materials process design and control laboratory finite element modeling of the deformation of 3d...
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Finite Element Modeling of the Deformation Finite Element Modeling of the Deformation of 3D Polycrystals Including the Effect of of 3D Polycrystals Including the Effect of
Grain SizeGrain Size
Wei Li and Nicholas ZabarasMaterials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering, Cornell University
URL: http://mpdc.mae.cornell.edu
April 25, 2008
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Grain/crystal
Inter-grain slip
Grain boundary
Twinning
MacroMeso
Mechanical properties of material are extremely essential to the quality of products
Preference on material properties requires efficient modeling and designing in virtual environment
Considerably advantageous to traditional error-correction method
Motivations
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Motivations
Adequate description of material properties using appropriate mathematical and physical models
Use appropriate model to capture the plastic slip in polycrystals and simulate the mechanical properties of the material.
Work as a Point Simulator in a multiscale framework
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CCOORRNNEELLLL U N I V E R S I T Y
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Outlines
Problem definition
Constitutive Model with homogenization method
Grain size effect model
Geometric processing techniques
Verifications, results and discussions
Conclusions
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CCOORRNNEELLLL U N I V E R S I T Y
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Strain
Str
ess
(Mp
a)
0 0.05 0.1 0.15 0.2
100
200
300
400
500
Modeling of realistic 3D polycrystalline microstructure
Voronoi Tessellation
Conforming grid generation
Virtual interrogation of microstructure, constitutive model
Mechanical response
Deformed microstructure
X Y
Z
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CCOORRNNEELLLL U N I V E R S I T Y
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Generate microstructure(Voronoi Tessellation)
Mesh the microstructure
Mechanical response and deformed microstructure
Constitutive model considering grain size effect
Homogenization boundary condition
Domain decomposition(Efficient parallel computation)
3D interrogation of microstructure
Procedures
Geometry processing techniques
Physical models implementation
Construct the relation between the microstructure and its mechanical properties
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Virtual Compression Test
The cubic region is compressed in one direction and stretch in the other two uniformly.
Initial conditions:(1) Prescribed velocity gradient on boundary;(2) Each grain has a random orientation.
0.5 0 0
0 1.0 0
0 0 0.5
L r t
L : Velocity gradientr : Strain rateΔt : Time step
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Z
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Constitutive Model
1 1
1 , 1, 2,3
nn n n
i ii
F FL FF F F L t I F
tDis x F x i
Steps:
Boundary conditions
Known conditions: all the parameters in previous time step, e.g. deformation gradient and its elastic and plastic components etc.
e pF F F
1
n
etrial pF F F
Tetrial etrial etrialC F F
1
2etrial etrialE C I
trial e etrialT L E
Crystal/lattice
reference frame
e1^
e2^
Sample reference
frame
e’1^
e’2^
crystalcrystal
e’3^
e3^
0trial trialT S
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Quaternion method
A quaternion method is adopted here to transform the orientation expressed in Rodrigues-Frank space to transformation matrix
tan2
r n
e1e’1
e2
e’2
e3
^ ^
^
^
^e’3^
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
1 2 3 4, , , sin , sin , sin ,cos2 2 2 2
q q q q q q q u v w
3
2 2 2 24 1 2 3 4
1
2 2ij ij i j ijk kk
a q q q q q q q q
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{ | ( )}
1 : systems
, , ( )
trial
trial
PA s t
m active slip
x A b x b s t
0 0sgn sgn
( )
trial trial e etrial
trial
A h t S L sym C S
b s t
x
Constitutive model: active slip systems
, Active
s s t h for all
1
1.0 for coplanar slip systemswhere
1.4 for noncoplanar slip systems
h q q h
q
0 1a
s
sh h
s
Constitutive Model
where
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Constitutive model (continued):
0
1
sgn( ) n
p trial p
Active
e p
F I S F
F F F
1( )
2e e T eE F F I
[ ]e eT L E
1( )
dete e T
eT F T F
F
det( ) TP F TF
Constitutive Model
Equivalent stress and strain by averaging over all elements
1 1
0
1
31
2
3
e
p P Pnn n
p p p
p p
t
eff
F FL F F F
t
D sym L tr L I
D D D dVV
D Ddt
1
1'
3
3' '
2
e
total
total total
eff
T T TdVV
T sym T tr T I
T T
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Lattice incompatibility
The material deformation gradient is composed of a plastic part due to slips in crystals and an elastic part that accounts for lattice distortion and rotation. This assumes that the lattice only distorts elastically.
Elastic distortion generally is not compatible with a regular displacement field, so it is natural to use elastic deformation gradient Fe (or (Fe)-1) as a measure of lattice incompatibility.
Grain size effect
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Grain size effect
Grain size effect and dislocation density
Lattice incompatibility is coupled with the evolution of dislocation density, which is highly intense on grain boundary.
This is because dislocation line can not move across grain boundary. Whenever dislocation is generated, it will assemble there and cause lattice distortion, which leads to lattice incompatibility.
Due to the restriction on grain boundaries, the grains can not deform as they wanted to and thus no gap or overlap occurs.
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1 1
, ,
e e
ij k ik jF F
0 1 2Active
k k kb
1n n
dt
10 2
00
ˆ ˆˆ ˆ ,
ˆ ˆb
bb
Bailey-Hirsch relationship:
Lattice incompatibility:
Shear strain rate:
Dislocation density:
Grain size effect
0 0n n Magnitude of lattice incompatibility in a slip system:
The first term considers the relation between lattice incompatibility and dislocation density.
The k1 and k2 are two experience functions coming from experiments.
