Primary creep in nickel: Experiments and theory

~~1-6160~87$3.00+ 0.00 Copyright Q 1987Pergamon Journals Ltd

Acta metali. Vol. 35, No. 7, pp. 1499-1514,1987 Printed in Great Britain. Al1rights reserved

PRIMARY

CREEP IN NICKEL: AND THEORY

EXPERIMENTS

A. S. ARGON and A. K. BHATTACHARYAf Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received 7 August 1986; in revised form 22 November 1986)

Abatraet-Primary creep in both polycrystaIline and single crystalline pure nickel has been studied at a temperature range of 0.54-0.62 T,, and stress range of 1-5 x 10-4a/p. Although constant strain rate experiments and stress jump experiments in primary creep suggest that dynamic recovery controls fast deformation, diffusion controlled static recovery was found dominant at all but the very earliest stages of primary creep. Both deformation resistance and internal stress was found to evolve systematically during primary creep. While such evolution reached steady state in single crystals at steady state in creep, this was not the case for polycrystals at the minimum creep rate. Theoretical scaling-law models are provided for the primary creep strain rate based on a combination of dynamic and static recovery for an evolving defo~ation resistance. These scaling laws, normalized with steady state properties, and in dimensionless time, demonstrate that while the time constants for dynamic recovery controlled creep are typically of only IO*s duration, those for static recovery controlled creep are of the order of IO4s, in the range of stress and temperature investigated here. These scaling laws also provide a general behavior pattern which can cope with the creep strain portion of any transient during primary creep. Fast, dynamic recovery controlled plastic portions of transient deformation require further study. Andrade’s famous t -*I3 time law for creep strain rate is shown to be a general by-product of dynamic recovery controlled transient defo~ation. R&mm&-Nous avons &udiC le fluage primaire dans Ie nickel pur, polycristallin et monocristallin, pour des tempbratures comprises entre 0,54 et 0,62 I&, et des contraintes variant entre 1 et 5 x 10e4 a/p. Bien que des experiences g vitesse de deformation constante, et d’autres avec sauts de contraintes en fiuage primaire, suggkrent que la restauration dynamique contrBle la dkformation rapide, nous avons trouvC que c’btait la restauration statique contrblke par diffusion qui pridominait ii tous les stades du fluage primaire i I’exception des tous premiers. La resistance ;i la diformation et la contrainte interne Cvoluent syst~matiquement pendant C fluage primaire. Alors qu’une telle tvolution atteint un rbgime stationnaire dans les mon~ristaux lorsque le fluage devient stationnaire, ce n’est pas le cas pour les polycristaux pour la vitesse de fluage la plus basse. Nous proposons des modtles theoriques en loi d’kchelle pour la vitesse de dkformation en fluage primaire, modbles bases sure une combinaison de restaurations dynamique et statique pour une rksistance ii la d&formation qui Cvolue. Ces lois d%chelle, normalis&es par des proprittts g l’btat stationnaire, et dans un temps sans dimension, dtmontrent que, tandis que les constantes de temps du fluage contrBl& par la restauration dynamique ne sont typiquement que de 10’s, celles du fluage contrd% par la restauration statique sont de I’ordre de lo4 s, dans les domaines de contraintes et de temp&ratures oit nous avons travailI& Ces lois d’bhelle foumis~nt bgalement un schima de comportement g&n&al qui peut rendre compte d’une partie de la deformation de fluage de n’importe quel domaine transitoire pendant le fluage primaire. Les parties plastiques rapides, contrBltes par la restauration dynamique, de la d&formation transitoire &essitent une ttude ultkrieure. Nous montrons que la fameuse loi d’Andrade, en c-*‘~, pour d&ire la vitesse de d&formation de fluage, est un sous-produit de la deformation transitoire contrBl&e par la restauration dynamique. Zusammenfasang-An poly- und einkristallinem reinem Nickel wurde das primire Kriechem im Tem~rat~~reich zwischen 0,54 und 0,62 T, und im Spannungs~reich zwischen l-5 x 10-j D/P untersucht. Obwohl Experimente mit konstanter Dehnungsrate und mit Spannung~pr~n~en im primgren Kriechbereich nahetegen, dai? die rasche Verformung von dynamischer Erholung gesteuert wird, wurde gefunden, da0 die diffusionsgesleuerte statische Erholung in slmtlichen Bereichen des prim&en Kriechens, a&r dem allerersten, iiberwog. Verformungswiderstand und innere Spannungen entwickelten sich systematisch wHhrend des primgren Kriechens. In den Einkristallen erreichte diese Entwicklung einen stationlren &stand im station&en Kriechzustand, dagegen war dieses bei den Polykristallen bei der kleinsten Kriechrate nicht der Fall. Fiir die Rate des primlren Kriechens werden theoretische Skalierungsmoddle angegeben, die auf einer Kombination von dyn~ischer und stat&her Erholung bei einem sich entwickelnden Kriechwiderstand aufbauen. Die Skalierungsgesetze sind auf die Eigenschaften des stationtren Zustandes notmalisiert und dimensionslos in der Zeit. Sie zeigen, daB die Zeitkonstanten des durch die dynamische Erholung gesteuerten Kriechens typischerweise nur lo2 Sekunden sind, wohingegen sich im Falle des durch statische Erholung gesteuerten Kriechens die GriiBenordnung lo4 ergibt (fiir den hier untersuchten Spannungs- und Temperaturbereich). Diese Skalierungsgesetze ergeben such eine allgemeine Beschreibung des Verhaltens, welche such den Teil der Kriechverformung beschreiben kann, der mit irgendeiner Transienten wahrend des primiiren Kriechens ~usammenh~n~. Schnelle, von der dynamischen Erhoiung gesteuerte plastische Teile einer transienten Verformung erfordern noch weitere Untersuchungen. Andrades beriihmtes t - ‘j3-Zeitgesetz fiir die Kriechrate ergibt sich als ein allgemeines Nebenergebnis der von der dynamischen Erholung gesteuerten transienten Verformung. TPresent address: Department A.M.35174

of Materials Science and Engineering, Stanford University, Stanford, CA 94305, U.S.A. 1499

1500

ARGON

and BHA’ITACHARYA:

PRIMARY

1. INTRODUCTION

Since the pioneering researches of Andrade [l], it has been established that creep under constant stress can exhibit a significant amount of transient deformation before settling down to a steady state. Changes in the stress state can evoke new transients. While some complex engineering alloys often show only small amounts of such transient creep, pure f.c.c. and b.c.c. metals, show important amounts of primary creep. In spite of this, and the generally acknowledged prominence of this type of creep in service, in many alloy systems, the vast majority of all creep studies have been devoted in the past to secondary, or so-called steady state creep. The functional dependences of the steady state creep rate on stress, temperature, and other micro-structural parameters have been summarized in a number of recent reviews [24]. Out of these studies, two different points of view have emerged. The studies of Dorn and co-workers [2,3] on pure metals, have led to an expression for the steady state tensile creep rate i* that is

i*=A(~)(-$)‘G)(~)”

(1)

where 0 is the applied tensile stress, D is the self diffusion constant with an exponential temperature dependence, and x is the stacking fault energy. The other terms, such as p, Cl, b, and kT have their usual meaning, while A and m are material constants, of which the latter is typically around 5 for pure metals. Of the various factors that govern the steady state creep rate, the most important two are the absolute temperature and the tensile stress. The former acts through the self diffusion constant and indicates that steady state creep is possible through diffusion controlled static recovery. While there is no generally accepted explanation for the stress dependence of the creep rate, it indicates a highly nonlinear behavior. The dependence of the creep rate on the stacking fault energy x [5] is considered to result from the difficulty of climb of extended jogs along dislocations [6]. This form of the steady state creep has been generally accepted as fundamental in the field of materials science. The second point of view, widely used in the engineering and the applied mechanics field, represents steady state creep in a simpler phenomenological form of i*=i

