University of Southern California Dissertations and Theses

Long range internal stresses in deformed single crystal copper

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Long range internal stresses in deformed single crystal copper

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Content LONG RANGE INTERNAL STRESSES IN DEFORMED SINGLE CRYSTAL
COPPER
by
Peter T. Geantil
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2013
Copyright 2013 Peter T. Geantil
Tomyparents: ThomasandRobynGeantil.
ii
Contents
Dedication ii
List of Figures vi
List of Tables x
Acknowledgements xi
Abstract xii
Chapter 1 : Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Crystal Structure, Dislocations and Plastic Deformation . . . . 2
1.2.2 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Composite Model . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Previous Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 In-Situ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.2 CBED Measurements . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.3 Dipole height Measurements . . . . . . . . . . . . . . . . . . . 26
1.3.4 X-ray Line Profile Asymmetry . . . . . . . . . . . . . . . . . . 29
1.3.5 In-situ vs. Unloaded Samples . . . . . . . . . . . . . . . . . . 33
1.3.6 Microbeam Measurements . . . . . . . . . . . . . . . . . . . . 34
1.3.7 LRIS Summary Tables . . . . . . . . . . . . . . . . . . . . . . 36
1.4 DAXM - Differential Aperture X-ray Microscopy . . . . . . . . . . . . 39
1.4.1 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.4.2 DAXM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter 2 : X-ray Microbeam Measurements 47
2.0.3 Q-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1 Initial Cell Interior Measurements . . . . . . . . . . . . . . . . . . . . 49
iii
2.1.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1.2 Experimental Procedure and Results . . . . . . . . . . . . . . . 49
2.1.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.4 Strain Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1.5 Cell Interior Sub-Profile Reconstruction . . . . . . . . . . . . . 56
2.1.6 Large Cell FWHM Analysis . . . . . . . . . . . . . . . . . . . 59
2.2 Dislocation Cell Interior and Wall Measurements . . . . . . . . . . . . 60
2.2.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3 Sub-Profile Reconstruction Using Small Sample V olume . . . . . . . . 70
2.3.1 Sub-Profile Reconstruction . . . . . . . . . . . . . . . . . . . . 71
2.3.2 Bulk Line Profile Reconstruction Using Sub-Profiles . . . . . . 74
2.3.3 X-Ray Attenuation Correction . . . . . . . . . . . . . . . . . . 76
2.3.4 Sub-Profile Subtraction . . . . . . . . . . . . . . . . . . . . . . 77
2.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 Sub-Profile Reconstruction Using Large Data Set . . . . . . . . . . . . 83
2.4.1 Sub-Profile Decomposition Comparison . . . . . . . . . . . . . 84
2.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5 Full Strain Tensor Measurements . . . . . . . . . . . . . . . . . . . . . 87
2.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.5.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 89
2.5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 91
2.6 Deviatoric and Hydrostatic Strain Tensor . . . . . . . . . . . . . . . . . 93
Chapter 3 : Dislocation Dynamics Simulation 96
3.1 MicroMegas (mM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.1 Sub-Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.1.3 Simulation Time and Dislocation Motion . . . . . . . . . . . . 98
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4.1 Localized Strains . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.2 Cell Interior and Wall Average Stress . . . . . . . . . . . . . . 108
3.4.3 LRIS Histograms . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.4 Dislocation Analysis . . . . . . . . . . . . . . . . . . . . . . . 115
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
iv
Chapter 4 : Transmission Electron Microscopy 125
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.1.1 Previous TEM Studies . . . . . . . . . . . . . . . . . . . . . . 126
4.1.2 TEM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.1.3 Dark Field and Weak Beam Diffraction . . . . . . . . . . . . . 131
4.1.4 Dislocation Identification Usinggb Contrast . . . . . . . . . . 134
4.1.5 Kikuchi Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.2 Transmission Electron Microscopy . . . . . . . . . . . . . . . 137
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Chapter 5 : Conclusion 143
5.1 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Reference List 146
Appendix A : Experimental Data 153
A.1 Dislocation Cell Interior and Wall Data . . . . . . . . . . . . . . . . . 153
A.2 Strain Tensor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Appendix B : Simulation Code 168
B.1 Fortran code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.2 LRIS codes in Igor Pro . . . . . . . . . . . . . . . . . . . . . . . . . . 185
v
List of Figures
1.1 Burgers Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Screw Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Crystal Structure and Slip Planes . . . . . . . . . . . . . . . . . . . . . 5
1.4 Slip Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Copper Dislocation Microstructure . . . . . . . . . . . . . . . . . . . . 10
1.6 LRIS Composite Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Idealized Dislocation Cell . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.9 Long Range Internal Stresses and the Pileup Model . . . . . . . . . . . 18
1.10 Backstress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.11 Bauschinger Effect in Zinc . . . . . . . . . . . . . . . . . . . . . . . . 20
1.12 LRIS Measurements from Dislocation Bowing . . . . . . . . . . . . . . 24
1.13 Calculated LRIS from Neutron Pinned Dislocations . . . . . . . . . . . 25
1.14 CBED LRIS measurements . . . . . . . . . . . . . . . . . . . . . . . . 27
1.15 X-ray Sub-profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.16 Compression/Tension Line Profiles . . . . . . . . . . . . . . . . . . . . 32
1.17 Kirkpatrick-Baez setup . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.18 Miller Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.19 DAXM Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
vi
1.20 CCD Diffraction Image . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 Cell Interior Line Profile . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 FWHM vs. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 Diffraction Intensity vs. Strain . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Cell Strain vs. Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Cell Interior Strains and Single Wall Strain . . . . . . . . . . . . . . . . 53
2.6 Strain Surface Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 Line Profile Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8 Large Cell FWHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.9 Beam Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.10 Stress/Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.11 Cell Wall / Interior Depth Study . . . . . . . . . . . . . . . . . . . . . 64
2.12 FWHM vs. Strain - 10-2007 Data . . . . . . . . . . . . . . . . . . . . 66
2.13 Cell Wall and Interior Stress vs. V olume Fraction . . . . . . . . . . . . 67
2.14 Cell Wall and Cell Interior Ratio . . . . . . . . . . . . . . . . . . . . . 68
2.15 X-Ray Line Profiles of V oxels and Respective Sum . . . . . . . . . . . 71
2.16 X-Ray line profile of bulk sample and voxel sum . . . . . . . . . . . . 72
2.17 Cell Wall and Interior X-ray Line Profiles . . . . . . . . . . . . . . . . 73
2.18 Sub-Profile Scaling and Sum . . . . . . . . . . . . . . . . . . . . . . . 74
2.19 Profile Reconstructions Using Different Cell/Wall Ratios . . . . . . . . 75
2.20 Attenuation Corrected V oxel Profiles . . . . . . . . . . . . . . . . . . . 76
2.21 Attenuation Corrected Cell/Wall Profiles . . . . . . . . . . . . . . . . . 77
2.22 Measured Line Profiles and Cell Interior Profiles . . . . . . . . . . . . 78
2.23 Summed Energy-Wire Profile and Summed Cell Interior Profile . . . . 79
2.24 Bulk Line Profile and Cell Interior and Wall Sub-Profiles . . . . . . . . 80
vii
2.25 Subtracted Wall Sub-Profile . . . . . . . . . . . . . . . . . . . . . . . . 81
2.26 Bulk Line Profile Reconstruction . . . . . . . . . . . . . . . . . . . . . 83
2.27 Summed Sub-Profiles vs. Extracted Sub-Profiles . . . . . . . . . . . . 85
2.28 Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.29 Strain Tensor Experimental Setup . . . . . . . . . . . . . . . . . . . . 90
2.30 Deviatoric and Dilatational Strain Tensor Illustration . . . . . . . . . . 94
3.1 Dislocation Cell Forming Stress . . . . . . . . . . . . . . . . . . . . . 102
3.2 Initial Dislocation Cell Structure . . . . . . . . . . . . . . . . . . . . . 103
3.3 Simulated Stress vs. Strain . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4 Slip System Dislocation Density . . . . . . . . . . . . . . . . . . . . . 104
3.5 Strain(
) Per Slip System . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.6 Plastic Strain in Cell Wall and Interior . . . . . . . . . . . . . . . . . . 107
3.7 LRIS Contour in Dislocation Cell . . . . . . . . . . . . . . . . . . . . 108
3.8 Simulated Stress Contours . . . . . . . . . . . . . . . . . . . . . . . . 111
3.9 Cell Interior LRIS Histogram . . . . . . . . . . . . . . . . . . . . . . . 112
3.10 Parallel Cell Wall Histogram . . . . . . . . . . . . . . . . . . . . . . . 113
3.11 Perpendicular Cell Wall Histogram . . . . . . . . . . . . . . . . . . . . 114
3.12 Single Layer Cell Wall Histogram . . . . . . . . . . . . . . . . . . . . 114
3.13 Cell Interior and [001] Cell Wall :
11
Tensor Component . . . . . . . . 116
3.14 Dislocation Analysis: Slip System 1 . . . . . . . . . . . . . . . . . . . 117
3.15 Dislocation Analysis: Slip System 3 . . . . . . . . . . . . . . . . . . . 118
3.16 Dislocation Analysis: Slip System 10 . . . . . . . . . . . . . . . . . . 119
3.17 Dislocation Analysis: Slip System 11 . . . . . . . . . . . . . . . . . . 120
3.18 LRIS Per Slip System . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.1 Ewald’s Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
viii
4.2 Select Diffraction Patterns . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3 Dark Field and Weak Beam Technique . . . . . . . . . . . . . . . . . . 132
4.4 Dislocation Contrasting : Weak Beam andgb . . . . . . . . . . . . . 133
4.5 Dark Field and Weak Beam Comparison . . . . . . . . . . . . . . . . . 134
4.6 Kikuchi Line Diagram and Micrograph . . . . . . . . . . . . . . . . . 136
4.7 Dislocation Characterization . . . . . . . . . . . . . . . . . . . . . . . 140
4.8 Dislocation Characterization: Dislocation Wall . . . . . . . . . . . . . 141
ix
List of Tables
1.1 Dipole Height Measurements . . . . . . . . . . . . . . . . . . . . . . . 28
1.2 Initial microbeam results from the Levine et al. [39] measurements.
A number of dislocation cell interiors were measured showing LRIS
ranged from 0-50% the applied stress. . . . . . . . . . . . . . . . . . . 35
1.3 Selected LRIS Measurements . . . . . . . . . . . . . . . . . . . . . . . 37
1.4 Selected LRIS Measurements (continued) . . . . . . . . . . . . . . . . 38
2.1 Calculated Strain Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1 Diffraction vector (g) and burgers vector (b) table used forgb analysis
by G¨ ottler in [13]. Theh100i burgers vectors are from Hirth locks. . . . 126
4.2 The 6 burger’s vectors and 6 diffraction vectors used for thegb analysis.
All diffraction vectors can be found at the [111] zone axis. . . . . . . . 136
4.3 Dislocation Characterization Results . . . . . . . . . . . . . . . . . . . 139
A.1 Cell Interior Data 08-2007 . . . . . . . . . . . . . . . . . . . . . . . . 153
A.2 Cell Interior Data 10-2007 . . . . . . . . . . . . . . . . . . . . . . . . 154
A.3 Cell Wall Data 10-2007 . . . . . . . . . . . . . . . . . . . . . . . . . . 157
A.4 Cell Interior Data 03-2008 . . . . . . . . . . . . . . . . . . . . . . . . 159
A.5 Cell Wall Data 03-2008 . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.6 Strain Tensor Data 11-2009 . . . . . . . . . . . . . . . . . . . . . . . . 166
x
Acknowledgements
I would first and foremost like to thank my advisor Dr. Michael E. Kassner for his
support, guidance, and understanding. He also encouraged and provided a number of
outstanding opportunities, for which I am forever grateful. I would also like to thank
my co-advisor Dr. Lyle Levine. I certainly would not have such a good understanding
of the material within this thesis without his knowledge, expertise, and patience. I must
also thank Dr. Benoit Devincre, who provided guidance and expertise on all dislocation
modeling work.
Finally I want to thank my colleagues Peter Veloo, Enoch Dames, Saro Nikki, and
I-Fang Lee, for thoughtful discussions and friendship.
xi
Abstract
The subject of backstresses or long range internal stresses (LRIS) in plastically deformed
crystalline materials has been investigated extensively over the past 40 years. It is cur-
rently believed that elevated stresses are present in regions of elevated dislocation den-
sity or dislocation heterogeneities in deformed dislocation microstructures. The hetero-
geneities include dislocation pile-ups, edge dislocation dipole bundles and dislocation
dense cell walls in monotonically and cyclically deformed materials. Understanding the
magnitude of long range internal stresses (LRIS) is especially important for the under-
standing of cyclic deformation and monotonic deformation. First, the fundamental theo-
ries needed to understand the subject matter are presented. Following this, is a summary
of select previous experiments aimed at assessing LRIS. This will then be followed by
my results from my experiments and analysis.
A number of experiments were performed at the Advanced Photon Source at Argonne
National Labs in an effort to measure LRIS within deformed single crystal copper. Mea-
surements were taken from a large number of locations within the dislocation microstruc-
ture, both in dislocation dense regions, and regions with low dislocation density. The
results from these experiments, as well as additional analysis relating to the nature of
the LRIS present, will be discussed.
Finally, the question as to the origin of LRIS will be discussed. It’s clear that the
dislocation microstructure is responsible for the stresses present, but it is unclear as to
xii
the configuration of the dislocations within the microstructure (ie. Burgers vector and
sign) which causes the LRIS. Theories present possible microstructural configurations
which may produce LRIS, and these will be compared to a TEM g b analysis of the
dislocation microstructure. Additionally, dislocation dynamics simulations (DDS) were
performed in an effort to model LRIS and the microstructure responsible. These models
will be analyzed and the result discussed in detail.
xiii
Chapter 1
Introduction
1.1 Motivation
The current method for developing new materials is centuries old, and is generally done
through the process of trial and error. Elements are mined and mixed, and different
procedures such as casting or cold working are used to produced new materials, but it
is generally inefficient and development costs are high. It is the use of trial and error
that makes the process of creating new materials both time consuming and costly. The
research presented here aims to understand a fundamental phenomenon related to mate-
rial behavior, in hopes that eventually the trial and error of new material development
can one day be removed.
Without an understanding of the principles of stresses and strains or statics and
dynamics, a good bridge is simply one that does not fall down. This metric, not break-
ing, is incapable of predicting how much of a load the bridge can handle, how much
material is required for construction, or how long it will last. These practical consider-
ations are taken for granted today, as we are able to design bridges. We can calculate
all of the necessary requirements and thresholds, once the underlying fundamentals are
well understood.
The goal of designing materials is of course a large undertaking and requires a deep
understanding of the deformation mechanics of materials. The research performed and
discussed here provides understanding of a part of these underlying mechanics. When
a crystalline metallic material is deformed, a dislocation microstructure develops within
1
the material. Along with this comes a heterogenous stress state which correlates directly
with this microstructure. These stresses are known as long range internal stresses (LRIS)
and it is this phenomenon which will be discussed in great depth.
Along with the larger goals, there are also more tangible and immediate questions
for which this research will provide insight. The Bauschinger Effect, which is the low
yield stress present when reversing the deformation direction, may have some origins in
LRIS. Additionally the underlying physics responsible for spring back in metal form-
ing may have some relation to LRIS. Perhaps most importantly, the research presented
here provides important insight into the deformation mechanics of crystalline material
deformation mechanics in general.
1.2 Background
In this section, we begin by presenting the basic principles of deformation in a metal-
lic crystal. These principles are then followed by an explanation of the microstruc-
tures which develop during deformation and the corresponding stresses (LRIS). A back-
ground on specific models developed to explain these stresses is then presented. Finally,
there is a detailed explanation of the Bauschinger Effect.
1.2.1 Crystal Structure, Dislocations and Plastic Deformation
To understand the mechanics of crystalline material deformation, it is first important to
understand that there are two types, elastic and plastic. Elastic deformation is reversible,
a simple flexing of the material (or more specifically, the spacing between the atoms
increases and decreases). Plastic deformation is non-reversible, and is caused by atomic
planes slipping to different crystallographic locations. These movements occur through
crystal defects which have specific properties, and are referred to as dislocations. There
2
(a)
(b)
! ! !
Figure 1.1: Burgers Vector
(a) The Figure on the (left) illustrates a perfect crystal lattice in 2D. The Figure on the
right is a crystal with an extra half plane of atoms inserted. Using the RHFS (Right
hand/finish to start), we assume the sense of the dislocation is out of the page. This
means a circuit is drawn counter-clockwise on the perfect crystal (left). The same circuit
is then drawn around the dislocation (right), the path drawn on the right does not end
where it started. The vector required to go from the finish point to the start point of the
circuit is known as the Burgers vector (illustrated by the thick red arrow).
(b) A shear force is applied to the material, causing the edge dislocation to glide
through, producing plastic deformation
are essentially three types of dislocations, edge, screw, and mixed (mixed having both
edge and screw character).
An important characterization of dislocations is the Burgers vector. To obtain the
Burgers vector from a given dislocation you must begin by choosing a convention. In
this case we will use the right hand/finish to start (RHFS) convention. First choose a
sense to the dislocation, in the case of Figure 1.1 the sense of the dislocation is chosen
as going out of the page. A square closed circuit is then drawn in the perfect crystal
3
!
Dislocation line
Dislocation loop
Burgers vector
Figure 1.2: Screw Dislocation
A Screw dislocation showing the Burgers vector, dislocation line, and shear force ()
required to move the dislocation. Using the RHFS convention, the dislocation sense
is chosen as the up direction, meaning the right-handed circuit has the Burgers vector
pointing in the downward direction. Notice the dislocation line and burgers vector are
parallel.
(jumping from lattice point to lattice point). This circuit is then drawn around the dislo-
cation on the imperfect crystal. Notice that the circuit in an imperfect crystal does not
end where it starts. The vector needed to close the circuit is known as the Burgers vector.
Figure 1.1(a) illustrates the basic principle of a Burgers vector for an edge dislocation.
Essentially, there is an extra half plane of atoms in the upper portion of the Figure. If
you imagine this in three dimensions, the edge dislocation stretches back into the page
and creates a dislocation line that is perpendicular to the burgers vector.
The principles are the same for screw dislocations and are shown in Figure 1.2. For
this dislocation we choose the sense as going up the page. Notice that the Burgers
vector is parallel to the dislocation line. This is in contrast to edge dislocations where
the Burgers vector is perpendicular.
4
Face Centered Cubic
Crystal Structure
[100]
[010]
[001]
(111) Slip Plane
[011] Direction
(for Burgers Vector)
Figure 1.3: Crystal Structure and Slip Planes
An illustration of the f.c.c. crystal structure on the left. Each sphere represents an atom.
Repeat this structure in all directions and an fcc crystal lattice is created. On the left is
an illustration of a slip plane, and an associated Burgers vector.
During deformation, dislocations move through the material causing small amounts
of plastic deformation (or plastic strain). An illustration of this process is shown in Fig-
ure 1.1(b) and Figure 1.2 where a shear stress is applied. The edge dislocation glides
through the material from right to left, and the screw dislocation travels from the mid-
dle to the upper right. These Figures show a simple cubic structure, while the material
researched in this thesis is copper, which has a face centered cubic (fcc) structure.
In three dimensional crystal structures, slip generally occurs (but not always) along
the close packed planes where the atoms are closest to each other. The slip direction on
these planes occurs in the direction of the smallest lattice translation vector, meaning in
the direction where the burgers vector is smallest. For copper, which is fcc, slip occurs
along the close packed planes of thef111g type, and the Burgers vector is in theh110i
directions. An illustration of the fcc structure, as well as the slip plane and burgers
vector is shown in Figure 1.3.
5
The applied force divided by the cross-sectional area (F=A) is known as the stress
. This applied force (and stress) will inevitably be at some angle to the slip planes
and slip direction. This means there is some component of the applied stress that acts
as a shearing stress along the slip plane. It is this stress along the slip plane and in the
slip direction which moves dislocations and causes plastic deformation. To calculate the
stress on the slip plane, from the applied force, we use the following equation,
=
F
a
A
cos cos; (1.1)
whereF
a
is the applied force,A is the cross sectional area on which the force is applied,
is the angle between the slip plane normal and the applied force, and is the angle
between the slip direction and the applied force. This equation effectively resolves the
shear stress onto the slip plane in the slip direction. The coefficient (cos cos) is
known as the Schmid factor, and is a measure of how much of the applied force causes
slip to occur. Figure 1.4 shows in detail the physical interpretation of these parameters.
Dislocations require a shear stress to conservatively glide through a material, in addi-
tion to creating stress fields of their own. By nature, dislocations cause short range
internal stresses within a crystal lattice (nanometer scale). Equations (1.2)-(1.4) show
the stress field associated with an edge dislocation. Notice there is noz component. An
edge dislocation creates no stress in this direction as it is parallel to the dislocation line.

xx
=
Gb
2(1)
y(3x
2
+y
2
)
(x
2
+y
2
)
2
: (1.2)

yy
=
Gb
2(1)
y(x
2
y
2
)
(x
2
+y
2
)
2
: (1.3)

xy
=
Gb
2(1)
x(x
2
y
2
)
(x
2
+y
2
)
2
: (1.4)
6
Slip plane normal
!
Slip direction
"
Applied force: F=#A
Cross sectional area: A
Slip plane
Figure 1.4: Slip Geometry
A forceF is applied and the resolved shear stress on a specific slip plane, in a specific
slip direction, can be calculated using the information shown.
Equation (1.5) describes the stress field associated with a screw dislocation. Screw
dislocations, having radial symmetry, can be expressed much more simply. [20]

z
=
Gb
2r
; (1.5)
whereG is the shear modulus, is Poisson’s ratio,b is the Burgers vector,x andy are
cartesian coordinates, andr is the radius in cylindrical coordinates. Notice the equations
for edge dislocations do not have az component, as the stress field only exists in thex
andy directions. These equations are for elastically isotropic materials (having the same
elastic constants in all directions) which copper is not, but they are a good approximation
7
and all that is needed for the purpose of understanding this thesis. Note also that the 1=r
dictates that stress fields fall off quickly when moving away from a dislocation.
The stress fields interact with each other in specific ways. Dislocations with the
same burgers vector but of opposite sign attract each other and dislocations of the same
burgers vector and same sign repel. If they are on the same plane, opposite signed
burgers vector dislocations will annihilate, however if they are not and simply approach
each other, a dipole pair can form. Dipole pairs have been used in an attempt to measure
LRIS, and this will be discussed later. Equations (1.6)-(1.7) show the force between
two dislocations of the same sign. If the dislocations are of opposite sign, the forces are
reversed.
F
x
=
Gb
2
2(1)
x(x
2
y
2
)
(x
2
+y
2
)
2
: (1.6)
F
y
=
Gb
2
2(1)
y(3x
2
+y
2
)
(x
2
+y
2
)
2
: (1.7)
It’s apparent from these equations that the stress fields created by dislocations inter-
act with each other. If obstacles are present in the crystal lattice it becomes more diffi-
cult for dislocations to glide through the material. Dislocations act as obstacles to other
dislocations and as plastic deformation continues they may begin to tangle together.
Additionally as plastic deformation occurs, the number of dislocations increases dra-
matically. This in turn increases the difficulty with which they can move freely, having
the effect of strengthening the material (also known as strain hardening). As these pro-
cesses continue with ever increasing numbers of dislocations and interactions, disloca-
tion microstructures emerge within the material, consisting of dislocation dense regions,
and regions with little or no dislocations present. It is these dislocation microstructures
8
which are integral to the understanding of LRIS and will be discussed in depth in the
following section.
1.2.2 Microstructure
During deformation, dislocations glide through the material, interacting with each other
and eventually forming a heterogeneous microstructure. This microstructure consists
of regions with high and low dislocation densities. In monotonic deformation, when
multiple slip systems are active, the microstructure takes on a particular form. Regions
of dislocation free (or relatively free) cell interiors are surrounded by dislocation dense
cell walls. In single slip deformation, a microstructure develops consisting of elongated
clusters of dislocations. At higher temperatures cell walls may take the form of subgrain
boundaries. Persistent slip bands of dipole walls form if cyclically deformed [47]. Fig-
ure 1.5 shows the dislocation microstructure of single crystal copper deformed by about
30% true strain along the [100] direction, and multiple slip systems are active. This
type of deformation produces a heterogenous structure of dislocation free cell interiors
surrounded by dislocation dense cell walls. All experiments performed for this thesis
are on this type of microstructure (i.e. copper deformed monotonically with multiple
slip systems active), therefore it is the only one which will be discussed.
The existence of LRIS being present in heterogeneous dislocation microstructures
has been a source of discussion for over 40 years. The main issue being whether
after deformation there are stresses present within the microstructure which exist on
the micron length scale. A number of experiments have been done in an attempt to
measure and prove the existence of these LRIS, namely X-ray line profiling, dislocation
pinning, CBED measurements, and X-ray microbeam diffraction.
9
Figure 1.5: Copper Dislocation Microstructure
A bright-field TEM image of the dislocation microstructure of copper uni-axially
deformed by roughly 30%. The TEM sample was cut perpendicular to the strain axis of
the bulk sample. The dislocation dense ’cell wall’ regions and the dislocation free ’cell
interior’ regions can be clearly seen. Additionally the subgrains are relatively equiaxed
in all directions.
Recently X-ray microbeam experiments have been able to non-destructively directly
measure the existence of LRIS in dislocation cell interiors [39]. These results defini-
tively show that long range internal stresses exist in monotonically deformed single
crystal copper and are frozen into the microstructure after deformation. Previous experi-
ments have been either ambiguous as to the value of internal stresses, or ambiguous as to
their existence at all. The microbeam results and the methods used to obtain them serve
as the foundation of this thesis, from which many questions can and will be answered
regarding the nature of these internal stresses. Before delving into the experimental
history however, a discussion on models developed to explain LRIS is insightful.
10
1.2.3 Composite Model
This section will present an influential model which may explain the origins of LRIS
using both the nature of dislocations and the microstructure present. An explanation of
the composite model will provide an excellent guide as to what to expect when examin-
ing LRIS experimental results.
The composite model was first proposed by Mughrabi [47] and Pedersen et al. [59]
as a physical model used to explain LRIS. It also proposed the stresses were of relatively
high value (often 3 times higher than the applied stress). Mughrabi presented the sim-
ple case where hard (high dislocation density walls) and soft (low dislocation density
channel or cell interior) sections of the microstructure are compatibly sheared. Due to
the higher yield stress of the walls, the soft sections deform plastically through dislo-
cation movement, while the hard wall sections deform elastically. This deformation is
allowed (compatible) due to a buildup of dislocations at the interface. Of course, higher
deformation stress then begins to deform both sections plastically.
The different regions each yield plastically at different stresses, depending on the
dislocation density and arrangements, and it is suggested that the composite structure
is under a heterogeneous stress-state. The high-dislocation density regions experience
a larger magnitude stress than the average or applied stress, and the interiors a smaller
magnitude stress than the applied stress.
Consider the applied flow stress
a
which deforms the sample. Due to the large
difference in dislocation densities, different flow stresses are required to deform the cell
wall (
w
) and cell interior (
I
). Once the applied stress reaches that of the cell interior
flow stress
I
, the cell interior regions begin to deform plastically while the walls are still
in the elastic regime. As the applied flow stress continues to increase and reaches that
of the wall flow stress
w
, the bulk material begins to flow plastically with cell interior
11

6

Figure 3. The composite model illustrating the Bauschinger effect. The different stress
versus strain behaviors of the cell walls and the cell interiors are illustrated in (a), while the
stress versus strain behavior of the composite is illustrated in (b). When the composite is
completely unloaded, the low-dislocation density cell interior region is under compressive
(back) stress. This leads to a yielding of this “softer” component in compression at a
“macroscopic” stress less than -!
avg
.