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1
2
2 200 1
20
2 20
00 0
1ˆ
2ˆ ˆ
ˆ ˆ2 2 2
ˆ ˆ
ˆ ˆ ˆ ˆ2
Active Active
s
Active Actives
b
k b bkk
k b
11 ˆ ˆ
ˆ ˆ ˆ ˆn n
n n dtdt
Shear resistance:
Updated shear stress:
Grain size effect
1ˆn is used as the plastic resistance on the slip systems to judge whether plastic deformation occurs
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Voronoi Tessellation method and microstructures
X
0
0.2
0.4
0.6
0.8
1
Y
0
0.2
0.4
0.6
0.8
1
Z
0
0.2
0.4
0.6
0.8
1
X Y
Z
X
0
0.2
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1
Y
0
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1
Z
0
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1
X Y
Z
X
0
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1
Y
0
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0.8
1
Z
0
0.2
0.4
0.6
0.8
1
X Y
Z
(a) (b) (c)
Steps:
1. Sample a set of points, say 5, in the domain;
2. Calculate the grain boundaries with V.T.;
3. Generate the grains.
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X
0
0.2
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0.8
1
Y
0
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0.8
1
Z
0
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0.4
0.6
0.8
1
X Y
Z
X
0
0.2
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0.8
1
Y
0
0.2
0.4
0.6
0.8
1Z
0
0.2
0.4
0.6
0.8
1
X Y
Z
Methods – Mesh generation
Advantages:
No restriction on grains
Fully adaptive to microstructure geometries
Element numbers manageable
Simulate the “real” microstructures without assuming unrealistic grain boundaries
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X
0
0.2
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0.8
1
Y
0
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0.8
1
Z
0
0.2
0.4
0.6
0.8
1
X Y
Z
X
0
0.2
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1
Y
0
0.2
0.4
0.6
0.8
1
Z
0
0.2
0.4
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0.8
1
X Y
Z
X
0
0.2
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0.6
0.8
1
Y
0
0.2
0.4
0.6
0.8
1
Z
0
0.2
0.4
0.6
0.8
1
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Z
Methods – Mesh generation
Conforming grids with 4097 elements
Pixel grids with 20×20×20 elements
Pixel grids with 70×70×70 elements
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Mesh Generation and Domain Decomposition
X Y
Z
X Y
Z
Mesh the grains
Split into brick elementsDomain decomposition
CD
GI
A
C D
G
J
MN
K
C
G
N
O
D
GL M
O I
G
H
J
MN
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X Y
Z
Domain decomposition
The whole region is decomposed into continuous sub-regions
Each sub-region is individually processed by one processor in parallel computation
Faster than using the indices to assign the elements to the processors
Divide the region into 32 parts, use 8 nodes (32 processors)
Speed: 33,000 time steps ~ 6hr, comparing with 17,000 time steps~ 6hr previously, approximately 50% increased
Each separate region is “continuous”, when integrating the local matrices, processors just need to communicate when doing calculation on the boundaries.
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Verifications – comparison with simulated result
Strain
Str
ess
(Mp
a)
0 0.2 0.4 0.6 0.8 1
100
200
300
400
This work (Taylor)
This work (Homogenization)
Anand and Kothari (1996)
Constitutive model, without considering grain size effect
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Verifications – comparison with experimental result
Constitutive model, grain size effect included
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Grains configuration in non-conforming grid
Since the adopted constitutive model is a non-scale model, grain size can not be altered by changing the calculated region.
In non-conforming grids, grain size is determined by the way of specifying the grains.
If each 1 element is seen as a grain, the average grain size is just the size of a single element.
If each 8(=23) elements are seen to compose a grain, the size will be twice as much.
Larger grain sizes can be obtained with similar method.
X Y
Z
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0 10 20 30 40 50 60100
150
200
250
300
350
400
450
500
550
600
1/D (mm-1)
Str
ess
(M
pa
)
5%
10%
15%
20%
Results and discussions – nonconforming grids
Comparison with the experimental results (Narutani and Takamura, 1991)
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Results and discussions – nonconforming grids
Comparison with the experimental results (Narutani and Takamura, 1991)
Stress-strain curves of different grain sizes
1/6, 1/8, 1/12, 1/24mm 24×24×24 elements
1/36mm 36×36×36 elements
1/48mm 48×48×48 elements
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StrainS
tres
s(M
pa)
0 0.05 0.1 0.15 0.2
100
200
300
400
500
Mechanical responseEquivalent stress field
Grain size: 1/36mm
Results and discussions – nonconforming grids
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Results and discussions – conforming grids
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2
0.4
0.6
0.8
1
z
0
0.2
0.4
0.6
0.8
1
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Z
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2
0.4
0.6
0.8
1
z
0
0.2
0.4
0.6
0.8
1
X Y
Z
Microstructure deformation
Equivalent Stress fieldMechanical response
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(a) (c) (e)
(b) (f)(d)
Results and discussions
Displacement field
Equivalent stress field
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X Y
Z
X Y
Z
20, 50 and 100 grains
Results and discussions
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Results and discussions
Mechanical responses of three different grain sizes
Comparison with experimental results
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Conclusions
(1) A finite element analysis of large deformation of 3D polycrystals is presented. The effect of grain size is included by considering a physically motivated measure of lattice incompatibility.
(2) A domain decomposition method, Voronoi Tessellation method and conforming grids generation technique are developed.
(3) Calculated mechanical properties of polycrystals are shown to be consistent with experimental results.
(4) Conforming grids method is adopted to investigate the strengthening effect of grain sizes.
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Information
Relevant publication
W. Li and N. Zabaras, “A virtual environment for the interrogation of 3D polycrystals including grain size effects”, Computational Materials Science, to be submitted
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
101 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801Email: [email protected]
URL: http://mpdc.mae.cornell.edu/
Prof. Nicholas Zabaras
Contact information