Impression creep of β-tin single crystals

Impression creep of β-tin single crystals

Materials Science and Engineering, 39 (1979) 1 - 10 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands 1 Impression Creep of ~-Tin Sing...

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Materials Science and Engineering, 39 (1979) 1 - 10 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

1

Impression Creep of ~-Tin Single Crystals

S. N. G. CHU and J. C. M. LI Materials Science Program, Department of Mechanical and Aerospace Sciences, University of Rochester, Rochester, N.Y. 14627 (U.S.A.) (Received August 17, 1978}

SUMMARY The impression creep o f H-tin single crystals was studied in three orientations: [001], [100] and [110]. The temperature dependence o f the steady state impression velocity showed two parallel processes. The activation enthalpy for the high temperature process was 25 - 26 kcal mo1-1 for all three orientations and showed no obvious effects o f stress. The value is comparable to that for self-diffusion, 25.6 kcal tool -1 . For the low temperature process, the activation enthalpy was stress dependent. Some values are 8.2 kcal mo1-1 for the [001] orientation at 1 6 - 20 M N m -2, 10 kcal tool -1 for [100] at 1 2 - 16 M N m -2, and 9.4 kcal mo1-1 for [110] at 1 6 - 20 M N m -2. For all orientations and all temperatures, the stress dependence o f the impression velocity obeyed a power law with stress exponents between 3.6 and 5.0, in general agreement with previously reported results from conventional creep tests. Slip lines developed around the impression were more numerous at low temperatures than at high temperatures. Pencil slip was observed at low temperatures but not at high temperatures in the [001] orientation. It is suggested that the low temperature process involves dislocation slip and the high tempera ture process involves dislocation climb.

1. INTRODUCTION Impression creep is a new creep test [ 1, 2 ] in which a cylindrical indenter with a flat end is pushed into the surface of a solid under constant load to make an impression. When the depth of the impression is recorded as a function of time, a transient stage and a steady state stage appear, just as in conven-

tional tensile or compression creep tests. For molecular crystals [1], LiF [3], and Cu-Ni single crystals [4], it was found that both the stress and temperature dependences of the steady state impressing velocity agreed with the corresponding dependences of the creep rate in conventional creep tests. Thus impression creep has been established as a convenient way of obtaining extensive creep information from a limited supply of materials. The earliest studies on the creep of H-tin were probably those of Shoji about 40 years ago, as mentioned by Tyte [5], for polycrystalline wires under constant load. Later Breen and Weertman [6] reported the tensile creep of/~-tin of two grain sizes, 0.14 and 0.037 mm, and found two activation enthalpies. They also replotted Tyte's data to demonstrate the existence of a bend in the plot of the logarithm of creep rate versus reciprocal temperature. According to Breen and Weertman, the activation enthalpy in the high temperature regime was 26 kcal mo1-1 while that in the low temperature regime was 11 kcal mo1-1. The transition temperature between the two regimes varied between 90 and 160 °C in the stress range 4.3 - 9.6 MN m -2. Later they studied single crystals [7] with [100] and [110] tensile directions and again found two activation enthalpies, 22 and 11 kcal mo1-1 at stresses of 4.5 - 6.0 MN m -2 with the transition temperatures varying between 110 and 130 °C. Still later, Weertman [8] reported compression creep tests along the [001] direction and again found 24.5 and 12 kcal mo1-1 with a transition temperature at about 100 °C at a stress of 6.3 MN m -2. These results indicate that there are two parallel processes taking place simultaneously in the creep of t~-tin but that grain boundary sliding is not one of them.

LOADING PLATFORM

LVDT ISOLATION RING

S

GUIDING SHAFT

BALL BUSHINGS DRIVING MOTOR GEAR

CHUCK PUNCH SAMPLE CONTROL T.C.

MEASUREMENT T.C. FURNACE CERAMIC SUPPORT

Fig. 1. Schematic diagram for vacuum creep apparatus.

