Acta metall. Vol. 37, No. 12, pp. 3255-3262, 1989
Printed in Great Britain. All rights reserved
0001-6160/89 $3.00 + 0.00
Copyright © 1989 Pergamon Press plc
TEMPERATURE A N D RATE D E P E N D E N C E OF
SATURATION STRESS FOR LOW AMPLITUDE FATIGUE
OF Cu CRYSTALS BETWEEN 4.2 A N D 350 K
Z. S. BASINSKI and S. J. BASINSKI
Department of Materials Science and Engineering and Institute for Materials Research,
McMaster University, Hamilton, Ontario, Canada L8S 4MI
(Received 2 May 1989)
Abstract--Copper single crystals oriented for single glide were fatigued at temperatures from 4.2 to 350 K
in inert atmosphere. Fatigue behaviour was similar at all temperatures. Saturation stress decreased
smoothly with increasing fatigue temperature from ~85 MPa at 4.2 K to ~27 MPa at 350 K. The effect
of cycling rate on saturation stress was also measured. The observations suggest that the same fundamental
mechanism, which must be mechanical in nature, operates throughout the temperature interval studied.
The activation energy of the operating mechanism, calculated from the observed temperature and rate
dependence of the saturation stress, increases linearly with temperature, from ~0.175eV at 77.4 K to
~0.7 eV at room temperature. At 77.4 K (as at room temperature) fatigue life is shorter in reactive than
in inert environment.
R6sum~-Des monocristaux de cuivre orient6s pour un glissement simple sont soumis ~ des essais de
fatigue fi des temp6ratures de 4,2 ~. 350 K sous atmosph6re inerte. Le comportement en fatigue est
identique ~i toutes les temp6ratures. Lorsque la temp6rature des essais croit, la contrainte de saturation
d6croit lentement, passant de _---85MPa fi 4,2 K ~. ---27 MPa 5. 350 K. L'effet de la vitesse de cyclage sur
la eontrainte de saturation est aussi mesur& Les observations sugg6rent que le m6me m6canisme
fondamental, qui doit &re de nature m6canique, op6re dans tout l'intervalle de temp6rature &udi&
L'6nergie d'activation du m6canisme actif, calcul6e ~i partir de la temp6rature d'essai et de la sensibilit6
de la contrainte de saturation ~i la vitesse, augmente lin6airement avec la temp6rature, passant de
=0,175 eV ~i 77,4 K ~ -~0,7 eV ~ la temp6rature ambiante. A 77,4 K (comme ~, la temp6rature ambiante)
la dur6e de vie en fatigue de l'6chantillon est plus courte dans une atmosph6re r6active que dans une
atmosph6re inerte.
Zusammenfassung--Fiir Einfachgleitung orientierte Kupfer-Einkristalle wurden bei Temperaturen zwischen 4,2 und 350 K in inerter Atmosph~ire ermfidet. Bei siimtlichen Temperaturen war das Ermfidungsverhalten/ihnlich. Die Sfittigungsspannung nahm mit zunehmender Ermfidungstemperatur von ~ 85 MPa
bei 4,2 K auf ~ 27 MPa bei 350 K langsam ab. Der EinfluB der Zyklenfrequenz auf die Sfittigungsspannung wurde aul3erdem gemessen. Die Beobachtungen legen die Annahme nahe, dab derselbe Grundmechanismus, der von mechanischer Natur sein mul3, im gesamten Temperaturintervall vorliegt. Die
Aktivierungsenergie dieses Mechanismus wird aus der beobachteten Temperatur- und Ratenabh~ingigkeit
berechnet; sie nimmt linear mit der Temperatur zu, von ~0,175eV bei 77,4K auf ~0,7eV bei
Raumtemperatur. Bei 77,4 K (wie bei Raumtemperatur) ist die Ermfidungsstandzeit in reaktiver Umgebung kfirzer als in inerter.
1. INTRODUCTION
Fatigue of f.c.c, single crystals in the low amplitude
region, for which the saturation peak stress is independent of applied constant plastic strain amplitude,
has received much attention in recent years. Most of
the reported observations, however, refer to fatigue at
r o o m temperature in air, where both microstructure
and mechanical properties are well documented.
There has been no systematic study of the dependence
of single crystal fatigue properties on temperature.
Earlier work has shown that the flow stress also
saturates at 77.4 K [1, 2] and 4.2 K [1], and that both
the saturation flow stress and the length of the
presaturation region are larger the lower the fatigue
temperature. At 4.2 K [1] the saturation stress is
approximately triple and at 7 7 . 4 K [1, 2] approximately double that at room temperature, while at
425 K [3] saturation stress is lower than at r o o m
temperature. At 77.4 K as at room temperature, the
saturation stress is independent of applied constant
plastic strain amplitude over a limited range of
amplitudes.
