
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00098684/00001
Material Information
 Title:
 Simulation of defects in crystals by point force arrays
 Creator:
 Georges, JeanPierre Jacques, 1946
 Publication Date:
 1972
 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 )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis  University of Florida.
 Bibliography:
 Bibliography: leaves 126127.
 General Note:
 Manuscript copy.
 General Note:
 Vita.
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 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 022656599 ( AlephBibNum )
13923616 ( OCLC ) ADA4861 ( NOTIS )

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Full Text 
SIMULATION OF DEFECTS IN CRYSTALS
BY POINT FORCE ARRAYS
By
JEANPIERRE 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 . . . . . . . . .
SelfEnergy 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 . . . . . . . . .
SelfEnergy 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 . . . . . . . . .
SelfEnergy 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 Selfenergy 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 (rr')
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 Halfwidth 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 halfwidth 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
JeanPierre 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 selfenergy 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 [13J.
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 [413]). 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 selfenergy 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 34 ij 1 X ix(3
G.. (R) + (3)
Gij) L 1 61J (31
16ng 1v R 1v 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.
6I 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 56V (
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 selfenergy 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 (rr'),
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 selfenergies 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(rr'). (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 selfenergy
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 selfenergy 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. (rr'), (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/21
u (r) = 2 R E 2 + l G (x pa,x qa,
p=R/2 q=L/2 q=(L/21)
a a
x3) G2m (xpa,x2qa,x3
14
15
10
4o
''I
\' Y \ \^.
^ ^ i
V 'H
La a La a
G3m (x1a, x22 3 2) + G3m(lpa, x2+ x+
La a La a(21)
+ G3m(x1pa, x2 x3 ) + G3m(xlpa, 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 selfenergy 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+lp)W (22)
Row pInt
p=l
W and W have the following expressions, respectively,
Row Int
Row 5 L1 + + 24 C33(O,0,a) (21) 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 (Lq) 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(2L1) 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 (Lq) 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 N1 N1 Nl
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
Nl Nl L
+ N Cf(N,q) + (R,q) + (R + I f(p,y) dy
q=l p=l N
NI 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(lV) 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(2v) k L2 r 1 I_ 1) 3.3903 .6872v
+ 1 vJ 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/21 +"
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)
p0 (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(x1pa) dp (32)
1 a
u2(x1,x2,x3) = g(xl) + g(xlRa) (33)
u (x1,x2,x3) g'(x1Ra) g'(x1)] (34)
3 a 3I
u2(x1'x2x3) 2880 L (3)(xRa) g(3)(Xl,3 (35)
etc., where
2 a 2
(x pa) + (x3 +
g(x pa) = n (36)
(x1pa) + (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 nae Ra
Sg(napa) dp = lim J g(napa) dp + J g(napa) dp (37)
0 e0 0 na+
For this specific case
2 2
g(napa)= n (napa) + a (38)
(na pa)
Since
Sn (napa)2 dp = 2 [(np) &n (napa) (np) (39)
the integral is equal to zero for p=n, and so
Ra
Sg(napa) dp = G(naRa) 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+ x3 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 wellknown 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 ta2l1 + (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 22~ 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
14
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(napa,qa,0) G22(napa,aa) (54)
p=0 q=
changing variables by setting u = np 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 halfwidth 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+iTT 2 (62
1 x
1
By comparison of Equations (61) and (62), the halfwidth 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 xl6ax +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 '
SelfEnergy 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 (xpa) + (x ) (xpa) + (x3 
(84)
R x3 + x3 
2 1 2
2. 2 2 (85)
p=0 (xpa) + (x3+ ) (x1_pa)2 + (x32
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 tan1 (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
FT pa1 2 47 +A2 + 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 selfenergies 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(1V)
For v = 1/3
E 2
On (95)
La 2n .2631a
if
C2 = 3.945a .
From Equation (91), the selfenergy 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 cutoff 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 cutoff 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 cutoff 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 semiinfinite 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',x2qa,x3
q=0
2
G2i (x+a,x2qa,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,x2qa,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,x2Na,x3 
2
G2i(x1+a,x2Na,x3+) + [3i(x1+a,x2Na,x3 
+0 1a 3 2)]x2+Na1x2 a a
+ G3i(x+a,x2Nax3 ) 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(x2qa) f(x2
^(SK) ('1 s2' F qa)
=0
2
a 1 a x2
ja 1 3'2___
32n 1V L 2 2 + 3/2
Lx +a) + x2 + a
vL (xl+a) + (x2qa)2
34 1
1 (x2qa)2a)
+2(1v) ir 22 ,23/2
(x+a) + (x 2qa) J
1
) 2 2 +2
(x1+a) + (x qa) + a
(x2qa)2
x+a)2 + (x2qa)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(lvI) 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(lv) 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 34v
2 n wn 2(1V) 2 3/ C2 16 1V
(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
(xqa) 1 1 f(x2Na
f(x2qa) yf(x2Na) 2 f(x2+Na)
3 x2Na x +Na
S32n(lv) 3/2 13 2
+ 327 ) x1+a) +(x2Na) 2+a (+a) +(x +Na) +aJ3/
(106)
with
34\. 1 1
f(x qa) = 2i
S(x +a)2+(x2qa)2 /(x +a) +(x2qa) +a
1 (x2qa) 2 (x2qa)2
2(1) p3/2 2 23/2
2( ) x1+a) +(x2qa)2 (x+a) 2+ (x2qa) 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(Nn
+ 3 a Nn N+n + a2 a 34\
+ 32TT(1v) + C 2 (
LL(Nn)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 32n 1M "
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 selfenergy 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 (4N1) 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 2N1
a G23(0,2Na,a)+ 2 a (2Nq) [G22(0,qa,0) G22(0,qa,a)
a G23(q G22(0,qa~a)
q=l
2 4 2N1 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+N1
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/2N1 L/2+N1 G
8N Z +2 (L+2N2q 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.
