ANALYSIS OF SLIP
TRANSFER
IN TI
-
5AL
-
2.5 (WT. %) AT TWO TEMPERATURES IN
COMPARISON TO PURE ALUMINUM
By
Chelsea
M.
Edge
A THESIS
Submitted to
Michigan State University
i
n partial fulfillment of the requirements
f
or the
degree of
M
aterial Science and Engineering
Master of Science
2020
ABSTRACT
ANALYSIS
OF SLIP TRANSFER
IN TI
-
5AL
-
2.5 (WT. %) AT TWO TEMPERATURES IN
COMPARISON TO PURE ALUMINUM
By
Chelsea
M
.
Edge
Understanding the deformation mechanisms present near grain
bou
ndaries in
polycrystalline hexagonal alloys will aid in improving modeling methods. Ti
-
5Al
-
2.5Sn samples
were tensile tested at 296K and 728K, and slip behavior was assessed near grain boundari
es.
From the EBSD measurements of grain orientations, vario
us
metrics related to the slip systems,
traces, residual Burgers vectors, and grain boundary misorientation were computed for
boundaries showing evidence of slip transfer and boundaries showing no
evidence of slip
transfer. This work is compared to a simi
lar
study of an Aluminum oligo
-
crystal to aid in
understanding the differences in slip behavior near grain boundaries in HCP and FCC crystal
structures. Slip transfer in Ti525 was generally obser
ved in less geometrically compatible
conditions than Al, and
sl
ip transfer occurs at high misorientation angles in Ti
-
5Al
-
2.5Sn much
more frequently than in Al.
iii
AKNOWLEDGEMENTS
First, I would like to thank my advisor, Dr. Thomas R. Bieler, for his continued support
throughout my two years at Michigan State University. He taught me most of what I know about
cryst
all
ography, and the processes used to further understand how material
s behave. His
previously developed code in Matlab, was the building block for my research, and his patience in
helping me understand it was pivotal to my work. Dr. Bieler created a space
for
us to have open
discussions about science and his approachability
made for a great learning environment.
I would also like to thank Dr. Phillip Eisenlohr for his
invaluable insights
, specifically
regarding
programming.
He provided a critical role i
n h
elp
ing me understand dislocations and
other crystallographic topics.
His detailed technical support and knowledge greatly improved the
quality of the work presented.
I would also like to express my gratitude toward my committee member Dr. Carl
Boehlert, f
or
his participation in my thesis discussion, and constructive critic
ism he has provided
to
complete the work.
In addition, Dr. Per Askeland was a great help to me with the completion
of EBSD analysis and use of the SEMs.
Additionally, I would like to tha
nk
Hongmei Li, whom I have not met, but know through
her work on Titanium alloys, which inspired this project.
I would also like to thank the team of graduate students within the metals group that
sacrificed their time to help me with my work and understan
din
g of concepts.
Specifically,
David
Hernandez Escobar
for his help with EBSD analysis
and Harsha Phukan
for h
is help on
understanding crystallography.
iv
Finally, I would like to express my deepest gratitude to my family
, friends, and Alex,
for
their const
ant
support and encouragement throughout the course of this endeavor.
v
TABLE OF CONTENTS
LIST OF TABLES
................................
................................
................................
......................
vii
LIST OF FIGURES
................................
................................
................................
...................
viii
KEY TO ABBREVIATIONS
................................
................................
................................
....
xii
Chapter 1: Introduction
................................
................................
................................
...............
1
1.1
Rationale and research objective
................................
................................
...........................
1
Chapter 2:
Literature Review
................................
................................
................................
......
3
2.1
Fundamentals of Titanium alloys
................................
................................
..........................
3
2.1.1
Titanium allotro
pes
................................
................................
................................
......................
3
2.1.2
Dislocations
................................
................................
................................
................................
..
3
2.1.3
Crystal Structure
................................
................................
................................
..........................
4
2.1.4
Deformation mechanisms
................................
................................
................................
............
5
2.2 Dislocations and slip tr
ansfer across grain boundaries
................................
.........................
6
2.2.1
Slip transmission
................................
................................
................................
..........................
6
2.2.2
Metrics used to analyze slip transfer
................................
................................
............................
8
2.3 Previous work
................................
................................
................................
......................
12
2.3.1
Titanium
................................
................................
................................
................................
.....
12
2.3.2
Aluminum
................................
................................
................................
................................
..
12
2.3.3
Aluminum and Titanium Crystal structure differences
................................
..............................
13
Chapter 3: Experimental Procedure
................................
................................
.........................
15
3.1 Material
................................
................................
................................
...............................
15
3.2
Mechanical testing
................................
................................
................................
...............
15
3.2.1 Sample preparation
................................
................................
................................
....................
15
3.2.2 Test set up
................................
................................
................................
................................
..
16
3.2.3 Tensile Test Results
................................
................................
................................
...................
17
3.3 Microstructural Observations
................................
................................
..............................
18
3.3.1 Orientation evaluation
................................
................................
................................
................
19
3.4 Grain pair analysis
................................
................................
................................
...............
21
3.4.1 Slip transfer
................................
................................
................................
................................
21
3.4.2 Slip transfer analysis
................................
................................
................................
..................
30
Chapter 4: Results
................................
................................
................................
.......................
33
4.1 Misorientation angle histogram
................................
................................
...........................
33
4.2 Metric combinations
................................
................................
................................
............
35
4.3 Misorientation angle
vs.
................................
................................
................................
.
37
4.4
vs.
residual Burger
s vector (
)
................................
................................
.....................
38
4.5 Misorientation angle
vs.
the sum of the Schmid factors
................................
.....................
40
4.6 Misorientation angle
vs.
................................
................................
............................
43
4.7
Slip systems present
................................
................................
................................
............
45
vi
Chapter 5: Discussion
................................
................................
................................
.................
52
5.1
vs.
................................
................................
................................
..............................
52
5.2 Misorientation angle
vs.
................................
................................
................................
.
55
5.3 Misorientation angle
vs.
the sum of Schmid factors
................................
...........................
57
5.4 Misorientation angle
vs.
................................
................................
...........................
58
5.5 Slip system analysis
................................
................................
................................
............
60
5.6 Comparison between measured weakly textured grain orientations to random populations
................................
................................
................................
................................
...................
62
Chapter 6:
Summary and conclusions
................................
................................
......................
69
6.1
Summary
................................
................................
................................
.............................
69
6.2
Conclusions
................................
................................
................................
.........................
69
6.3 Future work suggestions
................................
................................
................................
......
71
APPENDICIES
................................
................................
................................
...........................
72
APPENDIX A: Matlab hexagonal orientation code
(written by Thomas R. Bieler)
................
73
APPENDIX B: Matlab MPR hexagonal grain pair analysis code (written by Thomas R. Bieler)
................................
................................
................................
................................
...................
83
APPENDIX C: Sample reconstructed boundary and grain files
................................
...............
99
BIBLIOGRAPHY
................................
................................
................................
.....................
101
vii
LIST OF TABLES
Table 3.1: Measured composition of polycrystal Ti525 (Li, 2013)
................................
..............
15
Table 3.2: Ti525 tensile properties obtained from room temperature and high temperature tensile
tests (Li, 2013).
................................
................................
................................
.............................
18
Table 3.3: m'
values in the upper left
-
hand corner of the table indicates the slip system pair that is mo
st likely
transmitting slip.
................................
................................
................................
...........................
25
Table 3.4: Comparison of random poll of data from each sample, and the number/percentages of
the ful
ly analyzed GBs used in the analysis.
................................
................................
.................
31
aries,
53
Table 5.2: Temperature comparison for Al and Ti525. RT Al and HT Ti525 have similar
homologous temperatures.
................................
................................
................................
............
54
Table 5.3: Compensated Strength
s and values used to obtain them. The pure aluminum
compensated strength is much lower than the Titanium alloy (Aluminum 1100
-
O and Li 2013).
................................
................................
................................
................................
.......................
55
viii
LIST OF FIGURES
Figure 1
.1: Schematic of a turbo pump showing where Ti525 is commonly used. The highlighted
area points toward the part in the turbo pump that
is typically Ti525 (Sutto
n, 2017).
...................
2
Figure 2.1: (a) Disloc
ation present within the Burgers circuit. (b) Burgers vector (purple arrow)
from c to b in a perfe
ct crystal.
................................
................................
................................
.......
4
Figure 2.2
: Hexagonal crystal structure with shaded area indicating different type
s of slip planes.
The purple line represents the <
> direction in which prism, basal and pyramidal planes can
slip. The orange line
represents the <
> direction in which pyramida
l planes can slip.
.........
5
Figure 2.3: Slip traces interacting with a grain boundary. Focusing on case (b), the dislocations
are transmi
tted from grai
n 1 into grain 2, and a residual part of the dislocation in grain 1 is
retained in the grain boundary (Bayerschen, 2016).
................................
................................
.......
7
Figure 2.4: Representation of slip planes (in blue and green) and the
angles associated with them
used to calculate various factors (Abuzaid, 2016).
................................
................................
.........
8
Figure 2.5: Representation of slip in a cylindrical crystal and the associated angles describing the
slip plane a
nd slip direction used to compute the Schmid factor.
................................
.................
10
Fig
ure 3.1: Geometry of tensile samples electro
-
discharge machined out of Ti525 (Li, 2013).
..
16
Figure 3.2: Tensile test set up within the Tescan Mira3 SEM (Li, 2013).
................................
....
17
Figure 3.3: Stress vs. displacement curve for the room temperature (296K) and high temperature
(728K
) samples. Load drops are stress relaxation points when the test was paused for imaging
(Li, 2013).
................................
................................
................................
................................
.....
18
Figure 3.4: SEM image showing the microstructure of (a) the 296K and (b) the 728K tensile
t
ested Ti525 sample with fiduciary mark.
................................
................................
.....................
19
Figure 3.5: EBSD
IPF map of the deformed Ti525 tensile samples. The intentional scratch is the
fiduciary mark, which is visible in the IPF map
for (b)
the 728K sample, and is located out of the
frame in (a) the 296K sample (black rectangular areas are eliminated data that
have very small
grains/questionable indexing).
................................
................................
................................
......
20
Figure 3.6: {
} and
{
} p
ole figures for (a) the 296K and (b) the 728K Ti525 samples.
The black curved lines represent the 30° cones alon
g the major axis. The material has
orientations that are around 3 times random in both samples. (Li, 2013).
................................
...
21
ix
Figure 3.7: Computed slip system possibilities and corresponding slip traces for (a
) the left grain
and (b) the right grain, drawn in actual orientation present in th
e grain.
................................
......
23
Figure 3.8: Comparison of computed slip traces and observed slip traces in the SEM image.
....
24
Figure 3.9: Calculated prisms of each grain drawn relative to each other.
................................
...
26
Figure 3.10: Topography at the grain boundary in (a) is small
indicating that there is little to no
heterogenous strain on either side. Topography at the grain boundary in (b) is large indicating
that there is hetero
genous strain on both sides.
................................
................................
.............
27
Figure
3.11: (a) The red arrows indicate an upward step sense in the grain (light slip traces in the
lower part of grain 1 and dark traces in
grain 2 and the upper corner of grain 1).
(b) Sketch of
cross section along the orange line showing opposite directional
sense of shear (blue arrows).
Though slip traces may appear to be correlated, their slip planes are far from parallel, indicating
that
slip transfer cannot account for the obser
ved slip traces.
................................
......................
29
Figure 3.12: (a) Example of a grain boundary that is categorized as slip transfer, meaning that all
five criter
ia are met. (b) Exam
ple of a grain boundary that is categorized as no slip transfer,
meaning all five criteria for slip transfer were not met
.
................................
................................
30
Figure 3.13: Probability density (the integral is 1) for the Ti525 7
28K and 296K data, and
McKenzie probability distribution data for randomly generated hexagonal crystal orientations.
32
temperature (728K) sample and (b) the room temperature (296K) sample, binned by
misor
ientation
angle. The slip transfer category for both samples have higher values at lower
misorientation angles.
................................
................................
................................
...................
34
Figure 4.2
................................
...................
36
r (a) the room temperature (296K) sample and (b) the
high temperature (728K) s
ample. There is a decreasing trend for misorientation angles below 20°
in both samples. Above 20°, the trend is no l
onger observed.
................................
......................
38
room temperature (296K) Ti525 sample
and (b) the high t
dotted lines, have high m' parameters and low
residual Burgers vectors. The dashed black lines
represent the boundary where
are inside and outside of the box respectively.
................................
................................
.............
40
x
Figure 4.5: The
LG
+SF
RG
)
parameter
vs.
misorientation angle for
(a) th
e room temperature
(296K) Ti525 sample, and (b) the high temperature (728K) sample. High
LG
+SF
RG
)
is
ransfe
misorientation angle.
................................
................................
................................
.....................
42
Figure 4.6:
vs.
misorientation angle for (a) the
room temp
erature (296K) sample and (b)
the high temperature (728K) sample. In
both samples, a strong decreasing
................................
................................
.............
44
Figure 4.
vs.
misorientation angle represented with computed s
lip
systems for each grain pair in (a) the room temperature (296K) sample and (a) the high
temper
ature (728K) sample. There is a cluster of prism to prism slip in grain pai
rs with
misorientation angle less than 30° in the high temperature sample and
between 15° and 30° in the
room temperature sample.
................................
................................
................................
.............
46
vs.
misorientation angle
represented with observed active
slip systems for each grain pair for (a) the room temperature (296K) tensile te
sted sample and for
(b) the high temperature (728K) tensile tested sample. Few clusters are observed in both data
sets.
................................
................................
................................
................................
................
48
Figure 4.9
: Representation of slip systems prevalent in the high temperature (728K) and room
temperature sample. The most prevalent slip system pair in the room temperature (296K) data
is
basal to prism. The slip s
ystem pair that shows up the least in all the data i
s pyramidal to
pyramidal.
................................
................................
................................
...........................
50
Figure 5.1:
vs.
data for the (a) room temperature (296K) and (b) the high temperature
(7
28K) tens
ile tested Ti525 sample. Green boxes indicate comparable Al oligo
-
crystal tensile
results at room temperature for both (a) and (b).
................................
................................
..........
53
Figure 5.2: Aluminum oligo
-
data for (a) the room temperature (2
96K) sample and (b) the high temperature (728K) sample.
to the aluminum data.
................................
................................
................................
....................
57
Figure 5.3: Mi
LG
+SF
RG
) for the (a) room temperature Ti525 sample
and t
he (b) high temperature Ti525 sample. The aluminum boundaries represented
by the green
line show the difference between the aluminum data set which has a steep slope and th
e titanium
data set which slope is flatter.
................................
................................
................................
.......
58
(b) the high temperature (728K) sample. The red sh
aded area represents the aluminum oligo
-
xi
e shaded area
represents
the aluminum oligo
-
crystal
...........
60
Figure 5.5: Misorientation angle cumulative p
ercentages for Ti525 and
randomly generated data
sets separated by type of slip system observed in the grains. Pyramidal slip is behaving counter
-
intuitively with respect to more restrictive conditions.
................................
................................
.
64
Figure 5.6: O
verall Ti525 statistics in comparison to a random population
. Random population
data sets have a high frequency for pyramidal to pyramidal where experimental
results are low for the same category.
................................
................................
...........................
65
Figure 5.7: Relative frequency of data represented as half
-
circles. Larger area half
-
circle
represents a higher frequency. (a) Comparison of experimental Ti525 at room temperature and
high temperature. (b) Compariso
n of randomly generated d
ata sets
................................
............
66
Figure 5.8: Randomly generated orientation pairs with the most l
ikely slip systems present that
facilitates slip across the grain boundaries. (a) Unmodified
Schmid factors. (b) Schmi
d factor for
pyramidal is halved. A high frequency is present in slip system pairs with pyramidal slip
systems in both cases.
................................
................................
................................
...................
68
xii
KEY TO
ABBREVIATIONS
Al
Aluminum
ASTM
American Society for Testing and
Materials
BSE
Back
scattered electron
CP
Commercially Pure
CPFE
Crystal Pla
sticity Finite Element Modeling
CRSS
Critically Resolved Shear Stress
EBSD
Electron Backscatter Diffraction
EDM
Electro discharge machined
FCC
Face centered cubic
Fe
Iron
GBS
Grain Bou
ndary Sliding
HCP
Hexagonal close
-
packed
HT
High Tempera
ture
IPF
Inverse Pole Figure
m
Geometric compatibility factor
Ni
Nickel
RT
Room Temperature
SE
Secondary electron
SEM
Scanning Electron Microscopy
SF
Schmid factor
SFLG
Schmid fac
tor left grain
SFRG
Schmid factor right grain
xiii
SiC
Silico
n Carbon
Sn
Tin
Ti
Titanium
Ti525
Ti
-
5
-
Al
-
2.5Sn
Wt.%
Weight percent
XRD
X
-
ray Diffraction
Zn
Zinc
Residual
B
urgers vector
1
Chapter 1:
Introduction
1.1
R
ationale and research objective
The
overarching goal of the research is to further understand th
e deformation
mechanisms in titanium alloys, specifically Ti
-
5Al
-
2.5Sn (Ti525). Knowledge of the
deformations mechanisms can aid in
improving
the modeling meth
ods of titanium alloys such as
crysta
l plasticity finite element (CPFE) modeling, to
enable more
predictive ability
to model
heterogeneous strain near
grain boundar
i
es.
Titanium is a good candidate for the aerospace industry for many factors. One of the mo
st
important factors are the possibil
ities for weight reduction in an engine. Since titanium has
a high
strength
-
to
-
weight ratio, components traditionally made of steel or aluminum could be replaced
with titanium, depending on the application. Titanium
is
also
corrosion resistant
meaning
that
coatings are not
usually
required unless it is
in contact w
ith an
aluminum or a steel component in
which galvanic corrosion may occur. Although titanium has many factors that make it appealing,
it can cost from 3 to 10
times as much as aluminum or steel
(
Boyer, 1996)
.
Ti525 is mainly used in cryogenic application
s. It does not exhibit a ductile
-
to
-
brittle
transition which aids in high ductility and fracture toughness at low temperatures. Ti525 has
been used in the
space shuttle, on the hydrogen side of th
e high
-
pressure fuel turbo
-
pump
(Boyer
1996)
.
Figure 1.1
sh
ows a schematic of a turbo
-
pump, and the location of where Ti525 is
commonly used (Sutton, 2017).
At elevated temperatures Ti525 has excellent creep resist
ance
(Boyer, 1996)
.
2
Figure
1
.
1
:
Schematic of a tu
rbo pump showing where Ti525 is commonly used.
The highlighted
area points toward the part in the turbo pump that is typically Ti525 (Sutton, 2017).
3
Chapter 2:
Lit
erature
Review
2.1
Fundamentals of Titanium a
lloys
2.1.1
Titanium allotropes
allotrope
s
,
-
nominally
co
mmercially pure titanium, whereas
near
-
-
stabilizers such
as Sn, or Al
(
Li
,
2013)
. For the purposes of this paper, near
-
-
alloy studied is Ti
-
5Al
-
2.5Sn (Ti525).
2.1.2
Dislocations
Two im
portant concepts
that need to be u
nderstood
before moving forward in the paper
are the
dislocation
and
Burgers vectors.
Dislocations are line defects within a crystal. The
B
urgers vector
describes the relative displacement
on
the slipped plane
. The Burgers
circuit is a
pathway along atoms,
that form a closed loop within which dislocations are located.
Figure 2.1
(
a
)
shows
the Burgers circuit (a
-
b
-
c
-
d
-
e) and a defect within the Burgers circuit. When you
present the same circuit in a perfect crystal, there is
a loop closure failure. In order
to complete
the circuit, a vector from c to b would be necessary, see
Figure 2.1
(
b
)
.
This vector is the Burgers
vector (
Hull, 2011)
.
4
Figure
2
.
1
: (a) Dislocation present
within the Burgers circuit. (b) Bu
rgers
vector
(purple arrow)
from c to b in
a
perfect crystal
.
2.1.3
Crystal Structure
The crystal structure of
the
titanium alpha phase is hexagonal close
packed
(HCP)
.
Figure
2.2
shows
the hexagonal structure. The blue shaded a
rea represents the basal planes. T
he red
shaded area represents prism planes. The yellow/green shaded area represents pyramidal planes.
The a
1
, a
2
, a
3,
and c directions are represented by black arrows.
Slip occurs
on close packed
planes, in close packed di
rections. For hexagonal crystal st
ructures
slip has been observed in the
following
directions:
<
>
on
{
}, {
}, and {
} planes
, and <
3
> on {
}
plane
(
Li
, 2013
and Bridier, 2005
)
.
5
Figure
2
.
2
: Hexagonal cryst
al structure with shaded area indicating different types of slip planes.
The purple line represents the
<
> direction in which prism, basal and pyramidal planes can
slip. The orange line represents the <
> direc
tion in which pyramidal planes can
slip.
2.1.4
Defo
r
mation m
echanisms
The critical resolved shear stress (CRSS)
determines the ease of slip on a system.
Generally, w
ith a higher CRSS, the eas
e
of movement on that plane
de
creases.
Hongemei et.
a
l.
found that
at
both
296K and 728K
for the Ti5
25 alloy, prismatic slip was
more easily activated
than all other slip systems, including basal, since prismatic slip was found at low Schmid factors.
Basal
slip
was then considered to be the next slip system easily activa
ted in both samples
(
Li
,
20
13)
.
With different alloys,
different slip systems become more active. In near
-
slip
is
more active compared to commercially pure (CP) Ti. This is due to the addition of Al and
c
6
Sn, in which both increase the c/a ratio from 1.587 in CP Ti to
about 1.60, to
wards the idea
l c/a
ratio (1.633)
.
With a higher c/a ratio, a more closely packed basal plane results, which increases
the eas
e
of
basal
slip
.
In addition, with different temperatures, different slip systems become
more active. With Al alloyed Ti, a decrease in the CRSS
for both basal and prismatic slip
systems is indicated with an increase of temperature.
(
Li
, 2013)
.
At
high temperatures, climb is enabled. Climb is the process in which a dislocation
moves out of its slip plane
with the aid of diffusion
(Hull, 20
13).
This means that a dislocation
can move to another, possibly more favorable slip plane during high temperature deformation.
When dislocations move through the material, in one grain slip is
often
prevalent on a
specific slip system with a spe
cific Bur
gers vector.
If a
dislocation moves across a grain
boundary into another grain, it travels on another slip system with a different Burgers vector
in
the neighboring grain
. Burgers vectors have a direction and
length
, depending on if the Burgers
v
ector is
an vector, vector or a vector, the
length
will vary. For the purpose of
this
thesis
,
only the
Burgers vector
direction
s
were analyzed. That is, the Burgers vectors used
are unit vectors.
2.2
Dislocations
and
slip transfer
across
grain boundarie
s
2.2.1
Slip transmission
One of
the
goals
for this
thesis
is
to identify and analyze
slip transfer across grain
boundaries. Slip traces interact with grain boundaries in a few different ways in polycrystalline
metal
s.
Figure 2.3
depicts possible cases in which slip is transmitted. In case (a) dislocations are
stopped by the grain boundary, do not transmit, but pile up at the grain boundary. Case (b) (our
focus) is whe
n dislocations are emitted from grain 1
in
to
grain 2
through
the grain boundary
, and
7
residual Burgers vector content is left in the boundary
. In another example,
dislocations
can be
dissociated into the grain boundary, leaving no dislocation
s
in grain 2 (c
ase (c)). Case (d) shows
full transmission o
f slip. If a full transmission of slip has occurred (i.e. perfect alignment of the
slip directions), theory suggests that no residual Burgers
vector
is left in the grain boundary.
Case (e) shows where two disloca
tions meet at a grain boundary and generate
a new dislocation
in the grain boundary. Case (f) depicts a reflection of the dislocation in grain 1 back into grain 1
and leaving a dislocation in the grain boundary
(Bayerschen
, 2016
).
Figure
2
.
3
: Slip traces interacting with a grain boundary.
Focusing on c
ase (b), the dislocations
are
trans
mitted
from grain 1
in
to grain 2, and
a residual
part of the dislocation in grain 1 is
retained in
the grain boundary
(Bayerschen, 2016)
.
The case that wil
l be discussed is case (b) where
perfect
slip transfer does not occur
across the grain boundary
, implying that the Burgers vector changes direction
from one grain to
the next,
resulting in
some residual Burgers vector
(
) left in the grain boundary (see
Figure 2.3
(
b
)
).
The
better
the
alignment of the slip systems in grain 1 and grain 2, the more likely the
transmission of the dislocation
in
to grain 2
will occur
, and a
smaller
residual
component
of slip
would
be
retained in
the grain boundary. This
suggests that
minimizing the size of the residual
8
Burgers vector
would make a
slip transmission
event
more likely
, as observed and argued by
Shen et al. (1989) and Lee
et al.
(1988)
.
2.2.2
Metrics used to analyze slip transfer
The direction and relative size of the residual Burgers vector
can be
estimated using
data
obtained
from electron backscatter diffraction (EBSD) analysis.
Knowing the grain orientation
,
the Burgers vec
tor of
the left
grain (
)
,
which
we assume to be the
initiating
slip
system,
and the
right
grain
Burgers
vector
(
)
is
the transmitted slip
,
the residual Burgers vector,
is
estimated
by the following dislocation reaction equation
:
(1)
Figure 2.4
shows the Burgers vectors (
)
and
(
) on two different slip planes, and the
angles associated with the transmission.
Figure
2
.
4
: Representation of slip planes (in blue and green) and the angles
associated with them
used to calculate various factors
(Abuzaid, 2016)
.
Grain
Boundary
9
The geometric compatibility factor,
, is used as
a criterion
to determine if
slip transfer
is likely to occur. The factor can be calculated from two angles
and
, the angle between the
normal to the slip plane, and the angle between the two slip directions in grains 1 and 2,
respectively (see
Figure 2.4
). Therefore, the factor resolves the strain from
th
e
slip on grain 1
onto grain 2. The factor is calculated as follows
(Luster, 1994)
:
(2)
An
value closer to 1 would
imply that
slip
transfer
is more likely to occur on the
specified slip systems
, as they would be nearly
co
l
linear
, as opposed to a slip system pair that has
The misorientation angle
is
determined from the orientations of each grain through EBSD
analysis.
The misorientation between two grains is computed
based upon
the crystal orientations
o
f each grain,
g
A
for the crystal orientation of grain 1 and
g
B
for the crystal orientation of grain 2.