The work reported here, which is part of a wider
study of low amplitude fatigue in copper crystals, is
concerned with the effect of temperature and rate of
cycling on the saturation flow stress, with some
comments on the influence of environment on fatigue
behaviour. An outline of the programme appears in
the proceedings of ICSMA-8 [4].
3255
3256
BASINSKI AND BASINSKI: SATURATION STRESS OF CU CRYSTALS
2. E X P E R I M E N T A L
The specimens were 99.999% pure copper crystals
oriented for single glide, square in cross section
(4 × 4 mm2), with long axis along [321] and (17~5) and
(ITi) faces. Total length was ~4.5cm, and gauge
length 1.2 cm.
The crystals were prepared for fatigue as follows:
initially all faces were smoothed using a modified
version of a method originally formulated [5] for
producing flat shiny surfaces on CuA1 alloy crystals,
now generally known as Mitchell polish. The polish
obtained with pure copper is of the same quality, but
the crystals are much softer and must therefore be
handled with caution. A saturated solution of anhydrous cupric chloride in concentrated hydrochloric
acid is poured onto a fine pure cotton (i.e. smooth
and acid-resistant) cloth stretched over a flat glass
plate. A few drops of ethylene glycol are added,
forming a golden-coloured solution, and the crystal
face gently rubbed back and forth over the cloth.
When the crystal no longer glides smoothly and/or
the cloth appears black, the polishing process can be
renewed by adding a few more drops of ethylene
glycol. From time to time all liquid should be scraped
off the plate and new solution applied. The crystal
face will soon become smooth and shiny. When
adjacent faces are polished, the solution will stain the
already-polished faces but will not affect their general
flatness. (The combination of solution and rubbing
gives a shiny finish, solution alone etches and stains
the crystal.) If required, the polished faces may be
reshined by carefully rubbing on a clean almost dry
cloth over which ethylene glycol and a few drops of
fresh solution have been spread and scraped off.
To produce suitably shaped fatigue specimens, the
cross sectional area within the gauge length was
reduced by holding the crystal, mounted in a jig to
prevent bending, against a rotating cloth-covered
wheel 30 cm in diameter which dipped into a bath of
Mitchell polishing solution. Two opposite faces were
treated this way, usually the cross glide faces. Polishing with this set-up was reproducible enough to allow
rough calibration, the required shape could be obtained by polishing for a predetermined length of
time. The method produced damage-free specimens,
avoiding the stretching which can occur in crystals
grown in shaped moulds.
The freshly polished crystals, washed with ethylene
glycol followed by a stream of distilled water, will
appear clean, but contaminants remaining on their
surfaces may interfere with subsequent chemical
treatments. Any attempt to etch with Livingston's
etchant will meet with failure, and electropolishing in
aqueous phosphoric acid will produce a dark coating
which then disintegrates, leaving an uneven surface.
Washing in dilute (5-10%) ammonia solution followed by a stream of deionised distilled water appears
to remove the Mitchell polishing contaminants, the
crystal may then be etched or electropolished without
problem. Livingston's etchant, like Mitchell's solution, hinders any subsequent attempt to electropolish,
but the same washing procedure is effective in removing the contaminants. The problem in both cases
arises because cuprous halides are almost insoluble in
water, both the chloride and bromide, however, are
soluble in ammonia, forming complexes of the type
[Cu(NH3)2]C1 [e.g. 6].
After being electropolished, the crystals were fatigued in push-pull at constant plastic strain amplitude in a fatigue machine equipped with a cryostat.
A Lakeshore controller regulated the temperature via
a sensing element positioned on the inner cryostat
can; the specimen temperature was measured by
another sensor placed in a hole in the lower grip and
contacting the specimen through a thin film of silicone grease. Strain was measured at the specimen
using a temperature-independent clip-on extensometer.
3. E X P E R I M E N T A L O B S E R V A T I O N S A N D
DISCUSSION
3. I. Effect o f temperature on mechanical behaviour
For this part of the work, all crystals were cycled
at ~ 5 Hz at a constant plastic strain amplitude of
either + 10 3 or + 2 × 10 -3 at temperatures from 4.2
to 350 K. Use of the cryostat enabled control not only
of temperature but also of fatigue environment. In
some cases the crystals were fatigued under vacuum
of 1 mPa or better, but, more often, pure helium gas
was admitted into the specimen chamber to a pressure of about one atmosphere. The exchange gas
provided an inert environment while facilitating temperature control. At 4.2 and 77.4K some crystals
were fatigued while directly immersed in liquid
helium or nitrogen.
