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Dislocation Mobility in Pure Copper Single Crystals
Citation
Greenman, William Franklin
(1967)
Dislocation Mobility in Pure Copper Single Crystals.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/072F-RC63.
Abstract
This thesis presents the results of an experimental investigation of the stress dependence of dislocation velocity in 99.999 per cent copper single crystals. Dislocation displacements were detected by etch pitting dislocation sites on {100} planes. Stress pulses of microsecond duration were applied to single crystal test specimens by means of a torsion impulse machine. Maximum applied, resolved shear stresses ranged from 29 g/mm² to 236 g/mm², and calculated dislocation velocities ranged from 160 cm/sec to 710 cm/sec. The dislocations were presumably predominantly edge-oriented.
The growth of copper single crystals, the spark and chemical machining of single crystal test specimens, and the behavior of the etchants which reveal dislocation sites on {100} planes are also discussed.
The experimental data have been found to obey a linear relation between dislocation velocity and applied, resolved shear stress. This finding does not correlate with the explanation of the low strain rate sensitivity of the flow stress in copper as proposed by Cottrell (46)*, which predicts that dislocation velocity should be proportional to stress raised to a power of about 200. The low strain rate sensitivity of the flow stress in copper is explained by the high velocity of dislocations at low stresses and the strong stress dependence of the mobile dislocation density. This high velocity is interpreted as enabling the strain essentially to achieve its equilibrium value even at relatively high strain rates.
*One- and two-digit numbers appearing in parentheses indicate references listed at the end of the thesis.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Materials Science and Economics)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Minor Option:
Economics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Vreeland, Thad
Thesis Committee:
Unknown, Unknown
Defense Date:
29 July 1966
Record Number:
CaltechETD:etd-09192002-153519
Persistent URL:
DOI:
10.7907/072F-RC63
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3634
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DISLOCATION MOBILITY IN PURE COPPER
SINGLE CRYSTALS
Thesis by
William Franklin Greenman
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1967
(Submitted July 29, 1966)
-ii-
ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to
Professor Thad Vreeland, Jr., who directed this research and whose
aid and encouragement made the work possible. Throughout the
course of the research, many valuable comments and suggestions
were proffered by Professors David S. Wood and Donald S. Clark.
Messrs. G R. May and R. C. Blish were of great assistance in the
experimentation.
The author is very grateful for fellowships provided by the
Garrett Corporation and the R. C. Baker Foundation.
The torsion impulse machine was constructed and this re-
search was performed under a contract with the United States Atomic
Energy Commission.
Finally, the author wishes to express his appreciation to his
wife, Penelope, for her patience, wisdom, and encouragement which
also made this work possible.
-iii-
ABSTRACT
This thesis presents the results of an experimental investiga-
tion of the stress dependence of dislocation velocity in 99. 999 per
cent copper single crystals, Dislocation displacements were detected
by etch pitting dislocation sites on {100} planes. Stress pulses of
microsecond duration were applied to single crystal test specimens
by means of a torsion impulse machine. Maximum applied, resolved
shear stresses ranged from 29 g/mm* to 236 g/mm*, and calculated
dislocation velocities ranged from 160 cm/sec to 710 cm/sec. The
dislocations were presumably predominantly edge-oriented.
The growth of copper single crystals, the spark and chemical
machining of single crystal test specimens, and the behavior of the
etchants which reveal dislocation sites on {100} planes are also
discussed.
The experimental data have been found to obey a linear rela-
tion between dislocation velocity and applied, resolved shear stress.
This finding does not correlate with the explanation of the low strain
rate sensitivity of the flow stress in copper as proposed by Cottrell
(46)*, which predicts that dislocation velocity should be proportional
to stress raised to a power of about 200. The low strain rate sensi-
tivity of the flow stress in copper is explained by the high velocity
of dislocations at low stresses and the strong stress dependence of
the mobile dislocation density. This high velocity is interpreted as
enabling the strain essentially to achieve its equilibrium value even
at relatively high strain rates.
*One- and two-digit numbers appearing in parentheses indicate
references listed at the end of the thesis.
PART
II.
Til.
IV.
Vi.
Vil.
-iv-
TABLE OF CONTENTS
TITLE
Acknowledgments .....-+-+-ee.
Abstract . 6.6 6 «© eo wwe
Table of Contents. ....+.-++s .
List of Figures .......
Introduction. .
Experimental Equipment and Techniques ...
Scratching apparatus. .
moo OB >
- Specimen surface record
Preparation of Test Specimens
A. Crystal growth.....
Torsion impulse machine
S.e
B. Spark and acid machining
C. Amnealing. ....-.. »
Experimental Procedure . ,.
Experimental Results. ...
A. Specimen substructure. .
B. Stress pulse analysis. .
C, Etching observations. .
_D. Dislocation displacement
Discussion . ....-+-e-ee
Summary and Conclusions .
Appendix A..
- Specimen polishing and etching
os ¢ 89 ¢ e@ @
Stress and time measurement ........
PAGE
li
L1i
iv
vi
13
13
22
24
24
26
28
28
30
37
38
41
4]
44
58
66
76
96
98
Figure
10
11
12
13
14
15,
16
17
18
19
20
-vi-
LIST OF FIGURES
Inverse Strain Rate Sensitivity Versus Resolved
Shear Strain for Copper from Data of Conrad (26). :
Schematic of Torsion Pulse System.
Specimen Mounted on Thermal Buffer and Specimen
Holder.
Lower Section of Torsion Rod.
Specimen, Thermal Buffer, and Specimen Holder
Combination in Scratching Apparatus.
Crystal Segment on Goniometer in X-Ray Fixture.
Trepanning Operation in Agietron Machine.
Typical Specimen in Acid Lathe.
As Annealed and Scratched Dislocation Configura-
tion, Typical of Specimens from Crystal 3.
As Annealed Dislocation Configurations, Typical
of Specimens from Crystal 6.
As Annealed and Scratched Dislocation Configuration,
Exhibiting Spark Machining Damage.
Typical Torsion Machine Record.
Position-Time Diagram for Torsion Waves in
Thermal Buffer and Specimen.
Stress Pulse at Fixed End of Specimen 6-3-1.
Stress Pulse at Fixed End of Specimen 3-2-1.
Stress Pulse at Fixed End of Specimen 3-3-1.
Stress Pulse at Fixed End of Specimen 6-2-1.
Etched {100} Surface of Specimen 6-3-1.
Etched {100} Surface of Specimen 3-2-1.
Etched {100} Surface of Specimen 3-3-1.
20
23
25
33
34
36
42
A3
47
51
54
55
56
57
60
63
65
Figure
21
22
23
24
25
26
27
28
29
A-1
A-2
A-3
B-l
-vii-~
Etched {100} Surface of Specimen 6~2-1.
Dislocation Displacement Versus Distance from
Free End for Specimen 6-3-1, T= 28.7 g/mm™.
Dislocation Displacement Versus Distance from
Free End for Specimen 3-2-1, Tmax 7 68.8 g/mm °
Dislocation Displacement Versus Distance from
Free End for Specimen 3-3-1, T ax 7 134 g/mm*.
Dislocation Displacement Versus Distance from
Free End for Specimen 6-2-1, T wax 7 236 g/mm”.
Reduced Dislocation Velocity Versus Maximum
Resolved Shear Stress for m= 0.7.
Reduced Dislocation Velocity Versus Maximum
Resolved Shear Stress for m= 1.0.
Reduced Dislocation Velocity Versus Reciprocal
of Maximum Resolved Shear Stress.
Analysis of Data According to Theory of
Fleischer (41).
[111] Specimen Orientation.
[110] Specimen Orientation.
[100] Specimen Orientation.
Schematic Representation of Stress Pulse.
82
83
85
87
108
109
110
ll2
-]-
IL INTRODUCTION
Measurement of dislocation velocity provides information
from which theories of the influence of strain rate effect upon mechan-
ical properties of crystalline solids may be derived. Further, such
measurements provide information concerning the dynamic interac-
tions between moving dislocations and the crystal lattice, other dis-
locations, and other lattice defects. This thesis presents the first
direct measurements of dislocation velocity in a face-centered cubic
metal, copper.
A theory of the influence of strain rate upon mechanical prop-
erties can be derived from a knowledge of the dynamical behavior of
dislocations in crystals because of the direct dependence of plastic
strain rate on dislocation velocity. This dependence is due to the
fact that dislocation motion results in plastic deformation as ex-
pressed by the relation
y. = Ah/V = Ab C1]
where Yp is the plastic shear strain resulting from dislocations of
Burgers vector b sweeping out a total slip plane area of A in acrys-
tal of volume V. (A, is then the area per unil volume swept oul by
the dislocations. ) Equation 1 relates a macroscopic quantity, plastic
shear strain, to the microscopic dislocation parameters Ag and b
for the case of single slip. Differentiation of Equation 1 with respect
to time gives the rate of plastic deformation for single slip,
Yp = Ab = b O vat [2]
where v is the velocity of an element dt of dislocation line and the
-2-
integral is taken over all the dislocations in a unit volume. If the in-
tegral in Equation 2 is replaced by a product of the average velocity v
and the length of dislocation line per unit volume p, there results
the relation
= pbv . [3]
The dislocation density p could be the total length of all dislocations
per unit volume, and v, the average velocity of all dislocations. How-
ever, if all those dislocations which move do so with velocities of the
same order of magnitude, then it is more useful to take p and v to be
the density and average velocity, respectively, of only those disloca-
tions which actually move. The dislocation density and the average
dislocation velocity, and, consequently, the strain rate, are functions
of stress and strain.
Equation 3, which relates the macroscopic shear strain rate to
the dislocation parameters of density, Burgers vector, and velocity,
may be extended to the case in which slip on different slip systems is
occurring simultaneously. The contribution to the total deformation
rate from slip on a given slip system is given by Equation 3 modified
by an appropriate orientation factor. Gilman has pointed out that the
description of plastic deformation in terms of dislocation mechanics
represents a significant advance over classical plasticity theory (1).
When the variables v and p can be described as functions of the ap-
plied stress and the instantaneous strain, the macroscopic plastic
strain rate can then be related to the macroscopic stress and strain.
This relation permits the response of the crystal to be predicted for
-3-
any stress-time history, or alternately, it permits calculation of the
stresses required to deform a crystal on a given strain-time path.
The dislocation dynamics description of plastic deformation
has been employed to predict the dynamic stress-strain response of a
crystal (2), to describe transient creep behavior (3), and to explain
some observations in high-velocity impact experiments (4), none of
which could be adequately predicted by the classical plasticity theo-
ries. However, extension of the dislocation dynamics description of
plastic deformation to single crystals of materials other than those
for which velocity and density data exist or to polycrystalline aggre-
gates, if feasible at all, awaits experimental data for a greater range
of materiais, both single crystal and polycrystalline, and a statistical
approach to cope with the orientation factors in a multiple slip system.
Experimental techniques which permit dislocation velocity to
be measured directly have become available only recently. ‘Direct
experiments'! is a term used here to denote those experiments in
which the position of a dislocation is observed either continuously or
intermittently with time. The early information about dislocation ve-
locity was based on indirect measurements in internal friction exper-
iments, and the information obtained from these experiments depended
upon the model used to interpret the data. No model of or major pre-
supposition about dislocation behavior is necessary in order to inter-~
pret the data in the direct experiments on dislocation motion employ-
ing transmission electron microscopy, dislocation etch pitting, or
x-ray diffraction, although there is no certainty that dislocation be-
havior in the thin specimens employed in transmission electron
-4.
microscopy is identical to that in thick samples. Dislocation etch pit-
ting has been the technique used most extensively in direct experi-
ments on dislocation mobility. Until very recently, the x-ray dif-
fraction techniques had not heen employed for dislocation mobility
studies (5,6). While the continuous observation of moving disloca-
tions is only possible in the techniques of electron microscopy and
continuous etching, both of these techniques are limited to measuring
low dislocation velocities. The intermittent observation of dislocation
position as a function of time by means of etch pitting has, therefore,
been employed in most studies of dislocation velocity and mobility.
The present knowledge of dislocation mobility in single crys~
tals of several materials has resulted from experiments in whicha
stress pulse of known magnitude and duration was applied to a single
crystal specimen and dislocation displacement was detected by etch
pitting before and after the application of the pulse. Various minor
assumptions are implicit in the etch-pulse~etch experiments, but no
model is needed to interpret the results. For example, it is as-
sumed that the dislocation velocity is in phase with the stress, l.e.,
that at every instant of time the dislocation velocity at a given stress
is equal to the equilibrium or steady-state value of the velocity at that
stress. Obviously, this assumption is not valid during the time of
acceleration of the dislocation to the steady state velocity. There-
fore, the effect or magnitude of the acceleration time must be deter-
mined.
The stress dependence of dislocation velocity may be deter-
mined most easily if the applied resolved shear stress on the disloca-~
-5-
tion(s) is constant over a given duration of time, i.e., when an es-
sentially "square" (actually trapezoid-shaped) stress pulse is applied.
Then the velocity of the dislocation is just the dislocation displace-
ment divided by the time at constant stress. However, if the accel-
eration time of the dislocation is longer than the rise time of the
stress pulse, the above calculation of velocity can lead to erroneous
results. It is also important that the specimen be subjected to only
one known stress pulse and to no unknown stresses which would add to
or subtract from the dislocation displacements produced by the pulse
and upon which the velocity is calculated.
