Experiments and Simulations in Plasticity- From Atoms to

Experiments and Simulations in PlasticityExperiments and Simulations in Plasticity--From Atoms to ContinuumFrom Atoms to Continuum

Huseyin SehitogluHuseyin SehitogluDepartment of Mechanical Science and Engineering Department of Mechanical Science and Engineering

University of Illinois at UrbanaUniversity of Illinois at Urbana--ChampaignChampaignSymposium in Honor of David McDowell, Symposium in Honor of David McDowell, St.ThomasSt.Thomas, January 7, 2009, January 7, 2009

Collaborators: S. Kibey, D. Johnson, Collaborators: S. Kibey, D. Johnson, J.B.LiuJ.B.Liu, C. , C. EfstathiouEfstathiou, , M.SangidM.Sangid

Funding: NSFFunding: NSF--DMR Metals ProgramDMR Metals Program

3

Overview of the PresentationOverview of the Presentation

Stacking faultsDeformation twins

fcc

fcc

twin

Whelan et al., Proc. Roy. Soc. London (1957).

face centered cubic (fcc) metals and alloys

Karaman-Sehitoglu et al., Acta Mater (2000).

Sehitoglu et al. Acta Mat., APL, (2006-2008)

face centered cubic (fcc) metals and alloys

4

OutlineOutline

• Stacking faults in fcc materials.– Energy landscape/pathway (GSFE) – atomic level.

• Summary

• Deformation twinning in fcc metals.– Energy landscape/pathway (GPFE).– Mesoscale twinning stress model

• Stacking faults in fcc materials.

• Material Design (Cu-Al, Hadfield Steel with Nitrogen)

5

Plastic flow in fcc materials: slip and crossPlastic flow in fcc materials: slip and cross--slipslip

Polycrystalline material

Single crystal/grain

twinning

sliplow SFE metal e.g.: pure Ag

stacking fault ribbons

TEM image from: Whelan, Hirsch, Horne and Bollmann, Proc. Roy. Soc. London (1957).Karaman et al., Acta Mater (2001).

dislocation arraysFuji et al., Mater. Sci. Engg. A 319 (2001) 415-461.

Dislocation cells

low SFE alloys e.g.: nitrogen steels

strain

Stage I

Stage IStag

e II

twinning starts

stre

ss

Stage I

I

twin-twin, slip-twin

interaction

Stage III

medium/high SFE metal e.g.: pure Al

cross-slip

6

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

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) 1345-1359C. Efstathiou, H. Sehitoglu (2008), Unpublished work

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

7

The main principle of the DIC (Digital Image Correlation) technique is shown. Small square regions illustrate a subset. Displacement gradients are noted in the figure.

8

Plastic deformation due to slipPlastic deformation due to slip

Slip due to a perfect

dislocation

Callister (2000)

slipped stateIntrinsic stacking fault

t2

t1l

b1

b2

extended dislocationA perfect dislocation may split into partial dislocations…

Lee et al., Acta Mater (2001)

Intrinsic stacking fault

9

fccuzfcc

0.5

usγu

unstable

1.0 1.5 2.0 2.5 3.0

[ ]111

112⎡ ⎤⎣ ⎦

A

A

BC

BC

A

fcc

Energy pathway for a stacking faultEnergy pathway for a stacking fault

hcp

isfγs

isf

ABC

AC

A

intrinsic stacking fault (isf)

B

Generalized stacking fault energy (GSFE)

(Vitek, 1968)

12

bp bp

maximum

maxγ

m

AB

AA

C

BC

12

bp

10

Energy landscape for a stacking fault (Energy landscape for a stacking fault (γγ--surface)surface)

xu1<110>2

zu1<112>6

isfγ

maxγ

S. Kibey, J.B. Liu, M. W. Curtis, D. D. Johnson and H. Sehitoglu, Acta Mater. 54 (2006) 2991-3001

usγunstable stacking fault energy (Rice,1992)

