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Materials Science on CD-ROM User Guide

Dislocations

Version 2.1

John Humphreys, UMIST/University of Manchester
Boban Tanovic, MATTER

Assumed Pre-knowledge

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:

• The Nature of Dislocations
• The Movement of Dislocations
• Interactions of Dislocations
• Bowing of Dislocations

The Nature of Dislocations

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.

Dislocation Movement

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:

1. An edge dislocation with the option to permit climb (and thus to allow movement out of its slip plane).
2. A screw dislocation where the user can choose whether cross-slip will occur. If cross slip is operating the dislocation changes its slip plane half way through the crystal.
3. A mixed dislocation, which can only glide.

Interaction of Dislocations

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:

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:

Five pages of this section concentrate on how jogs are formed when dislocations interact

Dislocation pile-up

The significance of a pile-up

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 pile-up 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 pile-ups 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 Hall-Petch relationship:

The constant s_i is sometimes called the ‘friction stress.’

Work hardening

During the subsequent deformation of the material, dislocation pile-ups will occur not only at grain boundaries, but at other obstacles formed during the deformation, e.g. sessile dislocations such as the Lomer-Cottrell 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 pile-ups have a long-range stress field which affects the mobility of dislocations on other slip planes. Several work hardening theories based on the long-range stresses from dislocation pile-ups 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 pile-up 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):

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 short-range force on this dislocation, and not to affect the other dislocations.

3. Further dislocations are automatically emitted from the source s until the back-stress 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 pile-up n
• stress at the head of the pile-up 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 pile-up 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:

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 pile-up.

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:

If L is equated with the grain size d, then this equation becomes the Hall-Petch 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 short-range interactions such as the intersection of forest dislocations, and long-range 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 long-range 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:

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

1. Manual.
2. Semi-auto. A single run is made, in which the stress is raised automatically
3. 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:

1. applied stress
2. number of dislocations, N
3. vertical separation of dislocations (if N = 2)
4. number of runs made, R
5. 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:

Bowing of Dislocations

The interaction of a dislocation with obstacles in its glide plane, such as point defects, forest dislocations or second-phase 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 Frank-Read 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 non-equilibrium 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 non-linear 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 Gb²/2,

Two animations, presenting regenerative multiplication of dislocations by Frank-Read 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: