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Simulation of defects in crystals by point force arrays

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Title:
Simulation of defects in crystals by point force arrays
Creator:
Georges, Jean-Pierre Jacques, 1946-
Publication Date:
Copyright Date:
1972
Language:
English
Physical Description:
xi, 128 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Approximate values ( jstor )
Atoms ( jstor )
Crystals ( jstor )
Cubic crystal habits ( jstor )
Edge dislocations ( jstor )
Energy ( jstor )
Energy value ( jstor )
Mathematical constants ( jstor )
Modeling ( jstor )
Screw dislocations ( jstor )
Crystals -- Defects ( lcsh )
Dislocations in crystals ( lcsh )
Dissertations, Academic -- Engineering Sciences -- UF
Engineering Sciences thesis Ph. D
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 126-127.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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13923616 ( OCLC )
ADA4861 ( NOTIS )

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SIMULATION OF DEFECTS IN CRYSTALS
BY POINT FORCE ARRAYS














By

JEAN-PIERRE JACQUES GEORGES


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UXNITESITY OF FLORIDA IN PARTIAL
FLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1972



















ACKNOWLEDGMENTS


The author wishes to express his deep appreciation to

Dr. C. S. Hartley, Associate Professor of Engineering Science, Mechanics

and Aerospace Engineering, and chairman of the supervisory committee,

for guidance and counsel during this research.

The author also wishes to express his appreciation to Dr. L. E.

Malvern, Professor of Engineering Science, Mechanics and Aerospace

Engineering, to Dr. M. A. Eisenberg, Associate Professor of Engineering

Science, Mechanics and Aerospace Engineering, to Dr. J. J. Hren,

Professor of Materials Science and Engineering, and to Dr. J. B. Conklin,

Jr., Associate Professor of Physics, for serving on the supervisory

committee. Special thanks are due to Dr. S. B. Trickey for his helpful

assistance.

The author wishes to express his special gratitude to Dr. A. K.

Head, Chief Scientific Officer at the Commonwealth Scientific Indus-

trial Research Organization, Melbourne, Australia, for his very perti-

nent comments. Thanks are also due to Mrs. Edna Larrick for the typing

of this manuscript.

This research has been sponsored by the National Science

Foundation under the Grant GK 24360.


















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . . .

LIST OF TABLES . . . . .


LIST OF FIGURES . . . . . . . . . . .

KEY TO SYMBOLS . . . . . . . . . . . .

ABSTRACT . . . . . . . . . . . . .

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

2 BASIC CONCEPTS . . . . . . . . . .

Point Force . . . . . . . . . . .
Double Force . . . . . . . . .
Primitive Dislocation Loops . . . . . . .

3 RECTANGULAR DISLOCATION LOOP IN SIMPLE CUBIC CRYSTAL

Displacement Field . . . . . . . . .
Elastic Potential Energy . . . . . . . .

4 SCREW DISLOCATION IN SIMPLE CUBIC CRYSTAL . . . .

Displacement Field . . . . . . . . .
Self-Energy of the Screw Dislocation . . . . .
Single and Double Kinks in a Screw Dislocation . .

5 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED
FROM AN ARRAY OF SHEAR LOOPS . . . . . . .

Displacement Field . . . . . . . . .
Self-Energy of the Edge Dislocation . . . . .
Single and Double Kinks in an Edge Dislocation . .

6 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED
FROM AN ARRAY OF PRISMATIC LOOPS . . . . . .

Displacement Field . . . . . . . . .
Self-Energy of the Edge Dislocation . . . . .


Page

. . . . . . . ii

. . . . . . . v


vi


viii

xi



1

4

4
7
8

14

14
16

20

20
36
43



64

64
82
91



105

105
120


iii













TABLE OF CONTENTS (CONTINUED)

CHAPTER Page

7 CONCLUSIONS . . . . . . . . . . 124

BIBLIOGRAPHY .......................... 126

BIOGRAPHICAL SKETCH ...................... 128


















LIST OF TABLES


Table Page

1 Relative Displacement of Atoms Across the Slip Plane
for a Screw Dislocation . . . . . . . . 26

2 Variation of Force Constant C_ with the Atomic
Positions in a Screw Dislocation ........... 30

3 Atomic Displacements for a Single Kink in a Screw
Dislocation . . . . . . . . . . .49

4 Atomic Displacements for a Double Kink of Length 2a
in a Screw Dislocation . . . . . . . .. 51

5 Atomic Displacements for a Double Kink of Length 4a
in a Screw Dislocation . . . . . . ... 51

6 Atomic Displacements for a Double Kink of Length 6a
in a Screw Dislocation . . . . . . . .. 52

7 Relative Displacement Across the Slip plane for an
Edge Dislocation . . . . . . . . ... 75

8 Relative Displacements and Force Constants C2 at
Singular Points for an Edge Dislocation .. .. 80

9 Atomic Displacements for a Single Kink in an
Edge Dislocation . . . . . . . . ... 95

10 Atomic Displacements for a Double Kink of Length 2a
in an Edge Dislocation . . . . . . . . 96

11 Atomic Displacements for a Double Kink of Length 4a
in an Edge Dislocation . . . . . . ... 96

12 Atomic Displacements for a Double Kink of Length 6a
in an Edge Dislocation . . . . . . ... 96

13 Displacements and Force Constants C1 at Singular
Points for an Edge Dislocation . . . . . .. 116


















LIST OF FIGURES


Figure

1



2

3

4

5


Page

Coordinate System to Evaluate the Core Region
Around a Point Force . . . . . . . .. 10

Prismatic Loop in Simple Cubic Crystal . . . .. 10

Shear Loop in Simple Cubic Crystal . . . . .. 13

Shear Loop with Principal Axes . . . . ... 13

Rectangular Array of Shear Loops . . . . .. 15


6 Simulation of a Screw Dislocation . . . . .. 21

7 Relative Displacement Near the Core of a Screw
Dislocation . . . . . . . . ... . . 27

8 Relative Displacement Between -4a and 4a for a
Screw Dislocation . . . . . . . . ... .28

a
9 Atomic Arrangement in Planes x3 = of a
Screw Dislocation . . . . . . . . . 32

10 Distribution Function of a Screw Dislocation ... . 35

11 Region Where the Correction Energy Applies for
a Screw Dislocation . . . . . . .... 37

12 Array of Forces for a Single Kink in a Screw
Dislocation . . . . . . . . .. .. . 44

13 Array of Forces for a Double Kink in a Screw
Dislocation . . . . . . . . .. .. . 44


14 Atomic Arrangement Around
a Screw Dislocation .

15 Atomic Arrangement Around
in a Screw Dislocation

16 Atomic Arrangement Around
in a Screw Dislocation

17 Atomic Arrangement Around
in a Screw Dislocation


a Single Kink in


a Double Kink of Length



a Double Kink of Length



a Double Kink of Length

. . . . . . .


2a



4a



Ga













LIST OF FIGURES (CONTINUED)


Figure Page

18 Region of High Strain for a Double Kink in
a Screw Dislocation . . . . . . . ... 60

19 Atomic Relaxation for a Double Kink in a Screw
Dislocation . . . . . . . . ... . . 60

20 Array of Forces Simulating an Edge Dislocation . . 65

21 Atomic Arrangement in xl = 0 Plane for an Edge
Dislocation . . . . . . . . . . . 73

22 Relative Displacement Close to the Core of
an Edge Dislocation . . . . . . . ... 76

23 Relative Displacement for an Edge Dislocation ... . 77

24 Distribution Function for an Edge Dislocation ... . 83

25 Region Where the Correction Energy is Computed
for an Edge Dislocation . . . . . . ... 85

26 Array of Forces for Single Kink in an Edge Dislocation. 92

27 Array of Forces for Double Kink in an Edge Dislocation 92

28 Atomic Arrangement for a Single Kink in an
Edge Dislocation . . . . . . . . ... .98

29 Atomic Arrangement for a Double Kink of Length 2a
in an Edge Dislocation . . . . . . ... 99

30 Atomic Arrangement for a Double Kink of Length 4a
in an Edge Dislocation . . . . . . . .. 100

31 Atomic Arrangement for a Double Kink of Length 6a
in an Edge Dislocation . . . . . . . .. 101

32 Array of Prismatic Loop in x = 0 Plane . . . .. 106

33 Array of Prismatic Loop in xl = 0 Plane . . . .. 106

34 Relative Displacement of an Edge Dislocation . ... 118

35 Atomic Arrangement in x2 = 0 Plane of an
Edge Dislocation . . . . . . . . ... 119


vii

















KEY TO SYMBOLS


A Constant defined in Equation (72)

a Lattice parameter

b Burgers vector

C. Force constant corresponding to a point force acting in the
I

x. direction
1

Cijkl Components of the elastic constant tensor


d Constant defined in Equation (60)

dV- Element of volume at the point r
r

E Correction energy
c

EE Energy of an edge dislocation

E Energy defined in Equation (9)

ES Self-energy of a point force or energy of a screw dislocation

E Total energy of dislocation loop

F General symbol for a point force

f General symbol for any function

f. Component of a general force distribution

G General symbol for a point force

G. Component of Green's tensor
13

g General symbol for any function

h Vector separating points of application of the two point forces

forming a double force


viii













L Dimension defined in Figure 5 or Figure 33

n Normal at r of a surface

Pk Component of the dipole tensor

R Dimension defined in Figure 5 or Figure 33

R Vector defined by (r-r')

r Point where the displacement field is computed

r' Point of application of a point force or a double force

rO Constant defined in Equation (6), Equation (93) or Equation (195)

r' Constant defined in Equation (72)

r" Constant defined in Equation (88)
0

u General displacement field

j th order when computing the displacement field u

u' Perturbation of the displacement field due to the introduction

of a general kink in the crystal

u) Perturbation of the displacement field due to the introduction
(DK)
of a double kink in the crystal

u' S Perturbation of the displacement field due to the introduction
(SK)
of a single kink in the crystal

v Corrected displacement field between the planes of forces

W Energy of the system of forces

W Energy of the array of F forces in the simulation of an edge

dislocation by primitive prismatic loops

WFG Interaction energy between the arrays of F and G forces in the

simulation of an edge dislocation by primitive prismatic loops

WG Energy of the array of G forces in the simulation of an edge

dislocation by primitive prismatic loops

ix












Wnt Energy defined in Equation (26)

ow Energy defined in Equation (23)
Row

w Half-width for a screw or an edge dislocation

X. Component of the vector R
1

x. Component of the vector r
Distribution function for an edge dislocation

'22 Distribution function for an edge dislocation
a22 Distribution function for a screw dislocation

th
a.. K order term in the computation of the distribution function




Au2 Relative displacement across the slip plane


Au2 J order term in computation of the relative displacement Au2

6.. Kornecker delta


6(R) Dirac delta function

C Variable tending to zero

e. Component defined on page 16
1

S Peierls' symbol for the half-width of a dislocation

X Lame's constant

p Shear modulus

V Poisson's ratio

C.. Stress tensor component corresponding to the displacement

field u

7ij Stress tensor component corresponding to the displacement

field v

e Angle defined in Figure 1













Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy


SIMULATION OF DEFECTS IN CRYSTALS
BY POINT FORCE ARRAYS

By

Jean-Pierre Jacques Georges

December, 1972


Chairman: Dr. C. S. Hartley
Major Department: Engineering Science, Mechanics and Aerospace
Engineering

A new approach for analyzing dislocations and kinks in disloca-

tions in simple cubic crystals is presented. The crystal is considered

to be a continuum where defects are simulated by arrays of point forces

acting on the centers of atoms in the immediate neighborhood of the

defect. The magnitude of these forces is determined by the condition

that they have the same displacement field as the corresponding defect

in the ordinary continuum model. Infinitesimal prismatic and shear

loops are constructed for simple cubic crystals and used to construct

screw and edge dislocations. The arrangement of the atoms in the

vicinity of the dislocation line is obtained and compared to Peierls'

model. The self-energy of these dislocations is found to be of the

correct form provided the force cbnstants are correctly determined.

Atomic arrangements around kinks in screw and edge dislocations

have been computed and are presented. The model developed promises to

be of great value in studying atomic displacements in the vicinity of

the dislocation.

















CHAPTER 1


INTRODUCTION




A thorough comprehension of the nature of defects in crystalline

materials and especially in metals is fundamental in order to explain

many of the properties and the behavior of these solids. In particular,

vacancies, interstitials and dislocations cannot be ignored when diffu-

sion, mechanical behavior, electrical, optical and magnetic properties

are studied.

The usual theory of lattice defects assumes a "local" continuum

model. The matter concentrated in the atoms is supposed to be uniformly

distributed over the whole space occupied by the crystal. The local

atomic arrangement is ignored and the defect is replaced by a singular

line, point or surface in a continuum body [1-3J.

This model has proved to be extremely valuable for studying

properties which are not sensitive to the atomic configuration in the

vicinity of the defect, but it is limited by the discrete nature of the

atomic array. Consequently, it is always necessary that expressions

for the displacement field of the defect be terminated at some distance

from it. Furthermore, since the continuum approximation ignores the

local atomic arrangement around defects, it disregards the short range

anisotropy of the displacement field.

To remedy these shortcomings, atomistic computations have been

attempted. Atomic positions and interactions are considered explicitly












in the core region of the defect, with some laws defining the pairwise

atomic potentials (see [4-13]). Further from the center of the defect,

continuum theory is assumed to hold, so that the only atoms which need

to be considered are those whose positions are necessary for calcula-

tions of energies for the core region. Such a method involves first

the construction of a suitable interatomic potential, and, secondly,

sums over a large number of lattice points which have to be carried out

numerically. It is undoubtedly the best existing method of determining

the local atomic arrangement around defects, but it is costly and very

sensitive to the chosen size of the core region [14]. Furthermore, it

involves convergence problems, and the manner in which boundary condi-

tions are imposed is very delicate.

It is therefore worth exploring methods refining the ordinary

continuum model by introducing the atomic arrangement of the crystal,

but with a minimum increase in computational effort. In such a model,

the atoms will be considered to be embedded in a continuum and the

defect formed by the placement of suitable point forces at positions

corresponding to atomic sites close to the defect [15]. The result-

ing displacement field is the sum of the displacement fields of all

the point forces and is taken as the displacement field around the

defect. Examples of such constructions by superposition of infini-

tesimal loops have been given by Koehler [16], Groves and Bacon [17]

and Kroupa [181 for local continuum models.

In this present study, we shall concern ourselves in examining

straight screw and edge dislocations in simple cubic crystals. First,

a brief description of point forces and infinitesimal primitive loops







3




will permit us to analyze the displacement field and self-energy of dis-

locations. The screw dislocation will be constructed from an array of

primitive shear loops, and the edge dislocation from an array of prim-

itive shear luops and prismatic loops. Comparison will be made between

both models in the case of the edge dislocation. Furthermore, calcula-

tions of atomic displacements around kinks will be attempted for both

dislocations.
















CHAPTER 2


BASIC CONCEPTS



Point Force

A point force F is a highly localized body force distribution

applied to a material point in a continuum.

F.(r') = f. 6(R) dV (1)



where R = r r' and 6(R) is the Dirac delta function. The displace-

ment field at r, u.(r), due to such a point force at r', can be

obtained from the equilibrium equations of elasticity and Hooke's law

[2] in the form


u.(r) = F (r')G ..(R) (2)
3 1 13


where for an infinite isotropic body


I 3-4 ij 1 X ix(3
G.. (R) + (3)
Gij) L 1- 61J (31
16ng 1-v R 1-v R 3

G..(R) is the Green's tensor response function for a point force.
ij
It is the component parallel to x. of the displacement field at r' due
i
to a unit point force parallel to x. at r. It can be shown that the

Green's tensor is symmetric.

As we see from the expression for G..(R), this function is not
defined for = and we are unable to determine the displacement of
defined for R = 0 and we are unable to determine the displacement of











the point of application of the point force from Equation (2). To

remove this mathematical divergence, we shall associate a finite dis-

placement u(r') with the point of application of the point force.

This value u(r') can be considered as being the average resultant dis-

placement of points on a surface surrounding the point of application

of the point force. This surface is determined such that u(r') is the

mean value of the vector displacements of two points symmetrical with

respect to r'. This vector is acting in the same direction as the

point force.

u(r') = (r + r) + r (4)


Using polar coordinates as shown in Figure 1, the absolute value of the

displacement u(r') takes the following form.


6-I l r r (5)


So, for a definite value of lu(r')( we can define a surface of revolu-

tion about the direction of F, on which all the points have displace-

ment components (u(r')I along F. This surface surrounds a volume which

can be considered as a core surrounding the point of application of the

point force. The core can be interpreted as the volume where Equation

(2) for the displacement field is ro longer valid. The size of the

core depends directly on the value of Iu(r')J assigned.

It must be pointed out that the average value of the radius

vector, jrf, of the core is equal to the radius r of the sphere on

which the average displacement of its points is equal to /u(r')j,

that is












F 5-6V (
r- (6)
0 24nr(u(r;')| 1V-


On the other hand, ju(r')| can be related to a force constant. By anal-

ogy with a discrete lattice model, such a point force applied on an

atom causes an equal and opposite resisting force proportional to the

displacement of the atom given by


I = C u ) (7)

C is known as a force constant, and is the force acting on an atom

required to produce a unit displacement. In other terms, its inverse

is the displacement of the atom caused by a unit force acting on it.

This force constant is the parameter we shall use in the follow-

ing problems encountered. It will be determined for each special case

by requiring that our mathematical model obeys certain physical imper-

atives. It will be straightforward to deduce lu(r')l and the size of

the core from the value of C.

The self-energy of a point force is defined as being the work

done by this force against interatomic reaction forces when it is

introduced into the continuum. So, using Equation (7)


2
1 -'-i 1 F
ES = Fu(r')I = (8)
S 2 ur 2 C

(1) -
The interaction energy between two point forces F (r') and
(2) s
F (r) is
m


(1) (2) (1) (2)
S= F (r) u (r) (= F () F (r') G (r-r'),
I k k k m km












where the sign is determined following Cottrell's convention for dis-

location interaction energies, i.e., it is the work done by external

forces when the second force is applied in the presence of the first,

or vice versa. The total elastic potential energy of the system is

the sum of the self-energies of the two point forces less the pairwise

interaction between them.


Double Force

A double force is constructed from two equal and opposite point

forces F applied at points separated by a vector h. If the forces

are collinear, we have a double force without moment, otherwise with

moment. The strength of a double force is defined as


Pk = lim (hFk). (10)

M -0



The displacement field is obtained by superposition. If the

separation distance between the forces is very small, we can expand

the displacement field of each in a Taylor series about the midpoint

of h. Keeping only the first order terms, we are led to the displace-

ment field mentioned by Kroner [2]


u() = Pk (r') Gkj,L(r-r'). (1)



As for a single point force, the displacement of the point of

application is undefined, but this divergence can be removed in the

same way as before by introducing the concept of a core surrounding

a double force.












Primitive Dislocation Loops

Following Kroner's definition [2], an infinitesimal dislocation

loop in a continuum is the boundary of a microscopic surface which

separates regions in the continuum which have suffered a relative dis-

placement b. The Burgers vector of the loop is defined as the line

integral of the elastic displacement u around a circuit containing the

dislocation. The displacement field, at a point r, of such a loop of

surface dS with normal vector n and centered at r' is found to be


u (r) = b.n. cij G dS (12)
m I j ijkL km,L


where cijkL are the elastic constants.

The similarity between this expression and the displacement

field of a double force (Equation (11)) leads us to consider the infin-

itesimal dislocation loop as a nucleus of strain with the fundamental

double force tensor


Pk = cijk bin dS (13)


or, for an isotropic continuum,


Pk = C(6ikj + if jk) + ij6 k binj dS (14)


Up to this point, we have completely ignored the local atomic

arrangement around the loop. In real crystals the interatomic reaction

forces, developed when the atoms are displaced to form the defect, are

the physical origin of the double force tensor characteristic of the

dislocation loop. So it seems logical to construct such a loop by

applying point forces in the continuum, but at points corresponding

to atomic positions located immediately around the defect. The












displacement field of the loop is then the superposition of the dis-

placement field of each point force. Each primitive loop has the

character of a "unit cell" for the defect. These "unit cells" can be

assembled to form a more complicated defect like dilatation centers or

dislocations. So logically we can characterize the surface of the loop

dS such that the produce b dSI equals one atomic volume in the crystal

structure considered. This procedure will be analyzed more specifically

for simple cubic crystals.

(a) Primitive prismatic loop in simple cubic crystals

The arrangement of the first neighbors of a vacancy loop in

simple cubic crystal is shown in Figure 2. A primitive prismatic loop

is constructed in the following steps. First a vacancy is created by

removing an atom from the lattice. This vacancy is simulated by apply-

ing on its first neighbors forces of magnitude F directed towards the

vacancy center. In the second step, two extra forces, G, are applied

in a direction normal to the (001} plane, on the atoms in the (001)

direction, towards the center vacancy in order to collapse the configur-

ation onto the [001} plane. In this manner, we have set up three

double forces, all without moment, leading to the diagonal dipole

tensor


P11 = 22 = 2Fa,
and (15)

P33= 2a(F+G),


where a is the lattice parameter of the simple cubic crystal.

Though each pair of forces is clearly separated by a distance

2a, at distances from the loop large with respect to the interatomic


















FM
3






















FF 2


e .
xx



















Figure 2. Prismatic Loop in Simple Cubic Crystal












distance they appear as three double forces which can be identified

with a dislocation loop as described above. The Burgers vector of

this loop must represent the collapse of the atoms in the (001)

direction, whose relative displacement must be a in order to create

a new regular arrangement of the atomic planes. As stated previously,

the surface dS is chosen such that b dSI is equal to a here. So,

following Equation (14)

3
P11 22 = a
and (16)

P33 = (X+2p) a,


The forces applied on the atoms can now be obtained by comparing

Equations (15) and (16). The displacement field and the self-energy

of the loop can easily be deduced.

(b) Shear loop in simple cubic crystal

A primitive shear loop in a simple cubic lattice (Figure 3)

is constructed as follows. The forces F applied on the atoms impose

the direction of the shear. Since the loop must be kept in equilib-

rium with respect to its center, additional forces G have to be applied,

forming a couple whose moment about the center of the loop counter-

balances that of the shear forces F. The Burgers vector of the loop

is the smallest shift allowed by the atomic arrangement. By the same

method as for the prismatic loop, it is found that the only nonvanish-

ing components of the dipole tensor are

3
P12 = P = Ia = 2Fa = 2Ga, (17)
12 21












so the magnitude of the forces has the value

2
a
F = G = -- (18)


The shear loop can be represented with respect to its principal axes

x' and x' (Figure 4) leading to the dipole tensor


P = 2aF = P (19)


which represents two double forces without moment, perpendicular to

each other and acting in opposite senses.

(c) Conclusion

The primitive dislocation loops as described above are the

basic elements for our process of simulating larger defects, especi-

ally dislocations. We shall see that a suitable array of shear loops

can describe either an edge or a screw dislocation, but that an array

of prismatic loops can only simulate an edge dislocation.













S----
2.
! -- *.--- Ix
& ^(


X,


Figure 3. Shear Loop in Simple Cubic Crystal


X I1








u/ a




Figure 4. Shear Loop with Principal Axes


T-- F


















CHAPTER 3


RECTANGULAR DISLOCATION LOOP IN
SIMPLE CUBIC CRYSTAL




A rectangular dislocation loop having a Burgers vector a(100)

can be simulated by a rectangular array of primitive shear loops,

stacked as shown in Figure 5. The dimensions of the array are con-

sidered to be very large compared to the atomic distance. The axes

of reference are shown in Figure 5 with their origin at the center of

the loop. In this chapter, we are only interested in obtaining prop-

erties of the rectangular dislocation loop related to our main inter-

est, the displacement field and self-energy of the pure screw and edge

dislocations.


Displacement Field

The displacement field at any points of this array is simply the

sum of the displacement field of each point force


u (r)= F.(r') G. (r-r'), (20)
m 1i im
m i,r' m


where G. (R) is defined in Equation (3). Developing this sum leads to
im

the general expression


2 R/2 L/2 L/2-1
u (r) = 2 R E 2 + l G (x -pa,x -qa,
p=-R/2 q=-L/2 q=-(L/2-1)


a a
x3) G2m (x-pa,x2-qa,x3

14






15















-10


4o
'-'I










\' Y \ \-^.




^ ^ i

V 'H











La a La a
G3m (x1-a, x22 3 2) + G3m(l-pa, x2+ x+


La a La a(21)
+ G3m(x1-pa, x2 x3 -) + G3m(xl-pa, x2 x3 ) (21)



These equations for each component are valid everywhere in the con-

tinuum. They can be simplified for each particular region of the loop.

The regions of the continuum where a pure screw dislocation

is simulated correspond to




S and
x = 8 x = 2
2 2 2 2


where e1 and C2 are small compared to Ra and La, respectively.

Similarly, regions where the loops have a pure edge character corre-

spond to



{ l and
La La
I2 2 2- 2 2

where el and e2 are small compared to Ra and La, respectively.

Each particular case will be considered in the following chapters.