Mechanical behaviour in the initial stage of fatigue
of virgin crystals at liquid helium temperature was
erratic, sudden drops in load made amplitude control
virtually impossible. The severity of the instability
decreased with increasing cumulative strain; after the
onset of saturation, instabilities were not observed.
As a result of this type of behaviour, crystals with
initially flat faces became irregular in shape, with
undulating surfaces, illustrated by the profile in
Fig. 1. Since fatigue curves are smoother at ~ 15 K,
the low temperature instabilities are probably enhanced by adiabatic heating effects [7]. Unstable
deformation at very low temperatures could be prevented by prefatiguing the crystals in the presaturation region at room temperature. The crystals were
usually given about half the number of cycles which
would be required to achieve saturation. This simple
and convenient treatment hardened the crystals
enough to eliminate erratic behaviour at low temperature without introducing room temperature PSBs
into the dislocation substructure. In some cases,
crystals to be fatigued at temperatures not complicated by the occurrence of mechanical instabilities
BASINSKI and BASINSKI: SATURATION STRESS OF Cu CRYSTALS
Fig. 1. Profile resulting from unstable deformation in a
crystal fatigued at 4.2 K, all faces were originally fiat. Views
from opposite edges of the cross glide face (1II); above,
adjacent to (I43); below, adjacent to (1~5). (4.2 K, _+10 -3,
225 kilocycles).
were lightly prefatigued at room temperature to
reduce the risk of extraneous damage which might
occur during sample preparation and handling the
soft virgin crystals. The use of prestrained crystals
also reduced the possibility of deformation occurring
while the cryostat was being cooled. Although the
sensitivity of the extensometer is temperature-independent, its zero drifts during cooling because of
different thermal expansion of the crystal and extensometer members, which may result in specimen
deformation when cooling under constant strain control. Zero load control, which is not easily maintained
with extremely soft specimens, was therefore used
while cooling.
Apart from instabilities at very low temperatures,
the mechanical progress of fatigue at all temperatures
studied was similar to the behaviour observed at
room temperature in air. Figure 2 is a group of
representative cyclic hardening curves showing peak
4.2
8O
14
g_
77.4
hto
40
230
I
I
I0
I02
CUMULATIVE STRAIN
I
I05
Fig. 2. Cyclic hardening curves for virgin crystals; fatigue
temperatures are shown in degrees Kelvin. Curve marked
with arrow, + 10-3, all others _+2 x 10 -3.
3257
stress per cycle as a function of cumulative strain for
virgin (i.e. not prefatigued at room temperature)
crystals fatigued at the temperatures indicated. The
stresses are the mean of those in the tensile and
compressive half cycles. The curves represent crystals
cycled at + 2 x 10 -3 except for one cycled at + 10 3
at 4.2 K, marked with an arrow. With increasing
numbers of cycles, the peak flow stress per cycle at
first increased rapidly and then more slowly, forming
the presaturation region; the crystals then entered the
long period of saturation, sometimes called the equilibrium region. For most of the crystals shown in Fig.
2, cycling was terminated when the cumulative strain
reached 400-500, but tests at 4.2 and 305 K were
continued to a cumulative strain at ~4000. For
crystals cycled at room temperature in vacuum, the
presaturation region represents only a fraction of a
percent of the total fatigue life. In inert environment
the peak stress per cycle is remarkably constant in the
saturation region for fatigue near room temperature
and at very low temperatures, but at intermediate
temperatures (e.g. at 77.4 K) there is a steady drop in
saturation stress, reminiscent of that at room temperature in air. This will be further discussed in a
following paper [8]. For fatigue at a given plastic
strain amplitude, the approach to saturation is slower
the lower the fatigue temperature. Virgin crystals
cycled at + 2 × 10 3 will reach saturation in 1-2
kilocycles at room temperature, 4-6 kilocycles at
liquid nitrogen temperature, and at 15-20 kilocycles
at liquid helium temperature. At 4.2 K, crystals cycled at + 2 x 10 -3 reach saturation much earlier than
those fatigued at half this amplitude, as shown in Fig.
2; this effect is not apparent at 77.4 K and at room
temperature. At these temperatures, much lower amplitudes are probably necessary before delayed saturation appears. A maximum in the fatigue hardening
curve was usually observed at the beginning of saturation for temperatures down to about 60 K similar
to those found by Mughrabi [9] for room temperature
fatigue in air and by Woods [10] in dilute CuAI alloys.
No maxima were observed in the protracted approach to saturation at very low temperatures.
