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X-ray microbeam diffraction measurements of long range internal stresses in equal channel angular pressed aluminum; & Mechanical behavior of an Fe-based bulk metallic glass
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X-ray microbeam diffraction measurements of long range internal stresses in equal channel angular pressed aluminum; & Mechanical behavior of an Fe-based bulk metallic glass
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Content
X-RAY MICROBEAM DIFFRACTION MEASUREMENTS OF LONG RANGE
INTERNAL STRESSES IN EQUAL CHANNEL ANGULAR PRESSED
ALUMINUM
&
MECHANICAL BEHAVIOR OF AN FE-BASED BULK METALLIC GLASS
by
Thien Q. Phan
__________________________________________
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2015
i
To my parents: Hung Phan and Ngoc Nguyen
I could not have done this without you.
Thank you for your love and support throughout all these years.
ii
Acknowledgments
I would like to express my deepest gratitude toward my PhD advisors, Professor Michael
Kassner and Dr. Lyle Levine. Professor Kassner has provided me with all the support that I
needed may it be academic, financial, or emotional. He is one of the most honest people that I've
met and his integrity is beyond reproach. Over the course of my PhD, he has continued to push
me to be a better researcher and a better person. I am very fortunate to have learned from him
and had him for an advisor. Professor Kassner also introduced me to Dr. Lyle Levine at NIST for
the collaboration which made this research possible. Dr. Levine is one of the most
knowledgeable people I've ever met. I am grateful for the countless discussions I've had with Dr.
Levine regarding research as well as personal curiosities. Dr. Levine is also one of the most
careful researchers that I've met. I've learned a great deal regarding error analysis from Dr.
Levine. I am extremely fortunate to have the opportunity to continue working as a Post-doc
under Dr. Levine after my PhD. I could not have imagined having better advisors for my PhD
study.
I would like to specifically thank Professor Andrea Hodge. She has supported me even
before the start of my PhD at USC, by allowing me to work in her lab when I was pursuing my
master's degree. Furthermore, she found and recommended me a job during the time between my
master and my PhD. When it was time for me to apply to the PhD program, she introduced me to
my advisor, Professor Kassner. Furthermore, she continued to support me financially by offering
me research assistantships and fellowships during my PhD. Without Professor Andrea Hodge,
this would not have been possible.
I would like to thank my qualifying and defense committee members: Professor
Veronica Eliasson, Professor Edward Goo, Professor Andrea Hodge, Professor Steven Nutt, and
the late Dr. Oliver Franke for their encouragement and feedback. Their insightful comments and
suggestions have helped me tremendously in defining better directions for research and
improving the quality of my work.
On top of the support by advisors and faculty members, I am thankful for the support
from friends and colleagues at USC, specifically, members of the Hodge Research Lab and
Kassner Research Lab. They have always been there for me during both good and hard times.
They have never hesitated to help me throughout my PhD. I thank them for their help with the
materials characterization equipments and the many stimulating discussions. I am grateful to
have such good friends in my life.
Beside friends and faculty members, I would also like to thank Kimberly Needham for
her love and support. She has given me the much needed happiness and companionship that
helped make the PhD process much more enjoyable. The adventures that we have embarked on
and the fun that resulted have created a much needed work-life balance. She was extremely
iii
helpful during difficult times, such as during the periods before the screening and qualifying
exams, and the thesis defense. My journey through the PhD process would have been much more
difficult without her. I am grateful to have her be a part of my life.
And finally, I would like to thank my parents, Hung Phan, and Ngoc Nguyen. None of
this would be have been possible without them. Fifteen years ago, they left behind everything
they had and everyone they knew in Vietnam and moved to the United States in hope of a better
future for me. And since moving to the US, they have always encouraged me to pursue my love
of learning and higher education, and have given me every opportunity to do so. Because of
them, I was able to attend UC Berkeley for my undergrad and USC for graduate school. Their
sacrifices have allowed me to be where I am today. I know that no matter what happens, I will
always be able to count on them for love and support. I would like to thank them for supporting
me emotionally and spiritually throughout the writing of this thesis and in life in general. For
everything they have done for me, I am extremely grateful.
iv
Table of Contents
Acknowledgements ii
Chapter 1. Introduction 1
1.1. Motivation 1
1.2. Background 2
1.2.1. Composite Model 2
1.2.2. Bauschinger Effect 6
1.3. Differential Aperture X-Ray Microscopy 8
1.3.1. X-Ray Diffraction 8
1.3.2. Differential Aperture X-Ray Microscopy (DAXM) 10
1.4. Evidence of Long Range Internal Stresses 16
1.4.1. X-Ray Line Profile Asymmetry 16
1.4.2. CBED Measurements 18
1.4.3. Past Microbeam Measurements 19
1.5. Equal Channel Angular Pressing 25
1.5.1. Equal Channel Angular Pressing Processing 25
1.5.2. ECAP Microstructures and Properties 28
Chapter 2. Single Reflection (Direction) X-Ray Microbeam Measurements 34
of Long Range Internal Stresses
2.1. ECAP AA1050 34
2.1.1. Microbeam Sample Preparations 36
2.1.2. Powder Diffraction 37
2.1.3. Microbeam Measurement Set Up 39
2.1.4. Results 40
2.1.4.1. Grain/Subgrain Interior Measurements 40
for ECAP AA1050 Single Pass
2.1.4.2. Grain/Subgrain Interior Measurements 42
for ECAP AA1050 Multiple Passes
2.1.5. Discussion 45
2.1.5.1. Grain/Subgrain Microstructure in ECAP AA1050 45
2.1.5.2. LRIS in Grain/Subgrain Interiors 46
2.1.5.2.1. ECAP AA1050 Single Pass 46
2.1.5.2.2. ECAP AA1050 Multiple Passes 48
2.1.5.3. LRIS in Grain/Subgrain Boundaries or Cell Walls 49
2.1.5.3.1 ECAP AA1050 Single Pass 49
2.1.5.3.2 ECAP AA1050 Multiple Passes 51
2.2. ECAP AA6005 52
2.2.1. Microbeam Sample Preparation 53
2.2.2. ECAP AA6005 Powder Diffraction 53
2.2.3. Microbeam Measurement Set Up 54
v
2.2.4. Results 55
2.2.4.1 Grain/Subgrain Interior Measurements for ECAP AA6005 55
2.2.5. Discussion 56
2.2.5.1 LRIS in Grain/Subgrain Interiors of ECAP AA6005 56
2.3. Transmission Electron Microscopy 59
2.3.1. Background 59
2.3.2. Sample Preparations 59
2.3.3. Results 63
Chapter 3. Full Elastic Strain/Stress Tensor X-ray Microbeam Measurements 66
3.1. ECAP AA1050 66
3.1.1. Background 66
3.1.2. Microbeam Sample Preparations 67
3.1.3. Powder Diffraction 67
3.1.4. Microbeam X-Ray Diffraction Measurements 68
3.1.5. Full Elastic Strain/Stress Tensor Calculations 71
3.1.6. Transformation of Strain and Stress Tensors 73
3.1.7. Principal Strains and Stresses 76
3.1.8. Uncertainty Analysis 76
3.1.9. Results and Discussion 77
3.1.10. Conclusions 95
3.2. Single Crystal Cu 99
3.2.1. Algorithm for Full Elastic Strain/Stress Tensor Measurements 99
and Calculations
Chapter 4. Fracture Toughness and Flexural Modulus Measurements 104
of an Fe-Based Bulk Metallic Glass (SAM2X5)
4.1. Introduction 104
4.2. Experimental Procedures 107
4.3. Vickers and Nano-Indentation 109
4.4. Three-point Bending Measurements 110
4.5. Results 111
4.6. Discussion 117
4.7. Conclusions 120
References 121
Appendix A. Experimental Data 125
A1. ECAP AA1050 1P Matlab Outputs 125
A2. ECAP AA1050 2P Matlab Outputs 137
A3. ECAP AA1050 8P Matlab Outputs 153
Appendix B. Matlab Code 167
vi
B1. Matlab Code for Full Elastic Strain and Stress Tensors 167
for ECAP Al Calculations
B2. Reciprocal/Directional Vector Calculations 180
B3. Matrix Rotation Using Rodriguez Formula 181
B4. Matlab Code for Full Elastic Strain and Stress Tensors 183
for Single Crystal Cu
1
Chapter 1. Introduction
1.1 Motivation
As a crystalline material is deformed, dislocation formation leads to heterogeneity within
the bulk material. This heterogeneity creates microstructures which are made up of high and low
dislocation density areas. The range of microstructure varies greatly with the type of deformation
and the type of slip system present in the material. If the sample is cyclically deformed,
dislocation vein dipole bundles and persistent slip bands (PSBs) are formed. However, for a
multiple slip system, the dislocation heterogeneity exists in forms of three dimensional
dislocation cells made up of dislocation rich areas (cell walls), and dislocation poor areas (cell
interiors) [1].
It is understood that dislocations produce local and short range internal stresses, such as
with individual dislocations. What has been debated however, is the existence of long range
internal stresses (LRIS) in deformed crystalline materials, as well as the magnitude of these
stresses. Understanding LRIS is of great importance for understanding phenomenon such as the
Bauschinger effect, cyclic deformation, metal spring back, and most importantly, the basis of
plastic deformation. Previous attempts have been made in order to measure the LRIS using X-
Ray line profiling, dislocation pinning, convergent beam electron diffraction (CBED), and most
recently microbeam X-Ray diffraction.
Recent efforts in X-Ray microbeam measurements of internal elastic strains within
deformed single crystal Cu have been significant in the ability to non-destructively measure and
validate the existence of LRIS in dislocation cell interiors of monotonically deformed single
crystal copper [2-4]. Furthermore, elastic strains within cell walls have been calculated from
2
dislocation volume measurements and selective masking of diffraction data [2]. The results from
these experiments show internal elastic strains of opposite signs in the cell walls compared to the
cell interiors, thus validating the Composite Model proposed by Mughrabi [1]. However, these
LRIS are generally less than other investigations, which is very important.
However, there has been a lack of basic research toward the plastic deformation and
existence of LRIS within severely plastic deformed (SPD) metals. One such process is equal
channel angular pressing (ECAP). Due to the grain refinement produced by SPD processing,
dislocation movement and evolution in ECAP materials create high and low angle grain
boundaries. These boundaries are thought to be nonequilibrium with emanating extrinsic
dislocations. LRIS maybe higher in ECAP samples. The processing of materials via ECAP
ideally deforms metal via simple shear. The theoretical complete strain tensor for each pass can
be modeled. Here, we use X-Ray microbeam from a synchrotron to measure the internal elastic
strains of aluminum alloys after various ECAP passes. Additionally, the full elastic strain/stress
tensors are measured for ECAP Aluminum alloy 1050 of various passes. Aluminum alloy 1050
was used instead of pure Al due to the impurities in aluminum 1050 suppressing static and
dynamic recrystallization and grain growth.
1.2 Background
1.2.1 Composite Model
The Composite Model refers to Mughrabi’s [1, 5, 6] physical model to explain long range
internal stresses. When materials are deformed, “hard” and “soft” areas evolve from the
movement of dislocations. Areas which have a high density of dislocations are “hard” areas and
areas which have a low density of dislocations are “soft” areas. In a material such as deformed
3
single crystal copper, dislocation cell structures form; the dislocation rich areas are called cell
walls and dislocation poor areas are called cell interiors. In the Composite Model, these hard and
soft areas are compatibly sheared [1, 7]. From the heterogeneity of the dislocation distribution,
each component yields at a different flow stress depending on dislocation density and
orientation. Figure 1 shows the Composite Model with dislocation rich (cell walls) and
dislocation poor areas (cell interiors).
Figure 1. Composite Model, with hard (cell walls) dislocation rich areas, and soft (cell interiors)
dislocation poor areas. Interfacial dislocations allow elastic compatibility and form dipole pairs
across the wall
The Composite Model was suggested to explain the Bauschinger effect. First, assume that
the composite substructures exist within the sample. Next, consider the applied stress required to
deform the sample; as the applied stress increases and reaches the flow stress of the softer
regions, the cell interior starts to deform plastically (elastic-perfectly plastic is assumed). At this
point, the stresses within the cell walls are still within the elastic regime (Figure 2). Once the
4
applied stress exceeds that of the cell walls, the whole sample deforms plastically. This leads to
two regions of plastic yielding: micro yielding, where cell interiors yield plastically, and macro
yielding, where the bulk sample as a whole yields plastically[1].
Figure 2. a) Yielding behavior described by a composite of the stress and strain curves for
dislocation rich and poor areas. b) Upon unloading, back stresses within cell interiors cause the
bulk to yield at a lower stress [1, 8]
Equation 1 describes the applied stress as a function of stresses within cell walls and cell
interiors:
(1)
5
where
is the applied stress,
is the volume fraction of cell walls, and
is the
volume fraction of the cell interiors. Long range internal stresses (LRIS) can then be described as
the deviation from the applied stress as described by Equation 2
(2a)
(2b)
where
is the applied stress and
and
are the long range internal stresses present
in the cell interior and cell wall respectively.
Referring to Figure 2, as the sample is unloaded from tension, the soft and hard regions
are unloaded in parallel. Since the elastic regime of the cell walls is larger than that of the cell
interiors, as the (tensile) applied stressed is removed, the cell walls impose a compressive stress
on the softer areas, while stresses within cell walls remain positive. When the applied, or
average, stress is zero, the volume fraction weighted sum of the stresses must be zero. Equation 3
describes the equilibrium of stresses within the bulk:
(3)
From the equilibrium equation (Eq 3), stresses within the cell walls and cell interiors will
be opposite in signs. In other words, if the applied stress on the bulk is tensile, once the applied
stress is removed, the long range internal stress within the cell walls will be tensile, and the stress
in the cell interior will be compressive. This is known as back stress [1].
6
1.2.2 Bauschinger Effect
In monotonic deformation of a sample, or the first quarter cycle of cyclic deformation,
the material strain hardens as it is plastically deformed. The Bauschinger effect refers to what
happens upon reversal of the direction of straining: the material yields plastically at a lower
stress magnitude compared to the forward direction, in contrast to what is expected from
isotropic hardening (Figure 3). This type of hardening is often referred to as kinematic
hardening.
Figure 3. Stress vs. strain curve showing the Bauschinger effect
One of the earlier explanations for the Bauschinger Effect is from the pile-up model by
Seeger [9]. In the pile-up model, dislocations pile up due to heterogeneity of the dislocation
substructures (dislocation barriers, cell walls, etc). Dislocation pile-ups are responsible for the
long range internal stresses within the sample [9]. The stress caused by dislocation pile-ups is
described as follows (Equation 4):
7
(4)
Where
is the stress cause by n number of dislocations multiplied by the applied stress.
Since
at the dislocation barriers must be balanced for local equilibrium by -
, LRIS would
then exist. Seeger [9] and Mott [10] suggested that upon load reversal, this LRIS -
would aid
in lowering the yield stress.
Another possible explanation of the Bauschinger Effect has been due to the back stress
present in the cell interiors due to the Composite Model. Recalling the Composite Model
discussed in the previous section, dislocation heterogeneity leads to two distinct substructures of
dislocation rich regions and dislocation poor regions. Due to the different yield strengths, upon
unloading of the tensile applied load, hard regions place soft regions into a compressive state.
This creates the back stress present in cell interiors [1, 5]. As the strain direction reverses, the
LRIS already present in the cell interiors causes the cell interiors to plastically yield at a stress of
(Equation 5):
(5)
This micro yielding regime happens at a lower stress in the reverse direction than the
forward direction. This behavior would correspond to the so-called Bauschinger Effect.
One other possible explanation for the existence of the Bauschinger effect, which does
not involve LRIS, is caused by the quasi-reversible dislocation mechanism [11]. In the case of
monotonic deformation of the sample, in the forward loading direction, dislocations impediment
are caused by obstacles such as other dislocations that were formed. However, as the load is
reversed, dislocations in cell walls are able to easily reverse their directions into the lower
8
dislocation density cell interiors. This also leads to a reduction in dislocation density within the
cell walls, resulting in cell walls softening. Sleeswyk et al. [11] argues that the Bauschinger
Effect can thus be modeled with a “reversible” strain that can be subtracted from the total work
hardening strain, caused by the reversibility of the dislocation movement. Note that this model
used to explain the Bauschinger Effect relies largely on the reversible strain caused by
dislocation movement, thus, it results from the configuration of dislocations, rather than from
dislocation heterogeneity that cause a back stress [8].
1.3. Differential Aperture X-Ray Microscopy
1.3.1. X-Ray Diffraction
In order to understand differential aperture X-ray microscopy (DAXM) we first must
explain X-Ray diffraction. In contrast to transmission electron microscopy (TEM), which
requires sample thinning, X-Ray diffraction measurement of the lattice parameters is a
nondestructive measurement technique. X-Ray diffraction refers to the diffraction of X-Ray
when a beam is incident on a crystal lattice. In this particular case, X-Ray is diffracted off FCC
metals such as Cu and Al. When an X-Ray beam is incident on an atom, elastic scattering occurs,
constructive and destructive interference between diffracted X-Rays within the crystal lattice
lead to the observed diffraction pattern. Figure 4 shows the basis of X-Ray diffraction.
9
Figure 4. Elastic scattering geometry of X-ray diffraction, represents the diffracting angle,
and is the wavelength of the beam
Diffraction occurs according to Bragg's Law (Equation 6):
(6)
where is the wave length , or energy, of the incident beam , d is the lattice spacing of a
particular direction (reflection) [hkl], and is the diffraction angle. According to Bragg's law
(Equation 6), if we know the wavelength, or energy, of the incident beam and the diffracting
angle, the lattice spacings along a certain direction can be calculated. In measuring the lattice
spacings of both the undeformed sample and the deformed sample and comparing the two
values, strains can, thus, be calculated. Stresses can then be calculated based on the measured
strain values. In order to measure the LRIS within the grain/subgrain of the ECAP Al samples,
spatial resolution must be high, while keeping error in measurements low. This is possible due to
recent advancement in X-ray diffraction technique, called differential X-ray microscopy
(DAXM) [12].
10
1.3.2. Differential Aperture X-Ray Microscopy (DAXM)
Recent development in microbeam X-Ray diffraction techniques by Larson et al. [12] has
been crucial in the effort to measure long range internal stresses within deformed metals. DAXM
is able to examine individual 3D volumes of materials deep within the sample (tens to hundreds
of microns from the sample surface). DAXM measures internal elastic strain via accurately
determining diffraction angle of a focused X-Ray beam using triangulation. The set-up consists
of an incident microbeam X-Ray, the sample to be measured, a Platinum (Pt) scanning wire, and
an X-Ray detector.
Figure 5 shows the X-Ray microbeam focusing set up at the APS [12]. X-Ray beam from
the synchrotron is focused via Kirkpatrick - Baez mirrors, which uses two orthogonal parabolic
X-Ray mirrors to focus the beam in the X and Y directions independently to approximately 500
nm x 500 nm in cross section. The usage of the mirrors is of importance due to the nature of the
focusing. In using mirrors, different wavelengths are focused to the same focal point. This cannot
be achieved with lenses since they focus different wavelengths to different focal points. This is
important since DAXM requires using both the white (polychromatic) beam and monochromatic
beam for strain measurements. Monochromatic beam is possible due to the translating
monochromator, which acts as a wavelength (energy) filter, allowing only the specified
wavelength to pass through.
11
Figure 5. X-Ray microbeam set up. X-Ray beam is focused by Kirkpatrick-Baez mirrors to a
beam width of about 500nm in both directions.
As discussed in the previous section, diffraction measurement of elastic strain depends on
using the diffraction angle to calculate lattice spacings. However, since microbeam X-Ray has
the ability to penetrate deeply into the sample (tens to hundreds of microns depending on the
materials), the location, or depth, of the diffracting volume needs to be known in order to have an
accurate angle measurement. Without the wire, only the incident X-Ray beam and the diffraction
spot on the detector are known from machine calibration. However, the exact location of the
diffracting volume within the sample is not known due to the beam penetration. The diffracting
volume can be located anywhere along the penetrating X-Ray beam (from the surface of the
sample down to hundreds of microns below the surface). In order to more accurately determine
the location (depth) of the diffracting volume, and ultimately measure strain/stress, the use of a
triangulation wire is needed. Figure 6 shows the DAXM set up at the APS [4]. A cylindrical
Platinum (Pt) scanning wire is used to scan parallel to the surface of the sample in small discrete
12
steps, blocking diffraction from the sample at each step. Pt is used since it is X-Ray opaque. The
wire circular cross section is selected for the ease in calculating X-Ray absorption from different
angles. It is important to note that the straightness and roundness of the wire is crucial since any
small deviation in the wire would lead to large errors in the perceived depth. Figure 6 shows the
DAXM set up[4].
Figure 6. Differential aperture X-ray microscopy set up. X-ray beam hits the sample and is
diffracted to the sensor. The Pt profiler is used to determine exact position (depth) of the
diffracting volume [4].
Since diffraction spots are from various volumes along the beam at various depths, a wire
scan can be performed and the data collected can be used to reconstruct for depth resolved
diffraction images. In this process, a diffraction image is taken each time the wire is moved
13
across the surface. The complete set of wire scan images are then processed to reconstruct for
diffraction images at particular depths.
Additionally, an energy wire (EW) scan, a 2-dimensional scan, is necessary for the
measurements of LRIS. Cell and grain/subgrain interiors are separated by high dislocation
density areas, which may result in each diffraction volume showing several sub peaks depending
on the grain/subgrain microstructures. A pseudo white beam diffraction image can be
reconstructed from an energy wire scan, where a 2-dimensional scan is performed, scanning over
both energies and wire positions. Diffraction images from each energy are summed to create a
pseudo white (polychromatic) image. Figure 7 is an example of a pseudo white beam image
reconstructed from diffraction data obtained from an ECAP AA1050 sample processed through 2
pass [13].
Figure 7. Pseudo white beam image reconstructed from a large grain scan of an ECAP AA1050 2
pass sample. Peaks are diffraction from low dislocation density areas and smeared diffracted
intensities are caused by high dislocation density areas [13].
14
From the above image, several peaks can be observed. These peaks are caused by
diffraction of low dislocation density areas, which diffract more coherently, as opposed to the
diffused areas. These diffused diffraction areas are caused by high dislocation density areas.
Once the diffraction peaks are imaged, the next step in measuring strains is to process the raw
data (images) to calculate and measure the deformed lattice spacing.
We shall now discuss how lattice spacings are measured from X-Ray diffraction peaks.
The deformed lattice spacing is measured by performing an energy scan. When a volume of the
deformed sample is probed by X-Ray diffraction, a line profile is created from the energy scan
data, and from that, the lattice spacing can be calculated. The line profile is graphed from an
energy scan of a region of interest (ROI) on the detector. Once a region of interest is selected (of
a diffraction peak), an energy scan is performed. This is done by taking images each beam
energy step. An example of an X-Ray line profile is shown in Figure 8, where a graph of
intensity vs. q (the reciprocal lattice spacing) is shown. The X-Ray beam energy is scanned over
a range that would reveal the entire peak from the diffracted volume at step size of 3 eV. The
reciprocal lattice spacing (q) is calculated for each pixel at each energy step. The calculated data
is then grouped into discrete bins of a certain q range and the intensity from each group is added
together, giving a complete graph of intensity vs q [Figure 8].
15
Figure 8. Example of X-Ray line profile from an energy scan
The line profile is then fitted with either a Gaussian or Pseudo-Voigt curve in order to
determine the position of the peak. Lattice spacing d can then be calculated from q. Since X-Ray
diffraction measures lattice spacing in terms of a reciprocal space lattice spacing, q, we must
convert q to the real space lattice spacing, d. The relationship between q-vector and lattice
spacing d is as follows (Equation 7)
(7)
Where is the wavelength and is the Braggs scattering angle. This lattice spacing d
represents the lattice spacing of the coherently diffracting volume with an area the size of the
beam and depth determined by triangulation using the scanning wire.
16
1.4. Evidence of Long Range Internal Stress
1.4.1. X-Ray Line Profile Asymmetry
Some of the supporting evidence for long range internal stresses has come from
experiments showing the broadening and asymmetric nature of the X-ray line profile for
deformed single crystals [6, 14] [Figure 9]. As a crystal is deformed, its X-Ray line profile
broadens and develops an asymmetric shape. The peak of the X-ray line profile also shifts
compared to the undeformed sample line profile. Depending on the direction of strain the sample
underwent, compressive or tensile, the peak shifts in different directions. It is observed that the
higher the resolved shear stress from deformation, the broader and more asymmetrical the line
profile [6].
Figure 9. X-ray line profile of deformed [001] oriented copper single crystal. Note the increased
broadening and asymmetry with increased deformation [1]
17
From the Composite Model proposed by Mughrabi [1], this broadened and asymmetric
line profile can be decomposed into two independent and (assumed) symmetric sub-peaks, each
from a particular microstructure: cell walls, and cell interiors. Cell walls peaks, due to the high
density of dislocations, would be broader, or have a larger full width half maximum (FWHM),
and have a lower peak magnitude. Cell interior peaks, on the other hand, were assumed to have a
smaller FWHM and a higher peak magnitude. Figure 10 shows the X-Ray line profiles for the
deformed single crystal Cu sample as well as the decomposed line profiles from cell walls and
cell interiors [2].
Figure 10. X-Ray line profile for the deformed single crystal Cu sample as well as the
decomposed line profiles for dislocation cell walls and dislocation cell interiors [2]
According to Figure 10, line profiles are measured along the direction of deformation. X-
Ray line profile measurements are sensitive to the direction due to the Poisson ratio. Measuring
18
X-Ray line profiles along directions other than the strain direction would result in changes in the
line profile. The degree of asymmetry, peak broadening, as well as direction of peak shifting
would also change based on the direction of measurement. Furthermore, since different materials
have different microstructures (cell walls volume percent, grain/subgrain microstructures), it is
expected that the X-Ray line profiles will change based on the material even if they are deformed
at the same strain.
1.4.2. CBED Measurements
Previous researchers have tried to measure LRIS within single crystal copper through
dipole separations and convergent beam electron diffraction (CBED). CBED used a probe size of
80nm to assess the LRIS in unloaded monotonically and cyclically deformed Cu and creep
deformed Al and Cu [15]. CBED measurements were made near the cell/subgrain walls and
subgrain interiors of copper and aluminum samples deformed by creep testing. After creep
deformation, samples were thinned into TEM foils. Measurements of lattice parameters were
made at various locations within the channels and near the dislocation bundles in attempt to
measure the LRIS. Measured stress values remain close to zero and thus show no evidence to
support the existence of LRIS. However, these measurements had an uncertainty value of 0.8 σ
a
(applied stress) for the Cu experiment and 1.1 σ
a
for the Al experiment. Furthermore, sample
preparations of TEM foils might have relaxed internal stresses within the thin foils. Since the
errors in measurements are roughly the same order of magnitude as the applied stress, any
internal stress at the same order of magnitude at or less than the applied stress would not have
been resolved. Therefore, it appears that CBED is not the appropriate approach to measure stress
distribution within the cell or grain/subgrain, nor near the boundaries.
19
1.4.3 Past Microbeam Measurements
DAXM measurements of deformed single crystal Cu have been performed by Levine et
al. [4]. Axial [006] reflections were used to probe the elastic strain within dislocation cells along
the <001> deformation direction. The X-Ray microbeam was focused to about 0.5 μm diameter.
Copper samples were deformed in both tension and compression to a true strain of ~30%, with a
flow stress of ~200MPa. Depth resolution is provided by a 50 micron diameter Pt scanning wire.
Diffraction volumes were measured at depths up to 50 microns below the surface at an X-Ray
energy of ~14keV. X-Ray line profiles were measured using the XOR/UNI beamline 33BM at
the APS. The zero position of the scattering vector q was determined using the X-Ray line
profile measured from an undeformed sample. Figure 11 below shows the X-Ray line profiles
from the compression and tensile samples.
20
Figure 11 a) X-Ray line profile from <001> compression sample. b) X-ray line profile from
<001> tension sample. The red lines represent X-ray line profiles and vertical blue lines
represent diffraction peak from dislocation cell interiors measured by X-ray microbeam[4].
Note that both of these profiles exhibit the well documented asymmetry that is associated
with deformed copper. Axial lattice spacing strains measured from individual cell interiors via
DAXM are shown by vertical blue lines. A notable observation is that the results of the
compression sample show all cell interior lattice spacing measurements shifting to a smaller q, or
larger d. Correspondingly, in contrast to the results of the compressive sample, the results of the
tensile sample show lattice spacing measurements shifting in the other direction, larger q and
smaller d. The opposite sign of the stress in cell interiors compared to the applied stress is direct
evidence of the back stresses in dislocation cell interiors. These results agree with the predicted
results from the composite model by Mughrabi. LRIS within the cell interiors of compressive
deformed Cu were tensile in nature and the LRIS of cell interiors of tensile deformed Cu were
21
compressive in nature. The measured mean elastic strain for compression and tensile samples
were 8.5 +/- 1.1 x 10
-4
and -5.2 +/- 0.9 x10
-4
, respectively. These measurements were the first
spatially resolved elastic strain measurements that provide the first direct and quantitative test of
the composite model suggested by Mughrabi.
A key finding from Levine et al. [4] however, is the large variation of the measured
strains (and thus stresses) from cell to cell within the sample. Using the elastic modulus along the
<001> direction E = 66.6 +/- 0.5 GPa, stress values range from essentially zero up to 0.5 of the
applied stress. This variation in strains contributes significantly to the shape and the width of the
diffraction line profile. Previous sub-peaks decomposition analysis did not take this strain
variation into account. The assumption was that both line broadening and asymmetry are due to
dislocations alone and that the cell to cell variations are negligible. Similarly, variations in cell
walls strains are expected to be just as large.
Results by Levine et al. in 2011 [3] further examine the LRIS distribution in deformed
copper. DAXM was used to measure axial elastic strain (and thus stresses) in dislocation cell
interiors and cell walls in Cu samples deformed to a true strain of 28 +/- 0.1% and a true flow
stress of 210 MPa. The sample was once again <001> oriented 99.999% Cu single crystal. In this
experiment, Levine et al. set out to confirm the earlier work that i) average cell walls stress and
average cell interiors stress are opposite in sign, and ii) to determine the statistical distributions
of these stresses. Beam sizes were 330nm x 460nm and 460nm x 600nm for their two studies.
From the collected intensities, subsets of pixels from a single depth were associated with
microstructures within the sample. Sharp peaks are associated with low dislocation density areas
such as cell interiors and diffused scattering areas are associated with high dislocation density
areas such as cell walls. By summing contributions from the pixels of each subset, separate line
22
profiles for each microstructure can be produced. Strains and stresses from cell interiors and cell
walls can be measured even though cell walls are typically smaller than the spatial resolution of
the equipment. To show that the volume probed by microbeam diffraction provided an adequate
sampling of the microstructure, integrated microbeam line profiles were compared to
conventional volume average line profile measurements (Figure 12).
Figure 12. X-Ray line profiles from macroscopic beam, integrated microbeam and their
resolution function
The microbeam curves are obtained by integrating all of the depth resolved data for a
particular sample position. These curves compared very well with the line profile produced from
large volume X-Ray diffraction. All profiles exhibit the well known broadened and asymmetric
peak shape. Results from cell interior and cell wall strains are below.
23
Figure 13. Distributions of peak widths as a function of mean elastic strain for all measured cell
walls and interiors show two distinct populations[3].
From Figure 13, cell wall and cell interior strains exhibit a large variation in elastic strain,
corresponding to a stress range of about 100 MPa. Together, stress variations in cell interiors and
cell walls add up to about 200MPa, comparable to the 210MPa applied stress. Figure 14 shows
the result of the volume-normalized stress distribution function,
24
Figure 14. Measured distributions for cell interiors (red) and cell walls (blue). The mean cell
interior and cell wall stresses are separated by about 40 MPa and the average sample stress of
4MPa, within measurement uncertainty of zero [3].
Microbeam measurements show stresses in cell walls and cell interiors are roughly equal
in magnitude and opposite in sign. The cell walls exhibit significant compressive stress
compared to cell interiors. The magnitude of the difference is about 40MPa (or internal stress ~
±0.10 σ
a
), or one fifth of the applied stress. Since the sample is unloaded, force balancing
requires the total volume integral of stress to be zero. From TEM images, volume fraction of
each component was estimated. The walls were found to make up 55 +/- 10% of the sample
volume. From this estimate, the integrated stress gave a value of 4MPa, which is within the
measurement uncertainty of zero.
25
1.5. Equal Channel Angular Pressing
1.5.1 Equal Channel Angular Pressing Processing
Equal Channel Angular Pressing (ECAP) is a common type of severe plastic deformation
method used for grain refinement of relatively small billets of materials [16-18]. In a
polycrystalline material, yield stress varies with grain sizes according to the Hall-Petch
relationship, which is given by the following (Equation 8):
(8)
where
is the yield strength,
is the lattice friction stress,
is a constant of yielding,
and d is the grain size. According to Equation 8, smaller grain sizes would thus lead to an
increase in yield strength. Furthermore, at high temperatures, smaller grain sizes can lead to
superplasticity [19, 20]. Experiments have shown that ECAP can produce ultrafine grains ~200
nm (depends on the definition of grain boundaries in terms of misorientation angle) [21-24].
Furthermore, ECAP produces homogeneous grain sizes and uniform mechanical properties
throughout the billet. Figure 15 shows a schematic used to describe the ECAP process [25].
26
Figure 15. Principle of processing by ECAP
ECAP uses a die with a channel that is bent through an angle near the center of the die.
The sample is then machined to fit the channel and a plunger is used to press the sample through
the die. The channel is described by two angles: Φ is used to describe the angle between the two
channels, and the second angle Ψ is used to describe the angle of the outer arc of curvature where
the two channels meet. The ECAP processing introduces severe plastic deformation through
simple shear. For simple shear, the engineering shear strain as described by Xia et al. [26] in
Equation 9:
(9)
where is the engineering shear strain, and is the angular distortion caused by shear.
For the ideal situation, total shear strain in ECAP is described by Equation 10 [26]:
(10)
where is the angle between the two ECAP channels. The total equivalent strain after
one pass of ECAP is described by Equation 11:
27
11)
Plastic strain is introduced into a sample through shear deformation, leading to grain
elongation and texture formation after a single pass. Assuming zero strain in the x-axis (pressing)
direction of Figure 15, the shear deformation takes place under plane-strain conditions in the x–z
plane. Thus, the shape change under deformation in the x–z plane (the plane of two pressing
directions) can be simplified to the shearing of a circle. Figure 16 illustrates that a circle deforms
into an ellipse in the ECAP die with Φ = 90°.
Figure 16 A circle is deformed into an ellipse as it passes through the ECAP die. The long axis
(+22.5°) of the ellipse indicates the direction of maximum tensile plastic-strain and the short axis
indicates the direction of maximum compressive plastic-strain (−67.5°). Strain along the pressing
axis (0°) is approximately zero [26].
Figure 16 is based on the work of Xia and Wang [26]. The principle strains can be
obtained by comparing the radius of the circle and the long and short axes of the ellipse. Based
on a geometric analysis for the ECAP die (Φ = 90°), the maximum tensile plastic -strain, ε
True
=
0.88, will be oriented +22.5° from the x axis in the x–z plane and the maximum compressive
plastic-strain, ε
True
= −0.88, will be rotated −67.5° from the x axis within the same plane.
28
Since cross sectional area remains constant with ECAP, the process can be easily
repeated with various amount of passes and change in orientation (rotation) of the sample
between passes [16]. Several processing routes are used. Route A describes the process without
sample rotations in between passes, route B
A
prescribes a sample rotation of 90 degrees in
alternate directions between passes, route B
C
prescribes a sample rotation of 90 degrees in the
same direction between passes, and route C prescribes a 180 degrees rotation between passes
[27, 28]. Experiments have shown that the optimum processing route to produce equiaxed
ultrafine grains separated by high angle grain boundaries for FCC materials (such as aluminum)
is route B
C
[29].
1.5.2. ECAP Microstructures and Properties
Microstructural evolution in ECAP FCC materials such as Aluminum is well
documented. The following is an Orientation Imaging Microscopy (OIM) image of high purity
ECAP Aluminum from 1 up to 12 passes from Kawasaki et al. [30] Figure 17.
29
Figure 17. Microstructural evolution of high purity Al processed via ECAP [30]
The non-ECAP sample has grain sizes of roughly 1mm (Figure 17a) and was processed
through an ECAP die of geometry = 90 and = 20 degrees via route B
C
. The colors within
the grains show the orientations of each grain. Black grain boundaries have misorientation angles
between 2
o
and 15
o
and red grain boundaries have misorientation angles larger than 15
o
and are
30
considered high angle grain (HAB) boundaries. For the case of high purity aluminum, subgrains
become elongated and are separated by low angle boundaries with an average width of 2.9 μm
after the first pass. Grains are further refined to 1.8 μm after 2 passes. Equiaxed grains with
many high angles of misorientation are observed after 4 passes. After 8 and 12 passes, grain sizes
are roughly 1 μm. No change in grain size is observed after 8 passes of ECAP [30].
Figure 18. Microstructural evolution of Al-1%Mg solid solution alloy processed via ECAP[31]
31
Figure 18 shows that with the same processing condition as high purity Al, for Al-1%Mg
solid solution alloy, initial grain sizes are about 350 μm. Grain sizes are refined to about 700 nm
after 8 and 12 passes.
For the case of the high purity Al, the fraction of high angle grain boundaries (HAB),
boundaries that have 15
o
or more angle of misorientation, starts out low at lower number of
ECAP passes, and increases to ~50% after 4 passes and 53% after 8 passes [30]. Figure 19 shows
the distribution of grain boundary misorientation in high purity Al processed by ECAP [30].
Similarly, Figure 20 shows the distribution of grain boundary misorientation in Al-1%Mg alloy
processed by ECAP[31]. HAGs are 12%, 15%, 25%, 65% after 1, 2, 4, and 8 passes.
32
Figure 19. Evolution of grain boundary misorientations for high purity Al processed by ECAP
[30].
33
Figure 20. Evolution of grain boundary misorientations for Al-1%Mg alloy processed by ECAP
[31].
34
Chapter 2. Single Reflection (Direction) X-ray Microbeam
Measurements of Long Range Internal Stress
2.1 ECAP AA1050
It is widely believed that most of the high-angle boundaries (HABs) that lead to a refined
grain size are produced as a result of dislocation accumulation and reaction [32]. Many believe
that these boundaries are non-equilibrium with many extrinsic dislocations emanating from the
boundaries [33] and [34]. Thus, high LRISs may exist in these highly deformed metals. As
previously stated, understanding the LRISs in deformed metals, in general, is important since
they can influence a variety of important phenomena including the Bauschinger effect (important
for cyclic deformation, fatigue and metal forming) and metal springback (metal forming).
Alhajeri et al. [35] recently found the presence of elastic strains (and thus stresses) with
compressive and shear components in ECAP 1050 aluminum alloy in close proximity to
grain/subgrain boundaries (the beam diameter was ∼20 nm) using convergent-beam electron
diffraction (CBED).
The term “grain/subgrain boundary” is used here to reflect the observation that SPD -
induced boundaries exhibit a wide distribution of misorientations, from < 1° to misorientations
associated with HABs (e.g. > 15°). Lower misorientation boundaries are frequently classified as
“cell walls” or “subgrain boundaries”, depending on the degree of order or energy of the
boundary. Cell walls are often more diffuse and thicker at several tens of nanometers. Subgrain
boundaries can be lower energy (tilt, twist or mixed) boundaries and have thicknesses
comparable to HABs (e.g. < 1 nm).
