4
LETTERS TO THE EDITOR Stress dependence of dislocation velocity from stress relaxation experiments* Direct measurements,(1~2) using etch pit techniques, show that the variation of dislocation velocity with stress can be represented by the empirical relation V = (T/T&m (1) where v is the dislocation velocity, 7 the resolved shear stress, -r,, the resolved shear stress required to produce a dislocation velocity of 1 cm/see, and m a constant which is characteristic of the temperature and the material. For iron-3.25% silicon, Stein and LOW(~) found m to be 35 at 293°K. An indirect method of evaluating m was suggested by Guardc3) based on measurements of the strain rate sensitivity, such that a In i m’ = 2 ln-7 = 13 In v + amp _____ ~ a In 7 a In 7 (2) where p is the density of moving dislocations. Thus, from equation (1) m’=m+-aInp a In 7 (3) Guard found that, in single crystals of iron-3.25% silicon, m’ was 100 at small strains and between 60 and 80 for strains between 0.02 and 0.07 and suggested that the difference between m’ and m was due to a change in the number of moving dislocations when the strain rate is changed. Johnston and SteinC4) repeated Guard’s experiments and found, contrary to Guard, that m’ increased progressively with increasing strain and that when A In </A In 7 is extrapolated to zero strain, m’ =I m, is 45 which is close to the value obtained by direct measurement. These conflicting results arise almost certainly from limitations in the method. However, the indirect method has many attractions and needs further examination. An alter- native method is suggested here which avoids possible effects associated with transients in the machine and is used to elucidate the previous observations in iron-3.25 ‘A silicon alloy. When a tensile test in a hard beam machine is stopped the stress relaxes because the specimen continues to flow plastically. A typical relaxation curve is shown in Fig. 1. An equation for this curve may be derived from the usual dislocation parameters. The imposed strain rate during a constant strain test can be divided into elastic and plastic components. et = i, + i, (4) When the test is stopped i, = 0, and so where E combines the elastic modulus of the specimen and the machine. The stress on the specimen relaxes according to equation (5). From dislocation theory the plastic tensile strain rate is EII . = 0.5 bpv (6) where b is the Burgers vector of the dislocations. If we assume that v = (~/o,,)~ as found and Low in silicon iron then ED=---= cm da --Ad” at by Stein (7) 1 FIG. 1. STRAIN TIME Illustration of stress-relsxation experiment. ACTA METALLURGICA, VOL. 12, SEPTEMBER 1964 1089

Stress dependence of dislocation velocity from stress relaxation experiments

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Page 1: Stress dependence of dislocation velocity from stress relaxation experiments

LETTERS TO THE EDITOR

Stress dependence of dislocation velocity from stress relaxation experiments*

Direct measurements,(1~2) using etch pit techniques,

show that the variation of dislocation velocity with

stress can be represented by the empirical relation

V = (T/T&m (1)

where v is the dislocation velocity, 7 the resolved

shear stress, -r,, the resolved shear stress required to

produce a dislocation velocity of 1 cm/see, and m a

constant which is characteristic of the temperature

and the material. For iron-3.25% silicon, Stein and

LOW(~) found m to be 35 at 293°K. An indirect

method of evaluating m was suggested by Guardc3)

based on measurements of the strain rate sensitivity,

such that

a In i

m’ = 2 ln-7 =

13 In v + amp _____ ~ a In 7 a In 7

(2)

where p is the density of moving dislocations. Thus,

from equation (1)

m’=m+-aInp a In 7

(3)

Guard found that, in single crystals of iron-3.25%

silicon, m’ was 100 at small strains and between 60 and

80 for strains between 0.02 and 0.07 and suggested

that the difference between m’ and m was due to a

change in the number of moving dislocations when

the strain rate is changed. Johnston and SteinC4)

repeated Guard’s experiments and found, contrary to

Guard, that m’ increased progressively with increasing

strain and that when A In </A In 7 is extrapolated to

zero strain, m’ =I m, is 45 which is close to the value

obtained by direct measurement. These conflicting

results arise almost certainly from limitations in the

method. However, the indirect method has many

attractions and needs further examination. An alter-

native method is suggested here which avoids possible

effects associated with transients in the machine and

is used to elucidate the previous observations in

iron-3.25 ‘A silicon alloy. When a tensile test in a hard beam machine is

stopped the stress relaxes because the specimen

continues to flow plastically. A typical relaxation

curve is shown in Fig. 1. An equation for this curve

may be derived from the usual dislocation parameters.

