AECL-6442
ATOMIC ENERGY MË^H L'ÉNERGIE ATOMIQUEOF CANADA UMITEO U Ë J F OU CANADA LIMITEE
NON-LINEAR INTERNAL FRICTION IN A
SINGLE CRYSTAL OF ZIRCONIUM
FRICTION INTERNE NON LINEAIRE DANS UN CRISTAL
SIMPLE DE ZIRCONIUM
I. G. Ritchie, A. Atrens, K. W. Sprungmann
Whiteshell Nuclear Research Etablissement de RecherchesEstablishment Nucléaires de Whiteshell
Pinawa, Manitoba ROE 1LOApril 1980 avril
ATOMIC ENERGY OF CANADA LIMITED
NON-LINEAR INTERNAL FRICTION IN ASINGLE CRYSTAL OF ZIRCONIUM
by
I.C. Ritchie, A. Atrens and K.W. Sprtingmann
Whiteshell Nuclear Research EstablishmentPinawa, Manitoba ROE 1L0
1980 April
AECL-6442
FRICTION INTERNE NON LINEAIRE DANS UN CRISTALSIMPLE OE ZIRCONIUM
par
I.G. Ritchie, A. Atrens et K.W. Sprur.gmann
RESUME
C'est une étude du phénomène de friction interne non linéaire
dans un cristal simple de zirconium; ce qui couvre les interactions en-
tre les dislocations et les obstacles immobiles et celles entre les dis-
locations et les points d'attache mobiles. On y indique que la vibra-
tion et le recuit programmé à vibrations peuvent servir 3 séparer les
composantes de la friction interne dépendant du temps et de l'amplitude
de la sollicitation.
Un effet de montée maximale des impuretés a été obtenue en mo-
difiant la concentration effective des obstacles par changement graduel
de la vibration et de l'amplitude de la sollicitation. La répétition du
cycle thermique aux faibles amplitudes de la sollicitation et au-delà
de la limite finale de solubilité dans la phase solide comme dans le cas
de l'hydrogène dans le zirconium, ne résultt pas en une augmentation
cumulative de la densité de la dislocation que l'on observe lorsque des
échantillons polycristallins sont semblablement traités.
L'Energie Atomique du Canada LimitéeEtablissement de Recherches Nucléaires de Whiteshell
Pinawa, Manitoba ROE 1L01980 avril
AECL-6442
NON-LINEAR INTERNAL FRICTION IN ASINGLE CRYSTAL OF ZIRCONIUM
by
I.G. Ritchie, A. Atrens and K.W. Sprungmann
ABSTRACT
Non-linear internal friction phenomena in a single crystal of
zirconium are Investigated. Both the interactions between dislocations
and immobile obstacles and between dislocations and mobile pinning
points are involved. It is shown that vibration conditioning and pro-
grammed vibration annealing can be used to separate the time-dependent
and strain-amplitude-dependent components of the internal friction.
An impurity peaking effect has been generated by altering the
effective concentration of obstacles by step changes in strain amplitude
and vibration conditioning. Repeated thermal cycling at low strain am-
plitudes, through the terminal solid solubility boundary for hydrogen in
zirconium, does not lead to the cumulative increase in dislocation den-
sity observed when polycrystalline samples are treated similarly.
Atomic Energy of Canada LimitedWhiteshell Nuclear Research Establishment
Pinawa, Mani coba ROE 1LG1980 April
AECL-6442
CONTENTS
Page
1. INTRODUCTION 1
2. EXPERIMENTAL PROCEDURE 3
3. RESULTS 5
4. QUALITATIVE INTERPRETATION 9
5. QUANTITATIVE INTERPRETATION 14
6. CONCLUSIONS 18
REFERENCES 20
TABLES 24
FIGURES 26
1. INTRODUCTION
In the temperature range of technological Importance for zir-
conium and its alloys, the initiation of dislocation movement Is con-
trolled by the interactions between dislocations and obstacles in the
crystalline lattice. These obstacles to dislocation motion may be in-
trinsic or extrinsic in nature. For example, point defects, their clus-
ters, jogs, fine precipitate particles, and the trees' of the disloca-
tion 'forest' all form more or less localized, effective obstacles to
the motion of a particular dislocation segment. In zirconium, Impurity
interstitials such as hydrogen, oxygen and nitrogen all interact strongly
with non-scrtrf dislocations and can be considered, from the practical
point of view, as intrinsic point defects, since samples with less than
200 atomic ppm of these impurities are extremely difficult to prepare
and maintain at this level of purity during testing.
Internal friction is a very sensitive technique for the study
of the interaction between dislocations and pinning points or obstacles.
Three important classes of internal friction phenomena appear due to
these interactions:
1. Dislocation drag. A vibrating dislocation segment experiences
a drag due to its interactions with immobile obstacles dis-
tributed in a glide plane.
2. Dislocation unpinning. Stress-induced, thermally assisted
unpinning can tear a dislocation segment away from its segre-
gated cloud of impurities, into a relatively obstacle free
lattice, for part of each cycle of the applied stress.
3. Obstacle drag. At elevated temperatures, stress—Induced re-
arrangement of obstacles that are mobile within the disloca-
tion core can take place; this is equivalent to dragging of
the obstacles by the vibrating dislocation segment.
- 2 -
The phenomena in class 1 are expected to be observed in
freshly deformed, dilute, random solid solutions at temperatures where
the impurity atoms are immobile. However, after annealing at tempera-
tures at which the impurity point defectB migrate, an extended atmos-
phere will be formed around the dislocation and the phenomena in clafs 2
are expected to be observed. At temperatures at which the Impurity
point defects become mobile in the dislocation core, class 3 phenomena
are expected to control the observed internal friction as the impurities
undergo stress-induced rearrangement during each cycle.
Unfortunately, all three classes of phenomena can, under cer-
tain circumstances, give rise to amplitude-independent relaxation pea'cs
of internal friction and, under different circumstances, strain-ampli-
tude-dependent internal friction. Thus, although the technique of i.i-
ternal friction is certainly the most sensitive technique presently
available i<-ï the study of dislocation-pinning point interactions. Its
selectivity leaves much to be desired. Furthermore, the three clat&es
can be interrelated, with one phenomenon triggering another. For ex-
ample, stress-induced core rearrangement of pinning points can lead to
the appearance of long, free dislocation segments which, in turn, trig-
ger unpinning of the dislocation. Consequently, these phenomena, are
difficult to identify even when they occur separately, and the inter-
pretation of the results in the temperature-stress domain, w.iere they
overlap, is extremely difficult. In zirconium, these problem*, are com-
pounded by the presence of two types of impurity interstitial with
widely differing mobilities, i.e., hydrogen and oxygen interstitials.
