materials process design and control laboratory 1 grain-size effect in 3d polycrystalline...

CCOORRNNEELLLL U N I V E R S I T Y


Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory 11

Grain-size effect in 3D polycrystalline microstructure including texture

evolution

Bin Wen and Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

101 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/



Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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.

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.

Couple the macro-scale finite element simulation with underlying meso-scale constitutive model.




Outlines

Constitutive Model based on crystal slip theory

Texture evolution of polycrystalline material

Grain size effect model

Geometric processing techniques

Multiscale simulation with homogenization method

Conclusions




Strain

Str

ess

0 0.2 0.4 0.6 0.8 1

100

200

300

Modeling of realistic 3D polycrystalline microstructure

Mesh

Load

Virtual interrogation of microstructure

Mechanical response

Deformed microstructure




Microstructure constitutive model

The meso-scale microstructure is represented with a polycrystal aggregate. Each microstructure contains several grains having different orientations.

The initial orientation is assigned randomly and the constitutive model of the crystal is using the rate independent continuum slip theory developed by Anand. Total Lagrangian algorithm is adopted.

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 0

trial trialT S




{ | ( )}

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

Active slip systems is determined by comparing trial shear stress with slip resistance.

The update of the slip resistance is based on shear strain increment

Where the hardening matrix is

1

1.0 for coplanar slip systemswhere

1.4 for noncoplanar slip systems

h q q h

q

0 1a

s

sh h

s

, Active

s s t h for all





The plastic and elastic deformation gradient can then be updated. Cauchy stress and PK-I stress are also ready to be calculated as well as homogenized equivalent value.

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


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




Strain

Str

ess

(Mp

a)

0 0.2 0.4 0.6 0.8 10

100

200

300

400

Example: a uniaxial compression virtual test of a cubic microstructure.

Verification of constitutive model




Texture Evolution in a discrete manner

The texture is represented by various orientations of grains in microstruture. Properties of polycrystals (such as strength, heat conductivity, etc ) are highly dependent on the texture.

Crystals with random texture generally demonstrate isotropic characters while those having preferable texture distribution show anisotropic.

0

0

e

e T

m F m

n F n

The evolution of texture along with microstructure deformation can be tracked by the elastic twist while assuming plastic deformation causes glide on slip planes only. The change of slip system is evaluated as




The orientation of a grain is described by a rotation around a specific axis in the real space. This rotation can be conveniently expressed using vector (or point) in 3D Rodrigues space.

1 2 3{ , , } { tan , tan , tan }2 2 2x y zr r r r n n n

Considering the symmetries of a crystal (cubic structure for FCC), the Rodrigues space is able to be contracted into a finite fundamental zone. The texture is represented using discrete Orientation Distribution Function (ODF) in that fundamental zone.

Rodrigues representation and ODF




Initiated with random texture, a microstructure subjected to different deformation form (boundary condition) gives distinct evolution of textures.

{111}{110}

{111}{110}

Initial random texture

simple compression, szz=1.0

Texture evolution




{111}{110}

{110} {111}

Plane strain in y-z plane, szz=1.0

Simple shear, syz=0.6

Texture evolution




Grain size effect

Resistance to plastic flow in crystals is dominated by dislocation density. The presence and motion of dislocations lead to permanent deformation and strain-hardening and can cause incompatibility in crystals.

As the lattice incompatibility can be measured by elastic deformation gradient, it is reasonable to quantifies the incompatibility in Fe (Acharya and Bassani, 2000)

1 1, ,( )e e

ij k ik j i j kF F e e e

A evaluation form of dislocation density is considered as

0 1 2Active

k k kb

Where slip system lattice incompatiblity

ˆ ˆ: :n n

n̂is the unique skew symmetric tensor defined by slip normal.




Bailey-Hirsch relationship:

0ˆ ˆ b

Differentiate both sides with respect to time, a isotropic single crystal hardening law is obtained

1 2 2

00 122

0

ˆ ˆ1ˆ

ˆ ˆ2 2 2 2Active Active

k b bkb k

2 2

00

0 0

ˆ ˆˆ

ˆ ˆ ˆ ˆ2s

Active Actives

k b

Substitute k1 and k2 with initial strain-hardening rate 0

Grain size effect

The first term considers hardening by strain, and the second term considers the effect from strain gradient, which is affected majorly by grain size.




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12Z

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24x24x24 grid with cubic grain12x12x12 grid with cubic grain

24x24x24 grid with phase field grain

Examples with different grain size




12x12x12 elements

Stress-strain curves using grain size effect model. All of the microstructures are subjected to compression in z direction and stretch in the other two.

Examples of different grain size




X

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equiv_stress

2402202001801601401201008060

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64 grains

Domain decomposition

Stress field

Realistic grain




Voronoi Tessellation method and microstructures

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X Y

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(a) (b) (c)

Steps:1. Sample a set of points;2. Calculate the grain boundaries with V.T.;3. Generate the grains.




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Conforming mesh generation

Advantages:No restriction on grains Fully adaptive to microstructure geometriesElement numbers manageableSimulate the “real” microstructures without assuming unrealistic grain boundaries




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Conforming mesh generation

Conforming grids with 4097 elements

Pixel grids with 20×20×20 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




Conforming grids example

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Microstructure deformation

Equivalent Stress fieldMechanical response




Multi-scale simulation

Given the previous microstructure constitutive models, a multi-scale simulation can be naturally implemented. The microstructure is coupled with macrostructure through homogenization assumption.

In material processing, the mechanical response of a work piece is highly interested. To get an reliable prediction of material property in processing, accurate micro-scale constitutive model is needed.

x

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1

1

macro micro micro

B

macro micro microB

T T dV TV

T T TdV

F V F F

The macro-scale values are obtained from the microstructure in the form of volume average. In this work, Cauchy stress and its derivative with respect to deformation gradient are returned, and the PK stress in macro-continuum is calculated using deformation gradient at that point.

(det )

(det ) (det ) (det )

Tmacro macro micro macro

T T Tmacro macro micro macro macro micro macro macro micro macro

P F T F

dP d F T F F d T F F T dF

where

1 1 1 micro micro micromicro micro micro macro

B B B

T T Td T d T dV dT dV dF dV dF dF

V V V F F F

Multi-scale homogenization




Some examples

A cubic macrostructure consisting of 6x6x6 elements is compressed along z direction and stretched in the other two. The orientations of grains in all the microstructures are randomly assigned.

x

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x

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equiv_strain

0.2040.20350.2030.20250.2020.20150.2010.20050.20.19950.199

x

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equiv_stress

28026024022020018016014012010080

strain

stre

ss(M

pa

)

0 0.05 0.1 0.15 0.20

50

100

150

200

250

x

0

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Strain field Stress field

Examples




x

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equiv_strain

0.096650.09660.096550.09650.096450.09640.096350.09630.096250.09620.096150.09610.096050.0960.095950.09590.095850.09580.09575

x

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equiv_stress

188186184182180178176174172170168166164162160158156154152

All microstructures have the same texture.

Strain field Stress field

Examples

materials process design and control laboratory 1 grain-size effect in 3d polycrystalline...

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