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Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 1 / 38
Modeling the Austenite FerriteTransformation by Cellular AutomatonImproving Interface StabilityDelft University of Technology
Mathias Mul
September 19, 2014
Outline
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 2 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 3 / 38
Steel microstructure
Microstructure determines mechanical properties of steel.
Ferrite/Pearlitemicrostructure Iron atom lattices
Ferrite nucleation andgrowth (by Kees Bos, Principalresearcher at TATA Steel)
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 4 / 38
fasetransformatie.aviMedia File (video/avi)
CoolingHigh temperature: austenite (γ)
Low temperature: ferrite (α)
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 5 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 6 / 38
Moving boundary problemThe problem of the moving interface S can be stated as
vn = M∆G (Xγs ) the normal velocity of S
∂X∂t = ∇(D(X , z)∇X ) in Ω
γ , t > 0∂X∂n = 0 on ∂Ω∂X∂n = −(X
γs − Xα)vn on S
X (t = 0) = X0 on Ω
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 7 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 8 / 38
Cellular AutomatonModel built of cells with properties
F state
F neighbourhood
F transformation rule
example:
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 9 / 38
jellyfish.aviMedia File (video/avi)
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 10 / 38
Model outline
1 Compute carbon concentration at interface cells
2 Compute growth velocity of interface cells
3 Compute growth length of interface cells
4 Transform cells according to a transformation rule
5 Redistribute excess carbon from newly transformed cells
6 Solve a time step of carbon diffusion in austenite
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 11 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 12 / 38
Growth dynamicsFor every interface cell i we define:
Growth length `i ≥ 0Growth velocity vi ≥ 0Inward growth λi ≥ 0The velocity v is calculated according to the classical equation
v = M ∆G (X interface,T )︸ ︷︷ ︸driving force
, where ∆G : R3 → R,
and M the interface mobility.
λi =∑j∈Mi
wji`j
wji =1√k
where cells i and j are k-level neighbours
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 13 / 38
How to determine X interface ?
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 14 / 38
Growth dynamics(2)
Transformation rule:` Transformation rule:λ
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 15 / 38
Carbon Redistribution Mechanics
Xj = Xj +Xi − Xα∑
n wni· wji
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 16 / 38
Diffusion Time StepFind X (t + ∆t) on Ωγ(t) such that{
∂X∂t = ∇ · (D(z)∇X ) in Ω
γ(t), t < t̃ ≤ t + ∆t∂X∂n = 0 on ∂Ω
γ(t)
given X (t) on Ωγ and D(z) on Ω.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 17 / 38
Interface Carbon Smoothing
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 18 / 38
Increased Diffusion at Interface
D = D0 · e−Q(z)RT
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 19 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 20 / 38
1-dim CA in comparison to Murray-Landis
CA: Interface S always lies on pre-set pointsML: Interface S may freely move
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 21 / 38
AnimationCA.aviMedia File (video/avi)
AnimationML.aviMedia File (video/avi)
Unstable interfaces → Dendrites
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 22 / 38
AnimationNOTHING.aviMedia File (video/avi)
Unknown parameters
• Mobility M0 · e−Qα,γ
RT
• Nucleation process• Increased interface growth at boundaries• Smoothe range/Increased diffusion factor• Initial austenitic structure
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 23 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 24 / 38
Comparison: CA to Murray-Landis
∆z → 0, ∆t = 0.9 ∆zvmax
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 25 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 26 / 38
Inward growth results
M0 = 0.1
M0 = 0.6
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 27 / 38
Carbon smoothing results
M0 = 0.6
M0 = 1.5
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 28 / 38
Combined results:Inward growth & Carbon smoothing
M0 = 1.5
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 29 / 38
Fast Interface Diffusion
Test Example: Unwanted behaviour for M0 = 0.5
A wobbly shape fromthe outside.
A look from the insidereveals the dendriticstructure.
Slices of the grain.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 30 / 38
Fast Interface Diffusion
D = D0 · e−ρQγ
RT
5× higher diffusion coefficient
Outer grain view, ρ = 0.9. Inner grain view, ρ = 0.9.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 31 / 38
Fast Interface Diffusion
D = D0 · e−ρQγ
RT
30× higher diffusion coefficient
Outer grain view, ρ = 0.8. Inner grain view, ρ = 0.8.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 32 / 38
Next Subsection
1 IntroductionMicrostructureThe moving boundary problem
2 Model and MethodsCellular automatonModel outlineImplementationProblems
3 ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
4 Conclusions
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 33 / 38
Fraction Curve Fitting
The modeled fraction curve and the experimental fraction curve.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 34 / 38
Fraction Curve Fitting
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 35 / 38
Conclusions
F Inward growth seems to reduce dendritic growth and results inless extreme grain shapes
F Carbon smoothing reduces dendritic growth, smoothing area canbe scaled up at higher computational costs
F An increased interface diffusion coefficient reduces dendriticgrowth in an easy-to-implement way, at higher computationalcosts
F Cellular Automaton is a useful framework for phasetransformation models with local concentration differences.
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 36 / 38
Future Research
? Experimentally determine parameters for mobility and interfacediffusion.
? Adaptive grid refinements for a thinner interface
? Finite Elements for a better conditioned problem
? Parallel implementation for parts of the linear solver
? Develop cellular automaton hardware on a chip for fastcomputation and communication between cells
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 37 / 38
Steel structure by Olafur EliassonSource: www.mymodernmet.com
Mathias Mul (TU Delft) Austenite to Ferrite Transformation September 19, 2014 38 / 38
IntroductionMicrostructureThe moving boundary problem
Model and MethodsCellular automatonModel outlineImplementationProblems
ResultsConvergence of CA to Murray-Landis methodImproving interface stabilityFraction curves
Conclusions