 

WORK HARDENING AND DEFORMANON STRUCTURE
IN LITHIUM FLUORIDE SINGLE CRYSTALS

Thesis for the Degree of Ph; D.’ i
' MICHIGAN STATE UNIVERSITY
Karafholuvu Narayanasamy SUbramanian '

.1 9.66

  
   
   

  

L I B R A R Y
Michigan State
University

This is to certify that the

thesis entitled
WORK HARDENIM AND DEFORMATION STRUCTURE

IN LITHIUM FLUORIDE SINGLE CRYSTALS

presented by

Karatholuvu Narayanasamy Subramanian

has been accepted towards fulfillment
of the requirements for

_Eh.D._ degree in My

654% JUL“ 7’ (dish.

Major professjar

 

Date. “/1/3 5/44, 1/ /" 1/3517

1’ 6’

i
\
O~169 l

ABSTRACT

WORK HARDENING AND DEFORMATION STRUCTURE
IN LITHIUM FLUORIDE SINGLE CRYSTALS

By Karatholuvu Narayanasamy Subramanian

Lithium fluoride crystals having a dislocation density of
approximately 104 lines per square centimeter were successfully
grown by using the Czochralski method at the rate of 1/4 to 3
inches per hour utilizing off-centering and necking techniques. The
deformation studies of such crystals showed that in combined fatigue
and latent hardening experiments, the yield strength of the specimen
in the latent slip systems was as high as thirty fold compared with
the yield strength of the crystal in the as grown condition provided the
specimen underwent a complete reversal of the resolved shear stresses
acting in the primary slip systems in the fatigue cycles. The growth
of the slip bands during deformation and the stress pattern in the
deformed crystals were investigated with polarized light microscopy.
A band-like structure, which might be called deformation bands,
was observed to develop in the slip planes usually in a direction
perpendicular to the slip direction. There were few instances where
they were parallel to the line of intersection of the primary slip
planes with the non-conjugate slip planes. These bands were of
constant width of approximately 20 microns. Bulk specimens work

hardened rapidly during deformation and deve10ped such band

Karatholuvu Nar ayana 5 amy Sub r amanian

structures whereas thin Specimens did not develop the band
structure during deformation and were not work hardened. A
theory of deformation band formation based on interactions of
dislocation loops originated from randomly distributed sources
is pr0posed. The various hardening phenomena observed in
lithium fluoride single crystals are explained in terms of the
relative orientation of the slip planes and the slip directions

with respect to that of the deformation bands deve10ped.

WORK HAR DENING AND DEFORMATION STRUCTURE

IN LITHIUM FLUORIDE SINGLE CRYSTALS

BY

Kar atholuvu Narayana 3 amy Sub ramanian

A T HESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of
Metallurgy, Mechanics and Materials Science

1966

AC KNOW LE DGEMEN T5

The author wishes to express gratitude to Dr. C. T. Wei
for the guidance in this work. The helpful suggestions and
encouragements given by Dr. A. J. Smith, Dr. T. Triffet and
Dr. J. T. Pindera are also gratefully acknowledged. The entire
staff of the Department of Metallurgy, Mechanics and Materials
Science are thanked for their assistance and helpful discussions.
Finally, he would like to thank the Atomic Energy Commission

for the financial help granted under the contract no. AT (11 —l)-lO42.

ii

TABLE OF C ONT ENTS

Page

INTRODUCTION................. 1

EXPERIMENTAL TECHNIQUES. . . . . . . . . 16

RESULTS . . . . . . . . . . . . . . . ..... 25
DISCUSSION . . . . . . . ....... . . . . . 57
I. Crystal growth . ..... . . . . . . . . 57

II. Principles of polarized light microsc0py. 62.
III. Slip distribution and work hardening . . . 71
IV. Theory of band formation . . . . . . . . . 73

V. Relation between deformation structure and
workhardening.............. 105

CONCLUSIONS........... ..... .. 114

REFERENCES..................116

iii

LIST OF FIGURES

Figure Page
1 Projection of { 110} slip planes onto
(001) surface . . . . . . . . . . . . . . . . . . 15
2 Czochralski furnace for growing LiF single
crystals..................... 23
3 Arrangement for polarized light microsc0py . . Z4
4 Dislocation distribution in crystals grown by
Bridgman method: . . . . . . . . . . . . . . . . 34

(a) Cleaved Specimen,
(b) Acid-saw cut specimen.

5 (a), (b) Sub-grain boundary distribution in a
crystal grown by Czochralski method . . . . . . 35
(c) A region of low dislocation density in a
crystal grown by Czochralski method.

6 Effect of off- -centering on sub-boundary
distributionz... 36
(a) Before off— —centering,
(b) After off- -centering.

7 Divergent beam X—ray picture of LiF crystal

grown by Czochralski method (using off-

centering principles) . . . . . . . . . . . . . . 37
8 A typical stress-strain curve for a fatigue—

latent hardening experiment . . . . . . . . . . . 4O
9 Orthogonal slip bands on (100) surface as

observed at various orientations of the slip
vector with the direction of polarization in
darkfield:..... . 41
(a) Direction of polarization vertical,
(b) Direction of polarization 300 to the vertical,
(c) Direction of polarization 500 to the vertical,
(d) Direction of polarization 700 to the vertical.
10 Slip band growth observed on (100) plane during
unreversed cyclic loading . . . . . . . . . . . . 42
(a) After the first cycle,
(b) After the second cycle.

iv

Figure

11

12

13

14

15

16

17

LIST OF FIGURES (continued)
Page

Slip band growth observed on (100) plane
during reversed cyclic loading. . . . . . . 43
(a) After the first half of first cycle,
(b) After the second half of the first cycle
with reversed shear in the same slip
planes deformed in the first half cycle.

Slip distribution observed on a (100) plane of
a crystal deformed in the latent direction . . . 44

Orthogonal slip bands on the (100) surface of
a specimen prepared for the observation of
damage in the (011) slip plane . . . . . . . . . 45

Band structure observed in the slip plane of
the Specimen shown in figure 13, linearly
polarized light dark field observations: . . . . 46

(a) Direction of polarization vertical,

(b) Direction of polarization 300 to the vertical,

(c) Direction of polarization 450 to the vertical,

((1) Direction of polarization 650 to the vertical.

Polarized light observations of the bands in
the slip plane: . . . . . . ........ 47 and 48

(a) Bands making 100 to the direction of
polarization,

(b) Bands making 150 to the direction of
polarization, .

(c) Bands making 200 to the direction of
polarization,

((1) Bands making 250 to the direction of
polarization, 0

(e) Bands making 30 to the direction of
polarization,

(f) Bands making 350 to the direction of
polarization,

(g) Bands making 400 to the direction of
polarization,

(h) Bands making 450 t0 the direction of
polarization.

Band structure at various stages of deformation in
three-point bending; Burgers vector parallel to the
direction of polarization . . . . . . . . . . . . 49

Band structure at various stages of deformation
in three-point bending; Burgers vector making
500 to the direction of polarization. . . . . . . 50

Figure

18

19

20

21

22

23

24

25
26
27

28

29

LIST OF FIGURES (continued)

Network observed in a lithium fluoride
crystal . . . .

Band structure observed along non-orthogonal
slip plane intersections in a thick specimen
deformed in three-point bending; Burgers vector
making 200 to the direction of polarization .

Band structure in a thick sodium chloride

specimen deformed in uniaxial compression;

direction of polarization vertical . . . .
(a) and (b) are two different orientations
of the Burgers vector with respect to the
direction of polarization.

Band structure in the sodium chloride specimen
observed in figure 20 after thinning down. .
(a) and (b) reveal 100ps in these bands,
(c) and (d) reveal the stress concentration
caused by non-orthogonal slip band
intersections.

Transmission electron micrographs of an MgO
single crystal fatigued in cantilever bending at
room temperature . . . . . . . . . . . .....
Edge dislocation dipoles and debris observed
in transmission electron micrographs of
fatigued MgO specimens . .

Dependence of contrast on the orientation of
the slip vector with respect to the direction

of polarization .

General mechanism of band formation . . .
Model for formation of screw-lock

Stability of screw-lock . . . . . . .

Forces associated with the formation of
screw-lock . .

Model for the force field at the central
portion of the screw-lock .

Vi

Page

51

52

53

54

55

56

70

95
96

97

98

99

LIST OF FIGURES (continued)

Figure Page
30 Interaction of two rows of edge dislocations . . . . 100
31 Interaction of edge dislocation arrays in

nearby slip planes . . . . . . . . . . . . . . . . . 101
32 Stability of interacting dislocation 100ps . . . . . 102
33 Total strain energy of a dipole of unit length

making various angles with the Burgers vector . . 103
34 Mechanism of band formation due to

intersecting slip . . . . . . . . . . . . . . . . . . 104
35 Conjugate slip intersection . . . . . . . . . . . . 112
36 Non-conjugate slip intersection . . . . ...... 113

vii

INTRODUC TION

Work hardening in materials remains one of the basic
problems in plastic deformation. This phenomenon depends on
many variables such as chemical composition, crystal structure,
elastic properties, history of the specimen and imperfections.
Experimental observations on the hardening behavior of
materials with different crystal structures and under various
conditions have shown the complexity of this problem. For a
particular, chemically pure, and nearly perfect single crystal
the dislocation structure appears to be the most important
single factor.

The purpose of this work was to grow reasonably pure
lithium fluoride single crystals having low dislocation density
and to study their deformation behavior. The deformation
studies were directed towards the advancement of the under-
standing of work hardening. Although there has been a large
amount of research done on ionic crystals, the overall picture
of work hardening in this type of crystal as well as in other

materials is yet far from being clearly understood.

Theories of Work Hardening

 

Various theories have been advanced to explain the work
hardening of materials. According to Taylor's theory of work
hardening, slip commences at many position throughout a

crystal by the multiplication of positive and negative dislocations.

These dislocations stOp within the crystal due to the interaction of
their stress fields, and thus build up a 'superlattice' array of
dislocations of both signs. At any given strain, the stress field
associated with such an array Opposes the movement of other
dislocations so that the stress has to be increased to produce
further plastic flow. This theory gives a parabolic relationship
between stress and strain.

Mott's theory of work hardening is an extension of Taylor's

2

theory. He proposes that dislocations generated from a
Frank-Read source in a Slip plane may pile up against sessile
dislocations in the same slip plane. The dislocation pile-up so
formed at an obstacle consists of dislocations of the same Sign.
Due to the stress field around the piled up groups of dislocations
sessile dislocations are formed between the edge parts of the
dislocations in a group and the edge parts of the dislocation 100ps
generated from sources in some other slip systems. As a result,
when the stress is removed, the dislocations in the piled up groups
are prevented from being forced back to their sources by the back
stress from the pile-up. According to this mechanism hardening
is due to the stresses around the piled up groups which may be
regarded as 'super-dislocations' of strength 11?, where n is
the number of dislocations in a pile-up, and .1; is the Burgers
vector. Mott's theory differs from Taylor's in suggesting that

the piled up groups are locked in position and have n times as

much strength as those considered by Taylor. However, this

theory also gives a parabolic relationship between stress and strain.

The results of experiments with single crystals of face
centered cubic metals have shown evidence of three stages of
hardening under Specific conditions. 6—14 In order to explain
this many theories have been advanced. Mott has suggested
that during stage I, also known as easy glide, most dislocations
originated from Frank-Read sources leave the crystal. 2 The
fact that the dislocations of the first Operative slip system have
to cut through the dislocations of the Frank net intersecting
the slip plane has in itself been considered as a cause of

3’ 4’ 5 Koehler's view

hardening by Mott, Cottrell, and Friedel.
is that easy glide is an exhaustion hardening process, by which
he means that, at first, the longest Frank-Read sources come
into operation and as the dislocations generated from these are
held up by the obstacles in the slip plane, shorter and shorter
sources are activated.

