Using X-ray microbeam diffraction to study the long range internal stresses in plastically deformed materials

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Content USING X-RAY MICROBEAM DIFFRACTION TO STUDY THE LONG RANGE
INTERNAL STRESSES IN PLASTICALLY DEFORMED MATERIALS
by
I-Fang Lee
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATERIALS SCIENCE)
May 2015
Copyright 2015 I-Fang Lee
To my family
ii
Contents
Dedication ii
List of Figures viii
List of Tables xii
Acknowledgments xiv
Abstract xv
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Dislocations, Metal Crystal Structures and Slip Systems . . . . 3
1.1.1.1 Dislocations . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1.2 Metal Crystal Structures and Slip Systems . . . . . . 3
1.1.2 Plastic Deformation and Microstructures of Dislocations . . . . 5
1.1.2.1 Single Slip Deformation (Formation of Persistent Slip
Bands) . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2.2 Multiple Slip (Unloaded [001]-oriented Single-crystal
Cu Monotonic Deformation) . . . . . . . . . . . . . 7
1.1.2.3 Single-crystal Cu Cyclic Deformed to Pre-saturation . 8
1.1.2.4 Uniaxial Compression Creep Test of Polycrystalline
Copper . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.3.1 Internal Stress Theory of Bauschinger Effect . . . . . 11
1.1.3.2 Dislocation Theory- Pile-up Model . . . . . . . . . . 12
1.1.3.3 Dislocation Theory- Non-LRIS Approach . . . . . . 13
1.1.3.4 Bauschinger Effect- Composite Model . . . . . . . . 14
1.2 Experimental Evidence of LRIS . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 LRIS in Multipolar Edge-dislocation Wall Structures . . . . . . 16
1.2.2 Asymmetry of X-ray Diffraction Line Profiles (Multiple-Slip
Deformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3 Convergent Beam Electron Diffraction (CBED) Measurements . 19
1.2.4 Microbeam Measurements- Monotonic Deformation of Copper 21
1.3 Summary of LRIS in Previous Experiments . . . . . . . . . . . . . . . 22
iii
1.3.1 The Magnitude of LRIS . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 LRIS Dependence on Cell Interior Size . . . . . . . . . . . . . 26
2 Experimental Procedure 28
2.1 Introduction of Synchrotron Radiation . . . . . . . . . . . . . . . . . . 28
2.1.1 Synchrotron Radiation and Emission Mechanism . . . . . . . . 28
2.1.2 Properties of Synchrotron Radiation . . . . . . . . . . . . . . . 29
2.1.3 Synchrotron Radiation at Argonne National Laboratory . . . . . 29
2.1.3.1 Linear Accelerator . . . . . . . . . . . . . . . . . . . 29
2.1.3.2 Booster Synchrotron . . . . . . . . . . . . . . . . . . 30
2.1.3.3 The Electron Storage Ring . . . . . . . . . . . . . . 30
2.1.3.4 Insertion Devices . . . . . . . . . . . . . . . . . . . 30
2.2 X-ray Microbeam Measurements . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Three-dimensional X-ray Microbeam Diffraction . . . . . . . . 32
2.2.2.1 Differential-Aperture X-ray Microscopy (DAXM) . . 32
2.2.2.2 Three-dimensional Resolution of the X-ray Microbeam 34
2.2.2.3 Uncertainty of Single Component Strain . . . . . . . 34
2.2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3.1 Strain Data Analysis . . . . . . . . . . . . . . . . . . 36
2.2.3.2 Strain in the Cell Interiors and Cell Walls . . . . . . . 36
2.3 X-ray Powder Diffraction Measurements . . . . . . . . . . . . . . . . . 38
2.3.1 11-BM Beamline Characteristics at the APS . . . . . . . . . . . 38
2.3.2 Calibration of 11-BM Beamline . . . . . . . . . . . . . . . . . 39
2.3.3 Sample Preparation for 11-BM beamline X-ray Powder Diffrac-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Previous Experiment: Strain Measurements of Monotonically Deformed
Single-crystal Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Proof of the Composite Model . . . . . . . . . . . . . . . . . . 42
3 LRIS in Aluminum Processed by ECAP 44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Severe Plastic Deformation (SPD) Techniques . . . . . . . . . . . . . . 45
3.3 Mechanical Properties versus Grain Size . . . . . . . . . . . . . . . . . 46
3.4 Non-equilibrium Grain Boundaries . . . . . . . . . . . . . . . . . . . . 47
3.5 Equal-Channel Angular Processing (ECAP) . . . . . . . . . . . . . . . 47
3.5.1 Microstructure of ECAP Metals . . . . . . . . . . . . . . . . . 51
3.5.2 Shear Strain in ECAP . . . . . . . . . . . . . . . . . . . . . . 52
3.5.3 Principal Strains and Direction in ECAP . . . . . . . . . . . . . 57
3.5.3.1 Geometric Analysis for Principal Strains after 1-pass
ECAP . . . . . . . . . . . . . . . . . . . . . . . . . 57
iv
3.5.3.2 Matrix Algebraic Analysis for Principal Strains after
1-pass ECAP . . . . . . . . . . . . . . . . . . . . . . 60
3.5.4 Grain Elongation for ECAP Multiple Passes . . . . . . . . . . . 63
3.5.5 Mechanical Properties of ECAP AA1050 . . . . . . . . . . . . 66
3.5.6 Grain Size and Grain Boundaries of ECAP Aluminum . . . . . 66
3.6 LRIS in ECAP Metals by CBED/HOLZ Method . . . . . . . . . . . . . 68
3.7 Sample Preparation and Experimental Procedures . . . . . . . . . . . . 71
3.7.1 ECAP AA1050 via RouteB
C
. . . . . . . . . . . . . . . . . . 71
3.7.2 ECAP AA6005 via RouteC . . . . . . . . . . . . . . . . . . . 73
3.7.2.1 Mechanical Properties of AA6005 via ECAP RouteC 74
3.7.3 Sample Preparation of X-ray Powder Diffraction Measurement . 74
3.7.4 Sample Preparation of X-ray Microbeam Measurement . . . . . 75
3.7.4.1 Surface Electropolishing for X-ray Microbeam Diffrac-
tion Measurement . . . . . . . . . . . . . . . . . . . 76
3.7.5 X-ray Microbeam Measurement Setup . . . . . . . . . . . . . . 77
3.7.5.1 Experimental Procedures of X-ray Microbeam Mea-
surements . . . . . . . . . . . . . . . . . . . . . . . 77
3.7.5.2 Defining the Region of Interest (ROI) for the ECAP
sample . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7.5.3 Obtaining Accurate Bragg Angle for Fine-grain Sam-
ples . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.7.5.4 ECAP AA1050 1-pass . . . . . . . . . . . . . . . . . 80
3.7.5.5 ECAP AA1050 Multiple-pass via RouteB
C
. . . . . 81
3.7.5.6 ECAP AA6005 1-pass and 2-pass via RouteC . . . . 82
3.7.5.7 Rotation Angles of Reflection Planes . . . . . . . . . 82
3.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.8.1 Microstructure of ECAP AA1050 via RouteB
C
. . . . . . . . . 86
3.8.2 Microstructure of ECAP AA6005 via RouteC . . . . . . . . . 88
3.8.3 Powder Diffraction of ECAP AA1050 via RouteB
C
with Dif-
ferent Passes . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8.4 Powder Diffraction of ECAP AA6005 via RouteC with Differ-
ent Passes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.8.5 Grain/subgrain Interior Strains of ECAP AA1050 after 1-pass . 93
3.8.6 Grain/subgrain Interior Strains of ECAP AA1050 Multiple-pass
via RouteB
C
. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.8.7 Grain/subgrain Interior Strains of ECAP AA6005 1-pass and 2-
pass via RouteC . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.9.1 LRIS in Grain/subgrain Interiors for ECAP AA1050 1-pass . . 98
3.9.1.1 Estimate of High and Low Dislocation Ratio . . . . . 100
3.9.1.2 Composite Model . . . . . . . . . . . . . . . . . . . 101
v
3.9.2 LRIS in Grain/subgrain Interiors for ECAP AA1050 Multiple-
pass via RouteB
C
. . . . . . . . . . . . . . . . . . . . . . . . 103
3.9.3 LRIS in Grain/subgrain Interiors for ECAP AA6005 1-pass and
2-pass via RouteC . . . . . . . . . . . . . . . . . . . . . . . . 104
3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.10.1 ECAP AA1050 1-pass . . . . . . . . . . . . . . . . . . . . . . 104
3.10.2 ECAP AA1050 Multiple-pass via RouteB
C
. . . . . . . . . . . 105
3.10.3 ECAP AA6005 1-pass and 2-pass via RouteC . . . . . . . . . 105
4 LRIS in Deformed Single Crystal Copper 120
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3 X-ray Macrobeams versus X-ray Microbeams . . . . . . . . . . . . . . 122
4.4 X-ray Line Profile Asymmetry . . . . . . . . . . . . . . . . . . . . . . 123
4.5 Cell Interior and Wall X-ray Subprofiles . . . . . . . . . . . . . . . . . 124
4.6 Decomposition of Asymmetric X-ray Line Profiles by Mughrabi . . . . 126
4.7 Reconstruction of X-ray Subprofiles Measured by X-ray Microbeams . 127
4.8 Determine Cell/wall Ratio and Cell/wall Strains by Fitting the Asym-
metric Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.9 Summary Results of Various Fitting Methods . . . . . . . . . . . . . . 133
4.10 Sample Preparation and Experimental Procedures . . . . . . . . . . . . 134
4.10.1 Sample Preparation and X-ray Microbeam Measurement . . . . 135
4.10.2 Sample Preparation of TEM Microstructure Analysis . . . . . . 136
4.11 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.11.1 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.11.2 The Reconstruction of an Asymmetric X-ray Line Profiles by
X-ray Microbeam Diffraction . . . . . . . . . . . . . . . . . . 141
4.11.3 The Reconstruction of Cell Interior Subprofiles . . . . . . . . . 144
4.11.4 The Reconstruction of Cell Wall Subprofiles . . . . . . . . . . 146
4.11.5 Mean Value of Lattice Spacings . . . . . . . . . . . . . . . . . 150
4.11.6 Residual Strain/stress . . . . . . . . . . . . . . . . . . . . . . . 152
4.12 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.12.1 The Ratio of Cell Interiors and Walls . . . . . . . . . . . . . . 154
4.12.2 The Strains/stresses of Cell Interiors and Walls . . . . . . . . . 157
4.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5 Summary 164
Reference List 166
A Appendix1 174
A.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
vi
B Appendix2 177
B.1 Geometry File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
vii
List of Figures
1.1 Definition of the Burgers vector and Burgers circuit . . . . . . . . . . . 4
1.2 A dislocation loop in a slip plane in a deformed crystal [9] . . . . . . . 4
1.3 Schematics of FCC conventional unit cell and slip systems . . . . . . . 5
1.4 Deformation of copper by single slip and the formation microstructure . 7
1.5 Composite structure of dislocation cell interiors and cell walls of [001]
oriented Cu single crystal under monotonic deformation . . . . . . . . 8
1.6 Transmission electron microscopy (TEM) image of single-crystal Cu
deformed cyclically to pre-saturation [2] . . . . . . . . . . . . . . . . 9
1.7 TEM images of Cu deformed at different conditions . . . . . . . . . . . 10
1.8 A schematic showing the Bauschinger effect [21] . . . . . . . . . . . . 12
1.9 A schematic of dislocation pile-up in grains/subgrains . . . . . . . . . . 13
1.10 Stress-strain curve of (alpha) Zn single crystal . . . . . . . . . . . . . 14
1.11 The composite of heterogeneous dislocation microstructures . . . . . . 15
1.12 Stress-strain curve of cells and walls in the composite model . . . . . . 16
1.13 Local stresses in the channel of PSBs . . . . . . . . . . . . . . . . . . . 18
1.14 Composite model of the stressed dislocation cell/wall structure for multiple-
slip deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.15 Line broadening and asymmetry increases with increasing deformation . 20
1.16 Bright-field TEM image of a deformed copper polycrystal . . . . . . . 21
1.17 Laue zero-order CBED pattern . . . . . . . . . . . . . . . . . . . . . . 22
1.18 Cell interior strains under tensile or compression deformation . . . . . . 23
1.19 Strain distribution histogram . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 A schematic of an insertion device of undulator . . . . . . . . . . . . . 31
2.2 A schematic of differential-aperture X-ray microscopy (DAXM) [31] . . 33
2.3 A schematic of triangulation of microbeam diffraction [31] . . . . . . . 34
2.4 Laue pattern and X-ray line profiles of cell interiors and walls . . . . . 37
2.5 Sample holder for the 11-BM beamline X-ray powder diffraction . . . . 39
2.6 A TEM micrograph of a [001] deformed single-crystal Cu [5] . . . . . 41
2.7 A schematic of an X-ray microbeam wire scan . . . . . . . . . . . . . . 42
2.8 Stress distribution of cell interiors and walls of compression deformed
Cu single crystal along a [001] crystal axis . . . . . . . . . . . . . . . . 43
3.1 Schematics of severe plastic deformation techniques . . . . . . . . . . . 46
3.2 TEM micrographs of non-equilibrium grain boundaries . . . . . . . . . 48
viii
3.3 The four fundamental processing routes of ECAP . . . . . . . . . . . . 49
3.4 The material is deformed through simple shear . . . . . . . . . . . . . 50
3.5 Schematics of shear directions and planes for different ECAP routes [70] 51
3.6 Microstructure of pure aluminum after varying ECAP routes . . . . . . 53
3.7 Simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 An infinitesimal shear deformation ofd under simple shear [72] . . . . 55
3.9 A simple shear model of ECAP . . . . . . . . . . . . . . . . . . . . . . 57
3.10 A parallelogramBAab in the coordinateX
1
O
1
Y
1
deforms intoB
0
A
0
a
0
b
0
in the coordinateX
2
O
2
Y
2
in ECAP [72] . . . . . . . . . . . . . . . . . 58
3.11 A deformation model for ECAP [72] . . . . . . . . . . . . . . . . . . . 61
3.12 Principal direction and shear direction in ECAP . . . . . . . . . . . . . 62
3.13 A schematic shows the principal strains, principal directions, and the
direction of shear for the ECAP die of =

2
[72] . . . . . . . . . . . 63
3.14 Shape change of a circular element in the X-Z plane for material pro-
cessed by routeA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.15 True stress-true strain curves of the ECAP-processed AA1050 . . . . . 67
3.16 EBSD mapping of ECAP AA1050 with processing of routeB
C
. . . . . 68
3.17 A schematic of zero-order Laue zone and higher-order Laue zone . . . . 69
3.18 A TEM image of AA1050 after 2-pass ECAP via routeB
C
[56] . . . . 71
3.19 A TEM image of AA1050 after 4-pass ECAP via routeB
C
[56] . . . . 72
3.20 Geometry of the ECAP die for AA1050 . . . . . . . . . . . . . . . . . 72
3.21 The electropolishing schematic . . . . . . . . . . . . . . . . . . . . . . 76
3.22 Back-reflection Laue diffraction of ECAP Al sample . . . . . . . . . . 78
3.23 A region of interest (ROI) for a Laue pattern . . . . . . . . . . . . . . . 79
3.24 Wire scan to obtain an accurate Bragg’s angle. . . . . . . . . . . . . . . 80
3.25 A schematic view of the X-ray microbeam setup for AA1050 1-pass ECAP 81
3.26 A schematic of rotation angles, RotX and RotZ with respect to the 45

plane normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.27 The measured strains along the pressing direction versus RotX and RotZ
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.28 The measured strains along +22:5

with respect to pressing direction
versus RotX and RotZ angles . . . . . . . . . . . . . . . . . . . . . . . 86
3.29 The measured strains along22:5

with respect to pressing direction
versus RotX and RotZ angles . . . . . . . . . . . . . . . . . . . . . . . 87
3.30 A TEM image of AA1050 after 1-pass ECAP . . . . . . . . . . . . . . 88
3.31 TEM images of ECAP AA1050 after 2, 4 and 8 passes via route B
C
,
respectively from (a)-(c) . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.32 TEM bright field image of ECAP AA6005 via routeC . . . . . . . . . 90
3.33 Shear band formation around Fe-rich intermetallics in ECAP AA6005
via routeC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.34 Dislocation interaction with Mg
2
Si particle after 2-pass ECAP routeC
of AA6005
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
ix
3.35 High resolution X-ray powder diffraction spectrum of ECAP AA1050
via routeB
C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.36 X-ray powder diffraction (331) peak of AA1050 for different ECAP
passes using routeB
C
. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.37 X-ray powder diffraction (420) peak of AA1050 for different ECAP
passes using routeB
C
. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.38 X-ray powder diffraction (531) peak of AA1050 for different ECAP
passes using routeB
C
. . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.39 High resolution powder diffraction spectrum of ECAP AA6005 using
RouteC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.40 X-ray powder diffraction (331) peak of ECAP AA6005 via RouteC for
different passes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.41 X-ray powder diffraction (420) peak of ECAP AA6005 using RouteC
for different passes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.42 Powder diffraction (531) peak of AA6005 for different ECAP passes . . 113
3.43 Strain distribution of grain/subgrain interior elastic strains of AA1050
1-pass ECAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.44 Strains in the grain/subgrain interiors of ECAP AA1050 via route B
C
for different number of passes near the pressing direction (+4:9

) . . . . 115
3.45 LRIS-to-flow stress ratio in the grain/subgrain interiors of ECAP AA1050
versus ECAP number of passes . . . . . . . . . . . . . . . . . . . . . . 116
3.46 The internal strain distribution of AA6005 after and 2-pass ECAP routeC117
3.47 Laue diffraction pattern of AA6005 after ECAP 2-pass routeC . . . . . 118
3.48 Laue diffraction pattern of 2-pass ECAP AA1050 via routeB
C
. . . . . 119
4.1 Integrated microbeam (006) line profiles from deformed Cu versus the
X-ray macrobeam profile . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 Asymmetric X-ray line profiles . . . . . . . . . . . . . . . . . . . . . . 125
4.3 The decomposition of asymmetric X-ray line profiles . . . . . . . . . . 126
4.4 Progressive decomposition method of the asymmetric line profiles into
cell interior and wall subprofiles . . . . . . . . . . . . . . . . . . . . . 128
4.5 X-ray line profiles measured by X-ray microbeam diffraction . . . . . . 129
4.6 The assembled X-ray line profile from cell interior and wall subprofiles
measured by X-ray microbeams . . . . . . . . . . . . . . . . . . . . . 130
4.7 Comparison between a progressive method for derived subprofiles and
microbeam measured subprofiles . . . . . . . . . . . . . . . . . . . . . 131
4.8 Using different fitting functions to determine the cell interior and wall
ratios and subprofiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.9 Stereographic projection of a crystal . . . . . . . . . . . . . . . . . . . 136
4.10 A schematic of jet electropolishing . . . . . . . . . . . . . . . . . . . . 137
4.11 Bright field TEM image of sample BE1 at two different magnifications . 139
4.12 Bright field TEM image of sample BE2 . . . . . . . . . . . . . . . . . 140
x
4.13 Bright field TEM image of sample OA deformation . . . . . . . . . . . 141
4.14 Cell interior and cell wall volume ratio calculation . . . . . . . . . . . . 142
4.15 X-ray microbeam integrated (006) line profile for sample BE1 condition 143
4.16 X-ray microbeam integrated (006) line profile for sample BE2 condition 144
4.17 X-ray microbeam integrated (515) line profile for sample OA condition 145
4.18 The reconstructed cell interior profile of sample BE1 . . . . . . . . . . 146
4.19 The reconstructed cell interior profile of sample BE2 . . . . . . . . . . 147
4.20 The reconstructed cell interior profile of sample OA . . . . . . . . . . . 148
4.21 The decomposition of wall subprofile for sample BE1 . . . . . . . . . . 149
4.22 The decomposition of wall subprofile for sample BE2 . . . . . . . . . . 150
4.23 The decomposition of wall subprofile for sample OA . . . . . . . . . . 151
4.24 The reconstruction of wall subprofiles with different cell interior-wall
ratios for sample BE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.25 The reconstruction of wall subprofiles with different cell interior-wall
ratios for sample BE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.26 The reconstruction of wall subprofiles with different cell interior-wall
ratios for sample OA . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.27 The stress-strain curve of Cu sample BE1 . . . . . . . . . . . . . . . . 159
4.28 The stress-strain curve of Cu sample BE2 . . . . . . . . . . . . . . . . 160
4.29 The stress-strain curve of Cu sample OA . . . . . . . . . . . . . . . . . 161
xi
List of Tables
1.1 Common slip system of metals [7–9] . . . . . . . . . . . . . . . . . . . 6
1.2 Selected LRIS measurements [21] . . . . . . . . . . . . . . . . . . . . 24
1.3 Selected LRIS measurements (continued) . . . . . . . . . . . . . . . . 25
2.1 Summary of 11-BM technical specifications [42] . . . . . . . . . . . . 38
3.1 Calculated angles () between grain elongation direction and the extru-
sion x-axis [68] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Grain/subgrain boundary properties and dislocation density of ECAP
1-pass AA1050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Chemical composition of the AA6005 . . . . . . . . . . . . . . . . . . 73
3.4 Grain/subgrain sizes after ECAP routeC AA6005 . . . . . . . . . . . . 74
3.5 Hardness (HV) of AA6005 via ECAP RouteC . . . . . . . . . . . . . 74
3.6 Tensile test (MPa) of AA6005 via ECAP RouteC . . . . . . . . . . . . 74
3.7 Electro-chemically polishing solution and conditions of ECAP AA1050
and AA6005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8 Peak position and measured lattice spacing analysis of ECAP AA1050
using routeB
C
for different passes by X-ray powder diffraction . . . . 92
3.9 Measured lattice constants of ECAP AA1050 via route B
C
by X-ray
powder diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.10 Lattice constants of ECAP AA6005 via RouteC for different passes . . 93
3.11 Internal strain summary of ECAP AA1050 1-pass . . . . . . . . . . . . 95
3.12 Internal strain summary for ECAP AA6005 2-pass via routeC along the
pressing direction in grain/subgrain interiors . . . . . . . . . . . . . . . 97
4.1 Peak parameters for the subprofiles fitted by different functions [43] . 134
4.2 Mean value ofq and d-spacing . . . . . . . . . . . . . . . . . . . . . . 152
4.3 Summary of residual strain/stress in Cu samples . . . . . . . . . . . . . 153
4.4 The summary of cell and wall strain from peak position of fitting profiles 158
4.5 The summary of cell interior and wall stress from peak position of fitting
profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.1 AA1050 1-pass ECAP along pressing direction grain/subgrain interior
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
A.2 AA1050 1-pass ECAP tiled +22:5

with respect to the pressing direc-
tion grain/subgrain interior data . . . . . . . . . . . . . . . . . . . . . . 175
xii
A.3 AA1050 1-pass ECAP tiled22:5

with respect to the pressing direc-
tion grain/subgrain interior data . . . . . . . . . . . . . . . . . . . . . . 175
A.4 AA1050 multiple-pass ECAP via routeB
C
grain/subgrain interiors data 176
A.5 AA6005 2-pass ECAP routeC grain/subgrain interiors data . . . . . . . 176
xiii
Acknowledgments
I would like to thank my advisor, Dr. Michael E. Kassner for the continuous support
and guidance of my Ph.D. study, for his patience and understanding. I would also like
to thank my co-advisor Dr. Lyle E. Levine. His guidance helps me complete my Ph.D.
study and inspires me to solve many critical challenges. I wish to thank Prof. Andrea
Hodge, Prof. Steve Nutt and Andy Chen for encouraging me to continue studying and
always being supportive and the generous support from WISE.
Additional thanks are extended to my committee members and faculty members,
Prof. Edward Goo, Prof. Veronica Eliasson, Prof. Ping Wang and Prof. Terence Lang-
don for their guidance and insightful comments. I greatly appreciate all the supports
from Dr. Jon Tischler and Dr. Ruqing Xu at the APS at Argonne National Labora-
tory. I deeply appreciate my colleagues Peter Gentile, Yifu Zhou, Daoru Han, Thien
Phan and Kamia Smith who made themselves available for my questions and numerous
discussions.
Some of the chapters included in this thesis were the product of collaborative work.
All of those who contributed to the completion of these chapters are acknowledged.
xiv
Abstract
One of the essential fundamental physical properties of deformed metals is the ori-
gin of long-range internal stresses (LRIS) and the magnitude of LRIS which is related to
the reduction of the reverse yield and associated with the Bauschinger effect (important
for cyclic deformation and fatigue properties) and metal springback.
In this thesis, LRIS are examined for four particular cases, including a copper sin-
gle crystal deformed in compression by 28% along the [001], an identically deformed
copper single crystal with an additional 2% reverse tensile deformation, a copper single
crystal deformed by 28% along a non-symmetric crystal axis, and several equal-channel
angular pressing (ECAP) processed polycrystalline aluminum.
In the symmetric [001] experiments, the 2% tensile deformation along the [001]
was performed in order to investigate the effect of reverse stress. Compared with the
28% compressive deformed sample, the LRIS in the cell interiors were found to reduce
from +21:5 MPa to +13:3 MPa and from13:1 MPa to7:1 MPa in the cell walls. In
addition, the average stress difference between the cell interiors and walls also decreases
from +34:6 MPa to +20:4 MPa. Thus, the application of a small reverse strain caused
the LRIS in the cell interiors and walls to be reduced by about 40%, but retain the same
signs. It appears that the reduction of LRIS produces a partial change of dislocation
configuration in the cell interior and wall heterogeneous microstructures.
In addition, elevated LRIS was found in compressively deformed single-crystal
copper along a non-symmetric crystal axis <0.184, 0.107, 0.177>, compared with the
deformation along the [001]. Both LRIS in the cell interiors and walls increase. The
xv
LRIS in the cell interiors increases from +21:5 MPa to +24:9 MPa and in the cell walls
from13:1 MPa to17:9 MPa and the stress difference is about 42:8 MPa, which
is about 20% higher than 34:6 MPa (deformation along the [001]). The monotonic
deformation along a non-symmetric crystal axes other than the [001] is expected to
produce geometrically necessary boundaries (GNBs) with higher misorientation in the
walls, which may lead to higher LRIS in both cell interiors and walls.
For ultrafine grain metals (e.g. AA1050) processed by ECAP deformation, high
ratios of high angle boundaries were reported as 25% after 1-pass ECAP and
70% after 8-pass. LRIS were observed about19 MPa (0:1
a
) along the maximum
tensile plastic strain direction within the grain/subgrain interiors of 1-pass AA1050. The
magnitude of LRIS in ECAP AA1050 using routeB
C
is estimated from 0:1
a
to 0:2
a
in the grain/subgrain interiors. In ECAP AA6005, the LRIS are also observed and the
magnitude of 2-pass via routeC has a similar value of AA1050.
xvi
Chapter 1
Introduction
Numerous techniques have been developed for metal deformation and processing
to improve the mechanical properties for versatile use. Although the macroscopic prop-
erties of deformed metals have been thoroughly investigated in terms of yield strength,
deformation behavior, fatigue and creep, less work has been done on their fundamental
physical basis. One of the important fundamental physical properties of deformed met-
als is related to the origin of long-range internal stress (LRIS). Understanding the LRIS
in deformed metals is important since it can influence a variety of important phenom-
ena including the Bauschinger effect (important for cyclic deformation and fatigue) and
metal springback (metal forming).
Kassner et al. [1–4] have recently studied LRIS in cyclically deformed single-
crystal copper and aluminum by means of convergent beam electron diffraction (CBED),
and the dipole separation measurements. Levine et al. [5, 6] investigated LRIS in
monotonically deformed single-crystal copper by three-dimensional X-ray microbeam
diffraction. LRIS were evident in X-ray microbeam diffraction studies for deformed Cu
along [001]. The measured elastic stains (stresses) in the cell interiors are the oppo-
site signs to the applied compressive plastic strains and thus the results are consistent
with the composite model for LRIS. To further understand LRIS in copper in different
stress states, plastic strains were applied along different compression axes than [001].
1
Also, a small reverse tensile strain was applied along the [001] on a [001] compression
deformed single-crystal Cu (
loc
=
a
' 0:1 where
loc
is the local stress and
a
is the
applied stress).
Additionally, there is a lack of research regarding the LRIS associated with severe
plastic deformation (SPD). SPD metals are interesting because of their superior mechan-
ical properties (e.g. high strength) due to the refined grain size. For these highly
deformed metals, high angle boundaries (HABs) are usually observed and many believe
that these are non-equilibrium boundaries with many extrinsic dislocations emanating
from the boundaries. Thus, high LRIS may exist in these highly deformed metals.
High resolution X-ray microbeam diffraction was utilized to study the lattice strains
(stresses) in the monotonic deformed single-crystal copper and equal-channel angular
pressing (ECAP) processed polycrystalline aluminum alloys. The goal of this study was
to understand the LRIS associated with dislocation microstructures or grain boundaries
that are not produced by [001] axis tension/compression in Cu, where the misorientation
of the cell walls are relatively low at< 0:9

