Modeling of Twinning Stress Modeling of Twinning Stress H. Sehitoglu, S. Kibey, D. JohnsonH. Sehitoglu, S. Kibey, D. Johnson11,J.B.Liu,J.B.Liu11

University of Illinois, Department of Mechanical Science and University of Illinois, Department of Mechanical Science and Engineering,Urbana, IL 61801Engineering,Urbana, IL 61801

11Department of Materials Science and Engineering, Urbana, ILDepartment of Materials Science and Engineering, Urbana, IL

Presentation in honor of Prof. Alan Needleman, SES Meeting, Presentation in honor of Prof. Alan Needleman, SES Meeting, August 14August 14--16, 200616, 2006

Presentation OutlinePresentation Outline

• The need for a Twinning (nucleation) Stress Model• Previous Twinning Models• Fault Energies from DFT (Density Functional Theory)• Twin Nucleus as 3-layers• Energy Minimization (the Cu-Al example)• Anti-twinning• Conclusions

Hadfield Steel (fcc FeHadfield Steel (fcc Fe--MnMn--C steel)C steel)-- [111] Orientation[111] Orientation800

600

400

200

0

Axial Stress, MPa

0.50.40.30.20.10.0

Axial Strain

Hadfield Manganese Steel[111] OrientationDeformed under Tension at RTExperiment and Simulation

Experiment Simulation

Experiment, 12%

Experiment, 27%

Simulation, 12%

Simulation, 27%

I. Karaman, H. Sehitoglu,A. Beaudoin, Y.Chumlyakov, H.J.Maier, C. Tome, Acta Mat. 48 (2000) 2031-2047I.Karaman, H. Sehitoglu,K.Gall, Y. Chumlyakov, H.J. Maier, Acta Mat. 48 (2000) 1345-1359

The twinning (nucleation) stress is currently obtained from experiments. A theory to obtain this quantity from first principles (for metals and alloys) is needed.

twinning stress

Twin Nucleation StressTwin Nucleation Stress

dfdt

(β)

≥ 0 if τ twin(β) Σ,f (β)( )= τcrit− twin(β) f (β)( )

Nucleation Criteria used in Crystal Plasticity Models

The twin volume fraction evolves when the resolved shear exceeds the twinning nucleation stress.

Deformation by slip Deformation by twinning

( )0αs

( )0αn

s0(β)

n0(β)

matrix

twinned region

Aα

Aα AC

AC

αC

αCtwin

Pole mechanism for formation of a deformation twin

lenticular twinplano-convex twin

Venables, Deformation Twinning, Eds. Reed-Hill,Hirth and Rogers (1964)

Classical twin nucleation models for fcc alloys (contd.)Classical twin nucleation models for fcc alloys (contd.)

Venables, Deformation Twinning, Eds. Reed-Hill,Hirth and Rogers (1964)

nτ crit =

γ isfbp

+Gb2a0

1− 2θ2β

⎛⎝⎜

⎞⎠⎟

+ Kθ 2τ crit⎡

⎣⎢

⎤

⎦⎥τ crit =

γ isfbp

1θ β= =

Classical twin nucleation models (contd.)Classical twin nucleation models (contd.)

Pileup behind the twin Forest dislocations

Twinning partial

B

C

C

A

B

A

C

Lattice structure of deformation twins in fcc alloysLattice structure of deformation twins in fcc alloys

fcc

fcc

twin

Deformation twin in Hadfield steel [001] orientation 3% strain

I. Karaman, H. Sehitoglu, K. Gall, Y. I. Chumlyakov and H. J. Maier, Acta Mater.(2000).

fcc

fcc

twin

1 2 3 4 5

γ

ux /|1/6|

B

Generalized planar fault energy (GPFE) curve Generalized planar fault energy (GPFE) curve

t4 t5

A

112⎡ ⎤⎣ ⎦

[ ]111

pb

ABC

BC

ABC

fcc

tsf2γ

utγtwin nucleation

t2

A

A

ABC

BC

C

B

2 layer twin

t3

B 3 layertwin

ABC

AC

ABC

ISF formation

usγ

isfγ

u

s

ABC

AC

A

ABC

isf

Kibey, Liu, Curtis, Johnson, Sehitoglu Acta Mater. 54 (2006) 2991-3001

Formation of multiFormation of multi--layer twins in Culayer twins in Cu--5.0%Al5.0%Al

B

ABC

AC

ABC

A

112⎡ ⎤⎣ ⎦

[ ]111

pb

ABC

BC

ABC

fcc

A

A

ABC

BC

C

B

2 layer twin

ABC

AC

A

ABC

isf

3 layertwin

Twining in Cu-5.0at.%Al

Passage of 1/6 Shockley partials to produce 2 and 3 layer twins successively. The selected Al atom positions in the layers 2 and 6 within the 10 layer supercell permit a continuous shear to generate multiple twins with high symmetry.

