MICROMECHANICS OF POLYCRYSTALLINE TI-5AL-2.5SN USING DIFFERENTIAL

APERTURE X-RAY MICROSCOPY AND CRYSTAL PLASTICITY SIMULATION

By

Chen Zhang

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Materials Science and Engineering – Doctor of Philosophy

2018

ABSTRACT

MICROMECHANICS OF POLYCRYSTALLINE TI-5AL-2.5SN USING DIFFERENTIAL

APERTURE X-RAY MICROSCOPY AND CRYSTAL PLASTICITY SIMULATION

By

Chen Zhang

Understanding the deformation processes of crystalline materials at the microscale, also known
as micromechanics, is crucial for improving the reliability of structural materials. Therefore, it
is essential to find a robust and reliable methodology that can be used to facilitate the study of
micromechanics.

To this end, near surface plastic deformation processes of a Ti-5Al-2.5Sn sample deformed
under uniaxial tension at ambient temperature were analyzed using differential aperture X-ray
microscopy (DAXM) and computational models based on crystal plasticity theory. A series of
crystal plasticity simulations were conducted in this study using 3D microstructures reconstructed
from DAXM data, the results of which indicate that accurate 3D grain morphologies are more
important than fine tuning the boundary conditions and the constitutive model when it comes to
improving the accuracy of the simulated local kinematic responses (crystal orientation evolution).
Furthermore, the effect of 3D grain morphologies on the simulated local stress-strain responses was
evaluated by comparing them with residual lattice strain/stress tensors extracted from DAXM data,
the results of which suggest that the accuracy of simulated local stress state scales with the fidelity
of the surrounding reconstructed microstructures. The customized computational toolkits used to
extract deviatoric residual lattice strain/stress tensors from DAXM data were also assessed with
numerical studies, validating the robustness and the reliability of the customized computational
toolkits. Besides reconstructing 3D microstructure and validating simulation results, this work
also explores the potentials of extracting dislocation content from DAXM data, which yields a
new technique, a Frank-Bilby equation based streak analysis that is capable of providing misfit
dislocation density profile with high spatial resolution.

Overall, the work presented in this study demonstrates the potential of the methodology that
combines crystal plasticity simulations with synchrotron radiation techniques for the study of mi-
cromechanics. Future development of this methodology should focus on physics-based constitutive
model development assisted/validated by the DAXM characterization, which can be of great help
for the field of micromechanics.

Copyright by
CHEN ZHANG
2018

ACKNOWLEDGEMENTS

I would like to thank Dr. Bieler, my primary advisor, for the opportunity to work on this project
as well as the many opportunities to present my work in peer-reviewed journals and conferences.
His kind encouragement and patient instructions helped me overcome many academic and technical
hurdles during my graduate study. I will always remember his mentoring and pass it on as I progress
through my career.

Next, I would like to extend my gratitude to Dr. Eisenlohr, who has helped me with my research
on numerous occasions. Most of the findings present in this work are the direct results of many
intense debates with him, the memory of which I will always treasure.

I would also like to express my sincere thanks to Dr. Crimp, Dr. Boehlert, and Dr. Owen,
who provided me with much valuable advice and guidance. Without their support, it would be
impossible for me to present the research shown in this work.

Besides my committee members, I would also like to express my gratitude to Dr. Dierk Raabe
for an extended summer stay at Max-Planck-Institut für Eisenforschung, Düsseldorf, and the many
inspiring discussions with members of his department.

My sincere gratitude also goes to Dr. Hongmei Li for the in-situ tensile experiment and EBSD
characterization, Dr. Shanoob Nair for the excellent TEM analysis, Dr. Ruqing Xu and Dr. Tischler
at the Advanced Photon Source for their valuable insights on DAXM experiment, Dr. Barabash
at Oak Ridge National Lab for the fundamentals of streak analysis, and Dr. Martin Diehl at Max-
Planck-Institut für Eisenforschung for the insightful discussions and suggestions about DAMASK.
Last but not least, I would also like to thank my colleagues, Harsha Phukan, Aritra Chakraborty,
Mingmin Wang, Zhuowen Zhao and many other graduate students in the metal group for the
stimulating debates and discussions that help sustain this research.

This work present here was supported by the US Department of Energy, Office of Basic Energy
Science through grant No. DE-FG02-10ER46637 and in part by Michigan State University through

v

computational resources provided by the Institute for Cyber-Enabled Research. The Ti- 5Al-2.5Sn
alloy used in this study is provided by Mr. T. Van Daam of Pratt & Whitney, Rocketdyne (now
Aerojet Rocketdyne). The differential aperture X-ray microscopy (DAXM) characterization was
performed at Beamline 34-ID-E of the Advanced Photon Source. The use of the Advanced Photon
Source at Argonne National Laboratory was supported by the U.S. Department of Energy, Office
of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

vi

TABLE OF CONTENTS

.
.

.

.
.

.

.
.

.

.
.
.
.

. .
.

.
LIST OF TABLES .
LIST OF FIGURES .
.
KEY TO ABBREVIATIONS .
.
KEY TO SYMBOLS . .
CHAPTER 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. xxiii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
1
.
5
.
5
2.1 Deformation mechanisms in crystalline materials . . . . . . . . . . . . . . . . .
5
2.1.1 Dislocation slip .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Mechanical twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3 Deformation mechanism in Ti-5Al-2.5Sn . . . . . . . . . . . . . . . . . 10
2.2 Synchrotron X-rays diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 High energy X-ray diffraction microscopy . . . . . . . . . . . . . . . . . 15
2.2.2 Differential aperture X-ray microscopy . . . . . . . . . . . . . . . . . . 17
2.3 Crystal plasticity simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Kinematic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Constitutive description . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Microstructure input
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . 26
. 30
3.1 Sample specifications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Surface characterization with in-situ SEM and EBSD . . . . . . . . . . . . . . 30
3.3 Subsurface characterization with DAXM . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 First set of DAXM scans . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2
Second set of DAXM scans . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.3 Third set of DAXM scan . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3D MICROSTRUCTURE RECONSTRUCTION . . . . . . . . . . . . . 42
4.1 Manual reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Auto reconstruction .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Microstructure reconstruction with Barycentric interpolation . . . . . 47
4.2.2 Reconstruction of second generation 3D microstructure . . . . . . . . . 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER 3 EXPERIMENT CHARACTERIZATION . . . . . . . . . . . . . . . .

2.4 Critical assessment of the state of the art

4.3 Summary .

CHAPTER 4

.

.

.

.

.

.

.

.

.

.

CHAPTER 5 EXTRACTING LATTICE STRAIN FROM MICRO-LAUE

DIFFRACTION PATTERNS . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Fundamental theory of strain quantification . . . . . . . . . . . . . . . . . . . . 56

vii

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHAPTER 7 DAXM ASSISTED CRYSTAL PLASTICITY SIMULATIONS . . . .

5.2
Inherent accuracy limit of strain quantification technique . . . . . . . . . . . . 59
5.3 Effect of white noise on strain quantification accuracy . . . . . . . . . . . . . . 65
5.4 Summary .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
CHAPTER 6 ANALYZING DISLOCATION CONTENT USING STREAK ANALYSIS 70
6.1 Single voxel streak analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Enhanced streak analysis using Frank-Bilby equation . . . . . . . . . . . . . . 78
6.3 Summary .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
. 89
7.1 A phenomenological power-law based crystal plasticity model . . . . . . . . . . 90
7.1.1 Constitutive model description . . . . . . . . . . . . . . . . . . . . . . . 90
7.1.2 Constitutive model calibration . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Study of the effect of realistic 3D microstructure . . . . . . . . . . . . . . . . . 92
7.2.1 Microstructure reconstruction . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.2 Crystal plasticity simulation configurations . . . . . . . . . . . . . . . . 94
7.2.3 Effects on lattice reorientation . . . . . . . . . . . . . . . . . . . . . . . 97
7.2.4 Effects on stress response . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Comparison of simulated and measured residual lattice stress . . . . . . . . . . 105
7.3.1
Simulation environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3.2 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
. 114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Microstructure reconstruction requirement . . . . . . . . . . . . . . . . . . . . 115
8.2
Interpretation of mesoscale results . . . . . . . . . . . . . . . . . . . . . . . . . 116
. . . . . . . . . . . . . . . . . . . . . . . . 117
Interpretation of microscale results
8.3
CHAPTER 9 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . 119
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.1 Conclusions .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2 Future work .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
.
.
TERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . 123
EXPLORE THE POSSIBILITY OF RECOVERING UNDE-
FORMED CRYSTAL ORIENTATION USING CRYSTAL
PLASTICITY SIMULATION . . . . . . . . . . . . . . . . . .
MENTAT PROCEDURE FILE FOR 3D MODEL CON-
STRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
NOISE REMOVAL FOR MICROSTRUCTURE RECON-
STRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
MICROSTRUCTURE
BARYCENTRIC INTERPOLATION . . . . . . . . . . . . . . . 133
PYTHON IMPLEMENTATION OF THE VIRTUAL DAXM
EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

APPENDICES .
APPENDIX A
APPENDIX B

RECONSTRUCTION

APPENDIX C

APPENDIX D

APPENDIX E

APPENDIX F

USING

7.4 Summary .

.
CHAPTER 8 DISCUSSION .

.

.

.

.

.

.

. 127

.
.
.

.
.
.

.

.

.

.
.
.

.
.
.

.
.
.

viii

APPENDIX G

APPENDIX H

APPENDIX I

APPENDIX J

APPENDIX K

APPENDIX L

. 136

PYTHON IMPLEMENTATION OF THE SINGLE VOXEL
STREAK ANALYSIS . . . . . . . . . . . . . . . . . . . . . . .
ESTIMATE TOTAL DISLOCATION DENSITY WITH A
SINGLE WALL OF EDGE DISLOCATIONS . . . . . . . . . . 137
PYTHON IMPLEMENTATION OF THE FBE STREAK
ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
CALIBRATION OF THE CONSTITUTIVE MODEL US-
ING NELDER-MEAD BASED SIMPLEX SEARCH . . . . . . . 139
AN EFFICIENT TAYLOR GRADIENT ENHANCED PHE-
NOMENOLOGICAL CRYSTAL PLASTICITY MODEL . . . . 140
ISOTROPIC BUFFER EFFECT ON THE ACCURACY OF
LOCAL MICROMECHANICAL RESPONSES . . . . . . . . . 154

BIBLIOGRAPHY .

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

ix

LIST OF TABLES

Table 2.1: Common mechanical twinning models for titanium and its alloys . . . . . . . . . 13

Table 3.1: Chemical composition of the as received Ti-5Al-2.5Sn plate measured using

. .

. .

ICP-MS .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 7.1: Comparison of the first generation and second generation microstrucures

. . . . 107

Table K.1: Illustration of orientation relationships for seven different bi-crystal case stud-
ies with top-down view of hexagonal unit cell (tensile axis along the column
direction) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

. .

. .

.

x

LIST OF FIGURES

Figure 1.1: Schematics of the relationship between differential aperture X-ray microscopy
and crystal plasticity simulation. The grayed-out paths represent topics be-
yond the scope of this thesis.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1: Schematic illustrations of plastic deformation induced by the glide of edge

(top and middle) and screw (bottom) dislocations

. . . . . . . . . . . . . . . .

Figure 2.2: Schematics of the effect of different deformation mode (elastic, plastic, twin-
ning) on the lattice structure. The capital letter is used to refer to the original
framework whereas the corresponding lowercase letter represents the rotated
framework due to deformation.

. . . . . . . . . . . . . . . . . . . . . . . . . .

4

6

9

Figure 2.3: Schematic of the four geometrical elements of twinning (K1, η1, K2, and η2) . . 10

Figure 2.4: Primary dislocation slip systems for α-phase in Ti and its alloys (from left
to right): basal ({0 0 0 1}(cid:104)2 1 1 0(cid:105)) , prismatic ({1 0 1 0}(cid:104)2 1 1 0(cid:105)), pyramidal
<a> ({1 0 1 1}(cid:104)2 1 1 0(cid:105)), pyramidal <c + a> ({1 0 1 1}(cid:104)1 2 1 3(cid:105)). The arrows
indicates the Burgers vectors that facilitate dislocation slip for given slip
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
system.

.

.

.

.

.

.

.

.

. 11

Figure 2.5: Mechanical

twinning systems in α-phase Ti and its alloy: T1 twin-
ning ({1 0 1 2}(cid:104)1 0 1 1(cid:105)), T2 twinning ({1 1 2 1}(cid:104)1 1 2 6(cid:105)), C1 twinning
({1 1 2 2}(cid:104)1 1 2 3(cid:105)), C2 twinning ({1 0 1 1}(cid:104)1 0 1 2(cid:105)). The arrows indicates
the Burgers vector of the dislocation that facilitate twinning during plastic
deformation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

. .

. .

Figure 2.6: The continuous increase of the brilliance of X-ray sources developed in the
past hundred years, especially since the introduction of synchrotron radiation
(1st generation), established (synchrotron) X-ray as a invaluable radiation
source for material characterization.

. . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.7: A schematic diagram of the structure of a synchrotron facility, where high
energy polychromatic X-ray is generated by passing the high energy charged
particles (electrons or positrons) in a storage ring through an undulator (lattice
of magnets). The resulting white beam (polychromatic) can be further ad-
justed by adding various optic elements, such as a monochromator, focusing
devices, etc., in the optic path before delivered to the sample.

. . . . . . . . . . 17

xi

Figure 2.8: Schematic of a HEDM experiment layout (left: far-field HEDM, right: near-
field HEDM) with x-axis along incident beam direction, y-axis perpendicular
to the beam direction within the beam plane, and z-axis normal to the beam
plane. The distance between the sample and the detector, L, can vary de-
pending on the nature of the HEDM experiment (far-field or near-field). . . . . . 18

Figure 2.9: Schematic diagram of the DAXM structural microscope layout. The remov-
able/insertable micro-monochromator allows users to switch to monochro-
matic beam for energy scan, which is necessary for estimating the hydrostatic
component of the lattice strain tensor. The Kirkpatrick-Baez mirror (K-B
mirrors) are used to focus the incident beam to ∼0.5 µm in diameter. The Pt
wire is used to block diffraction signal from a specific subsurface volume, the
voxel diffraction pattern from which will be recovered through the deconvo-
lution of patterns collected by the CCD detector (charge coupled device) and
subsequently reconstructed and indexed.

. . . . . . . . . . . . . . . . . . . .

. 19

Figure 3.1: Top view of the dog-bone tensile sample (undeformed) cut from a commercial

Ti-5Al-2.5Sn plate.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 3.2: Secondary electron contrast image at approximately 3.5% deformation of the
tensile sample, showing various linear features such as slip traces and surface
ledges. The β-phase (bright) is scattered among the α-phase matrix.

. . . . . . 31

Figure 3.3: EBSD measured pole figures indicate a relatively weak texture, roughly about

2x randomness.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 3.4:

In situ (dotted) and approximated (solid) stress–strain response of the Ti-5Al-
2.5Sn sample. Stress relaxation during SEM image acquisition caused the
two intermittent drops in the dotted curve.

. . . . . . . . . . . . . . . . . . . . 32

Figure 3.5: Schematic representation of the standard DAXM experiment configuration

with single overhead detector at beamline 34-ID-E, APS.

. . . . . . . . . . . . 33

Figure 3.6: The first DAXM scan across a grain boundary between the grain of interest
and one of its neighbor. The scan position (dashed line) on the surface is a
rough estimate due to the uneven surface topography after the deformation. . . . 34

Figure 3.7: The second DAXM scan across the grain boundaries between the grain of

interest and two of its neighbors.

. . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.8: The third DAXM scan across the grain boundary between the grain of interest

and one of its neighbor.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xii

Figure 3.9: The fourth DAXM scan across the grain boundaries between the grain of
interest and two of its neighbors. This orientation map has a 45° inclination
to the sample surface due to its scanning path parallel to the x-axis.

. . . . . . . 36

Figure 3.10: Subsurface orientation map with tensile IPF colormap generated from the
first of the second set of DAXM scans, illustrating the inclined subsurface
grain boundary between two neighbors of the grain of interest

. . . . . . . . . 37

Figure 3.11: Subsurface orientation map with tensile IPF colormap generated from the
second of the second set of DAXM scans, illustrating the subsurface grain
morphology of a neighboring grain (yellow)

. . . . . . . . . . . . . . . . . .

. 38

Figure 3.12: Subsurface orientation map with tensile IPF colormap generated from the
third of the second set of DAXM scans, illustrating the subsurface grain
morphology of the same yellow neighboring grain along a different scanning
path that is perpendicular to the one used in Figure 3.11 . . . . . . . . . . . .

. 38

Figure 3.13: Subsurface orientation map with tensile IPF colormap generated from the
fourth of the second set of DAXM scans, illustrating the subsurface grain
morphology of a neighboring grain (brown)

. . . . . . . . . . . . . . . . . . . 39

Figure 3.14: Subsurface orientation map with tensile IPF colormap generated from the
fifth of the second set of DAXM scans, illustrating the subsurface grain
morphology of a neighboring grain located in north-west of the grain of
interest

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

.

.

.

.

.

.

.

.

Figure 3.15: Subsurface orientation map with tensile IPF colormap generated from the first
of the third set of DAXM scans, illustrating the floating neighboring grain
(yellow)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

. .

. .

.

.

Figure 3.16: Subsurface orientation map with tensile IPF colormap generated from the
second of the third set of DAXM scans, illustrating the variation of subsurface
grain morphology below the floating grain (yellow)

. . . . . . . . . . . . . . . 41

Figure 3.17: Subsurface orientation map with tensile IPF colormap generated from the
third of the third set of DAXM scans, confirming that the floating grain
(yellow) does not extend further under the sample surface . . . . . . . . . . . . 41

Figure 4.1: Schematic illustration of two partial DAXM characterization technique (cross-
blade scans and serial probing) applied to a synthetic microstructure generated
using Voronoi tessellation, the grain ID of which is represented with color.

. . . 43

Figure 4.2: EBSD map of the area of interest using the tensile IPF colormap, along with
the corresponding SEM image with the surface positions of the four DAXM
scans (post deformation).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xiii

Figure 4.3: DAXM blade scans used for the reconstruction of first generation 3D mi-

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

crostructure .

. .

Figure 4.4: The paper model (center) composed through manual alignment of the surface
EBSD map (top) and the four DAXM blade scans (H1, H2, H3 and X1) using
SketchUp®. The blade scan H1 was obscured by H3 as indicated by the gray
dashed line.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

.

.

.

.

.

.

Figure 4.5: The first generation 3D microstructure (134.5 × 145 × 60 µm) reconstructed
from the paper model (Figure 4.4) using MSC.Marc® Mentat 2012 consists
of 29 grains, 11 of which are subsurface grains. The resulting FEM mesh
is discretized with about 50 000 tetrahedral elements, which have four nodes
and ten integration points. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 46

Figure 4.6: Schematic demonstration of using nearest interpolation method (Voronoi tes-
sellation) to infer grain boundaries (dashed lines) using seed points measured
by DAXM. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

.

.

.

.

.

.

Figure 4.7: Schematic demonstration of using Barycentric interpolation built on top of
the Delaunay triangulation (blue dashed line) to infer grain shapes (red and
blue) using seed points measured by DAXM. The gray dashed lines are the
partition lines generated based on Barycentric coordinates. Only two grains
were drawn here due to lack of sufficient seed points to reconstruct other
grains using Barycentric interpolation.

. . . . . . . . . . . . . . . . . . . . .

Figure 4.8:

Illustration of reconstructed grain boundaries from three seed points using
Barycentric interpolation (left) and nearest interpolation (right). Two of the
seed points (seed 2 and seed 3) share the same crystallographic information,
which results in the difference of inferred grain boundary (dashed white lines)
when using different reconstruction method.

. . . . . . . . . . . . . . . . . .

. 49

. 50

Figure 4.9: Reconstructed 2D microstructure using Barycentric interpolation (bottom
left) and nearest interpolation (bottom right) from mocked 2D blade scan
data (top right) extracted from a surface EBSD map (top left) of the same
Ti-5Al-2.5Sn sample used in this study. Some of the differences in the
reconstructed microstructure are highlighted with white arrows. . . . . . . . . . 51

Figure 4.10: The comparison of the original 2D microstructure and the one reconstructed
using Barycentric interpolation with curvature flow (right), which is approx-
imated using five iterations of Gaussian blur with σ = 1 voxel. The halo
around the grain boundaries are the artifacts of blurring process due to using
pixel color as the crystallographic information to be interpolated. Mocked
2D DAXM blade scan is overlaid on top of both figures, along with markers
to indicate regions with high (•) and low (◦) reconstruction quality.

. . . . . . . 52

xiv

Figure 4.11: Combined seed points (EBSD and DAXM) for the reconstruction of second
generation 3D microstructure visualized using two IPF colormaps. Each
voxel shown here has a dimensionality of 1 × 1 × 1 µm3. . . . . . . . . . . . . . 53

Figure 4.12: The final second generation 3D microstructure (right) reconstructed from the
pruned seed points (left) consists of 54 grains, along with a layer of air voxels
overlaid on top to approximate the free surface. About a quarter of the voxels
in the final reconstructed microstructure were made transparent to showcase
the subsurface grain morphology.

. . . . . . . . . . . . . . . . . . . . . . . . 54

Figure 4.13: Grain morphology of the surface central grain (grain of interest) of the second

generation 3D microstructure.

. . . . . . . . . . . . . . . . . . . . . . . . . . 55

Figure 5.1: Schematic representation of the virtual diffraction configuration. The red cap
represents the circular detector covering a 45° detecting range similar to the
single detector configuration commonly used at beamline 34-ID-E, APS.

. . . . 60

Figure 5.2: Reconstruction accuracy of the deviatoric lattice deformation gradient ex-
tracted from regular micro-Laue diffraction patterns (n = 0) with the L2
and opt methods as a function of the number N of indexed peaks (left), the
lattice stretch magnitude ||U|| (middle), and the incremental lattice rotation
angle θ (right). Shaded band in the left diagram corresponds to standard
deviation around median value illustrated by a solid line. Middle and right
diagram depict probability density maps on a logarithmically equi-spaced
grid. Distribution of underlying data is presented in the top and right margin.

. 62

Figure 5.3: Reconstruction accuracy (left) and cumulative distribution of the number of
iterations required (right) to extract the deviatoric lattice deformation gradient
from about 104 sets of scattering vectors using opt with different virtual
detector sizes having cone angles of 45°, 90°, or 180° (from dark to light
blue). Computing times were measured for an implementation based on
Scientific Python (scipy 0.19.0) running in single-thread mode on a 14-core
2.4 GHz Intel Xeon E5-2680v4.

. . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 5.4: Reconstruction accuracy of the full lattice deformation gradient resulting
from the L2and optmethods (red and blue) as a function of the number N of
indexed peaks and the number n ∈ [1, N] of peaks with known length, i.e., for
which an energy scan is available.

. . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 5.5: Reconstruction accuracy of the deviatoric lattice deformation gradient result-
ing from noisy synthetic micro-Laue diffraction patterns using the standard
and spherical coverage of the reciprocal space. A total of 50 peaks were used
for both detector sizes (standard and spherical detector) to isolate the effect
of the detector coverage from the number of peaks indexed.

. . . . . . . . . . . 66

xv

Figure 5.6:

In the standard DAXM configuration (left, small detector), the accuracy of
deviatoric lattice deformation gradient extracted from synthetic noisy micro-
Laue diffraction patterns using the enhanced strain quantification method can
be improved by down-selecting the first M sets of scattering vectors that
are closest to their strain-free counterparts, regardless of the actual lattice
strain level (four levels: 0, 10−4, 10−3, 10−2). Such benefits from the down-
selecting are missing for configurations with a larger detector (right), except
for the trivial case (zero strain).

. . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 6.1: Three-dimensional view (bottom right) of the analyzed grain patch with gray
boundaries on the sample surface and subsurface DAXM scan volume show-
ing the grain structure as colored points on a ∼1 µm thick plane perpendicular
to the surface, with the X-ray microbeam entering the sample 45° from the
surface. The FIB-extracted TEM foil (gray rectangle perpendicular to the sur-
face) passes through the DAXM scan; a low magnification image of the foil
(top left) identifies the region analyzed by TEM (dotted frame). Backscattered
electron image (bottom left, dashed frame) shows that the surface slip steps of
the analyzed (central) grain are consistent with traces expected for basal slip
(blue) or pyramidal (cid:104)c + a(cid:105) slip (yellow) but not necessarily with prismatic
slip (red). Micro-Laue diffraction pattern (top right) originates from the voxel
of interest (located at the center of the larger red sphere close to TEM foil
location, roughly 3 µm below the sample surface), where the streak indicates
about 0.5° lattice orientation spread.

. . . . . . . . . . . . . . . . . . . . . . . 73

Figure 6.2: Bright-field transmission electron micrographs imaged in two beam condi-
tions, with marked diffraction vectors g. The top two conditions reveal that no
(cid:104)c + a(cid:105) type dislocations are present in the TEM foil. The bottom three con-
ditions demonstrate that the Burgers vectors of the dominant slip systems are
3[1 1 2 0]. Combining the g · b = 0
specifically a1 = ±1
and slip trace analysis, the dislocations marked by blue and red arrows in
(a) lie on the (0 0 0 1) plane with Burgers vectors of a3 and a1. The prisms
provide the perspective from the nearest zone axis B from which the foil was
tilted a few degrees to obtain a two-beam condition.

3[2 1 1 0] and a3 = ±1

. . . . . . . . . . . . . . . 74

xvi

Figure 6.3: Colored lines on concentric rings show streak directions that theoretically
result from edge dislocation content on basal (blue), prismatic (red), pyra-
midal (cid:104)a(cid:105) (green), and pyramidal (cid:104)c + a(cid:105) (orange) slip systems. The upper
left image provides an interpretive key. In the other three images the streak
opacity is scaled with global Schmid factor and the streak length is scaled to
reflect the relative projected streak length on the DAXM detector (all have
similar lengths). The streak direction of all three diffraction peaks is consis-
tent with slip on the a3 basal system, which is one of the three active slip
systems identified by TEM analysis (indicated in the top right). Note that all
three (cid:104)a(cid:105)/prismatic plane dislocation variants cause lattice rotation about the
same axis (i.e., (cid:104)c(cid:105)-axis), resulting in a common streak direction. All three
diffraction peaks emanate from the same DAXM voxel.

. . . . . . . . . . . . . 76

Figure 6.4: Homogeneously plastically sheared inclusion (along green arrow in middle
image) has geometrically necessary dislocations in an ellipsoid interface and
requires a homogeneous shear stress and lattice rotation to fit it into the non-
sheared neighborhood. The geometrical arrangement of dislocations (screw
character in the red box, edge character in the blue box) bridges the same
misorientation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

.

.

.

Figure 6.5: By tracking the (3 0 3 4) diffraction peak positions while scanning through
neighboring voxels (top) with a 2 µm thick sampling box, a sampling direction
that provides an orientation gradient similar to the one leading to the peak
streak is located along the (yellow triangle in the bottom image). In the bottom
image, (3 0 3 4) peaks from neighboring voxels (colored disks) are overlaid
on top of the (3 0 3 4) peak streak from the voxel in the center of the FBE
streak analysis region (white square), along with colored lines representing
the scanning path.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure 6.6: Various peak shifting paths and peak streaks for other diffraction peaks from

the same voxel of interest.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. 83

Figure 6.7: The per slip system density profile (dα
i

) with respect to η indicates the main
cause of unrealistic large dislocation density at low η is due to the presence of
negative density from a negative Burgers vector. This configuration provides
the linear system with more flexibility in finding a dislocation configuration
that helps reduce ∆T (black line, right axis), especially when the total dis-
location density is unconstrained due to low η. The blue curve labeled d3
i
corresponds to the basal a3 dislocations observed in the TEM foil.

Figure 6.8: The η profile with total dislocation density
 |di| and mismatch of Frank–

. . . . . . . 84

Bilby tensor ∆T shows that η ∈ (10−3
FBE based streak analysis.

,10−2) is a reasonable choice for the

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xvii

Figure 6.9: The FBE based streak analysis shows that a3 basal slip system exhibits the
lowest dislocation spacing di (highest area density) among all 24 slip systems,
indicating its dominance for this voxel. The second most active slip systems,
a2 basal slip and a3 pyramidal slip, exhibit about one order of magnitude
larger dislocation spacing than that for a3 basal slip.
In this dislocation
spacing plot, the slip families are represented in different colors, and the
shape of the symbol represents the sign of Burgers vector ((cid:78): positive, (cid:72):
negative)

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 6.10: Subsurface dislocation spacing (|di| in Burgers vector) maps illustrate the
basal and prismatic slip activity for the grain of interest, along with the
associated DAXM orientation map (top row). The density distribution shown
here is extrapolated using e−r2, which leads to the artifact (black blocks)
where no DAXM data is available. The relatively uniform distribution of
basal dislocations in the central grain suggests strong basal slip activity,
which is largely in agreement with the TEM analysis (Figure 6.2).

. . . . . . . 87

Figure 7.1: Experimental stress–strain curve (dashed) compared to simulation (solid)
resulting from optimized values of variable constitutive parameters listed
in Figure 7.2. The initial population of constitutive parameter sets results in
stress-strain responses within the shaded range. After Nelder–Mead (simplex)
optimization, each population member produces stress-strain responses very
similar to the exemplary cases PS1 and PS2. The observed effective hardening
range results from the (nonlinear) superposition of hardening ranges per
individual slip family illustrated in Figure 7.2.

. . . . . . . . . . . . . . . . . . 93

Figure 7.2: Visualization of the initial (s0, (cid:78)) and saturated (ss, (cid:72)) critical resolved shear
stresses (CRSS; vertical range is between (cid:78) and (cid:72) as s0 → ss) and hardening
slope (h0) for constitutive parameter sets PS1 and PS2. Both PS1 and PS2 are
an outcome of the calibration process and have volume-averaged stress-strain
behavior that fits well with most of the experimental reference as shown in
Figure 7.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

.

.

.

.

.

.

Figure 7.3: Grain construction based solely on surface EBSD data of the undeformed
sample (a and c) compared to reconstruction using additional subsurface
DAXM scans (b and d). Note the central grain on the surface (purple) in the
3D microstructure is much thicker on one side (positive y) than the other due
to the existence of a subsurface grain (green) right beneath it (see also e).

. . . 95

xviii

Figure 7.4: Schematic representation of two boundary conditions representing the ex-
treme cases of the loading environment of the microstructure volume.
In
the soft boundary condition, all faces except the two normal to y are uncon-
strained. In the hard boundary condition, all faces except the top (surface) are
forced to remain planar during deformation. For both boundary conditions,
all nodes on the two y faces have prescribed displacement to approximate the
tensile loading, and one node in the corner was constrained in the transverse
(x-axis) and vertical directions (z-axis) to provide numerical stability.

. . . . . 96

Figure 7.5: Simulated lattice reorientation maps for patches with different boundary con-
dition, grain morphology, and constitutive parameters compared to the exper-
imental reference.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

.

.

Figure 7.6: Histograms of point-by-point difference in lattice orientation at the surface
when altering the boundary condition (column 1), microstructure (column 2),
or constitutive model parameters (column 3). The averaged influence of each
factor on simulated lattice reorientation is compared in the last histogram
(column 4), which shows that a change in the boundary condition or grain
morphology would more likely lead to substantial change in the simulated
lattice reorientation than changing the constitutive parameters. The change
of lattice reorientation ∆θ shown here is in degrees.

. . . . . . . . . . . . . . . 100

Figure 7.7: Von Mises stress–strain responses of patches for soft (a) and hard (b) bound-
ary condition. Blue and red (or initially higher and lower) curves reflect
constitutive parameter sets PS1 and PS2, respectively. Solid and dotted lines
correspond to 3D and quasi-3D (columnar) grain morphologies, respectively.

. 101

Figure 7.8: Histograms of the average relative change (see Equation (7.8)) in surface
normal stress along the loading axis (σ22, top row) and transverse to it (σ11,
bottom row) when changing the boundary condition (∆boundary condition,
column 1), microstructure (∆morphology, column 2), or constitutive model
parameters (∆constitutive parameters, column 3). Fractional distribution
variation (in the bin of smallest relative changes) among the remaining two
variables, e.g. boundary condition and constitutive parameter set for a switch
of grain morphology (center column), are indicated by brackets for σ11.
Shaded interval (−2,0) corresponds to inaccessible values of average relative
changes.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

.

.

.

Figure 7.9: Principal stress vector field plot comparing the simulated surface stress ten-
sor evolution with strain between realistic and columnar grain structure for
constitutive parameter set PS1 and hard boundary condition.

. . . . . . . . . . 104

xix

Figure 7.10: The second generation 3D microstructure reconstructed from 12 DAXM blade
scans using the Barycentric interpolation method contains 54 grains. An air
layer is added at the top of the microstructure to approximate the free surface
condition during the tensile experiment of the original sample.

. . . . . . . . . 106

Figure 7.11: The overall stress-strain curve of the CPFFT simulation resulted from the

proposed load cases.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 7.12: The simulated surface lattice reorientation maps at 4 % tensile strain using
CPFE (left) and CPFFT (middle) are compared with the EBSD measurements.
The two simulated surface lattice reorientation maps are from crystal plasticity
simulations using the same constitute model and material parameters, but
different solver.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 7.13: Comparison of the magnitude of deviatoric residual lattice stress (|σD|) be-
tween CPFFT simulation (left) and those extracted from strain quantification
(right) with a log scale colormap. Similar patterns between two maps (high-
lighted with arrows) are observed in the region with most DAXM blade scans
(highlighted with white dashed box).

.

.

.

. 109

. . . . . . . . . . . . . . . . . . . . . . . 110

Figure 7.14: Comparison of the diagonal components of the deviatoric residual lattice
D|) between CPFFT simulation (left column) and those ex-
stress tensor (|σii
tracted from strain quantification (right column) with a log scale colormap.
Patterns observed in both maps are marked with white boxes whereas the
discrepancy between two maps is highlighted with red boxes.

. . . . . . . . . . 111

Figure 7.15: Comparison of the shear components of the deviatoric residual lattice stress
tensor (|σii
D|) between CPFFT simulation (left column) and those extracted
from strain quantification (right column) with a log scale colormap. Patterns
observed in both maps are marked with white boxes whereas discrepancy the
between two maps is highlighted with red boxes.

. . . . . . . . . . . . . . . . 112

Figure B.1: Pole figure of a-axis and c-axis reorientation of the central grain shows that the
measured undeformed orientation (EBSD) is different from the one recovered
through reverse loading (CPFE).

. . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure K.1: The shape function used for mapping the strain modifier κ from the geometri-
cal compatibility measurement for slip transfer (m(cid:48)) between two neighboring
material points.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

. .

xx

Figure K.2: The composite mesh used to visualize the data from the bi-crystal study is
constructed such that both spatial (tensile direction, normal to grain bound-
ary plane) and temporal (time) data can be analyzed within the same graph.
Considering the local Taylor factor is an important parameter in TGCP, the
composite mesh is also warped by the local Taylor factor such that the dif-
ference in deformation mode between two grains can be easily inspected.
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

.

.

.

.

.

.

.

.

.

.

.

Figure K.3: The zero gradient of the local Taylor factor near the interface turns off the
nonlocal calculation for both grain 1 and grain 2, which results in a transparent
grain boundary. The colored tensor glyphs hovering above the composite
surface are close to single straight lines, indicating the local stress state in the
bi-crystal remain close to pure uniaxial during the deformation.

. . . . . . . . . 148

Figure K.4: Due to the selective damping on[1 2 1 0](1 0 1 0) (left) for near grain boundary
material points in Grain 1, the deformation mode of the near grain boundary
material points in Grain 1 gradually shifted from dual slip mode to single slip
mode, which is reflected as a monotonically decreasing of local Taylor factor
(M) along time axis. The selective damping also reduces the corresponding
accumulated shear, leading to an increased slip activity for [1 1 2 0](1 1 0 0)
(right) so as to accommodate for the local deformation.

. . . . . . . . . . . . . 149

Figure K.5: In case 2, the simulated grain boundary effect in TGCP is not strong enough
to suppress a dominant slip system, but it does provide a small amount of aid
in helping non-suppressed slip systems indirectly.

. . . . . . . . . . . . . . .

. 150

Figure K.6: In case 3, the simulated grain boundary effect in TGCP visibly enhanced the
dual slip condition (increases in local Taylor factor), which leads to a evident
"ridge" along the grain boundary in Grain 1.

. . . . . . . . . . . . . . . . . .

. 150

Figure K.7: Case 5 demonstrates how complicated grain boundary effect can be simulated
with TGCP when multiple slip systems are active at the same time. Due to
the initial high local Taylor factor value, opacity mapping is applied to the
composite mesh to add transparency to material points with near zero values.

. 151

Figure K.8: The stress state in case 5 is more complicated than dual slip conditions. Due
to the slip activity of multiple slip systems, the local stress state deviates from
the global uniaxial loading condition. However, the grain boundary effect
does not drastically alter the stress state near the grain boundary as the shape
and orientation of the tensor glyph near the grain boundary are similar to
those from grain interior.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Figure L.1: A subset volume enclosed in an isotropic buffer (rim) is extracted from a
synthetic microstructure consisting of 8000 grains (semi-transparent, 128 ×
128 × 128 grids).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

.

.

xxi

Figure L.2: The deviation due to using isotropic rim with various thickness expressed in

terms of Cauchy stress (left) and strain (right).

. . . . . . . . . . . . . . . . . . 156

Figure L.3: The deviation due to using isotropic rim with various thickness expressed in
terms of Cauchy stress (left) and strain(right) discretized with respect to the
proximity to the interface to buffer (x-axis, 0 means at the interface to buffer
and 0.5 represents the center of the extracted volume).

. . . . . . . . . . . . . 157

Figure L.4: The deviation due to using isotropic rim with various thickness expressed
in terms of Piola-Kirchhoff stress (left) and deformation gradient deviation
(right) discretized with respect to the proximity to the interface to buffer (x-
axis, 0 means at the interface to buffer and 0.5 represents the center of the
extracted volume).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

.

