Effects of the Peierls stress on the transition from power law creep to Harper-Dorn creep

Acta metall, mater. Vol. 43, No. 4, pp. 1415-1419, 1995

~

Pergamon

Copyright © 1995ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/95$9.50+0.00

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E F F E C T S OF THE PEIERLS STRESS O N THE T R A N S I T I O N F R O M P O W E R LAW CREEP TO H A R P E R - D O R N C R E E P J. N. WANGt:~ and T. G. NIEH Chemistry and Materials Science, L-370, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550, U.S.A. (Received 16 February 1994; in revised.form 20 July 1994)

Abstract--Using extensive data on metals, ceramics and silicates, it is demonstrated that power law dislocation creep transits to Harper-Dorn creep at the Peierls stress of a crystal. This transition at such a stress level is perhaps induced by the dislocation density being dependent upon the applied stress in power law creep but determined by the Peierls stress in Harper-Dorn creep.

1. INTRODUCTION

2. TRANSITIONFROM P-L CREEPTO H-D CREEP

It is well known that steady-state creep in coarse polycrystalline materials is determined by dislocation creep. In this case, grain size is too large for deformation processes such as grain boundary sliding or diffusional creep (i.e. Nabarro-Herring (N-H creep) creep and Coble creep [1-3]) to be dominant. There are two kinds of dislocation creep. One is power law (P-L) creep [4] with a stress exponent n generally equal to 3-5, and the other is Harper-Dorn (H-D) creep [5] with n = 1. Experiments have shown that at a fixed temperature, P-L creep is rate-controlling at higher stresses whereas H - D creep is dominant at lower stresses. Data on H - D creep has been extended recently. Now this creep has been observed not only for metals and alloys but also for ceramics and silicates. Based on experimental data on different kinds of materials, it was shown that the dislocation density in H - D creep is determined by the Peierls stress of the crystal, zp [6, 7] rather than by applied stress a, as in P-L creep. It was suggested early that the rate-controlling flow process in H - D creep in A1 may be dislocation climb under saturated conditions [8, 9]. It was showed recently that this flow process in A1 probably also applies to other materials (metals, ceramics and silicates) [10]. Based upon the fact that both experimental data and theoretical analyses showed that the dependence of dislocation density (upon a or zp) changes at a stress level of zp [7], it is inferred that the transition from P-L creep to H - D creep may occur also at this stress level. The present study is to substantiate this inference.

The stress marking the transition from P-L creep to H - D creep, a,, can be determined from experimental data of strain rate g and stress a. This may be done by making a conventional plot of log (~) against log(a) for a given material. The intersection of the line best fitting the low stress data (n = 1) and that best fitting the high stress data (n > 1) represents the a~. All materials which have shown H - D creep and their at data are listed in the left half of Table 1, in which Tm is the absolute melting temperature, d is the grain size, and E is taken as the dynamic, average, unrelaxed, polycrystalline, Young's modulus of the material. For some of the listed materials, the applied stress is noted to be relatively low to induce P-L creep, the exact transition stress must be larger than the maximum experimental stress. By definition, zp is the minimum shear stress required to cause dislocation glide in a crystal lattice. Although experimental data on zp are not available for most materials, they may be estimated approximately from the theoretical equation [61].

tFormerly with Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan. :~To whom all correspondence should be addressed.

rp ~

20 e x p ~ - ~ )

(1)

where L is the distance between the atomic planes, and q is a material constant given by 3-2v r/= 4(l--v)

(2)

where v is the Poisson's ratio. To estimate rp for a crystal, L is taken as the distance between the most closely packed atomic planes in the lattice, b as the spacing between atoms in the most closely packed direction on the most closely packed plane, and v as the average value of the dynamic, unrelaxed Poisson's ratio, as in earlier analyses [6, 7, 10]. This procedure leads to values of Zp which

1415

1.55 x 105 (1073 K) [4] 1.19 x 10~ (873 K) [4] 5.10 x IlY (823 K) [4]

5.62 x 104 (1058 K) [4]

2.03 x 104 (973 K) [4]

fl-Co

~t-Ti

NaCI

2.06 x 105 (1273 K) [13]

1.42 x 105 (1372 K ) 1.35 x 105 (1673 K) [14] 3.54 x 105 (1473 K) [15] 3.04 x 105 (1723 K) [4] 1.92 x 105 (1423 K)* 7.84x 104 (1000 K) [12] 2.78 x 105 ( 1549 K) [ 16]

TiO2

Mn05 Zn0.s Fe20~

CaTiO3

KTaO3

(Coos Mg0+s)O KZBF3

A1203

BeO

1.31 x 10~ (1573 K) [12]

