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Centre Sciences des matériaux et des structuresDépartement Rhéologie, Microstructures, Thermomécanique
FR CNRS 3410 – CIMReV
UMR CNRS 5307Laboratoire Georges FRIEDEL
11&12 Sept. 2014 (v1) David PIOT 1
Workshop onMean Field Modelling for
Discontinuous Dynamic Recrystallization
Fréjus Summer SchoolRecrystallization Mechanisms in
Materials
David PIOT 2
Workshop on Mean-Field Modelling Introduction
Motivation+ Illustration of mean-field modelling dedicated
to discontinuous dynamic recrystallization (DDRX)
+ Theoretical derivations related to ergodicity
Outline+ How to average dislocation densities? How to
keep constant the volume?+ How to test an assumption about the
dependency of parameters?+ Impact of the constitutive equation choice
David PIOT 3
Abstract 1/3Structure of a mean-field model for DDRX
Mean-field = mesoscopic description+ Description at the grain scale+ Inhomogeneities at microscopic scale are averaged+ Dislocation density homogeneous within each grain+ Localization / Homogenization
Assumptions to simplify (but not mandatory)+ No topological features+ Distribution of spherical grains of various diameters+ Localization: Taylor assumption
+ Homogenization: b
David PIOT 4
Abstract 2/3Structure of a mean-field model for DDRX
Variables for describing microstrcurure+ As no stochastic is considered, all grains of a given
age have the same diameter and dislocation density because they have undergone identical evolution → one-parameter (nucleation time ) distributions (for non initial grains)
+
Grain number nucleated at time : ,
Plastic strain (stain rate for each grain)
, d
Dislocation density: ,
Grain diameter: ,
t
N t
t u u t
t
D t
&
& &
David PIOT 5
Abstract 3/3Structure of a mean-field model for DDRX
Evolution of grain-property distributions+ 1. Equation for strain hardening and dynamic
recovery giving the evolution of dislocation densities
+ 2. Equation for the grain-boundary migration governing grain growth or shrinkage
+ 3. A nucleation model predicting the rate of new grains
+ 4. Disappearance of the oldest grains included in (2) when their diameter vanishes
David PIOT 6
1. Strain hardening anddynamic recovery
Constitutive model for+ Strain hardening+ Dynamic recovery+ In the absence of recrystallization
General equation+ Each grain behaviour is described by the same equation
+ Several laws can be used, e.g.:
+ The parameters are temperature and strain-rate dependent
dd
Hi
1
Yoshie Laasraoui Jonas
Kocks Mecking
Power law
h r
h r
H
H
H
H
i
i
i
David PIOT 7
2. Grain-boundary migration
Mean-field model+ Each grain is inter-
acting with an equiv-alent homogeneousmatrix
Migration equation
+
+ M grain-boundary mobility, T line energy of dislocations
: growth
matrix
D
matrix
: shrinkage
, 2 ,
Dt M t t
tT
David PIOT 8
3. Nucleation equation
Various nucleation models available “Simplest” equation tentative
+ Nucleation of new grains ( = t) is assumed to be proportional to the grain-boundary surface
+
+ Here, p = 3 is assumed It is the unique integer value for p compatible with
experimental Derby exponent d in the relationship between grain size and stress at steady state using the closed-form equation between p and d in the power law case
,N
t t f t S tt
NIn practice, is specified as pff k
David PIOT 9
Exercise 1 1/3Mean dislocation-density
Discrete description of grains (Di)
3+Volume is kept const t ani
i
V D
David PIOT 10
Exercise 1 1/3Mean dislocation-density
Discrete description of grains (Di)
3+Volume is kept const anti
i
V D
2+By derivatio n, 0i i
i
D D &
David PIOT 11
Exercise 1 1/3Mean dislocation-density
Discrete description of grains (Di)
3+Volume is kept const anti
i
V D
2+By derivatio n, 0i i
i
D D &
2+With migration eq., 0 i i
i
D
David PIOT 12
Exercise 1 1/3Mean dislocation-density
Discrete description of grains (Di)
+ I.e. average weighted by the grain-boundary area
3+Volume is kept const anti
i
V D
2+By derivatio n, 0i i
i
D D &
2+With migration eq., 0 i i
i
D 2
2+Leading to the definition, i i
i
ii
D
D
Annex: On the rush…
What about grain growth?+ Hillert (Acta Metall. 1965)
2
1
1 10
N
i ii
D D DD D
& &
Annex: On the rush…
What about grain growth?+ Hillert (Acta Metall. 1965)
2
12
2
1 1
1 10
1
N
i ii
N Ni
i ii i i
D D DD D
DD D D
D D
& &
Annex: On the rush…
What about grain growth?+ Hillert (Acta Metall. 1965)
Mixed formulation+ With stored energy: average dislocation-
density+ With surface energy: average grain-size
2
12
2
1 1
1 10
1
N
i ii
N Ni
i ii i i
D D DD D
DD D D
D D
& &
David PIOT 16
Exercise 1 2/3Mean dislocation-density
Continuous description for a volume unit+ After vanishing of the initial grains 3
0+
6 , , , d 1
tt N t D t
David PIOT 17
Exercise 1 2/3Mean dislocation-density
Continuous description for a volume unit+ After vanishing of the initial grains 3
0+
6 , , , d 1
tt N t D t
3
3 2
0
+ , , 0, 0 with 0
, , 3 , , , d 0t
N t t D t t t nuclei D
DNt D t N t t D t
t t
David PIOT 18
Exercise 1 2/3Mean dislocation-density
Continuous description for a volume unit+ After vanishing of the initial grains
Nucleation is ocurring (t = ) and D = 0 Disappearance of old grains (t = + tend) and also D =
0
3
0+
6 , , , d 1
tt N t D t
3
3 2
0
+ , , 0, 0 with 0
, , 3 , , , d 0t
N t t D t t t nuclei D
DNt D t N t t D t
t t
0 if onlyNt
David PIOT 20
Exercise 1 3/3Mean dislocation-density
Volume constancy
2
0 , , , d 0
t DN t t D t
t
2 (migra t n io )D MT&
David PIOT 21
Exercise 1 3/3Mean dislocation-density
Volume constancy
2
0 , , , d 0
t DN t t D t
t
2 (migra t n io )D MT&
20
20
, , d
, , , d
t
t
N t t D t
N t t D t
David PIOT 22
Exercise 1 3/3Mean dislocation-density
Volume constancy
2
0 , , , d 0
t DN t t D t
t
2 (migra t n io )D MT&
20
20
, , d
, , , d
t
t
N t t D t
N t t D t
2
0
2
0
, , , d
, , d
t
t
t N t D tt
N t D t
David PIOT 23
Exercise 2 1/2Ergodicity and averages
Steady state = dynamic equilibrium+ Ergodicity postulate when S. S. is
established + Averages over the system = averages over
time for a typical element of the system
+ All characteristic and their distribution does not depend on time and the only variable to label grains is their strain/age (current – nucleation time)
end
01 end
1 1 X d
N t
ii
X X t tN t
?