Elastic Potential Energy

The work done by the forces comprising the array, that is, the

energy of the system, is the sum of the self-energy of each point force,

minus the pairwise interaction energies, as defined by Equations (8)

and (9). We call W the energy of a row of forces, that is, two lines
Row









of forces parallel to the x2 axis for a given xl coordinate, and we call

W nt the interaction energy between two rows of forces, as defined above.
Int
Following these appellations, the total energy of the system has the form
R
W = (R+1)W + 2 (R+l-p)W (22)
Row pInt
p=l

W and W have the following expressions, respectively,
Row Int

Row 5 L1 + + 24 C33(O,0,a) (2-1) G22(0,0,a)
"o 2 3 2
2 4
+ -- [G22(0,La,0) G22(0,La,a) G33(0,La,0) G33(O,La,a)j

L
S22a4 (L-q) CG22(O,qa,0) G22(O,qa,a)
q=1
2 4 2 4 L
4a G23(0Laa) 2Pa4 Z G23(O,qa,a) (23)
q=l

WInt 2a4(2L-1) G22(pa,0,0) -G22(pa,0,a)] + .2a4 G a33(pa,0,)+G33(pa,O,a)

+ 2a4 LG22(pa,La,0) G22(pa,La,a) G33(pa,La,O) G33(pa,La,a)

24 24 L
2p a G23(pa,La,a) 424 a S G23(pa,qa,a)
q=l
L (24)
+ 42a4 E (L-q) CG22(pa,qa,0) G22(pa,qa,a) (24)
q=l

Following Equation (22), W becomes

2L4 L
W 2a4(R+1)L G (0,0,a)+ 2 S G2(O,qa,0) G22(,qa,a)
C2 q=l
R R L
+ 2 [G22(pa,0,0) -G22(pa,0,a' +4 S S G22(pa,qa,0)
p=l p=l q=l

G2(pa,qa,a) + 2a4(R+1) 1 1)+ G (0,0,a) + G33(0,0,a)


+ G G22(0,La,0) -G22(0,La,a) -G33(0,La,0) -G33(0,La,a]

L
2 z q [G22(0,qa,0) G22(0,qa,a)
q=l










R
[G22(pa,0,0) -G22(pa,0,a)+ G33(pa,0,0) + G33(pa,0,a)
p=l
R
+ F G22(pa,La,0) -G22(pa,La,a) -G33(pa,La,0) -G33(pa,La,a)
p=l
R L
4 Z S q [G22(pa,qa,0) -G22(pa,qa,a) G23(0,La,a
p=l q=l
L R+1 R+1 L
2 Z G23(0,qa,a) -2 G23(pa,La,a) -4 Z G23(pa,qa,a)
q=l p=l p=l q=l


+ a4L -2 E p 22(pa,0,0) -G22(pa,0,a)]
p=l
R L
4 s p [G22(pa,qa,0) G22(pa,qa,a)
p=l q=1


+ .2a4 pG22(pa,0,0) -G2(pa,0,a) -G3(pa,0,0) -G33(pa,0,a)]
p=1
R
Ep G22(pa,La,0) -G22(pa,La,a) -G33 (pa,La, 0) -G33(pa,La,a)
p=l
R L
+ 4 ZS pq GG22(pa,qa,0) G22(pa,qa,a)]
pUq-
p=l q=l
R+1 L R+I
+ 2 Z p G23(pa,La,a) + 4 S E p G23(pa,qa,a (25)
p=l q=l p=l

The only mathematical difficulty in the computation of such a formidable

expression lies in computing the single and double sums. The first terms

are computed, up to a chosen integer N (usually N=20), and the rest of the

terms are approximated by an integral. The following approximation follows:
L L
L 1 1 d (26)
Z f(q) f(q) +2- f(N) + f(L) + f(x) dx, (26)
q=l N
and for a double sum:
R L N-1 N-1 N-l
f S f(p,q) E S f(p,q) + f(p,N) + f(p,L)
p=l q=l p=l q=l p=l
N-l N-l L
+ N- Cf(N,q) + (R,q) + (R + I f(p,y) dy
q=l p=l N








N-I R L R
+ E "S f(x,q) dx + I j f(N,y) dy + -' f(x,N) dx
q=l N N N




L R
+ f(N,N) + f(R,L) + f(x,y) dx dy (27)
N N

These approximations give the correct form of the divergence
4
in R and L for divergent sums, and give an accuracy of 1 part in 10

when the sum converges, which is sufficient for the model employed.

The final expression for W/pa becomes

W (R+1)L F4 a 2.8545 2.36767 R+1 2RL
3 -4T C1 v 2n(l-V) -2
a R+ R +L
L + R+1 1-- i 1F 1.2979 -.6458v
L /-n+- 4 a (3- 2+ I---
L + 2R +L

L ,.6420+ .3580v 2(2-v) k L2 r 1 I_ 1) 3.3903 .6872v
+ 1 v-J 1- v 4 + 4 2n, F3 C2 4n(2-)) 6 (28)


W is not the energy of the physical dislocation loop. It con-

tains an extra strain energy in the region bounded by the plane on which

forces are applied and the boundaries of the array, which does not

account for the relaxation of the atoms. In the way the forces have been

applied, a relative displacement greater than a/2 has been created across

most of the slip plane for the atoms reaching their final configuration.

It is from this final configuration that the relative displacement proce-

dure must be measured in order to calculate the actual strain energy

stored between the planes of forces. Such a correction energy will be

computed in each case, for the pure screw and pure edge dislocation.

We shall not examine all the properties of the dislocation loop

here, since our purpose is to treat this loop as an intermediate step

in simulating screw and edge dislocations.

















CHAPTER 4


SCREW DISLOCATION IN SIMPLE CUBIC CRYSTAL



From the results found in the previous chapter, the displacement

field, the relative displacement across the slip plane, and the self-

energy of the screw dislocation in simple cubic crystal will be obtained.


Displacement Field

As seen previously, the regions of the dislocation loop having

the characteristics of a pure screw dislocation correspond to

Ra Ra
xl 2 + x = T -

and


2 e 2 and 2 2::


where e1 and e2 are small compared to Ra and La, respectively.

Since both regions represent two identical dislocations, but of

opposite signs, we shall only consider the first one, and translate the

x2 axis by an amount of -Ra/2 so that it becomes the boundary of the

array. We shall keep the same symbols xl and x2 for the new variables.

(See Figure 6.)

The only change in the expression for the displacement field as

written in Equation (21) is that p is now summed from zero to R. Since

an analytical expression is desired for um, an approximation different














from that given by Equation (26) will be employed to compute the dis-

crete sums. Euler's formula [19] is most suitable for this case:

b b
Sf(p) = f(x) dx + Cf(a) + f(b) + f' (b) f(a
a a


[. (b)- f" 3024 )(b) f(5)(a)


+ ... (29)

The accuracy of the approximation depends on the number of terms used.

The advantage of this method is that at most of the points where the

displacement field is computed, the three first terms are sufficient
-3
for the accuracy required, i.e., a relative error of 10- is accepted.

The first summations which have to be computed are the two

summations on q. These sums can be computed exactly because we always

consider x2 small with respect to La, or in other words, our range of

interest is far from both ends of the dislocation line. So we shall

have
L/2 L/2-1 +"
S f(q) = E f(q) = f(x) dx (30)
q=-L/2 q=-(L/2+1) -m


where f represents the whole expression to be summed. The following

components of the displacement field have been found:


Ul(x1',2,X) = 0
2 a 2
R (x -pa) + (x )2
a 1 J 2
u (x1,x2,x2 = (31)
p-0 (x -pa)2 + (x3 2


u3(xl'x2,x3) = 0











We can already notice that some of the characteristics of the displace-

ment field around a screw dislocation are displayed, that is, the

components ul and u3 are equal to zero, and u2 is independent of x2.

Euler's formula can be applied a second time to obtain a final

expression for u2, the accuracy depending on the number of terms retained.

We shall label the different terms composing u2 in the following way,


u0(x1x2,x3) -a g(x1-pa) dp (32)


1 a
u2(x1,x2,x3) = g(xl) + g(xl-Ra) (33)



u (x1,x2,x3) g'(x1-Ra) g'(x1)] (34)

3 a 3I
u2(x1'x2x3) 2880 L (3)(x-Ra) g(3)(Xl,3 (35)


etc., where

2 a 2
(x -pa) + (x3 +
g(x -pa) = n (36)
(x1-pa) + (x3- 2)


The computation of u2 raises a mathematical problem, since we

are integrating over a region where the integrand contains singular

points for certain values of x1 and x3 (xl = na, x3 = a/2). For these

points, the integral can be broken into two parts:

Ra na-e Ra
Sg(na-pa) dp = lim |J g(na-pa) dp + J g(na-pa) dp (37)
0 e-0 0 na+

For this specific case

2 2
g(na-pa)= n (na-pa) + a (38)
(na -pa)











Since

Sn (na-pa)2 dp = -2 [(n-p) &n (na-pa) (n-p) (39)

the integral is equal to zero for p=n, and so

Ra
Sg(na-pa) dp = G(na-Ra) G(na). (40)
0

The integration makes all the singular points vanish, except

for those at x = 0 and x3 = a/2. So the final analytic expressions
S3
for u2 become

0 a x3 1 X -1 xl
u2 (X1 X3) a 2 t + tan + tan (41)
-TT3 + x3 2

2 a
S x -1 x x + (x3
x3 ta atan + xa 2 2-
x3+ x-3 x1 + (x3

2 a(x + 2
1 a x1 3+ 3(4
u2 (1'X3 Sn 2 a 2 (42)
x + (x --)
1 3 2

2 x x
2 Ia 1 (43
U2 (Xl'X3) = -2 a 2 2 (43)
2 x3 24nT 2 a2 2 a2 (43)
1 + (x ) X + (x3+2)

and


a4 x1 3(x3 2 3 2 x -x2
u2(xl' 3 730n 2 a 23 2
+ (x3 +
(44)

Then, since u2(x ,x3) is the sum of u (x ,x3) for every i,
0 1 2
u2( x3) = u20(x 3) + u2I(x 3) + u2(xx3

3
+ u2(x,x3) + .. (45)
2 1'x345












Equation (41) takes three different forms, following the region where

x3 is computed: T corresponds to x3 > a/2, 2r(x3/a) to -a/2 < x3 < a/2,

-n to x3 < -a/2.

An asymptotic expression for u2 can be obtained when x2 and x3

are considered large with respect to the atomic distance, but still far

from the ends of the dislocation line.


u2(xx3 + tan1 x3 (46)



This is the well-known expression obtained from the Volterra solution

for a screw dislocation [20].

The relative displacement across the slip plane is defined as

a a a (47)
Au2(x1) = u2(x ) u(x, ) = 2u2(x1,).


It can be directly deduced from Equation (31) or Equation (42)

to (45):
2 2
m (x -pa) + a
Au2(x,) = (48)
2 1 2TT p=0 (x -pa)2


or

0 1 2 3
Au2(x1) = Au2(x ) +. u2(x1) + Au2(x1) + Au2(x) + ... (49)
2 1 2 1 2 4 2 1 2 1

with

0 a -1 X1 X1 1 +
Au2(xI) 2 tan ---a+ a (50)
Au (x 2 ta2l--1- + (50)



2 2
1 a (51)
Au2(x1) (51)
x1











n2 a i[ X_1 i2 (52)

3 a 1 1
and
3. a4 1 xl(X1 3a )
Au (x) 0 2-2~ 1 (53)
2 1 60x + a


The more and more precise expressions for u 2(x1) are plotted

in Figures 7 and 8, and listed partially in Table 1. A remarkable

precision is obtained for the regions where Ix1 a after evaluating

only a few terms. On another hand, the only term which is not singular

at x = 0 is Au (x ). It seems to deviate significantly from the correct

curve for u 2(x ). However, the order of magnitude of the real relative

displacement at xl can be obtained approximately by interpolation. So

the relative displacement of atoms above and below the slip plane is

known everywhere except at x1 = 0.



TABLE 1. Relative Displacement of Atoms Across the Slip Plane for
a Screw Dislocation


0 1 2 3
x /a Au2/a ... + Au/a ...+ Au2/a ... + Au /a
1 2 1 2 2

.0000 .0000 .0000 .0000

5.0 .0316 .0347 .0349 .0349
4.0 .0394 .0442 .0446 .0446
3.0 .0521 .0605 .0614 .0613
2.0 .0766 .0943 .0970 .0968
1.0 .1397 .1948 .2081 .2074
.5 .2243 .3524 .3949 .3890
+ .5 .7757 .9037 .8613 .8696
+ 1.0 .8603 .9155 .9022 .9033
+ 2.0 .9234 .9412 .9385 .9387
+ 3.0 .9479 .9563 .9554 .9554
+ 4.0 .9606 .9654 .9651 .9651
+ 5.0 .9684 .9715 .9713 .9713
+ m 1.0000 1.0000 1.0000 1.0000




















li i
I 2 I

. 23






'. i / X




I .
II








/ '/ .




/-/ /
/" /
/' / .
'I
I.,i
,: / .


Figure 7. Relative Displacement Near the Core of
a Screw Dislocation








28

X



I)






r4

0
o
(r









0





(o










0

0
a












-f







CO
O




















1-4
01
T-<


<3






3 3




S30


* I











Another way of computing the relative displacement at atomic

points is to go back to the definition of the displacement field by

Green's functions.


u2 (na) = 2a2 G22(na-pa,qa,0) -G22(na-pa,aa) (54)
p=0 q=--


changing variables by setting u = n-p changes Equation (54) into






2 2
Au (na) = 24a : SG 2 2G (ua ,qa,0) -G 2(ua,qa,a)\
l u= q=-m



+ 4pa 2 2 2G2(Oqa,0) G22(O,qa,a)]
q=l



+ Cn) -22pa2 G22(O,0,a) (55)


where C2(n) is the force constant defined in Equation (7) for a point

force acting on a point at a distance na from the origin. After computa-

tion of the sums as presented in Equation (26), Au (na) becomes

2 mr2
2ua2 a .866 2.368V u + 1
Lu2(na) = C (n) 2 1 2 (56)
2 u=n+l a 5


For each value of n, a direct comparison can be made between

the values of Au (na) from Equations (49) to (53) on one hand, and

Equation (56) on the other hand. Since both ought to be identical,

a value of C2(n) will be obtained for each value of n. Table 2 lists

the different values of Au2(n) and C (n) for n=0 to 5 and for n= .

The values of C (n) are very nearly constant, and it seems reasonable

to expect a value of C2(0) very close to C2(m). This extrapolation

permits us to evaluate u2 (0).












TABLE 2. Variation of Force Constant C2 with the
Atomic Positions in a Screw Dislocation


Au2 2u 2
n -(n)- --(x) C(n)/pa
a C2 a 1
X1 =na


0 .2885
1 .3988 .9033 .3964
2 .4343 .9387 .3965
3 .4511 .9554 .3966
4 .4607 .9651 .3966
5 .4670 .9713 .3966

.4957 1.0000 .3966


Replacing

and noticing that


C (n) by C (m) in Equation (56) for any values of n

for large n


2
A n 2ua a 2.866 2.368V
u (na) = a = +
2 C2(n) 2r 1 -V


a very simple expression for the relative displacement across the

slip plane for n 0 is found:


m 2
a u + 1
Au (na) = a -2n 2
n+1 u


In particular for n=0, Au2(na) = .794a. This value fits very well on

the interpolated curve for Au (x ), as shown in Figure 8. For the

atomic points corresponding to xl= -na, with n >0 a direct transfor-

mation of Equation (48) leads to

2
a u +1
Au2(-na) - E n&-u (59)
n a












It is striking to notice that the symmetry of the screw dislocation

displacement field is preserved at the atomic points. The dislocation

line, in the continuum sense, lies exactly at x= -a/2. This result

could have been guessed earlier by simply considering an oriented path

around each loop. By adding the loops together, the only remaining part

of the path would be a straight line at x =-a/2. This method can be

generalized for determining dislocation lines in more complicated cases.

This symmetry does not appear in Equations (49) to (53) because of the

divergence of these expressions at x = 0.

A mapping of the atomic displacements in atomic planes imme-

diately above and below the slip plane is shown in Figure 9.

The width of the screw dislocation is defined to be the region

in which the relative displacement is comprised between a/4 and 3a/4.

Since the region where the relative displacement is equal to 3a/4

cannot be known exactly, the value 2w = .73a for the width can be

obtained by rough measurements on Figures 7 and 8.

The relative displacement described above can be compared to

the results obtained by Peierls [20,21]. Fitting the expression for

relative displacement obtained from the Peierls model to our results

leads to the expression

a a -1 Xl + d
u2(x) = + tan (60)


where w is the half-width and d a translation parameter fixing the

center of symmetry of Au2. For large xl, this expression can be

expanded in consecutive powers of 1/x1,












i

,'e t


a- --/ 0--.


K


D0


c


Q )


L
L


0-


.j


()

s_












a w a wd
Au2(x1) = a + +2 (61)
S1 x
1


The same series expansion holds for Equations (49) to (53) and leads to

2
a a a a
u2(x1) = a 2 x + + (62)
2 1 2Trx 1+i-TT 2 (62
1 x
1

By comparison of Equations (61) and (62), the half-width and the

translation parameter are

a a
w d (63)
2' 2


These values correspond to the Peierls' model when the origin is taken

at x1 = -a/2. Although the width of the dislocation in our model is

slightly smaller than in Peierls model, the two expressions for the

relative displacement across the slip plane are exactly the same for

large values of xl, as shown in Figure 8.

Finally, following Eshelby's suggestion [1 ], a distribution

function for infinitesimal dislocations in the glide plane can be defined.

Instead of being the result of a singularity concentrated on the x2 axis,

the straight screw dislocation is considered to be composed of a contin-

uous distribution of infinitesimal dislocations. This distribution

function is, in fact, the component o22 of the dislocation density tensor

as defined by Kroner [2]. It is to be found equal to

d(Au2)
22(1 dx (64)


Differentiating Equations (49) to (53) with respect to xl leads to











2 2
0 1 x1 + a
a22 = 2 --2- (65)
x1

3
1 1 a
22(x1) = 2 2 2 (66)
x (x +a )



2 x a
-2 ( a 1-- (67)
x1 (x1+a)


and

F 4 22 4
4 xl-6ax +a
3 a 1 1
22 (x) 120- 2 2 4 4 (68)
(x + a ) x1


It can be easily verified that

+"
f a22(x1) dx1 = a (69)


A plot of the successive approximations is made in Figure 10,

emphasizing the values of 22(x ) at the atomic positions. Symmetry

of the distribution function at these points with respect to xl=-a/2

is evident.

In summary, we shall emphasize that this model is in perfect

agreement with the previous techniques employed for obtaining the dis-

placement field of a straight screw dislocation. An improvement has

been made in the present case, obtaining a simple analytic expression

for the atomic displacements around the defect without any exceptions.

Such atomic displacements can be obtained by lattice dynamics computa-

tion, but these have the disadvantages of being difficult to use and

being an entire numerical method [13].






35



,i I





II

I.
// I
d) b

/ 0o






---0 oE


-7f 1 .I



0
'-4
-/ / H,








(/ i a








Ix
-I I
j"--. -- I






SI '






T I d '












Self-Energy of the Screw Dislocation

As it has been explained in the previous chapter, such an array

of point forces can simulate a system composed of two infinitely long

parallel screw dislocations of opposite signs if the length La is taken

much larger than the separation distance, or width of the array, Ra.

In this case, the energy of the system per unit length of screw disloca-

tion becomes W/La when L is large.


W R+1 2 pa ] R (70,
Ta a 4TT -I C+ (70)
772 0r
La 4] 2 r


where the constants A and r; are defined by


S2.8545 2.3676v
A = (71)
V

and
1
-n = 1.179 (72)
r;
0


Taking the special case of v = 1/3 leads to


W R+1 2 3097 2 R
=a a 41 + 3.097 3076 (73)
a +~ 3 2r <.3076


As already mentioned, this energy is higher than the energy of

the system composed of two parallel screw dislocations of opposite sign,

because of the nonrealistic strain energy stored in the region between

the planes of forces. The region of the continuum where the strain is

larger than 1/2 is shown in Figure 11 and has the following boundaries:

{ a ,a
25 x3 _2
(74)
-.35a xl Ra + .35a.










37
















-H
CZ

0
r-











0












41



r(
(U












P4
e4














-4





-4
+>
0,










+1 .
<;











The points x = -.35a and x = Ra + .35a are those where Au2(x1)= a/2.

Outside these limits, u 2(x1) is smaller than a/2. The correction

energy which has to be subtracted from W/La is composed of the strain

energy per unit length of screw dislocation due to the displacement

field u(x1,x3), minus the strain energy per unit length of screw dis-

location due to the displacement field measured relative to the final

atomic positions inside the slab, v(x1,x3) described as follows:

v1(x X1 ) = 0


v2(X1,X3) = u2(x1,x3) x3 (75)

V3(XX3) = 0 .

The stress fields associated with these two displacement fields

are

a12= x
1
= (76)
u2
\23 = 3-


and

Sv2 Bu2
12 = x= ~L x
1 1
= / (77)
v2 u2
2T3 3 x )3


The correction energy is the following integral computed over the volume

mentioned above.

E .. dV (78)
c 2 j ij ij j










Using Green's theorem, this volume integral can be transformed into

a surface integral,

E 1 (T.. U v.) n dS, (79)
c 1j 1 ij 1 3

where u is the normal to the surface S. Using Equations (75) and (76)

leads to


c 2 2 Tx 2 in1 2x 2 \2 ) n dS (80)

or


E = 3 x n + x x3 + u n dS (81)
c 2 J 3 ox 3 ox 3 2
S 1 3

where the surface, S, is composed of the areas



a a
S : x -35a x Ra+ .35a;


a a
x I: -.35a, -< x <


S3: x1 = Ra + .35a, -:5 x < .


Taking the symmetry with respect to x = Ra/2 into account, and the fact

that the integrand is not dependent on x2, lead us to the final formal

expression for E /La:


E a/2 u2
c x3 (-.35a, x3) d3
a/2 1

Ra/2 a u2 a a a
+ 2p 3a2 T (xl') + u2(x1' dx1 (82)
.35a 3

The analytical expressions for u2, 6u2/ax1, du 2/x3 are obtained

from Equation (31) and its derivatives:


_ __











R (x -pa)2 + (x +)2
u2(xx3) =2 a 2
p=0 (x -pa) + (x --)
1 3 2


6u2
x3 ;- (Xl'X3) =
1


and

u2 a
Tx- (x1'x3= 2
3


a 1 pa x -pa
2 x3 Pa 2 (x 2 2
p=0 (x-pa) + (x ) (x-pa) + (x3 -
(84)


R x3 + x3 -


2 1 2
2. 2 2 (85)
p=0 (x-pa) + (x3+ ) (x1_pa)2 + (x3-2


The mathematical problem of integrating a function over a region where

the integrand has singular points for certain values of x1 is removed

by the same argument as the one used in the previous section.

After integration, Equation (82) becomes


E 2 R 2
c S a F(p + .35) (p+'35) + 1 + 2 tan-1 (p+.35
La @ 2 + 2 tan
La p=0 (p+.35)2


2 R
35 + (86)


The summation over p is computed in the sameway as before using Equa-

tion (26), which gives finally for the correction energy

E 2 2
c Ia R a (87
= R 2r (87)
La 2 n r,


kn r0 = .3129.
0


The total energy of the system composed of two antiparallel

screw dislocations is













ET W Ec
T W c (89)
La La La

or

F-T pa1 2 47 +A-2 + R 2 R. (90)

L2 (r")2 2
0


From the usual continuum theory of dislocations, this energy has to be

equal to the sum of the self-energies of both screw dislocations, minus

their interaction energy


E 2 2
T ua r 4a r
=2 -- n En (91)
La 4rr r0 2rn R


The requirement that our expression (90) has to be identical in form

to Equation (91), will force us to choose the still unknown parameter

C2 such that the term divergent with R vanishes. Thus,

E 2
ET a R, (92)
La 2n r

with

1
2- = 1.3367, (93)
r

if

a 1 2.8545 2.3676 (94)
C 2 4T(1--V)

For v = 1/3


E 2
On (95)
La 2n .2631a

if


C2 = 3.945a .












From Equation (91), the self-energy of a pure single screw dislocation

can be written

E 2
S a r (97)
La= 4n .263a


Two important remarks can be made here. First, the value found

for C2 is very close to those found from the displacement field computa-

tions. This proves the consistency of the correction energy with the

displacement field included from the array of forces. Secondly, it is

found that the core parameter in Equation (97) is independent of V,

Poisson's ratio. This is in complete agreement with Peierls' result,

which gives a value of r0 equal to a/e, that is, equal to .37a, where

e is the naperian base of logarithms.

So both models give nearly identical results, but with a slightly

different r .

For a direct comparison with Volterra's dislocation model, the

core radius r in Volterra's model has to be reinterpreted and cannot be

considered anymore as a cut-off core radius where Hooke's law does not

apply. It is rather a constant containing all the constant terms aris-

ing in the computation of the core energy. The Volterra cut-off radius

could be evaluated in a better way by defining the region where Hooke's

law does not apply, e.g., where the strain is larger than .10. From

Figure 8, such a cut-off radius can be approximated as being about

r0 = 1.5a.

So, the technique of simulating dislocations by a point force

array seems to be very successful in describing the principal features

of the defect, even though a complete accuracy in the computation of the











atomic displacements cannot be reached because of the elastic and

isotropic approximation.


Single and Double Kinks in
a Screw Dislocation

A single or a double kink can be simulated by simply adding to

the array of forces an extra row of shear loops, parallel to the screw

dislocation line, and a semi-infinite or finite extent, respectively.

A representation of the modified array is sketched in Figures 12 and 13.

The displacement field and the energy of these defects are handled in

the same way as for the straight screw dislocation.


A. Displacement field

The displacement field of these defects is obtained by adding

to the displacement field of the pure screw dislocation, the displace-

ment field resulting from the extra row of forces. The latter, u', has

the following expressions for, respectively, a single kink and a double

kink of length 2Na.


u ( ) = pa2 CG2i(x+a',x2-qa,x3
q=0

2
G2i (x+a,x2-qa,x+3 2 r.2i(x+ax2,x3


2
-G2i(x2+a,x2,x3+ C3i(x+ax -P

2+ G3i(x x2'x3+ 3i x2,3
+ G 3i(x1+a,x2,x, ) (9+)











i












X,



Figure 12. Array of Forces for a Single Kink
in a Screw Dislocation


X,
X,


Figure 13. Array of Forces for a Double Kink
in a Screw Dislocation










N
2 a
i(D 1 = a CG2i(x+a,x2-qa,x -)
i (D x2," x,3) =a2 F 2 1 2 a 32
q=-N

2
2(x1+a,x 2qax3 CGi(x+a,x+na,x 3


G2i(x+a,x+Na,x3+ ) + Gi(x +a,x2-Na,x3 -

2
G2i(x1+a,x2-Na,x3+) + [3i(x1+a,x2-Na,x3 -

+0 1a 3 2)]x2+Na1x2- a a
+ G3i(x+a,x2-Nax3 ) G3(x+a,+Nax3 -)


G3i(x+ax2+Na,x3 + ) (99)


Computations of u' and u3 for both kinds show that these displacements
1 3
are very small, even of the region of high distortion in the x2 direc-

tion. They reach a magnitude of a few thousandths of an atomic distance.

Therefore, we shall concentrate our attention on the u' component of u'

and, more specifically, on the atomic displacements in the planes just

above and below the slip plane (x = a/2), since this is the region of

highest distortion.

Because of the existence of singular points at the points of

application of the point forces, several special cases will be consid-

ered. As a first step, we shall restrict our range of interest by notic-

ing the various symmetries in the expression of u'. It shows an odd

symmetry with respect to x3 = 0, and an even symmetry with respect to

1 = -a for both cases, single and double kinks. It shows an even

symmetry with respect to x2 = 0 in the special case of the double kink.

