Evidence for Nabarro-Herring creep in metals: fiction or reality?

Materials Science and Engineering, A 165 (1993) 133-141

133

Evidence for Nabarro-Herring creep in metals: fiction or reality? O. A . R u a n o

CENIM, C.S.L C., Av. de Gregorio del Amo 8, 28040 Madrid (Spain) J. W a d s w o r t h

Lawrence Livermore National Laboratory, Chemistry and Materials Science, P.O. Box 808, L-353, Livermore, CA 94550 (USA) J. W o l f e n s t i n e

Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92717 (USA) O. D. S h e r b y

Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 (USA) (Received June 25, 1992; in revised form December 11, 1992)

Abstract It is shown that all creep data in metals ascribed to the Nabarro-Herring (N-H) diffusional creep mechanism cannot in fact be definitively described by the N - H creep theory. Rather, the creep mechanism is associated with either HarperDorn (H-D) dislocation creep or grain boundary sliding (GBS). Specific responses are presented to work by Fiala and Langdon (F-L) who have presented opposing views to the above conclusions. Creep data for copper used by F-L to support N - H creep are shown to be incompatible with the theory, since the experimental creep rates are about three orders of magnitude higher than predicted. The copper creep data, however, can be predicted by phenomenological equations for GBS and H - D creep. It is concluded that as long as rigorous and physically sound theories for creep controlled by GBS and H - D creep are not forthcoming, researchers will continue to use, erroneously in the authors' view, the N - H diffusional creep mechanism to explain low stress creep behavior of polycrystalline solids.

I. Introduction

Nabarro-Herring (N-H) creep is the most rigorous as well as the most elegant of all creep theories developed to date [1, 2]. In the theory, creep occurs as a result of stress-directed lattice diffusion of atoms wherein grain boundaries and surfaces are considered as sources and sinks of vacancies. N - H creep is often called diffusional creep, and does not require dislocation motion. (Coble [3] introduced grain boundary diffusion as an important contributor to diffusional creep at low homologous temperature.) Over the years it has been assumed that the N - H creep mechanism is invariably the correct one to describe low stress creep. Evidence has also been compiled to suggest that the N - H model quantitatively predicts the creep rate of many metals when tests are performed at low stresses and high homologous temperatures [4, 5]. More recently, however, reanalyses of these early data [4] and analyses of other data [6-10] have led to the conclusion that there is in fact virtually no convincing evidence for 0921-5093/93/$6.00

N - H creep in metals. Furthermore, it has been concluded that all these early data could be described more appropriately by a diffusion-controlled dislocation-creep mechanism, namely Harper-Dorn (H-D) creep [11]. Both N - H and H - D creep predict Newtonian-viscous flow (i.e. the creep rate is directly proportional to stress) and lattice diffusion controls the rate of the deformation process in both cases. In a recent paper [12], Fiala and Langdon (F-L) have rejected the conclusions of Ruano et al. [7] and stated that in their own analyses a "very careful evaluation of these early data shows that these experiments were conducted under conditions where the dominant mechanism is Nabarro-Herring rather than HarperDorn creep". In their paper, F-L claim that when "accepted and unambiguous" and "widely cited" data are used then "the conclusions of Ruano et al. are in error". The purpose of this paper is to respond to the arguments of F-L, to show that their data are inconsistent with their interpretation, and to reaffirm, with yet more © 1993 - Elsevier Sequoia. All rights reserved

134

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Evidencefor Nabarro-Herring creep in metals

evidence, that H - D dislocation creep is the probable dominant deformation mechanism at low stresses (where Newtonian viscous flow is observed) rather than the diffusional creep processes of either Nabarro-Herring or Coble.

