1m material modelling at various length scales · m3. strain gradient crystal plasticity...

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

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)