HETEROGENEOUS DEFORMATION IN
POLYCRYSTALLINE COMMERCIAL PURITY TITANIUM
By
Yiyi Yang

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Materials Science and Engineering
2011

ABSTRACT
HETEROGENEOUS DEFORMATION IN
POLYCRYSTALLINE COMMERCIAL PURITY TITANIUM
By
Yiyi Yang
Heterogeneous deformation is commonly viewed as a precursor to damage
nucleation. Only if the heterogeneous deformation is reliably modeled can location
where cracks form be properly predicted. In this research project, commercial purity
titanium (hexagonal close-packed (hcp) metal) has been used as a model material to
study different scales of heterogeneous deformation in four-point bending specimens
and fatigued-to-crack specimens.

A series of comprehensive experimental and

modeling methods have been conducted to characterize the heterogeneous
deformation within different microstructure patches.
Heterogeneous deformation within a microstructure patch was quantified by a
new combined technique using atomic force microscopy (AFM), backscattered electron
(BSE) imaging, and electron-backscattered diffraction (EBSD). Local shear distribution
maps derived from z-displacement data of slip steps and/or twins measured by AFM
were directly compared to results of a crystal plasticity finite element (CPFE) simulation
that incorporates a phenomenological model of the deformation processes to evaluate
the ability of the CPFE model to match the experimental observations. The CPFE
model successfully predicted most types of active dislocation slip systems within the
grains at the correct magnitudes, but the spatial distribution of the strains within grains
differed between the measurements and the simulation. Since the difficulty of directly
measuring the critical resolved shear stress (CRSS) of different deformation systems in

	
  

titanium is a major cause of the deficiency of the match between CPFE modeling results
and experimental measurements, nano-indentation technique is introduced to help
study the values of CRSS and saturation resistance of different deformation systems for
commercial purity titanium.

Arrays of conical nano-indentations were performed on

patches of grains. AFM scans and EBSD measurements on the patches indicate that
the surface pile-up topography was strongly crystallographically dependent.

By

optimizing the modeling results of the nano-indentation behavior in different grains, a
more reliable set of values for CRSS and saturation resistance has been determined
using a non-linear optimization procedure.

On a larger scale, a novel approach

including BSE imaging and discrete Fourier transformation (DFT) processing is
established to determine plastic zone shape and to measure plastic strain distribution
within the plastic zone around a fatigue crack tip. The measured plastic zone size is in
good correlation with the theoretically predicted zone sizes. This approach also holds
promise to measure local heterogeneous deformation in other kinds of deformed
samples.
This work has helped identity the needs for future experiments focused on the
observation of dislocation activity near grain boundaries using ECCI, and a statistical
study on the relationship of heterogeneous deformation in neighboring grains and
damage nucleation at grain boundaries.

Future simulation work should focus on

developing a more comprehensive CPFE model, considering the information of slip/twin
transfer and grain boundaries resistance, in order to simulate heterogeneous
deformation within grain patches more reliably.

	
  

Copyright by
YIYI YANG
2011

	
  

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor, Dr. Martin Crimp, and
other committee members, Dr. Thomas Bieler, Dr. Carl Boehlert, Dr. James Lucas, and
Dr. Viktor Astakhov.

Their wisdom, knowledge, and commitment to the highest

standards inspired and motivated me. I would like to thank Dr. Philip Eisenlohr and
many other collaborators in Max-Planck-Institut für Eisenforschung GmbH for their
invaluable discussions and suggestions. I would also like to show my special gratitude
to Dr. Baokang Bi for his assistance with the AFM work, Dr. Zambaldi for the help on the
simulation work of nano-indentation, Dr. Rachel Tomlinson, Dr. Eann Patterson, and
Mark Dawson for their help with BSE-DFT work, and all of those who supported me in
any respect during the completion of the project. Most especially, I owe my deepest
gratitude to my parents and Zhongping who have been always supporting, encouraging
and believing in me.
Most of the research work in this dissertation has been conducted with funding
from The National Science Foundation (grant #DMR-0710570) and Deutsche
Forschungsgemeinschaft (grant #EL-681/2-1).

	
  

v	
  

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ ix
LIST OF FIGURES ........................................................................................................... x
LIST OF SYMBOLS AND ABBREVIATIONS ................................................................ xvi
CHAPTER ONE
INTRODUCTION ..............................................................................................................1
1.1 Importance of and Challenges for Measuring and Simulating Heterogeneous
Deformation and Predicting Micro-crack Initiation ..................................................1
1.2 Basic Approaches to Understanding Heterogeneous Deformation ........................3
1.2.1 Slip Transfer Criteria ......................................................................................3
1.2.2 Geometrically Necessary Dislocations...........................................................6
1.3 Experimental Characterizations and Simulations on Heterogeneous Deformation 9
1.3.1 Experimental Characterizations .....................................................................9
1.3.1.1 Single crystals and bi-crystals...........................................................10
1.3.1.2 Poly-crystals......................................................................................14
1.3.2 Simulations of Heterogeneous Deformation ................................................18
1.4 Current Understanding of the Relationship between Heterogeneous Deformation
and Damage Nucleation........................................................................................23
1.5 Studies on Commercial Purity Titanium and Other Hexagonal Materials .............27
1.5 Motivations for This Study.....................................................................................32
1.6 Overview ...............................................................................................................34
CHAPTER TWO
MATERIAL, EXPERIMENTAL DETAILS, AND SIMULATION FRAMEWORK ..............37
2.1 Material .................................................................................................................37
2.1.1 Material A .....................................................................................................37
2.1.2 Material B .....................................................................................................40
2.2 Sample Preparation ..............................................................................................40
2.3 Experiment Details................................................................................................45
2.3.1 Scanning Electron Microscopy.....................................................................45
2.3.1.1 Electron Imaging ...............................................................................45
2.3.1.2 EBSD/OIM measurement .................................................................45
2.3.2 Four-point Bending Testing..........................................................................49
2.3.3 Atomic Force Microscopy.............................................................................49
2.3.4 Nano-indentation..........................................................................................52
2.3.5 Fatigue Testing and Thermoelastic Stress Analysis Measurement .............55
2.4 Slip/twin Plane Trace Analysis..............................................................................56
2.5 Constitutive Law in Crystal Plasticity Finite Element Method ...............................57

	
  

vi	
  

CHAPTER THREE
MEASURING AND CPFEM SIMULATING MICRO-SCALE HETEROGENEOUS
DEFORMATION .............................................................................................................60
3.1 Microstructure in a Four-point Bending Sample....................................................61
3.1.1 Slip Lines and Twins ....................................................................................61
3.1.2 Grain Boundary Ledge Formation................................................................61
3.1.3 A Selected Microstructure Patch for Further Studies ...................................64
3.2 Quantitative Measurement of Heterogeneous Deformation within a Grain Patch 71
3.2.1 AFM Measurement and Accuracy Check Using T1 Twins...........................71
3.2.2 Generation of Local Shear Distribution Maps ..............................................75
3.2.3 Local Shear Distribution Evolution at Different Strain Level ........................77
3.3 Crystal Plasticity Finite Element Modeling ............................................................81
3.3.1 Constitutive Law...........................................................................................81
3.3.2 Mesh Generation and Boundary Condition ..................................................82
3.3.3 Modeling Results..........................................................................................85
3.4 Discussion.............................................................................................................88
3.4.1 Comparison of Experimental Observations and CPFE Simulations ............88
3.4.2 Effect of Modeled Area Size in Simulation Results ......................................90
3.4.3 Effect of Rim Crystal Orientation in Simulation Results ...............................91
3.5 Conclusions ..........................................................................................................93
CHAPTER FOUR
CALIBRATING PARAMETERS FOR CPFE MODELS USING NANO-INDENTATION .95
4.1 Importance and Advantages of Measuring CRSS Using Nano-indentation..........96
4.2 Grain Orientation Dependent Behavior during Nano-indentation .........................98
4.2.1 Relationship between Hardness and Grain Orientation ...............................98
4.2.2 The Relationship between Pile-up Topographies and Grain Orientation ...101
4.3 CPFEM Simulation of Indentation and CRSS Optimization................................105
4.3.1 Constitutive law ..........................................................................................105
4.3.2 Mesh Generation and Boundary Conditions ..............................................107
4.3.3 Crystal Plasticity Parameter Optimization Procedures ..............................107
4.3.4 Modeling Results........................................................................................109
4.4 Discussion...........................................................................................................112
4.4.1 Automated Identified Constitutive Parameter Values ................................112
4.4.2 Comparison of Simulation Results using Two Sets of CRSS ....................115
4.5 Conclusions ........................................................................................................117
CHAPTER FIVE
MEASURING MESO-SCALE HETEROGENEOUS DEFORMATION IN COMMERCIAL
PURITY TITANIUM ......................................................................................................118
5.1 Observation of Microstructure around a Fatigue Crack Tip using BSE Images..119
5.2 Image Analysis Technique Using DFT................................................................122
5.2.1 Basic Information of DFT ...........................................................................122
5.2.2 DFT Image Data Processing......................................................................122
5.3 Quantifying the Relation between FWHM and Strain Using In-situ Tensile Test 127
5.4 Residual Strain Map around a Crack Tip in the Fatigue Specimen ....................131

	
  

vii	
  

5.5 Discussion...........................................................................................................131
5.5.1 The Influence of Brightness, Contrast, Magnification, and Calibration
Procedures on the Strain Map ............................................................................131
5.5.2 Comparison between the Plastic Zone Size Measured by BSE-DFT Method
and Those Proposed by Other Works.................................................................137
5.5.3 The Application of BSE-DFT Method .........................................................143
5.6 Conclusions ........................................................................................................144
CHAPTER SIX
BROAD IMPACTS OF THE STUDY ............................................................................146
CHAPTER SEVEN
CONCLUSIONS AND FUTURE WORK.......................................................................149
7.1 Conclusions ........................................................................................................149
7.2 Future work .........................................................................................................151
APPENDIX A ................................................................................................................154
APPENDIX B ................................................................................................................157
APPENDIX C................................................................................................................171
REFERENCES .............................................................................................................174

	
  

viii	
  

LIST OF TABLES

Table 1.1 CRSS of different slip modes reported for α-titanium.
Table 1.2 Deformation twinning modes in α-titanium.
Table 3.1 Deformation systems in grains within the solid-line box in Figure 3.3(a) and
3.4(a)-(b) at global strain level 1.5%, 3%, and 6%.
Table 3.2 Hardening parameters used to describe evolution of the critical resolved
shear stress.
Table 4.1 The indentation axis in crystal coordinate system, the angle between c-axis
and indentation axis (η) of selected indented grains in specimen S0-2, and
the maximum residual depth (hres) of the indents in the center of these
grains.
Table 4.2 Optimized constitutive parameters for prismatic, basal and pyramidal <c+a>
slip systems. The other constant parameters for the three systems in the
simulations were chosen as: m = 20, h0 = 200 MPa, a = 2.

	
  

ix	
  

LIST OF FIGURES

Figure 1.1 Three-dimensional slip transfer geometry at grain boundaries.
Figure 1.2 If each grain of a poly-crystal deforms in a uniform manner, overlap and
voids appear (b). This can be corrected by introducing geometrically
necessary dislocations, shown in (c) and (d).
Figure 1.3 (a) Orientations of the two crystals in relation to the compression direction
Z. GND distribution in a 10% (b) and a 30% (c) strained aluminum bicrystal.
Figure 1.4 A schematic drawing of the setup of 3D-XRD.
Figure 1.5 Two vertical XRD scans (line 1 and 2) compared with the corresponding
EBSD scan data. All curves display the total misorientation relative to the
base orientation of the pillar plotted as a function of distance from the pillar
base.
Figure 1.6 (a) SACP composite from the deformed grain (~1% strain) shown in (b).
The ‘‘×’’ indicates the electro optic axis at zero sample tilt. (b) Channeling
contrast image of the deformation defects in a grain of TiAl with the
calculated {111} plane traces. Arrowheads indicate bands of dislocations
piled up in the neighboring grain at points of intersection of deformation
twins with the grain boundary.
Figure 1.7 The various mesh types used in Ref. [61]: (a) real 3D mesh (31,780
nodes), (b) real 3D mesh (130,818 nodes), (c) quasi-3D mesh (14, 076
nodes).
Figure 1.8 (a) Backscattered electron image of γ-γ TiAl grain boundary damage
initiation due to twin interaction with the grain boundary; (b) 2D schematic
of crack nucleation at a concentrated band of slip or twinning.
Figure 1.9 Slip systems for α-titanium: (a) prismatic slip, (b) basal slip, (c) pyramidal
<a> slip, and (d) pyramidal <c+a> slip. The arrows only represent the slip
directions, not the magnitudes of the dislocation slip.
Figure 1.10 Twinning systems for α-titanium: (a) T1 twinning, (b) T2 twinning, (c) C1
twinning, and (d) C2 twinning. The arrows only represent the twinning
directions, not the magnitudes of the twinning.
Figure 2.1 {0001} and {10 10} pole figures of (a) specimen S0, and (b) specimen S45.

	
  

x	
  

Figure 2.2 Specimen orientations in Material A plate (left), and the size of specimens
for four-point bending tests and nano-indentation tests (right).
Figure 2.3 {0001} and {10 10} pole figures of Material B.
Figure 2.4 Size and shape of specimens for in-situ tensile tests cut from Material B.
Figure 2.5 Size and shape of specimens for fatigue tests cut from Material B plate.
Figure 2.6 Backscattered electron image of a microstructure patch with deposited fine
tungsten cross patterns.
Figure 2.7 A typical setup of an EBSD/OIM system in SEM.
Figure 2.8 Deformation stage for in-situ tensile test.
Figure 2.9 Four-point bending stage with a bended specimen loaded.
Figure 2.10 Schematic drawing (a) and an SEM image (b) showing the size and shape
of the AFM probe tips used in this project.
Figure 2.11 BSE image of the indented microstructure patch in specimen S0-2.
Figure 2.12 BSE image of the indented microstructure patch in specimen S0-3.
Figure 3.1 A typical microstructure patch from S45-1 at 1.5% strain. (a) shows a grain
with dense slip lines, while (b) shows a grain with much less slip lines.
Twins also shows in grains, as illustrated in (c).
Figure 3.2 Secondary electron images show ledges (circled) along grain boundaries in
S0-1under 8% global strain; (b) was taken of the same microstructural
patch with the whole sample rotated 180 degrees from (a). Circled
boundaries showing reverse contrast in two images, indicating these are
ledges and not cracks.
Figure 3.3 AFM “Section” scans shown ledges with different topography at grain
boundaries in S0-2 under 1.5% global strain, indicating that heterogeneous
deformation occurs in small deformation.
Figure 3.4 Microstructure patch of specimen S45-1 at ~1.5% tensile strain. The patch
within the solid outlined box was quantitatively analyzed using AFM-based
method, and the patch within the dashed outlined box was modeled by
CPFEM. Inverse pole figures of the grain orientation changes within the
solid outlined box before and after deformation are shown in (b) and (c),
respectively. Directions of slip lines and twins in each grain are highlighted
by white lines. These activated deformation systems were identified using
trace analysis based on EBSD data.

	
  

xi	
  

Figure 3.5 The same microstructure patch in S45-1 specimen as shown in Figure
3.4(a) following further deformed to ~3% and ~6% tensile strains, shown in
(a) and (b) respectively. The patch within the solid outlined box was
quantitatively analyzed using AFM-based method. The growth of five T1
twins in grain 2 with the increase of the global strain can be easily
observed. Furthermore, secondary deformation systems, highlighted by
white solid lines in (b), were observed in several hard grains, such as grain
2, at 6% strain level.
Figure 3.6 BSE image illustrating the area characterized using AFM and CPFEM at
1.5% global strain. Different activated deformation systems characterized
using trace analysis are denoted by color lines.
Figure 3.7 Examples of surface topography in grain 2 (left) and grain 3 (right)
measured by AFM after deformation. Line section profiles reveal that much
larger and more homogeneous surface steps result from twins (left) than
from slip lines (right). Dashed line is interpreted as the undistorted surface
inclination and serves as the basis for evaluating the overall height change
along the section.
Figure 3.8 Schematic representation of a cross-section through a twinned portion of a
grain. The twin trace on the sample surface is normal to the sketch (i.e.,
parallel to the Y direction). The surface displacement h due to twinning is
determined from an AFM line section in the X direction. tx and ttrue are the
twin width in the X direction (projected twin width) and the twin thickness
along the twin plane normal direction, respectively.
Figure 3.9 Schematic drawing showing the parameters associated with calculation of
the local average shear caused by the activated deformation system with
the slip plane normal n and Burgers’ vector direction b in sample coordinate
system. h is the surface height change caused by this deformation system.
Xm(n-1), Xmn, and Xm(n+1) are reference grid increments in the x direction.
Figure 3.10 The calculated local shear deviation at 1.5% global strain based on the
measurements using both horizontal and vertical AFM “Section” lines for 50
randomly chosen tiles. The measurements agree within about 10% in both
directions for all tiles.
Figure 3.11 The AFM-based experimentally measured sum of all individual local shear
distribution map of the highly characterized microstructure patch shown in
Fig. 3.4(a) at global strain level of (a) 1.5%, (b) 3%, and (3) 6%.
Figure 3.12 Individual local shear associated with different types of deformation
systems in the microstructure patch shown in Fig. 3.4(a). The experimental
measurements show that, in the early deformation stage, the local shear
	
  

xii	
  

increases almost proportional to the overall tensile strain in all grains, while
during further deformation (3% to 6%), the local shear in grains with
pyramidal slip shows a much larger increase rate than that in grains with
prism or basal slip.
Figure 3.13 (a) – (c) Three different sizes of 3D meshes of the grain patch
corresponding to the microstructure within the dashed box in Fig. 3.4(a).
Figure 3.14 (a) – (c) The summed individual shears for the top surface resulting from
simulated tensile loading to 1.5% engineering strain using the three
meshes shown in Figure 3.13(a) – (c), respectively.
Figure 3.15 The comparison of the summed individual shears distribution on the frontside and back-side of the meshes shown in Figure 3.13(b).
Figure 3.16 Comparison of experimental measured (shown in row (a)) and FEM
simulated (using mesh (c)), shown in row (b)) individual local shear
associated with different types of deformation systems in the microstructure
patch shown in Figure 3.4(a) at 1.5% strain. FEM simulation results
illustrate that the spatial distribution of local shear caused by prismatic and
basal slip agrees well with experimental measurement. Most of the local
shear caused by T1 twinning is observed in grain 2. Simulation does not
successfully simulate the pyramidal <c+a> slip in grain 10 where AFM
measurement reveals around 0.01 local shear.
Figure 3.17 CPFE model illustrating how the surrounding orientation (indicated with
Euler angles) affects local shear distributions at 2.2% global strain.
Figure 4.1 Material pile-up (left) and sink-in (right) increases or decreases the contact
height, hc, during indentation to the maximum indentation depth, hmax.
Figure 4.2 The array of indents applied on the microstructure in specimen S0-2.
Indentions were spaced by 20 µm × 16 µm. A semi-transparent OIM map
with inverse pole figure coloring is superimposed on a backscattered
electron image.
Figure 4.3 Color-scaled map of the maximum indentation depth (unit: nm) for the
indentation array shown in Figure 4.2. Each tile represents one indentation
in the middle of this tile.
Figure 4.4 Force-displacement curves from indentations conducted in the middle of
soft grains 5 and 10, and hard grain 2. The maximum load is 6 mN. The
lower inclination of the curve during loading stage for grain 2 indicates that
grain 2 is more resistant to plastic deformation.

	
  

xiii	
  

Figure 4.5 Surface topography scanned by AFM for indents in soft grain G7 (left) and
hard grain G2 (right) indicating the grain orientation dependent behavior of
pile-ups.
Figure 4.6 Pile-up topographies measured by AFM positioned in the inverse pole
figure (IPF) according to the corresponding indentation axes. The indents
are rotated according to the convention defined in [117]. The indents in the
upper half of the IPF are also displayed in the lower half by a mirroring
operation (improper rotation) with respect of (1 0 -1 0) plane and marked
there with an asterisk.
Figure 4.7 A deformed (indented) finite element mesh of hexahedral elements.
Figure 4.8 Experimental, CPFE simulated surface topographies, as well as the Δzi, j
map, of indents 6-a and 13-o, which were selected for the identification of
the constitutive parameters. The size of each map is 2.5 µm × 2.5 µm.
Figure 4.9 Experimental, CPFE simulated surface topographies, as well as the Δzi, j
map, of indents 6-o and 13-i, which were selected for the validation of the
set of optimized parameters. The size of each map is 2.5 µm × 2.5 µm.
Figure 4.10 Twenty-two simulated pile-up topographies measured by AFM positioned in
the IPF according to their indentation axes. The pile-ups are shown in gray
scale, and the regions below zero are shown in blue. The contour lines are
starting at -0.04 µm and 0.04 µm for below and above zero regions, and
drawn every 0.04 µm. Similar trends of pile-up features were found in
CPFE simulations as that shown in the experimental measurements.
Figure 5.1 The around-crack-tip region was divided by a 15 × 15 (225 in total) grid as
illustrated, and each BSE image was taken from the middle part of each
tile.
Figure 5.2 Upper array (shown in Figure 5.1) of backscattered electron images around
the crack tip at a center-to center spacing of 187 µm. The images in nearcrack-tip region show more lattice rotation and contrast variation, indicating
more residual plastic deformation.
Figure 5.3 (a) BSE images h5, h8, h11, and h14 from Figure 5.2; (b) their
corresponding DFTs and (c) a horizontal line profile across the center of
each DFT and showing a decrease in the center peak width with increasing
distance from the crack tip to the image location.
Figure 5.4 Schematic plots showing the process of fitting line profiles and measuring
the full width at half the maximum of the profile. Line profiles were first

	
  

xiv	
  

smoothed using an adjacent-averaging five-point window. The left half of a
smoothed profile was then fitted to an exponential curve.
Figure 5.5 The full width at half maximum of the FFT center peak versus the distance
from the crack tip at (a) θ = 0° (b) θ = 45°
Figure 5.6 Engineering stress-strain curve obtained for the commercially pure
titanium; the arrows indicate the strain increments at which the in-situ
uniaxial tensile test was interrupted to collect BSE images.
Figure 5.7 BSE images of a grain patch at different global strain levels during in-situ
tensile test indicating the evolution of microstructure features.
Figure 5.8 The full width of half the maximum of the DFT as a function of the
engineering strain based on images recorded at increments of plastic strain
during an in-situ tensile test.
Figure 5.9 Map of plastic strain around the crack tip (the area shown in Figure 5.1)
based on the FWHM of the DFTs of BSE images.
Figure 5.10 BSE images from the same microstructure patch and their corresponding
DFT line profiles showing that the change of brightness does not affect the
FWHM of the line profiles.
Figure 5.11 BSE images from the same microstructure patch and their corresponding
DFT line profiles showing that the change of contrast does not affect the
FWHM of the line profiles.
Figure 5.12 Phase difference and magnitude of the TSA signal as a function of distance
along a line through the tip of the fatigue crack in the direction of crack
growth. The extent of the plastic zone found from the phase difference map
is superimposed as a black area.
Figure 5.13 Map of plastic strain around the crack tip based on the FWHM of the DFTs
of images from the region shown in Figure 5.1. The white line shows the
measurement of plastic zone using TSA data. The green and blue solid
lines indicate the theoretical estimations of the plastic zone size based on
Dugdale’s and Irwin’s approaches, respectively.
	
