STUDY OF HETEROGENEOUS DEFORMATION IN HEXAGONAL TITANIUM USING
HIGH ENERGY X-RAY DIFFRACTION
By
Harsha J. Phukan
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Materials Science and Engineering – Doctor of Philosophy
2021
ABSTRACT
STUDY OF HETEROGENEOUS DEFORMATION IN HEXAGONAL TITANIUM USING
HIGH ENERGY X-RAY DIFFRACTION
By
Harsha J. Phukan
It is well known that grain boundaries have a beneficial effect on strengthening properties in crys-
talline materials. There is however, a much lesser degree of understanding on how grain boundaries
determine the propensity of a material for damage nucleation. In polycrystalline aggregates the me-
chanical response of individual grains to loading conditions is highly heterogeneous and dependent
upon the morphology and relative orientation of the neighboring grains. Surface characteriza-
tion methods such as electron backscatter diffraction (EBSD) present only a partial understanding
of this heterogeneous deformation behavior. X-ray diffraction using high brilliance beams from
synchrotron sources provide a powerful non destructive means to characterize the subsurface me-
chanical response of grains. Such high energy x-ray diffraction microscopy (HEDM) methods
can measure changes in grain orientation, morphology and local strain evolution with high spatial
resolution.
The low crystal symmetry of hexagonal metals make them ideal candidates to study individual
slip systems and identify specific types of dislocations. Deformation in hexagonal materials is
strongly dependent upon the relative orientation between the c-axis and the loading direction.
Conditions that lead to the nucleation of deformation twinning in hexagonal metals are not well
understood. This makes it challenging for physically based crystal plasticity models to predict
twinning events. Another important consideration in polycrystalline deformation is the dependence
of the local stress state on the geometrically necessary dislocation (GND) density.
In this work, two different samples of polycrystalline pure titanium having textures were char-
acterized using two different HEDM techniques. The first specimen has a predominantly "hard"
texture with respect to the loading direction. In-situ far field high energy diffraction microscopy
(FF HEDM) was used to characterize the formation of discrete twinning events during a tensile
test. The propensity for twin nucleation in a grain by slip transfer from neighboring grains for each
of the identified twinning events is evaluated using a geometrical parameter along with conditions
of spatial proximity. The second specimen has a largely "soft" texture with respect to the loading
direction. The evolution of local morphology and local stress state during a four point bending
experiment were captured using differential aperture x-ray microscopy (DAXM). The GND density
was estimated for each voxel from the orientation gradient.
In contrast to surface measurements on the same material deformed in bending, where prism
slip bands nucleated mechanical twins, a greater amount of pyramidal ⟨𝑐 + 𝑎⟩ slip was correlated
to twin formation in the interior of the specimen, though it is not clear whether the twin or the
⟨𝑐 + 𝑎⟩ slip initiated the correlated shear. Comparison with similar studies in titanium shows that
the type of slip system likely to nucleate a T1 twin is strongly dependent upon the loading direction
and initial texture.
In this work a quantitative matching of the FF HEDM data and more recently collected Near
Field (NF) HEDM dataset is done using criteria of maximum crystallographic misorientation and
Euclidean distance. Additionally, a comparison was made between the kinematic descriptor (lattice
reorientation as a function of load) and the grain averaged stress measures in FF HEDM. This
was done in order to determine the limits of FF HEDM for assessing complex mesoscopic loading
events such as deformation twinning.
A means to visualize the heterogeneity in the local stress state is enabled by DAXM charac-
terization. Moreover, the in-situ DAXM experiment enabled the estimation of the GND density
from the lattice rotation gradient. The analysis was able to identify the contributions to the total
GND density from individual slip systems. The local agglomeration of GND (pileups) is strongly
dependent upon the local stress state and the transmissivity of dislocations across grain boundaries.
In contrast, the global stress state does not have a strong correlation with local GND accumulation.
The present work is a step towards developing a better understanding of local mechanical
response in polycrystalline materials. It is expected that results from this study can help better inform
constitutive relations governing crystal plasticity based models that simulate material deformation.
Copyright by
HARSHA J. PHUKAN
2021
This thesis is dedicated to my parents.
v
ACKNOWLEDGEMENTS
At the very onset I take this opportunity to express my gratitude towards my advisor, Dr.
Thomas Bieler for his guidance, encouragement and patience throughout this work. I would also
like to thank the members of my dissertation committee: Drs. Carl Boehlert, Martin Crimp, Philip
Eisenlohr and Patrick Kwon, for their valuable insights and constructive criticism.
I would like to acknowledge the support of the Materials World Network (grant NSF-DMR-
1108211) and the Department of Energy (DE-FG02-10ER46637) for this work. Use of the Ad-
vanced Photon Source facilities was made possible with the support of the Department of Energy,
Office of Basic Energy Sciences under DOE/BES DE-FG02-10ER46637.
I would also like to thank Dr. Leyun Wang, faculty of Materials Science at Shanghai Jiao Tong
University, for his collaborative help in conducting beamline experiments at APS, and his technical
inputs.
I would be remiss if I did not express my heartfelt thanks to Drs. Peter Kenesei, Wenjun Liu,
Jun Sang Park, Hemant Sharma and Ruqing Xu for their help in conducting experiments at APS;
and subsequent post processing, interpretation and analysis of data.
Gratitude is also due to Professor Dierk Raabe and colleagues at the Max-Planck-Institut für
Eisenforschung for facilitating a wonderful learning experience during my visits there.
I also take this opportunity to thank my fellow graduate students (current and former) for their
help and encouragement; in particular Drs. Ajith Chakkedath, Markus Downey, Songyang Han,
Mingmin Wang, Chen Zhang, Quan Zhou, and Mr. Yang Su.
Finally I want to thank my wife Mollie for her patience and belief in me. I could not have come
this far without her support.
vi
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation for current work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Deformation systems in hexagonal titanium . . . . . . . . . . . . . . . . . . . . . 2
1.3 High energy x-ray diffraction (HEXRD) . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Basics of plastic deformation in crystalline materials . . . . . . . . . . . . . . . . . 6
2.1.1 Crystallographic Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Schmid’s law for slip in a single crystal . . . . . . . . . . . . . . . . . . . 7
2.1.3 Multiplication of dislocations . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.4 Generalized Schmid Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.5 Deformation twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Plastic deformation in hexagonal titanium . . . . . . . . . . . . . . . . . . . . . . 13
2.3 An overview of crystal plasticity principles . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Slip transfer and dislocation interactions at grain boundaries . . . . . . . . . . . . 23
2.4.1 Implications of slip transfer in polycrystalline materials . . . . . . . . . . . 25
2.5 Twin formation as a result of slip transfer from a neighboring grain (S+T twinning) 27
2.6 Geometrically necessary dislocations (GND) . . . . . . . . . . . . . . . . . . . . . 29
2.7 Review of HEXRD methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.1 3D XRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7.1.1 Indexing data obtained from 3D XRD . . . . . . . . . . . . . . . 39
2.7.1.2 Local strain tensor evaluation from FF HEDM . . . . . . . . . . 41
2.7.1.3 Accuracy in determination of grain positions using 3D XRD . . . 42
2.7.2 Differential aperture x-ray microscopy (DAXM) . . . . . . . . . . . . . . . 43
2.7.2.1 Local strain tensor evaluation from DAXM . . . . . . . . . . . . 47
2.7.2.2 Calculation of GND density from DAXM data . . . . . . . . . . 48
2.8 Opportunity for research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
CHAPTER 3 MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 Titanium specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.1 Tensile specimen for Far Field Experiment . . . . . . . . . . . . . . . . . . 52
3.1.2 Four point bending specimen for DAXM Experiment . . . . . . . . . . . . 54
3.2 Far Field HEXRD (Chapter 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Post deformation surface EBSD of the tensile specimen . . . . . . . . . . 55
vii
3.3 Experimental setup for in-situ DAXM characterization (Chapter 6) . . . . . . . . . 56
CHAPTER 4 IN-SITU FAR FIELD XRD CHARACTERIZATION OF TENSILE DE-
FORMATION OF A COMMERCIAL PURITY TITANIUM SPECIMEN . . 58
4.1 Analysis of the diffraction patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Determination of grain centers of mass and grain averaged strain tensors . . 59
4.1.2 Quantification of error in local strain calculations . . . . . . . . . . . . . . 61
4.2 3D Reconstructed model of the microstructure prior to loading . . . . . . . . . . . 62
4.2.1 Model parameters and simulation of tensile test . . . . . . . . . . . . . . . 62
4.2.2 Sensitivity of mechanical response of model to dilatational layer thickness . 64
4.3 Effect of local stress state on deformation systems . . . . . . . . . . . . . . . . . . 65
4.4 Identification of twinning events . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 Far Field HEDM Characterization . . . . . . . . . . . . . . . . . . . . . . 72
4.5.1.1 Assessment of slip transfer from neighboring grains . . . . . . . 72
4.5.1.2 Stress evolution in parent grains . . . . . . . . . . . . . . . . . . 78
4.6 Comparison of FF 3D XRD results with surface EBSD mapping . . . . . . . . . . 86
4.7 Cross validation of local stress tensor calculations and grain positions between
Fable and MIDAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.8 Evaluation of identified twins using Near Field HEDM . . . . . . . . . . . . . . . 91
CHAPTER 5 A COMPARATIVE STUDY OF THE ACCURACY OF FF HEDM METHOD
WITH RESPECT TO NF HEDM . . . . . . . . . . . . . . . . . . . . . . . 99
5.1 Classification of errors from the FF and NF HEDM experiments . . . . . . . . . . 99
5.2 Estimation of grain volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 NF HEDM: Reduction of data on the basis of Confidence Index . . . . . . . . . . . 102
5.4 Matching of centroids between NF and FF HEDM datasets . . . . . . . . . . . . . 102
5.4.1 Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4.2 Algorithms used for matching grains between FF-NF datasets . . . . . . . . 103
5.4.3 Grain size comparison between matched grains in FF/NF datasets . . . . . 105
5.4.4 Implementation of matching algorithm . . . . . . . . . . . . . . . . . . . . 107
5.4.5 Application of translational correction between FF/NF datasets . . . . . . . 109
5.4.6 Relationship between NF grain size and matching centroids in FF . . . . . 110
CHAPTER 6 COMPARISON OF KINEMATICS OF CRYSTAL ROTATION AND
GRAIN AVERAGED STRESS MEASUREMENTS OBTAINED FROM
FF HEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 Calculation of plastic spin axes for active slip systems . . . . . . . . . . . . . . . . 113
6.2 Comparison of slip system plastic spin axis and lattice reorientation . . . . . . . . 114
CHAPTER 7 IN-SITU DAXM CHARACTERIZATION OF A COMMERCIAL PU-
RITY TITANIUM SPECIMEN SUBJECTED TO A FOUR POINT BEND-
ING TEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1 Texture of CP-Ti specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 White beam diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
viii
7.2.1 Local deviatoric strain tensor estimation from DAXM data . . . . . . . . . 124
7.3 Geometrically necessary dislocation density calculation from DAXM data . . . . . 124
7.3.1 Rotation gradient calculation of DAXM dataset . . . . . . . . . . . . . . . 129
7.3.2 Calculation of dislocation density from the rotation gradient . . . . . . . . 130
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.4.1 Localized GND concentrations at grain boundaries . . . . . . . . . . . . . 134
7.4.2 Assessment of local stress heterogeneity using principal components . . . . 134
7.4.2.1 Local stress heterogeneity within and above the purple grain . . . 138
7.4.2.2 Local stress heterogeneity in region 2 triple junction with hard
and soft grains . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4.3 Description of parameters used in analysis of slip transfer at the voxel level 142
7.5 Summary assessment of inter-and intra-granular heterogeneous deformation . . . . 144
CHAPTER 8 CONCLUSIONS AND SCOPE FOR FUTURE WORK . . . . . . . . . . . 146
8.1 Far Field HEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.1.1 Twin identification and role of slip transfer in twin nucleation . . . . . . . . 146
8.1.2 Comparison of FF results with Near Field and surface EBSP mapping . . . 147
8.2 DAXM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2.1 Assessment of heterogeneity in stress state due to local constraints . . . . . 148
8.3 Scope for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
APPENDIX A ADDITIONAL FAR FIELD ANALYSIS RESULTS . . . . . . . . . 151
APPENDIX B PARENT/TWIN/NEIGHBOR SLIP TRANSFER ANALYSIS FROM
NF HEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
APPENDIX C STEREOGRAPHS OF EVOLUTION OF LATTICE PLASTIC
SPIN AXIS AS A FUNCTION OF BULK STRAIN . . . . . . . . . 158
APPENDIX D HYDROSTATIC STRAIN TENSOR MEASUREMENT USING
MONOCHROMATIC SETTING OF DAXM . . . . . . . . . . . . . 160
APPENDIX E PRINCIPAL STRESS AND PRINCIPAL DIRECTION MAPS
FOR REGIONS 3,4 AND 5 . . . . . . . . . . . . . . . . . . . . . . 166
APPENDIX F CORRELATION OF GND CONTENT WITH RESIDUAL BURG-
ERS VECTOR AND SLIP TRANSFER PARAMETER . . . . . . . 170
APPENDIX G SLIP TRANSFER PARAMETER V/S RESIDUAL BURGERS
VECTOR PLOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
APPENDIX H COORDINATE TRANSFORMATIONS . . . . . . . . . . . . . . . 205
APPENDIX I LIST OF PYTHON AND MATLAB SCRIPTS USED FOR DATA
ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
ix
LIST OF TABLES
Table 2.1: Parameters associated with the four twinning modes in CP-Ti (Wang, 2011) . . . 15
Table 2.2: Enumeration of the three different configurations for 3D XRD, based on sample
to detector distance (L). Adapted from Park et al. (Park et al., 2017). . . . . . . . 38
Table 3.1: Composition (wt%) of Grade 1 Titanium Plate. Adapted from Bieler et al. (2014) 52
Table 4.1: Parameters used in Crystal Plasticity model . . . . . . . . . . . . . . . . . . . . 64
Table 4.2: Summary of Twinning Events Observed in 14 Layers in two specimens . . . . . 80
Table 4.3: Summary of 13 twinning events and their stress history correlation with respect
to neighboring grains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Table 7.1: Slip Systems considered for GND density calculation . . . . . . . . . . . . . . 133
Table 7.2: Combinations of parameters used to assess relationship between residual Burg-
ers vector, 𝑚′ and GND density . . . . . . . . . . . . . . . . . . . . . . . . . . 144
x
LIST OF FIGURES
Figure 2.1: Schematic showing the geometry of edge and screw dislocations in a simple
cubic lattice. (a) perfect lattice, (b) an extra half-plane of atoms ABCD
inserted into the top half of the crystal. DC is a positive edge dislocation,
(c) left handed screw dislocation DC is formed by displacement of the faces
ABCD relative to each other in the direction AB, (c) right handed screw
dislocation DC,(e) atomic planes with spacing b in a perfect crystal, (f)
planes distorted by a right-handed screw dislocation. (Hull and Bacon, 2011) . 7
Figure 2.2: Schematic illustrating the concept of critical resolved shear stress for a single
crystal (Hull and Bacon, 2011). It is important to note that in general 𝜙 and
𝜆 are not coplanar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure 2.3: Schematic showing the process of regenerative multiplication of dislocations
by a Frank-Read source [adapted from Read (1953)]. . . . . . . . . . . . . . . . 9
Figure 2.4: Illustration of generalized stress state for an arbitrary grain in a polycrystal.
The local stress tensor corresponding to this generalized stress state is shown
as a function of X, Y and Z. T is the traction vector, while n and m are the
slip plane normal and slip direction respectively. . . . . . . . . . . . . . . . . . 10
Figure 2.5: Schematic showing the four twinning elements (Christian and Mahajan,
1995). 𝜿1 and 𝜿 2 are twinning and conjugate planes respectively. The
corresponding twinning and conjugate directions are 𝜼1 and 𝜼2 . 𝜼1 , 𝜼2 ,and
the normals to 𝜿1 and 𝜿 2 lie in shear plane P. . . . . . . . . . . . . . . . . . . 13
Figure 2.6: Schematic illustrating the major crystallographic planes and directions (a) ,
and the slip modes (b) associated with them (adapted from Wang (2011)). . . . 14
Figure 2.7: Schematic illustrating the geometrical relationship between a T1 twin and its
parent lattice. Adapted from (Bieler et al., 2014). . . . . . . . . . . . . . . . . 17
Figure 2.8: Schematic illustration of the concept of deformation gradient. The total
deformation gradient 𝐹 can be decomposed into the product of the elastic and
the plastic components(Roters et al., 2010). . . . . . . . . . . . . . . . . . . . 19
Figure 2.9: Illustration of dislocation pileup at a grain boundary between two crystals
𝐴 and 𝐵. Dislocations shown here are edge type; in reality they are a
combination of both edge and screw types. Adapted from Livingston and
Chalmers [1957] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
xi
Figure 2.10: Angles used to calculate the Luster-Morris parameter 𝑚′. 𝜅 is the angle
between slip directions; while 𝜓 is the angle between the slip plane normals.
𝜃 is the angle between the slip plane traces on the grain boundary plane.
Adapted from Bieler et al. (2009). . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 2.11: Use of 𝑚′ to determine the propensity of twin nucleation as a result of slip
transfer across a grain boundary. Twins can be observed in grain 2, where
the activated twin system has a high 𝑚′ (0.936) with respect to prism ⟨𝑎⟩ slip
system in grain 1. Adapted from Yang et al. (2011). . . . . . . . . . . . . . . . 28
Figure 2.12: Schematic diagram showing how GNDs of edge type (left) and screw type
(right) can accumulate in a plastically deformed crystal. Here 𝑏 indicated the
slip direction and 𝑛 denotes the slip plane normal. 𝑚 = 𝑏 × 𝑛 . Adapted from
Arsenlis and Parks [1999]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 2.13: Schematic illustrating a basic 3D XRD experimental setup. 𝑍 is the sample
rotation axis; 𝜔 is the angle of rotation about 𝑍; 𝐿 is the sample to detector
distance; 2𝜃 is the Bragg angle; 𝜂 is the azimuthal angle. Adapted from
Poulsen (Poulsen, 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 2.14: Schematic illustration of Debye-Scherrer patterns for a polycrystalline mate-
rial with (a) random texture (well annealed);(b) strong texture (Taddei, 2015). . 39
Figure 2.15: Schematic diagram illustrating the Laue condition. . . . . . . . . . . . . . . . . 44
Figure 2.16: Schematic representation of an Ewald sphere in two dimensions (a) Monochro-
matic beam, (b) Polychromatic beam containing a range of wavelengths. The
parallel set of crystallographic planes (hkl) that form a reciprocal lattice point
are shown on the top right inset (Ice and Pang, 2009) . . . . . . . . . . . . . . 45
Figure 2.17: Schematic of a DAXM experimental setup to interrogate a polycrystalline
specimen. An enlarged view of the differential aperture (profiler) Pt wire is
shown on the top right (Yang et al., 2004). . . . . . . . . . . . . . . . . . . . . 47
Figure 2.18: Example illustrating the importance of relative grain positions for viable S+T
twin nucleation. The left figure (a) shows two grains with a favorable relative
position (along with a high 𝑚′ value). Figure (b) shows the same grains with
relative positions that may not favor S+T twinning. . . . . . . . . . . . . . . . 49
Figure 3.1: Tensile specimen extracted from a larger sample that was previously subjected
to a bend test. Because the sample had a prior bending load, a macro residual
stress was initially present in the sample (Wang et al., 2014) . . . . . . . . . . . 53
xii
Figure 3.2: Right: Cross section of the illuminated volume of the tensile specimen (look-
ing down the tensile axis), showing the distribution of the normal component
of the strain tensor (𝜖 𝑧𝑧 ), prior to tensile loading. Left: the coordinates of the
strain map are shown in context of the tensile specimen. The tensile axis is
parallel to the 𝑍 direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 3.3: Evolution of global stress as function of engineering strain for the tensile
experiment. Inset figure shows a schematic representation of the experimental
setup for FF-HEDM. A schematic representation of the eleven layers examined
along the gage section of the tensile is shown on the right. Inset: Schematic
illustration of the load relaxation with respect to time is shown for three
different strain states–prior to yield (magenta), post yielding (green) and at
maximum stress (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 3.4: Surface tensile direction EBSD IPF map of the far side (facing detector) of the
gage section of the unloaded specimen shows several twins and orientation
gradients in some grains. The scanned area is highlighted in green. . . . . . . . 56
Figure 3.5: CP-Ti sample setup in the four point bending stage and dimensions are shown
on the bottom right of the figure. Experimental setup for the in-situ DAXM
characterization of four point bending is shown on the top left. The directions
of the incoming and diffracted beams are shown schematically. The beamline
coordinate system is shown on the top right: Z denotes the direction of the
incoming beam; F is anti-parallel to the surface normal (denoted by the green
vector) of the specimen; X is directed normal out of the page. Position and
displacement direction of the differential aperture (Pt wire) are shown on the
bottom left of the figure (Adapted from Larson et al. (Larson et al., 2002)). . . . 57
Figure 4.1: Outline of data analysis strategy that was implemented in a suite of MatLab
codes. The .log files (containing orientation and grain index information),
and .gve files (containing scattering vector information) are obtained using
FABLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 4.2: Left:Voronoi tessellated model generated from 3D XRD Data set. This
microstructure shown here represents the unstrained condition, prior to the
tensile test. A dilatational layer of one volume element thickness is used as
the surrounding medium along the 𝑋 and 𝑌 directions. The tensile direction
is parallel to the Z axis. Right: Evolution of normal component of stress
along the tensile direction 𝜎𝑧𝑧 in the crystal plasticity simulation of the tensile
loading. In the CP model, a uniform compressive strain of 10−3 is applied for
each of the first three load steps; followed by a uniform tensile strain of 10−3
for each of the subsequent load steps. . . . . . . . . . . . . . . . . . . . . . . . 63
xiii
Figure 4.3: Evolution of volume averaged equivalent (von Mises) stress of the far field
microstructure model as a function of bulk strain for four different thicknesses
of the surrounding free surface layer. It is seen that the mechanical response
is not sensitive to dilatational layer thickness. . . . . . . . . . . . . . . . . . . . 64
Figure 4.4: Cumulative distribution function (CDF) plots of Schmid Factors correspond-
ing to three load steps for basal ⟨𝑎⟩, prism ⟨𝑎⟩, pyramidal ⟨𝑎⟩, pyramidal
⟨𝑐 + 𝑎⟩ and T1 systems. The global Schmid factor (top row) is calculated
using the global uniaxial stress tensor; while the local grain averaged stress
tensor is used to obtain generalized Schmid factors (maximum 0.707) (middle
and bottom rows). The middle row shows the comparison of the experimen-
tally obtained local Schmid factors with an earlier CP model that imposes
only tensile strain without initial compression. The bottom row shows the
comparison of local Schmid factors with the results of the CP model with
initial compression. The Schmid factors in the CDF plots are the highest per
slip system family for each grain. . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 4.5: Neighborhood of all grains in layer 1 prior to deformation, at the load step
just before twinning at 1.3% engineering strain, and after a twin formed at
1.5% engineering strain in orientation space (top row). The spatial map of
the 2D slice is shown in the bottom row with grain identification numbers.
The bottom right figure shows an enlarged view of the region of interest for
the twin formation with grain unit cell orientations based upon geometrical
positions in the lower left map of the slice. Tensile direction is along z
direction (out of page). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure 4.6: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems in
the parent grains identified in layers 1, 2, 3, 4 and 5. Green arrows indicate the
load step where the twin was first identified, and filled red markers indicate
the observed twin variant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 4.7: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems
in the parent grains identified in layers 6, 7, 8, 9 and 10. Details of symbols
and color conventions are given in the caption of figure 4.6. . . . . . . . . . . . 71
Figure 4.8: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems
in the parent grain identified in layer 11. Details of symbols and color
conventions are given in the caption of figure 4.6. . . . . . . . . . . . . . . . . 72
xiv
Figure 4.9: Schematic outlining the criteria used to identify the neighbor grain associated
with the highest likelihood of slip transfer, thereby triggering a twin. 𝜃 is the
angle between the unit vector joining the centers of masses of the parent grain
and neighbor; and the unit vector normal to the slip plane in the neighbor.
The three parameters (Absolute value of the Schmid Factor, Absolute value
of sin 𝜃 (Spatial alignment factor SAF) and 𝑚′ can be plotted in a 3-D space.
A point at the origin (red) would have the worst possible spatial alignment
for slip transfer, while a point at the top right corner (green) would have the
highest likelihood for slip transfer. The purple slip planes have lower SAF
while the green slip plane has a higher SAF. . . . . . . . . . . . . . . . . . . . 73
Figure 4.10: The 𝑚′ relationship observed between the activated twin system and pyrami-
dal ⟨𝑐 + 𝑎⟩ system of a neighboring grain is shown for six different instances
(Layers 1,2, 4, 6, 9 and 10).The orientations of the parent, twin and neigh-
boring grains are viewed along the tensile Z axis, (left column) as well as the
beam X direction, (right column). The letters "P", "T" and "N" indicate the
parent, twin and neighbor grains respectively. The corresponding Bunge Eu-
ler angles and 𝑚′ values are also indicated for each instance.The blue vector
shows the slip direction from the blue dot away from it on the positive surface
of the plane, and a half length orange dot and vector indicates a twin system.
The plane normal points out of the page for beige slip planes and into the
page for gray planes. The dotted red, green and blue lines represent the a1 , a2
and a3 directions, respectively. Colors of the grains correspond to (0 0 0 1)
inverse pole figure representations of the orientations. . . . . . . . . . . . . . . 74
Figure 4.11: Summary of twinning events in the 11 layers as a function of 𝑚′: Size of
the markers are proportional to the Schmid factor rank of the twin system.
Filled circles (•) indicate geometrically more plausible instances, where slip
transfer may have resulted in nucleation of twins. Data from a previous study
(1.01,1.02 and 1.03 (Bieler et al., 2014) in table 4.2) are plotted as □. Cases
with SAF > 0.6 (𝜃 > 37°) are represented as filled colored markers, where the
colors represent different families of slip and twin systems. Among these,
the cases with a highest likelihood of S+T twinning with SAF > 0.8 (𝜃 > 53°)
are shown in a darker gold color, while the lighter gold symbols represents
instances where 37°< 𝜃 < 53°. Prism ⟨𝑎⟩ is represented by red, basal ⟨𝑎⟩ by
blue, pyramidal ⟨𝑎⟩ by green, pyramidal ⟨𝑐 + 𝑎⟩ by golden (transitions from
lighter gold to a darker shade for more plausible cases of S+T twinning); and
T1 twinning by black. Inset shows the linear regression plot, considering
only the highly plausible ⟨𝑐 + 𝑎⟩ (dark gold) points. . . . . . . . . . . . . . . 78
Figure 4.12: Evolution of normal component of local stress tensor (along the tensile di-
rection) for parent and neighboring grains (layers 1, 2, 3, 4 and 5). Prisms
indicate the Bunge Euler angle orientation of the grains. Transition in back-
ground shade from gray to white indicates the bulk strain at which the twin
was identified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xv
Figure 4.13: Evolution of normal component of local stress tensor (along the tensile direc-
tion) for parent and neighboring grains (layers 6, 7, 8, 9 and 10). Details on
symbols and color conventions are given in the caption in figure 4.12. . . . . . . 82
Figure 4.14: Evolution of normal component of local stress tensor (along the tensile direc-
tion) for parent and neighboring grains (Layer 11). Details on symbols and
color conventions are given in the caption in figure 4.12. . . . . . . . . . . . . . 83
Figure 4.15: Evolution of local average von Mises stress 𝜎𝑣𝑀 in identified parent and
selected neighbor grains across the 11 layers. The load steps corresponding
to twin identification is indicated by the gradual transition in background
color from gray to white. The measure of local stress deviation from ideal
uniaxial tension is given by cos 𝜃 = 𝜎.𝑈. . . . . . . . . . . . . . . . . . . . . 84
Figure 4.16: The top row shows the tensile sample and the region from which EBSD
measurements were made on the surface between the sample and the detector.
The top right figure shows the grain reference orientation deviation map where
the reference is the average grain orientation in the EBSD data. Two different
regions with twins (labeled a and b) are shown. in the second and third rows.
The green prisms show the relative positions of grain centers of mass from
the far field data (FF) in which twins were detected, and just beneath them, the
enlarged corresponding part of the EBSD maps; the green and black prisms
are not in good spatial agreement. Beneath the EBSD scans, misorientation
maps between the FF COM orientation and the 2-D EBSD grain orientations
show agreement within 10°. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 4.17: Histogram of Euclidean distance for grains consistent with indexation using
Fable and MIDAS (top row), and cumulative distribution plot of the same
grains (bottom row), for unstrained (prior to loading) and final unloaded
states respectively. The top right plot also shows the histogram of Euclidean
distance for grains consistent between FF analysis (MIDAS) for the final
unloaded state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Figure 4.18: Cumulative distribution function (CDF) plots of grain radii estimated by
MIDAS for the undeformed and final unloaded states respectively. Inset
shows a magnified view of the tail of the CDF plots. There is a significant
change in the slope towards the tail of curve, going from the state prior to
deformation to the final unloaded state. . . . . . . . . . . . . . . . . . . . . . . 91
Figure 4.19: Cumulative distribution function plots for equivalent strain (top row) and von
Mises stress (bottom row) evolution, showing the differences between two
indexing methods. Strain and stress evolution are shown for four different
load states: unstrained state prior to tensile loading, 75% of bulk strain prior
to yield, maximum load and final unloaded state. . . . . . . . . . . . . . . . . 92
xvi
Figure 4.20: Perspective (left) and XY plane (right) projections for the gage volume in-
terrogated by NF HEDM, shown with IPF colors. The data corresponds to
the final unloaded state in the tensile experiment (after dismounting of the
specimen from the tensile stage). The tensile direction was along the Z axis. . . 93
Figure 4.21: Analysis of the first parent, twin and high 𝑚′ grain from figure 4.10. NF
HEDM provides a more precise relative position and information on grain
shape. Bottom left: plot of the c-axis disorientation of the neighboring grain
as a function of bulk strain, along with the evolution of equivalent stress
in both parent and neighboring grains. Top right: Evolution of resolved
shear stress for 4 families of slip in the neighboring grain, and bottom right:
misorientation of the plastic spin axis for each of the 12 pyramidal⟨𝑐 + 𝑎⟩
slip systems as function of bulk strain. The color scheme used for the 12
pyramidal⟨𝑐 + 𝑎⟩ slip systems is the same for the top right and bottom right
plots. The symbol and color scheme for the slip systems is indicated in the
inset text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Figure 4.22: Revisit of the third parent, twin and high 𝑚′ grain from figure 4.10. NF
HEDM enables a more precise relative position and information on grain
shape. Bottom left: plot of c-axis disorientation of the neighboring grain as
a function of bulk strain, along with the evolution of equivalent stress in both
parent and neighboring grains. Top right: Evolution of resolved shear stress
for 4 families of slip in neighboring grain, and bottom right: misorientation
of the plastic spin axis for each of the 12 pyramidal⟨𝑐 + 𝑎⟩ slip systems as
function of bulk strain. The color scheme used for the 12 pyramidal⟨𝑐 + 𝑎⟩
slip systems is the same for the top right and bottom right plots. The symbol
and color scheme for the slip systems is indicated in the inset text. . . . . . . . . 96
Figure 4.23: Resolved shear stress plot as a function of 𝑚′ for the seven twinning events
validated with NF HEDM data plotted in the same manner as figure 4.11.
Here, all of the slip systems with a favorable geometric compatibility with
respect to the active twin variant in the parent grain are of pyramidal⟨𝑐 + 𝑎⟩
type. The data points from the FF HEDM analysis that correspond to high 𝑚′
relationship with respect to the active twin variant are indicated by ’𝑥’. The
size of the markers are proportional to the Schmid factor rank of the active
twin variant. The NF points that coincide with the FF results are indicated by
translucent circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Figure 5.1: Classification of errors from the FF and NF HEDM experiments. . . . . . . . . 100
Figure 5.2: Maps showing distribution of the confidence index (CI) in the interrogated
microstructure projected on the XY plane (looking down along tensile axis
over the entire interrogated volume). Cumulative distribution plots of CI
(unfiltered data and with different threshold values) are shown in bottom left.
The cumulative distribution curves converge at a CI value ∼ 0.95. . . . . . . . 101
xvii
Figure 5.3: Cumulative distribution plot of the disorientation between two different grain
averaging techniques for the reduction of the NF data. The same group of ori-
entations are averaged: the first approach being simple quaternion averaging,
and the second using the more optimized grain averaging algorithm available
in DAMASK. The maximum disorientation between the mean quaternion
approach and the DAMASK algorithm is less than 3°. At least 90% of the
points have a disorientation of 1.5°or less. The point cloud of the reduced NF
data is shown on the right with IPF colors denoting crystal orientation with
respect to the Z axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 5.4: Forward algorithm used for matching grains between the FF and NF datasets,
where the misorientation criterion is applied before the ball radius. . . . . . . . 105
Figure 5.5: Reverse algorithm used for matching grains between the FF and NF datasets,
where the ball radius criterion is applied first and the misorientation threshold
is applied in the second step. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 5.6: Cumulative distribution function plots for matched grains using forward al-
gorithm (left) and reverse algorithm (right), using a misorientation tolerance
of 5°. In both the matching operations, a ball radius of 1500 𝜇𝑚 was used
(entire domain of the NF dataset). Both the approaches yield identical results.
Inset on the right hand side shows CDF plot for a much smaller subset, i.e.
only centroids within a 200 𝜇𝑚 ball radius are considered. . . . . . . . . . . . . 107
Figure 5.7: Arrangement of voxels in a 2D slice of the NF HEDM data, looking down the
tensile (Z) axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Figure 5.8: Cumulative distribution plots for FF and NF grain radius value for the matched
grains with a cut-off ball radius of 200𝜇𝑚. The figure on the left shows the
comparison between the two datasets with the as-is FF grain radius values.
The figure on the right shows the comparison after the FF grain radius values
were multiplied by a factor of 2.9, which gives a closer match with the NF data. 108
Figure 5.9: Comparison of grain centroid positions from FF and ND HEDM results,
corresponding to the final unloaded state of the specimen. Only the centroids
with NF-FF Euclidean distance ≤ 200𝜇𝑚 are considered here. The left side
of the figure shows the connecting vectors scaled and shaded in proportion to
the calculated values of NF grain radius. The same set of plots are repeated
on the right side, with the connecting vectors scaled and shaded in proportion
to FF grain radius, without a corrective scaling factor. . . . . . . . . . . . . . . 109
xviii
Figure 5.10: Left: CDF plot of same data from Figure 5.9, with and without the correction
vector. From the leftward shift of the CDF curve, using the correction vector
results in a better spatial match between the two datasets. Right: projection of
the calculated correction vector in the YZ plane. Its magnitude and direction
are also indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 5.11: Replotting of the same data as in figure 5.9, except in this case the correction
vector is applied to the FF data points. The the FF data appears much better
aligned with the corresponding NF data points. . . . . . . . . . . . . . . . . . . 111
Figure 5.12: Top right: cumulative distribution plots of grain sizes, in terms of number of
constituent voxels. For the subset of grains matched with FF dataset, there is a
shift to right, indicating a larger average grain size. Left: For matched grains,
grain size plotted as a function of the unscaled grain radius. Bottom right:
variation of the correlation coefficient as a function of threshold number of voxels.112
Figure 6.1: Schematic representation showing the sense of shear and plastic spin effected
by crystallographic slip, that results in lattice rotation. 𝑛 denotes the slip
plane normal, while 𝑏 denotes the slip direction. . . . . . . . . . . . . . . . . . 114
Figure 6.2: The upper left prism identifies the initial grain orientation in the sample
frame with the three ⟨𝑎⟩ axes labeled. The top row shows three (0 0 0 1)
stereographic projections showing the lattice spin axis at each load step from
from the beginning of the plastic regime (white) to ∼ 3% (black). The
⟨𝑎⟩ axes of the crystal frame are identified at the center of the upper right
stereographic projection. Increasing the spacing 𝑖 between load steps from
1 to 5 (left to right) smooths the fluctuations between load steps. Middle
and bottom rows: Stereographic projections show the plastic spin axes for
(basal⟨𝑎⟩ (light blue circles on perimeter), prism⟨𝑎⟩ (red circles in the center)
and pyramidal⟨𝑐 + 𝑎⟩ (rainbow colors). The rainbow color scheme for the 12
pyramidal⟨𝑐 + 𝑎⟩ slip system spin axes are shown in the lower left-hand corner
(the two spin axes for different directions on the same plane are close together
with dark and light tones of the same color (plane), and more widely spaced
adjacent spin axes are for the same slip direction on different planes, which
have either a darker or lighter color of the two planes). The symbols for each
spin axis are scaled in proportion to their resolved shear stress magnitude.
The black point identifies the observed spin axis. A blue arc connects a blend
of the two most highly stress slip systems (largest symbols). The observed
spin axis has an arrow pointing to the blend of the top two slip systems that is
closest (largest dot product), In the middle and bottom rows, the most favored
slip systems are different, indicating significant fluctuations in the stress state,
but the black arrows indicate that that the observed spin axis is closer to the
blend of the highly stressed slip systems. With smoothing, the black arrow
gets longer (dot product becomes smaller). . . . . . . . . . . . . . . . . . . . . 115
xix
Figure 6.3: Maximum dot product between lattice spin axis and blended slip system plas-
tic spin axes (linear combination of the highest and second highest resolved
shear stress slip systems), for the twinned grain in figure 6.2. The effect of
smoothing (increasing 𝑖) leads to less fluctuation, and suggests alternating
sets of active slip systems. The colors of the markers indicate the fraction of
the slip system with the second highest resolved shear stress in the blended axis. 117
Figure 6.4: Maximum dot product between the observed lattice spin axes and blended
high resolved stress slip system axes (slip systems with highest and second
highest resolved shear stress), plotted as a function of bulk strain for six
members of three types of grains; twinned grains, neighbors to the twinned
grains, and arbitrarily chosen grains. For each plot, the average value of the
dot product is plotted as a dashed horizontal line and annotated in each plot. . . 119
Figure 6.5: Cumulative distribution plots for the maximum dot product for the three
categories of grains shown in 6.4. The effect of spacing between load steps
is more significant for the twinned grains than for the other two classes of grains. 120
Figure 7.1: (a) OIM Map of the undeformed microstructure of the 4 point bending spec-
imen prior to bending. The region interrogated by DAXM is outlined with a
dashed rectangle. Prisms indicating orientations of the soft and hard grains
are shown. (b) Optical image of the microstructure at the end of the final
bending increment (∼ 3.5% macroscopic strain). The soft and hard oriented
grains are outlined in blue and green respectively. . . . . . . . . . . . . . . . . 122
Figure 7.2: Surface global Schmid factor maps for specimen prior to bending. The overall
texture is conducive to basal and prism ⟨𝑎⟩ slip. . . . . . . . . . . . . . . . . . 122
Figure 7.3: Orientation maps from the indexed DAXM results. The undeformed specimen
was interrogated before mounting with a step size of 4𝜇m (a). For the
subsequent bending increments 1, 2, 3 and 4 ((b) through (e)) a coarser step
size (6𝜇m) was used. The vertical and horizontal lines (colored yellow)
overlaid on the coarse dataset in (b) indicate the locations of finer finer 2𝜇m
step size H and X scans that are shown beneath the coarser map. . . . . . . . . 123
xx
Figure 7.4: Local equivalent strain distribution map corresponding to the unstrained state
and after four subsequent bending increments. The cumulative distribution
function compares the equivalent strain corresponding to the unstrained state
prior to bending (blue), and the four subsequent bending increments. The
inset shows the evolution of the equivalent strain averaged over the entire
interrogated volume for five strain states. The lower row shows the spatially
resolved equivalent strain maps for the unstrained and four deformation states.
The IPF color map for the microstructure corresponding to the unstrained state
is shown on the left hand side of the top row. The unstrained scan was done
prior to the bending experiment and covered a larger volume with a finer step
size. The other scans were taken after each load step during the bending
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Figure 7.5: Schematic outlining strategy used to calculate GND density from DAXM dataset.127
Figure 7.6: Left: The sphere surrounding a kernel for an exemplary voxel within the
combined (Coarse scan + Fine scan) DAXM dataset for bend increment 2
is shown for ball radii range of 10-40 microns. Cumulative distribution
function (CDF) plots of the residual sum of squares error for each of the
three components of the rotation gradient tensor. The error increases with
increasing ball radius used for the nearest neighbor search. . . . . . . . . . . . . 128
Figure 7.7: Bend increment 2: Line scan DAXM dataset superposed on coarse serial
probed DAXM dataset. The two datasets are combined for purposes of the
lattice rotation gradient calculation. . . . . . . . . . . . . . . . . . . . . . . . . 130
Figure 7.8: Bend increment 2: Maps showing the distribution of the nine components of
the rotation gradient tensor for the fine scan grid. . . . . . . . . . . . . . . . . . 131
Figure 7.9: Bend increment 2: GND density maps, showing total GND density (b) and
GND densities specific to slip system families. Screw dislocation contribu-
tions of ⟨𝑎⟩ type are in (g). Comparing the total GND density with the IPF
map (a), it can be seen that the GND concentrations coincide with the grain
boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 7.10: Bend increment 2: Orientation map of the regions of interest where the
scaled slip transfer parameter is plotted as a function of residual Burgers
vector. Region 1 comprises a grain boundary and an intragranular region
with an orientation gradient; region 2, 3 and 5 are triple junctions; and region
4 is a grain boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xxi
Figure 7.11: Principal Stress distribution for the intragranular region in the purple grain
shows varying local stress states indicated by different "principal stress jacks"
that differ from the average stress tensor for the purple grain shown on the
left. The color of each line in the stress jack indicates the value of the
principal stress component in MPa, which are annotated on the average stress
jacks. The tensile direction is parallel to the 𝑌 axis in the coordinate system
shown while 𝑍 is the surface normal. The stress jacks are oriented according
to the XYZ axes, indicating that the largest compressive principal stress is
roughly perpendicular to the stress axis. The average principal stress jack
corresponding to the entire measured volume is also shown on the light gray
sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 7.12: Principal Stress map for the high angle grains near the boundaries above
the purple grain, including the corresponding grain average "principal stress
jacks" for the pink and lavender grains above the purple grain, which have a
significantly different stress state than the purple grain. The tensile axis is
parallel to the global 𝑌 direction (shown on the right of figure). The principal
stress values are annotated on the grain average and volume average stress
jacks. The stress in the pink and lavender grains are much different from the
purple grain, and the stress state near the boundaries differ significantly from
the grain interiors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Figure 7.13: Bend increment 2: Principal Stress maps for the entire vertical fin containing
the LAGB and HAGB of region 1, with each principal component shown
separately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Figure 7.14: Principal Stress direction map for the triple junction in region 2, where
the viewpoint is identified by the X-Y-Z arrows. The grain boundaries are
delineated by black lines. Principal stress jacks corresponding to average
stress tensor for each individual grain, along with their magnitudes and sense
are also shown for comparison. The tensile axis is parallel to the global 𝑌
axis shown on the left of the figure. . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 7.15: Orientation and total GND density maps for region 1. The voxels chosen for
slip transfer analysis in the high angle grain boundary (∼ 28 °misorientation)
and intragranular (low angle grain boundary with misorientation < 1.5°)
regions are shown (circled voxels). The Bunge Euler angles denoting the
average grain orientations are also noted here. It is important to note here
that prisms denoting the crystal orientations are drawn from the perspective
of the sample normal direction, with 𝑌 pointing vertically up and 𝑋 to the right. 143
xxii
Figure 7.16: Map showing spatially resolved deviation from the average grain orientation
in the lower grain of region 1 (Left). The cumulative distribution of misori-
entation deviation within this grain is shown on the right. The region with a
relatively higher deviation coincides with the high local GND accumulation
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Figure A.1: The 𝑚′ relationship observed between the activated twin system and pyramidal
⟨𝑐 + 𝑎⟩ system of a neighboring grain is shown for layers 3 and 5.The
orientations of the parent, twin and neighboring grains are viewed along the
tensile Z axis, (left column) as well as the beam X direction, (right column).
For description of the symbols, the reader is referred to figure 4.10. . . . . . . . 151
Figure A.2: The 𝑚′ relationship observed between the activated twin system and pyramidal
⟨𝑐 + 𝑎⟩ system of a neighboring grain is shown for layers 6, 7, 8 and 11.The
orientations of the parent, twin and neighboring grains are viewed along the
tensile Z axis, (left column) as well as the beam X direction, (right column).
For description of the symbols, the reader is referred to figure 4.10. . . . . . . . 152
Figure B.1: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 6 in FF HEDM and
validated by NF HEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Figure B.2: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and
validated by NF HEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure B.3: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 9 in FF HEDM and
validated by NF HEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Figure B.4: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and
validated by NF HEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure B.5: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and
validated by NF HEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure C.1: Stereographic projection of the evolution of the lattice plastic spin axis is
shown for different values of bulk strain for a twinned grain. . . . . . . . . . . . 158
Figure C.2: Stereographic projection of the evolution of the lattice plastic spin axis is
shown for different values of bulk strain for the neighbor with a high 𝑚′
relationship with respect to the active twin variant in the grain shown in figure C.1159
xxiii
Figure D.1: Laue patterns obtained from energy wire scans of (a) soft grain (unstrained
condition through bend increment 3), (b) hard grain (unstrained condition
through bend increment 4). The variation of reciprocal lattice vector mag-
nitude with depth is shown for each bend increment. The plots on the third
row of (a) and (b) show the intensity profiles as a function of the reciprocal
lattice vector magnitude. The approximate locations in the grain interior used
for the energy wire scans in the hard and soft grains are shown (marked in
red) as insets in the intensity plots. The depths at which the subsurface grain
boundary exists is shown by blue vertical lines in the 𝑄 v/s depth plots for
both the hard and soft orientations. . . . . . . . . . . . . . . . . . . . . . . . . 163
Figure D.2: Local hydrostatic strain maps for the soft (left) and hard (right) grains, shown
for bend increments 0 (unstrained state) to 3. The prisms indicating the
orientations of the hard and soft grains with respect to the loading direction
are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Figure E.1: Principal Stress direction map for the triple junction in region 3. The grain
boundaries are delineated by black lines. Principal stress jacks corresponding
to average stress tensor for each individual grain, along with their magnitudes
and sense are also shown for comparison. . . . . . . . . . . . . . . . . . . . . . 167
Figure E.2: Bend increment 2: Principal Stress distribution for the triple junction in region 5.168
Figure E.3: Bend increment 2: Principal Stress distribution for the triple junction in region 4.169
Figure F.1: Orientation and total GND density maps for regions 2, 3, 4 and 5, correspond-
ing to the second bending increment. The voxels on either side of the grain
boundary used for slip transfer parameter calculations are shown on the right
of each orientation map. The Bunge Euler angles denoting the average grain
orientations are also shown. Prisms denoting orientations are drawn from the
perspective of the sample normal, with 𝑌 pointing up and 𝑋 pointing to the
right (coordinate system shown on the right side of image). The sample coor-
dinate system is shown on the left hand side of image. The tensile direction
is parallel to the 𝑌 direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Figure F.2: The Luster-Morris parameter plotted as a function of magnitude of residual
Burgers vector for the LAGB (top row) and HAGB (bottom row) in region
1. Legends in the plots show the color/shape convention used to denote the
interacting slip system in a voxel pair. This convention is followed for the
subsequent plots. The symbol size for each datum point is scaled with the
sum of the local Schmid factors for the interacting slip systems. . . . . . . . . . 172
xxiv
Figure F.3: LAGB in region 1 scaled by GND density: Left column shows 𝑚′′ =
𝑚′ ′′
𝑚𝑎𝑥(𝜌 𝑠 ,𝜌 𝑠 ) plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 |; while 𝑚 =
𝛼 𝛽
𝑚′
𝑚𝑖𝑛(𝜌 𝑠 𝛼 ,𝜌 𝑠𝛽 ) as a function of 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots are shown on the
right column. Each plot corresponds to one of four transmitting slip systems
considered. Four voxel pairs along the grain boundary are considered here.
For each transmitting slip system type in each voxel pair, only the data points
with the five highest local Schmid factor sums are plotted (A total of 20 plots
per plot). The choice of the transmitting slip system is arbitrarily made as
described in section 7.4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Figure F.4: For four voxel pairs in the LAGB in region 1, each row of plots corresponds to
the initiator slip system families with the higher local Schmid factor consid-
ered as the transmitting system, leading to more datum points in the pyramidal
slip system plot as initiators than other systems. For each initiating slip
system family member in each voxel pair, only the datum points with the five
highest local Schmid factor sums are considered. The number of data points
plotted for each initiator slip system type are indicated. The number of datum
points per slip system family for the plots scaled by maximum and minimum
GND density (left and right columns respectively) are identical. . . . . . . . . . 179
Figure F.5: For four voxel pairs in the LAGB in region 1, each row of plots corresponds
to the initiator slip systems with the lower GND density is considered as the
transmitting system, leading to more datum points in the basal and prism⟨𝑎⟩
slip system plot as initiators than when the Schmid factor is considered in
figure F.4. For each voxel pair, only the data points with the five highest local
Schmid factor sums are considered. The number of data points plotted for
each initiator slip system type are indicated. . . . . . . . . . . . . . . . . . . . 180
Figure F.6: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown
in figure F.3, filtered using a criterion of 𝑚′>=0.8. . . . . . . . . . . . . . . . . 181
Figure F.7: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown
in figure F.4, filtered using a criterion of 𝑚′>=0.8. . . . . . . . . . . . . . . . . 182
Figure F.8: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown
in figure F.5, filtered using a criterion of 𝑚′>=0.8. . . . . . . . . . . . . . . . . 183
Figure F.9: Some instances of slip system interactions for basal, prism⟨𝑎⟩, pyramidal⟨𝑎⟩
and pyramidal⟨𝑐 + 𝑎⟩ initiating systems for the LAGB voxels in region 1 are
shown. The LSf values, which were calculated using the local stress tensor,
and the Bunge Euler angles and 𝑚′ values are noted. The size of the prisms
is proportional to the sum of the LSf values of the sum of the Schmid factors
for the slip system pair. Prisms are drawn from the perspective of the sample
normal, with 𝑌 pointing upwards and 𝑋 pointing towards the right. . . . . . . . 184
xxv
Figure F.10: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function of
scaled residual Burgers vector. The methodology used to plot the data is
identical to F.3. In this case the initiator in an interacting pair of slip systems
is chosen arbitrarily as described in section 7.4.3. . . . . . . . . . . . . . . . . 185
Figure F.11: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function of
scaled residual Burgers vector, using the maximum of LSf as criterion for
choosing the transmitting slip system. . . . . . . . . . . . . . . . . . . . . . . . 186
Figure F.12: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function
of scaled residual Burgers vector, using the minimum of GND density as
criterion for choosing the transmitting slip system. . . . . . . . . . . . . . . . . 187
Figure F.13: HAGB region 1: Same data as plotted in figure F.10: only points with 𝑚′>=0.8
are plotted here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Figure F.14: HAGB region 1: Same data as plotted in figure F.11: only points with 𝑚′>=0.8
are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Figure F.15: HAGB region 1: Same data as plotted in figure F.12: only points with 𝑚′>=0.8
are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Figure F.16: Using the minimum GND density criterion for the transmitting grain, 𝑚′′
is plotted as a function of scaled residual Burgers vector magnitude for the
boundary between the hard (beige) and soft oriented (blue) grains in region
2. Data plotted is for four pairs of voxels situated along the grain boundary.
Only the top five points (sorted by descending order of sum of LSf values) per
voxel pair are considered here. The locations of the points that correspond to
a high likelihood of slip transfer are encircled. . . . . . . . . . . . . . . . . . . 191
Figure F.17: Using the minimum GND density criterion for the transmitting grain, 𝑚′′ is
plotted as a function of scaled residual Burgers vector magnitude; for the
boundary formed by the two softer oriented grains (blue and purple) in region
2. This boundary has the lowest likelihood for slip transfer of the three. Data
plotted is for four pairs of voxels situated along the grain boundary. Only the
top five points (sorted by descending order of sum of LSf values) per voxel
pair are considered here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Figure F.18: Using the minimum GND density criterion for the transmitting grain, 𝑚′′
is plotted as a function of scaled residual Burgers vector magnitude for the
boundary between the hard beige and soft oriented purple grain in region
2. Data plotted is for four pairs of voxels situated along the grain boundary.
Only the top five points (sorted by descending order of sum of LSf values)
per voxel pair are considered here. . . . . . . . . . . . . . . . . . . . . . . . . 193
xxvi
Figure F.19: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ), considering
basal slip as the transmitting system. The transmitting slip system is chosen
on the basis on minimum slip system specific GND density in voxel pair.
The data plotted are from 12 grain boundaries encompassed by regions 1, 2,
3, 4 and 5. The envelope of data points for the 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case shows
an inverse correlation between GND density and slip transmissibility are
enclosed within the lines in each plot. The description of the symbol shape
and color scheme of the data points is explained in section F.0.1 . . . . . . . . . 194
Figure F.20: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ), considering
prism⟨𝑎⟩ slip as the transmitting system. The transmitting slip system is
chosen on the basis on minimum slip system specific GND density in voxel
pair. Data plotted from 12 grain boundaries encompassed by regions 1, 2,
3, 4 and 5. The envelope of data points for the 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case shows
an inverse correlation between GND density and slip transmissibility are
enclosed within the lines in each plot. . . . . . . . . . . . . . . . . . . . . . . . 195
Figure F.21: Bend increment 2: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ),
considering pyramidal⟨𝑎⟩ slip as the transmitting system. The transmitting
slip system is chosen on the basis on minimum slip system specific GND
density in voxel pair. Data plotted from 12 grain boundaries encompassed by
regions 1, 2, 3, 4 and 5. The envelope of data points for the 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case
shows an inverse correlation between GND density and slip transmissibility
are enclosed within the lines in each plot. . . . . . . . . . . . . . . . . . . . . . 195
Figure F.22: Bend increment 2: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ),
considering pyramidal⟨𝑐 + 𝑎⟩ slip as the transmitting system. The transmit-
ting slip system is chosen on the basis on minimum slip system specific GND
density in voxel pair. Data plotted from 12 grain boundaries encompassed by
regions 1, 2, 3, 4 and 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Figure G.1: Bend increment 2: Scaled slip transfer parameter plotted as a function of
scaled residual Burgers vector magnitude for the first grain boundary (shown
in the inset) constituting the triple junction in region 3 of microstructure
described in 7.4.3. The choice of the transmitting slip system is made on the
basis of minimum GND density of the voxel pair. . . . . . . . . . . . . . . . . 198
Figure G.2: Bend increment 2: Scaled slip transfer parameter plotted as a function of
scaled residual Burgers vector magnitude for the second grain boundary
(shown in the inset) constituting the triple junction in region 3 of microstruc-
ture described in 7.4.3. The choice of the transmitting slip system is made on
the basis of minimum GND density of the voxel pair. . . . . . . . . . . . . . . 199
xxvii
Figure G.3: Bend increment 2: Scaled slip transfer parameter plotted as a function of
scaled residual Burgers vector magnitude for the third grain boundary (shown
in the inset) constituting the triple junction in region 3 of microstructure
described in 7.4.3. The choice of the transmitting slip system is made on the
basis of minimum GND density of the voxel pair. . . . . . . . . . . . . . . . . 200
Figure G.4: Bend increment 2: Scaled slip transfer parameter plotted as a function of
scaled residual Burgers vector magnitude for the grain boundary in region 4
of microstructure described in 7.4.3. . . . . . . . . . . . . . . . . . . . . . . . 201
Figure G.5: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted
as a function of scaled residual Burgers vector magnitude for the first grain
boundary constituting the triple junction in region 5 of microstructure de-
scribed in 7.4.3. Each column of plots correspond to slip system interaction
at a grain boundary (inset). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Figure G.6: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted
as a function of scaled residual Burgers vector magnitude for the second
grain boundary constituting the triple junction in region 5 of microstructure
described in 7.4.3. Each column of plots correspond to slip system interaction
at a grain boundary (inset). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Figure G.7: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted
as a function of scaled residual Burgers vector magnitude for the third grain
boundary constituting the triple junction in region 5 of microstructure de-
scribed in 7.4.3. Each column of plots correspond to slip system interaction
at a grain boundary (inset). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Figure H.1: Schematic showing the transformation from the APS Beamline 34 Coordinate
system to TSL-OIM coordinate system. The two systems can be brought into
coincidence by rotating the APS system clockwise about the 𝑋 axis by 135°.
The transformation matrix (𝑔𝑇34𝐼
𝑆𝐿 ) is calculated from the direction cosines
𝐷𝐸
between the two coordinate systems. . . . . . . . . . . . . . . . . . . . . . . . 206
Figure H.2: Schematic showing the transformation from the APS Beamline 1 Coordinate
system to TSL-OIM coordinate system. The two systems can be brought into
coincidence by rotating the APS system counter clockwise about the 𝑋 axis
by 90°. The transformation matrix (𝑔𝑇𝐼 𝐷1𝑆𝐿 ) is calculated from the direction
cosines between the two coordinate systems. . . . . . . . . . . . . . . . . . . . 207
xxviii
KEY TO ABBREVIATIONS
3DXRD three-dimensional X-ray Diffraction
APS Advance Photon Source
CP Crystal Plasticity
CP-Ti Commercial purity titanium
CRSS Critical Resolved Shear Stress
DAXM Differential Aperture X-ray Microscopy
EBSD Electron Backscatter Diffraction
FFHEDM Far Field High Energy Diffraction Microscopy
GND Geometrically Necessary Dislocations
hcp hexagonal close packed
HEDM High Energy Diffraction Microscopy
Lsf Local Schmid factor
NFHEDM Near Field High Energy Diffraction Microscopy
OIM Orientation Imaging Microscopy
SAF Spatial Alignment Factor
SSD Statistically Stored Dislocations
XRD X-ray Diffracton
xxix
CHAPTER 1
INTRODUCTION
Titanium and its alloys find widespread use as structural materials due to their favorable strength
to weight ratio. Bulk deformation behavior of metals can be sufficiently predicted by continuum
based models that assume isotropic character. At the meso-scale however, mechanical behavior
of crystalline materials is governed by their anisotropic character. Furthermore, in a polycrystal
the morphology and stress state within each grain is strongly influenced by the deformation of its
neighbors. This has major implications for bulk strength, texture evolution and room temperature
formability. It is therefore imperative that material models at the meso scale are able to capture this
heterogeneity in strain response.
1.1 Motivation for current work
Mesoscale mechanical response of individual grains in polycrystals is strongly dependent on
the strain accommodation provided by the neighboring grains. Furthermore, conditions of local
subsurface stress states that could lead to nucleation of mechanical twins are not well under-
stood. Current crystal plasticity (CP) based material models that simulate mechanical behavior
do not have a reliable means of predicting twin nucleation. In light of these observations, a true
three-dimensional characterization of local stress state and morphology is necessary to reliably
map structure property relationships. Development of such a body of knowledge has significant
implications in material processing and furthering the understanding of damage nucleation. The
electron backscatter electron diffraction (EBSD) method is a well known means to characterize
the response of grains to local stress, but it only provides surface information. three-dimensional
EBSD (EBSD tomography) can provide 3D characterization at spatial resolution of the order of
tens of nanometers (Konijnenberg et al., 2015). This however, requires expensive focused ion
beam (sequential serial sectioning); or Laser (sequential layer ablation) infrastructures. Moreover
these methods are not non-destructive in nature. In this work, synchrotron based X-ray diffraction
1
is used to obtain three-dimensional characterization of polycrystals in a nondestructive manner at
high spatial resolutions.
In addition to characterization of the three-dimensional morphology of the constituent grains,
a second important question arises with regard to the local stress state and its effect on macroscale
mechanical response. This necessitates the measurement of local grain orientations and stress
states within a polycrystalline aggregate using reliable experimental techniques. Towards this end,
the present work quantifies the capability of two different high energy x-ray diffraction methods
(HEDM) in measuring grain orientations, positions and local stress states.
This work will help advance the understanding of the local stress state in polycrystalline aggre-
gates. It is expected to help identify conditions where twin nucleation occurs by slip transmission
across grain boundaries. Moreover, local grain averaged strain accommodation in the twin neigh-
borhood is also assessed. Furthermore, a correlation is identified between the local stress state
and the role of geometrically necessary dislocation (GND) density as an indicator of how strain is
accommodated in polycrystalline materials.
1.2 Deformation systems in hexagonal titanium
Plastic deformation in crystalline materials is the result of atomic scale displacements brought
about by slip mechanisms. At low homologous temperatures and strain rates, crystallographic
slip is the predominant mechanism of deformation, while mechanical twinning is dominant at low
temperatures and/or high strain rates. Hexagonal titanium has a c/a ratio of about 1.588, which is
less than the ideal value1. Under conditions where the loading direction is almost perpendicular
to the c-axis (soft orientation), prism ⟨𝑎⟩ slip is readily activated; whereas when the c-axis is
inclined between 20° and 70° to the loading direction, basal ⟨𝑎⟩ is favored. On the other hand,
if the loading direction is close to being parallel to the c-axis (hard orientation) then pyramidal
slip and mechanical twinning become favored –note that being favored does not imply that they
occur as they have higher CRSS values, so this leads to complex deformation conditions. There is
1 The ideal c/a ratio for the hexagonal close packed lattice is ∼1.633.
2
some agreement that nucleation of twins is a stochastic event, although slip transfer across grain
boundaries has been seen to nucleate twins in certain cases (Bieler et al., 2014; Wang, 2011).
1.3 High energy x-ray diffraction (HEXRD)
X-ray diffraction, specifically using high brilliance beams from synchrotron sources is a highly
effective way for three-dimensional characterization of meso scale deformation in polycrystals.
HEXRD methods enable non destructive characterization of samples in both in-situ and ex-situ
configurations. In the current work, the HEXRD techniques used can be broadly classified into
three types listed below.
1. Far field high energy diffraction microscopy (FF HEDM) , that uses a monochromatic beam
2. Near-field high energy diffraction microscopy (NF HEDM) using a monochromatic beam.
3. Differential aperture x-ray microscopy (DAXM), that uses white beam diffraction.
The first method is similar to powder diffraction (Poulsen, 2012), where each diffracting family
of crystallographic planes from the polycrystal show up as Debye Scherrer rings. Provided the
specimen has a coarse grain size, unique orientations can be identified with a resolution of 0.1◦
to 0.01◦ . Spatial resolution is ∼ 10 𝜇m, and grain averaged strain tensors can be evaluated with a
resolution of 10−4 (Park et al., 2017). The depth resolution is of the order of a millimeter (Park
et al., 2017).
The second method is conceptually similar to FF HEDM–the main difference here is in the
sample to detector distance. In the case of NF HEDM this distance is much closer (∼ 10 mm,
compared to ∼ 1 m for FF HEDM). This method is similar to a non-destructive 3D electron
backscatter diffraction characterization of a polycrystalline aggregate using a line focused beam
(Shastri et al., 2007; Park et al., 2017). This method is sensitive to orientation gradients–the angular
resolution is in the 0.01°-0.1°range, while the spatial resolution is of the order of units of microns
(Park et al., 2017). Strain measurements however, cannot be made using this method.
3
The third method is based on Laue diffraction (Ice and Pang, 2009) using a highly collimated
polychromatic beam. This method enables high spatial resolution and the ability to precisely
measure the angles between Laue spots. Comparison of the changes in inter-spot angles with respect
to the unstrained unit cell enables measurement of lattice distortion (Ice and Pang, 2009). Thus
both lattice orientation and deviatoric strain can be extracted with sub-micron spatial resolution
and angular resolution of the order of 0.01◦ . Furthermore, voxel specific values of local strain
can be obtained with a resolution of 10−4 (Larson et al., 2008). Since micro Laue diffraction
using a polychromatic beam does not yield the magnitude of the reciprocal lattice vectors, lattice
dilatation cannot be evaluated using this procedure. An additional procedure enables the use of a
monochromatic beam with a calibrated (known) energy that allows the estimation of the scattering
vector magnitude that enables identification of the local hydrostatic component of the strain tensor.
Therefore it is possible to obtain the full description of the local strain tensor using a combination
of the two procedures.
In the present work, two polycrystalline specimens of commercial purity titanium (CP-Ti)
were characterized using FF HEDM, NF HEDM and DAXM. The specimen evaluated with FF
HEDM/NF HEDM had a hard crystallographic texture and the sample evaluated with DAXM had
a predominantly soft texture with respect to the loading direction.
1.4 Structure of the dissertation
This dissertation is structured as follows. The current chapter provides an introduction to
the work and the motivation behind the work. Chapter 2 provides a review of the literature and
background that forms a basis for the present work. A detailed outline of the materials and methods
used is elucidated in chapter 3.
Chapter 4 describes an in-situ far field study of deformation twin evolution in hexagonal titanium
during a tensile test. Here, discrete twin-parent grain pairs are identified using geometrical and
spatial proximity conditions. Local strain accommodation observed in parent and neighboring
grains due to twin formation are discussed. A comparison is made between two different methods
4
used for post-processing (identification of individual grains) at different loading states of the
specimen. A comparison is made between grains identified during the far field experiment and
surface EBSD measurements. Additionally, the relative positions and morphology of the twinned
and candidate triggering grains are examined in light of the more recently collected near field high
energy diffraction microscopy (NF HEDM) data.
Chapter 5 provides a comparative statistical study between the FF HEDM and NF HEDM
methods. Two algorithms for matching FF HEDM and corresponding NF HEDM data are discussed
here. Furthermore, the relationship between reported grain radius values in FF HEDM and grain
size calculated from number of constituent voxels in NF HEDM is explored. A quantitative
comparison of the kinematics and grain averaged stress measurements from far field high energy
diffraction microscopy (FF HEDM) is presented in chapter 6.
Chapter 7 details an in-situ Differential aperture X-ray microscopy (DAXM) characterization of
a hexagonal titanium specimen subjected to a four point bending test. Local strain accommodation
and propensity of slip transfer between adjacent grains are discussed. The change in magnitude
and direction of local principal stress components are used to characterize the variation in stress
state from grain interior to the boundary and surface.
Finally, conclusions and scope for future work are discussed in chapter 8.
5
CHAPTER 2
LITERATURE REVIEW
2.1 Basics of plastic deformation in crystalline materials
2.1.1 Crystallographic Slip
Plastic deformation by slip in crystalline materials is typically brought about by atomic scale
displacements of line defects known as dislocations. These slip mechanisms are comprised of shear
displacements and are lattice invariant i.e. despite shape change, the crystal structure remains as
it was prior to deformation (Bhadeshia, 1996). Crystallographic slip takes place along specific
planes and in specific directions (usually those with the highest atomic density). Dislocations are
line defects, that cause local discontinuities in the natural atomic order of a crystal. A dislocation
is a line in the plane that separates the slipped and the unslipped parts of the crystal. One way to
visualize a dislocation is illustrated in figure 2.1, where an extra half plane of atoms is inserted into
the top half of a perfect crystal (b). This would result in a displacement of one atomic spacing of
the faces of the slot. The dislocation line DC locates the region where the perturbation of atoms
from their normal positions is the most. As one moves away from the dislocation line, the extent
of interatomic bond distortion decreases. The dislocation line DC shown in (b) is a positive edge
dislocation and is commonly denoted by the ⊥ symbol. If the extra half plane of atoms was inserted
in the bottom half of the crystal, DC would be a negative edge dislocation, denoted by ⊤.
In the case of screw dislocations, the relative displacement of one side of the crystal occurs
parallel to AB. Schematics for left and right handed screw dislocations are shown in figure 2.1 (c)
and (d) respectively. Screw dislocations can be represented as a single surface helicoid (Hull and
Bacon, 2011) as shown in figure 2.1 (f). In general, dislocations are of mixed character (they can
be decomposed into edge and screw components).
The direction and magnitude of shear of a dislocation is denoted by the Burgers vector. One
6
Figure 2.1: Schematic showing the geometry of edge and screw dislocations in a simple cubic
lattice. (a) perfect lattice, (b) an extra half-plane of atoms ABCD inserted into the top half of
the crystal. DC is a positive edge dislocation, (c) left handed screw dislocation DC is formed
by displacement of the faces ABCD relative to each other in the direction AB, (c) right handed
screw dislocation DC,(e) atomic planes with spacing b in a perfect crystal, (f) planes distorted by
a right-handed screw dislocation. (Hull and Bacon, 2011)
important feature of distinction between edge and screw dislocations is that the direction of the
Burgers vector is normal to the edge dislocation line; whereas it is parallel to the screw dislocation
line. A combination of a slip plane normal and slip direction (direction of Burgers vector) constitute
a slip system.
2.1.2 Schmid’s law for slip in a single crystal
The criterion for determining if a slip system will be activated in a single crystal is given by
Schmid’s law. Schmid proposed that a single crystal subjected to uniaxial load would begin to
deform only when the resolved shear stress exceeds a critical value. The concept of resolved shear
stress is illustrated in figure 2.2 using a schematic of a cylindrical single crystal of cross sectional
area A under a uniaxial tensile load F. The angle between the F and slip plane normal is denoted
7
Figure 2.2: Schematic illustrating the concept of critical resolved shear stress for a single crystal
(Hull and Bacon, 2011). It is important to note that in general 𝜙 and 𝜆 are not coplanar.
by 𝜙, while the angle between F and slip direction is denoted by 𝜆 (Hull and Bacon, 2011). The
equation for the shear stress resolved in the direction of slip is given as follows.
𝐹
𝜏= 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜆 (2.1)
𝐴
If in equation 2.1, the force term is substituted by the tensile force necessary to initiate slip 𝐹𝑐 ,
then 𝜏 becomes 𝜏𝑐 . This minimum value of shear stress, required to initiate plastic deformation is
defined as the critical resolved shear stress (CRSS). The product cos 𝜙 cos 𝜆 is called the Schmid
factor (𝑚). For a single crystal subjected to a uniaxial tensile load, the maximum value of Schmid
factor is 0.5. Clearly, the most active slip systems will have a higher value of 𝑚.
2.1.3 Multiplication of dislocations
Dislocations multiply in the bulk of a crystal by regenerative multiplication mechanisms such as
a Frank-Read type source (Hull and Bacon, 2011). A Frank-Read source is activated when a critical
8
2R
(a) (b) (c)
(d) (e) (f)
Figure 2.3: Schematic showing the process of regenerative multiplication of dislocations by a
Frank-Read source [adapted from Read (1953)].
shear stress is reached that causes a dislocation pinned between two pinning points to bow out into
a dislocation loop that can escape from the pinning points. An example of a Frank-Read source
is schematically shown for an edge dislocation in figure 2.3. Figure 2.3 (a) shows the segment of
dislocation pinned at its two ends by obstacles (which could be dislocation interactions or nodes,
jogs, precipitates etc.). The direction of the force acting on the dislocation, 𝐹 = 𝑏.𝜏 is shown by
the arrows, where 𝑏 is the Burgers vector and 𝜏 is the shear stress. As a result of this force the
dislocation bows out (figure 2.3 (b)). In order to arrive at the critical configuration of a semi-circle
a maximum shear stress 𝜏𝑚𝑎𝑥 = 𝐺𝑏/𝑅, where 𝑅 is the critical radius of curvature. As the stress
exceeds 𝜏𝑚𝑎𝑥 , the radius of curvature becomes smaller and the dislocation becomes unstable and
continues to bow out. Figure 2.3 (c) shows the bowing shortly before the curved portions of the
segment come in contact with each other. The parts that touch each other annihilate because their
Burgers vectors have opposite sign for the same line direction (figure 2.3 (e). Finally a strain
dislocation segment and a loop remain in the stable configuration (figure 2.3 (f)). Under the applied
stress the loop will expand and encounter other obstacles. Then the process (a)-(f) repeats.
2.1.4 Generalized Schmid Factor
The Schmid law applies to conditions where the stress tensor has only one non-zero diagonal
component and zeroes in off-diagonal components. Because neighboring grains usually deform
9
T
Z n
m
Y
X
http://www.esrf.eu/UsersAndScience/Experiments/StructMaterials/news/data-explosion Adapted from Euler Angle and Orientation Matrix Toolkit, Bieler 2014 [11]
Figure 2.4: Illustration of generalized stress state for an arbitrary grain in a polycrystal. The local
stress tensor corresponding to this generalized stress state is shown as a function of X, Y and Z. T
is the traction vector, while n and m are the slip plane normal and slip direction respectively.
heterogeneously (Ashby, 1970), the stress state becomes more complicated as shown in figure 2.4.
Even though the imposed global load may be uniaxial, the local stress state for an arbitrary grain
may have non-zero off diagonal components. The local stress tensor, slip direction and the slip
plane normal are used to derive a generalized Schmid factor (Soare, 2014). For a general stress
√
2
tensor, the maximum possible magnitude of the local Schmid factor is 2 (∼ 0.707) (Bieler, 2014).
As shown in figure 2.4, the local stress tensor 𝜎 acts on an arbitrary grain in the polycrystalline
aggregate. The traction vector 𝑻 acting on a unit area on an arbitrary plane is given as follows.
𝑻 = 𝝈𝒏 (2.2)
where 𝒏 is the unit plane normal vector. Crystallographic slip is typically expressed in the
crystal coordinate system. The first step in this process is to rotate the stress tensor onto the crystal
coordinate system as shown below.
𝝈′ = 𝑹𝑻 𝝈𝑅 (2.3)
where 𝝈′ is the rotated stress tensor and 𝑹 = 𝒈𝑻 is the transpose of the orthogonal orientation
matrix derived from the crystal orientation.
The shear stress 𝜏 acting on the plane defined by 𝒏 in the direction of slip 𝒎 is given by:
10
1
𝜏 = 𝑻.𝒎 = 𝝈′ 𝒏.𝒎 = 𝝈′ : 𝒏 ⊗ 𝒎 = 𝝈′ : (𝒏 ⊗ 𝒎 + 𝒎 ⊗ 𝒏) (2.4)
2
where 𝒎 is the slip direction parallel to the Burgers vector.
Now if 𝑷 = 12 (𝒏 ⊗ 𝒎 + 𝒎 ⊗ 𝒏), equation 2.4 can be rewritten as:
𝜏 = 𝝈′ : 𝑷 = 𝝈𝒊′ 𝒋 𝑷 𝒊 𝒋 (2.5)
If the stress tensor is now normalized i.e. ||𝝈′ || = 1, then equation 2.5 gives the generalized
Schmid factor that has to be evaluated for every slip system in order to determine which slip systems
are most favored for slip.
For polycrystalline materials, Sachs (Sachs, 1928) was the first to suggest a way to estimate
a composite Schmid factor by assuming that only one slip system is active per grain and then
averaging the Schmid factor values for all individual grains in the specimen. Since no constraint
is imposed on the deformation of individual grains by neighboring grains, this approach gives a
lower bound for the yield strength in uniaxial tension (Sachs, 1928).
Taylor proposed a model based on the von Mises criterion–which requires that at least five
independent slip systems must be active in order to ensure strain compatibility among the constituent
grains in a polycrystal (Taylor, 1938). The assumption made here is that these five systems
correspond to the least amount of work during the arbitrary shape change of the aggregate. Taylor’s
assumption can therefore be formalized as follows:
∑︁
𝝈𝑑𝝐 = 𝜏𝑐 𝑑𝜸𝑖
Í𝑖 (2.6)
𝑖 𝑑𝜸𝑖 1
𝝈 = 𝜏𝑐 = 𝜏𝑐 ,
𝑑𝝐 𝑚
Í
where, 𝝈 and 𝑑𝝐 are the macroscopic stress and strain increment in the uniaxial test and 𝑑𝜸 is the
1
summation of strain increments over all the active slip systems. 𝑚 is the reciprocal of the Schmid
factor.
11
2.1.5 Deformation twinning
Twinning is a secondary deformation mechanism that is common in low symmetry metals and
alloys (Christian and Mahajan, 1995). Twinning modes are activated when deformation by slip alone
does not provide an adequate number of independent slip systems for shape change. In deformation
twinning, reorientation of the parent lattice occurs by coordinated atomic displacements that are
equivalent to simple shear, whereby the original volume of the crystal is preserved (Bilby and
Crocker, 1965). This requirement of simple shear necessitates that the atomic displacements are
an integral fraction of the lattice points1 (Christian and Mahajan, 1995). This also ensures that the
crystal symmetry is retained for twin formation, where the crystal structure of the parent lattice is
preserved in the twin. This is in contrast to other shear transformation processes such as martensite
formation, where either the crystal structure or the c/a ratio changes in the transformed lattice (Such
a transformation does not require the atomic displacements to be a rational fraction of the lattice
points).
Figure 2.5 shows the four basic elements of a deformation twin. 𝜿1 is the invariant plane of
shear and the corresponding shear direction is 𝜼1 . The second undistorted plane is 𝜿 2 and the
conjugate shear direction 𝜼2 is given by the intersection of 𝜿 2 and the plane of shear P. Fixing 𝜿1
and 𝜼2 ; or 𝜿2 and 𝜼1 defines a twinning mode (Christian and Mahajan, 1995).
Twins are classified as type I, type II or compound–depending on which of the four basic
elements are rational. For type I twins, 𝜿1 and 𝜼2 constitute a rational plane and rational direction
respectively of the parent lattice. In type II twins, 𝜿2 and 𝜼1 form the rational plane and direction
respectively. For compound twins, all the four crystallographic elements are rational (Christian and
Mahajan, 1995).
1 The displacements are multiples of a rational number 𝑃
𝑄, where 𝑃 and 𝑄 are integers.
12
Figure 2.5: Schematic showing the four twinning elements (Christian and Mahajan, 1995). 𝜿1
and 𝜿2 are twinning and conjugate planes respectively. The corresponding twinning and conjugate
directions are 𝜼1 and 𝜼2 . 𝜼1 , 𝜼2 ,and the normals to 𝜿1 and 𝜿 2 lie in shear plane P.
2.2 Plastic deformation in hexagonal titanium
Hexagonal titanium has a less than the ideal c over a ratio (∼1.588). The most readily activated
slip plane is prism, followed by basal, both having the ⟨𝑎⟩ type Burgers vector (Lütjering and
Williams, 2007). Due to lower symmetry compared to cubic metals, the slip modes for plastic
deformation in hexagonal titanium are strongly dependent upon the orientation of the loading
direction with respect to the c-axis. When basal or prism slip systems are active, the grain is in a
"soft" orientation, but when they are not easily activated the crystal is in a "hard" orientation (Wang
et al., 2010a; Littlewood et al., 2011). Figure 2.6 illustrates the major planes and slip directions (a),
and the relevant slip modes (b) that are associated with them (Wang, 2011).
According to Paton et al. (Paton et al., 1973), basal slip ({0 0 0 1}<1 1 -2 0>) and prismatic slip
({1 0 1 0}<1 1 -2 0>) have historically been considered the most important deformation modes in
pure alpha titanium. Levine Levine [1966] determined that prism ⟨𝑎⟩ slip in the temperature range
0-210 𝐾 was activated by overcoming the Peierls barrier. Non-basal ⟨𝑐 + 𝑎⟩ slip was reported as a
major deformation mode in Ti-Al alloys by Cass (Cass, 1970). Furthermore Paton, and Backofen
(Paton and Backofen, 1970) found that mechanical twinning in pure titanium can occur at any
temperature, and is therefore an important mode of plastic deformation.
13
(a) (b)
Figure 2.6: Schematic illustrating the major crystallographic planes and directions (a) , and the slip
modes (b) associated with them (adapted from Wang (2011)).
More recently, micropillar compression tests conducted on commercial purity titanium (CP-Ti)
specimens oriented in the [2 1 1 0] direction confirm prism ⟨𝑎⟩ slip to be the primary slip mode
(Kishida et al., 2020). For specimens oriented along the [0 0 0 1] direction first order pyramidal
⟨𝑐 + 𝑎⟩ slip is observed. Calculations using inverse-power law relationship show that the CRSS
values for first order pyramidal ⟨𝑐 + 𝑎⟩ slip is higher than those of any other active slip modes at
room temperature (Kishida et al., 2020).
The primary slip directions in hexagonal titanium are the three ⟨1 1 2 0⟩ (⟨𝑎⟩) vectors. These
lie on three families of crystallographic planes– {0 0 0 1} planes; three {1 0 1 0} and six {1 0 1 1}
planes. Out of the twelve slip systems, only eight are independent. Furthermore, the shape changes
effected by slip systems {0 0 0 1}<1 1 -2 0> and {1 0 1 0}<1 1 -2 0> combined are equivalent to
the {1 0 1 1}<1 1 -2 0> slip system, only four of which are independent. (Lütjering and Williams,
2007).
Even when basal and prism slip are active, they do not satisfy the von Mises criterion, where a
minimum of five independent slip systems are needed to effect an arbitrary shape change by plastic
deformation (prism and basal slip do not enable a shape change along the the ⟨𝑐⟩ axis direction)
(Lütjering and Williams, 2007). Therefore, pyramidal ⟨𝑐 + 𝑎⟩ slip and deformation twinning are
often observed (Yu et al., 2013). These systems allow changes in the crystal dimension along the
14
Table 2.1: Parameters associated with the four twinning modes in CP-Ti (Wang, 2011)
Mode 𝜿1 𝜼1 𝜿2 𝜼2 s Misorientation2
(𝛾 2 −3)
T1 3 {1 0 1 2} ⟨1 0 1 1⟩ {1 0 1 2} ⟨1 0 1 1⟩ √ =0.174 85°
3𝛾
1
T2 {1 1 2 1} ⟨1 1 2 6⟩ {0 0 0 1} ⟨1 1 2 0⟩ 𝛾 =0.630 35°
2(𝛾 2 −2)
C1 4 {1 1 2 2} ⟨1 1 2 3⟩ {1 1 2 4} ⟨2 2 4 3⟩ 3𝛾 =0.219 65°
(4𝛾 2 −9)
C2 {1 0 1 1} ⟨1 0 1 2⟩ {1 0 1 3} ⟨3 0 3 2⟩ √ =0.099 54°
4 (3)𝛾
⟨𝑐⟩ axis, though they are less readily activated. It has been observed that the density of twins in
commercial purity titanium (CP-Ti) increased significantly with increasing strain rates, strain and
decreasing temperature (Christian and Mahajan, 1995; Nemat-Nasser et al., 1999).
Four twinning modes have been observed in hexagonal titanium. T1 and T2 accommodate
extension along the c-axis (extension twins), and C1 and C2 accommodate contraction along the
c-axis (compression twins). The 𝑐/𝑎 ratio (referred to as 𝛾 in table 2.1) is used to calculate the
shear s evolved due to a twinning mode. Table 2.1 summarizes the important parameters associated
with the four twin modes. Because the CRSS for ⟨𝑐 + 𝑎⟩ pyramidal slip is higher than twins, twins
can be activated when the crystal is in a hard orientation (Littlewood et al., 2011; Britton et al.,
2015).
Akhtar and Teghtsoonian (1975) observed a three-stage hardening behaviour during plastic
deformation of single crystal titanium. More recently, Chichili et al. (1999) showed that for
hexagonal titanium, although dislocation motion accounted for most of the plastic deformation,
twin-dislocation interactions play an important role during strain hardening at room temperature.
Therefore mechanical twinning can have a strong influence on macroscopic properties like ductility
in hexagonal titanium
T1 twins ({1 0 1 2}⟨1 0 1 1⟩) in titanium are also activated when extension along the c-axis is
required. The geometrical orientation relationship between a T1 twin and its parent lattice are
shown in figure 2.7, where the c-axis misorientation is 85° about the [1 1 2 0] axis. Moreover, the
2 c-axis misorientation between twin and parent lattice
3 T: Extension twin
4 C: Compression twin
15
parent and the twin lattices share a common ⟨𝑎⟩ axis. Twinning provides a necessary response to
strain compatibility and has been shown to enable higher ductility in hexagonal metals (Christian
and Mahajan, 1995). The exact mechanism behind twin nucleation is complex and is not clearly
understood. Prior work (Capolungo et al., 2009; Beyerlein and Tomé, 2010; Bieler et al., 2014) has
shown that twinning does not necessarily obey the Schmid law. It is also observed, that in hexagonal
titanium, the twin variant activated does not always correspond to the highest resolved shear stress.
In their study on deformation twinning in hexagonal titanium, Richeton et al. (2012) observed that
although all active variants had relatively high Schmid factors (>0.38), the active twins did not
necessarily correspond to the highest Schmid factor among the possible variants. Schuman et al.
(2011, 2012) proposed a variant selection criterion for T1 and C1 twins in T40 titanium, based
on deformation energy required for nucleation. This criterion was found to account for 85% of
the observed variants. It has been observed that deformation twinning plays an important role in
enhancing ductility in hexagonal metals. Dislocations at twin boundaries result in strain relaxation
and lowering of energy of the crystal (Kochmann and Le, 2009). From the standpoint of material
processing, room temperature formability of hexagonal metals is still a significant challenge.
Therefore, it is important to understand the relationship between energy dissipation from twin
formation and strain relaxation in the parent and neighboring grains. Moreover, several previous
studies have confirmed that deformation twins in hexagonal metals nucleate at grain boundaries
(Wang et al., 2009a; Beyerlein and Tomé, 2010; Christian and Mahajan, 1995). Therefore, their
role in strain accommodation is of particular interest in further developing the understanding of
intergranular crack nucleation.
Nucleation and growth of deformation twins in titanium is strongly dependent upon the loading
conditions; which can be more complex than simple monotonic loading when practical structural
applications or material processing operations are considered. Such complexity in loading con-
ditions can effect further changes within an already formed twin (Wang et al., 2014; Guan et al.,
2021). For example, removal or reversal of the direction of a monotonic load can result in detwin-
ning, thereby reducing the twinned volume. Moreover, under conditions where the loading is not
16
Figure 2.7: Schematic illustrating the geometrical relationship between a T1 twin and its parent
lattice. Adapted from (Bieler et al., 2014).
monotonic, new twins can form inside an existing twin (a phenomenon known as double twinning)
(Xu et al., 2018; Huang et al., 2019). More recently the kinematics and local stress evolution in
parent and twin grains in polycrystalline magnesium during a compression tension experiment were
studied in detail by Louca et al. (2021), using 3D Xray diffraction. It was observed that detwinning
was typically associated with reversal of sign in the resolved shear stress associated with the active
variant.
As mentioned earlier, the interactions of twins and dislocations play an important role in strain
hardening of hexagonal titanium. In terms of physics based modeling of deformation behavior,
commonly used crystal plasticity (CP) based schemes consider twinning as directionally constrained
slip systems (Kalidindi, 1998). Detailed studies of the stress and strain rates associated with a
nucleating twin are a subject of current investigations. Spatially resolved models for twin nucleation
and propagation have been developed recently for hexagonal and orthorhombic crystal structures
(Ardeljan et al., 2015; Guo et al., 2017; Cheng and Ghosh, 2017). Furthermore, numerical studies
of the stress field around deformation twins in magnesium have also been conducted (Arul Kumar
et al., 2015). Most of these mesoscale models are based on crystal plasticity (CP) principles that use
finite element or spectral based solvers. In most of these models,the underlying physics of plastic
17
deformation is typically captured by the phenomenological power law. In its basic formulation the
phenomenological model takes into account the kinematics of elasto-plastic deformation with the
evolving critical resolved shear stress as a state variable. Moreover, interactions between slip and
twin systems can also be considered. A more sophisticated physical basis is to use the dislocation
density as the state variable, which is implemented in the microstructure based formulation. More
recently, Wronski et al. used a visco-plastic self-consistent (VPSC) model to simulate the evolution
of {1 0 1 2} tensile and {1 1 2 2} compression twinning in hexagonal titanium. Additionally this
model also incorporated a simple Monte Carlo scheme to predict variant selection (Wronski et al.,
2018).
2.3 An overview of crystal plasticity principles
In this section the basics of deformation kinematics and its application to constitutive relations
governing crystal plasticity (CP) are reviewed. A fundamental quantity in the CP formulation is
the deformation gradient. The concept of the deformation gradient is illustrated in figure 2.8. The
position vector of a material point in a body in the reference configuration is represented by 𝑑𝑥.
The vector can be mapped in the deformed (current) configuration by 𝑑𝑦 = 𝑑𝑥 + 𝑑𝑢, where 𝑑𝑢 is the
differential total displacement vector (Roters et al., 2010). The current and reference configuration
vectors are related by the deformation gradient as follows.
𝜕𝑦 𝜕𝑢
𝑑𝑦 = 𝑑𝑥 = 𝑰 + 𝑑𝑥 = 𝑭𝑑𝑥 (2.7)
𝜕𝑥 𝜕𝑥
Where 𝑰 denotes the identity tensor of rank 2. Therefore the deformation gradient 𝑭 can be defined
as a second order tensor that maps the displacement of a material point from the undeformed
(reference) state to the deformed (current) state. Since the deformation gradient is an indicator of
shape change, 𝑭 can be expressed as the product of pure rotation 𝑹 and a symmetric tensor that
captures pure stretching. This can be represented in two ways ,depending on the order of operation.
𝑭 = 𝑹𝑼 = 𝑽 𝑹 (2.8)
18
Figure 2.8: Schematic illustration of the concept of deformation gradient. The total deformation
gradient 𝐹 can be decomposed into the product of the elastic and the plastic components(Roters
et al., 2010).
Where 𝑼 and 𝑽 are the symmetric right and left stretch tensors respectively. A fundamental
constitutive premise in crystal plasticity is the multiplicative decomposition of the total deformation
gradient into elastic and plastic components as follows.
𝑭 = 𝑭𝒆 𝑭 𝒑 (2.9)
where 𝑭𝒆 and 𝑭 𝒑 are respectively the elastic and plastic components of the deformation gradient.
As shown in figure 2.8, the elastoplastic deformation from the reference configuration can be
considered to occur in two stages. The first stage is the intermediate (relaxed) configuration, that
corresponds to the irreversible (plastic) response. This deformation, captured by 𝑭 𝒑 remains even
after all external forces and displacements causing it are removed. From this configuration, the
elastic deformation corresponding to 𝑭𝒆 is imposed to arrive at the current (deformed) configuration.
This two stage breakdown is a theoretical construct; as in reality the intermediate configuration
would require that the dislocations causing shape change would not reside within the material point
neighborhood (Roters et al., 2010).
The rate of evolution of plastic deformation can be expressed as follows.
𝑭¤𝒑 = 𝑳 𝒑 𝑭 𝒑 (2.10)
19
If dislocation slip is the only mode of plastic deformation, the velocity gradient 𝐿 𝑝 may be expressed
as the sum of the shear rates on all active slip systems (Roters et al., 2010).
∑︁𝑛
𝑳𝒑 = 𝛾¤𝛼 𝒎 𝜶 ⊗ 𝒏𝜶 (2.11)
𝛼=1
where 𝒎 𝜶 and 𝒏𝜶 are respectively the unit slip direction and slip plane normal vectors corresponding
to slip system 𝛼; 𝛾¤𝛼 is the shear rate corresponding to slip system 𝛼. The summation is made over
the total number of active slip systems 𝑛.
A constitutive model for deformation typically relates the shear rate to the external stress and
microstructural states of the material. There are two distinct approaches to developing constitutive
models–phenomenological and microstructure based. The former approach considers critical
resolved shear stress (CRSS) as a state variable for each slip system. The latter approach uses the
dislocation density as state variable to calculate flow rate.
In the phenomenological treatment, the shear rate is formulated as a function of the resolved
shear stress (RSS) and the evolution of material state is related to the total shear and the shear rate
(Roters et al., 2010). Rice (Rice, 1971) proposed a formulation on this basis for face centered cubic
materials, where the kinetic law on a slip system, relating the shear rate to the RSS 𝜏 𝛼 and slip
resistance 𝜏𝑐𝛼 is given as follows.
1
𝜏𝛼 𝑚
𝛾¤𝛼 = 𝛾¤0 𝛼 𝑠𝑔𝑛(𝜏 𝛼 ) (2.12)
𝜏𝑐
In equation 2.12, 𝛾¤0 and 𝜏𝑐𝛼 are material parameters.
The phenomenological model also accounts for the hardening caused by the interaction of a
fixed slip system 𝛼 with other slip systems (𝛽) (Roters et al., 2010).
𝑛
∑︁
𝜏¤𝑐𝛼 = ℎ𝛼𝛽 𝛾¤ 𝛽
𝛽=1
(2.13)
𝛽 𝑎
" ! #
𝜏
ℎ𝛼𝛽 = 𝑞 𝛼𝛽 ℎ0 1 − 𝑐
𝜏𝑠
20
Here ℎ𝛼𝛽 is known as the hardening matrix and it relates the influence of slip system 𝛽 to system 𝛼.
𝑞 𝛼𝛽 is called the latent hardening parameter, commonly taken to be 1.0 for co-planar slip systems
and 1.4 otherwise.
The formulations in equations 2.11, 2.12 and 2.13 can be adapted to account for mechanical
twinning as shown by Kalidindi [1998]. In this case, the twinned region is identified by its volume
fraction and appropriate boundary conditions are used. The effect of twinning is added as an
additional component by taking advantage of the analogy between slip and twin systems (Roters
et al., 2010).
A more physically realistic constitutive model for plastic deformation would use dislocation
density as the critical internal variable. Furthermore, a full characterization of the microstructure
requires specification of other variables such as grain size and shape, second phase fractions and
precipitate morphology (Roters et al., 2010). Naturally the complexity of the model increases with
the addition of more parameters; currently used models have incorporated only a few of them. Ma
and Roters [2004] proposed a model based on the density of mobile dislocation 𝜌𝑚 𝛼 that glide among
the slip system 𝛼 in order to accommodate plastic deformation. In order to accomplish this they
have to overcome the stress fields due to parallel dislocations (that cause the passing stress) and
forest dislocations that inhibit their motion, for each slip system 𝛼. Accordingly, for face centered
cubic crystals the total dislocation density can be decomposed into the parallel (glide, 𝜌 𝛼𝑃 ) and
forest components (𝜌 𝛼𝐹 ). This is related to the density of immobile (statistically stored) dislocations
𝛽
(𝜌 𝑆𝑆𝐷 ) by an interaction term 𝜒𝛼𝛽 (Roters et al., 2010) as follows.
𝑁
∑︁
𝛽
𝜌 𝛼𝐹 = 𝜒𝛼𝛽 𝜌 𝑆𝑆𝐷 | cos(𝑛𝛼 , 𝑡 𝛽 )| (2.14)
𝛽=1
𝑁
∑︁
𝛽
𝜌 𝛼𝑃 = 𝜒𝛼𝛽 𝜌 𝑆𝑆𝐷 | sin(𝑛𝛼 , 𝑡 𝛽 )| (2.15)
𝛽=1
Where 𝛽 is the slip system interacting with 𝛼, 𝑛𝛼 is the plane normal of slip systerm 𝛼, and
𝑡 𝛽 is the line direction corresponding to slip system 𝛽. It is important to mention that only edge
21
dislocations are considered in the formulations shown in equations 2.14 and 2.15, due to their
limited out of plane mobility (Roters et al., 2010).
The relationship between the shear rate and density of mobile dislocations can be expressed by
the Orowan equation.
𝛾¤𝛼 = 𝜌𝑚
𝛼
𝑏𝑣 𝛼 (2.16)
Here, 𝑏 is the magnitude of the Burgers vector and 𝑣 is the average velocity of the mobile
dislocations. The interaction strength term accounts for mutual interactions between slip systems
𝛼 and 𝛽 i.e. self interaction, co-planar interaction, cross-slip and Lomer Cottrell lock (Roters et al.,
2010).
An important distinction can be made between local and non-local constitutive models. In local
models, the state variables of a material point are not related to its neighboring points. Such a
scheme is adequate for prediction of stress-strain curves for texture in polycrystalline materials.
They are however, not as effective in predicting deformation behavior when the simulation scale
becomes considerably smaller, as in nanoindentation or micropillar compression (Roters et al.,
2010). In this situation, it is important to account for the relationship between grain size and plastic
deformation. Non-local models that relate state variables between a material point to its neighbors
are more suitable in these instances. The grain size dependence on flow stress is empirically given
by the well known Hall-Petch relation (Petch, 1953; Dieter, 2011).
An increase in flow stress with decreasing grain size is often attributed to an increase in pile
up of mobile dislocations, which in turn act as pinning sources for dislocations moving on other
planes. The average pinning distance of dislocations is inversely proportional to the square root of
dislocation density. Therefore, as the pinning distance decreases, the high dislocation density in turn
increases the stress required to operate a Frank-Read source and induce further plastic deformation.
Therefore pileups of dislocations act as barriers to plastic deformation. It is typically observed
that such pileups accumulate at grain boundaries, suggesting that they are strongly correlated with
local orientation gradients and contribute towards lattice curvature. In this context the dislocations
22
associated with these pileups are deemed as geometrically necessary. The analysis of geometrically
necessary dislocations (GND) is a focus of the present work; their presence is predicated on
the hypothesis that impeded dislocation systems contribute towards pileups. In this analysis, the
primary focus is on local accumulation of GNDs, and their long range effect on strain fields (non-
local effects) are not considered. Details of the mathematical basis for the GND density calculation
methodology used in this work are discussed in section 2.6.
2.4 Slip transfer and dislocation interactions at grain boundaries
At the microscopic level, a flaw can nucleate in a material when the local strain is unable to
accommodate changes in geometry (Bieler et al., 2009). In polycrystalline materials, these local
accommodations to shape change often occur at and near grain boundaries. This response to
mechanical loading is strongly dependent upon the deformation mechanism (crystallographic slip,
twinning etc); and the local strain accommodation by the neighborhood of an individual crystallite
Strain transfer across interfaces can occur by the transmission of dislocations. Livingston and
Chalmers [1957] studied heterogeneous strain evolution in aluminum bicrystals and observed that
on most of the slip systems, dislocations accumulate at the grain boundary. If a grain boundary
hinders the movement of a dislocation associated with a particular slip system, the dislocations pile
up against the boundary. A schematic representation of this phenomenon for a bicrystal (following
Livingston and Chalmers) is shown in figure 2.9. Such pileups are associated with large stress
concentrations at the grain boundaries. Furthermore, slip activation in the neighbor occurs when
these stress concentrations activate dislocation sources in the adjacent crystal. Bieler et al. [2009]
categorized three distinct types of deformation transfer that can occur across a grain boundary.
1. the grain boundary behaves as an opaque barrier that hinders transmission of slip systems.
As a result of this additional intragranular slip systems activate in order to facilitate strain
accommodation (Zaefferer et al., 2003).
2. the grain boundary is an imperfect barrier to dislocation. It allows partial transmission of
slip across it and residual dislocations are left behind.
23
A B
Figure 2.9: Illustration of dislocation pileup at a grain boundary between two crystals 𝐴 and 𝐵.
Dislocations shown here are edge type; in reality they are a combination of both edge and screw
types. Adapted from Livingston and Chalmers [1957]
3. the grain boundary allows slip transmission without (nearly) any hindrance. Such boundaries
provide little or no resistance to deformation.
From the above observations it is clear that slip transmission is most efficient for low angle
grain boundaries (LAGB) and high coincident site lattice (low Σ) boundaries. More recently,
Di Gioacchino et al. [2020] showed that strain transfer can also occur in high angle grain boundaries
(HAGB) by local lattice rotations.
Several metrics have been used in order to assess the degree of slip transmissibility across grain
boundaries. Most of these metrics use the geometry of the interacting slip systems to quantitatively
determine the propensity for slip transfer. Livingston and Chalmers developed the following
parameter that can be used to quantify slip transmissibility.
𝑁 = (𝑒𝑖𝑛 .𝑒 𝑜𝑢𝑡 ) ∗ (𝑔𝑖𝑛 .𝑔𝑜𝑢𝑡 ) + (𝑒𝑖𝑛 .𝑔𝑜𝑢𝑡 ) ∗ (𝑒 𝑜𝑢𝑡 .𝑔𝑖𝑛 ) (2.17)
In equation 2.17, 𝑒 and 𝑔 refer to the slip plane normal and slip direction respectively. The
subscripts "in" and "out" refer to the grains corresponding to the incoming and outgoing slip
systems respectively. The second term on the right hand side of the equation 2.17 must be
minimized for ease of slip transfer (Livingston and Chalmers, 1957; Kacher et al., 2014). For
example, in the ideal case, where the geometry is perfectly aligned for slip transfer, the second term
in equation 2.17 vanishes. The parameter proposed by Livingston and Chalmers (equation 2.17) is
poorly correlated with experimental observations (Di Gioacchino et al., 2020).
24
The magnitude of the Burgers vector of the residual dislocation is another commonly used
measure of slip transfer.
𝑔𝑏
𝑏𝑟 = 𝑏𝑖𝑛 − 𝑏 𝑜𝑢𝑡 (2.18)
𝑔𝑏
Here 𝑏𝑟 denotes the Burgers vector of the residual dislocation at the grain boundary, while 𝑏𝑖𝑛
and 𝑏 𝑜𝑢𝑡 refer to the Burgers vectors of the incoming and outgoing dislocations. This parameter is
associated with the strain energy density increase in the grain boundary. Therefore for ease of slip
𝑔𝑏
transmission the strain energy density increase |𝑏𝑟 | should be minimized (Kacher et al., 2014).
A third metric is the geometric criterion 𝑀𝐿𝑅𝐵 proposed by Lee et al. [1989], which is defined
as follows.
𝑀𝐿𝑅𝐵 = (𝑙𝑖𝑛 .𝑙 𝑜𝑢𝑡 ) ∗ (𝑔𝑖𝑛 .𝑔𝑜𝑢𝑡 ) = 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜅 (2.19)
In equation 2.19, 𝑙 denotes the line of intersection of the slip plane and the grain boundary, and 𝜃 is
the angle between 𝑙𝑖𝑛 and 𝑙 𝑜𝑢𝑡 . A schematic representation of the angles 𝜓, 𝜅 and 𝜃 that describe the
slip transfer geometry is shown in figure 2.10. The first dot product term (𝑙𝑖𝑛 .𝑙 𝑜𝑢𝑡 ) can be replaced
with (𝑒𝑖𝑛 .𝑒 𝑜𝑢𝑡 ) for an alternative form that does not require knowing the inclination of the grain
boundary, which is written as follows (Luster and Morris, 1995).
𝑚′ = 𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜅 (2.20)
𝑚′ is called the slip transfer parameter (or Luster-Morris parameter). 𝜓 is the angle between the
normals to the incoming and outgoing slip planes; while 𝜅 is the angle between the respective slip
directions.
2.4.1 Implications of slip transfer in polycrystalline materials
Slip transfer has major implications for fatigue behaviour and damage nucleation in metals.
An impediment to slip transfer can lead to stress concentrations from dislocation pileups that may
nucleate cracks at grain boundaries (Hémery et al., 2018). In contrast, conditions where impediment
to slip transfer is low result in more homogeneous plastic deformation. Consequently slip transfer
25
Figure 2.10: Angles used to calculate the Luster-Morris parameter 𝑚′. 𝜅 is the angle between slip
directions; while 𝜓 is the angle between the slip plane normals. 𝜃 is the angle between the slip
plane traces on the grain boundary plane. Adapted from Bieler et al. (2009).
is an important consideration for materials processing as well as structural health of in-service
components.
Some recent studies conducted on polycrystalline Al have shown that slip transfer is rare for
grains with near cubic orientation, where it was only observed for 𝑚′ >0.97 and low angle boundaries
(Bieler et al., 2019).
In two phase titanium alloys, it has been found that the propensity for slip transmission is
affected by the volume fraction of alpha phase in addition to the relative orientations of interacting
grains. Characterization of near alpha titanium (Ti-6242Si) subjected to low cycle fatigue using
EBSD and transmission electron microscopy (TEM) (Joseph et al., 2018) has shown that plastic
deformation occurs mainly by slip (basal ⟨𝑎⟩, prism ⟨𝑎⟩ and pyramidal ⟨𝑎⟩) in the primary 𝛼
grains. Furthermore, most of the slip transfer occurred between grains of similar orientation.
At the boundaries of soft and hard grains, slip transfer does not readily take place. The stress
concentrations at the grain boundary from the dislocation pileup result in activation of dislocation
sources within the hard oriented grain. Once nucleated, the dislocations in the hard grain propagate
via cross slip (Joseph et al., 2018).
Studies on in-situ scanning electron microscopy (SEM) characterization of Ti-6Al-4V have
identified both high values of 𝑚′ and resolved shear stress in the outgoing slip system to be
26
important for effective slip transfer (Hémery et al., 2018).
More recently, slip interactions at grain boundaries in commercial purity titanium were charac-
terized using electron channeling contrast imaging (ECCI) by Han and Crimp [2020]. Favorable
conditions for transfer of primary slip systems require high global Schmid factors, coupled with
small misalignment between slip planes and grain boundary intersection lines.
2.5 Twin formation as a result of slip transfer from a neighboring grain (S+T
twinning)
Slip transfer has been identified as a possible mechanism for strain accommodation in both
single phase (𝛼 (Guo et al., 2014; Wang et al., 2010a) and two-phase (𝛼 + 𝛽) titanium (Seal et al.,
2012). In polycrystalline materials there are additional constraints arising from the morphology
of the neighborhood of a grain under consideration. In hexagonal metals, it is important to take
into account the relative orientation between the c-axis and the loading direction. Deformation
twinning is an important mechanism that facilitates strain accommodation in hexagonal metals and
its relationship to slip transfer continues to be a topic of interest. Following Wang et al. [2010],
twins nucleated as a result of slip transfer across a grain boundary are defined as S+T twinning.
In hexagonal titanium, the likelihood of S+T twinning increases when the value of 𝑚′ approaches
one (Yang et al., 2011). An example is shown in figure 2.11, transfer of prism ⟨𝑎⟩ slip takes place
more readily across the boundary between grains 1 and 2 than across the boundary between grains 2
and 3. Boundary 1-2 has a higher value of 𝑚′ (0.936). Fewer geometrically necessary dislocations
were found in grain 1 than in grain 3, because of the lack of pileups in grain 1.
Richeton et al. [2012] studied the kinematic compatibility between crystallographic slip and
activated twin systems in hexagonal titanium. Based on imposing tangential continuity conditions
on plastic distortion at the twin-parent interface, they found that the twin modes observed had
compatibility with either prism ⟨𝑎⟩ or pyramidal ⟨𝑐 + 𝑎⟩ slip. Bieler et al. [2014] used HEXRD
to assess whether twin nucleation in subsurface grains in hexagonal titanium was caused by slip
transfer from a neighboring grain onto the parent grain. It was observed that at least some of
27
Figure 2.11: Use of 𝑚′ to determine the propensity of twin nucleation as a result of slip transfer
across a grain boundary. Twins can be observed in grain 2, where the activated twin system has a
high 𝑚′ (0.936) with respect to prism ⟨𝑎⟩ slip system in grain 1. Adapted from Yang et al. (2011).
the twins may have nucleated as a result of transmission of prism ⟨𝑎⟩ slip from a neighboring
grain. Nervo et al., [2016] studied twin evolution in Ti-4Al during a compression experiment
along a direction normal to the c-axis; and observed high compatibility between subsurface T1
twins and prism ⟨𝑎⟩ slip in neighboring grains. Wang et al. [2010] observed slip induced twin
nucleation (S+T) in about 25% of the twinned grains in a study with a pre-polished surface of
CP-Ti. Abdolvand et al. (Abdolvand et al., 2015a,b) conducted in-situ 3D XRD characterization
of tensile behaviour in polycrystalline zirconium. Using the local grain averaged stress tensor, the
generalized Schmid factor was calculated in this study. It was observed that in hard oriented grains
where twinning is favored, the local (generalized) Schmid factors of the six twinning variants have
comparable values. Furthermore, the activated system among the six twin variants did not always
have the highest local Schmid factor. It is therefore apparent that details of the loading history and
local stress state between parent and adjacent grains influence the likelihood of twin nucleation.
28
2.6 Geometrically necessary dislocations (GND)
The concept of geometrically necessary dislocations was first proposed by Nye in his seminal
work (Nye, 1953) and further developed by Ashby (Ashby, 1970). Work hardening in crystals occurs
when dislocation motion is inhibited. Dislocations that accumulate as a result of this inhibition
can be categorized into two types–geometrically necessary dislocations (GNDs), and statistically
stored dislocations (SSDs) in order to maintain strain compatibility. A schematic illustrating how
GNDs of both edge and screw type accumulate is shown in figure 2.12. In both cases a crystal
undergoing single slip is shown. If the crystal can be divided into three sections, whereby each
section is deformed independently of the others, each section undergoes plastic deformation by
expansion of dislocation loops. There is a linear increase of plastic strain (along the slip direction
for the edge case; and along direction 𝑚 for the screw case). For the edge dislocation case, the
screw components leave the material once they reach the boundaries of each section. Dipoles of
the edge components, however remain; and when the sections are assembled back together, there
are remnant negative edge dislocations that do not annihilate. A similar result can be drawn for
the case of screw dislocations: the edge components exit the material at the free boundaries for
each section and dipoles of positive screw dislocations remain. In both cases the non-annihilating
dislocations contribute to lattice curvature.
In contrast to GNDs, statistically stored dislocations (SSDs) are randomly trapped during plastic
deformation. Ashby [1970] defines this class of dislocations as "not geometrically necessary in the
uniform deformation of a pure single crystal". Rather, the accumulation of such dislocations can be
attributed to random interactions and trapping. Although SSDs contribute to the work hardening
of a crystal, they do not contribute to lattice curvature.
Nye (1953) introduced a non-symmetrical second order tensor to quantify the GND content in
a lattice. In this formulation only the lattice rotation gradient is considered and long range elastic
strain fields are neglected as shown in equation 2.21. Here, 𝑉 is the reference volume, 𝑑𝑠 is an
element of arc length along the dislocation line, and 𝐿 is the total dislocation line length contained
within 𝑉. A component of the Nye tensor 𝛼𝑖 𝑗 represents a dislocation with Burgers vector 𝑖 and line
29
b
b
Figure 2.12: Schematic diagram showing how GNDs of edge type (left) and screw type (right) can
accumulate in a plastically deformed crystal. Here 𝑏 indicated the slip direction and 𝑛 denotes the
slip plane normal. 𝑚 = 𝑏 × 𝑛 . Adapted from Arsenlis and Parks [1999].
direction 𝑗. Kröner and Ashby extended this formulation by considering the elastic strain gradients
(Arsenlis and Parks, 1999). It is important to note here that SSDs do not contribute to the Nye
tensor.
∫
1
𝛼𝑖 𝑗 = 𝑏𝑖 𝑡 𝑗 𝑑𝑠 (2.21)
𝑉 𝐿
A continuum based description of the Nye tensor and its use in calculation of the GND density is
based on the deformation gradient. In their formulation of deformation theory of plasticity, Fleck
and Hutchinson (Fleck et al., 1994) showed that the density of geometrically necessary dislocations
is proportional to the plastic strain gradient. Moreover lattice rotation and stretch occurs during
elastic deformation. Therefore the relative displacement 𝑑𝑢 of two material points can be resolved
into three components (equation 2.22).
𝑑𝑢 = 𝑑𝑢 𝑝 + 𝑑𝑢 𝑅 + 𝑑𝑢 𝑒 (2.22)
30
𝑝
𝑑𝑢 𝑝 = 𝛽 𝑘 𝑗 𝑑𝑋 𝑗
𝑑𝑢 𝑅 = 𝜔 𝑘 𝑗 𝑑𝑋 𝑗 (2.23)
𝑑𝑢 𝑒 = 𝜖 𝑘𝑒 𝑗 𝑓 𝑋 𝑗
where 𝑑𝑢 𝑝 is the relative displacement due to slip, 𝛽 𝑝 is the slip tensor, 𝑑𝑢 𝑅 is the relative
displacement due to lattice rotation 𝜔, and 𝑑𝑢 𝑒 is the relative displacement due to elastic strain 𝜖 𝑒 .
𝜆
The specific slip tensor 𝛽 𝑝 , is defined for a slip system 𝜆, defined by slip direction 𝑠𝜆 and slip
plane normal 𝑛𝜆 . The total slip tensor 𝛽 𝑝 takes into account the contribution from all of the active
slip systems (Das et al., 2018).
∑︁ 𝜆
𝛽𝑝 = 𝛽 𝑝 𝑠 𝜆 ⊗ 𝑛𝜆 (2.24)
𝜆
Furthermore, following Nye (Nye, 1953), crystallographic slip can be related to the resultant
Burgers vector, as a closure failure of the Burgers circuit 𝑐, on surface 𝑆 with plane normal 𝑁.
∮ ∮
𝑝 𝑝
< 𝐵 >𝑘 = 𝑑𝑢 𝑘 = 𝛽 𝑘 𝑗 𝑑𝑋 𝑗 (2.25)
𝑐 𝑐
The line integral in equation 2.25 can be converted into a integral over the surface 𝑆 by using
Stoke’s theorem as follows.
∬ ∬
𝑝 𝑁 𝑦𝑒
< 𝐵 >𝑘 = 𝜖𝑖 𝑗 𝑚 𝛽 𝑘 𝑗,𝑖 𝑁𝑚 𝑑𝑆 = 𝛼 𝑘𝑚 𝑁𝑚 𝑑𝑆, (2.26)
𝑆 𝑆
where 𝜖𝑖 𝑗 𝑚 is the permutation tensor, 𝛼 is the dislocation tensor (Nye, 1953). From Fleck and
Hutchinson (Fleck et al., 1994) the Nye tensor can be expressed as the curl of the slip tensor as
follows:
𝑁 𝑦𝑒 𝑝
𝛼𝑘𝑚 = 𝜖𝑖 𝑗 𝑚 𝛽 𝑘 𝑗,𝑙 (2.27)
The Nye tensor can now be related to the dislocation distribution inside the crystal. If the flux of
dislocations is defined as 𝑞, with Burgers vector 𝑏 and unit line direction 𝑙 cutting through the plane
with surface normal 𝑁, 𝛼 𝑁 𝑦𝑒 can be written as follows (Das et al., 2018):
𝑁 𝑦𝑒
𝛼 𝑘𝑚 = 𝑞𝑏 𝑘 𝑙 𝑚 (2.28)
31
𝑁 𝑦𝑒
It can be defined that 𝜌𝑚 = 𝑞𝑙 𝑚 so that 𝛼 𝑘𝑚 = 𝑏 𝑘 𝜌𝑚 . Therefore, for a general slip system 𝜆 the
dislocation tensor may be rewritten as follows.
∑︁
𝛼 𝑁 𝑦𝑒 = (𝑏𝜆 ⊗ 𝜌𝜆 ) (2.29)
𝜆
Since the total relative displacement 𝑑𝑢 along a closed circuit is zero, the closure failure of the
∮ 𝑝
Burgers circuit < 𝐵 > 𝑘 = 𝑐 𝑑𝑦 𝑘 must be balanced by the net sum of the displacements due to lattice
∮
rotation and elastic strain 𝑐 (𝑑𝑢 𝑘𝑅 + 𝑑𝑢 𝑒𝑘 ).
The deformation gradient (2.8) can be written as follows.
𝜕𝑥 𝜕𝑢
𝐹= =𝐼+ = 𝐼 + 𝛽, (2.30)
𝜕𝑋 𝜕𝑋
where 𝑢 is the displacement, 𝐼 is the identity matrix and 𝛽 is the displacement gradient. As seen in
section 2.3, 𝐹 can be decomposed into the elastic (𝐹 𝑒 and plastic (𝐹 𝑝 ) components. Crystallographic
slip is captured by the plastic component of 𝐹, which is equivalent to the information included in
the the slip tensor (equation 2.23). Since 𝐹 𝑝 = 𝐼 + 𝛽 𝑝 , the Nye tensor can be written as the curl of
the plastic deformation gradient.
𝑁 𝑦𝑒 𝑝
𝛼𝑚𝑘 = 𝜖𝑖 𝑗 𝑚 𝐹𝑘 𝑗,𝑖 (2.31)
If small strains are considered, the dislocation tensor can be written in terms of elastic (𝛽 𝑒 ) or
plastic (𝛽 𝑝 ) displacement gradients.
𝜕 𝛽13𝑝 𝑝 𝑝 𝑝 𝑝 𝑝
𝜕 𝛽12 𝜕 𝛽11 𝜕 𝛽13 𝜕 𝛽12 𝜕 𝛽11
𝜕𝑥2 − − −
𝜕𝑥3 𝜕𝑥 3 𝜕𝑥 1 𝜕𝑥1 𝜕𝑥2
𝑝 𝑝 𝑝 𝑝 𝑝 𝑝
𝛼 𝜕𝜕𝑥𝛽23 − 𝜕 𝛽22
𝜕𝑥3
𝜕 𝛽21
𝜕𝑥 3 −
𝜕 𝛽23
𝜕𝑥 1
𝜕 𝛽22
𝜕𝑥1 −
𝜕 𝛽21
𝜕𝑥2
(2.32)
𝑝2 𝑝 𝑝 𝑝 𝑝 𝑝
𝜕 𝛽33 𝜕 𝛽32 𝜕 𝛽31 𝜕 𝛽33 𝜕 𝛽32 𝜕 𝛽31
𝜕𝑥2 − 𝜕𝑥3 𝜕𝑥 3 − 𝜕𝑥 1 𝜕𝑥1 − 𝜕𝑥2
𝑒
𝜕 𝛽12 𝑒
𝜕 𝛽13 𝑒
𝜕 𝛽13 𝑒
𝜕 𝛽11 𝑒
𝜕 𝛽11 𝑒
𝜕 𝛽12
𝜕𝑥3 − 𝜕𝑥2 𝜕𝑥 1 − 𝜕𝑥 3 𝜕𝑥2 − 𝜕𝑥1
𝑒 𝑒 𝑒 𝑒 𝑒 𝑒
𝛼 𝜕𝜕𝑥𝛽22 − 𝜕𝜕𝑥𝛽23 𝜕𝜕𝑥𝛽23 − 𝜕𝜕𝑥𝛽21 𝜕𝜕𝑥𝛽21 − 𝜕𝜕𝑥𝛽22 (2.33)
𝑒3 2
𝑒 𝑒
1
𝑒
3 2
𝑒
1
𝑒
𝜕 𝛽32 𝜕 𝛽33 𝜕 𝛽33 𝜕 𝛽31 𝜕 𝛽31 𝜕 𝛽32
𝜕𝑥3 − 𝜕𝑥2 𝜕𝑥1 − 𝜕𝑥1 𝜕𝑥2 − 𝜕𝑥1
Furthermore, the displacement gradient can be resolved into the lattice rotation and lattice strain
components as
𝛽𝑒 = 𝜔𝑒 + 𝜖 𝑒 (2.34)
32
Then equation 2.33 can be written in the following form:
𝑒
𝜕𝜔12 𝑒
𝜕𝜔13 𝑒
𝜕𝜔13 𝑒
𝜕𝜔21 𝜕𝜖12𝑒 𝑒
𝜕𝜖 13 𝑒
𝜕𝜖13 𝑒
𝜕𝜖11 𝑒
𝜕𝜖 11 𝑒
𝜕𝜖12
𝜕𝑥3 − 𝜕𝑥 2 𝜕𝑥 1 𝜕𝑥1
𝜕𝑥3 −
𝜕𝑥2 𝜕𝑥 1 − 𝜕𝑥3 𝜕𝑥2 − 𝜕𝑥 1
𝜕𝜔 𝑒 𝑒
𝜕𝜔23 𝑒
𝜕𝜔21 𝑒
𝜕𝜔21
𝑒 𝑒
𝜕𝜖 23 𝑒
𝜕𝜖23 𝑒 𝑒 𝑒
𝛼 32
𝜕𝑥2 𝜕𝑥1 − 𝜕𝑥3 𝜕𝑥2
+ 𝜕𝜖 22
− −
𝜕𝜖21 𝜕𝜖 21
−
𝜕𝜖22 (2.35)
𝜕𝑥𝑒3 𝜕𝑥2 𝜕𝑥 1 𝜕𝑥3 𝜕𝑥2 𝜕𝑥 1
𝑒 𝑒 𝑒 𝑒
𝑒 𝑒 𝑒 𝑒 𝑒
𝜕𝜔32 𝜕𝜔13 𝜕𝜔31 𝜕𝜔32 𝜕𝜖32 𝜕𝜖 33 𝜕𝜖33 𝜕𝜖31 𝜕𝜖 31 𝜕𝜖32
𝜕𝑥 2 − 𝜕𝑥1 𝜕𝑥3 − − −
𝜕𝑥3 𝜕𝑥 3 𝜕𝑥2 𝜕𝑥 1 𝜕𝑥3 𝜕𝑥2 𝜕𝑥 1
In equation 2.35, the contribution to 𝛼 from lattice rotation is much larger than from elastic strain
gradients (Das et al., 2018). The relationship between the Nye tensor and dislocation density was
established in equation 2.29. It is possible to rewrite the 3 × 3 tensor as a 9 × 1 column vector.
Following Arsenlis et al. (Arsenlis and Parks, 1999), a linear operator 𝐴 (An 9 × 𝑘 matrix for 𝑘
types of dislocations) can now be defined. The 𝑘th column of 𝐴 contains the dyadic product of
the Burgers vectors and line direction of the 𝑘th dislocation type. The scalar values of dislocation
density for 𝑘 types of dislocations can be arranged in the form of a 𝑘 × 1 column vector 𝜌.
𝛼 = 𝐴𝜌 (2.36)
In explicit form, equation 2.36 can be written as follows (Das et al., 2018).
𝑏 21 𝑙12 𝑏 31 𝑙13 . . . 𝑏 1𝑘 𝑙1𝑘
11
𝑏 𝑙 𝛼11
11
𝑏 1 𝑙 1 𝑏 21 𝑙22 𝑏 31 𝑙23 . . . 𝑏 1𝑘 𝑙2𝑘
12 𝛼12
𝜌1
11
𝑏 1 𝑙3 2 2 3 3
𝑏1 𝑙3 𝑏1 𝑙3 . . . 𝑏1 𝑙3 𝑘 𝑘
𝛼13
𝜌
11 2 2 3 3 𝑘 𝑘
2
𝑏 𝑙 𝑏 2 𝑙 1 𝑏 2 𝑙1 . . . 𝑏 2 𝑙1 𝛼21
21
∑︁
𝑘 𝑘
𝜌3
(𝑏 ⊗ 𝜌 ) = 𝑏 12 𝑙21 𝑏 22 𝑙 22 𝑏 32 𝑙23 . . . 𝑏 2𝑘 𝑙2𝑘 = 𝛼
.. 22 (2.37)
𝑗 .
11 2 2 3 3 𝑘 𝑘
𝑏 2 𝑙3 𝑏2 𝑙3 𝑏2 𝑙3 . . . 𝑏2 𝑙3 . 𝛼23
..
3 3
11 2 2 𝑘 𝑘
𝑏 𝑙 𝑏 3 𝑙 1 𝑏 3 𝑙1 . . . 𝑏 3 𝑙1 𝛼31
31
𝜌𝑘
𝑏 1 𝑙 1
32 𝑏 23 𝑙 22 𝑏 33 𝑙23 . . . 𝑏 3𝑘 𝑙2𝑘 𝛼
32
11
𝑏 3 𝑙3 2 2 3 3
𝑏3 𝑙3 𝑏3 𝑙3 . . . 𝑏3 𝑙3 𝑘 𝑘
𝛼33
In general, 𝑘 > 9, so there exists no unique solution for 𝜌. If 𝛼 and 𝐴 are known, optimization
schemes can be used to solve for 𝜌. Arsenlis and Parks [1999] have outlined two possible optimiza-
tion methods for this purpose. The first method is the 𝐿 2 optimization scheme that minimizes the
squares of dislocation densities. In this approach it is necessary to include all possible slip systems
33
(including the ones least likely to activate), since the calculation is independent of the resolved
shear stress. One drawback of this method is the lack of any physical basis, in the sense that the
energetics of dislocations are not considered.
A more physically meaningful scheme is the 𝐿 1 approach that minimizes the total dislocation
elastic energy: (1 − 𝜈) −1 𝑘 𝜌 𝑘 + 𝑘 𝜌 𝑘𝑠𝑐𝑟𝑒𝑤 . Here 𝜈 is the Poisson’s ratio, and both screw
Í 𝑒𝑑𝑔𝑒 Í
and edge contributions from the different 𝑘 types of dislocations are taken into account. If it is
assumed that the magnitude of the Burgers vectors for all dislocations are the same; and the lattice
is elastically isotropic, the following relation holds true. Moreover, it is also assumed that the
dislocations are either pure edge or pure screw, where
𝐸 𝑒𝑑𝑔𝑒 1
= (2.38)
𝐸 𝑠𝑐𝑟𝑒𝑤 1 − 𝜈
Several surface and subsurface techniques have been used for experimental estimation of GND
density in polycrystalline materials. A popular surface method is EBSD, which enables the
estimation of lattice distortion from local crystal orientation. Ruggles and Fullwood [2013] used
high resolution EBSD (HR EBSD) to estimate bulk GND content in face centered cubic material
(nickel). This approach only allows for the estimation of the lattice distortion components of the
Nye tensor. Konijnenberg et al. [2015] have shown that this drawback can be overcome by using
three-dimensional electron backscatter diffraction (3D EBSD). Using serial sectioning to perform
3D EBSD, they were able to calculate GND density in a copper bicrystal during a micro cantilever
bending experiment.
DAXM methods have been used by Larson et al. [2008] to experimentally measure lattice
curvature and elastic strain in plastically deformed silicon. More recently, Guo et al. [2020] used
DAXM to determine spatially resolved values of the rotation gradient in order to estimate the
GND density in hexagonal titanium. The approach used by them is based on the methodologies
outlined by Arsenlis and Parks [1999] and Das et al. [2018]. Here the 𝐿 1 approach is used as the
optimization scheme to solve for dislocation density.
34
2.7 Review of HEXRD methods
X-ray beams from synchrotron sources are typically about fourteen orders of magnitude more
brilliant (photons/s/𝑚𝑚 2 /𝑚𝑟𝑎𝑑 2 /eV) compared to conventional sources (Ice and Pang, 2009). This
provides the means to obtain narrow micron or sub-micron beams. Two important advantages of
synchrotron based x-ray diffraction techniques are the ability to conduct subsurface characterization
non-destructively and minimal sample preparation requirements compared to surface methods such
as EBSD. Based on the type of beam used (monochromatic or polychromatic), beam energy and
the algorithms used for three-dimensional reconstruction, they can be broadly classified into the
following three categories.
1. three-dimensional x-ray diffraction (3D XRD) microscopy (also known as high energy diffrac-
tion microscopy (HEDM)): In this approach, a high energy (∼ 50 keV or higher) monochro-
matic beam is used to generate Debye-Scherrer diffraction patterns from a sample that is
rotated incrementally about a stationary axis. Thereafter, a 3D reconstruction is done using
tomography algorithms (Poulsen, 2012). Here, the primary focus is on a specific setting of
3D XRD, known as far-field HEDM (FF HEDM).
2. A second setting of 3D XRD used in this study is near field HEDM (NF HEDM). The basic
principle is the same as FF HEDM, except for the fact that the sample to detector distance
used here is much smaller (typically ∼ 10 mm). It is possible to obtain finer spatial resolution
(2.2) using this setting compared to FF HEDM.
3. Micro-Laue diffraction: Here, a strong polychromatic (white) beam is used to generate Laue
diffraction patterns. The reconstruction is typically done using triangulation algorithms with
the help of a differential aperture. The beam energy used in this method is generally lower (7-
30 keV) than for 3D XRD. The emphasis here is on a special case of micro-Laue diffraction,
known as differential aperture x-ray microscopy (DAXM).
35
In this section, the fundamental aspects of these techniques are discussed, along with their
applications in three-dimensional characterization of polycrystalline materials.
A basic underlying principle of x-ray diffraction is the Bragg condition. For a monochromatic
x-ray beam of wavelength 𝜆 incident on a set of parallel crystal planes, constructive interference of
the reflecting waves occurs when the path difference is an integer (𝑛) multiple of 𝜆.
2𝑑 sin 𝜃 = 𝑛𝜆 (2.39)
Where 𝑑 is the interplanar spacing and 𝜃 is the angle between the incident beam and the real space
set of atomic planes.
2.7.1 3D XRD
The first demonstrations of 3D XRD experiments were done in 1997 by Poulsen et al. (1997).
In general, four standard modes of operation have been defined by Poulsen (2012). Mode I is
used for quick statistical information of properties of individual crystallites e.g. grain volume,
grain averaged strain tensor, orientation, and phase. Mode II can be used to determine the three-
dimensional center of mass positions of individual grains, in addition to properties determined by
Mode I. Modes III and IV enable a complete volumetric mapping of grains and orientations for
undeformed and deformed specimens respectively (Poulsen, 2012).
The schematic representation of a typical 3D XRD experimental setup is shown in figure 2.13.
An almost parallel monochromatic x-ray is incident on the specimen. The specimen is mounted
on a stage that enables it to rotate about the Z-axis by the angle 𝜔 (normal to the beam direction).
It is possible to mount a deformation frame on a stage to enable translations in the X, Y and Z
directions, as well as additional rotation axes.
As the specimen rotates, there are parts of the illuminated material that satisfy the Bragg
condition and generate a diffracted beam. The diffraction patterns are recorded on a series of two
dimensional detectors. The rotation is typically done using a constant increment (Δ𝜔), for each
exposure. This provides a uniform sampling for the specimen and allows for the Bragg condition
36
Figure 2.13: Schematic illustrating a basic 3D XRD experimental setup. 𝑍 is the sample rotation
axis; 𝜔 is the angle of rotation about 𝑍; 𝐿 is the sample to detector distance; 2𝜃 is the Bragg angle;
𝜂 is the azimuthal angle. Adapted from Poulsen (Poulsen, 2012).
to be satisfied for a multitude of crystallographic planes for any crystal orientation. Use of multiple
two dimensional detectors positioned at different distances (𝐿) from the center of rotation allows
for three-dimensional characterization. Depending upon the range of sample to detector distance
(𝐿) used, the detector settings can be either near field (𝐿 ∼ 10 mm); far-field (𝐿 ∼ 1 m); or
very-far-field (𝐿 ∼ 5 m). For the three aforementioned configurations, there are differences in the
resolution and size of the detectors used. Detailed information about the configurations along with
information about their spatial, orientation and strain resolutions are enumerated in table 2.2. This
configuration is available at 1-ID beamline at the Advanced Photon Source (APS) at the Argonne
National Laboratory.
An important advantage of 3D XRD is the ability to perform in-situ analysis. This enables
the characterization of lattice rotation and strain evolution during mechanical deformation (e.g. a
tensile test). It is also possible to characterize phase transformations, nucleation and grain growth
during thermo-mechanical experiments (Poulsen, 2012).
Hereafter, the focus is on the Mode II type of experiment using FF HEDM as applied to
polycrystalline materials. In this approach it is possible to obtain information on phase, grain
center-of-mass position and volume, and the local grain average strain tensor. For coarse grained
specimens the far field setup is quite efficient, although the spatial resolution offered in this
37
Table 2.2: Enumeration of the three different configurations for 3D XRD, based on sample to
detector distance (L). Adapted from Park et al. (Park et al., 2017).
Experimental Resolution Remarks
Technique
Near-field Spatial resolution ∼ 1𝜇𝑚; Area detector (1.5𝜇𝑚 square pixels cov-
high energy 0.1°- 0.01° angular resolution ering ∼ 3.1 × 3.1𝑚𝑚 2 . Sample to de-
diffraction tector distance ∼ 10 mm. Enables 3D
microscopy reconstruction of polycrystalline struc-
(NF HEDM) ture. Not capable of providing local
strain information
Far-field Spatial resolution ∼ 10𝜇𝑚; Area detector resolution:
high energy 0.1°-0.01° crystallographic 200𝜇𝑚 square pixels, covering
diffraction orientation resolution; 10−4 ∼ 410𝑚𝑚 × 410𝑚𝑚. Sample to detec-
microscopy strain resolution tor distance ∼ 1 m. Capability of 3D
(FF HEDM) reconstruction, determining centers of
masses of crystallites, crystallographic
orientations and local elastic strain
tensor.
Very-far-field Angular resolution: ∼ 0.01° Area detector resolution: ∼ 60𝜇𝑚 pix-
high energy els, covering ∼ 50𝑚𝑚 × 30𝑚𝑚 area.
diffraction Capable of non destructive mapping of
microscopy individual grains in a polycrystal.
(VFF HEDM)
configuration is relatively low. On the other hand, the angular resolution is high, which makes
the method sensitive to elastic strain and crystal orientation (Park et al., 2017). This setup is less
sensitive to the grain shape, therefore only the average position of the grain center of mass (COM)
can be estimated.
In the present study, near field HEDM (NF HEDM) is also used to determine grain positions,
morphology and orientations in the illuminated microstructure. Since this technique enables finer
spatial resolution compared to FF HEDM, it is possible to capture grain morphology and orientation
gradients within an individual grain. In their work on characterization of gold oligocrystals,
Menasche et al. [2020] found the resolution in orientation measurement using NF HEDM is about
an order of magnitude better than surface electron backscatter diffraction (EBSD) measurements
38
Random texture
(a)
Strong texture
(b)
Figure 2.14: Schematic illustration of Debye-Scherrer patterns for a polycrystalline material with
(a) random texture (well annealed);(b) strong texture (Taddei, 2015).
that use a standard Hough transform based indexing method.
As mentioned earlier, in a far field experiment, when a particular crystallographic plane satisfies
the Bragg condition, the diffracted X-ray is recorded on an area detector as a Debye-Scherrer pattern
(Park et al., 2017). For a polycrystalline specimen, the patterns can be distinguished for two different
kinds of specimens. If the specimen is well annealed with a random texture, the Debye-Scherrer
patterns manifest as a set of concentric circles with peaks randomly located around the circle, where
each circle corresponds to a family of crystallographic planes. On the other hand, for a specimen
with a strong texture, the concentric circles have portions with no diffraction peaks because certain
ranges of crystal orientations are not present in the specimen, relative to the incident beam. Figure
2.14 illustrates the difference in the diffraction patterns for the two cases.
2.7.1.1 Indexing data obtained from 3D XRD
Indexing is the process of assigning unique grain identities to groups of diffraction peaks.
Indexing starts with the identification of diffraction peaks for each 𝜔 in detector space. The
first step in the peak searching process is to uniquely identify diffraction spots in detector space
(segmentation). This requires a sufficient number of distinct diffraction spots to ensure that there
is minimal overlapping of peak intensity from different grain orientations. Peaks are then filtered
by specifying a threshold (lower bound) of intensity. For deformed crystals that exhibit significant
39
peak broadening, multiple thresholds can be used in the segmentation process. The second step
in this process is to determine the directions of the diffracted X-rays corresponding to the peaks
(Poulsen, 2012). For FF HEDM, it is initially assumed that all of the grains are positioned at the
center of rotation. Broadly speaking, there are two basic approaches to indexing.
1. Forward projection: In this approach, the entire orientation space is traversed in increments.
At any given orientation, the theoretical scattering vectors (𝐺) are calculated. Then the
theoretical values are matched to the experimental 𝐺 vectors in the neighborhood. The
efficiency of matching is measured by a completeness criterion; which is the ratio of the
observed number of 𝐺 vectors to the expected number.
This approach works well for a smaller number of grains; however the likelihood of false
indices increases if the number of grains in the illuminated volume is large (Sharma et al.,
2012b). One way to overcome this drawback is to combine the orientation-space scan with
a spatial grid in the illuminated volume. At each point in the grid, all orientations are
simulated; which narrows the margins used for matching the observed and simulated peaks.
This method has been implemented in the Grainspotter program available under the FABLE
package developed by the Risø National Laboratory (Schmidt, 2014).
2. Backward projection: The basic idea of this approach is to group diffraction spots belonging
to the same orientation together. If the diffracting grain is located at the origin of the
laboratory coordinate frame, it is possible to assign a straight line for each corresponding
diffraction peak in Rodrigues-Frank (RF) orientation space. A unique orientation would
be identified from a point where several of these lines intersect. On the other hand if the
diffracting grain is not located at the origin of the laboratory frame, each of the diffraction
lines in RF space no longer intersect at a point. Rather they form an intersection volume that
can be identified as a unique orientation as long as the number of grains is small (Sharma
et al., 2012a,b).
One drawback of the methods enumerated above is the increase in the number of false iden-
40
tifications with increasing number of grains. This is a result of the higher possibility of overlap
of diffraction spots on the detector. This is more pronounced when individual grains have large
orientation spreads. Other factors that can lead to high overlap are differences in unit cell size in
different phases and strongly preferred orientations (sample texture) (Poulsen, 2012).
A second problem that arises during the indexing process is from the difficulty of identifying
smaller grains. It is typical for grain sizes to be lognormally distributed in a specimen. Moreover,
there is a sharp drop in integrated intensity of peaks with increasing value of the Bragg angle 2𝜃.
For grains that are coarse enough, all or most of the diffraction spots get correctly identified during
peak searching. Grains that are very small in size have peaks with lower intensities and do not
meet the minimum assigned threshold during peak search are not identified. Grains that have sizes
that fall between the two aforementioned categories can be identified if the Bragg angle is low.
This however, may require multiple thresholds to be imposed at different 2𝜃 values in an iterative
manner (Poulsen, 2012).
More recently, efforts have been made to overcome the drawbacks of forward and backward
projections by using a mapping mode that uses a reduced orientation space. This has been
incorporated in the MIDAS package developed at Argonne National Laboratory (Sharma et al.,
2012a,b; Wozniak et al., 2015). This method has been observed to yield better results for centers
of mass and orientation when the number of grains considered is higher. MIDAS has been used
for indexing data from a FF HEDM experiment for characterizing irradiated Fe alloys (Park et al.,
2015).
2.7.1.2 Local strain tensor evaluation from FF HEDM
The capability to extract local grain averaged strain tensor is an important feature of FF HEDM.
One approach is the methodology used by Margulies et al. to calculate lattice strain in a copper
polycrystal (Margulies et al., 2002). Here, the fundamental equation for strain calculation is based
on measurement of the relative change in interplanar spacing of selected crystallographic planes. A
41
given strain tensor component can be obtained by differentiation of the Bragg equation as follows.
𝑑 − 𝑑0
𝜖= = − cot 𝜃Δ𝜃 (2.40)
𝑑0
where 𝑑0 and 𝑑 are the interplanar spacings for unstrained and strained lattice respectively.
A specific strain tensor component 𝜖𝑖 resolved in the direction of the scattering vector identified
by the direction cosines 𝑙𝑖 , 𝑚𝑖 and 𝑛𝑖 can be written as follows.
©𝜖11 𝜖12 𝜖13 ª © 𝑙𝑖 ª
® ®
𝜖𝑖 = (𝑙𝑖 𝑚𝑖 𝑛𝑖 ) 𝜖21 𝜖22 𝜖23 ® 𝑚𝑖 ® (2.41)
® ®
® ®
® ®
𝜖31 𝜖32 𝜖33 𝑛𝑖
« ¬« ¬
Combining equations 2.40 and 2.41 , the six independent strain tensor components resolved in any
direction given by direction cosines 𝑙, 𝑚 and 𝑛 can be evaluated as follows.
𝜖 = 𝜖11 𝑙 2 + 𝜖 22 𝑚 2 + 𝜖33 𝑛2 + 𝜖12 𝑙𝑚 + 𝜖13 𝑙𝑛 + 𝜖23 𝑚𝑛 (2.42)
In order to solve for the six unknown strain tensor components a minimum of six independent
diffraction spots are necessary. In reality, the number of such measurements are greater than six, so
an overdetermined system of linear equations is obtained. This can be easily solved using singular
value decomposition (SVD) or a least squares scheme. Details of the algorithm used for grain
averaged strain tensor calculation are given in section 4.1.1.
2.7.1.3 Accuracy in determination of grain positions using 3D XRD
As enumerated in table 2.2, FF HEDM enables determination of grain positions (centers of
mass) with a resolution of 10𝜇𝑚 or better. On the other hand, NF HEDM enables a finer resolution
(1𝜇𝑚 or better), and provides information on grain morphology in three-dimensional space. The
accuracy in the determination of grain center of mass (COM) using FF HEDM is strongly dependent
on the experimental setup and the state of the illuminated volume in the specimen (Park et al.,
2017). For example, if there is a large amount of deformation or orientation gradients present in
42
the microstructure, smearing of the diffraction spots can occur, resulting in higher uncertainty in
determination of the grain position.
Based on the combined far field and near field HEDM characterization of an in-situ tensile test
in alpha titanium (Ti-7Al alloy), Turner et al. [2016] reported that the average difference in grain
positions between the two datasets is ∼ 16𝜇𝑚. In this experiment, the average grain size of the
interrogated microstructure was ∼ 100𝜇𝑚.
2.7.2 Differential aperture x-ray microscopy (DAXM)
DAXM is a special application of micro-Laue diffraction that enables depth resolved charac-
terization along the beam direction. This technique was developed by Larson et al. [2002]. A
fundamental difference from HEDM is that in this case, a white (polychromatic) beam is typically
used to interrogate the specimen. This technique enables characterization at submicron resolution.
A notable advantage of this technique is the high level of depth resolution; which is of the order
of hundreds of microns for low Z materials and tens of microns for high Z materials. Moreover,
orientation measurements can be made with high angular precision (Yang et al., 2004).
The von Laue formulation describes the conditions for constructive interference from a set of
parallel crystallographic planes in terms of the scattering vector. Let 𝐾𝑖 and 𝐾𝑜 denote the wave
vectors of the incoming and outgoing (reflected) x-ray beams respectively. If a monochromatic
(single wavelength) beam undergoing elastic scattering is considered, then |𝐾𝑖 | = |𝐾𝑜 | = 𝜆1 , where
𝜆 is the wavelength of the beam. The scattering vector is then given as follows.
Δ𝐾 = 𝐾𝑖 − 𝐾𝑜 (2.43)
For a crystal plane with Miller indices (ℎ 𝑘 𝑙), the Laue condition can be written as follows.
Δ𝐾 = ℎ𝑎 ∗ + 𝑘 𝑏 ∗ + 𝑙𝑐∗ , (2.44)
where 𝑎 ∗ , 𝑏 ∗ and 𝑐∗ are the basis vectors for the reciprocal lattice; and the term on the right
hand side of equation 2.44 is the reciprocal lattice vector corresponding to plane (ℎ 𝑘 𝑙). It can
43
Figure 2.15: Schematic diagram illustrating the Laue condition.
be shown that equation 2.44 is equivalent to the Bragg condition. Figure 2.15 shows a beam with
incoming wave vector 𝐾𝑖 impinging on a crystal plane at an angle 𝜃. 𝐾𝑜 denotes the outgoing wave
vector.
According to the Laue condition, the scattering vector → −𝑔 = →− −
→
𝐾𝑖 − 𝐾𝑜 = Δ𝐾 is equal to the
→
− −
→
reciprocal lattice vector corresponding to (ℎ 𝑘 𝑙). Since the scattering is elastic | 𝐾𝑖 | = | 𝐾𝑜 |. 𝐾𝑖 and
𝐾𝑜 make the same angle with the plane normal to → −𝑔 . Therefore, scattering is equivalent to a Bragg
reflection at an angle 𝜃 from a plane with reciprocal lattice vector ℎ𝑎 ∗ + 𝑘 𝑏 ∗ + 𝑙𝑐∗ .
If 𝑑 ℎ𝑘𝑙 is the interplanar spacing for planes parallel to (ℎ 𝑘 𝑙) and → −
𝑔0 is the shortest wave vector
parallel to →−𝑔 , then
→
− 1
𝑔0 = (2.45)
𝑑 ℎ𝑘𝑙
From equation 2.45 it follows that:
→
−𝑔 = 𝑛→−
𝑔0
|→
−𝑔 | = 𝑛 (2.46)
𝑑 ℎ𝑘𝑙
−𝑔 | = 2| −
|→
−→
𝐾 | sin 𝜃
𝑖𝑛
From figure 2.15,
−𝑔 | = 2| −
|→
−→
𝐾𝑖𝑛 | sin 𝜃 (2.47)
44
O
(a) (b)
Figure 2.16: Schematic representation of an Ewald sphere in two dimensions (a) Monochromatic
beam, (b) Polychromatic beam containing a range of wavelengths. The parallel set of crystallo-
graphic planes (hkl) that form a reciprocal lattice point are shown on the top right inset (Ice and
Pang, 2009)
From equations 2.46 and 2.47,
2 𝑛
sin 𝜃 =
𝜆 𝑑 ℎ𝑘𝑙 (2.48)
𝑛𝜆 = 2𝑑 ℎ𝑘𝑙 sin 𝜃
Equation 2.48 is the Bragg condition. If now a polychromatic beam containing wavelengths in
the range [𝜆 𝑚𝑖𝑛 , 𝜆 𝑚𝑎𝑥 ] is considered; the sets of crystal planes satisfying the Bragg condition can be
illustrated by the Ewald sphere in figure 2.16. Figure 2.16 (a) shows the case for a monochromatic
beam of wavelength 𝜆. The incident wave vector terminates at the origin 𝑂 of the reciprocal lattice.
A sphere can be drawn with center at the tail of the incident vector having a radius of 𝜆1 . The
sphere passes through the reciprocal lattice origin and any point hkl that it also intersects indicate
a set of planes (ℎ 𝑘 𝑙) that satisfy the Bragg condition. Figure 2.16(b) shows the Ewald spheres
for a polychromatic (white) beam containing wavelengths in the range [𝜆 𝑚𝑖𝑛 , 𝜆 𝑚𝑎𝑥 ]. In the limiting
cases, the wave vectors correspond to the minimum and maximum wavelengths 𝜆 𝑚𝑖𝑛 (outer sphere)
and 𝜆 𝑚𝑎𝑥 (inner sphere) respectively. The volume of reciprocal lattice in between these two extreme
values (region shaded in blue) will have multiple Ewald spheres intersecting the reciprocal lattice
at different points. Therefore, there will be several crystal planes for which the Bragg condition
will be satisfied, noting that there are many more points within the sphere than the circular cross
section depicted. This illustrates the basic principle behind micro-Laue diffraction.
The schematic illustration of the experimental setup for DAXM at ID-34E beamline at APS is
45
shown in figure 2.17. A polychromatic x-ray beam from the synchrotron source passes through a
removable micro-monochromator setting. The monochromator allows the passage of either a white
beam or reflects a monochromatic beam from the lower part of the undulator onto the Kirkpatrick-
Baez (K-B) mirrors; which collimate the beam to 1𝜇𝑚 2 cross section. The specimen is typically
positioned at 45° with respect to the incident beam direction (Yang et al., 2004).
The Laue diffraction diffraction patterns have a high range of angular divergence. In order to
profile them, a 50𝜇m Pt wire is used as a differential aperture (figure 2.17, top right). This wire
translates in submicron steps parallel to the specimen surface. Diffracted beams from the specimen
pass through the submicron sized opening formed between two successive positions of the profiler
wire. By subtracting the Laue patterns before and after each profiler step, the differential intensity
distribution of the diffracted beams passing through the aperture can be obtained. Reconstruction of
the complete diffraction patterns for voxels along the beam direction is done using algorithms that
triangulate from the position of the patterns on the detector to its source along the beam direction.
In order to achieve submicron depth resolution, it is important that the distance from the wire to
detector 𝐷 𝐷𝑒𝑡 is ∼ 200 times greater than that from wire to incident beam 𝐷 𝑋−𝑅 (Yang et al., 2004;
Ice and Pang, 2009).
Using DAXM it is possible to obtain structural information, orientation, and the strain tensor
for each material location along the direction of the incident beam with a positional accuracy of
∼ 0.1𝜇𝑚. Scans can be conducted as line scans where one surface coordinate is varied while
keeping the other constant; as well as in a raster mode. Scans with finer step sizes take longer and
synchrotron beam time becomes a constraining factor in the experiment.
For coarse grained microstructures, a serial probing method developed by Eisenlohr et al.
(Eisenlohr et al., 2017) is effective for depth resolved characterization. In this approach a regular
array of DAXM probes is constructed. Each probe in this grid scans the specimen to the desired
depth. The second step identifies individual grains based on their similarity of orientation in
physical space (Eisenlohr et al., 2017).
A more recent development in this area is dark field DAXM, which was first reported in a study
46
Figure 2.17: Schematic of a DAXM experimental setup to interrogate a polycrystalline specimen.
An enlarged view of the differential aperture (profiler) Pt wire is shown on the top right (Yang et al.,
2004).
conducted of recovery in tensile deformed aluminum (Simons et al., 2015). This technique uses
the monochromatic beam setting and diffraction angles (2𝜃) in the range of 10-30°, enabling spatial
resolution of the order of 100 𝑛𝑚.
2.7.2.1 Local strain tensor evaluation from DAXM
In contrast to HEDM that can yield only the grain average strain tensors, the local elastic strain
can be obtained with submicron resolution using DAXM. Depending on whether the deviatoric
or the full strain tensor is to be obtained, two settings of DAXM can be employed. With the
polychromatic beam setting the (deviatoric) shape change of the unit cell can be measured via the
change in lattice parameters. In addition, the hydrostatic component can be obtained using the
energy scan setting. Here, a monochromatic beam of known energy is used to obtain interplanar
spacing for a specific {ℎ 𝑘 𝑙}. This in turn can be compared to the lattice parameters of the unstrained
crystal to measure lattice dilatation (Yang et al., 2004).
47
Yang et al. [2004] have conducted full strain tensor measurements from DAXM analysis of a
bent cylindrical specimen of Si. Typically, spatially resolved elastic strains can be measured with
an accuracy of 10−4 . More recently, Zhang et al. [2018] have shown for a synthetic DAXM dataset
that the deviatoric or full deformation gradient can be obtained with an accuracy of 10−9 or better.
2.7.2.2 Calculation of GND density from DAXM data
Because lattice orientations can be measured with submicron spatial resolution using DAXM,
it enables the evaluation of local lattice curvature in three dimensions. Therefore the GND density
can be calculated in plastically deformed regions (Yang et al., 2004).
Larson et al. [2008] calculated the spatially resolved Nye tensor from lattice curvature mea-
surements using DAXM in thin deformed Si plates. More recently Guo et al. [2020] have used the
modified Nye-Kröner-Bilby equation to calculate GND density in plastically deformed hexagonal
titanium based on lattice rotation gradient measurement using DAXM.
2.8 Opportunity for research
Accommodation of strain during heterogeneous deformation in polycrystalline materials has a
strong influence on properties like strength and ductility. This in turn has critical implications in
microstructure design for room temperature formability and fatigue resistance. For example, poor
formability in low symmetry metals like magnesium is an impediment towards their widespread
use in structural applications. It is also crucial to understand the interaction of slip systems with
dislocation pileups at grain boundaries for improved design to withstand low cycle fatigue (Joseph
et al., 2018). One approach to assess the strain accommodation at the microstructure level in
polycrystalline materials is to use the slip transfer parameter as a quantitative marker. To this end,
it is important to be able to determine positions and shape information of constituent grains in a
polycrystal, along with the local stress state.
Geometrically compatible slip systems in neighboring grains can generate twinning partials
at the grain boundary, resulting in a stable twin nucleus (Wang et al., 2009a). The slip transfer
48
m'=0.9
(a) (b)
Figure 2.18: Example illustrating the importance of relative grain positions for viable S+T twin
nucleation. The left figure (a) shows two grains with a favorable relative position (along with a
high 𝑚′ value). Figure (b) shows the same grains with relative positions that may not favor S+T
twinning.
parameter 𝑚′ can be used as a criterion to assess the possibility of S+T twinning. A high value
of 𝑚′, however is not a sufficient condition to guarantee twin nucleation by slip transfer because it
is also important to also consider the relative positions of the grains and slip planes with respect
to each other in three-dimensional space. An illustrative example is shown in figure 2.18, where
the slip system in a grain has a favorable 𝑚′ value with respect to the activated twin variant in
its neighbor. Theoretically, shear transfer from the slip system in the adjacent grain is possible
across the grain boundary in both cases (figures 2.18(a) and (b)). It is however, the arrangement in
figure 2.18(a) that has a more favorable relative position for shear transfer, and thereby S+T twin
nucleation, as impingement of the slip system is more likely in this case. Therefore, based upon the
geometry, an attempt can be made to answer whether slip transfer plays a significant role in twin
formation.
FF HEDM characterization enables the determination of grain orientations and positions, along
with quantification of grain averaged stress tensors. It should be noted however, that the uncertainty
associated with the determination of the grain centers of mass using FF HEDM is relatively high
(most available literature puts it in the order of 10 𝜇𝑚 (Turner et al., 2016)). Moreover, it is
not possible to obtain information about grain morphology using FF HEDM, since only the grain
centroids are calculated. It is in this context that NF HEDM can be used to assess grain positions
determined using FF HEDM, since the positional error using the former method is at least an order
49
of magnitude smaller. Moreover, with the finer resolution data obtained from NF HEDM it is
possible to determine the morphology of individual grains within the interrogated volume.
An important goal of this work is to determine the limits of usability of FF HEDM data, mainly
in terms of the error associated with grain position. Using FF data in conjunction with NF analysis,
the relative difference in grain positions and orientations can be determined. This information,
combined with the grain averaged stress tensors can be invaluable in understanding local deforma-
tion behavior–for example mechanical twinning and the role of local strain accommodation in its
nucleation.
Using FF HEDM, a measure of the average local stress state, in the form of grain averaged
local Schmid factors can be obtained, along with average grain orientation. In order to make an
assessment of obstacles in the path of slip however, it is important to take into account the lattice
rotation gradients and spatially resolved local stress tensor. In this respect, DAXM proves to be a
very useful characterization tool. This would enable a better understanding of the interplay between
slip transmissibility and its correlation to slip system specific GND density. It is of particular interest
to understand the character of geometrically necessary dislocations piling up at grain boundaries,
and their effect on slip transfer. The broad underlying hypothesis here is that regions of local high
GND concentrations would act as or reflect impediments to efficient slip transfer. Conversely, low
local GND concentration regions near grain boundaries may reflect efficient slip transfer through
the boundary, or, a general lack of dislocation activity in the region. As DAXM can also be used to
obtain spatially resolved stress tensor information, lattice rotation gradient and grain morphology,
it should be possible to correctly interpret the significance of a low GND density near a grain
boundary.
The broader goal of this work is to develop a three-dimensional understanding of local deforma-
tion behavior near grain boundaries. The ability to predict twin nucleation can help in optimizing
microstructure design for room temperature formability. Additionally, prediction of crack initiation
sites due to cyclical loading conditions are a challenge in material and component design – local
heterogeneity in stress states is an important consideration here. It is well-known that realistic
50
information on the grain morphology is critical to developing reliable CP models (Zhang et al.,
2015). It is expected that a more detailed understanding of the local processes of deformation will
reveal relationships that can be introduced into models in order to develop better constitutive for-
mulations that can successfully predict the details of deformation processes near grain boundaries.
Such information will improve the ability to bridge the gap between modeling in the microstruc-
tural and bulk (continuum) scales. This is important for optimal microstructure design via material
processing, as well as going beyond a statistical damage model for prediction of catastrophic failure
of in-service structural components.
51
CHAPTER 3
MATERIALS AND METHODS
3.1 Titanium specimens
Commercial purity titanium specimens were used for both the far field HEDM and DAXM
experiments. The chemical composition of the specimens is given in table 3.1 (Bieler et al., 2014).
The source material had been annealed to obtain an average grain size of 100 𝜇𝑚.
3.1.1 Tensile specimen for Far Field Experiment
The tensile specimen used for the HEDM in-situ deformation experiment was cut out out of
a larger sample that was previously subjected to a four point bending test, as shown in figure 3.1.
Results of the characterization of this bend specimen has been discussed in a prior work (Wang
et al., 2009b). Figure 3.2 shows the distribution of the normal component of the strain tensor (𝜖 𝑧𝑧 )
within a cross section of the gage volume interrogated using FF HEDM. The top half of the cross
section shows a compressive residual character, while the bottom half shows a tensile character.
This strain state is typical of a cross section of a bent beam, where the neutral axis passes through
the center and has a strain approaching zero. The tensile specimen had a 1 mm square cross section
with a nominal gage length of 5 mm. Texture analysis of the source specimen revealed that a high
density of {0 0 0 1} poles were concentrated around a 30° cone centered about the tensile axis of
the bending test as described in previous work (Wang et al., 2010b). This preferred {𝑐} texture is
about 8 times random parallel to the tensile axis. Therefore, a majority of the grains had a hard
orientation with respect to the tensile direction.
Table 3.1: Composition (wt%) of Grade 1 Titanium Plate. Adapted from Bieler et al. (2014)
Element O Fe Al Cu C Ni S Cr N Ti
Fraction 0.169 0.049 0.017 0.017 0.015 0.013 0.011 0.011 0.004 bal.
52
Figure 3.1: Tensile specimen extracted from a larger sample that was previously subjected to a bend
test. Because the sample had a prior bending load, a macro residual stress was initially present in
the sample (Wang et al., 2014)
Z
Y
X
Figure 3.2: Right: Cross section of the illuminated volume of the tensile specimen (looking down
the tensile axis), showing the distribution of the normal component of the strain tensor (𝜖 𝑧𝑧 ), prior
to tensile loading. Left: the coordinates of the strain map are shown in context of the tensile
specimen. The tensile axis is parallel to the 𝑍 direction.
53
3.1.2 Four point bending specimen for DAXM Experiment
The texture of the four point bend specimen had a predominantly soft orientation with respect
to the loading direction. A schematic showing the dimensions of the specimen is shown in figure
3.5.
3.2 Far Field HEXRD (Chapter 4)
3.2.1 Experimental setup
The FF-HEXRD experiment was conducted at the beamline 1-ID of the Advanced Photon
Source (APS), Argonne National Laboratory. The specimen was mounted on a custom designed
load frame and deformed in 37 load steps. Figure 3.3 shows the experimental setup of the FF-
HEDM; along with the sequence of tensile loading to ∼ 3% engineering strain and subsequent
unloading. Elastic loading took place in the first seven steps, followed by plastic deformation
during the succeeding 23 steps. The last seven steps correspond to the unloading of the specimen.
It was anticipated that the initial texture, loading state and the relatively high resolved shear stress
on twin planes were conducive to the nucleation of {1 0 1 2}⟨1 0 1 1⟩ deformation twins.
Following each load step, the central gage section was illuminated by a high energy (∼ 65.4
keV) monochromatic X-ray beam. An area detector with resolution of 2048 × 2048 pixels, spanning
an area of 0.4 × 0.4𝑚 2 was placed about 1 m away from the specimen. After each load step, the
specimen was held at constant displacement and rotated about the tensile axis through an 𝜔 range
of 140° (20° to 160°). Diffraction (Debye-Scherrer) patterns were collected for each step of 1°. The
monochromatic beam was used to interrogate eleven ∼ 100𝜇𝑚 thick layers along the gage length.
Thus, for each of the 37 loading steps, a total of 140 diffraction patterns were collected for each
layer examined.
The inset of figure 3.3 shows a schematic illustration of the trend of the load time plot corre-
sponding to the tensile test. Significant relaxation of load was observed during the course of loading
and subsequent unloading. The FF scans for each load step were taken after most of the relaxation
54
Load
Load time
time
Load
time
Figure 3.3: Evolution of global stress as function of engineering strain for the tensile experiment.
Inset figure shows a schematic representation of the experimental setup for FF-HEDM. A schematic
representation of the eleven layers examined along the gage section of the tensile is shown on the
right. Inset: Schematic illustration of the load relaxation with respect to time is shown for three
different strain states–prior to yield (magenta), post yielding (green) and at maximum stress (red).
had taken place. Figure 3.3 (inset) shows the schematic illustration of the load time curves for
three different strain states: prior to yield ( where a 2-3% relaxation was observed, represented
by the magenta curve), post yield (6-8% relaxation: green curve) and at maximum stress (8-9%
relaxation: red curve).
3.2.2 Post deformation surface EBSD of the tensile specimen
After the tensile experiment was completed the far side (facing the detector) of the unloaded
specimen was lightly ground and electropolished. The surface was then examined using a Tescan
Mira 3 scanning electron microscope equipped with an Orientation Imaging MicroscopyTM system
(Ametek, Mahwah NJ). Electron backscattered diffraction (EBSD) maps were obtained of the
1000𝜇𝑚 × 1000𝜇𝑚 center of the gage length in the region examined using FF-HEDM. A step
size of 2𝜇𝑚 was used for the EBSD scan. Figure 3.4 shows the orientation map of the scanned
55
Figure 3.4: Surface tensile direction EBSD IPF map of the far side (facing detector) of the gage
section of the unloaded specimen shows several twins and orientation gradients in some grains.
The scanned area is highlighted in green.
region. Several discrete deformation twins and strong orientation gradients within some grains
were observed in the microstructure.
3.3 Experimental setup for in-situ DAXM characterization (Chapter 6)
The bend specimen was mounted on a custom built four pointing bending stage as shown in
figure 3.5. Deformation was carried out in four increments to a curvature on the tensile surface that
corresponded to a bulk strain of 3.5%. For each strain increment, a highly collimated polychromatic
beam of cross section 1𝜇𝑚 × 1𝜇𝑚 was used for a coarse DAXM characterization with a step size
of 6𝜇𝑚. A detailed description of this serial probing method is given in (Eisenlohr et al., 2017).
56
To detector H
Specimen surface normal
Incoming Z
X
beam 45 º
Frame holding
the Pt wire
(Differential
aperture)
F
25 mm
Pt-wire profiler
2 mm
Incoming microbeam m
3m
Figure 3.5: CP-Ti sample setup in the four point bending stage and dimensions are shown on the
bottom right of the figure. Experimental setup for the in-situ DAXM characterization of four point
bending is shown on the top left. The directions of the incoming and diffracted beams are shown
schematically. The beamline coordinate system is shown on the top right: Z denotes the direction
of the incoming beam; F is anti-parallel to the surface normal (denoted by the green vector) of
the specimen; X is directed normal out of the page. Position and displacement direction of the
differential aperture (Pt wire) are shown on the bottom left of the figure (Adapted from Larson et
al. (Larson et al., 2002)).
A volume of 600𝜇𝑚 × 400𝜇𝑚 × 325𝜇𝑚 was interrogated in this experiment. At the second strain
increment, a set of more finely spaced scans was done (step size of 1.5𝜇𝑚), along the X and H
directions. Laue diffraction patterns were collected on a system of area detectors of 2048 × 2048
pixel resolution.
Additionally, for each bending increment the beam was switched to monochromatic mode and
energy wire scans were conducted within two specific grains of interest. For the first grain, the
energy wire scan was done for all the four strain increments; for the second grain it was done until
the third increment due to beam time limitations. Details of these scans are elaborated in chapter 6.
57
CHAPTER 4
IN-SITU FAR FIELD XRD CHARACTERIZATION OF TENSILE DEFORMATION OF
A COMMERCIAL PURITY TITANIUM SPECIMEN
In this chapter the tools and methodologies use to analyze data the FF HEDM tensile exper-
iment are elaborated, followed by a detailed elucidation of the results and their discussion. The
methodology used to identify twinning events from the FF HEDM data is discussed in detail.
The possibility of deformation twinning nucleated as a result of slip transfer from a neighboring
grain (S+T twinning) is explored in terms of local stress evolution in the parent (twinned) and
neighboring grains, and geometrical consideration (based on slip transfer parameter 𝑚′ and relative
grain positions in 3D space). Furthermore, these relative grain positions and geometrical criteria
are re-evaluated in the context of more recently collected near field (NF) data corresponding to the
final unloaded stress state of the specimen.
A central question sought to be answered here is with respect to the robustness of the FF HEDM
data, i.e. how well do the kinematics in terms of grain disorientation as a function of load step,
match the grain averaged stress measurements.
Fable
Figure 4.1: Outline of data analysis strategy that was implemented in a suite of MatLab codes. The
.log files (containing orientation and grain index information), and .gve files (containing scattering
vector information) are obtained using FABLE.
58
4.1 Analysis of the diffraction patterns
The collected diffraction patterns were analyzed using FABLE, an open source software de-
veloped jointly by the European Synchrotron Research Facility (ESRF) and the Risø National
Laboratory Denmark. The analysis follows three fundamental steps. The first step identified peaks
that lie above a specified threshold value using the Peaksearch module. The second step assigned
reciprocal lattice vectors (𝑔 ℎ𝑘𝑙 ) to each of the peaks identified in the first step, using the Transfor-
mation module. Finally, the Grainspotter module indexed individual grains by correlating at least
30 𝑔 ℎ𝑘𝑙 with a given lattice orientation. This process was repeated for each load step for all 11
layers.
4.1.1 Determination of grain centers of mass and grain averaged strain tensors
The grain centers of mass (COM) and grain average strain tensors were calculated using the
least squares algorithm proposed by Margulies et al. (2002), and more recently implemented for
AZ31 by Aydıner et al. (2009).
Following the formulation in equation 2.42, a modified version is used after addition of two
correction terms that account for the offset in the sample coordinate system from the center of
rotation (Margulies et al., 2002). This is written as shown below.
©𝜖 11 𝜖12 𝜖 13 ª © 𝑙 𝑖 ª
® ® sin(𝜔𝑖 ) sin(𝜂𝑖 ) Δ𝑥 cos(𝜔𝑖 ) sin 𝜂𝑖 Δ𝑦
𝜖 𝑖 = (𝑙𝑖 𝑚 𝑖 𝑛𝑖 ) 𝜖 21 𝜖22 𝜖 23 ® 𝑚 𝑖 ® − cos(𝜔𝑖 ) +
® ® − sin(𝜔𝑖 ) + (4.1)
® ® tan(𝜃 𝑖 ) 𝐿 tan(𝜃 𝑖 ) 𝐿
«𝜖 31 𝜖32 𝜖 33 ¬ « 𝑛𝑖 ¬
Where Δ𝑥 and Δ𝑦 are the offsets from the center of rotation; 𝜔𝑖 is angle of rotation about the
tensile axis; 𝜂𝑖 the azimuthal angle; 𝜃 𝑖 the Bragg angle corresponding to diffraction vector 𝑖; and 𝐿
is the specimen to detector distance.
The lattice strain is calculated using the reciprocal lattice vectors for an ideal unstrained hexag-
onal titanium unit cell as reference. The lattice parameters used for the reference unit cell are
𝑎 0 = 𝑏 0 = 2.95 nm and 𝑐 0 = 4.683 nm; 𝛼 = 𝛽 = 90°and 𝛾 = 120°. The set of reciprocal lattice
vectors in fractional crystallographic coordinates to the corresponding vectors in the Cartesian
59
coordinate system by the following relation.
𝑔𝑐𝑎𝑟𝑡 = 𝑇 𝑔𝑐𝑟 𝑦 𝑜𝑟 𝑔𝑐𝑟 𝑦 = 𝑇 −1 𝑔𝑐𝑎𝑟𝑡 (4.2)
Where
©𝑎 0 𝑏 0 cos(𝛾) 𝑐 0 cos(𝛽) ª
®
𝑇 = 0 𝑏 0 sin(𝛾)
𝑐 0 (cos(𝛼)−cos(𝛽) cos(𝛾)) ® (4.3)
sin(𝛾)
®
®
®
𝑉
0 0 𝑎 0 𝑏 0 sin(𝛾)
« ¬
is the transformation matrix;
𝑉 = 𝑎 0 𝑏 0 𝑐 0 (1 − cos2 (𝛼) − cos2 (𝛽) − cos2 (𝛾) + 2 cos(𝛼) cos(𝛽) cos(𝛾) (4.4)
is the volume of the unstrained unit cell. 𝑔𝑐𝑎𝑟𝑡 and 𝑔𝑐𝑟 𝑦 are the set of vectors in the Cartesian
and fractional crystallographic coordinate systems respectively. For each indexed grain the change
in interplanar spacing is determined in terms of the difference between the measured and ideal
reciprocal lattice vectors.
Δ𝑔 ℎ𝑘𝑙 = ||(𝑔 ℎ𝑘𝑙 )𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 || − ||(𝑔 ℎ𝑘𝑙 )𝑖𝑑𝑒𝑎𝑙 || (4.5)
For each indexed grain, the elastic strain for each measured reflection (reciprocal lattice vector)
is given as follows.
Δ𝑔 ℎ𝑘𝑙
𝜖= (4.6)
(𝑔 ℎ𝑘𝑙 )𝑖𝑑𝑒𝑎𝑙
From equations 4.1 and 4.6, an overdetermined system of linear equations can be written as:
𝐴𝑋 = 𝑏 (4.7)
For 𝑛 reflections, 𝐴 is an 𝑛 × 8 matrix. Here each row of A (𝐴𝑖 ) contains strain terms and offset
from center of rotation corresponding to each measured g-vector 𝑖:
𝐴𝑖 = 𝑙𝑖2 , 𝑚𝑖2 , 𝑛𝑖2 , 2𝑙𝑖 𝑚𝑖 , 2𝑙𝑖 𝑛𝑖 , 2𝑚𝑖 𝑛𝑖 , Δ𝑥, Δ𝑦 .
60
𝑏 is an 𝑛 × 1 vector: 𝑏𝑖 = 𝜖𝑖 .
X is the vector containing the six symmetric components of strain tensor and center of mass
coordinates.
The system of equations 𝑋 = 𝐴−1 𝑏 can be solved by least squares or singular value decompo-
sition. A minimum of 17 independent reflections for each indexed orientation are necessary for a
reliable solution (with a fitting error of the order of 10−4 or lower) (Margulies et al., 2002).
The algorithm outlined above is implemented in a suite of MatLabTM codes. The grain average
stress tensor for each indexed grain is calculated as shown below. Here, 𝐶𝑖 𝑗 𝑘𝑙 is the fourth order
stiffness tensor for hexagonal titanium; 𝑋𝑌 𝑍 refers to the sample coordinate system and 123 denotes
the crystal coordinate system defined by the the orientation matrix g:
©𝜖 𝑋𝑋 𝜖 𝑋𝑌 𝜖 𝑋𝑍 ª ©𝜖 11 𝜖 12 𝜖 13 ª ©𝜎11 𝜎12 𝜎13 ª ©𝜎𝑋𝑋 𝜎𝑋𝑌 𝜎𝑋𝑍 ª
® 𝜖cry =g· 𝜖sam ·gT ® 𝜎𝑖 𝑗 =𝐶𝑖 𝑗 𝑘𝑙 · 𝜖𝑘𝑙 ® 𝜎sam =gT · 𝜎cry ·g ®
𝜖
𝑌𝑋 𝜖𝑌𝑌 𝜖 𝑋𝑍 ® −−−−−−−−−−−−→ 𝜖 21
® 𝜖 22 𝜖23 ® −−−−−−−−−−−−→ 𝜎21
® 𝜎22 𝜎23 ® −−−−−−−−−−−−−→ 𝜎𝑌 𝑋
® 𝜎𝑌𝑌 𝜎𝑌 𝑍 ®®
® ® ® ®
«𝜖 𝑍 𝑋 𝜖 𝑍𝑌 𝜖𝑍 𝑍 ¬ «𝜖 31 𝜖 32 𝜖33 ¬ «𝜎31 𝜎32 𝜎33 ¬ «𝜎𝑍 𝑋 𝜎𝑍𝑌 𝜎𝑍 𝑍 ¬
4.1.2 Quantification of error in local strain calculations
In this section, the procedure to calculate the fitting error in the least squares algorithm discussed
above is shown. Equation 4.7 can be rewritten as follows.
𝜖 = 𝜖11 𝑙 2 + 𝜖 22 𝑚 2 + 𝜖33 𝑛2 + 𝜖12 𝑙𝑚 + 𝜖13 𝑙𝑛 + 𝜖23 𝑚𝑛 (4.8)
Moreover, in terms of interplanar spacing the lattice strain 𝜖 can be written as follows (Margulies
et al., 2002).
𝑑 − 𝑑0
𝜖= (4.9)
𝑑0
The error of fit can be expressed as the residual of the sum of squares 𝑑𝜖.
𝑑𝜖 = Σ( 𝐴𝑋 − 𝑏) 2 (4.10)
61
Taking into account the error of fit, the six symmetric components of the strain tensor can be
written as follows.
𝜖± = [𝜖11 ± 𝑑𝜖, 𝜖22 ± 𝑑𝜖, 𝜖33 ± 𝑑𝜖, 𝜖12 ± 𝑑𝜖, 𝜖13 ± 𝑑𝜖, 𝜖23 ± 𝑑𝜖] (4.11)
As shown in 4.1.1, the anisotropic Hooke’s law can be used to determine the error in stress
measurement, which can be expressed in terms of an equivalent scalar value (von Mises stress).
The average equivalent stress error for the data evaluated in this work was found to be ∼ 10
MPa, which is within 10−4 in terms of equivalent strain.
4.2 3D Reconstructed model of the microstructure prior to loading
In order to facilitate the study of grain interactions in 3D space, it is important to obtain an
estimate of neighboring grains, grain morphology and their relative positions. Also, simulating the
experiment with a computational model enables assessment of material model assumptions by direct
comparison of the local stress-strain history in individual grains. To this end, a set of seed points
were generated using the COM and Bunge Euler angle information of identified grains at zero strain
load step for each of the 11 layers. Grain orientations that were misoriented by less than 5° and with
a Euclidean distance less than 100𝜇𝑚 between the centers of mass in two or three successive layers
were assigned the same grain number. The importance of closely capturing the three-dimensional
morphology of the actual microstructure has been shown in the work done by Zhang et al. (2015).
Using this set of seed points in 3D space, a non-periodic model of the experimentally investigated
microstructure was constructed (figure 4.2) with 106 volume elements, using a Voronoi tessellation
module available in DAMASK (Düsseldorf Advanced Materials Simulation Kit).
4.2.1 Model parameters and simulation of tensile test
The specimen used for the tensile experiment had several grains with strong compressive residual
stresses prior to loading. In order to capture this initial stress state, a uniform compressive strain
of 10−3 /s was applied for each of the first three load steps. This was based on the determination of
62
300
200
100
0
-100
-200
-300
-1.00 0.00 1.00 2.00 3.00 4.00
Bulk Strain/%
Figure 4.2: Left:Voronoi tessellated model generated from 3D XRD Data set. This microstructure
shown here represents the unstrained condition, prior to the tensile test. A dilatational layer of one
volume element thickness is used as the surrounding medium along the 𝑋 and 𝑌 directions. The
tensile direction is parallel to the Z axis. Right: Evolution of normal component of stress along
the tensile direction 𝜎𝑧𝑧 in the crystal plasticity simulation of the tensile loading. In the CP model,
a uniform compressive strain of 10−3 is applied for each of the first three load steps; followed by a
uniform tensile strain of 10−3 for each of the subsequent load steps.
the average initial equivalent strain in the volume interrogated by 3D XRD (which was of the order
of 10−3 ). Hereafter, a uniform tensile strain of 10−3 /s was applied for the subsequent 42 load steps
to impose a bulk tensile strain of ∼ 4%. The evolution of the normal component of the bulk stress
tensor (parallel to tensile axis) is shown in figure 4.2.
The crystal plasticity model was implemented using the DAMASK spectral solver (Eisenlohr
et al., 2013). Since the model is non-periodic and the spectral solver requires imposition of
periodic boundary conditions, the microstructure was sheathed in a dilatational layer (free surface)
of one volume element thickness along the 𝑋 and 𝑌 directions (figure 4.2, left). The constitutive
description of the model is based on the phenomenological power law described in section 2.3. The
material parameters used in the current model are enumerated in table 4.1.
63
Table 4.1: Parameters used in Crystal Plasticity model
Parameter Value Units
𝐶11 162.2 GPa
𝐶12 91.8 GPa
𝐶13 68.8 GPa
𝐶33 180.5 GPa
𝐶44 46.7 GPa
𝛾¤ 1 × 10−3 𝑠−1
𝑆𝑙𝑖 𝑝
𝜏0 (per family) [80,90,110,260,260,260] MPa
𝜏0𝑇 𝑤𝑖𝑛 (per family) [220,220,250,250] MPa
400
350
300
250
200 1 Voxel thickness 3 Voxel thickness
150
100 1 Voxel Thickness
3 Voxel Thickness
50 5 Voxel Thickness
10 Voxel Thickness
0
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Bulk Strain/% 5 Voxel thickness 10 Voxel thickness
Figure 4.3: Evolution of volume averaged equivalent (von Mises) stress of the far field microstruc-
ture model as a function of bulk strain for four different thicknesses of the surrounding free surface
layer. It is seen that the mechanical response is not sensitive to dilatational layer thickness.
4.2.2 Sensitivity of mechanical response of model to dilatational layer thickness
An important question arises with regard to the free surface layer used around the microstructure
model–how sensitive is the model response to the layer thickness? In order to assess this, the volume
averaged stress evolution of the model was evaluated for four different thicknesses of the dilatational
layer. It can be observed from equivalent stress plot in figure 4.3 that the effect of dilatational layer
thickness on the average mechanical response of the model is not significant.
64
4.3 Effect of local stress state on deformation systems
The top row of figure 4.4 shows the cumulative distribution of global Schmid factors corre-
sponding to three load steps for basal ⟨𝑎⟩, prism ⟨𝑎⟩, pyramidal ⟨𝑎⟩, pyramidal ⟨𝑐 + 𝑎⟩ and 𝑇1
twin systems. In this case the stress tensor is the same for all load steps (based on global uniaxial
tension). The small changes in the cumulative distribution profiles are purely due to small crystal
rotations with increasing strain.
The generalized Schmid factor, calculated using the average local stress tensor for each grain is
shown on the middle and bottom rows (Dashed lines indicate data from experiment, while the solid
lines correspond to the CP model). Using the local stresses, the CP model predicts lesser activity
than observed for all slip systems. The middle row shows the comparison with a model that did not
have the initial compressive strain imposed before tensile loading. Clearly, the strain history plays
an important role with regard to the slip systems that are likely to be activated, as the predicted
activity is closer to the observed activity for every slip system when the pre-compression takes
place. Compared to the global stress, the likelihood of pyramidal ⟨𝑐 + 𝑎⟩ slip and T1 twinning
being activated is much higher when the local stress state is considered. Clearly, the texture is
highly favorable for pyramidal ⟨𝑐 + 𝑎⟩ slip, while being less conducive to ⟨𝑎⟩ slip on basal, prism,
or pyramidal planes.
Several significant observations can be made with regard to the comparison between the ex-
perimental and model data. Prior to loading, the experimental data show a similar propensity for
pyramidal ⟨𝑐 + 𝑎⟩ slip and T1 twinning. The corresponding model data (with the initial com-
pression) indicates a much lower likelihood of T1 twinning at this stage, when compared with the
tension-only model and experiment. This indicates that the stress state of most grains in the model
is such that they are not able to overcome the initial critical stress necessary for twinning. Both
models show a similar lower likelihood of pyramidal ⟨𝑐 + 𝑎⟩ slip when compared with experiment
prior to tensile loading. For basal⟨𝑎⟩ and prism⟨𝑎⟩ slip, the model with initial compression tracks
the experimental data more closely than the tension-only model.
At approximately 0.4% bulk strain (about half way through yield), the model with initial com-
65
pression tracks the experimental data more closely for prism⟨𝑎⟩ and pyramidal⟨𝑎⟩ slip. It can also be
seen that the fraction of grains with a propensity to form twins has increased significantly compared
to the unstrained state. For T1 twinning, the tension-only model is closer to the experiment.
At 1.5% bulk strain, all of the grains should have experienced plastic deformation. The model
with initial compression captures the experimental trend for prism⟨𝑎⟩, pyramidal⟨𝑐 + 𝑎⟩ and T1
twinning better than the tension-only model. It can be seen that except for basal slip, the model
captures the local stress state quite well for the other slip and twin systems. On the other hand,
basal⟨𝑎⟩ slip tracks closer to the experiment for the tension-only model. It is apparent that once
initial compressive residual stresses are overcome, the model with initial compression matches the
experiment quite closely for most of the deformation systems.
4.4 Identification of twinning events
Twinning events during the in-situ tensile test were identified using the following misorientation
and spatial proximity criteria:
1. The twin and corresponding candidate parent must have a c-axis misorientation of 85° and
share a common a-axis.
2. Because the average grain size of the area of interest is about 100𝜇𝑚, the centers of mass
of the candidate twin and parent grain must be sufficiently close together for the twin to be
plausibly within the parent grain.
3. The twinned orientation must be present in at least two consecutive load steps.
This method of identifying a twin-parent pair is illustrated in figure 4.5. The three {0 0 0 1} pole
figures indicate the initial orientation of indexed grains in layer 1 (prior to tensile loading), at the
load step just before the twin was detected (1.3% bulk strain) and after 1.5% bulk strain respectively.
The large cluster of points close to the center of the {0 0 0 1} pole figures (where the sample tensile
axis is coincident with the center of of the pole figure) confirms the initial assumption that hard
orientations are predominant in the specimen. It is also clear from the pole figures that the twin is
66
Unstrained state (prior to tensile loading) Pre yield (~0.4% bulk strain) ~1.5% bulk strain
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3 Prism
Pyramidal
0.2 Pyramidal
T1 Twinning
0.1 Basal
0
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Global Schmid Factor
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
CP Model: Tension only
0.1
Dashed lines: Experiment
Solid lines: CP Model
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
CP Model: Ini�al compression
0.1
before tension
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Generalized Schmid Factor (based on grain averaged stress tensor)
Figure 4.4: Cumulative distribution function (CDF) plots of Schmid Factors corresponding to three
load steps for basal ⟨𝑎⟩, prism ⟨𝑎⟩, pyramidal ⟨𝑎⟩, pyramidal ⟨𝑐 + 𝑎⟩ and T1 systems. The global
Schmid factor (top row) is calculated using the global uniaxial stress tensor; while the local grain
averaged stress tensor is used to obtain generalized Schmid factors (maximum 0.707) (middle and
bottom rows). The middle row shows the comparison of the experimentally obtained local Schmid
factors with an earlier CP model that imposes only tensile strain without initial compression. The
bottom row shows the comparison of local Schmid factors with the results of the CP model with
initial compression. The Schmid factors in the CDF plots are the highest per slip system family for
each grain.
67
(0001) (0001) (0001)
90
0.0% Strain 1.3% Strain 1.5% Strain
100µm
100µm
Figure 4.5: Neighborhood of all grains in layer 1 prior to deformation, at the load step just
before twinning at 1.3% engineering strain, and after a twin formed at 1.5% engineering strain in
orientation space (top row). The spatial map of the 2D slice is shown in the bottom row with grain
identification numbers. The bottom right figure shows an enlarged view of the region of interest for
the twin formation with grain unit cell orientations based upon geometrical positions in the lower
left map of the slice. Tensile direction is along z direction (out of page).
not present in the load step immediately before 1.5%. The paired parent (grain in red numbered
12) and twin (magenta colored grain numbered 121) orientations are shown in the second pole
figure in figure (top row right), from the measurement at 1.5% strain. It is important to note here
that although there are other grains with orientations near the circumference of the pole figure that
appear to be twins, they were not present in at least two consecutive load steps. The corresponding
COM positions projected in 2D X-Y space are shown below the pole figures. An enlarged view of
the twin-parent pair in layer 1 and its immediate neighborhood is shown with prisms to illustrate
the orientations of grains in the vicinity (bottom right of figure). The common ⟨𝑎⟩ -axis of the twin
and its parent grain is indicated with a magenta line for the parent and blue for the twin.
68
Altogether, 13 twinning events were observed within the 11 layers (with 2 twins each in layers
3 and 6). The Schmid factors of the T1 twin systems were computed using the measured grain
averaged stress tensor. Because this stress tensor is a local stress within the microstructure and
not the global stress, the generalized Schmid factor is computed (for which the maximum possible
√
2
value is 2 ). The evolution of the resolved shear stress (RSS) on the six T1 twin variants in the
parent grains in the 11 layers are shown for each of the 13 twin events in figures 4.6, 4.7 and 4.8.
In each plot of the twinned grain, the observed twin variant is indicated with red dot symbols. In
the evolution of the RSS for the first layer, the activated twin variant had the highest RSS value.
This is however, not always the case, as observed from the RSS plots for parent grains observed in
some of the other layers (e.g 2,3 and 7), which indicates that the Schmid law (highest RSS) did not
account for twin nucleation.
4.5 Results and Discussion
The focal point of this work is to examine the propensity of deformation twins being nucleated
as a result of shear transfer from a compatible slip or twin system in a neighboring grain. This
assessment was done using a compatibility factor (Luster-Morris parameter 𝑚′)1.
Geometrically compatible slip systems in neighboring grains can generate twinning partials at
the grain boundary, resulting in a sustainable twin nucleus (Wang et al., 2009a). In this context, the
value of 𝑚′ can be used as a criterion to evaluate the possibility of slip transfer from a neighboring
grain to an identified parent grain that generated a twin. However, a high value of 𝑚′ alone is not a
sufficient condition to guarantee slip transfer. In addition to the Schmid factor of the slip system in
the neighbor grain and 𝑚′, a third parameter is considered. A spatial plausibility assessment can
be made by taking into consideration the relative position and elevation of the grains with respect
to each other. A quantitative criterion for determining a plausible high 𝑚′ neighbor is outlined in
figure 4.9. The likelihood of slip transfer to nucleate a twin will be high if the plane normal of the
slip system in the neighbor grain is highly inclined to the vector between centers of mass of the
1𝑚′ is only meaningful for active slip systems
69
400 400
Variant 1 Variant 1
Variant 2 Variant 2
350 350
Variant 3 Variant 3
Variant 4 Variant 4
Variant 5 Variant 5
300 300
Variant 6 Variant 6
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
150 150 Load step corresponding to twin
observation in Layer 2
Load step corresponding to twin
100 100
observation in Layer 1
50 50
0 0
-50 -50
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
(a) (b)
400 400
Variant 1 Variant 1
Variant 2 Variant 2
350 Variant 3 350 Variant 3
Variant 4 Variant 4
Variant 5 Variant 5
Variant 6 Variant 6
300 300
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
Load step corresponding to twin observation in
150 Layer 3 (1st observed twin) 150
100 100
Load step corresponding to twin observation in
50 50
Layer 3 (2nd observed twin)
0 0
-50 -50
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
(c) (d)
400 400
Variant 1
Variant 1
Variant 2
Variant 2 350
350 Variant 3
Variant 3 Variant 4
Variant 4 Variant 5
Variant 5 300 Variant 6
300
Variant 6
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
150 150
Load step corresponding to twin
100 100
observation in Layer 4
50 50
Load step corresponding to twin
0 0 observation in Layer 5
-50 -50
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
(e) (f)
Figure 4.6: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems in the
parent grains identified in layers 1, 2, 3, 4 and 5. Green arrows indicate the load step where the
twin was first identified, and filled red markers indicate the observed twin variant.
70
400 400
Variant 1
Variant 1
Variant 2
Variant 2
350 Variant 3 350 Variant 3
Variant 4 Variant 4
Variant 5 Variant 5
300 Variant 6 300 Variant 6
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
150 150 Load step corresponding to twin observation in
Layer 6 ( 2nd observed twin)
100 Load step corresponding to twin observation in 100
Layer 6 ( 1st observed twin)
50 50
0 0
-50 -50
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
(a) (b)
400 400
Variant 1
Variant 2
350 350 Variant 3
Variant 4
Variant 5
300 300
Variant 6
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
150 150
Load step corresponding to twin observation in
Load step corresponding to twin observation in
Layer 8
Layer 7
100 100
50 50
0 Variant 1 0
Variant 2
Variant 3
-50 Variant 4 -50
Variant 5
Variant 6
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
(c) (d)
400 400
Variant 1 Variant 1
Variant 2 Variant 2
350 350
Variant 3 Variant 3
Variant 4 Variant 4
Variant 5 300 Variant 5
300
Variant 6 Variant 6
250 250
Resolved Shear Stress/MPa Resolved Shear Stress/MPa
200 200
150 150
Load step corresponding to twin Load step corresponding to twin
100
observation in Layer 9 100 observation in Layer 10
50 50
0 0
-50
-50
-100
-100 0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Bulk Strain/%
Bulk Strain/%
(e) (f)
Figure 4.7: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems in the
parent grains identified in layers 6, 7, 8, 9 and 10. Details of symbols and color conventions are
given in the caption of figure 4.6.
71
400
Variant 1
Variant 2
350
Variant 3
Variant 4
300 Variant 5
Variant 6
250
Resolved Shear Stress/MPa
200
150
100 Load step corresponding to twin
observation in Layer 11
50
0
-50
-100
0 0.5 1 1.5 2 2.5 3
Bulk Strain/%
Figure 4.8: Evolution of resolved shear stress (RSS) for each of the six T1 twin systems in the
parent grain identified in layer 11. Details of symbols and color conventions are given in the caption
of figure 4.6.
two grains, as illustrated in figure 4.9. A high value of sin 𝜃 implies a high probability that a slip
vector trajectory will point into the neighboring grain to stimulate twin nucleation. A smaller value
of sin 𝜃 may induce a twin on one end of the grain boundary but be much less likely to stimulate
a twin on the other end of the same boundary. The absolute value of sin 𝜃, termed here as the
spatial alignment factor (SAF), is considered a metric of spatial alignment of the two grains and
the slip/twin systems. A SAF value of 1 indicates ideal spatial alignment, while a value of zero
corresponds to the lowest likelihood for slip transfer to occur.
4.5.1 Far Field HEDM Characterization
Detailed analysis of the twins identified in six of the 11 layers shown in figure 4.10 illustrate the
range of conditions observed. For details of the orientation and relative positions for the remaining
twinning events, the reader is referred to appendix A.1.
4.5.1.1 Assessment of slip transfer from neighboring grains
Six different twinning events and their compatibility analysis with respect to the neighboring
grains are shown in figure 4.10. The orientations (Bunge Euler angles) of the parent, twin and
neighboring grains are indicated for each case. The relative position and orientation relationships
72
SAF
Figure 4.9: Schematic outlining the criteria used to identify the neighbor grain associated with the
highest likelihood of slip transfer, thereby triggering a twin. 𝜃 is the angle between the unit vector
joining the centers of masses of the parent grain and neighbor; and the unit vector normal to the
slip plane in the neighbor. The three parameters (Absolute value of the Schmid Factor, Absolute
value of sin 𝜃 (Spatial alignment factor SAF) and 𝑚′ can be plotted in a 3-D space. A point at the
origin (red) would have the worst possible spatial alignment for slip transfer, while a point at the
top right corner (green) would have the highest likelihood for slip transfer. The purple slip planes
have lower SAF while the green slip plane has a higher SAF.
between the parent grain and neighbor of interest are represented in both the X-Y and X-Z planes,
where the Z-axis is the tensile direction. The relative positions of the grain prisms shown are
based on the COM values from which a Voronoi tessellated representation of the grain within the
microstructure was generated. A corresponding prism indicating the unit cell is overlaid for each
grain, with the slip and twin systems indicated by the shaded plane and Burgers vector residing
in the plane. The position of the COM of the twin is indicated by the prism without a slip plane
indicated. The red-green-blue 𝑎 1 -𝑎 2 -𝑎 3 coordinate directions are drawn inside each unit cell; one
of these directions is the common ⟨𝑎⟩ axis between the twin and parent grains, which allows the
rotation axis for the 85° c-axis misorientation of the twin to be visualized.
In layer 1, the twin was observed at approximately 1.5% global strain. The neighborhood of the
parent grain and nucleated twin, shown in figure, indicates that most of the grains in the vicinity of
the identified parent grain have hard orientations, where the direction of the tensile axis is normal
73
Z
Y
Y
X View along tensile axis (Z) View looking toward beam (X)
Layer 1
m'=0.96
P: 127.6, 1.0, 358.0
T: 66.2, 275, 61
N: 78.5,13.8, 226
T
T
P
N N P
Layer 2
m'=0.87
P: 175.9, 28.9, 260.4
T: 79.8, 81.2, 31.2
N: 12.4, 351.8, 156.4
T
T
P
P
N N
Layer 4, m'=0.94
P: 338, 169.3, 94
T: 245, 95.4, 250.6
N: 349.6, 146.3, 311.1
T
P
T
P
N N
Layer 6, m'=0.99
P: 90.5, 11.8, 16.0
T: 46, 274, 8.5
N: 353.9, 155, 16.8
T N
P T P
N
Layer 9, m'=0.97
P: 291, 178, 228.8
T: 62.3, 86.7, 358.4
N: 28, 154, 320 N N
T
T
P
P
Layer 10, m'=0.94
P: 29.7, 176.2, 21.7
T: 68.3, 82.3, 357.5
N: 0.7, 334.7, 124 N N
T T
P
P
Figure 4.10: The 𝑚′ relationship observed between the activated twin system and pyramidal ⟨𝑐 + 𝑎⟩
system of a neighboring grain is shown for six different instances (Layers 1,2, 4, 6, 9 and 10).The
orientations of the parent, twin and neighboring grains are viewed along the tensile Z axis, (left
column) as well as the beam X direction, (right column). The letters "P", "T" and "N" indicate
the parent, twin and neighbor grains respectively. The corresponding Bunge Euler angles and 𝑚′
values are also indicated for each instance.The blue vector shows the slip direction from the blue dot
away from it on the positive surface of the plane, and a half length orange dot and vector indicates
a twin system. The plane normal points out of the page for beige slip planes and into the page for
gray planes. The dotted red, green and blue lines represent the a1 , a2 and a3 directions, respectively.
Colors of the grains correspond to (0 0 0 1) inverse pole figure representations of the orientations.
74
out of the page. The likelihood of slip/shear transfer between the identified parent grain in layer 1
(grain number 12) with grains adjacent to it in the same layer are assessed using the slip transfer
parameter 𝑚′. The activated twin system in the parent grain was found to have high 𝑚′ relationship
(∼ 0.96) with a pyramidal ⟨𝑐 + 𝑎⟩ system in the neighboring grain number 24.
The relative positions of the two grains in layer 1 are visualized in the first row of figure
4.10, showing the parent grain along with a neighboring grain lying within the same layer; with a
pyramidal ⟨𝑐 + 𝑎⟩ slip system that has a high 𝑚′ value with respect to the activated twin system.
The location of the twin is on the other side of the neighboring grain. Furthermore, the direction
of shear in the neighboring grain is such that it is not likely to intersect the parent grain. Therefore
the likelihood of twin nucleation as a result of slip transfer is low in this instance. This can further
be verified by using the criteria outlined in figure 4.9. Despite having high values of 𝑚′ and
Schmid factor (0.96 and 0.55 respectively), the SAF is only 0.68, indicating poor spatial alignment
conditions for effective slip transfer.
On the other hand, in layer 2 (second row in figure 4.10), the value of 𝑚′ between the activated
twin system and pyramidal ⟨𝑐 + 𝑎⟩ system of the neighbor is relatively low (0.87). In this instance,
the SAF is very close to unity (0.99, or 82°). This indicates a high likelihood of shear transfer from
the neighboring grain that would result in S+T twin nucleation. Similarly, it can be observed that
the likelihood of S+T twinning with closely aligned pyramidal ⟨𝑐 + 𝑎⟩ systems is high for layers 4
(third row), 9 (fifth row) and 10 (sixth row) in figure 4.10.
The results of the slip transfer parameter calculations for 13 twinning events observed within
the 11 layers are enumerated in table 4.2. The first three lines identify twins observed in a different
sample of the same material examined in a prior work (Bieler et al., 2014). Layer 1 of the currently
analyzed sample is labeled 2.01, and the range of Schmid factors of the potential twin variants
plotted in figures 4.6, 4.7 and 4.8 for the parent grain is provided in square brackets. The rank of
the Schmid factor corresponding to the observed twin variant is also indicated for each twin (red
markers in figures 4.6, 4.7 and 4.8). Each row in the table identifies a neighbor grain slip system
that has a high 𝑚′ value with respect to the observed twin system in the parent grain. Moreover, the
75
relative position, the bulk strain at the point of first twin observation are also indicated. In layers
3 and 6 (2.03 and 2.06 in table 4.2), two twins were observed at different strains, so two ranges
of twin Schmid factors are provided. Most of the twinned grains have more than one neighboring
grain with a slip system having a high 𝑚′ relationship with the twin system. In such situations, an
SAF cut-off value of 0.8 (53°) is used to further reduce the list of candidates and determine whether
slip transfer is plausible in each case. Instances where the likelihood of S+T twinning is high are
in bold font. In most cases, the activated twin variants have high geometrical compatibility with
pyramidal ⟨𝑐 + 𝑎⟩ systems in the neighboring grains.
Figure 4.11 shows the resolved shear stress for the activated twin variant plotted as a function
of 𝑚′ for the 13 observed twinning events. Slip transfer parameters for slip systems in neighboring
grains for which S+T twinning is less plausible are plotted with open symbols. The size of the
markers are scaled in proportion to the Schmid factor rank of the activated twin variant on a scale
of 1 to 6 (6 being the highest among all the variants, 1 being the lowest). In some cases, more than
one slip system in neighboring grains could facilitate twin formation of the same twin (details are
in table 4.2). For example, there are four points corresponding to the twin in layer 1: pyramidal
⟨𝑐 + 𝑎⟩, T1 twin, basal and prism ⟨𝑎⟩. Similarly for the twin in layer 4, there are two points
corresponding to two different pyramidal ⟨𝑐 + 𝑎⟩ slip systems in the neighboring grain. Out of
these two, one has a high SAF value (0.98) while the other has an SAF value of 0.63.
It can be seen from figure 4.11 that prism ⟨𝑎⟩ slip systems in neighboring grains do not have a
strong geometrical relationship with the twin systems of any of the identified parent grains; having
both low values of 𝑚′ and poor spatial alignment. Therefore, the likelihood of the observed twin
nucleation by shear transfer from prismatic ⟨𝑎⟩ slip systems in neighboring grains is low. The
range of observed resolved shear stresses corresponding to the observed twins lie between 180
and 320 MPa. Eight of the 13 twins (∼ 61%) have plausible S+T twinning from ⟨𝑐 + 𝑎⟩ systems
in neighboring grains, indicated by filled gold symbols. There does not appear to be significant
differences in the resolved shear stress between twins with plausible S+T (filled markers), and those
where S+T is less plausible (open symbols). These data clearly illustrate that deformation twinning
76
does not necessarily comply with the Schmid law. Moreover, from the current observations, a high
likelihood of geometric compatibility is seen for the twin systems with pyramidal ⟨𝑐 + 𝑎⟩ slip
systems in adjacent grains.
These observations differ from some earlier studies. In an in-situ neutron scattering compression
experiment perpendicular to the preferred c-axis direction, Nervo et al. (Nervo et al., 2016) reported
high compatibility between observed clusters of subsurface T1 deformation twins and prismatic
⟨𝑎⟩ slip in neighboring grains during a compression experiment of a Ti-4Al alloy. Also, Wang et al.
(Wang et al., 2010a) used EBSD to characterize T1 extension twins in CPTi specimens deformed
in bending with a favored c-axis texture aligned with the tensile direction (the same material, and
indeed from the same specimen used in the present study); as well as with a different specimen of
the same material oriented with a softer orientation along the loading axis. It was observed that
only one instance of S+T twinning was apparently activated by slip transfer from pyramidal ⟨𝑐 + 𝑎⟩,
and the rest were ostensibly triggered by prism ⟨𝑎⟩ slip. Similarities with prior studies indicate that
the geometrical compatibility of active twin systems with dislocation slip in neighboring grains
is not strongly dependent on initial texture, directionality of loading (tensile or compressive) and
orientation of the loading direction. Clearly slip transfer is unable to account for nucleation of all
the observed twins, so it is likely that the non S+T twins may have resulted from grain boundary
defects due to local strain concentrations at boundaries and triple junctions or strain incompatibility
between the parent and adjacent grains.
A more significant observation from this study is the fact that the nature of twin nucleation from
shear transfer is different in the interior and surface; as this sample was taken from the undeformed
end of the same specimen where S+T twin nucleation apparently occurred as a result of shear
transfer from prism ⟨𝑎⟩ slip. Only pyramidal ⟨𝑐 + 𝑎⟩ could account for the most plausible instances
of S+T twins in the interior of the microstructure. This observation is supported by the fact that
the texture of the specimen prior to deformation is highly favorable for pyramidal ⟨𝑐 + 𝑎⟩ slip,
as evident from the cumulative distribution plot for local and global Schmid factor (figure 4.2).
While grains that favor prism ⟨𝑎⟩ slip are not dominant, they do exist; as at least 11 grains in the
77
300
300
200
τRSS/MPa
0.85 0.9 0.95 1
200 basal
prism
pyramidal
pyramidal
T1 Twin
0.7 0.8 0.9 1.0
m'
Figure 4.11: Summary of twinning events in the 11 layers as a function of 𝑚′: Size of the markers are
proportional to the Schmid factor rank of the twin system. Filled circles (•) indicate geometrically
more plausible instances, where slip transfer may have resulted in nucleation of twins. Data from
a previous study (1.01,1.02 and 1.03 (Bieler et al., 2014) in table 4.2) are plotted as □. Cases with
SAF > 0.6 (𝜃 > 37°) are represented as filled colored markers, where the colors represent different
families of slip and twin systems. Among these, the cases with a highest likelihood of S+T twinning
with SAF > 0.8 (𝜃 > 53°) are shown in a darker gold color, while the lighter gold symbols represents
instances where 37°< 𝜃 < 53°. Prism ⟨𝑎⟩ is represented by red, basal ⟨𝑎⟩ by blue, pyramidal ⟨𝑎⟩
by green, pyramidal ⟨𝑐 + 𝑎⟩ by golden (transitions from lighter gold to a darker shade for more
plausible cases of S+T twinning); and T1 twinning by black. Inset shows the linear regression plot,
considering only the highly plausible ⟨𝑐 + 𝑎⟩ (dark gold) points.
interrogated volume deformed primarily by prism slip (Wang et al., 2017). None of these 11 grains,
however, had a neighboring grain with a detectable twin.
4.5.1.2 Stress evolution in parent grains
The evolution of the normal component of the grain averaged stress tensor (𝜎𝑧𝑧 ) for all the 13
twinning events is plotted in figures 4.12, 4.13 and 4.14, where the parent grain is represented
with a blue line. Because deformation twinning is a response to strain accommodation between a
grain and its neighbors during plastic deformation, it can be expected that the twinned grain will
experience a stress relaxation upon nucleation of a twin. Therefore, the hypothesis that formation
78
of a twin relieves a stress buildup can be checked. Furthermore, if slip transfer occurred to nucleate
a twin, the stress state in the neighboring grain should nominally track that of the twinned grain.
This forms the basis of hypothesis(i), where stress evolution in a twinned grain and a neighbor
are correlated. Conversely, if slip transfer did not take place, the reduction in stress in the parent
grain should be compensated by an increase in stress in the neighboring grain (hypothesis(ii),
anti-correlated stress).
Table 4.3 summarizes an assessment of these hypotheses. At the strain where twins nucleated,
a drop in the 𝜎𝑧𝑧 stress component values is apparent (and greater than the uncertainty) in layers 2
and 4; while an increase is observed in layers 9 and 10. In layer 4, where geometrical plausibility
exists for S+T twinning, the 𝜎𝑧𝑧 stress (shown in the figure) is lower in the parent grain than in
the high 𝑚′ neighbor grain. There is a drop in stress with twin formation; after which the stress
in the parent grain approaches that of the neighboring grain. This observation is consistent with
hypothesis(i) and is indicated by a filled dot in table 4.3. Conversely in layer 10, where S+T
twinning is also deemed plausible, the stress in the parent and neighbor grain are similar. There
was however, a marked increase in stress post twin formation, and a subsequent drop below the
neighbor grain, suggesting an anti-correlated behavior to hypothesis(i), and is marked with an ×
in table 4.3. In layer 2, where the grain geometry favors S+T twinning the stress in the parent
grain was between the stress of two neighboring grains; and a drop is observed at the point of twin
nucleation. Post twin formation, there was a sharp increase in stress for one of the neighboring
grains; while a steady increase was noted for the high 𝑚′ grain and the parent grain itself. With
respect to the second twin observed in layer 6 (at 2.2% bulk strain), where S+T twinning is highly
likely, an increase in stress is observed in the parent grain. Concurrently, a decrease is noted in
stress for the high 𝑚′ neighbor grain in an anti-correlated manner consistent with hypothesis(ii).
Therefore in table 4.3 it is marked with an ×. For the twinned grain in layer 5, stress relaxation
at the point of twin formation is concurrently marked by a sharp increase in stress in the high 𝑚′
neighbor grain. S+T twinning here is deemed to be implausible based on geometry and spatial
criteria, which is consistent with hypothesis(ii). Based on the analysis of the 13 twinning events
79
Table 4.2: Summary of Twinning Events Observed in 14 Layers in two specimens
Layer ε 𝑚 self 𝑚 ′ (SAF,Angle) 𝑚 neighbor σ(τ), (MPa)
1.01 1.2% lowest 0.95 pyr⟨𝑎⟩ (250)
1.02 1.5% 3rd highest 0.95 pyr⟨𝑐 + 𝑎⟩ (290)
1.03 1.5% lowest 0.96 basal⟨𝑎⟩ (250)
2.01 [0.49-0.55]
Same Layer 1.5% highest 0.96 (0.68, 43°) 0.55 pyr⟨𝑐 + 𝑎⟩ 442 (245)
Above Parent 1.5% highest 0.98 (0.48, 29°) 0.5 T1 442 (245)
2.02 [0.34-0.47]
Same Layer 2.2% 2nd highest 0.95 (0.94, 70°) 0.28 pyr⟨𝑎⟩ 460 (301)
Below Parent 2.2% 2nd highest 0.99 (0.53, 32°) 0.54 T1 460 (301)
Above Parent 2.2% 2nd highest 0.87 (0.99, 82°) 0.44 pyr⟨𝑐 + 𝑎⟩ 460 (301)
2.03 [0.49-0.54], [0.3-0.48]
Below Parent 1.89% 3rd highest 0.93 (0.8, 53°) 0.44 pyr⟨𝑐 + 𝑎⟩ 453(267)
Same Layer 2.73% 4th highest 0.95 (0.99, 82°) 0.56 pyr⟨𝑐 + 𝑎⟩ 474 (204)
2.04 [0.36-0.42]
Same Layer 1.5% 5th highest 0.98 (0.63, 39°) 0.47 pyr⟨𝑐 + 𝑎⟩ 442 (240)
Below Parent 1.5% 5th highest 0.94 (0.98, 79°) 0.47 pyr⟨𝑐 + 𝑎⟩ 442(240)
2.05 [0.31-0.54]
Below Parent 1.89% 4th highest 0.86 (0.43, 25°) 0.32 basal 456 (180)
Below Parent 1.89% 4th highest 0.93 (0.48, 29°) 0.29 pyr⟨𝑐 + 𝑎⟩ 456 (180)
2.06 [0.39-0.59], [0.49-0.60]
Same Layer 2.04% 2nd highest 0.99 (0.89, 63°) 0.45 pyr⟨𝑐 + 𝑎⟩ 456(233)
Below Parent 2.2% 5th highest 0.99 (0.87, 60°) 0.59 pyr⟨𝑐 + 𝑎⟩ 460 (241)
2.07 [0.49-0.56]
Same Layer 2.73% highest 0.93 (0.73, 47°) 0.46 pyr⟨𝑐 + 𝑎⟩ 474 (273)
Below Parent 2.73% highest 0.96 (0.68, 43°) 0.36 pyr⟨𝑐 + 𝑎⟩ 474 (273)
2.08 [0.37-0.55]
Below Parent 2.73% 6th highest 0.93 (0.43, 25°) 0.62 pyr⟨𝑐 + 𝑎⟩ 474 (218)
Below Parent 2.73% 6th highest 0.96 (0.32, 19°) 0.45 basal 474 (218)
2.09 [0.52-0.57]
Above Parent 1.76% 6th highest 0.99 (0.66, 41°) 0.64 T1 449 (223)
Above Parent 1.76% 6th highest 0.97 (0.88, 62°) 0.57 pyr⟨𝑐 + 𝑎⟩ 449(223)
2.10 [0.54-0.63]
Above Parent 1.5% 2nd highest 0.94 (0.95, 72°) 0.49 pyr⟨𝑐 + 𝑎⟩ 442(238)
Below Parent 1.5% 2nd highest 0.96 (0.71, 45°) 0.54 pyr⟨𝑐 + 𝑎⟩ 442 (238)
Above Parent 1.5% 2nd highest 0.95 (0.63, 39°) 0.51 basal⟨𝑎⟩ 442 (238)
2.11 [0.48-0.56]
Same Layer 2.73% 4th highest 0.97 (0.66, 41°) 0.61 T1 474 (210)
Below Parent 2.73% 4th highest 0.93 (0.68, 43°) 0.39 pyr⟨𝑐 + 𝑎⟩ 474 (210)
Layer Interactions with neighboring grains in same layer and layers above or below,
ε engineering strain where twin was first observed,
𝑚 self Schmid factor [range] and rank for observed twin,
𝑚 neighbor Schmid Factor of Neighbor Slip/Twin System,
σ global stress from stress-strain curve (MPa),
τ resolved shear stress on neighbor slip system (MPa).
Rows in boldface indicate most plausible instances of S+T twinning.
across the 11 layers of the interrogated volume, it is observed that hypothesis(i) holds true for the
instances of S+T twin nucleation; while hypothesis(ii) is not conclusive for the non S+T cases.
With increasing global strain, there occurs redistribution of stress between the parent grains
and their neighbors, leading to changes in the local stress tensor field. This effect is more apparent
80
600 z y Layer 1 (S+0) Layer 2 (S+T)
x
Neighbor Grain
400 High m' Grain
200
σzz / MPa
High m' Grain
0
Parent Grain
Neighbor Grain
-200
Parent Grain
-400 Error based on 0.0001 strain
conversion using c-axis modulus
(a) (b)
600 Layer 3 (S+T)
Layer 3 (S+T)
Parent Grain
400
200
σzz / MPa
High m' Grain Parent Grain
High m' Grain
0
-200
-400
(c) (d)
600 Layer 4 (S+T)
Layer 5 (S+0)
400
High m' Grain
200
σzz / MPa
High m' Grain Parent Grain
Parent Grain
0
-200
-400
0 1 2 3 0 1 2 3
εbulk / % εbulk / %
(e) (f)
Figure 4.12: Evolution of normal component of local stress tensor (along the tensile direction) for
parent and neighboring grains (layers 1, 2, 3, 4 and 5). Prisms indicate the Bunge Euler angle
orientation of the grains. Transition in background shade from gray to white indicates the bulk
strain at which the twin was identified.
81
600 Layer 6 (S+T)
Layer 6 (S+T)
400
200
σzz / MPa
0
Parent Grain High m' Grain High m' Grain
Parent Grain
-200
-400
(a) (b)
600
Layer 8 (S+0)
400
Layer 7 (S+0)
200
σzz / MPa
High m' Grain Parent Grain
0
Parent Grain High m' Grain Neighbor Grain
-200
-400
(c) (d)
600 Layer 9 (S+T)
Layer 10 (S+T)
400 High m' Grain
Parent Grain
Parent Grain
200
σzz / MPa High m' Grain
0
Neighbor Grain
-200
-400
0 1 2 3 0 1 2 3
εbulk / % εbulk / %
(e) (f)
Figure 4.13: Evolution of normal component of local stress tensor (along the tensile direction) for
parent and neighboring grains (layers 6, 7, 8, 9 and 10). Details on symbols and color conventions
are given in the caption in figure 4.12.
82
600 Layer 11 (S+0)
400
Parent Grain
200
σzz / MPa
High m' Grain
0
-200
-400
0 1 2 3
εbulk / %
Figure 4.14: Evolution of normal component of local stress tensor (along the tensile direction) for
parent and neighboring grains (Layer 11). Details on symbols and color conventions are given in
the caption in figure 4.12.
Table 4.3: Summary of 13 twinning events and their stress history correlation with respect to
neighboring grains.
Layer 1 2 3 4 5 6 7 8 9 10 11
1st 2nd 1st 2nd
Mode S+0 S+T S+T S+T S+T S+0 S+T S+T S+0 S+0 S+T S+T S+0
Parent Δ𝜎 - ↓ ↓ ↓ ↓ ↓ ↓ ↑ ↓ ↓ ↑ ↑ ↑
Correlation ◦ ◦ ◦ • • x • x • x • x x
1st /2nd first/second twin in current layer
◦ correlated 𝜎𝑧𝑧 , with one neighbor
• correlated 𝜎𝑧𝑧 , with highest 𝑚 ′ neighbor
x anti-correlated
in figure 4.15, where the von Mises stress is plotted in conjunction with a metric that shows the
deviation of local stress from ideal uniaxial tension (expressed by the color of the symbols in the
plots). The deviation from a global uniaxial tensile stress state is assessed using a dimensionless
parameter (cos 𝜃), based on the work done by Schuren et al. (2015). The cos 𝜃 parameter is the
scalar product of the normalized vector containing the six symmetric components of the local grain
averaged stress tensor, and the the six component unit vector denoted ideal global uniaxial tension.
A value of +1 means that the local stress state is coincident with ideal uniaxial tension; whereas
−1 denotes an anti parallel (uniaxial compression) state. In layers 1, 3 (first observed twin), 4 and
9, the deviation of the local stress state from ideal uniaxial tension follows a similar trend. At the
start of the loading process, the stress state is more compressive, and it traces an increasing tensile
83
z Layer 1 (S+0) Layer 2 (S+T)
y
600 Parent Grain
x
σVM / MPa
400 High m' Grain High m' Grain
Neighbor Grain Neighbor
200 Parent Grain Grain
Typical error in plastic zone
0
Layer 3 (S+T)
Layer 3 (S+T)
600
σVM / MPa
400
Parent Grain Parent Grain High m' Grain
High m' Grain
200
0
Layer 4 (S+T) Layer 5 (S+0)
600
σVM / MPa
400
High m' Grain
Parent Grain High m' Grain
200
Parent Grain
0
Layer 6 (S+T) Layer 6 (S+T)
600
σVM / MPa
400
Parent Grain High m' Grain
High m' Grain Parent Grain
200
0
Layer 7 (S+0) Layer 8 (S+0)
600
σVM / MPa
400
Parent Grain High m' Grain Neighbor Grain
Parent Grain High m' Grain
200
0
Layer 9 (S+T) Layer 10 (S+T)
600
σVM / MPa
400
Parent
Parent Grain High m' Grain Neighbor Grain Grain High m' Grain
200
0
0 1 2 3 0 1 2 3
εbulk / % εbulk / %
Layer 11 (S+0)
600
σVM / MPa
400
Parent Grain High m' Grain
200
0
0 1 εbulk / % 2 3
-1 cos(θ) = σ·U 1
Figure 4.15: Evolution of local average von Mises stress 𝜎𝑣𝑀 in identified parent and selected
neighbor grains across the 11 layers. The load steps corresponding to twin identification is
indicated by the gradual transition in background color from gray to white. The measure of local
stress deviation from ideal uniaxial tension is given by cos 𝜃 = 𝜎.𝑈.
84
trajectory with load. Upon unloading, the stress state switches back to a compressive configuration.
On the other hand, the stress state of the grains in layers 7 and 11 do not exhibit any compressive
character at all. In these two layers, the stress state tends towards a strong tensile character with
increasing load, and the degree of which abates gradually with unloading. Similarly, in layer 5, the
stress state switches from compressive to a strongly tensile character with increasing load, before
falling back to a compressive state upon unloading.
The above observations are consistent with the loading history of the specimen. As mentioned
earlier, the tensile specimen was cut out of a larger sample used in a four point bending experiment.
It was also determined in an earlier study (Wang et al., 2014) that several grains in the microstructure
had a strong compressive residual stress state. This is probably a consequence of prior deformation
in the ends of the four point bending experiment of the source sample (Wang et al., 2010b). It is
however, unlikely that this played a significant role in nucleation of discrete twin during the tensile
loading. Clearly, grains that had a strong compressive residual stress state prior to the tensile
experiment did not register a very strong tensile character with increased load. On the other hand,
grains that did not have a compressive residual stress state to begin with developed a strong tensile
character with increasing load.
To summarize, several expected features associated with slip transfer induced twin nucleation
can be assessed using information obtained from a FF-HEDM experiment. Parallel testing of
hypotheses (i) and (ii) can be made for the parent and slip transfer (high 𝑚′) grains for each
twinning event. The correlated stress argument looks most consistent for the twin in layer 4, while
not as convincing for layer 9. The stress evolution for the twinned and slip transfer grains in layer
10 is anti correlated with hypothesis (i), suggesting that while conditions favoring S+T twinning
exist, the twin likely formed by a different mechanism. For the cases that are geometrically and
spatially less likely, there is mixed agreement with hypothesis (ii). Nevertheless, slip transfer does
appear to be a likely, but probably not dominant mechanism for twin nucleation.
85
4.6 Comparison of FF 3D XRD results with surface EBSD mapping
An important question that arises for FF characterization, is whether twin-parent pairs identified
can also be matched to surface EBSD maps. In this section, results from the comparison between FF
and surface EBSD measurements are discussed. The objective of this comparison is to determine
whether COM and orientation information obtained from FF analysis matches the surface EBSD
data taken from the gage section of the specimen. Several discrete twin-parent pairs were identified
on the surface of the specimen in the orientation map in figure 3.4. Three of these twins were
also detected in the FF-HEDM analysis, as shown in figure 4.16. The top row shows the inverse
pole figure and grain average orientation deviation (GROD) maps respectively, of the EBSD
measurement spanning the gage section of the specimen. The surface considered is opposite to
the side facing the incoming x-ray beam. Figure 4.16 (a) shows the comparison of the FF analysis
specific to layer 1 with the surface EBSD map. The same comparison for layers 10 and 11 is
shown in Figure 4.16 (b). In each case, the prisms shown in green denote the COM positions and
orientations of the grains identified using FF analysis. The matching grains on the surface EBSD
map are identified using black prisms. The third row of (a) and (b) shows the map of calculated
misorientation between the matched grains in FF and EBSD measurements.
It is apparent that the FF grain orientations are not positioned in the same spatial relationship
as the EBSD maps, suggesting uncertainty in the FF positions. It is worth noting however, that
the EBSD map is a 2-D slice, while the COM from FF represents the full volume of the grain, so
it is not likely that the positions would agree perfectly. It is generally known, but not commonly
quantified that the agreement between COM and 3-D sectioned grains are in better agreement near
the center than on the periphery (Park et al., 2017), due to the fact that peak intensity from grains
in the center are more uniformly detected than grains on the periphery, leading to fewer diffraction
peaks available for identification of the grain. Furthermore, it is likely that many of the actual
twins may not have been large enough to be detected by 3D XRD with a certainty of 5°or less. In
addition, the strong orientation gradients present in the specimen (figure 4.16, top left) could also
contribute to the uncertainty of the COM position in the FF data.
86
(a)
10º
Z
X
(b) 0º
Tensile Axis IPF Grain Average Orientation Deviation
(a) FF COM positions, orientations (b) FF COM Positions, orientations
T
P
P
T
T P
T P
P
T
T P
T P
P
T
T P
0 5 10
Figure 4.16: The top row shows the tensile sample and the region from which EBSD measurements
were made on the surface between the sample and the detector. The top right figure shows the
grain reference orientation deviation map where the reference is the average grain orientation in
the EBSD data. Two different regions with twins (labeled a and b) are shown. in the second and
third rows. The green prisms show the relative positions of grain centers of mass from the far field
data (FF) in which twins were detected, and just beneath them, the enlarged corresponding part of
the EBSD maps; the green and black prisms are not in good spatial agreement. Beneath the EBSD
scans, misorientation maps between the FF COM orientation and the 2-D EBSD grain orientations
show agreement within 10°.
The misorientation between the FF grain averaged orientation and each pixel in the corre-
sponding EBSD grains are plotted in the lower row of figure 4.16, where the color represents the
misorientation from the reference FF orientation. The grains with larger twins have orientations
that agree within 5° with respect to the two measurements. The misorientation range of some of
the neighboring grains have an orientation spread between 3° and 7°.
The orientation discrepancy was investigated to determine whether this spread in misorientation
87
could be attributed to a systematic error between the coordinate systems in the EBSD and FF XRD
experiment, but no systematic difference in the directions of crystal X, Y or Z directions were
identified. This implies that the differences in orientation also reflect the fact that the FF-HEDM
measurement is an average of all voxels in the grain; while the pixels in the EBSD measurement
identify a range of orientations present in only one plane of the grain. Because the actual orientations
are not spatially resolved within the grains, it is not possible to compare the two orientation
measurements in a one-to-one manner. Nevertheless, there is sufficiently good agreement between
the two measurements to be convinced that the same grain was identified with both methods.
4.7 Cross validation of local stress tensor calculations and grain positions
between Fable and MIDAS
In order to ensure consistency of the relative grain positions that were taken into account to
determine favorable conditions for S+T, the FF results were also analyzed using MIDAS. Also,
a near field (NF) scan was made on the deformed sample, but it was dismounted, so that the
absolute position of the sample in the goniometer was lost. A couple layers of the near field data
set were processed, but not sufficient to compare with far field data. During Summer 2020, near
field data were measured again and analyzed, and the centers of mass can now be compared with
the FF data for the final unloaded state in the next chapter. Also during Summer 2020, the FF
data were reanalyzed using a more advanced post processing package called MIDAS. The results of
comparison of the grain averaged stress tensor and grain centers of mass between Fable and MIDAS
are discussed in this section. Details of comparison of the FF data and the recently completed NF
analysis corresponding to the final unloaded state are discussed in chapter 5.
The centers of mass and local strain/stress tensors of grains indexed using Fable were compared
to the results obtained using MIDAS and a near-field measurement that provides the shapes and
boundary locations of each grain. At the unstrained state ∼ 90% of the grains indexed with Fable
match with the corresponding indexed results obtained using MIDAS. At the final unloaded state,
the match for FABLE indexed grains is only about 75%. The number of grains obtained using
88
MIDAS at the final state is almost double, indicating that MIDAS indexing algorithm is more
sensitive to sub-grain formation than Fable.
In order to analyze the variation of indexation results between Fable and MIDAS, the Euclidean
distance and misorientation (<=5°) between each pair of matched grains is used as a metric of error.
The distribution of the Euclidean distance for the matched grains, corresponding to two loading
states (unstrained prior to tensile loading, and final unloaded state) is shown in figure 4.17.
160
140
120
Number of grains
100
80
60
40
20
0
0 50 100 150 200 250 0 50 100 150 200 250
1
With Correction
0.9 Without Correction
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 50 100 150 200 250 0 50 100 150 200 250
Euclidean Distance/ m
Unstrained state Final unloaded state
Figure 4.17: Histogram of Euclidean distance for grains consistent with indexation using Fable and
MIDAS (top row), and cumulative distribution plot of the same grains (bottom row), for unstrained
(prior to loading) and final unloaded states respectively. The top right plot also shows the histogram
of Euclidean distance for grains consistent between FF analysis (MIDAS) for the final unloaded
state.
The error between the centers of masses is found to be systematic along one specific direction.
89
Therefore a correction vector is applied to the initial vector joining the centers of masses of each
matched pair of grains. With respect to the unstrained condition, approximately 88% of the matched
grains have Euclidean distances less than 100𝜇𝑚 (below the average grain size). The error is seen
to be higher at the final unloaded state as 75% of the matched grains have Euclidean distance below
the average grain size.
Upon applying the correction vector, the Euclidean distance follows a lognormal distribution for
both the unstrained (prior to tensile loading) and final unloaded states, as seen from the histograms
in figure 4.17.
Figure 4.18 shows cumulative distribution function (CDF) plot of grain radius for the unde-
formed and final unloaded states respectively. The grain radii decrease uniformly in the final
unloaded state. This can be attributed to an increase in volume fraction of twins in the aggregate, as
well as development of low angle boundaries within grains. Upon closer observation of the lower
tail of the CDF, it is evident that features of the distribution are preserved in the deformed material
(e.g. lower arrows). In the initial state, there is a notable change in slope at the upper arrows, which
is still there but less apparent in the deformed state, possibly indicating division of grains due to
twin formation.
The local grain averaged stress tensor calculations described in section 4.1.1 were compared
with those made using MIDAS. The magnitude of the mean difference in equivalent stress for the
matched grains obtained using the two approaches is ∼ 60 MPa (∼ 13% of the 450 MPa flow stress).
Figure 4.19 shows the cumulative distribution plots of strain/stress evolution for the matched grains,
corresponding to four different load states. Overall, the local equivalent strain and stress values
obtained from the two methods closely track each other, but the values of strains (and consequently,
stresses) are systematically higher with the MIDAS analysis prior to plastic deformation at low
elastic strains.
From the above observations, it is clear that the relative positions of the grains indexed using
Fable are consistent with the results obtained using MIDAS, especially at the unstrained condition
prior to tensile loading. This is important in terms of ensuring consistency of the relative grain
90
1
Final unloaded state
Initial State prior to deformation
0.8
F(x)
0.4
0
0 10 20 30 40 50 60
Grain Radius/ m
Figure 4.18: Cumulative distribution function (CDF) plots of grain radii estimated by MIDAS for
the undeformed and final unloaded states respectively. Inset shows a magnified view of the tail of
the CDF plots. There is a significant change in the slope towards the tail of curve, going from the
state prior to deformation to the final unloaded state.
positions considered to assess possibility of S+T twinning. Furthermore, as seen from the compar-
ative cumulative distribution plots in figure 4.19, the local equivalent strain/stress values obtained
from the two methods closely track each other.
4.8 Evaluation of identified twins using Near Field HEDM
As mentioned in the previous section, NF data was collected from the dismounted tensile
specimen. One advantage of the NF setting is the high spatial resolution, which is at least an
order of magnitude better than FF HEDM. For a detailed analysis of positional error between the
91
Unstrained state Pre-yield (75% of yield) Maximum load Final unloaded state
1
Fable
0.9
MIDAS
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -4 10 -3 10 -4 10 -3 10 -4 10 -3 10 -4 10 -3
Equivalent Strain
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
400 600 800 400 600 800 400 600 800 400 600800
Equivalent Stress/MPa
Figure 4.19: Cumulative distribution function plots for equivalent strain (top row) and von Mises
stress (bottom row) evolution, showing the differences between two indexing methods. Strain and
stress evolution are shown for four different load states: unstrained state prior to tensile loading,
75% of bulk strain prior to yield, maximum load and final unloaded state.
FF and NF HEDM datasets, the reader is referred to chapter 5. This enables the visualization of
the morphology of parent, twin and neighboring grains. In this section, the twinning events from
section 4.5.1, are re-examined in light of the NF data.
Figure 4.20 shows two projections: perspective (left) and XY plane (right), where the tensile
axis is parallel to the Z axis. Several T1 twins were identified based on the 85°c-axis misorientation
from the data, more than the 13 sets identified from FF HEDM. In fact, twins were found in 34%
of the grains.
For the FF data, the twinned/neighboring high 𝑚′ grains were examined using conditions of
geometric compatibility and relative positions in 3D space. As the NF data contains information
about the grain morphology, it is possible to identify neighboring grains that have interfacial contact
with the twin, thereby providing more clarity in terms of identifying candidate S+T grain pairs.
To check if pyramidal⟨𝑐 + 𝑎⟩ slip is responsible for plastic deformation in a grain, the evolution
92
100μm
Figure 4.20: Perspective (left) and XY plane (right) projections for the gage volume interrogated by
NF HEDM, shown with IPF colors. The data corresponds to the final unloaded state in the tensile
experiment (after dismounting of the specimen from the tensile stage). The tensile direction was
along the Z axis.
of the plastic spin axis associated with a given slip system can be assessed as a function of strain.
According to this hypothesis, plastic spin axis corresponding to the slip system most responsible
for plastic deformation should have the least relative misorientation. For this analysis, the plastic
spin axis for each slip system is calculated by taking the cross product of the slip plane normal and
Burgers vector and tracking its misorientation with respect to the axis orientation prior to tensile
deformation.
Taking the information about grain morphology from the NF HEDM data, the first twinned
grain considered in figure 4.10 is considered. When the shape of the twin formed is examined, it is
found that it touches the surface of the same neighboring grain that was identified as the ’triggering’
grain in FF HEDM. This is shown on the top left portion of figure 4.21.
The bottom left plot in figure 4.21 shows the evolution of the disorientation of the neighboring
grain as a function of bulk strain, along with the c-axis misorientation evolution. If pyramidal⟨𝑐 + 𝑎⟩
is indeed the driver of crystal rotation, then there should be less change in the c-axis disorientation
with increasing strain than the grain disorientation, because the rotation axes of ⟨𝑐 + 𝑎⟩ slip
are between the c-axis and the basal plane). There is an increasing deviation between the grain
disorientation and the c-axis disorientation starting at a strain a little below the value where the
93
400
350
P 300
m′= 0.96
T 250
/MPa
200
RSS 150
N
100
50
P: Parent (twinned) grain
N: Neighbor to twinned grain 0
T: Twin
-50
3 800 2
disorientation
Misorientation of Plastic Spin Axis/degrees
c-axis misorientation 1.8
700
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
1.4
2
500
1.2
/MPa
Degrees
Twin identified at this loadstep
1.5 400 1
vM
0.8
300
1
0.6
200
0.4
0.5
100 0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure 4.21: Analysis of the first parent, twin and high 𝑚′ grain from figure 4.10. NF HEDM
provides a more precise relative position and information on grain shape. Bottom left: plot of the
c-axis disorientation of the neighboring grain as a function of bulk strain, along with the evolution
of equivalent stress in both parent and neighboring grains. Top right: Evolution of resolved shear
stress for 4 families of slip in the neighboring grain, and bottom right: misorientation of the plastic
spin axis for each of the 12 pyramidal⟨𝑐 + 𝑎⟩ slip systems as function of bulk strain. The color
scheme used for the 12 pyramidal⟨𝑐 + 𝑎⟩ slip systems is the same for the top right and bottom right
plots. The symbol and color scheme for the slip systems is indicated in the inset text.
94
twin was observed.
The top right plot in figure 4.21 shows the evolution of the resolved shear stress of all four families
of slip systems (basal⟨𝑎⟩, prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩) in the neighboring grain
as a function of bulk strain. The resolved shear stresses of the pyramidal⟨𝑐 + 𝑎⟩ slip family are
much higher than the other three families, further supporting the likelihood of pyramidal⟨𝑐 + 𝑎⟩
slip activity. The pyramidal⟨𝑐 + 𝑎⟩ slip system that has a high 𝑚′ value with respect to the activated
twin variant in the parent grain is indicated in this plot by the ’∇’ symbol.
In the lower right plot of figure 4.21, the evolution of the plastic spin axis misorientation
from the initial crystal orientation for all 12 pyramidal⟨𝑐 + 𝑎⟩ systems in the neighboring grain
is plotted with strain. For each load step, the misorientation of the plastic spin axis is calculated
by comparing with the unstrained orientation (prior to tensile loading). Starting around the load
step of twin identification, the misorientation of the plastic spin axis changes the least for the slip
system with high 𝑚′ relationship with respect to the active twin variant in the parent grain. This
least change in the rotation axis is consistent with the greatest activity of this slip system.
Another example of a twinned grain from the FF HEDM analysis is examined in the context of
NF HEDM data (The third example from figure 4.10). In this case, the neighboring grain in contact
with the twin surface is not the same grain as was identified in FF HEDM. The relative positions of
the parent, twin and neighboring grains for this example, along with the relevant plots are shown
in figure 4.22.
In this case, the disorientation of the grain is very small, suggesting balanced activity of slip
systems that result in little orientation change. However, the slip system with the highest 𝑚′
relationship with respect to the active twin variant does have the highest resolved shear stress, but
it also has the greatest misorientation of the plastic spin axis, from which it is difficult to conclude
that its activity was primary. Indeed, the slip system that would balance the rotation caused by this
system is the light orange system (directly below the one with the dashed line in the key), and it
has the second highest resolved shear stress and its rotation axis tracks very closely with the purple
system.
95
400
350
N 300
250
200
/MPa
m′= 0.93
150
RSS 100
P T
50
0
P: Parent (twinned) grain -50
N: Neighbor to twinned grain -100
T: Twin -150
3 800
2
disorientation
Misorientation of Plastic Spin Axis/degrees
c-axis misorientation 700 1.8
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
2 1.4
500
1.2
/MPa
Degrees
Twin identified at this loadstep
1.5 400 1
vM
300 0.8
1
0.6
200
0.4
0.5
100 0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure 4.22: Revisit of the third parent, twin and high 𝑚′ grain from figure 4.10. NF HEDM
enables a more precise relative position and information on grain shape. Bottom left: plot of c-axis
disorientation of the neighboring grain as a function of bulk strain, along with the evolution of
equivalent stress in both parent and neighboring grains. Top right: Evolution of resolved shear
stress for 4 families of slip in neighboring grain, and bottom right: misorientation of the plastic
spin axis for each of the 12 pyramidal⟨𝑐 + 𝑎⟩ slip systems as function of bulk strain. The color
scheme used for the 12 pyramidal⟨𝑐 + 𝑎⟩ slip systems is the same for the top right and bottom right
plots. The symbol and color scheme for the slip systems is indicated in the inset text.
96
Each of the 13 twinned grains were examined in light of the NF HEDM characterization done
for the final unloaded step. In two of the instances (twins in layers 5 and 8, as identified during
the FF HEDM analysis), it was not possible to identify the twins in the corresponding NF HEDM
data. This could be a result of detwinning during the unloading process or very small twin volume
fraction within the parent grains. In case of four others (twins in layers 2, 3 and the first instance in
layer 6 identified in the FF HEDM analysis), the twin was embedded inside the parent grain without
contacting any neighboring grain. Therefore a total of 7 twinning events were examined using the
approach outlined in figures 4.21 and 4.22.
Similar to figure 4.11, the data from the 7 twinning instances examined in light of the NF HEDM
data are plotted in figure 4.23 (solid circles) with the data from figure 4.11. Only the points where
pyramidal⟨𝑐 + 𝑎⟩ has high 𝑚′ with respect to the activated twin variant are shown. As in 4.11, the
points are scaled in proportion to the Schmid factor rank of the active twin variant. There are three
points that match with the previous data, as shown by the translucent points from NF that coincide
with the FF points. It can be observed that the more confident NF values are in the same range as
those observed with the FF analysis. The difference with respect to the other points is because of
the choice of different neighbor grains based on grain position and morphology information from
NF HEDM. From the standpoint of grain position and geometrical alignment of pyramidal⟨𝑐 + 𝑎⟩
slip system with the active twin variant, all look to be plausible candidates for slip transfer for
the latter seven cases. However, it cannot be said with certainty that the twins observed indeed
nucleated as a result of pyramidal⟨𝑐 + 𝑎⟩ slip transfer.
It can therefore be seen that it is important to quantify the uncertainty associated with grain
positions in the FF HEDM analysis. Moreover, in order to make an assessments of complex
mesoscopic loading situations such as S+T twinning, it is critical to understand the limits of
grain averaged stress measurement. A detailed analysis of results from matching (grain position
and crystallographic orientation) FF HEDM data with corresponding NF HEDM data is made in
chapter 5. Additionally, a quantitative comparison of kinematic (lattice rotation) and grain averaged
stress measurements of the FF HEDM technique is made in chapter 6.
97
300
Resolved Shear Stress/MPa
200
pyramidal
0.7 0.75 0.8 0.85 0.9 0.95 1
Slip Transfer Parameter(m' )
Figure 4.23: Resolved shear stress plot as a function of 𝑚′ for the seven twinning events validated
with NF HEDM data plotted in the same manner as figure 4.11. Here, all of the slip systems with a
favorable geometric compatibility with respect to the active twin variant in the parent grain are of
pyramidal⟨𝑐 + 𝑎⟩ type. The data points from the FF HEDM analysis that correspond to high 𝑚′
relationship with respect to the active twin variant are indicated by ’𝑥’. The size of the markers are
proportional to the Schmid factor rank of the active twin variant. The NF points that coincide with
the FF results are indicated by translucent circles.
98
CHAPTER 5
A COMPARATIVE STUDY OF THE ACCURACY OF FF HEDM METHOD WITH
RESPECT TO NF HEDM
In this chapter, FF HEDM data on grain positions and orientation are compared with corre-
sponding data from the NF HEDM experiment. As explained in chapters 2 and 4, the uncertainty
associated with grain positions in 3D space using FF HEDM is relatively high (of the order of
10𝜇𝑚). Therefore a systematic quantification of the positional error becomes important for valida-
tion of the slip transfer analysis conducted in chapter 4. In this context, it is known that NF HEDM
is able to determine grain positions with a better uncertainty than FF HEDM by at least an order of
magnitude. Hence, comparison of the grain positions using both FF and NF HEDM techniques is
a good approach for cross validation.
The NF HEDM data used for comparison in this study was collected for the load step corre-
sponding to the final unloaded condition, after the specimen was removed from the stage. As such,
the NF HEDM data was measured seven years after the FF HEDM characterization was completed.
As no changes in the room temperature microstructure are expected, the NF dataset serves as a
useful benchmark for comparison of the FF HEDM data.
The different sources of position and orientation errors for the FF and the NF datasets are dis-
cussed. Details of an algorithm used to compare the 3D positions and crystallographic orientations
of the FF and NF datasets are laid out. Additionally, a comparison of size estimations of matched
grains (FF with NF) is made for each technique, using the grain radius as a metric.
5.1 Classification of errors from the FF and NF HEDM experiments
The types of the errors present in identifying grain positions and unique crystal orientations using
FF and NF HEDM methods can be classified into the following three categories. In this chapter,
the terms ’centers of mass’ and ’centroids’ have the same meaning. A schematic representation of
the error classification is shown in Figure 5.1.
99
Figure 5.1: Classification of errors from the FF and NF HEDM experiments.
1. Intrinsic error from the FF measurements (𝛿): This error quantifies the uncertainty associated
with grain position (center of mass) determination in FF HEDM. Based on literature, this
error is ∼ 10 𝜇m (Park et al., 2017).
2. Intrinsic error from the NF measurements (expressed in terms of confidence index, 𝐶 𝐼): de-
fined as the fraction of simulated Bragg peaks from a data point that matches the experimental
scattering data. This term is a function of both the position in 3D space as well as crystal
orientation. A simulated Bragg peak from a data point (voxel) is considered to be qualified
if it intercepts the detector at multiple positions. For a crystal orientation to be considered,
the number of qualified peaks (𝑁 𝑞𝑢𝑎𝑙 ), is a function of 𝑄 𝑚𝑎𝑥 = 𝑚𝑎𝑥|𝑔 ℎ𝑘𝑙 |, which imposes a
limit on the set of {ℎ 𝑘 𝑙} and number of diffraction peaks considered (Li and Suter, 2013).
The resultant value of 𝐶 𝐼 is a floating point number between 0 and 1 for each NF data point
(Li and Suter, 2013; Turner et al., 2016).
3. Extrinsic error, which relates the NF and FF datasets: This can be subdivided into two cate-
gories. The first is the rotation necessary to bring the crystal orientation of the corresponding
FF and NF grains into coincidence. The second category is the translational error between
respective centroids in the NF and FF datasets, quantified using Euclidean distance (Δ).
5.2 Estimation of grain volume
Sharma et al. (2012b) have shown that the volume of an indexed grain (𝑉𝑔 ) in FF HEDM can
be estimated using the following relation.
100
𝑚 ℎ𝑘𝑙 𝐼𝑔
𝑉𝑔 = 𝑉𝑔𝑎𝑔𝑒 ΔΘ (5.1)
2 𝑘 𝐼𝑝
ΔΘ = 𝑎𝑟𝑐𝑠𝑖𝑛[sin 𝜃 cos(Δ𝜔 + cos 𝜃| sin 𝜂| sin(Δ𝜔)] − 𝜃 (5.2)
where, 𝑚 ℎ𝑘𝑙 : multiplicity of crystallographic plane, 𝑘: number of diffraction images in which
the diffraction spot was present, 𝑉𝑔𝑎𝑔𝑒 : illuminated volume, 𝐼𝑔 : integrated intensity of diffraction
spot, 𝐼 𝑝 : intensity per diffraction ring on the image if the sample were a powder.
Assuming a spherical shape, the grain radius can then be estimated as follows.
√︂
3 3𝑉𝑔
𝑅𝑔 = (5.3)
4𝜋
1.00
All data
≥ 0.1
≥ 0.2
≥ 0.3
cumulative probability
≥ 0.4
0.75 ≥ 0.5
0.50
0.25
0.00
0 0.5 1
Confidence Index
Figure 5.2: Maps showing distribution of the confidence index (CI) in the interrogated microstruc-
ture projected on the XY plane (looking down along tensile axis over the entire interrogated volume).
Cumulative distribution plots of CI (unfiltered data and with different threshold values) are shown
in bottom left. The cumulative distribution curves converge at a CI value ∼ 0.95.
101
5.3 NF HEDM: Reduction of data on the basis of Confidence Index
The purpose of this step is to filter out voxels that constitute noise in the NF HEDM dataset.
Figure 5.2 shows maps illustrating the spatial distribution of the confidence index (CI) for the gage
volume (the projection shown is on the XY plane, looking down along the tensile axis). Using
thresholding values in an iterative manner (in this case, they are arbitrarily chosen in increments of
0.1), the noise threshold in the NF data can be identified. The cumulative distribution curves for the
reduced data with different threshold values from 0 to 0.5 converge at a CI value of approximately
0.95. By inspection, if a CI threshold greater than 0.5 is used, significant voids are created in the
microstructure, indicating a loss of meaningful data points. Therefore, a cut-off CI value of 0.5
was chosen to filter the dataset. This resulted in a 21% reduction of data points: from ∼ 8.9 × 106
voxels to ∼ 7 × 106 voxels.
5.4 Matching of centroids between NF and FF HEDM datasets
As mentioned in section 5.1, there are two sources of extrinsic variation between the FF and
NF datasets–in orientation space and in real 3D space. In this study, coincidence of the datasets in
orientation space was accomplished by a counter clockwise rotation of the FF data by 20°about the
Z axis. Matching of centroids between FF and NF was done by considering both crystal orientation
and position in 3D space.
5.4.1 Data reduction
Prior to the matching process, it is important to consider the difference in size between the two
datasets. The number of points (indexed voxels with 3D positions and orientations) for the NF data
is three orders of magnitude larger than the corresponding FF data. Therefore, it is necessary to
reduce the size of the NF data in order to make a meaningful comparison. Accordingly, the average
3D coordinate and average orientation for each grain in the NF data was calculated. This results in
a comparable number of data points in the both FF (1702) and NF (1562) datasets. It is pertinent to
mention here that during the averaging process of the NF data, a grain that is split by an embedded
102
twin is considered to be a single grain instead of two separate grains.
Two approaches for calculating the average crystallographic orientation were used for data
reduction. The first approach, is based on the methodology outlined in the work by Cho et al.
(2005),where the list of orientations are reduced to the fundamental zone and converted to quater-
nion space. Then the average orientation is given by a resultant quaternion comprising the arithmetic
mean of each individual component of the input orientations. One drawback of this averaging ap-
proach is that symmetrically equivalent orientations in the boundaries of the fundamental zone can
be considered highly disoriented, leading to erroneous results for average orientations.
The Python based module for calculating average orientation available under DAMASK has an
optimized algorithm that accounts for symmetrically equivalent orientations. For the reduced FF
and NF HEDM datasets, the results using both approaches yield the same result with the maximum
disorientation between the two methods not exceeding 3°. Figure 5.3 shows the cumulative
distribution function plot of the disorientation between the two orientation averaging approaches
for the reduction of the NF dataset. Here the same population of voxels are averaged: first using
the quaternion averaging method, and then the DAMASK averaging algorithm. At least 90% of
the points have disorientations ≤ 1.5°. It should also be noted that no apparent difference was
observed in matched grains between the NF and FF datasets using either method of averaging
crystal orientation.
5.4.2 Algorithms used for matching grains between FF-NF datasets
Two different approaches were used to search for matching grains between the two datasets.
The two (forward and reverse) algorithms are summarized in figures 5.4 and 5.5 respectively. In
both cases, a point cloud of NF centroids around each FF data point is considered. Once a NF point
is matched to a FF grain, the NF voxel list is updated, so that the same voxel is not considered when
matching another FF grain.
In the first approach (figure 5.4), the misorientation of a given FF data point is calculated with
respect to the entire set of NF data points. Then a misorientation tolerance criterion is applied, i.e.
103
Distribution of disorientation in reduced NF dataset
(average orientation calculated from mean quaternion v/s DAMASK algorithm) Point cloud of reduced NF data with IPF colors
1.00
cumulative probability
0.75
0.50
Z
Y
0.25
X
0.00
0 1.5 3
Disorientation (degrees)
Figure 5.3: Cumulative distribution plot of the disorientation between two different grain aver-
aging techniques for the reduction of the NF data. The same group of orientations are averaged:
the first approach being simple quaternion averaging, and the second using the more optimized
grain averaging algorithm available in DAMASK. The maximum disorientation between the mean
quaternion approach and the DAMASK algorithm is less than 3°. At least 90% of the points have
a disorientation of 1.5°or less. The point cloud of the reduced NF data is shown on the right with
IPF colors denoting crystal orientation with respect to the Z axis.
the list of NF data points with a misorientation less than or equal to the tolerance is generated. If
this list is non-empty, then the Euclidean distances between the FF point and points in this reduced
list are calculated. The NF data point in this list that has the smallest Euclidean distance is chosen
to be the matching grain. This process is iterated over the entire set of points in the FF dataset.
In the second approach (figure 5.5), the nearest neighbor point cloud using a specified ball
radius is generated first. The misorientation tolerance is then applied, and if the list of filtered
points is non-empty, the point with the smallest Euclidean distance is chosen as the matching grain.
In order to assess whether the results of the matching process would be different in each
approach, the datasets were analyzed using both forward and reverse algorithms.
The Euclidean distance distributions for the matching grains using the two different algorithms
are shown in figure 5.6. For both cases, a misorientation tolerance of 5°is applied. Clearly, the
results of the matching operation is independent of the type of algorithm (forward or reverse) used
to implement it. The CDF plot for a smaller subset of the data, i.e. only those centroid pairs with
104
Misorientation
tolerance
Far field dataset Near field dataset
VFFi
VFF = (VFF1, . . . VFFm) VNF = (VNF1, . . . VNFn)
Generate list of NF
voxels satisfying
misorientation criterion
Move to nextFF voxel
Update NF voxel list
No
Non-empty list?
Yes
Calculate Euclidean
distance
Reduced list Ball radius
min(Euclidean distance)
Matched pair
(VFFi, VNFk )
Figure 5.4: Forward algorithm used for matching grains between the FF and NF datasets, where
the misorientation criterion is applied before the ball radius.
FF-NF Euclidean distance of ≤ 200 𝜇𝑚 is shown in the inset on the right hand side of figure 5.6.
5.4.3 Grain size comparison between matched grains in FF/NF datasets
In addition to position and crystal orientation, an important consideration in the matching
exercise is the grain size. This involves examining how well the reported grain radii from FF data
match the grain sizes from the NF data. In order to make an equivalent comparison, the sizes of
the matched grains in the NF dataset are expressed in terms of grain radius as follows.
Figure 5.7 shows the arrangement of voxels in a 2D slide of the NF HEDM dataset. A series of
lines can be drawn to reduce it to a hexagonal grid arrangement. If a regular hexagonal comprising
six voxels is considered, the distance between two adjacent voxels is ∼ 2.88𝜇𝑚. The series of lines
is drawn such that each voxel is encloses in 2D space by an equilateral triangle.
105
Specify ball radius
Far field dataset Near field dataset
VFFi
VFF = (VFF1, . . . VFFm) VNF = (VNF1, . . . VNFn)
Generate pointcloud
Move to nextFF voxel
No Misorientation
Non-empty list?
tolerance
Update NF voxel list
min(Euclidean distance)
Yes
Matched pair
(VFFi, VNFk )
Figure 5.5: Reverse algorithm used for matching grains between the FF and NF datasets, where the
ball radius criterion is applied first and the misorientation threshold is applied in the second step.
The length of a side of the enclosing equilateral triangle, 𝐿 is given as follows.
2.88 + (0.5 × 2.88)
𝐿= (5.4)
𝑠𝑖𝑛( 𝜋3 )
From the above equation, the value of 𝐿 comes out to 5𝜇𝑚. Assuming a thickness of 10𝜇𝑚
for a slice of the NF HEDM data, the volume of an triangular element is given as: 𝑉0𝑁 𝐹 =
1
2 × 𝐿 (2.88 + (0.5 × 2.88)) × 10 = 108.28(𝜇𝑚) 3 .
If the number of constituent voxels in a grain is 𝑁, then the total volume of the grain is given
as: 𝑉𝑔𝑁 𝐹 = 𝑁 × 𝑉0𝑁 𝐹 .
Therefore, assuming a spherical grain shape the radius can be calculated as follows.
√︄
3 3𝑉𝑔𝑁 𝐹
𝑅𝑔𝑁 𝐹 = (5.5)
4𝜋
106
Forward algorithm Reverse algorithm
1.00 1.00
cumulative probability
0.75 0.75
0.50 0.50 1.00
0.75
0.50
0.25 0.25
0.25
0.00
0 100 200
0.00 0.00
0 750 1500 0 750 1500
Δ(μm)
Figure 5.6: Cumulative distribution function plots for matched grains using forward algorithm
(left) and reverse algorithm (right), using a misorientation tolerance of 5°. In both the matching
operations, a ball radius of 1500 𝜇𝑚 was used (entire domain of the NF dataset). Both the
approaches yield identical results. Inset on the right hand side shows CDF plot for a much smaller
subset, i.e. only centroids within a 200 𝜇𝑚 ball radius are considered.
Figure 5.8 shows the cumulative distribution plots of the grain radii for the FF and NF datasets
corresponding to the final unloaded stress state. The reported FF grain radii do not show a good
match with the calculated grain radii for the matched NF grains. The median value of the FF
grain radii is smaller than the NF grain radii by a factor ∼ 2.9 (15.5𝜇𝑚 in FF and 44.3𝜇𝑚 in NF).
Moreover, the FF grain radius values show a wider variance than the corresponding NF values. If
the FF radius values are multiplied by a factor of 2.9, a better match between the two datasets is
observed.
5.4.4 Implementation of matching algorithm
Figure 5.9, shows the lines connecting the centroids of the matched FF and NF grain pairs using
a maximum ball radius of 200𝜇𝑚. The plot on the left side has the connecting vectors scaled and
shaded in proportion to the calculated NF grain radius (𝑅𝑔𝑁 𝐹 ), i.e. a smaller NF grain radius has
107
L
2.88μm
Figure 5.7: Arrangement of voxels in a 2D slice of the NF HEDM data, looking down the tensile
(Z) axis.
FF grain radii (as-is ) FF grain radii factored by 2.9
1.00
cumulative probability
0.75
FF FF
0.50
NF NF
0.25
0.00
1 101 102 1 101 102
Grain radius(µm) Grain radius(µm)
Figure 5.8: Cumulative distribution plots for FF and NF grain radius value for the matched grains
with a cut-off ball radius of 200𝜇𝑚. The figure on the left shows the comparison between the two
datasets with the as-is FF grain radius values. The figure on the right shows the comparison after
the FF grain radius values were multiplied by a factor of 2.9, which gives a closer match with the
NF data.
108
Vectors scaled by NF Grain Radius Vectors scaled by FF Grain Radius
100
80
GrainRadius(μm)
Projection on XY Plane Projection on XY Plane
(looking down tensile axis) 50 (looking down tensile axis)
20
Projection on YZ Plane 0 Projection on YZ Plane
Figure 5.9: Comparison of grain centroid positions from FF and ND HEDM results, corresponding
to the final unloaded state of the specimen. Only the centroids with NF-FF Euclidean distance
≤ 200𝜇𝑚 are considered here. The left side of the figure shows the connecting vectors scaled and
shaded in proportion to the calculated values of NF grain radius. The same set of plots are repeated
on the right side, with the connecting vectors scaled and shaded in proportion to FF grain radius,
without a corrective scaling factor.
a thinner darker line. The right plot has the connecting vectors scaled and shaded in proportion
to the reported FF grain radius without the correction factor of 2. The more similar shade of gray
illustrates the narrower distribution of FF grain sizes. Also, the projections along the Z and X axes
shows a dominant directionality between the two data sets.
5.4.5 Application of translational correction between FF/NF datasets
From the projection of the 3D plot in figure 5.9 on the YZ plane, a systematic translational shift
of the NF data can be seen with respect to the FF points. Therefore, the average translation vector
was used to improve the spatial match of the two datasets. Accordingly, the vector sum of all the
connecting lines were calculated, which is then divided by the total number of matching pairs of
−
→
centroids (821). The resultant vector (Δ𝑟 𝑢𝑛𝑖𝑡 ) and its magnitude, along with its projection in the YZ
plane are shown in figure 5.10. From the CDF plot, translating all of the FF centroids along the
direction and by the magnitude of the correction vector results in an improved spatial match of the
109
1.00
cumulative probability
0.75
0.50
0.25
0.00
1 101 102
Euclidean distance (µm)
Figure 5.10: Left: CDF plot of same data from Figure 5.9, with and without the correction vector.
From the leftward shift of the CDF curve, using the correction vector results in a better spatial
match between the two datasets. Right: projection of the calculated correction vector in the YZ
plane. Its magnitude and direction are also indicated.
FF/NF datasets and illustrated in 5.11.
5.4.6 Relationship between NF grain size and matching centroids in FF
An important outcome of matching of the FF/NF datasets is to identify the relationship between
the NF grain size and the corresponding matched grains in FF. A working hypothesis here is that
there should be a positive correlation between the reported FF grain radius and the number of
constituent voxels in the matched NF grain.
Comparing the distribution of grain size in the NF dataset to the subset of FF grains that were
matched, the FF grains have a larger size, as shown in the CDF plot in figure 5.12 (top right). For
the matched grains, a weak linear correlation (𝑅 2 = 0.2061) can be seen. The motivation behind
applying a threshold voxel size of NF grains is based on the hypothesis that smaller grains may
contribute towards noise during the matching process. If a threshold is applied, the value of the
correlation coefficient (𝑅 2 ), i.e. smaller grains below a given number of voxels are not considered.
The value of 𝑅 2 increases up to a maximum value (0.4634) for a threshold of 400 NF voxels, and
110
Vectors scaled by NF Grain Radius Vectors scaled by FF Grain Radius
100
80
GrainRadius(μm)
Projection on XY Plane Projection on XY Plane
(looking down tensile axis) 50 (looking down tensile axis)
20
Projection on YZ Plane 0 Projection on YZ Plane
Figure 5.11: Replotting of the same data as in figure 5.9, except in this case the correction vector is
applied to the FF data points. The the FF data appears much better aligned with the corresponding
NF data points.
then decreases after that. Therefore the best fit for linear correlation is obtained if grains larger
than 400 voxels are considered, but no improvement in 𝑅 2 is achieved if the threshold is increased
beyond this value.
The weak linear correlation between the grain size of the matched FF and NF grain datasets
suggests that the reported values of FF grain radius is not a close approximation of the actual
grain size. Since the resolution of the NF characterization is finer (2.5𝜇𝑚), the metrics used for
estimating grain size from this dataset (number of constituent voxels or calculated grain radius)
provides a more credible estimate of the grain size.
In terms of the algorithms used to match between the FF and NF datasets, both the forward
and reverse approaches converge to the same result. Therefore, it does not matter whether the
misorientation or the ball radius criteria are applied first.
Additionally, the relative error in grain position from FF HEDM shows a strong inverse rela-
tionship as a function of grain size. At load steps corresponding to significant plastic deformation,
the relative positional error becomes significant in smaller sized grains. For the grains that were
matched between the FF and NF datasets corresponding to the final unloaded state, the reported
111
105 1.00
l og10(Nu mVo x) = 0.8857 × l og10(GR a d i u s) + 2.3625
R 2 = 0.2061
cumulative probability
104
0.75
Number of Voxels (NF)
103
0.50
102
0.25
101
100 0.00
105
1 102 104
l og10(Nu mVo x) = 1.3641 × l og10(GR a d i u s) + 1.9233
R 2 = 0.4634
Number of voxels
0.45
0.4
Number of Voxels (NF)
2
104
Correlation coefficient: R
0.35
0.3
103 0.25
Threshold of best fit
10 15 20 25 30 35 40 45
Grain Radius/ m (FF) 0 500 1000 1500 2000
Threshold Numberof Voxels (NF)
Figure 5.12: Top right: cumulative distribution plots of grain sizes, in terms of number of constituent
voxels. For the subset of grains matched with FF dataset, there is a shift to right, indicating a larger
average grain size. Left: For matched grains, grain size plotted as a function of the unscaled grain
radius. Bottom right: variation of the correlation coefficient as a function of threshold number of
voxels.
grain radius values from FF varies from the corresponding NF values by about a factor of 2.
Moreover, at larger plastic strains, smearing of diffraction patterns could result in larger relative
positional errors. Therefore, when using grain position data from FF HEDM, it is important to
consider the deformation state of the specimen.
112
CHAPTER 6
COMPARISON OF KINEMATICS OF CRYSTAL ROTATION AND GRAIN AVERAGED
STRESS MEASUREMENTS OBTAINED FROM FF HEDM
The motivation behind this chapter stems from the need to assess the reliability of grain averaged
stress measures in predicting complex mechanical events such as slip induced twin nucleation (S+T
twinning). For a given grain, FF HEDM enables the calculation of relative lattice reorientation
as a function of strain. Here, the objective is to quantify how well this kinematic descriptor of
plastic deformation can be reconciled with the grain averaged stress measurements obtained using
FF HEDM. From the standpoint of kinematics, the relative lattice reorientation can be obtained
from FF HEDM as a function of load step. On the other hand, based on lattice strain, the grain
averaged stress tensor can also be calculated. The latter can be projected onto the slip systems that
are most likely to play a role in crystal rotation, and compared with the observed crystal rotation.
Quantification of how well these two measurements match each other is an important step towards
assessing the limits of the FF HEDM technique.
6.1 Calculation of plastic spin axes for active slip systems
A useful way to determine if a particular slip system is active during plastic deformation is to
check how well aligned the corresponding plastic spin axis is to the lattice reorientation at that load
step. If 𝑛 and 𝑏 are the slip plane normal and the slip direction respectively, then the plastic spin
axis is given as follows.
− →
→ − −
𝐿 = 𝑏 ×→ 𝑛
(6.1)
←− − → −
𝑜𝑟 𝐿 = → 𝑛 × 𝑏
The direction of the resultant plastic spin axis has to correctly reflect the sense of lattice rotation
brought about by the slip system considered, as can be seen from the schematic representation of
the sense of shear effecting lattice rotation due to slip (figure 6.1). To be consistent with observed
113
Figure 6.1: Schematic representation showing the sense of shear and plastic spin effected by
crystallographic slip, that results in lattice rotation. 𝑛 denotes the slip plane normal, while 𝑏
denotes the slip direction.
rotations and the direction of shear informed by the resolved shear stress, equation 6.1 is modified
as follows.
→− →
− −
𝐿 = 𝑠𝑔𝑛(𝑅𝑆𝑆)( 𝑏 × → 𝑛) (6.2)
where 𝑅𝑆𝑆 is the resolved shear stress for the considered slip system, calculated using the grain
averaged stress tensor.
6.2 Comparison of slip system plastic spin axis and lattice reorientation
This section details the methodology used to quantify how well the measured kinematic ori-
entation evolution compares with the expected rotation based upon a simple estimate based upon
the most highly stressed slip systems using the stress obtained from FF HEDM stress. For the
purposes of this study only prism⟨𝑎⟩, basal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩ slip systems are considered
to be active, and to simplify this analysis as starting point, the critical resolved shear stress for all
of these families are consider equal.
The prism denoting the lattice orientation prior to tensile loading in the sample frame is shown
on the top left of figure 6.2. The top row of stereographs shows the lattice spin axis for each load step
in the plastic regime up to the point of maximum tensile load (∼ 3% bulk strain). These spin axes
vary with increasing strain, and the progression is indicated by a change in the shade of gray from
114
Initial crystal orientation in (0 0 0 1) Stereographic Projection (Crystal frame)
sample frame i=3
Y
i=1 i=5
a3 a2
X
a1 a3 a2
a1
Load step prior to
maximum tensile strain
ϵ = 2.73 %
12 Pyramidal
slip systems
Prism
Basal ϵ = 2.93 % (Maximum tensile strain)
Figure 6.2: The upper left prism identifies the initial grain orientation in the sample frame with
the three ⟨𝑎⟩ axes labeled. The top row shows three (0 0 0 1) stereographic projections showing
the lattice spin axis at each load step from from the beginning of the plastic regime (white) to
∼ 3% (black). The ⟨𝑎⟩ axes of the crystal frame are identified at the center of the upper right
stereographic projection. Increasing the spacing 𝑖 between load steps from 1 to 5 (left to right)
smooths the fluctuations between load steps. Middle and bottom rows: Stereographic projections
show the plastic spin axes for (basal⟨𝑎⟩ (light blue circles on perimeter), prism⟨𝑎⟩ (red circles
in the center) and pyramidal⟨𝑐 + 𝑎⟩ (rainbow colors). The rainbow color scheme for the 12
pyramidal⟨𝑐 + 𝑎⟩ slip system spin axes are shown in the lower left-hand corner (the two spin axes
for different directions on the same plane are close together with dark and light tones of the same
color (plane), and more widely spaced adjacent spin axes are for the same slip direction on different
planes, which have either a darker or lighter color of the two planes). The symbols for each spin
axis are scaled in proportion to their resolved shear stress magnitude. The black point identifies the
observed spin axis. A blue arc connects a blend of the two most highly stress slip systems (largest
symbols). The observed spin axis has an arrow pointing to the blend of the top two slip systems that
is closest (largest dot product), In the middle and bottom rows, the most favored slip systems are
different, indicating significant fluctuations in the stress state, but the black arrows indicate that that
the observed spin axis is closer to the blend of the highly stressed slip systems. With smoothing,
the black arrow gets longer (dot product becomes smaller).
115
white to black, where the lightest points represent the transition from elastic to plastic deformation.
The darkest point identifies the rotation axis during the final tensile load step prior to unloading.
In the column with 𝑖 = 3, the crystal rotation axis is computed based upon the orientations before
and after the reference load step, to increase the range of the orientation change. Similarly, for
𝑖 = 5, the rotation axis between two load steps before and two load steps after the reference load
step is computed. As 𝑖 is increased, the oscillation of the grain plastic spin axes decrease as the
fluctuations between load steps are smoothed out.
For a given load step, the two most favored slip systems are chosen based on the magnitudes of
the resolved shear stress values (calculated using the grain averaged stress tensor). Then the two
slip system plastic spin axes for the two ’most favored’ slip systems are calculated using equation
6.2. Assuming that the actual slip activity could result from a linear combination of these two slip
systems, a series of intermediate spin axes resulting from different weights of the two are generated.
These intermediate axes are then compared to the observed lattice spin axis using a dot product to
see how close they are to each other. For each grain, the dot product of the observed lattice plastic
spin axis at a given load step is calculated with respect to the blend of the two most highly stressed
plastic spin axes. The maximum value of this dot product and the corresponding fraction of the
highest resolved shear stress slip system in the blended pair are recorded.
An example of this exercise for a twinned grain at the last two load steps corresponding to
bulk strain values of ∼ 2.73% and ∼ 2.93% is shown in the middle and bottom rows of figure
6.2. The three families of slip systems (basal⟨𝑎⟩, prism⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩) are plotted on
stereographs in the crystal frame for each of the three values of 𝑖. In the middle row, the two slip
systems with the highest magnitudes of resolved shear stress are pyramidal ⟨𝑐 + 𝑎⟩ slip systems
with spin axes that are on different sides of the stereograph (the large yellow and red circles). The
black point representing the crystal spin axis is near the center of the stereograph, and an arrow
points to the intermediate combination of the red and yellow slip systems with the minimum dot
product. In contrast, at the last loading step, the two most highly stressed systems have very similar
spin axes (two pyramidal⟨𝑐 + 𝑎⟩ systems on the same plane indicated by two large blue points).
116
i=1 i=3 i=5
1 1
Fraction of 2nd highest RSS slip system
0.8 0.9
0.6 0.8
0.4
Dot Product
0.7
0.2 0.6
0 0.5
-0.2 0.4
-0.4
0.3
-0.6
0.2
-0.8
0.1
-1
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0
Bulk Strain/%
Figure 6.3: Maximum dot product between lattice spin axis and blended slip system plastic spin
axes (linear combination of the highest and second highest resolved shear stress slip systems), for
the twinned grain in figure 6.2. The effect of smoothing (increasing 𝑖) leads to less fluctuation, and
suggests alternating sets of active slip systems. The colors of the markers indicate the fraction of
the slip system with the second highest resolved shear stress in the blended axis.
The lattice spin axis for the final load step (colored black) is different from the prior load step,
and is closer to these two blue slip system spin axes than other ⟨𝑐 + 𝑎⟩ slip systems. Clearly, the
average stress tensor differs between these two load steps, and the observed rotation axis changes
in a direction that reflects the change in the stress tensor.
Smoothing resulting from increasing the load step increment used to compute the observed
lattice spin axis leads to a more concentrated set of spin axis points (less fluctuation). However,
as the value of 𝑖 increases, the distance between the observed spin axis and the most favored slip
system spin axes increases. This indicates that the last load step differed from the trend of plastic
spin in the prior part of the deformation process.
The results of this analysis for the twinned grain are shown in the plots in figure 6.3. Here, the
maximum dot product is plotted as a function of bulk strain. Each datum point is colored by the
fraction of the slip system along the blended axis as indicated by the color bar. In the twinned grain,
a little less than half of the dot products are high values, indicating that there are large fluctuations
in the evolving stress tensor. This indicates that the concept of this analysis potentially able to relate
the observed grain spin axis history to favored slip systems, and refinements to this approach may
provide an improved agreement between this analysis strategy and the observations.
The process outlined in figure 6.3 is used for each load step in the plastic regime for three
117
different classes of grains represented in three rows (figure 6.4). The first type of grains are those
where deformation twins were formed. The second type of grains are neighbors to the twinned
grains that have a favorable geometrical and spatial alignment for slip transfer (high 𝑚′ grains). The
third type of grains are arbitrarily chosen from the gage volume interrogated by FF HEDM. Each
plot has the history of observed vs. highly stressed slip systems dot products from six grains from
each of the three categories are chosen for this comparison. For each class of grains, and for each
value of 𝑖, the average value of the dot product is calculated and identified with a horizontal dashed
line. The average value of the dot product increases slightly with increasing 𝑖 for the twinned grains.
In case of the other two classes of grains, there is a slight decrease in the average values for 𝑖 = 3,
but it increases for 𝑖 = 5.
These results are further summarized in the cumulative distribution plots of the maximum value
of the dot product for the same three categories of grains in figure 6.5, indicating the considerable
spread in the values of the dot product. Nevertheless, the twinned grains have more values of the
dot product that are closer to one. This can also be seen in the cumulative distribution plot in figure
6.5 where at least 46% of the points have values ≥ 0.8. If this blend of the two most favored slip
systems dominates the reorientation, the values of the CDF should be low until the dot product
approaches 1, which is most clearly observed in the twinned grains. On the other hand, figure 6.5
shows that for the two other categories of grains (neighboring and arbitrarily chosen grains), the
effect of smoothing is not as significant.
Additionally, the effect of changing the spacing between load steps is more profound in case
of the twinned grains category as well. As the spacing is increased from 1 to 5, the percentage
of data points with a dot product of 0.8 or higher increases. This indicates that despite the
complexity associated with twinning, this method may be able to account for significant activity of
pyramidal⟨𝑐 + 𝑎⟩ slip that is associated with twinning, as this analysis currently tends to ignore the
contributions of basal or prism slip, which may be more significant in the randomly chosen grains.
From the analysis shown in this chapter, it is clear that the variation between the kinematics of
plastic deformation (as measured by lattice spin), and the grain averaged stress tensor measurement
118
1
i=1 i=3 i=5
0.8
0.6
0.459
Twinned grains
0.4 0.517
0.439
0.2
0
-0.2
-0.4
-0.6 1
Fraction of 2nd highest RSS slip system
-0.8
-1 0.9
1
Neighboring grains
0.8 0.8
Dot Product
0.6
0.7
0.4
0.451 0.423 0.461
0.2 0.6
0
-0.2
0.5
-0.4
0.4
-0.6
-0.8 0.3
-1
0.2
1
0.8 0.1
0.6
0
Arbitrary grains
0.4
0.2 0.281 0.266 0.296
0
-0.2
-0.4
-0.6
-0.8
-1
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/%
Figure 6.4: Maximum dot product between the observed lattice spin axes and blended high resolved
stress slip system axes (slip systems with highest and second highest resolved shear stress), plotted
as a function of bulk strain for six members of three types of grains; twinned grains, neighbors
to the twinned grains, and arbitrarily chosen grains. For each plot, the average value of the dot
product is plotted as a dashed horizontal line and annotated in each plot.
from FF HEDM is quite significant. Therefore, analysis of complex load transfer events in a
polycrystal such as deformation twinning, the use of grain averaged measurements may not be
sufficient for understanding what takes place, particularly in regions near grain boundaries where
even greater variations in local stress state are expected. To gain better understanding of the
influences of slip in one grain on its neighbor, the ability to measure the local stress tensor
with a finer spatial resolution, such as the capability offered by DAXM is required for improved
understanding of local plastic deformation processes.
119
Twinned grains Neighboring grains Arbitrary grains
1
n=1
0.9 n=3
cumulative probability
n=5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Dot Product
Figure 6.5: Cumulative distribution plots for the maximum dot product for the three categories of
grains shown in 6.4. The effect of spacing between load steps is more significant for the twinned
grains than for the other two classes of grains.
120
CHAPTER 7
IN-SITU DAXM CHARACTERIZATION OF A COMMERCIAL PURITY TITANIUM
SPECIMEN SUBJECTED TO A FOUR POINT BENDING TEST
The approaches used for analysis of data obtained from the in-situ DAXM characterization
at different load steps are described. The experimental setup and overall geometry for the in-situ
characterization of the bending experiment are shown in figure 3.5. The spatially resolved deviatoric
strain tensor is calculated for each bending increment. In addition, a methodology to calculate slip
system specific GND density based on lattice rotation gradient is outlined. This is followed by a
detailed description of the results and their discussion in section 7.4. The heterogeneity of stress
state at grain boundaries is illustrated by tracking the variation in spatially resolved principal stress
and principal directions for the second bending increment (section 7.4.2). Finally the relationship
between local accumulation of GND density and slip transmissibility at the second bend increment
is explored in section 7.4.3.
7.1 Texture of CP-Ti specimen
The four point bending specimen used for this experiment has a predominantly soft texture with
respect to the loading direction i.e. prism slip is readily activated (Figure 7.1). Within the region
examined by DAXM, two grains are considered to be of particular interest: a hard oriented grain,
where prism slip is not expected to be readily activated; and a soft oriented grain adjacent to it.
Indeed, evaluation of the surface global Schmid factor indicates that prism ⟨𝑎⟩ slip is less likely to
be active in the hard grain; while it is highly favored for the soft grain (Figure 7.2).
7.2 White beam diffraction
The experimental setup for the in-situ bending experiment has been discussed in section 3.3.
Prior to bending, a DAXM scan of the region of interest was done with a step size of 4𝜇m.
After the specimen was placed in the bending stage, DAXM scans were done following each
121
100 um
Loading Direc�on
HG
SG HG
SG
SG: Soft grain
(a) (b) HG: Hard grain
Figure 7.1: (a) OIM Map of the undeformed microstructure of the 4 point bending specimen prior to
bending. The region interrogated by DAXM is outlined with a dashed rectangle. Prisms indicating
orientations of the soft and hard grains are shown. (b) Optical image of the microstructure at the
end of the final bending increment (∼ 3.5% macroscopic strain). The soft and hard oriented grains
are outlined in blue and green respectively.
Figure 7.2: Surface global Schmid factor maps for specimen prior to bending. The overall texture
is conducive to basal and prism ⟨𝑎⟩ slip.
122
bending increment. The orientation maps of the scanned regions for undeformed and four bending
increments are shown in figure 7.3 (a)-(e). At the second bend increment (∼ 1.7% bulk strain), line
scans were made along the H and X directions with a step size of 1.5𝜇m (figure 7.3(c)). All of the
work described hereafter is from this second increment of strain.
Loading direction
(a) Undeformed specimen prior to mounting
(d) Bend increment 3
(b) Bend increment 1
H
X
(c) Bend increment 2
(e) Bend increment 4
Figure 7.3: Orientation maps from the indexed DAXM results. The undeformed specimen was
interrogated before mounting with a step size of 4𝜇m (a). For the subsequent bending increments
1, 2, 3 and 4 ((b) through (e)) a coarser step size (6𝜇m) was used. The vertical and horizontal lines
(colored yellow) overlaid on the coarse dataset in (b) indicate the locations of finer finer 2𝜇m step
size H and X scans that are shown beneath the coarser map.
123
7.2.1 Local deviatoric strain tensor estimation from DAXM data
The Laue reflections obtained from white beam diffraction enables the mapping of the shape
change of the unit cell with respect to the undistorted lattice. Optimization schemes were used to
evaluate the deformation gradient which forms the basis for calculating the deviatoric strain tensor.
This is implemented in the LaueGo module developed at the APS, which was used to analyze the
data in an IGOR Pro environment.
Figure 7.4 shows the spatially resolved equivalent strain distribution for the DAXM measured
volume for the unstrained state and the four subsequent bending increments (Interrogated volumes
shown in figure 7.3). Below the plot, maps at a particular cross section are shown (the unstrained
state has a vertical step size of 4𝜇m and the four bending increments have a coarser 6𝜇m step
size). In the unstrained condition, the outlines of the grain boundaries are also shown in figure
7.4 (left, bottom row). While the sample is unstrained, there are clearly varying internal strains
within the grains as indicated by the color gradients, where local accumulation of high equivalent
strain above 0.001 can be observed in the vicinity of grain boundaries and triple junctions. With
loading, the local strain state becomes more uniform, with subtle variations in color where grain
boundaries are located, and they become more apparent with increasing strain. The topic of
stress/strain heterogeneity near grain boundaries is examined in more detail in section 7.4.2. In the
first, second and third bending increments at least 97% of the voxels have equivalent strains less
than 0.005, 0.008 and 0.015 respectively. The inset shows the equivalent strain at each increment,
indicating that after the fourth increment, an overall relaxation of the equivalent elastic strain took
place (purple datum point). This indicates that a redistribution of elastic strain occurred within the
sample.
7.3 Geometrically necessary dislocation density calculation from DAXM
data
In this section the methodology used to calculate the GND density from DAXM data is detailed.
This follows the mathematical and physical basis laid out by Das et al. and Guo et al. (Das et al.,
124
Cumulative probability
Inc0 Inc1 Inc2 Inc3 Inc4
Unstrained Bend4
Figure 7.4: Local equivalent strain distribution map corresponding to the unstrained state and after
four subsequent bending increments. The cumulative distribution function compares the equivalent
strain corresponding to the unstrained state prior to bending (blue), and the four subsequent
bending increments. The inset shows the evolution of the equivalent strain averaged over the entire
interrogated volume for five strain states. The lower row shows the spatially resolved equivalent
strain maps for the unstrained and four deformation states. The IPF color map for the microstructure
corresponding to the unstrained state is shown on the left hand side of the top row. The unstrained
scan was done prior to the bending experiment and covered a larger volume with a finer step size.
The other scans were taken after each load step during the bending experiment.
125
2018; Guo et al., 2020) in section 2.6. The DAXM dataset used for GND calculation is the fine grid
of H-X scans collected at the second bending increment (∼ 1.7% bulk strain). The approach used
for GND calculation is summarized in figure 7.5. First, the reciprocal lattice vectors for each voxel
obtained from the reconstructed DAXM data are used to calculate the real space lattice vectors and
the orthogonal orientation matrix. The rotation gradient with respect to neighboring voxels in the
three principal (orthogonal) directions is obtained. Finally, the optimization function is set up to
minimize the total elastic energy of dislocations.
For each voxel, a ball of surrounding voxels is identified using a K-dimension nearest neighbor
search algorithm. In the current work, the nearest neighbor search is implemented using the KD
(K-dimensional) tree module available in SciPy. The disorientation matrix between each voxel and
each of the voxels within the ball is computed as follows,
𝜔𝑖 𝑗 = 𝑅𝐶𝑒𝑛𝑡𝑒𝑟 .𝑅𝑇𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 (7.1)
where 𝑅𝐶𝑒𝑛𝑡𝑒𝑟 and 𝑅 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 are the orthogonal orientation matrices for the kernel and neighboring
voxel respectively. The disorientation values are used to calculate the lattice rotation gradient tensor
by a least squares fit onto a hyperplane of dimension 𝑛 − 1.
1 1
𝑤 𝑋 𝑋 1 𝑋 1 𝜕𝑤
1 2 3 𝜕 𝑋1
. . . .
.. = ..
.. .. 𝜕𝑤
𝜕 𝑋2 (7.2)
𝑤 𝑋1 𝑋2𝑛 𝑋3𝑛 𝜕𝜕𝑤
𝑛 𝑛
𝑋3
For each neighboring voxel 𝑛, 𝑤 𝑛 = (𝑤 1𝑛 , 𝑤 2𝑛 , 𝑤 3𝑛 ) are the components of the infinitesimal lattice
(𝜔32 −𝜔23 ) (𝜔13 −𝜔31 ) (𝜔21 −𝜔12 )
rotation vector such that 𝑤 1𝑛 = 2 , 𝑤 2𝑛 = 2 and 𝑤 3𝑛 = 2 . (𝑋1𝑛 , 𝑋2𝑛 , 𝑋3𝑛 ) are the
coordinates of the 𝑛𝑡ℎ nearest neighbor voxel in three-dimensional space.
For each voxel 𝑝 in the DAXM dataset, 𝜕𝜕𝑤 𝑋1 , 𝜕𝑤
𝜕 𝑋2 and 𝜕𝑤
𝜕 𝑋3 are the lattice rotation
𝑝 𝑝 𝑝
gradients. The lattice rotation gradient tensor for a given voxel in the dataset is written as follows.
𝜕𝑤 1 𝜕𝑤 2 𝜕𝑤 3
𝜕 𝑋1 𝜕 𝑋1 𝜕 𝑋1
𝜔
𝛼𝑖 𝑗 = 𝜕𝑤 1
𝜕 𝑋2
𝜕𝑤 2
𝜕 𝑋2
𝜕𝑤 3
𝜕 𝑋2
(7.3)
𝜕𝑤 1 𝜕𝑤 2 𝜕𝑤 3
𝜕 𝑋3 𝜕 𝑋3 𝜕 𝑋3
126
Reconstructed DAXM Data
Reciprocal lattice
vectors (a*,b*,c*)
Reciprocal lattice
vectors (a, b, c)
Orientation matrix R
for each voxel
KD Tree
Nearest Neighbor
Search (ball radius=10um)
Disorientation matrix
for each voxel
Rotation gradient calculation
(least squares fit)
Lattice rotation
gradient tensor
Constraints
Linear programming problem
Figure 7.5: Schematic outlining strategy used to calculate GND density from DAXM dataset.
127
1.0
10 m
Disorientation matrix: 0.8
20 m
30 m
40 m
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
X1
X3
0.2
X2
0.0 0 50 100 150 200 250 300 350 400
Figure 7.6: Left: The sphere surrounding a kernel for an exemplary voxel within the combined
(Coarse scan + Fine scan) DAXM dataset for bend increment 2 is shown for ball radii range of
10-40 microns. Cumulative distribution function (CDF) plots of the residual sum of squares error
for each of the three components of the rotation gradient tensor. The error increases with increasing
ball radius used for the nearest neighbor search.
Figure 7.6 shows the cumulative distribution plot of the the error of fit (residual) for each component
of the rotation gradient tensor, based upon the four different values of ball radius.
The error of fit (residual) for each voxel is calculated by evaluating the square of the 𝐿 2 norm
using the following equation.
𝑅 𝜕𝑤 = (||𝑊 − 𝑃|| 2 ) 2 (7.4)
𝜕𝑋
128
Where,
1
𝑤
.
𝑊 = ..
𝑛
𝑤
and
1
𝑋 𝑋 1 𝑋 1 𝜕𝑤
1 2 3 𝜕 𝑋1
. . .
𝑃 = .. .. .. 𝜕𝑤
𝜕 𝑋2
𝑋1 𝑋2𝑛 𝑋3𝑛 𝜕𝜕𝑤
𝑛
𝑋3
The cumulative distribution plots in figure 7.6 show that the residual along each principal
direction is sensitive to the ball radius chosen for the nearest neighbor search. Also, the density
of voxels along the vertical (𝑋3) direction is greater than in the rest of the surrounding volume of
the sphere due to the combination of fine and coarse datasets. Clearly, the error increases with
increasing ball radius. For the present work, a ball radius of 10𝜇m is used for nearest neighbor
search. In theory, using a ball radius below 10𝜇m would yield even smaller residual values. The
limitation to using a smaller ball radius however, arises due to the (6𝜇m) resolution of the coarser
DAXM dataset. A ball radius less than 6𝜇m would yield a sparser point cloud leading to highly
skewed values of rotation gradients discussed next.
7.3.1 Rotation gradient calculation of DAXM dataset
The GND analysis is focused along the fine (H-X) line scan data collected at the second bend
increment (∼ 1.7% global strain). For calculating the lattice rotation gradient, the H-X scan data
are used in combination with the coarser dataset for the same load step as shown in figure 7.7.
Using only the fine resolution data around a kernel with a sparse number of voxels along the 𝑋1
direction leads to skewed values of rotation gradients. As seen from the different sphere sizes in
figure 7.6, the density of voxels is lower along the 𝑋1 direction, compared to 𝑋2 and 𝑋3. Using a
minimum ball radius of 10 𝜇m includes at least two sets of three or more voxels from the coarse
dataset along the 𝑋1 or 𝑋2 directions. A kernel with ball radius below 6 𝜇m has a lower density
129
of voxels along 𝑋1 or 𝑋2. Contributions from the sparse data set were weighted the same as from
the fine data set.
Figure 7.7: Bend increment 2: Line scan DAXM dataset superposed on coarse serial probed DAXM
dataset. The two datasets are combined for purposes of the lattice rotation gradient calculation.
Figure 7.8 shows the nine components of the rotation gradient tensor for the fine scan dataset
at the second bending increment. As expected, the magnitude of the rotation gradient components
is high in the vicinity of grain boundaries.
7.3.2 Calculation of dislocation density from the rotation gradient
The current work uses the formulation of the Nye-Kröner-Bilby (NKB) equation that follows
from equations 2.34 and 2.35. In equation 2.35, the contribution to the dislocation tensor 𝛼 from the
lattice rotation gradient is considered to be much greater than the elastic strain gradient. Therefore
the second term can be neglected here.
Equations 2.36 and 2.37 are rewritten below in terms of the components of the lattice rotation
gradient tensor.
130
6e-3
Radian/µm
0
-6e-3
Figure 7.8: Bend increment 2: Maps showing the distribution of the nine components of the
rotation gradient tensor for the fine scan grid.
11
𝑏 21 𝑙12 𝑏 31 𝑙13 . . . 𝑏 1𝑘 𝑙 1𝑘
𝜕𝑤 3 𝜕𝑤 2
𝜕 𝑋3 − 𝜕 𝑋2
𝑏 𝑙 𝛼11
11
𝑏 1 𝑙 1 𝑏 21 𝑙22 𝑏 31 𝑙23 . . . 𝑏 1𝑘 𝑙 2𝑘 𝛼 𝜕𝑤 2
12 12 𝜕 𝑋1
11
𝑏 21 𝑙32 𝑏 31 𝑙33 . . . 𝑏 1𝑘 𝑙3𝑘
𝜕𝑤 3
𝑏 1 𝑙3 𝜌1 𝛼13 𝜕 𝑋1
11 2 2 3 3 𝑘 𝑘
𝜕𝑤 1
𝑏 𝑙 𝑏 2 𝑙1 𝑏 2 𝑙1 . . . 𝑏 2 𝑙1 𝜌2 𝛼21
∑︁ 21 𝜕 𝑋2
𝑘 𝑘
(𝑏 ⊗ 𝜌 ) = 𝑏 12 𝑙21 𝑏 22 𝑙22 𝑏 32 𝑙23 . . . 𝑏 2𝑘 𝑙2𝑘 𝜌 = 𝛼 = 𝜕𝑤 1 − 𝜕𝑤 3
3 22 𝜕 𝑋1 𝜕 𝑋3 (7.5)
𝑗
11 2 2 3 3 𝑘 𝑘 .. 𝜕𝑤 3
𝑏 2 𝑙3 𝑏2 𝑙3 𝑏2 𝑙3 . . . 𝑏2 𝑙3 . 𝛼23 𝜕 𝑋2
11 2 2 3 3 𝑘 𝑘
𝜕𝑤 1
𝑏 𝑙 𝑏 3 𝑙 1 𝑏 3 𝑙1 . . . 𝑏 3 𝑙1 𝜌 𝑘 𝛼31
31 𝜕 𝑋3
𝑏 1 𝑙 1 𝑏 23 𝑙 22 𝑏 33 𝑙23 . . . 𝑏 3𝑘 𝑙2𝑘 𝛼 𝜕𝑤 2
32 32
𝜕 𝑋3
11 2 2 3 3 𝜕𝑤 2 𝜕𝑤 1
𝑘 𝑘 𝛼33 𝜕 𝑋2 − 𝜕 𝑋1
𝑏 3 𝑙3 𝑏3 𝑙3 𝑏3 𝑙3 . . . 𝑏3 𝑙3
𝐴𝜌 = 𝛼 (7.6)
where,
h i𝑇
𝛼 = 𝛼11 𝛼12 𝛼13 𝛼21 𝛼22 𝛼23 𝛼31 𝛼32 𝛼33 ,
131
𝑏 21 𝑙12 𝑏 31 𝑙13 . . . 𝑏 1𝑘 𝑙 1𝑘
11
𝑏 𝑙
11
𝑏 1 𝑙 1
12 𝑏 21 𝑙22 𝑏 31 𝑙23 . . . 𝑏 1𝑘 𝑙 2𝑘
11
𝑏 1 𝑙3 2 2 3 3
𝑏1 𝑙3 𝑏1 𝑙3 . . . 𝑏1 𝑙3 𝑘 𝑘
3 3
11 2 2 𝑘 𝑘
𝑏 𝑙
21 𝑏 2 𝑙1 𝑏 2 𝑙1 . . . 𝑏 2 𝑙1
𝐴 = 𝑏 12 𝑙21 𝑏 22 𝑙22 𝑏 32 𝑙23 . . . 𝑏 2𝑘 𝑙2𝑘 ,
11
𝑏 2 𝑙3 2 2 3 3
𝑏2 𝑙3 𝑏2 𝑙3 . . . 𝑏2 𝑙3 𝑘 𝑘
3 3
11 2 2 𝑘 𝑘
𝑏 𝑙 𝑏 3 𝑙 1 𝑏 3 𝑙1 . . . 𝑏 3 𝑙1
31
𝑏 1 𝑙 1
32 𝑏 23 𝑙 22 𝑏 33 𝑙23 . . . 𝑏 3𝑘 𝑙2𝑘
11
𝑏 3 𝑙3 2 2 3 3
𝑏3 𝑙3 𝑏3 𝑙3 . . . 𝑏3 𝑙3 𝑘 𝑘
and
h i𝑇
𝜌 = 𝜌1 𝜌2 𝜌3 . . . 𝜌 𝑘
Here 𝜌 𝑘 is the density corresponding to the 𝑘 𝑡ℎ type of dislocation. 𝑏𝑖𝑘 and 𝑙 𝑘𝑗 are the components
of the Burgers vector and line direction of the 𝑘 𝑡ℎ type of dislocation. The total dislocation density
Í
therefore, is given by 1𝑘 𝜌. The coefficient matrix 𝐴 with dimensions 9 × 𝑘 is built by taking into
consideration four families of slip systems–basal,prism⟨𝑎⟩,pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩.
Twin systems or secondary slip systems are not considered in this formulation. Here, 𝑘 is the total
number of slip systems considered, while 𝑖, 𝑗 = 1, 2, 3 are the three components of the Burgers
vector 𝑏 𝑘 and line direction 𝑙 𝑘 in 3D space.
As mentioned in section 2.6, if 𝑘 > 9, a unique solution for the dislocation density 𝜌 does not
exist. Since 𝐴 and 𝛼 are known, equation 7.5 can be solved using 𝐿 1 optimization. Here the
total elastic energy of the dislocations is minimized. The elastic energies of the edge and screw
dislocation are related to the Poisson ratio 𝜈 by the following relation.
𝐸 𝑒𝑑𝑔𝑒 1
= (7.7)
𝐸 𝑠𝑐𝑟𝑒𝑤 1 − 𝜈
1 If edge/screw contribution from the slip system is uniquely identified it is marked with an ’ ’. Screw dislocations
from basal ⟨𝑎⟩, prism ⟨𝑎⟩ and pyramidal ⟨𝑎⟩, that cannot be explicitly isolated are marked with ’*’
132
Table 7.1: Slip Systems considered for GND density calculation
Slip System Type Slip Plane Normal/Slip Direction Edge1 Screw 1
Basal ⟨𝑎⟩ (0 0 0 1) [2 1 1 0] *
Basal ⟨𝑎⟩ (0 0 0 1) [1 2 1 0] *
Basal ⟨𝑎⟩ (0 0 0 1) [1 1 2 0] *
Prism ⟨𝑎⟩ (0 1 1 0) [2 1 1 0] *
Prism ⟨𝑎⟩ (1 0 1 0) [1 2 1 0] *
Prism ⟨𝑎⟩ (1 1 0 0) [1 1 2 0] *
Pyramidal ⟨𝑎⟩ (0 1 1 1) [2 1 1 0] *
Pyramidal ⟨𝑎⟩ (1 0 1 1) [1 2 1 0] *
Pyramidal ⟨𝑎⟩ (1 1 0 1) [1 1 2 0] *
Pyramidal ⟨𝑎⟩ (1 1 0 1) [1 1 2 0] *
Pyramidal ⟨𝑎⟩ (0 1 1 1) [2 1 1 0] *
Pyramidal ⟨𝑎⟩ (1 0 1 1) [1 2 1 0] *
Pyramidal ⟨𝑐 + 𝑎⟩ (1 1 0 1) [2 1 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 1 0 1) [1 2 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 0 1 1) [1 1 2 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 0 1 1) [2 1 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (0 1 1 1) [1 2 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (0 1 1 1) [1 1 2 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 1 0 1) [2 1 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 1 0 1) [1 2 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 0 1 1) [1 1 2 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (1 0 1 1) [2 1 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (0 1 1 1) [1 2 1 3]
Pyramidal ⟨𝑐 + 𝑎⟩ (0 1 1 1) [1 1 2 3]
Therefore the objective function 𝐹 to be minimized can be expressed as follows (Das et al., 2018;
Guo et al., 2020).
∑︁ ∑︁
𝐹 = (1 − 𝜈) −1
𝑒𝑑𝑔𝑒
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝜌𝑘 + 𝜌 𝑘𝑠𝑐𝑟𝑒𝑤 (7.8)
𝑘 𝑘
subject to the constraints defined by equations 7.5 and 7.6; where 𝜌 𝑒𝑑𝑔𝑒 = (𝜌 𝑒𝑑𝑔𝑒 ) ⟨𝑎⟩ + (𝜌 𝑒𝑑𝑔𝑒 ) ⟨𝑐 + 𝑎⟩
denotes the edge dislocation contribution (including ⟨𝑎⟩ and ⟨𝑐 + 𝑎⟩ slip systems). Similarly
𝜌 𝑠𝑐𝑟𝑒𝑤 = (𝜌 𝑠𝑐𝑟𝑒𝑤 ) ⟨𝑎⟩ + (𝜌 𝑠𝑐𝑟𝑒𝑤 ) ⟨𝑐 + 𝑎⟩ represents the screw dislocation contributions. The linear
programming problem (equation 7.8) can be solved using standard solvers. In this particular case,
the LINPROG module available in MatLab was used. An important assumption is that dislocations
133
are either of pure edge or pure screw type.
In the case of the screw dislocation contributions from basal⟨𝑎⟩, prism⟨𝑎⟩ and pyramidal⟨𝑎⟩
slip, an aggregate screw dislocation density for each ⟨𝑎⟩ direction is obtained but cannot be explicitly
isolated in terms of individual slip systems.
Table 7.1 lists the slip systems considered for the constraint equation 7.5. If the edge or screw
dislocation contributions from the slip system can be uniquely identified, the column is marked
with an ’ ’. Screw dislocations from ⟨𝑎⟩ type of slip systems (basal, prism and pyramidal), that
cannot be explicitly isolated are marked by ’*’.
7.4 Results and discussion
Figure 7.9 shows the total and slip system family subset GND maps from each family for the
fine scan data corresponding to the second bending increment. Local agglomeration of both total
and slip system family GND density can be observed near grain boundaries and triple junctions.
7.4.1 Localized GND concentrations at grain boundaries
The localized GND concentrations were analyzed for twelve grain boundaries (including three
triple junctions) within the microstructure of the finer scan DAXM dataset (the 5 regions encom-
passing these boundaries are shown in figure 7.10). Region 1 is a vertical fin comprising of two
boundaries of interest: one low angle grain grain boundary within the same grain, and the other
being a high angle grain boundary (purple and pink colored grain). Region 2 encompasses a triple
junction, where one of the constituent grains has a hard orientation with respect to the loading axis
(beige grain on the right). Regions 3 and 5 also contain triple junctions, while region 4 is a high
angle grain boundary.
7.4.2 Assessment of local stress heterogeneity using principal components
An important consideration to understand the meso-scale mechanical response is the spatial
variation in the local stress state. For example, it is especially interesting to assess how the spatially
134
(c) (d)
(a)
basal (Edge dislocation contribution) prism (Edge dislocation contribution)
(e)
(f)
(b)
pyramidal (Edge dislocation contribution) pyramidal (Edge dislocation contribution)
(g) (h)
⍴/m3
Total GND
basal, prism, pyramidal (Screw dislocations) pyramidal (Screw dislocation contribution)
Figure 7.9: Bend increment 2: GND density maps, showing total GND density (b) and GND
densities specific to slip system families. Screw dislocation contributions of ⟨𝑎⟩ type are in (g).
Comparing the total GND density with the IPF map (a), it can be seen that the GND concentrations
coincide with the grain boundaries.
135
resolved stress tensor varies with respect to grain averaged values in the vicinity of grain boundaries.
Thus, the principal values and directions of the local stress tensor are used to quantify heterogeneity
of the driving force for mechanical response within the polycrystal. This is done by calculating the
Eigen values and Eigen vectors of the local stress tensor and comparing with the equivalent results
from the grain averaged stress tensor. Specifically, five distinct regions from the microstructure
interrogated by the line DAXM scans are examined (figure 7.10). These regions include three triple
junctions (regions 2, 3, and 5) and three other grain boundaries (two high angle grain boundaries
(HAGB) and one low angle grain boundary (LAGB) ).
The variation in magnitude and direction of the principal stress is visualized using "stress
jacks"; where the three principal directions are superposed onto a colored sphere that represents
each voxel, where the color represents the local crystal orientation. The magnitude and sense of
each principal stress component are denoted by the color of the respective vector. The principal
stress and directions for each voxel in two exemplary areas involving the purple grain are shown
in figures 7.11 and 7.12. For comparison, the corresponding grain averaged stress tensors and the
principal components for the purple grain and its neighbors above it also computed,
Í𝑛
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑘=1 𝜎𝑖𝑘𝑗
𝜎𝑖 𝑗 = (7.9)
𝑛
where 𝑛 is the total number of constituent voxels in the grain, and 𝑖, 𝑗 = 1, 2, 3. Only the voxels
from the coarse resolution dataset were used to calculate the average stress state of each grain, to
eliminate overemphasis from the fine scan regions. Similarly, the individual components of the
stress tensor averaged over the entire volume interrogated in the coarse DAXM data set comprising
about 70 grains is determined, indicating that the principal stress directions are not perfectly aligned
with the assumed global uniaxial tension.
The principal components in this analysis are referred to as first, second and third in order
of decreasing magnitude, and they are labeled for the average stress tensors in figures 7.11 and
7.12. To put these two maps into a larger context, maps of about six grains showing these three
components are presented in figure 7.13. It is noteworthy that the first principal stress component
136
1
2
5
3
4
Figure 7.10: Bend increment 2: Orientation map of the regions of interest where the scaled slip
transfer parameter is plotted as a function of residual Burgers vector. Region 1 comprises a grain
boundary and an intragranular region with an orientation gradient; region 2, 3 and 5 are triple
junctions; and region 4 is a grain boundary.
137
is mostly compressive, but the second component is mostly tensile, and has similar magnitude,
indicating that the stress state is largely bi-axial.
7.4.2.1 Local stress heterogeneity within and above the purple grain
The principal stress map for a region within the purple grain is shown in figure 7.11. The
intragranular zone between the black lines indicates a region with locally varying stresses that
differ significantly from neighboring voxels as well as the average stress (only the right side of this
region has a stress close to the grain average). Above and below the two lines, the stress jacks are
more uniformly oriented. The region along the grain boundaries with two neighboring grains above
is shown in figure 7.12. The local variation in stress tensor among the three grains in the triple
junction (pink, lavender, and purple grains) indicate that all three grains have significantly different
stress states near the boundary, and that these stresses differ significantly from their respective
grain averaged stress jacks. Some voxels adjacent to grain boundaries have high stress values.
The region in the purple grain below the grain boundary has a different stress state than the more
uniform stresses several voxels below the grain boundary.
Examining the principal stress maps in figure 7.13 identified by the three components, it can be
seen that there is significant variation in both direction and sense of the first (maximum) principal
component (left) in the vicinity of the grain boundary above the purple grain and even within the
intragranular region identified in figure 7.11 with variable stress tensors. The sense of the first
principal stress changes from compressive near the center of both grains to tensile along the grain
boundary and some locations within the intragranular zone of high local stress variation within
the purple grain. This variation in the first principal direction corresponds with high GND density
locations described later in section 7.4.3.
Considering the third principal stress and direction, the stress is mostly tensile but lower
in magnitude than the second component in most of the region. The sense of stress switches
from tensile near the grain centers to compressive in the vicinity of the grain boundary and the
intragranular zone with high local GND density. Moreover, taking all three principal stresses and
138
Grain averaged
stress state: purple grain
175
-255
83
Principal stress jack based on
volume averaged stress tensor
Figure 7.11: Principal Stress distribution for the intragranular region in the purple grain shows
varying local stress states indicated by different "principal stress jacks" that differ from the average
stress tensor for the purple grain shown on the left. The color of each line in the stress jack indicates
the value of the principal stress component in MPa, which are annotated on the average stress
jacks. The tensile direction is parallel to the 𝑌 axis in the coordinate system shown while 𝑍 is the
surface normal. The stress jacks are oriented according to the XYZ axes, indicating that the largest
compressive principal stress is roughly perpendicular to the stress axis. The average principal stress
jack corresponding to the entire measured volume is also shown on the light gray sphere.
directions into account, intragranular transition zones are observed in most of the grains where
waviness is apparent.
7.4.2.2 Local stress heterogeneity in region 2 triple junction with hard and soft grains
Examining the triple junction in region 2 between the purple grain and the hard (beige) and
soft (blue) grains (figure 7.14), it can be seen that the local principal stress and directions align
closely with the grain averaged state, except for the row of voxels that form the boundaries with
the blue and beige neighboring grains. In the soft (blue) grain, the directions of the compressive
(blue) principal component gradually changes as the grain boundary is approached from the left.
The positive principal direction becomes much larger as one gets closer to the grain boundary and
139
Grain averaged stress state:
lavender grain above purple grain
Grain averaged
stress state: pink grain
above grain boundary
Grain averaged
stress state: purple grain
175
-255
83
Principal stress jack based on
volume averaged stress tensor
Figure 7.12: Principal Stress map for the high angle grains near the boundaries above the purple
grain, including the corresponding grain average "principal stress jacks" for the pink and lavender
grains above the purple grain, which have a significantly different stress state than the purple grain.
The tensile axis is parallel to the global 𝑌 direction (shown on the right of figure). The principal
stress values are annotated on the grain average and volume average stress jacks. The stress in the
pink and lavender grains are much different from the purple grain, and the stress state near the
boundaries differ significantly from the grain interiors.
Principal Stress Components: First Second Third
Figure 7.13: Bend increment 2: Principal Stress maps for the entire vertical fin containing the
LAGB and HAGB of region 1, with each principal component shown separately.
140
Region 2
175
-255
83
Principal stress jack based on
volume averaged stress tensor
Figure 7.14: Principal Stress direction map for the triple junction in region 2, where the viewpoint is
identified by the X-Y-Z arrows. The grain boundaries are delineated by black lines. Principal stress
jacks corresponding to average stress tensor for each individual grain, along with their magnitudes
and sense are also shown for comparison. The tensile axis is parallel to the global 𝑌 axis shown on
the left of the figure.
triple junction. The stress tensor in the beige (hard) grain orientation is quite uniform and similar
to the grain averaged state. For the purple grain, the stress is close to the average stress tensor
except near the triple junction. The principal stress and principal direction maps of the remaining
three regions considered in this analysis are shown in appendix E.
From the above observations it can be surmised that changes in sense and direction of local
principal stress can be used to identify regions where local strain accommodation at grain boundaries
may involve stresses that activate slip systems that differ significantly from regions in the grain
interior. There are regions where significant deviation from the grain averaged stress state occur
also tend to have high local GND agglomeration, as described in section 7.4.3. Moreover, grains
with soft orientations noticeably have strong deviations from the average stress state in the vicinity
of grain boundaries. On the other hand, the local stress in the hard oriented grain in region 2 does
141
not deviate significantly from the average stress state.
7.4.3 Description of parameters used in analysis of slip transfer at the voxel level
Figures 7.15 shows the total GND density map for the intergranular (grain boundary) and
intragranular voxels for region 1. The GND map of Region 1 (figure 7.15) shows a band of
high GND density in the middle of the purple grain. The spatially resolved deviation from the
average grain orientation in figure 7.16 shows that this region is disoriented from the average grain
orientation by about a degree, and is hence labeled a low angle grain boundary (LAGB) region.
Specific regions such as the pairs of voxels circled are analyzed using strategies described below,
and provided in appendix F.
An important underlying assumption in this analysis is that 𝑚′ would have an inverse relationship
with respect to the magnitude of residual Burgers vector, i.e. instances where slip transfer readily
takes place (𝑚′ values approaching 1) would correspond to lower magnitudes of residual Burgers
vector. plotted as a function of residual Burgers vector are discussed in detail for specific regions
in the H-X scan dataset.
Let 𝛼 and 𝛽 be two adjacent voxels and 𝑠𝛼 and 𝑠 𝛽 be two interacting slip systems in grains 𝛼
and 𝛽, respectively. If slip transmission from 𝛼 to 𝛽 is considered, the residual Burgers vector left
within a grain boundary can be expressed by the following equation.
𝑠
𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑏𝑂𝑢𝑡 𝛽
− 𝑏 𝑠𝐼𝑛𝛼 (7.10)
𝑠
where 𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 is the residual Burgers vector; 𝑏𝑂𝑢𝑡 𝛽
and 𝑏 𝑠𝐼𝑛𝛼 are the Burgers vectors corresponding
to the transmitting and receiving voxels respectively.
The second parameter used to assess the likelihood of slip transfer is the Luster-Morris parameter
(𝑚′). The third parameter is the generalized Schmid factor on each slip system as evaluated using
the stress tensor from each voxel.
In the present analysis, several different combinations of parameters are used to plot the re-
lationship between |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 |, 𝑚′ and slip system specific GND density for a pair of interacting
142
[82,60,347]
[98,70,302]
Y
Intergranular region (HAGB)
X
[98,70,302]
in
ad
Lo
gd
on
cti
ire Intragranular region (LAGB)
Total GND (⍴/m 3)
[98,70,3.5]
Figure 7.15: Orientation and total GND density maps for region 1. The voxels chosen for slip
transfer analysis in the high angle grain boundary (∼ 28 °misorientation) and intragranular (low
angle grain boundary with misorientation < 1.5°) regions are shown (circled voxels). The Bunge
Euler angles denoting the average grain orientations are also noted here. It is important to note here
that prisms denoting the crystal orientations are drawn from the perspective of the sample normal
direction, with 𝑌 pointing vertically up and 𝑋 to the right.
CDF of misorientation deviation from average grain orientation
1
0.5
0
0 0.8 1.6
Misorientation(degrees)
Figure 7.16: Map showing spatially resolved deviation from the average grain orientation in the
lower grain of region 1 (Left). The cumulative distribution of misorientation deviation within this
grain is shown on the right. The region with a relatively higher deviation coincides with the high
local GND accumulation region
143
voxels. These combinations are listed in table 7.2. The independent variable used here is the
magnitude of the residual Burgers vector 𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 , while 𝑚′ is the dependent variable. Both of
these values can be scaled by the maximum or minimum values of the slip system specific GND
density (𝑚𝑎𝑥(𝜌𝛼 , 𝜌 𝑏𝑒𝑡𝑎 ) or 𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝛽 )) for the voxel pair.
Table 7.2: Combinations of parameters used to assess relationship between residual Burgers vector,
𝑚′ and GND density
Independent Variable Dependent Variable Basis of choice of transmitting voxel
|𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | 𝑚′ Arbitrary
𝑚′
𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝑏𝑒𝑡𝑎 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | 𝑚𝑖𝑛(𝜌 𝛼 ,𝜌 𝑏𝑒𝑡 𝑎 ) Arbitrary or 𝑚𝑎𝑥(𝐿𝑆 𝑓 )
𝑚′
𝑚𝑎𝑥(𝜌𝛼 , 𝜌 𝑏𝑒𝑡𝑎 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | 𝑚𝑎𝑥(𝜌 𝛼 ,𝜌 𝑏𝑒𝑡 𝑎 ) Arbitrary or 𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝑏𝑒𝑡𝑎 )
An important question arises here, as to the basis for choosing the transmitting slip system for a
pair of voxels. In this study three different approaches are used–the first being an arbitrary choice
of voxels on one side of the grain boundary as transmitters. For the second approach the slip system
with the greater local Schmid factor (LSf) is considered to be the transmitter. The third approach
uses the minimum value of slip system specific GND density as the criterion for the transmitting
voxel based upon the assumption that the slip system with the lower GND density will transmit
more readily. The reader is referred to appendix F for more details of this analysis and preliminary
outcomes.