The misorientation is defined as the rotation needed to bring the orientation of grain A into
coincidence with the orientation
of grain B. The calculatio
n for misorientation (
)
is as
follows:
about which the rotation happens ([n
1
,
n
2
,
n
3
]) is calculated, shown
below
(
Koc
ks, 1998
)
.
10
For hexagonal crystal structures, there are
12 symmetric orientation matrices. The OIM
software, from the above
([n
1
,
n
2
,
n
3
]) for all 12 variants of symmetric rotations axes. The smallest misorientation angle is
then chosen as
the misorientation angle with the associated rotation axes.
In pre
vious work, the resolved shear stress
based upon the global stress tensor
was
used
as a criterion
to
determin
e
if slip is likely. The Schmid factor is used as a
metric
to determine the
mos
t likely slip system
s that are activated in
the two
neighboring
grains. A higher Schmid factor
implies
that
slip
is
more likely to occur. The Schmid factor is determined by two angles,
,
when there is unidirectional stress
as described in
Figure 2.5
(Hull, 2011)
.
Figure
2
.
5
: Representation of slip in a cylindrical crystal and the associated angles
describing the
slip plane and slip direction
used
to compute the Schmid
factor.
11
The Schmid factor is calculated by the following equation, using the angles described i
n
Figure
2.5
(Hull, 2011)
:
(3)
Slip systems with higher Schmid factors will facilitate slip more easily. Since slip will be likely
with a high
and high Schmid factors
based upon
another metric worth
considering (
Bieler,
2019, Aliz
adeh 2020)
,
given
by:
(4)
where
SF
LG
is the Schmid factor in the left grain, and
SF
RG
is the Schmid factor in the right grain.
Another useful parameter is
, as slip transfer is
expected to be
facilitated with high
and low
, where
is the residual
B
urgers vector
(Alizadeh 2020
,
Bayerschen 2016, Shen
1988
, Lee 1989
)
.
12
2.3
Previous work
2.3.1
Titanium
The work done in this
thesis is built on
the work published by Hongemei Li
(2013)
,
r of the hexagonal close
-
packed alpha phase in titanium
Tension and creep samples were tested at low a
nd high temperatures
for a
variety of
alloys
.
For the purpose of this paper, the interest lies in the Ti525 alloy.
For each sample, d
istributions of active slip systems
in each grain
were calculated.
Basal
and prism slip were
found to be
dominant in the
Ti525 alloy. With
the
increase in tem
perature
,
basal active slip systems increased
in frequency
.
This
increase in ease of slip in basal slip
systems correlates to a lower CRSS
. Prismatic slip systems for Ti
-
CRSS with high tem
perature.
Building
upon
Hongmei
Li
w
ork, further investigation of
the tensile tested Ti525
sample at 296K and 728K was considered.
The focus was to look at grain
pairs and
determine if
there was slip transfer across the grain boundary.
Dividing the grain pairs into
slip transfer
and
no slip
transfer
categories
and using
the
various parameters
that have been discussed
,
allowed for
trends in
slip transfer
to be uncovered.
2.3.2
Aluminum
A similar activity was done by Alizadeh
(2020)
deh found that
slip
transfer
usually was associated with high
values.
Slip transfer is favored by low angle bou
ndaries, which have high
values
due to the low
misorientation geometry
.
Further
,
Alizadeh assessed
the
LG
+SF
RG
)
,
and the
par
ameter
s
vs.
the misorientation angle to observed trends.
Also, Linne and Daly examined a
similar specimen of
pure Al at 190
C using high resolution differential image correlation (DIC)
13
(
Linne
2020).
It is hypothesized that similar trends observed in the
Al data set
s
will be present in
the
Ti525
data set, with more complicated slip systems.
The difference in cry
stal structure
may
limit the similarities noticed between the Al
face centered cubic (
FCC
)
crystal structure and the
Titanium
hexagonal close pack
ed
(
HCP
)
crystal structure.
2.3.3
Aluminum and Titanium Crystal structure differences
Aluminum has face
-
centered cubic (FCC) crystal structure. Slip occurs on the close
-
packed planes along the close
-
packed directions, which in FCC is {111} and <110> respectivel
y.
Therefore, there are four close packed planes, each with 6 <110> directions, b
ut
positive and
negative directional sense are not distinguish
ed
, meaning
a reduction of a factor of t
wo.
Therefore
,
there
are 12 slip systems in FCC crystal structure, all with equal ease of operating
(
Jackson, 1991
)
. In contrast, titanium has a hexagonal crystal structure, where slip occurs on
basal, pyramidal, pyramidal and prism
slip systems
. There are thre
e slip systems each
on basal and prism planes that are easily activated (
Li
, 2013
), but they do not enable changes in
crystal dimension in the direction, so easy slip cannot enable needed shape changes
in the
direction
. There are an additio
nal 18 slip systems on pyramidal
planes but
slip on pyramidal
planes is much mor
e difficult to activate, especially in the directions, so higher resolved
shear stresses are required
(
Li
, 2013)
. Therefore, titanium alloys
have
fewer
slip system
s
that
facilitate
eas
y
slip
compared to
FCC crys
ta
ls
,
even though there are
many more slip systems
available.
To assess the difference between Al and Ti alloy slip transfer behavior, the methodology
developed by Alizadeh et al. is used on a
polycrystalli
ne
Ti alloy. This work will also examine if
this methodology can
yield simi
lar statistical information
in a standard polycrystal sample rather
than an oligo
-
crystal
foil
.
14
The following chapters will explain the experiments done, the results obtained, and
the
observed trends in the
296
K and
728
K tensile tested Ti525 samples. Analysis was done on slip
transfer data to further the investigation that Hongmei
Li
presented. The t
rends observed in the
lts of an aluminum alloy slip
transfer
data to compare differences in
F
CC and HCP materials.
15
Chapter 3:
Experimental Procedure
The description of the material and tensile
tests on the Ti
-
5
-
Al
-
2.5Sn alloy is summarized
blications.
3.1
Material
The Ti
-
5
-
Al
-
2.5Sn (Ti525) alloy was provided by Pratt & Whitney, Rocketdyne. It was
rial was then annealed at 1127K for 1
hour for recrystallization, followed directly by air cooling, followed by a
vacuum annealing
process at 1033K for 4 hours to reduce the hydrogen content.
Table 3.1
denotes the measured
bulk composition of the alloy in weight percent (
Li
, 2013).
Table
3
.
1
:
Measured composition of
polycrystal Ti525 (
Li,
2013)
Element
Al
Sn
Fe
Zn
Ti
Weight %
4.7
2.7
0.2
0.1
balance
3.2
Mechanical testing
3.2.1
Sample preparation
Two specimens were prepared by
a mechanical polish, using 400, 600, 1200, 2400 and
4000 Silicon Carb
ide
(SiC) grinding papers sequentially for 5 to 10 minutes each. Between each
grinding step, the sample was rinsed with water. After grinding, a final polishing step with
polishing clo
th from Buehler (catalog No. M500
-
12PS) and five parts colloidal silica wit
h
0.06µm particle size and one part 30% hydrogen peroxide. This step took around an hour to
obtain the desired finish. Water was again used to rinse the sample after polishing then
16
ultrasonically cleaned with acetone and methanol respectively to remove th
e colloidal silica (
Li
,
2013).
Multiple dog
-
bone samples were electro
-
discharge
machined
(EDM)
from the forging
with a gage width of 3mm
and 10mm length.
Figure 3.1
shows the geometry of the EDM
samples.
Figure
3
.
1
: Geometry of tensile samples electro
-
discharge machined out of Ti525 (
Li
, 2013
)
.
3.2.2
Test set up
Th
e samples examined were
in
-
situ
tensile tested
within the Tescan Mira3 SEM, with a
displacement rate of 0.004 mm/s (approximate strain rate of 10
-
3
s
-
1
). The tensile stage was built
by Ernest F. Fullam,
Incorporated. Displacement time and load data was rec
orded using the
MTESTW (version F 8.83) control software. To obtain the desired temperature of the sample, a
6mm diameter tungsten
-
based heating unit was used. One sample was tested at 296K and another
a
t 728K. The sample was held at this temperature for 3
0 minutes prior to loading. During testing,
the temperature of the sample was monitored using a thermocouple that was welded to the side
of the gage section. Throughout the test, a vacuum below 2
×10
-
6
To
rr was maintained.
17
Figure
3
.
2
: Tensile test set up within the
Tescan
Mira
3
SEM
(
Li
, 2013)
.
3.2.3
Tensile Test
Results
Figure 3.2
shows the set up within the Tescan Mira3.
The stress
-
displacement cur
ves are
presented in
Figure
3
.
3
. The load drops shown are due to pausing the test
for image acquisitions
.
Table 3.2
gives
the yield strength and m
aximum stress obtained from the stress displacement
curves
(
Li
, 2013).
18
Figure
3
.
3
: Stress
vs.
displacement curve for the room temperature (296K) and high temperature
(728K) samples. Load drops are st
ress relaxation points when the test was paused for imaging
(
Li
, 2013).
Table
3
.
2
: Ti525 tensile properties obtained from room temperature and high temperature tensile
tests
(
Li
, 2013)
.
Tempera
ture
Yield Strength (MPa)
Max Stress (MPa)
296K
~660
769
728K
~3
0
0
434
3.3
Microstructural Observations
Figure 3.4
(
a
)
and
(
b
)
show representative SE SEM images of the deformed Ti525 in
-
situ
296
K
and
728
K
tensile tested sample. The scratch in the figure is a fiduciary mark so the
location of the EBSD analyzed area was easier to find.
19
Figure
3
.
4
: SEM image showing the microstructure of
(a)
the
296
K
and (b) the
728
K
tensile tested Ti525 sample with fiduciary mark.
3.3.1
Orientation evaluation
After deformation, EBSD orientation maps were created from a portion of the gage
section, marked with a fiduciary mark. The maps were obtained
using
OIM Analysis
TM
Version
7.3.1, by EDAX.
Figure 3.5
(
a
)
and
(
b
)
show the inverse pole figure (IPF) orientation maps with
a variety of colors observed, meaning the samples are not st
rongly textured. For the room
temperature sample, (
Figure 3.5
(
a
)
) the
fiduciary mark is located just out of
the
frame
to
the left
of the image
and in
Figure 3.5
(b), the high temperature sample, the fiduciary mark is cropped
out
(shown as a black stripe)
. The black rectangles
in the room temperature sample
are areas
where the EBSD software
determined many small grai
ns
that were probably artifacts of poor
indexing
, so they were cropped out of the analysis.
Clean
-
up
procedures including one or two
rounds of near
-
neighbor correlation and one round of dilation was used to eliminate small grains
from the analysis.
This
provided
a convenient reconstructed boundary file to analyze.
Fiduciary mark
s
(a)
(b)
20
Figure
3
.
5
: EBSD IPF map of the
deformed
Ti525 tensile samples.
The intentional
scratch is the fiduciary mark
, which is visible
in t
he IPF
map for (b) the 728K sample, and is
located out of
the
frame in (a) the 296K sample (black rectangular areas are eliminated data that
have very small grains/questionable indexing)
.
Fiduciary mark
(a)
(b)
21
Figure 3.6
(a
) and (b) shows the pole figures for both the 296K and 728K data,
respectively. The dark lines represent the 30° cones along the major axis. The maximum value
in the legend indicates that the orientation of the material after deformation is ~3 times ran
do
m
for both samples.
Figure
3
.
6
:
{
} and {
}
p
ole figures for (a) the 296K and (b) the 728K Ti525 samples.
The black curved lines represent the
30° cones along the major axis
. The material has
or
ientations that are around 3 times random in both samples.
(
L
i, 2013).
3.4
Grain pair analysis
3.4.1
Slip
transfer
To determine if slip transfer was present through grain boundaries, an in
-
depth look at
slip traces near the grain boundary was necessary. If t
here were correlated slip traces in one grain
(
a
)
Room
Temperature
(
b
) High Temperature
22
and a neighboring grain, a set of criteria were considered to deter
mine if slip transfer accounted
for the correlated slip traces. These criteria are:
(1)
D
etermination of probable slip systems which would be
consistent with observed
slip traces
.
(2)
A
value associated with the observed correlated slip systems gen
erally larger
than 0.7
.
(3)
T
he residual burgers vector (
) associated with the observed correlated slip
systems is generally smaller than
0.5b
.
(
4
)
T
he Schmid factor of each slip system is generally larger than 0.25, (
5
) the
topography at the
grain boundary is small indicating that the boundary does not
lead to heterogenous strain on both sides
.
(
6
)
T
he observed slip traces on each sid
e have a topographical directional sense that
implies that the slip planes are
approximately
parallel.
To determine if probable slip systems are consistent with observed slip traces (criterion
(1)), the Euler angles for
each grain in a grain pair were ent
ered into a Matlab code (shown in
Appendix A
: Matlab hexagonal orientation code (written by Thomas R. Bieler)
)
.
This code
draws the unit cell in the actual orientation that is
present in the grain. The unit c
ell is drawn
multiple times with different slip systems in order of decreasing Schmid factor. It also draws
the
slip trace th
at would be expected if this slip system was active.
Figure 3.7
sh
ows the output of
this code for two grains
.
23
Figure
3
.
7
:
Computed
slip system possibilities
and corresponding slip traces
for (a) the left grain
and (b) the righ
t grain
, drawn in actual orientation present in the grain.
Code
generated
s
lip
traces
are
then compared to the SEM image of the grain, to
determine what computed slip trace
best matches the observed slip trace in the SEM image.
Figure 3.8
shows an example of the comparison, where the
trace of slip system number 11
24
matches
closely with the observed slip traces in the left grain and similarly, the
trace of slip
system number 5
matches the obser
ved slip traces in the right grain. In cases where there are
multiple slip systems on the same plane (on pyramidal and basal planes), the slip
direction
with
the highest Schmid factor is chosen.
Figure
3
.
8
:
Comparison of computed slip traces and observed slip traces in the SEM
image.
criteria (4) (low residual
B
urgers vectors) are met, another code was used
(also
written by Dr.
Thomas
R.
Bieler
, shown in
Appendix B
). The code reads the reconstructed boundary file (which
includes the misorientation of each boundary) and grain file generated by
the
OIM Analysis
software
. Sample fil
es are shown in
Appendix C
.
The probable slip conditions present and the
Schmid factors are determined
for each grain. Next, the code computes grain boundary
parameters,
values and residual Burgers vectors (
) for ea
ch slip system pair possibility.
Thresholds values are set up to filter out slip system possibilities that are unlikely to be
25
meaningful, such as Schmid factors lower than 0.2 and
values below 0.6
.
Tabl
e 3.3
is then
generated for each slip system pair in order of decreasing Schmid factor. The slip system
number is shown on the far most left column for the left grain, and upper most row
for the right
grain. The corresponding number (right or below, respe
ctively) is the corresponding Schmid
factor. The values in the middle of the table are the
metrics for each corresponding slip
system pair, and high values of
values are bolded. Values that are bolded, and in the upper
left
-
hand corner of the t
able, are the best candidates for slip transfer because they have high
values, low
values, and high Schmid factors. As with the other code, prisms of each grain are
drawn in their relative positions to each other, and slip systems are illustrated for
slip system
pairs in order of decreasing
values
, as illustrated for the grain pair
in
Figure 3.9
. I
f there
are multiple slip systems that have slip traces that look like they could be present in th
e SEM
image of the grain, this table is used to determine which one is most likel
y to account for the
observation.
Table
3
.
3
:
v
alues in the upper left
-
hand corner of the table indicates the slip system pair that is most likely
transmitting slip
.
26
Figure
3
.
9
:
Calculated
p
risms of each grain drawn relative to each other.
To det
ermine if criterion (5) was met, further observations of the grain boundary were
made to determine if a large ledge (indicated by topographic contrast) or small ledge was
present.
Figure 3.10
(a) shows an
example of a small (or non
-
observable) ledge at the grain
boundary, while
Figure 3.10
(b) shows an example of a larger ledge at the grain boundary that
indicates that heterogeneous strain occurred on bo
th sides.
27
Figure
3
.
10
: Topography at the grain boundary in (a) is small indicating that there is little to no
heterogenous strain on either side. Topography at the grain boundary in
(b) is large indi
cating
that there is heterogenous strain on both sides.
To determine
if
the observed slip traces on each side have a topographical directional
sense that implies that the slip planes are nearly parallel (criteria (
6
)), the surface topogra
phy of
the neighb
oring grains w
as
assessed further.
By considering surface topography, from the sense
1
, the topographic shape of the slip steps
was identified in each grain. If the slip steps did not appear parallel, le
ading to contrasting dark
1
or the image.
Gra
in boundary
(large ledge)
Grain bou
ndary
(small ledge)
(a)
(b)
28
and light steps on either side of the boundary, the likelihood of slip transfer was low because the
sense of shea
r is in different directions
as illustrated in
Figure 3.11
(a) and (b), and ther
efore this
criteria was not met
.
Figure 3.11
(a) shows the top view of two grains, as seen in the SEM. The
red lines indicate the upward direction of steps. The orange line shows where a cross section is
depicted in
Figure 3.11
(b), where the same red arrows show upward steps. It also shows the
sense of shear in each grain which are far from parallel. If the slip steps were generally parallel,
the criteria would be met.
29
Figure
3
.
11
: (a) The red arrows indicate an upward step sense in the grain (light slip traces in the
lower part of grain 1 and dark traces in grain 2 and the upper corner of grain 1). (b) Sketch of
cross
section along the orange line showing opposite directional sense of shear (blue arrows).
Though slip traces may appear to be correlated, their slip planes are far from parallel, indicating
that slip transfer cannot account for the observed slip trace
s.
Grain 1
Grain 2
Grain Boundary
(a)
(
b
)
30
If
all five
criteri
a
were
met, the grain boundary was categorized as a case of slip
transfer. If one
or two criteria were not met, the boundary was categorized as possible slip
transfer,
or
if
more criteri
a
w
ere
not met, the grain boundary was categorized
as a case of
no slip transfer.
An example of a case of slip transfer
is shown in
Figure 3.12
(
a
)
.
Figure 3.
12
(
b
)
shows an example of a boundary where all five criteria are not met, therefor it is categorized
as a case of no slip transfer.
Figure
3
.
12
:
(a) Example of a grain boundary that is categorized as slip transfer, meaning that all
five criteria are met. (b)
Example of a grain boundary that is categorized as no slip transfer,
meaning all five criteria for slip tra
nsfer were not met.
3.4.2
Slip
transf
er
analysis
To obtain a representative set of data from the samples, it was necessary to understand
what types of grain boundaries were present in
each
sample. 50 random grain boundaries were
chosen and identified as
transfer
slip transfer
ip
transfer
as
explained in section
3.4
:
Grain pair analysis
. Of these, 8%
was
determined
to fall into the
for
both the room temperature and high temperature samples. The
percentages of the random poll of grain b
oundaries is shown
in
Table 3.4
.
Grain boundaries were
fully analyzed until the percentages matched the random sampling data
, to ensure
that
the
analyzed grain boundaries were representative of the w
hole sample
.
Grain Boundary
Grain Boundary
(a)
(
b
)
31
In the room temperature sample, 146 grains were analyzed and
divided into the three
groups
,
maybe slip
transferred
study and consisted of
10
grain boundaries. 73 grain boundaries were
identified
as slip transfer
.
The not
slip transfer
group
consisted
of around 6
8
grain boundaries.
In the high temperature sample, around 173 grain boundaries were analyzed and d
ivided
into the three groups. The
transferred
grain boundary group consisted of 93 grain
boundaries.
transferred
78
gra
in boundaries.
The rest of the grains in the
transferred
was
not used in the
data
analysis.
Table
3
.
4
: Comparison of random
poll
of data
f
r
o
m each sample, and
the number/percentages of
the fully analyzed GBs used in the analysis.
Slip
transf
er
GBs
Slip
transfer
GBs (%)
Not slip
transfer
GBs
Not slip
transfer
GBs (%)
RT Random Poll
from Ti525
24
52%
22
48%
RT Ti525 fully
analyzed
73
52%
68
48%
HT Random
Poll
from Ti525
25
54%
21
46%
HT Ti525 fully
analyzed
93
54%
78
46%
Figure 3.13
shows a probability of the misorientation angles collected from the Ti525
high temperature and room temperature data. In addition, random orientations likely for
hexagonal crystal
structure were generated and
plotted against the Ti525 data.
T
here are some
differences
, specifically in misorientation angles less
than
40°, with the Ti525 data having a
higher probability.
32
Figure
3
.
13
:
Probability
density (the
inte
gral
is
1)
for the Ti525
728K
and
296K
data, and
McKenzie
probability
distribution data for randomly generated hexagonal crystal
orientations.
33
Chapter 4:
Results
This chapter presents the data from
the grain boundaries that were fully analyzed,
including metrics di
scussed in the
Chapter 2:
Literature Review
.
4.1
Misorientation angle histogram
This section presents the data collected from the room temperature
(296K) tensile tested
sample and the high temperature (728K) tensile tested sample. From the orientation maps, the
misorientation angles,
values, Schmid factors, and residual Burgers vec
tors (
) for the grain
pairs were extracted as explained in
Chapter 3: Experimental Procedure
. Each grain boundary
was sorted
to
either
slip transfer
or
no
slip transfer
categories.
Figure 4.1
shows a histogram of
the
slip transfer
and
no slip transfer
points
plotted
with respect to the misorientation angle between grains for (a) the room temperature Ti525
sample, and (b) the high
temperature sample. Below 30° misorientation,
slip transfer
data are
significantly higher
than
no slip transfer
data in both sa
mples. The cumulative percentage lines
slip transfer
no slip transfer
tation
angles.
34
Figure
4
.
1
:
Histograms representing
the
slip
transfer
and
no slip transfer
data for the (a) high
temperature (728K) sample and (b) the room temperature (296K) sample, binned by
miso
rientation angle.
The s
lip transfer
category
for both sample
s have higher values at lower
misorientation angles.
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
0
2
4
6
8
10
12
14
16
10
20
30
40
50
60
70
80
90
100
Misorientation angle (deg)
(a) Room temperature sample
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
0
2
4
6
8
10
12
14
16
10
20
30
40
50
60
70
80
90
100
Number
Misorientation angle (deg)
(b) High temperature sample
35
4.2
M
etric combinations
T
o determine
which metric
combinations were of interest, a matrix of all the combinations
was created
.
Figure 4.2
shows
a matrix with the possible correlations for
(a) the room
temperature Ti525 sample, and (b) the
high temperature sample. In both matrices, metric pairs
with interesting trends
include
vs.
,
vs.
misorientation angle, and
vs.
misorientation
angle.
36
Figure
4
.
2
: Possible combinations of factors, with
(a)
(
b
)
37
4.3
Misorien
tation angle
vs.
Figure 4.3
shows the trend
of the
slip transfer
and
no slip transfer
data
for
misorientation
vs.
the geometric compatibility factor
(
), for (a) the room temperature (296K)
sampl
e and (b) the high temperature (728K) sample. Below 20° misorientation angle in both the
downward as the misorientat
ion angle increases. Above 20°
,
the points
begi
n to spread out, and
the downward sloping trend is no longer noticeable. The room temperature sample has 3 points
with
a
sample does not have as strong of a trend
comp
no
slip transfer
no slip transfer
data otherwise.
38
Figure
4
.
3
:
vs.
misorientation angle for (a) the room temperature (296K) sample and (b) the
high temperature (728K) sample. There is a decreasing trend for misorientation angles below 20°
in both
samples. Above 20°
, the trend is no longer observed.
4.4
vs.
resi
dual
B
urgers
vector
(
)
The Luster
-
Morris parameter (
)
vs.
the residual Burgers vector
(
)
is plotted in
Figure
4.4
for (a) the
room
temperature
(296K)
sample and (b) the
high
temper
ature
(728K)
sample.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
20
40
60
80
100
m'
Misorientation angle (deg)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
20
40
60
80
100
m'
Misorientation angle (deg)
(a)
Room
t
emperature
(b)
High
t
emperature
39
The black dotted
lines are plotted such that the maximum number of
slip transfer
points are
inside the box and the maximum number of
no slip transfer
point
s are outside the box. For the
room temperature sample
,
64% of the
slip tra
nsfer
points are located inside the black dashed
box, and 94% of the
no slip transfer
points are outside the dashed box. For the high
temperature sample
,
91% of the
slip
transf
er
points are located inside the black dashed box,
and 68% of the
no slip
transfer
points are outside the dashed box.
The high temperature data
shows a much larger black dotted box compared to the room temperature data
, and the relative
percentag
es of
the two populations inside and outside the box are reversed
.
slip trans
fer
data are clustered in the lower right
-
hand corner of the graph,
i.e. at
no slip transfer
out, and
there are many points
within the lower right box for the high temp
erature data.
This indicates that the slip transfer
happens more
frequently
residual
Burger
vectors in the high
temperature sample.
40
Figure
4
.
4
:
vs.
the residual Burgers
vector for (a) the room temperature (296K) Ti525 sample
and (b) the high temperature (728K) sample. The cluster
slip transfer
dotted lines, have high m'
parameters and low residual Burgers vectors. The dashed
black lines
represen
t the boundary where the maximum number of
slip transfer
and
no slip transfer
data
are inside and outside of the box respectively.
4.5
Misorientation angle
vs.
the sum of the Schm
id factors
As described in
Chapter 2:
Literature Review
, at high
and high Schmid factors, slip
transfer
is likely to occur.
The sum of the t
(
LG
+SF
RG
)
) is plotte
d against the misorientation angle
in
Figure 4.5
(
a) the room
temperature (296K) data and (b) the high temperature (728K)
data,
where SF
LG
and SF
RG
is the
Schmid factor of the left grain and right grain r
espectively. In the room temperature sample
above the purple lin
e lies the maximum
slip transfer
data (~75%) and below lies the maximum
no slip transfer
data (~82%).
In the
high temperature sample
above the purple solid line lies
the maximum
s
lip transfer
data (~94%) and below lies the maximum
no slip transfer
data
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
b
m'
(a) Room temperature
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
b
m'
(b) High temperature
41
(~56%). Both thresholds
in the high temperature and room temperature sample are increasing
lines, lyin
g between 0.4 and 0.8
LG
+SF
RG
)
no slip transfer
a larger
spread of
LG
+SF
RG
)
slip transfer
LG
+SF
RG
)
closer to 1 for
slip
transfer
between
35° and 50° misorientation that have a
LG
+SF
R
G
)
value below 0.6. This cluster is not
present in the room temperature data.