David PIOT 24
Exercise 2 1/2Ergodicity and averages
Steady state = dynamic equilibrium+ Ergodicity postulate when S. S. is
established + Averages over the system (constant) =
averages over time for a typical element of the system
end
01 end
1 1 X d
N t
ii
X X t tN t
end
end
22
1 0
220
1
d
d
Nt
i ii
N t
ii
D t D t t
D t tD
2014 David PIOT 25
Exercise 2 2/2Ergodicity and averages
n: average dislocation-density weighted by Dn
+ + Steady-state case
2
end
end
0
0
d
d
t n
n t n
D t
D t
David PIOT 26
Exercise 2 2/2Ergodicity and averages
n: average dislocation-density weighted by Dn
+ + Steady-state case
2
end
end
0
0
d
d
t n
n t n
D t
D t
end end
0 0 d d
2
t tn nDD t D t
MT
&
David PIOT 27
Exercise 2 2/2Ergodicity and averages
n: average dislocation-density weighted by Dn
+ + Steady-state case
2
end
end
0
0
d
d
t n
n t n
D t
D t
end end
end end
0 0
0 0
d d2
d d 2
t tn n
t D tn n
D
DD t D t
M
D t D D M
T
T
&
David PIOT 28
Exercise 2 2/2Ergodicity and averages
n: average dislocation-density weighted by Dn
+ + Steady-state case
2
end
end
0
0
d
d
t n
n t n
D t
D t
end end
end end
0 0
0 0
d d2
d d 2
t tn n
t D tn n
D
DD t D t
M
D t D D M
T
T
&
, nn
David PIOT 29
Exercise 2 2/2Ergodicity and averages
n: average dislocation-density weighted by Dn
+ + Steady-state case
2
end
end
0
0
d
d
t n
n t n
D t
D t
end end
end end
0 0
0 0
d d2
d d 2
t tn n
t D tn n
D
DD t D t
M
D t D D M
T
T
&
end
0 end 0 , 1 d (steady state only)
t
nn t t
David PIOT 30
Exercise 3 1/3Strain-hardening law influence
Comparison YLJ / PW (/KM)+ PW tractable with closed forms+ Physically still questionable+ Easy to switch data from one to another law
MONTHEILLET et al. (Metall. and Mater. Trans. A, 2014)
David PIOT 32
Exercise 3 3/3Strain-hardening law influence
Alternative codes, both for nickel+ DDRX_YLJ+ DDRX_PW+ Parameters in drx.par
Pure nickel strained at 900 °C and 0.1 s–1
For YLJ: example For PW: example Grain-boundary mobility and nucleation
parameter obtained (direct closed form for PW) from steady-state flow-stress and steady-state average grain-size
Comparison ReX Frac. / Soft. Frac.
It depends on… Nb content and what else?
0.0 0.2 0.4 0.6 0.8 1.0Stra in
0.0
0.2
0.4
0.6
0.8
1.0Pure N i0 .1 s -1
900 °C
F ractionR ecrysta llized
Fractiona l Soften ing
0 1 2 3Stra in
0.0
0.2
0.4
0.6
0.8
1.0N i-0.1 N b0.1 s -1
900 °C
F ractionR ecrysta llized
Fractiona l Soften ing
Exercise 4 1/1Impact of the initial microstructure
Comparison quasi Dirac / lognormal+ Initial average grain-size : 500 µm+ Flag 0
Initial grain-size distribution: Gaussian “Standard deviation”: Variation coefficient (SD/mean) Quasi Dirac : variation coefficient 0.05 (already done)
+ Flag 1 Initial grain-size distribution: lognormal “Standard deviation”: ln-of-D SD (usual definition,
dimensionless) Parametric study (e.g. 0.1, 0.25, 0.5, 1)
Exercise 5 1/1Test of models for parameters
Mean field models+ Relevant tools to test assumptions for
modelling the dependence of parameters with straining conditions
Exemple : strain-rate sensitivity+ Rough trial
GB mobility, nucleation, recovery, only depend on temperature
Strain hardening: power law
+ Screening by comparing 0.1 with 0.01 and 1 s–1
0 0
mh h &&