CaO

MgO

MgCh 6HzO UO2

at-Zr

~t-Fe

2.01 x 105 (1573 K) [4] 2.00 x l0 s (1973 K) [4]

1.24 x 104 (554 K) [4] 2.90 x 10+ (495 K) [4]

Pb

Sn

4.27 x 104 (913 K) [4]

AI

Material

E (MPa) (at T)

1373 ~ 1473

0.61 ~ 0 . 6 6

0.87 ~ 0.99

0.87~0.98

999~1115

1464 ~ 1633

0.66~0.71

0.74

0.53

0.72 ~ 0 . 8 8

0.51 ~ 0 . 6 2

0.54

0.62 ~ 0 . 6 6

0.51

0.75 ~ 0 . 9 6

0.91

0.55

0.44

0.48

0.59 ~ 0 . 6 4

0.98

0.92 ~ 0 . 9 8

> 0.98

TIT,

1373~ 1473

1723

1473

1373 ~ 1673

7 9 5 ~ 1031

1473

1948 ~ 2073

1573

293 ~ 373

973

1058

823

873

1039~ 1127

473

554 ~ 587

> 640

T (K)

single crystal single crystal 8 . 0 ~ 12.6

10~20

25 ~ 306

28 ~ 8 9

100 ~ 1000

20 & single crystal single crystal

single crystal single crystal

single crystal 100 ~ 1000

100 ~ 4 4 3

125 ~ 342

123 ~ 4 7 8

140~ 569

2000

1500

> 3300 & single crystal

d (gm)

10 -4

m > 13 M P a

> 3.06 x

> 1 . 2 8 x 10 -4

> 2 . 6 0 x 10 +

> 1.64 x 10 -4

5.63 x 10 -5 (1423 K ) 2.22 x 10 -~ (1673 K ) > 1.95 x 19 -4

> 3 . 4 x 10 -+

2.56 x 10 + 1.28 x 10-+

1.62 x 10 -4 2.43 x 10 -4 1.00 x 10 -4

at = 4 M P a

4.09 x 10 -+

4.64 x 10 -~

9.19 x 10 s

3.56 x 10 s

3.23 x 10 -s

8.06 x 19--6

8.00 x 10 -+

3.00 x 10 -6

m/E

Beauchesne & Poirier [16] Y a m a d a [45], W a n g et al. [30]

Fryxell & Chandler [15], W a n g [41] Lessing & G o r d o n [42], W a n g [41] D i m o s & Kohlstedt [43], W a n g et al. [30] Poirier et al. [44]

Hirthe & Brittain [37], K r i s h n a m a c h a r i et al. [38], W a n g [39] Nishikawa et al. [14], W a n g [40]

Banerdt & Sammis [28] U r a i [29], W a n g et al. [30 Seltzer et al. [31], R u a n o et al. [32] Wolfenstine [33], R a m e s h et al. [34], R o u t b o r t [35] Dixon-Stubbs & Wilshire [36]

Malakondaiah & R a m o R a o [27]

N o v o t u y et al. [26]

Malakondaiah & R a m o R a o [24] Fiala et al. [25]

M o h a m e d et al. [21]

H a r p e r & D o r n [5], Barrett et at. [20], M o h a m e d et al. [21] M o h a m e d & Ginter [22], Yavari et al. [23] M o h a m e d et al. [21]

D a t a source for trt or for finding at

0.24 [121 0.24 [12] 0.25 [12] 0.29 [12] 0.29 [12] set as 0.24

set as 0.33

0,3 [12]

0.22 [12]

0.33 [12] 0.25 [12]

0.33 set as a-Zr 0.26 [12]

0.32 [65] 0.29 [65] 0.33 [65]

0.33 [641 0.33 [651

0.34 [64]

v

x/2 1.41

x/2

cubic

orthorhombic

x/2

cubic

cubic

1.36

hexagonal

1.70 x 10 -3

1.45 × 10 -3

1.45 × 10 -3

1.91 x 10 -3

2.21 x 10 -3

5.41 x 10 4

1.27 x 10 3

x/2

1.63

2.75 x 10 -3

1.75 x 10 -3

6.11 x 10 -4

1.15 x 10 -3

7.23 x 10 -s

4.93 x 10 +

4.78 x 10 -4

4.49 x 10 +

5.41 x 1 9 - 5

1.11 x 10 -~

5.12 x 10 s

5.12 x 10 s

rv/G at 0 K

1.29

hexagonal

body central tetragonal cubic

x/2

x/2

cubic

cubic

1.44

2

1.59

1.59

1.63

2

1.86

2

2

L/b

tetragonal

cubic (f.c.c.) body centered tetragonal cubic (f.c.c.) cubic (b.c.c.) close packed hexagonal close packed hexagonal cubic (f.e.c.)

cubic (f.c.c.)