35
In the work of Alhajeri et al. [35], the strain orientations were not determined. However,
an estimate of the magnitude of the long-range internal stresses based on the maximum principal
stress in the vicinity of the boundary was about 112 MPa. Based on stress vs. strain curves
reported in Refs. [36] and [37], this is roughly 0.75 of the 150 MPa flow stress, σ
a
. CBED is not
able to measure elastic strains from regions of lower stress such as grain/subgrain interiors [8,
15]. It also suffers from possible measurement uncertainty from relaxation after unloading,
especially considering the very thin sections of the transmission electron microscopy (TEM) foils
that are typically examined using CBED [15]. In addition, it is difficult to correlate the strains
measured using CBED diffraction patterns with specific sample orientations, especially for ultra
fined grain (UFG) samples.
We utilized high-resolution X-ray microbeam diffraction here to examine the lattice
strains of low-dislocation density regions within the grain/subgrain interiors in ECAP-deformed
1050 Aluminum. These strains were measured along specific directions with reference to the
ECAP die. These measurements complement the earlier work by Alhajeri et al. [35] to more
fully assess LRIS in ECAP Aluminum.
The specimens used for LRIS measurements were identical to those in the Alhajeri et al.
CBED experiment [35]. The as-received ECAP AA1050 aluminum billet contained 99.5 wt.%
aluminum with Fe 0.26; Si 0.14; Mg < 0.05; Cu < 0.05; Mn < 0.05; Zn < 0.05; V < 0.05 and Ti <
0.03 as the major impurities, and it was machined to 10.0 mm in diameter and 65.0 mm in length
for ECAP processing. These impurities suppress static and dynamic recrystallization and grain
growth[38, 39]. ECAP was conducted at ambient temperature with a pressing speed of 0.5 mm/s.
The solid ECAP die was made with an internal channel bent into an abrupt angle of Φ = 90°,
with the outer arc curvature of the channel joint rounded with Ψ = 20°, as shown in Figure 15.
36
The ECAP die imposes an equivalent uniaxial strain of ≈ 0.88 with each pass [26, 40]. Between
each pass, specimens were rotated 90° (route B
C
) [24, 27, 30].
Three directions were examined for the ECAP AA1050 1P sample: a direction near the
maximum tensile plastic-strain (+27.3°), near zero strain (+4.9°) and a compressive strain
(−17.5°, about 0.53 of maximum compressive true strain). For the multiple pass ECAP AA1050
samples (2, 4, and 8 passes), only the near zero strain direction was examined (+4.9
o
).
For the 1P sample measurements, the direction of theoretical maximum compressive
plastic-strain (about −67.5°) could not be m easured because the sample angles would have
caused the platinum wire to collide with the sample surface. Instead, we probed the elastic strains
in a direction where the sample is under a smaller-magnitude (0.53 maximum) compressive
plastic-strain at −17.5 °. The elastic strains were measured within grains/subgrains using well-
defined peaks as shown in Fig. 12. We use, hereinafter, the notation +27.3°(H), +4.9°(M) and
−17.5°(L) referring to “high”, “medium” and “low” strains. According to microstructural
analysis, the average grain elongation angle with respect to the pressing direction is typically
between 20° and 30° after one pass of ECAP for a similar die geometry (Φ = 90°) [21], [41] and
[42]. Thus, the theoretical predictions and experimental analyses are consistent in this respect.
2.1.1 Microbeam Sample Preparations
After ECAP pressing, cylindrical rods of approximately 8.0 mm height were produced
using a cross-sectional cut through each billet by electrical discharge machining (EDM). The
surface of the disc was then mechanically polished using silicon carbide grit paper (to 2400 grit)
and then electro-chemically polished in a perchloric acid solution to provide a flat, damage-free,
37
surface. X-Ray microbeam diffraction was used to measure elastic strains from grain/subgrain
interiors at depths ranging from 0 µm to 150 µm below the polished surface.
2.1.2 Powder Diffraction
Strain determination requires an accurate value for the unstrained lattice parameter of our
AA1050 alloy. Some of the solutes present in AA1050 can increase (e.g. Mg) the lattice
parameter while others decrease the parameter (e.g. Si, Fe). Furthermore, the equilibrium
solubilities of some of these elements (e.g. Si and Fe) in Al may be negligible while others can
be more substantial (e.g. Mg). A further complication is that non-equilibrium solute
concentrations may be present depending on the processing history. Consequently, we believe
the best way to assess the lattice parameter of this alloy was direct measurement since prediction
based on composition appears impossible. We directly measured the unstrained lattice parameter
using the x-ray powder diffraction instrument on the 11-BM beamline at the Advanced Photon
Source (APS), at Argonne National Laboratory. The 11-BM beamline operates a high resolution
powder diffractometer that provides high-accuracy measurements of the average lattice
parameter and peak broadening. The diffractometer on 11-BM was specifically designed for such
measurements and it is carefully calibrated using National Institute of Standards and Technology
(NIST) standard reference materials (SRMs) (e.g. a mixture of ≈ 70 wt.% Si (SRM 640d) and ≈
30 wt.% Al
2
O
3
(SRM 676)). This allows the lattice constant to be measured with a one standard
deviation uncertainty of 0.0001 Å. A cylindrical rod, 2.0 mm in diameter and approximately 8.0
mm in length, was cut from the center axis of the 8.0 mm tall cylindrical rods cut from the ECAP
sample. The 2.0 mm rod is cut by EDM and chemically etched to about 0.8 mm diameter. This
rod was then fitted into a Kapton capillary tube and sent to the 11-BM beamline for high
38
resolution powder diffraction measurements (Figure 21). The powder diffraction beam is ~ 1.5
mm x 0.5 mm.
Figure 21 Powder diffraction sample geometry
Figure 22 Powder diffraction X-Ray line profiles for ECAP 1050, (331) reflection left, and (420)
reflection right. Note the consistency in lattice parameters measured
It is not possible to measure the average lattice parameter accurately using the microbeam
diffraction instrument because the lattice parameter varies with position and the sample volumes
we can average over are too small to reduce the uncertainties. Instead, the microbeam instrument
is calibrated using a pure (un-doped) Si single crystal, which typically allows calibration to
within a few times 10
-5
strain. This is much smaller than the stated statistical uncertainty of 10
-4
39
strain which mostly originates from our ability to locate the peak centers of the broadened Al
reflections. The as-received lattice parameter of AA1050 at ambient temperature was calculated
from 30 keV powder diffraction measurements of approximately 23 reflections. Figure 22 shows
X-Ray line profiles from (331) and (420) reflections for different ECAP passes. Lattice
parameter values remain consistent between all samples as received, 1, 2, 4, and 8 passes, with
the largest variation of 2.0 × 10
-4
Å, which suggests that these samples exhibit negligible local
residual stresses on length scales that are large compared to dislocation microstructures and
small compared to sample sizes. The ambient temperature lattice parameter was measured as
4.05000(10) Å for ECAP AA1050 as received sample, which compares with a value of
4.04950(15) Å for pure Al. The Alhajeri et al.[35] strain values assumed pure Al lattice
parameters, producing an artificial strain of about 1.2 × 10
-4
. However, since this strain is much
smaller than their CBED measurement resolution, the original reported data by Alhajeri et
al.[35] were used for our strain and stress calculations.
2.1.3 Microbeam Measurement Set Up
X-ray microbeam diffraction measurements were conducted on beamline 34-ID-E at the
APS. The monochromatic x-ray beam was focused to ≈ 1 µm using Kirkpatrick –Baez mirror
optics, which provided high spatial resolution in the two axes perpendicular to the incident x-ray
beam. The spatial resolution along the beam path was determined by the grain/subgrain size. The
energy was scanned over reflections in the vicinity of 14 keV using steps of 3 eV, which matches
the resolution of the instrument. The diffracted beams were detected on an amorphous-Si area
detector. Depth information was obtained by translating a 50 µm diameter platinum wire profiler
parallel to the sample surface, blocking diffracted X-Rays from the sample; the origin of the
diffracted X-Rays was then determined by triangulation [12]. <531> reflections were used to
40
obtain lattice spacings for suitably oriented Al grains/subgrains for each measurement. The
geometry of the X-Ray microbeam setup is illustrated in Figure 23. The d-spacings were
converted into elastic strains using the unstrained lattice parameter obtained from the powder
diffraction measurements.
Figure 23 Geometry of ECAP 1050 microbeam experiment set up. Samples were measured along
the zero strain direction, and approximately along the +/- 25
o
off axis
2.1.4 Results
2.1.4.1 Grain/Subgrain Interior Measurements for ECAP AA1050 Single Pass
Most of the elastic strain values are negative (compressive) at +4.9°(M) (close to the
pressing direction) (Figure 24). The mean strain of the measured grain/subgrain interiors (at
+4.9°) is about -1.9 × 10
-4
, which converts to about -13.6 MPa long-range internal stress
(Young’s modulus along the <531> direction ≈ 71 GPa for aluminum [43]). It is important to
41
note that any stress value is only approximate since we measure only a single component of the
strain tensor.
Figure 24. Strain distribution of ECAP AA1050 1 pass sample along various direction off of the
pressing axis. Note the stars represent the mean values.
Only negative strains (compressive strain) are observed in the +27.3
o
(H) direction, close
to the theoretical maximum tensile plastic-strain in the specimen. The maximum and minimum
compressive elastic strains are -4.8 × 10
-4
and -1.1 × 10
-4
, respectively, and the mean strain is -
2.7 × 10
-4
, indicating an average long-range internal stress of about -19.0 MPa. The magnitude of
this stress is about 0.13 σ
a
[36].
Finally, for the -17.5°(L) direction, the mean strain is around -6.4 × 10
-5
, or -4.5 MPa in
internal stress. Note that our one standard deviation uncertainty for an individual measurement is
42
1.0 × 10
-4
. This includes a systematic error from uncertainties in the instrument calibration and
statistical uncertainties due to determination of the diffraction peak centers. We estimate the
uncertainty of the mean strain values to be 5×10
-5
. Thus, the measured mean strain of -6.4×10
-5
for the -17.5°(L) direction is not statistically very different from zero strain.
Note although the most right-hand data point for the +4.9°(M) direction is anomalous, but
this measured value is correct. It is possible that the corresponding sample volume resides very
near to a grain/subgrain boundary, and thus belongs to a different strain population. If this data
point is removed from the data set, the mean strain becomes -2.4 × 10
-4
which is within the
measurement uncertainty of the mean strain for the +27.3°(H) direction. Interestingly, the
variation in the strain for the -17.5°(L) direction is similar to the variation in the other two
directions (although the averages are different). This suggests that the sample conditions, such as
strain uniformity in grain/subgrain interiors, are similar for all three directions. Overall, the
trends are consistent with the Composite Model.
2.1.4.2 Grain/Subgrain Interior Measurements for ECAP AA1050 Multiple
Passes
Long range internal stresses (LRIS) were also assessed in AA1050 equal channel angular
pressed (ECAP) using route B
C
for 2, 4, and 8 passes along the near zero strain direction. LRIS
were measured in the grain/subgrain interiors by X-Ray microbeam diffraction.
The ECAP AA1050 samples were processed via route B
C
, with samples rotated by 90
degrees between each pass [24, 27, 30]. Samples of 2, 4, and 8 passes were examined. Again, the
zero strain direction is along the axial direction or pressing direction.
43
Figure 25 The distribution of low dislocation areas strains of AA1050 after 1, 2, 4, and 8 ECAP
passes. The strains are characterized at the center of the sample nearly along the pressing
direction (4.9 degrees from pressing direction). Each star represents the average value for each
set of data
Figure 25 shows that most of the measured elastic strain values are negative except for 1
value from the 1 pass sample. Table 1 shows the mean strain of the measured grain interiors for
the multiple pass samples, the standard deviation, and the uncertainty, or standard error, of the
mean value (standard deviation divided by the square root of the number of data points used to
calculate mean). While the mean values of elastic strain for various passes remain negative, from
the standard deviation of the spread of data from Table 1, the distribution of strain for the 1 pass
sample is significantly (about 2 times) larger than for the multiple pass samples. The standard
deviation for the 1 pass sample is 2.2 × 10
-4
compared to roughly 1 × 10
-4
for the samples which
have undergone multiple ECAP passes. There is no significant difference between the spread of
44
the strain distribution of the 2, 4, and 8 pass samples. The mean elastic strain values of all
samples vary from -3.1 × 10
-4
to -1.9 × 10
-4
with the uncertainty ranging from 3.0 × 10
-5
to 6.7 ×
10
-5
.
Table 1 Standard deviation and uncertainty of mean values for measured strain of low dislocation
regions for AA1050 aluminum after 1, 2, 4, and 8 ECAP passes along the pressing direction,
illustrated by Figure 25.
1 Pass 2 Pass 4 Pass 8 Pass
Mean elastic
strain
-1.9 x 10
-4
-2.99 x 10
-4
-2.21 x 10
-4
-3.09 x 10
-4
Standard
deviation
2.24 x 10
-4
1.04 x 10
-4
9.34 x 10
-5
1.20 x 10
-4
Uncertainty of
mean value
6.74 x 10
-5
3.28 x 10
-5
2.95 x 10
-5
3.79 x 10
-5
There is no clear correlation between mean strain and the number of ECAP passes
between 1, 2, 4, and 8 pass samples along the pressing direction. Measured strain values are
calculated based on the as-received lattice parameter (unstrained) from Beam line 11 at the
Advance Photon Source. The measured mean elastic strain of the cell interiors at 4.9
o
for the 1,
2, 4, and 8 pass samples are -1.9 × 10
-4
, -3.0 × 10
-4
, -2.2 × 10
-4
, and -3.1 × 10
-4
, respectively.
These values convert to about -13.6 MPa, -21.2 MPa, -15.7 MPa, and -21.9 MPa long range
internal stresses for 1, 2, 4, and 8 passes, respectively (Young's Modulus along the <531>
direction ≈ 71 GPa ). This equates to a magnitude of 0.09 σ
a
, 0.14 σ
a
, 0.08 σ
a
, and 0.10 σ
a
for 1, 2,
4, and 8 passes (flow stress σ
a
~ 148 MPa, 150 MPa, 180 MPa, and 200 MPa, for 1, 2, 4, and 8
passes)[36].
45
2.1.5 Discussion
2.1.5.1 Grain/Subgrain Microstructure in ECAP AA1050
According to microstructural analysis using TEM and EBSD, the grain/subgrain size is
small at about 1 μm [44-47], with about 10 % to 20 % of the boundaries being of high angle, i.e.
having misorientations greater than 15° [30, 46, 47]. The grain/subgrain interiors have a
dislocation density of about 6.6 × 10
13
m
-2
based on other work [48]. The annealed Al has a
dislocation density of about 10
11
m
-2
[49]. Large strain deformation often produces substructure
where “crystallites” with mixed high and low misorientation boundaries are observed [32, 47,
49]. The average grain size (considering only high angle boundaries) is in a wide range of about
8 µm to 26 µm (e.g. 8.2 µm at the top edge of the ECAP billet cross section, 26 µm at the center
and 19 µm at the bottom edge)[46]. After ECAP of 1, 2, 4, and 8 passes, the microstructure
consists of grains having boundaries with mixed high and low misorientation angles. For the 1
pass condition, about 10% - 20% of the boundaries are high angle grain boundaries (HAGBs)
having misorientation greater than 15º, after 2 passes, HAGB fraction is 25%, after 4 passes, the
HAGB fraction is ~50%, and after 8 passes, HAGBs fraction is ~53%[30, 46] based on EBSD.
According to TEM analysis, average grain/subgrain size formed by high and low angle grain
boundaries is about 1µm for 2 pass, 900 nm for 4 pass, and 680 nm for 8 passes [35].
Dislocations are distributed heterogeneously within grains. Figure 26 shows the TEM images of
the microstructures for ECAP AA1050 with 2 passes and above.
46
Figure 26 TEM Images of ECAP AA1050 2, 4, and 8 passes. Images from Alan Fox at Asian
University
2.1.5.2. LRIS in Grain/Subgrain Interiors
2.1.5.2.1. ECAP AA1050 Single Pass
Deformation-induced internal stresses have been investigated extensively using different
techniques[8]. However, the results vary with investigators, and appear to vary with the
deformation process, as well. Mughrabi et al.[14] reported relatively high LRIS of about 1.0 σ
a
in persistent slip band (PSB) walls in cyclically deformed copper. Conventional x-ray diffraction
line profile analysis by the same group found that the LRIS in a monotonically deformed single
crystal Cu was about 0.4 σ
a
in cell walls and about 0.1 σ
a
in cell interiors. These findings were
rationalized in terms of a “Composite Model”[14, 50]. As the material is deformed, compatibility
dislocations accumulate between soft (cell interiors, etc.) and hard (cell walls, non-equilibrium
grain boundaries, etc.) regions which lead to different local stresses in the walls and cell
interiors. Mechanical equilibrium requires that the volume fraction weighted sum of the stresses
47
in the cell interiors and the cell walls must be equal to zero in an unloaded specimen. Our group
(Levine et al.[3]) also investigated long-range internal stresses in compressively deformed
copper using a more reliable x-ray microprobe. We showed that LRIS are present (but lower) in
dislocation cell interiors +0.1 σ
a
, while balancing stresses are present in the cell walls -0.1 σ
a
[3,
4] (the volume fraction of cell walls was about 0.5).
Kassner et al.[8] and Legros et al.[51] showed an absence of internal stress near and
away from the dislocation dipole bundles (veins) in Cu and Si single crystals, cyclically
deformed in single slip to presaturation (no PSBs). There are also other cases (e.g. creep) where
internal stresses have not been found in materials using the CBED technique[51]. This may be
due to poor resolution, as compared to other techniques such as x-rays, and relaxation that can
occur in thin foils[8, 35, 52]. Thus, much of the most recent work found LRIS in plastically
deformed materials to be modest. In severely plastically deformed materials such as ECAP
aluminum alloy, however Alhajeri et al. used CBED[35] to show that there were internal stresses
near grain/subgrain boundaries. As mentioned earlier, ECAP may produce dislocation
configurations (e.g. boundaries) that produce relatively high LRIS. The CBED of Alhajeri et al.
did not determine the orientation of the stresses to the pressing geometry. Therefore, we utilized
x-ray microbeam technique to buttress the CBED limitations of LRIS research on ECAP
aluminum.
Our results demonstrate that compressive internal elastic strains (stresses) are present in
the +27.3°(H) direction. The mean value of this internal strain is around -2.7 × 10
-4
(-19.0 MPa),
which corresponds roughly to -0.13 σ
a
. Near the pressing direction (+4.9°(M)), compressive
internal strains (stresses) are measured to be slightly smaller than those in the +27.3°
direction,
and the average strain is -1.9 × 10
-4
(-13.6 MPa) or about -0.09 σ
a
. The smallest mean strain
48
value (close to zero, at -0.6 × 10
-4
) is measured in the -17.5°(L) direction. We expect a
compressive internal elastic strain within low-dislocation density regions in the grain/subgrain
interiors along the +27.3°(H) according to the Composite Model. This elastic strain should trend
towards a tensile strain as the orientation changes to +4.9°(M), and then to -17.5°(L), consistent
with the Composite Model. Although the mean compressive elastic strain decreased in
magnitude, it never quite became tensile. Tensile strains might have been observed if we had
been able to examine strains in the -67.5° direction, where the plastic-strain is most compressive.
It must be mentioned, again, that the data presented is extracted from those regions within
the grains/subgrains that have relatively low-dislocation densities. This point will be discussed
further where stresses in the higher dislocation density regions of the grain/subgrain interiors are
considered.
2.1.5.2.2 ECAP AA1050 Multiple Passes
There is no clear trend between the number of passes and the normalized long range
internal stresses as a fraction of the flow stress for different ECAP passes. The variations in
magnitude of the normalized long range internal stresses are not significant and these values
remain roughly the same for all samples.
Our results indicate that compressive internal elastic strains are present along the pressing
direction (4.9
o
) for all different ECAP passes. The internal elastic strains for the 2, 4, and 8
passes remain relatively consistent with the internal elastic strain for the 1 pass sample along the
+4.9
o
direction. Since the 4.9 degree direction is near the pressing direction, or the direction of
zero strain, we expect the 1 pass internal elastic strain to be near zero. This however, was not the
case and we did not observe zero internal elastic strain for our 1 pass sample, nor for any of our
49
multiple pass samples since the mean elastic strain were all compressive and range from -1.9 x
10
-4
to -3.1 x 10
-4
for 1P and 8P. Since the processing history of the starting billet was not
known, and was not annealed before ECAP processing, LRIS stress could have existed within
the grain/subgrain interior in the as-received billet from prior material processing.
2.1.5.3 LRIS in Grain/Subgrain Boundaries or Cell Walls
2.1.5.3.1 ECAP AA1050 Single Pass
All of the powder diffraction data for different ECAP passes (for the unstrained specimen
and up to 8-pass) give almost identical values for the lattice constant, a
o
(the difference is about
0.0002 Å, or about 5 × 10
-5
strain) with a powder diffraction beam size (1.5 mm × 0.5 mm). This
demonstrates that these samples exhibit negligible local residual strains (stresses) over length
scales that are small compared to the sample dimensions and large compared with the dislocation
microstructure.
Stress equilibrium dictates that the weighted sum of the stresses in the grain/subgrain
interiors and boundaries must be equal to zero in the unloaded specimens. Here, we consider the
width of a non-equilibrium grain/subgrain boundary to be the distance over which all of the
extraneous dislocations of the boundary are contained. This width must be approximated in order
to calculate the volume fraction and thus the resulting mean grain/subgrain interior and boundary
loads. The volume fraction of grain/subgrain boundaries can be calculated by a simple model.
We assume the sample is composed of stacked cubical grains/subgrains with a side length of a
(the distance from one grain/subgrain boundary center to the next), and the grain/subgrain
boundary width is b. The percent volume fraction of the grain/subgrain boundary is, then, [a
3
–
(a – b)
3
] / a
3
. According to the works of Ding et al.[33] and Horita et al.[34] non-equilibrium
50
grain boundary widths vary from 5 nm to 10 nm, although these values are for samples deformed
to a strain of 7, while the specimens of this study were only strained to ≈ 1. In addition, for SPD
metals, extraneous dislocations are observed in both high angle and low angle grain/subgrain
boundaries by TEM [53, 54]. We assume that there are similar extrinsic dislocation features for
grain and subgrain boundaries. For our grain/subgrain size (1 µm), the volume fraction of the
boundaries varies between 1.5 % (5 nm) and 3.0 % (10 nm). The work of Alhajeri et al.[35]
reports a maximum axial strain value of 1.1 × 10
-3
, which converts into a maximum axial stress
of 112 MPa
(about 0.75 σ
a
). However, this strain was measured using a beam diameter of 20 nm
immediately adjacent to the grain/subgrain boundary. Thus, for the purposes of a force-balance
“check”, a boundary width of about 40 nm must be assumed. That is, the strain they measure is
an average over a width of 40 nm, which may exceed the true boundary width.
To balance this stress at the grain/subgrain boundary, the mean stress in our 1-pass
grain/subgrain interior should be ≈ -14.6 MPa (about -0.097 σ
a
, converted into a strain of about -
2.1 × 10
-4
), which is the average force (stress) over low-dislocation density interior and high-
dislocation density interior regions that balance the assessed force over a boundary region of 40
nm width. According to our measurements, the average strain in the low-dislocation density
interiors is about -2.7 × 10
-4
(-0.13 σ
a
).
To partition the LRIS in the high- and low-dislocation density regions of the
grain/subgrain interiors, we need an approximate ratio for the high-dislocation and low-
dislocation volumes. We determined the minimum number of counts per pixel in the large grain
data and subtracted this from each pixel to obtain an approximate background-subtracted image.
We then took the large grain diffraction data and masked out the peak positions. Next we
estimated the diffuse scattering level in the nearby regions and subtracted this off to obtain an
51
estimate of the total diffracted intensity from the low-dislocation volumes. The remaining
intensity should come from the high-dislocation density volumes. Within the grain/subgrain
boundaries, we obtained, roughly, 55 % ± 10 % high-dislocation density volume within the
grain/subgrain interior. The stress in the high-dislocation density region from a force balance
would be roughly -9.0 MPa (-0.06 σ
a
) to -12.1 MPa (-0.08 σ
a
). Thus, the interiors have low LRIS
balanced by higher LRIS in the boundaries. Thus, in summary, for the boundary region, the
stress is measured by CBED as 0.75 σ
a
, for the low-dislocation density grain/subgrain interior,
the stress as measured by x-ray microbeam is -0.13 σ
a
and by measurement of the high-
dislocation density volume fraction within
the grain/subgrain interior, the stress by mechanical
equilibrium is about -0.07 σ
a
.
2.1.5.3.2 ECAP AA1050 Multiple Passes
Grain/subgrain sizes for 2, 4, and 8 pass ECAP AA1050 are roughly 1 micron, 900 nm,
and 680 nm [35] based on TEM. According to Alhajeri et al., for the 2 pass sample, the
maximum axial strain of 1.1 x 10
-3
is measured by CBED adjacent to the nonequilibrium grain
boundary (which may have extrinsic emanating dislocations), which converts to a maximum
axial stress of 112 MPa (about 0.75 σ
a
, [flow stress = 150MPa for 2 pass]). For a 4 pass sample,
a maximum axial strain of -1.8 10
-3
was reported, which converts to a maximum axial stress of
147 MPa (about 0.82 σ
a
, [flow stress = 180 MPa for 4 pass]). Alhajeri et al. did not report a value
for the 8 pass sample due to Kikuchi lines being too blurred. Assuming a 40nm thick
nonequilibrium grain boundary, the percent volume of this boundary is 12% for 2 pass and 13%
for 4 pass. The orientation of these stresses was not assessed. To balance these stresses in the
grain/subgrain boundaries, the mean stress in our 2 pass grain/subgrain interior is -14.6 MPa, or
about -0.097σ
a
. For our 4 pass sample, the mean stress in the grain/subgrain interior is -21.5
52
MPa, or about -0.119σ
a
. These grain/subgrain interior strains represent the volume weighted
average stress over both the high and low dislocation density areas. From these values and our
microbeam measured low dislocation density areas elastic strains, stresses in the high dislocation
areas can be calculated. The volume percentage of high dislocation density areas must be
approximated in order to calculate the elastic strains within these regions. A large grain scan
image is used to approximate volume percent from our previous experiment from Lee et al. [13].
A 55% +/- 10% volume ratio is obtained for high dislocation areas within the grain/subgrain.
Table 2 summarizes the long range internal stress within high dislocation density areas.
Table 2 Long range internal stresses within high dislocation areas at 45% and 65% (55 ± 10 %)
high dislocation density volume fraction.
Number of
passes
LRIS in high dislocation areas
(45% cell walls) (MPa)
LRIS in high dislocation
areas (65% cell walls) (MPa)
Normalized LRIS
(σ/σ
a
)
1 -15.9
-15.1
-0.11 to -0.10 σ
a
2 -6.5
-11.0
-0.04 to -0.07 σ
a
4 -28.6
-24.6
-0.16 to -0.14 σ
a
2.2 ECAP AA6005
Along with Aluminum Alloy 1050, Aluminum Alloy 6005 was also processed by ECAP
and its LRIS measured. ECAP AA6005 specimens were produced starting with an as-received
billet aluminum containing 98.2 wt. % Al, and Si 0.88, Mg 0.60, Fe 0.27, Mn 0.10, Ti 0.015, Cr
0.010, Cu 0.005, and Zr <0.005 as the major impurities. A T4 temper condition was imposed in
order to induce a complete precipitation of secondary phase particles and stabilize the
microstructure prior to ECAP, the alloy was homogenized at 530ºC/3h and slowly cooled to
room temperature (at a rate of 80ºC/h). AA6005 samples underwent ECAP via route C at room
temperature. The ECAP die consists of two cylindrical channels bent at an angle of 90º and an
53
outer curvature radius of 20º. Samples are rotated by 180º between passes with route C
processing.
2.2.1 Microbeam Sample Preparation
Microbeam sample preparation for ECAP AA6005 is identical to AA1050 samples in the
previous section.
2.2.2 ECAP AA6005 Powder Diffraction
Samples preparation and conditions for powder diffraction were identical to conditions
for ECAP AA1050. Once again, samples were prepared to fit in Kapton tubes and sent to
beamline 11 for powder diffraction measurement. Figure 27 details the results from (331) and
(420) reflections.
Figure 27. Powder diffraction X-ray line profile for ECAP AA6005 1 and 2 Pass for (331) and
(420) reflection. Note the differences in lattice parameters measured between the 1 pass and the 2
pass sample. This suggests the existences of residual stress within the sample
54
From powder diffraction data, in contrast to values from the 1050 sample, there is a
marked difference in measured lattice parameters between 1 and 2 pass samples for 6005. Lattice
parameter for 1 pass sample was 4.050369 Ǻ and 4.051286 Ǻ for the 2 pass. This represents a
difference of .000917Ǻ between the 1P and 2P sample, an equivalent of 2.3 x 10
-4
strain. This is
a significant difference
For the 1 pass sample, only the a
o
value measured by powder diffraction was used to
calculate strain. For the 2 pass sample, in addition to the lattice parameter measured by powder
diffraction, a large grain scan was performed near the location where measurements were taken
so as to measure a "local a
0
". This large grain scan contains diffraction information from all
features inside the grain: low dislocation areas, high dislocation areas, and grain boundaries. The
lattice parameter for the 2P measured by the large local grain scan was 4.051766 Ǻ, compare d to
the value measured by powder diffraction 4.051286 Ǻ, which signifies a difference of 0.00048
Ǻ, or a strain value of 1.2 x 10
-4
. This difference between this local lattice parameter and that
from X-ray microbeam is the LRIS. A large grain scan was not performed on the 1P sample due
to time restriction. However, based on the results obtained from the 2P local a
o
grain scan,
significant macro-scale residual stress is present in ECAP AA6005 samples can be concluded.
2.2.3 Microbeam Measurement Set Up
The microbeam measurement set up at beam line 34 is similar to the set up for the
measurements of ECAP AA1050. Strain measurements are made near the pressing axis for
ECAP AA6005 1P and 2P samples.
55
2.2.4 Results
2.2.4.1 Grain/Subgrain Interior Measurements for ECAP AA6005
Figure 28 shows the results of the internal elastic strains within the low dislocation density
of the grain/subgrain interiors for ECAP AA6005. Elastic strains are calculated based on the
measured lattice spacings compared to the average lattice parameter from powder diffraction for
the 1P sample. However, due to the inconsistency between powder diffraction lattice parameters,
accurate elastic strains cannot be calculated since the measured lattice spacings contain strains
due to both the ECAP process as well as macro-scale residual stresses. Nevertheless, strains for
the 1P sample are calculated based on the powder diffraction measured a
o
and may be inaccurate,
and strains for the 2P sample are calculated based on both the powder diffraction lattice
parameter and the local average lattice parameter (Figure 28).
Figure 28 The distribution of cell interior strains of 6005 Al after 1 and 2 ECAP passes. The
strains are assessed at the center of the sample near the pressing direction.
56
Table 3 Standard deviation and uncertainty of mean values of measured strains for ECAP 6005 1
pass and 2 passes
1 Pass (based on
powder a
o
)
2 Pass (based
on local a
o
)
2 Pass (based on
powder a
o
)
Mean strain -2.92E-04 -3.82E-04 -2.62E-04
Standard deviation 3.14E-04 2.04E-04 2.04E-04
Uncertainty of mean value 9.94E-05 6.44E-05 6.44E-05
Figure 28 shows the strain measured at various cell interiors for ECAP aluminum 6005 for
1 and 2 passes using different a
o
, both based on powder diffraction and large grain scan for the 2
pass. <531> lattice spacings along the 4.9 degree off the pressing direction were measured based
on the microbeam x-ray diffraction of relatively dislocation free regions within the cell interiors
after 1 and 2 passes. Strain values were measured based on well-defined peaks of low dislocation
areas. The strain values were measured near the pressing direction. The measured strain values
from 1P and 2P samples could not be directly compared since they are calculated based off
different values of a
o
. Once again, similar to the AA1050 strain values, the standard deviations of
strain values for AA6005 (Table 3) show that the distribution of strains for the 1 pass sample are
much broader than the 2 pass sample, with a standard deviation of 3 x 10
-4
for the 1 pass and 2 x
10
-4
for the 2 pass sample.
2.2.5 Discussion
2.2.5.1 LRIS in Grain/Subgrain Interior of ECAP AA6005
Low dislocation density in the grain/subgrain internal stresses are calculated by
comparing measured lattice parameters and comparing them to lattice parameters from powder
diffraction measurements from beamline 11 as well as the large grain local a
o
. In contrast to
57
AA1050 samples, powder diffraction lattice parameters vary markedly between the 1 pass and 2
pass sample for AA6005 samples. Once again, a difference of .000917Ǻ between the 1 pass and
2 pass is observed, an equivalent of 2.3 x 10
-4
strain. This strain due to the variation between
powder diffraction lattice parameters for 1P and 2P samples is in the same order of magnitude as
our measured internal elastic strain. Local residual strains (stress) is thus considered to be present
over length scales that are small compared to the sample dimensions but large compared with the
microstructure and is the cause of the variations. Because of this, mean values for strain of
sample 1 and 2 cannot be directly compared with each other since strain values are calculated
based on different lattice parameters. For our 1 pass sample, elastic strains were calculated based
on the powder diffraction. For our 2 pass sample, elastic strains were calculated based on both
powder diffraction data and the local a
o
from a large grain scan. This large grain scan includes
diffraction from all features of the grain/subgrain, including the grain/subgrain boundary, high
dislocation density regions, and low dislocation density regions. In this case, we can assume
(using the local measured a
o
) that the average residual strains in the local area including grain
interiors, high dislocation areas and low dislocation areas integrate to zero. Thus, using this local
a
o
would give us more accurate internal elastic strain measurements for the 2 pass. The lattice
parameter for the 2P measured by a local grain scan was 4.051766 Ǻ, compared to the value
measured by powder diffraction 4.051286 Ǻ, which signifies a difference of 0.00048 Ǻ, or a
strain value of 1.2 x 10
-4
, also on the same order of magnitude as the measured mean internal
elastic strain. We may expect variation on the same order of magnitude between the internal
elastic strains calculated based on lattice parameter from powder diffraction versus by a local
large grain scan for the 1 pass condition for AA6005. It is then unreasonable to compare reported
LRIS within cell interiors for the 1 pass sample to the 2 pass sample. A local a
o
was not
58
measured for the 1 pass sample due to the fact that a large local grain was not found for
measurement due to time restriction.
Note from Figure 29 the high number of dislocations that exist within the grain/subgrain
of ECAP AA6005 compared to that of ECAP AA1050 (Figure 26). Grain sizes are about 500 nm
compared to 1 µm of the 1050 sample. Furthermore, grain interiors of AA6005 contain many
more dislocations which in turn results in much weaker diffraction spots. It was fortunate that
we found a large grain which diffracts strongly for our 2 pass sample; however, it was
unfortunate that none could be found for the 1 pass.
Figure 29 TEM image of ECAP AA6005 2 pass. Image from Alan Fox at Asian University
59
2.3. Transmission Electron Microscopy
2.3.1. Background
TEM imaging was performed on ECAP AA1050 1P and 8P samples at USC. While there
exist studies on grain elongation of the ECAP process using optical microscopy [21, 42]. Since
the focus of this study is on the full elastic strain tensor measurements of ECAP Al samples, it is
important to keep track of the directions of maximum and minimum elastic strains in order to
compare them with results from microbeam measurements. Again, according to the simple shear
deformation behavior based on ECAP theoretical models and optical studies, it is expected to
observe grain elongation along the +22.5
o
off the pressing axis (according to theory) [26] and
+25
o
to +30
o
off the pressing axis (according to optical studies) [21, 42].
2.3.2. Sample Preparations
ECAP AA1050 samples of 1 and 8 passes were prepared for TEM analysis. Slices were
cut from the center of the ECAP samples using a wiresaw (with a 800 grit SiC oil slurry)
following the geometries described in Figure 30. All straight cuts were performed using the
wiresaw.
60
Figure 30. TEM disk preparations from the bulk ECAP sample. Wiresaw/ultrasonic cuts are in
red and scribe marks are in green. Four main stages are illustrated. Step 1 shows the bulk ECAP
sample cut into a cylinder. Step 2 shows the cylinder cut into thin slices. Step 3 shows the scribe
marks used for sample tracking. Step 4 shows both the ultrasonic cuts (red) and scribe marks
(green) on the TEM disks. Sample coordinate convention is similar to that used for X-Ray
diffraction measurements with the +P direction denoting the pressing direction.
A cylindrical section was cut from the bulk ECAP sample (step 1). With the top (+Y
direction) and pressing axis (+P direction) identified, a slice was cut along the y-z plane
approximately 380 μm thick (step 2 of Figure 30). Scribe marks were then used to identify the
top and pressing directions with respect to the bulk sample (step 3). The slice was then polished
using SiC grit paper (800 and 1200 grit) down to a thickness of ~280 μm. 3mm-diameter disks
were then partially cut using an ultrasonic cutter down to a depth of ~ 100 μm. Marks along the
+Y and +P directions were made using a scribe in order to keep track of the disk orientations
according to step 4 of Figure 30. Once the marks were made, ultrasonic cutting was continued
until the disks are completely cut through. Disks were then inserted in the sample holder and
electro-polished using a Fischione Model 110 Automatic Twin-Jet Polisher.
61
After much trial and error, a polishing recipe consisted of 75% methanol and 25% nitric
acid was found to be effective for both commercially available Al foils and ECAP AA1050
samples. Optimum polishing conditions were found as follows: the electrolytic fluid was cooled
by pouring liquid nitrogen (LN) directly into the methanol-nitric acid mixture until ice forms.
Electro-polishing was performed immediately once the ice melted with a jet speed set between 5
and 6 (arbitrary values, jet speed setting of the jet polisher ranges from 0-10) with a controlled
voltage between 10-12V, and a resulting polishing current between 15-20 mA. The twin-jet
polisher has an alarm (optical sensor) set to detect the presence of a hole in the TEM foil and
polishing is turned off. The average polishing time for ECAP AA1050 samples ~280 μm thick is
about 20 minutes. Resulting holes were approximately 150 μm in diameter. The sample holder is
removed from the electrolytic solution and rinsed in a methanol bath by repeatedly dipping the
holder and sample in methanol 15 times. Once the 3mm-disk is jet-polished, it is removed from
the sample holder and air dried. Scribe marks made along the perimeter of the disks are not
affected by the jet polishing process since they are covered by the sample holder and do not
come into contact with the electrolytic solution.
Optical images of the polished disks are taken at various magnification to track the
orientation of the hole with respect to the directional scribe marks along the disk perimeter. This
is done so that image rotation in the TEM can be tracked using the features of the hole at various
magnifications.
TEM images were taken on a JEOL JEM 2100F microscope at USC Center for Electron
Microscopy and Microanalysis (CEMMA). Samples were inserted into the holder with the
directional marks noted. The entire hole was first imaged at low magnifications. This allows for
the orientation to be tracked between the optical and TEM images. Figure 31 shows both a)
62
optical and b) TEM images of the jet-polished hole for the ECAP AA1050 8P sample. Note that
the two holes are mirror images of each other due to the different point of views. The optical
image is viewed from the "top" and TEM image is viewed from the "bottom". After the TEM
image is mirrored to match the point of view of the optical image, the rotation between the two
images are calculated by calculating the rotation angle between lines drawn between two
identical locations on each image.