42
Figure
4
.
5
:
The
LG
+SF
RG
)
parameter
vs.
mi
sorientation angle for (a) the room
temperature (
296K) Ti525 sample, and (b)
the high temperature (728K) sample. High
LG
+SF
RG
)
is noted for
slip transfer
data in both cases. There is a decreasing trend in
slip
transfer
data below 30° misorientatio
no slip transfer
above 30° misorienta
tion angle.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
20
40
60
80
100
LG
+SF
RG
)
Misorientation angle (deg)
75%
82%
(a) Room temperature
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
20
40
60
80
100
LG
+SF
RG
)
Misorientation angle (deg)
94%
56%
43
4.6
Misorientation angle
vs.
Chapter 3
identifies
another factor worth looking at graphically,
, where
is the
residual Burgers
vector.
Figure 4.6
shows
the relationship between mis
orientation angle and
. In
Figure 4.6
(a) the room temperature (296K) sample, above the purple line (which is
constructed t
o have
maximum
slip transfer
points above, and maximum
no slip transfer
points below) is 90% of the
slip transfer
data, and below to which has 60% of the
no slip
tr
ansfer
data.
Figure 4.6
(b), the
high
temperature (
728
K) sample, above the purple solid line is
63% o
f the
slip transfer
data, and below the purple solid line is 84% of the
no
slip transfer
data. There is a decreasing
trend for
slip transfer
data below the 30° misorientation
angle for both the high temperature (728K) data
and the room temperature (295K) data
. The high
temperature data set is observed to have a stronger trend in this region, as the
data is more
no slip transfer
slip transfer
data
have a larger spread but there appe
ars to be a threshold for the
slip transfer
data
(above
= 1
), while the
no slip transfer
data are sca
no slip
transfer
slip transfer
to the high temperature data.
44
Figure
4
.
6
:
vs.
misorientation angle for (a) the room temperature (296K) sample and (b)
the high temperature (728K)
sample. In both samples, a strong decreasing
trend for
slip
transfer
data below 30° misorientation angle
.
0.001
0.01
0.1
1
10
100
0
20
40
60
80
100
Misorientation angle (deg)
(a) Room Temperature
74%
78%
0.001
0.01
0.1
1
10
100
0
20
40
60
80
100
m'/
Misorientation angle (deg)
(b) High Temperature
76%
80%
45
4.7
Slip systems present
Each datum point
presented thus far
has two grains associated with it, each with its own
favored/active slip systems.
Figure 4.7
and
Figure 4.8
represents the same data presented in
Figure 4.6
but with the
slip
systems
computed to be present
distinguished
by color and shape.
In
Figure 4.7
(a
),
the
slip transfer
data for the room temperature (296K) sample, there is a cluster
of prism to prism slip transfer between 15° and 30° misorientation angle is noted. A
cluster of prism to basal slip above 80° misorientat
ion angle is also noted. Basal to basal
favor
ed slip is common at lower misorientation angles. In
Figure 4.7
(b), the
slip transfer
for
the high
temperature
(728K)
sample, there is a cluster of prism
to prism
slip transfer in
grain pairs with misorientation angle less than 45°. There is an observed cluster of prism to
pyramidal slip
transfer
between 20° and 30° misorientation angle. Around 45°
misorientation
angle there is another cluster of
prism to pyramidal. A cluster of basal to pyramidal
between misorientation angles 45
°
to 60° is also observed
.
46
Figure
4
.
7
Slip transfer
vs.
misorientation angle
represented with
computed
slip
systems for each grain pair in (a) the
room
temperature
(296K)
sample and (
a
) the
high
temperature
(728K)
sample. There is a cluster of prism to prism slip in grain pairs with
mi
sorientation angle less than 30° in the high temperature sample and between 15° and 30° in the
room temperature sample.
0.1
1
10
100
0
10
20
30
40
50
60
70
80
90
100
m'/
b
Misorientation angle (deg)
basal
basal + prism
basal + pyr
basal + pyr
prism
prism + pyr
prism + pyr
pyr
(a) Room temperature
0.1
1
10
100
0
10
20
30
40
50
60
70
80
90
100
m'/
b
Misorientation angle (deg)
basal
basal + prism
basal + pyr
basal + pyr
prism
prism + pyr
prism + pyr
pyr
pyr + pyr
pyr
(b) High
t
emperature
47
Figure 4.8
no slip transfer
with colors
and shapes
representin
g
observed
slip systems in the two grains
for (a) the room temperature (
2
96K) sample, and (b) the
high tempera
ture (728K) sample
.
For the room temperature sample
Figure 4.8
(a), there is a
cluster of pyram
idal slip
systems
between 40° and 50° misorientation angle. There are two
clusters of prism + pyramidal between 50° and 60° miso
rientation angle and 75° to 90°
misorientation angle. In the high temperature sample
Figure 4.8
(b), below 25° misorientation
angle, both points are prism to prism slip. There is no other trend observed in the data,
as the data are sporadic above 25° misorientation ang
le.
48
Figure
4
.
8
:
No slip transfer
vs.
misorientation angle represented with observed active
slip systems for each grain pair for (a) the room temperature (296K) tensile tested sample and for
(b) the high temperature (728K) ten
sile tested sample. Few clusters are observed in both data
sets.
0.001
0.01
0.1
1
10
100
0
20
40
60
80
100
Misorientation angle (deg)
basal
basal + prism
basal + pyr
basal + pyr
prism
prism + pyr
prism + pyr
pyr
pyr + pyr
pyr
(a) Room temperature
0.001
0.01
0.1
1
10
100
0
20
40
60
80
100
m'/
b
Misorientation angle (deg)
basal
basal + prism
basal + pyr
basal + pyr
prism
prism + pyr
prism + pyr
pyr
pyr + pyr
pyr
(b) High
t
emperature
49
Figure 4.9
presents statistics comparing prevalent slip systems for
slip transfer
and
no
slip transfer
data in the
(a)
room
tem
perature (
296
K) sample and
(b)
high
temperature
(728K)
samples.
The relative area of the half circle
represents the frequency of that slip system pair.
For
the room temperature
sample, in both the
slip transfer
and
no slip transfer
data sets, basal to
prism has the highest frequency. For the h
igh temperature
slip transfer
data set, prism to prism
and prism to pyramidal are both the most prevalent.
In t
he high temperature
no slip
transfer
data, prism to pyramidal
is most prevalent. As a group, the most prevalent slip
system pair is basal to prism and the least prevalent slip system pair is pyramidal to
pyramidal.
The high temperature sample has a significant increase
in pyramidal slip
transfer data c
ompared to the ro
om temperature sample.
50
Figure
4
.
9
:
R
e
presentation of slip systems prevalent in the high temperature (728K) and room
temperature sample.
The most prevalent slip
system
pair
in the room temperature
(296K)
data is
basal to prism. The slip
system pair that shows up the least in all the data is pyramidal
to
pyramidal.
51
The chapter has provided an extensive
assessment
of
slip transfer in
the high temperature
and
room temperature tensile tested Ti525 alloy. Using
parameters such as the misorientation
angle
(
)
, Schmid
factor, and residual Burgers vector (
)
,
the
slip transfer
and
no
slip
transfer
data
were compared
. Low
, and high
grain pairs are more li
kely to have slip
transfer present. The
parameter
vs.
the misorientation angle shows a strong decreasing trend
for
slip transfer
data below
30° misorientation angle. The trends found in this section have
some differences from r
oom temperature to high
temperature tensile testing
, which
includes the
high temperature sample having a general larger spread of data.
52
Chapter 5:
Discussion
This chapter provides
an
analysis of the results presented in
Chapter 4:
.
This
chapter will
compare the results of the high temperature tensile tested sample, to the room temperature tensile
tested sample. In addition, the results obtained in the Ti525 room temperature tensile (29
6K)
sample and high temperature (728K) sample are espe
cially useful when compared to the results
investigation in an aluminum oligo
-
crystal tensile sample t
ested at room temperature.
Comparison between the two
crystal structures will shed new understanding about what
facilitates slip transfer.
5.1
vs
.
Figure 5.1
(
a
)
and (b)
compare
the
vs
.
residual Burg
ers vector (
) for the room
temperature (296K) sample and high temperature (728K) sample, respectively. The green box
identifies the cluster boundary determined by
Alizadeh between
slip transfer
and
no slip
transfer
for the Aluminum oligo
-
crystal. Ali
zadeh maximized the percentage of
slip transfer
points within the box and percentage of
no slip transfer
points outside the box. Within the box
lies 93% of
slip transfer
data for the Al sample. Outside the box lies
86% of
no slip transfer
data
for
the aluminum sample. A similar activity was completed for the Titanium sample, as
explained in
Chapter
4
: Results
. A comparison of Al maximized boundaries and Titanium
maximized boundaries
in
Table 5.1
.
The aluminum sample has a boundary of lower
and
Figure 5.1
(b)). In
comparing the room tem
perature Ti525
sample and the room temperature aluminum sample
(
Figure 5.1
(a)), the boundaries are very similar indicating their data sets are similar. Thi
s shows
the temperature dependence of the
vs.
factors. Higher
values and lower
values
53
enable slip transfer when the temperature in the material is hotter. In comparing the titanium
room temperature sample to the titanium high temperature sample, more points with lower
values are observed
.
Figure
5
.
1
:
vs.
data for the (a) room temperature (296K) and (b) the high temperature
(728K) tensile tested Ti525 sample.
Green box
es
indicate
comparable
Al oligo
-
crystal
tensile
results
at room t
emperature fo
r both (a) and (b)
.
Table
5
.
1
:
Percentages of
slip transfer
and
no
inside and outside of boundaries,
respectively. RT and HT Ti5252 relationships are flipped with high
percentage of
no slip
transfer
outside of the box in RT and high percentage of
slip transfer
inside the box for HT.
Slip transfer
(Inside)
No
slip transfer
(Outside)
RT Ti525
64%
94%
HT Ti525
91%
68%
RT Al
93%
86%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
b
m'
(b) High temperature
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
b
m'
(a) Room temperature
54
Dislocation climb in titanium
alloys is facilitated by higher temperatures. If dislocations
can climb near the boundary to align themselves with a lower
geometry partner, then slip
transfer is enabled under less favorable conditions. Higher diffusion r
ates
enable
recovery
processes
to take place
near and
within grain boundaries, so that residual Burgers vector debris is
more easily absorbed.
The homologous temperatures of the two materials are different from each other. The
melting
temperature for Ti525 is 1863K (1590
C) and the m
elting temperature for aluminum is
933K (660
C). Room temperature is 0.16
T
m
for Ti525
and
0.32T
m
for aluminum. In addition,
the high temperature Ti525 experiment was performed at 0.39
T
m
(728K) which is equivalent to
364K (90
C) for Al. This fractio
n is closer to that of the room temperature aluminum
homologous
temperature
.
Table 5.2
shows these relations.
Table
5
.
2
: Temperature comparison for Al and Ti
525. RT Al and HT Ti525 have similar
homologous
temperatures.
Homologous
Temperature
Melting Temperature
(Kelvin)
Test Temperature
(Kelvin)
RT Ti525
0.16
T
m
1863
296
RT Al
0.32
T
m
933
296
HT Ti525
0.39
T
m
1863
728
Given that the slip transfer b
ehavior in aluminum and Ti525 are more similar to each
other at room temperature, the effect of alloying elements and/or the much lower CRSS in pure
aluminum may lead to less stress assisted climb forces for a similarly high h
omologous
temperature, such th
at the higher stresses in Ti525 facilitated climb more effectively than in Al.
The
stiffness
normalized
stre
ngth
in Ti525 was
0.006
for the room temperature sample and
0.003
55
for the high temperature sample
while it wa
s
0.00036
in pure Al.
Table 5.3
shows the
normalized
strength and
the values used to obtain the metric. The aluminum compensated strength is
significantly
lower
than the Titanium
alloy
,
due to the
effects of alloying
.
Table
5
.
3
: Compensated Strengths and values used to obtain them. The
pure
aluminum
compensated strength is much
lower
than
the
Titanium
alloy
(
Aluminum 1100
-
O and
Li
2013)
.
Modulus (GPa)
Yield
Strength
(MPa)
Compensated Strength (
Yield
Strength /
RT Ti525
110
660
0.006
HT Ti525
110
330
0.003
RT Al
70
25
0.0003
6
5.2
Misorientation angle
vs.
Figure 5.2
shows
the misorientation angle
vs.
categories
for both the titanium oligo
-
crystal and the aluminum alloy
samples
.
T
he room
temperature (296K) Ti525 tensile sample
(a)
, and the high temperature (728K) Ti525 tensile
sample
(b) are overlaid with t
he shaded areas
represent
ing
the locus of
most
of the
points in the
aluminum polycrystal. Since aluminum has a cubic structure
with a maximum
d
isorientations
of
63°
there is a much larger range of misorientation as well as
value
s for Ti525
slip transfer
points
.
The black box shows the
comparable
area
from
the aluminum
data
. Alizadeh identified a
threshold
of about 20
that best separated the
slip transfer
and
no
slip
transfer
categories
,
which occurred in the mid
dle of the
slip transfer
region (blue region).
The titanium
alloy
shows similar
behavior, but the threshold is not as
distinct
,
as most observations are at
misorientations
larger than
20°. This disparity could
arise from the fewer easy slip systems in
hexagonal crystal structures
that lead to
more heterogeneous stress states in titanium.
Another
possibility is that the aluminum oligo
-
crystal grains have mostly free surfaces while the Ti525
56
sample is a polycryst
al, where only one side of the grain has a
free surface, which makes the
stress state more complex.
Also,
unlike the Ti alloy,
the aluminum
oligo
-
crystal
has
a strong texture, so that most
grains had a more similar stress state
and strain response
.
W
ith
fewer
easy slip systems available
in the
Ti525
, slip may be required on slip systems that do not facilitate slip easily, leading to a
wider
variation in the local stress state, which would lead to more spread in the data, as the
Schmid factor is based
upon the assumption of a uniform uniaxial st
ress. Furthermore, the
no
slip transfer
points are not
present
in the
room
temperature titanium sample below ~30°
misorientation, while
in the aluminum and
high
temperature Ti525 data sets, they are
present
at
misorientation
s as low as
~10°
in Al
and th
e high temperature
Ti525 data. These differences
imply that differences in crystal structure and geometrical limitations need to be considered.
57
Figure
5
.
2
:
Aluminum oligo
-
crystal
slip transfer
(blue
shaded area) and
no slip transfer
(red
shaded area) compared to data of the titanium polycrystal
slip transfer
and
no slip transfer
data for (a) the room temperature (296K) sample and (b) the high temperature (728K) sample.
Titanium
slip transfer
and
no slip transfer
points do not follow as strict of a trend compared
to the aluminum data.
5.3
Misorientation angle
vs.
the sum of
Schmid factors
Figure 5.3
shows comparison between the misorientation
angle
vs.
LG
+SF
RG
)
for
the (a) room temperature Ti525 alloy, (b) high temperature Ti525 alloy. The green solid line
represents the
thresholds for aluminum, where
slip transfer
points
was prevalent above the
green line (for
strongly textured Al with
two different tensile a
xis directions
). The same process
done with the titanium data set as described in
Chapter 4:
Results
. The trends between the
boun
daries of the aluminum and titanium
slip tra
nsfer
data are significantly different, in that a
shallow
positive slope
best separates prevalent
slip
transfer
and
no slip transfer
populations
,
and
the threshold is much
lower threshold than that for Al.
The boundaries for the high
temperature and room temperature samples in the titanium alloy are very similar, but lower for
58
high temperature data. This also indicates that a threshold for
LG
+SF
RG
)
vs.
misorientation
an
gle is heavily dependent on materi
al and crystal structure. Clearly, the geometrical constraints
for slip transfer are much smaller in the hexagonal crystal structure than in Al.
Figure
5
.
3
:
Misorientation angle
vs.
LG
+SF
RG
)
for th
e (a)
room
temperature Ti525 sample
and the (b)
high
temperature Ti525 sample. The aluminum boundaries
represented by the green
line show the difference between the
aluminum data set
which has a steep slope
and the titanium
data set
which slope is flatter
.
5.4
Misorientation angle
vs.
Figure 5.4
presents
the
misorientation
angle
vs.
for
(a)
the
room
temperature
tensile tested sample
and (b) the high temperature sample
.
The shaded areas are approximate
representations for mi
sorientation angle
vs.
for the aluminum oligo
-
crystal. The blue
shaded region for the
aluminum oligo
-
crystal
slip transfer
data and the blue cluster of data
59
f
r
om the titanium alloy below 30° mi
sorientation angle line up well, indicating a strong
co
rrelation between the two data sets
in this range
. The red shaded area representing the
aluminum
no slip transfer
data expands greatly below
= 1
, as do
the red
x markers
representing the titanium
no slip transfer
data. The black rectangle represe
nts the data bounds
for the FCC aluminum data set.
Figure
5
.
4
also shows
that there are less geometrical constraints
for
slip transfer in
the
hexagonal Ti alloy than
the Aluminum FCC material. In comparing
the
296K to 728K Ti525 data, the boundary lines separating maximum
slip transfer
and
no slip
transfer
are
ne
arly the same, but the higher temperature has a slightly smaller slope
indicat
ing
that temperature does not have a great effect on
b
vs.
the misorientation angle.
60
Figure
5
.
4
:
Misorientation
vs.
for (a) the room temperature (296K)
tensile test sample and
(b) t
he high temperature (728K) sample. The red shaded area repres
ents the aluminum oligo
-
crystal
no slip transfer
data set. The blue shaded area represents the aluminum oligo
-
crystal
slip transfer
data set and line up with the
slip transfer
points in the t
itanium data sets.
5.5
Slip system analysis
Figure 4.6
shows the favored slip systems for grain pairs that have
slip transfer
in (a)
the room temperature (296K) Ti525 sample and in (b) the high temperature (728K) Ti525
sample. The clusters of slip systems described in the section c
ould be attributed to geometric
conditions that the hexagonal crystal system
imposes on the slip tr
ansfer process
. At
lower
misorientation ranges,
slip transfer between all
slip systems are likely to
be favored
. The cluster
of basal to pyramidal slip tr
ansfer between misorientation angles 45° and 60° noted in the
slip transfer
high temperature (728
K) sample can be explained by the angle between the two
61
slip planes. At a misorientation of 45° to 60° the basal plane of one crystal
is more likely to be
al
ign
ed
with the pyramidal plane of the second crystal.
Figure 4.7
shows the slip systems in grain pairs that
slip transfer
for (a) the
room temperature, and (b) the high temperature
Ti525 sample.
T
here are
no
strong trends
observed
in either graph
. This
is consistent with that there should be many
no slip transfer
conditions between mismatched slip systems of any family
,
su
ch that
non
-
favored slip systems
is ob
served
.
Figure 4.8
presents the statistics around the slip systems observed in the
slip transfer
data and
no slip transfer
data. The main
slip systems activated are basal and prism
, consistent
with
the fact that these are the most easily activated systems
. This is
consis
tent with
literature
that examines the relative activity of slip systems in
Li
, 2013
).
62
5.6
Comparison
between measured weakly textured grain orientations
to random
populations
5000 orientations pairs for a hexagonal crystal structure were randomly generated. In each
pair
the
probable slip system
s
for uniaxial tension was determined for each grain. For each grain,
the Schmid factors w
ere
calculated for each slip system
and i
n each grain pair,
was
calculated. Two different filters were applied
to 5000 randomly generated orientation pairs to
obtain two populations. The first (Ran 0.9) collects
slip system
pairs with an
value above
the
threshold of 0.9
and
grain pairs that have
Schmid factors
of
a mini
mum of 0.3 for each grain.
The second data
d a Schmid factor
greater than 0.25, yielding a much larger set of nearly 25,000 data.
Figure 5.5
s
hows the
corresponding cumulative
misorientation
fraction
for the
experimental and randomly generated data, separated by slip system pair type.
In basal to basal
slip,
a
ll t
he Ran 0.9 data is below ~20° while all the Ran 0.8 data lies below ~35°. The
green
arrow indicates which direction the Ran 0.9 data set lies in comparison to the Ran 0.8 data set. It
is known that Ran 0.9 is more restrictive than Ran 0.8.
As i
t is hyp
othe
sized that the room
temperature Ti525 sample (Exp
-
RT) is more restrictive than the high temperature Ti525 sample
(Exp
-
HT)
, the hypothesis is supported by the room temperature data being mostly to the left
(lower misorientations)
of the high
temperature
obs
ervations, but the two cross at the highest
misorientations
. The arrow is
green
because the experimental data does follow the same trend
(the more restrictive data on the
left
) compared to the randomly generated data
until about 60° in
which the Exp
-
RT an
d Exp
-
HT switch
.
In
prism to prism slip, the experimental data sets are
much closer to each other, and to the simulation, but
contrary to the hypothesis,
the higher
temperature data appear to be m
ore restr
icted than lower temperature data (for most of the
misorientation range
)
. In
pyramidal to pyramidal slip, the high temperature Ti525, Ran
63
0.9, a
nd Ran 0.8 data all follow a same general trend
, and the small number of room temperature
observations makes it im
possible to compare with the high temperature data, but the simulations
show a similar
trend
. In
pyramidal to
pyramidal slip, there are two observed peaks
in the randomly generated data, one around 25° misorientation angle, and the other around 75°
misorientation angle. Interestingly
,
the experimental
data
shows one peak between
the randomly
generated
peaks
,
at around 45° misorientation angle. The purple arrow depicts this discrepancy.
For the mixed slip system slip transfer,
basal to prism
slip
shows
the more restrictive
of
the random data sets (Ran 0.9)
at highest misorientations (
on the right
)
, and the hypothesized
more restricted of the experimental data (Exp
-
RT)
is
also
to
the right
of the higher temperature
data
. The green arrow depicts this
, but the spread of slip tr
ansfer misorientations of these two slip
systems is much greater in the experimental data
.
In b
asal to pyramidal slip,
all data sets
have a similar trend.
In b
asal to pyramidal slip,
below 50°
the more restrictive of the
randomly generated data is on the right, while the hypothesized more restrictive experimental
data set (Exp
-
RT) is on the left. Above 50°, the
more restrictive randomly generated data set is
now on the left, while the room temperature experimental data is on the right. This is depicted by
the red arrows on the graph
, indicating trends opposing the hypothesis
.
In
prism to pyramidal
slip,
Ran 0.9 to the left of Ran 0.8, and Exp
-
RT to the right of Exp
-
HT,
again contrary to the
hypothesis
.
In p
rism to pyramidal slip,
the more restrictive Ran 0.9 to the right of Ran 0.8,
and the more restrictive Exp
-
RT to the left of Exp
-
HT (except for between 50° and 70°).
In
pyramidal to pyramidal slip
there
is
a relationship opposite of expected when
concerning the restrictiveness of the data sets.
64
Figure
5
.
5
: Misorientation angle
cumulative percentages for Ti525 and randomly generated data
sets
separated b
y type of slip system observed in the grains
. Pyramidal slip is behaving counter
-
intuitively with respect t
o
more restrictive conditions.
It is important to note that the experim
ental data does have a 3 times random texture,
while the randomly generated d
ata has a true to random texture. This difference could account
for the behavior contrary to the
hypothesis in
Figure 5.5
.
In
general, pyramidal slip
behav
es
counter
-
intuitively with respect to more restrictive conditions.
Figure 5.6
p
resents the overall Ti525 statistics for the type of slip
system observed in the
data
compared to
the
two randomly generated data sets
.
It is hypothesized that the room
temperature data would have more restrictive slip transfer conditions than the high temperature
data set due
to the higher ease of movement of dislocations by climb in the high temperature
exp
erimental data set.
B
asal
-
basal
slip and basal
-
prism slip
transfer have a higher number of
observations at room temperature than elevated temperature, which could be related to the ease
65
of t
hese slip systems. The variance in the relative activity of different combinations
of slip
systems may reflect different temperature dependence on the critical resolved shear stress for the
four families of slip systems.
The randomly generated data sets have a larger percentage shown
in the pyramidal to pyramidal and pyramidal to pyramidal categories, but
there are few experimental observations of these slip systems. The experimental data shows
a
large percentage with basal to prism slip while the random data sets have low percentages for
this category.
Figure
5
.
6
: Overa
l
l Ti525 statistics in comparison to a random population
. Random population
data sets have a high frequency for pyramidal to pyramidal where experimental
results are low for the sam
e category.
Figure 5.7
show
s
the same data, but the frequencies are repre
sented as half
-
circles. The
larger the area of the half
-
circle, th
e
higher
the frequency
for that slip system pair.
In comparing
the two data sets, the randoml
y
generated data favors the lower right
-
hand corner, while the
experimental data favors the upper left
-
hand corner. The randomly generated data
hav
e
a large
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
Frequency
(a)
66
frequency of
slip system pairs including a pyramidal plane,
and no distinction about the critical
resolved shear stress,
which
could
account for the large values for pyramidal
slip systems
.
Figure
5
.
7
:
Relative frequency of data represented as half
-
circles. Larger area half
-
circle
represents a higher frequency. (a) Comparis
on of experimental Ti525 at
room temperature and
high temperature. (b) Comparison of randomly generated data sets
67
Another analysis was done with the randomly generated data, to find the most likely slip
system
pair
that would be present in the grain pairs.
To determine the most likely slip system
pair,
a
factor was calculated as follows:
(5)
The slip system pair with the highest factor for each orientation pair was deemed the slip
system pair to most likely be present and facilitate slip.
Figure 5.8
(a)
shows the frequency
of each slip
system pair represented by relative area of a
circle.
There is a
larger frequency
associated with the right
side
of the chart,
especially
the
pyramidal slip system.
The multiplicity of the pyramidal plane give
s a
significant
advantage
to slip
transfer
possibilities
.
To
compensate for this, the Schmid factor for the
pyramidal slip system was halved
, which
would simulate a higher CRSS for this specific
slip plane
, and
shown in
Figu
re 5.8
(b)
.
Reducing the likelihood for slip
caused the
generated data
to have
a similar distribution to
the
experimental data, with basal to prism
, and
prism to pyramidal
sli
p system pairs having a large frequenc
y
, and
a smaller frequency
in the
lower
right
-
hand
corner of the chart
in both sets
.
This shows that the
factor in Equation 5
captures the overall phenomena that is observed experimentally.
68
Figure
5
.
