Thermal-Metallurgical-MechanicalInteractions

Michel Bellet

Non Linear Computational Mechanics

Athens Spring Week, 2009

22NLCM March 2009

The Context: Transformation of Metallic Alloys

Solidification

Heat Treatments

WeldingHeat Transfer

Mechanics

Microstructure

Non-Linearities…

33NLCM March 2009

Thermal-Metallurgical-Mechanical Interactions

Heat TransferTemperature

MechanicsDeformation, Stress

Liquid flow

MicrostructurePhase fractions

Phase changes: liquid-solid; solid-solid

Thermophysical properties depend on µstructureLatent heat of transformations

Mechanical propertiesdepend

on µstructure

Deformations

associated

with

phase change

Change ofconfiguration (contacts, gaps…

)

Mechanical pow

er

Convection effects(liquid

flow)

T –dependance

ofmaterial behaviour,

Thermal expansion

Phase transform. depend

on stress state

44NLCM March 2009

Outline

• Energy conservation

Some reminders about heat equation

Extension to the multiphase material

Numerical treatment and non-linearities

• Interaction heat transfer metallurgy

Solid state phase transformations

• Interaction metallurgy mechanics

Transformation plasticity

• Application to the modelling of heat treatment processes

• Non-Linearities arising from liquid-solid phase change

• Energy conservation with liquid-solid transformation

• Numerical treatment

• Interaction mechanics heat transfer

• Application to the modelling of solidification processes

55NLCM March 2009

ρ specific mass

e internal energy

v velocity field

f mass density of volume forces

T stress vector (surface force) along the surface of ω

r volumetric density of input heat

q surface density of input heat

Some Reminders about Energy Conservation

∫∫∫∫∫ ∂∂++⋅+⋅=+

ωωωωωρρ qdSrdVdSdVdVe

tvTvfv )

2

1(

d

d 2

1st principle of thermodynamics :

for any domain ω of a studied system,

Variation of energy Power of external forces Heat input power

∫∫ ⋅+ωω

ρ dVt

dV vv

vεσd

d)(: &

∫ω ρ dVtd

d

2

1 2v

∫ω ρ dV

t

2

2

1

d

dv

∫∫∫∫ ∂++=

ωωωωρ qdSrdVdVdV

t

e)(:

d

dvεσ &

F. Fer, Thermodynamique macroscopique, Tome 1 : systèmes fermés, Gordon & Breach (1970)H. Ziegler, An introduction to thermomechanics, North-Holland (1977)P. Germain, Mécanique, Tome 1, Ellipses (1986)

Theorem of kinetic energy(or virtual work principle)

66NLCM March 2009

Energy Conservation

Fourier law :

∫∫∫∫ ∂++=

ωωωωρ qdSrdVdVdV

t

e)(:

d

dvεσ &

nnq ⋅∇−−=⋅−= )( Tkq

∫∫∫∫ ∇⋅∇++=ωωωω

ρ dVTkrdVdVdVt

e)()(:

d

dvεσ &

)(:d

dTkr

t

e ∇⋅∇++= εσ &ρ

For any ω :

)(: Tkret

e ∇⋅∇++=∇⋅+∂∂

εσv &ρρ

)(:)()(

Tkret

e ∇⋅∇++=⋅∇+∂

∂εσv &ρρ

k thermal conductivity

n outward unit normal vector

T temperature

77NLCM March 2009

Energy Conservation

)(:d

dTkrp

t

ph

t

h ∇⋅∇+++

⋅∇+=∇⋅+∂∂

εσvv &ρρ

Considering pressure ~ constant (ok for condensed matter),

and a single phase medium,

∫=T

Tp dch

0

)( ττ specific heat

rTkTct

Tc

t

Tc ppp +=∇⋅∇−∇⋅+

∂∂= εsv &:)(

d

d ρρρ

pcT

h =∂∂

ρp

eh +=Enthalpy per unit of mass

rTkht

h +=∇⋅∇−∇⋅+∂∂

εsv &:)(ρρ

(using mass conservation)

Isσ p−=

rTkt

h +=∇⋅∇− εs &:)(d

dρ

rTkht

h +=∇⋅∇−⋅∇+∂

∂εsv &:)()(

)( ρρ

88NLCM March 2009

Energy Conservation for a Multiphase Material

rTkht

h +=∇⋅∇−⋅∇+∂

∂εsv &:)()(

)( ρρis satisfied in any phase k of a representative elementary volume (REV) of the multiphase material

