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Mechanical Twin Nucleation, Propagation And
Mechanical Twinning During Creep Deformation In TiAl

presented by

Zhe Jin

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MECHANICAL TWIN NUCLEATION, PROPAGATION
AND MECHANICAL TWINNING DURING

CREEP DEFORMATION IN TiAl

By

Zhe Jin

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

1994

ABSTRACT
MECHANICAL TWIN NUCLEATION, PROPAGATION
AND MECHANICAL TWINNING DURING CREEP DEFORMATION IN TiAl
By

Zhe Jin

Gamma titanium aluminide (TiAl) is a promising candidate material for high
temperature and high performance applications because of its high strength, low density,
good oxidation resistance, and excellent high temperature mechanical properties. In
recent years, much work has been done in order to understand its deformation
mechanisms at room temperature and high temperatures. The experimental results have
shown that mechanical twinning is an important deformation mechanism in TiAl both at
room temperature and at high temperature. Even under creep deformation conditions the
mechanical twinning still has a significant contribution to the creep deformation.
Therefore, in order to understand the mechanical twinning behavior during creep
deformation of TiAl, mechanical twin nucleation and propagation mechanisms and the

mechanical twinning contribution to the creep deformation in TiAl were studied.

Mechanical twin nucleation and propagation were in situ observed in creep
deformed specimens using electron beam illumination in TEM. The mechanical twins
nucleated either at grain boundaries or at twin interfaces due to the local stress
concentration. The nucleus was either a superlattice intrinsic stacking fault that was a
result of emission of one twinning dislocation from the grain boundary or the twin

interface, or an extrinsic superlattice stacking fault resulting from the emission of two

twinning dislocations. The twinning dislocation was identified to be Shockley partial
1/6< 115] and the twinning plane was identified as (111). The twinning propagation
mechanism was a homogeneous glide of twinning dislocations on every adjacent twinning

plane.

The stress state necessary for twin propagation is that the principal tensile stress
axis must be orientated in the vicinity of [SST] crystal orientation. The stresses in the
twin layers are classified into three categories: forward stress, back stress and external
stress, (the definition of each stress is referred to section 4.4). A stress analysis on a
thin twin layer shows that both the forward stress and back stress are very large at the
twin tip, but they decrease as the distance from the twin tip increases. At very large
distances from the twin tip, both stresses tend to be constant. The external stress is

constant along the twin layer investigated.

Both true-twinning and pseudo-twinning configurations are observed in creep
deformed specimens. The analyses of mechanical twin (true-twin and pseudo-twin)
formation during creep of TiAl show that mechanical twinning in TiAl obeys the

maximum resolved shear stress criterion.

Stress concentrations at grain triple points are found to be accommodated by
formation of fine mechanical twins. These fine mechanical twins result in zigzagged
grain boundaries at their intersections with the grain boundaries and hence inhibit the
grain boundary sliding. The formation of fine mechanical twins are probably controlled

more by the local stress concentration than by an externally applied stress.

Copyright by

ZHEJIN

1994

To My Parents

Zhong-Zhi J in and Yu-J in Cui

ACKNOWLEDGENIENTS

I sincerely appreciate the help of my major professor, Dr. Thomas R. Bieler. He
has been a good friend and has provided me with an effective and encouraging

environment to finish this research.

I am also very grateful to my graduate committee, Prof. Martin A. Crimp, Prof.
David Grummon, Department of Materials Science and Mechanics, and Prof. Carl L.
Foiles of Physics and Astronomy Department. They have provided many effective
suggestions in selecting the research topic and the facilities necessary to pursue this

research.

Particularly, I would like to express my sincere appreciation to Howmet
Corporation, Whitehall, Michigan for the financial and material support to make this
research possible. I would like to express my thanks to Dr. B. London, Dr. T. Thom,

Dr. NE Paton, Dr. D.A. Weeler for their contributions in pursuing this research.

I would also like to express my gratitude to my wife, Jiyan An, for her patient

support and encouragement during my entire graduate study.

vi

TABLE OF CONTENTS

LIST OF TABLES ........................................................................ xiv

LIST OF FIGURES ...................................................................... xvi
CHAPTER ONE. INTRODUCTION ................................................. 1
CHAPTER TWO. THEORY ........................................................... 7
2.1. Titanium Aluminide TiAl ......................................................... 7
2.1.1. Crystal Structure ........................................................ 7
2.1.2. Stacking Faults .......................................................... 9

(a) APB (Antiphase Boundary) ........................................ 11

(b) SISF (Superlattice Intrinsic Stacking Fault) .................... 13

(c) CSF (Complex Stacking Fault) ................................... 13

(d) SESF (Superlattice Extrinsic Stacking Fault) .................. 14

2.1.3. Dislocations and Dislocation Core Structures ...................... 14

(a) 1/2 <110] Type ..................................................... 15

(b) <101] Type ......................................................... l6

vii

(c) 1/2 <112] Type .................................................... 16

(d) <100> Type ....................................................... 17

2.1.4. Dislocation Blocking Mechanisms ................................... 17
(a) Roof-Type Blocking ................................................ 18

(b) Kear-Wilsdorf Blocking ........................................... 20

(0) Deep Peierls Valley Blocking .................................... 21

(d) SSE—Tube Type Blocking .......................................... 22

(e) Dislocation Interaction Blocking ................................. 26

2.1.5. Lamellar Structure and Its Formation ............................... 28
(a) Type-I Lamellar Structure ......................................... 28

(b) Type-II Lamellar Structure ........................................ 29

2.1.6. Phase Transformation Mechanisms .................................. 31
2.2. Mechanical Twinning Theory .................................................... 33
2.2.1. Definition ................................................................. 33
2.2.2. Seven Twinning Classes ............................................... 35
2.2.3. Classic Twinning Modes ............................................... 37
2.2.4. Crystallography of Mechanical Twinning ........................... 38
2.2.5. Atomic Movement in Twinning Shear .............................. 45
2.2.6. Strain in Twinning ...................................................... 48
2.2.7. Prediction of Twinning Elements (Twinning Criterion) .......... 52
2.2.8. Twinning in Superlattice ............................................... 53

viii

2.2.9. Mechanical Twin Nucleation and Propagation ..................... 61

(a) Dislocation Pole Mechanism in bcc Metals ................... 62
(b) Dislocation Pole Mechanism in hcp Metals ................... 63
(c) Dislocation Pole Mechanism in fcc Metals .................... 66
2.2.10. Twin Shape ............................................................. 68

(a) A Model Concerning Energy Associated
With Twinning .................................................. 68

(b) A Model Based on Twinning Dislocation

Interaction Forces ............................................... 72

2.3. Creep Theory ........................................................................ 75
2.3.1. Microstructural Correspondences to Creep ......................... 76

2.3.2. Mechanical Model ...................................................... 77
2.3.3. Microstructural Model ................................................. 80

CHAPTER THREE. MECHANICAL TWIN NUCLEATION

AND PROPAGATION IN TiAl ............................... 82
3.1. Materials and Experimental Procedure ......................................... 83
3.2. Results and Analyses ............................................................... 85

3.2.1. In Situ Observations of Mechanical Twin

Nucleation and Propagation ........................................... 85
(a) Twin Nucleation at Grain Boundaries .......................... 85
(b) Twin Nucleation at Twin Interfacies ........................... 100
(c) Twin Layer Morphology Near The Twin Tip ................. 107

3.2.2. Post-Mortem Observation of Mechanical Twinning

During Creep Deformation ........................................... 107

3.3. Discussion ........................................................................... 113
3.3.1. Mechanical Twin Nucleation in TiAl ................................ 113

3.3.2. Mechanical Twinning Mechanisms in TiAl ......................... 116

3.3.3. On The DO19 phase in Fine Mechanical Twins .................... 121

3.3.4. Twin Morphology ....................................................... 122

3.4. Summary ............................................................................. 123

CHAPTER FOUR. FORCE AND STRESS ANALYSES

IN A THIN TWIN LAYER ..................................... 126

4.1. Theory ................................................................................ 127
4.1.1. Forces on Each Twinning Dislocation ............................... 127

4.1.2. Dislocation Interaction Forces ........................................ 131

4.1.3. Internal Friction Force ................................................. 134
4.1.4. The Force Due to The Stacking Faults ............................. 136
4.1.5. Applied Force ........................................................... 137

4.1.6. Definitions of External Force, Forward Force

and Backward Force ................................................... 138

4.2. Twinning Dislocation Distribution Within A Thin Twin Layer ............ 139
4.3. Force Calculation .................................................................. 144
4.3.1. Calculation of Forward Force Ff ..................................... 144

4.3.2. Calculation of Backward Force Fb ................................... 148

4.3.3. Calculation of External Force Fa .................................... 155

4.4. Stresses in The Thin Twin Layer ................................................ 157
4.5. Simplified Equations for Ff, F, and F“ ......................................... 162
4.6. Discussion ........................................................................... 168
4.6.1. Stress and Force Distributions Within The Twin layer .......... 168

(a) External Stress 1,, ................................................... 168

(b) Forward Stress r, .................................................... 169

(0) Back Stress 1,, ........................................................ 170

4.6.2. Dislocation Distribution in The Case of Equal

Backward Force on Each Twinning Dislocation ................... 175
4.6.3. Effect of Dislocation Location on The Stress Distribution ....... 176
4.7. Summary ............................................................................. 178

CHAPTER FIVE. MECHANICAL TWINNING DURING

CREEP DEFORMATION IN TiAl ......................... 183

5.1. Introduction .......................................................................... 183
5.2. Material and Experimental Procedure .......................................... 184
5.3. Results ................................................................................ 186
5.3.1. Optical Microstructure ................................................. 186

5.3.2. Cross Twinning and Parallel Twinning in Lamellar Grains ..... 187

5.3.3. Fine Mechanical Wins at Grain Triple Points .................... 190

5.4. Analysis and Discussion ........................................................... 194
5.4.1. Cross Twinning in Lamellar Grains ................................. 194

5.4.2. Cross Twinning in Equiaxed ‘y Grains .............................. 202

5.4.3. Fine Mechanical Twins at Equiaxed 7 Grain Triple Points ..... 203

xii

(a) Fine Mechanical Twins Formed by Accommodation
of Stress Concentration at Equiaxed 7 Grain
Triple Points ........................................................ 203

(b) Formation of Fine Mechanical Twins Prevents

Grain Boundary Sliding .......................................... 204
(c) On DO“, Crystal Structure in Fine Mechanical Twins ...... 207
5.5. Conclusions .......................................................................... 209

CHAPTER SIX. CONCLUDING REMARKS AND

RECOMMENDED FUTURE WORK ......................... 211

LIST OF REFERENCES ................................................................ 214

xiii

Table 2.1.
Table 2.2.
Table 2.3.
Table 2.4.
Table 2.5.
Table 2.6.
Table 2.7.
Table 3.1.
Table 3.2.

Table 3.3

Table 4.1.
Table 4.2
Table 4.3
Table 4.4

Table 4.5

Table 5.1.

Table 5.2

LIST OF TABLES

Dislocation behavior in TiAl at different temperature ranges ...... 18
Twinning modes in cubic lattices ........................................ 36
Twinning modes in cubic lattices ........................................ 58
Possible true twinning modes in cubic superlattices .................. 60
Twinning modes in non-cubic superlattices ............................ 61
Twinning modes in hcp metals ........................................... 66
Creep Deformation Mechanisms At Temperature T/Tm 2 0.4 79

Composition of the 'y TiAl specimen ................................... 83
The mechanical twinning elements in Fig. 3.1 ........................ 93
Schmid factors on all possible twinning systems within

the (111) plane ........................................................... 93
The distance of twinning dislocations from the twin tip ............. 143
The forces on each twinning dislocation ............................... 149
The stresses at each twinning dislocation .............................. 160
The forces calculated using simplified equations ..................... 166

The modified stresses and forces for the first three
twinning dislocations .................................................... 171

Schmid Factors for Mechanical Twinning Systems
in 7/7 Lamellae .......................................................... 199

Misorientation and Grain Boundary Structure of Grains at the
Grain Triple Point ....................................................... 205

xiv

Figure 2.1 -
Figure 2.2 -

Figure 2.3 -

Figure 2.4 -

Figure 2.5 -

Figure 2.6 -

Figure 2.7 -

Figure 2.8 -

Figure 2.9 -

Figure 2.10 -

Figure 2.11 -

Figure 2.12 -

LIST OF FIGURES

Unit cell of superlattice L10 .................................................... 7

Phase diagram of TiAl .......................................................... 8

Stacking fault configurations as viewed along [1T0]. (a) APB,

(b) SISF, (c) CSF, (d) SESF(I), and (e) SESF(II) ......................... 9

Crystal projection as viewed along [111]. bA is a crystal vector
which results in an APB, bC results in a CSF, and bS results

in a SISF ........................................................................... 12
Thompson tetrahedron. The primed letters indicate the antisites

with respect to the unprimed letters ........................................... 12
Dislocations in superlattice Ll0 ................................................. 15
Glissile configurations of superdislocations. The single line is

a SSF, the jagged line is an APB, and the double line is a CSF ........ 19
Roof-type blocking configurations. The single line is a SSF,

the jagged line is an APB, and the double line is a CSF .............. 19
Kear-Wilsdorf blocking configurations. The dashed line is the

APB in the {001} plane, and the single line is the SSF in the

{1 1 1} plane ........................................................................ 20
The Peierls relief along different crystallographic directions

in TiAl ............................................................................. 22
Schematic diagram of the row of broken Ti-Ti bonds .................... 23
Formation of SSF—tube blocking. (a) Formation of nonaligned

jogs on a superdislocatiog; (b,c) formation of a dipole;

(d) a SSF—tube along [101] direction .......................................... 24

XV

Figure 2.13 -
Figure 2.14 -
Figure 2.15 -
Figure 2.16 -

Figure 2.17 -

Figure 2.18 -

Figure 2.19 -

Figure 2.20 -

Figure 2.21 -

Figure 2.22 -

Figure 2.23 -

Figure 3.1 -

Figure 3.2 -

Definition of orientation variants in (‘y + 02) lamellae ..................... 30
Relations between ci and pi .................................................... 39
The sign of four twinning elements .......................................... 41
The rationality of K1 in type-I twinning ..................................... 42
Schematic illustration of lattice shuffling during type-I twinning

for the of q = 4 .................................................................. 46
lattice point displacement during twinning shear ......................... 49
The extension and contraction of sample length

for twinning in zinc .............................................................. 51
The structure and interchange shuffles during (120) twinning

in D0, .............................................................................. 5 6
Dislocation pole mechanism for twinning in bcc lattice .................. 64
Q02) twinning in zirconium. Projection of the lattice on the

(1210) plane. Circles are in plane of the paper. Squares are
a/2 above and below the paper. Solid symbols indicate atom
positions in the twin ............................................................. 65

Dislocation pole mechanism in fee lattice ................................... 67

Mechanical twinning propagation configuration which was

observed in situ in TiAl: (a) the tilted image of fine mechanical

twins within a grain interior; (b) the image of fine mechanical

twins viewed parallel to the twinning plane; (c) the diffraction

pattern across several fine twin layers; and (d) indexes of the

diffraction pattern which shows a twin relationship across the

twin-matrix interfaces ........................................................... 86

Schematic diagram of mechanical twinning nucleation and
propagation procedure in TiAl, which shows that the mechanical
twinning nucleates at a grain boundary and propagates into the
grain interior. (a) and (b) are the configuration of mechanical
twinning nucleation at the grain boundary with the emission of
one twinning dislocation from the grain boundary; (c) and (d) are
the same configuration as (a) and (b) except that two twinning
dislocations are emitted from the grain boundary simultaneously;

xvi

Figure 3.3 -

Figure 3.4 -

Figure 3.5 —

Figure 3.6 -

Figure 3.7 -

Figure 3.8 -

Figure 3.9 -

(e) is an intermediate state of twinning propagation; and (f) is
the final state in which the twinning propagation ceases in the
grain interior as twin layers "A" and "B" in Fig. 4.1 (a) ................ 89

Schematic diagram of the incoherent twin boundary structure in

terms of individual twinning dislocations, which is viewed in

[110] direction. The boundary between the first dislocation and

the second dislocation at the twin tip is an intrinsic stacking fault,

the boundary between the second and the third dislocations is an

extrinsic stacking fault, and the boundary between any two adjacent
dislocations after the third dislocation is a twin plane. The

twinning dislocations are denoted by numbers: the leading

dislocation at the twin tip is counted as the first dislocation, the

one just behind it is the second one, and so on ............................ 92

Crystal orientation relationships of the twinning elements and
the possible twinning directions in (111) plane. The cross mark
indicates the direction of creep tensile axis ................................. 94

Schmid firctor contours with respect to the twinning system of

(111)[1 12], which indicates that a tensile principal stress should

be in the vicinity of [551] crystal direction in order to operate

the (111)[112] twinning system ................................................ 96

Mechanical twinning propagation mechanism in TiAl, which
shows the twin layers resulting from the glide of (a) one, (b) two,
(0) three, and ((1) four twinning dislocations in the (111) twinning
plane during the twinning propagation. The left side column is
the glide sequence of l/6[112] twinning dislocations. The middle
cglumn is the atomic arrangements of resultant twins viewed in
[110] direction, in which the open circles indicate the atoms
located in the plane of the paper and the shaded squares indicate
the atoms in the plane just beneath the paper. The right side
column is the atomic stacking sequences along [111] direction
before and after twinning, in which the primed letters indicate

the sheared planes ................................................................ 97
The sequences of mechanical twinning nucleation and propagation

in TiAl ............................................................................. 101
(a) The side view of twin layers A, B, C, and D in Figure 4.7,

and (b) the diffraction pattern taken across these twin layers ............ 104
The thin twin morphology near the twin tip ................................ 108

xvii

Figure 3.10 -

Figure 3.11 -

Figure 4.1 -

Figure 4.2 -

Figure 4.3 -

Figure 4.4 -

Figure 4.5 -

Figure 4.6 -

The configuration of fine mechanical twins resulting from the
accommodation of local stress concentration at a grain triple

point (a). Diffraction pattern across the interfaces (b) shows

that these fine laths are twin-related with matrix and the existence
of fine DO19 phase between them (0). Diffraction pattern taken in
untwinned area within the same grain without tilting the sample
((1) shows that the untwinned region has the same crystal

orientation as the matrix of fine mechanical twins as shown in (b).

(e) is the indexes of (d). It also shows that the growth of fine
mechanical twins as pointed out with an arrow between the
completely developed fine twins and the matrix stopped in the

grain interior ....................................................................

(a) shows three thin twin layers indicated by letters "A", "B"
and "C" within a y grain interior, which indicates that the
nuclei of these twin layers were formed inside the 7 grain;

(b) is a diffraction pattern taken across the twin layer ...................

Forces on individual twinning dislocation in the thin twin layer.
(a) On the first twinning dislocation, (b) on the second one,

(c) on the third, and (d) on the fourth ......................................

(a) Variation of Peierls energy as a function of transverse
displacement (u), (b) Variation of the lattice friction stress.

(w, :5 b2) .....................................................................

(a) is the amplified image of a thin twin layer and (b) is a
schematic drawing of the twinning dislocation distribution in

the thin twin layer .............................................................
The distance of twinning dislocations from the twin tip .................
The forces distribution along the twin layer ...............................

Shear stresses distribution in the thin twin layer ..........................

Figure 4.7 - Comparisons of results obtained from the original equations with

Figure 4.8 - The modified stress distribution along the thin twin layer ...............

Figure 4.9 - The modified force distribution along the thin twin layer ................

results obtained from the simplified equations ............................

xviii

.. 110

119

.. 129

.. 135

.. 141

145

.. 150

. 161

.. 167

172

173

Figure 4.10 -

Figure 4.11 -

Figure 5.1 -

Figure 5.2 -

Figure 5.3 -

Figure 5.4 -

Figure 5.5 -

Figure 5.6 -

Figure 5.7 -

Figure 5.8 -

Figure 5.9 -

Figure 5.10 —

Figure 5.11 -

Dislocation distribution at equal backward force on every
twinning dislocation. (Fb/Gb = 1.18 x 10'2) ................................ 177

Effect of the twinning dislocation location on the backward stress.
(a) For the second dislocation, (b) for the third dislocation, and
(c) for the tenth dislocation .................................................... 179

Optical micrographs of (a) initial microstructure and (b) after
creep deformation ................................................................ 186

(a) Cross twinning configuration in a lamellar grain, (b) diffraction
pattern of the original lamellae, and (c) orientations of the original

lamellae ............................................................................ 188
Cross twinning configuration within a large equiaxed 7 grain .......... 191
Fine mechanical twin configurations at equiaxed 7 grain triple

points ............................................................................... 192
Fine mechanical twins (a), diffraction pattern across several fine

twin interfaces (b), and twin relation in the diffraction pattern (0) ..... 193
Atomic arrangement of true-twinning (a) and pseudo-twinning (b) ..... 196

Stereographic projection of possible twinning systems along
[110] direction .................................................................... 198

Schematic mode of cross twinning: (a) initial condition of two
lamellae; (b) metastable condition of twinning in lamellae 1;
(c) final configuration of cross twins as seen in Fig. 5 .2 (a) ............ 200

Fine mechanical twins from accommodation of local stress

concentration (a), twin end configuration at equiaxed 7 grain

boundary (b), and dislocation slip near zigzagged boundary (c) ........ 206
Boundary between an equiaxed 7 grain and a lamellar grain ........... 208

Optical microstructure showing the roughness of lamellar grain
boundaries ......................................................................... 208

CHAPTER ONE

INTRODUCTION

Gamma titanium aluminide (TiAl) is a promising candidate material in high
temperature and high performance applications for its high strength, low density, good
oxidation resistance, and excellent high temperature mechanical properties. The main
barrier for application in practice is its brittleness at room temperature “'3'. In the last
decade, most of the research effort in TiAl has been concerned with fundamental studies
of dislocation structures in single phase materials [“0'. More recent studies on Ti-rich
alloys consisting of two phases (7+a2) which present better room temperature ductility
have also been focused mainly on the dislocation structures “”6'. However, experimental
results have shown that mechanical twinning is an important deformation mechanism in
TiAl both at room and high temperatures “7'2”. Particularly, under creep deformation

conditions, mechanical twinning contributes significantly to the creep deformation “’23‘3”.

Titanium aluminide TiAl has a L10 ordered face centered tetragonal structure.

The mechanical twins observed are of the {111} < 112]' type which are the variants in

 

"‘ The notations " < " and " > " in the indexes indicate that the two indexes nearest to " < " or
" > " are permutable one to another without changing its properties.

2

which the L10 superlattices are mechanically twinned without disturbing the L1o crystal
structure. It has been reported that when the L10 structure is homogeneously sheared by
the variants other than {111} < 112] type, the L10 structure will be changed into a L1l
superlattice structure “1. Based on this concept, the twinning operation by the variants
other than {111}<112] is thought to be forbidden ”'3‘. The details of mechanical
twinning behavior and its contribution to the deformation in TiAl are not completely

understood at present.

No fundamental investigations on mechanical twinning in TiAl have been carried
out until the very recent work done by Farenc, Coujou and Couret [32‘ on twin
propagation in TiAl, by Wardle, Phan and Hug ‘33] and by Sun, Hazzledine and Christian
[3‘5" on twin intersections. Farenc, Coujou and Couret first investigated in situ
mechanical twin propagation at room temperature and at 400 °C. They found that " twins
are formed by a/6<112] partial dislocations gliding in octahedral planes" and the twin
propagation is due to "a partial dislocation turning around a perfect dislocation".
However, the material used in their study was an aluminum rich Ti..,,Als4 single 7 phase
alloy. The basic properties of Al-rich alloys (single 7 phase) are inherent brittleness at
room temperature “‘11", but the 7 phase in the 7+0:2 lamellar structure has been found
to be intrinsically ductile ‘3‘”. Therefore, the mechanical twinning behavior and its role
in the deformation in Al-rich single phase alloys may differ from those in Ti-rich two
phase alloys. This is because the stacking fault energys of 7 phase in Al-rich single 7
phase and in Ti-rich 7+0:2 lamellar grain are different [3°]. The twin intersection has

been comprehensively studied by Wardle, Phan and Hug [33‘ for both Al-rich and Ti-rich

3

alloys at room temperature and by Sun, Hazzledine and Christian [34’3“ for an alloy of
stoichiometric composition (50Ti - 50A] at. %) in the temperature range -196 °C to 900
°C, respectively. In Wardle, Phan and Hug’s work ‘33', two different twin intersections
were observed and it was found that the structures of twin intersections depended on "the
orientation of the common direction of the twin habit planes". Similarly, in Sun,
Hazzledine and Christian’s work ”4”“, two types of twin intersections have been found:
type I intersection occurs along < 110] directions while type II intersection occurs along
<011] directions. In the type I intersection, two different configurations are observed
depending on the deformation temperatures: (i) At room temperature, one twin band is
deflected and the other remains rectilinear. The lattice in the intersection region remains
in L10 structure and is congruent with the lattice of the deflected twin; (ii) at high
temperature, both twin bands are deflected. In type II intersection, both twin bands are
deflected at both room and high temperatures. Type II intersection has been found to be

associated with 'A < 110] dislocations in the twin-matrix interface.

However, all these fundamental studies on mechanical twinning have been carried
out on the specimens deformed at common tensile testing strain rates, for example, the
strain rate in [34] was 5 x 105 s", which is much faster than normal creep strain rates.

Therefore, these observations may not be directly applicable to creep deformation.

For mechanical twinning in creep deformation, very limited research has been
reported [22. 2”“37'39'. Also, there are two differing results concerning twinning in creep

deformation of TiAl. Loiseau and Lasalmonie [22' investigated creep deformation of

4

equiaxed single phase 7 Ti46A154 and found that twinning was an important creep
deformation mechanism at deformation temperatures up to 800 °C. 1 in and Bieler [28'3”
have found that mechanical twinning is significant in creep deformation at 765 °C for Ti-
rich alloy (Ti-48Al—2Nb-2Cr). However, Huang and Kim [39‘ studied creep behavior of
two phase 7+a2 alloy with composition of Ti-47.0Al-1.0Cr—1.0V-2.5Nb at 900 °C and
observed no evidence of twinning in creep deformation. Pseudo-twinning, which was
thought to be "forbidden" in the literature ”5"“ and it will be discussed later, was found

in a creep deformed TiAl specimen '2‘”.

In addition, the creep deformation mechanisms are not clear at present. The
activation energies for creep of TiAl are much larger than those for self-diffusion and
interdiffusion in TiAl [39:41:42], which indicate that the creep rate may be controlled by
some processes other than diffusion. The reported values of the stress exponents vary
widely from about 2 to 8 [434"]. This indicates that several deformation mechanisms may
be involved in creep deformation of TiAl. Mechanical twinning has been found to be
an important creep deformation mechanism [22. "*3" 37'3”, but mechanical twinning is
influenced very little by diffusion processes, and dislocation configurations in creep
specimens are similar to those observed in short term tensile specimens “'9‘. All these
indicate that mechanical twinning may play an important role in the creep deformation

of TiAl.

