A THEORETICAL STUDY OF THE
INTERACTEGN 0F AN EQGE DISLOCATION
WITH A CGATED C§Y§TAL BOUNDARY

TIxasls Ior I'Iw Dogma o-I DIV. D.
MICHIGAN STATE UNIVERSITY

Gary Hamilton Conners
1962

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This is to certify that the

thesis entitled

A THEORETICAL STUDY OF THE
INTERACTION OF AN EDGE DISLOCATION
WITH A COATED CRYSTAL BOUNDARY

presented by

Gary H. Conners

has been accepted towards fulfillment
of the requirements for

Ph.D. degree inAEElied Mechanics

H . ‘: MINA

Major prof&9)r

 

 

Date (C >Jr0~v $113

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Michigan Stab
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ABSTRACT

A THEORETICAL STUDY OF THE INTERACTION OF AN EDGE
DISLOCATION WITH A COATED CRYSTAL BOUNDARY

by Gary Hamilton Conners

Numerous experimental results have indicated that a surface
layer can act as a barrier to the egress of edge dislocations from
the interior of a crystal. In the present investigation, a mathematical
model for this barrier effect is developed using complex variable
methods of continuum elasticity theory.

A solution of the elasticity problem is first found for the case
of an edge dislocation in a homogeneous half plane by employing an
image dislocation. An approximate solution for the case of a half
plane coated with a uniform layer of material having a different
Young's modulus is then obtained by modifying this image through the
introduction of a magnification factor.

This solution is employed to determine curves of force per unit
length of dislocation line as a function of distance from the interface
for various ratios of Young's moduli in surface layer and substrate
material. In all cases this force is attractive toward the interface
at a sufficiently great depth. For ratios of Young's moduli above a
certain value, however, the curves pass through an equilibrium
position at which the force vanishes and take on increasingrepulsive
values as the interface is approached. 7 The theory thus predicts the
existence of a barrier effect, and provides an expression for the shape

of the potential barrier.

A THEORETICAL STUDY OF THE INTERACTION OF AN EDGE
DISLOCATION WITH A COATED CRYSTAL BOUNDARY

BY

Gary Hamilton Conners

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics
and Mate rials Science

1962

ACKNOWLEDGMENTS

I wish to express my sincere thanks to my thesis adviser
Dr. T. Triffet for acquainting me with the problem and for his
excellent advice and encouragement during its investigation.
I am also grateful to the other members of my committee for
their interest.

Thanks are also due to my wife Gwenyth for her patience
and help in preparing the manuscript, to Mr. Charles Newman
for his valuable assistance with the numerical calculations, and
to Delco Appliance Division of General Motors for providing for
my support during the course of this work.

***************

ii

TA BLE OF CONTENTS

CHAPTER Page
1 INTRODUCTION ................... 1

l. 1 Preliminary Remarks ........... 1

l. 2 Experimental Background ......... 2

l. 3 Analysis of Experiments .......... 5

1.4 Objectives ................. 8

2 THEORETICAL BACKGROUND ........... 10

Z. 1 Complex Variable Formulation of Plane

Elasticity Theory. . . ........... 10

2. 2 Elastic Dislocation Theory; General
Concepts......... ......... l4

2. 3 Elastic Dislocation Theory; Complex
Methods ........... - ........ 18
3 BOUNDARY VALUE PROBLEMS .......... 21
3. 1 Homogeneous Half Plane .......... 21
3. 2 The Coated Plane Problem ......... 2.7

4 COATED PLANE SOLUTION AND APPLICATIONS. 33

4. 1 Solution ................... 33

4. 2 Applications ................. 37

5 CONCLUSION . . . ......... . ....... 47
BIBLIOGRAPHY ......................... 50
APPENDIX ........................... 53

iii

LIST OF TABLES

TABLE Page
I Experimental Results of Roscoe ............ 3

11 Experimental Results of Westwood ........... 4

III Crystal Properties ................... 41

iv

FIGURES

10.

11.

LIST OF FIGURES

Arc geometry used in deriving stress formulas. . . .
The formation of an edge dislocation. . . . ......

The geometry associated with an edge dislocation in
a homogeneous half plane ..... . . . . .......

The geometry associated with an edge dislocation in
acoatedhalfplane .....

Magnification factor N vs. dimensionless distance
frorq: surface for various ratios of Young's Modu-
_._l-

lus E0. .......... . ..............

Dimensionless equilibrium distance from surface
Ao/h as a function of the ratio El; . . . . . ......
Force F per unit length on an edge dislocation in. KCl
as a function of distance from interface for various
coatings. All coatings 50 R in thickness ......

Force F per unit length on an edge dislocation in LiF
as a function of distance 1 from interface for various
coatings. All coatings 50 X in thickness .......

Force F per unit length on an edge dislocation in LiF
as a function of distance 1 from interface, for
various thicknesses of MgO coating ..........

Force F per unit length on an edge dislocation in Al
as a function of distance from interface I with alumi-
num oxide coating. Coating is a umed 25 R in thick-
ness, and two possible ratios of EL are used . . .
o
:4:
Dimensionless stress 0" for an edge dislocation
. exx . X.
With Burgers vect’qr ByJ, as a function of for .

various values of— . . . . ........
By

' Page

12

15

22

28

39

40

43

44

45

46

54

LIST OF FIGURES - Continued

FIGURE

12.

13.

Page
a:
Dimensionless stress 0' yy for an edge dislocation
with Burgers vector B J, as a function of X— for
Y B
various values of—. . . . . . . . . . . .Y ..... 55
B
Y
Dimensionless stress 0’x for an edge dislocation
with Burgers vector Byj , as a function of %— for
various values of— ............. Y . . . . 56
BY

vi

a}:
“.7”,
4:

LIST OF SYMBOLS

Distance of dislocation from surface
Normal stress components

Shear stress

Displacement components

Complex stress functions

Complex stress functions, E = (p.537: )9'
Lamé elastic constants

Young's modulus

Poisson's ratio

Burgers vector

Burger 8 vector components

. Dislocation core radius

complex variable, . z 1‘ x + iy
Position variable on real axis

Position variable on substrate--surface coating interface
Substrate region

Surface coating region

Coating thickness

Di stance of dislocation from interface

Force per unit length on dislocation. line

*
, D1mens1onless stres ses

vii

CHAPTER 1

INTRODUCTION

1. 1 Preliminary Remarks

It is generally accepted that dislocations play a crucial role in
the deformation characteristics of crystalline solids. It follows,
therefore, that any factor which significantly affects the behavior of
dislocations may also be of importance in determining these character-
istics for a particular solid.

The property of dislocations which will be of interest here is the
stress field around them; only edge dislocations, which may be
treated as producing a state of plane strain, will be considered.

Since the stress components for an edge dislocation vary approximately
as 1/r for large r, where r is the distance from the dislocation line (1),
the stress field may be said to be a long range field (see Appendix for
stress field plots). This being the case, it is clear that interactions
with the surface may be of importance even-for dislocations well in the
interior of a crystal.

As a first step, therefore, it is reasonable to attempt to determine
the stress field around an edge dislocation as a function of its distance
from an idealized plane boundary, assuming in particular that the
elastic properties of the material do not change up to the boundary.
Such a model for a boundary is certainly a very unrealistic one, however,
for it is well-known that the properties at the surface of real materials
may be vastly different from those in the interior. Oxide layers,
adsorbed gas films, layers work hardened by machining, and various

other factors serve to modify the properties of surface regions.

