On the natural law of steady state creep

Scripta METALLURGICA

Vol.

23, pp. 1419-1424, 1989 Printed in the U.S.A.

Pergamon Press plc All rights reserved

ON THE NATURAL LAW OF STEADY STATE CREEP

M. Biberger and W. Blum Institut fiJr Werkstoffwissenschaften, Lehrstuhl I University of Erlangen - Niirnberg, D 8520 Erlangen. FRG

(Received June 6, 1989

Introduction High temperature steady state creep of pure crystalline materials m generally assumed to be controlled by climb of edge dislocations. Various models of climb controlled dislocation motion lead to the socailed "natural" creep law describing the creep rate t as a function of (normal) stress a and temperature T: t = A1 ( D G b / k T ) ( a / G ) 3

(1)

D is the coefficient of self diffusion, G is the shear modulus, b is the length of the Burgers vector, k the Boltzmann constant and A1 a constant of order 0.5 (1/. However, the natural creep law seems to be in conflict with experiment (see e.g. 1,2). Frost and Ashby (3) have collected data for the stress exponent of the steady state creep rate of a large number of crystalline materials. They conclude that materials which obey equation (1) "are the exception rather than the rule .... Over a limited range of stress, up to roughly 10-ZG, experiments are well described by a modification of equation (1 / (Mukherjee et al. (4)) with an exponent, n, which varies from 3 to about 10:" t = A~(DGb/kT)(a/G) n (2) "None can convincingly explain the observed values of n; and the large values of the dimensionless constant A~ (up to 1015) strongly suggest that some important physical quantity is missing from the equation in its present form" (5,6 / . This is an adequate description of the present state of knowledge. It is unsatisfactory that the natural creep law does not hold in the range of medium and high stresses. It is even more disturbing that it does not even seem to be valid in the limit of low stresses, although dislocation creep is generally believed to be diffusion controlled under these circumstances. However, inspection of experimental data, as collected in (11, indicates that this point has not been conclusively clarified. In the present work we report on a detailed study of the steady state creep rate of LiF as a function of stress and temperature. The results show that the natural creep law is in fact obeyed at low stresses. Experimental Compression creep tests have been done on single crystals of LiF of high purity (less than 1 ppm cationic impurities) purchased from Dr. K. Korth (Kiel). The samples were nearly cubic in shape (7.0 ram- 7.0 mm cross section, 6.85 mm + 0.05 mm height) in order to enhance multiple slip which is a prerequisite for the establishment of a steady state subgrain structure. Compression was done in the I1001 - direction. A kinked lever arm (7) allowed to hold the compressive stress ¢r (force per average cross section) constant during creep. Usually the deviations in stress were below 1% of tile average value up to a compressive strain of 0.4. The compression and the force acting on the sample were recorded at fixed intervals of strain (< 10 -4) by a data acquisition system.

1419 0036-9748/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press plc

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The compressive strain was calculated as e = -ln(l/Io), where I and Io are the current and the initial height of the sample at test temperature, respectively. The accuracy of strain and stress determination was better than 3 . 1 0 -5 and 0.02 MPa, respectively. The compressive creep rate ~ was determined by linear regression analysis from the strain - time data pairs obtained in a certain strain interval, which was usually equal to 5 • 10-4. The temperature was held constant within 4-0.25 K. The material parameters necessary to calculate the constants A1 and A2 from equations (1) and (2) were taken

from Frost and Ashby (3) asD=7.4.10-am2s-lexp(-214kJmol-1/kT),G=45.8GPa melting temperature Tm= 1140 K, b = 0.284 nm.

