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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
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
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
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