/,.\m

v

0

0-

=0

which is known as Norton’s law [7]. In this point of view, a more fundamental form, such as the Dom equation given above, is often acknowledged, but the simple form given by equation (2) is preferred for its operational ease. Brown and Ashby [4] have pointed out that the

CREEP

IN NICKEL

form of the expression given by equation (2) is not only simpler to use, but differs prominently from the Dom expression, in that the stress is normalized not with the shear modulus but with a reference stress oo. When the Norton equation is used to correlate the steady state creep rate (minimum creep rate) of large groups of solids, it is found that it leads to a tighter grouping of the non-Arrhenian portion of the coefficient of io. More interestingly, however, it leads to the discovery that the reference stress 0, is found to be the plastic deformation resistance of the solid at room temperature. Thus, this form has clear implications for extension into primary creep, which we will develop in this communication. Most earlier studies of primary creep itself have either taken the basic form for steady state and have associated it with a time-hardening function (for comprehensive discussions of such procedures, see Garofalo [8]), or have tried to furnish a basis for the different time laws [8]. Two prominent time laws have been identified. It was recognized quite early, that at low temperatures where deformation can be thermally activated without any recovery controlled structural alterations, the creep strain is a logarithmic function of time [9]. Alternatively, at somewhat higher temperatures or flow stress levels, the creep strain rate is found to be proportional to the inverse 2/3 power of time. This time law, which was first discovered by Andrade [l] and investigated by him over many years [lo], has been identified to be affected by thermal recovery in a general way [9]. It is found at any temperature in the Stage III work hardening behavior region of f.c.c. metals, and at elevated temperatures in primary creep. Outside an interesting theoretical attempt to relate it to recovery [ 111, the Andrade time law has not received a satisfactory explanation. Somewhat different and enlightening approaches on the nature of primary creep were made by Dorn et al. [12]. They have demonstrated that at a given constant stress, the strain rate during primary creep is governed by a combined time-temperature parameter 0 = t exp( - Q/kT), where Q is the activation energy for self diffusion that affects the steady state creep rate. Indeed, at the same constant stress level, all primary creep curves were shown to collapse on one another, when plotted as a function of 0. This indicates that the kinetics of primary creep is governed by the same processes that govern steady is state creep, which at elevated temperatures diffusion controlled static recovery. On a somewhat further reaching study, Amin et al. [13] have likened primary creep to the rate of approach to equilibrium in a unimolecular reaction in which time is measured by the steady state creep rate, and demonstrated the validity of this approach in a large collection of pure metals and simple solid solution alloys. This demonstrates that during primary creep, both the temperature and the stress dependence is the same as that in steady state creep.

ARGON and BHATTACHARYA: Table 1. Composition

PRIMARY

1501

CREEP IN NICKEL

of nickel used in experiments (in wt%)

C

S

P

Ti

Al

Mn

Cu

Fe

Co

Mg

Cr

0.01


<0.005


0.04

0.003


0.004

0.001

0.001

< 0.002

In more recent times, many investigators [14-181 have recognized that inelastic constitutive behavior in any form of deformation, including primary creep, needs for its description internal parameters, or “state variables” in addition to the applied driving forces of stress and temperature. These variables characterize the current state of the solid, and require associated considerations for their evolution, as the solid strains or as time passes. In view of these developments, the observation of Brown and Ashby [4] on a, in the Norton steady state creep law takes on deeper meaning. Thus, their observation suggests that u,, is a state variable, and that the Norton creep law, or some modification of it, also represents primary creep where, however, the deformation resistance cr,, evolves into a steady state value when steady state creep is achieved. In this communication, we explore this possibility both experimentally on pure nickel, and also theoretically. 2. EXPERIMENTAL

specimens, all tests reported here were carried out at constant load. Since the total strains recorded did not exceed 34% during primary creep, the elevation of stress was of the same magnitude, producing changes in creep rate of about 15% in the constant load experiments. Stresses were applied by means of dead load increments on a load pan attached to the long end of a load amplification bar of the creep machine. The pan was initially supported by an extended hydraulic jack without any load going into the specimen. Finally, the loads were applied to the specimens smoothly, without overshoot, by lowering the jack over a period of about 5 s. All time measurements reported start from the point of initial application of the stress. The creep chamber was equipped with a helium gas quenching system which could readily achieve initial cooling rates of 200”C/min in the specimens. These rates were fast enough to cool specimens to room temperature without any significant alteration in the high temperature state of the samples.

2.1. Material

2.3. Other tests

Since many superalloys are of nickel base, and since pure nickel itself is an attractive high temperature material, it was chosen for the experimental program. Commercially pure, polycrystalline nickel bars with the composition given in Table 1 were obtained from the International Nickel Company Research Laboratories.? Most of the experiments reported here were carried out on polycrystalline samples with a grain size of about 100 pm. Preliminary experiments indicated that at temperatures below 800°C grain growth is insignificant during the creep times of approximately l&20 h that were of interest. Some seeded single crystals of (111) orientations were also grown from the same material by an electron beam melting procedure using an MRC/EBZ unit. Polycrystalline specimens were machined into button ended shapes that could be readily gripped in TZM key-hole type grips. The single crystal specimens were clamped in special friction grips.

The strain rate sensitivity of the flow stress was measured separately in the same range of temperatures by strain rate change experiments in an Instron servo-hydraulic machine equipped with a similar vacuum chamber. The same information during primary creep was also obtained in the creep experiments by performing stress jumps and recording associated sudden strain rate changes. Reference deformation resistances were measured at room temperature on creep samples in interrupted primary creep experiments. These resistances were measured most often by means of micro-hardness experiments, but also by means of tension re-yield experiments. Stress dip experiments were performed during primary creep simply by removing smoothly a small portion of the dead load to achieve a short period of zero strain rate. Strains were measured by a strain gage equipped extensometer attached to the moving portion of the upper load train. Thus, the measured extensions included load train stretches inside the furnace. These, however, together with the elastic extension in the specimen at the test temperature, were subtracted from the overall extensions to arrive to the inelastic strains. By necessity, the measured inelastic strains reported here, include also time dependent plastic strains, and transient response controlled by dynamic recovery. As will be discussed below, this complicates the interpretation of the creep strains at very short times and the strain surges during stress jumps.

2.2. Creep testing equipment All creep experiments were carried out in a special constant load (or constant stress) creep machine equipped with a Centorr vacuum chamber operating at a pressure less than 4 x 10e6 torr. Although the creep machine could maintain constant stress in tWe are grateful to Dr Steven Floreen, formerly with that laboratory, for supplying this material.

ARGON and BHATTACHARYA:

1502

c ._ e ;i t; jj 0.10 ui

PRIMARY CREEP IN NICKEL

4 660C ; 25MPo v 7 IOC ; 21.17 MPo

o 760C ; 18.4 MPo o 800C ; 18.4 MPo o 8OOC ; 21.17 MPa

Time, seconds

Fig. 1. Uninterrupted primary creep curves of nickel poIycrystals at four different temperatures and stresses. The different behavior at very short time is attributed to rapid plastic extension, but may also be at&&d by an ambiguity in the origin of time.

