Creep rate stress exponents, the friction stress (αo) and the Lagneborg particle by-passing model

Scripta

METALLURGICA

Vol. 14, pp. 1 4 9 - 1 5 4 , 1980 P r i n t e d in the U . S . A .

P e r g a m o n P r e s s Ltd. All rights reserved

CREEP RATE STRESS EXPONENTS, THE FRICTION STRESS (00) AND THE LAGNEBORG PARTICLE BY-PASSING MODEL by W. J. Evans, A. W. Beale and G. F. Harrison N.G.T.E., Farnborough, U.K. (Received

November

6,

1979)

For some time there has been much speculation about the nature of the stress dependence of creep rates in precipitation hardened alloys (1-3). One difficulty is that although steady state creep rate data, ~s' can be described by the equation es

=

A°n

.... (I)

frequently the stress exponent n changes with increasing stress (o). For instance it is typical for n to have values of 4 to 5 at low stress but to be greater than 8 at higher stresses. Such variability has complicated the formulation of theoretical models. Recently, however, many of the conceptual problems have been resolved by the development of the friction stress, o o, approach to oreep behaviour. According to this approach, the difficulties with precipitation hardened alloys can be resolved by considering that creep deformation occurs not under the full applied stress but under the effective stress (o - Oo) (4-6). On this basis it has been demonstrated that the creep rate stress dependence has the form ~s

=

A

(o - Oo) p

.... (2)

with p having a constant value of 3.5 to 4 over the entire stress/temperature regime where a range of n values is necessary if equation i is used. A value for p of this magnitude can be accounted for by models based on the recovery and work hardening behaviour of dislocation networks (7,8), ie in essence by the models used to describe creep in pure metals. The effective stress approach, therefore, encourages the view that the rate determining mechanisms are basically similar for complex precipitation hardened alloys and for pure metals. It also rationalises the observed variations in n by demonstrating that they are the result of the way in which ~o varies with applied stress in precipitation hardened alloys. Thus at low stress when n is approximately 4, Oo increases linearly with o but, when n is about 8, oo becomes almost stress independent (5-6). The reasons for these changes in o o are not fully understood although it has been suggested that they are determined by the way in which dislocations bypass precipitates (4). It is interesting, therefore, that more recently Lagneborg, working on the basis of electron microscope observations of dislocations, has derived an effective stress equation similar to Equation (2) to account for creep in precipitation hardened alloys (9-12). His friction stress is designated Op and contributes to the creep flow stress through the equation: = sGb /P0 + Op .... (3) with 0 the total dislocation density and ~, G, b having their usual meaning. In this model it is proposed that both the magnitude and behaviour of Op are functions of the particle bypassing mode. At low stresses Op is considered to be the stress required for dislocations to climb over particles and an analysis of the climb mechanism predicts that o_ should increase linearly with applied stress and that n in Equation (I) should be approximately 4. As the stress continues to increase it is shown that Op eventually reaches a level at which alternative by-passing modes such as Orowan bowing or particle shearing occur. Lagneborg considers that at this stage ~- should become independent of stress and n in Equation (i) should change to a higher valuE. Qualitatively the similarity in the response of the experimentally determined o o and the theoretically derived Op to increases in applied stress is striking. So far, though, a

149 0036-9748/80/010149-06502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n P r e s s

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quantitative comparison between the theoretical and experimental values has not been made. In the present paper the opportunity to rectify this situation has been taken. An extensive evaluation of the creep behaviour of the nickel base precipitation hardened alloy, Nimonic 90*, has been made in which both a o values and particle sizes and dispersions have been determined. Much of the information from this work has been published previously (13-15) but it was considered worthwhile to use it to affect a comparison between go and ~p. The results of this exercise are reported below. Before considering the outcome of the work in detail it is necessary to be aware of two problems associated with using Nimonic 90 data:- (I) the precipitate particles are spherical whereas Lagneborg carried out his analysis for specifically orientated cubes, (2) misfit strains are not considered in the model whereas they are relatively large in Nimonic 90 (16). On the other hand, Lagneborg considers that the model can be applied to alloys with spherical particles and appears to demonstrate for a number of alloys that the errors are relatively small (ii). On the second issue, he acknowledges the misfit limitation to the model but also uses alloys with greater misfit strains than Nimonic 90 to demonstrate its applicability. In practice it is likely that a significant factor which allows a comparison to be made is the observation of Deker et al (16) that, as far as the flow stress is concerned, misfit probably becomes unimportant for temperatures above 0.6 Tm. In essence, therefore, although it is important to be aware of these limitations in reality their effects are probably only marginal. Experimental procedure Details on composition and heat treatment of Nimonic 90 have been given previously (13,14). The precipitate particles are spherical and measurements of their size and volume fraction were made following creep testing at 1073 and I173K. In each case the specimens were rapidly cooled from the steady state stage. The measurements involved determination of linear intercepts and conversion of these to spatial values in the manner suggested by Exner (17). Although reported previously (14) the I0 per cent, average and 90 per cent particle diameters, are recorded for convenience in Table I. Creep tests were conducted at three temperatures (I123K in addition to those mentioned above) under constant stress. Progressive load reductions were carried out at these temperatures to determine o o values. The experimental procedure has been described previously (13,14). Calculation of ~p The theoretical friction stress values were obtained using a computer routine to solve numerically the Lagneborg equations (9). Essentially Lagneborg derives an equation for the friction or back stress of the form:Gb d ~2x2 1 Tback 2l dx ty-~J .... (4) with G the shear modulus, h the Burgers vector, I the interparticle spacing, y the length of dislocation that leaves the slip plane as climb occurs and x the distance the dislocation needs to advance between the particles for the climb process to be completed. The relationship between y and x has the form:-

