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’ ICHIGAN STATE
EXIST LANSING, MICH 48824-1048

This is to certify that the
dissertation entitled

STUDIES OF CRACK PROPAGATION AND MICROCRACK
INITIATION IN A NEAR-GAMMA TIAL ALLOY

presented by
BOON-CHAI NG

has been accepted towards fulfillment
of the requirements for the

PhD. degree in Materials Science

firm? 73-8688

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5th May 2005

Date

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STUDIES OF CRACK PROPAGATION AND MICROCRACK INITIATION
IN A NEAR-GAMMA TIAL ALLOY

By
Boon-Chai Ng

A DISSERTATION
Submitted to
. Michigan State University
in partial fulfillment for the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Chemical Engineering and Materials Science

2005

ABSTRACT

STUDIES OF CRACK PROPAGATION AND MICROCRACK INITIATION
IN A NEAR-GAMMA TIAL ALLOY

By
Boon-Chai Ng

Fracture in a duplex y -TiAl alloy with equiaxed grains has been studied. The crack
path in a notched Mode 1 crack growth specimen was analyzed using a combination of
orientation imaging microscopy and selected angle channeling patterns as well as
electron channeling contrast imaging techniques.

A fracture propagation parameter, incorporating the contributions from deformation
twinning and ordinary dislocation systems, has been developed. This parameter, F ,4 n.3, is
capable of identifying boundaries that are likely to nucleate microcracks. It was observed
that microcracking is less likely when F A +3 < 1.0 but is more likely when F A_,B > 1.0.

The primary crack has a tendency to propagate towards clusters of weak boundaries
(identified by the fracture propagation parameter). Where it is not possible to follow a
boundary directly, the crack will cleave through a grain, via the shortest distance, towards

the next cluster of weak boundaries.

TO GOD
BE THE GLORY

iii

ACKNOWLEDGEMENTS

This dissertation would never have been made possible if not for the many
individuals who were there to help and guide me in one way or another. Among the
many individuals, I would like to single out a few to express my sincere thanks.

I would like to thank members of my Ph.D. committee, Drs. Thomas Bieler,
Martin Crimp, David Grummon and Dr. Duncan Sibley for their time, guidance and their
patience. Special mention goes to both my co-advisors Drs Martin Crimp and Thomas
Bieler. My skill as an electron microscopist is credited to Dr. Crimp. His dedication,
easy going and helpful suggestions on microscopy techniques have made me a better
microscopist. Dr. Bieler was instrumental in helping me obtain the texture analysis for
the material and had contributed many valuable suggestions during the drafting of this
dissertation. The painstaking review of this dissertation was undoubtedly credited to Dr.
Bieler.

I would also like to thank Dr. Darren Mason for providing the computational
programs needed to speed up the grain orientation analysis. Drs. Jimmy Kijai and
Sammy Chuah, both from Andrews University, for their helpful discussion on the use of
statistical tools for analysis and Dr. Benjamin Sirnkin, who has been a good fi‘iend and
help in developing the technique of electron channeling contrast imaging as well as
lengthy discussion on the TiAl alloys.

The financial supported by the Air Force Office of Scientific Research under
grant # AFRL no. F49620-01-1-0116, monitored by Dr. Craig Hartley is gratefully

acknowledged.

iv

Last but not least, I would like to thank my wife, Belle, and daughter, Dominique
for their help, support and patience throughout this long journey to complete this

dissertation.

TABLES OF CONTENTS

LIST OF TABLES ................................................................................. viii
LIST OF FIGURES ............................................................................... xi
CHAPTER lleTRODUCTION

1.1. General overview .................................................................... 1

CHAPTER 2:LITERATURE SEARCH

2.1. Overview .............................................................................. 5
2.2. Microstructures and phase relationships ......................................... 5
2.3. Effects of alloying elements ........................................................ 9
2.4. Deformation systems ............................................................... 10
2.5. Deformation at interfaces .......................................................... 11
2.5.1. y—y interfaces ............................................................ 11
2.5.2. Ordered domain interface ............................................. 11
2.5.3. Twin interface ........................................................... 12
2.5.4. Large angle y—y Grain Boundaries ................................... 12
2.6. Geometric compatibility factor ................................................... 13
2.7. Parameter for the nucleation of grain boundary cracks ........................ 13
2.8. Microscopy techniques ............................................................ 14
2.9. Electron channeling ................................................................ 14
2.10.Electron channeling patterns ...................................................... 15
2.11.Selected area channeling patterns (SACP) ...................................... 16
2.12.Electron channeling contrast imaging (ECCI) .................................. 20
2.13.Electron backseattercd diffraction patterns (EBSP) ............................ 22
CHAPTER 3:EXPERIMENTAL PROCEDURE
3.1. Material and Sample Preparation ................................................. 29
3.2. Microscopy .......................................................................... 31
3.3. Initial condition of near-y TiAl alloy ............................................. 32
3.4. X-ray —texture analysis ............................................................ 34
3.5. Sample loading in the SEM ....................................................... 34
3.6. Grain Examination .................................................................. 35
3.7. Grain Orientation ................................................................... 38
3.7.1. Use of EBSP and SACP to determine grain orientation .......... 38
3.7.2. Computations of Stereographic Projection ......................... 39
3.7.3. Plane Trace Computations ............................................ 39
3.8. Analysis of Grain Orientation .................................................... 39
3.8.1. Use of EBSP and SACP to determine grain orientation .......... 39
3.8.2. Use of Schmid factor and trace analysis to determine grain
orientation ............................................................... 47

vi

CHAPTER 4: RESULTS FROM INITIAL CRACK

4.1. Overview ............................................................................ 53
4.2. X-ray Analysis of near y-TiAl alloy ............................................. 53
4.3. Initial loading of the 4-point bend specimen ................................... 54
4.4. Width of twins and shear strains ................................................. 58
4.5. General observation of deformation twins and dislocations .................. 62
4.6. Reloading of the cracked specimen .............................................. 64
4.7. Examination of grains along the crack path .................................... 66
CHAPTER 5:ANALYSIS OF ARRESTED CRACK
5.1. Overview ............................................................................. 76
5.2. Crack path and microcrack nucleation ........................................... 76
5.3. Compatibility Factor for Active Twin Systems .................................. 77
5.4. Microcrack formation ............................................................... 80
5.5. Subsequent Re-Loading ............................................................. 84
5.6. Examination of microcracks along various grain boundaries .................. 88
5.7. Fracture Propagation Parameter ................................................... 88
CHAPTER 6: RESULTS OF MICROCRACK EXTENSION
6.1. Overview .............................................................................. 99
6.2. Sample A ............................................................................. 99
6.3. Sample B .............................................................................. 113
CHAPTER 7:DISCUSSION
7.1 . Overview .............................................................................. 1 18
7.2. Sample A ............................................................................. 1 18
7.3. Sample B ............................................................................. 122
7.4. Evaluating the Fracture Propagation Parameter for Other Samples... . . .. 126
7.5. Analysis of intact boundaries that have high FA_,B values ..................... 130
7.6. Effect of Ti3Al on the crack propagation ......................................... 133
CHAPTER 8: CONCLUSIONS ................................................................. 135
REFERENCE ....................................................................................... 138
APPENDD( l DETERMINATION OF GRAIN ORIENTATION ........................ 145
APPENDD( 2 IMAGES OF GRAIN INTERACTIONS .................................... 173
APPENDIX 3 COMPUTATIONAL PROGRAMS USED IN THE ANALYSIS ........ 182

APPENDIX 4 USING OTHER VARIABLES IN THE FRACTURE
PROPAGATION PARAMETER ................................................................ 201

vii

LIST OF TABLES

Table 4.1: Grain Orientations .................................................................. 70
Table 4.2: Luster and Morris Slip Compatibility Factors for grain B and C .............. 73

Table 5.1: Schmid factor and the geometric compatibility factor 111’ between active
twin systems in grain l and 2 (refer to figures 4.5 and 4.6) ................................. 79

Table 5.2: Schmid factor and the geometric compatibility factor m’ between active
twin systems in grain 1 and 4 .................................................................... 87

Table 5.3: Schmid factor and the geometric compatibility factor m’ between active
slip and twin systems in grain Y and Y’ (Idealized Case with misalignment of ~8
degrees) ............................................................................................. 92

Table 5.4: Schmid factor and the geometric compatibility factor m’ between active
slip and twin systems in grain K and J .......................................................... 93

Table 5.5: Values of the Fracture Propagation Parameters assigned to each set of
grains, depending on the direction of the crack path as it moves into these grain

boundaries. The selected F value is displayed on the far right hand side .................. 97
Table 5.6: t-test analysis for the intact and cracked boundary (precracked) .............. 98
Table 5.7: Intact Grain Boundaries with high F values ....................................... 98
Table 7.1: t-test analyses for the intact and cracked boundaries (post cracked) .......... 121

Table 7.2: Effect of dominant crystal orientation on deformation along notch root. . . 124
Table 7.3: t-test analyses for intact and cracked population means (Sample B). . . . . 126

Table 7.4: Simkin’s dataset using both the Fracture Propagation Parameter and
Fracture Initiation Parameter. ................................................................... 128

Table 7.5: t-test Analysis for Simkin’s specimen using Fracture Initiation
Parameter ........................................................................................... 129

Table 7.6: t-Test Analysis for Simkin’s specimen using Fracture Propagation
Parameter ........................................................................................... 130

Table A4-1: t-test for population means using the absolute values for the interaction
between two twinning vectors .................................................................... 202

viii

Table A4.2: Selected F“; values (Using absolute values for the computation between
twinning vectors) ................................................................................... 203

Table A4-3: t-test for population means using the “positive values set = zero” values
for the interaction between two twinning vectors .............................................. 204

Table A4z4: Selected F 4.3 values (where any “positive values were set to zero
values” for the computation between twinning vectors) ....................................... 205

ix

LIST OF FIGURES

Figure 2.1. Phase diagram of a binary TiAl Alloy [8] ........................................

Figure 2.2. Unit cell of L10 ordered TiAl with slip directions of the (111) slip
plane .................................................................................................

Figure 2.3. Unit cell of the D019 structure [21] ..............................................

Figure 2.4. Electron channeling is an effect of the different interactions of the
electrons with the crystalline material and depends on crystal
orientation ...........................................................................................

Figure 2.5. Formation of a selected area channeling pattern (SACP) .....................

Figure 2.6. SACPs with variation in the focal point (crossover) of the scan relative to
the specimen surface (a-c) and SACP image corrected dynamically (d) ...................

Figure 2.7. SACP composite reveals superlattice bands thereby making it easy to
correctly identify the grain orientation: (2,17,-14) ............................................

Figure 2.8. Near surface defects will cause a change in the BSE yield due to local
lattice variations fiom the Bragg condition. 91, is the Bragg angle and or is less or
greater than the Bragg angle .....................................................................

Figure 2.9. Alignment of a band to the microscope axis by tilting the
specimen ............................................................................................

Figure 2.10. Example of dislocation imaging using ECCI technique. a) Secondary
Electron image of microcracks. b) ECCI image reveals dislocations lying almost
parallel to the surface as well as end-on. Sample is a single crystal NiAl alloy [44]...

Figure 2.11. Schematic of electron diffraction in silicon with crystal planes and
associated diffraction bands shown [50] ........................................................

Figure 2.12. Schematic diagram illustrating the relationship between the diffracted
pattern and the tilted specimen ..................................................................

Figure 2.13. An EBSD pattern of TiAl at 25 kV and the corresponding result of the
computerized indexing showing the poles/zones. There is one chance in three that it
is correctly indexed ................................................................................

Figure 2.14. Inverse Pole Figure Maps (a) color (b) grayscale .............................

6

8

8

16

17

18

19

20

21

22

24

25

26

27

Figure 2.15. OIM map and its corresponding BSE image containing grains 4 and 7.
Note that the set of Euler angle colors are similar in both grain 4 and 7, indicating that
the EBSP technique was not able to distinguish between these two grains ................

Figure 2.16. Two similar sets of SACP composites. Careful examination revealed
superlattice [110> bands (marked) in Grain 7 whereas Grain 4 has one of the other
[011> bands .......................................................................................

Figure 3.1. A portion of the investment cast plate used in this study. Schematic
diagram showing orientation of sample A and sample B with respect to the
investment cast plate is shown ..................................................................

Figure 3.2. TiAl alloy was bonded to an aluminum backing to facilitate the controlled
crack extension .....................................................................................

Figure 3.3. Microstructure of the near-D-TiAl showing the equiaxed y grains and
small clusters of a2 colonies. ....................................................................

Figure 3.4. Undeformed near-y TiAl alloy showing pro-existing deformation twins.
In some regions, a higher density of the pre-existing twins was observed (bottom right
edge of the image). The densities of the pre-existing deformation twins vary from

grainto grain. ......................................................................................

Figure 3.5 Undeformed near-y TiAl alloy showing pre-existing deformation twins.
The densities of the pre-existing twins vary from grain to grain. But overall, the pre-
existing twins were uniformly distributed throughout the grain ............................

Figure 3.6. Fullam Deformation stage fixed with a 4-point bend fixture (shown at
higher magnification on the right side) and notched specimen in place ....................

Figure 3.7. ECCI images of two different twinning systems, each displaying 2 fine
micro-twins separated by approximately 1 micron. a) Aligning the beam axis (+) to
one edge of the {111} band. (note contrast from dislocations). b) Aligning the beam
axis (+) to the center of the band. The {111} twins corresponding to the {111}
channeling band are in sharp contrast when the beam is aligned to the center of the
band. ................................................................................................

Figure 3.8. A portion of the 4-point bend specimen aligned in relation to the
microscope screen as shown. ...................................................................

Figure 3.9. A computed stereograph with the grain normal (3,2, -21) and a horizontal

tensile loading direction [4,15,2] and its corresponding color code labeled in the
inserted table. ......................................................................................

xi

28

28

3O

3O

32

33

33

35

37

38

41

 

Figure 3.10. Plot of the plane trace with respect to the grain normal, (3,2, -21), and
tensile loading direction, [4,15,2] .................................................................

Figure 3.11. In order to confirm orientation information obtained from a proposed
stereograph, the specimen must be rotated and tilted in order to obtain SACP patterns
that can be compared to the three stereographic projections obtained from OIM data.
If the stereograph was viewed from the grain normal (marked “+”) towards the left,
then the specimen had to be rotated such that the notch was at the bottom of the
sample and tilted to the required angle to obtain the needed
information ..........................................................................................

Figure 3.12. The tilt mechanism allows only tilt in one direction. Thus it was
necessary to rotate the sample to a specific orientation first before tilting to the correct
angle. ................................................................................................

Figure 3.13. Three possible stereographic projections obtained from an EBSP scan,
and a SACP composite of grain 4. The SACP composite reveals a superlattice band,
approximately 25 degrees from the grain normal. This matches stereograph b having
grain normal (2,17, -14). .........................................................................

Figure 3.14. Three possible stereographic projections (a-c) and the observed [110>
superlattice band observed in the SACP when the specimen was rotated such that the
notch is at the bottom and the specimen is tilted ~5 degree as
shown ................................................................................................

Figure 3.15. Three possible stereographic projections for grain 8. The band a from
the top SACP corresponds to the superlattice band a ’ shown in the stereographic
projection a and band c from the bottom SACP corresponds to superlattice band c ’
(stereographic projection c). Since both bands 0 and c do not show any superlattice
information, stereographic projections a and c are incorrect and are eliminated.
Stereographic projection b is therefore the correct solution for grain 8 ....................

Figure 3.16. Diagram for calculating the Schmid factor .....................................

Figure 3.17. Comparison between pre-existing and deformation twins due to the
induced 4-point bend loading .....................................................................

Figure 3.18. BSE image of grain I with one dominant twinning plane (colored blue).
Note how the dominant plane traces propagate across another set of preexisting twins
Inserts a—c are plots of the three possible plane trace solutions with Schmid factors
indicated for the true twinning system on that plane. The observed dominant
twinning system corresponds well with plane trace plot 0, (grain Ic) displaying high
Schmid factors (0.41) for the dominant plane traces. Plane trace plots a and b ((grain
Ia and grain Ib respectively) had negative Schmid factors for the observed dominant
twin traces ...........................................................................................

xii

42

43

43

45

46

47

48

50

52

Figure 4.1 . X-ray pole-figure scan of specimen b with the orientation indicated
reveals a strong heterogeneous texture. The effect of oscillation alters the texture
measurement slightly, and inverse pole figures show that there are no highly preferred
crystal orientations. Great circle passing through {101} and {111} poles are shown,
and the little nub shown in the pole figures indicate 112 directions ........................

Figure 4.2. Secondary electron (SE) image of the crack running approximately 1.2mm
from the root of the notch. .......................................................................

Figure 4.3. High magnification ECCI images of the crack tip showing extensive
plastic deformation ahead of the crack tip. Both twin and dislocation contrast is
observed. ...........................................................................................

Figure 4.4. A backseattered electron (BSE) image of the crack path showing how the
crack moved from intergranular to transgranular and back to intergranular. The
intergranular crack regions are highlighted and numbered 1 — 3. The notch is located
at the top of the image while the arrested crack tip is located at the bottom of the
image. A schematic of the layout of the grains is shown on the right. The tensile
direction is horizontal. ...........................................................................

Figure 4.5. BSE image showing the arrested crack tip and the surrounding grains
1,2, and 4. The tensile axis is horizontal. The local strain along the grain boundary
between grain 1 and grain 2 was studied between the indicated arrows showing the
start and the end of the strain computation. ...................................................

Figure 4.6. ECCI image shows the leading (-111) twins from grain 1 and the
corresponding plane traces in grain 2. Microcracks (numbered 1 — 6) were observed
along the grain boundary. Image is rotated relative to figure 4.5.

Figure 4.7. High magnification ECCI image of twins. Dimensions were measured
directly from this image. The highlighted region (rectangular) was analyzed using the
NIH Image Software as shown in figure 4.8 and resulted in comparable FWHM
thickness measurements. .........................................................................

Figure 4.8. Examples of two twin measured using the NIH software to quantify the
gray scale. The intensity plot was superimposed over the high magnification BSE
image. ..............................................................................................

Figure 4.9. Plot of twin width vs. position along the grain boundary. The indexed

numbers correspond to the observed microcracks along the grain boundary.
Microcrack #7 was not shown in figure 4.6 ....................................................

xiii

55

56

56

57

59

60

6O

61

61

Figure 4.10. High magnification ECCI image of two microcracks, 5 and 6, (see
figures 4.5 and 4.6 for location of these microcracks). Both microcracks opened
asymmetrically in relation to the twin generated at the gain boundary ....................

Figure 4.11. BSE image of neighboring gains x and gain y. The region of interest is
highlighted and shown in figure 4.13 .........................................................

Figure 4.12. ECCI image of neighboring gains x and y that were highlighted in
figure 4.12. Note that the end of twin plane yl (at the gain boundary) showed a large
amount of plastic deformation in gain x that gives high contrast. Also, there is
correlation between twin plane y2 to twin plane x1. ........................................

Figure 4.13. A high magnification ECCI image of the highlighted region in figure
4.13 reveals high contrast variations resulting from a high dislocation density in gain
x ahead of the twin plane yl. ...................................................................

Figure 4.14. a) Secondary electron image showing surface features that diminish with
distance fiom the gain boundary. b) ECCI image reveals the heavily twinned planes
in gain 1 on the right and moderate twin activity in gain 2. Note that microcracks
have gown compared with figure 4.6. Some twins terminate in the gain in the
boxed region, shown in more detail in figure 4.15. Tensile axis is horizontal ............

Figure 4.15. Inset from figure 4.15b shows traces of twinning planes terminating in
the interior of the gain (arrows). ...............................................................

Figure 4.16. Microcracks nucleated along the gain boundary between gains 1 and
gain 4 near the triple point with gain 2. ....................................................

Figure 4.17. Orientation image mapping (OIM) image of crack path, shown in a
normal direction inverse pole figure map. .....................................................

Figure 4.18. BSE image of the crack with the notch on the top and the crack tip
located towards the bottom of the image. The gains that are close to the crack path
are labeled as shown. .............................................................................

Figure 4.19. BSE image of gains B and C with the plane traces and corresponding
Schmid factors superimposed on the gains ....................................................

Figure 4.20. BSE image of gains H, I, and HJ with the plane traces and
corresponding Schmid factors superimposed on the gains. Higher Magnification
BSE image of highlighted area A is shown in Figure 21 .....................................

Figure 4.21. Higher magrification of the highlighted region marked A in figure 4.20
showed microcracks along the gain boundary between gains H and HJ .................

xiv

62

63

63

64

65

65

66

68

69

72

73

74

Figure 4.22. BSE image of grains L and LA with a portion of the OIM map (see
figure 4.17) inserted. Primary crack changes its path several times as it cleaves
within gain LA. Microcracks are evident within gain LA (arrows) and at the gain
boundary region between gain L and LA as shown within the highlighted circle ....... 75

Figure 5.1. The computed projection of the (-111) plane based upon the known
crystal orientation of gain 1 (normal direction [1,5,6], and the horizontal tensile
direction (stage rotation axis), [3, -3,2]), shows superdislocation directions, (solid
lines), the ordinary dislocation direction (dashed) and the plane normal (bold arrow)
that has a component pointing out of the page. The [-11-2] twinning direction bisects
the triangle and is projected into the page ...................................................... 81

Figure 5.2. The projection of the (-111) twin plane superimposed on gain 1 provides
a means to visualize how twinning vector (bisects the triangle) moves towards the left
and into the page ................................................................................... 81

Figure 5.3. BSE image of a crack opening. Projections of the corresponding twin
planes are shown on the top 'of the BSE image. Solid arrows have component out of
page while the dotted arrow has a component going into the page ........................ 81

Figure 5.4. a) Thin twin shear. b) Thicker twin shear causes a local compression-
tension strain. This strain resulted in a local tension opening force and hence, a
microcrack opening between gain 1 and 2. ................................................... 84

Figure 5.5. Slip transfer by twinning requires that the absolute direction of the
twinning shear be not opposed in a twin system in the neighboring gain ................. 89

Figure 5.6. Directionality is important in evaluating the deformation of gains in the
path of the crack. A crack approaching gain A before gain B will cause more
deformation in gain A then in gain B. ...................................................... 89

Figure 5.7. Histogam of intact and cracked boundaries using Fa.» values ............... 96

Figure 6.1 . The undamaged microstructure below the primary crack was analyzed for
gain orientations, and the gains were labeled numerically ................................. 100

Figure 6.2. Schematic diagam shows the FA_,.3 values, the directional arrows (black
or gay), and the weak boundaries highlighted in thick black color ........................ 101

Figure 6.3. Higher magrification of the boxed region shown in figure 6.2. The
fracture propagation parameters are computed for each gain boundary and the weak
boundaries are identified with thick black lines ............................................... 102

Figure 6.4. Based on the observed weak boundaries, crack path was predicted to
move in the direction as shown in red .......................................................... 103

XV

Figure 6.5. When the sample was further loaded, the microcracks along 1-2 boundary
as well as 1-4 boundary linked up. Microcracks were also observed farther down the

1—4 gain boundary ................................................................................. 105
Figure 6.6. The crack path changed direction from the 1-2 boundary towards the 1-4
boundary ............................................................................................ 106

Figure 6.7. The crack was arrested near some Ti3Al particles at the 1-4 gain
boundary while a new crack had resurface further along the 1-4 boundary and cleaved
into gain 4 .......................................................................................... 106

Figure 6.8. The crack extended towards a goup of weak boundaries around gain
9 ...................................................................................................... 107

Figure 6.9. Pre-existing twins in undeformed gain 9 include the annealing twin that
divides gain 9 into two smaller grains ......................................................... 107

Figure 6.10. Microcracks developed as the crack was further loaded. Arrows
indicate the locations of fine microcracks. Higher magrification images of these
microcracks are shown in figures 6.10 through 6.13 .......................................... 108

Figure 6.11. Microcracks (arrowed) along the 49a—49b gain boundary. On the lefi of
gain 49b is a platelet of Ti3Al ............................................................... 109

Figure 6.12. Microcracks developed along gain boundary between gain 9 and gain
10a. The microcrack developed in a Ti3Al gain boundary precipitate. .................. 109

Figure 6.13. Microcrack occurred between the gain boundary bordering gain 10a
and gain 33c when the specimen was further loaded. The microcrack developed in a
Ti3Al gain boundary precipitate ................................................................ 110

Figure 6.14. Microcracks formed along the gain boundary Ti3Al between gain 10b
and 33a, gain 13 and gain 33a, gain 13 and gain 33 as well as between gain 32
and gain 33 when the specimen was further stressed via 4-point bending ................ 110

Figure 6.15. Path of the propagated crack with predicted path (white arrows)
superimposed over it. The black arrow indicates the gain boundary cracking in the
1-4 boundary and the gey arrow shows where the crack deviated by cleavage of gain
4 towards the weak boundaries in gain 9. Insert shows an image of the region before
the crack propagated through gain 4. ......................................................... 112

Figure 6.16. Histogam of intact and cracked boundaries of the lower portion of the
crack path ........................................................................................... 1 13

xvi

Figure 6.17. Back Scattered Electron image of the crack path of sample b. The SE
image is on the right top corner. The crack propagated completely through the
sample. The boxed region was analyzed and is shown in figure 6.18 ...................... 115

Figure 6.18. a) OIM image of the crack path. b) Corresponding BSE image. A large
amount of Ti3Al was observed along the crack path. A small cluster of gains were
labeled and analyzed using the fracture propagation parameter ............................ 116

Figure 6.19. Schematic diagam of the area of interest shows the F A93 values, the
directional arrows (black or gray), and the weak boundaries highlighted in thick black
color .................................................................................................. l 17

Figure 7.1. Relationship between texture and deformation processes in the direction
of the notch root are examined in Table 7.2 by considering maxima for Young’s
modulus and Schmid factor maxima for twins and ordinary dislocations .................. 123

Figure 7.2. Examination of specimen surfaces. 3) Ng’s specimen b)Sirnkin’s
specimen ............................................................................................ 127

Figure 7.3. Interaction between gain 13 and gain 16. The dominant plane traces for
both gains, their schmid factor and their F493 values were inserted. [16]. 131

Figure 7.4. The gain boundary between gain 6 and gain 8 shows intact boundary
[16] ................................................................................................... 132

Figure Al-l. Three possible stereogaphic projections and a SACP pattern of gain
A. The superlattice band from the SACP pattern matches stereogaph (b) having
gain normal (91,3,6) .............................................................................. 147

Figure A1-2. Three possible stereogaphic projections and a SACP pattern of gain
B. The superlattice band from the SACP pattern matches stereogaph (b) having
gain normal
(8,3,10) .............................................................................................. 148

Figure A1-3. Stereogaph and composite SACP pattern of gain BB. The superlattice
band from the composite SACP pattern matches stereogaph with gain normal
(11,10,-3) ........................................................................................... 149

Figure A1-4. Three possible stereogaphic projections and a SACP pattern of gain
BE. The superlattice band fi'om the SACP pattern matches stereogaph (c) having
gain normal
(6,4,1) ............................................................................................... 150

Figure A1-5. Three possible stereogaphic projections and a SACP pattern of gain
C. The superlattice band from the SACP pattern matches stereogaph (b having gain
normal (100,26,-75) ............................................................................... 151

xvii

Figure A1-6. Three possible stereogaphic projections and a SACP pattern of gain
D. The superlattice band from the SACP pattern matches stereogaph (c) having
gain normal (9,11,3) ............................................................................ 152

Figure A1-7. Stereogaph and composite SACP pattern of gain E. The superlattice
band from the composite SACP pattern corresponds well with stereograph with gain
normal (6,16,-17) ................................................................................... 153

Figure A1-8. Stereogaph and composite SACP pattern of gain FK. The superlattice
band from the composite SACP pattern corresponds well with stereogaph with gain
normal (1,1,-3) ..................................................................................... 154

