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M
nm1
mm1
• plastic anisotropy and non-uniform loading paths (macro-scale)Choi, Y, Walter, M.E., Lee, J.K., and Han, C.-S., 2005. Int. J. Solids & Structures, in review. Choi, Y., Han, C.-S., Lee, J.K., and Wagoner, R.H., 2005. Int. J. Plasticity, in review.Han, C.-S., Lee, M.-G., Chung, K., Wagoner, R.H., 2003. Commu. Num. Meth. Engng., 19 (6), 473-490.Han, C.-S., Chung, K., Wagoner, R.H., and Oh, S.-I., 2003. Int. J. Plasticity, 19 (2), 197-211.Han, C.-S., Choi, Y., Lee, J.K., and Wagoner, R.H., 2002. Int. J. Solids & Structures, 39, 5123-5141.
• crystal plasticity and composites (meso-scale ~ micron and above)Han, C-S., Kim, J.-H., and Chung, K., 2005. Accepted in Int. J. Solids & Structures.Han, C.-S., Wagoner, R.H., and Barlat, F., 2004. Int. J. Plasticity, 20, 1441-1461. Han, C.-S., Wagoner, R.H., and Barlat, F., 2004. Int. J. Plasticity, 20, 477-494.
• strain gradients plasticity and size dependence (micron to submicron scale)Han, C.-S., Gao, H., Huang, Y., and Nix, W.D., 2005. J. Mech. Phys. Solids, 53, 1188-1203. Han, C.-S., Gao, H., Huang, Y., and Nix, W.D., 2005. J. Mech. Phys. Solids, 53, 1204-1222.Han, C.-S., Ma, A., Roters, F., and Raabe, D., 2005. In preparation.Han, C.-S., Roters, F., and Raabe, D., 2005. In preparation.Zaafarani, N., Han, C.-S., Nikolov, S. And Raabe, D., 2005. Work in progress.
• dislocation theory and boundary effects (submicron to nanometer scale)Han, C.-S., Hartmaier, A., Gao, H., and Huang, Y., 2005. Accepted in Materials Science and Engineering A.
m1µ
Material modelling at various length scalesm1
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Plastic anisotropy evolution of rolled sheet metals determined by tensile tests
experimentsM1. Macro-plasticity
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150
200
250
300
350
400
450
- 45 0 45 90 135
0
0.06
0.14
0.220.36
ε
RD TD
new axis of symmetry
Orientation with respect to RD
Elas
tic li
mit
[MPa
]
θ
tensile testing
mild steel- Boehler & Koss (1991)- Kim & Yin (1997) - Choi/Walter/Lee/Han (2003)
stretch in 45 degrees from RD
testing orientation to RD
Yie
ld s
tress
[MP
a]
experimental observations
Rotation of symmetry axes observed by tensile tests and pole figures
M1. Macro-plasticity
Data from Boehler & Koss (1991)
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Rotational Hardening / Rotation of Anisotropy Axes
anisotropic yield function
isotropic kinematic rotational
M1. Macro-plasticity modeling
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Multiplicative decomposition & rotations
peFFF =peFFF =
pmp FRF = eV
peFVF =
B~
B
mRB
oB
φieφ
oie
φi
~e
φie
pxTxe FRRFF =
pFdecomposition is not unique
additional constitutive equation is necessary
pepeTee FVFRRFF ==
M1. Macro-plasticity modeling
Loret 1983, Dafalias 1985, 2000, Zbib & Aifantis 1988,Van der Giessen 1991, Bunge & Nielsen 1997, Levitas 1998,Hill 2001, Truong Qui & Lippmann 2001, Kowalczyka & Gambina 2004
plastic spin models:
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Tensile stretch tests
0
10
20
30
40
0 2 4 6 8 10
30 Degree
Engineering Strain (%)
Experiment(Kim & Yin '97)
FEM
-50
-40
-30
-20
-10
0
0 2 4 6 8 10
45 Degree
Engieering Strain (%)
Experiment(Kim & Yin '97)
FEM
-40
-30
-20
-10
0
0 2 4 6 8 10
60 Degree
Engineering Strain (%)
Experiment(Kim & Yin '97)
FEM
)tan(c ϑ=µ τφ φ
ϑ min. angle between EV of
and symmetry axes
pd
Experimental data by Kim & Yin 1997 for mild steel
Young’s Modulus E = 206 GPaPoisson’s ratio 3.0=ν