The so-called double etch technique uses the characteristics
of certain etchants to facilitate determination of dislocation displace-
ment. The specimen is etched before application of the stress pulse,
the pulse is applied, and the specimen is re-etched without first re-
moving the previous dislocation etch pits by chemical or electrolytic
polishing. The re-etching produces small, sharp-bottomed pits at the
new sites of dislocations which have moved during the stress pulse
and large flat-bottomed pits at their previous sites. Large, sharp-
bottomed pits form at the sites of dislocations which did not move.
Thus, a sharp~-bottomed pit and a flat-bottomed pit lying adjacent to
one another along the direction of slip give the amount of dislocation
displacement. Difficulties in determining the initial and final posi-
tions of a dislocation arise when dislocation displacements are great-
er than one-half the initial dislocation spacing and when dislocations
are moving on more than one slip system. |
Johnston and Gilman (7) were the first investigators to deter-
-6-
-_ mine dislocation mobility in a material, the ionic compound lithium
fluoride, with the technique of etch pitting. The dislocation mobility
data in lithium fluoride is quite extensive, velocity having been
studied as a function of applied stress, temperature, annealing treat-
ment, and irradiation. Dislocation velocity has also been determined
as a function of applied stress for silicon-iron (8,9), sodium chloride
(10), tungsten (11), and various semiconductor crystals (12, 13).
Adams (14) has investigated basal and non-basal dislocation mobility
in high purity and impurity-doped zinc, Although dislocation mobility
data exist for a variety of materials, no studies have yet produced
any direct information about dislocation velocity in the face-centered
cubic metals.
Among the facc-centercd cubic mctals of high purity (99. 999
per cent), there are only two for which dislocation etchants exist,
silver (15) and copper (16-19). Utilizing the etch pitting technique to
determine dislocation density, Young (20), Livingston (18), and
Hordon (21) have performed tension tests on high-purity copper
single crystals of various orientations to determine the relationships
among dislocation density, resolved shear stress, and shear strain
for such tests. Studies of dislocation motion in high-purity copper
single crystals have been made by Young (22, 23), Livingston (17, 24),
and Petroff (25), but no quantitative work on the stress dependence of
dislocation velocity was reported. From the results of indirect ex-
periments by Conrad (26) on low-purity (99.98 per cent) copper, as
analyzed according to the following treatment, it might be inferred
that the dislocation velocity in copper is a very sensitive function of
the applied stress.
Subsequent to the dislocation mobility studies of Johnston and
Gilman (7) and Stein and Low (8), Guard (27) suggested an indirect
method of determining the stress dependence of dislocation velocity
for those materials for which dislocation etchants did not exist --
the change of strain rate test during the usual tension or compression
test. How this technique might be useful in determining the depend-
ence of dislocation velocity on applied stress may be seen from Equa-
tion 3. Taking logarithms of both sides of the equation, differentiat-
ing with respect to the logarithm of the applicd rcsolved shcar stress
t , and defining the quantity m!' as with Johnston and Stein (28),
bint, _ Sinv , dtnp [4]
dint B2ntr dent *
m! =
_ Now m! or (8 ny, )/(2 4n7) can be approximated by
tn(¥,/¥,)
Arlt
where vy is the strain rate before the change, Yo is the strain rate
after the change, and At is the change of stress accompanying the
change of strain rate (At <<). Over most of the range of disloca-
tion velocities in. the materials mentioned above, the data may be
represented by the relation
vou Ar’, [5]
where A and m are constants for a given material and temperature.
Equation 4 then reduces to
at
m! = m+ 3pt ° [6]
If the moving dislocation density is independent of or rather insensi-
-8-
tive to the change of stress, the inverse strain rate sensitivity m!
should be approximately equal to the mability exponent m. Guard (27)
found no agreement between m! and the directly measured m (8) for
change of strain rate tests in silicon-iron, indicating that the moving
dislocation density is sensitive to the change of stress. Johnston and
Stein (28) reopened the question of the validity of strain rate tests,
reasoning that for very small strains m!' should equal m or, corre-
spondingly, extrapolating a curve of m!' versus strain to zero strain
would give m' =m. Using this technique, they obtained reasonable
agreement between m!' at zero strain for change of strain rate tests
and the directly measured values of m for lithium fluoride and sili-
con-iron. Of course, a major assumption in this change of strain
rate treatment is that the dislocation velocity in the material has the
functional dependence on stress as expressed by Equation 5.
Conrad (26) has performed such change of strain rate tests in
copper; however, the tensile axis of the single crystals was such as
to produce multiple slip initially,and also the specimens were sub-
jected to very large strains. These facts alone might preclude the
use of the data from these tests in order to determine the mobility
exponent. The inverse strain rate sensitivity, m', calculated from
the results of the experiments by Conrad, is plotted versus shear
strain in Figure 1. The strain rate ratio was 10. If any curve can be
fitted to the data at all, it would at least indicate a value of m large
relative to those obtained for other materials (lithium fluoride - 15 to
25, silicon-iron - 40), It is clear from the scatter of the data that a
change of strain rate test in copper might not allow an unambiguous
(92) petuoyd fo ejeq worjy taddoy s0F7 urerig
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4 “ALIAILISNSS SLVY NIVYLS ASYSANI
-10-
determination of the stress dependence of dislocation velocity. More-
over, it is not certain that the extrapolation method of Johnston and
Stein gives the mobility exponent, for Adams (14) found that in zinc
single crystals, change of strain rate tests gave m' = 83 and direct
measurements gave m= 5. Also, Schadler (11) found that the ex-
trapolation procedure was inaccurate for tungsten single crystals.
With the validity of indirect tests of dislocation mobility in
copper in question and with the lack of any quantitative data obtained
directly, it was decided to embark upon a program to determine the
stress dependence of dislocation velocity in copper as a representa-
tive face-centered cubic metal. It was, therefore, the purpose of the
present work to make direct measurements of dislocation velocity by
determining dislocation displacement produced by the application of a
stress pulse of known magnitude and duration. All test specimens
were to be of high-purity (99. 999 per cent) copper, and the tests en-
visioned above were to be conducted at room and several lower tem-
peratures. The data from the low-temperature tests were not avail-
able at the time this thesis was written but are to be included ina
later report. Finally, the experimental results were to be compared
with the stress dependence of dislocation velocity as predicted by
various current theories in an attempt to determine which theory or
theories give the most valid prediction of the actual behavior.
Various considerations enter into the experiments outlined
above. Single crystals should be employed such that the applied re-
solved shear stress may be known with as little uncertainty as possi-
ble; and to minimize the effects of unknown impurities, the purity of
-ll-
the single crystals should be as high as possible (in this case, 99.999
per cent). Since etch pitting was to be employed to determine initial
and final dislocation positions, the spacing between dislocation etch
pits should be at least twice the contemplated dislocation displace-
ment due to the stress pulse in order to know with a fair degree of
certainty the initial and final position of the dislocation. A restriction
of low dislocation density was thereby imposed upon the single-crystal
test specimens. It was assumed that a stress pulse of microsecond
duration would be required, since other investigators in copper had
available to therm testing machines capable of producing square stress
pulses of millisecond, second, and longer time duration and had re-
ported no quantitative data on dislocation velocity. Also, the experi-
ence of Adams (14) with zinc, the only soft metal yet studied, would
indicate that dislocations might move very fast in copper. For speci-
mens of reasonable dimensions, the microsecond duration dictated
application of the stress pulse by a wave propagation technique. Since
the validity of the etch-pulse-etch technique depends to a large extent
upon the application of a square or trapezoid-shaped stress pulse, the
rise and decay time of the stress pulse should be short, here, less
than five microseconds; and, therefore, the wave propagation should
be non-dispersive. The orientation dependence of the available dislo-
cation etchants for copper restricted the surface on which dislocation
displacement could be observed to within a few degrees of {111},
{100}, or {110} surfaces. These etchants are for both fresh, undec-~
orated dislocations and aged dislocations; no aging treatment is re-
quired after dislocation displacement.
-12-
Section Il of this thesis is concerned with experimental equip-
ment and techniques. The testing machine used to apply a stress
pulse to a single-crystal specimen is described. The measurement
of stress and time and the techniques employed to determine disloca-
tion displacement are also described. Section III deals with the prep-
aration of single-crystal test specimens with the proper orientation
and shape from large single crystals. Section IV describes the ex-
perimental procedure followed in applying the stress pulse to a speci-
men and in determining dislocation displacement. Section V presents
the experimental results of this investigation, the mcans of obtaining
the data, and the calculations to reduce the data. Section VI analyzes
the results in light of current theories and discusses physical mech-
anisms which might govern dislocation mobility in copper. Section
VII is a summary and presents conclusions derived from this investi-
gation.
-13-
Ol, EXPERIMENTAL EQUIPMENT AND TECHNIQUES
A. Torsion Impulse Machine
The most difficult of the requirements outlined above for the
proposed project would appear to be the application of the stress
pulse. A wave propagation technique must be employed for such load-
ing, and the type and mode of the waves must be such that the pulse is
not distorted as it propagates through the loading fixture and the spec-
imen. These conditions are satisfied by zero-order mode torsional
waves in isotropic, cylindrical bars, for which the phase and group
velocities are independent of frequency (29). A testing machine
capable of applying a single, square stress pulse of microsecond du-
ration to a cylindrical, single-crystal specimen of proper crystallo-
graphic orientation has been developed at the W. M. Keck Laboratory
of Engineering Materials at the California Institute of Technology (30).
The torsion testing machine generates stress pulses in the fol-
lowing manner. An initial static torque is applied to a section of a
cylindrical rod which is a part of the torsion rod or load train shown
schematically in Figure 2. The torque is applied to the top of the sec-
tion by dead weight loading of a cranking disc attached to the rod
through a rubber sleeve. A thin glass disc fastened to the bottom of
the section transmits this torque to a bakelite fixture which is at-
tached to a fixed bearing tube surrounding the rod. A 0.006 mm
thick aluminum foil is attached between the glass disc and bakelite
with Eastman 910 adhesive. The lower section of the torsion rod or
load train is attached to the opposite side of the glass disc by means
_cti4-
ate ~
DAMPING
| {” SECTION
| |
ad
CRANKING a
DISC - a
STATICALLY
\ TORQUED
SECTION
ALUMINUM . J
FOIL ie »)
GLASS
DISC
: DYNAMICALLY
- LOADED
- SECTION
THERMAL
BUFFER >
SPECIMEN ————L J
Figure 2, Schematic of Torsion Pulse System. |
~15-
of a butt adhesive bond. This section of the torsion rod is coaxial
with the upper section but does not carry any static torque. The
specimen is attached to the bottom end of this lower section of the
torsion rod, and.the bottom end of the specimen is the free end of the
torsion rod system. The section of the rod above the cranking disc is
coated with a viscoelastic material and attenuates the waves propa-
gating away fromthe specimen. The application of the stress pulse
to the specimen is initiated by a high-voltage capacitor discharge
through the aluminum foil. Explosion of the foil releases the static
torque and results in elastic waves which propagate away from the
glass disc interface of the torsion rod. The duration of the stress
pulse at any point in the specimen is the time required for the wave to
propagate from that point to the free end and return. The amplitude of
- the dynamic torque in the lower section of the torsion rod is one-half
the initial static torque applied to the machine and, therefore, pro-
portional to the weight hung on the cranking disc.
The requirements imposed upon the crystallographic orienta-
tion of the test specimens by the orientation dependence of the dislo-
cation etch and the characteristics of the torsion machine could for-
tunately be reconciled. The effect of elastic anisotropy of a single
crystal specimen upon the dispersion of the torsion pulse and the
stresses produced thereby have been calculated and are presented in
Appendix A. The form of the single-crystal test specimens chosen
for this investigation was that of a right circular cylinder, the cylin-
der axis and torsion axis being coincident with the <100> direction of
the crystal. This specimen orientation was not dispersive of the
-16-
stress pulse and provided four {100} observation surfaces upon which
a double etch pit technique could be employcd.
The testing system described above and in reference 30, with
the modifications described below, was used to apply the stress pulse
to test specimens. The governing consideration in altering this ma-
‘chine to test copper single crystals was that the acoustic impedance
at the interfaces between different materials in the load train or tor-
sion rod should be equal. The acoustic impedance for torsional
waves in circular rods is given by (31)
wa
2, °-> Vue [7]
where
a = radius of rod,
uu = shear modulus, or, for <100> oriented specimens,
“55! = “66! = “44,
p = density (mass/unit volume).
It was decided that the diameter of the torsion rods should be main-
tained at one-half inch in order to minimize the alteration of the ma-
chine. In this way, only the torsion rods needed to be changed; that
is, only the material of which they were made needed to be changed;
and modifications of other parts of the machine, such as the pillow
blocks and bearing which supported the torsion rods, were avoided.
Acoustic impedance can be made equal or matched for materials of
slightly different shear impedance Vue (32) by adjusting the diame-
ters of the rods; however, a large change of diameter at the torsion
rod = specimen interface was avoided by selecting a specimen diame-
-17-
ter of approximately one-half inch. It was felt that this diameter
would allow sufficient surface area to observe dislocation motion.
The torsion rods were made of steel (SAE 1040), a material which
closely meets the requirement of shear impedance matching with the
specimen.
n The shear impedance for a copper single crystal of <100> tor-
sion axis orientation may be calculated from
4, > ur [8]
where
551 = Cy4 = 7.51% io1} dynes /cm* » (33)
8. 93 e/com? :
Hac
which gives
(Z.) = 25.9% 10° g/cm*=sec
cu
The subscript "cu" refers to the specimen. Correspondingly, for
SAE 1040 steel (Young's modulus = 29 xX 10° Ib /in”) ’
u=11.3x 10° ib/in® = 7. 75x 101} dynes /cem* ’
p = 7. 84 g/cm” ’
and
25) 40 = 24, 6 x 10° g/cm”-sec ;
where the subscript '1040" refers to the torsion rod. From equality
of acoustic impedance across the interface, the following relation is
derived.
