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i.J'UTNAM, I.G. RITCHIE tfnd>J. HUGHES
a 0
^""Whiis^heJI-Myde^r
A PRELIMINARY SURVEY Of LOW FREQUENCY INTERNAL FRICTION IN
SOME ZIRCONIUM ALLOYS USING AN IMPROVED TORSION PENDULUM
by
J . Putnam*, I .G. R i t c h i e and F .J . Hughes
* Summer student from the University of Manitoba
ABSTRACT
This report outlines the basic approach to a study of the low frequencyinternal friction of zirconium and i ts alloys. To investigate point defectrelaxation peaks associated with alloy additions the number of grain boundarieswas reduced by using wire specimens with a bamboo structure. To test thesespecimens, an improved torsion pendulum was developed with associated electronicdata handling facil i t ies. Its unique features include the following:
(1) The specimen deflection is measured using a non-contacting fibreoptic transducer; the use and calibration of this device are described.
(2) A servo -mechanism enables the pendulum to be driven at constantstrain amplitude. Details of this novel approach are given and a direct relation-ship is shown to exist between the drive current and the damping of the specimen.
(3) A programmable temperature controller, in combination with theconstant amplitude drive function, enables damping versus temperature curves to begenerated automatically.
(4) A PDP 8/E computer is interfaced with the pendulum for data loggingand ini t ia l data reduction and curve fitting.
Damping peaks in zirconium have been measured from room temperature to700°C at a frequency of 1 to 3 Hz. Damping has been measured using the pendulumin both i ts decay and i t s constant strain amplitude modes, and shown to be quan-titatively similar. The experimental errors of both these modes have been analyzedandr found to be similar.
Damping peaks due to the a/3 phase transformation, grain boundarysliding and stress-induced reorientation of substitutional-interstitial atom paxrshave been observed. These results are discussed and their importance to possiblestrengthening mechanisms in zirconium alloys is indicated.
The improved torsion pendulum has proved to be ideal for studying pointdefect relaxation peaks in zirconium alloys, and recommendations are made fordetailed investigation of this phenomenon.
Atomic Energy of Canada Limited
Whitesftell Nuclear Research Establishment
Pinawa, Manitoba, ROE 1L0
September 1972
AECL-3953
Enquête préliminaire sur la friction interne à basse
Fréquence dans quelques alliages de zirconium employant
un pendule de torsion amélioré
par
J. Putnam*, I.G. Ritchie et F.J. Hughes
*Etudiant de l'université du Manitoba en stage d'été à l'EACL
Résumé
Ce rapport donne un aperçu de la façon d'aborder une étude de lafriction interne à basse fréquence du zirconium et de ses alliages.Afin de pouvoir enquêter sur les pics de relaxation des défauts depoint associés aux additions d'alliages,le nombre des limites de graina été réduit en utilisant des spécimens de fil ayant une structure debambou. Afin de mettre ces spécimens à l'essai, on a développé unpendule de torsion amélioré comportant des moyens électroniques pourtraiter les données. Ce pendule a des caractéristiques exceptionnelles:
1) La déflexion des spécimens est mesurée au moyen d'untransducteur optique à fibre non-contactante; on décrit l'utilisationet l'étalonnage de ce dispositif.
t L i
2) Un servomécanisme permet au pendule d'etre entraîné à uneamplitude constante d'efforts. On décrit cette nouvelle méthode eton montre qu'il existe une relation directe entre le courant d'entraînementet l'humidification du spécimen-,
3) Un contrôleur programmable de température fonctionnant encombinaison avec la fonction d'entraînement à amplitude constante permetaux courbes donnant l'humidification en fonction de la températured'etre engendrées automatiquement.
4) Un ordinateur PDP 8/E est relié au pendule pour_1'enregistrementdes données, la réduction initiale des données et L'ajustement descourbes..
Les.pics d'humidification ont été mesurés dans le zirconium à partirde la température ambiante, jusqu'à 700f>c à une fréquence de 1 à 3 Hz._L"*humidification a été. mesurée au moyen du pendule aussi bien dans sonmode de décroissance que dans son mode d'amplitude constante d'"effoTts etoh a constaté qu'il est quantitativement semblable. L'es erreursexpérimentales de ces deux modes ont,été analysées,et elles se sont- _avérées semblables. - ' -,
', Les pics d'humidification dus à. la transformation de phase o/g, • •le glissement des limitas de grains et une réorientation, engendrée parla contrainte, de paires d'atomes substitutionnels-interstitiels ont étéobservés. Ces résultats font l'objet de commentaires et leur importance -pour des mécanismes possibles de renforcement dans les alliagés^de . , 'zirconium est mentionnée.
Le pendule a torsion amélioré s'est,avéré idéal pour étudier les %pics de relaxation des défauts de point dans les alliages de zirconium.Les'auteurs recommandent l'étude détaillée"de ce phénomène. "- " _ . " •
L'Energie Atomique du Canada, Limitée ._ Etablissement de Recherches Nucléaires de Whiteshell-
Pinawa, Manitoba ROE 1L0Septembre 1972
AECL-3953
CONTENTS
Page
1.
3-
4.
5.
6.
7.
INTRODUCTION1.1 THEORY
1.2 PREVIOUS WORK
APPARATUS
EXPERIMENTAL TECHNIQUE
RESULTS
DISCUSSION AND RECOMMENDATIONS
REFERENCES
ACKNOWLEDGEMENTS
1
1
4
6
9
11
13
15
16
TABLE 1 Specimen Data
TABLE 2 Grain Size^ Damping and Oxygen Analysis after Testing
TABLE 3 Oxygen Peaks in the Zr-0.18wtZCu Bamboo Specimenr 10 T'ollowing Secondary Anneals before Testing
FIGURES
16
17
17
18
APPENDIX 1 Vibration Isolation-Consideration of the Existing Sub-- Base and General Recommendations and Cr i t e r i a for theConstruction of Sub-Bases for Low Frequency InternalFr ic t ion Apparatus
APPENDIX 2 Calculation of the Relationship between Drive Current( I o ) and Logarithmic Decrement (A)
APPENDIX 3 Estimation of Errors in the Logarithmic Decrement (A)due to Experimental Errors in the Measured Amplitudesof Free Decay
APPENDIX 4 'Calculat ion of Surface Strain
27
29
34
35
- 1 -
1. INTRODUCTION
Low frequency internal friction studies should provide a
useful technique for the study of strengthening mechanisms in zirconium
and its alloys. A torsion pendulum with a unique electronic servo system
has been built, which sustains the oscillation of the specimens at constant
amplitude. High precision measurements of internal friction in zirconium-
copper alloys between 1 and 3 Hz over the temperature range of room Leu.-
perature to 7OC°C have demonstrated the performance of the equipment and
the potential of the method.