3. ~XP~IM~AL

RE!WLTs

3.1. uninterrupted creep tests

Because grain growth occurs above 8OO”C, all creep experiments were confined to a temperature range of 660-800°C. This corresponds to 0.54-0.62 of the absolute melting point, and is well into the diffusion controlled creep regime. Experiments of Fetz [19] indicate that pure Ni with moderate cold work (25%) recovers fully in 30 min at around 600°C. All uninterrupted creep experiments at various stresses were taken beyond the point of minimum creep rate to determine also steady state creep prop erties for purposes of no~ali~tion of the p~rna~ creep information. All stresses reported in this study are shear stresses, and all strains are shear strains obtained by a Mises criterion. Thus, to obtain tensile stresses and resistances, all stresses can be multiplied by ,,,& and to arrive at normal strains, all strains can be divided by $ by the interested reader. Figure I shows five uninterrupted creep curves in the temperature range of interest, plotted on logarithmic coordinates. At very small times, there is a turn-down in the creep-time law, which is most likely affected by the peculiarities of the application of the load and ambi~ity about the origin in time. The total strain rates, however, do not suffer from this difficulty and can be seen to be of the order of lo-‘s-l at the start of primary creep. In all cases, a minimum creep rate without any perceptible necking is achieved at total strains of less than 0.05 and in times of the order of several thousand seconds. Figure 2 shows the results on the stress dependence of the minimum creep rate at three different temperatures, which was obtained from the uninterrupted creep experiments. The power-law stress exponents m range from 4.87 at WC down to 4.62 at 800°C and are very close to values given by Dom and co-workers [2], but somewhat lower than the value given by Brown and Ashby [4]. The temperature dependence of the minimum creep rate for two stress levels is given in Fig. 3, from which an

bp

I 10-6

I

I I IO‘5 IO.6 Sheor Strom Role, set-’ I

10-7 f”. Mmimum

t lo- 4

Fig. 2. Shear stress, shear strain rate relations for steady state creep in nickel at three temperatures.

lo5’i 0

2

4

6

i/T,

8

IO

l2xlo-’

K-’

Fig. 3. Temperature dependence of minimum creep rate at two different stress levels in nickel. An activation energy of 69 kcal/mol can be calculated from the data.

ARGON and BHATTACHARYA: activation energy of 69 kcal/mol can be calculated. This compares very well with the published values of the activation energy for self diffusion in nickel.

Table 3. Evolution m = 4.71;

3.2. Strain rate change tests

l(S)

It has often been assumed that constant strain rate experiments at high temperatures provide the same information obtainable from creep experiments, and that the kinetic law for the strain rate is the same for both. To probe this possibility, constant strain rate experiments were carried out on polycrystalline nickel, and the stress dependence of the strain rate was determined from strain rate changes imposed during the stress strain experiment. A corresponding set of experiments were carried out by producing stress jumps during primary creep resulting in sudden changes in creep strain rate. Of particular interest in these experiments is the stress exponent n of the strain rate, defined as d In i) n=dIn

(3)

The values of n that have been determined from the strain rate change and stress jump experiments are listed in Table 2. The values of n obtained from the strain rate change experiments during a tension test show a slight scatter, but no systematic variation as a function of strain, while the values obtained from the stress jump experiments in primary creep increase systematically by about 25% (at 800°C) to 40% (at 710°C) between the start of primary creep and eventual steady state. This increase is quite consistent with kinetics of cutting an ever densening forest of dislocation trees that results when the deformation resistance steadily increases. Clearly, the measurements indicate that the kinetic law in strain rate changes at constant structure is very different from that in structure evolution given in Fig. 2. 3.3. Evolution of deformation resistance during primary creep That the dislocation density systematically increases in primary creep is well known [20]. The mechanical consequences of this were measured by interrupting primary creep experiments at several points. This was achieved by rapidly cooling the sample under stress to room temperature by turning the heating element off and flooding the chamber with purified helium. The deformation resistance itself was measured by either a microhardness experiment or a tension re-yield experiment at room temperature. The specimen was then inserted back into the creep chamber and heated up rapidly under the Table 2. Stress exponent Temperature 660 710 800

(“C)

n = (d In i/d In u) in jump experiments Strain rate change

Stress jump

40 36.85 5 1.68 27.78 + 0.27

30.4-42.6 26-32

1503

PRIMARY CREEP IN NICKEL

f(S)

0 600 1200 1760 2400 4320 4450

resistance

in primary

T = 710°C a=21.17MPa; )i*=SxlO-‘s-l; r = 9.35 x 105 s VHN (kg/mn?) e 0 0.054 0.252 0.468

4 4200 7800 m ~4.66;

of deformation

creep

s$=44.26MPa; sH (MPa) 33.19 38.73 42.93 44.04

17.25 20.12 22.31 22.88

T = 760°C o = 18.4MPa; v* = 8.5 x lo-‘s-l; s,!, =41.4 MPa; s: = 42.1 MPa; 7 = 1.21 x lo6 s 0 VHN (kg/mm2)S, (MPa) Y(MPa) S, (MPa)

0 0.102 0.204 0.299 0.408 0.734 0.756

17.74 20.08

34.16

20.96 21.39

40.36 41.40

38.67

45.26 60.22

26.13 34.77

68.26

39.41

72.63

41.94

previous stress to continue the creep experiment by a further strain increment. The results of such experiments are given in Table 3 for both 710 and 760°C together with some other processed data to be discussed later below. The times indicated represent the total integrated time measured from the beginning of the creep experiment, and not counting periods of rapid heating, cooling, and time spent at room temperature. The Vickers hardness numbers (VHN) were obtained by a Leitz micro-hardness indenter under a load of 25 g. These values were used to calculate a Mises deformation resistance su obtained by dividing the hardness pressure by 3&. The additional data for 760°C gives the tensile re-yield stress Y at room temperature, and the shear resistance sy obtained from it by dividing it by $. The parameters 0 and r are dimensionless time and time constant respectively, that will be introduced in Sections 4 below, while j* is the steady state creep strain rate obtained from the uninterrupted reference creep curves, r~represents the shear stress level under which the experiment was carried out. The remaining parameters sr*i,st, and m, are the values of the shear deformation resistance (measured from hardness or yield experiments) at steady state creep, and the stress exponent at steady state, respectively. The measured deformation resistances are summarized in Fig. 4. 3.4. Evolution of internal stress during primary creep The reduced stress level at which a previously creeping sample will show no initial creep rate has been called the internal stress, and has been measured by many investigators [20] as a state variable characterizing steady state creep response. Although the interpretation of this internal stress has created some controversy, it is clearly a measure of the polarity not included in the deformation resistance, and its evolution in primary creep is of interest. Therefore, the internal stress criwas measured by the standard stress dip test at several stages in primary creep. The results for two stress levels each of two temperatures are given in Table 4, where y’ represents the creep strain

ARGON and BHA’ITACHARYA:

PRIMARY

CREEP IN NICKEL

mary creep, and this fraction is larger at lower temperatures. The results are also shown summarized in Fig. S(a) and (b), where again, the point of the minimum creep rate is marked with a cross. Clearly, in three out of the four curves in these figures, the internal stress gives no indication of having reached a stationary state at the minimum creep rate. We will probe the consequences of this later below.