(1) Displacement field of the single kink

The single kink will be the first case considered. For most

of the values taken by xl and x2, u2(SK' has the following expression:
1 ii 2(8bK)











U2(SK)(cx1,x22) = f(x2-qa) f(x2
^(SK) ('1 s2' F -qa)
=0

2
a 1 a x2
ja 1 3'2___
32n 1-V L 2 2 + 3/2
Lx +a) + x2 + a


vL (xl+a) + (x2-qa)2
3-4 1

1 (x2-qa)2a)



+2(1-v) ir 22 ,23/2
(x+a) + (x 2-qa) J


1
) 2 2 +2
(x1+a) + (x -qa) + a


(x2-qa)2

x+a)2 + (x2-qa)2 +


a2]3/2


(101)


On atomic positions, different expressions apply because of the singular

points situated on x = -a:

(a) x2 = na and x1 -a


So
a a'
U2(SXK) (x na, 8
q=1l


- 2 f(na
2


+ 2 f(qa)
q=0/


3
a na
32(l-vI) x +a)2 + na + a2


(b) x2 = -na for any xl


(102)


U(SK) (X,-na,-)


S- s f(qa) + f(na)
1=1 q= V 2


3
a na
32n(l-v) Qx1a)2 + n2a2 + a2]3/2
x1+a) + na + a


(100)


(103)











(c) x = na with n > 0, and x = -a


a an
u(S)(-a,na, ) = + 2(1-) 3/2
U2(SK) q q 2l3
q +1 ( +1)


1 1 1 n+l ia 2s a 3-4v
-2 n wn 2(1-V) 2 3/ C2 16- 1-V
(n +1)


(d) x = 0 and x = -a


a a 1 1 1 1
(K) q=l q (1- +1 (q +1)



+ 2 3a 13- 4V (105)
+ 2C- 321--


In all these cases, f has the same form as in Equation (101).

C2 has the same meaning as before. Since it has been noticed that the

force constant is very nearly constant for all rows of point forces, we

shall give it the value found from Equation (57). The atomic displace-

ments are tabulated in Table 3 for the following values of x1 and x2:

x = -2a, -a, 0, a

x2 = -5a, -4a, ..., 0, ..., 4a, 5a

To these values have to be added the corresponding atomic dis-

placements due to the straight screw dislocations. A mapping of the

atomic configuration above and below the slip plane is shown in

Figure 14. This sketch shows a very high distortion around the point

(x = -a, x2 = 0), but after a few atomic distances from this point,






j 48










.
1 4


-x 0
j j




rl




S0












'I' r=
U ]
r'> '^ ^s
C) (p
P ^ ]
K. 0~ ^ J.
y To 0
'-/ ," ^-4









>-' 0-
00' 0U R
I I Q
0 ,Q U]












the atomic configuration shows very little difference from the configu-

ration of the pure screw dislocation. Even if the actual width of the

kink cannot be expressed analytically, one can conclude that the

defect is very localized.


TABLE 3. Atomic Displacements for a Single Kink in a
Screw Dislocation


U2(SK)/a
x2/a
x = -2a x = -a x = 0 x = a
1 1 1 1


-5 .0019 .0020 .0019 .0016
-4 .0028 .0030 .0028 .0023
-3 .0046 .0052 .0046 .0034
-2 .0086 .0111 .0086 .0052
-1 .0173 .0341 .0173 .0073
0 .0347 .1473 .0347 .0102
1 .0374 .2605 .0374 .0105
2 .0461 .2834 .0461 .0126
3 .0501 .2893 .0501 .0163
4 .0519 .2915 .0519 .0154
5 .0529 .2926 .0529 .0161


(2) Displacement field

We shall consider here

stations have the same form for

ment field at a general point ii

expression:
2 N
u' (x a a
2(DK) 1'x2) 8 8T N
q=-N


of the double kink

a double kink of length 2Na. The compu-

a kink of length (2N+l)a. The displace-

n the plane x3 = a/2 has the following



(x-qa) 1 1 f(x2-Na
f(x2-qa) -yf(x2-Na) 2 f(x2+Na)


3 x2-Na x +Na

S32n(l-v) 3/2 13 2
+ 327 ) x1+a) +(x2-Na) 2+a (+a) +(x +Na) +aJ3/


(106)










with
3-4\. 1 1
f(x -qa) = 2i-
S(x +a)2+(x2-qa)2 /(x +a) +(x2-qa) +a


1 (x2-qa) 2 (x2-qa)2
2(1-) p3/2 2 23/2
2( ) x1+a) +(x2-qa)2 (x+a) 2+ (x2-qa) 2+a2S


(107)

The points where this expression do not apply are the points of

application of the point forces on the row x, = -a. Two separate

cases are considered.

(a) x = -a and x2 = na for n < N

a a +n N- I1
u (DK(-a Ina, ) + n N f(q) f(N+n) f(N-n


+ 3 a N-n N+n + a2 a 3-4\
+ 32TT(1-v) + C 2 (
LL(N-n)2 +i/2 LN+n)2+1lj3 C 2 16- -V
(108)

(b) x = -a and x2 = Na


2N 1 N
u (-, f(q)- f(2N3/2
q=1 (4N +1)

2 \
2C2 a321 /(-4) (109)
+ 2C 32-n 1-M "

with, in both cases,

f(q) 2 1 (110)
S2 + 2 (1-) 3/2 (110)
q +1/ (q2+1)












The sizes of double kinks have been considered, corresponding

to the values of N equal to 1, 2, and 3. Due to the symmetry already

mentioned, only the values of uDK) for the following values of x and
2(DK) 1
x2 have been computed:

x = -a, 0, a, 2a

x2 = 0, a, 2a, 3a, 4a, 5a

These values of u' D)are tabulated in Tables 4, 5, and 6,
2(DK)

corresponding, respectively, to N = 1, 2, and 3.


TABLE 4. Atomic Displacements for a Double Kink of Length 2a
in a Screw Dislocation


2(DK)/
x2/a
x = -2a x = -a x = 0 x = a x = 2a
1 1 1 1 1


0 .0201 .2374 .0201 .0031 .0009
1 .0188 .1363 .188 .0037 .0011
2 .0127 .0288 .0127 .0038 .0014
3 .0058 .0081 .0058 .0028 .0013
4 .0028 .0033 .0028 .0018 .0011
5 .0015 .0012 .0015 .0011 .0008




TABLE 5. Atomic Displacements for a Double Kink of Length 4a
in a Screw Dislocation


2(DK)/
x2/a
x = -2a x = -a x = 0 x = a x = 2a
1 1 1 1 1


0 .0375 .2781 .0375 .0074 .0022
1 .0328 .2661 .0328 .0080 .0023
2 .0246 .1444 .0246 .0066 .0024
3 .0155 .0321 .0155 .0057 .0024
4 .0073 .0097 .0073 .0040 .0020
5 .0037 .0042 .0037 .0005 .0006












TABLE 6. Atomic Displacements for a Double Kink of Length 6a
in a Screw Dislocation



u/(DK)/a
x2/a
S= -2a x =-a x = 0 x 1= a x = 2a



0 .0455 .2872 .0455 .0109 .0036
1 .0429 .2861 .0433 .0103 .0035
2 .0356 .2694 .0356 .0088 .0033
3 .0261 .1461 .0261 .0077 .0031
4 .0164 .0330 .0164 .0064 .0029
5 .0089 .0103 .0089 .0044 .0024





A mapping of the atomic arrangement is attempted in Figures

15, 16, and 17, corresponding to the cases N = 1, 2, and 3, respectively.

The same remarks can be made about the double kink configuration as has

been made for the single kink. Once again, the kink seems to be

a very localized defect for a structure like a simple cubic crystal.

In the case of N = 3, that is, where the length of the double kink is

6a, each single kink part of the whole kink seems to behave like a pure

single kink, which is to be expected when the double kink grows in size.

This argument will be used to extrapolate the energy of a single kink

from the energy of a very long double kink.


B. Energy of a double and a single kink

The energy of a double kink is defined as being the difference

between the energy of the modified 'array of forces and the energy of the

rectangular dislocation loop. As has been developed for the screw

dislocation, a correction energy term will be introduced to take into

account the actual strain field across the slip plane.























0 o


0*
8



8





-* OP


d


/ 4

N~


3
0






0
m
0)









-4
a 0


ID o



Ul
U ,




0


V0
I"


I


. 1











,0,



0.


'-




*'. -'(


, 1'
0



1








I




So
\ 0
'C


Q

0


-


0












Q Q










-4- C

r= C*
L .,, L











10






S1


-H
Q Is









F--'
O 0^


'S .-
k3 y
O T^\





-/ -s -k c
-I '

*3 /rs -^











In a first step, the difference between the energies of the

systems of forces has to be computed. It is simply the self-energy of

the extra double row of forces of length 2Na, and the interaction energy

between this double row of forces and the dislocation loop array.

Sa 1 2a 1
W = a (4N-1) -G22(,0,a + 2 + G33(0,0,a]
2 2 2 3

2 4
+ ,-- G22(0,2Na,0) -G2(0,2Na,a) -G33(0,2Na,0) -G33(0,2Na,a)


2 4 2 4 2N-1
a G23(0,2Na,a)+ 2 a (2N-q) [G22(0,qa,0) -G22(0,qa,a)
-a G23(-q -G22(0,qa~a)
q=l

2 4 2N-1 2 4 R+l La
2p2 a G23(0,qa,a) + a E G2 (pa,- -Na,0)
q=l p=l

La La La
G22(pa,--Na,a) + G22(pa,-+Na,0) G22(pa,+ Na, a)


La La La
+ G33(pa, -Na,0) +G33(pa,2 -Na,a) -G33(pa,--+Na,0)

L/2+N-1
Ggg(pa,-+Na,a) -2G23(pa, +Na,a) 2 G23(pa,qa,a)
q=- L+N+1

+ 4N [G22(pa,0,0) G22(pa,O,a)]


[ L/2-N-1 L/2+N-1 -G
8N Z +2 (L+2N--2q G(Paqa,0)-G(paqaa .
q=l L
L l q=L -N (I)
2 (111)

We shall consider cases for which the length of the double kink

is much smaller than the length of the dislocation line. Under this

condition, T expression of W is greatly simplified.











3-41) i 2N-1
1 =N a 1 3-4v 1 2 1 1 1
--a 8- 2(l~~- 4 q[2 2 )+ 2(1-v)) (2+
Pa + q=1 2q +1

3 2
1 3-4 +1 1 2 1 1 12
+ E & +--- -+--- ---7
S2 1-v I2 +1 q 72 ;3/2
q=l q 2+1 (q


Pa +a 1 1 1 7-8V 1 1 4N
2C2 2C 32n 1-- 32 1_- 2 1-- = (V 2 3/2
2 3 4N (4NO +1)



1 q q .(112)
-N + 2(1-9) (2 3/2 (1
q=l /q2 +1

The force constant C2 has been determined from previous computations

for the screw dislocation. Assuming that the slight modification of

the array does not have any influence on the value of C2, it is deter-

mined such that

A 1 (113)
C 42 2

with
a 2
1 3-4v 1 3-4V 1 1 1 3-4 q +1
A = 1 \-- +--
16 l-V 8r l-v q s 1- q1 -
q=1 q2+1 q=1 q


1 1 1
+ 4 --- (114)
q=1 q +1

So, W/pa3 becomes, after transformation,


W N 1 1 1 1 1
2N [2 -
3 n q 2 (1-) 2 3/2
a 2N q2+(q











pa 1 1 1 7-8v 1 1 4N
2C 2C 32n 1-V 32 1- -- l-v 3/2
2 3 4N+1 (4N-+1)


2N-1
1 -- + (115)
q=1 V+ (q2+1)



If Na is sufficiently large, but still smaller than the length

of the dislocation line, the corresponding double kink will behave like

two separate single kinks. Each will have an energy equal to half the

energy of the whole kink, that is,


W N, +a 1 1 1 q
2 2C2 2C3 32n 1-v 2n q1



1 q 1 (116)
2(1-2) 2 3/2]
(q +l)


The difference between twice the energy of a single kink and the energy

of a double kink represents the interaction energy between the kinks.

Two remarks are necessary here. First, a term proportional to the

length of the kink is included in the final expression, and is expected

to cancel out with a similar term in the correction energy. Secondly,

the force constant C3 appears. This has to be considered an unknown

parameter, since there is no physical condition which can be applied to

evaluate it. An approximation could be made by setting C3 equal to C2,

but there seems to be no particular justification for such an assumption.

The second step needed to obtain the final expression for the

energy of a double kink is the computation of the difference between

the correction energy for the pure screw dislocation and the correction











energy for the kinked screw dislocation. Since the former has already

been computed, we shall focus our attention mainly on the latter.

The region that suffers a strain larger than one-half is bounded

by the planes x3 = a/2, the surface parallel to x3 where the relative

displacement is equal to a/2 and the plane x = Ra/2 (see Figure 18).

The region situated between x1 = Ra/2 and xl = Ra will not be con-

sidered, since its deformation is the same as for the straight screw

dislocation.

We have already seen that the displacement field caused by the

array of forces is


ul (1,X2,X3)


2 (X 3)+ U'(x1x2X3) (117)

U(x 1x2,x 3x )


By taking into account the fact that after relaxation, the atomic

positions need to be referred to their closest neighbors (see Figure 19),

we are led to choose as actual displacement field across the slip plane

the following expressions:


v1(xy2,x3) = u (x1,x2,x3

v2(xl1,2,x3) = u(x,3) + u(XX2,X3) x (118)


v3(x1Y2,x3) = u(x 'x2x3

where y2 is related to x2 such that
2a
x3 > 0 Y = x2 +

a
x < 0 y = x 2
32 2






60




.'









Ra/- a----i- -- -- --- --








Figure 18. Region of High Strain for a Double Kink
in a Screw Dislocation


..x.


Figure 19. Atomic Relaxation for a Double Kink
in a Screw Dislocation











The stress fields corresponding to u and v have the following

components c.. and 'T., respectively,


a11(x1,x2,x3) =



C22(x1,x2,x3) =



a33(x1,X2,X3) =



C12(XlX2,x3) =



c13(x1x2,'3) =




a23(X1,X2,X3) =




T11(x1Y2,x3) =


T22(x1,Y2,x3) =

S33(x ,y ,x ) =


T12(x1,Y2,x3) =

T13(xlY2,x'3) =


723(x1lY2,x3) =


6ul (u62 3 u3
(2+20) + x +


6u, au, au,
2 1 3
(X+2p) + + +






S + x



11 l0 3 +2) )





\JX3 OX3 X2C}





11 (x1'x2'x3

C22(x1,x2,x3)


C33(X1,x2,x3)

C12 (x1,x2,x3)


Cl3(x1 x2,x3


C23(x ,x2,x3) -


(119)


(120)


Recalling Equation (79), the general form for the correction

energy is the difference between the strain energies in the region

of interest,











E = y u, n dS -1 S .. v. dS'
c 2 S 1 3 2


(121)


The surfaces S and S' differ only by the range of integration over x2

and y2, respectively. When x3 is positive, y2 is defined between

-La/2 + a/2 and La/2 + a/2, and when x3 is negative, y2 is defined

between -La/2 a/2 and La/2 a/2. Under these conditions, the inte-

grals involved with functions of y2 can be considered as integrals of

functions of x2 with different limits. The various identities follow:


La/2+a/2
Y f(y2) dy2
-La/2+a/2


La/-a/2

-La/2-a/2


La/2
5 f(x2) dx2
-La/2


La/2
f(y2) dy2 = f(x2)
-La/2


So E can be written as a surface integral
c
on a unique variable x2.


E = (.ij .i Tij v.) n dS
2 S 13 1 13 1

Replacing T and v. by their expressions
ij i
respectively, leads to


Ec = x3 21 nl + (x322 + 3n2
S


for x3 > 0,


dx2 for x < 0 .


(122)



(123)


over a function, depending


(124)


as functions of j.. and u.,




+ P(U2-x3) + X323 n3} dS,

(125)


a/2
E = dx23 x321 n1 + (x32 + u3)n2 ds
-a/2 C

a/2 La/2
-f dx3 5 x3c21 (~,x2,x3) dx2 dx3 +
-a/2 -La/2












La/2 Ra/2
+ j dx2 r (xx 2, Y 2 3 2 23(x1,x2,'Z dx1
-La/2 X1(x2)
(126)

In these integrals, C is the curve defined by


a a a
u2(x1,) + u2( x2 = (127)



that is, the "line of the kink," and X (x ) corresponds to a point

(x1,x2) on this curve.

The evaluation of these integrals is a complicated mathematical

problem which has to be solved numerically. Unfortunately, such numer-

ical computations have not been possible to achieve yet, mainly because

of the very complicated expressions for the displacement and strain

fields. However, further research on this mathematical problem can be

carried out and will lead to the correct answer for the energy of

a double kink. The final expression for'the energy would be obtained by

subtracting the difference between the correction energies given by

Equation (115). The linear term is expected to cancel, so that the

energy is a finite number. The limit of this number when N becomes

large would be twice the energy of the single kink.
















CHAPTER 5


EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED
FROM AN ARRAY OF SHEAR LOOPS



The same procedure as followed in the case of the screw disloca-

tion will be used in the case of the edge dislocation.


Displacement Field

As seen previously in the chapter concerning the whole rectan-

gular array of point forces, the regions where the dislocation loop has

a pure edge character are delimited by



and 2
La La
2 2 +e2 2 2 2


where e1 and e2 are small compared to Ra and La, respectively. Since

both regions simulate two identical parallel dislocations of opposite

sense, we shall only consider the first one. A translation of the x2

axis to the first row of forces x2 = La/2 will simplify the expres-

sions. The same symbols xl and x2 will be kept, having now the meaning

of 61 and e2, respectively. This part of the array of forces is repre-

sented in Figure 20. The only difference with Equation (21) is now the
L L-l
summation of q will be S and S So, u (r) becomes:
m
q=O q=l

2 R/2 L L-l
u (r) = Z + G m(x -pa,x2-qa,x -) -
p= -R/2 q=0 q=1








65















r
0

c
0





















o
+-4




bi
*0
a





-4
























S x
-l




03


0-



) i
/ m2











Ks











Gml-pa,x2-qax3+ G3m(x1-pa,x2x3-)

+ G3m (xl-pa,x2,x3+ + G3m(l-Pa,x'2-La,x3-)


+ G3m(x1-pa,x2-La,x3 + (128)


The summations on p and q will be accomplished as before, with

the help of Euler's formula (Equation (29)). The first summation

carried out will be on p, since the displacement field of the edge dis-

location ought to be independent of xl. Euler's formula becomes simply,

R/2 +m
E f(p) = f(x) dx (129)
p= -R/2 -

where f is a symbol for the whole expression to be summed. The dis-

placement field takes the form:

u1(x1,x2,'3) = 0 (130)

_2 (x a 2
a L (x2-qa) + a2
u2(x 1x2,x3) 1a6 ( + -2(1-) 2- a 2
q=0 q=l (x2-qa) + (x3-


1 (x3 a, (x 2
+x 3 + a 2 ( x -a
+-v 2 a2 (x3 2 a 2
(x2-qa) + (x 3 ) (x2-qa) (x3 2


a x2(x3 -) x2(x3+
16T(1-v) 2 x a 2 2 2 a 2 (131)



a
x2 + (x- ) x2 + (x3+2)



a L L-1 (x2-qa) (x -)
U3(X1,x2,x3) = 16i(1-V) (2' a,2
\= 2










(x2-qa) (x3 + 2)a R L2
(x2-qa)(x3 a 1-2V R/2 + +L2
2 ( a 2 1 4+ 1-v R
(x2-qa) + (x3 )


a 3-4v L + a 3-4V x 2 +( a,2 +
Sr 1- L 32n 1-v 2+


2 a a 1 (x3 -) (x3 2 +
+ n X2 + (x3 6 T 1-, -v -2 a 2 2 +a
x2+ (X X +
2 3 7) x2 + (x3 2
(132)

Unfortunately, these expressions are much less simple than those

found for the screw dislocation. But some of the essential features of

the displacement field of the edge dislocation can be noticed already,

that is, the lack of displacement in the x1 direction and the fact that

u2 and u3 are independent of x1.

Euler's formula applied a second time will give us the final

expression for the displacement field. The mathematical difficulty,

arising because of the singular points in the integrand, is overcome in

the same way as for the screw dislocation. The same symbols, u2 and

u3, will be used for the successive terms added to approximate u2 and
3 2
u3, respectively. For u2(x1,x2,x3) we find
L 2 13 22

a a -1 x x -a
( 1 a 2 2 -2
u2 (X x3= IT 2x3 + (x3 ) tan tan-1 ----
x-a x3 2 3

(x-a---x -)
(x3-2 tan -- tan
x3 3 2_ J
2 a 2 2 a
1 3-4v x2 + (x3 )2 + -a (X3 2
32 -TT 2 a2 -a + (x2-a) n 2 -- 2
x2+ (x3 (x2-a) + (x3 )
2 3 2j










a 1 x2(x3 ) x2(x3+ 2)
16-T 1-v a a 2 + 2 2 (133)
2 + (x3 2+ (x3

r 2 + a 2 -2 + a 21
1 a 3-4v x2 + (3 (x2-) + (x3
2(1,x2'x3 = n 1-v 2 2 a 2
+ (x3 -) (x2-a)2+ (x3 -
2 3
a )2 a 2 a 2
a 1 (x3 (X -) (x3 +
+ 32Tn 1- v 2 a2 2 a 2 22 a 2
x2 + (x3 ) (x2-a) + (x3 ) x2+(x3

(x3 + 2
2 aa2 (134)
(x2-a) + (X +



2 a2 3-4v x2 x2-a
u2(x12x2'x3 192 1-9 2 a 2 a 2
[2 +(x3-) (x2-a) + (x3

a 2
x2 x2-a a2 1 2(3-
2 a 2 2 a 2 96 1- 3 2
2+ (x3 a (x2-a) + (x3 +(x3
a2 2 2 2 (

(x2-a)(x3-)2 x2(x3+)2 (x -a)(x3+ )2

x2-a)2+ (x3-) x (x3 )2 x2-a)2+(x3+)

(135)
3 a5 3-4 2 x-3(x3 2
u(x xx)- 3-4v) x2 +
u2(x1'x2'x3 5760n 1- 3r2 a23 +
Lx2+(x3

(x2-a) x2-a)2-3(x3 )2] x2 3-3(x3 )2
S3 )233
x2-a)2+(x3 2 x2+(x3 2

(x2-a) (x2-a)2-3(x3+) 2 a4
2 23 1 480TT 1-
(x2-a)2+(x3 2 J










Sr2 2 2 a) 2
a 2 2 x-(x3)2 a 2 x2 LX2 (x3+
(x-2) 2 (x3



a 2 (x2-a) (x2-a)2 (x3 )2
+ (x3 4
(x2_a) 2 a2 2
(x-a)2 + (x3 a2

(x++2 22]
(x,-a) [x2-a)2 (x a2(1
( 23 2 (136)
x2-a)2 + (x3 2

with
0 1 2 3
u2 = u2 + u2 + u2 + u2 + (137)


For u (x xx3) we find
3 1' 2 x3
/-2-2
0 (1-2v)a R/2 + R /4+L2 1 a
u3 4(1-v) RLa + 32n(1-v) 3


(-v) x3 2+ x +r 22 + a) 2 (x3 a


S(x3 2 +x3 [(x3 2+ (x2a)2
+ 23 +a 2 + x2] +Oxn3 2 + 2+ (x-a) 2





a (x3 2 (x3 2
16n(1-v) 2 2 2 a 2138)
x2+(x ) x2+(x3


1 a X2(x3 -) x2(x3
u3(x1,x2x3) 32Tn(-v) 2 2 2 a 2
x+(x3 ) X2+(X3+2)










(x2-a)(x3 -) (x2-a)(x3+)
S(-a)2 (x a)2 (x-a) ( )
(x2-a) + (x3 -P (x2-a) + (x3


2 2 a
u3(l'2'3)= -192T(1-)) 3 -2


(x2-a) (x3 2 -
+
(x2-a)2 + (x3 2
2 a 2
(x2-a) (x-3+

x2-a)2 + (x3


3 a4 a
U3(x1,x2,x3) 1920TT(1-v) x3


x2(x3 -)2



2 a(x 2
2 (x3 '
(x x + a) 2 2
32 2 + (x +) 2 3
< .X2> r-!i~i


(140)


4 -a2 x2 a )
S + (x3 2-
[ + (X3-;2]


(x2-a)4 6(x3-)2(x2-a)2 + (x
+- 22]4 2
x2-a)2 + (x3 -)2 4

4 a 2 2 a 4
a 2 6( + ) x2 + (x3 +
- (x3 4
4 _a 2 4



(x 2-I4 6(x3+ (2x2-a)2+ (X34 -
(x2-a)2 + (x3 )2


0 1 2 3
u3 = u3 + u3 + u3 + u3 + .


with


(141)


(142)


(139)











The constant in the expression of u3 means only that the point of

non-bending of the lattice planes is set at x2 = La/2. However, the

relative positions of the atoms with respect to each other are inde-

pendent of this constant, and the stress ana strain fields will not

depend on the constant terms.

The component u2(x ,x2x3 ) has four singular points (x2 = 0,

x2 = a with x3 = a/2) and u3(x1,x2,x3) has two singular points

(x = 0 with x = a/2). This is not surprising, since the whole

array of forces is a superposition of two arrays of forces having

magnitude pa2/2, one starting at x2 = 0 and one starting at x2 = a.

An asymptotic expression for u2 and u3 can be computed by

considering x2 and x3 large with respect to the atomic distance.

The following expressions are found:

ul(x ,x2,x) = 0, (143)


a n/2 -1 x2 1 x2x3
u2(x,xx3) + tan x 2- -2 (144)
21 x2'x3 27 x 2(1-v) 2 2
-n/2 3 x2 + x3

2
a 1-2v 2 2 1 x3
n (x + )
u3(x12x2,x3) 2 n (x23 2(l-v) x2 2-



These equations can be compared to those arising from the

ordinary continuum model for an edge dislocation of the same sign [20]:

ul(x1,x2,x3) = 0, (146)


3) = -1 x3 x2x3
u2(x x2x tan + 2 2 (147)
2 2(1-N)(x2+x3)
i_ ^2 3











2 2

u3(1 ,x2,x3 4( x2+x3 2 22 (148)
) (x2+x3) + 4(1-v)(x2+x3)


Both sets of equations are identical when one is aware that u3

is determined only up to a constant. Adding the expression a/8n(1-v)

to Equation (145) leads automatically to Equation (148); this physically

means a change in the "cut plane." One can notice too that these
-0
asymptotic expressions for u come only from the first approximation u

This means that for a point situated at a large distance from the dis-

location line, the discrete array of point forces appears to be a con-

tinuous distribution of force on the two planes x3 = a/2.

A mapping of the atomic displacements, except for the singular

points, is shown in Figure 21. Obviously, this model shows a strong

dissymmetry with respect to the extra half plane (x2 = .75a, with

x3 < 0) in the region of the core.

