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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 Efstathiou, H. Sehitoglu (2008), Unpublished work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:
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
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:
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
21
Twinning stress equationTwinning stress equation
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
For a stable twin configuration:
22
Twin nucleus shapeTwin nucleus shape
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
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
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
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.
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Acta Materialia 55 (2007) 6843-6851
26
Predicted twinning stresses for fcc metals (contd.)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 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.)
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Appl. Phys. Lett. (2006)
34
Predicted fault energies for CuPredicted fault energies for Cu--xxAlAl
S. Kibey, J.B. Liu, D.D. Johnson and H. Sehitoglu, Appl. Phys. Lett. (2006)
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.
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