Non-Linear Computational Mechanics ATHENS week March 2016...

Thermal-Metallurgical-Mechanical

Interactions Michel Bellet

Non-Linear Computational Mechanics – ATHENS week – March 2016

NLCM - Michel Bellet 2016-03 2

The Context: Transformation of Metallic Alloys

Solidification

Heat Treatments

Welding

Heat Transfer

Mechanics

Microstructure

Non-Linearities…


Thermal-Metallurgical-Mechanical Interactions

Heat Transfer Temperature

Mechanics Deformation, Stress

Liquid flow

Microstructure Phase fractions

Phase changes: liquid-solid; solid-solid

Thermophysical properties depend on mstructure Latent heat of transformations


• Heat treatment of steels:

Fast cooling: martensite needles

Slow cooling: pearlite lamellae (a-

ferrite + Fe3C). Cementite appears in

bright. Allain et al., J Mater Sci 46 (2011) 2764–2770

• Solidification of aluminium:

Dendritic growth Billia et al., Mcwasp (2006) 359-366


Outline

• Energy conservation

– Some reminders about heat equation

– Extension to the multiphase material (spatial averaging method)

• Interaction heat transfer metallurgy

– Solid state phase transformations

• Interaction metallurgy mechanics

– Transformation plasticity

• Application to the modelling of heat treatment processes

– Numerical treatment and non-linearities

• 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


r specific mass (density)

e internal energy

v velocity field

f mass density of volume forces

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

r volumetric density of heat input

q surface density of heat input (heat flux)

s Cauchy stress tensor

strain rate tensor

Some Reminders about Energy Conservation

wwwwwrr SqVrSVVe

tddddd)

2

1(

d

d 2vTvfv

1st principle of thermodynamics:

for any domain w of a studied system,

Variation of energy

(internal + kinetic) Power of external forces Heat input power

ww

r Vt

V dd

dd)(: v

vvεσ

w r Vt

dd

d

2

1 2v

w r V

td

2

1

d

d 2v

wwwwr SqVrVV

t

eddd)(:d

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

(virtual work principle)

ε T)(2

1vvε


Energy Conservation

Fourier law:

wwwwr SqVrVV

t

eddd)(:d

d

dvεσ

nnq )( Tkq

wwww

r VTkVrVVt

ed)(dd)(:d

d

dvεσ

)(:d

dTkr

t

e εσ r

For any w,

)(: Tkret

e

εσv rr

)(:)()(

Tkret

e

εσv r

r

k thermal conductivity

n outward unit normal vector

T temperature

n

Tkq

w

Divergence

theorem


Energy Conservation

Considering pressure ~ constant (ok for condensed matter),

and a single phase medium,

T

Tp dch

0

)( specific heat

rTkt

TcT

t

Tc pp

εsv :)(

d

drr

pcT

h

r

peh Enthalpy per unit of mass

rTket

e

εσv :)()(

)(r

r1st principle of thermodynamics:

rTkt

ph

t

h

εsv :)(

d

drr

mass conservation

Iσs p

rTkt

ph

t

h

εsv :)(

d

d)(

)(r

r

Interaction with

mechanics

0)(

vr

r

t


Energy Conservation for a Multiphase Material

rTkht

h

εsv :)()(

)(r

ris satisfied in any phase k of a

representative elementary volume

(REV) of the multiphase material

b a

REV

Looking for an averaged conservation equation on the REV

The spatial averaging method

For any scalar function Y defined on the REV,

– Average in phase k:

– Mixture average:

– Two theorems:

volume fraction

of phase k

k

kk

k

kk

k

k gg

kk

k

kV

VVk

k

ggV

VV

k

k

k

k

VV

V

VV

d)(

d)(d)()(

1

11

0

000

x

xxx

k

kk

0

1

phase outside

phase inside

"intrinsic" average

in phase k

M. Rappaz, M. Bellet, M. Deville, Numerical Modelling in Materials Science and Engineering, Springer (2003)

*/ baa

aa

nvtt

*/ baaaa n


Energy Conservation for a Multiphase Material

b a

REV

rTkht

h

εsv :r

r

Average volumetric enthalpy

Average energy flux vector

Average mechanical power

Average volumetric heat input

Average thermal conductivity

k

kkkkk hghgh rrr )(

kk hgh )( vv rr

kkg ):(: εsεs

kkrgr

kkkgk

rTkHt

H

εsv :

k

kkkk HgHghH r

kk HgH )( vv


Solid State Phase Changes (Metallurgy Heat Transfer)

