Crystal plasticity modeling at small scales and at large scales

raffleescargatoireMechanics

Jul 18, 2012 (5 years and 28 days ago)

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Max-Planck-Institut für Eisenforschung GmbH
Dierk Raabe, FranzRoters, Stefan Zaefferer, Dirk Ponge
Crystal plasticity modeling
at small scales and at large scales
Crystal plasticity modeling
Crystal plasticity modeling
at small scales and at large scales
at small scales and at large scales
GeoMat2005, June7 -10, Aachen
www.mpie.de
edoc.mpg.de
contact
•Professor Dierk Raabe, Dr. Franz Roters
•Max-Planck-Institut fuer Eisenforschung
Max-Planck-Str. 1
40237 Duesseldorf
Germany
Tel+49 (0)211-6792-340 / -278 / -393
Emailraabe@mpie.de, roters@mpie.de
•www
http://www.mpie.de/
www
http://edoc.mpg.de/
￿Crystal plasticity FEM
￿Small scale
￿Single Crystals
￿Bicrystals
￿Large scale
￿Bulk Polycrystals
Outline
Outline
￿Crystal plasticity FEM
￿Small scale
￿Single Crystals
￿Bicrystals
￿Large scale
￿Bulk Polycrystals
Outline
Outline
{}
1FF
2
1
EwithCET
FFF
*****
P*
T
1
−==
=

crystalplasticity:
continuummechanics:
α
0
*αα
crit
αα
α
α0
α0
α0
α0
αP
PPP
STτwith)τ,f(τγ
nmSwithSγL
FLF
⋅≈=
⊗==
=

&
&
&
(
)
γ

γ,g
τ
α
crit
&
=
Crystal plasticity FEM
Crystal plasticity FEM
(
)
α
m1
α
crit
α
0
α
τsign
τ
τ
γγ
&&
=
()()
a
s
β
crit
0
ββαβαββ
β
αβα
crit
τ
τ
1hh,hqh,
γ
h
τ






−===

&&
hardeninglaw:strainrate law:
Crystal plasticity FEM
Crystal plasticity FEM
￿Crystal plasticity FEM
￿Small scale
￿Single Crystals
￿Bicrystals
￿Large scale
￿Bulk Polycrystals
Outline
Outline
crystal plasticity FEM at small scales
crystal plasticity FEM at small scales
Nanoindentation
Nanoindentation
, Cu Single Crystals
, Cu Single Crystals
experiment
simulation
Simple
Simple
Shear
Shear
, Al Single Crystals
, Al Single Crystals
￿Crystal plasticity FEM
￿Small scale
￿Single Crystals
￿Bicrystals
￿Large scale
￿Bulk Polycrystals
Outline
Outline
Deformation Behavior of Al
Deformation Behavior of Al
-
-
bicrystals
bicrystals
(with friction)
(with friction)
Deformation experiments
Channeldie experiment:
plane straindefor-
mation(30%
thicknessreduction)
Bicrystalswith<112> tiltboundaries
7.8°15.0°
30.7°
{111}
{111}
{111}
8%
y
x
z
von-Mises strain
Kleine Skalen: Oligokristalle, Al, ebene Dehnung
Kleine Skalen: Oligokristalle, Al, ebene Dehnung
15°, 30% reduction
30.7°, 30% reduction
experiment
simulation
eq. strain
7.8°, 30% reduction
Plane Strain Deformation of Al
Plane Strain Deformation of Al
-
-
bicrystals
bicrystals
7.4°, experiment
Deformation Behavior of Al
Deformation Behavior of Al
-
-
bicrystals
bicrystals
(no friction)
(no friction)
33.2°, experiment
7.4°, simulation
33.2°, simulation
V. Schulz, Plastic Deformation: Constitutive Description
Encyclopedia of Materials: Science and Technology
Microstructure
Microstructure
Dislocation classes
Dislocation classes
￿
Mobile dislocations
Mobile dislocations
accommodate the external plastic deformation
￿
￿
Immobile dislocations
Immobile dislocations
work hardening, including locks and dipoles
￿
Geometrically necessary dislocations
Geometrically necessary dislocations
preserve the lattice continuity
SSDIISSDISSDIISSDISSD
)()()()(
−−++
+++=
ρρρρρ
&&&&&
Rate equations
Rate equations
Rate equations are formulated for the immobile dislocation densi
Rate equations are formulated for the immobile dislocation densi
ties:
ties:
Four physical mechanisms will be used:
Four physical mechanisms will be used:
mechanism 1: lock formation
mechanism 2: dipole formation
mechanism 3: athermalannihilation
mechanism 4: thermally activated annihilation by climb
Locks
Dipoles
Athermal
Thermal
λ
ρρ
1
I
⋅⋅∝
+
v
m
&
dipolemm
dv⋅⋅⋅∝
+
ρρρ
II
&
γρρ
&&
mdipole
dc
6II
=
+
...
111
otherdis
++=
λλλ
γρρ
&&
⋅=
+
F4I
c
τυπ
1
)1(16
3


