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Deformation Twinning – Mechanisms and Modeling in FCC,
BCC Metals and SMAs Huseyin Sehitoglu
Mechanical Science and Engineering August 26, 2015
Grad. Students and Collaborators: K.Gall, I.Karaman, D.
Canadinc, J. Wang, T. Ezaz, A. Ohja, L. Patriarcha,
P.Chowdhury, S. Kibey, W.Abuzaid, M.Sangid, H.J. Maier , Y.
Chumlyakov
http://html.mechse.illinois.edu
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Background § Deformation modes in metals and alloys § Twinning
in fcc metals (Part 1) § Twinning in bcc metals (Part 2)
Twinning stress in SMAs-Twin nucleation model- §
Peierls-Nabarro (P-N) formulation § Energy landscape (GPFE) in
Ni2FeGa § Twin nucleation model based on P-N formulation
2
Outline
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3
Plas%c flow in fcc materials: slip
and cross-‐slip
Polycrystalline material
Single crystal/grain
twinning
slip low SFE metal e.g.: pure
Ag
stacking fault ribbons
TEM image from: Whelan, Hirsch,
Horne and Bollmann, Proc. Roy.
Soc. London (1957). Karaman-‐Sehitoglu,
Acta Mater (2001).
dislocaNon arrays Fuji et al.,
Mater. Sci. Engg. A 319 (2001)
415-‐461.
DislocaNon cells
low SFE alloys e.g.: nitrogen
steels
strain
stress
Stage I
Stage I
twinning starts
Stage III
medium/high SFE metal e.g.: pure
Al
cross-‐slip
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Deformation by Twin (fcc)
Deformation twin in Fe-Mn-C steel [001] orientation 3%
strain
I. Karaman- Sehitoglu et al, Acta Mater.(2000).
fcc
fcc
twin
B
C
C
A
B
A
C
fcc
fcc
twin
Mirror symmetry is seen across the twin boundary.
Twin boundary
Twinning : mechanism of plastic deformation at crystal
level.
twin boundary
twinning shear
a a
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Deformation by Slip Slip due to a perfect dislocation
Polycrystalline alloy
slip
Single crystal/grain
Karaman, Canadinc, Sehitoglu et al. Acta Mater (2001-2006).
dislocation
arrays
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Plas%c deforma%on due to slip
Slip due to a perfect
dislocaNon
Callister (2000)
slipped state Intrinsic stacking fault
t2 t1 l
b1
b2
extended dislocaNon A perfect dislocaNon
may split into parNal dislocaNons…
Lee et al., Acta Mater (2001)
Intrinsic stacking fault
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fcc uz fcc
0.5
usγu
unstable
1.0 1.5 2.0 2.5 3.0
[ ]111
112⎡ ⎤⎣ ⎦
A
A
B C
B C
A
fcc
primiNve cell
p q
r s
(111)
Energy pathway for a stacking
fault
hcp
isfγs
isf
ABC
AC
A
intrinsic stacking fault (isf)
B
Generalized stacking fault energy (GSFE)
(Vitek, 1968)
12bp bp
maximum
maxγ
m
AB
AA
C
BC
12bp
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Energy landscape for a stacking
fault (g-‐surface)
xu12
zu16
isfγ
maxγ
S. Kibey, J.B. Liu, M. W.
CurNs, D. D. Johnson and H.
Sehitoglu, Acta Mater. 54 (2006)
2991-‐3001
usγunstable stacking fault energy
(Rice,1992)
A
C
s
B u
m Energy for SF formaNon during
passage of a Shockley parNal=
area under this surface
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Classical twin nuclea%on model
Venables, DeformaNon Twinning, Eds.
Reed-‐Hill,Hirth and Rogers (1964)
crit crit2 isf
p
1 22 b
K⎡ ⎤⎛ ⎞ γ− + =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
θ θ τ τβ
1= =θ β
fiNng parameters: K, q and
b
Classical twinning stress equa%on:
Calibra%on of fiNng parameters for
different alloys is required.
need a more fundamental approach
to predict twinning stress.