  
	
  

	
  

xv	
  

LIST OF SYMBOLS AND ABBREVIATIONS
	
  
3D-XRD

Three-dimensional X-ray diffraction

AFM

Atomic force microscopy

b

Burgers vector of a given dislocation type

BSE

Backscattered electron

CPFE

Crystal plasticity finite element

CRSS

Critical resolved shear stress

DFT

Discrete Fourier transformation

ez

Specimen surface normal

EBSD

Electron-backscattered diffraction

ECCI

Electron-channeling contrast imaging

fip

Fracture initiation parameter

F

Deformation gradient

e

F

p

Elastic part of deformation gradient

F

Plastic part of deformation gradient

FWHM

Full width at half maximum

fcc

Face-centered cubic

FEM

Finite element method

FFT

Fast Fourier transformation

FIB

Focused ion beam

Gtopo

The topographic contribution to the objective fuction

GND

Geometrically necessary dislocation

h

Surface step height

	
  

xvi	
  

hcp

Hexagonal close-packed

KI

Applied stress intensity factor

p

L

Plastic velocity gradient

m

Dislocation slip direction

m’

Slip transfer parameter

MPODM

Multiple point over-deterministic method

n

Dislcation slip plane normal direction

N

Number of displaced twin planes

OIM

Orientation imaging microscopy

r

Plastic zone radius

R

Rotation matrix

s

Shear resistance of a given dislocation type

S

Schmid matrix for a given dislocation type

SACP

Selected-area channel pattern

SE

Secondary electron

SEM

Scanning electron microscopy

t

Direction of the traction force acting on a grain boundary

tx

Projected twin thickness

ttrue

True twin thickness

TEM

Transmission electron microscopy

TSA

Thermal-elastic stress analysis

Xmn

AFM section line length

z

Dislocation line direction of a given dislocation type

α

A given dislocation type

	
  

xvii	
  

α

Dislocation tensor

Δzi, j

Height difference between the measured and simulated topography

Δε

Cyclic strain

ΔKI

Cyclic stress intensity factor

ε

Applied strain

φ1 ,Φ, φ2

Euler angles

γ

Slip rate of a given slip system

κ

Angle between the two Burgers vectors of activated slip systems in two
neighboring grains

τ

Resolved shear stress on a given deformation system

ρ

Dislocation density of a given dislocation type

σy

Yield stress of a material

ωm , ωn

Complex roots of unity

Ψ

Angle between the two slip plane normals of activated slip systems in two
neighboring grains

	
  	
  	
  	
  	
  	
  	
  	
  	
  

	
  

xviii	
  

CHAPTER ONE
INTRODUCTION

1.1 Importance of and Challenges for Measuring and Simulating Heterogeneous
Deformation and Predicting Micro-crack Initiation
Heterogeneous deformation and local damage nucleation in metals have been
crucial topics for decades.

Simulations based on continuum fracture mechanics,

assuming an average value of both mechanical and crystallographic properties over
whole bulk materials, have been used to model macroscopic properties of bulk
materials, such as ductility, fatigue life, and fracture toughness [1-3]. Heterogeneous
deformation is commonly viewed as a precursor to damage nucleation. Only if the
heterogeneous deformation is reliably modeled can locations where cracks form in the
first place then can be properly predicted. Furthermore, from an engineering point of
view, knowledge of the micro-crack initiation locations can help to develop processing
routes, such as rolling and extrusion, to strengthen metals and alloys by avoiding the
development of these locations.
Although many experiments and simulations of localized damage nucleation in
polycrystalline materials during plastic deformation have been conducted in past
decades [4-7], few fundamental breakthroughs in predicting nucleation locations have
been achieved due to the complex behavior of grain boundaries.

It is generally

recognized that cracks prefer to nucleate at discontinuous interfaces, such as grain
boundaries [8, 9] and phase boundaries [10, 11], and that the plastic deformation in one
grain must transfer to another through grain boundaries in polycrystalline materials

	
  

1	
  

during deformation. During this process, some boundaries may initiate micro-cracks
easily — these boundaries are called “weak” boundaries, while those where damage
nucleation is relatively difficult are referred to as “strong” boundaries. Several factors,
however, can influence the behavior of grain boundaries, adding confusion to the
distinction between strong versus weak boundaries. The first factor is the orientations
of the two grains on the opposite sides of the grain boundary, which strongly influences
the transfer of dislocations across the boundary [12], and the kind and amount of
dislocations subsequently left in or near the grain boundary.

This factor also has

influence on the activated deformation systems in these grains. Details concerning this
factor will be provided in the following sections. The boundary energy is another factor
that should be considered. Miura et al., have shown that the higher the grain boundary
energy, the more easily the grain boundary fractures [13]. The boundary energy is,
however, closely related to the grain boundary structure, especially the cohesive
properties.

For example, low Σ boundaries may have more resistance to damage

nucleation due to their lower boundary energy and lower ability to absorb residual
dislocations than random boundaries [14]. Finally, yet importantly, the strain gradient
history in two adjacent grains also plays a crucial role in determining if a grain boundary
tends to crack. Recent research in finite element models is seeking ways to effectively
model strain gradients [7, 15, 16]. The same microstructure patch can display different
strain distributions after experiencing two different sets of deformation processes in
simulations, which may directly leads to the crack initiation at different grain boundaries.
To date, there has been little work combining all of these factors, leaving general rules
for predicting damage nucleation sites in poly-crystals undeveloped.

	
  

2	
  

1.2 Basic Approaches to Understanding Heterogeneous Deformation
Heterogeneous deformation usually results from two facts [17-19]. One is that
some “soft” grains are much more easily deformed than other grains, which means that
one or more deformation systems in the “soft” grains can be more easily activated than
those in “hard” grains under certain stress states, leading to large strain differences
between soft and hard grains. Secondly, equilibrium across the grain boundary requires
an additional stress field along the boundary, resulting from the various constraints
along the grain boundary of a given grain from different neighboring grains during
deformation process. Consequently, these influences from neighboring grains can lead
to heterogeneous deformation and associated strain gradients within the given grain.
Two basic theories have been reviewed in this section to help better understand the
heterogeneity within metallic materials during deformation processes.

1.2.1 Slip Transfer Criteria
An important approach to understanding the existence of heterogeneous
deformation in polycrystalline materials is the “slip transfer criteria”. During continuous
plastic deformation, strain needs to be transferred from one grain to another. This
transfer process often leaves residual dislocations in the grain boundaries and changes
the directions of Burgers vectors and slip planes of the original dislocations. To identify
which slip systems in neighboring grains will be activated during slip transfer processes,
the geometry of slip directions and planes, in relation to the misorientation between two
grains, has been studied by several authors. Luster et al. [12] presented a simple
parameter to describe the slip transfer event at grain boundaries:

	
  

3	
  

(1.1)
where, as shown in Figure 1.1, κ is the angle of a Burgers’ vector in one grain with
respect to the Burger’s vector in the neighboring grain, and Ψ is the angle of a slip
plane normal with respect of the normal in the other grain.

In addition, the angle

between the intersection line of one slip plane at grain boundary and the other
intersection line of the other slip plane in a neighboring grain is defined as θ. Luster’s
slip transfer criteria are built on three general rules for slip transfer in poly-crystals [20,
22]: (1) cosθ must be at a maximum; (2) the magnitude of Burgers vector of residual
dislocations in grain boundary must be at a minimum; (3) the resolved shear stress on
the outgoing slip system must be at a maximum. For example, if there are two slip
systems that are perfectly aligned, the parameter equals one, which means that the
grain boundary is transparent to dislocations (no heterogeneity existing near this
boundary). The parameter, m’, becomes zero when the slip planes or the slip directions
of two grains are perpendicular, which indicates that the edge dislocations cannot
transfer across the grain boundary (cross slip may still occur when the two plane
normals are perpendicular to each other).
Based on the m’ slip transfer criteria, Ashmawi and Zikry [21] further identified
the critical piling-up dislocation density near the boundary for slip transfer into the
adjacent grain. Instead of using Luster’s parameter, they use the slip transfer criteria
from Clark [22]:
(1.2)

	
  

4	
  

Figure 1.1 Three-dimensional slip transfer geometry at grain boundaries [6].
	
  

	
  

5	
  

which provides a framework to predict how heterogeneous deformation evolves as a
function of grain orientation and structure. Similar to m’, the smaller the parameter, ε,
is, the more difficult it is for dislocations to transfer, and thus the more dislocations will
pile up near the boundary (or transfer across with residual dislocation left in the
boundary), producing a larger heterogeneous stress field.
These simple rules provided some understanding of the heterogeneous strain
between adjacent grains near grain boundaries. They are, however, not sufficient to
describe heterogeneous deformation, since: (1) the process of transfer in the early
stage of plastic deformation affects subsequent deformation between two grains [21,
23]; and (2) it over-simplifies as they only consider one slip system activated in each
grain [20]. Also, it is still not clear how the heterogeneous strain is correlated to the
damage nucleation. For example, in random boundaries, heterogeneous strain may
result in large tensile stresses between two grains, which may lead to damage
nucleation [19, 21, 24].

Strongly cohesive boundaries, however, may force

heterogeneous strain in one or both grains in order to maintain compatibility [25].

1.2.2 Geometrically Necessary Dislocations
Another common approach to quantitatively measure heterogeneous deformation
is to measure the local distribution of the density of geometrically necessary dislocations
(GNDs).

Because of the heterogeneous deformation between grains during global

deformation processes in poly-crystalline specimens, as shown in Figure 1.2, GNDs are
necessary to avoid the generation of overlaps and voids at grain boundaries in order to
avoid micro-damage in the early stage of deformation. Ashby [26] and Busso et al. [27]

	
  

6	
  

	
  

	
  
Figure 1.2 If each grain of a poly-crystal deforms in a uniform manner, overlap and
voids appear (b). This can be corrected by introducing geometrically necessary
dislocations, shown in (c) and (d). [26]

	
  

7	
  

showed that the density of GNDs resulting from the existence of plastic strain gradients
in a given grain is completely geometrically determined. Nye [28] showed that there
exits a precise relationship between the dislocation tensor (α ), in other words, the field
of GNDs at a given point in the material and the dislocation density in its neighborhood,
expressed by
S

α = ∑ ρ (s)b(s) ⊗ z (s)

(1.3)

s=1

(s)

where ρ

(s)

is the dislocation density of dislocation type s, b

type s, and z

(s)

is the Burgers vector of the

is the dislocation line direction of type S. S represents the set of all

possible dislocations that could exist in the given crystal.
Because there are too many combinations of Burgers vectors and line directions
that can give an arbitrary α tensor, infinite sets of combinations can form the same
lattice rotation. Usually, to narrow down the number of solutions, certain assumptions
can be added to the general Equation 1.3 according to the nature of the materials of
interest. In one of the Nye’s papers [28], only nine dislocation types with Burgers vector
(s)

b

// <100> and z

(s)

// <100> were considered, which includes three pure screw and six

pure edge dislocations. In this case, Equation 1.3 can be written as a second-order
linear system form:
9

α ij = ∑ ρ (s)bi(s) z (s)
j

(1.4)

s=1

(s)

from which the unique or finite set(s) of solution of the dislocation density ρ

can be

solved. Sun et al. [29] only considered {111}<110> slip systems for face-centered cubic

	
  

8	
  

(fcc) metals, which contain 12 pure screw dislocations and 24 pure edge dislocations.
They solved for the dislocation density using Equation 1.3 by applying fcc dislocation
tensor deconstruction method [30]. In other works [31, 32], only edge dislocation types
were considered in order to simplify the calculation processes.
According to Ashby, the field of GNDs is totally lattice curvature determined,
therefore, the dislocation tensor (α ) can be experimentally determined by measuring the
local lattice rotations using transmission electron microscopy (TEM), electronbackscattered diffraction (EBSD), or X-ray diffraction. By comparing the variation of
GND density between and/or within different grains, the heterogeneity of plastic
deformation in these grains can be observed.
The relationships between micro damage initiation and the GND density,
however, are still unknown. First of all, most of the GND density calculations are based
on certain simplification assumptions, which may lead to the difference between
calculated results and the real microstructure environment. Also, a great change in
GND density across a grain boundary or within a grain does not necessary mean that
damage will initiate at this spot. A grain may generate large amounts of GNDs at a local
compression stress state, which generally does not tend to crack during deformation.

1.3 Experimental Characterizations and Simulations on Heterogeneous Deformation
1.3.1 Experimental Characterizations
Based on the theories discussed in Section 1.2, in order to fully understand the
heterogeneous deformation in different microstructure environments, including single,

	
  

9	
  

bi-, and poly- crystals, a large number of experimental studies have been conducted
over the decades.
1.3.1.1 Single crystals and bi-crystals
Single-crystal materials, such as silicon and niobium, are used in different kinds
of industries. A number of studies [33-36] on single-crystals have been conducted to
study heterogeneous deformation processes within a single grain due to different global
and/or local stress constraint. Also, since single crystals fully eliminate grain boundary
effects, many previous studies [37-39] have used single crystals as model materials to
study the relation between the activation of deformation systems and grain orientations
relative to uniaxial stress states, such as tension and compression.
Quantitative experimental measurements of heterogeneous deformation can be
traced back to Livingston and Chalmers’ work on cubic bi-crystals in 1950s [20]. Bicrystal specimens serve as another simplified prototype to model heterogeneous
deformation, because they have relatively simple stress states near the boundary and
the orientation of the grain boundary with respect to the applied stress can be easily
controlled. In bi-crystals, there is usually only one slip system in the interior part of the
grains under global uniaxial stress states. It has been shown, however, by Livingston et
al. that macroscopic continuity at the boundary will in general require the activation of at
least four slip systems near the grain boundary in a bi-crystal, even under uniaxial
stress states, because of the change in the grain boundary shape due to dislocation
slip. This result indicates that the local stress state in the near-grain-boundary region
can be much more complicated than in the interior parts of the grains, even in bicrystals. In other words, heterogeneous deformation exists in these bi-crystal tensile

	
  

10	
  

specimens.
To further quantity the dislocation activity in single crystals and bi-crystals during
deformation processes, the GND density is often calculated based on the theory
discussed in previous section using lattice rotation data measured by EBSD and/or Xray diffraction.
EBSDs are Kikuchi patterns that form as the backscattered electrons from a
stationary electron beam diffract when they exit the crystal.

EBSD provides a

convenient way to observe the surface heterogeneous deformation (especially local
crystal lattice rotations [40-42] and texture changes [43, 44]) on the surface of
crystalline materials. Advanced EBSD equipment has the advantage of being able to
collect information from small areas under the electron beam, with a resolution in the
range of hundreds of nanometers.

However, the quantification becomes difficult at

large strain levels. Sun et al. [29] have attempted to quantify the density of GNDs within
an aluminum bi-crystal and at the grain boundary, by extracting lattice curvatures using
EBSD and employing the geometrical relationship between strain gradients and GND
density. Their results show that GND density and distribution are highly dependent on
applied plastic strain level and grain boundary structure, as illustrated in Figure 1.3.
The recently developed three-dimensional X-ray diffraction (3D-XRD) technique
is able to characterize local orientations and stress states in bulk materials in a
nondestructive manner. The basic setup of the 3D-XRD is illustrated in Figure 1.4. By
analyzing the streak direction of a specific Laue reflection from a small volume (about
1 µm × 1 µm ×1 µm) of the material, the Burgers vectors associated with local GNDs
might be identified [31]. 3D-XRD studies on Cu single-crystal micro-pillars [42], and Ni

	
  

11	
  

	
  
	
  
Figure 1.3 (a) Orientations of the two crystals in relation to the compression direction Z.
GND distribution in a 10% (b) and a 30% (c) strained aluminum bi-crystal [29].

	
  

12	
  

Figure 1.4 A schematic drawing of the setup of 3D-XRD [45]. (For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic
version of this dissertation.)	
  	
  

	
  

13	
  

bi-crystals [31] have quantitatively characterized the strain heterogeneity of local lattice
rotation within/between grains. The in-situ 3D-XRD quantitatively measured the total
lattice rotation of Cu single crystal micro-pillar oriented for double slip activation during
compression testing [42], which was demonstrated by an EBSD scan from the same
crystal, as shown in Figure 1.5.

This work showed that lattice rotation occurred

heterogeneously along the vertical axis of pillars during the compression test.
Although single- and bi- crystals provide some basic understanding of grain
lattice rotation and dislocation transfer across grain boundaries, many observed
deformation behaviors in poly-crystals could not be simplified satisfactorily to the bicrystal deformation processes. Bi-crystals generally activate only one slip system with
the highest value of Schmid factor in the interior part of grains, where grains behave like
single crystals; in contrast, poly-crystals may require more activated slip systems during
plastic deformation processes to keep the grain boundaries cohesive according to the
von Mises criteria [35]. Also, poly-crystals can activate even more slip systems near
grain boundaries than in the interior of the grains [20], because of the complex stress
states and constraints near the boundaries.

1.3.1.2 Poly-crystals
Based on the studies on plastic deformation of single- and bi- crystals, a number
of extensive experimental studies of plastic heterogeneities have been carried out on
cubic or near-cubic system poly-crystalline metals in recent years.
Scanning electron microscopy (SEM) has been used to observe and measure
the plastic deformation to assess how the local lattice curvature and dilation associated

	
  

14	
  

Figure 1.5 Two vertical XRD scans (line 1 and 2) compared with the corresponding
EBSD scan data.

All curves display the total misorientation relative to the base

orientation of the pillar plotted as a function of distance from the pillar base [42].

	
  

15	
  

with GNDs affect the lattice interaction with incident electron beams. Selected-area
channel patterns (SACPs), formed from the variation in intensity of backscattered
electrons resulting from rocking the electron beam about a point on the surface of a
single crystal or grain, are well known to deteriorate with increases in plastic strain [4648].

Many studies have quantitatively elucidated the dislocation density and lattice

rotation behavior near crack tips in a poly-crystalline superalloy [49, 50], and an
aluminum alloy [51] during plastic deformation using SACPs.
One of the further developments of the application of SACPs is electronchanneling contrast imaging (ECCI), which allows the direct observation of dislocations
and other micro-defects near sample surfaces without destruction of bulk samples [52].
This technique can be conducted by aligning the optic axis of the microscope to the
edge of a strong channeling band using SACPs to set up “two-beam” conditions similar
to that in TEM. This non-destructive technique is ideal for observing the heterogeneous
behavior during the early stages of plastic deformation in crystalline materials [53, 54].
As shown in Figure 1.6, Simkin et al. [55] revealed dislocation pile-ups in a given grain
near grain boundary at which twins in the neighboring grain intersected with this grain
boundary. This observation is solid experimental evidence of the dislocation transfer
across grain boundary in poly-crystal environment.

Also, ECCI allows quantitative

measurements of dislocation density [56] from a much larger area (compared with TEM)
in different grains in bulk samples, which can reveal the heterogeneous deformation in
these grains. These two techniques, however, are usually limited to the study of rather
course grained polycrystalline materials or single crystals, because a clearly defined
SACP requires the capability to rock the beam only in an area of the tens to hundreds of

	
  

16	
  

Figure 1.6 (a) SACP composite from a deformed grain (~1% strain) shown in (b). The
‘‘×’’ indicates the electro optic axis at zero sample tilt. (b) Channeling contrast image of
the deformation defects in a grain of TiAl with the calculated {111} plane traces.
Arrowheads indicate bands of dislocations piled up in the neighboring grain at points of
intersection of deformation twins with the grain boundary. The bright contrast regions,
pointed by the black arrows in (b), near the grain boundary represents high density of
dislocations [55].

	
  

17	
  

microns in diameter, due to spherical aberration and focusing errors.
Other experimental studies, such as Delaire et al. [19], have shown that slip
systems with high Schmid factors (calculated based on the grain orientation collected by
EBSD and the global stress state) tend to extend across grains, while slip systems with
moderate Schmid factors had slip traces that extended part way into the grain interior in
polycrystalline columnar-grained pure copper specimen under uniaxial tensile state.
Another study has focused on the micro-scale distribution of GNDs during plastic
deformation. According to Barabash, et al. [57], in the near-surface zone of severely
deformed polycrystalline Ti after friction stir processing, the GND density shows a
heterogeneous distribution between the thermal mechanical affected zone and the heat
affected zone, between different grains in a given zone, and even within a particular
grain. The study of hexagonal close-packed (hcp) metals, however, is limited, which will
be discussed in Section 1.5.

1.3.2 Simulations of Heterogeneous Deformation
The crystal plasticity finite element (CPFE) method is often used to simulate the
plastic deformation processes that occur in three-dimensional microstructures, where
the heterogeneous deformation between and within grains co-exists during deformation
[19]. In CPFE simulation, the deformation gradient F is defined as the combination of
e

p

elastic (F ) and plastic (F ) parts, which is the same as conventional FEM:

F = F eF p

(1.5)

and by applying the flow rule,


F p = Lp F p

	
  

(1.6)

18	
  

p

The plastic velocity gradient (L ) is composed of the shear contributions of all slip
systems [58] in CPFE modeling,

(

L p = ∑ γ α mα ⊗ nα
S

)

(1.7)

where Sα = mα ⊗ nα is the Schmid matrix for a given deformation system. This is one
of the key equations that combine the crystallographic nature of dislocation slip with
conventional finite element method. The slip rate of a given deformation system ( γ α ) is
also closely related to the crystallographic nature of the materials, such as critical shear
stresses (CRSS) and hardening parameters. When the resolved shear stress of a given
system is above its CRSS, the given system will contribute to the total slip rate. On the
other hand, if the local stress becomes lower than the CRSS after work hardening, this
system will be shut down in the simulation. There are different ways to express the slip
rate:
n

τα
γ = γ 0 α sgn τ α [59]
s
α

( )

(1.8)

or

γ α =

τ α − xα − r α
K

n

(

)

sgn τ α − xα [60]

(1.9)

where τ α is the resolved shear stress on the given deformation system. sα , xα , rα are
related to CRSS and hardening parameters, the values are time- and/or straindependent to reflect the strain hardening process.

	
  

19	
  

Since the CPFE method is usually used to simulate microstructure scale
deformation processes, it is also important to experimentally identify the grain
orientations and shapes within the microstructral patches of interest before simulation.
EBSD and Orientation imaging microscopy (OIM) have been widely used to identify
crystal orientations, helping to generate meshes for CPFE simulations. Two kinds of
three-dimensional meshes can be found among these works. One is called a quasi-3D
mesh that represents the three-dimensional microstructure by extending vertically the
surface grain information perpendicular to the interior of the grain (forming a set of
columnar grains) [61, 62]. The other type of mesh is a real 3-D mesh that includes real
crystal information, shape, and orientation from beneath sample surface [63, 64]. To
develop such meshes, electro-polishing has often been used to remove surface layers.
However, the shortcoming is that it is hard to control the thickness of the removed layer,
which may result in errors in the construction of 3-D meshes. More recent studies [65,
66] have used 3D-EBSD (a combination technique of focused ion beam (FIB) and
EBSD)), during which the removed layer can be more easily controlled, leading to more
accurate 3D-mesh simulation studies.
To date, most CPFE microstructure modeling efforts have focused on cubic and
near-cubic materials such as aluminum [59, 67, 68], copper [19, 61], stainless steel [69],
and TiAl [70]. Some of the microstructural simulations results have been compared to
experimental data to assess the accuracy of these models. In a study of high-purity
columnar-grain aluminum under plane-strain compression [59], the quasi-3D CPFE
model was able to successfully predict both the average deformation texture and the
local grain rotations, but it failed to predict the morphology of the “deformation bands”.