A better model is, therefore, one for which a layer of material
with constant thickness and elastic constants different from those of
the bulk material is superimposed on the plane boundary of the simple
model. . Such a picture is still only an approximation, since the
properties of actual surface layers may vary continuously through
their thickness. The introduction of the inhomogeneity into the model
as represented by the idealized surface layer, however, greatly
increases the probability of predicting any interesting effects which real
surface layers may produce. Furthermore, for dislocations at some
distance from the boundary a complicated variation in elastic properties
may produce about the same overall effects as an equivalent stepwise

variation.

1. 2 Experimental Background

Before undertaking a theoretical study of the models mentioned in
the preceding section, it is important to note that ample experimental
evidence exists to indicate that surface conditions can, indeed, affect
the overall mechanical behavior of crystalline solids. The magnitude
of the observed effects is much larger than can be accounted for by the
methods of classical continuum mechanics as employed, for example, in
strength of materials. It therefore seems reasonable to hypothesize
that the effects are caused by the interaction of the surface with the
dislocation stress field; the analysis of the theoretical models is for this
reason a matter of definite practical importance.

The experimental investigations to be discussed below seem to be
of particular interest. In 1936, Roscoe reported that the presence of
an oxide film less than twenty atoms in thickness on cadmium increased
the critical shear stress by fifty per cent for a single crystal wire of
1/4 millimeter radius (2). Some of Roscoe's results are tabulated below.

The quantity So in this table is the critical shear stress.

Table 1. Experimental Results of Roscoe

i f
_ M-

 

Oxide Thickness So
0 50 gm/mmz
20 atoms 75
400 atoms 84
1200 atoms 122

 

The thickness of the oxide layer was estimated by comparing its color
with that of standard samples.

Roscoe found that his results were not dependent on the impurity
content of his samples over a wide range of purities, but did depend
somewhat on their crystallographic orientation. In addition he found
that even at large strains where the film was apparently completely
destroyed and incapable of supporting any stress itself, the stresses
remained at much higher levels than those observed with unoxidized
samples. This led him to the opinionthat "the crystal is affected by the
oxide to some considerable depth below its surface, " an opinion which
lends support to the assumption of a dislocation mechanism.

Following Roscoe's early work, numerous investigations of the
so-called Roscoe effect have been carried out. They constitute the first
class of experiments which are of interest here. The results of Andrade
and his co-workers in the period around 1950 provide some of the best
examples.

In 1951 Andrade and Henderson reported a marked increase in the
critical shear stress of silver single crystals heated in air, from a
value of about 25 gm/mmz at room temperature to about, 125 'gm/mmz
at 5000 C. (-3). No suchchange, Waslobs'erved with crystals heated in '
vacuum, indicating that the increase was probably caused by the

formation of an oxide layer. . The layer which produced this

marked effect, however, remained too thin to be visually observed.
Also of interest is the work reported by Westwood in 1960 on the
effect of surface condition on the properties of ionic crystals, in
particular lithium fluoride (4). Coatings were applied to compression
specimens with cross sectional areas of about 75 mm2 by vacuum
deposition of chromium and by chemical reaction with magnesium oxide
powder. The reacted coatings were found to have the most significant
effect on the mechanical properties; some of Westwood's results with

these coatings are given below, the data being for two different samples:

Table 11. Experimental Results of Westwood

 

m ‘

 

Coating Thickness So So
0 120 gm/mmz 156 gm/mmz
> 10 1.1 149 I ---
> 50 u --- 256

 

The reacted layer was believed to be an alloy layer of magnesium
in lithium fluoride. In addition to raising the critical shear stress, this
layer markedly affected the average yield stress, average stress at
fracture, average strain at fracture, and produced a "catastrophic stress
drop" in the case of thin (> 10 u)films when the critical shear stress
was reached.

Other important Roscoe effect experiments have been those of
Mukai in 1958 which demonstrated an increase in the tensile strength of
aluminum crystals which was proportional to the thickness of the oxide
coating up to a thickness of 700 X (5), of Menter and Hall in 1950 on the
behavior of cadmium crystals (6) supplementing the findings of Roscoe
and of Evans and Schwarzenberger in 1959 on slip processes in crystalline

substrates with various coatings (7).

A second class of experiments pertinent to this study deals with
the emission of acoustic pulses by deformed crystalline materials.
Acoustic emissions have been extensively studied in recent years by
Schofield (8), and by Tatro and his students at Michigan State University (9).

, Some acoustic emissions, in particular those associated with twin-
ning deformations as in the case of the well-known "tin cry" phenomenon,
have well-established origins (8). Of greater interest are the much
lower energy pulses observed during the deformation of single crystals
and polycrystalline samples of various metals, including aluminum and
zinc, which can not be traced to known sources.

Typical results of Schofield for an aluminum single crystal in
tension show some emissions beginning at very low stresses, with the
maximum emission rate of about 4, 000 counts per second being attained
slightly below the yield value. These emissions are of very small
amplitude, requiring amplification of 106 to 107, times for detection and
study (8).

Very recent experiments by Tatro and Liptai indicate that surface
treatments, such as sand blasting or polishing, can alter markedly the.
emission characteristics of an aluminum specimen (10). These results
demonstrate a strong dependence of acoustic emission behavior on

surface propertie s .

l. 3 Analysis of Experiments

While very little has been done in a quantitative way to explain the
phenomena described in the preceding section, a qualitative explanation
has been developed which seems adequate to explain at least the principal
features of the Roscoe effect. This theory has been extended here to
provide a possible explanation of acoustic emission effects as well.

It must be remembered, however, that coupled with the effects on critical

stress observed in the Roscoe effect are alterations in many of the other

significant mechanical properties. The complete picture for any one
material is very complex, and it is unlikely that any one mechanism
could be prOposed which would account for all the effects in every type
of material. Bearing this in mind, we can proceed to the discussion
of the process which has been suggested as the dominant mechanism
in the Roscoe effect.

As stated by Westwood, the central idea in this theory is that
"a surface coating can act as a stable barrier to dislocations, causing
pile-up, coalescence, and the formation of crack nuclei beneath the
film" (4). This hypothesis was verified by his experiments with lithium
fluoride, and he concluded on the basis of metallographic evidence that
"surface coatings can act as barriers to the egress of edge dislocations"
(4).

The suggested explanation of a surface layer's ability to function
as a dislocation barrier involves consideration of the elastic strain
energy stored in the crystal system. For the purposes of this discussion,
a system composed of a single dislocation located a distance I. from the
plane boundary of a semi-infinite, homogenous, isotropic medium coated
with a layer of material with different elastic constants will be con-
sidered. The slip plane in which the dislocation line is located will be
allowed to have an arbitrary orientation with respect to the boundary.

Since the stress field in such a system will depend upon the distance
X, clearly the elastic strain energy stored in the dislocation field, which

we will call V will also be a function of x. Then if a value in exists

D!
such that
V
X h=Xo

and if this extremum corresponds to a minimum in the strain energy,
10 will correspond to an equilibrium position for the dislocation, i. e. ,

a dislocation at 10 will lie in a potential well.

. The functional dependence of VD on x will obviously be related to
the elastic constants of the substrate and surface. Hence the possibility
of the existence of a potential well will also depend upon these constants.
Head investigated this possibility for a screw dislocation and found that
for constants such as those for aluminum with an aluminum oxide layer,
a potential minimum did exist for x approximately equal to h, where h
is the thickness of the layer (11). He also suggested that on the basis of
physical considerations such a minimum could be expected for an edge
dislocation.

