upscaling defects in steelmotivation from tata steel: recipe for strong and light steel difficult...
TRANSCRIPT
07/12/2011
Upscaling defects in steel
Lucia Scardia
Project and Motivation
Interdisciplinary project:
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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!
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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
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Influence of the microstructure
Ingredients: • Exploit the microstructure • Careful modelling Microstructure
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Influence of the microstructure
Different microstructure (grain size) corresponds to different macroscopic properties Example (a) coarser (b) finer
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Influence of the microstructure
Different microstructure (grain size) corresponds to different macroscopic properties Example (a) coarser (b) finer (b) is stronger than (a). Why?
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Influence of the microstructure
There are defects – dislocations - in the arrangement of atoms
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Influence of the microstructure
There are defects – dislocations - in the arrangement of atoms
Dislocations motion is responsible for deformation
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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
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Multi-scale Problem
Application scale
MACRO model
TATA
Grain/phase size
MESO model
Bridging MICRO/MESO scales
Dislocation scale
MICRO model
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Micro-meso upscaling
Bridging
MICRO/MESO
scales
Meso-model from fundamental dislocation model
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Micro-meso upscaling
Bridging
MICRO/MESO
scales
Meso-model from fundamental dislocation model
Zoo of mesoscopic models available
• Groma, Evers-Geers, Eshelby-Frank-Nabarro
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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
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• 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
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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
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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
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, ,b h Density profile depends on !
Micro-meso derivation: Discrete-to-Continuum
• Introduce a non-dimensional scaling parameter
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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)
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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
•
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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
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Thank you!
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