( dou ) - (2 i040 97
43 040 (25) |
s cu
where
-18-
d= diamctcr of copper specimen,
cu
d = diameter of steel torsion rod = 0.499 in.
1040
As calculated from Equation 9, the diameter required for the speci-
men is
dau = 0.493 in.
To avoid spurious stresses in the specimen produced by dif-
ferential thermal expansion of the steel and copper, a thermal buffer
was placed between the steel torsion rod and the copper specimen.
This thermal buffer would have ideally been another copper single-
crystal specimen hardened such that its yield stress was much larger
than the highest stress contemplated (about 400 g/mm* resolved
shear stress), thus preventing dispersion of the torsion pulse due to
plastic flow in the thermal buffer. Since such hardening was deemed
unfeasible and since thermal expansion is isotropic in cubic sym-
metry crystals, the thermal butfer was made of polycrystalline,
99. 9 per cent copper rod, which remained elastic at the stresses for
testing single crystals. To match the acoustic impedance of the
thermal buffer to the steel, the calculation given above is repeated
for polycrystalline copper where
u = 4.6x 101 dynes /cm* ,
p=8.9 g/cm” ,
and .
(Z.) = 20.2X 10° g/cm*=sec (31)
tb
(the "'tb"' subscript refers to the thermal buffer). Using Equation 9
again, d,, = 0.525 in. In matching the acoustic impedance between
tb
-19-
polycrystalline and single-crystal copper, it can be seen that dau =
0. 493 in-,as before. The length of the thermal buffer was 0.91 in,,a
length judged sufficient to attenuate the thermal stresses arising at the
steel - copper interface. |
As in reference 30, a 30° taper joint 3 in. from the end of the
torsion rod was employed to facilitate removal of a specimen for pol-
ishing and etching. A photograph of a specimen attached to a thermal
buffer, which is, in turn, attached to the end of the torsion rod, is
shown in Figure 3. This part of the torsion rod, one end of which is
tapered to fit the taper joint, will be referred to subsequently as the
specimen holder. The end of the specimen which is attached to the
thermal buffer will be referred to subsequently as the fixed end of the
specimen.
Another modification on the torsion machine was made in the
damping rod, the upper section of the torsion rod system in which the
torsional wave traveling away from the specimen is attenuated. On
the original version of the torsion machine, the damping material
- was Solithane 113, a polymer produced by Thiokol Chemical Company,
Trenton, New Jersey, which was composed of 50 per cent prepolymer
and 50 per cent catalyst by volume. In changing the torsion machine
in order to test copper crystals, the damping material was also
Solithane 113, but the ratio of components was 60 per cent prepoly-
mer and, 40 per cent catalyst by volume. Although this mixture was
thought to attenuate the outgoing wave more effectively, it was, in
fact, aless effective damping material. Nevertheless, the effect of
the reflected pulses was negligible, as will be demonstrated later,
-~20=
Figure 3. Specimen Mounted on Thermal Buffer and Specimen
Holder.
~21]-
Two aspects of the operation of the torsion machine should be
mentioned here. First, it was found in this investigation that a rise
time for the stress pulse of 2 usec was quite reproducible. This val-
ue is very close to the time which a shear wave in steel requires to
travel the radius of the rod (2 usec) Secondly, it would seem likely
that the release of the static torque might produce torsional wave
modes of higher order than zero. After giving a solution of the Poch-
hammer equations for torsional waves in cylindrical bars, Mason
states (34) ‘
Tf, however, the frequencies of all the components of
the signal are below the critical frequency f, , all the
higher modes are highly attenuated and after a short
distance the only mode transmitted is the zero-order
mode which does not have any dispersion (34).
He states an inequality for the maximum frequency such that higher
order modes are attenuated,
3 _ 3. 5.136 V
f£< gf, = a* ta [10]
where V is the torsional wave velocity, which in steel is 0.315 X
10° cm/sec; a is again the radius of the rod, in this case, 6.35 mm;
and the numerical factor 5.136 is the first root of a secular equation.
Using the above values, inequality 10 becomes
f£ < 1.5X10° cps . cil]
An indication of the torsion pulse component with the maximum fre-
quency is given by the rise time, 2\1sec. The period of this com-
ponent would then be 8 usec, and the frequency, 1. 25x 10° cps, satis-
fying inequality 11. If higher modes are produced in releasing the
static torque, they should be greatly attenuated before the torsion
pulse enters the specimen, which is about 24 in. from the place of
~22-
initiation of the torsion pulse.
B. Stress and. Time Measurement
Measurement of the stress pulse amplitude and duration was
accomplished by means of silicon strain-gage circuits whose output
was displayed on the screen of a Tektronix type 555 oscilloscope.
The strain gages, which were 0.050 in. long and had a 500 Q nominal
resistance and a gage factor of 140, were obtained from Micro-
systems, Inc.,; Pasadena, California, and were cemented (with
epoxy cement) to the steel torsion rod at a point 7.125 in. from the
end. A drawing of the lower section of the torsion rod is shown in
Figure 4. They were placed at 45° to the cylindrical axis and at po-
sitions 90° apart around the periphery of the rod. The strain gages
were connected in a voltage divider circuit such that bending and ex-
tensional strains were cancelled. The input voltage to the divider
circuit was supplied by a 1/2 per cent regulated D.C. power supply,
and the output voltage was conducted to the oscilloscope amplifiers
through an electrical noise filter (1000 Q - 270 pF). The oscilloscope
sweep circuit was triggered by the capacitor discharge, as in refer-
ence 30.
The calibrated sweep rates of the oscilloscope provided a
time base from which to determine stress pulse duration and the oc-
currence of other events. The strain-gage output voltage was dis-
played on both beams of the oscilloscope. The upper beam was trig-
peced by the capacitor discharge and had a sweep rate of 200 usec/cem
for atime span of 2000 usec, thus providing information about any re-
flected stress pulses applied to the specimen. The initiation of the
GLASS DISK
OY
(oO
STRAIN
GAGES ——__}
7.125 in..:
| Figure 4. Lower Section of Torsion Rod. |
~24=
lower trace was delayed about 116 usec from that of the upper trace,
and the trace covered atime span of 200 usec, thus providing more
detailed information about the stress pulse as detected by the strain
gages. The oscilloscope screen was photographed with a Polaroid
camera.
C. Scratching Apparatus
When fresh dislocations were introduced into a specimen, they
were created by scratching the {100} observation surface with a dia-
mond phonograph stylus or an alumina whisker while the specimen
was in a special scratching apparatus. The scratching element was
at the end of a lever arm on which a weight might be placed such that
a known force was exerted by the scratching element. The specimen,
attached to the thermal buffer and specimen holder, was moved slowly
under the scratching element such that the scratch began at the fixed
end of the specimen and ended at the free end. A photograph of a
specimen, thermal buffer, and specimen holder combination in the
scratching apparatus is shown in Figure 5.
D. Specimen Polishing and Etching
The chemical polish for copper employed in this investigation
was reported by Livingston (24) and consisted of the following in vol-
ume per cent (acid concentration in parentheses),
20°/o HNO, (70°/o)
fo) fe)
24°/o HC,H,O, (99.7 /o}
oO oO
52-/o H,P0, (85 -/o)
2°lo HCL (38°/0)
2°/o H,O .
~25—
Figure 5. Specimen, Thermal Buffer, and Specimen Holder
Combination in Scratching Apparatus.
-~26-
The removal rate of this polish, whether a quiescent or agitatcd so-
lution, was 1 to 2 u/min.
Two etches were used to observe dislocations at the {100} ob-
servation surfaces. One etchant, designated EB1, was reported by
Livingston (24) and is composed to 50 g of anhydrous ferric chloride
(FeCt.) in 100 ml of hydrobromic acid (HBr - 48 per cent), The oth-
er etchant, designated EB2, was discovered by Young (19) and contains
the following concentrations of components:
2M FeCt, e 6 H,0
7.8M HBr
Immersion in either etchant was followed by a rinse in hydrobromic
acid. The etching or removal rate of EB1 was about twice as fast as
that of EB2 as judged from the fact that an etching time of 5 sec was
_ required to produce the same pit size with EB2, whereas 3 sec was
required with EB1. The £100} observation surfaces were etched be-
fore and after the application of a stress pulse to a specimen. [If the
observation surfaces were scratched, they were etched after scratch-
ing and aftcr application of a stress pulse.
E. Specimen Surface Records
A record of an as-etched surface was obtained by making a
replica of the surface on cellulose acetate film with Tadd replicating
solution, manufactured by Ladd Research Institute, Inc., Burlington, |
_ Vermont. After drying, the replica was peeled from the surface and
mounted on a stiff Mylar sheet with Scotch Brand double~stick tape.
The Mylar sheet prevented curling of the replica. After further dry-
ing for 24h, the replica was made opaque and reflecting by vacuum
~27-
depositing aluminum on the replica surface. This process rendered
the replicas suitable for examination in a metallurgical microscope.
A microscope system for comparing the "before" and "after"
replicas which has been described previously (35) was used here to
determine dislocation displacement.
~28-
IIL PREPARATION OF TEST SPECIMENS
A. Crystal Growth
The single crystal specimens used in this investigation were
machined by various methods from much larger single crystals.
These parent single crystals were grown by a modified Bridgman
technique according to a procedure developed by Young and Savage
(36). Their technique for producing low dislocation density crystals
was to employ a very smooth crucible wall and to adjust the tempera-
ture gradients such that the gradient along the crucible axis was the
major one.
The procedure described in reference 36 was modified for use
in this investigation, but the essential features were maintained. A
graphite crucible, similar to that described in reference 36, was used
to contain the initial material charge, the molten material, and the
resulting single crystal.
Crucibles were machined from a block of graphile which is
designated 3499 by the manufacturer, Speer Carbon Company, St.
Marys, Pennsylvania. The maximum ash content of this graphite was
0.1 per cent, which was considerably greater than that for the graph-
ite crucibles used by Young and Savage. They reported that the
smoothness of the crucible wall was due to the purity of the graphite
from which the crucible was machined. The walls of the crucibles,
as machined, which were used in this investigation were not smooth,
but an attempt was made to increase the smoothness by coating the in-
side of the crucibles with Dag Dispersion No. 154, a suspension of
colloidal graphite in alcohol produced by Acheson Colloids Company,
~29-
Port Huron, Michigan.
The graphite crucible rested inside a Mullite tube closed at
one end and with an inside diameter of 1 $ in. This assembly was en-
closed in a Vycor tube which was connected to a vacuum system
through a ground quartz joint. Induction heating of the graphite cru-
cible was used to melt the copper charge. A copper-tube induction
coil surrounding the Vycor tube and driven by an rf generator was
translated along the axis of the tube to provide the moving temperature
gradient. A chromel-alumel thermocouple whose bead was situated at
the tip of the crucible between the Mullite tube and the crucible and
whose output was read on a Leeds and Northrup chart recorder pro-
vided temperature measurement.
The procedure in crystal growth was as follows. A charge of
99. 999 per cent copper rod obtained from American Smelting and Re-
fining Company was placed in a''Dag" coated crucible. The copper
rod was etched in concentrated nitric acid before placing in the cruci-
ble. The Vycor tube enclosing the crucible and Mullite tube was
placed in the induction coil and connected to the vacuum system. Af-
ter a vacuum of 1x 107° mm Hg was reached, the power to the induc-
tion coil was inc reased from zero such that the pressure in the sys-
tem, as measured at the port of the vacuum system, never exceeded
5X 1074 mm Hg. The initial position of the induction coil was such
that the center of the coil was at the tip of the crucible. Heating and
melting of the charge took place while the coil was in this position.
After the thermocouple had indicated a temperature of 2200 to 2300°F
for about 1}h, the coil raising mechanism was started; and the coil
-30-
moved at 1.2 in/h. The coil continued moving until the bottom of the
coil passed the plane of the end of the crystal, at which time the coil
raising mechanism was stopped and the power to the coil was gradual-
ly lowered. When cooling was complete, the system was vented to at-
mosphere and disassembled; and the crystal was carefully withdrawn
from the crucible. The crystal was etched in concentrated nitric acid
to determine if more than one grain had nucleated. The crystals were
usually about 5 in. long. They seemed to have a preferred growth di-
rection of <110>.
About one-half of the test specimens in this investigation were
machined from crystals obtained from the Sandia Corporation, Albu-
querque, New Mexico. These crystals were grown in essentially the
same manner as above, except that the environment was one atmos -—
phere of helium.
B. Spark and Acid Machining
Major shaping operations to obtain the desired shape and ori-
entation of specimens were performed in spark erosion machines, the
- Servomet spark-erosion machine, Model C, manufactured by Metals
Research, Ltd., Cambridge, England, and the Agietron spark-erosion
machine, type AB, manufactured by AG. fur industrielle Elektronik,
Locarno, Switzerland.