Structure

Table 1. List of d a t a chosen and determined for calculating the transition stress and the Peierls stress

1.35 x 10 -4 (0.72 Tin) 1.11 × l0 -4 (0.88 7 . ) 7.85 × 10 5 (0.53 T , ) 2.40 x 10 4 (0.74 TIn) 2.14 x 1 9 - 4 (0.69 T . ) 2.91 x 10 -4 (0.93 T . ) 1.21 x 10 + (0.93 T . ) 2+08 x 10-' (0.63 T , )

3.51 x l0 -4 (0.6 Tin)

2.49 x 10 -4 (0.54 Tin)

1.73 x 10 + (0.51 Tin) 7.34 x 10 -5 (0.63 Try)

6.10 x 10 -6 (0.91 Tin)

6.90 x 10 -~ (0.55 T , )

6.82 x 10 -~ (0.61 Tin) 7.20 x 10 ~ (0.48 Tm) 8.36 x 10 (0.44 Tin)

4.00 x 10 -6 (0.09 Tm) 8.71 x 10 ~ (0.98 Tm)

4.00 x 10 -6 (0.98 T , )

r~/G at T

Z

0

>

O

t'3

O

O

-]

rn

z

> Z

.~ o~

Hot pressed: ~4.17 x 105 (1763 ~ 1603 K) [18] Sintered: .,~ 3.76 x l0 s (1763 1603 K) 119]

*Average o f E for CoO and M g O [12].

SiC

H20

fl-SiO2

• -SiO2

1.61 x 105 (1573 K ) [4] 1.44x 105 (1073 K, 1500 MPa) [121 1.44 x 105 (1073 K, 1500 M P a ) [121 6.33 × 103 (263 K) [4]

9.30 x 104 (1424 K ) [171 1.51 x 105 (1873 K ) [4]

feldspar

(Mg, F e b SiO,

7.95 x 104 (973 ~ 1173 K) [12]

CaCo3

Material

E (MPa) (at T)

1603 ~ 1763

253~271

1100 ~ 1300

9 7 3 ~ 1173

1573

1873

1288 ~ 1560

973~1173

T (K)

0.54~0.59

0.93~0.99

0.52~0.61

0.45 ~ 0.55

0.74

0.88

0.79 ~ 0 . 9 6

0.60~0.72

T/T~

3.5~ 5

fin~e c~stal

3 0 ~ 100

200

15~70

single crystal single crystal

10,5~20

d (pm)

Hotpressed: 1.90 x 10 -3 (1763 K) 2.26 x 10 -3 (1703 K) 2.88 x 10 3 (1658 K) > 3 . 9 4 x 10 J (1603 K) Sintered: 1.98 x 10 3 (1763 K) >1.64×10 3 (1603 K)

> 1.58 × 10 -3

1.81 x 10 3

2.08 x 10 3

3.91 x 10 -4

4.17 x 10 4

7.92 x 10 -4 (973 K) 6.29 x 10 4 (1073 K) 4.64 x 10 -4 (1173 K) 8.60 x 10 4

a,/E

Steineman [56], Mellor & Testa [57], M u g u r u m a [58], W a n g [39] W a n g [59], Djemel et at. [60]

W a n g et al. [54], L u a n & Paterson [55]

0.17 [66]

set as 0.2

0.25 [12]

0.08 [12]

setas 0.24 0.24 [12]

W a n g & Toriumi [47] Kohlstedt & Goetze [48], Justice et al. [49], W a n g [50] K a r a t o et aL [51], W a n g [50], W a n g [41] W a n g [52], W a n g et al. [53]

0.32 [12]

v

Schmid et al. [46], W a n g [41]

D a t a source for tr~ or for finding a,

Table 1. Continued.

hexagonal

hexagonal

hexagonal

hexagonal

orthorhombic

tricfinic

trigonal

Structure

0.82

1.02

1.10

1.10

1.27

1.15

1.13

L/b

3.91 x 10 2

1.38 x 10 2

8.41 x 10 -3

1.05 x 10 -2

3.35 x 10 -3

6.59×10 ~

6.10 x 10 -3

~p/G at 0 K

5.07 x 10 -3 (0.59 T~) 5.25 x 10 -3 (0.57 Tin) 5.39x 10 3 (0.56 T~) 5.58 x 10-3 (0.54 T~)

1.11 x 10 -3 (0.96 T~)

1.18 x 10 3 (0.55 Tin)

7.82 x 10 4 (0.60 TIn) 7.09 x 10 -4 (0.66 T , ) 6.52 x 10 4 (0.72 T . ) 5.76 x 10 -4 (0.88 T~) 2.93 x 10 -4 (0.88 T~) 3.48 x 10 -4 3.48 x 10 -4 (0.74 T~) 1.62 x 10 -3 (0.50 T,D)

