Centre Sciences des matériaux et des structures Département Rhéologie, Microstructures,...

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


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


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



3+Volume is kept const anti

i

V D

2+By derivatio n, 0i i

i

D D &

David PIOT 11




i

V D


i

D D &

2+With migration eq., 0 i i

i

D

David PIOT 12



+ I.e. average weighted by the grain-boundary area


i

V D


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

& &



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

& &



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


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


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


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 19


Volume constancy

2

0 , , , d 0

t DN t t D t

t

David PIOT 20


Volume constancy

2

0 , , , d 0

t DN t t D t

t

2 (migra t n io )D MT&

David PIOT 21


Volume constancy

2

0 , , , d 0

t DN t t D t

t


20

20

, , d

, , , d

t

t

N t t D t

N t t D t

David PIOT 22


Volume constancy

2

0 , , , d 0

t DN t t D t

t


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


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


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




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




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




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




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 31


David PIOT 32


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 &&

Centre Sciences des matériaux et des structures Département Rhéologie, Microstructures,...

Documents

Transcript of Centre Sciences des matériaux et des structures Département Rhéologie, Microstructures,...