For future reference, some of the energy ranges and associ-
ated capacitances in the discharge circuit are given below for the
Servomet and Agietron machines. (The number of energy ranges on
the Agietron machine has been augmented from those existing on the
production machine. }
-3]-
SERVOMET
Range Capacitance (uF)
4 25 . .
increasing
5 2 25 energy and
6 .05 damage to
crystals
7 » Ol
AGIETRON
Range Capacitance (uF)
3 ol
2 - 03 . :
increasing
1 - Ol energy and
C . 005 damage to
crystals
B » 002
A - 0004
In order to determine the crystallographic orientation of the
crystal relative to its cylindrical axis, the tip of the crystal was cut
off perpendicular to the axis. This operation was performed by a
traveling wire electrode attachment in the Servomet machine operated
on range 6. The surface of the crystal thus exposed was etched in
concentrated nitric acid (to remove spark machining damage), anda
Laue back~reflection x-ray photograph of that surface was taken while
the crystal was in a V-shaped cradle which was interchangeable be-
tween an x-ray machine track and the Servomet machine. The crystal-
lographic orientation of the crystal was determined from the Laue pho~
tograph, and the crystal was cut along {100} planes with the wire at-
tachment to the Servomet machine operated on range 4 or 6.
The crystal was thus sliced into <100> oriented segments 15 to
20 mm long. A given segment was then mounted on a large goniometer
-32-
which was interchangeable between an x-ray machine track, the Ser-
vomet machine, and the Agietron machine. A photograph of a typical
segment mounted on the goniometer, which is on the adapter fixture to
the x-ray machine, is shown in Figure 6. With the aid of the goniom-
eter and a rotating electrode in the Servomet (a so-called planing
wheel), a wire-cut surface on a crystal segment could be planed flat
and parallel to a {100} surface within 1°
The next operation in specimen preparation from these oriented
segments was the machining of a right circular cylinder whose axis
was <100>. This was accomplished by using a thin-walled tube of
copper, brass, or more commonly, copper-tungsten alloy as the elec~
trode in either spark machine and "trepanning" with the electrode per-
pendicular to the {100} surface. The energy range used was 6 in the
Servomet machine, C on the Agietron machine. A photograph of the
trepanning operation in the Agietron machine is shown in Figure 7.
The diameter of the resulting cylinder was approximately 0. 525 in.
The rough specimen was then reduced in diameter on a chemi-~
cal or acid lathe of a design similar to that reported in reference 37.
The essential features were a rotating, cloth-covered Lucite wheel
which carried the solution to the specimen and a specimen-rotating
drive whose axis was accurately parallel to that of the latheing or acid
wheel. The specimen was placed at the end of a lever so that contact
with the acid wheel and consequent damage could be controlled. A
short piece of 3/8 in. copper rod whose ends were parallel was at-
tached to the spark-planed {100} end surface with Duco cement. The
3/8 in. copper rod facilitated attachment to the specimen drive. A
-336
Figure 6. Crystal Segment on Goniometer in X-Ray Fixture.
~34-
Figure 7. Trepanning Operation in Agietron Machine.
~35-
solution of 50 per cent (by volume) concentrated nitric acid and 50 per
cent water was first employed, but a saturated solution of cupric
chloride in concentrated hydrochloric acid proved more reliable and
facilitated a subsequent orientation procedure by producing etched
bands at the {100} surfaces on the cylindrical surface. A photograph
of a typical specimen being reduced in diameter in the acid lathe is
shown in Figure 8. The result of the acid latheing operation was a
right circular cylinder with a diameter 0. 495 to 0. 490 in., an axis
accurately <100>, one end accurately {100}, and the other end within
4° of {100}.
In order to increase the size of the {100} observation area, it
was decided to machine narrow flats at the {100} planes on the cylin-
drical surface of the specimens. These flats were, on the average,
about 3 mm wide and extended the length of the specimen. Fora
specimen with a diameter of 0.49 in. or 12.45 mm, the reduction in
diameter at the flats was 3 per cent. The variation of resolved shear
stress across the flats was, therefore, judged to be negligible.
To align the specimen such that these flats might be machined,
the specimen was first polished and etched to produce square pits on
the {100} end surfaces. The edges of these square pits were <100>.
A stainless steel block with accurately perpendicular faces was aligned
with the cross-hairs of an eyepiece in a metallurgical microscope.
The accurate {100} end of the specimen was attached to the steel
block with Duco cement, at the same time aligning the edges of the
square pits on the inaccurate {100} specimen end with the eyepiece
cross-hairs and, consequently, with the edges of the steel block.
-36-
Figure 8. Typical Specimen in Acid Lathe.
~37-
Using the steel block as a holder and alignment fixture for the speci-
men, the flats were then spark planed in the Servomet machine on
range 7. Attempts at acid or chemical planing of the flats were not
successful because the chemicals used degraded the Duco cement bond
to the extent that it failed, causing the crystal to drop off. Subsequent
to this writing, chemical planing of the flats has become possible.
The surfaces produced by spark planing of the flats were within 2° of
{100}. The other end of the specimen was then spark planed accur-
ately to {100} on ranges 6 and 7 of the Servomet machine. The re-
sulting specimen had accurately parallel {100} ends, flats at the {100}
observation surfaces, and approximately the proper diameter for
acoustic impedance matching in the torsion testing machine.
C. Anncalin g
After specimens were demounted from the stainless steel
blocks and polished for about 30 min in the chemical polish to remove
some spark=machining damage, they were placed on graphite in a
tube furnace through which dry hydrogen was passed. Specimens were
annealed in this fashion at 1020 - 1040°C for about 100 h. Dislocation
densities in the annealed specimens were not markedly decreased by
a slow, controlled cooling rate from the annealing temperature (about
55°C/h) so the normal furnace cooling rate was finally used. The
maximum cooling rate for this tube furnace was about 72°C/h.
Annealed specimens were kept in a dessicator until they were
to be tested. This procedure avoided contamination of the observation
surfaces.
-38-
IV. EXPERIMENTAL PROCEDURE
The sequence of operations in testing a given specimen was as
follows. First, a thermal buffer was aligned and attached to a speci-
men holder with Eastman 910 adhesive. Next, the specimen was at-
tached to the thermal buffer, also with Eastman 910. This last opera-
tion required some precaution. The alignment could not be accom-
plished in the customary V-block because of potential damage to the
specimen and contamination of the observation surfaces. The method
finally employed was to place the specimen on end on polyurethane
foam and to bring the thermal buffer and specimen holder down gently
on the specimen, aligning it by eye. After some practice, this could
be done with sufficient accuracy, the misalignment of axes being less
than 0.003 in. The flats at the {100} observation surfaces were then
identified by letters a,b,c,d written on the thermal buffer. The
thermal buffer was coated with Duco cement to prevent any contamina-
tion of the polishing and etching solutions used on the specimen.
Next, the specimen was chemically polished for 8 to 10 min ,
rinsed in distilled water, etched in either EB1 or EB2 etchants, and
rinsed in hydrobromic acid and distilled walter. Unless a specimen
were to have fresh dislocations produced at the observation surfaces,
that is, scratched, the etched surface produced in the above procedure
was replicated as described previously. This replica was the "before"
replica. -If, as in the case of the first two specimens tested, the spec-
imen were to be scratched, the first etch was removed by polishing for
8tol0min. Then the observation surfaces were scratched, the
scratch running almost parallel to the <100> direction. The specimen
~39-
was polished for about 10 sec, etched in EBI and rinsed in hydrobro-
mic acid and water; and the {100} observation surfaces were repli-
cated as described previously.
Next, the torsion machine was assembled and the static torque
applied. All adhesive bonds in the torsion machine of a temporary
nature were made with Eastman 910 adhesive. The torsion machine
was completely ready to operate before the specimen holder- thermal
buffer - specimen combination was attached to it. Immediately after
the torsion pulse was applied, the specimen holder was removed from
the load train by depolymerizing the Eastman 910 adhesive with heat
from a propane torch. No heating of even the thermal buffer could be
detected in this process, and the taper end of the specimen holder
was quickly cooled in water.
Immediately after testing, the specimen was rinsed in distilled
water, chemically polished for about 10 sec, etched in solutions EB1
or EB2, and rinsed in hydrobromic acid and distilled water. It should
be noted that a ten-second chemical polish was not performed to re-
move material but to prepare the {100} surfaces so that the etchant did
not produce general faceting and obscuring of the dislocation etch pits.
After the above etching, the observation surfaces were replicated.
This was the "after'' replica. In case general faceting did occur upon
etching, the specimen was chemically polished to remove that etch pat-
tern and then re-etched and replicated. This latter procedure pre-
cluded use of the double etch technique of discerning dislocation dis-
placement; however, dislocation movement could be determined from
the 'before' and 'after' replicas with the aid of the comparison micro-
-40-
scope. Techniques and procedures in the acquisition of stress and
dislocation displacement data are explained in the next section of
this thesis.
Four such tests were performed with nominal resolved shear
stresses ranging from 25 to 250 g/mm*. A different annealed crys-
tal was used in each test.
-41.
V. EXPERIMENTAL RESULTS
A. Specimen Substructure
There was a wide variation in dislocation substructure among
the specimens examined in this study. The type of substructure of a
specimen seems to be correlated with the crystal from which a spec-
imen was machined. Specimens were from two parent crystals:
crystal 3, obtained from Sandia Corporation; and crystal 6, grown as
described in Section Ill of this thesis. The first digit of a specimen
number is the number of the parent crystal.
The variation in dislocation substructure may be seen by com-
paring the photomicrographs of Figures 9 and 10. Figure 9 exhibits
the dislocation etch pit configuration typical of specimens derived
from crystal 3; and Figure 10, the configration typical of specimens
derived from crystal 6. Large, very low dislocation density sub-
grains with high density subboundaries could be obtained after spark
machining and annealing specimens from crystal 3, whereas speci-
mens from crystal 6 had amuch higher average dislocation density
with no or low density subboundaries. While the motion of fresh dis-
locations from a scratch was studied in the low dislocation density
crystal 3 specimens, only the motion of aged dislocations was studied
in crystal 6 specimens because the introduction of fresh dislocations
would have increased the density to the point that the dislocation dis-
placements could not be determined with any certainty. There was
found to be no significant difference between the motion of fresh and
aged dislocations.
It would seem that the higher dislocation density in crystal 6
-42—
Figure 9. As Annealed and Scratched Dislocation Configuration,
Typical of Specimens from Crystal 3. 100X.
Figure 10. As Annealed Dislocation Configurations, Typical of
Specimens from Crystal 6. 100X.
-44.
specimens is caused by spark machining. It is theorized that incrys-
tals like number 3, the stable substructure is grown into the crystal
and that this substructure limits motion and multiplication of the dis-
locations produced by spark machining. In crystals such as number
6, it is thought that the initial dislocation density is low with little or
unstable substructure. Therefore, in these crystals, dislocations
produced in the spark machining processes move large distances and
multiply many times.
A further example of spark machining damage in copper single
crystals is shown in the photomicrograph of Figure 11, which exhibits
an etched observation surface of specimen 3-2-1 after annealing and
scratching (the scratch damage extends across the top of the photomi-
crograph). The right edge of the photograph coincides with a spark-
planed end of the specimen. This end, a (100) surface, was spark
planed using ranges 6 and 7 in the Servomet machine. Spark machin-
ing damage extends some distance, approximately 450 yu, into the
crystal after removal of about 50 u by chemical polishing and after
annealing. Gniewek, et al. (38) have investigated spark machining
damage in copper due to spark planing on surfaces perpendicular to
{111} planes. This is the first study of dislocation motion in copper
where spark machining was the predominant means of shaping speci-
mens, for Young (20, 22, 23). and Petroff (25) used chemical machining.
B. Stress Pulse Analysis
Several features of the torsion machine records need to be ex-
plained before proceeding to calculations which yield the maximum ap-
plied, resolved shear stress, and the shape of the stress pulse for
-45.
Figure ll. ‘As Annealed and Scratched Dislocation Configuration,
Exhibiting Spark Machining Damage. 100X.
~46-
each test. A typical torsion machine record is shown in Figure 12,
This particular photograph is the record of the lest on specimen
6-2-1, where the resolved shear stress was calculated to be 236
g/mm*, the highest stress applied. Attention is called to several
features of the lowcr trace. The rise time of the pulse is about 2
usec. At about 60 usec later on the trace there is a dip which shows
the effect of an imperfectly matched taper joint where the specimen
holder connects to the lower section of the torsion rod. Only one
specimen holder was accurately fitted to the taper joint by lapping.
When that specimen holder was used in the torsion machine, no dip
was observed in the trace of the stress pulse. The effect of the im-
perfectly matched taper joint is not great since the stress rises to
the value which it had before the wave encountered the taper joint. At
about 110 usec on the lower trace there is a slight dip due to the
stress wave encountering the specimen holder - thermal buffer inter-
face and diameter change.
The end of the pulse detected by the strain gages shows the
- effect of the specimen. If the acoustic impedance of the thermal buf-
fer and the specimen were cqual and if the specimen remained elastic
upon application of the stress pulse, the trace would remain at that
level which it had before the wave encountered the specimen and then
would fall to zero in 2 usec. In this particular case the diameter of
the specimen was 0.491 in., less than the ideal 0.493 in. This de-
creased diameter should have caused only about 1 per cent decrease
in the signal if the specimen had remained elastic. In addition, if the
specimen had remained elastic, the signal would have fallen to the de-
~47 =
Figure 12. Typical Torsion Machine Record.
-48-
creased value, remained at that value for about 10 usec, and then de-
creased in about 2 «sce to the zero level of the signal. Evidence of
plastic flow in the specimen is seen in two features of the trace; the
greater than calculated decrease of the signal when the pulse trav-
ersed the specimen (12 per cent), and the long decay time of the
pulse. This trailing edge of the trace will be examined in greater de-
tail in connection with the analysis of the applied stress.