The imposed strain rate during a constant strain test

can be divided into elastic and plastic components.

et = i, + i, (4)

When the test is stopped i, = 0, and so

where E combines the elastic modulus of the specimen

and the machine. The stress on the specimen relaxes

according to equation (5). From dislocation theory

the plastic tensile strain rate is

EII . = 0.5 bpv (6)

where b is the Burgers vector of the dislocations.

If we assume that v = (~/o,,)~ as found

and Low in silicon iron then

ED=---= cm

da --Ad” at

by Stein

(7)

1 FIG. 1.

STRAIN TIME

Illustration of stress-relsxation experiment.

ACTA METALLURGICA, VOL. 12, SEPTEMBER 1964 1089

Page 2: Stress dependence of dislocation velocity from stress relaxation experiments

LOG TIME kecd

FIG. 2. Log. plot of stress-relaxation curves for silicon iron single crystals at 293°K.

where

0.5 bpE A=-_

00

and is a constant if p remains constant during the

relaxation.

Integrating

1 T m CT-~ = At + K

where h’ is the integration constant.

Taking logarithms,

(1 - m) In o = In (At + K)(m - 1)

therefore,

1 In G = (1 _ nz) In (Bt + C)

where I? = A(nr: - 1) and C = K(m - l), and

1 Ino=-- In l+it +

i 1

1 __- In C.

(1 -ml (1 - m)

When t = 0, o = of and

(1 -m)lno,=lnC

Therefore

1 ln (r == -~

(1 - m) In 1 + d t + In Us

( 1 (8)

Thus, plotting In G against In t should yield a straight

line of slope I/( 1 - m), when (B/C)t 3 1. Feltham(5) suggested that the stress-relaxation

curve could be represented by the empirical equation

~~-n=&ln(l+rct) (9)

where X, a are constants. This has been found to fit

the curves obtained in the present experiments very

well. It can be seen, however, that Feltham’s equation

can be derived from equation (8) by substituting

and assuming

for small values of Au/a,. Equation (8) then becomes

Au__“’ In l+g, i 1 (1-m) \

(10)

which is equivalent to equation (9) where

(m-l)z2+

1090 SCTA METSLLURGICA, VOL. 12, 1964

(11)

To test the validity of this approach and apply it

to comparable materials to those used by previous

workers experiments have been carried out on poly-

crystal and single crystal specimens of iron-3.25 %

silicon alloy. All tests were made at 293°K in a hard

beam autographically recording tensile machine.

Strain rates between 6.8 x lop5 and 1.8 x low3 were

used.

A typical result of the stress-relaxation method is

shown in Fig. 2 in which In o is plotted against In t according to equation (8). From this plot m = 64

and m = 76 for E = 0.027 and E = 0.052 respectively.

The results show that the relaxation is adequately

expressed by equation (8). To compare the value of

m determined by stress-relaxation with that obtained

from strain rate changing, a single crystal with a

tensile axis at 26” from [OOl] along the symmetry axis

[OOl] - [Oil] was used. Stress-relaxation and strain

rate changes were carried out alternately. The values

of m are given in Table 1 and indicate that comparable

values are given by both methods.

A more detailed series of stress-relaxation tests were

made on a second single crystal of similar orientation.

The variation of m with st’rain is given in Table 2.

TABLE 1. Comparison of m’ and m for single crystal (1)

Strain m’ E (?,/i, = 26.6) m

0.018 53 57 0.024 53 58 0.086 71 72 0.135 76 75

Page 3: Stress dependence of dislocation velocity from stress relaxation experiments

LETTERS TO THE EDITOR 1091

TABI;E 2. Variation of m with strain for crystal (2)

Strein of E kg/mm

.--__ 0.005 32.3 0.014 32.3 O.O?i 31.9 0.0<52 33.1 0.083 35.5 0.133 38.0 0.197 40.5 0.236 42.0

$78

64 64 61 76 59 80 81 85

A similar set of results for polycrystalline material

with an average grain diameter of 1.8 x 10P2 mm is

given in Table 3.