Figure 31. a) Optical micrograph of the hole near the center of the jet polished disk for ECAP
AA1050 8P sample. b) Low magnification TEM micrograph of the same hole of the jet polished
disk.
Next, thin (electron transparent) areas around the hole perimeter were identified for high
magnification imaging. Features near the thin areas are identified and tracked as the
magnification is increased. New smaller features are selected and tracked as the magnification is
increased and the field of view no longer contains previous, larger, features. Rotation angles
between two images of different magnifications are measured by tracking two points of the same
feature that appear in both images. Bright field (BF) and Dark field (DF) images of the
microstructures of the ECAP 1P and 8P samples were taken.
63
2.3.3. Results
Both 1P and 8P samples prepared exhibited a large amount of cracks along the perimeter
of the holes as well as specimen curling seen in both optical and TEM images after jet polishing
(Figure 31). These behaviors were not observed in pure Al foils prepared using identical
preparation techniques.
TEM images of the ECAP AA1050 1Pass sample exhibit large grains with elongation
approximately along the +25
o
off the pressing direction. This is consistent with theoretical model
and previous studies [21, 42]. Grain shapes and sizes appear to vary between different areas
around the hole as observed in Figure 32 a) and b). Figure 33 shows the a) bright field, and b)
dark field TEM micrographs, showing the microstructures of the 1 Pass samples consisted of
grains (defined by the large misorientation angles), and within the grains, high and low
dislocation density regions. This is consistent with TEM images done by our collaborator, Dr.
Alan Fox (Figure 26).
Figure 32. Bright field TEM micrographs of ECAP AA1050 1P sample from two separate
locations roughly 50 μm apart from the same sample. Large differences in grain size can be
observed between a) and b). Both exhibit grain elongation approximately +25
o
off the pressing
direction (+P)
64
Figure 33 a)Bright field and b) dark field TEM micrographs of ECAP AA1050 1P sample from
the same region. Dark field was performed to see individual grains
TEM images of the 8 Pass (Figure 34) sample show grains which appear much more
homogenous, less directional, and smaller compared to the 1P sample. The indicated directions
are for the final ECAP pass. Variations in grain shape and size between different locations
appear to be minimal in contrast to the 1P sample Figure 32. The observed microstructures
consisted of the grains (defined by large misorientations), and within the grains, low and high
dislocation density regions. It is interesting to note that in Figure 34 a) and b), there appears
slight grain elongation approximately +25
o
off the pressing axis (of the final pass). This is
similar to the observed grain elongation seen in the 1P sample. Figure 35 a) and b) show the
bright field and dark field TEM micrographs of an ECAP AA1050 8P sample. The
microstructures consists of grains (defined by high angle grain boundaries), and within the
grains, there are regions of high and low dislocation density.
65
Figure 34. Bright field TEM micrographs of ECAP AA1050 8P sample from two separate
locations roughly 50 μm apart from the same sample. Much smaller differences in grain size can
be seen between a) and b) when compared with the 1P sample. Both a) and b) exhibit slight grain
elongation approximately +25
o
off the pressing direction (+P) of the final pass
Figure 35. a) Bright field and b) dark field TEM micrographs of ECAP AA1050 8P sample from
the same region
66
Chapter 3. Full Elastic Strain/Stress Tensor X-ray Microbeam
Measurements
3.1 ECAP AA1050
3.1.1 Background
In this section, the full elastic strain and stress tensor measurements of the LRIS in ECAP
AA1050 are presented as a continuation of the earlier works [13, 55]. The full elastic strain/stress
tensors of low dislocation density areas within the grain/subgrain of ECAP AA1050 after 1, 2,
and 8 passes were measured using synchrotron X-ray microbeam diffraction at beam line 34ID-E
of the Advanced Photon Source (APS) at Argonne National Laboratory. The previous sections
and works [13, 55] only reported measurements along one direction (using a single reflection)
for each grain, and thus only the elastic strain/stress along one direction with respect to the
sample geometry was reported. In those studies, reflections were measured near (+5
o
) the
pressing direction, and off the pressing direction (+23.5
o
, and -17.5
o
) for a 1 pass ECAP AA1050
sample. Multiple pass samples (2, 4, and 8 pass) were measured only near the pressing direction
(+5
o
). Theoretically, the ECAP die imposes approximately +0.88 principal plastic strain for each
pass along the +22.5
o
direction with respect to the pressing direction, -0.88 plastic strain along
the -67.5
o
direction, and about zero plastic strain along the pressing direction [26]. Since only a
single reflection was measured in these earlier studies, the complete strain/stress tensors in the
sample coordinate system could not be determined, and, thus, the stress along arbitrary directions
in the sample could not be assessed. Also, the maximum LRIS may not be assessed
Three linearly independent reflections from each grain were assessed from the same
coherently diffracting volume (low dislocation density) within the grain/subgrains. These
reflections are used to calculate the full elastic strain and stress tensors, using the measured
67
unstrained lattice constant (which is different than the pure Al lattice parameter) and the Al
elastic constants. The measured elastic strain and stress tensors are then transformed (rotated)
into the sample reference frame (Fig.1). The measurements reported here complement the earlier
works by the authors [13, 55] and Alhajeri, et. al. [35] to fully assess the LRIS in ECAP
aluminum.
3.1.2 Microbeam Sample Preparations
The ECAP AA1050 specimens used were identical to those in the Alhajeri et. al. work
[35] which we also used in our previous studies [13, 55]. Samples were ECAP AA1050 after 1,
2, and 8 passes processed via route B
C
.
3.1.3 Powder Diffraction
Since strain tensor components are calculated based on a comparison between the
strained and unstrained lattice parameters, accurate strain/stress tensor measurements are highly
dependent on an accurate value for the unstrained lattice parameter (a
0
) of our 1050 alloy. For
reasons discussed in our earlier work [13], the unstrained parameter of pure Al is inadequate
since solutes in AA1050 can both increase and decrease the lattice parameter. The best way to
assess the lattice parameter of our samples is direct measurement. Unstrained lattice parameters
of AA1050 were measured using the X-Ray powder diffraction instrument on the 11-BM beam
line at the APS, at Argonne National Laboratory. The powder diffraction measurement
procedures were identical to those described earlier in this thesis and [13]. The measured lattice
parameter for the as-received (zero ECAP pass) was 4.05000(10) Å and was used as the
unstrained (baseline) lattice parameter a
0
for the strain tensor calculations. The unstrained (as-
received) sample exhibit no local residual stresses as discussed in earlier works [13, 55].
68
3.1.4 Microbeam X-Ray Diffraction Measurements
X-Ray microbeam diffraction measurements were performed on beamline 34-ID-E at the
APS using identical procedures as reported previously [13, 55]. Measurements for the current
study were made with a microbeam cross section of ≈ 0.4 μm × 1.4 μm. In contrast to the
previous works [13, 55], where only one reflection was measured for each grain, here, three
linearly independent reflections were used to obtain three independent lattice spacings from
different reflections and reflection directions associated with each grain/subgrain interior. The
full elastic strain and stress tensors from low dislocation volumes within the grain/subgrain of
ECAP AA1050 samples were extracted for ECAP AA1050 samples after 1, 2, and 8 passes. Six,
eight and seven grain/subgrain interiors were measured for the 1, 2, and 8 pass samples, for a
total of 63 measured reflections. Lattice parameters in the high dislocation density regions (cell
walls) were assessed on strained (0.30 strain) single crystal Cu in a previous work by the authors
[3]. There, the cell walls are diffuse and less well-defined, and X-ray diffraction corresponding
to the walls could be reasonable assessed. However, in ECAP, the grain/subgrain boundaries
have much higher dislocation densities which resulted in extremely diffuse diffraction peaks,
rendering lattice parameters measurements of the grain/subgrain boundaries extremely difficult.
While simple in theory, full strain tensor measurements are quite complicated in practice
and procedures have only recently become available. The reflections must originate from
identical volumes within the material for a legitimate full strain tensor calculation. In the past, an
imperfect depth resolving wire prevented an accurate determination of depth for peaks with wide
angular separations, rendering the matching of peaks to the same diffracting volume, and thus
full strain tensor measurement, impossible. However, with the recent development of a more
accurate and precise depth resolving wire assembly (varying by ≈ 10 0 nm over 5 mm), the
69
location of the diffracting volume can be trusted, enabling full strain tensor measurements. The
measured reflections were chosen from all three detectors to have the widest possible angular
range in order to minimize the uncertainty in the full strain tensor calculation.
In a polycrystalline sample, such as ECAP AA1050, the diffracting volumes often have
larger misorientations between grains compared with misorientations between dislocation cells in
a deformed single-crystal, such as with our earlier work in [3]. A white-beam diffraction image
of the bulk sample of AA1050 contains many diffraction patterns, each quite different from one
another due to the large misorientation between diffracting volumes in a polycrystalline sample.
Each diffraction pattern is unique and comes from a specific grain/subgrain along the beam path.
The depth profiling wire is used to measure the depth of the diffracting volume. The geometry
(lattice parameters and angles) of the strained unit cell and the crystallographic orientation
include 9 unknowns (6 independent components of the elastic strain tensor and 3 components
(Euler's angles) describing the unit cell orientation. By measuring the positions of three
independent diffraction spots on the detectors and measuring the corresponding lattice spacings
(d-spacings) by conducting energy scans, 9 parameters can be extracted allowing the complete
cell geometry and orientation to be determined.
Data collection started by performing a white beam wire scan using the “orange” detector
(top). Diffraction patterns from the white beam wire scan on the orange detector were
reconstructed for various depths. That is, diffraction patterns coming from each depth are
separated. Patterns of well-defined diffraction peaks from a well-defined diffraction volume are
then indexed from the depth resolved white beam images from the orange detector. White beam
images were then captured using the “purple” and “yellow” detectors (side detectors). By
matching the indexed patterns from the orange (top) detector to the purple and yellow (side)
70
detectors, separate diffraction peaks from the same diffraction volume are identified., Three
diffraction peaks are needed as described above. These diffraction peaks are chosen based on
their intensities and angular separations. Diffraction peaks are chosen from different detectors
when possible to have the widest angular range between each other, thereby decreasing the
uncertainties in the final strain and stress tensors. Diffraction peaks that are high in intensity and
well defined in shape result in a more accurate measurement of both the lattice spacing (from the
energy scans) as well as a more accurate determination of the lattice plane orientation (from the
peak centers). Here, the lattice plane orientations represent the plane normal directions of the
diffracted peaks. Energy scans are performed on each peak, once three linearly independent (hkl
reflections are independent) diffraction peaks are selected over an energy range large enough to
cover the full range of crystallographic orientations of the coherently diffracting volume within
the grain/subgrain.
The lattice spacing, Miller indices of the diffracted peak, and location of the peak centers
are determined from each energy scan. These steps are done in a similar fashion to those
described by the authors in [Levine]. Peak locations (on the detectors) are determined both by
fitting the peak with a Gaussian surface and calculating the "center-of-mass" of the region
surrounding the diffraction peak. A final assessment is made for the values of peak locations by
averaging of the results by peak fitting, calculating the "center of mass", and by visual inspection
[Levine]. For the majority of the cases, peak locations agree well with each other using the two
different peak-fitting assessments. The variations observed are used as uncertainties in peak
locations. Lattice spacings are determined as described by previous work by Levine et. al [3].
Briefly, the X-ray intensity at each pixel is recorded as a function of photon energy during an
energy scan. Using these data, a diffraction line profile is constructed for each pixel [3]. The line
71
profiles from all the pixels associated with a given diffraction peak on the detector are then
summed to form a composite line profile. The peak center of this line profile in reciprocal space
(not to be confused with the diffraction peak center on the detector) is found by fitting the profile
using a Gaussian, Lorentzian, or Pseudo-Voigt distribution, as appropriate depending on the best
fit.
3.1.5 Full Elastic Strain/Stress Tensor Calculations
Once the energy scans are performed and diffraction peak data analyzed, the lattice
spacings, peak locations, and reflection indices and the associated uncertainties of the three
peaks are used as inputs for a Matlab program written by the author (Thien Phan) (Appendix B)
to calculate the full strain tensor. The program reads in the instrument geometry calibration file
and the analyzed diffraction peak data to calculate the full strain tensor of the diffracting volume
in the crystallographic orientation. A Monte Carlo uncertainty analysis is used to propagate the
uncertainties from the inputs to calculate the uncertainty in each output.
First, the vectors from the diffraction volume to the diffraction peaks on the detectors are
determined. The geometry file describes each detector location in space as a series of translation
and rotation from a defined initial position. From the pixel locations and the geometry file, three
vectors from the diffraction volume to the diffraction peaks are calculated. From the incident
beam path and these three vectors, the plane normal vectors can be calculated and are
perpendicular to their respective diffraction planes. The plane normal vectors are then
normalized (in length) and multiplied by the reciprocal lattice spacing (q). These vectors are now
the reciprocal lattice vectors. The set of base vectors along the primary crystallographic
directions [100], [010], and [001] in reciprocal space are then calculated by Equation 12:
72
(12)
Equation 12. Algorithm to calculate the base vectors along the primary crystallographic
directions from 3 linearly independent vectors along arbitrary directions using Gaussian
Reduction
Here V
1
, V
2
, and V
3
(with components V
1x
, V
1y
, V
1z
, etc.) are the reciprocal lattice
vectors along their respective reflection (hkl), and V'
1
, V'
2
, V'
3
are the reciprocal lattice vectors
along the primary crystallographic directions [100], [010], and [001]. The real space unit cell
basis vectors are then converted from the reciprocal space vectors by Equation 13:
(13)
(13)
(13)
Equation 13. Equations to convert reciprocal space lattice vectors to real space lattice
vectors
where a
1
, a
2
, and a
3
are the real space unit cell basis vectors along the [100], [010], and
[001] directions. Once the real space unit cell basis vectors are calculated, the measured strained
unit cell parameters and angles are calculated. Unit cell lattice spacings are calculated by
measuring the lengths of the unit cell vectors (a
1
, a
2
, and a
3
). The angles α, β, and γ are
determined from the angles between the unit cell vectors. These values are then compared to the
unstrained lattice parameters of ECAP AA1050 measured from powder diffraction at beamline
11I-DE at the APS. The elastic strain tensor elements are calculated using the strained and
unstrained unit cell parameters following Equation 14:
73
(14)
The full elastic stress tensor is calculated by converting the full strain tensor to Voigt
notation (reduced engineering notation) and multiplying it by the appropriate stiffness tensor for
Al (Equation 15). The elastic constants used in the stiffness matrix [C] are assumed to be (C
11
=
108.2, C
12
= 61.3, C
44
= 28.5) [56]
(15)
where
3.1.6 Transformation of Strain and Stress Tensors
The calculated strain and stress tensors describe the full strain/stress state in the
coordinate system of the unit cell. However, for the case of ECAP processing, we are interested
in examining the full state of stress and strain in the sample coordinate system. First, the rotation
matrix used for rotating the strained unit cell into the lab orientation (Figure 36) was determined.
This would normally be sufficient to describe the transformation of the stress and strain tensors.
However, since the measured strained unit cell basis vectors are no longer orthogonal to each
other due to the shear components of the strain tensor, a new orthogonal basis system must be
defined for the crystallographic orientation. This new orthogonal basis system is defined in a
74
manner described by Levine et al. A new z' axis is defined as orthogonal to the strained x-y
plane, and the new y' axis is defined as orthogonal to the x-z' plane. The direction vectors along
the x, y', and z' axis of the unit cell are calculated. Next, the rotation matrix [R] to rotate
between the crystallographic reference frame and the lab frame is calculated. This rotation matrix
is then used to transform the strain and stress matrices from the crystallographic orientation into
the laboratory orientation (Equation 16). The laboratory reference frame is described by Figure
36.
(16)
Note that transformation of the full strain and stress tensors must be done independently
(separately); that is,
cannot be calculated from
using the original stiffness tensor [C]
before rotation ([C] is only appropriate for unit cell orientation). This is due to the fact that Al is
anisotropic and that the stiffness (fourth-rank) tensor is orientation dependent and varies with
crystallographic orientation.
Once the strain and stress tensors are rotated into the lab coordinate system, a second
transformation is used to rotate the strain and stress tensors from the lab coordinate system to the
sample coordinate system. This is calculated in a similar manner as the first transformation,
albeit, with a different rotation matrix, [R
1
]. The sample was positioned along the X-ray beam
with its surface normal oriented +45
o
from the X-ray direction as described by Figure 36. The
rotation matrix to transform the strain and stress tensors from the lab reference frame to the
sample reference frame is given by Equation 17:
(17)
75
The strain and stress tensors in the sample coordinate system are calculated by
substituting [R
1
] in the place of [R] into Equation 16. Again, the strain and stress tensors are
transformed independently (separately) from the lab orientation to the sample orientation
following the schematic in Figure 36. Note that the sample coordinate system is defined such that
the x-axis points in the same direction as the x-axis in the lab coordinate system, the z-axis points
along the pressing direction into the sample, and the x-y plane is the sample surface (Figure 36).
This is different from the ECAP coordinate system defined by Langdon et al. [16]. We chose to
define a new coordinate system due to the left-handed convention used by Langdon et al. [16].
Figure 36 shows the new coordinate system (X-ray convention) following the right-handed
convention compared with the coordinate system defined by Langdon (ECAP convention). From
this point on, the sample coordinate system and all directions will follow those described by the
"X-ray Sample Coordinate" from Figure 36.
Figure 36. Schematic showing the X-ray Microbeam procedure [13] and the various coordinate
systems. The ECAP, X-ray Laboratory, and X-ray Sample coordinate systems are shown in
relation to the ECAP sample. The x-direction points out of the page for the X-ray sample and
laboratory coordinate systems; the y-direction points out of the page for the ECAP coordinate
system. Note that the ECAP coordinate system uses a "left-handed" convention while the X-ray
Laboratory and Sample coordinate system use a "right-handed" convention.
76
3.1.7. Principal Strains and Stresses
The principal strains/stresses and their directions are calculated after the full elastic strain
and stress tensors are transformed into the sample coordinate system. This is done by
diagonalizing the elastic strain and stress tensors and calculating the eigenvalues and
eigenvectors. The eigenvalues are reported as the principal strains/stresses and the associated
eigenvectors are the principal strain/stress directions.
3.1.8. Uncertainty Analysis
The uncertainty analysis was handled in an identical way as described by Levine et al.
Uncertainties in the final output depend on the uncertainties in the instrument calibration,
diffraction peak position on the detector, determination of the center of the line profile (or lattice
spacing), and the unstrained lattice parameters. Instrument calibrations were done as described
by Levine to an energy resolution of E/E < 1×10
-4
, a root-mean-square angular uncertainty of
the peak positions of 0.005° (smaller than the subtended angle between each individual pixel),
and the wire angular deviations were less than 2 mrad. The instrument calibration uncertainties
were determined to be small compared to the uncertainties from the deformed ECAP AA1050
diffraction data and were not included in the uncertainty analysis.
The uncertainties in the diffraction peak locations (pixel locations on the detectors) and
line profile peaks (measured lattice spacings) were propagated through the analysis using a
Monte Carlo algorithm to calculate the uncertainties for each of the components of the full
elastic strain and stress tensors. For each measured input, an associated uncertainty (in the form
of one standard deviation) was determined. An input array of 10,000 numbers is then generated.
The input array has a Gaussian distribution with a center at the exact input, and a standard
77
deviation determined by the uncertainty. This is equivalent to generating 10,000 variants for each
set of diffraction measurements. The full elastic strain and stress tensors are then calculated
10,000 times, using the different variants of the measured inputs. Thus, there are 10,000 full
elastic strain and stress tensor outputs for each measured set of inputs. Each component of the
output elastic strain and stress tensors is analyzed and its standard deviation is calculated. This is
then reported as the uncertainty in the output value.
3.1.9 Results and Discussion
An example output of the Matlab program is shown in Table 4 for measurements made
from one grain/subgrain interior. The output includes the best values along with the calculated
uncertainties for the strained unit cell parameters (a, b, c, , , and ), the full elastic strain and
stress tensors in both crystallographic and sample orientations, the principal strains/stresses and
their directions, and the hydrostatic stress.
Table 4. Example output from Matlab with inputs from measurements of one low dislocation
density volume within a grain/subgrain of ECAP AA1050 1 Pass.
The measured lattice parameters are (in nm):
a = 4.0505063e-01 +/- 3.3750121e-05
b = 4.0483526e-01 +/- 4.4084291e-05
c = 4.0515898e-01 +/- 2.4056754e-05
alpha = 90.001526 +/- 0.009739
beta = 89.940184 +/- 0.011296
gamma = 89.977978 +/- 0.010207
The strain tensor components in sample coordinates:
e11 = 1.11e-05 +/- 9.56e-05
e22 = -3.88e-04 +/- 1.54e-04
e33 = 4.87e-04 +/- 6.43e-05
e23 = -3.47e-04 +/- 6.64e-05
e13 = -3.78e-04 +/- 6.99e-05
e12 = 1.36e-04 +/- 9.50e-05
The stress tensor components in sample coordinates (MPa):
s1 = 7.4 +/- 13.9
s2 = -14.4 +/- 19.4
s3 = 32.3 +/- 12.8
s4 = -16.8 +/- 3.5
s5 = -20.8 +/- 4.1
s6 = 10.1 +/- 4.9
78
The principal strains are:
Principal strain 1 = 8.11e-04 +/- 7.74e-05
Principal strain 2 = -1.91e-04 +/- 1.17e-04
Principal strain 3 = -5.09e-04 +/- 1.35e-04
The principal strain directions are:
v1 = (-0.449557, -0.294979, 0.843141)
v2 = (0.893173, -0.135895, 0.428690)
v3 = (0.011876, -0.945791, -0.324560)
The principal stresses are (MPa):
Principal stress 1 = 50.2 +/- 14.4
Principal stress 2 = -4.3 +/- 13.8
Principal stress 3 = -20.4 +/- 18.2
The principal stress directions are:
v1 = (-0.473848, -0.291316, 0.831025)
v2 = (0.864622, 0.025084, 0.501797)
v3 = (0.167027, -0.956298, -0.239993)
The hydrostatic stress is (MPa): 8.5 +/- 14.9
Table 5 lists the long range internal stresses along the principal stress directions of ECAP
AA1050 after 1 pass for 6 grain/subgrain interiors. The grains are relatively close together in this
and other experiments; tens of microns apart. From each grain interior, three principal stresses
are calculated. These long range internal stresses are the maximum normal stresses in an
orientation where the shear stresses are zero. Across all measurements from six different grain
interiors, the principal stresses (LRIS) range from about 50 MPa (tensile) to -100 MPa
(compressive), (0.34 σ
a
to -0.66 σ
a
), a spread of about 150 MPa, or roughly the magnitude of the
flow stress ( σ
a
= 148 MPa for 1 pass sample [36]). The uncertainty values for the principal
stresses, as evident in the example in Table 4, are roughly ±15 MPa. From the 18 principal stress
(LRIS) values (3 per grain/subgrain interior), only two values were tensile. The spread of
principal stresses (LRIS) measured within one grain interior is smaller, only about 70 MPa, or
about 0.5 σ
a
. This range (70 MPa) of principal stresses within each grain interior is roughly
consistent between different grain interiors for the 1 pass ECAP sample.
79
Table 5. Long range internal stresses of ECAP AA1050 after 1 Pass for 6 measured low
dislocation density volumes within the grains/subgrains along the principal stress directions. “G”
refers to different grain interiors. (σ
a
= 148 MPa for 1 pass sample [36])
ECAP AA1050 1 Pass
Principal
Stresses
G1 G2 G3 G4 G5 G6
σ1 (MPa) 50.2±14.4 12.3±14.5 -25.3±12.6 -15.8±15.0 -16.0±12.3 -20.8±16.5
σ2 (MPa) -4.4±13.8 -45.3±19.3 -63.9±19.8 -39.7±17.8 -42.3±18.0 -37.5±14.6
σ3 (MPa) -20.4±18.3 -48.2±19.9 -97.8±21.8 -71.5±18.1 -72.7±17.0 -69.5±21.7
σ1/σ
a
0.34±0.1 0.08±0.1 -0.17±0.09 -0.11±0.1 -0.11±0.08 -0.14±0.11
σ2/σ
a
-0.03±0.09 -0.31±0.13 -0.43±0.13 -0.27±0.12 -0.29±0.12 -0.25±0.1
σ3/σ
a
-0.14±0.12 -0.33±0.13 -0.66±0.15 -0.48±0.12 -0.49±0.11 -0.47±0.15
Table 6 lists the long range internal stresses along the principal stress directions of ECAP
AA1050 after 2 passes for 8 grain/subgrain interiors. Again, three principal stresses (LRIS) are
calculated from each grain/subgrain interior. The long range internal stresses range from 19 MPa
(tensile) to -160 MPa (compressive), (0.12 σ
a
to -1.07 σ
a
). This represents a range of ≈ 180 MPa,
or about 1.2 σ
a
( σ
a
= 150 MPa for 2 pass sample [36]). The range (difference between the largest
and smallest values) of the principal stresses within a particular grain interior is between 20 MPa
and 140 MPa.
Table 6. Long range internal stresses of ECAP AA1050 after 2 Pass for 9 measured low
dislocation density volumes within the grains/subgrains along the principal stress directions. "G"
refers to different grain interiors. (σ
a
= 150 MPa for 2 pass sample [36])
ECAP AA1050 2 Pass
Principal
Stresses
G1 G2 G3 G4 G5 G6 G7 G8
σ1 (MPa) 18.7±21.1 18.2±22.7 -4.8±17.9 34.5±17.5 -9.3±17.5 8.2±18.8 -18.3±16.9 -87.9±15.3
σ2 (MPa) -19.7±20.2 -14.2±18.6 -7.1±16.4 -29.9±15.1 -38.0±17.3 1.2±16.5 -27.7±12.7 -107.4±19.3
σ3 (MPa) -53.8±19.0 -49.4±19.2 -23.9±18.0 -101.5±18.1 -72.9±16.7 -11.1±17.4 -52.2±14.7 -160.8±21.9
σ1/σ
a
0.12±0.14 0.12±0.15 -0.03±0.12 0.23±0.12 -0.06±0.12 0.05±0.13 -0.12±0.11 -0.59±0.10
σ2/σ
a
-0.13±0.13 -0.09±0.12 -0.05±0.11 -0.2±0.10 -0.25±0.12 0.01±0.11 -0.18±0.08 -0.72±0.13
σ3/σ
a
-0.36±0.13 -0.33±0.13 -0.16±0.12 -0.68±0.12 -0.49±0.11 -0.07±0.12 -0.35±0.1 -1.07±0.15
80
Table 7 lists the principal stresses (LRIS) of ECAP AA1050 after 8 passes for 7
grain/subgrain interiors. The principal stress (LRIS) values range from 131 MPa (tensile) to -56
MPa (compressive), (0.65 σ
a
to -0.28 σ
a
) a range of about 190 MPa, or roughly 0.95 σ
a
( σ
a
= 200
MPa for 8 pass sample [36]). The range of principal stresses within a particular grain interior is
about 15 MPa to 100 MPa for the 8P sample. In this and the other samples (1P and 2P), there
often exists a significant degree of anisotropy in the stresses. This is not surprising considering
the fact that the applied deformation is highly anisotropic (ECAP deforms via simple shear). In 7
grain interiors, 2 have principal stresses that are all positive (tensile). This is not observed in the
1 and 2 pass cases.
Table 7. Long range internal stresses of ECAP AA1050 after 8 passes for 7 measured low
dislocation density volumes within the grains/subgrains along the principal stress directions. "G"
refers to different grain interiors. (σ
a
= 200 MPa for 8 pass sample [36])
ECAP AA1050 8 Pass
Principal
Stresses
G1 G2 G3 G4 G5 G6 G7
σ1 (MPa) -25.0±13.9 -11.3±15.4 23.2±17.1 4.2±12.4 68.0±19.3 130.6±21.9 0.5±16.7
σ2 (MPa) -43.8±14.9 -16.5±17.2 0.7±20.5 -15.6±15.7 56.4±18.8 78.6±14.5 -10.4±19.9
σ3 (MPa) -55.9±17.8 -27.1±17.5 -27.3±16.0 -27.6±17.2 21.9±17.7 32.4±16.1 -56.2±22.8
σ1/σ
a
-0.12±0.07 -0.06±0.08 0.12±0.09 0.02±0.06 0.34±0.10 0.65±0.11 0±0.08
σ2/σ
a
-0.22±0.07 -0.08±0.09 0±0.10 -0.08±0.08 0.28±0.09 0.39±0.07 -0.05±0.10
σ3/σ
a
-0.28±0.09 -0.14±0.09 -0.14±0.08 -0.14±0.09 0.11±0.09 0.16±0.08 -0.28±0.11
Figure 37 shows the stress tensor components of all measured low dislocation density
regions within the grain/subgrain interiors (transformed into the sample reference frame) from
the 1, 2, and 8 Pass samples. Corresponding stress components are grouped together. The shear
components ( σ
23
, σ
13
, and σ
12
) for all samples (1, 2, and 8 pass) are both smaller in magnitude
and exhibit less variation compared with the normal stress components ( σ
11
, σ
22
, and σ
33
).
Despite the ECAP process being that of simple shear, the measured shear components in the
sample reference frame remain much smaller compared with the normal components.
81
The 11 (x) and 22 (y) directions define the plane normal to the pressing axis (z, 33
direction). The 1P sample exhibits mostly negative stresses in the x and y direction, and on
average, zero stress along the pressing direction. The 2 pass sample exhibits mostly negative
stresses along all x, y, and z directions with the exception of two positive values along the y
direction. Normal stress components for the 8 Pass sample are slightly negative along the x
direction, while roughly zero along the y and z directions. The stresses in the 8P sample are more
isotropic than those in the 1P and 2P samples. Since the geometry of the grains/subgrains for the
1P and 2P samples are not fully isotropic (significant grain elongation and not fully
homogenous), and evolve to a more isotropic geometry with repeated ECAP passes with rotation
(route B
C
), the state of stress for the 8P sample is consistent with what we would expect for the
homogenized microstructures.
Figure 37. Full stress tensor components in the sample coordinate system for ECAP AA1050
after various passes. The 33 direction is along the ECAP axis.
82
Figure 38. Shows the elastic strain tensor components of all measured low dislocation
density regions within the grain/subgrain interiors (transformed into the sample reference frame)
from the 1, 2, and 8 pass samples. Corresponding strain components are grouped together.
Unlike the shear stress components in the stress tensor, the shear strain components ( ε
23
, ε
13
, and
ε
12
) for all samples (1, 2, and 8 passes) are roughly equal in magnitude and exhibit about the
same amount of variation compared with the normal strain components ( ε
11
, ε
22
, and ε
33
).
Figure 38. Full elastic strain tensor components in the sample coordinate system for ECAP
AA1050 after various passes. The 33 direction is along the ECAP axis.
This discrepancy between the magnitude in the shear components of the elastic strain
tensor and the stress tensor is a direct result of the generalized Hooke's Law and the stiffness
matrix for Al. Due to the matrix multiplication, normal components of the stress tensor are the
83
results of combinations of the products of the normal components of the strain tensor and C
11
and
C
12
. In contrast, shear components are only multiplied by C
44
. The larger magnitude of the shear
components in the elastic strain tensor is more consistent with the deformation process (simple
shear) imparted by ECAP. This is not observed in the associated stress tensor since the shear
stresses are much smaller in magnitude compared to the normal stresses.
It is important to note that since the calculation of the stress tensor requires the matrix
multiplication of the strain tensor by the stiffness matrix, each component of the stress tensor
depends on multiple values from the strain tensor. This means that it is impossible to accurately
predict the state of stress from incomplete strain tensor measurements, such as a single
measurements of strain along one reflection (direction), or from 2-dimensional measurements of
strains (Alhajeri's measurements [35] from CBED patterns).
Figure 39, Figure 40, and Figure 41 show the full elastic strain tensor components of all
measured low dislocation density regions within the grain/subgrain interiors (in the sample
reference frame) from the 1, 2, and 8 pass samples respectively. The average values are plotted
to the left of each strain component.
84
Figure 39. Full elastic strain tensor components in the sample coordinate system for ECAP
AA1050 after 1P. The 33 direction is along the ECAP axis.
Figure 40. Full elastic strain tensor components in the sample coordinate system for ECAP
AA1050 after 2P. The 33 direction is along the ECAP axis.
85
Figure 41. Full elastic strain tensor components in the sample coordinate system for ECAP
AA1050 after 8P. The 33 direction is along the ECAP axis.
The magnitudes of the average elastic strains along the ε
33
component from Figure 39,
Figure 40, and Figure 41 are along the comparable direction to elastic strain values reported in
the previous work by the author [55], where elastic strains were measured close to the pressing
directions (+4.9
o
off-axis) for ECAP AA1050 after various passes. The average elastic strains for
1 and 8 pass samples along the pressing direction are slightly positive and the average elastic
strain for the 2 pass sample along the pressing direction is slightly negative. These are different
than the reported negative elastic strains measured along the near pressing direction (+4.9
o
off-
axis) for all 1, 2, and 8 pass samples in the previous work [55].
Figure 42 shows all of the elastic strain components for the ECAP AA1050 1 pass sample
in two different reference frames. The squares are components of the elastic strain tensor in the
sample reference frame and the circles are the components of the elastic strain tensor in the
86
+22.5
o
off-axis reference frame, oriented along the predicted directions of maximum tensile and
compressive plastic strain.
Figure 42. Full elastic strain tensor components of ECAP AA1050 1 pass sample in the sample
reference frame and transformed (rotated) to +22.5
o
reference frame. Average values for each
component are plotted as solid triangles, red for the sample reference frame and blue for the
+22.5
o
reference frame. The red outline signifies the elastic strain components along the pressing
direction and along the +22.5
o
off-axis direction.
As expected, values along the 11 direction (x-axis) remain identical since the strain tensor
is transformed (rotated) along the y-z plane (ECAP pressing plane). The red box outlines
components along the z-axis for both the sample and +22.5
o
off-axis reference frames. These are
effectively the elastic strains along the pressing axis and along the +22.5
o
off-axis directions.
Values of the ε
33
component are plotted separately in detail in Figure 43.
87
Figure 43. Elastic strains of ECAP AA1050 sample after 1 pass along the pressing and +22.5 deg
off axis directions. Average values are plotted as solid triangles to the left of the elastic strain
values.
Figure 43 shows in detail the ε
33
components of the full elastic strain tensors transformed
along two different reference frames. Values plotted are the elastic strains along the pressing
direction and along the +22.5
o
off-axis direction. Again, the maximum (tensile) principal plastic
strain direction (+0.88) is along the +22.5
o
off axis, and zero strain along the pressing direction
based on theory [26]. We therefore, might expect a large difference between the measure elastic
strain between these two directions. However, only a small decrease (a strain of ≈ 6 10
-5
) is
observed from the average elastic strain along the pressing direction to the average elastic strain
along the +22.5
o
off-axis direction. The decrease in strain was similar to the observed decrease in
strain in a previous work [13], where a decrease of 8 10
-5
was reported from the near pressing
direction (+4.9
o
) to the +27.5
o
direction. However, both elastic strain average values measured in
88
the present work are positive, signifying a tensile strain along both the pressing and +22.5
o
directions, which does not agree with previous measurements [13], which were both negative.
If the composite model applies to the deformation of ECAP AA1050, and we assume a
two-component microstructure, then we should expect to see a compressive strain in the
measured elastic strain along the +22.5
o
direction. However this was not observed. Two possible
reasons could explain the discrepancy. One, the composite model does not apply to the
microstructure of deformed ECAP AA1050, that is, the grain/subgrain boundaries (defined by
high angle misorientations between grains with thicknesses of a couple nanometers) of ECAP
aluminum alloy does not act in a similar way as thick cell walls (high dislocation density
regions) in <100> deformed single crystal Cu [2]. Or two, that the assumption of a two-
component microstructure is insufficient, that the existence of high dislocation density regions
within the grain/subgrain interiors require a stress balance between three components (as is
assumed in previous works [13, 55]): the grain/subgrain boundaries, the low dislocation density
regions within the grain/subgrain interiors, and the high dislocation density regions within the
grain/subgrain interiors. Third, residual stresses may be present although undetected in our
powder experiment.
It is then of interest to investigate the directions of the principal elastic strains of the
measured low dislocation density regions within the grain/subgrain interiors and compare them
to the directions of the plastic strains imposed by the ECAP process. Figure 44 shows the
directions of the maximum and minimum principal elastic strains for all the measured low
dislocation density regions within the grain/subgrain interiors for 1, 2, and 8 pass samples. Each
blue arrow represents the direction of the calculated maximum elastic principal strain and each
89
red arrow represents the direction of the calculated minimum elastic principal strain for each low
dislocation density regions within the grain/subgrain interior.
Figure 44. Maximum (tensile) and minimum (compressive) elastic principal strain directions of
low dislocation density regions within the grain/subgrain interiors of ECAP AA1050 after
various passes. The coordinate system is identical to that of the "Sample X-ray Coordinate"
system described earlier in Figure 1.
From each full elastic strain tensor, the 3 principal strains are calculated by calculating
the three eigenvalues. Their associated eigenvectors are the three principal strain directions.
These three principal strain directions are perpendicular to each other and describe the
orientation where the elastic normal strain components are maximum, and elastic shear strain
components are zero. It is important to note that while the word "direction" is used, the arrows
only describe the orientation of principal strains.
90
It appears that the maximum and minimum elastic strain directions for the 1 pass sample
are the most directional (grouped together) compared to the directions of the 2 and 8 pass
samples. For the 1 pass sample, the maximum (tensile) elastic principal strain directions, on
average, point along the z-axis, or the pressing direction, with little out of ECAP-plane variations
(ECAP-plane is described by the y-z plane). Since ECAP processing deforms via simple shear
along the pressing plane (y-z) plane, it is expected to produce zero plastic strain after 1 pass in
the off-plane direction (x-axis). However, there is no expectation that the maximum elastic strain
direction is along the pressing direction.
It is observed that the maximum elastic strain directions for the 2P and 8P samples have
much larger variations as well as off-axis components (x-axis) compared with the 1P sample.
This is as expected due to the 90
o
sample rotation between each route-B
C
ECAP pass. The state
of strain within the 2P sample appears to be the least directional due to the large deformation
imparted by the ECAP process along two planes rotated by 90
o
. In contrast to the 2P sample, the
8P microstructures are already homogenized and the directionality of the arrows might be
reflective of the most recent ECAP passes. It is important to differentiate between the
directionality of principal strains and the isotropic nature of the state of strain. In order to gauge
the isotropy of strains, the magnitudes of all components of the strain tensor must be assessed.
While the 1P and the 8P samples both show (somewhat similar) directionality in the maximum
elastic strain direction based on Fig 3, the 8P sample exhibits a much more isotropic state of
strain when contrasting the full strain tensor components in Fig 7 and 9. There are much less
variations between the different strain components of the 8P sample compared with the 1P
sample.
91
It is observed that the minimum (compressive) elastic strain directions for all 1, 2, and 8
pass samples are less directional than the maximum (tensile) elastic strain directions. All samples
exhibit larger out of plane (x axis) components in the minimum (compressive) elastic strain
directions compared with the maximum elastic strain directions.