( 1 - 0 . 8 \ [T-3OOK~ rM 22,

Results Figure 1 shows creep curves in tile log ~ - e - representation as a flmction of stress at temperatures of 673 K and 1113 K, corresponding to 0.59 and 0.98 of the melting temperature. The slope Idlogt/d~l is a measure of the rate at which the crystals work harden. The rate of work hardening is large in the beginning of the creep test and decreases continuously, as steady state creep is approached. In the strain interval between 0.3 and 0.4 the decrease of i with e is rather small: at 43 MPa and 0.28 MPa t decreases by a factor of 1.7 and 1.2, respectively. There are two reasons for this decrease: Firstly, tllere may still be some work hardening. This is probably the case for tile highest stresses, where tile densest dislocation arrangement is built up in the crystals. Secondly, friction between the compression faces and tile specimen causes some apparent hardening at constant nominal stress ,as the area of the compression faces grows with strain. This is the reason why t decreases continuously even at low stresses, where the strain range of marked work hardening, i.e. the range of primary creep, is smaller than 0.03. While friction has a noticeable effect on the slope of the log ~ - • - curve, its influence on the absolute magnitude of (: is not too pronounced. This has been shown for lbolyerystals of AI by directly comparing the results from compression tests and tensile tests (8). In order to avoid a systematic error due to friction the stress dependence of the creep rate of LiF has been evaluated at a fixed strain of 0.40. At this strain work hardening has (nearly) ceased. Therefore we designate the creep rate at 0.40 as steady state creep rate. Note that these steady state creep rates are free from systematic errors due to plastic instability by necking. This is a significant advantage of testing in compression as compared to tension. Fignre 2 shows steady state t - (r - data in a double logarithmic plot for two temperatures. In both cases the data are well described by a straight line, in agreement with equation (2). At 923 K the stress exponent n is found to be 4,5 and the constant A2 is 1.12.106 (fig. 2a). Brown and Ashby (6) determined n = 4.5 and A2 = 3.4 - 106 from the results of Yu and Li (9) for LiF single crystals at the same temperature. The difference in A2 means that the steady state creep rates measured by Yu and Li at a given stress are by a factor of 3 larger than ours. Comparing the measured t - data one finds that this difference is found only in a limited stress range around 2 MPa. At 0.7 MPa our data are in close agreement with those of Yu and Li. Therefore the difference in the A2 - values must not be taken too seriously. At 673 K one determines from fig. 2b n = 6.6 and A2 = 6.75- 1013. Brown and Ashby (6) reported n = 6.6 and A2 = 2.6.1013, as derived from the experimental data of Cropper and Langdon (10) for polycrystals of LiF. The difference in the values of A2 by a factor of 2.6 means that the steady state creep rate for polycrystals is 2.6 times smaller than for the single crystals at the same applied stress. For a detailed comparison between single and polycrystals the shear strain rates "~ have to be compared at the same shear stress 7". For (100) - oriented single crystals the ratio M = ~/~ = cr/r equals 2; for polyerystals, on the other hand, M is the Taylor factor, which may take values between 2 and 3, depending on texture. Choosing M = 2.27 for the polycrystals one deduces from equation (2) that the steady state ;r(r) relation for single and polycrystals is the same. We conclude that the present data are in good agreement with the data of Cropper and Langdon (10). The fact that different stress exponents are found at different temperatures suggests that the stress exponent n in equation (2) is not constant at stresses _< 10-aG. This is in fact observed for LiF: Figure 3 shows all the steady state creep rates measured in this work. The relative scatter in ~ is usually less than about 20%. Our data agree well with those obtained by Streb and Reppich (11) from single crystals of similar purity. It is clearly seen that the stress exponent n, i.e. the slope of the curves in figure 3, decreases continuously with decreasing stress and finally reaches a value close to 3.

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STEADY STATE CREEP

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This has been checked by performing stress change experiments, an example of which is shown in figure 4. Stress changes have the advantage to allow the comparison of the creep rates for one specimen, thus reducing the scatter. The stress exponent n is determined in the way indicated in fig. 4, by interpolating the two branches at 1.1 MPa. The error in determining n is estimated to lie in the range of + 0.2 to 4- 0.3. Figure 5 shows the stress exponents determined from the stress change tests as a function of stress. The horizontal bars extend from the lower to the higher stress, the symbols are drawn at the average stress. Our results are in good agreement with those of Streb and Reppich (11) obtained at somewhat higher stresses. The other data from literature (9,10,12,13,14) included in fig. 5 show a rather large scatter, tlowever, the general trend that n increases with ¢r is confirmed. It should be mentioned that inspite of n being different in a given stress range, the t - values, which are determined by the combination of n and A2, may be similar, as it is the case for the data of Yu and Li (9) and of Cropper and Langdon (10) in comparison to ours (see above). In figure 6 the data of figure 3 are shown in a normalised form. As expected from the "Dorn - equation" (2), they all fall on a unique curve. Ilowever, as is to be expected from fig. 3, the slope of this curve is not constant at stresses below 10-3G, but decreases continously. In the limit of low stresses the data approach the straight line representing the natural creep law, equation (1), with n = 3 and A1 = 1. Discussion Ill the literature, tile values of the stress exponent n and the factor A2 are generally regarded to be constants in the "power law range" for a given material. It has been noted by Stocker and Ashby (5) and Brown and Ashby (6) that A2 and n are correlated. Fig. 7 shows the compilation of experimental data by Brown and Ashby for alkali halogenides and oxides. Similar compilations exist for metals (6). Each point in fig. 7 corresponds to a certain material. The values of n range from about 3 to 7. For instance, n = 4.5 is reported for single crystals of LiF, polycrystalline LiF is characterised by n = 6.6 (3). Figure 7 implies that different materials exhibit a different creep behaviour in the power law range. As outlined in the Introduction, this is hard to understand. IIowever, our results for LiF show that n and A2 are not constant, but depend on stress. Depending on the stress interval in which tile steady state creep rate is described by equation (2), different values of n and A2 are found (see fig. 2). The line shown in fig. 7 represents the A2 - n - relation determined from the experimental data of fig. 3. Thus the increase of n, accompanied by a steep increase in A2, is simply the result of the increasing deviation from the natural creep law with increasing stress (fig. 6): if the stress exponent n in equation (2) is raised, the factor A2 must also be raised in order to stay within the range of measured creep rates. The fact that tile A2 - n - data for single and polycrystals fall closely on the curve of fig. 7 means that the difference in the n - and A2 - values of the two materials can be attributed to the difference in the stresses where tile experiments have been done (see fig. 5). As the comparison with our data shows, there is no strong difference between single and polycrystals with respect to the steady state creep rates at a given stress. We have found that the deviation from the natural third power law is quite gradual (fig. 6) and therefore difficult to be detected. This means that tile term "power law breakdown", suggesting a kink in the steady state log t - log ~r - curve, is misleading. Rather it can be seen in the case of LiF that the PLB range, i.e. the stress range where the power law with a certain power n "breaks down", depends on the choice of n. For n = 5 one finds that the PLB range is about 8- 10-4G. For n = 4 the PLB range is near 3 . 1 0 - 4 G . For n = 3 the PLB range is lO-4G. The validity of the natural creep law in the limit of low stresses has important consequences. It means that the steady state creep rates of pure LiF carl be predicted quite accurately on the basis of the material constants D, b, and G, in the range of low stresses (fig. 6). Thus there is no need to introduce a constant back stress into equation (1) as it has been proposed by tIorita and Langdon (15) and by Derby and Ashby (16) in order to raise the power from 3 to a higher value. A further consequence of the validity of the natural creep law is that the problem of explaining the observed values of the stress exponent n and the large values of the constant A2 is reduced to the question for the causes of the deviation from the natural creep law at stresses above 10-4 G. Further work is necessary to answer this question. Observations made in transient creep tests (8,17) indicate that the processes of glide in the subgrain interior and of recovery at subgrain boundaries are coupled by internal