Based on the above circumstances, investigations of twin nucleation and

propagation in creep deformed TiAl and the contributions of mechanical twinning to

5
creep deformation of TiAl have been carried out in this study. At first, the basic

understanding of TiAl, the intensive twinning theory and a brief introduction of creep
theory has been summarized in Chapter Two. In Chapter Three, in situ observations of
mechanical twin nucleation and propagation in a creep deformed specimen are presented.
The mechanical twinning was observed to nucleate either at grain boundaries or at twin
interfaces due to the local stress concentration. The nucleus was either a superlattice
intrinsic stacking fault which resulted from the emission of one twinning dislocation from
the grain boundaries or the twin interfaces, or a superlattice extrinsic stacking fault
resulting from the emission of two twinning dislocations. The twinning dislocation was
identified to be Shockley partial 1/6 <11’2’] and twinning plane was identified as (111).
The twinning propagation mechanism was a homogeneous glide of twinning dislocation
on every adjacent twinning plane. The stress state necessary for twinning propagation
is that the principal stress axis must be orientated in the vicinity of [SST] crystal

orientation in the case of tensile stress state.

The force and stress analysis on a thin twin layer is presented in Chapter Four.
The result shows that both the forward stress and the back stress are very large at the
twin tip and decrease quickly as the distance from the twin tip increases. At very large
distances from the twin tip, both stresses tend to be constant. The external stress on the
investigated twin layer was found to be a residual stress in the matrix. In the case of no
external load and small residual stress, the twin propagation is controlled by the emission

of twinning dislocations.

6

In Chapter Five, some characterizations of mechanical twins in creep deformed
specimens are presented. The results show that mechanical twins are formed by either
true-twinning or pseudo-twinning. The mechanical twinning in creep deformation of
TiAl obeys the maximum resolved shear stress criterion. Stress concentrations at grain
triple points can be accommodated by forming fine mechanical twins. The zigzagged
grain boundary formed by mechanical twinning can inhibit grain boundary sliding. The
formation of fine mechanical twins are probably controlled more by the local stress

concentration than by an externally applied stress.

In the last chapter, Chapter Six, some general concluding remarks and some

recommended future work are stated.

CHAPTER TWO

THEORY

Some basic understanding of titanium aluminide TiAl, mechanical twinning theory

and creep theory will be summarized in this chapter.

2.1. Titanium Aluminide TiAl

2.1.1. Crystal Structure
Titanium aluminide TiAl has a Llo ordered face centered tetragonal structure with

a composition range of 49-66 atomic percent of aluminum, which varies depending on

[47491

temperature . The Titanium and aluminum atoms alternately stack in (002) plane,

 

 

 

 

 

 

 

 

o f
1 b I

———-—— - —___..__.

Figure 2.1 — Unit cell of superlattice L10.

8

as shown in Fig. 2.1. At the stoichiometric composition, the c/a ratio is 1.02 and the
tetragonality increases up to c/a= 1.03 with increasing aluminum concentration and
decreases to 1.01 with decreasing aluminum concentration [5°52]. At off stoichiometric
composition, excess titanium or aluminum atoms occupy antisites without creating
vacancies ”3‘. The TiAl phase remains ordered up to its melting temperature of about

1450 °C as shown in Fig. 2.2 ‘49], which shows a central portion of the equilibrium Ti-Al

 

 

 

 

 

phase diagram.
Weight Percent of A1
20 25 30 35 40 45 50
I l l l l I l
1600 ". L _ 2900
; f3 1400 — [3 E?
' Q, - 2500 L
l g 01 - 2300 E
8 1200 - a
8, / \ 2100 8.
E E
O)
E2 1000 _ 02 1900 H
_ 1700
800 — 1500
l 1 l l
1300
30 40 50

 

Atomic Percent of Al

Figure 2.2 - Phase diagram of TiAl.

2.1.2. Stacking Faults

Since TiAl has L10 structure with ratio c/a=1.02, it is very close to the fcc

structure. Like the fcc crystal, the geometric stacking sequence along <111> direction

is
ABCABCABCABC

There exist four types of stacking faults in the TiAl crystal structure, i.e., they are
antiphase boundary (APB), superlattice intrinsic stacking fault (SISF), complex stacking
fault (CSF) and superlattice extrinsic stacking fault (SESF). The atomic configurations

of these stacking faults are shown in Fig. 2.3, which are viewed along [1T0] direction.

[111]

 

—A'
-B'
-C'

l
u>nu

 

Figure 2.3 - Stacking fault configurations as viewed along [1T0]. (a) APB,
(b) SISF, (c) CSF, (d) SESF(I), and (e) SESF(II).

10

 

(Figure 2.3 continued)

11

Ill
can:

 

I
>nw>

(Figure 2.3 continued)

(a) APB (Antiphase Boundary)

The APB is created by the glide of a slip vector D’A (or CA), which is shown
in Fig. 2.4 by a crystal vector bA. Fig. 2.4 shows an atomic arrangement in a (111)
crystal plane where the stacking sequence is labeled with a, b, c. The slip vector D’A
(or CA) is shown in Thompson tetrahedron in Fig. 2.5. In this case the stacking

sequence across the APB changes into the following one:

ABCABCIA’B’ C’ A’B’ C’

where the primed letters indicate the displaced planes with respect to the unsheared

crystal that is indicated by letters without primes (it will be the same in the following).

12

[0“] [112] [101]

 

1 [1511

 

 

 

—->- [110]

Figure 2.4 - Crystal projection as viewed along [111]. bA is a crystal
vector which results in an APB, bC results in a CSF,
and b3 results in a SISF.

 

 

 

 

 

 

 

 

 

Dr
L__
Figure 2.5 - Thompson tetrahedron. The primed letters indicate the antisites with respect
to the unprimed letters.

13

It is more visible if one looks at the shear displacement in the [ITO] direction, which is
shown in Fig. 2.3. In Fig. 2.3 (a), one can easily find the antiphase boundary

configuration which lies in the (111) plane.

(b) SISF (Superlattice Intrinsic Stacking Fault)

The SISF is very common in TiAl. The SISF is formed by the glide of a slip
vector 3A in Fig. 2.5 or by a crystal vector bS in Fig. 2.4. If one looks at the stacking
sequence order, the SISF is formed by taking off one plane from the original stacking

sequence as the following:

ABCABCBCABC

In Fig. 2.3 (b), it can be seen that the atoms in the upper half of the crystal shift to the

left with respect to the lower half in an amount of BA.

(c) CSF (Complex Stacking Fault)
The CSF is created by the glide of HD’ (or BC’) in Fig. 2.5, which is shown in

Fig. 2.4 with a crystal vector be. The stacking sequence after shearing is

ABCABmerxnwr

From Fig. 2.3 (c), it is found that it has an antiphase boundary feature in addition to Fig.

2.3 (b) like configuration. Therefore, the CSF energy may be equal to APB energy plus

14

SISF energy.

((1) SESF (Superlattice Extrinsic Stacking Fault)
There are two possibilities to form the SESF. (i) The SESF is formed by the
glide of two BA in successive close-packed planes (type I). This type SESF looks like

the effect of inserting one extra plane between two original planes,

I
ABCABCBABCABC

which is shown in Fig. 2.3 ((1). (ii) The SESF is formed by the glide of slip vectors BD’
and BC’ (type II). The type H configuration is shown in Fig. 2.3 (e). Comparing type
I and type II configurations in Fig. 2.3 (d) and (e), it is easily found that type I is more

stable than type 11 because type 11 includes an APB feature.

2.1.3. Dislocations and Dislocation Core Structures

There are four types of dislocations in TiAl due to its ordered crystal structure,
that is, 1/2 < 110] type, < 101] type, 1/2 <112] type and <100> type, as shown in Fig.
2.6. Here 1/2 < 110] dislocations are normal dislocations but both < 101] and 1/2 < 112]
dislocations are superdislocations and < 100 > dislocations are cube dislocations.
Notations " < " and " > " in the indexes indicate that the two indexes nearest to " < " or
" > " are permutable one to another without changing its properties. The dislocation core

structures are very complicated and they depend on which plane they dissociate into

[011]

 

 

  

15

 

1/21112]

 

‘ O

Afr-72+.
------------
”-i-f-I-i-i-3

 

 

 

y 1/2[110]

Figure 2.6 - Dislocations in superlattice L10.

”“5”. The core structure dependence upon temperature is closely associated with the

temperature-dependent deformation behavior in TiAl 121. The typical dislocation

dissociation reactions of these four dislocations are listed in the following:

(a) 1/2 <110] Type

D’C’ --—> D’B + CSF + BC’

1/2[110] ---> 1/6[211] + CSF + 1/6[12I]

[3

 

Note that the marks "

D.
c;

 

 

 

C
30

is CSF,

is SISF,
is APB,
is SESF,

is no fault.

~ 16
(b) <1011TYP6
(i) 2 DA ---> D’A + APB + D’B + srsr + BA
---> D’B + csr + BA + APB + D’B + srsr + BA
[101] ---> 1/2[101] + APB + 1/6[211] + srsr + 1/6[1I2]

---> 1/6[211] + CSF + 1/6[1I2] + APB + 1/6[211] + SISF + l/6[1I2]

D'A D'E ‘ BA 1313 3A 1313 3A
O———O==OVWVV\AO

(ii) 2 D’A ---> D’C’ + 3 BA
---> D’C’ + BA + SESF(I) + 2 BA
---> D’C’ + 2 BA + SESF(II) + (D’B + C’B)
[101] ---> 1/2[110] + 1/2[1i2]
---> 1/2[110] + 1/6[1I2] + SESF(I) + 1/3[1'1'2]

---> 1/2[110] + 1/3[112] + SESF(II) + (1/6[211] + 1/6[El])

DC 3 BA D'C' BA 2 BA D'C 2 13A new]:
CD ------ O O -------- W o -------- W

(c) 1/2<112] Type
(1)3 BA ---> 2 BA + SISF + BA
1/2[1I2] ---> 1/3[1i2] + srsr + 1/6[1I2]
(ii) 3 BA ---> D’B + CSF + BA + APB + one + SISF + BA
1/2[11'2] ---> 1/6[211] + CSF + 1/6[1I2] + APB + 1/611'2'1] + SISF + 1/6[1I2]
(iii) 3 13A ---> BA + SESF + 2 BA
1/211'1'21---> 1/6[1I2] + SESF + 1/31112]
2 BA BA on BA C’B BA BA 2 BA

213A BA D'B BA C‘B BA BA 213A

 

 

17
(d) <100> Type

D’a’ ---> D’C’ + AB
---> D’fi + CSF + BC’ + A6 + CSF + (B
where "*" is not a Thompson notation.

[100] ---> 1/2[110] + 1/2[1I0]

or -—-> 1/6[211] + CSF + 1/6[12I] + 1/6[I2I] + CSF + 1/6[211]
DC‘ AB D'B BC A5 813
CD ------ O o———o ---------- o———o

The dislocation slip planes in TiAl are close packed {l l 1} planes. According to
the crystallography 0f TiAl, dislocations with 1/2 < 110] or <101] Burgers vectors can
slip in two different {111} planes while dislocations with 1/2 (112] Burgers vectors can
only slip in one {111} plane ‘5’. <100> dislocations may slip on the cube planes at

very high temperatures (about 1000 °C) [5".

It has been found that dislocation behavior is different at different temperatures.

Dislocations observed in different temperature ranges are summarized in table 2.1.

2.1.4. Dislocation Blocking Mechanisms

TiAl, like many intermetallic compounds, shoWs an anomalous yield stress-
temperature dependence, that is, unlike disordered metals and alloys, the yield stress of
TiAl increases as the temperature increases up to about 700 °C. The anomaly of yield
stress to temperature results from the glissile-sessile transformations of dislocations in.

different temperature ranges. The glissile dislocations have coplanar splitting

18

Table 2.1. Dislocation behavior in TiAl at different temperature ranges ‘5'°'°”

 

 

Dislocation Unblocked Blocked Unblocked

1/2 < 110] -196 to 100°C 200 to 540°C >600°C

1/2 <112] -196 to 100°C 200 to 600°C >700°C
(Not observed)

<011] -196 to 300°C 400 to 600°C > 700°C

< 100 > Not observed Not observed > 700°C

 

configurations as shown in Fig. 2.7. The sessile dislocations have various configurations
depending on their formation. Some dislocation blocking modes will be summarized in

the following.

(a) Roof-Type Blocking

The roof type blocking results from a resplitting of superdislocations from their
planar glissile configurations into two {111} noncoplanar sessile configurations (like a
peaked root), as shown in Fig. 2.8 ”4’5". Of these configurations, the one containing
SISF bands on both octahedral planes possesses the lowest energy compared to glissile

and other roof-type configurations [2'5“0'.

In addition, a planar sessile configuration was found for 1/2 <112] dislocations

19

<101> Dislocation 1/2<112> Dislocation

 

G 4>-O=O C? . OVV~O=O
5C B5 SC B5 SC A5 5C B8

 

 

 

 

 

f M $ M
5C B5 BC 5C A5 BC
0 4: c2 4)
8C B6+BC 8C A8+BC

Figure 2.7 - Glissile configurations of superdislocations. The single line is a SISF, the
jagged line is an APB, and the double line is a CSF.

5,: \N...

B6 01C 01C 8C B6

 

8C BC: 8C B5
c 1:3wa KM

Figure 2.8 - Roof-type blocking configurations. The single line rs a SISF, the jagged
line is an APB, and the double line is a CSF.

 

 

p..—

20
[5'61]. It contains a SESF band bounded by partial dislocations with parallel Burgers

vectors 1/6 < 112] and 1/3 < 112]. This configuration forms double layer twin in its core

and is strongly blocked.

(b) Kear-Wilsdorf Blocking
The Kear-Wilsdorf blocking occurs as a result of cross slip of superdislocations

into the cubic plane, as shown in Fig. 2.9 [59'6"].

Since the activation energy for the
destruction of Kear-Wilsdorf blocking is very high, the transformation of Kear-Wilsdorf
blocking to a glissile configuration is difficult. Taking 1/2<112] dislocation as an
example, this dislocation may dissociate according to the reaction (C’A + C’B)‘C"" =
C’A'C'“ + C’B‘C'A’, where the superscripts indicate the dislocation line directions.
Therefore, the screw type C’A‘C'“ can glide in (010) cube plane, and the other remains
blocked. The C’A‘C’“ partial gliding produces APB in (010) plane. Thus, Kear-Wilsdorf

blockings of <011] or 1/2 <112] dislocations can not be destroyed by cubic slip “”6”.

 

 

 

B01
8C B8
C} 9
a I
3 , C C O
1 (,2 3 8C B8
1 5c B8

———.———

Figure 2.9 - Kear-Wilsdorf blocking configurations. The dashed line is the APB in the
{001} plane, and the single line is the SISF in the {111} plane.

21
(c) Deep Peierls Valley Blocking

The covalent nature of interatomic bonds in TiAl is the cause of deep (compared
to metals) Peierls potential valleys. The deep Peierls valley blocking is based on the
formation of directional Ti-Ti bonds along < 110] directions in [001] plane that contains
Ti atoms. The slip planes {111} in gamma TiAl are equivalent, but the <1I0>
directions on this plane are not equivalent. As a result, two types of directions exist:
like atom direction and unlike atom direction as shown in Fig. 2.10 (a). Accordingly,
two types of dislocation families exist: one-color—set dislocations (the dislocation lines lie
along the rows of like atoms), and two-color-set dislocations (the dislocation lines are
parallel to the rows of unlike atoms). The presence of a dislocation of any set gives rise
to a row of broken bonds in the (111) slip plane as schematically drawn in Fig. 2.11.
However, the other row of covalent bonds lying in the slip plane is parallel to the
dislocation line for only the one-color—set. Fig. 2.10 (b) shows the relative Peierls
valleys of these two directions. The one-color—set dislocations are in deep energy

valleys, and hence are more stable than two-color—set ones which are in shallow valleys.

The dislocations of different sets differ also in the structure of double kinks. For
a one-color-set dislocation, the double kink consists of two equivalent kinks that are
oriented in two-color-set directions; for a two-color—set dislocation, the double kink
contains two nonequivalent kinks: one is in one-color—set direction and the other is in
two-color-set direction. Therefore, a transformation from the shallow valleys into the
deep valleys is thermodynamically preferred. At low temperatures, dislocations in deep

valleys are blocked. As temperature rises until the stress peak, the above transformation

22

 

 

 

 

 

>
X

 

(a) (b)

Figure 2.10 - The Peierls relief along different crystallographic directions in TiAl.

is thermally activated and more dislocations are blocked in the deep valleys so as to
increase the yield stress. Above the peak temperature, however, the deep valley blocking

is thermally destroyed so that the flow stress drops rapidly.

(d) SSF-Tube Type Blocking

The SSF-tube (Superlattice Stacking Fault tube) blocking is also an effective
blocking mode. The formation of such a configuration is schematically shown in Fig.
2.12 ‘9'. A slipping superdislocation intersects with the other dislocations to form a jog

in the second plane. As a result of splitting, the jog transforms into two nonaligned jogs

23

     

 

  
  
  

///Ae/
/{///

////

  

/
I////
//I

f
/¢
2
/%

///
7////////

        
  
  

%
//////
W

/

   

 

\\\\\\\V \\\“ .

      

 

  

 

 

Figure 2.11 — Schematic diagram of the row of broken Ti-Ti bonds. Al atoms are
removed for clarity. The gray shaded plane is a terminated half plane of
the dislocation and the cross-hatched plane is (111) plane.

on the trailing and leading dislocations. Between them, a SISF band is formed. On the

intersection edge of primary and secondary slip planes, stair-rod dislocations with
Burgers vector (AB+C6) are necessary, Fig. 2.12 (a). (Here we did not use the primed
letters as used in Fig. 2.5. This will not affect the analysis of SSF-tube type blocking.)
As the leading dislocation with Burgers vector 6C moves, a dipole arises. However, a
similar dipole cannot be formed on the trailing dislocation 26C, Fig. 2.12 (b). As some

part of the trailing dislocation comes into contact with the leading and the stair-rod

24

((1)
28C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“5 i!!! BC+BC
25C
(20
(4)
8C 25C
. — — _\
g \
l \
1 \
; . ”lflefl'r-CS n" \
‘1 A (b) \
' BC+BC
(b)

Figure 2.12 — Formation of SSF—tube blocking. (a) Formation of nonaligned jogs on_a
superdislocation; (b,c) formation of a dipole; (d) a SSE-tube along [101]
direction.

 

 

(d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

( A _1.
Vi Ap N I’
\' - “EL“ BC'I'BC
BC+B§
(C)
(d) 8C
25C
—--5c- —"- -"— -\
\
CB CB
25c \ 511 5A
: / ‘B+C5
E n
SAVE 8‘51
AB 4» CB Jwa’) BC
BC+BB
((1)

(Figure 2.12 continued)

26

dislocations, a reconstruction of the whole configuration takes place which results in a
dipole forming on the superdislocations. The dipole contains two SISF bands on the
parallel primary slip planes and one SISF band on the second plane, Fig. 2.12 (c). The
tube formation ends by the formation of APB bands which takes place during emission
of 1/2 <101] screw dislocations (vector CB in Fig. 2.12 (d)) in parallel cross-slip planes.
In practice, depending on the jog height, the tube may be observed either as two isolated

SISF’s or as a SESF (if the jog is not large).

(e) Dislocation Interaction Blocking

When two mobile dislocations are moving toward each other, these two
dislocations can react at their intersection and produce a third dislocation. This resultant
dislocation is either glissile on one of primary slip planes that the two unreacted
dislocations belong to, or glissible on the third slip plane, or completely sessile, i.e., to
form a stair rod dislocation. Either of the last two resultant dislocations can not move
in the primary slip planes of two reacting dislocations, and hence, blocks the two glissile
dislocations. Kawabata and Izurni [”1 made a good summary on some possible
dislocation reactions in TiAl. In addition to the dislocation reactions of Kawabata and
Izumi’s, some additional dislocation reactions in terms of < 100 > type dislocations are

summarized by the author. The added dislocation reactions are listed in the following.

Reactions between <100> and 1/2< 110] dislocations
[100100, + 1/2[i10](,,,, ---> 1/2[110](,,,,

[0011000, + 1/2[I10](,,,,---> 1 2 T12 my, #

27
[001],,,,, + 1/2['1'10](,,;, ---> 1/2 in up, #

[001](100, + 1/2[110](,;1, ---> 1/2 112 (11;) #

Reactions between <100> and <011] dislocations

[100].,,,, + [1011mm -—-> mung, + APB + 1/2[101]m,,
[100],,,,, + [011](,,;,---> 11231111115) + APB + 1/2[011](,,;, #
[001],,,,, + [10mm ---> 12mg“, + APB + 1/2[101],,,,,

[001],,,,, + [011],,,,, ---> 1/2[011](,,,, + APB + 1/2[01i](,,,,

Reactions between <100> and 1/2<112], 1/3<112], 1/6<112] dislocations
[100100, + 1/2[112](,;,, ---> mam,
[100100, + 1/3[112](,;,,---> 11312121521)
[100],,,,, + l/6[I12](,;,, ---> 1 12 5,2,
[001],,,,, + 1/2[112](,,,, ---> may,
[001],,,,, + 1/3[112],,,,, ---> W5”,

[001](100) + l/6[11§](1”) "'> 1/ 114 (51—1.)

Reactions between < 100 > dislocations
[1001(001) + [0101000) “‘> [1 101(001) #

[1001(001) + [0011(1oo)---> 1.1911501) #

[1001(001) + [0011(010) '"> [101](010) #

Here the underlines indicate resultant stair rod dislocations and the marks "11'" at the end

28

of equations indicate that the reactions are energetically equivalent in the forward and

backward directions.

2.1.5. Lamellar Structure and Its Formation

The lamellar structure in TiAl is formed by a phase transformation and it can be
classified into two types, type-I and type-II, according to 7 lath crystal orientation
relationships within lamellae. Both types of lamellar structures have alternating
a2(Ti3Al)/7(TiAl) plates at room temperature and the same crystallographic orientation
relationships between 012 and 7. The difference between these two types of lamellar
structures is that 7 plates in type-II have the same crystal orientations within a grain, but

7 plate crystal orientations in type-I vary from plate to plate.

(a) Type-I Lamellar Structure

The type-I lamellar structure is typically formed by the growth of 7 plates into
or phase or (12 phase in two phase regions: a+7 and 012+7 (see phase diagram shown in
Fig. 2.2). The TiAl (7) lamellar phase is formed on the basal plane of the Ti3Al (02) at
the expense of the Ti3Al phase and finally a lamellar structure consisting of lamellae of
TiAl and Ti3Al phases is formed in the two phase region ‘62]. The orientation relationship

between the TiAl and Ti3Al is the following 1°31,

{111}m // (0001)“...

<110>TiAl // <1120>,,,A,.

29
The < 110] and <101] in TiAl are not equivalent to each other, but the <1120>

directions in Ti3Al are all equivalent. Therefore, there exist six possible crystal

orientations, as shown in Fig. 2.13 "’4‘.

In Fig. 2.13, orientations E and F are
crystallographically identical to the orientations C and D, respectively. Therefore, four
distinguishable orientations exist in TiAl laths. With different combinations of two of
them, there exist four orientation relationships between two 7 plates: (1) when orientation
A in one 7 plate is parallel to orientation A in neighboring 7 plate, a translation order-
fault interface or no interface is formed between them; (2) when A/IC or A/lE, a 120°
rotational order-fault interface is formed where the c-axes of the two neighboring 7 plates
are perpendicular to each other; (3) when A//B, two 7 plates have true-twin relationship;
(4) when A/lD or A/IF, pseudo-twin relationship is formed. The pseudo-twin differs
from the true-twin. In the pseudo-twin, the atomic sites are in twin positions but these
sites are incorrectly occupied by anti-site atoms. Since the energy of lamellar interface
with true-twin orientation relationship is lower than those with other orientation relations,
the true-twin lamellar interface is more preferred. When the orientation relationships
other than the true-twin are observed, a thin lamella of Ti3Al is often found to be
sandwiched between the corresponding two TiAl lamellae. However, for mechanical
twinning, the energetic criterion is usually not suitable, particularly at high stress and low

temperature deformation. In this case a maximum resolved shear stress criterion

proposed by Jin is more suitable '2‘".

(b) Type-II Lamellar Structure

The type-II lamellar structure is formed by Ti3Al (01,) plate growing into TiAl (7)

30

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o 255,

Sémzvarfle

 

63683 Aae+$ E 35%? song—etc mo comma—Lon - 2 .N Eswmm

S8m2v~>86 Sémzvxrte
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31
phase. This lamellar structure has the orientation relationship between the TiAl and

Ti,Al as type-I does, that is, {111},.,,l // (0001)“,Al and <110>m // <1120>.,,,A,.
However, since all <1120 > directions in Ti3Al are equivalent to one another, there are
only one (12 crystal orientation and only one 7 crystal orientation within a grain, and the

7 crystal orientation is the orientation of the original 7 grain.

There are two ways to form either type-I or type-II lamellar structure, as shown

in the following.

For type-I, (i) a ---> or + 7,, ---> L(a/7) ---> I.(a2/7)

(ii) a "‘> (12 "'> (12 + 7p! "‘> 02 + 'Yp "‘> 14(02/7)

where 7,, stands for 7 plates, L(a/7) stands for lamellar structure of 7 phase and a phase,

7pt is 7 precipitates.

For type-II, (i) 7,, + aP ---> 7m +01p ---> L(7/a) ---> L(7/az)

(ii) 7111 -_-> 7m + C{pt ---> 7m + apl _‘-> 117/01) --_> 147/02)

where superscript "p" dedicates particles, subscript "pl" dedicates plates, subscript "m"

matrix and subscript "pt" precipitates.