It was shown previously [1] that for temperatures
down to 4.2 K there is a plateau region similar to the
low amplitude region at room temperature, in which
saturation stress is independent of plastic strain amplitude. In keeping with this, at all temperatures,
there was no systematic difference in saturation stress
for crystals fatigued at +_10 3 and at + 2 x 10 -3.
At all temperatures down to 4.2 K, fatigue loops
representing individual cycles followed the same progression of shapes as those observed at room temperature [e.g. 9]. The very earliest loops were almost
rectangular. With further cycling, the flow stress
increased continuously during each half cycle, so that
the loops became more pointed. Later, a flat portion,
which increased in length with further cycling, appeared at the end of each half cycle, the loops thus
slowly increased in area and eventually reassumed a
3258
BASINSKI AND BASINSKI: SATURATION STRESS OF Cu CRYSTALS
quasi-rectangular shape. At 296 K in air, the first
indication of flattening at the loop peak was correlated with appearance of the first PSB, which was
always a very fine line invisible to the naked eye [11].
One of the aims of the present work was to
establish the relationship between saturation stress
and temperature in the interval from 4.2 to 350 K. A
separate crystal was used for each constant temperature test, although this was not strictly necessary since
previous work has shown that when the fatigue
temperature is changed, saturation stress slowly
reaches a value characteristic of the current temperature, independent of thermal history. Fig. 3(a) shows
saturation stress as a function of temperature for
crystals cycled at ~ 5 Hz at constant plastic strain
amplitudes of ___ 10 .3 or + 2 x 1 0 - 3 in inert environment. The smooth curve was calculated using a cubic
least squares polynomial fit to the data. The broken
curve was obtained by correcting for temperature
dependence of the elastic modulus using Foreman's
energy factor for edge dislocations [12] and Overton
and Gaffney's elastic constants [13]. It is evident from
80
~4o
<
nL
I
I
IO0
200
TEMPERATURE ( ~
I
300
(a)
0.40
0.20
0~1
I
IO0
I
200
TEMPERATURE {K)
I
300
(b)
Fig. 3. (a) Saturation stress, %as a function of temperature.
Dashed curve, z*, saturation stress corrected for temperature dependence of the elastic modulus. (b) Rate of change
of modulus-corrected saturation stress with temperature.
the figure that saturation stress varies smoothly with
temperature over the interval studied, with no indication of any change in fatigue mechanism with temperature. The same stress-temperature information is
presented in Fig. 3(b) as the rate of change of
modulus-corrected saturation stress with temperature
calculated on the assumption that the curve in Fig.
3(a) accurately represents the data. Since for most
hardening mechanisms, flow stress is proportional to
elastic constants, the saturation stress corrected for
elastic modulus, z*, will be used when discussing the
implications of temperature and rate dependence of
the saturation stress; in particular, data in the form
shown in Fig. 3(b) are required in Section 3.3.
3.2. Effect of environment and temperature on fatigue
life
Once saturation is established, crystals fatigued at
room temperature in air inevitably show a gradual
small decrease in saturation peak stress with increasing cumulative strain, usually a decrease of < 5 %
throughout life. It has been shown [14, 15] that such
crystals contain a population of cracks which continue to grow in number and depth throughout the
saturation region until one or more reaches a critical
depth of ~ 100/~m, when failure ensues. The primary
glide path length in the crystals is ~ 5 mm, critical
cracks at each PSB face thus lead to an effective
reduction in cross-sectional area of 4%. The apparent
decline of the saturation peak stress is thus well
accounted for by crack growth and the consequent
reduction in effective crystal cross-section. No such
gradual drop in saturation stress is observed for
fatigue at room temperature in inert atmosphere, and
fatigue life is dramatically extended. For example, a
crystal cycled in vacuum at a constant plastic strain
amplitude of ___2.10 - 3 still showed a remarkably
steady saturation peak stress when the test was
terminated at 1.2 million cycles. It had thus already
endured six times the expected life of a crystal of the
same orientation fatigued at the same amplitude in
air, and still showed no sign of failure.