Figure A1-9. Three possible stereogaphic projections and a SACP pattern of gain
G. The band a‘ from the top SACP corresponds to the superlattice a shown in the
stereogaph (a) and while band b ’ corresponds to superlattice b (b). Since both bands
a’ and b’ do not show any superlattice information, stereogaphic projections a and b
are incorrect and are eliminated. Grain c with normal, (9,3,-4), is therefore the
correct solution for gain G ...................................................................... 155

Figure A1-10. Three possible stereogaphic projections and a SACP pattern of gain
H. The superlattice band from the SACP pattern matches stereogaph (c) having
gain normal (3,7,15) .............................................................................. 156

Figure Al-ll. BSE image of grain HJ (labeled) with three possible plane traces
superimposed on the image. The dominant twining plane corresponds well with the
positive Schmid factor displayed by the plane trace analysis (b), with gain normal
(31,25,-99) .......................................................................................... 157

Figure A1-12. Three possible stereogaphic projections and a SACP pattern of gain

J. The superlattice band fiom the SACP pattern matches stereogaph (c) having gain
normal

(7,8,10) .............................................................................................. 158

Figure Al-13. Three possible stereogaphic projections and a SACP pattern of gain
K. The superlattice band from the SACP pattern matches stereogaph (c) having
gain normal (7,8,10) .............................................................................. 159

Figure Al-14. Three possible stereogaphic projections and a SACP pattern of gain
L. The superlattice band fiom the SACP pattern matches stereogaph (c) having gain
normal (3,2,-21) .................................................................................... 160

Figure A1-15. Three possible stereogaphic projections and a composite SACP
pattern of gain 5. The band a’ from the composite SACP corresponds to the
superlattice (1 shown in the stereograph (a) while band b ’ corresponds to superlattice b
in stereogaph (b). Since both bands (I ’ and b’ do not show any superlattice

xviii

information, stereogaphic projections a and b are incorrect and are eliminated. Grain
5 with normal, (9,3,-10), is therefore the correct solution for gain
5 ...................................................................................................... 161

Figure A1-16. Three possible stereogaphic projections and a SACP pattern of gain
5A. The band a’ from the top SACP corresponds to the superlattice a shown in the
stereogaph (a). Since band a’ does not shown any superlattice information,
stereogaph (b) is eliminated. Further analysis is needed and is shown in figure A1-
17 ..................................................................................................... 162

Figure A1-17. BSE image of gain 5A with three possible plane traces superimposed
on the image. Since plane trace (b) had been eliminated earlier (see Figure A1-16)
and plane trace (a) showed negative Schmid factors for both the dominant twinning
planes, the plane trace (c) with normal (13,20,0) was selected as the correct gain
orientation .......................................................................................... 163

Figure A1-18. BSE image of gain 5B with three possible plane traces superimposed
on the image. From the BSE image, plane trace a has the characteristic of pre—existed
(evenly distributed twins) while plane trace b is the dominating deformation twin.
Plane trace analysis (a) with a gain normal (1,8,13) and a high positive Schmid factor
corresponding to band b is the most likely the correct gain orientation. Plane trace
analysis (b) and (0) both showed negative and low Schmid factor respectively for the
dominating deformation twin ..................................................................... 164

Figure A1-19. Three possible stereogaphic projections and a composite SACP
pattern of gain 1. The band a’ from the composite SACP corresponds to the
superlattice a shown in the stereogaph (a) while band b ’ corresponds to superlattice b
in stereogaph (b). Since both bands (1 ’ and b ’ do not show any superlattice
information, stereogaphic projections a and b are incorrect and are eliminated. Grain
1 with normal, (1,5,6), is therefore the correct solution for gain
1 ...................................................................................................... 165

Figure A1-20. Three possible stereogaphic projections and a composite SACP
pattern of gain 2. The band a’ from the composite SACP corresponds to the
superlattice a shown in the stereogaph (c) while band b ’ corresponds to superlattice b
in stereogaph (a). Since both bands (1 ’ and b ’ do not show any superlattice
information, stereogaphic projections a and b are incorrect and are eliminated. Grain
2 with normal, (100,0,-63), is therefore the correct
solution .............................................................................................. 166

Figure A1-21. Three possible stereogaphic projections and a composite SACP
pattern of gain ZW. The superlattice band a’ from the composite SACP corresponds
to the superlattice a shown in the stereogaph (c), confirming that gain ZW with
gain normal (0,2,3) is the correct orientation. ................................................ 167

xix

Figure A1-22. Three possible stereogaphic projections and a composite SACP
pattern of gain Z. The superlattice band a’ from the composite SACP corresponds to
the superlattice (1 shown in the stereogaph (c), confirming that gain ZW with gain
normal (7,20,-17) is the correct orientation ..................................................... 168

Figure A2-l. Intact boundary between gain A and gain BB .............................. 170

Figure A2-2. Interaction between gain B and BB is shaded gay. Microcracks
(arrowed) at the gain boundaries between C and BE and B and BE. Intact

boundaries between gains E and BE, BE and BB ............................................ 170
Figure A2-3. Intact gain boundary between gain C and gain D .......................... 171
Figure A2-4. Grain interaction between gains D-E-H ....................................... 171

Figure A2-5. Interaction at gain boundaries between gain D, F,FK,H,I and HJ. An
SE image had been superimposed over the BSE image to provide additional
information on the crack edges. Higher magrifications of the gain interaction are
shown in figure A2-6 and A2-7 .................................................................. 172
Figure A2-6. Intact boundaries between gains D and F, and between F and FK. 172
Figure A2-7. Intact boundaries between gains H —HJ and gains HJ- J .................. 173
Figure A2-8. Cracked boundary (arrowed) between gain I and gain H .................. 173

Figure A2-9. Interaction between gains FK - K- J. Intact boundary between gains
FK —J as well as between gains FK — K ....................................................... 174

Figure A2-10. Low magnification of gains K-J-L and LA. Higher magrification are
showninfigures A2-11-12 ....................................................................... 174

Figure A2-11. Evidences of cracked boundaries between gains K and J. The
observation of ‘stepped or jogged’ in the crack path strongly suggest the presence of

microcracks that resulted in the change in the crack path .................................... 175
Figure A2-12. Intact boundaries between gains K and LA ................................. 175
Figure A2-13. Intact boundaries between gains L and LA ................................. 176
Figure A2-14. Grain interactions between gains ZW-Z-5-5A-5B-1. Higher

magrification images are shown in figures A2-15 ............................................ 176
Figure A2-15. Intact boundaries between gains ZW and Z ................................ 177

XX

Figure A4-1. Histogam showing intact and cracked boundaries using F ,4.3 values
computed using absolute values for the interaction between two twinning vectors ...... 202

Figure A4-2 Histogam showing intact and cracked boundaries using F A-3 values

computed using “positive values set = zero” for the interaction between two twinning
vectors ............................................................................................... 205

Images in this thesis/dissertation are presented in color

Chapter 1: Introduction
1.1 General Overview

There continues to be a gowing demand for lightweight, high strength, and good
corrosion-resistant materials to be used in aerospace and automotive applications.
Gamma titanium aluminides are emerging as potential engineering materials for these
(aerospace and automotive) industries because of their low densities, high melting
temperatures, good elevated-temperature strengths and modulus retention, high resistance
to oxidation and hydrogen absorption, and excellent creep properties [1-5]. These alloys
have been aggessively pursued as an airfoil material, for the compressor and the low-
pressure turbine sections of gas turbine engines [6].

The gamma alloys developed so far consist of titanium, 46-52 at. % aluminum
and 1-10% at. % M, with M being at least one element from V, Cr, Mn, W, Mo, Nb and
Ta. [4]. These gamma TiAl alloys can be divided into either single-phase (gamma) alloys
or two phase (7 and (12) alloys [4]. The two-phase alloys with very small amounts of
Ti3Al have been found to exhibit higher yield and ultimate tensile strengths than single-
phase TiAl [7]. They were also more ductile and tougher than single phase TiAl [8].
However, these gamma alloys have yet to achieve their full potential as an engineering
material due to their low ductility and toughness at ambient temperatures, which, along
with poor formability, continue to plague their utilization [4]. Design engineers detest
such properties (low ductility and toughness) because of the perceived risk of
catastrophic failure [6].

The low toughness of these gamma alloys has been the primary focus in this

work. The toughness of a material is related to its ability to resist or arrest cracks.

Understanding why and how cracks propagate and arrest in these gamma alloys will
provide an important basis for the improvement of toughness in these alloys. A study of
the micro-structural conditions that enhance or suppress the initiation of microcracks in
the vicinity of the crack tips and along crack edges would thus provide valuable
information for improving the toughness of these alloys.

In order to characterize the micro-mechanisms associated with microcrack
initiation or enhanced deformation transfer, this study will focus on the propagation and
subsequent arrest of cracks in an equiaxed near-gamma TiAl alloy. Four-point bend
specimens were loaded in-situ in the scanning electron microscope (SEM) until a crack
propagated from a notch. Although the nature of the crack path and deformation defect
structures near crack tips have traditionally been examined via Transmission Electron
Microscopy (TEM) using thin foils made from the sample specimen, this study uses
electron channeling contrast imaging (ECCI) to observe these deformation defect
structures. This ECCI technique allows imaging of near surface dislocations and twins in
bulk specimens [9-13].

In many microscale or mesosacle deformation studies, it is important to know the
gain orientation. Active slip systems, gain boundary characteristics and elastic moduli
depend on gain orientation. The electron backseattercd diffraction (EBSD) technique,
performed using a scanning electron microscope, is commonly employed to determine
the gain orientation in bulk specimens. Alam and co-workers first developed this
technique in 1954 [14], calling them “high-angle Kikuchi patterns”, in recogrition of
related diffraction phenomena reported by Kikuchi [15] in the 19205. This technique,

although easy and quick, is not capable of distinguishing the c and a direction in

tetragonal structures with c/a ratio that differs from 1 by 5%. In such a tetragonal
structure, the EBSD analysis would result in three possible gain orientations. On the
other hand, selected area channeling patterns (SACP), a technique also performed using a
scanning electron microscope, can detect the superlattice bands fi'om an ordered
tetragonal structure and provide the correct gain orientation. This superlattice
information is rarely resolved with the EBSD technique. But the process of collecting
SACPs over various tilt positions, identifying the bands and zones in order to correctly
locate the gain normal, is very tedious and time consuming.

A new approach, incorporating the ease and quickness of EBSD analysis with the
unique capability of the SACP to differentiate the c and a direction had been used to
speed up the process of determining the gain orientation of tetragonal structures by a
factor of 10.

With correct gain orientations and the ability to observe microcracks at twin-
gain boundary intersections, it was possible to develop a fracture propagation parameter
that is able to identify how efficient slip can be transferred across the boundary. This
parameter incorporates the Schmid factor, the interaction of the leading twin vector with
the tensile direction, the ordinary dislocations, and the twinning vectors from the adjacent
gains.

The principle focus of this dissertation has been on a primary active crack as it
propagates through the y—y gain boundaries. This work has been built on the initial work
done by Ben Sirnkin [16] who developed a fracture initial parameter using a limited
number of data. By incorporating the analysis of more than 39 boundaries in gains

surrounding the primary fracture crack in the crack gowth specimen complimentary

fiacture propagation parameter has been developed. This fracture propagation parameter
only needs the primary tensile stress axis and spatially resolved gain orientations to
predict the path of the crack. The use of this parameter will provide the information
much needed by the desigr engineer as to where the crack would/might advance so that

designs can be modified to strengthen the product and to arrest any propagating crack.

Chapter 2: Literature Review

2.1. Overview

TiAl, an ordered intermetallic compound, has been viewed as a potential
candidate for high temperature structural material. This chapter reviews some of the
recent important developments of TiAl in the area of physical and mechanical properties
of TiAl. The microscopy techniques used by Sirnkin [16] on a similar research effort are

also reviewed at the end of this section.

2.2. Microstructures And Phase Relationships

A portion of the binary phase diagam [7] of the Ti-Al system is shown in figure
2.1. Depending on the composition, when the TiAl alloy is cooled from the melt, the
following reactions take place:

1. Liquid (L)—) (L + alpha ((1)) —) or phase.

2. Gamma (y) precipitates on basal planes within the (or + y) phase field.

3. As the material cools further, the or phase orders to a2 (D019 structure).

At room temperature in the as-cast condition (processes such as casting, ingot
metallurgy or powder metallurgy), the two-phase (TiAl + Ti3Al) TiAl alloys exhibit a
lamellar microstructure [7,17]. The y lamellae are formed in such a way that their closed
packed planes and directions are:

{111}TiAl // (0001) Ti3Al

<110> TiAl // <11-20> Ti3Al [6]

 

 

 

 

 

 

1600 ' r ' 1 r r ' 1 r
l.
L
1500* '-
r 9 \
1400- -
6
:. t
O
5 a
31300- -
8
E . .
'9 (1+7
1200~ y -
1100 r 02 _ ..
“2+1
1000 - x "‘
I J 1" L a J L l I
36 4o ' 44 48 52 56

Atomic Percent AI

Figure 2.1. Phase diagam of a binary TiAl Alloy [8]

While the three <11-20> directions on the (0001) basal plane are all equivalent in
the 02 phase (Ti3Al), the [110] direction is not equivalent to the two other <011]
directions on the (1 l 1) plane of the y phase (TiAl). Consequently, the y—phase can

display in six possible orientation relationships corresponding to the six possible

orientations of [110] on (111) on an (0001) plane in the a2 phase, consistent with the
ordering relationships given above [7,18,19].

A Although these lamellar structures may be beneficial for toughness and high-
temperature strength, they cause relatively poor room-temperature ductility [1,7,20].
Therefore, these alloys are usually further thermomechanically processed to achieve

desirable microstructures. The most desirable microstructures are the near-y

microstructtues, generally characterized by coarse y gains and banded regions consisting
of fine (y + a2) gains [7]. Near-y TiAl is obtained (after themomechanical processing)
by further annealing heat treatment at temperatures just above the eutectoid temperature
[7,20].

Near-y TiAl alloys, which are rich in titanium, are composed of two phases: y
phase (nearly stoichiometric TiAl) has the L10 face centered tetragonal (F CT) crystal
structure, and the (12 phase (Al-rich Ti3Al) has the D019 crystal structure (ordered
hexagonal) [21] as shown in figures 2.2 and 2.3 respectively. The face centered
tetragonal structure (similar to a face centered cubic (FCC) structure) consists of
alternating titanium and aluminum (002) planes stacked normal to the c-axis. At the
stoichiometric composition, the c/a ratio is 1.02 [4,5] (a=0.400nm and c=0.407nm) and
the tetragonality increases up to 1.03 as the aluminum concentration increases [4]. The
D019 structure has a=0.5775nm and c=0.4655nm. The two structures have regular
stackings of closed-packed planes. The (111) planes in the PCT and the (0002) bual

planes in the D019 structure are the closest-packed in their crystal structure.

 

Figure 2.2. Unit cell of L10 ordered TiAl with slip directions of the (111) slip plane.

[0001]
ll

7‘“ A

A

 

 

 

T T

Ego” c . K i
" h : E l , . 1

if il”fik‘ ' . ~' , [i2i0]

Vu/ hep: J

[2110] r 3D019

Figure 2.3. Unit cell ofthe D019 structure [21].

 

 

 

 

-4
- uh
<>C_

 

 

 

 

 

 

 

 

 

 

2.3. Effects of Alloying Elements

The microstructure and mechanical properties of TiAl alloys are influenced by
their chemical compositions. There are two goups of alloying elements. The first goup
of alloying elements provides precipitation strengthening or particle strengthening.
These alloys include boron, which results in the refinement of microstructures, and
silicon, which improves the creep and oxidation resistance [22]. The second goup of
alloying elements is associated with solid solution strengthening elements and includes
Cr, Mn, V, Nb, and Fe. [22].

Alloying additions of Cr, V, and Mn were reported to enhance the plasticity of
TiAl alloys [23,24]. It is believed that these additions enhance plasticity by stabilizing
thermal twins, which provide nucleation sites for twinning dislocations [22,25]. Fe, Cr,
V, and Nb had been reported to enhance the tensile strength at room temperature as well
as the retention of creep rupture life of TiAl [22]. Nb also improves the oxidation
resistance, elevated temperature strengths [22], and the yield strength of the alloy [26]. A
review of these alloying consideration can be found in [8].

In this study, 2 at. %Nb and 2 at %. Cr were added to binary TiAl. The addition
of Cr and Nb at this level does not change the essential two-phase character of the alloys
[27]. The solubility of Nb in 'y-TiAl is 6.86 at %. whereas only 3.07 at %. Cr can be
dissolved in y—TiAl. [22]. The alloying element Cr seems to be more likely to reside on
the Al sites in the gamma-TiAl and whereas the preferential site of the Nb is the Ti sites

[28-30].

2.4. Deformation Systems.

Deformation slip in ordered TiAl occurs exclusively on {111} planes [21] (Figure
2.2), either by dislocation slip or by twinning. However, unlike in FCC, the various slip
directions are strongly affected by the ordering [31]. Four physically different modes
have been distinguished, namely slip of v. <110]l ordinary dislocations, of <011]
superdislocations, l/2 <112] super-partial dislocations, and 1/6 <112] Shockley partial
dislocations leading to ordered twins. Slip along <110] directions does not affect the
ordering [31], so slip occurs by ordinary dislocations with 1/2 <110] Burgers vectors
moving on {111} planes. In contrast, ‘/2 <011] slip moves atoms to antisites positions
(see Figure 2.2) and thus two consecutive slip steps are necessary to preserve the order.
These dislocations exist as pairs and are called super-partial dislocations. Although these
super-partial dislocations typically glide on the closed packed planes {111}, Morris [32]
observed partial 1/2 [101] dislocations with edge character contained on (010) planes.
Superdislocations with 1/2<l 12] Burger vectors have been rarely observed, as might be
expected due to their larger Burgers vector [18,21].

Deformation twining is commonly observed in y-TiAl [1,4]. Unlike dislocation
motion, twin slip is polar, i.e. twins can only move in a single well-defined direction on a
given slip plane. There are only four 1/6 <11-2] type twin directions, one per {111) plane
that can be operated without destroying the order. Its slip vectors are of type 1/6<11-
2][21]. Other 1/2 <112] dislocations would result in psuedotwins with ordering

violations.

 

1. The mixed bracket < ] is used to recognize the symmetry in the first two indices whereas the third index
must be specified.

10

Ti3Al deforms predominantly by the motion of 2/3<11-20> superlattice

dislocations on the basal, prism and pyramidal planes [21].

2.5. Deformation at interfaces
25.1. Hillier-faces

The orientation relationship between two neighboring y lamellae can be described
by 180°, 60 ° and 120° rotations of the [110] direction in the interface plane [18] and
result in relationships called twin, pseudotwin and ordered-domain respectively [33].
The transfer mechanisms of the deformation at the interface (ordered domain and 180°
rotation) have been studied by [33,34] respectively and are summarized below.
2.5.2. Ordered domain interface

The deformation transfer mechanisms at y—y ordered domain interfaces were
observed to occur by two modes.

1. When deformation twins impinge on a gain boundary, 1/6<11-2] Shockley
partial dislocations entering the interface will result in the emission of l/2<110]
ordinary and sometimes <101] superlattice dislocations across the interface.
Because the superlattice dislocations have low mobility, it was speculated that
these superlattice dislocations accumulated in the interface [33].

2. When ordinary dislocations impinge on a gain boundary, they dissociate to form

Shockley dislocations and these dislocations are left in the interface [33].

11

2.5.3. Twin interface

For an 180° rotational twin interface, the crystal lattices on either side of the
interface share a common <110] crystal orientation. As such, it is possible for <110]
dislocations to be transferred through the interface without pinning and re-nucleation.

Pure screw dislocations had been observed to cross slip a twin-related interface

since both their Burgers vector and line direction lie in the interface plane.

The incident shearing can also be largely transferred through the twin interface by
the production twinning and ordinary dislocations. Non-screw dislocations will be
locked at the interface and no emission of dislocations will be observed in the
neighboring lamella.

2. 5.4. Large angle HGrain Boundaria

Gibson and Forwood [3 5] demonstrated that, at a large-angle y—y gain boundary,
the strain created by the deformation twin can be accommodated by the generation of
glide in both gains by the movement of 1/2 (110]-type dislocations, with a residue Burger
vector lefi at the common line of intersection. They did not observe any gain boundary
accommodation via the emission of <1 01] dislocations in their experiments.

Zghal et.al [33,34], on the other hand, reported both ordinary and
superdislocations were emitted in the second lamella when a twin crossed an ordered
domain interface. But these superdislocations were only observed at the gain boundary
interface due to their low mobility. Sirnkin [16] attributed these superdislocations near

the interface to the residual strain accommodation mentioned by Gibson and Forwood.

12

2.6. Geometric Compatibility Factor

Luster and Morris [3 6] proposed a geometrical slip compatibility factor to
describe the active deformation systems involved in strain transfer at a boundary in TiAl.
This factor is defined as m’=cothcosx, where 0 and x are the angles between the slip
plane normal and slip directions of a pair of deformation systems on either side of the
gain boundary. For slip systems in adjacent gains, the 111’ may vary between 0 and 1.
For m’=l , there is complete compatibility exist between the slip systems whereas for
m’=0, the slip systems are completely incompatible. Positive values of m’ denotes the
most probable activation of the deformation system. Although this parameter is effective
at describing activation of slip systems at or near boundaries, it does not correlate well

with the nucleation of microcracks at the gain boundaries [37,38].

2.7. Parameter for the nucleation of grain boundary microcracks

Sirnkin [16] proposed a parameter that tried to relate y—y gain boundary
microcrack nucleation to the global stress state and the individual gain orientation. By
incorporating the observations of Gibson and Forwood, [3 5] into his work, Simkin’s
parameter F was:

F = mtw lA’tw . lA’ord

 

l3... .ilz

ord

 

 

where mm is the Schmid factor for a specific deformation twinning system under the
global stress state and bi are unit vectors in the sample coordinate system that describe

the Burgers’ vector directions for the single twin system (tw) and all 8 of the ordinary

dislocation systems in each gain (ord). F is determined for each of the 8 twin systems in

13

A A

btw~t

 

 

the pair of gains. The component relates the twin shear direction to the

tensile direction, and the final summation measures the ability of the gain boundary to
conform to the twin displacement by emission of a/2<110] ordinary dislocations into both

bounding gains.

2.8. Microscopy techniques

Transmission electron microscopy, (TEM), is traditionally used to examine micro
mechanisms of plastic deformation in materials. But specimen preparation and the
possible artifacts associated with the thin foils preparation can limit this technique.

Electron channeling contrast imaging (ECCI), a scanning electron microscope
(SEM) technique has recently been used to examine deformation structures in the bulk
sample. This technique has been used to image near-surface crystal defects [for
reviews, see ref. 9-13]. The ECCI technique is best understood by examining the
source of the contrast, namely electron channeling, and electron channeling patterns.
This description represents a combination of prior work of others and illustrative

examples provided by the author.

2.9. Electron Channeling

When an energetic electron beam strikes a specimen, at a particular angle
where these electrons will be parallel to the crystal lattice planes, they will penetrate
deeper into the crystal, passing between the rows of atoms along “channels’. The
chances of these electrons escaping from the specimen will decrease with increasing

depth below the surface [39-40]. On the other hand, in crystals oriented slightly

l4

differently, electrons will interact strongly with the surface and will escape fiom the

specimen (figure 2.4).

2.10. Electron Channeling Patterns

In an SEM with a 2-Dimensional raster at low magnification, the normal action
of the scanning results in an angular beam deflection of approximately 8 degees from
the microscope axis [40]. Consequently other lattice planes also contribute to the
contrast and the resultant signal plot, the electron channeling pattern (ECP), shows
bands of contrast from all sets of planes normal, or close to normal, to the surface [39-
40]. The width of the band is twice the appropriate Bragg angle for the set of lattice
planes and is inversely proportional to the lattice plane spacing (d) and the accelerating
voltage. This ECP has the symmetry of the crystal lattice in the area examined. [39-
40]. But this technique requires the whole scanned area have the same crystal
orientation. So a single crystal is effectively analyzed with this technique.

When a polycrystalline material is imaged at low magnification, there are
discontinuities in the electron channeling patterns because each individual gain will
produce different signal level. The use of selected area channeling patterns (SACP)

overcomes this problem.

15

Electron Electron

 

Beam
Large 1]

O O O O
O O O O
O O O O
O O O

O O O O

O O O O O

 

Figure 2.4. Electron channeling is an effect of the different interactions of the
electrons with the crystalline material and depends on crystal orientation.
2.11. Selected Area Channeling Patterns (SACP)

In selected area channeling patterns, a charmeling pattern fiorn a small area is
formed by rocking the electron beam about two perpendicular axes on the area of interest,
with the resultant BSE signal as a function of tilt angles captured and displayed as shown
in figure 2.5 [10]. This technique of rocking the electron beam can be accomplished by
using a dedicated set of scan coils to rock the beam through the desired range of angles
while it is confined to a very small area on the specimen. Because the set of scan coils

mimic the action of a lens, it is important to readjust the focal point (cross-over) on the

are varied and finally dynamically corrected for the effects of spherical aberration. Joy et
al. used images of copper grids to show the effect of strengthening the scan coils and

correcting the spherical aberration effects [40].

I
I
Rocking : axis
angle \p I
I

 

Microscope
axis

 

 

 

Sample surface

Figure 2.5. Formation of a selected area channeling pattern (SACP).

Selected area channeling patterns, like ECPs, also show bands of contrast from all
planes normal to the surface [40] and are thus be able to reveal the crystal orientation of
the gain in question. By tilting a specimen through many orientations relative to the
beam (limited by the angle of tilt mechanism) a collection of SACPs can be assembled to
form a SACP composite map as shown in figure 2.7. The determination of crystal
orientation becomes one of pattern recoglition, comparing angles between bands and
angular bandwidths before confirming a solution. This technique is also capable of
detecting ‘superlattice’ (incomplete destructive interference) information as shown in

figure 2.7. However, the process of obtaining the SACPs through various tilts,

l7

assembling into a SACP composite map, identifying major zones, measuring band widths

and identifying bands to confirm a gain orientation, is tedious and laborious.

 

Figure 2.6. SACPs with variation in the focal point (crossover) of the scan relative to the
specimen surface (a-c) and SACP image corrected dynamically (d).

(ii1)
/

112 zone axis

   
  

213 zone axis

(202)

grain
normal

101 zone axis — .

. .
5i! .

\ (020)

Figure 2.7. SACP composite reveals superlattice bands thereby making it easy to
correctly identify the gain as having a gain normal: (2,17,-14).

l9

2.12. Electron Channeling Contrast Imaging (ECCI)

In the ECCI technique, the electron channeling contrast is produced when some
small near-surface volume of the crystal is on one side of some Bragg condition (e.g.
9 > 93), while the bulk of the crystal is at or opposite side of the Bragg condition (e.g.
0 < 03): the backseattered electron (BSE) yield tilted volume will be different from the
rest of the crystal, thus highlighting the local strain field (for instance) of a dislocation

(see Figure 2.8) [41-43].

   
 
   
 
 

BSE Yield .
Incrdalt

electron
beam

Specimen

 

Tilt volume

Figure 2.8. Near surface defects will cause a change in the BSE yield due to local lattice
variations from the Bragg condition. 01, is the Bragg angle and or is less or geater than
the Bragg angle.