[5, 6]. Other orientations of the stress are
expected to produce geometrically necessary boundaries (GNBs) with generally higher
misorientations than the incidental walls. These studies are important for understanding
the origin and magnitude of LRIS in deformed metals which are believed to be associ-
ated with many mechanical phenomena.
1.1 Background
Basic knowledge about plastic deformation, dislocation microstructures, types of
boundaries, models of LRIS and previous experimental evidence are introduced in this
section.
2
1.1.1 Dislocations, Metal Crystal Structures and Slip Systems
1.1.1.1 Dislocations
The stress-strain behavior of metals during deformation will be discussed. Metal
deformation usually includes two parts: an elastic regime and a plastic deformation
regime. For most metals, elastic deformation can only occur over a strain from zero
to about 0:002 0:005. When the metals are deformed beyond this value, the stress
is no longer proportional to the strain. This is a so-called "plastic deformation regime".
Plastic deformation is related to non-reversible atomic movements at a critical stress and
it is believed that dislocations mainly form in this regime.
To define the dislocation, Burgers vector b should be introduced. Burgers vector
b is defined as the magnitude and direction of a slip of a single dislocation. To define
this vector, a circuit loop is circled around the dislocation in the deformed crystal and
does not end where it starts. The vector needed to close the circuit is Burgers vector
b in Figure 1.1(a). For a pure edge dislocation in Figure 1.1(a), the Burgers vector is
perpendicular to the dislocation and indicates the slip direction, while for a pure screw
dislocation, it is parallel to the dislocation line (dash line in Figure 1.1(b)) [7–9]. Gener-
ally, dislocations are in the form of loops or networks in the three-dimensional crystals.
Figure 1.2 [7–9] illustrates a loop. The Burgers vector is constant along the entire dis-
location loop.
1.1.1.2 Metal Crystal Structures and Slip Systems
Dislocations prefer to move on specific planes and also move in specific directions
in metal crystals. The preferred planes are called the "slip planes" and the movement
directions are called "slip directions". The planes, combined with slip directions, are
called "slip systems". The formation of slip systems is strongly associated with metal
3
Figure 1.1: Definition of the Burgers vector and Burgers circuit
(a) Burgers circuit is performed around a positive edge dislocation (b) Burgers circuit
around a right-handed screw dislocation [7–9].
Figure 1.2: A dislocation loop in a slip plane in a deformed crystal [9]
crystal structures. In Figure 1.3, one of the {111} planes is shown for a face-centered
cubic (FCC) metal crystal, a conventional unit cell. The slip planes of the FCC crystal
are planes which are close-packed and slip directions are indicated by arrows, the short-
est distance in the slip plane [7–9]. Slip systems of different metals with FCC, body-
centered cubic (BCC) or hexagonal close-packed (HCP) crystal structures are shown in
Table 1.1 [7–9]. The direction of Burgers vector corresponds to the dislocation slip
4
direction, and its magnitude (moving or slipping distance) is equal to the unit slip dis-
tance. The distance is the shortest lattice translation vector.
Figure 1.3: Schematics of FCC conventional unit cell and slip systems
(a) FCC unit cell and {111}<110> slip system particularly at ambient temperature (b)
(111) slip plane of (a) is shown here and three <110> slip directions [7–9].
1.1.2 Plastic Deformation and Microstructures of Dislocations
With deformation, the density of dislocations increases and dislocations often form
low-energy configurations as cell walls or subgrain walls. It is believed by many that the
heterogeneous dislocation microstructures are associated with the LRIS in plastically
deformed metals [10–14]. Dislocations in deformed metals are frequently distributed
heterogeneously on a micrometer scale in alternating regions of high and low local dis-
location densities, respectively. In most common types of heterogeneous dislocation
distribution, dislocation cells develop under conditions of multiple slip (or primary slip
plus strong secondary-slip activity). Thus, during deformation, dislocations are gener-
ated with abundant gliding and become entangled into heterogeneous microstructures.
5
Table 1.1: Common slip system of metals [7–9]
Metals Slip Plane Slip Direction Number of Slip systems
Face-Centered Cubic
Cu, Al, Ni, Ag, Au {111} <1-10> 12
Body-Centered Cubic
-Fe, W, Mo {110} <-111> 12
-Fe, W {211} <-111> 12
-Fe, K {321} <-111> 24
Hexagonal Close Packed
Cd, Zn, Mg, Ti, Be {0001} <11-20> 3
Ti, Mg, Zr {10-10} <11-20> 3
Ti, Mg {10-11} <11-20> 6
It is believed that the heterogeneous distribution of dislocations can reduce the over-
all strain energy. In addition, through cyclic deformation under single slip conditions,
equal-space primary edge-dislocation bundles (veins) or multipoles are observed in Fig-
ure 1.4(a) [10].
It has been widely accepted that the driving force for the formation of the disloca-
tion walls and cell structures results in the reduction of the strain energy. LRIS has been
suggested in this heterogeneous dislocation microstructure by Mughrabi et al. [10–14].
Their studies provided evidence that LRIS exists in the loaded and unloaded states by
dislocation curvature measurements of dislocation loops "frozen” in place by neutron
irradiation. This may also indicate that LRIS is generated after a heterogeneous disloca-
tion microstructure forms from deformation. In Figure 1.4, the high dislocation density
region was estimated to have a LRIS of +2:0
a
while the interior was thought to have a
LRIS of0:5
a
[10].
1.1.2.1 Single Slip Deformation (Formation of Persistent Slip Bands)
Multipolar or bundles of edge-dislocation wall structures are shown in Figure 1.4.
To produce this kind of microstructure, single-crystal copper was orientated for single
6
Figure 1.4: Deformation of copper by single slip and the formation microstructure
(a) Dislocation wall structure of persistent slip bands (PSB) in the stress-applied state.
The image was taken on the glide plane. (b) A schematic of PSB walls and bowed-out
edge and screw dislocations [10].
slip by cyclic deformation [10]. These wall structures appear in cyclic saturation defor-
mation. Continued cyclic deformation can lead to an even more refined dipole structure
of persistent slip bands (PSBs). These dislocation walls are observed in the glide plane
and perpendicular to the primary Burgers vector. Some free dislocation segments bowed
out in the channels (low dislocation density regions).
1.1.2.2 Multiple Slip (Unloaded [001]-oriented Single-crystal Cu Monotonic
Deformation)
Tensile deformation of [001]-orientated single-crystal copper develops a typical
dislocation cell/wall structure illustrated in Figure 1.5 [14]. Multiple slip occurs during
deformation. A specimen was deformed with tensile stress in the [001] direction and
7
the image was taken along the [010] direction. Long-range internal stresses are sug-
gested in this multiple-slip cell wall/interior composite structure by using X-ray peak
asymmetry [15].
Figure 1.5: Composite structure of dislocation cell interiors and cell walls of [001]
oriented Cu single crystal under monotonic deformation
The image is taken in (010) section [14].
1.1.2.3 Single-crystal Cu Cyclic Deformed to Pre-saturation
Figure 1.6 [2] is the microstructure of single-crystal Cu produced by single slip
through cyclic deformation. The samples are deformed in tension and compression and
oriented for single slip with maximum and minimum Schmid factors of0:5 at 298 K
to 200 cycles. The plastic strain amplitude was about 1:25 10
3
. The strain rate was
2:510
3
s
1
. The process deformed the crystals to pre-saturation (half of the saturation
stress) at an axial stress of about 38 MPa. For this deformation stress condition, the
microstructure consists of dipole bundles (veins) and channels (low dislocation density).
8
There was no evidence of LRIS in the channels for the unloaded specimen (unloaded
[123] single crystal) by CBED analysis.
Figure 1.6: Transmission electron microscopy (TEM) image of single-crystal Cu
deformed cyclically to pre-saturation [2]
1.1.2.4 Uniaxial Compression Creep Test of Polycrystalline Copper
Copper rods were thermo-mechanically treated to produce a coarse grained poly-
crystalline material. The average grain size is 68 m. Cylindrical specimens were
deformed in uniaxial compression at temperatures of 298 K or 623 K. The microstruc-
tures of the deformed samples are shown in Figure 1.7 [16]. In Figure 1.7(a), the sample
is deformed to a strain of" = 0:08 at room temperature. Cell walls have a loose structure
(consisting of dislocation multipoles). If the sample is deformed to a strain of" = 0:25
at 623 K, a closed subgrain structure is created, as shown in Figure 1.7(b). The subgrain
9
boundaries consist of two-dimensional dislocation networks. There is only a small mis-
orientation between subgrains. CBED and X-rays are used to investigate long-range
internal stresses in the cell and subgrain interior. Both methods show that long-range
internal stresses exist in the cell as well as in the subgrain structures.
Figure 1.7: TEM images of Cu deformed at different conditions
(a) Cell structure of copper formed after deformation at = 180 MPa at room temper-
ature. (b) Subgrain structure forms after deformation at = 157 MPa and 623 K. TEM
micrographs taken with the foil normal parallel to load axis [16].
10
1.1.3 Bauschinger Effect
The Bauschinger effect [17–20] generally indicates a decrease of reverse yield
stress. If materials are under tensile deformation in one direction into the plastic regime,
then unloaded to zero stress, and reloaded compression in the reverse direction, they
may yield at a lower absolute value of the stress level. Figure 1.8 [21] shows a general
deformation behavior of the Bauschinger effect with the first cycle. This asymmetrical
yield behavior was first reported by Johann Bauschinger [19]. This effect has been stud-
ied extensively. The concept of backstresses or long-range internal stresses in materials
was first proposed in connection with the Bauschinger effect.
Often, the concept of "backstress" is used. The backstress here is expressed by the
following equation.

b
=
(
f
+
r
)
2
(1.1)
where
b
is the "backstress",
r
is the stress of reversal yielding, and
f
is the for-
ward stress on unloading. Several models or theories have been proposed to explain the
Bauschinger effect, such as by Sleeswyk et al. [22] and Mughrabi et al. [10, 11]
1.1.3.1 Internal Stress Theory of Bauschinger Effect
In 1918, Heyn [23] proposed a theory to explain lower reversal yield stress in
Bauschinger effect. Several assumptions have been made for his model.
1. The material is composed of small-volume elements. Each element of material
has ideal stress-strain behavior.
2. The elastic limits of various volume elements are not equal to each other.
11
Figure 1.8: A schematic showing the Bauschinger effect [21]
3. The absolute value of the elastic limit of any volume element is independent of
the direction of deformation, and it is the same in tension and in compression.
Upon loading, the stress distribution is not uniform. Therefore, residual stresses
will arise in the unloading condition. The residual stresses may be due to the incompat-
ibility between grains or individual elements. This may cause the stress to freeze in the
deformed metals ( e.g. Cu and Ni) [21]. However, the details of this mechanism are not
yet clear. The residual stress is responsible for the yield-lowering effect in the reverse
loading. Similar concepts were proposed to explain some experiments. For example,
Schimid et al. posited that the Bauschinger effect results from the action of internal
stresses locked up in cold worked metals [24].
1.1.3.2 Dislocation Theory- Pile-up Model
A different approach to explain the Bauschinger effect was originally proposed by
Mott and later developed by Seeger [25–27]. It suggested that during prestraining,
12
a long-range stress is built up through the formation of dislocation pile-up at barri-
ers (walls or grain boundaries). Figure 1.9 demonstrates dislocations piled up around
obstacles (barriers) and LRIS developed. The definition generally used for LRIS is:

l
=
a
+
i
(where
a
is the applied stress, and
l
is the local stress and
i
is the
LRIS). It is generally believed that the internal stresses exist in the loaded and unloaded
conditions.
Figure 1.9: A schematic of dislocation pile-up in grains/subgrains
(a) A schematic shows dislocation pile-up around the grain boundaries or dislocation
walls. (b) The variation of stress in a solid with
a
.
a
are the applied stress, and

i
is the local stress and
i
are the LRIS in the different heterogeneous dislocation
microstructures [21].
Another dislocation theory approach was proposed by Orowan and Sleeswyk et al.
[22, 28] and it describes dislocation movement differences in the forward and reverse
directions. It will be addressed in a later section.
1.1.3.3 Dislocation Theory- Non-LRIS Approach
Sleeswyk et al. came out with results that are similar to the Orowan type strain-
hardening mechanism which states that LRIS has less effect but a clear softening has
been observed in the reverse loading in Figure 1.10 [22, 28]. Dislocations have dif-
ficulty moving in the forward loading since the increase of obstacles. Those obstacles
13
Figure 1.10: Stress-strain curve of (alpha) Zn single crystal
For single-crystal, subtracting the strain from the cumulative strain after the first rever-
sal causes the reverse curve to approach closely the extrapolated forward curve [22].
come from the elevated dislocation density and the entangled dislocation will make the
dislocations difficult to move past. However, in reverse loading, dislocations easily
reverse to escape from the entangled walls and move across the cell interiors or grain
interiors, giving rise to a "reversible strain",.
1.1.3.4 Bauschinger Effect- Composite Model
Mughrabi and Pedersen et al. [10, 29] developed a composite model for LRIS. After
plastic deformation, heterogeneous dislocation microstructure will form. This heteroge-
neous structure will reduce the overall strain energy. In their model, in a deformed solid,
the material can be separated into two parts: cell walls and cell interiors. For example,
the walls have a higher yield stress (higher dislocation density) and the yield stress is
lower for the cell interiors (in Figure 1.11 and Figure 1.12). The elastic strain of the
walls is higher. Elastic compatibility can lead to LRIS with the opposite signs in the
cell interiors and walls. Mughrabi et al. mentioned that the interface of dislocations will
14
Figure 1.11: The composite of heterogeneous dislocation microstructures
Hard and soft regions are shown. The interface dislocation compensates the elastic strain
incompatibility between cell interiors and walls [10].
make up the elastic incompatibility between cell interiors and walls and give rise to the
LRIS.
The long-range internal stresses are defined consistent with equations,
I
=
a
+

I
, and
w
=
a
+
w
, where
I
and
w
are local stresses in the cell interiors and walls,

a
is the applied stress, and
w
and
I
are LRIS in the cell walls and interiors of the
composite substructure [10]. In Figure 1.12,
y
I
and
y
w
are defined as the yield stresses
of cell interiors and walls. Upon unloading, the force between cell interiors and walls
are balanced. The concept of heterogeneous stresses has also been widely embraced for
monotonic deformation including elevated-temperature creep deformation. Figure 1.12
explains how the Bauschinger effect occurs with the composite model.
15
Figure 1.12: Stress-strain curve of cells and walls in the composite model
Dislocation walls and cells form the composite structure. Dislocation walls and cells
have different yield stress. During the unloading process, the soft region is under com-
pression. This leads to lower macroscopic yield stress in the reverse loading [10].
1.2 Experimental Evidence of LRIS
As mentioned above, LRIS has been proposed to explain the Bauschinger effect in
reverse deformation. LRIS may not be observed in all the deformation microstructures
in which lower reversal yield stress occurs, although the LRIS concept has been well
accepted as one of fundamental reasons associated with the Bauschinger effect. In the
following sections, some experimental evidence for LRIS is introduced.
1.2.1 LRIS in Multipolar Edge-dislocation Wall Structures
Mughrabi reported the observation of multipolar dislocation walls after single slip
deformation. Multipolar dislocation walls are a dislocation microstructure where dif-
ferent signs of edge dislocation settle on the multiple parallel slip planes. Single-slip
16
deformation was performed by cyclic deformation at room temperature. In cyclic sat-
uration, persistent slip bands (PSBs) were observed (Figure 1.4). As mentioned in the
previous section, bowed-out dislocation segments were observed. The curvature of free
dislocation segments can be used to probe or calculate the local stress through the equa-
tion:

loc
=
T
br
(1.2)
where
loc
denotes the local shear stress,T represents the dislocation line tension,b is the
magnitude of the Burgers vector andr is the radius of dislocation line curvature. Close
to walls, short and curved dislocations are shown but less-curved dislocation segments
are mostly in the center region of channels.
According to above equation, the local shear stress
loc
, near the walls is approx-
imately twice as large as the applied peak stress which is 28 MPa in this case (Fig-
ure 1.13). The stresses near the walls are much higher than the applied stress. This
implies that there is a stress superimposed on the applied stress. This is significant evi-
dence of the existence of LRIS. However, the magnitude of LRIS reported is not very
accurate because the connection between the "pinned" dislocation configurations and the
elastic strains in the lattice is not as straightforward as in the above (1.2) stress equation
since interactions with dipoles and multipoles of the walls are considered important.
17
Figure 1.13: Local stresses in the channel of PSBs
Local stresses in different channel positions. The shear stresses are calculated from the
radius of bowed-out dislocation lines [10].
1.2.2 Asymmetry of X-ray Diffraction Line Profiles (Multiple-Slip Deformation)
Additional experimental evidence for LRIS was provided by examining copper
dislocation microstructures after multiple-slip deformation by X-ray diffraction. Fig-
ure 1.14 is the observed microstructure from deforming single-crystal copper along
[001] direction [10].
X-ray reflection line-broadening was measured on unloaded [001]-orientated
single-crystal copper. The X-ray line-profile broadening increased with the increased
strain and the asymmetric signature is observed. The line-profiles are shown in Fig-
ure 1.15 [10]. The line-profiles are asymmetrical, indicating that the profiles may be
composed of two symmetric profiles with different lattice parameter positions, result-
ing from different internal stresses. Thus, it is possible to decompose each one of the
18
Figure 1.14: Composite model of the stressed dislocation cell/wall structure for
multiple-slip deformation.
(a) Idealized dislocation cell with two symmetric active slip systems. The slip direction
is <110>. The glide planes are {111} (b) The resultant Burgers vectorb
res
is composed
ofb
1
andb
2
[10].
broadened asymmetric peaks into two broadened symmetric sub-peaks. The integrated
intensities of the sub-peaks are found to be mutually related by the same ratio as the
volume fractions of dislocation walls and cell interiors. The cell interior strain (stress)
shows the opposite sign to the applied stress; but the wall strain (stress) has the same
sign.
1.2.3 Convergent Beam Electron Diffraction (CBED) Measurements
Laue patterns can be used to simulate the unit cell geometry of undeformed or
deformed materials. The symmetry breaking of Laue patterns may indicate the presence
of crystal strains in the materials [16, 30]. In Straub’s studies [16], copper was deformed
in uniaxial compression creep tests at temperatures between 298 K and 633 K and then
19
Figure 1.15: Line broadening and asymmetry increases with increasing deformation
(002) X-ray diffraction line-profiles of tensile deformed [001]-orientated single-crystal
copper measured by conventional X-ray diffraction. Line broadening and asymmetry
increase with increasing deformation [10].
examined by the CBED. For an undistorted cubic crystal one expects three mirror planes
in a [111] zone-axis pattern and one mirror plane in [114] zone-axis patterns. First,
marked areas (A and B) are examined by CBED in the experiment. Figure 1.17(a) and
(b) reveals that all mirror planes are lost. Obviously, internal stresses are present within
the subgrain interiors (B region) [30].
In experimental [114] CBED patterns the absolute position of the HOLZ lines that
are most sensitive to lattice parameter changes can be determined quite exactly. The best
match between the simulated patterns and both the [111]- and the [114]-experimental
patterns that were recorded from the same location within the B central region of the
subgrain was obtained for a c and a, c b + 0:0001nm. Errors in lattice parameters
20
Figure 1.16: Bright-field TEM image of a deformed copper polycrystal
The sample is deformed by creep test with = 162 MPa at 573 K at/G(T ) = 4:3
10
3
, the ratio of compressive stress to the shear modulusG(T ). The foil normal is
parallel to the loading axis [16].
are of the order of 0:00005nm. These lattice parameters correspond to a local internal
stress of (25 20) MPa.
1.2.4 Microbeam Measurements- Monotonic Deformation of Copper
The 3D microbeam X-ray can measure a sample volume as small as (0:5)
3
m
3
.
The sample volume of this new technique is small enough to allow probing lattice spac-
ings within individual dislocation cell interiors and cell walls within a bulk deformed
specimen [5, 31]. Levine et al. [6] found that cell interior strains will change sign if the
external applied stress change from compression to tension in Figure 1.18. These phe-
nomena are observed by X-ray microbeam and conventional X-ray. Those strains are
measured by X-ray microbeam and taken as the direct evidence of the LRIS after plastic
deformation (vertical blue lines in Figure 1.18). Further, the sum of volume-weighted
21
Figure 1.17: Laue zero-order CBED pattern
(a) [111]-zone axis from position B central region (cell interiors) of the subgrain shown
in Figure 1.16 (b) [114]-pattern recorded from the same position. (c-d) the simulation
patterns of a and b for [111] and [114], respectively. [16]
strains/stresses of cell interiors and walls is close to zero consistent with the predictions
of the composite model which proposed by Mughrabi.
1.3 Summary of LRIS in Previous Experiments
There are some cases where LRIS may not be evident such as cyclic deformation
of copper without PSBs and steady-state creep test of copper [2, 3, 21]. However, LRIS
has been found in many experiment. It is believed that LRIS exists in some disloca-
tion heterostructures after deformation. In addition, some experiment results prove that
LRIS is present in loaded and unloaded states. LRIS develops during loading and many
remains ”frozen” in the unloading state. Kassner et al. [21] have summarized the mea-
sured LRIS in different deformation processes or stress conditions which are listed in
the following tables (Table 1.2 and Table 1.3).
22
Figure 1.18: Cell interior strains under tensile or compression deformation
(a) X-ray results from the <001> compression specimen. (b) Results from the <001>
tension specimen. Red curves represent the 006 line profile from conventional X-rays.
The vertical lines show the corresponding diffraction peak centers from individual dis-
location cell interiors, measured by X-ray microbeams [5].
1.3.1 The Magnitude of LRIS
Table 1.2 and Table 1.3 by Kassner et al. [21] lists many studies that attempted
to measure LRIS experimentally (and theoretically). The table represents typical values
of LRIS and the deformation process. LRIS values vary over the wide range from 0 to
2:0
a
(
a
= applied stress).
23
Table 1.2: Selected LRIS measurements [21]
Mat. Ref. Deformation
Mode
Strain Wall
LRIS
a
Interior
LRIS
a
Observation
Method
Temp. Notes
Cu Mughrabi [10] cyclic w/PSBs 2 0:5 in-situ neu-
tron irrad.
RT
Cu Mughrabi et
al. [12]
cyclic w/PSBs 1:3 0:37 in-situ neu-
tron irrad.
RT reanalysis of
above
Cu Mughrabi et
al. [11, 13, 14]
tension 0:4 0:1 X-ray peak
asymm.
unloaded
[001] oriented
Cu Lepinoux and
Kubin [32]
cyclic w/PSBs 2:5 0:5 in-situ
TEM
RT loaded single
crystal
Cu Kassner [2] cyclic no
PSBs
0 0 CBED RT unloaded
[123] oriented
Cu Kassner [2] cyclic no
PSBs
0 0 dipole sep. RT unloaded
[123] oriented
Cu Kassner [4] cyclic no
PSBs
0 0 X-ray
microbeam
RT unloaded
[123] single
crystal
Cu Borbely et al.
[33]
creep steady-
state
1 0:08 X-ray peak
asymm.
527 K loaded
Cu Kassner et al.
[3]
creep steady-
state
0 0 CBED 823 K unloaded
Cu Straub et al.
[16]
compression 0:3
0:6
0:05
0:08
X-ray peak
asymm.
RT and 633 K
24
Table 1.3: Selected LRIS measurements (continued)
Cu Straub et al.
[16]
compression observed CBED RT and 633 K
Cu Levine et al.
[5]
compression 0:29 X-ray
microbeam
RT unloaded
[001] oriented
Cu Levine et al.
[6]
compression 0:1 0:1 X-ray
microbeam
RT unloaded
[001] oriented
Si Legros et al.
[34]
cyclic 0 0 CBED RT unloaded
Ni Hecker et al.
[35]
cyclic w/PSBs 1:4
1:8
0:16
0:2
X-ray peak
asymm.
RT loaded
Ni Hecker et al.
[35]
cyclic no
PSBs
0 0 X-ray peak
asymm.
RT loaded
Al Kassner et al.
[1, 2]
cyclic no
PSBs
0 0 dipole sepa-
ration
77 K unloaded
[123] oriented
Al Kassner et al.
[3]
creep steady-
state
0 0 CBED 664 K Unloaded
Al
5%
Zn
Morris and
Martin [36]
creep steady-
state
25 1 precipitation
pinning
483-523 K
- Sedlacek et al.
[37]
creep 1:5
10:0
0:5 1:0 theoretical creep
- Gibeling et al.
[38], Nix et al.
[39]
creep 7:7 0:1
0:2
theoretical creep
25
1.3.2 LRIS Dependence on Cell Interior Size
Levine et al. [6] used the typical cell size distribution of deformed single-crystal
copper into the dipole stress model equation to calculate the elastic internal stress dis-
tribution of cell interiors. The results fit very well with measured strain (stress) data
fromX-ray microbeam diffraction. According to these results, LRIS dependence on cell
interior size can be defined. In the model, the stress at the center of the cell is given
approximately by (dipole field model):

c
(D) =
8b
d
w
(1) + (D + 2w)
=
A
D + 2w
(1.3)
whereb is the magnitude of the Burgers vector, is the shear modulus, is Poisson’s
ratio,
d
is the dipole density andw (thickness of cell wall) is about 150nm, (from TEM
images). Combining an equation of the volume fraction of the cell with diameterD (the
probability distribution of cell interior size is obtained from the micrographs), equation
(1.4) can be obtained:
V (D) =D
3
P
D
(D)dD =aD
3
dD (1.4)
This yields the prediction of stress distribution function:
P (
c
) =
B

