division of Basic Energy Sciences.

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