UIUC, August 6, 2008 Thermal-mechanical Behavior of...

ANNUAL REPORT 2008UIUC, August 6, 2008

Lance C. Hibbeler (MS Student)

Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-Champaign

Thermal-mechanical Behaviorof the Solidifying Shell

and Ideal Taper in a Funnel Mold

University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 2

The Problem

• Longitudinal Face Cracks (LFC)– Depression type

– More likely in transition region of the funnel mold

– Causes about 60% of breakouts at the Corus Direct Sheet Plant (IJmuiden, The Netherlands)

• Investigate effect of funnel mold design on LFC tendency and mechanisms using:– Plant experiments

– Computational models

100 mm


Thin-Slab Casting with Funnel Molds

Mold Narrow Face (NF)

Mold Wide Face (WF)

Solidifying Steel Strand

Submerged Entry Nozzle (SEN)

Funnel Region


Funnel MoldTerminology and Dimensions

Outer Funnel Width = 750 mm = 29.4”

Strand Width = 1200 mm (Variable)

Inner Funnel Width = 260 mm = 10.2”

Mold Wide Face

Mold Narrow Face Mold Narrow Face

Funnel Crown = 23.4 mm at top, 8 mm at bottom

Strand Thickness = 90 mm Funnel Opening = 136.8 mm

OuterFlat

InnerCurve

OuterCurve

InnerFlat

InnerCurve

OuterCurve

OuterFlat

Mold Wide Face

Funnel Length =

850 mm

= 33.5”

Mold Length =

1100 m

m =

43.3”

Centerline Slice

72 mm = 2.8”

r ≈ 23 m


0

2

4

6

8

10

12

14

16

18

20

22

0 100 200 300 400 500 600 700

Distance from Center (mm)

Pe

rce

nta

ge

of

To

tal L

FC

Bre

ak

ou

ts (

%) Inner Flat Inner Curve Outer Curve Outer Flat

January 2005 - August 2007

LFC Breakout LocationsData provided by A. Kamperman of Corus

Each interval is 10 mm. Shows the locations of depression-type LFC’s that caused breakouts

750 mm Outer Funnel Width


• Root cause is non-uniform heat transfer• Initiate nonuniformity (shell depression)

– Variations in slag rim thickness at meniscus– Gap from necking (self-correcting)– Gap from buckling (self-amplifying)

• Depression causes:– Lower heat flux– Higher shell temperature– Thinner shell– Grain growth (larger grains)– More brittle behavior– Stress and strain concentrations

• Tensile inelastic strain exceeds critical value cracks form

Longitudinal Facial Cracking: Depression Mechanism

Combination causes cracks

Amplifiesif buckling

1

2

3

45

Mo

ld W

all

Mo

ld W

all


Numerical Model

• Coupled thermal-stress analysis with ABAQUS 6.7– Transient solidification heat transfer in a moving 2D slice

– Special two-level integration scheme for elastic-viscoplastic mechanical behavior (implemented with a UMAT user subroutine)

• Temperature-dependent properties (isotropic)– Thermal conductivity, specific heat capacity

– Elastic modulus, coefficient of thermal expansion

– Inelastic strain depends on temperature, phase (L, δ, γ) and strain rate

• Mechanical contact– “Softened” exponential pressure-overclosure relationship

– Interfacial friction factor of µ = 0.16 [Meng et al., CMQ 45-1 pg. 79-94]

• Ferrostatic pressure

• 2D model includes the funnel shape and mold oscillations


Constitutive Equations

• Austenite (Kozlowski III model):

• δ-ferrite (Zhu modified power law):

• Liquid and mushy zone:– Treat as low yield stress, low elastic

modulus, perfectly-plastic solid

( ) ( ) ( ) ( ) ( ) ( )

( )( )( )

32

411

1

31

32

33

4 4 5 2

4.465 10s exp

130.5 5.128 10

0.6289 1.114 10

8.132 1.54 10

( ) 4.655 10 7.14 10 1.2 10

f Tf T K

f C f TT

f T T

f T T

f T T

f C C C

ε σ ε ε −−

−

−

−

⎛ ⎞×⎡ ⎤= − −⎜ ⎟⎜ ⎟⎣ ⎦ ⎝ ⎠= − ×

= − + ×

= − ×

= × + × + ×

( ) ( )

( ) ( )2

5.521

5.56 104

5

4

0.1 ( ) 300 (1 1000 )

1.3678 10

9.4156 10 0.34951 1.617 10 0.06166

nms f C T

f C C

m T

n T

ε σ ε−

−−

− ×

−

−

= +

= ×

= − × += × −

T in Kelvin, σ in MPa, C in weight % C

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6 7 8 9 10

Strain (%)

