Deformation Twinning – Mechanisms and Modeling in FCC, …E total =E edge energy contribution of...

University of Illinois at Urbana Champaign

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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


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


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


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

<112>

<111>

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


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


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


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


9

Energy landscape for a stacking fault (g-‐surface)

xu1<110>2

zu1<112>6

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 <112> u

m Energy for SF formaNon during passage of a Shockley parNal= area under this surface


10

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

isfγtsf2γ tsf2γ

tsf2γ tsf2γ

1Τ


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Energy pathway for twinning : pure Cu

usγutγ

utγutγ 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 tsf2γ 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 = Eedge

energy contribution of edge components

+ Escrew

energy contribution of screw components

− Wτ

work done byapplied stress

+ EGPFE

energy 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 tsf2γ 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

⎡ ⎤⎛ ⎞⎧ ⎫⎛ ⎞+ − −⎢ ⎥⎨ ⎬ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎢ ⎥⎩ ⎭ ⎝ ⎠⎣= +

+

⎡ ⎤⎛ ⎞ −⎜ ⎟⎢ ⎥

−

⎝ ⎠⎦ ⎦

− −

⎣

∫∫ τγ

υ

γ

π π

Total energy:

( )- 01

d

twin twinE N d dxγ γ= − ∫

- 0

d

SF SFE d dxγ γ= ∫

EGPFE

energy associated with twin-energy pathway

= Eγ -twin

energy 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 isf

btwin

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.



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


A theoretical model to predict the twinning stress has not been established.

Theoretical model is needed

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Deformation by Twinning (bcc)


DIC measurements: An example

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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


Generalized planar fault energy (GPFE) (MD calculations) for FeCr

We are concerned with the twin nucleation region of the GPFE.

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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


Experimental observations validate that three slip systems of symmetric configuration may be activated.

Why dislocation dissociation mechanism?

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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 2

A Ar r2B A B A

total rss A o rss B o GPFE SFo 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 2

mJ ( )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)

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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#

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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


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.

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Prediction of twin migration stress

Theoretical prediction is needed!


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


An incident twin is blocked because of the higher magnitude of residual dislocation at the twin boundary.

rb =1.0a

37


Slip-slip interaction

Slip-slip interaction

Twin-twin interaction

Twin-twin interaction Twin migration stresses

We try to predict these stresses



Twin systems analyzed in present work

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Christian and Mahajan, Progress in Materials Science,1995

Line of intersection of twins Example: <113> Intersection type

Cross product of n1 X n2

n1

n2

121( )× 112( )


τ 2

τ1

= τ M

τ T =

4b2(Gbr2 +

Gb22

4πln(

r2

w)+ γ modifiedGPFE

bN2

b( N2+1)

∫ )

2πb2

Gb12

4πln(

r2

w)

Minimization

∂Utotal

∂ς1

=∂Utotal

∂ς 2

= 0

Utotal = Einteraction/incident +Eline/incident +Eline/outgoing +Eresidual +EincidentGPFE +EmodifiedGPFE -Wincident -Woutgoing

= -Gb1

2

2πln[

rB - rA

ro

] + ln[rB

2ro

]+ ln[rA

ro

]⎧⎨⎩⎪

⎫⎬⎭⎪ζ1 +

Gb12

4πln(

Rw

) 2d +ζ1{ }+Gb2

2

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

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Residual dislocation

41

+Blockage

Incorporation+ Blockage

Incorporation+ Blockage


br =1.0a br = 0.8a

mτ =167 MPa mτ =144 MPaTheory

Theory Experiment

[101] Compression [111] Compression

<110> interesection

<210> intersection

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τ M

τ T = 0.83 τ M

τ T = 0.74


Results-Extended

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τ M

τ T

⎛⎝⎜

⎞⎠⎟

Harding , Proc. R. Soc.1967, 1968 Suzuki, J. Phys. Soc. Jpn, 1962


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 <110>,<210> and <113> cases causes the incident twin to be completely blocked.

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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

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Outline


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

[ ]010

Ti 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


50

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


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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( )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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( )( ) ( ) ( )

( )( )( )

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


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

Deformation Twinning – Mechanisms and Modeling in FCC, …E total =E edge energy contribution of...

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