8
:
Randomly generated
orientation pairs with the most likely slip systems present
that facilitates slip across the grain boundaries.
(a) Unmodified Schmid factors. (b) Schmid
factor for pyramidal is halved.
A h
igh f
requency is present in
slip system pairs with
pyramidal slip systems
in both cases
.
69
Chapter 6:
Summary and conclusions
6.1
Summary
This
thesis
describes the method and results of a study of Ti
-
5Al
-
2.5Sn defor
mation
behavior
focusing on slip transfer across grain boundaries
at two different temperatures (296K
and 728K).
The study investigates
across
grain boundaries and builds
upon
work done by Hongmei Li.
The re
sults
are
compared to a similar study that was done with
pure aluminum
by R. Alizadeh.
In comparing the titanium results to an aluminum sample,
some
trends are similar, such as favorable sli
p transfer of the same slip system family at low
misorientations.
D
etails
associated with
geometrical constraints of
more slip systems in the
hexagonal crystal structure
, but few facile ones,
versus face centered cubic crystal structure in
aluminum.
Thes
e results
show that slip transfer is much more commonly accomplis
hed in the
Ti
-
5Al
-
2.5Sn
deformation in and room and
to a greater extent at
high temperatures,
which
provides a basis for installing slip transfer criteria into
CPFE modeling of the alloy.
6.2
Co
nclusions
1.
Slip transfer
data is
more prevalent
than
no slip tr
ansfer
data
at misorientation angles
< 30°. At
these
low misorientation angles, slip traces are
categorized as
slip transfer
more than being categorized as
no slip transfer
at both
296K and 728K
.
2.
There is a
decreasing trend of
slip transfer
trace
s with increasing misorientation
below
20° for both Al and Ti525.
Many
slip transfer
cases
occur
at high
misorientations in Ti
tanium
.
3.
Literature suggests that h
igh
and low
enable
s
slip transfer.
This hypothesis was
found true in Ti525.
R
oom temperature
Ti525 and
room temperature
Al
uminum
show
a
vs.
70
4.
Considering
both
the Schmid factor
vs.
, and
vs.
, geometrical constraints for
slip transf
er
in
hexagonal crystal
structure
are smaller than
FCC.
5.
Misorientation angles that favor a specific slip system pair (angles between slip systems
are small) makes slip transfer easier and more prevalent.
6.
In slip s
ystem pairs where
ther
e is no slip transfer
, slip is not facilitated easily
as the
alignment of slip systems is low
(angles between slip systems are generally large).
7.
The b
asal + prism slip system
pair is most common in
Ti525.
This r
elates to
the low
CRSS of basal
and prism slip systems in Ti525
.
8.
Every type of slip system has a different kind of sensitivity with misorientation angle
regarding the experimental data
. This is partly due to the geometry of the slip systems,
and partly unexplained.
9.
Pyramidal slip
behav
es
counter
-
intuitively with respect to
high temperature and room
temperature restrictiveness.
10.
Using a random data set, t
he factor
and penalizing the
likelihood
for
slip by a factor of two
is able to capture the statistical trends similar
ly
to the
observed distribution of slip transfer observations.
71
6.3
Future
work suggestions
1.
A random generation o
f orientation pairs for the FCC crystal structure
to observe if
statistical trends are similar to that found in the Aluminum oligo
-
crystal experi
me
nts.
2.
Investigate grain boundary sliding in correlation with the residual Burges vector between
two grains an
d the associate
d
slip transfer
in the high temper
ature Ti525 data
.
An atomic
force microscope (
AFM
)
or
a profiliometer
could be used to identify the ledge h
eight at
the grain boundary.
3.
A similar investigation
with a
different deformation m
ode
, i.e.
creep o
r compression
testing to further the accuracy of CPFE modeling
by
creating rules around
slip transfer
across grain boundaries
.
4.
The work of Line et al. could be analyzed
with similar methods to determine the
slip
transfer
relationships between high temper
ature (
464K) and the room temperature (296K)
aluminum oligo
-
crystal
.
72
APPENDICIES
73
APPENDIX A: Matlab hexagonal orientation code (written by Thomas R. Bieler)
% HEXagonal orientation analysis by T.R. Bieler, November 22, 2010, updated 6
Nov 2018.
% No guarantees that it is 100% correct, but it seems so. If you use this for your work,
% it would be kind and ethical to acknowledge its use in published work.
clear; clc;
% dbstop if error;
% Imported grains file requires Bunge Euler an
gles in colu
mns 3:5, (x,y can be put in 1:2, for example)
% in as many rows as data set. Be wary about hidden assumptions;
% plotting uses x
--
> down, y right, so Euler angles are consistent with TSL maps,
a1=86.764;
b1=66.765;
c1=259.31;
%
3,4,5 are Eu
ler angles
Ang(1,:) = [1 0 a1+180 b1 c1];
ghkl = [
% for C3
-
1
-
1 2 8
0
-
1 1 7
1
-
2 1 8
0
-
1 1 4];
dum = size(ghkl); ghklrows = dum(1,1);
dum = size(Ang); Angrows = dum(1,1);
% Stress tensor is defined; put the one you
want in last positino, or make a new one.
ststens = [1 0 0 ; 0 0 0 ; 0 0 0]; LC =
'X'
;
% tension in X %
ststens = [0 0 0 ; 0 0 0 ; 0 0 1]; LC =
'Z'
;
% tension in Z %
ststens = [0
-
1 0 ;
-
1 0 0 ; 0
0 0]; LC =
'
-
XY'
;
% Shear in XY plane %
ststens = [0 1 0 ; 1 0 0 ; 0 0 0]; LC =
'+XY'
;
% Shear in XY plane %
ststens = [0 0 0 ; 0 0
-
1 ; 0
-
1 0]; LC =
'+XZ'
;
% Shear in XZ plane %
ststens = [1 0 0 ; 0 1 0 ; 0 0 0]; LC =
'BiXY'
;
% biaxial tension in X
-
Y %
ststens = [
-
1 0 0 ; 0 1 0
; 0 0 0]; LC =
'PSC
-
XY'
;
% Plane Strain Compression in XY plane %
ststens = [0 0 0 ; 0 1 0 ; 0 0 0]; LC =
'Y'
;
% tension in Y %
theta = 0;
Rst = [cosd(theta)
-
sind(theta) 0; sind(theta) cosd(theta) 0; 0 0 1];
% rotates stress about Z axis
%Rst
= [cosd(theta) 0
-
sind(theta); 0 1 0; sind(theta) 0 cosd(theta)]; % rotates stress about Y axis
%Rst = [1 0 0; 0 cosd(theta)
-
sind(theta); 0 sind(th
eta) cosd(theta)]; % rotates stress about X axis
ststens_R = Rst*ststens*Rst';
% This rotates a desired
stress tensor in the lab coord sys.
str2 = ststens_R*ststens_R';
ststens_mag = (str2(1,1)+str2(2,2)+str2(3,3))^.5;
ststens_n = ststens_R/ststens_mag;
% normalized stress tensor to get generalized Schmid factor
%
% Inverse pole figures can be drawn by s
etting values of ipfd; ipfd is
% inverse pole figure direction,
% iCTE is CTE direction, iEd is E direction, not checked recently, so bugs may exis
t.
stereo = 1;
% plots stereographic projection, otherwise direct projection
ipfd = 2;
%inverse pole fig
ure direction (x,y,z, = 1,2,3) will plot points from a group
ihex = 1;
% inverse pole figure for hexagonal crystal
iEd = ipfd;
% plot magnitude of E
in this direction as red (high)
-
blue (low)
% CTEd = ipfd; % plot magnitude of CTE in this direction as
gray scale within symbol
nslphex=69;
% Slip sys: b p paa pya 1c+a 2c+a T1 T2 C2 C1 3c+a in first row, set 0 to skip, 1 to plot
sschoice = [1 1 0 1 1 0 1 1 1 1 0
1 4 7 10 16 28 34 40 46 52 58
3 6 9 15 27 33
39 55 51 57 69 ];
sscol = 1;
for
isc = 1:1:nslphex
pssf(isc) = sschoice(1,sscol);
if
isc == sschoice(3,sscol)
sscol = sscol + 1;
end
end
%
% This section allows the point of view of the
unit cell to be changed.
Rotateview =
[1 0 0 ; 0 1 0 ; 0 0 1];
iRotatev = 0;
% Provide NUMBER of rotations
74
% rotation of observer from normal TSL point of view to another point of view (crystal stays put)
% This right hand rotation matrix has a 4th row
which has in columns 2,3, the angle a
nd axis of the rotation.
%Rotation(4,2:3,1) = [5,3]; % rotation to move axis to a tilted direction in the X
-
Y plane
%Rotation(4,2:3,1) = [
-
45,1]; % about the X axis to look through detector for DAXM
%Rotation(4,2
:3,1) = [180,2]; % rotate viewpoint
about vertical (x) axis + to view from above/right,
-
to view from above/left
Rotation(4,2:3,1) = [90,2];
% +90 to view from below,
-
90 to view from above
% Build the Euler angle rotation matrix starting with identit
y
if
iRotatev ~= 0
LC = [
'! rot !
'
LC];
for
i = 1:1:iRotatev
rang = Rotation(4,2,i);
if
Rotation(4,3,i) == 3
Rotation(1:3,1:3,i)=[cosd(rang),sind(rang),0;
-
sind(rang),cosd(rang),0;0,0,1];
elseif
Rotation(4,3,i) == 2
Rota
tion(1:3,1:3,i)=[cosd(rang),0,sind(rang);0,1,0;
-
sind(rang),0,cosd(rang)];
elseif
Rotation(4,3,i) == 1
Rotation(1:3
,1:3,i)=[1,0,0;0,cosd(rang),sind(rang);0,
-
sind(rang),cosd(rang)];
end
Rotateview = Rotateview*Rotation(1:3,1:3,i);
end
end
%
% Below, six points define plane, start point is start of Burgers vector b, 4th is end of b
% 1st to 2nd or 2nd to 3r
d or 3rd to 4th cross product identifines plane normal
c_a=1.59;
% 1.587 this is the last place for user input in this cell
...
slpsys = cell(4,nslphex);
O = [ 0 0 0 0];
% (I) (H)
A = [ 2
-
1
-
1 0]/3;
% C
------
-
B
B = [ 1 1
-
2 0]/3;
% /
\
/
\
C = [
-
1 2
-
1 0]/3;
% / a2 /
\
D = [
-
2 1 1 0]/3;
% /
\
/
\
E = [
-
1
-
1 2 0]/3;
% (J)D
------
O(P)
-
a1
-
> A(G)
F = [ 1
-
2 1 0]/3;
%
\
/
\
/
P = [ 0 0 0 1];
%
\
a3
\
/
G = [ 2
-
1
-
1 3]/3;
%
\
/
\
/
H = [ 1 1
-
2 3]/3;
% E
--------
F
I
= [
-
1 2
-
1 3]/3;
% (K) (L)
J = [
-
2 1 1 3]/3;
%
K = [
-
1
-
1 2 3]/3;
% where a1 = OA a2 = OC
a3 = OE
L = [ 1
-
2 1 3]/3;
% DA || a1, FC || a2, BE || a3
% Slip system definitions set as of 1 Nov 2019 to be consistent with DAMASK
%
Hex plane direction 1st and 4th point is Burgers vector
% basal
-
glide:&Be Mg Re Ti; Re
slpsys{
2,1} = [0 0 0 1; 2
-
1
-
1 0; D ; E ; F ; A ; B ; C ];
slpsys{2,2} = [0 0 0 1;
-
1 2
-
1 0; F ; A ; B ; C ; D ; E ];
slpsys{2,3} = [0 0 0 1;
-
1
-
1 2 0; B ; C ; D ; E ; F ; A ];
ibas = 1;
fbas = 3;
% prism
-
glide:Ti Zr RE; Be Re Mg
slpsys{2,4} = [ 0 1
-
1 0; 2
-
1
-
1 0; E ; E ; E ; F ; L ; K ];
slpsys{2,5} = [
-
1 0 1 0;
-
1 2
-
1 0; A ; A ; A ; B ; H ; G ];
slpsys{2,6} = [ 1
-
1 0 0;
-
1
-
1
2 0; C ; C ; C ; D ; J ; I ];
iprs = 4;
fprs = 6;
% prism
slpsys{2,7} = [ 2
-
1
-
1 0; 0
1
-
1 0; F; F; F; B ; H ; L ];
slpsys{2,8} = [
-
1 2
-
1 0;
-
1 0 1 0; B; B; B; D ; J ; H ];
slpsys{2,9} = [
-
1
-
1 2 0; 1
-
1 0 0; D; D; D; F ; L ; J ];
i2prs = 7
;
f2prs = 9;
% pyramidal
-
glide
--
**
--
CORRECTED
--
**
--
slpsys{2,10} = [ 1 0
-
1 1;
-
1 2
-
1 0; E ; E ; E ; D ; I ; L ];
slpsys{2,11} = [ 0 1
-
1 1;
-
2 1 1 0; F ; F ; F ; E ; J ; G ];
slpsys{2,12} = [
-
1 1 0 1;
-
1
-
1 2 0; A ; A ; A ; F ; K ; H ];
s
lpsys{2,13} = [
-
1 0 1 1; 1
-
2 1 0; B ; B ; B ; A ; L ; I ];
slpsys{2,14} = [ 0
-
1 1 1; 2
-
1
-
1 0; C ; C ; C ; B ; G ; J ];
slpsys{2,15} = [ 1
-
1 0 1; 1 1
-
2 0; D ; D ; D ; C ; H ; K ];
75
ipyra = 10;
fpyra = 15;
% pyramidal
-
glide:; all?
slpsys
{2,16} = [ 1 0
-
1 1;
-
2 1 1 3; A ; B ; I ; P ; L ; A ];
slpsys{2,17} = [ 1 0
-
1 1;
-
1
-
1
2 3; B ; B ; I ; P ; L ; A ];
slpsys{2,18} = [ 0 1
-
1 1;
-
1
-
1 2 3; B ; C ; J ; P ; G ; B ];
slpsys{2,19} = [ 0 1
-
1 1; 1
-
2 1 3; C ; C ; J ; P ; G ; B ];
sl
psys{2,20} = [
-
1 1 0 1; 1
-
2 1 3; C ; D ; K ; P ; H ; C ];
slpsys{2,21} = [
-
1 1 0 1; 2
-
1
-
1 3; D ; D ; K ; P ; H ; C ];
slpsys{2,22} = [
-
1 0 1 1; 2
-
1
-
1 3; D ; E ; L ; P ; I ; D ];
slpsys{2,23} = [
-
1 0 1 1; 1 1
-
2 3; E ; E ; L ; P ; I ; D
];
slpsys{2,24} = [ 0
-
1 1 1; 1 1
-
2 3; E ; F ; G ; P ; J ; E ];
slpsys{2,25} = [ 0
-
1 1
1;
-
1 2
-
1 3; F ; F ; G ; P ; J ; E ];
slpsys{2,26} = [ 1
-
1 0 1;
-
1 2
-
1 3; F ; A ; H ; P ; K ; F ];
slpsys{2,27} = [ 1
-
1 0 1;
-
2 1 1 3; A ; A ; H ; P ; K
; F ];
ipyrc = 16;
fpyrc = 27;
% pyramidal
-
2nd order glide
slpsys{2,28} = [ 1 1
-
2 2
;
-
1
-
1 2 3; (O+B)/2 ; C ; J ; (P+K)/2 ; L ; A];
slpsys{2,29} = [
-
1 2
-
1 2; 1
-
2 1 3; (O+C)/2 ; D ; K ; (P+L)/2 ; G ; B];
slpsys{2,30} = [
-
2 1 1 2; 2
-
1
-
1 3; (O+D)/2 ; E ; L ; (P+G)/2 ; H ; C];
slpsys{2,31} = [
-
1
-
1 2 2; 1 1
-
2 3; (O+E)/2 ; F
; G ; (P+H)/2 ; I ; D];
slpsys{2,32} = [ 1
-
2 1 2;
-
1 2
-
1 3; (O+F)/2 ; A ; H ; (P+I)/2 ; J ; E];
slpsys{2,33} = [ 2
-
1
-
1 2;
-
2 1 1 3; (O+A)/2 ; B ; I ; (P
+J)/2 ; K ; F];
i2pyrc = 28;
f2pyrc = 33;
% *** Twin directions are opposite in
Christian and Mahajan, and are not correcte to to be consistent with them
% FROM Kocks SXHEX plane direction 1st and 4th point is Burgers vector, order of C1 differs fro
m Kock's file
% {1012}<1011> T1 twins 0.17;
-
1.3 twins: all Twin Vector must g
o in the
% sense of shear, opposite C&M sense.
slpsys{2,34} = [ 1 0
-
1 2;
-
1 0 1 1; A ; B ; J ; K ; K ; K ];
slpsys{2,35} = [ 0 1
-
1 2; 0
-
1 1 1; B ; C ; K ; L ; L ; L
];
slpsys{2,36} = [
-
1 1 0 2; 1
-
1 0 1; C ; D ; L ; G ; G ; G ];
slpsys{2,3
7} = [
-
1 0 1 2; 1 0
-
1 1; D ; E ; G ; H ; H ; H ];
slpsys{2,38} = [ 0
-
1 1 2; 0 1
-
1 1; E ; F ; H ; I ; I ; I ];
slpsys{2,39} = [ 1
-
1 0 2;
-
1 1 0 1; F ; A ; I ; J
; J ; J ];
iT1 = 34;
fT1 = 39;
% {2111}<2116> T2 twins: 0.63;
-
0.4; Ti Zr Re
RE]; Also does not follow C&M definition for shear direction
slpsys{2,40} = [ 1 1
-
2 1;
-
1
-
1 2 6; (O+B)/2 ; C ; (J+I)/2 ; P ; (L+G)/2 ; A];
slpsys{2,41} = [
-
1 2
-
1 1; 1
-
2 1 6; (O+C)/2 ; D ; (K+J)/2 ; P ; (G+H)/2 ; B];
slpsys{2,42} = [
-
2 1 1 1;
2
-
1
-
1 6; (O+D)/2 ; E ; (L+K)/2 ; P ; (H+I)/2 ; C];
slpsys{2,43} = [
-
1
-
1 2 1; 1 1
-
2 6; (O+E)/2 ; F ; (G+L)/2 ; P ; (I+J)/2 ; D];
slpsys{2,44} = [ 1
-
2 1 1;
-
1 2
-
1 6
; (O+F)/2 ; A ; (H+G)/2 ; P ; (J+K)/2 ; E];
slpsys{2,45} = [ 2
-
1
-
1 1;
-
2 1 1 6; (O+A)/2 ; B ; (I+H)/2 ; P ; (K+L)/2 ; F];
iT2 = 40;
fT2 = 45;
% {1011}<101
-
2> C1 twins: 0.10; 1.1; Mg; Zr Ti]; agrees with C&M
slp
sys{2,46} = [ 1 0
-
1 1; 1 0
-
1
-
2;
P ; L ; A ; (A+B)/2 ; B ; I ];
slpsys{2,47} = [ 0 1
-
1 1; 0 1
-
1
-
2; P ; G ; B ; (B+C)/2 ; C ; J ];
slpsys{2,48} = [
-
1 1 0 1;
-
1 1 0
-
2; P ; H ; C ; (C+D)/2 ; D ; K ];
slpsys{2,49} = [
-
1 0 1 1;
-
1 0 1
-
2;
P ; I ; D ; (D+E)/2 ; E ; L ];
slpsy
s{2,50} = [ 0
-
1 1 1; 0
-
1 1
-
2; P ; J ; E ; (E+F)/2 ; F ; G ];
slpsys{2,51} = [ 1
-
1 0 1; 1
-
1 0
-
2; P ; K ; F ; (F+A)/2 ; A ; H ];
iC1 = 46;
fC1 = 51;
% {2112}<211
-
3> C2 twins:; 0.22; 1.2 Ti Zr Re];
agrees with C&M
slpsys{2,52} = [ 1 1
-
2 2; 1
1
-
2
-
3; (K+P)/2 ; L ; A ; (A+C)/2 ; C ; J];
slpsys{2,53} = [
-
1 2
-
1 2;
-
1 2
-
1
-
3; (L+P)/2 ; G ; B ; (B+D)/2 ; D ; K];
slpsys{2,54} = [
-
2 1 1 2;
-
2 1 1
-
3; (G+P)/2 ; H ; C ; (C+E)/2 ; E ; L];
slpsys
{2,55} = [
-
1
-
1 2 2;
-
1
-
1 2
-
3; (H+P)/2 ; I
; D ; (D+F)/2 ; F ; G];
slpsys{2,56} = [ 1
-
2 1 2; 1
-
2 1
-
3; (I+P)/2 ; J ; E ; (E+A)/2 ; A ; H];
slpsys{2,57} = [ 2
-
1
-
1 2; 2
-
1
-
1
-
3; (J+P)/2 ; K ; F ; (F+B)/2 ; B ; I];
iC2 = 52;
fC2 = 57;
%
-
3rd order glide;
slpsys{2,58} = [ 2
-
1
-
1
1;
-
1 2
-
1 3; F ; (K+L)/2 ; P ; P ; (I+H)/2 ; B ];
slpsys{2,59} = [ 2
-
1
-
1 1;
-
1
-
1 2 3; B ; (I+H)/2 ; P ; P ; (K+L)/2 ; F ];
slpsys{2,60} = [ 1 1
-
2 1;
-
2 1 1 3; A ; (L+G)/2 ; P ; P ; (J+I)/2 ; C ];
76
sl
psys{2,61} = [ 1 1
-
2 1; 1
-
2 1 3; C ; (J+I
)/2 ; P ; P ; (L+G)/2 ; A ];
slpsys{2,62} = [
-
1 2
-
1 1;
-
1
-
1 2 3; B ; (G+H)/2 ; P ; P ; (K+J)/2 ; D ];
slpsys{2,63} = [
-
1 2
-
1 1; 2
-
1
-
1 3; D ; (K+J)/2 ; P ; P ; (G+H)/2 ; B ];
slpsys{2,64} = [
-
2 1 1
1; 1
-
2 1 3; C ; (H+I)/2 ; P ; P ; (L+K)/2
; E ];
slpsys{2,65} = [
-
2 1 1 1; 1 1
-
2 3; E ; (L+K)/2 ; P ; P ; (H+I)/2 ; C ];
slpsys{2,66} = [
-
1
-
1 2 1; 2
-
1
-
1 3; D ; (I+J)/2 ; P ; P ; (G+L)/2 ; F ];
slpsys{2,67} = [
-
1
-
1 2 1;
-
1 2
-
1 3; F ; (G+L)/2 ; P ; P ; (I+J)/2 ; D ];
slpsys{2,68} =
[ 1
-
2 1 1; 1 1
-
2 3; E ; (J+K)/2 ; P ; P ; (H+G)/2 ; A ];
slpsys{2,69} = [ 1
-
2
1 1;
-
2 1 1 3; A ; (H+G)/2 ; P ; P ; (J+K)/2 ; E ];
i3pyrc = 58;
f3pyrc = 69;
% cell 3 = Cartesian slip system unit vectors, cell 4 = Schmid matrix %
for
gg = 1:1:g
hklrows
ghklC(gg,:) = [ghkl(gg,1), (ghkl(gg,1)+2*ghkl(gg,2))/sqrt(3), ghkl(gg,4)/c_a];
%plane normal in cartesian
unit_ghklC(gg,:) = ghklC(gg,:)/norm(ghklC(gg,:));
end
for
isc=1:1:nslphex;
% isc is slip system counter
n =[slpsys{2,isc}(1,1)
, (slpsy
s{2,isc}(1,1)+2*slpsys{2,isc}(1,2))/sqrt(3), slpsys{2,isc}(1,4)/c_a];
%plane normal in cartesian
m =[3*slpsys{2,isc}(2,1)/2, (slpsys{2,isc}(2,1)+2*slpsys{2,isc}(2,2))*sqrt(3)/2, slpsys{2,isc}(2,4)*c_a];
% slip direction in cartesian
p1=[3*sl
psys{2,i
sc}(3,1)/2, (slpsys{2,isc}(3,1)+2*slpsys{2,isc}(3,2))*sqrt(3)/2, slpsys{2,isc}(3,4)*c_a];
% point 1
p2=[3*slpsys{2,isc}(4,1)/2, (slpsys{2,isc}(4,1)+2*slpsys{2,isc}(4,2))*sqrt(3)/2, slpsys{2,isc}(4,4)*c_a];
% point 2
p3=[3*slpsys{2,isc}(5,1)
/2, (slp
sys{2,isc}(5,1)+2*slpsys{2,isc}(5,2))*sqrt(3)/2, slpsys{2,isc}(5,4)*c_a];
% point 3
p4=[3*slpsys{2,isc}(6,1)/2, (slpsys{2,isc}(6,1)+2*slpsys{2,isc}(6,2))*sqrt(3)/2, slpsys{2,isc}(6,4)*c_a];
% point 4
p5=[3*slpsys{2,isc}(7,1)/2, (slpsys{2,is
c}(7,1)+
2*slpsys{2,isc}(7,2))*sqrt(3)/2, slpsys{2,isc}(7,4)*c_a];
% point 5
p6=[3*slpsys{2,isc}(8,1)/2, (slpsys{2,isc}(8,1)+2*slpsys{2,isc}(8,2))*sqrt(3)/2, slpsys{2,isc}(8,4)*c_a];
% point 6
mag_m=(m(1,1)^2+m(1,2)^2+m(1,3)^2)^0.5;
mag_n=(n(1,
1)^2+n(1
,2)^2+n(1,3)^2)^0.5;
unit_m = m/mag_m;
unit_n = n/mag_n;
dot0(isc,1) = unit_m*unit_n';
slpsys{1,isc} = isc;
slpsys{3,isc} = [unit_n;unit_m;p1;p2;p3;p4;p5;p6];
% normal in first row, direction in next row, points in next 6 rows
slpsys{4,isc} = 0.5*(unit_m'*unit_n + unit_n'*unit_m);
% Schmid matrix
unit_tau(isc,:) = [isc, cross(unit_m, unit_n)];
% edge dislocation line direction tau unit vectors are generated
% if linedir(isc,2)<0
% linedir(isc,:) =
-
1.*lin
e
dir(isc,:);
% end
end
sortlinedir = sortrows(unit_tau,[
-
2,
-
3,
-
4]);
Schm_f_ss = zeros(nslphex,11,Angrows);
Schm_labvecA = zeros(nslphex+1,33);
sortmv = zeros(nslphex+1,33,Angrows);
xic = zeros(nslphex,3,ghklrows);
xicg =
zeros(nslphex,3,ghklrows);
xi
rg = zeros(nslphex,3,ghklrows);
xig = zeros(nslphex,3,ghklrows);
for
gg = 1:1:ghklrows
for
isc=1:1:nslphex;
xic(isc,1:3,gg) = cross(unit_tau(isc,2:4),unit_ghklC(gg,:));
% Streak direction vector in crystal frame,
not normalized!