The relative displacement across the slip plane in the direc-

tion x2 is simply a particular case of the expression of u 2(x ,x2,x3,

as obtained before for the screw dislocation:

a a a
u2(x2) = 2(x2') u2 -) = 2u2(x,) (149)



A suitable form foru2 (x2 ) can be obtained either from

Equation (131) or the set of Equations (133) to (136):

a L L1 3-4 (x 2-qa)2 + a2
Au (x) 7n a-) + 2
2q=0 q=l ( (x2-qa)

2 ax
1 a2 a ax2
S1- 2 2 8n(l-) 2 2 (150)
(x -qa) + a x2+ a
2 8 2
+aa









-T
o0 0 0

0 0 0 0
0


Q u


D_


0


0


S0

0 O


r0

0


In
<0_
3c


0 0O

O O o o o0

0 0ooo0


O 0O O 0 o


0

0

0

0


.m


-










or
0 1 2 3
u2(x) = Au(x2) + u (x) + Au2(x2) + Au (x2) + .. (151)

with
0 a 1 2 -1 -a a2 1 2
u2(x2) = A + tan1 + ta --- + 81- 2 2
x2+a

2 22 2
1 3-4v x2+a x2-a) +a
+16 1-v 2 n 2 + (x2-a) 2 (152)
a2 (x2-a)


r 22 2 2
Sa 3-4 v 2 2
u2(x2) = 32-n 1- + 2
2 (a2 a)

3
a 1 1 1
167n 1-v 2 2 2 (153)
2 a) + a


2 a2 3-4v 1 1 X2 x2a
[x2 x2+a (x- a) +a


a 4 1 x2 x2-a (1 5)
48TT 1- 22 2 2 22 '
x2+a (x a +a


3 a4 3-4v 1 1 x2(-3a2
Au2(X2 2880 1-v 3 3 2 2 3
2 x2 (x2-a) (2 + a )

(X -a _a) 2 2 2 2
(x2-a) [(x2- a a6 2-a

2 2 3 240T 1-v 2 2 2 4
(x2 -a) +a ]J 2 )

(x 2-a) (x 2-a)2-a2]
+ 4-a (155)
(x2-a)2 +2) a












The curves representing Au2 are plotted in Figures 22 and 23,

corresponding to the region close to the dislocation line and to a more

extended region, respectively. Only a few terms Au2 are needed to

-3
obtain relative precision equal to 10 in the region outside the points

x2 = -a and x2 = 2a. Unfortunately, the region in between is not known

and Au (0) and Au2(a) must be evaluated by an interpolation scheme.

Values of Au2 are listed in Table 7.


TABLE 7. Relative Displacement Across the Slip Plane
for an Edge Dislocation


x2/a Au2/a


5 .0512
4 .0626
3 .0805
2 .1123
1 .1817
0 .4
1 .68
2 .8280
3 .8921
4 .9221
5 .9391




The width of the dislocation can be reached by evaluating the

region where the relative displacement has values between a/4 and 3a/4.

From Figure 22, this region can be easily measured and has the value

W = 1.93a. This value is slightly higher than that obtained from

Peierls' model (W = 1.5a for V = 1/3). This would mean that our model

shows a dislocation slightly more extended than Peierls' model. This

result is opposite to what has been found for the screw dislocation.












































































a a'
4. 4






IJ I


> 'o
<*--






































N '0


I-
3 3






I
S J
' T) J


A
4



00


!rz

I0
kgw


ao 9


0
t o










(3

"
w

L
^ g
i





Is
co


J



S I












The relative displacement at atomic points can also be obtained

directly from Equation (128):



Au2(na) = a E + S 22(pa,na-qa,O)
p= \q=O q=1


G22(pa,na-qa,a)] G23(pa,na,0) + G23(pa,naa)) (156)


Replacing the variable (n-q) by u and making the singular point appear

at q=n, leads to the three following equations for u 2(na) correspond-

ing to n>l, n= and n=0, respectively.


+m n n-1 -
Au2 (na) = pai S + S + G 22(pa,ua,0)
p=- = u=1l u=l u=1

+m
G22(pa,ua,a) 2 G23(pa,na,a)
p= -

2o 2 2
+ 4pa2 E CG2(pa,0,O) -G22(pa,0 +O+2 pa2 G22(,a),
p=l 2
(157)



u2(a) = 2a2 S G 22(pa,ua,0) G22(pa,ua,a)
p= -_ u=l

+m
+ Ga [G22(pa,a,0) G22(pa,a,a) G23(pa,a,a)
p= -

22anO2 2
+ 44a2 S G22G(pa,0,0) G22(pa,O,a)] + 2a2 G22(0,0,a),
p=l 2
(158)


Au2(0) = 2pa2 u 1 G22(pa,ua,0) G22(pa,ua,a3
p= o

2
P=


+ 2pa2 G(p22(pa,0) 22(pa,0,a) + --a2 G(0,0,a)
p=l 2
(159)












After computation of the discrete sums using Equation (26),

these three equations become, in the same order:


u (na)= 2a2 +a 11.4392 9.4720V a 1 n
2 C 8rrT 1- iTT1-) 2
2 n +1

2 m

16 1-v 2 1-v 2
p=n p=n+l p p +1

2pa_ a 6.4060 5.6512V (161)
bu2(a) = --+ (161)
C S 1 v




2
pa a 11.4392 9.4720V
5u2(0) = + (162)



For each value of n, a direct comparison can be made between

Equations (160) to (162) on the one hand, and Equations (133) to (136)

on the other hand. Since both ought to be identical, C2 can be easily

deduced by subtraction. Values of Au2(na) computed in both ways and

the corresponding C2(n) are listed in Table 8. These values have been

computed for ) = 1/3. The force constants C2 seem to be much more sen-

sitive to n in the case of the edge dislocation than in the case of the

screw dislocation. Therefore the evaluation of the atomic displacements

at x2 = 0 and x2 = a, using the technique employed for the screw dislo-

cation is not possible due to the uncertainty of C2(0) and C2(1).

If the variation of C (n) is neglected after a few atomic distances

from the dislocation line, and C2(n) set equal to C2 () for every n,

the relative displacement becomes

a 3-4V p +1 1 2
Au (na) = a + p n- + [ 2
2 16 p=n p=n+1 1 2 1-v 2
L p p +1

a 1 n (163)
STT 1-v 2
n +1












Table 8. Relative Displacementsand Force Constants C2 at
Singular Points for an Edge Dislocation



n (Au(n) -2 Au (n)/a C
a 2 2 2 2


0 .253 a .4 E 6.80
1 .232 .68 m 4.40
2 .3625 .8280 4.2792
3 .4027 .8921 4.0867
4 .4243 .9221 4.0176
5 .4376 .9391 3.9872
6 .4466 .9500 3.9729
7 .4531 .9576 3.9635
8 .4580 .9632 3.9588
9 .4619 .9675 3.9557
10 .4649 .9709 3.9526



S.4927 1.0000 3.9432


The

negative xl


relative displacement at atomic points corresponding to

(x = -na) can be deduced directly from Equation (150):


a> m\ 2 2 1
a 3 3 Fl-4_ p l + 1 2
Au2(-na) = 16 i
p=n p=n+ p p +U


a 1 n
8r1 T-v 2
n+1


(164)


One can notice that adding Equations (163) and (164) leads simply

to a. This means that making the approximation that C2(n) is constant

and equal to C 2() for every n would imply that the relative displacement

is symmetrical with respect to xl = 0. But this is in contradiction with

the actual relative displacement computed from Equations (152) to (155)

and listed in Table 8, where a symmetry with respect to xl = a/2 is












clearly apparent. So, contrary to the case of the screw dislocation,

C2(n) cannot be replaced by C2(m), and their difference is sufficient

to shift the symmetry of the dislocation with respect to x1 = a/2 to

x1 = 0. In fact, it seems logical that in both cases, for the screw disloca-

tion as well as for the edge dislocation, the dislocation line lies

between two rows of atoms.

As mentioned for the screw dislocation, a distribution function

for infinitesimal dislocation loops can be introduced, following

Equation (64). Its expression is the component o21 of the dislocation

density tensor mentioned previously. So, by differentiating Equations

(152) to (155), this distribution function 21(x2) has the form

0 1 2 3
a21(x2) 21(x2) + 21(x2) + 21(x)+ 21(x2) + (165)

with
2 2
S(x2) 1 8(1) a2 + 2 +a
21( 2 8H(l-v) 2 212 1T- 1-1 2
+x2 a +(2-a) x2

2 2 2 2
(x2-a) +a 2 a -x2
+ -2 S- 1 -v 2 (166)
(x 2 871-a)2% 2 2 2
(x2-a) (x2+a2)



1 a 3-4v x2 + x2-a 1 1
a21(x2) 16T 1- 2 2 2 2 x
X2+a (x2-a) +a 2 2


a3 1 x2 x2-a
a+ 8l- 2 2 2 a 1)7
(x2+a2 (x2-a) +a



2 a2 3-49 1 a 12- 2
'21(x2) 96n 1- 2 2 + 22 + .2
2 x(a 2) a2+(x2-a)2











S2 2 2
4 a x2 a (x2-a)
a 1. 2 2 3 + (168)

(x2+a) (x2-a)2+a2]j



r 2 2 4 4
S 4 3a x-x-a
3 a 3-4v 1 1 22
21 960 1-v 4+ 4+ 2 2 4
2 2- (x2+a )

2 4 44 2 r 2 4 4
3a (x -a) (x2-a) -a a6 1 10x2a -5x2-a
x _) 24 + 240 1-v 2- 2,3
2 22 (x2+ a )
x2-a) +a L 2

2 2 4 4
10a (x 2-a) -5(x -a) -a1
+-. (169)
C(x2-a) 2+a2


Successive approximations of c21(x2) are plotted in Figure 14.

A direct comparison with Peierls' model cannot be achieved

successfully because of the lack of symmetry of the displacement field

around the core in our model. Both models give the same result for

points far from the dislocation line, but cannot be matched close to

the core region. We shall see that a description of the edge disloca-

tion from an array of prismatic loops is much more satisfactory and

more close to the real atomic arrangement at the center of the defect.


Self-Energy of the Edge Dislocation

As seen previously, two steps are required to obtain the final

energy of the edge dislocation. First the energy of the system of

forces will be computed, and then a correction energy term will be

introduced to subtract the excessive strain energy across the slip

plane, due to the system of forces itself.






83



I,







/0 0
11



/04
/
:2







-4o
I




I

--. a
~-^- -~-----------------'~'~. ~ ...'~"---- &


av * a a


















< *I LI
I ,' I
1 -
^''*S'^ .1 B










^ 0 0^ v'
a ^

j I p .1












When the rectangular array of forces is extended in the direc-

tion perpendicular to the Burgers vector, it simulates a system composed

of two antiparallel straight edge dislocations separated by La. The

energy per unit length of the array forces is the limit of W/Ra when

R is much larger than L. From Equation (28), it takes the form,


W La2 F A 2 2 r [1 1 \
Lpa Z_ + L + a22a [2


7.6170 2.8588v
4-(1 - ,j *(170)

with

2.860 2.368v
A = (171)
1 V


As previously seen for the energy of the kink in a screw dislocation,

C3 is an unknown parameter which cannot be obtained by a physical

argument as in the case of C2.

The choice of a suitable displacement field describing the

dislocation will be made as for the evaluation of the correction energy

of the double kink in a screw dislocation. The region where the relative

displacement across the slip plane is larger than a/2 is contained

between the planes x2 = a/2 and x2 = La a/2, as shown in Figure 25.

A displacement field chosen to approximate the actual relative

displacements of atoms across the slip plane is:

v1(X1,Y2,X3) = 0, (172)

v2(lx'y2,x3) = u2(x1lx2'x3) u3, (173)


v3(xlY2,'3) = u3(x1lx2,x3) (174)





j n


jo ____~_
X












depending on whether x3 is positive or negative, y2 is taken as:

a
Y2 = x2 + for x3 > 0 (175)

a
y2 = 2 for x3 < 0 (176)


The principal difficulty with this choice of v is that it is not

symmetric with respect to x3 = 0, since u2 varies with x2. We have:


v2(x1'2'X3) = u2(x'x2,x3) x3 (177)

and

v2(x1,Y2,-x3) = -u2(x1,x2+a,x3) + x3, (178)

where

I2 (x1,x2,x3)| i fu2(x1,x2+ax3) (179)


Since this lack of symmetry will have an effect only on the constant

terms which do not appear in the coefficients of R or n R, it will be

considered a sufficient approximation for our study.

The tensors a and T corresponding to the displacement fields

u and v, respectively, have the following components:


r u 2 Au3
011 Ex2 x3




2 3
22 = (a+2) 5 + 2 x3


Bu3 5u2

3 2


023 =32 i 2 + 6 ,3 (180)


___












11 (2,3) = "11(x2,x3)


722(y2,x3) = C22(x2,x3)


733(Y2,X3) = a33(x2,x3

723(2,3) = 023(x2,x3) p (181)


The correction energy is by definition the difference between the strain

energies associated with the above stress fields contained within the

volume of integration, i.e.,

E = 1 dV i e.. dV' (182)
c 2 Vij ij 2 13


By using Green's theorem, the volume integrals can be transformed into

surface integrals on the surfaces bounding V and V':


Ec = 1 0 u. n. dS 1 T.. v. n. dS', (183)
S2 1 S. 13 1 3


where n is the normal to the surface of integration. The differences

between S and S' are mainly due to a translation of the plane x3 = a/2

of an amount +a/2, and a translation of the plane x3 = -a/2 of an

amount -a/2. Because of these differences between the limits of

integration, all integrals over a function y2 can be transformed into

integrals over a function of x2:

La La-a/2
x3 > 0, f(y2) dy2 = f(x2) dx2, (184)
a a/2

La-a La-a/2
x3 < 1 f(y2) dy2 = f(x2) dx2 (185)
0 a/2


So the correction energy has the form










1
E = (a. u. T. v.) n. dS (186)
c 2 1 i 1 I

Replacing v and T in function of u and C, respectively, in Equation

(186) leads to


Ra = x322(,x3) + U3( x3) dx3
a/2


La-a/2
+ 23(x2') + 2(x2dx (187)
23 2 + 2 2'2 2J 'd2,
a/2

or
E a/2
R = 2 [x+2L) + u (( dx

La/2 ur u2(x2'
a/2 1 3

+ pa [u3(La/2,a/2) u3(a/2,a/2)] (188)

For computing these integrals, we shall use the expressions for

u. and its derivatives from Equations (131) and(132). The final summa-
1
tion with respect to q will be completed at the last step. Expressions

used for x3(au2/x 2) and 6u2/3x3 are listed below.

u2 ax3 L L- 1-4v x2-qa
x3 x- (x2,x Y16, 1-_ 2 a 2
2 q=0 q=l (x3-qa) +(

x2 -qa 2 (x a)

(x2-q a)2+(x3 -2)2 + 1-v I2 (x )2
x2 +3 (x2-qa) +(x3 )2

~\ a 2 2
(x2-qa) a3x ( _)2 2-
2-ax3 a (x3 ) x2

(x2-qa) 2+(x3 __ )2 j 162(1v-) x 2 a 2 32
2- fl x2 (3 5










a2 2 "3
a 3 a 2 2
+ (x +-) 2- ) (189)








3 23

(x2-qa)2+ (3 (x2-qa)2 + (x3 )
LxCcx3 2] )

3 f a
)2 a2 2+_ 2 16T(1-,)
C(x2-qa)2 + (x3 23


2 _a2 + (12 20
r 2 a 2 r a a2



x 2 3 2 x +2 3 2 2

We will not reproduce here the details of integration, but only write

the final results for each step of the computation. The correction

energy becomes

c a2 L= 1-2 1 -)2 +1 -1
R a -4 1 2 + 4 tan- (q
(q=0 q=1 (q -2)

~a 1-2V a as 1 5 7
2+ 1- 2 L L + 2Vn 1-2v 4


-1 1 16V2 18 + 5 M
+ TT tan 2 + (1- )(1-2V) + n 2(191)
(-2v(

Computing the single sums as mentioned in Equation (26) leads to the

final expression for the correction energy of the system, per unit

length of edge dislocation.




Full Text

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SIMULATION OF DEFECTS IN CRYSTALS BY POINT FORCE ARRAYS By JEAN-PIERRE JACQUES GEORGES A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 07 THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972

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ACKNOWLEDGMENTS The author wishes to express his deep appreciation to Dr. C. S. Hartley, Associate Professor of Engineering Science, Mechanics and Aerospace Engineering, and chairman of the supervisory committee, for guidance and counsel during this research. The author also wishes to express his appreciation to Dr. L. E. Malvern, Professor of Engineering Science, Mechanics and Aerospace Engineering, to Dr. M. A. Eisenberg, Associate Professor of Engineering Science, Mechanics and Aerospace Engineering, to Dr. J. J. Hren, Professor of Materials Science and Engineering, and to Dr. J. B. Conklin, Jr. , Associate Professor of Physics, for serving on the supervisory committee. Special thanks are due to Dr. S. B. Trickey for his helpful assistance. The author wishes to express his special gratitude to Dr. A. K. Head, Chief Scientific Officer at the Commonwealth Scientific Industrial Research Organization, Melbourne, Australia, for his very pertinent comments. Thanks are also due to Mrs. Edna Larrick for the typing of this manuscript. This research has been sponsored by the National Science Foundation under the Grant GK 24360.

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi KEY TO SYMBOLS viii ABSTRACT • • xi CHAPTER 1 INTRODUCTION 1 2 BASIC CONCEPTS 4 Point Force 4 Double Force ... 7 Primitive Dislocation Loops 8 3 RECTANGULAR DISLOCATION LOOP IN SIMPLE CUBIC CRYSTAL . 14 Displacement Field 14 Elastic Potential Energy 16 4 SCREW DISLOCATION IN SIMPLE CUBIC CRYSTAL 20 Displacement Field 20 Self-Energy of the Screw Dislocation 36 Single and Double Kinks in a Screw Dislocation .... 43 5 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED FROM AN ARRAY OF SHEAR LOOPS 64 Displacement Field 64 Self-Energy of the Edge Dislocation 82 Single and Double Kinks in an Edge Dislocation .... 91 6 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED FROM AN ARRAY OF PRISMATIC LOOPS 105 Displacement Field 105 Self-Energy of the Edge Dislocation 120

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TABLE OF CONTENTS (CONTINUED) CHAPTER Page 7 CONCLUSIONS 124 BIBLIOGRAPHY 126 BIOGRAPHICAL SKETCH 128 iv

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LIST OF TABLES Table Page 1 Relative Displacement of Atoms Across the Slip Plane for a Screw Dislocation 26 2 Variation of Force Constant C„ with the Atomic Positions in a Screw Dislocation 30 3 Atomic Displacements for a Single Kink in a Screw Dislocation 49 4 Atomic Displacements for a Double Kink of Length 2a in a Screw Dislocation 51 5 Atomic Displacements for a Double Kink of Length 4a in a Screw Dislocation 51 6 Atomic Displacements for a Double Kink of Length 6a in a Screw Dislocation 52 7 Relative Displacement Across the Slip plane for an Edge Dislocation 75 8 Relative Displacements and Force Constants C at Singular Points for an Edge Dislocation 80 9 Atomic Displacements for a Single Kink in an Edge Dislocation 95 10 Atomic Displacements for a Double Kink of Length 2a in an Edge Dislocation 96 11 Atomic Displacements for a Double Kink of Length 4a in an Edge Dislocation 96 12 Atomic Displacements for a Double Kink of Length 6a in an Edge Dislocation 96 13 Displacements and Force Constants C at Singular Points for an Edge Dislocation 116

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LIST OF FIGURES Figure Page 1 Coordinate System to Evaluate the Core Region Around a Point Force 10 2 Prismatic Loop in Simple Cubic Crystal 10 3 Shear Loop in Simple Cubic Crystal 13 4 Shear Loop with Principal Axes 13 5 Rectangular Array of Shear Loops ..... 15 6 Simulation of a Screw Dislocation 21 7 Relative Displacement Near the Core of a Screw Dislocation 27 8 Relative Displacement Between -4a and 4a for a Screw Dislocation 28 a 9 Atomic Arrangement in Planes x = ± — of a Screw Dislocation 32 10 Distribution Function of a Screw Dislocation 35 11 Region Where the Correction Energy Applies for a Screw Dislocation 37 12 Array of Forces for a Single Kink in a Screw Dislocation 44 13 Array of Forces for a Double Kink in a Screw Dislocation 44 14 Atomic Arrangement Around a Single Kink in a Screw Dislocation 48 15 Atomic Arrangement Around a Double Kink of Length 2a in a Screw Dislocation 53 16 Atomic Arrangement Around a Double Kink of Length 4a in a Screw Dislocation 54 17 Atomic Arrangement Around a Double Kink of Length 6a in a Screw Dislocation 55

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LIST OF FIGURES (CONTINUED) Figure Page 18 Region of High Strain for a Double Kink in a Screw Dislocation 60 19 Atomic Relaxation for a Double Kink in a Screw Dislocation 60 20 Array of Forces Simulating an Edge Dislocation .... 65 21 Atomic Arrangement in x = Plane for an Edge Dislocation 73 22 Relative Displacement Close to the Core of an Edge Dislocation 76 23 Relative Displacement for an Edge Dislocation 77 24 Distribution Function for an Edge Dislocation 83 25 Region Where the Correction Energy is Computed for an Edge Dislocation 85 26 Array of Forces for Single Kink in an Edge Dislocation. 92 27 Array of Forces for Double Kink in an Edge Dislocation . 92 28 Atomic Arrangement for a Single Kink in an Edge Dislocation 98 29 Atomic Arrangement for a Double Kink of Length 2a in an Edge Dislocation 99 30 Atomic Arrangement for a Double Kink of Length 4a in an Edge Dislocation 100 31 Atomic Arrangement for a Double Kink of Length 6a in an Edge Dislocation 101 32 Array of Prismatic Loop in x = Plane 106 33 Array of Prismatic Loop in x = Plane 106 34 Relative Displacement of an Edge Dislocation 118 35 Atomic Arrangement in x = Plane of an Edge Dislocation 119

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KEY TO SYMBOLS A Constant defined in Equation (72) a Lattice parameter b Burgers vector C. Force constant corresponding to a point force acting in the x. direction 1 C . ijkx Components of the elastic constant tensor d Constant defined in Equation (60) dV * Element of volume at the point r E Correction energy E Energy of an edge dislocation E E Energy defined in Equation (9) E Self-energy of a point force or energy of a screw dislocation E Total energy of dislocation loop F General symbol for a point force f General symbol for any function f . Component of a general force distribution G General symbol for a point force G. . Component of Green's tensor g General symbol for any function h Vector separating points of application of the two point forces forming a double force

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L Dimension defined in Figure 5 or Figure 33 n Normal at r of a surface P. Component of the dipole tensor R Dimension defined in Figure 5 or Figure 33 -4 -* — • R Vector defined by (r-r') r Point where the displacement field is computed r' Point of application of a point force or a double force Constant defined in Equation (6) , Equation (93) or Equation (195) Constant defined in Equation (72) Constant defined in Equation (88) (DK) General displacement field J order when computing the displacement field u Perturbation of the displacement field due to the introduction of a general kink in the crystal Perturbation of the displacement field due to the introduction of a double kink in the crystal u' Perturbation of the displacement field due to the introduction (SK) of a single kink in the crystal v Corrected displacement field between the planes of forces W Energy of the system of forces W Energy of the array of F forces in the simulation of an edge F dislocation by primitive prismatic loops W Interaction energy between the arrays of F and G forces in the FG simulation of an edge dislocation by primitive prismatic loops w Energy of the array of G forces in the simulation of an edge G dislocation by primitive prismatic loops

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W Energy defined in Equation (26) W Energy defined in Equation (23) w Half-width for a screw or an edge dislocation X. Component of the vector R x. Component of the vector r Q'„ 1 Distribution function for an edge dislocation a ^ Distribution function for a screw dislocation th a. . K order term in the computation of the distribution function ij a. . Au Relative displacement across the slip plane j th Au J order term in computation of the relative displacement Au 6 . . Kornecker delta 6 (R) Dirac delta function e Variable tending to zero e. Component defined on page 16 § Peierls' symbol for the half-width of a dislocation X Lame' s constant \x Shear modulus V Poisson' s ratio a. . Stress tensor component corresponding to the displacement field u Stress tensor component corresponding to the displacement field v Angle defined in Figure 1 iJ

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SIMULATION OF DEFECTS IN CRYSTALS BY POINT FORCE ARRAYS By Jean-Pierre Jacques Georges December, 1972 Chairman: Dr. C. S. Hartley Major Department: Engineering Science, Mechanics and Aerospace Engineering A new approach for analyzing dislocations and kinks in dislocations in simple cubic crystals is presented. The crystal is considered to be a continuum where defects are simulated by arrays of point forces acting on the centers of atoms in the immediate neighborhood of the defect. The magnitude of these forces is determined by the condition that they have the same displacement field as the corresponding defect in the ordinary continuum model. Infinitesimal prismatic and shear loops are constructed for simple cubic crystals and used to construct screw and edge dislocations. The arrangement of the atoms in the vicinity of the dislocation line is obtained and compared to Peierls' model. The self -energy of these dislocations is found to be of the correct form provided the force cbnstants are correctly determined. Atomic arrangements around kinks in screw and edge dislocations have been computed and are presented. The model developed promises to be of great value in studying atomic displacements in the vicinity of the dislocation. xi

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CHAPTER 1 INTRODUCTION A thorough comprehension of the nature of defects in crystalline materials and especially in metals is fundamental in order to explain many of the properties and the behavior of these solids. In particular, vacancies, interstitials and dislocations cannot be ignored when diffusion, mechanical behavior, electrical, optical and magnetic properties are studied. The usual theory of lattice defects assumes a "local" continuum model. The matter concentrated in the atoms is supposed to be uniformly distributed over the whole space occupied by the crystal. The local atomic arrangement is ignored and the defect is replaced by a singular line, point or surface in a continuum body [1-3J. This model has proved to be extremely valuable for studying properties which are not sensitive to the atomic configuration in the vicinity of the defect, but it is limited by the discrete nature of the atomic array. Consequently, it is always necessary that expressions for the displacement field of the defect be terminated at some distance from it. Furthermore, since the continuum approximation ignores the local atomic arrangement around defects, it disregards the short range anisotropy of the displacement field. To remedy these shortcomings, atomistic computations have been attempted. Atomic positions and interactions are considered explicitly

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in the core region of the defect, with some laws defining the pairwise atomic potentials (see [4-13]). Further from the center of the defect, continuum theory is assumed to hold, so that the only atoms which need to be considered are those whose positions are necessary for calculations of energies for the core region. Such a method involves first the construction of a suitable interatomic potential, and, secondly, sums over a large number of lattice points which have to be carried out numerically. It is undoubtedly the best existing method of determining the local atomic arrangement around defects, but it is costly and very sensitive to the chosen size of the core region [14]. Furthermore, it involves convergence problems, and the manner in which boundary conditions are imposed is very delicate. It is therefore worth exploring methods refining the ordinary continuum model by introducing the atomic arrangement of the crystal, but with a minimum increase in computational effort. In such a model, the atoms will be considered to be embedded in a continuum and the defect formed by the placement of suitable point forces at positions corresponding to atomic sites close to the defect [15]. The resulting displacement field is the sum of the displacement fields of all the point forces and is taken as the displacement field around the defect. Examples of such constructions by superposition of infinitesimal loops have been given by Koehler [16], Groves and Bacon [17] and Kroupa [18] for local continuum models. In this present study, we shall concern ourselves in examining straight screw and edge dislocations in simple cubic crystals. First, a brief description of point forces and infinitesimal primitive loops

PAGE 14

will permit us to analyze the displacement field and self-energy of dislocations. The screw dislocation will be constructed from an array of primitive shear loops, and the edge dislocation from an array of primitive shear loops and prismatic loops. Comparison will be made between both models in the case of the edge dislocation. Furthermore, calculations of atomic displacements around kinks will be attempted for both dislocations.