=
Gb
d
dipole
8
climb
c
B
K
DS
v
γτ
θ
&

8
2
BB
bulk
7II
)exp(
c
KK
Q
c
γρ
θ
τ
θ
ρ
&&
−−=

dv⋅⋅⋅∝

climbII
ρρρ
&
dv
m
⋅⋅⋅∝

ρρρ
I
&
constd
=
γρρ
&&
⋅−=

5I
c
Rate equations
Rate equations
Distinction of slip systems
Distinction of slip systems


=
=
×=
×=
N
N
ρ
ρ
1
P
1
F
),sin(
),cos(
β
ββαβαβα
β
ββαβαβα
ρκ
ρκ
dnn
dnn
A forest and a parallel dislocation density is calculated for
A forest and a parallel dislocation density is calculated for
each individual slip system
each individual slip system
F
m
F
B
mP
cbQ
K
bccQ
Gbc
ρ
ρ
ρ
ν
γθ
ρρτ












+++=
20slip
2
32
slip
1
2
ln
&
1
111







+−
−==
C
B
mP
B
F
V
K
Gbc
K
Q
c
bbv
θ
ρρτ
θ
υ
ρ
γ
1
slip
0
2
exp)exp(
 
 
ρ
ρρρρ
ρρρ
&
0
PF
,,
=








ρργ
ρ
τ
&
m
Roters: phys. stat. sol. b 240 (2003) 68-74
Ma, Roters: ActaMaterialia52 (2004) 3603–3612
Orowan
Orowan
equation
equation:
Calculation of the mobile dislocation density
Calculation of the mobile dislocation density
3
321
2
2/)(
Gbccc
K
B
BA
B
=
=
θ
044
222
=−−
PFmFm
ρρAρρAρ
0
!
m
ρ
τ
=


F
m
mP
ρ
ρA
ρρ
21
=
+
PFFFm
ρρρAAρAρ++=
222
22
101
101
5.01.0
1038.1
3
2
1
123
B
≤≤
≤≤
≤≤
×=
−−
c
c
c
JKK
for Al:
CC
Mb
GPa
G
°≤≤°
×=


45020
1086.2
19
10
θ
)10,10(
14−−
=A
PFm
B
ρρθρ
=
define:
For a homogenous dislocation structure…
For a homogenous dislocation structure…
Calculation of the mobile dislocation density
Calculation of the mobile dislocation density
Evolution law for GND dislocations
Evolution law for GND dislocations
Local control equations:
Local control equations:
Non local integration algorithm for FEM
Non local integration algorithm for FEM
Local algorithm:
Local algorithm:
Non local integration algorithm for FEM
Non local integration algorithm for FEM
(1) Use
i
SSD
ρ
and
i
GND
ρ
to calculate
1
~
+i
S
.
(2) Use
i
GND
ρ
and
1
~
+i
S
to update
1+
i
SSD
ρ
.
(3) Check convergence of
1+
i
SSD
ρ
, yes goto (4), no goto (1).

(4) Use
1
~
+
i
S
and
1+
i
SSD
ρ
to update
1+
i
GND
ρ
.
(5) Check convergence of
1+
i
GND
ρ
, yes goto (6), no goto (1).

(6) Finish iteration.
￿Separate the iteration number n into odd
and even
.
￿If n is odd, record the deformation gradient
, return one constant stress and
￿If n is even, call local algorithm
using recorded data, because we have
Global algorithm:
Global algorithm:
Non local integration algorithm for FEM
Non local integration algorithm for FEM
Grain boundary mechanism
Grain boundary mechanism
Slip systems for FCC crystal
Slip systems for FCC crystal
activation energy plots for
activation energy plots for
gb
gb
penetration
penetration
Simple shear of Al single / bi crystals
Simple shear of Al single / bi crystals
￿Single crystal test
￿Bi crystal with small angle grain boundary
￿Bi crystal with intermediate grain boundary
￿Bi crystal with large angle grain boundary
￿Material: Aluminum 99.999%
￿Test condition: 20°C, 0.001 1/s
experiments using GOM Aramis
Simulation by FEM
Simple shear of Al single crystal
Simple shear of Al single crystal
Fitting parameters
1.0
Self-
interaction
4.5
Lomer-
Cottrellock
1.6
Hirthlock
3.8
Glissile
junction
3.0
Cross slip
2.2
Coplanar
Interaction strength constant
Journal of the Mechanics
and Physics of Solids
50 (2002) 1979 –2009
0.3C8
1.0E10
10