A

C

s

B<112> u

mEnergy for SF formation during passage of a Shockley partial= area under this surface

11

Calculating the energy required to shear the latticeCalculating the energy required to shear the lattice

Initial fcc metallic supercell

periodicperiodic

periodic

periodic

1Τ

2Τ

[ ]111

112⎡ ⎤⎣ ⎦

top view of ‘A’ layer

side view

fcc lattice

primitive cell

p q

rs

(111)

pq

sr

112⎡ ⎤⎣ ⎦

isf fccisf

(111)

γA

E E−=stacking fault energy

fccE

(111)A

periodic

Supercell with an intrinsic stacking fault

periodic

periodic

isfE

periodic

Equiv. to passage of a Shockley partial

12

Energy required to twin the latticeEnergy required to twin the lattice

top view2Τ

BC

B

CB

A

A

BC

BC

BC

A

A

A

Intrinsic stacking fault

AC

A

32

a

BC

B

C

A

A

A

two layer fault

A

BA

3a

3Τ 3Τ3Τ

p2bpb

BC

B

A

C

three layer twin

AC

B

3Τ

p3b

A

A

next periodic supercell

[ ]111

11 2⎡ ⎤⎣ ⎦

B CBB

C A BA

fcc

B CC

A

Area under this curve is the required energy to twin the lattice by successive shear

usγutγ utγ

utγ utγ

isfγtsf2γ tsf2γ

tsf2γ tsf2γ

1Τ

13

Energy pathway for twinningEnergy pathway for twinning : pure Cu: 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. Lett. 89 (2006) 191911.

Fault energies converge after third layer sliding indicating the completion of twin nucleation.

TBMγ

TBF2= γ

14

Energy pathway for twinning : pure PbEnergy pathway for twinning : pure Pb

usγ utγ utγ utγ

isfγtsf2γ tsf2γ tsf2γ tsf2γ

utγ

twin nucleation twin growth

• VASP-PAW-GGA

• 8 x 8 x 4 k-point mesh

• 237.8 eV energy cutoff.

Convergence occurs after 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, ActaMaterialia 55 (2007) 6843-6851

15

Computed fault energies for fcc metalsComputed fault energies for fcc metals

The above table represents the most complete set of DFT-based theoretical calculations 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. Lett. (2005)c pair potential. Rautioaho, Phys. Status Sol. (1982).

(all energies in mJ/m2 )

S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851

16

Ideal strength prediction from GPFEIdeal strength prediction from GPFE

Ideal twinning stress can be related to the GPFE curves as follows:( ) ( )ut tsf

idealtwin

2b

x

x

uu

∂γ γ − γτ = − =

∂ π

Al

Cu

Simple shear case

ux/bp

However, non-ideal (real) twinning stresses of materials are of the order of MPa due to presence of defects.

S. Ogata, J. Li and S. Yip, Science (2002).

Can we predict realistic critical stresses using mesoscale models in conjunction with defect energy landscapes?

17

Classical twin nucleation modelClassical twin nucleation model

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

crit crit2 isf

p

1 22 b

K⎡ ⎤⎛ ⎞ γ−

+ =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

θ θ τ τβ

1= =θ β

fitting parameters: K, θ and β

Classical twinning stress equation:

Calibration of fitting parameters for different alloys is required.

need a more fundamental approach to predict twinning stress.