The stress dependence of creep rate was represented by a power law with exponents ranging from 3.6 to 5.1 in Weertman's results. Some deviations from the power law were observed by Hirokawa et al. [9] for tensile creep along the [010] direction in the low stress range 0.3 - 3 MN m -2. However, the latter data have too much scatter for meaningful activation enthalpies to be extracted from them. In view of its low melting point and the ease with which large single crystals may be grown,/]-tin is an ideal system for creep studies. The purposes of doing impression creep on/]-tin were as follows: (i) to see whether impression creep also gives two activation enthalpies with proper transition temperatures; (ii) to obtain the stress dependence of the transition temperature and compare it with theoretical predictions based on the assumption of two parallel processes; (iii) to study the punch stress dependence of impressing velocity and compare it with power law creep; (iv) to find punch size effect and determine the creep mechanisms; (v) to make microscopic observations in an a t t e m p t to identify both the high and low temperature processes.

2. EXPERIMENTAL TECHNIQUES

2.1. Sample preparation Single crystals of/]-tin were prepared by vacuum zone melting. The starting material was 99.97% pure granular tin from Mallinckrodt Chemical Works. The material was zone refined under 10 -5 Torr before crystal growth. Single crystal ingots of diameter 7/8 in and of length 8 in were grown and acid-sawn into cylinders of length 1 in. The orientations of the crystals were determined by X-ray back-reflection. The surfaces of the desired orientations were cut by acid sawing. The three orientations chosen for the experiment were [001], [100] and [110]. The cut surfaces were acid-planed flat and hand polished. 2.2. Vacuum creep apparatus Figure 1 is a schematic diagram for the impression creep apparatus. The testing sample is seated on the ceramic support inside a vertical split furnace. The temperature of the furnace is controlled to better than -+ 0.3 °C by a modified time-proportioning, zero-crossover firing, temperature controller. The load on the loading platform is transmitted via a case hardened steel shaft, guided by precision ball bushings, to the punch held by a chuck at the lower end of the shaft. The impressing depth is recorded by a linear variable differential transformer (LVDT) attached to the loading platform. Two m o t o r driven screws are used, one to lower the punch onto the sample surface, and a second to raise or lower the LVDT when the position of the moving core is outside the operating range. The entire apparatus is situated inside an 18 in diameter bell jar of a CVC CV-18 high vacuum system. A vacuum of 5 X 10 -s Torr can be attained. After the test, high purity helium gas is introduced into the bell jar to reduce the cooling time. 3. EXPERIMENTAL RESULTS

3.1. Impression depth-time relationship Figure 2 shows some typical impressing depth versus time curves. It is seen that t h e y resemble conventional creep curves with both transient and steady state stages. Because of geometric constraints, no tertiary stage ap-

T , 150

,e-TIN o o

E

A

J

[110] [001] [100]

I

f

SINGLE C R Y S T A L S

I

I

200 IIEI

119.9% 2.02x 107N/m 2 137.8°C 1.59 x IO 7 N / m 2 1 3 8 . 3 e C ~

°C

I

150

I r I E I'

I00 I

]

]

]

70 I

I

I

40 I

I

I

I

I

/

10-6

IIOC

10.7 E >l C

20

40

60 TIME

F i g . 2. T y p i c a l

impressing

,

80

i

I00

120

10-8

depth-time

relations

in

~-tin single crystals. T, 200

150

o

rain

% I00

70

40

£9 z to to uJ rr Q.

10-9

10-6

jO -Io

2.0

I O -7

2.5 I/T

E

Fig. 4. Temperature dependence of impressing velocity for [ 100 ] orientation in ~-tin single crystals.

>}u .

9

3.0 x 103 , °K -~

IO-e

grain or cell structure. As reported before [ 3 ], the plastic zone has a depth comparable with the diameter of the punch. The steady state microstructure is believed to be not very much different from that developed during conventional steady state creep.