% xi(i
sc,1:3,isc) = xi(isc,1:3,gg) / norm(xi(isc,1:3,gg));
end
end
% code to set up inverse pole figure and labeling, probably has some inconsistencies
sij = [0.9581
-
0.4623
-
0.1893 0.698 2.1413 2.8408]/100;
% for Ti from Si
mmons and Wang in units of 1
/GPa
Elow = 83.2; Ehigh = 145.5;
% Elastic Ccontants of Nb from Simmons and Wang for Ti
CTE = [15.4 15.4 30.6];
% this is for Sn, not a cubic material ... such as this inconsistency
f = figure(
'Position'
,
[0,0,500,500]); movegui(f,
'northwest'
); set(g
cf,
'Color'
, [1 1 1]); hold
on
;
hold
on
; axis
square
; xmax = 0;
if
c_a ~= 1
trideg = 45;
if
ihex == 1
trideg = 30;
end
angle = 0:1:trideg;
77
xang = cosd(angle); yang = sind(angle);
borderx = [0 xang 0]; bordery = [0 yang
0]; xmax = 1.02;
axis([0 xmax 0 xmax]),
% TickDir, 'out' ???
plot(borderx,bordery,
'k
-
'
);
edgecolor = [0,0,1];
% perimeter of plotting symbol
if
ipfd ==
2
edgecolor = [.9,.9,0];
end
if
ipfd == 1
edgecolor = [1,0,0];
end
for
p = 0:.05:1
% plot inverse pole figure gray scale symbols
plot(.03+p*xmax*.7,.95*xmax,
'o'
,
'LineWidth'
,4,
'MarkerEdgeColor'
,[1 1
-
p p],
...
'MarkerFaceColor'
,[1 1 1],
'MarkerSize'
,8)
if
c_a ~= 1
plot(.03+p*xmax*.7,.85*xmax,
'o'
,
'LineWidth
'
,1,
'MarkerEdgeColor'
,[p p p],
...
'MarkerFaceColor'
,[p p p],
'MarkerSize'
,6)
end
end
text(0.02,.9*xmax,
'E direction, yellow (low)
--
> magenta (high)'
);
if
c_a ~= 1
text(0.02,.8*xmax,
'CTE direction, black(low)
--
> white(high)'
);
end
i
f
stereo == 0
text(0.02,.7*xmax,
'Z projection, not stereographic'
);
else
text(0.02,.7*xmax,
'Stereographic projection'
);
end
text(0.02,.5*xmax,[
'IPF direction '
,num2str(ipfd)]);
text(0.02,.6*xmax,[
'c/a ratio '
,num2str(c_a)]);
% Generate orientat
ion matrices for each orientation in Ang
R45 = [1,0,0;0,cosd(45),
-
sind(45);0,sind(45),cosd(45)];
% Rotation of 45 deg about X axis
% %
for
iAng=1:1:Angrows;
%1;%4; %2; %
phidA = Ang(iAng,3:5);
phidA(1) = phidA(1);
%; +180% **** Rotating euler a
ngles (e.g. to correct for 180 rotation)
if
phidA(1)>360
phidA(1) = phidA(1)
-
360;
end
if
phidA(1)<0
phidA(1) = phidA(1)+360;
end
phisA = phidA*pi/180;
%Compute Bunge orientation matrix g
gphi1=[cos(
phisA(1,1)),sin(phisA(1,1)),0;
-
sin(phisA(1,1)),cos(phisA(1,1)),0;0,0,1];
gPhi=[1,0,0;0,cos(phisA(1,2)),sin(phisA(1,2));0,
-
sin(phisA(1,2)),cos(phisA(1,2))];
gphi2=[cos(phisA(1,3)),sin(phisA(1,3)),0;
-
sin(phisA(1,3)),cos(phisA(1,3)),0;0,0,
1];
gA=
gphi2*gPhi*gphi1; gAR = gA*Rotateview;
% to rotate point of view in plot
g(:,:,iAng)=gA; R(:,:,iAng)=gA';
gsgTA = gA*ststens_n*gA';
%rotated stress tensor
basalY(iAng,1) = abs(gA(3,2));
for
i = 1:1:3
CTEm(:,i) = gA(:,i).*CTE';
end
for
i = 1:1:3
% This gives the CTE in the x (1), y (2), z (3) directions
e1 = sij(1)*(gA(1,i)^4 + gA(2,i)^4) + sij(4)*gA(3,i)^4;
e2 = (2*sij(2) + sij(6))*(gA(1,i)^2 * gA(2,i)^2);
e
3 = (2*sij(3) + sij(5))*gA(3,i)^2 *
(gA(1,i)^2 + gA(2,i)^2);
EA(iAng,i)=1./(e1+e2+e3);
end
Eb = (EA(iAng,iEd)
-
Elow)/(Ehigh
-
Elow);
pv = gA(:,ipfd)'; pv = abs(pv);
if
c_a == 1
x(1) = median(pv);
x(2) = min(pv);
78
x(3) = max(pv);
else
x(1) = max(pv(1),pv(2));
x(2) = min(pv(1),pv(2));
x(3) = pv(3);
end
if
ihex == 1
ang = atand(x(2)/x(1));
if
ang > 30
% tangent of 30 deg
na
ng = 30
-
(ang
-
30);
radius = (x(1)^2+x(2)^2)^.5
;
x(1) = cosd(nang)*radius;
x(2) = sind(nang)*radius;
end
end
if
stereo == 1
plot(x(1)/(1+x(3)), x(2)/(1+x(3)),
'o'
,
'LineWidth'
,2,
'MarkerEdgeColor'
,
...
edgecolor,
'MarkerFaceColor'
,[
1 1
-
Eb Eb],
'MarkerSize'
,14)
text(x(1)/(1+x(3)), x(2)/(1+x(3))+.037,num2str(iAng))
else
plot(x(1),x(2),
'o'
,
'LineWidth'
,1,
'MarkerEdgeColor'
,edgecolor,
...
'MarkerFaceColor'
,[1 1
-
Eb Eb],
'MarkerSize'
,8)
text(x(1),x(2
)+.037,num2str(iAng))
en
d
for
isc=1:1:nslphex
Sf=0.;
for
i=1:1:3
for
j=1:1:3
% Compute Schmid Factor, slpsys{4 = Schmid matrix}
Sf=Sf+gsgTA(i,j)*slpsys{4,isc}(i,j);
end
en
d
if
i
sc >= 46 && Sf <0
Sf = 0.001 * Sf ;
% this is to prevent anti
-
twin shears from being seriously considered later
end
rot_nA = slpsys{3,isc}(1,:)*gAR;
rot_bA = slpsys{3,isc}(2,:)*gAR;
rot_p1 = slpsys{3,isc}(3,:)*gAR;
rot_p2 = slpsys{3,isc}(4,:)*gAR;
rot_p3 = slpsys{3,isc}(5,:)*gAR;
rot_p4 = slpsys{3,isc}(6,:)*gAR;
rot_p5 = slpsys{3,isc}(7,:)*gAR;
rot_p6 = slpsys{3,isc}(8,:)*gAR;
rot_tA =
cross(rot_bA',rot_nA);
% This variable has slip system number, Schmid Factor, plane and Burgers in sample coord syst, and hkl,uvw
Schm_f_ss(isc,1:1
1,iAng)=[isc, Sf, abs(Sf), slpsys{2,isc}(1,:),slpsys{2,isc}(2,:)];
% Schmid factors (1
-
3), rotated
plane normal (4
-
6), Computed rotated Burgers vector (7
-
9),
% , plane trace on Z surface (10
-
12) Computed rotated position vectors to points p1
-
p4 (13
-
24)
%
1 2 3 4
-
6 7
-
9 10
-
12
Schm_lab
vecA(isc,:) = [isc, Sf, abs(Sf), rot_nA, rot_bA, cross(rot_nA',[0,0,1]),
...
rot_p1, rot_p2, rot_p3, rot_p4, rot_p5, rot_p6, rot_tA];
end
%
13
-
15 16
-
18 19
-
21 22
-
24 25
-
27 28
-
30 31
-
33
%
% sortmv contains list from high to l
ow Schmid factor, with associated infomation to draw slip system in unit cell
% useful plotting unit cell vectors will sort to bottom row
Schm_labvecA(nsl
phex+1,:) = [0 1
-
1 [1 0 0]*gAR [0 1 0]*gAR [0 0 1]*gAR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
;
% 33 columns
sortmv(:,:,iAng) = sortrows(Schm_labvecA,
-
3);
if
iAng == 1
% Rotated xi (streak) vectors
for
gg = 1:1:ghklrows
ghklg(gg,1:3) = unit_ghklC(gg,:)*gAR;
for
isc=1:1:nslphex;
xicg(isc,1:3,gg
) = xic(isc,1:3,gg) * gA;
% streak direction vector, rotated to sample coordinates
xirg(isc,1:3,gg) = cross(Sc
hm_labvecA(isc,31:33),ghklg(gg,1:3));
% streak direction vector, from rotated sample coordinates
xig(isc,1:3,gg)
= xirg(isc,1:3,gg) * R45;
% rotated streak vector rotated from sample to detector coordinate system defined by R45
end
end
end
end
79
% plot the image of unit cell, slip vectors, planes, plane normals, and plane traces
% Then
choose what row (orientation) to analyze,
% and set iAng to this orientation (row), and run this second cell. X down
and Y to right!!!
iAng = 1;
ptpl = 1;
% 0 = don't plot plane traces,
ptss = 2;
% 0 just the unit cell, and 1 slip planes, 2 and d
irections, 3 and plane normals, 4 directions only,
pta123o = 1;
% 1 = a1 a2 a3 coordinates and origin (open circle),
nplots = nslphex;
%48; %1; % number of slip sorted systems to plot
% no further use
r input below here
% Dashed lines give plane tr
aces, (colors in groups based on slip
% families) Shorter plane traces imply that the slip plane is nearly
% parallel to page, longer traces imply that the plane is highly inclined.
% dotted red, green,
blue lines give the x,y,z edges of unit cell.
%
Turquiose/Teal line shows Burgers vector direction, with direction
% away from the ball end (
--
!
--
> adjusted for the sign of the Schmid factor <
--
!
--
).
% Slip plane is shaded light when the plane normal
has out
-
of
-
page
% component, darker when into the
page.
% Strategy: First extract useful vectors from slip system information to draw the hexagonal prisms
% positions in sortmv p1:13
-
15 p2:16
-
18 p3:19
-
21 p4:22
-
24 p5:25
-
27 p6:28
-
30
% positio
ns in pln p1:4
-
6 p2:7
-
9 p3:10
-
12 p4:13
-
15
p5:16
-
18 p6:18
-
21
for
isc = 1:1:nplots
if
sortmv(isc,1,iAng) == 1;
% locate the two basal planes
pln(1,4:21) = sortmv(isc,13:30,iAng);
% bottom basal plane
pln(2,4:21) = sortmv(isc,13:30,iAng);
% top basal plane
rotc = sortmv(isc,4:6,iAng)*c_a;
% basal plane normal * c/a
for
j = 4:3:19
pln(2,j:j+2) = pln(1,j:j+2) + rotc;
% move top plane up by a unit of c
end
a1 = sortmv(isc,7:9,iAng);
% locate a1 using SS1
elseif
sortmv(isc,1,iAng) == 2;
a2 = sortmv(isc,7:9,iAng);
% locate a2 using SS2
elseif
sortmv(isc,1,iAng) == 3;
a3 = sortmv(isc,7:9,iAng);
% locate a3 using SS3
end
end
for
isc = 1:1:nplots;
if
sortmv(isc,1,iAng) == 4;
% locate two prism planes on opposite sides using SS4
pln(3,4:21) = sortmv(isc,13:30,iAng);
for
j = 13:3:2
8
pln(4,j
-
9:j
-
7) = sortmv(isc,j:j+2,iAng) + a2
-
a3;
end
elseif
sortmv(isc,1,iAng) == 5;
% locate t
wo prism planes on opposite sides using SS5
pln(5,4:21) = sortmv(isc,13:30,iAng);
for
j = 13:3:28
pln
(6,j
-
9:j
-
7) = sortmv(isc,j:j+2,iAng) + a3
-
a1;
end
elseif
sortmv(isc,1,iAng) == 6;
% locate two prism planes o
n opposite sides using SS6
pln(7,4:21) = sortmv(isc,13:30,iAng);
for
j = 13:3:28
pln(8,j
-
9:j
-
7) = sor
tmv(isc,j:j+2,iAng) + a1
-
a2;
end
end
end
for
j = 1:1:2
% Find z elevation of basal planes
for
k = 1:1:3
pln(j,k) = (pln(j,3+k)+pln(j,6+k)+pln(j,9+k)+pln(j,12+k)+pln(j,15+k)+pln(j,18+k))/6;
end
end
center = (pln(1,1:3)+pln(2,1:3))/2;
for
j = 3:1:8
% Find z elevation of prism planes
pln(j,3) = (pln(j,12)+pln(j,15)+pln(j,18)+pln(j,21))/4;
end
so
rtpln = sortrows(pln,
-
3);
minx = 0; miny = 0; minz = 0; maxx = 0; maxy = 0; maxz = 0;
for
j = 1:1:8
% assemble vectors for plotting faces of hex prism
prsxplt(j,1:7) = [sortpln(j,4) sortpln(j,7) sortpln(j,10) sortpln(j,13) sortpln(j,16) sortpln(j,19)
s
ortpln(j,4)];
minx = min(minx,min(prsxplt(j,:))); maxx = max(maxx,max(prsxplt(j,:)));
80
prsyplt(j,1:7) = [sortpln(j,5) sortpln(j,8) sortpln(j,11) sortpln(j,14) sortpln(j,17) sortpln(j,20) sortpln(j,5)];
miny = min(miny,min(prsyplt(j,:))); maxy
=
max(maxy,max(prsyplt(j,:)));
prszplt(j,1:7) = [sortpln(j,6) sortpln(j,9) sortpln(j,12) sortpln(j,15) sortpln(j,18) sortpln(j,21) sortpln(j,6)];
minz = min(minz,min(prszplt(j,:))); maxz = max(maxz,max(prszplt(j,:)));
end
sscol = 1;
%decide if w
an
t to plot unit cells for this slip system
ipl =
-
8;
% Strategy: Next, start isc loop for plotting slip systems
ipc = 0;
% plot counter
for
isc = 1:1:nplots
ssn = sortmv(isc,1,iAng);
% slip system number
if
pssf(ssn) == 1
ipc =
ipc + 1;
if
ipl
-
ipc ==
-
9
% eight plots on a page
f = figure(
'Position'
, [0,0,1050,525]); movegui(f,
'northwest'
); set(gcf,
'Color'
, [1 1 1]); hold
on
;
ipl=ipl+8;
end
subplot(2,4,ipc
-
ipl)
hold
on
sp1 = sortmv(isc,13:15,iAng);
% beginning of Burgers vector
sp2 = sortmv(isc,16:18,iAng);
% extract plotted points on perimeter of the slip plane
sp3 = sortmv(isc,19:21,iAng);
sp4 = sortmv(isc,22:24,iAng);
sp
5 = sortmv(isc,25:27,iAng);
sp6 = sortmv(isc,28:30,iAng);
spx = [sp1(1) sp2(1) sp3(1) sp4(1) sp5(1) sp6(1) sp1(1)];
spy = [sp1(2) sp2(2) sp3(2) sp4(2) sp5(2) sp6(2) sp1(2)];
S
f = sortmv(isc,2,iAng);
% Schmid factor
Sfs = 1;
if
Sf < 0
Sfs =
-
1;
end
;
n = [0 0 0 sortmv(isc,4:6,iAng)];
% plane normal
b = [sp1 sp4];
% p1+sortmv(isc,7:9,iAng)]; % Burgers vector
nvec = sortmv(isc,4:6,iAng);
bvec = sortmv
(isc,7:9,iAng);
pt = sortmv(isc,10:12,iAng);
% plane trace
minx = min(minx, sp1(1)+n(4));
maxx = max(maxx, sp1(1)+n(4));
% find appropriate range of x and y for plot
miny = min(miny, sp1(2)+n(5));
maxy = max(
maxy, sp1(2)+n(5));
midx = (minx+maxx)/2;
midy = (miny+maxy)/2;
del = 2;
% These plots will match TSL with X down !!!! Plotting starts...
axis
square
set(gca ,
'ycolor'
,
'w'
); set(gca ,
'xcolor'
,
'w'
);
% make
axes white for ease in later arranging.
axis([midy
-
del midy+del
-
midx
-
del
-
midx+del])
if
pta123o > 0
if
Ang(iAng,4) < 90
% make the 3 coordinate axes
visible below slip planes
plot([0 a1(2)],
-
[0 a1(1)],
':'
,
'Lin
ewidth'
,3,
'Color'
,[1 0 .2]);
% plot x = red
plot([0 a2(2)],
-
[0 a2(1)],
':'
,
'Linewidth'
,3,
'Color'
,[.6 .8 0]);
% plot y = green
-
gold
plot([0 a3(2)],
-
[0 a3(1)],
':'
,
'Linewidth'
,3,
'Color'
,[0 0 1]);
% plot z = blue
end
end
if
ipl
-
ipc==
-
1
end
if
ptss > 0 && ptss < 4
% plot slip planes
if
sortmv(isc,6,iAng) > 0
% is k component of slip plane
normal positive or negative?
fill(spy,
-
spx, [.8 .8 .65])
% slip pl
ane filled warm gray
else
% slip plane filled cool gray if normal has neg z component
fill(spy,
-
spx, [.65 .65 .7])
%
plot([n(2) n(5)],
-
[n(1) n(4)],'Linewidth',3,'Color',[.65 .65 .7]);
end
if
n(6) > 0
pncolor = [0 0 0];
% positive plane normal color
else
pncolor = [.5 .5 .5];
% negative plane normal color
81
end
end
% plane is plotted
if
ptss == 2 || pt
ss == 4
% plot Burgers vectors
if
sortmv(isc,6,iAng) > 0
Bvcolor = [0 .7 .7];
if
ssn >= 55
Bvcolor = [.1 .6 0];
end
if
ssn >= 43 && ssn < 55
Bvcolor = [1 .6 0];
end
else
Bvcolor = [0 1 1];
if
ssn >= 55
Bvcolor = [.3 .9 0];
end
if
ssn >= 43 && ssn < 55
Bvcolor =
[1 .8 0];
end
end
Sf
s = 1;
if
ssn < 43
% will reverse the sign of Schmid factor for dislocations
end
if
Sf > 0
% plot Burgers vector direction
if
ssn >= 43
% this is for twins
-
the Burgers vector length is shown to
be 1/2 of the usual length in the unit cell
plot(b(2),
-
b(1),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) (b(2)+b(5))/2],
-
([b(1) (b(1)+b(4))/2]),
'Line
width'
,4,
'Color'
,Bvcolor)
else
plot(b(2),
-
b(1),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) b(5)],
-
[b(1) b(4)],
'Linewid
th'
,4,
'Color'
,Bvcolor)
% quiver(p1(1),p1(2),dp(1),dp(2)
,0,'Linewidth',2,'Color',Bvcolor)
end
else
% plot Burgers vector in opposite direction
if
ssn >= 43
% this is for twins
-
the
Burgers vector length is shown to be 1/2 of the usual length in the unit ce
ll
plot(b(2),
-
b(1),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) (2*b(2)+b(5))/3],
-
([b(1) (2*b(1)+b(4))/3]),
'Linewidth'
,4,
'Color'
,Bvcol
or)
else
plot(b(5),
-
b(4),
'.'
,
'MarkerSiz
e'
, 24,
'Color'
, Bvcolor)
plot([b(5) b(2)],
-
[b(4) b(1)],
'Linewidth'
,4,
'Color'
,Bvcolor)
%
quiver(p2(1),p2(2),
-
dp(1),
-
dp(2),0,'Linewidth',4,'Color',Bvcolor)
end
end
if
ptss ==3
plot([b(2) (b(2)+n(5))],
-
([b(1) (b(1)+n(4))]),
'Linewidth'
,4,
'Color'
,pncolor)
end
end
% Burger
s vector is plotted
for
j = 1:1:4
% plot the 4 top most surface prisms of the hex cell that have the highest z elevation
plot(prsyplt(j,:),
-
prsxplt(j,:),
'Linewidth'
,2,
'Color'
,[.0 .0 .0]);
end
if
pta123o > 0
if
Ang(iAng,4) >= 90
% make the 3 coordinate axes visible above slip planes
plot([0 a1(2)],
-
[0 a1(1)],
':'
,
'Lin
ewidth'
,3,
'Color'
,[1 0 .3]);
% plot x = red
plot([0 a2(2)],
-
[0 a2(1)],
':'
,
'Linewidth'
,3,
'Color'
,[.5 .6 0]);
% plot y = green
-
gold
plot([0 a3(2)],
-
[0 a3(1)],
':'
,
'Linewidth'
,3,
'Color'
,[0 0 1]);
% plot z = blue
end
end
if
ptpl == 1
% plot plane traces
if
ssn>=55
% compression twin plane traces green
plot([
-
pt(2) pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linewidth'
,3,
'Color'
,[.2 .8 0])
elseif
ssn>42 && ssn<55
% extens
ion twin plane traces
orange
plot([
-
pt(2) pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linewidth'
,3,
'Color'
,[1 .6 0])
elseif
ssn>12 && ssn<43
% plane traces green
-
gold
plot([
-
pt(2) pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linew
idth'
,3,
'Color'
,[.95 .85 0])
elseif
ssn<13 && ssn>6
% pyr green
plot([
-
pt(2) pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linewidth'
,3,
'Color'
,[0 .9 .5])
elseif
ssn<7 && ssn>3
% prism red
plot([
-
pt(2)
pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linewidth'
,3,
'Color'
,[1 .2 0])
else
% {medium slip systems} blue
plot([
-
pt(2) pt(2)],
-
[
-
pt(1) pt(1)],
'
--
'
,
'Linewidth'
,3,
'C
olor'
,[0 0 1])
end
%
----
> NOTE tha
t Schmid factor vector is plotted in correct direction,
82
end
% plot plane traces
if
pta123o == 1
% plot coordinate axes
plot(0,0,
'ko'
);
end
%
----
> Burgers vector is labeled and plotted with consistently signed b
vector direction.
if
ptss >= 1
title({[
'ssn'
num2str(ssn)
' n'
mat2str(slpsys{2,ssn}(1,:)) mat2str(Sfs*slpsys{2,ssn}(2,:))
'b'
],
...
[mat2str(sortmv(i
sc,4:6,iAng),3), mat2str(sortmv(isc,7:9,iAng),3)],
...
[
'Eule
rs = '
, mat2str(Ang(iAng,3:5),3)],
...
[
'c
-
axis = '
, mat2str(sortmv(nslphex+1,10:12,iAng),3)],
...
[LC
' Or
-
'
num2str(iAng)
' m'
num2str(isc)
' = '
num2str(Sfs*
Sf, 3) ]
})
end
end
end
83
A
PPENDIX
B:
Matlab
MP
R hexagonal grain pair analysis code
(written by Thomas R.
Bieler)
% T.R. Bieler
-
m' Schmid and fip calculator, 4 June
-
14 1 July 2013 with input from Adam L Pilchak
%
contains pieces from prior codes that are probably right, but no guarantees, use at yo
ur own risk.
% Written and used in Matlab release R2009b, and R2016a,
% substantially revised in 2018 during sabbatical at IMDEA Materiales
%
% Sources for these ideas
are discussed in
% Bieler et al. Int. J. Plasticity 25(9), 1655
1683, 2009,
and
% Kumar et al. J. Engineering and Materials Technology 130, 021012, 2008
% Bieler et al. Current Opinion in Solid State and Materials Science 18(4) 2014 212
-
226
% an
d related prior work.
% Important note
-
Euler angle computation coordinate system i
s not
% necessarily consistent with any other coordinate system. To obtain
% consistent results to make slip planes and Burgers vectors come out
% right, we pre
-
rotate a
cquired data by 180deg in first Euler angle so that
% the Euler angle coordinate syste
m has X down and Y right. Plotting is
% done with this perspective in mind. If your raw data is not pre
-
rotated
% then setting hkl = 1 and adjusting first Euler angle when computing g
% below is necessary.
% Input data are default type 2 grains file a
nd reconstructed grain boundary files from TSL
% The reconstructed boundary file does not contain phase ID, so the
grain
% file is also needed, and it is the primary source.
% This code expects phase ID to be 1 = hex, 2 = BCC, 3 = FCC, 4 = BCT (Sn).
% If
your Phase ID is different, then some adjustments need to be made
-
% probably easiest if you copy information
in column 10 of IDgr and put it in
% column 11, and then transform it to the phase number needed in this code in column 10.