Nevertheless, such temperature-stress regimes are often of real tech-
nological importance, and thus, any progress in unraveling the compet-
ing mechanisms is of interest. This is certainly the case for zirco-
nium and titanium in the temperature range where peaks or plateaus oc-
cur in the flow stress versus temperature curves and the observed mech-
anical behaviour is attributed to dynamic strain-aging.
- 3 -
In this report, our aim is to show that the complex, non-
linear internal friction observed in a single crystal of zirconium, in
the ambient to 350°C temperature range, can be explained in some detail,
at least qualitatively, by the three classes of phenomena outlined
above. In addition, we report some progress In separating the competing
phenomena, by using the techniques of 'vibration conditioning' and
'vibration annealing' at constant strain amplitude to obtain stabie,
reproducible dislocation-pinning point configurations.
2. EXPERIMENTAL PROCEDURE
The low-frequency, counterbalanced reed pendulum described byn 2)
Ritchie et al. ' was used to measure damping during free decay of
flexural oscillations. The measure of internal friction utied is the
logarithmic decrement, Û, given by 4 ° n" ln(A,/A , . ) , where A, and
A . are the vibration amplitudes during the first and n+lth oscilla-
tions, respectively.
Additional instrumentation has been developed , whereby the
pendulum oscillations can be maintained at a constant vibration ampli-
tude while the energy input per cycle is monitored. In this systen, the
voltage drive signal, D, applied to the coil-driving circuit, is directly
proportional to the current in the drive coil and, therefore, propor-
tional to the energy input to the pendulum. It can be shown that D and
A, at the same strain amplitude, are related simply by A = K I D ,
where I is the number of amperes per volt of drive signal applied to
the drive coil, D = D/N is the normalized drive signal (where N is the
nominal strain amplitude) and K is a constant obtained by calibration.
In practice, the amplitude at which the pendulum is maintained is preset
by adjusting a ten-turn potentiometer. Thus, for a particular sample,
it is convenient to refer to this turn-setting, N, as a nominal strain
amplitude.
- 4 -
The continuous drive technique Is particularly well suited to
the study of internal friction at constant strain amplitude, while an-
other parameter, such as time or temperature, varies. It has other im-
portant uses such as 'vibration conditioning' and 'vibration annealing1
(described below) and also allows a preset maximum amplitude to be at-
tained In programmed steps to avoid significant overshoot. This can be
important since, as pointed out by Atrens , the damping versus strain
amplitude curve in free decay in zirconium often depends upon the maxi-
mum amplitude and the time of vibration at this amplitude prior to the
start of a free ùecay.
The single crystal of zirconium tested in this study «as grown
by a floating-zone technique from crystal bar zirconium. The single
crystal rod was spark nachined to a rectangular prism, with gauge dimen-
sions 23.6 x 6.5 x 1.4 mm, suitable f^r testing in the pendulum. Laue
back-refleetion X-ray studies revealed tnat the orientation of the sin-
gle crystal specimen was such that the c-axis was parallel to the thick-
ness direction, and the longitudinal axis of the specimen made an angle
of 14.5° with an a-direction. After completion of the series of inter-
nal friction experiments, the specimen was analysed and found to contain
the following impurity concentrations (ug/g): Al, 17; Cu, 50; Cr, 45;
Fe, 89; Hf, 50; Mn, 7; Pb, 10; Si, 12-, Sb, 12; H, 34; 0, 200. Of these,
the relatively large atomic concentrations of oxygen and hydrogen are
the most significant impurities from the point of view of this study.
Prior to testing, an in situ anneal for one h^ur at 700°C was
performed to stabilize 'handling' strains introduced during specimen
mounting. All of the experiments were performed with the pendulum
chamber evacuated to % 10~ mm of Hg (0.13 mPa).
- 5 -
3. RESULTS
Initial exploratory experiments in the ambient to 400°C tem-
perature range (summarized in Figure 1) showed that the Internal fric-
tion was a complex function of the strain amplitude and the time of
vibration at that amplitude. In order to try to separate the components
of this complex, non-linear behaviour, it vas necessary to devise a
method cf arriving at a stable (essentially time-independent), repro-
ducible starting structure. Two methods were found which did not ne-
cessarily lead to the same structure:
1. Vibration conditioning (VC). This is simply continu.:' vibra-
tion at the chosen strain amplitude until there is no further
significant change in the measured internal friction with
time.
2. Programmed vibration annealing (PVA). This consists of con-
tinual vibration at the chosen strain amplitude, while -he
specimen is heated at a controlled rate to the prechosen an-
nealing temperature, held for the annealing time, then cooled
at a controlled rate to the next chosen test temperature.
A typical PVA used in this study is illustrated in Figure 2.
The specimen was heated from the previous test temperature to 410°C, at
2CC per min, held at that temperature for 2.5 h, and then cooled to the
next temperature at 0.5°C per min, all the while vibrating at a constant
strain amplitude of 3.1 x 10 (N = 2). This gave time-independent
damping, which was almost amplitude independent for N < 2, and a stable,
reproducible (+ 5 x 1C
properties for N > 2.
reproducible (+ 5 x 10 ) starting point for measurements of non-linear
Typical amplitude-dependent curves measured in free decay from
N = 7.5 at 186°C are shown in Figure 3. In this experiment, a PVA was
- 6 -
followed immediately by a step increase in amplitude from N - 2 to
N • 7. S and measurement of the free-decay curve (+). This was followed
by a return to N • 7.5 and a VC for one hour, after which the free-decay
curve (x) was measured. The VC at N - 7.5 was then continued until no
further evolution of the internal friction could be detected. The final
free-decay curve {•) from this state shows a knee at N • 2, as Indicated
in Figure 3. These three curves Indicate a large decrease in internal
friction with the time of vibration conditioning for amplitudes N > 2,
and a much smaller decrease for N < 2. In fact, the behaviour shows
that the system giving rise to the internal friction retains a memory of
the PVA at N - 2, even after a VC at N = 7.5.
The time dependence of the damping during a VC can be measured
directly, as shown in Figure 4. In this case, the test temperature was
238°C and a PVA at N « 2 was immediately followed by a VC at N - 7.5.
As shown in Figure 4, the initial increase in strain amplitude causes an
increase in the dampdig which gradually recovers over a period of about
five hours. A return to a lower strain amplitude (e.g., N = 2 as indi-
cated in Figure 4) does not significantly disturb the completely vibra-
tion-conditioned state.