£9 z to to h, (3:: 0-

i0-9

3.2. Temperature dependence

i 0 -10

2.0

2.5 I/T

3.0 x 10 3 , OK-'

Fig. 3. Temperature dependence of impressing velocity

for [001 ] orientation in ~-tin single crystals. pears and the deformation is always stable. Thus the secondary or steady state regime can be extended as much as needed. This is one of the m a n y advantages of impressing creep testing. The cause for the transient stage is believed to be the development of a steady state plastic zone under the punch, for example a sub-

The temperature dependences of the steady state impressing velocity are shown in Figs. 3 - 5 for the three orientations of the single crystals. It is seen that a bend appears in each, in agreement with the findings ~f Weertman and coworkers [ 6 - 8]. By assuming two parallel processes [10], the data can be analyzed to obtain the high and low temperature activation enthalpies. For the former, there was no obvious stress dependence of the activation enthalpy and hence an average was obtained, utilizing data at all stress levels. The results (kcal mo1-1) are 25 for [001], 26 for [100], and 26 for [110] orientations. These values are comparable with those obtained by Weertman and coworkers [6 - 8], and also with those obtained

4

T, °C 200

150

100

I

40

70

20 IO-e

I

'

I

I

I

I

[

/

I

I

I

I

2

3

4

5

0/

/

/

/ /

I0-"

E I--

I

[] I50°C n=4.1 ix 100% n-4.1 V 60°C n-4.3 0 40°C n=4A

10-7

>-

I

9 - T I N SINGLE CRYSTAL [001] Orientation o 194°C n-4.0

i 0 -e

iO-e

3lxl > I0 "9

a_ I0-~ iO-IO I

I

5

6 7 8 9107

I

I

II

PUNCHING STRESS,

i0-'o

N/m 2

Stress dependence of impressing velocity for [001 ] orientation in ~-tin single crystals.

Fig. 6. 2.0

2.5 I/T

3.0 x I03,

3.5

°K "l

Fig. 5. Temperature dependence of impressing veloc10-5

ity for [ 110] orientation in ~-tin single crystals.

i

I

I

I

I I

I

I

I

I

2

3

/~ - T I N SINGLE CRYSTAL

for self-diffusion [11, 12] which are (kcal mo1-1) 25.6 parallel to the c axis [11, 12] and 23.3 [11] or 25.1 [12] perpendicular to the c axis. Some discussion of the high temperature creep mechanism will be presented later. For the low temperature process, the activation enthalpy was somewhat stress dependent. Some values are (kcal mol-1): 8.2 for the [001] orientation at 16 - 20 MN m -2, 10 for the [100] at 12 - 16 MN m -2, and 9.4 for the [110] at 16 - 20 MN m -2. They are again comparable with those obtained by Weertman [6 - 8] for the low temperature processes. The stress dependence of the activation enthalpy will be discussed later.

[100]

tO-6

Orientation

o 2 0 8 °C n = 3 . 6 Q 150% n=3.9 A Io0Oc n=4.4 53 °C n = 4 . 5

E - ' 1 0 -7

o

i I0_8

lO-IC

3.3. Recrystallization during creep Recrystallization sometimes occurs during creep at high temperatures or stresses. In order to avoid recrystallization at low temperatures, measurements were performed in a limited stress range. In the high temperature regime, most of the data were obtained before recrystallization occurred. At temperatures above approximately 170 °C, the impressing velocity did not seem to vary with recrys-

4

1

I

5

6 7 8 9107

I

I

II

PUNCHING S T R E S S ,

-

N/m 2

Fig. 7. Stress dependence of impressing velocity for [100] orientation in ~-tin single crystals.

tallization and hence data were obtained during recrystallization, if it commenced in the transient creep stage.

i

10-6

i

i

i I I

i

2O " 2 0 0

[

B - TIN SINGLE CRYSTAL [I I 0 ] Orientation o 203 % n=4.5 o 150°C n=4.7 / /x I 0 0 °C n = 4 . 9 / / 6 0 °C n = 5 . 0 /

