upscaling defects in steelmotivation from tata steel: recipe for strong and light steel difficult...

07/12/2011

Upscaling defects in steel

Lucia Scardia

Project and Motivation

Interdisciplinary project:

2

Mathematics

Lucia

Mark Peletier

Adrian Muntean

Patrick van Meurs

Mech. Eng.

Marc Geers

Ron Peerlings

Cem Tasan

Michael Dogge

Industry

TATA Steel

Project and Motivation

Interdisciplinary project: Motivation from TATA Steel: Recipe for strong and light steel

DIFFICULT PROBLEM!

3

Mathematics

Lucia

Mark Peletier

Adrian Muntean

Patrick van Meurs

Mech. Eng.

Marc Geers

Ron Peerlings

Cem Tasan

Michael Dogge

Industry

TATA Steel

Influence of the microstructure

Ingredients: • Exploit the microstructure • Careful modelling

4

Influence of the microstructure

Ingredients: • Exploit the microstructure • Careful modelling Microstructure

5

Influence of the microstructure

Different microstructure (grain size) corresponds to different macroscopic properties Example (a) coarser (b) finer

6

Influence of the microstructure

Different microstructure (grain size) corresponds to different macroscopic properties Example (a) coarser (b) finer (b) is stronger than (a). Why?

7

Influence of the microstructure

There are defects – dislocations - in the arrangement of atoms

8

Influence of the microstructure

There are defects – dislocations - in the arrangement of atoms

Dislocations motion is responsible for deformation

9

Influence of the microstructure

There are defects – dislocations - in the arrangement of atoms

Dislocations motion is responsible for deformation Boundaries are a barrier smaller grains = more boundaries = harder material 10

11

Multi-scale Problem

Application scale

MACRO model

TATA

Grain/phase size

MESO model

Bridging MICRO/MESO scales

Dislocation scale

MICRO model

12

Micro-meso upscaling

Bridging

MICRO/MESO

scales

Meso-model from fundamental dislocation model

13

Micro-meso upscaling

Bridging

MICRO/MESO

scales

Meso-model from fundamental dislocation model

Zoo of mesoscopic models available

• Groma, Evers-Geers, Eshelby-Frank-Nabarro

14

Micro-meso upscaling

Bridging

MICRO/MESO

scales

Meso-model from fundamental dislocation model

Zoo of mesoscopic models available

• Groma, Evers-Geers, Eshelby-Frank-Nabarro

All phenomenological. How to choose one?

Micro-model: Pile-up of Dislocations

15

• Impenetrable grain

boundary

• Infinite vertical walls

of dislocations

• Spacing

• Externally applied

stress

Other groups: Ockendon, Mesarovic, Groma..

Micro-model: Discrete Dislocation Energy

interplanar distance

length Burgers vector

interaction energy

Poisson’s ratio

shear stress

shear modulus

16

0 11

( ) :2(1 ) j

n n ni j

n i

iij i

x xGbE x x

hV

h

b

G

V

0ix

Micro-meso derivation: Discrete-to-Continuum

Approach

Derive a continuum energy from in terms of a

continuum dislocation density for

via -convergence

17

0 11

( ) :2(1 ) j

n n ni j

n i

iij i

x xGbE x x

hV

nnE

Micro-meso derivation: Discrete-to-Continuum

18

, ,b h Density profile depends on !

Micro-meso derivation: Discrete-to-Continuum

• Introduce a non-dimensional scaling parameter

19

n

b

nh

0 11

1( ) : ( )

j

n n nn

n n i j i

iij i

E x x xn

V nn

x

0 11

( ) :2(1 ) j

n n ni j

n i

iij i

x xGbE x x

hV

Micro-meso derivation: Discrete-to-Continuum

(aspect ratio)

20

average dislocation distance~

interplanar distancen

0 11

1( ) : ( )

j

n n nn

n n i j i

iij i

E x x xn

V nn

x

Aspect ratio

SMALL

Aspect ratio

LARGE

Discrete model: Non-dimensional Energy

0 11

1( ) ( )

j

n n nn

n n i j i

iij i

E x x xn

V nn

x

• Subcritical regime

• First critical regime

• Intermediate regime

• Second critical regime

• Supercritical regime

1n

n

1~n

n

11n

n

~ 1n

1n

Aspect ratio

SMALL

Aspect ratio

LARGE

Aspect ratio ~ 1

Discrete-to-Continuum: Limit energies

THEOREM: The discrete energy Gamma-converges to

EFN - GROMA

NEW

EVERS-GEERS

NEW

NEW

0 01: ( ) (log )n xn E

0 0~1: ( ) ( )n V xn E

2

0 0 0)1 : (nn En V x

eff0 0

1~ 1: ( )n E V x

0 0)1: )( (n I xE

Micro-meso derivation: Discrete-to-Continuum

• NEW

•

23

eff0 0

1~ 1: ( )n F V dx x dx

eff

1

( ) ( )k

t kVV t

NN interactions

NNN interactions

1k

2k

Conclusions

• Rigorous mathematical derivation

• Physically-based meso-model

• Clear interpretation of the limit models

• Re-obtained well-known models without ad-hoc assumptions

• Obtained new models

24

Thank you!

25