The following theories have been prOposed for face
centered cubic materials to explain the stage II hardening, also
known as rapid hardening or linear hardening. Friedel ascribes
the rapid rate of work hardening in stage II to a particular
distribution of Lomer-Cottrell sessile dislocations formed by
the interaction of dislocations of the primary slip system with
those of the secondary systems. 7 He prOpounds that Lomer-
Cottrell sessile dislocations are formed around each active

source in at least two directions in the primary slip plane. The

piling up of dislocations against these Lomer-Cottrell barriers

surrounding each source increases the stress necessary to operate
the source. New sources begin to operate and are in turn blocked
by the formation of sessile dislocations. This theory, therefore,
assumes the operation of at least two secondary systems at the
beginning of stage II. He suggests that, as the piled up groups in
neighboring slip planes are closer to a particular source than
those in its own slip plane, and, as a result, a loop traversing
the primary slip plane has to overcome the variations of stress
produced by the neighboring groups, it is the neighboring groups
which are reSponsible for the hardening. Seeger assumes that

a continued formation of Lomer-Cottrell barriers throughout
stage II leads to a decrease of the slip distance with increasing
strain and to smaller dislocation groups of essentially constant
size, which are piled up against the barriers and more or less

randomly distributed. 8’ 9’ 10

Mott considers the secondary slip
to be reSponsible for most of the network of dislocations observed
in cold worked materials. 11 This network accounts for a consider-
able part of the flow stress. The secondary slip also hardens the
sources by forming glissile and non-glissile jogs; when the source
is stressed hard enough, the dislocations annihilate each other,
thus unlocking the source. The dislocations move until the jog
density becomes high.

The stage III hardening is generally attributed to cross-slip.
Seeger et al have suggested that stress relief in the piled up groups
is caused by dislocations leaking out of the pile-ups by cross-

8, 9,10

slip It is suggested that the stress necessary to cause the

screw components of these piled up dislocations to slip on the cross-
slip plane is less than that required for the dislocations to break
through the barrier. Friedel suggests that the leakage of disloca-
tions by cross-slip results in a relief of the stress in the pile-ups
and in the annihilation of dislocations by those of the Opposite Sign
from the neighboring pile-ups. 7 With further slip there is still
some hardening because the edge dislocations may continue to
pile up.

Basinski's hypothesis is that the internal stress varying
over distances longer than the separation between successive
dislocations emitted from the source will not contribute appreciably
to hardening and that the most important source of hardening is the
resistance encountered by the dislocations in cutting through the
dislocation forests. 12 He suggests that strain hardening is
primarily due to local interactions of intersecting dislocations.
Mitra and Dorn have shown that both the long range stress fields
and the short range stress fields contribute to the work hardening
of face centered cubic metals. 13 They reason that the long range
stresses due to piled up dislocation arrays account for the low
hardening rate in easy glide. Both the long range and localized
stress fields at the intersections of dislocation arrays contribute
to stage II hardening.

Kuhlman-Wilsdorf has suggested a work hardening mechanism
which is applicable to a wide variety of materials irrespective of the
type of their dislocation substructure. 14 According to this mechanism,

in the easy glide stage, dislocations multiply in a number of restricted

areas and penetrate from there into crystal regions still consider-
ably free of glide dislocations. This continues until a quasi-uniform
dislocation distribution is established. The cause of work hardening
in easy glide is primarily due to the accumulation of jogs on the
dislocations and the interaction between point defects and dislocations.
When there are no more areas left into which newly formed
dislocations have not yet penetrated rapid hardening ensues.
Irrespective of how the dislocations are arranged at the end of

easy glide, the stress necessary to overcome the line tension of
the dislocation segments must increase, because as the dislocation
density increases the average free dislocation length decreases.

The increase in the stress necessary to cause the dislocation
segments to bow out into 100ps is reSponsible for the major part

Of work hardening in stage II. Cross-slip and conservative climb
are the dominating factors in stage III.

Gilman has proposed a debris mechanism of strain hardening
based on the observations made on lithium fluoride and magnesium
oxide single crystals. 15 According to this mechanism damage in
the form of edge dislocation dipoles continuously builds up in
prOportion to the strain. These dipoles have the metastability
needed to account for the plastic strain. The debris, composed
Of small dislocation loops, acts as sources of internal stresses.

A considerable part of cross-slip which causes dislocation multi-
plication and hardening is initiated at the surface. Nabarro et al

suggest that the quantity of debris produced is proportional to the

number of dislocations already trapped in the crystal and hence the

density of debris is prOportional to the square of the strain.

Work Hardening Due to Deformation Bands

 

It has been suggested that deformation bands can cause work
hardening. Mott prOposes that the increase of applied stress required
to form stabilized lOOps, when the deformation bands become more
and more tightly packed with dislocations, manifests itself in the
form of work hardening.17 According to Doris Kuhlman, deformation
bands formed by the locking Of unlike dislocations act as strong
Obstacles against the movement Of dislocations, thus become the

source of work hardening.

Review of Deformation Bands

 

Deformation bands have been Observed in various metals like
iron, aluminum, zinc, etc. 19-24 The main features Of these bands
are the following: (1) each band is a thin sheet running through the
crystal and, at small strains, is almost a plane and perpendicular
to the active slip direction; (2) their width is approximately constant;
(3) they start being formed during the early stages of deformation;
(4) slip bands formed in the early stages bend themselves to follow
the curvature of the lattice and cross the deformation bands without
interruption, but those formed later do not penetrate the deformation
bands at all. 25

According to Mott, deformation bands are formed by walls of

positive dislocations on one side and negative dislocations on the

other. 2 In a deformation band locking of dislocations results from
slips in intersecting planes. When some dislocations are locked
due to the dislocations in an intersecting plane, the resulting stress
field acts as a barrier against dislocations moving in nearby planes
and stOps their motion. The dislocations so st0pped act in a
similar manner to stOp other dislocations in their neighboring
planes. Wei's mechanism of band formation in zinc single crystals
makes use of sources in parallel slip planes. 24 The interaction
through screw components of the dislocation loops makes them line
up in a direction perpendicular to the slip vector and stabilizes the

structure.

Experimental Limitations

 

Work hardening experiments are usually conducted by
deforming bulk crystals under uniaxial loading. The dislocation
structure of these crystals is mostly studied either at the surface
by using an etch-pit technique or by observing thin sections made
out of these deformed crystals under an electron microsc0pe.

Thus one is limited to either the surface or a thin section of the
bulk. These methods give only a partial picture of the dislocation
structure. To understand the work hardening behavior of materials,
it is necessary to know the three dimensional dislocation structure
in deformed bulk crystals. Polarized light microscopy of tran-
parent crystals, therefore, appears to be a suitable technique.
Lithium fluoride is a transparent crystal and is suitable for studies
under a polarized light microsc0pe. Besides, it has several other

advantages as described below.

Advantages of Lithium Fluoride

 

Lithium fluoride is an ionic crystal and has the NaCl
structure. The slip systems are one of the simplest. The primary
slip planes are the { 110} type planes and the secondary slip
planes are the { 001} type. The slip directions are <1I0> in
both cases. The projection of the six { 110} planes onto the
(001) plane is shown in figure 1. There is only one < 1T0>
direction in each { 110} slip plane. The slip bands due to slip
in planes perpendicular to the { 100} plane are known as 900
bands or orthogonal slip bands. On the other hand, the slip in
planes which are at an angle of 450 to the { 100} surface and
intersect along < 001> gives parallel slip bands or 450 bands.
There can be only one kind of mobile dislocation in each active
slip plane and hence the complication of dislocations with different
Burgers vectors to be found in the same Slip plane can be avoided.
Initiation of secondary slip is difficult due to the high stresses
needed for moving dislocations in the secondary slip planes. 26’ 27
In the NaCl structure no two primary slip planes have the same
Burgers vector, and hence cross-slip at room temperature is
improbable.

Lithium fluoride crystals having fairly low dislocation
density can be grown easily. The grown-in dislocations in these

28’ 29 The pinning

crystals are reportedly pinned by precipitates.
points are so close that these dislocations do not act as active

sources, but they may play an indirect role in the multiplication

10

process in these materials. 30 Specimens having { 100} bounding
surfaces can be cleaved easily and the damage introduced during

31’ 32 The damage introduced is mainly

fast cleavage is not severe.
on or near the surface and, therefore, can be polished off.

Besides the transmission electron microscopy and the etch-
pit method and their limitations mentioned previously a decoration
technique can be used to Obtain a three-dimensional picture of the
dislocation structure in ionic crystals when the density of dislocations
is low. 36 However, its usefulness is limited to small deformations,
and no further testing can be done once the crystals are decorated,
because the dislocations are pinned by impurities. The decoration
technique is, therefore, not suitable for studying the dislocation
substructure during various stages of loading. X-ray techniques
are non-destructive, but the low magnification obtainable does
not give enough information about the dislocation structure of
heavily deformed crystals.

A polarized light micrOSCOpy study of the Slip band growth
in a transparent bulk crystal can give a three—dimensional slip
distribution. 38—41 This method provides a magnification of the
order obtainable by the etchupit or decoration technique and is
free from the limitation Of the pinning of dislocations. However,
its limitations are not apparent without a knowledge of the nature

of polarized light and its relation with the stress field in the

crystal. These will be discussed in a later section.

11

Deformation Behavior of Ionic Crystals

 

Deformation studies Of ionic crystals have been helpful in
understanding the dislocation behavior in lithium fluoride. 28’ 44-46
The yield strength of lithium fluoride depends sensitively on the
impurity content. 47 A variation of 15 to 85 parts per million of

a divalent impurity like magnesium may increase the yield strength
by a factor of ten. Gilman has suggested a cross-slip mechanism
for dislocation multiplication in lithium fluoride crystals. 48’ 49
Another possible multiplication mechanism is based on the formation

28’ 50 According to this, there are

of jogs in screw dislocations.
two possible ways of multiplication depending on whether the jog is
large or small.

Cyclic stressing of lithium fluoride has shown that the
dislocations do not return to their original positions when the
stress is released. 51 Hardening in cyclic stressing is similar in
many respects to hardening in unidirectional stressing. 52 The
damage produced in the slip bands in fatigue stressed and unidirec-
tionally loaded magnesium oxide crystals is predominantly edge
dislocation dipoles. However, in the fatigue stressed Specimens
the density of the debris is lower compared to the unidirectionally
stressed specimens for comparable total strains. 53 The dislocation
density in a slip plane is independent of the number of fatigue cycles
indicating that slip in new planes is preferable to an increase in the
dislocation population within the same slip plane. Unlike materials

which fatigue harden in the early stages of fatigue deformation and

soften on further cycling, the materials, in which cross-slip is

12

. . . . - . . . 4,
d1ff1cult, fat1gue harden cont1nuously up to a certaln l1m1t. 5 55

The dynamic recovery of these materials on cyclic stressing is
suppressed by the absence of easy cross-slip, thus the materials
become stronger and stronger. 56 Consequently, if the material
does not fail during the first cycle at a particular stress level,

it would never fail at this stress level even after an infinite
number of cycles. Failure of the material occurs only during the
increase of the load but not during the fatiguing of the Specimen at

55’ 57 This

any part of the test if the load remains within a limit.
behavior is exhibited by lithium fluoride, sodium chloride and
magnesium oxide single crystals. Zinc also exhibits this
phenomenon at low temperatures. 58 The increase in the fatigue
strength of copper by the addition of zinc has been explained by the
reduction Of the tendency to cross Slip. 59 The necessity of cross-
slip for fatigue failure seems to favor Mott's mechanism rather
than a mechanism prOposed by Cottrell and Hull based on inter-
secting slips for its explanation. 60-62
McEvily Jr. and Machlin have demonstrated the necessity
Of cross-slip for fatigue failure. 55 Later Coronet and Gorum
obtained a conventional stress versus number of cycles curve for
magnesium oxide and, contrary to the above «~mentioned observations,
it showed fatigue failure. 63 McEvily Jr. and Machlin showed that
the fatigue failure of magnesium oxide suggested by Coronet and
Gorum was due to microcracks near the surface introduced by

cleavage that were not removed by polishing. 64 The author's

investigation Of fatigue damage in magnesium oxide crystals favors

13

the original proposal that there is no fatigue failure in these ionic

crystals. 53 Alden's work verifies and strongly supports the same

65, 66

Experiments with zinc and cadmium also Show the

67-69

necessity of cross-slip for fatigue failure. Of these two,

View.

cadmium is susceptible to fatigue failure because the primary
and secondary slip planes have a common Burgers vector and
consequently cross-slip is possible, whereas in zinc, cross-slip
as well as fatigue failure is absent. This method of "fatigue
hardening“ can increase the strength of ionic crystals five to six
times after millions of cycles with stepwise increase in load. 66
FATIGUE may be defined as the tendency of a material to break
under conditions of repeated cyclic stressing considerably below
the ultimate tensile strength. 70 It appears that this definition is
not appr0priate for ionic crystals and other materials which do
not have an endurance limit. So, in this article, the term fatigue
will be used in the sense of cyclic loading.