The results of experiments, such as those in Figure 4, showed
that the same reproducible starting state could be generated at any
temperature in the ambient to ^ 320°C temperature range by a PVA. A VC,
however, produced a stable state which depended upon the temperature.
This is underlined by the broad peak, Pa, in Figure 2, which occurred
after a VC at N = 7.5 and a temperature of 105cC. A similar peak was
observed during the heating branch of the PVA's for all the test tem-
peratures investigated, but the peak temperature increased almost lin-
early with the previous VC temperature. From these results, it can be
concluded, reasonably, that the stable state produced by a VC at N = 7.5
requires thermal activation to recover at N = 2.
- 7 -
Two regions of temperature were fcû-nd where a VC did not pro-
duce a unique final state. At temperatures below i 90°C and following
prolonged vibration at N » 7.5, the damping underwent slow, apparently
random excursions. In the 350 to 380°C temperature range, the phenom-
enon laoelled 'gasping" in Figure 2 (and the hatched region of curve (a)
in Figure 1) was ol srved, in which the internal friction cycles between
two levels, with a time constant that depends upon the temperature. The
term 'gasping' was first used by Takahashi to describe a similar phe-
nomenon in single crystals of zinc- (Cessler and Baxter and Wilks
have reported ar^logoiss phenomena in other materials. The gasping ob-
served in this study is of interest since it appears to be a result of a
dynamic, conditioned interaction between the pendulum drive system, at-
tempting to maintain the specimen at a constant strain amplitude, and an
induced periodic process in the specimen. The time constant of the
gasping correlates well with the relaxation time for che réorientation
of oxygen interstitial-subsCitutional Impurity pairs, wliiel» give rise to(9)a Snoek peak at ^ 45O°C in samples containing higher oxygen contents .
Three distinct sets of experiments were carried out to try to
separate and identify the mechanisms responsible for the not. linear pro-
perties of the internal friction. A preliminary outline of the results
and a tentative interpretation have been presented by Atrens et al. ,
here we will consider the results in more detail.
(i) Tests typified by Figure i were carried out at a series of
temperatures, and the time-dependent damping was analysed and
compared with various theoretical developments, as descriLad
in the next section.
(ii) At constant temperature, the recovery of the damping was moni-
tored after step increases in amplitude of 6N = 1, each step
in amplitude being preceded by the standard PVA at N = 2. The
results, measured at 186°C, are illustrated in Figure 5, where
only the damping values for t = 0, t = 1 h and the level after
complete recovery are shown for clarity. Also shown in Fig-
ure 5 are the levels of damping at nominal strain amplitude
N ° 2, after full recovery at several higher amplitudes (i.e.,
the starting point of the next PVA). Finally, a free-decay
curve from the completely vibration conditioned state at N - 9
is shown for comparison.
(iii) Again, following step increases in amplitude of 6N = 1 at con-
stant temperature, the recovery of the damping was monitored,
but this time wiunout the PVA between each step. In this
case, the starting state before each step increase in ampli-
tude was the state attained after one hoar of vibration condi-
tioning at the previous amplitude. The results, again at a
temperature of 186°C, are illustrated in Figure 6; tiity show
that when the amplitude steps are small and recovery is al-
lowed to take place at each amplitude, the magnitudes of the
time-dependent changes are also small. When the results are
taken in this sequence, a 'peaking' phenomenon is ob&erved.
For example, for the step increase from N = 6 to 7, the re-
covery does not proceed by a decrease in damping (as is the
case at low amplitudes), but the damping initially increases
and passes through a maximum during the VC (Figure 7 ) .
One of the moss, interesting conclusions from these experiments
is that, following a large step increase in amplitude (e.g., N = 2 to 9
in Figure 5 or N = 2 to 7.5 in Figure 3 ) , even after a VC at the higher
amplitude, the dissipating defects retain 3 memory of the previous PVA
at the lower amplitude. In contrast, as shown in Figure 6, this memory
effect is no longer observed if the amplitude is increased in small
steps and a vioration-conditioned recovery is allowed to take place at
each step.
- 9 -
4. QUALITATIVE INTERPRETATION
The non-llnaar internal friction phenomena observed in this
study are characteristic of the interactions between dislocations and
point defects. In fact, Wlnkler-Gnlewek et al. have shown that core
rearrangement of pinning points, or obstacle drag, can giva rise to both
amplitude- and time-dependent effects. However, it will be argued that
the results presented hire are more characteristic of two competing phe-
nomena: a strong amplitude dependence caused by thermally assisted un-
pinning, and a weaker amplitude dependence, together with time depend-
ence, caused by stress-induced core rearrangement of mobile pinning
points. As mentioned previously, a further complication is brought
about by the presence of at least two types of interstitial impurities
known to interact strongly rfith dislocations in the temperature range of
our experiments; these are hydrogen and oxygen.
According to Friedel , impurity interstitials are less mo-
bile in a dislocation core than they are in the lattice at a given tem-
perature. There is ample experimental evidence for this in the case of
impurity interst.ltials in the body-centred-cubic metals. For example,
in iron-nickel a'.lovs, the Snoek-Këster peak, attributed to the dragging
of a cloud of nitrogen interstitials by dislocations, occurs ^ 200°C
above fie temperature of the nitrogen Snoek peak. This is due to the
reorientation of nitrogen interstitials, which occurs by the primitive
lattice diffusion jump . In the zirconium lattice, because of their
site symmetry, neither single oxygen nor single hydrogen interstitials
can give rise to a Snoek peak; therefore, similar reasoning is not pos-
sible. However, a Snoek-type relaxation due to the stress-induced re-
orientation of di-interstitial impurities is possible and has been ob-
(14)served in the case of oxygen . In fact, in the temperature range
below 450°C, the diffusion parameters for oxygen in zirconium are in-
ferred rrom data concerning the stress-induced reorientation of di-in-
- 10 -
terstitHal oxygen or the reorientation of oxygen 'ntcrstitial-sub-
stituticnal atom pair6l ' . Thus, the diffusion parameters for single
interstltials of oxygen are somewhat uncertain. To date, no relaxation
peak in Eirconium-oxygen or zirconium-hydrogen alloys has been attrib-
uted to the analogue of the Snoek-KBster peak in iron-nickel alloys.
The diffusion data for hydrogen in zirconium, in the temperature range
of our results, are relatively uncertain also. Except fcr the work of
Gulbransen and Andrew^ in the temperature range 60° ':o 300°C, most
data have been obtained above 400°C. Nevertheless, al?. of the data
indicate that the lattice diffusion of hydrr>r.cn occurs very rapidly in
the temperature range of our experiments.