2
c
(
B

c
2w)
3
(1.5)
26
where
=
1
R
Dmax
D
min
D
3
d
(1.6)
and
B =
8b
d
w
(1)
(1.7)
The measured and predicted distribution functions agree quantitatively for stresses
higher than 25 MPa and they are in qualitative agreement at lower stresses (Figure 1.19).
The remarkably good agreement is shown between the measured and simulated distri-
bution functions.
Figure 1.19: Strain distribution histogram
Strain distribution (histogram) of cell interiors measured by X-ray microbeam diffrac-
tion of a sample deformed in compression. Dark curve is the modeling result [6].
27
Chapter 2
Experimental Procedure
2.1 Introduction of Synchrotron Radiation
2.1.1 Synchrotron Radiation and Emission Mechanism
Synchrotron radiation is electromagnetic radiation emitted when charged particles
or electrons are accelerated radially around a magnetic field. This X-ray radiation has
been observed in astronomy or produced by accelerators. Synchrotron radiation was
named after its discovery in a General Electric synchrotron accelerator built in 1946.
After that, more research emerged. Synchrotron radiation is usually produced through
acceleration of charged particles or electrons in the linear accelerator and storage ring.
When high-energy particles (usually electrons) are in rapid motion, forced to travel in a
curved path by a magnetic field (using bending magnets, undulators and/or wigglers to
high energy, typically in the GeV range), synchrotron radiation is produced. The radia-
tion produced in this way has a characteristic polarization and the frequencies generated
can range over the entire electromagnetic spectrum [40].
28
2.1.2 Properties of Synchrotron Radiation
Synchrotron radiation is the brightest artificial light source (e.g. X-rays) and it pos-
sesses many unique properties that can be utilized to characterize a variety of material
properties including [40]:
1. Broad wavelength range of spectrum (which covers from microwaves to hard X-
rays).
2. High photon flux/ high intensity
3. A small divergence and small size beam source
4. High energy/wave length resolution and stability
5. Polarization: both linear and circular
Thanks to these properties, synchrotron radiation has been successfully used to
characterize electronic structure, crystal microstructure, bonding properties, energy
band structure (absorption and emission), elastic strains and magnetic properties.
2.1.3 Synchrotron Radiation at Argonne National Laboratory
The Advanced Photon Source (APS) at Argonne National Laboratory provides the
brightest storage ring-generated X-ray beams in the United States. Several main com-
ponents/structures of the APS accelerator include as the following [41]:
2.1.3.1 Linear Accelerator
Electrons are accelerated in the linear accelerator to a high kinetic energy using a
high alternating electrical field. Selective phasing of the electric field accelerates the
electrons to 450 MeV . At 450 MeV , the electrons are relativistic: they are traveling close
to the speed of light.
29
2.1.3.2 Booster Synchrotron
Electrons are then injected from the linear accelerator into the booster synchrotron
device, a racetrack-shaped ring of electromagnets, and accelerated from 450 MeV to 7
GeV in one-half second. In order to maintain the orbital path of the electrons, the bend-
ing and focusing magnets also increase the electron field strength in synchronization
with the radio frequency field.
2.1.3.3 The Electron Storage Ring
After the electrons are boosted to 7 GeV , they are injected into the circumference
storage ring (1104 m circumference). At the APS, the storage ring is a circular structure
consisting of more than 1; 000 electromagnets and associated equipment. A powerful
electromagnetic field is generated to focus the electrons into a narrow beam that is bent
on a circular path as it orbits within vacuum chambers.
2.1.3.4 Insertion Devices
The APS is a "third-generation" hard X-ray light source that uses highly optimized
insertion devices to produce X-ray beams with high flux and brilliance. Flux is evaluated
by measuring the total number of photons per second in a narrow spectrum/energy band-
width and in a unit of solid angle in the horizontal and vertical directions with respect
to the whole unfocused X-ray beam. Brilliance describes the intensity and directional-
ity of an X-ray beam and it is one main index to determine the smallest focused beam
spot. Two kinds of insertion devices, the undulator and the wiggler, are commonly used.
An undulator is composed of a periodic dipole magnetic structure which produces a
magnetic field to deflect the electron beam sinusoidally. The static magnetic field is
designed along the length of the undulator with an alternating wavelength
u
. As the
30
electron beam passes through the periodic magnet structure, it is forced to oscillate and
radiate at specific harmonics. Thus, the X-ray intensity is amplified.
A wiggler is the other insertion device with a series of magnets designed to repeat-
edly, laterally, deflect a beam of charged particles inside a storage ring of a synchrotron.
These deflections cause X-ray emission simultaneously and thus the X-ray intensity
increases. A wiggler produces a broader spectrum of radiation than an undulator does
because the period and the strength of the magnetic field are not tuned to the frequency
of radiation produced by the electrons. Therefore, every electron in the wiggler radiates
independently. This generates a broader radiation spectrum. (Figure 2.1).
Figure 2.1: A schematic of an insertion device of undulator
It is an insertion device which is composed of an array of magnets. X-ray intensity is
amplified due to the oscillation of electrons. An undulator is specially designed with a
certain period of magnets to emit X-rays in a certain electron harmonic oscillation [41].
31
2.2 X-ray Microbeam Measurements
2.2.1 Introduction
Synchrotron X-ray microdiffraction and the differential-aperture X-ray microscopy
(DAXM) technique (beamline 34-ID-E at the Advanced Photon Source at Argonne
National Laboratory) is a novel analysis tool to explore strain (stress) distribution in
the materials in the submicron regime. This tool has led to a new view of observed
stress states in the dislocation microstructure in deformed metals and to new inspection
techniques of dislocation models. Therefore, X-ray microbeam diffraction opens a path-
way to understand the mechanical properties, stress and strain, in the submicron regime.
This allows us to conduct advanced research to investigate the correlation between dis-
location microstructure and stresses. The results of previous experiments, obtained from
the measurement of cell interior and wall strains in single-crystal copper, are introduced
from the three-dimensional X-ray microdiffraction method.
2.2.2 Three-dimensional X-ray Microbeam Diffraction
2.2.2.1 Differential-Aperture X-ray Microscopy (DAXM)
In 2002, Larson et al. [31] developed a polychromatic X-ray technique, using
white-beam differential-aperture X-ray microscopy (DAXM) technique, which allows
us to measure the crystallographic orientation and the internal shear stress state in rela-
tively small, spatially resolved volumes within a bulk specimen. Recently, this technique
has been extended by using a scanned monochromatic X-ray beam to measure absolute
lattice parameters and dilatational strains from deep submicrometer sample volumes.
The X-ray beam is focused to 0:5m 0:5m using Kirkpatrick-Baez focusing (KB)
32
mirrors (60 mm and 30 mm focal lengths) and undulator X-rays at the Advanced Photon
Source (APS) at Argonne National Laboratory (ANL).
Thus, the DAXM technique is used to measure strain (stress) in microstructure in
a sample with submicrometer spatial resolution. In addition, the DAXM exploits step
profiling of a 50 micrometer diameter platinum wire which moves across the sample
surface to provide a submicrometer-resolution X-ray slit (point-to-point resolution) in
Figure 2.2. Therefore, the diffraction signals at different depths can be separated. By
using the DAXM method, complete Laue diffraction patterns can be indexed as a func-
tion of depth along the penetration direction of the X-ray microbeam. This also allows
us to investigate single crystal and polycrystalline materials. More specifically, knowing
the diffraction spot location on the detector, the wire location that blocks this spot, and
the location of the incident X-ray beam, triangulation can be used to calculate the exact
location (submicron cubic area) along the incident beam of the corresponding diffract-
ing volume. Therefore, full 3-D resolution of microbeam diffraction can be obtained in
analyzed samples and full diffraction information (lattice spacings/strain tensor) from
submicrometer three-dimensional volumes in bulk materials can be accessed [5, 6, 31].
Figure 2.2: A schematic of differential-aperture X-ray microscopy (DAXM) [31]
33
2.2.2.2 Three-dimensional Resolution of the X-ray Microbeam
The DAXM method decomposes the overlapping profiles by scanning a platinum
wire through the diffracted beam path (in front of detector), and a triangulation method
determines the location of the diffraction volume along the incident X-ray beam. The
diffraction volume is determined by the intersection of the diffraction line that passes
through the diffraction spot on the detector and the leading edge of the wire and incident
X-ray beamline. Therefore, all the Laue diffraction patterns from different depths can
be analyzed (Figure 2.3).
Figure 2.3: A schematic of triangulation of microbeam diffraction [31]
2.2.2.3 Uncertainty of Single Component Strain
In this section, the measurement uncertainties using beamline 34-ID-E at the APS
will be discussed. The basic principle of d-spacing measurement is the Bragg equation:
34
2d sin = n. The uncertainty ofd would come from two factors, and, according
to the following equation:

4d
d

2
=

4

2
+

4 cot

2
(2.1)
In the 34-ID-E beam line setup the Bragg angle, 45

, and hence cot 1.
The uncertainty of X-ray wavelength (or X-ray energy, E) comes the Darwin
width of the reflection (diffraction width) used in the monochromator Si (111) and from
how well the monochromator is calibrated. A typical value4E=E is about 0:5 10
4
.
Uncertainty of the scattering angle has a few sources:
1. The geometry calibration (e.g. camera length, detector resolution and detector
geometry), dictates how accurately the angle of each pixel is known. This should
be very small, definitely smaller than4E=E according to the single-crystal Si
diffraction measurements, which are estimated to be about 0:1 0:2 10
4
.
2. The angular resolution of the detector is determined by the pixel size (0:2 mm) and
distance ( 500 mm from the sample surface). This is a square-shaped function
of width 4 10
4
in 2.
3. Divergence of the incident X-ray beam is determined by the size of the incident
slit (60 80m) and the focal length of the X-ray beam ( 130 mm). Hence it
has an angular width of 6 10
4
(the "angular width" mentioned above is not
the same as the "uncertainty"4).
In summary,4 < 2 10
4
and therefore,4d=d is 1 10
4
, in general, for single
measurement accuracy.
35
2.2.3 Data Analysis
2.2.3.1 Strain Data Analysis
The Figure 2.4(c) profile shows an example of an X-ray line profile probed from a
cell interior. A Gaussian function is used to fit the scattering curve (intensity versusq
vector) and the center of the Gaussian peak gives the average lattice spacing. Equations
(2.2) and (2.3) are used to convert the energies (wavelengths) into aq vector.
2d sin =n (2.2)
q = (4=) sin = 2=d (2.3)
Equation (2.2) is Bragg’s law in which is the Bragg scattering angle, is the wave-
length andd is the lattice spacing. From equation (2.3), the scattering vectorq is related
to lattice spacing. Once the centroid of the line profile is found, the average lattice
spacing can be obtained. Therefore, the strains can be calculated.
2.2.3.2 Strain in the Cell Interiors and Cell Walls
The diffraction (pixels) on the detector at a depth contain the diffraction signals
from the dislocation heterostructures including cell interiors and cell walls. The cell
interiors are the areas with lower dislocation densities, while the cell walls are areas
containing higher dislocation densities. Therefore, the crystal lattice of cell interiors
has fewer defects and will generate stronger diffraction (higher intensity pixels). The
crystal lattice from the walls will only generate smeared diffraction, which are dim and
diffuse on the detector. The diffraction from the cell interiors can be separated from the
36
cell walls by masking the high intensity area. The integrated line profile of this high
intensity area on the detector can be used to calculate the mean strain (stress) in the cell
interiors. The smeared diffractions can be integrated into diffraction line profiles of cell
walls (Figure 2.4).
Figure 2.4: Laue pattern and X-ray line profiles of cell interiors and walls
Laue diffraction of Cu cell interiors and cell walls at the depth of 4m. (a) The bright
spots represent the diffractions from cell interiors. The circled area is used for the
integrated line profile analysis of a cell interior. (b) The smeared diffraction signal is
attributed to a cell wall. The circled area is used for the integrated line profile of cell
walls. (c) theq profile of a cell interior which corresponds to the masking area in (a),
and the average strain is +3:4 10
4
. (d) theq profile of a smeared diffraction signal,
which corresponds to the masking area in (b) and the average strain is2:1 10
4
.
37
2.3 X-ray Powder Diffraction Measurements
2.3.1 11-BM Beamline Characteristics at the APS
Beamline 11-BM at the Advanced Photon Source at Argonne National Laboratory
is a dedicated X-ray powder diffraction instrument at the APS providing the highest
resolution powder diffraction data available in the United States. 11-BM beamline com-
bines the brilliance of an APS-bending magnet source with the efficiency of a focused
double-crystal Si monochromator to achieve world class sensitivity and resolution. The
diffractometer uses multiple single-crystal analyzer detectors (12 independent Si (111)
crystal analyzers and LaCl
3
scintillation detectors) to simultaneously offer high-speed
(1 hr) and high-resolution (q=q 2 10
4
) data collection. The beamline specifi-
cations are shown in Table 2.1. Typical samples are measured at 30 keV [42].
Table 2.1: Summary of 11-BM technical specifications [42]
Source Bending Magnet (BM). Critical Energy = 19:5 keV
Energy Range 15 35 keV ( = 1:0 0:34 Å)
Flux 5 10
11
phs/sec at 30 keV
Monochromator Si(111) double crystal (bounce up geometry)
Focusing Sagittally Bent Si (111) Crystal (Horz),
1 meter Si/Pt mirror (Vert)
Beam Size 1:5 mm (horizontal0:5 mm (vertical) focused at sample
Detection System 12 independent analyzer set with 2 separation of 2

Si(111) analyzer crystals and LaCl
3
scintillation detectors
Angular coverage 2 range: 0:5

130

at ambient temperature
(q
max
28 Å
1
at 30 keV)
Resolution q=q10
4
(min. 2 step size = 0:0001

)
38
Figure 2.5: Sample holder for the 11-BM beamline X-ray powder diffraction
The schematic shows the X-ray beam illuminating area. The sample for 11BM beamline
powder diffraction needs to be fitted into the kapton tube and plugged into the sample
base [42].
2.3.2 Calibration of 11-BM Beamline
Under ambient temperature conditions, the 12-analyzer detector system offers full
coverage of the 2 from 0:5

130

, with partial detector coverage extending an addi-
tional 22

in both directions. The beamlineq
max
is 28 Å
1
at 30 keV (equivalent to
25 Å). The minimum diffractometer 2 step size is 0:0001

, and a scan may contain up
to 160; 000 steps (12 intensity and 2 monitors values recorded at each step). The lattice
constant can be calculated from multi-reflections by powder diffraction data analyzing
software, GASA with an accuracy of approximately 0:0001 Å.
2.3.3 Sample Preparation for 11-BM beamline X-ray Powder Diffraction
The sample geometry is described in Figure 2.5 [42]. 11-BM rapid access mail-in
users are provided with Kapton tubes which are compatible with the mail-in mounting
bases. The inner diameter of the standard Kapton tube is about 0:8 mm. Samples were
39
machined to fit into the tube. Materials with high X-ray absorption coefficient need a
thinner sample thickness than the absorption length (attenuation length). The absorption
length is defined as the distance into a material where the X-ray beam intensity has
decreased to a value of 1=e ( 40%) of the incident beam intensity (I
0
).
2.4 Previous Experiment: Strain Measurements of Monotonically
Deformed Single-crystal Copper
Levine et al. [5, 6] investigated the long-range internal stresses in monotonically
deformed copper along the [001] direction. A typical TEM image of deformed copper
is shown in Figure 2.6. The wire scan of the microbeam is illustrated in Figure 2.7. The
wire profiler is typically moved parallel to the sample surface in steps of 0:7 m and
thus, it can resolve the diffraction of cell interiors and walls from different depths in the
experiment setup with a resolution of about 0:5m.
The process of microbeam diffraction measurements includes a 2-D scan of energy
and wire position (energy-wire scan). A broad enough range of X-ray microbeam ener-
gies is used to ensure diffraction from all of the diffraction volumes along the beam path.
The diffraction patterns from various subgrains/cells or dislocation walls would overlap
on the detector. At each energy of the energy scan, a complete wire scan with a 0:7m
step was performed by using a 50m diameter platinum wire profiler to block diffracted
X-rays from the sample. This provided a depth resolution of 0:5m. The average cell
interior size is about 1m. The profiler allows us to resolve the diffraction in the single
cell interior or in the area with the dense dislocation wall structure.
The plane spacing of the diffraction spot can be calculated by knowing the wave-
lengths and Bragg angle. Here, wavelengths are calculated from energies. In addition, a
platinum wire can block the diffraction spots on the detector. In this manner, the location
40
Figure 2.6: A TEM micrograph of a [001] deformed single-crystal Cu [5]
of the diffraction volume can be precisely determined. Energy is scanned up to 600 eV
with 2 eV steps. A series of lattice parameter values can be assigned to specific small
volumes of material leading to measurements of the average lattice parameter in a cell
interiors and walls of deformed single-crystal Cu.
The energy is scanned through the peak in 2 eV steps and the intensity versus
scattering energy is recorded. The energy at the centroid of the intensity diffraction peak
corresponds to the Bragg energy of the diffraction region, so the converted wavelength
that satisfies the Bragg diffraction can be calculated. In addition, the Bragg angle is
determined from the geometric calibration using the depth of the peak intensity and the
pixel position of the depth-resolved diffraction peak on the detector.
41
Figure 2.7: A schematic of an X-ray microbeam wire scan
This shows the incident microbeam drawn to the same scale as the TEM micrograph [5].
2.4.1 Proof of the Composite Model
Levine et al. [6] performed a compression test on the single-crystal copper along
the [001] crystal axis and measured the local strains. According to the method men-
tioned above, the small volume diffraction of X-ray microbeams consists of diffraction
signals from cell interiors and cell walls. Therefore, it can be decomposed into cell inte-
rior strains and cell wall strains. The strain distribution is obtained in Figure 2.8. The
positive strains (stresses) are obtained in the cell interiors but negative strains (stresses)
are observed in the cell walls in the unloaded sample. There is the same direction of
42
stress in the cell walls with the external stress. These results are consistent with the
prediction of composite model [10]. The LRIS is the ”residual stress” in the unloaded
sample.
Figure 2.8: Stress distribution of cell interiors and walls of compression deformed Cu
single crystal along a [001] crystal axis
The diffractions of dislocation heterostructure are used to calculate the stresses of cell
interiors and cell walls. The mean cell interior and cell wall stresses are separated by
40 MPa and the average sample stress of 4 MPa (dashed green line) is within the
measurement uncertainty [6].
43
Chapter 3
LRIS in Aluminum Processed by ECAP
3.1 Introduction
Severe plastic deformation (SPD) is one of the most common methods to refine
grain sizes [51–53]. The advantages of a refined microstructure are the improved
mechanical properties such as yield strength, hardness, ductility, fracture toughness,
fatigue resistance and low-temperature superplasticity. Equal-channel angular pressing
(ECAP) is a common SPD technique to attain a refined microstructure [52, 54, 55].
Experiments have shown that ECAP can lead to ultra-fine grains (UFG) 200nm, or
less (depending on the definition of a grain boundary in terms of misorientation angles),
after several passes for metals and alloys [52, 54–56]. Moreover, ECAP can produce
grain refinement while also providing homogeneous grain sizes, uniform mechanical
properties and maintaining the billet shape.
Little research has been done on the LRIS associated with SPD. SPD by ECAP
is a particularly interesting case since it is widely believed that most of the high-angle
boundaries (HABs) that lead to a refined grain size are produced as a result of dislocation
accumulation and reaction. Many believe that these boundaries are non-equilibrium with
many extrinsic dislocations emanating from the boundaries. Thus, high LRIS may exist
44
at boundaries in these highly deformed metals. In this chapter, LRIS in ECAP 1050 and
6005 aluminum alloys are investigated.
3.2 Severe Plastic Deformation (SPD) Techniques
Severe plastic deformation (SPD) techniques have been implemented or proposed
to produce micron/submicron- and even nano- crystalline metals and alloys. SPD tech-
niques are defined as methods which can impose a very high strain on metals often with-
out significantly changing the overall volume. Many different SPD methods have been
developed in the last 20 years mainly including high-pressure torsion (HPT), accumula-
tive roll bonding (ARB), and equal-channel angular pressing (ECAP). The basic princi-
ple of HPT is shown in Figure 3.1. A disk shaped specimen (current largest dimension
is 30 - 50 mm did disk and a thickness of about 10 mm) is pressed between two anvils
and with imposed hydrostatic pressure. The upper anvil is usually fixed, while the lower
one is rotated. Shear strain is imposed on the samples. For accumulative roll bonding
(ARB), two sheets are stacked together, rolled, cut into pieces, stacked together, again,
and rolled for additional cycles. A detailed description of this method can be found in
[57].
In ECAP, the material is pressed through a die which consists of two equal channels
with an angle of and, an associated arc of curvature, . The amount of imposed shear
strain per pass depends on the angle () between the two channels and it is about 1 for
90

ECAP die [57]. Many routes have been developed and the details of the deformation
mechanism will be introduced in the following sections.
45
Figure 3.1: Schematics of severe plastic deformation techniques
(a) High pressure torsion (HPT) (b) Accumulative roll bonding (ARB) (c) Equal channel
angular pressing (ECAP) [57].
3.3 Mechanical Properties versus Grain Size
The grain size of polycrystalline materials influences the mechanical properties
such as hardness, yield strength, tensile strength and fatigue strength. For many metals,
the yield strength
y
varies with grain size in accordance with the Hall-Petch equation.

y
=
0
+k
y
d
n
(3.1)
whered is the average grain diameter,n is approximately 0:5,k
y
is a material constant
and
0
is single crystal strength. Fine grained microstructures have drawn much interest
to dramatically improve the strength of metals. The Hall-Petch relationship breaks down
for very small grain sizes (typically at tens of nanometers). Yield strength increases with
decreasing grain size in the micron or submicron regime and then decreases with further
decreasing grain size (grain dimension of nanometers or less).
46
3.4 Non-equilibrium Grain Boundaries
After SPD, as shown in Figure 3.2, an ultra-fine grained structure is formed with
equiaxed grains. High percentages of high-angle boundaries in the microstructure are
usually observed. Valiev et al. [58] has suggested that these grain boundaries(HABs)
are usually nonequilibrium grain boundaries which are evidenced by the distorted shape
of boundaries in Figure 3.2. Also, high elastic strains may be produced in SPD sam-
ples and suggested to be associated with these nonequilibrium boundaries. For coarse-
grained metals, dislocation movement and twinning are the primary deformation mech-
anisms under plastic deformation (note that twinning usually does not occur in Al).
Ultra-fine, equiaxed grains with high-angle grain boundaries impede the motion of dis-
locations and consequently enhance strength. In addition, the grain boundaries gener-
ated by SPD are usually in a nonequilibrium state, with many dislocations that are not
geometrically necessary to form the grain boundary. These dislocations, as well as dis-
locations piled up near the grain boundaries, could move to facilitate grain boundary
sliding and grain rotation, and therefore increase the ductility. This deformation mech-
anism may be partially responsible for the switch from dislocation slip (or twinning) to
grain boundary sliding in SPD metals.
3.5 Equal-Channel Angular Processing (ECAP)
ECAP is a simple technique to deform materials into ultra-fine grained materials.
The ECAP die is shown in Figure 3.1(c) and the process is shown in Figure 3.3 [54, 59,
60]. Owing to different routes of processing, the resulting microstructures are different
and the deformation efficiency and uniformity are also different.
The ECAP die is composed of two channels with identical cross-section area and
shape, which are connected to each other at a certain angle. When the sample is pressed
47
Figure 3.2: TEM micrographs of non-equilibrium grain boundaries
(a) Cu after 16 passes ECAP, and (b) Ti after HPT (5 turns) and being heated at 250

C
for 10 min [58].
through the intersection plane of the two channels (the so-called shear plane, 45

with
respect to the pressing direction), the sample is deformed by means of simple shear and
is pushed out of the die (Figure 3.4) [59, 60].
Segal et al. [61] suggested that the ECAP deformation mechanism can be explained
by the simple shear (Figure 3.4). This kind of deformation can impose very high strain
into metal billets without changing the shape and size. At that time, the academic com-
munity did not pay much attention to this new deformation technique. Later, Valiev et
48
Figure 3.3: The four fundamental processing routes of ECAP
RouteA: the sample is not rotated for each pass. RouteB
C
: the sample is rotated 90

clockwise for each pass. RouteB
A
: the sample is rotated 90

clockwise and counter-
clockwise alternatively for each pass. Route C: the sample is rotated 180

for each
pass [59, 60].
al. [62] provided the details of grain size evolution of ECAP and revealed the potential
for using ECAP to produce UFG metals.
Different routes lead to different shear directions. Figure 3.5 illustrates the shear
directions/planes for different passes via different routes [60]. This simple model can
be used to explain the grain shape deformation evolution for different ECAP routes and
passes.
For different materials, the deformation effectiveness of grain refinement seems a
little different. However, it is generally accepted that using the concept of simple shear
plane direction and the interaction of the shear planes for different passes and rotation
angles can explain the formation of textured grains and grain refinement. Here, taking
Al or Al-alloy for example, Iwahashi et al. [63, 64] processed pure Al with ECAP
route A, B
C
and C using a die with = 90

and concluded that route B
C
leads to
49
Figure 3.4: The material is deformed through simple shear
Shear stress is imposed by ECAP die along the diagonal direction (45