Cu-‐based alloys
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Energy required to twin the laNce
top view 2Τ
B C
B
C B A
A
A
B C
B C
B C
A
A
A
Intrinsic stacking fault
A C
A
32
a
B C
B
C
A
A
A
two layer fault
A
B A
3a
3Τ 3Τ3Τ
p2bpb
B C
B
C
A
three layer twin
A C
B
3Τ
p3b
A
A
next periodic supercell
[ ]111
112⎡ ⎤⎣ ⎦
B C B B
C A B A
fcc
B C C
A
Area under this curve is the
required energy to twin the
laNce by successive shear
usγutγ utγ utγ utγ
isfγ tsf2γ tsf2γ tsf2γ tsf2γ
1Τ
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Energy pathway for twinning : pure
Cu
usγutγ utγ utγ utγ
isfγ
tsf2γtsf2γ tsf2γ tsf2γ
• VASP-‐PAW-‐GGA • 8 x 8 x
4 k-‐point mesh
• 273.2 eV energy cutoff.
S. Kibey, J.B. Liu, D.D. Johnson
and H. Sehitoglu, Appl. Phys.
Lec. 89 (2006) 191911.
Fault energies converge aYer third
layer sliding indica%ng the comple%on
of twin nuclea%on.
TBMγ
TBF2= γ
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Energy pathway for twinning : pure
Pb
usγ utγ utγ utγ
isfγtsf2γ tsf2γ tsf
2γ tsf2γ
utγ
twin nuclea%on twin growth
• VASP-‐PAW-‐GGA • 8 x 8 x
4 k-‐point mesh
• 237.8 eV energy cutoff.
Convergence occurs aYer the third
layer sliding for Pb as well.
Hence, a three-‐layer twin is
considered as the basic nucleus
in fcc metals.
S. Kibey, J.B. Liu, D.D. Johnson
and H. Sehitoglu, Acta Materialia
55 (2007) 6843-‐6851
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Computed fault energies for fcc
metals
The above table represents the
most complete set of DFT-‐based
theoreNcal calculaNons of fault
energies for fcc metals.
a fault energies from individual Refs.
in Table A-‐1, Hirth and Lothe
(1982). b fault energies computed
using SP-‐PAW-‐GGA. Siegel, Appl.
Phys. Lec. (2005) c pair potenNal.
RauNoaho, Phys. Status Sol. (1982).
H. Sehitoglu et al., Acta
Materialia 55 (2007) 6843-‐6851
(all energies in mJ/m2 )
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Mesoscale model for fcc twins
Total energy of the twin nucleus:
Etotal = Eedgeenergy contribution of edge components
+ Escrew
energy contribution of screw components
− Wτwork done byapplied stress
+ EGPFEenergy associated with twin-energy pathway
Mahajan and Chin, Acta Metallurgica
(1973)
DislocaNon configuraNon of the nucleus
( )( ) { } ( )
22
2
0
2
2 11 1
4 1 2 26 9
tw
s
i
e
n GPFE
totalGb d d d
N ln N ln ln
N d
Gb ddN ln
NN
N r
b
E d
E
,N −+ +
−
−
+
−−
⎡ ⎛ ⎞ ⎤⎜ ⎟⎢ ⎥
⎡ ⎛ ⎞ ⎤⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎦ ⎣
=⎠ ⎝ ⎠
τ
π υ π
Aδ
Bδ−
Bδ
Aδ
Cδ
Bδ
Bδ
[ ]111
⎡ ⎤⎣ ⎦211
⎡ ⎤⎣ ⎦011
A
C d
Total energy:
H. Sehitoglu et al.,Acta Materialia
55 (2007) 6843-‐6851
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Total energy of the twin nucleus
{ { {γ-energy required to
energy associat twin the latti
γ-
energy requiredto cross-slip
ed with twin-energy pathway c e
GPFE Stwin FE EE = −
usγ utγ utγ utγ
isfγtsf2γ tsf2γ tsf
2γ tsf2γ
utγ
twin nuclea%on
twin growth cross-‐slip
( ) ( ) ( )
( )
22
0 0
0
2
2
21 19
1
21
4 1 2 6dd
tw
total
twin F i
e
n
s
S
Gb d d dN ln N l
d dx
Gb ddN lnE n ln NN r
N
d ,N
N dd d
N
bx
⎡ ⎤⎛ ⎞⎧ ⎫⎛ ⎞+ − −⎢ ⎥⎨ ⎬ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎢ ⎥⎩ ⎭ ⎝ ⎠⎣= +
+
⎡ ⎤⎛ ⎞ −⎜ ⎟⎢ ⎥
−
⎝ ⎠⎦ ⎦
− −
⎣
∫∫ τγυ
γ
π π
Total energy:
( )- 01d
twin twinE N d dxγ γ= − ∫
- 0
d
SF SFE d dxγ γ= ∫
EGPFEenergy associated with twin-energy pathway
= Eγ -twinenergy required to twin the lattice
− Eγ -SF
energy requiredto cross-slip
Energy contribuNon of GPFE:
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Twinning stress equa%on
H. Sehitoglu et al.,Acta Materialia
55 (2007) 6843-‐6851
For a stable twin configuraNon:
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Predicted twinning stresses for fcc
metals
Twinning stress depends non-‐monotonically
on stacking fault energy.