To image defects using ECCI, it is necessary to set up the channeling
conditions, much like a selected area diffraction pattern (SAD) is used to set up a two
beam diffiacting condition for TEM imaging. An SACP (SAD mode) of the grain of
interest is first obtained and tilted until one of the bands in aligned to the microscope
axis as shown in figure 2.9. Switching back to the imaging mode, and refocusing at

high maglification, any defects can subsequently be irnaged-captured and integated

over multiple frames by the frame-store integal to the CamScan 44 FE SEM. Figure

20

2.10 shows an example of a secondary electron (SE) and ECCI images of a crack edge
of a single crystal NiAl alloy [44]. Microcracks can be observed in the SE image but
the ECCI image reveals dislocations lying parallel to the surface as well as end-on
along the side of these microcracks. These dislocations are observed to be in
bright/dark contrast. For end-on dislocations, with one end of each dislocation exiting
the crystal and the other end into the crystal, the imaging contrast decreases as the
dislocation penetrates deeper into the crystal. The reason for the near surface
sensitivity is that the electrons fi-om the incident beam loose coherency in penetrating
the crystal, provide less contrast as they travel further into the crystal, and thus
limiting the depth of detection [45-46]. Channeling contrast is highly dependent on
the angular relationship between the beam and the crystal and as such, sharp changes
in contrast may occur over angular changes less than 0.5 degees [39]. This also
means that any angular changes geater than 0.5 degees will result in a loss of defect
contrast in the crystal. Because 0.5 degees is a very small angle, it is usual to iterate

between tweaking the tilt control to align the band edge to the microscope axis in the

SAD mode, and then try imaging the dislocations in the imaging mode.

 

Figure 2.9. Alignment of a band to the microscope axis by tilting the specimen.

21

dislocatio--l

microcrackfl ~
‘ .

\v

   

Figure 2.10. Example of dislocation imaging using ECCI technique. a) Secondary
Electron image of microcracks. b) ECCI image reveals dislocations lying almost
parallel to the surface as well as end-on. Sample is a single crystal NiAl alloy [44].
2.13. Electron backseattered Diffraction

The electron backseattered diffiaction pattern, (EBSD), (also known as
Backscattered Kikuchi Diffraction, BKD), was first developed by Alam and co-workers
in 1954 [14], who called them “high-angle Kikuchi patterns ”, in recogrition of related
diffraction phenomena reported by Kikuchi [15] in the 19205 while studying diffraction
of electrons by thin films of mica. However, it was not until the 1970’s that Venables
and co-workers applied EBSD to metallurgical microcrystallogaphy, [47] paving the
way for a more widespread application of EBSD to the materials sciences. This
technique was further developed by Dingley and co-workers [48] to measure crystal
orientation in the SEM and subsequently refined for on-line pattern indexing with the aid
of a computer [49].

In the past 10 years, rapid developments in both hardware and software have

made EBSD an easy tool for the rapid analysis of microstructures in a range of crystalline

22

materials. It is a quantitative technique that reveals gain size, gain boundary character,
gain orientation, texture, as well as phase identity [50].

The collection of an electron backscatter diffraction pattern (EBSP) in the SEM is
relatively straightforward. A polished sample must be tilted to a relatively high angle
(typically 70°) inside the SEM. A collimated electron beam is then directed at the point
of interest on the sample surface: initial elastic scattering of the incident beam causes the
electrons to be deflected from a region just below the sample surface and to impinge
upon crystal planes in all directions. The atomic planes of the specimen are thus
showered with electron arriving fiom all direction and with a wide range of wavelengths.
For each set of planes in a crystalline specimen, there will always be some electrons that
will satisfy the Bragg equation for diffraction.

,1 = 2 (1 sin 0
where ,1 is the dominant electron wavelength, d is the spacing between the planes and 6 is
the incidence angle of the electrons with the planes.

The backscattered electrons collide with a phosphor screen mounted in the SEM
and create a diffraction pattern or “Kikuchi ban ”. These patterns consist of parallel
lines, one pair of each set of atomic planes, separated by a bright band as shown in figure
2.11. Intersections of these bands correspond to zone axes or poles. [51].

The resulting electron backscattered diffraction pattern (EBSP) is captured via the
phosphor screen interfaced to a low-light television or CCD (charged coupled device)
camera and digitized into the computer.

In order to interpret the diffraction pattern correctly, the position of the pattern

center, the distance between the specimen and the phosphor screen and the angle of tilt

23

must be determined or calibrated [52]. Figure 2.12 shows a schematic diagam of the
relationship between the diffracted pattern and the tilted specimen. The most convenient
method of determining the pattern center is by using a calibration specimen of known
orientation, such a piece of silicon (surface normal is [001]) with cleaved edge lying
along the [110] plane. [49,52]. The distance between the specimen and the screen can be
obtained using the relationship:
L = N/tan 43
where N and ¢ are shown in figure 2.12 and the angle of tilt is read off the tilt mechanism

located on the SEM.

 

Figure 2.11. Schematic of electron diffraction in silicon with crystal planes and
associated diffraction bands shown [50].

24

      
 

Rafi—9.179 .
-_¢ MYz center

xXMX

 

 

 

Figure 2.12. Schematic diagam illustrating the relationship between the diffracted
pattern and the tilted specimen.

Individual orientation measurements are made at discrete points on the specimen
defined by the beam controls. The locations of the points are defined by a grid area
prescribed by the user (both the length and breadth of the gid area as well as the step size
between the points). At each point in the gid, the backscatter Kikuchi diffraction pattern
is captured. By selecting the expected crystal phases flour a phase database, all possible
orientations of the crystal under the beam are automatically indexed. The orientation of
the area of interest is determined by considering the number, width and angle of the
bands at the intersection [53]. An example of the EBSD pattern and the corresponding
result of the computerized indexing are shown in figure 2.13. Euler angles that describe
the orientation are recorded along with the coordinates describing the position on the
sample surface. Thus it is possible to map the crystal orientation onto a color or
gayscale and shading each point on the gid according to some aspect of the crystal

orientation.

25

 

Figure 2.13. An EBSD pattern of TiAl at 25 kV and the corresponding result of the
computerized indexing showing the poles/zones. There is one chance in three that it is
correctly indexed.

The use of a highly collimated, stationary electron probe to focus on the specimen
permits precise placement of the electron probe and the diffraction pattern so formed
covers an angular range of more than 90 degees [54]. As such several major
crystallogaphic zone axes are usually included in the pattern. With advances in
hardware and software, the computer can now control the stage or the beam travel such
that the orientation at each point in a predetermined array can be automatically measured
and stored providing a means of making large number of individual lattice orientation
measurements.[14]. The resulting data set can be presented as maps containing
diffraction information (figure 2.14) or an orientation imaging microgaph (OIM)
[51,55]. This technique is known as Orientation Imaging Microscopy (OIM) and it has
the ability to image all boundaries exhibiting absolute rnisorientation exceeding 1 degee

[56].

26

 

Figure 2.14. Inverse Pole Figure Maps (3) color (b) gayscale.

Although this technique can measure approximately 1800 lattice orientation
within an hour [55] to determine gain orientations in cubic crystal, this technique does
have a drawback. It is not able to successfully distinguish the small c/a ratio in ordered
lattices, specifically y-TiAl. The c/a ratio is 1.02 resulting in inter-band angles within 0.5
degees of those for the cubic lattice [57]. Given that the EBSP pattern indexing has an
intrinsic uncertainity of ~ 1 degee, it is not possible to reliably distinguish features with
smaller angular variance. Thus the EBSP software will generate three orientations for
any given gain, represented by the three colors in maps. Figure 2.15 shows a portion of
the OIM and its corresponding BSE images containing gain 4 and 7. Note that the EBSP
was not able to resolve the differences between gain 4 and 7 (shown with the same Euler
angle colors). SACP composite maps, however, reveal differences between the two

gains as shown in figure 2.16 by the location of superlattice bands.

27

 

Figure 2.15. OIM map and its corresponding BSE image containing grains 4 and 7. Note
that the set of Euler angle colors are similar in both gain 4 and 7, indicating that the
EBSP technique was not able to distinguish between these two gains.

 

Figure 2.16. Two similar sets of SACP composites. Careful examination revealed
superlattice [110> bands (marked) in Grain 7 whereas Grain 4 has one ofthe other [011>
bands.

28

 

Chapter 3: Experimental Procedures

3.1 Material and Sample Preparation

The Ti-47.9Al-2Cr-2Nb TiAl alloy provided by GE Aircraft Engines (Cincinnati,
Ohio) was an investment cast plate produced by Howmet Corporation (Whitehall,
Michigan). The investment cast plate had been heat treated at 1093° C for 5 hr, followed
by a hot isostatic pressing (HIPing) process at 1205° C for 4 hr and further heat-treated at
1205° C for 2 additional hours before rapid cooling. Samples measuring 21 x 3.5 x 2.0
mm were cut from this investment cast plate (figure 3.1). The specimens were then
epoxy bonded to an aluminum backing material with similar dimensions to facilitate
controlled crack extension (this was necessary due to the low toughness of TiAl) as
shown in figure 3.2. The bonded specimens were pressed together with approximately 1
lb weight and cured in a vacuum oven maintaining a pressure of 250 psi and 80°C for 24
hours. A 1 mm notch was cut using a 0.3mm thick diamond saw and the surface was
then gound using a series of SiC ginding papers ranging fi'om 240 to 600 git size
followed by fine polishing using alumina powders ranging from 5 microns through 0.3
microns. The specimens were then electro-polished using a solution of 5% perchloric
acid, 30% butanol in methanol at ~ -60°C using 15 Volts for 20 minutes to achieve a

surface suitable for SACP, EBSP, and ECCI imaging.

29

i

A B

 

Figure 3.1. A portion of the investment cast plate used in this study. Schematic diagam
showing orientation of sample A and sample B with respect to the investment cast plate is

 

 

 

shown.
Notched
’ TiAl
EPOXY
binding" Al

 

 

 

TiAl : 3.5 x 2.0 x 2.1mm
Al : 3.5 x 3.0 x 2.1mm

I“igure 3.2. TiAl alloy was bonded to an aluminum backing to facilitate the controlled
clack extension.

30

3.2. Microscopy
All microscopy (SACP, ECCI, and EBSP) was carried out on a CamScan 44FE
SEM. The machine is fitted with a selected area channeling module for obtaining
selected area channeling patterns. Secondary electron siglals were captured using an
Everheart-Thornley (ET) detector mounted on the side of the chamber and the
backscattered (BSE) siglals were captured via a four quadrant polepiece-mounted silicon
diode type BSE detector. A CCD camera collected EBSP patterns via the phosphor
screen in a unit that projected into the chamber from the side of the microscope chamber.
All microscopy was carried out using an acceleration voltage of 25kV, a probe current
approximately 20-40nA, and a beam convergence angle (2a) of approximately 8mrad. A
working distance of 10 mm was maintained for all microscopy work except for EBSP
analysis, where the working distance was set at 31 mm. The HKL CHANNEL+
software package was used collected and interpret the diffraction patterns after it was
Calibrated with a cleaved {001} surface of a silicon wafer. The EBSP data thus obtained
Was then converted to an .ang file to be read using TSL-OIMm.3 software. This
50 ftware provided better tools for analyzing orientation and related aspects of crystalline
rrli<:rostr'uctures than the features in hkl version 4.2 software package. This software
Presents crystal orientation data in either euler angles or the (hkl)-[uvw] nomenclature
('3 - 3. gain normal and gain tensile loading direction or tilt axis).
All images (BSE, SE, SACP and ECCI) were captured and averaged, typically
using 8 frames, by the framestore feature that is integal to the CamScan 44FE SEM.
These images were stored as 640X480 pixel digital images using an external personal

coIliputer (PC) equipped with a flame gabber card.

31

3.3. Initial condition of near-y TiAl alloy
The microstructure of the near-y TiAl specimen consists primarily of y phase TiAl
and some partially decomposed ((12+y) colonies as shown in figure 3.3. An OIM scan
reveals approximately 92% TiAl and 8% Ti3Al. Close examination of the undeformed
alloy (figure 3.4 and 3.5) showed that while some gains showed little or no deformation
twins, high densities of deformation twins were observed in others. This variation in the
density of the deformation twins in the surrounding gains suggests that activation of
deformation twins during processing may have been sensitive to crystal orientation. In
general, these pro-existing twins (either little or high density) were uniformly distributed
vvithin the gain. A comparison of the preexisting twins before and after deformation due

to the 4-point bending will be discussed further in section 3.8.2.

 

Figure 3.3. Microstructure of the near-y-TiAl showing the equiaxed y gains and small
ClUSters of or; colonies.

32

 

 

Figure 3.4. Undeformed near-y TiAl alloy showing pro-existing deformation twins.

In some regions, a higher density of the pro-existing twins was observed (bottom right
edge of the image). The densities of the pre-existing deformation twins vary from gain
to gain.

'1
NJ
\., .

 

1 7°.
gmfisfl" . I

Figure 3.5. Undeformed near-y TiAl alloy showing pro-existing deformation twins. The
densities of the pro-existing twins vary fiom gain to gain. But overall, the pro-existing
twins were uniformly distributed throughout the gain.

33

3.4. X-ray -texture analysis

Texture analysis was performed using a Scintag XDS 2000 x-ray diffractometer
using 35kV, 40mA and a 2mm slit. The Preferred Orientation Package - Los Alamos
(popLA) software was used to obtain the necessary information. The process used to
obtain a texture measurement and analysis was similar to that described by Kallend et al.
[58] and a similar measurement of a specimen from the same casting is described in [59].
Basically this was a two steps process. A 2-theta scan was first performed on sample B
to obtain the appropriate peaks (Bragg) of the low indice planes (111, 002, 200, 022,
220). The peaks and suitable backgound 2-theta values were subsequently input into the

software for the subsequent pole figure scan.

3.5. Sample loading in the SEM

The electropolished samples were loaded in a 4-point bend fixture attached to an
E. Fullam deformation stage (see figure 3.6) mounted in the Camscan 44FE FEG-SEM.
Bending was performed at the minimum attainable crosshead rate (0.006 mm/s) and
stopped when a crack nucleated and propagated from the notch root. In one of the
samples, (sample A), the load was recorded at 123 lbs when the crack propagated

approximately 1.2 mm from the notch.

34

     
 

4-point
bend specimen

Figure 3.6. Fullam Deformation stage fixed with a 4-point bend fixture (shown at higher
magnification on the right side) and notched specimen in place.

3.6. Grain Examination

The cracked specimens were removed from the stage in air, and immediately
mounted on a specimen holder to permit easy tilting and subsequent analysis. The
samples were immediately returned to the SEM where they remained in vacuum during
all of subsequent analysis (though they was exposed to air when the chamber was opened
by other users several times per week). Following the bending and arrested fiacture, the
entire crack path of sample (A) was characterized and gains that were close to the crack
tip were examined in detail. Using a combination of EBSD/SACP, as well as Schmid
factor/plane trace analysis, the orientations of the gains and slip systems were
determined. The Schmid factor calculations were based on the assumption that the
notched 4-point bend configuration results in large tensile stresses normal to the notch at

the notch tip and subsequently at the primary crack tip. The technique of using EBSP

35

and SACP as well as combining the Schmid factor and plane trace analysis to determine
gain orientations will be discussed further in section 3.7.

The orientation and deformation relationships between adjacent gains at the
crack tip were analyzed using the Luster and Morris slip compatibility factor [3 6] for
active slip systems. To assess the local strain field due to twinning near the gain
boundaries in each gain, the true widths of the mechanical twins were determined by
tilting the specimen so that the beam axis was aligned parallel to the twinning plane. As
the twinning planes are of the {111} type, this is achieved by first obtaining a SACP of
the gain of interest and then tilting the specimen such that the appropriate {111}
channeling band is at the beam axis. The {111} SACP-band will be parallel with the
corresponding {111} plane trace. Although in the typical ECCI technique the beam axis
is aligled to one of the edges of the band (Bragg condition), to obtain good contrast of
the deformation defects (dislocations), as shown in figure 3.7a, this was not done in this
study. Instead it was found that the sharpest images of the twins were obtained when the
beam axis was centered in the {111} channeling band (figure 3.7b), which results in the
twins being imaged directly edge-on.

The relationship between the width of the twins and the observed microcracks
along the gain boundaries were analyzed. A sampling of the gain boundaries that were
intact or cracked near to/or in the path of the propagating crack was statistically analyzed
to allow the development of a crack gowth parameter.

After preliminary characterization, the specimen was reloaded to determine if the
modeling approaches described later could predict the crack path resulting from further

deformation. This modeling approach was then used to analyze the crack path on a

36

similar specimen, sample B and finally the Fracture Propagation Parameter and its
modeling approach were then compared with the Fracture Initiation Parameter using

Simkin’s existing sample data.

 

Figure 3.7. ECCI images of two different twinning systems, each displaying 2 fine
micro-twins separated by approximately 1 micron. a) Aligling the beam axis (+) to one
edge of the {111} band. (note contrast from dislocations). b) Aligning the beam axis (+)
to the center ofthe band. The {111} twins corresponding to the {111} channeling band
are in sharp contrast when the beam is aligned to the center of the band.

37

3.7. Grain Orientation
3. 7.1. Use of EBSP and SACP to determine grain orientation

Determination of the gain orientations was accomplished by using an approach
that combined the speed of Orientation Imaging Microscopy (OIM) and the unique ability
of SACP analysis to identify the correct crystal orientations. OIM will provide a large
number of orientation measurements in a relatively short period of time. Since three
orientations were commonly observed in a given gain based upon the EBSP pattern
indexing, the SACP technique was then used to identify which gain orientation was
correct, since the SACP can identify superlattice reflections.

All EBSP analysis was carried out with the specimens oriented according to the
sample space coordinate system shown in figure 3.8. The notch and crack gowth
direction is downwards along the negative y-axis. When the 4-point bend specimen was
tilted 70°, the x-axis was the rotational axis of the specimen and its orientation did not

change.

 

Figure 3.8. A portion of the 4-point bend specimen aligled in relation to the microscope
screen.

38

3. 7.2. Computations of Stereographic Projection

Given any set of gain normals and their tensile loading direction, it is easy to
compute and plot a stereogaphic projection with the principle poles and plane traces
projected in relation to the normal plane of the gain of interest. This is shown in figure
3.9. Each trace can be easily identified by a color code as shown. This progam
(stereogaphnb) was written by Dr. Mason, Adjunct Professor, Mechanical Engineering
department, MSU and was carried out using Mathematica 4.2.
3. 7.3. Plane Trace Computations

In order to identify the plane traces in this material, Dr. Mason also wrote a
progam that plots plane traces with respect to any given gain normal and tensile loading
direction. The mathematical details can be found in [15]. Plotting the plane trace (using
111_Trace_Projection.nb) was also carried out using Mathematica 4.2 and an example of
the plane trace with respect to the gain normal (3,2, -21) and tensile loading direction is

shown in figure 3.10. Each plane trace has been identified as shown in the figure.

3.8. Analysis of Grain Orientation
3. 8.1. Use of EBSP and SACP to determine grain orientation

The gain normal and rotational axis vectors obtained from the EBSP analysis
were input into the stereogaphic projection progam described in section 3.7.2 and
plotted as shown in figure 3.11. The gain normal was marked “+” in the stereogaphic
projection. The diffraction pattern obtained from an EBSP usually provided around 65
degees of pattern information whereas the angular spread for an SACP was only about 8

degees. However, this 8-degee limitation was overcome by tilting the specimen to

39

obtain similar information obtained in the EBSP patterns. However the stage can only be
tilted in one direction, so to achieve a larger angular view; the specimen was rotated and
tilted to obtain the necessary information using SACPs. By collecting SACPs at specific
tilt and rotation angles, composite SACP maps were obtained.

In figure 3.11 a stereogaphic projection is shown with a schematic of the
specimen and the tilt needed to obtain information in different parts of diffraction space
using SACPs. To view diffraction space above the pattern center (“+”) the specimen
must be aligied such that the notch is at the bottom of the specimen and tilted about the
specimen x axis to obtain the necessary diffraction information (see figure 3.11, sample
A with the direction of the tilt arrow). To obtain information viewed in the direction
towards the lefi, the specimen must be tilted as shown in figure 3.11, sample D, with the
direction of the tilt arrow indicated. Likewise, the same is done for the other two
directions as shown in samples B and C. The tilt mechanism that was fitted to the present
microscope (CamScan 44FE SEM) allows tilt in one specific direction. Consequently, in
order to achieve the desired tilt shown in figure 3.11 it was necessary to rotate the sample
first to the orientation that allows for tilting about the correct axis. Figure 3.12 shows
how sample D (shown in figure 3.11), was rotated through 90 degees, counterclockwise,
in order to allow proper SACP imaging of similar diffraction pattern shown by the

stereogaph.

4O

(V

 

 

 

Color Traces
Red (thin-solid) (100)
Purple (thin-solid) (010)

 

Orange (thin-dashed) (001)

 

Black (thick-solid) (1 10)

 

Blue (thick-solid) (1-10)

 

Green (thick-dashed) (101)

 

Cyan (thick-dashed) (-101)

 

Blue (thick-dotted) (011)

 

Magenta (thick-dotted) (01-1)

 

 

 

 

Figure 3.9. A computed stereogaphic projection with the gain normal (3,2, -21) and a
horizontal tensile loading direction [4,15,2] and its corresponding color code labeled in
the inserted table.

41

 

 

-0.75.
Plane Degrees \
(111) 120.73 ,1_
(.111) 36.67
(1.11) 23.26
{11.1) 117.93

 

Figure 3.10. Plot of the plane trace with respect to the gain normal, (3,2, -21), and
tensile loading direction, [4,15,2].

42

 

Figure 3.11. In order to confirm orientation information obtained from a proposed
stereogaphic projection, the specimen must be rotated and tilted in order to obtain SACP
patterns that can be compared to the three stereogaphic projections obtained from OIM
data. If the stereogaphic projection is viewed from the gain normal (marked “+”)
towards the left, then the specimen has to be rotated such that the notch is at the bottom
of the sample and tilt to the required angle to obtain the needed information.

    
  

90 degrees
counterclockwise
rotation

     

Figure 3.12. The tilt mechanism allows only tilt in one direction. Thus it was necessary
to rotate the sample to a specific orientation first before tilting to the correct angle.

43

As an example, to determine the orientation for gain 4, the analyzed EBSP data
provides three possible gain orientations (figure 2.15 of section 2.14) because of its
inability to resolve the small c/a ratio in the y-TiAl alloy. The stereogaphic projections
of these three potential orientations are plotted in figure 3.13. For the y-TiAl alloy, the
[110> bands are superlattice bands and are color coded in solid blue or black lines. The
SACP composite map of gain 4 reveals a superlattice band (highlighted with black bold
parallel lines) approximately 25 degees above the gain normal. This observation
corresponds with stereogaphic projection b, which has the blue [110> (superlattice band)
in the same orientation as observed in the SACP composite. Note that stereogaphic
projection a has a magenta dotted [011] color band and stereogaphic projection c has a
cyan dashed [-101] color band in the same direction thereby confirming that stereogaph
b is the correct orientation for gain 4 (gain normal (2,17, -14) with rotation axis [4,2,3]).

A SACP composite map was used in the above example to illustrate the location
of the gain normal with respect to the observed patterns and how the observed SACP
composite map relates to the stereogaph as the specimen (with the notch shown on the
top of the specimen) is tilted about the x-axis.

In the next example, the specimen had to be rotated (with the notch shown on the
left side of the specimen) and tilted in order to obtain the SACP for the superlattice
bands. Though a SACP composite map is shown (see figure 3.13), it is not necessary to
generate such a composite map, since collecting the critical observation of a superlattice
band in the microsc0pe (along with the tilt information) is sufficient to identify whether
stereogaphic projections a, b, or c is the correct gain orientation. Three possible

stereographic projections of a gain are shown in figure 3.14 along with an SACP image

44

obtained when the specimen was tilted ~ 5 degees with the notch positioned as shown in
the figure. The observed [110> superlattice band matches the traces in stereogaphic
projection c, thereby indicating the correct gain normal is (6,4,1) with a rotational axis of
[4, -7,4].

There are cases when it is not possible to image a particular superlattice band
because the required tilt is too high to generate a proper SACP imaging condition. In such
a case, the same logic can be used in reverse, to eliminate a stereogaph simply because it
predicted a superlattice band whereas the SACP did not reveal it. By eliminating two out
of the three stereogaphic projections, the third stereogaphic projection can be concluded
as having the correct orientation. This has been shown in figure 3.15.

superlattl :3
band

08
_.-—>

      

Grain 4a Grain 4b
(9,11 ,1 ,) [-7,5,8] (2,17,-14) [4,2,3]

  
 

grain normal

wy rilrég, \ .

Grain 4c
(1 r811 °)[4!'3I2]

Figure 3.13. Three possible stereogaphic projections obtained from an EBSP scan, and a
SACP composite of gain 4. The SACP composite reveals a superlattice band,
approximately 25 degees fiom the gain normal. This matches stereographic projection b
having gain normal (2,17, -14).

45

Graln a
(6.1.-4) [5,6,9] (9,3,13) [-11,7,6]

    

G
(6.4.1)[4-‘714]

Figure 3.14. Three possible stereogaphic projections (a-c) and the observed [110>
superlattice band observed in the SACP when the specimen was rotated such that the
notch is at the bottom and the specimen is tilted ~5 degree as shown.

46

 

no superlattice bands 3) (5,3,1) [1,4,2] (b) 8,1,6) [4,2,1]
observed in SACPs

’

 

«a
k
V

c) (7,1,-10) [10,20,91

Figure 3.15. Three possible stereogaphic projections for gain 8. The band a from the
top SACP corresponds to the superlattice band a’ shown in the stereogaphic projection a
and band c fiom the bottom SACP corresponds to superlattice band c’ (stereogaphic
projection c). Since both bands a and c do not show any superlattice information,
stereogaphic projections a and c are incorrect and are eliminated. Stereogaphic
projection b is therefore the correct solution for gain 8.
3. 8.2. Use of Schmid factor and trace analysis to determine grain orientation

With smaller or heavily deformed gains, it is difficult to obtain a good SACP
with the superlattice information necessary to determine the correct gain orientation. In
such cases, given the three possible gain orientation solutions, it is possible to plot the
plane traces and obtain each Schmid factor corresponding to the plane trace. This
information (Plane traces and Schmid factors) can be compared with the deformed gains

to match dominant twin traces with the highest Schmid factors.

The Schmid factor, 111, is based on the computation of the product of two cosine
angles; firstly, the angle between the normal of the twinning planes and the tensile
loading direction and secondly, the angle between the twin vector for that particular plane
and the tensile loading direction. A schematic is shown in figure 3.16. This calculation
required expressing the plane normal and slip directions in the specimen coordinate

system, and was incorporated into the Fracture Propagation Parameter Progam (see

__1_

Appendix 3).

 

 

    
  
 

Twin vector

 

‘ Normal to
.2 Twin Plane

Loading Direction

Figure 3.16. Diagam for calculating the Schmid factor.
In order to provide a good analysis of the plane traces in this alloy, the
characteristics of the pre-existing twins and the deformation twins is needed. A
comparison of the pre-existing twin and the deformation twins due to the induced 4-point

bend loading is shown in Figure 3.17. Figure 3.17a showed fairly visible preexisting

48

twins on several gains. Figure 3.17b showed the effect of the propagating crack as it
passes on the right of these images. Deformation twins were activated in most of the
gains but these deformation twins were not as uniformly distributed as previous noted
(figure 3.4 and 3.5).

Consider gains or, [3, 8, xand 1) in Figure 3.17. Most of these gains showed
increased twin activity within the gain afier deformation and the increased twin activities
were not uniformly distributed. In gain 4), the twin activity is very localized with most
of the deformation twins located nearer to the crack. Grain qwith a large annealing twin
in the gain had visible preexisting thin twins widely spaced out within the gain, as
shown in Figure 3.17a. But with the crack propagating nearby (to the right), these thin
twins had gown thicker and were highly intensified. The vaguely visible preexisting
twins in gain B were also highly intensified after the crack had propagated nearby. Grain
K was the only gain that showed no sign of any twin activity in either before or after the
crack had propagated nearby.