) of the intersec-
tion of two channels. After shearing, a simple schematic shows that the shape of shape
(1) is plastically deformed into that of shape (2) [59].
the most rapid grain refinement. For this reason,B
C
is recommended as the optimum
ECAP-processing route for grain refinement. Later, Oh-Ishi et al. [65] compared route
B
C
and routeB
A
for grain refinement and led to the conclusion that routeB
C
is more
effective than routeB
A
. In addition, Furukawa et al. [66] examined the shearing charac-
teristic associated with ECAP routes and the number of passes. The results showed that
evolution of the microstructure occurs most rapidly via routeB
C
and less rapidly with
routeC. The routesA orB
A
has the slowest evolution. As a result, Langdon et al. [67]
summarized the effectiveness of ECAP routes in grain refinement as routeB
C
>C >A,
B
A
[68].
The deformation mechanism of different ECAP routes is further understood in
many studies [63, 64, 69]. However, the fundamental role of shear planes which may
50
be associated with crystal structures in the grain refinement still have not been investi-
gated. Finally, Zhu et al. [68] concluded that the texture of microstructures and crystal
structures interacting with the shear plane may be the primary reasons for effective grain
refinement rather than the accumulative strains by investigating the fundamental mech-
anisms of grain refinement and analyzing the literature.
Figure 3.5: Schematics of shear directions and planes for different ECAP routes [70]
3.5.1 Microstructure of ECAP Metals
A TEM analysis of an ECAP-processed polycrystalline pure aluminum was
reported by Iwahashi et al., 1998 [54]. Iwahashi and his co-workers used an ECAP
die having = 90

and = 20

. This is the same geometry as that of the deformation
51
die for AA1050 of this study. It is shown in Figure 3.6 that the processing routes and
the number of processing passes significantly influence the microstructure [60]. It also
shows that the microstructure is not homogeneous, as can be observed from different
deformation planes.
The microstructure in the x-z plane of the polycrystalline aluminum clearly consists
of bands of grains/subgrains approximately oriented 45

with respect to the pressing
direction (parallel to the shearing plane of the ECAP die having = 90

). For multiple-
pass ECAP of pure aluminum, the microstructure differs with different routes. For route
A, the shape of the grains/subgrains viewed on the x-y plane and the y-z plane keeps
the elongated shape parallel with y axis and does not substantially change compared
with a one pass sample. The microstructure has the tendency to evolve into equiaxed
grains/subgrains via routeB
C
, and the equiaxed grain/subgrain microstructures can be
clearly observed from different deformation planes.
3.5.2 Shear Strain in ECAP
The shear strain for simple shear can be defined as
= tan where is the angular
distortion by the shear force. This can be observed in Figure 3.7(a). Iwahashi et al.
utilized
this definition to calculate the total shear strain in ECAP in Figure 3.7(b). For the ideal
situation, the total shear strain from ECAP die can be derived as [71, 72]

= tan =
(a
0
q)
(qd
0
)
(3.2)
52
Figure 3.6: Microstructure of pure aluminum after varying ECAP routes
Influence of the processing route and the number of processing passes on the microstruc-
ture [54].
53
Figure 3.7: Simple shear
(a) A square is deformed by tan by simple shear. (b) Deformation principle of equal-
channel angular pressing [72].
qd
0
=ad (3.3)
ab
0
=dc
0
=a
0
p =pq =ad cot

2
(3.4)
So,
a
0
q = 2ad cot

2
(3.5)

= 2 cot

2
(3.6)
where is the angle between two channels (shown in Figure 3.1(c)). The total equiva-
lent strain after 1-pass of ECAP has been calculated to be,
" =

p
3
=
2
p
3
cot

2
(3.7)
54
This simple approach does not consider the principal strains and also the rotation of the
principal strain axis. In addition, the equation (3.6) is only correct in describing the
small shear strain. Thus, the result of equation (3.7) is also questionable. Xia et al. [72]
thus proposed a general expression of calculating the shear strain for simple shear by
the integration method and the same approach is used to calculate the total shear strain.
Furthermore, the principal strain is calculated by a geometric method and matrix algebra
whereby, the equivalent strain is calculated. The methods are introduced in this section.
Figure 3.8 where a parallelogram OABC is deformed into OADE by a simple
shear. A plane strain condition is assumed. An infinitesimal change,d in the direc-
tion, occurs and the displacementu in x-axis andv in y-axes can be expressed as
Figure 3.8: An infinitesimal shear deformation ofd under simple shear [72]
u =y tan( +d)y tan() (3.8)
v = 0 (3.9)
Thus, the normal strain and shear strain increments along x- and y-axes can be expressed
as
55
d"
x
=
@u
@x
= 0 (3.10)
d"
y
=
@v
@y
= 0 (3.11)
d"
xy
=d"
yx
=
1
2
(
@u
@y
+
@v
@x
) =
1
2
d
cos
2

(3.12)
As is changed from
1
to
2
by simple shear, the accumulated shear strain can be
calculated by
"
xy
="
yx
=
Z

2

1
d"
xy
=
1
2
(tan
2
tan
1
) (3.13)
The above equation (3.13) can be used to calculate the total shear strain in ECAP. Fig-
ure 3.9 shown ECAP die, where
1
=

2
and
2
=

2
. Thus, the shear strain is
"
xy
="
yx
=
1
2
(cot(

2
) cot(

2
)) = cot(

2
) (3.14)
Thus, for =

2
, the total shear strain is

= 2"
xy
(3.15)
"
xy
=1 (3.16)
It turns out that the shear strain calculated by integrating infinitesimal strain increment
gives the same result as that calculated by a simple geometric derivation [71].
56
Figure 3.9: A simple shear model of ECAP
Shear directionsOo,Bb,Aa,B
0
b
0
,A
0
a
0
are parallel to each other in the schematic. The
angle between the two channels is [72].
3.5.3 Principal Strains and Direction in ECAP
After a single pass in ECAP, plastic strain is introduced into a sample through
shear deformation, leading to grain elongation and texture formation. The amount of
shear strain is directly linked to different passes and processing routes. Theoretical and
experimental evidence have proven that the grain elongation or texture formation in
ECAP is associated with the principal strains and the direction of principal strain. To
model this, a geometric analysis or matrix algebra analysis can be used, especially for
1-pass ECAP.
3.5.3.1 Geometric Analysis for Principal Strains after 1-pass ECAP
Assuming zero strain in the y direction, shear deformation takes place under plane
strain conditions in the x-z plane. Thus, the shape changes under shear deformation in
57
Figure 3.10: A parallelogramBAab in the coordinateX
1
O
1
Y
1
deforms intoB
0
A
0
a
0
b
0
in
the coordinateX
2
O
2
Y
2
in ECAP [72]
the x-z plane (the plane of two pressing directions) can be simplified to the shearing of
a circle.
Xia and Wang [72] have made a couple of assumptions: 1). volume is the same
throughout the two channels; and 2). the linesBb andAa in Figure 3.10 of the pressing
object needs to remain parallel to the diagonal line around the joint after passing through
the ECAP die. The translating coordination after passing through the ECAP die can then
be calculated through geometric analysis.
Figure 3.10 shows linesAa andBb in the entrance channel becomeA
0
a
0
andB
0
b
0
in the exit channel through 1-pass ECAP. All lines remain parallel to each other and
keep the same spacing between them. The coordinate system X
1
O
1
Y
1
and X
2
O
2
Y
2
are set up in the entrance of channel and exit of channel, respectively. P (x
1
;y
1
)
58
in X
1
O
1
Y
1
becomes P
0
(x
2
;y
2
) in X
2
O
2
Y
2
through 1-pass ECAP. Then, the relation
betweenP (x
1
;y
1
) andP
0
(x
2
;y
2
) can be expressed in the following:
x
1
=x
2
(3.17)
y
1
=y
2
2x
2
cot(

2
) (3.18)
Thus, we can assume a circle of radiusR
0
before shear deformation inX
1
O
1
Y
1
,
x
1
2
+y
1
2
=R
0
2
(3.19)
This circle will be transformed into ellipse,
(1 + 4 cot
2

2
)x
2
2
+y
2
2
4(cot

2
)x
2
2
y
2
2
=R
2
0
(3.20)
The long axis is rotated by an angle with respect toX
2
axis. The relation of angle
and the length along the long axis and short axis of the ellipse can be expressed as:
tan =
1 + cos(=2)
sin(=2)
(3.21)
R
1
=
R
0
sin(=2)
1 cos(=2)
(3.22)
R
2
=
R
0
sin(=2)
1 + cos(=2)
(3.23)
Then, comparing the length of ellipse with the circle radius, the principal strains can be
calculated.
59
"
1
=ln
R
1
R
0
=ln
sin(=2)
1 cos(=2)
(3.24)
"
2
=ln
R
2
R
0
=ln
sin(=2)
1 + cos(=2)
(3.25)
For =

2
, the principal strains are
"
1
= 0:881 (3.26)
"
2
=0:881 (3.27)
The long axis of the ellipse is tilted by an angle of 22:5

with respect to the pressing
direction of ECAP die. This is shown in Figure 3.11. A circle is deformed into an ellipse
as it passes through the ECAP die. The long axis (+22:5

) of the ellipse indicates the
direction of maximum tensile plastic-strain and the short axis indicates the direction of
maximum compressive plastic-strain (67:5

). Strain along the pressing axis (0

) is
approximately zero [72].
3.5.3.2 Matrix Algebraic Analysis for Principal Strains after 1-pass ECAP
Xia et al. [72] also used matrix algebra to analyze the principal strains and prin-
cipal strain directions after 1-pass ECAP. This method leads to the same results as the
previous geometric analysis. In this approach, plane strain condition is also assumed.
The coordinate transformation between P (x
1
;y
1
) in the entrance of the channel and
P
0
(x
2
;y
2
) in the exit of the channel can be defined as
60
Figure 3.11: A deformation model for ECAP [72]
2
4
x
1
y
1
3
5
=
2
4
cosh" + sinh" cos 2 sinh" sin 2
sinh" sin 2 cosh" sinh" cos 2
3
5
2
4
x
2
y
2
3
5
(3.28)
The transformation matrix corresponds to a pure shear with principal strains,", which
is along the x-direction at an angle of . This is shown in Figure 3.12(a).
The coordinate transformation can be also written as
2
4
x
1
y
1
3
5
=
2
4
cos sin
sin cos
3
5
2
4
1
0 1
3
5
2
4
x
2
y
2
3
5
(3.29)
The transformation matrix corresponds to a simple shear,
, in the direction of the x-
axis followed by a rotation of. The represented schematic is shown in Figure 3.12(b).
61
Figure 3.12: Principal direction and shear direction in ECAP
(a) Pure shear deformation due to the principal strains. It represents the transformation
matrix due to the principal strains. (b) Simple shear deformation. It represents the
transformation matrix corresponding to a simple shear,
, in the direction of x-axis and
followed by a rotation of [72].
By setting the two transformation matrices equal (eq.(3.28) and eq.(3.29)), the princi-
pal strains and the direction relation between principal strains and simple shear can be
resolved. Thus, the equations can be obtained
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
sinh" =

2
sin 2 =
2
p
4+
2
tan =

2
(3.30)
The angle between simple shear direction and principal strain is
= (3.31)
62
For =

2
and
=2, the same results with geometric analysis can be obtained,
8
>
>
>
>
>
<
>
>
>
>
>
:
"
1
= 0:881
"
2
=0:881
= 22:5

(3.32)
The detailed relations are shown in Figure 3.13.
Figure 3.13: A schematic shows the principal strains, principal directions, and the direc-
tion of shear for the ECAP die of =

2
[72]
3.5.4 Grain Elongation for ECAP Multiple Passes
Garcia-Infanta et al. [73] utilized a simple shear model in the ECAP die to predict
the grain elongation angle with respect to the x-axis for ECAP routeA. This is shown
63
in Figure 3.14. They simplified the transformation matrix by assuming the simple shear
direction is along the x-axis. The coordinate transformation matrix for the simple shear
model is written in the following:
Figure 3.14: Shape change of a circular element in the X-Z plane for material processed
by routeA
(a) Elliptical shape after one pass and four passes. (b) Predicted and experimental evo-
lution of inclination angle,, and direction of maximum elongation. (c) Predicted and
experimental value of relative width reduction,w=d, as functions of the shear strain
.
d is the original diameter of a circle andw is the short axis of an ellipse, which forms
through deformation. [73]
2
4
x
y
3
5
=
2
4
1
0 1
3
5
2
4
x
0
y
0
3
5
(3.33)
where
is the shear strain. A circle of diameter,d, would then become an ellipse in the
x-y plane after shear deformation according to
64
x
2
+ (1 +
2
)y
2
2
xy (
d
2
)
2
= 0 (3.34)
Furthermore, the angle of inclination,, of the principal strain direction with the axis of
the pressing direction is given as
= tan
1
[
1
2
(
p
(2 +
2
)
2
4
2
)] (3.35)
while the short axis length of the ellipse,w, is
w =
2d
q
2[2 +
2
+
p
(2 +
2
)
2
4]
(3.36)
This simple shear model can predict the grain elongation angle,, through ECAP (route
A). As the accumulative shear strains increase with the number of passes, the grain
inclination angle will decrease for ECAP routeA. The theoretical prediction matches
the experimental observation.
Zhu et al. [68] calculated the grain elongation angle,, based on Iwahashi et al.’s
[54] experimental results and deformation model for ECAP routesA,B
C
andC. For a
= 90

die, they have calculated the angles () between the grain elongation direction
and the pressing direction axis (x-axis) for varying ECAP routes and number of passes.
These values are presented in Table 3.1 [68].
Zhu et al. [68] made an assumption that the deformation texture orients the slip
plane between the shear direction and the grain-elongation direction. Such an assump-
tion agrees well with FCC metals such as Al-alloys and Cu, as well as HCP metals such
65
as Be, but may not agree with results for BCC metals. Still, however, these calculated
results match the experiment observation very well, especially for FCC metals through
different ECAP routes.
Based on Zhu et al.’s results, for ECAP routeA,B
C
, andB
A
, the maximum princi-
pal strains should be in the direction tilted 10

30

counterclockwise along the x-axis
for the initial few passes (withn < 4).
Table 3.1: Calculated angles () between grain elongation direction and the extrusion
x-axis [68]
Number of passes RouteA RouteB
C
RouteB
A
RouteC
() value (deg)
1 26:6 26:6 26:6 26:6
2 14:0 19:5 19:5 Equiaxed
3 9:5 26:6 12:6 26:6
4 7:1 Equiaxed 10:0 Equiaxed
3.5.5 Mechanical Properties of ECAP AA1050
The stress-strain curves by tension and compression tests are shown in Figure 3.15.
The yield stress of AA1050 after ECAP routeB
C
for different passes can be as high as
148 MPa for 1-pass, 150 MPa for 2-pass, 180 MPa for 4-pass and 200 MPa
for 8-pass [74, 75]. As the number of passes increases, the yield stress increases. This
is attributed to the high dislocation density and smaller grain sizes.
3.5.6 Grain Size and Grain Boundaries of ECAP Aluminum
The grain size and grain boundaries are analyzed by electron backscatter diffraction
(EBSD) shown in Figure 3.16 [76]. The light gray line indicates the subgrain bound-
aries (misorientation less than 15

). Owing to the resolution of this equipment, only
grain/subgrain boundary misorientations higher than 4

can be identified. The black
66
Figure 3.15: True stress-true strain curves of the ECAP-processed AA1050
(a) Tensile tests after processing by routeC. (b) Compression tests after processing by
routeC. (c) Tensile test processed by routeB
C
. (d) Compression tests after processing
by routeB
C
[75].
lines indicate boundary misorientations higher than 15

. For convenience, the bound-
aries are defined as high-angle grain boundaries (HABs), if the misorientation is higher
than 15

. If the misorientation is lower than 15

, the boundaries are defined as low-angle
grain boundaries (LABs).
The term grain/subgrain, is used to describe all the grain and subgrain boundaries
including high and low-angle boundaries. Here, the low angle boundaries can be even
lower than 2

, and can be characterized by TEM. After 1-pass ECAP, the grain/subgrain
sizes can be as small as 1:3m and the fraction of low angle boundaries can be about
75%. These values are shown in Table 3.2 [77]. The character of the boundaries have
67
Figure 3.16: EBSD mapping of ECAP AA1050 with processing of routeB
C
(a) Top area of billet cross-section (b) Center area (c) Bottom area. The black lines are
used to represent grain boundaries of which the misorientation angle is greater than 15

.
The light gray lines represent subgrain boundaries of which the misorientation angle is
smaller than 15

and greater than 4

[77].
been reported by several groups [63, 77–79]. In addition, as the number of passes
increases, the fraction of HABs increases and the grain size also decreases. For AA1050
via routeB
C
, the grain size (HABs only) is between 0:7m and 1m after 4 passes of
ECAP [63, 76, 77, 80].
Table 3.2: Grain/subgrain boundary properties and dislocation density of ECAP 1-pass
AA1050
represents dislocation density in the unrecrystallized grain.f
Sub
represents the fraction
of low angle boundaries. is the subgrain size.D is the average grain size. [77]
Parameter Top edge Centre Bottom edge
(10
14
m
2
) 4 4 4
f
Sub
0:473 0:746 0:73
(m) 1:3 1:3 1:3
D(m) 8:2 26 19
3.6 LRIS in ECAP Metals by CBED/HOLZ Method
The higher-order Laue zones (HOLZ) lines are simply the elastic part of the higher-
order Laue zone Kikuchi lines. These HOLZ lines are more sensitive to the changes in
68
Figure 3.17: A schematic of zero-order Laue zone and higher-order Laue zone
The Ewald sphere can intersect reciprocal-lattice points from planes not parallel to the
electron beam whoseq vectors are not normal to the beam. The sphere has an effective
thickness of 2 because of beam convergence and so intersects a range of these HOLZ
reciprocal-lattice points [81].
lattice parameter, since they come from much larger Bragg angles and q vectors. A
schematic of HOLZ is shown in Figure 3.17. The reasoning can be explained by the
following equation:
jqj =
2
d
(3.37)
jqj = 2
jdj
d
2
(3.38)
For smallerd values (lattice spacing), the value ofjqj is much larger withjdj remain-
ing constant and thus, a clear position change of HOLZ due to elastic strains can be
observed. In addition, HOLZ line patterns allow one to resolve whole lattice parameters
(a,b,c,,,
).
69
Ahajeri et al. [56] analyzed the LRIS by CBED/HOLZ method near the grain/-
subgrain boundaries and in the grain/subgrain interiors of AA1050 ECAP via routeB
C
samples. The average grain size after 2-pass ECAP is 1 m 0:66 m, and is
illustrated in Figure 3.18. By simulating the HOLZ lines, the lattice parameters were
obtained and compared with standard values, internal strains can be calculated. Near the
grain boundary 20nm (the beam size is 20nm), the lattice parameterc was 0:40289
nm (standard lattice parameter of pure aluminum is 0:40495 nm), which indicates a
compressive strain of approximately 0:1%. The angles and were also changed to
89:95

and 90:05

, respectively, indicating shear strains of approximately 0:044% and
0:044%, respectively. The other three parameters remained ata = b = 0:40329nm
and
= 90

. LRIS can only be measured in the 20nm region around the grain/subgrain
boundaries. While the electron beam position moves away from the 20nm area around
grain/subgrain boundaries, LRIS is not detectable.
For the 4-pass ECAP, near the grain/subgrain boundaries, the results show a =
b = 0:40329nm andc = 0:40289nm, = 89:8

, = 0:2

and
= 90

. A typical
TEM image is shown in Figure 3.19. These results correspond to a compressive strain of
approximately 0:1% and the shear strains are 0:175% and0:175% based on the angles
and, respectively.
Ahajeri et al. have evidenced for the first time that the maximum principal strain
around the grain/subgrain boundaries is about 1:1 10
3
and the maximum principal
stress can be estimated at 112:5 MPa for AA1050 2-pass ECAP via routeB
C
sample.
However, the internal strains/stresses were not measured away from 20nm area around
the grain/subgrain boundaries because of low strain values or strain relaxation in the thin
TEM foil.
70
Figure 3.18: A TEM image of AA1050 after 2-pass ECAP via routeB
C
[56]
3.7 Sample Preparation and Experimental Procedures
3.7.1 ECAP AA1050 via RouteB
C
The as-received ECAP AA1050 aluminum billet contained 99:5wt:% aluminum
with 0:26wt:% Fe and 0:14wt:% Si as the major impurities. It was machined to 10:0
mm in diameter and 65:0 mm in length for ECAP processing. ECAP route B
C
was
conducted at ambient temperature with a pressing speed of 0:5 mm s
1
. The solid
ECAP die was made with an internal channel bent into an abrupt angle of = 90

, with
the outer arc curvature of the channel joint rounded with = 20

(Figure 3.20)
1
.
1
ECAP AA1050 samples were provided by Dr. Yi Huang, Materials Research Group, Faculty of
Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK
71
Figure 3.19: A TEM image of AA1050 after 4-pass ECAP via routeB
C
[56]
Figure 3.20: Geometry of the ECAP die for AA1050
Abrupt angle = 90

outer arc of the channel, joint = 20

. The bold black line on
the sample indicates its upper surface.
72
3.7.2 ECAP AA6005 via RouteC
The high age hardening ability of 6XXX Al-Mg-Si alloys is caused by the con-
siderable precipitation during heat treatment [82]. Thus, the age hardening process has
attracted much attention to research on precipitation kinetics. These studies identified
Mg
2
Si and Si as the major and predominant second-phase precipitates [82]. Recently,
many refinement techniques such as ECAP have also been implemented to further
improve the mechanical properties. Here, the LRIS of AA6005 alloy after ECAP route
C was examined. The composition of AA6005 alloy is shown in Table 3.3. AA6005
samples of 10 mm in diameter and 100 mm in length were cut from the bars and salt-bath
annealed at 530

C for 3 hours and then slowly cooled to room temperature, at a rate of
80

C/h (T4 temper condition) to induce the complete precipitation of coarse secondary
phase particles. ECAP routeC was performed at room temperature
2
. The die geometry
is shown in Figure 3.20. Each pass can impose an equivalent strain" = 1:08 by consid-
ering abrupt angle = 90

and outer arc of the channel, joint = 20

in shear strain
calculation.
Table 3.3: Chemical composition of the AA6005
Si Mg Fe Mn Ti Cr Cu Zr Al
0:88 0:60 0:27 0:10 0:015 0:010 0:005 < 0:005 bal: wt:%
After the T4 process, the mean grain size was 78 2 m. After ECAP route
C, the grain and subgrain sizes are shown in the Table 3.4
2
.
LAB
, d
LAB
and f
LAB
refer to the mean boundary misorientation, sub grain size, and fraction of the low-angle
boundaries ( < 15

), i.e. subgrain, and
HAB
,d
HAB
andf
HAB
refer to mean boundary
misorientation, size, and fraction of the high-angle boundaries, i.e. grains.
2
ECAP AA6005 samples and the analysis of mechanical properties and microstructures were provided
by Dr. Marcello Carbibbo, Department of Mechanics, Polytechnic University of Marche, Italy
73
Table 3.4: Grain/subgrain sizes after ECAP routeC AA6005
ECAP pass Equivalent
LAB
d
LAB
f
LAB

HAB
d
HAB
f
HAB
strain (m) (m)
1-pass " = 1:08 4:7 2:25 0:72 28:5 6:55 0:28
2-pass " = 2:16 6:2 1:80 0:21 31:4 3:60 0:79
3.7.2.1 Mechanical Properties of AA6005 via ECAP RouteC
Microhardness and tensile test measurements were performed on AA6005-T4 after
ECAP routeC. Yield stress and ultimate strength were measured along ECAP pressing
direction. Tensile test specimens were rectangular with a thickness of 2 mm, width of 7
mm and length of 21 mm. Gauge length and width were 5 mm and 3 mm, respectively.
All tests were performed at room temperature. The hardness and tensile test results are
shown in the Table 3.5
2
and Table 3.6
2
.
Table 3.5: Hardness (HV) of AA6005 via ECAP RouteC
T4 (prior ECAP): 58 1 1-pass: 71 1 2-pass: 75 1
Table 3.6: Tensile test (MPa) of AA6005 via ECAP RouteC
T4 (prior ECAP) 1-pass 2-pass
Yield 140 165 172
UTS 210 232 248
3.7.3 Sample Preparation of X-ray Powder Diffraction Measurement
X-ray powder diffraction measurements were conducted on the 11-BM beamline
at the Advanced Photon Source (APS) at Argonne National Laboratory on ECAP sam-
ples. The 11-BM beamline operates a high-resolution X-ray powder diffractometer that
provides high-accuracy measurements of the average lattice parameter and peak broad-
ening. A cylindrical rod, 2:0 mm in diameter and approximately 8:0 mm in length, was
cut from the center axis of the ECAP sample by electrical discharge machining (EDM)
74
and chemically etched to about 0:8 mm diameter (30% nitric acid at 50

C). This rod
was then fitted into a kapton capillary tube for high-resolution X-ray powder diffraction
measurements. All of the X-ray samples were carefully prepared to avoid generating
any additional plastic strain (damage) other than that from the ECAP deformation. The
sample holder is shown in Figure 2.5. The ECAP samples can be fitted into the sample
base after being etched down to 0:8 mm (diameter). The beam size of 11BM beamline
is about 3 mm. The data obtained from this scan covers the 2 range 0:5

50

, with a
step size of 0:0001

, and using a wavelength of0:413315 Å (30 keV). Thus, the scan
covers a d-spacing range of 45 Å0:5 Å [42].
3.7.4 Sample Preparation of X-ray Microbeam Measurement
After ECAP pressing, discs of approximately 2:0 cm height were produced using
a cross-sectional cut through each ECAP billet by EDM. The surface of the disc was
then mechanically polished using silicon carbide grit paper (up to 2400 grit) and then
electro-chemically polished in a perchloric acid solution (Table 3.7) to provide a flat,
damage-free, surface. X-ray microbeam diffraction was used to measure elastic strains
from grain/subgrain interiors at depths ranging from 0m to 150m below the polished
surface.
Table 3.7: Electro-chemically polishing solution and conditions of ECAP AA1050 and
AA6005
Solution DC V oltage DC Current Temperature
5% perchloric acid 10 15 V 100 150 mA 0