Uniaxial loading of a lithium fluoride crystal along a < 100>
direction usually causes slip on one set of conjugate slip planes
only, in spite Of the fact that the resolved shear stresses on two
such sets Of planes are equal. 42’ 43 If the direction of loading is
changed after the original deformation to the direction perpendicular
to the plane containing the orthogonal slip bands, the crystal yields
at a much higher resolved shear stress. 71 This direction is known
as the latent direction or the hard direction. The hardening observed

in the latent direction depends on the plastic strain in the original

direction, and this phenomenon is called latent hardening. On the

14

other hand, if the crystal is stressed in the other remaining < 001>
direction perpendicular to the plane containing the parallel slip bands,
the material deforms easily. This direction is known as the soft

direction. Zinc crystals also exhibit the latent hardening property.

Purpose

In the present work, the combined effects of fatigue and
latent hardening in lithium fluoride crystals are investigated.
Besides, the effect of a complete reversal Of the stress in the
fatigue cycles is checked against the predictions of the debris
mechanism of hardening, according to which there should be no
difference in the behavior of the lithium fluoride crystals subjected
to a cyclic loading whether there is a complete reversal of stress
in each cycle or not. 64 The increase in yield strength in the latent
direction, which depends on the mechanical treatment given before
the testing in the latent direction, is compared with the yield
strength Of the initial slip systems of the same crystal. A study
of the deformation structure, Specifically the dislocation configuration
in the slip plane, is carried out and the theoretical basis for the
development of such a structure is discussed. The main objective
of this work is to correlate the dislocation structure with the work
hardening behavior of lithium fluoride crystals in the hope that the
results may shed some new light on the theory of work hardening

in general.

 

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EXPERIMENTAL TEC HNIQUES

Lithium fluoride crystals were initially grown by the
Birdgman method using a sharp tipped high density graphite
crucible. Lithium fluoride powder, obtained from Kawecki
Chemical Company, was used as the starting material and had
the chemical composition given in Table l. The crystals were
grown at the rate of one millimeter per hour. The graphite
crucible was fitted with a refractory sleeve to smooth out the
temperature gradient. Some of the crystals grown by this
method had approximately [ 100] as the direction of growth
and these were cleaved and used as seeds in a Czochralski
furnace shown in figure 2. All the crystals used in mechanical
testing were grown by the Czochralski method under the
conditions and improvisations described below for achieving
faster growth rates and lower dislocation densities. The high
density graphite crucible, holding the melt, and the seed were
rotated in Opposite directions at the rate of approximately one
to two revolutions per minute to obtain a homogeneous melt and a
planar solid-liquid interface. The furnace was water-cooled. The
rod holding the seed was also water-cooled to maintain a necessary
longitudinal temperature gradient. The heating element was made
of graphite. Two chromel-alumel thermocouples were placed at
diametrically Opposite points and a potential divider circuit was

used to find the average thermo~emf. This emf was fed to a

16

17

Minneapolis Honeywell controller to maintain the temperature of the
melt within 3: 20F of the set temperature which was usually 100 to
ZOOF above the melting point Of lithium fluoride. Nitrogen, helium
and argon were used to provide an inert furnace atmosphere. When
nitrogen was used it was passed through a purification train containing
alkaline pergallol, glass beads and phOSphorous pentoxide to
eliminate oxygen and water vapor. A small positive pressure of
this atmOSphere was maintained in the furnace when it was in
operation. A graphite sleeve having a half inch diameter hole in
the center was allowed to float on the top Of the melt to reduce
sudden temperature fluctuations during the growth of the crystals.
A secondary heating element was also used to reduce the radial
temperature gradient. The crystals were grown at the rate of

1/4” to 3'I per hour.

TO avoid sub-boundaries prOpagating from the seed during
the growth process Off-centering and necking methods were
employed. The growth prefers to follow the common axis of
rotation, thereby introducing an initial curved growth immediately
next to the seed if the seed is made in contact with the melt at a
point away from the axis of rotation. This is called the off-
centering. In the necking technique, the crystals were at first
pulled out much faster than the normal growth rate. This made
the crystals grow as thin as needles. After some time, the growth
rate was decreased and the temperature of the melt reduced in

order to Obtain crystals of the required size. The formation of a

l8

neck between the seed and the growing crystal prevented most of the
sub-boundaries in the seed from propagating into the grown crystal.

Fluoboric acid was used for rough polishing and distilled
water, for fine polishing. The etchant used was a very dilute
solution (approximately 10-4 molal solution) of ferric chloride in
distilled water. The dislocation etch-pits were counted using a
Baush and Lomb metallograph. For dislocation density studies,
the crystals were cleaved or cut with an acid-saw using fluoboric
acid.

The crystals were homogenized before testing by first
packing them with graphite powder in zirconia tubes and then
heating at 9000F for four days in an atmosphere of dry nitrogen.
After the heat-treatment, the furnace was cooled approximately at
the rate of 400}? per hour down to ZOOOF and thereafter cooled at a
very slow rate. Specimens (approximately 0. 5 cm cubes) were
cleaved out of this stock and first rough polished with fluoboric
acid and then fine polished with distilled water. The polished
crystals were sprinkled with alundum powder to introduce fresh
dislocations which acted as active sources. The testing was
always performed under compression with an Instron testing
machine using petroleum vaseline as the lubricant.

The deformation studies were focused on (i) the slip band
growth in the early stages of fatigue, with or without the reversal
of shear stresses in the fatigue cycles, and (ii) the effect of the
shear stress and strain in the initial slip systems, with or without

reversal, on the strength in the latent direction. The yield strength

19

Of a lithium fluoride crystal is considerably affected by the diavlent
impurities present in it. 47 Even specimens prepared from the
same single crystal gave scattered results. In order to avoid this
complication, the increase of strength Of a specimen in the latent
direction after a particular mechanical treatment was compared
with the yield strength of the initial Slip systems of the same
specimen. The reversal Of the resolved shear stress in the

initial Slip systems was achieved in the following way. The crystals
were compressed along an [ 100] direction and then turned around
and compressed in the soft direction instead of performing
alternate tension and compression experiments on the same
specimen. The latter would pose difficult grip problems. The
growth of the slip bands on the { 100} planes was studied with
linearly polarized light in a dark field. Detailed observations Of
the damage in the slip bands were made with monochromatic

light (mercury green) in a photoelastic set up and also with a
modified petrographic microscope.

In the slip plane the stress fields Of the dislocations are
such that they can be observed with polarized light. In the dark
field observations, where the directions of polarization of the
polarizer and the analyzer are perpendicular to each other, the
stress fields due to dislocations appear as bright lines provided
that the Slip vector has a certain orientation relationship with the
direction of polarization. Fairly thin Specimens were used since
the depth of focus was critical. The thin specimens were prepared

from bulk crystals, previously deformed in compression along the

 

20

[ 100] direction, by grinding them with emery paper so that the
broad surfaces were parallel to one of the slip planes active during
the deformation of the bulk crystal. The thin crystals were
subsequently polished with a metallographic polishing wheel and
then chemically polished with fluoboric acid at least for an hour.
Fluoboric acid polishes at a rate of approximately 20 microns
per minute, thus at least one millimeter of the crystal was
removed from each side. This prolonged chemical polishing
ensured the removal of the damage introduced during the grinding
and mechanical polishing. Finally the specimens were polished
with distilled water to obtain smooth surfaces. The thickness of
the specimens after the above treatments was of the order of one
thousandth of an inch. A petrographic microscope was used for
studying the damage in the slip plane. Observations were made
with linearly and circularly polarized monochromatic light in
both bright and dark fields. The microscope stage on which the
specimen was mounted was rotated to Obtain various relative
orientations of the slip vector with reSpect to the direction of
polarization. Long photographic exposures were required due
to the extremely low intensity of the lines observed in the slip
plane.

Lithium fluoride specimens having {110} planes as the
planes of observation were also deformed in three-point simple
bending to study the formation of a possible band-like structure.
The jig employed for bending the specimens was mounted on the

stage of the petrographic microsc0pe. Pictures were taken at

21

the successive stages of deformation using polarized light in the
dark field. The bending jig and the specimen were rotated together
to study the relationship between the direction of polarization and
the slip vector and its effect on the patterns. An optical condensing
system was used to enhance the light intensity in the field of
observation in order to reduce the exposure time for photographing.

The arrangement used for this purpose is shown in figure 3.

 

22

Table 1

 

 

Lithium ﬂuoride

H20

SO4

Acidity as HF

 

99.

200%

.700%
.050%
. 040%
.020%

. 005%

 

 

 

23

 

Fig. 2. Czochralski furnace for growing LiF single
crystals.

24

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RESULTS

A comparison of the etch-pit patterns of the surfaces of the
acid-saw cut and the cleaved crystals grown by the Bridgman
method showed that even a fast cleavage had increased the
dislocation density. Figures 4 (a) and (b) Show the typical
dislocation etch-pit patterns Observed in cleaved and acid-saw
cut crystals reSpectively. The cleaved crystals always had
comparatively more sub-grain boundaries and a larger number
of dislocations between the boundaries than the acid-saw cut
crystals. The average dislocation density in the cleaved crystals
was Of the order of 5. 4 x 104 lines per square centimeter, and
that in the acid-saw cut crystals was of the order of 2. 5 x 104
lines per square centimeter. Thus cleaving increased the
dislocation density by a factor of two.

The typical dislocation distribution in crystals grown by
the Czochralski method is shown in figures 5 (a), (b) and (c). The
sub-boundary distribution is shown in (a) and (b), whereas an area
of low dislocation density is shown in (c). These crystals were
cut with the acid-saw and polished with fluoboric acid to obtain
smooth surfaces. The ridges seen in these pictures are due to the
localized unevenness of the fluoboric acid polishing.

The necking and off-centering techniques were found to be

effective in eliminating the sub-boundaries propagating from the

seed. Figure 6 (a) shows the distribution of sub-boundaries grown

25

26

from the seed without using the Off-centering method. The absence
of sub —boundaries after using the off-centering method is shown in
figure 6 (b). The ridges in both these pictures are again due to
prolonged rough polishing with fluoboric acid. A pseudo Kossel
back reflection pattern of one such crystal taken with a micro-
focus X-ray unit is shown in figure 7. The absence of any visible
breaks in the lines observed in this pattern indicates that sub-
boundaries having 001' or greater disorientation were not present
in that crystal.

The results of the fatigue-latent hardening experiments
are tabulated in Table 2. A comparison of the yield stress in the
latent slip systems with the yield stress in the initial slip systems
of the crystals used in the fatigue -latent hardening experiments
Showed that the fatigue cycles with reversed shear were more
effective in increasing the strength than the unreversed fatigue
cycles. As this table shows, crystals fatigued under the
unreversed shear Show an increase in strength in the latent
direction by a factor of four or five. A typical stress-strain
curve for the fatigue -1atent hardening experiment with reversed
fatigue cycles is given in figure 8. It can be seen that in this
particular crystal the yield stress in the latent direction showed
a remarkable increase by a factor of approximately ten.

Polarized light observations of the slip bands on the { 100}
surfaces in the early stages of deformation are shown in figure 9.

The specimen used in this case was not etched. The series of

27

pictures in figure 9 were taken with a camera using a small aperture.
The polarizer and the analyzer were crossed and rotated together
clockwise as viewed from the camera. There was complete
extinction of the slip bands when the direction of polarization was
either parallel or perpendicular to the Burgers vector. When the
Slip vector made an angle of 450 with the direction of polarization

the slip bands were most distinctly visible. With the polarized

light micrOSCOpy only the growth of the orthogonal slip bands but

not the parallel slip bands on the { 100} surfaces can be

observed.

The slip band growth in a specimen subjected to an unreversed
cyclic loading is shown in figure 10. Pictures (a) and (b) Show
respectively the slip band distribution as observed on the { 100}
plane after the first and second cycles. It was Observed that
during the second cycle, very few new Slip bands were formed.
Figures 11 (a) and (b) Show the slip band distribution in a Specimen
subjected to reversed cyclic loading after the first and second half
cycles respectively. In this case the directions of the shear stresses,
in the Operating slip systems, during the second half cycle were
opposite to those during the first half cycle. Contrary to the former
case a large number of new slip bands were formed. Figure 12
shows the [ 100] observation of a crystal deformed in the latent
direction after being fatigued by the reversed shear. It appears
that the new slip bands have difficulty prOpagating throughthe

existing barriers.

28

Figure 13 shows the orthogonal slip bands in the (100.) plane
of one of the specimens prepared for the [ 110] observations. The
four pictures in figure 14 Show the effect of the orientation of the
direction of polarization with respect to the Burgers vector. In
figure 14 (a) the direction of polarization is perpendicular to the
Burgers vector whereas the direction of the analyzer is parallel to
it. This picture shows the regions of stress concentration within
the Slip plane, indicating that the slip is not uniform. But it does
not reveal any further details. More details of the stress
concentration areas were Obtained as the polarizer and the
analyzer were rotated together successively. In (b), (c) and (d),

a band-like structure becomes visible when the direction of
polarization makes an angle with the Burgers vector from 300 to
650, and can be best seen when this angle is 450. However, the
resolution was poor because the specimen was too thick. .