These images (other than gain 1:) are typical and are representative of the effect
of the crack on the activation of the deformation twins on the gains. In general,
preexisting twins were widely spaced. These preexisting twins were thin and were
uniform in thickness. Non-preexisting twins, on the other hand, tend to vary in twin
thickness, and/or tend to have a larger number of twins clustered together nearer to the

crack region.

49

 

Figure 3.17. Comparison between preexisting and deformation twins due to the induced
4-point bend loading.

With the backgound information on the pre-existing twins and the data from the
plane trace and Schmid factors, the gain orientations for gains whose SACP is not
possible can be analyzed. As an example, the plane traces of the three possible gain
orientations for gain I were plotted, and superimposed onto the gain as shown in figure
3.18. In the backscattered electron (BSE) image on the far right, the dominant twinning
system is observed to have propagated over another earlier set of twins. The observed
dominant plane traces (colored blue) correspond well with the high Schmid factors of
0.41 for the (-111) plane in gain Ic. Similar traces in potential orientations a and b
showed negative Schmid factors (-0.12 and —0.28 respectively) for the observed
dominant plane traces. On the other hand, another set of horizontal twins was highly
visible. From the characteristics of pre-existing twins and the observed location of these

horizontal twins with respect to the direction of the propagated crack, this set of twin is

50

most likely to have preexisted before the deformation. This is because with the crack
running from the top to the bottom of the image and if that (horizontal) twin was
activated, it would have been more localized closer to the crack. On the other hand, the
other set of twin (colored blue) showed higher twin trace intensity nearby the crack path
then away from the edge. This is evidence that this twin (colored blue) must be the
dominant twin that was activated. Thus the gain normal (2,8,5)[13, -7,6] is most likely
the correct orientation for gain I. The horizontal twins in the neighboring gain (gain H)
to the right have a Schmid factor of 0, and the vertical twins have a Schmid factor of
0.41. From a similar analysis presented later, it is evident that the horizontal twin
blocked the vertical twins, implying that they (the horizontal twins) were preexisting.

This method works very well with gains that are highly deformed, since the
highly deformed gains will show dominant twinning systems that allows comparison
with the computed plane traces. This method was used when the usual SACP method
failed to provide a good pattern to analyze the bands (due to a small gain size or large
amount of deformation. However, unlike the cases described above, the degee of
certainty is not 100%, because the local stress state may be different from the global
stress state, and/or pre-existing twins can complicate the analysis.

The tools and analytical techniques described in this chapter allow a systematic
approach to investigating the slip/twinning deformation process in detail. This will be

dealt with in the next chapter.

51

(11-1) 034 .-

     
  

 

(luminzmt .
planes trace

(1-11) 41.32, ., (11-1) -0.29

 

 

 

b) Grain Ib: (-60,30,-10)<10,l3,-21> C) Grain Ic: (2,8,5) <l3,-7,6>

Figure 3.18. BSE image of gain I with one dominant twinning plane (colored blue).
Note how the dominant plane traces propagate across another set of preexisting twins
Inserts a—c are plots of the three possible plane trace solutions with Schmid factors
indicated for the true twinning system on that plane. The observed dominant twinning
system corresponds well with plane trace plot c, (gain Ic) displaying high Schmid factors
(0.41) for the dominant plane traces. Plane trace plots a and b ((gain Ia and gain Ib
respectively) had negative Schmid factors for the observed dominant twin traces.

52

Chapter 4: Results from Initial Crack

4.1. Overview

A texture analysis was performed using x-ray diffraction to determine the
preferred crystal orientation of these y-TiAl alloys. Sample A was subjected to a 4-point
bend loading to initiate a crack to gow and propagate mid-way through the specimen
width before arresting the crack. Defect analysis of dislocations and twins was done to
determine the extent of plastic deformation within gains near the crack path. Twin
thickness was imaged using ECCI imaging and the thickness measured using the NIH
Image sofiware. Crystal orientations were determined using a combination of EBSP and

SACP techniques, as described in chapter 3.

4.2. X-ray Analysis of near-y TiAl alloy

An x-ray pole figure scan of Sample B (figure 4.1) showed that the specimen had
strong peaks from a few strongly diffracting (large) crystals, but with no obviously
preferred orientations, as indicated by the second set of pole figures where the specimen
was oscillated under the beam to improve sampling statistics. The oscillated pole figures
are more suitable for comparing the effect of texture on crack characteristics.

In view of this analysis and the orientation of samples A and B, both had the
dominant tensile stress in the same direction (TA). For Sample A, the TD direction at the
notch root contract due to the bending process (the back end will expand and get thicker).
Since there are more crystals near [111], the sfiflest orientation (218 GPa)[58], there will
be less elastic strain along the crack from (TD direction) for specimen A than in

specimen B, which would have more, as the ND direction is more compliant (139

53

GPa)[58] in [100] direction), allowing more elastic strain in the ND direction at the crack
tip, and perhaps, more elastic strain incompatibility.

In terms of plastic deformation, Specimen A needs to plastically expand in the TD
direction. In that direction, there is a small fiaction of material where ordinary
dislocations have a high Schmid factor, and a small fiaction of material where true tensile
twins have a high Schmid factor. In contrast, Specimen B needs to expand plastically in
the ND direction, and it has a higher volume fiaction of gains with high Schmid factors
for twins and ordinary dislocations, and comparatively less material in hard orientations
near 001 .

Taken together, it appears that more plastic deformation probably occurred in
specimen B than in A, leading to more work hardening, and hence higher stress before

the final crack propagated.

4.3. Initial loading of the 4-point bend specimen

Initial loading of the 4-point bend sample (a) resulted in mixed mode cracking
with the crack running approximately 1.2 mm from the notch root as shown in figure 4.4.
The load was 126lbs when the crack propagated from the notch. Extensive twin and
dislocation generation is apparent ahead of the crack tip as shown in figure 4.3. This
crack was observed to alternate between interganular fiacture and cleavage (Figure 4.4).

Areas where the crack was interganular are highlighted in figure 4.4.

54

Sample not oscillated
~ 34¢ . 111

 

Sample oscrlla

. «,4,- 4,

TA

Figure 4.1. X-ray pole-figure scan of specimen b with the orientation indicated reveals a
strong heterogeneous texture. The effect of oscillation alters the texture measurement
slightly, and inverse pole figures show that there are no highly preferred crystal
orientations. Great circle passing through {101} and {111} poles are shown, and the
little nub shown in the pole figures indicate 112 directions.

55

 

Figure 4.2. Secondary electron (SE) image of the crack running approximately 1.2mm

from the root of the notch.
. ‘ i I}

a

      

_.
r
1‘
t

r- '

Figure 4.3. High magnification ECCI images of the crack tip showing extensive plastic
deformation ahead of the crack tip. Both twin and dislocation contrast is observed.

56

 

Figure 4.4. A composite backscattered electron (BSE) image of the crack path showing
how the crack moved from interganular to transganular and back to interganular. The
interganular crack regions are highlighted and numbered 1 — 3. The notch is located at
the top of the image while the arrested crack tip is located at the bottom of the image. A
schematic of the layout of the gains is shown on the right. The tensile direction is
horizontal.

57

Large amounts of plastic deformation are also evident in the gains surrounding
the arrested crack tip shown in figure 4.5. In this figure, traces of different slip systems
are evident in the surrounding gains. The plane traces of gains 1, 2, and 4 are indexed.
The region indicated by arrows (Start and End) in figure 4.5 was further investigated at
higher magrification as shown in figure 4.6. This image (figure 4.6) is rotated relative to

figure 4.5 and shows microcracks generated along the gain boundary between gain 1

andgain2.

4.4. Width of twins and shear strains

The width of the twins was measured directly from higher magnification images
(for example, see figure 4.7) with an accuracy of +/- 20%. To check this approach,
selected images were scanned and analyzed using the NIH Image analysis software. Full-
Width-Half—Max (F WHM) measurement of intensity plots of selected areas (see figure
4.8) resulted in comparable twin thickness measm‘ement to those measured manually.

Figure 4.9 shows a plot of the twin thickness with respect to the location along the
gain boundary for gain 1. Twins that correlate with microcrack formation are labeled.
In general, twin widths geater than 175 nm were correlated with microcracks. The
exception is microcrack #3, which was associated with a goup of closely spaced finer
twins with a total width of 300 um. All the microcracks observed opened asymmetrically

in relation the twin generated at the gain boundary. This is shown in Figure 4.10.

58

‘ .. t ’
a 6'
I: \\ fig \
Hummus);
‘ l

.9 I'um

 

Figure 4.5. BSE image showing the arrested crack tip and the surrounding gains 1,2,
and 4. The tensile axis is horizontal. The local strain along the gain boundary between
gain 1 and gain 2 was studied between the indicated arrows showing the start and the
end of the strain computation.

59

 

 

Figure 4.6. ECCI image shows the leading (-111) twins from gain 1 and the
corresponding plane traces in gain 2. Microcracks (numbered 1 - 6) were observed
along the gain boundary. Image is rotated relative to figure 4.5.

 

ia- .

Figure 4.7. High magnification ECCI image of twins. Dimensions were measured
directly from this image. The highlighted region (rectangular) was analyzed using the
NIH Image Software as shown in figure 4.8 and resulted in compmble FWHM
thickness measmements.

60

 

 

200 .24

 

 

122.28
0

 

366

Pixels

 

 

 

Figure 4.8. Examples of two twin measured using the NIH sofiware to quantify the gay
scale. The intensity plot was superimposed over the high magnification BSE image.

TWIn Wldth

1hlckness(nm)
0185:8888
OOOOOOO

 

o

0 5 10 15 20 25 30 35 40 45
Dletanee along grain bou nduy

0 thickness H (n m)

Figure 4.9. Plot of twin width vs. position along the gain boundary. The indexed
numbers correspond to the observed microcracks along the gain boundary. Microcrack
#7 s not shown in figure 4.6.i

61

grainZ

 

microcracks

 

Figure 4.10. High magnification ECCI image of two microcracks, 5 and 6, (see figures
4.5 and 4.6 for location of these microcracks). Both microcracks opened asymmetrically
in relation to the twin generated at the gain boundary.
4.5. General observation of defamation twins and dislocations

It was generally observed that if strain fiom the twin shear within one gain was
accommodated or transferred to the adjacent gain, it was either by the generation of
dislocations and/or another set of twinning planes in the adjacent gain, as shown in
figures 4.11-4.13. Figure 4.11 shows two gains, (labeled x and y) located ~100 microns
fi'om gain 5A in sample A (see figure 4.4). A higher magrification image of the
highlighted region is shown in figure 4.12. In this figure (figure 4.12), two plane traces,
yl and y2 (from gain y), were observed to have some correlation with dislocation
activity in gain x. A higher magrification BSE image (figure 4.13) showed that the twin
plane yl resulted in the generation of dislocations across from the gain boundary, as

shown by the high contrast from a large dislocation density. Twin plane y2, on the other

62

hand resulted in the formation of deformation twins (plane x1) in the adjacent gain x

(figure 4.12).

;_-_-' grain y

 

' h
. 4' 4
.... 3., . ,
‘ 54‘; > "‘._ - .«A I]

Figure 4.11. BSE image of neighboring gains x and gain y. The region of interest is
highlighted and shown m figure 4.13.

twin plane yl

twin plane )2

grain y

 

Figure 4.12. ECCI image of neighboring gains x and y that were highlighted in figure
4.12. Note that the end of twin plane yl (at the gain boundary) showed a large amount of
plastic deformation in gain x that gives high contrast. Also, there is correlation between
twin plane y2 to twin plane x1.

63

's
twin plane )1 ‘,

grain _\'

 

Figure 4.13. A high magrification ECCI image of the highlighted region in figure 4.13
reveals high contrast variations resulting from a high dislocation density in gain x ahead
ofthe twin plane yl.

4.6. Reloading of the cracked specimen.

When this first specimen (sample A) was re-loaded in 4-point bending, the
microcracks at the boundary between gain 1 and gain 2 opened firrther. At the same
time, the surface relief from the deformation twins associated with the gain boundary
microcracks increased (Figure 4.14a). Examination of this surface relief using secondary
electron imaging reveals that the magritude of the deformation twin topogaphy
decreased with distance away from the gain boundary (Figure 4.14b). An ECCI image
of the same region also showed deformation twin contrast diminishing with distance from
the gain boundary (Figure 4.15). Microcracks also nucleated along the gain boundary
between gain I and gain 4 during the reloading of the specimen, as shown in figure

4.16.

 

Figure 4.14. a) Secondary electron image showing surface features that diminish with
distance from the gain boundary. b) ECCI image reveals the heavily twinned planes in
gain 1 on the right and moderate twin activity in gain 2. Note that microcracks have
gown compared with figure 4.6. Some twins terminate in the gain in the boxed region,
shown in more detail in figure 4.15. The tensile axis is horizontal.

 

Figure 4.15. Inset from figure 4.14b shows traces of twinning planes terminating in
the interior of the gain (arrows).

65

1') "
@CW? :51.
{gr—x U.) — - LJ

 

Figure 4.16. Microcracks nucleated along the gain boundary between gains 1 and
gain 4 near the triple point with gain 2.
4.7. Examination of grains along the crack path
An OIM scan of the crack path was performed on the arrested crack sample and is
shown in figure 4.17. As observed, different shades of color can be seen within each gain
due to EBSP sofiware’s inability to resolve the small c/a ratio. The corresponding BSE
image of the crack path is shown in figure 4.18. The gain orientations closest to the crack
path were determined using the technique of combining EBSP with SACP or the Schmid
factor values with plane trace analysis. These results are shown in Appendix A. The
gain orientations (gain normal and the stage tilt axis directions) are tabulated in Table
4.1.
Schmid factors for twinning and ordinary dislocation activity were determined for

each gain. For example, a BSE image of the interaction between gains B and C is

66

shown in figure 4.19. The crack path is to the right of both gains, propagating
downwards in the direction indicated in the image. The plane traces analysis and its
corresponding Schmid factors were superimposed over each gain. In gain B, two
dominant sets of twins were observed (one evenly distributed throughout the gain and
the second was more active near the crack path). These two sets of twins correspond well
with (-l l 1) and (1-11) plane traces, which have with similarly high Schmid factors of
0.38.

The gain boundary between grains B and C was intact and there is some degree
of correlation between the observed heavily twinned vertical (-111) planes in gain B and
both (11 l) and (-111) twinning systems in gain C. In gain C, however, it is difficult to
relate the observed twin traces with the low Schmid factors (including negative values).
The changes in localized stress states as the crack propagated may have accounted for the
activity of some of the twinning systems, or, these twins may have been pre-existing (e. g.
twins on (111) and 1-11) planes.

There is a remarkable degee of correlation between the vertical (-111) twins in
gain B and concentrated slip or twins on (-111) planes in gain C. The Luster and Morris
parameter (table 4.2) showed that there is a strong degee of slip compatibility between (-
111) twins in gain B and (-111) twins in gain C (m’=0.35) and (-111) ordinary
dislocations in gain C (m’=0.65). But the interpretation of slip in gain C is also

complicated by the annealing twin boundary in gain C near gain B.

67

    
    
    

I 50.00 um- — 5 steps

Figure 4.17 Orientation image mapping (OIM) image of crack path, shown in a normal
direction inverse pole figure map.

68

 

Figure 4.18. Composite BSE image of the crack with the notch on the top and the crack
tip located towards the bottom ofthe image. The gains that are close to the crack path
are labeled as shown.

69

Table 4.1: Grain Orientations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grain h k l u v w
A 1 3 6 ~15 1 3
B 8 3 ~10 -3 8 0

BB 11 1O -13 8 ~1 6
BE 6 4 1 4 -7 4
C 100 26 ~75 53 84 100
D 9 11 3 ~1 0 3
E 6 16 ~17 17 0 6
F 10 11 16 9 ~14 4
PK 1 1 ~3 15 ~9 2
G 9 3 -4 ~1 15 9
H 3 7 15 ~11 -6 5
HJ 31 25 ~99 100 40 41
I 32 59 ~99 50 25 31
J 7 8 1O 2 -3 1
K 7 10 6 -4 1 3
L 3 2 ~21 4 15 2
LA 6 6 -1 -2 3 6
ZW 0 2 3 1 ~12 8
Z 7 20 ~17 100 0 41
1 1 5 6 3 ~3 2
2 61 O 100 ~99 92 60
3 13 5 4 2 ~14 11
4 1 12 -10 4 3 4
5 4 13 ~12 3 0 1
5A 13 20 0 ~99 65 16
6 9 1 20 -8 12 3
7 9 11 0 ~11 9 13
8 8 1 6 -8 10 9
9 8 10 1 -4 3 2
9twln 10 2 11 -3 4 2
10a 10 1 7 -5 8 6
10b 1 3 0 ~25 5 -2
11 7 2 ~15 5 ~10 1
12 9 2 8 -4 ~10 7
13 6 1 -9 6 9 5
14 3 2 11 ~19 1 5
15 4 7 -8 -3 4 2
16 4 1 -6 10 14 9
17 6 1 4 ~8 12 9
18 4 1 -6 9 12 8
21 49 100 -48 100 -9 80
21a 12 1 19 -1 12 O

 

70

 

Table 4.1: Cont’d

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grain h K I u v w
22 70 40 ~41 0 1 1
23 81 17 100 ~25 100 4
23a 5 6 1 -8 5 1O
31 8 12 5 -4 1 4
32 2 3 ~18 9 0 1
33 5 3 ~3 3 11 16
33a 15 7 8 0 -4 3
33b 1 2 -7 -1 4 1
33¢ 10 9 14 -4 6 ~1
35 41 90 ~40 1 O 1
36 29 100 0 ~99 29
37 100 15 68 -48 100 49
49a 1 1 15 1 6 -5 9
49b 1 1 1 -9 3 3 4

 

 

In figure 4.20 the interaction between gains 1, H and HJ is shown. The path of
the crack is to the left of these gains. Both gains I and HJ show large amounts of
deformation twinning in the gains. There was a moderate amount of twin deformation in
gain H. Plane trace analysis and corresponding Schmid factors were superimposed on
each gain. A higher magnification BSE image of highlighted area A is shown in figure
4.21. Microcracks were observed along the gain boundary between adjacent gains HJ
and H as indicated by the arrows. Images of other gain boundary interactions with and
without microcracks are provided in figure 4.22 and in Appendix 2.

Figure 4.22 shows a BSE image of gains L and LA with a portion of the OIM
map (see figure 4.17) inserted on the left bottom corner to facilitate identifying the gain
boundary between these two gains. The primary crack propagating through grain LA
alternates between cleavage and jogging to a nearby intraganular microcrack, changing
its path several times. Two microcracks are evident at the gain boundary region between

gain L and LA as shown within the highlighted circle. These gain boundary

71

microcracks are identified by the sharp protrusion and are usually at an angle to the crack
path.
The presence of gain boundary cracking or the lack of it between adjacent gains

can thus be systematically investigated and will be described in the next chapter.
. (ll-1) 0.08
(1 1 1 ) 0.08

(1-11)0.3 4100.36,.) \’

Grain B
Grain C

,

(11Lr)0.14

 

Figure 4.19. BSE image of gains B and C with the plane traces and corresponding
Schmid factors superimposed on the gains.

72

w

Dire tion ofv ' _

Cra k Path

4—>

 

Figure 4.20. BSE image of gains H, I, and H] with the plane traces and corresponding
Schmid factors superimposed on the gains. Higher Magrification BSE image of
highlighted area A is shown in Figure 21.

Table 4.2: Luster and Morris Slip Compatibility Factors for grain B and C.

Schmid

 

73

.Grain‘l

,1

- icrocracks

 

Figure 4.21. Higher magrification of the highlighted region marked A in figure 4.20
showed microcracks along the gain boundary between gains H and HJ.

74

» . evidences of

t

E" \ microcracks ‘

 

Figure 4.22. BSE image of gains L and LA with a portion of the OIM map (see figure
4.17) inserted. Primary crack changes its path several times as it cleaves within gain
LA. Microcracks are evident within gain LA (arrows) and at the gain boundary region
between gain L and LA as shown within the highlighted circle.

75

Chapter 5: Analysis of Arrested Crack
5.] Overview
Luster and Morris (LM) geometric compatibility factor was used to provide some
insight into the slip transfer between gains along the initial crack path but the LM
numeric values were not wholly satisfactory in identifying or predicting the microcracks
at the gain boundaries. A fracture propagation parameter was developed to address this

problem (identify boundaries that are prone to microcracking).

5.2. Crack path and microcrack nucleation

The composite backscattered electron (BSE) image of the crack path (shown in
figure 4.6) that varies between transganular to interganular is consistent with
observation of mixed mode failure noted by other researchers [59,60]. The extensive
twin and dislocation generation observed ahead of the crack tip suggests that even though
the crack propagated mostly by cleavage, it generated plastic deformation ahead of the
crack tip. With sufficient plastic deformation in a gain ahead of the crack, the crack tip
blunted and stalled as shown figures 4.5 and 4.7. At the same time, microcracks
nucleated at the nearby gain boundary between gains 1 and 2 in figure 4.7 (arrows
indicate the region of interest) that indicates in a change in the crack path from
transganular to interganular. The nucleation/formation of microcracks ahead of the

crack tip is analyzed and discussed in more detail in sections 53-5.

76

5.3. Compatibility Factor for Active Twin Systems

With the ability to determine each gain orientation, plane traces can be easily
identified for deformed gains and the interactions between the adjacent gains can be
analyzed using the Luster and Morris compatibility factors. As the general state of stress
is known, it is possible to estimate the Schmid factors for the observed deformation
systems. Schmid factors were calculated based on the principal tensile stress direction,
and are only approximate since the actual stress condition changed as the crack gew. The
Schmid factors, along with the compatibility factors m’ between active twin systems in
gains 1 and 2, are shown in Table 5.1 in italics. Negative compatibility factors imply
that twinning on one system requires anti-twinning on the other, which is not probable,
while positive factors indicate various degees of twin compatibility, ranging up to a
value of 1 for full compatibility.

As shown in table 5.1, the (-111) twinning planes in gain 1 have the highest
Schmid factor (0.42) and this is consistent with the large numbers of twins observed in
gain 1 (see figure 4.5). [Traces of(1 1-1) and (1-1 1) twins are also observed in this gain,
fewer due to their lower Schmid factors. Figures 4.5 and 4.6 show that microcracks are
correlated with these active twin systems.

In gain 2, (figure 4.5) the most heavily twinned system was found to be (~111)
although the Schmid factor is -0.23, the lowest of the 4 {111} twin systems in the gain.
The reasons for this are not entirely clear. It is possible that the tensile loading direction
changed as the crack gew. Traces of the other twin systems are evident but to a lesser

amounts.

77

Table 5.1 shows that the highest value of m’ = 0.46 occurs for twin systems that
have negative Schmid factors. This illustrates how a high m’ value with low Schmid
factors does not necessarily mean that compatible slip transfer occurred.

The heavily twinned (~111) planes in gain 1 have poor geometric compatibility
with the (111) planes in gain 2 (m’=0.14), which also has a negative Schmid factor (-
0.11). However, in figure 4.6 there appears to be some correlation between these two
deformation systems even though the geometric compatibility is low. The (ll-1) traces
in gain 2 that have the highest twinning Schmid factor (0.21) and a negative
compatibility factor (~0.54) were not observed close to the boundary with gain 1. There
was better compatibility between the gain 1 dominant (~111) twinning system and
ordinary dislocation slip systems (including highly stressed (001)[110]) having high
Schmid factors in gain 2, with compatibility factors > 0.5.

The large number of twins observed in gain 1 relieved the large stress fields
ahead of the crack tip in gain 1. The deformation twin strain can either be dissipated
into other gains by strain accommodation at the gain boundaries via generation of
dislocations or twins in the adjacent gain, or stresses will accumulate at the gain
boundaries. If these stresses accumulate at the gain boundary, it may then result in
microcrack nucleation. Although there is twin activation in gain 2, there is weak
indication of twin to twin strain transfer from gain 1 to gain 2 (figure 4.6). This appears
to be consistent with the sigrificant numbers of microcracks opened up in direct
correlation to the most distinct (~111) twins in gain 1 (arrows in figure 4.6). However,
slip activation in gain 2 correlated with the dominant twins in gain 1 appears to be more

likely, on (~111) and (1-1 1) planes.

78

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79

In short, the Luster and Morris parameter offers some insight on the role of slip
transfer, but not on microcrack formation, as microcracks developed in a boundary where
twin to ordinary slip transfer occurred. Also the Luster-Morris parameter it is not wholly

satisfactory, because it is clear that the magritude of the Schmid factor is also important.

5.4 Microcrack formation

The microcracking between gains 1 and 2 can be directly correlated with the
geometry of the primary twinning system in gain 1. The shear developed by twinning on
the (~111) plane in the [~11-2] direction resulted in a local tensile stress on one side of the
twins where gain boundary microcracks nucleated. Figure 5.1 shows a projected image
of the (~111) twinning plane as it is tilted within the crystal. The (-111) plane is
represented by vectors for [110] ordinary dislocations (dashed line) and [011] and [10-1]
superdislocations (solid lines). The plane normal has a component out of the page and
the projection of the [~11-2] twining direction bisects the ‘triangle’ and is perpendicular
to the [110] ordinary dislocation direction. When this image is superimposed on gain 1,
(Figure 5.2), the direction of the twinning Burgers vector is to the left and into the gain

boundary.

80

  

1/6[-ll-2]

  

Figure 5.1. The computed projection of the (-111) plane based upon the known crystal
orientation of gain 1 (normal direction [1,5,6], and the horizontal tensile direction (stage
rotation axis), [3, -3,2]), shows superdislocation directions, (solid lines), the ordinary
dislocation direction (dashed) and the plane normal (bold arrow) that has a component
pointing out of the page. The [~11-2] twinning direction bisects the triangle and is
projected into the page.

1
O
I
1
1
1

 

Figure 5.2. The projection of the (-l l 1) twin plane superimposed on gain 1 provides a
means to visualize how twinning vector (bisects the triangle) moves towards the left and
into the page.

81

With the direction of the twinning vector defined, it is possible to explain the
asymmetric opening of the microcracks in relation to the twins. Figure 5.3 shows a crack
opening with the twinning plane fiom gain 1 perpendicular to the page. The projection
of this corresponding (~111) twin plane is inserted above and left of the image while the
projection of the (111) twin plane fi'om gain 2 (which has the highest geometric
compatibility factor of 0.14) is inserted above and right of the BSE image. The twinning
vector [~11-2] fiom gain 1 is projected into the page and upwards whereas the twinning
vector [1 1-2] from gain 2 is projected out of the page and to the left. A schematic thin
twin (-III) in gain 1 is shown in figure 5.4a. As the twin gew, the resulting twin vector
[~1 1-2] is projected upwards and into the paper on the left side of the twin. For a thin
twin, the adjacent gain could easily accommodate this small amount of shear elastically.
However, for a thicker twin, the shear at the boundary (see figure 5.4b) and the inability
of the adjacent gain to accommodate the larger amount of shear, a local compressive
stress would be generated on the left side of the twin . (Twinning on the opposite side of
this plane, i.e. (1-1-1) [1-12], would have equivalently resulted in a tensile stress on the
right side of the twin). Superposing the global tensile stress state on the local
twin/boundary stress state intensifies the tensile effects on the right side of the twin.
Thus the crack opened asymmetrically to the right of the twins, as observed. This
analysis further supports the observation that microcracks form as a direct consequence

ofthe (-111) twinning in gain 1.

82

Gram l , /

[4-10] 0’ I Grain 2

l/‘6[-l 1-2]

9

 

Figure 5.3. BSE image of a crack opening. Projections of the corresponding twin planes
are shown on the top of the BSE image. Solid arrows have component out of page while
the dotted arrow has a component going into the page.