Str

es

s (

MP

a)

Austenite

δ-Ferrite

Strain Rate = 10-4 s-1

Temperature = 1400 °CComposition = 0.045 %wt C


600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0.0 0.2 0.4 0.6 0.8 1.0 1.2Percent Weight Carbon

Te

mp

era

ture

(ºC

)

0.045 %C Steel Phases

[Won and Thomas, MTB, 2001]

Liquidus temperature 1530.45 ºCSolidus temperature 1507.51 ºC

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

600 800 1000 1200 1400 1600Temperature (ºC)

Ph

ase F

ractio

n (--

) Fraction α

Fraction δ

Fraction γ

Fraction L


Traveling Slice Analysis


Finite Element Mesh

Corner

Near inside of funnel

• Standard 4-node heat transfer elements

• Hybrid formulation of generalized plane strain mechanical elements

• Fixed mesh with over 16000 nodes

Solidifying Steel

Mold

Computational Domain


Process Parameters

0.045 %wtCarbon content

1.0 %/mNarrow face taper

104.2 mmMeniscus depth

10.86 sTime in mold

1200 mmStrand width

1545.0 ºCPour temperature

5.5 m/minCasting speed

2813 ºF

217 ipm

47”

4”


Heat Flux Profiles:Interfacial Thermal Boundary Condition

• Uniformly applied around most of the perimeter

• Decrease heat flux at corner (linear drop to 50% over 20 mm)

100%

100% 50%

20 mm

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

He

at

Flu

x (

MW

/m²)

0.0 2.2 4.4 6.6 8.8 11.0

Time Below Meniscus (s)

Q = 6.5*(t(s)+1.0)-0.5

Qavg = 2.92 MW/m²

Distance Below Meniscus (mm)


2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00

Casting Speed (m/min)

Av

era

ge

He

at

Flu

x (

MW

/m²)

New Plates Old Plates

CON1D New Plates CON1D Old Plates

Log. (New Plates) Log. (Old Plates)

Heat Flux Measurements

Simulated case

2.92 MW/m2

[Santillana et al., AISTech 2007]


Ferrostatic Pressure and Mold Wall Movement:Mechanical Boundary Conditions

Hot Face DisplacementFerrostatic Pressure

0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000

Pre

ss

ure

(k

Pa

)

0.0 2.2 4.4 6.6 8.8 11.0

Time Below Meniscus (s)


Delayed about one second to allow the outermost row of elements to solidify

0

2

4

6

8

10

12

14

0 200 400 600 800 1000

Dis

pla

ce

me

nt

(mm

)

0.0 2.2 4.4 6.6 8.8 11.0Time Below Meniscus (s)

Wide Face

Narrow Face



1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

1550

0 2 4 6 8 10 12 14 16 18 20

Distance Beneath Shell Surface (mm)

Te

mp

era

ture

(ºC

)

-21

-18

-15

-12

-9

-6

-3

0

3

6

9

12H

oo

p S

tre

ss

(M

Pa

)

Analytical Temperature

Numerical Temperature

Analytical Stress

Numerical Stress

2 s10 s30 s

Model Verification

ºC1494.48Liquidus temperature

ºC1494.38Solidus temperature

kPa35.0Yield stress in liquid material

GPa40.0Elastic modulus in solid

MPa20.0Yield stress at mold temp.

ºC1495.0Initial temperature

ºC1000.0Mold temperature

GPa14.0Elastic modulus in liquid

ValueProperty/Condition

kg/m37500.0Density

--0.3Poisson’s ratio

m/(m·ºC)20.0E-6Thermal expansion coefficient

W/(m·K)33.0Thermal conductivity

kJ/kg272.0Latent heat

J/(kg·K)661.0Specific heat

[Boley and Weiner, 1963]


Temperature Results

t = 10.86 s (mold exit)

Slight two-dimensional heat transfer in funnel transition region

Inner Curve Outer Curve

Solidus

9.6 mm


1000

1100

1200

1300

1400

1500

1600

0 2 4 6 8 10 12 14 16


Te

mp

era

ture

(ºC

)

L

δ

L + δ

γδ + γ

2 s 4 s 6 s 8 s 10 s

Temperature Solution


-12

-10

-8

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14 16


Str

es

s (

MP

a)

2 s4 s 6 s 8 s 10 s

Stress Profiles Through Thickness in Flat Regions

Compressive Stress

Tensile Stress


Stress Histories:Through Thickness in Flat Regions

-12

-10

-8

-6

-4

-2

0

2

4

6

0 1 2 3 4 5 6 7 8 9 10 11Time (s)

Str

es

s (

MP

a)

0.0mm 1.5mm 3.0mm 4.5mm 6.0mm 7.5mm

1.5 mm = 0.06”