0
1.0E-30C6
1.0E7C7
10.0C5
1C9
1.5E7C4
1.0C3
2.0C2
0.1C1
2.4E10
-19
JQbulk
3.0E10
-19
JQslip
10% 20% 30% 40% 50%
Simple shear of Al single crystal
Simple shear of Al single crystal
experiment
phenomenological
model (local)
physics-based
model (nonlocal)
Simple shear of Al bi crystal, 7.4°
Simple shear of Al bi crystal, 7.4°
10% 20% 30% 40% 50%
experiment
phenomenological
model (local)
physics-based
model (nonlocal)
Simple shear of Al bi crystal, 15.9°
Simple shear of Al bi crystal, 15.9°
10% 20% 30% 40% 50%
experiment
phenomenological
model (local)
physics-based
model (nonlocal)
Simple shear of Al bi crystal, 33.2°
Simple shear of Al bi crystal, 33.2°
10% 20% 30% 40% 50%
experiment
phenomenological
model (local)
physics-based
model (nonlocal)
Simple shear of Al bi crystals
Simple shear of Al bi crystals
7,4°15.9°33.2°
experiment
phenomenological
model (local)
physics-based
model (nonlocal)
￿Crystal plasticity FEM
￿Small scale
￿Single Crystals
￿Bicrystals
￿Large scale
￿Bulk Polycrystals
Outline
Outline
crystal plasticity FEM at large scales
crystal plasticity FEM at large scales
TCCP-FEM: the texture component crystal plasticity FEM
Common Solutions
Common Solutions
σ
σ
FEM
e.g. Hill
σ
e.g. TBH
σ
analyticalyieldsurface:
ui,j
M
FEM
e.g. TBH
ni
bj
γ
γγγ
gi
ui,j
M
FEM
e.g. TBH
ni
bj
γ
γγγ
gi
textureapproximationbysingleorientations:
Model
Model
Components
Components
),(
~
~
gg
cc
ωω
=
(
)
ω
~
cosexp)(
ccc
SNgf=
)()(
1
and
)2/cos(1
2ln
c
1
c
0
c
c
c
SISI
N
b
S

=

=
generalizedBessel functions
)(xI
l
valueisthehalfwidth
(meandiameterof a
sphericalcomponentin
orientationspace)
c
b
Lücke, Pospiech, Virnich, Jura: Acta metall. 29 (1980) p. 167
Helming, Schwarzer, Rauschenbach, Geier, Leiss, Wenk, Ullemeier, Heinitz: Z. Metallkd. 85 (1994) p. 545/554
1)(,
)()()(
00
C
0c
cc
C
1c
cc
==
=+=
∑∑
==
gfFw
gfwgfwFgf
gorientation
orientationdistributionfunction
Frandomtexturecomponent
volumeportionof all crystalswhich
belongto thetexturecomponentc
)(gf
c
w
Lücke, Pospiech, Virnich, Jura: Acta metall. 29 (1980) p. 167
Helming, Schwarzer, Rauschenbach, Geier, Leiss, Wenk, Ullemeier, Heinitz: Z. Metallkd. 85 (1994) p. 545/554
Texture Representation by Model
Texture Representation by Model
Components
Components
choiceof
normal direction
choiceof
prefereddirection
introductionof
samplesymmetry
MulTex2.0 K. Helming
Component
Component
Fit
Fit
experimental
pole figure
recalculated
pole figure
differential
pole figure
MulTex2.0 K. Helming
Component
Component
Fit
Fit
Decompose
Decompose
the
the
Component
Component
Local
Local
Homogenization
Homogenization
0,98
0,985
0,99
0,995
1
1,005
1,01
1,015
1,02
0102030405060708090
angle to RD
relative ear height
ideal random
1 random component
2 random components
4 random components
How
How
Random
Random
is
is
Random
Random
?
?
sample: Hydroaluminium (Bonn)
Application
Application
Aluminium Hot Band
Aluminium Hot Band
Experimental <111> Pole Figure
Recalculated <111> Pole Figure
Texture Component Fit
71 %29 %
Vol.bc
ϕ
ϕϕϕ
2
[°]φ φ φ φ [°]ϕ
ϕϕϕ1
[°]
-random
15.2245.06.5197.9
Aluminium Hot Band
Aluminium Hot Band
0153045607590
0,950,960,970,980,991,001,011,021,031,041,05
Simulation
Experiment