1.1 THEORY
Internal friction is a phenomenon whereby the mechanical
energy associated with the motion of a material subjected to an oscilla-
tory stress is converted into heat which is irreversibly transferred to
the surrounding environment. It occurs whenever the stress and strain
associated with a vibratory mode are out of phase with each other. Such
deviations from perfect elastic behaviour cause some of the mechanical
vibrational energy to be dissipated in each cycle resulting in a progres-
sive attenuation of the amplitude of vibration. This process is referred
to as damping. An experimental measure of damping can be expressed as
the natural logarithm of the ratio cf the amplitudes in two successive
vibrations. This is then called the logarithmic decrement (A) and gives
an indication of the fractional decrease in vibrational amplitude per
cycle. Neglecting amplitude dependence,
A = £ lJ^) [1]N \ A N /
when
N = number of oscillations between A o and AN,
- 2 -
A o = amplitude of first oscillation,
A N = amplitude of Nth oscillation.
From equation [1]
Equation [2] £s of J:he linear |prm,, Y;.=rmx)+ h,,, wheret, m,^Jhe
In A N vs N graph,,isminus .the. ,lpga.riJthM.q.s, ecreTOnt,.._, 8,.,,..^ ampli£u4e
falls off exponentially,with the. number ,p£ oscillations..;, ,; :J ,,, -? •
Another measure of internal friction is_rrthe fractional decrease
of vibrational energy per cycle.
A = -^il ;- [3]2W -fHu3H7 . - i'.f
whereW. ~ maiainum strain energy in one cycle per,,unit _yplume»
r 6W .= , lpssin strain energy.per, unit qlumein.that c^cle. :
Damping behaviour can result from several sources of anelastic strain.
This report deals with an investigation directed primarily towards an
understanding of internal friction due to the possible stress-induced
ordering of oxygen interstitial atoms in zirconium. The crystal structure,
of zirconium is close packed hexagonal (CPH) up to a temperature of 862°C,
where it transforms to the body centred cubic (BCC) phase. These are
referred to as the a and 6 phases respectively. In the work presented
here, we confined our attention to the low temperature phase where large
amounts (about 29 atomic %) of oxygen.can exist in solid solution in the
hexagonal lattice in interstitial sites. Of the two types of interstitial
positions available, the size of the oxygen atom requires that it generally
occupies the larger octahedral site. If the application of a stress
field renders some of the octahedral sites energetically more favourable
- 3 -
for oxygen atom residence, stress-induced ordering will occur with an
associated anelastie relaxation. In the presence of an oscillating stress,
such behaviour will give rise to damping.
'•JL'M: irajSPiIt^is^important«=to note thatrthe.stress-directed jump of the
Gxygeii^atbm" f ibm;qnefesitei tbtahbther issa?thermally. assistedT process and
therefbre^ atS anyS givenStemperature*thereiisia?"resonant" frequency of
vibraticii at which the!damping reaches;;a maximum. This coincides with the
maximum number of interstitial atoms that are excited, by the stress, over
the: activation barrier which must be surmounted during the jump process.
Below this maximum, the temperature of the material is such that the
average thermal energy packet that each atom is statistically receiving
during one stress cycle is insufficient in most cases, even with the assis-
tance of the applied stress, for the atom to complete a jump successfully,
and so the degree 6fJ damping is low.>;fln addition, above the maximum,
normal? thermal^ activation isi sufficient in Itself (without the aid of the
applied stress)" tb« overcome the jump barrier within a stress cycle and
hence, again, the dai^ing level %s;low. Such a phenomenon gives rise to
a peak in the damping :vs frequency curve at", a fixed temperature. A
corollary of this is that a peak occurs in the damping vs temperature
c u r v e 5 a t f - a ^ f i x e d ? * f r e q u e n c y i - • • l ' " r ^ ' r . : " . ? : i ' c : ' f j i ; : -'•••-• •
Although- this report is.•? directed at internal friction due to
stress-induced ordering of oxygen interstitials in zirconium, there are
other sources of internal friction in this system which cannot be neglected.
One is due to grain boundary relaxation. In poly crystalline materials,
this occurs through a process of thermally assisted sliding of one grain
over another, motion being confined to the grain boundary region. A
damping peak due to this phenomenon has beeri found in the 500°C to 600 G
range by several workers^1>z^. The other source of internal friction is
due to the a/3 transformation. It results from viscous flow of the
lattice during the transformation, with a maximum in the damping vs ;
temperature, curve occurring generally at about 860°Cs which corresponds
to the a/3 pha&~ transformation.temperature in zirconium. : There are also
other sources of damping which contribute to the total background damping,
e.g., that: due. to movement iand« unpinning: of ; dislocations^ though this i
contribution is apparently very small within the= experimental range inves-
tigated by us.
A theoretical expression of damping due to thermally activated
relaxation phenomena (e.g., stress-induced ordering and grain boundary
sliding) is found from classical rate theory which yields the following
equation relating damping to temperature (at fixed frequency):
rp
A = A -2- sech | (i - | ) [4]
where
A = logarithmic decrement or damping
• A- = amplitude of the damping temperature peak
T = peak' temperature (K)
T = temperature (K)
Q = activation energy of process
R = universal gas constant.
It should be noted that Q, the activation energy, is a quantity which
relates directly to the kinetics of the relaxation process, e.g., in the
case of stress-induced ordering, it is the enthalpy required for the
appropriate atomic jump; for graift boundary relaxation, it isn associated
with the details of the atomic and dislocation processes involved in grain
boundary-sliding, "-- ' - - - " "___.-'.
1.2 ' PREVIOUS WORK
' • Very little is,known about the a/B phase transformation damping
phenomenon. ,No theoretical equations have been developed for it. -• It is -
independent of frequency and therefore always peaks at the same~ temperature.
- 5 --
The major experimental difficulty to be resolved in an inves-
tigation of oxygen peaks in zirconium is that this peak is superimposed
on both the grain boundary and phase transformation peaks. In addition,
in the case of polycrystalline material, the comparative magnitude of fthe-
oxygen peak is generally small thus" combining to make resplutipn difficult.