	
  

20	
  

A dislocation-density-based finite element method (FEM) simulation results of the
heterogeneous strain field of a single-layer-grain copper sample plastically deformed by
uniaxial tension [19] matches well with the experiment results measured by grids
deposited on the polished prior to deformation sample surface.
In a study of polycrystalline copper by Musienko et al. [61] using both quasi-3D
and real 3D meshes, the simulation of local lattice rotation and local strain fields of a
100-grain patch during uniaxial tension based on the two kinds of meshes, as shown in
Figure 1.7, has been both compared to the experimental data measured by EBSD. The
results for the quasi-3D mesh retain some traces of the experimental behavior, but the
results are not as good as those obtained with the real 3-D mesh.
Nevertheless, unlike CPFE simulations of cubic metals, fewer simulation studies
of heterogeneous deformation of hexagonal metals and alloys have been reported [71,
72], and have not been adequately compared to experimental data. Details on the
simulation works of hcp materials will be discussed in Section 1.5.
Although various approaches of microstructural-scale computational simulations
of heterogeneous deformation have been conducted in recent years, it is still difficult to
simulate the behavior of grain boundaries and dislocations near grain boundaries,
resulting directly in difficulty in predicting damage nucleation at grain boundaries. In
most of the simulations discussed in the previous paragraphs, grain boundaries are
simply treated as if they are transparent, which means that dislocations can effectively
transfer across boundaries.

It is very difficult to simulate how deformation transfer

occurs by dislocation absorption and emission at grain boundaries due to the limited
understanding of the atomic details of grain boundaries and dislocation sources.

	
  

21	
  

Figure 1.7 The various mesh types used in Ref. [61]: (a) real 3D mesh (31,780 nodes),
(b) real 3D mesh (130,818 nodes), (c) quasi-3D mesh (14, 076 nodes).	
  

	
  

22	
  

Grain boundaries can be modeled as a location where interactions of a series of
Burgers vectors take place [73, 74]. In some simulation approaches, for example, low
angle boundaries were modeled as a series of edge dislocations existing in the lattice
[73]; while at high angle boundaries, the dislocation structures become much more
complicated, making interactions between boundaries and near boundary dislocations
unpredictable.

In other studies [75-77], special FEM elements (with different

mechanical and crystallographic properties from elements in the interior of grains) are
added at grain boundary regions to represent the different behavior of grain boundaries
(see Section 1.4).

1.4 Current Understanding of the Relationship between Heterogeneous Deformation
and Damage Nucleation
Many of previous studies have been focused on the parameters of crack growth
and final critical crack length, with the assumption that initial microcracks exist in the
materials, but these parameters are useless in predicting locations of damage
nucleation. Since the 1980s [75], more emphasis has been put on the initiation of
microcracks.

There are several mainstream approaches toward predicting damage

nucleation mechanisms at grain boundaries during plastic deformation at ambient
temperature based on the current understanding of heterogeneous deformation and slip
transfer mechanism.
The first paper on predicting and simulating the damage nucleation and the
development of the crack critical length was presented by Needleman [75] based on a
parameter called the cohesive interface energy. In this paper, void nucleation by atom

	
  

23	
  

de-bonding was correlated with normal and shear traction history in the grain boundary.
It was assumed that there exists a cohesive zone in front of the physical crack tip that
consists of upper and lower surfaces, called cohesive surfaces, held together by the
cohesive traction.

The cohesive traction is correlated to the separation distance

between the cohesive surfaces by the cohesive zone model.

The two cohesive

surfaces separate gradually under an external applied stress. The separation of these
surfaces at the tail of the cohesive zone will reach a critical value under a specific
stress, and the crack growth will stop. A fundamental parameter of the cohesive zone
model is the work of separation per unit area of cohesive surface, called the cohesive
energy density [76, 77]. This two-dimensional model was first developed for simulating
the micro-crack growth process using finite element method.
Later, the idea of energy equilibrium was adapted in damage nucleation models.
Hao et al. [78, 79] and Clayton and McDowell [80, 81] have further developed the model
to predict successfully to some extent local stress and local strain evolution history and
the boundary fracture. In their papers, each grain was modeled with a large number of
finite elements, permitting gradients of elastic and plastic deformation to develop, both
among different grains and within the same grains. Cohesive zone elements were
inserted at all grain boundaries, allowing both normal and tangential inter-granular
fracture initiation, and all the grain boundaries were assumed to have the same
strength. In the model, the heterogeneous deformation was simply correlated with the
residual lattice energy density between two grains. The locations of damage nucleation
in a material are based on how much residual energy could be released by voids at
grain boundaries under a certain applied stress level. The cohesive energy density at

	
  

24	
  

grain boundaries with different orientations varies with respect to applied stress.
Boundaries with higher cohesive energy, which are weakly bonded, may crack first.
This model, however, is not perfect, considering the lack of dependence on different
grain boundary strength arising from grain boundary structures and local grain
misorientation. Thereby, the predicted locations of void nucleation are less persuasive.
Recently, Simkin et al. [82] showed that in near-gamma polycrystalline TiAl (near
cubic structure), microcracks were observed where highly localized deformation twins
interacted with grain boundaries. It is the twin shear that causes a concentrated stress
on one side of the boundary (shown in Figure 1.8 (a)), which cannot be explained by the
theories introduced earlier. To fix the deficiency, Simkin et al. introduce a fracture
initiation parameter that contains more geometrical factors to interpret this phenomenon
[83]. The fracture initiation parameter (fip) is defined as:

F = mtw btw ⋅ t

∑ ord btw ⋅ bord

(1.10)

where, as shown in Figure 1.8(b), mtw is the Schmid factor for a specific twinning
system in the grain, assuming a local stress state equal to the global stress state, btw
and bord are the unit Burgers vectors for the single twin system and ordinary dislocation
systems in each grain respectively, t the direction of the traction force acting on the
boundary, btw ⋅ t is an expression of the opening force acting on the boundary, and

∑ ord btw ⋅ bord

describes how well the local shear direction at the boundary can be

accommodated by dislocations in adjacent grain. The results for TiAl show that larger
values of F correspond to the experimentally observed locations for crack initiation,
while maxima of the components of fip show weak relationship among themselves and

	
  

25	
  

(a)

(b)
Figure 1.8. (a) Backscattered electron image of γ-γ TiAl grain boundary damage
initiation due to twin interaction with the grain boundary [82]; (b) 2D schematic of crack
nucleation at a concentrated band of slip or twinning [82, 83].
	
  

	
  

26	
  

fracture initiation. This indicates that a combination of several factors, rather than a
single factor, influences the damage behavior at grain boundaries.

By combining

several geometric factors, a fip can identify the “weak” grain boundary conditions in the
context of a specific state of applied stress [70]. The fip may also identify when plastic
shear at boundaries can cause partial slip transfer, which will leave the residual
dislocations in the boundary, leading to damage nucleation at these boundaries.
Nevertheless, the fip only considers slip/twinning directions with respect to the applied
stress, instead of both the orientations of the slip/twinning planes and directions,
implying that slip directions play more significant roles in crack nucleation. The fip
works quite well in TiAl system because of the highly localized twinning behavior during
deformation [82], but its credibility in other alloy systems remains unproved.

1.5 Studies on Commercial Purity Titanium and Other Hexagonal Materials
In commercial purity titanium, four dislocation slip systems have been reported
[85, 90], as shown in Figure 1.9. {1010} < 1210 > prismatic slip is the primary slip
system during room temperature deformation, but the other three slip systems,

{0001} < 1210 > basal slip, {1011} < 1210 > pyramidal <a> slip, and {1011} < 2113 >
pyramidal <c+a> slip, can be activated in some orientations as well as in order to satisfy
the von Mises criteria to permit an arbitrary plastic strain in polycrystalline titanium. The
CRSS of slip systems in α-titanium determined in different studies are summarized in
Table 1.1. Among all slip systems, prismatic slip has the lowest CRSS. There are also
four deformation twinning systems in α-titanium that can contribute to deformation
(detailed information is given in Table 1.2 [91]), as shown in Figure 1.10. When the
	
  

27	
  

	
  
Figure 1.9 Slip systems for α-titanium: (a) prismatic slip, (b) basal slip, (c) pyramidal <a> slip, and (d) pyramidal <c+a>
slip. The arrows only represent the slip directions, not the magnitudes of the dislocation slip.

Figure 1.10 Twinning systems for α-titanium: (a) T1 twinning, (b) T2 twinning, (c) C1 twinning, and (d) C2 twinning. The
arrows only represent the twinning directions, not the magnitudes of the twinning.

	
  

28	
  

Table 1.1 CRSS of different slip modes reported for α-titanium.

Prismatic slip

Basal slip

Pyramidal <c+a>

Pyramidal <a>

Refence [86]
(relative values)

1

3~6

6~15

3~6

Refence [87]

30 MPa

150 MPa

120 MPa

N/A

Refence [92]

37 MPa

49 MPa

197 MPa

N/A

	
  

	
  

29	
  

Table 1.2 Deformation twinning modes in α-titanium.

Mode

Twining plane and direction

Theoretical shear

T1

{10 12} < 1011>
{1121} < 1 126 >
{1122} < 1123 >
{10 11} < 10 12 >

0.171

T2
C1
C2

	
  

30	
  

0.629
0.221
0.101

maximum principal stress direction is close to the crystal c-axis orientation, either <c+a>
slip, tensile (or extension) twinning modes (T1 and T2), or compressive (or contraction)
twinning modes (C1 and C2) have high Schmid factors, and their operation often
contributes to the crystal shape change. Among these four twinning systems,

{1012} < 1011>

(T1 extension) twinning is the most commonly observed at room

temperature due to its relatively low magnitude of shear (0.17). In this project, “soft”
crystal orientations and/or grains in α-titanium are defined as those grains with high
Schmid factors for prismatic slip – those grains having their c-axis nearly perpendicular
to the applied uniaxial stress. Hard crystal orientations and/or grains, however, are
those grains with low Schmid factors for prismatic slip, with their c-axis parallel to
applied uniaxial stress.
Compared to cubic metals, fewer works have been reported on heterogeneous
deformation of polycrystalline hexagonal materials (Mg, Ti, Zr, etc.) than those for cubic
materials. Simulations of a polycrystalline magnesium alloy [71] indicated that a high
stress concentration always occurred near grain boundaries between an untwined grain
and a twinned grain, regardless of the amount of global strain. The results demonstrate
that it is important to include deformation twinning and twin-induced reorientation into
the CPFE model analysis of heterogeneous deformation in hexagonal materials. Dunne
et al. [72] demonstrated that elastic anisotropy in titanium alloys has a significant effect
on the local stress distribution and the magnitude of accumulated slip.

They also

showed that the inclusion of length-scale effects tended to lead to a more smooth
plastic strain distribution locally near grain boundary by generating GNDs. In a study of
columnar polycrystal zirconium under tension by Héripréa et al. [93], CPFE modeling

	
  

31	
  

results were compared to the experimental strain field obtained by digital image
correlation at the scale of the grains. They suggested that this comparison could be
used for identification of parameters of crystallographic constitutive laws, such as CRSS
and hardening parameters for CPFE simulations. To date, there is still no breakthrough
on predicting damage nucleation in hexagonal materials.

1.5 Motivations for This Study
Hexagonal polycrystals offer a variety of opportunities to study heterogeneous
deformation and damage nucleation research. First of all, hexagonal metals have not
been studied as much as cubic materials, mainly because the process of modeling
hexagonal metals is more complicated. The CRSS and hardening parameters of the
various deformation systems are difficult to determine experimentally for these
materials.

For cubic metals, uniaixal tension and/or compression tests on single

crystals are often used to measure the values of CRSS for deformation systems. Work
by Williams et al. [84] has used hexagonal close-packed (hcp) TiAl alloy single crystals
to measure the CRSS of the material. However, for hexagonal metals, the values of
CRSS for the different systems are more often measured by carrying out statistical
analysis of slip traces in polycrystals [85] or curve fitting in association with CPFE
simulations [86]. Consequently, the published values of the CRSS of hexagonal metals
vary significantly [84-87]. Secondly, the lower crystal symmetry of hexagonal materials
is expected to result in more deformation heterogeneity than cubic metals. In general,
the c axis of hexagonal crystals is inextensible, except by the pyramidal <c+a> slip or
twinning.

	
  

In contract to cubic metals, in hexagonal materials, there are fewer slip

32	
  

systems in the most easily activated categories of slip systems (one slip plane and three
slip directions for basal slip; three slip planes and one slip direction on each plane for
prismatic slip) than those in cubic materials. This makes it easier to identify specific
activated slip or twin systems using trace analysis [55] under defined stress states.
Thirdly, hexagonal crystal materials also show strong texture after rolling processing.
Those hexagonal metals having higher values of the c/a ratio such as zinc or cadmium
reveal textures with basal poles tilted about ±15° to ±25° from the normal direction
towards the rolling direction [88]. Metals with c/a ratio lower than the ideal value, like
titanium, show a splitting of the basal poles in the transverse direction [88, 89]. With the
help of the EBSD technique, it is easy to recognize grain boundaries with high c-axis
misorientation, where strong deformation heterogeneity might exist.
Based on the discussions above, the long-term goals of this study is to
understand the relationships between activated slip systems and grain boundaries, and
to model their inflences on damage nucleation at grain boundaries in commercial purity
titanium. Indeed, the ability to predict damage nucleation is also one of the major goals
of computational plasticity. Since heterogeneous deformation is commonly viewed as a
precursor to damage nucleation, this long-term goal is actually based on accurately
measuring heterogeneous deformation in polycrystalline specimens, and developing
more reliable multi-scale models that can accurately model conditions that favor
damage nucleation in interfaces.

	
  

33	
  

1.6 Overview
In this study, commercial purity titanium (hcp metal) was used as a model
material to study different scales of heterogeneous deformation in fatigue crack samples
and four-point bending samples. A series of comprehensive experimental and modeling
analysis have been conducted to characterize the heterogeneous deformation within
different scales of microstructure patches.
Chapter Two provides all the material parameters and experimental details and
constitutive model using CPFE method including in this project.
In Chapter Three, the heterogeneous deformation within micro-scale patches is
presented. Based on a series of observations of the activation of deformation systems,
the formation of grain boundary ledges, and grain rotations within poly-crystalline αtitanium, the heterogeneous deformation within a carefully chosen microstructure patch
(including about 20 grains) was studied in detail. The local dislocation activity of this
patch was fully quantified using a new combined technique using atomic force
microscopy (AFM), Backscattered electron (BSE) imaging, and EBSD. EBSD data was
collected before and after four-point bending to facilitate trace analysis of deformation
activity in grains. Distribution maps of local shear caused by dislocation slip derived
from z-displacement data measured by AFM were then directly compared to results of
CPFE simulations that incorporate a phenomenological model of the deformation
processes to evaluate the ability of the CPFE model to match the experimental
observations. The existing CPFE model successfully predicted most types of active
dislocation slip systems within the grains at correct magnitudes, but the spatial

	
  

34	
  

distribution of strains within grains differed between the measurements and the
simulation.
As the difficulty of directly measuring the CRSS of different deformation systems
in titanium can be a major cause of the deficiency of the match between CPFE
modeling results and experimental measurements, in Chapter Four, a nano-indentation
technique is introduced to examine the CRSS and flow laws of commercial purity
titanium. Arrays of conical nano-indentations were made on patches of grains, and
AFM and EBSD scans of on the patch indicates that the surface pile-up topography was
strongly crystallographically dependent.

Corresponding CPFE simulations using a

similar constitutive model to that used in Chapter Three predicted the pile-up patterns in
good agreement with the experimental measurements. The CRSS values for prismatic,
basal, and pyramidal <c+a> slip of the CPFE model were identified by optimizing the
simulation results (load–displacement and residual pile-up pattern) of the indentation
process in different grain orientations using a non-linear optimization method. A rerun
of the CPFE simulation of the same microstructure patch studies in Chapter Three with
the optimized parameters, however, showed only minor differences, which indicates
more quantitative studies on strain hardening behavior and slip transfer across grain
boundaries are strongly needed.
In Chapter Five, a novel approach using BSE imaging and discrete Fourier
transformation (DFT) processing was established to measure the meso-scale plastic
strain distribution within plastic zone around a fatigue crack tip. An in-situ uniaxial
tensile calibration specimen indicates that the central peak widths of DFT of BSE
images have a close relationship with the residual strain with the grain patches. The

	
  

35	
  

measurement results were demonstrated to have good agreement with the plastic zone
size and shape measured using thermoelastic stress analysis (TSA). This approach
should prove to be an effective way to measure local heterogeneous deformation in
other kinds of deformed samples.
In the future, ECCI will be used to observe the dislocation activity near grain
boundaries to help further understanding the mechanism of heterogeneous deformation.
Future experiments will also be focused on establishing a new experimental
characterization method to facilitate the building of a real-3D mesh, and to compare the
CPFE simulation results of the real-3D mesh with the existing results from the quasi-3D
mesh. Future simulation work will be focused on developing a more advanced CPFE
model, considering the information of slip transfer across grain boundaries, to simulate
heterogeneous deformation within grain patches more reliably.

	
  

36	
  

CHAPTER TWO
MATERIAL, EXPERIMENTAL DETAILS, AND SIMULATION FRAMEWORK

2.1 Material
Two different sources of commercial purity titanium were used for different
aspects of this program. These will be designated as Material A and Material B in this
dissertation. Detailed information about the two sources of commercial purity titanium is
provided below.

2.1.1 Material A
Material A was provided by Max-Planck-Institut für Eisenforschung, Düsseldorf,
Germany (MPIE) as a plate approximately 15 cm × 20 cm × 5 cm in dimension.
Unfortunately, the original supplier and processing history of the plate are not known. It
had a moderately strong texture (about eight times random, as shown in Figure 2.1) and
an average grain size of about 80 microns. X-ray diffraction studies done in MPIE prior
to deformation indicated that the grains generally contained a low density of
dislocations, though the internal stress fields could be significant in minority grain
orientations, arising from the anisotropic coefficient of thermal expansion.
Specimens with dimensions of 25 mm × 3 mm × 2.5 mm (shown in Figure 2.2) for
four-point bending tests and nano-indentation tests were cut from the as-received
annealed plate with their longitudinal direction oriented at 0° and 45° from the plate
rolling direction, denoted as specimen series S0 and S45, respectively.

	
  

37	
  

	
  
Figure 2.1 {0001} and {10 10} pole figures of (a) specimen S0, and (b) specimen S45.

	
  

38	
  

Figure 2.2 Specimen orientations in Material A plate (left), and the size of specimens for
four-point bending tests and nano-indentation tests (right).	
  

	
  

39	
  

2.1.2 Material B
The other commercially purity titanium plate, denoted as Material B, had a weak
texture (about four times random, as shown in Figure 2.3) with an average grain size of
about 20 microns. Specimens for in-situ tensile tests and fatigue tests were cut from
the as-received annealed plate. The sizes and shapes of the two types of specimens
are shown in Figure 2.4 and Figure 2.5, respectively.

2.2 Sample Preparation
Prior to deformation, the specimens were mechanically polished. The specimens
were first ground using SiC grinding papers through 240, 400, 800, 1200, 2500, and
4000 grits (European units). Each grinding step took about 10 minutes, and specimens
were rinsed using water and methanol between each step. The pre-ground specimens
were then polished on a specific polishing cloth (Struers, Code: MECHE, Category No.
40400094) with 0.05 µm colloidal silica plus 30% hydrogen peroxide. The samples
were then dipped into glycerin 68% + nitric acid 16% + hydrofluoric acid 16% for 4 to 5
seconds at room temperature, and rinsed with distilled water after etching. Polishing
with colloidal silica and etching was repeated three to four times to remove the thin
deformation layer resulting from sample cutting and pre-grinding of the specimen
surfaces. The polished surfaces were considered to be acceptable when high-quality
EBSD patterns (confidence index greater than 0.8 for most of the patterns) could be
obtained from the specimen surface prior to deformation. To carry out differential image
correlation studies, fine tungsten cross patterns were deposited using FIB on some
areas of some Material A specimens prior to deformation, as shown in Figure 2.6.

	
  

40	
  

Figure 2.3 {0001} and {10 10} pole figures of Material B.	
  

	
  

41	
  

Figure 2.4 Size and shape of specimens for in-situ tensile tests cut from Material B.
	
  

	
  

42	
  

Figure 2.5 Size and shape of specimens for fatigue tests cut from Material B plate.
	
  

	
  

43	
  

Figure 2.6 Backscattered electron image of a microstructure patch with deposited fine
tungsten cross patterns.

	
  

44	
  

2.3 Experiment Details
2.3.1 Scanning Electron Microscopy
2.3.1.1 Electron Imaging
Two SEMs were used in this project to conduct imaging before and after
deformation. One was a CamScan 44FE field emission scanning electron microscope
(CamScan, Cambridge, UK) with a polepiece-mounted, four-quadrant, silicon diode type
BSE detector, located in Department of Chemical Engineering and Materials Science,
Michigan State University (MSU).

Backscattered electron (BSE) and secondary

electron (SE) imaging were conducted at an accelerating voltage of 25 kV, with a
working distance of 15 mm.

Images were captured using an external personal

computer with 640 × 480 pixel digital resolution.
The other SEM was a high-resolution JEOL JSM 6500 F SEM, located in
Department for Microstructure Physics and Metal Forming, MPIE. SE imaging was
conducted at an accelerating voltage of 15 kV, and captured with 1024 × 768 pixel
digital resolution.

2.3.1.2 EBSD/OIM measurement
Grain orientations were determined using identical orientation imaging
microscopy (OIMTM) systems (TSL, Draper, UT, USA) installed on both the CamScan
44FE and the JSM 6500. A schematic diagram of a typical EBSD system is shown in
Figure 2.7, and detailed information about the system can be found in a review paper of
Humphreys [94]. EBSD was carried out at a working distance of 33 mm on specimens
that were tilted 70 degrees from the horizontal surface. All of the EBSD data in this

	
  

45	
  

	
  
Figure 2.7 A typical setup of an EBSD/OIM system in SEM [94].

	
  

46	
  

project was collected using beam-scanning mode, during which the electron beam spot
scanning processes are controlled by the EBSD acquisition software. The EBSD data
was collected using OIM Collection v.OIMDC 4.51 (TSL, Draper, UT, USA), and the
EBSD data processing software used in this project was OIM Analysis v.5.31 (TSL,
Draper, UT, USA).

2.3.1.3 In-situ Tensile Testing
In-situ uniaxial tensile tests were conducted on the tensile specimen, shown
schematically in Figure 2.4(a). The commercial 100 lb in-situ tensile stage (E. Fullam,
Latham, NY, USA), shown in Figure 2.8, was installed within the CamScan 44FE
scanning electron microscope.