The two pulses occurring at the end of the upper trace are
caused by reflection of the stress wave from the damping section of
the load train. The incomplete attenuation which causes these reflec-
tions is due to the fact that the composition of the Solithane visco-
elastic material did not provide complete attenuation of the pulse
traveling away from the specimen. That such long rise-time pulses
cause a negligible increase to the resolved shear stress in the speci-
men can be seen by the following argument. If the rise time of a
stress pulse is sufficiently long, it can be visualized that the stress
in a short specimen might never reach the maximum stress of the
- pulse because the unloading stress wave propagating from the free
end begins to decrease the stress before the maximum stress is
reached. It can be shown that
A'/A)(2z/V
at, AV AN22/V oq) 23
. t'.
rise
where
Tr = maximum stress produced by a pulse,
A = amplitude of a pulse,
axial distance from the free end of the specimen,
XS
Ht
~49~
tise = rise time of a reflected pulse,
aud the prime superscripts refer to a stress wave reflected [rom the
damping section of the load train. Calculated in this manner, the
maximum stress applied to the specimen by the reflected pulse was
only about 3 per cent of the stress applied by the trapezoidal-shaped
pulse and, therefore, was deemed negligible.
The analysis of the stress pulse requires further explanation
before proceeding to the tabulation of the results of this investigation.
First, a general analysis of the stresses transmitted by torsional
waves in circular cross-section rods across interfaces between dif-
ferent diameters and materials will be presented. The boundary con-
ditions at an interface, in this case between the thermal buffer and
the specimen, are that the torque, ‘I’, and the angular velocity, 6 ,
‘be continuous.
Te = vou
SD = bau
[13]
where the subscripts th and cu again refer to the thermal buffer and
the specimen, respectively. Then the assumption is made that the
torsion shear stress distribution and its dependence on applied torque
which has been derived for the static case is true for the dynamic
case, i.@.,
t = Tr/J [14]
where
- = torsion shear stress,
r = radial position of an clemental volume in the rod,
-50-
J = polar moment of the cross section = nd */32 °
Then the boundary conditions in Equations 13 evaluated at the lateral
surface of the thermal buffer or specimen become
(Sy = [15]
tb cu
and.
Vv = Sth. Vv
tb a cu ’
cu
where a is the radius of a cross section, v is the linear velocity of
an elemental volume at the periphery of the cross section, and
v=zaé. [16]
In addition, from equating the impulse applied to an elemental volume
to its increased momentum,
Ar = pV Av [17]
where Af is the change in stress across the wave front, p is the ma-
terial density, V is the torsional wave velocity, and Av is the change
in linear velocity of an elemental volume due to passage of the wave
front. Figure 13 shows a typical position - time diagram which is
helpful in visualizing the stresses applied to the specimen and which
defines the numeral subscripts in the equations to follow.
From Equations 15 and 17 the following relations may be de-
h - coat” |
(eVI).
However, the product pVJ may be seen to be the acoustic impedance
rived:
AT
> = Ay [18]
for torsional waves, Z. Therefore, Equation 18 becomes
/-51-:
uswitoedg pue razmg |
ON3 Tewuisyy Ul SoAeM UOTSAOT, 1OjJ wiearseIg SWIL-uUoTHIsOg “ET 2an8T,7 |
q3u3 NSWI03dS SOVANSLNI Y333N8 IWWYSHL
NOILISOd 5 :
i mh wii (i ei cm ee
AWIL
~52-
Ta-Zz./zZ)
tb cu .
Ar, = At [19]
2 1 a +272)
and -
2(dy,/4..,)°
At; = AT) | TPE 7z [20]
a th “cu
r. 3 . ,
An. = 5 1 i‘ + a qa + a A)
At, = -AT i . [22]
6 1|/ (+ Z/4 0 + Z oy! Zip) .
It may seem that making the acoustic impedance Z continuous across
an interface, as was done in Section Hl, is equivalent to requiring that
AT, = 0 (no stress reflected at the interface) and to deriving AT 3 by
matching torque at the interface since ZplZ oy = 1, thus making
Ar, = Ar, (dia/4.,.) .
The source of AT) in the thermal buffer was the torsion stress
pulse created in the steel torsion rod. The torsion stress in the steel
rod was calculated from Equation 14 where r = d/2 and T, the torque,
was one-half of the static torque applied to the rod, which was, in
turn, equal to the product of the radius of the cranking disc and the
weight hung on it. The stress in the steel rod was translated into At,
in the thermal buffer by means of Equation 20.
The end of the stress pulse as detected by the strain gages
shows the effect of the specimen. More specifically, the end of the
pulse reveals the form of the stress pulse at the thermal buffer -
specimen interface because there is no distortion of the pulse between
that interface and the location of the strain gages, everything being
-5 36
elastic. However, there is some plastic flow in the specimen, even
apparently at the lowest stresses, and the stress pulse is thereby dis-~
torted. Considering this distortion, the only point on the specimen
where the form of the stress pulse is known with any degree of accu-
racy is at the thermal buffer - specimen interface (aside from the
free end), With the above considerations in mind, it was decided to
analyze the data in the following manner. The stress pulse at the in-
terface, on the fixed end of the specimen, was determined for each
test from enlargements of the torsion machine records (cf. Figure 12).
Graphs of the pulses are shown in Figures 14 through 17.
These pulses were analyzed in the following manner. The area
under a given curve, the impulse, was determined with a planimeter.
From the torsion machine record it was determined that the rise time
of the pulse was either 2 or 3 usec. Then a trapezoidal-shaped pulse
was formed from the impulse of the actual pulse. The height, dura-
tion of the plateau, and decay time of the trapezoidal pulse were ap-
portioned so as to conform as closely as possible to the shape of the
original pulse. These trapezoidal~shaped pulses are shown by dashed
lines in the Figures 14 through 17. The impulse contributed by any
secondary lobes or pulses was less than 10 per cent of that of the orig-
inal pulse and was not included in calculating the shape of the trape-
zoidal pulse.
If the specimens had remained elastic, the stresses AT, » etc,
could have been calculated from Equations 19 through 22. The ratio of
the decrease in the strain gage signal as the pulse encounters the
thermal buffer - specimen interface to the prior signal level is equal to
(54 |
Ob | oc
"[-€-9 Usuttoeds jo puy pextyz ye es—Mg ssozig “PT corns
(SpuodssossIWw) JWIL
(SHJOAIIIW) AFOVINIOA AWNOIS
Ol
al
|-55~
“"[-Z-€ wewtdedg yo puy pexty ye esTNg ssotig “GT enB1y
(SpuosasosoIw) JWIL
(SHOANIW) ZOVIIOA TWNOIS
(=56=
o¢
-(Spuoossosoiw) JWIL
02
"T-€-¢€ usuttsedg jo pug poxryz ye OFM Sserjg “g{T ean
Ol
— Pr CE oe ee OE
9¢
Ov
1 bb
8b
(SHJOAI}!W) JOVIIOA TVWNOIS
|-57-:
Ov
“[-¢-9 wauirsedg jo PUY PEXlT +e os—Ng ssaaig "TT OIn3t 7
(SpucossosIWw) JWIL
o¢
001
Ol
LL \ i 021
(SHOANIIW) JOVLIOA TWNOIS
-58-
ratio Ar,/Ar, .» if this ratio is calculated from Equation 19, assum-
ing that the specimen remained elastic, the signal decrease is less
than actually occurred. Therefore, the ratio of the signal decrease
to the prior signal level was determined and used in calculating
stresses in the specimen. This procedure is tantamount to determin-~
ing the effective acoustic impedance of the specimen. The solid lines
extending across the top of the graphs in Figures 14 through 17, for
example at 8.8 mV in Figure 14, represent the signal level due to the
stress pulse in the thermal buffer and the signal level which would
have been obtained if the acoustic impedance had been continuous
across the thermal buffer - specimen interface and/or plastic flow
had not occurred in the specimen. The decrease of the signal from
this level represents the stress Ar, reflecting from the interface.
The incident stress AT, is considered decreased by the ratio
of the signals after and before the pulse encounters the interface.
This decreased AT, is then used in calculating AT, » the stress in
the specimen, by means of assuming the torque is equal across the
thermal buffer - specimen interface. In resolving the stress on to
the {111} <110> slip system, AT, is decreased by a factor 1/Ve.
The resolved shear stresses, Tmax? for each test corresponding to
the plateau of the trapezoidal-shaped pulse are given in Table 1.
C. Etching Observations
Several features of the results using the {100} dislocation
etchants require some explanation since this is the first study to em-~
ploy them. In addition, since there is a wide variation in the features
of the etched {100} surfaces of the specimens tested, the photomicro-
~59-
graphs of these surfaces in Figures 18 through 21 need to be explained.
These photomicrographs display the dislocation configurations before
and after the application of a torsion stress pulse. The val! part ofa
figure refers to the configuration before testing; the "b'' part, to the
configuration after testing. In each micrograph the [100] specimen
axis is horizontal, and the free end of the specimen is toward the
right. The dislocation etch pits have a crystallographic shape, i.e.,
the edges are along <100> directions and the sides are {110} planes.
The <110> slip directions are, therefore, along the diagonals of the
square etch pits and at about 45° to the edges of the photographs.
Among these photomicrographs there is a variation in the
clarity with which etch pits reveal the dislocation configuration. This
variation is due to a problem with re-etching specimens after the ap-
‘plication of the stress pulse. The only explanation of the problem
which can be advanced is that some contaminant came in contact with
the {100} surfaces of the crystal at some time between the before and
and after etching. The contaminant altered the chemical reaction of
the etchant to the surface sufficiently that the dislocation etch pits
were less well-defined in relation to the general background. The
ten-second chemical polish after testing was not effective in removing
the contaminant. The only exception to the above remarks was that
specimen 3-3-1 (Figure 20) rc-ctched well, and the clarity of the pits
gave an excellent indication of the dislocation configuration.
The dislocation configuration in specimen 6-3-1, as observed
on flat a, is shown in Figure 18. These photographs were taken ata
point on the specimen where Z , the axial distance from the free end
- . -60-
Figure 18. Etched {100} Surface of Specimen 6-3-1, 250%.
-6 |=
of the specimen, was 17.2mm. The specimen was etched in both in-
stances in etchant EB2, for 2 sec in the case of Figure 18a, for 5 sec
in the case of Figure 18b. The dislocations revealed in Figure 18a
are dislocations which existed in the specimen after annealing; thus,
in this case, the motion of aged dislocations was studied.
At this point some observations will be made regarding the
etching of dislocation sites on {100} planes in copper. These re-
marks are generally valid for Figures 18 through 21 and for both
etchants. At least two types of pits may be distinguished in Figure
18a - small, light and small, dark pits. This difference in appear-
ance may be explained in several ways. The etchants may be dis-
tinguishing between positive and negative edge dislocations as Living-
ston found for a {111} dislocation etchant (39). However, since dis-
locations revealed by both light and dark pits move in the same
direction in the same slip system, it would seem that the differentia-
tion is not between positive and negative edge dislocations.
On the other hand, the etchants may be distinguishing between
screw- and edge-oriented dislocations. One fact which may militate
against the latter possibility is the following. For this specimen ori-
entation, there are three Burgers vectors connected with a given
{111} plane, one lying in the £100} observation surface and two in-
tersecting the surface at 45°, for which the resolved shear stress is
one-half as large as that for the Burgers vector lying in the observa-
tion surface. Dislocations whose Burgers vectors intersect at 45°
might reasonably be expected to be predominantly screw~oriented.
Now, although both light and dark pits were included in the determi-
-62=
nation of dislocation displacement, no systematic variation of dis-
placements between light and dark pits could be ascertained. If edge
and screw dislocations move at different velocities at the same stress
as Johnston and Gilman (7) found in lithium fluoride, a difference in
the displacements between screw and edge dislocations should be evi-
dent unless the effect is cancelled by the difference in resolved shear
stress described above. Therefore, it would seem either that the
etchants do not distinguish between edge and screw dislocations or
that screw dislocations move faster than edge dislocations in copper.
In Figure 18b can be seen another type of pit, a large, flat-
bottomed pit. This type is interpreted as the site where a dislocation
had been at the time of the first etch. The irregularity of the edges of
the pit is due to the 10 sec chemical polish which preceded the "after"
etching of the specimen. Also, due to this polishing, a dislocation
which did not move is manifested in the superposition of a small pit on
an irregular-shaped larger pit rathcr than a large, sharp-bottomed
pit which is normally seen in the double etch technique. Dislocation
motion can be seen at points A and B in Figure 18b.
The dislocation configuration in specimen 3-2-1, flata, is
shown in Figure 19, Etching was done in etchant EB! for 4 and 3 sec,
respectively. These photographs were taken at a point 11. 6 mm from
the free end of the specimen. The thick band of dislocation etch pits
extending horizontally across the photographs is the result of scratch-
ing this observation surface with an alumina whisker in the scratching
apparatus mentioned previously. The load on the whisker was 70 mg.
The line of etch pits extending from the top to the bottom of the photo-
-63-
Etched {100} Surface of Specimen 3-2-1, 250X.
Figure 19.
~64=
graphs is a subgrain boundary. It would appear that the density of dis-
locations has undergone a tremendous increase, but this is due toa
faceting of the background, presumably caused by some contaminant
which affects the etching behavior. This faceting produces a pebbled
surface and features, such as in the circle at A, which might be con-
fused with dislocation etch pits. Displacements of both aged and fresh
dislocations produced by the scratch were included in the data used in
determining the dislocation velocity. Dislocation motion can be seen
at points B and C in Figure 19.