2.1.6. Phase Transformation Mechanisms

Concerning phase transformation mechanisms, there is a well-accepted stacking

32
fault mechanism proposed by Blackburn “’5‘. A TiAl (L10) stacking fault has the

equivalent stacking sequence to the Ti3Al structure (D019). For example, in a SISF
(Superlattice Intrinsic Stacking Fault), the four atomic layers around L1o stacking fault

turns out to be DOl9 crystal structure,

LIOSISF
ABCABCABIABCABCABCABC

| D019 1

However, this mechanism is not suitable for other stacking faults existing in TiAl. In
the case of CSF (Complex Stacking Fault), even though the stacking sequence of four

atomic layers in CSF is similar to D0,, structure,

LloCSF
ABCABCABIABCABCABC

Ipseudo-DOl9 |

it is a pseudo-DO19 structure since it includes an anti-phase component, as shown in Fig.
2.3 (c). For the SESF (Superlattice Extrinsic Stacking Fault) and the APB (Antiphase
Boundary), the Blackburn mechanism is also not suitable, (see Fig. 2.3 (a), (d) and (e)).
But, since the SISF is the lowest interface energy condition of all possible stacking faults,
and therefore very common in TiAl, it can help explaining how the 0:2 precipites in the

7 matrix. Once a DA] phase nucleates at the SISF of TiAl, T13A1 grows along the

33
octahedral plane of TiAl by increasing the separation of the bounding Shockley partial

dislocations in the octahedral plane.
The 7 precipitation in 012 basal plane can also be explained by this mechanism in
the same way. In this case, glide of Shockley partials a/3 < 1010] in alternate basal

planes of Ti3Al matrix results in L1o atomic stacking sequence (TiAl),

ABABABABCABCABCABABAB

D019 I Ll0 I D019
m ABABABACBACBACBABABAB
D019 | L10 l D019

by operating opposite Shockley partials a/3 <I010] “55'6“.

2.2. Mechanical Twinning Theory

2.2.1. Definition

Mechanical twinning is generally defined as a homogeneous shape deformation
in which the deformed crystal structure is identical with that of the parent, but oriented
differently. If the product and parent are to remain in contact, the deformation must be
an invariant-plane strain. Since the two structures are identical, there can be no volume

change and the deformation must be a simple shear. In general, the twinning modes are

34

["7]. K, is the first invariant or

defined with four twinning elements: K,, K,, 1;, and ‘02
unrotated plane in the twinning operation; 17, is the direction of shear that is in the K,
plane. K, is called twinning plane and n, is called twinning direction. K2 is the second
undistorted (but rotated) plane and called conjugate twinning plane. The plane
perpendicular to K, and K2 and containing 1;, is called the plane of shear and denoted by
S. n2 is the intersection between K2 and S, and it is called conjugate twinning direction.
To any given twin mode, K,=(hkl), K2=(h’k’l’), 17,=[uvw] and n2=[u’v’w’], there
corresponds theoretically a conjugate or reciprocal mode K,’ =K2=(h’k’l’),

K; =K, =(hkl), n,’ =nz=[u’v’w’] and 172’ =17, =[uvw], with the same magnitude of shear

g.

When specifying a mechanical twinning mode, we also indicate the twinning shear
magnitude (g) and the shuffle parameters q and a in addition to the twinning elements.
Here q is the number of K, lattice planes crossed by a primitive lattice vector parallel to
172, and a is the number of K2 lattice planes crossed by a primitive lattice vector parallel

to 11,.

When describing the twinning crystallography, we also need to define the sign of
each twinning element. According to the Bilby-Crocker sign convention [°8l,the signs of
twinning elements are defined as the following. The relative signs are chosen such that
on twinning, the positive side of the K, plane shears in the positive 1;, direction and
similarly for the conjugate shear. The angle between the positive directions of n, and 712

is obtuse in the parent crystal, the angles between 11, and the normal to K2 plane and

35

between 712 and the normal to K, plane are both acute, and the directions of 17,, 772 and

the positive normal to S form a right handed set.

2.2.2. Seven Twinning Classes

Based on the above general twinning definition, mechanical twinning can be
classified into seven classes, as shown in table 2.2 “‘9'. Table 2.2 was obtained by using
the unimodular correspondence matrices given in table 2 of [70]. In table 2.2, m is the
reciprocal of the fraction of lattice points sheared directly to correct twin positions in a
primitive lattice, the remaining lattice points having to shuffle in order to get to the
correct twin positions, and n is the position number of the correspondence matrix in table
2 of [70]. Therefore, m.n indicates the twinning mode. The in and E in table 2.2
indicate that the correspondence matrix is the inverse of the preceding one. m, and 1nF
are the m values for bee and fee crystal structures respectively. The twinning shear
magnitude, g, is presented in ml’g2 form in order to get integers. The a and b in the
"class" column indicate two different relations between the twinning elements and the
asterisks indicate the indistinguishable subdivisions of the correspondence matrix.
Therefore, when the twinning modes with m = 1, m, = 1 and 111,. = l are operating,
no lattice point shuffles are needed for the simple cubic lattice, for the bcc lattice and for
the fcc lattice, respectively. For modes with m, m, or m, = 2, one half of the lattice
points must shuffle in the corresponding lattice. Similarly, for m, m, or m, = 4, three

quarters must shuffle.

36

Table 2.2. Twinning modes in cubic lattices

 

 

class M.n K, K2 17, 172 S ng2 m, n1F
1 2.7 1e+1 le'l 2e-2 2e+e 101 6 2 4
1 27- no 11f- 24f 44f+ 110 6 2 4
1 4.14111 1111 211 21111 011 18 8 2
1 IE 111 577 211 1455 011 18 8 2
2 1.3 100 111 011 211 011 2 2 1
2 1.6 100 524 2012 212 021 5 2 2
2 2.2 111 111 211 211 011 2 4 1
2 2.3 100 122 011 411 011 2 4 2
2 2.6 100 548 012 412 021 5 4 4
2 2.8 110 174 111 311 112 6 1 4
3 4.23 27+1 27-1 8j-4 8j+Z 102 34 8 4
4a 2.4 01c+ 01c 0c+1 Oc'l 100 2 4 4
4 2.21 01d 01d 0d1 0d1 100 2 4 4
4a 2.1_2. 0g+1 Og'l 01g+ 01g 100 10 4 4
4a 2.12 0311+ 0311- 0h+3 011-3 100 10 4 4
5b 4.4 2i-2 21+2 li+1 li‘l 101 4 4 4
5 4.28 k'2k“ k+2k- 1+41- 141+ 111 36 4 4

 

37

Table 2.2. Twinning modes in cubic lattices (continued)

 

 

 

class M.n K, K, 17, 17, S ng2 m, mF
6a 1.4 a+11 all all a+11 011 4 2 2
6a 1.7 0b+l our 016+ 01b 100 5 2 2
7a 1.2 100 120 010 210 001 1 2 2
7a 1.5 100 110 010 110 001 4 1 1
7a 2.1 100 140 010 410 001 1 4 4
7a 2.5 110 130 110 310 001 4 2 2
7 2.9 120 120 210 210 001 9 4 4
7a 2.10 100 310 010 430 001 9 4 4
7b 2.11210 251 245 201 122 9 4 4
a‘t = 2:2“; b=t = 315"; ci = 11:8“; (1 = 2“;

et = 4:1;12"; f=t = 12%;]:2; gi = 0'312; h2t = 40%;];1;
it = 8%:2; ji = 242:2; kt = 2*il; 1* = 2133*.

2.2.3. Classic Twinning Modes

Based on the classic twinning theory, the orientation relationships between the

parent and the twin are classified into four groups formed by

(i) reflection in K, plane;

(ii) rotation of 180° about 17,;

(iii) reflection in the plane normal to 17,;

(iv) rotation of 180° about the normal to K,.

[67] .

38

In these four orientation relations, at least two twinning elements must be rational. It
will be proved that modes (i) and (iv) or modes (ii) and (iii) are geometrically equivalent

to one another in the following section.

Since the twinning elements indexed by letters at to li in Table 2.2 are all

irrational, classes 3, 4, 5, 6 and some modes in class 1 are not classic twinning modes.

2.2.4. Crystallography of Mechanical Twinning

The earliest theory of mechanical twinning crystallography may be referred to
work done by Schmid and Boas in 1950 '7”. A few years later, the review papers by
Hall "2' and Cahn “’71, the work done by Kiho ”3'7", Jaswon and Dove ”5'7" described
several theories of mechanical twinning crystallography in their special cases. Several
years later in 1960’s, Billy and Crocker "’8‘, Bevis and Crocker “’9'7‘“ proposed a more
general and comprehensive theory. In this section, the crystallography of mechanical
twinning will be summarized. The twinning mode considered will be the classic twinning

mode.

Following the notation of tensor calculus, we define c, and p, (i= 1,2,3) are direct
primitive lattice bases to define the parent and the twin lattices, respectively, and
therefore the corresponding reciprocal lattice bases are defined by ci and p‘. The direct
primitive lattice bases c, and p, have the relations as shown in Fig. 2.14 corresponding
to the four orientation relationships of classic twinning modes. The p, is a reflection of

c, about K, plane in Fig. 2.14 (a); the p, in Fig. 2.14 (b) is obtained by 180° rotation of

39

c, about 17,; in Fig. 2.14 (c) the p, is the reflection of c, in the plane normal to 17,; and

in Fig. 2.14 (d) the p, is the 180° rotation of c, about the normal to K,.

 

 

Figure 2.14 - Relations between c, and p,.

4O

 

((1)

(Figure 2.14 continued)

41

In the notation of tensor calculus, we have the interplanar spacing of h, planes,
at = (cub/17]”, the identity distance along the u‘ direction, a = (cfiu’u’f’z, the unit vector
normal to K, plane, n‘c, = dcvhjc, = dhjc’, and the unit vector along 17, direction, b,c’ =
a"c,,u’ci = d’u’cj. The sign of these notations are shown in Fig. 2.15. According to Fig.
2.15, the twinning plane can be defined by K,, hi or n‘; the twinning direction is 17,, ui
or b,; the reciprocal twinning plane K,, k,- or g‘; the reciprocal twinning direction 17, or

v‘; the normal to the plane of shear S is s‘.

 

 

 

 

Figure 2.15 - The sign of four twinning elements.

Thus, the four orientation relationships of classic twinning modes can be

expressed in tensor notations as follows,

p,‘" = c, - Zdhzr’cj, (2.1)

42

Pf” = 20"b11/c1- ca, (2.2)
pf” = c, - 2a"b,u’c,, (2.3)
pi“) = zadhflCJ ' Ci. (2.4)

Here the superscripts (s=1,2,3,4) of p, in equations (2.1) to (2.4) indicate the four
corresponding twin relations. Investigating these four equations, we find that p,“’ = 11,“)
and 7),") = 11,“). So actually only two twin orientation relationships exist. Thus the
relations (i) and (iv) in the classic twinning modes shown in section 2.2.3 are classified

as type-I, and the relations (ii) and (iii) are classified as type-II.

Now let us look at the properties of type-I twinning. Fig. 2.16 shows the

geometry of twinning shear in K, plane. When the composition plane moves from a

 

 

 

 

 

Figure 2.16 - The rationality of K, in type-I twinning.

43

position containing the lattice point A at x‘c, to a position containing the origin O of the
parent lattice base c, and the lattice point B at -y‘p,, the shear of amount Gu‘ci is required

such that

m + Gu‘cs = -y‘Pt (2.5)
Here G = ga“dx‘h,-, and g is the twinning shear. The projection of x‘c, and y‘pi on the
normal to the twinning plane are clearly equal, so the vector (xi - y‘)c, in Fig. 2.16 must
lie in the composition plane. Since the xi and yi are the lattice points, they should be

integers, so (xi - y‘)c, is a rational vector. This vector can be one of an infinite number

of directions in the K, plane depending on the selection of x‘q, so K, must be rational.

Since p, = c, - 2dh,-rl’cj for type-I twinning, the equation (2.5) becomes

Gu‘ci = -(x‘+y‘)c, + 2dyjhjn‘ci (2.6)

Assume xi + yi = z‘, then we have

x‘hi + y‘h, = 2x’h, = 2y‘h, = z‘h, (2.7)

so we obtain

ga'ldzjhju‘ci = -2z‘ci + 2dzjhjn‘c, (2.8)

By using the orientation relationship of type-I, i.e., p, = c, - 2dhin5cj, equation (2.8) becomes

44
z‘c, + ga'1dzjhju‘ci = -z‘p, (2.9)
or

z‘ci + Gu‘ci = -z‘p, (2.10)

The equation (2.10) is identical to the equation (2.5). Thus, this equation indicates that
the parent lattice vector z‘c, is sheared to become the twin lattice vector -z‘p, with the

same indices. The vector z‘ci is thus undistorted and must lie in K, plane.

Since sic, is normal to the plane of shear, as shown in Fig. 2.15, we have by
definition s‘h, = c,,n‘si = c,,-u‘sj = 0. Taking the sealer product of each term in equation
(2.8) with 8%,, we obtain c,,-z‘si = 0. So z‘c, lies in the plane of shear and the rational
lattice vector z‘c, lies along the reciprocal twinning direction 17,. The same result holds
for the twin relationship (iv). Thus for type-I twinning, both K, plane and 17, direction
are rational. In the case that all four elements are rational, the twinning mode is

compound.

In a similar way, it can be proved that the K, plane and the 17, direction are
rational for type-II twinning mode, the details of which are in reference [68]. If the K,
plane and the 17, direction are also rational in the case of type-II, the twinning mode

becomes compound.

Therefore, the mechanical twinning modes can be classified as the following three

types:

45
Type-I: K, plane and 17, direction are rational;

Type-II: K, plane and 17, direction are rational;

Compound: all twinning elements K,, K,, 17, and 17, are rational.

2.2.5. Atomic Movement in Twinning Shear

In this section, we also use type-I twinning mode as an example to analyze the
atomic movement in twinning shear. We assume that v‘c, is the shortest lattice vector in
the direction of z‘c,, i.e., in the 17, direction, then zi = mv‘, where m is an integer. If
D is any lattice point of the parent in the plane through the lattice point v‘c, and parallel
to h, plane, as shown in Fig. 2.17, OD may be written (vi + f)c,, where f‘h, = 0. After

the twinning shear, OD becomes the vector

(vi + f)c, + ga"dv"h,u‘ci = -v‘p, + ft, (2.11)

Based on equation (2.1), we have

(vi + f‘)p, = (vi + f)c, + 2dvjhjn‘c, (2.12)
that is,

f‘p, = fc, (2.13)
So (2.11) becomes

(vi + f)c, + Gu‘c, = -v‘pI + fp, = (fi - v‘)p, (2.14)

Thus every point in this plane is also sheared to a twin lattice point.

 

 

=4.

Figure 2.17 - Schematic illustration of lattice shuffling during type-I twinning for the case of q

47

However, if the points on those hi planes lying between 0 and v‘, these points in
general will not shear directly to twin lattice points. In order to get to the correct twin
lattice points, these points must shuffle. If we assume that all lattice points on a given
h, plane shuffle in the same way, we need to consider only one lattice point on each

plane.

Let v‘hi = q, then a,-,-v‘ni = qd, where d is the interplanar spacing of plane 11,- and
q is the number of hi planes cutting v‘c,. If mici is a lattice vector connecting the origin
of primitive parent lattice base c, to any point on the h,- plane that has a distance d from
the origin 0, as shown in Fig. 2.17 (a) where q = 4, m‘h, will be equal to one and any
parent lattice point can be represented by rm‘q, where r is a positive integer and smaller
than q. During the twinning shear, the lattice points rm‘c, will move to positions
r(mi + ga“du‘)c, = r(mi + 2dni - 2q")c,, as shown in Fig. 2.17 (b). From Fig. 2.17 (b),
we see that not all r(mi + 2dni - 2q")c, positions are at the correct twin lattice points.
The correct twin lattice points relative to O in Fig. 2.17 are given by the lattice vectors
(smi - t‘)p,, where -t’p, is a lattice vector connecting O to any twin lattice point on the h,
plane and s is a positive integer less than q in the p lattice base. Therefore, the

difference

A = (smi - t‘)pl - r(mi + 2dni — 2q")c,

= [(s - r)mi + 2d(q - r - s)ni + (2rq"vi - t‘)]c, (2.15)

defines the shuffles necessary to take the sheared lattice points to the correct twin lattice

48

points. By taking the scalar product of A with h,c‘, we can see that the component of A
normal to h, is zero when 8 = q - r. Substituting 5 = q - r for s in above equation, we

obtain the shuffles in the h,- planes

A = [(q - 2r)mi - ti + 2rq“v‘]ci (2.16)

Therefore, the necessary condition for A = 0 is that

r=lnq an»

if we choose ti = vi and s = r. When q is even the lattice points on the hi plane defined
by r = 1/2 q need not shuffle. So if q = 2, no shuffles are needed for any parent lattice
point to move to the correct twin lattice point. But for q > 2, at least some lattice

points must shuffle as illustrated in Fig. 2.17.

For the multiple lattice structure, the shuffling is much more complicated than the
single lattice structure analyzed above. In the case of multiple lattice structure, the
shuffles both in the hi planes and in the directions perpendicular to hi planes are

necessary to move parent lattice points to the twin lattice positions.

2.2.6. Strain in Twinning
On the macroscopic scale, the twinning deformation consists of homogeneous
simple shear displacement parallel to the plane K, and in the direction 17,. The magnitude

of shear can be obtained by the following equation

49

s=2cot£n1n2=2tan2n1K2=2tanzn2K1 (2 . 18)

where < 17,17, is the acute angle between the two directions concerned, < 17,K, is the acute
angle between 17, and the normal to the plane K,, and < 17,K, is the acute angle between

the 17, and the normal to K,.

AZ P = (xyz) P' = (x'y'z')

 

 

“V

 

 

Figure 2.18 - Lattice point displacement during twinning shear.

Let us look how the specimen length changes corresponding to the twinning. We
choose the orthorgonal cartesian coordinate system such that the axes x and y in twinning
plane and y axis is in twinning direction, as shown in Fig. 2.18 "2’. The points P (x, y,
z) and P’ (x, y+sz, z) are lattice point coordinates before and after twinning, here s is

the magnitude of shear. Thus, the ratio of the lengths of the position vectors of P’ and

50

P is as follows

I 2 2 2 2 .1. _1
LL=( X "LY +225yz2+(sz +1)Z ) 2=(1+Zs=l=sin7ccosA+szsin2x) 2
x +y +z

(2.19)

where y = L cosh, z = L sinx. Since the maximum strain (extension or contraction in
length) is obtained in the plane of shear, i.e., when x = 0 or A = X, the maximum and
minimum values of the extension or contraction can be obtained by differentiation of

equation (2.19) and the result is expressed as

/
(—LL—)max=,/1+s*tanx (2.20)

The maximal extension (L’/L)°max and the maximal contraction (L’lL)°M in terms

of the magnitude of twinning shear (s) are expressed as follows

/
(_L_.)e =§+,’_§i+1 (2.21)
L m 2 4
L’ s 52
(_)C =—_+ __+1 (2.22)
L m 2 4

The extension and contraction of sample lengths are, therefore, conveniently
represented on the stereographic projection, as shown in Fig. 2.19 "2' which is for
twinning in a zinc crystal. Poles I to V1 indicate the six twinning planes {1012}. So if

the sample axis falls into the triangle A, contraction in length occurs for all six twinning

51

planes; if the sample axis falls in the triangle D, extension always occurs. For the
triangle B, the twinning planes II, III, V, VI result in contraction, but the planes I and
IV extension. In the triangle C, however, the planes II and V cause contraction but the

others extension.

The twinning deformation is relatively small compared to the deformation by slip.
The twinning shear is only a fraction of the lattice parameter, but the shear can be

unlimited in slip.

 

 

 

 

 

Figure 2.19 - The extension and contraction of sample length for twinning in zinc.

52
2.2.7. Prediction of Twinning Elements (Twinning Criterion)

According to the analysis in the previous sections, the twinning criterion should
include the following points "’3':

(i) the twinning shear should be small;

(ii) the shuffle mechanism should be simple, that is, q or m should be small;

(iii) the shuffle magnitudes should be small;

(iv) shuffles should be parallel to the twinning direction rather than perpendicular

to this direction.

The twinning criterion can be expressed in the matrix form as follows 170'

U,U, 5 gm + 3 (2. 23)

where 82.11.; is some maximum value of shear, and U,- is correspondence matrix for cubic

lattice that must satisfy

8’ = UyUg' - 3 (2.24)

32 = UIyUlg ' 3 (2.25)
where g is the twinning shear, and U“,- is the inverse of U,,-. So, for a given value of
gm, we can determine the correspondence matrix U,,. This correspondence matrix is

used to determine the twinning elements.

53

The twinning criterion can also be expressed in the notation of tensor calculus

as“

«dirt/papmfl) < q2(g2,,,, + 4) (2.26)

where q is the shuffle parameter described in section 2.2.1, and g2w_ is a given
maximum value of shear. So, for given values of q and gum, the values of h, and vi
which satisfy above inequality can be determined for a lattice defined by cij or c,,-. Thus

the twinning mode with the smallest possible g may be specified.

2.2.8 Twinning in Superlattice

Since the mechanical twinning in a superlattice was reported by Laves in 1952 “3',
many experimental results on mechanical twinning in superlattices have been reported,
such as the mechanical twinning in the superlattices L1, "9'3”, B, ”3'3“, D03 ”3"", L10
“33122341263”, D0,, m9”, DQ919132]. More recently, the review papers by Christian and
Laughlin [931, Yoo ‘9“, and Yoo, Fu and Lee [25‘ have also been published for various
superlattice structures. It is commonly believed that mechanical twinning makes many
important contributions to mechanical properties of ordered intermetallic alloys ‘9". In
this section, the mechanical twinning modes in superlattice structures will be

summarized.

The main difference between the twinning in superlattices and that in disordered

structures is that the twinning shear of superlattice creates not only a true-twin in some

54

twinning variants but also a pseudo-twin in other twinning variants. In the pseudo-twin,
the atomic sites are in twin positions but these sites are incorrectly occupied by anti—site
atoms. In the true-twin, however, the twin relations of the atomic sites are not only
satisfied crystallographically but also true in chemistry. True-twins may be further
classified into following three types [93:95],

(i) type—I/II twin, in which the direct variant gives a twin with a type-I
orientation relation (equivalent to orientation relations (2. 1) and (2.4)), and
the conjugate variant gives a type-II orientation relation (equivalent to
orientation relations (2.2) and (2.3)),

(ii) type-II/I twin, which is the reversed of type-I/II, and

(iii) combined twin, in which all four classical orientation relations are equivalent.

If K, plane is a mirror plane in the parent structure, the type-I/II is defined as type-I
since the conjugate modes do not actually represent twins. Similarly, if 17, is a two-fold

axis in the parent structure, the type-II/I becomes type-II.

The twinning modes discussed in the previous sections are referred to the
disordered structure. When the normal twinning mode of the disordered structure is
applied to the superlattice, it frequently leads to the incorrectly ordered product, and the
true-twinning mode requires a larger shear. This applies to all variants of the normal
twinning mode in almost all cubic superlattices, but only to some of the variants in a

non—cubic superlattice '93].

In general, for the twinning of a single lattice structure in which the primitive unit

55

cell contains only one atom, when the shuffle parameters q and q are both less than or
equal to 2, the superlattice twinning leads to a true-twin; otherwise, if both q and q are
larger than 2, the product of twinning is a pseudo-twin ‘93]. For the twinning of a
multiple superlattice structure in which the primitive unit cell contains more than one
atom, however, a structure shuffle is required. In addition, for the twinning of both
single and multiple superlattice structures, an interchange shuffle (or an order shuffle)

is needed to form true-twin.

If we define a unit cell in the parent in a such way that the primitive lattice
vectors are parallel to 17,, 17, and the positive normal to the plane of shear S, the frame
of this unit cell is sheared into the cell whose frame is twin related with the parent cell.
However, the further relative displacement of the atoms within the unit cell is required
to complete the twinning operation and restore the original structure. This atomic
displacement in the unit cell is called structure shuffle. The structure shuffle can only
complete the twinning operation in crystallography but not in chemistry. 80 various
atomic interchange shuffles in the unit cell are also necessary to obtain a true-twin. The
structure shuffle and interchange shuffle are schematically shown in Fig. 2.20 which
shows the formation of (120) twin in D03 '93]. Fig. 2.20 (a) is the parent unit cell. Fig.
2.20 (b) is the unit cell after twinning shear, which shows that the crystal structure of the
parent has been changed by the simple shear. In order to finish the twinning operation,
the structure shuffle is needed to restore the parent crystal structure, which is shown in
Fig. 2.20 (c). Fig. 2.20 ((1) shows a possible interchange shuffle configuration to

produce a true-twin as shown in Fig. 2.20 (e).

56

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

./ if M? / 12M” /

Figure 2. 20- The sctru ture and inte eerchang suh ffles du uring (120) twinning in D0,.

57

 

 

 

 

 

 

 

 

 

 

 

(d)

 

 

 

 

 

 

 

 

 

 

 

 

 

(6)

(Figure 2.20 continued)

Table 2.3 [93' lists some twinning modes in cubic superlattice structures which is
directly derived from the modes in disordered cubic lattice structures listed in table 2.2
and from some other references 17°96]. The shuffles of one half of the atoms are allowed

in table 2.3. The twinning modes in table 2.3 are listed in the order of increment of

twinning shear.

58

Table 2.3. Twinning modes in cubic lattices

 

 

Mode No.(m.n) g2 m, Twin type
4.2 1/ 8 8 combined
4.2” 1/8 2 combined
4.5 3/ 8 2 II
4.5T 3/ 8 8 I
2.2 1/2 4 combined
2.2T 1/2 1 combined
2.3 1/2 4 combined
2.3T 1/2 2 combined
4.10 7/ 8 8 I
4.10T 7/ 8 2 II
1.2 1 2 combined
2.5 l 2 combined
2.8 3/2 1 I
2.8T 3/2 4 II

 

59

Table 2.3. Twinning modes in cubic lattices. (Continued)

 

 

Mode No.(m.n) g2 m m, mF Twin type
1.3 2 1 2 1 combined
1.3T 2 1 1 2 combined
2.18 7/2 2 4 1 HI
2.18T 7/2 2 1 4 II-I
1.9 8 1 l 1 combined
1.9T 8 1 1 1 combined

 

* m.nT is the "transposed " mode obtained by interchanging K, and 17,, K, and 17,,
and changing the signs of the new K, and 17, referred to preceding twinning
mode.

For the cubic superlattices B2, D03 and L1,, a true-twinning mode without shuffles is
predicted if a 1 appears in the shuffle columns of table 2.3 for both the disordered and
the superlattice structures. For B32, however, additional examination is required in
addition to the shuffle parameter equal to 1 since the primitive unit cell contains two B

atoms. The possible true-twinning modes in cubic superlattices are listed in table 2.4.

For non cubic superlattices, the primary and complementary twinning modes of
L10 (the tetragonal), D0,2 (the tetragonal) and D0,, (the hexagonal) are listed in table

2.5 '9". In table 2.5, A is the ratio of c/a, which is based on disordered cell dimensions,

60

and e/2 is defined in equation b, = (e/2) 17 in which bz is the Burgers vector of a zonal

twinning dislocation '9'".

structure is 1/ 3.