τcrit ∼ K
γ isfbtwin
does not hold !
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Predicted twinning stresses for fcc
metals (contd.)
Twinning stress depends monotonically on
unstable twin SFE .
UnstableTwin Energy governs the physics of
twin nucleaNon.
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Predicted twinning stresses for fcc
metals (contd.)
S. Kibey, J.B. Liu, D.D. Johnson
and H. Sehitoglu, Acta Materialia
55 (2007) 6843-‐6851
b Bolling, Casey and Richman, Phil.
Mag. (1965). c Suzuki and Barrec,
Acta Metall. (1958). d Narita et
al., J. Japan Inst. Metals
(1978). e Yamamato et al., J.
Japan Inst. Metals (1983).
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Part 1-‐Summary
• Presented a hierarchical, mulNscale,
adjustable parameter-‐free approach for
twin nucleaNon in fcc metals
and alloys.
• Predicted twinning stresses are in
excellent agreement with available
experimental data.
• Our theory inherently accounts for
direcNonal nature of twinning.
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Background § Deformation modes in metals and alloys § Twinning
in fcc metals (Part 1) § Twinning in bcc metals (Part 2)
Twinning stress in SMAs-Twin nucleation model- §
Peierls-Nabarro (P-N) formulation § Energy landscape (GPFE) in
Ni2FeGa § Twin nucleation model based on P-N formulation
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Outline
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A theoretical model to predict the twinning stress has not been
established.
Theoretical model is needed
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Deformation by Twinning (bcc)
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DIC measurements: An example
24
σAfter =180 MPa σAfter = 220 MPa σAfter = 260 MPa
180 MPa
220 MPa
260 MPa
FeCr [010] compression High resolution DIC images (5X) allow to
capture the residual strain field after each loading stage allowing
to pinpoint the slip or twinning stress precisely.
(%)ε
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Generalized planar fault energy (GPFE) (MD calculations) for
FeCr
We are concerned with the twin nucleation region of the
GPFE.
25
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Other theoretical model
Too high!
A better model to predict the experimentally measured twinning
stress is lacking.
html.mechse.illinois.edu 26
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University of Illinois at Urbana Champaign Twinning mechanisms
in bcc
Mechanism
Pole mechanism1
Disloca%on core dissocia%on2
Slip disloca%on interac%on3
→ ×a a[111] 3 [111]2 6screw
Co`rell, A.H., Bilby, B.A., 1951.
Priestner, R., Leslie, W.C.,
1965. Sleeswyk, A.W.,1963.
. Lagerlof, K.P.D., 1993.
27
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Experimental observations validate that three slip systems of
symmetric configuration may be activated.
Why dislocation dissociation mechanism?
28
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Incorporate
Area under the GPFE gives the energy barrier to nucleate a twin.
We consider a three layer twin nucleus.
Model development
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University of Illinois at Urbana Champaign Prediction of
twinning stress
Total energy of dislocation configuration is written as:
0 0
- ( [ ] [ ] [ ]) - ( ) - ( ) ( d dα τ τ γ γπ
= + + + + ∫ ∫2ln ln ln - )2 2A Ar r2
B A B Atotal rss A o rss B o GPFE SF
o o o
r - r r rGbE n Gb b r - r b r - 2r x xr r r
'1 γ ) τπ
⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭
critical twinGb
b d
2 3 -2 3(2 ( 3 -1)
' ( ) [ ] ) [ ]twinγ γγ γ π γ γ π+− += −1sin 2 2.5 1.21 (2 2 sin
2 1.22 where N = 3
2 4UT SF
UT UT TSF- N -
G is calculated from MD d is the distance b/w dislocations A and
B and can be calculated from above equation
Interaction energy
No empiricism in the model
Line energies Work done GPFE
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35 40 45 50 550
200
400
600
Modeling (Present study)
γ TBM 2mJ ( )m
Experiments (including present study on Fe50Cr)
Fe-25Ni (Nilles et.al.)