In some systems, i t is possible to separate superimposed peaks,byS-a*.
frequency change; unfortunately, in our case, the-activation energies of
the grain boundary and oxygen peak seem to be sufficiently similar in
magnitude that they can only be shifted as a pair. Also, the phase trans-
formation; peak; is so; dominant that i t : i s unavoidably a niajor contribution
to the background level;.
Four experimental approaches seem available to overcome this
difficulty.
(1) The grain boundary peak can be reduced by reducing the
number of grain boundaries within the material, i . e . , approach single
crystal conditions. •.',-. ";
(2) I t is also possible to reduce the grain boundary peak by
reducing grain boundary sliding through impurity or alloy addition.; <<3)s. The height of the oxygen peak can be increased by in-
creasingthe oxygen content; There is an optimum concentration of dis-
solved oxygen above which saturation of nearest-neighbour octahedral sites
occurs, therefore reducing the probability of a jump occurring and thus
reducing the damping level.
(4) Gupta and Weinig^3^ have shown that in the case of
titanium.-.(similar in. many respects-to zirconium), the oxygen peak is
enhanced by the addition of a substitutional impurity.
Studies of internal friction due to interst i t ial oxygen relaxa-
tion- ;in metals at high temperatures have been limited, especially in
zirconium alloys.. Gupta and Weinig*3-* and Pratt et a l . ^ have studied
the alloy system Ti-X-0 and found interst i t ial peaks (X represents a sub-
stitutional alloying element). Bisogni et al , ( 5^ have observed a damping
peak at about 480"C (0.9 Hz) in Hf-Zr-0 alloys.
- 6 -
Within the last year, two reports have been published concern-
ing such phenomena in zirconium alloys. Mishra and Asundi^6' have studied
zirconium containing tungsten, iron and silicon together with oxygen.
Oxygen"peaks were found in most cases ranging from 410°C to 495°C. These
workers conclude that it is necessary to introduce a substitutional atom
into the zirconium lattice in order for a relaxation process to be obser-
\'able. Relaxation is thought to be due to the stress-directed ordering of
the strain dipole,created by the interstitial/substitutional pair.
••••-•i -•-=>- Browne??- also-reports aCpeak in zirconium containing 2.4% '
hafnium and 0.01% oxygen. He finds a peak at 460°G 0si93 Hz)- and 442"C;
(0.75 Hz). Browne suggests that the peak mechanism could be due to an
interstitial-interstit-ial: pairsas^weir^as- the ititerstitial-substitutional
pair of the type described by Gupta and Weinig^3\ Bratina and
however, studied Zr containing Hf and a multitude of other impurities.
They found only a transformation peak, a grain boundary peak and an
anomalous peak below 350°C.
2. APPARATUS
The inverted torsional pendulum used in this study is shown
schematically in Figure 1 [see also Figure 2 (a) and (b)]. Torsional
motion is initiated in the vertically suspended wire specimen by momen-
tarily pulsing the electro-magnets which attract soft iron rings attached
to the end of the inertia bar. Lateral oscillations, brought about by
slight misalignment of the electro-magnets, are rapidly damped out by a
suitably located dashpot. The ensuing decay in the amplitude of the
torsional vibrations is continuously monitored by a Fotonic Sensor*. The
Fotonic Sensor is an optically coupled displacement transducer. This
transducer, and its application to the measurement of damping of low
frequency oscillationst has been described in detail by Sprungmann and
* Trade name, MTI Instruments Division.
- 7 -
Ritchie . The frequency of the pendulum may be conveniently changed by
repositioning the balance weights on the inertia bar. The range of fre-
quencies available, however, is mainly controlled by the dimensions of the
specimen used.
" - "*"= • Results5at- frequencies in the range 0.8 to 2.25 Hz are; presented
in this report.- The electronic data-logging equipment used to measure and
displaythe vibration amplitudes issdescribed in reference (8). A PDP 8/E
computer is; used for initial data reduction and curve-fitting.
The pendulum chamber can be evacuated to a pressure of less
than ; 10~" mm Hg by a conventional vacuum system. This vacuum can be
maintained up to temperatures of 700°C. Damping due to residual atmosphere
in the pendulum is less than the background damping (in the alloys tested)
for pressures; below 10~- mm Hg. Temperature is adjusted'manually by con-
trolling the current supplied to a three-zone, non-induetively wound
resistance furnacei
In order to isolate the specimen from spurious building vibra-
tions, the pendulum assembly is mounted on a solid concrete base supported
by wooden beam absorbers. Further discussion of the problems associated
with this structure is presented in Appendix 1.
Having outlined the apparatus in general, certain components
and systems will be discussed more fully with an emphasis on any signifi-
cant improvements which have been made. The existing furnace arid manual
controller are only just adequate for the preliminary study reported here.
Temperatures as high as 1000°C have been reached with the existing system,
demonstrating that the heating capacity is more than adequate. However,
the temperature gradient observed during three random tests was found to
be 2.5 degC over the specimen length. This is unacceptably high for precise
damping measurements. Such gradients were measured using three thermo-
couples silver-soldered to a dummy specimen.
To improve the temperature control system, the torsion
pendulum furnace was connected to an automatic temperature controller.
The temperature controller has been used to control temperature and tem-
perature gradients in a reed pendulum apparatus. It has been described
in detail by Sprungmann and Ritchie^. With this system, the temperature
gradient £ w a s a ^ -35,0- t o =
70Q°G. Most, of the^results contained in this.report were^ obtained using
i t h i s = s y s t e m s ,;.-.•.. .••••- J j . v . / ' i H ; i ' •-:•' -• ••• ••'• •"*•>•'•••' ? ^ , 1;- ••-; • — ^ ; - •
During the:-l;atter part ;o£ this study, a.different technique
for measuring internal friction was attempted. The torsiohal pendulum
was driven at constant strain amplitude by the non-linear servo-mechanism
shown in the b lode?'diagram of Figure 3. Internal1 friction in this case
is measured by the energy input to the system as described in the second
method (eqn [%¥) in Section 1.1*
The Fotonic Sensor detects the instantanebus position of the
inertia arm. Its output is connected to the peak (Tetectibn circuit de-
scribed in reference (8) , and to an integrator. The integrator shifts the
phase of the displacement signal by 90° to generate a signal of the proper
phase to cancel the damping terms in the equation of motion. The output
of the peak detector is updated every cycle and is subtracted from the
reference amplitude to produce an error signal. Thus the error signal
changes in steps. This signal is fed into a proportional plus integral
control circuit, the output of which is multiplied by the integral of the
motion to form a current drive signal. Coupling to the pendulum is mag-
netic in nature. The drive current is fed to a pair of Helmholtz coils
which produce a region of uniform magnetic field. This in turn exerts a
couple on a small cylindrical permanent magnet inserted in the pendulum
rod. The magnet insert is situated so that its axis is perpendicular to
the pendulum axis. The photographs in Figure 2(a) and (b) show the posi-
tion of the Helmholtz coils.