The numerical value of (e/2) for the twinning mode in cubic

Table 2.4. Possible true-twinning modes in cubic superlattices

 

 

Mode No. S K, K, 17, 17, g2 Structures
(m.n)

(a) Modes without Shuffles

1.3 110 111 001 112 110 2 L12

1.3T 110 112 110 111 001 2 B2

1.9 110 112 001 111 110 8 B2,B32,DO3,L1,

1.9T 110 111 110 112 001 8 B2,B32,DO,,L1,

(b) Modes with 50% (non interchange) shuffles

2.3T 110 114 110 221 001 1/2 B2

1.2 001 120 100 210 010 1 B2,Ll,

2.5 0 0 1 1 3 0 1 1 0 3 1 0 1 1 0 1 B2,B32,no,

1.3 110 111 001 112 110 2 B2,B32,Do,

1.3T 110 112 110 111 001 2 L1,

 

61

Table 2.5. Twinning modes in non-cubic superlattices

 

 

 

e/2

Structure q S K, K2 in '12 8

Mn (#1) ('12)

type
110 P 2 1/2(110) {111} {111} 1/2(112) 1/2(112) min/(V2) (2A2-1)/(2>\2+1)

C 1 1/2(110) {11 1} (001) 1/2(112) 1/2(110) 1/2/1. 1/(2>.2 +1) 1
130,, P 2 l/2<1_10) {111} {111} 1/2(1_12> 1/2(112) (2A2-l)/(X\/2) (2)3-1)/(2>.2+1)

C 2 1/2(110) {111} (001) 1/2(112) (110) V2». 2/(2x2+1) 1

130,, (a) 4 i1/3(1120) {1102} {1102} 10101) i<l101> game/om) (axe/(ans)

(b) 2 (1100) {1121} (0001) 1/3(1126) 1/3(1120) l/x 1/(4x2+1) 1

 

2.2.9. Mechanical Twin Nucleation and Propagation

Mechanical twinning mechanisms can be classified into three categories, that is,
(1) dislocation pole mechanism ”“0“, (2) twinning dislocation homogeneous glide
mechanism [101,104], and (3) twin-matrix tilt boundary migration mechanism “05'. The main
differences between them are the details in the mechanical twin nucleation and
propagation. In the third model in which the twin interface is thought to be a tilt
boundary, the interface dislocations glide in the planes almost normal to the twin
interface. It was supposed that once such a tilt boundary was established, it could
migrate easily under a very small shear stress. However, the most intensively studied
model is the dislocation pole mechanism. This mechanism describes the twinning as a
result of a twinning dislocation rotation around an existing pole dislocation whose screw

component is perpendicular to the twinning plane and the component Burgers vector is

62
equal to the spacing of the planes parallel to the twinning plane 198:9”. Much

experimental work has been done to try to validate this model. However, some
experimental results had indicated that no pole dislocations existed during mechanical
twinning, but mechanical twinning was found to occur by a homogeneous shearing of the

twinning portion of the crystal with respect to the matrix (100,101).

The dislocation pole mechanisms were first proposed by Cottrell and Bilby [98' in
1951 for bcc crystals, Thompson '99] in 1952 for hcp crystals and Venables “0‘" in 1961
for fee crystals. Since the first observation of mechanical twinning in fcc metal (in
copper) was in 1957 "0", the twinning theory in fcc was developed much latter than those

in bee and hop.

Since the dislocation pole mechanism has been well developed, in what follows,
the dislocation pole mechanism in three common crystal structures, bcc, fee and hop, will
be analyzed in order to illustrate the mechanical twin nucleation and propagation

phenomena.

(a) Dislocation Pole Mechanism in bcc Metals
The dislocation pole mechanism in bcc metals is based on the dissociation of
normal dislocation b = 1/2[111] in a (112) plane into two partial dislocations: a sessile

mum and a glissile 1/6[111], i.e.,

1/2 [111] ---> 1/3 [112] + 1/6 [111] (2.27)

63
Since the partial dislocation 1/6[111] is also glissile in the (121) plane, it can cross-glide

onto the (121) plane. The rotation of this glissile partial dislocation 1/6[111] about the
normal dislocation 1/2[111] creates a monolayer twin. Since the screw part of the
normal dislocation in [121] direction has a Burgers vector equal to the interplanar spacing
of (121) plane in length, each complete rotation of 1/6[111] partial dislocation about
1/2[111] normal dislocation in the (121) plane will bring the partial dislocation onto the
next adjacent (121) plane. Continuous rotation of this partial dislocation about the
normal dislocation thickens the twin layer. So the partial dislocation 1/6[111] is called
twinning dislocation and the normal dislocation 1/2[111] is called twinning pole. The
twinning plane in this case is then the (121) plane. The propagation of the twin layer in
a direction parallel to the twinning plane occurs by the glide of twinning dislocations on

every twinning plane.

The twin nucleation and propagation model in bcc is schematically illustrated in
Fig. 2.21. One complete rotation of twinning dislocation 0F in Fig. 2.21 (a) about the
twinning pole 0A creates a monolayer twin, i.e., a twinning nucleus. The glide of
twinning dislocations forming incoherent twin boundary, as shown in Fig. 2.21 (b),

results in the propagation of twin layer along the twinning plane.

(b) Dislocation Pole Mechanism in hcp Metals
The twinning modes in hcp structures depend on the c/a ratio (A). Table 2.6
summarizes some twinning modes in hcp metals “m‘m'. Since the (1012) twinning is a

common twinning mode in hop, as shown in table 2.6, the dislocation pole mechanism

 

 

 

 

 

 

Figure 2.21 - Dislocation pole mechanism for twinning in bcc lattice.

in hep is illustrated based on this twinning mode. In this mechanism, the homogeneous
shear occurs on every other corrugated plane while the atoms between these planes are
involved in a shuffle, as shown in Fig. 2.22. The Burgers vector of the zonal twinning

dislocation in this case would be (>3-3)/(>3+3) [1011] ""1. Fig. 2.22 shows two

65

Burgers Vector of Twinning Dislocation

 

 

> [1011]

0

Figure 2.22 - (1012) twinning in zirconium. Projection of the lattice on the (1210)
plane. Circles are in the plane of paper. Squares are a/2 above and below
the paper. Solid symbols indicate atom positions in the twin.

possibilities for the formation of zonal twinning dislocation:

(i) a [0001] dislocation dissociates into the zonal twinning dislocation and a
[1010]' pole dislocation (the asterisk indicates that the indexes refer to the
twinned lattice);

(ii) a [1010] dislocation dissociates into a zonal twinning dislocation and a [00011'

pole dislocation.
Thus, as the zonal twinning dislocation finishes a revolution about the pole dislocation,

it will be on the next adjacent twinning plane.

66

Table 2.6. Twinning modes in hcp metals

 

 

 

q S K1 K2 7’1 712 g cm

4 1210 1012 105 1011 1011 (A2-3)/(A\/3)’ all

8 1210 1011 105 1012 3052 (412-9)/(4>.V3) zl.633

6 1100 1122 11% 115 2213 (2)3-4)/3>. (1.633

2 1100 1121 0002 H26 1120 l/)\ (1.633
*A = c/a

(c) Dislocation Pole Mechanism in fee Metals

The dissociation of normal dislocation 1/2[110] in fee lattice is as follows:

1/2 [110] ---> 1/3 [111] + 1/6 [112]

(2.28)

But rotation of the 1/6[112] twinning dislocation about the nodes would be confined to

the (111) plane, and after the formation of a monolayer twin by one revolution, the

twinning dislocation would reunite with the sessile 1/3[111] dislocation to reform

1/2[110] dislocation. This reformed 1/2[110] dislocation would glide onto the next (111)

plane and dissociate into 1/3[111] + 1/6[112] again, so that the revolution of twinning

dislocation 1/6[1 12] is in a new (111) plane. Repetition of this dissociation-revolution-

reunion procedure thickens the twin layer. This mechanism is schematically shown in

Fig. 2.23 “°°'. At beginning, a normal dislocation AC lies in plane (a) between the nodes

67

 

 

 

 

 

 

Figure 2.23 - Dislocation pole mechanism in fee lattice.

68
N, and N,, and the poles X and Y lie on plane (b), as shown in Fig. 2.23 (a). The

normal dislocation AC dissociates into A0: + aC on (a) plane as shown in Fig. 2.23 (b),
and when the twinning dislocation line reaches a semicircle it is unstable and quickly
extends to the configuration as shown in Fig. 2.23 (c). The partials marked "up" and
"down" in Fig. 2.23 (c) reunite with A01 to form AC, and a unit jog traveling along the
AC from the node N, (or N,) move the AC into the next (a) plane and repeat the
revolution as shown in Fig. 2.23 (d) and (e). In this way, a lenticular twin can be built
up, as shown in Fig. 2.23 (f). The twin thickness is linrited by the backward stress and

by the length of the pole dislocation.

2.2.10. Twin Shape

There are two models to predict the twin shape: one model considers system
energy change associated with twin nucleation and propagation, and the other model
considers forces acting on individual twinning dislocations. These two models have been

frequently used in modelling the mechanical twinning behavior and predicting the twin

Shape [112-114].

(a) A Model Concerning The Energy Change Associated With Twinning

The energy associated with twinning was expressed by Cooper in 1965 “‘5‘.
When we predict the equilibrium shape of a twin, we usually consider the minimum total
energy change in the specimen during the twinning. This can be done by differentiation

of an expression for the total energy of the twinned specimen.

69

The energy terms considered in this model are:

(i) twin boundary energy, which corresponds to the twin boundary surface

tension;

(ii) twinning dislocation interaction energy, which corresponds to the
mutual repulsive forces between the twinning dislocations making up
the incoherent twin boundaries;

(iii) the elastic strain energy due to the external stresses (the externally

applied stress and the local stress concentration);

(iv) dislocation line energy due to the dislocation line tension.

The expressions of these four energy terms reported by Cooper “‘5' are briefly

summarized in the following section.

(1) Twin Boundary Energy

The twin boundary energy is equal to the twin boundary surface energy multiplied

by the total twin boundary area:

B, = A7h(p-l) (2.29)

where A is a twin shape constant, A = 1 when one side boundary of twin is considered,
A = 2 when considering both side boundaries; h is the spacing between neighboring
twinning dislocations in the incoherent twin boundary; p is the total number of twinning
dislocations in one side of a twin boundary. When the twin boundary consists of a large

number of twinning dislocations, the twin boundary energy term can be expressed as

70
E, = A7hp. (2.30)

(2) Twinning Dislocation Interaction Energy

Since the length and the width of a twin are much lager than the spacing between
the twinning dislocations which form the incoherent twin boundary, the twinning
dislocations can be thought to be parallelly distributed. In addition, the following two
assumptions are made for the expression of dislocation interaction energy term: (i) any
pair of twinning dislocations will have zero interaction energy when separated by a
suitably large distance L; (ii) the twin is reasonably thin compared with its length so that
the dislocations behave as if all are in the same glide plane but there is no interaction
between one boundary and the other. The twinning dislocation interaction energy (E,)

for one side of twin boundary is as follows,

.. sz , £_
2 ———2n(1_v) [p lnh K(p)] (2.31)
where
K(p) =ln(p-2) !+ 1,2— [ln(p-l) 1+ln(i—1) 1] (2.32)
1:2

G is the elastic shear modulus, b is the Burgers vector of a twinning dislocation, and v

is Poisson’s ratio.

(3) Strain Energy Due to The External Stress

The work done by the externally applied shear stress on the formation of a twin,

71

i.e., the strain energy, is expressed as

E3=thoap2f(a)h (2.33)

where g is the twinning shear,
To is the applied shear stress,

a is the interplanar spacing of the twin composition plane, and

f(a)=a‘2(e'“-1)+%. (2.34)

If a = 0, 1(0) = l, and if a is large, f(oz) ---> Has.

(4) Twinning Dislocation Line Energy

The total dislocation line energy on one side of a twin boundary is

E, = 0.5pr2. (2.35)

Therefore, the total energy on one side of a twin boundary associated with

twinning is the summation of all these four energy terms, that is,

E = E, + E, + B, +13, (2.36)

01'

72

2
E=Yh(p-1) + Gb [pzlné-Iflp) 1 +291,ap2f(a) ling-13Gb?

211(1-v) h

(2.37)
It must be noted that in this model the energy due to the lattice friction on twinning

dislocations is not considered.

(b) A Model Based on Twinning Dislocation Interaction Forces
Cooper “‘6' first proposed a model based on twinning dislocation interaction forces
in 1966. Marcinkowski and Sree Harsha “”1 that used this model to predict the twin

shape. This model has also been used by others to predict twin shape “mm.

The forces considered in this model include
(i) the force due to the surface tension of twin interface,
(ii) the force due to the twinning dislocation line tension,
(iii) the force due to the applied stress
(iv) the force due to the repulsion between twinning dislocations.
This model can give more detailed results than the model considering the system energy
change. The expression of each force in this model is briefly summarized in the

following.

(1) The Force Due to The Surface Tension

The force due to the surface tension, F”, is numerically equal to the specific

surface energy,

73
Far. = "Y (2- 38)

where 7 is the twin boundary energy. The negative sign in the equation indicates that
the positive direction of this force points to the outside of the surface considered. The

unit is dyn/cm or N/m.

The twin plane energy is approximately equal to one half of the intrinsic stacking
fault energy. This is because the number of mis-stacking with respect to the second
neighbor planes across the intrinsic stacking fault has twice as many as that across the

twin plane.

(2) The Force Due to The Twinning Dislocation Line Tension
The line tension of a dislocation half loop causes a force (per unit length), Fm,

on the edge part of the loop,

FL.T. = ‘ Zsz/l (3.39)

where G = the shear modulus,
b = the Burgers vector,

1 = the radius of curvature of the dislocation loop at the twin tip.

(3) The Force Due to The Applied Stress

This force (per unit length), F

I,

is equal to the resolved shear stress times the

74

Burgers vector,

F, = 1b. (2.40)

The applied stress can be either the externally applied stress, or the locally concentrated

stress, or the residual stress in the matrix, or all of them.

(4) The Force Due to The Twinning Dislocation Interaction

Since the twinning dislocations in a twin boundary are all the same type of
dislocations, i.e., they have the same Burgers vector, the interaction force between them
is repulsive. The dislocation interaction force, F,, between two twinning dislocations,

b, and b,, can be expressed as follows

G

= . G . .
F, 2n(1_v)R[b,xC) (b,xt>1+ [(blC)*(b,C)] (2.41)

211R

 

 

where 3' is the dislocation line direction,

R is the distance between two dislocations,

G is the shear modulus,

v is Poison’s ratio.
Since many twinning dislocations are involved in a twin boundary, the total interaction
force on a twinning dislocation (on the mth twinning dislocation in this case) will be the

sum of all interaction forces,

75

n
F,,.=-—G— 11%;(b,,xt)a(b,x()+(b,,,<>*<b,o<)1. (2.42)

=1
#117

NH

The total force on a twinning dislocation (on the mth twinning dislocation) is

equal to the sum of all these forces,

F =FS.T.+FL.T.+Fa+Fd.m. (2°43)

m

2.3. Creep Theory

Creep is defined as a time dependent deformation. Creep deformation
mechanisms are strongly dependent upon temperature and applied stress. At sufficiently
low temperatures (T < 0.3Tm) ('I‘m is the melting temperature of the materials concerned),
when the applied stress is less than the elastic limit of a specimen, only elastic strain
occurs; when a higher stress (less than a stress nwded to cause immediate fracture) is
applied to a ductile specimen, both elastic and plastic strains occur within the specimen
on loading, and a subsequent time dependent strain accumulates while the stress is
maintained on the specimen, which is called logarithmic creep. The logarithmic creep
results in only very limited creep strains (< 1%) and does not cause eventual fracture.
However, as the temperature increases to above 0.4T,” the deformation appears as a
normal creep, the curve of which usually shows primary, secondary and tertiary stages.
Since logarithmic creep is generally of very limited practical significance, normal creep

(to be called creep hereafter) will be discussed in the following sections.

76

2.3.1. Microstructural Correspondences to Creep

Microstructural response to creep deformation varies with stress, strain,
temperature and materials. During creep of most metallic materials, particularly those
characterized by a high stacking fault energy, a relatively uniform dislocation
arrangement initially observed changes gradually as a result of the formation of a
subgrain structure. In the early primary stage, a heterogeneous subgrain structure forms,
that is, a high density of subgrains form in some region but not in others. With
increasing primary creep strain, the subgrain structure becomes more homogeneous; the
average misorientation across the sub-boundaries increases and remains constant at about
1°-2° throughout the secondary stage “18"‘9’. Although the average subgrain size at a
certain stress is observed to remain constant in the secondary stage, the locations of sub-

boundaries do not appear to be fixed.

The total dislocation density appears to increase with increasing strain in the
primary stage and to become constant during the secondary stage. It has also been
reported for various materials that coarse slip bands form during creep deformation "2°“

”2'. The average spacing of slip bands decreases with increasing stress.

Grain boundary sliding is thought to be significant particularly for creep at high
temperatures and low stress and for small grain size materials. The activation energy of
grain boundary sliding is between the values for self-diffusion in the lattice and along
grain boundaries “23'. In general, at the same stress and temperature, a decrease in

average grain size decreases creep resistance “24”“. The creep rate is insensitive to

77

grain size for dislocation creep except with very fine grain size, for which the increase
in creep rate is attributed to an increase in the contribution of grain boundary sliding to
the overall strain rate “2””. However, in the low stress or diffusional creep regime,

the creep rate increases rapidly with decreasing grain size.

2.3.2. Mechanical Model
The creep deformation at elevated temperature, T 2 0.4T,” is generally

expressed in a form of “27'

 

GbDO b p o n _ 2 44
kT (3) (6) exp( Qc/RT) ( . )

e55 =A
where 6,, = the steady state creep rate,
A = the pre-exponential (structural) constant,
G = the shear modulus,
D0 = the frequency factor,
k = Boltzmann’s constant,
T = the absolute temperature,
b = the Burgers vector,
(1 = the grain size,
p = the grain size exponent,
a = the imposed stress,
11 = the creep stress exponent,

Qc = the activation energy for creep,

78

R = the ideal gas constant.
In this equation, the stress exponent "11" indicates a specific creep deformation
mechanism, the activation energy "Q," indicates the rate limiting deformation process of

creep, and the grain size exponent p indicates the extent of grain size dependence of

creep.

Therefore, the creep deformation mechanisms, in general, can be characterized
by these three values, 0,, n and p. A number of creep deformation mechanisms are
summarized in table 2.7 in terms of these three values. In table 2.7, Q,, Q,,,, Qp and Qi
are the activation energies for lattice self diffusion, grain boundary diffusion, pipe

diffusion and interdiffusion of solute atoms, respectively, and B is a constant.

Diffusional creep (Nabarro-Herring or Coble creep) is considered to be important
at stresses that are too low for dislocation creep processes to be significant. Nabarro-
Herring creep is characterized that the steady state creep rate is controlled by diffusion
through the lattice “23"29'. In Coble creep, the creep rate is determined by grain

boundary diffusion “30‘.

Harper-Dom creep is characterized by a stress and temperature dependence of the
steady state creep rate comparable with the values expected for diffusional creep
processes, but deformation occurs by the generation and movement of dislocations “3”.

This indicates that stress exponents close to unity are not necessarily indication of

diffusional creep.

79

Table 2.7. Creep Deformation Mechanisms At Temperature T 2 0.4Tm

 

 

Temperature Stress 11 0c Mechanism
Low
Low 1 0,, Coble diffusion creep
1 Q1 Harper-Dom creep
Intermediate 7 0p Climb recovery (L.T.)
High é =A*exp(Bor) Q,(?) Power-law breakdown
High
Low 1 Q, Nabarro-Herring diffusion
creep
1 Q1 Harper-Dom creep
Intermediate 5 Q, Climb recovery (HT)
3 Q1 Viscous glide
High é =A*exp(Bar) Q,(?) Power—law breakdown

 

Dislocation creep is classified into two types, alloy type (Class I or Class A) and

pure metal type (Class II or Class M) “32'. In general, the creep of Class I or alloy type

materials is thought to be governed by the dislocation glide that becomes the slowest step

due to the viscous drag exerting on a moving dislocation by its solute atmosphere

[133,134]
9

where n is equal to 3. However, for pure metal type or Class II materials, the creep rate

is controlled by dislocation climb processes in which n is generally close to 5.

80
A change in p value from 2 to 3 should accompany the change in Q, value from

that for lattice diffusion to that for preferential diffusion along grain boundaries, as a
transition from Nabarro-Herring creep to Coble creep occurs with decreasing

temperature.

2.3.3. Microstructural Model

Diffusional creep occurs by stress induced vacancy flow at low stresses. The
vacancy flow from boundaries experiencing a tensile stress to those experiencing
compression may occur predominantly through the lattice or along the grain boundaries
depending on the stress and temperature applied. In this case the grain boundaries are
assumed to be perfect sources and sinks for vacancies. In the case of high dislocation
density and large grain size conditions, the vacancy flow along the dislocation cores, the

pipe diffusion may be the significant creep deformation mechanism.

For dislocation creep, two mechanisms exist: dislocation climb controlled creep
and dislocation glide controlled creep. Dislocation climb controlled creep (e. g. recovery
creep, jerky glide creep, network growth theory) is based on a three dimensional

dislocation network observed in steady state creep “35"3‘“.

Under the applied stress,
dislocations are generated to form dislocation network. Some dislocations in the network
can break away due to thermal fluctuations. The released dislocation can move through
the network and at the same time refine the dimension of the network. This process is
responsible for the work hardening stage of creep. The recovery process takes place by

the annihilation of dislocations in the network walls. Recovery results in a time

81

dependent increase in the dimension of network and decrease in dislocation density. At
a sufficiently long time, the hardening and recovery processes will balance to give a
constant creep rate. In this model, the slip process is athermal and can not occur until

the recovery has changed the substructure sufficiently.

Contrary to the recovery creep, in dislocation glide controlled creep (or viscous
glide creep) the rate controlling process is the rate of dislocation movement in their slip
planes. The slip rate is controlled by the diffusion of solute atoms or by the non-

s “3743’". At high temperatures, where

conservative motion of jogs on screw dislocation
the jogs on dislocations will be saturated with vacancies, the rate of movement of the
dislocations will depend on the rate of emission and absorption of vacancies that is equal

to the rate of vacancy diffusion from the dislocations to the sinks.

CHAPTER THREE

MECHANICAL TWIN NUCLEATION AND PROPAGATION IN TiAl

Mechanical twinning has long been thought to occur in crystalline materials only
at low temperature and high strain rate deformation conditions. However, several recent
studies in intermetallic compound TiAl have shown that the mechanical twinning not only
occurs at room temperature under the normal tensile testing conditions “'2" but also in
creep deformation conditions ”23°38’39““, i.e. , at intermediate temperatures and very low
strain rates. The contributions of mechanical twinning to the deformation of TiAl have

been shown to be significant.

In addition, Jin [23’ analyzed the mechanical twinning behavior in creep
deformation and found that pseudo-twinning obeyed a maximum resolved shear stress
criterion. Mechanical twinning accommodation of the stress concentration at grain triple
Points has also been observed '3‘". This mechanical twinning led to formation of
Zigzagged grain boundaries that would resist grain boundary sliding during creep

deformation.

However, the details about mechanical twinning in TiAl, such as how the

lMechanical twins nucleate, how they grow and what type of morphology they have, have

82

83

not been clearly understood. In order to understand the mechanical twinning mechanisms
in TiAl, the mechanical twin nucleation and propagation phenomena are studied using the
electron beam illuminating method in this chapter. The electron beam illuminating
method permits the investigation of mechanical twin nucleation and propagation processes
in situ. The results are analyzed in terms of the maximum resolved shear stress
criterion. The twin nucleation and propagation mechanisms are discussed. Based on the

observed results, a mechanical twin nucleation and propagation mechanism in TiAl is

Proposed-

3.l. Material and Experimental Procedure

The material used in this study was investment cast 7 TiAl made by Vacuum Arc
Remelting (VAR) and casting at Howmet Corp., Whitehall, MI. Cast rods (152 mm in
length and 16 mm in diameter) were HIP’ed (Hot Isostatic Pressed) at 1260 °C, 172 MPa
for 4 hours to eliminate solidification porosity. The composition of this material is

shown in table 3.1. The material was then heat treated in inert Ar atmosphere at 1300°C

Table 3.1 Composition of the 7 TiAl specimens

Ti Al Nb Cr Fe Cu Si O N H
(Wt%) bal 32.85 4.47 2.95 0.03 <0.01 0.02 555ppm 53ppm 23ppm

(311%) bal 47.4 1.9 2.2

84

for 20 hours and gas fan cooled with a cooling rate of 65 °C/min to produce a duplex
nricrostructure with (7+a,) lamellar grains plus equiaxed 7 grains. The rods were
machined into 62 mm long tensile specimens having a 25 mm gage length and diameter

of5 mm.

The specimens were deformed in tension in an ATS Stress-Relaxation/ Creep
testing machine. Strain was measured using an external extensometer. TWO specimens
were investigated for the study of twin nucleation and propagation mechanisms:

(i) The first specimen investigated was creep deformed under a constant stress,

176 MPa, at 765 °C in air to the strain of 4%, near the end of primary creep,
and furnace cooled under the load.

(ii) The second specimen investigated was deformed under a multi-stress-jump
creep condition at 765 °C in air with an initial stress of 290 MPa, a final
stress 103 MPa and a total strain of about 20 % . The specimen was furnace
cooled to room temperature at the final stress.

The temperature gradient along the gage length was less than 3 °C for both specimens.

For TEM investigation, 0.7 mm thick slices were cut in both longitudinal and
transverse directions from both crept specimens. 3 mm diameter disks were cut from the
slices using an ultrasonic cutting machine and ground to about 0.1 mm thick. The disks
were finally thinned using a double jets electropolishing system using a 10 % sulfuric acid
+ methanol solution at -20 °C. The electron beam illuminating technique was used in

order to directly observe the mechanical twin nucleation and propagation behavior. The

85

investigation was performed in a HITACHI H800 transmission electron microscope at

200 kV.

3.2. Results and Analyses

3.2.1. In Situ Observations of Mechanical Twin nucleation and Propagation

(a) Twin nucleation at Grain Boundaries
The electron beam illuminating method is a useful technique for investigating the
dislocation movement in situ in a normal transmission electron microscope. When the
electron beam hits the sample in TEM, the sample is heated due to the dissipated energy
from the electron beam. Therefore, the dislocation motion can be thermally activated.
If the local stress concentration exists in the place where the electron beam is
illuminating, the local stress may be relaxed by emission of dislocations. Since the creep
Specimen investigated was cooled under the applied creep stress, the local stress state at
grain triple points and grain boundaries at the applied creep stress was frozen. When the
sample was heated by the electron beam without any external stress, the frozen local

residual stress could be relaxed by the emission of dislocations at the grain boundaries.