% To run o
n particular boundary with two orientations, put values into IDGR
% with estimates of relative grain center posi
tions, and skip the file input in the next block
% [ _ 25 13] is maximum Sf in single slip in Z
clc; clear;
IDgr = [1 8.0421 8
9.3031 354.1300 10 10 60.2 0.364 1.3 0 1
% 40.6+180 8.5 296 159.4, 8.8, 203.7, 36 IDgr = [1
0 0 0 354.27
88.49 352.07 60.2 0.364 0 0 1 % 40.6+180 8.5 296 159.4, 8.8, 203.7, 37
2 2.6823 101.5918 354.2043
15 4 60.2 0.364 1.3 0 1]
% 270.6, 14.3, 96
delx = IDgr(2,5)
-
IDgr(1,5); dely = IDgr(2,6)
-
IDgr(1,6);
mid = [(IDgr(2,5)+IDgr(1,5)) (IDgr(2,6)+IDgr(1,6))]*.5; gbvec = [0 1;
-
1 0]*[delx;dely]*.5;
x1 = mid(1,1)+gbvec(1,1); y1 = mid(1,2)+gbv
ec(2,1); x2 = mid(1,1)
-
gbvec(1,1); y2 = mid(1,2)
-
gbvec(2,1);
trang = atand(
-
(y2
-
y1) / (x2
-
x1) );
if
trang < 0
trang = trang + 180;
end
RBdyn = [ 1 2 3 4 5 6 (delx^2+dely^2)^.5 trang x1 y1 x2 y2 2 1]
% Y goes down; length, angle, x1 y1 x2 y
2 right grain # , left grain #
fnameRCB =
'just_one_pair_of_points_RB.txt'
;
% The code reads #(1), orientation(2:4), grain center(5,6), phase ID(10) from IDgr, and
% grain boundary inclination from horizontal(8) and segment position(9:12) and right, left
grain #s (13,14)
% are read from the rec
onstructed grain boundary file
% If you want to enter your own data to explore using this code, you will want to populate the above
% two variables in the same way as Grain ID and Recrystructed Boundary files ar
e, as follows:
% In IDgr, Column 1 is grai
n number, columns 2
-
4 are Bunge Euler angles with X down and Y to the right
% columns 5 and 6 contain x and y positions of the data points (normally grain center in Grains file)
% In RBdy, in in row 1 (which is bou
ndary #1) euler angles for the two grains
identified in colums 13 and 14
% are provided in colums 1
-
6. colum 7 is grain boundary length, column 8 (H in excel), is the angle from the horizontal
% to the apparent grain boundary direction e.g. 90deg if you
want a vertical boundary. columns 9
-
12 have start and
% end positions of the grain boundary. In columns 13 and 14 (J&K) put the
% right,left grain numbers on either side of boundary 1 in that
order (yes, the right one first).
%%
84
clc; clear;
G
F2HeaderLines = 16;
% 11 for single phase, 12 for two phase
RCBHeaderLines = 11;
% total number of header lines for the Reconstructed boundary file
fnameGF2 =
'C:
\
Users
\
chels
\
Desktop
\
Titanium
\
Ten
sion
\
Ti525_#10_EBSD Scans
\
Ti525_10_GF.txt'
; GF2HeaderLines =
17;
fnameRCB =
'C:
\
Users
\
chels
\
Desktop
\
Titanium
\
Tension
\
Ti525_#10_EBSD Scans
\
Ti525_10_RB.txt'
; RCBHeaderLines = 11;
GF2HeaderLines = 12;
RCBHeaderLines = 8;
fileID =
fopen(fnameGF2)
dataGF = importdata(fnameGF2,
' '
, GF2HeaderLines)
IDgr = dataGF.data;
f
ileID = fopen(fnameRCB)
dataRB = importdata(fnameRCB,
' '
, RCBHeaderLines)
RBdyn = dataRB.data;
fclose(
'all'
)
fnlengthRCB = length(fnameRCB); chr=char(fnameRCB);
%%
Sfthr = 0.2;
% higher schmid tolerance value used to limit serach for high m' values
SflimL = 0.0;
% lower schmid tolerance value used to limit filling out table
mpthr = 0.75;
% threshold v
alue for considering m' values to be meaningful
hkl = 1;
% flag used to decide whether to adjust first euler angle for various reasons..
. see below
nslphex = 39; c_a_hex = 1.59;
% 1.587 for pure Ti ... not turning on compression twinning
in Hexagonal
nslpbcc = 24; c_a_bcc = 1.0;
% could differ for metastable phases... not turning on 123 slip
nslpfcc = 12; c_a_fcc = 1.0;
% ~1.02 for TiAl% For FCC, it is 18 if cube slip is included.
nslpbct = 32; c_a_bct = 0.5456;
% for Sn
nslp = [nslphex, nslpbcc, nslpfcc, nslpbct];
% number of slip systems used for phases 1, 2, 3, 4
c_a = [c_a_hex, c_a_bcc, c_a_fcc, c_a_bct];
% If your data is SINGLE PHASE, then you must put the correct phase number into the variable one_ss, HERE
one_ss = 1;
% e.g. , the if statement below will set phase = 1 for Hex, 2 for BCC(3 for FCC) in IDgr file column 10:
numPhases = max(IDgr(:,10));
%
check the number of phases in the dataset
if
numPhases >0 && (chr(fnlengthRCB
-
4) ==
'B'
|| chr(fnlengthR
CB
-
4) ==
'b'
)
% do nothing
else
% for a grain boundary trace, need to set IDGR to one_ss
IDgr(:,10) = one_ss;
% single phase,
---
!!!
set to 1 for HEX or 2 for BCC above !!!
---
for
i = 1:1:4
% else if it's two phase, do
nothing, if phase 1 is hex, 2 is bcc.
if
i ~= one_ss
nslp(i) = 0;
end
end
end
% Stress tensor is define
d using TSL convensions with x down !!! put the one you want last
sigma = [0,0,0; 0,1,0; 0,0,0];
nste
n = 1;
for
i = 1:1:nsten
str2 = sigma(:,:,i)*sigma(:,:,i)';
ststens_mag = (str2(1,1)+str2(2,2)+str2(3,3))^.5;
sigma_n(:,:,i) = sigma(:,:,i)/s
tstens_mag;
% normalized stress tensor to get generalized Schmid factor
sigma_v(:,i) = [sigma(1,1) sigma(2,2) sigma(3,3)]';
% vectorized version of trace
end
dIDgr = size(IDgr);
% This is a default type 2 grain file, both are needed.
dRBdyn = siz
e(RBdyn);
% This is a reconstructed grain boundary file
good = 0;
if
dRB
dyn(1,2) == 21;
for
ii = 1:1:dRBdyn(1,1)
end
else
RBdy = RBdyn;
end
dRBdy = size(RBdy);
% E(X,Y,Z) will be calculated from [S11 S12 S13 S33 S44 S66] in 1/GPa;
% hexago
nal stiffness chosen:
sij(1,:) = [0.9581
-
0.4623
-
0.1893 0.698 2.1413 2.408]/100;
% for Ti from Simmons and Wang in units of 1/GPa
85
% cubic stiffness chosen:
sij(2,:) = [0.6862
-
0.2581
-
0.2581 0.6862 1.2123 1.2123]/100;
% for Ta from Simmons and Wang in
units of 1/GPa
% dis
p('E(X,Y,Z) in GPa; S11 S12 S13 S33 S44 S66 210 Rayne, J.A. and B.S. Chandrasekhar,
% Elastic Ccontants of beta tin from 4.2K to 300K, Phys Rev. 118, 1545
-
49, 1960
sij(4,:) = [4.3627
-
3.3893
-
0.394 1.4501 4.5393 4.1667]/100;
% for
Sn in units of 1/GPa
% Ti
-
6Al, Ti
-
15Cr, alpha/beta in Ti6242 from J. Kim and S.I. Rokhlin, J. Acoust. Soc. Am. 126
-
6 dec 2009
% % Set up vectors useful for plotting unit cells with slip systems
O = [ 0 0 0 0];
% (I) (H)
A = [
2
-
1
-
1 0]/3;
% C
-------
B
B = [ 1 1
-
2 0]/3;
% /
\
/
\
C = [
-
1 2
-
1 0]/3;
% / a2 /
\
D = [
-
2 1 1 0]/3;
% /
\
/
\
E = [
-
1
-
1 2 0]/3;
% (J)D
------
O(P)
-
a1
-
> A(G)
--
> x
F = [ 1
-
2 1 0]
/3;
%
\
/
\
/
P = [ 0 0 0 1];
%
\
a3
\
/
G = [ 2
-
1
-
1 3]/3;
%
\
/
\
/
H = [ 1 1
-
2 3]/3;
% E
--------
F
I = [
-
1 2
-
1 3]/3;
% (K) (L)
J = [
-
2 1 1 3]/3;
%
K = [
-
1
-
1 2 3]/
3;
% where a1 = OA a2 = OC a3 = OE
L = [ 1
-
2 1 3]/3;
% DA || a1, FC || a2, BE || a3
% Slip system definitions set as of 1 Nov 2019 to be consistent with DAMASK
% Hex plane direction 1st and 4th point is Burgers vect
or
% basal
-
g
lide:&Be Mg Re Ti; Re
sshex(:,:,1) = [0 0 0 1; 2
-
1
-
1 0; D ; E ; F ; A ; B ; C ];
sshex(:,:,2) = [0 0 0 1;
-
1 2
-
1 0; F ; A ; B ; C ; D ; E ];
sshex(:,:,3) = [0 0 0 1;
-
1
-
1 2 0; B ; C ; D ; E ; F ; A ];
ibas = 1;
fbas = 3;
%
prism
-
glide:Ti Zr
RE; Be Re Mg
sshex(:,:,4) = [ 0 1
-
1 0; 2
-
1
-
1 0; E ; E ; E ; F ; L ; K ];
sshex(:,:,5) = [
-
1 0 1 0;
-
1 2
-
1 0; A ; A ; A ; B ; H ; G ];
sshex(:,:,6) = [ 1
-
1 0 0;
-
1
-
1 2 0; C ; C ; C ; D ; J ; I ];
iprs = 4;
fprs = 6;
% pris
m
sshex(:,:,7) =
[ 2
-
1
-
1 0; 0 1
-
1 0; F; F; F; B ; H ; L ];
sshex(:,:,8) = [
-
1 2
-
1 0;
-
1 0 1 0; B; B; B; D ; J ; H ];
sshex(:,:,9) = [
-
1
-
1 2 0; 1
-
1 0 0; D; D; D; F ; L ; J ];
i2prs = 7;
f2prs = 9;
% pyramidal
-
glide
--
**
--
CORRECTED
-
-
**
--
sshex(:,:,10) =
[ 1 0
-
1 1;
-
1 2
-
1 0; E ; E ; E ; D ; I ; L ];
sshex(:,:,11) = [ 0 1
-
1 1;
-
2 1 1 0; F ; F ; F ; E ; J ; G ];
sshex(:,:,12) = [
-
1 1 0 1;
-
1
-
1 2 0; A ; A ; A ; F ; K ; H ];
sshex(:,:,13) = [
-
1 0 1 1; 1
-
2 1 0; B ; B ; B ;
A ; L ; I ];
sshex(:,
:,14) = [ 0
-
1 1 1; 2
-
1
-
1 0; C ; C ; C ; B ; G ; J ];
sshex(:,:,15) = [ 1
-
1 0 1; 1 1
-
2 0; D ; D ; D ; C ; H ; K ];
ipyra = 10;
fpyra = 15;
% pyramidal
-
glide:; all?
sshex(:,:,16) = [ 1 0
-
1 1;
-
2 1 1 3; A ; B ; I ;
P ; L ; A ];
sshex(:,:,
17) = [ 1 0
-
1 1;
-
1
-
1 2 3; B ; B ; I ; P ; L ; A ];
sshex(:,:,18) = [ 0 1
-
1 1;
-
1
-
1 2 3; B ; C ; J ; P ; G ; B ];
sshex(:,:,19) = [ 0 1
-
1 1; 1
-
2 1 3; C ; C ; J ; P ; G ; B ];
sshex(:,:,20) = [
-
1 1 0 1; 1
-
2 1 3; C ;
D ; K ; P ; H ; C ];
ss
hex(:,:,21) = [
-
1 1 0 1; 2
-
1
-
1 3; D ; D ; K ; P ; H ; C ];
sshex(:,:,22) = [
-
1 0 1 1; 2
-
1
-
1 3; D ; E ; L ; P ; I ; D ];
sshex(:,:,23) = [
-
1 0 1 1; 1 1
-
2 3; E ; E ; L ; P ; I ; D ];
sshex(:,:,24) = [ 0
-
1 1 1; 1 1
-
2 3; E ; F ; G ; P ; J
; E ];
sshex(:,:,25) = [ 0
-
1 1 1;
-
1 2
-
1 3; F ; F ; G ; P ; J ; E ];
sshex(:,:,26) = [ 1
-
1 0 1;
-
1 2
-
1 3; F ; A ; H ; P ; K ; F ];
sshex(:,:,27) = [ 1
-
1 0 1;
-
2 1 1 3; A ; A ; H ; P ; K ; F ];
ipyrc = 16;
fpyrc = 27;
86
%
pyramidal
-
2nd ord
er glide
sshex(:,:,28) = [ 1 1
-
2 2;
-
1
-
1 2 3; (O+B)/2 ; C ; J ; (P+K)/2 ; L ; A];
sshex(:,:,29) = [
-
1 2
-
1 2; 1
-
2 1 3; (O+C)/2 ; D ; K ; (P+L)/2 ; G ; B];
sshex(:,:,30) = [
-
2 1 1 2; 2
-
1
-
1 3; (O+D)/2 ; E ; L ; (P+G)/2
; H ; C];
sshex(:,:,31)
= [
-
1
-
1 2 2; 1 1
-
2 3; (O+E)/2 ; F ; G ; (P+H)/2 ; I ; D];
sshex(:,:,32) = [ 1
-
2 1 2;
-
1 2
-
1 3; (O+F)/2 ; A ; H ; (P+I)/2 ; J ; E];
sshex(:,:,33) = [ 2
-
1
-
1 2;
-
2 1 1 3; (O+A)/2 ; B ; I ; (P+J)/2 ; K ; F];
i2pyrc = 28
;
f2pyrc = 33;
% *** Twi
n directions are opposite in Christian and Mahajan, and are not correcte to to be consistent with them
% FROM Kocks SXHEX plane direction 1st and 4th point is Burgers vector, order of C1 differs from Kock's file
%
{1012}<1011> T1 twins 0.17;
-
1.3 t
wins: all Twin Vector must go in the
% sense of shear, opposite C&M sense.
sshex(:,:,34) = [ 1 0
-
1 2;
-
1 0 1 1; A ; B ; J ; K ; K ; K ];
sshex(:,:,35) = [ 0 1
-
1 2; 0
-
1 1 1; B ; C ; K ; L ; L ; L ];
sshex(:,:,
36) = [
-
1 1 0 2; 1
-
1 0 1; C ;
D ; L ; G ; G ; G ];
sshex(:,:,37) = [
-
1 0 1 2; 1 0
-
1 1; D ; E ; G ; H ; H ; H ];
sshex(:,:,38) = [ 0
-
1 1 2; 0 1
-
1 1; E ; F ; H ; I ; I ; I ];
sshex(:,:,39) = [ 1
-
1 0 2;
-
1 1 0 1; F ; A ; I ; J ; J ; J ];
iT1 = 34;
fT1 = 39;
% {2111}<211
6> T2 twins: 0.63;
-
0.4; Ti Zr Re RE]; Also does not follow C&M definition for shear direction
sshex(:,:,40) = [ 1 1
-
2 1;
-
1
-
1 2 6; (O+B)/2 ; C ; (J+I)/2 ; P ; (L+G)/2 ; A];
sshex(:,:,41) = [
-
1 2
-
1 1; 1
-
2 1 6;
(O+C)/2 ; D ; (K+J)/2 ; P ; (G+H)/2
; B];
sshex(:,:,42) = [
-
2 1 1 1; 2
-
1
-
1 6; (O+D)/2 ; E ; (L+K)/2 ; P ; (H+I)/2 ; C];
sshex(:,:,43) = [
-
1
-
1 2 1; 1 1
-
2 6; (O+E)/2 ; F ; (G+L)/2 ; P ; (I+J)/2 ; D];
sshex(:,:,44) = [ 1
-
2 1 1;
-
1 2
-
1 6; (O+F)/2 ; A ; (H+G)/2 ; P ; (J+K)/2 ; E
];
sshex(:,:,45) = [ 2
-
1
-
1 1;
-
2 1 1 6; (O+A)/2 ; B ; (I+H)/2 ; P ; (K+L)/2 ; F];
iT2
= 40;
fT2 = 45;
% {1011}<101
-
2> C1 twins: 0.10; 1.1; Mg; Zr Ti]; agrees with C&M
sshex(:,:,46) = [ 1 0
-
1 1; 1 0
-
1
-
2; P ; L ; A ; (A+B)/2 ; B ; I ];
sshex(:
,:,47) = [ 0 1
-
1 1; 0 1
-
1
-
2; P ; G ; B ; (B+C)/2 ; C ; J ];
sshex(:,:,48) = [
-
1 1 0
1;
-
1 1 0
-
2; P ; H ; C ; (C+D)/2 ; D ; K ];
sshex(:,:,49) = [
-
1 0 1 1;
-
1 0 1
-
2; P ; I ; D ; (D+E)/2 ; E ; L ];
sshex(:,:,50) = [ 0
-
1 1 1; 0
-
1 1
-
2;
P ; J ; E ; (E+F)/2 ; F ; G ];
sshex(:,:,51) = [ 1
-
1 0 1; 1
-
1 0
-
2; P ; K ; F ; (F+A)
/2 ; A ; H ];
iC1 = 46;
fC1 = 51;
% {2112}<211
-
3> C2 twins:; 0.22; 1.2 Ti Zr Re]; agrees with C&M
sshex(:,:,52) = [ 1 1
-
2 2; 1 1
-
2
-
3; (K+P)/2 ; L ; A ; (A+C
)/2 ; C ; J];
sshex(:,:,53) = [
-
1 2
-
1 2;
-
1 2
-
1
-
3; (L+P)/2 ; G ; B ; (B+D)/2 ; D ; K];
sshex(:,:,54) = [
-
2 1 1 2;
-
2 1 1
-
3; (G+P)/2 ; H ; C ; (C+E)/2 ; E ; L];
sshex(:,:,55) = [
-
1
-
1 2 2;
-
1
-
1 2
-
3; (H+P)/2 ; I ; D ; (D+F)/2 ; F ; G];
sshe
x(:,:,56) = [ 1
-
2 1 2; 1
-
2 1
-
3; (I+P)/2 ; J ; E ; (E+A)/2 ; A ; H];
sshex(:,:,57) = [ 2
-
1
-
1 2; 2
-
1
-
1
-
3; (J+P)/2 ; K ; F ; (F+B)/2 ; B ; I];
iC2 = 52;
fC2 = 57;
mnslp = max(nslp);
ss = zeros(8,3,mnslp,4);
for
i=1:1:mnslp
% Change n & m t
o unit
vector,
if
i <= nslphex
n=[sshex(1,1,i) (sshex(1,2,i)*2+sshex(1,1,i))/3^.5 sshex(1,4,i)/c_a_hex];
% Plane normal /c_a_hex
m=[sshex(2,1,i)*1.5 3^.5/2*(sshex(2,2,i)*2+sshex(2,1,i)) sshex(2,4,i)*c_a_hex];
% Slip direction *c_a_h
ex
ss(1,:,i,1) = n/norm(n);
% alpha plane
ss(2,:,i,1) = m/norm(m);
% alpha direction PHASE 1 is HEX
ss(3,:,i,1) = [3*sshex(3,1,i)/2, (sshex(3,1,i)+2*sshex(3,2,i))*sqrt(3)/2, sshex(3,4,i)*c_a_hex];
%
hpoint
1
ss(4,:,i,1) = [3*sshex(4,1,i)/2, (sshex(4,1,i)+2*sshex(4,2,i))*sqrt(3)/2, sshex(4,4,i)*c_a_hex];
% hpoint 2
ss(5,:,i,1) = [3*sshex(5,1,i)/2, (sshex(5,1,i)+2*sshex(5,2,i))*sqrt(3)/2, sshex(5,4,i)*c_a_hex];
% hpoint 3
ss(6,:,
i,1) =
[3*sshex(6,1,i)/2, (sshex(6,1,i)+2*sshex(6,2,i))*sqrt(3)/2, sshex(6,4,i)*c_a_hex];
% hpoint 4
ss(7,:,i,1) = [3*sshex(7,1,i)/2, (sshex(7,1,i)+2*sshex(7,2,i))*sqrt(3)/2, sshex(7,4,i)*c_a_hex];
% hpoint 5
ss(8,:,i,1) = [3*sshex(8,1,i)/2
, (sshe
x(8,1,i)+2*sshex(8,2,i))*sqrt(3)/2, sshex(8,4,i)*c_a_hex];
% hpoint 6
end
if
i <= nslpbcc
n = [ssbcc(1,1,i),ssbcc(1,2,i),ssbcc(1,3,i)/c_a_bcc];
% slightly tetragonal has c/a <> 1.0
m = [ssbcc(2,1,i),ssbcc(2,2,i),ssbcc(2,3
,i)*c_a_bcc];
ss(1,:,i,2) = n/norm(n);
% bcc plane PHASE 2 is BCC
87
ss(2,:,i,2) = m/norm(m);
% bcc direction
ss(3,:,i,2) = [ssbcc(3,1,i),ssbcc(3,2,
i),ssbcc(3,3,i)*c_a_bcc];
% point 1
ss(4
,:,i,2) = [ssbcc(4,1,i),ssbcc(4,2,i),ssbcc(4,3,i)*c_a_bcc];
% point 2
ss(5,:,i,2) = [ssbcc(5,1,i),ssbcc(5,2,i),ssbcc(5,3,i)*c_a_bcc];
% point 3
ss(6,:,i,2) = [ssbcc(6,1,i),ssbcc(6,2,i),ssbcc(6
,3,i)*c_a_bcc];
% point 4
ss(7,:,i,2) =
[ssbcc(7,1,i),ssbcc(7,2,i),ssbcc(7,3,i)*c_a_bcc];
% point 5
ss(8,:,i,2) = [ssbcc(8,1,i),ssbcc(8,2,i),ssbcc(8,3,i)*c_a_bcc];
% point 6
end
if
i <= nslpfcc
n = [ssfcc(1,1,i),ssfcc(1,2,i),ssfcc(1,3,i)/c_a_fcc];
% slightly tetrag
onal has c/a <> 1.0
m = [ssfcc(2,1,i),ssfcc(2,2,i),ssfcc(2,3,i)*c_a_fcc];
ss(1,:,i,3) = n/norm(n);
% fcc plane PHASE 3 is
FCC
ss(2,:,i,3) = m/norm(m);
% fcc direction
ss(3,:,i,3) = [ssf
cc(3,1,i),ssfcc(3,2,i),ssfcc(3,3,i)*c_a_fcc];
% point 1
ss(4,:,i,3) = [ssfcc(4,1,i),ssfcc(4,2,i),ssfcc(4,3,i)*c_a_fcc];
% point 2
ss(5,:,i,3) = [ssfcc(5,1,i),s
sfcc(5,2,i),ssfcc(5,3,i)*c_a_fcc];
% point 3
ss(6,:,i,3) = [ssfcc(6,1,i),
ssfcc(6,2,i),ssfcc(6,3,i)*c_a_fcc];
% point 4
ss(7,:,i,3) = [ssfcc(7,1,i),ssfcc(7,2,i),ssfcc(7,3,i)*c_a_fcc];
% point 5
ss(8,:,i,3) = [ssfcc(8,1,i),ssfcc(8,2,i
),ssfcc(8,3,i)*c_a_fcc];
% point 6
end
if
i <= nslpbct
n = [
ssbct(1,1,i),ssbct(1,2,i),ssbct(1,3,i)/c_a_bct];
m = [ssbct(2,1,i),ssbct(2,2,i),ssbct(2,3,i)*c_a_bct];
ss(1,:,i,4) = n/norm(n);
% bct plane PHASE 4 is BCT
ss(2,:,i,4) = m/norm(m);
% bct direct
ion
ss(3,:,i,4) = [ssbct(3,1,i),ssbct(3,2,i),ssbct(3,3,i)*c_a_bct];
% point 1
ss(4,:,i,4) = [ssbc
t(4,1,i),ssbct(4,2,i),ssbct(4,3,i)*c_a_bct];
% point 2
ss(5,:,i,4) = [ssbct(5,1,i),ssbct(5,2,i),ssbct(5,3,i)*c_a_bct];
% point 3
ss(6,:,i,4) = [ssbct(6,1,i),ssbct(6,2,i),ssbct(6,3,i)*c_a_bct];
% point 4
ss(7,:,i,4) = [ssbct(7,1,i),ssbct(7,2,i),ssbct(7,3,i)*c_a_bct];
% point 5
ss(8,:,i,4) = [ssbct(8,1,i),ssbct(8,2,i),ssbct(8,3,i)*c_a_bct];
% point 6
end
end
%% Loop for grains to establish slip conditions for each grain
EY = zeros(int16(dIDgr(1,1)*1.1),3);
Sfplbv = zeros(mnslp+1,30);
sortmv = zeros(mnslp+1,30,int16(dIDgr(1,1)*1.1));
grcen = zeros(int16(dIDgr(1,1)*1.1));
listSf = zeros(8,dIDgr(1,1));
lists
s = zeros(8,dIDgr(1,1));
grcen(:,1) =
-
1;
% that is a little bigger that needed because some grain numbers are skipped,
% and are thus marked with
-
1. Grains are processed by grain number, not array location
grmax = 0; ngc
ount =
0; fprintf(
'Numbers and vectors computed for Grain # '
);
fnlengthRCB = length(fnameRCB); chr=char(fnameRCB);
Xr = [0;1;2;3];
for
ng=1:1:dIDgr(1,1);
% generalized Schmid factor calculation loop for each grain ng
if
ng>ngcount+dIDgr/10;
n
gcount=ngcount+dIDgr/10;
fprintf(
' %d '
,ng);
end
ig = IDgr(ng,1);
if
ig > grmax
grmax = ig;
end
if
ig > 0
% phase(1) grain center(2,3) eulers(4:6) grain ID
grcen(ig,
1:7) = [IDgr(ng,10) IDgr(ng,5:6
) IDgr(ng,2:4) IDgr(ng,1)];
phid = grcen(ig,4:6);
% phid is Euler phi angles in degrees
ph = grcen(ig,1);
% phase ID set
if
hkl == 1
phid(1) = phid(1)+ 180 ;
% + 180 or +90 t
o convert hkl to TSL software d
efault
if
phid(1)>360
% or +180 to modify TSL Euler angle coordinate system to have X down and Y right;
phid(1) = phid(1)
-
360;
88
elseif
phid(1) < 0
phid(1) = phid(1) + 360;
end
end
g1=[cosd(phid(1)),sind(phid(1)),0;
-
sind(phid(1)),cosd(phid(1)),0; 0,0,1];
g2=[1,0,0; 0,cosd(phid(2)),sind(phid(2)); 0,
-
sind(phid(2)),cosd(phid(2))];
g3=[cosd(ph
id(3)),sind(phid(3)),0;
-
sind(phid(3)),cosd(phid(3)),0;
0,0,1];
g=g3*g2*g1;
if
nsten == 1
sigma_n(:,:,ig) = sigma_n(:,:,1);
sigma_v(:,ig) = sigma_v(:,1);
end
gsgT = g*sigma_n(:,:,ig)*g';
%rotated stress tensor
grcen(ig,7:9) = [0 0 1]*g;
%c
-
axis dir
ection
grcen(ig,10:18) = [g(1,:) g(2,:) g(3,:)];
% Orientation matrix is stored
% calculate elastic modulus to find compliance mismatch
in three principal directions (from Nye textbook on Anisotropy)
if
ph == 5
% needs a differe
nt structure for the sij matrix
-
needs more terms, not working in this version.
elseif
ph == 4
for
i = 1:1:3
% This
gives the modulus in the x (1), y (2), z (3) directions (as looped by i)
e1 = sij(ph,1)*(g(1,i)^4 + g(2
,i)^4) + sij(ph,4)*g(3,i)^4;
e2 = (2*sij(ph,2) + sij(ph,6))*(g(1,i)^2 * g(2,i)^2);
e3 = (2*sij(ph,3) + sij(ph,5))*g(3
,i)^2 * (g(1,i)^2 + g(2,i)^2);
EY(ig,i)=1./(e1+e2+e3);
% NOTE: slip system and plane information not
installed in this version for Sn or TiAl
end
elseif
ph == 2 || ph == 3
for
i = 1:1:3
EY(ig,i) = 1
./(sij(ph,1)
-
2*(sij(ph,1)
-
sij(ph,2)
-
sij(ph,5)/2) *
...