The observation that environment influences
fatigue life is by no means new. As early as 1932,
Gough and Sopwith [16, 17] recognised that the
fatigue limit in air may not be an appropriate standard for comparing different materials. Their experiments with commercial aircraft materials in air and
in partial vacuum indicated that environment affects
fatigue resistance, however, since "the quantitative
effect was rather obscured by the nonuniformity of
the materials" they continued their investigations
using purer samples, including annealed copper. They
were able to conclude that the detrimental agent in air
is oxygen. In 1956, Thompson, Wadsworth and
Louat [18], found that polycrystalline copper specimens contained in a polythene bag filled with dry
oxygen-free nitrogen gas had fatigue lives 5~5 times
that of similar specimens in air. In 1958, Wadsworth
and Hutchings [19] found that for copper polycrys-
BASINSK1 and BASINSKI: SATURATION STRESS OF Cu CRYSTALS
tals, fatigue life is longer the lower the partial vapour
pressure of oxygen; the effect was more marked for
aluminum, but there was no significant difference in
fatigue life for gold in air, water vapour or dry
nitrogen gas. Wang et al. [20] found fatigue life in
high vacuum to be 15-30 times longer than in air for
copper single crystals, and noted that the saturation
stress increases very late in fatigue life, beyond the
point at which the tests were terminated in the present
work. Neumann [21] and Hunsche and Neumann
[22], also found greatly extended life for Cu crystals
fatigued in a vacuum of 1 mPa and better, and, at
very high cumulative strains (>50,000), a gradual
increase in peak stress with increasing cumulative
strain, which they ascribe to dislocation cell formation.
At room temperature, crystals of the same orientation as those used here, but fatigued in air, will fail
at a cumulative strain of ~ 1600 due to the slow
growth of cracks to critical depth [14, 15]. Crystals
fatigued in vacuum to this cumulative strain would be
at the beginning of fatigue life. At comparable cumulative strains the general features of PSBs (excluding
cracks) at the surface of specimens fatigued in air and
in vacuum are indistinguishable. It is thus likely that
fatigue life and failure are dependent not on PSB
structure itself, but rather on the propagation of
cracks, and that crack growth is accelerated in air.
Wang et al. [20] and Hunsche and Neumann [22]
arrived at a similar conclusion, however, they interpret the observations in terms of crack growth being
retarded in vacuum rather than accelerated in air.
Crack growth in inert and active environment will be
illustrated in a later paper [8]. For crystals fatigued at
room temperature in vacuum in the present work, the
pressure in the cryostat was 1 mPa or better. Data
from this and other specimens indicate that a fatigue
environment of about this quality was necessary to
prevent premature failure. Neumann [21] and Hunsche and Neumann [22] note that the effects of the
environment have been eliminated in a vacuum of
1 mPa, and found no further improvement in ultrahigh vacuum.
McCammon and Rosenberg [23] noted that fatigue
life of polycrystalline copper is many times longer at
4.2 K than at room temperature. This however, may
be largely due to the influence of environment, since
at room temperature the specimens were fatigued in
air, but at 4.2 K they were immersed in liquid helium.
However, although the temperature dependence of
fatigue life was not systematically studied, the present
work has shown that both environment and temperature are determining factors. Fatigue lives in vacuo
at room temperature and at 4.2 K were so long that
it was impractical to measure them, but for crystals
cycled at liquid nitrogen temperature in inert environment, the peak stress per cycle declined throughout
the saturation region and lives were much shorter.
Thus, in the absence of reactive environment, fatigue
life expectancy is high for very low temperature,
3259
decreases at intermediate temperatures, then increases again at room temperature; possible reasons
for this will be discussed in the light of information
on PSb surface geometry at various temperatures, to
be presented in an accompanying paper [8]. Crystals
fatigued while directly immersed in liquid nitrogen
had much shorter lives than those fatigued at the
same temperature in helium gas atmosphere, and the
rate of decrease of saturation peak stress with cycling
was the fastest ever observed. The directly immersed
crystals soon showed severe cracks. This invites comparison with fatigue behaviour in vacuo and in air at
room temperature, and suggests that oxygen, inevitably present in liquid nitrogen, provides a driving
force for cracking at lower temperature also.
3.3. Effect of Jrequency o f cycling
Saturation stress is dependent not only on fatigue
temperature (Fig. 3) but also on frequency of cycling.
Decrease in temperature or increase in strain rate will
cause an increase in flow stress. The total effect of an
abrupt change in temperature or strain rate comprises
an immediate change in flow stress followed by an
asymptotic approach to a saturation peak stress
characteristic of the new conditions. Presumably, the
same reversible and irreversible flow stress changes
well-established in the case of deformation in tension,
also apply to cyclic deformation. In this case, the
instantaneous effect on the flow stress represents a
reversible change at constant substructure, while the
asymptotic approach to a new saturation stress is
irreversible, associated with rearrangement of the
dislocation substructure. Some of the difficulties involved in both measuring and interpreting flow stress
changes in fatigue are discussed in an earlier paper [1].
In the present work, the quantitative relation between
cycling frequency and equilibrium saturation peak
flow stress was investigated by measuring the total
flow stress change (including both reversible and
irreversible components) caused by an abrupt change
in cycling frequency. Reversible changes are currently
being studied and will be the subject of a future
publication.