In the following, we assume that dislocation core diffusion of
the impurity interstitials, oxygen and hydrogen, is slower than lattice
diffusion. Thus, in the temperature range 100°C to 32O°C, core rear-
rangement of hydrogen defects during the period of the applied stress is
possible, whereas neither lattice diffusion, nor, a priori, core rear-
rangement of oxygen interstitials can occur in the same period. This
means essentially that hydrogen interstitials 'feel' the oscillating
applied stress, whereas oxygen interstitials feel only the time average
of the applied stress. The onset of the gasping phenomenon marks the
temperature where the movement of oxygen interstitials begins to contrib-
ute to the measured damping.
The decrease in internal friction with time (Figure 4), after
a step increase in strain amplitude, could be attributed to mree basic
mechanisms :
(i) If fresh dislocation segments are created by the sudden in-
crease in amplitude, the subsequent time dependence could sig-
nal the arrival of hydrogen pinning points, either by core
diffusion from pre-existing parts of the network, or by lat-
tice diffusion from the bulk.
- .11 -
(ii) If the sudden increase in strain amplitude causes unpinning of
the dislocation segments from their stabilized atmospheres
produced by the PVA, the subsequent time dependence could be
attributed to the gradual repinning of the freed segments,
either by core rearrangement of the existing atmosphere, or by
lattice diffusion from the bulk, or by both at the same time.
(iii) Assuming that dynamic distribution of pins in the stabilized
pinning atmosphere depends upon the amplitude of the applied
stress, and that the sudden increase in amplitude simply drags
the mobile pinning points back and forth over a larger ampli-
tude, the subsequent time dependence may be the development of
the new, dynamic distribution of mobile pinning points by core
rearrangement.
"he evidence for the unpinning of dislocations during the step
increase in amplitude, without any significant generation of fresh dis-
locations, is very persuasive.
(a) Optical examination of the specimen before and after testing
revealed no significant increase in the numbers of surface
etch pits, slip lines or microtwins. Subsequent electron mi-
croscopy revealed a very low density (< 10 cm ) of relatively
.'ong dislocations, mainly associated with hydride needles.
(b) The increase in damping after a step increase in amplitude was
reproducible and dependent upon the size of the step. For ex-
ample, the results of repeated tests at 186°C reproduced the
magnitude of the damping changes and the shape of the curve
shown in Figure 5. Such reproducibility is not consistent
with dislocation multiplication and indicates that the an-
elastic limit was not exceeded in these tests. Atrens
has shown that the onset of microplasticity in zirconium is
accompanied by very large amplitude-dependent changes in
damping, of the type investigated by Peguin et al.
- 12 -
(c) Both the maximum value, A , and the magnitude of the change in
damping, 6A, are surprisingly small (A < 2 x 10~ and-3
da < 1 x 10 ) and not at all consistent with dislocation
multiplication.
The apparent constancy of the structure of the dislocation-pin
system in these experiments is very surprising and puzzling. According
to an extrapolation of the terminal solid solubility curve for hydrogen(19)
in a-zirconium reported by Kearns and more recent data at lower(20 21)
hydrogen concentrations ' , the solvis boundary for the concentra-
tion of 34 yg/g of hydrogen measured in our single crystal occurs at
^ 288°C. However, no evidence of the production of fresh dislocations,
or the precipitation of liydrides, was observed on any of the cooling
branches of the PVA's (such as Figure 2) as the specimen was cooled
through the solvus boundary. This is in direct contradiction to our re-
sults on polycrystalline samples of comparable purity and hydrogen con-(22)
tent , where repeated thermal cycling generated enormous increases in
the dislocation density, leading to anomalously large increases in in-
ternal friction and modulus defect. These latter results are in agree-(23)
ment with the conclusion of Bell and Sawatzky that thermal cycling
under stress causes a cumulative strain due to the multiplication of
freshly generated hydride dislocations. Thus we conclude that in the
present experiments on a single crystal, there was no cumulative in-
crease in the density of fresh dislocation.» produced by cycling through
the terminal solid solubility boundary. Moreover, the peak temperature,
Pa, observed on the heating branches of the PVA's, varied with the tem-
perature of the preceding VC and, consequently, could not be attributed
directly to hydride resolution. A tentative explanation of this unex-
pected behaviour can be advanced, if it is assumed that the continued
vibration of the dislocations during hydride resolution maintains the
dislocation-pin structure, so that significant dislocation annihilation
does not occur. Then, it is probable that the hydrides will reprecipi-
tate at the same place in the lattice where the dislocation structure
- 13 -
has been maintained. Electron microscopic evidence of the nucleation of
hydrides at the dislocation structure remaining after dissolution of(24)
previous hydrides has altXoly been reported by Carpenter and Watters
and Nath et al. ( 2 5 ).
Having ruled out (i), the pinning of fresrily generated dis-
locations, we can present very strong evidence in favour of a combina-
tion of (ii) and (iii): i.e., dislocation unpinning caused by the large
step increase in strain amplitude, together with dragging of the mobile
pinning points. This evidence is contained in Figures 3 and 5, where
the data show that when the amplitude step is large enough, the system
of pins and dislocations retains a memory (indicated by the knee in the
damping versus amplitude curves) of the final structure induced by the
PVA at N = 2, even after many hours of vibration conditioning at N = 7.5.
In a region of the temperature-stress plane where some of the pins (hy-
drogen interstitials) are mobile and some of the pins (oxygen intersti-
tials) are immobile, an obvious way in which such a memory effect can be
retained is if the dislocations are torn away from the immobile pins by
the sudden large increase in amplitude. Then the rearrangement of the
mobile pins, in response to the increased amplitude, gives rise to the
observed time-dependent damping, while the recapture of the dislocation
segments, at the low-amplitude end of a free decay, by the immobile
pins, yields a memory effect. This picture is supported by the results
of Figure 6, which show that if the same high amplitude is reached in
small steps, with a VC at each step, the large scale unpinning, and con-
sequent large increase in damping at high amplitudes, can be avoided.
This is because enough time is allowed at each amplitude for the mobile
pin distribution to respond to the new increment in stress.
Although the damping was stable for any amplitude level < eH
after a VC at £„, the damping level upon return to N = 2 was always
slightly greater after the VC at N > 2, in spite of the memory effect
(see, for example, Figure 5). This is evidence of an irreversible
change in the mobile pin arrangement, at the constant test temperature,
probably due to the involvement of some new pins encountered by the un-
pinned dislocations. Peak Pa appears to mark the recovery of the ori-
ginal N « 2 structure during subsequent heating. Unfortunately, peak Pa
is too small and broad to attempt a detailed analysis.