Figure 45 shows the directions of the maximum and minimum principal stresses for all
the measured low dislocation density regions within the grain/subgrain interiors for 1, 2, and 8
pass samples. Each blue arrow represents the direction of the calculated maximum principal
stress (most tensile or least compressive) and each red arrow represents the direction of the
calculated minimum principal stress (most compressive) for each low dislocation density regions
within the grain/subgrain interior.
As expected, Figure 45 shows that the principal stress directions for all samples are
similar, though not exactly identical, to the principal elastic strain directions in Figure 44. This is
due to the anisotropy of the Al crystal. All observations regarding the directionality of the
principal elastic strain directions apply to the directionality of the principal stress directions.
Even with the high directionality of the maximum principal stress/strain directions for the 1P
sample (Fig.12 and 13), measurements made along a single reflection in the ≈ ±20
o
along the
pressing direction would be insufficient to detect any noticeable difference in strains and
stresses. This may be the reason why relatively little change was observed by the previous work
[13] in single reflection measurements along the pressing axis.
92
Figure 45. Maximum (tensile) and minimum (compressive) principal stress directions of low
dislocation density regions within the grain/subgrain interiors of ECAP AA1050 after various
passes. The coordinate system is identical to that of the "Sample X-ray Coordinate" system
described earlier in Figure 1.
Figure 46. shows the LRIS along the maximum (blue arrows) and minimum (red arrows)
principal stress orientations for individual grain/subgrain interiors for the ECAP AA1050 1 pass
sample.
93
Figure 46. LRIS within the 6 grain/subgrain interiors along the principal stress orientations of
ECAP AA1050 after 1 pass. The arrows describe the orientation of the principal stresses. The
blue arrows illustrate the orientations of the maximum (most tensile/least compressive) principal
stresses, and the red arrows illustrate the orientations of the minimum (most compressive/least
tensile) principal stresses. a) shows the side view, with the ECAP plane parallel to the page and
pressing direction pointing toward the right. b) shows the cross sectional view, with the pressing
axis pointing out of the page. The "*" next to the maximum LRIS in grain 5 signifies consistency
with the simple 2-component composite model. The coordinate system is consistent with that of
the "Sample X-ray Coordinate" system described earlier.
Figure 46. shows the LRISs along the principal stress orientations of the 1 pass sample
for 6 separate grain/subgrain interiors. Since the principal stress orientations are in 3-dimensions,
94
Figure 46 illustrates the LRISs and their orientations along two different points of view: a) with
the ECAP plane parallel to the page and the pressing direction pointing toward the right, and b)
with the pressing direction pointing out of the page. The arrows describe only the orientations
and their lengths are normalized to 1. For each grain/subgrain interior, the blue arrows describe
the orientations of the maximum (most tensile/least compressive) principal stress, and the red
arrows illustrate the orientations of the minimum (most compressive/least tensile) principal
stress. The directions of the arrows describe whether the LRIS is compressive (pointing toward
each other), or tensile (pointing away from each other). The magnitudes of the LRISs are listed
in corresponding colors to the arrows. LRISs are listed in MPa at the bottom of each
grain/subgrain plot, and in fraction of the flow stress (σ
a
= 148 MPa for 1P sample[36]) next to
the corresponding arrows. The LRIS values in Figure 46 are identical to the magnitudes of the
principal stresses listed as " σ1" (for blue values), and " σ3" (for red values) in Table 2. Again, the
maximum (most tensile/least compressive) principal stress orientations (blue arrows) for the 1P
sample align well with the pressing axis. Only 1 out of the 6 blue arrows (G6) has any significant
variations in the x-axis (out of ECAP-plane variation). Out of the 6 grains, only the LRIS along
the maximum (most tensile/least compressive) stress direction of "G4" is consistent with the
simple 2-component composite model. Since the blue arrow (of G4) points in the slightly
positive off-pressing-axis direction, this is a tensile plastic strain direction. According to the
simple 2-component composite model, the associated LRIS along the blue arrow directions
should be compressive, as was observed for 1 out of 6 grain interior). The LRIS along the
minimum (most compressive) principal stress orientations (red arrows) are all compressive. This
is in direct disagreement with the simple 2-component composite model. Again, based on theory,
the direction of maximum compressive plastic strain imparted by ECAP processing is along the -
95
67.5
o
off the pressing axis. With a simple composite model assuming a 2-component
microstructure and uniaxial deformation, we would expect a tensile elastic strain within the
grain/subgrain interiors, however, all the LRISs close to this direction are compressive. Overall,
Figure 46 shows that the LRISs within the grain/subgrain interiors are not consistent with the
composite model (assuming uniaxial deformation and 2-component microstructure)
It is important to note that Figure 46 shows 4 out of 6 LRISs are compressive close to the
pressing direction for the 1 pass sample. This is in contrast to the positive (tensile) elastic strain
values along the principal strain directions (also close to the pressing directions). While elastic
strains are positive (tensile) along the pressing direction, the LRIS can actually be negative
(compressive) along the maximum (most tensile/least compressive) principal directions, as is the
case reported here for 4 out of 6 grain/subgrain interiors. Therefore, while Figure 43 shows the
elastic strains along the pressing and +22.5
o
off-axis directions being positive (tensile), the
corresponding stresses are actually close to zero and slightly negative (compressive) in value.
3.1.10. Conclusions
Measured elastic strain/stress values for the full tensor assessments were calculated using
Matlab and the results have been rigorously checked for errors. However, there were
inconsistencies compared with previous measurements (single reflection) made on the same 1P
sample. Two reasons may account for the differences observed: the existence of residual stresses
at length scales smaller than the powder diffraction beam size, or that the LRIS in the
grain/subgrain boundaries caused by extrinsic dislocations vary in signs from one grain to the
next along a particular direction.
96
Since the full strain tensor measurements were assessed for grain/subgrains within tens of
microns in location, but differ in location from single-reflection measurements (also measured
within tens of microns in location), this could suggest the existence of residual stresses within the
sample. Although powder patterns indicated an absence of residual stresses in ECAP AA1050
samples, it is only valid for residual stresses with length scales larger than the powder diffraction
beam size (0.5 mm). Thus, this inconsistency (in measured elastic strains) may be attributed to
residual stresses with length scales smaller than the powder diffraction beam size (0.5 mm), and
larger than the microstructures (~1 μm). This suggests that the elastic strain and stress
measurements are convolutions of both LRIS and residual stresses. In order to deconvolute the
LRIS and the residual stress, we must be able to approximate the value of the residual stress.
The existence of residual stress was indicated by differences in baseline a
0
between
different samples measured by powder diffraction for the ECAP AA6005 samples. It was
determined that a strain difference of 1.2 x 10
-4
exist between the local large grain a
0
and the
global powder diffraction a
0
. To our best approximation, the residual strain assumed to exist in
ECAP AA1050 may be of the same magnitude as that of the ECAP AA6005. LRISs can then be
approximated by adding and subtracting the assumed residual stress to the x-ray measured
internal stresses to get a range of the LRISs. The maximum (in magnitude) LRISs are present in
ECAP specimens on the order of -0.37 to -0.49 σ
a
(compressive) and are roughly along the
vertical direction (y-axis) for the 1 pass sample. Maximum (in magnitude) LRISs are about -0.38
to -0.5 σ
a
(compressive) and are along random directions for 2P. Maximum (in magnitude)
LRISs are about 0.08 to 0.2 σ
a
(tensile) and are roughly along the vertical direction (y-axis) for
the 8 pass sample.
97
Values for the 1P and 2P samples are larger than values measured in previous works of ~
0.12 σ
a
(measured along the pressing direction) and values for the 8 pass sample are roughly
comparable to values measured in previous work of ~ 0.10 σ
a
(measured along the pressing
direction). The directions of the principal LRISs show that elastic strains and LRISs in
grain/subgrain interiors are most grouped together for the 1 pass sample, and are relatively
random for the 2 and 8 pass sample.
LRIS values measured suggest inconsistencies with the 2-component composite model.
This suggests that the simple 2-component composite model may not apply to the microstructure
of deformed ECAP AA1050. That is, the grain/subgrain boundaries (defined by high angle
misorientations between grains with thicknesses on the scale of a nanometer, more or less a “2D”
boundary) of ECAP aluminum alloy may not act in a similar way as thick cell walls (high
dislocation density regions, or 3D boundaries) in deformed single-crystal Cu [19]. The LRIS in
the thin ECAP boundaries may be the result of extrinsic dislocations and the signs of the stresses
in these (and grain/subgrain interiors) may change in sign from one grain interior to the next
along a particular direction.
While there may be inconsistency with the simple 2-component composite model as
described by Mughrabi for the uniaxial multi-slip deformation, a composite stress balance must
still be satisfied. That is, stress balance in ECAP AA1050 may need to be described using a 3-
component stress balance, that the stresses in the low dislocation density regions within
grain/subgrain interiors must be balanced by the stresses within the high dislocation density
regions within the grain/subgrain interiors and the grain/subgrain boundaries. The variations in
orientations of the principal elastic strains and stresses within the grain interiors suggest that the
98
elastic strains/LRIS within the high dislocation density regions and the grain/subgrain boundaries
also vary in orientations and may change in sign from grain to the next.
Measured elastic strain and LRIS values for different components of the tensor are much
more similar in magnitude in the 8P sample compared with the 1P sample. This suggests that
LRISs and elastic strains become more isotropic with increasing number of ECAP passes. While
other works have shown microstructure (grain size and shape) homogenization [11, 12, 20],
measurements reported here are the first confirmation that the local LRIS stresses and strains
also become increasingly isotropic after a large number of ECAP passes with sample rotation
using route B
c
.
It is important to discuss the accuracy of the LRISs assessed using 1D and 2D methods
such as x-ray diffraction along a single reflection and CBED measurements respectively. We
have observed a change in sign between the elastic strain and LRIS along a particular direction is
due to the nature of the tensor. By previously using a single reflection strain measurement
(microbeam diffraction), LRIS values can only be approximated by multiplying the strain by the
Young’s modulus o f the material, leading to inaccurate values. Furthermore, values measured
near the grain/subgrain boundaries by Alhajeri et. al.[6] using CBED which we utilized for a
stress balance analysis in our previous work [7] have their fair share of inaccuracies.
Measurements made by Alhajeri [6] gave a maximum principal LRIS of 0.75 σ
a
near the
grain/subgrain boundaries. On top of the possible stress relaxation caused by TEM foil
preparations, the orientation and location of the 2D strain tensor measured in [6] was not
specified. The full LRIS tensor measured in the present work shows the importance of
orientation since the LRISs vary drastically in magnitude and sign along different orientations.
This suggests that values measured by CBED [6] may be inaccurate and values measured are
99
inappropriate for stress balance purpose. Assessment of LRIS is best accomplished with the
analysis of the full elastic strain tensor.
Finally, the LRIS values reported here have been calculated using Matlab and the results
rigorously checked for errors. The inconsistency in elastic strain values may be due to residual
stresses that cannot be resolved using powder diffraction, or due to the changing in signs of the
LRIS caused by extrinsic dislocations at the grain/subgrain boundaries. The elastic long range
internal strain and stress tensors have been successfully assessed for small volumes within the
low dislocation density regions within the grain/subgrain interiors of ECAP AA1050. Compared
with results from previous measurements, values reported here may be the most accurate
measurements of LRISs within ECAP AA1050 samples.
3.2. Single Crystal Cu
3.2.1. Algorithm for Full Elastic Strain/Stress Tensor Measurements and
Calculations
Due to the differences in microstructure, the methods to measure the full strain tensor of
deformed Cu single crystal is different from those used to find the full elastic strain tensor of
ECAP AA1050. The small angle of misorientation between the cells of the deformed single
crystal Cu sample resulted in only one diffraction pattern under white beam (polychromatic).
Each diffraction "spot" however, is much more diffused (larger) than those observed in ECAP Al
and is made up of many smaller diffraction spots each from a different dislocation cell interior at
various depths. The small misorientations between dislocation cells result in very slight shifts
between diffraction patterns of the dislocation cells. This means that a very straight and well
calibrated depth profiling wire is needed for accurate depth reconstruction of the diffraction
100
patterns. Accurate depth reconstruction is important since the full strain tensor calculation
requires that the measured diffraction spots from different detectors must be associated with only
one dislocation cell interior. Strain calculations from spots which come from different cell
interiors but incorrectly assumed to be from the same diffraction volume can lead to
unreasonable large stress values (GPa range).
The unstrained lattice parameters of Cu used to calculate the strain tensor from
diffraction peak measurements are taken to be a
o
= 0.361496 nm [Wyckoff, 1963]. This is in
contrast to ECAP AA1050, which has solutes which change the lattice parameter, thus requiring
the use of powder diffraction to measure the unstrained lattice parameter. The Cu sample is a
high purity single crystal sample, enabling the use of reported lattice parameters of Cu for
unstrained lattice parameter.
While the method to measure the full strain tensor from polycrystalline samples such as
ECAP AA1050 requires 3 energy wire (EW) scans of three linearly independent diffraction
peaks to solve for the 9 unknowns in the full elastic strain tensor, these unknowns can also be
calculated by measuring 4 diffraction peaks. Full strain measurements of single crystal Cu
requires the measurements of at least 4 diffraction peaks, 3 of which are linearly independent and
are measured for directions (white beam wire scan), and the 1 one is measured for both direction
and energy (EW scan). This process allows for the construction of the unit cell by measuring the
directions of 4 reflections where 1 of those reflections are measured for the lattice spacing. This
effectively measures the full elastic strain tensor by measuring the deviatoric and hydrostatic
strain tensor separately. Information from the white beam wire scan from 4 linearly independent
diffraction peaks gives the directions and angles of diffracted planes but not lattice spacings.
This allows for the deviatoric strain tensor to be calculated using an arbitrary lattice spacing of 1.
101
The energy scan of the 4th diffraction peak allows for the measurement of one lattice plane
spacing and thus the hydrostatic components of the full strain tensor. The measurements are
completed according to the following steps:
First, a white beam wire scan is performed for all three detectors. This allows for the
reconstruction of the depth resolved white beam images for all diffraction peaks on all detectors.
That is, the diffraction patterns from dislocation cell interiors at various depths along the X-Ray
beam are reconstructed. The brightest 4 widely positioned diffraction peaks are selected and
matched to the dislocation cell interior from the reconstructed white beam images. The location
of each diffraction peak is measured on its respective detector.
Second, the lattice spacing must be measured for at least one additional reflection (hkl),
usually one of those used for the previous calculation. This is achieved by performing an energy
wire scan over the Orange detector. Due to its location right over the top of the sample,
diffraction peaks on the Orange detector are usually the brightest). The reconstruction of the
energy wire scan results in depth resolved energy wire scans. A diffraction peak associated with
the same white beam measured dislocation cell interior is chosen and its peak center and the
lattice spacing are measured.
The processes for obtaining peak centers and qs are identical to those used for ECAP Al
described in Section 1.2.4.2. Once the peak centers of all 4 reflections and q of 1 are determined,
the strain and unit cell basis can be calculated. The following calculations are done using Matlab
code similar to the one used for the full elastic strain/stress tensor calculations for Al. From each
reflection, several pieces of information can be calculated. For the 3 (directional) peaks that were
not energy scanned, the peak locations and the reflections (hkl) are measured. For the diffraction
102
peak that was energy scanned, the peak location, the reflection (hkl), and q are measured. Based
on this information, 4 vectors can be calculated, with 3 being plane normal directional vectors
and 1 being the reciprocal lattice vector. All 4 vector directions are determined from their peak
centers. While directions for all 4 vectors can be determined from peak centers, only the length
of the energy scanned vector is measured. The three directional vectors are given an arbitrary
length of 1. The next step is to determine the lengths of the 3 directional vectors. Using the
indices of each vector, a linear combination of the three directional vectors' indices is formed to
be equal to the reciprocal lattice vector's indices, or in matrix form (Equation 19):
(19)
Where h
i
k
i
l
i
for i = 1:3 are the reflection indices for the directional vectors and h
4
k
4
l
4
are the indices for the reciprocal lattice vector. This system of equations is solved for the scalar
multiple n
i
of each directional vector. Each directional vector is multiplied by its respective
scalar multiple and add up to be equal to the reciprocal lattice vector, and in matrix form
(Equation 18):
(18)
Where a
i
, b
i
, and c
i
are the x, y, and z components of directional vector i for i = 1, 2, and
3. a
4
, b
4
, and c
4
are the x, y, and z components of the reciprocal lattice vector. This system of
equations can then be solved for the lengths of the directional vectors s
1,2,3
. Once the lengths are
calculated, the 3 normalized directional vectors can be multiplied by their respective lengths to
make them reciprocal lattice vectors. At this point, any three of the four reciprocal lattice vectors
103
can be chosen to calculate the unit cell basis vectors and compare them to the unstrained lattice
parameters for the full strain/stress tensor calculation. The process to calculate the full elastic
strain/stress tensor and strain/stress transformation are identical to that used for ECAP Al as
described in Section 3.1.5 and 3.1.6.
104
Chapter 4. Fracture Toughness and Flexural Modulus
Measurements of an Fe-Based Bulk Metallic Glass (SAM2X5)
This section describes the supplemental work done in addition to the LRIS measurements.
4.1 Introduction
Bulk metallic glasses (BMGs) have been shown to have overall higher hardness and yield
strength compared to crystalline solids [57-59]. However, low ductility and fracture toughness
are their main drawbacks and efforts are being made to improve the fracture toughness [60-64].
Thus, accurate fracture toughness measurements of BMG samples need to be performed. While
numerous methods to measure mechanical properties of macro-scale samples exist, standard
testing procedures are often impossible to implement for bulk metallic glasses (BMGs). This is
primarily due to insufficient sample size restricted by sample processing limitations. Non-
standard tests can be used for mechanical testing but they can lead to artificial variations in
mechanical properties such as yield strength, modulus, and fracture toughness. Recently, fracture
toughness of Fe-based BMGs was measured with indentation, the more popular technique due to
sample size restrictions [65-67], and also notch toughness testing [62, 68, 69], although on non-
standard (ASTM) sample geometries (1.5 - 2 mm diameter rods from copper mold castings).
However, there are inherent problems with using indentation for fracture toughness
measurements. Indentation utilizes much smaller testing volumes compared to standard (e.g.
three-point bending) fracture toughness tests. Since the effective testing volumes are small,
effects of inclusions and voids on mechanical properties may not be fully realized. Furthermore,
indentation toughness values are calculated based on symmetric crack geometries produced by
Vickers indentations. Therefore, measurement accuracy is highly dependent on producing
symmetric cracks that are often not observed in actual testing. Even if the proper (symmetric)
105
crack geometries are produced, the measured indentation toughness values can vary significantly
from accepted K
IC
values [70, 71], as will be addressed later. Consequently, it may be more
accurate to measure fracture toughness via a more direct method such as three-point bending
which more directly measures the crack extension force.
Current measurements for fracture toughness using three-point bending of BMGs show a
large scatter in the data, even within the same composition Zr-based BMG from an isolated
research group [72]. Fe-based BMG toughness measurements also vary with testing procedure
and sample size. While there have been studies that measured fracture toughness of Fe-based
BMGs using three-point bending, fracture toughness specimens tested in those studies were
cylindrical rods with a relatively large notch root radius of 110 µm [62, 68, 69]. Specific testing
procedures and calculations were not described in detail. Since the sample geometry is not
standard, the toughness values calculated may deviate from values obtained by an accepted
standard (ASTM E399). By using specimens with cylindrical cross sections in earlier studies [62,
68, 69], the effective width of the specimen at the notch root varies with notch depth. Since
fracture toughness is calculated based on the geometry of the notch, this introduced another
variable that may lead to inaccurate fracture toughness measurements.
106
Table 8 summarizes the reported fracture (notch) toughness values (this and other
studies) measured from Fe-based BMG samples of various chemical compositions. Toughness
values range from ~ 3 to about 50 MPa√m [62, 69], and standard deviations of ~ ±10% for
fracture toughness values for each composition are observed. In earlier studies by the same
authors [73, 74] on Zr-based BMGs, fracture toughness values of samples with notch root radius
of 110 μm measured 5X higher (~ 100 MPa√m) compared with fatigue pre -cracked samples (~
20 MPa√) . Thus, a small (sharp) notch root radius (approaching that of a fatigue pre-crack), as
used in the present work, is desirable for fracture toughness measurements as it appears to be
more accurate.
Others have even performed toughness tests with notches machined using electric
discharge machining (EDM) for other types of BMGs [72, 75]. Local heating by EDM can
crystallize the volumes surrounding the notch, and may lead to inaccurate fracture toughness
values. In the present work, methods to test Fe-based BMGs were devised to improve the
accuracy of fracture toughness and modulus measurements. Using spark plasma sintering (SPS),
bulk SAM2X5 samples (with composition Fe
49.7
Cr
17.1
Mn
1.9
Mo
7.4
W
1.6
B
15.2
C
3.8
Si
2.
4) at the tens of
millimeter scale were synthesized. The relatively large bulk sample size in the present work
allowed for the preparations of test specimens in the millimeter scale with well-controlled
sample dimensions following closer to ASTM specifications. Wiresaw cutting allowed for a
relatively sharp notch radius (3X smaller than previous studies by others) and minimal sample
damage caused by heating and plasticity as suggested by results from XRD discussed
subsequently. These specimens were used in three-point bending for flexural modulus and
fracture toughness tests using a custom built mechanical testing machine. Values for modulus
107
and fracture toughness were also measured using indentation techniques (Vickers and nano-
indentation) in order to compare with values measured using three-point bending tests.
4.2 Experimental Procedures
Bulk SAM2X5 samples were produced by spark plasma sintering as reported by Graeve
et.al [76]. These as-received samples were disc shaped with dimensions ~ 20 mm in diameter
and ~ 2 mm in thickness. SAM2X5 sample density was measured by Archimedes principle. X-
Ray diffraction (XRD) profiles from a Rigaku Ultima IV at 1
o
/s using a Cu k-alpha X-Ray
source were used to assess the amorphous structure. XRD of the bulk samples was performed for
the as-received, rough ground (400 grit), and diamond lapped discs (to assess the effect of
mechanical grinding and diamond lapping on sample crystallization). TEM imaging with SAED
was performed using a JEM 2100F (JEOL) on thin foil samples prepared by dimpling and ion-
milling.
Two bulk SAM2X5 samples with amorphous morphologies (confirmed by XRD) were
selected for mechanical testing ( ≥95% amorphous). In order to mitigate the effects of heat and
plasticity during specimen preparation, surface lapping and wiresaw cutting were utilized due to
their low heat and low abrasive nature. The two bulk samples were first lapped using diamond
paste on both surfaces until the samples were 1860 μm in thickness and surfaces were parallel to
within 1 μm over 20 mm . These samples were then cut into smaller specimens by a wiresaw
using a SiC slurry for modulus and fracture toughness tests. One bulk sample was cut into six
flexural modulus specimens and four fracture toughness specimens. The second bulk sample was
cut into three flexural modulus specimens and three fracture toughness specimens.
Flexural modulus specimens were rectangular in shape and had cross sections ~ 1860 µm
x 500 µm in width and thickness, with test gauge lengths of 10 mm. Fracture toughness
108
specimens followed ASTM E399 recommended ratios (about a factor of two smaller) with cross
sections ~1 mm x 1860 μ m (width and thickness respectively), and gauge lengths of 8 mm.
Notching of the fracture toughness specimens was accomplished using the wiresaw, which used
a 50 μm diameter wire and a cutting slurry composed of 800 grit SiC powder (suspended in oil)
with a median particle size of 6.5 μm. The resulting notches were ~ 900 μm deep (about half the
specimen thickness), ~100 μm wide, and had notch root radii of ~ 40 μm.
A small scale mechanical testing machine was custom designed and built by the
investigators to test the modulus and fracture toughness (Figure 47) of SAM2X5 specimens. This
is similar to the machine used by Eberl et al. [77], which was used for testing thin films (micron-
scale in size) of yttrium-stabilized-zirconia (YSZ) deposited by electron-beam physical vapor
deposition (EB-PDV). The machine was designed to accept samples at the millimeter scale.
Figure 47. Micro-bending machine, close up of fracture toughness stages and sample.
109
Figure 47 illustrates the testing machine with a frame, two linear actuators, alignment
stage, load cell, safety springs, and a camera for digital image correlation (DIC) to assess strain.
Stage movement was achieved by actuators that were controlled by MatLab software. The DIC
software was obtained from Prof. Dr. Christoph Eberl of the University of Freiburg, Germany.
Young’s m odulus values were measured by three-point bending. Additionally, fatigue pre-cracks
(as specified ASTM E399) were not achieved due to early fracture. Therefore, toughness values
reported are for notch toughness.
4.3. Vickers and Nano-Indentation
Indentation (Vickers and nano-indentation) was performed to complement the modulus
and toughness data obtained by bending tests. Bulk SAM2X5 samples were ground using 200,
400, 600, 800, and 1200 grit SiC paper and polished using a 1 µm diamond particle suspension
for Vickers and nano-indentation. Vickers indentation was performed on the polished bulk
samples using a Leco LM100 indenter. For each bulk sample, 9 Vickers indentations were made
at a load of 300g and a dwell time of 10 seconds.
Indentation toughness was calculated using optical measurements of the crack lengths
emanating from the indent corners of 9 Vickers indents per bulk sample. One of the most
commonly used models to approximate fracture toughness of BMGs is that of Anstis et.al [78],
where a half-penny crack geometry is assumed. Indentation toughness was estimated based on
Equation. 20 [78]:
(20)
110
where K
c
is the toughness calculated from indentation, α (0.016 ± 0.004) is the calibration
constant (empirical constant), P is the indentation load, H is the hardness, E is the modulus, and
c
o
is the half crack length.
Nano-indentation was used to measure hardness and Young’s modulus, and
measurements were made using a Hysitron Triboindenter. A total of 36 nano-indents were made
for each sample. Nano-indentation was performed using a Berkovich tip (radius of ~ 200 nm)
with a constant loading rate of 1000 N/s and a maximum load of 7500 N. Hardness was
calculated from the maximum force divided by the indentation contact area. Reduced modulus
measured by nano-indentation was calculated based on the stiffness of the unloading curve and
the projected contact area. Young’s modulus was then calculated from the reduce d modulus
(assuming υ = 0.3).
4.4. Three-point Bending Measurements
Young’s modulus measurements of the flexural specimens were performed using our
testing machine in flexure by three-point bending. The cross-head stage movement rate was
5µm/s. Displacement values were determined from the movement of the piezo-actuator.
Combined with the information from the load cell, a load-displacement curve was constructed
for each modulus test, as illustrated in Figure 49, by subtracting the compliance of the machine
that was separately measured. The Young's modulus was calculated based on the beam bending
theory of a simply supported rectangular cross-section prismatic beam using Equation. 21:
(21)
111
where L is the length of the support, m is the slope of the load-displacement curve, b is
the width of the specimen, and d is the thickness of the specimen. The machine and testing
procedures were calibrated using a 6061-T6 Al specimen with a similar geometry to the
SAM2X5 specimens. Young's modulus values measured from our testing device for 6061-T6 Al
were within a 3% error of accepted values of 69 GPa [79].
In contrast to the standard practice of measuring K
IC
using a uniaxial tension test, three-
point bending was employed to measure fracture toughness of BMGs in this study and by others
due to difficulty in producing the correct sample geometries for uniaxial tension tests. Figure 47
illustrates the top surface of the sample and the method by which it was loaded into the stages.
Three point bending tests were performed by moving the cross-head stage at a rate of 2 μm/s.
Crack-mouth opening displacement was tracked using DIC from a camera mounted above the
stages. A load versus crack mouth opening displacement curve was reconstructed by matching
load cell data to DIC data. Recorded peak loads for each specimen were used to calculate
fracture toughness following ASTM E399 standards.
4.5 Results
The bulk sample density matches that of the fully dense theoretical value ~ 7.9g/cm
3
(similar to that of pure Fe). Optical and SEM imaging confirmed a fully dense cross section.
EDAX analysis revealed some Ni rich inclusions, ~ 20 µm in diameter, and oxygen-rich
boundary layers. The Ni inclusions are suspected to have originated from impurities in the as-
received powder, and oxygen boundaries likely originated from oxide layers of the as-received
powder particles.
112
Figure 48. TEM image, inset SAED pattern, and inset XRD plots of SAM 2X5 BMGs indicating
amorphous morphology
The lower inset of Figure 48 shows XRD profiles of SAM2X5 samples, and they have
wide, broad peaks, which confirms an amorphous structure. XRD was performed on sample
surfaces of as-received, after grinding (400 grit), and after diamond paste lapping. The XRD
plots all show identical curves, corresponding to amorphous structures. X-Ray penetration from a
Cu source K-alpha with an energy of 8.06 KeV has an attenuation length of ~ 4 m for pure Fe.
This means that the collected diffraction data originates from a surface layer with thickness
roughly half that distance (~ 2 μm ). It is reasonable to infer that penetration depth for Fe based
BMGs is similar and likely ~ 2 μm. Since XRD profiles on the three different bulk sample
preparations all indicate amorphous morphology, any recrystallization of the sample was
undetectable by XRD and was limited to a fraction of a couple of microns. Furthermore, since
the wiresaw cutting is both slower and is expected to produces less heat than rough grinding, it
113
is reasonable to infer that the crystallization and plasticity layer caused by wire saw cutting is of
negligible depth.
Figure 48 shows the amorphous structure of a bulk SAM2X5 sample cross section
captured via bright field TEM. The inset SAED pattern of the corresponding region shows broad
diffuse rings with a small amount of diffraction peaks. The results from XRD and TEM are
consistent with SAM2X5 microstructure being mainly amorphous with a small amount of
crystallinity ( ≥95% amorphous).
Average Vickers hardness values were 11.16±0.21 and 12.21±0.21 GPa, or about 1138
HV and 1245 HV for the two bulk samples. Hardness data was consistent with values reported
for other Fe-based BMGs [66, 80, 81]. Average hardness values obtained by nano-indentation
were 18.0±0.51 and 19.1±0.46 GPa which are higher than Vickers values. Material “pile -ups”
around nano-indents were observed (not shown). These are made by nano-indentation
(significant in size compared with the nano-indents) in contrast to Vickers indents, where no
significant pile-up (compared with the indent size) was observed. Nano-indentation
measurements are sensitive to material pile-ups since they change the effective contact area of
the nano-indents. Therefore, nano-indentation can overestimate the hardness (up to 50%) [82] if
the effect of pile-ups is not taken into account. Since Vickers hardness is calculated directly from
the measured indent areas, hardness values measured by Vickers indentation are perhaps more
accurate compared with values obtained by nano-indentation for SAM2X5.
Flexural modulus measurements were used due to gripping and brittle behavior
limitations associated with traditional tension testing. Others have measured elastic modulus of
brittle BMGs via other methods, such as compression of micro-pillars, nano-indentation, and
ultrasonic measurements [66, 80, 81, 83, 84]. In the present work, we aim to measure the bulk
114
properties of Fe-based BMGs. Modulus values were measured by two methods: three-point
bending, and nano-indentation.
Figure 49. Flexural Modulus and Fracture Toughness load versus displacement curves for
SAM2X5 samples tested by three-point bending
Figure 49 shows a load-displacement curve for a flexural modulus test. All flexural
modulus curves show an abrupt brittle failure. No significant plastic deformation was observed
in any of the modulus test specimens.
115
Table 8 summarizes the mechanical properties of SAM2X5 examined in this study and
other Fe-based BMGs from other studies measured by various testing methods.
The average Young's modulus values measured by three-point bending are 227.3±21.3
and 232.3±22.2 GPa from the six specimens cut from sample 1 and three specimens from sample
2 (SAM2X5),. Average Young’s modulus values obtained by nano-indentation are 276.5±9.4
and 303.3±9.1 GPa. Results from nano-indentation are higher than those from three-point
bending by about 25%. Young’s modulus values for both testing procedures used in the present
work are slightly higher than reported values of other Fe-based BMGs [84-86]. Young's modulus
of SAM2X5 obtained by three-point bending are about 10% higher than values measured by
resonant ultrasound spectroscopy (RUS) for other Fe-based BMGs of various compositions [62].
About the same variation (10%) is reported for the Young's modulus values measured by RUS
between different composition Fe-based BMGs [62]. Similar to the case of hardness, modulus
values obtained by nano-indentation are calculated based on the load and contact area of the
indent tip. Deviations in tip contact areas caused by material pile-up can lead to an
overestimation of modulus values (by up to 50%) [82]. The higher measured values of Young’s
modulus using nano-indentation compared with three-point bending are consistent with the
observed pile-ups around nano-indents. Furthermore, as discussed earlier, since three-point
bending test specimens are much larger compared to nano-indentation, values measured by
three-point bending would include the effects of inclusions and voids. Modulus values measured
by three-point bending are perhaps more accurate.
116
Table 8. Mechanical properties of various Fe-based BMGs obtained by different testing methods
Material E (GPa) Method
K
C
(MPa√m)
Method Notch Radius
Fe
51
Mn
10
Mo
14
Cr
4
C
15
B
6
[69] 200
Resonant
Ultrasound
Spectroscopy
3.5±0.8
Three-point
bending
110 µm
Fe
48
Mo
14
Cr
15
Y
2
C
15
B
6
[69] 215 12.5±4.5
Fe
49
Cr
15
Mo
14
C
19
B
2
Er
1
[62] 209 27.1±6.4
Fe
59
Cr
6
Mo
14
C
15
B
6
[62] 204 52.8±5.1
Fe
75
Mo
5
P
10
C
8.3
B
1.7
[68] 27
Three-point
bending
Fe
48
Cr
15
Mo
14
Er
2
C
15
B
6
[67] 3.2±0.3
Vickers
indentation
Fe
41
Co
7
Cr
15
Mo
14
C
15
B
6
Y
2
[66] 225
Resonant
Ultrasound
Spectroscopy
2.2
Vickers
indentation
Fe
52
Cr
15
Mo
9
Er
3
C
15
B
6
[86] 281
Nano-
indentation
Fe
41
Co
7
Cr
15
Mo
14
Y
2
C
15
B
6
[80] 265
Nano-
indentation
Fe
48
Cr
15
Mo
14
C
15
B
6
Gd
2
[85] 180
Nano-
indentation
Sample 1 (SAM2X5)
(This study)
227.3±21.3 5.0±0.7
Three-point
bending
40 µm
276.5±9.4
Nano-
indentation
2.5±0.14
Vickers
indentation
Sample 2 (SAM2X5)
(This study)
232.3±22.2 4.7±0.8
Three-point
bending
40 µm
303.3±9.1
Nano-
indentation
2.2±0.10
Vickers
indentation
117
Table 8. summarizes the average toughness values for samples 1 and 2 measured by
three-point bending and indentation. Average fracture toughness values measured by three-point
bending are 5.0±0.69 and 4.7±0.84 MPa√m for samples 1 and 2. These values are about twice as
high as values measured by Vickers indentation. The difference in fracture toughness values
measured by Vickers indentation and three-point bending can be attributed to either inaccuracies
in approximating indentation toughness, or the effect of the lack of fatigue pre-cracks in the
three-point bending specimens.
4.6 Discussion
Compared to fracture toughness tests such as a three-point bending test, indentation
toughness tests are burdened with a considerable amount of error [87]. For example, the assumed
half penny crack geometries are often difficult to obtain, and the calculation does not account for
the existence of irregular and secondary cracks (crack branching). Indentation fracture toughness
values are only accurate to around ±40% based on work by Antsis [78]. The accuracy of
indentation results have been shown to deviate about 50% from standard ASTM K
IC
measurements of SiC samples [70, 71].
It has been shown that sample size affects the mechanical properties in Zr-based BMGs.
Conner et. al. [88, 89] showed significant increase in bend ductility for Zr-based BMG wires and
foils smaller than 1 mm in size. That is, plastic strain to fracture in bending increased
significantly with decreasing sample size below a sample thickness of about 1 mm. Therefore, in
using fracture toughness specimens with dimensions larger than 1 mm, measured fracture
toughness values in three-point bending may be more comparable to values measured by large
scale testing (per ASTM), and that different (non representative of bulk properties) toughness
118
values are measured by indentation compared with three-point bending test as in this present
work.
Disparate results were reported on the effect of notched versus fatigue pre-cracked
specimens on the fracture toughness measurements of Zr-based BMGs. Toughness values of
notched samples of various widths were reported to exhibit higher (2X) fracture toughness
values compared to those of fatigue pre-cracked samples [73, 74]. The discrepancy in toughness
values was partly attributed to the differences in the failure mechanism at the notch root. The
fatigue pre-cracked samples in [73] had planar crack fronts while notched samples exhibited
large amounts of shear banding at the notch root as well as significant crack bifurcations. The
authors posited that energy absorption from shear banding and crack bifurcations in notched
samples may well explain the differences in measured toughness values. At the same time, others
have also shown no noticeable difference in measured fracture toughness of notched versus
fatigue pre-cracked samples [72, 90] in Zr-based and Ti-based BMGs. In those studies, both the
notched and fatigue pre-cracked samples exhibited extreme ductility, revealing significant shear
banding for both types of samples. Since failure mechanisms for both types of samples were
similar, energy absorption and thus measured toughness values were similar.
Currently, there has been no reported study done to compare toughness values of notched
versus fatigue pre-cracked (according to ASTM E399) for extremely brittle BMGs such as Fe-
based BMGs, due to the inability to fatigue pre-crack such brittle samples. Efforts made to
fatigue pre-crack SAM2X5 samples all resulted in catastrophic failures of the specimen. Note
that notch toughness specimens tested in this present work all exhibited planar crack fronts and
are absent of any shear banding or crack bifurcation, suggesting failure mechanisms similar to
those of fatigue pre-cracked specimens reported in [73] on Zr-based BMGs. Therefore, while the
119
lack of fatigue pre-cracks in toughness specimens in the present work prevents reported
toughness values from being more “standard fracture toughness values”, the manner in which
cracks propagate suggests that notch toughness values would be close to toughness values for
fatigue pre-cracked SAM2X5 specimens.
The effect of notch root radius on fracture toughness of a fine-grained ceramic (with
similar fracture toughness values to SAM2X5) was investigated by [91]. In a similar procedure
to the current study, sintered 1 μm grain -size Al
2
O
3
were tested using notches of various root
radii (3μm to 100μm) . The authors in [91] found that fracture toughness decreased with
decreasing notch root radius to 10 μm, below which values were consistent. The difference
between 40 μm (as in the present study) and 10 μm was a bout a 25% decrease. Another study
[92] measured indentation (Vickers) toughness of dental ceramics (with similar fracture
toughness values to SAM2X5) and compared them with values obtained by three-point bending
using specimens with sharp notch radii as in the previous study [91] (notch radius 15 μm). The
results found that indentation toughness values were unreliable as the calibration "constant"
(0.016) actually varies (~50%) within a given material to bring the two measurements into
coincidence.
These studies may demonstrate the inaccuracy of indentation toughness tests and
quantified the effect of the notch root radius on three-point bending tests for ceramics with
similar fracture toughness to SAM2X5. If we extend the findings in fracture toughness of
ceramics to SAM2X5, indentation toughness values may be unreliable and three-point bending
notch toughness values are more accurate if not slightly higher than the “true” fracture toughness
values.