ββββ αααα

REV

Looking for an averaged conservation equation on the REVThe spatial averaging method

For any scalar function Ψ defined on the REV,

Average in phase k:

Mixture average:

volume fraction

of phase k

∑∑∑ ===k

kk

k

kk

k

k gg ψψψψ

kkk

kV

VV kk

ggdV

dVdV

k

k

k

k

VV

V

VV

ψψψ

ψχψψ

===

==

∫

∫∫

)(

)()()(

1

11

0

00 0

x

xxx

=k

kk

0

1

phase outside

phase insideχ

"intrinsic" average

in phase k

M. Rappaz, M. Bellet, M. Deville, Numerical modelling in materials science and engineering, Springer (2003)

99NLCM March 2009

Energy Conservation for a Multiphase Material

ββββ αααα

REV

( ) rTkht

h+=∇⋅∇−⋅∇+

∂∂

εsv &:ρρ

Average volumetric enthalpy

Average energy flux vector

Average mechanical power

Average volumetric heat input

Average thermal conductivity

∑==k

kkkkk hghgh ρρρ )(

kk hgh )( vv ρρ =

kkg ):(: εsεs && =

kkrgr =

kkkgk =

( ) rTkHt

H+=∇⋅∇−⋅∇+

∂∂

εsv &:

∑===k

kkkk HgHghH ρ

1010NLCM March 2009

Solid State Phase Changes

( )

( ) 0)()(

0

=∇⋅∇−∂

∂

=∇⋅∇−∂

∂

Tkgt

Hg

Tkt

H

kkkk

Spatial averaging method

t

Tc

t

Hkp

k

∂∂=

∂∂

)(ρ

∫=T

Tkpk dcH

0

)()( ττρ

Simplifying assumptions• Advection neglected• Mechanical power neglected• Volumetric heat source r = 0

For each phase k,

( ) 0)( =∇⋅∇−∂∂+

∂∂

Tkt

TcgH

t

gkpkk

k ρ

( ) kk

p Ht

gTk

t

Tc

∂∂−=∇⋅∇−

∂∂ρ

∑∑≠

→≠

→ −=∂

∂kj

jk

ki

kik ggt

g&& when phase i is partially transformed into phase k0>→kig&

0=→kig& otherwise

( ) ∑ −=∇⋅∇−∂∂

→),(

)(ji

jijip HHgTkt

Tc &ρ

1111NLCM March 2009

Energy Equation in case of Solid State Phase ChangesNumerical Treatment (Finite Element Method)

∫ ∫ΩΩΩ

=∇⋅∇−∂∂∀ ∫ dVrdVTkdVt

Tcp ϕϕϕρϕ ')(Weak form

∫ ∫ ∫ΩΩΩ∂Ω

=∇⋅∇+⋅∇−∂∂

∫ dVrdVTkdSTkdVt

Tcp ϕϕϕϕρ 'n

nqn ⋅−=⋅∇Tk heat flux through> 0 if inwardgoverned by boundary conditions

Ω∂

∫ ∫ ∫ ∫ ∫ΩΩ∂Ω∂Ω∂ΩΩ

=−−+−+∇⋅∇+∂∂

∫ dVrdSqdSTTdSTThdVTkdVt

Tc

frc

impextBextTp ϕϕϕεσϕϕϕρ ')()(44

convection radiation imposed heat flux

( ) ∑ −=∇⋅∇−∂∂

→),(

)(ji

jijip HHgTkt

Tc &ρ

Simplifying assumptions• Advection neglected• Mechanical power neglected• Volumetric heat source r = 0

ϕ∀

1212NLCM March 2009

Finite Element Discretization (Galerkin formulation)

FKTTC =+&

0)()()(1

)( =−+−∆

∆+∆+∆+∆+∆+ ttttttttttt

tTFTTKTTTC

• Time integration scheme: implicit Euler type

0)( =∆+ ttTR

NON LINEARITIES arise from radiation and possibly convection,

and from the temperature dependent thermophysical properties

Solution using the Newton-Raphson method

∫Ω

= dVNNcC jipij ρ

∫∫∫Ω∂Ω∂Ω

++++∇⋅∇= dSNNTTTTdSNNhdVNNkK jiextextBjiTjiij ))((22εσ

∫∫∫∫ΩΩ∂Ω∂Ω∂

+++++= dVNrdSNqdSNTTTTTdSNThF iiimpiextextextBiextTi '))((22εσ

=M

&

M

&iTT

=M

M

iTT

NON LINEAR VECTOR EQUATION:

1313NLCM March 2009

Newton Raphson's Solution Algorithm

0)( =TR

TT

RTRTTR δδ

∂∂+=+ )()(

Algorithm :

Loop whileconvεν >)( )(

TR

End loop

Init )0(0 T=ν

)( )1(

)1(

−−

−=∆

∂∂ ν

ν

TRTT

R

TTT ∆+= − )1()( νν

1+=νν

Solution of a linear set of equations:- directs solvers (Gauss elimination technique)- iterative solvers (preconditioned conjugate gradient)

Objective :

( )

( )k

ij

k

ij

ik

t

jj

k

ij

ik

k

i

ijij

t

jjiji

T

FT

T

KKTT

T

C

tCtT

R

FTKTTCt

R

∂∂−

∂∂

++−∂∂

∆+

∆=

∂∂

−+−∆

=

11

1

1414NLCM March 2009

Interaction Heat Transfer Metallurgy

• Phase transformations, precipitation phenomena…

• Example of austenite decomposition for steels: heat treatment, welding…

• Two types of phase transformation

Displacive transformations: austenite ferrite, pearlite, bainite

• In isothermal conditions, such transformations are characterized by TTT diagrams(Time, Temperature, Transformation).

• A specific approach is developed to extend the modelling to non-isothermalconditions: see next slides.

Massive transformation: the martensitic transformation

• The transformed phase fraction directly depends on temperature (local atomicrearrangement in ferrite oversaturated with carbon):

( )TM

msegg−−−= β

γ 1MS martensite start temperature

Koïstinen, Marbürger, Acta Metall. 1959

1515NLCM March 2009

Transformations of Low-Alloyed SteelsFe-C Equilibrium Phase Diagram

A1

A3

γ

α α + γ

α-ferrite+

pearlite(α-ferrite + Fe3C)

C [wt%]

T [°C]

Liq

γ

Liq+γ

γ + Fe3C

α + Fe3C

γ + Fe3C723 °C

1616NLCM March 2009

400

500

600

700

800

900

1 10 100 1000 10000

début

10%

90%

fin

( )Tτ

Displacive Transformations of SteelsIsothermal Conditions: TTT Diagrams

kg

( ) )(

max, )(exp1Tn

kkkktTbgg −−=

1) Nucleation

2) Growth, Avrami's law:

Time [s]

Temperature

[°C]

Time [s]

Tstart

end

Conjugate effects ofthermodynamical unbalance, anddiffusion of chemical species

Time – Temperature – Transformation

1717NLCM March 2009

• Model based on the additivity principle: Decomposition of a non-isothermal history in a succession of isothermal

incremental steps, and summation of the incremental contributions

• Nucleation Sum of Scheil. Transformation starts when:

• Growth

• NB: Alternative models exist in the literature:

Displacive Transformations of Steels:Non-Isothermal Conditions

1)(

=∆∑i i

i

T

t

τ

...),,,,( Cik wGgTTfg γ&& =

Fernandes, Denis, Simon, Mat. Sci. Tech. 1986

Leblond et al. 1985 ; Waeckel et al. 1995

1818NLCM March 2009

Interaction Metallurgy Heat Transfer

• Heat source term associated with solid state phase change

• Averaged thermophysical properties depend on phase fractions

kkkgk =

kpkp cgc )(ρρ =

∑ −= →),(

)('ji

jiji HHgr &

1919NLCM March 2009

Interaction Metallurgy Mechanics

γgσ ρρ =+⋅∇Conservation of momentum

Spatial averaging

Constitutive equation of the multiphase solid material ?

2020NLCM March 2009

Interaction Metallurgy MechanicsSolid Multiphase Material

• Decomposition of the strain-rate tensor of the multiphase material

• Direct expressions of the deformations arising from phase transformations

Volume change

Transformation plasticity

tptrthvpelεεεεεε &&&&&& ++++=

sε

= ∑

→→→

ji

jijji

tpggK && )('

2

3 φ

Iε

−−= ∑

→→

ji

ji

i

ijtrg&&

ρρρ

3

1

Leblond et al., Int. J. Plasticity, 1989Desalos, Giusti 1982Fischer, Acta Metall Mater 1990

Temperature [°C]Temperature [°C]

Def

orm

atio

n[%

]

Def

orm

atio

n[%

]

With applied stressFree dilatometry

From Coret, 2001

γ

αγ

γα + Fe3C

2121NLCM March 2009

Interaction Metallurgy MechanicsSolid Multiphase Material

• Homogenization procedure: Taylor's assumption

• Constitutive models of the phases:

Lemaître & Chaboche evp model, for instance

Models may be different for each phase

( )tptr

k εεεEε &&&&& +−==ε&

localizationkε&

kσ σhomogenization

constitutivemodel

of phase k

kkg σσ =

0=+⋅∇ gσ ρ

should checkthe weak form of:

(Virtual Work Principle)

( ) σDε &&1−= elel

thvpel εεεε &&&& ++=

)()()(J

)(J

1

2

31

2

2

XsXs

Xsε −

+−−−

=m

yvp

K

Rσ&

εRQbR && )( −=

)(:)(2

3)(J2 XsXsXs −−=−

vpCεX && =

Iε Tth && α=

2222NLCM March 2009

ENERGY conservation

Non linear global solution

Solution Algorithm on a Time Increment

T

MICROSTRUCTURE evolutionModels for transformation kinetics

Local nodal solution

MOMENTUM & MASS conservation

Non linear global solution

p,v( ) 0, =pmech

vR

kg,...),,( TTgfg kk&=

Case of Solid StateTransformations only

( ) 0=TtherR

2323NLCM March 2009

Application to Solid State Transformations

2424NLCM March 2009

Validation vs Instrumented Test

Material : steel 16MnNiMo5

Initial conditions :

T0 = 20°C ; gbainite = 1

Boundary conditions:Surface heat source, Gaussian model:

R0 = 38 mm Q = 1200 W during 75 s

Convection and radiation on external faces:hconvection = 5 W m-2 K-1

qrayonnement = σε(T4-Text4) with ε = 0.7Text = 20 °C

⋅−⋅⋅⋅=

2

0

2

0

3exp

3)(

R

r

R

Qrq

π

Test "INZAT" developed at Insa- Lyon(N.Cavallo et al., 1998)

2525NLCM March 2009

Temperature evolution at different radial locations

Austenite fraction (at the end of heating)

12 mm12 mm14.5 mm14 mmZPA

9.5 mm9 mm12.5 mm12 mmZTA

TransWeldMeasuredTransWeldMeasured

Lower FaceUpper Facezones

TransWeld results

Comparison of the size of HAZ

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200Temps (s)

Tem

péra

ture

(°C

)

Inf: r=0mmInf: r=10mmInf: r=20mmInf: r=30mm

Cavallo[1998]

Evolution of phase fractions

Time [s]

Pha

se fr

actio

n [-

]

Time [s]

Tem

pera

ture

[°C]

Tem

pera

ture

[°C]

Time [s]

2626NLCM March 2009

Vertical Displacement of lower face, at r = 10 mm

Residual hoop stress, on the lower face

von Mises stress [Pa]

pressure [Pa]

Experimental [Cavallo, 1998] TransWeld

Dis

plac

emen

t[m

m]

Time [s]

M. Hamide, Ph.D. Thesis, Mines-ParisTech (2008)

Radius [mm]Radius [mm]

Hoo

pst

ress

[MP

a]D

ispl

acem

ent[

mm

]

Time [s]

2727NLCM March 2009

Air Cooling of a Rail Coupon(Eutectoid steel 0.8wt%C)

Time [s]

Deflection[m

m]

Fraction of pearlite

C. Aliaga, Ph.D. Thesis,Ecole des Mines de Paris (2000)

Fraction of pearlite

Time [s] Time [s]

Deflection[m

m]

2828NLCM March 2009

Considering Liquid-Solid Phase Change

2929NLCM March 2009

Liquid-Solid Phase Change

( )Tkht

h∇⋅∇=⋅∇+

∂∂

vρρ

Spatial averaging method

++=+= ∫∫

lsT

Tlpll

T

Tspss

l

l

s

s Ldcgdcghghgh /

,, )(00

ρτρτρρρρ

t

gL

t

Tc

t

hlls

p ∂∂+

∂∂=

∂∂

/)(ρρρ

ls

l

T

Tp Lgdc /)(

0

ρτρ += ∫

If the solidification path is a function of the temperature only, then )(Tgl

t

T

T

gLc

t

hlls

p ∂∂

∂∂+=

∂∂

/)(ρρρ

eq

pcρEquivalent heat capacity

)()( TcTc p

eq

p ρρ and

Heat equation can be solved directly for T• Convenient for a direct adaptation of commercial codes that solve diffusion problems,

• But can generate strong non-linearities,• and can make the numerical scheme non conservative