The in-situ tensile testing provided a calibration

standard for Material B, which will be discussed in detail later in Chapter Five. Another
separate continuous tensile test (non-stop test from 0% strain to specimen fracture) was
conducted to measure the stress-strain curve for material B. The strain increments at
which the in-situ test was stopped and images are taken were marked out by black
arrows on the stress-strain curve, as shown in Figure 5.6. BSE images were taken from
six areas in the middle of the specimen at each deformation step, under the identical
imaging conditions (brightness, contrast, magnification, and working distance) as those
for imaging the fatigue specimen of Material B. Because the brightness and contrast
knobs in the CamScan are not number-based, the positions of these knobs were
carefully noted when changing the specimens to guarantee the same brightness,
contrast, and working distance.

	
  

47	
  

Figure 2.8 Deformation stage for in-situ tensile test.

	
  

48	
  

2.3.2 Four-point Bending Testing
Four-point bending tests produce maximum tensile stress states at the directly
observable surface of the deformed specimens, also eliminating the triaxial stress states
present in uniaxial tensile tests. Also, according to an elastic-plastic continuum FEM
simulation of this geometry [95], the plastic strain at the surface is approximately
uniform in the region between the two inner bending pins spaced 5 mm. Therefore, the
assumption of uniaxial tensile stress can be made for the surface region within the two
inner bending pins during bending tests.
The bending stage shown in Figure 2.9 was designed so that a strained but still
loaded specimen could be investigated in the SEMs as well as the 3-D XRD facility at
Argonne National Laboratory (ANL). The polished samples examined in this project
were deformed by four-point bending to achieve a surface tensile strain of about 8%
and 15% for S0-1, 1.5%, 3%, and 5% for S45-1, and 1.5% and 4% for S0-2,
respectively.

Microstructure investigations were concentrated in a 2.5mm × 2.5mm

surface area located near the center of the bend specimens, where the stress state was
well approximated as uniform tension.

2.3.3 Atomic Force Microscopy
AFM was performed in “tapping” mode using two Dimension 3100 AFMs (Digital
Instruments, Plainview, NY, USA) in this work. One was located at MSU and the other
at MPIE. The AFM probes used in this particular project were TESP sharp silicon
probes (Bruker, Billerica, Massachusetts, USA). Figure 2.10 shows both a schematic
drawing (a) and an SEM image (b) of an AFM probe tip. The AFM data was collected

	
  

49	
  

Figure 2.9 Four-point bending stage with a bended specimen loaded.

	
  

50	
  

Figure 2.10 Schematic drawing (a) and an SEM image (b) showing the size and shape
of the AFM probe tips used in this project [96].

	
  

51	
  

	
  

and processed using the software Nanoscope III v. 5.31 (Digital Instruments, Plainview,
NY, USA).
In “tapping” mode, the cantilever oscillates at about its resonance frequency with
an amplitude typically from 100 nm to 200 nm. When the cantilever comes close to the
sample surface, the amplitude of the oscillation will be decrease, mainly due to Van der
Waals force and electrostatic forces. An electronic servo will adjust the height of the
cantilever to maintain a constant oscillation amplitude while the cantilever scans across
the sample. Consequently, a tapping-mode AFM scan is produced as a result of the
height change of the cantilever.

2.3.4 Nano-indentation
Nano-indentation was carried out with the help of Claudio Zambaldi at the MPIE
using a TriboScope 900 (Hysitron, Minneapolis, MN, USA). The nano-indentation in this
work was carried out under load-controlled conditions with a load of 6 mN.
Instrumented indentation was performed with an indenter of sphero-conical tip
geometry.
Rectangular arrays of indentations were made on specimens S0-2 (see Figure
2.11) and S0-3 (see Figure 2.12). Each region included about 20 grains with both soft
(c-axis parallel to the specimen surface) and hard grain orientations (c-axis
perpendicular to the specimen surface), with a total of 223 indents for specimen S0-2
and 220 for specimen S0-3. The residual indentation topographies were analyzed using
a Dimension 3100 (Digital Instruments, Plainview, NY, USA) atomic force microscope
outlined in Section 2.3.3.

	
  

52	
  

Figure 2.11 BSE image of the indented microstructure patch in specimen S0-2.

	
  

53	
  

Figure 2.12 BSE image of the indented microstructure patch in specimen S0-3.

	
  

54	
  

2.3.5 Fatigue Testing and Thermoelastic Stress Analysis Measurement
Before fatigue testing, the polished surface of the fatigue specimens (see Figure
2.5) were sprayed with a thin coat of matte black paint using a fine airbrush, in order to
provide uniform emissivity for the TSA measurement before fatigue test. The fatigue
tests were performed on a 50kN MTS servohydraulic loading frame in air, with mean
load = 600 N, and load amplitude = 150 N. The test frequency was 20 Hz, and the tests
were stopped after 27,200 cycles. During the fatigue crack growth, TSA data were
recorded at regular intervals.
The thermoelasticity analysis technique is based on the principle that under
adiabatic and reversible conditions, a cyclically loaded structure experiences in-phase
temperature variations that are proportional to the sum of the principal stresses, which
are measured using a sensitive infrared detector. Around the tip of a fatigue crack
during fatigue tests, however, there is a heat generation process due to the plastic
deformation/work, which is related to the dislocation motion and grain boundary sliding.
Heat evolved as a result of the relative phase difference due to these non-adiabatic
effects has been shown to be useful for dynamically imaging the plastic zone size [97,
98]. The TSA technique allows the plastic zone to be monitored in near real-time, in a
non-contacting manner. The phase of each TSA image was shifted so that the region
far away from the crack tip (far field) was zero in order to represent the phase difference
relative to the far field. The non-adiabatic region is thus represented by a negative
phase difference, and therefore the size and shape of the plastic zone can be quantified
using a binary filter on each image [98]. The fatigue tests and TSA measurements were

	
  

55	
  

conducted with the help of Rachel Tomlinson from Shefield University, while visiting
MSU.
Following fatigue testing, the matte black paint was removed by ultrasonic
cleaning in methanol bath for about three minutes. Subsequent SEM imaging reveled
the cleaning procedures allowed the matte black paint to be removed to reveal surfaces
consistent with pre-painted polishing conditions.

2.4 Slip/twin Plane Trace Analysis
When a grain orientation is measured by EBSD, described as three Euler angles
(φ1, Φ, φ2), the rotation matrix (R) of this orientation is given by:
⎡ cos φ
2
⎢
R = Rφ 2 RΦ Rφ1 = ⎢ sin φ
2
⎢
⎢ 0
⎣

− sin φ2
cos φ2
0

0 ⎤⎡ 1
0
0
⎥⎢
⎥ ⎢ 0 cos Φ − sin Φ
0
⎥⎢
1 ⎥ ⎣ 0 sin Φ cos Φ
⎦

⎤ ⎡ cos φ1 − sin φ1 0 ⎤
⎥
⎥⎢
⎢ sin φ
cos φ1 0 ⎥
⎥⎢
1
⎥
⎥⎢
⎦⎣ 0
0
1 ⎥
⎦

(2.1)

For a given slip/ twin system (hkl)[uvw], the slip/twin direction (m) and plane normal (n)
in the sample/lab coordinate system can be then identified as:

T

⎡ u ⎤
m=R ⎢ v ⎥
⎢
⎥
⎢ w ⎥
⎣
⎦

(2.2)

⎡ h ⎤
n= R ⎢ k ⎥
⎢
⎥
⎢ l ⎥
⎣
⎦

(2.3)

T

All of the possible traces of deformation systems in a given grain orientation were
calculated and drawn using a MATLAB (MathWorks, Natick, MA, U.S.A) code (see
Appendix I), based on the principle discussed above. The deformation systems in grain

	
  

56	
  

patches were then effectively identified by comparing their slip/twin plane traces in the
BSE images with calculated traces, because of the limited number of slip directions on
each deformation plane in α-titanium with hcp structure.
In this project, all the Schmid factors were calculated based on the assumption of
global uniaxial stress states:

m = cos φ cos λ

(2.4)

where φ is the angle between the stress axis and the slip plane normal, and λ is the
angle between the stress axis and the slip direction.

In the trace analysis, for

deformation systems having very similar plane traces, two factors were considered to
determine the deformation systems. First, the system with the highest Schmid factor
was chosen. This criterion has its limitations, because of the difference between the
assumption of global uniaxial stress states and the actual complicated local stress state
in a given grain. Therefore, if the two systems in a given grain have both similar plane
trace and similar high Schimid factors (both above or around 0.4), the second factor,
slip transfer criteria (see Section 1.2.1), was considered. If the deformation system(s) in
its neighboring grain have been determined, the deformation system in the given grain
having higher m’ with the determined system in neighboring was chosen.

2.5 Constitutive Law in Crystal Plasticity Finite Element Method
CPFE method simulations were conducted to model the deformation processes
of uniaxial tension and nano-indentation, and the results will be discussed in detail in
Chapter Three and Chapter Four. The basic constitutive description framework of the
material in these two simulations is the same, which are both based on a crystal

	
  

57	
  

plasticity formulation using the multiplicative decomposition of the total deformation
gradient and considering the anisotropic elastic constants of α-titanium, which have
been addressed in detail in many prior theoretical works [58, 59]. A brief summary of
the theoretical formulations is given as following:
The deformation gradient, denoted as F, is decomposed into two parts in a finite
e

p

deformation framework, the elastic gradient, F , and the plastic, F :

F = F eF p

(2.5)

The elastic strain part can be then expressed as:

Ee =

(

1 eT e
F F −I
2

)

(2.6)


I is a second-order identity tensor. The evolution of plastic gradient, F p , is given by

F p = Lp F p

(2.7)

where the plastic velocity gradient L resulting from activity on all deformation systems
p

is described [59] as
N

L =
p

def .sys.

∑
α

Pα γ

τ
0

α

sα

n

sgn (τ α )

(2.8)

with Pα = mα ⊗ nα as the Schmid matrices with respect to the undeformed state,

γ 0 = 10 −3 s −1 as reference shear rate, n the constant stress exponent, τ α the resolved
shear stress, and sα the shear resistance (in other word, CRSS). The evolution of sα
during deformation is written as

	
  

58	
  

N


sα =

def .sys.

∑
β

a

⎛
⎞
sα ⎟
β⎜
h 1−
γ β
0 ⎜
α⎟
s ⎠
⎝
s

(2.9)

β
with h0 , a, and sα the three hardening parameters.
s

The CRSS and hardening

parameters were based on prior published experimental results [85, 86, 92] and
adjusted slightly to enhance numerical stability and to better reflect the real deformation
process during simulations.

The chosen values for these parameters for each

simulation are listed in Chapter Three and Chapter Four.

	
  

59	
  

CHAPTER THREE
MEASURING AND CPFEM SIMULATING MICRO-SCALE HETEROGENEOUS
DEFORMATION

In this chapter, a novel technique that combines AFM, BSE imaging, and EBSD
is developed to allow quantitative analysis of the active deformation systems in different
grains in a selected microstructural patch of polycrystalline titanium of commercial
purity.

The specimens were deformed by four-point bending, and EBSD data was

collected before and after bending tests to facilitate trace analysis of deformation activity
in grains.

After each step of deformation, the surface step height changes due to

dislocation slip and twinning activity in a grain patch were analyzed using AFM. Maps
of local shear at three different strain levels (1.5%, 3%, and 6%) were generated to help
visualize the heterogeneous deformation in this particular microstructure patch.
Furthermore, by comparing these maps to the corresponding CPFE results, the quality
of the constitutive model employed and CPFE mesh effects can be critically assessed,
with the ultimate goal of accurately modeling the influence of grain boundaries on
damage nucleation.

	
  

60	
  

3.1 Microstructure in a Four-point Bending Sample
3.1.1 Slip Lines and Twins
Five specimens were loaded by four-point bending to various amounts of global
strains. On the tensile surface of deformed specimens, which was polished prior to
deformation, the surface topography change of the grains varies from one grain to
another, due to the differences in the grain orientation and the number density of
dislocations on activated deformation systems.

Figure 3.1 shows a typical

microstructure patch of a deformed specimen after 1.5% global strain, and three highmagnification images of three grains from the patch with different topography. Figure
3.1(a) shows a grain with dense and deep slip lines, while in Figure 3.1(b), the density
of slip lines in the grain is much lower. Also, a thin twin is shown in Figure 3.1(c).
	
  
3.1.2 Grain Boundary Ledge Formation
Since under a given stress state, the deformation systems that can be activated
in a given grain orientation in α-titanium are limited, additional plastic deformation
mechanisms may be necessary for polycrystalline compatability according to von Mises
criteria [99].

In this case, grain boundary ledges were also found under various

amounts of tensile strain. In sample S0-1 after 8% tensile deformation, SEM images
gave the appearance of grain boundary cracks, but upon closer inspection, these were
found to be ledges. As shown in Figure 3.2, when sample is rotated 180 degrees in the
SEM, the circled grain boundaries show reverse contrast in two images, indicating ledge
formation along these boundaries rather than cracks.

	
  

61	
  

	
  
Figure 3.1 A typical microstructure patch from S45-1 at 1.5% strain. (a) shows a grain
with dense slip lines, while (b) shows a grain with much less slip lines. Twins also
shows in grains, as illustrated in (c).

	
  
	
  

62	
  

Figure 3.2 Secondary electron images show ledges (circled) along grain boundaries in
S0-1under 8% global strain; (b) was taken of the same microstructural patch with the
whole sample rotated 180 degrees from (a).

Circled boundaries showing reverse

contrast in two images, indicating these are ledges and not cracks.
	
  

	
  

63	
  

Even at small strain levels, as low as 1% to 2%, some grain boundaries in
polycrystalline titanium begin to show the development of grain boundary ledges, which
can be clearly observed using AFM.

Figure 3.3 shows AFM scans of three

characteristic grain boundary ledges with different surface topography. Figure 3.3(a)
and (c) shows sheared grain boundary ledges. Local surface rotation is found in the
right grain in Figure 3.3(a), in (c), however, no obvious local surface rotation is apparent
in both grains. In Figure 3.3(b), the surface topography across the grain boundary is
much flatter than the other two. These different features of grain boundaries may result
from heterogeneous strain field near these grain boundaries.

3.1.3 A Selected Microstructure Patch for Further Studies
As discussed in Section 1.5, dense slip lines usually shows up in those soft
grains with their c-axis almost perpendicular to the uniaxial tensile stress. However,
less deformation may occur in those hard grains with their grains with their c-axis close
to the tensile stress. To better understand the heterogeneous deformation in α-titanium,
detailed studies of grain patches that have a mixture of both soft and hard grains are
essential.

To develop this understanding, a microstructure patch was selected in

sample S45-1 prior to deformation.
Figure 3.4(a) shows the selected microstructure patch at about 1.5% global
tensile strain.

Grains 3, 6, and 9 show the most obvious slip traces, while less

pronounced slip bands are found in grains 8 and 10. T1 twinning in grain 2 is confirmed
by EBSD (the misorientation angle between T1 twin and matrix lattice is about 85
degrees [91]). Figures 3.4(b) and (c) show OIM maps of the dashed box in Figure

	
  

64	
  

Figure 3.3 AFM “Section” scans shown ledges with different topography at grain boundaries in S0-2 under 1.5% global
strain, indicating that heterogeneous deformation occurs in small deformation.	
  

	
  

65	
  

3.4(a) before and after deformation, which provides orientation and shape change
information. The variation of deformation activity (Figure 3.4(a)) and crystal rotations
(Figure 3.4(c)) results from heterogeneous deformation among and within these grains.
Two further deformation steps were conducted on specimen S45-1 to obtain
global strain level of about 3% and 6% respectively. At 3% global strain, as shown in
Figure 3.5(a), the growth of the twins in grain 2 can be easily observed, while no
obvious secondary slip and twinning were found at this stage. At 6% strain, in addition
to the growth of twins, secondary deformation systems began to show up in several
hard grains, as illustrated by the white lines in Figure 3.5(b). Also, sharp ledges formed
at the boundaries of grains 8-9, 9-10, and 3-8 boundaries.
The deformation systems for all slip bands/twins in each grain were effectively
identified by the trace analysis discussed in Section 2.4. Three types of dislocation slip
systems (prismatic, basal, and pyramidal <c+a> slip) and one T1 twinning system (in
grain 2) were identified in the grain patch at 1.5% global strain, as illustrated in Figure
3.6. Prismatic slip dominates most of the soft grains, such as grains 1, 3, 6, 9, and 14,
while basal slip were found in grains with relatively hard orientations, such as grains 5,
7, 11, 13. Pyramidal <c+a> slip was only identified in grain 10, and no obvious slip
traces were observed in grain 4.

Secondary deformation systems, in paticular

pyramidal <c+a> slip and T1 twin, were identified in hard grains at 6% strain level.
Detailed deformation system information in each grain is shown in Table 3.1.

	
  

66	
  

	
  
Figure 3.4 Microstructure patch of specimen S45-1 at ~1.5% tensile strain. The patch
within the solid outlined box was quantitatively analyzed using AFM-based method, and
the patch within the dashed outlined box was modeled by CPFEM. Inverse pole figures
of the grain orientation changes within the solid outlined box before and after
deformation are shown in (b) and (c), respectively. Directions of slip lines and twins in
each grain are highlighted by white lines. These activated deformation systems were
identified using trace analysis based on EBSD data.
	
  
	
  

67	
  

	
  
Figure 3.5 The same microstructure patch in S45-1 specimen as shown in Figure 3.3(a)
following further deformed to ~3% and ~6% tensile strains, shown in (a) and (b)
respectively. The patch within the solid outlined box was quantitatively analyzed using
the AFM-based method. The growth of five T1 twins in grain 2 with the increase of the
global strain can be observed.

Furthermore, secondary deformation systems,

highlighted by white solid lines in (b), were observed in several hard grains, such as
grain 2, at 6% strain level.
	
  

68	
  

Figure 3.6 BSE image illustrating the area characterized using AFM and CPFEM at
1.5% global strain. Different activated deformation systems characterized using trace
analysis are denoted by color lines.

	
  

69	
  

Table 3.1 Deformation systems in grains within the solid-line box in Figure 3.3(a) and
3.4(a)-(b) at global strain level 1.5%, 3%, and 6%.

Grain number

Identified deformation systems using trace analysis
1.5% strain

3% strain

6% strain

1

(10 10)[12 10]

(10 10)[12 10]

(10 10)[12 10]

2

(10 12)[1011]

(10 12)[1011]

3

(10 10)[12 10]

(10 10)[12 10]

(10 10)[12 10]

4

N/A

N/A

N/A

5

(0001)[12 10]

(0001)[12 10]

(0001)[12 10]

6

(10 10)[12 10]

(10 10)[12 10]

(10 10)[12 10]

7

(0001)[12 10]

(0001)[12 10]

(0001)[12 10]

8

(01 10)[2 1 10]

(01 10)[2 1 10]

9

(10 10)[12 10]

(10 10)[12 10]

(10 10)[12 10]

(0001)[12 10]

(0001)[12 10]

(0001)[12 10]

(10 11)[1123]

(10 11)[1123]

(10 11)[1123]

11

(0001)[12 10]

(0001)[12 10]

(0001)[12 10]

12

(0001)[2 1 10]

(0001)[2 1 10]

(0001)[2 1 10]

13

(0001)[1 120]

(0001)[1 120]

(0001)[1 120]

14

(10 10)[12 10]

(10 10)[12 10]

(10 10)[12 10]

10

	
  

	
  

70	
  

(10 12)[1011]
(1101)[12 13]

(01 10)[2 1 10]
(0 111)[1213]

3.2 Quantitative Measurement of Heterogeneous Deformation within a Grain Patch
AFM is an effective tool for characterizing fine-scale topography induced by
different deformation processes. Since the 1990s [100], AFM has been used to study
surface relief produced during fatigue damage evolution in both face-centered cubic
(fcc) [101,102] and body-centered cubic (bcc) [103-105] metals, providing quantitative
data about surface relief evolution and the dislocation structures of persistent slip
bands.

AFM has been used to investigate the deformation process of nickel-base

superalloys during compression tests [106] and duplex stainless steel in in-situ tensile
tests [107], and the number of dislocations responsible for a slip step has been
determined based upon AFM data [107]. However, detailed systematic AFM data has
not been used to quantify deformation system activity and local shear strains in a
polycrystalline array in previous works. In the following sections, a newly developed
technique combining AFM and EBSD-based trace analysis was introduced, to quantify
the heterogeneous deformation system activity in different grains within microstructure
patches.
	
  
3.2.1 AFM Measurement and Accuracy Check Using T1 Twins
Two examples of AFM scans collected from grains 1 and 6 after ~1.5% strain are
shown in Figure 3.7. Twins and slip traces are clearly visible in the topography maps.
The surface steps in Figure 3.7(left) correspond to the (1012)[1011] T1 twin system
identified by EBSD orientation analysis and slip trace analysis. The twins cause more
uniform surface height change than the dislocation slip shown in Figure 3.7(right).

	
  

71	
  

	
  
Figure 3.7 Examples of surface topography in grain 2 (left) and grain 3 (right) measured
by AFM after deformation.

Line section profiles reveal that much larger and more

homogeneous surface steps result from twins (left) than from slip lines (right). Dashed
line is interpreted as the undistorted surface inclination and serves as the basis for
evaluating the overall height change along the section.

	
  

72	
  

Prior to calculating the amount of plastic deformation from surface topography
the accuracy of the AFM measurements needed to be checked. For this purpose, T1
twins (in grain 2) were chosen, since, in contrast to dislocation slip where many
dislocations may traverse a given slip plane, only a single twinning dislocation with
Burgers vector b passes on each adjacent twinning plane (separated by d along plane
normal n ) during the formation of a twin, thus resulting in a precisely known amount of
shear (0.17 for T1 twins [91]).

Therefore, the resulting surface step height is fully

determined by the crystallography and the thickness of a twin measured along n .
Figure 3.8 illustrates the principle of the accuracy check. For a given twin system
the number of displaced twin planes N follows from the measured surface step h and
the twin Burgers vector b projected onto the surface normal ez:

N=

h
b⋅e

(3.1)
z

This can be compared to the alternative derivation based on the projected twin
thickness tx, the twin plane normal n, and the inter-plane spacing d:

N=

t ⋅n
x

d


t ⋅n

≈ x
d

(3.2)

The true projected twin thickness tx is approximately equal to the apparent, i.e.,



measured, projected twin thickness t , if h << t . Using the AFM “Section” data of the
x

x

very left twin shown in Figure 3.7(left), the two alternatively derived values of N agree to
within 2%. This shows that the AFM measurement process is robust, so that a similar
analysis can be used to calculate the number of slip dislocations responsible for a

	
  

73	
  

	
  
	
  

	
  
Figure 3.8 Schematic representation of a cross-section through a twinned portion of a
grain. The twin trace on the sample surface is normal to the sketch (i.e., parallel to the Y
direction). The surface displacement h due to twinning is determined from an AFM line
section in the X direction. tx and ttrue are the twin width in the X direction (projected twin
width) and the twin thickness along the twin plane normal direction, respectively.
	
  

	
  

74	
  

series of parallel slip bands in a grain, such as the slip bands shown in Figure 3.7(right)
that create relatively small steps on the surface.

