Title: Simulation of defects in crystals by point force arrays
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 Material Information
Title: Simulation of defects in crystals by point force arrays
Physical Description: xi, 128 leaves. : illus. ; 28 cm.
Language: English
Creator: Georges, Jean-Pierre Jacques, 1946-
Publication Date: 1972
Copyright Date: 1972
 Subjects
Subject: Dislocations in crystals   ( lcsh )
Crystals -- Defects   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 126-127.
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00098684
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577167
oclc - 13923616
notis - ADA4861

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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.




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Last updated October 10, 2010 - - mvs