Discussion

In a recent letter@) Christian has suggested a

number of reasons why the density of mobile dislo-

cations remains effectively constant during stress

increments at constant strain associated with a change

in strain rate, as in the tests of Guard, etc. This, of

course, is a necessary assumption in the derivation of

equation (8) and the good agreement of this equation

with the experimental results justifies this assumption.

ff this conclusion is accepted then the values of m.

obtained in the present experiments are related

directly to changes in dislocation velocity with stress

and must be explained in these terms.

The criterion by which the significance of these

indirect measurements of m must be judged is whether

the macroscopic deformation processes occurring

during relaxation are controlled by the motion of

dislocations into undeformed material, since this is

the process to which “direct” values of m refer.@)

Obviously, as pointed out by Stein and Johnson, in

the hypothetical case of one dislocation moving

through a crystal in the very earliest stages of defor-

mation, the above situation exists. It is equally clear

that during the work hardening region deformation

proceeds by the movement of many dislocations

through already deformed material. Thus it does not

necessarily folIow that values of m in the earliest

stages of deformat,ion can be obtained by extrapolation

from values obtained in the work hardening region.

The deformation of single crystals of silicon-iron is

characterised by a Liiders region of deformation at

constant stress before work hardening. The Liiders

region propagates by the growth of slip lines into

undeformed material until the entire gauge length is

covered by macroscopically uniform slip. It follows

that at this stage of deformation at least part of the

strain is due to dislocations moving into undeformed

crystal. However, the vaIues of m obtained during

TABLE 3. Variation of wz with strain for a polycrystslline

specimen

Strain cr, f %/mm ,I)#

__~__---------- 0.005 57.4 55 0.014 57 89 0.030 5 7 :: 89 0.048 60.2 106 0.080 66.0 112 0.111 70.2 111 0.206 77.9 I12

Liiders propagation are higher than the valuesobtained

by Stein and Low and this can be interpreted by

supposing that there is a large contribution to the

strain rate by dislocation movement within the slip

lines. In the single crystals m remained approximately

constant throughout the Liiders propagation and

increased sharply at the onset of homogeneous work

hardening. The rise in m during work hardening

indicates that the ~ontribut,ioIl of dislocations at the

tips of advancing slip lines to the Liiders strain rate

is significant. Since the Ltiders region propagates at

a constant stress it may be assumed that8 the propa-

gation is such that a constant density and distribution

of moving dislocations is maintained wit.hin the

advancing front. If this is the case no change in 1%

is expected in the Liiders region once a stable

propagation has been established.

Thus, the sharp discontinuity in the Liiders and

work hardening values of m. and the Liiders band

observation discussed above appear to cast doubt on

the validity of the extrapolation method in Fe-Si

single crystals, at least as regards the Liiders and

work hardening regions. While the Ltiders propa-

gation is being established, however, there is

undoubtedly a greater contribution to the strain rate

from dislocations moving into undeformed material

particularly at lower strains and at zero strain the

true value of m should be obtained. It is suggested,

therefore, that extra,polat#ion may be justified in the

pre-yield region. Relaxation tests carried out at the

yield point give similar values of rta to those carried

out at larger Liiders strains.

In theexperimentalwork which led to the expression

21 = (o/gJ”, cr was taken as the externally applied

stress necessary to move the dislocations against the

frictional stress provided by the undeformed lattice.

In the work hardened material, the applied stress

required for any given velocity will be greater than

the stress required in the annealed material by an

amount CTD the effective back stress due to work hardening. The relationship

Page 4: Stress dependence of dislocation velocity from stress relaxation experiments

1092 ACTA METALLURGICA, VOL. 12, 1964

should be re-written

v= (12)

where of is the externally applied stress to give a

velocity v, and af, is the externally applied stress to

give unit velocity. Since this will apply in the indirect

measurements of m in the work hardening region,

equation (11) should be re-written

2.3 m = 1 + s (of - (Tb) (13)

If we assume that ( af - ob) is approximately COnStEd

in the work hardening region, m will vary with strain

as s varies with strain and s should be constant with

strain for constant m. This is not exactly the case,

but it is thought that a back stress argument gives a

clearer picture of why m varies with strain in work

hardening than simply using the externally applied

stress and extrapolating the artificially high values of

m to zero strain. It is tempting to explain away the

high values of m in the Liiders region in terms of the

back st’ress operating on dislocations moving within

the slip bands. Unfortunately, the back stress required

to give the correct values of m would slow down the

dislocations in the bands to such an extent that their

contrib-ution to a relaxation test would be almost zero.