120
4.7 Conclusions
Flexural modulus and fracture (notch) toughness for SAM2X5 were measured as 230
GPa and 4.9 MPa√m , respectively. Flexural modulus values are lower than those measured by
nano-indentation and appear to be more accurate due to the presence of material pile-ups around
the nano-indents. Fracture toughness specimens and testing conditions were kept as close to
ASTM E399 specifications as possible. Specimen sizes are roughly half the recommended values
with all ratios as recommended. Measured fracture toughness values of notch samples (without
fatigue pre-cracks) with relatively sharp notch root radii (40 µm) are about a factor of two larger
than (Vickers) indentation tests and appear to be more reliable.
121
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Appendix A. Experimental Data
A1. ECAP AA1050 1P Matlab Outputs
The input peak values from "G1.txt" were:
P1: (1779.15,893.57) qc 102.8460 with hkl (2 2 6) on detector 0
P2: (355.57,392.51) qc 134.3610 with hkl (1 7 5) on detector 2
P3: (224.53,760.28) qc 151.9220 with hkl (8 4 4) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:48:09
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0505063e-01
b = 4.0483526e-01
c = 4.0515898e-01
alpha = 90.001526
beta = 89.940184
gamma = 89.977978
Best strain tensor (crystal coordinates):
1.24393e-04 -1.92200e-04 5.22056e-04
-1.92200e-04 -4.06757e-04 -1.33103e-05
5.22056e-04 -1.33103e-05 3.92552e-04
Best stress tensor (crystal coordinates)(MPa):
12.58856 -10.95541 29.75721
-10.95541 -12.32235 -0.75869
29.75721 -0.75869 25.16525
Best strain tensor (sample coordinates):
1.11229e-05 1.36538e-04 -3.78829e-04
1.36538e-04 -3.88745e-04 -3.47010e-04
-3.78829e-04 -3.47010e-04 4.87810e-04
Best stress tensor (sample coordinates)(MPa):
7.43731 10.09434 -20.84654
10.09434 -14.40138 -16.89152
-20.84654 -16.89152 32.39554
Best principal strains are:
e1 = 8.11203e-04
e2 = -1.91475e-04
e3 = -5.09540e-04
Best principal stresses are (MPa):
sig1 = 50.20349
sig2 = -4.36846
sig3 = -20.40356
Best hydrostatic stress is (MPa):
8.47716
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0505063e-01 +/- 3.3750121e-05
126
b = 4.0483526e-01 +/- 4.4084291e-05
c = 4.0515898e-01 +/- 2.4056754e-05
alpha = 90.001526 +/- 0.009739
beta = 89.940184 +/- 0.011296
gamma = 89.977978 +/- 0.010207
The strain tensor components in crystallographic coordinates:
e11 = 1.2439267e-04 +/- 8.6902667e-05
e22 = -4.0675674e-04 +/- 1.1160669e-04
e33 = 3.9255242e-04 +/- 6.4321585e-05
e23 = -1.3310274e-05 +/- 8.4954853e-05
e13 = 5.2205638e-04 +/- 9.8566788e-05
e12 = -1.9220025e-04 +/- 8.9074029e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = 12.58856 +/- 14.88858
s2 = -12.32235 +/- 16.07229
s3 = 25.16525 +/- 14.40800
s4 = -0.75869 +/- 4.84243
s5 = 29.75721 +/- 5.61831
s6 = -10.95541 +/- 5.07722
The strain tensor components in sample coordinates:
e11 = 1.1122924e-05 +/- 9.5669769e-05
e22 = -3.8874481e-04 +/- 1.5443486e-04
e33 = 4.8781023e-04 +/- 6.4357491e-05
e23 = -3.4700976e-04 +/- 6.6401936e-05
e13 = -3.7882863e-04 +/- 6.9952634e-05
e12 = 1.3653758e-04 +/- 9.5092100e-05
The stress tensor components in sample coordinates (MPa):
s1 = 7.43731 +/- 13.95330
s2 = -14.40138 +/- 19.47837
s3 = 32.39554 +/- 12.89756
s4 = -16.89152 +/- 3.55599
s5 = -20.84654 +/- 4.13975
s6 = 10.09434 +/- 4.95186
The principal strains are:
Principal strain 1 = 8.1120305e-04 +/- 7.7437625e-05
Principal strain 2 = -1.9147478e-04 +/- 1.1759867e-04
Principal strain 3 = -5.0953992e-04 +/- 1.3538028e-04
The principal strain directions are:
v1 = (-0.449557, -0.294979, 0.843141)
v2 = (0.893173, -0.135895, 0.428690)
v3 = (0.011876, -0.945791, -0.324560)
The principal stresses are (MPa):
Principal stress 1 = 50.20349 +/- 14.40712
Principal stress 2 = -4.36846 +/- 13.82745
Principal stress 3 = -20.40356 +/- 18.28630
The principal stress directions are:
v1 = (-0.473848, -0.291316, 0.831025)
v2 = (0.864622, 0.025084, 0.501797)
127
v3 = (0.167027, -0.956298, -0.239993)
The hydrostatic stress is (MPa): 8.47716 +/- 14.85256
The input peak values from "G2.txt" were:
P1: (1618.40,1499.22) qc 116.0330 with hkl (2 -4 6) on detector 0
P2: (719.13,436.98) qc 134.3480 with hkl (7 -1 5) on detector 2
P3: (502.15,677.60) qc 126.9790 with hkl (3 -7 3) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:48:31
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0480790e-01
b = 4.0491030e-01
c = 4.0513957e-01
alpha = 90.047339
beta = 89.972882
gamma = 89.986715
Best strain tensor (crystal coordinates):
-4.74461e-04 -1.15880e-04 2.36536e-04
-1.15880e-04 -2.21811e-04 -4.13018e-04
2.36536e-04 -4.13018e-04 3.44626e-04
Best stress tensor (crystal coordinates)(MPa):
-43.80810 -6.60517 13.48255
-6.60517 -31.95885 -23.54205
13.48255 -23.54205 -5.39292
Best strain tensor (sample coordinates):
-4.57181e-04 -2.33297e-05 -9.27021e-05
-2.33297e-05 -4.57614e-04 -2.44097e-04
-9.27021e-05 -2.44097e-04 5.63149e-04
Best stress tensor (sample coordinates)(MPa):
-44.87816 0.34253 -5.54514
0.34253 -46.23130 -10.41066
-5.54514 -10.41066 9.94959
Best principal strains are:
e1 = 6.25240e-04
e2 = -4.38821e-04
e3 = -5.38066e-04
Best principal stresses are (MPa):
sig1 = 12.34893
sig2 = -45.26626
sig3 = -48.24253
Best hydrostatic stress is (MPa):
-27.05329
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0480790e-01 +/- 4.8926994e-05
b = 4.0491030e-01 +/- 3.3355365e-05
c = 4.0513957e-01 +/- 5.1668293e-05
128
alpha = 90.047339 +/- 0.013177
beta = 89.972882 +/- 0.010766
gamma = 89.986715 +/- 0.010507
The strain tensor components in crystallographic coordinates:
e11 = -4.7446072e-04 +/- 1.2343357e-04
e22 = -2.2181147e-04 +/- 8.6001477e-05
e33 = 3.4462632e-04 +/- 1.2991690e-04
e23 = -4.1301839e-04 +/- 1.1494690e-04
e13 = 2.3653597e-04 +/- 9.3881236e-05
e12 = -1.1588009e-04 +/- 9.1650495e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -43.80810 +/- 17.69271
s2 = -31.95885 +/- 17.01262
s3 = -5.39292 +/- 19.16681
s4 = -23.54205 +/- 6.55197
s5 = 13.48255 +/- 5.35123
s6 = -6.60517 +/- 5.22408
The strain tensor components in sample coordinates:
e11 = -4.5718057e-04 +/- 1.0674355e-04
e22 = -4.5761414e-04 +/- 1.7354707e-04
e33 = 5.6314884e-04 +/- 6.9098366e-05
e23 = -2.4409693e-04 +/- 9.6423747e-05
e13 = -9.2702084e-05 +/- 6.2793386e-05
e12 = -2.3329695e-05 +/- 1.1730644e-04
The stress tensor components in sample coordinates (MPa):
s1 = -44.87816 +/- 17.70872
s2 = -46.23130 +/- 22.62668
s3 = 9.94959 +/- 13.84153
s4 = -10.41066 +/- 4.92061
s5 = -5.54514 +/- 3.68331
s6 = 0.34253 +/- 6.10854
The principal strains are:
Principal strain 1 = 6.2524049e-04 +/- 4.7189653e-05
Principal strain 2 = -4.3882072e-04 +/- 1.2559326e-04
Principal strain 3 = -5.3806563e-04 +/- 1.2747090e-04
The principal strain directions are:
v1 = (-0.078629, -0.217609, 0.972864)
v2 = (0.861705, -0.505547, -0.043435)
v3 = (0.501280, 0.834906, 0.227265)
The principal stresses are (MPa):
Principal stress 1 = 12.34893 +/- 14.45285
Principal stress 2 = -45.26626 +/- 19.28826
Principal stress 3 = -48.24253 +/- 19.93772
The principal stress directions are:
v1 = (-0.095998, -0.174711, 0.979929)
v2 = (-0.970971, 0.233125, -0.053557)
v3 = (-0.219089, -0.956624, -0.192018)
129
The hydrostatic stress is (MPa): -27.05329 +/- 17.52093
The input peak values from "G3.txt" were:
P1: (1659.46,1578.47) qc 91.7552 with hkl (3 1 5) on detector 0
P2: (690.01,767.97) qc 142.1490 with hkl (8 2 4) on detector 1
P3: (746.34,234.59) qc 134.3550 with hkl (1 7 5) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:48:53
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0498892e-01
b = 4.0475699e-01
c = 4.0492618e-01
alpha = 89.949192
beta = 89.955837
gamma = 90.004738
Best strain tensor (crystal coordinates):
-2.76683e-05 4.13434e-05 3.85381e-04
4.13434e-05 -6.00424e-04 4.43116e-04
3.85381e-04 4.43116e-04 -1.82280e-04
Best stress tensor (crystal coordinates)(MPa):
-50.97345 2.35658 21.96673
2.35658 -77.83567 25.25763
21.96673 25.25763 -58.22475
Best strain tensor (sample coordinates):
-2.92539e-04 -1.54689e-04 -1.22564e-04
-1.54689e-04 -8.70979e-04 1.42943e-04
-1.22564e-04 1.42943e-04 3.53145e-04
Best stress tensor (sample coordinates)(MPa):
-63.88101 -6.47752 -5.48596
-6.47752 -95.71566 8.93600
-5.48596 8.93600 -27.43720
Best principal strains are:
e1 = 3.98398e-04
e2 = -2.89443e-04
e3 = -9.19327e-04
Best principal stresses are (MPa):
sig1 = -25.25768
sig2 = -63.92984
sig3 = -97.84635
Best hydrostatic stress is (MPa):
-62.34462
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0498892e-01 +/- 3.5892633e-05
b = 4.0475699e-01 +/- 4.8007542e-05
c = 4.0492618e-01 +/- 4.6415211e-05
alpha = 89.949192 +/- 0.010505
130
beta = 89.955837 +/- 0.013812
gamma = 90.004738 +/- 0.010214
The strain tensor components in crystallographic coordinates:
e11 = -2.7668255e-05 +/- 9.2318726e-05
e22 = -6.0042355e-04 +/- 1.2110469e-04
e33 = -1.8228023e-04 +/- 1.1733288e-04
e23 = 4.4311635e-04 +/- 9.1571480e-05
e13 = 3.8538125e-04 +/- 1.2049417e-04
e12 = 4.1343435e-05 +/- 8.9128725e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -50.97345 +/- 17.45579
s2 = -77.83567 +/- 17.20362
s3 = -58.22475 +/- 18.84023
s4 = 25.25763 +/- 5.21957
s5 = 21.96673 +/- 6.86817
s6 = 2.35658 +/- 5.08034
The strain tensor components in sample coordinates:
e11 = -2.9253865e-04 +/- 1.1785550e-04
e22 = -8.7097851e-04 +/- 1.5841633e-04
e33 = 3.5314513e-04 +/- 5.8506488e-05
e23 = 1.4294276e-04 +/- 9.3108398e-05
e13 = -1.2256394e-04 +/- 4.9883445e-05
e12 = -1.5468870e-04 +/- 1.3024037e-04
The stress tensor components in sample coordinates (MPa):
s1 = -63.88101 +/- 18.80776
s2 = -95.71566 +/- 21.84052
s3 = -27.43720 +/- 12.87964
s4 = 8.93600 +/- 4.95046
s5 = -5.48596 +/- 3.15722
s6 = -6.47752 +/- 6.72652
The principal strains are:
Principal strain 1 = 3.9839769e-04 +/- 7.5924230e-05
Principal strain 2 = -2.8944250e-04 +/- 1.5258872e-04
Principal strain 3 = -9.1932722e-04 +/- 1.6338140e-04
The principal strain directions are:
v1 = (-0.202070, 0.133875, 0.970178)
v2 = (-0.953762, 0.198151, -0.225994)
v3 = (-0.222497, -0.970986, 0.087645)
The principal stresses are (MPa):
Principal stress 1 = -25.25768 +/- 12.64877
Principal stress 2 = -63.92984 +/- 19.84241
Principal stress 3 = -97.84635 +/- 21.79369
The principal stress directions are:
v1 = (-0.162046, 0.138804, 0.976972)
v2 = (-0.972235, 0.146924, -0.182134)
v3 = (-0.168822, -0.979360, 0.111142)
The hydrostatic stress is (MPa): -62.34462 +/- 17.43553
131
The input peak values from "G4.txt" were:
P1: (1855.80,990.27) qc 102.8770 with hkl (2 2 6) on detector 0
P2: (677.95,181.51) qc 111.8630 with hkl (0 6 4) on detector 2
P3: (350.96,812.18) qc 152.0050 with hkl (8 4 4) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:49:15
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0479866e-01
b = 4.0495461e-01
c = 4.0502397e-01
alpha = 89.981156
beta = 89.954658
gamma = 89.995169
Best strain tensor (crystal coordinates):
-4.97460e-04 -4.21347e-05 3.95490e-04
-4.21347e-05 -1.12127e-04 1.64424e-04
3.95490e-04 1.64424e-04 5.91738e-05
Best stress tensor (crystal coordinates)(MPa):
-57.07119 -2.40168 22.54295
-2.40168 -38.99907 9.37214
22.54295 9.37214 -30.96507
Best strain tensor (sample coordinates):
-4.29088e-04 3.01039e-04 -2.37301e-05
3.01039e-04 -4.15646e-04 -6.36747e-05
-2.37301e-05 -6.36747e-05 2.94321e-04
Best stress tensor (sample coordinates)(MPa):
-54.23717 16.15864 -2.83556
16.15864 -56.39800 -2.32467
-2.83556 -2.32467 -16.40016
Best principal strains are:
e1 = 3.04130e-04
e2 = -1.30312e-04
e3 = -7.24231e-04
Best principal stresses are (MPa):
sig1 = -15.82453
sig2 = -39.69745
sig3 = -71.51335
Best hydrostatic stress is (MPa):
-42.34511
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0479866e-01 +/- 3.5385733e-05
b = 4.0495461e-01 +/- 5.1048517e-05
c = 4.0502397e-01 +/- 2.4768695e-05
alpha = 89.981156 +/- 0.012065
beta = 89.954658 +/- 0.010978
132
gamma = 89.995169 +/- 0.012156
The strain tensor components in crystallographic coordinates:
e11 = -4.9745989e-04 +/- 9.0604828e-05
e22 = -1.1212691e-04 +/- 1.2888730e-04
e33 = 5.9173757e-05 +/- 6.6022750e-05
e23 = 1.6442350e-04 +/- 1.0525465e-04
e13 = 3.9549031e-04 +/- 9.5744400e-05
e12 = -4.2134697e-05 +/- 1.0602639e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -57.07119 +/- 16.03993
s2 = -38.99907 +/- 18.22601
s3 = -30.96507 +/- 15.73261
s4 = 9.37214 +/- 5.99951
s5 = 22.54295 +/- 5.45743
s6 = -2.40168 +/- 6.04350
The strain tensor components in sample coordinates:
e11 = -4.2908834e-04 +/- 8.9538367e-05
e22 = -4.1564551e-04 +/- 1.8555886e-04
e33 = 2.9432081e-04 +/- 6.8827244e-05
e23 = -6.3674712e-05 +/- 6.8649419e-05
e13 = -2.3730090e-05 +/- 8.8814647e-05
e12 = 3.0103856e-04 +/- 9.9448915e-05
The stress tensor components in sample coordinates (MPa):
s1 = -54.23717 +/- 14.08041
s2 = -56.39800 +/- 22.44631
s3 = -16.40016 +/- 14.35368
s4 = -2.32467 +/- 3.68380
s5 = -2.83556 +/- 5.15707
s6 = 16.15864 +/- 5.15634
The principal strains are:
Principal strain 1 = 3.0412999e-04 +/- 6.5740941e-05
Principal strain 2 = -1.3031244e-04 +/- 1.3196466e-04
Principal strain 3 = -7.2423060e-04 +/- 1.2742323e-04
The principal strain directions are:
v1 = (-0.082026, -0.121812, 0.989158)
v2 = (0.696628, 0.702768, 0.144312)
v3 = (-0.712728, 0.700913, 0.027213)
The principal stresses are (MPa):
Principal stress 1 = -15.82453 +/- 15.03535
Principal stress 2 = -39.69745 +/- 17.79041
Principal stress 3 = -71.51335 +/- 18.14720
The principal stress directions are:
v1 = (-0.116201, -0.102879, 0.987883)
v2 = (0.720673, 0.675694, 0.155137)
v3 = (-0.683467, 0.729968, -0.004374)
The hydrostatic stress is (MPa): -42.34511 +/- 16.39189
133
The input peak values from "G5.txt" were:
P1: (1481.02,981.50) qc 126.9590 with hkl (3 3 7) on detector 0
P2: (286.89,106.52) qc 160.4590 with hkl (1 9 5) on detector 2
P3: (500.47,866.32) qc 127.0240 with hkl (7 3 3) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:49:37
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0475449e-01
b = 4.0498841e-01
c = 4.0502754e-01
alpha = 89.982663
beta = 89.961742
gamma = 90.017516
Best strain tensor (crystal coordinates):
-6.06459e-04 1.52762e-04 3.33660e-04
1.52762e-04 -2.86685e-05 1.51289e-04
3.33660e-04 1.51289e-04 6.79885e-05
Best stress tensor (crystal coordinates)(MPa):
-63.20857 8.70742 19.01865
8.70742 -36.11018 8.62350
19.01865 8.62350 -31.57697
Best strain tensor (sample coordinates):
-6.32640e-04 2.42606e-04 6.81563e-05
2.42606e-04 -2.49135e-04 -4.58742e-06
6.81563e-05 -4.58742e-06 3.14635e-04
Best stress tensor (sample coordinates)(MPa):
-66.43458 12.26919 1.68221
12.26919 -48.35742 1.02714
1.68221 1.02714 -16.10372
Best principal strains are:
e1 = 3.19834e-04
e2 = -1.33092e-04
e3 = -7.53882e-04
Best principal stresses are (MPa):
sig1 = -15.97747
sig2 = -42.26411
sig3 = -72.65414
Best hydrostatic stress is (MPa):
-43.63191
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0475449e-01 +/- 3.1984117e-05
b = 4.0498841e-01 +/- 3.5093281e-05
c = 4.0502754e-01 +/- 3.4867446e-05
alpha = 89.982663 +/- 0.010297
beta = 89.961742 +/- 0.012240
134
gamma = 90.017516 +/- 0.009028
The strain tensor components in crystallographic coordinates:
e11 = -6.0645918e-04 +/- 8.2912375e-05
e22 = -2.8668460e-05 +/- 9.0080318e-05
e33 = 6.7988472e-05 +/- 8.9368449e-05
e23 = 1.5128949e-04 +/- 8.9846496e-05
e13 = 3.3366045e-04 +/- 1.0673635e-04
e12 = 1.5276175e-04 +/- 7.8738810e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -63.20857 +/- 14.80152
s2 = -36.11018 +/- 14.91345
s3 = -31.57697 +/- 15.74747
s4 = 8.62350 +/- 5.12125
s5 = 19.01865 +/- 6.08397
s6 = 8.70742 +/- 4.48811
The strain tensor components in sample coordinates:
e11 = -6.3263972e-04 +/- 9.9748399e-05
e22 = -2.4913491e-04 +/- 1.4908396e-04
e33 = 3.1463546e-04 +/- 6.3661571e-05
e23 = -4.5874176e-06 +/- 6.5624719e-05
e13 = 6.8156327e-05 +/- 6.8026321e-05
e12 = 2.4260629e-04 +/- 1.0046252e-04
The stress tensor components in sample coordinates (MPa):
s1 = -66.43458 +/- 14.94100
s2 = -48.35742 +/- 19.23761
s3 = -16.10372 +/- 12.23149
s4 = 1.02714 +/- 3.48628
s5 = 1.68221 +/- 4.04175
s6 = 12.26919 +/- 5.30468
The principal strains are:
Principal strain 1 = 3.1983421e-04 +/- 5.7134582e-05
Principal strain 2 = -1.3309165e-04 +/- 1.4944005e-04
Principal strain 3 = -7.5388173e-04 +/- 1.3403636e-04
The principal strain directions are:
v1 = (-0.077711, -0.025100, -0.996660)
v2 = (0.429951, 0.901100, -0.056218)
v3 = (-0.899502, 0.432883, 0.059234)
The principal stresses are (MPa):
Principal stress 1 = -15.97747 +/- 12.27173
Principal stress 2 = -42.26411 +/- 18.00818
Principal stress 3 = -72.65414 +/- 17.01415
The principal stress directions are:
v1 = (0.045119, 0.048748, 0.997792)
v2 = (0.448176, 0.891664, -0.063829)
v3 = (-0.892806, 0.450066, 0.018384)
The hydrostatic stress is (MPa): -43.63191 +/- 14.90483
135
The input peak values from "G6.txt" were:
P1: (1898.96,1064.33) qc 80.6114 with hkl (-1 -1 5) on detector 0
P2: (733.87,729.96) qc 110.7760 with hkl (-1 -5 5) on detector 2
P3: (651.15,903.19) qc 134.3550 with hkl (-7 -1 5) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_wire-correction2.txt" geometry file
Calculations were done on 13-Jul-2015 14:49:58
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0494809e-01
b = 4.0487447e-01
c = 4.0495340e-01
alpha = 90.034255
beta = 90.004249
gamma = 89.965697
Best strain tensor (crystal coordinates):
-1.28356e-04 -2.99308e-04 -3.70746e-05
-2.99308e-04 -3.10129e-04 -2.98840e-04
-3.70746e-05 -2.98840e-04 -1.15058e-04
Best stress tensor (crystal coordinates)(MPa):
-39.95204 -17.06056 -2.11325
-17.06056 -48.47719 -17.03386
-2.11325 -17.03386 -39.32837
Best strain tensor (sample coordinates):
8.06610e-05 -1.58536e-04 1.25525e-04
-1.58536e-04 -6.30343e-04 -1.98680e-05
1.25525e-04 -1.98680e-05 -3.86020e-06
Best stress tensor (sample coordinates)(MPa):
-27.13924 -8.21771 7.56944
-8.21771 -67.84603 -0.70133
7.56944 -0.70133 -32.77233
Best principal strains are:
e1 = 1.95301e-04
e2 = -8.46823e-05
e3 = -6.64161e-04
Best principal stresses are (MPa):
sig1 = -20.77069
sig2 = -37.52841
sig3 = -69.45849
Best hydrostatic stress is (MPa):
-42.58587
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0494809e-01 +/- 4.4347379e-05
b = 4.0487447e-01 +/- 6.7849524e-05
c = 4.0495340e-01 +/- 1.5923928e-05
alpha = 90.034255 +/- 0.011360
beta = 90.004249 +/- 0.010521
136
gamma = 89.965697 +/- 0.011427
The strain tensor components in crystallographic coordinates:
e11 = -1.2835578e-04 +/- 1.1236944e-04
e22 = -3.1012873e-04 +/- 1.6937295e-04
e33 = -1.1505794e-04 +/- 4.6559934e-05
e23 = -2.9883965e-04 +/- 9.9058023e-05
e13 = -3.7074614e-05 +/- 9.1798728e-05
e12 = -2.9930803e-04 +/- 9.9709082e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -39.95204 +/- 16.62403
s2 = -48.47719 +/- 20.29126
s3 = -39.32837 +/- 15.63017
s4 = -17.03386 +/- 5.64631
s5 = -2.11325 +/- 5.23253
s6 = -17.06056 +/- 5.68342
The strain tensor components in sample coordinates:
e11 = 8.0660989e-05 +/- 1.3062767e-04
e22 = -6.3034324e-04 +/- 1.5430867e-04
e33 = -3.8601958e-06 +/- 4.6709452e-05
e23 = -1.9867968e-05 +/- 6.7961931e-05
e13 = 1.2552509e-04 +/- 4.7245022e-05
e12 = -1.5853551e-04 +/- 1.4684960e-04
The stress tensor components in sample coordinates (MPa):
s1 = -27.13924 +/- 18.55565
s2 = -67.84603 +/- 21.06225
s3 = -32.77233 +/- 13.00341
s4 = -0.70133 +/- 3.89493
s5 = 7.56944 +/- 3.15591
s6 = -8.21771 +/- 7.36084
The principal strains are:
Principal strain 1 = 1.9530079e-04 +/- 1.1005970e-04
Principal strain 2 = -8.4682263e-05 +/- 4.5647905e-05
Principal strain 3 = -6.6416097e-04 +/- 1.7318147e-04
The principal strain directions are:
v1 = (-0.825653, 0.171471, -0.537489)
v2 = (-0.523693, 0.121451, 0.843205)
v3 = (-0.209864, -0.977674, 0.010478)
The principal stresses are (MPa):
Principal stress 1 = -20.77069 +/- 16.52281
Principal stress 2 = -37.52841 +/- 14.56344
Principal stress 3 = -69.45849 +/- 21.73203
The principal stress directions are:
v1 = (-0.831783, 0.153149, -0.533556)
v2 = (-0.520001, 0.121390, 0.845496)
v3 = (-0.194255, -0.980719, 0.021332)
The hydrostatic stress is (MPa): -42.58587 +/- 17.05670
137
A2. ECAP AA1050 2P Matlab Outputs
The input peak values from "2P_G2_A.txt" were:
P1: (1616.83,858.62) qc 91.7731 with hkl (-1 3 5) on detector 0
P2: (500.78,187.77) qc 124.0930 with hkl (0 0 8) on detector 1
P3: (874.21,310.31) qc 119.1760 with hkl (3 5 5) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:50:20
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0499014e-01
b = 4.0485105e-01
c = 4.0506309e-01
alpha = 89.987635
beta = 89.962902
gamma = 90.056248
Best strain tensor (crystal coordinates):
-2.50403e-05 4.90841e-04 3.23731e-04
4.90841e-04 -3.67791e-04 1.07867e-04
3.23731e-04 1.07867e-04 1.55781e-04
Best stress tensor (crystal coordinates)(MPa):
-15.70560 27.97796 18.45265
27.97796 -31.78060 6.14839
18.45265 6.14839 -7.22509
Best strain tensor (sample coordinates):
-3.25076e-05 3.44375e-04 9.50585e-05
3.44375e-04 -2.78344e-04 5.17518e-04
9.50585e-05 5.17518e-04 7.38006e-05
Best stress tensor (sample coordinates)(MPa):
-13.97478 18.68368 7.15794
18.68368 -29.74501 28.54390
7.15794 28.54390 -10.99150
Best principal strains are:
e1 = 5.79526e-04
e2 = -8.76326e-05
e3 = -7.28944e-04
Best principal stresses are (MPa):
sig1 = 18.73737
sig2 = -19.65724
sig3 = -53.79142
Best hydrostatic stress is (MPa):
-18.23710
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0499014e-01 +/- 8.7363869e-05
b = 4.0485105e-01 +/- 6.8546597e-05
c = 4.0506309e-01 +/- 1.6833960e-06
alpha = 89.987635 +/- 0.006522
138
beta = 89.962902 +/- 0.009293
gamma = 90.056248 +/- 0.012404
The strain tensor components in crystallographic coordinates:
e11 = -2.5040334e-05 +/- 2.1697838e-04
e22 = -3.6779096e-04 +/- 1.7137083e-04
e33 = 1.5578064e-04 +/- 2.4641957e-05
e23 = 1.0786651e-04 +/- 5.6872680e-05
e13 = 3.2373072e-04 +/- 8.1073357e-05
e12 = 4.9084146e-04 +/- 1.0824515e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -15.70560 +/- 23.98967
s2 = -31.78060 +/- 20.86327
s3 = -7.22509 +/- 15.92410
s4 = 6.14839 +/- 3.24174
s5 = 18.45265 +/- 4.62118
s6 = 27.97796 +/- 6.16997
The strain tensor components in sample coordinates:
e11 = -3.2507645e-05 +/- 1.9316724e-04
e22 = -2.7834360e-04 +/- 1.5785194e-04
e33 = 7.3800592e-05 +/- 3.3457304e-05
e23 = 5.1751839e-04 +/- 9.9168104e-05
e13 = 9.5058496e-05 +/- 6.8831838e-05
e12 = 3.4437464e-04 +/- 1.1752332e-04
The stress tensor components in sample coordinates (MPa):
s1 = -13.97478 +/- 23.95957
s2 = -29.74501 +/- 21.32441
s3 = -10.99150 +/- 14.86768
s4 = 28.54390 +/- 5.71186
s5 = 7.15794 +/- 3.54875
s6 = 18.68368 +/- 5.65620
The principal strains are:
Principal strain 1 = 5.7952589e-04 +/- 1.4642344e-04
Principal strain 2 = -8.7632584e-05 +/- 1.0810587e-04
Principal strain 3 = -7.2894396e-04 +/- 1.1897964e-04
The principal strain directions are:
v1 = (-0.436099, -0.586698, -0.682351)
v2 = (-0.835420, -0.017902, 0.549320)
v3 = (-0.334501, 0.809608, -0.482332)
The principal stresses are (MPa):
Principal stress 1 = 18.73737 +/- 21.18090
Principal stress 2 = -19.65724 +/- 20.44224
Principal stress 3 = -53.79142 +/- 19.03975
The principal stress directions are:
v1 = (-0.474346, -0.575176, -0.666460)
v2 = (-0.829688, 0.039000, 0.556863)
v3 = (-0.294302, 0.817100, -0.495716)
The hydrostatic stress is (MPa): -18.23710 +/- 19.51089
139
The input peak values from "2P_G2_B.txt" were:
P1: (1604.56,856.31) qc 91.7755 with hkl (-1 3 5) on detector 0
P2: (488.50,189.01) qc 124.1040 with hkl (0 0 8) on detector 1
P3: (872.00,304.57) qc 119.1880 with hkl (3 5 5) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:50:42
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0502499e-01
b = 4.0486857e-01
c = 4.0502713e-01
alpha = 89.989884
beta = 89.977166
gamma = 90.059551
Best strain tensor (crystal coordinates):
6.10874e-05 5.19712e-04 1.99280e-04
5.19712e-04 -3.24530e-04 8.82519e-05
1.99280e-04 8.82519e-05 6.69935e-05
Best stress tensor (crystal coordinates)(MPa):
-9.17731 29.62357 11.35894
29.62357 -27.26275 5.03036
11.35894 5.03036 -8.90031
Best strain tensor (sample coordinates):
7.56082e-05 4.06066e-04 7.13570e-05
4.06066e-04 -2.78636e-04 4.03349e-04
7.13570e-05 4.03349e-04 6.57929e-06
Best stress tensor (sample coordinates)(MPa):
-5.69567 21.91611 5.41191
21.91611 -27.97235 22.36204
5.41191 22.36204 -11.67235
Best principal strains are:
e1 = 5.23273e-04
e2 = -3.12822e-05
e3 = -6.88439e-04
Best principal stresses are (MPa):
sig1 = 18.19802
sig2 = -14.15193
sig3 = -49.38646
Best hydrostatic stress is (MPa):
-15.11346
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0502499e-01 +/- 8.8154304e-05
b = 4.0486857e-01 +/- 6.9060694e-05
c = 4.0502713e-01 +/- 1.3192677e-06
alpha = 89.989884 +/- 0.006510
beta = 89.977166 +/- 0.009537
140
gamma = 90.059551 +/- 0.012711
The strain tensor components in crystallographic coordinates:
e11 = 6.1087380e-05 +/- 2.1878085e-04
e22 = -3.2452963e-04 +/- 1.7219168e-04
e33 = 6.6993522e-05 +/- 2.4539327e-05
e23 = 8.8251904e-05 +/- 5.6774305e-05
e13 = 1.9927971e-04 +/- 8.3218885e-05
e12 = 5.1971170e-04 +/- 1.1093848e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -9.17731 +/- 23.86481
s2 = -27.26275 +/- 20.61565
s3 = -8.90031 +/- 15.72641
s4 = 5.03036 +/- 3.23614
s5 = 11.35894 +/- 4.74348
s6 = 29.62357 +/- 6.32349
The strain tensor components in sample coordinates:
e11 = 7.5608227e-05 +/- 1.9305330e-04
e22 = -2.7863624e-04 +/- 1.5876484e-04
e33 = 6.5792898e-06 +/- 3.4398442e-05
e23 = 4.0334919e-04 +/- 1.0181267e-04
e13 = 7.1356959e-05 +/- 7.0448440e-05
e12 = 4.0606562e-04 +/- 1.1985896e-04
The stress tensor components in sample coordinates (MPa):
s1 = -5.69567 +/- 23.71674
s2 = -27.97235 +/- 21.14920
s3 = -11.67235 +/- 14.70547
s4 = 22.36204 +/- 5.86149
s5 = 5.41191 +/- 3.63462
s6 = 21.91611 +/- 5.77357
The principal strains are:
Principal strain 1 = 5.2327266e-04 +/- 1.7428429e-04
Principal strain 2 = -3.1282171e-05 +/- 7.1356457e-05
Principal strain 3 = -6.8843921e-04 +/- 1.1622003e-04
The principal strain directions are:
v1 = (-0.611965, -0.580213, -0.537449)
v2 = (0.687112, -0.053544, -0.724576)
v3 = (0.391631, -0.812703, 0.431438)
The principal stresses are (MPa):
Principal stress 1 = 18.19802 +/- 22.64442
Principal stress 2 = -14.15193 +/- 18.35776
Principal stress 3 = -49.38646 +/- 18.84969
The principal stress directions are:
v1 = (-0.634089, -0.559503, -0.533748)
v2 = (-0.684673, 0.085440, 0.723825)
v3 = (-0.359379, 0.824413, -0.437253)
The hydrostatic stress is (MPa): -15.11346 +/- 19.28829
141
The input peak values from "2P_G3.txt" were:
P1: (1901.78,480.30) qc 110.7950 with hkl (-1 5 5) on detector 0
P2: (470.83,486.07) qc 119.1600 with hkl (3 5 5) on detector 2
P3: (538.32,607.98) qc 124.1100 with hkl (0 0 8) on detector 1
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:51:04
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0502796e-01
b = 4.0490183e-01
c = 4.0500752e-01
alpha = 89.988125
beta = 89.995489
gamma = 89.994729
Best strain tensor (crystal coordinates):
6.90331e-05 -4.60052e-05 3.93721e-05
-4.60052e-05 -2.42412e-04 1.03600e-04
3.93721e-05 1.03600e-04 1.85756e-05
Best stress tensor (crystal coordinates)(MPa):
-6.25182 -2.62229 2.24421
-2.62229 -20.85861 5.90518
2.24421 5.90518 -8.61827
Best strain tensor (sample coordinates):
-2.45695e-04 1.00155e-04 -5.22612e-05
1.00155e-04 5.17961e-05 3.58585e-05
-5.22612e-05 3.58585e-05 3.90951e-05
Best stress tensor (sample coordinates)(MPa):
-22.20097 4.76141 -1.90480
4.76141 -6.77890 1.58551
-1.90480 1.58551 -6.74882
Best principal strains are:
e1 = 9.10082e-05
e2 = 4.16832e-05
e3 = -2.87495e-04
Best principal stresses are (MPa):
sig1 = -4.81772
sig2 = -7.05743
sig3 = -23.85355
Best hydrostatic stress is (MPa):
-11.90957
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0502796e-01 +/- 8.4286473e-05
b = 4.0490183e-01 +/- 4.8007374e-05
c = 4.0500752e-01 +/- 2.7646976e-06
alpha = 89.988125 +/- 0.006559
beta = 89.995489 +/- 0.010026
142
gamma = 89.994729 +/- 0.008919
The strain tensor components in crystallographic coordinates:
e11 = 6.9033062e-05 +/- 2.0969871e-04
e22 = -2.4241234e-04 +/- 1.2153690e-04
e33 = 1.8575570e-05 +/- 2.5917867e-05
e23 = 1.0359966e-04 +/- 5.7212406e-05
e13 = 3.9372132e-05 +/- 8.7491549e-05
e12 = -4.6005161e-05 +/- 7.7841711e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -6.25182 +/- 22.44206
s2 = -20.85861 +/- 16.48079
s3 = -8.61827 +/- 14.11396
s4 = 5.90518 +/- 3.26111
s5 = 2.24421 +/- 4.98702
s6 = -2.62229 +/- 4.43698
The strain tensor components in sample coordinates:
e11 = -2.4569492e-04 +/- 1.2317039e-04
e22 = 5.1796117e-05 +/- 1.4159989e-04
e33 = 3.9095096e-05 +/- 3.8003188e-05
e23 = 3.5858485e-05 +/- 9.9095748e-05
e13 = -5.2261186e-05 +/- 6.4879189e-05
e12 = 1.0015496e-04 +/- 1.1970865e-04
The stress tensor components in sample coordinates (MPa):
s1 = -22.20097 +/- 18.89102
s2 = -6.77890 +/- 19.71014
s3 = -6.74882 +/- 13.58117
s4 = 1.58551 +/- 5.63260
s5 = -1.90480 +/- 3.39951
s6 = 4.76141 +/- 5.85056
The principal strains are:
Principal strain 1 = 9.1008182e-05 +/- 1.2789926e-04
Principal strain 2 = 4.1683217e-05 +/- 8.7740297e-05
Principal strain 3 = -2.8749511e-04 +/- 1.2774278e-04
The principal strain directions are:
v1 = (-0.200402, -0.888790, -0.412178)
v2 = (-0.284262, -0.349865, 0.892631)
v3 = (0.937568, -0.296051, 0.182536)
The principal stresses are (MPa):
Principal stress 1 = -4.81772 +/- 18.07834
Principal stress 2 = -7.05743 +/- 16.75896
Principal stress 3 = -23.85355 +/- 18.21730
The principal stress directions are:
v1 = (-0.172824, -0.838008, -0.517566)
v2 = (-0.254046, -0.469767, 0.845446)
v3 = (0.951626, -0.277599, 0.131706)
The hydrostatic stress is (MPa): -11.90957 +/- 17.12811
143
The input peak values from "2P_G7.txt" were:
P1: (1912.30,1001.39) qc 126.9910 with hkl (3 -3 7) on detector 0
P2: (175.10,507.28) qc 111.8700 with hkl (0 -6 4) on detector 1
P3: (150.26,692.13) qc 119.1650 with hkl (5 -5 3) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:51:25
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0445488e-01
b = 4.0525241e-01
c = 4.0512323e-01
alpha = 89.938636
beta = 89.996230
gamma = 90.070545
Best strain tensor (crystal coordinates):
-1.34674e-03 6.14795e-04 3.28512e-05
6.14795e-04 6.22649e-04 5.35832e-04
3.28512e-05 5.35832e-04 3.04271e-04
Best stress tensor (crystal coordinates)(MPa):
-88.89697 35.04331 1.87252
35.04331 3.46733 30.54242
1.87252 30.54242 -11.46458
Best strain tensor (sample coordinates):
-8.88450e-04 -5.83956e-04 -6.53181e-04
-5.83956e-04 9.59839e-04 -2.63717e-04
-6.53181e-04 -2.63717e-04 -4.91208e-04
Best stress tensor (sample coordinates)(MPa):
-65.33555 -31.77681 -31.05100
-31.77681 23.86520 -16.07700
-31.05100 -16.07700 -55.42387
Best principal strains are:
e1 = 1.13255e-03
e2 = -2.16254e-05
e3 = -1.53074e-03
Best principal stresses are (MPa):
sig1 = 34.46103
sig2 = -29.89898
sig3 = -101.45628
Best hydrostatic stress is (MPa):
-32.29808
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0445488e-01 +/- 5.0816733e-05
b = 4.0525241e-01 +/- 3.2630532e-05
c = 4.0512323e-01 +/- 3.6420128e-05
alpha = 89.938636 +/- 0.008140
beta = 89.996230 +/- 0.012488
144
gamma = 90.070545 +/- 0.009624
The strain tensor components in crystallographic coordinates:
e11 = -1.3467392e-03 +/- 1.2777310e-04
e22 = 6.2264889e-04 +/- 8.4160087e-05
e33 = 3.0427127e-04 +/- 9.2799714e-05
e23 = 5.3583187e-04 +/- 7.1118675e-05
e13 = 3.2851232e-05 +/- 1.0883523e-04
e12 = 6.1479487e-04 +/- 8.3824634e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -88.89697 +/- 17.52381
s2 = 3.46733 +/- 15.06151
s3 = -11.46458 +/- 15.87118
s4 = 30.54242 +/- 4.05376
s5 = 1.87252 +/- 6.20361
s6 = 35.04331 +/- 4.77800
The strain tensor components in sample coordinates:
e11 = -8.8844986e-04 +/- 1.3193696e-04
e22 = 9.5983932e-04 +/- 1.3510837e-04
e33 = -4.9120846e-04 +/- 6.7974998e-05