In order to complete the picture, the effect of applying an external
stress field to the system must be considered. In such a case, the total

elastic strain energy will have the form

V=VD+VE+VI

where VB is the contribution from the external field, and VI is the energy
of interaction of the dislocation and applied fields. The position of the
energy minimum, if it exists, will then be a function of the applied
stresses as well as the other variables previously mentioned; for
example, one might expect that by increasing the applied stress, the
equilibrium position for a dislocation trapped near a boundary would be
forced closer to the boundary.

When the portion of the strain energy stored in the surface region
reaches some critical value VC’ it might be expected that the nature of
the boundary conditions at the interface between the substrate and the
surface layer would be altered. Implicit in the preceding discussion
has been the assumption that the layer was bonded to the substrate, 1. e. ,
attached so that the traction and displacements were continuous across

the boundary. (Traction is defined as the normal and shear stresses

acting on a surface.) Clearly two types of post-critical behavior can

 

. u

E 4%“

occur in which both layer and substrate deformations remain elastic;
either the layer may completely separate from the substrate over an
area near the dislocation, thus ceasing to play any role in the stress
picture, or it may begin to slide on the substrate, in which case only
the displacement component normal to the interface must be continuous.

Either of these types of transitions will cause an immediate
reduction in the strain energy, and most probably an altered dependence
of V on x in which no minimum may exist. In order for energy to be
conserved, then, the stored strain energy must be converted to other
energy forms.

Part of this energy will be dissipated by frictional forces acting
on the dislocation as it moves out to the surface. Another portion may
go into surface energy necessary to create new surface areas. The
remainder of the energy could be converted into an elastic wave which
will be propagated through the crystal; this wave may be detected as
an acoustic emission.

This completes the qualitative model for the observed surface
effects. To summarize, the presence of a surface layer causes a dis-
location to be trapped in a potential well below the surface, thus
inhibiting macroscopic slip. When the applied stresses reach some
critical value the strain energy is reduced due to a change in conditions
at the boundary, and the dislocation is released from the barrier. The
elastic strain energy stored in the dislocation field is partially con-
verted into an elastic wave which propagates through the crystal, and

is detected as an acoustic emission.

1. 4 Objectives

The initial difficulty which must be overcome in attempting to

extend the qualitative picture discussed in the preceding section to a

quantitative model which can be compared with experimental results
is that of determining the stress field around a dislocation near a
boundary, with various boundary conditions. Once this problem has
been solved, the behavior of the dislocation can be determined for
various appropriate values of the parameters involved.

The primary goal of this investigation is to solve the stress
boundary value problems for the case of a free boundary and a boundary
with a bonded layer. These problems will be formulated for a dislocation
with arbitrary orientation, but specific solutions will be sought only in
the case of the Burgers vector normal to the boundary. The complex
variable methods developed for plane, small strain elasticity problems
by Muskhelishvili and the Russian school of elasticians will be em-
ployed (12).

The results will then be used to obtain some quantitative estimates
of the behavior of dislocations for certain particular cases; conditions
under which a barrier effect can exist and the nature of the barrier will
be investigated. Clearly this program constitutes only a first step
toward a complete quantitative picture of such a complex phenomenon
as acoustic emission. It is hoped that the results will lead to some
understanding of the barrier effect, however, and will provide a foun-

dation for further theoretical and experimental investigations.

CHAPTER 2
THEORETICAL BACKGROU ND

2. 1 Complex Variable Formulation of Plane
Elasticity Theory
It is well-known that in plane elasticity problems, the stress
components 0’ , a’ and a’ can be obtained from a single function,
xx YY xy
the Airy stress function (13). If the stress function is represented by

U, the formulas for the stresses are:

2122-- : fi- 0- : gall—— (2.1)

0’“ gyz ' JYY axz ' xv -axay

The function U must be a biharmonic function, that is, it must be a

solution of the biharmonic equation

4 4 4
V‘U = LL;- + Z izU—fl + i2;- = 0 (2.2)
ex 33‘ 3y 9V

in order that the stresses will satisfy the conditions of compatibility.
It has been shown by Muskhelishvili that every biharmonic

function U(x, y) may be represented by two functions of the complex

variable 2‘. = x + iy (12). Denoting these functions by ¢(z) and X(z),

this representation is
2U = Z ¢(z) + z ¢(z) + X(z) + X(z) (2.3)

where the bar notation (e. g. , 3) indicates complex conjugation.
Muskhelishvili further demonstrated that both of the functions ¢(z)
and X(z) must be holomorphic, that is, single-valued and analytic, in

any simply connected region.
9 U
DY

and

 

By differentiation the following expressions for

are found:

10

ll

 

 

 

2 g: = ¢<z> + chz) + ¢(z) + we) + X'(z) + X'(z)
3U _ __ (2.4)

2 -—3Y- = i[-¢(z)+ z ¢'(z)+ ¢(z) - z¢'(z) + X'(z) - X'(z)]
These results can then be combined to give a simple expression for
£2 . i .231 .
9x 9y '

3:: + 113—27] = ¢(z.)+ z¢'(z) + wz) (2.5)
where )Wz) = if

Formulas for the stresses in terms of the holomorphic functions
Hz) and ¢'(z) can be obtained by considering the force acting on an
element of arc ds in the x, y ,plane with unit normal vector 3 and unit
tangent vector f, as shown in Figure 1.

If the force acting on ds has components (finds, U;nds) these will

be related to the stresses by:

 

0'“ = DY DXDY
0- = 0" cos(n,x) + 0' cos(n, y) = - QZU cos(n, x) +g-ZJZL -
Yn vx YY 9x av 9x
c08(n. Y)

Note that a’xy = a’yx because of the symmetry of the stress tensor.

Now cos (n,x) = dy/ds and cos (n, y) = -dx/ds.

 

Hence,
In complex form,
(o'xn + i o'yn)ds = - id (gli- + i -%_:'l) (2.7)

or, by means of equation (2. 5),

(a'xn + i a’yn)ds = - id [¢(z) + z ¢'(z) + Hz” (2.8)

 

z Z
”L. Cos(n,x) + 0;;y CoS(n.Y) = i‘tzj' cos(n,x) " 2—9— cos (n, y)

(2.6)

_ :a
I

1“"

 

 

 

12

3.:

 

 

Figure l. Arc geometry used in deriving stress formulas.

 

 

 

l3

Letting n be in the direction of the x- axis,

0’ +1 a“ = ¢1<z>+ ¢'(‘z')' -‘z¢"‘<z")' - x???) (2.9)
xx XY \

Similarly, by letting n be in the direction of the y' axis,

0’yy ' 5' 03,3, = ¢'(z) + <I>'(2) + ’z¢"(2) + )Nz) (2.10)

The most useful representations for the stresses are obtained by com-

bining the final two results. It is also convenient to adopt the notation

@(z)=¢'(z) and Y(z)= )Nz). (2.11)

The following formulas are then obtained:

axx + 0-yy= 2[§(z)+§(z) ]=4Re§(z)
(2.12)

O’yy - rxx+ 2 i a"):y = 2 [Z §'(z) +1212) ]

Plane elasticity problems involving only stresses thus reduce to
finding complex function O (z) and Y(z) which will give stresses
satisfying particular boundary conditions. . These functions, therefore,
are analogous to the Airy stress function in the more familiar formu-
lation of elasticity theory. . If displacements are also of interest, as,
for example, in cases where displacement boundary conditions are
given, the functions ¢(z) and ”(2) are also needed. The formula
derived by Muskhelishvili for the displacement components (u, v) has

the form

 

2 p (u +iv) = K¢(z) - z ¢'(z) - Hz) (2.13)

where p. and K are elastic constants for a homogeneous, isotropic

substanc e .