(g(1,i)^2*g(2,i)^2 + g(2,i)^2*g(3,i)^2
+ g(3,i)^2*g(1,i)^2) );
end
elseif
ph == 1
for
i = 1:1:3
EY(ig,i) = 1./(sij(ph,1) * (1
-
g(3,i)^2)^
2 + sij(ph,4) * g(3,i)^4 +
...
(sij(ph,5) + 2*sij(ph,3))*(1
-
g(3,i)^2)*g(3,i)^2 );
end
end
for
j=1:1:nslp(ph)
% direction plane Sfplbv means Schmid factor, plane and Burgers vector (and points on
plane)
Sfplbv(j,1) = j;
% m * sigma * n
Sfplbv(j,2) = ss(2,:,j,ph)*gsgT*ss(1,:,j,ph)';
%
generalized Schmid factor
if
ph == 1 && j>27 && Sfplbv(j,2)<0
Sfplbv(j,2) = 0.001*Sfplbv(j,2) ;
% this is to prevent anti
-
twin shears from being seriously considered later
end
Sfplbv(j,3) = abs(Sfplb
v(j,2));
% abs(generalized schmid factor)
Sfplbv(j,4:6) = g'*s
s(1,:,j,ph)';
% plane normal in lab coords
Sfplbv(j,7:9) = g'*ss(2,:,j,ph)';
% bv direction in lab coords
Sfplbv(j,10:12) = cross(Sfplbv(j,4:6),[0,0,1]);
%
plane trace
for
k = 1:1:6
is = 3*k+10;
ie = is+2;
Sfplbv(j,is:ie) = g'*ss(k+2,:,j,ph)';
% plane plotting vectors from origin to points in cell, in lab coords
end
end
%useful plotting for hexahedral tetragonal unit cell
vectors that sort to bottom row
Sfplbv(mnslp+1,:) = [ph 1
-
1 [1 0 0]*g [0 1 0]*g [0 0 1*c_a(ph)]*g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0];
% don't change g to g' here!
otherwise i
t may make incorrect cubic prisms
sortmv(:,:,ig) = sortro
ws(Sfplbv,
-
3);
% Sort slip systems by Schimd factor
% find slope of first four slip systems; y
-
intercept is close to max Schmid factor
% XRank = [1 0; 1 1; 1 2; 1 3
]; BL = XRank
\
YLSf; BR = XRank
\
YRSf;
YSf(1:4,1) = sort
mv(1:4,3,ig); mdl = fitlm(Xr,YSf);
grcen(ig,19:21) = [mdl.Coefficients{1:2,{
'Estimate'
}}' mdl.Rsquared.Ordinary ];
listSf(1:8,ig) = sortmv(1:8,3,ig); listss(1:8,ig) = sor
tmv(1:8,1,ig);
% end
end
% ig > 0 check
end
% ng
loop
fprintf(
' %d
\
n '
, ng);
for
ii = 1:1:dRBdy(1,1);
gblist(ii,1) = ii;
89
end
% to modify gblist with a short list, paste desired list into this variable as column and delete remaining r
ows
%% Now start processing by grain boundary...
gbcount = 0;
BL = [0 0 0]; BR = [0 0 0];
gbnorm = zeros(dRBdy(1,1),3);
gbtrac = zeros(dRBdy(1,1),3);
damp = zeros(24,dRBdy(1,1));
mpr = zeros(mnslp+2,mnslp+2,dRBdy(1,1));
pln = zeros(mnslp+2,mns
lp+2,dRBdy(1,1));
Bvd =
zeros(mnslp+2,mnslp+2,dRBdy(1,1));
rbvm = zeros(mnslp+2,mnslp+2,dRBdy(1,1));
rbvec = zeros(mnslp+2,3,mnslp+2,dRBdy(1,1));
nbcount = 0; fprintf(
'Computing grain boundary parameters for GB # '
);
jk4max = 1; top3d3 = 0; top3d2 =
0; top3d1 = 0; top3d0 = 0;
gbcheck = sortrows(gblist,1);
while
gbcount < length(gbcheck)
gbcount = gbcount+1;
gbnum = gbcheck(gbcount,1);
% for gbnum = 1:1:dRBdy(1,1); %gbnum is grain boundary number, will calculate m' and other damage parameter
s
if
gbnum>nbcount+dRBd
y/10;
nbcount=nbcount+dRBdy/10;
fprintf(
' %d '
,gbnum);
%, jk4max
end
grL = RBdy(gbnum,13); grR = RBdy(gbnum,14);
% check and reset RBdy to 13 (left grain) and 14 (right grain)
if
grL > 0 &&
grR > 0 && grL <= grmax && grR <= grmax
% dIDgr(1,1) && grR <= dIDgr(1,1)
if
grcen(grL,1) > 0 && grcen(grR,1) > 0
% this is to correct if left and right grains are wrong
if
grcen(grR,2) low tolerance
jk4 = 1;
% counter for
the number of m' calculations made where Schmid factors are > high tolerance
mpmax = 0; mploc = 0;
dpsum = 0; dpsum4 = 0;
mpsum = 0; mpsum4 = 0;
damage = 0; damage4 = 0;
mpr(2,1,gbnu
m) = grL; mpr(1,2,gbnum) = grR; mpr(2,2,gbnum) = gbnum;
% m
-
prime table label for grain numbers in mpr()
if
strcmp(num2str(sigma_v(:,ig)),num2str([1 0 0]'))
% stress axis || [100]
(X)
EgrL = EY(grL,1); EgrR = EY(grR,1);
elseif
strcmp(num2str(sigma_v(:,ig)),num2str([0 1 0]'))
% stress axis || [010] (Y)
EgrL = EY(grL,2); EgrR = EY(grR,2);
elseif
strcmp(num2
str(sigma_v(:,ig)),num2str([0 0 1]'))
% stress axis || [001] (
Z)
EgrL = EY(grL,3); EgrR = EY(grR,3);
elseif
trace(sigma_n(:,:,ig)) == 0
% Crude estimate of shear effects follows, may not be meaningful
EgrL = EY(grL,3)*abs(sigma_n(1,2,ig)) + EY(grL,2
)*abs(sigma_n(1,3,ig)) + EY(grL,1)*abs(sigma_n(2,3,ig));
EgrR = EY(grR,3)*abs(sigma_n(1,2,ig)) + EY(grR,2)*abs(sigma_n(1,3,ig)) + EY(grR,1)*abs(sigma_n(2,3,ig));
else
fprintf(
'Can''t calculate modulus for th
is stress state
\
r'
);
EgrL = 1; EgrR = 1
%pause
end
Eratio = min(EgrL,EgrR)/max(EgrL,EgrR);
% always use Emin/Emax!
F1A = zeros(1,mnslp);
% F1 FIP, Simkin et al. 2003 for grain A
%
F14A = 0; % F1 FIP w/ restriction on Schmid factor value for grain A
F1B = zeros(1,mnslp);
% F1 FIP for grain B
% F14B = 0; % F1 FIP w/ restriction on Schmid factor value for grain B
F1 = 0;
% F1 for grainA/grainB
F14A = zeros(1,mnslp);
% F14 for grainA (with restriction on Schmid fa
ctor value)
F14B = zeros(1,mnslp);
avgdp4 = .75;
% These values indicate instances where values
avgmp4 = .55;
% of dm' or m' are too l
ow to take seriously
% RBdy(gbnum,1:6) = (180/pi).*RBdy(gbnum,1:6);
90
gbnorm(gbnum,:) = [cosd(RBdy(gbnum,8)) sind(RBdy(gbnum,8)) 0];
gbtrac(gbnum,:) = sigma_n(:,:,grL)*gbnorm(gbnum,:)';
for
k = 1:1:mnslp
% Build table of m' values for each grain pair
SchmL = sortmv(k,2,grL);
% Use the correctly signed version of Schmid factor in position 2
mpr(k+2,1,gbnum) = sortmv(k,1,grL) ; mpr(k+2,2,gbnum) = abs(Schm
L) ;
% Schmid factor header grL is down (k goes with L)
if
abs(SchmL) > Sfthr
kmax = k;
end
for
j = 1:1:mn
slp
% j goes across (with R grain in columns)
SchmR = sortm
v(j,2,grR);
% Use the correctly signed version of Schmid factor in position 2
if
k == 1 && abs(SchmR) > Sfthr
% find size of upper left corner of m' mat
rix for which Schmid factors are higher than threshold
j
max = j;
end
mpr(1,j+2,gbnum) = sortmv(j,1,grR) ; mpr(2,j+2,gbnum) = abs(SchmR) ;
% grL goes down, grR across, (j goes with R)
if
abs(SchmL) > SflimL && abs(SchmR) > SflimL
mpl = sortmv(j,4:6,grR)*sortmv(
k,4:6,grL)';
% plane This is a dot product
mbv = sortmv(j,7:9,grR)*sortmv(k,7:9,grL)'*sign(SchmL*SchmR);
% direction This is a dot product
mprime = mpl*mbv;
% if
the b directions are pointing si
milarly mb is positive
mpr(k+2,j+2,gbnum) = mprime; pln(k+2,j+2,gbnum) = mpl; Bvd(k+2,j+2,gbnum) = mbv;
% m', plane and Burgers vector table
is filled in,
rbva = sortmv(j
,7:9,grR)
-
sortmv(k,7:9,grL);
%
store residual Burgers vector
rbvb = sortmv(j,7:9,grR) + sortmv(k,7:9,grL);
% store residual Burgers vector
if
norm(rbva) > norm(rbvb)
rbv = rbvb
;
else
rbv = rbva;
end
rbvm(k+2,j+2,gbnum) = norm(rbv);
rbvec(k+2,1:3,j+2,gbnum) = rbv;
% schmA
F1A(k) = F1A(k)+abs(sortmv(k,3,grL)*dot(sortmv(k,7:9,grL)',sigma_v(:,ig))*dot(sortmv(k,7:9,grL)',sortmv(j,7:9,grR)'));
% F1
for grain A
F1B(j) = F1B(j)+a
bs(sortmv(j,3,grR)*dot(sortmv(j,7:9,grR)',sigma_v(:,ig))*dot(sortmv(j,7:9,grR)',sortmv(k,7:9,grL)'));
% F1 for
grain B
if
mprime > mpthr
% assumes no slip transfer (or damage) occurs if m' < m' threshold or when m'
= 1
dampar = mprime + 0.1;
% pushes m' up by 0.1, so that 1 is worst case (i.e. m'=0.9 is most damaging condition).
if
dampar > 1
dampar = 2
-
dampa
r;
% ass
ume dampar < 1 means less likely to generate damage due to less slip transmission activity
end
dpsum = dpsum + dampar;
% damage parameter only, for all slip systems where m'
> mpthr
mpsum = mpsum + abs(mprime);
% m' only, for all slip systems where m' > mpthr
damage = damage + dampar*max(abs(SchmL),abs(SchmR));
% damage parameter modified by schmid factor
if
(abs(SchmL) > Sfthr && abs(SchmR) > Sfthr) || abs(SchmL) < 0.001 || abs(SchmR) < 0.001
% latt
er condition is for twins,
to enable seeing them later
% assumes that slip transfer happens only if Sf > 0.Sfthr and both
have high schmid factor and capturing effect of anti
-
twin
himp4(jk4,:
,gbnum) = [k j abs(mprime), mprime SchmL SchmR];
% location and values of high m' values for grain pair;
k(rows) is from left grain; j(columns) is from rig
ht grain
dpsum4 = dpsum4 + dampar;
% running sum of slip system in
teractions that have Sf > 0.4
mpsum4 = mpsum4 + abs(mprime);
% running sum of m' for slip system interactions that have
Sf > Sfthr
damage4 = damage4 + dampar*max(abs(SchmL),abs(SchmR));
%
damage parameter modified by schmid factor
if
jk4 > jk4max
jk4max = jk4;
end
jk4 = jk4 + 1;
F14A(k) = F14A(k)+abs(sortmv(k,3,grL)*dot(sortmv(k,7:9,grL)',sigma_v(:,ig))*dot(sortmv(k,7:9,grL)',sortmv(j,7:9,grR)'));
% F14 for grain A
F14B(j
) = F14B(j)+abs(sortmv(j,3,grR)*dot(sortmv(j,7:9,grR)',sigma_v(:,ig))*dot(sortmv(j,7:9,grR)',sortmv(k,7:9,grL)'));
%
F14 for grain B
end
if
mpmax < abs(mpri
me)
% record maximum
m' value % and its location in matrix ?
mpmax = abs(mprime);
end
jk = jk + 1;
end
% of m' > mpthr if statement
end
%
of if statement for values with schmid factors > SflimL
end
% slip system j loop for each B grain in pair
end
% slip system k loop for each A grain in pair
91
% Variables evaluated above in the l
oops: dpsum mp
sum jk4 dpsum4 mpsum4 F1 F14
gbodam = damage*abs(norm(gbtrac(gbnum,:))) ;
% damage parameter modified by apparent GB inclination
gbodam4 = damage4*abs(norm(gbtrac(gbnum,:))) ;
% damage para
meter for high schmid
modified by GB inclination
caxmis = (1
-
(grcen(grL,7:9)*grcen(grR,7:9)')^2)^.5;
% misorientation of c
-
axes (not meaningful for cubic)
avgdp = dpsum/(jk
-
1);
% average value of dm'
avgmp = mpsum/(jk
-
1);
% average value of m'
F1Asort = sort(F1A,
'descend'
); F1Bsort = sort(F1B,
'descend'
);
F14Asort = sort(F14A,
'descend'
); F14Bsort = sort(F14B,
'descend'
);
maxF1 = max(F1Asort(1),F1Bsort(1));
maxF14 = max(
F14Asort(1),F14Bsort(1));
if
jk4> 1
avgdp4 = dpsum4/(jk4
-
1);
%average value of damage for slip systems with Sf > 0.4
avgmp4 = mpsum4/(jk4
-
1);
%average value of m' for slip systems with Sf > 0.4
end
jk6 = 0; jktop3or6 = 0;
% counters for number of m' values to average later.
top3mpn = 0; top6mpn = 0; t3mpthr = 0; maxmpkj = 0; maxSfk = 0; maxSfj = 0; ma
xkjSfs = 0; Sfsum = 2;
pairsum = zeros(mnslp:mnslp); pairpro
d = zeros(mnslp:mnslp);
mpSflist = zeros(kmax*jmax:6); mpSfshlist = zeros(1,36); ikj = 1;
for
k = 1:1:kmax
% Rather than finding all m' values for Sf >
tolH, look only for top 3 or top 6 Sf value pairs
for
j =
1:1:jmax
% find sum of schmid factors for each element of mpr array and put in pairsum(), similarly for pairprod(uct)
pairsum(k,j) = mpr(k+2,2,gbnum) + mpr(2,j
+2,gbnum); pairprod(k,j) = mpr(k+2,2,gbnum) * mpr(2,j+2,gbnum);
mpSflist(ikj,1:6) = [k j abs(mpr(k+2,2,gbnum)) abs(mpr(2,j+2,gbnum)) abs(mpr(k+2,j+2,gbnum)) rbvm(k+2,j+2,gbnum)];
ikj = ikj + 1;
end
end
sortmplist = sortrows(mpSflist,
-
5);
for
ii = 1:1:6
mpSfshlist(1,6*(ii
-
1)+1:6*ii) = sortmplist(ii,1:6);
end
% parameters for max m' are in first row; k j Sf(kL) Sf(jR) m' rbvm
maxSfk = mpSfshlist(1,1); maxSfj = mpSfshlist(1,2); maxmpkj = mpSfshl
ist(1,5);
maxkjSfs = mpSfshlist(1,3)+mpSfshlist(1,4); maxkjSfp = mpSfshlist(1,3)*mpSfshlist(
1,4);
maxkjsmp = maxkjSfs*maxmpkj; maxkjpmp = maxkjSfp*maxmpkj; maxmprbv = mpSfshlist(1,6);
while
(jk6 < 6 || jk
top3or6 < 3) && Sfsum > 0
kjmax = [0 0 0 0];
for
k = 1:1:kmax
% find location of highest pairsum (and product) in current pairsum array
for
j = 1:1:jmax
if
pairsum(k,j)>kjmax(1)
%finds largest Spair value in current kmax x jmax Pairsum() array
kjmax = [pairsum(k,j) k j pairprod(k,j)];
%identifies location k j of sum of Schmid factors
end
end
end
Sfsum = kjmax(1);
mpchk = abs(mpr(kjmax(2)+2
,kjmax(3)+2,gbnum));
% puts (next) m' into mpchk
if
jk6 < 6 && Sfsum > 0
top6mpn = top6mpn + mpchk;
pairsum(kjmax(2),kj
max(3)) =
-
1;
% Now that the highest pairsum value is found and added to sum, m
ake it unfindable
jk6 = jk6 + 1;
if
jk6 == 3;
top3mpn = top6mpn;
% capture top 3 values in this variable
end
end
if
jktop3or6 < 3 && Sfsum > 0
pairsum(kjmax(2),kjmax(3)) = pairsum(kjmax(2),kjmax(3))
-
1;
% mark position with
-
2 if inside this query
if
mpchk > mpthr
t3mpthr = t3mpthr + mpchk;
% more stringe
nt criterion for m' only > mpthr used.
jktop3or6 = jktop3or6 + 1;
% [gbnum t3mpthr mpchk jktop3or6 maxkj]
end
end
end
top6mpn = top6mpn/jk6;
% These average values of m' are without regard to magnitude of m'
if
top3mpn > 0
top3mpn = top3mpn/3;
else
top3mpn = top6mpn;
end
if
jktop
3or6 == 3
92
t3mpthr = t3mpthr/jktop3or6;
top3d3 = top3d3 + 1;
elseif
jktop3or6 == 2
t3mpthr = t3mpthr/jktop3or6;
top3d2 = top3d2 + 1;
elseif
jktop3or6 ==
1
t3mpthr = t3mpthr/jktop3or6;
top3d1 = top3d1 + 1;
elseif
jktop3or6 == 0
t3mpthr = 0;
top3d0 = top3d0 + 1;
end
mpr(1,1,gbnum) = mpmax;
% GB#
, left gr, right gr, angle/axis of misorientation, average of top 3 m' values in upper kj box, maximum m' in upper kjbox, co
rresponding
Schmid factor sum,
% product between Schmid factor sum and maximum m' value, row for k (left grain), col
umn for j (rig
ht grain),
% left grain y
-
intercept, slope, and correlation coefficient, right grain y
-
intercept, slope, and correlation coefficient.
slpint = [
-
min(grcen(grL,20),grcen(grR,20))
-
max(grcen(grL,20),grcen(grR,20))
...
% slope, inte
rcept, and R^2
values for left and right
grains
max(grcen(grL,19),grcen(grR,19)), min(grcen(grL,19),grcen(grR,19)) grcen(grL,19:21) grcen(grR,19:21)];
mpmaxgrp = [t3mpthr maxmpkj maxkjSfs maxkjsmp maxkjSfp maxkjpmp maxmprbv maxSfk maxSfj];
tablepub(gbnum,:) = [gbnum grL grR RBdyn(gbnum, 7:10) mpmaxgrp slpint mpSfshlist] ;
damp(:,gbnum) = [maxSfk maxmpkj maxSfj dpsum da
mage gbodam caxmis dpsum4 damage4 gbodam4 avgdp avgdp4 avgmp avgmp4
...
maxF1 maxF14 ma
xF1*Eratio maxF14*Eratio Eratio top3mpn top6mpn t3mpthr kmax jmax];
% this is a summary matrix used for
plotting
end
% of if for grains with positive ID
end
% of if for valid grain pair
end
% of gbnum loop
gbcount = 1;
for
ii = 1:1:length
(tablepub)
if
tablepub(ii,1) > 0
tablepubshort(gbcount,:) = tablepub(ii,:); gbcount = gbcount+1;
end
end
fprin
tf(
' %d
\
n '
, gbnum);
plotname = {
' mp5k '
,
' maxmpkj '
,
' mp5j '
,
' dm''sum '
,
' m*dm''sum '
,
' gbo*m*dm''sum '
,
' cax
-
mis '
,
'
dm''sum4 '
,
...
' m*dm''4 '
,
' gbo*dam4 '
,
' norm dm'' '
,
' norm dm''4 '
,
' norm m'' '
,
' norm m''4 '
,
...
' max(F1A,F1B) '
,
' max(F14A,F14B) '
,
' Emax(F1A,F1B) '
,
' Emax(F14A,F14B) '
,
' Eratio '
,
'top3mpn'
,
'top6mpn'
,
't3mpt
hr'
};
%% Choose what to plot along the grain boundary map from variables 4
-
19 noted above
mpthr = mpthr; chsize = 16;
plist = [22 2];
%13 14 11 12 15 16 17 18 19 20 21
for
k = 1:1:length(plist);
% 1:1:length(plist); % 4:4:4 %
plnx = plist(k);
if
plnx == 13 || plnx == 14
bins = [.60 .64 .68 .72 .76 .80 .84 .88 .92 .96];
% for mp or mp4 or Eratio
elseif
plnx == 11 || plnx == 12
bins = [.80 .82 .84 .86 .88 .90 .92 .94 .96 .98];
% for normp or normp4
elseif
plnx == 7
bins = [0 0.156 0.309 0.454 0.588 0.707 0.809 0.891 0.951 0.988 1.000];
% for c
-
axis plot
% Corresponding degrees for bins above = [0 9 18 27 36 45 54 63 72 81 90]; % for c
-
axis plot
elseif
plnx == 22 || plnx == 2
bins = mpt
hr
:(1
-
mpthr)/10:1;
else
bins = linspace(min(nonzeros(damp(plnx,:))), max(damp(plnx,:)), 11);
% auto bin size for anything
end
binsdat = bins;
% temporary storage while setting up plot scale
max1 = max(RBdy(:,9));
% max
x
value for grR
max2 = max(RBdy(:,11));
% max x value for grL
maxx = max([max1; max2]);
may1 = max(RBdy(:,10));
% max y value for grR
may2 = max(RBdy(:,12));
% max y value for grL
maxy = max([may1; may2]);
f = fig
ure(
'Position'
, [0,0,750,500]); movegui(f,
'northwest'
); set(gcf,
'Color'
, [1 1 1]); hold
on
;
% plot is based upon TV rastering, as given by TSL
axis([0 maxx*2
-
maxy .15*maxy ]); axis
image
;
%equal;
set(gcf,
'Color'
,[1,1
,1])
%
surrounding field is this color
text(maxx*
-
0.12, 0.11*maxy,
'm'' '
,
'Fontsize'
, chsize);
93
text(maxx*
-
0.04, 0.11*maxy, num2str(binsdat(1,1),
'%4.3f'
),
'Fontsize'
, chsize);
bins = [10 20 30 40 50 60 70 80 90 100];
% To set color key fo
r boundarie
s
bcnt = 2;
for
gbnum = 1:1:100;
wid = 10; widk = 2;
% thickness for width bar
if
fix(gbnum/10) 90;
wid = 4; widk=4;
elseif
gbnum > 80;
wid = 3; widk=3;
end
else
text(maxx/100*(gbnum
-
1)
-
maxx*0.03, 0.11*maxy, num2str(binsdat(1,bcnt),
'%4.3f'
),
'Fontsize'
, chsize); bcnt = bcnt + 1;
end
vec = gbnum;
if
vec purple
elseif
vec>=bins(1) && vec blue
elseif
vec>=bins(2) && vec turquoise
elseif
vec>=bins(3) && vec dk
grn
elseif
vec>=bins(4) && vec green
elseif
vec>=bins(5) && vec yellow
elseif
vec>=bins(6) && vec orange
elseif
vec>=bins(7) && vec red
elseif
vec>=bins(8) && vec pink
elseif
vec>=bins(9) && ve
c
pink
elseif
vec > bins(10);
vRGB = [.6 0. .2];
%[.6 0. .3] m
auv
e bins(9)
end
plot(maxx/100*[gbnum
-
1; gbnum],[0.06*maxy; 0.06*maxy],
'Linewidth'
,wid,
'color'
,vRGB);
% place color key x coordinate at suitable place...
end
bins = binsdat;
% replacing the necessary bins parameters for plotting
along
grain boundaries
for
gbnum = 1:1:dRBdy(1,1);
% Plot parameter values on grain boundaries
if
RBdy(gbnum,13) > 0 && RBdy(gbnum,14) > 0 && RBdy(gbnum,13) <= grmax && RBdy(gbnum,14) <= grmax
% wid = (RBdy(iGB,16)
-
.95*minF1m)/(
maxF1
m
-
minF1m)*4;
wid = 2;
if
damp(plnx,gbnum) > bins(9)
wid = 4;
elseif
damp(plnx,gbnum) > bins(8)
wid = 3;
end
kind = damp(1,gbnum); jind = damp(1,gbnum);
kss = mpr(kind,1,gbnum); jss = mpr(1,jind,gbnum);
if
one_ss == 1 && kss < 7 && jss < 7
linetype =
':'
;
% dotted lines for hexagonal slip systems of basal and/or prism
elseif
one_ss == 1 && kss < 7 || one_ss =
= 1 &
& jss < 7
linetype =
'
-
.'