dx

-ffff.-

-

d

os

re

sin

.

=

0

x ~ + y2) Lagneborg points out that the maximum value of ~back ie ~ / 2 occurs either when x = d/2 with d the particle diameter or when y = I~2 whichever alternatige is reached first. For the present material the y = I/2 situation was achieved first except for the highest stresses. Another limitation is that meaningful values of back stress can be derived only when arc sin TI/Gb < i. This sets up an upper limit for ~back max at each temperature.

a.

Experimental data Creep rate data and measurement of ~p.

The published work on Nimonic 90 clearly demonstrates the change in creep rate stress exponents referred to in association with Equation (I). To illustrate this secondary creep *Trade name Henry Wiggin & Co Ltd

i

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rate data at I073K have been plotted against stress in Figure I. The change in exponent is apparent with n approximately 4.5 at low stress but increasing to about 8 above the transition stress. In addition Figure i contains initial creep rate data measured immediately after application of the load. These data demonstrate an important aspect about creep in Nimonic 90 which is that the initial creep rates do not exhibit the change in stress exponent which is characteristic of secondary behaviour so that over the entire stress range considered the stress exponent is approximately 4.5. Similar differences between initial and secondary creep rates were observed for the other temperatures (14). It has also been confirmed for Nimonic 90 that the o o values change their stress dependence when the secondary creep rate stress exponents change. Typical variations in o o at 1073, 1123 and I173K are shown in Figure 2. For low stresses, over the region where n = 4.5, o o is linearly dependent on applied stress but, when n changes to 8, o o becomes independent of o. The linear dependence can be represented by the equation: °o

=

[°o] in + K~

.... (6)

The intercept ~ in is called the inherent or initial friction stress(14). Recently it has been demonstra{e~ that [o~ in is a significant creep parameter as far as the relationship between initial and secondary creep rates is concerned. Thus if at a given temperature the initial creep rates are plotted against (o - ~ i n ~ a n d secondary rates against (o - o o) then both sets of data superimpose. The data from F~gur~ 1 are used to demonstrate this point in Figure 3. On the basis of the behaviour at the three temperatures it has been suggested that an appropriate equation to describe primary and secondary creep is: *,

,c

:

A*'

o-

[ O o ] (t

p exp

U- Q~/RT]

_

....(7)

with A constant, Qc the activation energy for creep at constant ~ 0 - [00-] (t)], p a constant with a value in the range 3.5 to 4 and [ o ~ (t) representing the fu~ctional-rel~tionship between o o and time during primary and ~ec~fidary creep. b.