In one type of experiment, the equilibrium saturation peak stress was measured before and after
changing the fatigue frequency at several fixed temperatures. A portion of a fatigue curve in Fig. 4(a)
shows frequency rate changes by a factor of 10
between 5 and 0.5 Hz at 200 and 300 K. The principal
sources of error in this type of measurement arise
from the relatively small flow stress changes involved,
the slow approach to a new equilibrium flow stress
level, especially at lower strain rates, and the effect of
small variations in temperature while the crystal is
fatiguing. Temperature variations were due mainly to
the increased rate of generation of heat which accompanies an increase in fatigue rate; heat is not readily
dissipated in an evacuated cryostat, so that time is
required to re-establish the required temperature.
Between 77.4 and 300 K, increasing the strain rate by
3260
BASINSKI ANn BASINSKI: SATURATION STRESS OF Cu CRYSTALS
49
~DOK
0.SHz
-I0
-48
183.5K
0.5Hz
200K
-47
J
•4 6
134.9 K
5 Hz
300K
5Hz
300K
03Hz
"7
-45
1t8.8K
0,5 Hz
300K
5Hz
-6
-
.44
135.6K
5Hz
300K
O-SHz
-5
'43
KCS
~-
c
(a)
-<
Kcs
-<
(b)
Fig. 4. Fatigue chart showing peak stress in tension (t) and
compression (c) as a function of cycle number. (a) Effect of
strain rate changes by a factor of 10 at 200 and at 300 K.
(b) Simultaneous strain rate and temperature changes.
Arrows indicate direction of increasing stress.
flow stress is thus accompanied by extensive rearrangements in the dislocation substructure. Presumably, corresponding adjustments are associated
with the dependence of saturation stress on cycling
rate, however, they would be difficult to measure
since, in the absence of temperature change, the
differences in stress level, and therefore the effect on
the substructure, would be small. There is also evidence that the crystal responds differently to decreases in fatigue temperature than to increases [1].
After decrease in fatigue temperature from 295 to
77.4 K, the average ladder spacing in PSBs decreases,
however, after temperature increase form 77.4 to
295 K, new PSBs with larger average ladder spacing
than the pre-existing ones, form in the matrix.
Figure 5 shows the collected data from both types
of experiment. For the constant temperature experiments Fig. 5 shows the data as measured. To obtain
the rate of change of saturation stress with strain rate
from the second type of experiment, which involved
temperature change, Fig. 3 was assumed to represent
accurately the relation between saturation stress and
temperature, and the mean of the two fatigue temperatures was taken as the effective temperature.
Changes in cycling rate were usually by a factor of 10
between 5 and 0.5 Hz, data from smaller rate changes
were normalized to a frequency ratio of 10. The curve
is a cubic least squares polynomial fit to the data.
Since crystals subjected to low amplitude cyclic
deformation exist in a state of apparent dynamic
equilibrium for hundreds of thousands of cycles,
theoretical treatments have concentrated on explanation of the saturation phenomenon. More than thirty
years ago, Backofen [24] concluded that in saturation
a state of dynamic equilibrium would exist in the
creation and annihilation of both screw and edge
dislocations. More recently, however, consideration
has been given to the possible role played by point
defect mechanisms. Properties such as cyclic hardening curves, sequence of fatigue loop shapes, division
of substructure into matrix and PSBs, are similar
from 4.2 to 350 K. Also, the temperature and rate
a factor of 10 caused the flow stress to increase by
only 3-5%, depending on temperature, so that large
errors could be introduced into the flow stress ratios
by uncertainty as to whether the asymptotic value of
the new saturation stress had been reached. An
alternative approach was to make smaller rate
changes, so that equilibrium was reached in a shorter
time, however, in this case the flow stress changes
were correspondingly smaller. In another type of
experiment, the strain rate change data were used in
conjunction with saturation stress-temperature data
to cancel, as far as possible, the effect of a rate change
by a simultaneous application of temperature change.
The change of temperature which would cause a
given change in saturation stress at any temperature
in the region studied can be estimated from Fig. 3.
Figure 4(b) shows some examples of simultaneous
rate and temperature changes. The first two sets were
chosen to have approximately equal and opposite
effects on the saturation stress level, while the next r"Y
•
pair of changes, to obtain data in a different tempero.olc
ature region, are such that the saturation stress must
approach a new value, and is accompanied by a
gradual decrease of stress over thousands of cycles. _c
The advantage of the experiments involving balanced
rate and temperature changes is that there is little or -I~- o.oo~
no change in saturation stress level, so that the new
values may be read with reasonable confidence without delay.