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stresses. The understanding of this coupling may provide the basis for understanding the deviations from the natural creep law. Tile observations made for LiF in the present work lead to the question, whether other pure materials, e.g. metals, behave in the same way. Experiments to clarify the situation for AI are underway. Conclusions 1. At high temperatures and stresses < 10-4 G the steady state creep rate is described by the natural creep law, equation (1), with a stress exponent of 3. 2. With increasing stress there is an increasing deviation from the natural creep law leading to an increase in the stress exponent n and the Dorn constant As with increasing stress, which can explain the correlation between As and n reported in the literature. Acknowledsments Thanks are due to the Deutsche Forschungsgemeinschaft for support of this work. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

W.D. Nix and B. Ilschner, in P. IIaasen, V. Gerold and G. Kostorz (eds.), Proc. 5th Int. Conf. on the Strength of Metals and Alloys, Vol. 3, p.1503, Pergamon Press, Oxford (1979) ,7. Cadek, Creep in Metallic Materials, Materials Science Monographs, Vol. 48, Elsevier, Amsterdam (1988) II.J. Frost and M.F. Ashby, Deformation Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, Pergamon Press (1982) A.K. Mukherjee, J.E. Bird and J.E. Dorn, ASM Trans. Quart., 62, 155 (1969) R.L. Stocker and M.F. Ashby, Seripta metall., 7, 115 (1973) A.M. Brown and M.F. Ashby, On the power law creep equation, Scripta metall., 14, 1297 (1980) M. Biberger, Diploma Thesis 1986, University of Erlangen - Niirnberg W. Blum, S.' Vogler, M. Biberger and A.K. Mukherjee, Stress Dependence of the Creep Rate at Constant Dislocation Structure, Mater. Sci. Engg., (1989) E.C. Yu and J.C.M. Li, Phil. Mag., 36, 811 (1977) D.R. Cropper and T.G. Langdon, Phil. Mag., 18, 1181 (1968) G. Streb and B. Reppich, Phys. Stat. Sol (a)., la, 493 (1973) C.V.S.IL Narayan Rao and A.L. Ruoff, J. appl. Phys., 43, 1437 (1972) D.R. Cropper and J.A. Pask, Phil. Mag., 27, 1105 (1973) A.L. Ruoff and C.V.S.IL Narayan Rao, J. Am. Ceram. Sot., 58, 503 (1975) Z. Horita and T.G. Langdon, in B. Wilshire and D.R.J. Owen (eds.), Proc. 2nd Int. Conf. Creep and Fracture in Engineering Materials and Structures, p.75, Pineridge Press, Swansea (1984) B. Derby and M.F. Ashby, Seripta metall., 18, 1079 (1984) M. Biberger, Current work

Vol.

23,

No,

8

STEADY

STATE

CREEP

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1424

STEADY STATE CREEP

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