Materials Science on CDROM User Guide
Dislocations
Version 2.1
John Humphreys, UMIST/University of Manchester
Boban Tanovic, MATTER
Before starting this module, it is assumed that the student is familiar with the
crystalline nature of metals and alloys. He/she should also be familiar with the following
terminology: stress, strain, elastic, plastic.
Module Structure
The module comprises 4 sections:
This section starts by explaining lattice distortions associated with dislocations and
showing how these result in stress fields. On the following pages the user can study the
dislocation Burgers vector and see how new dislocations may be created in reactions.
This routine consists of four pages which demonstrate the effect on a crystal of the
passage of a dislocation. The first page introduces the concept of slip, which is the most
common manifestation of deformation in crystalline solids. Animated graphics on the
following three pages show dislocations propagating from one side of the crystal to the
other, resulting in the displacement of one half of the crystal with respect to the other
by the Burgers vector. These three pages show:
 An edge dislocation with the option to permit climb (and thus to allow movement out of
its slip plane).
 A screw dislocation where the user can choose whether crossslip will occur. If cross
slip is operating the dislocation changes its slip plane half way through the crystal.
 A mixed dislocation, which can only glide.
This section starts by explaining that the nature of the interaction depends on the
relative positions of the dislocations, their Burgers vectors; and their edge or screw
character. On the following pages the user can examine the elastic interaction between
straight, parallel edge or screw dislocations with parallel Burgers vectors. This section
permits pair of dislocations to be positioned relative to one another before the program
demonstrates their motion to a stable position.
On page 5 of this section the simple elastic interaction equations are given for two
parallel edge dislocations, and the glide force is plotted as a function of separation for
dislocations of the same sign and of the opposite sign. In the equations n is Poisson's ratio, x is the separation of the dislocation
in the slip plane and y is their separation normal to the slip plane.
Stress fields of two parallel edge dislocations
This simulation computes and displays any chosen component of the stress field of a
specified dislocation. It assumes isotropic elasticity. The display always shows a section
perpendicular to the dislocation line which lies along z axis. The data required
with the default values, are shown in the following table:
Parameter 
Default value 
Burgers vector a/n [hkl] 
a/2[110] 
Lattice parameter in nm 
0.3 
Line direction [uvw] 
111 
Shear Modulus in GPa 
50 
Contour interval in MPa 
2 
Poisson’s ratio 
0.3 
Interaction of many edge dislocations
In this simulation, the user can examine the interaction of up to 100 edge
dislocations:
 The dislocations are placed randomly on the screen.
 Forces on each dislocation due to all the others within a certain radius are calculated.
 Each dislocation moves only one step at a time but the program allows those with the
largest forces on them to move first.
 The dislocations are constrained to stay within the background rectangle. This is of
course not totally realistic and occasionally some strange effects may occur. These are
generally easily recognised and as such should be ignored.
Parameters which may be altered by the user are:
Parameter 
Default Value 
Number of dislocations (2 to 150) 
50 
Sign of dislocations (1 or both) 
both (equal mixture) 
Glide or climb+glide 
c + g 
Five pages of this section concentrate on how jogs are formed when dislocations
interact
Dislocation pileup
The significance of a pileup
When a gliding dislocation encounters a large obstacle which it cannot pass, such as a
grain boundary, it will stop, and dislocations emitted from the same source will pile up
behind each other. The stress at the head of the pileup will then increase until it
reaches a critical value, at which point the stress concentration will be relieved by some
form of plastic relaxation.
Dislocation pileups are believed to play an important role during the initial plastic
yield of crystalline materials, in work hardening, and in fracture.
Yield of polycrystal
If a stress is applied to a polycrystal, dislocations will first move in a grain having
a large critical resolved shear stress. However, as dislocations cannot in general cross a
grain boundary, they will pile up at the boundary until the stress there is sufficient to
generate slip in an adjacent grain. At this point, general plastic flow can occur, and the
material is said to have yielded.
The yield stress s_{y} is related to the grain size d
by the HallPetch relationship:
s_{y}= s_{i
}+ kd ^{1/2}