1 I f 1 I ~ I I o-=7.92 MN/m 2

E - I 0 0 -- [I I0] O r i e n t u l l u l / *10 / x

7

-

-

-

/

uJ >

70

~

5 0

40

3 - 30

L) i0 7

"~

E

o~°~o~8w°~>

//

o

n~

~ , ~ 3 ° +

0_

I--

/

I O. I

°C

I 0.2

I

I I II 0.4 0.7 1.0

PUNCH RADIUS,

mm

Fig. 9. Effect of punch radius on impressing velocity in ~-tin single crystals.

10-9

,SH • &Ho-/~oI0

I0 l0

oJ

/

8 = ~'H°-~H o"

~9

7

iO-H 5

, , J iI 6 7 8 9107 PUNCHING

I 2

I 3

STRESS, N/m2

= 1.29 xlO-4m3/mole = 214 ,~3

T < 65°C

6 0

I10

I

20 PUNCHING S T R E S S ,

I

30

40 MN/m 2

Fig. 8. Stress dependence of impressing veloeity for [ 110] orientation in ~-tin single crystals.

Fig. 10. Effect of stress on activation enthalpy for [001] orientation in ~-tin single crystals.

3.4. Stress dependence The stress dependence of impressing velocity is shown in Figs. 6 - 8. The stress exponents have values between 3.6 and 5.0, again comparable with those obtained b y Weertman [6, 7]. As a function of temperature, they are generally higher for low temperatures than for high temperatures in all three orientations. However, the difference is not sufficiently large to be used for process characterization. Similar stress exponents merely indicate similar activation areas for the two dislocation processes.

3.6. Stress dependence o f activation enthalpy Breen and Weertman [6] found in their studies a stress-dependent activation enthalpy in the high temperature regime. However, since they did not analyze their data from the viewpoint of parallel processes and did not attempt any consistency correlations between stress and transition temperature, their stress dependence was somewhat uncertain. In this study, an analysis from the viewpoint of parallel processes [10] did not reveal any stress dependence of activation enthalpy in the high temperature regime. Hence an average activation enthalpy was obtained using ilaformation at all stress levels. In the low temperature regime, there was some stress dependence especially in the [001 ] direction as shown in Fig. 10. It is seen that, within the stress range studied, a linear relation is obtained which can be used to extrapolate the activation enthalpy to zero stress, giving a value of 8.8 kcal mo1-1 . The negative slope of the line is a b o u t 214 A 3. Aside from a factor which converts the punching stress into some effective shear

3.5. Punch size dependence As one of the unique features of impression creep [1] the punch size dependence of the steady state impressing velocity can serve as a way to indicate the dominant creep mechanism. Figure 9 shows the steady state impressing velocity as a function of punch radius in both low and high temperature regimes in the [100] orientation. The straight lines of slope unity indicate that the dominant mechanism is dislocation creep as opposed to bulk or surface diffusion of vacancies.

50

J

I I I ~ I /9-TIN SINGLE CRYSTALS Orientation SLOPE R(nl- n2)/(AHI-~Hz) o [001] -4.1 xlO-S°K -I -4.7xlO-S°K -t A [110] -63X10"5 °K "~ -6.0x IO-S °K -= o ~100] -6.1 xlO -5 °K'= -6.2xlO-5*K -I

2.44 T o

r

2.40 x UJ ¢r

ItY /J

LU

~ 2.36

~OlO]'

(

[T011

[too]

Z I...-

Slip Systems in a-Tin Single Crystal (BCT Structure C/a = 0.5457 )

~ 2.32

Fig. 12. Slip systems in ~-tin single crystals.

~- 2.28

I

I

6

7

I

8

II

9

,!s

i0 7

PUNCHING STRESS,

~

2s

N/m z

Fig. 11. Effect of stress on the transition temperature in ~-tin single crystals. stress in the plastic zone, such a slope is sometimes misleadingly taken as the activation strain volume (or the Burgers vector times the activation area). The correct activation strain volume can be calculated [13] from the stress exponent n. Apart from the same factor just mentioned, it is nkT/a, which for n = 4 is about 700 A 3 at 40 °C and 25 MN m -2. It is seen that this is different from the stress coefficient of activation enthalpy.

3.7. Stress dependence of transition temperature The transition temperature Tt is defined as the temperature at which the two processes produce the same impressing velocity if each is operating alone. If the two parallel processes have different stress exponents, the transition temperature will vary with stress. A t h e r m o d y n a m i c relationship can be derived [101

dTt_ do

RTt2 ( n~ -- n2 o

-

-

)

(1)

AH 2

which can be integrated to yield 1 Tt

1 -R( Tt0

nl --n2

AH-11 -

-

AH2

] ~ ( l n a - - l n o o ) (2)

This relationship can be used as a consistency check for data analysis. On plotting the reciprocal absolute transition temperature versus the logarithm of the punching stress, as shown in Fig. 11 for all three orientations, the slopes of the lines must be in agreement with the calculated values from the expression R(nl -- n2)/ (AH1 - - A H 2 ) . The agreement seems satisfactory.

3.8. Slip line observations Slip lines were observed around the impression in all three orientations at room temperature. Owing to the anisotropic nature of the /3-tin crystal, the bulged areas around the punch were not cylindrically symmetric. The family of slip systems in a ~-tin single crystal at low temperatures [14, 15] is shown in Fig. 12. There are a total of six different families of slip systems: (110) [ f l l ] , (110) [001], (100) [010], (100) [001], (101) [ i 0 1 ] and (121) [ i 0 1 ] . From the directions of the slip lines around the impression, it can be deduced that only certain slip systems are possible. For example, the slip lines in the two bulged areas around the impressions on the (100) surface are all parallel to the [001] direction, as shown in Fig. 13. The only possible slip systems producing these slip lines are ( l i 0 ) [111], (110) [001] and (010) [100]. Similarly, Figs. 14 and 15 indicate that the only_ possible slip systems operating are (110) [111] and (110) [001] on the (001) surface and (100) [010] and (110) [111] on the (110) surface. The slip line observations on the (100) and (110) surfaces are consistent with Weertman's findings for the surfaces of the bulk samples after conventional creep.

[O01]

l

--

[OlO]

Fig. 13. Slip lines developed on the (100) surface at room temperature in H-tin single crystals. However, from an X-ray topography study of H-tin single crystals Fiedler and Lang [16] observed only dislocations of Burgers vectors of [001] and 1/2 < 1 1 1 > types. Fiedler and Vagera [17] calculated the energy of dislocations using the anisotropic elasticity results of Stroh [18]. They found that the energy of dislocations of the [001] and 1/2 < 1 1 1 > types was much lower than that of other types. These findings are consistent with the slip line observations made in this study and further restrict the number of possible slip systems for each orientation. The slip lines did n o t appear at high temperatures. Figure 16 shows the bulged regions on the (100) surface around the impression formed at 195 °C and at a b o u t the same impressing velocity as that of Fig. 13 at room temperature. No obvious slip lines are discernible. Instead, sub-boundaries appear in the bulged region. Figure 17 shows the dislocation etch pits (developed during polishing with 10% HNOs) of the sub-boundary structure. Partially polygonized dislocation walls could be taken as indication of dislocation climb processes at high temperatures.

The fact that the absence of slip lines is not due to surface diffusion is shown by the following observation. After an impression creep test made at a low temperature, so that the slip lines were developed, a new impression was created on the same crystal in a high temperature test. After the second test, while the slip lines around the old impression were still visible, there were no slip lines around the new impression. This shows that had slip lines developed around the new impression, they would not have disappeared as a result of surface diffusion.