e23 = -2.6371748e-04 +/- 5.3213712e-05
e13 = -6.5318101e-04 +/- 1.1304285e-04
e12 = -5.8395605e-04 +/- 6.4632308e-05
The stress tensor components in sample coordinates (MPa):
s1 = -65.33555 +/- 17.54357
s2 = 23.86520 +/- 18.35269
s3 = -55.42387 +/- 13.46421
s4 = -16.07700 +/- 2.62228
s5 = -31.05100 +/- 6.19916
s6 = -31.77681 +/- 3.51914
The principal strains are:
Principal strain 1 = 1.1325502e-03 +/- 1.2624293e-04
Principal strain 2 = -2.1625354e-05 +/- 9.7338050e-05
Principal strain 3 = -1.5307439e-03 +/- 1.5018311e-04
The principal strain directions are:
v1 = (-0.261931, 0.963730, -0.051156)
v2 = (-0.551842, -0.106077, 0.827175)
v3 = (-0.791747, -0.244892, -0.559612)
The principal stresses are (MPa):
Principal stress 1 = 34.46103 +/- 17.17720
Principal stress 2 = -29.89898 +/- 14.89445
Principal stress 3 = -101.45628 +/- 18.22910
The principal stress directions are:
v1 = (0.281653, -0.956673, 0.073815)
v2 = (0.595065, 0.113807, -0.795579)
v3 = (0.752708, 0.268001, 0.601337)
The hydrostatic stress is (MPa): -32.29808 +/- 15.77136
145
The input peak values from "2P_G9.txt" were:
P1: (1961.50,1331.06) qc 116.0670 with hkl (2 6 4) on detector 0
P2: (624.87,480.65) qc 158.2340 with hkl (2 10 0) on detector 1
P3: (410.18,423.33) qc 101.7570 with hkl (-3 5 3) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:51:48
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0473573e-01
b = 4.0490938e-01
c = 4.0514405e-01
alpha = 89.972328
beta = 89.993090
gamma = 89.968336
Best strain tensor (crystal coordinates):
-6.52676e-04 -2.76142e-04 6.02587e-05
-2.76142e-04 -2.23863e-04 2.41430e-04
6.02587e-05 2.41430e-04 3.55676e-04
Best stress tensor (crystal coordinates)(MPa):
-62.53949 -15.74009 3.43475
-15.74009 -42.42817 13.76152
3.43475 13.76152 -15.24779
Best strain tensor (sample coordinates):
-4.92841e-04 -2.99050e-04 4.55364e-04
-2.99050e-04 -1.07555e-04 -1.10892e-04
4.55364e-04 -1.10892e-04 7.95318e-05
Best stress tensor (sample coordinates)(MPa):
-57.54868 -12.65168 24.01438
-12.65168 -36.20830 -4.75574
24.01438 -4.75574 -26.45848
Best principal strains are:
e1 = 4.44476e-04
e2 = -1.54452e-04
e3 = -8.10888e-04
Best principal stresses are (MPa):
sig1 = -9.31719
sig2 = -38.04705
sig3 = -72.85122
Best hydrostatic stress is (MPa):
-40.07182
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0473573e-01 +/- 6.0066605e-05
b = 4.0490938e-01 +/- 1.1557815e-05
c = 4.0514405e-01 +/- 8.1205488e-05
alpha = 89.972328 +/- 0.007599
beta = 89.993090 +/- 0.009917
146
gamma = 89.968336 +/- 0.006116
The strain tensor components in crystallographic coordinates:
e11 = -6.5267629e-04 +/- 1.5046668e-04
e22 = -2.2386350e-04 +/- 3.7662060e-05
e33 = 3.5567559e-04 +/- 2.0201856e-04
e23 = 2.4143017e-04 +/- 6.6292910e-05
e13 = 6.0258719e-05 +/- 8.6485877e-05
e12 = -2.7614196e-04 +/- 5.3346318e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -62.53949 +/- 17.61880
s2 = -42.42817 +/- 14.07192
s3 = -15.24779 +/- 21.62849
s4 = 13.76152 +/- 3.77870
s5 = 3.43475 +/- 4.92969
s6 = -15.74009 +/- 3.04074
The strain tensor components in sample coordinates:
e11 = -4.9284061e-04 +/- 1.1612958e-04
e22 = -1.0755542e-04 +/- 1.3403680e-04
e33 = 7.9531828e-05 +/- 3.5161350e-05
e23 = -1.1089171e-04 +/- 5.9740162e-05
e13 = 4.5536443e-04 +/- 8.1888226e-05
e12 = -2.9904979e-04 +/- 1.4320341e-04
The stress tensor components in sample coordinates (MPa):
s1 = -57.54868 +/- 18.61361
s2 = -36.20830 +/- 19.95107
s3 = -26.45848 +/- 13.14533
s4 = -4.75574 +/- 3.13135
s5 = 24.01438 +/- 4.38406
s6 = -12.65168 +/- 6.94140
The principal strains are:
Principal strain 1 = 4.4447620e-04 +/- 1.3738974e-04
Principal strain 2 = -1.5445211e-04 +/- 7.5200068e-05
Principal strain 3 = -8.1088829e-04 +/- 1.0267915e-04
The principal strain directions are:
v1 = (-0.501588, 0.423283, -0.754481)
v2 = (0.058029, -0.853699, -0.517524)
v3 = (-0.863158, -0.303365, 0.403642)
The principal stresses are (MPa):
Principal stress 1 = -9.31719 +/- 17.76469
Principal stress 2 = -38.04705 +/- 17.49639
Principal stress 3 = -72.85122 +/- 16.82245
The principal stress directions are:
v1 = (0.490040, -0.370128, 0.789219)
v2 = (-0.036175, 0.895964, 0.442651)
v3 = (-0.870949, -0.245466, 0.425669)
The hydrostatic stress is (MPa): -40.07182 +/- 17.02395
147
The input peak values from "2P_G11.txt" were:
P1: (1286.82,1266.47) qc 87.7681 with hkl (0 4 4) on detector 0
P2: (115.37,700.63) qc 131.6350 with hkl (-2 8 2) on detector 1
P3: (515.91,367.34) qc 167.1010 with hkl (-6 4 8) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:52:09
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0507254e-01
b = 4.0501628e-01
c = 4.0490837e-01
alpha = 89.999063
beta = 89.998912
gamma = 90.002506
Best strain tensor (crystal coordinates):
1.79115e-04 2.18733e-05 9.49823e-06
2.18733e-05 4.01908e-05 8.17773e-06
9.49823e-06 8.17773e-06 -2.26242e-04
Best stress tensor (crystal coordinates)(MPa):
7.97537 1.24678 0.54140
1.24678 1.45980 0.46613
0.54140 0.46613 -11.03589
Best strain tensor (sample coordinates):
-1.83718e-05 1.25126e-04 -1.14925e-04
1.25126e-04 1.04704e-04 6.49080e-05
-1.14925e-04 6.49080e-05 -9.32675e-05
Best stress tensor (sample coordinates)(MPa):
-1.26608 5.97454 -5.48884
5.97454 4.42654 2.84140
-5.48884 2.84140 -4.76118
Best principal strains are:
e1 = 1.82754e-04
e2 = 3.69953e-05
e3 = -2.26685e-04
Best principal stresses are (MPa):
sig1 = 8.22558
sig2 = 1.23989
sig3 = -11.06619
Best hydrostatic stress is (MPa):
-0.53357
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0507254e-01 +/- 6.9245353e-05
b = 4.0501628e-01 +/- 1.9186932e-05
c = 4.0490837e-01 +/- 5.1992757e-05
alpha = 89.999063 +/- 0.008438
beta = 89.998912 +/- 0.009563
148
gamma = 90.002506 +/- 0.010961
The strain tensor components in crystallographic coordinates:
e11 = 1.7911547e-04 +/- 1.7283870e-04
e22 = 4.0190841e-05 +/- 5.3470411e-05
e33 = -2.2624183e-04 +/- 1.3121180e-04
e23 = 8.1777322e-06 +/- 7.3640901e-05
e13 = 9.4982335e-06 +/- 8.3468377e-05
e12 = 2.1873274e-05 +/- 9.5664796e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = 7.97537 +/- 20.30696
s2 = 1.45980 +/- 15.31987
s3 = -11.03589 +/- 17.66182
s4 = 0.46613 +/- 4.19753
s5 = 0.54140 +/- 4.75770
s6 = 1.24678 +/- 5.45289
The strain tensor components in sample coordinates:
e11 = -1.8371790e-05 +/- 1.0286671e-04
e22 = 1.0470380e-04 +/- 1.7971710e-04
e33 = -9.3267522e-05 +/- 3.2107720e-05
e23 = 6.4907964e-05 +/- 1.2737699e-04
e13 = -1.1492547e-04 +/- 2.9994266e-05
e12 = 1.2512600e-04 +/- 8.6264593e-05
The stress tensor components in sample coordinates (MPa):
s1 = -1.26608 +/- 17.05522
s2 = 4.42654 +/- 22.02416
s3 = -4.76118 +/- 13.61815
s4 = 2.84140 +/- 6.93314
s5 = -5.48884 +/- 1.57903
s6 = 5.97454 +/- 4.27176
The principal strains are:
Principal strain 1 = 1.8275448e-04 +/- 1.3900500e-04
Principal strain 2 = 3.6995321e-05 +/- 7.5753077e-05
Principal strain 3 = -2.2668532e-04 +/- 1.1543501e-04
The principal strain directions are:
v1 = (-0.538883, -0.841967, 0.026378)
v2 = (-0.582323, 0.394965, 0.710565)
v3 = (0.608691, -0.367551, 0.703137)
The principal stresses are (MPa):
Principal stress 1 = 8.22558 +/- 18.91613
Principal stress 2 = 1.23989 +/- 16.59193
Principal stress 3 = -11.06619 +/- 17.47377
The principal stress directions are:
v1 = (-0.553341, -0.831331, 0.051980)
v2 = (-0.562465, 0.418951, 0.712821)
v3 = (0.614367, -0.365196, 0.699417)
The hydrostatic stress is (MPa): -0.53357 +/- 17.18251
149
The input peak values from "2P_G12.txt" were:
P1: (1345.40,1143.78) qc 87.7643 with hkl (4 0 4) on detector 0
P2: (241.25,974.27) qc 141.3390 with hkl (9 -1 1) on detector 1
P3: (170.37,84.47) qc 145.5720 with hkl (6 6 4) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:52:31
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0501510e-01
b = 4.0478788e-01
c = 4.0502475e-01
alpha = 90.011383
beta = 90.010401
gamma = 89.995153
Best strain tensor (crystal coordinates):
3.72613e-05 -4.23009e-05 -9.07672e-05
-4.23009e-05 -5.23785e-04 -9.92808e-05
-9.07672e-05 -9.92808e-05 6.11074e-05
Best stress tensor (crystal coordinates)(MPa):
-24.33045 -2.41115 -5.17373
-2.41115 -50.64350 -5.65901
-5.17373 -5.65901 -23.21206
Best strain tensor (sample coordinates):
-3.61477e-04 -2.73934e-04 -1.12588e-04
-2.73934e-04 -4.48685e-06 -6.92484e-05
-1.12588e-04 -6.92484e-05 -5.94526e-05
Best stress tensor (sample coordinates)(MPa):
-43.08119 -13.00366 -6.12128
-13.00366 -25.18700 -3.61699
-6.12128 -3.61699 -29.91781
Best principal strains are:
e1 = 1.44270e-04
e2 = -2.39236e-05
e3 = -5.45762e-04
Best principal stresses are (MPa):
sig1 = -18.33729
sig2 = -27.66507
sig3 = -52.18365
Best hydrostatic stress is (MPa):
-32.72867
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0501510e-01 +/- 9.8171405e-06
b = 4.0478788e-01 +/- 4.9034695e-05
c = 4.0502475e-01 +/- 5.2106811e-05
alpha = 90.011383 +/- 0.010688
beta = 90.010401 +/- 0.008366
150
gamma = 89.995153 +/- 0.006179
The strain tensor components in crystallographic coordinates:
e11 = 3.7261256e-05 +/- 3.4654492e-05
e22 = -5.2378465e-04 +/- 1.2348200e-04
e33 = 6.1107426e-05 +/- 1.3098482e-04
e23 = -9.9280850e-05 +/- 9.3232754e-05
e13 = -9.0767184e-05 +/- 7.3010509e-05
e12 = -4.2300874e-05 +/- 5.3920779e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -24.33045 +/- 12.31545
s2 = -50.64350 +/- 15.47341
s3 = -23.21206 +/- 16.57125
s4 = -5.65901 +/- 5.31427
s5 = -5.17373 +/- 4.16160
s6 = -2.41115 +/- 3.07348
The strain tensor components in sample coordinates:
e11 = -3.6147654e-04 +/- 6.2597779e-05
e22 = -4.4868510e-06 +/- 1.2643680e-04
e33 = -5.9452575e-05 +/- 2.9242022e-05
e23 = -6.9248415e-05 +/- 5.5841217e-05
e13 = -1.1258842e-04 +/- 8.1038950e-05
e12 = -2.7393368e-04 +/- 1.1682341e-04
The stress tensor components in sample coordinates (MPa):
s1 = -43.08119 +/- 14.27505
s2 = -25.18700 +/- 17.77570
s3 = -29.91781 +/- 11.27339
s4 = -3.61699 +/- 2.88252
s5 = -6.12128 +/- 4.51629
s6 = -13.00366 +/- 5.97312
The principal strains are:
Principal strain 1 = 1.4426970e-04 +/- 1.2882048e-04
Principal strain 2 = -2.3923579e-05 +/- 5.4969971e-05
Principal strain 3 = -5.4576209e-04 +/- 1.0818937e-04
The principal strain directions are:
v1 = (0.468828, -0.882346, 0.040822)
v2 = (-0.250007, -0.088229, 0.964216)
v3 = (0.847170, 0.462257, 0.261957)
The principal stresses are (MPa):
Principal stress 1 = -18.33729 +/- 16.91399
Principal stress 2 = -27.66507 +/- 12.79786
Principal stress 3 = -52.18365 +/- 14.78907
The principal stress directions are:
v1 = (0.458055, -0.888223, 0.035302)
v2 = (0.285679, 0.109485, -0.952051)
v3 = (0.841768, 0.446176, 0.303897)
The hydrostatic stress is (MPa): -32.72867 +/- 14.34424
151
The input peak values from "2P_G13_A.txt" were:
P1: (1744.39,875.73) qc 87.7523 with hkl (4 0 4) on detector 0
P2: (653.32,500.22) qc 110.8110 with hkl (7 -1 1) on detector 1
P3: (342.45,775.77) qc 127.9410 with hkl (6 4 4) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:52:53
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0494058e-01
b = 4.0445905e-01
c = 4.0497558e-01
alpha = 89.982155
beta = 89.977218
gamma = 89.980033
Best strain tensor (crystal coordinates):
-1.46855e-04 -1.74216e-04 1.98785e-04
-1.74216e-04 -1.33572e-03 1.55521e-04
1.98785e-04 1.55521e-04 -6.02863e-05
Best stress tensor (crystal coordinates)(MPa):
-101.46470 -9.93032 11.33074
-9.93032 -157.22233 8.86472
11.33074 8.86472 -97.40463
Best strain tensor (sample coordinates):
-7.45083e-04 -5.66349e-04 -3.10608e-05
-5.66349e-04 -8.59817e-04 -1.66934e-04
-3.10608e-05 -1.66934e-04 6.20415e-05
Best stress tensor (sample coordinates)(MPa):
-127.98917 -26.29409 -1.49632
-26.29409 -138.28034 -8.79870
-1.49632 -8.79870 -89.82215
Best principal strains are:
e1 = 9.99792e-05
e2 = -2.56620e-04
e3 = -1.38622e-03
Best principal stresses are (MPa):
sig1 = -87.92056
sig2 = -107.39402
sig3 = -160.77707
Best hydrostatic stress is (MPa):
-118.69722
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0494058e-01 +/- 1.3920194e-05
b = 4.0445905e-01 +/- 7.6442705e-05
c = 4.0497558e-01 +/- 5.9339672e-05
alpha = 89.982155 +/- 0.012823
beta = 89.977218 +/- 0.009911
152
gamma = 89.980033 +/- 0.007437
The strain tensor components in crystallographic coordinates:
e11 = -1.4685485e-04 +/- 4.2424671e-05
e22 = -1.3357170e-03 +/- 1.9059969e-04
e33 = -6.0286313e-05 +/- 1.4868736e-04
e23 = 1.5552149e-04 +/- 1.1174061e-04
e13 = 1.9878490e-04 +/- 8.6474085e-05
e12 = -1.7421609e-04 +/- 6.4894249e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = -101.46470 +/- 15.36007
s2 = -157.22233 +/- 21.56567
s3 = -97.40463 +/- 19.82131
s4 = 8.86472 +/- 6.36921
s5 = 11.33074 +/- 4.92902
s6 = -9.93032 +/- 3.69897
The strain tensor components in sample coordinates:
e11 = -7.4508316e-04 +/- 1.0146710e-04
e22 = -8.5981651e-04 +/- 1.5500506e-04
e33 = 6.2041528e-05 +/- 3.1421417e-05
e23 = -1.6693431e-04 +/- 5.7604802e-05
e13 = -3.1060827e-05 +/- 9.2557111e-05
e12 = -5.6634870e-04 +/- 1.5718999e-04
The stress tensor components in sample coordinates (MPa):
s1 = -127.98917 +/- 19.00164
s2 = -138.28034 +/- 22.22173
s3 = -89.82215 +/- 14.09614
s4 = -8.79870 +/- 3.16212
s5 = -1.49632 +/- 5.16742
s6 = -26.29409 +/- 7.86689
The principal strains are:
Principal strain 1 = 9.9979205e-05 +/- 5.3667206e-05
Principal strain 2 = -2.5662014e-04 +/- 1.4978963e-04
Principal strain 3 = -1.3862172e-03 +/- 2.0195935e-04
The principal strain directions are:
v1 = (0.126980, -0.242224, 0.961875)
v2 = (-0.739164, 0.623549, 0.254604)
v3 = (-0.661447, -0.743313, -0.099864)
The principal stresses are (MPa):
Principal stress 1 = -87.92056 +/- 15.52092
Principal stress 2 = -107.39402 +/- 19.62695
Principal stress 3 = -160.77707 +/- 22.29298
The principal stress directions are:
v1 = (0.113715, -0.228312, 0.966924)
v2 = (-0.772368, 0.591844, 0.230582)
v3 = (-0.624913, -0.773042, -0.109039)
The hydrostatic stress is (MPa): -118.69722 +/- 18.28303
153
A3. ECAP AA1050 8P Matlab Outputs
The input peak values from "8P_G1.txt" were:
P1: (1599.18,380.52) qc 80.6295 with hkl (3 -3 3) on detector 0
P2: (661.55,655.43) qc 127.9170 with hkl (0 -2 8) on detector 1
P3: (157.97,94.02) qc 119.1900 with hkl (1 -7 3) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:53:15
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0483269e-01
b = 4.0489024e-01
c = 4.0505823e-01
alpha = 89.999554
beta = 90.012289
gamma = 90.009181
Best strain tensor (crystal coordinates):
-4.13142e-04 8.00840e-05 -1.07197e-04
8.00840e-05 -2.71002e-04 3.89499e-06
-1.07197e-04 3.89499e-06 1.43784e-04
Best stress tensor (crystal coordinates)(MPa):
-52.50048 4.56479 -6.11024
4.56479 -45.83409 0.22201
-6.11024 0.22201 -26.38064
Best strain tensor (sample coordinates):
-9.83330e-05 -3.63313e-05 -2.46562e-04
-3.63313e-05 -2.66087e-04 1.80264e-04
-2.46562e-04 1.80264e-04 -1.75941e-04
Best stress tensor (sample coordinates)(MPa):
-37.62805 -1.96811 -11.86006
-1.96811 -44.62198 9.22208
-11.86006 9.22208 -42.46518
Best principal strains are:
e1 = 1.63979e-04
e2 = -2.38800e-04
e3 = -4.65540e-04
Best principal stresses are (MPa):
sig1 = -24.99219
sig2 = -43.84091
sig3 = -55.88211
Best hydrostatic stress is (MPa):
-41.57174
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0483269e-01 +/- 5.4049139e-05
b = 4.0489024e-01 +/- 2.4629766e-05
c = 4.0505823e-01 +/- 1.5039707e-05
alpha = 89.999554 +/- 0.008495
154
beta = 90.012289 +/- 0.011161
gamma = 90.009181 +/- 0.011525
The strain tensor components in crystallographic coordinates:
e11 = -4.1314233e-04 +/- 1.3569426e-04
e22 = -2.7100199e-04 +/- 6.5514138e-05
e33 = 1.4378380e-04 +/- 4.4455777e-05
e23 = 3.8949896e-06 +/- 7.4111626e-05
e13 = -1.0719711e-04 +/- 9.7359201e-05
e12 = 8.0083954e-05 +/- 1.0052521e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -52.50048 +/- 17.65829
s2 = -45.83409 +/- 14.39854
s3 = -26.38064 +/- 13.31630
s4 = 0.22201 +/- 4.22436
s5 = -6.11024 +/- 5.54947
s6 = 4.56479 +/- 5.72994
The strain tensor components in sample coordinates:
e11 = -9.8332987e-05 +/- 1.1653752e-04
e22 = -2.6608676e-04 +/- 1.2923471e-04
e33 = -1.7594077e-04 +/- 1.0559630e-04
e23 = 1.8026390e-04 +/- 8.3354704e-05
e13 = -2.4656169e-04 +/- 5.9834120e-05
e12 = -3.6331338e-05 +/- 7.8499506e-05
The stress tensor components in sample coordinates (MPa):
s1 = -37.62805 +/- 17.42537
s2 = -44.62198 +/- 15.69453
s3 = -42.46518 +/- 14.50543
s4 = 9.22208 +/- 4.66409
s5 = -11.86006 +/- 3.21770
s6 = -1.96811 +/- 4.16925
The principal strains are:
Principal strain 1 = 1.6397950e-04 +/- 6.1025970e-05
Principal strain 2 = -2.3880024e-04 +/- 7.9028928e-05
Principal strain 3 = -4.6553977e-04 +/- 1.3830085e-04
The principal strain directions are:
v1 = (-0.669621, 0.334512, 0.663106)
v2 = (-0.612042, -0.754309, -0.237534)
v3 = (0.420729, -0.564906, 0.709836)
The principal stresses are (MPa):
Principal stress 1 = -24.99219 +/- 13.81058
Principal stress 2 = -43.84091 +/- 14.73743
Principal stress 3 = -55.88211 +/- 17.68405
The principal stress directions are:
v1 = (-0.665601, 0.370967, 0.647579)
v2 = (-0.615998, -0.762954, -0.196081)
v3 = (0.421334, -0.529419, 0.736338)
The hydrostatic stress is (MPa): -41.57174 +/- 14.94241
155
The input peak values from "8P_G2.txt" were:
P1: (1722.75,1601.38) qc 80.6160 with hkl (-1 1 5) on detector 0
P2: (158.26,765.45) qc 119.1710 with hkl (-5 3 5) on detector 1
P3: (278.70,815.33) qc 141.3330 with hkl (-5 -3 7) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:53:37
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0500907e-01
b = 4.0490641e-01
c = 4.0498825e-01
alpha = 90.003893
beta = 90.005713
gamma = 89.992712
Best strain tensor (crystal coordinates):
2.23900e-05 -6.35974e-05 -4.98607e-05
-6.35974e-05 -2.31087e-04 -3.39608e-05
-4.98607e-05 -3.39608e-05 -2.90106e-05
Best stress tensor (crystal coordinates)(MPa):
-13.52140 -3.62505 -2.84206
-3.62505 -25.40948 -1.93577
-2.84206 -1.93577 -15.93208
Best strain tensor (sample coordinates):
-2.20263e-04 -8.46372e-05 1.01769e-05
-8.46372e-05 -2.41967e-05 5.67073e-05
1.01769e-05 5.67073e-05 6.75190e-06
Best stress tensor (sample coordinates)(MPa):
-24.80403 -4.65645 0.56873
-4.65645 -16.11084 3.07179
0.56873 3.07179 -13.94809
Best principal strains are:
e1 = 5.80191e-05
e2 = -4.05813e-05
e3 = -2.55146e-04
Best principal stresses are (MPa):
sig1 = -11.32341
sig2 = -16.47043
sig3 = -27.06913
Best hydrostatic stress is (MPa):
-18.28766
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0500907e-01 +/- 4.8856408e-05
b = 4.0490641e-01 +/- 5.5821257e-05
c = 4.0498825e-01 +/- 1.8629485e-05
alpha = 90.003893 +/- 0.009327
beta = 90.005713 +/- 0.008576
gamma = 89.992712 +/- 0.011645
156
The strain tensor components in crystallographic coordinates:
e11 = 2.2390032e-05 +/- 1.2303125e-04
e22 = -2.3108729e-04 +/- 1.3990969e-04
e33 = -2.9010562e-05 +/- 5.2291544e-05
e23 = -3.3960846e-05 +/- 8.1377706e-05
e13 = -4.9860661e-05 +/- 7.4833005e-05
e12 = -6.3597357e-05 +/- 1.0162362e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -13.52140 +/- 17.37208
s2 = -25.40948 +/- 17.82596
s3 = -15.93208 +/- 14.45476
s4 = -1.93577 +/- 4.63853
s5 = -2.84206 +/- 4.26548
s6 = -3.62505 +/- 5.79255
The strain tensor components in sample coordinates:
e11 = -2.2026300e-04 +/- 1.4117665e-04
e22 = -2.4196713e-05 +/- 1.5436016e-04
e33 = 6.7518956e-06 +/- 3.1994976e-05
e23 = 5.6707290e-05 +/- 4.1662341e-05
e13 = 1.0176935e-05 +/- 1.1093519e-04
e12 = -8.4637243e-05 +/- 6.9532609e-05
The stress tensor components in sample coordinates (MPa):
s1 = -24.80403 +/- 17.88180
s2 = -16.11084 +/- 19.30405
s3 = -13.94809 +/- 12.62962
s4 = 3.07179 +/- 2.44975
s5 = 0.56873 +/- 6.31254
s6 = -4.65645 +/- 3.96138
The principal strains are:
Principal strain 1 = 5.8019124e-05 +/- 8.7191074e-05
Principal strain 2 = -4.0581305e-05 +/- 9.9089969e-05
Principal strain 3 = -2.5514564e-04 +/- 1.1839883e-04
The principal strain directions are:
v1 = (-0.179893, 0.677259, 0.713414)
v2 = (-0.339620, 0.637889, -0.691199)
v3 = (-0.923199, -0.366631, 0.115259)
The principal stresses are (MPa):
Principal stress 1 = -11.32341 +/- 15.27167
Principal stress 2 = -16.47043 +/- 17.05126
Principal stress 3 = -27.06913 +/- 17.31818
The principal stress directions are:
v1 = (-0.196509, 0.657718, 0.727180)
v2 = (-0.395406, 0.625504, -0.672606)
v3 = (-0.897239, -0.419704, 0.137148)
The hydrostatic stress is (MPa): -18.28766 +/- 16.10657
157
The input peak values from "8P_G4.txt" were:
P1: (1986.33,1362.42) qc 110.7520 with hkl (-1 1 7) on detector 0
P2: (333.67,146.47) qc 147.9960 with hkl (3 -1 9) on detector 1
P3: (662.03,147.33) qc 119.1830 with hkl (3 5 5) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:53:59
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0492777e-01
b = 4.0496469e-01
c = 4.0510176e-01
alpha = 89.989312
beta = 90.040176
gamma = 90.020038
Best strain tensor (crystal coordinates):
-1.78650e-04 1.74834e-04 -3.50539e-04
1.74834e-04 -8.72056e-05 9.32647e-05
-3.50539e-04 9.32647e-05 2.51255e-04
Best stress tensor (crystal coordinates)(MPa):
-9.27364 9.96554 -19.98073
9.96554 -4.98491 5.31609
-19.98073 5.31609 10.88891
Best strain tensor (sample coordinates):
-9.04624e-05 2.49645e-04 7.72131e-06
2.49645e-04 -1.60292e-04 -3.25980e-04
7.72131e-06 -3.25980e-04 2.36154e-04
Best stress tensor (sample coordinates)(MPa):
-5.36246 14.22631 1.23129
14.22631 -8.24121 -18.35492
1.23129 -18.35492 10.23403
Best principal strains are:
e1 = 4.47532e-04
e2 = 1.37475e-05
e3 = -4.75880e-04
Best principal stresses are (MPa):
sig1 = 23.20672
sig2 = 0.69967
sig3 = -27.27603
Best hydrostatic stress is (MPa):
-1.12321
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0492777e-01 +/- 6.1778659e-05
b = 4.0496469e-01 +/- 5.7227187e-05
c = 4.0510176e-01 +/- 8.6668753e-06
alpha = 89.989312 +/- 0.006839
beta = 90.040176 +/- 0.004274
gamma = 90.020038 +/- 0.016055
158
The strain tensor components in crystallographic coordinates:
e11 = -1.7864964e-04 +/- 1.5467824e-04
e22 = -8.7205617e-05 +/- 1.4371446e-04
e33 = 2.5125543e-04 +/- 3.2295668e-05
e23 = 9.3264743e-05 +/- 5.9670440e-05
e13 = -3.5053919e-04 +/- 3.7316089e-05
e12 = 1.7483396e-04 +/- 1.4008937e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -9.27364 +/- 19.79356
s2 = -4.98491 +/- 19.22164
s3 = 10.88891 +/- 13.81694
s4 = 5.31609 +/- 3.40122
s5 = -19.98073 +/- 2.12702
s6 = 9.96554 +/- 7.98509
The strain tensor components in sample coordinates:
e11 = -9.0462416e-05 +/- 1.4842119e-04
e22 = -1.6029162e-04 +/- 1.5186871e-04
e33 = 2.3615421e-04 +/- 3.5210472e-05
e23 = -3.2597955e-04 +/- 5.1698239e-05
e13 = 7.7213124e-06 +/- 6.8746958e-05
e12 = 2.4964522e-04 +/- 1.2973252e-04
The stress tensor components in sample coordinates (MPa):
s1 = -5.36246 +/- 19.53713
s2 = -8.24121 +/- 19.50543
s3 = 10.23403 +/- 13.93768
s4 = -18.35492 +/- 2.93441
s5 = 1.23129 +/- 3.76051
s6 = 14.22631 +/- 7.41585
The principal strains are:
Principal strain 1 = 4.4753237e-04 +/- 7.2077657e-05
Principal strain 2 = 1.3747507e-05 +/- 1.6882058e-04
Principal strain 3 = -4.7587970e-04 +/- 1.0469817e-04
The principal strain directions are:
v1 = (-0.235515, -0.532685, 0.812884)
v2 = (0.826170, 0.330763, 0.456114)
v3 = (0.511836, -0.779001, -0.362189)
The principal stresses are (MPa):
Principal stress 1 = 23.20672 +/- 17.01769
Principal stress 2 = 0.69967 +/- 20.43513
Principal stress 3 = -27.27603 +/- 16.13686
The principal stress directions are:
v1 = (-0.250084, -0.569958, 0.782691)
v2 = (0.818810, 0.306924, 0.485127)
v3 = (0.516729, -0.762198, -0.389931)
The hydrostatic stress is (MPa): -1.12321 +/- 17.20951
159
The input peak values from "8P_G6.txt" were:
P1: (1751.69,1173.35) qc 80.6016 with hkl (1 1 5) on detector 0
P2: (235.31,934.42) qc 135.2120 with hkl (-2 6 6) on detector 1
P3: (463.76,440.88) qc 119.1600 with hkl (-3 -1 7) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:54:20
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0491490e-01
b = 4.0497145e-01
c = 4.0504518e-01
alpha = 89.974144
beta = 90.007880
gamma = 90.008796
Best strain tensor (crystal coordinates):
-2.10135e-04 7.67442e-05 -6.87518e-05
7.67442e-05 -7.05843e-05 2.25620e-04
-6.87518e-05 2.25620e-04 1.11556e-04
Best stress tensor (crystal coordinates)(MPa):
-20.22499 4.37442 -3.91885
4.37442 -13.68009 12.86035
-3.91885 12.86035 -5.13771
Best strain tensor (sample coordinates):
-1.11343e-04 5.69352e-05 -1.49511e-04
5.69352e-05 -2.48147e-04 1.07789e-04
-1.49511e-04 1.07789e-04 1.90327e-04
Best stress tensor (sample coordinates)(MPa):
-15.96876 3.63274 -8.89320
3.63274 -22.50300 6.16727
-8.89320 6.16727 -0.57103
Best principal strains are:
e1 = 2.64229e-04
e2 = -1.14487e-04
e3 = -3.18906e-04
Best principal stresses are (MPa):
sig1 = 4.15870
sig2 = -15.60596
sig3 = -27.59554
Best hydrostatic stress is (MPa):
-13.01426
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0491490e-01 +/- 6.0454685e-05
b = 4.0497145e-01 +/- 4.1335800e-05
c = 4.0504518e-01 +/- 1.0278982e-05
alpha = 89.974144 +/- 0.006845
beta = 90.007880 +/- 0.003627
gamma = 90.008796 +/- 0.013515
160
The strain tensor components in crystallographic coordinates:
e11 = -2.1013455e-04 +/- 1.5095623e-04
e22 = -7.0584252e-05 +/- 1.0514011e-04
e33 = 1.1155592e-04 +/- 3.5528467e-05
e23 = 2.2562019e-04 +/- 5.9715095e-05
e13 = -6.8751765e-05 +/- 3.1636473e-05
e12 = 7.6744236e-05 +/- 1.1791500e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -20.22499 +/- 17.70624
s2 = -13.68009 +/- 15.37773
s3 = -5.13771 +/- 11.76413
s4 = 12.86035 +/- 3.40376
s5 = -3.91885 +/- 1.80328
s6 = 4.37442 +/- 6.72115
The strain tensor components in sample coordinates:
e11 = -1.1134309e-04 +/- 1.2554197e-04
e22 = -2.4814655e-04 +/- 1.6538842e-04
e33 = 1.9032677e-04 +/- 2.9146684e-05
e23 = 1.0778857e-04 +/- 5.6549260e-05
e13 = -1.4951092e-04 +/- 3.3091645e-05
e12 = 5.6935213e-05 +/- 9.8736641e-05
The stress tensor components in sample coordinates (MPa):
s1 = -15.96876 +/- 15.87507
s2 = -22.50300 +/- 18.34874
s3 = -0.57103 +/- 11.47047
s4 = 6.16727 +/- 3.23895
s5 = -8.89320 +/- 1.91638
s6 = 3.63274 +/- 5.46867
The principal strains are:
Principal strain 1 = 2.6422949e-04 +/- 4.4976496e-05
Principal strain 2 = -1.1448662e-04 +/- 1.1151802e-04
Principal strain 3 = -3.1890575e-04 +/- 1.2690444e-04
The principal strain directions are:
v1 = (0.344749, -0.156404, -0.925573)
v2 = (0.822396, 0.525705, 0.217485)
v3 = (-0.452563, 0.836165, -0.309863)
The principal stresses are (MPa):
Principal stress 1 = 4.15870 +/- 12.28369
Principal stress 2 = -15.60596 +/- 15.69605
Principal stress 3 = -27.59554 +/- 17.09318
The principal stress directions are:
v1 = (-0.374595, 0.160207, 0.913242)
v2 = (-0.773369, -0.597302, -0.212439)
v3 = (0.511447, -0.785853, 0.347646)
The hydrostatic stress is (MPa): -13.01426 +/- 14.57724
161
The input peak values from "8P_G7.txt" were:
P1: (1877.74,1297.73) qc 91.7462 with hkl (1 -3 5) on detector 0
P2: (65.89,211.50) qc 124.1020 with hkl (0 0 8) on detector 1
P3: (377.87,955.99) qc 76.0055 with hkl (-2 -2 4) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:54:42
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0509558e-01
b = 4.0512746e-01
c = 4.0503375e-01
alpha = 90.016683
beta = 89.957440
gamma = 90.010333
Best strain tensor (crystal coordinates):
2.35699e-04 9.01913e-05 3.71497e-04
9.01913e-05 3.14676e-04 -1.45635e-04
3.71497e-04 -1.45635e-04 8.33336e-05
Best stress tensor (crystal coordinates)(MPa):
49.90055 5.14091 21.17534
5.14091 53.60458 -8.30117
21.17534 -8.30117 42.75463
Best strain tensor (sample coordinates):
4.44883e-04 1.39616e-05 -9.85798e-05
1.39616e-05 -4.76701e-05 -3.31872e-04
-9.85798e-05 -3.31872e-04 2.36495e-04
Best stress tensor (sample coordinates)(MPa):
61.68553 0.96438 -6.24156
0.96438 33.54480 -18.23192
-6.24156 -18.23192 51.02944
Best principal strains are:
e1 = 5.41348e-04
e2 = 3.61541e-04
e3 = -2.69181e-04
Best principal stresses are (MPa):
sig1 = 67.97062
sig2 = 56.38321
sig3 = 21.90593
Best hydrostatic stress is (MPa):
48.75325
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0509558e-01 +/- 7.6015496e-05
b = 4.0512746e-01 +/- 7.4841051e-05
c = 4.0503375e-01 +/- 4.8772174e-06
alpha = 90.016683 +/- 0.006012
beta = 89.957440 +/- 0.006852
gamma = 90.010333 +/- 0.010798
162
The strain tensor components in crystallographic coordinates:
e11 = 2.3569859e-04 +/- 1.8972966e-04
e22 = 3.1467567e-04 +/- 1.8650552e-04
e33 = 8.3333564e-05 +/- 2.7496389e-05
e23 = -1.4563465e-04 +/- 5.2460851e-05
e13 = 3.7149720e-04 +/- 5.9846606e-05
e12 = 9.0191350e-05 +/- 9.4257064e-05
The stress tensor components in crystallographic coordinates (MPa):
s1 = 49.90055 +/- 21.31616
s2 = 53.60458 +/- 21.08049
s3 = 42.75463 +/- 15.08190
s4 = -8.30117 +/- 2.99027
s5 = 21.17534 +/- 3.41126
s6 = 5.14091 +/- 5.37265
The strain tensor components in sample coordinates:
e11 = 4.4488322e-04 +/- 1.4372634e-04
e22 = -4.7670099e-05 +/- 1.3980597e-04
e33 = 2.3649470e-04 +/- 2.8846503e-05
e23 = -3.3187212e-04 +/- 6.5405511e-05
e13 = -9.8579823e-05 +/- 7.5835620e-05
e12 = 1.3961588e-05 +/- 1.4301032e-04
The stress tensor components in sample coordinates (MPa):
s1 = 61.68553 +/- 20.91200
s2 = 33.54480 +/- 20.78909
s3 = 51.02944 +/- 14.27359
s4 = -18.23192 +/- 3.50436
s5 = -6.24156 +/- 4.14351
s6 = 0.96438 +/- 6.82090
The principal strains are:
Principal strain 1 = 5.4134769e-04 +/- 1.2702485e-04
Principal strain 2 = 3.6154079e-04 +/- 1.1653656e-04
Principal strain 3 = -2.6918066e-04 +/- 7.4184908e-05
The principal strain directions are:
v1 = (0.690192, 0.367562, -0.623325)
v2 = (-0.721090, 0.421409, -0.549949)
v3 = (-0.060535, -0.829043, -0.555898)
The principal stresses are (MPa):
Principal stress 1 = 67.97062 +/- 19.37905
Principal stress 2 = 56.38321 +/- 19.01109
Principal stress 3 = 21.90593 +/- 17.87125
The principal stress directions are:
v1 = (0.685482, 0.355663, -0.635310)
v2 = (-0.725247, 0.410574, -0.552672)
v3 = (-0.064277, -0.839603, -0.539384)
The hydrostatic stress is (MPa): 48.75325 +/- 18.35397
163
The input peak values from "8P_G8.txt" were:
P1: (1959.34,693.97) qc 111.8200 with hkl (4 0 6) on detector 0
P2: (330.62,733.94) qc 141.3290 with hkl (7 5 3) on detector 1
P3: (338.67,678.26) qc 142.1900 with hkl (2 4 8) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:55:04
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0520887e-01
b = 4.0525316e-01
c = 4.0496229e-01
alpha = 90.017163
beta = 89.952408
gamma = 90.076485
Best strain tensor (crystal coordinates):
5.14490e-04 6.67805e-04 4.15531e-04
6.67805e-04 6.25034e-04 -1.49872e-04
4.15531e-04 -1.49872e-04 -9.30997e-05
Best stress tensor (crystal coordinates)(MPa):
88.27543 38.06489 23.68529
38.06489 93.45992 -8.54269
23.68529 -8.54269 59.77946
Best strain tensor (sample coordinates):
-2.37592e-05 -5.75288e-04 -5.25467e-04
-5.75288e-04 3.91338e-04 2.42418e-04
-5.25467e-04 2.42418e-04 6.78846e-04
Best stress tensor (sample coordinates)(MPa):
60.12186 -33.05437 -26.93015
-33.05437 80.24936 13.16116
-26.93015 13.16116 101.14359
Best principal strains are:
e1 = 1.26358e-03
e2 = 2.95164e-04
e3 = -5.12317e-04
Best principal stresses are (MPa):
sig1 = 130.56267
sig2 = 78.55101
sig3 = 32.40113
Best hydrostatic stress is (MPa):
80.50494
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0520887e-01 +/- 5.6282816e-05
b = 4.0525316e-01 +/- 5.2646789e-05
c = 4.0496229e-01 +/- 2.8747964e-05
alpha = 90.017163 +/- 0.008280
beta = 89.952408 +/- 0.007909
gamma = 90.076485 +/- 0.012455
164
The strain tensor components in crystallographic coordinates:
e11 = 5.1449042e-04 +/- 1.4102501e-04
e22 = 6.2503390e-04 +/- 1.3217039e-04
e33 = -9.3099668e-05 +/- 7.5252608e-05
e23 = -1.4987180e-04 +/- 7.2304797e-05
e13 = 4.1553149e-04 +/- 6.9020355e-05
e12 = 6.6780508e-04 +/- 1.0882286e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = 88.27543 +/- 18.49797
s2 = 93.45992 +/- 17.96487
s3 = 59.77946 +/- 14.81170
s4 = -8.54269 +/- 4.12137
s5 = 23.68529 +/- 3.93416
s6 = 38.06489 +/- 6.20290
The strain tensor components in sample coordinates:
e11 = -2.3759230e-05 +/- 1.5242422e-04
e22 = 3.9133777e-04 +/- 1.2049335e-04
e33 = 6.7884611e-04 +/- 6.7800500e-05
e23 = 2.4241796e-04 +/- 9.9935544e-05
e13 = -5.2546698e-04 +/- 9.3283980e-05
e12 = -5.7528805e-04 +/- 5.9263606e-05
The stress tensor components in sample coordinates (MPa):
s1 = 60.12186 +/- 19.96297
s2 = 80.24936 +/- 16.57151
s3 = 101.14359 +/- 14.97958
s4 = 13.16116 +/- 5.69299
s5 = -26.93015 +/- 4.56294
s6 = -33.05437 +/- 3.34318
The principal strains are:
Principal strain 1 = 1.2635775e-03 +/- 1.9779381e-04
Principal strain 2 = 2.9516402e-04 +/- 1.1052268e-04
Principal strain 3 = -5.1231688e-04 +/- 9.4447376e-05
The principal strain directions are:
v1 = (-0.512241, 0.526444, 0.678577)
v2 = (-0.167221, 0.713850, -0.680040)
v3 = (0.842405, 0.461817, 0.277631)
The principal stresses are (MPa):
Principal stress 1 = 130.56267 +/- 21.96112
Principal stress 2 = 78.55101 +/- 14.47168
Principal stress 3 = 32.40113 +/- 16.30788
The principal stress directions are:
v1 = (-0.505952, 0.513658, 0.692941)
v2 = (-0.237019, 0.689637, -0.684268)
v3 = (0.829358, 0.510447, 0.227176)
The hydrostatic stress is (MPa): 80.50494 +/- 16.56988
165
The input peak values from "8P_G11.txt" were:
P1: (1739.36,1294.61) qc 110.7610 with hkl (-1 -1 7) on detector 0
P2: (334.87,164.33) qc 141.3550 with hkl (-3 -5 7) on detector 1
P3: (125.70,632.83) qc 141.3410 with hkl (3 -5 7) on detector 2
The results and uncertainty were calculated based on 10000 iterations,
and "geoN_2013-11-16_00-46-00_cor2.txt" geometry file
Calculations were done on 13-Jul-2015 14:55:26
---------------------------------------------------------
The best lattice parameters are (in nm and degrees):
a = 4.0491564e-01
b = 4.0484397e-01
c = 4.0512449e-01
alpha = 89.986147
beta = 90.020035
gamma = 90.043712
Best strain tensor (crystal coordinates):
-2.08636e-04 3.81383e-04 -1.74799e-04
3.81383e-04 -3.85276e-04 1.20843e-04
-1.74799e-04 1.20843e-04 3.07375e-04
Best stress tensor (crystal coordinates)(MPa):
-27.34979 21.73884 -9.96356
21.73884 -35.63421 6.88805
-9.96356 6.88805 -3.14888
Best strain tensor (sample coordinates):
-1.68212e-04 -3.18343e-04 -2.29393e-04
-3.18343e-04 -3.25520e-04 -3.02119e-04
-2.29393e-04 -3.02119e-04 2.07194e-04
Best stress tensor (sample coordinates)(MPa):
-25.06096 -18.17904 -13.48020
-18.17904 -32.80737 -15.31746
-13.48020 -15.31746 -8.26456
Best principal strains are:
e1 = 3.61040e-04
e2 = 8.09579e-05
e3 = -7.28535e-04
Best principal stresses are (MPa):
sig1 = 0.46282
sig2 = -10.40020
sig3 = -56.19551
Best hydrostatic stress is (MPa):
-22.04430
---------------------------------------------------------
The measured lattice parameters are (in nm):
a = 4.0491564e-01 +/- 6.2629202e-05
b = 4.0484397e-01 +/- 5.9309687e-05
c = 4.0512449e-01 +/- 1.5054994e-05
alpha = 89.986147 +/- 0.008502
beta = 90.020035 +/- 0.010405
gamma = 90.043712 +/- 0.014517
166
The strain tensor components in crystallographic coordinates:
e11 = -2.0863618e-04 +/- 1.5622062e-04
e22 = -3.8527628e-04 +/- 1.4846432e-04
e33 = 3.0737483e-04 +/- 4.4475757e-05
e23 = 1.2084292e-04 +/- 7.4184357e-05
e13 = -1.7479923e-04 +/- 9.0775388e-05
e12 = 3.8138311e-04 +/- 1.2665121e-04
The stress tensor components in crystallographic coordinates (MPa):
s1 = -27.34979 +/- 20.81705
s2 = -35.63421 +/- 20.60829
s3 = -3.14888 +/- 16.46092
s4 = 6.88805 +/- 4.22851
s5 = -9.96356 +/- 5.17420
s6 = 21.73884 +/- 7.21912
The strain tensor components in sample coordinates:
e11 = -1.6821250e-04 +/- 1.5365437e-04
e22 = -3.2551963e-04 +/- 1.7928453e-04
e33 = 2.0719450e-04 +/- 2.9421076e-05
e23 = -3.0211923e-04 +/- 3.7103502e-05
e13 = -2.2939336e-04 +/- 1.2425342e-04
e12 = -3.1834349e-04 +/- 9.4038051e-05
The stress tensor components in sample coordinates (MPa):
s1 = -25.06096 +/- 20.68052
s2 = -32.80737 +/- 22.69860
s3 = -8.26456 +/- 14.57806
s4 = -15.31746 +/- 2.24048
s5 = -13.48020 +/- 7.07897
s6 = -18.17904 +/- 5.32466
The principal strains are:
Principal strain 1 = 3.6103988e-04 +/- 6.5799463e-05
Principal strain 2 = 8.0957925e-05 +/- 1.2631124e-04
Principal strain 3 = -7.2853544e-04 +/- 1.7671472e-04
The principal strain directions are:
v1 = (0.216832, 0.307212, -0.926609)
v2 = (0.793036, -0.608956, -0.016321)
v3 = (0.569278, 0.731295, 0.375672)
The principal stresses are (MPa):
Principal stress 1 = 0.46282 +/- 16.59769
Principal stress 2 = -10.40020 +/- 19.67874
Principal stress 3 = -56.19551 +/- 22.50515
The principal stress directions are:
v1 = (-0.300929, -0.258223, 0.918021)
v2 = (0.753541, -0.654381, 0.062946)
v3 = (0.584481, 0.710708, 0.391504)
The hydrostatic stress is (MPa): -22.04430 +/- 18.85504
167
Appendix B. Matlab Code
B1. Matlab Code for Full Elastic Strain and Stress Tensor for ECAP Al
Calculations
function [] = StrainCalcMC (peakfilename, geofilename, n)
% StrainCalcMC will calculate full strain tensor given 3 linearly
% independent diffraction peaks with known qc. The full strain/stress
% tensors are calculated for the crystal coordinates and rotated to sample
% coordinates. Uncertainty is calculated using Monte Carlo error
% propagation. Input of program n is the number of iterations used for
% error analysis. Mean values are calculated directly from the best input
% values and are not approximated.
%
% Program reads in peak input file in text format in identical format as
% Dr. Levine's C code input file, that is:
% Peakname, peak number, px, s_px, py, s_py, qc, s_qc, h, k, l, detector,
% depth(in microns), FWHM, and intensity
% Only px, py, spx, spy, qc, sqc, hkl, detector, and depth are used in full
% strain calculations
%
% Geometry information is read in from geometry file produced from APS
% equipments. Specifically, the translation (P) and rotation vector (R) for
% the three different detectors
%
% Vector from diffraction volume to pixel location is calculated by
% Pixel2XYZ, which calls on Rodriguez to apply the rotation transformation
%
% Define material parameters, change these values to reflect the unit cell
% parameters of the appropriate material
a0 = 0.405000; %lattice parameter as measured from powder diffraction for
ECAP AA1050 in nm
s_a0 = 0.00001; % Added 5/8/15 for uncertainty analysis
c_11 = 108.2;
c_12 = 61.3;
c_44 = 28.5;
% Read in values from input peak file, here the file is opened and read in
% line by line to get the appropriate information
% peakfilename = input('What is the input file? ');
fid = fopen(peakfilename);
tline = fgetl(fid);
firstpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px1 = firstpeak{3};
s_px1 = firstpeak{4};
py1 = firstpeak{5};
s_py1 = firstpeak{6};
qc1 = firstpeak{7};
s_qc1 = firstpeak{8};
h1 = firstpeak{9};
k1 = firstpeak{10};
l1 = firstpeak{11};
168
detector1 = firstpeak{12};
depth1 = firstpeak{13};
FWHM1 = firstpeak{14};
tline = fgetl(fid);
secondpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px2 = secondpeak{3};
s_px2 = secondpeak{4};
py2 = secondpeak{5};
s_py2 = secondpeak{6};
qc2 = secondpeak{7};
s_qc2 = secondpeak{8};
h2 = secondpeak{9};
k2 = secondpeak{10};
l2 = secondpeak{11};
detector2 = secondpeak{12};
depth2 = secondpeak{13};
FWHM2 = secondpeak{14};
tline = fgetl(fid);
thirdpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px3 = thirdpeak{3};
s_px3 = thirdpeak{4};
py3 = thirdpeak{5};
s_py3 = thirdpeak{6};
qc3 = thirdpeak{7};
s_qc3 = thirdpeak{8};
h3 = thirdpeak{9};
k3 = thirdpeak{10};
l3 = thirdpeak{11};
detector3 = thirdpeak{12};
depth3 = thirdpeak{13};
FWHM3 = thirdpeak{14};
% Ask for geometry file
% geofilename = input('What is the geometry file? ');
fid = fopen(geofilename);
% Read in rotation and translation vectors
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
169
O_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
OR(1) = O_R {1};
OR(2) = O_R {2};
OR(3) = O_R {3};
tline = fgetl(fid);
O_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
OP(1) = O_P {1};
OP(2) = O_P {2};
OP(3) = O_P {3};
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
Y_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
YR(1) = Y_R {1};
YR(2) = Y_R {2};
YR(3) = Y_R {3};
tline = fgetl(fid);
Y_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
YP(1) = Y_P {1};
YP(2) = Y_P {2};
YP(3) = Y_P {3};
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
P_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
PR(1) = P_R {1};
PR(2) = P_R {2};
PR(3) = P_R {3};
tline = fgetl(fid);
P_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
PP(1) = P_P {1};
PP(2) = P_P {2};
PP(3) = P_P {3};
% First, calculate the mean values using the best input (no Monte Carlo)
% Define the three vectors, Pixel2XYZ is used to calculate the plane normal
% vector from the diffraction volume to the pixel locations in lab
% coordinates.
A = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector1,px1,py1,qc1,depth1);
E = [h1 k1 l1];
B = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector2,px2,py2,qc2,depth2);
F = [h2 k2 l2];
C = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector3,px3,py3,qc3,depth3);
G = [h3 k3 l3];
170
% Calculate base vectors from hkl vectors and diffraction vectors by
% Gaussian reduction (rref). Matrix M is structured as such:
% M = h1 k1 l1| v1x v1y v1z
% h2 k2 l2| v2x v2y v2z
% h3 k3 l3| v3x v3y v3z
% as we reduce the left half of the matrix to the identity matrix, the
% corresponding right half of M now represents the real world reciprocal
% vectors for the directions corresponding to the left side, which is 100
% 010 and 001 respectively, which is the unit cell basis vectors
M = [E A; F B; G C];
M = rref(M);
% Solve for reciprocal unit cell basis vectors (selecting the appropriate
% components of the matrix)
Vb1 = M(1,4:6);
Vb2 = M(2,4:6);
Vb3 = M(3,4:6);
% Solve for the real space unit cell basis vectors by calculating the
% real space vectors given the reciprocal space vectors
V1r = 2*pi*(cross(Vb2,Vb3)/(dot(Vb1,cross(Vb2,Vb3))));
V2r = 2*pi*(cross(Vb3,Vb1)/(dot(Vb2,cross(Vb3,Vb1))));
V3r = 2*pi*(cross(Vb1,Vb2)/(dot(Vb3,cross(Vb1,Vb2))));
% Create new real space unit cell basis ORTHOGONAL vectors Vs, where new z'
% is perpendicular to x-y plane, and y' is perpendicular to x-z' plane. The
% rotation matrix (Rot2LabMatrix) is the inv of Vs. This is for the purpose
% of determining the orientation of the strained lattice cell, enabling for
% the calculation of the stress/strain tensor in the lab and sample
% orientation
Vsx = V1r;
Vsz = cross(V1r, V2r);
Vsy = cross(Vsz, Vsx);
Vs = [Vsx/norm(Vsx); Vsy/norm(Vsy); Vsz/norm(Vsz)];
Rot2LabMatrix = transpose(Vs); %changed from inv(Vs) to transpose(Vs) 5/1/15
% Unit cell lengths (original/best values)
or_a1 = norm(V1r);
or_b1 = norm(V2r);
or_c1 = norm(V3r);
% Angle calculations from basis vectors
or_alpha1 = atan2(norm(cross(V2r,V3r)), dot(V2r,V3r))*180/pi;
or_beta1 = atan2(norm(cross(V1r,V3r)), dot(V1r,V3r))*180/pi;
or_gamma1 = atan2(norm(cross(V1r,V2r)), dot(V1r,V2r))*180/pi;
% Strain tensor calculations
e(1,1) = (or_a1/a0)*sind(or_beta1)*sind(or_gamma1)-1;
e(1,2) = -(or_a1/(2*a0))*sind(or_beta1)*cosd(or_gamma1);
e(1,3) = (or_a1/(2*a0))*cosd(or_beta1);
e(2,2) = (or_b1/a0)*sind(or_alpha1)-1;
e(2,3) = (or_b1/(2*a0))*cosd(or_alpha1);
e(3,3) = (or_c1/a0)-1;
e(2,1) = e(1,2);
e(3,1) = e(1,3);
171
e(3,2) = e(2,3);
Or_strain = e;
% Stress tensor calculations
e = [e(1,1);e(2,2);e(3,3);2*e(2,3);2*e(1,3);2*e(1,2)];
stiffness = [c_11 c_12 c_12 0 0 0;
c_12 c_11 c_12 0 0 0;
c_12 c_12 c_11 0 0 0;
0 0 0 c_44 0 0;
0 0 0 0 c_44 0;
0 0 0 0 0 c_44];
s = stiffness*e;
sig(1,1) = s(1);
sig(2,2) = s(2);
sig(3,3) = s(3);
sig(2,3) = s(4);
sig(1,3) = s(5);
sig(1,2) = s(6);
sig(3,2) = s(4);
sig(3,1) = s(5);
sig(2,1) = s(6);
Or_stress = sig;
% Calculate hydrostatic stresses (sum of the diagonals/3)
hydro_stress = (Or_stress(1,1) + Or_stress(2,2) + Or_stress(3,3))/3;
% Calculate the strain and stress tensor in lab coordinates
Lab_strain_org = Rot2LabMatrix*Or_strain*transpose(Rot2LabMatrix);
Lab_stress_org = Rot2LabMatrix*Or_stress*transpose(Rot2LabMatrix);
% Define rotation matrix to rotate stress/strain tensor from lab
% coordinates to sample coordinates. This matrix rotates the tensor 45
% degrees counter clockwise about the x axis so that the z axis points
% along the sample direction
Rot2SamMatrix(1,1) = 1;
Rot2SamMatrix(2,2) = cosd(45);
Rot2SamMatrix(2,3) = sind(45);
Rot2SamMatrix(3,2) = -sind(45);
Rot2SamMatrix(3,3) = cosd(45);
Sample_strain_org = Rot2SamMatrix*Lab_strain_org*transpose(Rot2SamMatrix);
Sample_stress_org = Rot2SamMatrix*Lab_stress_org*transpose(Rot2SamMatrix);
% This is done because of all the rotations and sine and cosine
% approximation,at this point, strain and stress tensor became
% infinitesimally assymetric, rendering problems for the diagonalization
% process
Sample_strain_org(1,2) = Sample_strain_org(2,1);
Sample_strain_org(1,3) = Sample_strain_org(3,1);
Sample_strain_org(2,3) = Sample_strain_org(3,2);
Sample_stress_org(1,2) = Sample_stress_org(2,1);
Sample_stress_org(1,3) = Sample_stress_org(3,1);
Sample_stress_org(2,3) = Sample_stress_org(3,2);
% Calculate principal strains by diagonalizing the strain tensor, the
% eigenvalues are the principal strains. Eigs finds the eigenvalues and
172
% sort them, the '3' refers to the # of eigenvalues displayed, and 'la'
% means to sort the eigenvalues from largest to smallest based on algebraic
% value. If 'la' is left out, the default choice for sorting is largest to
% smallest in magnitude
[strain_direction,principal_strain] = eigs(Sample_strain_org,3,'la');
% Calculate principal stresses by diagonalizing the stress tensor, the
% eigenvalues are the principal stresses
[stress_direction,principal_stress] = eigs(Sample_stress_org,3,'la');
% This ends the calculation portion to find best/mean values without error
% analysis
%-------------------------------------------------------------------------
% Second, generate arrays of each input with a random Gaussian distribution
% based on the standard deviation for each input to iterate and run Monte
% Carlo
a0 = a0 + s_a0.*randn(n,1);
px1 = px1 + s_px1.*randn(n,1);
py1 = py1 + s_py1.*randn(n,1);
qc1 = qc1 + s_qc1.*randn(n,1);
px2 = px2 + s_px2.*randn(n,1);
py2 = py2 + s_py2.*randn(n,1);
qc2 = qc2 + s_qc2.*randn(n,1);
px3 = px3 + s_px3.*randn(n,1);
py3 = py3 + s_py3.*randn(n,1);
qc3 = qc3 + s_qc3.*randn(n,1);
% Iterate through n number of times to get n sets of answers so standard
% deviations can be extracted
% Preallocations for speed. Saved 1.5 seconds, took 5 mins to write :(
an = zeros(1,n);
bn = zeros(1,n);
cn = zeros(1,n);
alphan = zeros(1,n);
betan = zeros(1,n);
gamman = zeros(1,n);
e1 = zeros(1,n);
e2 = zeros(1,n);
e3 = zeros(1,n);
e4 = zeros(1,n);
e5 = zeros(1,n);
e6 = zeros(1,n);
sig1 = zeros(1,n);
sig2 = zeros(1,n);
sig3 = zeros(1,n);
sig4 = zeros(1,n);
173
sig5 = zeros(1,n);
sig6 = zeros(1,n);
Lab_e1 = zeros(1,n);
Lab_e2 = zeros(1,n);
Lab_e3 = zeros(1,n);
Lab_e4 = zeros(1,n);
Lab_e5 = zeros(1,n);
Lab_e6 = zeros(1,n);
Lab_sig1 = zeros(1,n);
Lab_sig2 = zeros(1,n);
Lab_sig3 = zeros(1,n);
Lab_sig4 = zeros(1,n);
Lab_sig5 = zeros(1,n);
Lab_sig6 = zeros(1,n);
Sam_e1 = zeros(1,n);
Sam_e2 = zeros(1,n);
Sam_e3 = zeros(1,n);
Sam_e4 = zeros(1,n);
Sam_e5 = zeros(1,n);
Sam_e6 = zeros(1,n);
Sam_sig1 = zeros(1,n);
Sam_sig2 = zeros(1,n);
Sam_sig3 = zeros(1,n);
Sam_sig4 = zeros(1,n);
Sam_sig5 = zeros(1,n);
Sam_sig6 = zeros(1,n);
p_strain1 = zeros(1,n);
p_strain2 = zeros(1,n);
p_strain3 = zeros(1,n);
p_stress1 = zeros(1,n);
p_stress2 = zeros(1,n);
p_stress3 = zeros(1,n);
hydrostatic_stress = zeros(1,n);
% Loop is similar to the mean calculation, except for keeping track of all
% the iterations of variables
for i = 1:n
A = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector1,px1(i),py1(i),qc1(i),depth1);
B = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector2,px2(i),py2(i),qc2(i),depth2);
C = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector3,px3(i),py3(i),qc3(i),depth3);
% Define the three vectors
E = [h1 k1 l1];
F = [h2 k2 l2];
G = [h3 k3 l3];
% Calculate base vectors from hkl vectors and diffraction vectors by
% Gaussian reduction
M = [E A; F B; G C];
M = rref(M);
% Solve for reciprocal unit cell basis vectors
Vb1 = M(1,4:6);
Vb2 = M(2,4:6);
Vb3 = M(3,4:6);
174
% Solve for the real space unit cell basis vectors
V1r = 2*pi*(cross(Vb2,Vb3)/(dot(Vb1,cross(Vb2,Vb3))));
V2r = 2*pi*(cross(Vb3,Vb1)/(dot(Vb2,cross(Vb3,Vb1))));
V3r = 2*pi*(cross(Vb1,Vb2)/(dot(Vb3,cross(Vb1,Vb2))));
% Create new real space unit cell basis ORTHOGONAL vectors Vs, where new
% z'
% is perpendicular to x-y plane, and y' is perpendicular to x-z' plane.
% Vs is the rotation matrix
Vsx = V1r;
Vsz = cross(V1r, V2r);
Vsy = cross(Vsz, Vsx);
Vs = [Vsx/norm(Vsx); Vsy/norm(Vsy); Vsz/norm(Vsz)];
Rot2LabMatrix = inv(Vs);
% Unit cell lengths
a1 = norm(V1r);
b1 = norm(V2r);
c1 = norm(V3r);
% Set an (a new) to collect variations of a1 (lattice spacing) into an
% arry, same with bn and cn
an(i) = a1;
bn(i) = b1;
cn(i) = c1;
% Angle calculations from basis vectors
alpha1 = atan2(norm(cross(V2r,V3r)), dot(V2r,V3r))*180/pi;
beta1 = atan2(norm(cross(V1r,V3r)), dot(V1r,V3r))*180/pi;
gamma1 = atan2(norm(cross(V1r,V2r)), dot(V1r,V2r))*180/pi;
% Collect values of angles into arrays of alphan, betan, and gamman
alphan(i) = alpha1;
betan(i) = beta1;
gamman(i) = gamma1;
% Strain tensor calculations
e = zeros (3);
e(1,1) = (a1/a0(i))*sind(beta1)*sind(gamma1)-1;
e(1,2) = -(a1/(2*a0(i)))*sind(beta1)*cosd(gamma1);
e(1,3) = (a1/(2*a0(i)))*cosd(beta1);
e(2,2) = (b1/a0(i))*sind(alpha1)-1;
e(2,3) = (b1/(2*a0(i)))*cosd(alpha1);
e(3,3) = (c1/a0(i))-1;
e(2,1) = e(1,2);
e(3,1) = e(1,3);
e(3,2) = e(2,3);
% Generate arrays of strain data for each strain value
e1(i) = e(1,1);
e2(i) = e(2,2);
e3(i) = e(3,3);
e4(i) = e(2,3);
e5(i) = e(1,3);
e6(i) = e(1,2);
strain_tensor = e;
175
% Stress tensor calculations
e_eng = [e(1,1);e(2,2);e(3,3);2*e(2,3);2*e(1,3);2*e(1,2)];
s = [c_11 c_12 c_12 0 0 0;
c_12 c_11 c_12 0 0 0;
c_12 c_12 c_11 0 0 0;
0 0 0 c_44 0 0;
0 0 0 0 c_44 0;
0 0 0 0 0 c_44]*e_eng;
sig(1,1) = s(1);
sig(2,2) = s(2);
sig(3,3) = s(3);
sig(2,3) = s(4);
sig(1,3) = s(5);
sig(1,2) = s(6);
sig(3,2) = sig(2,3);
sig(3,1) = sig(1,3);
sig(2,1) = sig(1,2);
sig1(i) = sig(1,1);
sig2(i) = sig(2,2);
sig3(i) = sig(3,3);
sig4(i) = sig(2,3);
sig5(i) = sig(1,3);
sig6(i) = sig(1,2);
stress_tensor = sig;
% Calculate the hydrostatic stress
hydrostatic_stress(i) = (sig1(i)+sig2(i)+sig3(i))/3;
% Calculate the strain and stress tensors in lab coordinate, generate
% array of strain data for stress and strain in lab coordinates
Lab_strain = Rot2LabMatrix*strain_tensor*transpose(Rot2LabMatrix);
Lab_stress = Rot2LabMatrix*stress_tensor*transpose(Rot2LabMatrix);
Lab_e1(i) = Lab_strain(1,1);
Lab_e2(i) = Lab_strain(2,2);
Lab_e3(i) = Lab_strain(3,3);
Lab_e4(i) = Lab_strain(2,3);
Lab_e5(i) = Lab_strain(1,3);
Lab_e6(i) = Lab_strain(1,2);
Lab_sig1(i) = Lab_stress(1,1);
Lab_sig2(i) = Lab_stress(2,2);
Lab_sig3(i) = Lab_stress(3,3);
Lab_sig4(i) = Lab_stress(2,3);
Lab_sig5(i) = Lab_stress(1,3);
Lab_sig6(i) = Lab_stress(1,2);
% Calculate the strain and stress tensors in sample coordinate, generate
% array of strain data for stress and strain in sample coordinates
Rot2SamMatrix(1,1) = 1;
Rot2SamMatrix(2,2) = cosd(45);
Rot2SamMatrix(2,3) = sind(45);
Rot2SamMatrix(3,2) = -sind(45);
Rot2SamMatrix(3,3) = cosd(45);
Sample_strain = Rot2SamMatrix*Lab_strain*transpose(Rot2SamMatrix);
Sample_stress = Rot2SamMatrix*Lab_stress*transpose(Rot2SamMatrix);
176
Sample_strain(1,2) = Sample_strain(2,1);
Sample_strain(1,3) = Sample_strain(3,1);
Sample_strain(2,3) = Sample_strain(3,2);
Sample_stress(1,2) = Sample_stress(2,1);
Sample_stress(1,3) = Sample_stress(3,1);
Sample_stress(2,3) = Sample_stress(3,2);
Sam_e1(i) = Sample_strain(1,1);
Sam_e2(i) = Sample_strain(2,2);
Sam_e3(i) = Sample_strain(3,3);
Sam_e4(i) = Sample_strain(2,3);
Sam_e5(i) = Sample_strain(1,3);
Sam_e6(i) = Sample_strain(1,2);
Sam_sig1(i) = Sample_stress(1,1);
Sam_sig2(i) = Sample_stress(2,2);
Sam_sig3(i) = Sample_stress(3,3);
Sam_sig4(i) = Sample_stress(2,3);
Sam_sig5(i) = Sample_stress(1,3);
Sam_sig6(i) = Sample_stress(1,2);
% Calculate the principal strains. Calculating the principal strains
% from the sample reference frame ensures that the principal strain
% directions are with respect to the sample coordiate system
princ_strain = eigs(Sample_strain,3,'la');
p_strain1(i) = princ_strain(1);
p_strain2(i) = princ_strain(2);
p_strain3(i) = princ_strain(3);
% Calculate the principal stresses. Calculating the principal stresses
% from the sample reference frame ensures that the principal stress
% directions are with respect to the sample coordiate system
princ_stress = eigs(Sample_stress,3,'la');
p_stress1(i) = princ_stress(1);
p_stress2(i) = princ_stress(2);
p_stress3(i) = princ_stress(3);
end
% Saving outputs onto file with name of input peak file. Details include
% number of iterations and geometry file used as well as the time of
% calculation. Input peak values were also added in the output file to keep
% track of input data used.
filename = sprintf('%s output',peakfilename);
fileID = fopen(filename,'w');
fprintf(fileID,'The input peak values from "%s" were:\n', peakfilename);
fprintf(fileID,'P1: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',firstpeak{3},firstpeak{5},firstpeak{7},firstpeak{9},firstpeak{10},first
peak{11},firstpeak{12});
fprintf(fileID,'P2: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',secondpeak{3},secondpeak{5},secondpeak{7},secondpeak{9},secondpeak{10},
secondpeak{11},secondpeak{12});
fprintf(fileID,'P3: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',thirdpeak{3},thirdpeak{5},thirdpeak{7},thirdpeak{9},thirdpeak{10},third
peak{11},thirdpeak{12});
177
fprintf(fileID,'The results and uncertainty were calculated based on %d
iterations,\n',n);
fprintf(fileID,'and "%s" geometry file\n', geofilename);
fprintf(fileID,'Calculations were done on %s\n', datestr(now));
fprintf(fileID,'---------------------------------------------------------
\n');
fprintf(fileID,'The best lattice parameters are (in nm and degrees):\n');
fprintf(fileID,'a = %10.7e\n', or_a1);
fprintf(fileID,'b = %10.7e\n', or_b1);
fprintf(fileID,'c = %10.7e\n', or_c1);
fprintf(fileID,'alpha = %f\n', or_alpha1);
fprintf(fileID,'beta = %f\n', or_beta1);
fprintf(fileID,'gamma = %f\n', or_gamma1);
fprintf(fileID,'\n');
fprintf(fileID,'Best strain tensor (crystal coordinates):\n');
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(1,1),Or_strain(1,2),Or_strain(1,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(2,1),Or_strain(2,2),Or_strain(2,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(3,1),Or_strain(3,2),Or_strain(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best stress tensor (crystal coordinates)(MPa):\n');
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(1,1)*1000,Or_stress(1,2)*1000,Or_stress(1,3)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(2,1)*1000,Or_stress(2,2)*1000,Or_stress(2,3)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(3,1)*1000,Or_stress(3,2)*1000,Or_stress(3,3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best strain tensor (sample coordinates):\n');
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(1,1),Sample_strain_org(1,2),Sample_strain_org(1,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(2,1),Sample_strain_org(2,2),Sample_strain_org(2,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(3,1),Sample_strain_org(3,2),Sample_strain_org(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best stress tensor (sample coordinates)(MPa):\n');
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(1,1)*1000,Sample_stress_org(1,2)*1000,Sample_stress_org(1,3
)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(2,1)*1000,Sample_stress_org(2,2)*1000,Sample_stress_org(2,3
)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(3,1)*1000,Sample_stress_org(3,2)*1000,Sample_stress_org(3,3
)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best principal strains are:\n');
fprintf(fileID,'e1 = %10.5e\n',principal_strain(1,1));
fprintf(fileID,'e2 = %10.5e\n',principal_strain(2,2));
fprintf(fileID,'e3 = %10.5e\n',principal_strain(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best principal stresses are (MPa):\n');
fprintf(fileID,'sig1 = %5.5f\n',principal_stress(1,1)*1000);
fprintf(fileID,'sig2 = %5.5f\n',principal_stress(2,2)*1000);
178
fprintf(fileID,'sig3 = %5.5f\n',principal_stress(3,3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best hydrostatic stress is (MPa):\n');
fprintf(fileID,'%5.5f\n',hydro_stress*1000);
fprintf(fileID,'---------------------------------------------------------
\n');
fprintf(fileID,'The measured lattice parameters are (in nm):\n');
fprintf(fileID,'a = %10.7e +/- %10.7e\n', or_a1, std(an));
fprintf(fileID,'b = %10.7e +/- %10.7e\n', or_b1, std(bn));
fprintf(fileID,'c = %10.7e +/- %10.7e\n', or_c1, std(cn));
fprintf(fileID,'alpha = %f +/- %f\n', or_alpha1, std(alphan));
fprintf(fileID,'beta = %f +/- %f\n', or_beta1, std(betan));
fprintf(fileID,'gamma = %f +/- %f\n', or_gamma1, std(gamman));
fprintf(fileID,'\n');
fprintf(fileID,'The strain tensor components in crystallographic
coordinates:\n');
fprintf(fileID,'e11 = %10.7e +/- %10.7e\n', Or_strain(1,1), std(e1));
fprintf(fileID,'e22 = %10.7e +/- %10.7e\n', Or_strain(2,2), std(e2));
fprintf(fileID,'e33 = %10.7e +/- %10.7e\n', Or_strain(3,3), std(e3));
fprintf(fileID,'e23 = %10.7e +/- %10.7e\n', Or_strain(2,3), std(e4));
fprintf(fileID,'e13 = %10.7e +/- %10.7e\n', Or_strain(1,3), std(e5));
fprintf(fileID,'e12 = %10.7e +/- %10.7e\n', Or_strain(1,2), std(e6));
fprintf(fileID,'\n');
fprintf(fileID,'The stress tensor components in crystallographic coordinates
(MPa):\n');
fprintf(fileID,'s1 = %5.5f +/- %5.5f\n', Or_stress(1,1)*1000,
std(sig1)*1000);
fprintf(fileID,'s2 = %5.5f +/- %5.5f\n', Or_stress(2,2)*1000,
std(sig2)*1000);
fprintf(fileID,'s3 = %5.5f +/- %5.5f\n', Or_stress(3,3)*1000,
std(sig3)*1000);
fprintf(fileID,'s4 = %5.5f +/- %5.5f\n', Or_stress(2,3)*1000,
std(sig4)*1000);
fprintf(fileID,'s5 = %5.5f +/- %5.5f\n', Or_stress(1,3)*1000,
std(sig5)*1000);
fprintf(fileID,'s6 = %5.5f +/- %5.5f\n', Or_stress(1,2)*1000,
std(sig6)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The strain tensor components in sample coordinates:\n');
fprintf(fileID,'e11 = %10.7e +/- %10.7e\n', Sample_strain_org(1,1),
std(Sam_e1));
fprintf(fileID,'e22 = %10.7e +/- %10.7e\n', Sample_strain_org(2,2),
std(Sam_e2));
fprintf(fileID,'e33 = %10.7e +/- %10.7e\n', Sample_strain_org(3,3),
std(Sam_e3));
fprintf(fileID,'e23 = %10.7e +/- %10.7e\n', Sample_strain_org(2,3),
std(Sam_e4));
fprintf(fileID,'e13 = %10.7e +/- %10.7e\n', Sample_strain_org(1,3),
std(Sam_e5));
fprintf(fileID,'e12 = %10.7e +/- %10.7e\n', Sample_strain_org(1,2),
std(Sam_e6));
fprintf(fileID,'\n');
fprintf(fileID,'The stress tensor components in sample coordinates
(MPa):\n');
fprintf(fileID,'s1 = %5.5f +/- %5.5f\n', Sample_stress_org(1,1)*1000,
std(Sam_sig1)*1000);
179
fprintf(fileID,'s2 = %5.5f +/- %5.5f\n', Sample_stress_org(2,2)*1000,
std(Sam_sig2)*1000);
fprintf(fileID,'s3 = %5.5f +/- %5.5f\n', Sample_stress_org(3,3)*1000,
std(Sam_sig3)*1000);
fprintf(fileID,'s4 = %5.5f +/- %5.5f\n', Sample_stress_org(2,3)*1000,
std(Sam_sig4)*1000);
fprintf(fileID,'s5 = %5.5f +/- %5.5f\n', Sample_stress_org(1,3)*1000,
std(Sam_sig5)*1000);
fprintf(fileID,'s6 = %5.5f +/- %5.5f\n', Sample_stress_org(1,2)*1000,
std(Sam_sig6)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The principal strains are:\n');
fprintf(fileID,'Principal strain 1 = %10.7e +/- %10.7e\n',
principal_strain(1,1), std(p_strain1));
fprintf(fileID,'Principal strain 2 = %10.7e +/- %10.7e\n',
principal_strain(2,2), std(p_strain2));
fprintf(fileID,'Principal strain 3 = %10.7e +/- %10.7e\n',
principal_strain(3,3), std(p_strain3));
fprintf(fileID,'\n');
fprintf(fileID,'The principal strain directions are: \n');
fprintf(fileID,'v1 = (%f, %f, %f)\n',
strain_direction(1,1),strain_direction(2,1),strain_direction(3,1));
fprintf(fileID,'v2 = (%f, %f, %f)\n',
strain_direction(1,2),strain_direction(2,2),strain_direction(3,2));
fprintf(fileID,'v3 = (%f, %f, %f)\n',
strain_direction(1,3),strain_direction(2,3),strain_direction(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'The principal stresses are (MPa):\n');
fprintf(fileID,'Principal stress 1 = %5.5f +/- %5.5f\n',
principal_stress(1,1)*1000, std(p_stress1)*1000);
fprintf(fileID,'Principal stress 2 = %5.5f +/- %5.5f\n',
principal_stress(2,2)*1000, std(p_stress2)*1000);
fprintf(fileID,'Principal stress 3 = %5.5f +/- %5.5f\n',
principal_stress(3,3)*1000, std(p_stress3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The principal stress directions are: \n');
fprintf(fileID,'v1 = (%f, %f, %f)\n',
stress_direction(1,1),stress_direction(2,1),stress_direction(3,1));
fprintf(fileID,'v2 = (%f, %f, %f)\n',
stress_direction(1,2),stress_direction(2,2),stress_direction(3,2));
fprintf(fileID,'v3 = (%f, %f, %f)\n',
stress_direction(1,3),stress_direction(2,3),stress_direction(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'The hydrostatic stress is (MPa): %5.5f +/- %5.5f\n',
hydro_stress*1000, std(hydrostatic_stress)*1000);
fclose(fileID);
end
180
B2. Reciprocal/Directional Vector Calculations
function [V_f] = Pixel2XYZ(OR,OP,YR,YP,PR,PP,d,px,py,qc,depth)
% Define detector and sizes
% Orange detector
OX = 409.600; % size of Orange detector in the X direction in mm
OY = 409.600;
ONx = 2048; % Number of Orange detectors in pixels
ONy = 2048;
% Yellow detector
YX = 204.800;
YY = 204.800;
YNx = 1024;
YNy = 1024;
%Purple detector
PX = 204.800;
PY = 204.800;
PNx = 1024;
PNy = 1024;
% Calculate pixel size, since pixels are square, pixel size in the x and y
% directions are the same
Ops = OX/ONx; % Ops = Orange pixel size
Yps = YX/YNx; % Yps = Yellow pixel size
Pps = PX/PNx; % Pps = Purple pixel size
if d == 0;
Nx = ONx;
Ny = ONy;
ps = Ops;
R = OR;
P = OP;
if qc == 0
qc = 1;
end
elseif d == 1;
Nx = YNx;
Ny = YNy;
ps = Yps;
R = YR;
P = YP;
if qc == 0
qc = 1;
end
elseif d == 2;
Nx = PNx;
Ny = PNy;
ps = Pps;
R = PR;
P = PP;
181
if qc == 0
qc = 1;
end
end
Vx = (px-0.5*(Nx-1))*ps;
Vy = (py-0.5*(Ny-1))*ps;
Vz = 0;
V = [Vx Vy Vz];
% Applying translating vector
V = V + P;
% Applying rotation vector using Rodriguez formula
V = Rodriguez(V,R);
V = transpose(V);
% Add depth information in the z-direction
V(3) = V(3)-depth/1000;
% Calculate directional reciprocal vector
V_n = V/norm(V);
V_r(1) = V_n(1);
V_r(2) = V_n(2);
V_r(3) = V_n(3) - 1;
V_r = [V_r(1) V_r(2) V_r(3)];
% Normalized reciprocal vector
V_r = V_r/norm(V_r);
% Full reciprocal vector
V_f = V_r*qc;
end
B3. Matrix Rotation Using Rodriguez Formula
function [Vr] = Rodriguez(v,R)
theta = norm(R); % angle of rotation = length of vector
R = R/norm(R); % normalize the rotation vector
% For easy reading
x = R(1);
y = R(2);
z = R(3);
c = cos(theta);
s = sin(theta);
c1 = 1-c;
% Matrix from Rodriguez formula
R(1,1) = c+x^2*c1;
R(1,2) = x*y*c1-z*s;
R(1,3) = y*s+x*z*c1;
182
R(2,1) = z*s+x*y*c1;
R(2,2) = c+y^2*c1;
R(2,3) = -x*s+y*z*c1;
R(3,1) = -y*s+x*z*c1;
R(3,2) = x*s+y*z*c1;
R(3,3) = c+z^2*c1;
Vr = R*transpose(v);
end
183
B4. Matlab Code for Full Elastic Strain and Stress Tensor for Single Crystal
Cu
function [] = StrainTensorCalcCu (n)
% StrainTensorCalcCu will calculate full strain tensor given 4 diffraction
% vectors directions, 3 of which is linearly independent but length is not
% known and 1 with known length. NOTE: For intput file, put the vector with
% known length last. The full strain/stress tensors are calculated for the
% crystal coordinates and rotated to sample coordinates. Uncertainty is
% calculated using Monte Carlo error propagation. Input of program n is the
% number of iterations used for error analysis.