Calculation of op

[ Taking the precipitate data in Table 1 and the Lagneborg relationship X = d/f 2, with f the volume fraction of precipitate, values of op were determined for the climb model described by Equations (4) and (5). Both average and 90 per cent particle diameters were used in the calculations and the resultant dependence of Op on applied stress is shown in Figure 4. The 90 per cent diameters were included in the calculation to represent weaker regions of the alloy where, for instance, particle sizes and spacings are greater than average. At I173K there is little difference between the Op values for average and 90 per cent particle diameters so that only data for the former have been plotted (in fact the maximum value of Op allowed by Equation (5) is about 17 per cent lower for the 90 per cent diameters). At I073K, however, there is a significant difference between the op values for the two particle diameters as Figure 4 demonstrates. Summarising, it would appear that at low applied stress Op values are similar for both temperatures and for the two precipitate sizes at each temperature but, with increasing stress the data for the various particle conditions becomes progressively different. In fact it is noticeable that while ~p is virtually linearly dependent on oA at low stress, there is a distinct upwards curve to the data line as stress increases. If a comparison is made at constant stress, Op tends to be larger for the larger precipitate diameters but the maximum ~p allowed by the arc sin T%/Gb restriction decreases with increasing particle size. For average particle diameters the maximum Op are 278 and 68 Mn/m 2 respectively at 1073 and I173K. To illustrate how the calculated op and measured o o compare, Op data for average particle diameters at IO73K have been superlmposed on a graph containing °o values at 1073, 1123 and I173K in Figure 5. Calculated o for only one temperature have been used in order to avoid P . . . . . . a confused mass of data; IO73K was chosen as thls temperature because, wlthln the llmltatlons apparent from Figure 4, these o o values appear to give a reasonable approximation of behaviour for average particle sizes over-the whole temperature range considered here. On this basis, it is apparent that the calculated Op show somewhat large differences from the measured o o and are at variance with the extrapolations used to obtain [ O ~ i n as shown by the dotted lines. The calculations and measurements correlate best at I173K but here it must be remembered that the maximum permissible ~ is 68 MN/m at an applied stress of 85 MN/m 2 . This is only slightly greater than the stress a~ which the first °o measurement was made (77 MN/m2). Finally, it

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has been noted already that ~ tends to increase at a given stress with increasing temperature. This contrasts with the behavlour of ao where a significant decrease wit~ £ncreasing temperature is observed. Discussion It is evident from Figures 4 and 5 that the calculated friction stresses although showing the general trend do not relate closely to the measured ~o" Furthermore there is a continuous upwards curve to the stress dependence of ~p which is at variance with the experimental observations on G o. On closer examination it is apparent also that the theory fails to achieve one of its more important objectives which is the prediction of the stress levels at which there is a change in creep rate stress exponent. For instance with the I173K results the calculations based on the average particle diameters show that the maximum allowable c_ is 68 MN/m2 at an applied stress of 85 MN/m2 . At higher applied stress the theory requires t~at alternative particle by-passing modes should operate thereby causing a change in creep rate stress exponent. Both the secondary creep rate data and the o o measurements show that the change in exponent actually oecurs between 145 and 155 MN/m2 which is almost twice the value predicted by the Lagneborg model. A similar difference between prediction and measurement also exists for the data at IO73K, although here the magnitude of the discrepancy is reduced. Another point of some concern relates to the high apparent activation energies for creep in nickel base alloys that are obtained when the measurements are carried out at constant stress. A strength of the ~o approach is the fact that if the activation energies are evaluated at constant (oo - Go) this anomaly is removed and the alloys behave in a similar manner to pure metals with activation energies close to those for self-diffusion. This implies that the discrepancy occurs because determinations at constant stress make no allowance for the temperature dependence of friction stress. However an examination of Figures 4 and 5 reveals that for Nimonic 90 the ~p values cannot affect the same magnitude of temperature compensation that is obtained with G o. At low stress ~p is almost independent of temperature but at higher stresses there is a progressive deviation in which ~p becomes greater the higher the temperature. This contrasts with the measured G o in which there are significant reductions in magnitude with increasing temperature. The Lagneborg model also does not take account of the inherent friction stress term [ Go] in obtained by extrapolation of measured G 0 data to zero applied stress. Earlier it was shown that [ Oo] in can be used to affect a super-posltlon of initial and secondary creep rates by plotting the former against (~ - [ Go] in) and the latter against (~ - Go). Elsewhere evidence has been put forward to support the [ Go] in concept by demonstrating a direct llnk between [ ~ in and ~' precipitate in Nimonic 90 (14), The fact that the calculated ~p extrapolate through the origin, therefore, would seam to argue against the viability of this model in its present form and in fact Lagneborg concedes that a term such as [ Go]in should be included particularly for alloys with spherical particles. The final point at issue concerns the fact that the Lagneborg model does not account for the experimental observation that the stress exponent for initial creep rates does not change even though there is a marked transition from 4.5 to 8 for the secondary data. The stress exponent for the initial creep rates is 4.5 which, according to the model, implies that climb is the particle passing mode. The change in stress exponent for the secondary creep rates is attributable to ap becoming so large that climb gives way to an alternative by-passing mode. On this basis the lack of a ehange in exponent for initial rates must be due to ~_ not being sufficiently large for the alternative mode to occur. This implies that the mode~ requires ap to increase during primary creep. If Equation (3) correctly accounts for the flow stress there must be a simultaneous decrease in the total dislocation density during the primary stage. This is at variance with electron microscope measurements which show that the total dislocation density increases during the primary stage as Lagneborg has pointed out in a review on creep behaviour (18). Concludin ~ remarks The Lagneborg model on particle by-passing mechanisms provides a good qualitative description of secondary creep behaviour in precipitation hardened alloys. Thus it accounts for the changes in creep rate stress exponent that are observed while it also invokes the existence of a friction stress which appears to behave in a similar way to friction stresses that have been determined experimentally. However, a detailed assessment of the model shows it to be quantitatively inadequate in a number of important areas:-

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i. It does not predict sufficiently accurately the magnitude of the experimentally determined friction stresses (Oo). ii.