341) i 2N1
1 =N a 1 34v 1 2 1 1 1
a 8 2(l~~ 4 q[2 2 )+ 2(1v)) (2+
Pa + q=1 2q +1
3 2
1 34 +1 1 2 1 1 12
+ E & + + 7
S2 1v I2 +1 q 72 ;3/2
q=l q 2+1 (q
Pa +a 1 1 1 78V 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(19) (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 34v 1 34V 1 1 1 34 q +1
A = 1 \ +
16 lV 8r lv 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 78v 1 1 4N
2C 2C 32n 1V 32 1  lv 3/2
2 3 4N+1 (4N+1)
2N1
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 1v 2n q1
1 q 1 (116)
2(12) 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 onehalf 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/ ai    
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/2a/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(U2x3) + 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 Ll
summation of q will be S and S So, u (r) becomes:
m
q=O q=l
2 R/2 L Ll
u (r) = Z + G m(x pa,x2qa,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
Gmlpa,x2qax3+ G3m(x1pa,x2x3)
+ G3m (xlpa,x2,x3+ + G3m(lPa,x'2La,x3)
+ G3m(x1pa,x2La,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 (x2qa) + a2
u2(x 1x2,x3) 1a6 ( + 2(1) 2 a 2
q=0 q=l (x2qa) + (x3
1 (x3 a, (x 2
+x 3 + a 2 ( x a
+v 2 a2 (x3 2 a 2
(x2qa) + (x 3 ) (x2qa) (x3 2
a x2(x3 ) x2(x3+
16T(1v) 2 x a 2 2 2 a 2 (131)
a
x2 + (x ) x2 + (x3+2)
a L L1 (x2qa) (x )
U3(X1,x2,x3) = 16i(1V) (2' a,2
\= 2
(x2qa) (x3 + 2)a R L2
(x2qa)(x3 a 12V R/2 + +L2
2 ( a 2 1 4+ 1v R
(x2qa) + (x3 )
a 34v L + a 34V x 2 +( a,2 +
Sr 1 L 32n 1v 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 tan1 
xa x3 2 3
(xax )
(x32 tan  tan
x3 3 2_ J
2 a 2 2 a
1 34v x2 + (x3 )2 + a (X3 2
32 TT 2 a2 a + (x2a) n 2  2
x2+ (x3 (x2a) + (x3 )
2 3 2j
a 1 x2(x3 ) x2(x3+ 2)
16T 1v a a 2 + 2 2 (133)
2 + (x3 2+ (x3
r 2 + a 2 2 + a 21
1 a 34v x2 + (3 (x2) + (x3
2(1,x2'x3 = n 1v 2 2 a 2
+ (x3 ) (x2a)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 ) (x2a) + (x3 ) x2+(x3
(x3 + 2
2 aa2 (134)
(x2a) + (X +
2 a2 34v x2 x2a
u2(x12x2'x3 192 19 2 a 2 a 2
[2 +(x3) (x2a) + (x3
a 2
x2 x2a a2 1 2(3
2 a 2 2 a 2 96 1 3 2
2+ (x3 a (x2a) + (x3 +(x3
a2 2 2 2 (
(x2a)(x3)2 x2(x3+)2 (x a)(x3+ )2
x2a)2+ (x3) x (x3 )2 x2a)2+(x3+)
(135)
3 a5 34 2 x3(x3 2
u(x xx) 34v) x2 +
u2(x1'x2'x3 5760n 1 3r2 a23 +
Lx2+(x3
(x2a) x2a)23(x3 )2] x2 33(x3 )2
S3 )233
x2a)2+(x3 2 x2+(x3 2
(x2a) (x2a)23(x3+) 2 a4
2 23 1 480TT 1
(x2a)2+(x3 2 J
Sr2 2 2 a) 2
a 2 2 x(x3)2 a 2 x2 LX2 (x3+
(x2) 2 (x3
a 2 (x2a) (x2a)2 (x3 )2
+ (x3 4
(x2_a) 2 a2 2
(xa)2 + (x3 a2
(x++2 22]
(x,a) [x2a)2 (x a2(1
( 23 2 (136)
x2a)2 + (x3 2
with
0 1 2 3
u2 = u2 + u2 + u2 + u2 + (137)
For u (x xx3) we find
3 1' 2 x3
/22
0 (12v)a R/2 + R /4+L2 1 a
u3 4(1v) RLa + 32n(1v) 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+ (xa) 2
a (x3 2 (x3 2
16n(1v) 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)
(x2a)(x3 ) (x2a)(x3+)
S(a)2 (x a)2 (xa) ( )
(x2a) + (x3 P (x2a) + (x3
2 2 a
u3(l'2'3)= 192T(1)) 3 2
(x2a) (x3 2 
+
(x2a)2 + (x3 2
2 a 2
(x2a) (x3+
x2a)2 + (x3
3 a4 a
U3(x1,x2,x3) 1920TT(1v) 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]
(x2a)4 6(x3)2(x2a)2 + (x
+ 22]4 2
x2a)2 + (x3 )2 4
4 a 2 2 a 4
a 2 6( + ) x2 + (x3 +
 (x3 4
4 _a 2 4
(x 2I4 6(x3+ (2x2a)2+ (X34 
(x2a)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
nonbending 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(1v) 2 2
n/2 3 x2 + x3
2
a 12v 2 2 1 x3
n (x + )
u3(x12x2,x3) 2 n (x23 2(lv) 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(1N)(x2+x3)
i_ ^2 3
2 2
u3(1 ,x2,x3 4( x2+x3 2 22 (148)
) (x2+x3) + 4(1v)(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(1v)
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 34 (x 2qa)2 + a2
Au (x) 7n a) + 2
2q=0 q=l ( (x2qa)
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 34v x2+a x2a) +a
+16 1v 2 n 2 + (x2a) 2 (152)
a2 (x2a)
r 22 2 2
Sa 34 v 2 2
u2(x2) = 32n 1 + 2
2 (a2 a)
3
a 1 1 1
167n 1v 2 2 2 (153)
2 a) + a
2 a2 34v 1 1 X2 x2a
[x2 x2+a (x a) +a
a 4 1 x2 x2a (1 5)
48TT 1 22 2 2 22 '
x2+a (x a +a
3 a4 34v 1 1 x2(3a2
Au2(X2 2880 1v 3 3 2 2 3
2 x2 (x2a) (2 + a )
(X a _a) 2 2 2 2
(x2a) [(x2 a a6 2a
2 2 3 240T 1v 2 2 2 4
(x2 a) +a ]J 2 )
(x 2a) (x 2a)2a2]
+ 4a (155)
(x2a)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,naqa,O)
p= \q=O q=1
G22(pa,naqa,a)] G23(pa,na,0) + G23(pa,naa)) (156)
Replacing the variable (nq) 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 n1 
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 1v 2 1v 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 34V p +1 1 2
Au (na) = a + p n + [ 2
2 16 p=n p=n+1 1 2 1v 2
L p p +1
a 1 n (163)
STT 1v 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 Fl4_ p l + 1 2
Au2(na) = 16 i
p=n p=n+ p p +U
a 1 n
8r1 Tv 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(lv) 2 212 1T 11 2
+x2 a +(2a) x2
2 2 2 2