3030NLCM March 2009

Liquid-Solid Phase Change

( )Tkht

h∇⋅∇=⋅∇+

∂∂

vρρ

present if at leastone phase is moving

a liquid phase

large deformations of solid

vvv == βα ( )vv hh ρρ ⋅∇=⋅∇

If the densities ρα and ρβ are close enough and quasi constant, 0≈⋅∇ v

vv ⋅∇≈⋅∇ hh ρρ

( )Tkt

hh

t

h∇⋅∇==⋅∇+

∂∂

d

dρρρ v

0=αv and ρα, ρβ are close enough and quasi constant,

mass conservation: 0=⋅∇ v

ββ vv g=

vv ⋅∇≈⋅∇ βρρ hh

( )Tkht

h∇⋅∇=⋅∇+

∂∂

vβρρOnly the heat of phase β istransported with the averagevelocity

Two limiting cases

3131NLCM March 2009

Numerical Treatment (Finite Element Method)

Presented here under simplifying assumptions:• Advection effects neglected, or • Solidification path depending on the temperature only (gl(T))• ρα, ρβ are close enough and quasi constant

vvv == βα ( ) 0d

d=∇⋅∇− Tk

t

hρ

ls

l

T

Tp Lfdch /

0

)( += ∫ ττ

∫ 0)(d

d=∇⋅∇−∀

ΩΩ∫ dVTkdV

t

hϕϕρϕWeak form

∫ ∫ 0d

d=∇⋅∇+⋅∇−

ΩΩ∂Ω∫ dVTkdSTkdV

t

hϕϕϕρ n

nqn ⋅−=⋅∇Tk heat flux through> 0 if inwardgoverned by boundary conditions

Ω∂

∫ ∫ ∫ ∫ 0)()(d

d 44 =−−+−+∇⋅∇+Ω∂Ω∂Ω∂ΩΩ

∫frc

dSqdSTTdSTThdVTkdVt

himpextBextT ϕϕεσϕϕϕρ

convection radiation imposed heat flux

3232NLCM March 2009

Finite Element Discretization

FKTHC =+&

0)()()(1

)( =−+−∆

∆+∆+∆+∆+∆+ ttttttttttt

tTFTTKHHTC

Time integration scheme: implicit Euler type

0)( =∆+ ttTR

Solution using the Newton-Raphson method

0)( =∆+ ttHRor

∫Ω

= dVNNC jiij ρ

∫∫∫Ω∂Ω∂Ω

++++∇⋅∇= dSNNTTTTdSNNhdVNNkK jiextextBjiTjiij ))((22εσ

∫∫∫Ω∂Ω∂Ω∂

++++= dSNqdSNTTTTTdSNThF iimpiextextextBiextTi ))((22εσ

=⋅

M

M

&i

hH

=M

M

iTT

NON LINEAR VECTOR EQUATION:

NON LINEARITIES arise from radiation and possibly convection,

and from the temperature dependent thermophysical properties,

and from the enthalpy-temperature relation

3333NLCM March 2009

• is known if there is a direct relation between and T (i.e. is known)

• or should be calculated by a model linking to T

Newton Raphson's Solution Algorithm

Algorithm :

Loop whileconvεν >)( )(

HR

End loop

Init )0(0 H=ν

)( )1(

)1(

−−

−=∆

∂∂ ν

ν

HRHH

R

HHH ∆+= − )1()( νν

1+=νν

)( )()( νν HTT =

( )

( )43421434214342143421

summednotsummednotsummednotsummednot

11

1

k

kk

itt

j

k

kk

ij

k

k

ik

t

jj

k

kk

ij

ik

k

i

ijij

t

jjiji

H

T

T

FT

H

T

T

K

H

TKHH

H

T

T

C

tCtH

R

FTKHHCt

R

∂∂

∂∂−

∂∂

∂∂

+

∂∂+−

∂∂

∂∂

∆+

∆=

∂∂

−+−∆

=

∆+

h

T

∂∂ h )(Tgl

h

3434NLCM March 2009

Solidification: Transport of Chemical Species

l s

REV

0=∇⋅∇−⋅∇+∂

∂cDc

t

cv

c Volumetric concentration

D Diffusion coefficient

Spatial averaging

At the process scale, diffusion of chemical species is extremely slow (typically > 105 s/cm)

0=⋅∇+∂

∂vc

t

c

If transport and diffusion are negligible, we get the equation for a closed system: 0=∂

∂t

c

This equation, along with the thermodynamic phase diagram, affects the transformation path("trajectory" of a REV in the phase diagram). Limit cases: Lever Rule, and Scheil-Gulliver.