10 "4

I

I

Copper

o D • a v

~ t3

2. Evaluation of the F-L analyses on copper 10 "7

g=14~-~ ~ k T ] ~

(1)

where g is the creep rate, D E is the lattice diffusion coefficient (DE= 2.9 X 10 -13 m 2 s-~ at 1040 °C [14]), d is the true grain size ( d = 1 . 7 7 6 £ [15]), E is the dynamic unrelaxed polycrystalline Young's modulus, b is the Burgers' vector (b = 2.56 x 10 -1° m for copper), k is Boltzmann's constant, o/E is the modulus-compensated stress and T is the absolute temperature. The broken line in Fig. 1 is the predicted curve from the N - H diffusional creep theory (eqn. (1)). As can be seen, the N - H creep rate is 400-2000 (i.e. about three orders of magnitude) slower than the actual creep rates. This is direct proof that the creep behavior of the copper tested by Pines and Sirenko [13] is not controlled by N - H creep. The conclusion of F-L on evidence for N - H creep in copper, therefore, cannot be correct. The key to an understanding of the creep behavior of the copper tested by Pines and Sirenko lies, inter-

%--

T = 1313 K (1040°C) t~ = 0.245 MPa

10 -5

10 -6

The major accepted difference between N - H and H - D creep, as noted by F-L, is the effect of grain size on the creep rate for each deformation mechanism. F-L selectively illustrated this difference by using the extensive creep data of Pines and Sirenko [13] for copper tested at 1040 °C (TITm = 0.97 where Tm is the absolute melting temperature). In Fig. 1 the F-L plot of creep rate for copper as a function of Burgers' vectorcompensated linear intercept grain size Lib (their Fig. 3) is reproduced (£ is the mean linear intercept grain size). F-L concluded that the strong dependence of the creep rate on grain size from 35 to 2300/~m (£/b from about 105 to 9x106) (Fig. 1), where the grain size exponent is -1.7, is evidence for diffusional creep since the N - H creep theory predicts a slope of - 2 . 0 [1, 2]. At grain sizes greater than 2300/am, no grain size effect is observed. F-L associate this coarse-grainsize region with H - D creep since no intrinsic grain size effect is generally included in the H - D equation. What F-L did not do, surprisingly, was to compare the actual experimental creep rates with those predicted by the N - H theory for the stress and temperature conditions used by Pines and Sirenko [13]. That prediction can be made readily from the N - H creep relation

I

(L/b)crit

S-I

\

10 "8

\

\

Powder metal strip Rolled strip Rolled sheet Wire Single or bicrystal

v

\

\

\

\

10 -9

\

10 -10

\

\

\

\

\\

N-H PREDICTION

\ 10-11

\ 10 -12

\

I 105

106

10 7

10 8

10 9

L/b Fig. 1. Influence of normalized grain size on the creep rate of copper at 1313 K (data of Pines and Sirenko [13]), reproduced from Fiala and Langdon [12]).

estingly, in the Langdon-Mohamed grain-size/stress deformation mechanism map developed by Ruano et al. [7] and reproduced in the F-L paper [12] (their Fig. 1). At the stress level used by Pines and Sirenko (o = 0.245 MPa or modulus-compensated stress o/E=3.9x 10-6), the principal deformation mechanism is predicted to be grain boundary sliding (GBS). It is therefore appropriate to analyze the data of Pines and Sirenko on the assumption that GBS and H - D creep are the two independent deformation mechanisms contributing to the creep of copper at o = 0.245 MPa. The phenomenological relations for H - D creep and GBS were used to predict the copper creep data of Pines and Sirenko. The H - D creep relation was taken from Wu and Sherby [6] and is given as eqn. (4) in this paper. The GBS creep relation is taken from Sherby and Wadsworth [16] and is given as gGBS= 2 x 10 9 ~

(2)

The predictions for GBS and H - D creep for copper are shown and compared with the Pines and Sirenko data in Fig. 2. The predicted curves, given by solid

O. A. Ruano et al. 10 4

-

-

F

T

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Evidence for Nabarro-Herring creep in metals F ~ /