Note that the yield stress on reversal is relatable to the LRIS in the
composite, but "
b
of Eq. (2) is not equal to the LRIS. Also note that elastic
perfectly plastic behavior is assumed as well. As Mughrabi points out, the
disparate stresses suggest an elastic incompatibility and further suggests the
necessity for dislocations at the interface. Of course, the LRIS must
ultimately result from the stress-field summations of all of the dislocations in
a single-phase material. Said another way, the stress-states (LRIS) of each
element in the structure can be, in principle, assessed based on the
knowledge of the precise location of all dislocations. These interfacial
dislocations of this model are the principal basis of the LRIS.

walls or dipole
bundles
channels or
cell interiors
τ τ
τ
τ
τ τ τ
τ
γ
τ
τ τ
γ
Figure 1.6: LRIS Composite Model
(a) The cell interior yields plastically at
l
while the wall continues to yield elastically
until
w
which is also the macro-yield stress. When the applied stress is removed, the
walls relax along with the cell interiors. Because the elastic regime is larger in the walls,
the interiors are put into compression.
(b) An illustration of the micro, and macro yield regimes according to the composite
model.[52]
and wall deforming in parallel. The relationship between the cell walls, cell interiors,
and applied stress can be expressed as,

a
=f
w

w
+f
I

I
; (1.8)
where
a
is the applied stress,f
w
is the area fraction of the wall, andf
I
the area fractions
of the interior along the plastically deforming glide plane. Consequently the stresses
12
present in the microstructure are dependent on the fraction of cell wall and interior
material.
It’s also important to note that when the applied stress is removed a back-stress is
created in the cell interiors as the cell walls elastically relax. This is due to the elastic
regime being larger for the cell walls than the interiors. Along with the back-stress is a
balancing forward stress in the cell walls. With no applied stress, the sum of all LRIS
must be zero. Essentially, the LRIS can be considered a deviation from the applied
stress. Equations (1.9) and (1.10) illustrate this principle.

w
=
a
+
w
; (1.9)
and

I
=
a
+
I
; (1.10)
where
w
is the stress present in the walls under load,
a
is the applied stress, and
I
is the stress present in the cell interiors under load. Once the applied stress has been
removed, the bulk material must be in equilibrium and have a net internal stress of zero.
Combining Equation (1.8) with (1.9) and (1.10) and assuming no applied stress, we have
the following,
f
w

w
+f
I

I
= 0 (1.11)
To explain qualitatively, if a crystalline material is deformed in compression, the
walls would experience an internal stress greater (in magnitude) than the applied stress
due to the higher yield stress, while the cell interiors would experience a lower (in
magnitude) stress than the applied. Once the applied stress is removed, these LRIS
are still present in the microstructure. Using equation (1.11) we see that this means the
13
!!c < 0
!
Cell Wall
Cell Interior
!!w > 0
! !
Figure 1.7: Idealized Dislocation Cell
The left most diagram shows the accumulation of dislocations inside the walls as well
as at the interface. The second panel shows the sum of the dislocations, allowing for
compatible deformation of the walls and cell interiors, as well as a source of internal
stress. The third panel shows the resulting internal stresses present in the microstructure
[48].
walls still experience a compressive LRIS, and to maintain equilibrium, the cell interiors
experience a tensile LRIS, also known as a back stress [47].
It is also important again to note that the cell interior and wall must deform com-
patibly. Figure 1.7 illustrates a simplified two dimensional dislocation cell. The
crosshatched lines represent the simplified slip system consisting of two active slip
planes. Dislocations present on these slip planes can be seen in the left most panel
accumulating at the walls as they act as obstacles to the dislocation glide. Some disloca-
tions pass through the walls, while others are deposited onto the wall. The middle panel
shows the effective sum of the dislocations gliding and being deposited onto the wall.
The final structure illustrates the corresponding LRIS present. Significantly, this type
14
of dislocation arrangement allows for deformation compatibility between the walls and
cell interior (one can deform elastically, while the other plastically). Additionally, the
sum of the dislocations provide a tensile internal stress in the dislocation dense walls,
and a compressive stress in the cell interiors, as our previous discussion of the composite
model predicts.
1.2.4 Bauschinger Effect
The composite model, with it’s predictions of backstresses in deformed metal materials,
may have a connection to the Bauschinger effect [48]. When a crystalline metal material
is deformed monotonically, the material hardens considerably. If the applied stress is
reversed, the material plastically deforms at a stress less in magnitude than if it were
deformed monotonically. This is in contrast to what is expected based on isotropic
hardening. A sample Bauschinger effect is illustrated in Figure 1.8 and equation (1.12)
helps to quantify the deformation in relation to backstresses [31].

b
=
(
f
+
r
)
2
: (1.12)
In this equation
b
is the backstress,
r
is the yield stress on reversal of the direction
of straining, and
f
is the stress in the forward sense, just prior to the reversal. It is
important to note that
b
is often related to LRIS by other investigators, but as will
be discussed, it is unclear if this is true. Backstresses have sometimes been explained
by kinematic hardening (or a translation of the yield surface). If
b
= 0, there is no
backstress and all hardening is isotropic. With immediate yielding on reversal,
b
=
f
and all hardening is kinematic and not isotropic. Not only is the flow stress lower on
reversal but the hardening features are different as well.
15
!f
!r
Stress
Strain
-!f
isotropic hardening
lost strength
Figure 1.8: Bauschinger Effect
An illustration of the Bauschinger effect. Notice the material yields at a stress that
is much lower than expected for isotropic hardening with reversal of the direction of
straining.
The Bauschinger effect is important as it appears to be the basis for low hardening
rates and low saturation stresses (and failure stresses) in cyclic deformation and fatigued
materials. The Bauschinger effect is also critically important for metal forming. This is
shown in Figure 1.10 and as will be discussed later,
b
is not related to a LRIS.
Pile-up Model
One of the early physically-based explanations for the Bauschinger effect was that dis-
location pile-ups, as shown in Figure 1.9(b), lead to long range internal stresses [64].
The stress,
p
, ahead of the pile-ups is estimated by the following equation.
16

p
=n
a
; (1.13)
wheren is the number of dislocations in the pile-up and
a
the applied stress [77]. Thus,
in the presence of pile-ups, high local stress develops.
Local equilibrium is achieved when the high stress against a barrier (e.g. cell walls),

p
, is balanced by a stress,
p
. Seeger et al. [64] and Mott [46] suggested that on
reversal of the direction of straining, this internal stress,
p
, assists the plasticity on
reversal and decreases the applied stress for reverse plasticity, leading to a Bauschinger
effect.
Composite Model
The composite model discussed previously and shown in Figure 1.6 was also suggested
as an underlying cause of the Bauschinger effect. As soft and hard regions are unloaded
in parallel from tension, the hard region places the soft region in compression while the
stress in the hard region is still positive. When the average stress is zero, the stress in the
hard region is positive while negative in the soft region. Thus it can be argued that the
backstress creates a Bauschinger effect due to reverse plasticity in the soft region. The
composite model for backstress is illustrated in Figure 1.6b. Note that the yield stress
on strain reversal is related to the long range internal stress in the composite, but
b
of
equation (1.12) is not equal to the LRIS. Also note that elastic perfectly plastic behavior
is assumed as well. As Mughrabi points out, the disparate stresses suggest an elastic
incompatibility and further suggests the necessity for dislocations at the interface. Of
course, the LRIS must ultimately result from the stress-field summations of all of the
dislocations in a single-phase material. Said another way, the stress-states (LRIS) of
each element in the structure can be, in principle, assessed based on the knowledge of
17
a
x
LRIS
´
a
b
Figure 1.9: Long Range Internal Stresses and the Pileup Model
(a) A profile of long range internal stresses as they would relate to the microstructure
shown. (b) An illustration of the pile-up model, with blue dislocation walls acting as
obstacles to dislocation motion.
the precise location of all dislocations. These interfacial dislocations are the principal
basis for the LRIS, as was illustrated in Figure 1.7.
Non-LRIS Factors
Looking at other factors is important in order to understand the effects of LRIS on the
Bauschinger effect. In an early experiment performed by Kassner et al. [31] a nearly
random dislocation arrangement in monotonically deformed stainless steel produced
nearly the same elevated temperature Bauschinger effect as one with cells and/or sub-
grains where LRIS could be significant, according to the composite model [31]. This
18
a
b
Figure 1.10: Backstress
(a) The backstress defined and calculated by equation (1.12) above for 304 stainless
steel at creep temperatures, as a function of monotonic pre-strain based on Bauschinger
tests. [31] and (b), the normalized backstress as a function of strain.
is illustrated in Figure 1.10. Additionally, it can be noted from the aluminum experi-
ments of Kassner et al. [29, 28] that a very pronounced Bauschinger effect is evident in
the first cycle (just 0.0005 monotonic plastic strain) at 77 K when a dislocation cellular
microstructure is not expected to be present during the very early Stage I deformation
(Schmid factor = 0.5). The Bauschinger effect is comparable to the case where a het-
erogeneous vein/channel substructure developed after hundreds of cycles. These exper-
iments suggest that the Bauschinger effect is largely independent of micro-structural
heterogeneities. Since LRIS are only present in such heterogeneous microstructures,
the Bauschinger Effect does not appear to measure nor be reflective of LRIS.
Sleeswyk et al. [65] analyzed the Bauschinger effect in several metals at ambient
temperature. They found that the hardening behavior on reversal can be modeled by
that of the monotonic case provided a small (e.g., 0.01- 0.05) reversible strain is sub-
tracted from the early plastic strain associated with each reversal (in Figure 1.11 this
19

8

Figure 5. A zinc single crystal is deformed at ambient temperature in tension to a strain of
.07, then unloaded from 0.54 MPa and compressed (although only the magnitutde stress is
reported in the figure and is always positive in sign). The material yields at ! 0.1 (negative)
MPa and gives rise to a Bauschinger effect. The hardening on reversal approaches that for
continued monotonic deformation if a “reversible” strain ! is subtracted to the dashed
vertical line. (From Sleeswyk et al., 1978)

2.1.3 Non-LRIS Explanation for BE

In early work performed by (Kassner et al., 1985) a nearly random
dislocation arrangement in monotonically deformed stainless steel produced
nearly the same elevated temperature Bauschinger effect as one with cells
and/or subgrains where LRIS could be significant according to the composite
model (Kassner et al., 1985). This is illustrated in Fig. 4. Also, it can be
noted from the aluminum experiments of Kassner et al. (1997, 1999) that a
very pronounced Bauschinger effect is evident in the first cycle (just 0.0005
monotonic plastic strain) at 77 K when a cellular substructure is not expected
to be evident during the very early Stage I deformation (Schmid factor =
0.5). The Bauschinger effect is comparable to the case where a
heterogeneous vein/channel substructure developed after hundreds or
thousands of cycles. These experiments suggest that the Bauschinger effect
is largely independent of such pronounced heterogeneities as cells or
subgrains. Inasmuch as LRIS are related to such heterogeneities perhaps the
Bauschinger Effect does not measure nor is especially reflective of LRIS.

Sleeswyk et al. (1978) analyzed the Bauschinger Effect in several metals at
ambient temperature. They found that the hardening behavior on reversal
can be modeled by that of the monotonic case provided a small (e.g., 0.01-
0.05) “reversible” strain is subtracted from the early plastic strain associated
Figure 1.11: Bauschinger Effect in Zinc
A zinc single crystal is deformed at ambient temperature in tension to a strain of 0.07,
then unloaded from 0.54 MPa and compressed (although only the magnitude stress is
reported in the Figure and is always positive in sign). The material yields at

= 0.1 (neg-
ative) MPa and gives rise to a Bauschinger effect. The hardening on reversal approaches
that for continued monotonic deformation if a reversible strain is subtracted to the
dashed vertical line. [65]
is approximately). This led to the conclusion of an Orowan-type mechanism (long-
range internal stresses are not especially important) [57] with dislocations easily revers-
ing their motion (e.g across cells or from dislocation tangles or reversing from pile-ups).
Sleeswyk et al. [65] suggested gliding dislocations during work hardening encounter
increasingly effective obstacles and the stress necessary to activate further dislocation
motion or plasticity continually increases (work-hardening). On reversal of the direc-
tion of straining from a forward sense, under, the dislocations will easily move past
those, non-regularly- spaced, obstacles that have already been surmounted. Thus, the
flow stress on reversal is initially relatively low,<. There is a relatively large amount
of plastic strain on reversal to( +d) in comparison to the strain associated with
an incremental increase in stress to ( +d) in the forward direction. This is referred
20
to as the Orowan-Sleeswyk explanation for the Bauschinger effect. There is also the
implication that the obstacles to the gliding dislocations are non-uniformly distributed
in the slip planes, which is expected. An example is cell interiors having a relatively low
obstacle density, while cell-walls, dislocation tangles, etc. have a much higher density.
Sleeswyk et al. [65] suggested that the reversible strain,, is related to the cell size; dis-
locations have a lower obstacle density on reversal into the cell interior from cell walls.
This suggests that a substantial component to the Bauschinger effect or the reversible
strain,, is configurational, rather than relating to LRIS.
The non-configurational sources for the reversible strain on reverse loading include
dislocation loop unbowing and re-bowing in the opposite direction, plasticity associated
with the long range internal backstress, and recovery on reversal of applied stress (mean-
ing recovery of plastic strain due to dislocation annihilation) [23, 27, 44]. These phe-
nomena could be responsible for the anelasticity observed in unloading as well. Obser-
vations of anelasticity in single crystals of 610
4
[11] and polycrystals of 10
3
[71, 70]
have been observed in unloaded specimen at temperature, during creep.
A maximum unbowing of dislocations for a cubic grid of dislocations can be esti-
mated using the following,

A

=
b
p

8
p
6
; (1.14)
where
A
is the strain caused from the unbowing, is dislocation density andb is the
Burgers vector. If

= 10
14
10
15
m
2
for cyclically and monotonically deformed Cu
single crystal [26, 42], an upper limit for
A
is 10
3
. Not surprisingly, these bowed, or
flexing dislocations also can give rise to apparent decreases in the elastic modulus dur-
ing strain-rate increase tests in heavily deformed material [21]. The sum of maximum
recoverable axial plastic strains from various elementsi, under load from backstresses
is
21

B
=
n
X
1

i

m
E

=

a
E
; (1.15)
whereE is the elastic modulus,
m
is the maximum stress and
i
is a local stress prior
to unloading. In this case for simplicity, we assume
B

=
a
E
where
a
is the applied
stress. That portion of or
B
due to recovery, which we will call
c
cannot be easily
predicted as this is a contribution to based on dislocation density changes. However,
if we sum the strain contributions from these various processes, it is small for cyclic and
monotonic deformation of Cu, roughly 10
3
to 3 10
3
. The observed values of are
roughly an order of magnitude too large. What this means is that the principal factors
resulting in the Bauschinger effect most likely need to include the Orowan-Sleeswyk
model in addition to any early micro-yielding from LRIS. In-situ reverse deformation
experiments in the high voltage transmission election microscope (HVEM) by Kassner
et al. [30] on pure Al with a cellular dislocation microstructure suggested that on reversal
the cell wall formed in monotonic deformation shows some unraveling, consistent with
the Orowan-Sleeswyk explanation.
1.3 Previous Experiments
This section contains a review of previous LRIS experiments. Attempts to character-
ize LRIS have been ongoing for decades, and experimental results very substantially.
Various different methods were used, but only the most important and influential exper-
iments will be discussed here.
1.3.1 In-Situ
A number of in-situ experiments have been performed attempting to evaluate whether
long range internal stresses exist and if so, measure the resulting magnitudes of these
22
stresses. Morris and Martin [45] quenched a creep-deformed solid solution Al-Zn alloy
that behaves essentially identically to pure Al at elevated temperature. When cooling
to ambient temperature however, precipitates form that allegedly pin dislocations (and
dislocation loops) in place. The Orowan bowing equation,

loc
=
T
br
; (1.16)
where is the local shear stress,T is the dislocation line tension,b the modulus of the
Burgers vector, and r is the radius of curvature, was applied and suggested enormous
long range internal stresses near dislocation walls. Nabarro et al. [54] noted that the
curvature of the dislocation segments may not reflect the LRIS but instead, the curva-
ture was increased by the chemical pressure of the super-saturated zinc. This may imply
that with low temperature precipitation, stress fields from the precipitate (coherency
or dilatational) or the mass transfer from the matrix to the precipitate (leaving vacan-
cies behind) may create a hydrostatic stress gradient leaving bowed dislocations whose
configuration is independent of any long range internal stress. The results from this
experiment are shown in Figure 1.12
Lepinoux and Kubin [38] performed in-situ deformation experiments using electron
microscopy on copper single crystals. Dislocations were observed in the persistent slip
band channels and were examined under stress. Equation (1.16) calculates the local
stresses in the persistent slip band channels using the measured curvature of dislocations
present. Results showed evidence of varying stress fields within the microstructure,
which may have meant LRIS were present at up to three times the applied stress.
A more influential experiment was conducted by Mughrabi [47] with results shown
in Figure 1.13. Here, deformed single crystal Cu specimens were irradiated with neu-
trons to pin the dislocations before thinning for TEM studies. This research (and the
preceding study mentioned [38]) have long been cited as proof of long-range internal
23
Al-5at% Zn
7 Û& ǅ =3.3 MPa
Ƹ =4.4x10 s
Ƹ =7%
-6 -1 •
70
60
50
40
30
20
10
0
0 4 8
ǅ (MPa)
eff
Distance ( m)
ǅ a
Figure 1.12: LRIS Measurements from Dislocation Bowing
Long range internal stresses calculated from dislocation loop-radii. The dislocations
were pinned under load using precipitates in creep deformed Al-Zn alloy with developed
subgrains. The high stresses are very close to the subgrain walls. [45]
stresses in cyclically deformed metals. These two studies assessed long range internal
stresses by measuring dislocation loop radii as a function of position within the het-
erogeneous microstructure, again in cyclically deformed Cu single crystal to saturation
(PSBs). After measuring the radii, the standard Orowan bowing equation was used to
calculate the internal stresses.
Observations confirmed that the dislocations were bowed more drastically near the
dislocation walls than in the center of the channels. After using equation (1.16), values
next to the walls again were calculated at up to almost three the applied stress, while
in the channels about half the applied stress. Using a spacial average, the authors cal-
culated that in the walls, the stress should be three times the applied stress. In a recent
recalculation and publication of the data, the cell channel stresses were raised and the
24
23 screw dislocations evaluated
58 edge dislocations evaluated
applied stress=28 MPa
0.0 0.5 1.0
80
70
60
50
40
30
20
10
x/d
c
τ
loc
[MPa]
Cu single crystal
d =PSB channel width
c
Original
Modified
Figure 1.13: Calculated LRIS from Neutron Pinned Dislocations
Calculated long range internal stresses from dislocation bowing in TEM samples. The
samples were neutron irradiated cyclically-deformed Cu single crystals under load with
persistent slip bands present. The original data is show from from [47] as well as the
recent modifications by the same investigators (dashed line) [51]. The high local stresses
correspond to regions with high dislocation density (persistent slip band walls).
wall stresses were lowered to 0.63
a
and 2
a
respectively [51] (LRIS - 1.5
a
). Correc-
tions were necessary because the interpretation of the pinned dislocation configurations
to elastic strains in the lattice is not as straightforward as equation (1.16), especially
since interactions with dipoles and multipoles of the wall are considered important and
equation (1.16) does not allow for these effects.
The recent corrections by Mughrabi are also shown in Figure 1.13. Even though the
calculated wall stresses were lowered from their original values, it appears that cyclic
deformation to saturation (where PSBs form) may be a case where LRIS are relatively
high at 1:5
a
. As will be discussed later, alternate microstructural heterogeneities appear
to have lower values of LRIS. A final point to make is that the in-situ experiments done
25
by Lepinoux and Kubin [38] calculated stresses using equation (1.16) and estimates
of the internal stresses are similar to the original values of Mughrabi et al. [47] in
Figure 1.13, but suffer from the same limitations as the original Mughrabi estimates
(pre-correction).
1.3.2 CBED Measurements
An alternate method used to measure lattice strains in subgrain boundaries of an
aluminum sample, and subgrain dipole bundles of a copper polycrystal, was CBED
(convergent-beam electron diffraction). After creep deformation, the samples were
unloaded and electropolished into thin foils. Results for the lattice parameter mea-
surements showed some scatter, but no evidence emerged of a residual stress in the
aluminum grain boundaries within the margin of error8 MPa (applied creep stress of
7MPa). The results were similar for the copper samples. No evidence emerged from the
lattice measurements within the margin of error30 MPa (applied creep stress of 20 and
40 MPa). CBED however, does not appear to be well suited for this type of experiment,
as the margin of error is of the same order as the applied stress. The internal stresses can
not be resolved if they are of the same order, or less than the applied stress. Additionally
the areas where measurements were taken was very thin, and surface relaxation effects
are unknown [25]. Results are shown in Figure 1.14.
1.3.3 Dipole height Measurements
As a metal is cyclically deformed, it hardens until a saturation of the flow stress is
reached. Up until this point there is often no formation of the PSBs. Instead, there is a
uniform vein like microstructure, where high dislocation density dipole bundles are evi-
dent [1, 34]. Generally, persistent slip band walls have higher dislocation densities than
veins. We are interested in dipoles as a measurement of maximum dipole heights may
26

23

Table 2. Three single crystals cyclically deformed to pre-saturation. Dipole heights were
measured using TEM. Observed (!
a
) stresses and calculated (!
d
from Eq. 6)
stresses based on observed primary “h” values at ambient temperature and 77 K
(Kassner et al., 2000).

2.3.2.2 CBED

Lattice parameter measurements were made near (within 80 nm) dipole
bundles and within the channels in Cu cyclically deformed to presaturation.
The uncertainty, however, was equal to the flow stress for the cyclically
deformed Cu examined. No evidence of long-range internal stress was
noticed. Recent work by Legros et al. (2008) on cyclically deformed silicon
to presaturation, using CBED also failed to detect LRIS to within 14% of the
applied stress. Of course, another difficulty with these experiments (besides
uncertainties) is that the region of the foils examined in CBED is fairly thin,
under 100 nm, so relaxation of LRIS could have occurred in the unloaded
specimens.

h (nm) T/T
m
!
d
!
a

SFE
mJ/m
Al
(77 K)
9.6
31.0
(ave)
(max) (h
c
)
0.12 51
16
20
20
200
Ni

4.1
5.7
(ave)
(max) (h
c
)
0.17 280
196
50
50
130
Cu 5.2
13.5
(ave) primary
(max) primary (h
c
)
0.22 132
52
19
19
60
Figure 1.14: CBED LRIS measurements
TEM micrograph of a dislocation dipole channel and corresponding dislocation dipole-
bundle walls. Locations where CBED was used are indicated, and an attempt to deter-
mine the lattice parameter was made. From the lattice parameter information, an assess-
ment of any long-range internal stresses was done. The black dots also project the area
of the volume of material from which the CBED determination was made [25].
allow the prediction of local stresses in cyclically deformed materials. The approximate
stress to separate a dipole of height h can be calculated from the elastically isotropic
equation,

d
=
Gb
8(1)h
c
; (1.17)
where G is the shear modulus, b the Burgers vector, and is Poissons ratio. Wider
separation dipoles thanh
c
are unstable under
d
.
Cyclic deformation experiments on aluminum and copper single crystals (oriented
for single slip) deformed to about half the saturation stress by Kassner et al. [28] showed
27
Table 1.1: Dipole Height Measurements
Three single crystals cyclically deformed to pre-saturation. Dipole heights were mea-
sured using TEM. Observed applied (
a
) stresses and calculated (
d
from equation
(1.17)) stresses based on observed primaryh values at ambient temperature and 77 K
[26].
h(nm) T=T
m

d

a
SFEmJ=m
Al (77 K) 9.6 (ave) 0.12 51 20 200
31.0 (max) 16 20
Ni 4.1 (ave) 0.17 280 50 130
5.7 (max) 196 50
Cu 5.2 (ave) primary 0.22 132 19 60
13.5 (max) primary (h
c
) 52 19
that the average dipole heights in the pre-saturation microstructure are also approxi-
mately independent of location, being equal in the dipole bundles and channels, con-
sistent with other Ni work [72, 4] (although with PSBs). This could suggest a uniform
stress state across the microstructure, which would mean no long range internal stresses
exist.
The maximum local dipole heights are the widest stable dipoles and suggest through
equation (1.17), a maximum local stress in the vicinity of the dipole. The maximum
dipole height translates to a stress, according to equation (1.17), that is about equal to
the applied cyclic stress for Al. Results are shown in Table 1.1.
These results appear to show no evidence of long-range internal stresses in Al single
crystals, which were cyclically deformed to pre-saturation. However, the stress required
to separate the widest dipoles in Cu is 2-3 times the applied stress, while for Ni is
roughly four times the applied stress [28, 72, 4, 26]. The issue, however, is that there
appears to be no fluctuation in the calculated stresses when going from channels to
bundles, meaning it is unclear if a heterogenous stress state exists. It is clear that a
homogeneous stress state of two times the applied stress can not exist. The high stresses
calculated may be caused by thermal activation. The dipole heights are not expected to
28
change significantly after unloading. It may also be dislocation pile-ups (just a few dis-
locations) in the lower stacking fault energy Cu and Ni which explain narrower dipoles.
Generally speaking, the CBED results are ambiguous when trying to assess the magni-
tude and nature of LRIS.
1.3.4 X-ray Line Profile Asymmetry
X-ray line profiles are obtained when measuring a Bragg peak, and the profile of it’s
diffraction spot. X-ray line profiles of undeformed single crystal materials are essen-
tially delta functions, in that the width of the line profile is very narrow, and the inten-
sity or amplitude of the diffraction signal is high. When a material is deformed, there
are characteristic changes to the line profile, that have been thought to indicate the exis-
tence of LRIS [58, 74]. When a perfect crystal is deformed, the diffraction line profile
(or rocking curve) broadens and develops a characteristic asymmetric shape. There is
also a change in the location of the maximum of the line profile, as it shifts away from
the perfect crystal peak. The peak shift seems to indicate that there is some amount of
material in the sample that is intensely diffracting, but with a different lattice parameter
than the undeformed sample (ie. elastically strained). The other side of the diffrac-
tion profile slopes down more slowly, indicating there is possibly other material in the
sample experiencing a lattice parameter shift as well.
According to the previously discussed composite model, the asymmetric peak results
from two separate diffraction sub-peaks coming from two regions within the microstruc-
ture (dislocation dense walls, and low density cell interiors or channels). Each of these
two regions diffracts in a characteristic way, such that the superposition of these two line
profiles results in the measured X-ray line profile. Experiments were performed where
line profiles were measured on deformed samples, and attempts were made to decom-
pose the profile into two sub-profiles [74]. An example of this type of decomposition
29
Figure 1.15: X-ray Sub-profiles
The red line is a diffraction profile from monotonically deformed single crystal copper
deformed along the [100] direction to a true strain of 0:3. The blue and green profiles
are possible sub-profiles which, added together, equate the red line profile. The blue pro-
file would be from the the dislocation cell interiors, and the green from the dislocation
cell walls [40].
is shown in Figure 1.15. The experiments were successful in that values for LRIS were
obtained, but the major flaw here is that given a single line profile, there are infinite
combinations of sub-profiles which could sum to this.
Additional observations regarding the asymmetry of the line profile were made,
which strengthen the evidence for long range internal stresses. Line profiles were taken
along the same axis as the deformation axis, and perpendicular to it. If the sample was
deformed along the [100] direction, the line profile is first measured along the [100]
direction. Thus the measurement is sensitive to the lattice parameter along the deforma-
tion axis. When the line profile is taken along the [010] direction, the profile asymmetry
flips (meaning the internal stresses change sign). The peak shift also flips sign relative
to that of the perfect crystal peak, and the shift offset also decreases. These phenomena
30
draw strong connections to the Poisson effect, and are evidence of the presence of LRIS
[52].
There are additional factors which can affect the shape of the line profile as well.
The asymmetry of the line profile appears to be strain dependent. If monotonic strain is
increased, the asymmetry and peak broadening increase. Additionally, if we plastically
deform a different crystalline material, we obtain a different line profile that is poten-
tially more or less asymmetric. In single crystal aluminum, for example, there is very
little asymmetry, however in copper the asymmetry is quite pronounced [74, 52, 16].
Finally, the asymmetry of the line profiles switch depending on whether the sam-
ple is deformed in compression or tension. Recent measurements by Levine et al.
[39] illustrate this phenomenon and the line profile measurements are shown in Fig-
ure 1.16. Importantly, this behavior is expected in terms of the composite model. The
LRIS present in the cell walls and interiors should switch sign, thus switching the bulk
X-ray line profile.
A few studies have been conducted on cyclically deformed Ni and Cu. Investigators
[17, 2] unloaded their samples from various points in the hysteresis loop and obtained
X-ray asymmetry results on single-crystal Ni, and polycrystalline Cu. An interesting
phenomenon is the observation of asymmetry for saturation, where persistent slip bands
are present, but an absence for presaturation (no persistent slip bands). This led Hecker
et al. [17] to conclude the presence of long range internal stresses only when persistent
slip bands are present in cyclically deformed Ni. This is consistent with the Kassner et
al. [26] and Legros et al. [37] CBED work on long range internal stresses in cyclically
deformed, pre-saturation, microstructures.
In the Ungar et al. study [74], single crystal Cu was deformed in tension, and the
[002] line profile was measured. The sub-profiles or sub-peaks were assumed to be
symmetric. Additionally, the integral of the intensity of each sub-profile was assumed
31
Axial 006
Line Profile
Unstrained
30 % strain
30 % strain
Figure 1.16: Compression/Tension Line Profiles
X-ray line profiles from a compression and tension sample, both deformed 30%. Notice
the shift in asymmetry from Compression to tension [39].
to be proportional to the values used in the stress equilibrium equation (1.8). Values for
the volume fraction of cell interior to cell wall material was obtained using TEM. The
larger amplitude and smaller full-width half-maximum (FWHM) peak was proposed to
come from the dislocation cell interiors, while the shorter, broader peak was a result of
the cell walls. Once the sub-peaks were decomposed from the bulk sample peak, the
centroids of the sub-peaks were found, and attributed to the average lattice parameter
32
of each region. Using this calculated lattice parameter the internal stress of each region
can be solved using the following,
d
w
d
=

w
E
; (1.18)
and
d
i
d
=

i
E
; (1.19)
where d
w
is the deviation in lattice spacing for the wall profile, and d
i
for the cell
interior profile. The final value for the internal stress in the walls was roughly 2
a
(or
twice the applied stress), while the cell interior value was roughly 0:2
a
. A later study
performed by Ungar et al. [75] used single crystal Cu compression deformed samples,
and suggested lower magnitude values of 0:1
a
in the cell interiors and 0:4
a
in the cell
walls [75].
1.3.5 In-situ vs. Unloaded Samples
A final issue which must be addressed is how the measured stresses from an unloaded
specimen relate to internal stresses in a sample under load. It is important to know
if there is any change that occurs when the sample is unloaded, or if is it simply an
elastic process. Borbely et al. measured the line profiles on single and polycrystalline
copper samples both under load and after unloading [3]. The X-ray diffraction line pro-
files provided the expected asymmetry under stress at room temperature and at elevated
temperatures (up to 633 K).
After unloading, the full-width half maximum of the peaks decreased by 20% for the
room temperature samples, and 30% for the samples deformed at 527 K. The dislocation
density was evaluated using the Fourier coefficients method [15, 73] and results show a
calculated decrease in dislocation density of 17-30%. The asymmetric line profiles were
33
decomposed into two sub-profiles for the loaded and unloaded samples, and there was a
calculated decrease in the long range internal stresses by roughly 1/3.
1.3.6 Microbeam Measurements
Results from the first microbeam measurements of cell interiors were published by
Levine et al. [39]. A compressive and tensile specimen were prepared by monoton-
ically deforming single crystal Cu along the [100] axis to a true strain of 30%. The
flow stresses (
a
) reached were 196 MPa and 185 MPa respectively. The specimens
were then cut with an acid saw to avoid any mechanical or thermal damage. Microbeam
measurements were taken near the center of the gauge.
The microbeam was used to scan the interior of the samples at up to 50m in depth.
A number of diffraction measurements were taken from the [006] planes perpendicular
to the deformation axis, meaning the lattice parameter parallel to the deformation axis
was being measured. This is important as the experiment only measured uniaxial stress
along the same line as the applied deformation stress. The measured lattice parameters
for diffracting dislocation cells within the samples is shown in table 1.2. Note all of
the strains have been converted into stresses using the elastic modulus along theh100i
direction using the stress-strain relationship,

11
=E
[100]