-16.0

-12.0

-8.0

-4.0

0.0

4.0

8.0

0 50 100 150 200 250 300 350 400 450 500 550 600 650

Distance from Mold Centerline (mm)

Sh

ell

Su

rfac

e S

tres

s (M

Pa)

750 mm Outer Funnel Width950 mm Outer Funnel Width

2 s

4 s

6 s

8 s

10 s

Surface Stress Around Perimeter:Effect of Funnel Width


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 50 100 150 200 250 300 350 400 450 500 550 600 650


Max

imu

m S

hel

l S

tres

s (M

Pa)

750 mm Outer Funnel Width950 mm Outer Funnel Width

2 s

4 s

6 s

8 s

10 s

Stress Near Solidification Front:Effect of Funnel Width

Peak stress (in γ phase)


• A unique attribute of funnel molds is that the steel shell is significantly bent as it slides down the mold

Funnel Bending Effect

Tension

Compression

Compression

Tension

y

x

w

2·h

This end pinned, u = 0M

εmax, compressive

εmax, tensile

x

y

Rε = −

[R.C. Hibbeler, Mechanics of Materials, 5e, 2003]

• Beam theory from solid mechanics can elucidate the important parameters in the phenomenon


Analytical Bending Model:Comparison with Numerical Model

• Take the difference between bending a beam to the funnel radius at the meniscus and the funnel radius at some other depth:

δ = distance from neutral axis ≈ shell thickness

• Compare with results of 2D model with the thermal effects subtracted

( ) ( )( )

( )( ) ( ) ( ) ( )

( ) ( )meniscus

bendingmeniscus meniscus

z z r z r zz z

r z r z r z r z

δ δε δ

⎛ ⎞−= − = ⎜ ⎟⎜ ⎟

⎝ ⎠

Bending Strain on Solidification Front

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 200 400 600 800 1000

Me

ch

an

ica

l Be

nd

ing

Str

ain

(%

)

0.0 2.2 4.4 6.6 8.8 11.0Time Below Meniscus (s)

Analytical ModelNumerical Model

En

d o

f fu

nn

el l

en

gth


750 mm OuterFunnel Width

950 mm OuterFunnel Width

( )2( )( )

4 16 ( )outer funnel width inner funnel widthcrown z

r zcrown z

−= +

⋅


Analytical Bending Model:Effect of Outer Funnel Width

Larger outer funnel width = Wider funnel = Larger radius = Lower bending strain and strain rate

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 200 400 600 800 1000

750 mm Outer Funnel Width850 mm Outer Funnel Width950 mm Outer Funnel Width

En

d o

ffu

nn

el l

en

gth


Me

ch

an

ica

l B

en

din

g S

tra

in (

%)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 200 400 600 800 1000

750 mm Outer Funnel Width850 mm Outer Funnel Width950 mm Outer Funnel Width

En

d o

ffu

nn

el l

en

gth

Distance Below Meniscus (mm)M

ec

ha

nic

al

Be

nd

ing

Str

ain

Ra

te (

%/s

)

All cases with 260 mm inner funnel width


Analytical Bending Model:Effect of Inner Funnel Width

Smaller inner funnel width = Wider funnel = Larger radius = Lower bending strain and strain rate

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 200 400 600 800 1000

260 mm Inner Funnel Width130 mm Inner Funnel Width0 mm Inner Funnel Width

En

d o

ffu

nn

el l

en

gth


Me

ch

an

ica

l B

en

din

g S

tra

in (

%)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 200 400 600 800 1000

260 mm Inner Funnel Width130 mm Inner Funnel Width0 mm Inner Funnel Width

En

d o

ffu

nn

el l

en

gth


Me

ch

an

ica

l B

en

din

g S

tra

in R

ate

(%

/s)

All cases with 750 mm outer funnel width


Analytical Bending Model:Effect of Crown

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 200 400 600 800 1000

30 mm Crown at Top20 mm Crown at Top10 mm Crown at Top


Me

ch

an

ica

l B

en

din

g S

tra

in (

%)

En

d o

ffu

nn

el l

en

gth

0.00

0.03

0.06

0.09

0.12

0.15

0 200 400 600 800 1000

30 mm Crown at Top20 mm Crown at Top10 mm Crown at Top

Distance Below Meniscus (mm)M

ec

ha

nic

al

Be

nd

ing

Str

ain

Ra

te (

%/s

)

En

d o

ffu

nn

el l

en

gth

Smaller crown = Shallower funnel = Larger radius = Lower bending strain and strain rate

Crown at bottom = 8 mm. The difference between the crown at top and bottom is the key dimension here.


