Bastiaan van der Boor Grain growth by...

Tata Steel SlideCurvature-driven grain growth

Bastiaan van der Boor

Grain growth by curvature


Content

2

1 Fundamentals

2 Micro structure models

3 Curvature methods

4 Results

5 Preliminary conclusions & Further research


Micro-structure

3

Micro-structure determines the mechanical properties of steel

Example of grain growth in a micro-structure of austenite at 1200°C


Three processes that determines the micro-structure

1. Phase transformations

• Austenite to ferrite transformation

2. Recrystallization & Recovery

• Austenite to austenite

3. Grain growth by curvature

• Austenite to austenite

4

Goal of this master thesis is to extend the existing model of

Tata Steel with grain growth by curvature

T.B.D.


Soap froth

5

The growth of metal grains is similar to soap bubbles

)11

(yx RR

G +=∆ γ

xRMv

γα

2=

Young-Laplace equation


Curvature

6

An arbitrarily simple, closed curve with a point P on it, there is a unique

circle which most closely approximates the curve near P

R

1=κ


Triple points, where three grains meet

7

Every triple point seeks it equilibrium state, which depends on the grain

boundary energy γγγγ

231312 γγγ ==

°=120iθ 3,2,1=iEquilibrium if


Examples

8

A grain with 6 corners is in equilibrium state if the angle is 120 degrees

Two vertices determine the curvature of a grainboundary


Structure of Literature Study

9

Micro-Structure

Models

Conclusion

Tests

Curvature

MethodsCounting

Cell

Vertex

Methods

Nippon Kawasaki Lazar

Hillert Level

Set

Phase

Field

Cellular

Automata

Embedded

Polygon with

virtual vertices

Embedded

PolygonSimple Polygon


Hillert

10

A very good approximation of the average grain size distribution

Assumption

Grain Size Distribution (GSD)

( )( )

−

−

−= +

uu

ueuP

2

2exp

22)(

2

βββ

β

TATA 3F63 Temp: 1200 C

0

50

100

150

200

0 100 200 300 400 500 600 700

time [sec]

D [micrometer]


Cellular Automata

11

CA model first developed by John von Neumann

� State

� Neighborhood definition

� Transformation rule

Each cell has assigned a



12

Micro-Structure

Models

Conclusion

Tests

Curvature

MethodsCounting

Cell

Vertex

Methods

Nippon Kawasaki Lazar Placeholder

text

Hillert Level

Set

Phase

Field

Cellular

Automata

Placeholder

text

Embedded

Polygon with

virtual vertices

Embedded



Counting cell method

13

Due to the sharp interface, interpolation is needed. Hence, a lot of

calculations have to be made

Computational costly


Vertex method

14

Using vertices is the most efficient method to calculate the effect

of curvature

2-D Voronoi with 50 nucleation points

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

5

10

15

20

25

30

35

40

45

50

� Less points are needed to calculate

� Vertex points are most influential in grain growth

� When more points are needed, virtual vertices can be added

Advantages

A hybrid model

� Run CA model

� Extract vertices

� Run vertex method

� Update states using the position of the vertices


Vertex by Nippon Steel

15

Method that is heavily dependent on the specific grain boundary energy

iiivirtual Mv κγ=,

∑=

=3

1

,

j ij

ij

ijitripler

rMv γ


Vertex method based on Minimization of Grain Boundary Energy (1/2)

16

Method first introduced by Kawasaki (1952), based on idea

from Phase Field Method

Dissipation Energy + Potential Energy = 0

0)()(

2

1 2

=+ ∫∫ dssdsM

sv

GBGB GB

γ

Minimize energy over position


Vertex method based on Minimization of Grain Boundary Energy (2/2)

17

Governing equations

∑

∑

∑

=

=

−

−=

−=

)(

)(

2

2

)(

3

1

2

1

i

j ij

ij

iji

i

j

iji

ijijij

ijijij

ijij

ij

i

j

jijiii

r

rf

DD

xyx

yxy

rMD

vDfvD

γ


Neumann-Mullins (1/3)

18

2D Neumann-Mullins (1956) revived by expansion to

3D in 2007 by MacPherson and Srolovitz

� Closed curve, enclosing area grows with the same rate

� Growth of a grain enclosed by others:

� Grain growth of a grain in 3D:

2D: Neumann-Mullins 3D MacPherson & Srolovitz

)6(3

nMdt

dA−−=

πγ



19

Governing equation of a virtual vertex (arbitrary point between only

two grains), satisfies Neumann-Mullins relation

πα

γα

21

21

21

=

×+

∆=

∑=

n

i

i

iee

eetMv



20

Governing equations of a triple point, who satisfy the Neumann-

Mullins relation

−−

−−

−

−

−∆=

−

3

301

102

2

1

1

32

21

πα

παγ

ee

eetMvi



21

Micro-Structure

Models

Conclusion

Tests

Curvature

MethodsCounting

Cell

Vertex

Methods

Nippon Kawasaki Lazar

Hillert Level

Set

Phase

Field

Cellular

Automata

Embedded

Polygon with

virtual vertices

Embedded



Tests

22

Three polygon have been constructed to test the different methods

on their performance


Results (1/3)

23

Embedded Polygon: Kawasaki vs exact Neumann-Mullins relation


Results (2/3)

24

Embedded Polygon: Lazar vs exact Neumann-Mullins relation


Results (3/3)

25

Problem in Kawasaki: the connection of a triple point with a virtual vertex


Overview & Conclusion

26

The method by Lazar shows the best results

++

+

--

?

NM, variable n, virtualson a straight line

MethodComputation

timeCircle

Neumann-Mullins n=6

NM variable n virtuals virtues on the circle

Counting Cell -- ? ? ?

Vertex methods:

Nippon ++ + -- --

Kawasaki ++ +/- ++ --

Lazar ++ ++ ++ ++


Further Research

27

Ready to implement in cellular automata model of Tata Steel

� Implementation of 2D and 3D in cellular automata model of Tata Steel

� Extract vertices from CA grid

� Solve motion equations for vertices

� Update CA grid from vertices motion

� Anisotropic variant of Lazars method


Bastiaan van der Boor

Grain growth by curvature

Bastiaan van der Boor Grain growth by...

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