The 'peaking-effect', shown in Figure 7, represents a complex
situation where either the initial step in amplitude has caused some
unpinning, which 1 B gradually suppressed by core rearrangement of the
mobile pins, or core rearrangement during the initial stages of the test
leads to the production of long free segments which trigger some unpin-
ning, followed by repinning. This peaking phenomenon can be predicted
theoretically using the theories of thermally assisted unpinning ,(27)
the zig-zag motion of a dislocation in a distribution of point defects ,128)
or the viscous drag of Simpson and Sosin , provided the concentration
of pins is J.ncre?.i»ed or decreased in a regular manner. The results in
Figure 7 represent the first demonstration of the peaking effect by
effectively changing the local concentration of impurity pinning points.
Previous observations of the peaking effect have all been associated
with the gradual increase in the concentrât!' if intrinsic point de-
fects by Irradiation.
5, QUANTITATIVE INTERPRETATION
In the qualitative model outlined above, the non-linear inter-
nal friction contains contributions from thermally assisted dislocation
unpinning and from stress-induced core redistribution of mobile pins.
At the present time, no theoretical framework has been developed which
can be applied to the quantitative interpretation of this complex situ-
ation. Nevertheless, many authors have treated the unpinning and core
redistribution problems separately; for example, stress-induced core re-(29-31)
distribution, or obstacle drag, has been thoroughly investigated ,
culminating with the detailed calculations of Winkler-Gniewek et al. ,
- 15 -
and several groups have investigated thermally assisted unpinning, viz:( 32)
for equal dislocation loop lengths , for an exponential distribution
of loop Ieng:h6 , or for the zig-zag motion of a dislocation in a(27)
field of obstacles . Nor.-lir.ear internal friction measurements have
been reviewed by de Batist and Perez et al,
Krishtal and Vyboishchik' measured identical dislocation
parameters fiom time-dependent damping in the amplitude-dependent and
amplitude-independent ranges for copper-arsepic alloys. Consequent., y,
a similar analysis was attempted on the results reported here. It is
postulated that the internal fricrio:'. can be describe.! by:
A(t) - A = (A. - A ) exp (- .") (1)o n o ^
where A(t) is the damping at time t, dM and A are the boundary values
when t » 0 and t •• « respectively, ersd r is an ordering parameter. !'
can be expressed as:
r - (t/t)n (2)
where n is an undetermined exponent and T is the relaxation ti&e. In
the case of Snoek ordering (e.g., Quist and Carpenter ), n is 1/3 for
impurity interstitials and T is ths average time for one jump of the
species undergoing ordering. In the case of the rearrangement of sub-
stitutional impurities in the cores of dislocations in copper, Oren et
al. found n = 1 at low temperatures with a deviation from this law
at higher temperatures, while Krishtal and Vyboishchik also found
n = 1 for substitutional impurities in copper. In addition, there is
both experimental and theoretical evidence (see the review by Bullough )
that n = 1/2 applies to the initial stages of the drift of vacancies to
a dislocation.
From equations (1) and (2), it can be seen that a plot of
In r(= lnln[(Ax, - A )/(A(t) - A )]) versus In t should be a straightMo o
- 16 -
line, of slope n. A typical curve is shown in Figure 8 and the results
of the analysis are listed in Table 1. The results indicate at least
two separate stages, with n varying between 1/2 and 7/4 for the first
stage, and, interestingly enough, being quite close to 1/2 for the final
stage, for all of the temperatures investigated. Thus, an attempt was
made to determine the activation energy of the process from the inter-
cepts of plots, such as Figure 8, for the final stage (n ̂ 1/2). These
intercepts yield n In T and, assuming that i = T exp Q/kT where Q is
the activation energy, k is Boltzmann's constant and T is the absolute
temperature, Q/k is readily determined from the slope of a plot of In T
versus 1/T. However, the resulting Arrhenius plot, Figure 9, is linear
only for temperatures above 180°C. The relaxation parameters obtained
are Q » 0.16 eV/atoa and T » 98.7 s. For a process involving single° -14interstitial jumps, a T of 10 s is expected. Consequently, this
enormous value of x is difficult to interpret. For example, if the
normal relationship between T and D is invoked (i.e., D Q - 6a /IQ,
where 8 is a constant of the order of unity and a is the lattice para--17 2 -1
meter), D Q ̂ 1 x 10 cm *s is obtained indicating that, if core dif-
fusion of interstitials is involved, the appropriate diffusion equation
is D = 1 x 10"17 exp (- 0.16/kT). However, the extremely high value of
T obtained from this analysis probably means that a much more complex
process is involved. Moreover, the method of analysis used in this case
may be invalid. For example, the n ̂ 1/2 process may not stem from time
zero, but may be triggered by the initial process.
A number of internal friction peaks attributed to the presence
of hydrogen in zirconium-hydrogen alloys occur in the temperature range
from just below ambient to 100°C . These thermally activated pro-
cesses have been attributed to various mechanisms, including the stress-
induced jumping of hydrogen interstitials from hydride particles into
the matrix and back. However, these relaxation peaks were only of a
readily observable intensity in alloys containing more than 1 at.% H.
The peak temperatures of the internal friction peaks (Figure 1), ob-
- 17 -
served at 7 Hz in this study, do not correlate with the activation para-
meters of any of those peaks previously reported in the zirconium-hydro-
gen system. In fact, the results shown in Figure 1 are very similar to(43)
those reported by Loh for a series of zirconium alloys containing
small additions of the rare earths, neodytnium and dysprosium. Following
Loh , we have attempted to interpret the low-temperature side of the
pesk (curve (a)) in Figure 1 in terms of the obstacle drag theory pro-(44)
posed by Schoeck . Snoek-KBster peaks of this kind are well known
only in the case of interstitial impurities and, in addition, we believe
that hydrogen defect mobility in this temperature range is far more
likely than the movement of substitutional impurities suggested by(A3)
Loh . Unfortunately, Loh did not analyse his samples for hydrogen
content.
The relaxation time, T, given by Schoeck is:
T - [akT CQ L*/(Gb3 Do)] exp [(QM + EgJ/kTJ (3)
where a is a constant of the order of unity, b is the Burgers vector of
the dislocations involved, G is the shear modulus, L is the average
network length, C is the concentration of obstacles in the lattice, (X.
is their migration energy and E is the binding energy between the
obstacles and the dislocations. Since the dislocations involved are
associated with the hydride precipitate particles, we assume that the
hydrogen concentration close to the precipitate particle is given by the
solubility of hydrogen at that temperature. Thus, according to refer-
ence 20, C in equation (3) can be written as :
C =3.13 exp (- 0.33/kT) (4)o
Experimental values of T can be estimated by assuming that the network
lengths, Lj, are exponentially distributed, and by fitting the experi-
mental curve with a peak computed numerically from Schoeck's expression(44)
for the relaxation strength . Then, if the temperature dependence of
- 18 -
C over the range of the peak is ignored, equation (3) shows that a plot
of In (T/T) VS. 1/7 should yield a straight line with a slope equal to
(QM + EB - 0.33)/k.
