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• 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
Plastic anisotropy evolution of rolled sheet metals determined by tensile tests
experimentsM1. Macro-plasticity
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)
Rotational Hardening / Rotation of Anisotropy Axes
anisotropic yield function
isotropic kinematic rotational
M1. Macro-plasticity modeling
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:
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
simulationM1. Macro-plasticity
Draw bead simulation with rotational hardening
orientation to rolling direction:
rotation angle (o)
o30Ψ =
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
Crystal plasticity
M2. Crystal plasticity (meso scale)
Slip systems of an FCC crystal
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)
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
Platelet precipitates Spherical precipitates
Tensile stresses
Tensile back stress
11ε
11ε
11ε
11ε
11τ
11x 11x
11τ
tensile stretch testM2. Crystal plasticity (meso scale)
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
Von Mises stress
0.11 =u75.01 =u
5.01 =u25.01 =u
numerical exampleM2. Crystal plasticity (meso scale)
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
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
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
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
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
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
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
0 surface sources 5 surface sources 10 surface sources
1t
2t
3t
4t5t
simulationsM4. Dislocation dynamics (submicron-nanometer scale)
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)
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)
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)
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)
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)