„Implementation of a particle pinning model in a cellular automaton ...

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„Implementation of a particle
pinning model in a cellular
automaton based simulation of
primary recrystallization“
Luc HANTCHERLI
TFE (Diploma/ Master thesis)-May-August 2004
MPIE Düsseldorf
Structureof thepresentation
•Part I:
Introductionin cellularautomatontheorywithparticular
referenceto theoriginal model
•Part II:
Implementationof Zenerpinningeffectsin theoriginal
program
•Part III:
Simulationswithand withoutparticleeffects
Computer simulationof recrystallization
•Drivenbytwoneeds
:
–Engineering orientedneed
:
•Simulationsas quantitative predictions
•Descriptionof averagetextureorgrainsizeas output
–Research orientedneed
:
•Simulationsas waysto improveunderstandingof
recrystallization phenomena
Simulation of recrystallization
•Stronglysimulatedin theearly1980s
•Threecurrentmethodsformesoscopic
simulationof RX:
–vertexmethods
–Monte Carle Potts models
–CellularAutomata
•Otherpossiblemodels:
–phase-fieldmodels
–finite elementmethods
Cellularautomaton(CA) theory
•Basic ideaof CA
:
–to explaincomplexsystemsbyapplyinga setof
transformationrulesto cellsof lattice.
•3
importantkeyfeatures:
–Cell/ Neighboringcells
–States/ Attributes
–Transformation rule(s)
Exampleof 1D CA algorithm
•Fibonacci rabbitsor
Pascal Triangle
•Evolution
laws:
–1 monthto mature
–2 monthsto producea
newlybornpair
•Transformation rule:
p(x)=p(x-1)+p(x-2) (x>1)
Exampleof 2D/3D RX model(Raabe 1999)
•Cell: spacediscretization
•Attribute: orientation, dislocationdensity
•State: „recrystallized“or„non-
recrystallized“
•Switchingrule: probabilisticformulationof
theTurnbullequation
Transformation rule(1/4)
•Turnbullequation:
•New formulationwith
2 contributions:
–Deterministicpart:
–Probabilisticpart:
nnvp
kT
Q
mmp






−==exp
0
w
0
vv
&
=






−=
=
kT
Q
kT
pb
w
b
kTm
exp
3
3
0
nv
0
&
Transformation rule(2/4)
•Scalingprocedurein order to obtainscale-
independentgrainboundaryvelocities:
•Scalingprocedurein order to usea wholerange
of mobilitiesand drivingforces:
(
)
m
m
b
kTm
Vwith
Vww
λ
λ
3
0
=
=
=
nvv
0
&
()
()







=
=
=








=
kT
Q
V
pm
w
Vwith
w
V
V
Vw
m
m
m
exp
ˆ
ˆ
ˆ
ˆ
0
0
0
0
0
λ
λ
λ
nv
vnv
0
0
&
&
Transformation rule(3/4)
•Determination of V
0
byusingthefactthatprobabilities
cannotbelarger than1:
•So that:
1exp
ˆ
min
min
0
maxmax
0
max










=
kT
Q
V
pm
w
m
λ









=
kT
Q
pm
V
m
min
maxmax
0
min
0
exp
λ
Transformation rule(4/4)
•Switchingprobabilityof celli dueto a cellj is
givenby:
•Monte Carlo Decision:
maxmaxmin
0
0
exp
ˆ
pm
pm
kT
Q
V
pm
w
ijijij
m
ijij
ij
=









=
λ



>


ij
ij
wrifrejectedswitch
wrifacceptedswitch
r
ˆ
ˆ
]1,0[
Algorithmstructure: input-outputinformation
Reviewof forcesactingon grainboundaries
•Drivingforces:
–Dueto thedifferencein storedenergiesbetween
neighboringgrains
–Classicallyevaluatedby:
•Resistiveforces:
–Opposegrainboundarymigration
–Possiblesource: Particles
2
2
1
Gbp
ρ
=
Pinningbya stablearrayof incoherent
particles
•Magnitudeof thepinningforce
calculatedbyZenerisgiven
by:
•Comparisonbetweenp and
pZ:
r
f
p
Z
γ
2
3