A B C A

Converged supercell sizes for CuConverged supercell sizes for Cu--xxAl Al

[ ]111

112⎡ ⎤⎣ ⎦

A

AB

C

A

BC

AB

C

Al

Cu-5.0at.%Al

10 layer supercell

pb

AB

C

A

BC

AB

C

Cu-8.3at.%Al

Al

9 layer supercell

• VASP-PAW-GGA

• 8 x 8 x 4 k-point mesh with 273.2 eV energy cutoff.

VASPVASP--PAW fault energies for CuPAW fault energies for Cu--xxAlAl

L. E. Murr, Interfacial Phenomena in Metals and Alloys (1975).

S. Ogata, J. Li and S. Yip, Phys. Rev. B, 71, 224102 (2005).

C. B. Carter and I. L. Ray, Phil. Mag., 35, 189 (1977).W. B. Pearson in: G. V. Raynor (Ed.), A Handbook of Lattice Spacings and Structures of Metals and Alloys (1958).

all energies in mJ/m2

GPFE curve for pure CuGPFE curve for pure Cu

usγutγ utγ utγ utγ

isfγ

tsf2γtsf2γ tsf2γ tsf2γ

GPFE curves for CuGPFE curves for Cu--xxAl alloysAl alloys

S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitolgu, submitted to Appl. Phys. Lett. (2006)

GPFE curves for CuGPFE curves for Cu--xxAl alloys (contd.)Al alloys (contd.)

S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, submitted to Appl. Phys. Lett. (2006)

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Bδ

Aδ

τ

τ

Bδ

Cδ

Formation of the Twin NucleusFormation of the Twin Nucleus

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

Intrinsic SFs on {111} plane interact to form a twin nucleus

A

BCδ

(111) plane

Formation of constriction Formation of constriction ABAB

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Bδ

Aδ

τ

τ

Bδ

Cδ

AB

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

leading trailing

B A ABδ δ+ ⎯⎯→

constriction

AB AB dissociates into an unstable, high energy faultdissociates into an unstable, high energy fault

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Aδ

δ B

τ

τ

Bδ

Cδ

Unstable high energy fault

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

leading trailing

AB A B

δ δ⎯⎯→ +

The above sequence of leading and trailing partials is inconsistent with Thompson’s rule and forms a high energy unstable fault, which will dissociate into an extrinsic stacking fault.

High energy fault dissociates into a stable extrinsic SFHigh energy fault dissociates into a stable extrinsic SF

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

τ

τ

Bδ

Cδ

extrinsic stacking fault

Aδ

Bδ

CδCδ−

annihilation

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

B A CA B Cδ δ δ

δ δ δ

⎯⎯→ +

⎯⎯→ +

Fault pair formation after annihilationFault pair formation after annihilation

fault pair acts as a precursor to twin nucleation

According to Mahajan and Chin (1973), two identical fault pairs interact to produce a 3 layer twin nucleus.

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Bδ

Aδ

Bδ

Cδ

fault pair

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

Fault pairs interact to form a 3 layer twin nucleusFault pairs interact to form a 3 layer twin nucleus

Fault pairs interact to

give 3 layer twin nucleus

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Bδ

Aδ

BδCδ

⎡ ⎤⎣ ⎦011

⎡ ⎤⎣ ⎦211

[ ]111

Bδ

Aδ

Bδ

Cδ

Aδ

Bδ−

Bδ

Aδ

Cδ

Bδ

Bδ

[ ]111

⎡ ⎤⎣ ⎦211

⎡ ⎤⎣ ⎦011

Mahajan and Chin, Acta Metallurgica, vol. 21 (1973) 1353-1363.

Mahajan and Chin (1973) have also proposed a 3 layer twin as the nucleus for twinning.

Proposed twin nucleusProposed twin nucleus

Our GPFE convergence trends are in agreement with above proposition.

Aδ

Bδ−

Bδ

Aδ

Cδ

Bδ

Bδ

[ ]111

⎡ ⎤⎣ ⎦211

⎡ ⎤⎣ ⎦011

Growth of a twin from the nucleusGrowth of a twin from the nucleus

•These 3 layer nuclei grow into each other to form a finite sized twin.

We can determine the aspect ratio that will minimize the total energy of the twin. Because the twin thickness is small with respect to its width, Friedel (1964) treats the dislocations as co-planar.

y,

Bδ

Bδ

Bδ

Bδ−

Aδ

Cδ

[ ]111

⎡ ⎤⎣ ⎦211X,

Bδ

Bδ

Bδ

Bδ−

Aδ

Cδ

BδBδ

Bδ

Bδ−Aδ

Cδ

Immobile trailing partials

mobile twinning partials

h

d

a hNa

=

N layer twin

Total energy of the twin nucleusTotal energy of the twin nucleus

To determine non-ideal twinning stress required to nucleate a twin, minimize the total energy of the 3-layer twin nucleus.