More details of these band formations in one of the stress
concentration regions of figure 14 were revealed by reducing the
thickness of the specimen and by studying it with a petrographic
microscope using the plane polarized light. The results are shown
in figures 15 (a) - (h). The crystal was mounted on the microscope
stage and rotated successively while the polarizer and the analyzer
were kept fixed to obtain different orientations of the Burgers vector
with respect to the direction of polarization. The parallel set of
bright lines in these pictures are due to the slip in the slip planes
which are the conjugate of the slip plane under Observation. The

conjugate slip bands appear to be Spaced more or less evenly. The

29

striking feature of this set of pictures is the band structure along
the direction A'A" in each picture. These bands represent a
certain stress pattern and are in the plane of observation. They
are different from the parallel set of bright lines due to the
intersecting slip planes. Each of these bands appear to be made
up of loops that have lined up in a direction perpendicular to the
Slip vector. These lOOps become prominent only when the
orientation of the Burgers vector is in the range Of 200 to 300
with respect to the direction of polarization.

Thin specimens were found to be extremely ductile when
deformed by using a three-point bending jig. No apparent
hardening was observed in them and no band structure was
detected either. In thick Specimens, because of a complicated
stress distribution, slip took place not only in the plane under
observation and its conjugate but also in planes non-conjugate to
it. This facilitated the study of the role of the intersecting slip
in the formation of these bands and the interaction of non-conjugate
slip with the band structure. Figures 16 and 17 are plane polarized
monochromatic light dark field pictures Of a region Of the specimen
at various stages Of progressive deformation designated as I, II
and III. Two sets of figures are presented in order to show the
dependence of the contrast of the band structure on the orientation
of the Burgers vector with respect to the direction of polarization.
The band structure is not at all visible in figure 16, where the
Burgers vector is parallel to the direction of polarization. The

band structure is visible in figure 17 where the direction of

30

polarization is inclined at 500 with the slip vector. In figure l7-II
the bands just begin to form and the stress birefringence shows
high stress concentration. In figure 17-III the bands are well-
developed and further deformation of the Specimen is difficult.

The pictures in these two sets and figure 15 indicate that these
bands are of approximately constant width of 20 microns. They are
evenly spaced and are usually perpendicular to the slip vector.
Figure 18 shows an enlarged portion of the left side of figure 17-III.
The interesting feature revealed by this picture is the alignment

of the loops along directions which deviate gradually by approxi-
mately 200 on both sides Of the direction of growth.

In a few instances the loops lined up along the line of inter-
section of the slip plane under observation with various other
primary slip planes as shown in figure 19 where the bands BB' are
along the line Of intersection of a non-conjugate Slip plane with the
slip plane under observation. These bands are not well—developed,
and give a lower contrast than the bands formed along the direction
perpendicular to the Burgers vector, i. e. the line Of intersection
between the conjugate slip system and the system under observation.
The Specimen used in this particular case was about 2. 5 mm thick,
and the knife edges through which the bending load was applied were
close together. As a result the stress pattern in the Specimen
became very complicated. The slight curvature of the band
structure perpendicular to the slip vector is due to the curvature
Of the tOp surface of the specimen caused by the lateral compressive
stresses along the axis of bending, resulted from the complicated

stress pattern in such a thick Specimen.

 

31

The band structure in the active slip plane of a sodium
chloride crystal deformed under compression along [ 100]
direction is shown in figure 20. This specimen was 2. 4 mm
thick and not thin enough for a clear observation. The band
structure in the same Specimen after thinning is Shown in
figures 21 (a) and (b) which reveal the 100ps associated with
these bands. Figures 21 (c) and (d) Show the stress concen-
tration due to the intersection of nonuconjugate slip with this
band structure.

The essential features of the band structure Observed in
the slip plane are summarized as following:

(1) The bands were usually perpendicular to the slip

vector.

(2) There was some evidence of bands developing along
the line of intersection between the non-conjugate
slip planes and the slip plane under observation.

(3) The bands were typically Of constant width Of
approximately 20 microns.

(4) They were made up of IOOps usually interlinked by
segments parallel to the slip vector.

(5) They were distributed non-uniformly throughout the
slip planes.

(6) They did not develop in thin specimens which did not
work harden.

(7) The existing bands in a direction perpendicular to the
slip vector acted as strong barriers for non—conjugate

slip.

32

Although lithium fluoride is well suited for studies using the
etch—pit, polarized light, X-ray and decoration techniques, a direct
Observation of the dislocations in lithium fluoride using electron
transmission microsc0py is difficult if not impossible, because of
its poor thermal conductivity and low melting point. Vibration of
the specimens due to electrostatic charges is also a problem. On
the other hand, magnesium Oxide, which has the same crystal
structure as lithium fluoride and exhibits a behavior similar to
lithium fluoride in fatigue and latent hardening, is well suited for
transmission electron microscopy. Figures 21 and 22 are electron
transmission micrographs of magnesium oxide single crystals
deformed in fatigue at room temperature. The damage in the slip
bands, which happen to be the parallel slip bands in all these
pictures, consists predominantly Of edge dislocation dipoles
in the shape of elongated lOOps. Some of the observations made
in these micrographs are summerized below. The points A, B, C
and D whereat there are sharp bends on the dislocation lines, are
collinear in figure 22 (a). The tOp and bottom segments of the
dislocations at B, C and D are respectively more or less parallel
tO each other. The same details can also be seen at points A and
B in figure 22 (b). Figure 23 (a) shows five dislocations A - E
which are appoximately parallel to one another. From the geometry
of these dislocation lines they appear to have been generated by a
Sirgle source. The center portion of the dislocation CC' bends around
an elongated dislocation lOOp. At point E there is a dipole of

approximately edge orientation and at point E' a dipole of approximately

 

 

33

screw orientation. Both these dipoles are partly made of the same
dislocation EE'. Figure 23 (b) gives a much clearer picture of the
formation of these dipoles. At points X and Y there are edge and
screw dipoles respectively. It appears that these dipoles are formed
as a result of the interaction Of the lOOps AA‘, BB' and CC'. These
dipoles need not always be edge dipoles or close to the edge
orientation. AS this picture shows they can be Of various orientations.
Figure 23 (c) shows edge dislocation dipoles which appear to be
formed by the interaction of two dislocations from one source with
two from another. Figure 23 (d) shows the interaction of two

dislocations Of predominantly edge orientation.

34

 

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42

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Fig. 13. Orthogonal slip bands on the (100) surface of
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Fig. 14.

 

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DISC USSION

I. Crystal Growth

As it was shown in the previous chapter, sub-grain
boundaries and possibly individual dislocations originated in the
seed were effectively prevented from propagating into the growing
crystal by using either the off-centering method or the necking
technique. During the growth of a crystal, each dislocation line
that intersects the liquid-solid interface must extend into the
newly crystallized material. It has been experimentally observed
that if the dislocation density is low enough to make dislocation
interactions unimportant, each individual dislocation would
prOpagate along a path normal to the liquid-solid interface.
However, the precipitation of excess vacancies and the presence
of stresses in the growing crystal due to thermal gradients or
due to impurity segregation can cause the end of a dislocation
to move away from this path.

These ideas were utilized in the off-centering method.
Since both the crystal and the melt were rotated with reSpect to
the same axis, the growing crystal had a tendency to align itself
with the axis of rotation. The seed was placed in contact with the
melt at a point away from the axis of rotation, and hence it
developed a curved region before it grew along the axis of
rotation. Most of the dislocations extending from the seed termi-

nated at the crystal surface in the curved portion because of their

57

 

58

tendency to propagate along the direction normal to the solid —liquid
interface. The sub—boundaries and the dislocations propagating
from the seed were, therefore, practically eliminated.

The necking technique for reducing the number of dislocations
propagating from the seed is based on the following observations.
During fast growth dislocations tend to follow low index crystal-
lographic paths in their slip plane, whereas during slow growth
they tend to follow the temperature gradient. In the necking
technique the crystals were grown at a fast rate for some time
and then the growth rate was slowed down to produce crystals of
the required size. If [100] happens to be the growth direction,
four of the six { 110} slip planes are at 450 to the growth
direction and intersect the Surface of the crystal. Most of the
dislocations present in the seed will not propagate into the fast
growing crystal because they tend to remain in the slip plane. In
the necked section, the other two slip planes, which are parallel
to the direction of growth, occupy only a very small area compared
to the seed or the growing crystal. This also helps to reduce the
number of dislocations extending from the seed into the growing
crystal.

The dislocation density of the crystals grown was approxi-
mately 104 lines per square centimeter. It was not noticeably
affected by the growth rate in the range 6 to 75 mm per hour employed.

73’ 74 obtained crystals having 104 dislocations per

Washburn et a1
square centimeter at a growth rate of l to 10 mm per hour. Improved

growth conditions, such as the reduction of mechanical disturbances,

 

59

rotation of the crucible and the seed to give a planer interface and
homogeneity, and water cooling the seed rod to maintain a
longitudinal temperature gradient, helped to increase the growth
rate up to 75 mm per hour in the present work.

Thermal stresses and vacancy condensation are other
possible sources of dislocations. The vacancy condensation can
produce dislocation loops, but it probably cannot account for all
the grown—in dislocations observed in lithium fluoride crystals.
Washburn has suggested that internal stresses must be reduced
in order to avoid dislocation multiplication during growth. These
internal stresses are due to nonwlinear thermal gradients,
segregation of impurities which may cause local change in the
lattice parameters, inclusion of foreign particles, etc. In the
present work the impurity level should have been very low because
the crystals had a yield stress as low as that of pure crystals. The
major cause of the grown—in dislocations seems to be the non-linear
thermal gradients. The longitudinal temperature gradient was
fairly well controlled whereas the thermal gradient in the radial
direction was not. Attempts were made to use an additional
heating element just above the soliduliquid interface to make it
planar. In spite of maintaining the temperature difference between
the melt and this heating element at about lOOOF, no appreciable
difference in dislocation density could result. At smaller
temperature differences it was not possible to grow the crystal
because the temperature of the seed approached the melting point

thereby making the longitudinal temperature gradient too small.

 

 

60

Attempts to grow crystals with large cross-sections (greater
than 12 mm square) resulted in specimens with white inner cores.
The core of such a specimen was made up of a large number of
small crystallites. The region outside this core was a single
crystal. The crystallites were formed far above the interface in
a region of the crystal that had already grown. At the time of
their formation the region outside the core had completely
solidified and was shrinking while the core itSelf was still at a
temperature near the melting point and susceptible to plastic
deformation. The thermal stresses due to the shrinking of the
outer region might have cauSed the formation of the crystallites.
This effect would be severe in lithium fluoride because it is a
poor thermal conductor.

The above observation suggests the reason why it has not
been possible to reduce the dislocation density of lithium fluoride
single crystals below 104 lines per square centimeter. Most of
the dislocations are probably caused by the thermal stresses which
exist even in small crystals due to the temperature gradient present,
especially in lithium fluoride which has a poor thermal conductivity.
In large crystals the thermal streSSes are more pronounced. This
may introduce a large number of dislocations which form the
boundaries of the crystallites. In small crystals the effect is not
so severe, but it has not been possible to eliminate the thermal
stresses as well as dislocations completely with the present

technique.

 

61

The crystals grown by Czochralski method conformed to
crystallographic Symmetry (inspite of the rotation of the melt and
the seed either in the same direction or in opposite directions).

The crystals grown in a [ 100] direction had a fourfold symmetry
and the four surfaces were { 001} planes. The crystals grown

in a [ 110] direction had a two fold symmetry and the planes having
larger areas were { 001} ; the other two surfaces were { 1T0} .
When the growth direction was [ 111] the crystals had a threefold
symmetry with { 1—10} surfaces. This can be interpreted in

terms of surface energy. Since { 100} planes have less surface
energy than any other planes, they will have maximum area and
predominate. 75 In [ 100] crystals it is possible to have four

{ 001} planes parallel to the growth direction. These will form
the bounding surfaces. In [ 110] crystals it is possible to have
only two { 001} surfaces along the growth direction. These

will form the major portion of the surface area of the crystal.

The other two bounding surfaces are { 1T0} because they have

the next lowest energy. In [ lll] crystals there is no possibility
of having {100} surfaces. The {1T0} planes, which have the
next lowest energy, are the only ones that can form the bounding

surfaces.