It seems more likely that in a single slip band the

dislocations are moving at a wide range of velocities

due to the local variation in applied stress, and that

there is a significant contribution to the strain rate

from dislocations moving below the velocity of dislo-

cations at the tips of the slip bands.

The -polycrystalline results are very similar to the

single crystal results in that a discontinuity in m is

found between the Liiders and work hardening regions.

The results from relaxation tests close to the yield

point gave smaller values of m (Table 3), than tests

at larger Liider strains. The results at low strains are

probably due to the fact that the Liiders front is very

broad a,nd is not fully developed at these strains.

Department qf Metallurgy

The Uwiversity

Liverpool

F. W. NOBLE

D. HULL

References 1. W. G. JOHNSTON and J. J. GILMAN, J. AppZ. Phys. 30, 129

(1969). 2. D. F. STEIN and J. R. Low, ibid. 31, 362 (1960). 3. R. W. GUARD, Acta Met. 9, 163 (1961). 4. W. G. JOHNSTON and D.F. STEIN, ibid. 11, 317 (1963). 5. P. FELTHAM,J. Inst. Met. 89, 210 (1961). 6. J. W. CHRISTIAN, Acta Met. 12, 99 (1964).

* Received March 26, 1964.

Role of CuO whisker growth in the oxidation

kinetics of pure copper*

Sartell et aZ.(l) proposed recently a new oxidation

mechanism of pure copper in which CuO whisker

growth played an important role. According to this

mechanism, a change in plasticity of Cu,O at about

796°C causes the kinetics of the oxidation of copper

to change. Above 7OO”C, Cu,O is plastic, but below

this temperature growth stresses in the oxide layers

are sustained, sufficient in magnitude to extrude CuO

whiskers from the surface. When the CuO whiskers

are formed, they have many more vacancies than in

the original state, and hence it requires little thermal

activation to move the Cu,O/CuO interface into the

vacancy-rich CuO. Under these conditions, the

activation energy for diffusion of Cu+l is about

20 kcal/mole. At higher temperatures, there is no

vacancy-rich CuO into which the Cu,O/CuO interface

can move, and consequently a higher activation

energy of about 40 kcal/mole is observed. They

substantiated their interpretation by experimental

rate curves showing a change in slope at 700°C.

However, there is some evidence opposed to that.

Firstly, Lasko et aZ.t2) observed CuO whisker growth

at average whisker densities of about 106/cm2 at

800°C in oxygen. Secondly, the CuO whiskers,

according to Sartell et al., grow at the base rather

than the t’ip, but this seems unlikely in the case of

proper whiskers of oxides.(3) Inasmuch as the oxide

does not have negative surface energy, but metals

may have, growth stress must play an important role

in the growth process of the whisker.

Although decisive evidence for the growth stress

has not been obtained, strains epitaxially induced in

Cu,O films were foundt4) at lower temperatures. At

higher temperatures, above about 5OO”C, strains due

to the micro stress distribution were observed by

means of an X-ray diffraction technique, but uniform

strain due to the macro stress could not be

detected.t5)

The present experiments were undertaken to

measure quantitatively the strains in the Cu,O layer

at higher temperatures.

Copper specimens (99.999% Cu) were obtained by

cutting slices from a cold-rolled and annealed sheet

having the preferred orientation, in which the (100)

face was approximately parallel to the rolling plane.

The micro strain measurements were carried out by

means of a high-temperature X-ray diffraction

technique at temperatures of 310-76O”C, at atmos-

pheric pressure. Details of these procedures have

been described elsewhere. c5) Figure 1 shows a change

of the half-value width of Cu,O (Ill)-reflection, using