83

local
compression-tension
strain
gain 2 / /

 

 

 

/ / microcrack

gain 1

    

a) Twin Shear b) Thicker Twin Shear

Figure 5.4. a) Thin twin shear. b) Thicker twin shear causes a local compression-tension
strain. This strain resulted in a local tension opening force and hence, a microcrack
opening between gain 1 and 2.

From the direction of the twinning vectors projected in figure 5.3, it can be seen
that while the twinning vector (~111) [~11-2] fiom gain 1 had a component upward and
into the page, the twinning vector (111)[11-2] from gain 2 had a component out of the
page and to the left. Though the two vectors appear to have negative compatibility twin
direction, the planes normals also have a negative compatibility, making the
compatibility factor product positive. With the two twinning systems opposed to each

other, it is no wonder that the strain was not completely transferred across from gain 1 to

gain 2 by twinning.

5.5 Subsequent Re-Loading

When the specimen was re-loaded in 4~point bending after the characterization
described above, surface ledges developed along some of the (~111) twins in gain 1
(Figure 4.14). These features appear to be largest at or near the gain boundary and

diminish in the direction toward the center of the gain.

84

The twin contrast also appears strongest at the gain boundary and often
disappears toward the gain interior (see-highlighted region in figure 4.14(b) that is
magrified in figure 4.15. This suggests that the twins were activated at the boundary
between gains 1 and 2 and propagated into gain 1, but only some of the twins
propagated across the entire gain (Figure 4.5). This history implies that the operative
twinning system in gain 1 was the compliment to the one used in characterizing, i.e.
instead of a (~111) [~11-2] twinning system, it is more likely that the (1-1-1)[1~12]
twinning system operated, which moved mass away from the boundary on the side that
the crack opened up. Microcracks also nucleated along the gain boundary between
gains 1 and 4 during re-loading of the specimen, as shown in Figure 4.17. As noted in
Table 5.2, the heavily twinned (~111) planes in gain 1 have very poor geometric
compatibility with any of the twin systems in gain 4, with the highest factor m’ of 0.06
between a/6[-11-2](-111) twins of gain 1 and a/6[11~2](111) twins ofgain 4. With
such low compatibility between twinning systems between these two gains, it is not
surprising that the boundary cracked. Furthermore, only the most dominant twinning
(1 1-1) system in gain 4 interior has a positive, but moderate, Schmid factor. The
(111)a/6[11-2] and (~111)a/6[-11~2] twin systems that interact with the gain boundary
cracks, have negative Schmid factors implying a more complex local state of stress, were
operating, or that most of the twins were preexisting.

Interestingly, the Luster and Morris geometric compatibility factor showed that
the heavily twinned (-111) planes in gain 1 have a good geometric compatibility with the

(1-11)[110] ordinary dislocation in gain 4, having a high m’ value of 0.85. But even

85

with that high m’ value, microcracks were observed along the gain boundary between
gain 1 and gain 4.

As shown in figures 4.3, and 4.5, plastic deformation readily occurs near the crack
tip by available slip and twinning systems, but the ability to dissipate this deformation to
neighboring gains is hampered by the need for effective accommodation of concentrated
twin shear at the gain boundaries with neighboring gains. Table 5.2 shows that there
are no high compatibility values for twin transfer, though several slip systems have high
compatibility with the dominant (-111)[-11-2] twins in gain 1.

Luster and Morris examined the relationship between deformation systems across
y-TiAl gain boundaries using a geometric compatibility factor m’. Based on this
assumption, it would be reasonable to use this factor 111’ to serve as a microcrack
predictor or indicator, low m’ for microcrack possibility and high m’ for no microcrack
formation or intact boundaries. But judging from the two sets of gain boundary
interactions for gain boundaries 1~2 and 1-4, it is not entirely clear that the Luster-Morris
parameter can unambiguously predict conditions for microcracking. The high value of
m’ (0.85) between the dominant twin in gain 1 and the most highly stressed ordinary
dislocation system (1-11) [110] in gain 4, provides some indication that slip transfer
occurred across the boundary, but this does not explain the occurrence of microcracks at
these boundaries.

The ability to predict microcrack formation may be the key to understanding
toughness or the lack of it, in a material. Thus a new parametric approach is needed to

predict this microcrack formation.

86

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87

5.6 Examination of microcracks along various grain boundaries

Grain boundaries between two adjacent gains along the path of the crack were
examined for possible microcrack nucleation. In areas slightly away fi'om the crack
path, it is easy to spot the microcracks along the gain boundaries. On the other hand, it
is n7t possible to see microcracks if they gew to become part of the primary crack. In
such cases, it is important to observe ‘tell tale’ signs. These ‘telltale’ sigrs include
sharp protrusion of the microcracks at an angle with respect to the direction of the crack
path as shown in figure 4.22. It is common to see microcracks lying at an angle to the
primary crack along a gain boundary. In other cases, the slight change in the path of the
primary crack suggests that it had probably passed through gain boundary microcracks

that gew and eventually joined up similar to that observed in Figure 4.14b.

5.7 Fracture Propagation Parameter

A total of 39-gain boundaries surrounding the primary crack were analyzed to
allow the development of a fiacture propagation parameter analogous to Simkin’s
fracture initiation parameter (see section 2.7). In this study, the role of twinning as a
deformation accommodation mechanism in gain B due to twinning deformation in gain
A was examined (noting that the lack of compatible twinning may correlate with
microcracking). This requires consideration of directionality of twin to twin deformation
transfer that was not considered by Sirnkin, since deformation transfer by twinning
requires that the absolute direction of the twinning shear be not opposed to twinning in
the neighboring gain (as depicted in figure 5.5). Directionality is also relevant to the

process of crack propagation, because a crack approaching gain A before gain B will

88

cause more deformation in gain A than B (see figure 5.6). Thus, slip transfer is more

importantfi‘omgainAtoBthanBtoA.

 

Figure 5.5. Slip transfer by twinning requires that the absolute direction of the twinning
shear be not opposed in a twin system in the neighboring gain.

 

Figure 5.6. Directionality is important in evaluating the deformation of gains in the path
of the crack. A crack approaching gain A before gain B will cause more deformation in
gain A then in gain B.

89

An adaptation of Simkin’s fiacture initiation parameter that takes directionality into

account can be expressed as

A

b...- 13.-.0 + i 6 ~6...I ]

 

88W . l A“?

A A 2
F... = m...|b..-tl [2,.

The three parts of this parameter are as follows: mm. is the maximum Schmid

factor for a reference twin system in gain A; b AM “t identifies how well the reference

twin Burgers vector is aligred with the tensile traction unit vector i (for simplicity, in the
direction of tensile loading); the two sums of the dot products in large square brackets
describe how well the Burgers vector of this most highly stressed twin system in gain A
is aligred with the Burgers vectors for ordinary dislocations and/or twins in gain B,
indicating the possibility of slip transfer. Ordinary dislocations and the twins in gain A
are not considered in the sum because the dot product terms will always contribute the
same values to the sum. Absolute values are used to identify interactions that are slip
based, where either slip direction is equally probable. The second sum recognizes the
directionality of twin shear (Figure 5.5). It is also possible to relax the directionality
requirement but such attempts resulted in poor statistics as shown in Appendix 4.

Ideally, if two gains have a highly compatible pair of twin orientations, then one
of the twin system in gain Y will be more closely aligred with the burgers vector for
twins in gain Y’. For example, gain Y with orientation (7,20, ~17)[100,0,41] and gain
Y’ with orientation (7,20, -9)[100,0,78] are misoriented by 20.2° but there is an 8°
difference between (~111) plane normals). One of the twin system, (~111)[-11~2], in
gain Y is closely aligred with the burgers vector for twins (~111)[-11-2] in gain Y‘ than

the other twinning or dislocations systems (see Table 5.3) with m’ =0.93. The same twin

90

system (~111)[-11-2] in gain Y however, may not be as closely aligned with the other
three sets of slip systems in gain Y’. But this slip compatibility with only the most
. highly stressed twinning system in gain Y’ is sufficient to allow twin-based slip transfer
across that leaves only some residual strain in the gain boundary. Note that all of the
other Luster-Morris parameters have low values, making the overall sum rather low, too.
Taking into account the summation of all the interaction of the twin slip in gain Y with
the ordinary dislocations and the twin system in the adjacent gain (gain Y’), the
Fracture Propagation Parameter computation indicated a low F A03 value of 0.59.

On the other hand, if the twin system (1-11)[1~1-2] in gain K is not aligned with
any of the four twin system in gain J (see Table 5.4), then there will be more residual
strain left on the gain boundary, creating a highly stressed boundary that is more likely
to result in microcracks. The Fracture Propagation Parameter would have a much higher
value (1.31), due to many more combinations of slip systems that have modest values of
the Luster-Morris parameter.

With this fiacture propagation parameter established, two sets of the fiacture
propagation parameter values (abbreviated as F A 03 values) were computed for each set of
gain boundary interactions in Table 5.5. There are two sets of F A03 values simply
because of the directionality of the interaction between gain X to gain Y or gain Y to
gain X. If the direction of the crack is moving from gain X to gain Y, then the value of
x-y is used. But if both gain X and gain Y are lying side-by-side, then the maximum
value is selected. The final selection of the F 403 value for each gain boundary
interaction and its corresponding observation of cracked (denoted as 1) or intact (denoted

as 0) gain boundary is shown on the far right hand column.

91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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93

Population means fiom these two sets of FA.” values can be easily calculated. To
compare the population means of these two distribution to see if they are sigrificantly
different, the student t—test was used.

In a typical t-test, using a null hypothesis that the population means are equal, the
t-statistics, which involves the difference of the two sample means, was computed to see
how extreme it (t-statistic) is. If the t statistic is unusually large, which says that the
difference of means is geat compared to its estimated standard deviation, then the
difference is statistically significant.

The t-test analysis can be easily performed using the Data Analysis Tools found
in Microsoft Excel and the results are shown in Table 5.6

The t-test analysis showed that the sample mean and the standard deviation for
the intact boundary are 0.73 and 0.31 respectively while those of the cracked boundary is
1.23 and 0.15 respectively. Assuming a null hypothesis that the two sample means are
statistically equivalent, there is a 99% chance that t critical (for two-tail) will fall within
+/- 2.99. From the analysis, t-statistic was computed to be +/-5.47, a value that is much
larger than the t-critical value. This suggests that the difference of means is geat and is
statistically sigrificant. There is a 99% confidence that the cracked population, llAnc,
exceeded the intact population mean, 11,131, with individual confidence intervals (based on
the mean and +/- standard deviation) for 1.123,; and H.431 of (1.07, 1.38) and (0.42, 1.03),
respectively.

Figure 5.7 shows a histogam of both the intact and cracked F 4:3 values. It is
clear that microcracking is unlikely when F 403 < 1.0, but is more likely when FA.” >

1.0. Thus, this parameter may be used to predict a path for crack propagation (i.e., gain

94

boundaries with F A :3 > 1.0 are prone to fracture while boundaries with lower values
resist cracking).

As shown in figure 5.7, there were a number of intact values with high F A23
values. These adjacent gains were re-examined to seek possible reasons for the lack of
boundary cracking. Table 5.7 lists gain boundaries with high F 403 values that were
intact. The reasons for having intact boundaries while having a high F A 03 values are
listed on the right hand column of the table.

In gains BE and E (see appendix 2-Figure A2-2) the gain boundary was almost
at 90 degees to the path of the crack. The net tensile stress on this boundary is small
since the boundary is parallel to the stress axis.

There was only a small boundary length between gains L and J, compared to
their overall gain sizes and as such this boundary may have been insigiificant. This
boundary was also further away from the crack (see appendix 2-Figure A2-10).

For the other gain boundaries, it was observed that there were higher F04, values
near the regions of interest and the crack had continued to move towards those
boundaries with higher high Fa.» values instead. (See appendix 2- Figure A2-5).

The Luster-Morris compatibility factors provide information on the slip
compatibility between two adjacent gains but the numeral value thus obtained does not
provide a convincing way to describe the effectiveness of the deformation transfer across
the neighboring gain or prediction of microcracking at the gain boundaries. The newly
developed fracture propagation parameter, on the other hand, will be evaluated to

determine if it can predict intact or cracked boundaries. Knowledge about these intact or

95

cracked boundaries will lead to understanding how effective the deformation transfer had

been in these gains.

Intact Mean, Stdev: 0.73,0.30 1

Cracked Mean, Stdev: 1.23,0.15

 

 

 

 

 

 

 

 

 

 

.la5

13 -
133-

321 —.
3- -III-
0.llllllmnl .f—
1

 

0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Fracture Propatlon Parameter

1
1

1

1

1

1

1

I h_tact El cracgdg 1

L,

 

 

Figure 5.7. Histogam of intact and cracked boundaries using Fag-l, values.

96

Table 5.5: Values of the Fracture Propagation Parameters assigned to each set of
grains, depending on the direction of the crack path as it moves into these grain
boundaries. The selected F value is displayed on the far right hand side.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FA-e (x FA-a Intact-=0, Selected
Grain X Grain Y - [I (y - x) crack=1 F value
A BB 0.407 0.349 0 0.407
B C 0.755 0.517 0 0.755
BB BE 0.464 1.234 0 0.464
BE E 1.174 0.624 0 1.174
D E 0766 0.593 0 0.593
C E 0.528 0.576 0 0.576
C D 0.476 -1.11 0 0.476
D H -0.832 0.137 0 0.137
D F 0915 0.312 0 0.312
H J 1.26 1.468 0 1.26
G F 0.347 1.09 0 1.09
F FK 0.983 0.88 0 0.983
FK K 0.986 0.42 0 0.986
H HJ 1.066 -0.067 0 1.066
HJ J 1 .084 0.8532 0 1 .084
K LA 0.712 -0.398 0 0.712
J L 1.09 0.941 0 1.09
K L 0.44 0.709 0 0.709
L LA 0.699 -0.315 0 0.699
ZW Z 0.41 0.468 0 0.468
5 Z 0.72 0.53 0 0.72
5 58 0.44 0.6 0 0.6
zw 6 0.47 1.234 0 0.47
5 2 0.847 0.77 0 0.847
Z 6 0.32 0.765 0 0.32
BB B 0.4364 0.841 1 0.841
D I -0.82 1.309 1 1.309
H I 1.29 1.31 1 1.31
F I 1.42 1.43 1 1.43
I FK 1.28 1.02 1 1.28
HJ FK 1.399 1.12 1 1.399
FK J 0.93 1.08 1 1.08
K J 0.632 1.311 1 1.311
J LA 1.153 -0.35 1 1.153
5 5A 0.72 1.19 1 1.19
5 1 0.35 0.87 1 0.87
5A 1 1.13 1.011 1 1.13
1 2 1.15 0.668 1 1.15
4 1.38 0.895 1 1.38

 

97

 

Table 5.6: t-test analysis for the intact and cracked boundary (precracked).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t—Test: Two-Sample
Intact Cracked
Mean 0.73 1.23
Standard deviation 0.30 0.15
Variance 0.093 0.024
Observations 24 13
Pooled Variance 0.069
Hypothesized Mean Difference 0
df 35
t Stat -5.47
P(T<=Q one-tail 1 .90E—06
t Critical one-tail 2.72
P(T<=t) two-tail 3.81 E-06
t Critical two—tail 2.99

 

 

 

 

Table 5.7: Intact Grain Boundaries with high F ,4.” values

 

 

 

 

 

 

 

 

Grain X Grain Y F A.” v.1“, Comment
D H 1.278 Boundary with higher F value nearby
BE E 1.174 Grain boundary || stress direction
G F 1.09 Boundary with higher F value nearby
J L 1.09 Short grain boundary length, far from crack
HJ J 1.084 Boundary with higher F value nearby
H H] 1.066 Boundary with higher F value nearby

 

 

 

 

 

98

Chapter 6: Results of Microcrack Extension
6.1 Overview
In this section the Fracture Propagation Parameter was evaluated near the crack
tip to allow prediction of the path of the crack. This sample (A) was subsequently loaded
to induce further cracking and the predicted path and the actual path were compared.
This Fracture Propagation Parameter was also applied to another sample (B) to see if it

was able to account for the way the crack had propagated.

6.2 Sample A

The undamaged microstructure below the primary crack was analyzed and the
grains labeled numerically as shown in figure 6.1. Figure 6.2 shows a schematic
representation of the grain boundaries in this region with the F443 values indicated.
Black arrows were inserted to denote the likelihood of fracture for deformation increasing
in the direction of the arrows across the boundaries whereas gray arrows indicate
otherwise. Grain boundaries highlighted with thick black lines denote boundaries with
directional F values > 1, i.e. weak boundaries. A higher magnification of the boxed
region is shown in figure 6.3. From identification of weak boundaries the crack can be

predicted to follow the path highlighted in red as shown in figure 6.4.

99

 

Figure 6.1. The undamaged microstructure below the primary crack was analyzed for
grain orientations, and the grains were labeled numerically.

100

 

 

0.34 0.76
1 1 fl
0.71

0.74

 

 

 

 

1.14

 

Figure 6.2. Schematic diagram shows the F ,4.” values, the directional arrows (black or
gray), and the weak boundaries highlighted in thick black color.

101

   
  

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Figure 6.3. Higher magnification of the boxed region shown in figure 6.2. The fiacture
propagation parameters are computed for each grain boundary and the weak boundaries
are identified with thick black lines.

102

 

Figure 6.4. Based on the observed weak boundaries, crack path was predicted to move in
the direction as shown in red.

103

When the specimen was deformed further, the microcracks that formed along the
1-2 gain boundary as well as the 1-4 gain boundary linked up. Other microcracks were
generated farther down the 1-4 gain boundary, as shown in figure 6.5. With continued
deformation, the crack changed direction (see figure 6.6) to follow along the 1-4 gain
boundary until a jog in the boundary (see figure 6.7) caused the crack to extend by
cleavage through gain 4. A cluster of Ti3Al had diverted the propagating crack
downwards into the gain 4.

Figure 6.8 shows the primary crack moving towards a goup of weak boundaries
around gain 9, (figures 6.2 - 6.4). Grain 9 had a pre—existing annealing twin bisecting
the whole gain as shown in figure 6.9, as well as some pre-existing deformation twins.
New microcracks (arrowed) were observed ahead of the crack path. These cracks
occurred at the gain boundaries between 7-49a, 9-49a, and 49a-49b boundaries. As the
specimen continued to be stressed, microcracks continued to form as highlighted in black
arrows in figure 6.10 and higher magnification images of these gain boundaries where
the cracks formed are shown in figures 6.11 through 6.14.

Figure 6.1 shows the path of the crack as the specimen was stressed in the 4-point
bend fixture. The primary crack continued its path via cleavage in gain 4 towards gain
9, jogged just below but parallel to the annealing twin within gain 9 and continued to
cleave via gain 108 and gain 10b towards gain 13 (a gain similar to gain 9, having
weak boundaries and an annealing twin within). The primary crack was arrested in gain
31 as shown in the figure while a new crack initiated in the neighboring gain 32 and

continued down towards another cluster of weak boundaries at the junction of gains 21-

104

35-37-23a. The gain boundary between 31 and 32 has very low F443 values (figure

6.2).

grain 4

Tensile Axis

 

Figure 6.5. When the sample was further loaded, the microcracks along 1-2 boundary as
well as 1-4 boundary linked up. Microcracks were also observed farther down the 1-4

grain boundary.

105

 

Figure 6.6. The crack path changed direction from the 1-2 boundary towards the 1-4
boundary. '

 

Figure 6.7. The crack was arrested near some Ti3Al particles at the 1-4 grain boundary
while a new crack had resurface further along the 1-4 boundary and cleaved into gain 4.

106

 

Figure 6.9. Pre-existing twins in undeformed gain 9 include the annealing twin that
divides gain 9 into two smaller grains.

107

I -1: 5‘.

11.x
7” / 7

“‘9
-.' v,“
‘.- ~

 

Figure 6.10. Microcracks developed as the crack was further loaded. Arrows indicate
the locations of fine microcracks. Higher magnification images of these microcracks are
shown in figures 6.10 through 6.13.

108

 

Figure 6.11. Microcracks (arrowed) along the 49a-49b gain boundary. On the left of
gain 49b is a platelet of Ti3Al.

grain 9

 

Figure 6.12. Microcracks developed along gain boundary between gain 9 and gain
10a The microcrack developed in a Ti3Al gain boundary precipitate.

109

grain 33c

' 13111:"; ,

grain 10a

 

Figure 6.13. Microcrack occurred between the grain boundary bordering gain 10a and
gain 33c when the specimen was further loaded. The microcrack developed in a Ti3Al
gain boundary precipitate.

 

Figure 6.14. Microcracks formed along the gain boundary Ti3Al between gain 10b and
33a, gain 13 and grain 33a, gain 13 and gain 33 as well as between gain 32 and gain
33 when the specimen was further stressed via 4-point bending.

110

Even though microcracks nucleated in some of the weak boundaries ahead of the
crack tip, the crack did not divert directly towards these cracks, similar to the microcracks
in gains 1-2, and 1-4. Instead, the crack continued towards the next cluster of weaker
boundaries. Microcracks were observed in weak boundaries between gains 49a-49b,
10a-13, 13-33a, lOa-31, 31-35, and 35-37, and many of these boundaries had Ti3Al
precipitates.

Although the crack did not directly follow weak boundaries, perhaps due to the
influence of gains beneath the surface, the surface microcracks had a disproportionately
strong effect on the crack path.

Figure 6.16 shows a histogam of both intact and cracked boundaries for the lower

portion of the crack path. It is again clear that microcracking is less likely when F 4.,3 <

1.0 but is more likely when F A—)B > 1.0.

111

 

Figure 6.15. Path of the propagated crack with predicted path (white arrows)
superimposed over it. The black arrow indicates the gain boundary cracking in the 1-4
boundary and the gey arrow shows where the crack deviated by cleavage of gain 4
towards the weak boundaries in gain 9. Insert shows an image of the region before the
crack propagated through gain 4.

112

 

Intact Mean. Stdev: 0.64, 0.45
Cracked Mean, Stdev: 1.08, 0.35

U! 0:

&

Number of observatlona
N (.0

.5

 

O

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Fracture Propagatlon Parameter

 

 

I Intad CI cracked |

 

Figure 6.16. Histogam of intact and cracked boundaries of the lower portion of the
crack path.
6.3 Sample B

Sample B, from the same casting and with similar geometry, but a different
orientation, (see figure 3.1) was loaded in 4-point bending but the crack propagated
through the whole sample. As with sample A, sample B also had mixed mode cracking].
Figure 6.17 shows an SE image and its corresponding BSE (at higher magnification)
image of sample B. Because the crack had propagated through the specimen, it resulted
in a large crack opening displacement. Obtaining meaningful information near these
widely opened crack edges were more difficult since backscattered electrons escape via
these crack edges as the beam is rastered near to these edges [61]. A region nearer to the
crack tip region (where the opening of the crack was smaller) was thus examined and an

OIM scan was performed. This region (boxed) is shown in figure 6.18a.

 

1 The extensometer and the load cell used with the E. Fullam deformation stage was not working properly
and thus did not register the extension or the load when the crack propagated through the notch.

113

The OIM scan of the boxed region is shown in figure 6.17a. Different colors are
present within each gain due to the inability of EBSP to resolve the small c/a ratio. The
corresponding BSE image of the crack path is shown in figure 6.17b. One observation on
this portion of the sample was the presence of a large amount of Ti3Al along the crack
path. Since the Fracture Propagation Parameter was designed basically for TiAl alloys, it
is crucial to locate regions within the crack path that showed sufficient TiAl gains so that
this parameter may be applied without undue influence from Ti3Al. The crack path in
this sample could then be analyzed to see if the concept of this fi‘acture propagation
parameter (i.e., intact boundaries having low F .11-+3 values (<1) and cracked boundaries
having high F A _,3 values (>l)) could work with a differently oriented sample using the
same specimen geometry.

A portion of the cracked region (away from the high density Ti3Al region was
thus located and analyzed (see figure 6.17b). The crack cleaved through gain D5 before
terminating in gain D4, but a new crack had initiated at the gain boundaries between
D4-D1 and D1-D3 regions and moved downwards through the D4-D3 gain boundary.

Using the Fracture Propagation Parameter, this microstructural patch was labeled
with the directional arrows corresponding to the FA.” values as shown in figure 6.18.
The weak boundaries were also identified and highlighted in black bold lines. Similar to
the previous sample (A), it was observed that the crack in this region of the sample ran
through regions of weak boundaries or jumped from one region/cluster of weak
boundaries to another weak cluster of boundaries. Given that the orientation of the
specimen with respect to the texture is different, the fi‘acture propagation parameter

appears to be a robust predictor of crack propagation.

114

llHHI 1|II

 

Figure 6.17. Back scattered electron image of the crack path of sample b. The SE image
is on the right top comer. The crack propagated completely through the sample. The
boxed region was analyzed and is shown in figure 6.18.

115

;.\

I

I}. . Ia
{101111111

 

Figure 6.18. a) OIM image of the crack path. b) Corresponding BSE image. A large
amount of Ti3Al was observed along the crack path. A small cluster of gains were
labeled and analyzed using the fracture propagation parameter.

116

 

Figure 6.19. Schematic diagam of the area of interest shows the F443 values, the
directional arrows (black or gay), and the weak boundaries highlighted in thick black
color.

117

Chapter 7: Discussion
7 .1 Overview
A fracture propagation parameter was used to explain the observed crack path for
two samples with different orientations with respect to the casting (Samples A and B).
Boundaries with fiacture propagation parameter > 1 were identified as weak boundaries
and the primary crack had a tendency to propagate towards clusters of weak boundaries.
Where it is not possible to follow a boundary directly, the crack cleaved through a gain,

via the shortest distance towards the next cluster of weak boundaries.

7 .2 Sample A

Further deformation of sample A showed that the crack deviated fiom the 1-2
gain boundary towards the 1-4 gain boundary direction where microcracks had also
formed with the additional loading. The crack did not follow the 2-4 boundary that was
aligred with the gowing crack because the fracture propagation parameter between
gains 2 and 4 was F 2_4 = 0.99 and whereas was F1_4 = 1.39. Though the F 2_4 value is
~l , the weaker boundary between gains 2 and 4 provided a lower energy path for the
crack despite the fact that this boundary was nearly parallel to the maximum tensile
direction. In other words, the boundary between gains 1 and 4 is weaker than the
boundary between 2 and 4 and hence was easier to fracture. The crack sought weak
boundaries in preference to boundaries with high tensile stress acting on them, and thus,
followed a path that connected weak boundaries (high FA_B values).

The advancing crack shown in figure 6.5 was diverted towards the 1-4 gain

boundary. This crack was subsequently arrested as it approached some clusters of 012

118

 

(Ti3Al) alloy particles as shown in figure 6.6. The presence of these 012 particles has
previously been found to have some effect on the TiAll [8,62]. It was also observed that
sometimes cracks formed in the interface between 7 and 012 for reasons that are probably
similar to the reasons that are being sought for the boundaries in the 7 phase, but this is
beyond the scope of this project.

With continued loading, a new crack surfaced slightly further along the 1-4
gain boundary (see figure 6.7) but since it is not possible for cracks to move parallel to
the tensile loading direction indefinitely, the crack extended by cleavage through gain 4,
perhaps triggered by the microcrack adjacent to Ti3Al. The crack by-passed the
microcracks that had formed along the 1-4 boundary, (see figure 6.5) in the process. The
cleavage crack gew downward towards the next nearest cluster of weak boundaries
surrounding gain 9. The crack took the “shorter route” towards the next cluster of weak
boundaries surrounding gain 9.

The annealing twin interfaces within gain 9 (see figure 6.8) have high FA_B
values, and although the boundary was not highly stressed in Mode I, the boundary
fractured parallel to the stress axis along the lower twin boundary (see figure 6.15), and
then fractured by cleavage toward the gain boundary between gains 9 and 108 that has a
high F 9_10 value =l.47. The crack path is clearly heading towards the next cluster of
weak boundaries, which surround gain 13. The boundaries along gain 13 are apparently

very weak. showing FA_B values as high as 1.92.