.

xxii

KEY TO ABBREVIATIONS

2D
3D

Two-dimensional
Three-dimensional

3DXRD 3D X-Ray Diffraction Microscopy
APS
Al
ANL
BCC
BI

Advanced Photon Source
Aluminum
Argonne National Laboratory
Body-centered cubic microstructure
Barycentric interpolation strategy

COBYLA Constrained Optimization by Linear Approximation

Coherent X-ray diffraction imaging

CXDI
DAXM Differential Aperture X-ray Microscopy
DCT
DIC
EBSD
ECCI
ESRF

Diffraction contrast tomography
Digital image correlation
Electron backscatter diffraction
Electron channeling contrast imaging
European Synchrotron Radiation Facility

ff-HEDM Far field high energy X-ray diffraction microscopy

FBE
FIB

Frank–Bilby equation
Focused ion beam

Hexagonal close-packed microstructure

FWHM Full Width at Half Maximum
HCP
HEDM High energy X-ray diffraction microscopy
HREBSD high-resolution electron backscattering diffraction
ICP-MS

Inductively coupled plasma mass spectroscopy
Inverse pole figure

IPF

xxiii

Material genome initiative
Micro computed tomography

MGI
µ-CT
µ-Laue Micro-Laue diffraction
nf-HEDM Near field high energy X-ray diffraction microscopy

Nearest interpolation strategy
Representative volume element

NI
RVE
SRXM Scanning reflection X-ray microscopy
SEM
SVD
SVE
SAXS
Sn

Scanning electron microscopy
Singular value decomposition
Statistical volume element
Small-angle X-ray scattering
Tin
Taylor factor gradient enhanced crystal plasticity model
Transmission X-ray microscopy

TGCP
TXM

xxiv

KEY TO SYMBOLS

Diffracted wavevector

Burgers vector
Deformation gradient

Dislocation line direction
Dislocation slip plane normal
Frank–Bilby tensor

b
F
F(cid:63) Deformation gradient reciprocal dual
FD Deviatoric deformation gradient
k
ρi Dislocation density for slip system i
t
n
T
p(cid:63) Hydrostatic (spherical) component of deformation gradient
B0
k0
δi j Kronecker delta
Lattice curvature tensor
ki j
ω Lattice rotation tensor

i jk Levi-Civita operator
α Nye tensor
g

Ideal lattice basis
incident wavevector

Orientation matrix
Principal strain deviators
ei
Principal stress deviators
σi
B(cid:63) Reciprocal lattice basis
q
M Scattering matrix, column space of which are scattering vectors
m Schmid factor
e

Scattering vectors

strain deviator tensor
Streaking vector

ξ

xxv

CHAPTER 1

INTRODUCTION

The study of the deformation mechanisms in polycrystalline materials at the microscale, also known
as micromechanics, becomes more and more critical due to the increasing demand for precise control
of the structural and functional properties of heterogeneous materials at the level of the individual
grains. Many experimental characterization techniques that are designed to characterize different
aspects of material properties at the microscale have been developed to facilitate the research of
micromechanics.

Scanning electron microscopy (SEM) was initially developed to collect information about the
surface topography and composition [1]. When used in conjunction with electron backscatter
diffraction microscopy (EBSD), which provides microscopic crystallographic information, SEM-
based slip trace analysis can be used to identify the active slip systems facilitating the plastic flow
during the deformation at the microscale. The main application of this analysis is to provide valu-
able insights on how polycrystalline materials accommodate heterogeneous deformation through
dislocation activity under different loading environments [2–5]. For the static defects (dislocations
and stacking faults) close to the surface of bulk materials, electron channelling contrast imaging
(ECCI) can be used to provide direct observation of these defect structures [6–8], which serves as
an alternative to the conventional transmission electron microscopy (TEM) based characterization
techniques [9]. Additionally, electron microscopy can also be used to measure the surface strain
field, where the in-plane components of the surface strain tensor are estimated through digital image
correlation (DIC) [10–12], and the out-of-plane components are calculated based on the topography
measured by atomic force microscopy (AFM) for regions with characterizable slip traces [13–15].
Despite the seemingly unlimited potential of electron-microscopy-based techniques, it is funda-
mentally a surface technique due to the limited penetration depth of electrons1. Some researchers
1 The penetration depth of electrons depends on both the target materials and the accelerating
voltage, which often leads to a penetration depth of less than 10 µm with current technology [16].

1

have used serial polishing ( [17–20]) or selective milling with focused ion beam (FIB, [21]) to sys-
tematically remove surface materials such that the subsurface volume can be exposed to electrons
for characterization. However, the destructive nature of this approach limits its applications due to
the loss of sample after characterization.

The X-ray diffraction technique is by nature a non-destructive characterization technique that
suffers similar limitations due to its limited penetration depth in traditional settings. However, the
continuous development of synchrotron facilities in the past few decades provides researchers with
high energy X-rays that can penetrate hundreds of microns into the sample. Consequently, many 3D
characterization techniques utilizing the high energy X-rays were developed to investigate material
properties non-destructively, including high energy X-ray diffraction microscopy (HEDM) [22,23],
differential aperture X-ray microscopy (DAXM) [24, 25], coherent X-ray imaging [26], and micro
computed tomography (µ-CT) [27]. Although these synchrotron X-ray based techniques can provide
valuable information about material properties at the microscale non-destructively, they do require
access to synchrotron facilities, which only provide limited access to the scientific community
for a short period each year. Therefore, the inherent scarcity of synchrotron X-ray data requires
researchers to look for accompanying techniques that can make the most of the limited data available
from synchrotron X-ray characterization.

Computational models have been widely used to assist the research in micromechanics for many
decades [28–31]. During this time, many materials models have been developed to simulate material
behavior under various conditions at different scopes. Among these models, crystal plasticity theory
based models often provide more accurate predictions of local deformation history thanks to their
representative volume element (RVE) design, which makes them relatively easy to apply realistic
boundary conditions to realistic microstructures in the simulation framework [32]. Therefore,
models based on crystal plasticity theory have an inherent synergy with experiment characterization
techniques, making them ideal candidates to accompany the synchrotron X-ray diffraction based
techniques in the pursuit of advancing micromechanics of polycrystalline materials.

The objective of this dissertation is to demonstrate a synergistic approach for utilizing both syn-

2

chrotron radiation techniques and crystal plasticity simulations to further deepen the understanding
of the micromechanics of polycrystalline materials. To this end, a polycrystalline tensile sample
cut from a commercial Ti-5Al-2.5Sn plate was deformed up to ∼4 % tensile strain at ambient
temperature, the undeformed and deformed states of which were characterized with both EBSD
and DAXM. The local deformation process of a small subset of grains from a region of interest was
simulated using crystal plasticity simulations with the 3D microstructure reconstructed from EBSD
and DAXM data. The simulation results were then compared with experimental measurements
derived from EBSD and DAXM data, elucidating the critical factors for improving the accuracy of
predicted kinematic and constitutive responses. To provide a clear demonstration of the synergistic
approach described above (see Figure 1.1), this dissertation is divided into several chapters, the
details of which are listed below.

• Chapter 2 contains a literature review on the recent development of synchrotron X-ray
diffraction based characterization techniques and crystal plasticity theory based models,
along with a summary of the main deformation mechanism in titanium alloys.

• Chapter 3 details the surface and subsurface characterization process using EBSD and DAXM
respectively, along with the sample specification of the Ti-5Al-2.5Sn tensile sample used in
this study.

• Chapter 4 demonstrates two 3D microstructure reconstruction methods, the results of which

are used as the input for crystal plasticity simulations detailed in Chapter 7.

• Chapter 5 provides the fundamentals of the strain quantification process, through which the
(residual) lattice strain can be extracted from micro-Laue diffraction patterns collected during
DAXM experiment.

• Chapter 6 extends the application of an existing single DAXM voxel2 streak analysis from
edge-type GNDs to all GNDs, along with an improved streak analysis method that can provide

2 A DAXM voxel is the smallest characterization unit in DAXM characterization.

3

(misfit) dislocation density profiles for each DAXM voxel.

• Chapter 7 is a case study where the DAXM data is synergistically incorporated into different
stages of crystal plasticity simulations to reveal the critical factors of improving the accuracy
of the simulated kinematic and constitutive responses.

• Chapter 8 provides in-depth discussions of the finding from each chapter, and Chapter 9 sum-
marizes this study, followed by a brief discussion of possible future research opportunities.

Figure 1.1: Schematics of the relationship between differential aperture X-ray microscopy and
crystal plasticity simulation. The grayed-out paths represent topics beyond the scope of this thesis.

4

DAXMCrystalPlasticitysimulationcrystalorientationlatticestraindislocationcontentpatternindexationchp.5strainquantificationstreakanalysischp.6microstructurechp.4reconstructionchp.7calibrationvalidationmodeldevelopmentCHAPTER 2

LITERATURE REVIEW

In this chapter, two of the most common deformation mechanisms for crystalline materials under
ambient temperature are briefly reviewed, followed by a concise summary of modern synchrotron
radiation based characterization techniques that can be used to study the local deformation process.
Crystal plasticity simulations, the accompanying techniques for synchrotron radiation characteri-
zation in this study, are reviewed regarding their kinematic formulation, constitutive description,
and the associated microstructure input. An outline of this dissertation inspired by the reviewed
topics is provided at the end of the chapter, aiming at providing a more in-depth description of the
motivation of this study.

2.1 Deformation mechanisms in crystalline materials

Crystalline materials, particularly metallic materials, are crucial in most aspects of modern
engineering. One unique feature of crystalline materials are their variational malleability, which is
closely tied to the underlying deformation mechanisms. In this section, two of the most common
deformation mechanisms for crystalline materials at ambient temperature, dislocation slip, and
mechanical twinning, are briefly reviewed, followed by a concise description of the polycrystalline
titanium alloy used in this study.

2.1.1 Dislocation slip

Dislocation slip generally refers to the conservative motion (glide) of many dislocations in the plane
that contains the associated line and Burgers vector (Figure 2.1). At ambient temperature, where
non-conservation dislocation motion (climb) is difficult due to low diffusivity and non-equilibrium
point defect densities, dislocation slip is considered to be the primary carrier for plastic flow in
crystalline materials. Dislocation slip is defined by its slip plane and slip direction, the combination
of which is referred to as a slip system. The slip planes are normally planes with particular

5

crystallographic importance such as high atomic density. Similarly, the slip directions are the
direction in a given slip plane that corresponds to the shortest lattice translation vector (highest line
density).

Figure 2.1: Schematic illustrations of plastic deformation induced by the glide of edge (top and
middle) and screw (bottom) dislocations [9].

Conventionally, the activation of dislocation slip is described as the resolved shear stress acting
on the dislocation reaching a particular (critical) level, hence the term critical resolved shear stress.
In modern dislocation theory, the motion of dislocation, often in the form of dislocation slip, is
considered to result from two competing forces: the driving force (resolved shear stress) and the slip
resistance due to the presence of other dislocation structures such as forest dislocations, dislocation
cells, and low angle grain boundaries. Notably, the slip resistance for a single dislocation in an
otherwise perfect crystal, also known as the Peierls (or Peierls-Nabarro) stress is estimated based on
the energy barrier of the dislocation core structure, offering the lower bound of the slip resistance.
Some modern dislocation density based constitutive descriptions were derived from this theory,

6

examples of which are provided in Section 2.3.2.

For perfect dislocation slip, the associated dislocations would eventually either exit the material
or annihilate with other dislocations with opposite Burgers vectors inside the material, leaving no
evidence inside the material other than the macroscopic shape change. However, real dislocation
slip often leaves trails of residual dislocations behind while facilitating the plastic flow, forming
various static (sessile) dislocation structures such as dislocation loops, dislocation arrays, and
dislocation cells. Due to the static displacement field around these sessile dislocations, lattice
curvature naturally occurs around these dislocation structures. If multiple dislocation structures
with similar static displacement field, were concentrated in one location, they could reorangize into
a low angle grain boundary after the plastic deformation. However, if these dislocation structures
were spaced out within a grain, an orientation field can be observed after the plastic deformation.
In 1953, Nye [33] first provided a tensorial quantity to measure the lattice curvature induced by
static dislocation structures,

α ≈ ρ · b ⊗ t,

(2.1)

where α is the the Nye tensor, ρ, b, and t are the associated dislocation density, Burgers vector, and
the dislocation line direction. The corresponding lattice curvature ki j can then be calculated using

ki j = αji − 1

2 δi j αkk,

(2.2)

where δi j is the Kronecker delta. Around the same time, Frank and Bilby [34] provided another
dislocation structure model where the lattice curvature in a given crystal is represented by sets of
misfit dislocations residing in a single misfit plane. These misfit dislocations can be sampled along
an arbitrary direction x,

(N × ξi) · x

· bi = T · x,

(2.3)

where T = I − S−1 is the Frank-Bilby tensor, S is the transformation matrix corresponding to the
lattice orientation gradient, di is the spacing for each of the s sets of dislocations with Burgers
vector bi and line direction ξi, and N is the plane normal of the interface plane. Regardless of the
choice of the dislocation structure model, the net dislocations responsible for the lattice curvature

(cid:20) 1

di

s


i=1

(cid:21)

7

are generally termed as geometrically necessary dislocation (GND), as these dislocations help
the material to accommodate the plastic deformation by introducing small geometrical changes
(orientation change) at the microscale. However, GNDs are often a subset of the total dislocation
content, statistically stored dislocation (SSD), since some sessile dislocations generate opposite
lattice curvatures that cancel each other out. In other words, the GND content becomes increasingly
similar to the SSD content with decreasing analysis scope. Although the theoretical description
of these dislocation structures was established back in the 1950s, the associated characterization
remains challenging due to the inherent small scale of dislocations despite the recent development
in imaging techniques such as TEM and ECCI.

2.1.2 Mechanical twinning

Mechanical twinning generally refers to the formation process of a specific type of crystallographic
structure in a crystalline solid1, where two regions with special orientation relationship are con-
nected through a planar interface [35]. This planar interface is denoted as the twin boundary, which
can be either incoherent or coherent, depending on whether interpenetration occurs between the
parent (un-sheared) and the twin (sheared). Although comprehensive understanding of mechanical
twinning at the microscale is challenging [36–39], the formation of a mechanical twin is generally
considered as a sudden rotation of part of the material due to sufficient amount resolved shear
stress operating on a particular crystallographic plane (twinning plane) along a specific direction
(twinning shear direction). The most well-received theory of mechanical twinning suggests that the
formation and motion of pairs of partial twinning dislocations, one sessile and one glissile, are its
principal actors, along with necessary “atomic shuffling” to maintain the orientation relationships
between the parent and the twinned region [40]. Besides the formation process, mechanical twin-
ning is also distinguished from dislocation slip by the way it facilitates plastic flow (Figure 2.2).
More specifically, collective motion of twining dislocations result in a more homogeneous shear
1 The twin structure observed in crystalline materials can be induced by many external factors,

including thermal, electrical, chemical, and mechanical stimuli.

8

that preserves the specific orientation relationship between the parent and twin, regardless of the
magnitude of the external load, whereas the shear induced by dislocation slip is affected by both the
local dislocation structures and the external load. Furthermore, the direction of the slip-induced
shear can be bi-directional whereas mechanical twinning shear generally has a preferred direction
due to the much higher resistance of anti-twinning [41]. Dislocation slip is usually the primary
carrier of plastic deformation as long as a sufficient amount mobile dislocations is present in the
system experiencing deviatoric deformation above its elastic limit. However, crystals with a limited
number of independent slip systems or low stacking fault energies, such as crystals with hexagonal-
structured lattice, are considered to be prone to twin formation, primarily when the available slip
systems cannot accommodate the external load.

Figure 2.2: Schematics of the effect of different deformation modes (elastic, plastic, twinning) on
the lattice structure [42]. The capital letter is used to refer to the original framework whereas the
corresponding lowercase letter represents the rotated framework due to deformation.

The geometrical characteristics of (mechanical) twinning consist of four elements: K1, η1, K2,
and η2 (Figure 2.3). K1 denotes the habit plane, which is the planar interface separating the parent
and the twined region. The required resolved shear stress direction is labeled as η1, which also
resides in the habit plane. The plane with the normal vector S is defined by K1 and η1, which
contains the planar shear motion that leads to twin formation. The inclined plane denoted with K2
is the second undistorted plane, which represents the reoriented volume due to the simple shear
associated with twinning. The vector η2 is referred to as the conjugate shear direction, which shows
the intercept of the shear plane S and the second undistorted plane K2. In most published work,

9

these four twinning elements are often expressed in Miller index notation. As a convention, the
twinning system said to be type I if the associated {K1, η2} consist of rational numbers. Similarly,
type II twinning system contains rational {K2, η1}. In rare cases, all four elements of twinning can
consist of rational numbers, the associated twinning system of which is denoted as the compound
twinning system.

Figure 2.3: Schematic of the four geometrical elements of twinning (K1, η1, K2, and η2) [35].

2.1.3 Deformation mechanism in Ti-5Al-2.5Sn

The material used in this study, Ti-5Al-2.5Sn, is an α/near-α titanium alloy with a β-transus
temperature range between 1040 ◦C and 1090 ◦C [43]. Similar to pure titanium and other titanium
Ti-5Al-2.5Sn = 4.48 × 103 kg m−3) and has good corrosion
alloys, Ti-5Al-2.5Sn is low density (ρ
resistance [44]. Additionally, Ti-5Al-2.5Sn also has relatively high strength (σyield ≈ 0.8 GPa),
thanks to the solid-solution strengthening effect of tin (Sn) and aluminum (Al). Besides improving
strength through substitutional solid-solution strengthening, Al also serves as the main α-phase
stabilizer for Ti-5Al-2.5Sn. More specifically, the presence of Al in Ti-5Al-2.5Sn increases the
volume fraction of the α-phase as well as the β-transus temperature, stabilizing its mechanical
properties at elevated temperatures. Consequently, this particular titanium alloy is commonly
used in gas turbine engines and aircraft engines.
In addition to high-temperature applications,
Ti-5Al-2.5Sn is also used in aerospace structural parts that operate at cryogenic temperatures, as it
can retain adequate toughness at low temperatures due to its lack of a ductile-to-brittle transition
region [45,46].

10

Similar to other α/near-α titanium alloys, the mechanical properties of Ti-5Al-2.5Sn are pre-
dominantly determined by its main phase (α-phase)2, which has a hexagonal structured lattice3.
There are four primary slip families commonly considered when describing plastic flow induced
by dislocation slip in materials with a hexagonal structured lattice (Figure 2.4). [47] Depending on
the Burgers vector involved, these four slip families can be further categorized into two groups:
<a>-type and <c+a>-type.

−→a

−→a

−→a

−−−→c + a

Figure 2.4: Primary dislocation slip systems for α-phase in Ti and its alloys (from left to right): basal
({0 0 0 1}(cid:104)2 1 1 0(cid:105)) , prismatic ({1 0 1 0}(cid:104)2 1 1 0(cid:105)), pyramidal <a> ({1 0 1 1}(cid:104)2 1 1 0(cid:105)), pyramidal
<c + a> ({1 0 1 1}(cid:104)1 2 1 3(cid:105)). The arrows indicates the Burgers vectors that facilitate dislocation
slip for given slip system.

The <a>-type group consists of three slip families (basal, prismatic, and pyramidal <a>), the
slip systems of which shares the same type of Burgers vector (cid:104)2 1 1 0(cid:105) but different gliding planes
({0 0 0 1}, {1 0 1 0}, and {1 0 1 1}). Due to the crystal symmetry of the hexagonal structured lattice,
the number of crystallographic equivalent slip systems for the three <a>-type slip families are three,
three, and six respectively. Depending on the actual c/a ratio present in the alloy, the activation
energy associated with the critical resolved shear stress (CRSS)4 varies among three families. For
instance, pure titanium has a c/a ratio of 1.587, which results in higher atomistic density in prismatic
2 The secondary phase (β-phase, body-center cubic) is not discussed in this study due to the
small volume fraction (about one precent) of the β-phase present in the Ti-5Al-2.5Sn tensile sample.
3 Traditionally, the metals with hexagonal structured lattice are often referred to as “hexagonal
close-packed” (HCP) metals. However, this is an inaccurate terminology as it is common to have no
close-packed plane in a given titanium alloy due to the non-ideal c/a ratio ((c/a)Ti alloy ≈ 1.587 (cid:44)
1.633 = (c/a)ideal). Hence, the term “hexagonal-close-packed” (HCP) is replaced with “hexagonal
structured lattice” in this study.
4 The term critical resolved shear stress is initially defined for the single crystal that follows the
Schmid law, which assumes all dislocations remain still until the associated driving force, namely
the resolved shear stress, reaches a critical level.

11

T1

T2

C1

C2

Figure 2.5: Mechanical
T1 twinning
({1 0 1 2}(cid:104)1 0 1 1(cid:105)), T2 twinning ({1 1 2 1}(cid:104)1 1 2 6(cid:105)), C1 twinning ({1 1 2 2}(cid:104)1 1 2 3(cid:105)), C2 twin-
ning ({1 0 1 1}(cid:104)1 0 1 2(cid:105)). The arrows indicates the Burgers vector of the dislocation that facilitate
twinning during plastic deformation.

twinning systems in α-phase Ti and its alloy:

planes compared with other planes. Since it is relatively easier for dislocations to move/glide on
a denser plane (low activation energy), the CRSS for prismatic slip system is much lower than
those other slip systems in pure titanium. [48] On the other hand, the substitutional alloying atom,
Al, has a smaller radius than titanium (rAl = 0.125 nm < rTi = 0.176 nm), resulting in a more
densely packed basal plane in Ti-5Al-2.5Sn by increasing the c/a ratio slightly. Therefore, the
CRSS difference between the basal and prismatic slip systems are reduced in Ti-5Al-2.5Sn due to
the decrease of atomistic density difference between the two types of gliding planes [49].

The <c+a>-type slip systems consists of only one slip family with two crystallographically
equivalent <c + a> Burgers vectors and six pyramidal gliding planes, resulting in a total number
of twelve pyramidal <c+a> slip systems. Due to the relatively longer Burgers vector, dislocations
from the pyramidal <c+a> family often have higher activation energy than those of the <a>-type
slip families, resulting in much higher CRSS values observed in experiment [50,51].

Mechanical twinning (Table 2.1) is another essential deformation mechanism for materials with
hexagonal structured lattice. There are four common twinning systems (Figure 2.5) observed in
α-phase titanium and its alloys, the activation requirements of which are closely tied to the stress
state associated with the c-axis. More specifically, the tension (extension) twinning (T1, T2) modes
are activated when the crystal is under tension along its c-axis. Similarly, compression (contraction)
twinning (C1, C2) modes can be activated if the crystal experiences compression along its c-axis.
Although mechanical twinning plays a vital role in the plastic deformation of titanium and its

12

alloys, there is no twinning activity observed in the volume analyzed in this study. Therefore,
analysis regarding mechanical twining was not relevant in the characterization and crystal plasticity
simulations performed in this study.

Table 2.1: Common mechanical twinning models for titanium and its alloys

mode
T1
T2
C1
C2

twinning system misorientation
85°(cid:104)1 2 1 0(cid:105)
{1 0 1 2}(cid:104)1 0 1 1(cid:105)
{1 1 2 1}(cid:104)1 1 2 6(cid:105)
35°(cid:104)1 1 0 0(cid:105)
{1 1 2 2}(cid:104)1 1 2 3(cid:105)
65°(cid:104)1 1 0 0(cid:105)
{1 0 1 0}(cid:104)1 0 1 2(cid:105)
54°(cid:104)1 2 1 0(cid:105)

T1 is the most commonly observed twining events in titanium and its alloys.

twinning shear

0.171
0.629
0.221
0.101

2.2 Synchrotron X-rays diffraction

The discovery of X-rays by Röntgen in 1895 and the subsequent X-ray diffraction experiments
of crystals performed by the Braggs (W.H. Bragg and W.L. Bragg) established X-ray diffraction
based characterization techniques as the foundation of crystallography, a field of studies focusing
on relating the properties of crystalline materials with its structure (unit cell and basis). Various
laboratory X-ray sources (tubes and rotating anodes) have been developed over the years to pro-
vide reliable X-rays, facilitating the development of crystallography, metallurgy, and eventually
materials science.
In the 1970s, it was discovered that the quality of X-ray beams, often mea-
sured in brilliance5, can be increased by about seven orders of magnitude when switching from
laboratory X-ray sources to a synchrotron source (Figure 2.6). These synchrotron X-rays are often
acquired by bending/oscillating high energy charged particles (traveling at relativistic speeds) with
a bending-magnet (BM) or insertion-device (ID, undulator) in a synchrotron facility (Figure 2.7).

5 The brilliance, Bx-ray is a unit defined as

Bx-ray =

(cid:219)nphoton
αAsource f

where (cid:219)nphoton denotes the number of photons emitted per second, α represents the collimation of
the beam (in milli-radian), Asource is the area of the X-ray source (in mm), and f ≈ 0.1 % is the
fixed relative energy bandwidth arbitrarily chosen to represent the average photon energy.

13

The resulting high energy polychromatic X-rays can be further adjusted through various optical
elements along their path before they reach the sample, which allows researchers to develop dif-
ferent characterization techniques that utilize the high brilliance of synchrotron X-ray beams. For
instance, the following synchrotron X-ray based characterization techniques utilize the flexible
configuration of synchrotron X-rays to provide microstructure related properties over a wide range
of length scales.

• Micro computed axial tomography (µ-CT)

Micro computed axial tomography, commonly known as µ-CT, is a 3D imaging technique
that is used in many fields, including biomedical [52], geosciences [53], and materials
science [54]. The fundamental mechanism of µ-CT is similar to the conventional computed
axial tomography (CT), initially developed by Hounsfield [55], where the 3D structure is
reconstructed from radiographic projections of the target over a wide range of projection
angles using the Fourier slice theorem. By using (sub)-micron collimated synchrotron X-
rays, the spatial resolution of the structure reconstructed with µ-CT can reach as small as
1 µm3. However, additional filters (smoothing, noise-reduction, etc.) are often required to
improve the fidelity of the reconstructed structure when processing real-life data.

• Diffraction contrast tomography (DCT)

Diffraction contrast tomography is a near-field X-ray diffraction technique where the projec-
tions of each grain are collected at various projection angles [56–58]. The grain morphology
is recovered using an algorithm similar to µ-CT, and the crystal orientations are acquired by
indexing the recorded diffraction pattern. Since the near-field diffraction signals are used
for both morphological and crystallographic reconstruction, DCT is restricted to unstrained
polycrystalline samples with low plastic deformation to reduce reconstruction error induced
by the overlap and distortion of the diffraction spots.

• Coherent X-ray diffraction imaging (CXDI)

CXDI is a “lensless” imaging technique that is often used to characterize the structure of

14

samples at the nanoscale [26, 59]. The fundamental mechanism of this technique is to use
the inverse Fourier transform to convert the diffraction patterns back to real space image,
analogous to what is done in transmission electron microscopy, where the missing phase
information is often recovered through an iterative approach based on the algorithm originally
developed by Miao [60]. Thanks to the “lensless” nature, this technique does not rely on
complex optical elements, which makes the captured image inherently aberration-free.

• Small-angle X-ray scattering (SAXS)

SAXS is a technique that is designed to capture the statistical representation of the particle
size in a given sample. More specifically, the synchrotron X-rays are first converted into
a monochromatic beam with a monochromator, then collimated using a set of X-ray optics
before hitting the sample. The elastic scattering that occurred while the X-rays traveled
through the sample can be recorded with a position sensitive detector, the resulting diffraction
angles between 0.1° and 10°. The corresponding diffraction patterns can then be used
to analyze the micro/nano-scale microstructure of the sample, which makes it a valuable
characterization technique for biomedical materials, including proteins [61, 62], viruses
[63,64], and DNA [65,66].

Besides the techniques mentioned above, there are two synchrotron X-ray based characterization
techniques that are of particular importance for this study: high energy X-ray diffraction microscopy
(HEDM) and differential aperture X-ray microscopy (DAXM). Since both techniques can provide
detailed crystallographic information about the material at the microscale, a more detailed summary
is provided for these two techniques in the rest of this section.

2.2.1 High energy X-ray diffraction microscopy

High energy X-ray diffraction microscopy (HEDM) is a full-field transmission X-ray microscopy
(TXM) technique developed at sector 1 (1-ID)6 of the Advanced Photon Source (APS) at Argonne
6 A similar technique, 3D X-Ray Diffraction Microscopy (3DXRD) was developed in parallel

at European Synchrotron Radiation Facility (ESRF). [68]

15

Figure 2.6: The continuous increase of the brilliance of X-ray sources developed in the past hun-
dred years, especially since the introduction of synchrotron radiation (1st generation), established
(synchrotron) X-ray as a invaluable radiation source for material characterization [67].

In the standard HEDM configuration (Figure 2.9), the high
National Laboratory (ANL) [69].
energy (≥ 50 KeV) monochromatic X-rays generated from synchrotron radiation are shaped into
a wide and narrow rectangular beam using slits in order to illuminate a slice of the sample while
the sample rotates around the beam plane normal. The resulting diffraction signals from the
illuminated grain ensembles are collected by an area detector, the position of which may vary
depending on the nature of the experiment (far/near-field HEDM abbreviated as ff/nf-HEDM). For
a typical ff-HEDM, the detector is positioned beyond the far-field (Fraunhofer) limit (L ≥ 1 m).
Consequently, the diffraction patterns collected from an ff-HEDM are inherently a representation
of the Fourier transform of the electron density of the illuminated grains. Therefore, diffraction
signals (peaks) from the same grain are crystallographically consistent in the detector space, which
makes it possible to extrapolate the center-of-mass locations and the average crystal orientation by
grouping and indexing associated peaks. Additionally, the radial deviations in the diffraction peak

16

Figure 2.7: A schematic diagram of the structure of a synchrotron facility [67], where high
energy polychromatic X-ray is generated by passing the high energy charged particles (electrons or
positrons) in a storage ring through an undulator (lattice of magnets). The resulting white beam
(polychromatic) can be further adjusted by adding various optic elements, such as a monochromator,
focusing devices, etc., in the optic path before delivered to the sample.

positions can also be used to infer the grain average strain, offering valuable insights into the local
deformation state inside the material [70,71]. In an nf-HEDM setting, multiple detectors are often
positioned within the near field (Fresnel regime, L ≤ 10 mm), encoding the projections of each
grain within the diffraction signals. The illuminated slice is partitioned into grids (voxels) to extract
the microstructure from sets of nf-HEDM diffraction patterns with the help of virtual diffraction
algorithms. By maximizing the overlap between the simulated and measured diffraction patterns,
the microstructure within the illuminated slice can be approximated, along with the associated
crystallographic orientation.

2.2.2 Differential aperture X-ray microscopy

Differential aperture X-ray microscopy (DAXM), also known as micro-Laue diffraction (µ-Laue),
is a scanning reflection X-ray microscopy (SRXM) technique that utilizes the high brilliance and

17

Figure 2.8: Schematic of a HEDM experiment layout (left: far-field HEDM, right: near-field
HEDM) with x-axis along incident beam direction, y-axis perpendicular to the beam direction
within the beam plane, and z-axis normal to the beam plane. [72, 73] The distance between the
sample and the detector, L, can vary depending on the nature of the HEDM experiment (far-field
or near-field).

highly focused polychromatic X-ray beams from synchrotron radiation to characterize the spatial
distribution of various properties related to the microstructure. As shown in Figure 2.9, a highly
focused high energy polychromatic incident beam (Eincident ∈ [7,30] KeV, dincident ≈ 0.5 µm) is
used to probe the subsurface microstructure, generating convoluted Laue patterns from all locations
along the beam direction until the X-ray attenuates. A Pt wire with a diameter of 50 µm7 is used
to block diffracted X-rays from various locations by scanning just above the sample surface, which
isolates diffraction from a particular location along the beam direction. Consequently, the Pt wire,
often referred to as the differential aperture, nominally provides 1 µm resolution along the sample
depth direction, making the DAXM characterization an innate 3D characterization technique with
a voxel probe size of 1 µm3.

After a DAXM experiment is performed, a reconstruction subroutine extracts the diffraction
patterns for each probed voxel through the deconvolution of the associated raw diffraction patterns.
These reconstructed diffraction patterns, termed as micro-Laue diffraction patterns in this study,
are then passed to an indexation subroutine, which identifies the peak position using 2D Gaus-
sian/Lorentzian function. The identified peak positions from each micro-Laue diffraction pattern
are then compared with a reference pattern generated from a strain-free crystal, determining the
7 For microstructures with larger grains, a larger Pt wire or a wedge shaped aperture is often

recommended to avoid reconstruction artifacts.

18

Figure 2.9: Schematic diagram of the DAXM structural microscope layout [74]. The remov-
able/insertable micro-monochromator allows users to switch to monochromatic beam for energy
scan, which is necessary for estimating the hydrostatic component of the lattice strain tensor. The
Kirkpatrick-Baez mirror (K-B mirrors) are used to focus the incident beam to ∼0.5 µm in diameter.
The Pt wire is used to block diffraction signal from a specific subsurface volume, the voxel diffrac-
tion pattern from which will be recovered through the deconvolution of patterns collected by the
CCD detector (charge coupled device) and subsequently reconstructed and indexed.

most probable crystal orientation for each probed voxel. This indexing process provides researchers
with a 3D crystal orientation field of the characterized volume with high spatial (∼1 µm3) and angu-
lar (∼0.01°) resolution, which can be used to reconstruct the corresponding 3D microstructure for
subsequent analysis. Besides mapping the crystal orientation field, it is also possible to estimate the
(deviatoric) lattice strain tensor for each probed voxel by comparing the measured peak positions
with the reference (strain-free) ones [24], the process of which is further examined in Chapter 5.
Furthermore, the geometrically necessary dislocation content in each probed voxel can also be
identified by analyzing the streak direction of the diffraction peaks. This process, termed as streak
analysis in this study, will be further discussed in Chapter 68.

2.3 Crystal plasticity simulations

Crystal plasticity theory is designed to provide an elegant and robust description of both the
plastic anisotropy of polycrystalline materials [75]. The extrinsic aspects of the plastic anisotropy,
often attributed to the texture of the sample, are addressed through (spatial) discretization such
8 Due to the rarity of beam time for HEDM and the fact that only average strain information
is obtained from HEDM, only DAXM characterization was performed for the sample used in this
study.

19

that the constitutive responses from each material point9 can be calculated individually instead
of using a forced uniform strain/stress distribution within each grain, a conventional approach
for early plasticity models derived from the Taylor [28] and Sachs models [76]. The inter-grain
compatibility and the global equilibrium state are achieved through multi-level homogenization,
which also ensures that the local compatibility and equilibrium conditions are satisfied at each
material point. The intrinsic aspects, on the other hand, are addressed by modeling the stress-strain
contributions from each activated deformation mode (dislocation slip and twinning) such that
the plastic deformation can be resolved within the traditional continuum mechanics framework.
Although there are different ways to implement the crystal plasticity theory [77–79], the rest of
this section focuses on the most common realization of crystal plasticity theory, the variational
crystal plasticity framework, which extends the established continuum mechanics framework to
include the plastic deformation [32]. The review of the variational crystal plasticity framework is
described in three parts, the kinematic description, the constitutive description and the acquisition
of the microstructure required as an input.

2.3.1 Kinematic description

The kinematic description of the variational crystal plasticity framework provides a mathematical
representation of the shape change of a given crystal, which is realized through the multiplicative
decomposition of the total deformation gradient F,

F = Fe Fp ,

(2.4)

where Fe is the elastic deformation gradient and Fp is plastic deformation gradient. The elastic part
of the deformation falls back into the classic continuum mechanics domain where the stress-strain
responses can be resolved using generalized Hooke’s law. The plastic deformation, on the other
hand, is resolved by relating the evolution of plastic deformation gradient (cid:219)Fp

(2.5)
9 The term “material point” here refers to the primary unit used to discretize the material, the

(cid:219)Fp = Lp Fp

dimensionality of which depends on the scale of the analysis/simulation.

20

to the dislocation slip through the velocity gradient Lp

Lp = (cid:219)γαmα ⊗ nα,

(2.6)

where (cid:219)γα is the shear rate of slip system α with slip direction mα and slip plane normal nα 10.
It is important to recognize that Equation (2.6) is a first-order approximation of the deformation
gradient induced by edge dislocations, which is expressed as the dyadic product of slip direction mα
and slip plane normal nα. This approximation inevitably attributes all plastic deformation to edge
dislocation movement, indicating that the deformation gradient predicted by the variational crystal
plasticity framework is the lower bound where all screw dislocations are permanently pinned during
the deformation. However, it has been reported through both theoretical [80–83] and experimental
[84–86] research that edge dislocations can exhibit higher mobility during plastic deformation
than its screw counter parts for some materials. Therefore, the kinematic relationship established
in Equations (2.4) to (2.6), albeit its inherent flaws, can provide a meaningful representation of
the plastic shear related to mobile dislocations, which enables the development of different crystal
plasticity models by combining the kinematic description discussed above with different constitutive
descriptions.

2.3.2 Constitutive description

The constitutive description of the variational crystal plasticity framework links the deformation
of the materials (shape change) with the external stimuli, the governing equation of which is often
called the kinetic equations. Generally speaking, there are two main types of constitutive descrip-
tions: phenomenological [87–89] and physics-based [90–93]. The phenomenological constitutive
descriptions are the core of many phenomenological crystal plasticity models, which use state vari-
ables such as critical resolved shear stress and saturation stress to determine the plastic flow at each
material point. For instance, the phenomenological constitutive model used in this study is based
on an exponential formula derived from creep observations. More specifically, the plastic flow at

10 Einstein summation is used here for the implied summation over the superscript α.

21

each material point during the simulated plastic deformation is approximated by assuming an expo-
nential relationship between the shear rate, the resolved shear stress, and the critical resolved shear
stress, the evolution of which is also empirically modeled through another exponential function11.
Due to the abstraction provided by the state variables, the phenomenological formulations offer a
succinct representation of the constitutive relationship of a given system, making it computation-
ally efficient. However, the ignorance of the actual plastic flow carrier, dislocations, renders the
phenomenological constitutive models sensitive to materials, forcing users to calibrate the same
constitutive model for different materials or different deformation modes. Furthermore, the lack
of explicit representation of dislocation evolution in the phenomenological formulations forces the
phenomenological constitutive models to be length-scale independent, making them incapable of
simulating the highly organized and heterogenous patterns of dislocation cells and walls.

The physics-based constitutive models, on the other hand, provide direct representations of
dislocation behaviors such that the evolution of dislocations can be translated into the plastic flow.
The most common approach of realizing this translation is through the Orowan equation,

(cid:219)γα = ραbvα,

(2.7)

where ρα is the (mobile) dislocation density, b is the magnitude of Burgers vector and vα is the
average dislocation velocity for mobile dislocations. Depending on the assumed description of
the dislocation density, there are many different ways to model the evolution of dislocations. For
example, Ma et al. [92, 94] described the mobile dislocation density (ρα) as the combination of
forest dislocations (ρα
F

) and dislocations parallel to slip plane (ρα
P

),

ρα = BT

F ρα
ρα
P,

(2.8)

(cid:113)

22

where T is the temperature in Kelvin and B is a fitting parameter related to the shear modulus. The
associated effective shear stress is then defined as
|τα| − ταpass = |τα| − C
0

for |τα| − ταpass ≥ 0
otherwise

(cid:113)

ταeff =

ρα
F

+ ρα
P

(2.9)



11 More detailed description of this model can be found in Section 7.1.1.

where C is another fitting parameter that is related to the dislocation resistance ταpass. In parallel,
Alankar et al. [95,96] described the mobile dislocation density based on the associated processes:
generation, annihilation, and flux. Therefore, the evolution of dislocation density is defined as

(cid:219)ρα = (cid:219)ραgen + (cid:219)ραann + (cid:219)ραflux,

where

(cid:219)ραgen =

(cid:219)ρα−,gen|¯v−,gen|

¯l−,gen

+

(cid:219)ρα
+,gen|¯v+,gen|
¯l+,gen
+ ρα−R(|¯vα

(cid:219)ραann = −ρα

+| + |¯vα−|)

(2.10)

(2.11)

(2.12)

(cid:115)


where l is the average dislocation segment length and R is the critical capture radii for associated
dislocations. The dislocation resistance is approximated with a Tayler type (latent) hardening,

ταpass = µb

Gαβ ρβ,

(2.13)

β

where µ is the Poisson ratio and Gαβ is the latent hardening matrix. Theoretically, the nearly
countless ways to describe the dislocation content in a given system lead to many different dis-
location density based constitutive models, each one of which contains specific fitting parameters
that require additional model calibration [97–102]. Due to the lack of experiment characterization
techniques that can provide spatially resolved dislocation density directly, these physical-based
constitutive models are often calibrated at the constitutive level through the conventional curve
fitting, making it difficult the assess the validity of the dislocation content predicted from these
models.

2.3.3 Microstructure input

One of the advantages of using models based on crystal plasticity theory is its capability of
applying realistic boundary conditions on realistic microstructures, which enables researchers
to simulate and analyze the spatial heterogeneity of the plastic deformation at the microscale.
Although the first crystal plasticity simulation was performed for a single crystal [89], it was quickly

23

adapted to polycrystalline materials (2D and 3D). However, early polycrystal crystal plasticity
simulations were restricted to either the analysis of statistically representative microstructures or
planar stress-strain analyses of oligocrystalline samples12 due to the limitations of conventional
characterization techniques such as electron microscopy and optical microscopy [103, 104].
In
the early 1990s, crystal plasticity simulations of true 3D microstructure became possible, thanks
to the invention of the serial sectioning technique, which reveals the subsurface material for
characterization by systematic sample polishing [105]. Ever since then, research was conducted
to perfect this technique, leading to the creation of a computational toolkit DREAM.3D, which
specializes in 3D microstructure reconstruction from serial sectioning data [106, 107]. The recent
development of dual-beam microscope systems further improved the serial sectioning technique by
replacing the time-consuming sample polishing with focused ion beam milling, making it possible
to extract the 3D microstructure of a small volume without losing the entire sample [108]. Despite
the popularity and the continuous development of serial sectioning, it is fundamentally a destructive
characterization technique, upon the completion of which the volume of interest is inevitably lost.
Hence, it is strategically necessary to place serial sectioning as the last step in the characterization
pipeline, which limits its application in the field of micromechanics.

In contrast to serial sectioning, synchrotron radiation based characterization techniques are non-
destructive by nature, due to the high penetration depth of synchrotron X-rays. Differential aperture
X-ray microscopy (DAXM), high energy X-ray diffraction microscopy (HEDM), and diffraction
contrast tomography (DCT) are three of the most common examples of the non-destructive 3D
microstructure characterization techniques that utilize the high brilliance of synchrotron X-rays,
the descriptions of which are provided in Section 2.2. Although these synchrotron radiation based
techniques can provide 3D microstructures nondestructively, the financial cost and the overall time
consumption are significantly higher than serial sectioning, which prevents them from becoming
the standard 3D microstructure characterization technique for the microstructure characterization
community.

12 Oligocrystalline samples are ideally polycrystalline samples consisting of columnar grains

with grain boundary planes perpendicular to the sample surface.

24

Up to this point, it was tacitly assumed that crystal plasticity simulations require realistic 3D
microstructures that are faithfully (point-to-point) reconstructed based on the experimental charac-
terization. However, there are situations where the exact stress-strain profile from a given sample
is not as important as the statistical stress-strain responses from samples that are similar to the
one investigated. For such cases where the statistical description of the material is more meaning-
ful than a deterministic description, microstructures reconstructed following a statistical volume
element (SVE) design, also known as stochastic microstructures, are sufficient for the associated
crystal plasticity simulations than using a reconstructed representative volume element (RVE).
Typical stochastic microstructure reconstruction involves collecting 2D or 3D statistics of mor-
phological and crystallographical data, which are then used to build synthetic 3D microstructures
that are statistically equivalent to the samples of interest. For example, two or more orthogonal
observations of two-dimensional (2D) electron backscatter diffraction (EBSD) maps can be used to
generate a 3D stochastic microstructure by populating statistically equivalent ellipsoids within the
volume, which are then grown into grains using Voronoi tessellation [109,110]. These statistically
equivalent ellipsoids are often generated following special formalism, n-point statistics, where the
microstructure is described as a set of distributions of material states such as phase composition,
grain size, and crystal orientations [111].

It is important to recognize that the boundary between a real microstructure reconstruction and
the stochastic microstructure reconstruction has blurred in the past few years with the increasing
popularity of the machine-learning based microstructure reconstruction techniques [112]. In these
microstructure reconstruction techniques, full mapping of 3D microstructures using expensive
characterization techniques such as synchrotron radiation is replaced with strategic partial-mapping
to reduce the total cost. The resulting voids are filled following the general guideline of the stochastic
microstructure reconstruction, creating a hybrid type microstructure that can theoretically offer the
best of two reconstruction schemes.

Another aspect of the microstructure reconstruction that has been neglected so far is its realiza-
tion in practice. Traditionally, the finite element framework is the de facto machinery to implement

25

crystal plasticity models, the process of which often involves translating the constitutive model into
a user subroutine or material procedure file for commercial finite element software. This flexible
model, often termed the crystal plasticity finite element framework (CPFE), makes it possible for re-
searchers to extend the application of crystal plasticity theory from simulating stress-strain response
to mechanical twinning [39,113–115], grain boundary effects [116,117], and phase transformation
related processes [118–121]. The microstructure used in CPFE is often a literal translation of the
microstructure to finite element mesh, along with necessary material states tied to each element.
This realization of microstructure has an intrinsically high computational cost, which hinders its
capability in simulating the deformation history for a large volume. A parallel framework based
on a spectral method was developed to resolve this issue, where the equilibrium state is solved as a
sum of specific “basis functions” using fast Fourier transform (FFT) [122,123]. The microstructure
input required by this method is a voxelated mesh, which offers a more compact data structure than
its counterpart in CPFE.

2.4 Critical assessment of the state of the art

Reliable characterization and analysis of the deformation processes at the microscale is crucial
for advancing the field of micromechanics. To this end, two of the essential deformation mechanisms
for crystalline materials, dislocation slip, and mechanical twinning, were briefly reviewed in this
chapter. Although various optical and electrical microscopy based techniques have been developed
over the years to assist the study of these deformation mechanisms, the associated characterization
processes are either restricted to surface measurements (etch pits, ECCI, HREBSD) or destructive
by nature (TEM, 3DEBSD). Hence, the growing desire for three-dimensional and nondestructive
characterization techniques is leading to the development of modern synchrotron radiation based
methods, which were reviewed in the second part of this chapter. As discussed in Section 2.2,
synchrotron X-ray based characterization techniques can offer rich 3D information with high
spatial resolution (∼1 µm3) about the deformation process of crystalline materials nondestructively.
However, the characterization process itself is often time-consuming and expensive, mostly due to

26

its requirement of a synchrotron facility. Consequently, an accompanying method that can make
the most of the valuable data from synchrotron X-ray characterization is deemed desirable so that
the overall cost for 3D non-destructive analysis of the deformation processes at the microscale can
be brought down to a level feasible for more general application. For instance, a computational
model validated by synchrotron X-ray data can be used to study the local deformation process
without continuous access to a synchrotron facility, provided that the simulated local deformation
process is representative of real materials. Among the many possible choices, models based
on crystal plasticity theory exhibit a promising future due to the elegant and robust description
of the plastic anisotropy of crystalline materials, the details of which were reviewed regarding
the kinematic formulation, constitutive description and the associated microstructure input in
Section 2.3. Despite the rapid development in the field of synchrotron X-ray radiation and crystal
plasticity simulation respectively, synergistic application of both, in the pursuit of micromechanics,
remains a challenging topic due to three missing links.

This first missing link is between the 3D microstructure measured by synchrotron radiation
techniques and the 3D microstructure input required by crystal plasticity simulations. Ideally, a full
field mapping of the 3D microstructure, such as high volume raster scanning, is most desirable for
crystal plasticity simulations. However, the cost of full field mapping using synchrotron radiation,
especially for samples with hundreds of grains, is too high to be practical. Consequently, 3D
microstructures reconstructed from partial synchrotron X-ray characterization are used more often
when investigating the local deformation process with crystal plasticity simulation. Depending on
the quality of the partial synchrotron X-ray characterization and the reconstruction algorithm, the
fidelity of the reconstructed 3D microstructure may vary, which could, in turn, affect the accuracy
of the simulated local plastic deformation history. Therefore, understanding the relationship be-
tween the fidelity of the reconstructed 3D microstructure and the accuracy of the simulated local
deformation process is crucial for the synergistic application of synchrotron X-ray radiation and
crystal plasticity simulations in the field of micromechanics.

The second missing link is between the material descriptions in the crystal plasticity models and

27

the real materials. More specifically, the constitutive models in the crystal plasticity simulations
are often calibrated at the macroscopic level by fitting one or many stress-strain curves with cor-
responding experimental results. In recent years, an advanced calibration process was established
where the simulated surface topography of nanoindentations was compared with AFM measure-
ments to assist the calibration process [51]. Nevertheless, such curve fitting techniques are either
inherently calibrations at the macroscopic level where the material properties are homogenized or
at a relatively smaller scale capturing material properties for single crystals instead of polycrystals.
In other words, it is debatable whether the associated simulation results are accurate enough for
the analysis at the mesoscale or microscale. Therefore, additional validation at the mesoscale
with the help of synchrotron data is required to ensure the simulated local stress-strain profile is
representative for the analysis of the local deformation history.

The third missing link is between the virtual plastic flow carrier defined in crystal plastic-
ity simulation and the actual plastic flow carrier in the investigated materials. As discussed in
Section 2.3.2, a physics-based constitutive description can offer more direct insights into the lo-
cal deformation history at the microscale due to its inherent connection with dislocation density.
Therefore, physics-based crystal plasticity simulations are more suitable for the analysis of plastic
flow carrier, including both dislocation slip and deformation twinning, at the microscale. However,
these physics-based models are often calibrated at the constitutive level due to experimental con-
straints. Consequently, it is necessary to acquire spatially resolved experimental characterization
of dislocation density at the microscale such that the simulated dislocation density evolution can
be validated before being used to interpret the local deformation history at the microscale.

This dissertation aims to address the three missing links discussed above by demonstrating
a synergistic approach of utilizing both synchrotron radiation techniques and crystal plasticity
simulations to investigate the local deformation history of a Ti-5Al-2.5Sn tensile sample deformed
at ambient temperature. The 3D microstructure reconstructed from partial DAXM characterization
is detailed in Chapter 4, which consists of two different microstructure reconstruction methods.
Both versions of the 3D microstructures were used in the study of the effect of 3D microstructure on

28

the simulated local plastic deformation history using a phenomenological crystal plasticity model
(Chapter 7). The simulated mesoscale deviatoric residual lattice stress profile is also compared
with those extracted from the associated DAXM data, elucidating the relationship between the
fidelity of the reconstructed microstructure and the accuracy of the simulated stress-strain response.
The experimental residual lattice stress profile was extracted using a strain quantification method
proposed in Chapter 5, where the inherent limits and the associated noise resistance of the proposed
method are also discussed to evaluate its reliability. Chapter 6 demonstrates two different methods
that can be used to extract dislocation content from DAXM data. The first one extends the
application of an existing single DAXM voxel streak analysis from edge-type GNDs to all types
of GNDs, providing an efficient qualitative method to characterize the dislocation content in each
DAXM voxel. The second method is derived from the Frank-Bilby dislocation structure model,
providing a more comprehensive quantitative description of the dislocation content (the misfit
dislocation density profile) for the analyzed volume.
In-depth discussion of these findings are
provided in Chapter 8, followed by a concise summary and possible future research topics detailed
in Chapter 9.

29

CHAPTER 3

EXPERIMENT CHARACTERIZATION

The polycrystalline sample used in this study was a Ti-5Al-2.5Sn tensile sample, the specifications
of which are described in Section 3.1. The details about the EBSD characterization of the sample
surface, both before and after the tensile experiment are presented in Section 3.2, followed by the
subsurface characterization of the sample using DAXM (Section 3.3).

3.1 Sample specifications

The sample used in this study was a tensile specimen with a gauge section of 1.5 × 3 × 10 µm3
cut from a commercial Ti-5Al-2.5Sn plate with an average grain size of 45 µm (Figure 3.1).
The chemical composition of the bulk material is listed in Table 3.1, which was characterized
using inductively coupled plasma mass spectroscopy (ICP-MS). The bulk material has a near-α
microstructure with a small amount (1 volume fraction) of body-centered cubic (bcc) β-phase
dispersed at boundaries between the hexagonal α-phase (Figure 3.2).
Table 3.1: Chemical composition of the as received Ti-5Al-2.5Sn plate measured using ICP-MS

Element

Weight percent (%)

Ti
92.4

Al
4.7

Sn
2.7

Fe Zn
0.2 0.1

Cl
0.03

3.2 Surface characterization with in-situ SEM and EBSD

Before the tensile experiment, the surface of the tensile sample shown in Figure 3.1 was me-
chanically polished to a finish of 0.06 µm using colloidal silica to facilitate EBSD measurements in
a Camscan 44FE scanning electron microscope (SEM). The bulk material texture was then approx-
imated using EBSD data from the central region of the sample (area of interest), which indicates a
near random texture (maximum of 2 x random) present in the tensile specimen (Figure 3.3)1. After
1 The in-situ tensile experiment and surface characterization, including SEM and EBSD mea-

surements, were performed by former graduate student Dr. Hongmei Li.

30

Figure 3.1: Top view of the dog-bone tensile sample (undeformed) cut from a commercial Ti-5Al-
2.5Sn plate.

Figure 3.2: Secondary electron contrast image at approximately 3.5% deformation of the tensile
sample, showing various linear features such as slip traces and surface ledges. The β-phase (bright)
is scattered among the α-phase matrix.

31

510440.81.5r=2r=4unit:mm2222areaofinterestβ-phaseβ-phaseα-matrixα-matrixtensiledirectiontensiledirectiongrainofinterestgrainofinterestEBSD characterization, the tensile specimen was deformed inside a Carl Zeiss EVO LS25 SEM
using a screw-driven tensile stage (MTI/Fullam, Albany, NY) operating with a constant displace-
ment rate of 0.04 mm s−1 at ambient temperature. The load and displacement data were recorded
by MTESTW data acquisition and control software (Admet Inc., Norwood, MA, USA). The in-situ
test was paused at two strain levels (2 % and 3.5 %) to collect secondary electron images, which
resulted in the stress relaxation observed in Figure 3.4. The final local strain of the area of interest
was estimated as 3.5 % using a coarse digital image correlation (DIC) approach, where distinctive
topographic features were used to evaluate the strain along the tensile direction.

Figure 3.3: EBSD measured pole figures indicate a relatively weak texture, roughly about 2x
randomness.

0.5

1.0

2.0

Figure 3.4: In situ (dotted) and approximated (solid) stress–strain response of the Ti-5Al-2.5Sn
sample. Stress relaxation during SEM image acquisition caused the two intermittent drops in the
dotted curve.

32

00.20.40.60.8100.010.020.030.040.05strainstress / GPaimageacquisitions3.3 Subsurface characterization with DAXM

After the in-situ tensile experiment, the sample was taken to the beamline 34-ID-E at the
Advanced Photon Source (APS), Argonne National Laboratory (ANL) for subsurface character-
ization. A total number of 12 DAXM scans (Figures 3.6 to 3.17) were performed during three
different beamtimes (2011-11, 2012-2, 2012-7), resulting in three sets of DAXM data that were
then used for 3D microstructure reconstruction (Chapter 4), lattice strain quantification (Chapter 5),
and dislocation content characterization (Chapter 6). The standard configuration of the DAXM
characterization at the beamline 34-ID-E, APS was used for all 12 scans, using a single overhead
detector (2048 × 2048 pixels) and a polychromatic X-ray source with the energy range of 7 KeV to
30 KeV (Figure 3.5).

Figure 3.5: Schematic representation of the standard DAXM experiment configuration with single
overhead detector at beamline 34-ID-E, APS.

3.3.1 First set of DAXM scans

The first set of DAXM scans was performed during the third cycle of the beamtime at APS in 2011,
using a step size of 1 µm along both the sample surface (x, y) and the depth direction (z). Due

33

45O~44O409.6mm510.3mmincidentbeam7~30KeVCCDdetectorsamplexyz2048x2048to the weak diffracting signal from the deformed sample, the exposure time was set to 0.75 s per
voxel (about one frame per second), which limited the volume that could be characterized with the
available beam time. Consequently, only four scans were performed around a selected grain where
various slip traces and surface steps/ledges were observed after the in-situ tensile test (Figure 3.2).
Figures 3.6 to 3.9 illustrate the subsurface orientation distribution generated from the four
DAXM scans. These EBSD-like subsurface orientation maps are colored using the IPF colormap
along the tensile direction (y-axis) such that grains with hard orientation can be easily identified
through visual inspection of the red channel of each orientation map. The scattered voxels above
the sample surface are artifacts introduced by the indexing algorithm used at APS, whereas the
similar features near the bottom of each scan (z < −100 µm with zsurface = 0) are the result of
pattern quality degradation due to attenuation.

Figure 3.6: The first DAXM scan across a grain boundary between the grain of interest and one of
its neighbor. The scan position (dashed line) on the surface is a rough estimate due to the uneven
surface topography after the deformation.

3.3.2 Second set of DAXM scans

The first set of DAXM scans focused on characterizing a grain of interest whereas the second set of
DAXM scans focused on characterizing the 3D microstructure of the neighboring grains. During

34

(0001)(1010)(2110)141µmz(ND)y(TD)y(TD)x(RD)40µm40µmAAAABBBBtensiledirectiontensiledirectionFigure 3.7: The second DAXM scan across the grain boundaries between the grain of interest and
two of its neighbors.

Figure 3.8: The third DAXM scan across the grain boundary between the grain of interest and one
of its neighbor.

35

141µmz(ND)y(TD)NNMMMMNNy(TD)x(RD)40µm40µm(0001)(1010)(2110)tensiledirectiontensiledirection141µmz(ND)y(TD)OOPPy(TD)x(RD)OOPP40µm40µm(0001)(1010)(2110)tensiledirectiontensiledirectionFigure 3.9: The fourth DAXM scan across the grain boundaries between the grain of interest and
two of its neighbors. This orientation map has a 45° inclination to the sample surface due to its
scanning path parallel to the x-axis.

the second set of DAXM characterization occurred during the first cycle of 2012 beamtime, a total
number of six DAXM scans were performed2. To improve the overall pattern quality, a longer
exposure time (1 s per voxel) was used for all six scans performed during this cycle3. However, due
to the limited beamtime available, the spatial resolution of the second batch of DAXM scans was
reduced by increasing the surface scanning step size from 1 µm to 2 µm and cutting the probing
depth from ∼200 µm to ∼50 µm with a 1 µm depth resolution.

2 The two DAXM experiments were about three months apart, during which time the sample

was stored in a plastic sample container at ambient temperature.

3 Due to data corruption, one of the six DAXM scans was excluded from this study.

36

199µmyzx(RD)SSTTy(TD)x(RD)40µmy(TD)(0001)(1010)(2110)tensiledirectiontensiledirectionWith the longer exposure time, the noise level of the second batch of DAXM data is signif-
icantly lower than the previous ones (Figures 3.10 to 3.14). However, there are still significant
gaps (no indexing results) scattered across the orientation maps, especially in regions near grain
boundaries. These gaps are most likely due to the deformation induced micro-Laue diffraction
pattern degradation where the indexing algorithm was not able to identify the crystal orientations.

Figure 3.10: Subsurface orientation map with tensile IPF colormap generated from the first of
the second set of DAXM scans, illustrating the inclined subsurface grain boundary between two
neighbors of the grain of interest

37

66µmyzx(RD)y(TD)BBBBAAAAy(TD)x(RD)40µm(0001)(1010)(2110)tensiledirectiontensiledirectionFigure 3.11: Subsurface orientation map with tensile IPF colormap generated from the second of
the second set of DAXM scans, illustrating the subsurface grain morphology of a neighboring grain
(yellow)

Figure 3.12: Subsurface orientation map with tensile IPF colormap generated from the third of
the second set of DAXM scans, illustrating the subsurface grain morphology of the same yellow
neighboring grain along a different scanning path that is perpendicular to the one used in Figure 3.11

38

63µmyzx(RD)x(RD)y(TD)CCCCDDDDy(TD)x(RD)40µm(0001)(1010)(2110)tensiledirectiontensiledirection46µmz(ND)y(TD)EEEEEEEEFFFFFFFFy(TD)x(RD)40µm40µm(0001)(1010)(2110)(2110)tensiledirectiontensiledirectionFigure 3.13: Subsurface orientation map with tensile IPF colormap generated from the fourth of
the second set of DAXM scans, illustrating the subsurface grain morphology of a neighboring grain
(brown)

Figure 3.14: Subsurface orientation map with tensile IPF colormap generated from the fifth of the
second set of DAXM scans, illustrating the subsurface grain morphology of a neighboring grain
located in north-west of the grain of interest

39

69µmyzx(RD)y(TD)(0001)(1010)(2110)HHHHGGGGy(TD)x(RD)40µmtensiledirectiontensiledirection68µmyzx(RD)y(TD)IIIIJJJJy(TD)x(RD)40µm(0001)(1010)(2110)tensiledirectiontensiledirection3.3.3 Third set of DAXM scan

The first two sets of DAXM scans covered the grain of interest as well as most of its neighbors.
However, two small neighboring grains located to the lower right of the grain of interest were left
out. Therefore, a third DAXM experiment consisting of three short DAXM scans was performed
during the second beamtime cycle of 2012, aiming to measure the 3D microstructure of the two
small neighboring grains surrounded by the β-phase. A step size of 1 µm along both the scanning
(y-axis) and the depth direction (z-axis) was used for these scans to capture the fine features in
the small neighboring grains. With beamline upgrades that occurred after the previous second
experiment, a relatively shorter exposure time (0.5 s) was used without sacrificing pattern quality.
The resulting subsurface orientation maps are shown in Figures 3.15 to 3.17. The large volume
above the sample surface (dash line in Figures 3.15 to 3.17) are artifacts introduced by the indexing
algorithm where the subsurface volume was mirrored during pattern reconstruction, resulting in
seemingly meaningful indexing results for air.

Figure 3.15: Subsurface orientation map with tensile IPF colormap generated from the first of the
third set of DAXM scans, illustrating the floating neighboring grain (yellow)

40

z(ND)y(TD)QQQQQ'Q'Q'Q'y(TD)x(RD)40µm40µm(0001)(1010)(2110)tensiledirectiontensiledirectionFigure 3.16: Subsurface orientation map with tensile IPF colormap generated from the second of
the third set of DAXM scans, illustrating the variation of subsurface grain morphology below the
floating grain (yellow)

Figure 3.17: Subsurface orientation map with tensile IPF colormap generated from the third of the
third set of DAXM scans, confirming that the floating grain (yellow) does not extend further under
the sample surface

41

z(ND)y(TD)RRR'R'tensiledirectiontensiledirectiony(TD)x(RD)RRR'R'40µm40µm(0001)(1010)(2110)z(ND)y(TD)UUU'U'y(TD)x(RD)tensiledirectiontensiledirectionUUU'U'40µm40µm(0001)(1010)(2110)CHAPTER 4

3D MICROSTRUCTURE RECONSTRUCTION

DAXM characterization can provide 3D orientation information of the investigated region with a
reasonably high spatial resolution (∼1 µm) nondestructively. Ideally, the 3D microstructure of the
volume of interest could be acquired by raster scanning the entire volume of interest using DAXM.
However, this type of high-fidelity 3D mapping is rarely done using DAXM due to the limited
beamtime available at synchrotron sources1. Instead, it is more strategic to perform partial DAXM
characterization on the volume of interest using serial probing or cross-blade scanning (Figure 4.1),
followed by a manual or automated microstructure reconstruction process to approximate the 3D
grain morphology.

In this chapter, the procedure used to manually reconstruct the first generation 3D microstructure
of the volume of interest is detailed in Section 4.1, followed by an automated method that generates
a voxelated mesh that can be used by the spectral solver. Since the data from a blade scan can
also be used to analyze the lattice strain (Chapter 5) and the dislocation content (Chapter 6), the
microstructure reconstruction discussed in this chapter focuses on using the cross-blade scanning
technique. More information about the 3D microstructure reconstruction using the serial probing
technique can be found in reference [124].

4.1 Manual reconstruction

At the time of the construction of the first generation 3D microstructure, only four DAXM scans
focusing around a single grain of interest were thoroughly reconstructed and indexed. These four
DAXM scans from the first set of DAXM experiment were renamed as H3, H2, H1, and X1 from
the original labels2 for easy identification during the manual reconstruction (Figure 4.2).

1 For example, the typical amount of beam time available at APS is often less than five days per

cycle, and there are only three cycles per year.

2The original scan labels used the surface end node position to distinguish each scan, which can

be confusing during the manual reconstruction.

42

Figure 4.1: Schematic illustration of two partial DAXM characterization technique (cross-blade
scans and serial probing) applied to a synthetic microstructure generated using Voronoi tessellation,
the grain ID of which is represented with color.

Figure 4.2: EBSD map of the area of interest using the tensile IPF colormap, along with the
corresponding SEM image with the surface positions of the four DAXM scans (post deformation).

43

deepsub-surfacevolumenotcharacterizedwithDAXMserialprobingserialprobingcross-bladescanscross-bladescansTTSSPPOONNMMAABBBefore the manual reconstruction, the noise voxels in each DAXM scan were removed through
visual inspection, followed by manual identification of the grain boundaries (Figure 4.3). Then the
four cleaned DAXM scans, along with the surface EBSD map, were composed into a “paper model”
using the 3D model software SketchUp®provided by Google (Figure 4.4). During the composition
of the “paper model”, minor vertical adjustment (along the z-axis) was applied to each scan to
compensate for the rough surface topography induced by the plastic deformation. Based on the
spatial proximity and the crystal orientations identified from EBSD and DAXM characterization,
unique grain IDs were assigned to all grains, the crystal orientations of which are based on DAXM
and EBSD measurements. Since the crystal orientations measured from DAXM were from the
deformed sample and the microstructure reconstructed here was intended for CPFE simulation
of the tensile experiment, the orientations identified through surface EBSD measurements of the
undeformed sample were given higher priority than those determined from DAXM characterization
during the manual crystal orientation assignment. For example, there are five orientation candidates
for the surface purple grain in Figure 4.4, one coming from the surface EBSD measurements and the
other four derived from four DAXM scans, respectively3. The crystal orientation, identified from
EBSD measurement, was selected for this grain instead of the other four. However, there were also
many voxels that were only characterized by DAXM, the undeformed crystal orientation of which
cannot be recovered (Chapter B). Therefore, the average crystal orientation based on all the DAXM
measurements was used to approximate the crystal orientation of the undeformed subsurface grain.

Figure 4.3: DAXM blade scans used for the reconstruction of first generation 3D microstructure

3Due to the intragranular orientation gradient present in both the undeformed and deformed
sample, the average crystal orientations for each grain was used during the crystal orientation
assignment.

44

Figure 4.4: The paper model (center) composed through manual alignment of the surface EBSD
map (top) and the four DAXM blade scans (H1, H2, H3 and X1) using SketchUp®. The blade scan
H1 was obscured by H3 as indicated by the gray dashed line.

Using the coordinate picking tool provided by SketchUp®, the 3D coordinates of the grain
boundary intercepts, denoted as “binding points” in this study, can be extracted from the “paper
model”, along with the associated grain IDs. After all the binding points were recorded, the edges
connecting binding points were directly read off the “paper model”, the collection of which can be
used to infer the grain boundary plane. With the binding points and the associated connectivity
extracted from the “paper model”, MSC.Marc® Mentat 2012 was used to construct each grain
interactively. More specifically, the binding points of a particular grain were first added to the
model, followed by the connecting edges based on the recorded connectivity. Occasionally, extra
binding points and connecting edges were added to avoid extreme concave features. After all
the grain boundary planes were defined, 3D tessellation was performed to populate the volume
enclosed by the grain boundary planes with tetrahedral elements4. This process was repeated for
4The tetrahedron element (Element 127 in MSC.Marc® Mentat 2012) used here has four nodes

45

EBSDH2H1X1H3all 29 grains, resulting in a 3D FEM mesh shown in Figure 4.55.

Figure 4.5: The first generation 3D microstructure (134.5 × 145 × 60 µm) reconstructed from the
paper model (Figure 4.4) using MSC.Marc® Mentat 2012 consists of 29 grains, 11 of which are
subsurface grains. The resulting FEM mesh is discretized with about 50 000 tetrahedral elements,
which have four nodes and ten integration points.

4.2 Auto reconstruction

The manual 3D microstructure reconstruction process allows the user to adjust the shape of
each grain individually, making it possible to generate a full 3D microstructure with only four
DAXM blade scans. However, this process requires intensive human labor and a significant amount
of time while suffering from low reproducibility due to the additional binding points arbitrarily
introduced by the user to avoid unrealistic grain morphology. Therefore, an automated microstruc-
ture reconstruction method was developed to build the second generation 3D microstructure using
all 12 DAXM blade scans that were reconstructed and indexed by the time this reconstruction
and ten integration points.

5 More details about the commands used to generate the 3D FEM mesh using MSC.Marc®

Mentat 2012 can be found in Chapter C of the appendix.

46

commenced.

In this section, the mechanism of the Barycentric interpolation based microstructure recon-
struction method is first introduced using 2D examples, the reconstruction results of which were
compared with those from a traditional approach, which was initially designed for reconstruction
using serial probing data. Then the proposed reconstruction method was applied to the 12 DAXM
blade scans ( Figures 3.6 to 3.17 ) to build the second generation 3D microstructure, which was
used as the input for crystal plasticity simulation detailed in Chapter 7.

4.2.1 Microstructure reconstruction with Barycentric interpolation

The core of microstructure reconstruction from partial DAXM characterization is a set of rules
designed to infer the crystallographic information of the non-characterized voxels from its neigh-
boring environment. The most common practice is to use the nearest interpolation strategy (NI)
where the non-characterized voxels are assumed to have the same crystallographic features as their
most adjacent characterized neighboring voxel. Mathematically, this is equivalent to partitioning
the space using a Voronoi diagram with known voxels as seed points (Figure 4.6). The resulting
partition lines (dashed lines in Figure 4.6) becomes the inferred grain boundaries for the given
microstructure.

Figure 4.6: Schematic demonstration of using nearest interpolation method (Voronoi tessellation)
to infer grain boundaries (dashed lines) using seed points measured by DAXM.

Generally speaking, the NI based automated microstructure reconstruction works well for build-

47

seedpointinferredgrainboundarying microstructures known to have equiaxed grains, due to the natural equiaxed blocky feature in a
Voronoi Diagram. Furthermore, the blocky feature of the inferred grain boundaries can be some-
what improved (softened) using curvature flow based grain growth when necessary, which often
requires trial and error [124]. However, since not all microstructures consist of equiaxed grains, it
is difficult to use the NI based method as a universal tool for microstructure reconstruction. Addi-
tionally, the crystallographic information from DAXM blade scans has high spatial heterogeneity,
where most of the characterization is performed within a relatively small fraction of the total vol-
ume. Consequently, the microstructure reconstructed from blade scans with the NI method often
contains many extreme blocky features that may require many iterations of curvature flow, which
tends to skew the shape of the grain boundaries accurately measured in the blade scan. Therefore,
an alternative approach based on an online 3D model reconstruction algorithm [125] was used
in this study to resolve this issue. In this proposed reconstruction method, the crystallographic
information of non-characterized voxels is inferred from the simplex formed by its N + 1 nearest
characterized neighboring voxels, where N is the dimensionality. More specifically, the volume of
interest is first partitioned based on seed points from the blade scans using Delaunay triangulation
(blue lines in Figure 4.7), followed by a second partitioning step where all the Barycentric (gravity)
centers of each Delaunay triangle are connected to form the inferred grain boundaries (dashed gray
lines in Figure 4.7).6 In practice, the two-step partitioning is approximated by directly calculating
the Barycentric coordinates (λi) of the non-characterized voxel in the simplex formed by the N + 1
nearest characterized neighboring voxel using the linear system below (3D example)

=

,

(4.1)

where (x, y, z) is the cartesian coordinates of the non-characterized voxel and (xi, yi, zi) is the carte-
6 The barycentric coordinate is a geometric coordinate system that describes a given vertex as
the linear combination of a specific simplex. In the context of a triangle, the Barycentric coordinates
are often referred to as areal coordinates, which can be used to specify subdomains.

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

x1

y1

z1
1

x2

y2

z2
1

x3

y3

z3
1

x4

y4

z4
1

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

λ1

λ2

λ3

λ4

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

x

y

z
1

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

48

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) .

x

y
1

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

λ1

λ2

λ3

x1

y1
1

x2

y2
1

x3

y3
1

sian coordinates of its i-th nearest characterized neighboring voxel. For 2D cases, Equation (4.1)
degenerates to

(4.2)

Once λi is known, the crystallographic information from one of the N + 1 vertices with highest
weight is copied to the voxel of interest. If two or more vertices share the same crystallographic
information, their weights (λ) are combined before the evaluation. For example, seed 2 and seed 3 in
Figure 4.8 share the same crystallographic information, which is different from seed 1. When using
BI to reconstruct the grain boundary, a straight line is generated (left in Figure 4.8), partitioning the
space into two individual grains. However, a bent grain boundary with a sharp angle is generated
using NI, which would require a few iterations of curvature flow to reduce the grain boundary
energy (straightening the bent grain boundary).

Figure 4.7: Schematic demonstration of using Barycentric interpolation built on top of the Delaunay
triangulation (blue dashed line) to infer grain shapes (red and blue) using seed points measured by
DAXM. The gray dashed lines are the partition lines generated based on Barycentric coordinates.
Only two grains were drawn here due to lack of sufficient seed points to reconstruct other grains
using Barycentric interpolation.

To evaluate the quality of the microstructure reconstructed from DAXM blade scans using the
BI method, a set of seven mocked 2D blade scans were generated by slicing a surface EBSD map
sampled from the same Ti-5Al-2.5Sn tensile sample used in this study (top left in Figure 4.9). These
seven 2D blade scans (top right in Figure 4.9) were converted to seed points for the reconstruction

49

Figure 4.8: Illustration of reconstructed grain boundaries from three seed points using Barycentric
interpolation (left) and nearest interpolation (right). Two of the seed points (seed 2 and seed 3) share
the same crystallographic information, which results in the difference of inferred grain boundary
(dashed white lines) when using different reconstruction method.

using BI method, resulting in the reconstructed microstructure shown in the bottom left of Figure 4.9.
The same set of seed points were also passed to the NI method, resulting in a slightly different
microstructure shown in the bottom right of Figure 4.9, with some of the differences highlighted
with white arrows in both reconstructed microstructures. Generally speaking, the microstructure
reconstructed using BI method is similar to the one generated using NI, except for the fuzzier and
smoother boundaries, which may be beneficial for reducing the amount of curvature flow required
to achieve realistic-looking grain boundaries.

To make the reconstructed grain boundaries more realistic, a total number of five iterations of
curvature flow approximated using Gaussian blur with σ = 1 pixel were applied to the microstruc-
ture reconstructed using BI. The resulting microstructure is compared with the target (EBSD
measurements) one with the mocked 2D DAXM blade scans overlaid on top. Through visual
inspection, it is evident that grain boundaries reconstructed in the regions that are enclosed/semi-
enclosed by the 2D blade scans tend to have higher quality (• in Figure 4.10), whereas those in the
regions with fewer 2D blade scans tend to be lower (◦ in Figure 4.10). Therefore, it is recommended
that multiple DAXM blade scans should be performed around the grain(s) of interest with different

50

seed1seed1seed2seed2seed3seed3BarycentricinterpolationBarycentricinterpolationseed1seed1seed2seed2seed3seed3nearestinterpolationnearestinterpolationFigure 4.9: Reconstructed 2D microstructure using Barycentric interpolation (bottom left) and
nearest interpolation (bottom right) from mocked 2D blade scan data (top right) extracted from a
surface EBSD map (top left) of the same Ti-5Al-2.5Sn sample used in this study. Some of the
differences in the reconstructed microstructure are highlighted with white arrows.

scanning paths to improve the overall quality of the reconstructed grain morphology.

4.2.2 Reconstruction of second generation 3D microstructure

The second generation 3D microstructure was reconstructed from the 12 DAXM blade scans
(Figures 3.6 to 3.17) using the BI method described in the previous section. Similar to the manual
reconstruction procedure, the spatial position of each set of DAXM scans were adjusted to match

51

surfaceEBSDmapsampledfromtheTi-5Al-2.5SntensilesamplesurfaceEBSDmapsampledfromtheTi-5Al-2.5Sntensilesamplemocked2Dbladescansmocked2DbladescansBarycentricinterpolationBarycentricinterpolationNearestinterpolationNearestinterpolationFigure 4.10: The comparison of the original 2D microstructure and the one reconstructed using
Barycentric interpolation with curvature flow (right), which is approximated using five iterations of
Gaussian blur with σ = 1 voxel. The halo around the grain boundaries are the artifacts of blurring
process due to using pixel color as the crystallographic information to be interpolated. Mocked 2D
DAXM blade scan is overlaid on top of both figures, along with markers to indicate regions with
high (•) and low (◦) reconstruction quality.

the surface EBSD map using the following translation vector

vEBSD = (0.0,0.0,0.0)
vDAXM1 = (690.0,2638.0,−2395.0)
vDAXM2 = (210.0,3375.0,−2384.0)
vDAXM3 = (678.0,2757.0,−2007.0).

These translation vectors were acquired by manually arranging each set of DAXM scans with
respect to the surface EBSD map using the 3D visualization software ParaView®. The resulting
seed point map is shown in Figure 4.11. The crystal orientation of each DAXM voxel is visualized
using both tensile (TD, left) and normal (ND, right) IPF colormaps. The surface EBSD map was
repeated three times along the negative z-direction with 1 µm interspacing to alleviate the amount
of skew on the reconstructed surface due to subsequent curvature flow. A random down-sampling
was applied to the repeated surface seed points, bridging the subsurface DAXM voxels with surface
voxels during the microstructure reconstruction using BI method.

As mentioned in Section 4.1, there are noise voxels in each DAXM scan, due to the errors

52

surfaceEBSDmapsurfaceEBSDmapBarycentricinterpolationwithcurvatureflowBarycentricinterpolationwithcurvatureflowFigure 4.11: Combined seed points (EBSD and DAXM) for the reconstruction of second generation
3D microstructure visualized using two IPF colormaps. Each voxel shown here has a dimensionality
of 1 × 1 × 1 µm3.

occurring during micro-Laue diffraction pattern reconstruction and indexing. These noise voxels
often have random crystal orientations that are different from their corresponding neighboring
voxels. Since the presence of these noise voxels could potentially have a negative impact on the
reconstruction process, a pre-processing of the seed points was performed, which removes the
outlier voxels within 10 µm radius7. The resulting clean seed points (left in Figure 4.12) were then
used to generate a 3D microstructure using the BI method that was then processed with a total
number of four iterations of grain growth with different growth parameters8. The final second-
generation 3D microstructure contains 54 individual grains, along with a layer of air voxels overlaid
on top to approximate the actual free surface9.

4.3 Summary

Microstructure reconstruction is a challenging topic, especially when it comes to reconstructing
3D microstructure from partial information. The two methods presented in this chapter, manual
reconstruction and the Barycentric interpolation based automated reconstruction, are both capable

7 The Python code snippet used to remove the noise voxels can be found in Chapter D.
8 The details about the microstructure reconstruction implementation can be found in Chapter E.
9 The microstructure constructed here is a Voxelated mesh designed for the spectral solver from
DAMASK, which assumes a periodic microstructure. Therefore, a layer of air voxels is necessary
to break the periodicity induced connection between the top and bottom face.

53

131µmy(TD)z(ND)x(RD)140µmnoises270µmtensiledirection(0001)(1010)(2110)TD131µmy(TD)z(ND)x(RD)140µmnoises270µmtensiledirection(0001)(1010)(2110)NDFigure 4.12: The final second generation 3D microstructure (right) reconstructed from the pruned
seed points (left) consists of 54 grains, along with a layer of air voxels overlaid on top to approximate
the free surface. About a quarter of the voxels in the final reconstructed microstructure were made
transparent to showcase the subsurface grain morphology.

of reconstructing a 3D microstructure with realistic grain morphology from DAXM blade scans.
However, both methods still have their drawbacks, which cannot be solved within their domain in
most situations. For example, the direct human interpretation of the seed points prevents the creation
of irregular shaped grains. However, the amount of human labor related to this process limits its
application to the reconstruction of small volumes. On the other hand, the automated reconstruction
method can easily reconstruct the 3D microstructures in a larger volume. However, this lack of
human interpretation of particular cases can sometimes lead to unexpected grain morphologies. For
instance, six DAXM blade scans were carried out around a surface grain of interest, which allows
the automated reconstruction process to generate a realistic looking grain in the volume enclosed
by the blade scans. However, the lack of DAXM characterization in the subsurface neighborhood
tricks the automated reconstruction into assuming that the heavily characterized central grain also
extends into the uncharacterized volume, resulting in the artifacts shown in Figure 4.13.

Supervised machine learning provides an opportunity to take advantage of the best parts of the
two methods. The direct human interpretation can be used as training data to train an automated
reconstruction system to “intelligently” identify the unique situations demonstrated in Figure 4.13,
which could be a more robust and reliable approach for microstructure reconstruction from partial
DAXM characterization.

54

131µmy(TD)z(ND)x(RD)140µm270µmtensiledirection(0001)(1010)(2110)TD100µm15µmy(TD)z(ND)x(RD)140µm200µmtensiledirectionairairFigure 4.13: Grain morphology of the surface central grain (grain of interest) of the second
generation 3D microstructure.

55

25µmy(TD)x(RD)y(TD)z(ND)x(RD)artifactsduetolackofsufficientDAXMscansEXTRACTING LATTICE STRAIN FROM MICRO-LAUE DIFFRACTION PATTERNS

CHAPTER 5

The knowledge of microscale lattice strain at particular locations, such as at crack tips and grain
boundaries, can offer valuable insight about the local stress state and its potential impact on the
local deformation process. Many experimental characterization techniques have been developed
to fulfill this need, such as high-resolution electron backscattering diffraction (HREBSD) [126],
neutron diffraction [127], and synchrotron X-ray based diffraction strategies [128–132]. Among
these techniques, one type of synchrotron X-ray diffraction technique, differential aperture X-ray
microscopy (DAXM) has been used to non-destructively measure the microscale lattice strain
near or at selected locations, such as tin whiskers surroundings [133], crack tips [134], grain
boundaries [135], and dislocation cell walls [136].

In this chapter, the fundamental theory of DAXM based lattice strain characterization, denoted as
strain quantification, is briefly summarized (Section 5.1), followed by a study exploring the inherent
accuracy limits of this technique using synthetic diffraction data (Section 5.2)1. A preliminary study
of the effect of white noise on the accuracy of the extracted lattice strain is provided in Section 5.3,
the implication of which is summarized at the end of this chapter (Section 5.4).

5.1 Fundamental theory of strain quantification

In classic diffraction theory, the basis B0 = [a0,b0,c0] of a strain-free crystal can transform to

a deformed basis

B = F B0

(5.1)

due to a continuous elastic distortion characterized by the deformation gradient F2. For such a
strained lattice, the scattering vectors q(i), where each i = 1,2, . . . , N is associated with a diffracting
1 These two sections were adapted from an article originally published in Scripta, 2018 [137].
2 The lattice distortion measured here is at a much larger scale compared to a single dislocation.
Therefore, the continuous linear description F proposed here is still physically meaningful for
describing the local elastic distortion.

56

plane having Miller indices n(i) = (h k l)(i), are given by the difference between the incident
wavevector k(i)

0 and diffracted wavevector k(i) in reciprocal space as

k(i) − k(i)

0 = q(i) = B(cid:63) n(i)

or expressed in terms of real space quantities3 as

=

(cid:16)2πB-T(cid:17) n(i)
-T(cid:17) n(i)
= F-T(cid:16)2πB0

= 2π(F B0)-Tn(i)

.

(5.2)

Thus, scattering vectors of the distorted lattice can be calculated from their undistorted counterparts
q0, if the deformation gradient F-T = F(cid:63) is known:

q(i) = F(cid:63) B0(cid:63)n(i) = F(cid:63) q(i)
0

or in matrix form M = F(cid:63)M0,

(5.3a)

(5.3b)

where M = [q(1),q(2), . . . ,q(N)] and M0 = [q(1)
0 ,q(2)
0 , . . . ,q(N)
]. Conversely, the deformation
gradient that causes observed deviations between q(i)
0 and q(i) can be uniquely determined from any
set of three linearly independent scattering vectors q(1,2,3)
and their corresponding q(1,2,3) through
0
inversion of M0, i.e.

0

F(cid:63) = M M0

−1

.

In the case of noisy scattering vectors, the deformation gradient can be found from all (three or
more) scattering vectors using least-squares regression (L2)4:

M M0

T = F(cid:63) M0 M0
F(cid:63) =

(cid:16)M M0

T(cid:17)(cid:16)M0 M0

T

T(cid:17)−1

.

(5.4)

times 2π.

3A basis B transforms to reciprocal space as B(cid:63) = 2πB-T, i.e. the inverse of the transpose of B
4 The solution of the least square regression demonstrated here is acquired using pseudo-inverse,

which can also be achieved using SVD.

57

Up to this point, it was tacitly assumed that all scattering vectors have a measured length, i.e., the
incident wavelength that is causing the diffraction is known. However, DAXM characterization
is a white beam technique, and the area detector technology in current use does not identify the
wavelength of one incident photons in each diffraction peak. Without knowledge of the wavelength,
only the direction, but not the absolute magnitude, of scattering vectors can be established. In other
words, the standard DAXM experiment can only characterize which set of parallel lattice planes
are diffracting, but not their associated spacing. Consequently, the unit cell distortion, which can
be expressed as the lattice deformation gradient F extracted from white-beam DAXM data using
strain quantification is only meaningful regarding its deviatoric component:

FD = F/(det F)1
3 .

(5.5)

To acquire the full lattice deformation gradient, additional scans using monochromatic X-rays of
different energies, commonly referred to as “energy scans”, are performed in order to determine the
length of n selected scattering vector(s). During such an energy scan, the occurrence of a particular
peak of interest is carefully monitored while the energy of the incident beam is incrementally
changed. The time required to scan multiple diffraction peaks scales in proportion to n because
different diffraction peaks generally do not share the same narrow window of scanned X-ray energy.
Consequently, in the more typical case of an unknown scattering vector length q(i), the relationship
between a distorted and undistorted scattering vector becomes

(5.6)

where(cid:98)q denotes the unit vector ((cid:12)(cid:12)(cid:12)(cid:12)(cid:98)q(cid:12)(cid:12)(cid:12)(cid:12) = 1) aligned with q, and p(cid:63) and FD(cid:63) correspond, respectively,

individual
correction

factor

to the spherical (or hydrostatic) and deviatoric parts of F(cid:63). Because p(cid:63) becomes obscured by each
unknown scattering vector length q(i), it is impossible to deduce the spherical part of F(cid:63) from
relation (5.6). For the same reason, using the least-squares regression (5.4) on a set of scattering

0 (cid:99)q(i)
q(i)(cid:99)q(i) = F(cid:63) q(i)
FD(cid:63)(cid:99)q(i)
(cid:99)q(i) =(cid:169)(cid:173)(cid:171)q(i)
q(i) p(cid:63)(cid:170)(cid:174)(cid:172)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

0

0

0 ,

58

The function:

U2(cid:16)(cid:101)F(cid:63)(cid:17)

=

+

(cid:34) n

N


i=1

i=n+1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:30)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)q(i)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:19)
(cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)q(i) −(cid:101)F(cid:63) q(i)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:35)(cid:44)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:99)q(i) −(cid:101)F(cid:63) q(i)

N

0

0

vectors, where a part or all are of unit length (such that M = [q(1), . . . ,q(n),q(n+1), . . . ,(cid:100)q(N)] and
To find an approximation(cid:101)F(cid:63) to the solution F(cid:63) of Equation (5.3a) for the most general case

similarly for M0), results in a less exact solution for FD(cid:63) because the individual correction factors
that relate Equation (5.6) to Equation (5.3a) are necessarily dropped.

where (only) n ≥ 0 of the N scattering vectors have measured length, an iterative optimization
scheme (opt) implemented in Constrained Optimization by Linear Approximation (COBYLA) with
a tolerance of 10−14 is used in this study.

(5.7)

is designed to penalize relative deviations between the actually observed scattering vectors and those

predicted from the assumed(cid:101)F(cid:63)5. The square root U is used as fitness function in the optimization
-T should be converted to(cid:102)FD using
scattering vectors are unit vectors (n = 0), the resulting(cid:101)F =(cid:101)F(cid:63)

scheme to fully utilize the underlying machine precision. In the usual white-beam case where all

Equation (5.5)6.

5.2 Inherent accuracy limit of strain quantification technique

It is essential to understand the theoretical accuracy limits for any characterization technique,
especially when the underlining theory relies on an optimization-based algorithm like the one used
for strain quantification. The uncertainty of the lattice strain measured from DAXM experiment
has been equated to the variation (±10−4) observed for the deviatoric strain tensor components
extracted from a nominally strain-free Si single crystal [24]. The source of this uncertainty is often
speculated as the uncertainty of the peak position on the detector [138] or the limited reciprocal

5 Instead of(cid:101)F(cid:63), the better conditioned(cid:101)F(cid:63) − I is selected as the quantity to be optimized.

6 This optimization based strain quantification method is implemented as a member function

for the Voxel class from voxel module provided by the Python package daxmexplorer.

59

Figure 5.1: Schematic representation of the virtual diffraction configuration. The red cap represents
the circular detector covering a 45° detecting range similar to the single detector configuration
commonly used at beamline 34-ID-E, APS.

space coverage due to small detector size [139]. However, the exact source of the uncertainty in the
extracted lattice strain has rarely been discussed, leaving the possibility that other sources, such as
the numerical optimization process, could also contribute to the reported value. In this section, a
direct method based on least-squares regression (L2, Equation (5.4)) and an iterative optimization
based method (opt, Equation (5.7)) are compared regarding their accuracy in extracting (deviatoric)
lattice deformation gradients from virtual DAXM experiments, thus elucidating whether the choice
of method is a pressing concern.

The assessment of the two methods (L2 and opt) was performed using virtual DAXM experi-
ments that simulate the actual beam and detector geometry at beamline 34-ID-E of the Advanced
Photon Source (APS) at Argonne National Laboratory7. The geometry has both the incident beam
k0 (with an energy range of 7 keV to 30 keV) and the overhead area detector with unit normal nCCD
fixed relative to the lab frame, such that

e3 (cid:107) k0,

e2 (cid:107) nCCD,

e1 (cid:107) (e2 × e3) .

7 The Python implementation of this virtual diffraction experiment is provided in Chapter F.

60

virtualcrystalincidentX-raycirculardetectore1e2e3diffractedwavevector(hkl)A large number (9.5 × 104) of virtual single crystals with random lattice orientation were placed

at the origin of the laboratory frame (see Figure 5.1) and distorted by deformation gradients,

F = R U.

Each rotation R is about a random axis with a rotation angle θ chosen from a logarithmically
uniform distribution between 0.001° and 0.1°. The (symmetric) stretch,

(cid:16)(I + 
) + (I + 
)T(cid:17)(cid:14)2

U = (I + 
)sym =

is composed from I (second-order identity tensor) and 
 having nine components that are randomly
sampled from a logarithmically uniform distribution ranging from 1 × 10−5 to 1 × 10−3.
(The
symmetrization of I + 
 biases the expected off-diagonal (shear) components to be slightly larger
than the diagonal (normal) components.)

The virtual detector captures signals diffracting from the single crystal within a 45° cone (see
Figure 5.1). Out of all possible diffracted wavevectors k those with zero structure factor8 or with
Miller indices (h , k , l) exceeding ±20 are excluded, a condition that is typically observed in the
beamline 34-ID-E data. As not every collected diffraction peak can necessarily be successfully
indexed in a real DAXM experiment, N ∈ [3,30] wavevectors are randomly selected out of those
detected, where bias towards lower (h , k , l) mimics the indexing outcome from APS. Furthermore,
a random fraction of n/N of the indexed wavevectors is assigned a known wavelength, such that
the absolute length of their associated scattering vector(s) can be identified.

The N indexed diffraction peaks are termed a “regular micro-Laue diffraction pattern” if all
scattering vectors are of unit length, i.e., n = 0 scattering vector lengths are known, as would be the
case under polychromatic DAXM conditions. In contrast, an “augmented micro-Laue diffraction
pattern” contains n > 0 scattering vectors for which the exact length is identified (in reality by
performing n additional energy scans).

Full (or deviatoric) lattice deformation gradients were extracted from 92 108 augmented (or
5527 regular) micro-Laue diffraction patterns using L2 and opt. The resulting accuracy of lattice

8here based on the face-centered cubic lattice for simplicity

61

Figure 5.2: Reconstruction accuracy of the deviatoric lattice deformation gradient extracted from
regular micro-Laue diffraction patterns (n = 0) with the L2 and opt methods as a function of the
number N of indexed peaks (left), the lattice stretch magnitude ||U|| (middle), and the incremental
lattice rotation angle θ (right). Shaded band in the left diagram corresponds to standard deviation
around median value illustrated by a solid line. Middle and right diagram depict probability density
maps on a logarithmically equi-spaced grid. Distribution of underlying data is presented in the top
and right margin.

strain reconstruction was quantified by the magnitude (Frobenius norm) of the deviation,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101)F − F(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:102)FD − FD

∆F =

and ∆FD =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,

(5.8a)

(5.8b)

between the extracted (full or deviatoric) deformation gradient and the prescribed one.

Figure 5.2 compares the reconstruction accuracy of both methods applied to regular micro-Laue

62

10-1010-810-610-410-2330ΔFD10-510-3||U||N10-1θ/°L2optdiffraction patterns, i.e. under white-beam conditions, as a function of the number of indexed peaks
N, the magnitude ||U|| of the stretch component, and the lattice orientation deviation angle θ. The
direct inversion method L2 produces deviations ∆FD that fall in the same range as the selected
stretch magnitudes 10−5 ≤ ||U|| ≤ 10−3. The deviations show no notable dependence on either
N or θ, but exhibit a strong scaling with ||U||. This direct relation between the amount of lattice
stretch and the accuracy of its determination is understandable, as under white-beam conditions the
individual correction factors in Equation (5.6) are inherently q(i)
0 /q(i) − 1 ≡ 0, while the true value
is bound by ||U||. In contrast, the deviation between prescribed and identified lattice deformation
gradient values reconstructed by the opt method is mostly insensitive to both ||U|| and θ, i.e.,
to the quantities that are to be identified, but is strongly sensitive to N. With increasing N, the
deviation ∆FD rapidly drops to around 10−9 and continues to decrease further, approaching an
inverse proportionality ∆FD ∝ N−1 close to N = 30 (dashed line in Figure 5.2 left). According
to these results, the accuracy achievable for the present 34-ID-E detector size and typical numbers
N of indexed peaks should be better than 10−9, which is well below the accuracy limit of 10−6
to 10−4 asserted by Hofmann et.al. [139] based on an argument of limited coverage of reciprocal
space associated with small detector sizes.

To investigate the influence of detector size on the reconstruction accuracy that is attainable
by the opt method, two virtual detectors larger than the current one installed at beamline 34-ID-E
are imagined such that the detected scattering vectors cover a correspondingly larger solid angle.
While only limited improvement in accuracy to about 10−11 results (Figure 5.3 left), the number
of objective function (Equation (5.7)) iterations expended by the opt method does notably decrease
with increasing detector size (Figure 5.3 right). More specifically, the typical computation time
to optimize the lattice deformation gradient decreases by a factor of about 3 when increasing the
detector cone angle from 45° (34-ID-E detector geometry) to 90°, and by another factor of about
3 when increasing the cone angle further to 180°. The practicality of substantially increasing
the present detector size is, however, debatable given the complications associated with proper
alignment of multiple smaller detectors and current limitations in large-area detector technology.

63

Figure 5.3: Reconstruction accuracy (left) and cumulative distribution of the number of iterations
required (right) to extract the deviatoric lattice deformation gradient from about 104 sets of scattering
vectors using opt with different virtual detector sizes having cone angles of 45°, 90°, or 180° (from
dark to light blue). Computing times were measured for an implementation based on Scientific
Python (scipy 0.19.0) running in single-thread mode on a 14-core 2.4 GHz Intel Xeon E5-2680v4.

Figure 5.4: Reconstruction accuracy of the full lattice deformation gradient resulting from the
L2and optmethods (red and blue) as a function of the number N of indexed peaks and the number
n ∈ [1, N] of peaks with known length, i.e., for which an energy scan is available.

64

10-1210-1110-1010-910-8101NΔFD0122000.02105103iterationscomputingtime/scumulativeprobability10-310-510-710-910-1110-130102030051015202530NnL2optexactsolutionisfoundΔFDetermination of the full lattice deformation gradient based on the augmented micro-Laue
diffraction patterns (white-beam DAXM with additional energy scans, Figure 5.4) demonstrates an
advantage of opt over L2 similarly to the observations in the white-beam case. The opt method
quickly saturates at reconstruction accuracies ∆F < 10−11 for n ≥ 3, while the L2 method is
typically about a factor 103 less accurate—with the exception of n = N. In other words, when
all scattering vectors have known length, such as in the case of high-energy X-ray diffraction
microscopy (HEDM), the L2 method can provide the exact solution, whereas the opt method
consistently provides very accurate estimates with ∆F ≈ 10−12 over a wide range of N and n.9
5.3 Effect of white noise on strain quantification accuracy

The numerical study performed in the previous section established that the uncertainty from
numerical optimization is negligible. Furthermore, the algorithm itself does not require a broad
coverage of the reciprocal space for ideal diffracting conditions (noise-free). However, real micro-
Laue diffraction patterns acquired from DAXM experiments often contain noise from various
sources, resulting in measurement errors commonly denoted as experiment uncertainties. This
experimental uncertainty was reported to be around ±10−4 based on an auxiliary study using a
calibration silicon wafer measured using the standard DAXM configuration at APS [24]. In this
study, the equivalent level of white noise, about one-tenth of a pixel10 on the detector in the standard
DAXM configuration, was introduced into the synthetic micro-Laue diffraction patterns to analyze
its impact on the accuracy of the deviatoric lattice deformation gradient extracted using the strain
quantification technique.

To this end, a quaternion vector,

r(i) = (cos(0.5α

(i)),sin(0.5α

(i))v

(i)
x ,sin(0.5α

(i))v

(i)
y ,sin(0.5α

(i))v

(i)
z )

(5.9)

9 The most commonly adopted strain quantification method for HEDM is slightly different from
the one proposed in this chapter. Instead of using the full scattering vector, the deviation of the
scattering vectors, ∆θ(i) ≈ |q(i)
0 − q(i)| is measured, which leads to the requirement of (at least) six
independent measurements in order to find full lattice strain tensor for a particular material point.
10 The sub-pixel resolution is achieved by the automated peak indexation subroutine developed

at APS.

65

Figure 5.5: Reconstruction accuracy of the deviatoric lattice deformation gradient resulting from
noisy synthetic micro-Laue diffraction patterns using the standard and spherical coverage of the
reciprocal space. A total of 50 peaks were used for both detector sizes (standard and spherical
detector) to isolate the effect of the detector coverage from the number of peaks indexed.

(i)
x , v

(i)
y , v

(i)
z ) and a rotation angle α(i) randomly
was constructed from a random rotation axis ˆv(i) = (v
sampled from a normal distribution with a standard deviation of 0.002°, which is equivalent to the
one-tenth of a pixel uncertainty mentioned above. This quaternion vector r(i) was then used to
perturb an indexed scattering vector q(i), mimicking the white noise observed in the real DAXM
experiment. N = 50 is selected for this preliminary study, such that the effect of reciprocal space
coverage can be isolated from the influence of the number of indexed peaks. The resulting “noisy”
synthetic micro-Laue diffraction patterns11 was then used as input for the same strain quantification
process, the accuracy analysis results of which are shown in Figure 5.5.

Due to the added white noise in the synthetic micro-Laue diffraction patterns, the error in the
extracted deviatoric lattice deformation gradients increased from ∼10−9 to ∼10−4 for the standard
detector, and ∼10−13 to ∼10−5 for the idealized spherical detector. Although the coverage of
reciprocal space (detector size) has limited impact on the inherent accuracy limits of the strain
quantification method, it is evident that it can influence the effect of white noise on the accuracy
of the extracted deviatoric lattice deformation gradient using the proposed method. The results
11 As part of the preliminary study of the effect of white noise on strain quantification quality,

the following analysis was performed for white beam cases only.

66

standardsphericalΔFD10-310-410-510-6in Figure 5.5 suggest that micro-Laue diffraction patterns collected using a spherical detector are
better suited for lattice strain measurements compared to those from a standard detector since the
broader coverage of reciprocal space can significantly reduce the negative impact of white noise.
However, since a spherical detector is difficult to implement due to various issues discussed in the
previous section, an alternative approach inspired by dynamic programming [140] is proposed to
reduce the negative impact of white noise on extracted deviatoric lattice deformation gradients.

Figure 5.6: In the standard DAXM configuration (left, small detector), the accuracy of deviatoric
lattice deformation gradient extracted from synthetic noisy micro-Laue diffraction patterns using
the enhanced strain quantification method can be improved by down-selecting the first M sets of
scattering vectors that are closest to their strain-free counterparts, regardless of the actual lattice
strain level (four levels: 0, 10−4, 10−3, 10−2). Such benefits from the down-selecting are missing
for configurations with a larger detector (right), except for the trivial case (zero strain).

The proposed enhanced strain quantification method assumes that the deviation between the
calculated scattering vector and its strain-free position is statistically representative of the white
noises present in the measurement. In other words, the lattice deformation gradients, which are
often close to the identity, can be more accurately estimated using scattering vectors that are close
to their strain-free positions than using all measured q vectors. A numerical study was performed to
test this enhanced method. In this study, the scattering vectors from the synthetic noisy micro-Laue
diffraction patterns were first sorted according to their deviations from the corresponding strain-free

67

MΔFD10-310-410-4010-310-210-510-61020304050ΔFD10-310-410-510-61020304050010-410-310-2Mpositions. Then, the first M ∈ [4,50] scattering vectors were selected for the strain quantification
process, and the resulting accuracy of FD was shown in Figure 5.6. For the trivial case of zero
lattice strain (black curves in Figure 5.6), the deviation of q(i) from q(i)
0 is the amount of white
noise present in each virtual crystal. Hence, it is evident that the accuracy of FD can be improved
by down-selecting the first M scattering vectors with the least amount of white noise present for
both the standard detector and the spherical detector, as indicated by the checkmark black curves in
both figures. For cases with strained virtual crystals, the deviation between q(i) and q(i)
0 is merely
an approximation of the amount of white noise present. Therefore, down-selecting the first best
M scattering vectors only benefits the cases using the standard detector, whereas cases using a
spherical detector do not benefit from the proposed down-selecting. The relationship between the
detector size and the effectiveness of the proposed method is most likely because a broader spatial
spread of q allows the optimization subroutine to average out most white noises, which is by nature
a more efficient method than the proposed down-selecting. Nevertheless, this preliminary study
shows that it is possible to use a more advanced algorithm to improve the accuracy of the FD
extracted from micro-Laue diffraction patterns collected using the standard detector, which is more
feasible compared to constructing a spherical detector.

5.4 Summary

In this chapter, the inherent accuracy limits of the strain quantification method are explored using
synthetic micro-Laue diffraction patterns. The results indicate that the numerical optimization is
not sensitive to the reciprocal space coverage, and has a minimal uncertainty contribution to the
overall errors in the extracted (deviatoric) lattice deformation gradients (≤ 10−10). Subsequent
preliminary analysis of the white noise effect using synthetic noisy micro-Laue diffraction patterns
reveals that the accuracy of the strain quantification method is profoundly affected by the amount of
white noise present. The numerical study performed in this chapter shows that a 0.02° uncertainty
on the standard overhead detector used at APS can increase the uncertainty of the extracted FD
from 10−10 to 10−4, which is consistent with the reported values [139]. Theoretically, these noise

68

induced uncertainties can be brought down to 10−5 by using a spherical detector covering the entire
(or half) reciprocal space. However, a DAXM experiment setting that is equivalent to a spherical
detector is difficult to obtain due to the limitations in current detector technology. Consequently, an
enhanced strain quantification method is proposed to alleviate the negative impact of the white noise
by down-selecting the scattering vectors according to their deviations from strain-free positions.
The results show a noticeable improvement (about a factor of three) in the accuracy of the extracted
deviatoric lattice deformation gradient, which provides a promising direction to further the strain
quantification method as a trust-worthy technique to measure lattice strain non-destructively.

69

CHAPTER 6

ANALYZING DISLOCATION CONTENT USING STREAK ANALYSIS

The dislocation content is closely tied to the plastic deformation history in polycrystalline mate-
rials, the knowledge of which can further the understanding of the deformation mechanism at the
microscale. To this end, many experimental characterization techniques have been developed to
characterize the dislocation content, including EBSD for GND measurements [141,142], TEM [9]
and ECCI in near-surface regions of bulk samples [143]. Although electron based dislocation char-
acterization techniques can achieve high spatial resolution 10 ∼ 100 nm [144], these techniques are
often limited to near-surface regions, rely on destructive preparation, or image only small volumes.
In contrast, DAXM provides an opportunity to non-destructively characterize the subsurface dis-
location content up to depths of hundreds of microns with a relatively larger diffracting volume of
approximately 1 µm3.

In this chapter, a single DAXM voxel based technique initially developed to characterize edge
GNDs [145] is briefly revisited, followed by a cross-validation study exploring the possibility
of extending its application to characterize screw and mixed-type dislocations1. An alternative
approach based on the Frank-Bilby tensor is proposed in the second half of this chapter, aiming to
address the limitations inherent in the original single voxel based technique.

6.1 Single voxel streak analysis

In the original single voxel base streak analysis [145, 147], the orientation gradient present in
the voxel of interest is assumed to be the result of pure edge-type GNDs. If the dislocation density
of a particular edge-type GNDs is much larger than the rest, the Nye tensor [33] associated with
the orientation gradient within the voxel can be approximated as

1 This section was adapted from an article originally published in Scripta, 2018 [146].

α ≈ ρ · b ⊗ t,

(6.1)

70

where ρ, b, and t are the dislocation density, Burgers vector, and the dislocation line direction of
the dominant edge-type GNDs, respectively. The corresponding lattice curvature tensor, ki j, can
then be expressed in terms of the Nye tensor,

ki j = αji − 1

2 δi j αkk .

(6.2)

In order to make the derivation simple to follow, a slip system based reference configuration is used
for all subsequent derivations, where e1 is parallel to the Burgers vector b, e2 is parallel to the plane
normal n, and e3 is parallel to the dislocation line direction t2. Within the reference configuration
defined above, the Nye tensor and curvature tensor can be further simplified as

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

α ≈

0 0 ρ
0 0 0
0 0 0

ki j ≈

0 0 0
0 0 0
ρ 0 0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

and

respectively.

(6.3)

(6.4)

(6.6)

Considering the relationship between lattice curvature tensor, ki j, and the lattice rotation tensor,

ωi j,

ωi j = −
i jk θk = 
i jk kklel,

(6.5)

where 
i jk is the Levi-Civita operator, the explicit form of the lattice rotation tensor ωi j resulting
from the orientation gradient induced by the dominant edge-type GNDs can be written as

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

ω =

0
−ρe1
0

ρe1 0
0
0
0
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = ρe1

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

1 0
0
−1 0 0
0
0 0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = ρe1 · 
3jk e3 = ρe1 · 
3jkt.

2 For edge dislocations, the dislocation line direction t is normal to the plane defined by the

Burgers vector b and the slip plane normal n, namely t = b × n

71

Let k be the diffraction vector associated with a particular set of diffracting planes, and the peak

streak vector ξ can be found through

ξ = ω · k,

(6.7)

assuming that the diffraction vector is perturbed from its ideal Bragg condition due to the presence
of local lattice orientation gradient. Combine Equation (6.6) with Equation (6.7), the peak streak
vector can be expressed as

ξ = ρx1 · 
3jktk
= ρx1t × k.

(6.8)

(6.9)

Equation (6.9) shows that if the orientation gradient within a DAXM voxel can be attributed to the
dominant edge-type GNDs, the resulting streak direction is almost parallel to the normal of the
plane defined by the t and k of the dominant edge-type GNDs.

This unique relationship between ξ, t , and k shown in Equation (6.9) is the core of the
single voxel based streak analysis, which has been used to characterize dislocation content in many
studies [148–151] despite the strong edge-type assumption. Nevertheless, Bragg’s law states that
Laue diffraction peak shapes are determined by the lattice orientation distribution in the diffracting
volume, but the relationships between the shape of the peak and the dislocation content of the
diffracting volume have rarely been discussed. The objective of the remaining part of this section
is to examine whether the interpretation based on edge dislocation content described above can be
applied to structures with a significant amount of screw or mixed dislocations.

To this end, the same Ti-5Al-2.5Sn tensile sample deformed in uniaxial tension at ambient
temperature (see [152] for details) served as model material. A diffraction spot streak analysis
was carried out on a DAXM data set collected at the beamline 34-ID-E (Advanced Photon Source,
Argonne National Laboratory) and compared to the dislocation content assessed from a TEM foil
extracted with a FEI Helios DualBeam FIB/SEM focused ion beam from the same near-surface
region as studied in the prior DAXM experiment3. Figure 6.1 reveals the spatial relationship
3 This particular DAXM blade scan is from a different region of the same tensile sample used

in Chapters 4 and 7.

72

Figure 6.1: Three-dimensional view (bottom right) of the analyzed grain patch with gray boundaries
on the sample surface and subsurface DAXM scan volume showing the grain structure as colored
points on a ∼1 µm thick plane perpendicular to the surface, with the X-ray microbeam entering
the sample 45° from the surface. The FIB-extracted TEM foil (gray rectangle perpendicular to the
surface) passes through the DAXM scan; a low magnification image of the foil (top left) identifies
the region analyzed by TEM (dotted frame). Backscattered electron image (bottom left, dashed
frame) shows that the surface slip steps of the analyzed (central) grain are consistent with traces
expected for basal slip (blue) or pyramidal (cid:104)c + a(cid:105) slip (yellow) but not necessarily with prismatic
slip (red). Micro-Laue diffraction pattern (top right) originates from the voxel of interest (located
at the center of the larger red sphere close to TEM foil location, roughly 3 µm below the sample
surface), where the streak indicates about 0.5° lattice orientation spread.

between the locations of the DAXM dataset penetrating at an angle of 45° below the surface and
of the TEM foil identified by its edge A–B. One DAXM voxel lying close to the intersection with
the TEM foil and showing distinctly streaked peaks was selected for single voxel based streak
analysis (Figure 6.1 top right). The orientation of the TEM foil was chosen to allow access to a
sufficient number of diffraction vectors to reliably determine the nature of dislocations through g·b
invisibility assessment with a JEOL 100CX II TEM operating at 120 kV in conjunction with line
trace analysis.

The sequence of TEM images shown in Figure 6.2 reveals dislocations arranged in many bands

73

DAXMtraceFIBtracepyramidal<c+a>traceABanalyzedTEMareabasaltraceprismatictrace65µm4µmg

(1 2 1 2)
(0 0 0 2)
(0 1 1 2)
(1 0 1 2)
(1 1 0 1)

a1
•
◦
◦
•
•

(a)
(b)
(c)
(d)
(e)

Visibility
a2
•
◦
•
◦
•

a3
•
◦
•
•
◦

(cid:104)c + a(cid:105)

•
•
•
•
•

◦ invisible • visible

Figure 6.2: Bright-field transmission electron micrographs imaged in two beam conditions, with
marked diffraction vectors g. The top two conditions reveal that no (cid:104)c + a(cid:105) type dislocations
are present in the TEM foil. The bottom three conditions demonstrate that the Burgers vectors of
3[1 1 2 0]. Combining the
the dominant slip systems are specifically a1 = ±1
g · b = 0 and slip trace analysis, the dislocations marked by blue and red arrows in (a) lie on the
(0 0 0 1) plane with Burgers vectors of a3 and a1. The prisms provide the perspective from the
nearest zone axis B from which the foil was tilted a few degrees to obtain a two-beam condition.

3[2 1 1 0] and a3 = ±1

along traces of the basal plane (lower left to upper right), while only one slip band along the trace of
a prismatic plane is apparent, i.e. from upper left to lower right, both being consistent with the blue
and red traces in Figure 6.1 upper left (rotated roughly 60°)4. The dislocation contrast variation
under different diffraction conditions (shown in Figure 6.2) reveals that most of the dislocations in
3[1 1 2 0] (blue arrows in Figure 6.2a), while a smaller number
the basal slip bands have b = a3 = ±1
3[2 1 1 0] (red arrows in Figure 6.2a). In all of the five projections, the
of them have b = a1 = ±1
dislocations marked in blue run almost parallel to their a3 Burgers vector shown in blue on the
hexagonal unit cells, i.e. are of screw character. Similarly, the dislocations marked in red can be
4 The residing planes are estimated with the help of the prism shown in the lower left corner of

each TEM image.

74

B≈[2243](1212)(a)a3a2a1B≈[1120](b)(0002)a2a3a1(0112)B≈[2243](c)a3a2a1(d)1µma3a2a1(1012)B≈[2243]B≈[1120](1101)(e)a2a3a1identified as screw dislocations with a1 Burgers vector. Most of the dislocations in the prismatic
slip band (white arrows) were also identified to have b = a1, hence, would be mobile on the (0 1 1 0)
prism plane, with a smaller number having a3. Compared to the mostly straight dislocations seen
in the basal slip bands, the dislocations in the prismatic band are generally more tangled, which
may result from basal slip intersecting the prismatic slip band. The high (low) frequency of slip
band observations in the TEM foil of basal (prismatic) slip is consistent with the more narrowly
(widely) spaced surface slip traces evident in the SEM image in Figure 6.1.

To assess whether the mostly near-screw dislocation content observed in the TEM foil can be
identified by a (nominally pure edge dislocation-based) analysis of the streaked diffraction peaks,
a DAXM voxel (red sphere in Figure 6.1) that falls close to the TEM foil location was analyzed5.
The upper left part of Figure 6.3 shows the streaked (3 0 3 4) diffraction peak as an example
concentrically surrounded by all theoretically expected streak directions from edge dislocation
content. The three different (cid:104)a(cid:105) Burgers vectors are distinguished by line type (solid, dashed,
dotted) for the basal (blue), prism (red), pyramidal (green) systems, but no line type distinction is
made for (cid:104)c + a(cid:105) systems. The (3 0 3 4) peak streak direction is consistent with the presence of
a3 dislocations on the basal plane (dotted blue line). The observed streak direction is also close
to those expected from edge dislocations on two pyramidal (solid and dashed green lines) as well
as on one pyramidal (cid:104)c + a(cid:105) plane (orange line collinear with dashed green line). Analysis of
additional streaked peaks (shown in Figure 6.3), results in a3 dislocation content on basal planes
(dotted blue) as the only dislocation content consistent with the major streak direction in all three
analyzed peaks. This assessment is consistent with the slip activity anticipated using global Schmid
factors (unidirectional tensile stress state), as indicated by line opacity of the different theoretical
streak directions in Figure 6.3, as well as with the predominant dislocation content found in the
TEM analysis.

The critical aspect of this study is that although the peak streak analysis model is based on
an edge dislocation assumption, the analysis successfully determines the Burgers vectors of the
5 The Python codes used to generate the streak dials are detailed in Chapter G of the appendix.

75

(3 0 3 4)

(3 0 3 4)

(1 0 1 1)

(3 1 4 8)

Figure 6.3: Colored lines on concentric rings show streak directions that theoretically result from
edge dislocation content on basal (blue), prismatic (red), pyramidal (cid:104)a(cid:105) (green), and pyramidal
(cid:104)c + a(cid:105) (orange) slip systems. The upper left image provides an interpretive key. In the other
three images the streak opacity is scaled with global Schmid factor and the streak length is scaled
to reflect the relative projected streak length on the DAXM detector (all have similar lengths). The
streak direction of all three diffraction peaks is consistent with slip on the a3 basal system, which
is one of the three active slip systems identified by TEM analysis (indicated in the top right). Note
that all three (cid:104)a(cid:105)/prismatic plane dislocation variants cause lattice rotation about the same axis (i.e.,
(cid:104)c(cid:105)-axis), resulting in a common streak direction. All three diffraction peaks emanate from the
same DAXM voxel.

76

basalprismaticpyramidal<a>pyramidal<c+a>TEMTEMTEMFigure 6.4: Homogeneously plastically sheared inclusion (along green arrow in middle image) has
geometrically necessary dislocations in an ellipsoid interface and requires a homogeneous shear
stress and lattice rotation to fit it into the non-sheared neighborhood. The geometrical arrangement
of dislocations (screw character in the red box, edge character in the blue box) bridges the same
misorientation.

dominant dislocation content despite them being predominantly screw in character. This can be
rationalized with a thought experiment involving a spherical Eshelby inclusion (Figure 6.4 left) that
is homogeneously sheared into an ellipsoid by uniformly-spaced dislocation loops. To force this
deformed volume back into the originally occupied spherical volume, homogeneous shear stress
as well as a rigid-body rotation are required (Figure 6.4 right). Because this rotation is the same
everywhere within the sheared volume, any part of the interface has the same misorientation from
the surrounding volume. Thus, diffraction from any voxel containing a part of the interface, such
as the red and blue voxels indicated in Figure 6.4 that comprise parent and sheared material, would
result in the same diffraction spot splitting. During and after plastic deformation, such interfaces
are generally not sharp since dislocations are typically not aligned in perfect walls, resulting in
diffraction peaks that will be smoothly streaked according to the local orientation gradient.

According to the edge dislocation model for streaked peaks, edge dislocations with the same
Burgers vector but different line directions (for example an edge dislocation lying on basal versus
prismatic planes) will cause different streak directions. In contrast, screw dislocations do not have
a unique slip plane due to the collinearity of their Burgers vector and dislocation line direction.
Therefore, the model based on pure edge content should not be applicable. Nevertheless, the
results indicate that the Burgers vector of the screw dislocations present in the TEM foil can be
identified with the streak analysis model, and their spatial (planar) arrangement is consistent with

77

the slip planes defined by corresponding edge line directions and Burgers vectors. That is, the a3
dislocations identified by TEM are consistent with the streak analysis for a3 edge dislocations with
(cid:104)0 1 1 0(cid:105) line direction, which are restricted to (0 0 0 1) planes. As illustrated in Figure 6.2, the a3
screw dislocations are well aligned in (0 0 0 1) slip bands, consistent with the plane of the assumed
edge dislocation content. Thus, the analysis is reflective not only of the dislocation Burgers vectors
present, but also their particular planar arrangement, and hence, provides evidence of slip history.
Despite the fact that the majority dislocation content has been correctly identified by the
streak analysis, significant other dislocation content observed in the TEM foil has not, i.e., neither
prismatic nor basal a1 dislocation content resulted in identifiable streaked peaks (Figure 6.3).
Theoretically, if a diffracting voxel contains spatially isolated dislocation variants that each causes
different orientation spreads, distinct diffraction peak streaks should result, which might be analyzed
individually. If the dislocation content is integrated more homogeneously throughout a voxel, the
associated diffraction peaks will be more diffuse, as exemplified in Figure 6.3. Since the most
numerous dislocation variant will dominate the streak direction, while relatively lower numbers of
other dislocation variants will contribute shorter associated streaks, it is difficult to conclusively
deconvolute all but the most dominant dislocation variant from a diffusely streaked peak. A vital
piece of information that cannot be established from peak streak analysis of a single DAXM voxel
is the spatial gradient of the underlying lattice orientation. From the analysis of adjacent DAXM
voxels, the gradient can be obtained based on the orientation difference, making it possible to
calculate the proper mix of interfacial dislocation variants using the Frank–Bilby equation [34].
Thus, in addition to identifying the most dominant dislocation variant, the absolute dislocation
densities required to bridge the local orientation gradient can be found, which will be examined in
the next section.

6.2 Enhanced streak analysis using Frank-Bilby equation

The missing link between the orientation gradient in the reciprocal space and the real space
limits the original single voxel based streak analysis to the semi-quantitative characterization of the

78

dominant dislocation variant only. To address this limitation, an alternative streak analysis strategy
incorporating the Frank–Bilby equation (FBE) is described in this section.

Assuming that a smooth orientation gradient can be found among voxels within the same grain,
it is possible to infer the lattice orientation gradient within a voxel from the neighboring orientation
field. With the approximated in-voxel orientation gradient, the internal dislocation structure for
each voxel can be modeled with the FBE such that the in-voxel geometrically necessary dislocations
are approximated by sets of misfit dislocation arrays residing in an interface plane that corresponds
to the in-voxel orientation gradient [34, 153]. According to the FBE, the Burgers vector b of the
misfit dislocation arrays across an arbitrary direction x can be linked to the Frank–Bilby tensor
T = I − S−1 through

(cid:21)

(N × ξi) · x

· bi = T · x,

(6.10)

where S is the transformation matrix6 corresponding to the in-voxel lattice orientation gradient, di 7
is the spacing for each of the s sets of dislocations with Burgers vector bi and line direction ξi, and
N is the plane normal of the interface plane8.

Using the identity of a triple product

[a · (b × c)] · d = [d ⊗ (b × c)] · a,

Equation (6.10) can be converted into a standard linear system

(cid:20) 1

di

s


i=1

s


i=1

(6.11)

(6.12)

(6.13)

where the line direction in a given FBE framework can be easily identified with

[bi ⊗ (N × ξi)] · x = T · x

1
di

ξi = N × ni,

assuming that all misfit dislocations are constrained in the interface plane. To further simplify the
linear system in Equation (6.12), a new geometrical quantity ci is defined as

ci = (N · ni) · N − ni = (N ⊗ N) · ni − ni,

(6.14)
6 For a relatively small orientation gradient, S is usually close to the pure rotation transformation.
7 The unit of di is the number of Burgers vector between two dislocation lines. For example,
8 All calculations are performed in the coordinate system defined by the reference voxel.

di = 10 indicates the i-th set of misfit dislocations have a spacing of 10|b|.

79

such that Equation (6.12) can be further simplified into

using the triple cross product identity

i=1

(cid:32) s


(cid:33)

· x = 0

(bi ⊗ ci) − T

1
di

a × (b × c) = b(a · c) − c(a · b).

(6.15)

(6.16)

Similar to the original single voxel based streak analysis, the solution to Equation (6.15) is not
unique when more than two possible active slip systems are considered. However, it is possible
to find an estimate close to the lower bound of the true dislocation density by restricting the total
amount of misfit dislocation arrays used to facilitate the lattice curvature. Therefore, the numerical
solution to Equation (6.15) can be found using an optimization/minimization subroutine with the
following objective function

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s


i=1

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + η

s


i=1

F(ρi) =

ρi(bi ⊗ ci) − T

| ρi|,

(6.17)

i

where ρi = d−1
represents the area density of misfit dislocation arrays within the Frank-Bilby
interface plane for slip system i, and η is the Lagrange multiplier that scales the importance of
minimizing the total dislocation density.

An FBE-based streak analysis was conducted for the same slice of DAXM data used in the
previous single voxel based streak analysis to demonstrate its implementation. The selected DAXM
data set consists of 8072 voxels covering a region approximately 137 µm × 236 µm spanning 15
grains (Figure 6.1). Focusing on the same region within the pink grain analyzed in the previous
section, a particular voxel located at the intersection of the FIB trace and DAXM trace (red ball in
Figure 6.1) is considered in the context of 17 neighboring voxels. The interface plane containing
misfit dislocation arrays within each voxel of interest was inferred from the neighboring orientation
field, which was approximated by the collection of discrete crystal orientation measurements of all
the voxels within the region. More specifically, a diffraction peak with an evident streak is selected
as the reference. While scanning through the neighboring voxels along a linear path (colored dashed
lines in Figure 6.5), the center of peaks with the same index, if it exists, are overlaid on top of the

80

selected diffraction peak. If the diffraction peaks along a specific linear scanning path coincide
with the streak direction, the direction of this linear scan is then used to approximate the normal
of the interface plane containing misfit dislocation arrays, because the orientation change along
the selected scan path is similar to the orientation gradient within the voxel of interest. Ideally,
only one diffraction spot is required to identify the normal of the interface plane containing misfit
dislocation arrays for each voxel. However, the DAXM data used in this study is a single blade
scan, which limits the neighboring orientation field to 2D, confining the scan directions (interface
plane normal) to the DAXM scanning plane. In other words, the misfit plane normal identified
using this DAXM data set is a 2D projection of the correct misfit plane normal. Hence, three more
diffraction peaks (Figure 6.6) with visible streaks were selected to go through the identification
process to confirm the identified interface plane normal (yellow path with label 40°).

With the identified interface plane normal, N, along with the Frank–Bilby tensor T calculated
based on the in-voxel orientation gradient9, it is possible to calculate the misfit dislocation densities
using an optimization scheme with Equation (6.17). However, it is necessary to select a proper
Lagrange multiplier η such that stable and physically meaningful total misfit dislocation densities
can be achieved. To this end, systematic testing with η uniformly sampled from 10−5 to 105 in
log space was performed for the voxel of interest. The resulting profiles of η (Figures 6.7 and 6.8)
suggest that η ∈ (10−3
,10−2) is a reasonable range for FBE based streak analysis. Setting η below
the range identified above, the linear system used in the FEB analysis would put too little weight
in reducing the total dislocation density, resulting in unrealistically high total dislocation density.
In other words, a small η (η < 10−3) encourages the optimization subroutine to seek a dislocation
structure that would best minimize ∆T even if it means to use an unrealistically high amount of misfit
dislocations. On the other hand, using a high η (η > 102) would force the optimization subroutine
to reduce the total dislocation density as much as possible, which would effectively deviate the
optimization procedure from its primary purpose, finding a dislocation structure that can minimize
∆T. The proposed range of η offers a right balance between reducing ∆T and maintaining low total
9 The in-voxel orientation gradient is approximated by the two neighboring voxels whose

diffraction peak center located close to the end of the selected peak of interest.

81

Figure 6.5: By tracking the (3 0 3 4) diffraction peak positions while scanning through neighboring
voxels (top) with a 2 µm thick sampling box, a sampling direction that provides an orientation
gradient similar to the one leading to the peak streak is located along the (yellow triangle in the
bottom image). In the bottom image, (3 0 3 4) peaks from neighboring voxels (colored disks) are
overlaid on top of the (3 0 3 4) peak streak from the voxel in the center of the FBE streak analysis
region (white square), along with colored lines representing the scanning path.

82

Y(tensiledirection)Z10µmStreakanalysisregionYZ0º20º40º60º80º100º120º140º160º2µm(tensiledirection)60º40º0º160º120º(3034)Figure 6.6: Various peak shifting paths and peak streaks for other diffraction peaks from the same
voxel of interest.

dislocation density (
 | ρi|), with η ∈ (10−3

dislocation density. Hence, the value of η was selected to be 1 × 10−3 for the FBE based streak
analysis of the voxel of interest according to the numerical study above. The resulting total area
,10−2), is usually around 10−3 Burgers vector per m2,
which is close to the total dislocation density estimated using single wall of edge dislocations for
the given voxel.10

With η = 10−3, the FBE-based streak analysis11 shows that basal slip of a3 is the most active
slip system in the voxel of interest due to its low dislocation spacing, which is equivalent to a
high area density (Figure 6.9). This result is consistent with TEM observations where the basal
slip of a3 is qualitatively identified as the active slip system throughout the TEM foil. However,

10 More details about this estimates can be found in Chapter H.
11 The Python implementation of the FBE streak analysis can be found in Chapter I

83

120º40º60º(3145)60º40º0º120º160º(2135)40º60º(2023)) with respect to η indicates the main cause
Figure 6.7: The per slip system density profile (dα
i
of unrealistic large dislocation density at low η is due to the presence of negative density from
a negative Burgers vector. This configuration provides the linear system with more flexibility in
finding a dislocation configuration that helps reduce ∆T (black line, right axis), especially when
the total dislocation density is unconstrained due to low η. The blue curve labeled d3
corresponds
i
to the basal a3 dislocations observed in the TEM foil.

FBE based streak analysis reported the second most active slip system to be a basal slip of a2
and pyramidal (cid:104)a(cid:105) slip of a3 with the dislocation spacings about one order of magnitude more
significant than the dominant slip system (lower area density), which is different from the FIB-TEM
analysis. More specifically, the basal slip of a1 should be the second most active slip system
according to observation from the FIB-TEM analysis. One possible cause for this misidentification
of the second most active slip from FBE-based streak analysis could be the translation of GNDs
in a given voxel to the associated misfit dislocation arrays that reside in the corresponding misfit
plane. More specifically, the GNDs were computationally “straightened out” and moved into the
misfit plane. During this translation process, some non-reactive GNDs become reactive due to the
presence of the penalty term in Equation (6.17). For example, one can assume originally only a3

84

10410210010210410121010108106104102010210410610810101012di102101100101TTd1id2id3id4id5id6id7id8id9id10id11id12id13id14id15id16id17id18id19id20id21id22id23id24id3iFigure 6.8: The η profile with total dislocation density
 |di| and mismatch of Frank–Bilby tensor

,10−2) is a reasonable choice for the FBE based streak analysis.

∆T shows that η ∈ (10−3

and a1 basal dislocations were present in a given volume, and they were sparsely distributed. The
stable structure above is inevitably altered when all GNDs were moved into the same plane. The
initially separated a3 and a1 basal dislocations are now occupying the same plane, which leads to
the following imaginary dislocation reaction,

(−a3) + (−a1) → a2.

When the area density of a3 basal dislocations is significantly higher than a1 basal dislocations, most
of the a1 basal dislocations are converted to a2 basal dislocations, resulting in the misidentification
of the second most active slip system. Considering that the penalty term is crucial in preventing an
unrealistically high dislocation density, it is difficult to circumvent the uncertainty of the Burgers
vector due to the circular reaction in a hexagonal unit cell.

The other apparent disagreement between FIB-TEM analysis and the FBE-based streak analysis
is the density level of prismatic slip in the given TEM foil. From the FIB-TEM analysis, a band
of dislocations with all three a dislocations were observed lying on prismatic planes (Figure 6.2).

85

104102100102104101100101102103104105106107|di|102101100101TstableregionFigure 6.9: The FBE based streak analysis shows that a3 basal slip system exhibits the lowest
dislocation spacing di (highest area density) among all 24 slip systems, indicating its dominance
for this voxel. The second most active slip systems, a2 basal slip and a3 pyramidal slip, exhibit about
one order of magnitude larger dislocation spacing than that for a3 basal slip. In this dislocation
spacing plot, the slip families are represented in different colors, and the shape of the symbol
represents the sign of Burgers vector ((cid:78): positive, (cid:72): negative)

On the other hand, the FBE-based streak analysis reported very low misfit dislocation density
for all three prismatic slip systems for the voxel of interest (Figure 6.9). Considering that the
sampling size of the TEM foil (4 µm × 10 µm × 0.2 µm) is much larger than the voxel used in the
FBE (1 µm × 1 µm × 1 µm), the low density for prismatic slip reported by FBE is possibly caused
by sampling a small region of the TEM foil without the concentrated prismatic slip band. Further
analysis of the near surface region using FBE strengthens this hypothesis. As shown in Figure 6.10,
the density distribution of the basal slip system is more or less homogeneous in the grain of interest
(highlighted with a black box), which correspond to the uniformly distributed a3 (marked with blue
arrow) and a1 (marked with red arrow) basal dislocations in Figure 6.2. More specifically, the long
dislocation lines with an a1 Burgers vector on the basal plane are evenly distributed, with short
dislocation lines with Burgers vector of a3 on the same basal plane scattered in-between. This kind
of distribution of basal dislocations would result in an evenly shaded dislocation density map. On
the other hand, the concentrated prismatic slip band is sparsely distributed in the TEM foil, which

86

103104105106107108109|di|basala3a3a2a2prismaticpyramidal<a>pyramidal<c+a>SlipSystemsis consistent with the non-uniform distribution of dislocation density reported by FBE-based streak
analysis. This qualitative comparison of the dislocation density between the FBE and FIB-TEM
analysis shows that the FBE-based streak analysis can provide a spatially resolved representation
of the dislocation content without damaging the sample.

DAXM map (IPF, TD)

1. [2 1 1 0](0 0 0 1)
2. [1 2 1 0](0 0 0 1)
3. [1 1 2 0](0 0 0 1)
4. [2 1 1 0](0 1 1 0)
5. [1 2 1 0](1 0 1 0)
6. [1 1 2 0](1 1 0 0)

di/b

101

104

Figure 6.10: Subsurface dislocation spacing (|di| in Burgers vector) maps illustrate the basal and
prismatic slip activity for the grain of interest, along with the associated DAXM orientation map
(top row). The density distribution shown here is extrapolated using e−r2, which leads to the
artifact (black blocks) where no DAXM data is available. The relatively uniform distribution
of basal dislocations in the central grain suggests strong basal slip activity, which is largely in
agreement with the TEM analysis (Figure 6.2).

Thus, a FBE based streak analysis can provide a reasonable description of the spatially resolved
dislocation content in a deformed sample, when compared with FIB-TEM analysis. For slip systems
that operate on the same slip plane, the assumption of the minimum total dislocation length used
in the FBE based streak analysis prevents it from accurately extracting the Burgers vector for the
second most active slip system. It is also worth mentioning that the dislocation spacing maps in
Figure 6.10 are not an entirely accurate representation of the real dislocation content inside the
sample due to the lack of sufficient DAXM data to form an actual 3D orientation field for FBE
based streak analysis.

87

6.3 Summary

In this chapter, the per-voxel-based orientation gradient extracted from micro-Laue diffraction
patterns collected from DAXM experiment was used to estimate the geometrically necessary
dislocation content in each voxel. The original single voxel based streak analysis was validated
against FIB-TEM, the results of which also extend the application of the single voxel based streak
analysis to more than simple edge-type GNDs. However, the inherent disconnection of real space
and reciprocal space representations of the orientation gradient restricted its characterization to be
semi-quantitive at best. To overcome this limitation, a Frank-Bilby equation (FBE) based streak
analysis was proposed and validated using the same FIB-TEM analysis. The preliminary results
indicate that the FBE based streak analysis is a promising technique for characterizing dislocation
content in 3D non-destructively.

88

CHAPTER 7

DAXM ASSISTED CRYSTAL PLASTICITY SIMULATIONS

Detailed knowledge of the deformation process at the micro-scale is critical for furthering the
understanding of the micromechanics of polycrystalline materials. Due to the limitations in ex-
perimental characterization techniques, research on micromechanics of polycrystalline materials
often relies on computational models with different material descriptions [154–157]. Among these
computational models, crystal plasticity theory based models spatially discretize (representative)
volumes of microstructure and, thus, can directly apply realistic boundary conditions to realistic
microstructures [32, 90, 158–160]. Because of this more physically based structure, models based
on crystal plasticity have the potential to provide more accurate predictions of local deformation
history, as demonstrated by some studies [161–164].

In this chapter, a phenomenological power-law based crystal plasticity simulation implemented
in a finite element framework is performed to evaluate the effect of realistic 3D microstructure on
the simulated kinematic (lattice reorientation) and constitutive (stress-strain evolution) responses1.
The same phenomenological power-law based constitutive description is also used in a follow-
up study performed in a crystal plasticity fast-Fourier transform framework (CPFFT, DAMASK),
where the simulated deviatoric residual lattice stress in a relaxed Ti-5Al-2.5Sn tensile sample is
directly compared with those extracted from DAXM using the strain quantification detailed in
Chapter 5. The comparison between the simulated and measured crystal orientations and residual
lattice stress state provides valuable insights on the micromechanics of polycrystalline materials,
inspiring a pseudo nonlocal crystal plasticity model detailed in the appendix, Chapter K.

1 The sections related to the first generation 3D microstructure were adapted from an article

originally published in International Journal of Plasticity, 2015 [152]

89

7.1 A phenomenological power-law based crystal plasticity model

7.1.1 Constitutive model description

The phenomenological power-law based constitutive model used in this chapter follows the estab-
lished continuum mechanical framework of elastoplasticity at finite strain, as summarized in [32],
with a multiplicative decomposition of the total deformation gradient into elastic and plastic parts

F = Fe Fp

(7.1)

where Fe represents the elastic stretching and rotation of the crystal lattice and Fp denotes the
deformation gradient associated with dislocation slip. The elastic Green–Lagrangian strain tensor
can be calculated from the elastic stretching of the crystal lattice, i.e.

(cid:16)Fe

T Fe − I(cid:17)

Ee =

1
2

,

(7.2)

where I is the second rank identity tensor. The second Piola–Kirchhoff stress tensor, as the
work-conjugate to the Green–Lagrangian strain tensor, is defined through

S = C Ee = Fe−1(det Fe) σ Fe

-T

,

(7.3)

where C is the elastic stiffness tensor and σ is the Cauchy stress tensor. The evolution of the plastic
deformation gradient can be defined as

(cid:219)Fp = Lp Fp.

(7.4)

The plastic velocity gradient Lp is caused by dislocation slip and twinning activity and is described
by

Lp = (cid:219)γαmα ⊗ nα.

(7.5)
In the formula above, (cid:219)γα is the shear rate of slip system α with slip direction mα and slip plane
normal nα; summation is taken over the slip systems α = 1, . . . , Nslip. The shear rate due to
dislocation flow, (cid:219)γα, is defined as

(cid:219)γα = (cid:219)γ0

(7.6)

sgn(cid:0)τα(cid:1)

(cid:12)(cid:12)(cid:12)(cid:12) τα

sα

(cid:12)(cid:12)(cid:12)(cid:12)n

90

with a reference shear rate (cid:219)γ0 = 10−3 s−1, resolved shear stress τα = S · (mα ⊗ nα) of slip
system α, and corresponding critical resolved shear stress sα. The exponent n = 50 represents
the stress sensitivity (inverse rate sensitivity) of dislocation slip. The evolution of the saturation
stress for dislocation slip, i.e. work hardening, is central to the material model. The hardening rule
implemented in this study is defined as

(cid:18)

(cid:19)a

(cid:12)(cid:12)(cid:12)(cid:219)γ β(cid:12)(cid:12)(cid:12) ,

qαβ

(cid:219)sα = h0

1 − sα
ss

(7.7)

where h0 is the reference hardening parameter for dislocation slip interaction, ss is the saturation
stress that is taken as constant for each slip system family, parameters a = 1/3, qαα = 1, qαβ = 1.4
control the latent hardening, and summation is taken over β = 1, . . . , Nslip.

7.1.2 Constitutive model calibration

The parameters of the constitutive model were determined through an established curve fitting
technique [122], where the simulated global stress-strain curve was fitted to the measured stress-
strain behavior under the same boundary condition (uniaxial tension). A stochastic microstructure,
which consists of a periodic geometry discretized into a regular 10 × 10 × 10 grid, was used in
this calibration process. The texture measured by EBSD2 was discretized using 8000 distinct
orientations following the method detailed in reference [165]. This orientation population was
distributed in groups of eight orientations per grid point and homogenized assuming isostrain
conditions (i.e., Taylor assumption) at each grid point.

The calibration procedure minimizes the cumulative difference between the measured and
simulated uniaxial von Mises stress-strain curve up to a strain of 4 %. The minimization strategy is
based on an simplex algorithm [166] and implemented as a Python script that adjusts the constitutive
parameters and reruns the simulation until the stress deviation integrated along the strain falls below
the tolerance of 1 MPa.3

same material but measured by X-ray diffraction.

2 This (weak (cid:104)1 0 1 1(cid:105) fiber) texture differs markedly from the texture reported in [4] for the
3 The Python implementation of this calibration process can be found in the appendix, Chapter J.

91

This calibration procedure was performed eight times, starting with different initial sets (sim-
plexes) of constitutive parameters resulting in the shaded band of Figure 7.1. No unique (global)
optimum was observed, but each of the eight starting populations converged to a different set of
optimized constitutive parameters that all produce a reasonable approximation of the stress-strain
response of the tensile test. Among the eight outcomes, one set of constitutive parameters, denoted
as PS1 (parameter set 1), was selected based on previous statistical analysis of the critical resolved
shear stresses (CRSS) on the same sample by [50], where prism slip is slightly more easily ac-
tivated than the basal slip. To study the influence of constitutive parameters on the outcome of
CPFE simulations, a second set, termed PS2, of constitutive parameters, which have the largest
Euclidean distance from PS1 in the parameter space, was selected as an exemplary comparison
group. Figure 7.1 presents the von Mises stress-strain curves of both PS1 and PS2 in comparison
to the experimental measurement. The CRSS range and hardening slope of PS1 and PS2 are shown
in Figure 7.2. The parameters of set 2 reverse the relative ease of prism and basal slip and thus
provide an interesting computational experiment that assesses the significance of the CRSS ratios
between the two most easily activated slip systems.

7.2 Study of the effect of realistic 3D microstructure

7.2.1 Microstructure reconstruction

The 3D microstructure used in the original study published in 2015 [152] is the first-generation
3D microstructure, the reconstruction process of which is detailed in Section 4.1 of Chapter 4. A
quasi-3D microstructure with columnar grains was constructed as the control group to demonstrate
the effect of using a realistic 3D microstructure that contains many subsurface grains. More specifi-
cally, the quasi-3D microstructure was constructed by extending the surface grain morphology (Fig-
ure 7.3a) into the third dimension, yielding 18 columnar grains in a volume of 134.5×145×60 µm3
that was meshed by about 40 000 tetrahedral elements (Figure 7.3c). A geometrical quantity κ was
introduced to quantify the similarity between the 3D microstructure and its approximation by the
quasi-3D microstructure. This measurement κ is defined as the fraction over which the surface

92

Figure 7.1: Experimental stress–strain curve (dashed) compared to simulation (solid) resulting from
optimized values of variable constitutive parameters listed in Figure 7.2. The initial population
of constitutive parameter sets results in stress-strain responses within the shaded range. After
Nelder–Mead (simplex) optimization, each population member produces stress-strain responses
very similar to the exemplary cases PS1 and PS2. The observed effective hardening range results
from the (nonlinear) superposition of hardening ranges per individual slip family illustrated in
Figure 7.2.

orientation at position x, y extends into the depth

κ(x, y) =

d(x, y)

h

,

where d(x, y) is the depth at which the surface orientation changes due to the presence of a
subsurface grain and h is the overall depth of the modeled volume. The resulting map of the
structure approximation error is shown in Figure 7.3e. Each white-shaded grain is columnar in
both geometries (compare Figure 7.3c and d). The darker the shade, the earlier the surface grain
orientation is replaced by a subsurface grain along the depth (z) direction. It turns out that within a
volume fraction of about 0.6 the same lattice orientations (i.e., identical grain structure) are found
in the 3D microstructure and its quasi-3D approximation.

93

00.20.40.60.8100.010.020.030.040.05experimentstrainstress / GParange of initial guessesPS1PS2effectivehardening rangeoptimizedFigure 7.2: Visualization of the initial (s0, (cid:78)) and saturated (ss, (cid:72)) critical resolved shear stresses
(CRSS; vertical range is between (cid:78) and (cid:72) as s0 → ss) and hardening slope (h0) for constitutive
parameter sets PS1 and PS2. Both PS1 and PS2 are an outcome of the calibration process and
have volume-averaged stress-strain behavior that fits well with most of the experimental reference
as shown in Figure 7.1.

7.2.2 Crystal plasticity simulation configurations

This study of the effect of using realistic 3D microstructure was carried out in a crystal plasticity
finite element (CPFE) framework. More specifically, the actual simulations using the 3D and quasi-
3D microstructure shown in Figure 7.3 were carried out using the commercial FEM solver Marc
2012 (MSC Software Corporation, Newport Beach, CA 92660, USA), along with a phenomenolog-
ical power-law module from the Düsseldorf Advanced Material Simulation Kit (DAMASK, [167])
used as a user subroutine, the theory of which is detailed in Section 7.1.1. Since very little twinning
activity was observed in the sample and none in the modeled volume, four slip families (basal,
prism, first-order pyramidal (cid:104)a(cid:105) and (cid:104)c + a(cid:105)) with total 24 slip systems, were used as the sole
carrier of plastic flow.

A uniform single lattice orientation is assumed for all grains in both microstructures, although
various studies have shown that the initial intragranular misorientation can affect the crystal re-
orientation during plastic deformation [90, 158]. The assumption above is consistent with the
fact that the bulk material is known to be recrystallized [4] and with the EBSD analysis of the

94

10-11101PS1PS2s0, ss and h0 / GPabasalprismpyramidal〈a〉h0〈c+a〉(a) Surface EBSD map used to generate quasi-3D
microstructure.

(b) Surface EBSD map plus four DAXM maps ar-
ranged according to their 3D relationship.

(c) Exploded grain structure and corresponding quasi-
3D microstructure of columnar grains based on (a).

(d) Exploded grain structure and corresponding 3D
microstructure developed from (b).

(e) Fractional extent of the surface grain orientation
into the depth. White shade indicates constant orien-
tation through the thickness (i.e., a columnar grain).
The darker the shade, the closer a differing grain ori-
entation is to the surface.

Figure 7.3: Grain construction based solely on surface EBSD data of the undeformed sample (a
and c) compared to reconstruction using additional subsurface DAXM scans (b and d). Note the
central grain on the surface (purple) in the 3D microstructure is much thicker on one side (positive
y) than the other due to the existence of a subsurface grain (green) right beneath it (see also e).

95

undeformed material that shows that about 85 % of the grains have a maximum grain orientation
spread below 0.5°, which is close to the precision of standard Hough transformation-based EBSD
measurements [168].

Two different boundary conditions corresponding to two extreme cases (Figure 7.4) were used
to approximate the loading experienced by the modeled microstructure volume during the uniaxial
tensile test. For the “soft” boundary condition (Figure 7.4a), the constraint of the neighboring grains
was assumed to be negligible such that all faces except the two loading faces were unconstrained
during the deformation. The “hard” boundary condition (Figure 7.4b), on the other hand, assumed
that the influence of the neighboring grains was strong enough to force all faces except the top
free surface, to remain planar during the deformation. The realistic loading environment of the
microstructure volume should fall in between these soft and hard boundary conditions. For both
boundary conditions, the displacement of the node at (0,0,0) was partially constrained in the
transverse (x-axis) and vertical directions (z-axis) to provide numerical stability.

planar

load

free

z

x

y

(a) “Soft” boundary condition with four uncon-
strained faces.

(b) “Hard” boundary condition with only one
unconstrained face (the surface).

Figure 7.4: Schematic representation of two boundary conditions representing the extreme cases
of the loading environment of the microstructure volume. In the soft boundary condition, all faces
except the two normal to y are unconstrained. In the hard boundary condition, all faces except
the top (surface) are forced to remain planar during deformation. For both boundary conditions,
all nodes on the two y faces have prescribed displacement to approximate the tensile loading, and
one node in the corner was constrained in the transverse (x-axis) and vertical directions (z-axis) to
provide numerical stability.

In summary, the quasi-3D and 3D microstructures (see Figure 7.3) were deformed to 4 %
tensile strain under two alternative boundary conditions (soft and hard, see Figure 7.4) and two
alternative sets of constitutive parameters (PS1 and PS2, see Figure 7.1). A total number of eight
cases (two grain-morphologies under two different boundary conditions with two sets of constitutive

96

parameters) were evaluated so that the influence of the grain morphology, boundary conditions, and
constitutive model on the kinematic and constitutive response of the CPFE model can be assessed
systematically. The simulation results were analyzed with respect to lattice reorientation and stress
response. The simulated lattice reorientation data was used to identify the best combination of the
three factors above by comparing with the EBSD measurement. The simulated stress data were
analyzed at both macro (volume-average) and micro scale.

7.2.3 Effects on lattice reorientation

The crystal orientation of a given material point changes with plastic deformation from its initial
orientation O0 ∈ SO(3) to O. The resulting change in orientation (reorientation) is ∆O = O O0
T,
which can be conveniently transformed into a Rodrigues vector or quaternion to extract the lattice
reorientation angle θ. Surface maps of the lattice reorientation angle at 4 % simulated tensile strain
are shown in Figure 7.5 for the two different grain morphologies using both constitutive parameter
sets and the hard and soft boundary conditions. The EBSD-based measurement at 4 % tensile strain
is also included at the bottom of Figure 7.5 to facilitate the comparison of the kinematic responses
of the eight different simulation conditions with experiment.

The eight simulated lattice reorientation maps (upper two rows in Figure 7.5) show magnitudes
of rotation similar to those in the EBSD-based map (i.e., comparing maximum and minimum
brightness levels).4 However, there are significant differences in the details of the in-grain rotation
between the eight simulations and the EBSD lattice reorientation map. A distinct feature observable
in all four quasi-3D simulations is a band of larger reorientation that runs across multiple grains from
about the lower left to the upper right corner. This feature is not present in the EBSD measurement.
The EBSD map shows a distinct band of larger reorientation running almost vertically across the
central grain. This feature also appears in the simulations when using the 3D microstructure
(left two columns in upper two rows of Figure 7.5), but is entirely missing in the quasi-3D
4The maximum measurement error is less than 2°, based upon prior experience in obtaining the
same orientation in the same location on the same specimen mounted in the same microscope stage
by the same person a few days apart.

97

3D

quasi-3D

PS1

PS2

PS1

PS2

soft
BC

hard
BC

← EBSD map

∆O 0

20°

Figure 7.5: Simulated lattice reorientation maps for patches with different boundary condition,
grain morphology, and constitutive parameters compared to the experimental reference.

microstructure simulations (right two columns in top two rows of Figure 7.5). This relationship
between the lattice reorientation band and the grain morphology indicates that the simulated crystal
reorientation is notably affected by the morphology of the grains. A similar outcome was obtained
in a strictly computational model where the shapes of grains but not the neighbor orientations
were changed [169]. The fact that the reorientation band is present with two different boundary
conditions and two different parameter sets suggests that using microstructures with realistic grain
morphology may be as important as having an advanced constitutive model.

Furthermore, a closer inspection of the first two columns of Figure 7.5 reveals that the simulation
with the hard boundary condition and constitutive parameter set PS1 produces the closest match
to the EBSD map. The fact that simulation with the hard boundary condition produced a better
prediction of the lattice reorientation is expected since the selected volume is in the middle of the
sample where only its surface (top face) is truly free. In other words, the more constrained boundary

98

condition used in this study is more realistic than a simple uniaxial tension boundary condition.
Similarly, the relationship between the choice of the constitutive parameters and the quality of the
simulated lattice reorientation can be explained by the fact that PS1 is selected based on additional
statistical analysis of the same sample, such that the CRSS values for each slip system are closer to
reality than with PS2.

It is clear from this qualitative analysis that grain morphology, boundary conditions, and the
constitutive parameters affect the kinematic response of the CPFE simulation. In addition to the
qualitative analysis above, point-wise differences ∆θ(x) of the lattice reorientation were calculated
between corresponding members of groups of four shown in Figure 7.5 to quantify the influence of
these factors on the kinematic response of the simulation. For example, to quantify the correlation
between the choice of boundary condition and the simulated lattice reorientation, each of the four
maps in the first row is subtracted from the map below it such that boundary condition is the
only variable. Then the frequency of absolute lattice reorientation |∆θ| is binned every 2° and
normalized by the total number of data points. The resulting histograms for these four comparisons
are shown in the leftmost diagram in Figure 7.6, where the blue curves with different line styles
represent the reorientation change due to switching the boundary conditions with other conditions
fixed as labeled. Similarly, the middle two histograms in Figure 7.6 present the effect of changing
between the two grain-morphologies and the two constitutive parameter sets (black and red curves,
respectively).

It is seen from the four histograms of ∆boundary condition (|∆θ|hard−soft), that the majority of
the lattice reorientation changes are less than 4°. Nevertheless, high lattice reorientation changes
(larger than 4°) do exist in various cases with different constitutive parameters and grain morphology.
The general trend of the four curves of ∆boundary condition shows that changes in the boundary
condition do not lead to substantial lattice reorientation changes in the sample surface. A similar
conclusion can also be drawn for the grain morphology (black) and constitutive parameters (red)
curves. However, in the blue curves with “3D” labels and in the black curves with the “hard”
boundary condition, there are significant fractions of the populations with higher |∆θ| values that

99

Figure 7.6: Histograms of point-by-point difference in lattice orientation at the surface when altering
the boundary condition (column 1), microstructure (column 2), or constitutive model parameters
(column 3). The averaged influence of each factor on simulated lattice reorientation is compared
in the last histogram (column 4), which shows that a change in the boundary condition or grain
morphology would more likely lead to substantial change in the simulated lattice reorientation
than changing the constitutive parameters. The change of lattice reorientation ∆θ shown here is in
degrees.

extend to the right side of the histogram plots, indicating that there are more significant differences
in the lattice reorientation than in the quasi-3D simulations. The trend described above implies
that the 3D microstructure is more sensitive to the boundary condition change than the quasi-3D
microstructure. Also, the hard boundary condition is more sensitive to the effect of changing the
morphology. In contrast, the change in the constitutive parameters has a mild effect on the simulated
lattice reorientation, as all the curves are in the left half of the graph. Furthermore, there is no clear
correlation between the different morphologies or boundary conditions for the different constitutive
parameters, suggesting that lattice reorientation is mostly insensitive to the constitutive parameter
values. This point can be emphasized by averaging the curves in each of the three plots, where the
overall effect of each factor on the simulated lattice reorientation is compared in the rightmost plot
in Figure 7.6. Altering the boundary condition or grain morphology causes a similar and more
significant change in the lattice reorientation than the constitutive parameters.

7.2.4 Effects on stress response

The previous section discussed the effect of boundary condition, grain morphology, and constitutive
parameters on the lattice reorientation, a kinematic response. In this section, the influence of these

100

10-610                                                            20PS1,3DPS2, 3DPS1,quasi-3DPS2,quasi-3D||hard-softfractionboundary condition/ degree0                                                            20soft,PS1soft,PS2hard,PS2hard,PS1||3D-quasi3Dmorphology/ degree020soft,3Dhard,3Dhard,quasi-3Dsoft,quasi-3D||PS1−PS2constitutive parameters/ degree0                                                           20 morphology constitutive boundary condition/ degreefactors on the volume-averaged and the spatially resolved stress-strain response are examined.

Figure 7.7 presents the volume-averaged von Mises stress-strain curves for the columnar and
three-dimensional grain morphologies simulated using both constitutive parameter sets and both
boundary conditions. Recalling that the two constitutive parameter sets are calibrated against the
same stress-strain curve by homogenizing 8000 orientations, it is expected that the differences
between curves for the simulated aggregates of 20–30 grains should not be significant. The lack
of significant differences between curves for the same boundary condition indicates that the effect
of grain morphology and constitutive parameters is limited. This lack of substantial difference
may, in part, be a result of the series of calibrations. However, a comparison of curves with the
same constitutive parameters (same color) and grain morphology (same line style) in Figures 7.7a
and 7.7b reveals that the hard boundary condition systematically leads to a higher stress response
compared to the soft boundary condition. This higher stress response reflects the additional
constraints imposed by the hard boundary condition. Thus, the influence of the boundary condition
on the simulated volume-averaged stress-strain responses is stronger than grain morphology and
constitutive model after calibrating the model.

(a) Soft boundary condition

(b) Hard boundary condition

Figure 7.7: Von Mises stress–strain responses of patches for soft (a) and hard (b) boundary
condition. Blue and red (or initially higher and lower) curves reflect constitutive parameter sets
PS1 and PS2, respectively. Solid and dotted lines correspond to 3D and quasi-3D (columnar) grain
morphologies, respectively.

The relative importance of proper grain morphology and constitutive parameters on the sim-

101

00.20.40.60.800.010.020.030.040.05strainstress / GPaQuasi-3D (PS1)Quasi-3D (PS2)3D (PS1)3D (PS2)00.20.40.60.800.010.020.030.040.05strainstress / GPaQuasi-3D (PS1)Quasi-3D (PS2)3D (PS1)3D (PS2)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) +

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σA

i j(x)

i j(x) − σB
i j(x)
2 σA

i j(x)

i j(x) − σB
i j(x)
2 σB

(7.8)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)



ulated mechanical response of polycrystalline aggregates on the stress-strain response discussed
above is subtle. To evaluate the influence of the three factors on the local constitutive responses in
the CPFE simulations, an analysis similar to the one used for lattice reorientation was performed for
the in-plane components of the surface stress tensor at a strain of 4 %. To quantify the point-wise
difference in stress between any two cases, termed A and B, an ‘average relative change’ for each
stress component was calculated as:

avg. rel. change(cid:16)

σi j(x)(cid:17)

= sgn(cid:169)(cid:173)(cid:171) σA

i j(x)
i j(x)
σB

(cid:170)(cid:174)(cid:172)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σA



where x denotes the spatial location, and A, B correspond to two simulations for which one out of
the three investigated factors (boundary condition, grain morphology, or constitutive parameters)
has been changed with the other two fixed. The average relative change quantifies the significance
of a stress difference between a reference σA
after altering any one of the
three factors mentioned above.

i j and the modified σB
i j

Figure 7.8 presents histograms of the average relative change using bins of width 0.5 (i.e.
bin increments of 50 %) for both the normal stress components in the loading direction (σ22,
top) and transverse to the loading direction (σ11, bottom).5 A change of boundary condition,
grain morphology, or constitutive model causes about the same small relative variation in σ22,
as indicated by the narrow distribution in the top row of Figure 7.8. In each case, there are four
conditions plotted in the same way, as shown in Figure 7.6, but they are nearly indistinguishable.
This is because ∆σ22 is relatively small compared to σ22 ≈ 0.5 GPa, which pushes the distribution
curves towards the center (zero implies no difference). Furthermore, changes in the sign of σ22 are
improbable such that no data falls into the left half of any histogram in the top row of Figure 7.8.
In contrast, the histograms of the transverse normal stress σ11 (Figure 7.8 bottom) are much
broader because the magnitude of σ11 is much smaller than σ22. Hence, relative differences in
σ11 are larger, and differences in the sign occur, as indicated by negative values in the lower
set of histograms, where there cannot be data in the open interval (−2,0).6 Also, unlike the

5 The evaluation of σ12 is similar to σ11, so only one is shown.
6 There is no combination of values for which Equation (7.8) yields values in this interval.

102

Figure 7.8: Histograms of the average relative change (see Equation (7.8)) in surface normal stress
along the loading axis (σ22, top row) and transverse to it (σ11, bottom row) when changing the
boundary condition (∆boundary condition, column 1), microstructure (∆morphology, column 2),
or constitutive model parameters (∆constitutive parameters, column 3). Fractional distribution
variation (in the bin of smallest relative changes) among the remaining two variables, e.g. boundary
condition and constitutive parameter set for a switch of grain morphology (center column), are
indicated by brackets for σ11. Shaded interval (−2,0) corresponds to inaccessible values of average
relative changes.

distribution of the orientation changes in Figure 7.6, where there was almost no discernible change
for small variations, the stress differences for all combinations of fixed factors have different levels
of probability over the full range. Consequently, the differences in the most highly probable subset
of the distribution are examined next.

The spread among the three sets of four curves is identified using the gray bracket (Figure 7.8
bottom) for the largest (and most relevant) fraction of small differences in stress. This spread is
most significant for a change in the constitutive parameter set and smallest for a change in the grain
morphology. The smaller this spread, the less critical the values chosen for the two fixed factors

103

10-21-505avg. rel. change of σ22fractionboundarycondition-505avg. rel. change of σ22morphology-505avg. rel. change of σ22constitutiveparameters10-21-505avg. rel. change of σ11fractionPS1, 3DPS2, 3DPS1, quasi-3DPS2, quasi-3D-505avg. rel. change of σ11hard, PS2soft, PS2hard, PS1soft, PS1-505avg. rel. change of σ11soft, quasi-3Dhard, 3Dhard, quasi-3Dsoft, 3Dare, with respect to the alteration of the remaining varied factor. As this spread is smallest when
the grain morphology is varied, the boundary conditions and constitutive parameters have less of
an effect on the difference in the local stress state. Hence, as the other two plots show, the different
outcomes in surface stress are mostly dominated by the differences in grain morphology, while the
influence of boundary conditions and constitutive parameters are less significant and similar.

¯
 = 0.40 %

¯
 = 1.0 %

¯
 = 4.0 %

D
3

D
3
-
i
s
a
u
q

compression

tension

Figure 7.9: Principal stress vector field plot comparing the simulated surface stress tensor evolution
with strain between realistic and columnar grain structure for constitutive parameter set PS1 and
hard boundary condition.

To further analyze how the stress state is affected by the initial shapes of the grains, the simulated
surface stress tensor is visualized using vector field plots for the quasi-3D and 3D microstructure at
three levels of strain. The plots show the value and (projected) direction of principal surface stress
averaged over areas of 8 × 8 µm2. Due to the averaging process, some of the finer details of spatial
variations in the stress tensor are washed out, especially near grain boundaries. The hard boundary
condition and constitutive parameter set PS1 were used for this analysis since the combination of
these two factors resulted in better lattice reorientation prediction.

As shown in Figure 7.9, the dominant stress state just after yield ( ¯
 = 0.40 %) is uniaxial tension

104

xyzalong the horizontal y-axis, and there are no apparent differences arising from the different grain
morphologies. With increasing deformation, subtle differences start to emerge in several locations.
For example, at 1.0 % tensile strain, the upper central region of the quasi-3D microstructure starts
to show weak compressive stress along the (vertical) x-axis, whereas the same area of the 3D
microstructure remains in tension. With increasing strain, the local stress state in the 3D case
increasingly differs from the quasi-3D case up to 4 % tensile strain, where the magnitude of vertical
compression is much greater in the left side of the central grain.

From this analysis of the surface principal stress evolution, it is clear that the effects of grain
morphology gradually intensify with increasing strain level, especially in the central grain where
the DAXM characterization was more complete. Thus, the correct grain morphology becomes
increasingly important with increasing strain.

7.3 Comparison of simulated and measured residual lattice stress

In the study with the first generation 3D microstructure, it was demonstrated that the grain
morphology and the subsurface grains are very important for accurately simulating the local defor-
mation process, especially the lattice reorientation where an intergranular lattice reorientation band
was successfully captured. However, this study [152] did not include a direct comparison of the
residual lattice stress/strain between the crystal plasticity simulation and the DAXM characteriza-
tion, due to the lack of sufficient DAXM blade scans as well as a customizable strain quantification
toolkit at the time of the publication. Therefore, the following study was carried out, focusing on
the direct comparison of residual lattice strain between crystal plasticity simulations and DAXM
characterization, which utilizes the strain quantification method described in Chapter 5.

7.3.1 Simulation environment

The 3D microstructure used in this study is the second generation 3D microstructure reconstructed
from one surface EBSD map and twelve DAXM blade scans using the BI method (Figure 7.10), the
detailed process of which is covered in Section 4.2.2 of Chapter 4. Due to the larger reconstructed

105

volume of the second generation microstructure (see Table 7.1 for details), the simulation framework
is switched from the original CPFE to a crystal plasticity fast-Fourier transform framework (CPFFT),
which has been demonstrated to be more efficient for simulating the deformation process of larger
volumes at relatively small computational cost [122]. More specifically, the same tensile experiment
from Section 7.2 was simulated with the spectral solver provided by DAMASK in conjunction with
the phenomenological power-law based constitutive model detailed in Section 7.1.1. Since the
spectral solver from DAMASK assumes periodic microstructure in all three directions, a layer of
air voxels (15 µm) was added to the top of the spectral mesh (Figure 7.10) to break the forced
periodicity such that the surface of the spectral mesh remains a free surface during the simulation.
Contrary to common practice, no isotropic buffer volume was added to the other five faces as an
auxiliary study (see Chapter L) shows that the added buffer volume has only limited impact on the
simulated stress/strain responses of the center volume, where the grains of interests were located
in this study. The constitutive parameter set used in this study is the physically more meaningful
one from the calibration described in Section 7.1.2.

Figure 7.10: The second generation 3D microstructure reconstructed from 12 DAXM blade scans
using the Barycentric interpolation method contains 54 grains. An air layer is added at the top of
the microstructure to approximate the free surface condition during the tensile experiment of the
original sample.

The boundary condition used in this study is a volumetric average displacement controlled load

106

100µm15µmy(TD)z(ND)x(RD)140µm200µmtensiledirectionairair(cid:219)F =

0

∗
0
0 5 × 10−4 0
∗
0

0

0 ∗ ∗
∗ ∗ ∗
∗ ∗ 0

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , P =
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , P =

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(7.9)

(7.10)

Table 7.1: Comparison of the first generation and second generation microstrucures

dimensions/µm3 Nscans Ngrains Nsubsurface grains
134.5 × 145 × 60
140 × 200 × 100

11
30

29
54

4
12

1st-gen
2nd-gen

case where a fixed strain rate is specified at the start of each increment in the form of (cid:219)F and P,

where (cid:219)F is the global deformation gradient rate, P is the global Piola-Kirchhoff stress, and ∗
indicates complementary boundary conditions. A total number of 100 increments was used during
the tensile stage, followed by an unloading stage with another 100 increments using

(cid:219)F =

0

∗
0
0 −5 × 10−5 0
∗
0

0

0 ∗ ∗
∗ ∗ ∗
∗ ∗ 0

so that the resulting local stress-strain state is comparable to those from strain quantification since
the tensile sample is unloaded during the DAXM characterization. The final volume averaged
stress-strain curve resulting from this boundary condition (load cases) is shown in Figure 7.11.

7.3.2 Results and discussions

Before carrying out the direct comparison of lattice strain between the CPFFT simulation and
DAXM characterization, comparison of the lattice reorientation maps was performed to ensure
that the change of framework (CPFE → CPFFT) does not affect the simulation results. As shown
in Figure 7.12, the surface lattice reorientation band is present in all three maps with a slightly
different location. This discrepancy of the lattice reorientation band position is most likely due to
the difference in the 3D grain morphology. More specifically, the grain clusters in the central region
of the second microstructure are different from the previous one due to the additional eight DAXM

107

Figure 7.11: The overall stress-strain curve of the CPFFT simulation resulted from the proposed
load cases.

blade scans as well as the change of the reconstruction process (manual → auto). Furthermore, both
reconstructed microstructures are, at best, an approximation of the real microstructure, resulting in
the differences between the simulated lattice reorientation bands and the one calculated based on
EBSD measurements. The wavy features present in the middle map (CPFFT) are artifacts of the
spectral solver, which can be theoretically eliminated with a proper low-pass filter. Nevertheless,
no additional filtering is performed in this study for CPFFT simulation results to avoid introducing
new artifacts. Overall, most of the main features in lattice reorientation maps (see examples marked
with arrows in Figure 7.12) are present after switching from CPFE to CPFFT, indicating that the
change in the solver (finite element solver → spectral solver) should not be considered as a primary
source of discrepancy in the subsequent analysis.

Evaluation of the quality of the simulated residual lattice strain/stress was performed by compar-
ing the simulated deviatoric residual lattice stress tensor (σD) with those calculated from DAXM
data. There are two main reasons for using σD as the benchmark for this evaluation. First of all, the
DAXM characterization was performed after the tensile experiment. Therefore, the residual lattice
strain extracted from DAXM blade scans are elastic responses, which correspond to the residual

108

0.60.40.20.00.010.020.030.040.05σCauchy/GPaεCauchyCPFE

CPFFT

EBSD

0°

20°

Figure 7.12: The simulated surface lattice reorientation maps at 4 % tensile strain using CPFE
(left) and CPFFT (middle) are compared with the EBSD measurements. The two simulated surface
lattice reorientation maps are from crystal plasticity simulations using the same constitute model
and material parameters, but different solver.

stress present in the sample. Second, no energy scans were performed for any of the 12 DAXM
blade scans used in this study. Consequently, only the deviatoric components of the residual lattice
strain tensors extracted through strain quantification are meaningful for analysis. In practice, the
residual lattice strain tensor from DAXM data 
DAXM is calculated from the full lattice deformation
gradient extracted with strain quantification using eq. (7.2). The residual stress tensor σDAXM is
then approximated using Hooke’s law with the stiffness matrix of pure titanium,

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

CTi =

C11 C12 C13

C12 C11 C13

C13 C13 C33
0
0
0

0
0
0

0
0
0
0 C44
0
0

0
0
0
0
0 C44
0

0
0
0
0
0
0 C66

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

,

where C11 = 160 GPa, C33 = 181 GPa, C44 = 46.5 GPa, C12 = 90 GPa, C13 = 66 GPa, and
C66 = 0.5(C11 − C22) [170]. Finally, the deviatoric residual stress tensor σD,DAXM is calculated
by removing the hydrostatic components using

i j
D,DAXM = σ
σ

i j

DAXM − 1
3 σ

i j
DAXMδi j,

109

xyzgrainofinterestgrainofinterestwhere δi j is the Kronecker delta. The same formula is also used to remove the hydrostatic compo-
nents of the simulated residual stress tensor, which is readily available from CPFFT simulation.

0.01 GPa

1 GPa

Figure 7.13: Comparison of the magnitude of deviatoric residual lattice stress (|σD|) between
CPFFT simulation (left) and those extracted from strain quantification (right) with a log scale
colormap. Similar patterns between two maps (highlighted with arrows) are observed in the region
with most DAXM blade scans (highlighted with white dashed box).

Comparing the magnitude (fig. 7.13), normal components (fig. 7.14) and shear components
(fig. 7.15) of the deviatoric residual stress tensor of a single DAXM blade scan with the associated
simulation results indicates that the accuracy of the simulated σD scales with the fidelity of the
reconstructed microstructure. More specifically, the simulated patterns in the top center region
where DAXM characterizations were focused upon are similar to those from DAXM scan. For
example, a clear transition from low residual stress to high residual stress can be found in the top
center region along the negative z-direction in both maps, which suggests that the simulated local
stress tensors are representative of those in the sample. However, the exact values of σD, CPFFT
are rarely the same with σD, DAXM, which is somewhat expected considering no back-stress nor
grain boundary effects are present in the constitutive description used in this study. There are
regions with accurate pattern prediction (white boxes in figs. 7.14 and 7.15) that were also observed
in regions with fewer DAXM scans. However, the reliability of these predictions was often less
than desirable because failed predictions (red boxes in figs. 7.14 and 7.15) were often found in the
same neighborhood. Nevertheless, the comparison of deviatoric residual stress between simulation
and experiment indicates that the simulated local stress state are representative of those in the real

110

regionwithmostDAXMbladescansy(TD)z(ND)x(RD)CPFFTCPFFTDAXMsample, provided that a microstructure with high fidelity is used as input for the simulation.

CPFFT

DAXM

σxx
D

σ yy
D

σzz
D

0.01 GPa

1 GPa

Figure 7.14: Comparison of the diagonal components of the deviatoric residual lattice stress tensor
D|) between CPFFT simulation (left column) and those extracted from strain quantification
(|σii
(right column) with a log scale colormap. Patterns observed in both maps are marked with white
boxes whereas the discrepancy between two maps is highlighted with red boxes.

111

xyzCPFFT

DAXM

σ yz
D

σxz
D

σ

x y
D

0.01 GPa

1 GPa

Figure 7.15: Comparison of the shear components of the deviatoric residual lattice stress tensor
D|) between CPFFT simulation (left column) and those extracted from strain quantification
(|σii
(right column) with a log scale colormap. Patterns observed in both maps are marked with white
boxes whereas discrepancy the between two maps is highlighted with red boxes.

112

xyz7.4 Summary

In the study with the first generation 3D microstructure, it is demonstrated that the influence
of the geometric accuracy of the grain microstructure outweighs those of the choice of boundary
conditions and constitutive parameters. The comparison between the simulations with 3D and
quasi-3D microstructure revealed that a geometrically realistic grain morphology, as well as a
physically more realistic boundary condition, are more important than fine-tuning the constitutive
model parameters to simulate the heterogeneous deformation processes at the grain scale accurately.
A follow-up study shows that it is possible to estimate the local lattice strain/stress tensor in the
polycrystalline aggregate, the accuracy of which seems to scale with the fidelity of the reconstructed
3D grain morphology. Nevertheless, the study presented in this chapter shows that DAXM can be a
valuable bridge between (crystal plasticity) simulations and experimental characterization as it can
provide both the necessary crystallographic information for the 3D microstructure reconstruction
and the spatially resolved lattice strain distribution to assist interpretation of the results from
simulation and other experiments. Furthermore, the dislocation content extracted via streak analysis
outlined in Chapter 6 can potentially be useful for the development of a more physics-based
constitutive model, which provides new opportunities in the development of dislocation density
based constitutive models.

113

CHAPTER 8

DISCUSSION

Chapters 4 to 7 present a synergistic approach of using both differential aperture X-ray microscopy
and crystal plasticity simulations to investigate the local deformation processes of a Ti-5Al-2.5Sn
tensile sample deformed at ambient temperature. During the investigation, two 3D microstruc-
tures were constructed from DAXM blade scans using two reconstruction methods detailed in
Chapter 4. The two resulting 3D microstructures, termed the first and second generation 3D mi-
crostructure, were converted into finite element and spectral meshes respectively for corresponding
crystal plasticity simulations. The simulated kinematic response, represented by the surface lattice
reorientation, were compared with those derived from EBSD measurements and revealed that the
morphological and crystallographic fidelity of the input microstructure outweighs fine tuning of the
constitutive model and boundary conditions when it comes to improving the accuracy of the simu-
lated kinematic responses. Similarly, the simulated mesoscale constitutive responses, represented
by the simulated deviatoric lattice stress tensors were compared with those extracted from DAXM
data using the new technique detailed in Chapter 5. This comparison of mesoscale constitutive
responses between crystal plasticity simulations and experimental characterization indicates that
the accuracy of the simulated mesoscale constitutive responses scales positively with the fidelity
of the local reconstructed microstructure, where regions reconstructed with more DAXM data tend
to have more accurate prediction than those with fewer. In addition to the comparisons between
simulation and experiment results, the dislocation content of a near-surface region of the sample
was also extracted from the associated DAXM blade scan using the proposed streak analysis method
detailed in Chapter 6. The resulting spatially resolved 3D (misfit) dislocation density profile can be
used to facilitate the interpretation of the microscale responses in the future. Since focused discus-
sions for each specific study are provided in the corresponding chapter along with the associated
results, this chapter is dedicated to discussing how this project addresses the three missing links
identified at the end of Chapter 2.

114

8.1 Microstructure reconstruction requirement

As demonstrated in this study, the accuracy of the simulated kinematic and constitutive re-
sponses scales positively with the morphological and crystallographic fidelity of the reconstructed
microstructure. In other words, accurately capturing finer details using crystal plasticity simulation
requires a microstructure that has to be both morphologically and crystallographically accurate
within the neighborhood of the feature of interest. On the other hand, such highly accurate 3D
microstructures may be unnecessary if the feature of interest is at a relatively larger scale, in which
case, expensive beam time and computational resources spent on the microstructure reconstruction
should be allocated elsewhere to maximize overall efficiency. For instance, if the ultimate goal of
the research is to understand the grain-average stress-strain evolution in a polycrystalline sample
during deformation, a full field technique that provides the center-of-mass and the corresponding
crystallographic orientation, such as ff-HEDM [69, 71], 3DXRD [68] and DCT [56], is a better
choice than carefully mapping the entire volume using 3DEBSD or DAXM. Conversely, if the
strain patterning due to intra-grain dislocation structure is the feature of interest, it is necessary
to spend the time and resources to map out the grain and its neighbors carefully so that accurate
morphological and crystallographic representations of the volume of interest can be established for
the associated crystal plasticity simulations.

It is also worth pointing out that the effect of microstructure seemed to be limited to the vicinity
of one or two grains, as demonstrated in this study where relatively accurate predictions were
achieved by carefully reconstructing the grain of interest and its first order neighbors. Therefore,
it is theoretically possible to study the deformation mechanisms from regions with characteristic
features, termed Eigen features in this section, using microstructure reconstructed from DAXM or
3DEBSD data. Then, statistical analysis can be performed to link these Eigen features with the
mesoscale microstructure. These kinds of analysis can be done with a multi-scale 3D microstructure
where the mesoscale microstructure is captured by full-field technique, and the Eigen features
are carefully mapped using DAXM or nf-HEDM. The associated microstructure reconstruction
algorithm for this kind of hybrid data can be derived from the existing machine learning powered

115

microstructure reconstruction methods [112], which can also pull statistics from the full field
characterization data to facilitate the reconstruction of Eigen features.

8.2 Interpretation of mesoscale results

In this study, the simulated mesoscale responses, represented by the surface lattice reorientation
and the deviatoric stress tensors, are mostly in agreement with those derived from experimental
characterization. The accuracy of these mesoscale predictions seems to scale positively with the
morphological and crystallographic fidelity of the reconstructed microstructure. According to the
analysis presented in Chapter 7, the simulated mesoscale results are most sensitive to the microstruc-
ture change. In other words, the grain morphology and the crystallographic orientation field are the
dominant factors for the mesoscale results whereas the boundary condition and constitutive model
parameters seem to have a limited effect. The results presented above indicate that the multiplicative
decomposition based kinematic description (Equations (2.4) to (2.6)) are sufficient to capture the
mesoscale kinematic responses as long as accurate morphological and crystallographic data is used
as its input. Similarly, the constitutive parameters calibrated with a global stress-strain curve are
sufficient for the selected phenomenological power-law based constitutive model to approximate
the mesoscale stress-strain evolution correctly. Therefore, the established calibration scheme where
the constitutive model is often calibrated based on macroscopic data is shown to be adequate for
estimating the mesoscale responses, provided that accurate morphological and crystallographic
information is used as input.

The conclusion described above might only apply to the phenomenological constitutive de-
scriptions as the interpretation of the associated results are often limited to the mesoscale. For
physics-based constitutive description, where the results are often interpreted at the dislocation
structure level (microscale) [94,96], the associated characteristics of the plastic flow carriers (dislo-
cations) calibrated from macroscopic data might not be representative of the real material. In other
words, the dislocation activity interpreted from these physics-based crystal plasticity simulations
could be inaccurate despite accurate mesoscale responses being achieved. Consequently, it is still

116

necessary to find a method that can provide calibration data at the microscale such that the resulting
dislocation density evolution from the calibrated physics-based crystal plasticity simulations can
be used to interpret the dislocation activity with confidence.

8.3 Interpretation of microscale results

The Frank-Bilby equation based streak analysis detailed in Chapter 6 demonstrates that it is
possible to estimate the (misfit) dislocation density profile from DAXM data nondestructively. The
extracted dislocation density profile is qualitatively validated by the FIB-TEM analysis, the results
of which indicate that the extracted dislocation density profile is representative of the material.
Although direct comparison between the simulation and the extracted dislocation density profile
was not performed in this study due to the phenomenological nature of the selected constitutive
description, the results presented in Chapter 6 lay the foundation for studying the dislocation activity
using the proposed synergistic application of crystal plasticity simulation and synchrotron X-ray
based characterization. More specifically, the spatially resolved (misfit) dislocation density profile
can be used to validate existing physics-based crystal plasticity simulation results, elucidating
whether the current macroscopic calibration scheme can provide constitutive parameters that are
physically representative of the associated dislocation characteristics. Furthermore, the extracted
density profile can also be used to calibrate an existing dislocation density based constitutive model,
ensuring that the associated dislocation characteristics are appropriately calibrated at the dislocation
structure level. Furthermore, the extracted dislocation density profile can also serve as a starting
point for the development of new physics-based dislocation density model, or as a reference to
adjust existing models such that dislocation evolution can be more accurately captured through the
associated crystal plasticity simulations.

Despite the many promising applications of the proposed Frank-Bilby equation based streak
analysis, several challenging issues need to be addressed before it can be used to assist the devel-
opment of physics-based constitutive models.

• The first challenge lies in the data acquisition. As discussed in Chapter 6, the proposed FBE

117

analysis requires a large number of reconstructed micro-Laue diffraction patterns with high
spatial resolution (≤ 1 µm), the acquisition of which is expensive. Therefore, support from
a project such as the material genome initiative (MGI) is necessary so that characteristic
micro-Laue diffraction patterns for different materials can be systematically acquired and
converted into a public database, which would allow the general public to develop, calibrate
and validate their physics-based constitutive model without the need of a specific synchrotron
data set.

• The second challenge resides in the process of linking the orientation field in the reciprocal
space (micro-Laue diffraction patterns) with the corresponding one in real space (misfit
plane normal). In this study, this linking process was done manually, which is both time-
consuming and challenging to scale up. Furthermore, human interpretation of diffraction
patterns can be subjective, which reduces the accuracy of the associated misfit dislocation
density. Consequently, a computer vision based automated program is needed to automate
this process, relieving researchers from tedious labor while improving the reproducibility of
the reported dislocation density.

• The last challenge is related to the interpretation of the misfit dislocation density. Most
current dislocation density based constitutive models describe the dislocations as GNDs and
SSDs, which follows the dislocation structure model initially proposed by Nye [33]. However,
the proposed streak analysis is derived from a different dislocation structure model proposed
by Frank and Bilby [34]. Therefore, proper translation1 between misfit dislocation density
and the density of GNDs is necessary to use the extracted misfit dislocation to facilitate the
development of dislocation density based constitutive model.2

1 The translation would probably require to “straightened out” any spatial GND density to

compare against a planar representation as imagined for the Frank-Bilby analysis.

2 Alternatively, a dislocation density model based on the dislocation structure proposed by Frank
and Bilby can be developed to utilize the misfit dislocation density directly. However, this approach
would be more difficult than the one proposed in the main text.

118

CHAPTER 9

CONCLUSIONS AND FUTURE WORK

9.1 Conclusions

This thesis demonstrates a synergistic approach of using DAXM characterization and crystal
plasticity simulations to study the micromechanics of polycrystalline materials. More specifically,
the local deformation history of a near-surface volume from a Ti-5Al-2.5Sn tensile sample deformed
at ambient temperature was simulated using phenomenological crystal plasticity simulations. The
3D microstructures used as input for the crystal plasticity simulations were reconstructed from
DAXM blades scans using two different reconstruction methods detailed in Chapter 4. Comparison
of the kinematic and constitutive responses between crystal plasticity simulations and the experiment
characterizations were performed, revealing that the accuracy of the simulated local deformation
history scales positively with the morphological and crystallographic fidelity of the reconstructed
microstructure. Furthermore, spatially resolved (misfit) dislocation density profile of another near-
surface volume of the same sample was also extracted from the associated DAXM blade scan
using the proposed Frank-Bilby equation based streak analysis, paving the road for future studies
where the simulated dislocation content can be validated and used to interpret the local deformation
history.

9.2 Future work

The content of this thesis demonstrates how the combination of DAXM characterization and
crystal plasticity simulation can be used to further the study of micromechanics in polycrystalline
materials. However, it is by no means an exhaustive or comprehensive demonstration of all possible
ways to advance the study of micromechanics with DAXM and crystal plasticity simulation. Some
potential future research opportunities derived from this thesis are list below.

• Integrated crystal plasticity simulation environment with DAXM support

119

An integrated crystal plasticity simulation framework like DAMASK provides easy access to
crystal plasticity simulation for the research community. Extending the existing simulation
framework with DAXM support will provide researchers the opportunities to build and
validate crystal plasticity models with DAXM data within the same framework, which should
expedite the advancement in micromechanics of polycrystalline materials.

• Virtual X-ray diffraction lab for DAXM characterization

The limited access to a synchrotron facility often requires researchers to make critical deci-
sions for DAXM characterization at the beamline, which inevitably reduces the effectiveness
of actual beamtime used for characterization. A virtual micro-Laue diffraction lab that mim-
ics the input and output of the synchrotron facility would allow to make better experiment
plans prior to arriving at the beamline. Furthermore, this virtual X-ray diffraction lab should
also be customizable, which would allow beamline users to test their data processing al-
gorithm (indexation and reconstruction) without relying on the servers at the synchrotron
facility.

• Development of a new dislocation density model with FBE streak analysis

The FBE streak analysis provides a spatially resolved misfit dislocation density field in
3D. This information can be used to facilitate the development of next-generation dislocation
density based constitutive models, which would allow direct physical evidence of dislocations
to be used as input for crystal plasticity simulations.

• Crystal plasticity simulation assisted “smart” DAXM scan for in-situ experiments

The DAXM characterization often requires the beamline users to strategically place a fixed
scan path at the beginning of the in-situ experiment. Due to the limited beam time available,
beamline users often choose coarse scans to cover larger volume, leaving potential sites of
interests uncharacterized. A crystal plasticity simulation assisted “smart” scan would take
the initial DAXM scan data to build a 3D microstructure. Before the next straining step, a
crystal plasticity simulation performed using the 3D microstructure mentioned above could

120

be used to identify the location of potential sites of interest such as high-stress concentration
locations, large orientation gradient regions and potential damage nucleation sites. With the
location of potential sites of interests identified, the computer can automatically adjust the
scanning path, maximizing the effectiveness of beam time.

121

APPENDICES

122

APPENDIX A

TERMINOLOGY

Burgers vector

A vector defining the closure failure around a dislocation.

Crystal plasticity

A continuum-scale simulation paradigm for solid materials where the deformation is defined

through integrating dislocation slip induced shear and sometimes twinning induced shear.

Discrete dislocation dynamics

A micro-scale simulation paradigm where the deformation is defined through integrating stress-
induced equations of motion, which are defined by elastic interactions among discrete dislocation
line segments.

Electron backscatter diffraction (EBSD)

A method commonly used in SEM to capture and index the diffraction pattern from a given

volume, which can be automated into mapping the orientation of a prepared surface.

Fast Fourier Transform (FFT)

FFT is an image-based algorithm that solves the equations of equilibrium and compatibility in

the Fourier space.

Feature (Data Analysis)

Feature in data analysis is generally defined as a pattern/manifestation of correlations among

different components within the dataset.

123

Finite Element Method (FEM)

A continuum-scale simulation method where the body is discretized into small units. Each unit
has a simple geometrical shape and variable size, which is generally treated as a homogeneous
volume during deformation.

Grain

In polycrystalline materials, the distinct, approximately compact region which can be identified

with single lattice orientation is defined as a grain.

Grain boundary

The interface between any pair of grains in a polycrystalline specimen.

Kernel average misorientation (KAM)

At each point, the KAM is defined by averaging the misorientation between this point and a

kernel of points around it.

Lattice reorientation (LR)

The amount rotation induced by the plastic deformation, which is defined as the smallest rotation
angle required to bring frame associated with the deformed lattice into coincidence with the frame
associated with the undeformed lattice.

Low-angle grain boundary

Grain boundaries that result from dislocations whose cores do not overlap.

Micromechanics

Micromechanics refers to the analysis of heterogeneous materials on the level of the individual

constituents that constitute these materials.

124

Misorientation

The smallest rotation angle required to bring two lattices on either side of a boundary into

coincidence.

Orientation matrix (g)

The orientation at any point can be identified using an orientation matrix, which can rotate the

frame associated with the lattice into coincidence with the reference frame.

Orientation gradient

The gradient related to the distribution of orientations within given volume.

Residual stress

Residual stress refers to the stress that remains within a rigid body after manufacturing or
processing in the absence of external forces or thermal gradients. There are commonly three types
of residual stress

• Type I, the macro residual stress in a component on the scale larger than the grain size.

• Type II, the micro residual stress varying on the scale of the grain in the material.

• Type III, sub-micro residual stress varying within a grain due to lattice defects.

Schmid factor (m)

Dislocation slip related geometric factor that helps resolve far-field stress tensor, σ, onto a local

slip system (α) as a simple shear stress, τ.
Slip direction

For a given slip system, the slip direction is a unit vector that is parallel to the Burgers vector.

125

Slip plane

A glide plane for conservative motions of dislocations, which is usually defined through the

plane normal vector.

Slip system

The combination of slip direction and slip plane that defines the dislocation motions and

corresponding shear strain.

Solid solution strengthening

Solid solution strengthening is an alloying technique for strength improvement, the fundamental
mechanism of which relies on dislocation pinning through the distortion in lattice due to the presence
of alloying elements.

Strain quantification

The data analysis procedure where the elastic lattice strain is extracted by analyzing the deviation

in the micro-Laue diffraction pattern.

Streak analysis

The data analysis procedure where the dislocation content of the given volume is measure by
measure the diffraction peak streaking, which is the projection of the local orientation gradient in
reciprocal space.

Texture

In a polycrystalline specimen, grains usually exhibit a preference for certain orientations due
to the material processing. This preferred crystallographic orientation is commonly denoted as
the texture to provide a general sense of the crystal orientation distribution associated with given
specimen.

126

APPENDIX B

EXPLORE THE POSSIBILITY OF RECOVERING UNDEFORMED CRYSTAL

ORIENTATION USING CRYSTAL PLASTICITY SIMULATION

It has been established that CPFE simulation can accurately predict the crystal reorientation due
to plastic deformation when realistic 3D microstructure is used [152]. However, it is unclear
whether CPFE simulation can be used to recover the undeformed crystal orientation by reversing
the global load with the help of a realistic 3D microstructure. To this end, the same first-generation
3D microstructure reconstructed in Section 4.1 was adapted where the deformed surface crystal
orientation measured using EBSD was assigned to the corresponding elements. Then the 3D
microstructure was deformed using the same constitutive model detailed in [152] with the reverse
loading case, a uniaxial compression along the original tensile direction. Since the center grain
on the surface has the most realistic grain morphology compared to the rest, the associated sim-
ulated “undeformed” orientation is compared with the corresponding EBSD measurement of the
undeformed sample to evaluate whether it is possible to recover the undeformed crystal orientation
through reverse loading.

The pole figure of a-axis ([2 1 1 0]) and c-axis ([0 0 0 1]) of the simulated undeformed central
grain were plotted in Figure B.1, along with those from the EBSD measurements as well as the
deformed crystal. By connecting the a-pole and c-pole of the deformed crystal with EBSD/CPFE
results, the actual/recovered crystal reorientation path is visualized using a solid/dashed line. The
large angle between the two lines indicates that the proposed method, merely reversing the global
loading, cannot recover the correct undeformed crystal orientation, despite the usage of a realistic
3D microstructure. This result is not surprising since the strain hardening process is generally
considered as an irreversible process in most crystalline materials.1

1 Another possible explanation would be the asymmetric activation of slip system during plastic

deformation, which makes the associated lattice reorientation strain path dependent.

127

Figure B.1: Pole figure of a-axis and c-axis reorientation of the central grain shows that the
measured undeformed orientation (EBSD) is different from the one recovered through reverse
loading (CPFE).

128

a-axisc-axisdeformeddeformedundeformed(EBSD)undeformed(EBSD)undeformed(CPFE)undeformed(CPFE)xyAPPENDIX C

MENTAT PROCEDURE FILE FOR 3D MODEL CONSTRUCTION

Using the technique described in Section 4.1, a Mentat procedure file 1 was generated to build the
first-generation 3D model interactively using the procedure file reader provided by MSC.Marc®
Mentat 2012. When executed, the procedure file will first initialize the workspace in Mentat with
the following commands.

1 | Preparation
2 * new_model yes
3 * select_reset
4 * plot_reset
5 * expand_reset
6 * move_reset
7 * elements_solid * regenerate |** make elements visible **|
8 * set_curve_type line
9 * set_curve_div_type_fix_avgl |** make curve division by fix length **|
10 * set_curve_div_avgl
11 10
12 * set_nodes off * regenerate
13 * set_sweep_tolerance 1e-5
14
15 |Pre - clean_up
16 * clear_curve_divisions
17 all_existing
18 * remove_curves
19 all_existing
20 * remove_points
21 all_existing
22 * renumber_all

Then, the binding points for all grains are added to the new model one at a time using the

command below

1 * add_points x

y

z

|

pt -ID

Once all points are added to the model, the procedure file will connect the corresponding binding
points to form the intercepts of grain boundary planes using

1 The procedure file can be found on Github/KedoKudo/dissertationMSU/appendices/firstGen-

Model.proc.

129

1 * add_curves pt1 -ID

pt2 -ID

| edge -ID

Before building the 3D grain, a quick pruning is performed to remove duplicated edges using
*sweep_all, followed by a subdivision of all edges using

1 * apply_curve_divisions

all_existing

With all edges properly divided into ten segments, each grain can be systematically constructed

by repeating the following steps:

• Prepare

a

clean workspace

by making

all

3D elements

invisible

using

*invisible_all_sets.

• Generated 2D grain boundary plane that enclosing the grain using *af_planar_trimesh

edgeIDs.

• Prune the generated grain boundary planes using the following commands
1 * sweep_all
2 * sweep_nodes all_outline
3 * align_shells
4 100

• Grow tetrahedron elements inside the grain using *dt_tet_tri_mesh all_visible.

• Store generated 3D elements into a set.

• Repeat until all grains are stored as sets.

Once all grains are generated, the procedure file perform a final round of cleanup before

configuring the material properties for each grain using the following commands

1 * clear_curve_divisions
2 * remove_curves
3 * remove_points
4 * sweep_all
5 * renumber_all

all_existing
all_existing

all_existing

Finally, the procedure file generates a displacement controlled load curve and configures the job
properties, followed by saving the 3D mesh into a *.mud file along with all the simulation related
configurations.