IS= v 710ch1). o760C(H,, 0 fGOC(Y),

0

0.2

21.17 18.4 18.4

MPO

,6

1.0

Time

Fig. 4. Evolution of deformation resistance in shear at 710” and 760°C as measured by means of micro-hardness experiments and at 760°C as measured by tension yield experiments.

at which the measurement was made, yip represents the initial plastic strain determined according to a procedure to be discussed in Section 4 below. The symbol (*) indicates the position of the ~nirn~ creep rate and a: the internal stress at the minimum creep rate. The internal stress evolves into a significant fraction of the applied stress during pri-

Tabk 4. Evolution of internal (shear) stress during primry T =

(9)

4.1 x 5.5 x 7.0 x 8.4 x 9.5 x 10.7 x 11.3x

parameter

Although the time-law of creep is of less fundamental importance than the strain rate and its dependence on stress and temperature, it has been widely reported since the pioneering investigations of Andrade [l]. As we discussed in Sections 3.1 and 3.3 above, the proper presentation of the time law is beset with two problems. In the very beginning, there is an ~biguity about the origin in time, and in creep strain, as considerable additional rapid plastic extension occurs. Since the kinetics of this deformation relates less to static recovery and more to thermally activated glide, or at most to dynamic recovery, it must be separated from the creep strains. Although it is most desirable to make the separation in the strain components an the basis of different kinetics, this requires rapid and error free sampling in the early phases of creep, which could not be achieved here. Therefore, the kinks in the logarithmic plot of total

creep

750°C 0. IMPa)

2.1 I x 10’ 4.57 x IO’ 6.67 x, IO’ 10.63 x IO’ 13.99 x 103 18.00 x 10’ 20.40 x 10’

4. THEORY

4.1. Time-temperature

MPo

0.0

0.6

0.4

8, Dlmenslonless

MPa

10 -3 10-l IO-’ IO-’ 10-3 to-3 10 3

8.13 9.66 10.35 II.12 11.27 11.42 il.50

o;/a? 0.707 0.840 0.900 0.966 0.98 1 0.994 1.000

0.h 0.4416 0.525 0.563 0.604 0.613 0.621 0.625

T = 750°C (~,(;,24.84 MPa: Y,~a 22 x IO-‘; 0: = 16.41 MPa) YC o. (MPa) O./O? oh

(“)

t.1

_

2.16 x 6.28 x 11.68 x 23.56 x 27.02 x

lo’ IO’ 10% 10’ 10’

10-X 10.’ IO-” lo-’ 10 -’

14.41 15.79 16.41 16.71 16.84

0.878 0.963 1.000 1.018 1.026

T = 800°C (a = 18.4 MPa; yIP= 14.5 x IO-‘; a: = 9.20 MPa) b, (MPa) I(S) YE cl,la : 1.10 x 10’ 7.3 x 10-J 8.05 0.875 1.86 x 10” 9.7 x lo-’ 8.74 0.950 2.34 x lo’ II.2 x 10-j 9.05 0,984 2.94 x 103 12.4 x 10-j 9.20 1.Wtl T = 800°C (a = 25 MPa. o* = 13.04 MPa) a, (MPaj ’ 0: OJb

l(S)

(‘)

13.0 x 19.5 x 24.3 x 38.5 x 45.3 x

0.48 x 1.20 x 1.92 x 3.24 x

10’ IOJ 10’ lo’

11.4 1212 12.58 13.04

0.846 0.930 0.964 I.000

0.442 0.485 0.503 0.522

0.580 0.636 0.660 0.673 0.678

Time.

seconds

Time,

seconds

*Jo 0.437 0.475 0.492 0.500

Fig. 5. Evolution of internal stress: (a) at 750°C and two stress levek, and (b) at 800°C and two similar stress kvels. Note that the internal stress has not come to a stationary state at the minimum creep rate,

ARGON

and BHATTACHARYA:

PRIMARY

CREEP IN NICKEL

1505

brackets the time exponent for primary creep of l/3 reported by Andrade. 4.2. The primary creep processes

I

0

I

I

I

1

Sxld" 8=

t ex;(_&)

,",,,'

Fig. 6. Creep shear strain at two stress levels, plotted as a function of a single time-temperature parameter 0, indicating that the kinetics of the deformation is the same as that at steady state. inelastic strain as a function of time, shown in Fig. 1, were used arbitrarily as the origin for creep strain.

These kinks occur in nearly every case at inelastic strain rates somewhat in excess of low3 s-l. As we will see below, for times larger than these, the kinetics of the deformation is governed by static recovery. Therefore, at constant stress, the creep strain should be a function of a single time temperature parameter 0, as has been suggested by a number of investigators, including Dom [12]. The primary creep strain-at two different shear stress levels of 18.4 and 25 MPa in experiments at 750”, 772” and 8OO”C, have been plotted in Fig. 6 as a function of 9, in which the activation energy Q was taken to be that determined from the mimumum creep rate (i.e. 69 kcal/mol). Clearly, at constant stress, the kinetics of creep is indeed governed by static recovery and therefore, self diffusion. Figure 7, which shows the information of Fig. 6 plotted on logarithmic coordinates, indicates that the time law is a power function of 0, and that the power exponent for the two cases is 0.305 for 25 MPa and 0.412 for 18.4 MPa, respectively. This

I

I

IO-"

8 = t exp

I

t,o,,

10-10

(- & ) , seconds

Fig. 7. Same creep strain as timetemperature

relationship

given in Fig. 6. above, plotted on logarithmic coordinates, indicating that the data brackets the strain-time law of Andrade for dynamic recovery controlled creep.

The results presented in Section 3 and the timetemperature parameter correlation of primary creep strain demonstrated in Section 4.1 above, indicates a certain set of phenomena that occur during primary creep. Strain rate changes during constant strain rate experiments, and stress jumps imposed during primary creep evoke a thermally activated glide response, with dynamic recovery, that has been investigated by Mecking, Kocks and their co-workers [21-231, which we will analyze somewhat further in Section 4.3 below. The signature of this response is a phenomenological stress exponent of about 3@40 in the creep range of T x (0.5-0.65) T,,. In constant strain rate experiments in this creep range, all the indications are that a constant ratio is maintained between the flow stress and the current deformation resistance that is typical of thermally activated glide [24]. In a constant stress creep experiment, however, deformation starts out as thermally activated glide with strong dynamic recovery, but settles down quickly into a static recovery governed process, after the first l-5min, when the strain rate has dropped down to say, 5 x 10e4 s-l, while dynamic recovery presumably still continues, but to an ever decreasing extent. The transition from nondiffusive dynamic recovery to diffusive static recovery control of the creep rate is not too clear. What can be demonstrated readily is that dynamic recovery has a much shorter time constant than static recovery, and that the metastable dislocation configurations that can be removed by dynamic recovery are exhaustable and of a somewhat different kind than those that can be removed by diffusional processes. Although little is known of these key recovery processes, it is clear that they are not quasi-viscous where dislocation clusters shrink down in a self-similar manner at increasing rates, as it is often depicted in elementary models. Rather, all recovery, whether dynamic or static, are most likely of a jerky nature as studied in -situ in some detail by Prinz et al. [25]. In these direct observations, it was established that small sessile obstacles (most likely chopped up, faulted prismatic dipoles that effectively pin down large dislocation tangles) are removed “abruptly”, which then permits a glide controlled collapse of the tangles. It is likely that in dynamic recovery, cross slip is involved in the removal of the pinning obstacles either directly in the obstacle itself, or in the encounter of the obstacle with another glide dislocation that provides the proper combination of kinetic impulse or kinematic configuration. In both primary creep under a constant stress, and in a constant strain rate experiment under increasing stress, the dislocation density, and the reference

plastic resistance that it represents, increase mono-

1506

ARGON and BHATTACHARYA:

tonically. Since the effective “time constant” for static recovery decreases as the mean dislocation spacing decreases (i.e. as the deformation resistance increases), a steady state is finally reached where all measurable properties of the deformation state, including deformation resistance, internal stress, etc. attain stationary states. In the theoretical model to be presented below, we will have nothing new to add to the existing level of understanding to this steady state which is qualitatively clear, but not so mechanistically. We will instead use the properties of the steady state as normalization constants. This less than fully satisfactory way, nevertheless leads to a very satisfactory scaling law for primary creep and creep transients. 4.3. The primary creep equation On the basis of the relative magnitudes of the measured deformation resistance and internal stress that were presented in Section 3 above, we conclude that the deformation resitance in shear, s, is the principal state variable of interest for monotonic response, and represents the isotropic hardening effect of the average dislocation density p. The principal evolution equation for this dislocation density is then given by

PRIMARY

CREEP IN NICKEL

Since the current average dislocation density prescribes an isotropic deformation resistance s, by the well known relation, s =apb& the evolution law can be transformed

d(s/p)

1 ds .

dp-

(4)

where dp+ -=-. dt

dp dy

W -

-Kzp2

’

(5a)

and

dtor

dp-

dt-

- -2K,p3.