62

II. Principles of Polarized Light Microsc0py - Observations at
Various Orientations of the Slip Vector with Respect to
Direction of Polarization

In polarized light observations isoclinics, which appear as
dark lines, show the loci of the points where the direction of one of
the principal stresses is parallel to the axis of the polarizer and
the direction of the other, to that of the analyzer. Isochromatics,
which appear as bright lines, are due to principal stresses being
unequal, and their intensities are prOportional to the difference in
the principal stresses. In plane polarized light dark field
observations, both the isoclinics and isochromatics are present
while circularly polarized light dark field observations give the
isochromatics only. Therefore, by studying the fringe patterns
consisting of isoclinics and isochromatics, it is possible in
principle to follow the directions of the principal stresses from
point to point as well as to obtain the difference in the magnitudes
of the principal stresses.

In plane polarized light dark field observations the polarizer
and the analyzer were rotated simultaneously through the same
angles. As described previously there was complete extinction
of the patterns only when the polarizer was either parallel or
perpendicular to the orthogonal slip bands. Since these slip bands
were otherwise present in the plane polarized as well as the
circularly polarized light dark field observations, they were
isochromatics.

For an interpretation of the photoelastic patterns resulting

from a three-dimensional stress field the stress-optic law in three-

63

dimensions, which is connected with the concept of secondary
principal stresses, should be applied. Secondary principal stresses
for a given direction are defined as the principal stresses resulting
from the stress components which lie in a plane normal to this
direction. 76 There exists at the same point an infinite number of
secondary principal stresses, depending on the choice of the
direction through the given point, although for each direction, the
pair of secondary stresses is unique. The stress-optic law in
three-dimensions may be described by the following prOpositionsz
(a) A polarized beam of light on passing through a stressed medium
is resolved into components along the secondary principal stress
directions corresponding to the direction of the ray at the point of
entrance.

(b) The vibrations associated with the beam of light traveling through
the stressed body are at each point parallel to the directions of the
secondary principal stresses for the given ray.

(0) If, however, the directions of the secondary principal stresses
for the given ray rotate as the light advances, then the directions of
vibration of the components of the light vector also rotate through
the same angle.

(d) When the secondary principal stresses remain constant between
the point of entrance and the point of exit the retardation in wave

lengths or fringes is given by
n : C t (p —- q)

where C is the strain-optic coefficient

64

t is the actual distance traveled by the light and
p and q are the secondary principal stresses.
(e) If the directions of the secondary principal stresses remain

constant and the magnitudes vary, then
t
n = C I (p — q) dt .
0

When the directions of the secondary principal stresses rotate,

the rotation tends to increase the retardation; the exact relationship
between this rotation and its effect on the retardations is at present
not fully established.

(f) Normal stresses parallel to the ray produce no photoelastic
effects.

(g) Systems of shear stresses which are c0planar with the direction
of the ray, and the components of which are parallel and perpendicular
to the ray, produce no photoelastic effects.

There have been few attempts to interpret the observations in
photoelastic experiments with regards to the stress birefringence of
the dislocations and the orientation dependence of the intensity of the
birefringence patterns.NYe was the first one to point out that the
birefringence observed in the glide bands of deformed silver ch10ride
crystals is the direct consequence of the presence of a majority of
edge dislocations of one sign. 77 Such an array of edge dislocations
of one sign results in a stress pattern which changes from compression
to tension on crossing the slip plane. This stress pattern has been
shown by Nye to be reSponsible for the birefringent band as seen

under polarized light. Kear and Pratt found asymmetrically colored

FF

 

65

birefringence bands in the interior of quenched sodium chloride
crystals when a sensitive tinted plate was included in the Optical
system. 78 The asymmetry of the color on each side of the slip
plane was shown to be associated with the presence of edge
dislocations of a particular sign. Bond and Andrews made the
first significant observation of the birefringence pattern around
a single edge dislocation in silicon that it conformed to the stress
pattern predicted by the theory of elasticity. 7

Bullough has calculated the expected intensity distribution
in the immediate neighborhood of an edge dislocation in silicon
for both plane polarized and circularly polarized infrared light
based on the principal strains, and suggests that the birefringence
patterns observed by Bond and Andrews are due to macrodislocations
but not due to individual edge dislocations. 80 The principal strains

associated with an edge dislocation are

A
t - —--- .. ..
1‘
A
622 = —Z‘[X+(1~2V)Y]
1'
l 2 Z
ny

where 9 is defined as the angle between the slip direction (X-axis)

and the X' principal axis,

b

A=4n<1_u)

 

' l/ is Poisson's ratio

66

r2=x2+y‘2 .

Using these values he has obtained an expression for the transmitted

light intensity T as

2 2 2 2 2 2
T = —————4a A " t C X— sin2(26 +25)
)\ 4
r
where a is the amplitude of the incident plane polarized light
t is the thickness of the Specimen

C is the mean strain—optic coefficient
)x is the wave length of the incident light
6 is the angle between the polarizer and the slip
direction.
He has also plotted the intensity distribution versus the orientation
of the slip direction with respect to the polarizer.

Mendelson analyzed the birefringence due to the dislocations
in the glide bands of sodium chloride single crystals. 81 By
considering a slip band as being consisted of a row of equally spaced
edge dislocations he argues that the stress field between neighboring
edge dislocations in the slip band is such that the principal directions
are respectively parallel and perpendicular to the Burgers vector.
The shear stresses on the planes parallel and perpendicular to the
Burgers vector vanish completely midway between any two edge
dislocations in an array of edge dislocations of the same sign. This

is also true at the points lying directly above and below the dislocation

67

lines because the shear stresses on such surfaces all add up to

zero except for those dislocations which are near the two ends of the
array. At all other points the shear stresses on such surfaces

almost cancel each other. This leads to the conclusion that one of

the principal directions in the stress field of a row of edge dislocations
of the same sign is parallel or nearly parallel and the other direction,
perpendicular or nearly perpendicular to the slip vector. Therefore
when the plane of polarization is parallel to one of these principal
directions, most of the light does not go through the analyzer since the
Neumann-Maxwell law states "light passing through the stress region
is polarized parallel to the direction of the principal stresses in the
plane of the wave front at that point. " The component of the incident
light beam in the other principal direction or the direction of the
analyzer is zero and hence the intensity of the transmitted light

beam is exceedingly low in such a case.

Based on these concepts of photoelasticity, the photoelastic
observations made in this work may now be interpreted in the manner
given below:

(i) Screw dislocations

For a single screw dislocation, the difference in the secondary
principal stresses in { 100} and { 110} planes of observation
vanishes resulting in zero intensity of the stress birefringence
patterns.

(ii) Edge dislocations
The stress birefringes observed‘in { 100} and { 110}

planes must be due to the stress field of edge dislocations as

68

explained below:
(a) { lOO} observations

Figures 9 (a), (b) and (d) show individual Spots along the
slip bands, whereas figure 9 (c) shows complete extinction of the
pattern when the direction of polarization is parallel to the slip
vector. One cannot interpret these Spots as being due to individual
dislocations since were it true, the pattern would not vanish when
the direction of polarization is parallel to the slip vector according
to Bullough's explanation.

For complete extinction of a point along the isochromatics,
the principal directions of the stress field at that point should be
parallel to the directions of the polarizer and the analyzer
respectively. The present observations agree with the results
obtained by Mendelson with plane polarized light in dark field in
that there was complete extinction when the plane of polarization
was either parallel or perpendicular to the slip vector. The slip
bands observed by him were thick and intense, similar to those
shown in figure 10.

In the light of Mendelson's explanation one could overcome
the difficulty encountered in interpreting the extinction of the
photoelastic patterns by Bullough's model. Accordingly each spot
can be considered as being due to the stress field of an array of
edge dislocations. However, the possibility of having such edge
dislocations alone during deformation is very small. Thus the
stress field causing the birefringence is probably due to the edge
components of the arrays of dislocation loops in the slip plane as

shown in figure 24.

69

(b) { lIO} observations

In { llO} observations the stress field of the edge dislocation
arrays has its two secondary principal directions reSpectively parallel
and perpendicular to the Burgers vector, and therefore the maximum
intensity of the stress birefringence patterns is observed when the
plane of polarization is at 450 to the Burgers vector. There is
complete extinction when the plane of polarization is either parallel

or perpendicular to the Burgers vector.

(iii) Mixed dislocations

Mixed dislocations can be treated as being composed of edge
and screw components whose contribution to the birefringence patterns
varies with the orientation of the mixed dislocation with respect to
the Burgers vector. Therefore one may consider the stress
birefringence of mixed dislocations as being due to their edge
components of reduced strengths only.

In { 110} observations stress birefringence patterns are
seen to be consisted of 100ps usually lined up in a direction
perpendicular to the Burgers vector. These patterns are probably
caused by dislocation 100ps lying in the slip plane. Dislocation 100ps
are in general made up of edge, screw and mixed parts. The stress
field of the edge parts and that of the edge components of the mixed
parts may give rise to patterns not inconsistent with the observed

ones.

7O

 

 

   

No ext'nction

 

 

 

 

 

 

Complete extinction

 

 

 

 

 

Fig. No. 24
Dependence of contract on the orientation of the slip
vector with respect to the direction of polarization

 

71

III. Slip Distribution and Work Hardening

It has been experimentally proved by Stokes, Johnston and
Li that MgO specimens develOping block slip, in which a single slip
band expands laterally to fill the entire gage length, are extremely
ductile. 82 Bruneau and Pratt showed that specimens, which develop
conjugate as well as non—conjugate slip intersections, work harden
much more rapidly than specimens which deve10p conjugate slip
bands alone. 83 Nabarro lists the order of increasing rate of shear
hardening as (1) simple shear and kinking, (2) single slip, (3) double
slip, and (4) quadruple slip. 84 The increase in work hardening in
the specimens which deve10p non-conjugate slip is interpreted by
the above authors in terms of interactions of individual dislocations
moving in such non-conjugate slip planes that their Burgers vectors
intersect at an angle of 600. This intersection produces barriers
which inhibit further movement of other dislocations.

The results of the present investigation agree in general with
the above picture as far as the effects of the conjugate and the non-
conjugate slip intersections on work hardening are concerned.
However, it seems possible to interpret the work hardening in this
type of materials by means of the band structure observed in the
slip planes of the deformed bulk crystals rather tlan the interaction
of individual dislocations. The high ductility and the low but finite
work hardening of specimens deformed in single slip may be
attributed to the formation of the band structure in a single slip

system. The hardening in double slip, especially during a non—cyclic

72

loading, may be due to the interaction of dislocations moving in one
slip plane with the band structure in the conjugate slip plane. In

the case of non-conjugate slip, the dislocations moving in a slip
plane have to cut through many bands which are present in the
non-conjugate slip planes with difficulty, thereby causing a maximum

work hardening.

 

73

IV. Theory of Band Formation

The damage in the slip plane was made up of bands which
were resolved as being consisted of overlapping 100ps. An
interesting aspect of this band formation is that it is usually in
a direction perpendicular to the Burgers vector. These bands
are similar to the ones observed in zinc where they are classified
as deformation bands. 24 (It is to be discussed later in section (c)
of this chapter that intersecting slip may create collinear sources
which may contribute to the formation of the band structure. How-
ever, this mechanism cannot explain the band structure formation
in zinc because the basal plane is the only active slip plane at a
moderate stress level and room temperature. There Should be a
general mechanism of the band formation regardless of the
intersecting slip. Extending the idea originated by Frank, Wei
has prOpos ed such a mechanism based on the interaction of
dislocation 100ps originating from sources in neighboring atomic
planes. 24 The following discussion is intended to justify such a

mechanism from the interaction energy and force considerations.

(a) Assumptions

It is assumed that random dislocation networks which exist
in crystals before deformation starts are the active sources. The
grown-in edge dislocations in ionic crystals are pinned by
precipitates. These precipitates have been observed in the electron

Z8, 30 The

microsc0pic study of magnesium oxide single crystals.
pinning is so close that the grown-in edge dislocations cannot act

as sources, whereas screw dislocations, which are not pinned by

 

 

74

the impurities in the crystal, act as the sources in the early stages
of deformation.
(b) Single Slip

The sources distributed at random are most likely non-COplanar.
Some of them may lie in nearby parallel planes. In a set of parallel
slip planes in which dislocation lOOps are expanding due to the applied
stress, the lOOps should have the same Burgers vector. When these
expanding lOOps approach each other the interacting segments can be
either edges or screws of the Opposite signs or mixed segments
whose components are Of the Opposite signs as shown in figure 25.
(i) Screw Interaction

The interaction force per unit length between two screw

dislocations of Opposite signs is an attractive force given by

 

Z
_ p» b
Fr — 2 Ti' r
where r is the distance between the two dislocations,
H is the shear modulus, and
b is the Burgers vector.