 

1 It has been argued that the 01; phase scavenges interstitial impurities from the 7 phase, thereby lowering
its oxygen content, causing an increase in ductility in the 7 phase

119

The crack subsequently moved transganularly into gain 31 and was arrested.
Grain 31 had strong boundaries with all of its neighboring gains. But a new crack
initiated in the neighboring gain 32 and continued down the path towards another cluster
of weak boundaries at the junction of gain 21-35-37-23a. The cause for this
discontinuity in the path may be due to the fact that the boundary between gains 31 and
32 was strong. It is interesting to note that even though a large microcrack had gown in
the 35-37 grain boundary (see figure 6.9), the crack did not divert towards that weak
boundary but continued its path downwards, cleaving transganularly into gain 35. It
subsequently stalled at the gain boundary between gain 23 and gain 238; gain 238 was
heavily twinned similar to gain 1, where the crack was stalled earlier.

The observation of microcracks between gains 49a-9, 498-7, 49a-49b, 9-10a,
lOa-33c, 10b-33a, 13-33, 32-33 (see figure 6.9—6.13) ahead of the advancing primary
cracks corresponded very well with high F A}; values. The only exception was the
observation of a single tear along the boundary between gains 32 and 33, which had a
low PM; value of 0.43 (Figure 6.14). Even though microcracks nucleated in some of the
weak boundaries ahead of the crack tip, the crack did not divert directly towards these
microcracks (similar to the microcracks in the 1-2 and 1-4 gain boundaries) but rather, it
continued towards the next cluster of weaker boundaries.

The F A_B values for both the intact and cracked boundaries beyond the original
crack were further analyzed using the student t-test and the results are shown in Table
7.1. Again, assuming a null hypothesis that the two sample means are equal, there is a
99% chance that t critical (for two-tail) would fall within +/- 2.98. From the analysis, t-

statistic was computed to be +/-3.347, a value that is larger than the t-critical value. This

120

suggests that the difference of two sample population means is geat and is statistically
sigrificant. The mean value of F ,4 n3 for the cracked population, "ABC, exceeded the
intact population mean, 1.113,, with 99% confidence that uABc > ”A81: with individual

confidence intervals for Mac and 1111131 0f (0.73, 1.44) and (0.18, 1.09), respectively.

Table 7.1 t-test ana see for the intact and cracked boundaries st cracked .

 

 

 

 

 

 

 

 

 

 

 

t-Test: Two-Sample : intact and cracked boundaries
Intact Cracked
Mean 0.64 1.08
Standard deviation 0.45 0.35
Variance 0.20 0.12
Observations 12 27
Hypothesized Mean Difference 0
(if 37
t Stat -3.35
t Critical one-tail 2.71
t Critical two-tail 2.98

 

 

 

 

 

Based on the analyzed data, (gain orientation, dot product of the interaction
between the highly stressed twinning vector with the ordinary dislocations and twins
from the adjacent gain) and the observed microcracks, high F A_B values between two
adjacent gains, would most likely result in the creation of microcracks along the gain

boundaries. The crack, however, did not exactly follow boundaries that had microcracks.

121

Instead, it tended to cleave within a gain in order to connect regions having clusters of
weak boundaries. Hence, the path of a crack in a y-TiAl alloy could be predicted by
noting the locations of weak boundaries.

There were four cracked boundaries with low values. In figure 6.13 Ti3Al had
precipitated along the boundary. The microcrack formed between the Ti3Al/TiAl
interface in gain 33c. This microcrack may have developed due to the strain
incompatibility between these two phases since there are fewer facile slip systems in the
Ti3Al compared to TiAl.[63])]. This microcrack is close to another microcrack formed
nearby (in the 32-33 gain boundary) (see figure 6.14). The low F 4.3 value indicates a
strong boundary (32-33), suggesting that the microcracks observed in these two sets of
boundaries might have been connected below the surface.

Two cracked boundaries that were almost parallel to the tensile loading direction
with low F,” values (31-35, 32-35, Figure 6.2) opened up as the crack propagated
perpendicular to these boundaries. This also suggests that the crack may have been

connected below the surface,

7.3 Sample B

The concept that a primary crack, in a y—TiAl alloy will propagates via regions of
high FA_B values or weak boundaries has been demonstrated using sample A. To
determine if this hypothesis will work with any gamma-TiAl alloy with similar specimen
geometry and loading conditions, sample B was tested, which had texture differences

perpendicular to the loading axis.

122

In this sample, the primary crack ran almost entirely through the width of the
specimen. This might have been due to poor bonding of the epoxy resin, but there was
no evidence of delarnination between the TiAl and the aluminum backing. Thus,
assuming that the aluminum or bond did not account for the geater initial stored energy
prior to cracking, the elastic analysis based upon the texture is considered next.

As shown in figure 4.1, the x-ray analysis revealed that sample A and sample B
were oriented differently in terms of their orientation with respect to the notch root. Both
specimens have the same texture/properties along the primary tensile axis (TA). Figure
7.1 shows how elastic modulus and highly stressed slip and twinning systems are related
to crystal orientation. The effect of dominant crystal orientation on properties in the
notch direction is summarized in Table 7.2. This indicates that the geater amount of
strain energy release in Sample B (longer crack propagation) can be understood in terms

of elastic and plastic deformation processes that depend on texture.

130 (5138 220 0 I__ 0.5
Young's Schmid factor
Modulus for CD,

 
    

 
   

For TiAl twins

Figure 7 .1. Relationship between texture and deformation processes in the direction of
the notch root are examined in Table 7.2 by considering maxima for Young’s modulus
and Schmid factor maxima for twins and ordinary dislocations.

123

Table 7.2: Effect of dominant crystal orientation on deformation along notch root.

 

     

SanplaA

 

 

More <111> stiff orientations and <100>

compliant orientation parallel to notch root.

Less <111> and <100> orientations parallel
to notch root.

 

Heterogeneous elastic strains resulted from
the different sets of orientations.

More homogenous elastic strains since
there are no distinct set of orientations.

 

Have fewer orientations that have high
Schmid factors for twins and ordinary
dislocations.

More orientations with high Schmid factors
for twins and ordinary dislocations, i.e. a
high Schmid factor indicates that the
deformation system is able to operate to
carry the strain.

 

Due to differences in orientations (stiff and
compliant), there is modest plastic
deformation in the notch root with minimal
shape change.

Due to the more homogeneous orientations
and higher Schmid factors for twins and
ordinary dislocations, there are substantial
plastic deformations in the notch root
resulting in shape change.

 

With modest plastic deformation, local
stresses are built up at the grain boundaries
and fi'acture stress is reach with elastic
strain.

With substantial plastic deformation, the
material is also worked hardened, thereby
increasing its stored strain energy.

 

 

When fracture stress is reached, there is
less stored strain energy released with the
fracture.

 

When fracture stress is reached, there is a
huge amount of stored strain energy
released with the fracture.

 

An OIM scan was performed on a region closer to the crack tip since the width

(gap) of the crack at that region was smaller and ECCI/EBSP analysis was possible. As

shown in figure 6.17b, the presence of Ti3Al particles complicated the analysis of gain

orientation and plane traces. Thus, only regions with fewer TiaAl particles were selected

and analyzed.

The fracture propagation parameter can explain the behavior of this portion of the

crack path easily. Using the same concept of weak boundaries for high FA_3 values (>1),

it was observed that sample B followed similar fracture trends as sample A, i.e. the crack

propagated via weak boundaries or connected via weak boundaries.

124

 

The primary crack (shown in figure 6.16) cleaved through gain D5 and was
arrested in gain D4. But a new crack had surfaced along the boundary region between
gain D1 and D4. This region had a high FA_B value of 1.23. The crack then moved along
the gain boundary and connected with boundaries with the next highest FA_B values at
the boundary region between D7 and D3 (F A_B values = 1.15). From there, the crack
moved on to gain D7 and D8 which had a F A}; values of 1.39. Within this small region,
it was observed that the crack continued to move in a way consistent with the observed
crack path in sample A, that is, the crack sought out and linked the weakest boundaries as
it propagated.

The student t-test was performed on this small population to see if the difference
between sample means for both the intact and crack population were statistically
sigrificant. Again assuming a null hypothesis that the two population means were the
same, the analysis (Table 7.3) showed a 95% chance that the t critical (for two-tail)
would fall within +/- 2.84. From the analysis, t-statistic was computed to be +/-2.85, a
value that is slightly larger than the t—critical value. Hence the difference of two sample
population means is statistically sigrificant, despite the smaller population of
misorientations. The mean value of F “.3 for the cracked population, uABc, exceeded the
intact population mean, 111181, with 95% confidence that uABc > 11,131, with individual
confidence intervals for um and 111181 of (1.05, 1.34) and (0.60, 1.07), respectively.

However this analysis is based upon a much smaller population of datum points.

125

 

Table 7.3: t-test analyses for intact and cracked population means (Sample B)

 

 

 

 

 

 

 

 

 

 

 

t-Test: Two-Sample :intact and cracked boundaries
Intact Cracked
Mean 0.83 1.20
Standard deviation 0.23 0.15
Variance 0.05 0.02
Observations 5 4
Hypothesized Mean Differencer 0
df 7
t Stat -2.85
t Critical one-tail 2.36
t Critical two-tail 2.84

 

 

 

 

 

7.4 Evaluating the Fracture Propagation Parameter for Other Samples
Development of the fracture propagation parameter was based on examination of
the surface of the crack induced as result of a 4-point bend loading as shown in figure
7.23. To see if this fracture propagation parameter can be applied to a similar material
but with different loading geometry (see figure 7.2b), gain information from Simkin’s
thesis/experiment was used to generate the F A _,.3 values. Simkin’s studies involved
bending a smooth bar of the same y-TiAl alloy (see figure 7.2b) and examining the
tensioned surface for microcracks along the gain boundaries. With Simkin’s 4-point
loading condition, it can be assumed that all the gains on the tensile surface were
stretched similarly at the same time. Consequently, the selection of the FA.» values
between two adjacent gains were based on the larger of the two values (because there

was no reason to choose one gain as a reference gain without an existing crack), similar

126

to Simkin’s selection criteria. The F 4413 values and Simkin’s F values, together with the
condition of the boundaries (either cracked or intact) were tabulated as shown in Table
7.4. Both parameters showed similarly high F495 values for cracked boundaries but
intact boundaries showed otherwise a variety of values. While the fracture initiation
parameter used by Simkin produced generally low F values, ranging from 0.59 to 1.11,
the analysis using the fracture propagation parameter showed a wider range of EH19

values, ranging from 0.57 to 1.32.

 

Examine
Examine Load Surface region
crack tip region //\ /\/
,'/ V .1 / ’ V ' x I . 7
/ .1 ' / / )1 y/
1' // / " ,r"’\ /,/ , /;./
é 1~*f’r—~~fl A...
(J A. L)"

a b Load

Figure 7.2. Examination of specimen surfaces. 8) Ng’s specimen b)Sirnkin’s specimen.

The t-tests for both methods are shown in Tables 7.5 and Table 7.6. While there
is a sigiificant difference between the values of the cracked and intact boundaries using
Simkin’s Fracture Initiation Parameter (see Table 7.5, where the t Stat value of -3.24 is
outside the t Critical value of +/-2.44), the t-test for the values of the cracked and intact
values using the Fracture Propagation Parameter in Table 7.6 showed otherwise. The
sample mean and its standard deviation for the cracked boundaries were 1.08 and 0.30
respectively. These values are comparable with sample A. So the fracture propagation

parameter is fairly accurate in describing conditions where microcracks form. But the

127

sample mean and its standard deviation for the intact boundaries were 0.95 and 0.26

respectively. This value is slightly higher than the data set from sample A.

Table 7.4: Simkin’s dataset using both the Fracture Propagation Parameter
and Fracture Initiation Parameter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Propagation Initiation
Grain Grain Parameter Parameter Fracture?

1 3 1.12 0-685 Yes
7 8 1.18 1317 Yes
10 12 1.31 0363 Yes
14 15 1.61 1414 Yes
l8 19 0.92 1-06 Yes
28 29 1.04 1-155 Yes
32 33 0.8 1-124 Yes
34 35 1.1 1389 Yes
37 38 1.38 1467 Yes
45 47 0.62 0907 Yes
48 49 0.85 “03 Yes
1 5 0.57 0.376 No
6 8 1.32 1043 No
6 9 0.85 1.04 No
13 16 1.18 0749 No
14 17 0.81 1-119 No
19 20 1.25 0732 No
31 33 1.13 1-044 No
44 46 0.61 1039 No
n1 n2 0.6 0.715 No
n1 n3 0.77 0.593 No
n4 n5 0.96 0327 No

 

 

 

 

 

 

 

128

Assuming a null hypothesis and a 95 % confidence, t critical (for two-tail) would
fall within +/- 2.44 whereas the t-statistic was computed to be +/-1.06. Because the t-
statistic falls within the t critical value, the null hypothesis is true, that is, there is no
statistically sigiificant difference in the two sets of population means. Hence this
fracture propagation parameter does not work well with the specimen geometry shown in

figure 7 .2b. The reasons for this difference will be examined in the next section.

Table 7.5: t-test Analysis for Simkin’s specimen
using Fracture Initiation Parameter.

 

 

 

 

 

 

 

 

 

 

 

 

t-Test: Two-Sample: Intact and Cracked Boundaries
Intact Cracked
Mean 0.89 1.18
Standard deviation 0.1 8 0.21
Variance 0.03 0.04
Observations 10 10
Pooled Variance 0.04
Hypothesized Mean Difference 0
df 18
t Stat -3.24
t Critical one-tail 2.10
t Critical two-tail 2.44

 

 

 

 

 

129

Table 7.6: t-Test Analysis for Simkin’s specimen using
Fracture Propagation Parameter.

 

 

 

 

 

 

 

 

 

 

 

 

t-Test: Two-Sample : Intact and Cracked Boundaries
Intact Cracked
Mean 0.95 1.08
Standard deviation 0.260 0.30
Variance 0.06 0.09
Observations 10 10
Pooled Variance 0.08
Hypothesized Mean Difference 0
df 18
t Stat -1.06
t Critical one-tail 2.10
t Critical two-tail 2.44

 

 

 

 

 

7.5 Analysis of intact boundaries that have high FA.” values

One of the gain boundaries (gain l3-gain 16) that did not result in cracking
although registering a high F A93 value (1.18) is shown in figure 7.3. The dominant twin
traces for gains 13 and 16, their Schmid factors and the F A.” value are provided. As
shown at the top of the figure, the (-111) twin in gain 13 with the highest Schmid factor
of 0.49 and FA.» =1.18 might have resulted in the gain boundary cracking with gain 16,

but these twins were not observed near the boundary. Instead, the twinning system with

130

E4.» = 0.78 was most active at the boundary between gains 13 and 16, so the boundary

remained intact.

 

Figure 7.3. Interaction between gain 13 and gain 16. The dominant plane traces for
both gains, their schmid factor and their F .4418 values were inserted. [16].

Another gain boundary with a high FA.” value that remained intact is shown in
figure 7.4. The interaction between gains 8 and 6 was calculated to have a high EH);
value of 1.32 in the direction from gain 8 towards gain 6. Note that the boundary
segnent between gains 8 and gain 6 is relatively small. The stress and strain from
either gain 6 or gain 8 were probably absorbed/accommodated by gains 7 or 9 that
have a much larger contact with gains 6 and 8. The fact that a small microcrack

occurred in the 7-8 gain boundary may have relieved the stress in the gain 6-8

131

boundary. Interestingly, the propagation parameter identified the 6-8 boundary as weaker

than the 7-8 boundary, whereas the initiation parameter predicted the opposite.

 

Figure 7.4. The gain boundary between gain 6 and gain 8 shows intact boundary [16].

If three of the intact boundaries (with high F4” values) were eliminated from the
analysis, then there is a 90 % confidence that it will continue to follow the prediction of
the Fracture propagation parameter. These two observations of exceptional intact gain
boundaries indicates that the stress and strain shielding by neighboring gains is
significant, making prediction strategies more complex.

The fracture propagation parameter takes into account the directionality of the
twinning vector of the individual gain and their placement with respect to the direction

of the propagating crack, as well as allowing twinning and ordinary dislocations to

132

accommodate strain across a boundary. That provides a very strong sense of the
direction of the shearing strain. With Simkin’s test specimen, twinning as an
accommodation mechanism was not considered, suggesting that ordinary dislocation
accommodation is more significant than twinning for nucleating microcracks. With only
knowledge of the global tensile stress state, rather than the local stress state, this fracture
propagation parameter was not able to provide as robust of a prediction of the intact
boundaries, although the cracked boundaries were fairly consistent with the values found

in the other two samples.

7.6 Effect of Ti3Al on the crack propagation.

The volume fraction of Ti3Al particles in this y-TiAl alloy was about 8%. Due to
the small quantity and prohibitively small sizes for convenient crystallogaphic analysis,
the effects of this phase has been omitted in the fracture propagation parameter analysis.
But the observed interaction of the crack with the Ti3Al particle (see figure 7.1) which
resulted in stalling the primary crack (toughening mechanism) and the formation of
140microcracks (non-toughening mechanism) on the edge some of these Ti3Al particles
(figure 6.7, 6.9 — 6.13), suggests that these inclusions might play a dual role in the
fracture analysis. Further work is needed to examine how slip transfer does / does not
occur from y to the a; phase.

Samples A and sample B have slightly different geometry in terms of their
orientation with respect to the notch root and the fracture propagation factor was robust
enough to effectively predict the formation of microcracks and thereby provide the data

needed to predict the path of the crack. Microcracks may nucleate at the gain boundary

133

due to the high F 4.,3 value between the two adjacent gains but the primary crack may
not propagate through these microcracks. The primary crack, instead, will propagation
via weak boundaries through the shortest route possible, cleaving into a crystal in order to

move fiom one cluster of weak boundaries to another cluster of weak boundaries.

134

Chapter 8: Conclusions

This dissertation has focused primarily on active cracks as they propagated
through the y—y gain boundaries. Cracks propagating through bulk y-TiAl alloys were
studied using a combination of electron backscattered patterns, selected area channeling
patterns, electron channeling contrast imaging and x-ray techniques.

The Luster and Morris geometric compatibility factor was used to provide some
insights into the slip transfer fi'om one gain to another but the numeric value does not
provide a convincing way to describe the effectiveness of the deformation transfer across
the neighboring gain or to predict the opening of microcracks at the gain boundaries.

Microcracks nucleated at the gain boundaries as a result of the inability of the
neighboring gain B to effectively accommodate the shear generated by the dominant
deformation twins in gain A.

The microcracks observed at the y—y boundaries correlate well with a fracture

propagation parameter F .4493

Bad-l tw tw-l A”

A A 2 A A 4 A A
F... = m...lb...-tl[ 2: lb. - b...| :2 b ~b..| ]

The fracture propagation parameter takes into account the directionality of the
twinning vector of an individual gain and its orientation with respect to the direction of
the propagating crack, as well as allowing twinning and ordinary dislocations to
accommodate strain across a boundary.

With this fracture propagation parameter, it was observed that microcracking is

less likely when FA_,3 < 1.0 but is more likely when FA_,.B > 1.0. This fracture

135

propagation parameter was able to predict intact or cracked boundaries following further
crack gowth based on the F ,4 +3 values.

Although cracks nucleated in some of the weak boundaries, the primary crack did
not necessary divert towards these microcracks. The primary crack had a tendency to
propagate towards clusters of weak boundaries. Where it was not advantageous to follow
a boundary directly, the crack cleaved through a gain, via the shortest distance towards
the next cluster of weak boundaries.

This fracture propagation parameter was robust enough to consistently explain the
observed crack path for two samples with different orientations with respect to the
casting. Differences in texture in these two specimens had a large impact on the crack
arrest capability.

Further work can be done to continue to improve on this parameter. Ti3Al
particles account for more than 8% of the weight percent and it was observed that these
particles may either cause microcracks to form at the az—y interface, which
counterbalances the increase in ductility in the TiAl phase due to gettering effects. These
particles and their effects were left out of the formulation. The fi'acture propagation
parameter computed the relationship between the dominant twinning vector in one gain
and the other slip vectors in a neighboring gain.

The plane normal criterion present in the Luster-Morris parameter was not
considered in the computation of the fracture propagation parameter. Since directionality
is important in this parameter, the change in orientation of the accommodating plane with

respect to the dominant twinning plane might have some effect on this fracture

136

propagation parameter and the observed microcrack formation, and this warrants further

investigation.

137

10

11

12

REFERENCES

M. Yamaguchi, “High Temperature Intermetallics- with Particular Emphasis On
TiAl”, Materials Science and Technology, April 1992, vol.8, 299-307 (1992).

D.M. Dimiduk, “Gamma Titanium Aluminides-An Emerging Materials Technology”,
Gamma Titanium Auminides, Edited by Y-W. Kim, R. Wager, and M. Yamaguchi,
The Minerals, Metals & Materials Society, pp. 3-20 (1995).

D.M. Dimiduk, “Gamma Titanium Aluminide Alloys — An Assessment Within The
Competition Of Aerospace Structural Materials”, Materials Science and Engineering
A263 281-288 (1999).

Young-Won Kim, “Intermetallic Alloys Based On Gamma Titanium Aluminde”,
JOM, Vol. 41, No. 7, July 1989, pp. 24-30 (1989).

D. Shechtman, M.J. Blackburn and HA. Lipsitt, “The Plastic Deformation of TiAl”,
Metallurgical Transactions, Volume 5, June , pp.1373-1381 (1974).

S.M.L. Sastry and HA. Lipsitt, “Fatigue Deformation of TiAl Base Alloys”,
Metallurgical Transactions A, Vol. 8A, February 1977, pp 299-308 (1977).

M. Yamaguchi and H. Inui, “TiAl Compounds For Structural Applications”,
Structural Intermetallics, Ed by R. Darolia, J .J . Lewandowski, C.T. Liu, P.L. Martin,
D.B. Miracle and M.V. Nathal, The Minerals, Metals & Materials Society, pp. 127 —
142 (1993).

S. Huang, “Alloying Considerations in Gamma-Based Alloys”, Structural
Intermetallics, Ed by R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B.
Miracle and M.V. Nathal, The Minerals, Metals & Materials Society, pp 299-307
(1993).

J.T. Czernuszka, NJ. Long and RB. Hirsch, “Analysis of Defects in Bulk
Semiconductors Using Electron Channeling Contrast Imaging”, Inst. Phys. Conf.,
Ser. No 117: Section 10 (1991).

DC. Joy, “Direct defect imaging in the high resolution SEM,” MRS Proceeding, Vol.
183, Materials Research Society, pp. 199-209 (1990).

A.J. Wilkinson and PB. Hirsch, “The effects of surface stress relaxation on electron
channeling contrast imaging,” Philosophical Magazine A, 72, No. 1, pp. 81-103
(1995).

AJ. Wilkinson, G.R. Anstis, J.T. Czemuszka, N.J. Long and PB. Hirsch, “Electron

channeling contrast imaging (ECCI) of interfacial defects in strained silicon-
germanium layers on silicon,” Philosophical Magazine A, 68 pp. 59-80 (1993).

138

13

14

15

16

17

18

19

20

21

22

23

24

BC. Ng, B.A. Simkin and MA. Crimp, “Examination of dislocations near crack tips
in stoichiometric NiAl,” Deformation and Fracture of Ordered Intermetallic Materials
111, edited by W.O. Soboyejo, T.S. Srivatsan and H.L. Fraser, The Minerals, Metals
and Materials Society, pp. 337-348 (1996).

MN. Alam, M. Blackman and D.W. Pashley,”High-angle Kikuchi patterns,”
Proceeding of the Royal Society of London, Series A, Vol. 221, No, 1145, Jan. 21
1954, pp. 224-242 (1954).

S. Kikuchi, Japan Journal of Physic. 5, pp. 83-96 (1928).

BA. Sirnkin, “Tensile crack initiation in y—TiAl,” Ph.D. Thesis, Michigan State
University (2003).

Young-Won Kim, "Ordered Intermetallic Alloys, Part III: Gamma Titanium
Aluminides", JOM, July 1994, pp.30-39 (1994).

H. Inui, A. Nakamura, M.H. Oh and M. Yamaguchi, "Deformation Structures in Ti-
Rich TiAl Polysynthetically Twinned Crystals", Philosophical Magazine A, Vol. 66,
No.4, pp. 557-573 (1992).

H. Inui, M.H. Oh, A. Nakamura and M. Yamaguchi, "Ordered Domains in TiAl
Coexisting With Ti3Al In The Lamellar Structure of Ti-rich TiAl Compounds,"
Philosophical Magazine A, Vol. 66, No. 4, pp. 539-555 (1992).

R.LeHohn, H. Clemens and H.Kestler, “ Powder Metallurgy (PM) Gamma-Based
Titanium Aluminide Structures For Use in Various High Temperature Aerospace
Applications”, Gamma Titanium Aluminides 1999, Edited by Y-W. Kim, D.M.
Dimiduk, and M.H. Loretto, The Minerals, Metals & Materials Society, pp. 49-58
(1999)

M. Yamaguchi, and Y. Umakoshi, "The Deformation Behavior of Intermetallic
Superlattice Compounds", Progess in Materials Science Vol. 34, pp. 1-148 (1990).

Fu-Sheng Sun, C.X. Cao, M.G. Yan, Y.T. Lee and SE. Kim, “Solid solution
strengthening mechanism of TiAl alloy,” Gamma Titanium Auminides 1999, Edited
by Y-W. Kim, D.M. Dimiduk, and M.H. Loretto, TMS, pp. 415-422 (1999).

SC. Huang and EL. Hall, "The Effects Of Cr. Additions To Binary TiAl-Base
Alloys," Metallurgical Transactions A, Vol. 22A, pp. 2619-2627 (1991).

SC. Huang and EL. Hall, "Characterization Of The Effect Of Vanadium Additions

To TiAl Base Alloys," Acta Metallurgical Materialia, Vol. 39 No. 6 pp. 1053-1060
(1991)

139

25

26

27

28

29

3O

31

32

33

34

T. Tsujimoto and K. Hashimoto, "Structures And Properties of TiAl-based Alloys
Containing Mn," Materials Research Society Symposium. Proceeding, Vol. 133,
pp39l-396 (1989).

Yoo-Dong Hahn and Sung H. Whang, " Deformation And Its Structure In L10 Ti-Al-
Nb Compound Alloys," Materials Research Society Symposium. Proceeding, Vol.
133, pp. 385-390 (1989).

D.l-Iu, A.B. Godfrey and M.H. Loretto, "Characterization Of Thermally Exposed
TiAlCrNb Alloys," Gamma Titanium Aluminides 1999, Edited by Y-W. Kim, D.M.
Dimiduk, and M.H. Loretto, The Minerals, Metals & Materials Society, pp. 409-414
(1999) *

T. Sikora, M. Jaouen and G. Hug, “Bonding mechanisms in ternary g-TiAl + X
(X=Cr, Mn, Nb): a comparison between absorption near edge fine structures and
theoretical predictions,” Gamma Titanium Auminides 1999, Edited by Y-W. Kim, D.
M. Dimiduk, and M. H. Loretto, The Minerals, Metals & Materials Society, 1999, pp.
83-88 (1999).

T. Kawabata, T. Tamura and O. Izumi, “Parameters for ductility improvement in
TiAl,” High Temperature Ordered Intermetallic Alloys III, edited by CT Liu, A.I.
Taub, N.S. Stoloff and CC. Koch, Materials Research Society Symposium
Proceedings, Vol. 133, pp. 329-334 (1989).