Fe (Harding) Fe-50Cr
Fe-3V (Suzuki et. al.)
Application to other bcc alloys
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Close agreement!
Comparison with experiments
html.mechse.illinois.edu
a Harding (1967,1968) b Calcula%ons
based on Ogata et al. (2005)
c Nilles and Owen (1972) d
Suzuki et al. (1966)
32
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100 200 300 400 500 6005
6
7
8
9
10+
+
Modeling (Present study)
Experiments
Fe-3at.%V
V
Fe-25at.%Ni
Fe
Nb
30 γ ut − 2γ tsf( ) d1 + d2( )d{112} γ ut + γ sf( )
Ta Fe-3at.%Si
Mo
W
10#
10#
10#
10#
10#
10#
33
Harding , Proc.R. Soc.1967, 1968 Meyers et al.Acta Materialia,
2001 Narita and Takamura, Disloc. Solids, 1992 Nilles and Owen,The
Soc. of Metals, 1972
Sehitoglu et al., Phil Mag Letters, 2014
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Twin system-I
Twin-system-II
Twin Migration Stress
Twin migration stress is the stress required to move the
twinning dislocation along the twin boundary, thus translating it
by one layer.
39
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Prediction of twin migration stress
Theoretical prediction is needed!
html.mechse.illinois.edu 35
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University of Illinois at Urbana Champaign Twinning partial
y[010]
a (121)[111]6
Incident disloca%on a3× [111](121)6
a (121)[111]6
Twin
x[100]
= 1.0arbResidual dislocation plays an important role
What is twin migration?
r 1 2b = b - b36
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An incident twin is blocked because of the higher magnitude of
residual dislocation at the twin boundary.
rb =1.0a
37
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Slip-slip interaction
Slip-slip interaction
Twin-twin interaction
Twin-twin interaction Twin migration stresses
We try to predict these stresses
html.mechse.illinois.edu 38
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Twin systems analyzed in present work
39
Christian and Mahajan, Progress in Materials Science,1995
Line of intersection of twins Example: Intersection type
Cross product of n1 X n2
n1
n2
121( )× 112( )
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τ 2τ1
= τM
τ T=
4b2(Gbr2 +
Gb22
4πln(
r2w
)+ γ modifiedGPFEbN2
b( N2+1)
∫ )
2πb2Gb1
2
4πln(
r2w
)
Minimization
∂Utotal∂ς1
=∂Utotal∂ς 2
= 0
Utotal = Einteraction/incident +Eline/incident +Eline/outgoing
+Eresidual +EincidentGPFE +EmodifiedGPFE -Wincident -Woutgoing
= -Gb1
2
2πln[
rB - rAro
] + ln[rB2ro
]+ ln[rAro
]⎧⎨⎩⎪
⎫⎬⎭⎪ζ1 +
Gb12
4πln(
Rw
) 2d +ζ1{ }+ Gb22
4πln(
Rw
) l2 -ζ 2{ }+
Gbr
2
4πln(
Rw
)ζ 2 + A1 γ incidentGPFE0
N1
∫ dλ + A2 γ modifiedGPFEN2
N2+1
∫ dλ - τ1b1A1 - τ 2b2A2
40
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Residual dislocation
41
+Blockage
Incorporation+ Blockage
Incorporation+ Blockage
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br =1.0a br = 0.8a
mτ =167 MPa mτ =144 MPaTheory
Theory Experiment
[101] Compression [111] Compression
interesection
intersection
42
τ M
τ T= 0.83
τ M
τ T= 0.74
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Results-Extended
43
τ M
τ T⎛⎝⎜
⎞⎠⎟
Harding , Proc. R. Soc.1967, 1968 Suzuki, J. Phys. Soc. Jpn,
1962
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Conclusions-Part 2
ü Twin migration stress is lower than the twin nucleation
stress.
ü Residual dislocation affects twin migration stress. Higher the
magnitude of residual dislocation, higher is the twin migration
stress.
ü Intersection types of interacting twins is an important
parameter to predict the outcome of twin-twin interaction. Higher
magnitude of residual dislocation in , and cases causes the
incident twin to be completely blocked.