;
% dot dash lines for hexagonal slip systems if one ss is basal or prism
else
linetype =
'
-
'
;
end
vec = damp(plnx,gbnum);
if
vec purple
elseif
vec>=bins(1) && vec blue
elseif
vec>=bins(2) && vec turquoise
elseif
vec>=bins(3) && vec
dk grn
elseif
vec>=bins(4) && vec green
elseif
vec>=bins(5) && ve
c yellow
elseif
vec>=bins(6) && ve
c orange
elseif
vec>=bins(7) && vec red
elseif
vec>=bins(8) && vec pink
elseif
vec>=bins(9) && vec
pink
elseif
vec > bins(10);
vRGB = [.6 0. .2];
%[.6 0. .4] mauve bin
s(9)
en
d
plot([RBdy(gbnum,9);RBdy(gbnum,11)],[
-
RBdy(gbnum,10);
-
RBdy(gbnum,12)],linetype,
'Linewidth'
,wid,
'color'
,vRGB);
end
end
end
% k for plist
set(gca,
'FontSize'
,chsize);
xlabel(
'Position, microns'
);
ylabel(
'Position, microns'
);
%% pl
ot grain numbers
chsize = 14;
for
ng = 1:1:grmax
if
grcen(ng,1) == 1 || grcen(ng,1) == 4
text(grcen(ng,2),
-
grcen(ng,3),int2str(ng),
'color'
,[0 .5 1],
'FontWeight'
,
'bold'
,
'FontSize'
,chsize);
%[ng grcen(ng,4:6)]
elseif
grcen(ng,1) == 2
text(grcen(ng,2),
-
grcen(ng,3),int2str(ng),
'color'
,[0 .5 0],
'FontWeight'
,
'bold'
,
'FontSize'
,chsize);
elseif
grcen(ng,1) == 3
text(grcen(ng,2),
-
grcen(ng,3),int2str(ng),
'color'
,[.8 0 0],
'FontWeight'
,
'bold'
,
'F
ontSize'
,chsize);
end
end
% % plot gb numbers
for
gbnum = 1:1:dRBdy
% plot grain boundary numbers
if
RBdy(gbnum,13) > 0 && RBdy(gbnum,14) > 0 && RBdy(gbnum,13) <= grmax && RBdy(gbnum,14) <= grmax
text((2*RBdy(gbnum,9)+RBdy(gbnum,11))/3,
-
(2*RBdy(gbnum,10)+RBdy(gb
num,12))/3,
...
int2str(gbnum),
'color'
,[0 0 .5],
'FontWeight'
,
'light'
,
'F
ontSize'
,chsize);
end
end
%% Plot of rotated grain around grain 1... (for single intial grain pair with file of rotated orientations)
clear
m1
; clear
m2
; clear
m3
; symbsize = 15;
colorvec = [0 .2 .7 ; 1 0 0 ; 1 .8 0 ; .7 0 .6 ; 0 .8 0 ; 0 1 1 ; .6
0 .2 ; 0 0 1 ; 1 .4 0 ];
for
ii = 1:1:360
m1(ii,:) = mpr(3,3,ii) ;
m2(ii,:) = [mpr(3,4,ii) mpr(4,4,ii) mpr(4,3,ii) ];
m3(ii,:) = [mpr(3,5,ii) mpr(4,5,ii) mpr
(5,5,ii) mpr(5,4,ii) mpr(5,3,ii) ];
m1s(ii,:) = round( symbsize * mpr(3,2,ii) * mpr(2,3,ii) + 1 ) ;
m2s(ii,:) = [round( symbsize*1.3 * mpr(3,2,ii) * mpr(2,4,ii) + 1 )
...
r
ound( symbsize*1.0 * mpr(4,2,ii) * mpr(2,4,
ii) + 1 )
...
round( symbsize*0.7 * mpr(4,2,ii) * mpr(2,3,ii) + 1 ) ];
m3s(ii,:) = [round( symbsize*1.3 * mpr(3,2,ii) * mpr(2,5,ii) + 1 )
...
round( symbsize
*1.3 * mpr(4,2,ii) * mpr(2,5,ii) + 1 )
...
round( symbsize*1.0 * mpr(5,2,ii) * mpr(2,5,ii) + 1 )
...
round( symbsize*0.7 * mpr(5,2,ii) * mpr(2,4,ii) + 1 )
...
round( symbsize*0.7 * mpr(5,2,ii) * mpr(2,3
,ii) + 1 ) ];
95
end
f = figure(
'Position'
, [
0,0,800,600]); movegui(f,
'northwest'
); set(gcf,
'Color'
, [1 1 1]); hold
on
;
plot(m1,
'LineWidth'
,3)
plot(m2,
'
--
'
,
'LineWidth'
, 2)
plot(m3,
':'
,
'LineWidth'
, 2)
legend(
'mp11'
,
'mp12'
,
'mp22'
,
'mp21'
,
'mp13'
,
'mp23'
,
'mp33'
,
'mp32'
,
'mp31'
,
'Location'
,
'southw
est'
);
ylabel(
'm prime value'
);
xlabel( [mat2str(grcen(1,4:6))
' rotation incremented about Z axis of '
mat2str(grcen(2,4:6))] );
hold
off
;
f = figure(
'Position'
, [0,0,800,600]); movegui(f,
'northwest'
);
set(gcf,
'Color'
, [1 1 1]); hold
on
;
for
ii = 1
:1:360
plot(ii, m1(ii,1),
'+'
,
'MarkerSize'
, m1s(ii,1),
'color'
,colorvec(1,:));
for
jj = 1:1:3
plot(ii, m2(ii,jj),
'o'
,
'MarkerSize'
, m2s(ii,jj),
'color'
,colorvec(jj+1,:));
end
for
jj = 1:1:5
plot(ii, m3(ii,jj),
'^'
,
'MarkerSize'
,
m3s(ii,jj),
'color'
,colorvec(jj+4,:));
end
end
legend(
'mp11'
,
'mp12'
,
'mp22'
,
'mp21'
,
'mp13'
,
'mp23'
,
'mp33'
,
'mp32'
,
'mp31'
,
'Location'
,
'southwest'
);
ylabel(
'm prime value (size = sum of SF)'
);
xlabel( [mat2str(grcen(1,4:6))
' rotation incremented a
bout Z axis of '
mat2str(grcen(2,4:6))] );
hold
off
;
%% Choose your favorite grain boundary
---------------------------------------
gbnum = 276
mpr_cur = mpr(:,:,gbnum); rbvm_cur =
rbvm(:,:,gbnum); rbvec_cur = rbvec(:,:,:,gbnum); mplimit = .6
whiteanno
tation = 1;
% make gb trace and axes white
ptpl = 1;
% plots plane traces if = 1 Red dashed line in plot is perpendicular to the line connecting the centers of the two grains
prsxpl
t = zeros(8,7);
% Black solid line in plot is the RC b
oundary segment orientation
prsyplt = zeros(8,7);
% 13 and 14 lead to misplaced plane traces.
prszplt = zeros(8,7);
clear
Sflist
; clear
sortSflist
;
mp4gr = himp4(:,:,gbnum);
sorth
imp4 = sortrows(mp4gr,
-
4);
% This sorts on basis of actual m' value
mp
limit = min(mplimit, sorthimp4(1,4));
gbcen = [RBdy(gbnum,11)+RBdy(gbnum,9) RBdy(gbnum,12)+RBdy(gbnum,10)]/2;
v13cpos = [grcen(RBdy(gbnum,13),2) grcen(RBdy(gbnum,13),3)]
-
gbcen;
% find vector from center of GB to grain center in raster coordinates
v14cp
os = [grcen(RBdy(gbnum,14),2) grcen(RBdy(gbnum,14),3)]
-
gbcen;
v1314b = [(RBdy(gbnum,12)
-
RBdy(gbnum,10))
-
(RBd
y(gbnum,11)
-
RBdy(gbnum,9)) ];
% vector [dy,
-
dx] pointing perpendicular to GB
if
v13cpos * v1314b' < 0
v1314b =
-
v1314b;
end
v1314bn = v1314b/
norm(v1314b)
% find unit vector pointing perpendicular to GB in raster coords
plabel = [
'Sfthr k j: '
num2str(Sfthr,3)
' '
num2str(damp(23,gbnum),2)
' '
num2str(damp(24,gbnum),2)
' t3mpthr '
num2str(damp(22,gbnum),3)];
g_gb_g = [num2str(RBdy(gbnum,13))
' '
mat2str(grcen(RBdy(gbnum,13),7:9),4)
' '
num2str(gbnum)
' '
mat2str(grcen(RBdy(gbnum,14),7:9),4)
'
'
num2str(RBdy(gbnum,14)) ];
ipl =
-
6; imp = 0;
% Strategy: Next, start isc loop for plotting slip systems
while
sorthimp4(imp+1,4) >= mplimit
im
p = imp + 1;
if
sorthimp4(imp,1) ~= 0
% evaluate only for recorded values (m'>.6)
if
ipl
-
imp==
-
7
% six plots on a page
f = figure(
'Position'
, [0,0,1200,750]); movegui(f,
'northwest'
); set(gcf,
'Color'
, [1 1 1]); hold
on
;
ipl=ipl+6;
end
subplot(2,3,imp
-
ipl); hold
on
; set(gcf,
'Color'
, [1 1 1]);
if
whiteannotation == 1
set(gca ,
'ycolor'
,
'w'
); set(gca ,
'xcolor'
,
'w'
);
% make axes white for ease in later arranging.
else
plot([0 1.5*cosd(RBdy(
gbnum,8))], [0 1.5*sind(RBdy(gbnum,8))],
'
-
k'
);
% plots gb from map from angle given in normal x
-
y space
end
Sfsum = 0;
kind = sorthimp4(imp,1)+2; jind = sorthimp4(imp,2)+2;
kss = mpr(kind,
1,gbnum); jss = mpr(1,jind,gbnum);
96
for
igr = 13:1:14
% 1 i.e. first for the left grain in column 13, then the right grain in column 14 in checked Reconstructed Boundary file.
grnum = RBdy(gbnum,igr);
issr = sorthimp4(imp,i
gr
-
12);
% ss rank # in gr13 issr
is slip system Schmid factor order #
if
igr == 13
del = v1314bn;
cellcolor = [0 0 0];
%[.3 0 .5]; plot ([0 del(1)],
-
[0 del(2)])
else
% del i
s position vector from center of
gb to 13 in raster coord system (ydown)
del =
-
v1314bn;
cellcolor = [0 0 0];
%[0.5 0 0]; plot ([0 del(1)],
-
[0 del(2)])
end
% Plot the image of hexagonal unit
cell, slip vectors, planes, plane
normals, and plane traces
% Strategy: First extract useful vectors to draw the hexagonal prisms from slip system information
% positions in mvs p1:13
-
15 p2:16
-
18 p3:19
-
21 p4:22
-
24 p5:25
-
27 p6:28
-
30
% posi
tions in hpln p1:4
-
6 p2:7
-
9
p3:10
-
12 p4:13
-
15 p5:16
-
18 p6:18
-
21
for
isc = 1:1:nslphex
if
sortmv(isc,1,grnum) == 1;
% locate basal planes using SS1
hpln(1,4:21) = sortmv(i
sc,13:30,grnum);
% bottom basal plane
hpln(2,4:21) = sortmv(isc,13:30,grnum);
% top basal plane
rotc = sortmv(isc,4:6,grnum)*c_a_hex;
% basal plane norma
l * c/a
for
j = 4:3:19
hpln(2,j:j+2) = hpln(1,j:j+2) + rotc;
% move top plane up by a unit of c
end
a1 = sortmv(isc,7:9,grnum);
% locate a1 using SS1
elseif
sortmv(isc,1,grnum)
== 2;
a2 = sortmv(isc,7:9,grnum);
% locate a2 using SS2
elseif
sortmv(isc,1,grnum) == 3;
a3 = sortmv(isc,7:9,grnum);
% locate a3 using
SS3
end
end
for
isc = 1:1:nslphex
if
sortmv(isc,1,grnum) == 4;
% locate two prism planes on opposite sides using SS4
hpln(3,4:21) = sortmv(isc,13:30,grnum);
for
j = 13:3:28
hpln(4,j
-
9:j
-
7) = sortmv(isc,j:j+2,grnum) + a2
-
a3;
end
elseif
sortmv(isc,1,grn
um) == 5;
% locate two prism planes on opposite sides using SS5
hpln(5,4:21
) = sortmv(isc,13:30,grnum);
for
j = 13:3:28
hpln(6,j
-
9:j
-
7) = sortmv(isc,j:j+2,grnum) + a3
-
a1;
end
elseif
sortmv(isc,1,grnum) == 6;
% locate two prism pl
anes on opposite sides using SS6
hpln(7,4:21)
= sortmv(isc,13:30,grnum);
for
j = 13:3:28
hpln(8,j
-
9:j
-
7) = sortmv(isc,j:j+2,grnum) + a1
-
a2;
end
end
end
for
j = 1:1:2
% Find z elevation of basal planes
for
k = 1:1:3
hpln(j,k) = (hpln(j,3+k)+hpln(j,6+k)+hpln(j,9+k)+hpln(j,12+k)+hpln(j,15+k)+hpln(j,18+k))/6;
end
end
center = (h
pln(1,1:3)+pln(2,1:3))/2;
for
j = 3:1:8
% Find z elevation of prism planes
hpln(j,3) = (hpln(j,12)+hpln(j,15)+hpln(j,18)+hpln(j,21))/4;
end
sortpln = sortrows(hpln,
-
3);
minx = 0; miny = 0; minz = 0; maxx = 0; maxy = 0; maxz = 0;
for
j = 1:1:8
% assemble vectors for plotting faces of hex prism
prsxplt(j,1:7) = [sortpln
(j,4) sortpln(j,7) sortpln(j,10) sortpln(j,13) sortpln(j,16) sortpln(j,19) sortpln(j,4)];
minx = min(minx,min(prsxplt(j,:))); maxx = max(maxx,max(prsxplt(j,:)));
prsyplt(j
,1:7) = [sortpln(j,5) sortpln(j,8) sortpln(j,
11) sortpln(j,14) sortpln(j,17) sortpln(j,20) sortpln(j,5)];
miny = min(miny,min(prsyplt(j,:))); maxy = max(maxy,max(prsyplt(j,:)));
prszplt(j,1:7) = [sortpln(j,6) sortpln
(j,9) sortpln(j,12) sortpln(j,15) sortpln(j,1
8) sortpln(j,21) sortpln(j,6)];
minz = min(minz,min(prszplt(j,:))); maxz = max(maxz,max(prszplt(j,:)));
end
97
sp1 = sortmv(issr,13:15,grnum);
% identify
plotted points on the slip plane
sp2 = sortmv(issr,16:18,grnum);
sp3 = sortmv(issr,19:21,grnum);
sp4 = sortmv(issr,22:24,grnum);
sp5 = sortmv(issr,25:27,grnum);
sp6 = sortmv(is
sr,28:30,grnum);
spx = [sp1(1
) sp2(1) sp3(1) sp4(1) sp5(1) sp6(1) sp1(1)];
spy = [sp1(2) sp2(2) sp3(2) sp4(2) sp5(2) sp6(2) sp1(2)];
ssn = sortmv(issr,1,grnum);
% slip system number
Sf = sortmv(issr,2,grnum);
% Schmid
factor
Sfsum = Sfsum + abs(Sf);
n = [0 0 0 sortmv(issr,4:6,grnum)];
% plane normal
b = [sp1 sp4];
% p1+sortmv(issr,7:9,grnum)]; % Burgers vector
pt = sortmv(issr,10:12,grnum);
% plane
trace
midx = (minx+maxx)/2;
midy = (miny+maxy)/2;
cellcenter = [midx midy];
dx =(1.6*del(2)
-
cellcenter(1));
% del is raster, cellcenter is TSL coords
dy =(1.6*del(1)
-
cel
lcenter(2));
% so not dy = del(2)
-
cellcenter(2);
[dx dy];
% diagnostic
% These plots will match TSL with X down !!!! Plotting starts
if
grcen(grnum,5) < 90
% if PHI < 90, then make the 3 coo
rdinate axes visible below slip planes
plot([0 a1(2)]+dy,
-
([0 a1(1)]+dx),
':'
,
'Linewidth'
,3,
'Color'
,[1 0 .2]);
% plot x = red
plot([0 a2(2)]+dy,
-
([0 a2(1)]+dx),
':'
,
'Linewidth'
,3,
'Color'
,[.6 .8 0]);
% plot y = green
-
gold
plot([0 a3(2)]+dy,
-
([0 a3(1)]+dx),
':'
,
'Linewidth'
,
3,
'Color'
,[0 0 1]);
% plot z = blue
end
if
sortmv(issr,6,grnum)>0
% is k component of slip plane normal positive or negative?
fill(spy+dy,
-
(spx+dx), [.8 .8 .65])
% slip plane filled warm gray
% plot([n(2) n(5)],
-
[n(1) n(4)],'Linewidth',3,'Color',[.8 .8 .65]);
else
% slip plane filled cool gray if normal has neg z component
fill(spy+dy,
-
(spx+dx), [.65 .65 .7])
% plot([n(2) n(5)],
-
[n(1) n(4)],'Linewidth',3,'Color',[.65 .65 .7]);
end
if
sortmv(issr,6,grnum)>0
Bvcolor = [0 .7 .7];
if
ssn >= iC1
Bvcolor = [.1 .6 0];
end
if
ssn >= iT1 && ssn <= fT2
Bvcolor = [1 .6 0];
end
else
Bvco
lor = [0 1 1];
if
ssn >= iC1
Bvcolor = [.3 .9 0];
end
if
ssn >= iT1 && ssn <= fT2
Bvcolor = [1 .8 0];
en
d
end
Sfs = 1;
if
ssn < iT1
Sfs = sign(Sf);
end
if
Sf > 0
% plot Burgers vector direction
if
ssn >= iT1
% this is for twins
-
the Burgers vector length is
shown to be 1/2 of the usual length in the unit cell
plot(b(2)+dy,
-
(b(1)+dx),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) (b(2)+b(5))/2]+dy,
-
([b(1) (b
(1)+b(4))/2]+dx),
'Linewidth'
,4,
'Color'
,Bvcolor)
else
plot(b(2)+dy,
-
(b(1)+dx),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) b(5)]+dy,
-
([b(1) b(4)]+dx),
'Linewidth'
,4,
'Color'
,Bvcolor)
end
else
% plot Burgers vector in opposite direction
if
ssn >= iT1
% this is for twins
-
the Burgers vector length is shown to be 1/2 of the usual length in the unit cell
plot(b(2)+dy,
-
(b(1)+dx),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
plot([b(2) (2*b(2)+b(5))/3]+dy,
-
([b(1) (2*b(1)+b(4))/3]+dx),
'Linewidth'
,4,
'Color'
,Bvcolor)
else
plot(b(5)+dy,
-
(b(4)+dx),
'.'
,
'MarkerSize'
, 24,
'Color'
, Bvcolor)
98
plot([b(5) b(2)]+dy,
-
([b(4) b(1)]+dx),
'Linewidth'
,4,
'Color'
,Bvcolor)
end
end
for
j = 1:1:4
% plot the 4 top most surface prisms of the hex cell that have the highest z elevation
plot(prsyplt(j,:)+dy,
-
(prsxplt(j,:)+dx),
'Linewidth'
, 2,
'Color'
, cellcolor);
end
if
gr
cen(grnum,5) > 90
% if PHI < 90, make the 3 coordinate axes visible above slip planes
plot([0 a1(2)]+dy,
-
(
[0 a1(1)]+dx),
':'
,
'Linewidth'
,3,
'Color'
,[1 0 .2]);
% plot x = red
plot([0 a2(2)]+dy,
-
([0 a2(1)]+dx),
':'
,
'Linewidth'
,3,
'Color'
,[.6 .8 0]);
% plot y = green
-
gold
plot([0 a3(2)]+dy,
-
([0 a3(1)]+dx),
':'
,
'Linewidth'
,3,
'Color'
,[0 0 1]);
% plot z = blue
end
if
ptpl == 1
if
ssn >= iC1
ptrcolor = [.2 .8 0];
% compression twin plane traces green
elseif
ssn >= iT1 && ssn <= fT2
ptrcolor = [1 .6 0];
% extension twin pla
ne traces orange
elseif
ssn > ipyrc && ssn <= f2pyrc
ptrcolor = [.95 .85 0];
% plane traces green
-
gold
elseif
ss
n <= fpyra && ssn >= ipyra
ptrcolor = [0
.9 .5];
% pyr green
-
blue
elseif
ssn <= f2prs && ssn >= iprs
ptrcolor = [1 .2 0];
% prism red
else
ptrcolor = [0 0 1];
% basal blue
end
%
----
> NOTE that Schmid factor vector is plotted in correct direction,
plot([
-
pt(2) pt(2)]+dy,
-
([
-
pt(1) pt(1)]+
dx),
'
--
'
,
'Linewidth'
,3,
'Color'
,ptrcolor)
end
if
igr == 13
line1 = [
'L g'
num2str(grnum)
' m'
num2str(issr)
' = '
num2str(Sfs*Sf, 2)
' ss'
num2str(ssn)
...
' n'
mat
2str(sshex(1,:,ssn)) mat2str(Sfs*sshex(2,:,ssn))
'b'
];
Sflist(imp,1:3) = [grnum ssn Sf];
end
if
igr == 14
title({[line1] [
'R g'
num2str(grnum)
' m'
num2str(issr)
' = '
num2str(Sfs*Sf
, 2)
' ss'
num2str(ssn)
...
' n'
mat2str(sshex(1,:,ssn)) mat2str(Sfs*sshex(2,:,ssn))
'b'
] [g_gb_g] [plabel
' m'' = '
, num2str(sorthimp4(imp,4),3)]
...
[
'rBvec = '
mat2str(rbvec(kind,:,jind,gbnum),3)
'mag = '
num2s
tr(rbvm_cur(kind,jind),3)]} );
Sflist(imp,4:6) = [grnum ssn Sf];
end
end
% graincen for cubic or hex if (has middle else)
end
% igr
-
grains 13 and 14
rbvgry = [max
(0,1
-
rbvm_cur(k
ind,jind)) 0 0]
%; / max(max((rbvm_cur)));
plot(0, 0,
'k+'
);
% gray scale of residual Burgers vector
plot([0 rbvec(kind,2,jind,gbnum)]*2,
-
[0 rbvec(kind,1,jind,gbnum)]*2,
'
-
'
,
'Color'
, rbvgry,
'LineWidt
h'
, 2);
axis
square
;
axis([
-
3 3
-
3 3]*1.5*(1.4
-
Sfsum));
% plot representation of grain boundary 1.2 for cubic, 1.4 for hex
Sflist(imp,7:9) = [imp Sfsum sorthimp4(imp,4)];
else
fprintf(
'No slip system has
Schmid factors > %4.2f fo
r one of the two grains at GB %d %d
\
n'
, Sfthr, gbnum, imp)
end
% if statement for continuing the loop for non
-
zero himp4 values
end
% imp m' loop
sortSflist = sortrows(Sflist,
-
8); sortSfsize = size(sortSflist);
99
APPENDIX C: Sample reconstructe
d boundary and grain files
Sample reconstructed boundary
file
# Header: Project1::post scan rotated cleaned cropped::All data::Grain Size 2/12/2020
#
# Column 1
-
3: right hand
average orientation (phi1, PHI, phi2
in radians)
# Column 4
-
6: left hand average orientation (phi1, PHI, phi2 in radians)
# Column 7: Misorientation Angle
# Column 8
-
10: Misorientation Axis in Right Hand grain
# Column 11
-
13: Misorientation Axis in
Left Hand grain
# Column 14: lengt
h (in microns)
# Column 15: trace angle (in degrees)
# Column 16
-
19: x,y coordinates of endpoints (in microns)
# Column 20
-
21: IDs of right hand and left hand grains
1.367 1.287 4.784 0.863 1.274 5.
234 32.52
-
18 14
-
11
-
18 14
-
11 18.009 23.6 0.00 49.07 16.50 41.86 27 1
1.367 1.287 4.784 0.863 1.274 5.234 32.52
-
18 14
-
11
-
18 14
-
11 14.503 178.9 16.50 41.86 31.00
42.15 27 1
1.367 1.2
87 4.784 0.284 0.699 0.128 59.88
-
9 14 1
-
9 14 1 23.116 5.7 0.00 73.90 23.00 71.59 27 43
4.068 0.194 2.325 0.284 0.699 0.128 51.35
-
11 4
-
3
-
11 4
-
3 19.218 8.6
0.00 142.61 19.00 139.72 76 43
0.339 1.343 5.643 4.068 0.194 2.325 90.01 28
-
1
-
8 28
-
1
-
8 11.846 32.4 0.00 182.44 10.00 176.09 96 76
0.339 1.343 5.643 0.394
1.268 5.600 5.47 24
-
11 7 24
-
11 7
2.887 90.0 0.00 196.88 0.00 199.76 96 112
100
Sample of grain file
# Header: Project1::post scan rotated cleaned cropped::All data::Grain Size 2/12/2020
#
# Partition
Formula:
# Grain Tolerance Angle: 5.00
# Minimum Grain S
ize: 2
# Minimum Confidence Index: 0.00
# Multiple Rows Requirement: Off
# Column 1: Integer identifying grain
# Column 2
-
4: Average orientation (phi1, PHI, phi2) in degrees
# Column 5
-
6: Average Pos
ition (x, y) in microns
# Column 7: Average Image Quality
(IQ)
# Column 8: Average Confidence Index (CI)
# Column 9: Average Fit (degrees)
# Column 10: An integer identifying the phase
# 1
-
Titanium (Alpha)
# 2
-
Titanium (Beta)
# Co
lumn 11: Edge grain (1) or interior grain (0)
1 49
.464 72.992 299.89 18.911 20.765 234.9 0.470 0.80 1 1
2 105.059 4.763 247.52 54.452 3.723 216.5 0.623 1.06 1 1
3 199.783 57.925 157.96 81.238 12.2
49 176.8 0.539 1.18 1 1
4 190.524 86.037 1
66.98 93.086 3.490 222.9 0.515 1.05 1 1
5 59.181 67.583 287.44 108.472 27.021 183.5 0.497 0.93 1 1
6 138.098 80.576 209.67 129.297 6.448 303.9 0
.565 0.94 1 1
101
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