11
; (1.20)
where
11
is the internal stress calculated from the measured strain
11
, multiplied by
the elastic modulus. As copper is a highly anisotropic material, the elastic modulus
is different depending on the direction of compression or tension. In this experiment,
measurements were taken along the [001] direction, and the elastic modulus ofE
001
=
66:6 0:5 GPa was used.
34
Table 1.2: Initial microbeam results from the Levine et al. [39] measurements. A
number of dislocation cell interiors were measured showing LRIS ranged from 0-50%
the applied stress.
Long Range Internal Stress Measurements (MPa)
Compression 6 37 39 45 46 51 56 64 67 73 93 100
Tension -13 -16 -25 -26 -32 -39 -42 -45 -73
Applied flow stress
Compression -196 MPa
Tension 185 MPa
Average LRIS
Compression 56 MPa
Tension -34 MPa
A notable observation is the opposite sign of the cell interiors stresses with relation
to the applied flow stress. This is evidence that back stresses are present in the dis-
location cell interiors, as the composite model predicts. Additionally, rocking curves
(line profiles) were taken of the bulk samples from the [100] planes, and the asymmetry
displayed reverses when comparing the tensile and compression samples (discussed in
the previous section). This also supports the composite model. No measurements were
done in the walls at that time, but for the bulk material to have a net stress of zero as
there is no applied stress, the walls must contain a forward stress as the composite model
predicts.
An unexpected result of these measurements was the range of stresses seen within
the cell interiors. This experiment is particularly sensitive to these fluctuations, as the
experimenters were measuring individual dislocation cell interior lattice parameters.
The large strain fluctuations are certainly a new and important development in the study
of long range internal stresses, as they were neither predicted, nor expected.
The microstructure of a cyclically deformed specimen oriented for single slip was
also probed with the microbeam. Single crystal copper was deformed for 200 cycles for
a net strain of roughly 1%. The microbeam was used to scan the dipole bundles and
35
channels, being sensitive to the [220] lattice plane parameter. No strains were measured
within the margin of error 1 10
4
, which equates to roughly 7 MPa [39].
1.3.7 LRIS Summary Tables
The amount of research on the subject of long range internal stresses is quite remarkable.
A summary of selected experiments is shown in tables 1.3 and 1.4. The range of values
for long range internal stresses is large, and it is clear there is still a fair amount of
uncertainty as to the true nature (e.g. existence) of this phenomenon.
36
Table 1.3: Selected LRIS Measurements
Mat. Ref. Deformation
Mode
Strain Wall
LRIS
a
Interior
LRIS
a
Observation
Method
Temp. Notes
Cu Mughrabi [47] cyclic w/PSBs 2 0:5 in-situ neu-
tron irrad.
RT
Cu Mughrabi et
al. [51]
cyclic w/PSBs 1:3 0:37 in-situ neu-
tron irrad.
RT reanalysis of
above
Cu Mughrabi et
al. [52, 49, 50]
tension 0:4 0:1 X-ray peak
asymm.
unloaded
[001] oriented
Cu Lepinoux and
Kubin [38]
cyclic w/PSBs 2:5 0:5 in-situ
TEM
RT loaded single
crystal
Cu Kassner [26] cyclic no
PSBs
0 0 CBED RT unloaded
[123] oriented
Cu Kassner [26] cyclic no
PSBs
0 0 dipole sep. RT unloaded
[123] oriented
Cu Kassner [22] cyclic no
PSBs
0 0 X-ray
microbeam
RT unloaded
[123] single
crystal
Cu Borbely et al.
[3]
creep steady-
state
1 0:08 X-ray peak
asymm.
527 K loaded
Cu Kassner et al.
[24]
creep steady-
state
0 0 CBED 823 K unloaded
Cu Straub et al.
[68]
compression 0:3
0:6
0:05
0:08
X-ray peak
asymm.
RT 633 K
37
Table 1.4: Selected LRIS Measurements (continued)
Cu Straub et al.
[68]
compression observed CBED RT 633 K
Cu Levine et al.
[39]
compression 0:29 X-ray
microbeam
RT unloaded
[001] oriented
Si Legros et al.
[37]
cyclic 0 0 CBED RT unloaded
Ni Hecker et al.
[17]
cyclic w/PSBs 1:4
1:8
0:16
0:2
X-ray peak
asymm.
RT loaded
Ni Hecker et al.
[17]
cyclic no
PSBs
0 0 X-ray peak
asymm.
RT loaded
Al Kassner et al.
[29, 26]
cyclic no
PSBs
0 0 dipole sepa-
ration
77 K unloaded
[123] oriented
Al Kassner et al.
[24]
creep steady-
state
0 0 CBED 664 K Unloaded
Al
5%
Zn
Morris and
Martin [45]
creep steady-
state
25 1 precipitation
pinning
483-523 K
Sedlacek et al.
[63]
creep 1:5
10:0
0:5 1:0 theoretical creep
Gibeling et al.
[11], Nix et al.
[55]
creep 7:7 0:1
0:2
theoretical creep
38
1.4 DAXM - Differential Aperture X-ray Microscopy
The development of a specialized micro-diffraction tool by Larson et al. [35] at Argonne
National Labs, has been extremely important in the effort to characterize long range
internal stresses within deformed materials. It was used in a previously discussed exper-
iment by Levine et al. [39] which provided valuable and novel insight into the nature of
long range internal stresses.
Differential Aperture X-ray Microscopy (or DAXM) has the ability to probe materi-
als non-destructively, and measure lattice parameters in very small volumes. In princi-
ple it is able to resolve diffraction information from sub-micron volumes within a bulk
sample, thus enabling experimentalists to obtain important information on the crystallo-
graphic orientation, lattice parameter dilatation, and deviatoric strain of the sample. Just
as important, DAXM is completely non-destructive, unlike many other methods such as
TEM, which require thin film preparation. DAXM also has the ability to measure tens
of microns into the sample, negating any worries of extraneous surface effects that may
not be well understood. In fact, this type of technology can be used to study such surface
effects if present.
The sub-micron X-ray beam at Argonne National Labs is a substantial contribution
to what makes DAXM possible. The beam is focused down to roughly 0:5m 0:5m
using a Kirkpatrick-Baez mirror setup. Smaller focal sizes have been achieved and used,
but uncertainties in the mirror fabrication process make it difficult to obtain such small
spot sizes reliably. The setup consists of two parabolic orthogonal X-ray mirrors set
at grazing incidence to the X-ray beam. The mirrors focus the beam in the x and y
directions independently. Figure 1.17 is a schematic of the X-ray microbeam optical
setup.
Using this beam, a number of elements work synchronously to create the DAXM
technique. The basic theory behind these diffraction experiments is simple.
39
Polychromatic
X-ray beam from
undulator
Translating
Monochromator
Polychromatic or
Monochromatic
X-ray beam
Kirkpatrick-Baez
focusing mirrors
130 mm focal length
60 mm
focal length
0.5 µm
beam width
Slits
Figure 1.17: Kirkpatrick-Baez setup
Schematic of the KB mirror setup. A polychromatic X-ray beam enters through a slit
and is then converted to monochromatic, or not, depending on the experiment. The beam
is then further columnized through another slit, and is then focused using perpendicular
KB focusing mirrors.
1.4.1 X-ray Diffraction
To begin the discussion on diffraction, first recall section 1.2.1, which discusses crystal
structures and defects. The structure of greatest interest to this research is face centered
cubic (FCC) which is shown in Figure 1.3. The simplest way to describe a crystal
structure mathematically is by translation vectors. The fcc structure is simple, in that
the three translation vectors are orthogonal. The translation vectors also have the same
magnitude (hence the cubic part of face centered cubic). Generally these translation
vectors are denoted by the three primitive vectorsa
1
,a
2
, anda
3
.
In many diffraction experiments, it is the crystallographic planes that are of most
importance. Normally, the planes in a crystal are denoted by hkl, also known as the
Miller indices. Given a specific set of planes hkl, they intersect the three crystallo-
graphic axes ata
1
=h, a
2
=k, anda
3
=l. The spacing between the planes is given by the
40
a
a
a
1
2
3
h
a1
k
a2
l
a3
(a)
normal
O
b1
b2
s0
!
s
!
hkl
Hhkl
(b)
Figure 1.18: Miller Indices
The three crystallographic directionsa
1
,a
2
,a
3
and their relation to the crystallographic
planes, and Miller indices. The crystallographic directions can also be labeleda,b, and
c but this gets confusing when dealing with reciprocal vectors which are conventionally
written as b
1
, b
2
, and b
3
. The distance from the origin to the next plane along the
normal, is the spacing between the crystallographic planes.
fact that a plane goes through the origin, and another through the intersect points men-
tioned. Figure 1.18 shows the details of this principle.
Next we introduce the reciprocal lattice, which is the Fourier transform of the spatial
function of the crystal lattice. The reciprocal lattice primitive vectors are shown in
equations (1.21)-(1.23).
b
1
=
a
2
a
3
a
1
a
2
a
3
; (1.21)
b
2
=
a
3
a
1
a
1
a
2
a
3
; (1.22)
and
b
3
=
a
1
a
2
a
1
a
2
a
3
: (1.23)
41
Using these vectors, we can now describe the crystallographic planes more easily,
using a vectorH
hkl
defined in equation (1.24) where,
H
hkl
=hb
1
+kb
2
+lb
3
: (1.24)
H
hkl
has the property that it is normal to the planes, and it’s magnitude is the reciprocal
of the distance between the planes. If you map out the locations ofH
hkl
for all hkl, the
reciprocal lattice is drawn as shown in Figure 1.18(b).
Now that the crystal is defined in real space and reciprocal space, we can consider
the diffracting X-ray beam. First we define the unit beam vectors
0
, and the diffracting
beam unit vectors. If we draw a vector from the origin to any lattice point in reciprocal
space, this is anH
hkl
which meets the requirements for diffraction as shown in Figure
1.18. Additionally,H
hkl
= (ss
0
)=. Figure 1.18b illustrates the relationship between
the incident beam, diffracted beam, and lattice plane orientation and spacing [76].
This is generalized in the crystallographic real space using the Laue equations,
(ss
0
)a
1
=h; (1.25)
(ss
0
)a
2
=k; (1.26)
and
(ss
0
)a
3
=l: (1.27)
These give the relationship between the incident wave unit vector s
0
, the diffracting
wave unit vector s, the crystal lattice primitive vectors a
i
, the wavelength and the
Miller indicesh,k, andl [76]. For our purposes, these equations can be combined and
reduced into the well known Bragg’s Law, shown in equation (1.28).
42
n = 2d
[hkl]
sin; (1.28)
where is the wavelength,d the lattice spacing (between the diffracting crystallographic
planes), and the angle of the beam onto the sample. Therefore, the lattice parameter
d
hkl
can be solved given the wavelength (or energy) of the incident beam and incident
angle. Equation (1.20) will then yield the stress, given the lattice parameter and it’s
deviation from the unstrained value.
1.4.2 DAXM
A single wavelength () can be selected from a polychromatic source by use of a
monochromator as shown in Figure 1.17. can be determined from the position of the
diffracted X-rays onto the detector, the origin of the beam, and the sample diffraction
location. With this information equation (1.28) calculates thed-spacing.
The most challenging aspect of DAXM, and what sets it apart from other methods,
is how to decipher the location in the sample where diffraction occurs. Figure 1.19 is an
illustration of the experimental setup and an explanation follows below.
As the beam penetrates into the sample, a diffraction pattern is captured on the amor-
phous silicon area detector. Now, since the beam is going into the sample, there is a col-
umn of material (0:5m 0:5m) which is responsible for the diffraction pattern. A
platinum wire is then scanned in half micron steps across the sample, blocking diffrac-
tion from part of the diffracting column with each step. After each step of the wire an
image is taken. The image is then subtracted from the previous image, and the difference
of the two is the diffraction coming from the sample that was just occluded by the most
recent wire step. It’s important to also note that the wire radius is quite large (Figure
1.19 is not to scale), so it can be thought of as a knife-edge. Initial experiments assumed
43
Figure 1.19: DAXM Illustration
Schematic of the DAXM setup. The X-ray beam hits the sample and is diffracted onto
the CCD. A platinum wire scans across the sample in sub-micron steps, and provides
depth resolution to the measurements. The other two dimensions are provided by the
micron sized beam (and the focusing KB mirrors).
no diffraction was coming from behind the wire, but in fact there is a small amount,
which is accounted for in the reconstruction algorithms.
Because the wire position, location and direction (incident vector) of the incoming
X-rays, and location of the diffracted X-rays on the area detector are known, the loca-
tion of the diffracting material can be calculated. The lattice parameterd
hkl
can then be
calculated given the known monochromatic wavelength and crystal diffraction loca-
tion. With the lattice parameter now known and thus the strain, the LRIS can finally be
calculated.
44
Figure 1.20: CCD Diffraction Image
CCD image of diffraction coming from a 0:50:50:5m volume inside the sample.
The dark blue spots are diffraction from dislocation cell interiors. They are spatially
resolved from each other on the CCD due to the small differences in crystallographic
orientation. The diffuse diffraction is from dislocation dense walls, which connect the
cell interiors.
It is important to note that crystallographic rotations take place in the dislocation
microstructure when deforming the sample. As the sample is deformed, the dislocation
channels (in cyclically deformed) and cells (in monotonically deformed) may rotate in
various directions. With a monochromatic beam, the cells which diffract must satisfy
the Bragg condition, meaning they must have a specific crystallographic orientation. To
avoid this complication the experiment not only makes wire steps, but also steps the
monochromator, selecting discrete values of in 2-3 eV (electron-V olt) steps over a
range of 1-2 keV . Using this methodology, the Bragg condition will eventually be met
for each part of the dislocation microstructure, and all crystallographic orientations will
eventually provide diffraction information. Generally, locations where the crystallo-
graphic rotations are not large is preferred. This limits the range of wavelengths needed
to satisfy the Bragg condition for each cell rotation, and is a necessary efficiency.
45
The monochromator selects a wavelength, and the wire is stepped across the sample,
the next wavelength is selected, and once again the wire is stepped across the sample.
Eventually thousands of CCD images are taken. The wire images are subtracted and
combined with the energy scanned images to reconstruct the data into images that look
very similar, if not identical, to a white beam depth resolved X-ray diffraction image,
where all cell interiors are diffracting. Essentially, it is a reconstructed diffraction image
as if it were a white beam image, only it contains all of the information from the scanned
energy beam, as well as the depth resolution. With this information, we are able to
take depth steps into the sample, with sub-micron resolution, and calculate the lattice
parameter (or q-distribution) for any diffraction incident on the detector. An example of
a white beam reconstruction is shown in Figure 1.20. This exceptional technique was
developed by Larson et al. [35] and is further detailed there.
46
Chapter 2
X-ray Microbeam Measurements
The historical experimental evidence for LRIS was detailed previously in section 1.3 in
which a variety of experiments and their results were discussed. Results pointed to the
existence of LRIS, but the magnitude and exact nature of LRIS was still unknown. More
importantly, the first direct measurements of LRIS were made a few years ago by Levine
et al. [39] and discussed in section 1.3.6. There were 21 measurements from dislocation
cell interiors in deformed single crystal Cu samples. The work in the following chapter
builds upon these results.
It is also important to note that the results presented here are a first of their kind.
There are, therefore, a number of smaller explorations into the data which look for
answers to smaller questions. These are the first to be addressed. Following the smaller
results are much more important studies which help to understand much more pressing
questions related to LRIS.
2.0.3 Q-Space
Before presenting the experiments and results, a small amount of additional theory con-
cerning X-ray diffraction must be presented. When a cell interior is probed by the
microbeam, an X-ray line profile is extracted. An example profile is shown in Figure
2.1.
Once we obtain the line profile, we can then attribute an average lattice spacing to
the curve. A Gaussian is fit to the curve, and the center of the Gaussian peak gives
the average lattice spacing. Now, since the the line profile is in q-space, where q is
47
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Intensity
104.6 104.4 104.2 104.0 103.8
Q (nm
-1
)
C5_B_D2
root:C5_B_D2:
22Mar2008
Q
c
= 104.2454 (nm
-1
)
FWHM = 0.1
strain = 3.9e-04
depth = 0 µm
Figure 2.1: Cell Interior Line Profile
A line profile from a dislocation cell interior. It is plotted as intensity vs. q-vector. The
scattering vectorq is equal to 2=d or (4=) sin.
the scattering vector, we must convert this then into an average d-spacing, or lattice
parameter. The scattering vector q is related to the lattice spacing by the following
equation,
q =

4

sin =
2
d
; (2.1)
where is the wavelength, the Bragg scattering angle, andd the lattice spacing. Thus
upon finding the peak of the line profile fit, we can obtain an average lattice spacing.
Then by knowing the lattice spacing for pure unstrained copper, we can calculate the
strain and thus stress within the dislocation cell.
The lattice parameter of unstrained copper is d
0
= 0.3614960 nm and the elastic
modulus of copper along the [100] axis isE
100
=66.6 GPa 0.5 GPa [36].
48
The uncertainty in the strain measurements is roughly 1 10
4
which is equivalent
to 7 MPa stress. After calibration of the microbeam using a single Si crystal, the strain
uncertainty is very small, at roughly 2 3 10
5
. The main source of the uncertainty
is the detector pixel size, which controls the ability to accurately find diffraction peak
centers.
2.1 Initial Cell Interior Measurements
2.1.1 Sample
Initial DAXM measurements were done on a compression sample deformed to a final
flow stress of200 MPa and a true strain of 30%. The sample was 99.999+% Cu single
crystal and was cylindrical with a diameter of 10 mm and a length of 20 mm. After
deformation, the piece used for DAXM measurements was cut with an acid saw from
the center of the sample gauge.
2.1.2 Experimental Procedure and Results
A series of energy-wire scans were performed on the sample in a 3x4 grid set of loca-
tions. Each location provides a column of diffracting material that is resolved in three
dimensions, as described in section 1.4. In addition, dislocation cell interiors within the
same voxel can be resolved by their crystallographic orientations. After analyzing all
of the diffraction data, a total of 65 intense diffraction spots were found and analyzed.
These spots are attributed to 65 dislocation cell interiors. The analyzed cell interior data
is listed in appendix A.1.
In addition to the cell interiors found and analyzed, a search was performed for valid
cell wall data. Although unexpected, a single successful wall measurement was done,
49
however no others were found due to the low diffraction intensity. Most importantly, it
was a proof of concept, showing we are able to measure more diffuse diffraction from
cell wall material, with the help of some experimental refinements.
With these initial cell interior results, we were then able to look for basic correla-
tions. Figures 2.2 - 2.4 illustrate a first look. The single wall measured is omitted from
the data analysis.
2.1.3 Analysis
In Figure 2.2 the full-width half-maximum (FWHM) is plotted vs. the measured strain.
There appears to be no correlation between the two. This gives insight into two relevant
issues. First, the width of the diffraction line profiles are related to the dislocation con-
tent. This indicates that cell interiors have potentially non-zero dislocation content. Sec-
ond, the dislocation content in the cell interiors does not appear to affect the measured
strain values. These two points are complicated by an unknown factor however. When
measuring a cell interior, it’s difficult to know how much wall diffraction is present as
this could potentially cause an increase in the FWHM. The results are therefore not
conclusive, but do give some insight.
In Figure 2.3 the integrated intensity of the diffraction spot is plotted against the
strain. The X-ray line profile is first fitted with a gaussian peak. This gaussian is then
integrated using the following equation,
Z
1
1
ae
(x+b)
2
=c
2
dx =ajcj
p
; (2.2)
wherea is the amplitude andc the full-width at half-maximum. These results are then
plotted against the strain measurements. Since the size of the dislocation cell is related
to the integrated diffraction intensity, we hoped to see if there are were correlations.
50
0.25
0.20
0.15
FWHM
1.0x10
-3
0.8 0.6 0.4 0.2
Strain
Figure 2.2: FWHM vs. Strain
The width of the diffraction line profile plotted vs. the strain. No obvious correlations
can be seen, however the varying FWHM of the cell interiors seems to indicate that they
are not dislocation free.
This is somewhat complicated by the fact that there is no way to know the size of a
particular dislocation cell that is only clipped with the microbeam. Generally speaking,
more dislocation cells are clipped than whole ones diffracting. Thus the cell size is only
statistically connected to the integrated diffraction intensity, and not directly connected.
A TEM analysis of the cell structure shows the average cell size at roughly 0.5 m
which is the same size as the microbeam resolution. With all of this being considered,
there does appear to be some structure to this data but further work is needed and will
be discussed in section 2.2.
Finally the measured strains are plotted vs. the depth in Figure 2.4. The purpose
here is to assess whether the depth of the measurement has any effect on the strain, or if
there are any surface effects which affect the cell strains. The measurements in Figure
2.4 show no correlation between the depth and strain. It is possible that there are surface
effects not observed, but they must be on a smaller length scale than the experimental
resolution.
51
4
3
2
1
Integrated Diffraction Intensity
800x10
-6
600 400 200
Strain
Figure 2.3: Diffraction Intensity vs. Strain
The integrated X-ray diffraction line profile plotted vs. the strain. The distribution of
diffraction intensity being small at high strains indicates that small cell interiors are
probably what experience higher strains, although more data is needed, and will be
explored in more detail later.
1.0x10
-3
0.8
0.6
0.4
0.2
Strain
8 6 4 2 0
Depth (µm)
Figure 2.4: Cell Strain vs. Depth
The strain measured in a dislocation cell in relation to the depth. There appear to be no
surface effects present.
52
Figure 2.5: Cell Interior Strains and Single Wall Strain
A total of 52 dislocation cell interiors and 1 cell wall were analyzed and the results are
overlain on the X-ray diffraction line profile from the bulk sample.
What is most important with this new data, is the confirmation of the results in the
Levine et al. [39] study. The internal strains showed a high level of variation, exhibiting
internal stresses which raged from 0-50% the applied stress. The cell interiors also show
values which are consistent with the composite model. As the sample was deformed in
compression, all of the cell interior stresses are expected to be in tension. Additionally,
the single dislocation wall which was measured, exhibits a compressive stress, again
consistent with the composite model. To illustrate these principles more clearly, the
data is plotted overlain on the bulk X-ray line profile in Figure 2.5.
53
2.1.4 Strain Map
An additional study was done which looks closely at a single diffracting volume (or
voxel). The entire pseudo white beam image (displayed in Figure 1.20) was analyzed in
a grid pattern. Each section was analyzed and the resulting X-ray line profiles attributed
to average strains in those regions. The purpose being to look at the average strain asso-
ciated with different sections and features of the diffraction pattern. With the knowledge
gained from previous experiments, it is expected the intense regions of the CCD are
associated with cell interiors, while the more diffuse diffraction is from cell walls. Fig-
ure 2.6 displays the CCD pseudo white beam on the floor of the three dimensional box.
The contour graph floating above shows the relative strains associated. The blue on the
contour (upper peaks) represents a tensile strain, while the red (lower peaks) represents
a compressive strain. The white on the contour is the value of zero strain.
There are a few observations made from this Figure. There are locations where wall
data is expected to be present. Given two dislocation cell interiors, with two different
crystallographic orientations, it is necessary that dislocations be present between the
two, to account for the crystallographic orientation shift. Perhaps the most important
result from the strain mapping, is the location of the wall diffraction data. Normally,
it’s expected to be between two cell interior diffraction spots, yet in this case it is not
there (or can not be resolved). This means that any search for wall data, must include
looking at all of the diffraction data collected, not just diffraction that is between two
cell interior peaks.
In later experiments, the beam size is decreased, allowing for better analysis of
these types of structures. The CCD was also replaced with an amorphous-Si area detec-
tor, greatly improving the data acquisition rate. Because wall diffraction is so diffuse,
it’s easily lost in the intense cell interior diffraction, thus a sensitive detector and long
diffraction acquisition times are required.
54
Figure 2.6: Strain Surface Plot
A surface plot of strain values calculated at different regions on a single (pseudo) white
beam diffraction image. Under the surface plot is the white beam image of the diffrac-
tion pattern.
Another element explored was the grid pattern used for the scans. Since the scans
were done in a grid, an attempt was made to draw correlations between the different
adjacent energy-wire scans, and attempt to discern the actual size of larger dislocation
cell interiors was done. Most of the cell interior strain measurements are taken from
only the part of the cell illuminated by the microbeam. When looking at the grid of
scans, and looking for cell interiors large enough to be seen in a number of scans, the
results were unclear because of a small mount of sample drift. A single scan can take
anywhere from 4-8 hours, and over this period of time the drift is small enough that it
55
does not affect the results. However, over the time period two scans, there is enough
thermal drift such that correlating scans in a grid pattern becomes impossible.
2.1.5 Cell Interior Sub-Profile Reconstruction
Finally a rather important result was obtained and published in Levine et al. [40]. The
broad distribution of elastic strains shown in this data set, raises an important question
regarding the validity of the classical sub-profile analysis discussed in section 1.3.4.
The experiments discussed previously, draw conclusions about the distribution of dis-
locations and the internal stress values in plastically deformed metals. In those studies,
dislocation cell interior and cell wall sub-peaks are extracted from the bulk X-ray line
profile using a mirroring technique. The resulting profiles are then analyzed using a the-
oretical description of diffraction from the dislocation structures. Most importantly, the
dislocation diffraction theories used in these analyses assume all of the observed X-ray
broadening originates from the dislocations, and none comes from the large variations
in cell to cell elastic strains. This was reasonable at the time, as the large cell to cell
variations in strain were not explicitly measured, or even expected.
This leads to the question whether variation in cell to cell strains affects the shapes of
the sub-peaks, and if so, how much does it affect the shape of the bulk X-ray line profile.
The answer to these questions is not immediately clear. For example, the cell interior
sub-peak corresponds to a direct sum of the line profiles from all of the dislocation cell
interiors. The line profile from the bulk of the sample is then the sum of the two sub-
peaks. However, these sub-profiles are non-unique. Essentially, there are infinite pairs
of sub-peaks, which can be summed to form the bulk line profile.
To investigate this problem, a sub-profile was constructed using dislocation cell inte-
riors by summing 52 diffraction line profiles obtained from individual, spatially resolved
dislocation cells (listed in appendix A.1). The shape of the constructed profile was then
56
compared to those measured from individual dislocation cells. The aim was to look at
the contribution the cell to cell variation has on the width of the sub-profile.
The instrument resolution function was determined by measuring the 006 X-ray line
profile from an unstrainedh001i Cu single crystal under the same experimental condi-
tions. The instrument function is almost perfectly Lorentzian with a FWHM of (0.065
0.002) nm
1
. Scaling this by the magnitude of the [006] scattering vector,q (shown
in equation (2.1)) gives a FWHM of 6 10
4
in q=q. The FWHM of the instrument
profile is non negligible when compared with the FWHMs of the cell interior measured
profiles. Thus the actual peak widths (without instrumental broadening) must be deter-
mined using a deconvolution algorithm.
The measured profiles are combinations of Gaussian and Lorentzian line shapes and
may therefore be approximated as V oigt functions. Thus, the FWHMs of the approxi-
mately Gaussian intrinsic line shapes were determined using the following [56],
F
G
=
q
(F
V
0:5346F
L
)
2
0:2166F
2
L
; (2.3)
where F
G
, F
V
, and F
L
are the FWHM of the Gaussian, V oigt, and Lorentzian func-
tions (V oigt being the measured, Lorentzian being the instrumental broadening resolu-
tion function, and Gaussian being the intrinsic line profile).
The results show the line profile widths as varying by a factor of 3.5, and the inten-
sities by a factor of 17. It’s important to mention, again, that we are getting diffraction
from only the portions of dislocation cells hit with the microbeam, therefore the intensity
is only statistically connected to the size of the dislocation cells (and does not directly
reflect the size of the cell).
Three line profiles are displayed in Figure 2.7a which are more extreme examples of
X-ray line profiles. These illustrate the difference in intensity, width, and peak location
(strain) measured. It’s clear when looking at these, that a sub-profile deconvolution, as
57
Figure 2.7: Line Profile Comparisons
(a) Three line profiles from different cell interiors. The variation in intensity, FWHM,
and peak center (strains) are apparent. (b) The sum of all 52 cell interior line profiles and
a Gaussian fit to the sum. The three cell interior line profiles from (a) are also shown for
reference. They have relatively low intensity compared to the sum. (taken from [40])
was discussed in section 1.3.4, has no way to consider the varying nature of these profiles
which make up the sub-profile. In Figure 2.7b the sum of the 52 line profiles is shown,
along with the three line profiles from the previous figure, as well as a Gaussian fit to
the constructed profile. The constructed profile is smooth, confirming that a statistically
significant number of individual dislocation cell profiles was used. Additionally the
sub-profile is fairly symmetric, in agreement with the earlier deconvolution methods
discussed, although the symmetry requirement is only an assumption.
The peak location of the constructed sub-profile gives a mean elastic strain of 5.0
10
4
and FWHM of 1.5 10
3
. When directly averaging the 52 line profiles the elastic
strain is (5.3 0.3) 10
4
and the FWHM is (1.58 0.06) 10
3
which are close
to the other values. The reason these values are so close, is because the majority of the
52 line profiles fall between the extremes, and the average is dominated by dislocation
broadening (the FWHM of the profiles), rather than the varied strain values. A more
in depth comparison of this sub-profile reconstruction method with the extracted sub-
profile method will be discussed later in section 2.4.1.
58
2.1.6 Large Cell FWHM Analysis
Transmission Electron Microscopy (TEM) studies of deformed single crystal copper
generally show well-defined dislocation microstructures. A typical micrograph is show
in Figure 1.5. This microstructure is described as dislocation free cell interiors sur-
rounded by dislocation dense cell walls. The TEM micrograph reenforces this descrip-
tion. Unfortunately, the process of preparing TEM foils is destructive and it’s well
known dislocations can escape at the surface of the foil, thus the true nature of disloca-
tion content within the dislocation cell interiors may be non-zero. There is therefore a
motivation to look at large dislocation cell interior data, to see what, if anything, can be
discerned regarding the dislocation content of cell interiors.
Many dislocation cell interiors are larger than the 3D resolution of the microbeam.
Thus, when stepping through the material, the same cell interior is measured at different
spatially resolved points and we are able to step from one side of a dislocation cell
interior to the other. This provides the ability to look into statistics which may prove
informative. Figure 2.8 is a noisy plot showing the FWHM of a diffraction line profile
coming from a dislocation cell interior, compared to the location while moving across
the dislocation cell interior. The thick black line in the middle shows the average of the
data set.
As stated previously, the data is very noisy. So much so that it provides little infor-
mation, and the average of the data points is almost flat, but appears to have a slight dip
in the middle. In an idealized case, it could be argued that the FWHM would start high
near a wall due to increased dislocation broadening, then decrease as you move through
the cell, and finally increase again as you reach the other side of the cell. This pattern is
too difficult to see in the data except in the largest cell which spans 2.5m (solid pink
line with 6 data points). As stated before the average dips as well, but only slightly.
59
0.20
0.15
0.10
0.05
0.00
FWHM
1.0 0.8 0.6 0.4 0.2 0.0
% distance across dislocation cell Interior

Average
Figure 2.8: Large Cell FWHM
The FWHM of dislocation cell interiors that are at least 1.5 m or larger. The thick
black line in the middle shows the average of the lines.
A complicating factor is the inability to know if the microbeam is probing only the
center of the cell, or the center of the cell and part of a wall. However given that the
FWHMs for the cell interiors are all of the same magnitude, and are non-zero, it’s clear
that the dislocation content for these cell interiors is non-zero as well.
2.2 Dislocation Cell Interior and Wall Measurements
Now that the initial results for the cell interiors are well established, the next step is
to look for diffraction information from dislocation cell walls. To probe these smaller
sections within the microstructure, a smaller probe is advantageous. The microbeam
60
Figure 2.9: Beam Profile
The microbeam profile after focusing with the KB mirrors. The excellent spatial resolu-
tion allows for precise probing of the dislocation microstructure.
profile shown in Figure 2.9 is precisely this. The profiling wire has a programmed step
size of 0.7m along the sample surface, which projects onto the microbeam as a step
size of 0.5m. Thus the volume of each spatially resolved element is approximately
0.08m
3
.
Using these beam dimensions, an experiment in October 2007 using a new specimen,
provided excellent results. The new sample was prepared and stored in liquid nitrogen
61
as there is some concern as to whether LRISs relax over time in a sample stored at room
temperature. Recrystallization does in fact happen in 98% cold rolled high purity Cu
at room temperature[60]. Previous measurements have shown that maximum internal
stresses can be as high as 50% the applied stress which is also cause for concern. The
full set of diffraction data analyzed for both cell wall and interior strain data can be seen
in appendix A.1.
2.2.1 Sample Preparation
Theh001i oriented 99.999+% Cu single-crystal cylindrical compression specimen was
10 mm in diameter and 20 mm in length. It was deformed in compression at a con-
stant crosshead speed (with an initial strain rate of 4 10
4
s
1
) to a true flow stress
of 210 1 MPa and a true strain of 28.0 0.1% (see Figure 2.10). The unloaded
sample was cut in half perpendicular to the cylinder axis using wire electrical-discharge
machining (EDM). The top 120m of the sectioned surface of the sample was removed
by electropolishing to remove any EDM damage which may have occurred. All X-ray
measurements were made from a region near the center of the original gauge section.
2.2.2 Results
Diffraction data was obtained from 280 spatially resolved sample volumes (voxels)
within the sample. Within this data a total of 93 cell interiors and 56 cell walls were
found and analyzed. Many of the cells measured were larger than the beam spatial res-
olution, meaning there were independent measurements of the same cell interiors and
walls but in different locations (2 on average). When analyzing the diffraction images,
bright intense spots are designated cell interiors, and some diffuse scattering designated
as dislocation walls. The wall data was found in expected arrangements where two dis-
tinct intense spots (cell interiors) are connected by diffuse diffraction (wall material) all
62
Figure 2.10: Stress/Strain Curve
The true stress vs. true strain curve. A final flow stress of 210 MPa was achieved under
monotonic deformation along theh100i axis.
within a single voxel. Because two cell interiors are only resolved due to their slightly
different crystallographic orientations when in the same voxel, it logically implies that a
wall connecting the two must have an orientation which rotates from one cell interior to
the other. Additionally, other situations were present where diffuse scattering appeared
to have no connection to a cell interior diffraction spots, yet analysis showed these to
have the expected characteristics of cell walls.
Depth Resolved Cell Wall Study
Dislocation walls in the sample vary in thickness, but not substantially and are often
thinner than the beam resolution. Figure 1.5 shows a standard TEM micrograph with
a range of wall thicknesses. The micrograph is only a 2D slice however, and likely
the thick walls observed are actually thinner in the third dimension. Because our beam
resolution is often larger than the walls, there are only two ways to resolve them, the
63
Depth Resolved Cell Interior & Cell Wall Strain
Analysis
Depth Spot A Wall B Spot C
4 4.10E-04
4.5 3.60E-04 -5.40E-05
5 -1.30E-04 8.00E-04
A
B
C
Figure 2.11: Cell Wall / Interior Depth Study
Three depth steps with 0.5m resolution. A is a cell interior present in the first depth,
as we step deeper into the material, we seeA disappear and cellC arrive. Between these
two we know a wall must exist, so analysis of the diffuse diffraction in areaB shows
that there is a negative strain present. The strain values are displayed in the table below.
The wall is resolved both by crystallographic orientation and by depth resolution.
first is through crystallographic misorientations, as mentioned previously. The second
is through our depth resolution.
Figure 2.11 illustrates a special case. Two cell interiors are separated by a cell wall,
and the three are resolved both by crystallographic orientation and depth resolution. Cell
A appears 4m into the sample. Moving one depth step in, we begin to exit cellA and
enter cellC. One more 0.5m step into the sample and cellA has almost disappeared
while C has become much more prominent. Between the two cell interiors there is
faint diffuse diffraction. By analyzing the area designated byB, an X-ray line profile
is extracted with a large FWHM and indicates is a forward stress. It is reasonable to
assume then, that it is a dislocation cell wall. CellA has an internal stress of of 27.3
7 MPa, cellC an internal stress of 44.3 7 MPa, and wallB a stress of -8.6 7 MPa.
64
This study illustrates the ability of DAXM to probe the microstructures present
within our sample. Some of this of course depends on the signal to noise ratio, which
has to do with detector exposure times, but given enough time, the beam size is able
to resolve diffraction data from cell walls and interiors both spatially and by crystallo-
graphic orientation, and with this information, the LRIS present in each.
Cell Interior and Wall Statistics
The walls showed a variation in FWHM and strain. This is consistent with what was
discovered when cell interiors strains were first measured. These results showed stress
values for cell interiors ranging from 0 to 100 MPa, while the walls ranged from 20
to -100 MPa, or 50% the applied stress in both directions. Thus the maximum stress
differential present in our sample was 200 MPa, which is close to the final compressive
flow stress of the bulk sample (210 MPa).
When examining the FWHM of the peaks plotted against the axial elastic strain, as
illustrated in Fig. 2.12, it is very clear there are two groupings of data points. Dislocation
cell interiors show they consist of narrower peaks, while walls are much broader. All of
the FWHM measurements have been adjusted for instrument resolution using equation
(2.3), where the FWHM for the instrumentation (F
V
) was measured using an unstrained
piece of copper under identical experimental conditions. F
V
was measured as q=q =
6 10
4
.
There is a good chance that some wall profiles include scattering from more than
one dislocation wall, the measured centroid of the X-ray line profile gives a volume-
weighted average of the elastic strain in the respective cell walls for these cases. Sim-
ilarly, there are some cell interiors which have wall scattering in them. In these cases
however the wall diffraction has a negligible affect on the peak position and may only
65
0.4
0.3
0.2
0.1
FWHM
1.5x10
-3
1.0 0.5 0.0 -0.5 -1.0
Elastic Strain
Cell Interior
Cell Wall
Figure 2.12: FWHM vs. Strain - 10-2007 Data
A number of cell walls and interiors present in the CCD data were analyzed. Intense
diffraction spots, as well as diffuse sections were analyzed. There are two clear group-
ings of data points, which correspond to dislocation dense cell walls, and cell interiors.
slightly effect the centroid, as the diffraction intensity from wall material is much less
intense than cell interiors.
For small volumes in the kinematic scattering regime, the scattered intensities are
proportional to the volumes probed. By integrating the line profiles from individual
wall and cell interior measurements, we can examine the volume averaged stress distri-
butions within the sample. X-ray absorption must also be taken into account by using
the following equation,
I
t
=I
0
e