7.5 Summary assessment of inter-and intra-granular heterogeneous defor-
mation
From this preliminary assessment of heterogeneous stress and local GND density, it is evident
that local variations of stress, particularly in the vicinity of grain boundaries are significant, and
these variations have to be taken into account in order to describe the local dislocation activity.
Although the stress tensor averaged over the interrogated volume approaches ideal uniaxial tension
in some grains, it is clear that the local stress tensors vary widely from the global stress, especially
near grain boundaries. Assuming an average grain size of 100𝜇m, the investigated volume contains
approximately 70 grains: therefore it is a much smaller subset of the bulk volume of the specimen,
144
where the number of grains is of the order of 105 . The measurements and analysis possible with
DAXM show that true understanding of instances of local slip interactions at the grain boundary
cannot be examined effectively using even grain averaged stress tensors.
145
CHAPTER 8
CONCLUSIONS AND SCOPE FOR FUTURE WORK
8.1 Far Field HEDM
8.1.1 Twin identification and role of slip transfer in twin nucleation
As described in chapter 4, a total of 13 discrete twin-parent pairs were identified using FF
3D XRD. Among these twins, the activated variant did not necessarily have the highest local
Schmid factor values. From just the FF HEDM analysis, it would appear that five of the twins
may have nucleated as a result of slip transfer of pyramidal⟨𝑐 + 𝑎⟩ slip across the grain boundary.
When assessing the plausibility of slip transfer across a grain boundary, it is important to take into
consideration both the geometrical and spatial compatibility.
One important observation from this study is that the subsurface slip activity in a polycrystal
can be different from slip activity on the surface. Earlier surface studies pointed towards relatively
high prism ⟨𝑎⟩ slip activity leading to twin nucleation, whereas subsurface investigations reveal a
much higher likelihood of pyramidal ⟨𝑐 + 𝑎⟩ slip leading to twin nucleation. Indeed, prism ⟨𝑎⟩
slip systems were not found to have high geometric compatibility with the activated twin variants
observed. Comparison with similar studies in titanium shows that the type of slip system most
likely to nucleate a T1 twin is dependent on the loading direction and initial texture (Abdolvand
et al., 2015a; Nervo et al., 2016)
A semi quantitative metric of the initial local stress in each grain was obtained by comparing the
measured grain averaged stress tensor with ideal uniaxial tension. The stress state of several of the
parent grains approached the uniaxial tensile state following an initial nearly uniaxial compressive
state with increasing global load (which were present due to internal stresses in the sample). The
magnitude of stress in the analyzed twinned grains registers a drop or remains steady around the
point of twin formation. In some less common cases, an stress increase was observed. This confirms
146
that twin formation is often associated with stress relaxation in the parent grain and accommodation
of the stress drop by neighboring grains in the polycrystal. The final stress state of the parent grain
is dependent on how the accommodation of strain takes place in the neighborhood.
For the FF HEDM data,a quantified comparison between the kinematic descriptor (lattice
reorientation) and grain averaged stress state is made, indicating that the actual relationship is more
complex than the metric can identify. This also helps to identify the limits of using FF HEDM for
understanding complex mesoscopic loading states in polycrystalline materials.
8.1.2 Comparison of FF results with Near Field and surface EBSP mapping
Comparison of FF results with surface EBSD mapping at the final unloaded state yielded partial
matches for twin parent pairs and grains in their neighborhood. Only three twin-parent pairs were
matched within a misorientation threshold of 5°. Comparison of the angle-axis misorientation
between the XRD and EBSD did not reveal any systematic mounting error leading to variation in
misorientation for the twins identified. Large variations in spatial positions between the surface
EBSD and FF 3D XRD were noted indicating significant uncertainty in FF grain positions. To
ensure that the results of the FF dataset obtained using Fable were credible, the data were also
analyzed using the more sophisticated MIDAS package. A good match between the two sets of
results was obtained, in terms of grain center of mass positions and grain averaged strain/strain
tensors, but MIDAS provided a larger number of indexed grains (section 4.7). Thus, it is possible
that the large variations in misorientation and spatial positions could be an intrinsic outcome
of comparing a 2D EBSD slice with a 3D XRD dataset of a deformed sample. The strong
orientation gradients present in the specimen probably contributed to this wide range of orientation
variation. The results of matching with EBSD are not as favorable as some of the reported work on
polycrystalline hexagonal metals with more random texture, such as Louca and Abdolvand (2021).
The FF HEDM data corresponding to the final unloaded state was also compared with the NF
data, using criteria of Euclidean distance between the grain center of mass and the misorientation
between the two FF and NF orientations. Two different algorithmic approaches were used for the
147
comparison exercise. In the first approach, the misorientation criterion is applied before the distance
criterion, while the order was reversed for the second approach. Both algorithms converged to the
same result for matching grains in the two datasets.
For the matched set of grains between FF and NF HEDM, the median grain radius of the NF data
is greater than the reported FF grain radius by a factor of ∼ 3. There is a weak linear correlation
between the reported FF grain radius and the number of constituent voxels in the matched NF
grains. Overall, the grain volume estimated by NF is more credible compared to the FF grain
radius values due to the finer spatial resolution of the NF method. Many more twinned grains were
found in the NF dataset than the 13 twinning events identified in the FF data. Out of the 13 twins
identified from the FF HEDM analysis 11 twins were also found in the NF dataset. Out of this
population of 11 twins, 7 had interfacial contact with neighboring grains. Therefore slip transfer
analysis was repeated for only these 7 twinning instances. It is pertinent to mention here that out
of the 7 twins that were identified between FF and NF datasets, the neighboring grains deemed to
have good geometrical compatibility (high 𝑚′ with respect to activated twin variant) and spatial
proximity in FF analysis were the same for 3 instances in the NF dataset. During the NF analysis
the remaining 4 twins had different neighboring grains with interfacial contact with the twin. It
was observed from the NF data set indicated that ∼ 34% of the grains contain T1 twins.
8.2 DAXM analysis
8.2.1 Assessment of heterogeneity in stress state due to local constraints
Assessment of the spatially resolved principal stress and their directions suggest strong hetero-
geneity in the local stress magnitude and direction within an individual grain. This reveals strong
constraint effects from neighboring grains. The variation in principal stress magnitude, sense and
direction is more pronounced in the vicinity of grain boundaries. Therefore, it is highly likely that
the local stress state may have a greater influence on which slip systems are active as revealed by
the GND accumulation than the pure geometrical relationship between two grains.
148
8.3 Scope for future work
These data sets provided continuing opportunity for deeper analysis. Using the far field 3D
XRD analysis, several discrete pairs of twin-parent grain pairs were identified that were likely
active in slip transfer. However, the analysis conducted did not seek evidence for whether twins
initiated ⟨𝑐 + 𝑎⟩ slip or vice versa. From analysis of rotations in the grain, it is possible to discern
if ⟨𝑐 + 𝑎⟩ slip took place on the identified slip system using orientation changes such as that used
by Leyun Wang et al.. It is also however, important to consider combinations of slip systems that
could account for the observed crystal orientation evolution.
Also, the number of matched grains between 3D XRD and surface EBSD mapping is low
(<40%). It would be more efficient to develop an automated process to identify twin parent pairs
from the FF HEDM data, compared to the manual method employed in the current study. Though
there is no obvious way to do this conveniently, it would be helpful if in-situ EBSD could be
conducted concurrently with the 3D-XRD characterization. The NF data was only available for the
final unloaded state of the tensile experiment. For grain matching between FF and NF datasets, it
would be very helpful if a corresponding NF characterization of the data is also conducted prior to
tensile loading.
In the DAXM study, the GND content was calculated using the lattice rotation gradient and the
effect of long range elastic stress fields was not considered. Therefore the formulation is highly
localized. A non-local formulation would take into account the influence of elastic stress fields
that could result in a better resolved GND mapping. Moreover, this mesoscopic approach does not
capture the more complex mechanics of dislocations at the atomistic level. For completeness, a
multiscale approach associating the meso-scale results with atomistic simulations would be provide
further opportunity to explore the large amount detail available in this data set.
149
APPENDICES
150
APPENDIX A
ADDITIONAL FAR FIELD ANALYSIS RESULTS
A.1 Grain positions showing propensity of S+T twinning
Z
Y
Y
X View along tensile axis (Z) View looking toward beam (X)
Layer 3 (First observed twin)
m'=0.93 N
P: 57.9, 190.4, 149.2 P
T: 332.0, 86.3, 228.3 N
N: 349.5, 146.0, 71.1
P
T T
Layer 3 (Second observed twin)
m'=0.95
P: 151.3, 35.1, 255.4 P
T: 46.7, 86.2, 145.5 P
N: 102.7, 10.0, 294.3 N
N
T
T
Layer 5
m'=0.93 P
P: 332.7, 134.7, 340.0 P
T: 181.0, 127.0, 261.6
N: 164.0, 13.3, 198.0
N
N T
T
Figure A.1: The 𝑚′ relationship observed between the activated twin system and pyramidal ⟨𝑐 + 𝑎⟩
system of a neighboring grain is shown for layers 3 and 5.The orientations of the parent, twin
and neighboring grains are viewed along the tensile Z axis, (left column) as well as the beam X
direction, (right column). For description of the symbols, the reader is referred to figure 4.10.
151
Z
Y
Y
X View along tensile axis (Z) View looking toward beam (X)
Layer 6 (First observed twin)
m'=0.99 P
P: 354.0, 155.3, 256.7 P
T: 342.0, 70.0, 187.8
N: 270.3, 168.1, 224.0
N T
T N
Layer 7
m'=0.96 P P
P: 299.0, 17.8, 170.5
T: 109.3, 76.1, 243.1 N
N: 228.1, 168.0, 111.6 N
T T
Layer 8
m'=0.93
P: 97.6, 31.5, 337.0 P
T: 194.0, 82.0, 208.2 T
N: 85.0, 15.2, 95.6 T
N
N
P
Layer 11
m'=0.93
P: 349.2, 176.2, 156.2 N
T: 253.4, 95.2, 184.1 T P
N: 55.3, 173.8, 155.4
T
P N
Figure A.2: The 𝑚′ relationship observed between the activated twin system and pyramidal ⟨𝑐 + 𝑎⟩
system of a neighboring grain is shown for layers 6, 7, 8 and 11.The orientations of the parent, twin
and neighboring grains are viewed along the tensile Z axis, (left column) as well as the beam X
direction, (right column). For description of the symbols, the reader is referred to figure 4.10.
152
APPENDIX B
PARENT/TWIN/NEIGHBOR SLIP TRANSFER ANALYSIS FROM NF HEDM
In this part of the appendix, the remaining five twinning events identified in FF HEDM that
were validated by NF HEDM are shown. For description of the plotted parameters the reader is
referred to section 4.8
400
350
300
T m′= 0.99 250
/MPa 200
RSS 150
N
P 100
50
P: Parent (twinned) grain
N: Neighbor to twinned grain 0
T: Twin -50
3 800 2
disorientation
700 1.8
Plastic Spin Axis Misorientation/degrees
c-axis misorientation
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
1.4
2
500
1.2
/MPa
Twin identified at this loadstep
Degrees
1.5 400 1
vM
0.8
300
1
0.6
200
0.4
0.5
100 0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure B.1: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 6 in FF HEDM and validated
by NF HEDM.
153
400
350
300
P
250
T
m′=0.91
/MPa
200
N
RSS
150
100
P: Parent (twinned) grain 50
N: Neighbor to twinned grain
0
T: Twin
3 800 -50
disorientation 2
c-axis misorientation 700 1.8
Plastic Spin Axis Misorientation/degrees
2.5 vM Stress-Neighbor
vM Stress-Parent
600 1.6
2 1.4
500
/MPa
1.2
Degrees
1.5 400 Twin identified at this loadstep
vM 1
300 0.8
1
0.6
200
0.4
0.5
100
0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure B.2: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and validated
by NF HEDM.
154
400
350
N
m′= 0.87
300
250
/MPa 200
T RSS
P 150
100
50
P: Parent (twinned) grain
N: Neighbor to twinned grain 0
T: Twin -50
3 800
2
disorientation
c-axis misorientation 700 1.8
Plastic Spin Axis Misorientation/degrees
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
2 1.4
500
1.2
/MPa
Degrees
Twin identified at this loadstep
1.5 400 1
vM
300 0.8
1
0.6
200
0.4
0.5
100
0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure B.3: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 9 in FF HEDM and validated
by NF HEDM.
155
400
350
T 300
T P
P 250
m′= 0.93 N
/MPa
200
RSS 150
N
100
P: Parent (twinned) grain 50
N: Neighbor to twinned grain
0
T: Twin
-50
3 800 2
disorientation
c-axis misorientation 1.8
700
Plastic Spin Axis Misorientation/degrees
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
1.4
2
500 1.2
/MPa
Degrees
Twin identified at this loadstep
1.5 400 1
vM
0.8
300
1 0.6
200
0.4
0.5
100 0.2
0
0 0 0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Bulk Strain/%
Bulk Strain/%
Figure B.4: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and validated
by NF HEDM.
156
400
350
N
300
m′= 0.93
250
/MPa 200
T
RSS 150
P
100
50
P: Parent (twinned) grain
N: Neighbor to twinned grain 0
T: Twin
3 800 -50
2
disorientation
c-axis misorientation 700
Plastic Spin Axis Misorientation/degrees
1.8
2.5 vM Stress-Neighbor
vM Stress-Parent 1.6
600
2 1.4
500
1.2
/MPa
Degrees
1.5 400 Twin identified at this loadstep
1
vM
300 0.8
1
0.6
200
0.4
0.5
100 0.2
0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Bulk Strain/% Bulk Strain/%
Figure B.5: Parent/Twin/High 𝑚′ neighbor for twin identified in layer 7 in FF HEDM and validated
by NF HEDM.
157
APPENDIX C
STEREOGRAPHS OF EVOLUTION OF LATTICE PLASTIC SPIN AXIS AS A
FUNCTION OF BULK STRAIN
Here the evolution of the lattice plastic spin axis is shown as a function of bulk strain for a
twinned grain and its neighbor. For each load step, the arc comprising of plastic spin axes resulting
from linear combinations of the two slip systems with the highest resolved shear stress is also
shown. For details on the notations and symbols, reader is referred to 6.2.
ϵ = 0.31 % ϵ = 0.36 % ϵ = 0.37 % ϵ = 0.46 % ϵ = 0.53 % ϵ = 0.64 % ϵ = 0.72 %
ϵ = 0.85 % ϵ = 0.98 % ϵ = 1.11 % ϵ = 1.28 % ϵ = 1.48 % ϵ = 1.56 % ϵ = 1.76 %
ϵ = 1.89 % ϵ = 2.04 % ϵ = 2.17 % ϵ = 2.42 % ϵ = 2.5 % ϵ = 2.6 % ϵ = 2.73 %
ϵ = 2.93 %
i=1
Figure C.1: Stereographic projection of the evolution of the lattice plastic spin axis is shown for
different values of bulk strain for a twinned grain.
158
ϵ = 0.31 % ϵ = 0.36 % ϵ = 0.37 % ϵ = 0.46 % ϵ = 0.53 % ϵ = 0.64 % ϵ = 0.72 %
ϵ = 0.85 % ϵ = 0.98 % ϵ = 1.11 % ϵ = 1.28 % ϵ = 1.48 % ϵ = 1.56 % ϵ = 1.76 %
ϵ = 1.89 % ϵ = 2.04 % ϵ = 2.17 % ϵ = 2.42 % ϵ = 2.5 % ϵ = 2.6 % ϵ = 2.73 %
ϵ = 2.93 %
i=1
Figure C.2: Stereographic projection of the evolution of the lattice plastic spin axis is shown for
different values of bulk strain for the neighbor with a high 𝑚′ relationship with respect to the active
twin variant in the grain shown in figure C.1
159
APPENDIX D
HYDROSTATIC STRAIN TENSOR MEASUREMENT USING MONOCHROMATIC
SETTING OF DAXM
D.0.1 Hydrostatic strain
Since the magnitude of the reciprocal lattice vectors cannot be uniquely determined using
the polychromatic mode in DAXM, only the deviatoric component of the strain tensor can be
determined, since it is directly associated with the lattice curvature. By setting the beam to
monochromatic mode, an incident beam of known energy can be used to obtain the magnitude of
the reciprocal lattice vector for a given set of planes that can be indexed. If the deviatoric strain
tensor is known then the local hydrostatic component can be calculated as shown below.
The decomposition of the full strain tensor into its deviatoric and hydrostatic components can
be written as follows.
©𝜖11 𝜖12 𝜖13 ª ©𝜖 ℎ𝑦𝑑𝑟𝑜 0 0 ª
® ®
= 𝜖21 𝜖22 𝜖23 ® + 0 (D.1)
® ®
𝜖 𝐹𝑢𝑙𝑙 𝜖 ℎ𝑦𝑑𝑟𝑜 0 ®
® ®
® ®
𝜖31 𝜖32 𝜖33 0 0 𝜖 ℎ𝑦𝑑𝑟𝑜
« ¬ « ¬
where 𝜖 ℎ𝑦𝑑𝑟𝑜 is the hydrostatic component of the strain tensor. It is significant to mention here that
the hydrostatic strain is considered to be a scalar quantity in the above formulation. In general, the
hydrostatic strain can vary in the three orthogonal directions. A second important consideration
in this formulation is that the magnitude of the reciprocal lattice vector is determined based on a
single Laue peak at each strain state.
The full lattice distortion can be expressed as a function of 𝜖 ℎ𝑦𝑑𝑟𝑜 as follows.
©1 + 𝜖11 + 𝜖 ℎ𝑦𝑑𝑟𝑜 𝜖12 𝜖13 ª
®
𝐷 (𝜖 ℎ𝑦𝑑𝑟𝑜 ) = 𝐼 + 𝜖 𝐹𝑢𝑙𝑙 = (D.2)
®
𝜖21 1 + 𝜖22 + 𝜖 ℎ𝑦𝑑𝑟𝑜 𝜖23 ®
®
®
𝜖31 𝜖32 1 + 𝜖33 + 𝜖 ℎ𝑦𝑑𝑟𝑜
« ¬
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The lattice distortion tensor 𝐷 ℎ𝑦𝑑𝑟𝑜 can now be used to evaluate the real space lattice vector in
strained condition using the following relation.
−−−−−−−→ −
→
𝑅𝑆𝑡𝑟𝑎𝑖𝑛𝑒𝑑 = 𝐷 (𝜖 ℎ𝑦𝑑𝑟𝑜 ). 𝑅0 (D.3)
Equivalently, equation D.3 can be written in terms of the reciprocal lattice vectors.
−−−−−−−−−−−−−−−→ −−−−−−−−−→
𝑄 𝑆𝑡𝑟𝑎𝑖𝑛𝑒𝑑 (ℎ, 𝑘, 𝑙) = 𝐷 (𝜖 ℎ𝑦𝑑𝑟𝑜 ) −1 .𝑄 0 (ℎ, 𝑘, 𝑙) (D.4)
For a set of reciprocal lattice vectors 𝑎 ∗ , 𝑏 ∗ and 𝑐∗ the magnitude of the 𝑄 vector is given as
follows.
→
− →
− →
−
∥𝑄∥ = ℎ. 𝑎 ∗ + 𝑘. 𝑏 ∗ + 𝑙. 𝑐∗ (D.5)
The EW scan measures the magnitude of the 𝑄 vector. Therefore from equation D.5 , the
relationship between the measured 𝑄 vector and lattice distortion tensor can be written as:
𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = ∥𝑄 𝑆𝑡𝑟𝑎𝑖𝑛𝑒𝑑 (ℎ, 𝑘, 𝑙)∥ = 𝐷 (𝜖 ℎ𝑦𝑑𝑟𝑜 ) −1 .𝑄 0 (ℎ, 𝑘, 𝑙) (D.6)
Rearranging the terms in equation D.6, the fitness function is obtained as follows.
𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝐷 (𝜖 ℎ𝑦𝑑𝑟𝑜 ) −1 .𝑄 0 (ℎ, 𝑘, 𝑙) = 0 (D.7)
Equation D.7 is a function of a single scalar 𝜖 ℎ𝑦𝑑𝑟𝑜 . In order to solve the optimization problem
a continuous interval which contains a solution satisfying equation D.7 is considered. In this case
the interval [−0.02, 0.02] is considered to contain a solution that satisfies the fitness function, i.e.
the hydrostatic stress lies within this closed interval. The estimation of the hydrostatic strain can
be done using an implementation of Brent optimization algorithm available under SciPy.
D.0.2 Energy wire scan using monochromatic beam diffraction
In order to obtain the complete local strain tensor description with the hydrostatic component,
it is necessary to determine the magnitude of the reciprocal lattice vector for a given set of crys-
161
tallographic planes. This can be accomplished by making the incident x-ray beam monochromatic
for a range of known energies (resulting in a series of energy wire (EW) scans). This enables the
calculation of the hydrostatic strain component associated with lattice dilatation.
In this work, EW scans were done at the approximate surface centers of the hard and soft oriented
grains. For both grains, scans were done prior to bending, and at the same location following each
subsequent bend increments. The scan for the final bend increment in the soft grain could not be
completed due to beam time constraints. Figure D.1 shows the variation of the magnitude of the
reciprocal lattice vector with depth for the hard and soft grains at different bending increments.
As expected, peak broadening is observed for both grains with increasing bulk strain. In the
unstrained state, the peak intensity for the softer grain is an order of magnitude higher than for the
hard oriented grain, due to reflections from different crystallographic planes. With increasing bulk
strain the peak intensity decreases for the soft grain, indicating an increasing amount of defects
such as dislocations. For the hard grain, the peak intensities for bend increments 3 and 4 are more
similar to each other, indicating that there is some change in lattice orientation and defect content
at the final strain state but less than in the soft grain. For the first bending increment, a change
in energy is noted from the unstrained state for both the soft and hard oriented grains. In the soft
orientation, the intensity peak shift to the the right indicates an an increase in the magnitude of
the reciprocal lattice vector (|𝑄|), or compression in real space. The peak intensity for the soft
grain moves to the right after strain is applied. Thereafter, little change in |𝑄| is apparent for the
bend increments 2 and 3. On the other hand, the hard oriented grain shows a peak shift to the left,
indicating a tensile change in real space. There is a sequential shift of the intensity peak through
the first, second and third bending increments. For the final bending increment there is a noticeable
shift to the right, resulting in decrease of |𝑄|. These changes reflect effects of elastic anisotropy
during loading and changes in grain shape with plastic strain.
In the current study, the EW scans are taken from the center of a coarse grain in a single phase
material. Furthermore the specimen was subjected to a relatively low strain, whereby the lattice
dilatation would be very small. Therefore, within a single grain, the assumption of a scalar value of
162
(a) Soft Grain (b) Hard Grain
Bend Increment 1 Prior to Bending Bend Increment 1
Prior to Bending
Depth of subsurface grain boundary
Bend Increment 2 Bend Increment 2 Bend Increment 3
Bend Increment 3
Bend Increment 4
Figure D.1: Laue patterns obtained from energy wire scans of (a) soft grain (unstrained condition
through bend increment 3), (b) hard grain (unstrained condition through bend increment 4). The
variation of reciprocal lattice vector magnitude with depth is shown for each bend increment. The
plots on the third row of (a) and (b) show the intensity profiles as a function of the reciprocal lattice
vector magnitude. The approximate locations in the grain interior used for the energy wire scans
in the hard and soft grains are shown (marked in red) as insets in the intensity plots. The depths at
which the subsurface grain boundary exists is shown by blue vertical lines in the 𝑄 v/s depth plots
for both the hard and soft orientations.
hydrostatic strain is valid. Moreover, These results indicated that within a single grain, the spatially
resolved lattice dilatation can also be considered to be uniform.
As the interrogated volume approaches the grain boundaries the assumption of a scalar value
of hydrostatic strain is not valid, because of complex interactions with neighboring grains. This is
evident in the change in Q as one approaches the boundary in the hard grain at bend increment 1. In
bend increments 2-4, the peak intensity drops and becomes unmeasurable. Ideally, spatially resolved
EW scans have to be conducted to obtain the reciprocal lattice vector magnitude corresponding to
each voxel, which is a time-consuming exercise in an actual experiment. In the present work, EW
163
scans were conducted near the center of two grains. The important assumption made for subsequent
calculation of the hydrostatic strain is the absence of strong orientation and stress gradients in these
locations.
Y
Unstrained state Bend increment 1
X
Hydrostatic Strain Loading direction
Bend increment 2 Bend increment 3
Figure D.2: Local hydrostatic strain maps for the soft (left) and hard (right) grains, shown for bend
increments 0 (unstrained state) to 3. The prisms indicating the orientations of the hard and soft
grains with respect to the loading direction are also shown.
Figure D.2 shows the spatial distribution of hydrostatic strain in the hard and soft oriented
grains; corresponding to the unstrained state and first three bending increments. A larger spread
of local hydrostatic strain is observed for the soft grain compared to the hard grain. The character
of the hydrostatic strain for the softer orientation changes from near zero to compressive with
increasing bulk strain. In the case of the harder orientation, the character of the dilatation changes
from near zero to tensile at the first bend increment; and changes to compressive with increasing
164
bulk strain. In both cases the magnitudes of the hydrostatic strain are found to be small; therefore
lattice dilatation is not significant.
165
APPENDIX E
PRINCIPAL STRESS AND PRINCIPAL DIRECTION MAPS FOR REGIONS 3,4 AND 5
For the three grains associated with the triple junction in region 3 (figure E.1), a change in sense
of the first principal stress component from compressive to tensile (darker red) from the center
towards some of the grain boundary neighborhood voxels is observed in the lower two grains,
where the direction also changes as one gets closer to the boundary. It is also pertinent to note that
that average stress state in the lower left grain is nearly biaxial, since the magnitude of the third
principal component is relatively low. In the upper (interior) grain, there is a band that traverses
from left to right where there is a strong (more vertical) tensile component that is in a different
direction than the average, and there are more compressive regions in the upper left and lower right
regions close to grain boundaries.
Considering the grain boundary in region 4 (figure E.3), the first principal components for both
the top and bottom grain are characterized by a near-balance in tensile and compressive states,
where the compressive component is stronger in the upper (interior) grain and stronger tensile in
the lower (surface) grain. There is a band emanating from the grain boundary toward the upper
right that shows a different direction of the tensile component. For the bottom grain the tensile
component becomes stronger close to the grain boundary, along with small change in direction.
In the case of the triple junction in region 5 (figure E.2) the stress state in the left two grains
is quite uniform, and a near-balance of tensile and compression in the first two principal stresses.
There is a band of strong tensile stress near but away from the lower right grain boundary in the
lower left grain, and the stress along the boundary differs from the grain interior. The upper right
(interior) grain has isolated voxels with significantly different stress states, and there is a subtle
waver of different stress direction in the middle of the grain. The second component of the principal
stress for the top right grain changes significantly in direction and sense (compressive to tensile)
near the boundary with the top left grain. It can be seen that the grain averaged principal stress state
in the three grains is almost biaxial, as one of the three components has a much lower magnitude
166
compared to the other two.
Region 3
175
-255
83
Principal stress jack based on
volume averaged stress tensor
Figure E.1: Principal Stress direction map for the triple junction in region 3. The grain boundaries
are delineated by black lines. Principal stress jacks corresponding to average stress tensor for each
individual grain, along with their magnitudes and sense are also shown for comparison.
167
230 Region 5 211
-235
-5 -223 27
-225
43
175
-255
183
83
Principal stress jack based on
volume averaged stress tensor
Figure E.2: Bend increment 2: Principal Stress distribution for the triple junction in region 5.
168
Region 4
122
-366
246
175
-255
83
340
Principal stress jack based on
volume averaged stress tensor
72 -407
Figure E.3: Bend increment 2: Principal Stress distribution for the triple junction in region 4.
169
APPENDIX F
CORRELATION OF GND CONTENT WITH RESIDUAL BURGERS VECTOR AND
SLIP TRANSFER PARAMETER
F.0.1 Analysis of intragranular high local GND density accumulation area
It can be expected that instances where the likelihood of slip transfer is high (𝑚′ values closer
to 1), would correspond to lower magnitudes of residual Burgers vector. Correspondingly, a lower
value of 𝑚′ should correspond to higher magnitudes of 𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 . Figure F.2 shows the plots of
𝑚′ as a function of |𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | for the LAGB and HAGB of region 1. Each symbol in these plots
corresponds to one pair of slip system interactions for four adjacent voxel pairs. Each voxel in this
interacting pair has a unique crystallographic orientation and deviatoric stress tensor associated
with it. The face color and shape of each symbol indicate the transmitting slip system chosen (in
this case arbitrarily as described in section 7.4.3 and listed in table 7.2). The edge color of each
symbol denotes the slip system in the receiving voxel. As an example, if a transmitted basal slip
system interacts with a prism slip system in the receiving voxel, the symbol is a triangle with blue
face color and red edge color. In addition, the size of each symbol1 is scaled with the sum of the
local Schmid factors of the interacting slip systems for the voxel pair considered.
F.0.1.1 Analysis of 𝑚′ and |𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 |
Figure F.2 shows the results of four voxel pairs along the grain boundaries. For each type of
initiating slip system (basal, prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩) in each voxel pair, only
the instances with the five highest local Schmid factor sums are plotted (20 data points per plot).
From purely geometrical considerations, an inverse relationship should exist between the value
of 𝑚′ and |𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 |, which is evident in some of the plots, where slip transfer would be expected
for slip system pairs in the upper left corner, and blocked slip at the boundary in the lower right.
1 Symbol Size is proportional to the sum of the local Schmid factors (LSf); size =500 × 𝑠𝑢𝑚(𝐿𝑆 𝑓𝑠 𝛼 , 𝐿𝑆 𝑓𝑠𝛽 ) 3 )
170
Region 2 Region 3
[98,70, 2]
[85, 60, 3]
[345, 122, 77]
[56, 103,359]
[104, 88, 352]
[291, 103, 106]
Y
Total GND (⍴/m 3)
Prisms
Region 4 Region 5 X
[104, 88, 352]
[178, 76, 350] [252, 95, 79]
[57, 97, 315] [245, 80, 86]
Figure F.1: Orientation and total GND density maps for regions 2, 3, 4 and 5, corresponding
to the second bending increment. The voxels on either side of the grain boundary used for slip
transfer parameter calculations are shown on the right of each orientation map. The Bunge Euler
angles denoting the average grain orientations are also shown. Prisms denoting orientations are
drawn from the perspective of the sample normal, with 𝑌 pointing up and 𝑋 pointing to the right
(coordinate system shown on the right side of image). The sample coordinate system is shown on
the left hand side of image. The tensile direction is parallel to the 𝑌 direction.
However, there is no discernable trend in these plots, so the next step is to take into account the
role of slip system specific GND density that reflects accumulation of a high density of GNDs or
transmission of dislocations through the boundary, resulting in a low density.
F.0.1.2 Analysis of 𝑚′ and |𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | with GND density
The local slip system specific GND densities in each voxel pair can be considered by choosing
a scaling parameter for 𝑚′ based on either the maximum or the minimum values of GND density
pertinent to the slip system pair. Therefore, the scaled slip transfer parameter (𝑚′′) may be expressed
171
1
Basal-Basal Pyramidal -Basal
Basal-Prism Pyramidal -Prism
0.75 Pyramidal -Pyramidal
Basal-Pyramidal
Pyramidal -Pyramidal
Basal-Pyramidal
Pyramidal -Basal
Pyramidal -Prism
0.25
Pyramidal -Pyramidal
Pyramidal -Pyramidal
0
1
0.75
0.25
0
0.4 0.6 0.4 0.6 0.4 0.6 0.4 0.6
Figure F.2: The Luster-Morris parameter plotted as a function of magnitude of residual Burgers
vector for the LAGB (top row) and HAGB (bottom row) in region 1. Legends in the plots show the
color/shape convention used to denote the interacting slip system in a voxel pair. This convention
is followed for the subsequent plots. The symbol size for each datum point is scaled with the sum
of the local Schmid factors for the interacting slip systems.
by one of the following equations.
𝑚′𝛼𝛽 𝑚′𝛼𝛽
′′ ′′
𝑚 = 𝑜𝑟 𝑚 = (F.1)
𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 )
where 𝑚′𝛼𝛽 is the slip transfer parameter, considering slip systems 𝑠𝛼 and 𝑠 𝛽 for voxels 𝛼 and 𝛽
respectively.
Figure F.3 shows the plots of the scaled values of 𝑚′ and |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | for the LAGB in region
1. Plots using 𝑚𝑎𝑥(𝜌𝛼 , 𝜌 𝛽 ) and 𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝛽 ) as scaling parameters are shown on the left and
right columns respectively in figure F.3. Comparing with the LAGB row of plots in figure F.2,
use of these scaled parameters exaggerates the inverse relationship between the residual Burgers
vector magnitude and slip transfer parameter, such that a slope to the points is evident. Using the
minimum dislocation density causes points to be more concentrated. As discussed in section 7.4.3,
the transmitting slip systems are arbitrarily chosen to emanate from voxels on one side of the grain
boundary.
172
F.0.1.3 Analysis with GND density and choice of transmitting slip system by LSf
Another approach is to specifically choose the transmitting slip system as the one with the
higher local Schmid factor between 𝑠𝛼 and 𝑠 𝛽 based upon the assumption that the more favored
slip system is the initiating system. These data are then sorted by initiator slip system type (basal,
prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩). The same data as in figure F.3, are plotted using
this modified approach in figure F.4. In this case, the number of symbols plotted in each figure
will not necessarily be equal, as one grain may have more highly favored slip systems than the
other. This is evident in the fewer number of initiating systems on the basal plot, and the larger
number of initiating systems in the prism plot and pyramidal plots. The number of datum
points appearing in each plot are indicated in figure F.4.
F.0.1.4 Analysis with GND density and choice of transmitting slip system by 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 )
The third strategy for choosing the initiating slip system is based on the minimum value of slip
system specific GND density between the interacting systems. Figure F.5 shows the 𝑚′′ and scaled
|𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots generated for the LAGB in region 1 using this approach. For the LAGB using
𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) as the transmitter criterion results in higher number of data points plotted for basal
and pyramidal⟨𝑎⟩ slip compared with 𝑚𝑎𝑥(𝐿𝑆 𝑓 ) (figure F.4).
F.0.1.5 Analysis of slip systems filtered by 𝑚′ >= 0.8
The data plotted for the LAGB in region 1 (figures F.3, F.4 and F.5) are filtered based on a slip
transmissibility criterion (only points with 𝑚′>=0.8 are considered) and re-plotted in figures F.6,
F.7 and F.8 respectively.
For the LAGB in region 1, no basal slip system is found to have a favorable geometrical alignment
(𝑚′ >=0.8) for slip transfer. Similarly, there was only one plausible instance of pyramidal⟨𝑐 + 𝑎⟩
with a high likelihood for transfer. The number of basal and pyramidal⟨𝑐 + 𝑎⟩ systems likely to
transmit did not change with the criteria used for choice of the initiator (figures F.6, F.7 and F.8).
173
Clearly there is a significant variation in the number of high 𝑚′ prism⟨𝑎⟩ and pyramidal⟨𝑎⟩ across
the three choice criteria. In case of the arbitrary choice criterion, there are are nine instances of
favorable geometric alignment for transfer of prism⟨𝑎⟩ slip across the boundary (𝑚′ >= 0.8), out
of which four instances have near perfect geometrical alignment (𝑚′ ∼ 1.0). When the maximum
LSf is considered as the criterion for choice of transmitting slip system, the number of plausible
instances is found to be eight (with 2 having 𝑚′ ∼ 1.0). The number of plausible instances for
prism⟨𝑎⟩ slip increases to eleven (with four near perfect cases), when 𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝛽 ) is considered
as the choice criterion. All of the eight data points correspond to low values of GND density.
There are also seven possible instances where a high likelihood of pyramidal⟨𝑎⟩ slip transfer can
take place. When pyramidal⟨𝑎⟩ slip is considered the number of plausible slip transfer instances
using the three criteria are six (one near perfect), seven (two near perfect) and four (2 near perfect)
respectively.
It is worth noting that no favorable conditions for transfer of basal slip were determined using
both the maximum LSf and minimum GND density as choice criteria for the transmitting system.
When prism⟨𝑎⟩ slip is considered using maximum LSf and minimum GND density criteria yield
eight and eleven favorable instances respectively. Out of these six instances are identical between
the two choice criteria used. In the case of pyramidal⟨𝑎⟩ slip, the number of favorable instances
found using the two criteria are seven and four respectively, out of which three are identical. It
is pertinent to mention here that across the three criteria examined, the same slip system pair for
pyramidal⟨𝑐 + 𝑎⟩ is found to be the only plausible instance of slip transfer.
Because this neighborhood has very low misorientations, geometrically every slip system
should be highly favored for slip transfer, but the criteria of dislocation density and local stress
state provides additional constraints on what slip transfer processes are more likely to be active.
The lack of many slip systems participating in plausible slip transfer suggests that there is much
dislocation entanglement in this neighborhood.
Some of the slip system pair interactions for basal, prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩
initiating systems are illustrated in figure F.9. Here the size of each prism pair illustrating the
174
crystallographic orientations are scaled in proportion to the sum of the LSf of the interacting slip
systems. Although near-perfect geometrical alignment for basal slip transfer exists, the variation
in the local stress state makes the activation of basal slip less likely. From the standpoint of near
ideal geometrical alignment and local stress state, there is high likelihood for transfer of prism⟨𝑎⟩
and pyramidal⟨𝑎⟩ slip. The only favorable instance of pyramidal⟨𝑐 + 𝑎⟩ slip transfer is also shown
(bottom right of figure F.9).
F.0.2 Analysis of slip transfer in the HAGB
Plots of the scaled slip transfer parameter as a function of scaled magnitude of 𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 for
HAGB in region 1 using the three different criteria for transmitting slip choice are shown in figures
F.10,F.11 and F.12 respectively. The data is plotted in an identical manner as described for figures
F.3,F.4 and F.5 respectively for the LAGB case.
For the HAGB in region 1, five favorable instances of basal slip transfer, in terms of geometrical
alignment and local Schmid factor, are found irrespective of the criteria used for choice of transmit-
ting system. If the transmitting slip system is arbitrarily chosen, no favorably aligned candidates
for prism⟨𝑎⟩ slip transfer (𝑚′>=0.8) were found. When the maximum Schmid factor is considered
as the criterion for transmitting slip system choice, two plausible instances of prism⟨𝑎⟩ are found.
Using the criterion of minimum GND density, only one favorable instance for transfer of prism⟨𝑎⟩
slip is identified. Overall, the likelihood of prism⟨𝑎⟩ slip transferring across the grain boundary is
low. Similarly the likelihood for pyramidal⟨𝑎⟩ slip transfer is also low. Three favorable candidates
for possible transfer of pyramidal⟨𝑎⟩ slip are noted when the maximum Schmid factor criterion is
used to choose the transmitting system.
The five instances where basal slip transfer is favored are identical irrespective of the choice
criteria used. Two instances of favorable pyramidal⟨𝑎⟩ slip transfer are identical using both
maximum Schmid factor and minimum GND density conditions. The sole favorable instance of
pyramidal⟨𝑐 + 𝑎⟩ slip transfer is also identical using both conditions.
175
F.0.3 Analysis of slip transfer in triple junction grain boundaries
Figures F.16, F.17 and F.18 show the plots of 𝑚′′ as a function of scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | for each
of the three grain boundaries constituting the triple junction in region 2. For the analysis of this
region, 𝑚𝑖𝑛(𝜌𝛼 , 𝜌 𝛽 ) is used as the basis for determining the transmitting slip system in a pair of
voxels.
Figure F.16 shows the plots of 𝑚′′ plotted as a function of scaled 𝑏𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 for the grain boundaries
that constitute the triple junction in region 2. The first grain boundary (first column of plots in
figure F.16) is between a relatively hard oriented grain (beige, inset) and a soft oriented grain (blue,
inset). There were no instances that favored transfer of basal slip across the boundary. Only one
instance that could favor transfer of prism⟨𝑎⟩ slip was identified (𝑚′ ∼ 0.9). This is consistent with
very low likelihood of activation of prism⟨𝑎⟩ slip; especially if slip transfer is deemed to occur
from the hard oriented grain to the softer grain. There was one possible instance each, favoring
transfer of pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩ slip respectively. The locations of the datum points
corresponding to these favorably aligned instances of slip transfer are circled in figure F.16.
The second grain boundary in region 2 (figure F.17) is between the blue and purple grains
shown in the inset. There was only one favorable instance each for transfer of basal and prism⟨𝑎⟩
slip respectively. Four cases with favorable geometrical alignment for transfer of pyramidal⟨𝑎⟩;
and one such instance for pyramidal⟨𝑐 + 𝑎⟩ were found, but the plotted values are lower and to the
right than the other two grain boundaries.
The third grain boundary in region 2 ( F.18) is formed by the hard grain and softer oriented
purple grain. No instances favoring transfer of basal or pyramidal⟨𝑐 + 𝑎⟩ slip were identified, but
there were three scenarios where prism⟨𝑎⟩ slip appears likely. Four pyramidal⟨𝑎⟩ systems with
high likelihood of transmitting were also identified.
Plots for the grain boundaries in regions 3, 4 and 5 are included in the appendix G.
In several of the slip system pairs considered, it is observed that in cases where the likelihood
of slip transfer is high (𝑚>=0.9), the magnitude of slip system specific dislocation density is low.
Figures F.19, F.20, F.21 and F.22 show 𝑚′ plotted as a function of the maximum and minimum
176
values of slip system specific dislocation density, using voxel pair instances from the twelve
grain boundaries encompassed by regions 1, 2, 3, 4 and 5. Here the transmitting slip systems
(basal,prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩ respectively) are chosen using the minimum
GND density of the voxel pair.
In figure F.19, 𝑚′ and GND densities are plotted for cases where basal slip is considered as the
transmitting system. In the plot on the left hand side, where 𝑚′ varies with respect to the maximum
GND density, an envelope of data points can be found where 𝑚′ decreases with increasing values
of GND density. Furthermore, interactions with ⟨𝑐 + 𝑎⟩ have much lower m" values than potential
slip transfer with the other three slip systems. A similar trend can be seen for the case of prism⟨𝑎⟩
and pyramidal⟨𝑎⟩ initiator slip systems (figures F.20 and F.21 respectively). This trend is less
obvious when pyramidal⟨𝑐 + 𝑎⟩ slip is considered as the initiator in figure F.22, suggesting that
pyramidal⟨𝑐 + 𝑎⟩ slip as the initiation system is much less likely, which may reflect the preferred
soft orientations that predominate in this material.
177
GB misorienta�on<2∘
5 5
10 10
Basal-Basal
Basal-Prism
10-5 Basal-Pyramidal 10-5
Basal-Pyramidal
-15 -15
10 10
105 105
Prism -Prism
Prism -Basal
-5 -5
10 Prism -Pyramidal 10
Prism -Pyramidal
10-15 10-15
5 5
10 Pyramidal -Basal
10
Pyramidal -Prism
-5
10 Pyramidal -Pyramidal
Pyramidal -Pyramidal
10-5
-15 -15
10 10
105 Pyramidal -Basal
105
Pyramidal -Prism
10-5 10-5
Pyramidal -Pyramidal
Pyramidal -Pyramidal
10-15 10-15
10-20 100 1015 10-20 100 1015
𝑚′
Figure F.3: LAGB in region 1 scaled by GND density: Left column shows 𝑚′′ = 𝑚𝑎𝑥(𝜌 𝑠 𝛼 ,𝜌 𝑠𝛽 )
𝑚′
plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 |; while 𝑚′′ = 𝑚𝑖𝑛(𝜌 𝑠 𝛼 ,𝜌 𝑠𝛽 ) as a function of
𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) × |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots are shown on the right column. Each plot corresponds to one of
four transmitting slip systems considered. Four voxel pairs along the grain boundary are considered
here. For each transmitting slip system type in each voxel pair, only the data points with the five
highest local Schmid factor sums are plotted (A total of 20 plots per plot). The choice of the
transmitting slip system is arbitrarily made as described in section 7.4.3.
178
GB misorienta�on<2∘
5 5
10 10
10-5 10-5
6 Data points
-15 -15
10 10
5 5
10 10
-5 -5
10 10
19 Data points
-15 -15
10 10
5 5
10 10
-5 -5
10 10
43 Data points
-15 -15
10 10
5 5
10 10
-5 -5
10 10
12 Data points
-15 -15
10 10
-20 0 15 -20 0 15
10 10 10 10 10 10
Figure F.4: For four voxel pairs in the LAGB in region 1, each row of plots corresponds to the
initiator slip system families with the higher local Schmid factor considered as the transmitting
system, leading to more datum points in the pyramidal slip system plot as initiators than other
systems. For each initiating slip system family member in each voxel pair, only the datum points
with the five highest local Schmid factor sums are considered. The number of data points plotted
for each initiator slip system type are indicated. The number of datum points per slip system family
for the plots scaled by maximum and minimum GND density (left and right columns respectively)
are identical.
179
GB misorienta�on<2∘
5
10 105
10-5 10-5
18 Data points
-15
10 10-15
105 105
10-5 10-5
23 Data points
-15
10 10-15
105 105
10-5 10-5
28 Data points
10-15 10-15
105 105
10-5 10-5
11 Data points
-15
10 10-15
-20 0 15
10 10 10 10-20 100 1015
Figure F.5: For four voxel pairs in the LAGB in region 1, each row of plots corresponds to the
initiator slip systems with the lower GND density is considered as the transmitting system, leading
to more datum points in the basal and prism⟨𝑎⟩ slip system plot as initiators than when the Schmid
factor is considered in figure F.4. For each voxel pair, only the data points with the five highest
local Schmid factor sums are considered. The number of data points plotted for each initiator slip
system type are indicated.
180
GB misorienta�on<2∘ Arbitrary choice of initiator system
5 5
10 10
10-5 10-5
0 Data points
-15
10 10-15
105 105
10-5 10-5
9 Data points
-15
10 10-15
105 105
10-5 10-5
6 Data points
-15
10 10-15
105 105
10-5 10-5
1 Data point
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure F.6: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown in figure
F.3, filtered using a criterion of 𝑚′>=0.8.
181
GB misorienta�on<2∘ max(LSF) used as criterion for choice of initiator system
105 105
10-5 10-5
0 Data points
-15
10 10-15
105 105
10-5 10-5
8 Data points
-15
10 10-15
105 105
10-5 10-5
7 Data points
-15
10 10-15
105 105
10-5 10-5
1 Data point
-15
10 10-15
-20 0 15
10 10 10 10-20 100 1015
Figure F.7: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown in figure
F.4, filtered using a criterion of 𝑚′>=0.8.
182
5
GB misorienta�on<2∘ used as criterion for choice of initiator system
10 105
10-5 10-5
0 Data points
10-15 10-15
105 105
10-5 10-5
11 Data points
-15
10 10-15
105 105
10-5 10-5
4 Data points
10-15 10-15
105 105
10-5 10-5
1 Data point
-15
10 10-15
-20 0 15
10 10 10 10-20 100 1015
Figure F.8: LAGB region 1: 𝑚′′ v/s scaled |𝑏 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 | plots for the same data points shown in figure
F.5, filtered using a criterion of 𝑚′>=0.8.
183
[98 70 2] [98 70 3] Region 1: LAGB [98 70 2] [98 70 3]
LSF₁ = 0.36 LSF₂= 0.37 LSF₁ = 0.15 LSF₂= 0.16
LSF₁ = 0.57 LSF₂= 0.55 LSF₁ = 0.55 LSF₂= 0.60
m' ~ 1 m' ~ 1 m' = 0.878
m' ~ 1
Basal
Prism
[98 70 2] [98 70 3]
[98 70 2] [98 70 3]
LSF₁ = 0.58 LSF₂= 0.60
m' ~1 LSF₁ = 0.57 LSF₂= 0.60 LSF₁ = 0.4 LSF₂= 0.1
m' ~1 m' ~1
Pyramidal Pyramidal
Figure F.9: Some instances of slip system interactions for basal, prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and
pyramidal⟨𝑐 + 𝑎⟩ initiating systems for the LAGB voxels in region 1 are shown. The LSf values,
which were calculated using the local stress tensor, and the Bunge Euler angles and 𝑚′ values are
noted. The size of the prisms is proportional to the sum of the LSf values of the sum of the Schmid
factors for the slip system pair. Prisms are drawn from the perspective of the sample normal, with
𝑌 pointing upwards and 𝑋 pointing towards the right.
184
GB misorienta�on~28∘
5
10 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 -20 10 0
10 15
10 -20 10 0 10 15
Figure F.10: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function of scaled
residual Burgers vector. The methodology used to plot the data is identical to F.3. In this case the
initiator in an interacting pair of slip systems is chosen arbitrarily as described in section 7.4.3.
185
GB misorienta�on~28∘
5
10 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure F.11: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function of scaled
residual Burgers vector, using the maximum of LSf as criterion for choosing the transmitting slip
system.
186
GB misorienta�on~28∘
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure F.12: HAGB region 1: Scaled slip transfer parameter 𝑚′′ plotted as a function of scaled
residual Burgers vector, using the minimum of GND density as criterion for choosing the transmit-
ting slip system.
187
GB misorienta�on~28∘ Arbitrary choice of initiator system
5
10 105
10-5 10-5
5 Data points
-15
10 10-15
105 105
10-5 10-5
2 Data points
-15
10 10-15
105 105
10-5 10-5
1 Data points
10-15 10-15
105 105
10-5 10-5
1 Data point
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure F.13: HAGB region 1: Same data as plotted in figure F.10: only points with 𝑚′>=0.8 are
plotted here .
188
GB misorienta�on~28∘ max(LSF) used as criterion for choice of initiator system
105 105
10-5 10-5
5 Data points
-15
10 10-15
105 105
10-5 10-5
0 Data points
10-15 10-15
105 105
10-5 10-5
3 Data points
10-15 10-15
105 105
10-5 10-5
1 Data point
-15
10 10-15
10-20 100 1015 10-20 100 1015
Figure F.14: HAGB region 1: Same data as plotted in figure F.11: only points with 𝑚′>=0.8 are
plotted.
189
GB misorienta�on~28∘ used as criterion for choice of initiator system
105 105
10-5 10-5
5 Data points
10-15 10-15
105 105
10-5 10-5
1 Data point
-15
10 10-15
105 105
10-5 10-5
2 Data points
10-15 10-15
105 105
10-5 10-5
1 Data point
10-15 10-15
10-20 10 0
10 15
10-20 100 1015
Figure F.15: HAGB region 1: Same data as plotted in figure F.12: only points with 𝑚′>=0.8 are
plotted.
190
GB misorienta�on~66∘ used as criterion for choice of initiator system
5 5
10 10
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure F.16: Using the minimum GND density criterion for the transmitting grain, 𝑚′′ is plotted as
a function of scaled residual Burgers vector magnitude for the boundary between the hard (beige)
and soft oriented (blue) grains in region 2. Data plotted is for four pairs of voxels situated along
the grain boundary. Only the top five points (sorted by descending order of sum of LSf values) per
voxel pair are considered here. The locations of the points that correspond to a high likelihood of
slip transfer are encircled.