130

APPENDIX D

NOISE REMOVAL FOR MICROSTRUCTURE RECONSTRUCTION

The noise voxels in the seed points composed from surface EBSD map and DAXM blade scans
are the voxel with crystal orientations significantly deviate from the neighboring voxels. In other
words, the crystallographic orientation of the noise voxel is the outlier within a given search radius.
In theory, the disorientation between the voxel of interest and each one of its neighboring voxels
within the given radius (10 µm) need to be calculated to determine whether the voxel of interest is
noise voxel or not. However, this process is computationally expensive using existing Python toolkit
due to the involvement of instance initialization as well as the overhead of member function call.
Therefore, an alternative method using the IPF color vectors is used in this study to identify noise
voxel where the pair-wise total Euclidian distance of the IPF color vectors (IPF_001_hexagona
and IPF_010_hexagona) is performed using vector math. The Python code snippet related to this
particular calculation is shown below:

["{} _pos". format (i+1) for i in range (3)]
["{} _IPF_001_hexagonal ". format (i+1) for i in range (3)]
["{} _IPF_010_hexagonal ". format (i+1) for i in range (3)]

1 # seeds.txt is a DAMASK ASCII table containing all the seed points
2 seedsDF = pd. read_csv ("seeds.txt", sep=’\t’, skiprows =8)
3
4 lb_pos
=
5 lb_ipf
=
6 lb_ipf +=
7 lb_eulers = ["{} _eulers ". format (i+1) for i in range (3)]
8
9 allSeedsTree = spatial . KDTree ( seedsDF [ lb_pos ])
10
11 radius = 10
12 minimumNeighborWithSimilarColor = 10
13 maxIpfDiffAllowed = 0.1
14 notNoise = np.ones( seedsDF .shape [0])
15
16 # start pruning
17 for myIDX in range( seedsDF .shape [0]):
18
19
20
21
22

cntPT = seedsDF [ lb_pos ]. loc[myIDX , :]

if cntPT [’3_pos ’] > 0:

continue

# skip the surface point

131

23
24
25
26
27
28
29
30
31
32
33
34

:

neighborIDX = allSeedsTree . query_ball_point (cntPT , radius )

if len( neighborIDX ) < minimumNeighborWithSimilarColor :

notNoise [myIDX] = 0
continue

# outlier point is considered as noise

ipf_me = seedsDF [ lb_ipf ]. loc[myIDX , :]
ipf_neighbors = seedsDF [ lb_ipf ]. loc[ neighborIDX , :]

ipf_diff = np. linalg .norm( ipf_neighbors - ipf_me , axis =1)

if len( ipf_diff [ ipf_diff < 1e -5]) < minimumNeighborWithSimilarColor

if min( ipf_diff [ ipf_diff >= 1e -5]) > maxIpfDiffAllowed :

notNoise [myIDX] = 0

35
36
37
38 seedsDF [’notNoise ’] = notNoise
39
40 seedsClean = seedsDF [ seedsDF [’notNoise ’]>0]

132

MICROSTRUCTURE RECONSTRUCTION USING BARYCENTRIC INTERPOLATION

APPENDIX E

The core of Barycentric interpolation method is computing the Barycentric coordinates of the voxel
of interest within the simplex formed buy its N + 1 nearest characterized neighboring voxels. An
exemplary implementation of this method written in Python (BarycentricInterpolation.py)
is available on Github, which takes in a list of cleaned seed points (seeds_cleaned.txt) to
generate a 3D microstructure (vicBary.geom). Four iterations of curvature flow based grain
growth (grainGrowth.sh) are performed on the generated microstructure to achieve the final
realistic looking second generation 3D microstructure.

133

APPENDIX F

PYTHON IMPLEMENTATION OF THE VIRTUAL DAXM EXPERIMENT

This chapter presents a Python implementation of the virtual DAXM experiment described in
Chapter 5. A JSON file (see sample file in Listing F.1) is used to specify the virtual experiment
configurations, which can be parsed by a parser function (read_config(configJSON)) to set up
the virtual diffraction environment.

1 {
2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26
27 }

"job",
" test_idealDiffraction ",

" taskType ":
"name":
" dataFileName ": " test_idealDiffraction .dat",
" voxelArchive ": " test_idealDiffraction .h5",
" n_voxels ":
" labConfig ": {

10,

"k0":
"n_CCD":
"X-ray Energy (KeV)":
" detector angular range": 22.0

[0.0, 0.0, 1.0],
[0.0, -1.0, 0.0],
[7, 30],

},
" DAXMConfig ": {

" hkllist ":
" n_indexedPeaks ":
" n_fullQ ":
" peakPositionUncertainty /deg":
"angR range":
"magU range":

" hkls_FCC .txt",
[12],
[0],

[0.002],

[1e-3, 1e-1],
[1e-5, 1e-3]

},
" StrainCalcConfig ": {

" maxiter ":
"xtor":
"eps":

},
" monitor ": true

1e10,
1e-14,
1e-1

Listing F.1: Sample virtual diffraction configuration file

Since all the virtual crystallites used in this study are FCC crystals, a list of diffractable re-
ciprocal lattice points based on the structural factor of FCC crystal symmetry is generated and
cached in a text file using the function genHKLs(lattice). Subsequently, the virtual experi-

134

ment can be streamlined using a worker function (grad_student(jobConfig)), which auto-
matically configures the virtual diffraction, reads in the cached diffractable scattering vector list,
and perform the virtual diffraction experiment. All the FCC crystals used in the virtual DAXM
experiment are stored in an HDF5 archive, whereas the target deformation gradients and cal-
culated deformation gradients are stored in an ASCII table, along with the number of indexed
peaks (N) and the number of peaks characterized with energy scan (n). In this automatic worker
function, function make_random_defgrad(angR, magU) is used to generate a random deforma-
tion gradient to strain the virtual crystallite. Subsequently, the virtual (augmented) micro-Laue
diffraction patterns are made by calculating the visibility of all diffractable peaks using function
calc_visible_peaks(...), the N results of which are then randomly selected as the recorded
diffraction peaks in the pattern using function random_select_npeaks(visible_hkls, n).

For the virtual diffraction experiment with white noise, each scattering vector is perturbed
around a random axis at an arbitrary angle, the Python implementation of which is the function
perturb_vector(vector, ang, axis=None).

135

PYTHON IMPLEMENTATION OF THE SINGLE VOXEL STREAK ANALYSIS

APPENDIX G

This chapter presents a Python implementation of the single voxel streak analysis described in
Chapter 6, which produces streak dials that can be used to identify the dominant dislocation variant
in a given voxel.

Manual identification of the spatial position of the voxel of interest is required such that the
micro-Laue diffraction pattern (HDF file) and the associated indexation results (XML elements)
can be extracted and paired using the function find_element(h5ImageName, xmlFileName,
namespace). With the indexation results extracted, the streak dial can be generated using the
plot function plt_pattern(DAXM_Data), which can then be overlaid on top of the associated
diffraction peak during post processing using Affinity Designer.

136

APPENDIX H

ESTIMATE TOTAL DISLOCATION DENSITY WITH A SINGLE WALL OF EDGE

DISLOCATIONS

The total dislocation density in a given DAXM voxel can be estimated using a simple model where
the orientation gradient present is attributed to a dislocation wall consisting of one type of edge
dislocations. With the in-voxel orientation gradient extracted for a voxel of interest stored as a conf
file (Listing H.1), the associated dislocation density can be estimated with a Python script.

1 # NOTE:
2 # Data in here require manual update for difference case analysis.
3 # Data from reference voxel, extracted from the DAXM indexation file.
4 [reference]
5 coordinateAPS = 925.9
1668.8 -1108.2
6 eulersAPS = 214.7232
90.7724 200.2541
7 astar = 13.0036174 8.7033411 -18.8125672
8 bstar = -6.7861113 -5.0795173 -22.9543480
9 cstar = -7.5387731 10.8779542 -0.1784326
10 lattice = hexagonal
11 latticeParameters = 0.2965 0.2965 0.4747 90 90 120
12 H5_Image = 38077_59
13
14 # Data from the neighboring voxel providing similar orientation gradient to the
15 # local orientation gradient within the reference voxel.
16 [neighbor]
17 coordinateAPS = 925.9
1668.8 -1107.2
18 eulersAPS = 214.7267
90.9114 200.2316
19 astar = 13.0351170 8.6709235 -18.8057360
20 bstar = -6.7466193 -5.1205937 -22.9568627
21 cstar = -7.5391675 10.8771067 -0.2105406
22 lattice = hexagonal
23 latticeParameters = 0.2965 0.2965 0.4747 90 90 120
24 H5_Image = 38077_60
25
26 # Global stress tensor (necessary for Schmid factor analysis)
27 [load]
28 stressTSL = 0 0 0 0 1 0 0 0 0

Listing H.1: Single voxel data extracted for estimating the total dislocation density of a given
DAXM voxel

137

APPENDIX I

PYTHON IMPLEMENTATION OF THE FBE STREAK ANALYSIS

A python implementation of the FBE streak analysis can be found on Github.

138

APPENDIX J

CALIBRATION OF THE CONSTITUTIVE MODEL USING NELDER-MEAD BASED

SIMPLEX SEARCH

The calibration engine used to find the constitutive parameters for the Ti-5Al-2.5Sn tensile sample
can be found on Github, along with the shell script that performs the CPFFT simulation and
post-processing during each iteration of the simplex evaluation.

139

APPENDIX K

AN EFFICIENT TAYLOR GRADIENT ENHANCED PHENOMENOLOGICAL

CRYSTAL PLASTICITY MODEL

K.1 Motivation and background

The crystal plasticity framework pioneered by [171] provides a spatially resolved numerical
representation of the microstructure. The spatially resolved constitutive description of the mi-
crostructure, also known as the representative volume element (RVE) makes it relatively easier
to track the deformation history at a particular location. The introduction of RVE also allows
the crystal plasticity models to directly utilize more realistic microstructure from experimental
characterizations such as electron backscatter diffraction (EBSD), orientation imaging microscopy
(OIM™) [172–174] as well as more advanced 3D microstructure characterization techniques, in-
cluding 3D-EBSD [175, 176], high energy X-ray microscopy (HEXM) [22, 23], and differential
aperture X-ray microscopy [24,177]. The utilization of realistic microstructure as simulation input
usually helps crystal plasticity models provide a more accurate prediction of local deformation
history [157, 158, 160, 161]. Mainly, when a realistic 3D microstructure is used as input, a simple
phenomenological power-law based constitutive model can provide reasonably accurate kinematic
responses [152]. However, merely employing a realistic 3D microstructure cannot adequately com-
pensate for the oversimplification of the constitutive model. For example, the local description of
strain hardening in the phenomenological power-law based constitutive model allows each mate-
rial point to carry out its strain hardening without consulting its neighboring environment. This
simplification forces all grain boundaries to remain transparent to all dislocations within the vol-
ume of interest throughout the deformation. The forced total transparency of all grain boundaries
contradicts many observations where the grain boundaries are often observed to serve either as a
dislocation sink [178, 179] or source [180, 181] depending on the local deformation environment.
Hence it is necessary to include the grain boundary effect at the constitutive level such that more

140

accurate kinematic and constitutive responses can be simulated at the microscopic level. Many
constitutive models that are of the physics-based were developed in the past few years to address
the need to explicitly incorporate various nonlocal effect, particularly the grain boundary effect
on nearby dislocation activity. [182–184] The most common approach amongst all physical based
constitutive models is the discrete dislocation density model (DDD), which directly models the dis-
location evolution at each material point throughout the deformation history. Although DDD model
generally provides a more physically meaningful representation of dislocation activity during the
deformation [96,185], it is at the cost of high runtime memory consumption and total computational
time. Furthermore, a preliminary study also shows that complicated constitutive description like
DDD models tends to lead to numerical instability when applying to large-scale simulation, which
makes it difficult to simulate the grain boundary effect for an ensemble of hundreds of crystallites.
It is essential to find a middle ground where accurately simulating the grain boundary effect
can be achieved without spiking the computational cost. As demonstrated in [186], it is possible
to approximate the phase boundary effect using phenomenological power-law based constitutive
model by semi-manually adjusting the critical resolved shear stress for material points near the
phase boundary at the start of each iteration. Following a similar strategy, an efficient local Taylor
factor gradient enhanced phenomenological power-law based crystal plasticity model (TGCP) is
proposed in this study where the grain boundary effect is approximated by adjusting the shear rate
of each slip system according to its neighboring material points. Considering the range of grain
boundary effect varies with local deformation environment [187], an implicit grain boundaries
paradigm is used in this work where the grain boundaries are treated as a derived property of the
gradient of the local Taylor factor, which allows the grain boundary effect to propagate into grain
interior when necessary. The proposed constitutive description is implemented in an open source
crystal plasticity fast-Fourier-transform (CP-FFT) framework, DAMASK [122]. Considering the
elegance and robustness of bicrystal demonstrated by many studies [188–191], several numerical
analysis of bicrystals with different orientation pairs were performed to evaluate the simulated grain
boundary effect from TGCP.

141

K.2 Constitutive Framework of TGCP

Following the classic continuum mechanic framework, the total deformation gradient, F can be

separated into elastic and plastic portions through multiplicative decomposition

F = Fe Fp

(K.1)

where the elastic portion, Fe generally represents the elastic stretch and rotation of the crystal
lattice, and the plastic portion Fp denotes the associated plastic deformation due to dislocation slip.
The Green-Lagrangian strain tensor, Ee, can be directly calculated from the elastic portion of

the deformation gradient using

Ee =

1
2

(cid:16)Fe

T Fe − I(cid:17)

,

(K.2)

where I is the second rank identity tensor. The corresponding work-conjugate of Ee can be found
through

S = C Ee = Fe−1(det Fe) \ Fe
where C is the stiffness tensor and \ is the Cauchy stress tensor.

-T

,

The plastic deformation rate is defined with

(cid:219)Fp = Lp Fp.

where the plastic velocity gradient Lp is associated with dislocation slip through

Lp = (cid:219)γαmα ⊗ nα.

(K.3)

(K.4)

(K.5)

In the equation above, α is used to refer to the type of slip system, which is defined by its slip
direction mα and slip plane normal nα. The symbol (cid:219)γα represents the shear rate for slip system α,
which links the overall evolution of the deformation gradient with the activity of each slip system
at each material point.

In the basic phenomenological power–law model provided by DAMASK, (cid:219)γα is defined through

an exponential function (flow rule),

(cid:219)γα = (cid:219)γ0

sgn(cid:0)τα(cid:1)

(cid:12)(cid:12)(cid:12)(cid:12) τα

sα

(cid:12)(cid:12)(cid:12)(cid:12)n

142

where (cid:219)γ0 = 10−3 s−1 is the reference shear rate, τα = S · (mα ⊗ nα) is the resolved shear stress,
n = 50 controls the stress insensitivity for all dislocation slip, and sα represents the current slip
resistance, also known as the critical resolved shear stress. The evolution of sα is defined through
the hardening rule

(cid:18)

(cid:19)a

(cid:219)sα = h0

1 − sα
ss

qαβ (cid:219)γ β

where h0 is the reference hardening parameter for all dislocation slip interaction, ss is the saturation
stress for each slip family, parameters a = 1/3, qαα = 1, qαβ = 1.4 control the latent hardening
path, and summation is taken over β = 1, . . . , Nslip.

The flow rule and hardening rule described above effectively prescribe a particular strain
path for each slip system at each material point. This phenomenological description of strain
path avoids the expensive calculation of dislocation evolution, which results in a very efficient
constitutive model for evaluating the stress-strain responses during plastic deformation. However,
these simplified formulae above ignore the interaction between material points at the constitutive
level, effectually letting each material point deform without consulting its neighbors for the local
deformation environment. As mentioned earlier, the ignorance of local environment will lead to
the extreme situation where all dislocations can move freely within the simulated volume, which
contradicts many experimental observations where interfaces (grain boundaries, phase boundaries,
and sub-grain boundaries) can reduce slip activities in material points close to the interfaces. To
simulate the grain boundary effects on dislocation activities in the CP-FFT model, a new parameter,
strain path modifier (κα) is introduced into the flow rule,

(K.6)

(K.7)

(cid:219)γα = κα (cid:219)γ0

(cid:12)(cid:12)(cid:12)(cid:12)n
(cid:12)(cid:12)(cid:12)(cid:12) τα
(cid:19)a

sα

sgn(cid:0)τα(cid:1) ,
(cid:12)(cid:12)(cid:12)κ β (cid:219)γ β(cid:12)(cid:12)(cid:12) .

qαβ

1 − sα
ss

as well as the hardening rule,

(cid:18)

(cid:219)sα = h0

Assuming the dominant method for each material point to accommodate to its local deformation
environment is through slip transfer with its neighbors, the strain path modifier, κα can defined as

143

a function of slip transfer parameter (m(cid:48)α
i j

= max(mα
i

i

j ))
nβ

j · nα
mβ
i j ) − 0.93) + 1) + 0.9

κα(xi) = 0.1 · ERF(16.0(min(m(cid:48)α

(K.8)

where ERF(. . .) is the error function, xi is the position vector, and j = 1, . . . ,6 denotes the six
nearest neighbors for material point at xi. The physics behind the design of the strain path modifier
is that the presence of a grain boundary will impede the dislocation motion due to dislocation
pile-up unless a perfect slip transfer condition (m(cid:48) = 1.0) can be found for a given system. Hence
for situations where perfect slip transfer is not possible, the shear rate for given slip systems will
be slightly reduced according to the opacity of the grain boundary. A previous study shows slip
transfer is rarely observed for grain boundaries with m(cid:48) lower than 0.8. Thus, for cases where
m(cid:48) < 0.8, the amount of penalty on shear rate due to the presence of interface is capped at 20%,
which is arbitrarily chosen (Figure K.1).

Figure K.1: The shape function used for mapping the strain modifier κ from the geometrical
compatibility measurement for slip transfer (m(cid:48)) between two neighboring material points.

Although the efficiency of the phenomenological approach is high, the m(cid:48) calculation for each
neighbor and the local adjustment of the shear rate can significantly increase the computing time for
the proposed constitutive model, especially for simulations using implicit grain boundary paradigm

144

0.70.80.911.10.70.80.911.1m’κwhere the extra nonlocal calculation (Equation (K.8)) is carried out at each material point. As
a countermeasure for the inherent high computational cost, it is necessary to only perform the
non-local calculation for material points whose shear rate requires adjustment to accommodate its
local deformation environment such that the overall efficiency of the proposed pseudo-nonlocal
constitutive model stays close to the original phenomenological power-law model.

||Lp||Frob

Considering the dislocation slip is affected by both orientation gradient and local stress field, the
is proposed here to help
gradient of an accumulated shear based local Taylor factor, Mi =
identify material points that require slip transfer to accommodate local deformation environment.
For material points with zero local Taylor gradient (∇Mi = 0), the slip systems at xi have similar
efficiency in resolving local plastic strain as its neighbors. In other words, it is not necessary to
adjust the shear rate at xi as each material point within the neighborhood is capable of resolving
the plastic deformation without the help of slip transfer. On the other hand, if ∇Mi (cid:44) 0, dislocation
transfer among material points within the neighborhood is required to resolve the local plastic
strain, which means the nonlocal calculation is required at xi. Hence, the final formula for strain
path modifier in TGCP is defined as


 | (cid:219)γα|



κα
i =

0.1 · ERF(16.0(min(m(cid:48)α
1.0,

i j ) − 0.93) + 1) + 0.9,

if |∇Mi| ≥ mactive
if |∇Mi| < mactive

,

(K.9)

where mactive = 10−3 is the minimum local Taylor gradient required to perform nonlocal calculation
for material point at xi.
K.3 Application of TGCP for bicrystals of Ti-5Al-2.5Sn

To evaluate the simulated grain boundary effect from the proposed TGCP model, a virtual
bicrystal Ti-5Al-2.5Sn tensile sample consisting of 16x16x16 elements with periodic boundary
condition was virtually deformed up to 5% strain with the tensile axis perpendicular to the grain
boundary plane. The grain boundary effect in TGCP is interpreted by analyzing the difference in
simulated accumulated shear between TGCP and SDCP considering there is no spatially resolved
material property suitable for evaluating the grain boundary effect in the implicit grain boundary

145

paradigm. Various orientation pairs (Table K.1) were tested for the artificial bicrystal sample
to investigate how the simulated grain boundary effect varies under different local deformation
environment. A composite mesh (Figure K.2) is constructed such that the bi-crystal simulation
results are demonstrated along the tensile axis and temporal axis using the same graph. Scalar
properties from simulations are visualized with color while the tensorial properties like stress tensor
are visualized by a set of superquadric tensor glyph [192] hovering above the composite mesh. The
material parameters used in the simulation were from a previous study of the same material [152]
and the damping from strain path modifier, κ in TGCP is configured with the default cap value
(20%).

Table K.1: Illustration of orientation relationships for seven different bi-crystal case studies with
top-down view of hexagonal unit cell (tensile axis along the column direction)

case 1

case 2

case 3

case 4

case 5

As the current calculation of ∇M relies on a mesh-based neighbor search subroutine provided
by DAMASK, the volume affected by grain boundary is tied to the physical size of the elements.
Hence, the bi-crystal microstructure used here is equivalent to a cube of 16x16x16 µm, ensuring
that the grain is not too small to have the grain boundary effect penetrate the whole grain while
retaining at least 1 µm is reserved to characterize the grain boundary effect according to [193].

K.3.1 Bi-crystal with ∇M = 0 (trivial case)

In this special case (case 4 in Table K.1), the crystal orientation of grain 1 is set to (45°, 90°, 0°)
while the crystal orientation of grain 2 is set to (135°, 90°, 0°). In both grains, only one basal slip,

146

Figure K.2: The composite mesh used to visualize the data from the bi-crystal study is constructed
such that both spatial (tensile direction, normal to grain boundary plane) and temporal (time)
data can be analyzed within the same graph. Considering the local Taylor factor is an important
parameter in TGCP, the composite mesh is also warped by the local Taylor factor such that the
difference in deformation mode between two grains can be easily inspected.

[2 1 1 0](0 0 0 1) is active under the given load condition, which leads to a zero gradient of the local
Taylor factor near the grain boundary (Figure K.3). The zero gradient of the local Taylor factor
effectively turns off the nonlocal calculation in TGCP by directly bypassing the nonlocal calculation
for each material point near the grain boundary. This particular bi-crystal case demonstrates that
TGCP can degenerate back to SDCP, allowing each material point to carry out their hardening as
long as the local slip system has a similar capability regarding carrying out the plastic deformation
as its neighbors. For more general cases, the easy switch of nonlocal calculation can help speed
up the overall simulation by performing cheap evaluations of local hardening at material points in
grain interior with small orientation spread.

K.3.2 Bi-crystal with ∇M (cid:44) 0

For most grain boundary, the local Taylor gradient is not zero, which indicates that blockage of
dislocation slip due to the presence of grain boundary cannot be ignored. Case 1-3 from Table K.1
are simple cases where the top crystal (denoted as Grain 1) has a dual active slip mode, only one
of which has a high m’ with a slip system from the neighboring grain (denoted as Grain 2). Case 5
is a slightly more complicated case where more than two slip systems are active in both grains at
the same time, demonstrating how the simulated grain boundary in TGCP affects the deformation
history in a more sophisticated environment.

147

Figure K.3: The zero gradient of the local Taylor factor near the interface turns off the nonlocal
calculation for both grain 1 and grain 2, which results in a transparent grain boundary. The colored
tensor glyphs hovering above the composite surface are close to single straight lines, indicating the
local stress state in the bi-crystal remain close to pure uniaxial during the deformation.

In case 1 (Figure K.4), two slip systems, [1 2 1 0](1 0 1 0) and [1 1 2 0](1 1 0 0) are active with
similar Schmid factors under given load condition. Due to the non-zero local Taylor factor gradient,
both slip systems should be affected by the presence of a non-transparent grain boundary. However,
slip system [1 1 2 0](1 1 0 0) in Grain 1 has a high m’ (close to 1.0) with [2 1 1 0](0 0 0 1) in Grain 2.
The near-perfect slip transfer condition effectually negates the damping effect on [1 1 2 0](1 1 0 0),
allowing it to harden as if the grain boundary were transparent. On the other hand, slip system
[1 2 1 0](1 0 1 0) in Grain 1 lacks similar assisting neighboring slip system, forcing it to reduce its
accumulated shear for material points near the grain boundary. This nonlocal effect leads to an
increase in accumulated shear of [1 2 1 0](1 0 1 0) in the interior of Grain 1, mimicking an increase
in the density of [1 2 1 0](1 0 1 0) in the center of Grain 1 due to a non-transparent grain boundary.
This selective damping on [1 2 1 0](1 0 1 0) also indirectly forced [1 1 2 0](1 1 0 0) to be more active
for near grain boundary material points in Grain 1, providing a sense of dislocation flux without
explicit calculation(Figure K.4). Interestingly, the shape of the stress tensor does not seem to be
affected by the effect of grain boundary, remaining close to uniaxial stress state across the bi-crystal

148

during the entire simulation.

Figure K.4: Due to the selective damping on [1 2 1 0](1 0 1 0) (left) for near grain boundary material
points in Grain 1, the deformation mode of the near grain boundary material points in Grain 1
gradually shifted from dual slip mode to single slip mode, which is reflected as a monotonically
decreasing of local Taylor factor (M) along time axis. The selective damping also reduces the
corresponding accumulated shear, leading to an increased slip activity for [1 1 2 0](1 1 0 0) (right)
so as to accommodate for the local deformation.

The results of case 1 demonstrate that the presence of grain boundary could suppress one
slip system while encourages another for material points close to the grain boundary. However,
the magnitude of the grain boundary effect is sensitive to the initial slip condition near the grain
boundary. For example, case 2 (Table K.1) is very similar to case 1, with a slight rotation
around c-axis for Grain 2. However, this small rotation alters the local stress state, which makes
[1 2 1 0](1 0 1 0) the dominant slip system for the dual slip condition in Grain 1. As with case
1, the presence of the grain boundary impedes the slip activity for [1 2 1 0](1 0 1 0) while leaving
[1 1 2 0](1 1 0 0) in Grain 1 unscathed due to a perfect slip transfer condition (m’ = 1.0) with
[2 1 1 0](0 0 0 1) in Grain 2. Due to the dominant state of[1 2 1 0](1 0 1 0) in Grain 1, the suppression
from grain boundary does not visibly alter the overall deformation mode for Grain 1. On the other
hand, if the suppressed slip system is a non-dominant one (case 3 in Table K.1), the grain boundary
effect is such that clear difference in the local Taylor factor can be observed between the interior of
Grain 1 and regions close to grain boundary (Figure K.6).

Based on the results from simple situations (case 1-3), it is evident that TGCP approximates the
grain boundary effect by suppressing slip systems without good slip transfer support while indirectly

149

Figure K.5: In case 2, the simulated grain boundary effect in TGCP is not strong enough to suppress
a dominant slip system, but it does provide a small amount of aid in helping non-suppressed slip
systems indirectly.

Figure K.6: In case 3, the simulated grain boundary effect in TGCP visibly enhanced the dual
slip condition (increases in local Taylor factor), which leads to a evident "ridge" along the grain
boundary in Grain 1.

encouraging slip systems with high m’. To maintain the stress-strain balance at the grain level, the
influence of grain boundary can also propagate into grain interior through a “seesaw” effect where
the suppression of one slip system near the interface will lead to an increase of slip activity for the
same slip system in grain interior. This simple heterogeneous modification of slip activity based
on slip transfer condition with neighboring material points can also lead to more complex grain
boundary effect when multiple slip systems are involved. In case 5, two basal slip systems, three
prismatic slip systems and two pyramidal <c+a> slip systems are active with given load condition
(Figure K.7). Due to the complicated local stress state (Figure K.8), all slip systems in near grain
boundary material points are somewhat affected by the presence of the grain boundary. Unlike
the simple ”seesaw“ effect observed in case 1-3, the effect of grain boundary on each slip system

150

appears to be more “chaotic”. For example, [2 1 1 0](0 1 1 0) in Grain 1 is indirectly encouraged
due to high m’ with [2 1 1 3](1 0 1 1) in Grain 2.
If this were dual–slip condition, a monotonic
decreasing of slip activity for [2 1 1 0](0 1 1 0) in the interior of Grain 1 would have been observed
in the simulation results. Instead, the slip activity of [2 1 1 0](0 1 1 0) in the grain interior of Grain 1
gradually moves from being suppressed to encouraged and to suppressed again as the deformation
progressed, which could be the result of secondary slip system being activated and affected by grain
boundary.

Figure K.7: Case 5 demonstrates how complicated grain boundary effect can be simulated with
TGCP when multiple slip systems are active at the same time. Due to the initial high local Taylor
factor value, opacity mapping is applied to the composite mesh to add transparency to material
points with near zero values.

Due to the complicated composite effect from multiple active slip system, it’s hard to precisely
describe how the presence of grain boundary affects the hardening history of each active slip system.
However, it also demonstrates how a simple implementation of selective modification of hardening
rate can lead to various grain boundary effects, which not only depends on the slip system but the
neighboring deformation environment as well.

151

Figure K.8: The stress state in case 5 is more complicated than dual slip conditions. Due to the
slip activity of multiple slip systems, the local stress state deviates from the global uniaxial loading
condition. However, the grain boundary effect does not drastically alter the stress state near the
grain boundary as the shape and orientation of the tensor glyph near the grain boundary are similar
to those from grain interior.

K.4 Summary and future work

In this study, a modified phenomenological power-law based constitutive model TGCP is pro-
posed to simulate the effect of grain boundary during plastic deformation. By incorporating local
Taylor factor gradient and the slip transfer parameter (m’) into an existing standard phenomenologi-
cal power–law subroutine, various grain boundary effects were observed for five different bi-crystal
cases. For dual active slip condition, a “seesaw” type grain boundary effect is seen where the
suppression of one slip system near grain boundary could lead to an increase of slip activity for
the same slip system in the grain interior. Furthermore, any non-suppressed slip system due to slip
transfer support from neighboring material points (high m’) will usually get an indirect boost in slip
activity due to the suppression of another slip system. For situations where multiple slip systems
are active, the simulated grain boundary effect on each slip system is more chaotic and challenging
to deconvolute, especially when subsequent slip activation is involved. However, the simulation
results provide indirect evidence that it is possible to construct a simple model to describe grain
boundary effect despite various grain boundary effect are reported from experiment observation.

152

There are numerous opportunities for expanding the capability of TGCP model. For example,
the current model does not consider dislocation type, which can be easily implemented as the
interface plane normal between neighbors can be easily identified in the spectral mesh. Another
possible future development of this model can be focusing on applying it to microstructure con-
structed from real 3D data as well as removing its mesh sensitivity by providing a physical distance
based neighboring material points search subroutine.

153

APPENDIX L

ISOTROPIC BUFFER EFFECT ON THE ACCURACY OF LOCAL

MICROMECHANICAL RESPONSES

L.1 Introduction

Simulating local micro-mechanics using real microstructure often uses an isotropic buffer
volume, denoted as “rim” here, surrounding the microstructure mesh to approximate the long-
range effect from the matrix materials. In this study, the effectiveness of using isotropic buffers to
approximate the long-range effect from the matrix materials is investigated using CPFFT simulation
with synthetic microstructures. The results indicate that the long-range effect of the matrix materials
cannot be adequately captured using an isotropic buffer, and the benefits of using isotropic buffers
are mostly limited to the grains near the interface to the buffer, which is often not the grains of
interests.

L.2 Methodology

A synthetic microstructure consist of 8000 grains is partitioned in a 128 × 128 × 128 spectral
(voxelated) mesh. This microstructure denoted as “REF”, is deformed using a spectral solver up
to 10% uniaxial tensile strain using the same constitutive model described in Section 7.1 along
with the material parameter set 1 (PS1). One-eighth of the “REF” (opaque rainbow, Figure L.1) is
extracted and enclosed in an isotropic rim (dark blue, Figure L.1), then deformed under the same
condition. The stress/strain tensors of the extracted volume are compared between “REF” and those
enclosed in isotropic buffers with different thickness, the results of which are used to evaluate the
effectiveness of using isotropic buffers to approximate the loading environment provided by the
matrix material.

154

Figure L.1: A subset volume enclosed in an isotropic buffer (rim) is extracted from a synthetic
microstructure consisting of 8000 grains (semi-transparent, 128 × 128 × 128 grids).

L.3 Results

Figure L.2 shows the statistics of the Cauchy stress and strain extracted from both “REF” and
those with different rim size (in voxels) using box plots. Despite the substantial variation in the
rim sizes (0 → 64), the overall statistic distributions of Cauchy stress/strain from meshes with rims
remain roughly the same, about 10% lower than those from the correct/reference solution from
“REF”. In other words, there are no statistical benefits of using isotropic buffers to approximate
the matrix material as the resulting stress-strain responses will always be different from the exact
solution regardless of the thickness of the rims.

Although the rim size has almost no effect on the statistic distribution of the approximated
stress/strain, it does have some limited impact on the voxels close to the interface. To quantitatively
analyze this effect, a geometry based quantity, buffer proximity, is defined as the normalized
shortest distance between a voxel and the interface in all three directions, where 0 means right next
to the buffer/rim and 0.5 means right in the center of the volume. The distributions of Cauchy
stress and strain were then plotted with respect to this buffer proximity using band-plot introduced

155

Figure L.2: The deviation due to using isotropic rim with various thickness expressed in terms of
Cauchy stress (left) and strain (right).

in Chapter 5, revealing that the buffer size does have insufficient effect on voxels close to the
interface where a relatively thinner rim provide slightly better approximation than the thick ones
(Figure L.3). The same conclusion can also be drawn from a similar analysis of Piolar-Kirchoff
stress and deformation gradient deviation shown in Figure L.4.

L.4 Summary

The results from this study show that using isotropic buffers is not an effective way to approx-
imate the long-range effect from the matrix materials in CPFFT simulations. Regardless of the
thickness of the buffer, including zero thickness, there is always about 10% off in the central region
compared with the true stress-strain state.

156

0481624324864rimSize/voxel1061071081091010∆σCauchyσCauchy,ref0481624324864rimSize/voxel10-410-310-210-1100∆εCauchyεCauchy,refFigure L.3: The deviation due to using isotropic rim with various thickness expressed in terms
of Cauchy stress (left) and strain(right) discretized with respect to the proximity to the interface
to buffer (x-axis, 0 means at the interface to buffer and 0.5 represents the center of the extracted
volume).

Figure L.4: The deviation due to using isotropic rim with various thickness expressed in terms of
Piola-Kirchhoff stress (left) and deformation gradient deviation (right) discretized with respect to
the proximity to the interface to buffer (x-axis, 0 means at the interface to buffer and 0.5 represents
the center of the extracted volume).

157

0.00.10.20.30.40.5buffer proximity1071081091010σCauchyrefDelta_rim0Delta_rim4Delta_rim8Delta_rim16Delta_rim32Delta_rim640.00.10.20.30.40.5buffer proximity10-310-210-1100
CauchyrefDelta_rim0Delta_rim4Delta_rim8Delta_rim16Delta_rim32Delta_rim640.00.10.20.30.40.5buffer proximity108109∆prim0rim4rim8rim16rim32rim640.00.10.20.30.40.5buffer proximity10-210-1100∆frim0rim4rim8rim16rim32rim64BIBLIOGRAPHY

158

BIBLIOGRAPHY

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

S. Zaefferer and G. Habler. chapter Scanning electron microscopy and electron backscatter
diffraction, pages 37–95. Mineralogical Society of Great Britain & Ireland, 1st edition, 2017.

F. Bridier, P. Villechaise, and J. Mendez. Analysis of the different slip systems activated
by tension in a α/β titanium alloy in relation with local crystallographic orientation. Acta
Materialia, 53(3):555–567, 2005.
Z. Keshavarz and M. R. Barnett. Ebsd analysis of deformation modes in mg–3al–1zn. Scripta
Materialia, 55(10):915–918, 2006.
H. Li, C. J. Boehlert, T. R. Bieler, and M. A. Crimp. Analysis of slip activity and hetero-
geneous deformation in tension and tension-creep of ti–5al–2.5sn (wt %) using in-situ sem
experiments. Philosophical Magazine, 92(23):2923–2946, 2012.
J. R. Seal, M. A. Crimp, T. R. Bieler, and C. J. Boehlert. Analysis of slip transfer and
deformation behavior across the α/β interface in ti–5al–2.5sn (wt.%) with an equiaxed
microstructure. Materials Science and Engineering A, 552:61–68, 2012.
J. Ahmed, A. J. Wilkinson, and S. G. Roberts. Characterizing dislocation structures in
bulk fatigued copper single crystals using electron channelling contrast imaging (ecci).
Philosophical Magazine, pages 237–246, 1997.
J. Ahmed, A. J. Wilkinson, and S. G. Roberts. Study of dislocation structures near fatigue
cracks using electron channelling contrast imaging technique (ecci). Journal of Microscopy,
195(3):197–203, 1999.

J. R. Seal, T. R. Bieler, M. A. Crimp, T. B. Britton, and A. J. Wilkinson. Characterizing slip
transfer in commercially pure titanium using high resolution electron backscatter diffraction
(hr-ebsd) and electron channeling contrast imaging (ecci). Microscopy and Microanalysis,
18(S2):702–703, 2012.
D. Hull and D. J. Bacon. Introduction to Dislocations. Butterworth-Heinemann, 5th edition,
2011.

[10] M. Bertin, C. Du, J. P. M. Hoefnagels, and F. Hild. Crystal plasticity parameter identification
with 3d measurements and integrated digital image correlation. Acta Materialia, 116:321–
331, 2016.

[11] Z. Chen and S. H. Daly. Active slip system identification in polycrystalline metals by digital

image correlation (dic). Experimental Mechanics, 57(1):115–127, 2017.

[12] S-H Joo, J. K. Lee, J-M Koo, S. Lee, D-W Suh, and H. S. Kim. Method for measuring
nanoscale local strain in a dual phase steel using digital image correlation with nanodot
patterns. Scripta Materialia, 68(5):245–248, 2013.

159

[13] Y. Yang, L. Wang, T. R. Bieler, P. Eisenlohr, and M. A. Crimp. Quantitative atomic force
microscopy characterization and crystal plasticity finite element modeling of heterogeneous
deformation in commercial purity titanium. Metallurgical and Materials Transactions A,
42(3):636–644, 2011.

[14] F. Wang, S. Sandlöbes, M. Diehl, L. Sharma, F. Roters, and D. Raabe. In situ observation of
collective grain-scale mechanics in mg and mg–rare earth alloys. Acta Materialia, 80:77–93,
2014.

[15] T. A. Book and M. D. Sangid. Strain localization in ti-6al-4v widmanstätten microstructures

produced by additive manufacturing. Materials Characterization, 122:104–112, 2016.

[16] National Institute of Standards and Technology. Estar, stopping power and range tables for

electrons.

[17] F. Bachmann, R. Hielscher, and H. Schaeben. Grain detection from 2d and 3d ebsd data—

specification of the mtex algorithm. Ultramicroscopy, 111(12):1720–1733, 2011.

[18] M. Calcagnotto, D. Ponge, E. Demir, and D. Raabe. Orientation gradients and geometrically
necessary dislocations in ultrafine grained dual-phase steels studied by 2d and 3d ebsd.
Materials Science and Engineering A, 527(10-11):2738–2746, 2010.

[19] M. Diehl, D. An, P. Shanthraj, S. Zaefferer, F. Roters, and D. Raabe. Crystal plasticity study
on stress and strain partitioning in a measured 3d dual phase steel microstructure. Physical
Mesomechanics, 20(3):311–323, 2017.

[20] A. Khorashadizadeh, D. Raabe, S. Zaefferer, G. S. Rohrer, A. D. Rollett, and M. Winning.
Five-parameter grain boundary analysis by 3d ebsd of an ultra fine grained cuzr alloy
processed by equal channel angular pressing. Advanced Engineering Materials, 13(4):237–
244, 2011.

[21] L. A. Giannuzzi and F. A. Stevie. A review of focused ion beam milling techniques for tem

specimen preparation. Micron, 30(3):197–204, 1999.

[22] H. F. Poulsen, S. Garbe, T. Lorentzen, D. Juul Jensen, F. W. Poulsen, N. H. Andersen,
T. Frello, R. Feidenhans’l, and H. Graafsma. Applications of high-energy synchrotron
radiation for structural studies of polycrystalline materials. Journal of Synchrotron Radiation,
4(3):147–154, 1997.