C
3% nitric acid
92% ethyl alcohol
75
Figure 3.21: The electropolishing schematic
3.7.4.1 Surface Electropolishing for X-ray Microbeam Diffraction Measurement
Electropolishing, also known as electrochemical polishing, is an electrochemical
process that removes material from metallic materials (e.g. Al, Al alloy or Cu). For
X-ray microbeam diffraction experiments, Laue diffraction and intensity are sensitive
to the flatness of the sample surface and artificial strains can result from the machining
process. Thus, an electropolishing process is usually adopted to flatten the surface and
remove several tens of micrometers to a hundred micrometers of material on the surface.
Typically, the microbeam sample surface is immersed in a temperature-controlled
bath of electrolyte and is applied with positive DC voltage (anode). A cold bath is usu-
ally needed for the aluminum alloys. The negative terminal is attached to another metal
(e.g., copper or graphite). A current passes from the anode, where metal on the surface is
dissolved in the electrolyte. At the cathode, a reduction reaction occurs, which normally
produces hydrogen. Electrolytes used for electropolishing are most often concentrated
acid solutions with a high viscosity. To achieve electropolishing of a rough surface, the
76
protruding parts of the surface profile must dissolve faster than the recesses. This pro-
cess, referred to as anodic leveling, is achieved by a mass transport limited-dissolution
reaction. Anodic dissolution under electropolishing conditions flattens metal objects
due to increased current or charge density on corners and sharp areas. The schematic is
shown in Figure 3.21.
3.7.5 X-ray Microbeam Measurement Setup
3.7.5.1 Experimental Procedures of X-ray Microbeam Measurements
X-ray microbeam diffraction measurements were conducted on beamline 34-ID-
E at the APS. The monochromatic X-ray beam was focused to 0:5
2
m
2
using
Kirkpatrick-Baez mirror optics, which provided high spatial resolution in the two axes
perpendicular to the incident X-ray beam. The spatial resolution along the beam path
was determined by the grain/subgrain size. The energy was scanned over reflections in
the vicinity of 14 keV using steps of 3 eV , which matches the resolution of the instru-
ment. The diffracted beams were detected on an amorphous-Si area detector. Depth
information was obtained by translating a 50 m diameter platinum wire profiler par-
allel to the sample surface, blocking diffracted X-rays from the sample; the origin of
the diffracted X-rays was then determined by triangulation [31]. {531} reflections were
measured to obtain lattice spacings for suitably oriented Al grains/subgrains for each
measurement.
3.7.5.2 Defining the Region of Interest (ROI) for the ECAP sample
After 1-pass ECAP, grain/subgrain sizes of AA1050 are only about one microm-
eter. Grain/subgrains are composed of high-angle grain boundaries (> 15

) and low-
angle grain boundaries (< 15

). With 1-pass ECAP, about 25% of the grain boundaries
are high-angle grain boundaries [77]. In the experiment, a monochromatic incident
77
Figure 3.22: Back-reflection Laue diffraction of ECAP Al sample
beam energy of about 14 keV was chosen. All the grains/subgrains along the beam
path which meet the Bragg reflection condition with incident beam energy will lead to
bright diffraction spots on the detector. There are usually several diffraction rings shown
on the detector due to the misorientation among the grains/subgrains. Diffraction rings
are made up of diffraction spots from multiple grain/subgrain interiors with the same
reciprocalq vector, wherejqj = 2/d
hkl
. The back-reflection Laue diffraction method
is designed for X-ray microbeams at the APS (Figure 3.22). In order to obtain accu-
rate lattice spacing for diffraction sports on detector, a small scan area, or the region of
interest (ROI), where a higher intensity spot is visible need to be identified. An energy
scan can then be performed in which the range can show the "whole" spot in steps of 3
eV . A schematic, in Figure 3.23, shows typical diffraction rings for an ultra-fine grain
material. The region of interest is defined by the dash square.
78
Figure 3.23: A region of interest (ROI) for a Laue pattern
3.7.5.3 Obtaining Accurate Bragg Angle for Fine-grain Samples
To calculate the accurate Bragg angle, the depth of diffraction volume needs to
be located. In this case, the platinum wire was manually brought to block the whole
diffraction spot (Figure 3.24). Thus, from the center of the diffraction spot, platinum
wire position and incident beam path, the depth of diffraction volume from the surface
can be calculated through a triangulation method. By knowing the depth of diffraction
volume in the sample, an accurate Bragg angle can be obtained. This is due to the fact
that the Bragg angle calculation is directly calculated from the intersection point of the
incident and diffraction beam, not from the sample surface.
Then, an accurate q vector for different incident beam energies can be derived
in the ROI. Combining all the data of intensity versus q vector for the energy scan, a
scattering profile can be obtained. A Gaussian function is adopted to fit the scattering
data. The peak position of the Gaussian curve is used to determine the averageq values
(or d-spacing) for the diffraction volume.
79
Figure 3.24: Wire scan to obtain an accurate Bragg’s angle.
(a) The platinum wire is moved to block the diffraction spot. (b) Diffraction signals
in the ROI are integrated and converted into an X-rayq-profile and fitted by Gaussian
function.
3.7.5.4 ECAP AA1050 1-pass
The geometry of the X-ray microbeam setup is illustrated in Figure 3.25. Initially,
the sample was oriented at an angle of 45

with respect to the incident beam, so the lat-
tice spacings were measured for atomic planes that were approximately perpendicular
to the ECAP pressing direction. In addition, the sample was tilted by22.5

relative
to the pressing direction, allowing measurement of the d-spacings close to the direc-
tion of theoretical maximum tensile plastic-strain (+22:5

) or towards a relatively large
compressive plastic-strain (22.5

) 0:5 of maximum compressive true strain. The
actual orientations of the measured plane normals were approximately +27:3

, +4:9

,
and17:5

off the pressing direction, when taking into account the measured positions
of the diffraction spots on the detector. Finally, the d-spacings were converted into elas-
tic strains using the lattice parameter which was obtained from the powder diffraction
measurements. In the experiment, {531} planes were measured for strain calculation of
80
grain/subgrain interiors since a monochromatic incident beam energy of about 14 keV
was chosen and Bragg angle was set to about 45

.
Figure 3.25: A schematic view of the X-ray microbeam setup for AA1050 1-pass ECAP
The elastic strains ( lattice spacing) of grains/subgrains along the ECAP pressing direc-
tion are measured in the illustrated configuration. Bragg angle is set as 45

and
energy is set around 14 keV . The direction of dash line (45

counterclockwise titled
from Z axis) is roughly the strain measured direction. The inset shows that the sample is
tilted by +22:5

(left) and22:5

(right). The strains in the directions along +22:5

and
22:5

with respect to the ECAP pressing direction can be measured. The bold black
line on the sample indicates the upper surface which is close to the top surface center of
ECAP exit channel.
3.7.5.5 ECAP AA1050 Multiple-pass via RouteB
C
For multiple passes, a monochromatic incident beam energy of 14 keV was
chosen and the lattice spacings of {531} were measured along the pressing direction.
The sample surface is tilted 45

with respect to the incident beam to obtain the strains
along the pressing direction. Since the ECAP sample is rotated for 90

during each pass
for routeB
C
, accumulated plastic strains by increasing the number of passes cannot be
81
easily calculated along a specific direction and the maximum principal strain direction
may be rotated. Thus, only strains along the pressing direction are measured at the
center position for multiple passes.
3.7.5.6 ECAP AA6005 1-pass and 2-pass via RouteC
For the AA6005 samples, the lattice spacings of {531} are measured along the
pressing direction shown as in Figure 3.25. The sample surface is tilted 45

with respect
to the incident beam. All strains are measured at the center position of samples since
the maximum principal strain direction may be rotated after 2-pass route C.
3.7.5.7 Rotation Angles of Reflection Planes
In Figure 3.25, the microbeam measurement setup schematic shows that the angle
between the reflection plane and the incident beam is shown as 45

, and the angle
between the reflection beam and the incident beam is 90

for strain measurement of
the ECAP sample. In this case, the lattice spacings along the pressing direction can
be measured. For the same settings, the ECAP sample can be tilted to measure the
lattice spacing along other directions with respect to the pressing direction. Experi-
mentally, the reflection spots (e.g. {531}) on the detector which come from reflection
beam roughly perpendicular to the incident beam are chosen and indexed to calculate
the crystal strains. However, the angle between the reflection beam and the incident
beam is not exactly 90

and two beams are not in the y-z plane (shown in Figure 3.26).
Therefore, the measured reflection plane normal is not exactly parallel to the direction,
45

with respect to the incident beam in the y-z plane. Thus, a small angle is present for
the measured strains in the measurement directions (e.g. pressing direction or22:5

).
This issue may increase the divergence of the measured strain distribution and suggests
82
the need for a detailed study to allow the strains to be compared for different measuring
directions.
Figure 3.26: A schematic of rotation angles, RotX and RotZ with respect to the 45

plane normal
The angles by rotating around X axis and Z axis to match the
!
OB (reference vector)
with the measured plane normal (
!
ON) are defined as RotX and RotZ. The angle unit
is degree. The rotation direction of negative angles, RotX(neg) and RotZ(neg) are indi-
cated. The inset (right) shows a schematic of the sample position with respect to the
incident beam and reflection beam
To examine this problem, the strain values versus the reflection beam rotation angle
needs to be investigated. As discussed previously, if a diffraction spot (point C) is
directly above the diffracting volume (pointO) in the sample (in Figure 3.26), then the
diffracting planes are perpendicular to lineOB which bisects angleAOC. This is our
reference direction (
!
OB). In reality, the diffraction spots are generally not directly above
the diffracting volume and the actual orientation of the lattice planes is characterized
83
using two rotation angles, RotX and RotZ. RotX is a rotation about the laboratory X
axis (see Figure 3.26) and RotZ is a rotation about the Z axis, defined by the incident
beam direction. To find these angles, the location of a diffraction spot on the detector is
found, allowing us to determine the vector perpendicular to the lattice planes. Since the
reflections examined often come from anywhere along a diffraction ring on the detector,
RotZ can have a wide range of values. RotX is more constrained since we use {531}
reflections with a fairly constant photon energy. RotZ can be found by rotating the lattice
plane normal about the Z axis onto planeOABC. The vector is then rotated around the
X axis until it is along the
!
OB. This defines RotX. These calculations all assumed that
the diffracted beam came from a depth in the sample of 100 microns; however, changing
this depth makes only a minuscule change in the calculated angles (much smaller than
0:1

). The measured strains as a function of RotX and RotZ for the untilted sample are
shown in Figure 3.27. As shown in Figure 3.25 inset, the ECAP sample is titled22:5

to measure the strains along22:5

directions with respect to the pressing direction. The
measured strains as a function of RotX and RotZ along +22:5

and22:5

directions
are shown in Figure 3.28 and Figure 3.29.
For strain measurements in ECAP samples, the lattice spacings of {531} of grain/-
subgrain interiors whose normals align close to three setting directions (0

and22:5

with respect to pressing direction) can be measured. However, grains/subgrains are ran-
domly distributed in the samples, so the plane normal directions do not exactly match
the setting directions. For the angles, the actual measurement directions can be calcu-
lated. The distribution of RotX encompasses a narrow range and the average angles of
RotX are about +4:9

for the pressing direction, +4:8

for the +22:5

direction and +5

for the22:5

direction, respectively. Thus, the actual measured strain directions can
be adjusted to +4:9

, +27:3

and17:5

with respect to pressing direction. In addition,
by the analysis of strains versus RotZ (titled angles from y-z plane), the results conclude
84
Figure 3.27: The measured strains along the pressing direction versus RotX and RotZ
angles
The ECAP sample surface is set 45

with respect to the incident beam to measure the
strains along the ECAP pressing direction. The schematic is shown in Figure 3.25.
that the measured strain distribution does not vary with RotZ. This analysis also sup-
ports the that the measured strains for three directions can represent the strain values in
ECAP sample along different directions.
85
Figure 3.28: The measured strains along +22:5

with respect to pressing direction ver-
sus RotX and RotZ angles
The measured strain direction is set along +22:5

with respect to the ECAP pressing
direction. The schematic is shown in Figure 3.25.
3.8 Results
3.8.1 Microstructure of ECAP AA1050 via RouteB
C
The grain size before ECAP processing is about 64 m. After 1-pass ECAP, the
average grain/subgrain size is dramatically reduced to 0:67m 0:44m in the trans-
verse plane (x-z plane defined in Figure 3.20) in Figure 3.30 [83]. The shape elongates
86
Figure 3.29: The measured strains along22:5

with respect to pressing direction ver-
sus RotX and RotZ angles
The measured strain direction is set along22:5

direction with respect to the ECAP
pressing direction. The schematic is shown in Figure 3.25.
along the maximum principal stress. Clear grain/subgrain boundaries can be observed
and emanating dislocations and dislocation bundles are evident.
As the number of ECAP passes increases, the grain boundaries are better defined
with high misorientations and the dislocation density in the grain/subgrain interior
decreases. These results are consistent with Murata et al.’s results [84] in which the
maximum dislocation density about 5 10
13
m
2
(grain boundaries and grain interiors)
87
in Al after ECAP is at 1:5 equivalent shear strain and then the dislocation density
decreases. The grain size does not change substantially after 2 passes (Figure 3.31)
3
.
Figure 3.30: A TEM image of AA1050 after 1-pass ECAP
The coordinate is defined in Figure 3.20 and the image was taken in the x-z plane (trans-
verse direction) [83].
3.8.2 Microstructure of ECAP AA6005 via RouteC
After the heat treatment process, silicon, Mg
2
Si and Al-Fe-Mn-Si intermetallic
particles form in 6XXX aluminum alloys. [82] In addition to precipitation hardening,
mechanical properties can be further improved by refining the grain sizes. In this sec-
tion, microstructures of AA6005 after ECAP are described. After 2-pass ECAP routeC,
the average grain/subgrain size reduces from 78m to about 1:8m (Table 3.4). Grain
shape is elongated after 1-pass ECAP and tends to be equaxied after 2-pass shown in
Figure 3.32
2
. Higher dislocation density in the grain/subgrain interiors is also observed
in the 2-pass sample than that in 1-pass sample in Figure 3.32. Si and Mg
2
Si particles
are able to be sheared. Once the size is smaller than 45 nm and 80 nm, respectively,
3
The AA1050 ECAP 2, 4 and 8 passes TEM mircographs were provided by Dr. Yi Huang, Materials
Research Group, Faculty of Engineering and the Environment, University of Southampton, UK and Dr.
Alan G. Fox, Mechanical Engineering Department, Asian University, Thailand
88
Figure 3.31: TEM images of ECAP AA1050 after 2, 4 and 8 passes via route B
C
,
respectively from (a)-(c)
The image were taken in the x-y plane (transverse direction). The coordinate is defined
in Figure 3.20
3
.
precipitate particles can no longer be sheared and their shapes do not change substan-
tially after ECAP. Al-Fe-Mn-Si intermetallic particles are not able to be sheared. In
Figure 3.33
2
, shear bands formed around intermetallic particles after 1-pass and 2-pass
of ECAP. In Figure 3.34
2
shows that dislocations entangled with Mg
2
Si particles after
ECAP
2
.
3.8.3 Powder Diffraction of ECAP AA1050 via RouteB
C
with Different Passes
All of the X-ray powder diffraction peaks for different ECAP passes of AA1050
(for the as-received specimen and up to 8-passes) have almost identical positions, which
give the similar lattice constant,a
0
. In Figure 3.36, Figure 3.37, and Figure 3.38, three
reflection peaks, {331}, {420}, and {531} are shown and the peak positions of three
reflection peaks for different ECAP passes are almost in the same positions. The lattice
spacings of these three reflections also have been measured by microbeam diffraction for
the grain/subgrain strains and thus the peak positions are carefully examined. The lattice
89
Figure 3.32: TEM bright field image of ECAP AA6005 via routeC
(a) AA6005 after 1-pass ECAP routeC. (b) AA6005 after 2-pass ECAP routeC.
2
parameter for the as-received AA1050 was 4:05000(10) Å, for 1-pass: 4:05020(10) Å,
for 2-pass: 4:05010(10) Å, for 4-pass: 4:05010(10) Å and for 8-pass: 4:05010(10) Å.
These lattice parameters are analyzed by GASA for 23 X-ray reflection peaks. The dif-
ference is about 0:0001 Å, or less than 310
5
strain. This demonstrates that these sam-
ples exhibit negligible local residual strains (stresses) over length scales that are small
compared to the sample dimensions and large compared with the dislocation microstruc-
ture. Thus, the average residual strains (stresses) in the local area including grain/sub-
grain interiors and grain/subgrain boundaries (or walls) are about zero (Figure 3.35,
Figure 3.36, Figure 3.37, and Figure 3.38; Table 3.8 and Table 3.9).
In addition, this lattice parameter is slightly different from the pure Al lattice
parameter of 4:049050(15) Å that is generally used in other AA1050 LRIS studies.
The difference in lattice parameter from pure Al and AA1050 translates to an equiva-
lent strain error of +10
4
, which is significant. The radii of Al, Si and Fe are 0:118,
0:111, and 0:156nm respectively. The composition of AA1050 is 0:25% Si and 0:4%
90
Figure 3.33: Shear band formation around Fe-rich intermetallics in ECAP AA6005 via
routeC
(a) AA6005 after 1-pass ECAP routeC. (b) AA6005 after 2-pass ECAP routeC.
2
Figure 3.34: Dislocation interaction with Mg
2
Si particle after 2-pass ECAP routeC of
AA6005
2
Fe. Using a simple weighted average and a hard sphere model, at least a 10
4
increase in
a
0
is expected. Moreover, there are no extra reflection peaks observed in high-resolution
powder diffraction done at 11BM beamline at the APS other than aluminum. Thus, all
91
Table 3.8: Peak position and measured lattice spacing analysis of ECAP AA1050 using
routeB
C
for different passes by X-ray powder diffraction
Sample Planes 2 (deg) d (Å) FWHM
1050 as-received (311) 25:704 0:929076 0:018
(420) 26:382 0:905606 0:018
1050 1-pass (311) 25:703 0:929111 0:027
(420) 26:382 0:905606 0:028
1050 2-pass (311) 25:703 0:929111 0:025
(420) 26:382 0:925606 0:025
1050 4-pass (311) 25:703 0:929111 0:022
(420) 26:382 0:905606 0:024
1050 8-pass (311) 25:703 0:929111 0:025
(420) 26:382 0:905606 0:025
Table 3.9: Measured lattice constants of ECAP AA1050 via routeB
C
by X-ray powder
diffraction
Sample GASA data Uncertainty Approximatea
0
(Å)
1050 as-received 4:049992 0:0001 Å 4:05000(10)
1050 1-pass 4:050199 0:0001 Å 4:05020(10)
1050 2-pass 4:050134 0:0001Å 4:05010(10)
1050 4-pass 4:050083 0:0001Å 4:05010(10)
1050 8-pass 4:050112 0:0001Å 4:05010(10)
the solutes in AA1050 occupy the same sites with Al, if atom sizes are comparable.
They reasonably increase the lattice constant of AA1050 compared with pure Al.
3.8.4 Powder Diffraction of ECAP AA6005 via RouteC with Different Passes
The lattice parameter for the AA6005 was 4:05040(10) Å for 1-pass and
4:05130(10) Å for 2-pass as shown in Table 3.10. These lattice parameters are ana-
lyzed by GASA for 23 X-ray reflections. Figure 3.39, Figure 3.40, Figure 3.41, and
Figure 3.42 display reflection peaks that are shifted to the smaller 2 angle for 2-pass.
The average lattice constant difference between 1-pass and 2-pass is about 0:0009 Å
which equates a strain of 2:2 10
4
. This indicates that there is a residual stress in the
ECAP AA6005 samples. Therefore, this may be caused by non-uniform deformation.
92
Table 3.10: Lattice constants of ECAP AA6005 via RouteC for different passes
GASA data Uncertainty Approximatea
0
(Å)
1-pass 4:050369 0:0001 Å 4:05040(10)
2-pass 4:051286 0:0001 Å 4:05130(10)
3.8.5 Grain/subgrain Interior Strains of ECAP AA1050 after 1-pass
{531} lattice spacings along directions approximately +27:3

, +4:9

, and17:5

off the pressing direction (in the plane of the die channels) were measured by X-ray
microbeam diffraction within low-dislocation regions of grain/subgrain interiors after
ECAP processing. The measured lattice spacings were compared with those of the as-
received samples (measured by powder diffraction), allowing the elastic strains to be
calculated. The directions of measured strains are defined with respect to the pressing
direction.
The long-range internal strains within the grain/subgrain interiors as measured by
X-ray microbeam diffraction are illustrated in Figure 3.43. As described above, the
grain/subgrain interior elastic strains of the ECAP samples are measured along orienta-
tions +27:3

, +4:9

, and17:5

off the pressing (axial) direction in the plane defined
by the two pressing axes. In this experimental setup, internal strains were measured
close to the directions where the sample experienced the theoretical maximum tensile
plastic-strain (+27:3

) and approximately zero strain (+4:9

). The direction of theoreti-
cal maximum compressive plastic-strain (about67:5

) could not be measured because
the sample angles would have caused the platinum wire to collide with the sample sur-
face. Instead, the grain/subgrain interior elastic strains in a direction where the sample is
under a smaller-magnitude (0:53 maximum) compressive plastic-strain at17:5

were
probed.
93
Figure 3.43 shows that most of the elastic strain values are negative (compressive)
at +4:9

(close to the pressing direction). The mean strain of the measured grain/sub-
grain interiors (at +4:9

) is about1:9 10
4
, which converts to about13:6 MPa
long-range internal stress (Young’s modulus along the <531> direction 71 GPa for
aluminum [85]). It is important to note that any stress value is only approximate since
we measure only a single component of the strain tensor. Only negative strains (com-
pressive strain) are observed in the +27:3

direction, close to the theoretical maximum
tensile plastic- strain in the specimen. The maximum and minimum compressive elastic
strains are4:810
4
and1:110
4
, respectively, and the mean strain is2:710
4
,
indicating an average long-range internal stress of about19:0 MPa. The magnitude of
this stress is about 0:13
a
. Finally, for the17:5

direction, the mean strain is around
6:4 10
5
, or4:5 MPa internal stress. Note that our one standard deviation uncer-
tainty for an individual measurement is about 1:0 10
4
. This includes a systematic
error from uncertainties in the instrument calibration and statistical uncertainties due to
determination of the diffraction peak centers. We estimate the uncertainty of the mean
strain values to be about 5 10
5
. Thus, the measured mean strain of6:4 10
5
for
the17:5

direction is not statistically very different from zero strain.
Although the most positive data point for the +4:9

direction is anomalous, this
measured value is correct. It is possible that the corresponding sample volume resides
very near to a grain/subgrain boundary, and thus belongs to a different strain population.
This possibility will be further explored in the Discussion section. If this data point is
removed from the data set, the mean strain becomes2:4 10
4
, this is still within
the measurement uncertainty of the mean strain for the +27:3

direction. Interestingly,
the variation in the strain for the17:5

direction is similar to the distribution in the
other two directions (although the averages are different). This suggests that the sample
conditions, such as strain uniformity in grain/subgrain interiors, are similar for all three
94
directions. The difference between the mean strain values is caused by the magnitude
(or strength) of the net plastic strains that the sample experienced in the different exam-
ined directions. Overall, the trends are consistent with the composite model as will be
discussed in the next section. (Table 3.11)
Table 3.11: Internal strain summary of ECAP AA1050 1-pass
Strains in the AA1050 for 1-pass ECAP along different direction with respect to
pressing direction.
Direction Strain range Mean strain Corresponding stress (MPa)
+27:3