3.9. Pencil slip Body-center cubic iron [ 1 9 ] , mercury [20] and silver chloride [ 21] sometimes have corrugated slip surfaces made up of strips of slip planes b o u n d e d by a c o m m o n slip direction. On a macroscopic scale the apparent slip surface may n o t be a crystallographic plane, and is determined b y the maximum resolved shear stress. Taylor [19] was the first to model the slip b y a bundle of rods slipped along their length. Later Nye [21] named such deformation "pencil slip". In this work, pencil slip was

=

[ttO]

Fig. 14. Slip lines developed on the (001) surface at r o o m t e m p e r a t u r e in H-tin single crystals.

observed for the first time for/~-tin single crystals. For the [001] orientation, as the impression was formed during the test, the b o t t o m surface directly under the impression began to bulge o u t within a spot of the same shape and approximately the same size as the punch. The thickness of the sample was about 2 cm. Details of pencil slip will be published later.

3.10. Impressing velocity-strain rate comparisons An etch pit study of the plastic zone underneath the punch in LiF [3] showed t h a t the plastic zone depth was approximately the same as the punch diameter. The deformation at any instant thus can be regarded as the compression of a cylinder of length 2a. The q u a n t i t y v/2a, where v is the impressing velocity, can be regarded as the instantaneous

creep rate of the plastic zone. The punching stress used to reach a certain creep rate o f the plastic zone can be compared with the creep stress in a conventional unidirectional creep test for the same creep rate. It has been shown t h a t the ratio of the punching stress to the creep stress in the conventional unidirectional tests at the same creep rate is around 3.1 in succinonitrile crystals [ 1]. This value is consistent with the value of the usual hardness to yield stress ratio. In this study of H-tin single crystals, data by Weertman and coworkers [7, 8] on unidirectional tensile creep tests were used for comparison. The aforementioned ratio in the [110] orientation varied from 3.5 at 60 to 3.9 at 203 °C. Owing to the scatter of their data for the [100] orientation, no comparison was made in that direction. For the [001] orientation, a ratio of 2.8 was obtained in the temperatures below 100 °C at stresses of 6.3 MN m - 2 and in the temperature above 120 °C at a stress of 2.5 MN m -2. Their high temperature data at a stress of 6.3 MN m - 2 were n o t reliable because the stress exponent was 7.5, which seemed too high for tin. It is seen that the effective stress for compression creep in the impression creep test for H-tin is also about one-third of the punching stress, consistent with the usual hardness to yield stress ratio.

3.11. Creep mechanisms The punch size effect shown in Fig. 9 suggests that both the high and low temperature processes are due to dislocation creep rather than to bulk or surface diffusion [ 1 ]. The slip line observations and the pencil slip phenomenon confirm this suggestion. The fact that slip lines do not appear at high temperatures suggests further that the dislocation process at high temperatures is different from that at low temperatures. The absence of pencil slip at high temperatures also supports this suggestion. Since the activation enthalpy for high temperature creep is comparable to t h a t for self-diffusion, a possible creep mechanism at high temperatures is dislocation climb as opposed to slip at low temperatures. However, from the nature of the ends in Figs. 3 - 5 the low temperature process must take plac e in parallel with the high temperature process.

l [OOl] -~ [T I0"I

Fig. 15. Slip lines developed on the (110) surface at room temperature in a ~-tin single crystal.

t [001] ~ [0 I0.] Fig. 16. Absence of slip lines on the (100) surface at high temperature.

4. CONCLUSIONS

Consistent data were obtained by impression creep testing o f ~-tin single crystals in all three orientations [001], [100] and [110]. Only one crystal was needed for each orientation so t h a t sample to sample variations were avoided.

The temperature dependence of the steady state impressing velocity in the stress range 7.5 - 20 MN m -2 showed the same kind of bend in all three orientations as observed by Weertman and coworkers. From the nature of the bend, two parallel processes were assumed for each stress level. The activation enthalpy for the high temperature process (25 - 26 kcal

10

the plastic zone. Comparisons could then be made with the conventional creep results. The ratio of the punching to the creep stress in the conventional tests at the same creep rate was a b o u t 3.5 to 3.9 in the [110] and 2.8 in the [001] orientation. These values are consistent with the usual hardness to yield stress ratio.