A signal, proportional to the rectified signal in the Helmholtz
coils (and hence proportional to the damping), is applied to the Y axis
of an X-Y plotter, while the signal from the controlling thermocouple is
applied to the X axis. In this way damping versus temperature curves at
- 9 -
constant strain amplitude are automatically displayed as the test proceeds.
Derivation of the relationship between logarithmic decrement and drive
signal is presented in Appendix 2.
Tests with the drive system outlined above have been very
successful. The X-Y plots indicate very little background noise and tests
can be run unattended overnight, when background noise is at a minimum.
The system exhibits remarkable stability even under adverse conditions and
responds rapidly to a step change in the reference amplitude. A maximum
error of about 1.5% has been estimated in driving at constant amplitude.
This is acceptable and compares favourably with errors involved in measur-
ing free decay (see Appendix 3).
The effect of the length of the suspension fibre on stability
is worthy of note. With the suspension fibre about 5 inches long, it was
shown that the torsional and lateral rigidities of the pendulum were
about equal. Under these circumstances, slight instabilities introduced
lateral vibrations. This problem was solved by increasing the flexural
rigidity of the system by reducing the length of the suspension fibre to
about Xh inches.
Since some specimens exhibited torsional permanent set after
in-situ annealing, a DC motor drive was attached to the Fotonic Sensor
holder, so that the gap between the sensor and inertia arm could be adjusted
in vacuo. This has proved to be an important modification. The DC motor
and Fotonic Sensor holder are shown in Figure 2(a) and (b).
3. EXPERIMENTAL TECHNIQUE
Apart from pure crystal bar zirconium used for this investiga-
tion f-3 alloys were prepared using a vacuum arc melting furnace. The
alloying elements chosen were copper, titanium,, aluminum, iron and yttrium,
- 10 -
the exact composition being given in Tables 1 and 2. No deliberate
attempt was made to add oxygen to the alloys, the oxygen contents
reported in Table 2 being of parasitic origin. The small ingots of alloys
were then cold swaged and cold drawn with intermediate anneals into wire
form with-a diameter of about^6 .05 :incHesy; five-inch ilbng specimens were
c u t f r o m t h i s w i r e * ' f ' « ST7;i7': T ' — " -?--.->•*•" J - -^••---••s - -'•- <*>•---<•
The effective length between the grips -wa's abbut 3*5 inches,
resxilting in a frequency of - 2 Hz. -- This frequency was chosen because the
drive system was designed for a frequency range from 2 to 30 Hzv At
frequencies much greater than 2 Hz, thermally, activated peaks," such as the
oxygen peak and the grain boundary peak, move to higher temperatures.'
This leads to further masking of the oxygen peaks by the tail of the
transformation peak, which does not shift with frequency. At frequencies
much less than 1Hz* the; time required to -log a given data point becomes ;
significant and improved temperaturestability Is requiredV: The, surface • -
strain amplitude^ e y is also determined by the dimensions and geometry of
the specimen, e is derived in Appendix 4; The^rarige -of e values used ;iri:'
these tests was 4.0x10" 6 to 3.5x3.0~5; For this surface strain ampilitude
range, damping was found to be essentially strain amplitude independents
Bamboo structures were "obtained using" the following thermo-
cycling heat treatments on specimens wrapped in tantalum foil and sealed
in Vycor tubes containing Vs of ah atmosphere of argon.
3 consecutive cycles
Before testing, the specimens were annealed at 700°C in the
torsional pendulum to remove handling strains.
Tests were carried out with the apparatus evacuated to an
initial pressure of less than lxlO~ 3mm Hg. At this pressure^ the lowest
damping measured was A = 4X10"1*. A discussion of the errors incurred in
840
840
1200
840
°C
°G
°C
°C
for
for
for
for
74
6
7
days
days
hours
days
- 11 -
measuring the logarithmic decrement by the decay method is presented in
Appendix 3. The maximum percentage error in A is estimated to be less
than 2.5%.
-,'-.-• A number of programs7were devised to facili tate on-line data
reduction and curve fi t t ing 'with a VW: Q/E computer.
4 . RtSULTS
Some of the results obtained by the decay technique for nine
specimens are tabulated in Tables 1, 2 and 3 together with details of
heat treatments $ grain size and oxygen contents (measured after testing).
FigureT4 shows three tests oil pure Zr specimens. Curves (1), (2) and (3)
are for small-grained, large-grained and "bamboo"-grained structures
respectively. The results for curve (3) were obtained at a frequency of
0.82 Hz compared with frequencies of 1.86 Hz and 1.73 Hz for curves (1)
and (2) respectively. For" a meaningful comparison of these curves, the
results must be normalized to the same frequency. After normalization of
the results to a frequency of 1.85 Hz, it was found that, at the high
temperature end^f the curves, the damping decreased as the grain size
increased, while at the low temperature end of the curves the background
damping increased slightly as the grain size increased. No oxygen peak
was detected in these specimens."
Results for two tests on a Zr-0.18 wt% Cu alloy are shown in
Figure 5. Curve (1) is for a specimen with a relatively small grain size,
while curve (2) is for a specimen with a very much larger grain size.
The former has an oxygen peak at 450 °C and the latter shows a peak at
455°G. Figure 6 shows damping vs temperature curves for the other four
Zr alloys tested. No oxygen peak was observed in these alloys.
- 12 -
Figure 7 is an example of a computer pr int-plot generated by
the PDP 8/E computer. The data in this figure are the same as Figure 6,
curve (d) (specimen number 5) , together with a theoretical grain boundary
damping curve. The parameters cfioTsen" for the generation of the theoretical
grain boundary peak J^& iistSdibnjtKei computer ^ r in t^plo t i "^Background'
dampihg^and peak damping values) are-arbitrary:> estimates, while the rvalue-.'•> <
of 52 kcal mol"1 used for the activation energy was taken from the work(B)of Bungardt and Preisendanzv '.