In these relations, (dp/dy) is the effective rate of accumulation of dislocation density in strain hardening, which is expected to be influenced by dynamic recovery. The two expressions in (5b) and (5~) are two alternative possibilities for diffusion controlled decrease of dislocation density in static recovery, which were considered by Prinz et al. [25]. The first is valid when volume controlled diffusion governs, while the second is one where core diffusion governs. The temperature dependence of K2 and K3, which has been discussed in detail by these authors, is not of interest here, since it will be absorbed into the behavior at steady state creep. On the basis of the observations of Prinz et al. [25] of static recovery in the electron microscope, we prefer the form of (5~). Thus

dp dp . -2K,p3.

x=dry

into

4

dt

(84 which is recognized to be of the form derived earlier by Bailey [26] and Orowan [27]. In equation (8a), 8 is the effective strain hardening rate incorporating possible dynamic recovery effects, making it strain rate dependent, and the constant C with the dimension of reciprocal time is an abbreviation of the collection of terms in equation (8). We note that in the first term on the RHS of equation (8), the strain rate acts as a kinematic “converter” of hardening in strain increments to hardening in time increments. We now assume that at any time, the creep rate (like the strain rate in thermally activated glide) can be given by an equation similar to equation (2), i.e. i) = fo(a/s)”

dp dp+ -=x+x dt

(7)

(9)

where s is the current shear deformation resistance discussed above, u the shear stress, and v0 is the principal scale factor of the strain rate, and contains the temperature dependence of the rate controlling process, which is assumed to be diffusion controlled static recovery for creep. In equation (9), the exponent m is considered to be the same as that in steady state creep. The main justification for this comes from the time-temperature correlation at constant stress that was demonstrated in Section 4.1 above. When s is replaced with s*, its value at steady state, the strain rate should be that of steady state creep, i.e. y *. A phenomenological form of this type also holds for thermally activated glide, where the exponent n, however, is much higher, ranging from about 3&40 in the creep range to all the way up to 300 at low temperatures, where even dynamic recovery is absent [22]. In thermally activated glide, a kinetic law of the form of equation (9) can be readily shown to be a good phenomenological approximation to the exponential expression for the strain rate, with an activation free enthalpy that is a function of a/s. That this form also survives in creep is revealing. Here, however, we will merely accept the form without attempting to derive it from fundamental considerations. Thus, the change in the strain rate during primary creep at constant stress and temperature results primarily from the evolution of deformation resistance s, which leads from equation (9) to

$&&)yp)

(10)

ARGON and BHATTACHARYA:

which upon substitution of equation further use of equation (9) gives

Who) -=-zto dt

m

0 1’

(8a) and the

w+IYrn

(fJ/P) %

We note that at steady state, the term in the brackets of the RHS of equation (11) must vanish. This permits replacing some of the rate terms in equation (11) by the steady state creep rate 3 *

which also implies reaching mation resitance s*

Substitution of equation (12a) into (11) and normalizing the creep strain rate now with the steady state creep rate, we obtain the general form of the primary creep equation

*I

1507

CREEP IN NICKEL

at a reference strain rate of 1.6 x 10-5s-‘, to other strain rates and noting the initial strain hardening rate imm~iately after the strain rate change. Their measurements (their Fig. 9) has been re-plotted in Fig. 8. The figure indicates that at very high strain rates, the normalized hardening rate reaches an athermal plateau at 4.18 x 10-3, where presumably, the hardening behavior reverts from Stage III back to Stage II. At very low strain rates, less than 10-5s-‘, the effective strain hardening rate goes to zero and becomes even negative, indicating the dominance of a static recovery process over strain hardening. They have suggested a functional dependence for the effective normalized strain hardening rate Oj,u that is

a steady state defor-

Wb)

&f/if -=dt

PRIMARY

mjo(O /p) (y *~~o)(~+ ‘jirn (o/P) x(-?L~+l~‘m[~

-(g+““m]

(13)

where we expect, nevertheless, that the effective strain hardening rate Q may itsetf be a function of strain rate, as a result of dynamic recovery. 4.4. Creep with both dynamic and static recovery Mecking and Kocks [22] have recently summarized their earlier studies [21] on dynamic recovery in Al single crystals at 453 K (0.486 T,) presenting data on the strain rate dependence of the effective strain hardening rate. This was obtained by making strain rate changes from a reference flow stress at 10 MPa,

in which 6+,/p is the athermal hardening rate of Stage 11 and ~0”and p are constants, although the actual data indicates that both $0” and p vary markedly, going from p = 1 and yz = 2.33 x lo-‘s-l near the athermal transition to p= l/3 and jg = 1.4 x 10-‘2s-’ at the region near the steady state of deformation. Taking the form given in equation (14), we note that equation (13) can provide in principle a possibility of steady state by dynamic recovery alone when static recovery is absent. Thus, assuming that this is indeed possible and that the information in equation (14) truly pertains to nondiffusive processes, we write a new differential equation for the strain rate by starting with equation @a), where we set C = 0, and substitute equation (14) for the effective strain hardening rate. We now consider an alternative kinetic law for thermally activated glide li = $@/s)

(15)

where s has the same meaning as before, but FZ is the temperature dependent stress exponent of thermally activated glide [equation (3)], and $i is the strain rate coefficient that gives the best fit to the actual Arrhenius expression. With these, the alternative differential equation for the strain rate where dynamic recovery controls the evolution of the reference deformation resistance becomes

W/?j;)

---_=.....P

j;dt

0,

i’

(o/p) ()Op

li;l

n

(2n + I)in

-

x [l-~~~~].

(16)

We note that under these conditions, steady state deformation is possible at a strain rate of 10-5

10-4

f,

10-3

10-2

l/P

set“

Fig. 8. Strain rate dependence of normalized effective strain hardening rate in Al single crystals at 453 K. Data re-plotted from Mecking and Kocks [22].