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CHAPTER 2 BASIC CONCEPTS Point Force A point force F is a highly localized body force distribution applied to a material point in a continuum. F.(r') = f. f 6(R) dV(1) J J v r where R = r r' and 6 (R) is the Dirac delta function. The displacement field at r, u (r) , due to such a point force at r' , can be i obtained from the equilibrium equations of elasticity and Hooke 1 s law [2] in the form u.(r) = F.(r' , )G. . (R) (2) J i ij where for an infinite isotropic body G..(R)= * ij 16rrp, ^ |r| + ^ |r| 3 _ (3) G (R) is the Green 1 s tensor response function for a point force, ij It is the component parallel to x . of the displacement field at r' due to a unit point force parallel to x. at r. It can be shown that the Green' s tensor is symmetric. As we see from the expression for G . (R) , this function is not U defined for R = and we are unable to determine the displacement of

PAGE 16

the point of application of the point force from Equation (2). To remove this mathematical divergence, we shall associate a finite displacement u(r') with the point of application of the point force. This value u(r') can be considered as being the average resultant displacement of points on a surface surrounding the point of application of the point force. This surface is determined such that u(r' ) is the mean value of the vector displacements of two points symmetrical with respect to r' . This vector is acting in the same direction as the point force. u(r') = fpCr' + r ) + u(r' rY] . (4) Using polar coordinates as shown in Figure 1, the absolute value of the displacement u(r' ) takes the following form. 2„-l |u(?')| = 16TTH 3-4V 1 1 cos — + (5) l-v r lv r So, for a definite value of ju(r')] we can define a surface of revolution about the direction of F, on which all the points have displacement components |u(r')| along F. This surface surrounds a volume which can be considered as a core surrounding the point of application of the point force. The core can be interpreted as the volume where Equation (2) for the displacement field is r.o longer valid. The size of the core depends directly on the value of Ju(r')J assigned. It must be pointed out that the average value of the radius vector, rj , of the core is equal to the radius r of the sphere on which the average displacement of its points is equal to Ju(r')|, that is

PAGE 17

° 24TT^|u(r')| On the other hand, |u(r')| can be related to a force constant. By analogy with a discrete lattice model, such a point force applied on an atom causes an equal and opposite resisting force proportional to the displacement of the atom given by |f| = c|u(r*')| . (7) C is known as a force constant, and is the force acting on an atom required to produce a unit displacement. In other terms, its inverse is the displacement of the atom caused by a unit force acting on it. This force constant is the parameter we shall use in the following problems encountered. It will be determined for each special case by requiring that our mathematical model obeys certain physical imperatives. It will be straightforward to deduce |u(r')| and the size of the core from the value of C. The self-energy of a point force is defined as being the work done by this force against interatomic reaction forces when it is introduced into the continuum. So, using Equation (7) E s = |f|u(?')| =|L.. (8) The interaction energy between two point forces F (r 1 ) and F C2) (r') is m E T = F, (1) (r) u (2) (r') = F (1) (?) F (2) (r*') G (?-?'), (9) I k k k m km

PAGE 18

where the sign is determined following Cottrell's convention for dislocation interaction energies, i.e. , it is the work done by external forces when the second force is applied in the presence of the first, or vice versa. The total elastic potential energy of the system is the sum of the self-energies of the two point forces less the pairwise interaction between them. Double Force A double force is constructed from two eqiial and opposite point forces ± F applied at points separated by a vector h. If the forces are collinear, we have a double force without moment, otherwise with moment. The strength of a double force is defined as P * k = lira (h iV(10) |h|-o |f| — The displacement field is obtained by superposition. If the separation distance between the forces is very small, we can expand the displacement field of each in a Taylor series about the midpoint of h. Keeping only the first order terms, we are led to the displacement field mentioned by Kroner [2] u.(7) =P £k (?<) G k .y ?-?•). (id As for a single point force, the displacement of the point of application is undefined, but this divergence can be removed in the same way as before by introducing the concept of a core surrounding a double force.

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Primitive Dislocation Loops Following Kroner's definition [2], an infinitesimal dislocation loop in a continuum is the boundary of a microscopic surface which separates regions in the continuum which have suffered a relative displacement b. The Burgers vector of the loop is defined as the line integral of the elastic displacement u around a circuit containing the dislocation. The displacement field, at a point r, of such a loop of surface dS with normal vector n and centered at r' is found to be n (r) « b.n. c... . C. . dS (12) m 1 j ijk£ km,x where c. ., . are the elastic constants, ljki The similarity between this expression and the displacement field of a double force (Equation (11)) leads us to consider the infinitesimal dislocation loop as a nucleus of strain with the fundamental double force tensor P. # = c ... b.n. dS (13) kX ljkX i j or, for an isotropic continuum, \i CXVji + 6 i/j*) + x Vk J Vj ds (14) Up to this point, we have completely ignored the local atomic arrangement around the loop. In real crystals the interatomic reaction forces, developed when the atoms are displaced to form the defect, are the physical origin of the double force tensor characteristic of the dislocation loop. So it seems logical to construct such a loop by applying point forces in the continuum, but at points corresponding to atomic positions located immediately around the defect. The

PAGE 20

displacement field of the loop is then the superposition of the displacement field of each point force. Each primitive loop has the character of a "unit cell" for the defect. These "unit cells" can be assembled to form a more complicated defect like dilatation centers or dislocations. So logically we can characterize the surface of the loop dS such that the produce |b • dS| equals one atomic volume in the crystal structure considered. This procedure will be analyzed more specifically for simple cubic crystals. (a) Prim itive prismatic loop in simple cubic crystals The arrangement of the first neighbors of a vacancy loop in simple cubic crystal is shown in Figure 2. A primitive prismatic loop is constructed in the following steps. First a vacancy is created by removing an atom from the lattice. This vacancy is simulated by applying on its first neighbors forces of magnitude F directed towards the vacancy center. In the second step, two extra forces, ± G, are applied in a direction normal to the {00l} plane, on the atoms in the (OOl) direction, towards the center vacancy in order to collapse the configuration onto the {001} plane. In this manner, we have set up three double forces, all without moment, leading to the diagonal dipole tensor P ll = P 22 = 2Fa ' and (15) P 33 = 2a(F+ G), where a is the lattice parameter of the simple cubic crystal. Though each pair of forces is clearly separated by a distance 2a, at distances from the loop large with respect to the interatomic

PAGE 21

10 Figure 1. Coordinate System to Evaluate the Core Region Around a Point Force 3 / / x. 01 F+G Figure 2. Prismatic Loop in Simple Cubic Crystal

PAGE 22

11 distance they appear as three double forces which can be identified with a dislocation loop as described above. The Burgers vector of this loop must represent the collapse of the atoms in the (OOl) direction, whose relative displacement must be a in order to create a new regular arrangement of the atomic planes. As stated previously, the surface dS is chosen such that |b • dS| is equal to a here. So, following Equation (14) P ll = P 22 = X * 3 and (16) P 33 = a + 2 ^ } a ' The forces applied on the atoms can now be obtained by comparing Equations (15) and (16). The displacement field and the self-energy of the loop can easily be deduced. (b) Shear loop in simple cubic crystal A primitive shear loop in a simple cubic lattice (Figure 3) is constructed as follows. The forces F applied on the atoms impose the direction of the shear. Since the loop must be kept in equilibrium with respect to its center, additional forces G have to be applied, forming a couple whose moment about the center of the loop counterbalances that of the shear forces F. The Burgers vector of the loop is the smallest shift allowed by the atomic arrangement. By the same method as for the prismatic loop, it is found that the only nonvanishing components of the dipole tensor are P 12 = P 21 = ^ = 2Fa = 2Ga ' (17)

PAGE 23

12 so the magnitude of the forces has the value 2 F = G = p, ~ . (18) The shear loop can be represented with respect to its principal axes x' and x' (Figure 4) leading to the dipole tensor 1 ^ P il = 2aF = " P 22 • (19) which represents two double forces without moment, perpendicular to each other and acting in opposite senses, (c) Conclusion The primitive dislocation loops as described above are the basic elements for our process of simulating larger defects, especially dislocations. We shall see that a suitable array of shear loops can describe either an edge or a screw dislocation, but that an array of prismatic loops can only simulate an edge dislocation.

PAGE 24

13 ,-*-fr« s%~\~1f' X, Figure 3. Shear Loop in Simple Cubic Crystal /. \ y * p / / & / / p\ / \ Figure 4. Shear Loop with Principal Axes

PAGE 25

CHAPTER 3 RECTANGULAR DISLOCATION LOOP IN SIMPLE CUBIC CRYSTAL A rectangular dislocation loop having a Burgers vector a(lOO) can be simulated by a rectangular array of primitive shear loops, stacked as shown in Figure 5. The dimensions of the array are considered to be very large compared to the atomic distance. The axes of reference are shown in Figure 5 with their origin at the center of the loop. In this chapter, we are only interested in obtaining properties of the rectangular dislocation loop related to our main interest, the displacement field and self -energy of the pure screw and edge dislocations. Displacement Field The displacement field at any points of this array is simply the sum of the displacement field of each point force u (?) = S F.(r') G. (r-r* ), (20) m . -\ i 1m where G (R) is defined in Equation (3). Developing this sum leads to im the general expression 2 R/2 I / L/2 L/2-1 E / [ S + E p=-R/2| lq=-L/2 q=-(L/2-l) u (r) = u. -gZ (I s + z G 2m (x r pa ' X 2qa ' 14

PAGE 26

15

PAGE 27

16 G 3m (X r pa ' V•' X 3-| } + G 3m ( V pa ' X 2 + •' X 3 + f > ' l G 3m (X r pa > X 2"•' X 3-f } + G 3 m ( V Pa ' X 2"•' X 3 + l } (21) These equations for each component are valid everywhere in the continuum. They can be simplified for each particular region of the loop. The regions of the continuum where a pure screw dislocation is simulated correspond to x i = " T + e i X 2 2 and Ra X l = T ~ € i X 2 = 6 2 where e and e are small compared to Ra and La, respectively. Similarly, regions where the loops have a pure edge character correspond to and X l= e i La X 2 = T " 6 2 where e and e are small compared to Ra and La, respectively. Each particular case will be considered in the following chapters. Elastic Potential Energy The work done by the forces comprising the array, that is, the energy of the system, is the sum of the self-energy of each point force, minus the pairwise interaction energies, as defined by Equations (8) and (9). We call W the energy of a row of forces, that is, two lines Row

PAGE 28

17 of forces parallel to the x axis for a given x coordinate, and we call W the interaction energy between two rows of forces, as defined above. Int Following these appellations, the total energy of the system has the form R W = (R+l)W n + E (R+1-P)W T Row „ Int P=l (22) W and W have the following expressions, respectively, 3 5 (J, a Row 2L-1 L C 2 C 3J 2 4 + 1 Sr-C G 33 (0 ' ' a) " (2L 1} G 22 ( °'°' a D + Ji~(g (O.La.O) G 22 (0, La, a) G 33 (0,La,0) G 33 (0,La,a)] 2 4 L r -i + 2(13 E (L-q) [_G 22 (0,qa,0) G 22 (0,qa,a)J q=l 2 4 2 4 n a G 23 (0,La,a) 2^ a E G 23 (0,qa,a) q=l (23) Int ^ 2 a 4 (2L-l) (G 22 (pa,0,0) G 22 (pa,0,a)] + p, 2 a 4 Q} 33 (pa,0 ,0)+G 33 (pa,0,a)] + ua Q} 22 (pa,La,0) G 22 (pa,La,a) G 33 (pa,La,0) G 33 (pa,La,a)J 2 4 2 4 2p, a G „(pa,La,a) 4p, a E G (pa,qa,a) 23 L Q=l 23 2 4 r~ ~\ + 4|j, a E (L-q) [G ipa,qa,0) G 22 (pa,qa,a)J . q=l Following Equation (22) , W becomes W= n, 2 a 4 (R+l)L^ G 22 (0,0,a)f2 E [g^O, qa,0) 0^(0, qa, a)] (24) q=i R L + 2 E [G 22 (pa,0,0) -G 22 (pa,0,a2)+4 E E (jJ^Cpa.qa.O) p=l " ' p=l q=l " G 22 (pa ' qa ' a [l} + ^ 4 ^D V?(^~C^) + l&22 (0 '°' a) +G 33 ( °'°' a >] + § (G 22 (0,La,0) -G 22 (0,La,a) -G 33 (0,La,0) G 33 (0,La, a)] 2 E q [p 22 (0,qa,0) G 29 (0,qa,af| Q=l

PAGE 29

18 R £ (G 22 (pa,0,0) -G 22 (pa,0,a) + G (pa.0,0) +G 3 (pa,0 f a)) P=l R + £ QG 22 (pa,La,0) G 22 (pa,La,a) G^Cpa ,La,0) -G (pa.La.af) P=l R L 4 E E q QG 22 (pa,qa,0) G 22 (pa, qa, a)J -G 23 (0,La,a) p=l q=l L R+l R+l L -2 E G 23 (0,qa,a) -2 £ G 23 (pa ,La, a) 4 £ E G 23 (pa,qa,a) q=l " * p=l p=l q=l + ^a 4 L<-2 Ep [G 22 (pa,0,0) -G 22 (pa,0,a)] L p=l R L "\ -4 E E p [p 22 (pa,qa,0) G^pa.qa, af) > P=l q=l "' J + n a < E p[G 22 (pa,0,0) -G 22 (pa,0,a) -G 33 (pa,0,0) -G 33 (pa,0,a)] E pQ} 22 (pa,La,0) -G 22 (pa,La,a) -G 33 (pa,La,0) -G 33 (pa,La,a)J P=l R L + 4 E E pq [_G 22 (pa,qa,0) G 22 (pa,qa,a)J p=l q=l R+l L R+l + 2 E p G 23 (pa,La,a) + 4 E E p G 23 (pa,qa,a) ) . (25) p=l " q=l p=l The only mathematical difficulty in the computation of such a formidable expression lies in computing the single and double sums. The first terms are computed, up to a chosen integer N (usually N=20), and the rest of the terms are approximated by an integral. The following approximation follows: L L E f (q) « f(q) +| f (N) +| f(L) +J f (x) dx, (26) q=l " "" N and for a double sum: R L N-l N-l N-l E E f(p,q) m E E f(p,q)+| E Qf(p,N) + f(p,L)) p=l q-=l p=l q=l p=l N-l _ N-l L + | E ff(N,q) + e(R,qf| + E J f (p,y) dy q=l ^~ p=l N

PAGE 30

19 + E J f (x,q) dx + i f f (N,y) dy + i J f (x,N) dx q=l N N N L R .. + ± f f (R,y) dy + i j f (x,L) dx + [_f (N,L) + f (R,N) N N L R + f (N,N) + f (R,L)j + J J i (x,y) dx dy . (27) These approximations give the correct form of the divergence 4 in R and L for divergent sums, and give an accuracy of 1 part in 10 when the sum converges, which is sufficient for the model employed. The final expression for W/ua becomes (R+1)L . ua 2.8545 2.3676V R+l pa L + 44tt „ ua 2. 8545 2.3676V 4tt J — + C 2 1 V 2tt(1-v) 2n 2RL 2n C~ + «/R +l R+l 4tt R + ^ 2 2 +L . 1 1 \ 1.2979 -.6458V 1.6420 + .3580V 1 V 2(2-y) Jr +L 1 V 4tt /X 2 ' fl 1 A 3.3903 .6872V 2ttu^---) 4tt(1-v) (28) W is not the energy of the physical dislocation loop. It contains an extra strain energy in the region bounded by the plane on which forces are applied and the boundaries of the array, which does not account for the relaxation of the atoms. In the way the forces have been applied, a relative displacement greater than a/2 has been created across most of the slip plane for the atoms reaching their final configuration. It is from this final configuration that the relative displacement procedure must be measured in order to calculate the actual strain energy stored between the planes of forces. Such a correction energy will be computed in each case, for the pure screw and pure edge dislocation. We shall not examine all the properties of the dislocation loop here, since our purpose is to treat this loop as an intermediate step in simulating screw and edge dislocations.

PAGE 31

CHAPTER 4 SCREW DISLOCATION IN SIMPLE CUBIC CRYSTAL From the results found in the previous chapter, the displacement field, the relative displacement across the slip plane, and the selfenergy of the screw dislocation in simple cubic crystal will be obtained. Displacement Field As seen previously, the regions of the dislocation loop having the characteristics of a pure screw dislocation correspond to r Ra T + e i Ra x i = T ~ e i and < 2 2 V where 6 and e are small compared to Ra and La, respectively. Since both regions represent two identical dislocations, but of opposite signs, we shall only consider the first one, and translate the x axis by an amount of -Ra/2 so that it becomes the boundary of the array. We shall keep the same symbols x and x for the new variables. (See Figure 6. ) The only change in the expression for the displacement field as written in Equation (21) is that p is now summed from zero to R. Since an analytical expression is desired for u , an approximation different 20

PAGE 32

21

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22 from that given by Equation (26) will be employed to compute the discrete sums. Euler's formula [191 is most suitable for this case: b b E f (P) = J f (x) dx + (_f (a) + f (bf] + — (V (b) f ' (a)J a a .^[ f ... (b) . f ... (a)]+s _|_Q f «) (b) .^ u) 3 + ... . (29) The accuracy of the approximation depends on the number of terms used. The advantage of this method is that at most of the points where the displacement field is computed, the three first terms are sufficient -3 for the accuracy required, i.e., a relative error of 10 is accepted. The first summations which have to be computed are the two summations on q. These sums can be computed exactly because we always consider x small with respect to La, or in other words, our range of interest is far from both ends of the dislocation line. So we shall have L/2 L/2-1 + °° 2 f(q) = S f (q) = J f (x) dx (30) q=-L/2 q=-(L/2+l) -« where f represents the whole expression to be summed. The following components of the displacement field have been found: r WW = ° < WW = 4W \^ oW w = ° (x -pa) + (x 3 +2-) 2 a~~2 p-0 (x i -pa) + (x 3 --) R a „ (31)

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23 We can already notice that some of the characteristics of the displacement field around a screw dislocation are displayed, that is, the components u, and u„ are equal to zero, and u is independent of x . Euler' s formula can be applied a second time to obtain a final expression for u , the accuracy depending on the number of terms retained. We shall label the different terms composing u in the following way, u 2 (x i'W = ^{ g( V pa) dp (32) vwv = £? [ g(x i } + g(x r Ra D (33) U 2 (X 1' W = 48^ ( V Ra) g ' (X 1 } ] (34) U 2 3(X 1' X 2' X 3 ) " 28^ & (3)
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24 Since J 0n (na-pa) dp = -2 f(n-p) Ski (na-pa) (n-p)~J , (39) the integral is equal to zero for p = n, and so Ra J g(na-pa) dp = G(na-Ra) G(na). (40) The integration makes all the singular points vanish, except for those at x =0 and x = ± a/2. So the final analytic expressions for u become U 2 = 4^ 2n^ . -1 1 .-11 + tan + tan 3 2 (41) + 2x, ft 1 Xl tan L X 3 + J tan 2 , a 2 X l + < x 3 + 2> "\ c i fe "2 — ; — T2 >. X, + (X„ -75-) J 3 2' 2 , a,2 1 , , * X l + U 3 + 2 } u 2 (x i'V = i^ ^-§ — ; — rs X l + (X 3-2 } (42) U 2 (X 1' X 3 } = " 24^ r 2 a s 2 2 a x 2 X l + (X 3-2 } X l + (X 3 + 2 ) J (43) and VW = " 730TT 4 KC«*b-5» -4D *i[ 3 "vf> -O (44) Then, since u (x x ) is the sum of u (x ,x ) for every i, £t X o *^ X o U 2 (X 1' X 3 ) = U 2 (X 1' X 3 } + "2^1'V + U 2 (X 1' X 3 } + u 2 (x llX3 ) + (45)

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25 Equation (41) takes three different forms, following the region where x is computed: rr corresponds to x > a/2, 2tt(x /a) to -a/2 < x < a/2, -TT to x < -a/2. An asyn.ptotic expression for u can be obtained when x and x are considered large with respect to the atomic distance, but still far from the ends of the dislocation line. WV = 2^ , TT ,-11 ± — + tan — 2 X 3J (46) This is the well-known expression obtained from the Volterra solution for a screw dislocation [20] . The relative displacement across the slip plane is defined as A W = V x i'2> w-^ = 2 V x i'2> (47) It can be directly deduced from Equation (31) or Equation (42) to (45): Au (x ) = ±_ S Sm 21 2n n _ 2 2 (x -pa) + a p=0 (x 1 ~pa) (48) 12 3 Au (x ) = Au (x ) +.Au 2 (x ) + Au (x ) + Au (x ) + (49) with AU 2 (X 1 } = 2^ 2 2 , x x x + a -11 1 n 1 2 tan + rr + — m = a a 2 x 1 -J (50) 2 2 1 a X l + 3 AU 2 (X 1 } = 4^ 2— (51)

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26 A VV = m 2 2 x + a 1 J (52) and AU 2 (X 1 } = 36Cm , 2 x i (x 1 3a 2 ) n 3 / 2 2\3 C l ( X l + 3 ) (53) The more and more precise expressions for Au (x ) are plotted in Figures 7 and 8, and listed partially in Table 1. A remarkable precision is obtained for the regions where |x | ^ a after evaluating only a few terms. On another hand, the only term which is not singular at x = is Au (x ). It seems to deviate significantly from the correct curve for Au (x ). However, the order of magnitude of the real relative displacement at x can be obtained approximately by interpolation. So the relative displacement of atoms above and below the slip plane is known everywhere except at x = 0. TABLE 1. Relative Displacement of Atoms Across the Slip Plane for a Screw Dislocation x /a Au°/a + Au /a + Au /a 3 + Au 2 /a 0000 .0000 5.0

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27 Au. A^,AZ Figure 7. Relative Displacement Near the Core of a Screw Dislocation

PAGE 39

28

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29 Another way of computing the relative displacement at atomic points is to go back to the definition of the displacement field by Green' s functions. Au (na)=2|ia I £ f~G (na-pa , qa ,0) G (na-pa, qa, a)~j (54) p=0 q=-°° changing variables by setting u = n-p changes Equation (54) into 2 / n °°\ + " r -i Au 2 (na) = 2pa J E + E j E ^G 22 (ua ,qa,0) G 22 (ua,qa, a)J \u=l u=l/ q=-=° + 4^ 2 Z QG 22 (0,qa,0) G 22 (0,qa,a)] q=l + cTnT' 2 ^ 2 G 22 ( °'°' a) ' (55) where C (n) is the force constant defined in Equation (7) for a point force acting on a point at a distance na from the origin. After computation of the sums as presented in Equation (26) , Au (na) becomes n 2 Au„(na) = 2 V C„(n) 2rr 2 2.866 2.368V _ . u + 1 L m 7i — 1 v u=n+l (56) For each value of n, a direct comparison can be made between the values of Au (na) from Equations (49) to (53) on one hand, and Equation (56) on the other hand. Since both ought to be identical, a value of C„(n) will be obtained for each value of n. Table 2 lists the different values of Au (n) and C (n) for n = to 5 and for n = °°. The values of C (n) are very nearly constant, and it seems reasonable to expect a value of C (0) very close to C (°°) . This extrapolation permits us to evaluate Au (0) .

PAGE 41

TABLE 2. Variation of Force Constant C with the Atomic Positions in a Screw Dislocation 30 Au r -(n) 2^ia C„ Au, ' (X 1 ) C 2 (n)/\iA 2885 3988 4343 4511 4607 4670 4957 .9033

PAGE 42

31 It is striking to notice that the symmetry of the screw dislocation displacement field is preserved at the atomic points. The dislocation line, in the continuum sense, lies exactly at x = -a/2. This result could have been guessed earlier by simply Considering an oriented path around each loop. By adding the loops together, the only remaining part of the path would be a straight line at x = -a/2. This method can be generalized for determining dislocation lines in more complicated cases. This symmetry does not appear in Equations (49) to (53) because of the divergence of these expressions at x = 0. A mapping of the atomic displacements in atomic planes immediately above and below the slip plane is shown in Figure 9. The width of the screw dislocation is defined to be the region in which the relative displacement is comprised between a/4 and 3a/4. Since the region where the relative displacement is equal to 3a/4 cannot be known exactly, the value 2w = .73a for the width can be obtained by rough measurements on Figures 7 and 8. The relative displacement described above can be compared to the results obtained by Peierls [20,21]. Fitting the expression for relative displacement obtained from the Peierls model to our results leads to the expression -1 x i + d Au (x ) = | + tan — , (60) 2 1 2 TT w ' where w is the half-width and d a translation parameter fixing the center of symmetry of Au . For large x , this expression can be expanded in consecutive powers of 1/x ,

PAGE 43

32 © O *© c© e !« ce> © c© i° (© x -ea-ea-f3B-— oo ote co < < o :: ©; e ($> ' J o CBlCSJ II c c CO Si <

PAGE 44

33 a w a wd , _ .. . AU 2 (X 1 } = a -^3T + n + T ••• • (61) 1 X l The same series expansion holds for Equations (49) to (53) and leads to 2 A W = a "2^x+ 4i^ + ••• • (62) 1 X l By comparison of Equations (61) and (62) , the half-width and the translation parameter are w = | ; d = | . (63) These values correspond to the Peierls' model when the origin is taken at x = -a/2. Although the width of the dislocation in our model is slightly smaller than in Peierls model, the two expressions for the relative displacement across the slip plane are exactly the same for large values of x , as shown in Figure 8. Finally, following Eshelby's suggestion [1 ], a distribution function for infinitesimal dislocations in the glide plane can be defined. Instead of being the result of a singularity concentrated on the x axis, the straight screw dislocation is considered to be composed of a continuous distribution of infinitesimal dislocations. This distribution function is, in fact, the component a of the dislocation density tensor as defined by Kroner [2], It is to be found equal to d(Au ) *22 (X 1> = -dx7~ • (64) Differentiating Equations (49) to (53) with respect to x leads to

PAGE 45

34 2 2 *22 = 2^ 2— (65) 1 , n 1 22 v V ' 2TT ,22' x (x +a ) (66) and 2 i ^ a » no (xJ 2 2 1 X l a 2* 2 2 2 L _x i (x i+ a ) _ 4 2 2 4 x 6a x + a 22 1' 120tt jl_ 2 2.4 4 L. ^l^ } X l-» (67) (68) It can be easily verified that + 00 I ^2 < V dX l = (69) A plot of the successive approximations is made in Figure 10, emphasizing the values of a (x ) at the atomic positions. Symmetry of the distribution function at these points with respect to x = -a/2 is evident. In summary, we shall emphasize that this model is in perfect agreement with the previous techniques employed for obtaining the displacement field of a straight screw dislocation. An improvement has been made in the present case, obtaining a simple analytic expression for the atomic displacements around the defect without any exceptions. Such atomic displacements can be obtained by lattice dynamics computation, but these have the disadvantages of being difficult to use and being an entire numerical method [13].