14

The constant (.1 has its usual significance as one of the Lamé
constants. The constant K is related to the Lame constants by
+ 3 p
K: A (2. 14)
A + H
in the case of plane strain. The Lamé constants >0 and P may be
obtained from the more common elastic constants E, Young's modulus,

and v Poisson's ratio: , by the following formulas:

 

' _ E v
A ‘ (1+ v)(1- 2v) (2.15)
“L z z (1E+ v) (2.16)

Having established the basic equations of the complex formulation
of elasticity theory, we can proceed to investigate methods by which
this formulation can be applied to dislocation problems. It will be
necessary first, however, to discuss some of the fundamental ideas of
the theory of dislocations which will be of importance in the subsequent

development.

2. 2 Elastic Dislocation Theory; General Concepts

Edge dislocations of the type to be considered here may be
pictured as being produced by cutting a thick-walled cylinder as shown
in Figure 2 along the plane AB, displacing the upper cut surface a
distance Bx in the negative x direction while leaving the lower surface
fixed, and then rejoining the cut surfaces (1). . Clearly the displacements
in the cylinder will not be single valued after this process is completed,
since u slightly above the plane AB will differ from u slightly below
this plane by an amount Bx.

. Non- single-valued displacements of this type constitute one of the

types of deformations first studied in detail by Volterra and termed

15

 

 

4.53.903me ompo no mo aoflmEHOH 2:.

.N 0.“;th

 

f'\
g

 

16

dislocations by Love (14); expressions for the stresses obtained by
Volterra and others, with the assumption that the material is homo-

geneous and isotropic, are, as given by Cottrell (l):

_ KB
0'... - W (-3... - fl
0' = K (sz - Y’) (Z 17)
W (x+ ) '
a' = K (x3 - xy‘)

xy (x + )
The constant K in these formulas is given by

K: (2.18)

217 (1-v) °

It is apparent from the discussion of the Volterra dislocation that
such a dislocation could not occur in a simply connected medium.
Furthermore, since the stress components as given by equations (2.17)
approach infinity near the origin of coordinates, these equations could
not be valid if there were material present in this region. Volterra
dislocation theory can, therefore, only be applied to dislocations in
crystals if it is assumed that a "core" of crystalline material has been
removed along the longitudinal axis of the dislocation. The radius of
this core must be sufficiently large so that the stresses and strains on
its boundary will not exceed the range of validity of linear elasticity
theory.

Physically the core represents the region where the concept of
stress in an elastic continuum ceases to be valid, and must be replaced
by detailed considerations of interparticle forces. The core radius,

I’o, may be taken to be less than 10 Bx- . Since core structure will not be
taken into account in the applications discussed here, this distance
places a lower bound on the range of applicability of the results to be

obtained .

17

In addition, a considerable simplification in the application of
dislocation theory to crystals is obtained by assuming that the crystalline
material is isotropic. This assumption, which will be made here, is
not a necessary one in principle, however, and some work in dislocation
theory has been carried out without making it. Dislocation stress fields
in anisotropic media have been studied by several investigators, notably
Burgers (15) and Eshelby (16). . No fundamentally different results would
be expected for the problems to be considered in this study if the added
complication of anisotropy were to be taken into account.

As a final consideration in the application of continuum dislocation
theory to crystals, it is convenient to be able to represent the discontinuity
in displacement by a vector making an arbitrary angle with the x axis,
rather than restricting it to the x direction. If this vector is represented
by Ii, it may be written in terms of its components as:

A “ A
b = Bxi + Byj
Since the region of interest is that where linear elasticity theory applies,
limay be regarded as the result of superimposing two separate dis-
locations, one produced by the cutting, displacing and rejoining process
discussed at the beginning of this section and the other by repeating this
process with displacements in the y direction.

The vector E is referred to in crystal dislocation theory as the
Burgers vector (1). It lies in the slip plane, and points in the direction
of slip. Since the Burgers vector may have any direction in the (x, y)
plane, the slip direction may take on any value in this range. The
direction and magnitude of the Burgers vector will be determined by the
crystal structure of a particular material, the slip planes and directions
being in general planes and directions of highest atomic density while

the magnitude will be equal to the lattice spacing in the slip direction.

18

Expressions for the stress components of a dislocation having both

x and y components are as follows:

K
O’xxi W [BXI-3XZY ‘ Y3) + By (43 + XYZII

K .
O’yy = (1:24.33)! [Bxisz ' Y3) + BY (-x3 - 3xy2)] (2.19)
0.x)! = (x +31f) . [Bx(x3 " XVI) + BY (‘XZY + Y3”

It should also be noted that the core radius must now depend on Ibl rather
on Bx only. The value ,0 = 10 [El will be assumed here, unless other—

wise indicated.

2. 3 - Elastic Dislocation Theory; Complex Methods

In applications of complex elasticity theory to regions which are
not simply connected, as in the case of a half plane with a dislocation
core removed, the functions Q (2) and Y(z) may be such that they have
singularities in the core region. Further, it is found by Muskhelishvili
that if these functions are to correspond to displacements which are not
single valued and stresses which are single valued, they must have the

form

toe): ““0 = Yam: “yd (2.20)

 

 

where a, b, c and d are real constants. . By using the stress formulas
(2.11) and (2. 12) the following expressions for the stress components

are readily obtained:

1
axx = W [(3a - c) x3 + (5b - d)x2y + (-a-c)xyz + (b-dIY3] (2-31)

Vyy = W [(a+C)x3 + (-b+d)xzy + (5a + c)xyz + (3b + d) 7"] (2.22)

19

on = m [<-b+d)x3 + (3a + c)x’y + (3b +8)va + (-a-C>Y’l (2- 23)

These expressions will agree with those of Cottrell for an edge dislocation

located at x = o, y = o if the constants are assigned the following values:

 

 

a=-§—z§1- b:-KB5‘
(2.24)
c:-§—2B—¥— dz+KB2x

Thus the complex stress functions which describe the stress field of an

edge dislocation at the origin of coordinates are as follows:

K B +'B
(2.25)
K B -v'B
You): _ 2' . _IE_1_J$_

It is evident from equations (2. 19) that the stresses given by these
functions vanish at infinity; they are, therefore, satisfactory for
describing the stress field of a dislocation in an infinite medium with
no traction on the boundaries. In the cases of present interest where
semi-infinite media will be dealt with, the functions §(z) and Y(z) must
be modified so that the stresses calculated-from them will satisfy boundary
conditions on the boundaries which are not at infinity.

This will be attempted here by assuming that
*
a.) = <1 .(z) + Q (z)

YIz) = 31(2) + 39%)

an: *
where § (z) and Y (z) are functions holomorphic in the region of
7 interest. These functions correspond to a stress field superimposed on

that of the dislocation but since they are holomorphic do not represent

20

any new dislocation in the material. The problems to be studied here
. *
involve determining what form q (z) and Y (z) must have in order

to satisfy particular boundary conditions.

