Materials Process Design and Control Laboratory Finite Element Modeling of the Deformation of 3D Polycrystals Including the Effect of Grain Size Wei Li.

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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




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




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




Outlines

Problem definition

Constitutive Model with homogenization method

Grain size effect model

Geometric processing techniques

Verifications, results and discussions

Conclusions




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




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




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

X Y

Z




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




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




{ | ( )}

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




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




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




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.




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.




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




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

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

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

(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.




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

0.4

0.6

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




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

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

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

Methods – Mesh generation

Conforming grids with 4097 elements

Pixel grids with 20×20×20 elements

Pixel grids with 70×70×70 elements




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




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.




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




Verifications – comparison with experimental result

Constitutive model, grain size effect included




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




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)





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




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 – 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

X Y

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




(a) (c) (e)

(b) (f)(d)

Results and discussions

Displacement field

Equivalent stress field




X Y

Z

X Y

Z

20, 50 and 100 grains






Mechanical responses of three different grain sizes

Comparison with experimental results




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.




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

Materials Process Design and Control Laboratory Finite Element Modeling of the Deformation of 3D Polycrystals Including the Effect of Grain Size Wei Li.

Documents