relative ear hight [1]
angle to rolling direction [°]
Aluminium Hot Band
Aluminium Hot Band
Aluminium Cold Band
Aluminium Cold Band
Experimental <111> Pole Figure
Recalculated <111> Pole Figure
Texture Component Fit
38 %16.810.637.5141.1
32 %30 %
Vol.bc
[°]
ϕ
ϕϕϕ
2
[°]φ φ φ φ [°]ϕ
ϕϕϕ1
[°]
-random
14.534.229.3108.9
0153045607590
0,940,950,960,970,980,991,001,011,021,031,04
Experiment
Simulation
Gauss only


relative ear hight [1]
angle to rolling direction [°]
Aluminium Cold Band
Aluminium Cold Band
0153045607590
0,940,950,960,970,980,991,001,011,021,031,04
Simulation
Gauss + fibre
Experiment
Simulation
Gauss only


relative ear hight [1]
angle to rolling direction [°]
Aluminium Cold Band
Aluminium Cold Band
0255075100
0.51.01.52.0
0.20.40.60.8
Ear area(mm
2
)

Ear height(mm)
Volume fraction of Cube [%]




0153045607590
5.05.56.06.57.07.5
Cube50% S50%
S100%
Cube100%
Cube25% S75%
Cube37.5% S62.5%

Earing height (mm)
Angle to rolling direction [
O
]
Mixing
Mixing
S and
S and
Cube
Cube
components
components
Quantativeanalysis
Earprofileswithdifferent volumefraction
InverseEngineering
Cube37.5% + S 62.5% with15°
°°°scatterwidth
Inverse
Inverse
Anisotropy
Anisotropy
Engineering
Engineering
Conclusions
Conclusions
￿
￿
Small scale
Small scale
￿
￿
Constitutive law based on dislocation densities
Constitutive law based on dislocation densities
including geometrically necessary dislocations
including geometrically necessary dislocations
￿
￿
Grain boundary mechanism
Grain boundary mechanism
￿
￿
Non local integration algorithm
Non local integration algorithm
￿
￿
Large scale
Large scale
￿
￿
Texture Component method for the mapping of
Texture Component method for the mapping of
crystallographic texture
crystallographic texture


enhanced prediction of local deformation behaviour of
enhanced prediction of local deformation behaviour of
bicrystals
bicrystals


enhanced prediction of anisotropy, e.g.
enhanced prediction of anisotropy, e.g.
earing
earing
in cup drawing
in cup drawing
References
•D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer: Acta Materialia 49 (2001) 3433–3441,
„Micromechanical and macromechanical effects in grain scale polycrystal plasticity
experimentation and simulation”
•D. Raabe, Z. Zhao, S.–J. Park, F. Roters: Acta Materialia 50 (2002) 421–440, „Theory of
orientation gradients in plastically strained crystals”
•Z. Zhao, F. Roters, W. Mao, D. Raabe: Adv. Eng. Mater. 3 (2001) p.984/990, „Introduction of A
Texture Component Crystal Plasticity Finite Element Method for Industry-Scale Anisotropy
Simulations”
•M. Sachtleber, Z. Zhao, D. Raabe: Materials Science and Engineering A 336 (2002) 81–87,
“Experimental investigation of plastic grain interaction”
•D. Raabe, P. Klose, B. Engl, K.-P. Imlau, F. Friedel, F. Roters: Advanced Engineering Materials 4
(2002) 169-180, „Concepts for integrating plastic anisotropy into metal forming simulations”
•D. Raabe, Z. Zhao, W. Mao: Acta Materialia 50 (2002) 4379–4394, „On the dependence of in-grain
subdivision and deformation texture of aluminium on grain interaction”
•D. Raabe and F. Roters: International Journal of Plasticity 20 (2004) p. 339-361, „Using texture
components in crystal plasticity finite element simulations”
•S. Zaefferer, J.-C. Kuo, Z. Zhao, M. Winning, D. Raabe: Acta Materialia 51 (2003)4719-4735., „On
the influence of the grain boundary misorientation on the plastic deformation of aluminum
bicrystals”
•Z. Zhao, W. Mao, F. Roters, D. Raabe: Acta Materialia 52 (2004) 1003–1012, „A texture
optimization study for minimum earing in aluminium by use of a texture component crystal plasticity
finite element method”
•Y. Wang, D. Raabe, C. Klüber, F. Roters, Acta Mater., 52 (2004) 2229-2238.,“Orientation
dependence of nanoindentation pile-up patterns and of nanoindentation microtextures in copper
single crystals”