Mechanical twin nucleation and propagation were observed in situ using the
electron beam illuminating method. Fig. 3.1 shows images of an intermediate stage of

a twin propagation event. Fig. 3.1 (a) is the tilted image of fine mechanical twins within

86

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88
a grain interior, and Fig. 3.1 (b) is the image of fine mechanical twins viewed parallel

to the twinning plane. The sequential images of twin nucleation and propagation
resulting in the current twin configuration were not recorded although it was observed
in situ. The diffraction pattern taken across the thin twin layers is shown in Fig. 3.1 (c).
The indexed diffraction pattern is shown in Fig. 3.1 ((1). These fine twin layers
originated at a grain boundary and propagated toward the grain interior by the emission

of twinning dislocations at the grain boundary.

The observed formation of the fine twin layers is schematically drawn in Fig. 3.2.
At first, a twinning dislocation was formed at a grain boundary, and then, this dislocation
gradually bowed out from the grain boundary, Fig. 3.2 (a). After the dislocation bowed
out to a certain radius, it left the grain boundary and moved very quickly toward the
grain interior. This dislocation left a stacking fault behind it, Fig. 3.2 (b), which was
a twin nucleus. This twin nucleus did not grow further until the second twinning
dislocation was emitted from the grain boundary. The second twinning dislocation
caught up to the first one quickly after it left the grain boundary, and it appeared to push
the leading twinning dislocation further away from the grain boundary. In some
instances, two twinning dislocations, which might be in two adjacent twinning planes and
were very close to each other, were observed to bow out simultaneously, Fig. 3.2 (c),
and propagated together, Fig. 3.2 (d). The emission of subsequent individual twinning
dislocations following the second one was similar to that of the second one and the
trailing twinning dislocations appeared to push the leading twinning dislocations to move

forward. The twin propagation occurred by the continuing emission of twinning

89

 

 

(a)

 

 

 

 

(e)

(d)

 

 

 

 

Figure 3.2 -

 

 

Schematic diagram of mechanical twin nucleation and propagation
procedure in TiAl, which shows that the mechanical twin nucleates at
a grain boundary and propagates into the grain interior. (a) and (b) are
the configuration of mechanical twin nucleation at the grain boundary
with the emission of one twinning dislocation from the grain boundary; (c)
and (d) are the same configuration as (a) and (b) except that two twinning
dislocations are emitted from the grain boundary simultaneously; (e) is an
intermediate state of twin propagation; and (i) is the final state in
which the twin propagation ceases in the grain interior as twin layers
A and B in Fig. 3.1 (a).

90
dislocations from the grain boundary, Fig. 3.2 (e) and (1).

Once a twinning dislocation was emitted, it glided quickly in its own slip plane
to a position near but behind the previously emitted twinning dislocations. The effect of
this phenomenon is evident in the twin layers indicated by letters A and B in Fig. 3.1
(a), which were at an intermediate stage of twin propagation. The spacing between two
twinning dislocations varied depending on their locations with reference to the leading
dislocation, which is indicated by an arrow in Fig. 3.1 (a) and Fig. 3.2 (t). The
dislocations located near the leading dislocation are more closely spaced than the

dislocations farther away from the leading dislocation.

From analysis of crystal orientations of matrix and twins in Fig. 3.1 (a) and (b)
according to the diffraction pattern given in Fig. 3.1 (c) and (d), the interface between
the thin twin layer and the matrix, i.e., the twin plane, was determined to be a (111)
plane, and the twin propagation direction in Fig. 3.1 was determined to be the [112]
direction. It is easily seen in Fig. 3.1 (c) and (d) that the crystal direction [110] in the
matrix is parallel to the crystal direction [110] of the twin. This is the case of the
antiparallel (A // A’) orientation relationship between two 7 laths, which gives a true
twin relation between two adjacent 7 lamellae, as defined by Yamaguchi and Umakoshi
"4". It is also the true twin condition in which the orientation variant A, [110], in one
7 lath is parallel to the orientation variant B, [110], in the other 7 lath (A // B) as
defined by Yang and Wu “’4'. Therefore, these thin twins in Fig. 3.1 are true twins. The

twinning system of the thin twin layers in Fig. 3.1 thus must be (lll)[ll2], and the

91

Burgers vector of the twinning dislocations must be 1/6[112] [23:39”, a Burgers vector
of Shockley partials. In order to form a true twin , each twinning dislocation must glide
on each adjacent (111) plane. Therefore, we can determine that the mechanical twin
nucleus in Fig. 3.2 (b) is an intrinsic stacking fault. Similarly, in the case that the two
twinning dislocations emitted from the grain boundary as in Fig. 3.2 (d), the twin nucleus
is an extrinsic stacking fault including two adjacent (111) planes. However, after the
third twinning dislocation emission, the twin layer does not possess any specific stacking
fault feature, but it does create a four atomic layer twin. Subsequent twinning dislocation
emission on the next twinning plane causes the twin to become thicker. These emitted
twinning dislocations form an incoherent twin boundary at the edge of the twin layer, and
the migration of this incoherent twin boundary results in the propagation of twin layer.
Therefore, the incoherent twin boundary of twin layers A and B in Fig. 3.1 can be
Characterized as shown in Fig. 3.3. The boundary between any two adjacent twinning
dislocations beyond the third twinning dislocation are identical and have the feature of
(111) twin plane. According to the incoherent twin boundary structure shown in Fig.
3 .3, the true-twin plane energy can be estimated to be about one half of the intrinsic or
extrinsic stacking fault energy in TiAl, assuming that the intrinsic stacking fault energy
and the extrinsic stacking fault energy are equal to one another. Based on the intrinsic
Stacking fault energy existing in the literature, 70 mJ/ 1n2 “’1, the true twin plane energy

in TiAl is about 35 mJ/m’.

The twinning elements for twins in Fig. 3.1 are easily characterized, and the

result is listed in table 3.2. The relative orientations between them are shown in Fig.

92

y“ (— Twinning direction [112]
.................................................................. (.1224.

 

    

(l)

x

X

Leading twinning dislocation

W Intrinsic stacking fault
s\\\\\\\\\\\\\N Extrinsic stacking fault
Twin interface

 

Figure 3.3 - Schematic diagram of the incoherent twin boundary sy'ucture in terms of
individual twinning dislocations, which is viewed in [110] direction. The
boundary between the first dislocation and the second dislocation at the
twin tip is an intrinsic stacking fault, the boundary between the second and
the third dislocations is an extrinsic stacking fault, and the boundary
between any two adjacent dislocations after the third dislocation is a twin
plane. The twinning dislocations are denoted by numbers: the leading
dislocation at the twin tip is counted as the first dislocation, the one just
behind it is the second one, and so on.

93
3.4, which is viewed along [110] direction. The dashed line in Fig. 3.4 indicates the

position of K, plane after true-twinning.

Table 3.2. The mechanical twinning elements in Fig. 3.1

 

K1 K2 711 ’72 Shear

(111) (111) [112] [112] 0.707

 

The Schmid factors of all possible twinning directions in the (111) twin plane
were calculated with respect to the tensile axis of the creep specimen. The result in table

3.3 shows that the operating twinning system, the twinning system (111)[112], in this

Table. 3.3 Schmid factors on all possible twinning systems within the (111) plane

 

 

Twinning System Schmid Factor
(111)1112] 0.0364
(111)[211] -0.4372
(111)[121] 0.3783

 

Case is not the one with the highest Schmid factor with respect to the applied creep stress

State. This indicates that the mechanical twin nucleation and propagation in Fig. 3.1 is

94

Trace of K1 plane
111: [112]

K2: (III) V

Trace of K2 plane

 

Kr = (111)

Tensile "
stress axis 121] Trace 0f K2 plane

712 = [112] after twinning

[112]

Figure 3.4 - Crystal orientation relationships of the twinning elements and the possible
twinning directions in (111) plane. The cross mark indicates the direction
of creep tensile axis.

95

not related to the applied creep stress but it is a result of the local stress concentration.
Based on this result, a local principal stress state that caused the (111)[112] twinning is
determined by assuming that the tensile local principal stress is such that it makes the
Schmid factor on (111)[112] twinning system the largest, as shown in Fig. 3.5, although
the actual local stress state might be much more complicated. The equal Schmid factor
contour lines in Fig. 3.5 are plotted with respect to the (111)[112] twinning system. This
shows that the principal axis of the local tensile stress for (111)[112] twinning in TiAl
should be in the vicinity of the crystal direction of [551] within the grain where the

twinning occurs.

Twinning dislocations were observed to be emitted from the grain boundary in
this case. But investigating the grain boundary by tilting the sample in TEM, we did not
find any evidence of existence of dislocation poles in the grain boundary as indicated in
the dislocation pole mechanism [98"°°’”2’. Based on the experimental observation that the
mechanical twins in Fig. 3.1 were formed by glide of 1/6[112] twinning dislocations,
it can be proposed that the twin propagation observed in this case occurs by a 1/6[112]
twinning dislocation homogeneous glide mechanism, that is, l/6[112] twinning
dislocations glide homogeneously on every adjacent (111) twinning plane with one
1/6[112] twinning dislocation on each (111) twinning plane. This twin propagation
mechanism is schematically illustrated in Fig. 3.6. The left side column in Fig. 3.6
shows the glide sequence of l/6[112] twinning dislocations in the (111) twinning plane
during the twin propagation. The middle column shows the atomic arrangements of

resultant twins corresponding to the twinning dislocation glide, in which the open circles

96

 

_ 112
111 .9
_221
331
SST 321
110 ' 0.49 ') 2'11
551 320 0.43 3,-1-

 

  
 
 
 
  
 

0.32 1

 

 

 

331 210
221 310 0°17
/‘
100
w 111 ‘ 110

Figure 3.5 - Schmid_factor contours with respect to the true-twinning system of
(1 11)[l 12], whigh indicates that a tensile principal stress should be in the
vicinity of [551] crystal direction in order to operate the (lll)[112]
twinning system.

97

 

 

 

 

 

Allertwhnln‘ Beforetwillln‘
C — B
- - B — A
[110] 112 .
I 1 Matrix A C
C' -— B
111 T .
( ) wrn A A
Matrix C '— C
B —- B
A -— A
(a)
Stackingeequcncc

 

 

 

 

B —- C

_ A — B

Menu C _ A

on) 11611151; Twin 3' g
(111 r 112

2 '51 I: A '— A

Matrix C — C

B — B

A -— A

 

Figure 3.6 - Mechanical twin propagation mechanism in TiAl, which shows the twin
layers resulting from the glide of (a) one, (b) two, (c) three, and ((1) four
twinning dislocations in the (111) twinning plane during the twinning
propagation. The left side column is the glide sequence of l/6[112]
twinning dislocations. Th_e middle column is the atomic arrangements of
resultant twins viewed in [110] direction, in which the open circles indicate
the atoms located in the plane of the paper and the shaded squares indicate
the atoms in the plane just beneath the paper. The right side column is the
atomic stacking sequences along [111] direction before and after twinning,
in which the primed letters indicate the sheared planes.

98

 

 

 

 

 

Stocking-equates
Anatwhnlng Beforetwimh.
A — A
. <3 --- (2
Matrix
B -— B
.. A' ._ A
111 wrm‘. . B' C
(111) 11611121; Tm C' — B
(111) 11501214.. A _ A
insurer (3 ‘3
B — B
A — A
(C)
Steekingeequence
lkflertuflnndng lacronstunruun.
C
B
A
111) (r
L A.

 

(111)
(111)
(111)

 

cg
>wo>uo>wn>w

((1)

(Figure 3.6 continued)

99

indicate the atoms located in the plane of the paper and the shaded squares indicate the

atoms in the plane just beneath the paper. The atomic arrangements are viewed in [110]
direction. The right side column shows the atomic stacking sequences along [111]
direction before and after twinning, in which the primed letters indicate the sheared
planes. Therefore, the glide of the first twinning dislocation (the leading twinning
dislocation in Fig. 3.1) creates a twin of two atomic layers, as shown in Fig. 3.6 (a).
Investigating the atomic stacking sequence change after the twinning, one can see that
this two atomic layer twin actually is an intrinsic stacking fault. The glide of the two
twinning dislocations in two adjacent (111) planes creates a twin with three atomic
layers, see Fig. 3.6 (b), which is an extrinsic stacking fault as shown in the right side

column. After the glide of the third twinning dislocation on the next adjacent (111) plane
with respect to the first two twinning dislocations, a four atomic layer twin is formed,
and it has no single stacking fault feature, but it does have two twin planes, see Fig. 3.6
(c). Therefore, following the second twinning dislocation, the glide of any 1/6[112]
twinning dislocation on the (111) twinning plane does not change the features of twin

interfaces but it thickens the twin with one more atomic plane, see Fig. 3.6 (c) and (d).

The thickness of the thin twin layers can be calculated based on the twin
nucleation and propagation mechanism proposed above. Although the crystal structure
of TiAl is anisotropic, the atomic stacking along any <111 > directions is similar. This
means that the spacings of all {111} planes are identical. Therefore, the thin twin layer
thickness can be calculated by counting the number of emitted twinning dislocations,

assuming that the observed twin layers are perfect twins. For the longer twin A in Fig.

100
3.1 (a), the thickness is 7.2 nm, and the thickness of the shorter one B in Fig. 3.1 (a)

is 5 .8 nm. For the rest of the twins in Fig. 3.1 (a), since most twinning dislocations

have already propagated farther away from the region shown in Fig. 3.1 (a), the

thicknesses can not be calculated.

(b) Twin Nucleation at Twin Interfaces

The long twin layers in Fig. 3.1 were continuously illuminated by the electron
beam, and it was found that these twin layers were continuously growing into the grain
interior. These growing twin layers then met an existing thin twin lath (called "vertical
twin" in this case), whose twin plane was (111) plane, and crossed the vertical twin by
the initiation of new twinning dislocations on the other side of the vertical twin. This

is the case of twin nucleation at twin interfacies.

Fig. 3.7 shows the entire sequences of twin nucleation and propagation procedure.
In Fig. 3.7, the left sides are the TEM micrographs, which were taken sequentially
during the twin intersection, twinning dislocation emission and twin propagation; the
right side column is a schematic illustration of twin nucleation and propagation sequence
corresponding to the left side pictures. Fig. 3.8 (a) is the thin twin image taken by
tilting the specimen such that the electron beam was parallel to the twin layer interfaces.

The diffraction pattern taken across these thin layers is shown in Fig. 3.8 (b).

The four thin twin layers in Fig. 3.7 are denoted by C, D, E and F. The

configuration shown in Fig. 3.7 (a) is the starting state of these sequential images, where

101

 

Figure 3.7 — The sequences of mechanical twin nucleation and propagation in TiAl.

102

 

(Figure 3.7 continued)

103

 

(Figure 3.7 continued)

 

Figure 3.8 - (a) The side view of twin layers C, D, E, and F in Figure 3.7, and (b) the
diffraction pattern taken across these twin layers.

the thin twin layers C, D, E and F in (a) originated from the grain boundary, which was
located to the right and below the picture. At the beginning, twin C was in the
intermediate state of propagation and did not reach the existing vertical twin, as shown
in Fig. 3.7 (a). While twin layers E and F had intersected with the vertical twin lath,
and the high density twinning dislocation pile-ups had formed within these two twin
layers at the intersection, which indicates that the existing twin lath was acting as a
barrier for the propagation of twins E and F. However, for twin D, in addition to the
high density of twinning dislocation pile-up in the twin layer at the intersection, this twin
had crossed the vertical twin lath and extended to a certain length in the other side of the

vertical twin. Investigating the intersection portion within the vertical twin lath, there

105

is no evidence of structure and crystal orientation changes within it.

As time passed, the twin C propagated and met the vertical twin, and the twin D’
further elongated, as shown in Fig. 3.7 (b). The twinning dislocations within twin C
were not densely piled up at the intersection, indicating that a stress concentration at the
intersection was not formed at this moment. It was found that the propagation of twin
D’ occurred as the twinning dislocations were emitted from the interface of vertical twin.

So more dislocations can be seen in twin D’ in Fig. 3.7 (b).

With the longer time, as shown in Fig. 3.7 (c), the twin D’ further propagated
so that the twin tip was out of this picture; more and more twinning dislocations within
the twin C came and piled up against the vertical twin interface resulting in the stress

concentration at the intersection as indicated by an arrow in Fig. 3 .7 (c).

In Fig. 3.7 (d), the stress concentration resulting from the twinning dislocation
pile-up at the intersection within the twin C was so large that the two twinning
dislocations were emitted on the other side of the vertical twin lath. The sequence of
these two twinning dislocation emission is as follows: Two twinning dislocations were
first gradually bowing out from the interface of vertical twin, and then, after the radius
of dislocation lines reached a certain value, they jumped into the matrix very quickly and
propagated to a certain distance. The layer formed by the glide of these two twinning
dislocations is a three atomic plane twin, as shown in Fig. 3.2 (c) and (d) and Fig. 3.6

(c), and is called the twin nucleus as mentioned in the previous section.

106

As more and more twinning dislocations were emitted from the vertical twin
interface, this twin nucleus grew into the matrix and became a thicker twin layer. The
propagation of this twin layer is clearly seen from Fig. 3.7 (e) and (1). Taking an
existing dislocation line in the matrix as a reference, the relative locations of twin layer
C’ with respect to the reference dislocation line in Fig. 3.7 (e) and Fig. 3.7 (f) indicate
that twin C’ in Fig. 3.7 (e) has propagated to a longer twin layer C’ as shown in Fig.

3.7 (1).

However, it was found that the twin layer could not propagate continuously until
more twinning dislocations were emitted from the interface. It appeared that the trailing
twinning dislocations were pushing the front twinning dislocations to move forward and
this push resulted in the continuous propagation of the twin layers. Once a twinning
dislocation was pushed out from the source, the twin interface at the intersection in this
case, it caught up to the front dislocations very quickly compared to its bowing out
period. This indicates that the twin propagation might be controlled by the twinning
dislocation emission procedure, not by the twinning dislocation glide in the twinning
plane. Since twin D’ had extensively propagated so that the twin tip was far away from
the location where the images were taken, very few twinning dislocations can be seen in

twin D’ in Fig. 3.7 (e) and (1).

During the investigation, the twin E and the twin F were not found to cross the
vertical twin lath even though the dislocation pile-ups were seen within them at the

intersections. This was probably because no more twinning dislocations were emitted

107

within these two twin layers from the original twinning dislocation source, i.e. , the grain
boundary. Therefore, the stress concentration caused by the dislocation pile-up was not

large enough to transfer the twinning strain across the vertical twin lath.

(c) Twin Layer Morphology Near The Twin Tip

According to the twin nucleation and propagation procedure analyzed above, the
morphology of thin twin layers shown in Fig. 3.1 and in Fig. 3.7 can be easily
determined. Fig. 3.9 shows a schematic three dimensional diagram of a thin twin layer.
The curved surface near the twin tip (with darker shading) is an incoherent twin
boundary, which is composed of twinning dislocations lying in the boundary. On an
atomic scale, this twin boundary is not a smooth interface but has a stair-like shape with
the stair height equal to the interplanar spacing of twin planes. The top and bottom flat
interfaces (with diagonal line shading) are coherent with the matrix, which are normal
twin planes. The lower diagram in Fig. 3.9 is a side view (cross section) of the thin

twin layer. Therefore, the shape of thin twin layer near the twin tip is semi-lenticular.

3.2.2. Post-Mortem Observation of Mechanical Twinning
During Creep Deformation

A post-mortem investigation of a creep specimen deformed up to the end of
primary creep state (the first specimen as indicated in the experimental section) indicates
that fine mechanical twins nucleated at grain boundaries and propagated into the grain
interior during the early stage of creep deformation in 7 grains in a similar way as

described in the previous section. Fig. 3.10 is an example which shows that a large

      

108

f—— Twinning direction

Twinnig dislocation lines

Incoherent twin interface

% Coherent twin interface

Cross section of twin layer
Coherent twin interfaces

Figure 3.9 - The thin twin morphology near the twin tip.

109

number of fine mechanical twins were formed by the mechanism described above. In
this case, the fine mechanical twins formed to accommodate the local stress concentration
at a grain triple point [3°]. Looking at the area between the fine mechanical twins and the
untwinned matrix, one can easily see that some fine mechanical twins did not propagate
all the way across the grain, as indicated by an arrow in Fig. 3.10. Diffraction patterns
taken within the fine twin area and within the untwinned matrix region, as shown in Fig.
3.10 (b) and ((1) respectively, show that the matrix within this fine mechanical twin
region has the same orientation as the equiaxed 7 grain within which the fine mechanical
twins formed. This indicates that fine mechanical twins in Fig. 3.10 (a) are formed by

deforming the 7 grain.

If we assume that the volume percent of twins in the fine twin region in Fig. 3.10
(a) is about 50%, the shear strain in this region is a half of the twin shear (0.707), i.e.,
the shear strain is equal to 0.354. The distance between two points "P" and "R" in Fig.
3.10 (a) is about 10 pm. So, the relative shear displacement of point "P" to point "R"
in the direction parallel to the twin-matrix interfaces is about 3.54 um, if we assume that
all fine twins sheared in the same direction. In a similar way, by counting the number
of laths between points "P" and "R" and assuming that the thicknesses of all laths are
roughly equal and one half of these laths are fine twins, we calculated the thickness of

a fine twin lath to be typically about 50 nm '3“.

In the diffraction pattern shown in Fig. 3.10 (b), an extra set of diffraction spots

(dots in the figure) in addition to the spots from fine twins and the matrix indicates that

110

 

(a)

Figure 3.10 - The configuration of fine mechanical twins resulting from the
accommodation of local stress concentration at a grain triple point (a).
Diffraction pattern across the interfaces (b) shows that these fine laths are
twin—related with matrix and the existence of fine D0,, phase between
them (0). Diffraction pattern taken in untwinned area within the same
grain without tilting the sample ((1) shows that the untwinned region has
the same crystal orientation as the matrix of fine mechanical twins as
shown in (b). (e) is the indexes of (d). It also shows that the growth of
fine mechanical twins as pointed out with an anew between the
completely developed fine twins and the matrix stopped in the grain
interior.

111

 

(Figure 3.10 continued)

112

 

(9)

(Figure 3.10 continued)

113

there exists a third phase in this fine twin region shown in Fig. 3.10 (a). An analysis
of these diffraction spots showed that these diffraction spots resulted from a D0,, type

structure.

3.3. Discussion

3.3.1. Mechanical Twin Nucleation in TiAl

Mechanical twinning was observed to nucleate by bowing out of one (or two)
l/6[112] twinning dislocation(s) from the grain boundaries and the twin interfaces in this
study. The twin nucleus can be either an intrinsic stacking fault or an extrinsic stacking
fault. However, the size of twin nucleus, i.e. , the distance that leading twinning
dislocation propagated into the grain interior before the second twinning dislocation was
emitted, was observed to vary for each twin. Since the size of electron beam used in
TEM was very small, the volume illuminated by electron beam was small too. The
thermal activation in the region near the leading twinning dislocation resulting from the
electron beam illumination varied from place to place. In addition, the local stress state
was probably not the same along the grain boundary, and it would change with the
emission of twinning dislocations from the grain boundary. Therefore, both the thermal
and the stress conditions at each location along the grain boundary were different.
However, a certain amount of local stress with the principal stress axis in the vicinity of
[551] direction in the case of tensile stress state is necessary for the (lll)[112] twinning

to nucleate.

114

As the twin nucleus grows, i.e., as the leading twinning dislocation propagates
into the grain interior, the region of stacking fault behind the leading dislocation
increases, and therefore a backward force on the leading dislocation develops in the
direction opposite to its propagating direction (The detailed analysis on the backward
force will be presented in the next chapter). When the twin nucleus reaches a certain
size, the backward force will balance the driving force on the leading dislocation and the
motion will cease. With the emission of the second dislocation, the twin propagates
farther, but the second dislocation can not reach or overtake the leading dislocation since
they repel each other due to the same type of l/6[112] dislocations "‘2'. Therefore, the
second dislocation appears to push the leading dislocation forward. However, since the
stacking fault behind the second dislocation is the extrinsic stacking fault (see Fig. 3.3),
which possesses a similar stacking fault energy to the intrinsic stacking fault, the
twinning dislocations only move slightly farther into the grain. After glide of the third
twinning dislocation, the interface between the twin and the matrix is a twin plane, whose
interface energy is about a half of the intrinsic stacking fault energy. The backward
force on the twinning dislocations following the second one is approximately one half of
that on the first two twinning dislocations. Therefore, the twin propagation is easier than

the nucleation.

Since the effect of a local stress concentration is limited, the local stress state
could not directly provide the twinning driving force to cross the entire grain. A possible
scenario for a transgranular twinning is the following: The twinning dislocations are

emitted from the grain boundaries and the twin interfaces due to the local stress

115

concentrations and forced to move forward to the limited range within which the local
stress is effective. Subsequent twinning dislocations push the front twinning dislocations
beyond the local stress field. In this case, the twin propagation occurs under the
continuous emission of twinning dislocations from the grain boundaries and the twin

interfaces.

Since the nucleus of mechanical twin is formed by bowing out a twinning
dislocation from the dislocation source, the driving force arising from the local stress
concentration must be large enough to overcome another backward force resulting from
the dislocation line tension. As the twinning dislocation bows out, the radius of the
twinning dislocation line decreases. This results in an increase in the backward force due
to the dislocation line tension. After the radius of twinning dislocation line reaches a
minimum value, the radius will increase as the twin nucleus grows, that is, the backward
force due to the dislocation line tension decreases with twin nucleus growth. Therefore,
there should exist a maximum backward force on leading twinning dislocation
corresponding to the minimum radius of twinning dislocation line during twin nucleation.
In order to form a twin nucleus, the driving force resulting from local stress state must
be large enough to compensate this maximum backward force besides the backward force
due to the increment of stacking fault area. So, to emit twinning dislocations
continuously at the grain boundaries and the twin interfaces, the local stress must
maintain being larger than a certain value, the critical stress for twinning dislocation
emission from the grain boundaries and the twin interfaces. If the magnitude of local

stress is lower than this critical stress, according to the twin propagation mechanism

116

proposed in the previous section, the twin could not propagate since no more twinning

dislocations would be emitted.

3.3.2. Mechanical Twinning Mechanisms in TiAl

Mechanical twinning mechanisms can be classified into three categories, that is,
(1) dislocation pole mechanism [98"°°', (2) twinning dislocation homogeneous glide
mechanism “0””, and (3) twin-matrix tilt boundary migration mechanism “03’. The main
differences between them are the details in the mechanical twin nucleation and
propagation. In the third model in which a twin interface is thought to be a tilt
boundary, the interface dislocations glide in the planes almost normal to the twin
interface. This theory assumes that once such a tilt boundary was established, it could
migrate easily under a very small shear stress. However, the most intensively studied
model is the dislocation pole mechanism. This mechanism describes the twinning as a
result of a twinning dislocation rotating around an existing pole dislocation whose screw
component is perpendicular to the twinning plane and the component Burgers vector is
equal to the spacing of the planes parallel to the twinning plane ‘98'100'. Much
experimental work has been done to try to validate this model. However, some
experimental results indicated that mechanical twinning was found to occur by a
homogeneous shear of the twinning portion of crystal with respect to the matrix “°"m‘.
This homogeneous shear mechanism requires that the individual twinning dislocations

glide successively on every neighboring plane parallel to the twinning plane.