__.

GBS

10 5

° = 0.245MPa

"

o/E

o

= 3 . 9 x 1 0 .6

1040°C

10 6

\ ' ~

.~ p=Z,.2xlO'°m "z

S'I 0,5

10 7

~ = 1.5 x 101°m-2

10,8

H-D CREEP

II)4

3. Evaluation of the F-L analyses on the critical grain size separating N - H from H - D creep

Copper

o

10 9

135

I

,

105

106

__

i

~

~

107

108

109

_%

L/b

Fig. 2. Prediction of the creep rate-normalized grain size data of Pines and Sirenko [13] from phenomenoiogical equations for GBS (eqn. (2)) and H-D creep (eqn. (4)). Symbols are the same as in Fig. 1.

lines, fit the experimental data quite well. This correlation gives strong support for GBS and H - D creep as the principal deformation mechanisms in this stress and temperature regime. The GBS line is within an order of magnitude of the Pines and Sirenko data on powder metallurgy strip material (given by circles) where many grains are present in the cross-section of samples, thereby allowing GBS to occur. Two lines are given for H - D creep, representing two different dislocation densities (H-D creep is accepted to be a function of the dislocation density [7, 17-23] and the specific equation used to obtain the lines is eqn. (4)). The dislocation densities p given in Fig. 2 are typical of those observed in metals where H - D creep has been identified as the deformation process [7-11, 18]. The dashed lines for H - D creep extended to the left of the GBS prediction range were drawn to indicate the predicted behavior if GBS could not occur. This is appropriate for considering tensile creep experiments involving fine wires in which the grain boundaries form a bamboo structure; in these cases GBS cannot occur since no shear stresses can exist at the grain boundaries. With regard to the Pines and Sirenko copper data, please note that the stress exponent will not be one in the fine-grain size region. This is because, according to eqn. (2), when GBS is controlling the creep process the stress exponent will be two. This prediction could not be verified because Pines and Sirenko performed studies at only a single stress (o = 0.245 MPa).

F-L criticized the calculation made by Ruano et al. in determining the critical grain size separating N - H from H - D creep. The calculation centers on a choice of the numerical constant for each creep process (AnD and A~H, as defined by F-L [12]). Ruano et al. [7] selected AHD = 1.5 x 10-10 as a typical value for analyzing creep data near the melting point and A~H = 14. The choice of these constants led to the conclusion that the critical grain size is at a value of d / b = 3 x 10 5 (i.e. d = 90 pm when b is selected to be 3 x 10- ~0 m). Since all grain sizes studied in the low stress creep experiments performed at temperatures near Tm are coarser than 90/~m, Ruano et al. concluded that H - D creep, and not N - H creep, was the rate-controlling deformation process. In order to alter the critical grain size to a coarser value than that calculated by Ruano et al., it is necessary to increase the value of A~H or decrease the value of A HDor do both. F-L made such selections, assigning A ~ n = 2 8 and A H D = I . 7 × 1 0 11 These selections yielded a value of d / b = 1.3 × 106, i.e. d = 400 pm. Their choice of constants, however, cannot be accepted for the following reasons. The value of A~H = 28 used by F-L was obtained from the relation ANH--O.7ANH ' where Ayn was assumed to be 40. But ANn = 40 is not representative of the N - H relation. F-L selected this value because Harris [24] used it, but Harris used this value in order to make experimental data fit with the N - H theory! This approach is obviously not appropriate. (Quoting from ref. 24, Harris stated: " . . . it is first necessary to assume a value for A NH"Herring derived A NH= 10 but experimental values are consistently higher for both pure metals and solid solutions typically in the range ANH = 30-50; we will assume ANH = 40".) With respect to the constant AnD, F-L selected AnD= 17 x 10 -tl, which again is not appropriate for a number of reasons. This value was based solely on creep data for aluminum. They did tabulate values of AHD for a few other metals (their Table 1), but did not use them for their selection of AnD. It must be emphasized that A Ht~is not a fundamental constant. It is accepted that A HD increases with an increase in dislocation density, as pointed out by a number of investigators [17-23]. Ruano et aL [7] recognized this structural variable, showed that it is a function of the dislocation density /9, and selected an average A ni~ value near the melting temperature for all metals studied; among these metals were those that crept faster than predicted by the N - H creep theory (r-Co, 6-Fe, Ni, Mo, Cu, Cr and Ag) and therefore were logically ascribed to H - D creep. The value of