191
5
GB misorienta�on~22∘ used as criterion for choice of initiator system
10 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 10 0
10 15
10-20 100 1015
Figure F.17: Using the minimum GND density criterion for the transmitting grain, 𝑚′′ is plotted as
a function of scaled residual Burgers vector magnitude; for the boundary formed by the two softer
oriented grains (blue and purple) in region 2. This boundary has the lowest likelihood for slip
transfer of the three. Data plotted is for four pairs of voxels situated along the grain boundary. Only
the top five points (sorted by descending order of sum of LSf values) per voxel pair are considered
here.
192
GB misorienta�on~63∘ used as criterion for choice of initiator system
5 5
10 10
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 -20 10 0 10 15 10 -20 10 0 10 15
Figure F.18: Using the minimum GND density criterion for the transmitting grain, 𝑚′′ is plotted
as a function of scaled residual Burgers vector magnitude for the boundary between the hard beige
and soft oriented purple grain in region 2. Data plotted is for four pairs of voxels situated along
the grain boundary. Only the top five points (sorted by descending order of sum of LSf values) per
voxel pair are considered here.
193
Basal Slip as Initiator
1
0.75
0.5
0.25
0 -5
10 100 105 1010 1015 10-5 100 105 1010 1015
Figure F.19: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ), considering basal slip as
the transmitting system. The transmitting slip system is chosen on the basis on minimum slip system
specific GND density in voxel pair. The data plotted are from 12 grain boundaries encompassed
by regions 1, 2, 3, 4 and 5. The envelope of data points for the 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case shows an
inverse correlation between GND density and slip transmissibility are enclosed within the lines in
each plot. The description of the symbol shape and color scheme of the data points is explained in
section F.0.1
194
Prism Slip as Initiator
1
0.75
0.5
0.25
0 -5
10 100 105 1010 101510-5 100 105 1010 1015
Figure F.20: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ), considering prism⟨𝑎⟩
slip as the transmitting system. The transmitting slip system is chosen on the basis on minimum slip
system specific GND density in voxel pair. Data plotted from 12 grain boundaries encompassed by
regions 1, 2, 3, 4 and 5. The envelope of data points for the 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case shows an inverse
correlation between GND density and slip transmissibility are enclosed within the lines in each
plot.
Pyramidal Slip as Initiator
1
0.75
0.5
0.25
0
-5
10-5 100 105 1010 1015 10 100 105 1010 1015
Figure F.21: Bend increment 2: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ),
considering pyramidal⟨𝑎⟩ slip as the transmitting system. The transmitting slip system is chosen
on the basis on minimum slip system specific GND density in voxel pair. Data plotted from 12
grain boundaries encompassed by regions 1, 2, 3, 4 and 5. The envelope of data points for the
𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) case shows an inverse correlation between GND density and slip transmissibility
are enclosed within the lines in each plot.
195
1
0.75
0.5
0.25
0 -5
10 100 105 1010 1015 10-5 100 105 1010 1015
Figure F.22: Bend increment 2: 𝑚′ plotted as a function of 𝑚𝑎𝑥(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ) and 𝑚𝑖𝑛(𝜌 𝑠 𝛼 , 𝜌 𝑠 𝛽 ),
considering pyramidal⟨𝑐 + 𝑎⟩ slip as the transmitting system. The transmitting slip system is
chosen on the basis on minimum slip system specific GND density in voxel pair. Data plotted from
12 grain boundaries encompassed by regions 1, 2, 3, 4 and 5.
196
APPENDIX G
SLIP TRANSFER PARAMETER V/S RESIDUAL BURGERS VECTOR PLOTS
In this section, the plots of scaled values of the Luster Morris parameter as a function of
scaled magnitude of residual Burgers vector are shown for regions 3, 4 and 5 of the microstructure
described in 7.4.3. Details of the parameter descriptions can be found in 7.4.3 and F.0.1. The plots
shown are for interacting slip systems in four voxel pairs on either side of a grain boundary. For
details on the symbol conventions used, the reader is referred to F.0.1. In all of the plots shown in
this section, the initiating slip system for each voxel pair is chosen based on the minimum value of
the slip system specific GND density.
The plots for the three constituent grain boundaries in the triple junction for region 3 are shown
in figures G.1, G.2 and G.3. For the first grain boundary shown in figure G.1, only one basal⟨𝑎⟩
system was found to have a high likelihood of slip transfer (𝑚′>=0.8); whereas no instances of
prism⟨𝑎⟩, pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩ are likely to transfer.
In case of the second grain boundary in region 3 (figure G.2), the only likely slip system to
transfer is pyramidal⟨𝑎⟩ (3 instances). For the third grain boundary in region 3 (figure G.3), the
number of slip systems with favorable geometrical compatibility for transfer of basal⟨𝑎⟩, prism⟨𝑎⟩
and pyramidal⟨𝑐 + 𝑎⟩ are 2, 1 and 2 respectively.
For the grain boundary in region 4, the only geometrically favorable instances of slip transfer
correspond to pyramidal⟨𝑎⟩ and pyramidal⟨𝑐 + 𝑎⟩ (1 each) (figure G.4).
For the first grain boundary of the triple junction in region 5 (figure G.5), the only favorable
instances of slip transfer correspond to basal⟨𝑎⟩ (2) and pyramidal⟨𝑐 + 𝑎⟩ (2) systems. With
respect to the second grain boundary in region 5 (figure G.6), the favorable instances for slip
transfer correspond to basal⟨𝑎⟩ (3), prism⟨𝑎⟩ (4) and pyramidal⟨𝑎⟩ (2). There were no instances
with a favorable geometrical compatibility for slip transfer for the third grain boundary in region 5
(figure G.7).
197
GB misorienta�on~62∘ used as criterion for choice of initiator system
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure G.1: Bend increment 2: Scaled slip transfer parameter plotted as a function of scaled
residual Burgers vector magnitude for the first grain boundary (shown in the inset) constituting the
triple junction in region 3 of microstructure described in 7.4.3. The choice of the transmitting slip
system is made on the basis of minimum GND density of the voxel pair.
198
GB misorienta�on~36∘ used as criterion for choice of initiator system
5 5
10 10
-5 -5
10 10
-15 -15
10 10
5 5
10 10
-5 -5
10 10
-15 -15
10 10
105 105
-5 -5
10 10
10-15 10-15
5 5
10 10
10-5 10-5
-15 -15
10 10
-20 0 15 -20 0 15
10 10 10 10 10 10
Figure G.2: Bend increment 2: Scaled slip transfer parameter plotted as a function of scaled
residual Burgers vector magnitude for the second grain boundary (shown in the inset) constituting
the triple junction in region 3 of microstructure described in 7.4.3. The choice of the transmitting
slip system is made on the basis of minimum GND density of the voxel pair.
199
GB misorienta�on~51∘ used as criterion for choice of initiator system
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 -20 10 0 10 15 10 -20 10 0 10 15
Figure G.3: Bend increment 2: Scaled slip transfer parameter plotted as a function of scaled
residual Burgers vector magnitude for the third grain boundary (shown in the inset) constituting the
triple junction in region 3 of microstructure described in 7.4.3. The choice of the transmitting slip
system is made on the basis of minimum GND density of the voxel pair.
200
GB misorienta�on~54∘ used as criterion for choice of initiator system
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure G.4: Bend increment 2: Scaled slip transfer parameter plotted as a function of scaled
residual Burgers vector magnitude for the grain boundary in region 4 of microstructure described
in 7.4.3.
201
GB misorienta�on~19∘ used as criterion for choice of initiator system
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 100 1015 10-20 100 1015
Figure G.5: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted as a
function of scaled residual Burgers vector magnitude for the first grain boundary constituting the
triple junction in region 5 of microstructure described in 7.4.3. Each column of plots correspond
to slip system interaction at a grain boundary (inset).
202
GB misorienta�on~80∘ used as criterion for choice of initiator system
105 105
-5 -5
10 10
10-15 10-15
105 105
10-5 10-5
10-15 10-15
105 105
-5 -5
10 10
10-15 10-15
105 105
10-5 10-5
10-15 10-15
10-20 10 0
10 15
10-20 100 1015
Figure G.6: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted as a
function of scaled residual Burgers vector magnitude for the second grain boundary constituting
the triple junction in region 5 of microstructure described in 7.4.3. Each column of plots correspond
to slip system interaction at a grain boundary (inset).
203
GB misorienta�on~66∘ used as criterion for choice of initiator system
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 5 10 5
10 -5 10 -5
10 -15 10 -15
10 -20 10 0 10 15 10 -20 10 0 10 15
Figure G.7: Bend increment 2: Bend increment 2: Scaled slip transfer parameter plotted as a
function of scaled residual Burgers vector magnitude for the third grain boundary constituting the
triple junction in region 5 of microstructure described in 7.4.3. Each column of plots correspond
to slip system interaction at a grain boundary (inset).
204
APPENDIX H
COORDINATE TRANSFORMATIONS
The following sections show the steps necessary for transformation to TSL-OIM coordinate
system from APS beamlines 1 and 34 ID-E respectively. It is important to note here that the final
TSL coordinate system for both transformations has 𝑋 pointing upwards, 𝑌 to the left and 𝑍 coming
out of the page.
H.1 34 ID-E beamline coordinate system to TSL (OIM) coordinate system
Figure H.1 shows the steps required to calculate the rotation matrix for coordinate transformation
from APS beamline 34 ID-E to TSL (OIM) system. The two coordinate systems have a common
𝑋 axis (Normal to the plane of the page). The direction of the incoming beam is coincident with
the beamline 𝑍 axis. The APS system can be brought into coincidence with the TSL system by a
clockwise rotation of 135°about the 𝑋 axis. The transformation matrix to accomplish this rotation
(𝑔𝑇34𝐼
𝑆𝐿 ) is calculated from the the direction cosines between the two coordinate systems.
𝐷𝐸
H.2 ID-1 beamline coordinate system to TSL (OIM) coordinate system
The transformation from the ID1 coordinate system to the TSL system is shown in figure H.2.
The direction of the beam is anti-parallel to the ID1 𝑋 axis. In order to bring the ID1 system into
coincidence with the TSL coordinate system, a counter clockwise rotation of 90°about the common
𝑋 axis is required. The corresponding transformation matrix 𝑔𝑇𝐼 𝐷1 𝑆𝐿 is also shown in figure H.2.
205
Y34IDE
CW ZTSL
45 º
X34IDE 135º
Z34IDE Sa
(Direction of incoming beam) m ple
Su
r fa
ce
XTSL
YTSL
Figure H.1: Schematic showing the transformation from the APS Beamline 34 Coordinate system
to TSL-OIM coordinate system. The two systems can be brought into coincidence by rotating the
APS system clockwise about the 𝑋 axis by 135°. The transformation matrix (𝑔𝑇34𝐼
𝑆𝐿 ) is calculated
𝐷𝐸
from the direction cosines between the two coordinate systems.
206
YID1
XTSL
YTSL
CCW
90 º
ZID1
XID1
(Beam Direction is anti-parallel to X-axis )
ZTSL
Figure H.2: Schematic showing the transformation from the APS Beamline 1 Coordinate system
to TSL-OIM coordinate system. The two systems can be brought into coincidence by rotating
the APS system counter clockwise about the 𝑋 axis by 90°. The transformation matrix (𝑔𝑇𝐼 𝐷1 𝑆𝐿 ) is
calculated from the direction cosines between the two coordinate systems.
207
APPENDIX I
LIST OF PYTHON AND MATLAB SCRIPTS USED FOR DATA ANALYSIS
#Suite of functions for analyzing Data containing Spatial and
Crystallographic Orientation Information
import damask
import numpy as np
from numba import njit
from scipy import spatial
import math
from scipy import linalg as La
def COMCoords(XYZ):
"""Find the geometric mean COM from a list of coordinates"""
Xmean=XYZ[:,0].mean()
Ymean=XYZ[:,1].mean()
Zmean=XYZ[:,2].mean()
return np.array([Xmean,Ymean,Zmean])
def AxisRotMat(RotString,Angle):
"""Function to generate rotation matrix: given
rotation angle in degrees (+ve for CCW, -ve for CW),
string specifying rotation axis"""
Ang=np.radians(Angle)
sA=np.sin(Ang)
cA=np.cos(Ang)
if RotString==’X’ or RotString==’x’:
208
R=np.array([[1.,0.,0.],[0.,cA,-sA],[0.,sA,cA]])
elif RotString==’Y’ or RotString==’y’:
R=np.array([[cA,0.,sA],[0.,1.,0.],[-sA,0.,cA]])
else:
R=np.array([[cA,-sA,0.],[sA,cA,0.],[0.,0.,1.]])
return R
def AvgOrientation(EulerList,unit=’radians’):
"""Function to determine average orientation given list of
Bunge Euler angles (nx3 numpy array)
in radians (default).The orientations input must be reduced to
fundamental zone apriori.
Based on Cho,Rollett and Oh,
Metallurgical and Materials Transactions A,
Volume 36A,3427-3438,December 2005 """
if unit==’radians’:
eulerList=EulerList
else:
eulerList=EulerList*np.pi/180.
QuatList=damask.Rotation.from_Euler_angles(eulerList).as_quaternion()
QuatAvg=np.array([np.sum(QuatList[:,0]),np.sum(QuatList[:,1]),\
np.sum(QuatList[:,2]),np.sum(QuatList[:,3])])
QuatAvg/=La.norm(QuatAvg)
return QuatAvg
@njit
209
def OrMat(Euler):
"""Function to calculate orientation matrix
given Bunge Euler Angles in degrees"""
sphi1=np.sin(np.deg2rad(Euler[0]))
sphi=np.sin(np.deg2rad(Euler[1]))
sphi2=np.sin(np.deg2rad(Euler[2]))
cphi1=np.cos(np.deg2rad(Euler[0])),
cphi=np.cos(np.deg2rad(Euler[1]))
cphi2=np.cos(np.deg2rad(Euler[2]))
gphi1=np.array([[cphi1,sphi1,0.],[-sphi1,cphi1,0.],[0.,0.,1.]])
gphi=np.array([[1.,0.,0.],[0.,cphi,sphi],[0.,-sphi,cphi]])
gphi2=np.array([[cphi2,sphi2,0.],[-sphi2,cphi2,0.],[0.,0.,1.]])
g=np.dot(gphi,gphi1)
g=np.dot(gphi2,g)
return g
@njit
def HexSymm(g):
"""Function to generate hexagonal symmetry operators
given orientation matrix"""
# 12 Symmetry Operation Matrices for hcp systems:
#Adapted from Hagege et al. 1980
S01=np.array([[1.,0.,0.],[0.,1.,0.],[0.,0.,1.]])
S02=np.array([[0.5,0.866,0.],[-0.866,0.5,0.],[0.,0.,1.]])
S03=np.array([[-0.5,0.866,0.],[-0.866,-0.5,0.],[0.,0.,1.]])
S04=np.array([[-1.,0.,0.],[0.,-1.,0.],[0.,0.,1.]])
S05=np.array([[-0.5,-0.866,0.],[0.866,-0.5,0.],[0.,0.,1.]])
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S06=np.array([[0.5,-0.866,0.],[0.866,0.5,0.],[0.,0.,1.]])
S07=np.array([[1.,0.,0.],[0.,-1.,0.],[0.,0.,-1.]])
S08=np.array([[0.5,0.866,0.],[0.866,-0.5,0.],[0.,0.,-1.]])
S09=np.array([[-0.5,0.866,0.],[0.866,0.5,0.],[0.,0.,-1.]])
S10=np.array([[-1.,0.,0.],[0.,1.,0.],[0.,0.,-1.]])
S11=np.array([[-0.5,-0.866,0.],[-0.866,0.5,0.],[0.,0.,-1.]])
S12=np.array([[0.5,-0.866,0.],[-0.866,-0.5,0.],[0.,0.,-1.]])
GMat=np.zeros((12,3,3))
GMat[0,:,:]=np.dot(S01,g)
GMat[1,:,:]=np.dot(S02,g)
GMat[2:,:]=np.dot(S03,g)
GMat[3,:,:]=np.dot(S04,g)
GMat[4,:,:]=np.dot(S05,g)
GMat[5,:,:]=np.dot(S06,g)
GMat[6,:,:]=np.dot(S07,g)
GMat[7,:,:]=np.dot(S08,g)
GMat[8,:,:]=np.dot(S09,g)
GMat[9,:,:]=np.dot(S10,g)
GMat[10,:,:]=np.dot(S11,g)
GMat[11,:,:]=np.dot(S12,g)
return GMat
@njit
def HexMisor(Euler1,Euler2):
"""Function to calculate misorientation between two hcp
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orientations; given
their Bunge Euler Angles in degrees"""
g1=OrMat(Euler1)
g2=OrMat(Euler2)
g=np.dot(g1,g2.T)
GM=HexSymm(g)
Ang=np.zeros((12,)) # Initialization of Angle Array
# Rounding to 2 Decimal places is done in
#order to ensure
#that values stay within [-1,1] interval!
Ang[0]=(GM[0,0,0]+GM[0,1,1]+GM[0,2,2]-1.)*0.5
Ang[1]=(GM[1,0,0]+GM[1,1,1]+GM[1,2,2]-1.)*0.5
Ang[2]=(GM[2,0,0]+GM[2,1,1]+GM[2,2,2]-1.)*0.5
Ang[3]=(GM[3,0,0]+GM[3,1,1]+GM[3,2,2]-1.)*0.5
Ang[4]=(GM[4,0,0]+GM[4,1,1]+GM[4,2,2]-1.)*0.5
Ang[5]=(GM[5,0,0]+GM[5,1,1]+GM[5,2,2]-1.)*0.5
Ang[6]=(GM[6,0,0]+GM[6,1,1]+GM[6,2,2]-1.)*0.5
Ang[7]=(GM[7,0,0]+GM[7,1,1]+GM[7,2,2]-1.)*0.5
Ang[8]=(GM[8,0,0]+GM[8,1,1]+GM[8,2,2]-1.)*0.5
Ang[9]=(GM[9,0,0]+GM[9,1,1]+GM[9,2,2]-1.)*0.5
Ang[10]=(GM[10,0,0]+GM[10,1,1]+GM[10,2,2]-1.)*0.5
Ang[11]=(GM[11,0,0]+GM[11,1,1]+GM[11,2,2]-1.)*0.5
for i in range(len(Ang)):
if Ang[i]>1:
Ang[i]=0.9999999999999
elif Ang[i]<-1:
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Ang[i]=round(Ang[i],2)
return min(np.arccos(Ang)*180./np.pi)
@njit
def Euclidean(Coord1,Coord2,Dim3D=True):
"""Function to return Euclidean distance
between two points given coords in 2D or 3D"""
if Dim3D:
Euc=(Coord1[0]-Coord2[0])**2+(Coord1[1]-Coord2[1])**2+\
(Coord1[2]-Coord2[2])**2
else:
Euc=(Coord1[0]-Coord2[0])**2+(Coord1[1]-Coord2[1])**2
return np.sqrt(Euc)
@njit
def MisOrList(Euler,EulerList,XYZ,XYZList):
MisOr=np.empty((EulerList.shape[0],))
EucDist=np.empty((XYZList.shape[0],))
for i,j in enumerate(EulerList):
MisOr[i]=HexMisor(Euler,j)
EucDist[i]=Euclidean(XYZ,XYZList[i,:])
return MisOr,EucDist
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@njit
def MisOrIndexList(VoxelNum,CoordList,EulerList,Dim=False):
"""Calculates misorientation of a list of voxels w.r.t
a given voxel and
returns a nx3 array:
(0th col: VoxelNumber, 1st col: misorientation,
2nd col: Euclidean distance)"""
EulerRef=EulerList[VoxelNum,:]
CoordRef=CoordList[VoxelNum,:]
MisOrList=np.empty((EulerList.shape[0],3))
for i,j in enumerate(EulerList):
MisOrList[i,0]=i
MisOrList[i,1]=HexMisor(EulerRef,j)
MisOrList[i,2]=Euclidean(CoordRef,CoordList[i,:],Dim3D=Dim)
return MisOrList
@njit
def MisOr(Ang,AngList,Tol=5):
"""Returns array of misorientations given an Euler angle
in degrees and list of EulerAngles below tolerance"""
Misor=np.empty((AngList.shape[0],))
for i,j in enumerate(AngList):
Misor[i]=HexMisor(j,Ang)
I=np.where(Misor<=Tol)[0]
return I
def ReduceOr(OrList,degrees=True,lattice=’hP’):
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"""Function to reduce list of orientations to fundamental zone.
Returns list of orientations
in terms of Bunge Euler angles in degrees"""
if degrees==True:
Angs=OrList
else:
Angs=np.degrees(OrList)
#Convert Euler angles to orientation matrices
gMat=np.empty((Angs.shape[0],3,3))
for i in range(Angs.shape[0]):
gMat[i,:,:]=OrMat(Angs[i,:])
#Change orientation matrices to quaternion
Quat=damask.Rotation.from_matrix(gMat).
as_quaternion()
AvgOr=damask.Orientation(Quat,lattice=lattice)
Arr=np.array(AvgOr.reduced) #Reduce orientations to fundamental zone
Angs=damask.Rotation.from_quaternion(Arr).
as_Euler_angles(degrees=True)
return Angs
def AverageOr(OrList,degrees=True,lattice=’hP’):
"""Function to calculate average orientation (Bunge Euler angles),
given list of Euler angles (numpy array)"""
if degrees==True:
Angs=OrList
else:
215
Angs=np.degrees(OrList)
#Convert Euler angles to orientation matrices
gMat=np.empty((Angs.shape[0],3,3))
for i in range(Angs.shape[0]):
gMat[i,:,:]=OrMat(Angs[i,:])
#Change orientation matrices to quaternion
Quat=damask.Rotation.from_matrix(gMat).
as_quaternion()
AvgOr=damask.Orientation(Quat,lattice=lattice) #
Arr=np.array(AvgOr.reduced) #Reduce orientations to fundamental zone
Q=damask.Rotation.from_quaternion(Arr).average()
return Q.as_Euler_angles(degrees=True)
@njit
def gMatList(OrList,degrees=True):
"""Function to calculate equivalent rotation
matrices given list of orientations"""
if degrees==True:
EulerList=OrList
else:
EulerList=np.degrees(OrList)
gMat=np.empty((EulerList.shape[0],3,3))
for i in range(EulerList.shape[0]):
gMat[i,:,:]=OrMat(EulerList[i,:])
return gMat
def Euclid(p,Lp):
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"""Method to calculate Euclidean distance between a given point and
a list of neighboring points in 3dimensions."""
Euc=[math.sqrt((p[0]-i[0])**2+(p[1]-i[1])**2+\
(p[2]-i[2])**2) for i in Lp]
return Euc
def KDTreeNN(Coords,VoxelNum,r=2,Print=False):
"""Function to perform nearest neighbor search for
a chosen voxel and output
the voxel numbers coordinates for streak analysis
Print option included for output arguments """
L=Coords.shape[0]
tree=spatial.cKDTree(Coords)
idx = tree.query_ball_point(Coords[VoxelNum,:],r)
E=Euclid(Coords[VoxelNum,:],Coords[idx,:])
E=np.asarray(E)
E=E.reshape(len(E),1)
idx=np.asarray(idx)
idx=idx.reshape(len(idx),1)
C=np.hstack((idx,E))
C = C[C[:,1].argsort()] #Sorting by Euclidean Distance
217
IdxSorted=C[:,0].astype(int)
#Optional print arguments
if Print:
print("Total number of Voxels in this file\n")
print(L)
print("Coords[IdxSorted]\n")
print (Coords[IdxSorted])
print("idx\n")
print(idx)
print("VoxelIndex Euclidean Dist\n")
print(C)
print("IdxSorted\n")
print (IdxSorted)
return IdxSorted
##########################
#Code to reduce NF HEDM Data to Centroids:
#This is pre-processing step prior to
#comparison with FF HEDM Data
import damask
import numpy as np
import pyvista as pv
from scipy import spatial
import math
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from scipy import linalg as La
#Load NF HEDM Orientation and spatial data
Arr=np.loadtxt(’NFAffine25CWXDownData.txt’)
print(Arr.shape)
CoordsAll=Arr[:,:3]
EulerAngsAll=Arr[:,3:6]
print(EulerAngsAll.shape)
##########################
#-------------------------
#Specify ball radius
Radius=160
#Immutable list of all voxels
AllIndices=np.arange(CoordsAll.shape[0]).
astype(int)
#Mutable list (Starts same as AllIndices)
IndexList=np.arange(CoordsAll.shape[0]).
astype(int)
#1st iteration
#-------------
Coords=CoordsAll[IndexList,:]
EulerList=EulerAngsAll[IndexList,:]
VoxelNum=IndexList[0]
IdxSorted= KDTreeNN(Coords,VoxelNum,r=Radius,Print=True)
#Further pruning the NN list on the basis of
#given misorientation tolerance
AngList=EulerList[IdxSorted,:]
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MO=MisOr(AngList[0,:],AngList,Tol=5)
print(MO)
print(MO.shape)
Ids=IdxSorted[MO]
print(Ids)
###########
CC=Coords[Ids,:]
print(CC.shape[0])
#########
KK=np.isin(Coords.sum(axis=1),CC.sum(axis=1))
print(KK.shape)
Inds=np.where(KK==False)[0]
print(Inds.shape)
#########
#Lists to store average centers of mass,
#grain size (number of voxels) and avg orientation information!
COMList=[]
EulerMeanList=[]
NumVoxList=[]
COMList.append(COMCoords(CC)) #Avg Center-of-Mass
EulerMeanList.append(AverageOr(AngList[MO,:])) #Avg orientation
NumVoxList.append(CC.shape[0]) #Number of voxels per grain
#--------------------------------
#Ready for next iteration
Coords=Coords[Inds,:]
print(Coords.shape)
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EulerList=EulerAngsAll[Inds,:]
print(EulerList.shape)
#---------------------------------
count=1
while np.size(Coords)>0:
IndexList=np.arange(Coords.shape[0]).astype(int)
print(IndexList.shape[0])
VoxelNum=IndexList.min()
IdxSorted= KDTreeNN(Coords,VoxelNum,r=Radius,Print=True)
print(IdxSorted)
print(IndexList[IdxSorted])
#Further pruning the NN list on the basis of
#given misorientation tolerance
AngList=EulerList[IdxSorted,:]
MO=MisOr(AngList[0,:],AngList,Tol=5)
print(MO)
print(MO.shape)
Ids=IdxSorted[MO]
print(Ids)
CC=Coords[Ids,:]
KK=np.isin(Coords.sum(axis=1),CC.sum(axis=1))
print(KK.shape)
Inds=np.where(KK==False)[0]
print(Inds.shape)
Coords=Coords[Inds,:] #Ready for next iteration
print(Coords.shape)
EulerList=EulerList[Inds,:]
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print(EulerList.shape)
COMList.append(COMCoords(CC))
EulerMeanList.append(AverageOr(AngList[MO,:])) #Avg orientation
NumVoxList.append(CC.shape[0]) #Number of voxels per grain
count+=1
print(count)
KK=AngList[MO,:]
print(’Count {}, Misor {}’.\
format(count-1,HexMisor\ (EulerMeanList[count-1],KK[-1])))
print(count)
###################
#Class to Perform Comparison analysis between kinematic and
#local stress descriptions in FF HEDM dataset (Chapter 6)
import os
import plotly.express as px
import damask
import pandas as pd
import numpy as np
from scipy import linalg as La
class FFHEDMQuality:
def __init__(self,RSSArr,n,LoadStep,EulerArr):
self.RSSArr=RSSArr
self.n=n
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self.LoadStep=LoadStep
self.EulerArr=EulerArr
def VecToTens(self,STensVec):
"""Method to convert 6 component stress
vector into 3x3 symmetric tensor"""
self.STensVec=STensVec
Tens=np.zeros((3,3))
Tens[0,0],Tens[1,1],Tens[2,2]=STensVec[0],STensVec[1],STensVec[2]
Tens[1,0],Tens[0,2],Tens[2,1]=STensVec[5],STensVec[4],STensVec[3]
Tens[0,1],Tens[2,0],Tens[1,2]=Tens[1,0],Tens[0,2],Tens[2,1]
return Tens
def BlendAxes(self,maxangle,a,b,DirArr,Nmix=10,degrees=True):
self.maxangle=maxangle
self.a=a
self.b=b
self.DirArr=DirArr
self.Nmix=Nmix
self.degrees=degrees
if degrees==True:
maxang=np.radians(5)
else:
maxang=maxangle
Stereomixaxes = np.zeros((Nmix,2))
mixaxes = np.zeros((Nmix,3))
blend = np.linspace(0,maxang,Nmix)
223
blendRev=np.zeros((blend.shape[0],))
blendFrac=np.zeros((blend.shape[0],))
for i in range(Nmix):
Stereomixaxes[i] = damask.util.project_stereographic((
damask.Rotation.from_axis_angle\
(list(self.DirArr[self.a])+\
[blend[i]],normalize=True) *
damask.Rotation.from_axis_angle\
(list(self.DirArr[self.b])+\
[blend[Nmix-i-1]],normalize=True)).\
as_axis_angle(pair=True)[0])
mixaxes[i]=(damask.Rotation.from_axis_angle\
(list(self.DirArr[self.a])+[blend[i]],normalize=True) *
damask.Rotation.from_axis_angle\
(list(self.DirArr[self.b])+\
[blend[Nmix-i-1]],normalize=True)
).as_axis_angle(pair=True)[0]
blendRev[i]=blend[Nmix-i-1]
blendFrac[i]=blend[i]/(blend[i]+blendRev[i])
return mixaxes,Stereomixaxes,blendFrac
def SlipDirPlane(self,SlipSysNum,lattice=’hP’,c=1.58):
self.SlipSysNum=SlipSysNum
c=damask.Crystal(lattice=’hP’,c=1.58)
k = c.kinematics(’slip’)
b=k[’direction’][self.SlipSysNum]
n=k[’plane’][self.SlipSysNum]
224
return b,n
def SchmidFactor(self,STens,n,b):
"""Method to calculate Schmid factor given
3x3 symmetric tensor, slip plane normal (n)
and slip direction"""
self.STens=STens
self.n=n
self.b=b
STensN=self.STens/La.norm(self.STens)
nNorm=self.n/La.norm(self.n)
bNorm=self.b/La.norm(self.b)
#Tensordot can be used to obtain same result
SF = np.einsum(’ij,i,j’,\
STensN,bNorm,nNorm)
return SF
def ColorString(self,RGBArr,print=False):
"""Method to generate plotly compatible color string,
given list of colors as an nx3 array of RGB values"""
self.RGBArr=RGBArr
self.print=print
CL=[]
for i in self.RGBArr:
rgb=tuple(int(x) for x in i[0:3])
rgba=’rgb’+str(rgb)
CL.append(rgba)
225
if self.print==True:
print(CL)
return CL
def StereoSlipAxes(self,systemNum,nSlip,SFList):
"""Function to generate 3D Axis and 2D
stereographic coordinates for a given
category of slip systems for hcp
system: category of slip system
(0: basal, 1: prism, 2: 2nd order prism, 3: pyramidal,
4:pyramidal)
nSlip: number of slip systems in given category
SF:Schmid factor
(sgn(SF)is multiplied to bxn to obtain slip
system reorientation axis)"""
self.systemNum=systemNum
self.nSlip=nSlip
self.SFList=SFList
c=damask.Crystal(lattice=’hP’,c=1.58)
k = c.kinematics(’slip’)
LineDir=np.zeros((self.nSlip,3))
axes = np.zeros((len(k[’direction’][self.systemNum]),2))
for i,(m,n) in enumerate(zip(k[’direction’]\
[self.systemNum],k[’plane’][self.systemNum])):
axes[i] = damask.util.project_stereographic\
(np.sign(self.SFList[i])*np.cross\
(c.to_frame(uvw=m),c.to_frame(hkl=n)))
226
LineDir[i,:]=np.sign(self.SFList[i])*\
np.cross(c.to_frame(uvw=m),c.to_frame(hkl=n))
LineDir[i,:]=LineDir[i,:]/La.norm(LineDir[i,:])
return LineDir,axes
def CollectSlipSys(self,nBasal,nPrism,nPyrca,mBasal,mPrism,mPyrca):
"""Accumulates the slip plane normals
and slip directions for basal,prism
and pyramidal in that order"""
self.nBasal=nBasal
self.nPrism=nPrism
self.nPyrca=nPyrca
self.mBasal=mBasal
self.mPrism=mPrism
self.mPyrca=mPyrca
nAll=np.vstack((self.nBasal,self.nPrism,self.nPyrca))
mAll=np.vstack((self.mBasal,self.mPrism,self.mPyrca))
return nAll,mAll
def CollectSlipAxes(self,LineDirBasal,\
StereoBasal,LineDirPrism,StereoPrism,LineDirPyrca,StereoPyrca):
"""Accumulates the line directions for
basal,prism and pyramidal in that order"""
self.LineDirBasal=LineDirBasal
self.LineDirPrism=LineDirPrism
self.LineDirPyrca=LineDirPyrca
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self.StereoBasal=StereoBasal
self.StereoPrism=StereoPrism
self.StereoPyrca=StereoPyrca
StereoAll=np.vstack((self.StereoBasal,\
self.StereoPrism,self.StereoPyrca))
LineDirAll=np.vstack((self.LineDirBasal,\
self.LineDirPrism,self.LineDirPyrca))
return LineDirAll,StereoAll
def StereoAxis(self,OrList,n):
self.OrList=OrList
self.n=n
N=self.OrList.shape[0]
Axis=np.zeros((N,3))
AxisStereo=np.zeros((N,2))
for i in range(n,N-n):
Axis[i,:]=damask.Orientation.from_Euler_angles\
(phi=self.OrList[i-self.n,:],
degrees=True,family=’hexagonal’)\
.misorientation\
(damask.Orientation.\
from_Euler_angles\
(phi=self.OrList[i+n,:],degrees=True,family=’hexagonal’))\
.as_axis_angle(pair=True)[0]
AxisStereo[i,:]=damask.util.project_stereographic(Axis[i,:])
228
return Axis[self.n:N-n,:],AxisStereo[self.n:N-n,:]
def TopTwoRSSMag(self,RSSArr,LoadStep):
"""Return the indices of the two highest stressed slip systems"""
self.RSSArr=RSSArr
self.LoadStep=LoadStep
AbsRSSA=np.abs(self.RSSArr)
q=2
idx = np.argpartition(AbsRSSA[self.LoadStep], -q)[-q:]
indices = idx[np.argsort((-AbsRSSA[self.LoadStep])[idx])]
return indices
def MaxRSS(self,RSSArr,LoadStep,print=False):
"""Given a load step and the full RSS array, finds the Slip system
numbers with the highest magnitude for that loadstep"""
self.RSSArr=RSSArr
self.LoadStep=LoadStep
self.print=print
Basal=np.abs(self.RSSArr[self.LoadStep,:3])
Prism=np.abs(self.RSSArr[self.LoadStep,3:6])
Pyrca=np.abs(self.RSSArr[self.LoadStep,6:])
MaxBasal=np.where(Basal==Basal.max())[0][0]
MaxPrism=np.where(Prism==Prism.max())[0][0]
MaxPyrca=np.where(Pyrca==Pyrca.max())[0][0]
if self.print==True:
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print(’MaxPrism={}’.format(MaxPrism))
print(’MaxBasal={}’.format(MaxBasal))
print(’MaxPyrca={}’.format(MaxPyrca))
return MaxBasal,MaxPrism,MaxPyrca
def VecBlender(self,a,b,LineDirA,LineDirB,misor=5,Nmix=10):
self.misor=misor
self.Nmix=Nmix
maxangle = np.radians(self.misor)
self.a=a
self.b=b
self.LineDirA=LineDirA
self.LineDirB=LineDirB
mixaxes = np.zeros((self.Nmix,2))
mixVec=np.zeros((self.Nmix,3))
blend = np.linspace(0,maxangle,Nmix)
N=blend.shape[0]
for i in range(self.Nmix):
mixVec[i]=(damask.Rotation.from_axis_angle\
(list(self.LineDirA[self.a])+[blend[1]],normalize=True) *
damask.Rotation.\
from_axis_angle(list(self.LineDirB[self.b])+\
[blend[1]],normalize=True)
).as_axis_angle(pair=True)[0]
mixaxes[i] = damask.util.project_stereographic((
damask.Rotation.\
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from_axis_angle(list(self.LineDirA[self.a])\
+[blend[1]],normalize=True) *
damask.Rotation.\
from_axis_angle(list(self.LineDirB[self.b])\
+[blend[1]],normalize=True)
).as_axis_angle(pair=True)[0])
return mixVec,mixaxes
def StereoPlotter(self,rPrism,rBasal,rPyrca,\
axesPyrca,axesP,axesB,mixaxes,CL,AxisStereo,AnnStr):
self.rPrism=rPrism
self.rBasal=rBasal
self.axesPyrca=axesPyrca
self.axesP=axesP
self.axesB=axesB
self.mixaxes=mixaxes
self.CL=CL
self.AxisStereo=AxisStereo
N=self.AxisStereo.shape[0]
df=pd.DataFrame(dict(x=self.axesPyrca[:,0],
y=self.axesPyrca[:,1],
size=15*rPyrca/rPyrca.max(),
level=self.CL))
df1=pd.DataFrame(dict(x=self.AxisStereo[:,0],
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y=self.AxisStereo[:,1],
S=10,
level=range(self.AxisStereo.shape[0])))
cir=np.linspace(0,2*np.pi,1000)
L=list(range(N))
#Pyramidal slip
p=px.scatter(df,
x=’x’,
y=’y’,
color=’level’,
size=’size’,
color_discrete_map=’identity’,
range_x=[-1.2,1.2],
range_y=[-1.2,1.2],
width=400,
height=370)
#title=AnnStr)
#Prism slip (red markers)
p.add_scatter(x=axesP[:,0],
y=self.axesP[:,1],
marker=dict(size=15*rPrism/rPrism.max(),\
color="rgb(255,0,0)"),
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mode=’markers’)
#Basal slip (sky blue markers)
p.add_scatter(x=self.axesB[:,0],
y=axesB[:,1],
marker=dict(size=15*rBasal/rBasal.max(),\
color="rgb(135,206,235)"),
mode=’markers’)
p.add_scatter(x=self.mixaxes[[0,-1],0],
y=self.mixaxes[[0,-1],1],
marker=dict(size=4,color="rgb(0,0,255)"),
mode=’lines’)
#p.update_traces(marker={’size’: 10})
p.add_scatter(x=self.AxisStereo[:,0],
y=self.AxisStereo[:,1],
marker=dict(size=10, color=L,colorscale=’greys’),
mode=’markers’)
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p.add_scatter(x=np.cos(cir),
y=np.sin(cir),
marker=dict(size=1,color="rgb(0,0,0)"),
mode=’markers’)
p.update(layout_showlegend=False)
p.show()
return p
###################################################
#Script for matching FF data with NF data based on
#Orientation threshold and Euclidean ball radius specified
#Reverse algorithm
import os
import numpy as np
import damask
#NFEulerAngs=np.loadtxt(’NFReducedEulerAngs.txt’)
NFCoords=np.loadtxt(’NFReducedCoords.txt’)
FFDataPath=’/Users/harshaphukan/Desktop/Dissertation/NFFFMatching/’
FFDataFileName=’FFReducedRotated.txt’
FFData=np.loadtxt(os.path.join(FFDataPath,FFDataFileName))
FFCoords=FFData[:,:3]
FFEulerAngs=FFData[:,3:6]
NFEulerAngsRot=np.loadtxt(’NFEulerAngsRot.txt’)
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FFIndexList=range(FFCoords.shape[0])
MisOrTol=5
EucTol=1500
#Immutable list of all voxels (Near Field dataset)
AllIndicesNF=np.arange(NFCoords.shape[0]).astype(int)
# mutable number of voxels in Near Field
#Dataset (starts same as AllIndicesNF)
NFIndexList=np.arange(NFCoords.shape[0]).astype(int)
#Initialize list of Near field coordinates and Euler angles
NFCoordList=NFCoords
NFAngList=NFEulerAngsRot
MatchList=[]
NFMatchCoords=[]
for ii in FFIndexList:
TempCoords=np.vstack((FFCoords[ii,:],NFCoordList))
IdSorted,EucDist=KDTreeNN(TempCoords,0,r=EucTol,Print=True)
Ids=IdSorted[1:]-1
print(Ids)
MisOrient=np.empty((Ids.shape[0],))
for i,j in enumerate(Ids):
MisOrient[i]=HexMisor(FFEulerAngs[ii,:],NFAngList[j,:])
print(MisOrient)
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I=np.where(MisOrient<=MisOrTol)
print(I[0])
L=len(I[0])
print(np.size(I[0]))
print(Ids[I[0]])
print(EucDist[I[0]+1])
print(MisOrient[I[0]])
if np.size(I[0])>0:
HArr=np.hstack((Ids[I[0]].reshape(L,1),EucDist[I[0]+1]\
.reshape(L,1),MisOrient[I[0]].reshape(L,1)))
print(HArr)
H=HArr[0,:]
H=np.hstack((ii,H))
MinIndex=H[1].astype(int)
print(MinIndex)
CC=NFCoordList[MinIndex,:]
print(CC)
#Remove voxel from NF datalist
KK=np.isin(NFCoordList.sum(axis=1),CC.sum())
print(KK.shape)
Inds=np.where(KK==False)[0]
print(Inds.shape)
NFCoordList=NFCoordList[Inds,:]
NFAngList=NFAngList[Inds,:]
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MatchList.append(H)
NFMatchCoords.append(CC)
print("Voxel {},Coords {} FF: Coords {} NF".\
format(ii,FFCoords[ii,:],CC))
MatchList=np.array(MatchList)
Euc=np.array(MatchList[:,-2])
print(MatchList)
#####################
%{
MatLab Function to output grain positions, crystallographic
orientations and averaged stress tensors of indexed grains
from FF HEDM data: Takes the .log and g vector files as input
%}
% Note that the Euler angles are corrected by adding 30-degrees to phi2 in
% the Result
function [GrainArray]=LoadDataSameDir(FilePath,LogFile)
%Routine to extract strain/stress tensors and positional data of grains
%from .gve and .log file. Developed by Beaudoin and Wang: Uses least
%squaresapproach proposed by Marguiles et al. [2002]
fPrefixGVE=FilePath;%input(’Type in path for .gve file:’,’s’);
fPrefixLOG=FilePath;%input(’Type in path for .log file:’,’s’);
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file_nameLog=LogFile;%input(’Enter log filename, e.g xyz.log:’,’s’);
log_data=loadGrainSpotterLog(file_nameLog,fPrefixLOG);
S1=strsplit(file_nameLog,’_’);
file_nameGVE=strcat(S1{1},’_’,S1{2},’_’,S1{3},’.gve’);...
%input(’Enter GVE filename, e.g xyz.gve:’,’s’);
fPostfix = ’’;
[name,ext]=fileparts(file_nameLog);
logStem=strtok(ext,’_’);
fLogStem=strcat(logStem,’_’,’00’);
fGVEstem=fLogStem;
[nameG,extG]=fileparts(file_nameGVE);
[a,fPostfixGVE]=strtok(extG,’t’);
fPostfixGVE=strcat(’_’,fPostfixGVE);
[b,fPostfixLOG]=strtok(ext,’t’);
fPostfixLOG=strcat(’_’,fPostfixLOG);
N_grain=length(log_data);
238
%% This part extracts the Image Number from the .log filename
if length(ext)==22
Image_string=strcat(ext(9),ext(10),ext(11));
elseif length(ext)==21
Image_string=strcat(ext(8),ext(9),ext(10));
else
print(’Please rename your .log and .gve files to ...
fit the compatible format.\n’)
end
if Image_string(1)==’0’
ImageString=strcat(Image_string(2),Image_string(3));
else
ImageString=Image_string;
end
Image_number=str2num(ImageString);
scanNo = Image_number*ones(N_grain,1);
grainNo = 1:N_grain;
strain=zeros(3,3);
[g0Cry, g_idl] = g0hcp(2.95, 4.683); % New parameter
239
g_ideal = g_idl;
EulerAng=zeros(length(scanNo),3);
GrainArray=zeros(length(scanNo),12);
for scan = 1 : length(scanNo)
filelog = parseGrainSpotterLog([fPrefixLOG fLogStem num2str...
(scanNo(scan)) fPostfixLOG ’.log’]); % fPostfix ’.log’]);
% Read the corresponding Fable g-vector file
peaks_gve_dat = loadGVE([fPrefixGVE fGVEstem ...
num2str(scanNo(scan)) fPostfixGVE ’.gve’]);
% find g-vectors for grains in the log file,
%using indices to the g-vector file
clear strn stddevs
igrain = 0;
for ig=grainNo(scan):grainNo(scan)
igrain = igrain + 1;
240
clear gve gveID A B hkl2_
% Find the number of g-vectors in the log file
Ngvec = length(filelog(ig).refl...
(:,1));
cnt = 0;
for i=1:Ngvec
% Only include if below some threshold internal angle
if filelog(ig).refl(i,22)...
< 2.0
cnt = cnt+1;
gveID(cnt) = find(peaks_gve_dat(:,9)==...
filelog(ig).refl(i,3));
% The measured 2-theta, omega and eta for this g-vector
toe(cnt,:) = filelog(ig).refl(i,[13 16 19]);
hklS(cnt, :) = filelog(ig).refl(i,4:6);
end
end
Ngvec = cnt;
% Load in the g-vector and it’s norm (or 1/d-spacing)
gve = peaks_gve_dat(gveID,[1:3 6]);
% The following was a check, to make certain that we did not have
241
% Friedel pairs
% close(1)
%figure(1)
%plot(gve(:,2),gve(:,3),’+’)
%hold on
%plot(-gve(:,2),-gve(:,3),’o’);
FriedelPairs = zeros(Ngvec,1);
for i=1:Ngvec
for j=1:Ngvec
if i ~= j
if isequal(hklS(i,1:3), -hklS(j,1:3))
% mag = (dot(a,b)/(norm(a).^2));
% if mag>0.998 && mag<1.001
FriedelPairs(i) = 1;
FriedelPairs(j) = 1;
end
end
end
end
% FIX-UP DISTANCE
% gve = 0.999*gve;
242
% idx = find(FriedelPairs==1);
%
% [Ngvec length(idx)]
%
% Ngvec = length(idx);
% gve = gve(idx,:);
OutSide = zeros(Ngvec,1);
for i=1:Ngvec
OutSide(i) = (norm(gve(i,1:3)) > 0.92)...
& (norm(gve(i,1:3)) < 0.95) ; % 0.795;
end
idx = find(OutSide ~= 1);
[Ngvec length(idx)];
Ngvec = length(idx);
gve = gve(idx,:);
% Now, the least squares problem
% A*X = B, or X = A\B
243
B = zeros(Ngvec,1);
for i=1:Ngvec
% The delta spacing will be a minimum with
%corresponding ideal hkl
gs = norm(gve(i,1:3))-g_ideal;
[val,midx] = min(abs(gs));
% B is the measured elastic strain for
%each reflection (i.e. g vector)
B(i,1) = -gs(midx)/...
g_ideal(midx);
end
for i=1:Ngvec
% lmn is the g vector in sample coordinate system
lmn = gve(i,1:3)/norm(gve(i,1:3));
% new parameter
dx_term = -( cosd(toe(i,2)) +(sind(toe(i,2))...
*sind(toe(i,3)))/tand(toe(i,1)/1.0) )/999654.8;
% new parameter from Ti_1516.par
dy_term = -( sind(toe(i,2)) +(cosd(toe(i,2))...
*sind(toe(i,3)))/...
244
tand(toe(i,1)/1.0) )/999654.8;
A(i,:) = [lmn(1)^2 lmn(2)^2 lmn(3)^2 2*lmn(1)*...
lmn(2) 2*lmn(1)*lmn(3) 2*lmn(2)*...
lmn(3) -dx_term -dy_term];
end
X =A\B; % matlab solve least square problem in one command!
%figure(4)
%plot(B-A*X);
fit_error = B - A*X;
strn(:,igrain) = A\B;
stddevs(igrain) = std(B-A*X);
% Filter results from the fit, here.
% mark’s 0.8e-3)
idx = find(abs(fit_error)<0.004);...
% .0018); % 0.8e-3);
B1 = B(idx);
A1 = A(idx,:);
245
strn(:,igrain) = A1\B1;
%figure(5)
%plot(B1-A1*strn(:,igrain))
stddevs(igrain) = std(B1-A1*strn(:,igrain));
%pause
end
Rowstrn(scan,:)=strn’; % display strain and position for all grains
%Strain tensor calculation
strain(1,1)=strn(1);
strain(2,2)=strn(2);
strain(3,3)=strn(3);
strain(1,2)=strn(4);
strain(2,1)=strn(4);strain(1,3)=strn(5);
strain(3,1)=strn(5);strain(2,3)=strn(6);strain(3,2)=strn(6);
% 90 deg rotation
filelog(ig).U=filelog(ig).U*[0 -1 0;1 0 0;0 0 1];
% strain tensor in crystal coordinate,
%filelog(ig).U is the R matrix of the grain
strain_c(:,:)=filelog(ig).U’*...
strain(:,:,igrain)*filelog(ig).U;
% stiffness tensor of Ti
C=[162.4e3, 92e3, 69e3, 0.0, 0.0, 0.0;
92e3, 162.2e3, 69e3, 0.0, 0.0, 0.0;
246
69e3, 69e3, 181.6e3, 0.0, 0.0, 0.0;
0.0, 0.0, 0.0, 47.2e3, 0.0, 0.0;
0.0, 0.0, 0.0, 0.0, 47.2e3, 0.0;
0.0, 0.0, 0.0,
0.0, 0.0, 35.2e3;];
strain_c_vec=[strain_c(1,1), strain_c(2,2),...
strain_c(3,3),strain_c(2,3)*2...
,strain_c(3,1)*2,strain_c(1,2)*2]’;
% Stress in the crystal coordinate system using General Hooke’s Law
stress_c_vec=C*...
strain_c_vec;
stress_crystal(scan,:)=stress_c_vec’;
stress_c=[stress_c_vec(1),stress_c_vec(6),stress_c_vec(5);
stress_c_vec(6),stress_c_vec(2),stress_c_vec(4);
stress_c_vec(5),stress_c_vec(4),stress_c_vec(3)];
% stress in the sample coordinate system
stress=filelog(ig).U*stress_c*...
filelog(ig).U’;
% stress in the sample coordinate system in vector form
stress_vec(scan,:)=[stress(1,1),stress(2,2),...
stress(3,3),stress(1,2),...
stress(1,3),...
stress(2,3)]’;
247
EulerAng(scan,1)=log_data(scan).euler(1);
EulerAng(scan,2)=log_data(scan).euler(2);
EulerAng(scan,3)=log_data(scan).euler(3)+30;
GrainArray(scan,1)=scan;
GrainArray(scan,2:3)=Rowstrn(scan,7:8);
GrainArray(scan,4:9)=stress_vec(scan,:);
GrainArray(scan,10:12)=EulerAng(scan,:);
end
end
248
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249
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