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1
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
University of Illinois at Urbana Champaign
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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6
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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7
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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8
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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11
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
22
Outline
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A theoretical model to predict the twinning stress has not been established.
Theoretical model is needed
23
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
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
29
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
30
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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
31
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Close agreement!
Comparison with experiments
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a Harding (1967,1968) b Calcula%ons based on Ogata et al. (2005) c Nilles and Owen (1972) d Suzuki et al. (1966)
32
University of Illinois at Urbana Champaign Current Results
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
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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47
Detwinning and Twinning of NiTi Martensite
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48
Compressive stress-strain response of Ni54Fe19Ga27 at temperature of -190 °C.
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49
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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51
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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52
( )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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54
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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56
Predicted and Experimental twinning stress versus unstable twin nucleation energy for SMAs
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57
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
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