Initial yield stress MPa06.1070 =τ
Hill’s [1950]yield function 3550.2
,0092.1,5837.0
66
2312
=β=β=β
Isotropichardening
25.0n,MPa544c isoiso ==
Plastic spinparameter 350c −=φ
( )τddτω ppp −µ= φ
,
M1. Macro-plasticity modeling
Han et al. 2002
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simulationM1. Macro-plasticity
Draw bead simulation with rotational hardening
orientation to rolling direction:
rotation angle (o)
o30Ψ =
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Springback height and twisting mode
• Unexpected twisting for isotropic (ISO) and isotropic - kinematic hardening (ANK)
• Best springback height prediction with rotational hardening
0
10
20
30
40
50
60
70
80
-30 -20 -10 0 10 20 30 40 50Z
- coo
rdin
ate
(mm
)
Y - coordinate (mm)
ISO
RIK
ANK
EXP
M1. Macro-plasticity spring-back example
orientation to rolling direction: o30Ψ =
springback
x
z
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Crystal plasticity
M2. Crystal plasticity (meso scale)
Slip systems of an FCC crystal
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Incorporation of Elastic Inclusion Model
modeling
IM f)f1( τττ +−=
pIe εKε = I
eeI εΓτ =
pεε =
Brown/Stobbs 1971
Bate/Roberts/Wilson 1981
• hard precipitates not subjected toplastic deformation
• homogeneously distributed precipitates
• interaction between precipitates negligible
• Eshelby approach yields useful approximation for precipitate strain
ΛIK −=
∑= )p()p()p(
f1 f ΛΛ
∑ =⊗⊗⊗Λ= 3
1ijkl)p(
l)p(
k)p(
j)p(
i)p(
ijkl)p( eeeeΛ
)p(lkij
)p(klji
)p(klij Λ=Λ=Λ
accommodation tensor:
Eshelby tensor:
)p(ie
)p(
pε
Ieε
M2. Crystal plasticity (meso scale)
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peFFF =
e** FRF =
1epe
1ee
1 −−− +== VLVVVFFl &&
∑ =αααα ⊗γ+=
n
1T**p msRRL &&∑ =α
ααα− ⊗γ==n
11
ppp~~~ msFFL &&
Kinematics αom
αos
αm
αs
αm~
αs~
αm
αsB~
B
*R
B
eV
oB
M2. Crystal plasticity (meso scale) kinematics
pF
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Platelet precipitates Spherical precipitates
Tensile stresses
Tensile back stress
11ε
11ε
11ε
11ε
11τ
11x 11x
11τ
tensile stretch testM2. Crystal plasticity (meso scale)
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Plane strain die channel compression
compression U1
load
F
)211(1 =x)011(3 =x
20
10
10
M2. Crystal plasticity (meso scale) numerical example
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Von Mises stress
0.11 =u75.01 =u
5.01 =u25.01 =u
numerical exampleM2. Crystal plasticity (meso scale)
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Indentation of Ag single crystals
data from Ma & Clarke 1995
Anisotropy of size effects in single crystals
)m(h1 1−µaging time / particle radius
yiel
d st
reng
th
θ′′ θ′ θGP zones
UAPA
OAr∝
r1
∝
<100 >< 010>
Al-3%Cu crystal (Barlat & Liu 1998)
<010> <100>
M3. Strain gradient crystal plasticity (micron/submicron scale)
2
oHH ⎟
⎠⎞⎜
⎝⎛
h d
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Micro and meso-scale deformation
α
α
γ∇
γ
plasticlattice distortion
pF
conventional crystal plasticity
strain gradient crystal plasticity
dislocation theory
++
M3. Strain gradient crystal plasticity (micron/submicron scale) modeling
==Geometrically Necessary
Dislocations
Acharya/Bassani 2000Aifantis 1987Evers et al. 2002,2004Groma 1997,2003 Gurtin 2002Menzel/Steinmann 2000Shizawa/Zbib 1999Shu/Fleck 1998
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Beam bending in plane strain
)sin/(cos2 ωωκ±=γα x
=A03i =εplain strain:
012 =ε
211 xκ=εpure bending:
incompressibility :