136

O. A. Ruano et aL

/

Evidence for Nabarro-Herring creep in metals

AHDof 1.5 x 10 -10 established by this approach is an order of magnitude higher than that selected by F-L. A value of AHD = 1.5 x l0 -1° would indicate that the dislocation density of most of the metals studied in the H - D region is higher than that observed in aluminum, where AHD = 1.7 X 10 -11. This difference was considered to be reasonable; it is discussed in detail below.

1013

'

'~'""]

'

where ApL is the constant for power-law creep, Deff is the effective diffusion coefficient which incorporates contributions from both lattice and dislocation pipe diffusion, n is the stress exponent for power law creep (selected as five) and oi/E is the modulus-compensated internal stress. Equation (3) can be rewritten to include a dislocation density term, since oi/E was shown to be 0.4bp 1/2 through the Taylor relation [6, 7]. For creep experiments performed at high homologous temperatures and low stresses, Deff = D L. When these relations are substituted, eqn. (3) becomes

_a

= 0"13ApL(bp)2DL E

(4)

The validity of eqn. (4) is established by showing that the dislocation density observed in the H - D regime is equal to that predicted from eqn. (4). This is possible since five different materials have been studied in the regime where H - D creep takes place and where dislocation densities have been measured. The experimentally-observed dislocation densities are plotted in Fig. 3 as a function of the predicted dislocation densities (eqn. (4)), for A1 [18, 25], NaC1 [26], Fe-3Si [8], A1-5Mg [10], and a - Z r [7, 27]. The good correlation obtained gives strong credence to the internal stress model for H - D creep. Only aluminum is seen to

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A model has been developed [6, 7] to describe H - D creep in which dislocation density and stacking fault energy are the principal microstructural variables. The model is based on the presence of an internal stress which both assists and inhibits diffusion-controlled power-law creep. The internal stress was shown to be related to the dislocation density within grains and subgrains. When the applied stress is equal to or below the internal stress, H - D creep is achieved, i.e. Newtonian viscous flow is observed. The resulting creep rate griDis given as

i

1012

t:l.t~ aa O

4. H - D creep as influenced by dislocation density and stacking fault energy

.......

CI

p A l ( l )

108 o A l ( 2 )

107

108

109

1010

1011

1012

1013

p, m "l

PREDICTED FROM H-D CREEP

Fig. 3. The observed dislocation density in the H-D creep regime compared to the predicted dislocation density by eqn. (4). A1 ( 1) is from Mohamed and Ginter [25]; A1 (2) is from Barrett et al. [18].

deviate from the predicted relation. In this case, the measured dislocation densities from two separate investigations are seen to be three and ten times lower than is predicted by eqn. (4). The dislocation densities were determined by etch pit techniques which commonly give lower values than those observed by transmission electron microscopy [28]. By use of eqn. (4), the importance of dislocation density in determining the critical grain size that separates N - H creep from H - D creep can be established. The critical grain size is calculated by equating the H - D creep relation (eqn. (4)) with the N - H creep relation (eqn. (1)), and requires selection of a specific value of A PL. The value of A PL, however, is well established to be a function of the stacking fault energy [29-31]. In the following analyses, two values of ApL were selected, one for high stacking fault energy materials and one for low stacking fault energy materials. An ApL value of 4 x 1012 is typical of high stacking fault energy metals (this is the average value based on aluminum and the b.c.c, metals a-Fe, Mo and Cr [7]). An ApL value 1.5 x 10 l° is typical of low stacking fault energy materials (e.g. Cu). Burgers' vector was selected to be 3 x 10-l0 m. The dimensionless constant in the modulus-compensated N - H relation Eb3/kT was made equal to 72, a value previously selected by Ruano et al. [7]. Figure 4 shows the critical grain size separating H - D creep from N - H creep as a function of dislocation density and stacking fault energy. Observing the