44
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Background § Deformation modes in metals and alloys § Twinning
in fcc metals (Part 1) § Twinning in bcc metals (Part 2)
Twinning stress in SMAs-Twin nucleation model (Part 3)- §
Peierls-Nabarro (P-N) formulation § Energy landscape (GPFE) in
Ni2FeGa § Twin nucleation model based on P-N formulation
45
Outline
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Shape Memory Alloys (SMAs)-Part 3
Darren Hartl, Aerospace applications of shape memory alloys
46
Applications of SMA including medical and aerospace.
Reduction of engine noise
SMA beams
Chevron
• Thermal Shape Recovery ü Shape Memory
• Elastic Shape Recovery ü Pseudoelasticity
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Detwinning and Twinning of NiTi
Martensite
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Compressive stress-strain response of Ni54Fe19Ga27 at
temperature of -190 °C.
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Energy Barrier of (100) Twin
2
mJm
γ ⎛ ⎞⎜ ⎟⎝ ⎠
xuc
241TMEmJm
γ =
[ ]00113.5
=M
cc
Generalized planar fault energy (GPFE)
[ ]100
[ ]001
[ ]010Ti Ni 0.46 A Shuffle in
Ti 0.23 A Shuffle in Ni
3. [001]9a
B19’ 3 layer twin aYer only
shear 3 layer twin aYer
shuffle
following shear
Shear DirecNon
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uA: atom displacement above slip plane (plane A) uB: atom
displacement below slip plane (plane B) f(x): disregistry or slip
distribution, uA-uB Solve f(x) using Force balance
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Hall, Bacon
Narrow disloca%ons are more difficult
to move than wide ones.
Disloca%ons with larger b are
more difficult to move. As
unstable fault energy increases, the
disloca%on width narrows.
Disregistery becomes complex for SMAs
(for slip and twinning)
Review of Peierls-Nabarro (P-N) model
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( )1 1 1 1 x N 1 db b x x d x 2df (x) tan + tan + tan +...+ tan2
N
− − − − − −⎡ ⎤⎛ ⎞− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟π ζ ζ ζ ζ⎝
⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
Due to the interaction of multiple twin dislocations, the
disregistry function f(x) is:
-40 -20 0 20 40 600.0
0.2
0.4
0.6
0.8
1.0
Twin nucleation Dislocation slip
f (x)b
xζ
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Lenticular twin
Twins nucleate from1-2 grain boundary and then grow toward 2-3
grain boundary.
Wang L et al, Metallurgical and Materials Transactions A,
2010.
Twin configuration
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GPFEETwin boundary energy (GPFE)
GPFE of L10 Ni2FeGa
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55
( )( ) ( ) ( )
( )( )( )
2 2
2
1 12 22 2
1 cos 21
4 1 2
12sin tan +...+ tan +...+ 2
tsf isfus isf ut
m
m
b N b Nshd N sh
Nmb d mb NdN mb d mb Nd
µ ν θ γ γτ γ γ γ
π ν
ζζζ ζ ζ ζ
=∞− −
=−∞
− ⎧ ⎫⎡ ⎤+⎛ ⎞⎪ ⎪= + − + − − ×⎨ ⎬⎢ ⎥⎜ ⎟− ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎧ ⎫⎡ ⎤ −
−⎛ ⎞ ⎛ ⎞− − −⎪ ⎪ ⎪ ⎪×⎨ ⎬ ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ + − + −⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎪ ⎪⎩ ⎭
⎩ ⎭
∑
Critical stress required to nucleate a twin:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
int
2 i=N-1
to GPFE
+ +
SF twinm=- m=-
lital ne
2
22
2
i=2
E = E
γ (f(mb - d)
E
µb L1- νcos θ N ln - ln N - 2 !+ ln N - i !+ ln i -1 !4π 1-
ν
)b + N -1 γ (f(mb - d)
+ + -E
Nµb 1- νco
d
W
Nτ)b
= +
+ s θ4π 1- ν
h- ds∞ ∞
∞ ∞
⎧ ⎫⎡ ⎤⎨ ⎬⎣ ⎦⎩ ⎭
∑ ∑
∑
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Predicted and Experimental twinning stress versus unstable twin
nucleation energy for SMAs
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ü Twinning stress calculated based on P-N formulation and GPFE
curves
provides an excellent basis for a theoretical study of the twin
nucleation in
SMAs.
ü The proposed twin nucleation model reveals that twinning
stress has an
overall monotonic dependence on the unstable twin nucleation
energy. To
achieve smaller twinning stress in SMAs, shorter Burgers
vectors, lower
unstable twin energies and larger interplanar distances are
desirable.
Part 3- Summary