0)()(

0

Tkgt

Hg

Tkt

H

kkkk

Spatial averaging method

t

Tc

t

Hkp

k

)(r

T

Tkpk dcH

0

)()( r

Simplifying assumptions

• Advection neglected

• Mechanical power neglected

• Volumetric heat source r = 0

For each phase k,

0)(

Tk

t

TcgH

t

gkpkk

k r

kk

p Ht

gTk

t

Tc

r

kj

jk

ki

kik ggt

g when phase i is partially transformed into phase k 0kig

0kig otherwise

),(

)(ji

jijip HHgTkt

Tc r

Interaction Metallurgy

Heat transfer


Interaction Heat Transfer Metallurgy

jig

• Phase transformations, precipitation phenomena…

• How to get the ?

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

– Two types of phase transformation

• Diffusive transformations: austenite ferrite, pearlite, bainite

• Displacive or massive transformation: the martensitic transformation


A1

A3

g

a a + g

a-ferrite +

pearlite (a-ferrite + Fe3C)

C [wt%]

T [C]

Liq

austenite

g

Liq+g

g + Fe3C

a + Fe3C

g + Fe3C

723 C

Transformations of Low-Alloyed Steels

Fe-C Equilibrium Phase Diagram

http://fr.wikipedia.org/wiki/Fichier:Diagramme_fer_carbone.svg

http://fr.wikipedia.org/wiki/Fichier:Diagramme_fer_carbone.svg


Transformations of Low-Alloyed Steels

Slow cooling: pearlite lamellae (a-ferrite

+ Fe3C). Cementite appears in bright.

Alla

in e

t al.,

J M

ate

r Sci

46 (

2011)

2764–2770

• In isothermal conditions, austenite

decomposition is characterized by TTT

diagrams (Time, Temperature,

Transformation).

• A specific approach is developed to

extend the modelling to non-isothermal

conditions: see next slides.


400

500

600

700

800

900

1 10 100 1000 10000

temps (s)

Te

mp

éra

ture

(°C

)

début

10%

90%

fin

T

kg )(

max, )(exp1Tn

kkkktTbgg

1) Nucleation starts at time

2) Growth: Avrami's law Time [s]

Te

mp

era

ture

[C

]

Time [s]

T start

end

Time – Temperature – Transformation

Isothermal Conditions: TTT Diagrams

Diffusive Transformations

T

• Out of equilibrium isothermal

transformations

• Conjugate effects of

thermodynamical unbalance,

and diffusion of chemical species


• Model based on the additivity principle:

– Decomposition of a non-isothermal history in a series of incremental isothermal

steps

– Summation of incremental contributions

• Nucleation

– Sum of Scheil. It is assumed that transformation starts when:

1)(

i i

i

T

t

Non-Isothermal Conditions


Time

Tem

pera

ture

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


• Growth

• NB. Alternative models exist in the literature:

...),,,,( Cik wGgTTfg g

Leblond et al. 1985 ; Waeckel et al. 1995

Non-Isothermal Conditions


Time

Tem

pera

ture


• At higher cooling rate, Carbon cannot diffuse

quickly enough to allow the growth of pearlite

(ferrite and cementite)

• Below a given temperature, the thermodynamic

unbalance is so large that austenite transforms

into a carbon-oversaturated ferrite: martensite.

• The transformed phase fraction directly

depends on temperature

• It induces both dilatation (CC less dense by

about 4%) and shear due to C insertion in CC

Displacive transformation: martensite formation

Fast cooling: martensite needles

Alla

in e

t al.,

J M

ate

r Sci

46 (

2011)

2764–2770

TM

msegg

b

g 1

MS martensite start temperature

Koïstinen & Marbürger, Acta Metall. (1959)


Interaction Metallurgy Mechanics

γgσ rr Conservation of momentum

Spatial averaging

Constitutive equation of the multiphase solid 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

Iε

ji

ji

i

ijtrg

r

rr

3

1

Leblond et al., Int. J. Plasticity (1989)

Desalos & Giusti (1982)

Fischer, Acta Metall Mater (1990)

Temperature [C] Temperature [C]

Defo

rmation [%

]

Defo

rmation [%

]

With applied stress Free dilatometry

From Coret,

Ph.D. (2001)

g

ag

ga + Fe3C

zzji

jijjizztp

σggKε

)('

One-dimensional expression,

along direction z:

Different expressions of K and in literature

Solid Multiphase Material



• Homogenization procedure: Taylor's assumption (or other choice !)