(17) permitting

a re-normalization

of equation

(16) by this

1508

ARGON and BHA’ITACHARYA:

strain rate to result in

dt

(fJ/P)

1

(+)‘“““‘[ 1 -(qJ.

x

CREEP IN NICKEL

(2411, in fully normalized forms as follows

n);;(QhI~)(~*‘/3@(“+“‘”

W/l’*‘) -=-

PRIMARY

(18)

Clearly, the differential equation for dynamic recovery is of a very similar type as that for static recovery given in equation (13), the two differ, of course, in the exponents n vs m, and in the steady state strain rate. Their most fundamental difference, however, is in their time constant. While the time constant r of equation (13) involves the kinetics of static recovery incorporated in i),,, that of dynamic recovery, r’ incorporates the kinetics of dynamic recovery (presumably cross slip) that controls 9;. Before we write these two differential equations and their time constants in final form, we perform a needed simplification on equation (13) to make it more parallel to equation (18). Noting that when static recovery is controlling, i.e. C # 0, dynamic recovery is almost certainly also present, we need to give an operationally simpler form for the effective strain hardening rate 0 /p in equation (13), than that of equation (14). Guided by the desire to also provide an explanation for the ubiquitous Andrade transient creep law, we suggest a simple, crude power-law fit to equation (14), in the form of Q

1’

0

‘I2

(19)

T

P

Yo

.

For the data of Mecking and Kocks on Al, the crude fit represents the slanted straight line in Fig. 8. It offers a reasonably simple fit to the actual data in the range near steady state by the dynamic recovery process alone. For the Mecking and Kocks data, using the steady state creep strain rate coefficient y. = 7.4 x 10e4 s-i (obtained from data presented by Brown and Ashby [4]) a = 5 x 10e3. With this simplification, we now give a final form for the differential equation for strain rate in primary creep by incorporating equation (19) with equation (12), which modifies equation (13) into the more convenient form of

1

m?jOa(i */.j,)om+ 2)/h

d(? -=_ M *) dt

(O/P) x

(q+y

1

_

(V>“” ““““I

(20)

with the new steady state strain rate as

LO1 c

y*=)jo7-

u

Yo P

5 1 -

Q

final

(Dynamic Recovery) 9 = _

-_=a

and nondimensional

WOm+ 10)

(21)

With this simplification accomplished, we now write the two separate differential equations for strain rate in dynamic recovery control [equation (22)] and in both dynamic and static recovery control [equation

e,

=

(LJ+“‘“[l _(!rJ]

(22)

nf;(QJ~)(j*‘/j~)(“+‘)“t

(23)

(U/P) (Dynamic & Static Recovery) y = _

(+)‘““‘“[

(T)(xnil’?-]

1 _

(24)

e = mvoa (v */Yo)‘3”+ 2)‘2m t .

(O/P)

(25)

In the above expressions, 6’ and 0 are dimensionless time for the dynamic recovery controlled creep and the combined dynamic and static recovery controlled creep processes respectively. Clearly, the multipliers of time in both of these expressions are the reciprocal time constants for these two processes, which we will examine later. The steady state creep rates that are used for normalization are given by equation (17) and (21) respectively where all singly primed rate quantities relate to thermally activated glide and all unprimed rate quantities to diffusion controlled creep (the quantity yl is a fitting constant, as is the exponent p). 4.5. The Andrade transient creep law The simplification of the dynamic recovery controlled strain hardening relation by equation (19) leads quite naturally to Andrade’s transient creep law. This law, given by an inverse t213power dependence of the creep strain rate is one of the most widespread observations that so far has not received a satisfactory explanation outside of an interesting attempt by Mott [ll], which, however, is not fully compatible with current understanding. Andrade’s time law should be most widely observed in the Stage III hardening region of dynamic recovery at intermediate temperatures, but requiring no diffusion. Thus, under these conditions, where static recovery is of little importance, the transient creep rate is given by equation (16), which we write through the use of equation (14) in its more primitive form of d(V/3@ = - ---&(;~+““(;) ~ 9; dt

(164

and replace the effective strain hardening with the simplified power-law form of equation (19) to obtain

W/SJ

’

-=__

f;dt

(Z)

.

0 ;

(5n + 2)/2n

(16b)

ARGON and BHA’ITACHARYA: with a,=a

gl li2

(4 YO

uw

used to relate the normalizing strain rate coefficient in the simplified form to the rate coefficient of thermally activated glide. Equation (16b) can be integrated directly to obtain the strain rate as a function of time, as

=t+c,

iti! (3n+2)‘2n

0 -rY

(26)

where

(27) and C’, is an integration constant. We note that typically o/p x 3 x 10m4, n x 35, M’= 1.15 x lo-2 (for it&= 4 x 10e3, y. = 7.4 x 10e4), as for Al, giving for z~ M 0.12 s. Consider further a typical initial condition in a transient creep experiment of yi= 5 x 10-3s-’ at t = 0. This gives for C,=O.71. Thus, for times much larger than CJ,, i.e. minutes or larger, C, is negligible and the creep rate is given by

1509

PRIMARY CREEP IN NICKEL &,= 3.35 x 10-5s-’ below) j* = 1.04.10-6s-’

(for a/s* = 0.478 and j* (experimental)

a = O(O.05) (as determined

from experiments,

see below). From these values, we calculate T’ = 262 s, and r = 1.97 x 104 s, respectively for the time constants for dynamic recovery and combined dynamic and state recovery controlled creep deformation. Clearly, although dynamic recovery most certainly governs during constant strain rate testing, as emphasized by Mecking and Kocks [21-231 and demonstrated by the high n values, it ceases to be of importance after several minutes into the creep experiment, as the correlation in Figs 6 and 7 and the kinks in the creep curves in Fig. 1 suggest. Therefore, we will seek a verification of the experimental primary creep data against equation (24) alone. To test the validity of equation (24), we consider two uninterrupted primary creep curves at 7’ = 710°C and a = 21.17 MPa, and at 760°C and cr = 18.4 MPa, respectively. The relevant information determined from steady state creep tests is as follows: T = 7lO”C, a/p = 4.1 x 10m4,

m =4.71, 1’*= 1.04 x IO-6s-‘, Since n is in the range of 35 or even larger, the effective exponent of time is very nearly 2/3, which is the signature of Andrade’s transient creep law. 5. EXPERIME~AL 5: 1. The primary

creep law

Prior to testing the primary creep law against the two parallel differential equations for dynamic recovery [equation (22)J and combined dynamic and static recovery [equation (24)], we evaluate the respective time constants r’ and T, given by

m = 4.66, y* = 2.81 x 10m6s-‘,

Figure 9 shows the decrease of the normalized creep rates for these two temperatures against time. If the primary creep equation given in equation (24) were valid, then the function F(j/j*)

= ma)i,(~ */yap+ 2)‘2m ‘in

(29)

(@i/J) r = ~~a ($ */j@)(*n+Zfi2m for Ni, at a temperature, say 710°C and stress level of 21.17 MPa. For this purpose, we use the following values for the various quantities b//i =4.1 x 10-4 n = 37 (from Table 2) y;t = 5 x lo-' s-’ (for a o/s ratio of 0.9 at n = 37) S/p = 5 x lo-’ (typical Stage II hardening Y‘*‘= 10-5s-’ (as for Al, from Mecking Kocks 1221) m = 4.71 (from Fig. 2)

T = 76O”C, a/p = 3.82 x 10e4,

j,, = 1.849 x 10m4s-‘, OLto be determined.

VERIFICATIONS

(a/u) ?’ = ni);(@‘Jru)(j*‘/j;)“+

PO= 3.354 x 10m5s-‘, LXto be determined.

rate) and

(Q/P)

(31)

should be constant throughout the creep experiment and be equal to the reciprocal of the time constant r given in equation (30). As seen on the top half of Fig. 9, this is indeed found to be the case. From here, we find that the time constant for 710 is z,,~ = 1.69 x lo4 s and for 760 r,60 = 0.59 x lo4 s, respectively. Since in either case, of all the parameters entering into r only 01is not known from independent measurement, but only guessed from the data of Mecking and Kocks on AI, we treat it as adjustable and calculate a values of 5.96 x lO’-2 for 710°C and 9.89 x lo-’ for 76O”C, respectively. Both of these values are higher than expected at first sight, but become acceptable when realized that to obtain the normalized hardening rate, they must be m~tiplied by the ratio of (v/&)“*, which is well less than unity.