Figures 8 and 9 show tracings of X-Y plots of temperature
against damping obtained by the drive technique for the Zr-0.18 wt% Cu
alloy. All of the damping vs temperature curves for large-grained speci-
mens of this alloy exhibited a peak in the temperature range 450 to 455°C.
The damping vs temperature curve for the Zr-0.18 wt% Al alloy also showed
a small inflection at about 445°C, but this result has not yet been con-
firmed. No indication of a peak at 450°C was found for the Zr-Ti, Zr-Fe
and Zr-Y alloys.
The strength of the oxygen peak in the Zr-0.18 wt% Cu bamboo
structured alloy seems to be dependent on the thermal history of a specimen
following the primary heat treatment required to produce a large grain
s ize . Thus, the data collected in Table 3 show that a prolonged vacuum
anneal at 700°C or above, in-s i tu prior to testing, increases the proba-
b i l i t y of the oxygen peak being observed. These secondary anneals were
introduced in an attempt to increase the concentration of oxygen in
solution in the alloys. As can be seen from Table 2, such 700°C anneals
were successful in increasing the oxygen content from 700 yg/g to
2,000 yg/g.
The two curves in Figure 9 i l l u s t r a t e the effect of increasing
the heating rate during testing. Curve (1) was generated in 5.7 hours and
curve (2) in 20 minutes. This is an important resul t which i l lus t ra tes
that damping vs temperature curves can probably be generated accurately in
a fraction of the time normally taken. Further studies should be in i t i a ted
to determine an optimum heating ra te for this study.
- 13 -
All tests, except those on large-grained and bamboo-structured
specimens, showed the low temperature tail of a grain boundary peak. The
low temperature tail of the peak due to the a/$ phase transformation is
apparent-t:on*all!tests*including; those on bamboo structure specimens;
5. DISCUSSION AND RECOMMENDATIONS
The experiments described in this report are of an exploratory
nature and therefore the data must be considered to be preliminary and
incomplete. However, some important observations have been made which
allow soine useful".* conclusions to be made immediately and which form' the
basis of a future detailed experimental programme -to investigate the
phenomenon more fully.
We have shown that within the temperature/frequency spectrum
and oxygen content range investigated, stress-induced ordering is not
possible in zirconium alloys in the presence of oxygen alone. Thus, a
comparison of the damping curves of pure zirconium and the zirconium
copper alloy show incontrovertibly that the presence of a substitutional
atom is required to produce an observable damping/temperature peak. The
strength of this peak can be enhanced by increasing the oxygen content
through repeated annealing treatments at 700°C. It is concluded that this
peak is due to the stress-directed ordering of the dipole formed by an
interstitial/substitutional atom pair. The existence of such a relaxation
centre is of considerable significance to the study of deformation pro-
cesses in zirconium alloys and to the technological feasibility of im-
proving the mechanical properties of zirconium by alloying additions. This
arises from the ability of a dipole to interact with any stress field
such as that associated with other defects (e.g., dislocations). Two
possibilities will be briefly mentioned to illustrate this point.
- 14 -
(1) The high temperature creep rate of zirconium alloys is
probably governed by the limiting velocity of mobile dislocations giving
rise to the creep deformation. The strong interaction of any relaxation
centre wLth-theses to^ - - - .-—
retard, dislocationmovement; -through^a, strain^ageing,,process, will, result
i n . g r e a t e r , , c r e e p ; r e s i s t a n c e . . , . • •.-- -•-•.; .-* • ^ u j , -.;.-•:: •••-., ;^,:.,,c>
(2). Neutron^irradia|:iSnion:jzd^cdniumalipyss;has been shown
to increase their creep rate and hence decrease the life potential of
reactor components. I t has been suggested, by several workers, that
creep., rate enhancement is caused.by the^migratipn,,gfr Jh^e;exce|s> vacancies,
produced during.irradiation, to, the, dislocations .thus jlnc^reasing, their ,
mobility. :_This effect would be^reduced if thelif | t ime of a vacancy was
increased such that the probability of i t s annihilation by a self inter-
s t i t i a l would be increased. The interstitial/substitutional dipole
detected in the present experiments could have such an effect by inter-
acting with the stress field of the vacancy, thus forming a metastable
complex which could act as a vacancy/interstitial recombination centre.
Within the experimental range investigated, the only alloy to
show a definite oxygen peak was Zr-0.18 wt% Al alloy. This latter system
would require further studies. The limitation of such behaviour to a
Zr-copper alloy may be the consequence of the fact that this alloy had
the largest grain size.of the group investigated. Thus further work seems
warranted .on the other systems to attempt to reduce the grain boundary
peak (at temperatures around 450"C) by selecting a suitable thermomechani-
cal treatment to increase the grain size. It is evident from;the data
presented in Table 3 that, in addition to the hsat- treatment schedule used
prior to testing, the annealing treatment given to the specimen following
the insertion into the apparatus may be of some importance to the experi-
ment. This is a significant result as i t reflects the variation of
strength of the oxygen peak with oxygen content, as controlled by, annealing
at 700°C.
The data are also probably limited by the maximum concentration
of oxygen in solution in the various alloy systems. Some attempt will be
- 15 -
made* in further investigations, to introduce greater quantities of oxygen
so that the magnitude of the relaxation peak may be increased. This
would also faci l i ta te the measurement of the activation energy and other
parameters associated with this behaviour.
6 . REFERENCES
1. W.J. Bratina and B.C. Winegard, Internal friction in zirconium,J. Metals J3, AIME Trans. 206_ (1956) 186.
2. K. Bungardt and H. Preisendanz, Damping and modulus of shearof zirconium and zirconium-hydrogen alloys, Z. Metallkunde51 (1960) 280. , - - .
3. D. Gupta and S. Weinig, Interactions between interstitial andsubstitutional solutes in an HCP lattice, Acta Met. 10_ (1962)296.
4. > ; j .N. Pratt et a l . , Internal friction in titanium and Htanivm-oxygen alloys* Ac_a Met. £ (1956) 203.
5. , E. Bisognl et al., diffusion of gases in special interstitialsites in hafnium, J. Less Common Metals ]_ (1964) 197.
6. S, Mishra and M.K. Asundi, Internal friction due to oxygen-substitutional impurii§txtom complexes in zirconium, Bhabha
—; Atomic Research Centre report 458, Bombay, India (1970).