% Program reads in peak input file in text format in identical format as
% Lyle's C code input file, that is:
% Peakname, peak number, px, s_px, py, s_py, qc, s_qc, h, k, l, detector,
% depth(in microns), FWHM, and intensity. For the 3 peaks whose qc is not
% measured, use an input of 0 for qc.
% Only px, py, spx, spy, qc, sqc, hkl, detector, and depth are used in full
% strain calculations
%
% Geometry information is read in from geometry file produced from APS
% equipments. Specifically, the translation (P) and rotation vector (R) for
% the three different detectors
%
% Vector from diffraction volume to pixel location is calculated by
% Pixel2XYZ, which calls on Rodriguez to apply the rotation transformation
%
% Define material parameters, change these values to reflect the unit cell
% parameters of the appropriate material
a0 = 0.361496; %for Cu single crystal(Wyckoff, 1963)
s_a0 = 0.000005;
c_11 = 169.1; %+/- 0.2
s_c11 = 0.2;
c_12 = 122.2; %+/- 0.3
s_c12 = 0.3;
c_44 = 75.41; %+/- 0.05
s_c44 = 0.05;
% Ask for peak input file
peakfilename = input('What is the peak input file? ');
fid = fopen(peakfilename);
% Read first input peak
tline = fgetl(fid);
firstpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px1 = firstpeak{3};
s_px1 = firstpeak{4};
py1 = firstpeak{5};
s_py1 = firstpeak{6};
qc1 = firstpeak{7};
s_qc1 = firstpeak{8};
h1 = firstpeak{9};
k1 = firstpeak{10};
184
l1 = firstpeak{11};
detector1 = firstpeak{12};
depth1 = firstpeak{13};
FWHM1 = firstpeak{14};
% Read second input peak
tline = fgetl(fid);
secondpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px2 = secondpeak{3};
s_px2 = secondpeak{4};
py2 = secondpeak{5};
s_py2 = secondpeak{6};
qc2 = secondpeak{7};
s_qc2 = secondpeak{8};
h2 = secondpeak{9};
k2 = secondpeak{10};
l2 = secondpeak{11};
detector2 = secondpeak{12};
depth2 = secondpeak{13};
FWHM2 = secondpeak{14};
% Read third input peak
tline = fgetl(fid);
thirdpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px3 = thirdpeak{3};
s_px3 = thirdpeak{4};
py3 = thirdpeak{5};
s_py3 = thirdpeak{6};
qc3 = thirdpeak{7};
s_qc3 = thirdpeak{8};
h3 = thirdpeak{9};
k3 = thirdpeak{10};
l3 = thirdpeak{11};
detector3 = thirdpeak{12};
depth3 = thirdpeak{13};
FWHM3 = thirdpeak{14};
% Read fourth input peak
tline = fgetl(fid);
fourthpeak = textscan(tline,'%s%f%f%f%f%f%f%f%f%f%f%f%f%f%f');
px4 = fourthpeak{3};
s_px4 = fourthpeak{4};
py4 = fourthpeak{5};
s_py4 = fourthpeak{6};
qc4 = fourthpeak{7};
s_qc4 = fourthpeak{8};
h4 = fourthpeak{9};
k4 = fourthpeak{10};
l4 = fourthpeak{11};
detector4 = fourthpeak{12};
depth4 = fourthpeak{13};
FWHM4 = fourthpeak{14};
% Ask for geometry file
geofilename = input('What is the geometry file? ');
185
fid = fopen(geofilename);
% Read in rotation and translation vectors
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
O_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
OR(1) = O_R {1};
OR(2) = O_R {2};
OR(3) = O_R {3};
tline = fgetl(fid);
O_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
OP(1) = O_P {1};
OP(2) = O_P {2};
OP(3) = O_P {3};
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
Y_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
YR(1) = Y_R {1};
YR(2) = Y_R {2};
YR(3) = Y_R {3};
tline = fgetl(fid);
Y_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
YP(1) = Y_P {1};
YP(2) = Y_P {2};
YP(3) = Y_P {3};
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
fgetl(fid);
tline = fgetl(fid);
186
P_R = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
PR(1) = P_R {1};
PR(2) = P_R {2};
PR(3) = P_R {3};
tline = fgetl(fid);
P_P = textscan(tline,'%*[^{]%*[{]%f%*[,]%f%*[,]%f');
PP(1) = P_P {1};
PP(2) = P_P {2};
PP(3) = P_P {3};
% Define the four vectors, Pixel2XYZ calculates the vectors from the
% diffraction volumes to the pixel locations, vectors A, B, and C are the
% direction vectors and D is the full length vector
A = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector1,px1,py1,qc1,depth1);
E = [h1 k1 l1];
B = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector2,px2,py2,qc2,depth2);
F = [h2 k2 l2];
C = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector3,px3,py3,qc3,depth3);
G = [h3 k3 l3];
D = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector4,px4,py4,qc4,depth4);
H = [h4 k4 l4];
% Normalize the 3 directional vectors
An = A/norm(A);
Bn = B/norm(B);
Cn = C/norm(C);
% Solve for N, linear combination of the 3 directional vectors to
% make the 4th vector, calculated based on hkl. This solves the following
% system of equations [HKL_V1-3][N] = [HKL_V4]
N = [transpose(E) transpose(F) transpose(G)]\transpose(H);
% Multiply normalized vector by the appropriate linear combo constants
V1 = An.*N(1);
V2 = Bn.*N(2);
V3 = Cn.*N(3);
% Solve for L, lengths of the 3 directional vectors
L = [transpose(V1) transpose(V2) transpose(V3)]\transpose(D);
% These vectors have the correct length
V1_l = An*L(1);
V2_l = Bn*L(2);
V3_l = Cn*L(3);
% Guassian Reduction of 3 vectors with lengths we just solved for, this
% step could be done with any combinations of any 3 vectors at this point,
% has been tested to give identical results
M = [E V1_l;F V2_l;G V3_l];
M = rref(M);
% Solve for reciprocal unit cell basis vectors
Vb1 = M(1,4:6);
Vb2 = M(2,4:6);
Vb3 = M(3,4:6);
187
% Solve for the real space unit cell basis vectors
V1r = 2*pi*(cross(Vb2,Vb3)/(dot(Vb1,cross(Vb2,Vb3))));
V2r = 2*pi*(cross(Vb3,Vb1)/(dot(Vb2,cross(Vb3,Vb1))));
V3r = 2*pi*(cross(Vb1,Vb2)/(dot(Vb3,cross(Vb1,Vb2))));
Vsx = V1r;
Vsz = cross(V1r, V2r);
Vsy = cross(Vsz, Vsx);
Vs = [Vsx/norm(Vsx); Vsy/norm(Vsy); Vsz/norm(Vsz)];
Rot2LabMatrix = transpose(Vs);
% Unit cell lengths
or_a1 = norm(V1r);
or_b1 = norm(V2r);
or_c1 = norm(V3r);
% Angle calculations from basis vectors
or_alpha1 = atan2(norm(cross(V2r,V3r)), dot(V2r,V3r))*180/pi;
or_beta1 = atan2(norm(cross(V1r,V3r)), dot(V1r,V3r))*180/pi;
or_gamma1 = atan2(norm(cross(V1r,V2r)), dot(V1r,V2r))*180/pi;
% Strain tensor calculations
e(1,1) = (or_a1/a0)*sind(or_beta1)*sind(or_gamma1)-1;
e(1,2) = -(or_a1/(2*a0))*sind(or_beta1)*cosd(or_gamma1);
e(1,3) = (or_a1/(2*a0))*cosd(or_beta1);
e(2,2) = (or_b1/a0)*sind(or_alpha1)-1;
e(2,3) = (or_b1/(2*a0))*cosd(or_alpha1);
e(3,3) = (or_c1/a0)-1;
e(2,1) = e(1,2);
e(3,1) = e(1,3);
e(3,2) = e(2,3);
Or_strain = e;
% Calculate principal strains by diagonalizing the strain tensor, the
% eigenvalues are the principal strains. Eigs finds the eigenvalues and
% sort them, the '3' refers to the # of eigenvalues displayed, and 'la'
% means to sort the eigenvalues from largest to smallest based on algebraic
% value. If 'la' is left out, the default choice for sorting is largest to
% smallest in magnitude
[strain_direction,principal_strain] = eigs(Or_strain,3,'la');
% Stress tensor calculations
e = [e(1,1);e(2,2);e(3,3);2*e(2,3);2*e(1,3);2*e(1,2)];
stiffness = [c_11 c_12 c_12 0 0 0;
c_12 c_11 c_12 0 0 0;
c_12 c_12 c_11 0 0 0;
0 0 0 c_44 0 0;
0 0 0 0 c_44 0;
0 0 0 0 0 c_44];
s = stiffness*e;
sig(1,1) = s(1);
sig(2,2) = s(2);
sig(3,3) = s(3);
sig(2,3) = s(4);
sig(1,3) = s(4);
sig(1,2) = s(4);
188
sig(3,2) = sig(2,3);
sig(3,1) = sig(1,3);
sig(2,1) = sig(1,2);
Or_stress = sig;
% Calculate principal stresses by diagonalizing the stress tensor, the
% eigenvalues are the principal stresses
[stress_direction,principal_stress] = eigs(Or_stress,3,'la');
% Calculate hydrostatic stresses (trace/3)
hydro_stress = (Or_stress(1,1) + Or_stress(2,2) + Or_stress(3,3))/3;
% Calculate the strain and stress tensor in lab coordinates
Lab_strain_org = Rot2LabMatrix*Or_strain*transpose(Rot2LabMatrix);
Lab_stress_org = Rot2LabMatrix*Or_stress*transpose(Rot2LabMatrix);
% Define rotation matrix to rotate stress/strain tensor from lab
% coordinates to sample coordinates. This matrix rotates the tensor 45
% degrees counter clockwise about the x axis so that the z axis points
% along the sample direction
Rot2SamMatrix(1,1) = 1;
Rot2SamMatrix(2,2) = cosd(45);
Rot2SamMatrix(2,3) = sind(45);
Rot2SamMatrix(3,2) = -sind(45);
Rot2SamMatrix(3,3) = cosd(45);
Sample_strain_org = Rot2SamMatrix*Lab_strain_org*transpose(Rot2SamMatrix);
Sample_stress_org = Rot2SamMatrix*Lab_stress_org*transpose(Rot2SamMatrix);
% This ends the straight calculation portion to find best/mean values
% Second, generate arrays of each input with a random Gaussian distribution
% based on the standard deviation for each input to iterate and run Monte
% Carlo
a0 = a0 + s_a0.*randn(n,1);
c_11 = c_11 + s_c11.*randn(n,1);
c_12 = c_12 + s_c12.*randn(n,1);
c_44 = c_44 + s_c44.*randn(n,1);
px1 = px1 + s_px1.*randn(n,1);
py1 = py1 + s_py1.*randn(n,1);
qc1 = zeros(n,1);
px2 = px2 + s_px2.*randn(n,1);
py2 = py2 + s_py2.*randn(n,1);
qc2 = zeros(n,1);
px3 = px3 + s_px3.*randn(n,1);
py3 = py3 + s_py3.*randn(n,1);
qc3 = zeros(n,1);
px4 = px4 + s_px4.*randn(n,1);
py4 = py4 + s_py4.*randn(n,1);
qc4 = qc4 + s_qc4.*randn(n,1);
189
% Iterate through n number of times to get n sets of answers so standard
% deviations can be extracted
% Preallocations for speed. Saved 1.5 seconds, took 5 mins to write :(
an = zeros(1,n);
bn = zeros(1,n);
cn = zeros(1,n);
alphan = zeros(1,n);
betan = zeros(1,n);
gamman = zeros(1,n);
e1 = zeros(1,n);
e2 = zeros(1,n);
e3 = zeros(1,n);
e4 = zeros(1,n);
e5 = zeros(1,n);
e6 = zeros(1,n);
sig1 = zeros(1,n);
sig2 = zeros(1,n);
sig3 = zeros(1,n);
sig4 = zeros(1,n);
sig5 = zeros(1,n);
sig6 = zeros(1,n);
Lab_e1 = zeros(1,n);
Lab_e2 = zeros(1,n);
Lab_e3 = zeros(1,n);
Lab_e4 = zeros(1,n);
Lab_e5 = zeros(1,n);
Lab_e6 = zeros(1,n);
Lab_sig1 = zeros(1,n);
Lab_sig2 = zeros(1,n);
Lab_sig3 = zeros(1,n);
Lab_sig4 = zeros(1,n);
Lab_sig5 = zeros(1,n);
Lab_sig6 = zeros(1,n);
Sam_e1 = zeros(1,n);
Sam_e2 = zeros(1,n);
Sam_e3 = zeros(1,n);
Sam_e4 = zeros(1,n);
Sam_e5 = zeros(1,n);
Sam_e6 = zeros(1,n);
Sam_sig1 = zeros(1,n);
Sam_sig2 = zeros(1,n);
Sam_sig3 = zeros(1,n);
Sam_sig4 = zeros(1,n);
Sam_sig5 = zeros(1,n);
Sam_sig6 = zeros(1,n);
p_strain1 = zeros(1,n);
p_strain2 = zeros(1,n);
p_strain3 = zeros(1,n);
p_stress1 = zeros(1,n);
p_stress2 = zeros(1,n);
p_stress3 = zeros(1,n);
hydrostatic_stress = zeros(1,n);
% Loop is similar to the mean calculation, except for keeping track of all
% the iterations of variables
190
for i = 1:n
% Calculate the 4 vectors and define their reflections
A = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector1,px1(i),py1(i),qc1(i),depth1);
B = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector2,px2(i),py2(i),qc2(i),depth2);
C = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector3,px3(i),py3(i),qc3(i),depth3);
D = Pixel2XYZ(OR,OP,YR,YP,PR,PP,detector4,px4(i),py4(i),qc4(i),depth3);
E = [h1 k1 l1];
F = [h2 k2 l2];
G = [h3 k3 l3];
H = [h4 k4 l4];
% Normalize the 3 directional vectors
An = A/norm(A);
Bn = B/norm(B);
Cn = C/norm(C);
% Solve for N, linear combination of the 3 directional vectors to
% make the 4th vector, calculated based on hkl. This solves the following
% system of equations [HKL_V1-3][N] = [HKL_V4]
N = [transpose(E) transpose(F) transpose(G)]\transpose(H);
% Multiply normalized vector by the appropriate linear combo constants
V1 = An.*N(1);
V2 = Bn.*N(2);
V3 = Cn.*N(3);
% Solve for L, lengths of the 3 directional vectors
L = [transpose(V1) transpose(V2) transpose(V3)]\transpose(D);
% These vectors have the correct length
V1_l = An*L(1);
V2_l = Bn*L(2);
V3_l = Cn*L(3);
% Guassian Reduction of 3 vectors with lengths we just solved for, this
% step could be done with any combinations of any 3 vectors at this
point,
% has been tested to give identical results
M = [E V1_l;F V2_l;G V3_l];
M = rref(M);
% Solve for reciprocal unit cell basis vectors
Vb1 = M(1,4:6);
Vb2 = M(2,4:6);
Vb3 = M(3,4:6);
% Solve for the real space unit cell basis vectors
V1r = 2*pi*(cross(Vb2,Vb3)/(dot(Vb1,cross(Vb2,Vb3))));
V2r = 2*pi*(cross(Vb3,Vb1)/(dot(Vb2,cross(Vb3,Vb1))));
V3r = 2*pi*(cross(Vb1,Vb2)/(dot(Vb3,cross(Vb1,Vb2))));
% Create new real space unit cell basis ORTHOGONAL vectors Vs, where new
z'
% is perpendicular to x-y plane, and y' is perpendicular to x-z' plane.
% Vs is the rotation matrix
Vsx = V1r;
Vsz = cross(V1r, V2r);
191
Vsy = cross(Vsz, Vsx);
Vs = [Vsx/norm(Vsx); Vsy/norm(Vsy); Vsz/norm(Vsz)];
Rot2LabMatrix = inv(Vs);
% Unit cell lengths
a1 = norm(V1r);
b1 = norm(V2r);
c1 = norm(V3r);
% Set an (a new) to collect variations of a1 (lattice spacing) into an
% array, same with bn and cn
an(i) = a1;
bn(i) = b1;
cn(i) = c1;
% Angle calculations from basis vectors
alpha1 = atan2(norm(cross(V2r,V3r)), dot(V2r,V3r))*180/pi;
beta1 = atan2(norm(cross(V1r,V3r)), dot(V1r,V3r))*180/pi;
gamma1 = atan2(norm(cross(V1r,V2r)), dot(V1r,V2r))*180/pi;
% Collect values of angles into arrays of alphan, betan, and gamman
alphan(i) = alpha1;
betan(i) = beta1;
gamman(i) = gamma1;
% Strain tensor calculations
e = zeros (3);
e(1,1) = (a1/a0(i))*sind(beta1)*sind(gamma1)-1;
e(1,2) = -(a1/(2*a0(i)))*sind(beta1)*cosd(gamma1);
e(1,3) = (a1/(2*a0(i)))*cosd(beta1);
e(2,2) = (b1/a0(i))*sind(alpha1)-1;
e(2,3) = (b1/(2*a0(i)))*cosd(alpha1);
e(3,3) = (c1/a0(i))-1;
e(2,1) = e(1,2);
e(3,1) = e(1,3);
e(3,2) = e(2,3);
% Generate arrays of strain data for each strain value
e1(i) = e(1,1);
e2(i) = e(2,2);
e3(i) = e(3,3);
e4(i) = e(2,3);
e5(i) = e(1,3);
e6(i) = e(1,2);
strain_tensor = e;
% Calculate the principal strains
princ_strain = eigs(strain_tensor,3,'la');
p_strain1(i) = princ_strain(1);
p_strain2(i) = princ_strain(2);
p_strain3(i) = princ_strain(3);
% Stress tensor calculations
e_eng = [e(1,1);e(2,2);e(3,3);2*e(2,3);2*e(1,3);2*e(1,2)];
s = [c_11(i) c_12(i) c_12(i) 0 0 0;
c_12(i) c_11(i) c_12(i) 0 0 0;
192
c_12(i) c_12(i) c_11(i) 0 0 0;
0 0 0 c_44(i) 0 0;
0 0 0 0 c_44(i) 0;
0 0 0 0 0 c_44(i)]*e_eng;
sig(1,1) = s(1);
sig(2,2) = s(2);
sig(3,3) = s(3);
sig(2,3) = s(4);
sig(1,3) = s(5);
sig(1,2) = s(6);
sig(3,2) = sig(2,3);
sig(3,1) = sig(1,3);
sig(2,1) = sig(1,2);
sig1(i) = sig(1,1);
sig2(i) = sig(2,2);
sig3(i) = sig(3,3);
sig4(i) = sig(2,3);
sig5(i) = sig(1,3);
sig6(i) = sig(1,2);
stress_tensor = sig;
% Calculate the principal stresses
princ_stress = eigs(stress_tensor,3,'la');
p_stress1(i) = princ_stress(1);
p_stress2(i) = princ_stress(2);
p_stress3(i) = princ_stress(3);
% Calculate the hydrostatic stress
hydrostatic_stress(i) = (sig1(i)+sig2(i)+sig3(i))/3;
% Calculate the strain and stress tensors in lab coordinate, generate
% array of strain data for stress and strain in lab coordinates
Lab_strain = Rot2LabMatrix*strain_tensor*transpose(Rot2LabMatrix);
Lab_stress = Rot2LabMatrix*stress_tensor*transpose(Rot2LabMatrix);
Lab_e1(i) = Lab_strain(1,1);
Lab_e2(i) = Lab_strain(2,2);
Lab_e3(i) = Lab_strain(3,3);
Lab_e4(i) = Lab_strain(2,3);
Lab_e5(i) = Lab_strain(1,3);
Lab_e6(i) = Lab_strain(1,2);
Lab_sig1(i) = Lab_stress(1,1);
Lab_sig2(i) = Lab_stress(2,2);
Lab_sig3(i) = Lab_stress(3,3);
Lab_sig4(i) = Lab_stress(2,3);
Lab_sig5(i) = Lab_stress(1,3);
Lab_sig6(i) = Lab_stress(1,2);
% Calculate the strain and stress tensors in lab coordinate, generate
% array of strain data for stress and strain in lab coordinates
Rot2SamMatrix(1,1) = 1;
Rot2SamMatrix(2,2) = cosd(45);
Rot2SamMatrix(2,3) = sind(45);
Rot2SamMatrix(3,2) = -sind(45);
Rot2SamMatrix(3,3) = cosd(45);
Sample_strain = Rot2SamMatrix*Lab_strain*transpose(Rot2SamMatrix);
193
Sample_stress = Rot2SamMatrix*Lab_stress*transpose(Rot2SamMatrix);
Sam_e1(i) = Sample_strain(1,1);
Sam_e2(i) = Sample_strain(2,2);
Sam_e3(i) = Sample_strain(3,3);
Sam_e4(i) = Sample_strain(2,3);
Sam_e5(i) = Sample_strain(1,3);
Sam_e6(i) = Sample_strain(1,2);
Sam_sig1(i) = Sample_stress(1,1);
Sam_sig2(i) = Sample_stress(2,2);
Sam_sig3(i) = Sample_stress(3,3);
Sam_sig4(i) = Sample_stress(2,3);
Sam_sig5(i) = Sample_stress(1,3);
Sam_sig6(i) = Sample_stress(1,2);
end
% Saving outputs onto file with name of input peak file. Details include
% number of iterations and geometry file used as well as the time of
% calculation. Input peak values were also added in the output file to keep
% track of input data used.
filename = sprintf('%s output',peakfilename);
fileID = fopen(filename,'w');
fprintf(fileID,'The input peak values from "%s" were:\n', peakfilename);
fprintf(fileID,'P1: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',firstpeak{3},firstpeak{5},firstpeak{7},firstpeak{9},firstpeak{10},first
peak{11},firstpeak{12});
fprintf(fileID,'P2: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',secondpeak{3},secondpeak{5},secondpeak{7},secondpeak{9},secondpeak{10},
secondpeak{11},secondpeak{12});
fprintf(fileID,'P3: (%5.2f,%5.2f) qc %5.4f with hkl (%d %d %d) on detector
%d\n',thirdpeak{3},thirdpeak{5},thirdpeak{7},thirdpeak{9},thirdpeak{10},third
peak{11},thirdpeak{12});
fprintf(fileID,'The results and uncertainty were calculated based on %d
iterations,\n',n);
fprintf(fileID,'and "%s" geometry file\n', geofilename);
fprintf(fileID,'Calculations were done on %s\n', datestr(now));
fprintf(fileID,'---------------------------------------------------------
\n');
fprintf(fileID,'The best lattice parameters are (in nm and degrees):\n');
fprintf(fileID,'a = %10.7e\n', or_a1);
fprintf(fileID,'b = %10.7e\n', or_b1);
fprintf(fileID,'c = %10.7e\n', or_c1);
fprintf(fileID,'alpha = %f\n', or_alpha1);
fprintf(fileID,'beta = %f\n', or_beta1);
fprintf(fileID,'gamma = %f\n', or_gamma1);
fprintf(fileID,'\n');
fprintf(fileID,'Best strain tensor (crystal coordinates):\n');
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(1,1),Or_strain(1,2),Or_strain(1,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(2,1),Or_strain(2,2),Or_strain(2,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Or_strain(3,1),Or_strain(3,2),Or_strain(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best stress tensor (crystal coordinates)(MPa):\n');
194
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(1,1)*1000,Or_stress(1,2)*1000,Or_stress(1,3)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(2,1)*1000,Or_stress(2,2)*1000,Or_stress(2,3)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Or_stress(3,1)*1000,Or_stress(3,2)*1000,Or_stress(3,3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best strain tensor (sample coordinates):\n');
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(1,1),Sample_strain_org(1,2),Sample_strain_org(1,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(2,1),Sample_strain_org(2,2),Sample_strain_org(2,3));
fprintf(fileID,'%10.5e %10.5e %10.5e\n',
Sample_strain_org(3,1),Sample_strain_org(3,2),Sample_strain_org(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best stress tensor (sample coordinates)(MPa):\n');
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(1,1)*1000,Sample_stress_org(1,2)*1000,Sample_stress_org(1,3
)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(2,1)*1000,Sample_stress_org(2,2)*1000,Sample_stress_org(2,3
)*1000);
fprintf(fileID,'%5.5f %5.5f %5.5f\n',
Sample_stress_org(3,1)*1000,Sample_stress_org(3,2)*1000,Sample_stress_org(3,3
)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best principal strains are:\n');
fprintf(fileID,'e1 = %10.5e\n',principal_strain(1,1));
fprintf(fileID,'e2 = %10.5e\n',principal_strain(2,2));
fprintf(fileID,'e3 = %10.5e\n',principal_strain(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'Best principal stresses are (MPa):\n');
fprintf(fileID,'sig1 = %5.5f\n',principal_stress(1,1)*1000);
fprintf(fileID,'sig2 = %5.5f\n',principal_stress(2,2)*1000);
fprintf(fileID,'sig3 = %5.5f\n',principal_stress(3,3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'Best hydrostatic stress is (MPa):\n');
fprintf(fileID,'%5.5f\n',hydro_stress*1000);
fprintf(fileID,'---------------------------------------------------------
\n');
fprintf(fileID,'The measured lattice parameters are (in nm):\n');
fprintf(fileID,'a = %10.7e +/- %10.7e\n', or_a1, std(an));
fprintf(fileID,'b = %10.7e +/- %10.7e\n', or_b1, std(bn));
fprintf(fileID,'c = %10.7e +/- %10.7e\n', or_c1, std(cn));
fprintf(fileID,'alpha = %f +/- %f\n', or_alpha1, std(alphan));
fprintf(fileID,'beta = %f +/- %f\n', or_beta1, std(betan));
fprintf(fileID,'gamma = %f +/- %f\n', or_gamma1, std(gamman));
fprintf(fileID,'\n');
fprintf(fileID,'The strain tensor components in crystallographic
coordinates:\n');
fprintf(fileID,'e11 = %10.7e +/- %10.7e\n', Or_strain(1,1), std(e1));
fprintf(fileID,'e22 = %10.7e +/- %10.7e\n', Or_strain(2,2), std(e2));
fprintf(fileID,'e33 = %10.7e +/- %10.7e\n', Or_strain(3,3), std(e3));
fprintf(fileID,'e23 = %10.7e +/- %10.7e\n', Or_strain(2,3), std(e4));
fprintf(fileID,'e13 = %10.7e +/- %10.7e\n', Or_strain(1,3), std(e5));
fprintf(fileID,'e12 = %10.7e +/- %10.7e\n', Or_strain(1,2), std(e6));
fprintf(fileID,'\n');
195
fprintf(fileID,'The stress tensor components in crystallographic coordinates
(MPa):\n');
fprintf(fileID,'s1 = %5.5f +/- %5.5f\n', Or_stress(1,1)*1000,
std(sig1)*1000);
fprintf(fileID,'s2 = %5.5f +/- %5.5f\n', Or_stress(2,2)*1000,
std(sig2)*1000);
fprintf(fileID,'s3 = %5.5f +/- %5.5f\n', Or_stress(3,3)*1000,
std(sig3)*1000);
fprintf(fileID,'s4 = %5.5f +/- %5.5f\n', Or_stress(2,3)*1000,
std(sig4)*1000);
fprintf(fileID,'s5 = %5.5f +/- %5.5f\n', Or_stress(1,3)*1000,
std(sig5)*1000);
fprintf(fileID,'s6 = %5.5f +/- %5.5f\n', Or_stress(1,2)*1000,
std(sig6)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The strain tensor components in sample coordinates:\n');
fprintf(fileID,'e11 = %10.7e +/- %10.7e\n', Sample_strain_org(1,1),
std(Sam_e1));
fprintf(fileID,'e22 = %10.7e +/- %10.7e\n', Sample_strain_org(2,2),
std(Sam_e2));
fprintf(fileID,'e33 = %10.7e +/- %10.7e\n', Sample_strain_org(3,3),
std(Sam_e3));
fprintf(fileID,'e23 = %10.7e +/- %10.7e\n', Sample_strain_org(2,3),
std(Sam_e4));
fprintf(fileID,'e13 = %10.7e +/- %10.7e\n', Sample_strain_org(1,3),
std(Sam_e5));
fprintf(fileID,'e12 = %10.7e +/- %10.7e\n', Sample_strain_org(1,2),
std(Sam_e6));
fprintf(fileID,'\n');
fprintf(fileID,'The stress tensor components in sample coordinates
(MPa):\n');
fprintf(fileID,'s1 = %5.5f +/- %5.5f\n', Sample_stress_org(1,1)*1000,
std(Sam_sig1)*1000);
fprintf(fileID,'s2 = %5.5f +/- %5.5f\n', Sample_stress_org(2,2)*1000,
std(Sam_sig2)*1000);
fprintf(fileID,'s3 = %5.5f +/- %5.5f\n', Sample_stress_org(3,3)*1000,
std(Sam_sig3)*1000);
fprintf(fileID,'s4 = %5.5f +/- %5.5f\n', Sample_stress_org(2,3)*1000,
std(Sam_sig4)*1000);
fprintf(fileID,'s5 = %5.5f +/- %5.5f\n', Sample_stress_org(1,3)*1000,
std(Sam_sig5)*1000);
fprintf(fileID,'s6 = %5.5f +/- %5.5f\n', Sample_stress_org(1,2)*1000,
std(Sam_sig6)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The principal strains are:\n');
fprintf(fileID,'Principal strain 1 = %10.7e +/- %10.7e\n',
principal_strain(1,1), std(p_strain1));
fprintf(fileID,'Principal strain 2 = %10.7e +/- %10.7e\n',
principal_strain(2,2), std(p_strain2));
fprintf(fileID,'Principal strain 3 = %10.7e +/- %10.7e\n',
principal_strain(3,3), std(p_strain3));
fprintf(fileID,'\n');
fprintf(fileID,'The principal strain directions are: \n');
fprintf(fileID,'v1 = (%f, %f, %f)\n',
strain_direction(1,1),strain_direction(2,1),strain_direction(3,1));
fprintf(fileID,'v2 = (%f, %f, %f)\n',
strain_direction(1,2),strain_direction(2,2),strain_direction(3,2));
196
fprintf(fileID,'v3 = (%f, %f, %f)\n',
strain_direction(1,3),strain_direction(2,3),strain_direction(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'The principal stresses are (MPa):\n');
fprintf(fileID,'Principal stress 1 = %5.5f +/- %5.5f\n',
principal_stress(1,1)*1000, std(p_stress1)*1000);
fprintf(fileID,'Principal stress 2 = %5.5f +/- %5.5f\n',
principal_stress(2,2)*1000, std(p_stress2)*1000);
fprintf(fileID,'Principal stress 3 = %5.5f +/- %5.5f\n',
principal_stress(3,3)*1000, std(p_stress3)*1000);
fprintf(fileID,'\n');
fprintf(fileID,'The principal stress directions are: \n');
fprintf(fileID,'v1 = (%f, %f, %f)\n',
strain_direction(1,1),stress_direction(2,1),stress_direction(3,1));
fprintf(fileID,'v2 = (%f, %f, %f)\n',
strain_direction(1,2),stress_direction(2,2),stress_direction(3,2));
fprintf(fileID,'v3 = (%f, %f, %f)\n',
strain_direction(1,3),stress_direction(2,3),stress_direction(3,3));
fprintf(fileID,'\n');
fprintf(fileID,'The hydrostatic stress is (MPa): %5.5f +/- %5.5f\n',
hydro_stress*1000, std(hydrostatic_stress)*1000);
fclose(fileID);
end
Abstract (if available)
Abstract
The long range internal stresses in equal channel angular pressed aluminum alloy 1050 and 6005 were measured using X-ray microbeam diffraction from a synchrotron. Measurements were made at the Advance Photon Source at Argonne National Lab. The full strain/stress tensors of submicron size volumes within the materials have been successfully assessed.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Phan, Thien Q.
(author)
Core Title
X-ray microbeam diffraction measurements of long range internal stresses in equal channel angular pressed aluminum; & Mechanical behavior of an Fe-based bulk metallic glass
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
09/04/2015
Defense Date
07/29/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
bulk metallic glass,equal channel angular pressing,long range internal stress,OAI-PMH Harvest,synchrotron X-ray,X-ray microbeam diffraction
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kassner, Michael E. (
committee chair
), Eliasson, Veronica (
committee member
), Goo, Edward (
committee member
), Hodge, Andrea M. (
committee member
), Levine, Lyle (
committee member
)
Creator Email
sirthienphan@gmail.com,thienqph@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-176788
Unique identifier
UC11273271
Identifier
etd-PhanThienQ-3882.pdf (filename),usctheses-c40-176788 (legacy record id)
Legacy Identifier
etd-PhanThienQ-3882.pdf
Dmrecord
176788
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Phan, Thien Q.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
bulk metallic glass
equal channel angular pressing
long range internal stress
synchrotron X-ray
X-ray microbeam diffraction