The dislocation configuration in specimen 3-3-1, flat d, is
shown in Figure 20. Etching was done in etchant EB1 for 3 sec in
each case. These photographs were taken at a point 13.5 mm from the
: free end of the specimen. As in Figure 19 the band of dislocation
etch pits extending horizontally across the photographs is the result of
scratching this observation surface. In this case, the scratching ele-
ment was a diamond phonograph stylus, and the force was 330 mg.
Note that the width of the band and, correspondingly, the damage done
by the scratching element are about the same in both Figures 19 and
20 in spite of the different forces and scratching elements. In gather-
ing data of dislocation displacement from this test, the motion of dis-
locations from the edge of the scratch was studied. In addition, dis-
placements of aged dislocations far from the scratch were measured
to determine the possible effect of residual stresses near the scratch
and to determine any differences between the motion of aged disloca-~
tions and that of fresh dislocations. Motion of dislocations from the
scratch is evident. In addition tothe motion of single dislocations as
-65-
Figure 20. Etched {100} Surface of Specimen 3-3-1, 250X.
-66-=
at B, there is also a cooperative motion of two and more dislocations
as al A.
The dislocation configuration in specimen 6-2-1, flat a, is
shown in Figure 21. This specimen was etched for 4 sec in EBI be-
fore testing. Since the pits were rather large and somewhat indistinct
(the sides of the pits are terraced), the specimen was etched for 5
sec in EB2 after the test. This etching was preceded by chemical
polishing for 8. min in order to remove the pits from the previous
etching. This was about a standard time to remove the pits from an
etching by chemical polishing; therefore, the pits were about 8 to 16
wu deep, according to the previously stated polishing rate of 1 to 2
/min. The use of the double etch technique was not possible in this
case because the "before" etch pits had to be removed; however, the
comparison microscope system made it possible to determine dislo-
cation displacements.
D. Dislocation Displacements
Acquisition of dislocation displacement data proceeded as fol-
lows. The 'before' and 'after' replicas of an observation surface for
a given test were aligned in the comparison microscope system using
the substructure as fiducial marks. Displacements were measured
with a filar eyepiece which had been calibrated with a stage microme-
ter, It is not at all clear from viewing the areas shown in Figures 18
through 21 whatthe before and after positions of a moving dislocation
might be. Indeed, it took considerable time examining the replicas of
the observation surfaces to determine the position from which even
one dislocation might have moved. In general, attention was focused
oe
9 Se
Figure 21. Etched {100} Surface of Specimen 6-2-1, 250X.
~68~
on a small, dark or light etch pit, and the problem of from where it
came was posed. In some cases, as with a single aged dislocation,
the dislocation could have moved from any one of four positions 90°
apart around the new position. However, in many cases, as with
dislocations moving from a scratch or from a line-up of aged dislo-
cations, the choice of before positions was restricted totwo. Even
when there were four positions from which the dislocation might have
moved, the ambiguity could be resolved by aligning one cross~hair of
the eyepiece along the diagonals of the etch pits, or along a pile-up of
of dislocations (and thus, along the slip line) and then by observing
which large, flat-bottomed etch pit lay along the slip line from the
small, dark or light pit by means of the translating cross-hair. None-
theless, much judgment was required even in the most obvious casc
shown in Figure 20.
After the problem of the before and after position of a disloca~
tion was solved, the displacement was measured. The displacement
was the distance between centers of the etch pits. It was determined
that the center of an etch pit could be reproducibly determined within
2.
Increasing the duration of the stress pulse should increase the
displacement of a dislocation in a linear manner. The duration of the
stress pulse in these experiments increases with increasing distance
“from the free end of the single crystal specimen. Dislocation dis-
placements were determined at a minimum of three different stations
along the axial length of the specimen. At each station an average of
24 measurements of dislocation displacement, d, and distance of the
-~69-
dislocation from the free end were taken. The mean of the displace-
menl measurements al each station versus distance from the free end
of the specimen, z, is shown in graphs in Figures 22 through 25.
Figure 22 is for the lowest stress test (28.7 e/mm*), and Figures 23,
24, and 25 arc for increasingly higher stress tests. The circled
points represent the mean of the displacement measurements, the
number in parentheses is the number of measurements taken, and the
length of the vertical lines through the points represents the standard
deviation. In Figure 24 at z= 12.8 mm, the point represents dis-
placements of aged dislocations situated at least 180 yy from the center
of the scratch damage on the observation surface. Due to the slight
difference between the displacements of aged and fresh dislocations at
the same stress and pulse duration, the displacement of aged disloca-
tions was measured in the high dislocation density specimens.
Several features of the graphs in Figures 22 through 25 are to
be noticed. First, since the units of the abscissae can be converted
into time at the constant stress, the slope of lines drawn through the
data points is proportional to the dislocation velocity at that stress.
Secondly, the rough linearity of the dependence of displacement on
distance from the free end, or time, indicates that dislocations began
to move and reached terminal velocity in a negligible time, that is, in
a timc negligible compared to the 2 or 3 usec rise time of the stress
pulse.
Further analysis of the data requires that the dislocation dis-
placement at the axial station corresponding to the fixed end of the
specimen be determined since the stress pulse applied to that point is
-70~
35 a
30 —
r | . | (25)
oat
< .
= 20 | . | ~
= |
a ;
OIF | (26) | |
=z
2)
te
© 10k 4
at
”) i
ay ; ©
a © |
(21) x
5i- “ +
‘@) | | | | | | | | | | |
o 2 4 6 8 10 12 14 16 18 20 22 24
DISTANCE FROM FREE END (millimeters)
Figure 22. Dislocation Displacement Versus Distance from Free End
: 2
£ = wa te = Py e
or Specimen 6-3-1, fT ax 28.7 g/mm
35 es a a a
30 _~
(23) .
wn
é a 4
FF
s 20 | —
= | |
9 .
os
3 bb (23) -
Ee
ss
9 lor _
9) i
— c
O° @
a]
5 - -
0 a Gs GO DG
0 2 4 6 8 10 12 14 16 18 20 22 24
DISTANCE FROM FREE END (millimeters)
‘Figure 23. Dislocation Displacement Versus Distance from Free End.
. 2
for Specimen 3~2-1, T nax = 68.8 g/mm”™.
68 a ee ee
64+ | ~
60h | | _
56 | | (24) 4
@ B2E | -
oO
BE (ay | 7
fi 40
O _. _
< 36
a. Lo i : _
a 32 A ) a
- eat | _
fe) | 4 (6) : |
a | oe
O (26)
O 20 c | =
I @
wo a)
lef - _
8r —
a NN RS OS eG
0 2 4 6 8 10 12 14 16 18 20 22 24
DISTANCE FROM FREE END (millimeters)
‘Figure 24. Dislocation Displacement Versus Distance from Free End
. 4 2
for Specimen 3-3-1, TO = 134 g/mm*.
90 a
8s0Fr- _
(23)
_ t0Or —
fo)
: .
€ 60}- | | =
ci (25)
= _
a §
Oo.
2 40-r | =
30k ~
rs)
A fc)
Be0k (24) 2 _
uU
Ra)
lO —
re) I i | | L | | | | i
0 2 4 6 8 10 12 14 16 18 20 22 24
DISTANCE FROM FREE END (millimeters)
|Figure 25. Dislocation Displacement Versus Distance from Free End
oo 2
for Specimen 6-2-1, Tax = 236 g/mm’.
-74-
‘is known. The dislocation displacement at the fixed end of a given
specimen can be obtained by extrapolation from the measured values
of displacement. The position of the fixed end for each specimen is
shown in the graphs in Figures 22 through 25. The displacement at the
fixed end was taken to be the mean of the values obtained by extrapo-
lating from pairs of points to the fixed end points. In Figure 23,
however, the mean value (36. 4) of the displacement at the fixed end
obtained in this way was judged too low. A linc drawn through the
three points yielded a value of dislocation displacement at the fixed
end of 40 4 which was considered more accurate. The values of the
dislocation displacement at the fixed end of the four specimens are
given in Table 1.
-75-
TABLE 1
Values of Stress and Dislocation Displacement, d, .
at Fixed End of Specimens
Specimen qd T ax ty ts , t,
Number (u) (¢/mm®) (usec) (usec) (usec)
6-3=1 31.5 28. 7 3 15 27
3-2-1 40 68. 8 2 10 23
3-3-1 55.9 134 3 14 18
6-2-1 110.3 236 2 12 ai
-76-
VI. DISCUSSION
The results presented in Table 1 are values of dislocation
displacement in copper due to application of a controlled stress
pulse. These results indicate in several ways that the velocity of
dislocations in copper is relatively well-defined. The graphs in
figures 22 through 25 demonstrate that dislocation displacement is
a relatively smooth, if not linear, function of time at constant
stress for stress pulses of the order of tens of microseconds and
displacements of from 10% to 110%. The maximum standard devia-
tion expressed as a percentage of the mean dislocation displacement
was 13% in this investigation whereas for the work of Johnston and
Gilman (7) it was 30%. Moreover, each velocity data point in this
study represents the result of some 70 dislocation displacement
measurements.
Young (22, 23) and Livingston (24) obtained dislocation dis-
placements comparable to those in this investigation for stress
pulses of somewhat lower magnitude but with durations of seconds.
The explanation offered here for this discrepancy is that the dislo-
cations were stopped by an internal stress barrier and actually
moved during only a fraction of the pulse duration. The examples
of dislocation displacement cited by Young and Livingston seem to
be isolated instances, in contrast to this study, in which the dis-
placements of many dislocations were averaged. Although in the
investigations by Young and Livingston the dislocation displacements
were generally less than 100u,, dislocations could move large dis-
tances, as predicted by the high velocities found in this study, when
-77-
barriers to these large displacements were small. For example,
in the work by Young (23), dislocations traveled the width of a sub-
grain, a distance of at least 600n before being stopped by a sub -
boundary when the crystal was subjected to a stress pulse of
70 g/mm*. Subboundaries in copper have been found to be effective
barriers to dislocation motion (17).
It is of interest to examine the results of this study in the
light of current theories of dislocation mobility to determine which
theory most accurately predicts the velocity-stress relation. How-
ever, since the stress pulse was trapezoidal-shaped and not "square"
i.e., since the applied stress was not constant during the entire
period of dislocation motion, it is necessary to take this stress
variation in time into account. Other investigators of dislocation
mobility using the etch-pulse~etch technique have assumed that the
dislocation velocity was adequately defined by dividing the disloca-
tion displacement by the duration or time at constant stress of the
stress pulse, thereby assuming a "square" stress pulse. It was
not felt that this procedure for obtaining velocity was meaningful
for the results of this investigation. Since the various formulae
for dislocation mobility express the average dislocation velocity v
as a function of applied shear stress T and other variables, the
problem of obtaining velocities from the data in Table 1 was posed.
These data are analyzed below to give dislocation velocity versus
stress. The procedure adopted was the following: A specific form
of the velocity-stress relation was assumed to apply. This relation
was integrated with respect to time to give dislocation displacement.
-78-
The functional dependence of stress on time was inserted in the
stress-dependent half of the equation. The dislocation displace-
ment as a function of the maximum resolved shear stress applied,
the shape of the stress pulse, and various parameters ina given
relation was thereby obtained. How closely a given velocity-stress
formula predicted the displacements obtained in this investigation
could be determined from this resulting equation for dislocation
displacement. This procedure assumes that the dislocation ve-
locity is in phase with the stress during the entire stress pulse,
i.e., that the time of acceleration of the dislocation is negligible,
The manner in which dislocation displacement increases with
time of loading in each test indicates that the time of accelera-
tion of the dislocation was negligible in this investigation.
The first representation of the stress dependence of dis-
location velocity was not derived from a theory but from an em-
pirical fit of the dislocation mobility data for lithium fluoride and
silicon-iron, This relation was given in Equation 5 but is re-
stated here in slightly different terms as
v= vo(t/t ys [23]
where v-is the dislocation velocity (equivalent to the v of Section I);
+t is the applied, resolved shear stress; To is the applied, resolved
shear stress required to produce a velocity of Yo which is taken
here as 1 cm/sec, and m is the mobility exponent. Assuming the
-79-
validity of Equation 23, an analysis is presented in Appendix B which
yiclds a formula for dislocation displaccmecnt due to the application
ofa trapezoidal - shaped stress pulse. This formula for the displace-
T ™Tt,tm(t,-t,)
a+ (“) = | . [24]
fe) mtil
The symbols Trax’? ty? tz? and t, are defined in Appendix B and
ment, d, is
Figure B-1. Having values of d, T, .., andt,, t,; t,, a value of
m was assumed, and TT, for each displacement and stress pulse was
calculated from Equation 24, The mean of the four values of To for
each assumed value of m was calculated. This mean T, was used
to calculate the displacement, d', by means of Equation 24. The
error in determining displacement in this way is defined as
Id-d' |
ar = @ 3; [25]
and the mean error e for each value of m assumed was calculated.
This procedure was followed for values of m = 0.5, 0.6, 0.7, 0.8,
and 1.0. The results are given in Table 2. The calculated value
of Te is quite sensitive to the value chosen for m. The mean error,
e, in calculating ‘d from Equation 24 assuming values of m and,
consequently, To? is minimized for m= 0.7. Therefore, the form
vs (t/t)o", [26]
where T= 0.0248 g/min” gives a "best fit". Equation 26 anda
reduced dislocation velocity which is defined as
-80-
Table 2
mean T.