The results of this analysis are given in Table 2 and Fig-
ure 10 and lead to a value of Q,, + £„ - 0.86 eV/atom. D can also be
estimated from equation (3) but the resulting value depends upon the-5 -4
average network length, L . Since L falls between 10 to 10 cm,
the corresponding limits of D are 0.009 and 0.91. These values are not
consistent with the most recent data for the diffusion of hydrogen in(45)zirconium; for example, Mazzolai and Ryll-Nardzewski obtained
D = 0.04 exp (- 0.59/kT) in a study of the Gorsky effect. If
Q » 0.59 eV/atom is accepted as the migration energy of hydrogen in
zirconium, then our results suggest a value of 0.27 eV/atom for the
binding energy between hydrogen interstitials and the dislocations.
This value is higher than would be expected for the Interaction between
a single hydrogen interstitial and a dislocation segment. It seems far
more likely that a more complex hydrogen obstacle is involved, probably
the di-interstitial. As mentioned above, the di-interstitial can under-
go stress-induced ordering and, therefore, give rise to a Snoek peak and
then, as is the case in body-centred-cubic metals, the same defect can
be dragged by the dislocations at a higher temperature, giving rise to a
Snoek-KSster peak. Moreover, Tung and Sommer studied the interac-
tion between hydrogen di-interstitials and dislocations in titanium.
They estimate 0.26 eV/atom as the binding energy between the di-inter-
stitials and dislocations, which is consistent with our result.
6. CONCLUSIONS
1. using the techniques of vibration conditioning and programmed vib-
ration annealing described in this report, the time- and amplitude-
dependent components of non-linear damping can be separated, to a
certain extent.
- 19 -
2. The same techniques can be used to attain a stable, reproducible
starting structure when tlie obstacles to the dislocation movement
are mobile.
3. In situations where there are both mobile and immobile obstacles,
dislocation unpinning, without dislocation multiplication, leaves
a memory of the pinned configuration (in the form of the immobile
pins) in the lattice.
4. An impurity peaking effect can be generated if the effective con-
centration of obstacles is altered by combinations of vibration
conditioning and step changes in the strain amplitude.
5. In a single crystal of zirconium of commercial purity, hydrogen
obstacles appear to be mobile in the temperature range investi-
gated, whereas oxygen obstacles are not.
6. The internal friction peak at ">• 110°C (7 Hz) can be attributed to
the dragging of hydrogen di-interstitials by dislocation segments
(i.e., a Snoek-KSster relaxation). An estimate of 0.35 eV/atom
is obtained for the binding energy.
7. At temperatures above 180"C, more complex clusters of hydrogen
interstitials appear to be formed and to undergo rearrangement on
the dislocation lines.
8. Repeated thermal cycling of a single crystal, at a low strain am-
plitude through the terminal solid solubility boundary, does not
lead to a cumulative increase in dislocation density from the gen-
eration of fresh hydride dislocations, observed when polycrystal—
line samples are thermally cycled at comparable strain amplitudes.
- 20 -
REFERENCES
1. I.C. Ritchie and K.W. Sprungtnann, "A Counterbalanced ReedPendulum Apparatus for Internal Friction Studies", J. Phys.E.: Sci. Inst. 5, 1168 (1972).
2. I.G. Ritchie, J.R. Saltvold, H.K. Schmidt and K.W. Sprungmann,"Instrumentation and Techniques for the Measurement of LowFrequency Internal Friction", J. Phys. E.: Sci. Inst. 6,341 (1973).
3. J.R. Saltvold and H.K. Schmidt, "A Servo System for MeasuringInternal Friction", Proceedings 22nd International Instrumen-tation Symposium, May 1976, San Diego, California, p. 619.
4. A. Atrens, "Dislocation Damping in Strained and Aged ZirconiumAlloys", J. Aust. Inst. of Metals 20, 5: (1975).
5. A. Akhtar and A. Teghtsoonian, "Plastic Deformation of Zirco-r/um Single Cr;'Btals", Acta Met. lj>, 655 (1971).
6. S. Takahashi, "Anelasticity of Zinc", J. Appl. Phys. 23, 866(1952).
7. J.O. Kessler, "Internal Friction and Defect Interaction inGermanium", Phys. Rev. JJtà, 646 (1957).
8. W.J. Baxter and J. Wilks, "Instabilities in the Internal Fric-tion of Some Specimens of Copper", Phil. Mag. _7> 427 (1962).
9. I.C-. Ritchie and A. Atrens, "The Diffusion of Oxygen in Alpha-Zirconium", J. Nucl. Mater. 67, 254 (1977).
10. A. Atrens, I.G. Ritchie and K.W. Sprungmann, "Time and Ampli-tude Dependent Damping in a Single Crystal of Zirconium", Pro-ceedings 6th International Conference on Internal Friction andUltrasonic Attenuation in Crystalline Solids, R.R. Hasigutiand N. Mikoshiba, eds., University of Tokyo Press, Tokyo,1977. p. 633
11. W. Winkler-Gniewek, S. Schlipf and R. Schindlmayr, "Disloca-tion Damping Due to Mobile Pinning Points", Proceedings 5thInternational Conference on Internal Friction and UltrasonicAttenuation in Crystalline Solids, D. Lenz and K. Liicke, eds.,Springer-Verlag, Berlin, 1975. Vol. II, p. 246.
12. J. Friedel, Dislocations, Pergamon Press, London, 1964. p. 291.
- 21 -
13. A.S. Nowick and B.S. Berry, Anelastic Relaxation in Crystal-line Solids, Academic Press, Hew York, 1972. p. 401.
14. J.L. Gacougnolle, S. Sarrazin and J. de Fouquet, "InternalFriction Due to Oxygen in Ultra-Pure Zirconium", Proceedings5th International Conference on Internal Friction and Ultra-sonic Attenuation in Crystalline Solids, D. Lsax and K. LUcke,eds., Springer-Verlag, Berlin, 1975. Vol. I, p. 343.
15. P.G. Fuller and D.R. Miller, "Internal Friction and OxygenPair Diffusion ii. Alpha-Zirconium", Met. Sci. 11, 16 (1977).