CHAPTER 3
BOUNDARY VALUE PROBLEMS

3. l Homogeneous Half Plane

The first problem to be considered is that of a dislocation in a
half plane for which the elastic constants do not vary up to the boundary.
. Figure 3 illustrates the geometry of the problem. An edge dislocation
with Burgers vector ‘1: is located in the point -i)\ in the complex plane.
It is assumed that the boundary y = o is traction free, i. e. , has no
normal or shear stress acting on it.

Since the dislocation is not located at the origin in this problem,
the stress functions Q 0(2) and X(z) must be changed so that they

have singularities at z = -i). . They become '

K (B +'B
2 2+1).
(3.1)

.5? K B - in
= - —-— g #—
0(2) 2 '2 +1).

* *
We must then seek functions Q (z) and Y (z) having no singularities
in the lower half plane, and such that when the stresses are calculated

using the combined stress functions

Q (2) = Q .(z) + 1 *(z)

and

Ym = You) + X’(z)
they will satisfy the stress boundary conditions

21

22

 

/ Free boundary
0 ‘I

 

' Slip plane Y

 

 

 

Figure 3. . The geometry associated with an edge dislocation in a
homogeneous half plane.

23

at y=0. (3.2)

A reasonable choice to investigate for the functions desired is

* _ K (1+ 16
351(71- ? ' 77?.— ,4
* K + '6 (3.3) 1
1171(2) = _. l__1_.

2 ’z - ix. I
These functions represent the stress field of an arbitrary i'image"
dislocation located at a point conjugate to the location of the real dis-
location.

Then evaluating the stresses at y = o and substituting in the

boundary conditions, one obtains

_ K 3
[O'yyly:o - Wu-ZBy-t cl“ 7 )x +

(213x+ (3-6)xe + (-6 BY +50. + 7)).7‘x

+ (.213x -3(3 - 6)).3] = o ' (3.4)

_ K 3
[Wee W “23x - W)“

(-23 - 3o. +v)>..x" +(-213x +35 + 5) xix

v
+ (sz + u + em] = o - (3.5)

Equations (3.4) and (3. 5) will be satisfied for all values of x if
and only if the coefficients of all powers of x in the terms in square
brackets vanish identically. This condition provides a set of eight linear
algebraic equations for the undetermined constants a, [3, 'y, and 6 as

follows:

24

a + y = 2 BY -30. + 'y = 2 By (3.6)
5n + ‘y = 6 By 0. + 'y - -2 By
- : .. .. '6 = ..
s a 2 13x (3 + z Bx (3.7)
—3(3— 5:23X 3p+5=sz

It is readily seen that no non-trivial solution exists for this set

of equations. A useful approximate solution, however, is obtained by

setting
0. = By 7 = By
(3.8)
fl = -Bx 6 = Bx

This choice of values signifies in physical terms that the Burgers
vector of the image dislocation be taken as the mirror image of the
Burgers vector of the real dislocation under reflection in the real axis.
Such a selection for the image is equivalent to an approximate solution
arrived at by Koehler (l7) and others by different methods. Its effect
is to satisfy the boundary conditions on the normal stress but to double
the shear stress on the boundary.

This failure of the image solution to satisfy all of the boundary
conditions on the problem can be remedied by superimposing an additional
solution on it. One would expect this to be the case on physical grounds,
since a dislocation of the type considered here can certainly exist in an
elastic material, indicating that a solution to the elasticity problem

exists.

25

It is shown by Muskelishvili (12) that in a half plane problem for
a homogeneous medium only one of the usual stress functions may be
considered as independent. If §:‘(z) and Yin) are taken as the stress
functions for the additional superposition field, then §:(z) may be
regarded as the independent function. Once §:(z) is found, V:(z)
can be determined from it by means of known formulas (12).

Letting t be the position variable along the real axis, the

a):
boundary conditions for the problem of determining O; (z) are

(3.9)

YYz

where the superscript 0 indicates the contribution from the dislocation
field, while the * terms with subscript 2 arise from the additional
superimposed stress field.

It can be shown that the solutions to this problem is given by the

Cauchy integral

 

>1: (1) °
_ 1 20' (t)
{2(2) _ 2“ Hi) ———JL—t ’_‘ 2 dt (3.10)

Since a’xyoh) is a real function and decreases as l/t for large t,

26

a):
T2 (t) as given by (3. 10) must necessarily be a holomorphic function.

The existence of a solution to the problem is therefore proven.
The integral in (3. 10) may readily be evaluated by the usual
method of closing the contour of integration with an infinite semicircle
in the lower half plane and calculating the residues at the poles inside
this contour. As may be seen from equation»(3. 5), there are poles
at -i A and at z, which may range over the lower half plane.
In calculating ind) ) for example, one has a pole of order three at -1.)

so that

* . _ (t+i>t )3
92"“) )" '2§['d't';' { (1+ 2)(1 +0.)

(3.11)

4(th3 - BY >, t2 - 13x )\2t + 13y )( 3)} ]t=_i

The shear stress evaluated at -i 2 is an important factor in
calculating the force on the real dislocation, as will be seen later.
The contribution to this from the stress function (3. 11) is found negligible
by Keehl'er; thus for purposes of calculating this force, it is sufficient

to consider only the stresses due to the image dislocation.

27

3. 2 The Coated Plane Problem

It is now of interest to consider the problem of the preceding
section when a layer of material of thickness h adjacent to the real
axis has elastic constants different from those in the remainder of
the lower half plane. These two regions will be referred was the
surface layer and the substrate; the geometry is shown in Figure 4.

For convenience the notation R0 will be used to represent
the substrate region, and R, will denote the surfacelayer which is
assumed to be bonded to R0. In addition, 3 will be employed as
the position variable on the interface, i. e. , on the common boundary
of R0 and R1. The variable t will be the position variable on the
real axis, as in the preceding section.

The conditions to be satisfied in this problem are that traction and

displacements must be continuous across the interface, and traction must

28

Free boundary

z“ .x

 

7 I

Surfac layer

 

y—q44-

R

Interfac e

 

Substrate

 

 

Figure 4. The geometry associated with an edge dislocation in a coated
half plane.

29

vanish on the free boundary. These conditions must be stated in terms
of the stress functions in the region R0 and R1. Before this can be
accomplished, some additional notation is required. Stress functions
for R0 will be indicated by the subscript 0 written on the left, while
those in R1 will have the subscript 1 written in the same way. Thus DO
and DY are the stress functions in R0, and IQ and yare the stress
functions for R1.