=
MPap
M
Pap Z
1.0
10


Particledescriptioninsidethealgorithm
•2 possibilitiesfordiscribing
particles:
–MODEL 1 :
•grainboundarieshavethesizeof
a cell
•f, r areconsideredas cell
attributes
–MODEL 2 :
•Grainboundarieshaveno size
and arerepresentedbyfaces
(segments) betweencells
•f,rareconsideredas grain
boundaryparameters
Algorithmictreatmentof particleseffects
insidetheprogrammainloop
•IFneighborcelljisrecrystallized
THENcalculationof misorientationangle betweeni and j
IFmisorientationangle >15°
THENf = 0 (eventuallyf
HAGB)
calculationof initialmisorientationangle betweeni
and j
IFinitialmisorientationangle >25°
THENf = fVHAGB
calculationof thecorrespondingswitching
probability
Problems in gettingrealisticvalues(1/3)
•Grainboundarymobility:
–assumedto dependonlyon
temperatureand misorientation
angle
–Lowangle grainboundaries
havenotmobility
–High angle grainboundaries
havea constantmobility
–M
0
and Q aretakenfromFe-
3%Si data
–No „specialgrainboundaries“
wereconsidered
Problems in gettingrealisticvalues(2/3)
•Grainboundaryenergy:
–LAGB energyisdescribedby
theReadequationbutisof no
interest
–HAGB energyissupposed
constantaround0.79 J/m²
–Theroleof coincidencesite
latticein grainboundary
energywas nottakeninto
account
Problems in gettingrealisticvalues(3/3)
•Dislocationdensity:
–As cellattribute, dislocationdensitiesmustbeevaluatedfrom
EBSD measurements
–Twopossibleways:
•Useof theImage Quality
•Useof theTaylor factor
–Dislocationdensityforeachcellisfinallyevaluatedby:










−=
minmax
min
1
²
20
IQIQ
IQIQ
Gb
i
i
ρ
Cold-rolledstateof oneIF-steelsample
Simulationswithdifferent nucleationcriteria
70% of ρmax
90% of ρmax
80% of ρmax
Discussionon thenucleation
•MechanicalInstabilitycriterion:
–Alwaysfulfilledbecauseof thechosennucleationmodel
–Notionof „potential nucleus“
•KineticInstabilitycriterion:
–Spontaneousgeneratednucleiareoftencloseto LAGB
–Notionof „succesfullnucleus“
•ThermodynamicalInstabilitycriterion:
–Criticalnucleussizegivenby:
–Around0.1 µm forthesimulation, whichisexactlythecellsize
N
crit
G
r
Δ
=
γ
2
Simulation withparticlepinningeffect
10
20
30
40
50
60
70
80
90
100
050100150200250
Annealing time [s]
Recrystallized volume fraction [%]
f = 0%
f = 28%
f = 28,5%
f = 29%
f = 29,5%
f = 30%
Discussionon thepinningeffect
•Particlesslowdown therecrystallization
–Inhibition of nucleiexpansion
–„successfullnuclei“cannotgrownormally
•Recrystallizationfront canbestopped, but
–f/r ratiosin theZenermodelmustbeVERY large, whichisrelativelyrare
–A grainmustbecompletelyhermetic
•Conclusion:
–ParticlesALONE cannotexplainpartiallyrecrystallisedmicrostructures
–Otherphenomenamustbetakenintoaccount
Simulationswithdifferent nucleationcriteria
70% of ρmax
90% of ρmax
80% of ρmax
References
•D. Raabe: Philosophical Magazine A, vol 79 (1999), No 10, 2339–2358,
„Introduction of a scaleable 3D cellular automaton with a probabilistic switching
rule for the discrete mesoscale simulation of recrystallization phenomena“
•D. Raabe, R. Becker: Modelling and Simulation in Materials Science and
Engineering 8 (2000) 445-462, „Coupling of a crystal plasticity finite element
model with a probabilistic cellular automaton for simulating primary static
recrystallization in aluminum“
•D. Raabe: Acta Materialia 48 (2000) 1617–1628, „Scaling Monte Carlo Kinetics
of the Potts model using rate theory”
•D. Raabe: Annual Review of Materials Research 32 (2002) p. 53-76, „Cellular
automata in materials science with particular reference to recrystallization
simulation”
•D. Raabe, Acta Materialia, 52 (2004) 2653-2664., “Mesoscale simulation of
spherulite growth during polymer crystallization by use of a cellular automaton”
•D. Raabe, L. Hantcherli, Computational Materials Science vol. 34, (2005) pages
299–313, “2D cellular automaton simulation of the recrystallization texture of an
IF sheet steel under consideration of Zener pinning”