Comparing our observations with the twinning mechanisms described above, the

117

twinning in TiAl is consistent with the twinning dislocation homogeneous glide
mechanism. Since the twin nuclei were formed by bowing out twinning dislocations at
the grain boundaries, it was not necessary to form a dissociated dislocation jog in the
twin plane. The normal dislocations and the superdislocations in TiAl are usually in the
state of dissociated forms due to the relatively low stacking fault energy of this material.
The more commonly dissociated configurations are such that the stacking faults are
bounded by the Shockley partial dislocations, i.e., the Shockley partial dislocations are
common and stable in TiAl '6'. Thus the emission of Shockley partial dislocations
(twinning dislocations) at the grain boundaries and the twin interfaces instead of perfect
dislocations (normal dislocations or superdislocations) is reasonable. It is due to the low
stacking fault energy that the twin nuclei can propagate easily by the successive glide of

these Shockley partial dislocations on every adjacent plane.

However, a recent study by Farenc, Coujou and Couret [32‘ indicates that a
dislocation pole mechanism was observed in situ in a TiAl specimen deformed at room
temperature in tension. These authors identified the twinning dislocations by
characterizing the stacking fault fringe configurations in the tilted twin images. The twin
propagation procedure is the same as the one observed in our study, but the twinning
dislocation source is within the grain interior and the twinning procwds by the rotation
of an a/6<112] type partial dislocation around a perfect dislocation. In our study on a
large strain specimen, which was deformed up to a tertiary state of creep deformation,
an evidence of twin nucleation within the grain interior was also found, as shown in Fig.

3.11. Three individual thin twin layers, indicated by letters "A", "B" and "C", can be

118

seen in Fig. 3.11. These twin layers are completely within a 7 grain interior. So the
nuclei of these twin layers were formed inside the 7 grain. The source of twinning
dislocations might be some defects in the grain. Because no inclusions and second
phases were found near the twin layers, the possible source of twinning dislocations
might be the dislocation jogs. Thus, the twin layers probably propagated by the

dislocation pole mechanism.

Therefore, the mechanical twinning mechanism in TiAl may depend on the
location of twin nucleation and the amount of strain. Since the large strain creates more
dislocations within the 7 grains, the possibility of formation of dislocation jogs and pole
dislocations within the 7 grains increases with increasing strain. So the dislocation pole
mechanism is reasonable at large strain. If the twin nuclei are formed at grain
boundaries, the twinning dislocation homogeneous glide mechanism is preferred, but in
the case that the twin nuclei are formed in the grain interior, the dislocation pole

mechanism is more probable.

It must be noted that the image of dissociated perfect dislocations in a tilted
microslip band is similar to that of the fine mechanical twins since the perfect
dislocations in TiAl easily dissociate into the partial dislocations with the stacking faults
between the partials. Therefore, it is necessary to identify whether the observed images
are fine mechanical twin layers or microslip bands before further analysis. A diffraction

pattern analysis is necessary to make this distinction.

119

 

(a)

Figure 3.11 - (a) shows three thin twin layers indicated by letters "A", "B" and "C"
within a 7 grain interior, which indicates that the nuclei of these twin
layers were formed inside the 7 grain; (b) is a diffraction pattern taken
across the twin layers.

120

 

 

 

 

 

0
° illT
-__ 111
111
l u
o X
000
D‘ 111
ill
0 1111 0
l3
1::
O 0

(Figure 3.11 continued)

121
3.3.3. On The D0,, Phase in Fine Mechanical Twins

The diffraction pattern in Fig. 3.10 (b) indicates that fine mechanical twin region
at grain triple point as shown in Fig. 3.10 (a) consists of three different layers: the
matrix, the fine twins and the D0,, structural phase. The formation of fine mechanical
twins has been analyzed in previous section. But the formation of DOI, layers
accompanying the twinning is not clear. No experimental results clearly show how the
D0,, layers form. Therefore, in the following, some possibilities of D0,, diffraction
spots are discussed. (a) If the stress concentration at the grain triple points and/or grain
boundaries is such that it does not emit 1/6<112] twinning dislocations on every (111)
plane, four atomic layers of D0,, structure, i.e. , a nucleus of the Ti3Al crystal structure,
will be formed by glide of a single 1/6 <112] twinning dislocation on a (111) plane “43'.
So, in the case that a large volume of 7 crystal is deformed by the heterogenous glide
of 1/6[112] partial dislocations, the diffraction spots resulting from the nuclei of DO,9
crystal structure may be visible. (b) The crystal structure across a true-twin plane in
TiAl is also a three atomic layer D0,, structure, see Fig. 3.6 (d). If one uses a large
size of aperture in selecting area diffraction, the total volume of D0,, layers in the fine
twin region will be large, and therefore, its contribution to D0,, diffraction spots is also

considerable.

It is worth noting that such a uniformly distributed thin lath configuration in Fig.
3.10 (a) is unlikely to be formed by phase transformations such as or, --- > a,+7 and or -
-- > a,+7 due to its large interfacial energy. Such a thin 7 phase layer could not be

thermodynamically stable during a phase transformation. Therefore, the thin 7 phase

122
layer should either grow and become thicker or be eliminated by the growth of adjacent

7 laths. The resultant configuration in the case of phase transformation should be the
lamellae containing coarse 7 laths with different thickness. The observation of fine twin
configurations only near the equiaxed 7 grain triple points in this study also provides an
evidence that the fine mechanical twins are formed due to the local stress concentration
at the equiaxed 7 grain triple points but not by the phase transformation. In addition,
the composition of the 7 phase is the same at the creep test temperature, 760 °C, as at
a higher phase transformation temperature, so the chemical driving force to decompose

7 phase is zero.

3.3.4. Twin Morphology

Twin morphology has long been described as either a lenticular shape or an
elliptic shape, particularly when modelling the mechanical properties, such as elastic
strain energy and stress field, of twins 11‘3"“:“5'. There are two methods that have been
generally used to predict twin shapes: one is based on the twinning energy calculation,
and the other is based on the elastic forces acting on individual twinning dislocations.
Both methods assumed that a twin has a symmetric shape so that the lenticular or elliptic
shape was predicted. However, the thin twin layers observed in this study do not have
a symmetric shape but a semi-lenticular shape near the twin tip, as shown in Fig. 3.9.
Since the energy of the flat coherent twin plane is lower than that of the curved
incoherent twin boundary and the strain energy resulting from the twinning shear can be
neglected for very thin twin layer “‘3', the semi-lenticular twin shape may be

energetically more stable than the fully elliptic twin shape.

123
Based on the traditional symmetric twin morphology, Marcinkowski and Sree

Harsha “‘7' first calculated the twinning dislocation distributions in incoherent twin
boundaries and the stress distribution surrounding the twin tip by considering the elastic
forces acting on the twinning dislocations. The result shows that the twinning
dislocations close to the twin tip are distributed more densely than those away from the
tip. This trend of dislocation distribution is consistent with the observations in this study
as shown in Fig. 3.1 and Fig. 3.7. But the twinning dislocations in the upper and the
lower incoherent twin boundaries of a symmetric twin layer tend to be vertically aligned
one to another “‘7‘. If this were true in this study, the overlapped stacking fault fringe
images and the coupled twinning dislocation lines would be seen. But the twin layer
images observed in this study show only one set of periodically changed stacking fault
fringes and the single dislocation lines. This indicates that the twin layers observed in
this study are thickened by the glide of twinning dislocations on only one side of the twin

layer and the shape near the twin tip is semi-lenticular.

3.4. Summary

The following summary can be made concerning the mechanical twin nucleation

and propagation in investment cast near gamma TiAl (Ti-48Al-2Nb—2Cr) specimens creep

deformed at 765 °C.

1. Mechanical twin is nucleated by the bowing out of twinning dislocations at the

grain boundaries and the twin interfaces due to the local stress concentration. The

124

nucleus for true-twinning observed in this study is either a superlattice intrinsic stacking
fault (SISF) or a superlattice extrinsic stacking fault (SESF). The superlattice intrinsic
stacking fault is formed by bowing out one l/6[112] twinning dislocation from the grain
boundaries or the twin interfaces. The superlattice extrinsic stacking fault is formed by
emission of two 1/6[112] twinning dislocations from the grain boundaries or the twin

interfaces on adjacent (111) planes.

2. The mechanical twin propagation mechanism in TiAl observed in this study

is a homogeneous glide of twinning dislocations on every adjacent twinning plane.

3. Mechanical twin propagation mechanisms in TiAl depend on the locations of
twin nucleation. If a twin nucleates at the grain boundary, it propagates by the twinning
dislocation homogeneous glide mechanism. However, if a twin nucleates within a 7

grain interior, the twin propagation is controlled by the dislocation pole mechanism.

4. The locations of twin nucleation seem to be affected by strain. In low strain
creep specimen (4%) investigated, we found that all fine mechanical twins resulting from
the accommodation of stress concentration at grain triple points initiated at grain
boundaries. But in a large strain creep specimen, twins originating in the grain interior

were observed.

5. The reason of occurrence of D0,, diffraction spots is not clear at present. It

seems to be the result of the heterogenous glide of 1/6[112] partial dislocations and the

125

existence of three layers of D0,, structure at each true-twin interface. Further work is

needed to test these hypotheses.

6. The thin twin morphology near the twin tip observed in this study has a semi-

lentucular shape rather than a lenticular or an elliptic shape.

CHAPTER FOUR

FORCE AND STRESS ANALYSES ON TWIN PROPAGATION

The stress required for mechanical twin propagation is supposed to be such that
the driving force for mechanical twin propagation must be equal to or larger than the
backward force (F,,) on the twinning dislocations. This backward force acts on the
twinning dislocations in an opposite direction to the twinning dislocation moving
direction, i.e. , the twin propagation direction. At an equilibrium condition, i.e. , at a
constant propagation rate, the backward force is equal to the driving force on each
twinning dislocation. Thus, the stress necessary for mechanical twin propagation must
be such that the driving force on the twinning dislocations at least equals the backward

force on the same twinning dislocations.

Because the twinning dislocations are the same type of 1/6[112] Shockley partial
dislocations (see section 3.2.1), a repulsive force exists between the twinning
dislocations. This repulsive interaction force results in a separation between the twinning
dislocations. But, the stress imposed on a twin layer results in a reduction in spacing
between twinning dislocations. So, we can obtain the stress distribution along a twin

layer by calculating the forces on the twinning dislocations in the twin layer.

126

127

4.1. Theory

4.1.1. Forces on Each Twinning Dislocation

Before the forces on each twinning dislocation are computed, the forces acting on

each twinning dislocation in a twin layer are identified. The incoherent twin boundary

structure shown in Fig. 3.3 is used for the force analysis. On the first (the leading)

twinning dislocation, four forces are identified, as shown in Fig. 4.1 (a):

(b):

(i) the forward dislocation interaction force, F”, which results from the repulsive

force of the twinning dislocations behind the leading dislocation;

(ii) the applied force, F” (or external force F“ as defined later), which results
from the externally applied stress, the local stress state (including residual
stress), and is in the same direction as the twinning propagation direction;

(iii) the internal friction force, Fm, which always acts in the opposite direction

to the twin propagation direction;

(iv) the force due to the superlattice intrinsic stacking fault behind the leading

twinning dislocation, Fsm, which pulls the leading twinning dislocation

backward.

On the second twinning dislocation, six forces are identified, as shown in Fig. 4.1

(i) the forward dislocation interaction force, F“, which is due to the dislocations

behind the second twinning dislocation within the twin layer;

(ii) the applied force, F,;

128

(iii) the force due to the superlattice intrinsic stacking fault, Fs,s,.., which pulls the
second twinning dislocation forward;

(iv) the backward dislocation interaction force, FM, due to the leading twinning
dislocation;

(v) the internal friction force, ch.

(vi) the force due to the superlattice extrinsic stacking fault, FSESF, which pulls

the second twinning dislocation backward.

0n the third twinning dislocation, six forces are identified, as shown in Fig. 4.1
(c).

(i) the forward dislocation interaction force F”;

(ii) the applied force F,;

(iii) the force due to superlattice extrinsic stacking fault, FSBF, which fulls the
third dislocation forward;

(iv) the backward dislocation interaction force, FM, due to the first two twinning
dislocations;

(v) the internal friction force, Fae;

(vi) the force due to the twinning plane surface tension, Fm, which pulls the third

twinning dislocation backward.

0n the fourth twinning dislocation, the forces F,, FM, FM, and FM are similar
to those defined for the third dislocation, but there are no stacking fault forces on this

dislocation. Forward and backward FM, forces exist on the fourth twinning dislocation,

129

as shown in Fig. 4.1 (d). However, since they are the same type of forces acting in the
opposite direction, the net force from them is assumed to be zero, so they can be

eliminated during the force calculation.

0n the rest of the twinning dislocations, the forces on each twinning dislocation

are exactly the same as those on the fourth one.

Twinning direction [113]
(

 

 

J. (111) plane

(a)

Figure 4.1 - Forces on each twinning dislocation in the thin twin layer. (a) On the
first twinning dislocation, (b) on the second one, (c) on the third, and (d)
on the fourth.

130

Twinning direction [11.2.]

 

 

.I.

(111) plane

0))

Twinning direction [112]

 

 

 

 

 

 

<
.1.
Fa ( -—'+ Ffric
Ff.d. ( ——) Fb.d.
FSESF ‘ ) Ftp.
.I. (111) plane

(C)

(Figure 4.1 continued)

131

Twinning direction [112]
(

 

 

 

J— (111) plane

((1)

(Figure 4.1 continued).

4.1.2. Dislocation Interaction Forces

Since all the twinning dislocations within the twin layer shown in Fig. 3.1 are the
same type of dislocations, i.e., they are all the Shockley partials of l/6<112], the
interaction force between any two twinning dislocations is repulsive. If we calculate the
interaction force resulting from dislocations located in front of a specific dislocation, the
result will be the backward dislocation interaction force (FM). Similarly, the force
calculated from the dislocations behind a specific dislocation is the forward dislocation

interaction force (FM).

132

Since the distance between two neighboring twinning dislocations near the twin
tip are small compared to their lengths as shown in Fig. 3.1, the twinning dislocations
are considered as parallel dislocations, and the dislocation end effects can be neglected.
Therefore, the repulsive interaction force between two twinning dislocations can be

expressed as

_ G . G . .
Fd- 2n(1_v)R [b,xC) (b,><C)l+ 2with, C) *(b, ()1. (4.1)

 

 

where F, is a twinning dislocation interaction force between two dislocations per unit
length in the direction perpendicular to the dislocation line,
b, and b, are Burgers vectors of two twinning dislocations,
3’ is a dislocation line direction,
R is a distance between two dislocations,
G is the shear modulus,
v is Poison’s ratio.
The first term in the right hand side is due to the edge component of the dislocations, and

the second term is due to the screw components.

Since Fm, and FM are the forces referring to the interactions of a specific
dislocation with many other twinning dislocations behind and ahead of it in the twin
layer, respectively, FM and FM should be the summations of all Fd’s for the twinning

dislocations behind the dislocation considered and of those ahead of it, respectively.

133

Using a cartesian orthogonal coordinates as shown in Fig. 3.3 and considering the
general situation based upon mixed dislocations, the forward dislocation interaction force

F“, for the mth twinning dislocation is expressed as follows,

 

 

Ff.d.= f b*sin2a (Xi-Km)*[(X,--Xm)2-(d,,p,*(i-m))2]
Gb i=m.121!<1-v> t<x,-x,)2+(d,,,,*(i-m)>212

 

n —
+ 2 b*cosza (X,- Xm) ' . (4.2)
i=m+1 2“ (Xi-Xm)2+[dt,p,*(1-m)]2

where, a is the angle between the Burgers vector and the dislocation line,
d”, is the interplanar spacing of the twinning plane,
x,,, is the location of the mth twinning dislocation,
x,- is the location of the ith twirming dislocation that is behind the mth twinning
dislocation,
and the forward dislocation interaction force FM is normalized with Gb so that F,,,,,/Gb

is dimensionless.

For the backward dislocation interaction force FM, we can get a similar
expression, but only difference is that the dislocations ahead of the mth dislocation should

be considered in this case:

134

F... j“ mine, (xm-xa * 1 (xm-x.) 2- (d... * (In-1') >21

Gb 1:, 2r(1-v> [(xm-X,)2+(d,,p,*(m-i) >212

 

 

 

:2: “€032“ (Km-Xi) . (4 .3)
1:1 21! (Xm-x2)2+[dt.p.*(m_i)]2

4.1.3. Internal Friction Force
For the internal friction force, we consider only a lattice friction force that is due
to the Peierls (or Peierls-Nabarro) stress, Up, on the twinning dislocations. The stress

necessary for a dislocation to pass from one Peierls potential valley to the next has been

calculated by Peierls ““1 and Nabarro “‘71 as
a = orp sin(21ru/a) (4.4)

where a, is the Peierls stress,
11 is the displacement of a dislocation perpendicular to itself,
a is the distance between one close-packed atomic row and the next.
The Peierls stress is directly related to the Peierls energy, as shown in Fig. 4.2, and it

is expressed as

=2Wp= 29 -_4"5 4 5
ap ab (1_v)exp( ). ( . )

where Wp = the Peierls energy,

135

 

 

 

0'
Gph
° ° Viv.) as “’b
_Gp._

 

Figure 4.2 - (a) Variation of Peierls energy as a function of transverse displacement (u),
(b) Variation of the lattice friction stress. (W o z Gb’). (After Fantozzi,
Esnouf, Benoit and Ritchie “‘31).

136
r = the half-width of the dislocation characterizing its degree of delocalization,

r = d / [2(l-v)] for an edge dislocation, and 5' = d / 2 for a screw dislocation,

d = the spacing of the glide plane.

According to the above equation for the Peierls stress, it can be concluded that
the Peierls stress for a partial dislocation is much smaller than that for a perfect
dislocation because the Burgers vector effect on the Peierls stress is in the negative

exponential ““"49‘.

The internal friction force F,ric on a dislocation per unit length then can be

calculated in a normalized form

Firic.= 2 ' _4“C 4 7
Gb (1_v)31n(21tu/a)exp( —b ). ( . )

 

4.1.4. The Force Due to The Stacking Faults

The forces due to the stacking faults, such as F5”, FSESF and F, are all

P-’
numerically equal to the corresponding stacking fault energies, 75,“, 7SESF and 7”,,
respectively. Since the intrinsic stacking fault energy is approximately equal to the

extrinsic stacking fault “5°"5”, we have

I33131? z Fsrasr- (4- 8)

137

For the force F”, exerted by the twinning plane surface tension, since the twinning plane
energy is about one half of the stacking fault energy “5145”, it is numerically equal to one

half of the magnitude of the force resulting from the stacking fault (FSISF or F353,):

F”, = 1/2 FSISF = 1/2 FSESF. (4.9)

4.1.5. Applied Force

The applied force is expressed as the following:
Fa=oa(me*be+ms*bs) (4.10)
or

Fa=aab (cosa *cosp *cosy +cosa *cosy *C080) (4 . 11)

where or, = the externally applied stress,
m, = the Schmid factor for edge dislocation component,
m, = the Schmid factor for screw dislocation component,
bc = the edge component of the twinning dislocation,
b, = the screw component of the twinning dislocation,
or = the acute angle between the Burgers vector and the dislocation line,
,8 = the angle between the applied tensile axis and the direction of b,,
7 = the angle between the applied tensile axis and the twinning plane normal,

0 = the angle between the applied tensile axis and the direction of b,.

138

Thus in the case of an equilibrium condition, since the net total force imposed on

a twinning dislocation is zero, the following equation is established:

Fe + Ff.d. + Ffl'ic. + Fb.d. + Fsrsr + FSESF + Ft.p. = 0 (4-12)

4.1.6. Definitions of External Force, Forward Force and Backward Force

For the analysis convenience, we define some new forces in terms of their acting
directions and sources. At first, we define the force resulting from the externally applied
stress, the locally concentrated stress and the residual stress as an external force, F“.
Here, we assume that all these external stresses result in forces acting on a twinning
dislocation in the same direction as the twin propagation direction, so that the external
force, F“, is always parallel to the twin propagation direction in this analysis. Secondly,
we define the sum of forces acting in the twin propagation direction, except for the
external force, as a internal forward force or simply a forward force, F,. Finally, we
define the sum of forces acting in the opposite direction to the twin propagation direction
as an internal backward force or simply a backward force, F,,. The terminology "internal
force" means that this force is an intrinsic character of a twin, in other wards, the
internal force coexists with the twin, i.e. , no twin then no internal force. While the
external force is different, whether a twin exists or not, this force still exists in the

matrix.

Thus we have the following equations for each twinning dislocation according to

the previous analysis. For the first dislocation,

139

F,, = F,, (4.13)

F, — F,,, (4.14)

F, — F,,, + F,,SF (4.15)
For the second dislocation,

F,, = F,, (4.16)

F, = F,,, + F,,,F, (4.17)

F, = F,,, + F,,S, + F,,, (4.18)
For the third dislocation,

F,, = F,, (4.19)

F, = F,,, + F539,, (4.20)

F, = F,,, + F,, + F,,, (4.21)
For the dislocations beyond the third one,

F,,, = F,, (4.22)

Fr = Fm. + Ftp.’ (4-23)

F, = F,,, + F,,. + F,,, (4.24)

In what follows, we will calculate all these three forces, F,, F, and F,,, within

mmAdmg3un

4.2. Twinning Dislocation Distribution Within A Thin Twin Layer

Let us take a thin twin layer A in Fig. 3.1 as an example to see how the twinning

dislocations are distributed within the thin twin layer. For the determination of twinning

140

dislocation position, the image of this thin twin layer is further amplified to a magnitude
of 120,000X, as shown in Fig. 4.3 (a) and it is schematically illustrated in Fig. 4.3 (b).

The twinning dislocations in this twin layer are labeled using numbers as shown in Fig.

4.3 (b).

The distances of these twinning dislocations from the leading twinning dislocation
are measured and the results are listed in table 4.1. For the second twinning dislocation,
the distance was determined by taking the one-third of the distance of the third twinning
dislocation that was directly measured from the image, since the exact location of the
second dislocation line was hard to be determined from the image. This estimation will
be evaluated in section 4.6.1. The distances of twinning dislocations from the leading
twinning dislocation were measured up to the sixteenth dislocation. Since the seventeenth
dislocation is blocked by an extrinsic dislocation as shown in Fig. 4.3 (b), the distance
of the dislocations beyond the sixteenth are not simply related to the forces analyzed
before, and a force due to the dislocation reaction between the seventeenth and the
extrinsic dislocations should be considered for the seventeenth dislocation. This effect
can be seen in Fig. 4.3 (a) in which the spacings between the twinning dislocations
behind the seventeenth are less than those just ahead of it due to the extrinsic dislocation
blocking. So the extrapolated positions of dislocations beyond the sixteenth are

considered when calculating the forward forces on the dislocations from 11 to 16.

The directly measured dislocation locations vs. the dislocation number is plotted

in Fig. 4.4, and the data fit the following equation,

141

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Table 4.1. The distance of twinning dislocations from the twin tip

 

 

Dislocation Distance (A) Dislocation Distance (A)
l 0 12 2803
2 48 13 3247
3 145 14 3787
4 290 15 4333
5 480 16 5167
6 694 17 6000
7 916 18 7000
8 1249 19 8200
9 1582 20 9600
10 1915 21 11400

11 2359

 

144

25:163.04-179.69*i+66.71*iZ-4.47*i3+0.14*i4. (4.25)

where xi

the distance of the ith dislocation from the twin tip,

i = the dislocation number.

Thus the extrapolated positions of the dislocations beyond the sixteenth dislocation are

obtained by extrapolating the plot in Fig. 4.4. The extrapolated distances of the

dislocations beyond the sixteenth are also listed in table 4.1.

4.3. Force Calculation

4.3.1. Calculation of Forward Force F,

According to the previous analysis, the forward force on each twinning dislocation

can be expressed as follows.

On the leading twinning dislocation,

Far = Ff.d. = b*sin2a Z: (xi-X1) *1 (X,-x,)2-d2,,,,,*(i—1)2]
Gb Gb 211(1-v) 1=2 [(xi-x1)2+d2(111)*(i-1))212

n
+ b*cosza

(x,-x,)
2n

7 , (4.26)
1=2 (xi—x1)2+d2(111,*(1-1)2

0n the second twinning dislocation,

145

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F£.2 = Ff.d. + Fsrsp': b*sin2a Zn: (Xi-X2) * [ (Xi-X2) 2-d2(111) * (i-2) 2]

 

G'b Gb Gb 21t(1-v) 1:3 [(Xi-X2)2+d2(111)*(i-2))2]2
b__*_2_cosza ____12:3(X (Xi-x2) + YSISF (4.27)
= X1""’{2)2+d2(111)*(in?)2 Gb

On the third twinning dislocation,

 

 

 

Ff.3=Ff.d.+FSESF_ b*sin2a n 4(Xi‘x3)*[(Xi-X3)2‘d2(111,*(i-3)2]
Gb 019 Gb 21t(1-V)l [(xi_x3)2+d2(111)*(i-3))2]2

n
+ b*cosza (Xi-x3) + YSESF _ (4 . 28)
21: 1=4 (Xi-X3)2"'d2(111)”Hi-3)2 Gb

 

On the mth twinning dislocation, i.e., on any twinning dislocation beyond the

third twinning dislocation,

 

 

Em: Fm. + Ftp. = b*sin2a f (xi-x» * t (xi-x,» 2‘d2(111>* <i-m> 2:
Gb Gb Gb 21I(l-V) i=m+1 [(xi_xm)2+d2(111)*(i_m))ZJZ

n
+ b*cosza (xi ‘Xm) . + YSISF . (4 . 29)
2“ i=m+1 (Xi-Xm)2+d2(111)*(l-m)2 26b

 

where m 2 4.

147

In the previous chapter, it has been proved that the twinning dislocations in the
twin layer shown in Fig. 4.3 (a) are edge Shockley partials 1/6[112], so the angle
between the Burgers vector and the twinning dislocation line (a) is 90°, and the
magnitude of the Burgers vector of twinning dislocations (b) is 1.633 A. The interplanar
spacing of the twinning plane (dam) is 2.3166 A for the stoichiometric composition.
The shear modulus G = 69620 MPa, and Poison’s ratio r = 0.265 are from [154]. The
superlattice intrinsic stacking fault energy (7313,.) is equal to 70 mJ/m2 according to the
work done by Hug, Loiseau and Veyssiere '6'. Therefore, the superlattice extrinsic

stacking fault energy (735”) is 70 mJ/mz, and the twinning plane energy (7”,) is 35 mJ/mz.