Kirchhoff condition:
0ii =ε∑ 222 xκ−=ε
( )∑α
ααα ⊗γ= Smsε &&
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
κ− 00000000
M3. Strain gradient crystal plasticity (micron/submicron scale) example
)(f|| ω≠κ∝A
|cos|G ωκ=ηα
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hκ
oMM
o15=ω
5.0=β25.0=β125.0=β
0=β
hl
=β0.1=β
Beam bendingmaximal lattice distortion
0→ω
αs
maxG →ρα
decreasing size
hκ
o75=ω
oMM
minimal lattice distortiono90→ω
αs
minG →ρα
M3. Strain gradient crystal plasticity (micron/submicron scale) example
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Depth dependent deformation via discrete dislocation dynamics
free surface
glide planes o45±
σ σ
time
σ
M4. Dislocation dynamics (submicron-nanometer scale) simulations
applieed stress
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0 surface dislocation sources free surface
0
- 200
- 400
- 600
Dislocation dynamics (submicron-nanometer scale)
depth in nmy
Discrete dislocation dynamics simulation
simulations
symm
etric boundary
sym
met
ric b
ound
ary
Double click on movie
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10 surface dislocation sources free surface
0
- 200
- 400
- 600
Dislocation dynamics (submicron-nanometer scale)
depth in nmy
Discrete dislocation dynamics simulation
simulations
symm
etric boundary
sym
met
ric b
ound
ary
Double click on movie
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0 surface sources 5 surface sources 10 surface sources
1t
2t
3t
4t5t
simulationsM4. Dislocation dynamics (submicron-nanometer scale)
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0 surface sources
Peach-Koehler force
dislocation speed
dislocation density
total toward surface into interior
source density in interior: 25*1/µm2
mmp vb ρ=ε&Orowan relation:
simulationsM4. Dislocation dynamics (submicron-nanometer scale)
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dislocation density
total toward surface into interior
source density in interior: 25*1/µm2
10 surface sources
Peach-Koehler force
dislocation speed
simulations
mmp vb ρ=ε&Orowan relation:
M4. Dislocation dynamics (submicron-nanometer scale)
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surface: number of sources: 3nucleation stress: 0.5
interior: number of sources: 50nucleation stress: 0.5
pεpε
y y
relation of plastic deformation between surface and interior materialis dependent on dislocation sources
surface: number of sources: 13nucleation stress: 0.1
interior: number of sources: 50nucleation stress: 0.5
Depth dependent straindepthdepth
M4. Dislocation dynamics (submicron-nanometer scale)
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Tensile stress simulations for free standing thin film
τ
pε
h1t2t
3t
4t
5t
6t
7t
8t
9t
pε
free surfaces
nm1000=h
nm1000=hnm125=h
nm125=h
simulations
in agreement with thickness dependence of tensile stretch experiments
(Kalkmann et al. 2002, Espinosa et al. 2004)
M4. Dislocation dynamics (submicron-nanometer scale)
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Case 1:
• many defects
• high dislocation source density
•
• higher flow stress near surface than in interior
• stronger nano-hardness for high defect density
Case 2:
• hardly any defects in interior
• low dislocation source density
•
•• lower flow stress near surface than in interior
• weaker nano-hardness for low defect density
bulknuc
surfacenuc τ≤τ
bulknuc
surfacenuc τ<<τ
free surface
free surface
Consequence for flow stress near surface
M4. Dislocation dynamics (submicron-nanometer scale)