Since cross-slip is difficult in ionic crystals, this attractive force
will cause the two screw components to arrange themselves parallel
to one another in a plane perpendicular to the slip planes when the
separation is a minimum. In the case Of a large number Of
dislocation loops being generated from two neighboring sources

and interacting in their screw components, all the screw components
will line up in similarly parallel pairs Of Opposite signs resulting in

the formation of a screw-lock. The early stage of the screw-lock

75

formation can be accomplished relatively easily by the movement Of
a few pairs. However, after a large number Of pairs have been
locked, any further addition Of the pairs becomes difficult because
it requires the movement and rearrangement of all the pairs locked
already. The screw-lock becomes a barrier against which the
subsequent dislocations may pile up thereby making the source
inoperative and causing some degree Of work hardening. It seems
that screw-locking is a possible cause of band formation in
materials that do not have easy cross-slip. It explains why the
loops Observed in these bands usually line -up in a direction

perpendicular to the Burgers vector.

Dislocation Model and Simple Mathematical Analysis of Screw-Lock
A simple mathematical analysis of the screw-lock is presented

in this section. This analysis gives only a qualitative or a semi-
quantitative picture because of the following assumptions and
approximations: (i) the distance between any two dislocations in

each slip plane is constant, (ii) there is an equal number Of
dislocations N in each slip plane, (iii) second order effects are
neglected enabling the use Of the principle Of superposition to

calculate the strain energy and the total force for the system.

Formation of the Screw-lock -- Strain Energy Considerations
The self energy of a single screw dislocation is given by the

formula

 

76

where r1 is the radius of the surface free Of stress

rO is the radius Of the dislocation core.

Consider an isolated region away from the free surface as shown in
figure 26 (a), and assume that there is no interaction between the
tOp and bottom rows before the screw -lOck is formed. The total
strain energy U“) for this arrangement having N dislocations

in each slip plane can be calculated by the following equation:

2 r
_ Zpr l
Um _ ——————4TT [loge(—r0) —1] _+2UO

where U is a term accounting for the interaction energy between

0

those dislocations within the same slip plane whose spacing does
not change during the formation of the screw-lock. The radius

r is very large compared with the dimensions of the screw-lock,

l

and can, therefore, be treated as a constant. The difference
between r1 and r0 is about seven order so that small variations

inr

1 will not affect the calculations appreciably.

Figure 26 (b) shows the dislocation distribution after the
screw-lock is formed. In this arrangement, the interaction
between dislocations in parallel slip planes cannot be neglected.

This interaction energy, US for a pair Of unlike screw dislocations

I 5
is given by
s E b2 rl
U1: - 2n [10ge(_r—)_l]

where r is the distance between the dislocations. The total strain

energy, U( of the screw-lock having N pairs Of screw dislocations

f)’

of Opposite signs, becomes

77

ZNEbZ 1”1 s
Um : 4n [loge(;;)-l] +2UO+NUI+AUO

where A U0 is the decrease in strain energy due to the interaction
Of the individual dislocations in the screw-lock with dislocations Of
the opposite sign in the neighboring pairs. A U0 —~ 2 U0 when the
separation of the slip planes is very small compared to the distances

between the neighboring dislocations in each slip plane. It approaches

zero when the slip planes are separated by a large distance. The

quantity N UI5 is negative which lowers the energy of the system ,1
thereby making the screw—lock configuration more stable. To (it)
r ,
estimate N UIS , let N = 100 and _1 = 106, one Obtains N UIS’: 150
r

ev per atomic plane threaded by the screw-lock having 100 pairs.

Stability Of the Screw-lock

Once the screw-lock is formed it does not dislodge by itself
because it requires an increase in energy to do so. To prove this,
consider a screw-lock having N dislocations in each slip plane as
shown in figure 27 (i). Let one pair move out leaving (N-l) pairs
in the screw-lock as shown in figure 27 (ii). These two configu—
rations have (N-l) common pairs whose self energy as well as
interaction energy remain constant in the two cases. The difference
in strain energy between these two configurations is the difference
between the interaction energy Of one extra pair A — B in
configuration (i) and the interaction energy of A — D separated
by a distance r3 in configuration (ii). All other interaction
energies are the same in the two cases.

The total strain energy U“) of configuration (i) is

78

 

H 2 r1
_ l _ —— _
and that of configuration (ii) U(ii) is
2 r
_ . _ H b. _.1_ _
U(ii) — U 2 TI" [loge (r3) 1]
where U' is a constant which includes the self energy Of ZN

dislocations and the interaction energy Of (N—l)
pairs within themselves and with A - B or A - D,
r2 is the distance between the two slip planes under
consideration,
is the distance between the dislocations A and D
in configuration (ii).
Therefore the difference A U in total strain energy in both the

configurations is given by

AU : Uni) ' Um
2 r r
H b __1_ _ __1_
_ 2 [loge (r ) 10ge (r )]
3 2
2 r
_ _ LL .2:
— 2 Tr 10ge ( )
3
2. r
LIL .2
2 1T 10ge (r2)
Assuming r2 = 10 b and r3 = 1000 b, this gives an increase in

strain energy Of approximately 4 ev per atomic plane threaded by
the screw-lock. SO the first configuration is stable, and once the

screw-lock is formed it will not dislodge itself.

 

 

lulll.|l..I|I||‘l|I|.il| II. ..

 

79

Forces associated with the formation of the screw-lock

Consider two rows of screw dislocations coming from two
different SOurces lying in different slip planes near each other and
separated by a distance y as shown in figure 28 (a). Each row
has N dislocations uniformly separated by a distance x. Due to
the applied shear stress 7', the top row is moving to the left and
the bottom row is moving to the right which is designated as the
positive direction. Only the forces acting in the X—direction will
be responsible for the formation of the lock.

Consider figure 28 (b) in which two rows having N dislocations
each overlap each other by N' number Of dislocations. The force

FX acting on the leading dilecation of the bottom row is given by

 

 

2 N‘-l N-N'
E b
FXZZW ”‘2 12X 2]+[ 2 2X 2]}
n21 (nx) + y 11:1 (nx) + y
b2 N-l l
+ 4*— [ z _] + 7b
2 TT x _ n
n—l
'—
where _ E b2 N21 nx is the -=X component Of the
Zn n:l (n )2 + Z
X Y attractive forces due to the
dislocations to the left Of
the leading dislocation in
the top row,
_ l
+ p. b2 NZN nx is the +X component Of the
Zn

attractive forces due to the
dislocations to the right Of
the leading dislocation in

the top row,

 

 

80
2 N-l is the +X component Of the
11:1 n repulsive forces due to the
dislocations lying to the left
of the leading dislocation in
the same slip plane, and
’Tb is the force due to the
applied stress.
When the leading dislocation Of the two rows approach each other as
shown in figure 28 (a) the first component is zero, and all the remaining
three are directed in the +X direction and, therefore, help the l

formation Of a screw-lock. When the first pair in the screw-lock is

formed
_ b2 N-l nx b2 12-1 1
Fx_ ZTr[ Z 2 Z]+2Trx[ H]+Tb'
n=l (nx) +y n=l

As the rows move passing each other further, the —X component
Of the attractive forces due to the dislocation to the left of the leading
dislocation in the top row becomes appreciable and thus reduces the
value of Ex. When the two rows overlap by 1; dislocations in each
row the positive and negative chomponents of the attractive forces
due respectively tO the dislocations lying to the right and left cancel
each other. After the leading dislocation crosses this point, the net
X-component Of the attractive forces will be in the -X direction and
thus Oppose the applied force and the +X component Of the repulsive
forces. However, the X—component Of the net resultant force will be

in the +X direction, facilitating the formation Of as many pairs as

possible before the leading dislocation stops.

Force Field Components in the Central Portion of the Screw-lock
Consider the central portion Of the screw—lock consisting of

three pairs Of dislocations as shown in figure 29 (a) with no applied

 

 

 

 

u l.i.l1- .

...l. |.|Iik. .

81

force. Dislocation l experiences interaction forces due to dislocations

2 through 6. For any pair Of screw dislocations the radial interaction force

2
F : _H_b
r 2 ‘IT r
has components
F _ p. b2 x
X

ZTr (xZ +y2)

F : __H_b:_Y_Z
Y 2 1T (x + y )
The repulsive‘forces between dislocations l and 3 and l and
2 are Of equal mangitude and Opposite in sign. They will cancel 7.
each other. The X—components Of the attractive forces between 1
and 4 and l and 6 are Of equal magnitude and Opposite in sign.
They too will cancel each other. The attractive force between 1
and 5 does not have any components in the X-direction. Thus the
dislocation l, in the central portion Of the screw—lock experiences
no net resultant force in the X-direction due to the immediate
neighboring pairs. The force due to interaction between this
dislocation and the next pairs On either side or the pairs beyond
will be very weak. Generalizing this idea, it can be said that most
of the dislocations in the central portion, except the ones near the
Outer edges of the screw-lock, experience no or very little net
force in the X—direction.
The resultant force Fy on dislocation l Of a screw-lock
having (ZN + 1) pairs of dislocations is
N b2
2

b2
F _ M { -—2 +2 22 T—
y “:1 (DX) +Y

y Z'rT

 

 

82

2.
1 —-—1-)—7 both x and y are greater than b

In the series 2
(nx) +Y

M42

and n is a finite number having integral values between 1 and N.
The denominator (nx)2 + y2 is always greater than the numerator
b2. Therefore, the series has a finite sum, and it converges fast.
Assuming that y = 10 b and x : 100 b , the error in considering
the first term alone is less than three percent and the error in
considering the first two terms is less than one percent. Thus for
all practical purposes the first term will be sufficient to give the
approximate force in the Y-direction.

The magnitude of Fy determines if cross—slip will take
place. For ionic crystals the stress required for cross-slip through
the secondary slip plane is ten to twelve times the stress required
for primary slip. Knowing this one can determine a maximum value
Of y beyond which cross-slip will not take place. The maximum
value so obtained is Of the order Of 10 b.

(ii) Edge Interaction
The interaction force field between two edge dislocations of

Opposite signs has the following components:

 

 

Z 2 2

F : P‘ b X(X ”Xi
x 211' (l - U) (X2 + YZ)2

2. 2 2

F 2 H b y(3X +Y)
y 2 TT (1 —- V) (x2 + yZ)2

where x and y are the coordinates of one of the dislocations with
respect tO the other at the origin, with X~axis along the direction of

the Burgers vector and Y—axis perpendicular to the slip plane. The

 

83

component FX is attractive when lxl > lyl and is repulsive
when I xl < I yl . In the absence Of applied stresses and dislocation
climb the equilibrium configurations of these dislocations are at

l XI = ] yI and at x = 0. Under an applied shear stress '7' acting
on the slip plane in the slip direction and for a particular value of
the distance between the slip planes y, the value of x will depend
on the applied stress '7'. When '7' is greater than Hb/8rr(l -'1/)y ,
the dislocations will cross Over and move away from each other,
otherwise, they will attain an equilibrium configuration when the
repulsive force component becomes equal and opposite to the
applied force resulting in a plus -minus pair.

In the early stage of deformation when the randomly
distributed sources are Operating, the sources will probably be in
planes which are far apart. There will be no severe interaction
between the individual dislocations due to their short range stress
fields. If the approaching segments are edge components of
Opposite signs, they can cross over easily. However, when two
arrays of edge dislocations in parallel planes approach each other
this will not be the case because of their long range stress fields.

Consider the interaction of a positive and a negative edge
dislocation arrays moving in slip planes separated by a large
distance a . The stress field of a finite set of uniformly spaced
horizontal edge dislocation array has the following stress components

at distances comparable to its finite width 2L. 85

 

 

 

 

 

 

 

 

 

 

 

 

0' = - 2 tan 1 ZLY
xx 2
x + y - L
x—L x + L
' Y[ 2 2 ]
(x-L) + y (x+L) + y
0_ : Y[ X - L _ X + L ]
W <x-L)‘2 + y (x+L)2 + y2
__ -1 2L y
O'ZZ - - Z 1/ tan 2 2 Z
x + y _. L
-L +
0' = - ElOgJ-EX )2 X2
xy l(x+L) + y
L.
2 Z
_ Y + l
2 2
(x-L) + y (x+L) + y
a = 0' = 0
yz xz
in units of
p. b
2n h (l - V)
. . . Nh
where L IS the half Width Of the array, Viz, L = -2—-
N is the total number of dislocations
h is the Spacing between dislocations in the array.

