Modeling of Twinning Stresshtml.mechse.illinois.edu/files/presentations/twin_20... · 2008. 4....
Transcript of Modeling of Twinning Stresshtml.mechse.illinois.edu/files/presentations/twin_20... · 2008. 4....
-
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