C. Woodward, S.A. Kajihara, S.I. Rao, and D.M. Dimiduk, " The Influence Of Solid
Solutions On Flow Behavior In Gamma-TiAl," Gamma Titanium Aluminides 1999,
Edited by Y-W. Kim, D.M. Dimiduk, and M.H. Loretto, The Minerals, Metals &
Materials Society, pp. 49-58 (1999).

H. Mecking, CH.Hartig, and U. F. Kocks, “Deformation Modes in y-TiAl As Derived
From The Single Crystal Yield Surface”, Acta Materialia, Vol. 44, No. 4, pp. 1300-
1321 (1996).

MA. Morris, “Dislocation Configurations In Two-Phase Ti-Al Alloys.III.
Mechanisms Producing Anomalous Flow Stress Dependence On Temperature”,
Philosophical Magazine A, Vol. 69, No. 1, pp. 129-150 (1994).

S. Zghal, A. Coujou and A. Couret, "Transmission Of The Deformation Through y—y
Interface In A Polysynthetically Twinned TiAl Alloy. I. Ordered Domain Interfaces
1200 rotational)", Philosophical Magazine A, Vol. 81, No. 2, pp. 345-364 (2001).

S. Zghal and A. Couret, " Transmission Of The Deformation Through y-y Interface In

A Polysynthetically Twinned TiAl Alloy. II. Twin Interface (180° rotational)",
Philosophical Magazine A, Vol. 81, No. 2, pp. 365-382 (2001).

140

35

36

37

38

39

40

41

42

43

45

Gibson MA, Forwood CT, “Slip transfer of deformation twins in duplex gamma -
based Ti-Al alloys. III. Transfer across general large-angle gamma - gamma gain
boundaries,” Philosophical Magazine A. Vol. 82, 1381- 1404 (2002).

J. Luster, and M.A. Morris, "Compatibility of Deformation in Two-Phase TiAl
Alloys: Dependence on Microstructure and Orientation Relationships", Metallurgical
and Materials Transactions A, Vol. 26A, July 1995, pp.1745-1756 (1995).

BA Sirnkin, TR Bieler and M.A. Crimp. “Tensile Microcrack Initiation at
y—y Boundaries in TiAl”, ”, Mechanisms and Mechanics of Fracture: The John Knott
Symposium, eds. W.O. Soboyejo, J .J . Lewandowski, and RD. Ritchie, The Minerals,
Metals & Materials Society, Warrendale, PA. pp. 309-314 (2002).

B.C.Ng, T.R. Bieler, M.A, Crimp, “Effect of Crystal Orientation on Crack Nucleation
and Arrest in a Near-Gamma TiAl Alloy”, Mechanisms and Mechanics of Fracture:
The John Knott Symposium, eds. W.O. Soboyejo, J.J. Lewandowski, and KO.
Ritchie, The Minerals, Metals & Materials Society, Warrendale, PA. pp. 303-308
(2002)

Dale E Newbury, “ The origin, detection, and uses of electron channeling contrast,”
Scanning Electron Microscopy, pp. 1047-1054 (1974).

David Joy, Dale E Newbury, David Davidson, “Electron channeling patterns in the
scanning electron microscope,” Journal of Applied Physics, Vol. 53, No. 8, August
1982 pp. 81-122 (1982).

T. Ichinokawa, M. Nishimura, H. Wada, “Contrast reversals of pseudo-Kikuchi
bands and lines due to detector position in scanning electron microscopy,” Journal of
the Physical Society of Japan, 36 No. 1, pp. 221-226 (1974).

D.G. Coates, “Kikuchi-like reflection patterns obtained with the SEM,” Philosophical
Magazine, 16 (1967), pp. 1179-1184 (1967).

G.R.Booker, A.M.B. Shaw, M.J. Whelan, and PB. Hirsch, "Some Comments on the
Interpretation of 'Kikuchi Like Reflection Pattems' Observed by Scanning Electron
Microscopy," Philosophical Magazine, 16 , pp. 1185-1191 (1967).

BC. Ng, B. Ghosh and M.A. Crimp, "Electron Channeling Contrast Imaging of
dislocations near crack tips in bulk NiAl single crystals", Interstitial and
Substitutional Effects in Intermetallics, 1998 TMS Proceedings, editied by 1. Baker,
R.D. Noebe and BF. George, pp. 163-176 (1998).

BA. Sirnkin, M.A. Crimp, “Dislocation detection depth measurements in silicon

using electron channeling contrast imaging,” Proceedings, Microscopy and
Microanalysis, edited by G.W. Bailey, J .M. Corbett, R.V.W. Dimlich, J .R. Michael
and NJ. Zaluzec, 1996, pp. 140-141 (1996).

141

46 J.T. Czemuszka, N.J. Long, E.D. Boyes, and PB. Hirsch, "Electron Channeling
Contrast Imaging (ECCI) of Dislocations in Bulk Specimens, Materials Research
Society Symposium Proceeding, Vol. 209, pp. 289-292 (1991).

47 J .A. Venables and OJ. Harland, “Electron back-scattering patterns - A new technique
for obtaining crystallogaphic information in the scanning electron microscope,”
Philosophical Magazine, 27, 1193-1200 (1973).

48 DJ. Dingley, “Diffraction fi'om sub-micron areas using electron backscattering in a

scanning electron microscope,” Scanning Electron Microscope. 1984 11: 569-575
(1984).

49 D.J.Dingley and Karim Baba-Kishi, “Use of electron back scatter diffraction patterns

for determination of crystal symmetry elements,” Scanning Electron Microscopy, vol.
II, pp. 383-391 (1989).

50 Tim Maitland, “Electron Backseattered Diffraction”, Advanced Materials &
Processes, May 2004, pp. 34-36 (2004).

51 Stuart 1. Wright and Brent L. Adams, “Automatic analysis of electron backscatter
diffraction patterns,” Metallurgical Transactions A, vol.23A, March 1992, pp. 759-
767 (1992).

52 DJ. Dingley and V Randle, “Review microtexture determination by electron back-
scatter diffraction,” Journal of Materials Science, 27 pp.4545-4566 (1992).

53 RN. Quested, P.J. Henderson and M. Mclean, “Observations of deformation and
fiacture heterogeneities in a nickel-base superalloy using electron back scattering
patterns,” Acta Metallurgical, Vol. 36, No. 10. pp. 2743-2752 (1988).

54 J .R. Michael and RP. Goehner, “ Crystallogaphic Phase Identification in the
Scanning Electron Microscope: Backseattered Electron Kikuchi Patterns Imaged with
a CCD-Based Detector”, MSA Bulletin Vol. 23, No. 2, pp. 168-175 (1993).

55 Stuart 1. Wright, Brent L. Adams and Karsten Kunze, “Application of a new
automatic lattice orientation measurement technique to polycrystalline aluminum,”
Materials Science and Engineering A160 pp.229-240 (1993).

56 Brent L. Adams, Stuart 1. Wright and Karsten Kunze, “Orientation imaging: The
emergence of a new microscopy,” Metallurgical Transaction A, Vol. 24A, April
1993, pp.819-831 (1993).

5 7 Michael J. R, Sandia National Laboratories, private communication, Nov. 2001.

142

58 J .S. Kallend, et. al, “Operational Texture Analysis”, Materials Science and

Engineering A132, pp.1- 11 (1991).

59 Y.He, et. al., “Elastic constants and thermal expansion of single crystal g-TiAl from

300 to 750 K,” Materials Science and Engineering A239-240, pp. 157-163 (1997).

60 J. Zhang, et. al., “A fractogaphic study on tensile fiacture surface of duplex

61

62

63

microstructure TiAl alloy,” Scripta Materialia, 32 (12) 1815-1818 (1996).

BA. Simkin, M.A. Crimp, T.R. Bieler and DE. Mason, “The Effect of Crystal
Orientation on Deformation Transfer at 'y-y Boundaries in a Near-y TiAl Based
Alloy,” Structural Intermetallics 2001, eds. K.J. Hemker, D.M. Dimiduk, et al., TMS,
Warrendale, PA, pp. 391-400 (2001).

BC. Ng , B.A, Sirnkin and M.A, Crimp, “Application of the Electron Channeling
Contrast Imaging Technique to the Study of Dislocations Associated with Cracks in
Bulk Specimens,” Ultramicroscopy 75, pp. 137-145 (1998).

V.K. Vasudevan, M.A Stucke, S.A. Court and H.L. Fraser, “The influence of second
phase Ti3Al on the deformation mechanism in TiAl”, Phil. Mag. Letter, 59, (6) p.
299-307 (1989).

Y.H. Lu, Y.G. Zhang, L.J. Qiao, Y.B. Wang, C.Q.Chen, W.Y. Chu, “In-Situ TEM

study of fracture mechanisms of polysynthetically twinned (PST) crystals of TiAl
alloys”, Materials Science and Engineering A289, pp. 91-98 (2000).

143

Appendices

144

Appendix 1
Determination of Grain Orientation

145

 

Color Traces

 

Red (thin-solid) (100)

 

Purple (thin-solid) (010)

 

Orange (thin-dashed) (001)

 

 

 

 

 

Black (thick-solid) (l 10)
Blue (thick-solid) (1-10)
Green (thick-dashed) (101)
Cyan (thick-dashed) (-101)
Blue (thick-dotted) (01 l)

 

Magenta (thick-dotted) (01-1)

 

 

 

 

Reprint from figure 3.9.
The traces in the computed stereogaphs followed the above color code.

146

«w

a) (21111-7) ['151412] C) (6131'1) [31'1115]

superlattice

 

b) 91,3,6) [-15,-1,3] Grain A

Figure A1-1. Three possible stereogaphs and a SACP pattern of gain A. The
superlattice band from the SACP pattern matches stereograph (b) having gain normal
(91,3,6).

147

 

a) (3,10,—15) {22,-1.7} c) (7,9,3) [4,-1.8]

    

b)(8,3.-10) [4.8.0]

superlattice Grain B
bands

Figure A1-2. Three possible stereographs and a SACP pattern of gain B. The
superlattice band from the SACP pattern matches stereogaph (b) having gain normal
(8,3,-10).

148

  
 

superlattice

bands

\

(11,10,-13)[81 '116]

Grain BB

Figure A1-3. Stereogaph and composite SACP pattern of grain BB. The superlattice
band from the composite SACP pattern matches stereogaph with grain normal (11,10,-
13).

149

’4

j"
a (9. 3, 13) ; [-ll. 7. 61 c (a, 4, r); [4, -7, 41\

superlattice bands

V

 

b (6’13‘4); [596r9]

Figure A1-4. Three possible stereogaphs and a SACP pattern of gain BE. The
superlattice band from the SACP pattern matches stereogaph (c) having gain normal
(6,4,1)-

150

a)(7,5,1) [3,-5.4] c)(3,1,4) [-11,9,6]

superlattice
bands

 

b)(100,26,—75)[53,84,100]
Grain C

Figure A1-5. Three possible stereographs and a SACP pattern of grain C. The
superlattice band fi'om the SACP pattern matches stereogaph (b having gain normal
(100,26,-75).

151

    

superlattice
bands

O
V

b) (2.6.-5) [23,-1,3] GrainD .A'\:I.'.

Figure A1-6. Three possible stereogaphs and a SACP pattern of gain D. The
superlattice band from the SACP pattern matches stereogaph (c) having gain normal
(9,11,3).

152

         

superlattice§§>
‘T‘F-

   

(6:161‘17) [17,016]

Grain E

Figure A1-7. Stereogaph and composite SACP pattern of gain E. The superlattice band
from the composite SACP pattern corresponds well with stereogaph with gain normal
(6,16,-1 7).

153

   
  

superlattice
bands

 
  
 

(1,1,-3)[15,-9,2] Grain FK

Figure Al-8. Stereogaph and composite SACP pattern of gain FK. The superlattice
band from the composite SACP pattern corresponds well with stereogaph with grain
normal (1,1,-3).

154

     

a)(4,3,10)[-11,18,-1] c)(9,3,-4)[-1,15,9]

superlattlce "0 superlattlce Concluded

bands
Grain G

 

b)(9,4,3)[0,-3,4]

Figure A1-9. Three possible stereographs and a SACP pattern of gain G. The band a‘
from the top SACP corresponds to the superlattice (1 shown in the stereogaph (a) and
while band b’ corresponds to superlattice b (b). Since both bands a’ and b ’ do not show
any superlattice information, stereogaphs a and b are incorrect and are eliminated. Grain
c with normal, (9,3,—4), is therefore the correct solution for gain G.

155

'45.-

a) (Allin-17,88]

    
 

( superlattic
A band
I

Grain H

b) (5929-1)[79'9917i

Figure A1-10. Three possible stereogaphs and a SACP pattern of grain H. The
superlattice band from the SACP pattern matches stereogaph (c) having gain normal
(3,7,15).

156

I Grain HJ 2’ ,1 3...... .9... ..

”Ammant plane trace

   

 

. 1 :
"ll)o"a("4@ (-lll)
(ll-|)~0J2 ” (l.||).o_og H

 

a mu. (II-mus

    

 

 

 

a-)(19.5.7i[-1.1.21 b) (3125-95) [100.40.411| c)(17,4,5)[-8,9,2o]

 

Figure Al-l 1. BSE image of gain HJ (labeled) with three possible plane traces
superimposed on the image. The dominant twining plane corresponds well with the
positive Schmid factor displayed by the plane trace analysis (b), with gain normal

(31,25,-99).

157

 

: . [75,31,100] c) (7,8,10) [2,-3,1]

.33

b) (5,6,4) [43.2.7] .

    

\3‘3.‘ .

Figure Al-12. Three possible stereogaphs and a SACP pattern of gain J. The

superlattice band fiom the SACP pattern matches stereogaph (c) having gain normal
(7,8,10).

   

158

;%

a)(3,5,-4) [11.3.12] c) (7.10.6)1-4.1.3l

   
  
   

superlattice
bands

Grain K

 

b) (6.7.10) [3,-4.1]

Figure Al-l3. Three possible stereogaphs and a SACP pattern of gain K. The
superlattice band from the SACP pattern matches stereogaph (c) having gain normal

(7,8,10).

159

     
  

a(21,2,3) [-2,15,4] c) (3,2,-21) [4,15, '

Grain L

 

b) (331,2) [4,-2,15]

Figure A1-14. Three possible stereogaphs and a SACP pattern of gain L. The
superlattice band from the SACP pattern matches stereogaph (c) having gain normal
(3,2,-21).

160

  
  

C) (9’31'10) ['11310]

" concluded

b) (88,100.33) [-39,1,100] \' a. Grain 5

Figure A1-15. Three possible stereogaphs and a composite SACP pattern of gain 5.
The band a’ from the composite SACP corresponds to the superlattice (1 shown in the
stereogaph (a) while band b’ corresponds to superlattice b in sterogaph (b). Since both
bands a’ and b’ do not show any superlattice information, stereographs a and b are
incorrect and are eliminated. Grain 5 with normal, (9,3,-10), is therefore the correct
solution for gain 5.

161

superlattice

‘
A a
*6 ”'3

a. (150,33) [8,33,50] b. (1,16,25) [17,-99,63]

     

no superlattices
. bands

c. (13,20,0) [-99,65,16]
Grain 5A

Figure Al-l6. Three possible stereogaphs and a SACP pattern of grain 5A. The band
a‘from the top SACP corresponds to the superlattice a shown in the stereogaph (a).
Since band a’ does not shown any superlattice information, stereogaph (b) is eliminated.
Further analysis is needed and is shown in figure A1-17.

162

 

(111) 0.02

 

 

 

 

 

a. (1,50-33) [8,33,50] b. (1,15,25)‘ [17,-99,63] c. (13,503) [-99,65,16]

Figure A] -17. BSE image of gain 5A with three possible plane traces superimposed on
the image. Since plane trace (b) had been eliminated earlier (see Figure Al -l6) and plane
nace (a) showed negative Schmid factors for both the dominant twinning planes, the
plane trace (c) with normal (13,20,0) was selected as the correct gain orientation.

163

I

.3” _ ,
ram 5b '7 '-

n - '
A » “ P
'( q- 1,.

     

(-111)I-0.34

~ ." u' -_ >1
..
‘I \t
:1 A I
.q

 

     

49s

a) (1,8,13) b) (1,15,-11) C) (10,16,1)

 

Figure A1-18. BSE image of gain SB with three possible plane traces superimposed on
the image. From the BSE image, plane trace a has the characteristic of pre-existed
(evenly distributed twins) while plane trace b is the dominating deformation twin. Plane
trace analysis (a) with a gain normal (1,8,13) and a high positive Schmid factor
corresponding to band b is the most likely the correct gain orientation. Plane trace
analysis (b) and (c) both showed negative and low Schmid factor respectively for the
dominating deformation twin.

164

I
aa\

4P“ \
A§ ‘

I (LIL-9) [8.5.71

Grain 1

 
 

A»?

1: (1.5.6) [3.-3.2[

 

5 (12.132) H.691

concluded

Figure A1-19. Three possible stereogaphs and a composite SACP pattern of gain 1.
The band a’ from the composite SACP corresponds to the superlattice (1 shown in the
stereogaph (a) while band b’ corresponds to superlattice b in sterogaph (b). Since both
bands a’ and b ’ do not show any superlattice information, stereographs a and b are
incorrect and are eliminated. Grain l with normal, (1,5,6), is therefore the correct
solution for gain 1.

165

.) (100,61,1)[2o,33,31[

,9
4.,

b)(l00.0,-63) [63,97,100]

 

Figure A1-20. Three possible stereogaphs and a composite SACP pattern of gain 2.
The band a’ from the composite SACP corresponds to the superlattice a shown in the
stereogaph (c) while band b’ corresponds to superlattice b in sterogaph (a). Since both
bands a’ and b’ do not show any superlattice information, stereogaphs a and b are
incorrect and are eliminated. Grain 2 with normal, (100,0,-63), is therefore the correct
solution.

166

superlattice bands

   

 

a) (3,5,0) [45.9.1] c) (0,2,3) [1,-12,8]

   

Grain ZW

b) (0,5,-3) [6,61,100]

Figure A1-21. Three possible stereographs and a composite SACP pattern of grain ZW.
The superlattice band a’ from the composite SACP corresponds to the superlattice a
shown in the stereogaph (c), confirming that gain ZW with gain normal (0,2,3) is the
correct orientation.

167

   
 

c) (7,20,-17) [100,0,41[

b) (21,25,8) [-38,1,100]

Grain Z

Figure A1-22. Three possible stereographs and a composite SACP pattern of gain Z.
The superlattice band a’ from the composite SACP corresponds to the superlattice 0
shown in the stereogaph (c), confirming that gain ZW with gain normal (7,20,-17) is
the correct orientation.

168

Appendix 2
Images of Grain interactions

169

50 [1m .2
<—> ,

 

 

. » , 4'0 ‘ ‘fT‘I: ‘
Figure A2-2. Interaction between grains B and BB is shaded gray. Microcracks

(arrowed) at the gain boundaries between C and BE and B and BE. Intact boundaries
between gains E and BE, BE and BB.

\‘ 1 “ ' \

170

“1111111“
"’ 11212532
I , 5,. ‘

 

Figure A2-4. Grain interaction between gains D-E-H

171

 

Figure A2-5. Interaction at gain boundaries between gains D, F,FK,H,I and HJ. An SB
image had been superimposed over the BSE image to provide additional information on
the crack edges. Higher magnifications of the gain interaction are shown in figure A2-6
and A2-7.

 

Figure A2-6. Intact boundaries between gains D and F, and between F and FK.

172

   

Figure A2-8. Cracked boundary (arrowed) between gain I and gain H.

173

 

shown in figures A2-11-12.

174

 

Figure A2-11. Evidences of cracked boundaries between gains K and J. The
observation of ‘stepped or jogged’ in the crack path strongly suggest the presence of
microcracks that resulted in the change in the crack path.

 

Figure A2-12. Intact boundaries between gains K and LA.

175

“11):(111) _
11

(111)

 

Figure A2-14. Grain interactions between gains ZW-Z—5-5A-5B-1. Higher
magnification images are shown in figures A2-15.

176

    

Figure A2-15. Intact boundaries between gains ZW and Z.

177

Appendix 3
Computational progarns used in the analysis

178

 

Program #1

Fracture Propagation Parameter Progam.
Run via MathCad Software.

Grain A and Grain B
Crystal Information / Lattice Parameters
1

a:=l
b:=l
c:=1.02

Unit Vector WRT Orthonormal CS Given Coordinates WRT Tetragonal CS
0

 

M...).=[(.0.)2.(.1..)2.(.,.)2

C1 := Al-Bl

a-go

bg1

cgz

unit(g) I: m

C; = —1.00

179

Grain Stu'face Normal/1‘ ensile Axis Information; column i corresponds to the surface normal and tensile
axis for that gain with respect to the fct coordinate system.

INDICES ARE INPUTTED VERTICALLY

(h)
(surface normals, tensile directions) (k)
(l)
f 17 17 3
Ngzz 7 43 4
(10 24 l
(146 497 19
1,, := —146 499 —18
(-146 542 15

 

Conversion Of Normal/Tensile Directions To Unit Vectors in Real Space

N1 ;= (unit(Ng(O>))

T = (unit(’l‘g(0>))
T2 = unit(Tg(l>))
T6 := (unit(Tg<2>))
S] := Tl x N1
82 := T2 x N2
S6 := T6 x N6

* We do not use N6 or the third grain.

Back-calculation Of Rotation Matrices From Grain To Lab Basis. The transformation operates under the
assumptions that N corresponds to k, T corresponds to j, and TxN corresponds to jxk = i.

(1406111420 0 o 0 o o)
0 O 0 N0 N1 N2 0 0 O
0 O 0 0 0 0 N0 N1 N2
T0 T1 T2 0 0 O 0 0 0
A(N,T,S):= 0 0 0 T0 T1 T2 0 0 0
0 O O 0 0 0 T0 T1 T2
s0 s1 82 O 0 0 O O 0
0 0 0 SO 8182 0 0 0
\O O 0 O O 0 s0 8182)

 

 

(X X X

0 l 2
0091= x3 x4 ‘5'

(x6 x7 x8

5'
ll
o—‘OF-‘O—‘O

 

 

\Oi
Q

(A(N1,T1,sr)‘ 1.1)
Q(A(N2,r2,sz)'1.1)

Or I
Q2:
06 := Q(A(N6,T6,S6)' 1-6)

Check Orthogonality:
1.00 0.00 0.00 1.00 —0.00 —0.01
Qr-Q1T= 0.00 1.00 0.01 QzT-Qz= —0.00 1.01 -0.00
0.00 0.01 1.00 —0.01 —0.00 0.99

1.00 —o.oo —o.oo
Q6TQé= —0.00 1.00 —o.oo
—o.oo —0.00 1.00

Check Mapping of Basis Vectors:
1.00
Ql-(Tl x N1) = 0.00
0.00
0.00 0.00
0,11 = 1.00 Ql-Nl = 0.00
0.00 1.00

181

1.00
Qz-(TZ x N2) = 0.00
0.00
0.00 0.00
Qz-T2= 1.00 Qz-NZ = 0,00
0.00 1.00
1.00
Q6-(T6 x N6) = 0.00
0.00
0.00 0.00
Q6-T6 = 1.00 Qa-N6 = 0.00
0.00 1.00

It follows that rotation Qi carries a vector written in the crystal basis into its equivalent vector in the lab
basis.

Define all possible slip systems:

DEFORMATION SYSTEMS;
M=planes, D=deformation vector.

1 1 l 1 —1 —1 —1 —1 1 1 l 1 —1 -l —1 —1
M:= l 1 l 1 1 l 1 1 —1 -l -l —1 —l —1 —1 —l
1 l 1 1 1 1 1 1 1 1 l 1 1 1 I 1

1 1 1 0 —1 l l 0 1 1 l 0 —l 1 1 O
Dz: 1 —1 O 1 l l 0 1 —l 1 0 1—1 —1 01
—2 0 —1 —1 —2 0 l -1 —2 0 —1 1 —2 O 1 1

182

Define functions to transform slip system normal and direction vectors into lab basis:

m1(i) == Qr-(unit(M<i>))

m2(i) 021611119»
"160) 1= Q6-(nnit(M<i)))
410) == Q1 -(unit(D®»
‘120) := Qz~(unit(D<i>))
060) == QG'(|lllit(D<i>»

The tensile direction or the tilt direction in lab space ('1')

T := 01.11

Calculate compatibilty factors for all possible combinations of slip system combinations between

grains 1 and 2 : (mu)
Ill1120.1) := (m1(i)-m2(1))(d1(i)-d2(1))
M]; := mm(16, 16,6112)

SLIP SYSTEM NUMBERS FOR THE lst GRAIN VERTICAL,
FOR THE 2nd HORIZ.