4:8 to1:1 10
4
2:7 10
4
19:0
+4:9

4:9 to 3:2 10
4
1:9 10
4
13:6
17:5

2:6 to 1:6 10
4
0:6 10
4
4:5
3.8.6 Grain/subgrain Interior Strains of ECAP AA1050 Multiple-pass via Route
B
C
The distribution of grain/subgrain interior elastic strains of AA1050 with multiple
passes is shown in Figure 3.44. These strains are characterized at the center of the
sample near the pressing direction (4:9

from axial direction). The ECAP die can impose
roughly +0:19 of the maximum true plastic strain on the sample for this measurement
direction which is about 21% of the maximum plastic strain after 1-pass ECAP. Then,
ECAP AA1050 samples were processed via routeB
C
, in which the samples are rotated
by 90

between each pass.
Most of the measured elastic strains are negative except for one value from the 1-
pass sample. The width of the strain distribution for the 1-pass specimen is significantly
larger (about 2 times) than for the multiple pass specimens. The standard deviation for
the 1-pass specimen is 2:2 10
4
compared to roughly 1 10
4
for the samples which
have undergone multiple ECAP passes.
95
The LRIS increases slightly with the number of passes and the mean axial strain
of the cell interiors for the 1, 2, 4 and 8 pass samples are1:9 10
4
,3:0 10
4
,
2:2 10
4
and3:1 10
4
, respectively. These values convert to about13:6 MPa,
21:2 MPa,15:7 MPa and21:9 MPa long-range internal stresses for 1, 2, 4 and
8 passes, respectively (Young’s Modulus along the <531> direction 71 GPa). This
equates to 0:09, 0:14, 0:08 and 0:10
a
for 1, 2, 4 and 8 passes (flow stress
a
148
MPa, 150 MPa, 180 MPa, and 200 MPa for 1, 2, 4 and 8 passes, respectively [74]).
There is no clear trend between the number of passes for the long-range internal stresses
as a fraction of the flow stress for a different number of ECAP passes (Figure 3.45).
3.8.7 Grain/subgrain Interior Strains of ECAP AA6005 1-pass and 2-pass via
RouteC
In Figure 3.46, interior strains are shown for 2-pass ECAP AA6005 via routeC.
The measurement setup is the same as the ECAP AA1050. {531} lattice spacings
about 4:9

off the pressing direction were measured in the center position of the ECAP
samples. TEM images show that heterogeneous dislocation microstructures form after
ECAP (Figure 3.32) and dislocations are entangled with Mg
2
Si precipate particles, and
Fe-rich intermetallics (Figure 3.33 and Figure 3.34).
Strain values were calculated on well-defined peaks from low dislocation areas and
some typical Laue diffraction spots are shown in Figure 3.47. Powder diffraction data
in Figure 3.40, Figure 3.41 and Figure 3.42 show a peak shift after 1-pass and 2-pass
ECAP. The lattice parameter for 1-pass was 4.05040(10) Å and 4.05130(10) for the 2-
pass sample, respectively, a difference of 0.0009 Å, which equates to a strain of 2:2
10
4
. For AA6005-T4 samples, there is no consistent lattice constant for different ECAP
passes, which indicates that the deformation is not homogeneously uniform throughout
96
the samples. In this case, no standard lattice constant can be used for different passes to
calculate the long range internal stress that results from the ECAP deformation.
Other than a lattice constant obtained by powder diffraction, a large grain scan was
performed near the internal strain measuring area in order to measure a local "lattice
constant" that is influenced by residual stress. This is represented in Figure 3.47. With
that, the local residual stress effect might be further minimized. This large grain scan
contains diffraction information from all features inside the grain including low dislo-
cation density areas (similar to cell interiors), high dislocation density areas (similar to
cell walls), and grain/subgrain boundaries.
The lattice constant for 2-pass AA6005 measured by a local large grain scan was
4:05180(10) Å. Compared to the value measured by powder diffraction, 4:05130(10) Å,
the 2-pass AA6005 value differs by 0:0005 Å, or a strain value of 1:210
4
. The differ-
ence between the local lattice constant and powder-diffraction measured value indicates
that the residual stress exists in AA6005.To investigate the small grain/subgrain inter-
nal strains, the local average lattice constant is adopted to avoid non-uniformity in the
sample. The internal strains measured by microbeam diffraction with the local lattice
constant are all negative in Figure 3.46. The mean internal elastic strain is3:8 10
4
.
Table 3.12 shows the standard deviations of strain value for AA6005 2-pass microbeam
sample. This mean internal strain can be converted to an estimated stress of27:2 MPa
for 2-pass (local microbeam diffraction), which equates to 0:16
a
.
Table 3.12: Internal strain summary for ECAP AA6005 2-pass via route C along the
pressing direction in grain/subgrain interiors
2-pass
Mean value 3:82 10
4
Standard deviation 2:04 10
4
Uncertainty of mean value 6:44 10
5
97
3.9 Discussion
3.9.1 LRIS in Grain/subgrain Interiors for ECAP AA1050 1-pass
Deformation-induced internal stresses have been investigated extensively using dif-
ferent techniques [21]. However, the results vary with investigators, and appear to vary
with the deformation techniques as well. Mughrabi et al. [11] reported relatively high
LRIS of about 1:3
a
in persistent slip band (PSB) walls and0:37
a
in cell interiors
in cyclically deformed copper. Conventional X-ray diffraction line profile analysis by
the same group found that the LRIS in a monotonically deformed single crystal Cu was
about 0:4
a
in cell walls and about0:1
a
in cell interiors. These findings were ratio-
nalized in terms of a ”composite model” [11, 30]. As the material is deformed, com-
patibility dislocations accumulate between soft (cell interiors, etc.) and hard (cell walls,
non-equilibrium grain boundaries, etc.) regions which lead to different local stresses in
the walls and cell interiors. Mechanical equilibrium requires that the volume fraction
weighted sum of the stresses in the cell interiors and the cell walls must be equal to zero
in an unloaded specimen. Levine et al. [6] also investigated long-range internal stresses
in compressively deformed copper using a more reliable X-ray microprobe. Results by
Levine et al. showed that LRIS is present (but lower) in dislocation cell interiors +0:1
a
while balancing stresses are present in the cell walls0:1
a
[5, 6](the volume fraction
of cell walls was about 0:5).
Kassner et al. [21] and Legros et al. [76] showed an absence of internal stress
near and away from the dislocation dipole bundles (veins) in Cu and Si single crystals,
cyclically deformed in single slip to presaturation (no PSBs). There are other cases
(e.g. creep) in which internal stresses were not observed in materials using the CBED
technique [76]. This, as mentioned earlier, may be due to poor resolution, as com-
pared to other techniques such as X-rays, and relaxation that can occur in thin foils
98
[21, 56, 86, 87]. In severely plastically deformed materials such as ECAP aluminum
alloy, however Alhajeri et al. [56] recently used CBED to show that there were internal
stresses near grain/subgrain boundaries. As mentioned earlier, ECAP may produce dis-
location configurations (e.g. boundaries) that produce relatively high LRIS. The CBED
of Alhajeri et al. did not determine the orientation of the stresses to the pressing geom-
etry. Therefore, this study utilized X-ray microbeam technique to buttress the CBED
limitations of Alhajeri et al. LRIS research on ECAP aluminum.
The current study demonstrates that compressive internal elastic strains (stresses)
are present in the +27:3

direction. The mean value of this internal strain is about
2:7 10
4
(19:0 MPa), which corresponds roughly to0:13
a
. Near the press-
ing direction (+4:9

), compressive internal strains (stresses) are measured to be slightly
smaller than those in the +27:3

direction, and the average strain is1:9 10
4
(13:6
MPa) or about0:09
a
. The smallest mean strain value (close to zero, at0:6 10
4
)
is measured in the17:5

direction. We expect a compressive internal elastic strain
within low-dislocation density regions in the grain/subgrain interiors along the +27:3

according to the composite model. This elastic strain should trend towards a tensile
strain as the orientation changes from +4:9

to17:5

, also consistent with the com-
posite model. Although the mean compressive elastic strain decreased in magnitude, it
never quite became tensile. Tensile elastic strains might have been observed if we had
been able to examine strains in the67:5

direction, where a compressive plastic-strain
is expected.
It must be mentioned that the data presented is extracted from those regions within
the grains/subgrains that have relatively low dislocation densities. This point will be
discussed further in the next section where stresses in the higher dislocation density
regions of the grain/subgrain interiors are considered.
99
3.9.1.1 Estimate of High and Low Dislocation Ratio
Figure 3.48 shows a false-color image of the diffracted intensity on the detector
from a single grain/subgrain. The grain/subgrain is approximately 1:0 m 3:0 m
3:0 m in size, at a depth of 70:0 m below the sample surface, and the diffracted
intensity is from the d-spacing-equivalent {551} or {711} reflections. The scan volume
was somewhat smaller than the grain size, at 1:0m 0:6m 3:0m. These data
were obtained by scanning the energy from 16:800 keV to 16:908 keV in 3 eV steps
and summing the resulting images. Figure 3.48 shows three distinct diffraction peaks
from low-dislocation regions within the grain/subgrain, and smeared intensity from the
dislocation-rich volumes. Hence, the dislocation density (or dislocation distribution)
within the grain/subgrain interior is heterogeneous.
To partition the LRIS in the high- and low-dislocation density regions of the grain/-
subgrain interiors, an approximate ratio for the high-dislocation and low-dislocation
volumes is required. The minimum number of counts per pixel in the large grain data
was determined (Figure 3.48) and subtracted from each pixel to obtain an approximate
background-subtracted image. The peak positions were masked out from the large
grain diffraction data. Next, the diffuse scattering level was estimated in the nearby
regions and this was subtracted to obtain an estimate of the total diffracted intensity
from the low-dislocation volumes. The remaining intensity should originate from the
high-dislocation density volumes. Roughly, 55%10% relatively high-dislocation den-
sity volume was identified within the grain/subgrain interior. Therefore, there are three
types of volumes: 1.) grain/subgrain boundaries or walls, 2.) high dislocation density
interiors, and 3.) low dislocation density interiors.
100
3.9.1.2 Composite Model
Stress equilibrium dictates that the weighted sum of the stresses in the grain/sub-
grain interiors and boundaries must be equal to zero in the unloaded specimens. The
width of a non-equilibrium grain/subgrain boundary is the distance over which all of
the extraneous dislocations of the boundary are contained. This width must be approx-
imated in order to calculate the volume fraction and thus the resulting mean grain/sub-
grain interior and boundary stresses. The volume fraction of grain/subgrain boundaries
can be calculated by a simple model. We assume the sample is composed of stacked
cubical grains/subgrains with a side length ofa (the distance from one grain/subgrain
boundary center to the next), and a grain/subgrain boundary width of b. The percent
volume fraction of the grain/subgrain boundary is, then,
[a
3
(ab)
3
]
a
3
(3.39)
According to the works of Ding et al. [88], Horita et al. [89], and Hoeppel [90],
non-equilibrium grain boundary widths vary from 5nm to 10nm. These values are for
samples deformed to a strain of 7, and the specimens of this study were strained only
to 1. For SPD metals, extraneous dislocations are observed in both high-angle and
low-angle grain/subgrain boundaries by TEM [90–92]. It is assumed that there are sim-
ilar extrinsic dislocation features for boundaries formed at low and high strains. For our
grain/subgrain size (1m), the volume fraction of the boundaries varies between 1:5%
(5nm) and 3:0% (10nm). The work of Alhajeri et al. reports a maximum axial strain
value of 1:1 10
3
, which converts into a maximum axial stress of 112 MPa (about
0:75
a
). However, this strain was measured using a beam diameter of 20nm immedi-
ately adjacent to the grain/subgrain boundary. Thus, for the purposes of a force-balance
101
"check," a boundary width of about 40 nm must be assumed. That is, the measured
strain is an average over a width of 40nm ( the volume fraction is 11:5%) , which may
exceed the true boundary width. To balance this stress at the grain/subgrain boundary,
the mean stress in our 1-pass grain/subgrain interior should be14:6 MPa (about
0:097
a
, converted into a strain of about2:1 10
4
). This is the average force
(stress) over the low-dislocation density interior and the high-dislocation density inte-
rior regions that balance the assessed force over a boundary region of 40 nm width.
According to our measurements, the average strain in the low-dislocation density inte-
rior is about2:710
4
(0:13
a
). These interior strain (stress) values would be below
the measurement limit of CBED and, therefore, could not be measured by Alhajeri et
al. [56].
The ratio of high/low dislocation density areas within the grain/subgrain interiors
can be obtained, and the stress can be calculated. The stress in the high-dislocation
density region from a force balance would be roughly9:0 MPa (0:06
a
, for 45%)
to12:1 MPa (0:08
a
, for 65%). Thus, the interiors have low LRIS balanced by
higher LRIS in the boundaries. Thus, in summary, for the boundary region, the stress
is measured by CBED as 0:75
a
, for the low-dislocation density grain/subgrain interior,
the stress as measured by X-ray microbeam is0:13
a
and by measurement of the
high-dislocation density volume fraction within the grain/subgrain interior, the stress by
mechanical equilibrium is about0:07
a
. The stress balance equations are shown in
the following:

GB
f
GB
+
int:
f
int:
= 0 (3.40)

int:
=
lowdis:density
f
lowdis:density
+
highdis:density
f
highdis:density
(3.41)
102
Where,
GB
( 0:75
a
) is the stress around grain/subgrain boundaries (20nm around
boundaries), f
GB
11:5% is the grain/subgrain boundary volume ratio, and f
int:

88:5% is the grain/subgrain interior volume ratio. In addition,
lowdis:density
( 0:13
a
)
is measured by X-ray micirobeam diffraction andf
lowdis:density
andf
highdis:density
are
45% and 55%, respectively.
3.9.2 LRIS in Grain/subgrain Interiors for ECAP AA1050 Multiple-pass via
RouteB
C
X-ray microbeam diffraction measurements of a commercial-purity AA1050 alloy
processed by multiple-pass ECAP consistently reveal negative elastic strains within
+4:9

of the pressing axis direction within grain/subgrain interiors in Figure 3.43.
The average elastic strains for 1, 2, 4 and 8 passes were1:9 10
4
,3:0 10
4
,
2:2 10
4
and3:1 10
4
which equate to approximately 0:09
a
, 0:14
a
, 0:08
a
and 0:10
a
, respectively as illustrated in Figure 3.44 and Figure 3.45. There is a slight
trend of increasing elastic strain with the number of ECAP passes along the +4:9

direc-
tion. The normalized LRIS is approximately independent of strain. The present work
suggests that the LRIS only modestly increases with plastic strain and the normalized
LRIS is essentially constant. The results also show that the LRIS in non-equilibrium
boundaries with many extraneous dislocations may lead to relatively high (0:75
a
) LRIS
and may exceed those in other deformation-induced boundaries where the LRIS in the
dislocation walls was found to be relatively low. The volume and LRIS of the interior
compensates the volume and LRIS of the walls for a zero net stress in the unloaded
material. Due to the polycrystalline nature of the specimen, friction effects, etc., it is
ideal to measure strains along different directions and extract the full strain tensor. An
additional effort is underway to conduct such measurements.
103
3.9.3 LRIS in Grain/subgrain Interiors for ECAP AA6005 1-pass and 2-pass via
RouteC
For different ECAP AA6005 passes, a clear shift in powder diffraction peaks
between 1-pass and 2-pass ECAP samples was observed. Thus, a residual stress prevails
in the diffraction volume (0:8 mm diameter 3 mm rod). The local lattice constant is
used to eliminate the influence of the long-range (local) residual stresses in the interi-
ors. The normalized internal stresses are 0:16
a
. These normalized internal stresses of
AA6005 are similar to the values measured in AA1050.
3.10 Conclusions
3.10.1 ECAP AA1050 1-pass
1. X-ray microdiffraction of a commercial purity AA1050 alloy revealed long-range
internal stresses in grain/subgrain interiors. Compressive strains (stresses) are
observed near the pressing direction +4:9

and in the off-pressing direction
+27:3

. In the17:5

off-pressing direction, the mean strain is approximately
zero.
2. The average internal stress in the maximum tensile strain direction (+27:3

) is
approximately19:0 MPa or about0:13 of the applied stress for the low-
dislocation density regions within the grain/subgrain interiors. As roughly
55%10% of the interiors may be of a higher dislocation density, the stress within
these regions is estimated to be0:06 to0:08 of the applied stress.
3. These results, together with the work of Alhajeri et al. [56]who used CBED to
assess the long-range internal stresses near grain/subgrain boundaries, suggest
significant LRIS in SPD ECAP AA1050 that are approximately consistent with
104
the classic composite model. The ECAP may represent a case where the stress
near the boundaries may be relatively high, on the order of 0:75
a
.
3.10.2 ECAP AA1050 Multiple-pass via RouteB
C
1. X-ray microbeam diffraction measurements of commercial purity AA1050 alloy
processed by multiple-pass ECAP consistently reveals negative elastic strains
within +4:9

of the pressing axis direction within grain/subgrain interiors.
2. The average elastic strains for 1-, 2-, 4- and 8-passes were1:9 10
4
,3:0
10
4
,2:210
4
and3:110
4
which equate to approximately 0:09
a
, 0:14
a
,
0:08
a
, and 0:10
a
respectively.
3. There is a slight trend of increasing elastic strain with the number of ECAP passes
along the +4:9

direction.
3.10.3 ECAP AA6005 1-pass and 2-pass via RouteC
1. Compressive strains/stresses are measured within +4:9

of the pressing direction
in AA6005 grain/subgrain interiors after ECAP via route C. LRIS in AA6005
grain/subgrain interiors is opposite of the external plastic strains/stresses and is
consistent with the prediction of the composite model.
2. Unlike AA1050, residual stresses were observed in AA6005 as a result of plas-
ticity. This complicated assessing LRIS. In the case of AA6005, an observed
lattice parameter for a large grain near the region of the X-ray microbeam mea-
surement is made. The difference between thea
0
from the large grain scan and
the microbeam lattice parameters allowed an estimate of LRIS.
3. The magnitude of LRIS in AA6005 2-pass is 0:16
a
, similar to the LRIS magni-
tude in AA1050.
105
Figure 3.35: High resolution X-ray powder diffraction spectrum of ECAP AA1050 via
routeB
C
Intensity versus 2 for AA1050 as-received, 1-pass, 2-pass, 4-pass and 8-pass.
106
Figure 3.36: X-ray powder diffraction (331) peak of AA1050 for different ECAP passes
using routeB
C
107
Figure 3.37: X-ray powder diffraction (420) peak of AA1050 for different ECAP passes
using routeB
C
108
Figure 3.38: X-ray powder diffraction (531) peak of AA1050 for different ECAP passes
using routeB
C
109
Figure 3.39: High resolution powder diffraction spectrum of ECAP AA6005 using
RouteC
Intensity versus 2 for AA6005 1-and 2-pass.
110
Figure 3.40: X-ray powder diffraction (331) peak of ECAP AA6005 via Route C for
different passes
111
Figure 3.41: X-ray powder diffraction (420) peak of ECAP AA6005 using RouteC for
different passes
112
Figure 3.42: Powder diffraction (531) peak of AA6005 for different ECAP passes
113
Figure 3.43: Strain distribution of grain/subgrain interior elastic strains of AA1050 1-
pass ECAP
These strains are characterized at the center of the sample near the pressing direction
(+4:9

from axial direction) and tilted +27:3

and17:5

relative to the pressing direc-
tion, within the plane defined by the two pressing axes. The one standard deviation
measurement uncertainty for an individual measurement is approximately1:0 10
4
.
114
Figure 3.44: Strains in the grain/subgrain interiors of ECAP AA1050 via routeB
C
for
different number of passes near the pressing direction (+4:9

)
115
Figure 3.45: LRIS-to-flow stress ratio in the grain/subgrain interiors of ECAP AA1050
versus ECAP number of passes
116
Figure 3.46: The internal strain distribution of AA6005 after and 2-pass ECAP routeC
The strains are characterized at the center of the sample near the pressing direction
(tilted +4:9

from the pressing direction). The local standard lattice parameter (a
0
=
4:05180(10) Å) is obtained by measuring a large area in Figure 3.47 near the microbeam
measurement positions.
117
Figure 3.47: Laue diffraction pattern of AA6005 after ECAP 2-pass routeC
The scanning area is over large length scale to include interiors and dislocation area
signals. The diffracted intensity is from the d-spacing-equivalent {533} reflections. The
scan volume was 1:0m 0:6m 4:3m at a depth of 10:0m below the sample
surface. This data was obtained by scanning the energy from 16:636 keV to 17:556 keV
in 3 eV steps and summing the resulting images. The local lattice constant is estimated
to be 4:05180(10) Å, that includes residual stress.
118
Figure 3.48: Laue diffraction pattern of 2-pass ECAP AA1050 via routeB
C
False-color image of the energy-integrated diffracted intensity from an individual grain/-
subgrian in a 2-pass sample. The peaks are from low-dislocation density regions and the
smeared intensity is from the high-dislocation density volumes within the same grain/-
subgrain.
119
Chapter 4
LRIS in Deformed Single Crystal Copper
4.1 Introduction
In the previous chapters, experiments have shown evidence of long-range internal
stresses, including X-ray peak asymmetry (conventional X-ray diffraction), transmission
electron microscopy (TEM)/convergent beam electron diffraction (CBED), and disloca-
tion bowing. Mughrabi (1983) [10] first proposed a composite model to explain the
formation of asymmetric X-ray line profiles and the progressive decomposition method
to extract the subprofiles of cell interiors and walls from the asymmetric X-ray line pro-
file to calculate LRIS in the cell interiors and walls. The composite model simplifies the
dislocation microstructures in monotonically plastic deformed Cu with two components,
cell interiors and cell walls. In addition, the composite model assumes high dislocation
density areas (cell walls) have a higher flow stress than the low dislocation density area
(cell interiors). During plastic deformation, different stresses exist in the cell interi-
ors and in the cell walls. Additionally, upon elastic unloading, LRIS of opposite signs
appear in these two components.
Therefore, owing to different elastic strains in the cell interiors and walls, differ-
ent diffraction peaks will be produced in the reciprocal space (q vector), and assemble
120
into an asymmetric X-ray line profile. The composite model can explain the LRIS for-
mation very well. To calculate the average strain/stress from the broadened X-ray line
profiles for the cell interiors and walls, the asymmetric profile is decomposed into the
cell interior and wall subprofiles. Levine et al. [5] directly measured the cell interior
and wall X-ray line profiles from the micrometer volumes. Cell interiors show stronger
diffraction intensity on the detectors, while the walls only show smeared signals due to
their high dislocation density. This was the first time ever that the cell interior and wall
strains could be measured directly.
In this chapter, the methods which are proposed by Levine et al. [6] are introduced
as well as a modified method to resolve the cell wall subprofiles from the asymmetric
X-ray profile by X-ray microbeam diffraction. In addition, a discussion of how LRIS
varies with the deformation procedures will be investigated. Monotonic deformation is
performed along different crystal axes and the resulting LRIS are evaluated.
4.2 Motivation
Levine et al. [6] measured the LRIS in the monotonically deformed single-crystal
copper oriented along [001] axis and produced uniform, equiaxed low misorientation
cell interiors (see Figure 2.6) and indicated that the magnitude of LRIS in the cell inte-
riors and walls is about0:1
a
and +0:1
a
(
a
is applied stress) in compression. LRIS
in the cell interior is balanced by that in the cell walls. In addition, there is a stress dif-
ference 40 MPa between the cell interior and cell walls, when the crystal is deformed
to about 210 MPa to a strain of0:28. However, monotonic deformation along [001] in
single crystal is a simple case. Usually, for most of forming processes, the deformation
procedures are more complicated (e.g. cold work, rolling, bending, SPD, stamping and
shearing). Most processing procedures can include the stress reversal (cyclic stress) and
complex stress states, leading to complex slip systems (e.g. unlike [001] in tension or
121
compression in single crystal, as discussed). In those cases, LRIS are believed to exist
but the LRIS values are unknown. The existence of the LRIS has been suggested to be
associated with many mechanical properties such as fatigue, the Bauschinger effect or
metal forming (metal springback).
4.3 X-ray Macrobeams versus X-ray Microbeams
Conventional synchrotron diffraction line profile measurements (large sample vol-
ume) performed on the compressive deformed Cu sample along [001] crystal axis at
XOR/UNI 33-BM beamline at the Advanced Photon Source (APS) at Argonne National
Laboratory are compared with integrated X-ray microbeam profiles measured at 34-ID-
E at the APS (Levine et al. [6]). Ideally, the beam size of 33-BM beamline is about
1:5 mm 0:25 mm and diffraction signals are collected on the detector, similar to the
approach of others utilizing one-dimensional X-ray detectors (conventional experiment
for X-raymacrobeam). Two different beamlines are used to measure the (006) reflection
on deformed single-crystal copper. These results are plotted as a function of the wave
vector,q, in Figure 4.1. The assembled integrated X-ray profiles by X-ray microbeam
are very similar to the profile obtained by X-ray macrobeam which measures a much
larger sample volume. The sampling by X-ray microbeams over different positions and
depths can reasonably measure the diffraction volume which includes large amount of
cell interiors and walls and contributes to the asymmetric X-ray profile. Thus, the data
measured by X-ray microbeam can be well aligned with X-ray macrobeam experiments.
For the macrobeam and microbeam instrument calibration, high-resolution line profile
measurements were performed first on the peak maximum region using a Si (111) ana-
lyzer crystal to calibrate the detector. Then, measurements were also performed on the
122
(006) reflection profile of a undeformed Cu single crystal to determine the overallq vec-
tor resolution of the measurement. The resolution functions for both X-ray microbeam
and macrobeam are shown in Figure 4.1 [6].
Figure 4.1: Integrated microbeam (006) line profiles from deformed Cu versus the X-ray
macrobeam profile
Five integrated microbeam line profiles over a larger sampling volumes, including cell
interior and cell wall diffraction signals are presented and matched well with the X-ray
macrobeam profile (gray thick line). The resolution functions for both microbeam and
macrobeam are shown as well [6].
4.4 X-ray Line Profile Asymmetry
As crystalline materials are plastically deformed (such as copper), multiple slip
occurs and heterogeneous dislocation structures (cell interior and wall composite) will
form. The dislocation heterogeneous microstructure has been observed in deformed
single- and polycrystalline copper. Owing to the deformation, several characteristics are
123
shown in the X-ray line profiles such as peak broadening and asymmetric X-ray line
profiles. Peak broadening is associated with increasing defects. Asymmetry of X-ray
line profiles is considered as the evidence of internal stresses in which the magnitude is
different in the cell interiors from in the cell walls, subgrains, and high angle boundaries.
Figure 4.2(a) shows the broadened (006) reflection line profile, from single crystal Cu
compressed by 30% along the [001] axis. For the comparison, X-ray line profile for the
undeformed copper is also shown in the Figure 4.2(a), and Figure 4.2(b) shows the (006)
reflection line profile from a Cu single crystal with 30% tensile strain along a [001] axis.
While the sample is compressively deformed, the peak position of X-ray line profile (q
value) will shift to lower q value. However, if the sample is under tension, the peak
position will shift to the higherq value. It has been speculated that the sharp side and
peak position of the X-ray asymmetric peak (q profile) are close to the shape and peak
position of the cell interior subprofile. For the X-ray asymmetric q profile, the X-ray
profile is decomposed into two sides based on the peak position, sharp and shallow sides.
According to the results in Figure 4.2, this means that the elastic strains/stresses in the
cell interiors are in the opposite signs to the external stresses. After tensile deformation,
a compressive elastic strain will exist in the cell interiors. On the other hand, after
compressive deformation, a tensile elastic strain will show in the cell interiors.
4.5 Cell Interior and Wall X-ray Subprofiles
Figure 4.3 shows that an asymmetric X-ray line profile which was measured by
conventional X-ray can be separated into cell interior and wall subprofiles. The cell
interior subprofile is close to the sharp side of the asymmetric profile. The cell wall
subprofile is more broadened than the cell interior subprofile [11, 43]. Many models
have been proposed to explain how dislocation density influences the X-ray peak width
and intensity and how to calculate the dislocation density from the X-ray subprofiles
124
Figure 4.2: Asymmetric X-ray line profiles
Conventional X-ray line profiles after compressive plastic deformation (a) (30%)
and tensile plastic deformation (b) ( +30%). Asymmetric curves are shown for both
conditions. An unstrained peak profile is shown in (a) [5].
[44, 45]. To investigate those properties from cell interior and wall subprofiles, sev-
eral assumptions are made by Mughrabi et al. regarding the decomposition of cell and
wall subprofiles: a.) the subprofiles are symmetric, b.) the area under the subprofiles
(q-space) are proportional to the cell interior and wall volumes. (The microstructure
volume can be obtained by TEM), c.) the asymmetric X-ray profile is composed of two
symmetric subprofiles, and d.) the peak broadening is mainly contributed by disloca-
tion broadening. Based on these assumptions, an asymmetric line profile was decom-
posed into subprofiles to analyze the average strains/stresses for cell interiors and walls
[10, 11, 43, 46].
125
Figure 4.3: The decomposition of asymmetric X-ray line profiles
(001) line profile from a tensile deformed Cu single crystal along a [001] crystal axis by
conventional X-ray diffraction. It shows an asymmetric feature. The asymmetric X-ray
profile can be decomposed into cell interior and wall subprofiles [10].
4.6 Decomposition of Asymmetric X-ray Line Profiles by Mughrabi
To extract the subprofiles from the asymmetric X-ray line profile, the progressive
decomposition method is used [10, 11, 43]. The subprofile of the cell interior will adopt
the sharp side of the asymmetric X-ray line profile due to less diffraction scattering by
dislocations. First, the peak position of the asymmetric profile is located and then the
sharp side of the asymmetric profile is reflected around the peak position as shown in
Figure 4.4. Second, the 1
st
order cell wall profile can be obtained by subtracting the
asymmetric X-ray line profile and the symmetric line profile. The previous step allows
getting the right side of the 1
st
order cell wall profile. To get the whole symmetric profile
126
of the 1
st
cell wall subprofile, a reflection around the peak position of the subtracted
profile is made (green line in Figure 4.4). To obtain the 1
st
order cell interior subprofile
(blue line in Figure 4.4), the subtraction between asymmetric profile and 1
st
order cell
wall subprofile is implemented again. The first order cell interior profile is very close
to a symmetric profile. Similar procedures can be adopted to get the 2
nd
order cell wall
profile. The progressive procedures can be operated several times until both cell interior
and wall subprofiles are close to symmetric and also no obvious changes (e.g. intensity
and shape) of subprofiles between the consequent deriv. The progressive method can be
used to decompose an asymmetric X-ray line profile measured by X-ray macrobeam or
microbeam. This method was used by Mughrabi to estimate the average cell interior and
wall strains in the deformed single crystal copper from an asymmetric X-ray macrobeam
line profile [11].
4.7 Reconstruction of X-ray Subprofiles Measured by X-ray
Microbeams
As mentioned earlier, cell interior and wall X-ray profiles are measured by X-ray
microbeam diffraction from about a 0:5m cubic volume [5, 43]. The individual inte-
grated X-ray line profiles (q profiles) of cell interior and wall are obtained by masking
clear diffraction spots or smeared diffraction signals on the Laue diffraction images and
then are fitted by a pseudo-V oigt function (The measured individual cell interior pro-
files are combinations of Gaussian and Lorentzian function line shapes and therefore be
approximated as pseudo-V oigt function). In Figure 4.5[43], 97 cell interior and 49 wall-
fitted X-ray line profiles are plotted and it shows that the full width at half maximum
(FWHM) of cell interior line profiles are smaller due to their low dislocation density and
peaks shift to left side in Figure 4.5(a). In addition, tensile strains are shown in most
127
Figure 4.4: Progressive decomposition method of the asymmetric line profiles into cell
interior and wall subprofiles
Dark line is an asymmetric X-ray line profile from 28% compression single-crystal cop-
per along a [001] crystal axis. Red line is the symmetric line profile which adopts the
sharp side of asymmetric peak and reflects around the peak position. 1
st
order cell wall
subprofile comes from the subtraction between asymmetric peak and symmetric peak.
1
st
order cell interior subprofile comes from the subtraction between asymmetric peak
and 1
st
order cell wall subprofile.
of the cell interiors, while compressive strains are shown in the walls. This sample is
deformed compressively to 30% true strain along a [001] axis.
The summed cell interior and wall subprofiles are plotted in Figure 4.6. The rela-
tive areas of the subprofiles in Figure 4.6 can be determined by the relative area in the
microstructure. According to TEM micrographs, the cell interior-to-wall ratio is about
45=55 (45% cell interiors and 55% walls). After assigning the area ratio to two subpro-
files, two subprofiles are assembled into an asymmetric X-ray line profiles (dark line).
128
Figure 4.5: X-ray line profiles measured by X-ray microbeam diffraction
(a) 97 cell-interior X-ray line profiles (b) 47 cell wall X-ray line profiles. [43]
In addition, a typical X-ray line profile measured by macrobeam X-rays is shown in
Figure 4.6. The subprofile assembled line profile and macrobeam profile match well.
This indicates that the sampling of microbeam line profiles can present the diffraction
properties of the whole sample.
129
Figure 4.6: The assembled X-ray line profile from cell interior and wall subprofiles
measured by X-ray microbeams
Two reconstructed subprofiles are shown in the figure. The green dash line is mea-
sured by conventional X-ray diffraction. The assembled X-ray profile obtained from
microbeam measurement can fit conventional X-ray line profile well. [43]
4.8 Determine Cell/wall Ratio and Cell/wall Strains by Fitting the
Asymmetric Profile
To extract the subprofiles from the asymmetric X-ray line profile, a variety of meth-
ods or functions can be utilized to fit the asymmetric curve and extract the subprofiles.
In Figure 4.7 the progressive derived subprofiles (Mughrabi’s method) are plotted with
the microbeam measured subprofiles. It shows a huge difference in the peak position of
cell wall subprofile and the area under the subprofiles. The progressive method cannot
give accurate results regarding average strain values (peak positions) and cell interior-
to-wall ratio (the area ratios between cell interiors and walls) which should be very close
to the cell interior-to-wall area ratio analysis from TEM image analysis. Thus, Levine
130
et al. [43] used different functions such as the Gaussian function and the pseudo-V oigt
function to fit the asymmetric line profile to extract the cell interior and wall subprofiles
which can be used to calculate the average strains of cell interiors and walls and cell
interior-to wall ratios. The results are shown in Figure 4.8. In the fitting, Mughrabi’s
assumptions are followed. (e.g. symmetric functions for subprofiles, an asymmetric
X-ray line profile is composed of two symmetric subprofiles and area ratio of subpro-
files is proportional to the ratio in the microstructure). Both Gaussian and pseudo-V oigt
function can fit the asymmetric line profile well, but the results of peak position and
area under the subprofiles are very different. Summarized results are shown in the next
section.
Figure 4.7: Comparison between a progressive method for derived subprofiles and
microbeam measured subprofiles
The 2
nd
subprofiles (Mughrabi’s progressive method) are plotted with microbeam mea-
sured subprofiles. After twice of progressive procedure, the cell interior and wall sub-
profiles show similar line profiles. [43]
131
Figure 4.8: Using different fitting functions to determine the cell interior and wall ratios
and subprofiles
An asymmetric X-ray line profile measured by convention X-ray is fitted by symmetric
functions to extract symmetric cell interior and wall subprofiles. The fitted cell interior
and wall subprofiles are summed to compare with the asymmetric X-ray line profile.
(a) Gaussian functions, (b) pseudo-V oigt functions, and (c) pseudo-V oigt functions with
45=55% cell interior-to-wall volume ratio constrained. The microbeam measured sub-
profiles are also plotted in (c) for comparison. [43]
132
4.9 Summary Results of Various Fitting Methods
Table 4.1 [43] lists the peak positions of cell interior and wall subprofiles and wall
ratios which are fitted by several methods, including the progressive method, Gaussian
fit and pseudo-V oigt fit. All the fitted results cannot match the microbeam measured
results. Theq difference between cell interior and wall subprofiles is overestimated. In
addition, the accurate volume ratio of cell interior-to-wall cannot correctly be calculated.
Even though a constrained volume condition (45%=55% cell interior-to-wall ratio) is
imposed to the pseudo-V oigt fit, q (related to the strain difference between cell inte-
riors and walls) is still overestimated. This indicates that by fitting the asymmetric line
profile, the magnitude of LRIS is usually overestimated. The reason why a unique and
accurate solution could not be found was that an infinite number of nearly symmetric
subprofiles can be found to fit the asymmetric profile. There are no rational ways to
decide the most correct fitting results. For example, two sides of the asymmetric profile
cannot fit well at the same time. Thus, different constraints of fitting functions (e.g.
progressive method, Gaussian and pseudo-V oigt functions) can produce very different
solutions. Typically, sharper subprofiles will contribute more to the asymmetric peak.
Nevertheless, it was determined that including the correct cell-wall and cell-interior vol-
ume fractions as obtained from TEM micrographs was a critically important constraint
factor. If the volume ratio, obtained from TEM micrograph analysis, is applied as the
constraint (limited condition) for the fitting functions such as pseudo-V oigt function in
Figure 4.8, the fitting curve by pseudo-V oigt function can match the measured cell inte-
rior subprofile better and also at least resemble the measured wall subprofile. Therefore,
to extract more accurate subprofiles, accurate cell interior/wall volume ratios should be
the most important factor for the fitting method.
133
Table 4.1: Peak parameters for the subprofiles fitted by different functions [43]
(q
i
: the peak positionq value of cell interior subprofile,q
w
: the peak positionq value of
cell wall subprofile, q: the difference ofq values between peak positions of cell
interior and cell wall subprofiles,Wall%: the wall ratio is defined by the ratio of area
between cell wall subprofile and cell interior subprofile.)
q
i
(nm
1
) q
w
(nm
1
) q (nm
1
) Wall%
Measured subprofiles 104:248(3) 104:309(5) 0:061(6) 55(10)
Progressive method 104:240 104:382 0:142 35
Gaussian fit 104:241 104:315 0:074 67
pseudo-V oigt fit 104:237 104:374 0:137 38
Constrained pseudo-V oigt 104:232 104:333 0:101 55
4.10 Sample Preparation and Experimental Procedures
The <0 0 1>-oriented 99:999 + % pure Cu single-crystal cylindrical compression
specimen had nominal dimensions of 10 mm in diameter by 20 mm in length. This
sample was compressively deformed (28%) along [001] crystal axis and used as our
experimental control sample. Previously, Levine et al. [6] already measured LRIS
by microbeam diffraction which are about +0:1
a
for cell interiors and0:1
a
for
cell walls (
a
is applied stress 210 MPa) and the average LRIS difference between
cell interiors and walls is about 40 MPa. To know how the LRIS evolves for differ-
ent deformation procedures, we studied the LRIS in the more complicated deformation
processes where more complicated microstructures developed. In [001] deformed Cu
single-crystal, only low misorientation cell walls develop ( < 1