It is not surprising that temperature changes are
I
I
I
I00
200
300
followed by asymptotic approach to the new saturaTEMPERATURE
(K)
tion stress level. Saturation stress and PSB ladder
spacing are inversely related [1], approach to a new
Fig. 5. Strain rate sensitivity of the saturation stress.
,
BASINSKI and BASINSKI: SATURATION STRESS OF Cu CRYSTALS
dependence of the saturation stress are described by
smooth functions in this temperature interval. Limited determinations of PSB ladder spacing indicate
that this too varies with temperature only in scale but
not in kind. None of the observations suggest any
change in mechanism, it therefore seems reasonable
to assume that the same fundamental mechanism
operates throughout the temperature region studied.
At room temperature, a variety of theoretical models
might be envisaged to explain saturation, since defects may disappear in many ways, each of which can
lead to a system of balanced defect acquisition and
loss. At very low temperatures the choice of mechanisms is limited, making this region more suitable for
study of the fatigue process. The low temperature
region also has the advantage that correction for
temperature dependence of the elastic modulus is
small.
In view of the lack of systematic low temperature
experimental data, saturation cannot be discussed
with any confidence in terms of a specific detailed
model, but it is possible to reach some general
conclusions by examining the limited data on temperature and rate dependence of the saturation stress
presented above.
The activation energy, E, for the process which
determines the saturation stress can be obtained from
the equation [25, 26]
E
(~r*/?~T):+
T
1/T(?,z*/~ ln);)r
The ratio of data for (Or*/gT).; [Fig. 3(b)] and
1/T(~z*/~ In ?)T (Fig. 5) is surprisingly constant in
the temperature interval studied, having a value of
~25. The activation energy for the process, calculated from the data, shown as a function of absolute
temperature in Fig. 6, depends linearly on temperature having a value of ~0.175eV at 77.4K and
reaching ~0.7 eV at 295 K. The fundamental fatigue
mechanism is thus stress (i.e. temperature) dependent.
According to the review of Balluffi and Granato [27],
0.8
0 .zl
t-
/://
I
I
IOO
200
T E M P E R A T U R E (K)
•J
3261
activation energy for vacancy migration near room
temperature is 0.7 eV, vacancy mechanisms are thus
possible at this temperature. However, at 100 K the
observed activation energy is only ~0.2 eV, which is
lower than the migration energy of mono-, di- and
trivacancies, ruling out any saturation mechanism
involving vacancy movement. Recovery mechanisms
requiring migration of point defects with even lower
migration energies, for example interstitials, are unlikely, since electrical resistivity measurements show
that essentially no recovery occurs below 80-100 K
[28, 291.
The data thus indicate that saturation stress varies
smoothly with temperature down to 4.2 K, that there
is no indication of change of fundamental mechanism
in the temperature interval 4.2-350 K, and that
activation energy for the fundamental process
increases linearly with temperature. The fundamental
process governing saturation must therefore be
mechanical in nature, most likely a dislocation
mechanism.
It has been pointed out [e.g. 30, 31] that for copper
single crystals, the flow stress for the onset of Stage
III in tension (r3) corresponds well with the fatigue
saturation stress. This correspondence appears to
extend down to very low temperature. Superimposed
plots of resolved shear stress and rate of hardening
for deformation in tension (Fig. 2 of Ref. [32]) show
that after the constant Stage II region the hardening
rate begins to decline at a flow stress which is about
the same as the fatigue saturation stress at 295, 77.4
and 4.2 K. The fundamental dislocation processes
may thus be similar in the two cases. Since
stress-strain curves gradually bend over after the
linear region, T3 is not a precisely determinable quantity. Even where the differential (rate of hardening)
curve is used+ which more clearly shows changes in
hardening rate, the complication as to how to calculate the stress-strain curve arises [32]. If fatigue
saturation and Stage lII of work hardening are
indeed related, then data on saturation stress, and its
dependence on temperature, would be helpful in
deformation studies in that they provide a reliable
value for r~.
|
4. SUMMARY AND CONCLUSIONS
I
300
Fig. 6. Activation energy of the process limiting saturation
stress, as a function of temperature.
In the temperature interval 4.2-350 K:
Fatigue hardening curves are similar at all temperatures (Fig. 2). There is a smooth relation between
saturation stress and temperature (Fig. 3). Saturation
stress is rate-dependent, increase in cycling rate results in increase in saturation peak stress (Fig. 5). All
observations suggest that over the whole temperature
interval studied, saturation is governed by the same
fundamental mechanism, which must therefore be
mechanical in nature. The activation energy for the
basic mechanism increases linearly with temperature
from ~0.175eV at 77.4K to ~0.7eV at room
3262
BASINSKI AND BASINSKI: SATURATION STRESS OF Cu CRYSTALS
temperature (Fig. 6). Fatigue life is dependent both
on temperature and environment.