~p/G at T

>

7~

70

©

O

70 rn ,-0

rn 70 t-

70 ©

z

,..]

z

-]

.:V.

va

z

O

1418

WANG and NIEH: TRANSITION FROM POWER LAW CREEP TO HARPER-DORN CREEP

represent the lower bound limits of the true Peierls stresses of crystals at T = 0 K. These estimated stresses are in reasonable agreement with most reported experimental measurements. For example, the estimated zp/G for A1 at T = 0 K is 5.12x 10-5, which agrees well with the experimentally-determined value, 3 x 10-5 [62], at T= 11 K. Due to thermal vibration, Dietze [63] estimated that the value of zp/G would decrease by a factor of ~ 13 between 0 K and the melting point. The value of %/G at a given temperature can be estimated by assuming a linear relationship between "~p/G and T. The estimated values of %/G for a number of materials showing H - D creep are included in the right-half of Table 1. Figure 1 is a plot ofa~/E vs %/G. It is clearly shown that a, is linearly proportional to %. That is, the larger the %, the larger the at. After converting the axial transition stress into shear transition stress, zt (at=~/3 rt), it is found that zt~rp. This result is consistent with that reported early [11], based on few experimental data available at that time. 3. DISCUSSION It is illustrated in the preceding section that the transition from P-L creep to H - D creep occurs at the Peierls stress of a crystal. The transition at such a stress level may be a result of the fact that the dislocation density changes from a dependence on the applied stress (P-L creep) to a dependence on the Peierls stress (H-D creep) at a stress that is equal to the Peierls stress [7]. It was pointed out that dislocation density is independent of the applied stress when the stress level is lower than the Peierls stress at which H - D creep starts to operate. The transition of P-L creep to H - D creep at the Peierls stress suggests that pure dislocation glide is impossible in H - D creep. This is supported by a recent 10-2 . . . . . . . . .

,

...........................

a-SiO2 (p) / ~'SiO2(P) I / . ~

.~o(,) l i / 1 1

plagioelase(s) N~LanI~ U/ oli,,~ (, + p) \ "/ M k lint"

t°-3

O't 10_4 E

]l[ll

"f sic (p)

ct-Z~Cp) - - - - - - - ~ _ \'c.OCs) ~x-'ri(p) ~ ~ uOz(s) Pb

/

(p)

/,~

q

~-Fe

(p)

Al(s+p)

10-6

1o"6 . . . . . . . l;-~ . . . . . . i6-' . . . . . . . i8"3 . . . . . . . io'Z rp/a

Fig. I. Dependence of the transition stress from power law creep to H-D creep on the Peierls stress in the single crystalline(s) and polycrystalline(p) materials listed in Table I. It is indicated that the transition probably occurs at the Peierls stress. (at: axial transition stress; ~t: equivalent shear transition stress.)

TEM observation in quartz [52-54]. It was observed that in the H - D creep regime in quartz, free dislocations were generally curved, long and homogeneously distributed in the interiors of original grains or subgrains. Many bowed towards the same direction. This is in contrast to that often observed in specimens deformed in the P-L creep regime, in which curved free dislocations were short whereas straight free dislocations found in some grains were long and strongly crystallographically controlled. These crystallographically controlled dislocations, suggesting the occurrence of pure glide, have not yet been observed in the H - D creep regime. At a relatively high temperature (generally> 0.5 Tin), as the grain size decreases, H - D creep which is grain size independent may transit to diffusional creep (e.g. N - H creep or Coble creep) which is grain size sensitive. It was analyzed and also illustrated using experimental data that the grain size marking the transition between these two kinds of Newtonian viscous creep is reversely proportional to the Peierls stress [67]. This result on the transition grain size and the present one on the transition stress indicate that H - D creep is dominant over other creep over wide ranges of stress and grain size in crystalline materials with high Peierls stresses. 4. CONCLUSION Examination of existing data from metals, ceramics and silicates, it is shown that power law dislocation creep transits to H - D creep at a stress that is of the same magnitude as the Peierls stress of a crystal. This transition at such a stress may be a result of the fact dislocation density becomes independent of the applied stress at stresses below the Peierls stress. Acknowledgements--J. N. Wang would like to acknowledge post-doctoral support from the Japan Society for the Promotion of Science. This work was, in part, performed under the auspices of the U.S. Department of Energy by LLNL under contract No. W-7405-Eng-48. REFERENCES

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WANG and NIEH:

TRANSITION FROM POWER LAW CREEP TO HARPER-DORN CREEP

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1419

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