If we replace all these values and dislocation positions in table 4.1 into above

equations, we have

 

n 2 '
F x»: x--S.3666* 1-1 2
f.1=0.3537 J. [21 ( ) 1 (4.30)
Gb 12:2 [Xi+5.3666*(i—1)212

Ff-2=o.3537zn: (xi-48)*[(xi_48)2-5'3666*(i-2)2] +6 1571x10'3
Gb 1:3 [(x3-48)2+5.3666*(i-2)2]2

 

(4.31)

n 2 - 2
x--145 * x.-145 -5.3666* 1-3
( 1 ) [( 1 ) ( )] +6.1571x10'3,

F
£3=o.3537 .
ab 1:4 [(x3-145)2+5.3666*(1-3)2]2

 

(4. 32)

148

’1 (xi-xm)*[(xi-xm)2-5.3666*(.i-m)2]

h=o.3537 E .
Gb i=m+1 [(Xi‘Xm)2+5.3666*(1-m)2]2

+3.0786x10‘3.

(4.33)

where m 2 4.

When calculating the forward forces, we considered only six dislocations beyond the
dislocation concerned, that is, n - m = 6 in the above equations, because the interaction
force between two dislocations separated with a large distance can be neglected. The
calculated results are tabulated as shown in table 4.2 and plotted in Fig. 4.5. The
calculation was carried out up to the sixteenth twinning dislocation in the twin layer

Shown in Fig. 4.3. For the last six dislocations, dislocations 11 to 16, we used some

eXtrapolated dislocation positions in the calculation.

4-3 .2. Calculation of Backward Force F,,

Before calculating the backward force, let’s first calculate the internal friction

fOl‘ce Fm. The internal friction force on a unit length dislocation is

Ffric.= 2 sin(2nu/a)exp(-ig—c-) . (4.34)

Gb (1-v)

Here the internal friction force is normalized with Gb. If we take the maximum value

of sin(21ru/a), i.e., assume sin(21ru/a) = 1, the equation (4.34) can be rewritten as

149

Table 4.2 The forces on each twinning dislocation

 

 

Dislocation Normalized Normalized Normalized

force (F f/ Gb) force (Fb/Gb) force (F,,/Gb)

1 1.2598 x 10'2 6.1718 x 10'3 -6.4262 x 10’3
2 1.3324 x 10'2 1.3560 x 10'2 2.3550 x 10“
3 1.1319 x 10‘2 1.2843 x 10‘2 1.5236 x 10‘3
4 7.2392 x 10‘3 8.2083 x 10‘3 9.6910 x 10“
5 6.7571 x 10‘3 7.5630 x 10'3 8.0630 x 10“
6 6.3764 x 10'3 7.3210 x 10'3 9.4460 x 104
7 5.6098 x 10'3 7.3130 x 10‘3 1.7032 x 10'3
8 5.5341 x 10‘3 6.2352 x 10'3 7.0110 x 10“
9 5.3869 x 10‘3 5.9253 x 103 5.3830 x 10“
10 4.9830 x 10‘3 5.7939 x 10'3 8.1090 x 10“
l 1 4.9235 x 10'3 5.3094 x 10'3 3.8590 x 104
12 4.8103 x 10'3 5.1606 x 10'3 3.5030 x 10“
13 4.5376 x 10'3 5.0948 x 10‘3 5.5720 x 10“
14 4.4683 x 10‘3 4.8440 x 10‘3 3.7570 x 10“
15 4.0562 x 10'3 4.7521 x 10'3 6.9590 x 10“
16 3.9493 x 10‘3 4.3423 x 10’3 3.9300 x 10“

I

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151
FfIiC. = 2 _ 431‘ 4 35
Gb 1-vexp( b)' (' )
For an edge dislocation, g' = d / (2—2v), then the equation (4.35) becomes

F'frric.= 2 exp[- 4nd ] (4.36)

Gb 1-v 2b(1—v)

 

We know that v = 0.265, d = 2.3166 A, b = 1.633 A, so the magnitude of normalized
internal friction force (F,,,c,/Gb) is equal to 1.4723 x 10‘s. The direction of Fm, is

Opposite to the twin propagation direction. This result indicates that the internal friction

force is a constant for any twinning dislocation.

The forces due to SISF, SESF and twin interface tension have been calculated in

the previous section, that is,

FSISF FSESF YSISF -
= = =6.1571X10 3. (4.37)
Gb Gb Gb

Ft-P- = YSISF=3.O786X10‘3. (4-38)
Gb 2Gb

In the following, we will calculate the backward dislocation interaction force.
Since the first (leading) twinning dislocation does not have any twinning dislocations

ahead of it, there is no backward dislocation interaction force on the first twinning

dislocation.

152

According to the equation (4.3), the backward dislocation interaction force on the

second twinning dislocation is

Fb.d.2 = b*sin2a (Xz—Xl)*[(X2—X1)2—d2(111)(2‘1)2]
Gb 2n(1-v) [(xz-x1)2+d2(111, <2-1>212

 

+ bcosza (X2_X1)

. (4.39)
21: ("‘2'x1)2+dz(111)"‘(2‘1)2

 

Since 0: = 90°, b = 1.633 A, ,, = 0.265, d0“, = 2.3166 A, x, = 48 A, x, =0,

equation (4.39) becomes

Fb.d.2= 1.633 48*(482—2.31662)
Gb 2*3.14*(1-0.265) (482+2.31662)2

 

 

=7 . 3877.3(10'3 .

For the third twinning dislocation,

Fb.d.3= b (X3-X2) *[(x3—x2)2_d2(1rr)]
Gb 2N(l-V) [(X3_x2)2+d2(111)]2

 

+ (x3-x1) *[(x3-x1)2-d2‘11“*22]

(4.40)
[(X3 _X1) 2+d2(111)*22] 2

 

Putting the known numerical values into (4.40), we have

153

 

Fb.d.3=0 353.7“4.8=1=(482-2.31662)+ 145* (1452-4*2.31662)
Gb ' (482+2.31662)2 (1452+4*2.31662)2
=9.74.93x10'3

Similarly, for the fourth twinning dislocation, we have

Fbg.d.4 =0 . 3537 *{ (X4'X3) * [ (X4'X3) 2'd2(111)]

b [(x4-x3)2+d2(111)]2

 

+ (X4_X2) * [ (X4'X2) 2-4""dz(111)]
[(X4-X2)2+4*d2(111)]2

 

+ (X4-x1) * [ (X4"X1)2_9*d2(111)]}

(4 .41)
[(x4-X1) 2+9 ’"dz(111)]2

 

Here x1 = 0, x2 = 48, x3 = 145, and x4 = 290. So the backward dislocation interaction
force on the fourth twinning dislocation is equal to
F

b.d.4_ -2
—— —1 . 4461X10
Gb

In general, the backward dislocation interaction force on the mth twinning

dislocation (m 2 4) can be written as

“7’1 (xm-xi) * [ (xm-x,) 2-5 . 3666* (m-i) 2]
1:1 [(x,,,—x,)2+5.3666=«(m—i)2]2

 

3E2L9=0.3537*
Gb

(4.42)

154

Therefore, the backward force for each twinning dislocation in the twin layer

shown in Fig. 4.3 can be calculated as follows.

Fb.1 ___ FSISF’+ Ffric.

Gb Gb Gb

 

=6 .1571X10’3+1.4723x10‘5

=6.1718X10‘3

Fb.2 = Fb.d.2 + FSESF+ Ffric.
Gb Gb Gb Gb

 

=7 .3877x10'3+6 .1571x10'3+1.4723x10'5

=1 . 356 OX10‘2

Fb.3 Fb.d.3 + Ft.p. + Ffric.
Gb Gb G’b Gb

=9 .7493X10'3+3 . 0786X10’3+1 .4723X10‘5

=1 . 284,3)(10'2

Fb.m= Fl).d.m+ Ft.p. + Ffric.
Gb Gb Gb Gb

(4.43)

155

m-l 2 ' 2
x -x. * x -x. -5.3666* m-1

1:1 [(x,,,-xi)2+5.3666*(In-i)2]2

 

+3 .O786X10‘3+1.4723X10’5

m—l

x -X. * x -x. 2-5.3666* m-i 2
=0.3537 (m 1) [( m 1) ( )1

1:1 [(x,,,-xi)2+5.3666=l=(m-i)2]2

 

+3 .0933X10'3.

where m 2 4. The calculated backward forces are tabulated in table 4.2 and plotted in

Fig. 4.5.

4.3.3. Calculation of External Force F,,,

According to the definition of the external force in section 4.1.6, we have

g+m+n=0 MM)
01'

F,, = Fb - F, (4.45)

that is, the external force is numerically equal to the magnitude of backward force minus
the magnitude of corresponding forward force. Thus, the magnitude of external force,

F,,, on each twinning dislocation is expressed as follows,

Fex.1= Fb.1'Fr.1

Gb Gb

 

_ ._ 2_ 2
_0_ 3537231 (145 x)*[(145 x) 5. 3666*(3-1) ]

156

 

 

n 2 - 2
x * x- -5.3666* 1-1
=6.1718x10'3-O.3537 1 ( 1 ,( ) . (4.46)
1:2 [x1.2+5.3666*(.1-1)2]2
Fex.2=Fb.2'Ft.2
Gb G'b
118-xi»: 48-X1-2-5.3666* 2-1' 2
-o 35371::l( ) [( ) ( )]. (4.47)

 

[(48-xi.)2+5 3666*(2- -i)2]2
i¢2

 

Fex.3 = Fb.3"Fr.3
Gb Gb

-3 . 07 86X10‘3 .

 

[(145—x,.)2+5 3666*(3-132]2
i¢3

 

(4.48)
Fex.m_, Fb.m-Ff.m
Gb Gb
- 2- 2
035372 (xm MX)*[(X X1) 5.-3666*(m 1)], (4.49)

 

[(xm -x )2+5. 3666*(m- -.z')2]2
1991!?

where m 2 4. The subscripts' 'm" and " 1" represent the corresponding dislocation

number.

157
The calculated results are listed in table 4.2 and plotted in Fig. 4.5. From the

results, we can see that the external force on the first twinning dislocation is negative.

This will be discussed later.

4.4. Stresses in The Thin Twin Layer

Stresses in the thin twin layer are classified into three categories similar to the
forces on the twinning dislocation: (i) forward shear stress, 1,, that produces the forward
force (Ff); (ii) backward shear stress (or back stress), 1., corresponding to the backward
force (F,,); and (iii) external shear stress, 1w corresponding to the external force (F,,).
The external shear stress includes the externally applied stress, the locally concentrated
stress and the residual stress within the matrix. Since the effect of stress normal to the
twin plane on the twinning dislocation glide are negligible comparing to that of the shear
stresses “5", we will not consider the normal stress in the stress calculation. Therefore,
the above three shear stresses are simply called forward stress, back stress and external
stress, respectively. The terminology "external" here means that the stress is from the
sources outside the twin layer, that is, it exists in the matrix even before the formation
of the twin layer. However, the internal stress (either forward stress or back stress) is
an intrinsic stress that is related to the formation of the twin, that is, it results from the
internal sources such as twinning dislocation interaction, twin interface tension, and

lattice friction on twinning dislocation motion.

The shear stress and the force on a dislocation per unit length, which is

158

perpendicular to the dislocation line, are correlated in the form of

F=T'b (4.50)

Therefore, the shear stress 1' can be easily obtained by dividing the magnitude of force

(F) by the length of Burgers vector (b), i.e.,

1=Wb AM)

This calculated shear stress is the stress at the location of the corresponding twinning

dislocation.

In what follows, we will list all formulas used in the calculation of shear stresses.

The calculated results will be shown in table 4.3 and Fig. 4.6.

For the forward stress 1,

 

 

n 2 - 2
x * x--5.3666* 1-1
c,,,=24624.29 1 [21 ( ) 1, (4,52)
1:2 [X1+5.3666*(i—1)2]2
’1 (x,-48)*[(xg-48)2-5.3666*(1-2)21+428 66

t =24624.29
f.2 12:3 [(xi—4,8)2+5.3666=I‘(i-2)2]2

(4.53)

159

 

 

’1 (xi-145)*[(xi-145)2-5.3666*(i-3)2]
rf3=24624.29 , +428.66,
' 1:4 [(xi-145)2+5.3666*(1-3)2]2
(4.54)
’1 (x.-x ) 4 [ (x.-x )2-5.3666* (i-m)2]
tfm=24624.29 2 1 m 1 m , +214.33,
' i=m+1 [(Xi‘Xm)2+5.3666*(1-m)2]2
(4.55)

where m 2 4. The subscripts indicate the correspondent twinning dislocation numbers.
This notation is also applicable to the following formulas. The unit of stress is MPa, and

this is the same in the following equations.

For the back stress 1,,

=Fb.1

=429.68(MPa), (4.56)

 

1'61

b‘

"11

b'2=944.01(MPa), (4.57)

 

1"19.2

b‘

F
Tb.3= b'3=894.10(MPa) , (4,53)

 

(3‘

m-l (Km-Xi) *[(xm-xi)2-5.3666*(m-i)2]
1:1 [(Km-xi)2+5.3666=I=(m-i)2]2

 

tbom=24624.29 +215.36,

(4.59)

160

Table 4.3 The stresses at each twinning dislocation

 

 

Dislocation 1, (MPa) 1., (MPa) 1,, (MPa)
1 877.07 429.68 -447 .39
2 926.92 944.01 17.09
3 781.97 894.10 112.04
4 503.99 571.46 70.94
5 470.43 526.54 57.10
6 443.92 509.69 65.68
7 390.55 509.13 118.12
8 385.28 434.09 48.09
9 375.04 412.51 36.40
10 346.92 399.55 55.44
11 342.77 369.64 25.84
12 334.89 359.28 23.36
13 315.91 354.70 37.77
14 311.08 337.24 30.33
15 282.39 330.84 47.41
16 274.95 302.31 26.33

 

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162

where m 2 4.

For the external stress 1“

n 2 - 2
X-* x. -5.3666* 1-1

tx1=429.68-24624.292: 1 (21 ,( 2: .
1:2 [X1 +5.3666*(1-1) ]

 

(4. 60)

418-31. * 48-x- 2-5.3666* 2-1’ 2
44621291131311‘ 1) H 1) ( H

ex 2 [(48-xi)2+5.3666*(2-i)212
1'42

 

I

(4.61)

145-x. ... 145—x. 2-5.3666* 3-1‘ 2
=.24624 291;: ( 1) H 1) .( ) 1-214.33,
3 [(145-Xi)2+5.3666*(3-1)2]2
i?

 

1’-ex.3

 

(4.62)
_ 2- 2
{24624 2912 (X, -x,-,,,)*[(x x) 5. 3666*(m- 1) ]
=1 [(29,721)‘°‘+5.3666=o=(m-1‘)2]2
11m
(4.63)

4.5. Simplified Equations for F,, F,, and F,,,

Let’s see an example how the equations are simplified. The equation (4.26) can

be rewritten as

163

_ dz<111)"‘(-7:-1)2

 

 

 

 

 

 

 

 

 

Ff.1_ b*sin2a n 1 (Xi-X1)2
Gb 21t(l-V)12X--X1 [1+ d2(111)*(i—1)2]2
(Xi—X1)2
n
+ b*cosza Z 1 1
2n 1=2 xi-Xl 1+dz(111)*(-7:-1)2 (4'64)
(Xi-X1)2
Since
d2(111)*(i-1)2
(xi-x1)2
the equation (4.64) can be reduced as
_F__f.1_[ b*31n2a batcos2 a] (4.65)
Gb 21t(1-v) 1:21:

In a similar way, we can simplify all equations used for the calculation of Ff, Fb

and Fe, as follows.

164
For the forward force Ff

       

n1
.7312=:2?,

 

8 +6 .1571x10‘3,

F n 1
455:0.35372 _
1=3 X1

 

 

 

 

 

F
M=0. 3537 2 .0786x10‘3.
Gb i_m+1x
For the backward force F,,
F 111-1 1
“=0 . 3537 E +3.0933x10'3.

where m 2 4.
For the external force Fe,

II
.1_ -3 1
——-6.l718X10 -O.3537 —
Gb 12:2 X1

F13 1 -
b 2 x.-145

(4.67)

(4.68)

(4.69)

(4.70)

(4.71)

165

 

 

n
Fexz l
—-—=o.3537 , (4.72)
i¢2
n
F
453:0.35372 _L_—3.0786x1o-3, (4.73)
G'b 1=1 145-x,-
i¢3
n
F
ex.m=0.3537z 1 ’ (4.74)
Gb 1=1xm-xi
iatm

wherei _>_ 4.

The results calculated using these simplified equations are shown in table 4.4.
The comparison between the results calculated from the original equations and the results
from the simplified equations, as shown in Fig. 4.7, indicates that the results from the
simplified equations can perfectly represent the results obtained from the original
equations. The deviations between the results from the simplified equations and the
results from the original equations are very small, only about :I: 0.024. So we can
directly use the simplified equations in the calculation of forces on the twinning

dislocations.

166

Table 4.4 The forces calculated using simplified equations

 

 

Dislocation F r/ Gb Fb/Gb F,,/Gb
1 1.2660 x 10'2 6.1718 x 10‘3 -6.4882 x 10'3
2 1.3039 x 10'2 1.3644 x 10'2 6.0500 x 10“
3 1.1322 x 10'2 1.2901 x 10'2 1.5790 x 10“
4 7.2409 x 10'3 8.2139 x 10'3 9.7300 x 104
5 6.7583 x 10‘3 7.5663 x 10‘3 8.0800 x 10“
6 6.3773 x 10'3 7.3230 x 10'3 9.4570 x 10‘4
7 5.6102 x 10'3 7.3151 x 10‘3 1.7049 x 10'3
8 5.5345 x 10'3 6.2365 x 10'3 7.0200 x 10“
9 5.3871 x 10‘3 5.9257 x 10'3 5.3860 x 104
10 4.9831 x 10'3 5.7944 x 10‘3 8.1130 x 10“
11 4.9235 x 10'3 5.3096 x 10‘3 3.8610 x 10“
12 4.8104 x 10'3 5.1607 x 10‘3 3.5030 x 10“
13 4.5376 x 10‘3 5.0950 x 10'3 5.5740 x 10“
14 4.3936 x 10'3 4.8442 x 10‘3 4.5060 x 104
15 4.0562 x 10'3 4.752 x 10'3 6.9580 x 104
16 3.9493 x 10‘3 4.3423 x 10'3 3.9300 x 10“

 

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4.6. Discussion

4.6.1. Stress and Force Distributions Within The Twin Layer
From the previous results, we can see that the force distributions along the twin
layer is similar to the stress distributions. So we will analyze only the stress distributions

in this section, and this analysis is also applicable to the force distributions.

(3) External Stress 7,,

External stress has been defined as the stress resulting from the externally applied
load, the local stress concentration and the residual stress within the matrix. Since the
specimen was free from external load during the TEM investigation, the externally
applied stress term can be eliminated from the external stress. Investigating the external
stress line in Fig. 4.6, we can see that the external stress is more or less evenly
distributed along the thin twin layer, which indicates that no stress concentration exists
within the region of the twin layer investigated. Also, no stress concentration would be
expected in the middle of a grain. This means that the external stress calculated in this
study may be purely a residual stress within the matrix where the twin layer was
growing. It is possible that the twin layer investigated has propagated long enough so
that the front portion of twin layer, which we are studying, has been out of the range of
local stress concentration that is at the grain boundary, as shown in Chapter Three. The
negative value of external stress at the twin tip may be due to the wrong values of back
stress and forward stress, which will be analyzed in the following section. So taking an

average of the calculated external stresses with neglect of the stress at the fist twinning

169

dislocation, the residual stress (1,.) in the matrix is

7,, = [Tex],,,_ = 51 MPa (4.75)

where [1”],v, means the average value of external stresses except for the stress at the first
twinning dislocation. We assume now that the external stress, equal to the residual stress
in this case, is distributed evenly along the thin twin layer including the first twinning
dislocation. Then the driving force resulting from the external stress is the same for all

the twinning dislocations in the thin twin layer of Fig. 4.3 (a).

(b) Forward Stress r,

The forward stress, which is an internal stress resulting from the dislocation
interaction and the twin interface tension, decreases as the distance from the twin tip
increases, as shown in Fig. 4.6. This decrement of forward stress is remarkable near
the twin tip up to about 1000 A, which corresponds to the location of the seventh
twinning dislocation, and the decrease becomes gentle beyond about 1000 A. The
forward stress tends toward a constant value far from the twin tip. This indicates that
the driving force for the twinning dislocation glide resulting from the forward stress is
different for the dislocations located at different places in the twin layer. The driving
force necessary for the twinning dislocation glide is very large near the twin tip, and it
drops quickly as the distance increases. Therefore, the twinning dislocations having large
distance from the twin tip glide forward more easily than the twinning dislocations near

the twin tip.

170

The forward stress at the first twinning dislocation is smaller than that at the
second twinning dislocation, as seen in Fig 4.6 and in table 4.3. This is possibly due to
the incorrect approximation of location of the second twinning dislocation. So if we
extrapolate the plot of forward stress using the data from the third dislocation to the sixth
dislocation, we have the forward stress at the first and the second twinning dislocations
equal to 1245.6 MPa and 1059.4 MPa, respectively. Thus the modified forward stress
distribution along the thin twin layer is as shown in Fig 4.8. The corresponding forward

force distribution is shown in Fig. 4.9.

(c) Back Stress 1,,

The back stress is an internal stress, which results from the dislocation
interaction, the twin interface tension and the internal friction on the twinning
dislocations. The change of back stress with the distance is similar to that of forward
stress, that is, back stress drops very quickly with the distance near the twin tip, and
decreases gently with the distance when the distance is large. Since the back stress
results in a force (backward force) opposite to the twin propagation direction, the back
stress resists the twinning dislocation glide, so it must be overcome in order for twinning
dislocations to move forward during the twin propagation. So the resistance to
dislocation glide in the twin layer is very large near the twin tip and relatively small at
a location far away from the twin tip. In other words, the trailing twinning dislocations
have less resistance for glide than the leading twinning dislocations, so the trailing
twinning dislocations more easily glide comparing to the leading twinning dislocations.

This conclusion is the same as that drawn from the forward stress analysis.

171

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Comparing the back stress with the forward stress on the first twinning dislocation
in the twin layer as shown in Fig. 4.6 and in table 4.3, we can see that the back stress
is smaller than the forward stress. If this were true, the first twinning dislocation should
glide further forward until the forward stress at the first twinning dislocation was equal
to the back stress. But this is not what we observed since the twin tip was stationary.
Therefore, there may exist a very large internal friction on the first twinning dislocation
(or say for the twinning dislocations near the twin tip). This internal friction on the
dislocations near the twin tip should be much larger than that on the dislocations located
far away from the twin tip. The magnitude of internal friction stress at the dislocations
near the twin tip are equal to the difference between the forward stress, the residual

stress and the stress due to the stacking fault (1”), the back stress (1“) i.e.,

Tfricm'p = 7-f + Trs - TS.F. - Tb.d.’ (476)

where mm, is the internal friction stress on the twinning dislocations near the twin tip.

The corresponding internal friction force on the dislocations near the twin tip is equal to

the difference between the corresponding driving force (the forward force plus the force

due to the residual stress) and the backward forces due to the stacking fault and the

dislocation interaction,

Ffric,tip = Ff + Frs - FS.F. ' Fb.d., (477)

where PM,” is the internal friction force on the twinning dislocations at the twin tip, F,,

175

is the force resulting from the residual stress 1“,. So the modified back stress and the

modified backward force should be

Tb,tip = Twp + Tex = TfJip + Tm (4.78)

Fb,tip = Fftip + F", (4.79)
01'

Fb.tip = b Tb.lip' (4.80)

Here the subscripts "f.tip" indicates the modified forward stress and modified
forward force at the twin tip. The modified forward stresses at the twin tip have been
calculated for the first three twinning dislocations in the previous section, so the modified
back stresses and the modified backward forces for the first three twinning dislocations
can be easily calculated using equations (4.78) and (4.80). The results of the modified
stresses and the modified forces for the first three twinning dislocations are listed in table
4.5. The plots for modified stresses are shown in Fig. 4.8 and for modified forces in

Fig. 4.9.

4.6.2. Dislocation Distribution in The Case of Equal Backward Force or Equal
Driving Force on Each Twinning Dislocation
At an equilibrium condition, the backward force on a twinning dislocation should
be equal to the driving force on the same twinning dislocation. Thus the equal backward

force condition is equivalent to the situation of equal driving force when we consider the

176

twinning dislocation distribution in a twin layer.

If we assume the backward force on any twinning dislocation is constant, then the
twinning dislocation distribution within a twin layer will be uniform, i.e., the spacing
between any two adjacent twinning dislocations will be constant, as shown in Fig. 4.10.
Here we selected a normalized backward force equal to 1.18 x 102. The result shows
that if the backward force on every twinning dislocation is equal to 1.18x10F2 Gb, the
spacing between any two adjacent twinning dislocations, except for the that two twinning
dislocations, is equal to 164 A. This result is inconsistent with the experimental
observation shown in Fig. 3.1 and Fig. 3.5. So this result indirectly proves that the

backward force (and the forward force) along a twin layer is not constant.

4.6.3. Effect of Dislocation Location on The Stress Distribution

Fig. 4.11 (a) shows how much the deviation of the second twinning dislocation
location from the measured position affects the back stress on this dislocation. We
selected :1: 10 A deviation from the measured value 48 A. This deviation of the
dislocation location results in either 131 MPa stress increment when the dislocation is at
the left side of the measured position or 87 MPa stress decrement when the dislocation
is located at the right side of the measured position. This indicates that a small deviation
of the dislocation location results in a large variation in stress on the second dislocation.
Location of the third twinning dislocation has a similar effect on the stress as the second
one, as shown in Fig. 4.11 (b), however, the effect is less than the second one. In Fig.

4.11 (b), the deviation of j: 10 from the measured value, 145 A, results in either 42 MPa

177

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178

increase in stress on the right side of the measured position or 34 MPa decrease in stress
on the right side of the measured position. If we look at the tenth dislocation, as shown
in Fig. 11 (c), the effect of deviation of dislocation location from the measured position
on the stress is very small. :th deviation results in only +4 or -3 variation in stress.
Even at the deviation of $50 for the tenth twinning dislocation location, the stress
change is only +20 or -15 MPa that is still smaller than those for the second and the
third twinning dislocations at the deviation of j; 10. Therefore, the determination of
twinning dislocation positions is very important for the dislocations near the twin tip.
For the dislocations far away from the twin tip, a slight deviation from the correct
position does not change the stress much. This is reason why the stress curves in Fig.