At larger distances compared with the width of the array,
the above stress components reduce to those of a single "super-
dislocation" Of strength Nb situated at the center Of the array.
In such a case their interaction can be considered as that of two
super-dislocations of strength Nb each, which have long range

stress fields instead of the short range stress fields of single

 

85

edge dislocations as discussed above. The minimum shear stress
7' required to cross over will depend on the separation a of the

two slip planes and is given by

 

_ H b'
7 8n(1 .. u) o.
where
b' : Nb

When a is small the shear stress 7' is very large and the cross
over does not take place. The dislocations, then, reach an
equilibrium configuration having a horizontal separation x as a
function of a . In the absence Of applied shear stresses and
dislocation climb, the dislocation arrays would reach an equilibrium
configuration when the line connecting their centers of gravity makes
an angle Of 450 with the slip plane.

As deformation proceeds, the dislocations moving in nearby
planes will approach each other resulting in a configuration shown
in figure 30 (b) which may be transformed into the configuration
shown in figure 30 (c) by rearrangement of the dislocations through
a climb process — a model for the deformation bands as suggested
by Mott.

In the case Of single edge dislocations moving in nearby slip
planes the short range forces play a prominent role. For slip planes
approximately 10 A: apart the dislocations cannot pass over each
other since the applied stress required is very high, being Of the
order Of one hundredth of the theoretical shear strength. Therefore,

they will form a plus uminus pair which is sometimes termed as debris.

 

 

86

For the interaction of two arrays of edge dislocations in
nearby slip planes as shown in figure 31 (a), the plus -minus pair
formed by the two leading dislocations as indicated above, acts
as a barrier against which the rest Of the dislocations pile up.
The net force FXA acting on the leading dislocation A may be
calculated by considering one array as a pile-up and the other
having uniformly spaced dislocations with 6 being the space

betWeen neighboring dislocations is

 

 

FX 2 (N-1)'rb 4“,}
A
+ u b2 §[<n-1)6+x]{[(n-1>6+x]2 -y2}
2n (1 —V) 2 2 2
n=2 {[(n-l)6+x] +y}
2 2 2
u b X(x -L)
+Tb+ Z'rr(l -1/) (X2+y2)2

where the first term accounts for the stress concentration, the
second term includes the interaction force components due to
dislocations in the other plane with the dislocation A and the
third term is the force due to the applied stress. If N is large,
FX may become a driving force sufficiently high enabling A to
cross over.

The process may repeat, and after two dislocations have
formed pairs as shown in figure 31 (b) the net force Fx acting on

the second dislocation is

 

87

u b2 N {(n—né +x]i[<n-1>6 +1.12 ' Y2}
2’1““— 2 Z 2 2
TT( " 7/) n:3 { [ (n-1)5 + X] + y }

u b2 x (x - ya)
er(l-1/) (X2 + y2)2

+’Tb+

The contribution due to interaction with the leading pair containing
dislocation A is, however, negligible. If this process is repeated
until N' pairs have formed, the net force acting on the N'th

dislocation is

 

F : (N _Nl) 'Tb+ (1 b2 g [(n'1)6+x]{[(n21)6+X]2‘YZ}
XN' 2"”7’) n=N'+1 {[(n-1)6+x] +y2} Z

Z Z 2
Mb X(x -y)
21—1/ 2 22

+7b+

It is evident from this expression that as N' increases F
NI

decreases, and eventually the two rows will stOp against each other.
The above considerations lead to the conclusion that the
leading dislocations, by forming one Or more plus -minus pairs,
hold the rest of the dislocations against each other resulting in a
back stress. When a sufficient number of edge dislocations have
piled up against the plus -minus pairs the back stress will build upto

a level making the sources inoperative.

 

 

.IIIhII ll. in . I'll III-I Illicit—I I311-

88
(iii) Interaction of Two Sets of Dislocation LOOps on Parallel
Slip Planes

When the expanding dislocation lOOps lying on parallel slip
planes approach each other, the interacting segments can be either
edges or screws of Opposite signs or mixed as shown in figure 25.
As discussed earlier, the interaction Of the screw components will
form the screw-lock which is a stable configuration and causes work
hardening. The interaction of the edge components will lead to the
formation of plus -minus pairs and dislocation pile-ups. The sources
will soon become inoperative due to the back stress of the pile-ups.
It can be shown that the interaction between mixed dislocations may
also lead to the formation Of screw-lock, because it has the lowest
strain energy.

Consider two dislocation lOOps originated from two different
sources lying in different slip planes, approach and interact with
each other. For simplicity of discussion Of the mechanism, assume
the two 100ps are semi-circular Of equal diameters and form either
a mixed, an edge, or a screw dipole as shown in figures 32 (a), (b)
or (c) reSpectively. The total dislocation length and the slipped
area are the same in all three configurations. The dislocation
segments other than the dipole segments are also Of the same length
irreSpective Of the type of dipole under consideration and will thus
have approximately the same energy. Hence the difference in the
strain energies of these configurations is due to the difference in
the strain energy Of the dipole segments only. The strain energy

Of the dipole segment depends on its orientation with respect to the

 

 

89

Burgers vector. It is sufficient, therefore, tO consider the strain
energy per unit length Of the dipole as a function of its orientation
because all the three dipoles have the same length.

The self energy per unit length of an edge dislocation is

Z r

e N H b 1
Us '" 4Tr(l-1/) [loge(rO)-1]
and that of a screw dislocation is
2 r
8 ~ [.1 b 1
US _ 4rr [10ge(rO)-1]

The mixed dislocation can be treated as a superposition Of an edge
dislocation and a screw dislocation having their Burgers vector
perpendicular tO each other. The strength Of the edge component
is b sin 9 and that of the screw component is b cos 9 , where

0 is the angle between the dislocation line and the Burgers vector.

The self energy per unit length Of the mixed dislocation is

 

Z r .2
m ~“b 1 Sin 0 2
US _ 4n [10ge(rO)-1][———1_V+cos 0]
2 r
_ Mb 1 Z
— 4Tl’(1-=V) [loge(r0)_1][1_v cos 0]

The interaction energy per unit length between two edge

dislocations is

2

. r
e _ b __1_
U1 ‘ ‘ 21r(l-1/) [10ge(r

)+%—cosZ¢-l]

where r and (1) are the polar coordinates of one of the dislocations

with respect to the other at the origin.

 

 

90

For a stable configuration Of the edge dipole <1) is equal to 450. The

interaction energy per unit length of the edge dipole is thus

2 r

_e_ Mb 1
U1 - - mllogei‘rl'” °

Similarly the interaction energy per unit length Of the screw dipole is

2 r
s_ Mb _l_
UI- -21r [10ge(r

)-1]

Therefore, the interaction energy per unit length Of a mixed dipole is

 

 

Z r .2
m_ “b 1 8111 9 2
U1 - - a. [loge<—,>-11[———(1_W +cos e]
2 r
Mb i 2
- - 21r(1-1/) [loge(r)‘1][1-l/ cos 0]

m
UT I
2 r

: m [loge ($13) -1][1 -V COSZ 9]

2 r
- --——-—— [loge (—:—) -l][l -UCOSZG]

$177 [10ge 63)][1 -1/ cos2 0]

When 9 is zero, it gives the total energy of the screw dipole U?

and when 9 is 900, it gives the total energy of the edge dipole U; .

m

T
. . .m o o
increasmg value Of UT as 9 changes from 0 to 90 . Therefore,

A plot Of U versus 9 as shown in figure 33 gives a monotonically

the screw dipole having 0 : 00 has the lowest energy and hence is

 

 

 

91

the most stable configuration. The lepe Of this curve gives the
torque acting on the dipole and it is zero when the dipole is in
either the edge or the screw orientation. It has a maximum value
when the dipole is at an angle of 450 with the Burgers vector.

This torque will cause a mixed dipole to reorient to the screw
orientation and facilitate the formation Of the screw-lock. The
distance between the two dislocation segments of the edge dipole
will be J2 times the distance between the two Of the screw dipole.
This leads to a small increase in the difference in strain energy
between the two configurations.

This difference will be enhanced further when N dislocation
loops generated from each Of two neighboring sources in two parallel
slip planes interact. When the interaction is of the screw type the
total strain energy will be that of N screw dipoles and thus it will
be N times the total energy Of an individual screw dipole. For
the edge type interaction, however, the total strain energy will be the
energy of the first few edge dipoles plus the Self energies of the
dislocations held up against the edge dipoles. Interaction between
dislocations of Opposite signs always reduces the total energy
thereby making the total energy Of the screw type interaction lower
than that of the edge type interaction where the lowering is due to
one Or a few dipoles only. This will make the difference between the
total strain energy of the screw interaction and the edge interaction
still greater. These arguments lead to the conclusion that the screw
interaction is the most stable configuration and can, therefore, be

called screw-lock. SO the mixed components should reorient them-

 

 

92

selves to form the screw-lock as long as the energy decrease due
tO reorientation is greater than the energy required to increase the

dislocation length sufficiently to make this configuration possible.

(iv) Development of Deformation Bands

As discussed above, the interaction Of dislocation loops
generated from sources in neighboring parallel planes may give rise
to screw -locks which, in turn, result in the formation Of a band
structure in a general direction perpendicular to the slip vector.
The screw -lock formed in this manner should be the major factor
contributing to the band formation in crystals which deform by a
single slip system.

The lithium fluoride crystals had a grown-in dislocation
density Of about 2. 5 x 104 dislocations per square centimeter.
Assuming an even distribution, the distance between two neighboring
sources should be approximately 60 microns. This should produce
lOOps Of approximately 60 microns diameter, which is of the same
order Of magnitude as the observed value of 20 microns.

The defect structure shown in figure 18 appears like three
dimensional dislocation networks and it resembles the dislocation
networks Observed in sodium chloride crystals by the decoration
techniques. 36 These networks are probably not resulted from
deformation since in such a case dislocation lOOps usually arrange
themselves along a direction perpendicular to the Burgers vector
or along the non-conjugate slip intersections as has been Observed
throughout this investigation. This suggests that they are the grown-

in three—dimensional dislocation networks. In addition the spacing

 

 

 

93

between the dislocations in this picture is of the same order as the

Spacing calculated by assuming a uniform distribution of the grown—

in dislocations using the dilecation density data Obtained by the

etch-pit technique.

(c) Multiple Slip — Development Of Deformation Bands Parallel to
the Line Of Intersection of the Slip Planes.

The mechanism of band formation discussed so far should
have general applicability to most materials. It can be the controlling
mechanism in materials with no intersecting primary slip planes such
as zinc. However, for ionic crystals there seems to be acting at the WW
same time an alternate mechanism as Observed in lithium fluoride and
discussed below. As shown in figure 19 the band structure may also
be considered as being formed along the direction of the intersection
of the primary slip planes.

Consider two intersecting conjugate slip planes A and B
as shown in figure 34. In plane A there are two dislocation loops
(I and II) expanding due to applied stress. The two loops may, how-
ever, be in nearby parallel planes. In the intersecting slip plane
B a source is Operating and is generating dislocation 100ps which
intersect with the expanding loops in plane A. The loops I and II
will thus acquire jogs which become larger and larger as more
intersections take place. The jog acquired by loop I will be larger
than that of loop II because it is intersected by more dislocations.
These jogs act as pinning points and the segments in the planes A‘
and A" may act as new sources of dislocations as shown in figure

34 (c) and (d). Loops generated from these sources may develop

94

into the Observed band structure.

The sources will Operate in parallel planes A', A", etc.
which, when viewed in a direction normal to these sets Of planes,
will appear to lie along the line Of intersection Of planes A and B.
This line is perpendicular to the slip vector for the conjugate slip
intersection. For non-conjugate slip planes the intersection line
is inclined to the slip vector as seen in figure 19. From this point
on the general mechanism Of loop interaction discussed above takes
over and screw-locks are formed in the case of conjugate slip plane
intersections. In the case of non-conjugate slip plane intersections
the interacting dislocations are mixed in nature. Since the dislocation
lOOps are aligned before the interaction takes place, they would have
to increase their lengths by a large proportion to form screw-locks.
SO the interacting dislocations may stay as mixed dipoles when the
band structure is not fully developed.

The band formation in directions other than the one perpen-
dicular to the slip vector was Observed in one Specimen only. This
specimen was thick and had a complicated stress pattern. In a
number of specimens where non-conjugate slip was present, no band
structure was developed along the line Of intersection. The bands
shown in figure 19, which are due tO non-conjugate slip intersection,
have poor contrast compared to the bands Observed along the line of
intersection Of the conjugate slip systems indicating that the general
mechanism Of band formation in a direction perpendicular to the slip

vector is perhaps more important than the mechanism discussed here.