x
; (2.4)
whereI
0
is the incident beam intensity,= the mass attenuation coefficient, andx the
mass thickness, which is the thickness multiplied by the material density. Additionally,
the National Institute of Standards and Technology (NIST) maintains a thorough list of
66
Compressive Tensile
4 MPa
40 MPa
Average stress
Figure 2.13: Cell Wall and Interior Stress vs. V olume Fraction
Stress distribution of cell wall and interior measurements relating to their volume frac-
tions. The difference between the volume averaged stress in cell walls and cell interiors
is 40 MPa, with a total average of 4 MPa (Figure from [42]).
absorption coefficients [19]. With this information, we are able to explore the statistical
aspects of LRIS present within the sample.
The volume averaged distribution of stresses is shown in Figure 2.13 and was first
reported in Levine et al. [42]. Cell wall diffraction is more diffuse in real space (larger
FWHM) and the integrated intensity of the measurements was lower than for the cell
interiors. Thus distributions in Figure 2.13 for cell interiors and cell walls have been
individually normalized.
It is important to note that the sample was unloaded during these measurements.
Assuming that the sample volume measured is large enough to be considered represen-
tative of the bulk sample (which it is, as can be seen in [42]), equation (1.11) must be
satisfied. Essentially, we know the stress distributions of cell interiors and cell walls,
so it only remains to match this with the observed ratio of cell interior to wall material.
67
cell interior and cell wall stresses would be expected to be
approximately equal and symmetric about zero. The sepa-
rated,bipolarstraindistributionfunctionsshowninFig.4a
are therefore direct evidence that the cell wall dislocation
structures develop dipolarstresses during deformation; this
is in agreement with the basic concept of the composite
model.
The above discussion establishes the bipolar nature of
the stress fields and the applicability of the composite
model for large strains; however, it is equally important
to determine the underlyingprocesses driving the evolution
of the measured broad distributions of elastic strain. Feasi-
ble mechanisms include stochastic processes operating dur-
ing the evolution of individual dislocation walls and
Fig. 5. (a) Typical TEM micrograph of the Cu compression sample. (b) The same micrograph with the dislocation walls shaded in gray. The shaded area
comprises!55% of the micrograph.
c
< 0
w
> 0
(b) (a) (c)
Fig.6. Idealizedillustrationofthecompositemodel.(a)Duringtensiledeformationdislocationsbuildupalongdislocationdensewalls.(b)Thesetrapped
dislocations produce dipolar stress fields in the cell walls. (c) Resultant long-range internal stresses with back stresses in the cell interiors and forward
stresses in the cell walls.
L.E. Levine et al./Acta Materialia 59 (2011) 5803–5811 5809
Figure 2.14: Cell Wall and Cell Interior Ratio
(a) A TEM micrograph of our Cu compression sample deformed 28%. (b) A coloring
in of wall material shows it comprises roughly 55% of the image. (Figure from [42]).
An analysis was done in [42] and is shown in Figure 2.14. A simple coloring in of the
wall material in a TEM image shows the wall material comprises 55 10% and cell
interior 45 10%. Using these estimates, the average stress present in the sample is
4 MPa, which is smaller than the uncertainty of7 MPa.
2.2.3 Discussion
There are a number of important points to be made regarding these results. First, the
cell wall and interior stresses vary greatly from about -50% to 50% the applied stress.
Additionally the distribution of these stresses is asymmetric. Most of the LRIS present
is closer in magnitude to 10% the applied stress, with long tails extending into the larger
positive and negative stresses. Additionally the asymmetric distribution is mirrored from
68
cell interior to cell wall, as shown in Figure 2.13. The difference between the average
stresses in the walls and interiors is 40 MPa. This is roughly 20% the applied stress.
The confirmation of backstress in the cell interiors and forward stress in the cell walls
gives insight into the two LRIS models discussed previously. The pile-up model shown
in Figure 1.9 and first introduced by Seeger et al. [64] assumes dislocations pile up on
dislocation walls, however the sign and type of dislocations are random, and thus the
LRIS present in the cell interiors and walls can be of both positive and negative values.
Additionally under this model the stresses present in the cell interiors are expected to
sum to zero, and have a symmetric distribution about zero, with the same situation
existing for cell walls. Our results rule out this type of behavior.
The composite model, discussed at length in section 1.2.3, produces only back
stresses in cell interiors and forward stresses in cell walls. This is precisely the behavior
we observe in our results and is a strong indication that the composite model is accurate.
Remember that the composite model predicts dislocations of one sign (effectively) col-
lect on one side of a dislocation wall, while dislocations of the opposite sign collect on
the other side. Thus there is an effective net dipole character to the wall, which is respon-
sible for the LRIS present. In an effort to characterize this effect, a very simple model
was introduced by Levine [42] which has qualitative agreement to the data obtained,
although details of this model are beyond the scope of this thesis there is one final note.
The model predicts the walls only require a small number of effective dipoles, with a
density of 4.0 m
1
, to create the stress distributions seen here. This is an important
result, and is something that can be referenced when doing TEM analysis of dislocation
walls and when using other modeling approaches.
69
2.3 Sub-Profile Reconstruction Using Small Sample
Volume
A final important study can be performed using the cell interior and wall data obtained.
Similar to the reconstruction performed in section 2.1.5, the current data may allow a
reconstruction of not only a cell interior sub-profile, but a cell wall sub-profile as well.
In fact, because we have individual line profiles for both cell interior and wall elements,
an attempt at constructing an asymmetric line profile from the ground up provides very
valuable insight.
The first step is investigation of the amount of diffracting material which is statisti-
cally adequate. Figure 2.15a shows line profiles from 14 adjacent voxels. These X-ray
line profiles were obtained in a single energy-wire scan and are resolved spatially using
the platinum profiling wire illustrated in Figure 1.19. It’s clear that the line profiles
shown vary in shape and size, and are far from resembling a characteristic bulk X-ray
line profile as shown in Figure 1.16. Thus, the volume of a single voxel is inadequate
for this study. If we sum the 14 adjacent voxels, shown in Figure 2.15b, it appears to be
more representative of the bulk material. The summed voxel profile is smooth and shows
the characteristic asymmetry and deformation broadening expected of a monotonically
deformed single crystal Cu sample.
A direct comparison between an X-ray line profile from a bulk sample and the
summed voxel line profile is shown in Figure 2.16. The bulk line profile was obtained at
the Advanced Photon Source (APS) on beamline 33-BM at Argonne National Labs. The
sample was single crystal Cu deformed to roughly 30% along the [100] axis. The line
profile was obtained by rocking the sample through the [006] reflection and diffraction
data was gathered on a charged coupled device (CCD).
70
20
15
10
5
0
Intensity
104.8 104.4 104.0 103.6
Q (nm
-1
)
Voxel line profiles 150
100
50
0
104.8 104.4 104.0 103.6
Q (nm
-1
)
(a) (b)
Voxel line
profile sum
Figure 2.15: X-Ray Line Profiles of V oxels and Respective Sum
(a) The X-ray line profiles of 14 adjacent voxels taken in a single energy-wire scan.
Each voxel has a volume of 0.08m
3
. (b) The sum of the profiles, showing a smooth
asymmetric line profile similar to that of the bulk Cu sample.
Although the profiles are not identical, they are close considering the summed voxel
profile comes from only 1.04 m
3
of material. Most importantly the voxel-summed
profile is smooth and shows some of the asymmetry and peak broadening present in the
bulk line profile. Based on these assessments, this small amount of material will provide
insight when performing a sub-profile reconstruction.
2.3.1 Sub-Profile Reconstruction
The 14 voxels present were analyzed and a total of 14 cell interiors and 11 cell walls
were found. The cell interiors had strains ranging from 0 to 1:1 10
3
(0 to 73 MPa)
and the cell walls have a strain range of 0 to1:4 10
3
(0 to 93 MPa). The X-ray line
profiles for the cell walls are shown in Figure 2.17a and the cell interiors are shown in
Figure 2.17c. The profiles were interpolated and summed with the corresponding sums
shown in Figures 2.17 b and d respectively.
71
Q (nm
-1
)
Intensity
Bulk line profile
Sum of voxels
Figure 2.16: X-Ray line profile of bulk sample and voxel sum
The line profile from the sum of the 14 voxels is overlain on the bulk X-ray line profile.
The summed profile is not as asymmetric, but it is an excellent match when compared
with the individual voxel line profiles. It is smooth and exhibits some of the asymmetry
associated with uni-axial monotonic deformation.
The cell interior line profiles contain two profiles of high intensity. This is attributed
to a fairly large cell interior, and is confirmed upon inspection of the detector data. The
cell wall line profiles are not as smooth as the cell interior profiles due to the weak
signal to noise ratio, which can’t be avoided in this case. Surprisingly, considering these
factors and the small sample size, the sub-profiles in Figure 2.17 (b) and (d) are smooth
and fairly symmetric. The pseudo-V oigt functions fit well and give LRIS values of -15
MPa 7 MPa (strain of2:3 10
4
) for the cell wall sub-peak, and 34 MPa 7 MPa
(strain of 5:2 10
4
) for the cell interior sub-peak.
Pseudo-V oigt functions were used to fit the data, as the instrument resolution func-
tion was shown to be Lorentzian, and the diffraction spots are Gaussian. The convolution
of these two waveforms is a V oigt function.
Given the uncertainties, these numbers are actually in good agreement with the val-
ues in Figure 2.13. Not only the values given by the sub-peaks, but also the difference
between them is roughly 40 MPa (given the uncertainties). The sample set is very small
72
7
6
5
4
3
2
1
0
105.5 105.0 104.5 104.0 103.5
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Intensity
105.5 105.0 104.5 104.0 103.5 103.0
Q (nm
-1
)
Dislocation cell wall
line profiles
Cell wall line profile sum
(a)
(b)
12
10
8
6
4
2
0
104.6 104.4 104.2 104.0 103.8
Dislocation cell
interior line profiles
40
30
20
10
0
104.8 104.6 104.4 104.2 104.0 103.8 103.6
Cell interior line
profile sum
(c) (d)
Q (nm
-1
)
Q (nm
-1
) Q (nm
-1
)
Intensity
Figure 2.17: Cell Wall and Interior X-ray Line Profiles
(a) X-ray line profiles from 11 dislocation cell walls. (b) The sum of the wall profiles,
along with a pseudo-V oigt curve fit. (c) X-ray line profiles from 14 cell interiors. (d)
The sum of the cell interior and wall profiles and the corresponding pseudo-V oigt curve
fit (in dashed bold red). The number of dislocation cell wall and interior profiles is
small, yet the summed line profiles are fairly symmetric and fit well to the pseudo-V oigt
functions.
but this lends validity to the statistical legitimacy of the sub-peak reconstructions per-
formed here. On a final note, the low signal to noise ratio of the cell walls may mean the
statistics are somewhat lacking as the curve does appear slightly warped in Figure 2.18,
but this will be addressed in the subsequent section.
73
150
100
50
0
Intensity
105.0 104.8 104.6 104.4 104.2 104.0 103.8 103.6
Sum of voxel
line profiles
Cell Interior
subpeak
Cell wall subpeak
(a) (b)
150
100
50
0
104.8 104.6 104.4 104.2 104.0 103.8 103.6
Q (nm
-1
)
Reconstructed sum
Cell interior subpeak
Cell wall subpeak
Bulk Line Profile
55% Cell Wall
45% Cell Interior
Q (nm
-1
)
Figure 2.18: Sub-Profile Scaling and Sum
(a) The three summed line profiles from Figures 2.16 and 2.17. The profiles are shown
with their relative intensities to illustrate the small amount of cell wall and interior
diffraction compared to the total collected from the voxels. (b) The summed sub-profiles
for the cell interiors and walls are scaled up and adjusted to be in agreement with the
TEM micrograph in Figure 2.14. The wall sub-profile comprises 55% of the diffraction
signal, and the cell interior sub-profile 45%. This sum is then compared with the line
profile from the bulk sample.
2.3.2 Bulk Line Profile Reconstruction Using Sub-Profiles
Now that the sub-profiles have been established, a reconstruction of the bulk X-ray line
profile can be performed. Figure 2.18a shows the relative intensities of the cell wall,
cell interior, and voxel summed profiles. The varying intensities are clearly drastic, and
simply summing of the two sub-profiles would not be effective.
To scale the profiles appropriately, the ratio of cell wall to cell interior material was
used from Figure 2.14, where the walls comprise roughly 55% of the material, and
the cell interiors 45% (and thus these amounts of diffracted intensity). The results of
this scaling are shown in Figure 2.18 and compared with the bulk X-ray line profile.
Surprisingly, the reconstructed profile is fairly successful here. The peak is slightly
shifted and it is not a perfect fit, but it is quite favorable considering the small amount
of diffracting material used.
74
150
100
50
0
104.6 104.4 104.2 104.0 103.8 103.6
50% Cell Wall
50% Cell Interior
150
100
50
0
104.8 104.6 104.4 104.2 104.0 103.8 103.6
Q (nm
-1
)
Reconstructed sum
Cell interior subpeak
Cell wall subpeak
Bulk Line Profile
55% Cell Wall
45% Cell Interior
150
100
50
0
104.8 104.6 104.4 104.2 104.0 103.8 103.6
65% Cell Wall
35% Cell Interior
Intensity
Intensity
Q (nm
-1
) Q (nm
-1
)
(a)
(b) (c)
Figure 2.19: Profile Reconstructions Using Different Cell/Wall Ratios
The dislocation cell interior and cell wall sub-profile ratios were adjusted to create three
X-ray line profiles. They are then compared to the bulk X-ray line profile in an attempt
to see which ratio of cell wall to cell interior best fits.
The sample volume used in this study is quite small, at only 1.08m
3
, so there is a
possibility of the cell interior/wall volume fraction ratio varying from the TEM study. To
investigate this, the relative sizes of the cell wall and interior sub-profiles were adjusted
to two additional values, 50% wall and cell interior, and 65% wall 35% interior. The
results of these adjustments are displayed in Figure 2.19.
Based on the profiles reconstructed in Figure 2.19 the ideal ratio is roughly 45%
cell interior material and 55% cell wall material, which matches the TEM estimates.
Using this ratio, the constructed profile best fits the bulk X-ray line profile, although as
75
30
25
20
15
10
5
0
104.8 104.6 104.4 104.2 104.0 103.8
Q(nm
-1
)
Attenuation Corrected
Voxel Line Profiles
300
250
200
150
100
50
0
Intensity
104.8 104.6 104.4 104.2 104.0 103.8 103.6
Q (nm
-1
)
Uncorrected
Absorption
corrected
Figure 2.20: Attenuation Corrected V oxel Profiles
The left panel shows the attenuation corrected line profiles for the voxels. The right
panel shows the sum of these profiles overlain on the uncorrected sum (which has been
scaled up to match). They are essentially identical.
mentioned before, it is not an exact fit. Notably, this best fit ratio has wall material at a
higher percentage than what is usually assumed [47, 52, 73]. However, those studies did
not deal with such heavily deformed Cu samples, and most importantly, our sample was
oriented for deformation along the [001] axis, meaning the dislocation walls are thick
and fuzzy.
2.3.3 X-Ray Attenuation Correction
When adjusting the data for X-ray absorption and scattering using equation 2.4, there
is no difference in the shape of the voxel-summed line profile. Figure 2.20 shows the
attenuation corrected voxel line profiles on the left, and the sum of these profiles on the
right, as well as the uncorrected profile for comparison.
When correcting the dislocation cell interior and cell wall data, there is also no
difference seen. The summed profiles for the cell interiors and walls are shown in Figure
2.21 with the uncorrected profiles for comparison. Given that the measured strains don’t
76
80
60
40
20
0
Intensity
104.8 104.6 104.4 104.2 104.0 103.8
Q(nm
-1
)
Cell Interiors
8
6
4
2
0
105.0 104.5 104.0 103.5
Q(nm
-1
)
Cell Walls
Uncorrected
Attenuation Corrected
Figure 2.21: Attenuation Corrected Cell/Wall Profiles
The summed line profiles from dislocation cell interiors and cell walls compared with
the uncorrected line profiles. Again the uncorrected line profile has been scaled to match
the corrected profile.
correlate with depth (as was seen in Figure 2.4) this result is not particularly surprising.
It is worth noting that the small sample volume, small amount of cell wall data, and small
amount of cell interior data, provide sub-profiles that don’t change when accounting for
X-ray attenuation. This result lends support to the statistical validity of the study.
2.3.4 Sub-Profile Subtraction
Since the cell interior data is much easier to locate and measure, another method for
deducing wall strains may be by subtracting the cell interior sub-profile from the bulk
profile. Efforts made using this approach in the previous section were unsuccessful. A
more thorough study using many more cell interiors (and thus a smoother sub-profile)
may provide additional insight. This section investigates LRIS using this method.
Five energy wire scans were performed on a single trip to the Advanced Light Source
at Argonne National Labs. Using these 5 scans, a total of 86 cell interiors were extracted
for the purpose of this study (all data is shown in table A.4). The cell interiors from each
77
60
50
40
30
20
10
0
Intensity
104.6 104.4 104.2 104.0 103.8
Q(nm
-1
)
Individual scans
Cell interior sums
Figure 2.22: Measured Line Profiles and Cell Interior Profiles
Line profiles from five different energy-wire scans, and the sum of the measured cell
interiors from each wire scan.
scan were then summed, and compared with the sum of each wire scan. This data is
shown in figure 2.22. Notice the measured cell interior diffraction profiles are of various
sizes and shapes, with each one consisting of between 13-21 individually measured cell
interiors. Notice also that the bulk profiles from each wire scan exhibit different shapes
as well (although when scaled, they are surprisingly similar as was shown in Levine et
al. [42]).
The data was then summed and is shown in figure 2.23. The percentage of cell
interior diffraction compared to the summed wire-scans, is not appreciably different
than in the previous section where only a single energy-wire scan was used. The cell
interior sub-profile is noticeably more smooth due to the improved statistics.
Next the cell interior data was scaled up to 45% of the diffraction intensity, and
subtracted from the measured bulk line profile (which is again a sum of the 5 energy-
wire scans). The result is shown in figure 2.24. Finally, the calculated wall profile is
shown in figure 2.25 with a corresponding pseudo-V oigt fit.
78
200
150
100
50
0
Intensity
104.8 104.6 104.4 104.2 104.0 103.8 103.6 103.4
Q(nm
-1
)
Bulk X-ray line profile
Cell interior subpeak

Figure 2.23: Summed Energy-Wire Profile and Summed Cell Interior Profile
The line profiles coming from the sum of the 5 energy-wire scans, and 86 cell interiors.
The cell interior has a peak location of q = 104:227 with a strain of 5:6 10
4
and stress of 37 MPa 7 MPa. This is consistent with the previous section, although
slightly higher. The subtracted profile attributed to the dislocation cell wall, has a peak fit
location ofq = 104:296 meaning a strain of 110
4
and stress of -6 MPa. The location
of the maximum of the wall sub-profile curve has a higher strain of3:5 10
4
and
stress of -23 MPa.
All of these results show that creating of a wall sub-profile using this method is only
somewhat successful. When analyzing the wall sub-profile stress, the peak position
is ambiguous. The data suggests average wall stresses are somewhere between 6-23
MPa, which is consistent with previous findings. This study therefore strengthens the
previous findings, although it does not provide any new information. It’s important to
note however, that new findings are not the primary purpose of this section. The intent is
more to validate the measured wall sub-profile from the previous section, as the amount
of wall diffraction collected is very small compared to cell interior values. In fact, figure
79
200
150
100
50
0
Intensity
104.8 104.4 104.0 103.6
Q(nm
-1
)
Bulk profile
Cell interior profile
Difference (wall)
Figure 2.24: Bulk Line Profile and Cell Interior and Wall Sub-Profiles
The bulk line profile from the summed energy-wire scans, along with the cell interior
sub-profile. The difference between the two is shown and should be representative of
the wall sub-profile.
2.25 compares the summed wall sub-profile with the subtraction sub-profile, and shows
a good agreement between the two.
2.3.5 Discussion
When cell interiors are analyzed, the line profiles often exhibit some small amount of
asymmetry in the tails. This is generally believed to be caused by overlapping wall
diffraction data, although there is simulated evidence some may be caused by Burger’s
vector polarization [14]. In fact, the most difficult issue in collecting wall data, is it’s
relatively diffuse and weak diffraction. The low signal strength is very often completely
covered by the more intense cell interior diffraction. This problem in and of itself cre-
ates a motivation for attempting sub-peak reconstructions. The dislocation cell interior
data is extremely reliable. The peaks are sharp and smooth with only a small amount of
wall diffraction present. This small amount causes a very slight asymmetry to the profile
80
140
120
100
80
60
40
20
Intensity
104.8 104.6 104.4 104.2 104.0 103.8 103.6
Q(nm
-1
)
Wall subprofile
Pseudo-Voigt fit
Summed wall sub-profile
Figure 2.25: Subtracted Wall Sub-Profile
The wall sub-profile along with a pseudo-V oigt fit. The peak location of the fit curve
shows an average stress of -6 MPa, while the peak of the sub-profile has an associated
stress of -23 MPa.
which can be seen in Figure 2.1 in the tails of the line profile, but doesn’t affect the mea-
surement of the strain present in the cell interior. However, the cell wall data collected is
from comparatively very weak signals, as can be seen in Figure 2.18a. Therefore what a
sub-profile reconstruction provides in this study, is validation of the wall strain data. If
a reconstructed bulk line profile can be made accurately using dislocation wall data, it
shows the weak cell wall measurements are representative of the wall strains in the bulk
material.
In the study of summed sub-profiles presented here, the wall data is incredibly weak
when compared to the cell interior data and the summed voxel profile. However when
scaling up the signal a reconstruction of the asymmetric bulk X-ray line profile was sur-
prisingly successful. The main flaw is perhaps a slight undulation preventing a perfect
fit. This is likely due to too few cell interior and/or wall profiles being measured.
The sub-profiles reconstructed (both summed and subtracted) do show good agree-
ment with the earlier statistical studies done in section 2.2 which uses all of the cell wall
81
and interior data measured. The strains associated with the sub-profiles are in relatively
good agreement. This again seems to suggest that the sub-profiles provide a good shape
and proper asymmetry to the re-constructed bulk X-ray line profile.
Perhaps most importantly, the sub-profiles can be compared with earlier studies,
which attempted to extract the sub-profiles from the bulk line profile [47, 52, 74, 75].
The end results for these studies (shown in table 1.3) were cell wall and cell interior
strains that are too large. They were later lowered to strains of 0.4 the applied stress
in the cell walls, and -0.1 the applied stress in the cell interiors. Part of this is due
to different assumptions regarding cell interior/wall ratios. In Mughrabi et al. [52],
the resolved sheer stresses applied to the samples ranged from 26-75 MPa (75 MPa
resolved shear stress is 185 MPa applied stress along the [001] axis), and the volume
fractions of the walls ranged from 15% 30%, which is quite different than the
55% wall ratio estimated here. Essentially, the best way to assess the validity of the
decomposition of a bulk X-ray line profile into two sub-profiles, is by performing a sub-
profile decomposition on the bulk line profile presented here, and compare it directly
to the constructed sub-profiles. This was performed by Levine et al. [41] and will be
discussed in the next section.
The study by Levine [41] used all of the available cell wall and interior data shown in
Figure 2.12 in an effort to reconstruct the bulk X-ray line profile of the sample. Instead
of using the raw line profile data as was used in the studies presented here, the line
profiles for the cell interiors and walls were fit with pseudo-V oigt functions and these
fits were used in the sub-profile sums. What follows is a summary of the results from
this broader reaching study.
82
Figure 2.26: Bulk Line Profile Reconstruction
The summed pseudo-V oigt line profiles from dislocation cell interiors and cell walls
create the two sub-profiles shown. The sub-profiles are then summed to create the solid
black line, which is then compared with the measured sample bulk X-ray line profile.
The Figure is from Levine et al. [41].
2.4 Sub-Profile Reconstruction Using Large Data Set
A total of 97 cell interior and 47 cell wall line profiles were used. The peaks were
approximated using pseudo-V oigt functions and the amplitudes were adjusted for X-ray
attenuation. The instrument resolution function was not deconvolved as it is very small
compared to the profile widths (q=q = 6 10
4
). The profiles were then summed to
create two sub-profiles. These were then scaled so that the wall sub-profile consisted of
55% of the diffraction intensity, and the cell interior consisted of the other 45%. This
cell/wall ratio is the same as that of the TEM micrograph shown in Figure 2.14. The two
sub-profiles were then summed and compared with the line profile from the bulk X-ray
line profile, as shown in Figure 2.26.
The constructed profile provides an excellent fit to the bulk X-ray line profile. The
sub-profiles are smooth and symmetric, and the sum shows the characteristic asymmetry
83
caused by deformation. A number of different cell/wall ratios were calculated, and it
was found that the ratio of 55% cell wall to 45% cell interior was the best fit [41]. It is
also worth noting that the symmetry of the sub-profiles shown here, is much better than
in the small volume study. This is due to both the larger number of cell and wall data
used, and the fact that the profiles were pseudo-V oigt fits. As can be seen in Figure 2.17
b and d, there are slight asymmetries at the base of each sub-profile, which is due to the
issues discussed previously in section 2.3.5.
2.4.1 Sub-Profile Decomposition Comparison
When deconstructing an X-ray line profile into two sub-profiles, there are an unlim-
ited combination of sub-profiles which can be used. To narrow down the possibilities,
Mughrabi et al. [52] provide the following procedure. The profile is first mirrored
around it’s peak. The mirrored tail is then subtracted from the other tail, which decays
more slowly. The resulting peak is then mirrored to obtain a symmetric sub-profile,
considered to be the wall sub-profile. This profile is then subtracted from the bulk line
profile, and is attributed to the cell interior sub-profile. There are also a few assump-
tions made to better refine this method. First, the profiles are symmetric, and second,
the profiles are proportional to the cell wall/interior material in the sample.
The sub-profiles in Figure 2.26 are symmetric, and scaling them based on the
cell/wall volume fractions found in the TEM sample, is consistent with the two cri-
teria of the decomposition method. In this case, the main factor which contributes to
the symmetry of the sub-peaks is the FWHM of the individual cell wall and interior line
profiles. The FWHM of the distribution of stresses for the cell interiors is roughly 0.03
nm
1
, while the average FWHM for the individual line profiles is four times larger, at
0.12nm
1
. The cell walls have a FWHM for the stress distribution of 0.04nm
1
while
the average line profile FWHM is 0.34nm
1
[41]. Essentially, the line profiles are much
84
Figure 2.27: Summed Sub-Profiles vs. Extracted Sub-Profiles
The summed pseudo-V oigt line profiles from dislocation cell interiors and cell walls
create the solid line sub-profiles shown. The extracted sub-profiles using the mirrored
iterative technique are shown with the dotted lines. The Figure is from Levine et al.
[41].
larger than the shifts in the line profiles due to the LRIS present. This is only partly in
agreement with the decomposition methods, as they assumed all broadening was due to
dislocations, and not strain variations.
Sub-profiles were obtained using the mirroring method described previously (with
a second iteration done, a third iteration produced essentially no difference), and are
shown in Figure 2.27. The profiles are displayed along with the sub-profiles constructed
from the sum of the individual cell wall and interior line profiles. There is a clear
difference in the shape and size of the two sets of profiles. The important effect here
being that the difference between the cell interior and wall stresses is overestimated by
a value of 2.3, compared to the ones measured here [41].
85
2.4.2 Discussion
The constructed sub-profiles, when scaled for volume fractions measured using TEM
data and then summed to create the bulk X-ray line profile, provide an excellent fit.
These sub-profiles show that LRIS measurements using other methods of decomposition
clearly overestimate the LRIS present. Additionally, the excellent if of the constructed
bulk X-ray line profile further reenforces the validity of the microbeam measurements
used to create the sub-profiles.
The main problem with the decomposition method discussed here is the initial
premise. The asymmetry on the steep side of the bulk line profile is assumed to come
from the cell interior sub-profile, while the more gently sloped side is a result of the
wall sub-profile. This premise is ultimately flawed. When looking at Figure 2.26 it’s
clear the left side of the line profile is a combination of both sub-profiles. This makes it
essentially impossible to know (without measuring LRIS explicitly within cell interiors
or cell walls) what the two sub-profiles should look like [41].
Finally, the goal of producing robust dislocation cell interior and cell wall sub-
profiles has been achieved. The earlier study used a small sample volume, with a small
number of cell walls and cell interiors, while this study used a large amount and obtained
essentially identical results. Perhaps more importantly, the Levine et al. [41] study
assessed the validity of the long used decomposition method detailed in section 2.4.1.
It’s now known that TEM studies must be done to assess the correct volume fractions of
cell interior and wall material present. Even with this information, it is still impossible
to assess the values of the LRIS present in the sample.
The next hurdle is to obtain full strain tensors, instead of simply uniaxial strains in
the Cu samples. The next section deals with current efforts to measure full strain tensors
within dislocation cell interiors in deformed single crystal Cu samples.
86
2.5 Full Strain Tensor Measurements
An important detail in all of the LRIS measurements performed, is the fact that they are
only uniaxial measurements. That is, only one component of the strain tensor is being
measured,
33
, the one along the axis of deformation. In principle, the current setup
should allow a measure of the complete strain tensor. What follows in this section is
some background and then a detail of the methods employed to measure the full strain
tensor.
2.5.1 Background
To understand the following sections, it is first important to understand what a strain
tensor is, and the different forms it can take. The general meaning behind the strain
tensor, is to provide a three dimensional mathematical description for deformation. That
is, given a point A in a body, if the body is moved and A moved within the body, the
strain tensor describes the local deviation of A within the body. Essentially, the motion
of A is broken into pieces. There are rotational and translational components (for the
body movement), and a strain tensor (for the motion of A within the body).
In elastic theory, there are two strain tensors we generally start with, the Lagrangian
and Eulerian strain tensors. Generally speaking the Lagrangian (Green) tensor is defined
in terms of the undeformed coordinates, while the Eulerian (Almansi) tensor is defined
in terms of the deformed coordinates.
The Lagrangian tensor is well suited for material deformation, and is commonly
defined as,

ij
=
1
2

u
i
a
j
+
u
j
a
i
+
u
k
a
i
u
k
a
j

; (2.5)
87
whereu is the vector from the initial point to final point, anda is the vector from the
origin to the initial point. However our strains are quite small, thus we can neglect the
second order term, leaving only,

ij
=
1
2

u
i
a
j
+
u
j
a
i

; (2.6)
which is fundamentally the same as the Eulerian tensor for small strains, and is called
the infinitesimal strain tensor.
Once this is solved the strain tensor is produced and is of second order and sym-
metric, meaning there are 9 components, of which only 6 are unique. The tensor in it’s
general form is written as,

ij
=
0
B
B
B
@

11

12

13

21

22

23

31

32

33
1
C
C
C
A
; (2.7)
wheree
12
= e
21
,e
13
= e
31
, ande
32
= e
23
. In all of the previous experiments, we have
only been measuring thee
33
component. It’s simple to see that this is only a part of the
picture when measuring LRIS.
A minimum of three Laue spots are needed to solve for the complete strain ten-
sor. The strain tensor contains six unknowns, so six independent measurements must be
taken. This can be achieved by knowing the lattice parameters of three linearly inde-
pendent sets of crystallographic planes. In addition, the angles of the diffracting planes
must be known. Once the lattice parameters and angles are known for the measured
Laue spots, the lattice parameters and angles of the unit cell can be solved, and from
this the full strain tensor. The equations used to solve for the (Linear Lagrange) strain
tensor elements are,
88
a
1
b
1
c
1
!1
!1
"1
Unit cell
Figure 2.28: Unit Cell
The unit cell showing the relationship between the lattice parametersa,b, andc and the
angles,, and
.
2
6
6
6
4