• Constitutive models of the phases:

– Lemaître & Chaboche EVP model, for instance

– Models may be different for each phase

tptr

k εεεEε ε

localization kε

kσ σhomogenization

constitutive

model

of phase k

kkg σσ

0 gσ r

should check

the weak form of:

(Virtual Work Principle)

σDε 1

elel

thvpel εεεε

)()()(J

)(J

1

2

31

2

2

XsXs

Xsε

myvp

K

Rs

εRQbR )(

)(:)(2

3)(J2 XsXsXs

Iε Tth a

Solid Multiphase Material

XεX g vpC3

2


Numerical Treatment

VrVTkV

t

Tcp d'd)(d rWeak form

VrVTkSTkV

t

Tcp d'ddd r n

nqn Tk heat flux through

> 0 if inward

governed by boundary conditions

VrSqSTTSTThVTkV

t

Tc

frc

impextBextTp d'dd)(d)(dd44 sr

convection radiation imposed heat flux

),(

)(ji

jijip HHgTkt

Tc r

Simplifying assumptions

• Advection neglected

• Mechanical power neglected

• Volumetric heat source r = 0

In the solid state, and using finite elements


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

VNNcC jipij dr

SNNTTTTSNNhVNNkK jiextextBjiTjiij d))((dd22s

VNrSNqSNTTTTTSNThF iiimpiextextextBiextTi d'dd))((d22s

iTT

iTT

NON LINEAR VECTOR EQUATION:


Newton Raphson's Solution Algorithm

0)( TR

TT

RTRTTR

)()(

Algorithm:

Loop while conv )( )(TR

End loop

Init )0(0 T

)( )1(

)1(

TRTT

R

TTT )1()(

1

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

tC

tT

R

FTKTTCt

R

11

1


ENERGY conservation

Non-linear global resolution

Solution Algorithm on a Time Increment

T

MICROSTRUCTURE evolution

Models for transformation kinetics

Local nodal resolution, at each node

MOMENTUM & MASS conservation

Non-linear global resolution

p,v 0, pmech vR

kg,...),,( TTgfg kk

0TtherR

Applications: Solid State

Transformations


Solid State Transformations by Laser Heating

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

qradiation = s(T4-Text4) with = 0.7

Text = 20°C

2

0

2

0

3exp

3)(

R

r

R

Qrq

Test "INZAT" developed at INSA-Lyon

(J.-F. Jullien et al.)


Temperature evolution at different radial locations

Austenite fraction (at the end of heating)

zones Upper Face Lower Face

Measured TransWeld Measured TransWeld

ZTA 12 mm 12.5 mm 9 mm 9.5 mm

ZPA 14 mm 14.5 mm 12 mm 12 mm

TransWeld results

Comparison of the size of the heat affected zone (HAZ)

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200Temps (s)

Tem

pératu

re (

°C

)

Inf: r=0mm

Inf: r=10mm

Inf: r=20mm

Inf: r=30mm

Cavallo [1998]

Evolution of phase fractions

Time [s]

Ph

ase

fra

ction

[-]

Time [s]

Te

mp

era

ture

[C

]

Te

mp

era

ture

[C

]

Time [s]

M. Hamide, Ph.D. Thesis, MINES ParisTech (2008)


Vertical displacement of lower face, at r = 10 mm

Residual hoop stress, on the lower face

Experimental [Cavallo, 1998] TransWeld

Time [s]

Hamide, Massoni, Bellet, Int J Numerical Methods Engineering 73 (2008) 624-641

Radius [mm] Radius [mm]

Hoo

p s

tress [M

Pa

] D

isp

lace

ment [m

m]

Time [s]

80 0

5. E+08

von Mises stress [Pa]

pressure [Pa] 2.9 E+07

-2.8 E+08


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)

Fra

ctio

n o

f p

ea

rlite

Time [s] Time [s]

De

fle

ction

[m

m]

1 2

3 4

Considering Liquid-Solid Phase

Change


Solidification: Phenomena at Different Scales

macro

(~ 0.1 m)

s

l

s+l meso

(~ 0.1 to 10 mm)

s

l

s l

micro

(~ 10 to 100 mm)

REV of a mushy material


Liquid-Solid Phase Change: Enthalpy - Temperature

Tkht

h

vr

rSpatial averaging method

lsl

T

T plsT

T lpll

T

T spssl

ls

s LgcLcgcghghgh //,, )(d)(dd

000

rrrrrrrr

HYP: gl(T ). The "solidification path" depends on T only

and is known a priori

FE discretization: 0)( TR non-linear function

Newton-Raphson iterative resolution

NB 1: (cf slide 10)

the discretization of the transport term may take very different forms, according to velocities vl

and vs.