ACKNOWLEDGMENTS

Fig. 17. Etch pits of partial dislocation walls developed at high temperatures.

This work was supported b y the U.S. Department of Energy through contract EY-76S-02-2296,001.

REFERENCES

mo1-1) agreed with that for self-diffusion; that for the low temperature process was in the range 8 - 10 kcal mo1-1. All values were comparable with those obtained b y Weertman. The stress dependence of the steady state impressing velocity showed the usual p o w e r law in all three orientations. The stress exponents were between 3.6 and 5.0, again comparable with those obtained by Weertman. The steady state impressing velocity was found to be proportional to the punch size for the same punching stress, which shows that dislocation rather than diffusional processes control the impression creep of r-tin in the stress and temperature ranges covered. Slip line observations at low temperatures indicated certain possible operating slip systems in each of the three orientations. The absence of slip lines and the appearance of partially polygonized sub-boundaries at high temperatures suggest dislocation climb as a high temperature process. Pencil slip in the [001] orientation was established b y the development at the b o t t o m surface of the crystal of a bulge of the same size and shape as the impression. The activation enthalpy for pencil slip decreased linearly with punching stress below 65 °C. The extrapolated value at zero stress was 8.8 kcal mo1-1 . By dividing the impressing velocity b y the plastic zone depth, which is taken as the punch diameter, the impressing velocity was converted into an instantaneous creep rate of

1 S. N. G. Chu and J. C. M. Li, J. Mater. Sci., 12 (1977) 220. 2 H. Y. Yu and J. C. M. Li, J. Mater. Sci., 12 (1977) 2214. 3 E. C. Yu and J. C. M. Li, Philos. Mag., 36 (1977} 811. 4 E. C. Yu, Ph.D. Thesis, University of Rochester, N.Y., U.S.A., 1977. 5 L. C. Tyte, Proc. Phys. Soc. A, 278 (1938) 153. 6 J. E. Breen and J. Weertman, J. Met., 7 (1955) 1230. 7 J. Weertman and J. E. Breen, J. Appl. Phys., 27 (1956) 1189. 8 J. Weertman, J. Appl. Phys., 28 (1957) 196. 9 T. Hirokawa, K. Yomogita and K. Honda, Proc. Int, Conf. Strength o f Metals and Alloys, Tokyo, 1967; Trans. Jpn. Inst. Met., 9 Suppl. (1968) 387. 10 J. C. M. Li, in J. C. M. Li and A. K. Mukherjee (eds), Rate Processes in Plastic Deformation o f Materials, Am. Soc. Met., Metals Park, Ohio, 1975, p. 479. 11 J. D. Meakin and E. Klokholm, Trans. Metall. Soc. AIME, 218 (1960) 463. 12 C. Coston and N. H. Nachtrieb, J. Phys. Chem., 68 (1964) 2219. 13 J. C. M. Li, Trans. Metall. Soc. AIME, 233 (1965) 219. 14 E. Schmid and W. Boas, Plasticity, Chapman and Hall, London, 1968. 15 M. Lorenz, Z. Metallkd., 59 (1968) 419. 16 R. Fiedler and A. R. Lang, J. Mater. Sci., 7 (1972) 531. 17 R. Fiedler and I. Vagera, Phys. Status Solidi A, 32 (1975) 419. 18 A. N. Stroh, Philos. Mag., 3 (1958) 625. 19 G. I. Taylor and C. F. Elam, Proc. R. Soc. London, Ser. A, 112 (1926) 337. 20 K. M. Greenland, Proc. R. Soc. London, Set. A, 163 (1937) 28. 21 J. F. Nye, Proc. R. Soc. London, Ser. A, 198 (1949) 191.