7. KrM. Browne, Stress induced diffusion of oxygen in alpha- zirconium, Scripta Met. 5_ (1971) 519.
8. K.W. Sprungmann and I.G. Ritchie, An improved reed petidulum.. apparatus and techniques for the study of internal frvctton• of ceramic single crystals, Atomic Energy of Canada Limited
report AECL-3794, Pinawa, Manitoba (1971).
- 16 -
^.ACKNOWLEDGEMENTS
The authors wish to acknowledge the assistance of A.R. Reich
and D.P, McCooeye in preparing the specimens and K.W. Sprungmann and
H.K. Schmidt for their contributions to the development of the experimental
apparatus and technique.
TABLE 1
SPECIMEN DATA
SPECIMEN
Pure Zr
Pure Zr
Pure Zr
Zr-0.18wt%Cu
Zr-0.18wt%Cu
Zr-O.lwt%Ti'
Zr-0.3wt%Al
Zr-0.02wt%Fe
Zr-0.05wt%Y
SPECIMENNUMBER
3
8
9
- 2
10
4 • '
5
6
7
FIGURE
4(1)
4(2)
4(3)
5(1)
5(2)
6(5)
6(d)
6(b)
6(c)
MEDIANFREQ.(Hz)
1.86
1.73
0.82
2.18 -
2/38 '
2.24
2.34
2.23
2.20
BACKGROUNDAT 350°C(x 103)
0.536
0.988
1.312
-- 0.541- -
0.569- -
1.667-
0.908 ,
1.914
0.988
OXYGENPEAKTEMP.
None
None
None
450 °C
455°C •-
None
445°C
None
t
None
HEAT TREATMENT
Annealed at 760°Cfor 12 hours
Thermocycled-annealedat 735°C and cooled
Thermocycled-annealedat 735°C and cooled
•Annealed at 680-700°Cfor several hours
Thermocycled-annealedat 729°C and cooled
Annealed at - 650°C
Annealed at- 790°Cand cooled
Annealed at 745°Cand cooled
Annealed at 770°C-and copied ,
- 17 -
TABLE 2
GRAIN SIZE, DAMPING AND OXYGEN ANALYSIS AFTER TESTING
SPECIMEN
Pure Zr
Pure Zr
Pure Zr
Zr-0.18wt%Cu
Zr-0.18wt%Cu
Zr-0.1wt%Ti
Zr-0.3wt%Al
Zr-0.02wt%Fe
Zr-0.05wt%Y
SPECIMEN "NUMBER
3
8
9
2
10
4
5
6
7
GRAIN SIZE(ram)
0.18
>0.50
BambooStructure
0.06
BambooStructure
0.01
0.04
0.03
0.03
DAMPINGAT 450*0(xlO3)
1.40
1.50
2.70
5.00
; 1.55
25.00
3.00
8.00
4.00
Vg 02/g±50 yg
550
1900
1710
700
,2000
950
450
550
475
TABLE 3
OXYGEN PEAKS IN THE Zr-Q.18wt%Cu BAMBOO SPECIMEN NUMBER 10
FOLLOWING SECONDARY ANNEALS BEFORE TESTING
FIGURE NUMBER
8 (a)
8(b)
8(c)
9
700°C
700°C750 °C
for
forfor
700-783°C
700oC for
HEAT TREATMENT
1 hour at 5 x 10~2 torr
17 hours at 1.0 atm1 hour at 10"1* torr
for 2 hours at 10~3 torr
16 hours at 10~3 torr
OXYGEN PEAK;f
None
Small peak atabout 450"C
Same
Same
- 18 -
HIGH VACUUMGAUGE HEAD
VALVE
SERVICE PORTS
FOTONIC SENSORCONNECTION
MAGNETIC RINGS
INERTIA BAR
BALANCE WEIGHTS
STAINLESS STEELBASE PLATE
FURNACE CASING
COUNTER BALANCE
BELL JAR
DASH POT
: i.6w VACUUM
GAUGE HEAD
FOTONIC SENSOR
ELECTROMAGNETS
VACUUM BOX
STAINLESS STEELBASE
\ :CONCRETEBASE >
FIGURE I : TORSION PENDULUM FOR DECAY MEASUREMENTS
- 19 -
(a)
(b)
FIGURE 2: TORSION PENDULUM AND DRIVE SYSTEM
REFERENCE PEAKAMPLITUDE
V + _ACTUALPEAK
AMPLITUDE
INTEGRATOR
INTEGRAL PLUSPROPORTIONAL CIRCUIT
I
L
"J X Sin Wot
- K , Xo C M Wot"Wo" COUNTER
BALANCE
MULTIPLIER—»>
DRIVER
I|
PEAK DETECTION CIRCUIT
AVOLTMETER
HELMHOLTZCOILS,
INERTIABAR- A
THERMOCOUPLE
FOTONIC'SENSOR
ORIVESIGNAL
.(PROPORTIONALTO DAMPING}
FIGURE 3 SCHEMATIC DIAGRAM OF TORSIONPENDULUM DRIVE SYSTEM
X-Y PLOTTER
Y
DAMPING r\xfc
-JU—SPECIMEN
TEMPERATURE
DCVOLTMETER
TEMPERATURE
o
15
n13
'2
II
10
« 9
Z 8
g ,L
a
4
3
2
IF
MATERIAL
FINE GRAINLARGE GRAINBAMBOO
SPEC.
369
NO.
i i
CURVE NO.
1
3
MEDIAN FREQUENCY
'•" - ' i . JB'e ' •"i.r?0.82
(HZ)
350
a-
400 450TEMPERATURE (°C)
500
2 H
550
I
to
FIGURE i4 EFFECTS OF GRAIN SIZE ON DAMPING IN PURE ZIRCONIUM (DECAY METHOD)
too
£3
0
!, MATERIAL
FINE GRAINLARGE GRAIN
SPEC. NO. f" p • " •I *
10
CURVE12
350 400 500
to
550
TEMPERATURE &C)
FIGURE 5 EFFECT OF GRAIN SIZE ON DAMPING IN Zr-0.18 wt% Cu ALLOY (DECAYMETHOD)
(0
90
eo
70
60
50
40
30
20
10
COMPOSITION
Zr-OI wf %Ti
Zr-Q02wt%Fe
Zr-0.05wt%Y
Zr-0.3wt%AI
v ;:
SPEC. NO.