By means of equation (2. 10), the condition for continuity of the

traction across the interface can be written in the form:

01> (s) + J; (a) + s 015's) + owe) =
11> (s)+.l$ (.)+s, '(s)+.§’(s)

(3.12)

Similarly, by means of (2. 13) the condition for continuity of the displace-

ments can be written as

 

1 ‘
"Z‘TLO— I K0 049(9) - SOTIS) '- OHSI. l =
(3.13)

3—i— IK1 I‘NS) - 81§(S) - 1H8) I

where the subscripts on the constants indicate the region in which they are
appropriate.
A more useful equation is obtained by differentiation of (3. 13),

which yields

1 [no clue) - 0iii?) - soy—((3 - 0—337.) 1 z

‘7‘ “° (3.14)

711-);— [ K1 @(8) - J05) - 815'“ - 1117(3) 1'

 

Equations (3. 12) and (3.14) can then be combined to obtain

30

I QC?) = “0 + ”1 Kg 0§(s) +

 

 

 

 

 

 

 

 

 

Ho (1 '1’ K4)
(3.15)
E- " I‘- m t ’—
11:” Em [05(6) + s o (S) + oY(SII
It is now convenient to let
K = EO+ [£le
1 Ho (1 + K1)
and
___ |-}_o " E1
K‘ m1 + (t...)
Then one has
1 @(s) = K, 0Q05) + K, [05(5) + s 0963(9) + 037(3) 1‘ " (3.16)
Equation-(3. 16) can then be combined with ,3. 12) to determine an
expression for 1Y(s). It is found to be:
1? (s) = ms) + (1 - K.) [.ms) + @(s) + some.) 1
(3.17)

-Kz [2R8 o§(s) + 037(5) + So§'(3) +- 5.3: { 0?“) "I“

Eofi'isl + omsl I

Before continuing the analysis, it is useful to express the constants
K1 and K2 in terms of familiar quantities, by means of equations (2. 14)
through (2. 16). These constants have a particularly simple form in the
case when Poisson's ratio v has the same value in both R0 and R1. In this
case, which will be closely approximated in most physical circumstances,

ED+ E1(3 - 41’)

K1 2 4E0 (1 - v)

 

(3. 18)

E0 - E1
4Eo (1 - v)

 

K2:

31

where E0 and E1 are Young's moduli in R0 and R1 respectively. For

convenience in making numerical calculations we will assume v = — °

 

 

 

4 ,
then
_ 1 §3_1_
K1 — -3- (1 + 2 E0)
K .1.” 5.11.) (3.19)
z ‘ 3 ' E0
. E1
Letting ET- = l + 3 e, we have
0
K1 = 1 “I" 26
(3.20)
K2 = " 6
We may then write from equations (3. 16) and (3. 17)
15(8) = OTIS) + s F(s)
(3.21)
{37(8) = 3(8) + e G(s)
where F(s) and G(s) are given by
its) = 2.15s) - 0%) - sows) - 02%)
(3.22)

G(s)= -2Re{o§(s) - 0 (5)}‘1’31m { so§'(s) }
C1 _..._..
+ 83—; i (@(S) + so§'(8) + 0Y(S) }

 

Equations (3. 21) express the continuity conditions in terms of the para-
meter 6, which is a measure of the discontinuity in the material

properties. Clearly in the case 5 = 0, that is, when the surface layer
has-the same elastic constants as the substrate, these conditions reduce

simply to

32

19(5) 0 Q“)

II

(3.23)

1Y(S) 19.218)

The problem which must now be solved is to determine 1 @(z)
and 1W2) such that the condition that the free boundary must be
traction free will be satisfied, in addition to satisfying the continuity
conditions (3. 21). By means of equation (2. 10), the condition at the

free boundary may be expressed in the form

 

 

,§(.)+,§(t) +t1Q'It) + film = o (3.24)

Equations (3. 21) and (3. 24), along with the usual conditions that
all functions vanish as Izl approaches we , constitute the complete
statement of the boundary conditions on the problem. One would expect,
however, that although a solution probably exists which will satisfy
these conditions completely, such a solution would probably be too
unwieldy for practical applications. The proc edure which will be
followed in solving the coated plane problem will, therefore, be to
weaken the boundary conditions by retaining only their real parts when
the complete conditions lead to excessive complications. By analogy
with the preceding problem, such a procedure may be expected to provide
a good approximation for the stresses in) the neighborhood of the dis-

location.

CHAPTER 4
COATED PLANE SOLUTION AND APPLICATIONS

4 . 1 Solution

We begin by rewriting the first of equations (3. 21) in the form

@(s) = K. @(s) - eF.(s) (4.1)

where

F1(S)=--F(SI + Zo§(8) (4.2)

The conditions at the interface may then be expressed in their weakened
form as
1§(3) = K1 oQiS) - 6131(9))
(4. 3)
19:18) = 337(8) + 6 PzIS)

where
131(8) = Re F1(s)
(4.4)
132(5) = Re 0(8)

Equations (4. 3) will be satisfied if in the region R; the functions

1E“) and M2) are as follows:

1§(z) = K; o§(z) - 6 P1(z)
(4.5)

inz) = m2) ‘1' é 132(2)

33

34

In these equations, P1(z) and Pz(z) are functions whose real parts have
the boundary values p1(s) and pz(s) respectively on the interface.

These functions may readily be constructed by means of a power series
expansion; such a construction will clearly be a useful one here since h,
the thickness of the surface layer, is almost always very small in cases
of physical interest, so that only the first few terms in the expansion
will need to be retained.

Writing s explicitly as (x-ih), the series representations are as

follows:
P1(x, y) = p1(x-ih) + i(h+y). EELS—xiii)-
(4.6)
(My)2 - 92p (x-ih) . (h+y)3 33B (x-ih)
- Z . 31 x - 1 O . 91); +____
and
1320‘. Y) = pz(x-ih) + i(h+y). L_P:;:1h.)
(4. 7)

 

1111;); , eZpr-ih) _ i(thyF' ,e’2;(x-ih) +

3x2 9x

It can be shown by applying the Cauchy-Riemann conditions that functions
obtained in this manner are analytic.

An important goal has now been reached. The functions 1§(z) and
3(2) can be completely determined by means of (4. 5), (4. 6), and (4. 7)
in region R1 if o§(z) and OISE-(z) are prescribed in R0; that is, for a
particular stress distribution in the substrate, the distribution in the
surface layer can be calculated. The problem is then to find o§(z) and
0Y(z) such that the stress on the free boundary will satisfy condition
(3. 24), or a weakened form of this condition.

As in the case of the homogeneous half plane, one has

.§(z) = 0970(2) + oQM)

317(2) = .Y.(z) + 09”“(2)

(4 . 8)

35

where o§o(z) and Mb) are the stress functions representing the dis-
location stress field as given by equations (3. 1), and 0§*(z) and olfiz)
represent the superposition field. Also as in that problem, the super-
position field will be obtained by an image method. The classical
method of attacking a problem of this type involving two plane
boundaries is to use an infinite series of images, the location of each
being determined by analogy with the corresponding optical problem.
We will proceed in a somewhat different manner, however, and seek a

solution by means of a single image described by the following functions:

K NB - 1 MB
of“) z _ , 3! AL

2 'z -1).
(4.9)

 

15. NBv+iMBx
2 'z-i).

*
oYIZ) =

These functions represent a mirror image dislocation with the x and y
components of the Burgers vector multiplied by magnification factors

M and N respectively. The solution of the problem will then consist of
expressions for M and N as functions of x and 6, these quantities being
determined in such a way that the conditions at the free boundary are
satisfied.

Before attempting to find a specific solution, certain additional
simplifications will be made. First, the dislocation will be taken as
oriented so that its Burgers vector is normal to the boundary, so that
Bx: 0. This means that only N( 1 , 6) must be found. Second, the surface
layer will be assumed to be thin enough so that it will be sufficient to
retain only the first term of each of the series (4. 6) and (4. 7). . Finally,
we will weaken the boundary condition (3. 24) as in the preceding problem
by discarding its imaginary part; in physical terms, this means that the
condition on the shear stress will be ignored. The first two of these

simplifications are made solely for the purpose of reducing the algebraic

36

difficulties involved in finding a solution. The final one is of a different
character; as in the former problem, it is highly improbable that a practical
solution could be found without making it.

On substitution of the stress functions into the boundary condition

equation (3. 24), the following algebraic expression is obtained:

JEL- [2x2(-1+N) + 6 12 (-1+N)] - (x2 + 1.2)(1 - N) =

Kyl
4 h
(«Mia-61,3) (1+ 3-3347) + (4.10)

41h
z z z
N(x- X +6Xz)(l m)

Here kl=x-h and 1,: X+h.

To obtain a reasonable form for the solution, we will only attempt
to satisfy (4. 10) in the region lx|<< ). , that is, near the intersection of
the slip plane with the boundary. This is clearly the region of greatest

importance. The explicit solution obtained in this range is

 

 

 

K3Kl1 " 16 3)} ' 2
N : 1 - (4.11)
_2&_.p4h 2
K1 - 1 x
This may be rewritten as:
3+u 16h)2
N ___ I 5 (4.12)

3+(1+8%12€

As a check on this solution, one may note that in the case 6 = 0
it yields N = 1. Thus the previously obtained approximate solution to the
homogeneous half plane problem is included in the present solution as a
special case. Of particular interest is the fact that in this case the solu-
tion is exact in the sense that it satisfies the weakened boundary con-

ditions for all values of x, and not only for le<< 1.

37

It is readily apparent that in general (4. 10) would be satisfied
for all x only if N were a function of x as well as the other parameters.
Since such a form for N is not allowable in our model, we are restricted
to an approximate solution which can be made to fit in the region of
greatest interest. . This limitation is a result of using a- single image
rather than a series of images; the reduction in algebra and the useful
form of the result obtained by the present method, however, more than
made up for its approximate nature. The factor N may be regarded as
representing the perturbing effect on the homogeneous solution of the
introduction of the inhomogeneity. It is not difficult to see then Lwhy the
present result should be exact for all x in the case €=?0. In general the
smaller the value of e, the larger the range of x values over which this
solution will closely approach the exact solution.

As a final comment on the result, it should be noted that when the
dislocation core approaches the interface between the surface layer and
the substrate, interactions may occur which will depend primarily on
the microscopic structure of the two media. These interactions clearly
can not be predicted by the present continuum theory. Thus the validity
of the approach employed here becomes doubtful in the region
1 -h < 10 (E) .

In the following section, some of the implications of equation
(4. 12) with respect to the behavior of dislocations in real materials will
be explored. These implications are of particular interest with regard
to the experimental results and their qualitative interpretation discussed

in Chapte r 1.

4. 2 Applications

Using equation (4. 12) values of the magnification factor N can be

computed for particular values of the dimensionless quantities ‘g'l, the
o

38

ratio of the Young's moduli in R1 and R0, and -%1- , the ratio of dis-
location distance from the free boundary to the film thickness. The
results of some of these calculations are shown in Figure 5 in the form

of a family of curves of N vs. -%- for various values of PL

 

E0 .
. . E
The most important feature of these curves 18 that for EL above a
o
certain value all of the curves cross the 7):: axis at some value :10 ,

where 2:9 > 1. At these crossing points N is equal to zero; physically
this means that the dislocation is in equilibrium at X =. X 0° The theory
thus predicts that the equilibrium positions for edge dislocations sug-
gested in Chapter 1 can exist in the presence of surface layers with

c ertain pr ope rtie s .

 

are

In Figure 6 the dimensionless equilibrium distances :10

plotted as a function of % . This curve shows that as one might
0
E1

intuitively expect the equilibrium distances increase as E increases.
0

The slope of the curve decreases monotonically, and it can be shown

that -2‘—9 approaches the limiting value 16 for very large values of EL

h E0.
From Figure 6 the minimum value of 1%— for which an equilibrium
0
position exists can also be determined. This limit, which we will call
X o
h

p031t10n at the interface. As can be seen from the curve, (EL )9! l. 3.
o

 

(1%; c’. corresponds to the value = 1; that is, it yields an equilibrium
This value probably does not have a great deal of real quantitative sig-
nificance, however, since it is found in a region where the application
of continuum methods is of doubtful validity, as mentioned in the preced-
ing section.

Another useful application of equation (4. 12) is in the calculation
of the force F per unit length acting in the slip direction on edge dis-
locations as a function of their distance I: X -h from the interface
between R1 and R0. According to well-known theory discussed by Cottrell
(1), this force will have the magnitude By 0;Y+(o, -X) where a'xy*(°' -X)

is the shear stress due to the image dislocation, evaluated at the

o

 

 

 

 

 

 

 

 

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40

eux

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41

location of the real dislocation. In the sign convention to be employed
here, positive values of F will represent attraction to the boundary,
while negative F indicates repulsion.

Clearly F can be calculated in terms of the parameters h, By,
and fig by means of (49), (4. 12) and the last of the stress field equations
(2. 19)? in which Bx is taken as zero. It was desired, however, to obtain
numerical values for F applicable to actual materials; to do this, numerical
values were assigned to the parameters.

For face-centered cubic crystals, which includes all the substrate
materials for which calculations were made, the magnitude of the Burgers
vector and in this case BY may be taken as E- where a is the lattice
constant. The values for a used here were given by Kittel (21), and are
summarized in Table 3.

Since the materials of interest are not isotropic, it was necessary

to use elastic compliances reported for single crystals of each substance

to obtain approximate values for the Young's moduli. For this purpose

it is reasonable to take E0 = ——1— and E1 = 4—1—— , where the Sn are
with I511)1

the usual elastic compliances as discussed by Huntington (22), and the
subscripts 1 and 0 have their usual significance. The values of Sn used
in the computations are all as reported by Huntington (22), and are dis-

played in Table 3.

Table 3. Crystal Properties

 

 

E M

Material 5,; a

LiF 1.14 x lO'lzcmz/dyne 4.02 .8

KCl 2.62. 6.28

Al 1. 57 4.04

Au 2. 33 ___

Cu 1. 50 ---

Gc 0. 978 -....

N1 0.734 -_-

M80 0.408 ---

 

42

As a final step necessary for making numerical calculations,
particular values were assigned to the surface layer thickness h.
. Figures 7 and 8 show curves of F in dynes/X x 10'7 vs. I in X for
LiF and KCl substrates having various coatings, with h taken as 50 X.
This value of h is in the range which would be of interest for experi-
mental investigations of surface film effects, and also is small enough
so that the assumptions made about film thickness in the theoretical
development should be reasonably well satisfied.

. Figures 7 and 8 indicate that for certain combinations of materials
equilibrium positions corresponding to values F = 0 occur, while for
others they do not. It is of particular interest, however, that as the
equilibrium positions become farther removed from the interface the
slope of the F vs I curves at the equilibrium positions becomes increas-
ingly smaller. Such an effect is also observed when the equilibrium
position is removed farther from the interface not by increasing E, but
by increasing h. This is shown in Figure 9, where plots of F vs for
various thicknesses of MgO on LiF are presented.

As a final application of the theory, plots of F vs 1 for aluminum
with an aluminum oxide coating are shown in Figure 10, the values
% = 3 and $71) = 4 being selected. These values of 1%; are suggested by
Head (11) as reasonable limits for the two substances. The thickness is
taken as 25 X, a figure in the range reported on the basis of experimental
findings by Cabrera and Mott (23), and others. . Figure 10 indicates
that an equilibrium position may be expected at a depth of 100 to 125 X,
or about 4 to 5 times h, below the interface. . For a screw dislocation
under the same conditions, Head (11) computed an equilibrium distance
about equal to h.

This completes our work with applications of the theory. In the
concluding chapter, certain features of the problem will be discussed
more fully and some areas in which the results may be extended will be

considered.