present if at leastone phase is moving

a liquid phase

large deformations of solid

3535NLCM March 2009

Solidification Path: Lever Rule (perfect diffusion)

ssll cccc == **

l

s

l

sls

lf

sslsssll

c

c

c

ck

mcTT

cgcgcgcgcc

==

+=+−=+==

*

*

/

0 )1(

Perfect diffusion:

A) Closed system assumption:

TT

TcT

kg

f

L

lss −−

−= )(

1

1 0

/

B) Open system assumption:

A direct relation between gs and T

( )

+=

+=−+=+=

∫ Lfdch

mcTT

ckggcgcgc

l

T

Tp

lf

lllssll

0

)(

)1(

ττknown areandIf hc

h

TcgT ll ∂

∂,,,

non linearsolution

ssll

ll

gg

g

ρρρ

+

Th andbetween

relationdirect A

Solute concentration

Tem

pera

ture

[°C]

Ts

l

sl +

TLSlope m

0csc lc

3636NLCM March 2009

Solidification Path: Scheil-Gulliver solution(perfect diffusion in liquid, no diffusion in solid)

ll cc =*Perfect diffusion in liquid:No diffusion in solid

A) Closed system assumption:

lsk

Lf

f

sTT

TTg

/1

1

1−−

−−

−=

B) Open system assumption:

A direct relation between gs and TA direct relation

between h and T

Vdt

dgcVc

dt

dV

dt

d ss

s *

solidinsolute )()( ==

dt

dgck

dt

cds

l

ls

s

/=

( )

+=

=

+=−+=+=

∫ Lfdch

dt

dgck

dt

cd

mcTT

ckggcgcgc

l

T

Tp

sl

ls

s

lf

lllssll

0

)(

)1(

/

ττknown are

and

andIf

)( ss gc

hc

h

TcgT ll ∂

∂,,,

non linearsolution

l

s

l

sls

c

c

c

ck

*

*

*

/ ==

Sol

ute

conc

entr

atio

n

Phase fraction

*

sc

lc

3737NLCM March 2009

Interaction Solidification or Metallurgy Mechanics

γgσ ρρ =+⋅∇Conservation of momentum

Spatial averaging

3838NLCM March 2009

Interaction Solidification or Metallurgy Mechanics

Different formulations of interest:

F2 Solid phase + liquid phase : Continuous casting of steel, mechanics of mushy zone0≈sv

( )lllllllll

l ggt

g vvvgMσ ×⋅∇+∂∂=++⋅∇ ρρρ )(

0=+−⋅∇ gMσ ss

s g ρExchange of momentum (Darcy's law)

( )lllllllll

l ggt

g vvvgMσ ×⋅∇+∂∂=++⋅∇ ρρρ )(

F3 Solid phase + liquid phase : Casting, modelling of residual distortions+stressessl vv ≈

0=+⋅∇ gσ ρ

Constitutive equation of the multiphase solid material ? Cf. previous slide

γgσ ρρ =+⋅∇

Constitutive equation of the semi-solid material (from liquid to solid) ?

F4 Solid multiphase material : Solid state phase changes in steel and alloys

F1 Solid phase fixed and rigid + liquid phase : Ingot casting, modelling of liquid convection

3939NLCM March 2009

Interactions Mechanics Heat Transfer

• The domain studied evolves

• Convection effects, due to fluid flow

• Thermal boundary conditions depend on distortions and stresses

( )nT

BATBA

fh

TThqq

T,

)(

δ=

−−=−=

βα nrefTT hh T+= ,1

11

))((22

−+

+++=

BA

BABAgas

T

TTTTkh

εε

σδ

][pressureContact PanT

n

TBv

Av

BA vv −

τTnT

A

B

δ

][widthGap mδ

T,refh

4040NLCM March 2009

Solution Algorithm on a Time Increment

ENERGY conservation

Non linear global solution

h

MOMENTUM & MASS conservation

Non linear global solution

( ) 0=htherR

Case withliquid/solid phase change

Microsegregation model

h

T

∂∂

MICRO-segregation

Local solution

( ) 0,, =ll cgTRh

TccgT sll ∂

∂,,,,

SOLUTE conservation

Linear global solution

c( ) 0=csolR

lls p,, vv( ) 0,, =llsmechpvvR

Not present if gl(T ) is given(direct relation enthalpy – temperature)