• Simple forms for simple cases: vs = 0, or vl = vs

• More complex forms otherwise… (and need to solve for both vs and vl !)

NB 2: Enthalpies and densities of phases can be obtained as a function of temperature through

coupling with thermodynamic databases, using software like Thermo-Calc® or JMatPro®

ssll hghgh )()( vvv rrr

)(Tfh r


Comment on Solidification Path

• gl(T) is a first-order approximation

• Solidification kinetics actually depends on the transport of chemical species at the scale of the microstructure (dendrite arms):

– Partition coefficients (equilibrium diagrams)

– Diffusion kinetics in solid and liquid phase

(during a finite process time out of equilibrium)

MICROSEGREGATION

• At the scale of the VER, we have:

• Note also that transport of chemical species occur at the scale of the whole part

MACROSEGREGATION

h

TccgT silil

,,,, ,,ich , Microsegregation Model

0

vc

t

c

~~ Segregation issues are not addressed in this lecture ~~


Interaction Solidification Mechanics

γgσ rr Conservation of momentum

Spatial averaging

Different formulations are possible, depending on the focus

• Liquid flow (assuming the solid phase fixed and rigid)

• Distortions and stresses in solid regions (ignoring fluid flow)

• Full interaction: deformable solid + fluid flow


Constitutive Equations: from Liquid to Solid State

liquid solid

Temperature

liquid fraction 0 1

TS TL

Elastic-ViscoPlastic Models

mushy mushy

TC =TS

ViscoPlastic Model

thvpεεε

sεmvp

K

1

2

3

Iε TT

th

r

r3

1

mKs mn

y KH ss

thvpelεεεε

sεmn

yvp

K

H1

)(

2

3 ss

s

Iε TT

th

r

r3

1

σDε 1][ elel

M. Bellet, V.D. Fachinotti, Comput. Meth. Appl. Mech. Eng. 193 (2004) 4355-4381

M. Bellet, O. Jaouen, I. Poitrault, Int. J. Num. Meth. Heat Fluid Flow 15 (2005) 120-142

+ ALE FEM Formulation


Interactions Mechanics Heat Transfer

• Convection effects, due to fluid flow

• Change of contact conditions

– Gaps may form

– Contact pressure may change

• Thermal boundary conditions depend on distortions and stresses

)( BATBA TThqq

ba nrefTT hh T ,

111

))((22

BA

BABAgas

T

TTTTkh

s

[Pa]pressureContact nT

n

TBv

Av

BA vv

TnT

A

B

[m]widthGap

T,refh

nT fh T,

Heat exchange coefficient

h [W m-2 K-1]

Applications: Solidification


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

Modelling

of filling


Octogonal Ingot 3.3 tons

Depth

measured

80 mm

calculated

65 mm

Gap

measured

30 mm

calculated

25 mm

1 0 liquid fraction

30 min 50 min 3 h


Sand Casting of a Braking Disc (Grey Iron)

part

core Two half-moulds


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

Measurement by X-ray diffraction

Simulation

Heat Transfer

Residual Stresses

Disc

Core

s rrs


Direct Chill Casting of Aluminium Alloys

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

S+L

Solid

V

Bottom block

mould

Primary

cooling

Secondary

cooling

3 Butt curl

2 Pull in

1 Residual

stress



Lihua Jing, post-master, CEMEF (2008)

Temperature [C] Liquid fraction [-]



Casting time: 1254 s, Ingot length: 1.34 m

yys

y

x

z

xxs

Horizontal

compressive stresses

along small face

Horizontal

tensile stresses

inside

Horizontal

tensile stresses in

solidified shell in

mould region

Horizontal

compressive

stresses along

rolling face

[Pa]

Lihua Jing, post-master Compumech, CEMEF (2008)


Arc Welding Process

Small distance interactions

around the Fusion Zone

Long distance

interactions

at the scale of

the assembly

[source: TWI]

• 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

Gas Metal Arc Welding


Stress formation

• Axial stress

• Transverse stress

../../../VIDEOS/soudage/Plaque316NLavecApport/Szz_Apport_plan.AVI

../../../VIDEOS/soudage/Plaque316NLavecApport/Temp_Szz_Plans.AVI


Comparison Calculations vs Experiments

GMA Welding

Metal deposition on steel plates

equipped with thermocouples and

displacement sensors

TC @ -5 mm and -7 mm under weld

surface

Plate 316NL, 10 mm thick

Electrode

Plate 8 to 10 mm

6 LVDT

12 TC


Essais 316LN (lvdt: 5, 6)