3.2.2 Generation of Local Shear Distribution Maps
Figure 3.9 illustrates the procedure for calculating shear on deformation system α
from the individual surface steps along an AFM section line. By identifying the system α
related to each surface step along a section line of length Xmn and cumulating the
overall height change per system, the number of individual displacements occurring
along Xmn can be calculated according to Equation 3.1.

Provided that the overall

surface height change across Xmn is small compared to Xmn, the average shear per
deformation system is given by

(

γ α = bα N α

)(

X

mn

)

⋅ nα = (bα

hα

)

α

b ⋅e

z

(X

mn

⋅ nα

)

(3.3)

The area enclosed by the solid box in Figure 3.4(a) was scanned using AFM between
each deformation step during four-point bending. The surface height changes caused
by deformation systems (identified independently by SEM/EBSD trace analysis) in the
microstructure patch within the solid-line box in Figure 3.4(a) at 1.5%, 3%, 6% global
strain level were measured by an array of horizontal AFM “Section” lines passing
through the center of each of 25 × 25 tiles spaced 10 µm × 10 µm.

The typical

measured step height is about two orders of magnitude smaller than this grid size. The
measurement of local shears at different strain levels were all checked by repeating
measurements using a vertical AFM “Section” line on 50 randomly chosen tiles, with

	
  

75	
  

	
  
Figure 3.9 Schematic drawing showing the parameters associated with calculation of
the local average shear caused by the activated deformation system with the slip plane
normal n and Burgers’ vector direction b in sample coordinate system. h is the surface
height change caused by this deformation system.
reference grid increments in the X direction.
	
  

	
  

76	
  

Xm(n-1), Xmn, and Xm(n+1) are

measurements agreeing to within about 10% in the X and Y direction for each tile, as
shown in Figure 3.10.
The sum of all individual local shears ( ∑ γ α ) at 1.5% global strain is plotted for
α

each tile in the map shown in Figure 3.11(a). While the maximum shear is 0.17, the
color scale of Figure 3.11(a) – (c) is capped at 0.14 (yellow) in order to reveal details
and spatial heterogeneity in the local shear distribution caused by dislocation slip. The
summed local shears in grains 3, 8, and 9, which deformed primarily by prism slip, is
higher than strains observed due to basal or pyramidal slip in grains 5, 7, and 10.
Shear maps for the individual deformation system types in the same microstructure
patch at 1.5% strain are shown in the first row of Figure 3.12.

Shear caused by

prismatic slip in grains 3, 8, and 9 varied from 0.02 to 0.08, while in grain 5, 7, and 10,
where basal and pyramidal <c+a> slip were favored, the shear varied between 0.005
and 0.02. Grains with dominant prismatic slip exhibited about three times higher shear
strains than grains with either basal or pyramidal slip activity. The shear in the grain
with the T1 twin varied from 0 to 0.17, consistent with the range expected for the
theoretical shear of the T1 twin being 0.17. Locally smaller values of shear arose due to
the fact that tiles do not necessarily comprise the same (full) fraction of a twin; hence
most of the tiles exhibited a shear less than 0.17.

3.2.3 Local Shear Distribution Evolution at Different Strain Level
The local shear distribution evolution with the increase of global tensile strain can
be observed by comparing the sum of all individual local shear distribution map shown
in Figure 3.11(a) - (c). Detailed comparison the local shear caused by each kind of
	
  

77	
  

Figure 3.10 The calculated local shear deviation at 1.5% global strain based on the
measurements using both horizontal and vertical AFM “Section” lines for 50 randomly
chosen tiles. The measurements agree within about 10% in both directions for all tiles.

	
  

78	
  

Figure 3.11 The AFM-based experimentally measured sum of all individual local shear
distribution map of the highly characterized microstructure patch shown in Fig. 3.4(a) at
global strain level of (a) 1.5%, (b) 3%, and (3) 6%.

	
  

79	
  

	
  
Figure 3.12 Individual local shear associated with different types of deformation systems in the microstructure patch
shown in Fig. 3.4(a).

The experimental measurements show that, in the early deformation stage, the local shear

increases almost proportional to the overall tensile strain in all grains, while during further deformation (3% to 6%), the
local shear in grains with pyramidal slip shows a much larger increase rate than that in grains with prism or basal slip.	
  
	
  

80	
  

deformation system at different strain levels is shown in Figure 3.12.
In the early stages of deformation (1.5% to 3%), the local shear of all identified
deformation systems increased almost proportional to the overall tensile strain in all the
grains. As observed from Figure 3.12, the average local shear caused by prismatic slip
in grains 1, 3, and 9 increased from about 0.04 to about 0.08 during this stage, and the
average basal and pyramidal slip in grains 5, 7, and 10 also increased about 0.01.
During further deformation (3% to 6%), the local shear in grains with pyramidal slip
exhibited a much larger increase rate than that in grains with prism or basal slip. Figure
3.12 shows that the average local shear of prismatic slip only increase about 0.01, and
basal almost no increase, while the local shear of pyramidal <c+a> slip increased 100%
in grain 10 (i.e. from an average shear of 0.005 to 0.012). This may be because the
work hardening and dislocation saturation in the grains showing prismatic and basal slip
in earlier deformation stages [108-110].

3.3 Crystal Plasticity Finite Element Modeling
3.3.1 Constitutive Law
The constitutive description of the material is based on a crystal plasticity
formulation using the multiplicative decomposition of the total deformation gradient and
considering the anisotropic elastic constants of α-titanium, as described in Section 2.5.
Dislocation slip systems in the particular model used in this work were treated as
bidirectional deformation systems.

Twinning systems were incorporated as

unidirectional dislocation slip systems, which results in diffuse straining, contrary to the
discretely sheared twins observed in the microstructure. The grain boundaries were

	
  

81	
  

transparent to slip (i.e., no specific resistance to slip at a grain boundary), which means
that local shears (dislocations) can effectively transfer across boundaries. The values
of CRSS and hardening parameters, as listed in Table 3.2, were chosen based on
literature [85, 86, 92], and adjusted slightly to increase the stability of the calculation
processes. The material model was implemented into the commercial FEM package
MSC.Marc.

3.3.2 Mesh Generation and Boundary Condition
Three different sizes of quasi-3-D meshes based on the microstructure in the
dashed box in Figure 3.4(a) were generated and are shown in Figure 3.13(a) – (c),
respectively. The meshes were developed by distributing nodes along the straight grain
boundary traces, planar surface meshing of the enclosed grains, and expansion by
50µm into the third dimension, and then evenly discretizing by five elements. Hence, all
grain boundaries are perpendicular to the surface in this approximation. To replicate
the constraint from the surrounding bulk material, the simulated microstructure patches
shown in Figure 3.13(a) and (b) were surrounded by a rectangular-frame shape rim,
while the mesh shown Figure 3.13(c) was placed in a rectangular pan-like rim (to apply
a the constraint from the material beneath the modeled the microstructure patch). The
crystal orientation identified as dominant from macro-texture measurement

(c-axis

lined up 45° from the tensile axis) was assigned to the rim. A tensile face load was
imposed on the right surface of the rim, with the left surface constrained in the X
direction and the back surface constrained in the Z direction for all the three meshes.

	
  

82	
  

Table 3.2 Hardening parameters used to describe evolution of the critical resolved
shear stress.

Deformation
System

{hkl}<uvw>

sα / MPa

α
sS / MPa

α
h0 / MPa

a

Prismatic

{1010} < 1210 >

60

200

100

2

Basal

{0001} < 1210 >

120

300

150

2

Pyramidal
<c+a>

{1011} < 2113 >

180

500

300

2

T1 twin

{10 12} < 1011>

125

200

150

2

	
  

	
  

83	
  

	
  

	
  
Figure 3.13(a) – (c) Three different sizes of 3D meshes of the grain patch corresponding
to the microstructure within the dashed box in Fig. 3.4(a).

	
  

84	
  

3.3.3 Modeling Results
Figure 3.14(a) – (c) maps the summed individual shears for the top surface
resulting from simulated tensile loading to 1.5% engineering strain using the three
meshes shown in Figure 3.13, respectively.
The simulations using meshes (a) and (b) show a large deformation band across
grains 6 and 9, while there is little deformation in other grains. This pattern of strain was
similar on the front-side and back-side, but strains were slightly larger on the backside,
despite the fact that the geometry of the mesh is the same, as shown in Figure 3.15.
These two simulation results, as shown in Figure 3.14(a) and (b), do not reflect the
experimentally observed heterogeneous deformation in this grain patch. For example,
only grain 3 and 9 show about 0.1 prismatic dislocation shear at 1.5% global strain, in
which local shear varied from 0.02 to 0.08 in experimental measurements. In other
grains, of which the deformation is dominated by basal, pyramidal <c+a>, and T1 twin
dislocation shears, almost no dislocation shear was found in the simulation.
The simulation results using mesh (c) shown in Figure 3.14(c) is much more
realistic local shear distribution. Large plastic strain was found in grains 1, 3, 9, and 14.
Grains 2, 8, 10, 11, and 13 have harder orientations, and show comparatively little
shear. The highest local shear in the simulation of around 0.14 is found in grain 14.
However, the local shear in grain 14 needs to be treated with caution because the large
strain concentration in grain 14 might result from an edge effect from the surrounding
rim. Since the artificially added rim was directly in contact with the right part of grain 14,
and also grain 14 is close to the corner of the mesh, the deformation of grain 14 could
have been easily influenced by the deformation process of this artificial rim, which is

	
  

85	
  

Figure 3.14(a) – (c) The summed individual shears for the top surface resulting from
simulated tensile loading to 1.5% engineering strain using the three meshes shown in
Figure 3.13(a) – (c), respectively.

	
  

86	
  

	
  
Figure 3.15 The comparison of the summed individual shears distribution on the frontside and back-side of the meshes shown in Figure 3.13(b).

	
  

87	
  

different from neighboring grain deformation process.

Figure 3.16(b) shows the

distribution of simulated shears from prismatic slip, basal slip, pyramidal <c+a> slip, and
T1 twinning systems (note different scales) using the mesh shown in Figure 3.13(c).
The shear caused by prismatic slip is mainly concentrated in grains 3, 6, and 8. In
grains 3 and 9, prismatic shear is higher (with a maximum value of 0.052) in the right of
the grain than in the left. Basal shear in the left of grain 10 and grain 11 varies from
0.02 to 0.03. Basal activity is also found in the corners of grains 5 and 7. The twinning
activity in grain 2 was distributed similarly to the experimental observations; high activity
correlates with twins in the bottom and center part of grain 2 (see Figure 3.4(a)). The
CPFE simulation does not account for the (long-range) twin habit- plane effect present
in the real specimen, where twins were primarily extended along their habit plane, or for
the change in crystal orientation resulting from twinning, thus, a direct match of twin
geometry between experiment and simulation was not expected. Pyramidal <c+a> slip
activity occurred mainly along the boundary between grains 3 and 9 in this
microstructure patch.

Detailed comparison between experimentally measured and

simulated dislocation shear distributions of this microstructure patch will be discussed in
Section 3.4.3.

3.4 Discussion
3.4.1 Comparison of Experimental Observations and CPFE Simulations
The CPFE modeling results can be directly compared to the details of the
experimentally measured shear distribution, as illustrated in Figure 3.16. The AFMderived maps of individual shears are effectively reproduced in the simulation, but the

	
  

88	
  

Figure 3.16 Comparison of experimental measured (shown in row (a)) and FEM simulated (using mesh (c)), shown in row
(b)) individual local shear associated with different types of deformation systems in the microstructure patch shown in
Figure 3.4(a) at 1.5% strain. FEM simulation results illustrate that the spatial distribution of local shear caused by
prismatic and basal slip agrees well with experimental measurement. Most of the local shear caused by T1 twinning is
observed in grain 2.

Simulation does not successfully simulate the pyramidal <c+a> slip in grain 10 where AFM

measurement reveals around 0.01 local shear.
	
  

89	
  

finer details differ. The highest shear value in the simulation is about the same as that
in the AFM measurement for the seven grains (2, 3, 6, 8, 9, 10, and 13) in the center of
the patch. From the AFM measurements, however, the highest shear is in the left of
grain 3, while the CPFE model showed this region to have less strain than the lower
right of grain 14. Secondly, Figure 3.16(a) shows that near the boundary between
grains 3 and 8, the shear caused by prismatic slip in grain 3 is about 0.05 – 0.07, while
its value is lower in the simulation. The shear activity of basal slip varying from 0.005 to
0.02 in grains 5 and 7 is well captured. The simulation also successfully captured the
basal activity in grain 10 both spatially and quantitatively. The simulation did not predict
the activity of pyramidal <c+a> slip in the right of grain 10, which contributed a shear of
about 0.01 in the experimental measurements. The twinning activity in grain 2 was
distributed similarly to the actual specimen; the twins are thicker near the boundary of
grains 1 and 2, where the twin was activated by slip–twin transfer [111].

3.4.2 Effect of Modeled Area Size in Simulation Results
As shown in Figure 3.14, the FEM simulations do not successfully reflect the real
local shear distribution of the microstructure patch using the mesh (a) and (b). There
are several probable reasons for the differences between the experimental results and
the CPFE modeling results. The most important among these is the modeled area size
effect. The interested microstructure patch has a relatively “soft” grain cluster in the
right side, making it highly possible that the plastic deformation tends to be
concentrated in this band in the simulation if no surrounding grain orientation and shape
information were provided. Two choices can be made to potentially solve this problem.

	
  

90	
  

The easier one is to extend the modeled area and to add an artificial pan-shape rim
around the modeled microstructure patch to simulate the constraint from the
neighboring microstructures, which has been demonstrated in the simulation results
using mesh (c). The local shear distribution in the middle part of the mesh is relatively
successfully modeled, as discussed in the previous section. The edge effect resulting
from the rim, however, still existed (see grain 14). This suggests that a mesh that can
correctly reflect the surrounding environment may need to include at least one layer of
real surrounding grain information, which leads to the second potential method to
increase the accuracy of CPFE simulation, i.e. including more surrounding real grain
information (both around and beneath the microstructure patch of interest) into the
mesh. The grain information beneath the surface can be collected using the EBSD-FIB
technique, i.e. 3D-EBSD, as described in Section 1.3.2. Also, a more efficient and
stable constitutive model is needed if a real 3D mesh is used in the simulation. In
cooperation with MPIE, future work has been focused on developing a new CPFE
constitutive model that uses parallel calculation processes to largely increase the
simulation efficiency.
	
  
3.4.3 Effect of Rim Crystal Orientation in Simulation Results
Three simulation results (to 2.2% global strain) with different crystal orientations
chosen for the rim are shown in Figure 3.17 to examine the effect of rim crystal
orientation on the simulation results in details. The orientation in the simulation on the
left is closest to the actual average grain orientation in the material measured using
EBSD, and is the orientation used in Figure 3.14 and 3.16(b). The maximum shear,

	
  

91	
  

Figure 3.17 CPFE model illustrating how the surrounding orientation (indicated with Euler angles) affects local shear
distributions at 2.2% global strain.

	
  

92	
  

shown in the top row in Figure 3.17, does not appear to be strongly influenced by the
chosen crystal orientation.

However, upon closer inspection, the orientation of rim

orientation does affect the shear on each particular family of slip systems, as illustrated
by the second, third, and forth rows, where prismatic, basal, and pyramidal <c+a> slip
activities differ in some locations, especially in several edge grains (for instance grains
5 and 11). When the rim has a hard grain orientation, a small increase of the local
shear of prismatic, basal, and T1 twin can be observed in the simulation, especially the
shear distributions in the grains near the patch edges. Less influence of the crystal
orientation of the rim can be found on the simulated deformation activity in the center
grains of the patch. The results suggests that accurate modeling of deformation system
activity in polycrystalline patches requires at least one layer of correctly oriented and
shaped grains as a buffer zone between the rim and grains of interest in the center.

3.5 Conclusions
Heterogeneous deformation activity within a polycrystalline microstructure of αtitanium with a series of tensile strain of about 1.5%, 3%, and 6% was quantitatively
measured using a technique combining AFM, EBSD, and BSE imaging. AFM provides
high-resolution measurement of the z-components of the slip and/or twin displacements
on sample surfaces, allowing fine-scale analysis of dislocation activity and derivation of
corresponding local shears averaged on a 25 × 25 grid of square tiles.

The

experimental result at 1.5% global tensile stress was used to effectively assess the
accuracy of CPFE simulations of deformation activity on different dislocation slip and
twinning systems. The phenomenological model examined successfully predicted most
	
  

93	
  

of active dislocation slip and twinning systems. In both experiment and simulation,
grains typically exhibited a single dominant slip system. Grains with dominant prismatic
slip exhibited about three times higher shears than grains with either basal or pyramidal
slip activity.

While the magnitude of shears was reasonably predicted within most

grains, their spatial distribution frequently differed from the measurement. The crystal
orientation of the rim was found to have a modest influence on the simulated dislocation
activity in the grains in the patch center. Better agreement with the measurements may
result from (1) improved three-dimensional representation of grain geometry; (2)
experimentally measuring critical resolve shear stresses and hardening parameters for
different deformation systems in this material; and (3) including realistic slip resistance
at boundaries into the model.

	
  

94	
  

CHAPTER FOUR
CALIBRATING PARAMETERS FOR CPFE MODELS USING NANO-INDENTATION

In this chapter, a combined study of the anisotropic nano-indentation response of
α-titanium was conducted using nano-indentation, EBSD, AFM, and CPFE simulations
to quantitatively determine the CRSS for different slip systems. This information is
critical for developing more reliable values of the key parameters in CPFE simulations.
Experimental nano-indentation studies, along with CPFE simulations of the nanoindentation response, provide an alternative opportunity to study the behavior of single
crystals in a polycrystalline environment. Because in most cases the grain size is much
larger than the size of the indentations, the nano-indention can be treated as the
deformation of a constrained single crystal. Such experiments allow the separation of
the influence of intrinsic crystal properties, such as grain orientations, from the influence
of polycrystalline effects, such as grain boundaries and neighboring grain orientations.
Analysis of the residual surface topography after indentation revealed the crystalorientation-dependent pile-up behavior of α-titanium. The CRSS values of different slip
systems were then determined using a non-linear optimization procedure [120, 121].
The calculated CRSS were 150±4 MPa, 349±10 MPa, and 1107±39 MPa for prismatic,
basal, and pyramidal <c+a> slip, respectively. These values were then used to simulate
the same microstructure patch discussed in Chapter Three, and no significant
differences were found between the two simulation results, indicating that more
parameters need to be considered to further improve the simulations.

	
  

95	
  

4.1 Importance and Advantages of Measuring CRSS Using Nano-indentation
As mentioned in the conclusions of Chapter Three, accurate values of the CRSS
and hardening parameters of a material are one of the critical requirements to improve
the CPFEM modeling results. One conventional way to measure the CRSS for many
crystalline materials is to do a series of uniaxial tension tests on dog-bone shaped
single-crystal specimens of the material with different orientations [112-114].

The

orientations of the specimens are usually carefully chosen to only allow a single
deformation system to be activated during the tensile tests so that the value of CRSS
(the yield stress on the stress-strain curve) for this particular system can be measured.
It is, however, difficult to apply this methodology to measure the CRSS of hcp metals,
i.e. α-titanium, because slip systems with similar slip planes and/or slip directions, as
well as twinning, in these metals make it hard to guarantee the activation of only one
deformation system.

This is the main reason for the large variation in the

experimentally measured CRSS values of α-titanium in the literature, as shown in Table
1.1. Furthermore, the single crystals of titanium and titanium alloys are also expensive
and difficult to produce.
Indentation (hardness) testing is an alternative tool for measuring the mechanical
response of crystalline materials. Some early literature [115-116] has demonstrated the
anisotropic hardness responses of crystalline materials.

The residual surface

topography, known as the pile-up and/or sink-in, around the indent after indentation, as
shown in Figure 4.1, was treated as a fingerprint of the deformation processes, which is
closely related to the orientation of the indented single grains [117].

Recently, the

development of the AFM and nano-indentation techniques provides a much more

	
  

96	
  

	
  
Figure 4.1 Material pile-up (left) and sink-in (right) increases or decreases the contact
height, hc, during indentation to the maximum indentation depth, hmax. [117]

	
  

97	
  

convenient and precise way to characterize the mechanical responses and pile-up
and/or sink-in profiles of different grain orientations within a polycrystalline environment
[118,119]. Since the average grain size in the tested material in this work (Material A,
see Section 2.2.1) is around 80 µm, which is much larger compared to the size of the
nano-indenter (around 1 to 2 microns), the mechanical responses during nanoindentation in the middle of grains can be treated as a single crystal response.

4.2 Grain Orientation Dependent Behavior during Nano-indentation
4.2.1 Relationship between Hardness and Grain Orientation
A rectangular array of applied nano-indentations on the microstructure patch in
specimen S0-2 is shown in Figure 4.2, on which the corresponding OIM map is overlaid.
Both soft (with c-axis perpendicular to the indentation axis) and hard (with c-axis parallel
to the indentation axis) grain orientations are present in this patch, which provides a
wide range of grain orientations to study. Figure 4.3 shows a color-scale map indicating
the maximum indentation depth (with respect of the original surface height) resulting
from all the indents within this region. The measured maximum depths were almost
homogeneous inside a given grain, with only small scatter. The standard deviation of
the maximum indentation depths was around 5% of the mean value within a given grain.
However, the heterogeneous responses of the hard and soft grains can also be easily
observed from Figure 4.3.

Grain 2, with the hardest grain orientation in this

microstructure patch, shows the lowest average maximum residual depth (274 nm),
which is only about two thirds of the average maximum residual depth in the softest
grain (420 nm). Also, the indentations close to grain boundaries usually show medium

	
  

98	
  

	
  
	
  
Figure 4.2 The array of indents applied on the microstructure in specimen S0-2.
Indentions were spaced by 20 µm × 16 µm. A semi-transparent OIM map with inverse
pole figure coloring is superimposed on a backscattered electron image.

	
  

99	
  

Figure 4.3 Color-scaled map of the maximum indentation depth (unit: nm) for the
indentation array shown in Figure 4.2.

Each tile represents one indentation in the

middle of this tile.

	
  

100	
  

	
  

values of the maximum depth of the two grains in both sides of the boundaries. Several
load-displacement curves recorded from the indentations in the middle different grains
are shown in Figure 4.4.

These indentation curves also clearly indicate significant

differences in the mechanical responses of these grains during indentation. The higher
slope of the loading indentation curve for grain 2 shows that the grains with their c-axis
nearly parallel to the indentation axis are more resistant to plastic deformation.
Table 4.1 shows the maximum residual depths of selected indents from grains,
along with different angles between their c-axis and indentation axis. These indents
were chosen from the center of each grain to prevent any influence from grain
boundaries. However, it cannot be guaranteed that there is no grain boundary existing
close to the polished surface below the indents. The distance from the chosen indent to
the grain boundary below it should be maximized if the grains are assumed to be
equiaxed, which is reasonable according to the SEM images. A general trend of the
increase of maximum residual depth, i.e. the decrease of hardness, was observed with
the indentation axis rotating away from the c-axis (basal plane (0 0 0 1) normal).