4141NLCM March 2009

Solidification: Macrosegregation

4242NLCM March 2009

Macrosegregation during Ingot Casting

adiabatic

adiabatic

adiabatic

adiabatic

Pb-48wt%Sn alloy

Simplifying Hypotheses

Equations

)(~)( vvv

gvv ×⋅∇+∂

∂=+

Κ−∇−∇⋅∇

l

lllgt

ggpgρρρµµ

0=⋅∇ v

0)( =∇⋅∇−⋅∇+∂

∂TkTc

t

hp vρρ

0)( =∇⋅∇−∇⋅+∂

∂llll cDgc

t

cv

llg vv =

Microsegregationmodel: lever rule

))()(1(~0ccTT lcrefT −−−−= ββρρ

2

322

)1(180 l

l

g

g

−=Κ

λ

1

0

=+==

=

sl

sl

l

gg

ρρρv

)(tt ∆−×⋅∇ vv

Linearization:Carman-Kozeny's modelfor Darcy permeability:

Non-linear: NR withh

T

∂∂

Non-linear: NR withc

cl

∂∂

73000 nodes,380000 elts

4343NLCM March 2009

c - c0 [%] gl

t = 200 s

t = 100 s t = 100 s t = 200 s

t = 800 st = 200 s

B. Rivaux, Ph.D. work (2009)

4444NLCM March 2009

Comparison with MeasurementsExperiments: Hebditch & Hunt, 1974

-8

-4

0

4

8

12

16

0 0.02 0.04 0.06 0.08 0.1

Distance au refroidisseur (m)

W-W

0 (w

t.pct

)

Hebditch & HuntCIMLIB_3DR2SOLCIMLIB_2D

z = 5 mm

-8

-4

0

4

8

12

16

0 0.02 0.04 0.06 0.08 0.1

Distance au refroidisseur (m)

W-W

0 (w

t.pct

)

Hebditch & HuntCIMLIB_3DR2SOLCIMLIB_2D

z = 25 mm

-8

-4

0

4

8

12

16

0 0.02 0.04 0.06 0.08 0.1

Distance au refroidisseur (m)

W-W

0 (w

t.pct

)

Hebditch & Hunt

CIMLIB_3D

R2SOLCIMLIB_2D

-8

-4

0

4

8

12

16

0 0.02 0.04 0.06 0.08 0.1

Distance au refroidisseur (m)

W-W

0 (w

t.pct

)

Hebditch & HuntCIMLIB_3DR2SOLCIMLIB_2D

z = 35 mm z = 55 mm

Distance to cooled face [m]

Distance to cooled face [m] Distance to cooled face [m]

Distance to cooled face [m]

c–c 0

[%]

c–c 0

[%]

c–c 0

[%]

c–c 0

[%]

4545NLCM March 2009

Solidification: Residual Stresses and Distortions

4646NLCM March 2009

Constitutive Equations: from Liquid to Solid StateNB: in formulation F3, solid phase + liquid phase, in the mushy state

liquidsolid

Temperature

liquid fraction0 1

TS TL

Elastic-ViscoPlastic Models

mushymushy

TC =TS

ViscoPlastic Model

thvp εεε &&& +=

sεmvp

K

−= 1

2

3 ε&&

Iε TT

th &&∂∂−= ρ

ρ3

1

mKεσ &=mn

y KH εεσσ &++=

thvpelεεεε &&&& ++=

sεmn

yvp

K

H1

)(

2

3 εσσσ

+−=&

Iε TT

th &&∂∂−= ρ

ρ3

1

σDε && 1][ −= elel

ls vv ≈

Bellet, Jaouen, Fachinotti (2005)+ ALE FEM Formulation

4747NLCM March 2009

Solidification of a 65-ton Steel Ingot

Solidified ingot before forging (diameter: 1.8 m)

Powder

Hot top

Moulds

Cast iron plate

Powder

Hot top

Moulds

Cast iron plate

Air gapPrimary skrinkageHot top

Liquid steel

Solid steel

Temperature during cooling

lg

1500°C

20°C

Liquid fraction during cooling

Modellingof filling

4848NLCM March 2009

Octogonal Ingot 3.3 tons

Depthmeasured80 mm

calculated65 mm

Gapmeasured30 mm

calculated25 mm

1 0liquid fraction

30 min 50 min 3 h

4949NLCM March 2009

Sand Casting of a Braking Disc (Grey Iron)

part

core Two half-moulds

5050NLCM March 2009

Comparison with measurements on plain discs[S. David & P. Auburtin, Matériaux2002, Tours, France, 2002]