4.6 are rough near the origin of the distance axis (near the twin tip).

4.7. Summary

The distributions of forces and stresses along a thin twin layer was numerically
calculated based on the dislocation theory and the morphology of twin nucleation and
propagation analyzed in previous chapter. The forces acting on a twinning dislocation
are classified into three categories: forward force, backward force and external force.
The corresponding stresses are forward stress, back stress and external stress. The
calculation was carried out based on the twinning dislocation locations in an
experimentally observed thin twin layer. The results show that both forward stress and
back stress are very large at the twin tip and drop quickly as the distance from the twin

tip increases. The external stress is the residual stress in the matrix, which is uniformly

179

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distributed along the twin layer. Comparison between the original equations and the
simplified equations for the calculation of forces and stresses shows that there is a very

small difference between them.

CHAPTERFIVE

NIECHANICAL TWINNING DURING CREEP DEFORMATION IN TiAl

5 .1. Introduction

In recent years, much work has been done to understand deformation mechanisms
of TiAl in short term properties both at room temperature and at high temperature """55‘.
But for long term creep deformation, much less information is available in the literature.
Also, there are two differing results concerning mechanical twinning in creep
deformation of TiAl. Loiseau and Lasalmonie 92' investigated creep deformation of
equiaxed single phase 7 Ti.,.,A154 and found that mechanical twinning was an important
creep deformation mechanism at deformation temperatures up to 800 °C. Huang and
Kim [39‘ studied creep behavior of two phase 7+oz2 alloy with composition of Ti—47.0Al—
1.0Cr-1 .0V-2.5Nb at 900 °C and observed no evidence of mechanical twinning in creep
deformation. Since one goal of TiAl components will be to replace nickel and cobalt
base superalloys in aircraft applications, understanding creep deformation at high
temperatures is necessary. However, creep deformation mechanisms are not clearly
identified: (1) The activation energies for creep are much larger than those for self-
diffusion and interdiffusion in TiAl [37""”', which suggests that the creep rate may be

controlled by processes other than the usual lattice diffusion mechanism; (2) The reported

183

184

value of the stress exponent varies widely from about 2 to 8 “3'44"56'157'. This indicates
that several deformation mechanisms are involved in the creep of TiAl. The details of
deformation mechanisms are not known and the limited results in the literature are not
consistent with the creep theory. Mechanical twinning has been found to be an important
component of creep deformation in TiAl ”2'2””. It is possible that a parallel creep
deformation mechanism exists in the case when the mechanical twinning and some other
creep deformation mechanisms such as dislocation movement, diffusion, recovery, and
so on, occur independently. Therefore, it is important to understand the mechanical
twinning behavior during creep deformation and its contribution to the creep deformation

in order to understand and therefore improve our ability to optimize this material.

In this chapter, observations of mechanical twinning behavior during creep and
some contributions of mechanical twinning to creep deformation in near-y TiAl are
presented. The results are analyzed based on the twin nucleation and propagation theory
proposed in the previous chapters and in terms of a maximum resolved shear stress

criterion for mechanical twinning proposed in this chapter.

5.2. Material and Experimental Procedure

The creep specimens were produced and machined at Howmet Corp. , Whitehall,
Michigan. Investment cast test bars, 16 mm in diameter and 125 mm long, were HIP’ed
before specimens with 25 mm gage length and 5 mm diameter were machined from the

bars. After HIP’ing, the test bars were heat treated at 1300 °C for 20 hrs in Ar

185

atmosphere, and cooled at 65 °C/min in argon to produce the equiaxed + lamellar

microstructure. The nominal composition was Ti-48Al-2Nb—2Cr atomic percent.

A constant stress test and a multi-stress drop test were conducted at 765 °C in air.
A constant stress creep test at 176 MPa was interrupted at 4% strain, near the end of
primary creep. The multi-stress drop test started at 276 MPa and the stress was dropped
in increments to a final stress of 103 MPa at about 20% strain. The specimens were
fumace cooled to room temperature while maintaining the final stresses. The

temperature difference between the two ends of the specimens was less than 3 oC.

Microstructural investigation on the specimens before and after deformation was
carried out using optical and transmission electron microscopy. Specimens for optical
microstructure were prepared by making longitudinal sections from the deformed
specimens and the original bars. The specimens were etched using Kroll’s reagent to
view grain structures after the specimens were ground and polished. For TEM
investigation, however, 0.7 mm thick slices were cut in both longitudinal and transverse
directions from deformed and undeformed samples. 3 mm diameter disks were cut from
the slices using an ultrasonic cutting machine and ground to about 0.1 mm thick. The
disks were finally thinned in a double jets electropolishing system using a 10% sulfuric
acid + methanol solution at 20 °C. The TEM investigation was performed on a
HITACHI H800 transmission electron microscope with an accelerating voltage of 200

kV.

186

5.3. Results

5.3.1. Optical Microstructure
The initial microstructure after heat treatment was a duplex structure with (7+a2)

lamellar colonies and equiaxed 7 grains, as shown in Fig. 5.1 (a). After creep

 

Figure 5.1 - Optical microstructures (a) before creep deformation and (b) after creep
deformation.

deformation, cross twinning configurations were observed within lamellar colonies as
indicated by the arrows in Fig. 5.1 (b). One possible form of mechanical twinning

occurs parallel to the existing lamellar interfaces, which is denoted as "parallel twinning"

187
hereafter. But it is difficult to distinguish parallel twins from original lamellae from their

images since both have the same image characteristics.

5.3.2. Cross Twinning and Parallel Twinning in Lamellar Grains

The investigation in lamellar grains exhibited many cross twinning configurations.
The cross twinning was found to occur when two differently oriented lamellae meet and
either or both of the lamellae are oriented in such directions that twinning occurs parallel
to the lamellar interfaces, as shown in Fig. 5.2 (a). Lamellar orientations with respect
to the tensile axis were determined. For the inclined lamellae in Fig. 5 .2 (a), the
interfaces were tilted 42° from the tensile axis. For the vertical lamellae, lamellar
interfaces are tilted 5° away from the tensile axis. Many similar configurations were

found in the multi-stress—jump creep specimen.

At the intersection region of two lamellae in Fig. 5.2, many relatively fine twins
were formed. Investigating the coarse lamellae that are located away from the
intersection region, the orientation relationship between the two lamellae has true-twin
relationship, as shown in Fig. 5.2 (b). For convenience, we denote "Lamellae 1" to the
inclined lamellae and "Lamellae 2" to the vertical lamellae in the following. These two
sets of lamellae are indexed in Fig. 5.2 (c) by taking the 7 phase matrix as a reference
orientation and the indices are referred to 7 laths in lamellae. Therefore, the interfaces
of Lamellae l are (111)l (subscription 1 indicating Lamellae 1); the interfaces of
lamellae 2 are (l ll)2; and the intersection axis of two lamellar sets is perpendicular to

the page and in the [110] direction, as shown in Fig. 5 .2 (c).

188

 

(a)

Figure 5.2 - (a) Cross twinning configuration in a lamellar grain, (b) diffraction pattern
of the original lamellae, and (c) orientations of the original lamellae.

189

 

(Figure 5.2 continued)

190

Lamellae 2
F

[11111

   
 

 

 

Lamellae l

[1'i'0]

(C)

(Figure 5 .2 continued)

The cross twinning configurations shown in Fig. 5.3 is a different situation from
that described above. The cross twinning in Fig. 5.3 were formed in an equiaxed 7
grain, and both approximately vertically oriented twins and slightly inclined twins were

formed during creep deformation, which will be verified later.

5.3.3. Fine Mechanical Twins at Grain Triple Points

In the specimen deformed to a strain close to the end of primary creep, we found
that very fine mechanical twins frequently formed at equiaxed 7 grain triple points. Fig.
5.4 shows that these fine mechanical twins formed at the grain triple points within an
equiaxed 7 grain, as indicated by letters A, B and C. The thickness of the fine twins in

Fig. 5.5 (a) is about 50 nm. These fine mechanical twins are much thinner than typical

 

Figure 5.3 - Cross twinning configuration within a large equiaxed 7 grain.

192

 

 

Figure 5.4 — Fine mechanical twin configurations at equiaxed 7 grain triple points.

193

 

 

 

 

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(C)

(b)

Figure 5.5 — Fine mechanical twins (a), diffraction pattern across several fine twin interfaces (b), and twin relation in the

diffraction pattern (c).

194

lamellar laths that form in phase transformation ‘22]. The diffraction pattern across these
fine mechanical twins is shown in Fig. 5.5 (b) and (c). Fig. 5.5 (c) is a schematic
drawing of the diffraction pattern of Fig. 5.5 (b). The open circles in Fig. 5 .5 (c) are
the diffraction spots from the 7 matrix, and the rhombuses are the twin spots. In
addition to these two spot sets, there also exist very faint spots in Fig. 5.5 (b), which are
designated by dots in Fig. 5.5 (c). This set of diffraction spots represents the D019
crystal structure. This result indicates that two crystal structures exist in Fig. 5.5 (a):
one is twin related 7 phase layers (the fine mechanical twins and the untwinned matrix),
and the other is fine DO19 structure layers. The locations of these D0,, structure layers
are not known in this case. These fine lath configurations will be called fine mechanical

twins in this study.

5.4. Analysis and Discussion

5.4.1. Cross Twinning in Lamellar Grains

If we look at a 7/7 lamellar interface, there exist six possible orientation relation-
ships between the two adjacent 7 lamellae that can be described in terms of < 110>
directions “53””. Among these six relations there are two kinds of twin relationships:
if [110] directions in two 7 laths are anti-parallel to one another in the lamellar interface,
these two adjacent 7 lamellae have a {111} < 112] true-twin relationship; if [110]
direction in one 7 lath is anti-parallel to [011] or [101] direction in an adjacent 7 lath,

a pseudo-twin will form ”‘3'”9'.

195

In a cast and heat treated specimen, the 7/7 lamellar interfaces with a
{111} < 112] true-twin relationship are more favored than those with a pseudo-twin
relationship, since true-twin relation has lower interface energy and is thermodynamically
preferred during solidification and phase transformation. Fig. 5.6 shows interface
structures of a true-twin in (a) and a pseudo-twin in (b). In Fig. 5.6 (a), the interface
is a 23 boundary with a (111) boundary plane, and two laths are fully symmetric in the
boundary plane (111). The interface of pseudo-twin in Fig. 5 .6 (b) is a 226 boundary,
that is, only atoms of every second (111) plane are in completely symmetrical locations
with respect to the boundary plane (111), while atoms of other (111) planes are in anti-
symmetric positions, in which perfect symmetric sites are occupied by atoms of different
type. This even 2 boundary contains antiphase boundary elements “6‘“. Therefore, the
energy of lamellar interfaces with pseudo-twin relation is higher than those with true-twin
relation. This is consistent with our observations. We found that original lamellae had

true-twin relationship in the specimens investigated, as shown in Fig. 5.2 (b).

For mechanical twinning, there are the same kinds of configurations as analyzed
above. General twinning systems in 7 phase are in the form of {111} < ll2> , the same
as for disordered fcc crystals. However, because of its anisotropic crystal structure, the
different twinning systems result in either true-twins or pseudo-twins during deformation.
If the twinning shear occurs in the direction of [112] on a (111) plane, the deformed
crystal will be exactly symmetric to the undeformed crystal with respect to the (111) twin
plane, which is a true-twin, as in Fig. 5.6 (a). If a crystal is deformed by either of

(111x511) or (111)[1§1], the result will be a pseudo-twin, as shown in Fig. 5.6 (b). In

196

 

(b)

Figure 5.6 - Atomic arrangement of true twinning (a) and pseudo twinning (b).

197

order to understand the twinning behavior in the larger strain creep specimen , the
possible twinning systems in Fig. 5.2 (a) are plotted in a stereographic projection along
[110] direction by considering the tensile axis orientation, as shown in Fig. 5.7, where
the twinning system is assumed to be the {111} < 112> type. A resolved shear stress
on each twinning system in Fig. 5 .7 is considered to determine the probable twinning
systems in this particular orientation. Table 5 .1 shows the computed Schmid factors for
these possible twinning systems, where d) is the angle between the tensile axis and the
normal to the twin plane, A is the angle between the tensile axis and the twinning
direction. In Lamellae 1, both (lll)[211] and (lll)[l2l] have relatively large Schmid
factors, and are more likely to operate during creep deformation. In lamellae 2, all
Schmid factors are small, so twinning is less likely. Therefore, the mechanical twins in
Fig. 5.2 (a) are formed by operating either (lll)[2ll] or (lll)[l2l] or both twinning
systems in lamellae 1. The mechanical twins in this case are pseudo-twins, and they
were clearly formed during creep deformation, since this kind of twinning is not

observed in undeformed specimens.

Since these pseudo-twins are not thermodynamically favored, an interfacial energy
criterion of mechanical twinning is unsuitable in this case. A maximum resolved shear
stress criterion for mechanical twinning is appropriate. This criterion is that mechanical
twinning operates in the twinning system with the highest resolved shear stress.
However, if the maximum resolved shear stress is less than a certain critical value, the

interfacial energy criterion may play a role.

198

 

Figure 5.7 - Stereographic projection of possible twinning systems along [110] direction.

199

Table 5.1. Schmid Factors for Mechanical Twinning Systems
in Lamellae 1 and Lamellae 2.

Lamellae l
(lll)<112> <15 A COM
(1 1 1)[1 12] 48° 84° 0.070
(lll)[211] 48° 45° 0473*
(lll)[l21] 48° 52° 0412*

(lll)[112] 48° 96° 0.070

 

lamellae 2
(111)<112> <15 A cosmos);
(lll)[fi] 85° 53° 0.052
(lll)[fii] 85° 20° 0.082
(lll)[fil] 85° 66° 0.035

(lll)[112] 85° 127° 0052

 

* Probable twinning system

Fig. 5.8 schematically shows the formation of cross twins in Fig. 5.2 (a)
according to the maximum resolved shear stress criterion. Fig. 5 .8 (a) is a starting
situation before twinning. Fig. 5.8 (b) is a metastable situation showing growth of
mechanical twins in lamellae 1 into lamellae 2. To accommodate this growth,
mechanical twinning on the other twinning system with a high Schmid factor is needed

as shown in Fig. 5.8 (c). Most thick twins can easily traverse through existing thin

200

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(b)

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metastable condition of twinning in lamellae l; (c) final configuration of
cross twins as seen in Fig. 5.2 (a).

 

 

 

 

 

 

 

(c)

(Figure 5.8 continued)

vertical twins. Therefore, mechanical twinning in Fig. 5.2 (a) occurred in the inclined
lamellar colony parallel to the lamellar interfaces and across the vertical lamellar colony
by twinning shear in (111)[2i_1] and (lll)[12l] systems. Therefore, this is a parallel
twinning with respect to the inclined lamellae, but it is also a cross twinning with respect

to the vertical lamellar colony.

In Fig. 5.2 (a) the inclined lamellar laths become finer after twinning. This
indicates that extensive mechanical twinning occurred during creep deformation in this

lamellae. But vertical lamellar laths are unchanged. This is because the inclined

202

lamellar laths of lamellae 1 contain two possible twinning systems having the highest
Schmid factor and the vertical lamellar laths of lamellae 2 do not have high Schmid
factor twinning systems. This is consistent with the maximum resolved shear stress

criterion.

5.4.2. Cross Twinning in Equiaxed 7 Grains

The cross twinning configuration in Fig. 5.3 shows a different situation than that
in Fig. 5.2 (a). In the case of Fig. 5.3, the cross twins were formed in an equiaxed 7
grain, and both approximately vertically oriented twins and slightly inclined twins are
formed during creep deformation, which is easily verified by carefully investigating the
twinning sequence of these two sets of mechanical twins. The inclined twins are sheared
by the formation of vertical twins at their intersections. This indicates that the inclined
twins formed before the vertical twins. However, the inclined twins are tapered as they
grow into the grain interior, which means that these twins are formed by the twinning
mechanism described in chapter three. A possible scenario for this sequence is as
follows: The inclined twins were first formed due to a local stress concentration, and
the vertical twins were generated by the operation of twinning system with a maximum

resolved shear stress resulting from the external tensile creep stress ”8'.

Comparing our result with others ”2'3” indicates that mechanical twinning is
playing an important role in creep deformation at the combination of stress and
temperature. Mechanical twinning reduces a stress concentration so as to maintain strain

continuity. There may be a twinning transition temperature, below which mechanical

 

203

twinning occurs. In Loiseau and Lasalmonie’s study 1221, the transition temperature is
about 800 °C for a complete equiaxed single 7 phase. But this transition temperature
may change depending upon the composition and microstructure. Our observations of
mechanical twinning in creep specimens are consistent with Loiseau and Lasalmonie’s
result where the equiaxed 7 grain structure was investigated. In Huang and Kim’s study
‘39], the material was in the two phase 7+oz2 lamellar condition; the mechanical twinning
did not operate, possibly due to the deformation temperature higher than the twinning
transition temperature. At high temperature, since dislocations are much more mobile
and some cube <100> dislocations become mobile, mechanical twinning is less

favorable than that at temperatures below the twinning transition temperature.

5.4.3. Fine Mechanical Twins at Equiaxed 7 Grain Triple Points

(a) Fine Mechanical Twins Formed by Accommodation of Stress Concentration

at Equiaxed 7 Grain Triple Points

Comparing a crystal orientation in the untwinned area with that in the fine twin
region, we found that the matrix orientation in the fine mechanical twin region in Fig.
5.4 was exactly the orientation of the untwinned area. This suggests that the fine
mechanical twins are formed by deforming the 7 phase. This is also evident in the
configuration of the fine mechanical twins growing toward the 7 phase grain interior,
which is located between the fine twin region and the untwinned region as indicated by

arrows in Fig. 5.4.

204

The configurations of twinning accommodation of stress concentrations at
equiaxed 7 grain triple points were also occasionally found in the specimens before creep
deformation. However, these accommodation twins are widely spaced compared to the
fine mechanical twins in crept specimens. These accommodation twins at the grain triple
points in the specimens before creep deformation are probably formed either by the
previous HIP’ing process or by the heat treatment or both. The deformation mechanisms
in HIP’ing process are similar to the creep deformation mechanisms, for example, the
grain boundary sliding and the diffusion accommodated deformation “6”. Therefore, the
stress concentration at the grain triple points during HIP’ing process is probable. Since
the cooling rate after heat treatment is relatively fast (65 °C/ min), the thermal stress
concentration at equiaxed 7 grain triple points is also possible due to the anisotropic
thermal contraction in 7 phase. However, the stress concentration formed during the
HIP’ing or the cooling could not result in so intensive twinning as observed in the crept
specimens. The shear displacement along the twin interfaces between two points "M"
and "N" in Fig. 5.5 (a) is calculated to be 1.18 pm. Such a large displacement could
not be caused by the thermal stress arising from the thermal contraction mismatch. The
observed fine mechanical twins at equiaxed 7 grain triple points in this study are
probably formed either by refining the largely spaced accommodation twins existing in
the specimens before creep deformation, or by twinning in untwinned grains due to local

stress concentrations at equiaxed 7 grain triple points during creep deformation.

(b) Formation of Fine Mechanical Twins Prevents Grain Boundary Sliding

Fig. 5.9 is an another example of fine mechanical twins formed at an equiaxed

205

7 grain triple point. These fine mechanical twins formed to accommodate the stress
concentration caused by grain boundary sliding and/or the deformation of adjacent
equiaxed 7 grains at the grain triple point (Fig. 5.9 (a)). However, this accommodation
twinning caused the grain boundaries to become zigzagged instead of smooth (Fig. 5.9
(b)). The presence of curled dislocations in the untwinned neighbor equiaxed 7 grain
suggests that subsequent creep deformation occurred by dislocation slip in the untwinned
equiaxed 7 grain interior near the zigzagged boundaries (Fig. 5.9 (c)). All three of the
grain boundaries that form the equiaxed 7 grain triple point in Fig. 5.9 (a) are large
angle grain boundaries. The grain orientation relationships and boundary structures of
these three grains (designated as grain I, II, III in Fig. 5 .9 (a)) were determined and are
shown in table 5.2. The nearest coincidence boundary and the deviation from this

boundary are indicated. The fact that these boundaries are high angle, and not low
energy special boundaries, indicates that grain boundary sliding would have been
relatively easy "‘2‘. These grain boundaries were smooth before twinning since they were
equiaxed 7 grain boundaries, which can be seen if one looks at the same grain boundary

in untwinned region.

Table 5.2. Misorientation and Grain Boundary Structure of Grains
at the Grain Triple Point.

 

Grain boundary I/II II/III III/I
Misorientation [120]/45° [164]/150° [in/132°

Boundary structure 3° from 215 7° from 233,, 7° from 2331,,

 

206

 

Figure 5.9 - Fine mechanical twins from accommodation of local stress concentration
(a), twin end configuration at equiaxed 7 grain boundary (b), and
dislocation slip near the zigzagged boundary (c).

207
The zigzagged grain boundary resulting from mechanical twinning is much

different from the serrated lamellar grain boundary in their morphology. The
" serrations" of the zigzagged grain boundary formed by mechanical twinning are
regularly distributed along the trace of the previous equiaxed 7 grain boundary, and the
" serrations" are small, as shown in Fig. 5.9 (b). However, the typical lamellar grain
boundary, the boundary between an equiaxed 7 grain and a lamellar grain or the
boundary between two lamellar grains, formed from phase transformation, are roughly

serrated, and the "serrations" are large and irregular, as shown in Fig. 5 .10 and 5.11.

(c) On D0,, Crystal Structure in Fine Mechanical Twins

The twinning mechanism in TiAl observed in this study is a homogeneous shear
of a crystal by the glide of 1/6<112] twinning dislocations on every (111) close packed
plane. However, if the stress concentration at the grain triple points and/or grain
boundaries was such that it did not create 1/ 6 < 112] twinning dislocations on every (111)
plane, four atomic layers of D0,,, structure, i.e. , a nucleus of the Ti3Al crystal structure,
could be formed by glide of a 1/6< 112] twinning dislocation on a (111) plane “‘5'. In
addition to this, the crystal structure across the true—twin plane in TiAl is also a three
atomic layer DO19 structure. Therefore, the occurrence of DO19 diffraction spots with
the twin related TiAl diffraction spots in the diffraction pattern in Fig. 5.5 (b) is
reasonable. It is worth noting that such a uniformly distributed thin lath configuration
in Fig. 5.5 (a) is very difficult to form through the phase transformation such as 012 --- >
a2+7 or a --- > a2+7. Such a thin new phase layer, 7 phase in this case, could not be

thermodynamically stable in the case of phase transformation, and therefore, the thin 7

208

 

Figure 5.11 - Optical microstructure showing the roughness of lamellar grain boundary.

209
phase layer should either grow and become thicker or be eliminated by growth of the

adjacent 7 laths. The resultant configuration should be the lamellae containing coarse
7 laths with different thickness. The observation of fine twin configurations only near
the equiaxed 7 grain triple points in this study also provides an evidence that the fine
mechanical twins are formed due to the local stress concentration at the equiaxed 7 grain

triple points but not by the phase transformation.

5 .5. Conclusions

1. Mechanical twinning is an important deformation mechanism in creep

deformation of TiAl intermetach compound.

2. An energetic criterion is not suitable for mechanical twinning in this study.
Above a certain value of stress (at constant temperature), the twinning transition stress

is apparently temperature dependent.

3. Mechanical twins are formed by either true-twinning or pseudo-twinning. The
formation of mechanical twins during the creep deformation of TiAl follows the

maximum resolved shear stress criterion.

4. The mechanical twins observed in this study are propagated by homogeneous
glide of 1/6<112] twinning dislocations on every close packed (111) plane, that is, by

a homogeneous shearing mechanism.

 

210

5. The stress concentration at the grain triple points can be accommodated by the
formation of fine mechanical twins. The zigzagged grain boundaries formed by
mechanical twinning can inhibit grain boundary sliding. The formation of fine
mechanical twins is probably controlled more by the local stress concentration than by

an externally applied stress.

CHAPTERSIX

CONCLUDING REMARKS AND RECOMMENDED FUTURE WORK

The in situ investigations of mechanical twin nucleation and propagation presented
here have demonstrated that mechanical twin nucleates by bowing out of twinning
dislocations at grain boundaries due to the local stress concentration and propagates by
continuous emission of twinning dislocations from the grain boundaries and glide of
twinning dislocations on every adjacent twinning plane. The nucleus of true-twinning is
found to be either a superlattice intrinsic stacking fault (SISF) in the case that one
1/6[112] twinning dislocation is emitted from the grain boundaries in a (111) plane, or
a superlattice extrinsic stacking fault (SESF) when two 1/6[112] twinning dislocations are
emitted from the grain boundaries in two adjacent (111) planes. The mechanical twin
propagation observed in this study occurs by trailing twinnning dislocations pushing

leading twinning dislocations in the twin layer.

The stress analysis on a thin twin layer has shown that the stress distribution
along the thin twin layer is not even. The changes of forward stress and back stress are
similar. The magnitudes of forward stress and back stress are very large near the twin

tip and drop very quickly as the distance from the twin tip increases. At very large

211

212

distance from the twin tip, the stress changes are very small and both stresses tend to
become constant. The calculation of external stress has shown that the magnitude of
external stress is small and evenly distributed along the thin twin layer investigated. The
external stress has been identified as a residual stress within the grain where the thin twin
layer is located. The result of stress analysis indicates that the trailing twinning
dislocations glide forward more easily in the twin layer than the leading twinning
dislocations. This result is consistent with the experimental observation of the trailing

dislocation pushing the leading dislocation phenomenon.

The formation of various mechanical twin configurations observed in creep
deformed specimens has been well understood, and interpreted using the twinning
mechanism identified and the maximum resolved shear stress criterion proposed in this
study. The cross twinning occurs when two differently oriented lamellar colonies meet
in such a way that one lamellar colony with a larger Schmid factor generates twins across
the other one. However, the cross twinning configurations are also observed in the large
equiaxed 7 grains: one set of twins, which initiate at grain boundaries, are formed due
to the local stress concentration and the other set of twins, which cross the first set of
twins, are formed by the external loading. Analysis of the cross twinning in the lamellar
colonies indicates that an energetic criterion for mechanical twinning is not suitable in
this study, and the pseudo-twinning obeys the maximum resolved shear stress criterion.
The fine mechanical twins at the equiaxed 7 grain triple points are due to the

accommodation of local stress concentration during the creep deformation.

213

to fully understand the mechanical twinning and creep deformation in TiAl:

1. Correlations between mechanical twinning and creep deformation in TiAl, or

the role of mechanical twinning in the creep deformation of TiAl.

2. What type of grain boundary structure is preferred for mechanical twin

nucleation at the grain boundary.

3. Investigation of creep deformed single crystal TiAl in which the Schmid law

can be precisely applied.

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