(x2a) +a 2 a x2
+ 2 S 1 v 2 (166)
(x 2 871a)2% 2 2 2
(x2a) (x2+a2)
1 a 34v x2 + x2a 1 1
a21(x2) 16T 1 2 2 2 2 x
X2+a (x2a) +a 2 2
a3 1 x2 x2a
a+ 8l 2 2 2 a 1)7
(x2+a2 (x2a) +a
2 a2 349 1 a 12 2
'21(x2) 96n 1 2 2 + 22 + .2
2 x(a 2) a2+(x2a)2
S2 2 2
4 a x2 a (x2a)
a 1. 2 2 3 + (168)
(x2+a) (x2a)2+a2]j
r 2 2 4 4
S 4 3a xxa
3 a 34v 1 1 22
21 960 1v 4+ 4+ 2 2 4
2 2 (x2+a )
2 4 44 2 r 2 4 4
3a (x a) (x2a) a a6 1 10x2a 5x2a
x _) 24 + 240 1v 2 2,3
2 22 (x2+ a )
x2a) +a L 2
2 2 4 4
10a (x 2a) 5(x a) a1
+. (169)
C(x2a) 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.
SelfEnergy 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 Laa/2
x3 > 0, f(y2) dy2 = f(x2) dx2, (184)
a a/2
Laa Laa/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
Laa/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 14v x2qa
x3 x (x2,x Y16, 1_ 2 a 2
2 q=0 q=l (x3qa) +(
x2 qa 2 (x a)
(x2q a)2+(x3 2)2 + 1v I2 (x )2
x2 +3 (x2qa) +(x3 )2
~\ a 2 2
(x2qa) a3x ( _)2 2
2ax3 a (x3 ) x2
(x2qa) 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
(x2qa)2+ (3 (x2qa)2 + (x3 )
LxCcx3 2] )
3 f a
)2 a2 2+_ 2 16T(1,)
C(x2qa)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= 12 1 )2 +1 1
R a 4 1 2 + 4 tan (q
(q=0 q=1 (q 2)
~a 12V a as 1 5 7
2+ 1 2 L L + 2Vn 12v 4
1 1 16V2 18 + 5 M
+ TT tan 2 + (1 )(12V) + 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 
PAGE 1
SIMULATION OF DEFECTS IN CRYSTALS BY POINT FORCE ARRAYS By JEANPIERRE 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
PAGE 2
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.
PAGE 3
TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi KEY TO SYMBOLS viii ABSTRACT Â• Â• 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 SelfEnergy 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 SelfEnergy 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 SelfEnergy of the Edge Dislocation 120
PAGE 4
TABLE OF CONTENTS (CONTINUED) CHAPTER Page 7 CONCLUSIONS 124 BIBLIOGRAPHY 126 BIOGRAPHICAL SKETCH 128 iv
PAGE 5
LIST OF TABLES Table Page 1 Relative Displacement of Atoms Across the Slip Plane for a Screw Dislocation 26 2 Variation of Force Constant CÂ„ 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
PAGE 6
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 = Â± Â— 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 Selfenergy 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 * Â— Â• R Vector defined by (rr') 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 Halfwidth for a screw or an edge dislocation X. Component of the vector R x. Component of the vector r Q'Â„ 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 Â§ Peierls' symbol for the halfwidth 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 JeanPierre 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 [13J. 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 [413]). 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
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will permit us to analyze the displacement field and selfenergy 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
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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. 2Â„l u(?') = 16TTH 34V 1 1 cos Â— + (5) lv 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
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Â° 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 = cu(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 selfenergy 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 = fu(?') =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
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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 selfenergies 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 Â± 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) ho f Â— 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 Â£k (?<) G k .y ??Â•). (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) Â« b.n. c... . C. . dS (12) m 1 j ijkÂ£ 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
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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 Â• 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, Â± 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
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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
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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 Â• 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 selfenergy 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)
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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 Â• (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.