O. A. Ruano et aL

/

high stacking fault energy line (bottom line in Fig. 4), one notes that when the dislocation density is extremely low, e.g. 106 m-2, N - H creep will dominate the creep process for grain sizes as coarse as 150000/xm (15 cm!). On the other hand, if the dislocation density is a typical high value, e.g. 10 ~2 m -2, N - H creep will not dominate the creep process until the grain size is less than 0.15/~m. The upper curve in Fig. 4 is for low stacking fault energy materials. It is observed that a decrease in stacking fault energy favors N-H creep. This effect, however, is relatively small. For example, at a dislocation density of 1012 m -2, N - H creep will not dominate until the grain size is less than 2.5 #m. The results shown in Fig. 4 indicate the futility of describing a unique critical grain size separating N - H from H - D creep, as done by F-L. Separation of the microstructural features influencing N - H and H - D creep involves a dilemma, because when a coarse grain size is achieved, for example by long-time heating at very high temperature, the dislocation density simultaneously decreases. Therefore, a decrease in creep rate is predicted for both N - H and H - D creep when the annealing time prior to testing is increased. Remarkably, the Pines and Sirenko data selected by F-L on copper (Figs. 1 and 2), together with other data on Newtonian viscous flow of copper [32, 33], allow just such an investigation of the predicted trend with annealing time prior to creep testing. Figure 5 illustrates the influence of soaking (or annealing) time on the creep rate of copper at o = 0.245 MPa and 1040 °C. The data are from Pines and Sirenko (Fig. 1) (excluding the powder metallurgy strip data in which GBS is rate controlling) and from creep tests on wires [32, 33]. The data in Fig. 5 show

that the creep rate decreases with increasing soaking or annealing time; this result is expected in H - D creep since the dislocation density will decrease. The grain size, in micrometers, is indicated beside each point. As can be seen, n o trend is evident with grain size; the fine grain size wire data studied by Udin et al. [32] and Pranatis and Pound [33] are contained within the band of coarse grain size data obtained by Pines and Sirenko. This correlation represents powerful experi= mental evidence that the dislocation density is influencing H - D creep, and refutes the existence of N - H creep. A similar argument was presented [34] in a discussion of Newtonian viscous creep of uranium dioxide.

5. Analysis of Jones' study on evidence for N - H creep The analyses presented in this paper and in earlier papers [7-10] suggest that no substantial evidence is available to support the idea that N - H creep is the dominant deformation mechanism for many metals at high homologous temperatures and low stresses. Nevertheless, it is worth considering any specific rare cases where the N - H model may appear to predict quantitatively the creep rate of certain metals. The analysis by Jones [4] shown in Fig. 6 illustrates those metals where Newtonian viscous flow was observed

10 -6

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Copper 1040°C ¢s 0.245 MPa =

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[] [1825)

k

DATAFROM

X X

SlRENKO EXCEPTWHERE INDICATED

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PINES and

(]~00}~

DOMINATES

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~, {670)

X X~

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et al

(800)

N~O~

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din

~, 'N,tU

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137

Evidence for Nabarro-Herring creep in metals

.%/ "%

DOMINATES

aPnrTl~t°iSnZ

"(~8)

(2051~

( ) Grain size in p.m

iO °

10-8

, 0

I0 ]

~

105

,

~

k

I 5

i

,

~

L

I 10

,

*

i

i 15

,I

106

107

108

109

ifll 0

l0 [ I

i(}12

1013

p, 111-2

Fig. 4. Critical grain size separating H-D creep from N-H creep as a function of dislocation density and stackingfault energy.