 

 

95

‘ ,
Edge Interaction

 

 

 

.‘O
> Screw Interaction
.r

 

 

Symbol
0 Right Handed ecrew

G). Lett Handed Screw

 

 

 

 

 

 

 

Mixed Interaction

 

Fig. No.2 5

General Mechanism of Bond Formation

 

 

96

 

 

 

 

 

 

 

 

 

m

Fig. No. ,2 6

Model for formation of Screw—Lock

”‘1
H

m@ 91>

 

 

97

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c‘
e e e e e O
o o O o o o
____ D
(i)
e e e r e o 5
L..3_n
(2 o o o o 0
(ii)
. Fig. No. 27-

Stability. of Screw-lock

00

   
 

 

a.“

 

 

A LM+ x —.i
6 9 @1—
Y
o o o F * «L
I N "
Leading dislocation
'5
. l N
i e e e
O O O—->- I;
' N
Fig. No. 2 8

Forces associated with the formation of

Screw Lock

 

 

 

 

 

 

 

 

99

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g...
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17

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e——>-<

<-

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w $ ‘

. \ W x
G-ﬁ—yQHG-ﬁ—v \ HGHQ-ﬁ—ryGé

-N -l l N
HgNa29

Model for the Force Field at the central portion
of the Screw-Lock "

+

O

 

 

 

 

100

.i..|..t.
T'r'r """""
TT'I’ 4""
(0)
-‘--‘-"' 1'1"!"
4-4-"' 1'1'1- (b)
4-“ 71"?
.L
‘- 1'
.l.
L T (c)
T
J- 1'
4- 1-
1.
Fig.No.3O

Interaction of two rows of edge dislocations

 

 

 

101

Fig. No. 3 I

Interaction of edge dislocation arrays
present in nearby slip planes.

 

(a)

(b)

 

102

Fig. No.32.

Stability of‘ interacting

 

 

 

 

dislocation loops

 
      
 

. --)-

Screw dipole

   
   
 

 

—+

 

 

103

 

 

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Fig. No. 3 4

 

104

 

 

 

 

 

 

 

 

Mechanism of Bond Formation due to lntersecting ‘

Slip

 

105

V. Relation Between Deformation Structure and Work Hardening
The debris mechanism of hardening prOposed by Gilman
suggests that the debris left behind by moving dislocations can impede
their motion. 15 As dislocations move through longer distances, more
and more debris is built up imposing greater resistance to dislocation
motion and causing hardening. As Observed in the present investi-
gation and by other investigators, non-conjugate slip which manifests
itself as latent hardening is by far more difficult tO take place than
the conjugate slip, and hence ionic crystals usually deform only in
one set of conjugate slip planes when deformed in uniaxial stressing
along the < 001> direction. According to the debris mechanism,
the debris should impede the motion of the dislocations irrespective
of their motion being in the conjugate or non-conjugate slip planes.
Thus it appears difficult to explain the latent hardening phenomenon
and the fact that conjugate slip is preferred with the debris mechanism.
Furthermore, according to Gilman's interpretation the debris
is presumably metastable, and as a result the material becomes
harder and harder during cyclic stressing. For a particular total
strain, irrespective of the type of fatigue procedure --- tension-
tension, tension-compression or compression-compression — the
resultant damage in the slip plane should be the same since debris
concentration should build up due to the motion of dislocations
regardless of the method Of testing. 64 In the present set of
experiments fatigue and latent hardenings were coupled, and contrary
to the predictions Of Gilman's mechanism it was Observed that hardening
in the latent direction depended on whether the fatigue cycle was completely

reversed or not.

 

 

106

All experimental Observations Of the debris so far have been
limited either to thin foils by using electron transmission microsc0py
or to surfaces of bulk crystals by using the etch-pit technique. 30’ 35' 52’ 86
It is uncertain that such Observed debris represent the defect structure
Of the deformed bulk materials.

The following is a prOposed model for work hardening based
on the band structure formed in the slip planes. Since the specimens
were deformed in the bulk form and the Observations were made in
specimens much thicker than those used for electron transmission
microsc0py, the band structure Observed here is a bulk phenomenon.
As the microsc0pe was not focused on the surface and what was
Observed was a two-dimensional projection of the three -dimensional
structure, it could not be a surface phenomenon. Besides, the
deformation studies have shown that this kind Of band formation
was absent in thin specimens where there was no appreciable

hardening either. Thus it seems that there is a close connection

between the hardening of the bulk crystals and the band formation.

(a) Single Slip System

As deformation proceeds, more and more randomly distributed
dislocation sources become active at various stress levels. Consider
a number Of sources Operating at a certain stress level. The
dislocation lOOps generated by these sources grow in size until
their growth is stopped by the interaction with loOps originated from
other sources. This will lead to the formation of screw-locks and

edge dipoles as discussed earlier. The initial phase Of the formation

 

 

107

of screw-locks should be completed with relative ease. But as more
dislocations are locked together, it becomes increasingly difficult
for additional pairs to enter the locks. The locks thus act as
barriers against which the dislocations pile up creating a back
stress, and making the dislocation sources inoperative at this
stress level. The material appears hardened. TO make these
sources active, the stress level has to be increased. New sources
in different, undeformed regions and previously inoperative will
become active. More screw—locks will be formed creating thus

the band structure discussed earlier. This process continues until
the active Slip planes are filled with the band structure. The stress
field associated with these bands resists the generation Of new
dislocations. This may be the major factor contributing to work
hardening in the case of single slip.

The resulting hardening does not disappear upon removal of
the applied stress because the screw-locks prevent the dislocations
in the locks from dislocking. The debris left behind by the dislocation
loops during their expansion will also prevent their contraction, but

this may not be a major contributing factor in hardening.

(b) Conjugate Slip Systems

It has been observed, as shown in figure 9, that in the case
of ionic crystals deformed under uniaxial stress along a < 100>
direction, deformation takes place Only in one Of the two possible
Sets Of conjugate slip planes having the same resolved shear stress.

Figure 35 illustrates the interaction of conjugate slip with the band

108

structure existing in a set of parallel Slip planes. It can be seen
that the band structure in these planes does not affect the motion of
dislocations in the conjugate planes to any great extent because the
bands are parallel to the intersecting slip planes. Conjugate slip
is easy because it can proceed through the gaps existing between
the bands in the intersecting planes without cutting through many
dislocations. If a band structure develops in one Of the planes in
the early stage of plastic deformation, this band structure does

not prevent its conjugate plane from becoming Operative and hence
acquiring a similar band structure. However, when the material
is considerably deformed, both the slip planes will have a tightly
packed band structure which should act as a barrier against further

conjugate slip.

(c) Latent Hardening -- Three or More Slip Systems

Latent hardening has been explained by Alden in terms of
individual dislocation interactions. 71 Since this is a bulk phenomenon,
it would be more apprOpriate to explain this phenomenon by using the
overall damage Observed in the slip planes of deformed bulk crystals.
When a crystal, deformed under uniaxial stress and having deformation
bands in a set of conjugate slip planes, is stressed again uniaxially
in a direction parallel to the slip planes having the band structure
there is no resolved shear stress acting in these originally activated
slip planes. New sets of slip planes, wherein the resolved shear
stress reaches the critical value, should become Operative. The

intersection of this non-conjugate slip with the existing slip bands is

 

 

 

 

109

difficult as illustrated in figure 36 because it has to cut thrOugh the
existing band structure in these slip planes. This necessitates a
high stress for further deformation manifested as latent hardening.
Figures 21 (c) and (d) Show the high stress concentration at the
intersections Of non-conjugate slip with the band structure and

explain why non-conjugate slip could not proceed easily.

(d) Fatigue Hardening

Materials which exhibit easy cross—slip characteristics such
as f. c. c. metals often show fatigue softening and fatigue failure,
whereas materials which do not cross slip easily show fatigue
hardening and no fatigue failure. 55 When cross-slip is possible
the screw components of the screw-lock may cross Slip, and
annihilate each other. This leaves a row of edge dislocations of
one Sign on one side and another row of opposite Sign on the other
side, a configuration on which Mott has based his discussion Of
deformation bands. 2 The screw -lOcks are probably absent in
theSe materials and hence would not contribute to any hardening.
In the materials such as LiF that do not cross slip easily, such
annihilation is unimportant. The screw—locks will acquire more
dislocation pairs as the cyclic stressing proceeds. New sources
become Operative, and new bands are formed as the stress level
is increased. The material will be hardened by the formation of
these bands.

The above mentioned results are in agreement with thOSe

predicated by Mott's mechanism Of fatigue failure based on cross-slip.

 

   

 

110

Materials which do not cross slip easily are not hardened
continuously in simple tension or compression. They have a
definite fracture strength. It has been Observed that the slip bands
in fatigue test and in simple tension or compression test differ
considerably. 53 Slip bands Obtained in fatigue test are wide,
whereas those Obtained in simple tension or compression are narrow
and intense. The dislocation density in the slip bands in the fatigue
specimens was much lower than the density of dislocations in the slip
bands in the tension or compression specimens. In the latter case
a tightly packed band structure in these slip planes develops and may

act as a barrier even for conjugate slip.

(e) Fatigue -Latent Hardening

The Observed characteristics of the growth of the slip bands
in the Specimens subjected to reversed or unreversed cyclic
stressing helps one to understand some specific features Of fatigue-
latent hardening. Specimens under reversed shear showed much
greater hardening in the latent direction than those under unreversed
shear. The slip band growth on { 100} surfaces caused by such
cyclic stressing is shown in figures 10 and 11. Under unreversed
shear very few new slip bands are formed in the second cycle. On
the other hand, under reversed shear, new slip bands are formed
in the second half cycle.

A dislocation loop will expand or contract, depending on the
type of applied stress and the direction of its Burgers vector. In

the case Of cyclic stressing under unreversed shear dislocation loops

111

having parallel Burgers vectors in parallel slip planes multiply.
Under reversed Shear, during the second half cycle, slip takes
place in different areas of the specimen where the dislocation
loops have their Burgers vectors antiparallel to those Of the
Operative dislocation loops in the first half cycle. Therefore, in
the latter case dislocation loops with parallel and antiparallel
Burgers vectors are present. Although the distribution of these
loops may not be uniform within any individual slip plane, the
overall distribution of both kinds of dislocation loops may be
considered uniform for the whole specimen.

The structure Obtained from such reversed cyclic stressing
may act as a strong barrier to a dislocation moving in the non—
conjugate Slip plane during deformation in the latent direction. The
whole specimen is interlaced with slip dislocations having parallel
and antiparallel Burgers vectors, and slip in the latent systems
becomes hard. On the other hand, the slip band distribution Obtained
from the unreversed cyclic stressing may not present difficulty Of
this kind. The dislocation loops have mainly parallel Burgers vectors.
The presence Of some unslipped regions allows to certain extent the
propagation of non-conjugate Slip during deformation in the latent

direction.

 

 

112

 

 

 

 

 

 

 

 

 

Fig No.35

Conjucate Slip Intersection

 

 

 

 

Fig. No. 3 6

(Non Conjugate Slip Intersection

 

 

 

 

C ONC LU SIONS

1. By using the Czochralski method and Off-centering and necking
techniques it was possible to grow at rates as high as 75 mm/hr
oriented LiF single crystals which were substantially free from
subboundaries with a dislocation density Of approximately 104/cm .
Pseudo Kossel patterns of the crystals taken with a divergent
X-ray beam generated from a micro focus unit revealed no sub-

boundaries with disorientations higher than one minute Of arc.

2. The results of fatigue experiments with cleaved LiF Specimens
showed that the yield strength in the latent direction increased by a
factor Of five compared with the initial yield strength after the
crystals had been deformed by repeatedly unidirectional loading

to a total strain of 16%, whereas a complete reversal Of the
resolved shear stresses acting on the initial slip systems during
each successive loading resulted in a thirty fold increase in the

yield strength in the latent direction for 20% total initial strain.

3. A band-like structure consisted of overlapping lOOpS aligned in

a direction perpendicular to the slip direction was observed with
polarized light in the slip planes Of specimens cut from the deformed
crystals. Similar bands were occasionally Observed to lie parallel

to the line of intersection Of slip planes also.

4. The experimental results also showed that the difference in the

work hardening behaviors Of the initial conjugate slip systems and

114

 

a.

115

the latent slip systems appeared tO have its origin in the relative
orientation of the bands with respect to the intersecting slip planes
whether the bands were parallel to the line of intersection Of the

slip planes or Oblique.

5. A theory Of formation Of the bands was deve10ped based on the
interaction of dislocation loOps in nearby parallel slip planes

showing such bands were in a low energy state and hence stable.

 

 

10.

11.

12.

13.

G.

. Friedel, 'Discussion on Work Hardening and Fatigue in Metals,

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