Cu-based alloys

18

Present ApproachPresent Approach

•energy to twin the lattice

Density functional theory

01 1⎡ ⎤⎣ ⎦

Aδ 211⎡ ⎤⎣ ⎦

[ ]111

mesoscale

Elastic dislocation theory •twinning

stress

atomic scale

19

Mesoscale model for fcc twinsMesoscale model for fcc twins

Total energy of the twin nucleus:

energy contribution of screw c

work denergy contribution of e

one byapplied stress

energy associated with twin-edge co nergyomponenmponents pathwat y s

ed s GPFEcrge et wotal EE EWE= + − +τ

Mahajan and Chin, Acta Metallurgica (1973)

Dislocation configuration 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

Cδ

Total energy:


20

Total energy of the twin nucleusTotal energy of the twin nucleus

γ-

energy required to

energy associat twin the latti

γ-

energy requiredto cross-slip

ed with twin-energy pathway ce

GPFE Stwin FE EE = −

usγ utγ utγ utγ

isfγtsf2γ tsf2γ tsf2γ tsf2γ

utγ

twin nucleation

twin growthcross-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= ∫

γ-

energy required to

energy associat twin the latti

γ-

energy requiredto cross-slip

ed with twin-energy pathway ce

GPFE Stwin FE EE = −

Energy contribution of GPFE:


21

Twinning stress equationTwinning stress equation


For a stable twin configuration:

22

Twin nucleus shapeTwin nucleus shape


Karaman –Sehitoglu et al., Acta Mater (2001)

316 stainless steel at 3% straintw

ins

23

For Thin twinsFor Thin twins

( ) ( )tsf isfcrit ut us isf

twin twin

22 3 21

3 b 4 2 3 b

γ + γτ = − γ + − γ + γ⎡ ⎤⎛ ⎞

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦N

N N

Increases Critical Twin StressDecreasesCritical TwinStress

24

Predicted twinning stresses for fcc metalsPredicted twinning stresses for fcc metals


Twinning stress depends non-monotonically on stacking fault energy.

isfcrit

twinbK γ

τ ∼

does not hold !

25

Predicted twinning stresses for fcc metals (contd.)Predicted twinning stresses for fcc metals (contd.)

Twinning stress depends monotonically on unstable twin SFE .

γut governs the physics of twin nucleation.


26

Predicted twinning stresses for fcc metals (contd.)Predicted twinning stresses for fcc metals (contd.)


b Bolling, Casey and Richman, Phil. Mag. (1965).c Suzuki and Barrett, Acta Metall. (1958).d Narita et al., J. Japan Inst. Metals (1978).e Yamamato et al., J. Japan Inst. Metals (1983).

27

Twinning is Twinning is directionaldirectional

/ 1 6 112xu ⎡ ⎤⎣ ⎦

isfγ

usγ utγ

tsf2γ isfγ

*usγ

tsf2γ

*utγ

/ 1 3 112xu ⎡ ⎤⎣ ⎦

[ ]111

Energetically favorable and observed Energetically unfavorable and not observed

1 112⎡ ⎤= ⎣ ⎦η

A

A

BC

BC

A

ABC

A

A

BC

twin12 1123

⎡ ⎤− = ⎣ ⎦b

( )1 111=κ

[ ]111

1 112⎡ ⎤= ⎣ ⎦η

twin1 1126

⎡ ⎤= ⎣ ⎦b

( )1 111=κ

fcc structure fcc structure( )111 112 ⎡ ⎤⎣ ⎦ twin ( )111 112 ⎡ ⎤⎣ ⎦ twin

28

isf us

us ut

1.136 0.151

1

T

T

⎡ ⎤γ γ= −⎢ ⎥γ γ⎣ ⎦≥ ⇒ twin nucleation

Twinning directionalityTwinning directionality

S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, APL,91,181916 (2007)(all energies in mJ/m2 & stresses in MPa )

[ ]111

1 112⎡ ⎤= ⎣ ⎦η

A

A

BC

BC

A

A

A

BC

BC

Atwin

12 1123

⎡ ⎤− = ⎣ ⎦b

( )1 111=κ

[ ]111

1 112⎡ ⎤= ⎣ ⎦η

twin1 1 126

⎡ ⎤= ⎣ ⎦b

( )1 111=κ

fcc structure fcc structure( )111 112 ⎡ ⎤⎣ ⎦ twin ( )111 112 ⎡ ⎤⎣ ⎦ twin

Bernstein and TadmorPhys. Rev. B (2004)

29

SummarySummary

• Presented a hierarchical, multiscale, adjustable parameter-free approach for twin nucleation in fcc metals and alloys.