11
=
a
1
a
0
sin
1
sin
1
1
12
=
21
=
a
1
2a
0
sin
1
cos
1

13
=
31
=
a
1
2a
0
cos
1

22
=
b
1
b
0
sin
1
1
23
=
32
=
b
1
2b
0
cos
1

33
=
c
1
c
0
1
3
7
7
7
5
;
(2.8)
where a
0
, b
0
, and c
0
are the unstrained lattice spacings of the unit cell, a
1
, b
1
, and c
1
are the measured lattice spacings, and;; and
are the angles between the respective
vectors [61]. Figure 2.28 is an illustration of these relationships. Note that angle is
the angle between lattice vectorsb andc, while is betweena andc, and
betweena
andb.
2.5.2 Experimental Procedure
The first method employed when attempting to measure the full strain tensor of a dislo-
cation cell interior, was energy-wire scans of three linearly independent Laue peaks. A
basic diagram of the new setup is detailed in Figure 2.29. The two side detectors were
not operational, so only the (new) amorphous Si main detector was used to gather data.
89
Amorphous Si Detector
X-Ray Microbeam
Profiling Wire
Figure 2.29: Strain Tensor Experimental Setup
The left panel shows the experimental setup with three new detectors installed. Only the
main detector was operational at the time of this experiment. The right panel is the Laue
pattern coming from our sample incident on the main detector. The three circled Laue
spots are the ones used in our energy-wire scans.
A sample pattern is shown to the right of the experimental setup, and the Laue pattern
can clearly be seen, as well as various peaks within each diffraction spot from the dislo-
cation cell interiors. The circled Laue spots are the three for which detailed diffraction
data was taken.
After acquiring the diffraction data, a few issues became apparent. The wire, which
is used to give depth resolution, is not straight. This translates to small depth offsets
for each Laue spot that does not hit the exact center of the wire. The calibration of the
experimental setup is done using only the center of the wire, first with a single crystal
Si sample, and then single crystal Cu. For the side peaks measured, there turned out to
be small offsets of -0.5m for the [028] and +0.5m for the [

208]. The depth offsets
were fairly easily deduced by looking at when the diffraction intensity first occurs on
90
the detector as the wire is moved. This is not, however, an ideal situation and will be
discussed more later.
An additional problem, is the close proximity of the Laue spots to one another. It is
an advantage when dealing with the non-straight profiling wire, but unfortunately, the
smaller angles between the Laue spots generate larger uncertainties in the calculation of
the strain tensor. Given that the two side detectors were non-operational at the time of the
experiment, the peaks chosen were the widest choice possible. They had an additional
bonus, in that the data for each could be collected simultaneously, as the X-ray energies
needed to meet the Bragg condition were the same.
2.5.3 Results and Discussion
A total of 26 sample volumes were investigated and diffraction data collected from the
three Laue spots mentioned previously. A total of 7 dislocation cell interiors were found,
with some big enough to span 3 depth steps (1.5m). A problem which is not insignifi-
cant, is the correlation of cell interior diffraction data from each Laue spot. In this scan,
the issue was avoided, as only one or two cell interiors were diffracting at any given
depth. Additionally there were easily recognizable patterns to the diffraction data. Each
cell interior was followed by another, so even with two cell interiors diffracting in the
same voxel, they were depth resolved the step before, where only one was present. All
of the cell interior data extracted from the diffraction data can be seen in appendix A.2.
From this data, and using equation (2.8), we can calculate the 6 independent strain
tensor components. Table 2.1 shows the calculated strain tensor components for all of
the dislocation cell interiors.
Many of these strain values are unreasonably large, so measurement uncertainties
must be considered. The uncertainties originate from two sources, the strain uncer-
tainty, and the angular uncertainty associated with the center of the diffraction peak.
91
Table 2.1: Calculated Strain Tensors
e11 e22 e33 e12 e13 e23
1.216E-03 1.252E-03 4.700E-04 -7.922E-04 2.380E-05 1.325E-03
8.940E-04 1.252E-03 4.700E-04 -7.919E-04 2.380E-05 1.325E-03
2.772E-03 1.076E-03 5.800E-04 1.080E-04 3.969E-04 1.376E-03
3.075E-03 3.168E-04 5.800E-04 -3.660E-05 3.700E-04 1.340E-03
4.083E-03 -1.063E-04 5.500E-04 8.109E-04 5.418E-04 1.470E-03
1.311E-03 2.236E-03 3.900E-04 -9.670E-05 -3.256E-04 1.795E-03
7.478E-04 2.236E-03 4.100E-04 4.140E-05 -3.254E-04 1.767E-03
6.369E-03 -1.212E-02 6.000E-04 -4.936E-03 2.610E-04 3.347E-03
2.157E-02 -3.366E-03 8.700E-04 5.200E-03 2.442E-03 2.467E-03
3.368E-03 2.525E-04 6.700E-04 -3.564E-04 -7.340E-05 1.749E-03
3.451E-03 -3.318E-04 6.400E-04 -6.958E-04 2.520E-05 1.881E-03
3.856E-03 -5.675E-04 6.200E-04 -6.604E-04 -7.040E-05 1.929E-03
Strain uncertainties were first calculated for each vector based on the diffraction angles
(cot()d), and then broken into cartesian components. These were then propagated to
the calculated lattice parameters. The uncertainty from this, without the angular uncer-
tainties, was large enough to make the strain tensor calculations unreliable.
The uncertainties calculated were a
1
=a
0
=2 10
3
, b
1
=b
0
=2 10
3
, and
c
1
=c
0
=2 10
4
. The c
1
=c
0
is fairly good, as it was directly measured using the
[006] peak. Unfortunately the other uncertainties are too large to provide meaningful
data. This does, however, provide a proof of concept. The ability to measure full strain
tensors using the current setup is possible, given some refinements.
A second much more thorough attempt was made using this method, yet another
issue arose. The energy-wire scans take a long time and there is a small amount of
thermal drift present. In the previous experiment this was not an issue, as only two
scans were done. One for the [006] peak, and another which included both the [024]
and the [

204] peaks. The reason they can not all be done in a single scan, is due to
the scanning of the X-ray energies and necessity for meeting the Bragg condition. It
92
is also worth noting that the undulator tracks along with the monochrometer energies,
to maintain the highest intensity X-ray beam possible. Thus if the energies are too far
apart, a different harmonic of the output energies must be chosen.
Our second attempt contained the Laue peaks [006]; [117]; [1

57]; [

406]; [

315], and
[

519]. These reflections give more reasonable uncertainties, as they provide large angu-
lar differences, and more than three peaks reduces the uncertainties even further. Unfor-
tunately, this requires a number of energy-wire scans to be performed.
Upon analysis of the diffraction data, it was apparent that the sample drift was sim-
ply too large for the beam to stay in one location over the period of time required. Thus
correlation between a large number of energy wire scans becomes impossible. A choice
must be made, a thermal control upgrade of the X-ray hutch, or a different approach. The
new approach is both much faster, and provides more reasonable experimental uncer-
tainties.
2.6 Deviatoric and Hydrostatic Strain Tensor
The new method for obtaining the full strain tensor contains two parts. First, obtain
the dilatational element of the strain tensor, and second, the deviatoric element. The
deviatoric strain tensor gives the angles between the lattice parameters and the relative
sizes of the lattice parameters, while the dilatational strain tensor gives the volumetric
change in the unit cell. Figure 2.30 illustrates these relationships and how they relate.
Finally, combining the deviatoric strain tensor and the dilatational strain tensor gives the
full strain tensor.
Experimentally, this translates into two separate experimental measurements. First,
an energy-wire scan must be done to obtain the hydrostatic (dilatational) element of
the complete strain tensor, and second a white beam wire scan must be performed in
93
Shear strain Relative Strain Volumetric Strain
Deviatoric Strains
Dilatational/Hydrostatic
Strains
!"
!(a/b)
a
b
!a/a= !b/b
Figure 2.30: Deviatoric and Dilatational Strain Tensor Illustration
Three types of strain. Pure shear strain and strain which changes the relative lengths of
the lattice parameters are described by the deviatoric strain tensor. V olumetric strain is
described by the Hydrostatic strain tensor.
the same location, to obtain the deviatoric strain tensor. Of course, once the first step
is completed, the sample will have shifted slightly due to a small amount of thermal
drift. There is also a small offset in position between the monobeam and the polybeam
with thermal drift being a smaller factor. Therefore the sample is re-aligned using live
white beam diffraction from the synchrotron and is compared to a reconstructed white
beam image from the previously performed energy-wire scan. Using this technique, the
two are compared and the sample is adjusted until we are completely sure the locations
are identical. A white beam wire scan is then performed on the sample. Finally, it is
important to point out that because the white beam scan does not need a scan of energies,
it requires only 1-2 hours (as opposed to 6-8 hours).
Although this experiment has been attempted twice, it has been unsuccessful due to
one final issue. The wire used to provide depth resolution is not perfectly straight. This
was not an issue in the earlier experiment as the angle between the diffracting beams
was not large, and compensating for the depth offsets could be done. The diffraction
94
spots were also bright, so discerning the offsets was simple. With the new method,
some Laue spots do not diffract strongly so solving for the depth offsets is non-trivial
and in some cases impossible. To further exacerbate the problem, the angles are larger
and the depth offsets are larger. Finally, to complicate matters further, there are multiple
cell interiors diffracting at any one time, thus correlation of the cell interiors becomes
essentially impossible when the depth information is not reliable.
To proceed with the experiment a ’perfectly’ straight wire must first be manufactured
and installed at beamline 34-IDE. This is currently in the process of being done, but is
not yet complete. Once complete, it will ideally allow for the measurement of full strain
tensors within dislocation cell interiors in our samples. This project is certainly the next
step in the evolution of LRIS experimentation and exploration.
95
Chapter 3
Dislocation Dynamics Simulation
The experimental results presented thus far give a good understanding of the magni-
tude and distribution of LRIS in single crystal copper uni-axially deformed along the
[100] axis. An important step in exploring the subject matter further is to examine the
dislocations present in the heterogenous microstructure. The Composite Model accu-
rately describes the experimentally observed forward and back stresses which exist in
heterogenous dislocation microstructures. The model also makes predictions regard-
ing dislocation motion, deposition, and the developing microstructure. Examination
of the dislocation microstructure using dislocation dynamics simulations (DDS) may
potentially verify the composite model’s predictions, as well as provide validity for the
modeling approach used. This chapter will present a background on DDS, followed by
results from DDS of a dislocation cellular microstructure in copper.
3.1 MicroMegas (mM)
All modeling was done using the DDS software microMegas, which is developed at the
Laboratoire d’Etude des Microstructures (LEM) under the leadership of Benoit Devincre
and Ladislas Kubin in Chˆ atillon, France. A brief summary of the modeling software is
discussed here, and extensive details can be found in Devincre et al. [7, 8].
96
3.1.1 Sub-Lattice
There are two fundamentally different approaches used in DDS today. One model cal-
culates dislocations on a continuum, while the other uses a discrete sub-lattice. The
continuum allows for more accurate modeling of dislocation cores and details of ele-
mentary configurations, while discrete lattices are more computationally efficient and
therefore better suited for larger scale simulations and higher dislocation counts. The
DDS software microMegas is of the second kind (although somewhat of a hybrid), using
a sub-lattice, or nodal structure on which it models dislocation interactions and stress
calculations. The actual positions of the dislocations are modeled on a continuum, and
this will be discussed more later.
The model discretizes dislocations onto the underlying sub-lattice using the follow-
ing mathematical approximations. Each slip system contains eight fundamental disloca-
tion types, two edge dislocations, two screw dislocations, and four of mixed character.
These eight can describe a dislocation loop as an octagon in the smallest sense, but the
scaling of the underlying sub-lattice can be such that a single Burgers vector is larger
than multiple sub-lattice translation vectors. Thus dislocation loops can be described
well using the eight dislocation types depending on the scaling. The scaling is a consid-
eration when establishing a simulation of a specific size and scope. In fact, the scaling
can be changed such that results can be compared to atomistic simulations. The code
is also able to simulate glissile dissociations, however it was not implemented for the
simulations presented in this thesis.
A major advantage of lattice based DDS, is tabulation of calculations and data,
which reduces the number of operations and improves computing performance. The
finite number of dislocation segment directions mean there are a finite set of stress field
calculations. Additionally most of the calculations are carried out on integers and are
therefore exact, and avoid non-rational fractions. To benefit from the tabulations, it is
97
critical the setup of the simulation sub-lattice and the associated displacement vectors
and lengths be considered. There are currently tabulations for many different crystal
structures including f.c.c., which is needed for copper.
3.1.2 Force
The configurational force calculated onto each dislocation line segment is known as the
Peach-Koehler force and is expressed byf
PK
= (b), whereb is the burgers vec-
tor and is a unit vector tangent to the dislocation line. The is broken up in to four
contributions,
applied
the applied stress,
int
the stress from the dislocation microstruc-
ture,
line
the stress from the line tension, and in some cases a fourth image correction
stress from surfaces or interfaces.
The calculation of these forces is the most CPU intensive procedure, and some opti-
mizations are used to make the simulation code more efficient. First, if the simulation
volume has periodic boundary conditions, then only one or two periods are used for the
calculation of these forces. Similarly, in the results discussed later in this chapter, only
one period was used in calculating the LRIS, as additional periods produced negligible
change. Secondly, contributions from local dislocations are calculated explicitly, while
dislocations which are farther away are compartmentalized into domains. The net stress
for the domain is calculated at it’s center, and then this is used for calculations on dis-
locations which are farther away. This optimizes the stress calculations, which would
otherwise be prohibitively large in number.
3.1.3 Simulation Time and Dislocation Motion
With the forces calculated, the next step is assessing the motion of the dislocations,
where calculated velocities and simulation time steps must be considered. A line seg-
ment velocity is determined locally by a balance between the Peach-Koehler elastic
98
forces and a resistive dissipative force. The resistive force is derived from thermally
activated motion, phonon drag, and other elementary phenomena [8]. Thus, the veloc-
ities are heavily material dependent, so different velocity laws are developed for each
material.
The discrete nature of lattice based simulations may create an issue with slow mov-
ing dislocations. If a dislocation moves too slowly, it is possible that it become artifi-
cially pinned. MicroMegas overcomes this obstacle by having the position of all dis-
locations on a continuum in real space. In fact, it is only the force calculations and
detection of intersection reactions that use the discretized positions. Additionally, the
force calculations for slow moving dislocations are not done every time step, but are
done less frequently depending on how slow they are moving, to further optimize the
force calculations.
As two segments approach each other, the interaction strength increases rapidly. If
the dislocations attract, then faster velocities are expected. Fast velocities require a
reduction in the time step, as well as the computing efficiency. To cope with this, there
is a maximum glide allowed, as well as a saturation stress.
More information on the specifics of microMegas can be found in Devincre et al.
[7, 8], but the brief summary provided here is sufficient for understanding the following
simulation results and discussion.
3.2 Background
Efforts to model LRIS in dislocation microstructures present in f.c.c. materials in multi-
slip deformation has been successfully done in earlier studies [5, 43]. In these studies,
dislocation dynamics (DD) simulations were used to reproduce the genesis of a disloca-
tion microstructure at small plastic strain ( 2 10
3
). The simulations were seeded
99
with dislocations, and, after a small amount of plastic strain, a loose heterogenous dislo-
cation microstructure was observed. However, it’s important to note that no simulation
at this point can produce a well developed dislocation microstructure as is present in
Cu deformed by 30%. The strain requirements are too large, and the amount time
required to produce such a simulation is prohibitively long.
Dislocation patterning is a systematic phenomenon observed in DD simulations and
is produced by the short-range and contact interactions between dislocations [6, 12]. It is
not entirely clear if LRIS contribute to the formation of dislocation patterning, however
they are found to appear in dislocation structures like the heterogenous dislocation cell
structures. In the latter case, LRIS has been found in qualitative agreement with the
composite model predictions, meaning back-stresses and forward stresses appear in the
expected configurations [43].
For the present calculations, an original shortcut procedure was used to quickly
develop a copper dislocation cell structure as close as possible to the one discussed
in the previous chapter (i.e. Cu single crystal deformed by 30%, or to a final flow
stress of roughly 150-200 MPa). The cell structure, once formed, was then deformed
in tension at a flow stress of roughly 150 MPa, similar to that used in the microbeam
measurements. During deformation, the fully developed dislocation microstructure was
analyzed to examine plastic strain, internal stress, and dislocation movement and con-
figuration.
3.3 Methodology
Rectangular prism simulation volumes containing only a single dislocation wall were
first attempted. The simulated volume quickly rotated such that one slip system became
dominant and made the simulation meaningless. Note that there are no ’grips’ in the
100
simulation, so the system can rotate under stress towards the slip system which provides
the lowest deformation stress. A subroutine was implemented which torques the simu-
lated sample slightly, in an attempt to prevent the rotation, but the small single walled
sample volume was simply too unstable. The second geometry attempted was a rectan-
gular volume containing two dislocation cell walls. Once again, it rotated and was too
unstable.
Finally, a larger more complete structure was used. Although it increased the amount
of computing time needed, it was the smallest stable volume element. The simulated
copper volume element was roughly the size of one dislocation cell interior/wall volume
with dimensions 0:43 0:3 0:33m and periodic boundary conditions. This volume
was first seeded with a random distribution of dislocation prismatic loops on four active
slip systems with an initial density of 4 10
14
m
2
. Only four slip systems were used
because TEM work on deformed single crystal Cu shows that this is generally the case
[32, 33]. Then, to quickly generate a starting cell structure, a tensile load of 200 MPa
was applied to only a section of the total simulation volume, as illustrated in figure
3.1. During this step, dislocation interactions were ’turned off’ in the loaded region,
thus allowing the dislocations to glide out of the center of simulated volume and form
dislocation cell walls on the borders, where interactions and junctions occurred. This
process created a single dislocation cell interior with dislocation dense cell walls on 3
sides, and a cell interior to cell wall ratio of 45% and 55% respectively. This ratio is
similar to the TEM work discussed previously. Finally, due to the periodic boundary
conditions, the cell interior is actually surrounded on all sides by dislocation cell walls.
Once the cell structure was formed, all interactions in the simulated volume are
turned on and the dislocation microstructure was relaxed at zero applied stress to check
for stability. The numerous junctions and joints successfully created a stable 3D dislo-
cation network. At this point the total dislocation density has increased to 1 10
15
m
2
.
101
Random distribution of dislocation dipolar loops
(periodic boundary conditions create a full cell structure) 0.43 x 0.33 x 0.3 microns
Figure 3.1: Dislocation Cell Forming Stress
A localized stress was applied to the initial random distribution of dislocations. Dis-
location interactions in the stressed area were turned off allowing the dislocations to
glide freely out of this zone. The resulting microstructure exhibits a well developed
dislocation microstructure.
Slices of the microstructure resulting from this initial shortcut are shown in figure 3.2
from the [100] and [111] directions.
With the microstructure clearly developed, a tensile deformation force was applied to
the structure along the [001] direction. The simulation attempted to deform the sample
at a constant strain rate, and due to the dislocation content, a stress of 150 MPa was
applied. Figure 3.3 is a graph of the stress applied during the simulation. The stress
fluctuated and increased over time as the dislocation density increased. During the entire
simulation run, the cell structure remained stable.
3.4 Results
Figures 3.3 - 3.5 show the simulated stress/strain relationship, the dislocation density
on the four pre-populated slip systems, and the plastic strain per slip system. When
the simulation initially begins to ramp up the applied stress, the dislocation structure is
102
100 nm
Figure 3.2: Initial Dislocation Cell Structure
Slices of the initial dislocation cell structure. The first is viewed along the [100] axis,
with the [100] wall removed. The second is a slice viewed perpendicular to the the [111]
axis. Due to the periodic boundary conditions, the walls are only required to be on one
side of the simulation box in each direction. A clearly defined cell interior is present.
coming from a relaxed state, and thus there is a general re-organization of dislocations
in the wall, which accounts for the initial drop in dislocation density seen in figure
3.4. After approximately 0.04 strain, the dislocation density increases. The other slip
systems available in the simulation (12 total slip systems) had essentially no contribution
to the strain, as the dislocation densities were zero.
In the initial setup, there was an equal dislocation density on each of the four slip
systems, however in the the process of forming the dislocation cell microstructure, the
dislocations multiplied at different rates. The larger dislocation densities on slip systems
1 and 10 have an obvious effect on figure 3.5 making them responsible for the majority
of the strain achieved in the simulation.
As the developed cell structure began to deform, emission of mobile dislocations
from the cell walls was observed. These dislocations travel through the cell interior,
and are deposited on the opposite side either at the wall interface, or inside the wall.
Concurrently, the dislocation arrangement adjusted in the walls, but only occasionally
103
150
100
50
0
Applied Stress (MPa)
0.20 0.15 0.10 0.05 0.00
Strain
Figure 3.3: Simulated Stress vs. Strain
The stress vs. strain of the simulated volume. The simulation adjusts the applied stress
to produce a steady rate of deformation
450x10
12
400
350
300
250
200
Dislocation Density
0.20 0.15 0.10 0.05 0.00
Strain
Slip System 1
Slip System 3
Slip System 10
Slip System 11
Figure 3.4: Slip System Dislocation Density
The dislocation density on the 4 active slip systems. After 0.10 strain, the dislocation
densities begin to slowly increase.
104
-3x10
-3
-2
-1
0
Gamma Per Slip Plane
0.20 0.15 0.10 0.05 0.00
Strain
Slip System 1
Slip System 3
Slip System 10
Slip System 11
Figure 3.5: Strain(
) Per Slip System
The strain per active slip system. The slip systems with more dislocation density are
responsible for a larger percentage of the overall strain.
did a dislocation pass completely through wall material. Plastic deformation of this
nature would, in principle, cause more deformation in the cell interior vs. the cell wall.
A subroutine was developed in an attempt to measure the localized strain values within
the simulated cell structure.
3.4.1 Localized Strains
To measure the strains occurring in the cell interior relative to the cell walls, the simu-
lated volume was first subdivided into 7 7 9 boxes. The strain tensor was calculated
for each box using the code in appendix B.1. Note that only plastic strain is being con-
sidered, as the simulation is only concerned with dislocation motion. The calculations
do not take into account elastic strains.
105
The strain tensors in the selected regions were then used to examine the average
strains in the cell walls and cell interiors. When looking at a column of boxes going
through the center of the simulation volume, figure 3.2 shows clearly that the column
would travel through the cell interior, and one of the dislocation cell walls. Results from
a single column do not provide adequate statistics however, and the data was far too
noisy.
Fortunately, this column also has adjacent columns which pass only through a single
dislocation wall. Using the simulation visualizations, columns of boxes were chosen in
each direction ([100], [010], and [001]) such that they travel through the cell interior and
then only one of the three walls. The strain tensors from these columns of boxes were
then averaged and the results are displayed in figure 3.6.
From these graphs, it is apparent that there is generally more plastic strain occurring
in the cell interiors along the strain axis, the e
33
component of the strain tensor, than in
the cell walls. This finding is consistent with the composite model predictions, where
the cell interior has a lower dislocation density, and thus a lower yield stress.
It’s important to note, the data in these plots has been smoothed. The cell walls actu-
ally exhibited fluctuations of positive and negative plastic deformation that were rela-
tively large. This is attributed to a reorganization of dislocations. However, smoothing
by ’nearest neighbors’ allows a better visualization of the net effect of the deformation
present in the walls. Essentially, the positive and negative strains present in the dislo-
cation cell walls cancels out, while the cell interior exhibits a larger cumulative plastic
deformation along the stress axis. This is an important result as it reinforces one of
the composite model’s predictions of cell interior sections providing more of the plas-
tic deformation. These areas have fewer obstacles and therefore lower flow stresses,
meaning dislocations travel more easily through them.
106
-0.4
-0.2
0.0
0.2
0.4
Strain
7 6 5 4 3 2 1 0
Simulation box subdivision
010 Dislocation Wall
e11
e22
e33
-0.4
-0.2
0.0
0.2
0.4
Strain
7 6 5 4 3 2 1 0
Simulation box subdivision
e11
e22
e33
100 Dislocation Wall -0.4
-0.2
0.0
0.2
0.4
Strain
8 6 4 2 0
Simulation box subdivision
e11
e22
e33
001 Dislocation Wall
Wall
Interior
Wall
Interior
Wall
Interior
[010] Wall Interior
Figure 3.6: Plastic Strain in Cell Wall and Interior
Strain measurements for the e
11
, e
22
, and e
33
strain tensor components evaluated in the
cell walls and cell interior. The bottom left box shows the volume in the simulation box
used to calculate the strains from the [010] wall and cell interior. The x-axis is spatial on
each data plot and corresponds to the earlier described subdivisions, with the dislocation
walls being between roughly 0-3 and, due to the periodic boundary conditions, also on
the far side of the plots. Based on the e
33
component, it is clear that the cell interior is
producing more deformation than the cell walls.
In addition to these results, the presence of LRIS was explored within the microstruc-
ture. The next section discusses the state of internal stress in detail. As a general intro-
duction, figure 3.7 shows the dislocation microstructure with the calculations of the
33
component of the stress tensor done on a plane within the simulation cell, after 0:5%
strain. The plane is perpendicular to the stress axis, and is a good visual reference for
the following results. It relates directly to figure 3.8 which is a series of planes where
stress contours are shown, similar to this one.
107
Figure 3.7: LRIS Contour in Dislocation Cell
The dislocation cell simulation volume with a contour showing the calculated LRIS
perpendicular to the plane along the strain axis. The average internal stress of the cell
interior in this slice is -19 MPa, which is roughly -0.1
a
. This value is similar to the
volume averaged back stress measured in previously discussed microbeam experiments.
3.4.2 Cell Interior and Wall Average Stress
Stress tensors within the sample volume were calculated on planes along a specified
direction. After some experimentation, it was found that a series of 9 planes perpen-
dicular to the strain axis and cutting at equal distances through the simulation volume,
was an adequate sampling. To check this, an average of all measured stresses was done
for 7, 10, 40, and 80 layers on the relaxed cell structure. Naturally the simulation is
constrained in that there should be a net stress of zero within the structure, with no load
applied. The average calculated stress along the [001] direction fluctuated by5 MPa
around zero, when averaging the different numbers of layers. A small technicality is
108
the location of the sampling points used when calculating the stress. Having sampling
points near dislocation cores can cause very large stresses, and this can skew the aver-
age. This is partially responsible for the shifting average. MicroMegas caps the output
stress, to keep the results within reason.
Another consideration is the resolution of the sampling layer. The resolution (i.e. the
number of sampling points on the plane) must be chosen such that it provides adequate
statistics. The results presented here used a grid of 41 x 37 sampling points. A similar
approach was used to determine this resolution, where increasing sampling points didn’t
appear to have a meaningful effect on stress averages. Finally, the stress values were
calculated considering all of the dislocations in the simulated volume and an additional
period of simulation boxes (i.e. a total volume of 27 simulation boxes). Increasing
the periodicity further to account for more dislocation volume replicas did not alter the
calculated stresses.
A measurement of the LRIS in the dislocation cell interior was performed after the
initial formation of the microstructure. Additionally the simulation was allowed to relax
to zero applied stress. The LRIS in the cell interior was measured at -99 MPa. This
is not of particular importance however, as it is due to the dislocation cell formation
process. When forming the dislocation cell structure, the system attempts to reach an
equilibrium, which results in a large LRIS. The simulation was then run for one month
of computing time with a tensile load (
a
) of roughly of 150 MPa.
After 20,000 time steps the simulation was relaxed briefly under no load. The struc-
ture was then analyzed using the 9 planes shown in figure 3.8. A direct average was
done on points within the dislocation cell interior and the average stress in this region
was measured at
33
=28 MPa, which is approximately 18% the applied stress but
109
in the opposite direction. Importantly, the magnitude and sign of this stress is in accor-
dance with microbeam measurements reported previously as well as composite model
predictions.
The [010] wall has an average LRIS value of 25 MPa, the [100] wall a LRIS of
6 MPa, and the [001] a value of -10 MPa. These results are once again similar to
microbeam results and composite model predictions. It is somewhat unexpected that
the [001] top/bottom wall shows a backstress. This value was obtained by averaging the
three layers 7, 8, and 9 shown in figure 3.8. Examining the average LRIS from each
layer individually, the values are 0, -18, and -10 MPa respectively. These layers show
elevated levels of dislocations, with layer 8 having the highest density and a LRIS of
-18 MPa. It is therefore clear that a backstress is present in this dislocation wall, which
somewhat unexpected.
3.4.3 LRIS Histograms
A direct average of the
33
stress tensor values may not be the best method for deter-
mining LRIS values. This is due to the random large stress values which appear when a
stress measurement node is near a dislocation core. A more insightful measure of LRIS
may come from stress value histograms. Although they are clearly not the same, they
are somewhat analogous to an x-ray line profile. Occasional large strain values near dis-
location cores do not affect the general shape and peak of a histogram, just as the Fourier
transform of a point defect in a crystal creates only diffuse x-ray diffraction (although
this is perhaps where the direct comparisons end). In this simulation, the stress measure-
ments are evenly spaced, thus higher counts of a particular stress is directly related to
the amount of material which is experiencing the measured stress. Thus, curve fittings
to the histograms should provide a good measure of the LRIS states.
110
-200 -100 0 100 200
Stress (MPa)
Layer 1 (bottom Layer)
0.33 µm
0.3 µm
Layer 2
0.33 µm
Layer 3
0.33 µm
0.33 µm
0.3 µm
Layer 4 Layer 5
0.33 µm 0.33 µm
Layer 6
0.33 µm 0.33 µm 0.33 µm
0.3 µm
Layer 7 Layer 8 Layer 9
Figure 3.8: Simulated Stress Contours
The complete stress tensor was analyzed for 9 equally spaced planes cut perpendicular
to the deformation axis. Only the stress tensor component parallel to the deformation
axis (
33
) is shown. The layers were spaced evenly within the simulation space, at a
distance of 0.05m apart. The morphology of the stresses make it simple to see the cell
interior area vs. the cell wall area.
111
250
200
150
100
50
0
Number of Counts
-400 -200 0 200 400
Stress (MPa)
Histogram of
Cell Interior
-13.6 MPa

FWHM : 97 MPa
Figure 3.9: Cell Interior LRIS Histogram
A histogram of the axial
33
stress values present in the dislocation cell interior. The
curve fit shows an average stress of -13.6 MPa. This is smaller than the direct compu-
tation of the average from the previous section, and still in agreement with composite
model predictions and microbeam measurements.
A histogram of the cell interior
33
stress data is shown in figure 3.9. A Lorentzian
was fit to the data and has a FWHM of 97 MPa and is centered at -13.6 MPa, which is
9% the applied stress. The previously calculated average stress is twice as large. This
value is still well within the measured values of LRIS for cell interiors in Cu deformed
at 150-200 MPa.
A histogram of the stresses measured in the [100] and [010] walls parallel to the
strain axis is shown in figure 3.10. The fig has a peak at 17 MPa and is twice as broad
as the cell interior histogram with a FWHM of 181 MPa. In the simulation volume, the
walls take up a fair portion of the overall simulation space. Given that the wall stress is
larger than the cell interior stress, and they are of comparable volume, it appears the top
wall will have to compensate for the additional backstress, to create a net stress of zero
in the simulation.
112
500
400
300
200
100
Number of Counts
-600 -400 -200 0 200 400 600
Stress (MPa)
17 MPa
Histogram of
Cell Wall Parallel
to Strain Direction