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13 ,*frÂ« s%~\~1f' X, Figure 3. Shear Loop in Simple Cubic Crystal /. \ y * p / / & / / p\ / \ Figure 4. Shear Loop with Principal Axes
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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. (rr* ), (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/21 E / [ S + E p=R/2 lq=L/2 q=(L/2l) u (r) = u. gZ (I s + z G 2m (x r pa ' X 2qa ' 14
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15
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16 G 3m (X r pa ' Vâ€¢' X 3 } + G 3m ( V pa ' X 2 + â€¢' X 3 + f > ' l G 3m (X r pa > X 2"â€¢' X 3f } + G 3 m ( V Pa ' X 2"â€¢' 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 ~ Â€ 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 selfenergy 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
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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+1P)W T Row Â„ Int P=l (22) W and W have the following expressions, respectively, 3 5 (J, a Row 2L1 L C 2 C 3J 2 4 + 1 SrC G 33 (0 ' ' a) " (2L 1} G 22 ( Â°'Â°' 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 (Lq) [_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 (2Ll) (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 Â„(pa,La,a) 4p, a E G (pa,qa,a) 23 L Q=l 23 2 4 r~ ~\ + 4j, a E (Lq) [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 'Â°' a) +G 33 ( Â°'Â°' a >] + Â§ (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
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18 R Â£ (G 22 (pa,0,0) G 22 (pa,0,a) + G (pa.0,0) +G 3 (pa,0 f a)) P=l R + Â£ 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 Â£ G 23 (pa ,La, a) 4 Â£ 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) Â« f(q) + f (N) + f(L) +J f (x) dx, (26) q=l " "" N and for a double sum: R L Nl Nl Nl 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 Nl _ Nl L +  E ff(N,q) + e(R,qf + E J f (p,y) dy q=l ^~ p=l N
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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 .. + Â± 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 Â„ ua 2. 8545 2.3676V 4tt J Â— + C 2 1 V 2tt(1v) 2n 2RL 2n C~ + Â«/R +l R+l 4tt R + ^ 2 2 +L . 1 1 \ 1.2979 .6458V 1.6420 + .3580V 1 V 2(2y) Jr +L 1 V 4tt /X 2 ' fl 1 A 3.3903 .6872V 2ttu^) 4tt(1v) (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.
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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
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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] + Â— (V (b) f ' (a)J a a .^[ f ... (b) . f ... (a)]+s __Q f Â«) (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/21 + Â°Â° 2 f(q) = S f (q) = J f (x) dx (30) q=L/2 q=(L/2+l) Â« where f represents the whole expression to be summed. The following components of the displacement field have been found: r WW = Â° < WW = 4W \^ oW w = Â° (x pa) + (x 3 +2) 2 a~~2 p0 (x i pa) + (x 3 ) R a Â„ (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 uÂ„ 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 = Â£? [ 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 (napa) dp = 2 f(np) Ski (napa) (np)~J , (39) the integral is equal to zero for p = n, and so Ra J g(napa) dp = G(naRa) G(na). (40) The integration makes all the singular points vanish, except for those at x =0 and x = Â± 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 Â— ; Â— T2 >. X, + (XÂ„ 75) J 3 2' 2 , a,2 1 , , * X l + U 3 + 2 } u 2 (x i'V = i^ ^Â§ Â— ; Â— rs X l + (X 32 } (42) U 2 (X 1' X 3 } = " 24^ r 2 a s 2 2 a x 2 X l + (X 32 } X l + (X 3 + 2 ) J (43) and VW = " 730TT 4 KCÂ«*b5Â» 4D *i[ 3 "vf> O (44) Then, since u (x x ) is the sum of u (x ,x ) for every i, Â£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 Â± Â— + tan Â— 2 X 3J (46) This is the wellknown 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 ) = Â±_ 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 + Â— m = a a 2 x 1 J (50) 2 2 1 a X l + 3 AU 2 (X 1 } = 4^ 2Â— (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 AuÂ°/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
PAGE 40
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)=2ia I Â£ f~G (napa , qa ,0) G (napa, qa, a)~j (54) p=0 q=Â°Â° changing variables by setting u = np changes Equation (54) into 2 / n Â°Â°\ + " 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==Â° + 4^ 2 Z QG 22 (0,qa,0) G 22 (0,qa,a)] q=l + cTnT' 2 ^ 2 G 22 ( Â°'Â°' 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 AuÂ„(na) = 2 V CÂ„(n) 2rr 2 2.866 2.368V _ . u + 1 L m 7i Â— 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 CÂ„(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 = Â°Â°. The values of C (n) are very nearly constant, and it seems reasonable to expect a value of C (0) very close to C (Â°Â°) . 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 CÂ„ 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 Â— , (60) 2 1 2 TT w ' where w is the halfwidth 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 Â© O *Â© cÂ© e !Â« ce> Â© cÂ© iÂ° (Â© x eaeaf3BÂ— oo ote co < < o :: Â©; e ($> ' J o CBlCSJ II c c CO Si <
PAGE 44
33 a w a wd , _ .. . AU 2 (X 1 } = a ^3T + n + T Â•Â•Â• Â• (61) 1 X l The same series expansion holds for Equations (49) to (53) and leads to 2 A W = a "2^x+ 4i^ + Â•Â•Â• Â• (62) 1 X l By comparison of Equations (61) and (62) , the halfwidth 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~ Â• (64) Differentiating Equations (49) to (53) with respect to x leads to
PAGE 45
34 2 2 *22 = 2^ 2Â— (65) 1 , n 1 22 v V ' 2TT ,22' x (x +a ) (66) and 2 i ^ a Â» 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 lÂ» (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 SelfEnergy 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 &nÂ— = 1.179 r o (72) Taking the special case of V = 1/3 leads to _W _ R+l 2 La 4tt ^ 4tt Â£Â£ + 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) Â•.35a < x < Ra + .35a .