- +•
6
7
5
CURVE
a
b
c
d
350 400 500 550450
TEMPERATURE (°C)
FIGURE 6 DAMPING CURVES FOR VARIOUS ZIRCONIUM ALLOYS (DECAY METHOD)
Nw
- 24 -
GO.... THEORETICAL CURVE**** EXPERIMENTAL CURVE
NQTEt DAMPING,,. LE|S THAN,,70*10,t-3,: s:: J ::::: it: : s ii'ii'i-itzfi in HimA=JS70i.._ - . -
BACKGRD=:1.5 JQ=s 52000 .,.•"'"'TP=:805 J*"^ 5 ' 3"
*
*
• *? •
/ *
* «
*
_ *
FIGURE-7: COMPUTER PRINT-PLOTEXPERIMENTAL AND THEORETICAL DAMPING CURVES Zr-0.05wt%Y
01 -
100 200 300TEMPERATURE PC)
500
<2in
Ixl0.1 -
300 400 500TEMPERATURE PC)
6 0 0
0.B
0.7
0 6
0 5
0.4
0 3
02
0 1
-
-
-
-
3 0 0
-
i i400 500
/
/
i '600 700
TEMPERATURE CC)
FIGURE 8= DAMPING IK Zr-GI8 wt % Cu-X-Y PLOTS USING THE DRIVE METHOD
-26 -
6.8
CO
UJ
Q
. C U R V E I GENERATED IN- 5-7 HOURS
CURVE 2 GENERATED IN 20 MINUTES
0.7
0.6
% 0.5
0.4
0.2
0.1
300 400 500TEMPERATURE {°C)
600
FIGURE 9 EFFECT OF HEATING RATE CHANGE ON DAMPING IN Zr-O-I8wt%
- 27 -
APPENDIX 1
VIBRATION-ISOLATION-CONSIDERATION OF THE EXISTING SUB-BASE
AND GENERAL RECOMMENDATIONS AND CRITERIA FOR THE CONSTRUCTION OF
SUB-BASES FOR LOW FREQUENCY INTERNAL FRICTION APPARATUS
The objectives which must be kept in mind when designing a
sub-base are as follows:
(a) ISOLATION, I.e., the structure should absorb as l i t t l e
shock and vibration as possible from i t s surroundings.
(b) DAMPINQ i . e . , i t should damp those frequencies which
i t does absorb as quickly as possible.
Unfortunately, the fact that numerous variables are involved
makes mathematical analysis of structural vibrations extremely difficult.
However, several qualitative arguments can be listed which might prove
useful in redesigning the sub-base.
(1) All steel structures, welded rather than riveted, have
low damping, in the range 1 to 3% of cri t ical damping. Concrete structures
have higher damping capacity, up to 8% of critical damping.
(2) Damping capacity of most structures is considerably
increased when overstressed.
(3) Oh passing from a medium of high modulus ,of elasticity to
a medium of lower modulus, displacements and accelerations can be consider-
ably amplified.
The in i t i a l design, consisting of a concrete block supported
on springs, is considered to be inadequate because of insufficient attention
- 28 -
to objective (b), i . e . , although the springs help to isolate the sub-base,
they do not damp out lateral (horizontal) vibrations of the block rapidly
enough.
The sub-base should be constructed from highly prcstressed
concrete. An absorber should be used between the sub-base and floor. I t
should have a low modulus of elasticity compared to the concrete. Wood
has a Young's modulus of elasticity of 1.6 xlO6 lbf/in2 , rubber has an
even lower modulus, but may not satisfy objective (b), The number of
contact points between the sub-base and the floor should be reduced to a
minimum. This-helps to put the bearing points in a state of high compres-
sive i;stress - thereby; increasing jtheix; damping capacity^b;ut also, unfor-
tunate^^flncreaslng ;their capacity; to absorb »vibratipns;.j| Optimization
of the design will require a process of tr ial and error.
- 29 -
APPENDIX 2
CALCULATION OF THE RELATIONSHIP BETWEEN DRIVE CURRENT (IQ)
AND LOGARITHMIC DECREMENT (A)
Analysis of the conditions required tot drive a torsion pendulum
at constant strain amplitude has usually been carried out by assuming that
the equation of motion of the torsion pendulum is the same as that of a
simply damped harmonic oscillator, i.e.,
;; .,....,, ' A T + B"0 + C0 = 0 [2.1]
where A8 is the inertial torque, B6 is the damping torque and C8 the
elastic restoring torque. This would be the case if the damping were due
to residual atmospheric resistance to the motion. If, however, internal
friction mechanisms contribute to the damping, the equation of motion
becomes more complex.
If we consider that the specimen in the torsion pendulum
exhibits a single anelastic relaxation peak (e.g., a peak due to stress-
induced ordering of interstitial atoms) a realistic equation of motion for
the pendulum can be developed as outlined below.
The energy dissipation in metals due to anelastic relaxation
can be described' by defining an internal parameter u which is disturbed
from its equilibrium value by "a"change in strain (e.g., ]X might refer to
the stress-induced difference in chemical potential of solute atoms in two
types of interstitial positions). Then \i obeys the simple relaxation
equation < ' ,
[2-2]
where TQ is the time of relaxation at constant angle of twist e (i.e.,
constant shear^strain amplitude), and D is a constant. 9 is a linear
function of T, the torque acting on the specimen, and p, i.e.,
- 30 -
e = YR XT + Eu [2.3]
where YR is the elastic torque stiffness when the shear modulus G is
relaxed and E~is a constant. " " _ _ _ _ _ _ _ _
From equations [2.2] and [2.3] it can be shown that under con-•
ditions of constant stress (i.e., T = 0), T^ the time of relaxation at
constant torque is given by,
TT = T 8 (1 + DE) [2.4]
Also combining equations [2.2] afld [2.3]; as :&t:'—4^0 (t refers
to time) i t i s also found'that Yu the torque s t i f f ness for the unreiaxed
shear modulus ( i . e . , the elast ic shear modulus under conditions when no
relaxation has taken place) is given by,
Y u = Y R (1 + DE) [2.5]
Using equations [2.4] and [2.5], u can be eliminated from equations- [2.2]
and [2.3], to yield
T + T0T = YR O + TT8) [2.6]
This is the torque-twist relationship for a standard linear solid. It is
the most general equation in stress, as specified by torque T, and shear
strain, as specified by angle of twist 6, and their first time derivatives.
Nearly all crystalline materials behave approximately as standard linear
solids at low stress levels.