16. I.G. Ritchie, K.W. Sprungmann, A. Atrsns and H.E. Rosinger,"Anelastic Relaxation Peaks in Single Crystals of Zirconium-Oxygen Alloys", Proceedings 6th International Conference onInternal Friction and Ultrasonic Attenuation in CrystallineSolids, R.R. Hasiguti and H. Mikoshiba, eds., University ofTokyo Pressa Tokyo, 1977. p. 447.
17. E.A. Gulbransen and K. Andrew, "Crystal Structure and Thermo-dynamic Studies on the Zirconium-Hydrogen Alloys", J. Elec-trochem. Soc. JL01, 474 (1954).
18. P. Peguin, J. Perez and P. Cobin, "Amplitude-Dependent Part ofthe Internal Friction of Aluminum", Trans. Met. Soc. AIME "Î3JL»438 (1967).
19. J.J. Kearns, "Terminal Solubility and Partitioning of Hydrogenin the Alpha Phase of Zirconium, Zircaloy-2 and Zircaloy-4",J. Nucl. Mater. 22, 292 (1967).
20. CD. Cann, A. Atrens, E.E. Sexton, F. Havelock, I.C. Ritchieand K.W. Sprungmarn, "Determination of the Terminal SolidSolubility of Hydrogen in Zirconium at Low Hydrogen Concentra-tions", Atomic Energy of Canada Limited Report, AECL-5996 (1978).
21. C D . Cann and A. Atrens, J. Nucl. Mater., in press.
22. I.G. Ritchie, A. Atrens and D.G. Blair, "Anomalously Large Am-plitude Dependent Damping and Modulus Defect in Polycrystal-line Zirconium", Proceedings 6th International Conference onInternal Friction and Ultrasonic Attenuation in CrystallineSolids, R.R. Hasiguti and H. .Mikoshiba, eds., University ofTokyo Press, Tokyo, 1977. p. 639.
23. L.G. Bell and A. Sawatzky, "Strain Induced by Thermal Cyclingof Hydrided Ozhennite 0.5", Atomic Energy of Canada LimitedReport, AECL-4761 (1974).
- 22 -
24. G.J.C. Carpenter and J.F. Matters, "An In-Situ Study of theDissolution of Y-Zirconium Hydride in Zirconium", J. Nucl.Mater. 73.» I 9 0 (1978).
25. B. Nath, G.W. Lorimer and N. Ridley, "Effect of Hydrogen Con-centration and Cooling Rate on Hydride Precipitation ina-Zirconium", J. Nucl. Mater. 5J5, 153 (1975).
26. L.J. Teutonico, A.V. Granato and K. LUcke, "Theory of ThermalBreakaway of a Pinned Dislocation Line with Application toDamping Phenomena", J. Appl. Phys. 3J5, 220 (1964).
27. J. Schlipf and R. Schindlmayr, "Theory of Dislocation Dampingin Dilute Alloys", Proceedings 5th International Conference onInternal Friction and Ultrasonic Attenuation in CrystallineSolids, D. Lenz and K. Liicke, eds. , Springer-Verlag, Berlin,1975. Vol. II, p. 439.
28. H.M. Simpson and A. Sosin, ''Contribution of Defect Dragging toDislocation Damping", Phys. Rev. B 5, 1382 (1972).
29. C.L. Bauer, "The Free Energy of a Pinned Dislocation", Phil.Mag. 11, 827 (1965).
30. E. Bode, "Equilibrium Distribution of Pinning Points at Dis-locations Under an External Vibratlonal Stress of Small Ampli-tude", Phil. Mag. U, 275 (1966).
31. G. Alefeld, "Grouping of Pinning Points on Dislocation Lines",Phil. Mag. LI, 809 (1965).
32. D.G. Blair, T.S. Hutchison and D.H. Rogers, "Theory of DampingDue to Thermally assisted Unpinning of Dislocations", Can. J.Phys. 49, 633 (1971).
33. R. de Batist, Internal Friction of Structural Defects in Cris-talline Solids, North-Holland, Amsterdam, 1972. p. 347.
34. J. Perez, P. Peguin, G. Fantozzi and P. Gobin, "Anelastic Phe-nomena Resulting from the Interaction of Dislocations withPoint Defects", Ann. Phys. 5, 303 (1970).
35. M.A. Krishtal and M.A. Vyboishchik, "Effect of the Volume ofDiffusing Elements on the Parameters of Diffusion along aDislocation", Fiz. Met. Metalloved. 38, 118 (1974).
36. O.P. Quist and S.H. Carpenter, "Snoek Pinning of Dislocationsin High Purity Niobium", Acta Met. 22, 321 (1975).
- 23 -
37. E.C. Oren, N.F. Fiore and C.L. Bauer, "Solute Atom-DislocationBinding in Dilute Copper Alloys", Acta Met. 14, 245 (1966).
38. R. Bullough, "Influence of Dislocation-Point Defect Interac-tions on the Kinetics of Recovery Processes". Proceedings ofConference on the Interaction Between Dislocations and PointDefects, Harwell Report AERE-R 5944, 1>'68. Vol. 1, p. 22.
39. K. Bungardt and H. Prelsendanz, "Damping and Modulus of Shearof Zirconium and Zirconium-Hydrogen Alloys", Z. Metallk. 51,280 (1960).
40. F. Povolo and E.A. Bisogni, "Internal Friction in Zirconium-Hydrogen Alloys at Low Temperatures", J. Nucl. Mater. 29_,82 (1969).
41. V. Provenza.no, P. Schiller and A. Schneiders, "Internal Fric-tion Study of Zirconium Alloyed with Hydrogen and Deuterium",I. Nucl. Mater. 52, 75 (1974).
42. F.M. Mazzolai, J. Ryll-Nardzewski and C.J. Spears, "An Inves-tigation of the Zirconium-Hydrogen System by Internal Fric-tion'1, Nuovo Cimento 231, 251 (1976).
43. B.T. Loh, Ph.D. Thesis, Rensselaer Polytechnic Institute,Troy, New York, 1965 (Published by University Microfilms Inc.,Ann Arbor, Michigan, 1968).
44. G. Schoeck, "Internal Friction Due to the Interaction BetweenDislocations and Solute Atoms", Acta Met. 11, 617 (1963).
45. F.M. Mazzolai and J. Ryll-Nardzewski, "An Anelastic Study ofthe Diffusion Coefficient of Hydrogen in Alpha-Zirconium",J. Less-Common Metals, 49, 323 (1976).
46. P.P. Tung and A.W. Sommer, "A Study of Dislocation-HydrogenInteraction in a-Titanium via Internal Friction Measurements",Acta Met. 22, 191 (1974).