(g¢/mm?*)
0.00111
0.00673
0.0248
0.0675
0.281
® |
0.136
0.095
0.088
0.100
0,209
-8l-
t,tm(t,-t, )
y= a/ jae | [27]
with m = 0.7 are plotted in the graph of Figure 26. The vertical
lines through the reduced dislocation velocity points represent the
uncertainty in this quantity due to the uncertainty in the values of
dislocation displacement at the specimen fixed end introduced by
the extrapolation procedure.
In order to compare the experimental results of this inves-
tigation with theoretical predictions, it is of interest to fit the data
to a linear relation (m = 1 in Equation 23). Figure 27 is a graph of
this relation together with the reduced dislocation velocity defined
by Equation 27 with m=1. The vertical lines through the data
points again represent the uncertainty in reduced dislocation veloc-
ity. The mean error is larger with m = 1, but the uncertainty in
reduced velocities is such that the difference in e for m = 0.7 and
m = 1 is not significant. A mean value of TS of 0.28 g/mm* was
obtained with m= 1, giving
v = (1/0. 28)!" [28]
It can be seen that changing m from 0.7 to 1.0 has little effect on
the values of reduced dislocation velocity.
The reduced dislocation velocity, v', defined by Equation 27,
is the closest approximation which can be made from the displace-
ient-stress pulse data to the actual average velocity of dislocations,
v, given in Equation 23 and subsequent equations. Since the data
were represented better by the relation in Equation 28 than
“2 ‘O= Ww IOF sseajs reaug
. paajosey WINUIIXEW[ SNsisa AJIOOTeA UoTTed0TSIq pe2npey gz eans1z7.
| (,wu/5) SS3YLS YVSHS G3ATOSSY WNIWIXVAN
OOO! 0Ol Ol
a ar ee ee ee :
eo} |
| -82-
a ol S20'0/2) = 4
(9as/wio) ‘A‘ALIQOTSA NOlLVOONSIA GazonaaHY
"O°L = Wi ZOy ssarig re8eyg
peajosey wnwixeyy snsze, AzIOOTeA UOTTedOTSIG pecnNpey ‘Lz e1n317 |
(,ww/6) SSau1S YVAHS G3A10S3Y WNWIXVW
ooo 00! ol
rrtt tt l TTrTTT_ | 20!
| -83-
(oas/wo) MALIOOTSA NOILVOONSIG Qa0nasy
-84-
subsequent formulae, the reduced velocity for m = 1 was used as
the value of actual velocity in analyzing the validity of other mobility
formulae.
Another formulation of the stress-dislocation velocity relation
has been suggested by Gilman (40), He has applied reaction rate
theory to dislocation motion and has shown that when the activated
complex is allowed two degrees of freedom the relation
vV=v @7D/t
; [29]
results, where Yo is a limiting velocity and D is a characteristic
drag stress. The data shown in Table 1 were analyzed according to
this formulation to determine whether it was a better representation
of the data than Equation 28. The limiting velocity Yo is presumed
to be the velocity of shear waves propagating ina (ill) direction
with the shear stress polarized so that it acts ina (110) direction
ona {lll} plane. This velocity, evaluated from the stiffness moduli
of the [111] specimen orientation as given in Appendix A, is
2.14x 10° cm/sec.
As a first approximation to this analysis of the data, as it
relates to Equation 29, the reduced dislocation velocity from the
previous analysis (for m = 1) vs. 1/T ax is plotted in Figure 28.
The data as interpreted in this way is not represented by Equation
29. More detailed calculations were made to ascertain the validity
of the exponential formula for the data. Equation 29 was integrated
as described above, to obtain a formula for dislocation displacement,
and D was calculated for each test. The mean of the values for D
| ‘ssorjg reays poajosey UWINUWITXeYT
JO Tedordioey snsaze, A}DOTOEA UOT}EDO[SIG peconpey “gz eans1z
| | (B/swu , Ol) af
¢c¢ O¢ G2 02 om O1
| | | | |
CI [-
-85-
' o'82 0/2) = 4 WOuS A ©
— aya-2°A =A WOYS,4 8
I ! | | I
201
(988/W9) ALIDOTSA NOILVOO SIO azonaaY
-86-
was used to plot Equation 29, assuming the above value of Vor A
reduced dislocation velocity, v'', for this formulation docs not lic
_on the straight line predicted by Equation 29.
Another theory for the relationship between dislocation
velocity and stress has been presented by Fleischer (41). This
theory is based upon the assumption that the velocity of dislocations
is governed by their interaction with impurity atoms which are ran-
domly distributed throughout the crystal. The theory is suitable to
crystals in which the hardening is primarily due to tegragonal lattice
distortions such as result from the introduction of magnesium into
lithium fluoride. Although this mechanism is not thought to be oper-
ative in copper, the data were also analyzed on the basis of this
theory. The formula given by Fleischer for dislocation velocity is
3 2
v = £L exp{-F b [1-(+/7)) ] /kT} , [30]
where f£ = frequency of vibration of a dislocation segment
of length L,
F = the maximum force exerted on a dislocation due
‘to the lattice distortion by the defect,
T= the flow stress at 0° K,
k Boltzmann's constant,
and T absolute temperature.
The logarithm of the reduced dislocation velocity defined by
; iL
Equation 27 for m = 1 is plotted vs. the quantily [(t/45)-(4/7)7] in
Figure 29. If the data of this investigation fit Equation 30, they
should lie on a straight line in the graph; however, they do not.
108 I | l I ]
= —
Pp-
oe ra |
a) - / 4
~~ 7 /
ie /
as a -
Oo
oO /
iT /
Lid
> /
Sb Y :
g |
a f
a Lf 4
O /
a /
Liu
« 4 Lo
~ v' FROM v =(7/0.28) °
T= FLOW STRESS
AT O°K
102 {| l l l |
“oO | 4 2 7
2 3 4
(t/T))- (t/t)
Figure 29, Analysis of Data According to Theory of Fleischer (41).
-88-
Physical mechanisms which might influence dislocation
mobility in copper and their relation to the above theories will now
be discussed. First, consider the assumptions and parameters of
the velocity-stress relationships presented thus far. Several points
should be mentioned here regarding the range of the variables in
this study. The plastic strains imposed on the specimens were
small compared to the elastic strains, and dislocation displacements
were small, that is, small relative to the initial dislocation spacing.
The dislocations were generally moving into dislocation-free areas.
Different mechanisms and, consequently, a different dislocation
velocity-stress relation may be obtained at higher strains where
the dislocation displacement is equal to or greater than the initial
dislocation spacing. However, for small displacements the relation
of Equation 28 is valid, and the small value of To indicates a low
resistance of the lattice to the motion of dislocations.
The theories of Gilman (40) and Fleischer (41) involving
point defect drag and, consequently, short-range interactions and
small dislocation displacements would be attractive explanations
of dislocation mobility in copper. However, the data are not rep-
resented by the formulae of these theories. It is, therefore, con-
cluded that the type of thermally-assistcd motion which is assumed
by the theories is not applicable to dislocation mobility in copper.
The theory of Seeger et al. (42) assumes an Arrhenius rela-
tion for the strain rate as do those of Gilman (40) and Fleischer (41).
The activation energy for this assumed form consists of two parts,
one stress dependent, the other not. The data were analyzed
-~89-
according to this theory and were found not to obey the relation pre-
dicted. Tho work of Conrad (26) on copper, as analyzed according
to this theory, shows that the exponential term which should be de-
pendent upon stress and temperature is actually relatively temper-
ature independent.
. It would seem plausible that dislocation motion in copper for
the range of variables in this study need not depend upon assistance
from thermal vibrations. A Peierls barrier to motion should be
quite small, considering the extension of dislocations in copper,
and might be surmounted by zero-point energy.
The foregoing discussion indicates that mechanisms whereby
the velocity of dislocations is governed by thermally-activated sur-
mounting of internal barriers to dislocation motion are not consistent
with the experimental results obtained in this investigation. Another
class of theories of the velocity of dislocations is based upon the
assumption that the interaction between the strain field of the moving
dislocation and the thermal vibrations of the crystal lattice, com-
monly known as the phonon drag and phonon viscosity effect. The
most recent example of this type of theory is due to Lothe (43).
Lothe predicts that for a straight, freely moving dislocation in
metais, the anharmonicity in the core region and the phonon vis-
cosity effect (lattice vibrations considered as a viscous phonon gas)
give rise to a drag stress, Tas at ordinary temperatures T~ 0,
where 9 is the Debye temperature, of
l Vv
Ta~ap E &)> [31]
-90-
where € is the thermal energy density and V is again the shear wave
velocity. The scattering of phonons by the dislocation is considered
to produce a drag stress of the same order of magnitude so that the
total drag stress due to anharmonicity, phonon viscosity, and phonon
scattering is
d 5 vie
Note that the drag stress is linear in velocity according to this
theory. The thermal energy density is assumed to be the classical
relation
€= oer [33]
a” /z
where a = the lattice parameter = 3.6x 1078 cm
and z = the number of atoms per unit cubic lattice cell = 4,
The drag stress calculated in this way is at most 55 per cent of the
applied stress on the dislocation, being about 15 ¢ /mm* at the low-
est applied stress and velocity and about 68 g/mm* at the highest.
Mason (44) has investigated the effect of phonon viscosity on dislo-
cation motion, but this theory predicts, according to Suzuki et al.
(45), the same order of magnitude of the drag stress as does the
theory of Lothe.
The mechanisms due to lattice vibrations which might resist
dislocation motion do not account entirely for the results of this
study. Nonetheless, it is of interest to note that those theories
which depend upon thermally-assisted dislocation motion predict
that velocity will decrease with decreasing temperature whereas
-91-
those theories that depend upon the existence of a drag stress due
to phonon scattering and viscosity predict that the dislocation velocity
will increase with decreasing temperature. Stress pulse tests on
copper specimens at low temperatures, which are in progress,
should enable one better to select between the two types of theories
of dislocation mobility.
Consider now the ramifications of the mobility relation,
Equation 28, and the values of velocity and stress for copper. Table
3 gives mobility data for several materials, and one may see that
at the flow stress (considered here as that stress at which the first
deviation from linearity is observed in a stress-strain curve of the
material) dislocations in copper are moving at a velocity several
orders of magnitude greater than the velocity of dislocations at the
flow stress in the ionic compounds, body-centered cubic metals,
and semiconductor materials thus far studied. The easy glide flow
or yield stress of the copper crystals used in this study was not
measured, but Young (22, 23) obtained values of 35 g/mm” and
65 g/mm* for copper crystals of the same purity so an average of
these two values was used as the flow stress given in Table 3 for
the crystals of this investigation. Since the difference in dislocation
velocities in copper and the other materials shown is so great, the
comparison is not invalidated by an inexact value of the flow stress.
It is well-known that the strain rate sensitivity of the flow
stress in copper is small. This is evidenced by a stress-strain
relation which is very insensitive to the strain rate. Cottrell (46)
interprets this phenomenon as due to a large mobility exponent in
-92-
SE
LI
G2
8°L
qUOUO dxo
ATTTIQONT
00¢c~
(90009)
7-01 xg
xx e
9-01 gil
g-0l * 1
g-01 XI
(sespa)7_01 ¥ G‘T
(smoatos) p-0l x6
¢-Ol I
(908 /urD) ssoijs
MOTT Je APLOOTO A
Oog~
(D 9009)
(qtuury Teuotizodoid) 906Z
(qy8uea}s
preté JessTo %¢°O) COZZI
OLT
(Ajtand ysty) o¢
(sseajs reoys
PSATOSOT TBOTIITD) 006
(jtuty Teuotzz0dorzd) goFrL
wrur/S) S$Soez1is MO
2 tur /5) 18 MOTT
(peyou ssorun sanjyerodure} utoor 3)
STVILOILW Ter9A0eg Tog eyeq AlITIqow woT}e00TSTtG
€ gel
UOTJESTISOAUT STU, x
~araddon
(ZT) urmMtueulrey
(Q) WOAT-UOdTIIg
(OT) eptzoTyD urntpog
(LZ) Sptzony[Z cantyyry]
(TT) wey sun yz
(sosoujuered ut
SdZNOS 0} BOUSTOFoOr)
Tetroyeyw
~93-
copper, i.e., a small change in stress produces a large change in
dislocation velocity. Silicon-iron with a relatively large strain rate
sensitivity of the flow stress has a mobility exponent of 35, so that
by this reasoning, the mobility exponent of copper should be large
compared to 35. This investigation shows the mobility exponent in
copper to be small. The small rate sensitivity of the flow stress in
copper must therefore be due to a strong stress dependence of the
density of moving dislocations and to the relatively high velocities
which dislocations achieve at relatively low stresses. Such high
velocities permit rapid dislocation multiplication so that the stress
and strain are nearly in equilibrium even at relatively high strain
rates. The sequence of events when stress is applied to a copper
crystal is envisioned thusly: Dislocations are traveling at high veloci-
ties and multiplying at stresses below the macroscopic yield stress,
The percentage of moving dislocations increases with stress as this
author has observed qualitatively and Young (23) has checked semi-
quantitatively. Ina short time the dislocation density has increased
to the point that the internal stress (from the strain ficlds of othcr
dislocations) which opposes the motion of a dislocation equals the
applied stress and dislocation motion ceases,
It is of interest to consider the effect of the internal stress
further and to contrast the dynamical behavior of dislocations in a
high-mobility-exponent material like silicon-iron to that in the low-
exponent material copper. Vreeland (47) has studied the effect of
a periodic internal stress field imposed on the constant applied
stress level on the average velocity of a dislocation in a crystal,
-94-
considering the mobility exponent as a parameter. The stress in
the dislocation mobility relation is taken as the local stress on the
dislocation, not just the applied stress. He found that the form of
the internal stress field had little effect on the analysis. As might
be expected, the mobility exponent has a profound effect on the
average velocity-average stress relation. The dislocation ina
material with a high mobility exponent like silicon-iron spends
much more time in the troughs of the periodic stress field than on
the crests; consequently, as the amplitude of the internal stress
field increases, the average dislocation velocity decreases rapidly.