A closed form expression for total energy can be written with following assumptions:

Total energy of the twin nucleus:

( ) ( )2

2

0

1 14 1 2 6

eedge

Gb d d dE N ln N ln ln NN rπ υ

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

• Assuming a thin twin and treating the edge components as a single pileup

Chou and Li in ‘Mathematical Theory of Dislocations’, ed: T. Mura, ASME (NY) 1969, p. 116

Etotal = Eedgeenergy contribution of edge components

+ Escrewenergy contributionof screw components

+ Eγ -twinenergy requiredto pass successivetwinning partials

− Eγ -SFenergy requiredto create ISF

− Wτwork done by resolved shear stress

Total energy of the twin nucleus (contd.)Total energy of the twin nucleus (contd.)

• Since only Aδ and Cδ are mixed dislocations, the screw components can be treated as a finite sized vertical wall:

[ J.C.M. Li, Acta Metall., vol. 8 (1960) p 563-574 ]

Bδ

Bδ

Bδ

Bδ−

Aδ

Cδ

Immobile trailing partials

mobile twinning partials

22 1

9 2s

screwGb dE dN ln

Nπ⎡ ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )- 01d

twin twinE N d dxγ γ= − ∫

• The energy required to pass successive twinning partials is given by:

• The energy required for formation of the intrinsic SF and for cross-slip:

- 0

d

SF SFE d dxγ γ= ∫

( ) ( )

( )0

2

0

2

22

2

0

19

1 14 1 2

1

2

6total

t

d

SF

d

win

e

t n

s

wiN d d

Gb d d dN ln N ln ln N

Gb ddN ln

E

N d bdx

N

N

d

r

x

π υ

π

τγγ

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

⎡ ⎤⎛ ⎞ −⎜ ⎟⎢ ⎥⎝ ⎠

=

+

⎩ ⎭ ⎝ ⎠⎣ ⎦

⎣

−−+ −

⎦

∫ ∫

Total energy of the twin:

Critical twin size and twinning stress can be determined by minimizing Etotal relative to d and N.

Etotal = Eedge + Escrew + Eγ -twin − Eγ -SF − Wτ

Total energy of the twin nucleus (contd.)Total energy of the twin nucleus (contd.)

• Work done by resolved shear stress in displacing twinning partials :

2twinW N d bτ τ=

Twinning stress equationTwinning stress equation

Relation between twin size and twinning stress based on present analysis:

h N a=

d

τ

τ

τcrit =GNπ

be2

1- υ( )+ bs2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+2

3N3N4

− 1⎛⎝⎜

⎞⎠⎟

γ ut +γ tsf + γ isf( )

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1btwin

+1

6btwinγ ut −

γ tsf + γ isf( )2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

wd

⎛⎝⎜

⎞⎠⎟

lnd + d 2 + w2

w

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−N − 1( )3N

1btwin

γ ut −γ tsf + γ isf( )

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

wd

⎛⎝⎜

⎞⎠⎟

lnd + d 2 + w2

w

⎛

⎝⎜⎜

⎞

⎠⎟⎟

+d

d 2 + w2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

3Nγ us + γ isf

btwin

⎛

⎝⎜⎞

⎠⎟+

13N

γ us − γ isfbtwin

⎛

⎝⎜⎞

⎠⎟wd

⎛⎝⎜

⎞⎠⎟

lnd + d 2 + w2

w

⎛

⎝⎜⎜

⎞

⎠⎟⎟

+d

d 2 + w2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Predicted twinning stress for fcc alloysPredicted twinning stress for fcc alloys

Twinning stress vs. unstable twin SFE barrierTwinning stress vs. unstable twin SFE barrier

Narita and Takamura in ‘Dislocations in Solids’ ed: Nabarro vol. 9 (1992) p. 135-189.; Szczerba, Mater. Sci. Engg. A234-236 (1997) p 1057.; Szczerba, Phil. Mag. vol. 84 (2004) p. 481.

Twinnability of fcc alloysTwinnability of fcc alloys

Twinning tendency Tt :

trailing partialus

crittwinning partialut

tKTK

γ= = λ

γ

orientation dependent parameter

Twinnability T is the orientation average of twinning tendency over all possible orientations.

T = 1

ΩTt( )min∫ dαdβdθdφA, where, Tt( )min = λmin

γ usγ ut

Bernstein and Tadmor (2004) have given an approximate form for twinnability as

T = 1.136 − 0.151γ isfγ us

⎡

⎣⎢

⎤

⎦⎥

γ usγ ut

Directionality in twinningDirectionality in twinning

Energetically favorable and observed

Energetically unfavorable and not observed

T =1.12T =1.02

Comparison of predicted twinning stress Comparison of predicted twinning stress

τcrit=867 MPaτcrit =120 MPa

Present model for twinning stress predicts the twinning directionality correctly, while the twinnability criterion does not.

ConclusionsConclusions

• Our twinning model predicts that in fcc alloys, a 3 layer twin is the twin nucleus, and the twinning stress was computed by minimizing the total energy of the 3 layer twin nucleus.

• The model predicts that twinning stress depends on the unstable SFE and unstable twin SFE barriers, in addition to intrinsic SFE.

• The model rules out twinning in the anti-twinning direction.

Backup slide for twinning stressBackup slide for twinning stress

Twinning stress vs. unstable SFE barrierTwinning stress vs. unstable SFE barrier