(1) 
The constant s_{i} is sometimes called the
‘friction stress.’
Work hardening
During the subsequent deformation of the material, dislocation pileups will occur not
only at grain boundaries, but at other obstacles formed during the deformation, e.g.
sessile dislocations such as the LomerCottrell barrier in FCC materials. Not only will
such barriers force the generation of new dislocations, thus increasing the dislocation
density, and hence the flow stress, but the dislocation pileups have a longrange stress
field which affects the mobility of dislocations on other slip planes. Several work
hardening theories based on the longrange stresses from dislocation pileups have been
formulated.
At a sufficiently large applied stresses, the barriers to slip will either be overcome,
or broken by the dislocations, and the rate of work hardening will be reduced.
The Simulation
This simulation shows a dislocation pileup and enables its properties to be measured,
and compared with theory.
1. Edge dislocations are introduced into a slip plane of a crystal under a small
applied stress s_{y}. The friction stress is taken to
be zero. The force on each dislocation due to all of the others is calculated from the
following equation (with y = 0):

(2) 
The force due to the applied stress is added, and the dislocation is moved a distance
proportion to this force in the appropriate direction on the slip plane.
2. The leading dislocation is held up at a barrier, which is assumed to exert a
shortrange force on this dislocation, and not to affect the other dislocations.
3. Further dislocations are automatically emitted from the source s until the
backstress from the existing dislocations prevents it from operating.
4. The applied stress may be increased at any time, and the positions of the
dislocations may be tabulated.
5. The following parameters are displayed and updated:
 applied stress s
 number of dislocations in the pileup n
 stress at the head of the pileup s*
 length of the slip plane L
 back stress on the dislocation source.
6. The sequence terminates when s* reaches a critical value s*_{c }(arbitrarily taken).
7. An option to alter the length of the pileup between the limits of 50 and 255 is
given.
8. Units The unit of distance employed is pixel. The units of stress may be
taken as approximately:

(3) 
Note regarding quantitative measurements
This limits the accuracy of the various parameters and, particularly when the
dislocations are close together, there will be some scatter in the results.
Suggested use of program
As well as being useful for tutorials or lecture demonstrations, the program can be
used as an ‘experiment’ to determine the properties of a pileup.
For example, the student could measure s* and n as a
function of s and L, and should be able to show to
within the accuracy of the experiment, that the following relationships hold:
If it is assumed that yield occurs when s* = s*_{c}, then it may be seen that the yield stress s_{y} is given by:
s_{y} = K(s_{c}^{*}/L)^{1/2}

(5) 
If L is equated with the grain size d, then this equation becomes the
HallPetch relationship (s_{y} = s_{i}
+ kd ^{1/2}) for the case when s_{i} = 0.
The theoretical basis of these equations is discussed in references 1 and 3.
Work Hardening
Introduction
The work hardening of a crystalline material is a complex phenomenon, because the
stress necessary to move a dislocation depends both on shortrange interactions such as
the intersection of forest dislocations, and longrange interactions with both near and
distant dislocations. Despite a considerable research effort in this area, a complete
understanding of the subject has not yet been achieved, even for single crystals.
The problem is in two parts. First, the variation of dislocation content with strain
must be determined. This aspect is not addressed in this program, but it should be noted
that a relationship between dislocation density r and strain e of the form:
is frequently found in practice.
Second, the dependence of the flow stress s on the
dislocation content must be determined, and this program investigates this relationship.
The model used here is similar to that of the Taylor work hardening theory (see e.g.
ref.1), in which the hardening due to longrange stresses from parallel dislocations is
considered.
The simulation
This simulation investigates the movement of a dislocation by glide under the combined
effect of an applied stress and forces from other parallel dislocations.
A chosen number of dislocations of random sign are placed at random positions in a
crystal. These dislocations remain stationary.
A mobile dislocation is placed at the edge of the crystal, and a stress applied. The
force F_{x} in the slip direction, on this dislocation due to the other
dislocations (equation (7) , and to the applied stress, is calculated, and the dislocation
is moved accordingly:

(7) 
x is the separation in the slip plane.
y is the separation normal to the slip plane.
The dislocation continues to move until the net force becomes zero.
The stress may then be raised (manually or automatically) until the dislocation begins
to move again. The applied stress required to propagate the dislocation all the way across
the slip plane is then taken to be the flow stress.
Using the program
1. If only one dislocation is placed in the crystal, then the passing stress may be
measured as a function of the vertical separation H of the dislocations.
In accordance with theory, it should be found that the flow stress is inversely
proportion to H.
2. The number of dislocations may be increased to a maximum of 100, and the flow stress
measured as a function of dislocation density. It will be found that the flow stress is
proportional to the square root of the dislocation density. This relationship, together
with equation (6), suggests, as in the Taylor theory, that during work hardening the flow
stress is proportional to the square root of the strain, as is often observed in practice.
For small numbers of dislocations, the flow stress will depend strongly on the
dislocation positions. In order to obtain a statistically significant result, the
experiment should be repeated a number of times, and provision is made for this to be done
automatically.
Modes of operation
 Manual.
 Semiauto. A single run is made, in which the stress is raised automatically
 Auto. 20 runs are carried out for the same dislocation density. Pressing the
SPACE bar allows further increments of 20 runs to be made.
Display
The following data are displayed on the screen:
 applied stress
 number of dislocations, N
 vertical separation of dislocations (if N = 2)
 number of runs made, R
 mean value of flow stress and standard deviation (if R>1).
Units
The ‘crystal’ on the screen has dimensions of 440x210 pixels. If comparisons
of measured stresses are to be made with theory, then the unit of stress can be taken as
approximately:

(8) 
The interaction of a dislocation with obstacles in its glide plane, such as point
defects, forest dislocations or secondphase particles is the dominant strengthening
mechanism in many alloys.
This suite of programs examines several aspects of such interactions by simulating the
movement of a dislocation on its slip plane.
The simulation
The dislocation is represented as a number of discrete points (up to 100), the
positions of which are calculated on the basis of the positions of the neighbouring
points.
In the absence of stress, each point will take up a position midway between its
neighbours, i.e. the dislocation will tend to straighten.
However, an applied stress will cause the point to be displaced by an amount which
depends on the stress. Thus the row of points simulates a dislocation with a line tension
which does not vary with orientation. In this simulation the dislocation is pinned at its
ends. As the applied stress s is increased, the dislocation is
seen to bow out into a radius of curvature R until, at a critical stress s_{y} the segment becomes unstable, and expands without
further increase in the applied stress.
Although the program halts at this point, in a real material this process could
continue, resulting in the formation of a dislocation loop by the FrankRead mechanism.
The segment length L may be carried and, if sufficient measurements are taken,
the student should be able to demonstrate that the following important relationships are
obeyed:
and
CAUTIONARY NOTES
1. This simulation does not accurately reproduce the velocity of the dislocation under
nonequilibrium conditions, and it is essential that the applied stress is increased
slowly, and only when the dislocation is in equilibrium, otherwise an instability which is
not representative of dislocation behaviour may occur.
2. If the dislocation bows out too far, the points along the line become too widely
separated in which case saturation occurs, and an unrealistic flattened loop occurs.
3. In making quantitative measurements, it should be noted that the resolution of the
applied stress is poor below ~15 and, where possible, measurements should be made at
higher stresses.
Within these limitations, the programs provide a realistic simulation of the behaviour
of a bowing dislocation from which measurements may be made and compared with theory.
Units and terminology
Lengths The unit of length used in the calculations is the pixel, a dot which
represents the smallest graphic unit of measurement on a screen, and all distances are
given in this unit.
Stress The units of stress are consistent throughout the package, and may be taken
as approximately Gb/100.
Obstacle strength The obstacle strength displayed is in arbitrary nonlinear units.
Although useful for comparative purposes, this should not be used as a quantitative
measure of strength. The obstacle strength is best measured as the ‘breaking
angle’ f (Phi) which, as shown in Fig. 1, is the angle
between the segments of a pinned dislocation as the obstacle is overcome.
At this stage, the total force F exerted by the obstacle on the dislocation is
given by:
or, taking the line tension T as Gb^{2}/2,
F = Gb^{2} cos (f /2)

(12) 
Two animations, presenting regenerative multiplication of dislocations by FrankRead
type sources and schematic representation of dislocations curling round the stress fields
from precipitates are shown in this section.
Bibliography
The student is referred to the following resources in this module:
Hull, D. and Bacon D.J., Introduction to Dislocations, Pergamon, 1984 Order!
Friedel, J., Dislocations, Pergamon, 1964
Hirth, J.P. and Loathe, J., Theory of Dislocations, Wiley, 1982
Foreman, A.J.E. and Makin, M.J., Phil. Mag., 14, 911, 1966
Martin, J.W., Micromechanisms in particlehardened alloys, Cambridge, 1980