PAGE 46

35

PAGE 47

36 Self-Energy of the Screw Dislocation As it has been explained in the previous chapter, such an array of point forces can simulate a system composed of two infinitely long parallel screw dislocations of opposite signs if the length La is taken much larger than the separation distance, or width of the array, Ra. In this case, the energy of the system per unit length of screw dislocation becomes W/La when L is large. 2 _W_ _ R+l 2 La 4tt ^ 4tT^ + A 2 \& R 2tt r' ' (70) where the constants A and r' are defined by A = 2.8545 2.3676V 1 V (71) and &n— = 1.179 r o (72) Taking the special case of V = 1/3 leads to _W _ R+l 2 La 4tt ^ 4tt ££ + 3.0979 C 2 lia &r. R 3076 (73) As already mentioned, this energy is higher than the energy of the system composed of two parallel screw dislocations of opposite sign, because of the nonrealistic strain energy stored in the region between the planes of forces. The region of the continuum where the strain is larger than 1/2 is shown in Figure 11 and has the following boundaries: < x < 2 ~ 3 2 (74) •.35a < x < Ra + .35a .

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37 a < a c •i-i he «

PAGE 49

38 The points x = -.35a and x = Ra + .35a are those where Au (x ) = a/2. Outside these limits, Au (x ) is smaller than a/2. The correction energy which has to be subtracted from W/La is composed of the strain energy per unit length of screw dislocation due to the displacement field u(x ,x ), minus the strain energy per unit length of screw dislocation due to the displacement field measured relative to the final atomic positions inside the slab, v(x ,x ) described as follows: V 1 (X 1* X 3 ) /v 2 (x i ,x 3 ) = u 2 (x 1( x 3 ) x 3 (75) v 3 (x lF x 3 ) = The stress fields associated with these two displacement fields ^2 a i2 = v. ^ du 2 7 23 " * o^ (76) and ^2 ^2 T i2 * sq = »-sq SV 2 / 3U 2 \ (77) The correction energy is the following integral computed over the volume mentioned above. E =| J" (a.. e U . -r.. eM dV c 2 J \ ij ij ij ij/ (78)

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39 Using Green's theorem, this volume integral can be transformed into a surface integral , 1 i (o. . u. T . v ) n dS , (79) where u is the normal to the surface S. Using Equations (75) and (76) leads to 2 J S ' 2 ov. l 2 ^q ~ v 2 o^j ! 1 n. + u 2 ox r ° X 3/ 3 dS (80) ur du sL 3dx l x 3 ox 3 2 1 3 (81) where the surface, S, is composed of the areas S : x = ± -, -35a < x < Ra + .35a; S 2 : x x = -.35a, S x„ < -: 3 2' S : x = Ra + . 35a, ^ x n < 3 2 Taking the symmetry with respect to x = Ra '2 into account, and the fact that the integrand is not dependent on x , lead us to the final formal expression for E /La: c E a/2 du 9 u = »I „ x 3 ^r ( "35a > V dX 3 a/2 1 Ra/2 .35a a 2 a a , a. 2 ^ (X 1'2 } " 2 + U 2 (X r2 } dx. (82) The analytical expressions for u , du /dx , du /dx are obtained from Equation (31) and its derivatives:

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40 VW R Z 2n < ~, 2 / ^ 2 (x -pa) + (x 3 +2> p=0 (x i ~pa) + (x 3 --) (83) h 3x~ (X 1' X 3 ) = ^ Z n X 3 1 p=0 and x pa ^ , *n 2 , x 2 / a x 2 ( Xl -pa) + (x 3+ -) ( Xl -pa) +(x 3 -^) J 9U 2 a R ox„ 13 2tt p=0 a X 3 "2 t x 2 / a x 2 , n 2 , a x 2 (x^pa) + (x 3+ -) (x r pa) + (x 3 --) (84) (85) The mathematical problem of integrating a function over a region where the integrand has singular points for certain values of x is removed by the same argument as the one used in the previous section. After integration, Equation (82) becomes pa 2 R La p=0 (p + . 35) 2m (P+ ' 35) t 1 + 2 tan" 1 (p+. 35) (P+.35) V* (•-I)(86) The summation over p is computed in the same way as before using Equation (26) , which gives finally for the correction energy E c La [13. R \» rr r R (87) and 0n rJJ = .3129. (88) The total energy of the system composed of two antiparallel screw dislocations is

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41 _T _ _W_ c La _ La La (89) T _ R+l 2 La " 4tt ^ 4tt ^+ A-2tt 2 2 R-rl 2 (ja „_ [ia 2tt & <'o> 2 2 (90) From the usual continuum theory of dislocations, this energy has to be equal to the sum of the self -energies of both screw dislocations, minus their interaction energy La ua 4rr \» 2m . 2rr R (91) The requirement that our expression (90) has to be identical in form to Equation (91) , will force us to choose the still unknown parameter C_ such that the term divergent with R vanishes. Thus, with if E 2 La 2tt r 2m — = 1.3367, r o |ia _ 1 2.8545 2.3676V 4tt(1-V) For V = 1/3 T _ _ya_ La ~ 2tt 2r. Ra 2631a (92) (93) (94) (95) if C = 3.945(ia (96)

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42 From Equation (91) , the self-energy of a pure single screw dislocation can be written E„ 2 r~ = *T=-fa . . (97) La 4tt . 263a Two important remarks can be made here. First, the value found for C is very close to those found from the displacement field computations. This proves the consistency of the correction energy with the displacement field included from the array of forces. Secondly, it is found that the core parameter in Equation (97) is independent of V, Poisson' s ratio. This is in complete agreement with Peierls' result, which gives a value of r equal to a/e, that is, equal to .37a, where e is the naperian base of logarithms. So both models give nearly identical results, but with a slightly different r . For a direct comparison with Volterra' s dislocation model, the core radius r in Volterra' s model has to be reinterpreted and cannot be considered anymore as a cut-off core radius where Hooke' s law does not apply. It is rather a constant containing all the constant terms arising in the computation of the core energy. The Volterra cut-off radius could be evaluated in a better way by defining the region where Hooke' s law does not apply, e.g., where the strain is larger than .10. From Figure 8, such a cut-off radius can be approximated as being about r Q = 1.5a. So, the technique of simulating dislocations by a point force array seems to be very successful in describing the principal features of the defect, even though a complete accuracy in the computation of the

PAGE 54

43 atomic displacements cannot be reached because of the elastic and isotropic approximation. Single and Double Kinks in a Screw Dislocation A single or a double kink can be simulated by simply adding to the array of forces an extra row of shear loops, parallel to the screw dislocation line, and a semi-infinite or finite extent, respectively. A representation of the modified array is sketched in Figures 12 and 13. The displacement field and the energy of these defects are handled in the same way as for the straight screw dislocation. A. Displacement field The displacement field of these defects is obtained by adding to the displacement field of the pure screw dislocation, the displacement field resulting from the extra row of forces. The latter, u' , has the following expressions for, respectively, a single kink and a double kink of length 2Na. CO q=0 2 " G 2i ( V a ' X 2qa ' X 3 + lQ " ^ C G 2i (X l +a 'V X 3-f } 2 G 2i ( v a 'V x 3 + ti] ^Qwv^v^-f* + G 3i ( V a ' X 2' X 3 + l } 3 • (98)

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44 -©— ©— o iya A 2. ^>— e-^c^-^^y^^-^)Figure 12. Array of Forces for a Single Kink in a Screw Dislocation h I I -"-".a X, Figure 13. Array of Forces for a Double Kink , in a Screw Dislocation

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45 2 N U 1(DK) (X 1' X 2' X 3 ) = ** E C^i^l^'V* 18 '^-^ q=-N 2 G 2i (x i +a -v qa ' x 3 + l9 J rC G 2i ( v , 'y M ' x 3"3 ) G 2i (X l +a 'V Na ' X 3 + l ) + Si^l^'V^'^"^ 2 G 21 ( V a 'V Ha ' X 3 + l ) J + J f-C G 31 ( V a 'V Ha ' JI 3-| ) + G 3i (x i+ a,x 2 -Na,x 3+ J) Gg. (Y a 'V Na ' X 3 " | } Si'V^V^s + M • (99) Computations of u' and u' for both kinds show that these displacements are very small, even of the region of high distortion in the x direction. They reach a magnitude of a few thousandths of an atomic distance. Therefore, we shall concentrate our attention on the u' component of u' , and, more specifically, on the atomic displacements in the planes just above and below the slip plane (x = ± a/2) , since this is the region of highest distortion. Because of the existence of singular points at the points of application of the point forces, several special cases will be considered. As a first step, we shall restrict our range of interest by noticing the various symmetries in the expression of u' . It shows an odd symmetry with respect to x = , and an even symmetry with respect to x = -a for both cases, single and double kinks. It shows an even symmetry with respect to x = in the special case of the double kink. (1) Displacement field of the single kink The single kink will be the first case considered. For most of the values taken by x_ and x„ , u ' . _ . has the following expression: 1 Z
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46 ' ( a i _ a U 2(SK) U 1' X 2'2 ; 8^ Z f(x 2 -qa) f(x 2 ) Lq=0 2 a x„ _a 1_ 2 32tt 1-v r o . r,1 3 / 2 T . 2 2 2 T UCx +a) + x + a _J (100) with f(X 2qa) = 2(1^0 / 2 2 / 2 2 2 V(x +a) + (x -qa) V(x +a) + (x -qa) + a _ 2(l-v) (x 2 -qa)' (x 2 -qa) f 2 2~| 3/2 r 2 2 2i 3/2 _L(x +a) + (x -qa) J (Jx +a) + (x -qa) + a J (101) On atomic positions, different expressions apply because of the singular points situated on x = -a: (a) x = na and x / -a U 2(SK) (X l' na 'l ) = ^ » n E + E f (qa) » f (na) q=l q=0 v ' 32rr(l-v) C2 2 2 2 I x„ + a) + n a + a J 3/2 (102) (b) x = -na for any x « 1 ^(SK)^!'"" 3 '^ = ^ E E f(qa) + £ f(na) L\q=l q=' 32tt(1-v) f, ,2 2 2 2~] MX +a) + n a + a -J 3/2 (103)

PAGE 58

47 (c) x = na with n > 0, and x = -a , 2
PAGE 59

.X 48 Q 9 Q Q o o' Q 8 G 6 Q P Q Q § 8 $ 8 $ 'G e' d 9 H o a J8

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49 the atomic configuration shows very little difference from the configuration of the pure screw dislocation. Even if the actual width of the kink cannot be expressed analytically, one can conclude that the defect is very localized. TABLE 3. Atomic Displacements for a Single Kink in a Screw Dislocation x 2 /a

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50 with f(X 2" qa) = 2(1^) I o 2 ' 2 j/(x +a) + (x -qa) */(x +a) + (x 2 ~qa) 2 2 + a 2(1-V) (x 2 -qa) (x 2 -qa) r 2 2i 3/2 T 2 2 2~f /2 Jjx +a) +(x 2 -qa) J [jx^ + a^+U^qa) +a J (107) The points where this expression do not apply are the points of application of the point forces on the row x = -a. Two separate cases are considered. (a) x = -a and x = ± na for n < N 1 & ^(DK)*-*!* 18 '^ = 8t E + E ) f(q) ~ f(N+n) 32tt(1-V) \'+n N-n E + q=l q N-n N+n JjN-n) 2 +lJ QN+n) 2 +lJ 2 pa a C 2 -16 | f (N-n)J (108) (b) x = -a and x = ± Na u i(K)
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51 The sizes of double kinks have been considered, corresponding to the values of N equal to 1, 2, and 3. Due to the symmetry already mentioned, only the values of u' for the following values of x and 2 ( DK ) 1 x have been computed: x -a, 0, a, 2a x 9 = 0, a, 2a, 3a, 4a, 5a . These values of u' are tabulated in Tables 4, 5, and 6, 2d ( DK ) corresponding, respectively, to N = 1, 2, and 3. TABLE 4. Atomic Displacements for a Double Kink of Length 2a in a Screw Dislocation x 2 /a

PAGE 63

52 TABLE 6. Atomic Displacements for a Double Kink of Length 6a in a Screw Dislocation x 2 /a

PAGE 64

53 © a a Q a § a a a ,9 o o ' "\ ' / o ' « J 4 o a 9, O, X a © C 9 a N (3 7R a a JI 8 a © © a J •0 a 3 o a •p -P KS C O Q) O S »H as CO U -S a s o O CO •H S as o ii c

PAGE 65

54 13 © § \ § a 8 Q R o (D (3 fcJ a Q Q o J" b f^» b 3 O Q cS X! c 3 c h o < -H +> 4-> at c o
PAGE 66

-J 55 a 9 G Q -Or© Q Q P O Q e r * 5 8 $ 8 8 ® 6 i ei o 3

PAGE 67

56 In a first step, the difference between the energies of the systems of forces has to be computed. It is simply the self -energy of the extra double row of forces of length 2Na, and the interaction energy between this double row of forces and the dislocation loop array. W.JLL. (4N-1>| A-G 2a ( 0> f a) 2 4 \x a + G 33 (0,0^ 2 4 + —~(jG 22 (0,2Na,0) -G 22 (0,2Na,a) G 33 <0,2Na,0) G 33 (0 , 2Na, a)) 2 4 2 4 2N ~ 1 r 1 |* a G 93 (0,2Na,a) + 2\i a 2 (2N-q) |^G 2 (0,qa,0) G 2 (0 f qa,a)J q=i 2 4 2N_1 2 4 R+1 f La 2|iB 2 G 23 (0,qa,a) + (i a 2 < G 22 (pa,— Na,0) q=l " p=l L G 22 (pa,-^-Na,a) + G 22 (pa ,-— + Na,0) -G^Cpa,-^ Na,a) + G (pa.-^-Na.O) + G 33 (pa,^ Na, a) ~ G 33 (P a '-^+ Na,0) La La L/2+N-1 G 33 (pa,— + Na,a) 2G 23 (pa,2 ^-+ Na,a) -22 G 23 (pa,qa,a) + 4N j^G 22 (pa,0,0) G 22 (pa,0,a)T) L/2-N-1 L/2+N-1 8N 2 +22 (L+2N-l-2q) q=l " "2 q~-N Q} 22 (pa,qa,0) G 22 (pa,qa, a)J (111) We shall consider cases for which the length of the double kink is much smaller than the length of the dislocation line. Under this condition, t', expression of W is greatly simplified.

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57 (2N-1 V* 1 3 4v , 1 v el 8tt 2(1-v + 4tt L 2 q=l , ,l._i «/q +1 2(1 " V) (q 2 + l) 3/2 J _1_ 8" q=l 3-4V . q +1 IIS T^7^~2+ 1^1 "2 q +i l — + q Vq 2 +1 , 2 3/2 (q +D j pa i pa 1 1 1 2C + 2C^ + 32rr 1-v 32rr 7-8v 4N 1-v -s/4N +1 1-v 2 3/2 (4N +1) 2N-1 2^ S q=l 1 / 2 Vq +1 2(1-V) , 2 3/2 (q +1) J (112) The force constant C has been determined from previous computations for the screw dislocation. Assuming that the slight modification of the array does not have any influence on the value of C 2 , it is determined such that |ia A _ 1 S + 4n = 2 ' (113) with 1 3-4V _1_ 3-4V /l 16rr 1-V + 8tt 1-v , q q=i l 1 N

PAGE 69

58 \ia |ia 1 1 1 2C^ + 2C^ + 327f 1-V 327f 7-8v 1 1-V Jsi 1 1-V 4N 3/2 4N +1 (4N +1) 2N-1 2rr q=l ^ . 2(1-V) 2 3/2
PAGE 70

59 energy for the kinked screw dislocation. Since the former has alreadybeen computed, we shall focus our attention mainly on the latter. The region that suffers a strain larger than one-half is bounded by the planes x = ± a/2, the surface parallel to x where the relative displacement is equal to a/2 and the plane x = Ra/2 (see Figure 18). The region situated between x = Ra/2 and x = Ra will not be considered, since its deformation is the same as for the straight screw dislocation. We have already seen that the displacement field caused by the array of forces is '"l^i-W swv + ^wv (117) V. U 3 ( VW • By taking into account the fact that after relaxation, the atomic positions need to be referred to their closest neighbors (see Figure 19) , we are led to choose as actual displacement field across the slip plane the following expressions: ^v (x ,y ,x„) = u' (x ,x„,x„) l v l' J 2' 3 1 1' 2' 3 \ V 2 (X l' y 2' X 3 ) = U 2 (X 1' X 3 ) + ^l'W ~ X 3 (118) v v„(x ,y„,x„) = u' (x ,x„,x„) V. 3 l' J 2' 3 3 1' 2' 3 where v is related to x such that J 2 2 x 3 >0 x 3 <0 2 '

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60 _Rq/2. t x. Figure 18. Region of High Strain for a Double Kink in a Screw Dislocation Figure 19. Atomic Relaxation for a Double Kink in a Screw Dislocation

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61 The stress fields corresponding to u and v have the following components a. . and T. ., respectively, dug /^ ou 3 CT 22 (X 1' X 2' X 3 ) = (X+2 ^ o^ + M^ + o^ 33 v 1' 2' 3' J 2 Su 2 3U 1 J 12 (J VW ^ [o^ + o•^ + o^ ff 13 (X l' X 2' X 3 } 1 ^3 V 2 BU 2 3U 3 CT 23 (x i' x 2 ' x 3 ) = li \^q + ^q + ^q (119) and T ll (x i' y 2' X 3 ) = 'WWV WW = WWV t 33 (x ,y ,x ) = a 33 (x 1 ,x 2) x 3 ) T 12 (X l' y 2' X 3 ) = ^VW i* ~(x ,y„,x„) = a „(x ,x„,x„) 13 v 1 ,J 2' 3 13 V 1' 2' 3 T 23 (X l' y 2' X 3 ) = a 23 (X !' X 2' X 3 ) " » (120) Recalling Equation (79) , the general form for the correction energy is the difference between the strain energies in the region of interest,

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62 E = ^ f a. . u. n. dS -i f T. . v. n. dS' . (121) c 2 >s ij i j 2 <^, ij i j The surfaces S and S' differ only by the range of integration over x and y , respectively. When x is positive, y is defined between -La/2 + a/2 and La/2 + a/2, and when x is negative, y is defined between -La/2 a/2 and La/2 a/2. Under these conditions, the integrals involved with functions of y can be considered as integrals of functions of x with different limits. The various identities follow: La/2+a/2 La/2 / f(y ) dy = J f(x ) dx for x >0, (122) -La/2+a/2 * -La/2 La/-a/2 La/2 J f (y ) dy = J f (x ) dx for x < . (123) -La/2-a/2 -La/2 So E can be written as a surface integral over a function, depending on a unique variable x . E = i f (CT. . u. T. . v.) n dS . (124) c 2 ^ ij i ij i Replacing T. . and v. by their expressions as functions of a and u , ij i ij i respectively, leads to E = \ I ( X 3 C 21 "l + (X 3 CT 22 f ^3 )n 2 + OvV + X 3 CT 23 ] n 3 ) dS » (125) i V 2 E c = 2 J /0 dX 3 /C X 3 CT 21 n i + (X 3 C 22 + ^3 )n 2 ] dS -a/2 C a/2 La/2 "J dX 3 / X 3 G 21 ( ^T' X 2' X 3 ) dX 2 dX 3 + -a/2 J -La/2 J Z1 " Z J 2 3

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63 La/2 Ra/2 + / ,, dX 2 J\ . GVwt* " •* I + I ff 23 (x l'Vf } J dX l • -La/ .s X v^Q'' (126) In these integrals, C is the curve defined by Ws* + "J^i'Va? " 4 • (127) that is, the "line of the kink," and X (x ) corresponds to a point (x ,x ) on this curve. The evaluation of these integrals is a complicated mathematical problem which has to be solved numerically. Unfortunately, such numerical computations have not been possible to achieve yet, mainly because of the very complicated expressions for the displacement and strain fields. However, further research on this mathematical problem can be carried out and will lead to the correct answer for the energy of a double kink. The final expression for the energy would be obtained by subtracting the difference between the correction energies given by Equation (115). The linear term is expected to cancel, so that the energy is a finite number. The limit of this number when N becomes large would be twice the energy of the single kink.

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CHAPTER 5 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED FROM AN ARRAY OF SHEAR LOOPS The same procedure as followed in the case of the screw dislocation will be used in the case of the edge dislocation. Displacement Field As seen previously in the chapter concerning the whole rectangular array of point forces, the regions where the dislocation loop has a pure edge character are delimited by \ s and La _ — e 2 2 2 where e and e are small compared to Ra and La, respectively. Since both regions simulate two identical parallel dislocations of opposite sense, we shall only consider the first one. A translation of the x axis to the first row of forces x = La/2 will simplify the expressions. The same symbols x and x will be kept, having now the meaning of e and e respectively. This part of the array of forces is represented in Figure 20. The only difference with Equation (21) is now the L L-l summation of q will be Z and E . So, u (r) becomes: q=0 q=l 2 R/2 j /l L-l\ V ? > = "IT ^ R/9 K + \) C G 2m ( V Pa ' V^' X 3 -f> " p= -r/2 n,q=° q=iy 64

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65 0) be a w CI CO 0) u o o a u

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66 " 'WV^'V^S + fD " C G 3m (x i" pa ' 3S 2' x 3-5 ) + G 3m (X r Pa ' X 2' X 3 + IO + (?3a (x 1 " pa 'V La » JI 3-| ) + G 3m (x r pa 'V La ' X 3 + H>(128) The summations on p and q will be accomplished as before, with the help of Euler' s formula (Equation (29)). The first summation carried out will be on p, since the displacement field of the edge dislocation ought to be independent of x . Euler' s formula becomes simply, R/2 +=> 2 f(p) = / f (x) dx , p= R/2 co (129) where f is a symbol for the whole expression to be summed. The displacement field takes the form: WW ° \q=0 q=lj 1 (x 2 -qa) + (Xg-g) 1 1-V . a. 2 (x 3 + i) (X 3 " 2 } (x 2 -qa) + (x 3+ -) (x 2 -qa) + (Xg--) (130) 16rr(l-v) V^ ^ X 2 (X 3 + 2 ) 2 a,2 2 a 2 X 2 + (x 3'2 ) X 2 + (X 3 + 2 } (131) U o(x, ,X X ) 3 V 1' 2'" 3' 16tt(1-V) L L-l E + E q=0 q=l (x 2 -qa)(x 3 --) , % 2 t a ^ 2 (x 2 -qa) + (x 3 --)

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67 (x 9 -qa)(x 3 + -) a 1-2V . R S/r. /2 + -/R /4+L -2 ^ 2 4 " 1-V (x 2 -qa) + (x 3 + -) a 3-4v 8rr 1-v &r. L + a 3-4v 32tt 1-V L X 2 + (X 3-2 } J + + ^ a 2 (X 3 + 2 } 2 *S 2 2 , (132) Unfortunately, these expressions are much less simple than those found for the screw dislocation. But some of the essential features of the displacement field of the edge dislocation can be noticed already, that is, the lack of displacement in the x direction and the fact that u and u are independent of x . Euler's formula applied a second time will give us the final expression for the displacement field. The mathematical difficulty, arising because of the singular points in the integrand, is overcome in the same way as for the screw dislocation. The same symbols, u and u , will be used for the successive terms added to approximate u and 3 2 u , respectively. For u (x x x ) we find 0. . 1 WW =4^ " (X 3-2> 1 3-4v 32tt 1-v -1 tan X 2^ ^3 + (X 3 + I ) Ta

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68 _a 1_ 16tt 1-v V^S"^ X 2 (X 3 + 2 ) 2 a.2 T 2 a, 2 L^*^"^ X 2 +(X 3 + 2 ) . (133) 1, , a 3-4V u (x , X , X ) = 2 1' 2' 3' 16rr 1-v 2 a. 2 . .2 , a.2" „ X 2 +(X 3 + 2 ) „ (X 2" a) +(X 3 + 2 ) _a 1__ 32tt 1-V t a ^ 2 2 a 2 X 2 +(X 3-2 } i ^ 2 (X 3-2 } ^ , a ^ 2 (x 2 -a) + (x 3 --) (X 3 + 2 } 2 a.2 2 , a 2 2 , a 2 X 2 +(X 3-2 } ( V a) +(X 3-2 } X 2 +(X 3 + 2 } r 2 +< x 3 -f> 2 ] Cv^ 2 ] c^-> a+ ^ + f>^ X 2 [ X 2-^ X 3-f> 2 ] (135) 3 a 3-4V Lv^s-^ J ( x 2 a) g X2 a) 2 3 (x 3 -|) 2 ^ x 2 [x^-3(x 3 + |) 2 ] [( V a) 2 + ( X3 -|) 2 ] 3 " K + (x 3 + |) 2 ] 3 (yQ [(x 2 -a) 2 -3(x 3 + |) 2 ] ) ^^ 9 a 9^ 3 I 480TT ^ X C +(x 3 + l> ] J

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69 a 2 ~2 (X 3"2 ) " + (X 3" 2° S 2 CV ( VI )2 3 # B , 2 X 2 C X 2(X 3 + I )2 3 r 2 . a, 2 V ° f 2 . a, 2 "1 LV^S"^ J L X 2 +(X 3 + 2 ) J ( x£ a) [(x 2 -a) 2 (x 3 -|) 2 ] 4 [(x 2 -a) 2 + (x 3 -|) 2 ] (x 2 a) C ( v a)2 (x 3 + f )2 T l 2 a)2 + ( x s + f )2 3 4 J (136) with 12 3 U„ = U_. + U + U„ + U + 2 2 2 2 2 (137) For u (x x , x ) we find /2 2 (l-2v)a „ R/2 + VR /4+L 3 4rr(l-v) fe RLa ( x o -o) 32tt(1-V) P 3 2 „ p, a N 2 2n ' n a,2 , x 2~) ^Q^-^ +X 2 J + ^ L (x 3"2 ) +( V a) J „ (-. a x 2 2-1 „ r, a s 2 , n 2 1 ^[(x 3+ -) + xj + ^ [_(x 3 + -) + (x 2 -a) J ~ (x 3 + 2 ) a(332rr ^j* Cv (x 3 -I )2 3 + ^ Cv (x 3 + I )2 ]j 16rr(l-v) , ^ 2 (X 3-2 } / ^ 2 (X 3 + 2 } 2 a.2 2 a, 2 V (X 3"2 ) V (X 3 + 2 } (138) U o( X , . X o> X o) = 3 1' 2' 3' 32n(l-v) Vs"* X 2 (X 3 + 2 ) 2 a. 2 2 a,2 X 2 +(X 3-2 } V (X 3 + 2 )