4.2.2 The Relationship between Pile-up Topographies and Grain Orientation
To further examine the grain orientation dependent behavior of α-titanium, the
residual surface topography of the selected indents circled in Figure 4.2 were examined
using AFM. Two examples of the AFM scans for indents are shown in Figure 4.5, which
illustrates the different residual surface topography between soft and hard grains. To
study this phenomenon in a systematic manner, the surface topography of these
indents are presented on an inverse pole figure, and the position of each

	
  

101	
  

	
  
Figure 4.4 Force-displacement curves from indentations conducted in the middle of soft
grains 5 and 10, and hard grain 2. The maximum load is 6 mN. The lower inclination of
the curve during loading stage for grain 2 indicates that grain 2 is more resistant to
plastic deformation.

	
  

102	
  

Table 4.1 The indentation axis in the crystal coordinate system, the angle between the
c-axis and the indentation axis (η) and the maximum residual depth (hres) of the indents
in the center of the selected indented grains in specimen S0-2.

Grain

Indent

Euler angles,
Indentation axis

Number Number

η, degrees

hres, nm

degrees

1

53, 66, 329

[12 -6 -6 5]

66

251

2

13 – i

108, 26, 278

[3 2 -5 10]

26

188

6

10 – m

77, 81, 261

[14 -5 -9 2]

81

258

7

6–a

95, 86, 239

[26 1 -27 2]

86

252

8

6–o

186, 51, 169

[4 1 -5 4]

51

211

10

6–h

76, 87,277

[9 6 -15 1]

87

259

11

6–k

268, 56, 90

[6 -3 -3 4]

56

228

14

	
  

13 – o

1–a

102, 114, 221

[14 -5 -9 6]

66

241

103	
  

	
  
Figure 4.5 Indent surface topography scanned by AFM in soft grain G7 (left) and hard
grain G2 (right), illustrating the grain orientation dependent behavior of pile-ups.

	
  

104	
  

topography was determined by the indent axis of the corresponding grains, as
described in detail by Zambaldi et al. [117].
Figure 4.6 clearly shows the general trend that all indented grains with their caxis rotated away from the indentation axis exhibit pile-up behavior that exhibits two
dominant pile-up hillocks on opposite sides of the indents. It was also found that the
line connecting the two pile-up hillock maxima beside each of these indents was roughly
perpendicular to the c-axis of the corresponding indented grain. The most pronounced
pile-up was on the (0 0 0 1) great circle, and the heights of the pile-ups decreased with
decreasing angle between the indentation axis and the c-axis (η). Roughly, when the
angle η was smaller than 30°, no obvious pile-up behavior was observed near the
indents.

4.3 CPFEM Simulation of Indentation and CRSS Optimization
4.3.1 Constitutive law
The CPFEM simulations were run by Claudio Zambaldi in MPIE. The constitutive
description of the material was based on a crystal plasticity formulation using the
multiplicative decomposition of the total deformation gradient and considering the
anisotropic elastic constants of α-titanium, as described in Section 2.5.
Dislocation slip systems, including prismatic slip, basal slip, and pyramidal <c+a>
slip, in this particular model, used in this work, were treated as bidirectional deformation
systems. The CRSS values for dislocation slip systems were optimized following the
procedures discussed in Section 4.3.3.

Deformation twinning systems were not

included in the simulation, since no nano-indentation induced twinning was observed

	
  

105	
  

	
  
Figure 4.6 Indentation topography measured by AFM, with their corresponding
indentation axes, positioned on an inverse pole figure (IPF). The indents in the upper
half of the IPF are also displayed in the lower half by a mirroring operation with respect
to the plane (1 0 -1 0) and are marked with an asterisk.

	
  

106	
  

using both AFM and EBSD. The material model was integrated into the commercial
FEM package MSC.Marc by using the subroutine hypela2.

4.3.2 Mesh Generation and Boundary Conditions
Since the size of the indenter was much smaller than the grain size, grain
boundary effects were neglected when the indentation is in the center of the grains, and
the indented grains were treated as a single crystal.

The grain orientations of the

simulated grains were collected using EBSD, and the grains were discretized by 8-node
hexahedral elements. Figure 4.7 shows a deformed mesh that sliced perpendicularly
accross the center of the indent.

A total of 4320 elements were applied.

The

deformation of the sphero-conical indenter was neglected by assuming it as a rigid
body. The maximum loading force was 6mN, which is the same value as that used in
the experiments.

4.3.3 Crystal Plasticity Parameter Optimization Procedures
The values of the CRSS and saturation resistance for three slip types, i.e.
prismatic, basal, and pyramidal <c+a> slip, were chosen as the six free parameters. An
optimization algorithm of the six variables was carried out using the downhill simplex
method provided by Nelder and Mead [120,121].

The optimization of the crystal

plasticity parameters using this method has also been demonstrated by Grujicic and
Batchu [122].
The objective function, the function that is to be minimized by the optimization
routine, was defined as the deviation of the simulated topographies from the

	
  

107	
  

Figure 4.7 A deformed (indented) finite element mesh of hexahedral elements [117].
	
  

	
  

108	
  

experimentally measured ones. With the help of Gwyddion software [123], the surface
topography data obtained by AFM was extracted and regenerated to a square X-Y grid
domain of 2.5 µm × 2.5 µm with a step size of 0.1 µm, and the simulated CPFE
topographies were also exported to the same domain.

The lowest point in AFM

measured topography was aligned with the indentation axis in the simulation.
The topographic contribution to the objective function of one indentation, Gtopo,
was described as the sum of height differences between the measured and simulated
topography at all points in the square grid:
i, j
i, j
Gtopo = ∑ zexp − zsim = ∑ Δzi, j
i, j

(4.1)

i, j

For optimization of the six constitutive parameters using two indentation topographies in
different grain orientations simultaneously, the objective functions for each indentation
were summed.
The initial values of the six parameters were randomly chosen, and 300 rounds
of simulations were conducted to minimize the objective function, i.e., to converge the
six constitutive parameters to the optimum values.

More details can be found in

Zambaldi et al.[124].

4.3.4 Modeling Results
The simultaneous optimization process was carried out for two selected indents
6-a (indentation axis close to [1 0 1 0]) and 13-o (indentation axis close to [2 1 1 1] ).
Figure 4.8 shows the measured surface and simulated surface topographies using
optimized parameters, as well as the Δzi, j map, of the two indents. Table 4.2 shows
	
  

109	
  

Figure 4.8 Experimental (left), CPFE simulated (middle) surface topographies, as well
as Δzi, j (right) maps, of indents 6-a and 13-o, which were selected for the identification
of the constitutive parameters. The size of each map is 2.5 µm × 2.5 µm.

	
  

110	
  

Table 4.2 Optimized constitutive parameters for prismatic, basal and pyramidal <c+a>
slip systems. The other constant parameters for the three systems in the simulations
were chosen as: m = 20, h0 = 200 MPa, a = 2.

Slip systems

CRSS ratio

Saturation resistance ,MPa

Prismatic slip

150 ± 4

1

1502 ± 125

Basal slip

349 ± 10

2.3

568 ± 17

Pyramidal <c+a>

	
  

CRSS, MPa

1107 ± 39

7.4

3420 ± 202

111	
  

the optimized values of the CRSS and saturation resistance of prismatic, basal, and
pyramidal <c+a> slip after 300 iterations. Two other indents, 6-o and 13-i, were chosen
to validate the identified set of parameters, as shown in Figure 4.9. In both validation
simulations, the simulated topographies show good agreement with the measured ones.
The total height difference

∑ Δzi, j

is 10.9 µm for 6-o, and 9.4 µm for 13-i, respectively,

i, j

both of which are smaller than the total height differences for indents 6-a (12.2 µm) and
13-o (12.7 µm).
Figure 4.10 shows the simulated pile-up behavior of the commercial purity
titanium with respect to the grain orientation using the optimized values of constitutive
parameters. All the simulations were performed at 6 mN load. Good agreement was
found between the simulations and the experimental measurements. The maximum
heights of the pile-ups decrease for those indents with their indentation axis closer to
the c-axis. It was also found that the connection line of the two pile-up maxima for each
indent was roughly perpendicular to the c-axis of the grain.

4.4 Discussion
4.4.1 Automated Identified Constitutive Parameter Values
Initial optimization runs for parameter identification based on individual
experimental measurements of a single indent were found not so satisfying.

For

example, the optimization can show bad convergence behavior when the Schmid factor
for one slip system is too low, and consequently negligible amounts of shear were
expected. This phenomenon was observed when running the optimization procedure

	
  

112	
  

Figure 4.9 Experimental (left), CPFE simulated (middle) surface topographies, as well
as Δzi, j (right) map, of indents 6-o and 13-i, which were selected for the validation of
the set of optimized parameters. The size of each map is 2.5 µm × 2.5 µm.

	
  

113	
  

Figure 4.10 Twenty-two simulated pile-up topographies positioned on the IPF according
to their indentation axes. The pile-ups are shown in gray scale, and the regions below
zero are shown in blue. The contour lines are starting at -0.04 µm and 0.04 µm for
below and above zero regions, and drawn every 0.04 µm. Similar trends of pile-up
features were found in CPFE simulations as that shown in the experimental
measurements.

	
  

114	
  

using indent 6-a (in the grain with an orientation close to [10 1 0] ). Basal slip systems
in this grain orientation are difficult to activate because the basal plane is nearly parallel
to the indentation axis. The convergence result shows large oscillations for basal slip.
To avoid this problem, the sum of two indented surface topographies from two grains
(with one having a soft orientation and the other having a relatively hard orientation), i.e.
the sum of the two Gtopo functions of the two indents were chosen to run the
optimization procedure, as described in Section 4.3.3. The consideration of the two
surfaces in the same run guaranteed that all three slip systems have a relatively high
Schmid factor in at least one indent.
The identified CRSS ratios were within the range of the ratios found in the
literature [85,86,125,126]. The determined CRSS value for the most active slip system,
i.e. prismatic slip, was expected to have higher accuracy, since the surface topography
is mainly influenced by the most active systems. The identified values of those less
active systems, however, need to be treated with caution. In this study, to simplify the
optimization process, only six main constitutive parameters were chosen as free
parameters.

Future study can include more parameters, such as cross-hardening

coefficients, to increase the accuracy of the convergence process.

4.4.2 Comparison of Simulation Results using Two Sets of CRSS
A rerun of the CPFE simulation of the deformation of the microstructure patch in
specimen S45-1 discussed in Chapter Three was carried out. The originally chosen
and identified sets of constitutive parameters were listed in Table 3.2 and 4.2. The
rerun simulation was stopped at about 1% global strain due to the stability issue of the

	
  

115	
  

existing model.

This problem may be caused by the high ratio of the CRSS for

prismatic and pyramidal <c+a> slip in the indentified set of values, which may lead to
very high local stress state in hard grains. These elements with high stress state can
usually cause the calculation instability for the simulation. However, by comparing the
prismatic and basal slip shear in the two simulations at 1% strain, no significant
difference was found in the simulation results with identified parameters using nanoindentation. As discussed in previous section, six free variables may be enough for
optimizing the deformation procedure in single crystals.

However, in a more

complicated situation, i.e. poly-crystalline environment, more variables need to be
considered.

For example, the optimization procedure for nano-indentation did not

consider the T1 twin activity, which was found in poly-crystalline microstructure patches.
Therefore, a further study is under way to optimize the simulation results using more
free variables, i.e. the CRSS and saturation resistance of pyramidal <a> slip.
Two other possible reasons for the lack of improvement are that the loading
conditions for indentation contain significant hydrostatic compression, which may affect
slip resistance by non-Schmid stress components that affect slip activation [127].
Secondly, the lack of grain boundaries may also frustrate dislocation nucleation
processes for non-prism slip [121]. Furthermore, the CPFE simulation does not contain
any form of additional slip resistance across a grain boundary [62]. Simulation of the
sub-surface grain geometry with an accurate 3D mesh may also possibly affect model
accuracy. Improvement in modeling slip behavior near grain boundaries appears to be
necessary to improve the agreement between experiment and simulation.

	
  

116	
  

4.5 Conclusions
A nano-indentation technique is introduced to help research the values of CRSS
and saturated resistance of prismatic, basal, and pyramidal <c+a> slip systems in
commercial purity titanium.

Arrays of conical nano-indentations were performed on

patches of grains with a wide range of orientations, and AFM and EBSD scans of on the
patch indicates that the two-hill-lock shape pile-up topography was strongly
crystallographically dependent. Corresponding CPFE simulations predicted the pile-up
patterns in good agreement with the experimental measurements. The six constitutive
parameters were automatically identified by optimizing the simulation results of the
indentation process in different grain orientations using a non-linear optimization
method. A rerun of the CPFE simulation of the same microstructure patch studied in
Chapter Three with the optimized parameters, however, showed only minor differences,
which indicates that more quantitative studies on strain hardening behavior and slip
transfer across grain boundaries are strongly needed.

	
  

117	
  

CHAPTER FIVE
MEASURING MESO-SCALE HETEROGENEOUS DEFORMATION IN COMMERCIAL
PURITY TITANIUM

In this chapter, a novel approach is established to determine plastic zone shape
and meso-scale plastic strain distribution at the tip of a fatigue crack by combining BSE
imaging and discrete Fourier transformation (DFT) image processing.

An in-situ

uniaxial tensile calibration test indicates that the full width at half maximum (FWHM) of
the central peak of the DFT of BSE images has a close relationship with the induced
plastic strain within selected grain patches. The plastic zone size and residual strain
distribution around a crack tip were determined by applying this technique to an array of
BSE images around the tip of a fatigue crack in a unloaded commercial purity titanium
specimen. The measurement results were compared with the plastic zone size and
shape measured using thermoelastic stress analysis, which can measure plastic zone
size based on the heat generated during the plastic deformation processes.

The

experimental results were further compared to theoretical estimates of plastic zone size
proposed by Dugdale and Irwin theories. Because of the good agreements between the
two experimental results, as well as between the BSE-DFT method and the theoretical
estimates, the BSE-DFT approach has been shown to be an effective way to measure
meso-scale heterogeneous plastic deformation in deformed polycrystalline samples.

	
  

118	
  

5.1 Observation of Microstructure around a Fatigue Crack Tip using BSE Images
The fatigue specimen was cut out from Material B, and its configuration was
illustrated by Figure 2.5.

After 27,200 fatigue cycles, the fatigue specimen was

unloaded from the fatigue test instrument, and an array of 15 × 15 (225 in total) BSE
2

images, as shown in Figure 5.1, were taken to cover a square area (about 7.84 mm )
around the crack tip of the sample at an instrument magnification of ×500. Each image
was 115 μm × 115 μm with a center-to-center distance of 187 μm, and each image
contains about 150 grains. A gradual change in the sharpness of the BSE images with
distance from the crack tip can be observed, as shown in Figure 5.2. At large distances
from the crack tip, e.g. shown in Figure 5.2(m14), the images display sharp, welldefined features allowing easy identification of the individual grains, which tend to
exhibit uniform brightness levels that are distinctly different to their neighbors. Strong
contrast variations are evident between the grains in those regions far away from the
crack tip because the different grain orientations result in large differences in channeling
behavior. Images acquired close to the crack tip, however, show significant variation in
brightness within individual grains, making the individual grains less distinct, and
consequently the images display less sharp contrast, as shown in Figure 5.2(h4). This
is a result of significant local crystal rotations and residual plastic strain within these
grain patches, as GNDs and statistically stored dislocation (SSDs) accumulate within
grains. Grain boundary ledges, slip lines, and secondary inter- and/or trans- granular
cracks were also observed in the grains close to the fatigue crack. The apparent loss of
distinct individual grain definition, the decrease in image sharpness, and increase in the

	
  

119	
  

Figure 5.1 The around-crack-tip region was divided by a 15 × 15 (225 in total) grid as
illustrated, and each BSE image was taken from the middle part of each tile.

	
  

120	
  

Figure 5.2 Upper array (shown in Figure 5.1) of BSE images around the crack tip at a center-to center spacing of 187um.
The crack tip is shown in image h4. The images in near-crack-tip region show more lattice rotation and contrast variation,
indicating more residual plastic deformation.

	
  

121	
  

intra-granular features in the BSE images is consistent with increasing residual plastic
strain with proximity to the crack tip.

5.2 Image Analysis Technique Using DFT
5.2.1 Basic Information of DFT
The discrete Fourier transformation (DFT) provides a measurement of the
frequency content and distribution of a signal. In two-dimension situation s, i.e. BSE
images, the DFT of an m by n matrix X is given by another m by n matrix Y:

Yp+1,q+1 =

m−1 n−1

jp kq
∑ ∑ ω m ω n X j+1,k +1

(5.1)

j=0 k =0

where ω m = e−2π i/ m and ω n = e−2π i/ n are the complex roots of unity.

A Fourier

transformation of a two-dimensional image gives a complex-number image, i.e. two
images are generated, identified here, for the sake of ease of understanding, as the
magnitude map and the phase map. The magnitude of a DFT provides the distribution
of the frequency components in the original image, while the phase of the DFT gives the
location of this frequency component in the original image. In the proposed technique,
the magnitude of the DFT is considered and will be shown to reflect the contrast change
within microstructure patches resulting from plastic deformation.

5.2.2 DFT Image Data Processing
While the phenomenon of the contrast change in BSE images during plastic
deformation, as discussed in Section 5.1, has been widely observed [48, 128], it has
never been used to quantify the strain levels associated with localized plastic

	
  

122	
  

deformation. Nevertheless, these contrast changes in BSE images can eventually lead
to a change in the signal frequency distribution of features within the corresponding
images. Therefore, if the frequency distribution of the BSE images can be quantified,
this may be related directly to the extent of plastic deformation.
DFT was conducted on all BSE images from the array (Figure 5.1) using
MATLAB (MathWorks, Natick, MA, U.S.A) built-in functions. First, the MATLAB function
‘fft2’ was used to compute the complex-number DFT images of the BSE images using
the fast Fourier transformation (FFT) algorithm. Functions ‘fftshift’ and ‘log2’ functions
were then applied to improve the output image quality, shifting the zero frequency to the
DFT image center, and enlarging the detailed frequency information, respectively.
Figure 5.3(b) shows FFT magnitude images of four BSE images (Figure 5.3(a))
collected at 561 µm intervals directly ahead of the crack.

Figure 3(b) shows the

magnitude of the DFTs of the corresponding images, and a distinct trend in the extent of
the central peak is apparent. The full width at half maximum of the central peak in the
horizontal line profiles, shown in Figure 5.3(c), decreases with the increasing distances
from the crack tip, indicating a potential relationship between the central peak width and
residual strain level.
To more accurately quantify the change of DFT with respect of levels of residual
plastic strain, symmetry of the DFT plot was assumed and the left half of the line profiles
crossing the center of DFTs were smoothed using the adjacent-averaging method with
a five-point window. The smoothed profiles were then fitted to exponential curves in
order to measure the (FWHM), as illustrated in Figure 5.4. Figure 5.5(a) shows the
FWHM distribution directly in front of the crack tip, and Figure 5.5(b) illustrates the

	
  

123	
  

	
  

Figure 5.3 (a) BSE images h5, h8, h11, and h14 from Figure 5.2; (b) their corresponding DFTs and (c) a horizontal line
profile across the center of each DFT showing a decrease in the center peak width with increasing distance from the crack
tip to the image location.

	
  

124	
  

Figure 5.4 Schematic plots showing the process of fitting line profiles and measuring the
full width at half the maximum of the profile. Line profiles were first smoothed using an
adjacent-averaging five-point window. The left half of a smoothed profile was then fitted
to an exponential curve.	
  	
  
	
  	
  

	
  

125	
  

(a)

(b)
Figure 5.5 The full width at half maximum of the FFT center peak versus the distance
from the crack tip at (a) θ = 0° (b) θ = 45°
	
  

126	
  

FWHM distribution 45° away from the crack line. The FWHM decreases dramatically
with increasing distance from the crack tip, while the rate of change decreases with
increasing distance, which further indicates the potential relationship between the
FWHM in DFTs and the local residual strain levels where the corresponding BSE
images were taken.

5.3 Quantifying the Relation between FWHM and Strain Using In-situ Tensile Test
The results of the in-situ uniaxial tensile test were used to quantify the
relationship between the residual plastic strain level in a given microstructure patch and
the FWHM of the DFT magnitude.

In order to obtain a detailed stress-strain

relationship, a separate tensile test was carried out on the large dog-bone specimen
using a conventional servo-hydraulic test machine. The engineering stress-strain curve
generated (shown in Figure 5.6) shows that the material yields at about 0.4% global
strain. In figure 5.6 the arrows below the stress-strain curve indicate the five strain
levels at which the in-situ tensile test was interrupted to acquire the BSE images in six
microstructure patches. Figure 5.7 shows one set of BSE images of a microstructure
patch at different strain levels. These images were subsequently processed using the
procedure described in the previous paragraph. The mean FWHM of the DFT from six
microstructure patches at each strain increment were plotted, as shown in Figure 5.8,
and a polynomial fitted to create a calibration curve (shown as dotted line in Figure 5.8)
that could be used to determine the residual plastic strain field of the fatigued specimen.
The calibration curve in Figure 5.8 was used to estimate the level of plastic strain in
each of the 225 BSE images shown in Figure 5.2 after applying the DFT and calculating

	
  

127	
  

Figure 5.6 Engineering stress-strain curve obtained for the commercially pure titanium;
the arrows indicate the strain increments at which the in-situ uniaxial tensile test was
interrupted to collect BSE images.

	
  

128	
  

Figure 5.7 BSE images of a grain patch at different global strain levels during in-situ
tensile test indicating the evolution of microstructure features.

	
  

129	
  

Figure 5.8 The full width of half the maximum of the DFT as a function of the
engineering strain based on images recorded at increments of plastic strain during an
in-situ tensile test.
	
  

	
  

130	
  

their FWHM. The elastic strain contribution was ignored because its contribution to the
overall strain is very small compare to the measured plastic strains.

5.4 Residual Strain Map around a Crack Tip in the Fatigue Specimen
Figure 5.9 shows a map of residual plastic strain field in the fatigued specimen
based on the calibrated DFTs. Thus, each tile in the map represents an average
residual plastic strain for a 187 μm × 187 μm area.
This strain map clearly reveals the spatial heterogeneity of the local plastic strain
distribution around the crack tip in the fatigue specimen. The shape of the plastic zone
around the crack tip is an oval-shaped area with high strain extending about 1.5 mm
ahead of the crack and about 0.5 mm on both sides of the crack. Also, the plastic wake
strain is exhibited along the crack. The wake strain at the crack tip has a maximum
value of 18%, which decreases dramatically with distance from the crack tip.

5.5 Discussion
5.5.1 The Influence of Brightness, Contrast, Magnification, and Calibration Procedures
on the Strain Map
The effects of the instrument parameters associated with the electron imaging
conditions need to be assessed before further discussion. Theoretically, changes in
brightness will not change the shape of the peak in the DFT, because the frequency
distribution of the image remains the same. Increases in brightness will still increase
the counts within the DFT peak, which means that the peak will be extended along the
y-axis, but the FWHM will not be changed, as illustrated in Figure 5.10. An increase in

	
  

131	
  

Figure 5.9 Map of plastic strain around the crack tip (the area shown in Figure 5.1)
based on the FWHM of the DFTs of BSE images.

	
  

132	
  

Figure 5.10 BSE images from the same microstructure patch and their corresponding
DFT line profiles showing that the change of brightness does not affect the FWHM of
the line profiles.