One may contrast this behavior of dislocation velocity to that in
copper. When the velocity-stress relation is essentially linear,
the amplitude of the internal stress field has no effect on the average
velocity until it equals the applied stress when the average velocity
goes to zero.
In copper, dislocations move and multiply rapidly at low
stresses so that the dislocation density and, consequently, the
internal stress need not increase too much before the internal stress
amplitude equals the low applied stress and dislocation motion
ceases. Phrased differently, the strain can essentially maintain
its equilibrium value and be in phase with the stress at high strain
rates. Since dislocations in silicon-iron move slowly at relatively
high stresses, the dislocation density must become rather high be-
fore the internal stress amplitude approaches the applicd stress.
The strain cannot assume its equilibrium value and lags behind the
stress at high strain rates.
~95-
The theory of yielding of a single crystal proposed by Johns -
ton (2) states that for a small mobility cxponcnt and a low density
of moving dislocations, there will be a large yield drop when a
stress-strain test is performed on the material. The density of
mobile dislocations was assumed to be a function of strain but not
explicitly a function of stress, whereas the mobile dislocation
density must be strongly stress dependent in copper. Except in
special cases, there is no yield drop in a stress-strain test of a
copper crystal. Therefore, there must be a large density of moving
dislocations at the flow stress, i.e., there must be weak pinning
of aged dislocations compared to that provided by interstitials in
body-centered cubic metals and divalent ions in lithium fluoride
and sodium chloride. A cogent argument for this interpretation
of the behavior of copper in light of this yield theory is in the paper
by Cottrell (46) and is the stress-strain curve for a copper whisker
which exhibits the largest known yield drop. The whisker must
have been initially free of dislocations, and the upper yield stress
of about 63 kg/mm“ is interpreted as the stress necessary to create
dislocations in the undislocated lattice. The yield drop to 3 kg /mm*
is very abrupt because of the high velocity and rapid multiplication
of the dislocations. Thus, in the case where there are no mobile
dislocations or, more generally, a low density of moving disloca-
tions, the low mobility exponent can give rise to a large yield drop.
-96-
SUMMARY AND CONCLUSIONS
The results of this inve stigation of dislocation mobility in
copper and those of a previous one in zinc (14) indicate that the
present theories of dislocation mobility cannot adequately predict
the observed velocity-stress relations, Dislocations in copper
move at high velocities at low stresses, and the velocity-stress
relation is approximately linear. The low strain rate sensitivity
of the flow stress is attributed to a strong stress dependence of
the density of moving dislocations and to the high velocities at low
stresses, rather than to a large mobility exponent.
The present theorics of dislocation mobility may be divided
into two categories. One which assumes that dislocation motion
is thermally assisted; the other, that thermal assistance is negli-
gible and that interaction with lattice vibrations causes a major
portion of the drag force on a moving dislocation. The form of
the velocity-stress relationship observed is in qualitative agree-
ment with the latter, which predicts drag stresses of about 35%
of the measured values. Which mechanism is operative in copper
“may be determined by low temperature tests of dislocation mobil-
ity, which are in progress. With such information, a more valid
theory of dislocation mobility in FCC and HCP metals might be
forinulated. It would be of interest to extend the range of stresses
for which dislocation velocities have been measured in this study
by testing low dislocation density specimens in which large dislo-
cation displacements can be followed.
-97-
Another result of this investigation which can be of use in
other dislocation studies is that the dislocation etchants employed
reliably reveal the sites of dislocations intersecting {100} planes.
-98-
APPENDIX A
The elastic properties of cylindrical, cubic-symmetry single
crystals with three different crystallographic directions as the cylin-
der axis will be investigated in this appendix. Specifically, the elastic
properties of the crystals when subjected to torsion about each of
these three crystallographic directions will be examined, and it will be
shown that a crystal with a [ 100] torsion and cylindrical axis is non-
dispersive of a torsion stress pulse applied to it.
The generalized Hooke's law may be expressed in tensor nota-~
tion as
O35; > Sajna & 11
[A-1]
where ore is the stress, Ce ited are the elastic stiffness moduli, and
ey is the strain. The summation convention over repeated subscripts
applies. The general transformation formulas relating the elastic
moduli for a given Cartesian coordinate system which is rotated with
respect to the crystallographic coordinate system are
= 4 t,.4.,4
t -
“apys ai’ Bj “yk [A-2]
81 °ijkl
where the a, 8, y, & subscripts range from 1 to 3 in the given rotated
coordinate system, the i,j,k,1 subscripts range from 1 to 3 in the
crystallographic coordinate system, and Lai is the direction cosine
between the x axis and the x, axis. Equation A-2 can be simplified
by introduc ing the reduced subscript notation where Can Cri ’
os Os » and en egy according to the following prescription.
~99-
ij or kl m orn
11 1
22 2
33 3 [A~3]
23 or 32 4
31 or 13 5
12 or 21 6
In addition, there are only three independent stiffness moduli for
cubic symmetry crystals, C)ysCyo:Cyy° Equation A-2 reduces to
t = ~
CaBys | Lait Bi tits: -C + Aug y 6°12" AgvAgathaa By eng [A-4]
where Anup is the Kronecker delta, i.e.,;
1 if a=f6
{ 0 if a#f [A-da]
and
C= C1784 2744 °
The stiffness moduli in the rotated coordinate system Capys
(or c! \ in the reduced notation) will be determined from Equation
A-~4 for the three single-crystal specimen orientations below.
Case 1. Cylindrical and torsion axis [111] (Figure A-1).
xy axis - [111]
x5 axis - rotated angle § from [110]
x3 axis - rotated angle 6 from [112]
Direction cosines
-100-
*] 2 3
x! i/f3 1/3 ai/ys
x! (cos sQ+ —= sing cos6 = sing 2/3 siné
, =i Goss 282) ve sub)
x! wer - sin8) - cos 5 sing) 2/3 cos§
sea Ee
The stiffness moduli are
It
il
C/3+c
C/3t+c
12
12
C/3+ C1
+ 2c
44
C/2+ Cio + ZC 44
C/6 + Ci
Cocos 6
32
_ Csin Csin 6
32"
C/2+c
12
+ 2c
44
(1 - 2 cos 26)
(1+ 2 cos 26)
Case 2.
~101-
che = . Ecos 8 (a 2 cos 28)
chy = caine (1 + 2 cos 26)
3V2
1 ~
Cha = C/6+c,,
clos C sin 6 (1 + 2 cos 26)
45 32
Ccos 6 (1 - 2 cos 26)
C46 = 32
chs = C/3 egy
CEE = 0
COE = C/3 + C44
Cylindrical and torsion axis [110] (Figure A-2).
x, axis - [110]
x5 axis - rotated angle 6 from [T10]
_x! axis - rotated angle 6 from [001]
Direction cosines
The stiffness moduli are
ro. 4
chy = 3 (c,;+e 12t2¢ 44)
C 2
12 7 30S 89+c,5
tl
~102-
C.. 2
> sin Otc),
- ~sin 26
. 4
of cos *@) + Cio + 2c
2 44
- 2
CL: 2 sin’ §
5 Sin 29 cos 6-5)
36 55,2 28+¢
8 44
oS
-103-
Case 3. Cylindrical and torsion axis [100] (Figure A-3).
x4 axis - [100]
x! axis
' .
xX, axis
- rotated angle 6 from [010]
- rotated angle @ from [001]
Direction cosines
xy %2 X3
x} ] 0 0
x5 0 cos§ sin§
x4 ) -sin§ cos8
ll
C12 = C12
C13 = S12
C14 =
Cis *
ChE =
C5 = C(cos*9 + sin“) + C1 + 2044
C53 = © sin” 26+ c)5
chy = - ¥ sin 28 cos 28
C25 =
-104-
Ch = Clcos* + sin*9) + C13 + 2C a4
C34 = S sin 268 cos 286
Che = 0
chy = 0
C4 = ¢ sin* 26+ c44
chs = 0
C46 = 0
C55 = C44
cag = 8
C66 = “44
Whether any or all of the three specimen orientations is dis-
persive of the torsion pulse will be determined in the following man-
ner. The application of a pure twisting deformation, with no warping
of the cross section and no radial displacement, is presumed equiva-
lent to the application of the torsion stress pulse to the specimen.
The stresses resulting from the torsional shear strain, ¢, , will be
determined from the stiffness moduli calculated above for the three
specimen orientations. Dispersion of the stress pulse will result if
the initial o, is converted into other stresses due to the elastic an-
isotropy. This method of determining wave dispersion allows the
relative magnitude of the dispersive stresses to be calculated.
The stresses resulting from ¢} for the [111] torsion axis
orientation are
-105-
to t lox
Oy = Cysé5 = 0
Ccos 6
of = cligh = YSSS* (1 ~ 2 cos 28) e!
2 25°5 32 5
t = 1 ! = . Gcos @ ~ !
of cheb 3 a (1 - 2 cos 28)e,
1 = cot gt = Csing
o} cheek = (1+ 2 cos 28)e,
3 V2
-oh = ctigt = (pcre de
5 5575 3 44° "5
= 0
06 = C5685
Thus, 05 03: and O4 arise to give dispersion. The numerical val-
ues of the stiffness moduli for copper in units of io! dynes fern” (33)
are given below in order to be able to evaluate the relative magnitudes
of the above stresses.
Cy, = 17.02
Cy> = 12.3
Cy, = 7.51
The magnitudes of these stresses relative to on » evaluated at 6=0,
are
o! o}
= = — = 0.6
5 5
and
o!
4 = 0,
oS
This solution is not exact, as there are the stresses O52 OR. o' oc-
curring at the stress-free cylindrical surface. However, there is
evidence of transfer of energy from shear to dilatational waves and
-106-
consequent dispersion of the torsion stress pulse.
The stresses resulting from ep for the [110] torsion axis
orientation are
o1 = cse5 = 0
oD = cbse = ©
0, = Cane, = 0
4 = C45e5 = 0
C.. 2
t ~ at at = (¥ 1
O, = Cases = (Fsin et cy, )es
oh = chet = -(2 sin 28)e}
6 ~ ©5685 ~"'4 5
Thus, o6 arises to give dispersion. The magnitude of this stress
relative to Oo evaluated at 6 = 7/4, is
ms = ° 52 e
In addition, Of varies with 8 around the specimen axis. As in the
case of the [111] torsion axis, the occurrence of a stress, o% » at
the cylindrical surface, which was presumed stress-free, indicates
the solution is not exact.
The stresses resulting from 5 for the [100] torsion axis
orientation are
1 = ot pt =
Oy Cy pe 0
07 = Co5@5 = 9
05 = che, = 0
1 = t 1 =
oy = Canes 0
-107-
1 = gt at 1
os C55e5 ~ S44%s
fl
OF CEges = 0
In this case, no dispersion results, and there are four {100} surfaces
available for observation of dislocation motion.
The inverse problem of applying a pure torsional moment,
Mo = Mi ; about the specimen axis, thereby producing only the
torsional shear stress To, = OF » has been solved for the [110] ori-
entation by Wood (48). He calculated the compliance moduli, BN) ’
for this orientation, imposed the stress ot, determined strains and
displacements, and concluded that
"the stress distribution is the same as that for pure
torsion of a circular rod of isotropic elastic material.
The strain distribution, however, differs from that in
the isotropic rod, and, furthermore, the cross sections
are warped. '' (48).
It should be noted that the analysis applies to along, single crystal
rod in which the radial and axial displacements, uw. and ul must be
independent of the axial coordinate z and in which the tangential dis-
. placement Ug must be a linear function of z. Of course, this was
not the experimental situation in this investigation.
The above procedure developed by Wood was attempted for the
case of a [111] specimen and torsion axis. However, the resulling
differential equations involving the displacements appear intractable
and are perhaps insoluble. The results of applying this procedure to
the case of the [100] oriented specimen show that the stress and ~
strain distribution is the same as in the case of pure torsion of a
circular rod of elastically isotropic material.
| -108-
“x [100]
Figure A-1. f1lil] Specimen Orientation.
4371001
f |
Xo’
Ca 9 _—[T0]
i > x2-[010]
x,- [100] | | x/-[ 110 |
Figure A-2. [110] Specimen Orientation.
-110- |
K.x/-[100],
Figure A-3. [100] Specimen Orientation,
-llli-
| APPENDIX B
Assume that the stress dependence of dislocation velocity can
be expressed as
v= (n/t oy [B-1]
where v is the dislocation velocity, 7 is the applied stress, To is
the applied stress which produces a velocity of 1 cm/sec, and m is
the mobility exponent. Then the displacement, d, ofa dislocation
can be written as
d = fv dt = frm) dt . [B-2]
To
The stress as a function of time, t, fora trapezoidal-shaped stress
pulse is displayed in Figure B-1l. The functional dependence of stress
on time is given below.
e = eX 4 O
T= Tax ty sts to [B-4]
t,-t
T= T ax =) ts sts t, [B-5]
Subscripts are defined in Figure B-1. Substituting these functions in
Equation B-2, and integrating over the appropriate ranges, yields
_ Tmax on ts + m(t,-ty)
d= = ) mit ° fB-6]
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26
36
4.
5e
To
~ 10,
ll.
12.
13.
14.
-113-
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