• Predicted twinning stresses are in excellent agreement with available experimental data.

• Our theory inherently accounts for directional nature of twinning.

30

OutlineOutline

• Stacking faults in fcc materials.– Energy landscape/pathway (GSFE) – atomic level.

• Summary

• Deformation twinning in fcc metals.– Energy landscape/pathway (GPFE).– Mesoscale twinning stress model

• Stacking faults in fcc materials.

• Material Design (Cu-Al, Hadfield Steel with Nitrogen)

31

Supercell for CuSupercell for Cu--8.3at.%Al8.3at.%Al

top view of ‘A’ layer

1Τ

2Τ

BC

B

CBA

A

A

BC

BC

BC

A

A

A

Intrinsic stacking fault (isf)

A BA

32

a

BC

B

C

A

A

A

two layer fault

CB

A

3a

3Τ3Τ

3Τ

p2bpb

BC

B

C

A

three layer twin

AC

B3Τ

p3b

A

A

next periodic supercell

solute Al atom[ ]111

112⎡ ⎤⎣ ⎦

• VASP-PAW-GGA

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

fcc

32

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

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

Convergence at the third layer sliding is seen for Cu-5.0at.% Al as well. Hence, a three-layer twin is the basic nucleus in fcc alloys.

33

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


34

Predicted fault energies for CuPredicted fault energies for Cu--xxAlAl


a Murr, Interfacial Phenomena in Metals and Alloys (1975).b Ogata, Li and Yip Phys. Rev. B (2005).c Carter and Ray, Phil. Mag. (1977).d Pearson In: A Handbook of Lattice Spacings and Structures of Metals and Alloys (1958)

(all energies in mJ/m2 )

The only ab initio calculations reported for fcc Cu-Al alloys.

35

Prediction of twinning stresses in alloysPrediction of twinning stresses in alloys

Twinning stress depends non-monotonically on intrinsic SFE. However, within Cu-xAl, the variation is monotonic. The present hierarchical, theory of twinning stress holds for fcc metals and alloys.

S. Kibey, L. L. Wang, J.B. Liu, D.D. Johnson, H. T. Johnson and H. Sehitoglu, manuscript in preparation.

36

Prediction of twinning stresses in alloys (contd.)Prediction of twinning stresses in alloys (contd.)

Twinning stress for Cu-xAl depends monotonically on unstable twin SFE.

S. Kibey, L. L. Wang, J.B. Liu, D.D. Johnson, H. T. Johnson and H. Sehitoglu, manuscript in preparation.

37

Addition of nitrogen to FeAddition of nitrogen to Fe--based materials based materials

J. Reed, J.Metals, 1989. Rawers and Slavens (1995)

Max at 1%N

38

FeMnNFeMnN--Theory vs. Experiment Theory vs. Experiment -- [111] Orientation[111] Orientation

2000

1500

1000

500

0

True

Stre

ss (M

Pa)

0.200.150.100.050.00True Inelastic Strain

Hadfield Steel [111] Orientationunder Compression, T=293 K

Experiment Simulation

HNHS (1.06 wt.% N)

HSw/oN (0 wt.% N)

39

Extended edge dislocationExtended edge dislocationY

(111)

b

d

1b

2b

X, 110⎡ ⎤⎣ ⎦

Z, 112⎡ ⎤⎣ ⎦b = b1 + b2

12

110⎡⎣ ⎤⎦ =16

211⎡⎣ ⎤⎦ +16

121⎡⎣ ⎤⎦

Z direction is equivalent to <112> direction for GSFE

X direction equivalent to <110> direction for GSFE

40

Generalized PGeneralized P--N modelN model• Generalize to extended dislocations

• Edge components:

ux =

−12π

be1 tan−1 xw

⎛⎝⎜

⎞⎠⎟

+ be2 tan−1 x − dw

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

( ) ( )( )