	
  

133	
  

contrast will increase the difference in brightness level between bright and dark pixels,
but will not change the image frequency distribution as measured by the FWHMs, as
shown in Figure 5.11.
However, the magnification of a BSE image can strongly influence the DFT peak
because a different field of view will be present for the same size of image. Thus, a low
magnification BSE image will contain more grains and the channeling bands within the
grains will be finer in scale than for a higher magnification image of the same sample,
resulting in a wider DFT distribution for the lower magnification image. Of course, if the
sizes of features within an area of interest are finer than the pixel size in the image, then
information will be lost, because the frequency of those features is too high to be
differentiated using the DFT, i.e. Nyquist sampling errors will occur [129]. One the other
hand, when the magnification is too high, there will be insufficient grains sampled to
provide statistically significant data, and the DFT may become asymmetric about the
DFT image center or too rough. In this situation, the assumption of symmetric curves in
DFT images and the curve smoothing procedure using in Section 5.2 will be
inappropriate. Therefore, it is important that images are acquired at an appropriate
magnification for the grain size of the material being studied. Working distance will also
influence the DFT distribution. A longer working distance may result in smaller rocking
angle of the electron beam to image the same area, which leads to narrower SACP
bands for a given grain. As a result, for the same microstructure patch at a given plastic
strain, longer working distance will decrease the FWHM of DFT.

Thus, it is also

essential that the optical arrangements, i.e. magnification and working distance in the
SEM, remain constant for both the calibration and fatigue samples in order to avoid

	
  

134	
  

Figure 5.11 BSE images from the same microstructure patch and their corresponding
DFT line profiles showing that the change of contrast does not affect the FWHM of the
line profiles.

	
  

135	
  

introducing errors. It is also important to note that the strain determined is an average
for the field of view of the image so a smaller field of view would allow higher spatial
resolution for the evaluation of strain (with associated errors outlined above).
The strain calibration procedure is another aspect that needs to be considered
briefly.

The assumption in the DFT calibration presented in Section 5.3 is that an

equivalent breakdown in channeling conditions, associated with a given distribution of
GNDs, occurs for the same equivalent strain, regardless of the strain path experienced.
Indeed, the two strain paths in the present work are considerably different. A uniaxial
strain distribution with measurements was taken during monotonic loading for in-situ
tensile test in SEM. In fatigue testing, a tri-axial strain distribution at the crack tip was
induced by high-cycle tension-tension fatigue with measurements taken after loading in
the SEM. In a study of aluminum alloy systems, Davidson and Lankford [130] proposed
that the applied strain ( ε t ) – dislocation density ( ρt ) relationship in tensile test can be
described as

⎛ σ ρ s/ 2 ⎞
εt = ⎜ 1 t ⎟
⎜ σ 0 gs ⎟
⎝
⎠

m

(5.2)

where m has a value of 10 for aluminum alloys, g is a proportionality constant, and the
parameters s and

σ1

σ0

are dependent on the alloy systems and loading conditions. On

the other hand, the relationship between cyclic strain ( Δε ) applied N times and
dislocation density ( ρ f ) relationship in fatigue test was described as

	
  

136	
  

⎛ σ ρ p/ 2 ⎞
f
f ⎟
Δε = 2 ⎜
⎜ 2K gp ⎟
⎝
⎠

n

where parameters p and

(5.3)

σf
2K

are dependent on the alloy systems and loading conditions

[130]. From Equation 5.2 and 5.3, with the assumption of the same dislocation density
in the two kinds of deformation processes, the relationship between ε t and Δε can be
derived.

Along with another previous study [131], the results indicate that fatigue

specimens can be expected to have a higher dislocation density for an equivalent strain
level than uniaxial tension specimens. Thus, the strain deduced for the fatigue sample
based on the calibration using the uniaxial sample, is probably an over-estimate of the
actual strain to some small extent. However, the strain distribution observed directly
ahead of the crack tip shown in Figure 5.9 is very similar to the observations and
conclusions of Crompton and Martin [46].

5.5.2 Comparison between the Plastic Zone Size Measured by BSE-DFT Method and
Those Proposed by Other Works
Data obtained using thermoelastic stress analysis along a line through the crack
tip and along the crack path is shown in Figure 5.12. The black area indicates the
measured plastic zone shape ahead the crack tip.

The TSA signal and its phase

difference with respect to the distance from the crack tip are plotted. The phase signal
is noisy ahead of the crack tip but the far-field (about 2 mm away from the crack tip) has
a mean value that deviates by less than five degrees from zero, indicating essentially

	
  

137	
  

Figure 5.12 Phase difference and magnitude of the TSA signal as a function of distance
along a line through the tip of the fatigue crack in the direction of crack growth. The
extent of the plastic zone found from the phase difference map is superimposed as a
black area.

	
  

138	
  

adiabatic conditions, as would be expected for elastically deforming material. Close to
and ahead of the crack tip, the phase difference exhibits a large negative value with a
maximum difference of about 22 degrees, indicating large plastic deformation within this
region. Immediately behind the crack tip, the phase difference is positive with a peak of
about ten degrees. These two regions of phase difference with opposite sign have
been attributed [98] to heat generation occurring ahead of the crack tip in the loading
part of the cycle as a result of dislocation generation and movement and to heat
generation behind the crack tip in the unloading part of the cycle due to contact of the
recently-formed flanks.

Therefore, as suggested by Diaz et al. [97], the transition

between positive and negative phase difference provides an indication of the location of
the crack tip along the direction of expected crack growth. Patki and Patterson [98]
applied a binary filter to the phase difference and identified the shape and size of the
plastic zone from the region of significant phase difference ahead of the crack tip. This
is the approach adopted here and the black area superimposed on Figure 5.12 shows
the cyclic plastic zone shape. According to the TSA data, the plastic zone shape is
approximately circular with a radius about 0.7 mm. Patki and Patterson [98] compared
the plastic zone size rp estimated using TSA with those determined using the theory
proposed by Irwin [132]:

⎡
⎤
1 K
2r = ⎢ I ⎥
y π ⎢σ ⎥
⎣ y⎦

2

(5.4)

where 2ry is the Irwin plastic zone diameter, KI is the applied stress intensity factor, and
σy is the yield stress of the material. However, instead of using KI, ΔKI, the stress

	
  

139	
  

intensity factor range, can usually give a better estimation of the plastic zone size [98].
The agreement between the plastic zone size measured using TSA and the estimated
plastic zone size using Irwin’s expression (using ΔKI instead of KI) is usually within the
0.2mm, as found by Patki and Patterson [98]. This value of ΔKI was found using the
Multiple Point Over-Deterministic Method (MPODM) to fit a Muskhelishvili-type
description of the crack tip stress field to the measured TSA data following the method
described by Diaz et al. [97], which was implemented in the software algorithm,
FATCAT.
Thus, in Figure 5.13, a comparison has been made of the crack tip plastic zones
determined from the BSE-DFT and TSA data obtained in this study and based on the
expressions proposed by Irwin [132] (see Equation 5.4) and Dugdale [133] (as
described by Equation 5.5).
2

π ⎡ KI ⎤
⎥ [133]
2ry = ⎢
8 ⎢σ y ⎥
⎣
⎦

(5.5)

The plastic zone found from the TSA data is smaller than that determined from
the BSE images; but is very similar to the estimation using Irwin’s expression. The
boundary of the plastic zone measured by TSA is of the same order of magnitude with
its boundary in the direction perpendicular to crack growth, corresponding to about 4%
plastic strain measured from the BSE images. Ahead of the crack, the boundary of the
plastic zone found from the TSA data corresponds to about 8% plastic strain evaluated
from the BSE images.

	
  

140	
  

Figure 5.13 Map of plastic strain around the crack tip based on the FWHM of the DFTs
of images from the region shown in Figure 5.1. The white line shows the measurement
of plastic zone using TSA data. The green and blue solid lines indicate the theoretical
estimations of the plastic zone size based on Dugdale’s and Irwin’s approaches,
respectively.

	
  

141	
  

The two experimental approaches are based on different physical principles, so
differences in results would be expected. The data from the BSE-DFT method is based
on detecting the local rotations of the grains by the geometrically necessary dislocations
required for plastic deformation. In theory, this disruption in grains necessary for the
observations in the SEM may not disappear when the load is removed. The removal of
the loading, however, could have caused some reverse plasticity as a consequence of
compatibility requirements between the crack tip plastic zone and the surrounding
elastic region. This reverse plasticity can influence the measurement of the plastic
strain distribution in the region closest to the crack tip but will have less effect on the
plastic zone size. The evaluation of the plastic zone from the TSA data, on the other
hand, is based on identifying the region in which the generation of heat by dislocation
generation and movement is sufficiently large to cause adiabatic conditions to be lost,
i.e. a heat flux or transfer occurs. A higher density of dislocations will generate a larger
heat flux, which will be easier to detect as a phase difference, and this will tend to occur
closer to the crack tip.

At lower loading frequencies, though still high enough to

maintain adiabatic conditions in the elastic field, there is more time for heat transfer to
occur and so the TSA measurements, based on phase difference, will show greater
sensitivity to the heat generated by plastic deformation; consequently, a lower density of
dislocations is likely to generate sufficient heat to produce a measurable phase
difference and so lower levels of plasticity will be detectable. This agrees with the
observations by Tomlinson et al. [134] that the plastic zone size measured using TSA
was smaller at higher frequencies. So a smaller plastic zone size measured by TSA

	
  

142	
  

than that measured by BSE-DFT method could be expected for a high-frequency fatigue
specimen.
On the other hand, there is a close relation between the size of the plastic zone
found using BSE-DFT measurements and the size estimated using Dugdale’s approach
(Figure 5.13). Ahead of the crack, the boundary of the plastic zone boundary estimated
using Dugdale’s approach corresponds to about 5% plastic strain evaluated from the
BSE images. In the direction perpendicular to crack growth, the plastic zone boundary
is corresponding to about 2% plastic strain measured from the BSE images.

5.5.3 The Application of BSE-DFT Method
BSE imaging is a common imaging mode in the SEM for studies of metals, so
this approach might be easily applied to the measurement of meso-scale
heterogeneous plasticity in other kinds of deformation, such as bending, compression,
and in creep tests. Also, the image collection process has the potential to be easily
automated using a computer-controlled stage and automated image capture increase
the collection speed.
However, there are some restrictions in terms of the types of materials suitable
for this method. First of all, in order to acquire statistically significant information, there
needs to be sufficient grains in the field of view. Therefore, materials with large grain
sizes will decrease the spatial resolution of this method. Another consideration is that
the materials need to be single-phase, or multi-phase with large primary-phase grains
and very small secondary-phase particles along grain boundaries and/or in grains, so
that the DFT is effectively applied only to the primary-phase grains.

	
  

143	
  

Otherwise, in

materials with approximately equal amounts of two different phases, the contrast of BSE
images will tend to be dominated by the difference between the phases instead of the
differences in grain structure arising from GNDs, so that the DFT will be a function of
the shape and quantity of these phases, which will mask the contrast associated with
lattice curvature.

5.6 Conclusions
The plastic strain around a fatigue crack in commercial purity titanium has been
successfully evaluated using a novel technique based on using the discrete Fourier
transform to analyze the structure of BSE images. The full-width at half maximum of the
magnitude of the DFT of the BSE images was used as a measure of the level of
plasticity based on the distortion of grains caused by dislocation accumulation during
plastic deformation.

The technique was calibrated using images obtained from a

uniaxial tension test performed in-situ in the SEM so that the magnitude of the average
plastic strain in the field of view of an image could be related to the FWHM for the
image. After a fatigue crack was propagated to a length of 5.88 mm, 225 BSE images
115 µm × 115 µm were collected in an array occupying a 2.8 mm by 2.8 mm square
around the crack tip and a map of plastic strain evaluated for this area. The measured
plastic zone size and shape were assessed during the fatigue loading using TSA. The
results from the two techniques are in general agreement and correlate with estimates
of plastic zone size based on the theoretical expressions due to Irwin and Dugdale
evaluated using the measured range of the stress intensity factor during the fatigue
cycle of the stress intensity factor. This new approach may prove to be useful for

	
  

144	
  

measuring local heterogeneous plastic deformation in other applications when there is a
dominant primary phase present. Also, it has great potential in assisting identification of
the role of the crack tip plastic zone in crack closure and plasticity-induced shielding.

	
  

145	
  

CHAPTER SIX
BROAD IMPACTS OF THE STUDY

Multi-scale characterizations and simulations of deformation processes of
advanced materials, such as titanium, are strongly needed to support all other relevant
technological innovations.

Success in understanding damage nucleation in design

environments will lead to reduced waste, accelerated time to market for highly valueadded manufactured goods, improved safety and economy in all aspects of technology
and engineering.

The studies discussed in this dissertation combine several

experimental and computational efforts to study how microstructural details affect the
plastic deformation in polycrystalline specimens.

These studies provide a solid

foundation leading to the further understanding of the relationships between the
heterogeneous deformation, often considered as a precursor of damage, crack
nucleation and propagation. This research project has impacts in a number of important
areas of both materials science and mechanical engineering research, including grainboundary engineering, dislocation dynamics, multi-scale finite element modeling, and
fracture mechanics.
In past decades, there has been considerable interest in developing advanced
experimental

methods

and

computational

heterogeneous deformation in metallurgy.

models

to

investigate

micro-scale

In previous studies, heterogeneous

deformation was often assessed using tools such as orientation imaging microscopy
and strain mapping.

However, such studies have rarely analyzed the operating

deformation mechanisms in the context of dislocation activity. In this dissertation, new

	
  

146	
  

experimental methods (see Chapter Three) have been established to investigate the
strain distribution and surface topography in deformed specimens in terms of activated
dislocation systems, which provides an improved approach to study heterogeneous
deformation. Experimental characterization indicates that, at the micro-scale, grains
with different orientations respond to the same global stress states in rather different
ways, including activated dislocation systems and lattice rotation gradients. Also, the
experimental study on the values of CRSS and saturated resistance stresses for three
deformation systems (see Chapter Four), i.e. prismatic slip, basal slip, and pyramidal
<c+a> slip, is an important part of further understanding dislocation activity in hcp
metals.
Although many previous simulation works have been conducted to study metal
deformation processes, very few have been critically assessed by testing the simulation
against actual microstructural deformation. The experimental characterizations in this
dissertation have proved to be efficient tools to help assessment and development of
the existing CPFE models, which may lead to a deeper understanding of material
deformation and damage processes. With the help of these tools, the importance of
different efforts, i.e. building real 3-D meshes and adding grain boundary effects, can be
evaluated.

Once good agreement between experimental and computational

microstructural deformation is reached, the modeling can simulate details in the
deformation behavior that cannot be directly measured, such as the local stress tensor.
Thus, the details of deformation processes that occur prior to observed damage can be
identified, and hence, the cause of damage nucleation can be quantitatively identified.

	
  

147	
  

From a mechanical engineering point of view, the mechanical behavior of solids
on the macroscopic scale is usually concerned.

It sometimes ignores the discrete

nature of materials, which results from microstructural variations in real materials.
Some classical theories, such as classic identification of the plastic zone size in front of
a crack tip, may be found satisfactory when the dimensions of the bodies of interest can
be compared to the characteristic lengths, i.e. grain sizes. However, in poly-crystalline
environments, grain sizes are much smaller than the specimen size, so microstructural
variation must be considered to further understand mechanical behaviors of materials.
In this dissertation, a new experimental method (see Chapter Five) successfully related
micro-scale feature evolution, i.e. lattice rotation and dislocation activity, to the mesoscale plastic strain distribution. This method provides an opportunity to further develop
classical theories based on crystal nature of the metals.

Also, this method has a

potential opportunity to be automated to scan large areas to quick locate microstructure
patches of interest, i.e. strain concentrated area. These patches can be then further
studied using other micro-scale experimental characterization methods.

	
  

148	
  

CHAPTER SEVEN
CONCLUSIONS AND FUTURE WORK

7.1 Conclusions
Different scales of heterogeneous deformation in commercial purity titanium (αtitanium) have been studied in both four-point bending and fatigue specimens. A series
of comprehensive experimental and CPFE simulation methods have been conducted to
characterize and model the deformation behavior of different microstructure patches.
Two new experimental techniques were established in this study to assist
detailed observation and quantification of heterogeneous deformation in poly-crystalline
specimens:
•

A technique combining AFM, EBSD, and BSE imaging was successfully
applied to a polycrystalline microstructure to provide the fine-scale analysis
of dislocation activity and derivation of the evolution of the corresponding
local shears at different levels of global strains, based on the high-resolution
measurements of the z-component of slip and/or twin displacements on
specimen surface.
Publication: Yang, Y., Wang, L., Bieler, T.R., Eisenlohr, P., Crimp, M.A.,
2011, Quantitative AFM characterization and crystal plasticity finite element
modeling of heterogeneous deformation in commercial purity titanium,
Metall. Mater. Trans. A, 42A, pp. 636-644.

•

A technique based on the analyze of the frequency magnitude distribution in
BSE images, by applying the DFT to the images of polycrystalline patches,

	
  

149	
  

was introduced to map the meso-scale plastic strain distribution resulting
from heterogeneous plastic deformation in complex loading and component
geometries. The measured plastic zone size and shape from this technique
has good correlation with estimates of plastic zone size based on the
theoretical expressions of Irwin and Dugdale.
Publication: Yang, Y., Crimp, M.A., Tomlinson, R., Patterson, E.A., 2011,
Quantitative measurement of plastic strain field at a fatigue crack tip, Proc.
Roy. Soc. A, in revision.

Two sets of CPFE simulations in both single- and poly- crystal environments
were conducted in this study.

These simulation results were compared to the

experimental measurements to assess the accuracy of the existing constitutive model
and to reveal the possibility of further developments of the model:
•

The existing CPFE constitutive model examined, using a quasi – 3D mesh,
successfully predicted most of active dislocation slip and twinning systems
in a poly-crystalline microstructure patch. Most of the magnitude of
dislocation shears were reasonably well predicted on a grain-to-grain basis,
according to the comparison of experimentally measured and simulated
local shear distribution.

Simulations indicated that grains with dominant

prismatic slip show about three times higher local shears than grains with
either basal or pyramidal slip activity at 1.5% global strain. The dislocation
shear distribution within individual grain, however, sometimes differed from
the measurements. Also, the crystal orientation of the simulation mesh rim

	
  

150	
  

was found to have a small influence on the simulated dislocation activity in
the grains in the patch center.
Publication: Yang, Y., Wang, L., Bieler, T.R., Eisenlohr, P., Crimp, M.A.,
2011, Quantitative AFM characterization and crystal plasticity finite element
modeling of heterogeneous deformation in commercial purity titanium,
Metall. Mater. Trans. A, 42A, pp. 636-644.
•

CPFE simulations predicted the pile-up behavior in single-crystals during
nano-indentation in single-crystals in good agreement with the experimental
measurements within a wide range of grain orientations. The values of
CRSS and saturation resistance for prismatic, basal, and pyramidal <c+a>
slip of α-titanium were automatically identified by optimizing the simulation
results of the indent surface topographies in different grain orientations
using a non-linear optimization method. A rerun of the CPFE simulation for
the same poly-crystalline microstructure patch mentioned in previous
paragraph with the optimized parameters showed only minor differences in
results, which indicates further quantitative studies on strain hardening
behavior and slip transfer across grain boundaries are needed.
Publication: Zambaldi, C., Yang, Y., Bieler, T.R., Raabe, D., 2011,
Orientation informed nanoindentation of alpha-titanium: indentation pile-up
in hexagonal metals deforming by prismatic slip, J. Mater. Res., in print.

7.2 Future work
To gain a deeper understanding of heterogeneous deformation in poly-crystalline

	
  

151	
  

hcp materials, as well as to apply the obtained knowledge in this study to the prediction
of crack initiation sites, series of further experimental and simulation studies are
suggested:
In the experimental aspect, future work can be focused on the observation of
dislocation activity near grain boundaries using ECCI in the early stages of plastic
deformation, a technique providing a smaller scale observation of dislocation activities
near grain boundaries or within individual grains, compared to the techniques
introduced in this dissertation.

For example, ECCI images can be taken at grain

boundaries where grain boundary ledges form at small strain level (1-2% strain) to
observe the dislocation distribution near those ledges. Also, a statistical study on the
relationship of the grain boundaries cracked in the first place and the deformation
processes of the grains on the both sides of those boundaries is needed to build the
connection between heterogeneous deformation and crack initiation.
In the CPFE simulation aspect, a more comprehensive CPFE constitutive model
that considers the information of slip-slip and slip-twin transfer across grain boundaries,
as well as slip resistance at grain boundaries, is strongly needed to simulate
heterogeneous deformation within grain patches more reliably. Also, a more detailed
set of experiment is recommended to measure the critical resolve shear stresses and
other hardening parameters for different deformation systems in titanium. Thirdly, an
improved three-dimensional representation of grain geometry is also recommended to
further increase the accuracy of simulation results. Three-dimensional information of
grain patches, including both grain shape and orientation, can be obtained by 3D-EBSD
and 3D-EBSD techniques.