FWHM : 181 MPa
Figure 3.10: Parallel Cell Wall Histogram
Stress (
33
) value histogram from the [100] and [010] cell walls oriented parallel to the
strain axis. The fit shows an average stress of 17 MPa. If the individual wall stress mea-
surements from the previous section are averaged, this value is identical, and in agree-
ment with composite model predictions and microbeam measurements. The FWHM is
twice as large as the cell interior histogram in figure 3.9
Figure 3.11 is a histogram of the top three layers of stress measurements (layers 7,
8, and 9). The fitted curve has a peak at -17 MPa and a FWHM of 265 MPa, which
is much larger than the perpendicular walls and cell interior histograms. The average
from the previous section was -10 MPa, although the histogram appears to be a better
measurement, as the curve fits fairly well.
Finally, figure 3.12 is a histogram of the stresses measured on a single layer in the
wall perpendicular to the stress axis, layer 8. The data is more noisy than the previous
histograms, as less data was used. The curve fit has a peak at -24 MPa, although it is
perhaps not as reliable as the previous measurements, given the poor fit. The average
calculated in the previous section was -18 MPa, so the wall does appear to have a large
backstress when compared to the other walls.
113
250
200
150
100
50
Number of Counts
-600 -400 -200 0 200 400 600
Stress (MPa)
-17 MPa
Histogram of
Wall Perpendicular
to Strain Direction
FWHM : 265 MPa
Figure 3.11: Perpendicular Cell Wall Histogram
A histogram of the
33
stresses present in the cell wall perpendicular to the strain axis.
The data comes from layers 7, 8, and 9.
80
60
40
20
0
Number of Counts
-600 -400 -200 0 200 400 600
Stress (MPa)
FWHM : 313 MPa
-24 MPa
Histogram of a Layer 8
Figure 3.12: Single Layer Cell Wall Histogram
A histogram of the stresses present in layer 8 measured in the cell wall perpendicular to
the strain axis.
114
Based on these measurements, the cell interior clearly shows a backstress with values
well within actual microbeam measurements. In fact, the cell interior values measured
here are close to the statistical average of0:10
a
LRIS. The two walls parallel to the
deformation axis show a forward stress that is in agreement with experimental measure-
ments as well. These are definite successes of the simulation construction presented
here. The final wall [001] which is perpendicular to the stress axis, shows a backstress.
This wall is not expected to have a backstress. In fact, previous X-Ray line profile
experiments show that when the sample is tilted 90

, the asymmetry of the line profile is
reversed, and decreased in relative agreement with the poisson ratio [52]. This implies
that the walls have a smaller LRIS in this direction, but still exhibit a stress opposite that
of the cell interior. This being said, figure 2.12 does show some wall material exhibiting
backstresses. It is therefore difficult to make an absolute judgement on this aspect of the
simulation.
Given that the full strain tensor has been calculated, a rotation of this nature can
be investigated. Figure 3.13 displays histograms made from the
11
component of the
stress tensor. The peak fits show the LRIS to be relatively large, although the curve fits
are somewhat poor. Most importantly, the cell interior stress switches sign, while the
[001] cell wall does not. This supports the case where a sample rotated 90

would still
exhibit the expected asymmetries, with forward stresses in the walls, and backstresses
in the cell interiors. Thus, it is certainly possible that some walls perpendicular to the
axis of deformation exhibit backstresses along this same axis.
3.4.4 Dislocation Analysis
It is clear that the simulation at time step 20k is in a state where the internal stresses
are similar to those measured using the X-Ray microbeam (except perhaps for the [001]
wall). What remains to be seen, is if the dislocation content in the walls exhibits any
115
250
200
150
100
50
0
Number of Counts
-600 -400 -200 0 200 400 600
Stress (MPa)
FWHM : 209
28 MPa
Cell Interior
Stress Tensor
Component σ
11
400
300
200
100
0
Number of Counts
600 400 200 0 -200 -400 -600
Stress (MPa)
[001] Cell Wall
Stress Tensor
Component σ
11
FWHM : 264
-46 MPa
Figure 3.13: Cell Interior and [001] Cell Wall :
11
Tensor Component
Calculated stress histograms from the
11
tensor component for the dislocation cell inte-
rior, and the [001] cell wall. As expected, the cell interior shows a stress sign opposite
to
33
component shown earlier. Interestingly the dislocation cell wall exhibits a stress
opposite to that of the cell interior.
of the predictions discussed in the composite model. Perhaps the most important pre-
diction, is an excess of dislocations of one particular sign on the side of a dislocation
wall. To explore this, the simulation code was modified to show dislocations of only
one sign on only one active slip system at a time. These structures were then divided
into sections and analyzed for dislocation content. Figures 3.14 - 3.17 are images of
the dislocation structures, showing dislocation of only one sign, on one slip system at a
time. The highlighted areas contain elevated dislocation content of a particular sign.
For the [010] wall on slip system 1, the right side of the wall is 51% negatively signed
dislocations. This is not very significant, however the left side (including the right side
of the simulation box, due to the periodic boundary conditions) is 60% positively signed
dislocations, thus the wall has a more substantial net dipole character. The [100] wall is
58% positively signed dislocations on the left side, and 53% negatively signed disloca-
tions on the right side, thus also exhibiting a net dipole character. Note the ’inside’ of
the walls, next to the cell interior, both show an excess of negatively signed dislocations.
116
y
z
- +
[010] Wall
x
z
[100] Wall
Slip System 1
x
y
+ -
Figure 3.14: Dislocation Analysis: Slip System 1
Positive and negative dislocations on slip system 1. This slip system has the second
highest dislocation density. The highlighted areas show where dislocations of a particu-
lar sign have been deposited in the cell wall.
117
y
z
- +
[010] Wall
x
z
[100] Wall
Slip System 3
x
y
+ -
Figure 3.15: Dislocation Analysis: Slip System 3
Positive and negative dislocations on slip system 3. The highlighted areas show where
excess dislocations of a particular sign are present. Although the [010] appears to have
no obvious concentrations, dislocation analysis does show a slight polarization of the
wall.
118
y
z
- +
[010] Wall
x
z
[100] Wall
Slip System 10
x
y
+ -
Figure 3.16: Dislocation Analysis: Slip System 10
Positive and negative dislocations on slip system 10. This slip system has the highest
dislocation density. Pixel analysis shows the [010] wall has an even distribution of pos-
itive and negative dislocations. The [100] wall is more strongly polarized, with 55%
negatively signed dislocations on the left half of the wall, and 51% positively signed on
the right half.
119
y
z
- +
[010] Wall
x
z
[100] Wall
Slip System 11
x
y
+ -
Figure 3.17: Dislocation Analysis: Slip System 11
Positive and negative dislocations on slip system 11. There is a higher density of + sign
dislocations on this slip system. This is also the dislocation slip system with the lowest
overall dislocation density.
120
When looking at slip system 3, the dislocations in wall [010] don’t appear to have
any obvious concentrations, however upon analysis, the wall is 53% positively signed
dislocations on the left side of the wall, and 54% negatively signed dislocations on the
right side of the wall. A smaller polarity but certainly still measurable. Analysis of the
[100] wall shows a roughly even distribution of dislocations.
For slip system 10, pixel analysis shows the [010] wall has an even distribution of
positive and negative dislocations. Analysis of the [100] wall shows the left side of
the wall is 55% negatively signed dislocations, while the right side of the wall is 51%
positively signed.
Finally for slip system 11, the [010] wall is 65% positively signed dislocations on the
left, and 59% positively signed dislocations on the right. Thus there is a dipole character,
as well as a net positive sign to the entire wall. Given that this system has the lowest
density of dislocations, extra dislocations of a particular sign have a larger impact on
the percentages. The [100] wall has no excess dislocations of either sign on the left side,
and is 55% positively signed dislocations on the right.
To investigate this further, the stress tensor was solved for using the contribution
of only one single slip system at a time. The stress tensor was solved along 9 planes,
exactly as was done previously in figure 3.8. The
33
component was then mapped and
layers 2-6 were averaged. This gives a stress contour which is representative of the
LRIS present in the [100] and [010] cell walls, and the cell interior. The results of these
calculations are shown in figure 3.18.
Interestingly, slip system 1 shows the largest swing of LRIS, and has only the second
highest dislocation density. It also exhibits a dipole character in both walls. Slip sys-
tem 3 shows a stress fluctuation when moving away from the [010] wall, but relatively
no stress fluctuations when moving perpendicular to the [100] wall. This is again in
121
-20
-10
0
10
20
-20 -10 0 10 20
ave_z
-20
-10
0
10
20
-20 -10 0 10 20
ave_z
-20
-10
0
10
20
-20 -10 0 10 20
ave_z
-20
-10
0
10
20
-20 -10 0 10 20
ave_z
Slip System 1 Slip System 3
Slip System 10 Slip System 11
[100] Wall
[010] Wall
-200
-100
0
100
200
MPa
Figure 3.18: LRIS Per Slip System
The
33
stress component was calculated for each slip system independently along 9
co-planar slices perpendicular to the strain axis of [001]. The slices which were not in
the [001] wall were then averaged and are shown here.
agreement with the dislocation sign analysis. The [010] wall exhibits a dipole character,
while the [100] wall does not.
The stress mapping for slip system 10 is less convincing, but does appear to have
stress fluctuations which are oriented more in accordance with the [100] wall having a
dipole character. Finally slip system 11, which has the lowest dislocation density of the
4 active systems, shows a fairly mixed LRIS profile. The walls both exhibit an excess of
122
positively signed dislocations, with the [010] wall having them on the side opposite the
cell interior, while the [100] wall has them next to the cell interior.
3.5 Discussion
The simulation results presented here support the assumptions and models discussed
in previous sections. Although it is perhaps not as exciting, a clear and precise under-
standing is important, and requires the subject matter be investigated using a number
of different approaches. What is certainly of some interest, is the the case of the per-
pendicular [001] wall showing a backstress, while showing the appropriate LRIS values
for the
11
and
22
. This appears to be a case which, although unexpected, is compati-
ble with current experimental results. The composite model makes no assumptions on
what the LRIS should be, and experimental measurements show the same characteris-
tics when the sample is rotated 90

. All of the other results (LRIS values, no artificial
pinning points, correct cell wall/interior ratio) are unsurprising, but this is an important
factor for modeling. It must first get the known variables correct, and then other results
regarding unknown phenomena can be taken more seriously.
The dislocation sign measurements presented here also generally support the theory
that dislocations of opposite sign are deposited on opposite sides of the dislocation wall.
Slip system 1 shows dislocations of the same sign are deposited on the inside of the
walls, while the other side of the wall has dislocations of the opposite sign. This is the
expected configuration, based on composite model predictions. Slip system 3 is less
consistent, with only one wall showing concentrations of same sign dislocations, while
the other wall is homogenous. This same situation is observed for slip system 10. Finally
slip system 11 shows dislocation concentrations, but not in the expected configuration,
where the inside of each wall contains excess dislocations of the same sign.
123
Perhaps most importantly, the configuration of dislocations and dipole nature of the
walls matches quite well with the LRIS mappings in figure 3.18. Slip systems 1, 3, and
10 all match, with slip system 11 being difficult to interpret. These results point strongly
to a link between the dipole character of the walls, and the LRIS values present in the
microstructure.
124
Chapter 4
Transmission Electron Microscopy
The magnitude and distribution of LRIS has been fully characterized experimentally for
the uniaxial case. In addition the simulation results support the experimental charac-
terizations, as well as the assumptions and predictions made by the composite model.
Perhaps a final piece to the puzzle, is a transmission electron microscope (TEM) study
of the dislocation microstructure, and the dislocations present.
A number of TEM dislocation studies have been done on deformed Cu, with the
objective of characterizing the heterogenous dislocation microstructure. Studies looking
at the dislocation cell size, strain dependence, dislocation burgers vector, and even the
dislocation segment lengths have all been done [9, 13, 33, 62, 66]. All of these details are
of course important, but there are none which address the very particular need presented
here, a detailed study of the dislocation content in a cell wall and determination of the
dislocation signs.
4.1 Background
This section contains brief summaries of TEM studies done on pure Cu. They go to
great lengths to characterize the microstructure of deformed copper, doing everything
from burgers vector analysis, to strain/cell size relationships. It is important to note,
however, than none attempt to fully characterize a dislocation cell wall. Additionally a
small amount of diffraction theory is introduced, to help understand the TEM techniques
used.
125
Table 4.1: Diffraction vector (g) and burgers vector (b) table used forgb analysis by
G¨ ottler in [13]. Theh100i burgers vectors are from Hirth locks.
gnb [110] [1

10] [101] [10

1] [011] 01

1] [001] [010] [100]
(1

11) O X X O O X X X X
(11

1) X O O X O X X X X
(

111) O X O X X O X X X
(20

2) X X O X X X X O X
(02

2) X X X X O X X X O
(

220) O X X X X X O X X
4.1.1 Previous TEM Studies
Staker and Holt [66] looked at the relationship between the strain, dislocation density,
and cell size. They found that the mean cell diameter was inversely proportional to
the square root of the dislocation density . Their data gives a relationship of d =
16
1=2
for 99.92% pure OFHC (oxygen-free high thermal conductivity) Cu. They also
found that the dislocation cell size was was inversely proportional to shear flow stress
. Meaning the flow stress is proportional to the square root of the dislocation density.
They did not, however, characterize any dislocations present in the walls.
G¨ ottler performed one of the few complete burgers vector analysis studies on
deformed copper [13]. Single crystal Cu samples were deformed in tension along the
[100] axis. The samples were then neutron irradiated in an attempt to pin dislocations
and stabilize the dislocation microstructures. Samples were cut along the [111], [110],
and [100] axes. Most importantly, G¨ ottler found that dislocations of all 6 burgers vec-
tors occurred with the same frequency. This includes those which are not activated by
the externally applied force. G¨ ottler used the gb criteria shown in table 4.1 for his
dislocation analysis.
The diffraction vectors used in table 4.1 adequately identify the 6 burgers vectors
from the active slip systems, as well as burgers vectors arising from Lomer-Contrell
126
locks and other dislocation interactions [62]. Characterization of theh100i burgers vec-
tors is needed specifically for Hirth locks. These are present in low numbers in sam-
ples with low dislocation densities, but not in heavily deformed samples. This may
be because they dissociate into glissile dislocations at higher stresses, upon unloading.
Additionally they are only present in samples which have been neutron irradiated, and
otherwise possibly dissociate during TEM specimen preparation [69]. G¨ ottler also found
that the dislocation density is related to the resolved shear stress by : = 0:30Gb
p
.
He found that the mean cell diameter D is related to the deformation stress by :
D = 4:2Gb=. The ratio of cell interior to total area was found to be 0.55 and con-
stant. Finally the thickness of the cell walls was found to be roughly proportional to
1=.
Kocks looked at independent slip systems in FCC materials and found that using
selection rules, there are only 5 [32]. All of the TEM work was on single crystal Cu
inclined for single slip, or with two slip systems active. Therefore the initial deformation
parameters are different than those discussed in this thesis, where the crystal is set up
for multiple slip. The study showed that secondary dislocations (those that come, not
from the active slip system, but from another driven by internal stresses) are plentiful.
Kolkman showed that in Cu, deformed only a small amount, less than 6 slip systems
are active [33], although this had previously been observed by others [10]. This study
dealt with samples that were sliced very thin before deformation, which appear to have
slightly different deformation mechanics than cylindrical Cu samples.
A model for describing work hardening was developed based mostly on the work
done by G¨ ottler [13] discussed previously. Schwinck and G¨ ottler [62] provide a table of
dislocation interactions which is useful for identifying burgers vectors of expected wall
dislocations and junctions.
127
Finally a rather in depth study of the dislocation microstructure in deformed Cu
was done by Steeds [67]. The study looked at samples oriented for single and double
slip. Steeds used the [1

11], [

111], and [

1

11] reflections and the bright field technique
to characterize dislocations. Perhaps the most relevant finding for this thesis, is the
discovery of secondary slip occurring in stage 1 deformation, and becoming prolific in
stage two. This reinforces the findings of G¨ ottler (and predates it), that dislocations of
all burgers vectors are present. No attempt was made to look at dislocation signs.
All of these studies provide valuable insight into the deformation mechanics and
dislocation microstructure of deformed copper. However, when looking at a single dis-
location wall there are very specific techniques which must be employed, that were not
used in any of the studies discussed. First, weak-beam diffraction must be used. The
walls are very dense in samples deformed 30%. Using bright or dark field techniques
don’t allow the dislocations to be individually resolved if they are very close to each
other. Secondly using the contrast criteria from G¨ ottler [13] or Steeds [67] is not practi-
cal. The reflections used require large rotations of the crystal sample. Large rotations of
the crystal change the observation angle, and move the observed location of the dislo-
cations substantially. This can make identification of the dislocations impossible when
they are closely packed.
To understand the methods used in this research, a brief review of diffraction theory
relevant to transmission electron microscopy will be discussed. The methods employed
in this thesis to image dislocations is somewhat more difficult in practice compared to
bright and dark field imaging.
4.1.2 TEM Theory
One of the most important concepts when dealing with TEM is Ewald’s sphere. This
is a geometrical representation of the incident and scattered electron beams, reciprocal
128
C
k
I k
D
Laue Zones
-second order
-first order
-zero order
Ewald Circle
Figure 4.1: Ewald’s Circle
The two dimensional equivalent of the Ewald Sphere. The radius of the sphere is 2=,
which is the length of the vectork
I
, the incident beam. A diffracting beamk
D
is shown
where the circle intercepts the reciprocal lattice. There are other diffracting vectors at
other intersection points in the first and second order Laue zones. Also note that the
reciprocal lattice points are not points, but are elongated because of the TEM sample is
very thin.
lattice of the crystal, and the angles of diffraction. Figure 4.1 shows the two dimen-
sional equivalent. Remember that the grid points of a reciprocal lattice represent a set
of crystallographic planes in real space. Thus when the Ewald sphere intersects a point,
the planes this point represents will diffract.
In a perfect infinite crystal, the reciprocal lattice points would be just that, points.
However the TEM sample is actually quite thin, thus the points are elongated causing
a number of (extra) reciprocal lattice points to intersect the Ewald sphere (as seen in
figure 4.1). This is quite different than X-Ray diffraction. For one, the Ewald sphere for
X-Ray diffraction is substantially smaller than for an electron beam (as the radius of the
sphere is proportional to 2=, and secondly the crystals are also substantially thicker.
As an example, a 12 keV X-Ray wavelength is 103 picometers, while a 200 kV TEM
electron beam wavelength is roughly 2.5 picometers.
129
000
200 220
020
[001] [111]
000 022
202 220 242
422
000
111
220
[112]
311
[101]
000
111 111
020
202
Figure 4.2: Select Diffraction Patterns
Rotation of the fcc sample to the four zone axes [101], [112], [001], and [111] produces
the diffraction patterns shown.
Now if the incident beam hits the sample at specific points of symmetry (namely
zone axes), recognizable diffraction patterns are produced. Figure 4.2 shows a num-
ber of diffraction patterns from different zone axes for an fcc crystal. These zone axes
diffraction patterns are used for orientation of the crystal, and creating specific diffrac-
tion conditions. Most importantly, the diffraction patterns are used to select specific
diffraction beams/spots forgb dislocation characterizations (which will be discussed
in depth later).
Comparing the diffraction hkls with those needed by G¨ ottler [13], we see that the
crystal must be rotated to the [111], [101], and [011] (not shown) zone axes, to obtain
all of the requiredgb criteria. Similarly for the Steeds [67] study, the crystal must be
rotated to the [101] and [011] zone axes.
Notice the [001] diffraction pattern does not have diffraction spots corresponding
to [

100] or [010]. The absence of these diffraction spots is due to an extinction rule
130
for diffracting beams which is dependent on the crystal structure. For fcc crystals, the
following extinction equation applies,
(hkl) = 1 +e
i(h+k)
+e
i(h+l)
+e
i(k+l)
; (4.1)
wheree
in
is +1 if n is an even integer and -1 if n is an odd integer. The diffraction point
does not exist if (hkl) = 0 [53]. Using this rule, it’s simple to see the diffraction spot
for the (100) planes produces the extinction equation (100) = 1 1 1 + 1 = 0 and
thus does not exist.
When a zone axis is found, a special technique can be used to look at only a sin-
gle specific diffracting beam. The incident beam is tilted such that the Ewald sphere
intersects both the origin and the desired diffraction spot. This is known as a two beam
condition. An aperture can then be used to select only the incident beam for viewing
(bright field), or the diffracting beam (dark field). Although the actual lenses and gen-
eral construction of a TEM is not discussed here, it’s important to note at this time, that
when selecting one of these beams to pass through the aperture, the actual projection of
the sample onto the projection screen is then composed of only the incident beam minus
the diffraction (bright field), or conversely only the diffracted beam (dark field). If you
imagine a hole in the sample, a bright field image would show the bright backlight of
electrons as they pass through the hole, while a dark field image would show a black
spot, since none of the electrons are diffracted (as there is no crystal there).
4.1.3 Dark Field and Weak Beam Diffraction
Dark field and weak beam techniques are performed in a similar fashion. First the
sample is tilted into a two beam condition, where the incident beam and diffracted beam
are diffracting strongly. The optical axis of the microscope is then tilted so that the
131
0
Dark Field Weak Beam
Ewald Sphere
Ewald Sphere
0
3g
Optic axis Optic axis
g
g
+s
-3g
Figure 4.3: Dark Field and Weak Beam Technique
Dark field and weak beam diagrams. The sample is first tilted to get the appropriate
diffraction conditions. The optical axis is then tilted so that the desired diffraction spot
goes through the aperture. Note the 0 diffraction spot was originally aligned on the
optical axis. The weak beam diagram showss which is the excitation error.
diffracted beam is centered in the microscope. A small aperture is then used so that only
the diffracted beam is permitted to pass onto the film. For weak beam, the same is done,
except that for a g/3g condition (which is a standard weak beam condition), the sample
is tilted so the 3g spot is diffracting strongly, and the optical axis is tilted such that the g
diffracted beam is allowed through the aperture. The values is known as the excitation
error. The basic geometrical representations of these techniques can be seen in figure
4.3.
The purpose behind the weak beam technique is to cause only a small amount of
diffraction near the dislocation core. Because the crystal lattice planes are bent around
a dislocation, the additional tilting attempts to move the normal crystal planes away
from a diffraction condition, and the bent planes into one. This basic effect can be seen
in figure 4.4. The positive and negative excitation errors come from using the same
132
g • b = 0
Weak beam TEM
dislocation core diffraction
Dislocation contrast
criteria
g • b = n + s - s
g
b
b
• •
g
• •
0
0
Figure 4.4: Dislocation Contrasting : Weak Beam andgb
Dark field and weak beam diagrams. The sample is first tilted to get the appropriate
diffraction conditions. The optical axis is then tilted so that the desired diffraction spot
goes through the aperture. Note the 0 diffraction spot was originally aligned on the
optical axis. The weak beam diagram showss which is the excitation error.
diffraction vector, but tilting the sample towards 3g or -3g (shown in the right panel of
figure 4.3).
Once a weak beam condition has been met, the resolution of the dislocations greatly
improves. The dislocation contrast line also moves closer to the dislocation core. A
comparison of a dark field image and a weak beam image is shown in figure 4.5. This
comparison clearly illustrates the necessity of using the weak beam method for disloca-
tion characterization in dense dislocation walls. A dislocation image in dark field may
be 30 nm wide, while in weak beam they can be much sharper at only 5 nm. In the dense
dislocation walls, inter dislocation spacing can easily be smaller than 30 nm. An addi-
tional benefit is the disappearance of bend contours and dislocation line double images
from dislocations whosegb = 2.
133
g/3g
g = 022
-
Figure 4.5: Dark Field and Weak Beam Comparison
The top panel is the dark field image of a dislocation wall. The bottom panel shows
the same dislocation wall, but using the weak beam technique. The dislocations are
dramatically sharper, and bend contours and double contrast images disappear. Both
micrographs were taken on a JEOL JEM 2100 LaB6 at USC Health Sciences Campus.
4.1.4 Dislocation Identification Usinggb Contrast
As has been mentioned previously, dislocations can be identified usinggb experiments.
Different diffracting vectors can be chosen such that dislocations will appear and dis-
appear depending on the burgers vector. A basic diagram of this principle is shown in
134
figure 4.4. It may seem somewhat counter-intuitive that the diffracting planes are actu-
ally vertical and parallel to the incident beam, but realize TEM diffraction is analogous
to a slit experiment, with the distance between the vertical planes being the slits.
It’s also important to note that gb = 0 is actually a simplification which works
for screw dislocations, but not always for edge dislocations. The full equation for edge
dislocations is m = (1=8) (g (bu)), where g is the diffraction vector (normal to
diffracting planes), b the burgers vector, andu a unit vector along the dislocation line
[18]. If bu is parallel to the g vector, the dislocation line does not disappear, but
does take on a characteristic diffuse appearance, and can thus still be identified [67]. If
gb = 2 the dislocation appears as a double line, which can also aid in identification,
although when using the weak beam technique the extra dislocation image fades and
disappears the larger the excitation error.
A confusing factor when doing this type of study, is the elastically anisotropic behav-
ior of Cu. What this means, is that even ifgb = 0, the dislocation may not completely
disappear, because all of the atoms around the dislocation are warped. This should not
be a major issue, however, as the dislocation line will be very faint.
The measurements done in this thesis use thegb = 0 criteria shown in table 4.2. To
locate a single zone axis with the necessary diffraction conditions for identifying every
burgers vector, a solutions togza whereza is a selected zone axis, for each burgers
vector, must be found. The solutions will all be parallel to the g vectors needed to solve
gb = 0. Using this method with the major zone axes, [111] was found to work best.
4.1.5 Kikuchi Lines
When rotating the sample inside the TEM, the best method for determining the crystal-
lographic orientation is through the use of Kikuchi lines. Kikuchi lines are produced by
diffuse electron scattering as the electrons hit atoms in the sample. The electrons scatter
135
Table 4.2: The 6 burger’s vectors and 6 diffraction vectors used for thegb analysis. All
diffraction vectors can be found at the [111] zone axis.
gnb [110] [1

10] [101] [10

1] [011] [01

1]
20

2 2 2 0

4

2 2

2

24

4 0 2

6 2

6
4

2

2 2 6 2

6

4 0
2

20 0 4 2

2

2

6
2

42

2

6 4 0

2

6
0

22

2 2 2 2 0

4
(a) (b)
Figure 4.6: Kikuchi Line Diagram and Micrograph
(a) The [111] zone axis Kikuchi line illustration of an fcc crystal [53]. (b) Actual Kikuchi
lines from the [112] axis in a pure Cu sample.
and end up meeting every Bragg condition, and diffract off of every crystallographic
plane. This creates a complex pattern of lines which correspond to the crystallographic
planes. A diagram is shown in figure 4.6, as well as an actual TEM micrograph showing
the lines.
Kikuchi lines are far more pronounced in thicker samples, and in regions of perfect
crystal. It is therefore fairly easy to focus the beam down into a single dislocation cell
interior, to find the exact crystallographic orientation. Using this orientation, two beam
136
conditions can be set up, as well as weak beam conditions. Often two cell interiors have
slightly different orientations, so when analyzing a dislocation wall, it’s essential to use
Kikuchi lines to constantly be aware of the diffraction conditions. Kikuchi lines were
used extensively in the study presented here, to orient and set up all of the diffraction
conditions.
4.2 Methodology
4.2.1 Sample Preparation
A 2 mm slice was cut from the deformed single crystal Cu sample using a Struers
Minitom low speed diamond saw, parallel to a [111] plane. The samples were then hand
polished down to a thickness of 150-300 m thickness. It was found that a thickness
of 300m was actually needed, to adequately preserve the dislocation microstructure.
Disks were then punched from the thinned sample, and electro-polished using a the
Struers TenuPol-5 jet polisher, with an LT recirculating chiller.
After much trial and error, an effective electrolytic fluid was found consisting of 500
ml distilled water, 250 ml phosphoric acid, 250 ml ethanol, 50 ml propanol, and 5.0 g
urea. Optimal polishing conditions were found to be at 5

C, 5 volts, and a flow rate of
45 (or when the two jets meet, at the operating temperature of 5

C). Polishing time was
set to automatic, and generally took 10 minutes to complete.
4.2.2 Transmission Electron Microscopy
TEM images were taken using a JEOL JEM 2100 LaB6 microscope at the USC health
science campus. Samples were marked with a small dot of whiteout so that they could
be inserted into the high-stability goniometer stage with the exact same orientation each
137
time. The microstructure was then extensively searched for dislocation walls that had
the best chance of being characterized. Bright field and dark field images were taken of
the microstructures, and it became clear that these images did not contain enough detail
to characterize the closely packed dislocations.
Weak beam was used to perform thegb analysis. Once the burgers vector analysis
was complete, an analysis of the burgers vector sign was performed. The dislocation
was first imaged with a positive excitation vector using a weak beam g/3g condition as
shown in figure 4.3. The sample was then rotated to a (roughly) g/-2g condition, which
gives a negative excitation vector. The general idea behind this is shown in the left panel
of figure 4.4. With the changing of the sign of the excitation vector, the dislocation
moves either left or right, depending on the sign of the dislocation [18, 78]. Essentially,
the sign of (gb)s dictates which side the dislocation contrast appears, so switchings
from positive to negative, gives the sign of g. Alternatively the g vector can be switched,
but this requires a much larger rotation of the crystal, and the parallax shifting becomes
problematic.
Often this method is used to determine whether a dislocation loop is of the vacancy
or interstitial type. In the study presented here, it is arbitrary to define which side of
the loop has a positive burgers vector, and which side has a negative burgers vector,
only that the convention remains consistent. What is most important here is the relative
numbers of dislocations with a particular Burgers vector and sign.
4.3 Results and Discussion
A number of dislocation dense walls were investigated, yet because of the high density
of dislocations, generally only a very small number of dislocations could be character-
ized. Another complication, is the amount of time required to characterize a number of
138
Table 4.3: Dislocation Characterization Results
signnb [110] [

110] [101] [

101] [011] [0

11]
+ 5 3 1 3 4 1
2 5 0 2 2 0
dislocations in the same small area. The sample heats due from the electron beam, and
causes small micro-structural changes over time. One case was particularly successful
however. A dislocation dense wall with an associated less dense cluster was analyzed
in detail using the previously describedgb criteria and weak beam method. An exam-
ple of the diffraction contrasting encountered is shown in figure 4.7. The two vertical
dislocations are clearly present in all but one micrograph for each. The left dislocation
is missing in only theg = 2

20 diffraction condition, while the right dislocation is only
missing in theg =

2

24 condition. This means the left dislocation has a burgers vector
of type
a
2
[110] and the right dislocation a burgers vector of type
a
2
[

110], although the
magnitudes are unknown and assumed to bea
p
2. The relative dislocation signs were
also ascertained using the shifting of the dislocation contrast, when switching the sign of
the excitation vector. This technique is shown in the bottom two micrographs of figure
4.7.
The analysis shown in figure 4.7 was applied to the wall structure shown in figure
4.8. These efforts were met with limited success. A number of dislocations could be
accurately identified, yet due to the extremely dense nature of the wall, no amount of
tilting could produce contrast changes in a significant number of dislocation clusters.
With this in mind, 28 distinct dislocations were able to be identified and characterized,
the results are shown in table 4.3.
Efforts were made to produce 2.5D images to discern which slip systems the dis-
locations occupied. This process requires two images be taken with a parallax shift
139
g = 202 g = 224 g = 422
g = 220 g = 242 g = 022
- - - - -
- - -
50 nm
g = 224
- -
(g!b)s = + (g!b)s = -
Figure 4.7: Dislocation Characterization
The six top panels are an example of dislocation contrast under 6 different diffraction
vectors. The bottom two panels show the dislocation contrast from a positive and nega-
tive excitation vector. Notice the positives shows contrast tot he left of the dislocation,
while the negatives image shows contrast on the right.
between the two. Usually a rotation of anywhere from 5-10

is sufficient. Addition-
ally the images must be under the same diffraction conditions (i.e. rotations must be
done along the appropriate Kikuchi lines). A unique problem was encountered how-
ever, which prevented wide use of this technique. The dislocations are bundled close
together, and rotations caused enough parallax shifting, that keeping track of where dis-
locations move becomes very difficult. In addition, even when maintaining the same
140
g = 422
g = 422 g = 422
- - - -
Figure 4.8: Dislocation Characterization: Dislocation Wall
The left panel shows a weak beam image of a dislocation wall. This section was chosen
because it contains both a dense dislocation wall running along the bottom of the micro-
graph, and a less dense dislocation tangle above. The right panel shows individually
identified dislocations traced in red. If they could be characterized, they were num-
bered. Not all of them were able to be identified, particularly in the dense dislocation
clusters. The white box highlights the dislocations detailed in figure 4.7.
diffraction condition, there is a shift in where the contrast is coming from for each dislo-
cation. These effects meant only a few dislocations were able to be adequately resolved
in 3 dimensions. There are therefore no good statistics regarding the active slip systems.
This information is perhaps not as important however, as it has been documented in
other studies of less heavily deformed specimens [13].
An analysis of the dislocation signs shows some amount of deposition and asymme-
try in dislocation sign distribution, but unfortunately no conclusions can be made. The
statistics are simply too small to draw any meaningful conclusions. It is worth com-
paring the structure to the simulations presented in Levine et al. [41]. The simulation
predicts only a very small number of effective dipoles be present, with a density of 4.0
m
1
in the dislocation walls, to obtain stresses of 0.10
a
. The dislocation tangle is
roughly 250 nm wide, meaning it only needs a single effective dipole to be in line with
141
this prediction. Visual inspection of the dislocation structure shows effective dipoles of
this density appear to exist for the [