( )21

22 2 20

2 1ee

xy

b x db xbx,x w x d w

μσπ ν

⎡ ⎤−= − +⎢ ⎥

− + − +⎢ ⎥⎣ ⎦• Screw components:

1 11 2

12z s s

x x du b tan b tanw wπ

− −⎡ ⎤− −⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) ( )( )

2122 2 2

02

ssyz

b x db xbx,x w x d w

μσπ

⎡ ⎤−= − +⎢ ⎥

+ − +⎢ ⎥⎣ ⎦

41

Generalized PGeneralized P--N model (contd.)N model (contd.)

Etotal d( )= Eelastic d( )+ Emisfit d( )

2 22 21 21 2

211 2

1 2 2

211 2

1 2 2

4 (1 ) 2 4 (1 ) 2

14 (1 )

( ) ( )1 18 (1 )

e es s

e es s

e es s

b br r db ln b lnw w

b b r r db b ln tanw w

b b r d r d r db b ln tan lnw w

μ μπ ν π ν

μπ ν

μπ ν

−

−

⎛ ⎞ ⎛ ⎞ −⎡ ⎤ ⎡ ⎤= + + +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎡ ⎤⎛ ⎞ −⎛ ⎞+ + +⎜ ⎟ ⎢ ⎥ ⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎣ ⎦

⎡ ⎤⎛ ⎞ − − −⎛ ⎞+ + + + +⎜ ⎟ ⎢ ⎥ ⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎣ ⎦

21

2

r dtanw w

−⎧ ⎫⎡ ⎤ +⎪ ⎪⎛ ⎞⎨ ⎬⎢ ⎥ ⎜ ⎟

⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

( )r

misfit x zru dxE ,uγ

−= ∫

Generalized SFE

( ) ( )−

= +∫r

xy x yz zr

elastic u u xE dd σ σ

Minimize to obtain stable stacking fault width. Etotal d( )

42

GSFE curve along <112>GSFE curve along <112>

Z, 112⎡ ⎤⎣ ⎦

uγSFγ

maxγu

sb

m

b

c

u

s

m

b

b

a

FCC lattice

<112>

HCP

FCC

X, 110⎡ ⎤⎣ ⎦

060

030

1 2116

⎡ ⎤⎣ ⎦

1 1216

⎡ ⎤⎣ ⎦

1 1102

⎡ ⎤⎣ ⎦

Z, 1 12⎡ ⎤⎣ ⎦

γ -curve along <112> direction

600 symmetry of γ surface

43

Results of atomistic calculationsResults of atomistic calculations

Pure iron

Nitrogen 2 layers away from stacking fault

Nitrogen 1 layer away from stacking fault

uz

16

<112>

44

Fourier fit for Fourier fit for γγ--curve : pure Fecurve : pure Fe

zu1<112>6

45

Fourier fit for Fourier fit for γγ--curve : Fecurve : Fe--4 at.%N4 at.%N

zu1<112>6

46

Fourier fit for GSFE FeFourier fit for GSFE Fe--4 at.%N4 at.%N

ux

12

<110>

zu1<112>6

ux

12

<110>

47

Effect of stable SFE (note finite separation for negative valuesEffect of stable SFE (note finite separation for negative values))

6

5

4

3

2

1

0

d /

b

4002000-200-400

γSF (mJ/m2)

γu = 514 mJ/m2

γmax = 1998 mJ/m2

Volterra

Peierls

goes to infinityIncreasing nitrogen

1% wt.

48

Summary (Summary (ctdctd.).)

••The models developed for twinning stress and The models developed for twinning stress and stacking fault widths can be utilized to design new stacking fault widths can be utilized to design new alloysalloys..

•Stacking fault width is determined by γ–surface,not by intrinsic stacking fault energy alone.

Documents

Experiments and Simulations in Plasticity- From Atoms to