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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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 3
Thin-Slab Casting with Funnel Molds
Mold Narrow Face (NF)
Mold Wide Face (WF)
Solidifying Steel Strand
Submerged Entry Nozzle (SEN)
Funnel Region
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 4
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 5
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 6
• 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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 7
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 8
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 9
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 10
Traveling Slice Analysis
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 11
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 12
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”
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 13
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)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 14
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]
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 15
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)
Distance Below Meniscus (mm)
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
Distance Below Meniscus (mm)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 16
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]
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 17
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 18
1000
1100
1200
1300
1400
1500
1600
0 2 4 6 8 10 12 14 16
Distance Beneath Shell Surface (mm)
Te
mp
era
ture
(ºC
)
L
δ
L + δ
γδ + γ
2 s 4 s 6 s 8 s 10 s
Temperature Solution
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 19
-12
-10
-8
-6
-4
-2
0
2
4
6
0 2 4 6 8 10 12 14 16
Distance Beneath Shell Surface (mm)
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 20
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”
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 21
-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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 22
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
Distance from Mold Centerline (mm)
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)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 23
• 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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 24
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
Distance Below Meniscus (mm)
750 mm OuterFunnel Width
950 mm OuterFunnel Width
( )2( )( )
4 16 ( )outer funnel width inner funnel widthcrown z
r zcrown z
−= +
⋅
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 25
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
Distance Below Meniscus (mm)
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 26
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
Distance Below Meniscus (mm)
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
Distance Below Meniscus (mm)
Me
ch
an
ica
l B
en
din
g S
tra
in R
ate
(%
/s)
All cases with 750 mm outer funnel width
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 27
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
Distance Below Meniscus (mm)
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.
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 28
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
Distance Below Meniscus (mm)
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
950 mm Funnel Length
1050 mm Funnel Length
Distance Below Meniscus (mm)
Me
ch
an
ica
l B
en
din
g S
tra
in (
%)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 29
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 30
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 31
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 32
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
Distance from Mold Centerline (mm)
-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
)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 33
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
Distance from Mold Centerline (mm)
Reference Case
950 mm Outer Funnel Width0 mm Inner Funnel Width
985 mm Funnel Length
Time = s
Dif
fere
nce
(m
m)
Effect of Geometry on Perimeter Length Changes
Buckling potential less with larger width
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 34
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 35
-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
Distance from Mold Centerline (mm)
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 36
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
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler • 37
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)