- 24 -
T A B L E 1
A N A L Y S I S OF T I M E - D E P E N D E N T RECOVERY A T N - 7 . 5
Temperature°C
316
238
225
200
186
180
163
154
146
124
105
Tine Exponent
Stage 1
0.45
0.97
0.64
0.50
0.73
0.83
1.7
1.2
0.93
0.66
0.75
n of Recovery
Stage 2
0.45
0.57
0.43
0.50
0.49
0.53
0.33
0.55
0.43
0.45
0.49
i
- 25 -
T °C
70
80
90
100
108
120
1
2
2
2
2
2
.83
.23
.58
.87
.97
.86
A
X
X
X
X
X
X
ANALYSIS OF THE
ACCORDING TO
le"3
lu"3
lO"3
10"3
10" 3
10-3
0
0
0
0
0
0
.395
252
152
106
070
042
TABLE_ 2
PEAK (N =
SCHOECK'
0
0
0
0
0
0.
'eriod(s)
13520
13562
13605
13646
13680
13733
2) IN FICURE
S TIIEOKY
In
- 10
- 11
- 11
- 12
- 12
- 12
(44)
i/T
.61
.08
.61
.00
.43
.96
1
2
2
2
2
2
2
T"1
.92
.83
.76
.68
.62
.54
(K"1)
x 10
x lO"3
x 10
x 10
x lu"3
x 10
T These values were found by comparison of the experimentaldamping values with the result given by Sclioeck'4''' for anexponential distribution of dislocation network lengths.
3.0
O
a>Eo0)
oo
iS 1.0o
Heating
Surface Strain Amplitudes
Repeated Free Pendulum at Rest N = 1 1.55 x 10"6
Decays at 144OC at 232CC for 16 hour N = 2 3.10 * 10~6
N = 5 7.75 x 10"6
200
Temperature C
400
FIGURE 1: (a) Damping During Continuous Cooling from 700°C While the Reed Pendulum was MaintainedVibrating at N = 2. The hatched region is the temperature range vnere 'gasping' wasobserved.
(b) Variation in Damping Between Nominal Amplitudes, N = 5 and N = 2, at Selected Temper-atures in the Ambient to 450°C Temperature Range.
2 0.153
O
asin
0.05
Pendulum maintained at N = 2( Strain Amplitude 3.1 x 10"«)Coil Driver at 0.5 mA/V
Pa
100 200
previous test at 105°C
Heating at 2°C per minute
Cooling at 0.5°C per minute
Region o f
Gasping
2.0
o
1.0
TemperaîureX400
FIGURE 2: Damping Variation with Annealing Temperature During a Typical Programmed VibrationAnnealing.
- 28 -
Surface Strain Amplitude * 106
5 10
CO
X
e0)E£ 1.3a
CO
o
186°C
+
+ xx*x
xx
+xx ..* Memory
*x • • _+ .• Free
+ 0X 1
1
+ + +
+.+
x x » x X X i <
X *
at N = 2
Decaysminuteshour
• Final Value
i i
• • • * •
x X x x x
from N
1
+ + + '
X X X X
•
= 7.5
—
12 4Nominal Amplitude (N)
FIGURE 3: Amplitude-Dependent Damping as Measured by Free Decays from N = 7.5at Times t = 0, 1 h and t •+ <» During Vibration Conditioning at N = 7.5.
0.11
0.09
S
0.0'
»••„,
100
Temperature 238°C
Nominal Amplitude K = 7.5
(Strain Amplitude 1.16 x 1
300Time (minutes)
1.4
1.2
I
1.0
100
FIGURE 4: Typical Time-Dependent Recovery Curve During Vibration Conditioning at N = 7.5. An initialincrease in damping, caused by the step increase in strain amplitude, recovers graduallyover a period of about five hours.
- 30 -
Surface Strain Amplitude * 106
5 10
1.6
1.7
ox
£
0)
Oo
ito
o
1.5
1.4
1.3
1.2
I.I
1.0
0.9
1
— • — 0 minutes—X— 1 hour
— Final ValueFree Decay from N = 9
Temperature 186°C /
/
N Damping Levels at N = 2after Full Recovery at N
Memory at N = 2
4 6Nominal Amplitude
FIGURE 5: Damping Values at Varijus Amplitudes at t = 0, t = l h andt •* » Compared with a Free Decay from N = 9. Each step upin amplitude was proceeded by a programmed vibration anneal-ing. Also shown are several damping values at N = 2 aftercomplete vibration conditioning at the indicated higheramplitude.
CO
o
E
1.3
.2
I.I
Surface Strain Amplitude * 106
5 10
Q 1.0o
•"H 0 .9re
0.8
I
— X - -
— 0 minutes- 1 hour
Final Value
Temperature 186°C
Free Decay from FinalValue at N = 8
I I4 6
Nominal Amplitude8
FIGURE 6: Damping Values at Various Values of N for t = 0 and t = 1 h Compared with a FreeDecay from N = 8 after Complete Vibration Conditioning. No programmed vibrationannealing was followed between the step increases in amplitude.
).O9
H<BS00'Htfi
> (11•H Ul-l iH
a o
O O
Nor
mal
ized
0»
D/N
(V
Temperature 186°CNominal Amplitude N =
1
7 af;:er a
••
1
step increase from N = 6
Surface Strain Amplitudes
N = 6 9.3 x 10"^N = 7 1.09 x 10~5
!
1
150
- I.I
" 1.0 ho
I
0.92 0 0
FIGURE 7:
100 15Time (minutes)
Peaking Effect Observed During Vibration Conditioning Following a Step Increase from N = 6 to 7.
1 0
0.0
-1.0
-2.0
-3.0
TEMPERATURE 186°C
TIME-DEPENDENT DAMPING a t N = 7 . 5
QS slope 0.73
slope 0. '9
In Time (minutes)
FIGURE 8: Relationship Between In r and In t for a Typical Time-Dependent Recovery During VibrationConditioning at N = 7.5.
10
I L
Actii. En. 0.16 eV• 98.? s
I I
1.6 1.7 1.8 1-9 2.0 2.1 2.2 2.3 2.4 2.5
T"1 or1)
FIGURE 9: Relationship Between In r and T for the Intercepts of the n ^ 1/2 Kinetics
at Temperatures > 180°C.
- 35 -
-10
- I I
-12
-13
à
/
7
QM + EB = 0.53 eV/atomEB = 0.19 eV/atom
2.6 2.8T"1, (K"1), x 10
3.0-3
3.2
FIGURE 10: Relationship Between In (T/T) and T for the Low-Temperature Side ofCurve (a), Figure 1.
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