101] and [011] dislocations.
142
Chapter 5
Conclusion
Long range internal stresses have been a contentious and actively researched field for
the last 30-40 years. The first direct measurements of LRIS were done by Levine et al.
[39] in 2006, and consisted of only a small number of measurements. The LRIS results
presented here contain a number of new and useful characterizations. The cell wall and
interior stresses vary greatly from about -50% to 50% the applied stress. Additionally
the distribution of these stresses is asymmetric. The volume averaged value for LRIS is
roughly 0.10
a
, with long tails extending into the larger positive and negative stresses.
Additionally this asymmetry is mirrored from cell interior to cell wall, as shown in Fig-
ure 2.13. Finally, the difference between the average stresses in the walls and interiors
is 40 MPa. Which is roughly 0.20
a
.
The Seeger et al. [64] pile-up model is shown to be incorrect, as the LRIS present in
the cell interiors and walls can be of both positive and negative values. This is certainly
not the case for cell interiors, and is only occasionally the case for cell walls. Finally
the stresses present in the cell interiors are expected to sum to zero, independent of the
walls, and similarly the wall stresses are expected to sum to zero.
The composite model produces only back stresses in cell interiors and forward
stresses in cell walls. This is the behavior observed experimentally thus far. The addi-
tional prediction of dislocation collecting on one side of a dislocation wall, while dis-
locations of the opposite sign collect on the other side also appears to be correct. Sim-
ulation results appear to support this hypothesis. The simulation results reinforce the
143
expectation that the net effective dipole character in the walls is responsible for the
LRIS present.
Reconstruction of the cell wall and interior sub-profiles using measured dislocation
cell wall and interior data was a success. The sample volume used in the study was small,
at only 1.08m
3
, but effectively represented the average LRIS values, and reconstructed
the bulk X-ray line profile rather well.
Based on the profiles reconstructed in figure 2.19 the proper ratio for our sample is
45% cell interior material and 55% cell wall material. This ratio has wall material at a
higher percentage than what is usually assumed [47, 52, 73].
The main problem with the sub-profiles, is the incredibly weak signal strength of
the wall data when compared to the cell interior data. The study using additional cell
interiors to subtract out the wall sub-profile was an excellent validation, even though
the sub-profile is not perfect. When compared to the summed wall sub-profile, the two
profiles are remarkably similar in FWHM and peak position.
The simulation results were a test in two directions. First, the simulation was able
to accurately model the LRIS present in a well developed dislocation cell structure.
Secondly, the successful modeling of the LRIS and the cell walls and interior allowed a
deeper probing of the microstructure. The results expose the interesting situation of the
[001] dislocation wall exhibiting a backstress, but proper stresses along the
22
and
33
directions. Finally the configuration of dislocations and the dipole nature of the walls
point strongly to a link between the dipole character of the walls, and the LRIS values
present in the microstructure.
144
5.1 Future Works
Efforts thus far to obtain a complete strain tensor have failed. It is perhaps the most
important step left, for full understanding of the nature of LRIS. An added benefit, is the
procedures developed to take these types of measurements would be infinitely useful.
Measurement of strain tensors could be performed on alternate materials, and alternate
deformation configurations. Additionally, results from full strain tensor measurements
could validate (or invalidate) the results from the simulations presented here.
The characterization of LRIS in Cu single crystals deformed under multiple slip con-
ditions is fairly complete at this point. Alternate materials, and alternate deformation
techniques and conditions are certainly viable candidates for exploration. The defor-
mation mechanics of Cu oriented for single slip produces a very different dislocation
microstructure, and at this point it is not clear what the LRIS values would be. Cyclically
deformed materials are also good candidates for research, as is an in-situ experiment.
Many of these situations have been studied using older methods such as bulk X-ray line
profile deconvolution methods. The work done by Levine et al. [41] that is based on the
microbeam measurements presented here, effectively shows the inadequacies of bulk
X-ray line profile deconvolution methods. Thus explicit measurements of dislocation
microstructures is currently the only way to acquire accurate information.
Microbeam measurements on tin solder may also prove useful, as whisker growth
is a well documented problem associated with internal stresses. For that matter, nano-
machines are known to have issues with residual stresses, and microbeam studies may
prove useful. Basically now that the tools are developed, there is a fair amount of fun-
damental scientific knowledge that can be acquired by probing the microstructure of a
variety of materials in a variety of situations, non-destructively.
145
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Appendix A
Experimental Data
A.1 Dislocation Cell Interior and Wall Data
Table A.1: Cell Interior Data 08-2007
Strain FWHM Amp. Depth Strain FWHM Amp. Depth
4.2 0.293811 1.05 -51.3 2.7 0.16134 14 -49.9
4.5 0.143933 1.5 -50.6 0.29 0.241743 0.95 -52
5.6 0.255551 7.6 -51.3 4.1 0.153555 15 -49.9
5.6 0.259606 3.7 -51.3 3.8 0.147047 14 -45
6.7 0.269385 3.6 -49.9 5.9 0.216929 4.5 -48.5
3.8 0.209458 2.2 -49.2 16 0.11589 8.3 -50.6
4.7 0.265425 2.1 -48.5 7.2 0.166118 6.5 -49.2
5.2 0.291437 1.9 -48.5 7.2 0.166118 6.5 -49.2
6.2 0.26934 2.7 -47.8 5.9 0.189937 8 -48.5
3.6 0.180486 5.3 -47.1 3.9 0.123587 16 -45.7
4 0.267756 4 -46.4 7.7 0.169153 1.5 -54.1
4.1 0.248537 5.2 -45 6.8 0.204624 2.6 -53.4
5.1 0.241862 8.8 -44.3 7.4 0.203989 4.2 -52
5.1 0.251735 11 -51.3 5.5 0.231114 2.5 -51.3
6.2 0.200561 4.7 -50.6 1.6 0.209444 2.4 -51.3
2.2 0.21313 7.2 -49.9 5.7 0.230172 2.1 -49.9
4.3 0.206626 3.2 -49.9 2.4 0.236768 1.6 -49.2
8.4 0.244884 2.7 -49.2 0.3 0.248753 1.4 -48.5
5 0.205512 4.7 -48.5 7 0.155184 3.9 -54.1
7 0.232554 4 -47.1 7.4 0.220873 1.3 -53.4
6.2 0.170317 8.5 -46.4 5.5 0.199312 2.7 -52.7
5 0.153034 1.5 -45 4.8 0.221611 1.4 -51.3
4 0.229593 2.6 -52 7.4 0.213737 2 -52
5.8 0.206522 4.6 -51.3 7.4 0.213737 2 -52
5.8 0.172759 9 -49.9 7.2 0.145832 1.55 -51.3
6.6 0.146139 5 -50.6 5.7 0.191666 2 -49.2
7.8 0.172951 3 -48.5 3 0.143557 5.8 -53.4
153
7.6 0.154821 4.6 -46.4 6.4 0.20471 1.6 -53.4
5.1 0.162864 3.7 -45 7.5 0.230401 2 -52
0.29 0.241743 0.95 -52 6.7 0.190166 2.2 -48.5
3.4 0.115542 11 -45 6 0.19828 2.9 -47.1
2.8 0.177214 1.2 -48.5 2 0.246282 1.4 -50.6
Table A.2: Cell Interior Data 10-2007
Spot: Depth: Strain: FWHM: Amplitude
A1-1 5 8.10E-04 0.103 0.18
A1-2 4 1.60E-03 0.108 0.44
A1-2 4.5 1.10E-03 0.098 0.41
A1-3 4 6.80E-04 0.136 0.41
A1-7 2 3.80E-04 0.205 0.12
A1-8 -1 5.20E-04 0.09 1.65
A1-8 -0.5 4.20E-04 0.096 2.9
A1-9 -2.5 4.20E-04 0.107 0.94
A1-9 -2 4.60E-04 0.079 1.04
A1-10 -5 4.90E-04 0.117 1.32
A1-10 -4.5 4.40E-04 0.145 1.5
B1-1 -11 6.10E-04 0.134 0.34
B1-2 -10.5 7.80E-04 0.106 0.15
B1-3 -10 2.10E-04 0.113 0.17
B1-4 -8 6.10E-04 0.13 0.42
B1-4 -7.5 7.00E-04 0.117 0.61
B1-5 -8.5 4.40E-04 0.137 0.49
B1-5 -8 7.30E-05 0.116 0.82
B1-5 -7.5 5.80E-05 0.12 0.5
B1-6 -7.5 6.80E-04 0.101 0.76
B1-6 -7 4.70E-04 0.114 1
B1-7 -6 7.90E-04 0.181 0.39
B1-8 -2.5 7.50E-04 0.12 0.45
B1-8 -2 6.20E-04 0.113 0.37
B1-9 -2.5 1.20E-03 0.175 0.37
B1-10 -1.5 3.30E-04 0.167 0.35
B2-1 -8.5 2.70E-04 0.134 2.9
154
Spot: Depth: Strain: FWHM: Amplitude
B2-1 -8 1.40E-05 0.103 3.3
B2-2 -9.5 3.80E-04 0.092 0.74
B2-2 -9 3.70E-04 0.109 0.94
B2-3 -5.5 7.00E-04 0.081 0.25
B2-3 -5 3.90E-04 0.096 0.39
B2-3 -4.5 4.10E-04 0.128 0.19
B2-4 -5 9.10E-04 0.106 0.49
B2-5 -1 8.70E-04 0.117 0.55
B2-5 -0.5 8.00E-04 0.122 0.71
B2-5 0 7.10E-04 0.123 0.32
B3-1 -12.5 5.60E-04 0.135 0.69
B3-1 -12 7.50E-04 0.191 0.76
B3-1 -11.5 5.50E-04 0.15 0.64
B3-2 -11.5 9.20E-04 0.112 0.88
B3-2 -11 4.90E-04 0.091 3.1
B3-3 -10 5.10E-04 0.141 0.26
B3-3 -9.5 6.00E-04 0.119 0.45
B3-3 -9 7.70E-04 0.123 0.36
B3-4 -9.5 1.20E-04 0.109 0.15
B3-4 -9 2.30E-04 0.094 0.25
B3-4 -8.5 2.60E-04 0.086 0.25
B3-5 -8.5 5.10E-04 0.118 0.4
B3-5 -8 9.20E-05 0.098 0.76
B3-6 -7 8.80E-04 0.096 0.9
B3-6 -6.5 5.10E-04 0.079 0.62
B3-7 -7 4.80E-04 0.135 0.47
B3-7 -6.5 4.60E-04 0.123 0.94
B3-7 -6 9.50E-06 0.139 0.53
B3-8 -2 8.10E-04 0.126 0.28
B3-8 -1.5 7.70E-04 0.099 0.43
D1-1 -4.5 7.30E-04 0.0992 1.5
D1-1 -4 5.70E-04 0.0981 1.4
D1-2 -3 1.10E-03 0.0959 1.44
D1-2 -2.5 6.00E-04 0.133 0.68
D1-3 -2 2.40E-04 0.0783 1.2
D1-3 -1.5 1.60E-04 0.0869 1.6
D1-3 -1 1.90E-04 0.0993 1
D1-3 -0.5 2.40E-04 0.1215 0.4
D1-4 2 8.90E-04 0.1049 0.21
D1-5 7 2.50E-04 0.0981 0.82
155
Spot: Depth: Strain: FWHM: Amplitude
D1-5 7.5 2.40E-04 0.0999 1.28
D1-5 8 3.50E-04 0.0885 1.6
D1-5 8.5 3.40E-04 0.0895 1.6
D1-5 9 3.60E-04 0.0969 1.36
D1-5 9.5 3.20E-04 0.1089 0.7
D2-1 -1.5 2.60E-04 0.1034 3.9
D2-1 -1 2.20E-04 0.103 5.5
D2-1 -0.5 1.20E-04 0.113 4
D2-2 4 9.30E-04 0.155 0.56
D2-2 4.5 7.90E-04 0.185 0.43
D2-3 6 7.40E-04 0.198 0.22
D3-1 -3 9.40E-04 0.125 1
D3-1 -2.5 6.40E-04 0.118 0.92
D3-2 -3 9.70E-04 0.115 0.61
D3-2 -2.5 7.40E-04 0.167 0.63
D3-3 -1.5 4.70E-04 0.124 4.1
D3-3 -1 3.70E-04 0.109 6.2
D3-3 -0.5 2.30E-04 0.118 4.7
D3-3 0 6.90E-05 0.115 2.4
D3-4 1.5 3.10E-04 0.135 0.8
D3-4 2 1.90E-04 0.116 0.6
D3-5 3.5 9.40E-04 0.154 0.35
D3-5 4 5.30E-04 0.181 0.51
D3-5 4.5 4.40E-04 0.179 0.37
D4-1 -4.5 6.70E-04 0.09 0.56
D4-1 -4 6.60E-04 0.116 1.08
D4-2 -1.5 4.20E-04 0.129 4.2
D4-2 -1 3.70E-04 0.099 7.5
D4-2 -0.5 1.40E-04 0.108 4.9
D4-3 1 1.60E-04 0.111 1.75
D4-3 1.5 1.70E-04 0.01 2.5
D4-3 2 7.10E-05 0.086 1.8
D4-4 4.5 5.20E-04 0.21 0.25
156
Table A.3: Cell Wall Data 10-2007
Spot: Depth: Strain: FWHM: Amplitude
A1W-1 0 -6.90E-04 0.311516 0.24
A1W-2 5 -4.20E-05 0.355588 0.12
A1W-3 1.5 -3.10E-04 0.357502 0.18
A1W-4 3.5 5.60E-05 0.352763 0.7
A1W-5 4.5 -7.60E-04 0.324248 0.052
A2W-1 0 -1.00E-04 0.321992 0.54
A2W-2 6.5 1.90E-04 0.313124 0.13
A2W-3 -2.5 2.70E-06 0.307354 0.32
A2W-4 1.5 1.10E-05 0.322527 0.45
B1W-1 -8 -9.00E-04 0.324755 0.27
B1W-2 -8 -5.90E-04 0.302296 0.6
B1W-3 -6 -2.60E-04 0.413028 0.16
B1W-4 -0.5 -2.30E-04 0.344234 0.57
B1W-5 -1.5 -1.00E-03 0.312721 0.2
B2W-1 -11 -7.40E-04 0.373361 0.14
B2W-2 -7.5 -3.60E-04 0.32268 0.1
B2W-3 -4 -1.20E-03 0.341852 0.15
B2W-4 -9 -6.00E-05 0.292505 0.31
B3W-1 -10.5 -1.10E-03 0.448254 0.035
B3W-2 -10 -8.00E-04 0.377554 0.085
B3W-3 -7.5 -1.50E-04 0.323341 0.27
B3W-4 -4 1.10E-04 0.286983 0.18
B3W-5 -4 -3.30E-04 0.356976 0.09
B3W-6 -3 -3.00E-04 0.330678 0.23
B3W-7 -2 -1.10E-04 0.310368 0.085
D1W-1 -1 -1.40E-04 0.326276 0.1
D1W-2 0 -3.00E-04 0.338368 0.14
D1W-3 1 -2.50E-04 0.327683 0.15
D1W-4 2 -5.20E-04 0.363692 0.105
D1W-5 6.5 -2.70E-04 0.318286 0.05
D2W-1 -1 2.50E-04 0.302669 0.16
D2W-2 -1 -3.50E-04 0.416678 0.18
D2W-3 3 -1.10E-04 0.333672 1.2
D2W-4 6 -2.50E-04 0.341586 0.08
D2W-5 6 2.50E-05 0.349312 0.14
D2W-6 7.5 -2.90E-04 0.315737 0.2
D2W-7 8.5 -2.30E-04 0.330033 0.23
D3W-1 1.5 5.40E-05 0.33385 0.25
D3W-2 2.5 -1.40E-04 0.330361 0.95
157
Spot: Depth: Strain: FWHM: Amplitude
D3W-3 5 -3.50E-04 0.316587 0.18
D3W-4 6 -1.00E-05 0.275854 0.08
D3W-5 7.5 -4.60E-04 0.333629 0.21
D4W-1 -1 1.50E-04 0.34337 0.23
D4W-2 1.5 -1.60E-04 0.284511 0.79
D4W-3 3 -4.00E-04 0.360517 0.48
D4W-4 5 1.90E-04 0.359094 0.47
D4W-5 6 1.30E-05 0.348331 0.95
158
Table A.4: Cell Interior Data 03-2008
Spot: Depth: Strain: fwhm: Amp: Peak-x Peak-y P
A1-1 -53 2.40E-04 0.09 0.72 155.81 0.04 90.410.04 2
-52.5 4.20E-04 0.11 2.1 155.540.02 89.380.02 1
-52 3.20E-04 0.097 3.5 155.450.02 88.480.03 2
A1-2 -49.5 6.10E-04 0.21 0.34 144.280.07 158.510.09 2
A1-3 -47.5 8.20E-04 0.13 1.09 181.510.04 113.720.03 2
-47 5.60E-04 0.083 3.9 181.610.02 112.570.02 2
-46.5 5.40E-04 0.079 4 181.580.02 112.580.02 1
-46 5.70E-04 0.092 2.75 181.960.03 112.440.02 1
-45.5 5.10E-04 0.11 1.25 183.550.06 112.450.05 1
-45 3.70E-04 0.13 0.366 185.130.07 112.390.07 1
A1-4 -45 6.90E-04 0.11 0.31 189.790.09 117.220.09 2
A1-5 -48.5 4.80E-04 0.26 0.56 155.120.06 155.330.09 3
A1-6 -44 3.80E-04 0.14 0.19 176.180.07 157.920.09 2
A1-7 -43.5 4.10E-04 0.15 0.375 183.900.07 128.610.08 2
A1-8 -43 1.30E-03 0.16 0.19 209.750.05 113.120.04 2
-42.5 1.00E-03 0.15 0.23 210.290.05 112.240.05 2
A1-9 -42 5.60E-04 0.14 0.17 261.290.05 55.390.04 2
-41.5 2.80E-04 0.11 0.27 261.100.03 54.680.03 2
A2-3 -52.5 1.40E-04 0.1 0.33 155.120.03 89.250.03 2
-52 2.10E-04 0.11 0.434 155.120.03 89.250.03 2
-51.5 3.80E-04 0.11 0.35 155.120.03 89.250.03 1
A2-4 -51.5 5.30E-04 0.12 0.51 159.300.03 73.750.03 1
-51 2.90E-05 0.12 0.17 159.300.03 73.750.03 2
A2-5 -52 6.60E-04 0.088 0.46 224.490.03 72.570.03 1
-51.5 4.30E-04 0.088 0.72 224.490.03 72.570.03 1
-51 4.20E-04 0.11 0.37 224.490.03 72.570.03 1
A2-6 -51 1.10E-03 0.17 1.54 158.870.03 114.570.05 2
-50.5 7.30E-04 0.17 1.71 158.210.04 116.790.07 2
A2-7 -50.5 4.30E-04 0.16 0.44 138.590.06 145.270.05 2
-50 2.80E-04 0.18 0.83 138.030.07 145.360.06 1
-49.5 1.10E-04 0.2 0.715 136.550.06 146.150.06 2
A2-8 -48.5 5.10E-04 0.17 0.87 115.430.06 170.860.08 1
-48 3.80E-04 0.17 0.98 113.20.1 169.50.1 1
A2-9 -47 7.30E-04 0.16 0.53 181.600.03 112.160.04 2
-46.5 6.50E-04 0.12 1.55 181.430.03 112.470.03 1
-46 6.00E-04 0.11 1.81 181.440.03 112.360.03 1
-45.5 4.00E-04 0.12 0.89 181.730.04 112.670.04 1
A2-10 -44.5 6.30E-04 0.17 0.66 195.930.06 160.840.07 2
-44 2.70E-04 0.13 0.49 194.450.05 158.770.05 1
159
A2-11 -42.5 1.10E-03 0.11 0.26 213.170.04 97.070.04 1
-42 7.10E-04 0.16 0.13 214.010.04 95.230.06 1
A2-12 -41 7.60E-04 0.091 0.32 252.770.03 39.730.03 1
-40.5 6.60E-04 0.11 0.17 252.770.03 39.730.03 2
B-1 -55 2.30E-04 0.16 2.05 192.890.05 155.300.05 2
-54.5 -9.50E-05 0.23 1.23 193.100.05 153.290.05 2
B-2 -55 1.90E-04 0.18 0.824 178.590.06 172.570.06 1
-54.5 2.70E-04 0.2 0.779 178.590.06 172.570.06 1
-54 3.70E-04 0.21 0.39 179.650.06 170.410.08 1
-53.5 8.60E-04 0.13 0.454 179.650.06 170.410.08 1
B-3 -54 9.00E-04 0.15 2.03 204.080.05 110.810.11 2
-53.5 4.10E-04 0.17 3 203.420.03 105.600.05 2
-53 4.60E-04 0.18 1.08 202.520.04 102.520.05 2
B-4 -52.5 3.60E-04 0.15 0.52 156.170.10 89.960.07 1
B-5 -52.5 8.40E-04 0.097 0.5 150.060.12 91.490.09 2
-52 9.80E-04 0.094 1.876 148.390.04 91.450.04 2
-51.5 7.30E-04 0.089 1.775 147.790.03 90.620.04 2
B-7 -51 7.00E-04 0.19 0.575 137.510.06 108.830.10 3
B-8 -51 7.00E-04 0.11 0.44 151.560.06 155.710.06 1
-50 4.60E-04 0.17 0.379 151.560.06 155.710.06 2
B-9 -50.5 5.30E-04 0.2 0.41 143.160.07 159.980.10 2
-50 5.10E-04 0.18 0.342 143.160.07 159.980.10 2
B-10 -49.5 1.20E-04 0.21 1.18 153.700.06 178.350.07 1
B-11 -48.5 6.80E-04 0.13 0.98 166.920.07 136.880.08 2
-48 7.30E-04 0.14 1.21 166.920.07 136.880.08 2
B-12 -47 5.60E-04 0.093 0.543 158.890.04 109.380.04 2
-46.5 4.30E-04 0.1 0.468 158.890.04 109.380.04 1
-46 3.00E-04 0.12 0.26 158.890.04 109.380.04 1
B-13 -47 5.70E-04 0.093 0.999 181.990.04 112.640.04 1
-46.5 5.70E-04 0.099 1.65 183.260.04 111.530.04 1
-46 4.40E-04 0.093 1.793 184.590.03 111.240.03 2
-45.5 4.00E-04 0.1 1.13 184.780.04 111.580.04 2
-45 4.60E-04 0.11 0.34 184.720.04 112.270.04 2
B-14 -44 8.40E-04 0.14 0.427 182.440.04 132.820.05 1
-43.4 8.50E-04 0.13 0.571 182.440.04 132.820.05 1
C-1 -51 8.30E-04 0.17 3.05 202.130.03 113.170.04 3
-50.5 7.50E-04 0.12 1.61 202.130.03 113.170.04 3
C-2 -50.5 5.70E-04 0.18 0.82 158.000.07 87.120.07 2
-50 7.80E-04 0.15 2.1966 158.000.07 87.120.07 2
-49.5 3.70E-04 0.15 1.57 158.000.07 87.120.07 2
160
C-3 -49.5 8.20E-04 0.097 3.459 148.220.02 91.890.02 1
-49 6.30E-04 0.11 4.44 148.220.02 91.890.02 1
-48.5 6.80E-04 0.11 2.49 148.220.02 91.890.02 1
C-4 -45.5 1.10E-03 0.091 0.723 165.420.05 135.590.05 1
-45 8.80E-04 0.11 0.88 165.420.05 135.590.05 1
C-5 -44.5 5.30E-04 0.12 0.674 158.260.03 109.490.03 1
-44 5.20E-04 0.12 1.517 158.260.03 109.490.03 1
-43.4 4.30E-04 0.12 1.036 158.260.03 109.490.03 1
C-6 -44 6.90E-04 0.11 0.395 185.390.04 109.700.06 1
-43.5 7.00E-04 0.12 0.742 185.390.04 109.700.06 1
-43 6.20E-04 0.11 1.138 185.390.04 109.700.06 1
-42.5 5.40E-04 0.13 0.744 185.390.04 109.700.06 1
D1-1 -53.5 7.60E-05 0.12 1.604 194.820.03 137.860.04 2
-53 6.20E-04 0.16 2.258 194.820.03 137.860.04 2
-52.5 5.60E-04 0.2 1.572 194.820.03 137.860.04 3
D1-2 -52 8.00E-04 0.12 2.2 186.520.04 127.630.04 1
-51.5 8.40E-04 0.13 3.47 184.130.06 127.960.03 2
-51 3.80E-04 0.13 2.11 181.690.05 126.900.03 2
D1-3 -50.5 3.40E-04 0.099 2.93 154.660.02 90.560.03 1
-50 1.40E-04 0.11 3.202 154.660.02 90.560.03 1
-49.5 2.70E-04 0.12 1.565 154.660.02 90.560.03 1
D1-4 -49 5.50E-04 0.15 0.4996 137.490.07 84.930.07 2
D1-5 -49.5 1.90E-04 0.12 0.491 135.200.07 80.250.09 2
D1-6 -49 4.50E-04 0.14 0.4887 147.440.10 91.510.07 1
D1-7 -48.5 3.50E-04 0.19 0.5367 135.200.07 110.370.09 2
D1-8 -48 7.40E-04 0.2 0.665 138.230.05 130.250.07 3
-47.5 4.70E-04 0.2 0.4536 138.230.05 130.250.07 2
D1-9 -47.5 1.80E-04 0.19 0.659 132.800.11 141.940.10 1
-47 -8.80E-05 0.2 0.3739 132.800.11 141.940.10 1
D1-10 -47 8.70E-04 0.14 0.776 143.650.04 160.690.04 1
-46.5 7.50E-04 0.15 0.513 143.650.04 160.690.04 2
D1-11 -46.5 5.60E-04 0.17 0.369 151.380.08 152.930.10 2
D1-12 -46 2.90E-04 0.19 0.421 148.380.05 167.880.05 2
D1-13 -45 6.10E-04 0.16 0.4644 162.670.04 130.410.07 2
D1-14 -44 6.20E-04 0.094 0.3166 181.920.04 112.320.03 1
-43.5 6.00E-04 0.11 0.366 181.920.04 112.320.03 1
-43 6.60E-04 0.12 0.244 181.920.04 112.320.03 1
D1-15 -40 7.40E-04 0.069 0.318 248.090.03 43.690.02 2
-39.5 9.00E-04 0.12 0.4316 248.090.03 43.690.02 2
-39 9.20E-04 0.11 0.411 248.090.03 43.690.02 2
161
D2-1 -52 5.30E-04 0.21 12.0156 195.370.04 140.760.05 2
-51.5 4.70E-04 0.24 7.0108 195.370.04 140.760.05 3
D2-2 -50 2.20E-04 0.14 13.11 155.140.02 90.550.03 2
-49.5 9.30E-05 0.12 13.64 155.140.02 90.550.03 1
D2-3 -48.5 3.30E-04 0.17 1.945 141.910.08 115.320.09 2
D2-4 -48.5 4.50E-04 0.2 1.379 134.710.07 136.180.07 2
-48 1.40E-05 0.2 1.496 134.710.07 136.180.07 1
D2-5 -48 5.60E-04 0.12 1.77 124.10.1 143.10.1 2
-47.5 4.60E-04 0.17 1.53 124.10.1 143.10.1 2
D2-6 -48 5.10E-04 0.11 2.298 136.960.08 145.060.06 1
-47.5 4.30E-04 0.15 1.42 136.960.08 145.060.06 2
D2-7 -47 4.20E-04 0.18 2.098 120.110.07 159.690.11 2
D2-8 -46 7.90E-04 0.16 3.034 117.890.05 169.530.05 2
-45.5 7.00E-04 0.13 2.54 117.890.05 169.530.05 1
D2-9 -45.5 1.10E-03 0.1 2.7355 158.850.03 105.680.03 1
-45 6.30E-04 0.13 3.705 158.850.03 105.680.03 2
D2-10 -43.5 2.90E-04 0.18 1.57 174.010.05 93.130.05 2
-43 3.20E-04 0.17 0.946 174.010.05 93.130.05 2
D2-11 -44 5.60E-04 0.084 1.01 181.400.04 112.580.04 1
-43.5 5.30E-04 0.1 1.09 181.400.04 112.580.04 1
D2-12 -40 7.90E-04 0.081 0.849 249.240.04 44.820.04 1
-39.5 5.60E-04 0.11 1.146 249.240.04 44.820.04 1
-39 4.20E-04 0.099 1 249.240.04 44.820.04 1
162
Table A.5: Cell Wall Data 03-2008
Wall: Depth: Strain: FWHM: Amplitude: Profile
A-53.5-1 -53.5 -2.50E-04 0.31 0.256 2
A-53.5-3 -53.5 5.00E-04 0.24 0.09 2
A-53.5-4 -53.5 1.60E-04 0.18 0.17 2
A-53-1 -53 -1.70E-04 0.3 0.44 2
A-52.5-1 -52.5 -1.40E-04 0.29 0.17 2
A-52.5-2 -52.5 -2.20E-04 0.3 0.356 2
A-52.5-3 -52.5 3.90E-04 0.22 0.18 2
A-52.5-4 -52.5 -5.30E-05 0.32 0.22 2
A-52.5-5 -52.5 -2.10E-04 0.3 0.89 2
A-52-1 -52 -2.80E-04 0.32 0.4 2
A-52-2 -52 -3.10E-04 0.33 0.222 2
A-51.5-1 -51.5 5.60E-05 0.29 0.036 3
A-51.5-2 -51.5 -2.20E-04 0.3 0.41 2
A-51.5-3 -51.5 -3.80E-04 0.34 0.16 3
A-51-1 -51 -2.30E-05 0.38 0.46 1
A-51-2 -51 -2.80E-04 0.29 0.144 2
A-50.5-1 -50.5 1.40E-04 0.33 0.81 2
A-49.5-1 -49.5 2.60E-04 0.27 0.17 2
A-49.5-2 -49.5 3.70E-04 0.25 0.073 2
A-49-1 -49 3.00E-04 0.23 0.21 2
A-49-2 -49 2.00E-04 0.24 0.18 2
A-48.5-1 -48.5 -4.40E-04 0.32 0.12 2
A-48-1 -48 -4.00E-04 0.32 0.1 2
A-47-1 -47 -3.70E-04 0.32 0.26 2
A-45-1 -45 4.00E-04 0.3 0.22 1
A-43.5-1 -43.5 -1.40E-04 0.28 0.1 2
A-43-1 -43 1.70E-04 0.3 0.085 2
A-43-2 -43 4.20E-04 0.26 0.096 2
A-42.5-1 -42.5 2.60E-04 0.3 0.29 2
A-42-1 -42 -3.90E-05 0.29 0.38 2
A2-53.5-1 -53.5 2.00E-04 0.25 0.76 2
A2-53.5-2 -53.5 2.70E-04 0.26 0.2 2
A2-52.5-1 -52.5 1.70E-04 0.3 0.16 2
A2-52.5-2 -52.5 4.20E-05 0.31 2.18 2
A2-52-1 -52 -1.90E-04 0.34 0.11 2
A2-51.5-1 -51.5 -1.60E-05 0.35 0.1 2
A2-51-1 -51 5.90E-05 0.29 0.11 2
A2-51-2 -51 4.50E-04 0.31 0.12 2
A2-50.5 -50.5 -1.40E-04 0.31 0.22 1
163
A2-49.5-1 -49.5 -2.00E-04 0.22 0.17 3
A2-48.5-1 -48.5 -3.00E-05 0.32 0.43 1
A2-48-1 -48 -2.20E-05 0.32 0.39 1
A2-47.5 -47.5 1.50E-04 0.3 0.27 1
A2-47 -47 2.60E-04 0.25 0.19 2
A2-47-2 -47 -2.40E-04 0.29 0.23 2
A2-46.5-1 -46.5 -1.60E-04 0.32 0.16 1
A2-46-1 -46 -2.00E-04 0.3 0.21 1
A2-45.5-1 -45.5 -7.40E-05 0.28 0.2 2
A2-45.5-2 -45.5 4.60E-04 0.38 0.12 2
A2-45-1 -45 1.90E-04 0.3 0.11 2
A2-42.5-1 -42.5 6.50E-05 0.29 0.11 2
A2-42-1 -42 -2.10E-04 0.28 0.1 2
A2-42-2 -42 -5.10E-05 0.31 0.07 2
A2-42-3 -42 -2.00E-04 0.29 0.09 3
A2-41.5-1 -41.5 -4.30E-04 0.24 0.065 2
B-55-1 -55 3.70E-04 0.45 0.44 2
B-55-2 -55 -5.20E-05 0.29 0.2 2
B-54.5-1 -54.5 -1.80E-04 0.31 0.324 2
B-54.5-2 -54.5 -1.40E-04 0.35 0.405 3
B-54-1 -54 1.60E-04 0.32 0.329 2
B-54-2 -54 -1.70E-04 0.37 0.332 3
B-53.5-1 -53.5 -3.40E-05 0.3 0.297 2
B-53-1 -53 -1.80E-04 0.33 0.236 3
B-55-2 -53 -1.80E-04 0.32 0.628 3
B-55-3 -53 3.50E-04 0.22 0.235 3
B-52.5-1 -52.5 1.10E-04 0.26 0.308 3
B-52-1 -52 9.00E-05 0.25 0.213 3
B-52-2 -52 3.40E-04 0.3 0.237 3
B-51.5-1 -51.5 2.80E-04 0.27 0.345 2
B-51-1 -51 1.30E-04 0.28 0.896 2
B-50.5-1 -50.5 9.30E-05 0.29 0.767 2
B-50.5-2 -50.5 3.80E-04 0.24 0.1885 3
B-50-1 -50 -8.10E-05 0.28 0.356 2
B-47-1 -47 -1.50E-04 0.29 0.428 3
B-46.5-1 -46.5 -1.10E-05 0.3 0.104 2
B-46-1 -46 9.90E-05 0.31 0.053 2
C-50-1 -50 -3.60E-04 0.32 0.3089 2
C-50.5-1 -50.5 -4.60E-05 0.32 0.6772 3
D1-51.5-1 -51.5 -1.70E-05 0.38 0.1 3
164
D1-51-1 -51 -2.20E-04 0.36 0.183 3
D1-50.5-1 -50.5 -3.90E-04 0.38 0.13 3
D1-49-1 -49 -2.20E-04 0.33 0.144 3
D1-47.5-1 -47.5 -2.60E-04 0.21 0.0707 3
D1-46-1 -46 -2.50E-04 0.31 0.0703 2
D1-45.5-1 -45.5 -1.80E-04 0.27 0.163 2
D1-45.5-2 -45.5 -1.90E-05 0.31 0.179 2
D1-45-1 -45 -4.60E-05 0.29 0.186 2
D2-1 -51 -2.10E-04 0.34 1.21 3
D2-2 -50.5 -4.60E-04 0.37 1 3
D2-3 -50 -3.30E-04 0.37 0.544 3
D2-4 -49.5 -3.60E-04 0.39 0.4154 3
D2-6 -48.5 -1.10E-04 0.36 0.03455 3
D2-7 -48.5 -1.40E-03 0.69 0.0459 3
D2-9 -45.5 -9.90E-05 0.35 0.401 1
D2-10 -45 -2.20E-04 0.34 0.327 2
D2-11 -45 -2.10E-05 0.32 0.1875 2
D2-12 -44.5 -4.40E-05 0.28 0.1737 2
D2-13 -40 -5.30E-04 0.43 0.113 2
165
A.2 Strain Tensor Data
Table A.6: Strain Tensor Data 11-2009
Label num x-pixel x uncert. y pixel y uncert. Strain strain uncer. h k l Depth fwhm amp
A1 1 1812.32 0.05 1862.52 0.04 5.40E-04 1.40E-04 -2 0 8 -32.5 0.098 0.098
A2 2 1806.11 0.09 1858.62 0.05 5.60E-04 1.40E-04 -2 0 8 -32 0.082 0.19
A2 3 1804.85 0.08 1858.04 0.04 5.90E-04 1.40E-04 -2 0 8 -31.5 0.093 0.21
A2 4 1803.38 0.16 1857.52 0.07 5.40E-04 1.40E-04 -2 0 8 -31 0.1 0.14
A3 5 1806.86 0.09 1832.8 0.08 6.30E-04 1.40E-04 -2 0 8 -30.5 0.16 0.023
A5 6 1831.6 0.1 1776.1 0.1 8.60E-04 1.40E-04 -2 0 8 -23.5 0.13 0.025
A6 7 1834.9 0.2 1747 0.2 1.00E-03 1.40E-04 -2 0 8 -22 0.15 0.018
A7 8 1790.14 0.05 1737.64 0.03 9.10E-04 1.40E-04 -2 0 8 -21 0.13 0.074
A7 9 1789.76 0.06 1737.27 0.04 8.40E-04 1.40E-04 -2 0 8 -20.5 0.13 0.059
A7 10 1789.95 0.07 1737.23 0.06 8.90E-04 1.40E-04 -2 0 8 -20 0.14 0.037
B1 1 1766.22 0.07 750.23 0.07 6.40E-04 1.40E-04 0 2 8 -32.5 0.11 0.98
B2 2 1761.65 0.04 746.25 0.03 7.60E-04 1.40E-04 0 2 8 -32.5 0.083 0.27
B2 3 1761.16 0.04 745.68 0.03 7.00E-04 1.40E-04 0 2 8 -32 0.096 0.3
B2 4 1761.22 0.04 745.71 0.03 7.10E-04 1.40E-04 0 2 8 -31.5 0.091 0.17
B3 5 1783.3 0.08 724.52 0.06 8.50E-04 1.40E-04 0 2 8 -31 0.18 0.03
B5 6 1838.77 0.1 677.42 0.09 9.40E-04 1.40E-04 0 2 8 -24 0.13 0.025
B6 7 1868 0.1 650.7 0.1 1.30E-03 1.40E-04 0 2 8 -22.5 0.13 0.014
B7 8 1828.95 0.1 638.14 0.07 9.80E-04 1.40E-04 0 2 8 -21.5 0.14 0.07
166
B7 9 1828.84 0.05 637.62 0.04 9.80E-04 1.40E-04 0 2 8 -21 0.12 0.078
B7 10 1829.1 0.1 637.58 0.06 9.70E-04 1.40E-04 0 2 8 -20.5 0.12 0.048
C1 1 782.84 0.08 1290.91 0.06 4.70E-04 1.00E-04 0 0 6 -31 0.065 0.28
C2 2 776.81 0.04 1285.64 0.04 5.80E-04 1.00E-04 0 0 6 -30.5 0.061 1.08
C2 3 776.39 0.04 1284.22 0.04 5.80E-04 1.00E-04 0 0 6 -30 0.071 1.5
C2 4 774.83 0.05 1282.49 0.04 5.50E-04 1.00E-04 0 0 6 -29.5 0.078 1.2
C3 5 788.36 0.06 1258.34 0.05 4.10E-04 1.00E-04 0 0 6 -29 0.075 0.58
C5 6 835.4 0.2 1188 0.1 6.00E-04 1.00E-04 0 0 6 -21.5 0.14 0.06
C6 7 831.5 0.1 1148.8 0.1 8.70E-04 1.00E-04 0 0 6 -20.5 0.16 0.08
C7 8 805.19 0.03 1151.54 0.02 6.70E-04 1.00E-04 0 0 6 -19.5 0.06 0.59
C7 9 805.49 0.03 1150.58 0.03 6.40E-04 1.00E-04 0 0 6 -19 0.052 0.8
C7 10 805.62 0.03 1150.14 0.02 6.20E-04 1.00E-04 0 0 6 -18.5 0.053 0.62
167
Appendix B
Simulation Code
B.1 Fortran code
All Fortran code was written in collaboration with Benoit Devincre and two of his grad-
uate students Gael and Stephano.
Modification to toolbox.f90 to calculate individual slip system contributions to the
LRIS
! a test to keep only the contribution of one slip system
! IndexSeg = (int((seg(i)%veclin-1)/8)) + 1
! if (seg(i)%jonc) IndexSeg = 0
! if (IndexSeg /= 1) cycle
Calculation of deformation from strain tensors
case(14) in histo.f90
! 1) INPUTS
! We first select the part of the film.bin file we want to use for the stats
write(*,*) ”statistics start at step: ?”
read (*,*) debutaffich
write(*,*) ”statistics end at step: ?”
168
read (*,*) finaffich
write(*,*) ”Snapshot frequency: ?”
read (*,*) freqaffich
if (freqaffich ¡= 0)freqaffich = 1
! Definition of the axis
! First axis: the evolution will be visualized along these axis
!write(*,*) ”Please enter the visualisation axis, x, y, z”
!read (*,*) Axis(1:3)
! Seul le cas de la statistique le long d’une direction macro est prevu
!if (Axis(1)*Axis(2)/=0 .or. Axis(2)*Axis(3)/=0 .or. Axis(1)*Axis(3)/=0) then
! write(*,*) ”Only the case with a visualisation axis parallel to a box axis (kind ¡100¿)
is programed”
! write(*,*) ”Please reconsider the axis”
! exit
!endif
! Normalization
!Axis(:) = Axis(:) / sqrt(Axis(1)*Axis(1) + Axis(2)*Axis(2) + Axis(3)*Axis(3))
! The file we want to analyze
fichier = ”gammabox.bin”
open(61,file=’../out/’//fichier,FORM=’UNFORMATTED’,STATUS=’OLD’,IOSTAT=KODERR)
read(61) NBoites
read(61) NboitesX
169
read(61) NboitesY
read(61) NboitesZ
read(61) NTSG
max length = max(NboitesX,NBoitesY ,NBoitesZ)
allocate(somme(JK))
allocate(somme temp(JK))
allocate(Gammabox Table(NBoites,NTSG))
allocate(Gammabox Table temp(NBoites,NTSG))
allocate(Gamma res(2*max length,NTSG))
allocate(tens box(3,3,NBoites))
! Initializations
somme(:) = 0
somme temp(:) = 0
Gammabox Table(:,:) = 0
Gammabox Table temp(:,:) = 0
tens box(:,:,:) = 0
! 2) LECTURE OF FILM.BIN
! The results files
open(62,FILE=’../out/gammabox result.txt’,STATUS=’UNKNOWN’)
open(63,FILE=’../out/straintensor result.txt’,STATUS=’UNKNOWN’)
170
! READING OF GAMMABOX.BIN
! Initialization of parameters for the loop
kk=1
! start of the loop
ijk = -1
Steps loop 5: do while (kk.gt.0)
ijk=ijk+1
! A new simulation step
read(61,IOSTAT=KODERR) kk
print *,’kk=’,kk
if(KODERR ¡ 0) then
print*,” End of the gammabox.bin file”
stop
elseif(KODERR ¿ 0) then
print*,”Reading Error 1”,KODERR,ijk
kk=1
cycle Steps loop 5
endif
read(61,IOSTAT=KODERR) (somme temp(JK), JK=1,NTSG)
if(KODERR ¡ 0) then
print*,” End of the gammabox.bin file”
171
stop
elseif(KODERR ¿ 0) then
print*,”Reading Error 2”,KODERR,ijk
kk=1
cycle Steps loop 5
endif
! Calculation of the total gamma for each system at this step
do JK=1,NTSG
somme(JK) = somme(JK) + somme temp(JK)
enddo
! The segments micro structure is loaded
do ni=1,NBoites
read(61,IOSTAT=KODERR) (Gammabox Table temp(ni,JK),JK=1,NTSG)
enddo
if(KODERR ¡ 0) then
print*,” End of the gammabox.bin file”
stop
elseif(KODERR ¿ 0) then
print*,”Reading Error 3”,KODERR,ijk
cycle Steps loop 5
endif
172
! Calculation of the gamma on each box for each system at this step
do ni = 1,NBoites
do JK=1,NTSG
Gammabox Table(ni,JK) = Gammabox Table(ni,JK) + Gammabox Table temp(ni,JK)
enddo
enddo
! Calculation of the strain tensor for all boxes combined
do JK=1,NTSG
do i=1,3
do j=1,3
tens sum(i,j) = tens sum(i,j) + (somme(JK)*tens(i,j,JK))
enddo
enddo
enddo
! Calculation of the strain tensor for each box
do ni =1,NBoites
do JK=1,NTSG
do i=1,3
do j=1,3
tens box(i,j,ni) = tens box(i,j,ni) + (Gammabox Table(ni,JK)*tens(i,j,JK))
enddo
enddo
enddo
173
enddo
!3) ANALYSIS ON THE DESIRED STEPS
!The steps we want to use for statistic
if(kk ¿= debutaffich .and. kk ¡=finaffich .and. modulo(ijk, freqaffich) == 0) then ! Begin
of the ”Big If”
! Mean gamma is calculated along the desired axis
! Recall: the multipole boxes fit well the simulated volume dimensions
! Intializations
!number1 = 0
!number2 = 0
!Gamma res(:,:) = 0
!do IX = 1,NBoitesX
!do IY = 1,NBoitesY
!do IZ = 1,NBoitesZ
! Formule which gives the box number function of IX,IY ,IZ (similar to B1D 3D)
!I = IX + (IY-1) * NBoitesX + (IZ-1)* (NBoitesX*NBoitesY)
!do JK = 1,12
! Box organization along the desired axis (axis is of kind ¡100¿)
174
! J = int(IX * Axis(1) + IY * Axis(2) + IZ * Axis(3)) ! Box number j along the axis
! temp = Gammabox Table(I,JK) ! Gamma on the considered box
! test if: in order to know the number of layer along the desired axis
! if (((IX-1) * Axis(1) - (IX-1)) + ((IY-1) * Axis(2) - (IY-1)) + ((IZ-1) * Axis(3) -
(IZ-1)) == 0 &
! .and. (JK == 1)) then
! number1 = number1 + 1
! Gamma res(J,JK) = Gamma res(J,JK) + temp
! else
! Gamma res(J,JK) = Gamma res(J,JK) + temp
! endif
! test if: in order to know the number of boxes on each plan normal to the axis
! if (IX * Axis(1) + IY * Axis(2) + IZ * Axis(3) == 1 .and. JK == 1) then
! number2 = number2 + 1
! endif
!enddo
!enddo
!enddo
!enddo
!do I = 1,number1
! do JK = 1,12
! Gamma res(I,JK) = Gamma res(I,JK) / number2
! enddo
175
!enddo
! The output file is written
write(*,*) ”Writting of results on file gammabox result.txt”
write(62,’(a,I6)’) ’kk=’,kk
write(62,’(12(1x,a,I2))’) (’Gam on sys:’,JK,JK=1,NTSG)
write(62,’(12(2x,e11.4))’) (somme(JK), JK=1,NTSG)
write(62,*)
write(62,’(12(4x,a,I2))’) (’Gam box:’,JK, JK=1,NTSG)
do I=1,NBoites
write(62,’(12(6x,e11.4))’) (Gammabox Table(I,JK), JK=1,NTSG)
enddo
! The tensor output file is written
write(*,*) ”Writting of results on file straintensor result”
write(63,’(a,I6)’) ’kk=’,kk
write(63,’(a,I6)’) (’Strain Tensor on simulation box (e11, e12, e13, e21, e22, etc.:’)
do i=1,3
write(63,’(6(6x,e11.4))’) (tens sum(i,j), j=1,3)
enddo
write(63,’(a,I6)’) (’Strain Tensor on sub boxes:’)
do ni=1,NBoites
do i=1,3
176
write(63,’(6(6x,e11.4))’) (tens box(i,j,NBoites), j=1,3)
enddo
enddo
! END OF THE CALCULATIONS MADE ON THIS STEP
! ======¿¿¿¿¿
endif ! End of the ”big if” selecting the step
! 4) END
read(61,IOSTAT=KODERR) caractest
if(KODERR ¡ 0) then
print*,” End of the gammabox.bin file”
stop
elseif(KODERR ¿ 0) then
print*,”Reading error 4”,caractest
cycle Steps loop 5
endif
enddo Steps loop 5
deallocate(somme)
deallocate(somme temp)
deallocate(Gammabox Table)
deallocate(Gammabox Table temp)
177
deallocate(Gamma res)
close(61)
close(62)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
case default
print*,”bye bye...”
end select
enddo ! End of the main do while loop
stop
Modified following code to compute multiple planes of stress at specific intervals
! We first load information about the stress Grid
write(*,*) ”How many nodes per side shall we take for the squared stress grid”
read(*,*) Nnodes
if (Nnodes ¡ 5 .and. Nnodes ¿ 500 ) then
print*,’you should consider a different number of nodes, STOP!!!’
stop
178
endif
write(*,*) ”Aspect ratio between stress mapping and the simulation cell”
read(*,*) coeftaille
write(*,*) ”Number of replicates per side of the reference volume Nx, Ny, Nz ?”
read(*,*) NRep(1:3)
write(*,*) ”Enter the grid normal direction, x, y, z”
read(*,*) GridNormal(1:3)
GridNormalNorm = sqrt(GridNormal(1)**2+GridNormal(2)**2+GridNormal(3)**2)
GridNormal(1:3) = GridNormal(1:3)/GridNormalNorm
write(*,*) ”Enter the number of planes you want to analyze.”
read(*,*) Nplane
write(*,*) ”Enter the spacing between the planes in microns.”
read(*,*) PlaneSp
PlaneSp = PlaneSp * 1.D-6 / avalue
write(*,*) ”Enter the normal distance from the top most plane to the simulation cell
center (microns)”
read(*,*) GridDist
179
GridDist = GridDist * 1.D-6 / avalue
! Initialization
open(111,FILE=’../out/sigma11’,STATUS=’UNKNOWN’)
open(122,FILE=’../out/sigma22’,STATUS=’UNKNOWN’)
open(133,FILE=’../out/sigma33’,STATUS=’UNKNOWN’)
open(123,FILE=’../out/sigma23’,STATUS=’UNKNOWN’)
open(113,FILE=’../out/sigma13’,STATUS=’UNKNOWN’)
open(112,FILE=’../out/sigma12’,STATUS=’UNKNOWN’)
allocate (sigint(3,3,Nnodes,Nnodes))
allocate (sigint2(3,3,Nnodes,Nnodes))
sigint(:,:,:,:)=0.0D0
sigint2(:,:,:,:)=0.0D0
hide(1:nsegm) = .false.
! Important segments information is predefined
do K = 1,nsegm
! rotule segments can be eliminated from the calculations
if (seg(k)%norme == 0) hide(k)=.true.
! The segment vector direction
T(1:3,K) = bveclin(1:3,seg(k)%veclin)/normlin(seg(k)%veclin)
! The Burgers vector of i
indexB = (int((seg(k)%veclin-1)/8))*8 + 1
180
B(1:3,K) = bveclin(1:3,indexB)/normlin(indexB)
!B produit vectoriel T
bpvt(1,k) = b(2,k)*t(3,k)-b(3,k)*t(2,k)
bpvt(2,k) = b(3,k)*t(1,k)-b(1,k)*t(3,k)
bpvt(3,k) = b(1,k)*t(2,k)-b(2,k)*t(1,k)
enddo
! Two elementary vectors defining the stress plan of mapping
if(GridNormal(1)+GridNormal(2).eq.0.) then
xxp1= GridNormal(3)/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(3)*GridNormal(3))
yyp1=0
zzp1=-GridNormal(1)/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(3)*GridNormal(3))
xxp2=(GridNormal(1)*GridNormal(2))/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(3)*GridNormal(3))
yyp2=-sqrt(GridNormal(1)*GridNormal(1)+GridNormal(3)*GridNormal(3))
zzp2=(GridNormal(2)*GridNormal(3))/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(3)*GridNormal(3))
else
xxp1= GridNormal(2)/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(2)*GridNormal(2))
yyp1=-GridNormal(1)/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(2)*GridNormal(2))
zzp1=0
xxp2=(GridNormal(1)*GridNormal(3))/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(2)*GridNormal(2))
yyp2=(GridNormal(2)*GridNormal(3))/sqrt(GridNormal(1)*GridNormal(1)+GridNormal(2)*GridNormal(2))
zzp2=-sqrt(GridNormal(1)*GridNormal(1)+GridNormal(2)*GridNormal(2))
181
endif
! Mapping increment
! widthp= coeftaille*MIN(Modur(1),Modur(2),Modur(3))/Nnodes
widthp= coeftaille*MAX(Modur(1),Modur(2),Modur(3))/real(Nnodes)
! Calculations can be done on multiple (Nplane) planes automatically.
do KK = 1,Nplane
sigint(:,:,:,:) = 0
sigint2(:,:,:,:) = 0
! The Center of the stress mapping
MapCenter(1)=BoxCenter(1)+GridNormal(1)*(GridDist-(PlaneSp*(KK-1)))
MapCenter(2)=BoxCenter(2)+GridNormal(2)*(GridDist-(PlaneSp*(KK-1)))
MapCenter(3)=BoxCenter(3)+GridNormal(3)*(GridDist-(PlaneSp*(KK-1)))
do i=1,Nnodes
print*,i
rr1= mapcenter(1)+real((i-Nnodes/2),DP)*xxp1*widthp
rr2= mapcenter(2)+real((i-Nnodes/2),DP)*yyp1*widthp
rr3= mapcenter(3)+real((i-Nnodes/2),DP)*zzp1*widthp
do j=1,Nnodes
r(1)=rr1+real((j-Nnodes/2),DP)*xxp2*widthp
r(2)=rr2+real((j-Nnodes/2),DP)*yyp2*widthp
182
r(3)=rr3+real((j-Nnodes/2),DP)*zzp2*widthp
if ((r(1) ¿= 0) .and. (r(2) ¿= 0) .and. (r(3) ¿= 0) .and. &
& (r(1) ¡= Modur(1)) .and. (r(2) ¡= Modur(2)) .and. (r(3) ¡= modur(3))) then
call sigma int(nsegm,r,sigtmp,sigtmp2,NRep) ! Calculation of the stress tensor at
coordinate r
sigint(:,:,i,j)=sigtmp(:,:)
sigint2(:,:,i,j)=sigtmp2(:,:)
endif
end do
end do
! Results of the stress computations are saved
! Results of sigma are saved in MPa
99 format (1x)
write(111,*) ’layer:’ ,KK
write(112,*) ’layer:’ ,KK
write(122,*) ’layer:’ ,KK
write(113,*) ’layer:’ ,KK
write(123,*) ’layer:’ ,KK
write(133,*) ’layer:’ ,KK
do i=Nnodes,1,-1
do k=Nnodes,1,-1
183
write(111,*) Nnodes/2-i, Nnodes/2-k, sigint(1,1,k,i) * mu * 1D-6,sigint2(1,1,k,i) * mu
* 1D-6
write(112,*) Nnodes/2-i, Nnodes/2-k, sigint(1,2,k,i) * mu * 1D-6,sigint2(1,2,k,i) * mu
* 1D-6
write(122,*) Nnodes/2-i, Nnodes/2-k, sigint(2,2,k,i) * mu * 1D-6,sigint2(2,2,k,i) * mu
* 1D-6
write(113,*) Nnodes/2-i, Nnodes/2-k, sigint(1,3,k,i) * mu * 1D-6,sigint2(1,3,k,i) * mu
* 1D-6
write(123,*) Nnodes/2-i, Nnodes/2-k, sigint(2,3,k,i) * mu * 1D-6,sigint2(2,3,k,i) * mu
* 1D-6
write(133,*) Nnodes/2-i, Nnodes/2-k, sigint(3,3,k,i) * mu * 1D-6,sigint2(3,3,k,i) * mu
* 1D-6
end do
end do
write(111,*)
write(112,*)
write(122,*)
write(113,*)
write(123,*)
write(133,*)
end do
184
B.2 LRIS codes in Igor Pro
#pragma rtGlobals=1 // Use modern global access method.
// Returns full path to destination wave as a string.
Function/S WavesAverage()
String baseName // name for source wave
String rangeofwave // range of waves to average
String wave ave // name for destination wave
String wn // contains the name of a particular wave
String wl // contains a list of wave names Variable index // get list of waves whose
names start with baseName
Variable first, last
Prompt baseName, ”What is the base name of the waves to average?”
Prompt first, ”What is the first file in the average?”
Prompt last, ”What is the last file in the average?”
DoPrompt ”Basename”, basename
DoPrompt ”First”, first
DoPrompt ”Last”, last
wave ave = ”ave ”+baseName //Label target wave
index = (first-1)
wl = WaveList(baseName+”*”, ”;”, ””)
wn = StringFromList(0, wl)
Duplicate/O $wn, $wave ave
185
WA VE dest = $wave ave // create wave ref for destination
dest = 0
do
wn = StringFromList (index, wl) // get next wave
if (strlen(wn) == 0) //no more names in list, break out of loop
break
endif
WA VE source =$wn //create wave ref for source
dest += source //add source to dest
index += 1
while (index ¡ last) //loop until last slice is added
dest /= (1+(last-first)) //divide by number of waves
return GetWavesDataFolder (dest,2) //string is full path to wave
KillWaves wn, wl, baseName, wave ave
End
Function GraphAll()
String baseName //First part of wave name in series to be read in
string LRISwavelist //List of waves to be graphed
string theWave //Target wave to graph
Variable index=0
//make list of waves Prompt baseName, ”What is the base name of the waves
you want graph”
186
DoPrompt ”Basename”, basename
LRISwavelist = WaveList(baseName+”*”, ”;”, ””)
//graph all waves in list
do
//get the next wave name
theWave = StringFromList(index, LRISwavelist)
if (strlen (theWave) == 0) //end do loop when all waves are graphed
break
endif
Display y1 vs x1 // Style of graph
ModifyGraph mode=3,marker=16, msize=9, zColor(y1)=f$theWave,-
250,250,RedWhiteBlue,0g
Label bottom NameOfWave($theWave)
index += 1
while (1)
End
Function GraphOne(TarWave) //Graph a single slice
Wave TarWave
Display y1 vs x1
ModifyGraph mode=3,marker=16, msize=9, zColor(y1)=TarWave,-
250,250,RedWhiteBlue,0
Label bottom NameOfWave(TarWave)
End
Function LRIS calc(TarWave,xwave,ywave)
187
wave TarWave //this is the z wave to be ”averaged”
wave xwave //this is the x coordinate wave
wave ywave //this is the y coordinate wave
Variable xx,yy,zz //counters for loops
Variable index=0 //counter for position in wavez
Variable xstart, xend, ystart, yend
Variable LRIS
String Tempwave
Prompt xstart, ”What is the x-start for LRIS calculation” //get inputs from user on
the rectangular
DoPrompt ”xstart”, xstart //area you wish you get an average stress of
Prompt xend, ”what is the x-end for LRIS calculation”
DoPrompt ”xend”, xend
Prompt ystart, ”What is the y-start for LRIS calculation”
DoPrompt ”ystart”, ystart
Prompt yend, ”what is the y-end for LRIS calculation”
DoPrompt ”yend”, yend
for(xx=xwave(0); xx¡=xend;xx+=1)
if(xx¿=xstart)
for(yy=ywave(0);yy¡=(abs(ywave(0))-1);yy+=1)
if(yy¿=ystart && yy¡=yend)
LRIS += TarWave(index)
endif
188
index +=1
endfor
else
index += 2*abs(xwave(0))
endif
endfor
LRIS /= ((xend-xstart)+1)*((yend-ystart)+1)
print ”x-range = ”,xstart,”to”,xend
print ”y-range = ”,ystart,”to”,yend
print ”The average LRIS = ”,LRIS
end
Function histo calc(TarWave,xwave,ywave)
wave TarWave //this is the z wave to be ”averaged”
wave xwave //this is the x coordinate wave
wave ywave //this is the y coordinate wave
Variable xx,yy,zz //counters for loops
Variable index=0 //counter for position in wavez
Variable xstart, xend, ystart, yend
Variable wavelength
Variable index2=0
Prompt xstart, ”What is the x-start for LRIS calculation” //get inputs from user on
the retangular
DoPrompt ”xstart”, xstart //area you wish you get an average stress of
Prompt xend, ”what is the x-end for LRIS calculation”
189
DoPrompt ”xend”, xend
Prompt ystart, ”What is the y-start for LRIS calculation”
DoPrompt ”ystart”, ystart
Prompt yend, ”what is the y-end for LRIS calculation”
DoPrompt ”yend”, yend
wavelength = (xend - xstart+1)*(yend-ystart+1)
Make /N=(wavelength) Histo wave
Histo wave = 0
for(xx=xwave(0); xx¡=xend;xx+=1)
if(xx¿=xstart)
for(yy=ywave(0);yy¡=(abs(ywave(0))-1);yy+=1)
if(yy¿=ystart && yy¡=yend)
Histo wave[index2] = TarWave[index]
index2 +=1
endif
index +=1
endfor
else
index += 2*abs(xwave(0))
endif
endfor
print ”x-range = ”,xstart,”to”,xend
print ”y-range = ”,ystart,”to”,yend
190
Make/N=100/O Histo wave Hist;DelayUpdate
Histogram/B=1 Histo wave,Histo wave Hist
Display Histo wave Hist
SetAxis/A
ModifyGraph mode=5, rgb=(0,0,0)
end
191