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70 (x 2 -a)(x 3 --) (x 2 a)(X 3 + 2 ) 2 a 2 2 a 2 (x 2 -a) + (x 3 --) (x 2 -a) + (x 3+ -) (139) u„(x ,x ,x )= (x. -~) 3 1' 2' 3 192tt(1-V) \ 3 2 2 , a,2 C (X 2 +(X 3 _ I )2 3 < ^ 2 / S 2 (x -a) (x„ — ) 2 3 2 [< V a, 2 + (x 3 -|) 2 ]J (x 2 -a) (x 3 + -) + (X 3 + 2 } 2 a 2 X 2 " (X 3 + 2' LK + < x 3 + l» 2 ] [(x 2 -a) + (x 3 + |) J J (140) u_(x, ,x_,x_) (x -^) 3 1' 2' 3 y 1920tt(1-V) \ v 3 2 r ^ V

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71 The constant in the expression of u means only that the point of non-bending of the lattice planes is set at x = La/2. However, the relative positions of the atoms with respect to each other are independent of this constant, and the stress ana strain fields will not depend on the constant terms. The component u (x x x ) has four singular points (x = 0, o \ Z o £ x = a with x = ± a/2) and u (x ,x ,x ) has two singular points Z, o o X t* «j (x = with x = ± a/2). This is not surprising, since the whole array of forces is a superposition of two arrays of forces having 2 magnitude |ja /2, one starting at x = and one starting at x = a. An asymptotic expression for u and u can be computed by considering x and x large with respect to the atomic distance. The following expressions are found: u 1 (x 1 ,x 2 ,x 3 ) = 0, (143) WW = 2^ TT/2 + tan -1 2 -TT/2 <3 *«-«4.4 (144) U„(X X o' X o) = 3 1' 2' 3 2it U(l-V) 1-2V . , 2 2. ton ( x r,+ x ^) 2 3 2(l-v) 2 2 X 2 + X 3. (145) These equations can be compared to those arising from the ordinary continuum model for an edge dislocation of the same sign [20] : WW = °(146) WW = 2TT tan — + 2 2(l-v)(x^+x 3 ) (147)

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2 2 , 1-2V .,22, X 2 " X 3 3 V 1' 2' 3' 2tt 4(1-v) 2 3'',, w 2 2, i 4(l-v)(x 2 +x 3 ) 72 (148) Both sets of equations are identical when one is aware that u is determined only up to a constant. Adding the expression a/8rr(l-V) to Equation (145) leads automatically to Equation (148); this physically means a change in the "cut plane." One can notice too that these asymptotic expressions for u come only from the first approximation u . This means that for a point situated at a large distance from the dislocation line, the discrete array of point forces appears to be a continuous distribution of force on the two planes x = ± a/2. A mapping of the atomic displacements, except for the singular points, is shown in Figure 21. Obviously, this model shows a strong dissymmetry with respect to the extra half plane (x = .75a, with x < 0) in the region of the core. The relative displacement across the slip plane in the direction x is simply a particular case of the expression of u (x ,x ,x ), as obtained before for the screw dislocation: Au 2 (x 2 ) = u 2 (x 2 ,-) u 2 (x 2 , --) = 2u 2 (X 2 ,_) . (149) A suitable form for .Au (x ) can be obtained either from Equation (131) or the set of Equations (133) to (136): 3 " 4V an 2 (x -qa) + a 2 (x 2 -qa) (150)

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73 X f o o o in o o o O O G .rfl o o o IO r o * ^* j V o o ro O o po o o ' o o o o o o -O--0 o o o b o o o o o o o o bo w

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74 Au 2 (x 2 ) = Au°(x 2 ) + AuJ(x 2 ) + Au^(x 2 ) + Au^) + .. (151) with . a I -1 X 2 -1 X 2 a ^l a" 1 X 2 Au„(x ) = o~ S 17 + tan — + tan ) + s~ TT, ~^ — o 2 2 2rr I a a ( 8tt 1-V 2 2 2 2 ,2 2 1 3-4V J . V 3 , . . ( V a) +S 1ft? 1=7 X 2 *» —2+ (X 2" a) * "ITx 2 (x 2 -a) (152) , 1, . a 3-4V Au„(x„) 2 2 32tt 1-V 2 2 , x 2 2' X 2 +a ^ X 2~ 3 + 3 ^7! — ^— + Bn g — X 2 (X 2 " 3) a 3 1 16rr 1-v 2 2 ' ,2 2 Ux + a (x -a) + a J (153) . 2. a 3-4V A W = "96^1=7 x Q x -a 2 2 2 2 _2 2 x 2 +a ( x o~ a ) + a _ a 4 1 48tt 1-V l(£.») s ^V] J 3 a 3-4V A VV = 288^ 117 2 2 x 2 (x 2 -3a ) 3 ' 3 .2 2 3 2 (X 2" a) (X 2 + a } (X 2 S) C (X 2a)2 " 3a2 ]' \ a 6 J_ / VV a2) r, . 2 2 f I +240TT1 V U 2 2 ,\ 4 l_(x 2 -a) +a J J M^ x 2 +a ^ ( X2 _ a) [(x 2 -a) 2 -a 2 ]^ [(x 2 -a) 2 + a 2 ] 4 J ' (154) (155)

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75 The curves representing Au are plotted in Figures 22 and 23, corresponding to the region close to the dislocation line and to a more extended region, respectively. Only a few terms Au are needed to _3 obtain relative precision equal to 10 in the region outside the points x„ = -a and x = 2a. Unfortunately, the region in between is not known and Au (0) and Au (a) must be evaluated by an interpolation scheme. Values of Au are listed in Table 7. TABLE 7. Relative Displacement Across the Slip Plane for an Edge Dislocation X 2 /a AU 2 /a 5 .0512 4 .0626 3 .0805 2 . 1123 1 . 1817 « .4 1 « .68 2 . 8280 3 .8921 4 .9221 5 .9391 The width of the dislocation can be reached by evaluating the region where the relative displacement has values between a/4 and 3a/4. From Figure 22, this region can be easily measured and has the value W = 1.93a. This value is slightly higher than that obtained from Peierls' model (W = 1.5a for V = 1/3). This would mean that our model shows a dislocation slightly more extended than Peierls' model. This result is opposite to what has been found for the screw dislocation.

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76

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A 77 £

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78 The relative displacement at atomic points can also be obtained directly from Equation (128): 4. CO f f CO CO 2 + „ j q=0 q= Au (na) = pa 2 I + E f~G (pa,na-qa,0) p_ cc V \q=0 q=l/ G 22 (pa,na-qa,a)J (_G 23 (pa, na , 0) + G 23 (pa,na, a)J I . (156) Replacing the variable (n-q) by u and making the singular point appear at q = n, leads to the three following equations for Au (na) corresponding to n>l, n=l and n = 0, respectively. 2 + m f n n-1 °° \ Au 2 (na) = |ia E E+E+E QG 22 (pa,ua,0) p= °° Vu=l u=l u=l/ + co "1 2 G 22 (pa,ua,a)J -pa E G 23 (pa,na,a) p— _ CO 00 2 + 4pa E (^G 22 (pa,0,0) -G 22 (pa,0,a)] +^--2p^ 2 G 22 (0,0,a), p=l 2 (157) Au 2 (a) = 2 pa E E Q} (pa,ua,0) G ?2 (pa,ua , a)j p= co u= i 2 r "i + pa E |j3 22 (pa,a,0) G 22 (pa,a,a) G 23 (pa,a,a)J 2 P + 4pa 2 E QG 22 (pa,0,0) G 22 (pa,0,a7| + ^~ 2pa 2 G^CO.O.a), p=l 2 (158) _|_CO CO Au 2 (0) = 2pa E E Q} (pa.ua ,0) G 22 (pa ,ua, a)] — — CO P= ro 2 + 2pa E [c 22 (pa,0,0) G 22 (pa,0 f a)] + J£pa 2 G 22 (0,0,a) p=l 2 (159)

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79 After computation of the discrete sums using Equation (26) , these three equations become, in the same order: , , 2ua a 11.4392 9.4720V a 1 n Au 2 (na) = — 8tt l-v 8TT 1-V 2 , n +1 16rr E + E n p=n+l/ l-v ' 2 l-v 2 , P +1 (160) , „ 2ua a 6.4060 5.6512v Au 2 (a) = -ft + st r^ (161) . ,„. ua a 11.4392 9.4720V Au 2 (0) =— + — — (162) For each value of n, a direct comparison can be made between Equations (160) to (162) on the one hand, and Equations (133) to (136) on the other hand. Since both ought to be identical, C can be easily deduced by subtraction. Values of Au (na) computed in both ways and the corresponding C (n) are listed in Table 8. These values have been computed for v = 1/3. The force constants C seem to be much more sensitive to n in the case of the edge dislocation than in the case of the screw dislocation Therefore the evaluation of the atomic displacements at x = and x = a, using the technique employed for the screw dislocation is not possible due to the uncertainty of C_(0) and C (1). If the variation of C (n) is neglected after a few atomic distances from the dislocation line, and C (n) set equal to C „( co ) for every n, the relative displacement becomes Au (na) = a -^— I E + E 2 16tt 1 p=n p=n+l. 3-4v „ p +1 1 m — =— + l-v 1-V 2 ^ p +1 J a 1 8rr 1-V 2 n +1 (163)

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Table 8. Relative Displacements and Force Constants C at Singular Points for an Edge Dislocation 80 1 2 3 4 5 6 7 8 9 10 l^oo-Sg. 253 232 3625 4027 4243 4376 4466 4531 4580 4619 4649 Au (n)/a

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81 clearly apparent. So, contrary to the case of the screw dislocation, C„(n) cannot be replaced by C ( ro ) , and their difference is sufficient to shift the symmetry of the dislocation with respect to x = a/2 to x = 0. In fact, it seems logical that in both cases, for the screw dislocation as well as for the edge dislocation, the dislocation line lies between two rows of atoms. As mentioned for the screw dislocation, a distribution function for infinitesimal dislocation loops can be introduced, following Equation (64). Its expression is the component o , „ 1 of the dislocation density tensor mentioned previously. So, by differentiating Equations (152) to (155), this distribution function a (x ) has the form a 21 (X 2 ) = a 21 U 2 ) + Q '21 (X 2 ) + Q 21 (X 2 } + a 21 (X 2 ) + (165) with 1 Q '21 CX 2 ; 8tt(1-v) 2 2 2, 2 La -t-x a +(x -a) _ l 3-4V 16tt l-V Qm 2 2 x 2+ a + im 2 2. (x -a) +a 2~ (x 2 -a) o 2 2 2 a -x„ i l 2 8TT l-V 2 2 (x 2 +a ) (166) 1 , . a 3-4v °'21 (X 2 ) = 18n T37 2 + 2 2, ,2 2 u x +a (x -a) +a X 2~tj a 1_ 8rr l-V 2 2 2 , 2 2^ (x -a) +a _(x 2 +a ) 2 (167) 2 a 3-4V *21 ( V _ 96tt T=v" 2 2 a -x„ 2 , ,2 , 2 2 2 x 2 (x 2 -a) (a + x 2 ) 2 , ^ a (x 9 -a) [_a +(x 2 -a) J

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82 48rr 1-v 2 2 3 (x -a)' (x„+a ) r ,2 2"] 2 L (x 2 _a) +a J (168) a 3-4v 21 960tt 1-v 2 (X 2" a) „ 2 2 4 4 3a x -x -a 2 2 , 2 2 4 (x 2+ a ) 2 4 4 4' 3a (x -a) (x -a) -a Qx 2 -a) +a J 2 2 4 4' 10a (x -a) -5(x -a) -a Q(x 2 -a) +a J r 2 2 4 4 10x 2 a -5x 2 -a 2401-v , 2 2 3 (x 2 + a ) (169) Successive approximations of a ? (x ) are plotted in Figure 14. A direct comparison with Peierls' model cannot be achieved successfully because of the lack of symmetry of the displacement field around the core in our model. Both models give the same result for points far from the dislocation line, but cannot be matched close to the core region. We shall see that a description of the edge dislocation from an array of prismatic loops is much more satisfactory and more close to the real atomic arrangement at the center of the defect. Self-Energy of the Edge Dislocation As seen previously, two steps are required to obtain the final energy of the edge dislocation. First the energy of the system of forces will be computed, and then a correction energy term will be introduced to subtract the excessive strain energy across the slip plane, due to the system of forces itself.

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84 When the rectangular array of forces is extended in the direction perpendicular to the Burgers vector, it simulates a system composed of two antiparallel straight edge dislocations separated by La. The energy per unit length of the array forces is the limit of W/Ra when R is much larger than L. From Equation (28) , it takes the form, W 2 = Lpa Ra Ua L C 2 4tt |ia 2rr(l-v) £n L + \i& 2(ia 7.6170 2.8588V 4TT(1 V) with A = 2.860 2.368V 1 V C 2 C 3 (170) (171) As previously seen for the energy of the kink in a screw dislocation, C_ is an unknown parameter which cannot be obtained by a physical argument as in the case of C . The choice of a suitable displacement field describing the dislocation will be made as for the evaluation of the correction energy of the double kink in a screw dislocation. The region where the relative displacement across the slip plane is larger than a/2 is contained between the planes x = a/2 and x = La a/2, as shown in Figure 25. A displacement field chosen to approximate the actual relative displacements of atoms across the slip plane is: (172) v 1 (x i ,y 2 ,x 3 ) = 0, wvv = w w v v„(x ,y„,x„) = u„(x ,x„.x„) , 3 1 ,J 2' 3 3 1' 2' 3 ' (173) (174)

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85 X^

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86 depending on whether x is positive or negative, y is taken as: y 2 = X 2 + 2 a yo = x o o for x > for x < (175) (176) The principal difficulty with this choice of v is that it is not symmetric with respect to x = , since u varies with x . We have: and where VwV = WW v 2 (x l' y 2'" X 3 ) = U 2 (X l' X 2 +a ' X 3 ) + V |u 2 (x 1 ,x 2 ,x 3 )| 4 |u 2 (x 1> x 2+ a,x 3 )| . (177) (178) (179) Since this lack of symmetry will have an effect only on the constant terms which do not appear in the coefficients of R or £,n R, it will be considered a sufficient approximation for our study. The tensors a and T corresponding to the displacement fields u and v, respectively, have the following components: f , /S aU 3 22 '33 9u 2 Su 3 2 3 5u 9u (9+2^) ^± + X tt-=>2 + BU 3 dX 3 + ^ (180)

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87 f T ll (y 3'V = CT 11 (X 2' X 3 ) ' T 22 (y 2' X 3 ) = CT 22 (X 2' X 3 ) ' 33 y 2' 3 33 2' 3 ' V> T 23 (y 2' X 3 ) = a 23 (X 2' X 3 ) " * (181) The correction energy is by definition the difference between the strain energies associated with the above stress fields contained within the volume of integration, i.e., E = i f a. . e. . dV ~ f T. . e. . dV 1 . c 2j ij ij 2 ^, ij ij (182) By using Green's theorem, the volume integrals can be transformed into surface integrals on the surfaces bounding V and V 1 : = i f a. . u. n. dS i f T. . v. n. dS* , 2 «J ij i J 2 J , ij i j (183) where n is the normal to the surface of integration. The differences between S and S' are mainly due to a translation of the plane x = a/2 of an amount +a/2, and a translation of the plane x = -a/2 of an amount -a/2. Because of these differences between the limits of integration, all integrals over a function y can be transformed into integrals over a function of x : La La-a/2 x 3 >0, / f(y 2 ) dy 2 = J f(x 2 ) dx 2 , a a/2 (184) La-a La-a/2 X 3 < °' / f(y 2 ) dy 2 = / J * Z a/2 f(x 2 ) dx 2 (185) So the correction energy has the form

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88 E = f (cr . u T . v. ) n. dS c 2 g ij 1 ij i J (186) Replacing v and T in function of u and a, respectively, in Equation (186) leads to E a/2 c C Ra -a/2 C X 3 a 22 ( l' X 3 ) + ^Vl'VJ dX c La-a/2 + I CI CT 23 (X 2'f ) + ^2 (X 2'I ) " *T3 dx 2> a/2 (187) E a/2 -£ = 2 f Ra o L ou (X+2u) x (£,x.) + (^i-\) u„ (|,x„)| dx. 3 ^ '2' A 3 3 v 2' 3 La/2 . ,a a p " X a u 3 ( 2'2 } + ^ J 5U 2 ox 2 2 U 2 (X 2'2 ) 1 dx„ + ^a (~u (La/2, a/2) u 3 (a/2,a/2)J . (188) For computing these integrals, we shall use the expressions for u. and its derivatives from Equations (131) and (132) . The final summation with respect to q will be completed at the last step. Expressions used for x (cu /ox ) and Bu /ox are listed below. O w i._, *J du 2 ax 3 / L <3 (ST ( W = 16* E n 1 2 Vq=0 q=l x qa ^ . r 2 , <\ 2 T LV (X 3"2 > J

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89 + (X 3 + 2 } , a, 2 2 (X 3 + 2 } " X 2 C&'vW (189) 3112 t 1 a / n 2 , a N. 2 (x 2 -qa) + (x 3 --) 2 1-V /• n 2 / S 2 (x 2 -qa) +(x 3 + -) . a v. 3 (X 3 + 2 ) [(x -qa) 2 + (x + |) 2 J r a A 2 ( VJ J 16tt(1-V) [(x 2 -qa) 2 + (x 3 -|) 2 3 r 2 3x2-. r 2 / a. 2 -j •2 L X 2(x 3-5 ) J X 2L X 2 " (X 3 + 2 ) J r 2 a 2~] f 2 a,2n: (190) We will not reproduce here the details of integration, but only write the final results for each step of the computation. The correction energy becomes ' L E 2 c _ ua Ra ~ 4tt (q-i) 2 +l n + J^ (q "2 ) ^O Q=0 q=l/ (q-o> + 4 tan" (q --)} ua 1-2V 2tt 1-V Sm L p« Ua 2 L + 2n 1 Z 1 « 5 7 1^ 2 ^ 4 + 40 , -1 _ 1 16v 18V +5 V „ „ + tt tan 2 + — — -z-rr. „. , + #72 2 4 ( 1 v) ( 1 2 v) 1-2V J (191) Computing the single sums as mentioned in Equation (26) leads to the final expression for the correction energy of the system, per unit length of edge dislocation.

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90 E c ua 2 L i^a 2 1 . |ja 2 r . 5198 + 2.3167V .3182V' ^ L + -w L — Ra rr l-v~"~ ' 2rr L (l-v)(l-2v) Recalling Equation (170), the total energy per unit length becomes Ra Ra Ra ' J . (192) (193) that is, E Ra T 2 = L \xa \i& A 1 C 2 + 4TT " 2 2 , ua 1 2n 1-v &n L 2rr 2 r 4tt pa |i JL 2 3 4.3273 6.7297V + 2.5 406V (1-V)(1-2V) 2-1 (194) For E /Ra to represent the energy of the system, C has to be T « chosen such that the term linear in L vanishes. Under this condition, C takes the same value as for the screw dislocation. Unfortunately, C„ remains unknown. It is clear here that it cannot be set equal to C as it is a priori assumed for the energy of the kink in the screw dislocation. If it were, the core parameter would be unrealistically large if one expects the same order of magnitude as for the screw dislocation. We can even foresee that C has to take a value much larger than C ; this force constant C , computed from the displacement field, was increasing greatly for atoms close to the dislocation line. Since C corresponds to the closest atoms to the dislocation line, it can be expected to be of the order of magnitude of C (0) . However, the total energy of the system composed of two parallel edge dislocations, and of opposite signs, has the form expected. (ia 7 La Ra 2n(l-V) r (195)

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91 and the self-energy for a pure edge dislocation is simply !g = ^ r (k — . (196) Ra 4tt(1-v) r Q The fact that the essential features of the self-energy of the edge dislocation are reached with our model is very encouraging. The evaluation of the force constant C remains the only uncertainty in the energy of the edge dislocation. Single and Double Kinks in an Edge Dislocation Similar to kinks in a screw dislocation, a single kink or a double kink in an edge dislocation can be simulated by adding only one shear loop at the end, x =0, of each row in the region x < for 1 ^ a single kink and in the region -Na < x < Na for a double kink. The array of forces takes the configuration shown in Figures 26 and 27, corresponding to single and double kinks, respectively. A. Displacement field The displacement field corresponding to these defects is given by the superposition of the displacement field of the edge dislocation and the displacement field of the extra forces introduced to simulate the defect. The latter, u' , has the following expression, for a single kink or a double kink of length 2Na, respectively. U 1(SK) = ^ Z n f(p) (197) p=0 and u! (DK) = ±fI f(p) , (198) p=-N

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92 ^fe^S) Figure 26. Array of Forces for Single Kink in an Edge Dislocation -^j) — -£y-£3 ~ < T~ ( F Figure 27. Array of Forces for Double Kink in an Edge Dislocation

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where f(p) = G^x^pa.x^a.Xg--) G.^-pa^+a^ + -) + G„„(x -pa,x„ ,x„ — ) G„„(x -pa,x„,x„+— ) 22 1 F ' 2' 3 2 22 1 K ' 2' 3 2 G 23 (x 1 -pa l x 2+ a,x 3 -2) G^C^-pa, V a,x 3+ -) 93 + G 23 (x 1 -pa I x 2> x 3 -j) + G 2 3 ( V Pa ' X 2' X 3 + 2 ) (199) We shall only consider the displacements in the x direction, since we are mainly interested in the change of the atomic configuration in a plane parallel to the slip plane. Due to the symmetry of the extra rows of forces with respect to x = -a/2, we shall only take account of points corresponding to x > -a/2, for example. The expressions for the displacement of points of application of point forces will be different from those for other atomic point, since the former are singular points. The following equations obtain in the plane x = a/2 for single and double kinks, respectively. (a) All x except x = and x < U 2(SK) " 32tt u=n 3-4V 1-v /2~2 ; ~2~ /2~2~ ~2 2 a/u a +(x +a) a/u a ( x + a ) + a /"2~2 2 /2~2 2 2 a/u a +x a/u a +x +a 1 1-V (x 2 + a) f 2 2 i ^1 |u a +(x +a) J 3/2 (x 2 +a) (x 2 +2a) x 2 (x 2 -a) 3/2 3/2 r 2 2 2 1 ' r 2 2 , v 2 2T' r 2 2 2 2 ~| lu a +x J u a +(x +a) +a J [u a +x +a J 3/2 (200)

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94 (b) x = and x < V. Vu +1 Vu +2 l-v (u 2 +l) 3/2 I 3 " 4v I 1 2 2C„ ' (201) and for a double kink of length 2Na, (a) All x except x = and -Na < x < Na a a 2 g I 3-4v U 2(DK) CX 1' X 2'2 ; 32rr „S (l-v) p=-N I / 2 2 / (x 1 ~pa)"+(x 2 +a) y' 2 2 2 / 2 2 / (x -pa) +(x +a) +a (/(x -pa) +x V(x -pa) 2 2 2 (l-v) (x 2+ a) 3/2 r 2 2*1 f 2221 _[_(x -pa) +(x 2 +a) J ^(x^^-pa) +x 2 ) J ,3/2 (x 2 +a) (x 2 +2a) x 2 (x 2 -a) r 9 , 2 2-i 3/2 f. .2 2 2-| 3/2 |^(x -pa) +(x +a) +a J |_(x -pa) +x 2 +a J J (202) (b) x = and -Na < x < Na « a N a 2 J 3 " 4v ! U 2(nc) (na '°»2 ) a 55S \ T \~~T= p=-N i 1 l-v "1 V(n-p) +1 N+n N-n\ Q(n-p) 2 +l] 3/2 r 2 2-i 3/2 [_(n-p) +2a J \ u= 1 u= 1 / 2 (203)

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95 The force constant C used here ought to be C (0) , computed from Equation (162). It can only be approximated, since Au (0) for the edge dislocation is only known approximately as 0.4a. In any event, we shall use this value to obtain the general trend of the atomic configuration in the immediate vicinity of the kink. The atomic displacements are tabulated in Table 9 for the single kink, and in Tables 10 to 12 for the double kinks. TABLE 9. Atomic Displacements for a Single Kink in an Edge Dislocation x l

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96 TABLE 10. Atomic Displacements for a Double Kink of Length 2a in an Edge Dislocation

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97 Figures 28 to 31 show the atomic configurations of the defects in the planes x = ± a/2, corresponding to the values listed in Tables 9 to 12 in the same order. The same comments made for kinks in the screw dislocation apply here, that is, particularly the localization of the distortion in the immediate vicinity of the defect. B. Energy of the double and single kinks The energy of a double kink will be defined as the difference between the energy of the modified rectangular array of forces and the unmodified rectangular dislocation loop. First, the energy difference between the two arrays of forces is equal to the self -energy of the extra double row of forces added to their interaction energies with the rest of the array. Their expressions are 2 4 E n = (2N+1) |i a Row ^--G 22 (0 ( ( a^ L 2 2 4 + [i a 2 4 ^2N+1 R/2 E (4N+2-p) + E (2N+1) _p=l p=2N+2 2N+1 R/2 p, a j E P+ E (2N+1) _p=l p=2N+2 [G 22 (pa,0,0)-G 22 (pa,0,a)] QG 33 (pa,0,0)+G (pa.O.af], (204) S x = 2(2N+1) n 2 a 4 E / [G 22 (0,qa,0) 0^(0, qa, a)] q=i L R/2 -s + 2 E QG 22 (pa,qa,0) -G (pa.qa.am [2G 23 (pa,a,a) -G 22 (pa,a,0) 2 4 + |i a 2N+1 R/2 E P+ E (2N+1) -p=l p=2N+2 + G 22 (pa,a,a) + G 33 (pa,a,0) + G 33 (pa,a,a)l (205)

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98 3D OOOPQ3 O OOfcO O C© c© cooicro CO GDOpOO O i « c© •;© o (Q) C© Q (© c© coo o co <© obo o 00; © OO O -a w CS c g 0) to c U U <

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99 'ID •'O CO o CO o co a ;© co coo o •:o ooe :-?hco ce-e PC: O bo o bo o oo o 3 —' ^CFn-G ^= • — • v s ; V_ A., a V — y CO CO o CO CO O CO CO c © CO CI o:; o bo O bo o bo o Do o u c o o +» +j a c o 0) o 6 -H 0) CO 5 Q a U
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100 CO CO qpO O c co co cbo o •'© co obo o co ce-eteo o co co'cte o 0>C5 CNJ I CO a pec toe ;-0 OGfOO o •:o ce-c CO 'CO o Oj o oo o u c o o H -H •p -p a c o o o 5 rH O 03 bX) -H a q ^ 0) H 60 5 -a W O •H fl 6 a o co (_o obo o © CO ;f)0 O t

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101 c© .;© :: bo o CD CO o to o c© oo-e co c eo c oop o Ool'Qj O CO 00 o 1 OO rO; O CO ( CO COOrGD O ' fcH •'© CO C "©;; o bo o xi -p SP 8)
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102 with W Row I (206) In the expressions above, we always considered the length of the double kink to be much smaller than the length of the dislocation. Replacing the Green's functions by their explicit expressions leads to i I -„ , \ W = =r(2N+1) 16tt R/2 + 2 E X 2N+2 2 3/2 2 3/2 SP +2) (p +1) j 3-4V ua ua 3 2N+1 tMn fr^i 3-4V 1-V 1 + 2p J VP +^ 1-v 2 3/2 \(p +2) , 2 3/2 (P +D / (207) The term pa/C , depending on the force constant, can be replaced by its value, as defined in Equations (113) and (114). The final expression for the energy difference between the systems of forces becomes 3 I R/2 W p,a ~8t7 (2N+1) < 4tt + 2N+2 3-4v 1-v Jv +<

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103 + 1-V I , 2 3/2 2 3/2 (P +2) (p +1) Ita 3 2 g +1 3-4V I p ,\ + 1 ( 2 P (208) If N becomes large, but still smaller than the length of the edge dislocation, the double kink behaves like two separate single kinks. The corresponding energy of the system of forces for each single kink is the limit of W/2 when N increases: + T^ l7T^37g2 > 2 '^ \(P +2) (p +1) The interaction energy between the two kinks is simply (2W W) . As for the kinks in a screw dislocation, a correction energy term has to be introduced to remove the excess strain energy contained in the region between the planes of forces. Unfortunately, the same difficulties arise as before and we shall not be able to perform a complete analysis of the energy of a double kink. The computation of the strain energy can be done exactly in the same way as for the screw dislocation, and its development here would only be a repetition of the treatment of the correction energy for a double kink in a screw dislocation. We anticipate that in the complete expression for the energy of the kink, the coefficient of N can be made to vanish, leaving only a pure number. It is interesting to point out that Equations (208) and

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104 (209) have exactly the same form as Equations (115) and (116) , except for the presence of the force constant C . This is due mainly to the screw dislocation character of the kink segments. If a complete treatment of the correction energy can be achieved, the energy of a kink in an edge dislocation is thoroughly determined. The results already obtained are very encouraging and a complete treatment should lead to a better understanding of kinks in edge dislocations.

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CHAPTER 6 EDGE DISLOCATION IN SIMPLE CUBIC CRYSTAL CONSTRUCTED FROM AN ARRAY OF PRISMATIC LOOPS A rectangular dislocation loop formed by four segments of edge dislocations can be simulated by an array of primitive prismatic loops as defined in the first chapter. This is equivalent to a rectangular plane of vacancies plus an extra contraction in a direction normal to this plane. The loop is shown in Figures 32 and 33. The forces F and G are those computed from Equations (15) and (16) , and have the resepctive values: F = — ^ (210) and G = ^-Z a . (211) The same procedure as used in the previous chapters will be followed here to obtain the displacement field, width and self-energy of the pure edge dislocation. Displacement Field The loop considered will be rectangular, of dimensions La and Ra in directions parallel to x and x respectively. This means that the loop is composed of (R+l) (L+l) vacancies. The edge dislocation considered will have its line parallel to the x axis and located 105

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106 oooo®oooo o o ao~ k &u-@o *• 0© I 0O 00 t 00 00 r 0O 00 \ 00 000© t-eee^ 000060000 Figure 32. Array of Prismatic Loop in x = Plane QO$$.QLQC 0i 00* 01 O i m 1 Ra xa i 1 1 j' re Lo 1_1 T I I 1, I *0 © 1*0 O Figure 33. Array of Prismatic Loop in x. =0 Plane

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107 near x = 0. So the coordinates x and x will always be considered small with respect to La and Ra , respectively. Considering the displacement field in this region as the superposition of the displacement fields of all the point forces leads to the following formal expression in terms of Green's functions: Xa 2 L/2 f a WW = -r E ,„ " G 3i (x i' x 2 qa x 3-2 ) q= L/2 ^ . G 3i (X l' X 2qa ' X 3 + f ) + G 3i (X l' X 2qa ' X 3 + l +Ra) 3a 1 Xa 2 R+1 r la a + G 3i (X l' X 2qa ' X 3 + — +Ra) j + — ^ Ki (X l' X 2 + i-' X 3-2 + Ra) + G 2i (X l' X 2 + • +a ' X 3-f +ra) " G 2i (X r X 2-•' X 3-f +ra) La a -\ X + 2u 2 R +} L/ y 2 J r=l q= L/2 G ,(x +a,x -qa,x -~+ra) G (x -a,x -qa,x -+ ra) ) , (212) where G. . (X ,X , X ) is defined by Equation (3). ij X 2 3 Again, Euler's formula, Equation (29), will be used to evaluate the discrete summations. The first summation considered will be along the line of the dislocation. This sum is found to be identical to the first term of Euler's approximation: L/2 L/2 Z f(q) = J f (x) dx , (213) q= L/2 L/2 so an expression of u.(x x , x ) can be written as:

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108 u (x ,x ,x ) = 1 ' ' 8tt(1-v) (1-2V) X l (X 3-2 } X l (X 3 + 2 } 2 , a -> 2 2 / a x 2 V'^"^ X 1 +(X 3 + 2 ) R+l fa< n 2 / a ^ 2 , 3-4V (x r a) + (x 3 -+ ra) f 16tt A l-2v *" . ,2 a ,2 'I (x +a) +( x 3 _ 2 + > " a ^ 1-2V (x +a)' (x i -a)' /• x 2 / a n 2 n 2 , a x 2 (x^a) +(x 3 ~-+ra) (x -a) + (x 3 ~-+ra) (214) U 2 (X 1' X 2' X 3 ) = °« (215) u„(x ,x„,x„) = 3 V 1' 2' 3 J 16n(l-V) fo /~2 . 2 3-4V 1-2V 1-2V L/2 + VL /4+R a. 2 a. 2 4^R + ^.[x 1+ (x 3 -|) ] + Bn [x i+ (x 3 + |) ] 2 a,2 2 a.2 X l +(X 3-2 } V (X 3 + 2 ) R+l f(x 3 --+ ra) (x +a) (x 3 -+ ra)(x r a) 8n(l-2v) \ a 2 2 a2 2/ r=l i( x 3 -2 + ra > +(x +a) (x 3~2 + ra ^ +(x i~ a) I (216) The second step of the computation consists of applying Euler's formula a second time, giving a series of expressions which represent successively higher approximations to the displacement field of the edge dislocation. The mathematical difficulties encountered when computing integrals over a region containing singular points are solved in the same manner as for the screw dislocation in Chapter 3. The final analytic expressions for u and u take the form

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109 U = U + U + U + U + . . . , (217) with WW = is< h*i + ( v a) tan x7n (x r a) tan ^T 3-4V , a, ( V a) +(X 3 + 2 } (X„ +-) S.T, 16n(l-2v) 3 2' "" , x 2 , a x 2 (x r a) + (x 3 + -) ' Wl* X 1 (X 3 + I ) Jrr(l-V) (1-2V) A 2 , a, 2 2 . a, 2 l^x i+ (x 3 --) V (X 3 + 2^ (218) The terms -to, -ttx and rra in Equation (218) correspond to the regions x > a, -a < x < a and x < -a, respectively. 1, y a(3-4v) ( V a) +(X 3 + 2 } WW = 32tt(1-2v ^: ^— T2 (x r a) + (x 3 + -) 2 2 (x -a) (x +a) 16tt(1-2v) ) 2 a 2 2 a 2[ ' ^(x r a) +(x 3 + -) (x 1+ a) +(x 3+ 2)"j 2 ( , a 2 (3-4.) J V a/2 V a/2 u, (x 1 ,x_,x_) = (219) l v 1' 2' 3' 96n(l-2v) \ , ,2 a,2 ,2 a 2 I (x -a) +( x 3 +2^ (x +a) +( x 3 + ^ 2 f (x -a) 2 a , a -> / 1 (x + -) < jr48tt(1-2v) 3 2 j 2 a 2 "I' L[(V a) +(X 3 + 2 ) J (x+a) 2 *) = rr> , (220) [_(x 1+ a) 2 + (x 3+ |) 2 ]

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110 and 3, , a 4 (3-4v) , a f 3( -V a)2 (X 3 + f )2 U (X ,X ,X ) = — ; — — (X + — ) 1 1' 2' 3 2880tt(1-2v) 3 2 \ n n 3 a^ 2 L (x r a) +(x 3 + l } 3 3(x +a) -( x o + 2^ Q(x 1+ a) +(x 3 + |) ] V )2 [ ( V a)2 ( vi )2 ] (x + o) 240tt(1-2v 3 2' . LL
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Ill 1 x ,+ a 1, v a . a. ) 1 U„(X X X ) = — — — (X +— ) 3 1' 2' 3' 16tt(1-2v) 3 2' \ , .2 . a 2 ( Y a) +(x 3 + 2 ) (x r a) + (Xg + -) (224) 2 WW 96tt(1-2v) t 2 [(x^a) Q(x i+ a) -(x 3 + f) ] [(x i+ a) 2 + (x 3 + |) 2 ] ( x r a)[(x 1 -a) 2 -(x 3+ |) 2 3" > > [(x r a) 2 + (x 3 + |) 2 ] 2 J (225) and ^ n r, >4 „, 2, a. 2 , a,4l 3 a 4 |(x+a) Q(x+a) -6(x + a) (Xg + ? ) +0X3+-) J u (x , x , X ) = 3 l' 2' 3' 960tt(1-2V) \ _ _ -. 4 (x 1+ a) +(x 3 + 2) J ( v a) C ( v a)4 6( v a)2(x 3 + l )2+(x 3 + l )4 J > l a / • (226) [(x 1 -a) 2 + (x 3+ |) 2 ] The consistency of this displacement field with results alreadyobtained from the ordinary continuum model is checked by considering x and x to be far from the dislocation line, and by expanding u and u in a Taylor series, f( Y e i-VV = f(x r x 3 ) + e i o^ (x r x 3 ) + e 3 -&q ^i'V(227) where s and e are small with respect to x and x respectively. The following expressions are found:

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WW = 2r7 112 -TT/2 "\ -1 X 3 1 X 1 X 3 + tan — > + ,* . -4~ , (228) /0 x ( 4rr(l-vT 2 2' TT/2 1 j x i+ x 3 where -tt/2 and tt/2 correspond to x > and x < 0, respectively. Also U 2 (X 1' X 2' X 3 ) = ° * (229) W W= stTTT^T a { 2 0n L/24VW4+R 2 2 -^ ( V X I>? 4!^ "2^2 (230) J V X 3 These equations match exactly with the ordinary continuum model, Equations (146), (147) and (148), when one realizes that u (x ,x ,0) is equal to -a/4 from the way the displacement field has been defined. By contrast with the simulation of an edge dislocation by an array of shear loops, in the present case the displacement field is . symmetrical with respect to the plane x = 0. The only singular points occur for x = ± a and x = -na + a/2 (n > 1) , and x = with x = a/2. We shall demonstrate that the plane x = plays a unique role for the dislocation in the sense that the displacement of singular points will be shown from the displacement of the points symmetric with respect to x = 0. We have the fundamental relations: u 1 (x 1 ' x 2' na+ 2} + U l (K l' X 2~ na ~f^ = ~2' (231) and u 3 (x 1 ,x 2 ,na + -) = u 3 (x 1 ,x 2 ,-na --). (232)

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Let us first consider the special case of n = for u . From Equation (214) , we have 113 .a. va 1 a ^ VW^ = 8tt(i-*i-2v) "2~T + T&R E f(ra) ' x +a r=l (233) ax 0° V X ' X -§> = 8tt(1-v)(1-2v) T-2 + ife Sf(ra-a), x+a r=l (234) with , ,2 2 2 *t , 3-4V r (X r a) +r a 2 f(ra) = T^ fe " ^-T2 + T=S (x+a) +r a (X + a) , N 2 2 2 (x+a) +r a (x x -a) 2 2 2 (x -a) +r a (235) Noticing that E f(ra-a) = E f(ra) + f (0) , r=l r=l (236) the two identities follow: OO (-K — ?>1 , a a, a _ _ x a 3-4v l 1 ' VVVS* + u l (x l ,x 2 , --) =_ Ef (ra)+— _ -o>u -, Cx, ,x 0> -) = a 3-4v„ U r a) 072 1 1' 2' 2 1 1' 2'2 16rr 1-2 V*"' 2 (x^a) (238) On another hand, from Equations (218) to (221) , u (x ,x , 2> has the exact value

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114 -a/4 ax, . a _ \ ,. va 1_ VW ~2 } \ X l /4 + 8tt(1-V) (1-2V) 2 2 /A X 1 +a a/4 1 (3-4 V) a „ ( V a) + ^ ,, ^.., 071 32tt(1-2\» "" , ,2 ' (x +a) (239) where the terms -a/4, -x /4 and a/4 correspond to the region x > a, -a < x < a, and x < -a, respectively. Comparison with Equations (237) and (238) leads to u (x , x„ , ) + u (x ,x„,— ) -x /2 1 1' 2' 2 J V 1' 2'2 1 -a/2 a/2 (240) Accidentally, an interesting identity has been proved, that is, 2 CO 1 1 1 3-4v„ ( V a) 2 I6rr 1-2 v ""' N 2 8n (x x ~a) r=l Z f(ra) . (241) This result can easily be generalized to x = na + a/2 and x =-(na + — ) for any value of n. The displacements in the x direction for these points have the form u 1 (x lf x 2 ,na + |) = Va x na 8TT(1-V)(1-2V) \ 2 2 2 I x +n a x (na+a) ~~ 2 2 x + (na+a) + 16* E " E J f(ra) ' \r= 1 r= 1 / (242) when it is noticed that E f(ra+na) = E f(ra) E f (ra) , r=l r=l r=l (243)

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and 115 U l (X l' X 2'' na -2 ) 8tt(1-v)(1-2v) 2 2 2 x +n a x (na+a) x (na+a) 16tT E + E f(ra) r=l r=0/ (244) So, by adding Equations (242) and (244), the same expression as Equation (237) is found, which leads to u 1 (x i ,x 2 ,na + -) + Ui (x i( x 2 ,-na--) -a/2 a/2 (245) The same line of reasoning applies to u , leading to u 3 ( X;L ,x 2 ,na + -) = u 3 (x i ,x 2 ,-na--) (246) Equation (245) will help us to find the atomic displacements of the singular points and the force constant C . The latter can be obtained by expressing the displacement of these singular points in terms of Green's tensor: \a 2 L/2 , u (a,-na--)= — E ( G 31 (a,x 2 -qa, -na-a) q= L/2 R+l L/2 + G (a,x -qa,-naA + -~ a E E (g (2a, x -qa, -na-a+ra) 31 2 J 2 r=l -L/2 V11 l G (0,x -qa, -na-a+ra) ) 11 ' 2 ' J (247) After summing on q, Equation (247) becomes n+1 u^a.-na-f) = --^-J^—2 2 .(n+1) +1 n+1.

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116 \+2\i a 2 C (248) By comparing the values of u (a, -na, -a/2) from Equations (217) and (248) , C can be computed for each value of n. Several values are listed in Table 13. TABLE 13. Displacements and Force Constants C^ at Singular Points for an Edge Dislocation u (— na)/a c t (v 1/3) 4100 4511 4731 4833 4882 4910 ,4928 4940 .4948 ,4954 5000 4.692 4.688 4.679 4.674 4.672 4.671 4.669 4.669 4.669 4.669 4.666 The relative displacement across the slip plane can be easily deduced and has the form AU 1 (X 1 } = W2 } " W ) ^l'-^ (2<*9)

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117 Recalling Equation (239) , Au has the final form 9 a r > a 3 ~^r (X l +a) a V aX l Au i (x i } = TfrFl=2v fc ^-4^ (i-v)(i-2v) T-T • (250) (x -a) x +a Values of Au are plotted in Figure 34. Finally, a mapping on the x = plane of the atomic arrangement is shown in Figure 35. A direct comparison with Peierls' model can be attempted, since both models are symmetrical about x = 0. From Peierls' model, the relative displacement has the form a a -1 X 1 k(xj = tan X -J, (251) 1 1 2 TT i where § = 2T^r • (252) We shall expand Equations (249) and (250) in powers of 1,'x and shall select § so that both expressions match at large values of x . So first order this gives a 3-2 v a A VV ^ — r;< 253 > and for the Peierls' model a 5 Au (x ) = -2. (254) 11 IT X By matching Equations (252) and (253) Peierls' model is identical to ours for large x , if In contrast with the screw dislocation, this is not the value found by Peierls. Peierls' expressions for § = a/2(l-v) and §= (a/4) (3-2V) / (1-V)

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118 In 3 3 0) bo w Q. -H Q >

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119 o o

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120 are plotted in Figure 34. For V = 1/3, the values of § are .75 and .875, respectively. So our model for the edge dislocation is wider than Peierls' model. It is impossible to say which value is the actual one, since both models are completely different and involve different approximations. As a conclusion we would say that simulating edge dislocation by an array of prismatic loops gives a displacement field which exhibits the expected symmetry relative to the extra half-plane, in contrast to the result for the shear loops. Self-Energy of the Edge Dislocation The energy of the system of point forces is composed as usual of the sum of the self-energies of the point forces less the sum of their pairwise interaction energies. Because of the formidable length of the expression, we shall divide the whole equation into smaller subdivisions. (a) Energy of the array of G forces: 1 W_ = (R+1)(L+1)G G G (2a, 0,0) LI L 2 q=l R E q=l R L + 2G (R+l) Z (L-l-q) f~G (0,qa,0) G (2a,qa,0)l q=l 2 R + 2G (L+l) E (R+l-q) Pg (0,0,pa) G (2a,0 ,pa)] + 4G E £ (R+l-p) (L+l-q) [~G (0,qa,pa) G (2a,qa,pa)~] . p=l q=l (256)

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121 (b) Energy of the array of F forces; 2F 2 (L+1) . ow 2 _ ,„ _ . „_2 L w = F C + 2F (L+l) G 33 (0,0,a) + 2F E (L+l-q)[2G (0,qa,0) 3 "*" q=l + 2G 33 (0,qa,a) G 33 (0,qa,Ra) G (0 ,qa,Ra+2a) 2G 33 (0,qa,Ra+a)J (L+l) QsG^O , ,Ra+a) + G 33 (0,0,Ra) f G 33 (0,0,Ra+2a)"l + 2F ^ R+1) + 2F 2 (R + 1) G 22 (0,a,0) + 2F 2 E (R=l-P) Q2G 22 (0,0,pa) + 2G 22 (0,a,pa) P=l -G 22 (0,La,pa) G 22 (0,La+2a,pa) 2G 22 (0,La+a,parj (R+l) r"2G 22 (0,La+a,0) + G 22 (0,La,0) + G 22 (0 ,La+2a,ofj L R R 16 E E G_„(0,qa,pa) 8 E G (0,La+a,pa) q=l p=l p=l 8 E G (0,qa,Ra+a) 4G (0 ,La+a, Ra+a) ; q=l (257) (c) Interaction energy between both arrays: FG E (L+l) G (a,0,pa) + E (R+l) G l9 (a,qa,0) -< I"' 1 1^ Lp=l q=l C L R E (L+l-q) G ir! (a,qa,Ra+a) + E (R+l-p) G (a,La+a,pa) q=l p=l 4 ["(L+l) G (a, 0, Ra+a) + (R+a) G (a,La+a,0r) 16 R L E E (L+l-q) G (a,qa,pa) Lp=l q=l 13 R L + E E (R+l-p) G (a,qa,pa) P=l q=l (258)

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122 The total energy of the system of forces is the sura of these three equations. After computation of the various sums and collecting terms containing R and L, the following expression is found for the energy of the whole array of forces. W _ (R+l) (L+l) (1-v) . ua 3.3524 8.6586V 1 1 ,r -,N /) 2R L Z 2 -2 1 1 ,r, ,% /> 2RL L+ >/R +L /~2 2~ R+ VR +L 1 V " J^~ 2 2rr 2 (1-2V) (L+1) R_±VRJ±_ + (R+1) ^LWR + L /2 2^ + VR + (R+l+L+l) 1 (1-V) T8.0325 5.5644V 4rr (1-2V) r «.03Z5 5. 5644V -i 2L 1-V J 1 v 2 2TT (1-2V) 2

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123 o 2 ua (1-V) (I^l)a 4^ (1 _ 2v) 2 ^« + (R+D 4rrua 3.3524 8.6586V ~C~ + 1-V H,a 2 (1-V) 8.0325 5.5644V 4tt (1-2V) lia V r 4rr(ia _ 7.4947 9. 1180v ~j ua_ V 2 , |ia 1 TT 1-V (2.2148) (260) In the special case of V = 1/3, we have 2 *»r £ c i 2 W (1*1) a „H.a«)[^ + .6»8| -Jfc.&fc. 2TT 4rrixa 14.9274 (261) Both force constants C and C are unknown unless C is taken to be the 1 o Jvalue found from the displacement field computations. However, C 3 will still remain unknown, and the same difficulty as was encountered in the simulation from shear loops is encountered here. Furthermore, Equation (260) does not have the well-known form of the energy of such a system. This is due to the excess strain energy stored between the planes x = ± a/2 after deformation. All the material contained between the planes x = ± a before the introduction of the defect is compressed into a slab less twice the original width. Unfortunately, it has not been possible to propose a displacement field in this region which would permit one to obtain the correction energy for this system. The problem is much less simple than for the simulation from shear loops. So further research has to be done in this direction.

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CHAPTER 7 CONCLUSIONS The new approach presented here has given us very satisfying results for the computations of the small range displacement field and the self-energy of the various dislocations, especially for the case of the screw dislocation where a complete treatment could be done. For the long range displacement field, this model is in agreement with ordinary continuum model in all cases. When compared to Peierls' model, the screw dislocation has a narrower width in our model, although the edge dislocation has a broader width. The arrangement of atoms in the vicinity of the dislocation line is obtained for most of the cases and at distances closer to the dislocation line than the ordinary continuum model permits. It has not been possible to compare the atomic displacements with atomistic computations, since simple cubic structure is hypothetical. Its study, however, will be the guide of studies on real crystals. At this time, there are two principal difficulties in applying this model: (a) the evaluation of the force constant corresponding to the forces normal to the slip plane, and (b) the computation of the excess strain energy between the planes of forces. One way to solve these problems would be to compare with computer simulation models. 124

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125 Unfortunately, this can only be done on real crystal structures. So the next step for pursuing this study would be a direct application of the method employed here to reach the properties of dislocation and kinks in face centered cubic crystal, body centered cubic crystals or hexagonal compact crystals.

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BIBLIOGRAPHY 1. J. D. Eshelby, Solid State Physics , 3, 79 (1956). 2. E. Kroner, "Kontinuumstheorie der Versetzungen und Eigenspannungen,' E rgeb. der Angew. Math . , 5, Springer-Verlag, Berlin (1958). 3. R. de Wit, Solid State Physics , 10 , 249 (1960). 4. H. B. Huntington, J. E. Dickey, and R. Thompson, Phys. Rev . , 100 , 1117 (1955). 5. T. Kurosawa, J. Phys. Soc. Japan, 19 , 2096 (1904). 6. F. Granzer, G. Wagner, and J. Eisenblatter , Phys. Status. Solidi , 30, 587 (1968). 7. R. M. J. Cotterill and M. Doyama, Phys. Rev . , 145 , 465 (1966). 8. M. Doyama and R. M. J. Cotterill, Phys. Rev . , 150 , 448 (1966). 9. R. Chang and L. T. Graham, Phys. Status. Solidi , 18 , 99 (1966). 10. R. Chang, Phil. Mag . , 16, 1021 (1967). 11. R. Bullough and R. C. Perrin, Atomic Energy Research Establishment, Harwell Report No. T-P-292 , 1967 (unpublished). 12. R. Bullough and R. C. Perrin, in Dislocation Dynamics , edited by A. R. Rosenfield, G. T. Hahn, A. L. Bement , and R. I. Jaffee, McGraw-Hill, New York (1968). 13. P. C. Gehlen, A. R. Rosenfield, and G. T. Hahn, J. Appl. Phys . , 39, 5246 (1968). 14. P. C. Gehlen, J. R. Beeler, and R. I. Jaffee, -eds. , Interatomic Potentials and Simulation of Lattice Defects , to be published by Plenum, Boston, 1972. 15. C. S. Hartley and R. B. Bullough, "On the Description of Crystal Defects by Point Force Arrays," AERE-TP-490 , Harwell, U. K. , July, 1972. 16. J. S. Koehler, J. Appl. Phys., 37, 435 (1966). 126

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127 17. P. P. Groves and D. J. Bacon, J. Appl. Phys . , 40, 4207 (1969). 18. F. Kroupa, Czech. J. Phys . , B 12 , 191 (1962). 19. J. B. Scarborough, Numerical Mathematical Analysis , The Johns Hopkins Press, Baltimore (1955). 20. J. P. Hirth and J. Lothe, Theory of Dislocations , McGraw-Hill, New York (1968). ~ "~ 21. R. E. Peierls, Proc. Phys. Soc. , 52, 23 (1940).

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BIOGRAPHICAL SKETCH Jean-Pierre Georges was born in Les Lilas, France, on March 29, 1946. He completed his secondary education at the "Lycee Hoche" in Versailles; he passed the Baccalaureat in Mathematics in 1963; he attended the classes of "Mathematiques Superieures" and "Mathematiques Speciales" in the same college. He entered the "Ecole Centrale des Arts et Manufactures de la Ville de Paris" in 1966, and received the degree of "ingenieur des Arts et Manufactures" in July, 1969. From September, 1969, to the present time, he has pursued his studies at the University of Florida. Working as a graduate assistant in the Department of Metallurgical and Materials Engineering, he received the degree of Master of Science in Engineering in December, 1970. Since then, he worked as a graduate assistant in the Department of Engineering Science and Mechanics toward the degree of Doctor of Philosophy. Jean-Pierre Georges is a member of the "Association des Anciens El eves de 1' Ecole Centrale de Paris." 128

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Craigys. Hartley, Chairman Associate Professor of Engineering Science, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. > Lawrence E. Malvern, Professor of Engineering Science, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. lartin A. Eisenberg, Associate Professor of Engineering Science, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. CLoa^, John J. Hren, Professor of Materials Science and Engineering

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^5 Of). Vrvx^ ^(.W^-AW4X James B. Conklin, Jr. Professor of Physics Associate This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1972 Dean, ' College of Engineering Dean, Graduate School

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