) and cells are equiaxed.
Other deformation orientations may be more complicated with a different LRIS. Several
experiments were also designed to study the LRIS magnitude regarding how the reverse
stress influences the LRIS in addition to how the LRIS changes if plastic deformation is
made along different crystal axes than [100].
134
1. Cu-sample BE1: <0 0 1> Cu single crystal, deformed in compression to 28%.
This is similar experiment condition to that done by Levine et al. [5, 6]. This
sample is made into dogbone-shape specimens.
2. Cu-sample BE2: <0 0 1> Cu single crystal, deformed first in compression to 28%
and then in tension for approximately 2%. This sample is identical to sample BE1,
but with an additional applied plastic tensile strain to assess what this does to the
LRIS.
3. Cu-sample OA: <0.184, 0.107, 0.177> Cu single crystal, deformed in compres-
sion to around 28%.This sample is identical to sample BE1, but with a different
compression axis. The sample tensile axis is shown in the stereographic projection
diagram (Figure 4.9).
4.10.1 Sample Preparation and X-ray Microbeam Measurement
After deformation, the all unloaded samples (BE1, BE2 and OA) were 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
(the method is addressed in chapter 3) 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.
The spatially resolved scanning monochromatic DAXM measurements were con-
ducted using an X-ray microbeam focused to the area of 0:5
2
m
2
by two Kirkpatrick-
Baez focusing mirrors (KB mirrors) using the beamline 34-IDE at the APS. For depth
resolved scanning, the profiling wire is step-scanned parallel to the sample surface with
a step 0:7 m, which equates to a step of 0:5 m along the incident beam path.
135
Figure 4.9: Stereographic projection of a crystal
The tensile axis (0.184, 0.107, 0.177) of Cu-sample OA is plotted in the stereo projection
diagram (red dot).
The platinum wire sequentially blocks diffracted beams, which are incident on indi-
vidual pixels of the detector (diffraction spot) and allows determining the origin of the
diffraction volume by triangulation. For each wire position, the X-ray energy is scanned
around E 14 keV through the peak (e.g. (006)) in an energy step of 3 eV to determine
the energy and the depth at which the intensity is a maximum [5, 6].
4.10.2 Sample Preparation of TEM Microstructure Analysis
For all single-crystal copper samples (BE1, BE2 and OA), jet electropolising was
used to thin the disc type samples (with a diameter of about 3 mm and a thickness of
approximately 150 m) to make TEM plane-view samples. To prepare the 150 m
thick disc sample, first a 3 mm rod was made by electrical-discharge machining (EDM)
and then wire saw was used to cut the rod into disc shape which thickness is about
136
1 mm. Mechanical polishing was then performed to reduce the disc thickness to 150
m. The general procedure is to perform the polishing by silicon carbide grit paper
from 800 grit to 2400 grit. Then, the disc sample is detached from the holder in the
acetone. The smooth side is glued on the holder and the rough side of the disc sample is
polished by the same procedure until the total thickness is roughly 150m. Superglue
glues the sample instead of crystal glue to reduce the heating effect which can affect the
dislocation microstructures. Finally, electropolishing is used to flatten the surface and
get rid of damaged volume due to mechanical polishing.
Figure 4.10: A schematic of jet electropolishing
(a) The schematic of jet electropolishing (b) Typical scanning electron microscope
(SEM) images of a 3 mm copper disc. Struer, TenuPol-5 electropolisher is used for
Cu polishing.
Electrolytic thinning of conductive materials is an effective method of quickly pro-
ducing specimens for transmission electron microscopy (TEM) without any induced
artifacts. Thus, the pristine dislocation cell and wall structures after deformation can
be observed. Twin jets (Struers, TenuPol-5) simultaneously polish both sides of the
137
sample, creating electron transparent specimens within a few minutes. The electropol-
isher features easily adjustable electrolyte flow, polishing voltage, termination detection
sensitivity, and jet and specimen holder positions. This technique produces thinning
in the form of a dimple under moderate conditions with no mechanical damage. The
resulting dimple is perfectly polished and free of surface roughness. Strain hardening
and surface deformation during thinning are avoided. The specimen is easily handled
with the remaining thick and solid dimple edges. The basic schematic is shown in Fig-
ure 4.10(b). The copper discs were polished at 12 V in 30% H
3
PO
4
solution in H
2
O
at room temperature (double flow and pump flow rate: 12 (arbitrary units)).
4.11 Results
4.11.1 Microstructure
TEM characterization of three experimental conditions (BE1, BE2 and OA) are
shown in Figure 4.11, Figure 4.12 and Figure 4.13 The TEM plane-view images are
taken on the plane perpendicular to the deformation axis.
In Figure 4.11, sample BE1 is deformed along the <001> crystal axis of single-
crystal Cu, and the cell interior and cell wall heterogeneous dislocation microstructures
are observed. The cell size is roughly 0:8m and the cell interior size is about 0:5m.
The cell size is defined as the width of cell wall plus cell interior size. For sample BE2
(Figure 4.12), the deformation is similar to sample BE1 but an additional 2% strain is
applied in the reverse direction. The cell interior and cell wall dislocation microstruc-
tures are also observed for this sample. The cell size is also about 0:8m and the cell
interior size is around 0:4 m. For off-axis deformation (sample OA condition, Fig-
ure 4.13), the cell size is about 1:1m, and the cell interior size is around 0:7m. The
138
elongated cell interior shape is observed and many dislocations are randomly distributed
throughout the cell.
Figure 4.11: Bright field TEM image of sample BE1 at two different magnifications
The <001> Cu single crystal is deformed in compression to around 28%. The average
cell size is 0:8m and cell interior size is 0:5m. (b) is an enlarged image of (a) (dash
square area).
To estimate the cell/wall ratios for the aforementioned three conditions, wall areas
are colored with blue as shown in Figure 4.14. In central bright-field TEM images (two-
beam condition,
!
g = [022]) the walls show a dark contrast while the cells show a bright
contrast. Software, Image J, is used to calculate the cell interior and wall areas and
then the cell-interior/cell-wall ratio can be calculated. The cell/wall ratios are roughly
35%=65%, 25%=75%, and 35%=65% for sample BE1, sample BE2, and sample OA
conditions, respectively. Suitable adjustment of edge sharpness and contrast of wall area
contrast will roughly lead to a deviation of10% difference in the ratio calculations.
In the literature [6], the ratio of cell interior and wall for sample BE1 is around
45%=55%. Therefore, the cell-interior/cell wall ratio of BE1 will need to have further
study. In addition, the deformation along the non-symmetric crystal axis or to the high
139
Figure 4.12: Bright field TEM image of sample BE2
<001> Cu single crystal, deformed first in compression to around 28% and then in ten-
sion for approximately 2%. The average cell size is 0:8m and cell interior size is 0:4
m.
resolved shear stress should result in higher misorientations in walls and many geomet-
rically necessary boundaries (GNB) [47, 48]. In the TEM analysis, the cell walls seem
to be wider and the lower cell interiors ratios are obtained for all samples. These may
be attributed to the thicker TEM sample foils since cell walls are inclined with respect
to dislocation directions. Also, geometrically necessary boundaries are not observed in
TEM images of the OA sample as expected. Therefore, the obtained ratios for three con-
ditions are questionable. Further microstructure analysis such as cell size of larger areas
and misorientations between adjacent cells will be conducted in the future. In order to
show the decomposition of cell interior and cell wall subprofiles from X-ray line profile,
the obtained ratios from TEM analysis are still used for following microbeam analysis.
140
Figure 4.13: Bright field TEM image of sample OA deformation
<0.184, 0.107, 0.177> Cu single crystal, deformed in compression to around 28%. The
average cell size is about 1:1m and cell interior is 0:7m. (b) is an enlarged image of
(a) (dash square area).
4.11.2 The Reconstruction of an Asymmetric X-ray Line Profiles by X-ray
Microbeam Diffraction
X-ray integrated line profiles, including signals from cell interiors and cell walls
can be reconstructed by X-ray microbeam measurements. To attain reasonable sam-
pling, X-ray microbeam experiments are used to examine different sample positions
and depths. Energy-wire scan is applied for each scan and then an X-ray integrated
line profile can be obtained from a 0:5
3
m
3
diffraction volume. All X-ray line pro-
files from small sample volumes are summed to reconstruct the asymmetric X-ray line
profile, which has been proven to resemble the X-ray line profile measured by conven-
tional X-ray experiments by Levine et al. [6]. For three different deformation condi-
tions, the reconstructed integrated line profiles are shown in Figure 4.15, Figure 4.16
and Figure 4.17. The asymmetric signature is observed. Planes which are normal to the
141
Figure 4.14: Cell interior and cell wall volume ratio calculation
Dislocation walls are painted with blue color to estimate the cell and wall ratio. (a)
Bright field TEM image of sample BE1 (b) Cell walls in (a) are colored with blue color
and contrast is adjusted. (c, d) TEM images of sample BE2. (e, f) TEM images of
sample OA. The cell interior-to wall ratios are estimated at 35%=65%, 25%=75% and
35%=65% for sample BE1, BE2 and OA, respectively.
deformation axes of the Cu single-crystals are chosen for X-ray microbeam diffraction
measurements. For sample BE1 and BE2, reflection (006) is chosen since the deforma-
tion is made along [001] crystal axis.
Cu-sample OA was compressively deformed along <0.184, 0.107, 0.177> with the
true plastic strain 28%. With such a large plastic strain and non-symmetric tensile
axis, crystallographic rotation will occur. Therefore, after deformation, the sample axis
is not along the <0.184, 0.107, 0.177>. In the experiment, the plane normal of (515)
142
was observed to be the closest direction with the deformation axis. The angle deviation
between the final deformation axis and [515] is about 2:7

.
Figure 4.15: X-ray microbeam integrated (006) line profile for sample BE1 condition
It was deformed compressively to 28% along [001] crystal axis direction and then
unloaded. The X-ray line profile, (006) reflection, shows the asymmetric shape. The
X-ray line profile is the sum of the integrated line profiles measured from 2 sample
positions with 52 different depths. The diffraction volume of each depth position is
about 0:5
3
m
3
. The dash line indicates the position of Cu (006) standard q value,
q = 104:2864nm
1
(unstrained).
143
Figure 4.16: X-ray microbeam integrated (006) line profile for sample BE2 condition
It was deformed compressively to 28% along [001] crystal axis direction and then
applied tensile stress to a strain of 2%. The asymmetric X-ray line profile is the sum
of the integrated X-ray line profiles measured from 3 sample positions with 117 depths.
The diffraction volume of each depth position is about 0:5
3
m
3
. The dash line indicates
the position of Cu (006) standardq value,q = 104:2864nm
1
(unstrained).
4.11.3 The Reconstruction of Cell Interior Subprofiles
As mentioned in chapter 2, the extraction method of individual cell interior X-
ray line profiles from microbeam Laue diffraction images were introduced. By mask-
ing the diffraction spots (high intensity spot) on the Laue images in each depth, the
masked diffraction spots can be integrated into X-ray line profiles which can represent
the X-ray line profiles of individual cell interiors. The X-ray line profiles of individual
144
Figure 4.17: X-ray microbeam integrated (515) line profile for sample OA condition
<0.184, 0.107, 0.177> orientated Cu single crystal, deformed in compression to around
28%. The asymmetric X-ray profile is the sum of integrated X-ray line profiles measured
from 4 sample positions with 190 depths. The diffraction volume of each depth position
is about 0:5
3
m
3
. The dash line indicates the position of Cu (515) standard q value,
q = 124:1255nm
1
(unstrained).
cell interiors are generally symmetric and can be fitted by pseudo-V oigt function. The
bright Laue diffraction spots at different positions and depths are carefully examined
and integrated into cell interior X-ray line profiles. Large numbers of cell-interior line
profiles are shown for three deformation conditions in Figure 4.18(a), Figure 4.19(a)
and Figure 4.20(a). Thus, the summarized profile of all cell interior line profiles in Fig-
ure 4.18(b), Figure 4.19(b) and Figure 4.20(b) can represent integrated subprofiles of
cell interiors for the whole sample.
145
Figure 4.18: The reconstructed cell interior profile of sample BE1
(a) 102 individual cell interior profiles, fitted by pseudo-V oigt function are shown in the
same figures. (b) The cell interior subprofile is reconstructed by adding all the individual
cell interior profiles in (a).
4.11.4 The Reconstruction of Cell Wall Subprofiles
Levine et al. [5, 6] have directly reconstructed the wall X-ray line profile from the
smeared signals on the Laue diffraction images. The same methods have been used to
find the wall integrated line profiles at different depths and positions. The wall strains
which are calculated from peak positions of wall line profiles should be of the opposite
sign as those of the strains in the cell interiors. In the current experiments, compressive
stresses are applied on the samples and thus tensile strains should be expected to exist
in the most of the cell interiors, especially for sample BE1 and OA (only deformed with
compressive strain); negative strains should be found around the smeared signals, cell
walls. However, using the standardq value of copper, positive strains are mainly found
for both cell interiors and cell walls.
Based on the signatures mentioned above, we can conclude that residual stress
existed in the copper samples (BE1, BE2 and OA) after plastic deformation. In this
146
Figure 4.19: The reconstructed cell interior profile of sample BE2
(a) 391 individual cell interior profiles, fitted by Pseudo-V oigt function are shown in the
same figures. (b) The cell interior subprofile is reconstructed by adding all the individual
cell interior profiles in (a).
case, the standardq value cannot be used and also it is difficult to find the represented
wall strains from the smeared signals. To overcome this issue, the mean values of lattice
spacing from asymmetric microbeam X-ray line profiles which include signals from cell
interiors and walls were calculated. The mean value of lattice spacing of a reflection
peak will lead the integrated strain of the asymmetric X-ray line profile to zero. Thus,
the relative strains of cell interiors and walls to the mean value of lattice spacing then
can be calculated regardless of the residual stress which has been observed to affect the
lattice spacing in the sample [49]. The calculation will be introduced in the following
sections.
To reconstruct the wall subprofiles, the subtraction procedure between asymmet-
ric microbeam X-ray line profile and cell-interior subprofile can be applied but several
assumptions [11, 43, 50] need to be followed. First, it has been suggested that an asym-
metric X-ray line profile can be decomposed into two subprofiles of cell interiors and
walls. Second, subprofiles are symmetric and can be generally fitted by pseudo-V oigt
147
Figure 4.20: The reconstructed cell interior profile of sample OA
(a) 231 individual cell interior profiles, fitted by Pseudo-V oigt function are shown in the
same figures. (b) The cell interior subprofile is reconstructed by adding all the individual
cell interior profiles in (a).
function. Third, the area ratios between cell interior and wall line profiles are propor-
tional to the ratios between the cell interior and wall in microstructures. The volume
ratio of cell interior-to-wall in microstructure can be calculated from area ratio in a
TEM bright field image. Thus, to apply the subtraction procedure, the ratios between
cell interior and wall must be known.
In the previous section, the cell interior/wall ratios have been calculated from TEM
microstructure images. For sample BE1 and sample OA, the area of cell interiors is esti-
mate 35% and for sample BE2, the cell interior ratio is 25%. The calculation values from
TEM images are very rough but basically high wall ratios are observed for three defor-
mation conditions. Thus, it is assumed that the cell-interior ratio is about 35% for the
reconstruction of wall profiles for sample BE1 and OA in Figure 4.21 and Figure 4.23.
The cell interior ratio is 25% for sample BE2 and it is shown in Figure 4.22.
First, the X-ray line profile intensity is normalized by the peak intensity to give an
unit of 1 and then the area of X-ray line profile is calculated. Second, the reconstructed
148
Figure 4.21: The decomposition of wall subprofile for sample BE1
The wall subprofile is reconstructed by the subtraction between overall X-ray line profile
and cell-interior subprofile. The vertical dash line is the standardq value of unstrained
Cu single crystal (q(006,STD) = 104:2864nm
1
). The vertical solid line indicates the
q mean value of (006) asymmetric X-ray line profile (q(006, mean)= 104:2309nm
1
).
subprofile of cell interior is assigned the area under the line profile to 35% of the X-
ray line profile. In the following, the subtraction between the asymmetric X-ray line
profile and the cell-interior subprofile is implemented. Thus, a cell wall subprofile can
be obtained. Finally, pseudo-V oigt fitting is applied for this cell interior and cell wall
subprofiles. The average lattice spacing or strains of the cell interior and cell wall can
be estimated from the fitted subprofiles in Figure 4.21, Figure 4.22 and Figure 4.23.
149
Figure 4.22: The decomposition of wall subprofile for sample BE2
The wall subprofile is reconstructed by the subtraction between overall X-ray line profile
and cell-interior subprofile. The vertical dash line is the standardq value of unstrained
Cu single crystal (q(006,STD) = 104:2864nm
1
). The vertical solid line indicates the
q mean value of (006) asymmetric X-ray line profile (q(006, mean) = 104:2683nm
1
).
4.11.5 Mean Value of Lattice Spacings
The asymmetry of X-ray line profile after monotonic deformation has been demon-
strated as the evidence of the formation of opposite strain signs in the cell interiors and
walls. However, in some cases, a residual stress may exist and locally shift the mean
value of the q vector far from the standard q value. Thus, using the mean value of q
150
Figure 4.23: The decomposition of wall subprofile for sample OA
The wall subprofile is reconstructed by the subtraction between overall X-ray line profile
and cell-interior subprofile. The vertical dash line is the standardq value of unstrained
Cu single crystal (q(515,STD) = 124:12554nm
1
). The vertical solid line indicates the
q mean value of (515) asymmetric X-ray line profile (q(515, mean) = 124:0850nm
1
).
vectors (centroid) for the strain calculation should have the overall strain equal to zero
for the examined diffraction peaks. To calculate the mean q value for an asymmetric
diffraction peak, the center-of-mass equation is adopted.
q
average
=
P
qIntensity
P
Intensity
(4.1)
151
In the above equation (4.1), the volume ratio of each q value of a diffraction peak is
assumed to be proportional to the its integrated intensity.
The calculation is applied only for the area near major diffraction peak. For sample
BE1 and BE2, the meanq value of {006} is calculated betweenq = 103:6nm
1
and
q = 105 nm
1
and for sample OA, the mean q value of {515} is calculated between
q = 123:6nm
1
andq = 124:6nm
1
. The mean values are shown in Table 4.2.
Table 4.2: Mean value ofq and d-spacing
Condition q (nm
1
) d-spacing (nm)
Sample BE1,d
(006)
104:2309 0:060281
Sample BE2,d
(006)
104:2683 0:060260
Sample OA,d
(515)
124:0850 0:050634
4.11.6 Residual Strain/stress
As already mentioned, a residual strain/stress may exist in all the studied samples
(BE1, BE2 and OA) since positive strains are mainly measured in both cell interiors and
walls. After obtaining the q mean values from the asymmetric X-ray line profiles for
sample BE1, BE2 and OA, the residual strains, which prevail in the measurement area
and can be calculated (in Table 4.3) by equation,
=
q
standard
q
mean
q
mean
(4.2)
It was found that positive residual strains (stresses) are +5:3 10
4
(35:3 MPa),
+1:7 10
4
(11:3 MPa) and +3:3 10
4
(44:6 MPa), respectively. These strains
are larger than the strain resolution ( 1 10
4
) of X-ray microbeam beam and also
observed over a large diffraction volume. The diffraction volume of microbeam mea-
surement can be estimated by beam size (0:5
2
m
2
) times the total scanning depths.
152
Thus, the diffraction volumes for sample BE1, BE2 and OA are about 0:5 m 0:5
m 26 m, 0:5 m 0:5 m 58:5 m and 0:5 m 0:5 m 95 m. The
diffraction volumes are much bigger than the average cell interior and wall sizes (the
average cell interior diameter plus average width is roughly 0:8m for sample BE1 and
BE2 and 1:1m for OA). Therefore, residual strain/stress can be claimed to exist in the
studied samples over a large length scale. This residual strain/stress may result from the
non-uniform deformation of Cu single crystal. Sample BE2 were designed for tension/-
compression and thus had to be made in dog-bone shape. For comparison, all samples
were made in dog-bone shape. Deforming these samples in compression results in sub-
stantial boundary effects, and thus barreling will occur. Thus, an additional asymmetric
stress would be imposed on the samples and this might be the possible reason to cause
the residual stresses in the samples.
Therefore, to know the local strain/stress variation due to the formation of cell inte-
rior/wall dislocation microstructures, the meanq value must be used instead of standard
q value to calculate the local strain variation in the cell interiors and walls.
Table 4.3: Summary of residual strain/stress in Cu samples
Elastic modulus ofE
001
= 66:66 GPa;E
515
= 135:14 GPa
Condition Residual Strain Residual Stress Flow Stress
r
/
a
(
r
) MPa (
a
) MPa
Sample BE1, (006) 5:3 10
4
35.3 281 0.13
Sample BE2, (006) 1:7 10
4
11.3 139 0.08
Sample OA, (515) 3:3 10
4
44.6 183 0.24
153
4.12 Discussion
4.12.1 The Ratio of Cell Interiors and Walls
To examine the reliability of the cell interior and wall ratios calculated from TEM
micrographs, several other cell interior ratios are tested for the reconstruction of wall
subprofiles. In Figure 4.24, Figure 4.25 and Figure 4.26, different cell interior ratios
are tested for the extraction of wall subprofiles for three deformation conditions. For
sample BE1 and OA, the cell interior and wall subprofiles can be reasonably well-fitted
by pseudo-V oigt function by assigning 25%, 35% and 45% cell interior ratios. For
sample BE2, the cell interior and wall subprofiles can be also well fitted by pseudo-V oigt
function with several cell interior ratios. Therefore, it is difficult to judge the correct cell
interior ratio by the fitting results. As mentioned in the microstructure section, the cell
interior ratios obtained by analyzing the TEM images may not be able to provide the
sampling for three deformation conditions. Thus, further studies will be investigated in
the future.
154
Figure 4.24: The reconstruction of wall subprofiles with different cell interior-wall ratios
for sample BE1
(a) 25% cell interiors. (b) 35% cell interiors. (c) 45% cell interiors. (d) 55% cell
interiors.
155
Figure 4.25: The reconstruction of wall subprofiles with different cell interior-wall ratios
for sample BE2
(a) 15% cell interiors. (b) 25% cell interiors. (c) 35% cell interiors.
156
Figure 4.26: The reconstruction of wall subprofiles with different cell interior-wall ratios
for sample OA
(a) 25% cell interiors. (b) 35% cell interiors. (c) 45% cell interiors. (d) 55% cell
interiors.
4.12.2 The Strains/stresses of Cell Interiors and Walls
The strains/stresses of cell interiors and walls for the three deformation conditions
are summarized in Table 4.4 and Table 4.5. In addition, the stress-strain curves of
sample BE1, BE2 and OA are shown in Figure 4.27, Figure 4.28 and Figure 4.29 As
mentioned previously, the residual stress exists in the sample after deformation and thus
the standard copper lattice constant cannot be used to calculate the LRIS in cell interiors
157
Table 4.4: The summary of cell and wall strain from peak position of fitting profiles
Elastic modulus ofE
001
= 66:66 GPa;E
515
= 135:14 GPa
q
(006)
, sample BE1 mean value = 104:2309 nm
1
; q
(006)
, sample BE2 mean value =
104:2683nm
1
;
q
(515)
, sample OA mean value = 124:0850nm
1
.
% cell interior Cell fitting Wall fitting q Cell interior Cell wall
profile profile strain strain
(nm
1
) (nm
1
) (nm
1
)
Sample BE1 35% 104:1973 104:2514 0:054 3:2 10
4
2:0 10
4
Sample BE2 25% 104:2475 104:2793 0:032 2:0 10
4
1:1 10
4
Sample OA 35% 124:0622 124:1015 0:039 1:8 10
4
1:3 10
4
Table 4.5: The summary of cell interior and wall stress from peak position of fitting
profiles
The integrated stress is calculated by the equation,
i
f
interior
+
w
f
wall
, where