SOAKING TIME AT 1050°C, h Fig. 5. Influence of soaking time at 1050 °C on the creep rate of copper: data from Pines and Sirenko [13], Udin et al. [32], and Pranatis and Pound [33].

138

O. A. Ruano et aL

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Evidence for Nabarro-Herring creep in metals

25 o Cu • Cu •.o Cu o Ag

0

20

~ m Au

0 15

•

0

A Co • Y-Fe

DN-H

DTRACER 10

0 5mFe o Cr ÷ Mo

0 AtL a

AU

0A

0 aid[3A 0

o

2

3,

i

i

sb

o

P= ~DTRACER.t/lad

Fig. 6. Predicted diffusion coefficient from the N-H relation, divided by the tracer diffusion coefficient, as a function of the dimensionlessparameter P. and where N - H creep was considered to be the dominant deformation mechanism. The figure is a plot of the predicted diffusion coefficient from the N - H relation DN_n divided by the tracer diffusion coefficient Dtracer a s a function of a dimensionless parameter involving the diffusion coefficient, time of testing t and dimensions of the sample (a xl). These are the socalled "widely cited" data cited by F-L in their support of N - H creep. As can be seen, however, seven of the nine metals analyzed reveal that the N - H creep diffusion coefficients are higher than those obtained from radioactive tracer studies. These seven metals (Ni, Co, 6-Fe, Cr, Mo, Ag, and Cu) cannot be controlled by N - H creep, and their behavior was attributed to H - D creep based on an internal stress-assisted dislocation creep model [7]. Only Au and 6-Fe are seen to fall on the line where DN_H equals Dtracer. Even for the cases of 6-Fe and Au at the grain sizes studied, it is possible that the mechanism is not N - H creep but H - D creep. This possibility is strengthened if the dislocation density required to fit these creep data is a reasonable one based on the H - D relation (eqn. (4)). A dislocation density calculation was made for 6-Fe using the values of A PL' DL and E from a-Fe data [7], which is reasonable since 6- and a-Fe have the same crystal structure. By use of these values together with the low stress creep data for 6-Fe [35], a dislocation density of 8 x 10 s m - 2 w a s determined. This low value of the dislocation density is very reasonable for 6-Fe since the temperature of testing was 0.99 Tm and the diffusivity of iron in the delta region is very high because of its b.c.c, structure. Therefore H - D creep could be contributing significantly to the creep of 6-Fe. A similar calculation of the dislocation density was made for Au, utilizing the published data for power law

creep, diffusivity and modulus [36]; the A p L value for Au is 2.6 x 10 ~0. The corresponding dislocation density required in the H - D relation (eqn. (4)) to fit the viscous creep data of Alexander et aL for Au [37] is 4 x 10 l° m-2. This value, although higher by a factor of 50 than that calculated for 6-Fe, is quite reasonable for Au because of its low stacking fault energy, which makes dislocation annihilation difficult. It is thus concluded that all creep data for metals at high temperatures and low stresses, in the range where Newtonian viscous flow is observed, are readily and reasonably explained by H - D creep. Notwithstanding the present view that N - H creep has been proved not to be an important deformation mechanism in metals at low stresses, it remains as a possible mechanism under certain circumstances. As pointed out above, if the dislocation density is low and the grain size is fine, N - H creep will dominate over H - D creep (Fig. 4). Experiments with ultrafine wires (e.g. 50/zm in diameter or less) would result in a finegrained material with a bamboo structure. Such a material could exhibit creep controlled by a diffusional creep mechanism rather than by a dislocation mechanism in the Newtonian viscous region.

6. Denuded zones and particle-hardened materials Microstructural evidence in the form of denuded zones in zirconium hydride hardened magnesium alloys has been presented as proof that N - H creep dominates deformation in these materials [38]. These denuded zones appeared at transverse grain boundaries (where tension stresses were present) during tensile creep in what was believed to be the Newtonian viscous region. This is considered as proof that diffusional flow of magnesium atoms occurred from longitudinal grain boundaries (where no tensile stresses existed) to transverse grain boundaries. These results have been highly publicized as proof of the existence of N - H creep [5, 30, 39]. The observations of denuded zones in the zirconium hydride hardened magnesium alloys remain a specific piece of evidence supporting diffusional creep that requires explanation if diffusional creep is to be disregarded as the deformation mechanism in this alloy. In an analysis of these observations [40], it is stated that: "It is concluded that not only were Squires et al. correct in attributing the preferred orientation of the precipitate-free zones to directional diffusion but this process was the dominant deformation mode". Since these early observations, new evidence has been obtained which may cast doubt on the almost universal acceptance of denuded zones as compelling proof of diffusional creep. Specifically, Whittenberger [41] has

O. A. Ruano et al.

/

shown in nickel-based alloys that denuded zones appear only on some grain boundaries, and furthermore are seen on both longitudinal and transverse boundaries. These observations cannot be explained by the diffusional creep process. Wilshire [42] has also presented evidence that questions the diffusional creep process for commercial particle-hardened materials. He showed transmission electron micrographs illustrating considerable dislocation activity at grain boundaries and interiors in the Newtonian viscous region. Wilshire proposed that the power law and Newtonian viscous regions are controlled by the same dislocation creep process in a similar manner to that of the internal stress H - D model. In agreement with the present authors, Wilshire concluded that "the available experimental evidence does not justify the almost universal assumption that, at high temperatures, creep normally occurs at low stresses by stress-directed vacancy flow without dislocation movement". The dominant point being made throughout this paper is that the actual creep rate in metals is always higher than the rate predicted by N - H creep in the Newtonian viscous flow region of stresses (the only possible exceptions being 6-Fe and Au). In particlehardened metal systems, however, the observed creep rate is often lower than the creep rate predicted by the N - H model in the Newtonian viscous region [18, 42]. An example is the case of high purity aluminum containing 0.5% Fe, in which the iron is in the form of AI3Fe particles studied by Barrett et al. [18]. This polycrystalline material (grain size 300/~m) was studied from 0.95 to 0.99 Tm and at stresses where both power law creep and Newtonian viscous flow were observed. These data are plotted in Fig. 7 as diffusion-compensated creep rate as a function of modulus-compensated stress. The creep rate predicted by the N - H model is shown. The actual creep rate is 100 times lower than the rate predicted by N - H creep. A typical explanation proposed for this disagreement is that precipitates in some way inhibit the ability of grain boundaries to absorb or emit vacancies [24], and thus the diffusional creep rate is lower than that predicted by the N - H relation. Such an explanation, however, is at present too general to permit quantitative analyses. It is possible to predict the A1-Fe results based on the internal-stress-assisted power law creep model under conditions of constant structure [6]. For this condition, the relation is 1 mpL D e e f A 3 be

g=2

3

o

- -

a i

+ o-

o"i

139

Evidence for Nabarro-Herring creep in metals

(5)

In eqn. (5), it is the subgrain size or the principal barrier spacing to dislocation motion (whichever is smaller), A is a microstructure constant equal to 4, A eL

1010

. . . .

~

~ ,

,

AI - 0.5%Fe L = 300 l.tm

109

108

/

107 PREDICTED FROM N-H CREEP . . . . . .

~dDL 106 m

[

105 104 103

~ /

lO2

j

101

~

10-6

~ u

,

,

,

~

~ t~ t~ PREDICTED O"~ FROM liD ~ and POWER ~ _ L A W CREEP

,,JI

~

10-5

,

,

, , ,, 10-4

r~/E Fig. 7. Diffusion-compensated creep rate as a function of modulus-compensated stress for AI-0.5%Fe. Predictions from N-H and H-D models are shown.

is 2 × 1012 for pure aluminium and the stress exponent is eight. The solid curve shown in Fig. 7 is the predicted curve for it equal to 2.2/~m and oi/E equal to 1.15×10 -5 (which yields a dislocation density of p = 1 × 10 ~° m-2, from the Taylor relation). The values of it and p required for the prediction shown in Fig. 7 are reasonable. The value of it is in the order of the interparticle spacing cited by Barrett et al. [18] of 10/~m; the value of p is higher than that observed for aluminum but essentially equal to that observed in AI-5Mg (Fig. 3). The relatively high value of p for the particle-hardened aluminum material is reasonable since particles will inhibit annihilation of dislocations. Although the predicted curve does not follow the sharp change observed in the creep data from a stress exponent of one to a stress exponent of eight (Fig. 7), the predicted trend is encouraging.

7. Summary and conclusions Experimental evidence for N - H creep in metal-base materials appears to be non-existent. For the case of pure metals, the creep rate is almost invariably higher than that predicted by the N - H creep model (Fig. 6). For the case of particle-hardened metals, the creep rate is lower than that predicted by the N - H creep model

140

O. A. Ruano et al.

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Evidence for Nabarro-Herring creep in metals

(Fig. 7). It is concluded that all creep data for metals studied in the Newtonian viscous region are controlled by H - D dislocation creep. The principal microstructural features influencing H - D creep are the dislocation density and stacking fault energy (Fig. 4) and the barrier spacing for dislocation motion (Fig. 7). F-L have recently presented two examples that they claim to negate the conclusions of Ruano et al. regarding the lack of evidence for N - H creep. The evidence presented in the present paper, however, suggests that their statements are unwarranted. In their first example, F-L illustrate the influence of grain size on creep of copper as proof of the validity of N - H creep. In the present paper it is shown that these data cannot be associated with N - H creep since the creep rates are about three orders of magnitude higher than the N - H theory predicts (Fig. 1). Rather, these data are to be associated with the additive contribution of GBS and H - D creep to plastic flow (Fig. 2). In their second example, F-L claim that all grain sizes finer than 400/zm will be controlled by N - H creep and not by H - D creep. In this paper, however, it is shown that the critical grain size (separating N - H from H - D creep) is not fixed and depends strongly on the dislocation density and stacking fault energy (Fig. 4). A critical grain size of 400/zm can be expected only for the case of high stacking fault energy materials with a dislocation density of 3 x 108 m -2 (Fig. 4). Since dislocation densities are typically higher than 3 x 108 m -2 in the H - D creep regime (Fig. 3), the critical grain size will be finer than 400/tm. Soaking time prior to creep testing is shown to be an important variable in determination of the creep rate of copper in the Newtonian viscous region (Fig. 5). This effect is attributed to a decrease in dislocation density with increasing soaking time, which will decrease the creep rate when H - D dislocation creep is controlling the deformation process. It is suggested that N - H creep may be a rate-controlling creep process under conditions where the dislocation density is low and the grain size is fine (Fig. 4). Studies on ultrafine wires, which will lead to ultrafine grains, can help confirm the validity of N - H creep. In the case of particle-hardened materials, N - H creep theory must be modified to take account of the nature of grain boundaries as sources and sinks for vacancies if it is to become a realistic theory. The phenomenological equations developed for describing GBS (eqn. (2)) and H - D creep (eqns. (4) and (5)) are useful scientific tools in explaining and predicting the creep behavior of materials at elevated temperatures and low stresses. Theories of creep at the atomic level that closely approach these phenomenological relations are needed. As long as these theories are not forthcoming, researchers will continue to use,

in our opinion erroneously, the N - H diffusional creep mechanism to explain low stress creep behavior in polycrystalline solids.

Acknowledgments The authors gratefully acknowledge partial support of this program from the Office of Naval Research, Contract N00014-91-J-1197 and from the NATO Grant 0032/88, and for that part of the work performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48.

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