	
  

152	
  

APPENDICES

	
  

153	
  

APPENDIX A
MATLAB Code for Slip System Trace Analysis in Commercial Purity Ti

clc;
clear;
% basal <a>
ssa(:,:,1) = [0 0 0 1; 2 -1 -1 0];
ssa(:,:,2) = [0 0 0 1; -1 2 -1 0];
ssa(:,:,3) = [0 0 0 1; -1 -1 2 0];
ibas = 1;
fbas = 3;
% prism <a>
ssa(:,:,4) = [0 1 -1 0; 2 -1 -1 0];
ssa(:,:,5) = [1 0 -1 0; -1 2 -1 0];
ssa(:,:,6) = [-1 1 0 0; -1 -1 2 0];
iprs = 4;
fprs = 6;
% pyramidal <a>
ssa(:,:,7) = [0 1 -1 1; 2 -1 -1 0];
ssa(:,:,8) = [1 0 -1 1; -1 2 -1 0];
ssa(:,:,9) = [-1 1 0 1; -1 -1 2 0];
ssa(:,:,10) = [0 -1 1 1; 2 -1 -1 0];
ssa(:,:,11) = [-1 0 1 1; -1 2 -1 0];
ssa(:,:,12) = [1 -1 0 1; -1 -1 2 0];
ipyra = 7;
fpyra = 12;
% pyramidal <c+a>
ssa(:,:,13) = [1 0 -1 1; 2 -1 -1 -3];
ssa(:,:,14) = [1 0 -1 1; 1 1 -2 -3];
ssa(:,:,15) = [0 1 -1 1; 1 1 -2 -3];
ssa(:,:,16) = [0 1 -1 1; -1 2 -1 -3];
ssa(:,:,17) = [-1 1 0 1; -1 2 -1 -3];
ssa(:,:,18) = [-1 1 0 1; -2 1 1 -3];
ssa(:,:,19) = [-1 0 1 1; -2 1 1 -3];
ssa(:,:,20) = [-1 0 1 1; -1 -1 2 -3];
ssa(:,:,21) = [0 -1 1 1; -1 -1 2 -3];
ssa(:,:,22) = [0 -1 1 1; 1 -2 1 -3];
ssa(:,:,23) = [1 -1 0 1; 1 -2 1 -3];
ssa(:,:,24) = [1 -1 0 1; 2 -1 -1 -3];
euler = [0 0 0];
sliptrace = [0,0,0];
phi1 = euler(1,1);

	
  

% Euler = euler-[90,0,0] input crystal orientation

154	
  

PHI = euler(1,2);
phi2 = euler(1,3);
g11 = cosd(phi1)*cosd(phi2)-sind(phi1)*sind(phi2)*cosd(PHI);
g12 = sind(phi1)*cosd(phi2)+cosd(phi1)*sind(phi2)*cosd(PHI);
g13 = sind(phi2)*sind(PHI);
g21 = -cosd(phi1)*sind(phi2)-sind(phi1)*cosd(phi2)*cosd(PHI);
g22 = -sind(phi1)*sind(phi2)+cosd(phi1)*cosd(phi2)*cosd(PHI);
g23 = cosd(phi2)*sind(PHI);
g31 = sind(phi1)*sind(PHI);
g32 = -cosd(phi1)*sind(PHI);
g33 = cosd(PHI);
g = [g11,g12,g13;g21,g22,g23;g31,g32,g33];
sigma = [0,0,0;0,1,0;0,0,0];
c_a=1.59;

% rotation matrix

% input global stress state

for i = 1:1:24
% convert n & m to unit vector
n = [ssa(1,1,i) (ssa(1,2,i)*2+ssa(1,1,i))/3^.5 ssa(1,4,i)/c_a];
m = [ssa(2,1,i)*1.5 3^.5/2*(ssa(2,2,i)*2+ssa(2,1,i)) ssa(2,4,i)*c_a];
ss(1,:,i) = n/norm(n);
% alpha plane
ss(2,:,i) = m/norm(m);
% alpha direction
end
for k = 1:1:24
burger(1,:,k) = ss(2,:,k);
rot_burger(:,1,k) = g'*burger(1,:,k)';
z = [0 0 1];
burger_z(k) = rot_burger(:,1,k)'*z';
end

% rotate slip direction

for j = 1:1:24;
plane = ss(1,:,j);
plane = plane';
rot_plane = g'*plane;
z = [0,0,1];
sliptrace(:,:,j) = cross(rot_plane',z);
x = sliptrace(1,1,j);
y = sliptrace(1,2,j);
X = [0,x];
Y = [0,y];

% rotate slip plane normal

axis([-1 1 -1 1]);
if j < 4
k = j;
elseif j <= 6

	
  

155	
  

k = j-3;
elseif j <= 9;
k = j-6;
elseif j <= 12;
k = j-9;
elseif j <= 15;
k = j-12;
elseif j <= 18;
k = j-15;
elseif j<21;
k = j-18;
else
k = j-21;
end
axis square
axis([-1 1 -1 1])
plot(X,Y);
text(x-0.05*k,y,num2str(j));
hold on
end;

% plot traces

axis square
axis([-1 1 -1 1]);
SF = zeros(24,1);
sigma = g*sigma*g';
for i = 1:1:24
a = ss(1,:,i);
b = ss(2,:,i);
N = g'*a';
M = g'*b';
SF(i) = N(2)*M(2);
schmidfactor(i) = a*sigma*b';
end;

	
  

% Shimid factor calculation

% alpha plane
% alpha direction

156	
  

APPENDIX B
Procedure File for Simulating 3D Meshes for CPFE Modeling Shown in Figure
3.11(c)

*new_model yes *select_reset
*plot_reset *expand_reset *move_reset

365 237 0
371 220 0
430 225 0
40
130 0
138 179 0
272 143 0
296 138 0
354 168 0
370 144 0
430 155 0
215 104 0
267 126 0
296 110 0
308 87
0
359 108 0
401 74
0
430 110 0
40
40
0
220 40
0
305 40
0
410 40
0
430 40
0
0
530 0
40
530 0
430 530 0
470 530 0
0
0
0
40
0
0
430 0
0
470 0
0
*stop_procedure *fill_view
*set_curve_type line *add_curves
1
2
1
6
6
7
2
7
2
3
3
8

|++++++++++++++++++++++++++++++
| points used to build the sample
geometry
|++++++++++++++++++++++++++++++
*add_points
40
490 0
120 490 0
230 490 0
380 490 0
430 490 0
40
460 0
108 470 0
219 464 0
127 459 0
174 441 0
225 423 0
378 407 0
40
405 0
103 389 0
192 407 0
301 383 0
361 389 0
430 395 0
132 348 0
168 353 0
178 324 0
278 330 0
284 283 0
389 296 0
430 300 0
40
290 0
121 301 0
139 296 0
157 296 0
231 238 0

	
  

157	
  

8
9
7
8
11
16
12
4
3
4
5
12
6
13
9
14
19
15
10
11
20
21
16
17
24
23
22
24
18
13
26
19
27
28
21
29
23
31
32
37
36
30
32
25
26
34

	
  

10
10
9
11
16
17
17
12
4
5
18
18
13
14
14
19
20
20
15
15
21
22
22
24
31
31
23
25
25
26
27
27
28
29
29
30
30
32
38
38
37
36
33
33
34
35

28
35
35
41
41
42
36
42
42
43
37
43
38
39
39
40
33
40
34
48
48
49
41
49
49
50
44
50
43
44
44
45
39
45
45
46
40
47
46
47
46
51
50
51
47
52
51
52
53
54
54
55
55
56
1
54
5
55
53
57
56
60
48
58
52
59
57
58
58
59
59
60
*stop_procedure
*set_curve_tolerance_absolute
*redraw *remove_points all_existing
*sweep_all
*remove_unused_nodes
*remove_unused_points
*renumber_all
|++++++++++++++++++++++++++++++
| set divisions for each curve (line)

158	
  

|++++++++++++++++++++++++++++++
*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 1 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 12 # | End of List
*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 13 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 2 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 14 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 3 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 15
*apply_curve_divisions 15 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 4 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 16 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 11
*apply_curve_divisions 5 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 17 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 6 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 18 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 7 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 19 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 8 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 20 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 9 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 21 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 10 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 22 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 11 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4

	
  

159	
  

*apply_curve_divisions 23 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 35 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 24 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 11
*apply_curve_divisions 36 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 25 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 37 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 26 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 38 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 27 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 39 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 28 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 40 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 29 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 41 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 30 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 42 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 31 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 43 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 32 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 44 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 5
*apply_curve_divisions 33 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 45 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 34 # | End of List

	
  

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7

160	
  

*apply_curve_divisions 46 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 58 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 47 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 59 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 48 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 60 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 49 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 61 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 8
*apply_curve_divisions 50 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 9
*apply_curve_divisions 62 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 16
*apply_curve_divisions 51# | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 18
*apply_curve_divisions 63 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 11
*apply_curve_divisions 52 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 64 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 12
*apply_curve_divisions 53 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 9
*apply_curve_divisions 65 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 10
*apply_curve_divisions 54 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 66 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 55 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 67 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 56 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 68 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 57 # | End of List

	
  

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4

161	
  

*apply_curve_divisions 69 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 81 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 6
*apply_curve_divisions 70 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 53
*apply_curve_divisions 82 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 71 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 53
*apply_curve_divisions 83 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 72 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 84 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 3
*apply_curve_divisions 73 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 85 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 11
*apply_curve_divisions 74 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 86 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 7
*apply_curve_divisions 75 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 39
*apply_curve_divisions 87 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 2
*apply_curve_divisions 76 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 88 # | End of List

*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 77 # | End of List

|++++++++++++++++++++++++++++++
| partition elements
|++++++++++++++++++++++++++++++
*af_planar_quadmesh 1 2 3 4 # | End of
List
*select_elements all_existing
*store_elements G1 all_selected
*identify_sets *regen *select_elements
all_existing

*set_curve_div_type_fix_ndiv
*set_curve_div_num 39
*apply_curve_divisions 78 # | End of List
*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 79 # | End of List
*set_curve_div_type_fix_ndiv
*set_curve_div_num 4
*apply_curve_divisions 80 # | End of List

	
  

*af_planar_quadmesh 4 5 6 7 8 9 # |
End of List
*select_elements all_existing
*store_elements G2 all_selected

162	
  

*select_clear *select_elements
all_existing

*af_planar_quadmesh 12 30 31 32 33
29 # | End of List
*select_elements all_existing
*store_elements G9 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 6 15 14 13 12 11
10 # | End of List
*select_elements all_existing
*store_elements G3 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 13 18 35 34 30 #
| End of List
*select_elements all_existing
*store_elements G10 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 14 16 17 18 # |
End of List
*select_elements all_existing
*store_elements G4 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 36 20 22 38 37 #
| End of List
*select_elements all_existing
*store_elements G11 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 3 9 21 20 19 # |
End of List
*select_elements all_existing
*store_elements G5 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 23 27 41 40 39
38 # | End of List
*select_elements all_existing
*store_elements G12 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 8 25 24 23 22 21
# | End of List
*select_elements all_existing
*store_elements G6 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 28 33 43 42 41 #
| End of List
*select_elements all_existing
*store_elements G13 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 7 10 26 25 # |
End of List
*select_elements all_existing
*store_elements G7 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 43 32 44 45 46
47 48 # | End of List
*select_elements all_existing
*store_elements G14all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 11 29 28 27 24
26 # | End of List
*select_elements all_existing
*store_elements G8 all_selected
*select_clear *select_elements
all_existing

	
  

*af_planar_quadmesh 31 34 50 49 44
31 # | End of List
*select_elements all_existing
*store_elements G15 all_selected

163	
  

*select_clear *select_elements
all_existing

*af_planar_quadmesh 58 46 59 69 68
67 # | End of List
*select_elements all_existing
*store_elements G22 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 51 37 39 53 52 #
| End of List
*select_elements all_existing
*store_elements G16 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 60 71 72 70 69 #
| End of List
*select_elements all_existing
*store_elements G23 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 53 40 42 48 56
55 54 # | End of List
*select_elements all_existing
*store_elements G17 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 66 68 70 73 74 #
| End of List
*select_elements all_existing
*store_elements G24 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 45 49 61 60 59 #
| End of List
*select_elements all_existing
*store_elements G18 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 73 72 75 76 # |
End of List
*select_elements all_existing
*store_elements G25 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 62 52 54 64 63 #
| End of List
*select_elements all_existing
*store_elements G19 all_selected
*select_clear *select_elements
all_existing

*af_planar_quadmesh 1 5 15 16 81 78
80 # | End of List
*af_planar_quadmesh 80 2 19 36 51 62
84 86 82 77 # | End of List
*af_planar_quadmesh 81 79 83 88 85
75 71 61 50 35 17 # | End of List
*af_planar_quadmesh 84 63 65 74 76
85 87 # | End of List
*select_mode_invert
*select_elements all_existing
*store_elements rim all_selected
*select_clear *identify_sets *regen

*af_planar_quadmesh 64 55 57 67 66
65 # | End of List
*select_elements all_existing
*store_elements G20 all_selected
*select_clear *select_elements
all_existing
*af_planar_quadmesh 56 47 58 57 # |
End of List
*select_elements all_existing
*store_elements G21 all_selected
*select_clear *select_elements
all_existing

	
  

*sweep_all *remove_unused_nodes
*remove_unused_points
*renumber_all *check_distorted
*expand_reset

164	
  

*set_expand_translation z 10
*set_expand_repetitions 10
*expand_elements all_existing
*select_reset *select_method_box
*select_elements

*store_nodes left_nodes all_selected
*select_reset *select_clear
*select_method_box *select_nodes
-1
550
-1
550
-1
1
*store_nodes back_nodes all_selected
*select_clear
*select_method_box *select_faces
469.5 470.5
-1 530
-1 110
*store_faces right_faces all_selected
*select_clear
*select_method_box *select_nodes
-1
1
-1
11
-1
21
*store_nodes fixed_nodes all_selected

0 470
|x region
0 530
|y region
50 100
|z region
*store_elements backside all_selected
*stop_procedure *set_nodes off
*elements_solid *redraw
*stop_procedure
|++++++++++++++++++++++++++++++
| parameters definition for loading
|++++++++++++++++++++++++++++++
*define max_disp .25 *define max_time
150 *define n_steps 150 *define El_time
20
*define Pl_time 80 *define P2_time 150
*define El_load -250 *define Pl_load 300 *define P2_load -330

|++++++++++++++++++++++++++++++
| boundary conditions
|++++++++++++++++++++++++++++++
*apply_type face_load *apply_dof p
*apply_dof_value p *apply_dof_table p
load_time
*select_clear *select_faces *select_sets
right_faces
*add_apply_faces all_selected
*stop_procedure

*new_md_table 1 1 *set_md_table_type
1 time
*table_name load_time
*table_add 0 0
El_time El_load
|elastic loading, a point (x,y) in time-load
figure
Pl_time Pl_load
|plastic loading
P2_time P2_load

*new_apply
*apply_type fixed_displacement
*apply_dof x *apply_dof_value x
*add_apply_nodes left_nodes # |end list
*new_apply
*apply_type fixed_displacement
*apply_dof z *apply_dof_value z
*add_apply_nodes back_nodes # |end
list
*new_apply
*apply_dof x *apply_dof_value x
*apply_dof y *apply_dof_value y
*apply_dof z *apply_dof_value z
*add_apply_nodes fixed_nodes # |end
list

*table_fit *stop_procedure
*select_reset
|
define lower nodes, upper faces, back
nodes and fixed nodes
*select_clear *select_method_box
*select_nodes
-.5 .5
-1
550
|1<y<550
-1
110
|1<z<110

	
  

165	
  

*stop_procedure

*icond_dof_value var 4
*select_elements
*select_sets G4
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G1
|element
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 1
|element sequence
*select_elements
*select_sets G1
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G5
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 5
*select_elements
*select_sets G5
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G2
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 2
*select_elements
*select_sets G2
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G6
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 6
*select_elements
*select_sets G6
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G3
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 3
*select_elements
*select_sets G3
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G7
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 7
*select_elements
*select_sets G7
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G4
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2

	
  

*new_icond
*icond_type state_variable
*icond_name ic_G8
*icond_type state_variable

166	
  

*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 8
*select_elements
*select_sets G8
*add_icond_elements all_selected
*select_clear

*icond_name ic_G12
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 12
*select_elements
*select_sets G12
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G9
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 9
*select_elements
*select_sets G9
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G13
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 13
*select_elements
*select_sets G13
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G10
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 10
*select_elements
*select_sets G10
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G14
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 14
*select_elements
*select_sets G14
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G11
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 11
*select_elements
*select_sets G11
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G15
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 15
*select_elements
*select_sets G15
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable

	
  

167	
  

*new_icond
*icond_type state_variable
*icond_name ic_G16
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 16
*select_elements
*select_sets G16
*add_icond_elements all_selected
*select_clear

*select_clear
*new_icond
*icond_type state_variable
*icond_name ic_G20
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 20
*select_elements
*select_sets G20
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G17
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 17
*select_elements
*select_sets G17
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G21
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 21
*select_elements
*select_sets G21
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G18
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 18
*select_elements
*select_sets G18
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G22
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 22
*select_elements
*select_sets G22
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G19
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 19
*select_elements
*select_sets G19
*add_icond_elements all_selected

	
  

*new_icond
*icond_type state_variable
*icond_name ic_G23
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 23
*select_elements

168	
  

*select_sets G23
*add_icond_elements all_selected
*select_clear

*icond_dof_value var 27
*select_elements
*select_sets backside
*add_icond_elements all_selected
*select_clear

*new_icond
*icond_type state_variable
*icond_name ic_G24
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 24
*select_elements
*select_sets G24
*add_icond_elements all_selected
*select_clear

|++++++++++++++++++++++++++++++
| MATERIAL DATA
|++++++++++++++++++++++++++++++
*material_type mechanical:hypoelastic
*material_option
hypoelastic:method:hypela2
*material_option
hypoelastic:pass:def_rot
*add_material_elements
all_existing

*new_icond
*icond_type state_variable
*icond_name ic_G25
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 25
*select_elements
*select_sets G25
*add_icond_elements all_selected
*select_clear

|++++++++++++++++++++++++++++++
| GEOMETRIC PROPERTIES
|++++++++++++++++++++++++++++++
mech_three_solid
*geometry_option cdilatation:on
*geometry_option assumedstrn:off
*geometry_option ctemperature:on
*geometry_option red_integ_capacity:off
*add_geometry_elements
all_existing

*new_icond
*icond_type state_variable
*icond_name ic_rim
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2
*icond_dof_value var 26
*select_elements
*select_sets rim
*add_icond_elements all_selected
*select_clear

|++++++++++++++++++++++++++++++
| LOADCASES DEFINITION
|++++++++++++++++++++++++++++++
*loadcase_name tension
*loadcase_ctable
*loadcase_type static
*loadcase_value time max_time
*loadcase_value nsteps n_steps
*loadcase_value maxrec 20
*loadcase_value ntime_cuts | Number of
timestep cutbacks 30

*new_icond
*icond_type state_variable
*icond_name ic_backside
*icond_type state_variable
*icond_dof var *icond_dof_value var
*icond_param_value state_var_id 2

|++++++++++++++++++++++++++++++
| JOB DEFINITION
|++++++++++++++++++++++++++++++
*sweep_all *renumber_all
*surfaces_wireframe *regen *job_class
mechanical

	
  

169	
  

*add_job_loadcases tension
*add_job_iconds icond_mpie

*edit_post_var user8 gm B2
*add_post_var user9
*edit_post_var user9 gm B3
*add_post_var user10
*edit_post_var user10 gm P1
*add_post_var user11
*edit_post_var user11 gm P2
*add_post_var user12
*edit_post_var user12 gm P3
*add_post_var user13
*edit_post_var user13 gm Py1
*add_post_var user14
*edit_post_var user14 gm Py2
*add_post_var user15
*edit_post_var user15 gm Py3
*add_post_var user16
*edit_post_var user16 gm Py4
*add_post_var user17
*edit_post_var user17 gm Py5
*add_post_var user18
*edit_post_var user18 gm Py6
*add_post_var user19
*edit_post_var user19 gm Py7
*add_post_var user20
*edit_post_var user20 gm Py8
*add_post_var user21
*edit_post_var user21 gm Py9
*add_post_var user22
*edit_post_var user22 gm Py10
*add_post_var user23
*edit_post_var user23 gm Py11
*add_post_var user24
*edit_post_var user24 gm Py12

*job_option large:on
*job_option plasticity:l_strn_mn_add
*job_option cdilatation:on
*job_option update:on
*job_option finite:on
*job_option post ascii/binary | Write
Result File as formatted ASCII or binary
*job_param post 1 | write each ith
increment to *.t16 (binary) or *.t19 (ascii)
*add_post_var von_mises
*add_post_var eel_strain
*add_post_var tepl_strain |doesn't work
with genmat_cz.f
*add_post_var thickness
*add_post_var eq/yl_stress
*add_post_var pe_energy
*add_post_var pl_strain
*add_post_var stress_1
*add_post_var stress_2
*add_post_var stress_3
*add_post_var stress_4
*add_post_var stress_5
*add_post_var stress_6
*add_post_var user1
*edit_post_var user1 Approx. element
thickness
*add_post_var user2
*edit_post_var user2 phi1
*add_post_var user3
*edit_post_var user3 PHI
*add_post_var user4
*edit_post_var user4 phi2
*add_post_var user5
*edit_post_var user5 Misorientation
angle
*add_post_var user6
*edit_post_var user6 Accumulated slip
*add_post_var user7
*edit_post_var user7 gm B1
*add_post_var user8

	
  

*add_post_tensor strain
*add_post_tensor stress
*job_option solver_nonsym:on
*job_usersub_file mpie_marc_wly.f
*job_option user_source:compile_save
*job_option user_source:run_saved
*sweep_all
*renumber_all
*save_as_model 3-1 large patch with
backside.mfd yes

170	
  

APPENDIX C
Material File for Building 3D Meshes for CPFE Modeling Shown in Figure 3.11(c).

27
1
18
Data for G1
40, 40, 18, 1, 0
1, 117, 68.177, 335.31, 0.0, 0.05, 1
Data for G2
40, 40, 18, 1, 0
1, 150, 76.851, 345.12, 0.0, 0.05, 1
Data for G3
40, 40, 18, 1, 0
1, 63, 63.331, 31.25, 0.0, 0.05, 1
Data for G4
40, 40, 18, 1, 0
1, 320, 58.148, 160.62, 0.0, 0.05, 1
Data for G5
40, 40, 18, 1, 0
1, 114.478, 68.177, 335.31, 0.0, 0.05, 1
Data for G6
40, 40, 18, 1, 0
1, 120, 76.851, 345.12, 0.0, 0.05, 1
Data for G7
40, 40, 18, 1, 0
1, 63, 63.331, 31.25, 0.0, 0.05, 1
Data for G8
40, 40, 18, 1, 0
1, 304.988, 87.891, 146.90, 0.0, 0.05, 1
Data for G9
40, 40, 18, 1, 0
1, 7.760, 82.334, 98.33, 0.0, 0.05, 1
Data for G10
40, 40, 18, 1, 0
1, 355.113, 57.524, 111.67, 0.0, 0.05, 1
Data for G11
40, 40, 18, 1, 0
1, 180.553, 80.885, 275.94, 0.0, 0.05, 1
Data for G12
40, 40, 18, 1, 0
1, 260.318, 67.429, 196.78, 0.0, 0.05, 1
Data for G13

	
  

171	
  

40, 40, 18, 1, 0
1, 141.362, 40.626, 334.09, 0.0, 0.05, 1
Data for G14
40, 40, 18, 1, 0
1, 351.434, 87.058, 97.08, 0.0, 0.05, 1
Data for G15
40, 40, 18, 1, 0
1, 302.553, 77.612, 171.44, 0.0, 0.05, 1
Data for G16
40, 40, 18, 1, 0
1, 315.635, 82.277, 124.67, 0.0, 0.05, 1
Data for G17
40, 40, 18, 1, 0
1, 68.876, 87.371, 16.24, 0.0, 0.05, 1
Data for G18
40, 40, 18, 1, 0
1, 319.311, 60.032, 107.50, 0.0, 0.05, 1
Data for G19
40, 40, 18, 1, 0
1, 347.423, 64.738, 107.55, 0.0, 0.05, 1
Data for G20
40, 40, 18, 1, 0
1, 206.077, 82.379, 236.40, 0.0, 0.05, 1
Data for G21
40, 40, 18, 1, 0
1, 100.739, 86.154, 346.53, 0.0, 0.05, 1
Data for G22
40, 40, 18, 1, 0
1, 258.0, 114.3, 326.8, 0.0, 0.05, 1
Data for G23
40, 40, 18, 1, 0
1, 158.002, 62.846, 286.39, 0.0, 0.05, 1
Data for G24
40, 40, 18, 1, 0
1, 206.077, 82.379, 236.40, 0.0, 0.05, 1
Data for G25
40, 40, 18, 1, 0
1, 74.150, 80.321, 359.88, 0.0, 0.05, 1
Data for rim
40, 40, 18, 1, 0
1, -45, 90, 0, 0.0, 0.05, 1
Data for backside
40, 40, 18, 1, 0
1, -45, 90, 0, 0.0, 0.05, 1

	
  

172	
  

REFERENCES

	
  

173	
  

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