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. CHAPTER 5

C ONC LU SION

The theory developed in the preceding chapters has led to results
which provide a good picture of the behavior of a dislocation near a
coated boundary. By treating this situation mathematically as a modifi-
cation of the case of the homogeneous half plane, these results have
been obtained in a simple form which makes them especially useful for
applications to particular materials.

The curves shown in Figures 7 through 10 indicate clearly that
under certain conditions a surface layer can act as a "barrier to the
egress of edge dislocations, " in agreement with the suggestion of 7
Westwood (4) discussed in Chapter 1. . The approximate value (gr-)pl. 3,
obtained from Figure 6, provides a simple criterion for determining
under what conditions this barrier effect may be expected to occur.

. This criterion should be of help in the selection of materials for experi-
ments involving the barrier effect--for example, Roscoe effect investi-
gations.

. In addition to indicating the conditions in which a surface layer
can act as a barrier, the present results provide useful information about
the shape of the potential barrier. As previously noted, the slope of the
force-depth curves near the equilibrium position decreases as the
equilibrium depth increases. Physically this means that the effective
barrier becomes "softer" as the equilibrium depth is increased by
increasing either the thickness or Young's modulus of the surface layer.
It is apparent that as the barrier becomes softer the amplitude of the

displacements from equilibrium produced by thermal agitation, externally

47

48

applied forces, or the stress fields of other dislocations, will increase.
Consequently the softer the barrier, the smaller will be the probability
of finding a dislocation at the calculated equilibrium position.

. For small displacements from equilibrium, the force-depth curve
is approximately linear so that a dislocation in this region may be
regarded as being restrained by a linear spring. The possibility of
experimentally looking for acoustic absorption peaks at the resonant
frequencies of these oscillator systems therefore suggests itself.
Probably such an effect would be small because of the relatively small
number of dislocations which would be involved, and would require very
carefully controlled conditions in order to be observed.

With regard to extending the present theory, it would be of great
interest to develop a method for calculating the equilibrium positions of
each one of an array of several parallel edge dislocations on the same
slip plane. This might be accomplished by a» modification of the elegant
theory of Eshelby, Frank and Nabarro (24) in which the first dislocation
in an array is considered as being held against a perfectly steep barrier.
The effect of barrier softness should be to produce a more even spacing
of the equilibrium positions than in the case treated by Eshelby e_t a_1_l.(24).
This problem is of particular importance since its solution has the
possibility of being checked experimentally. The locations of dis-
locations in arrays piled up against boundaries have in fact already been
determined by etch pit methods by several investigators; the very recent
work of Mendelson (25) with NaCl crystals having ozonized surfaces is a
good example.

The results obtained here could also be extended by carrying out
explicitly the solution for the case Bx 31 0, in which case an expression
for the magnification factor M as a function of )t and 5 would also be
required, and by investigating the effect on the solution of retaining some
additional terms in the series expansions (4.6) and (4. 7). These con-

siderations could be of importance in applications of the theory to the

49

interpretation of particular experimental results. As examples, it
might be of interest to investigate the effect of crystal orientation on
the spacing of dislocations in piled up arrays, or to investigate the
film thickness dependence of the Roscoe and related effects up to thick-
nesses where higher order terms in (4.6) and (4. 7) would clearly not
be negligible.

As another consideration, it may be noted that certain other
boundary value problems of interest related to those studied here might
advantageously be investigated by similar methods. The problem of a
dislocation in a thin film with plane boundaries is one of these. Another,
more closely related to the present problem, is that of the substrate
with a freely sliding film. As indicated earlier, . an approximation to
this situation may occur after some critical point is reached in the
deformation of real materials. This problem will perhaps be more
satisfactory mathematically, in that the conditions at the interface will
be sufficiently weak to begin with not to need further weakening since
only one displacement component must be continuous across the inter-

face.

(1)

(Z)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

BIBLIOGRAPHY

Cottrell, A. H.
"Dislocations and Plastic Flow in Crystals, "
Clarendon Press, Oxford (1953).

Roscoe, R.
Phil. Mag., 31, 399 (1936).

Andrade, E. N. da C. and Henderson, C.
Phil. Trans. Roy. Soc. A, 244, 177 (1951).

Westwood, A. R. C.
Phil. Mag., 8th Ser., _5_, 981 (1960).

Mukai, M.
J. Sci. Hiroshima Univ. A, L2, No. 2, 99 (1958).

Menter, J. W. and Hall, E. 0.
Nature, 165, 611 (1950).

Evans, T. and Schwarzenberger, D. R.
Phil. Mag. 8th Ser., 1, 889 (1959).

Schofield, B. H.
. "Acoustic Emission Under Applied Stress, "
Lessels and Associates report (August 31, 1961).

Tatro, C.
. "Sonic Techniques in the Detection of Crystal Slip in Metals, "
Progress° report, Div. of Eng. Res., Mich. State'Univ.
(Jan. 1, 1959). O

Tatro, C. and Liptai, R.
Unpublished experimental results.

Head, A. K.
.Phil. Mag., 35, 92 (1953).

50

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

51

Muskhelishvili, N. I.
"Some Basic Problems of the Mathematical Theory of
Elasticity, " Noordhoff, Groningen, Holland (1953).

Timoshenko, S. and Goodies, J.. N.
"Theory of Elasticity, " McGraw-Hill, New York, 2nd
ed. (1951).

Love, A. E. H.
"A Treatise on the Mathematical Theory of Elasticity, "
4th ed. , Dover, New York» (1944).

Burgers, J. M.
Proc. Kon. Ned. Akad. Wet., .4_Z_, 293 (1939).

Eshelby, J. D.
Phil. Mag., 42, 903 (1949).

Koehler, .J. S.
Phys. Rev., 69, 397 (1941).

Churchill, R. V.
"Introduction to Complex Variables and Applications, "
McGraw-Hill, New York (1948).

Morse, P. M. and Feshbach, H.
"Methods of Theoretical Physics, Part I, " McGraw-Hill,
New York (1953).

Read, W. T.
"Dislocations in Crystals, " McGraw-Hill, New York (1953).

Kittel, C.
Introduction to Solid State Physics, 2nd ed. , Wiley,
New York (1961).

Huntington, H. B.
. "Solid State Physics, " vol. 7,. Seity and Turnbull eds. ,
Academic Press, New York (1958).

Cabrero, N. and Mott, N- F.
Rep. Prog. Phys., l_2_, 163 (1948-49).

52
(24) Eshelby, J. D.,. Frank, F. C. and Nabarro, F. R. N.
Phil. Mag., 42, 351 (1951).

(25) Mendelson, S.
J. Appl. Phys., 23, 2182 (July, 1962).

APPENDIX

It 18 of help in visualizing dislocation stress fields to have plots
of the stress components for the important case of a dislocation in an

infinite medium. . Curves of this type are provided in Figures 11
through 13.

 

*
These plots show the dimensionless stresses 0’ xx : 012xx ,
0" an:
0.31;) = —I%X , and a'xy = __xy as a function of the dimensionless
distance

.I
BX for a dislocation with Burgers vector b = By j . Plots are
given for various positive values of the dimensionless distance

, * *
Since (7' ' and 0'
xx x

a— ;
* Y
are odd functions of x while 0" is-
an even function of x, the form of the curves for negative values of
x .

—- is a arent.

B PP

Y

53

54

 

 

 

 

 

 

 

 

 

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x
By], as a function of £3; for various values of F; .

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