i
is the average LRIS in cell interiors,
w
is the average LRIS in cell walls, f
interior
is 35% andf
wall
is 65% for sample BE1 and OA. For sample BE2,f
interior
is 25% and
f
wall
is 75%.
% cell interior Flow Cell interior Cell wall Stress Integrated
Stress stress stress difference stress

a
(MPa) (MPa) (MPa) (MPa)
Sample BE1 35% 281 21:5 13:1 34:6 1:0
(0:08
a
) (0:05
a
)
Sample BE2 25% 139 13:3 7:1 20:3 2:0
(0:1
a
) (0:05
a
)
Sample OA 35% 183 24:9 17:9 42:8 2:9
(0:14
a
) (0:1
a
)
and walls. Instead, theq mean values of {006} and {515} are used for the strain calcula-
tion of sample BE1, sample BE2 and sample OA conditions. The samples are deformed
uniaxially and thus the internal stresses can be estimated by the equation,E
hkl
(E
hkl
:
elastic modulus for thehkl orientation and: strain).
To study the influence of reverse plastic stress on the LRIS, the results between
sample BE1 and sample BE2 can be compared. For sample BE2, the beginning deforma-
tion condition is similar to sample BE1 (28% compressive strain) but a smaller reverse
158
Figure 4.27: The stress-strain curve of Cu sample BE1
<001> single-crystal Cu was deformed in compression to 28%. The maximum true
stress is 281 MPa. The normalized LRIS in the cell interiors (
i
b
/
a
) is about 0:08. This
is a schematic of the loading and unloading processes for sample BE1(not actual data).
strain (2% tensile strain) is applied. The mean strain of cell interior for sample BE2
is about 2:0 10
4
which is smaller than the strain, 3:2 10
4
of sample BE1. By
assuming the cell interior/wall ratio is 35%=65% for sample BE1 and 25%=75% for
sample BE2, the wall subprofiles can be obtained and then to get the mean strain value
of walls. The wall strain of sample BE2 is about1:1 10
4
and the wall strain of
sample BE1 is about2:0 10
4
. In the beginning of 28% compressive plastic defor-
mation, a compressive LRIS in the walls and a tensile LRIS in the cell interiors can be
expected. Then a 2% tensile plastic strain is applied on the sample. In this case, two
159
Figure 4.28: The stress-strain curve of Cu sample BE2
<001> single-crystal Cu was deformed first in compression to 28% and then in tension
for 2%. The maximum reverse true stress is 139 MPa. The normalized LRIS in the cell
interiors (
i
b
/
a
) is about 0:1. This is a schematic of the loading and unloading processes
for sample BE2 (not actual data).
phenomena may happen because of the reversal stress: (1) the dislocation configuration
around walls changes completely. Therefore, the LRIS in cell walls become positive
but the LRIS in the cell interiors become negative; and (2) the dislocation configuration
partially changes due to the smaller reverse plastic strain so the LRIS in cell interiors
and cell walls are still the same signs but the magnitude of the LRIS reduces.
According to the results of sample BE2, lower LRIS in the cell interiors and walls
are observed. The LRIS in the cell interiors and walls keep the same signs with sample
BE1. In addition, the strain/stress difference between cell interiors and walls becomes
160
Figure 4.29: The stress-strain curve of Cu sample OA
<0.184, 0.107, 0.177> single-crystal Cu was deformed in compression to 28%.The max-
imum true stress is 183 MPa. The normalized LRIS in the cell interiors (
i
b
/
a
) is about
0:14. This is a schematic of the loading and unloading processes for sample OA (not
actual data).
smaller. Therefore, the partial change of dislocation configuration can be concluded to
cause the smaller LRIS in the cell interiors and walls.
For the deformation along different crystal axis (sample OA condition), the stress
difference between cell interiors and walls is larger than the strain difference for sample
BE1. The stress difference is about 42:8 MPa for sample OA which is larger than the
value for sample BE1, 34:6 MPa. In addition, the internal stresses in cell interiors
and walls are 24:9 MPa and17:9 MPa, respectively which are higher than the internal
161
stresses in cell interiors and cell walls of sample BE1. Along different deformation
axes, LRIS will vary due to the generation of different dislocation configuration. For the
sample OA condition, the deformation axis is roughly along [515] direction and higher
LRIS in the cell interiors and walls are observed. Deformation along the non-symmetric
crystal axis may result in more crystallographic rotations, leading to high misorientation
in the cell walls. Higher disorientation in the cell walls may lead to higher LRIS in the
cell interiors and walls.
4.13 Conclusions
Effect of the reverse stress:
1. Cell interior and wall strains/stresses remain the same signs with an additional 2%
reverse tensile strain.
2. Smaller LRIS in the cell interiors and walls are observed and the stress differ-
ence between cell interiors and walls decreases about 40% with an additional 2%
reverse strain.
3. The reverse plastic strain of 2% reduced the LRIS in the cell interiors and walls
without changing the signs, demonstrating that a partial change of the dislocation
heterogeneous microstructure was produced.
Effect of different deformation axes:
1. Higher LRIS in the cell interiors and walls are observed as the deformation is
along the non-symmetric crystal axis, <0.184, 0.107, 0.177> .
2. Larger stress difference is observed for the deformation along the non-symmetric
axis, <0.184, 0.107, 0.177> . It increases about 20% compared with the deforma-
tion along <001>.
162
3. Deformation along the non-symmetric axis, <0.184, 0.107, 0.177> may cause
many crystallographic rotations, which lead to higher misorientation in the cell
walls. Higher misorientation in cell walls may result in higher LRIS.
163
Chapter 5
Summary
These studies investigated the long-range internal stresses in metal materials in
two systems: (1) single-crystal copper deformed with a small reverse plastic strain (2%)
or along different crystal axes and (2) ultra-fine grain aluminum alloys (e.g. AA1050
and AA6005) processed by equal channel angular pressing (ECAP). Levine et al. have
experimentally proven that the LRIS exist in the cell interiors and walls in single-crystal
copper after monotonic deformation with the magnitude of about 0:1
a
in both cell
interiors and walls. The stress between cell interiors and walls is about 40 MPa. LRIS
associated with more complex deformation procedures attract our attention. For the
LRIS evolutions, resulting from an additional reverse plastic deformation or deformation
along different crystal axes, two interesting findings are evident. First, small reverse (e.g.
2%) plastic deformation may lead to the reduction of internal stress in the cell interiors
and walls. Second, deformation along the non-symmetric crystal axis may lead to higher
LRIS in both the cell interiors and walls.
The generation of LRIS in the deformed single-crystal copper has been suggested
to be associated with the formation of heterogeneous dislocation microstructures, cell
interiors and walls. According to our studies, the magnitude of LRIS in the deformed
single-crystal copper also evolves with the deformation procedures and deformation
164
axis. These changes are considered to be associated with the change of dislocation
microstructure and misorientation in walls.
The observation of LRIS in the deformed single-crystal copper is still limited
around low-angle boundaries and in the cell interior and wall microstructures. There-
fore, the LRIS in the polycrystalline metals with ultra fine grains and higher misorien-
tation were investigated. The magnitude of the LRIS in ECAP aluminum alloys (e.g.
AA1050 and AA6005) is roughly between 0:1
a
and 0:2
a
in the grains/subgrains for
different numbers of ECAP passes with the LRIS in boundaries uncertain, although
other CBED work suggests that these may be relatively high around
a
, or so.
165
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Appendix A
Appendix1
A.1 Experimental Data
Table A.1: AA1050 1-pass ECAP along pressing direction grain/subgrain interior data
Grain/subgrain Pixel X Pixel Y Depth RotZ RotX Strain
1 1354.68 425.56 66.29 13.5 5 3.9 10
4
2 1360.8 340.1 56.38 15.3 5 2.1 10
4
3 1339.57 576.59 286.07 10.3 4.9 4.9 10
4
4 1320.82 873.59 11.16 3.7 4.8 1.3 10
4
5 1334.4 1877 99.51 18.2 4.8 1.7 10
4
6 1323.72 1223.58 163.54 4.1 4.9 7.5 10
5
7 1330.51 1644.34 4.29 13.3 4.9 9.7 10
6
8 1335.59 1635.66 334.63 13.1 4.9 4.0 10
4
9 1341.46 679.22 2.39 -8 5 3.2 10
4
10 1348.7 465.8 300.52 -12.7 5 2.7 10
4
11 1305.73 1401.78 16.91 8.1 4.7 2.9 10
4
174
Table A.2: AA1050 1-pass ECAP tiled +22:5

with respect to the pressing direction
grain/subgrain interior data
Grain/subgrain Pixel X Pixel Y Depth RotZ RotX Strain
1 1350.1 230.9 2318 17.6 4.7 4.2 10
4
2 1265.47 1647.29 2228 13.4 4.1 1.4 10
4
3 1319.23 1071.24 2264.3 0.7 4.7 4.8 10
4
4 1338.83 440.13 2229 13.2 4.7 1.1 10
4
5 1339.49 439.69 2226 13.2 4.7 1.9 10
4
6 1339.11 440.05 2246 13.2 4.7 1.4 10
4
7 1327.87 824.64 2243 4.8 4.8 3.2 10
4
8 1324.8 807.7 2254.4 5.2 4.7 3.0 10
4
9 1322.6 753.7 2309.5 6.4 4.7 1.8 10
4
10 1321.4 973.6 2317.9 1.5 4.7 3.9 10
4
Table A.3: AA1050 1-pass ECAP tiled22:5

with respect to the pressing direction
grain/subgrain interior data
Grain/subgrain Pixel X Pixel Y Depth RotZ RotX Strain
1 1320.97 1504.35 2228.8 10.3 4.7 1.6 10
4
2 1322.8 841.1 2278 4:5 4.1 5.5 10
5
3 1340.4 1592 2260 12.2 4.9 1.4 10
4
4 1337.4 1470.1 2229 9.6 4.9 6.2 10
5
5 1322.3 1202.7 2291.7 3.7 4.7 1.9 10
4
6 1339.1 920.8 2259.3 2.7 4.9 4.4 10
5
7 1338.24 1002.94 2231.1 0.8 4.9 1.2 10
4
8 1348 591.1 2224 10 4.9 2.6 10
4
9 1366.2 509.4 2286 11.8 5.0 3.8 10
5
10 1372.3 394.1 2217 14.2 5.1 9.8 10
5
175
Table A.4: AA1050 multiple-pass ECAP via routeB
C
grain/subgrain interiors data
Grain/subgrain 2-pass 4-pass 8-pass
Strain Depth Strain Depth Strain Depth
1 1.2 10
4
90.28 2.6 10
4
18.11 2.2 10
4
86.57
2 1.7 10
4
180.1 3.5 10
4
18.24 2.7 10
4
83.76
3 3.4 10
4
74.2 1.7 10
4
40.74 4.8 10
4
307.62
4 2.4 10
4
28.04 2.5 10
4
-6.2 4.6 10
4
1.15
5 2.9 10
4
50.39 3.2 10
4
-7.2 2.2 10
4
178.38
6 4.0 10
4
3.41 1.9 10
4
-12.17 1.6 10
4
45.02
7 3.8 10
4
119.57 2.3 10
5
70.23 4.2 10
4
298.8
8 2.4 10
4
12.47 1.9 10
4
-14.06 1.6 10
4
179.16
9 4.1 10
4
80.49 1.6 10
4
17.6 3.5 10
4
273.18
10 4.0 10
4
125 2.6 10
4
244.39 3.5 10
4
172.28
Table A.5: AA6005 2-pass ECAP routeC grain/subgrain interiors data
Grain/subgrain 2-pass
Strain Depth
1 1.3 10
4
49.41
2 5.5 10
4
127.57
3 5.5 10
4
159.71
4 2.0 10
4
360.42
5 9.4 10
5
406.58
6 2.5 10
4
380.91
7 3.2 10
4
35.49
8 5.6 10
4
358.91
9 6.2 10
4
25.43
10 5.5 10
4
3.65
176
Appendix B
Appendix2
B.1 Geometry File
The geometry file used for the rotation angle calculation of ECAP grain/subgrain
reflections is listed below.
$ f i l e t y p e geom etr yF ile N
$ d a t e W r i t t e n Mon , Nov 26 , 2012
$ t i m e W r i t t e n 1 7 : 2 0 : 5 6 . 6 (6)
$EPOCH 3436795257 / / s e c o n d s from m i d n i g h t J a n u a r y 1 , 1904
/ / Sample
$SampleOrigin {8200.00 ,4758.83 ,4476.00}
/ / sample o r i g i n i n raw PM500 u n i t s ( micron )
$SampleRot {0.00600000 ,0.00600000 ,0.00001800}
/ / sample p o s i t i o n e r r o t a t i o n v e c t o r
( l e n g t h i s a n g l e i n r a d i a n s )
/ / D e t e c t o r s
$ N d e t e c t o r s 3
177
/ / number of d e t e c t o r s i n use , must be
<= MAX_Ndetectors
$d0_Nx 2048
/ / number of unb i nn ed p i x e l s i n f u l l d e t e c t o r
$d0_Ny 2048
$d0_sizeX 409.600 / / s i z e of CCD (mm)
$d0_sizeY 09.600
$d0_R {1.20165612 ,1.21265248 ,1.21799215}
/ / r o t a t i o n v e c t o r ( l e n g t h i s a n g l e i n r a d i a n s )
$d0_P {25.352 ,2.483 ,510.829}
/ / t r a n s l a t i o n v e c t o r (mm)
$d0_timeMeasured Mon , Nov 26 , 2012 , 1 6 : 2 9 : 1 5 (6)
/ / when t h i s geometry was c a l c u l a t e d
$d0_geoNote Optimized u s i n g C a l i b r a t i o n L i s t O r a n g e 0
$ d 0 _ d e t e c t o r I D PE1621 7233335 / / u n i qu e d e t e c t o r ID
$d1_Nx 1024
/ / number of unb i nn ed p i x e l s i n f u l l d e t e c t o r
$d1_Ny 1024
$d1_sizeX 204.800 / / s i z e of CCD (mm)
$d1_sizeY 204.800
$d1_R {1.76585667 ,0.73420326 ,1.75916408}
/ / r o t a t i o n v e c t o r ( l e n g t h i s a n g l e i n r a d i a n s )
$d1_P {142.279 ,2.643 ,412.000}
178
/ / t r a n s l a t i o n v e c t o r (mm)
$d1_timeMeasured Sat , Nov 10 , 2012 , 1 7 : 5 9 : 0 3 (6)
/ / when t h i s geometry was c a l c u l a t e d
$d1_geoNote Optimized u s i n g C a l i b r a t i o n L i s t Y e l l o w 1
$ d 1 _ d e t e c t o r I D PE0820 7631807 / / u ni qu e d e t e c t o r ID
$d2_Nx 1024
/ / number of unbi nn ed p i x e l s i n f u l l d e t e c t o r
$d2_Ny 1024
$d2_sizeX 204.800 / / s i z e of CCD (mm)
$d2_sizeY 204.800
$d2_R {0.61383828 ,1.50298645 ,0.62118817}
/ / r o t a t i o n v e c t o r ( l e n g t h i s a n g l e i n r a d i a n s )
$d2_P {142.818 ,2.971 ,417.118}
/ / t r a n s l a t i o n v e c t o r (mm)
$d2_timeMeasured Sat , Nov 10 , 2012 , 1 7 : 5 9 : 0 3 (6)
/ / when t h i s geometry was c a l c u l a t e d
$d2_geoNote Optimized u s i n g C a l i b r a t i o n L i s t P u r p l e 2
$ d 2 _ d e t e c t o r I D PE0820 7631850 / / u ni qu e d e t e c t o r ID
/ / Wire
$wireDia 52.50 / / d i a m e t e r of wi re ( micron )
$ w i r e K n i f e 0
/ / t r u e i f w ir e on a k n i f e edge ,
f a l s e f o r f r e es t a n d i n g wi re
$ w i r e O r i g i n { 2 . 5 0 , 0 . 0 0 , 0 . 0 0 }
179
/ / w ir e o r i g i n i n raw PM500 frame ( micron )
$wireRot {0.00450000 ,0.00684000 ,0.00003375}
/ / w ir e p o s i t i o n e r r o t a t i o n v e c t o r
( l e n g t h i s a n g l e i n r a d i a n s )
$wireAxis { 1 . 0 0 0 0 0 0 , 0 . 0 0 0 0 0 0 , 0 . 0 0 0 0 0 0 }
/ / u n i t v e c t o r a l o n g w ir e a x i s ,
u s u a l l y c l o s e t o ( 1 , 0 , 0 )
$wireF 0 . 0 0
/ / F of wir e f o r a c o n s t a n t F wir e sca n
( raw PM500 u n i t s )
180

Asset Metadata

Creator Lee, I-Fang (author)

Core Title Using X-ray microbeam diffraction to study the long range internal stresses in plastically deformed materials

Contributor Electronically uploaded by the author (provenance)

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Materials Science

Publication Date 04/28/2015

Defense Date 02/18/2015

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

Tag back‐stresses,dislocation heterogeneous microstructure,ECAP aluminum,long‐range internal stresses (LRIS),OAI-PMH Harvest,plastically deformed copper

Format application/pdf (imt)

Language English

Advisor Kassner, Michael E. (committee chair), Levine, Lyle E. (committee chair), Eliasson, Veronica (committee member), Goo, Edward K. (committee member)

Creator Email ifanglee@usc.edu,iflee325@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-562158

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Abstract (if available)

Abstract One of the essential fundamental physical properties of deformed metals is the origin of long‐range internal stresses (LRIS) and the magnitude of LRIS which is related to the reduction of the reverse yield and associated with the Bauschinger effect (important for cyclic deformation and fatigue properties) and metal springback. ❧ In this thesis, LRIS are examined for four particular cases, including a copper single crystal deformed in compression by 28% along the [001], an identically deformed copper single crystal with an additional 2% reverse tensile deformation, a copper single crystal deformed by 28% along a non‐symmetric crystal axis, and several equal‐channel angular pressing (ECAP) processed polycrystalline aluminum. ❧ In the symmetric [001] experiments, the 2% tensile deformation along the [001] was performed in order to investigate the effect of reverse stress. Compared with the 28% compressive deformed sample, the LRIS in the cell interiors were found to reduce from +21.5 MPa to +13.3 MPa and from −13.1 MPa to −7.1 MPa in the cell walls. In addition, the average stress difference between the cell interiors and walls also decreases from +34.6 MPa to +20.4 MPa. Thus, the application of a small reverse strain caused the LRIS in the cell interiors and walls to be reduced by about 40%, but retain the same signs. It appears that the reduction of LRIS produces a partial change of dislocation configuration in the cell interior and wall heterogeneous microstructures. ❧ In addition, elevated LRIS was found in compressively deformed single-crystal copper along a non-symmetric crystal axis <0.184, 0.107, 0.177>, compared with the deformation along the [001]. Both LRIS in the cell interiors and walls increase. The LRIS in the cell interiors increases from +21.5 MPa to +24.9 MPa and in the cell walls from −13.1 MPa to −17.9 MPa and the stress difference is about 42.8 MPa, which is about 20% higher than 34.6 MPa (deformation along the [001]). The monotonic deformation along a non‐symmetric crystal axes other than the [001] is expected to produce geometrically necessary boundaries (GNBs) with higher misorientation in the walls, which may lead to higher LRIS in both cell interiors and walls. ❧ For ultrafine grain metals (e.g. AA1050) processed by ECAP deformation, high ratios of high angle boundaries were reported as ∼ 25% after 1‐pass ECAP and ∼ 70% after 8‐pass. LRIS were observed about −19 MPa (−0.1σa) along the maximum tensile plastic strain direction within the grain/subgrain interiors of 1‐pass AA1050. The magnitude of LRIS in ECAP AA1050 using route BC is estimated from 0.1σa to 0.2σa in the grain/subgrain interiors. In ECAP AA6005, the LRIS are also observed and the magnitude of 2-pass via route C has a similar value of AA1050.