It does not predict the correct functional dependence of o o on applied stress.

iii. It does not predict the correct applied stress at which there is a change in creep rate stress exponent in Nimonic 90. iv. It does not account for anomalous activation energies for creep at constant stress in precipitation hardened alloys. v. It does not invoke values for the inherent friction stresses, [oo]in, that can be used to affect a direct correlation between initial and secondary creep rates. vi. It cannot account for the different stress dependences of initial and secondary creep rates observed in Nimonic 90. In essence, therefore, the model, whilst being an interesting first step towards a mathematical description which encompasses current ideas on creep, is not sufficiently comprehensive in its present form to provide an adequate description of creep in a precipitation hardened alloy such as Nimonic 90. Copyright

~

Controller HMSO London 1979

References 1. 2. 3. 4. 5. 6. 7. 8. 9. I0.

ii. 12. 13. 14. 15. 16. 17. 18.

Bo A. Wilcox and A. H. Clauer, Trans Met Soc AIME, 236, 578, (1969). K. E. Amin and J. E. Dorn, Acta Met, 17, 1429, (1969). D. Sidey and B. Wilshire, Met Sci J, 3, 56, (1969). K. R. Williams and B. Wilshire, Met Sci J, 7, 176, (1973). J. D. Parker and B. Wilshire, Met Sci J, 9, 248, (1975). W. J. Evans and G. F. Harrison, Met Sci J, iO, 307, (1976). D. Mclean, Trans Met Soc AIME, 242, 1193, (1968). P. W. Davies and B. Wilshire, Scripta Met, 5, 475$ (1971). R. Lagneborg, Scripta Met, 7, 605, (1973). R. Lagneborg and B. Bergman, Proceedings 3rd International Conference Strength of Metals and Alloys, Cambridge 316, (1973). R. Lagneborg and B. Bergman, Metal Sci J, I0, 20, (1976). B. Bergman and R. Lagneborg, Jernkontesets Ann, 155, 368, (1971). W. J. Evans and G. F. Harrison, Materials Sci and Eng, 37, 271, (1979). W. J. Evans and G. F. Harrison, Met Sci J, 13, (1979), W. J. Evans and G. F. Harrison, Scripta Met, 9, 239, (1975). R. F. Dekker and J. R. Mihalisin, Trans ASM, 62, 481, (1969). H. E. Exner, Int Met Reviews, 17, 25, (1972). R. Lagneborg, Int Met Reviews, 17, 130, (1972).

Table I Temperature

I073K

I173K ~E 600

10% of particle diameters under

0.028 ~m

0.130 ~m

50% of particle diameters under

0.056 ~m

O.194 ~m

90% of particle diameters under

0.085 ~ml 0.258 ~m

Average volume fraction f

0.29

''x /

150

.... ~

I

[

10 ~

0.32

/

10 .7

10-~ s ECS

Fig I

-

Stress dependence of initial and secondary creep rates at I073K

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CREEP RATE STRESS EXPONENTS

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400

1073

"e z ~2(X)

~o

./

IO73K O 20¢

~ o ~ °/~ i

~.~,~

OJ.~O

z

~r

~ioc ~

s~

'~

i

0~0

~

A/A-A~A~

,~

1123~

0

9

0

0

0

Yloo b

/

I

50 )0-8

I I0-5

10-?

O" M N I m 2

Fig 2

S

I 10-s

~z AND ~sSECS-, Fig 3

Stress dependence of measured friction stresses (Oo)

Superposition of initial and secondary creep rates using appropriate effective stresses

• AVERAGEPARTICLEDIAMETERSAT1073K t 90% PARTICLEDIAMETERSAT 1073K

A AVERAGEPARTICLEDIAMETERSAT1173K O'p

-I MAXIMUMPOSSIBLE

300

30(3

% (lO73~)

z

/

2O0 +

~

~

Oo

O

10o

100

MN/ mz

Fig 4

~,

Stress dependence of calculated friction

stresses,

Op

Fig 5

I

L

I

i

100

200 CF MN/mz

300

tOO

Comparison between measured o o and c a l c u l a t e d Op

I