-0,002

-0,0018

-0,0016

-0,0014

-0,0012

-0,001

-0,0008

-0,0006

-0,0004

-0,0002

0

0,0002

0 50 100 150 200 250 300 350 400

Temps (s)

Dé

pla

cem

en

t (m

)

Test3: lvdt5

Test3: lvdt6

C5 : FE

Essais 316LN (lvdt: 2, 4)

-0,001

-0,0008

-0,0006

-0,0004

-0,0002

0

0,0002

0,0004

0 100 200 300 400 500

Temps (s)

Dép

lace

men

t (m

)

Test3: lvdt2

Test3: lvdt4

C2: FE

Vertical Displacements


• [PhD O. Desmaison 2013] ANR "SISHYFE"

(Areva, Industeel)

• Non steady-state formulation

• Multiphysic approach, 3D FEM, level set

technique – Interactions with arc plasma and laser

radiation

– Modeling of material deposition

– Surface tension

– Solid mechanics

– Steel grade 18MND5

• Dynamic adaptive remeshing

Welding: Hybrid Arc-Laser Welding Process

Extension of welding pool (fusion

zone) along welding and transverse

direction

EXP: ICB LeCreusot Simulation


• Simulation of stress formation

– Evolution of stress distribution between pass 1.1 and 1.2

YYs-200 MPa 600 MPa

Pass 1.1 Welding Pass 1.1 Cooling

Pass 1.2 Welding Pass 1.2 Cooling



– Comparison Simulation / Measurements after deposit of 3 layers (6 passes)

Simulation Experiment: measurement by the

contour method (Areva)

EXP : méth. contours (Areva) Simulation



– Comparison Simulation / Experiment (Method of hole drilling)

Stress vs Depth


Smaller scale approaches

• So far, volume fractions of the different phases

• Smaller scale approaches are possible, modelling smaller scale structural features:

– Grains

– Dendrites, lamellae…

• Examples…


CA-FE Modelling (Cellular Automata – Finite Elements)

• Capture of the growth of grain envelopes

• « Macroscopic » mesh (FE, non structured) for heat transfer calculations

• « Microscopic » mesh (CA, structured) for grain growth calculation

• Application example:

– Solidification of a parallelepipedic specimen under constant T between its lateral sides

– Sn-3%wtPb

– 400 K/m; 0.03 K/s

85 000 nodes, 330 000 elements, 7 500 000 cells, 64 proc., 17 days.

Carozzani, Gandin, Digonnet, Bellet, Zaidat, Fautrelle, Metallurgical and Materials Transactions A 44, 2 (2013) 873-887


CA-FE Modelling – Application to multipass welding

• Duplex (ferrite / austenite)

stainless steel

Fe-0.02C-22Cr-2Ni-2Mn-0.45Mo-

0.2N (wt-pct) – 120 mm x 30 mm x 25 mm (FE)

– Initial grain density 1011 m-3 / (~4 cells /

grain)

• 3 passes with partial remelting – Power / Heat source

• 10 mm/s - 8000 W

– Added metal • Wire velocity 80 mm/s

• Wire radius 0.3 mm

Adaptative

FE mesh

Fixed

CA mesh

Desmaison, Bellet, Guillemot, Computational Materials Science 91 (2014) 240-250



Pass 1



Pass 2



Pass 3

Chen, Guillemot, Gandin, ISIJ Int. 54 (2014) 401


Phase field Modelling

• Calculation of phase change at the scale of dendritic arms (liquid/solid) or lamellae (solid/solid)

• Resolution of conservation equations (heat transfer, momentum, mass, solutes) and evolution of a phase function

• Application restricted to a limited number of crystals

• According to Moore’s law, ,

phase field modelling at the scale of a part should be available by the end of this century (2070-2100): Voller & Porté-Agel, J Comp Physics 179 (2002) 698

C. Sarkis, P. Laure, Ch.-A. Gandin (Cemef): 2.7 M elts, 0.47 M nodes, 56 h on 16 cores

Y

PP 3

2

0 2


Conclusions

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

– Highly coupled and non-linear problems

– Requiring robust numerical solvers, including mesh size and time step automatic adaptation

• The concurrent liquid-solid and solid state phase change can really increase the algorithmic 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…)

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