PAGE 48
37 a < a c Â•ii he Â«
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 = Â»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 Â° 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 = Â± , 35a < x < Ra + .35a; S 2 : x x = .35a, S xÂ„ < : 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 = Â»I Â„ 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:
PAGE 51
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 oxÂ„ 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* (Â•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 \Â» 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 ^+ A2tt 2 2 Rrl 2 (ja Â„_ [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 \Â» 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 Â— = 1.3367, r o ia _ 1 2.8545 2.3676V 4tt(1V) 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 selfenergy of a pure single screw dislocation can be written EÂ„ 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 cutoff 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 cutoff 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 cutoff 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 semiinfinite 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 3f } 2 G 2i ( v a 'V x 3 + ti] ^Qwv^v^f* + G 3i ( V a ' X 2' X 3 + l } 3 Â• (98)
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44 Â©Â— Â©Â— o iya A 2. ^>Â— 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 fC 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 Â• (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 = Â± 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 xÂ„ , 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 xÂ„ _a 1_ 2 32tt 1v 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(lv) (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 ) = ^ Â» n E + E f (qa) Â» f (na) q=l q=0 v ' 32rr(lv) C2 2 2 2 I xÂ„ + a) + n a + a J 3/2 (102) (b) x = na for any x Â« 1 ^(SK)^!'"" 3 '^ = ^ E E f(qa) + Â£ f(na) L\q=l q=' 32tt(1v) f, ,2 2 2 2~] MX +a) + n a + a J 3/2 (103)
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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 Â§ 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(1V) (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 = Â± na for n < N 1 & ^(DK)**!* 18 '^ = 8t E + E ) f(q) ~ f(N+n) 32tt(1V) \'+n Nn E + q=l q Nn N+n JjNn) 2 +lJ QN+n) 2 +lJ 2 pa a C 2 16  f (Nn)J (108) (b) x = a and x = Â± 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 Â© a a Q a Â§ a a a ,9 o o ' "\ ' / o ' Â« J 4 o a 9, O, X a Â© C 9 a N (3 7R a a JI 8 a Â© Â© a J Â•0 a 3 o a Â•p P KS C O Q) O S Â»H as CO U S a s o O CO Â•H S as o ii c
PAGE 65
54 13 Â© Â§ \ Â§ a 8 Q R o (D (3 fcJ a Q Q o J" b f^Â» b 3 O Q cS X! c 3 c h o < H +> 4> at c o
PAGE 66
J 55 a 9 G Q OrÂ© Q Q P O Q e r * 5 8 $ 8 8 Â® 6 i ei o 3
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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. (4N1> AG 2a ( 0> f a) 2 4 \x a + G 33 (0,0^ 2 4 + Â—~(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 (2Nq) ^G 2 (0,qa,0) G 2 (0 f qa,a)J q=i 2 4 2N_1 2 4 R+1 f La 2iB 2 G 23 (0,qa,a) + (i a 2 < G 22 (pa,Â— Na,0) q=l " p=l L G 22 (pa,^Na,a) + G 22 (pa ,Â— + 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+N1 G 33 (pa,Â— + 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/2N1 L/2+N1 8N 2 +22 (L+2Nl2q) 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 (2N1 V* 1 3 4v , 1 v el 8tt 2(1v + 4tt L 2 q=l , ,l._i Â«/q +1 2(1 " V) (q 2 + l) 3/2 J _1_ 8" q=l 34V . q +1 IIS T^7^~2+ 1^1 "2 q +i l Â— + q Vq 2 +1 , 2 3/2 (q +D j pa i pa 1 1 1 2C + 2C^ + 32rr 1v 32rr 78v 4N 1v s/4N +1 1v 2 3/2 (4N +1) 2N1 2^ S q=l 1 / 2 Vq +1 2(1V) , 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 34V _1_ 34V /l 16rr 1V + 8tt 1v , q q=i l 1 N
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58 \ia ia 1 1 1 2C^ + 2C^ + 327f 1V 327f 78v 1 1V Jsi 1 1V 4N 3/2 4N +1 (4N +1) 2N1 2rr q=l ^ . 2(1V) 2 3/2
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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 onehalf is bounded by the planes x = Â± 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^iW swv + ^wv (117) V. U 3 ( VW Â• 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 ,xÂ„) = u' (x ,xÂ„,xÂ„) 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 vÂ„(x ,yÂ„,xÂ„) = u' (x ,xÂ„,xÂ„) 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^ + oâ€¢^ + 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 ,yÂ„,xÂ„) = a Â„(x ,xÂ„,xÂ„) 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 ) " Â» (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/2a/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 Â» (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* " Â•* I + I ff 23 (x l'Vf } J dX l Â• La/ .s X v^Q'' (126) In these integrals, C is the curve defined by Ws* + "J^i'Va? " 4 Â• (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 _ Â— 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 Ll summation of q will be Z and E . So, u (r) becomes: q=0 q=l 2 R/2 j /l Ll\ V ? > = "IT ^ R/9 K + \) C G 2m ( V Pa ' V^' X 3 f> " p= r/2 n,q=Â° 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 35 ) + G 3m (X r Pa ' X 2' X 3 + IO + (?3a (x 1 " pa 'V La Â» 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 Â° \q=0 q=lj 1 (x 2 qa) + (Xgg) 1 1V . a. 2 (x 3 + i) (X 3 " 2 } (x 2 qa) + (x 3+ ) (x 2 qa) + (Xg) (130) 16rr(lv) 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(1V) L Ll 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 12V . R S/r. /2 + /R /4+L 2 ^ 2 4 " 1V (x 2 qa) + (x 3 + ) a 34v 8rr 1v &r. L + a 34v 32tt 1V L X 2 + (X 32 } 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 32> 1 34v 32tt 1v 1 tan X 2^ ^3 + (X 3 + I ) Ta