Analytical Bending Model:Effect of Funnel Length

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 200 400 600 800 1000

850 mm Funnel Length950 mm Funnel Length1050 mm Funnel Length


Me

ch

an

ica

l B

en

din

g S

tra

in R

ate

(%

/s)

Longer Funnel = Increases strain near bottom of funnel (larger radius for more time), but lowers strain rate

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 200 400 600 800 1000

850 mm Funnel Length




Me

ch

an

ica

l B

en

din

g S

tra

in (

%)


Pushing the Shell

• As the crown decreases, the steel shell is pushed inward to the SEN and outward to the narrow faces– Opposed by friction and solidification shrinkage

– Most noticeable at early times before opposing effects are strong, but always present

Decreasing crown

Shell pushed “uphill” by funnel

Opposed by shrinkage


Interfacial Gaps

• Predicted gaps are mechanical effects, due to no coupling of mechanical model to heat transfer model (coupled model would have larger gaps)

• Small gaps in the inner curve region– “Shell pushing” encourages good contact

• Larger gaps in the outer curve region– Shrinkage from outer flat region is resisted by “shell pushing”

• Both influenced by bending effect

• Shrinkage minus “pushing” meets somewhere in the middle of the funnel

0.000

0.010

0.020

0.030

0.040

0.050

0 100 200 300 400 500 600

Distance from Centerline (mm)

Inte

rfa

cia

l Ga

p (

mm

)

6 s 9 s 10.86 s

Inner Flat Inner Curve Outer Curve Outer Flat


Decomposing Perimeter Length Changes

1. Calculate the total arc length of the shell as a function of distance from the center of the mold

2. Subtract current perimeter length from initial (at meniscus) to get a measure of shrinkage = “Actual Funnel”

3. The geometry of the mold allows a closed-form expression of the same quantity; subtract this from the calculated shell shrinkage = “Adjusted Funnel”

4. Subtract the corresponding shrinkage of a parallel mold from the above quantity (“what happens to the shell” –“what the shell wants to do”) = “Difference”

5. Adjust for the fact that the shell in a funnel mold is slightly longer, and that longer pieces shrink more = “Adj. Difference”

6. Result is a quantification of “pushing” vs. shrinkage as well as a measure of buckling tendency in the funnel


Perimeter Length Changes

Inner funnel width = 260 mm, outer funnel width = 750 mm, crown at top = 23.4 mm, crown at bottom = 8 mm, funnel length = 850 mm

10.86

-8

-7

-6

-5

-4

-3

-2

-1

0

1

0 50 100 150 200 250 300 350 400 450 500 550 600


-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Differen

ce (mm

)

Actual FunnelAdjusted FunnelActual ParallelDifferenceAdjusted Difference

Time = s

Inner Flat Inner Curve Outer Curve Outer Flat

Per

imet

er L

eng

th C

han

ge

(mm

)


10.86

-0.20

-0.18

-0.15

-0.13

-0.10

-0.08

-0.05

-0.03

0.00

0.03

0.05

0 50 100 150 200 250 300 350 400 450 500 550 600


Reference Case

950 mm Outer Funnel Width0 mm Inner Funnel Width


Time = s

Dif

fere

nce

(m

m)

Effect of Geometry on Perimeter Length Changes

Buckling potential less with larger width


Tangential Displacement

• Project the incremental displacement vector onto a vector tangent to the mold surface– Neglects the “global” mold displacement, but

includes the effects of the shell sliding around the funnel “corners”

Mold

Shellux

uyu

t

u’x

y


-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 50 100 150 200 250 300 350 400 450 500 550 600 650


Tan

gen

tial

Dis

pla

cem

ent

(mm

)

750 mm Outer Funnel Width950 mm Outer Funnel WidthParallel Mold

2 s

6 s

10 s

Surface Displacement:Effect of Funnel Width


Conclusions

• A larger funnel radius provides:– More uniform heat transfer– Smaller bending effect in the transition region– Less bunching of the shell in the transition region

• A “better” funnel (with respect to depression-type LFCs) has a wide funnel and small crown

• Many other phenomena can cause LFCs

Decrease crown

Increase outer funnel width

Decrease inner funnel width


Acknowledgements

• Continuous Casting Consortium Members (Nucor, Postech, LWB Refractories, Corus, Labein, Goodrich, Arcelor-Mittal Riverdale, Baosteel, Steel Dynamics, ANSYS)

• Dr. Seid Koric, Dr. Chunsheng Li, Dr. Hong Zhu

• Begoña Santillana, Arie Hamoen, ArnoudKamperman, Dr. Jean Campaniello, Frank Grimberg

• National Center for Supercomputing Applications (NCSA) at UIUC – “Tungsten” and “Abe” clusters

• Dassault Simulia, Inc. (ABAQUS parent company)

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