1510

ARGON and BHA’ITACHARYA:

P 710-C. 0 760-c,

v =21.17MPo Q = 18.4 MPO

J

I

Time,

II

I04

I03 seconds

Fig. 9. Decrease of normalized creep strain rate as a function of time during primary creep in nickel at 710°C 21.17 MPa, and 76OC, 18.4 MPa. The upper part of the figure indicates that the overall kinetics is governed over the entire range in time by a dimensionless time function of stress and the steady state creep rate. We take for an instructive

the average of

measure

these values at di = 7.92 x 10e2. It is worth emphasizing that the agreement demonstrated here is for the diffusion controlled range of primary creep for times larger than lo2 s, and does not cover an earlier portion of rapid transient plastic extension, which is nearly as large as the total primary creep extension up to steady state deformation. In this portion where accurate data sampling is difficult, it is nevertheless clear that very substantial

Time,

PRIMARY

CREEP IN NICKEL

departures occur from the behavior characterized by equations (24) and (25). This must be born in mind for the discussion below on the additional transients during primary creep when the stress is suddently changed. The primary creep law is based on the evolution of internal deformation resistance s. Therefore, the strain rate should also at any time be related to the deformation resistance by satisfying the kinetic law given by equation (9). This can be verified by comparing the deformation resistance predicted from equation (9) with the actual measurements. This is demonstrated in Fig. 10, where the evolution of the normalized room temperature Vickers hardness (VHN/VHN*) and normalized shear deformation resistance (S/S*) obtained from tension yield experiments are plotted as a function of dimensionless time 0, during primary creep experiments, at 760°C and 18.4 MPa. The predicted evolution of s/s * according to equation (9) is shown by the diamonds. These predicted values lie closer to the values measured by the tension experiment than the hardness experiment. This might be expected since the hardness experiment perturbs the actual deformed state in a more major way than the tension re-yield experiment. 5.2. Stress jump transients during primary creep We interpret the basic differential equation (24) for the strain rate and its associated evolution of the deformation resistance on which it was built and represented by equation (9) as the master behavior pattern for primary creep. Figure 1l(a) and (b) give a schematic representation of the normalized behavior pattern y/v* and s/s* as a function of dimen-

seconds

OY

II ’

S/S* I.0

‘:’ /-

-

3

2

T=760C c = 16 4MPa

-

~fi+~~2, z+Y.$j

f 0

0.2

0.4 0.6 8. Dimensionless

0.6

1.0

Time

Fig. 10. Comparison of the evolution of the reference hardness and shear resistance in nickel in primary creep, with theoretical expectations.

0

I

1

62

61

63

e

Fig. Il. Schematic representation of decrease in normalized creep rate and associated evolution of normalized plastic shear resistance as a function of dimensionless time. The response of the material to a stress jump and stress decrease in primary creep are indicated by path 2 and path 3, respectively.

ARGON

and BHATTACHARYA:

PRIMARY

1511

CREEP IN NICKEL Time.

seconds

004-

$

0.03 -

6 b 8

0.02 -

F T =76OC Time,

L OO

I 0.5

I 1.0 Time,

1 1.5

I 20

25x10’

seconds

Fig. 12. Change in primary creep strain in response to a stress jump at 760°C in nickel.

sionless time 0. In an uninterrupted test, as s/s* increases, j/j* steadily decreases. Since the behavior is fully normalized, it should also represent the transients that would occur upon stress jumps or stress removals that are part of the diffusion controlled deformation, above the fast nondiffusive, plastic transients that cannot be represented by equation (24), as we already discussed above. On Fig. 11(a) and (b), the transients can be represented as follows: Consider an uninterrupted experiment under a stress u, from 0 = 0 to B = 0,. At this time, a sudden increase in the applied stress is produced. This, according to equation (12b) produces a sudden increase in the steady state deformation resistance from sf to st, that is the eventual target for evolution for s. Since s itself, just before and immediately after the stress jump remains continuous through the jump, a sudden decrease occurs in s/s * after the upward stress jump. This sets the evolution back to an earlier point in dimensionless time to t&, for which the corresponding strain rate has jumped from d, to j2, as shown by the set of arrows in Fig. 1l(a) and (b). Naturally, the actual increase in the strain rate upon stress increase will be larger because of the accompanying rapid plastic extension. When the stress is suddenly decreased to (r3at time el, an opposite response occurs. The asymptotic resistance s* decreases suddenly from ST to sf, and under the same current level of s, the evolution is set forward in dimensionless time to t& and the new strain rate is suddenly decreased to &. Two other considerations complicate the response upon stress removal. First, if the current level of s is already larger than the asymptotic value sf, the new strain rate & will be zero until the current s drops below s:. This would be indicated in Fig. 1l(b), where the new & goes beyond the point in 0 where steady state is achieved. In this case, at this stress level or any lower Table 5. Operational

seconds

Fig. 13. Same information shown in Fig. 12, presented on a logarithmic plot, showing the division of strain into initial plastic response and later creep response in both the parent transient curve, as well as in the second transient, after the stress jump. stress level time must pass for the current s to decrease down to the level s:. When this is achieved,

the deformation goes directly to steady state. Second, if a substantial level of internal stress crl is present, a similar rest period at zero strain rate results with smaller reductions of stress, while larger reductions of stress even evoke an initial negative strain rate response. An experiment was carried out to investigate the response to a stress increase. At 760°C a sample was crept at 18.4 MPa for 1200 s, at which time, the stress was increased to 2 1.17 MPa, and a new surge in creep resulted, as shown in Fig. 12. The total shear strain at 18.4MPa, and the shear strain increment at 21.17 MPa are plotted in Fig. 13 on logarithmic coordinates. Using once again the kinks in the creep curve and its increment to separate the rapid plastic strain from the slow creep strain, we find that transition to diffusion controlled creep occurs after 90 s into the creep curve. On the basis of this separation and the steady state information from other experiments that is given in Table 5 below, the decrease of the normalized creep rate with dimensionless time f3 is plotted in Fig. 14 by the curve with open circles for the first increment at 18.4 MPa. At 1200 s, or a dimensionless time 0 = 3.55 x 10-2, the stress is suddenly increased. A rapid plastic strain increment occurs for the first 90 s, as Fig. 13 demonstrates. Upon entry into the diffusion controlled creep regime, the normalized strain rate is at 18.97. Thus, we start the new increment of creep deformation from the point in 0 = 8.9 x low3 and v/v* = 18.97 in the old curve, consider this time as 0, given in Fig. 11, and consider the increments of time determined from Fig. 13 at 21.17 MPa as to be added to 0, to account for the new passage of time. The resulting curve with solid circles for this new incremental condition is shown also in Fig. 14. The agreement of results with

data for the stress jump

experiment

at 760°C

CT,= 18.4 MPa, a,//~ = 3.82 x IO-‘, a = 7.92 x 10m2 m =4.66, i,=8.89 x lO~‘s-I, i* =8.5 x IO-‘s-l, T =3.38x 104s a,=21.17MPa, a,//~ = 4.39 x 10-4, c[ = 7.92 x IO-* m = 4.66; &, = 8.89 x lo-‘s-‘, i* = 1.63 x 10m6 s-‘, 7 = 1.27 x 104s

1.512

ARGON and BHATTACHARYA:

PRIMARY CREEP IN NICKEL 0.08,

I

I

I

1

I

T = 760C

COll>

1

#A” 01

0

I

1

I

I

I

2 Time,

I

4

3 seconds

5x10s

Fig. 15. Comparison of creep curves in polycrystalline nickel with that in a single crystal with a (111) orientation.

o Q = 16.4MPo l

1.0 I

10-3

1

I

I

Q =2l.i7MPa

,,,,,I

I

10-2 8, Dimensionless

I

I!llllj 10-l

Time

Fig. 14. Change in normalized creep rate with dimensionless time for the stress jump experiment in Figs 12 and 13. is not as good as it should be. The reason, in part can be traced to a deviation of the initial curve at 18.4 MPa stress from normal behavior, as can be seen by comparing it (curve of open circles in Fig. 14) with the curve in Fig. 9.

expectations

6. DiSCUSSION 6.1. Nature of steady state in creep

Steady state creep data in the literature is often based on the correlations existing for the point of the minimum creep rate in a tension experiment. This information can often be misleading and not relate to a structural steady state. Thus, when a creep acceleration into tertiary behavior is present, the specimen is either beginning to neck or is undergoing interna fracture processes. In either case, this can occur well before a structural steady state is attained. This has been encountered in the present experiments, as is clear from Table 4, and directly visible in Fig. 5 for the evolution of internal stress. As the figure shows, a plateau for the internal stress is not reached at the minimum creep rate. Since intergranular cavitation can occur in polycrystalline samples even in the absence of necking which accelerates the creep rate, some experiments were carried out on electron beam grown Ni single crystals, seeded to have an orientation near (111). This orientation is texturally stable, and has strain hardening behavior very similar to that of a polycrystai [28I. Figure 15 shows the uninter~pted creep curves at 750°C and 18.4MPa Mises shear stress of a

o~entation single crystal in compa~son with that of a polycrystalline sample. We see from here quite graphically that while the polycrystal reaches a minimum creep rate point rather early in either time or strain, in comparison in the (111) oriented single crystal, a true steady state appears to be reached, followed by prolonged extension at this steady state. Figure 16 shows the results of uninterrupted stress dip experiments to measure the internal stress in a single crystal in comparison with that in a polycrystal. Clearly, in a polycrystal, the internal stress rises very rapidly and at the minimum creep rate attains an internal stress level of only 11.S MPa, while in the single crystal, a much longer time is needed to attain a true steady state at which the internal stress reaches a level of 12.6 MPa. Finally, the corresponding comparison for the evolution of the Vickers hardness between polycrystal and single crystal is shown in Fig. 17, where again, the finai level of resistance reached at steady state in the single crystal is 15% higher than that in the poiycrystal at the minimum creep rate. 6.2. Internal stress The theoretical models for primary creep that were presented in Sections 4.3 and 4.4 were based on the evolution of deformation resistance as the isotropic

515,

I

1

1

I

T = 750C ui*

= II .SMPO

P-Xtat

1s-xto1 z OoIJo5 tjl

Time,

I

seconds

Fig. 16. Comparison of internal stress (Mises shear) evolution in a nickel single crystal with that in a polycrystal during primary creep. The steady state internal stress in the single crystal is larger than the one in the polycrystal at the minimum creep rate.

ARGON and BHAT’IACHARYA:

26 -

VHN* =22.3

P-Xtal

VHN* = 25.6

S-X tal

I

I I Time,

2 seconds

I 3

4x105

Fig. 17. Evolution of Vickers hardness in a nickel single crystal during primary creep. The steady state deformation resistance in the nickel single crystal is larger than that achieved in the polycrystal at the minimum creeep rate. property of the average dislocation density. For monotonic deformation and for stress jumps, this picture furnishes a very satisfactory way of accounting for primary creep. Periods of zero strain rate and eventual negative strain rates as the applied stress is decreased suggests that the internal stress is not merely a manifestation of an anisotropy in the deformation resistance, but is indeed a polarized set of stresses in the deforming body that oppose the effect of the applied stress. Although there is no single satisfactory model for the internal stress, it is likely that it arises from some deformation induced misfit, such as bowing sub-grain boundaries [29], intergranular constraints [30,31] to deformation, etc. These misfits can usually be removed by small amounts of reverse deformation, and therefore, are of consequence only in repeated stress reversals. Nevertheless, a more complete model must incorporate the internal stress and furnish an evolution law for it. 6.3. Evolution during primary creep The theoretical developments proposed in Section 4 are only in the nature of scaling laws. The form of the basic kinetic law for primary creep given by equation (9), though plausible, could not be derived from more fundamental principles beyond an appeal to an extension of the form given by equation (15), which has its basis in thermally activated glide of dislocations over slip plane obstacles. The kinetic law is not independent of the evolution law of equation (8). In our developments, it was used primarily as a kinematic “converter” in equation (8) from hardening by strain increments to hardening by time increments. In a fully developed theory, steady state tit is necessary to point out here that in earlier developments [32,33], these restrictions were not fully appreciated and the kinetic law for thermally activated deformation [equation (1S)] was offered with a constant hardening rate to predict steady state. Regretably, such developments do not lead to satisfactory primary creep laws and must be abandoned.

PRIMARY CREEP IN NICKEL

1513

should be relatable to more independently prescribed statistical processes of hardening and recovery rather than through semi-empirical measures of effective hardening rates 0. Clearly, the scaling laws have been developed only unidirectionally and are offered merely for systematic interpolation procedures between an initial condition and a steady state. The steady state expressions given by equations (12a) and (12b) were suggested merely for replacement of some mechanistic parameters with experimentally measurable rate properties. They are not offered as useful relations of predictive value. Thus, e.g. the reintroduction of equation (12b) in the kinetic law to derive a more refined steady state creep law gives equation (12a) with a decrease of the stress exponent from m down to .5m/(5 + m), if 0 were treated as a constant, which of course, it is not.t The relatively abrupt transition from dynamic recovery control of evolution in the initial fast plastic transient to static recovery control throughout the rest of primary creep and reversion back to dynamic recovery in any upward stress change indicates a delicate balance in the complementary nature of these processes. As already pointed out above, we believe this indicates that dynamically removable dislocation clusters are finite in number and can be readily exhausted during the initial phases of primary creep. In constant strain rate deformation, they are constantly replenished. Since the time-constants of the two recovery processes are quite far apart, there must be clearly differentiable features between dynamically removable cluster configurations and configurations requiring diffusion for removal of key obstacles.

7.

CONCLUSIONS

During constant strain rate deformation in the creep range and during very early phases of primary creep, dynamic recovery controls evolution of deformation resistance. This is also true whenever an upward stress jump is imposed on the creep deformation. After a short period (ca lo2 s) of fast plastic deformation under dynamic recovery control, primary creep is controlled principally by static recovery. This permits scaling of all rate processes in primary creep by those at the corresponding steady state. During primary creep, both the deformation resistance and the internal stress systematically evolve according to a Bailey-Orowan law. While this evolution reaches a stationary state at steady state creep in single crystals, it is still in progress in polycrystals at the minimum creep rate. research has been supported by the MRL Division of NSF under grant DMR-84-18718 through the Center for Materials Science and Engineering at MIT. Additional equipment support was obtained from funds from the Quentin Berg Professorship of ASA.

Acknowledgements-This

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ARGON and BHATTACHARYA:

AS4, 11 (1910). and J. E. Dorn, in

Proc. R. Sot.

2. J. E. Bird, A. K. Mukherjee

CREEP IN NICKEL

16. A. R. S. Ponter and F. A. Leckie, J. Engng Mater. Tech.

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