Acknowledgements--This work was supported by a grant
from the National Science and Engineering Research Council of Canada.
REFERENCES
I. Z. S. Basinski, A. Korbel and S. J. Basinski, Acta metall.
28, 191 (1980).
2. C. E. Feltner, Phil. Mag. 12, 1229 (1965).
3. J. R. Hancock and J. C. Grosskreutz, Acta metall. 17,
77 (1969).
4. Z. S. Basinski and S. J. Basinski, Proc. 8th Int. Conf. on
the Strength o f Metals and Alloys, Tampere, Finland,
1988 (edited by P. O. Kettunen, T. K. Lepisto and
M. E. Lehtonen), Vol. 1, p. 13. Pergamon Press, Oxford
(1988).
5. J. W. Mitchell, J. C. Chevrier, B. J. Hockey and J. P.
Monaghan, Can. J. Phys. 45, 453 (1967).
6. N. V. Sidgwick, Chemical Elements and Their Compounds, Vol. 1, p. 120. Clarendon Press, Oxford (1950).
7. Z. S. Basinski, Proc. R. Soc. A240, 229 (1957).
8. Z. S. Basinski and S. J. Basinski, Acta metall. 37, 3263
(1989).
9. H. Mughrabi, Mater. Sci. Engng 33, 207 (1978).
10. P. J. Woods, Phil. Mag. 28, 155 (1973).
11. Z. S. Basinski and S. J. Basinski, unpublished (1980).
12. A. J. E. Foreman, Acta metall. 3, 322 (1955).
13. W. C. Overton and J. Gaffney, Phys. Rev. 98, 969
(1955).
14. Z. S. Basinski and S. J. Basinski, Acta metall. 33, 1319
(1985).
15. Z. S. Basinski and S. J. Basinski, Eshelby Memorial
Symposium, Sheffield, 1984, p. 583. Cambridge Univ.
Press (1985).
16. H. J. Gough and D. G. Sopwith, J. Inst. Metals 49, 93
(1932).
17. H. J. Gough and D. G. Sopwith, J. Inst. Metals 56, 55
(1935).
18. N. Thompson, N. J. Wadsworth and N. Louat, Phil.
Mag. 1, 113 (1956).
19. N. J. Wadsworth and J. Hutchings, Phil. Mag. 3, I154
(1958).
20. R. Wang, H. Mughrabi, S. McGovern and M. Rapp,
Mater. Sci. Engng 65, 219 (1984).
21. P. Neumann, in Physical Metallurgy (edited by R. W.
Cahn and P. Haasen), Vol. 2, p. 1553. Elsevier Science,
Amsterdam (1983).
22. A. Hunsch¢ and P. Neumann, ASTM STP 924 (edited
by J. T. Fong and R. J. Fields), Vol. I, p. 26 (1988).
23. R. D. McCammon and H. M. Rosenberg, Proc. R. Soc.
A242, 203 (1957).
24. W. A. Backofen, Fracture (edited by B. L. Averbach,
D. K. Felbeck, G. T. Hahn and D. A. Thomas), p. 435.
MIT Press, Cambridge, Mass; Wiley, New York; Chapman & Hall, London (1959).
25. Z. S. Basinski, Acta metall. 5, 684 (1957).
26. U. F. Kocks, A. S. Argon and M. F. Ashby, Prog.
Mater. Sci. 19, 1 (1975).
27. R. W. Balluffi and A. W. Granato, Dislocations in Solids
(edited by F. R. N. Nabarro), Vol. 4, p. 1, North
Holland, Amsterdam (1979).
28. E. W. Johnson and H. H. Johnson, Trans. metalL Soc.
A.I.M.E. 233, 1333 (1965).
29. Z. S. Basinski and S. J. Basinski, Acta metall. 37, 3275
(1989).
30. G. Rudolph, P. Haasen, B. L. Mordike and P. Neumann, Proc. Ist Int. Conf. on Fracture, Vol. 2, p. 501
(1965).
31. L. M. Brown, Proc. Ashby Symp., TMS Fall Meeting,
Chicago, Illinois (1988).
32. S. J. Basinski and Z. S. Basinski, Dislocations in Solids
(edited by F. R. N. Nabarro, Vol. 4, p. 261. North
Holland, Amsterdam (1979).