Measurement by X-ray diffraction

Simulation

Heat Transfer

Residual Stresses

Disc

Core

θθσ rrσ

5151NLCM March 2009

Continuous Casting

5252NLCM March 2009

Ladle

Tundish

Mould

Secondarycooling

Unbending

CuttingContinuousCaster

5353NLCM March 2009

Computational Approach

t = 0

Vc

Global non-steady state strategy:the mesh grows at the extractionvelocity

liquid

mushy

solid

mou

ld

gap

EULERIAN

ALE

LAGRANGIAN

Vc

Vc

Imposed normal stress(pressure)

5454NLCM March 2009

Direct Chill Casting of Aluminium

Schematics and photos from J.-M. Drezet (EPFL) and Alcan

y

S+L

Solid

V

Bottom block

mould

Primary

cooling

Secondary

cooling

3 Butt curl

2 Pull in

1residual

stresses

5555NLCM March 2009

Direct Chill Casting of Aluminium

Temperature [°C] Liquid fraction [-]

Lihua Jing, post-master formation Compumech, CEMEF, 2008

5656NLCM March 2009

Direct Chill Casting of Aluminium

Casting time: 1254 s, Ingot length: 1.34 m

yyσ

y

x

z

xxσ

Horizontalcompressive stresses

along small face

Horizontaltensile stresses

inside

Horizontaltensile stresses in solidified shell in

mould region

Horizontalcompressive stresses along

rolling face

[Pa]

5757NLCM March 2009

Steel Continuous CastingLiquid Flow and Solidificationin Mould Region

Liquid fractionTemperature T > TL1556°C

1526°C

1

0

Velocity vectors

M. Henri, Ph.D. work, 2008

5858NLCM March 2009

Consequence on thesurface slab temperature

Formation of Air GapInfluence of Mould Taper

Solidification and Air Gap Formation

5959NLCM March 2009

Stresses in Solidified Shell

Axial stress in the shellof a round billet

R = 106 mm

12.9 mm

Mouldexit

Zoom on mould exit

liquid solid

]Pa[zzσ

Mould exitz = - 0.6 m

Min mesh size:∆r = 0.4 mm

6060NLCM March 2009

Plastic Deformation and Distortion

17.8 mm

][−ε

z = - 1 m

Zoom on solid shell at z = -1 m

Min mesh size:∆r = 0.4 mm

Radial displacement at surface [m]

Distance from

meniscus

[m]

Fachinotti & Cardona

THERCAST (at 30 s)

THERCAST (at 44 s)

R2SOL (at 30 s)

R2SOL (at 44 s)

6161NLCM March 2009

Welding of Steels

6262NLCM March 2009

Arc Welding Process: Complex and Coupled Phenomena

Small distance interactionsaround the Fusion Zone

Long distanceinteractions

at the scale ofthe assembly

[source: TWI]

Gas Metal Arc Welding Process

• Rapid and localized heating• Fusion of filler metal and of base

metal (mixing)• Solidification and formation of the

weld bead• Metallurgical changes in the

neighbourhood of the Fusion Zone: Heat Affected Zone

• Deformations and stress, locally, and at the scale of the part assembly

• Often multipass

6363NLCM March 2009

Fraction of bainite

Adaptation based on T Adaptation based on T and gbainite Profile of bainite fraction in a transverse section, along the upper surface.

austenite

GTA fusion line on A508 steel plate10 V, 150 A, 1 mm / sSource : q(r, θ) = 13 MW m-2 for r < 5 mm, ∀θhT = 12 W m-2 K-1

qradiation = σε(T4-Text4) W m-2 with ε = 0.75 and Text = 25 °C

M. Hamide, M. Bellet, IJNME, 2008

6464NLCM March 2009

Steel S355 (0.1%C, 1.2%Mn, Nb, V, Ti)GMA welding, 34 V, 330 A, 8 mm/s (~1 kJ/mm)Wire 1.2 mm, 0.125 m/s

Axial stress Transverse stress

Temperature

6565NLCM March 2009

Conclusions

• During metal processing, interactions between heat transfer, metallurgy andmechanics are numerous

Highly coupled and non-linear problems

Requiring robust numerical solvers

Possibly (ideally ?!) requiring mesh size and time step adaptation

• The concurrent liquid-solid and solid state phase change can really increase thealgorithmic complexity

Depending on the objectives, different formulations can be envisaged

• Such coupled analyses require a lot of material and process data

Issue of characterization tests, and associated identification techniques

Issue of data bases (multiple information sources, merging…)