 

0

1

 

0.39

0.60

0.64

0.04

41.14

0.31

0.28

0.03

0.22

-0.20

 

-0.60

0.22

-0.42

-0.63

-0.52

-0.00

0.45

-0.45

0.22

0.00

 

0.04

0.63

0.35

—0.28

-0.38

0.26

0.46

-0.20

—0.09

-0.17

 

0.63

0.42

0.76

0.34

0.13

0.27

0.02

0.25

-0.30

-0.17

 

-0.22

-0.22

-0.30

-0.08

0.08

-0.00

-0.07

0.07

-1.00

0.01

 

-0.20

-0.00

-0.17

-0.17

-0.00

0.28

0.14

0.14

-0.01

-1.00

 

0.09

0.19

0.18

-0.01

-0.07

0.14

0.13

0.01

0.86

-0.50

 

M12 =

-0.29

-0.19

-0.35

-0.16

0.07

0.14

0.01

0.13

-0.87

-0.49

 

-0.15

0.53

0.13

-0.39

-0.66

-0.01

0.56

-0.57

0.08

—0.00

 

(Dm‘JOJQ-th-to

0.32

0.01

0.28

0.27

-0.01

0.82

0.41

0.40

0.00

0.28

 

.L
O

0.03

0.46

0.25

-0.20

-0.57

0.40

0.69

-0.30

0.07

0.14

 

.3
—l

0.28

-O.45

0.02

0.47

0.57

0.42

—0.29

0.70

-0.07

0.14

 

.a
N

0.08

-0.43

-0.15

0.28

.014

0.32

004

-0.28

-0.23

0.21

 

.s
00

0.44

-0.16

0.30

0.46

-0.53

-0.00

0.46

-0.46

0.23

0.00

 

..L
A

0.15

0.30

0.28

-0.02

-0.14

0.28

0.26

0.02

0.32

-0.18

 

.3
0|

 

 

-0.28

 

0.46

 

-0.02

 

-0.47

 

0.39

 

0.28

 

-0.20

 

0.48

 

0.09

 

-0.19

 

183

 

Calculate the schmid factor for grain 1(SF1x)
For (111) [1 1-2] with the tensile direction transformed to the lab y axis (010)

SF10= [1111(0) 01 T1)(0)][d1(QT1)]Qr

For (-1 l l) [~11-2] with the tensile direction transformed to the lab y axis (010)

Sr14== [1111(4) -(Q1 T1)(4][dr) (Q1T1)]

For (1-1 1) [1 1-2] with the tensile direction transformed to the lab y axis (010)

SF1s== [mr(3) (Qr T1)(3)][d1-(-Q1(T1)]

For (-1-1 1) [1-1-2] with the tensile direction transformed to the lab y axis (010)
Srrrz == [m1(12)-(Qr ~T1)][dr(12)-(Q1 'T1)]

Calculate the schmid factor for grain 2 (SF2x)

For (1 l l) [1 1-2] with the tensile direction transformed to the lab y axis (010)

SF20 :)=[m2(0 (02 (-'r2)(0)][dz “(02 12)]

For (-1 11) [-1 1-2] with the tensile direction transformed to the lab y axis (010)

SF24 :(=[m24) °'(Qz T2)(][dz4) (Q: n(-)]

For (1-1 1) [1 1-2] with the tensile direction transformed to the lab y axis (010)

Sm =[m2(3) -(-Qz-)(8)12][dz -(-Qz 12)]

For (-l-l 1) [1-1-2] with the tensile direction transformed to the lab y axis (010)
SF212 := [mz(12)-(Qz-T2)][dz(12)-(Qz~12)]

Compute the twin vector dot with the tensile direction, denotes as BO,B4,B8,B12 (Absolute Value)
B10 = )(d11 0))(01 '11))
314 = )(d1(8)(4))(01n)I

Bra = (d1(13)Q1T1|
B112 = )(dr(12))( (Q1 “)1

Compute the twin vector dot with the tensile direction, denotes as 80,84,383” Absolute Values
1320 = )(02(0))(Qz

T2)l

4==l(dz(4))(: T2)l

B28-- |(42(8))'(- T2)|
13212 = I 11202)) (02 “)1

184

Compute the sum of twin vector doted ordinary dislocations denoted as D0,D4,C8,D12 where the
twin will interact with all the ordinary dislocations on the other side of the boundary Absolute values

 

n11== W o))(.,( 1+))| [(1,(o).)(.1,(s))|+|(1,(o).1,()(9))[+[(.1.(o).)(1,(13))|
Du== |(dn( 4)-W4-()41(4)(821))|+|(41())(425)|+|( ).(.1,(9))[. (11(4))(1,(13))[
D1s== |(d1( 8-)) (82(1))I + |(d1(8) -(dz(5))| + |(d1(8))-(dz(9))| + (81(8) -(82(13))l
om:- Km (12)) (82(1))I + |(81(12))-(82(5))| + |(d1(12))-(dz(9))| + |(a1(12))-(82(13))|
821.- |(.1,(0.))(111(1))| + |(82(0))~(a1(5))| + |(82(0))-(81(9))| + (82(0))-(a1(13))|
D21== |(82(4))-(d1(1))| + |(82(4))-(d1(5))| + l(dz(4))-(d1(9))| +|(.12(4)).(.1,(13))|
Dzs== I(82( 8)) (81(1))I + |(dz(8))- 81(5))l + |(dz(8))-(d1(9))l + |(82(8))-(d1(13))|
Dzu== [(1,(..12))(.1,(1))[.[(.1,(12)).(.1,(s))[ [(1, ()12)(.1,()9)|+|(.1,(12)).(.1.(13))[

Compute the sum of twin vector dot other twins from adjacent grain Actual Value

8.1.1.111- -d1(0) 82( 0)+ d1( 0-4)82( )+ d1(0) 82(8) + 81(0)-dz(12)

Ea...14= -4.1,() (0+) 81(4) 42(4-)+d1(4)dz(8)+d1(4)-82(12)
8.10.12: -8¢1() d2+(0) 41(8-4)82()+d1(8)-d2(8)+d1(8)-82(12)
81..112= —a1(12) 1,.(0)..1,(12) 82(4)+a1(12)-82(8) 81(12)-82(12)
8.10.211:- -082() d10+() 82(0)-d1(4)+82(0)-d1(8)+dz(0)-d1(12)
EAos241='4‘12()d1+(0)112(4')111(4)+d2(4)'dl(8)+d2(4)'d1(12)
2,,,,,;= -d2( 8)d1( 0)+ 82(8 MW + 82(8)-d1(8) + 82(8)-d1(12)
81.612 (82(12)-81(0)+ 82(12)-a1(4)+ 82(12)-a1(8) + .1,(12).1,(12))

Using Actual Values for twin vector dotted with twin vectors on the other grain.
This subprogram will replace the negative values will be replaced with ZERO as it has not significant

meaning.

E1020 = (111(0))(dz(0))
E1024 = (81(0))-(42(4))
E11122 == (d1(0))-(dz(8))
1510212 = 01(0))-(dz(12))

EAOBO := “(E1020 < O , 0 , E1020)
1311034 == if (E1024 < 0101131024)
EA030 := “(E1023 < 0, 0, E1020)
E14031: == if(1510212 < 0 0.1310212)

111420= (01 (4412)” (0))

Ii31424-- —111(( 4))02( (4 4))
81.1211:= —d1(( 4)-dz)( (8))
E14212== ((11 (MN (12))

185

EA4BO := “(E1420 < 0, 0, E1420)
EA4B4== if (E1424 < 0 0 E1424)
EA4as= ifE E1428 < 0 0 E1428)
E44312=111E10212 <0 0 E14212)

E1820== (01(3))4121 (0))
E1824== (01(3)-02()( 4))
E1828== (4111324211 3))
812212: (41(8))-dz((12))

E11800 == if(131820 < 0.0,Erszo)
E14834 == if(1311124 < 0.01E1824)
11,1338 :=1r(r:1m < 0.0.1513”)
EA81312==1f(E18212 <0 0 E18212)

E11220== (01 (1212-42)” (0))
E11224= (0111 2)-¢12()( 4))
E112281=(dr(1 2))(!1213»
E112212=(d111212-))(<12(1212))

EArzarz == if(13112212 < 0 2 0113112212)
13.11230 == if (E11220 < 010. E11220)
E41234 == if(E11224 < 0.0. E11224)
EA1238 == if(E11220 < 0201131828)

anArz == if(1120112 < 0 1 0. Ezorrz)
E30110 ==1f(E2010 < 0101132010)
1130.44 == if(1)2014 < 0.0,1‘2014)
l":l!0148==11’(1‘32018 <0 0 E2018)

E2111 (dz( 4-))( (0))
E2414 'dz‘(( 4)-d)(1(4))
82412: (.1,( 4)-)(d1 (8))
E24112== (02(4))4111 1(12

))

186

 

15341112 == if(E24112 < 0201E24112)
E34140 == if(E2410 < 0201192410)
E3444 == ifEE2414 < 02 01532414)

EB4A8 == if E2418 < 0101132418)
E2810==(dz(3)-)(d1 (0 0))

11211 = (11(8) ) (11(4))

E2818 =(dz(8 ))(d1(8))
Fasuz =(d2 3)) d1(12))

EBsAlz == if(Eztmz < 0202E28112)
EBSAO == if(1921310 < 0102132810)
E38144 == if(E21314 < 020152814)
E88A8 == if(E2818 < 0102E2818)

E21210 =2=(dz(1 H1)( (0)
E21214== (02(12 ))(d1(4)
F2121812=(d2(1)°dl)( (3)

E212112-= (02(1 2)-)(¢11(1 ))

EBIZAIZ == if(E212112 < 010219212112)
E312Ao == if (E21210 < 02011721210)
EB12A4 == if (E21214 < 0101 E21214)
EmzAs == if (E21218 < 0 2 0 2 E21218)

Summing the leading vector dotted with the other vectors in the other grain
EAO == E4030 + E14034 + EAons + EAolm
E30 == E3040 + E3044 + EBOAS + EBoAlz

EM == E14430 + E14434 + EA4BS + EA4B|2
EB4 == E34140 + E34144 + EB4A8 + E34141:

EAs == EASBO + E14834 + EAsBs + EASBIZ
Ens == E38140 + E38144 + 33381“ + EBsAlz

EAIZ := EAuno + EAIZB4 + E41238 + EA12312
E312 == E31240 + E3121“ + EBIZAS + Emzmz

187

Compute the Factor formulated by Benjamin Sirnkin

30010 == Smo-Burclo
B¢n14== 81114-314014
Bell18 == SF18'318'C18
Bell112 == SFIIZ’BIIZ'CIH

Benzo == Sno-Ezo-Czo
Bell24 == Sm4'Bz4'C24
30028 == Sm-Ezs'Czs
B00212== Sm12'3212-C212

Compute Steve Factor Z which is a product of SF, B, D E

FAO == SFIO‘BIO'DIO‘EAO
FA41= SF14°314-Dr4'EA4
FA81= SF18'318'DIS°EA8
FA123= SFIIZ'BIIZ'DIIZ‘EAIZ
F30 == Sno-Bzo-Dzo'Eso

F34 == SF24'324'DZ4'EB4

F38 == Sns-st-Dzs-Ens
F312 == SFZIZ‘BZIZ‘DZIZ'EBIZ
($1410 B10 D10 EA0 36010 F140)
SFM B14 D14 EA4 Bell14 FA4
SF18 318 018 EA8 Benls FA8
SF112 B112 13112 EA12 Bell112 FA12
SF20 320 020 E30 130020 F130
Sm B24 l>24 E34 Benn F134
Sm 328 028 E38 30028 F38
($17212 B212 1)212 E312 Bell212 F312)

 

 

188

(-O.l6
001
0.30
-O.l6
—0.18
0.27
005

 

\—0.18

0.48
0.01
0.94
0.47
0.51
0.95
0.05
0.50

2.71
1.27
1.92
2.72
2.72
1.92
1.27
2.71

1.15
1.92
0.28
0.80
1.15
0.28
2.56
1.17

-0.30
—0.00
0.88
—0.30
—0.35
0.79
—0.01
—0.35

_0.24\
—0.00
0.16
—0.17
-0.28
0.14
—0.01
—0.29 )

 

189

Program #2

Stereographic Projection Plot
Run via Mathematica

<<Graphics‘ParametricPlotBD‘
(*uses normal and tilt axis*)
(* Initial Definitions *)
a=l ; b=l; c=1.;
mag[v_]:=Sqrt[(a V[[1]])“2 +(b V11211)“2 +(C V[[3]])“2];
unit[V_]:={a V[[1]]2b V[[2]]2c V[[3]]}/mag[V];
unit2[V_]=={V[[l]]2V[[2]]}/Sqrt[V[[l]]“2+V[[2]]“2];
Proj[n_,v_]:=v-(v.n)n;
(* Input appropriate crystal information *)
(* Define reference planes for projection *)
M={{1,0,0},{0,1,0},{0,0,l},{l,1,0},{l,-l,0},{1,0,1},{-
l,O,l},{O,1,l},{O,

11-1}};
m=Map[unit,M];
(* Define surface normal/in-plane orthogonal directions *)
(* Define matrix of all grain sample normals *)
GSN:={{30,42,11},{26,39,17},{3,4,1}};
(* Define matrix of all tensile directions *)
GTA:={{ll,-10,8},{-32,27,7},{l9,-l8,15}};

(* Define matrix of all corresponding rotation axes *)

GRA:={Cross[GTA[[1]],GSN[[1]]],Cross[GTA[[2]],GSN[[2]]],
CIOSS[GTA[[3]]1GSN[[3]]]};

snxloc={l,0,0}; snyloc={0,l,0}; snloc={0,0,l};

(* Function to rotate a vector v about axis n by an angle theat,
measured to be postive around n *)

ROTATE[n_,theta_,v_]:=
v + Sin[theta] Cross[n,v] + (l-Cos[theta]) (n (n.v) - v);

(* Great circle function wrt local coordinate system *)

v[phi_]:={Cos[phi],O,Sin[phi]};

190

(* Function to calculate components of v (written with respect to
crystal \
basis) with respect to the local coordinate system *)

CONVERT[V_]:=unit[{snx.v,sny.v,sn.v}];

PROJECT[V_]==(2/(l-V[[3]])) {V[[l]]1V[[2]]};

(* Denote normal/rotation axis for calculation *)
(* SN = GSN[[3]]; sn = unit[SN]; *)

SN {ll,3,-12}; sn = unit[SN];

(* R GRA[[3]]; snx = unit[R]; *)

R= {6,—10,3}; snx = unit[R];

SNY=Cross[SN,R]; sny=unit[SNY];

Pcircle =
ParametricPlot[{2 Cos[t],2 Sin[t]},{t,0,2 Pi},AspectRatio \[Rule]

12
DisplayFunction \[Rule] Identity, PlotRange \[Rule] {{-2,2},{-

212)}.
PlotStyle \[Rule] {{GrayLevel[O.5], Thickness[0.01]}}];

NewPlot = Pcircle;
Do[
(* Decision Tree For {100} Family Of Planes *)
If[i\[LessEqual] 3, hueswitch = 1, hueswitch = 0];
OldPlot = NewPlot;
If[hueswitch \[Equal] 1,
huesetting = (i-1)/3;
(4
Ensure that normal to crystal plane is directed into same \
half space as sample normal *)
If[m[[i]].sn >= 0, mnorm=m[[i]], mnorm=-m[[i]]];
(* Transform crystal plane normal mnorm into planenorm,
written with respect to the tn (tensile-

sample normal) coordinate system *)

planenorm =CONVERT[mnorm];

(it

191

Check to see if crystal plane is the same as the sample normal

*)
If[sn.mnorm \[Equal] l, CurrentPlot = Pcircle ,

(*
Calculate line of intersection between crystal plane and
sample surface in t-s local coordinate system *)

axisl = CONVERT[Cross[sn,mnorm]];

(*
Determine correct angle to rotate snxloc about sn so that
it properly aligns with the axisl *)

If[axisl.snyloc \[GreaterEqual] O, delta=ArcCos[axisl.snxloc],

delta=-ArcCos[axisl.snxloc]];
(*

Calculate acute angle between crystal plane normal and

sample normal.
This is the angle through which circular path,
after the first rotation,
is rotated about axisl to its final state. *)

psi = Pi/2-ArcCos[sn.mnorm];
(* Print[ToString[psi]]; *)
(1%
Calculate path of intersection of crystal plane with unit

sphere with respect to t-
3 local coordinate system *)

vt[phi_]:=ROTATE[axisl,psi,ROTATE[snloc,delta,v[phi]]];

(*
Check whether crystal plane is parallel to sample normal.
If it is,

then the projection is just the line of intersection between

the crystal plane and the sample surface. If it is not,
then we simply proceed with the algorithm *)

If[sn.mnorm\[Equal]O,
(* Check if axisl[[2]]=0. If so,

then the intersection is vertical. If not,
then proceed as usual. *)

192

 

If[axisl[[2]]\[Equal]O,
CurrentPlot =
ParametricPlot[{O,t},{t,-2,2},AspectRatio \[Rule] 1,

DisplayFunction \[Rule] Identity,
PlotRange \[RUle] {1-2I2}l{-2I2}}I

PlotStyle \[Rule] {{Hue[huesetting],
Dashing[{0.02,0.02}]}}];

upperlimit = 2/Sqrt[l+(axisl[[2]]/axisl[[l]])“2];
lowerlimit = -2/Sqrt[l+(axisl[[2]]/axisl[[l]])AZ];
CurrentPlot =

ParametricPlot[{t,(axisl[[2]]/axisl[[l]])
t},{t,lowerlimit,
upperlimit},AspectRatio \[Rule] 1,
DisplayFunction \[Rule] Identity,
PlotRange \[Rule] {{-2,2},{—2,2}},

PlotStyle \[Rule] {{Hue[huesetting],
Dashing[{0.05,0.05}]}}];

1;

(iv
Calculate projection of 3-
dimensional path into sample surface *)

rt[phi_]:=(2/(1-vt[phi][[3]]))
{vt[phi][[1]]2vt[phi][[2]]};

(* Generate plot of projection *)

CurrentPlot =
ParametricPlot[{rt[phi][[1]],rt[phi][[2]]},{phi,Pi, 2 Pi},
AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity,
PlotRange \[Rule] {{-2,2},{-2,2}},
PlotStyle \[Rule]
{{Hue[huesetting],Dashing[(0.02,0.02}]}}];

1;

193

I

(* Remainder Of Procedure Handles Non {100} family planes *)
huesetting = (i-4)/6;

(ir

Ensure that normal to crystal plane is directed into same half
space \
as sample normal *)

If[m[[i]].sn \[GreaterEqual] O, mnorm=m[[i]], mnorm=—m[[i]]];

(* Transform crystal plane normal into tn (tensile-
sample normal) coordinate system *)

planenorm = unit[{mnorm.snx,mnorm.sny,mnorm.sn}];

(* Check to see if crystal plane is the same as the sample normal

If[sn.mnorm \[Equal] l, CurrentPlot = Pcircle ,

(*
Calculate line of intersection between crystal plane and

sample \
surface in t-s local coordinate system *)

axisl = unit[Cross[snloc,planenorm]];

(1k
Determine correct angle to rotate snxloc about snloc so that
it \
properly aligns with the axisl *)

If[axisl.snyloc \[GreaterEqual] O, delta=ArcCos[axisl.snxloc],
delta=-ArcCos[axisl.snxloc]];

(*-
Calculate acute angle between crystal plane normal and sample
\
normal. This is the angle through which circular path,
after the first rotation,
is rotated about snloc x planenorm to its final state. *)

psi = Pi/2-ArcCos[snloc.planenorm];

(* Print[ToString[psi]]; *)

(1*
Calculate path of intersection of crystal plane with unit

sphere \
with respect to t-s local coordinate system *)

vta[phi_]:=ROTATE[snloc,delta,v[phi]];

194

vtb[phi_]:=ROTATE[axisl,psi,vta[phi]];

(* Check whether crystal plane is parallel to sample normal.
If it is,
then the projection is just the line of intersection between
the \
crystal plane and the sample surface. If it is not,
then we simply proceed with the algorithm *)

If[sn.mnorm\[Equal]O,

(* Check if axisl[[2]]=0. If so,
then the intersection is vertical. If not,
then proceed as usual. *)

If[axisl[[2]]\[Equal]0,

CurrentPlot =
ParametricPlot[{0,t},{t,-2,2},AspectRatio \[Rule] 1,
DisplayFunction \[Rule] Identity,
PlotRange \[Rule] {{-2,2},{-2,2}},
PlotStyle\[Rule]{Hue[huesetting]}];

upperlimit = 2/Sqrt[l+(axisl[[2]]/axisl[[l]])“21;
lowerlimit = -2/Sqrt[l+(axisl[[2]]/axisl[[1]])A2];
CurrentPlot =

ParametricPlot[{t,(axisl[[2]]/axisl[[l]])
t},{t,lowerlimit,
upperlimit},AspectRatio \[Rule] 1,
DisplayFunction \[Rule] Identity,
PlotRange \[Rule] {{-2,2},{-2,2}},
PlotStyle\[Rule]{Hue[huesetting]}];

(*
Calculate projection of 3-
dimensional path into sample surface *)

rt[Phi_]:=(2/(1-th[phi][[31]))
{vtb[phi][[1]]2th19hi1112111;

(* Generate plot of projection *)

195

PROJ

CurrentPlot =
ParametricPlot[{rt[phi][[1]],rt[phi][[2]]},{phi,Pi, 2 Pi},
AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity,
PlotRange \[Rule] {{-2,2},{-2,2}},
PlotStyle\[Rule]{Hue[huesetting]}];

];
1;
1;
NewPlot = Show[OldPlot, CurrentPlot];
I{illlg}];

= Show[NewPlot, DisplayFunction \[Rule] $DisplayFunction,

Axes\[Rule] False]

Print["Surface Normal = "<>ToString[SN]];
Print["Rotation Axis = "<>ToString[R]];
(* Color Key: Red (dashed) = (100), Green (dashed) = (010),
Blue (dashed) = (001), Red = (110), Yellow = (1-10), Green = (101),

Cyan = (-101), Blue = (011), magenta = (01-1) *)
% Start of Graphics

% End of Graphics
MathPictureEnd

\
\>"]

\[SkeletonIndicator]Graphics\[SkeletonIndicator]
Surface Normal = {11, 3, -12}
Rotation Axis = {6, -10, 3}

vec1:={0,1,0}; vecZ:={02l2l};

angle[a_,b_]:=ArcCos[unit[a].unit[b]];

angle[vec1,vec2]
0.785398

196

Progranfi3

111 trace projection
Run via Mathematica

<<Graphics‘ParametricPlot3D‘

<<Graphics‘Graphics‘

(* Initial Definitions *)

a=l ; b=l; c=l.;

mag[V_]==Sqrt[(a V[[1]])“2 +(b V[[2]])“2 +(C V113]])“2];

unit[V_]=={a V[[l]]2b V[[2]]10 V[[3]]}/mag[V];

(* Define reference planes for projection *)

—--—------PLANES *)
M:={{l,1,l},{-l,1,l},{l,-1,1},{1,1,-l}};
UM:=Map[unit,M];

SIZE = Length[UM];

(* Define Normal/Tensile Directions WRT Crystal Basis *)
----------- SURFACE NORMALS *)
NORMSET:={{30,42,11},{26,39,17},{3,4,1}};
UN:=Map[unit,NORMSET];

----------- TENSILE DIRECTIONS *)
TENSSET:={{283,—254,l98},{-592,355,91},{19,-18,15}};

UT:=Map[unit,TENSSET];

(* Select Normal Vector / Tensile Direction Pair For Calculation *)

GN:=1;

197

(* Define Tilt Angle *)

SAMPLE TILT *)

phi= 0 Degree;

NORM:=UN[[GN]]; TENS:= UT[[GN]]; LROT:=Cross[TENS,NORM];
(* Calculate Direction Vectors For Zero Degree Tilt Plane Traces *)
OLDLINES = (1;
Do[
DIRECTION = Cross[NORM,UM[[i]]];
LINES = Append[OLDLINES,DIRECTION];

OLDLINES = LINES;

I

{i211SIZE}];

LU:=Map[unit,LINES];

(* Do Routine To Generate Matrix Of Directions Of Trace Lines On
Projected \

Plane *)

PLOTNEW = {1;

anglelist = {1;

Do[

PLOTOLD = PLOTNEW;

(* Calculate Trace Of Each Intersection of a {111} plane w/ surface
*)

TRACE =
LU[[i]] - (Cos[phi] Sin[
phi] (Cross[NORM,ROT].LU[[i]]) NORM) - (Sin[phi])“2
(Cross[NORM,
ROT].LU[[i]]) Cross[NORM,ROT];
TRACENORM = unit[TRACE];

(* Calculate trace of tensile axis *)

198

TENTRACE =
TENS - (Cos[phi] Sin[
phi] (Cross[NORM,ROT].TENS) NORM) - (Sin[phi])“2
(Cross[NORM,
ROT].TENS) Cross[NORM,ROT];

(* Calculate cos“-l of dot product between positive rotation axis
and \
line direction L[[i]] *)

theta = ArcCos[ROT.TRACENORM];

(* Check to see if line direction TRACENORM is in quadrants 1-2 or
\

quadrants 3-4;
A value > 0 indicates that TRACENORM is in upper two quadrants

**)
checkvalue = TRACENORM.TENTRACE;

If[checkvalue \[GreaterEqual] 0,

direction = {Cos[theta],Sin[theta]};

direction = {-Cos[theta],Sin[theta]};
];
slope = direction[[2]]/direction[[1]];
newangle = ArcTan[direction[[2]]/direction[[l]]]/Degree;
If[newangle<0, angle = newangle + Pi/Degree, angle = newangle];
newanglelist = Append[anglelist,angle];
anglelist = newanglelist;

P[x_]:=slope x;

lwrlimit -l/Sqrt[1+slope“2];

uprlimit l/Sqrt[l+slope“2];

PLT=Plot[P[x],{x,lwrlimit,uprlimit},DisplayFunction \[Rule]
Identity,
PlotStyle\[Rule]{GrayLevel[(i—l)/SIZE], Thickness[0.01]}];
PLOTNEW = Append[PLOTOLD,PLT];
r{iIIISIZE}];

Show[PLOTNEW,DisplayFunction\[Rule]$DisplayFunction,AspectRatio \[Rule]
12

199

PlotRange\[Rule]{-l,1}];
Print[TableForm[anglelist]];

(* Order of shading is also order of crystallographic planes in the
matrix \
"M" above,

evolving from darker shading to lighter shading.

The angles listed below are given in the same order,

and indicate the angle that the line makes w/ the positive x-axis,

measured counterclockwise from that axis. Therefore,

they are given as numbers between zero and Pi. *)

GraphicsData["PostScript", "\<\

% End of Graphics
MathPictureEnd
\
\>"]
\!\(\*
InterpretationBox[GridBox[{
{"136.8674086569651‘"},
{"61.85578208388911‘"},
{"171.8296228125404‘"},
{"107.79361094154851‘"}
)2
RowSpacings->1,
ColumnSpacings->3,
RowAlignments—>Baseline,
ColumnAlignments->{Left}],
TableForm[ {136.8674086569651, 61.85578208388911,
171.82962281254041,
107.79361094154851}]]\)

200

Appendix 4
Using other variables in the Fracture Propagation Parameter

201

Fracture Propagation Parameter:

A A 2 A A 4 A A
F...=m...lb..:tl[B§..lb...° b...) +2 b ~12...) M]

Btw = 1 Atw

1. When the second sum is such that only the absolute values are used:

4 A A
Bug 1 bAlw . bBlw)

 

 

 

F
1

1

1

1 8

1 a 1
1 '6 g 6 1
1 3 E 4 1
1 S g 2 1
1 '= o ’
} 0

‘ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 J
} fracture propagation parameter

1 1in&a&?22i-2112@<01

 

Figure A4-1. Histogram showing intact and cracked boundaries using F 4.); values
computed using absolute values for the interaction between two twinning vectors.

Table A4-l: t-tost for population means using the absolute values for the interaction
between two twinning vectors

 

202

Table A4.2: Selected F M; values

(Using absolute values for the computation between twinning vectors)

 

Absolute
Values

Selected

0=lntact

 

Graln1

Gralnz

FA-B value

1=craek

 

A

0.55

 

0.84

 

BB

0.84

 

0.5

 

1.22

 

0.61

 

0.61

 

0.51

 

1.27

 

1 .278

 

1 .43

 

1.39

 

1.33

 

1 .43

 

1.41

 

1.41

 

1 .47

 

1.2

 

QE-mmmzzooooocg

1.24

 

'I'l
X

1 .49

 

1 .43

 

1.35

 

1.31

 

0.712

 

1.4

 

1.13

 

0.77

 

1.04

 

0.51

 

0.75

 

1.24

 

0.75

 

0.47

 

1.45

 

1.37

 

0.75

 

06

 

1.29

 

1.44

 

0.81

 

NN-b—tngmgmmmgf'XLc—XEXI

 

 

mac.moan—samg‘fiNNSI-r-ggac-Ec—xflg‘x‘m—L—'nI—Urnmrnfimo

 

0.67

 

DOA-bood-EOO-EOOOOO40040-30édOOdOAOO-EOOOOO-‘OO

 

203

 

2. When the second sum is such that only the negatives values were used and the positive
values were set to zero:

number of

observations
0 _. N 00 113- UI O) \l

4 A A
Bugletw.bBtwl If<0
, , WWW , , _ _ W ,WWW W
Intact: 0.43+l- 0.19 ‘
Cracked: 0.70+I-0.17
170113013)‘ ‘

ragged <12

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1
Fracture Propagation Parameter

Figure A4-2. Histogram showing intactanEngdboEdan‘es—u'smg—FAJ values

computed using “positive values set = zero” for the interaction between two twinning

vectors.

Table A4-3: t-test for population means using the “positive values set = zero” values
for the interaction between two twinning vectors

 

204

Table A4z4: Selected F 4.3 values (where any “positive values were set to zero
values” for the computation between twinning vectors)

 

Positive=Ze
Values Selected 0=lntact
Graln1 Gralnz 'FA-B value: 1=eraek
BB 0.407
0.755
0.841
0.464
1.174
0.593
0.576
0.476
1.309
1.278
0.312
1.31
1.26
1.43
1.09
0.983
1.28
1.399
0.986
1.08
1.066
1.311
1.084
0.712
1.153
1.09
0.709
0.699
0.468
0.72
1.19
0.44
0.47
0.87
1.13
0.54
0.32
1.491
1.38
1.07
1.45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QE—momxroooooofifiggm>

 

.n
X

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0:0: I 'n11-n _ _ _ tn
m4:Alcoa»44mm>NNSr1—;;LLL¢—xxxx‘n c. 11:: Ommmmmo
GOA—8004400-8OOOOO400-8O—hO-A—hOO—tO—EOO-AOOOOO-KOO

NN-t—Ingmgcnmmgl-XLBXEXI

 

 

 

 

 

 

205

   

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