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68 _a 1_ 16tt 1v V^S"^ X 2 (X 3 + 2 ) 2 a.2 T 2 a, 2 L^*^"^ X 2 +(X 3 + 2 ) . (133) 1, , a 34V u (x , X , X ) = 2 1' 2' 3' 16rr 1v 2 a. 2 . .2 , a.2" Â„ X 2 +(X 3 + 2 ) Â„ (X 2" a) +(X 3 + 2 ) _a 1__ 32tt 1V t a ^ 2 2 a 2 X 2 +(X 32 } i ^ 2 (X 32 } ^ , a ^ 2 (x 2 a) + (x 3 ) (X 3 + 2 } 2 a.2 2 , a 2 2 , a 2 X 2 +(X 32 } ( V a) +(X 32 } X 2 +(X 3 + 2 } r 2 +< x 3 f> 2 ] Cv^ 2 ] c^> a+ ^ + f>^ X 2 [ X 2^ X 3f> 2 ] (135) 3 a 34V 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" 2Â° S 2 CV ( VI )2 3 # B , 2 X 2 C X 2(X 3 + I )2 3 r 2 . a, 2 V Â° f 2 . a, 2 "1 LV^S"^ J L X 2 +(X 3 + 2 ) J ( xÂ£ 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 UÂ„ = U_. + U + UÂ„ + U + 2 2 2 2 2 (137) For u (x x , x ) we find /2 2 (l2v)a Â„ R/2 + VR /4+L 3 4rr(lv) fe RLa ( x o o) 32tt(1V) P 3 2 Â„ p, a N 2 2n ' n a,2 , x 2~) ^Q^^ +X 2 J + ^ L (x 3"2 ) +( V a) J Â„ (. a x 2 21 Â„ 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(lv) , ^ 2 (X 32 } / ^ 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(lv) Vs"* X 2 (X 3 + 2 ) 2 a. 2 2 a,2 X 2 +(X 32 } 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) uÂ„(x ,x ,x )= (x. ~) 3 1' 2' 3 192tt(1V) \ 3 2 2 , a,2 C (X 2 +(X 3 _ I )2 3 < ^ 2 / S 2 (x a) (xÂ„ Â— ) 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 + lÂ» 2 ] [(x 2 a) + (x 3 + ) J J (140) u_(x, ,x_,x_) (x ^) 3 1' 2' 3 y 1920tt(1V) \ v 3 2 r ^ V
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71 The constant in the expression of u means only that the point of nonbending 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 Â£ x = a with x = Â± a/2) and u (x ,x ,x ) has two singular points Z, o o X t* Â«j (x = with x = Â± 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 *Â«Â«4.4 (144) UÂ„(X X o' X o) = 3 1' 2' 3 2it U(lV) 12V . , 2 2. ton ( x r,+ x ^) 2 3 2(lv) 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 = Â°(146) WW = 2TT tan Â— + 2 2(lv)(x^+x 3 ) (147)
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2 2 , 12V .,22, X 2 " X 3 3 V 1' 2' 3' 2tt 4(1v) 2 3'',, w 2 2, i 4(lv)(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(lV) 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 = Â± 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 O0 o o o b o o o o o o o o bo w
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74 Au 2 (x 2 ) = AuÂ°(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 AuÂ„(x ) = o~ S 17 + tan Â— + tan ) + s~ TT, ~^ Â— o 2 2 2rr I a a ( 8tt 1V 2 2 2 2 ,2 2 1 34V J . V 3 , . . ( V a) +S 1ft? 1=7 X 2 *Â» Â—2+ (X 2" a) * "ITx 2 (x 2 a) (152) , 1, . a 34V AuÂ„(xÂ„) 2 2 32tt 1V 2 2 , x 2 2' X 2 +a ^ X 2~ 3 + 3 ^7! Â— ^Â— + Bn g Â— X 2 (X 2 " 3) a 3 1 16rr 1v 2 2 ' ,2 2 Ux + a (x a) + a J (153) . 2. a 34V 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 1V l(Â£.Â») s ^V] J 3 a 34V 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 xÂ„ = 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 Â« .4 1 Â« .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 Â£
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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 + Â„ j q=0 q= Au (na) = pa 2 I + E f~G (pa,naqa,0) p_ cc V \q=0 q=l/ G 22 (pa,naqa,a)J (_G 23 (pa, na , 0) + G 23 (pa,na, a)J I . (156) Replacing the variable (nq) 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 n1 Â°Â° \ Au 2 (na) = ia E E+E+E QG 22 (pa,ua,0) p= Â°Â° Vu=l u=l u=l/ + co "1 2 G 22 (pa,ua,a)J pa E G 23 (pa,na,a) pÂ— _ 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)] Â— Â— CO P= ro 2 + 2pa E [c 22 (pa,0,0) G 22 (pa,0 f a)] + JÂ£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) = Â— 8tt lv 8TT 1V 2 , n +1 16rr E + E n p=n+l/ lv ' 2 lv 2 , P +1 (160) , Â„ 2ua a 6.4060 5.6512v Au 2 (a) = ft + st r^ (161) . ,Â„. ua a 11.4392 9.4720V Au 2 (0) =Â— + Â— Â— (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 Â„( co ) for every n, the relative displacement becomes Au (na) = a ^Â— I E + E 2 16tt 1 p=n p=n+l. 34v Â„ p +1 1 m Â— =Â— + lv 1V 2 ^ p +1 J a 1 8rr 1V 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^ooSg. 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, CÂ„(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 , Â„ 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(1v) 2 2 2, 2 La tx a +(x a) _ l 34V 16tt lV Qm 2 2 x 2+ a + im 2 2. (x a) +a 2~ (x 2 a) o 2 2 2 a xÂ„ i l 2 8TT lV 2 2 (x 2 +a ) (166) 1 , . a 34v Â°'21 (X 2 ) = 18n T37 2 + 2 2, ,2 2 u x +a (x a) +a X 2~tj a 1_ 8rr lV 2 2 2 , 2 2^ (x a) +a _(x 2 +a ) 2 (167) 2 a 34V *21 ( V _ 96tt T=v" 2 2 a xÂ„ 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 1v 2 2 3 (x a)' (xÂ„+a ) r ,2 2"] 2 L (x 2 _a) +a J (168) a 34v 21 960tt 1v 2 (X 2" a) Â„ 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 2401v , 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. SelfEnergy 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(lv) Â£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 vÂ„(x ,yÂ„,xÂ„) = uÂ„(x ,xÂ„.xÂ„) , 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 Â£,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^) ^Â± + 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 Â«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 Laa/2 x 3 >0, / f(y 2 ) dy 2 = J f(x 2 ) dx 2 , a a/2 (184) Laa Laa/2 X 3 < Â°' / 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 Laa/2 + I CI CT 23 (X 2'f ) + ^2 (X 2'I ) " *T3 dx 2> a/2 (187) E a/2 Â£ = 2 f Ra o L ou (X+2u) x (Â£,x.) + (^i\) uÂ„ (,xÂ„) 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 dxÂ„ + ^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 1V /Â• 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(1V) [(x 2 qa) 2 + (x 3 ) 2 3 r 2 3x2. r 2 / a. 2 j Â•2 L X 2(x 35 ) 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 (qi) 2 +l n + J^ (q "2 ) ^O Q=0 q=l/ (qo> + 4 tan" (q )} ua 12V 2tt 1V Sm L pÂ« Ua 2 L + 2n 1 Z 1 Â« 5 7 1^ 2 ^ 4 + 40 , 1 _ 1 16v 18V +5 V Â„ Â„ + tt tan 2 + Â— Â— zrr. Â„. , + #72 2 4 ( 1 v) ( 1 2 v) 12V 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 Â— Ra rr lv~"~ ' 2rr L (lv)(l2v) 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 1v &n L 2rr 2 r 4tt pa i JL 2 3 4.3273 6.7297V + 2.5 406V (1V)(12V) 21 (194) For E /Ra to represent the energy of the system, C has to be T Â« chosen such that the term linear in L vanishes. Under this condition, C takes the same value as for the screw dislocation. Unfortunately, CÂ„ 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(lV) r (195)
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91 and the selfenergy for a pure edge dislocation is simply !g = ^ r (k Â— . (196) Ra 4tt(1v) r Q The fact that the essential features of the selfenergy 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) = Â±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) Â— Â£yÂ£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^ + ) + GÂ„Â„(x pa,xÂ„ ,xÂ„ Â— ) GÂ„Â„(x pa,xÂ„,xÂ„+Â— ) 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 34V 1v /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 1V (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 lv (u 2 +l) 3/2 I 3 " 4v I 1 2 2CÂ„ ' (201) and for a double kink of length 2Na, (a) All x except x = and Na < x < Na a a 2 g I 34v U 2(DK) CX 1' X 2'2 ; 32rr Â„S (lv) 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 (lv) (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 2i 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 Â« a N a 2 J 3 " 4v ! U 2(nc) (na 'Â°Â»2 ) a 55S \ T \~~T= p=N i 1 lv "1 V(np) +1 N+n Nn\ Q(np) 2 +l] 3/2 r 2 2i 3/2 [_(np) +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 = Â± 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+2p) + 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 CÂ© cÂ© cooicro CO GDOpOO O i Â« cÂ© Â•;Â© o (Q) CÂ© Q (Â© cÂ© coo o co <Â© obo o 00; Â© OO O a w CS c g 0) to c U U <
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99 'ID Â•'O CO o CO o co a ;Â© co coo o Â•:o ooe :?hco cee PC: O bo o bo o oo o 3 Â—' ^CFnG ^= Â• Â— Â• v s ; V_ A., a V Â— y CO CO o CO CO O CO CO c Â© CO CI o:; o bo O bo o bo o Do o u c o o +Â» +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 Â•'Â© co obo o co ceeteo o co co'cte o 0>C5 CNJ I CO a pec toe ;0 OGfOO o Â•:o cec CO 'CO o Oj o oo o u c o o H H Â•p p a c o o o 5 rH O 03 bX) H a q ^ 0) H 60 5 a W O Â•H fl 6 a o co (_o obo o Â© CO ;f)0 O t
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101 cÂ© .;Â© :: bo o CD CO o to o cÂ© ooe co c eo c oop o Ool'Qj O CO 00 o 1 OO rO; O CO ( CO COOrGD O ' fcH Â•'Â© CO C "Â©;; 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 Â„ , \ W = =r(2N+1) 16tt R/2 + 2 E X 2N+2 2 3/2 2 3/2 SP +2) (p +1) j 34V ua ua 3 2N+1 tMn fr^i 34V 1V 1 + 2p J VP +^ 1v 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 34v 1v Jv +<
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103 + 1V I , 2 3/2 2 3/2 (P +2) (p +1) Ita 3 2 g +1 34V 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 = Â— ^ (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 selfenergy 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 ooooÂ®oooo o o ao~ k &u@o *Â• 0Â© I 0O 00 t 00 00 r 0O 00 \ 00 000Â© teee^ 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 Â© 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 ,Â„ " G 3i (x i' x 2 qa x 32 ) 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 + Â— +Ra) j + Â— ^ Ki (X l' X 2 + i' X 32 + Ra) + G 2i (X l' X 2 + â€¢ +a ' X 3f +ra) " G 2i (X r X 2â€¢' X 3f +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(1v) (12V) X l (X 32 } 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 , 34V (x r a) + (x 3 + ra) f 16tt A l2v *" . ,2 a ,2 'I (x +a) +( x 3 _ 2 + > " a ^ 12V (x +a)' (x i a)' /Â• 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 ) = Â°Â« (215) uÂ„(x ,xÂ„,xÂ„) = 3 V 1' 2' 3 J 16n(lV) fo /~2 . 2 34V 12V 12V 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 32 } V (X 3 + 2 ) R+l f(x 3 + ra) (x +a) (x 3 + ra)(x r a) 8n(l2v) \ 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 34V , a, ( V a) +(X 3 + 2 } (XÂ„ +) S.T, 16n(l2v) 3 2' "" , x 2 , a x 2 (x r a) + (x 3 + ) ' Wl* X 1 (X 3 + I ) Jrr(lV) (12V) 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(34v) ( V a) +(X 3 + 2 } WW = 32tt(12v ^: ^Â— T2 (x r a) + (x 3 + ) 2 2 (x a) (x +a) 16tt(12v) ) 2 a 2 2 a 2[ ' ^(x r a) +(x 3 + ) (x 1+ a) +(x 3+ 2)"j 2 ( , a 2 (34.) J V a/2 V a/2 u, (x 1 ,x_,x_) = (219) l v 1' 2' 3' 96n(l2v) \ , ,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(12v) 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 (34v) , a f 3( V a)2 (X 3 + f )2 U (X ,X ,X ) = Â— ; Â— Â— (X + Â— ) 1 1' 2' 3 2880tt(12v) 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(12v 3 2' . LL
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Ill 1 x ,+ a 1, v a . a. ) 1 UÂ„(X X X ) = Â— Â— Â— (X +Â— ) 3 1' 2' 3' 16tt(12v) 3 2' \ , .2 . a 2 ( Y a) +(x 3 + 2 ) (x r a) + (Xg + ) (224) 2 WW 96tt(12v) 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 Â„, 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(12V) \ _ _ . 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 / Â• (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 iVV = 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 Â— > + ,* . 4~ , (228) /0 x ( 4rr(lvT 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 ) = Â° * (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 = Â± 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*i2v) "2~T + T&R E f(ra) ' x +a r=l (233) ax 0Â° V X ' X Â§> = 8tt(1v)(12v) T2 + ife Sf(raa), x+a r=l (234) with , ,2 2 2 *t , 34V 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(raa) = E f(ra) + f (0) , r=l r=l (236) the two identities follow: OO (K Â— ?>1 , a a, a _ _ x a 34v l 1 ' VVVS* + u l (x l ,x 2 , ) =_ Ef (ra)+Â— _ o>u , Cx, ,x 0> ) = a 34vÂ„ U r a) 072 1 1' 2' 2 1 1' 2'2 16rr 12 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(1V) (12V) 2 2 /A X 1 +a a/4 1 (34 V) a Â„ ( V a) + ^ ,, ^.., 071 32tt(12\Â» "" , ,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 , xÂ„ , ) + u (x ,xÂ„,Â— ) 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 34vÂ„ ( V a) 2 I6rr 12 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 + Â— ) 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(1V)(12V) \ 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(1v)(12v) 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)= Â— E ( G 31 (a,x 2 qa, naa) q= L/2 R+l L/2 + G (a,x qa,naA + ~ a E E (g (2a, x qa, naa+ra) 31 2 J 2 r=l L/2 V11 l G (0,x qa, naa+ra) ) 11 ' 2 ' J (247) After summing on q, Equation (247) becomes n+1 u^a.naf) = ^J^Â—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 (Â— 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^ (iv)(i2v) TT Â• (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 Â§ = 2T^r Â• (252) We shall expand Equations (249) and (250) in powers of 1,'x and shall select Â§ so that both expressions match at large values of x . So first order this gives a 32 v a A VV ^ Â— 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 Â§ = a/2(lv) and Â§= (a/4) (32V) / (1V)
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118 In 3 3 0) bo w Q. H Q >
PAGE 130
119 o o
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120 are plotted in Figure 34. For V = 1/3, the values of Â§ 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 halfplane, in contrast to the result for the shear loops. SelfEnergy of the Edge Dislocation The energy of the system of point forces is composed as usual of the sum of the selfenergies 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 (Llq) f~G (0,qa,0) G (2a,qa,0)l q=l 2 R + 2G (L+l) E (R+lq) Pg (0,0,pa) G (2a,0 ,pa)] + 4G E Â£ (R+lp) (L+lq) [~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 _ ,Â„ _ . Â„_2 L w = F C + 2F (L+l) G 33 (0,0,a) + 2F E (L+lq)[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=lP) 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_Â„(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+lq) G ir! (a,qa,Ra+a) + E (R+lp) 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+lq) G (a,qa,pa) Lp=l q=l 13 R L + E E (R+lp) 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) (1v) . 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 (12V) (L+1) R_Â±VRJÂ±_ + (R+1) ^LWR + L /2 2^ + VR + (R+l+L+l) 1 (1V) T8.0325 5.5644V 4rr (12V) r Â«.03Z5 5. 5644V i 2L 1V J 1 v 2 2TT (12V) 2
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123 o 2 ua (1V) (I^l)a 4^ (1 _ 2v) 2 ^Â« + (R+D 4rrua 3.3524 8.6586V ~C~ + 1V H,a 2 (1V) 8.0325 5.5644V 4tt (12V) lia V r 4rr(ia _ 7.4947 9. 1180v ~j ua_ V 2 , ia 1 TT 1V (2.2148) (260) In the special case of V = 1/3, we have 2 *Â»r Â£ c i 2 W (1*1) a Â„H.aÂ«)[^ + .6Â»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 wellknown form of the energy of such a system. This is due to the excess strain energy stored between the planes x = Â± a/2 after deformation. All the material contained between the planes x = Â± 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 selfenergy 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, SpringerVerlag, 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. TP292 , 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, McGrawHill, 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," AERETP490 , 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 , McGrawHill, New York (1968). ~ "~ 21. R. E. Peierls, Proc. Phys. Soc. , 52, 23 (1940).
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BIOGRAPHICAL SKETCH JeanPierre 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. JeanPierre 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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