For the torsion pendulum described in this report, it is clear
that the polar moment of inertia I of the pendulum rod and cross-piece
(inertia arm) is very much greater than that of the specimen. Then during
free decay Newton's second law gives
. • T - -16 ' '/ • [2.7]
- 31 -
and thereforeT 6 * = -Te*® [2.8]
Using the above equations, equation 2.6 becomes
-• - B -k'-\> i j / r J £ + , i £ p Y R ^ e + " Y R 6 ^ = ¥^ [2.9]
which is the equation of motion of the torsion pendulum for a standard
linear: solid, specimen.
An exponentially decaying sinusoidal solution to the above
equation, e.g.,
9 = 90 exp (-3t) cos bit . [2.10]
by substitution, u) is the angular frequency of the
3 is the modulus of decay and 90 is a constant. For [2.10]
to be the solution to the equation of motion, [2.9], it can be shown that
for small values of 3 the logarithmic decrement, A = 2ir3/o), is given by,
and the dynamic torque stiffness Y^ = Ico2 is given by
In the above equations ? = / T R T e and Y = /Y UY R. Consideration of
equation [2.9] shows that if the pendulum is to be driven at constant
angle of twist (i.e., constant surface shear strain amplitude), the current
in the Helmholtz coils must generate a torque on the pendulum equal to and
opposite to the torque causing damping, i.e.,
Drive torque = -[I TQ'S + YRTT6] [2.13]
When this is the case, the pendulum oscillates at constant amplitude
0 = 0Q cos tot and equation [2.13] becomes
- 32 -
Drive torque = 0)6o(sin ut) [YRTT - IU2TQ] [2.14]
Using equation [2.12], this can be rewritten as
Drive torque = 9Q (yu - YR) 1 ^%^'i- s l n <** [2-15(a) ]
= 9n X . A sin ut [2.15(b)]
Equations [2.14] and [2.15] show that the current11. in -the
Helmholtz coils must be 90° out of phase with the displacement or angle of
twist, i . e . ,
I = Io sin tot [2.16]
It can be shown that the resultant magnetic field at the centre of the
Helmholtz coils is given by
3 2 7 T n I [2.17]H =
where n is the number of turns in each coil; and &; Is the mean radius of
the coils. If m is the magnetic moment of, the magnet attached to the
pendulum, the drive torque exerted on the magnet is given by
Drive torque = mH cos 0 [2.18]
For the very s'mall angles of twist normally involved in torsion pendulum
experiments cos 9 5 1, and equations [2.15(a)] and [2.18] yield
327TmnI0
5a/5 -
or ' [2-193
I = Kl90yA
- 33 -
where Ki is a constant for a given pair of coils and magnet. It is easily
shown that the torque stiffness of a given specimen is given by,
Y ° HT . [2.20]
where I1 is the polar geometrical moment of inertia of the specimen cross-
section and % the length of the specimen. Further, it is shown in
Appendix.^ that; thersurfaci shear, strain amplitude is. given by e = r9Q/il.
H e n c e , , . , . •...:•..
I o = K2£GA = K3e (Gu - Ge) 1+^m)2 E2.21]
where K2 and K3 are constants for a given specimen.
The above result shows that the amplitude of the current
I o in the Helmholtz coils is proportional to the logarithmic decrement A,
when the pendulum is driven at constant surface shear strain amplitude e.
- 34 -
APPENDIX 3
ESTIMATION OF ERRORS IN THE LOGARITHMIC DECREMENT (A)DUE TO EXPERIMENTAL-ERRORS IN THE MEASURED AMPLITUDES ;OF FREE DECAY
"• Application^bf-thes' chain rule: to equation--[1] of the— ••-'•=•'
text yields the following equation for the fractional error 8 A /A in logar-
ithmic decrement due to errors 6"N, 6AQ and SAJJ in the observed quantities.
Under normal circumstances, there is no error in counting N and therefore
AN is zero. Nevertheless, it can be seen that the error in A is mainly
determined by the magnitude of N. To maintain a constant fractional or
percentage error in A over a range of A values, N must be increased (and
therefore the strain amplitude range) as A decreases. For the values of
A normally encountered, ^^/AJJ is much larger than SA.Q/AQ since pendulum
instabilities and transducer noise become more arid more noticeable as the
amplitude decreases. An estimate of the percentage error in A can be
obtained by computing the quantity 100 fiAjj/A^ NA. For a typical run, e.g.,
Figure 5, curve 2, the lowest damping recorded was 5.4 xlO"1*. This was
calculated from peak height values (proportional to displacement amplitude),
AQ = 5.017V, AJJ = 2.922V and N = 103. Taking SA0 = 6*AN = 0.05 yields a
value for the percentage error of 2.5%. The magnitude of this error is
typical of errors quoted for torsion pendulum results.
- 35 -
APPENDIX 4
CALCULATION OF SURFACE STRAIN
Pendulum geometry and specimen geometry for surface shears t ra in amplitude e are shown in Figure A(l) and (2). Consideration ofFigure A(2) shows tnat
r e o[4.1]
where r is the radius of the specimen, 2. its length and 8Q the angular
displacement of the pendulum. Figure A(l) shows that 6Q = ZQ/R for
small angular displacements. R is the horizontal displacement of the
Fotonic Sensor from the axis of the pendulum arid xQ is the displacement
amplitude as measured hy the Fotonic Sensor;• Equation [4.1] then becomes
e = ~%r = K2 *> I4-2
where Kz is constant for a given test.
The value of R is difficult to measure precisely due to
the diameters of the sensor and the pendulum red. K2 can be calculated
much more accurately by use of an optical" lever. A single measurement
would be necessary for each new specimen assembly. This measurement is
easily accomplished as illustrated in Figure A(l) by attaching a small
"mirror to the pendulum axis and positioning a lamp and scale, as shown,
U - **&> 14.3,
where L- is the distance of the lamp from the mirror and Y o the horizontal
deflection of the reflected light spot in the vertical plane containing
the lamp and scale. L can be made several meters long and K2, as given by
equation [4.3], more accurately determined.
- 36 -
•EQUILIBRIUM REST POSITION OF INERTIA BAR
// /
*o ' ' /
/FOTONIC SENSOR PROBE
LIGHT SOURCE
FIGURE A ( I ) :
PENDULUM GEOMETRY
POSITION OF INERTIA MEMBER ATMAXIMUM SHEAR STRAIN AMPLJTUDE
SCALE
FIGURE A (2) :
SPECIMEN GEOMETRY
FIGURE A = PENDULUM (I) AND SPECIMEN (2) GEOMETRYFOR CALCULATION OF SURFACE STRAINS
3 1'
, 1
A
5l
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