[23] H. F. Poulsen, S. F. Nielsen, E. M. Lauridsen, S. Schmidt, R. M. Suter, U. Lienert, L. Mar-
gulies, T. Lorentzen, and D. Juul Jensen. Three-dimensional maps of grain boundaries
and the stress state of individual grains in polycrystals and powders. Journal of Applied
Crystallography, 34(6):751–756, 2001.

[24] B. C. Larson, W. Yang, G. E. Ice, J. D. Budai, and J. Z. Tischler. Three-dimensional X-ray
structural microscopy with submicrometre resolution. Nature, 415(6874):887–890, 2002.
[25] Wenjun Liu, Gene E. Ice, Bennett C. Larson, Wenge Yang, Jonathan Z. Tischler, and
J. D. Budai. The three-dimensional x-ray crystal microscope: A new tool for materials
characterization. Metallurgical and Materials Transactions A, 35(7):1963–1967, 2004.

160

[26]

I. Robinson and R. Harder. Coherent x-ray diffraction imaging of strain at the nanoscale.
Nature Materials, 8(4):291–298, 2009.

[27] E. L. Ritman. Micro-computed tomography—current status and developments. Annual

Review of Biomedical Engineering, 6(1):185–208, 2004.

[28] G. I. Taylor. Plastic strain in metals. J. Inst. Metals, 62:307–324, 1938.
[29] K. Hwang, Y. Guo, H. Jiang, Y. Huang, and Z. Zhuang. The finite deforamtion theory of
Taylor-based nonlocal plasticity. International Journal of Plasticity, 20(4-5):831–839, 2004.
[30] F. Delaire, J. L. Raphanel, and C. Rey. Plastic heterogeneities of a copper multicrystal
deformed in uniaxial tension: experimental study and finite element simulations. Acta
Materialia, 48(5):1075–1087, 2000.

[31] T. R. Bieler, P. Eisenlohr, F. Roters, D. Kumar, D. E. Mason, M. A. Crimp, and D. Raabe.
The role of heterogeneous deformation on damage nucleation at grain boundaries in single
phase metals. International Journal of Plasticity, 25(9):1655–1683, 2009.

[32] F. Roters, P. Eisenlohr, L. Hantcherli, D. D. Tjahjanto, T. R. Bieler, and D. Raabe. Overview
of constitutive laws, kinematics, homogenization, and multiscale methods in crystal plasticity
finite element modeling: Theory, experiments, applications. Acta Materialia, 58:1152–1211,
2010.
J. F. Nye. Some geometrical relations in dislocated crystals. Acta Metallurgica, 1(2):153–
162, 1953.

[33]

[34] B. A. Bilby. Types of dislocation source. In Defects in crystalline solids. Report of the
Conference of the International Union of Physics, pages 124–133, London, 1955. The
Physical Society.
J. W. Christian and S. Mahajan. Deformation twinning. Progress in Materials Science,
39(1-2):1–157, 1995.

[35]

[36] L. Wang, P. Eisenlohr, Y. Yang, T. R. Bieler, and M. A. Crimp. Nucleation of paired twins

at grain boundaries in titanium. Scripta Materialia, 63:827–830, 2010.

[37] S. Z. Wu, H. W. Yen, M. X. Huang, and A. H. W. Ngan. Deformation twinning in submicron
and micron pillars of twinning-induced plasticity steel. Scripta Materialia, 67(7-8):641–644,
2012.

[38] L. Wang, R. Barabash, T. Bieler, W. Liu, and P. Eisenlohr. Study of {1 1 ¯2 1} twinning
in α-ti by ebsd and laue microdiffraction. Metallurgical and Materials Transactions A,
44(8):3664–3674, 2013.

[39] H. Abdolvand, M. Majkut, J. Oddershede, S. Schmidt, U. Lienert, B. J. Diak, P. J. Withers,
and M. R. Daymond. On the deformation twinning of mg az31b: A three-dimensional
synchrotron x-ray diffraction experiment and crystal plasticity finite element model. Inter-
national Journal of Plasticity, 70:77–97, 2015.

161

[40] B. A. Bilby and A. G. Crocker. The theory of the crystallography of deformation twinning.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
288(1413):240–255, 1965.

[41] A. T. Paxton, P. Gumbsch, and M. Methfessel. A quantum mechanical calculation of the

theoretical strength of metals. Philosophical Magazine Letters, 63(5):267–274, 1991.
J. D. Clayton. Nonlinear mechanics of crystals, volume 177, chapter Mechanical Twinning
in Crystal Plasticity, pages 379–421. Springer Netherlands, Dordrecht, 2011.

[42]

[43]

I. Weiss and S. L. Semiatin. Thermomechanical processing of alpha titanium alloys—an
overview. Materials Science and Engineering: A, 263(2):243–256, 1999.

[44] R. Boyer, G. Welsch, and E. W. Collings, editors. Materials Properties Handbook: Titanium

Alloys. ASM International, Metals Park, OH, 1995.

[45] Q. Y. Sun and H. C. Gu. Tensile and low-cycle fatigue behavior of commercially pure
titanium and ti–5al–2.5sn alloy at 293 and 77 k. Materials Science and Engineering: A,
316(1-2):80–86, 2001.

[46] Christoph Leyens and Manfred Peters. Titanium and Titanium Alloys. 2011.
[47] M. H. Yoo. Slip, twinning, and fracture in hexagonal close-packed metals. Metallurgical

Transactions A, 12A:409–418, 1981.

[48] H. Li, D. E. Mason, Y. Yang, T. R. Bieler, M. A. Crimp, and C. J. Boehlert. Comparison
of the deformation behaviour of commercially pure titanium and ti–5al–2.5sn(wt.%) at 296
and 728 k. Philosophical Magazine, 93(21):2875–2895, 2013.

[49] A. Poty, J-M Raulot, H. Xu, J. Bai, C. Schuman, J-S Lecomte, M-J Philippe, and C. Esling.
Classification of the critical resolved shear stress in the hexagonal-close-packed materials by
atomic simulation: Application to α-zirconium and α-titanium. Journal of Applied Physics,
110(1):014905–, 2011.

[50] H. Li, D. E. Mason, T. R. Bieler, C. J. Boehlert, and M. A. Crimp. Methodology for
estimating the critical resolved shear stress ratios of α-phase Ti using EBSD-based trace
analysis. Acta Materialia, 61(20):7555–7567, 2013.

[51] Y. Su, C. Zambaldi, D. Mercier, P. Eisenlohr, T. R. Bieler, and M. A. Crimp. Quantifying
deformation processes near grain boundaries in alpha-titanium using nanoindentation and
crystal plasticity modeling. International Journal of Plasticity, 86:170–186, 2016.

[52] M. L. Bouxsein, S. K. Boyd, B. A. Christiansen, R. E. Guldberg, K. J. Jepsen, and R. Müller.
Guidelines for assessment of bone microstructure in rodents using micro-computed tomog-
raphy. Journal of Bone and Mineral Research, 25(7):1468–1486, 2010.

[53] V. Cnudde and M. N. Boone. High-resolution x-ray computed tomography in geosciences:
A review of the current technology and applications. Earth-Science Reviews, 123:1–17,
2013.

162

[54] H. Bale, M. Blacklock, M. R. Begley, D. B. Marshall, B. N. Cox, and R. O. Ritchie.
Characterizing three-dimensional textile ceramic composites using synchrotron x-ray micro-
computed-tomography. Journal of the American Ceramic Society, 95(1):392–402, 2011.

[55] G. N. Hounsfield. Computerized transverse axial scanning (tomography): Part 1. description

of system. The British Journal of Radiology, 46(552):1016–1022, 1973.

[56] W. Ludwig, S. Schmidt, E. M. Lauridsen, and H. F. Poulsen. X-ray diffraction contrast
tomography: a novel technique for three-dimensional grain mapping of polycrystals. i. direct
beam case. Journal of Applied Crystallography, 41(2):302–309, 2008.

[57] P. Reischig, A. King, L. Nervo, N. Viganó, Y. Guilhem, W. J. Palenstijn, K. J. Batenburg,
M. Preuss, and W. Ludwig. Advances in x-ray diffraction contrast tomography: flexibility
in the setup geometry and application to multiphase materials. Journal of Applied Crystal-
lography, 46(2):297–311, 2013.

[58] M. Syha, A. Trenkle, B. Lödermann, A. Graff, W. Ludwig, D. Weygand, and P. Gumbsch.
Validation of three-dimensional diffraction contrast tomography reconstructions by means
of electron backscatter diffraction characterization. Journal of Applied Crystallography,
46(4):1145–1150, 2013.
J. Miao, R. L. Sandberg, and C. Song. Coherent x-ray diffraction imaging. IEEE Journal of
Selected Topics in Quantum Electronics, 18(1):399–410, 2012.
J. Miao, P. Charalambous, J. Kirz, and D. Sayre. Extending the methodology of x-ray
crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature,
400(6742):342–344, 1999.

[59]

[60]

[61] A. G. Kikhney and D. I. Svergun. A practical guide to small angle x-ray scattering (saxs)
of flexible and intrinsically disordered proteins. FEBS Letters, 589(19PartA):2570–2577,
2015.

[62] S. Skou, R. E. Gillilan, and N. Ando. Synchrotron-based small-angle x-ray scattering of

proteins in solution. Nature Protocols, 9(7):1727–1739, 2014.

[63] G. Zhu, W. G. Saw, A. Nalaparaju, G. Grüber, and L. Lu. Coarse-grained molecular modeling
of the solution structure ensemble of dengue virus nonstructural protein 5 with small-angle
x-ray scattering intensity. The Journal of Physical Chemistry B, 121(10):2252–2264, 2017.
[64] C. P. Jones, W. A. Cantara, E. D. Olson, and K. Musier-Forsyth. Small-angle x-ray scattering-
derived structure of the hiv-1 5 utr reveals 3d trna mimicry. Proceedings of the National
Academy of Sciences, 111(9):3395–3400, 2014.

[65] S. Fischer, C. Hartl, K. Frank, J. O. Rädler, T. Liedl, and B. Nickel. Shape and interhelical
spacing of dna origami nanostructures studied by small-angle x-ray scattering. Nano Letters,
16(7):4282–4287, 2016.

[66] L. K. Bruetzel, P. U. Walker, T. Gerling, H. Dietz, and J. Lipfert. Time-resolved small-
angle x-ray scattering reveals millisecond transitions of a dna origami switch. Nano Letters,
18(4):2672–2676, 2018.

163

[67]

Jens Als-Nielsen and Des McMorrow. Elements of Modern X-ray Physics. Wiley, Hoboken,
UNITED KINGDOM, 2011.

[68] H. Poulsen. Three-Dimensional X-Ray Diffraction Microscopy. Springer Berlin Heidelberg,

Berlin, Heidelberg, 2004.

[69] R. M. Suter, D. Hennessy, C. Xiao, and U. Lienert. Forward modeling method for mi-
crostructure reconstruction using x-ray diffraction microscopy: Single-crystal verification.
Review of Scientific Instruments, 77(12), 2006.

[70] U. Lienert, S. F. Li, C. M. Hefferan, J. Lind, R. M. Suter, J. V. Bernier, N. R. Barton, M. C.
Brandes, M. J. Mills, M. P. Miller, B. Jakobsen, and W. Pantleon. High-energy diffraction
microscopy at the advanced photon source. JOM, 63(7):70–77, 2011.
J. V. Bernier, N. R. Barton, U. Lienert, and M. P. Miller. Far-field high-energy diffraction
microscopy: a tool for intergranular orientation and strain analysis. The Journal of Strain
Analysis for Engineering Design, 46(7):527–547, 2011.

[71]

[72] S. Schmidt. Watching the growth of bulk grains during recrystallization of deformed metals.

Science, 305(5681):229–232, 2004.

[73] L. Wang, Z. Zheng, H. Phukan, P. Kenesei, J-S Park, J. Lind, R. M. Suter, and T. R.
Bieler. Direct measurement of critical resolved shear stress of prismatic and basal slip in
polycrystalline ti using high energy x-ray diffraction microscopy. Acta Materialia, 132:598–
610, 2017.

[74] W. Yang, B. C. Larson, J. Z. Tischler, G. E. Ice, J. D. Budai, and W. Liu. Differential-
aperture x-ray structural microscopy: a submicron-resolution three-dimensional probe of
local microstructure and strain. Micron, 35(6):431–439, 2004.

[75] P. R. Dawson. Computational crystal plasticity. International Journal of Solids and Struc-

tures, 37(1-2):115–130, 2000.

[76] G. Sachs. Mitteilungen der deutschen Materialprüfungsanstalten, chapter Zur Ableitung

einer Fließbedingung, pages 94–97. Springer, Berlin, Heidelberg, 1929.

[77] L. P. Evers, D. M. Parks, W. A. M. Brekelmans, and M. G. D. Geers. Crystal plasticity model
with enhanced hardening by geometrically necessary dislocation accumulation. Journal of
the Mechanics and Physics of Solids, 50:2403–2424, 2002.

[78] M. Knezevic and D. J. Savage. A high-performance computational framework for fast crystal

plasticity simulations. Computational Materials Science, 83:101–106, 2014.

[79] M. Zecevic, R. J. McCabe, and M. Knezevic. A new implementation of the spectral crystal
plasticity framework in implicit finite elements. Mechanics of Materials, 84:114–126, 2015.
[80] F R N Nabarro. Dislocations in a simple cubic lattice. Proceedings of the Physical Society,

59(2):256, 1947.

164

[81] Yuri S. Kivshar and David K. Campbell. Peierls-nabarro potential barrier for highly localized

nonlinear modes. Phys. Rev. E, 48:3077–3081, Oct 1993.

[82] B. Joós, Q. Ren, and M. S. Duesbery. Peierls-nabarro model of dislocations in silicon with

generalized stacking-fault restoring forces. Phys. Rev. B, 50:5890–5898, Sep 1994.

[83] P. Carrez, D. Ferré, and P. Cordier. Peierls–nabarro model for dislocations in mgsio3post-
perovskite calculated at 120 gpa from first principles. Philosophical Magazine, 87(22):3229–
3247, 2007.

[84] H. Saka and T. Imura. Direct measurement of mobility of edge and screw dislocations in
3% silicon-iron by high voltage transmission electron microscopy. Journal of the Physical
Society of Japan, 32(3):702–716, 1972.

[85] D. F. Stein and J. R. Low Jr. Mobility of edge dislocations in silicon-iron crystals. Journal

of Applied Physics, 31(2):362–369, 1960.

[86] P A Gordon, T Neeraj, Y Li, and J Li. Screw dislocation mobility in bcc metals: the role of
the compact core on double-kink nucleation. Modelling and Simulation in Materials Science
and Engineering, 18(8):085008, 2010.
J. R. Rice.
Inelastic constitutive relations for solids: an internal variable theory and its
application to metal plasticity. Journal of the Mechanics and Physics of Solids, 19:433–455,
1971.

[87]

[88]

J. W. Hutchinson. Bounds and self-consistent estimates for creep of polycrystalline materials.
Proceedings of the Royal Society A, 348:101–127, 1976.

[89] D. Peirce, R. J. Asaro, and A. Needleman. An analysis of nonuniform and localized defor-

mation in ductile single crystals. Acta Metallurgica, 30(6):1087–1119, 1982.

[90] K-S Cheong and E. P. Busso. Discrete dislocation density modelling of single phase FCC

polycrystal aggregates. Acta Materialia, 52(19):5665–5675, 2004.

[91] A. Ma and F. Roters. A constitutive model for fcc single crystals based on dislocation
densities and its application to uniaxial compression of aluminium single crystals. Acta
Materialia, 52(12):3603–3612, 2004.

[92] A. Ma, F. Roters, and D. Raabe. A dislocation density based constitutive model for crystal
plasticity fem including geometrically necessary dislocations. Acta Materialia, 54:2169–
2179, 2006.

[93] L. P. Evers, W. A. M. Brekelmans, and M. G. D. Geers. Scale dependent crystal plasticity
framework with dislocation density and grain boundary effects. International Journal of
Solids and Structures, 41(18-19):5209–5230, 2004.

[94] A. Ma, F. Roters, and D. Raabe. On the consideration of interactions between dislocations
and grain boundaries in crystal plasticity finite element modeling – theory, experiments, and
simulations. Acta Materialia, 54(8):2181–2194, 2006.

165

[95] A. Alankar, I. N. Mastorakos, and D. P. Field. A dislocation-density-based 3d crystal

plasticity model for pure aluminum. Acta Materialia, 59(19):5936–5946, 2009.

[96] A. Alankar, P. Eisenlohr, and D. Raabe. A dislocation density-based crystal plasticity
constitutive model for prismatic slip in α-titanium. Acta Materialia, 59(18):7003–7009,
2011.

[97] E. Van der Giessen and A. Needleman. Discrete dislocation plasticity: a simple planar
model. Modelling and Simulation in Materials Science and Engineering, 3:689–735, 1995.
[98] A. Arsenlis and D. M. Parks. Modeling the evolution of crystallographic dislocation density
in crystal plasticity. Journal of the Mechanics and Physics of Solids, 50(9):1979–2009, 2002.
[99] T. Hochrainer, S. Sandfeld, M. Zaiser, and P. Gumbsch. Continuum dislocation dynamics:
Towards a physical theory of crystal plasticity. Journal of the Mechanics and Physics of
Solids, 63:167–178, 2014.

[100] P-L Valdenaire, Y. Le Bouar, B. Appolaire, and A. Finel. Density-based crystal plasticity:

From the discrete to the continuum. Physical Review B, 93(21), 2016.

[101] K. Schulz, M. Sudmanns, and P. Gumbsch. Dislocation-density based description of the
deformation of a composite material. Modelling and Simulation in Materials Science and
Engineering, 25(6):064003–, 2017.

[102] N. Grilli, K. G. F. Janssens, J. Nellessen, S. Sandlöbes, and D. Raabe. Multiple slip disloca-
tion patterning in a dislocation-based crystal plasticity finite element method. International
Journal of Plasticity, 100:104–121, 2017.

[103] D. Raabe, M. Sachtleber, H. Weiland, G. Scheele, and Z. Zhao. Grain-scale micromechanics

of polycrystal surfaces during plastic straining. Acta Materialia, 51:1539–1560, 2003.

[104] L. Delannay and M. R. Barnett. Modelling the combined effect of grain size and grain shape

on plastic anisotropy of metals. International Journal of Plasticity, 32-33:70–84, 2012.

[105] R. T. DeHoff. Quantitative serial sectioning analysis: preview. Journal of Microscopy,

131(3):259–263, 2011.

[106] M. A. Groeber, B. K. Haley, M. D. Uchic, D. M. Dimiduk, and S. Ghosh. 3d reconstruction
and characterization of polycrystalline microstructures using a fib–sem system. Materials
Characterization, 57(4-5):259–273, 2006.

[107] M. A. Groeber and M. A. Jackson. Dream.3d: A digital representation environment for
the analysis of microstructure in 3d. Integrating Materials and Manufacturing Innovation,
3(1):5–, 2014.

[108] P. A. Shade, M. A. Groeber, J. C. Schuren, and M. D. Uchic. Experimental measurement
of surface strains and local lattice rotations combined with 3d microstructure reconstruction
from deformed polycrystalline ensembles at the micro-scale.
Integrating Materials and
Manufacturing Innovation, 2(1):5, 2013.

166

[109] A. Brahme, M. H. Alvi, D. Saylor, J. Fridy, and A. D. Rollett. 3d reconstruction of

microstructure in a commercial purity aluminum. Scripta Materialia, 55(1):75–80, 2006.

[110] D. M. Turner and S. R. Kalidindi. Statistical construction of 3-d microstructures from 2-d

exemplars collected on oblique sections. Acta Materialia, 102:136–148, 2016.

[111] D. T. Fullwood, S. R. Niezgoda, and S. R. Kalidindi. Microstructure reconstructions from
2-point statistics using phase-recovery algorithms. Acta Materialia, 56(5):942–948, 2008.
[112] R. Bostanabad, A. T. Bui, W. Xie, D. W. Apley, and W. Chen. Stochastic microstructure
characterization and reconstruction via supervised learning. Acta Materialia, 103:89–102,
2016.

[113] A. Staroselsky and L. Anand. A constitutive model for hcp materials deforming by slip and

twinning. International Journal of Plasticity, 19(10):1843–1864, 2003.

[114] H. Abdolvand, M. R. Daymond, and C. Mareau.

Incorporation of twinning into a crys-
tal plasticity finite element model: Evolution of lattice strains and texture in zircaloy-2.
International Journal of Plasticity, 27(11):1721–1738, 2011.

[115] J. Cheng, J. Shen, R. K. Mishra, and S. Ghosh. Discrete twin evolution in mg alloys using a

novel crystal plasticity finite element model. Acta Materialia, 149:142–153, 2018.

[116] M. Knezevic, B. Drach, M. Ardeljan, and I. J. Beyerlein. Three dimensional predictions
of grain scale plasticity and grain boundaries using crystal plasticity finite element models.
Computer Methods in Applied Mechanics and Engineering, 277:239–259, 2014.

[117] P. Zhang, M. Karimpour, D. Balint, J. Lin, and D. Farrugia. A controlled poisson voronoi
tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis.
Computational Materials Science, 64:84–89, 2012.

[118] M. Cherkaoui, M. Berveiller, and H. Sabar. Micromechanical modeling of martensitic
transformation induced plasticity (trip) in austenitic single crystals. International Journal of
Plasticity, 14(7):597–626, 1998.

[119] D. Raabe and R. C. Becker. Coupling of a crystal plasticity finite-element model with a
probabilistic cellular automaton for simulating primary static recrystallization in aluminium.
Modelling and Simulation in Materials Science and Engineering, 8(4):445, 2000.

[120] M-G Lee, S-J Kim, and H. N. Han. Crystal plasticity finite element modeling of mechanically
induced martensitic transformation (mimt) in metastable austenite. International Journal of
Plasticity, 26(5):688–710, 2010.

[121] S. Manchiraju and P. M. Anderson. Coupling between martensitic phase transformations and
plasticity: A microstructure-based finite element model. International Journal of Plasticity,
26(10):1508–1526, 2010.

[122] P. Eisenlohr, M. Diehl, R. A. Lebensohn, and F. Roters. A spectral method solution to crystal
elasto-viscoplasticity at finite strains. International Journal of Plasticity, 46:37–53, 2013.

167

[123] M. Malahe. An efficient spectral crystal plasticity solver for gpu architectures. Computational

Mechanics, 2018.

[124] P. Eisenlohr, P. Shanthraj, B. R. Vande Kieft, T. R. Bieler, W. Liu, and R. Xu. Subsurface grain
morphology reconstruction by differential aperture x-ray microscopy. JOM, 69(6):1100–
1105, 2017.

[125] A. Bermano, A. Vaxman, and C. Gotsman. Online reconstruction of 3d objects from arbitrary

cross-sections. ACM Transactions on Graphics, 30(5):1–11, 2011.

[126] T. B. Britton and A. J. Wilkinson. Measurement of residual elastic strain and lattice rotations
with high resolution electron backscatter diffraction. Ultramicroscopy, 111(8):1395–1404,
2011.

[127] A. J. Allen, M. T. Hutchings, C. G. Windsor, and C. Andreani. Neutron diffraction methods

for the study of residual stress fields. Advances in Physics, 34(4):445–473, 1985.

[128] D. A. Hall, A. Steuwer, B. Cherdhirunkorn, T. Mori, and P. J. Withers. A high energy
synchrotron x-ray study of crystallographic texture and lattice strain in soft lead zirconate
titanate ceramics. Journal of Applied Physics, 96(8):4245–4252, 2004.

[129] J. D. Almer and S. R. Stock. Internal strains and stresses measured in cortical bone via

high-energy X-ray diffraction. Journal of Structural Biology, 152(1):14–27, 2005.

[130] V. Biju, N. Sugathan, V. Vrinda, and S. L. Salini. Estimation of lattice strain in nanocrystalline
silver from X-ray diffraction line broadening. Journal of Materials Science, 43(4):1175–
1179, 2008.

[131] N. Jia, Z. H. Cong, X. Sun, S. Cheng, Z. H. Nie, Y. Ren, P. K. Liaw, and Y. D. Wang. An in
situ high-energy X-ray diffraction study of micromechanical behavior of multiple phases in
advanced high-strength steels. Acta Materialia, 57(13):3965–3977, 2009.

[132] J. K. Edmiston, N. R. Barton, J. V. Bernier, G. C. Johnson, and D. J. Steigmann. Precision
of lattice strain and orientation measurements using high-energy monochromatic X-ray
diffraction. Journal of Applied Crystallography, 44(2):299–312, 2011.

[133] M. Sobiech, M. Wohlschlögel, U. Welzel, E. J. Mittemeijer, W. Hügel, A. Seekamp, W. Liu,
and G. E. Ice. Local, submicron, strain gradients as the cause of sn whisker growth. Applied
Physics Letters, 94(22):221901–, 2009.

[134] S. Y. Lee, R. I. Barabash, J-S Chung, P. K. Liaw, H. Choo, Y. Sun, C. Fan, L. Li, D. W. Brown,
and G. E. Ice. Neutron and x-ray microbeam diffraction studies around a fatigue-crack tip
after overload. Metallurgical and Materials Transactions A, 39(13):3164–3169, 2008.

[135] Y. Guo, D. M. Collins, E. Tarleton, F. Hofmann, J. Tischler, W. Liu, R. Xu, A. J. Wilkinson,
and T. B. Britton. Measurements of stress fields near a grain boundary: Exploring blocked
arrays of dislocations in 3d. Acta Materialia, 96:229–236, 2015.

168

[136] L. E. Levine, P. Geantil, B. C. Larson, J. Z. Tischler, M. E. Kassner, and W. Liu. Validating
classical line profile analyses using microbeam diffraction from individual dislocation cell
walls and cell interiors. Journal of Applied Crystallography, 45(2):157–165, 2012.

[137] C. Zhang, T. R. Bieler, and P. Eisenlohr. Exploring the accuracy limits of lattice strain

quantification with synthetic diffraction data. Scripta Materialia, 154:127–130, 2018.

[138] A. Poshadel, P. Dawson, and G. Johnson. Assessment of deviatoric lattice strain uncertainty
for polychromatic X-ray microdiffraction experiments. Journal of Synchrotron Radiation,
19(2):237–244, 2012.

[139] F. Hofmann, B. Abbey, W. Liu, R. Xu, B. F. Usher, E. Balaur, and Y. Liu. X-ray micro-beam
characterization of lattice rotations and distortions due to an individual dislocation. Nature
Communications, 4:2774 EP, 2013.

[140] Richard Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, USA,

2010.

[141] T. J. Ruggles and D. T. Fullwood. Estimations of bulk geometrically necessary dislocation

density using high resolution EBSD. Ultramicroscopy, 133:8–15, 2013.

[142] T. B. Britton and A. J. Wilkinson. Stress fields and geometrically necessary dislocation
density distributions near the head of a blocked slip band. Acta Materialia, 60(16):5773–
5782, 2012.

[143] M. A. Crimp. Scanning Electron Microscopy Imaging of Dislocations in Bulk Materials,
Using Electron Channeling Contrast. Microscopy Research and Technique, 69:374–381,
2006.

[144] Pierre Rolland, Keith Graham Dicks, and R Ravel-Chapuis. Ebsd spatial resolution in the
sem when analyzing small grains or deformed material. Microscopy and microanalysis,
8(s02):670–671, 01 08.

[145] R. Barabash, G. E. Ice, B. C. Larson, G. M. Pharr, K.-S. Chung, and W. Yang. White
microbeam diffraction from distorted crystals. Applied Physics Letters, 79(6):749–751,
2001.

[146] C. Zhang, S. Balachandran, P. Eisenlohr, M. A. Crimp, C. J. Boehlert, R. Xu, and T. R.
Bieler. Comparison of dislocation content measured with transmission electron microscopy
and micro-laue diffraction based streak analysis. Scripta Materialia, 144:74–77, 2018.

[147] R. I. Barabash, G. E. Ice, and F. J. Walker. Quantitative microdiffraction from deformed
Journal of Applied Physics,

crystals with unpaired dislocations and dislocation walls.
93(3):1457–1464, 2003.

[148] R. I. Barabash, G. E. Ice, N. Tamura, B. C. Valek, J. C. Bravman, R. Spolenak, and
J. R. Patel. Quantitative characterization of electromigration-induced plastic deformation in
Al(0.5wt%Cu) interconnect. Microelectronic Engineering, 75(1):24–30, 2004.

169

[149] O. P. Karasevskaya, O. M. Ivasishin, S. L. Semiatin, and Y. V. Matviychuk. Deformation
behavior of beta-titanium alloys. Materials Science and Engineering: A, 354(1-2):121–132,
2003.

[150] R. Maaß, D. Grolimund, H. Van Swygenhoven, M. Willimann, M. Jensen, T. Lehnert,
M. A. M. Gijs, C. A. Volkert, E. T. Lilleodden, and R. Schwaiger. Defect structure in
micropillars using x-ray microdiffraction. Applied Physics Letters, 89(15):151905, 2006.

[151] B. Nickel, R. Barabash, R. Ruiz, N. Koch, A. Kahn, L. C. Feldman, R. F. Haglund,
and G. Scoles. Dislocation arrangements in pentacene thin films. Physical Review B,
70(12):125401, 2004.

[152] C. Zhang, H. Li, P. Eisenlohr, W. J. Liu, C. J. Boehlert, M. A. Crimp, and T. R. Bieler. Effect
of realistic 3d microstructure in crystal plasticity finite element analysis of polycrystalline
ti-5al-2.5sn. International Journal of Plasticity, 69:21–35, 2015.

[153] J.B. Yang, Y. Nagai, and M. Hasegawa. Use of the frank–bilby equation for calculating misfit

dislocation arrays in interfaces. Scripta Materialia, 62(7):458 – 461, 2010.

[154] K-H Jung, D-K Kim, Y-T Im, and Y-S Lee. Prediction of the effects of hardening and texture
heterogeneities by finite element analysis based on the Taylor model. International Journal
of Plasticity, 42:120–140, 2013.

[155] H. Wang, K. Hwang, Y. Huang, P. Wu, B. Liu, G. Ravichandran, C. S. Han, and H. Gao.
A conventional theory of strain gradient crystal plasticity based on the Taylor dislocation
model. International Journal of Plasticity, 23(9):1540–1554, 2007.

[156] X. Wu, S. R. Kalidindi, C. Necker, and A. A. Salem. Prediction of crystallographic texture
evolution and anisotropic stress–strain curves during large plastic strains in high purity
alpha-titanium using a Taylor-type crystal plasticity model. Acta Materialia, 55:423–432,
2007.

[157] C. Rehrl, B. Völker, S. Kleber, T. Antretter, and R. Pippan. Crystal orientation changes: A
comparison between a crystal plasticity finite element study and experimental results. Acta
Materialia, 60(5):2379–2386, 2012.

[158] K-S Cheong and E. P. Busso. Effects of lattice misorientations on strain heterogeneities in

FCC polycrystals. Journal of the Mechanics and Physics of Solids, 54(4):671–689, 2006.

[159] E. Busso, F. T. Meissonnier, and N. P. O’Dowd. Gradient-dependent deformation of two-
phase single crystals. Journal of the Mechanics and Physics of Solids, 48(11):2333–2361,
2000.

[160] Y. Liu and Y. Wei. A polycrystal based numerical investigation on the temperature depen-
dence of slip resistance and texture evolution in magnesium alloy AZ31B. International
Journal of Plasticity, 55:80–93, 2014.

[161] D. S. Li, S. Ahzi, S. M’Guil, W. Wen, C. Lavender, and M. A. Khaleel. Modeling of
deformation behavior and texture evolution in magnesium alloy using the intermediate theta-
model. International Journal of Plasticity, 52:77–94, 2014.

170

[162] H. Gao, Y. Huang, W. D. Nix, and J. W. Hutchinson. Mechanism-based strain gradient
plasticity—I. Theory. Journal of the Mechanics and Physics of Solids, 47(6):1239–1263,
1999.

[163] N. A. Fleck, G. M. Muller, M. F. Ashby, and J. W. Hutchinson. Strain gradient plasticity:

Theory and experiment. Acta Metallurgica et Materialia, 42(2):475–487, 1994.

[164] D. Kuhlmann-Wilsdorf. The theory of dislocation-based crystal plasticity. Philosophical

Magazine A, 79(4):955–1008, 1999.

[165] P. Eisenlohr and F. Roters. Selecting sets of discrete orientations for accurate texture

reconstruction. Computational Materials Science, 42(4):670–678, 2008.

[166] J. A. Nelder and R. Mead. A Simplex Method for Function Minimization. The Computer

Journal, 7(4):308–313, 1965.

[167] F. Roters, P. Eisenlohr, C. Kords, D. D. Tjahjanto, M. Diehl, and D. Raabe. DAMASK: the
Düsseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE
based or a spectral numerical solver. In O. Cazacu, editor, Procedia IUTAM: IUTAM Sym-
posium on Linking Scales in Computation: From Microstructure to Macroscale Properties,
volume 3, pages 3–10, Amsterdam, 2012. Elsevier.

[168] A. J. Schwartz, M. Kumar, B. L. Adams, and D. P. Field, editors. Electron Backscatter

Diffraction in Materials Science. Springer US, Boston, MA, 2009.

[169] S. L. Wong, J-S Park, M. P. Miller, and P. R. Dawson. A framework for generating synthetic
diffraction images from deforming polycrystals using crystal-based finite element formula-
tions. Computational Materials Science, 77:456–466, 2013.

[170] Desmond Tromans. Elastic anisotropy of hcp metal crystals and polycrystals. 6, 01 2011.
[171] R. J. Asaro and A. Needleman. Texture development and strain hardening in rate dependent

polycrystals. Acta Metallurgica, 33(6):923–953, 1985.

[172] C. Niederberger, W. M. Mook, X. Maeder, and J. Michler.

In situ electron backscatter
diffraction (ebsd) during the compression of micropillars. Materials Science and Engineering
A, 527(16-17):4306–4311, 2010.

[173] M. Gee, K. Mingard, and B. Roebuck. Application of ebsd to the evaluation of plastic de-
formation in the mechanical testing of wc/co hardmetal. International Journal of Refractory
Metals and Hard Materials, 27(2):300–312, 2009.

[174] S. Mandal, B. Gockel, S. Balachandran, D. Banerjee, and A. D. Rollett. Simulation of
plastic deformation in ti-5553 alloy using a self-consistent viscoplastic model. International
Journal of Plasticity, 2017.

[175] D. Raabe, D. Ma, and F. Roters. Effects of initial orientation, sample geometry and friction
on anisotropy and crystallographic orientation changes in single crystal microcompression
deformation: A crystal plasticity finite element study. Acta Materialia, 55(13):4567–4583,
2007.

171

[176] N. Zaafarani, D. Raabe, F. Roters, and S. Zaefferer. On the origin of deformation-induced

rotation patterns below nanoindents. Acta Materialia, 56(1):31–42, 2008.

[177] Y. Yang, L. Wang, C. Zambaldi, P. Eisenlohr, R. Barabash, W. Liu, M. R. Stoudt, M. A.
Crimp, and T. R. Bieler. Characterization and modeling of heterogeneous deformation in
commercial purity titanium. JOM, 63(9):66–73, 2011.

[178] A. Kunz, S. Pathak, and J. R. Greer. Size effects in al nanopillars: Single crystalline vs.

bicrystalline. Acta Materialia, 59(11):4416–4424, 2011.

[179] N. Kheradmand, H. Vehoff, and A. Barnoush. An insight into the role of the grain boundary
in plastic deformation by means of a bicrystalline pillar compression test and atomistic
simulation. Acta Materialia, 61(19):7454–7465, 2013.

[180] D. E. Spearot, K. I. Jacob, and D. L. McDowell. Dislocation nucleation from bicrystal
interfaces with dissociated structure. International Journal of Plasticity, 23:143–160, 2007.
[181] L. Capolungo, D. E. Spearot, M. Cherkaoui, D. L. McDowell, J. Qu, and K. I. Jacob.
Dislocation nucleation from bicrystal interfaces and grain boundary ledges: Relationship to
nanocrystalline deformation. Journal of the Mechanics and Physics of Solids, 55(11):2300–
2327, 2007.

[182] Y. Guo, Y. Huang, H. Gao, Z. Zhuang, and K. C. Hwang. Taylor-based nonlocal theory
of plasticity : numerical studies of the micro-indentation experiments and crack tip fields.
International Journal of Solids and Structures, 38:7447–7460, 2001.

[183] S. Yefimov, I. Groma, and E. van der Giessen. A comparison of a statistical-mechanics based
plasticity model with discrete dislocation plasticity calculations. Journal of the Mechanics
and Physics of Solids, 52(2):279–300, 2004.

[184] S. Yefimov and E. Van der Giessen. Multiple slip in a strain-gradient plasticity model
motivated by a statistical-mechanics description of dislocations. International Journal of
Solids and Structures, 42(11-12):3375–3394, 2005.

[185] C. Reuber, P. Eisenlohr, F. Roters, and D. Raabe. Dislocation density distribution around
an indent in single-crystalline nickel: Comparing nonlocal crystal plasticity finite element
predictions with experiments. Acta Materialia, 71:333–348, 2014.

[186] J. R. Mayeur, I. J. Beyerlein, C. A. Bronkhorst, and H. M. Mourad. Incorporating interface
affected zones into crystal plasticity. International Journal of Plasticity, 65:206–225, 2015.
[187] S. J. Vachhani, R. D. Doherty, and S. R. Kalidindi. Studies of grain boundary regions in
deformed polycrystalline aluminum using spherical nanoindentation. International Journal
of Plasticity, 81:87–101, 2016.

[188] L. C. Lim and R. Raj. Continuity of slip screw and mixed crystal dislocations across bicrystals

of nickel at 573 k. Acta Metallurgica, 33(8):1577–1583, 1985.

[189] J. D. Livingston and B. Chalmers. Multiple slip in bicrystal deformation. Acta Metallurgica,

5(6):322–327, 1957.

172

[190] T. Hirouchi and Y. Shibutani. Mechanical responses of copper bicrystalline micro pillars
with σ3 coherent twin boundaries by uniaxial compression tests. Materials Transactions,
55(1):52–57, 2014.

[191] C. S. Kaira, S. S. Singh, A. Kirubanandham, and N. Chawla. Microscale deformation behav-
ior of bicrystal boundaries in pure tin (sn) using micropillar compression. Acta Materialia,
120:56–67, 2016.

[192] G. Kindlmann. Superquadric tensor glyphs. In Proceedings of the Sixth Joint Eurograph-
ics – IEEE TCVG Conference on Visualization, Vissym’04, pages 147–154, Aire-la-Ville,
Switzerland, Switzerland, 2004. Eurographics Association.

[193] W. A. Soer, K. E. Aifantis, and J. T. M. De Hosson. Incipient plasticity during nanoindentation
at grain boundaries in body-centered cubic metals. Acta Materialia, 53(17):4665–4676,
2005.

173