Abstract (if available)

Abstract The subject of backstresses or long range internal stresses (LRIS) in plastically deformed crystalline materials has been investigated extensively over the past 40 years. It is currently believed that elevated stresses are present in regions of elevated dislocation density or dislocation heterogeneities in deformed dislocation microstructures. The heterogeneities include dislocation pile-ups, edge dislocation dipole bundles and dislocation dense cell walls in monotonically and cyclically deformed materials. Understanding the magnitude of long range internal stresses (LRIS) is especially important for the understanding of cyclic deformation and monotonic deformation. First, the fundamental theories needed to understand the subject matter are presented. Following this, is a summary of select previous experiments aimed at assessing LRIS. This will then be followed by my results from my experiments and analysis. ❧ A number of experiments were performed at the Advanced Photon Source at Argonne National Labs in an effort to measure LRIS within deformed single crystal copper. Measurements were taken from a large number of locations within the dislocation microstructure, both in dislocation dense regions, and regions with low dislocation density. The results from these experiments, as well as additional analysis relating to the nature of the LRIS present, will be discussed. ❧ Finally, the question as to the origin of LRIS will be discussed. It’s clear that the dislocation microstructure is responsible for the stresses present, but it is unclear as to the configuration of the dislocations within the microstructure (ie. Burgers vector and sign) which causes the LRIS. Theories present possible microstructural configurations which may produce LRIS, and these will be compared to a TEM g · b analysis of the dislocation microstructure. Additionally, dislocation dynamics simulations (DDS) were performed in an effort to model LRIS and the microstructure responsible. These models will be analyzed and the result discussed in detail.

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Asset Metadata

Creator Geantil, Peter T. (author)

Core Title Long range internal stresses in deformed single crystal copper

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 10/02/2013

Defense Date 09/09/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag long range internal stress,OAI-PMH Harvest,single crystal copper,X-ray diffraction

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kassner, Michael E. (committee chair), Goo, Edward K. (committee member), Hodge, Andrea M. (committee member), Levine, Lyle E. (committee member)

Creator Email petertrain@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-334478

Unique identifier UC11296561

Identifier etd-GeantilPet-2074.pdf (filename),usctheses-c3-334478 (legacy record id)

Legacy Identifier etd-GeantilPet-2074.pdf

Dmrecord 334478

Document Type Dissertation

Format application/pdf (imt)

Rights Geantil, Peter T.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA